%------------------------------------------------------------------------------
% File     : ITP286^4 : TPTP v8.2.0. Released v8.0.0.
% Domain   : Interactive Theorem Proving
% Problem  : Sledgehammer problem VEBT_SuccPredImperative 00347_018417
% 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    : 0094_VEBT_SuccPredImperative_00347_018417 [Des22]

% Status   : Theorem
% Rating   : 1.00 v8.1.0
% Syntax   : Number of formulae    : 10206 (3355 unt; 709 typ;   0 def)
%            Number of atoms       : 29478 (11024 equ;   0 cnn)
%            Maximal formula atoms :   71 (   3 avg)
%            Number of connectives : 214796 (2687   ~; 348   |;2098   &;196598   @)
%                                         (   0 <=>;13065  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   96 (   7 avg)
%            Number of types       :   20 (  19 usr)
%            Number of type conns  : 3926 (3926   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :  693 ( 690 usr;  18 con; 0-13 aty)
%            Number of variables   : 30140 (3191   ^;25563   !; 870   ?;30140   :)
%                                         ( 516  !>;   0  ?*;   0  @-;   0  @+)
% SPC      : TH1_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 20:22:37.369
%------------------------------------------------------------------------------
% Could-be-implicit typings (33)
thf(ty_t_VEBT__BuildupMemImp_OVEBTi,type,
    vEBT_VEBTi: $tType ).

thf(ty_t_VEBT__Definitions_OVEBT,type,
    vEBT_VEBT: $tType ).

thf(ty_t_Heap__Time__Monad_OHeap,type,
    heap_Time_Heap: $tType > $tType ).

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

thf(ty_t_Code__Evaluation_Oterm,type,
    code_term: $tType ).

thf(ty_t_Heap_Oheap_Oheap__ext,type,
    heap_ext: $tType > $tType ).

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

thf(ty_t_Product__Type_Oprod,type,
    product_prod: $tType > $tType > $tType ).

thf(ty_t_Numeral__Type_Onum1,type,
    numeral_num1: $tType ).

thf(ty_t_Numeral__Type_Obit1,type,
    numeral_bit1: $tType > $tType ).

thf(ty_t_Numeral__Type_Obit0,type,
    numeral_bit0: $tType > $tType ).

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

thf(ty_t_Multiset_Omultiset,type,
    multiset: $tType > $tType ).

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

thf(ty_t_Assertions_Oassn,type,
    assn: $tType ).

thf(ty_t_Enum_Ofinite__3,type,
    finite_3: $tType ).

thf(ty_t_Enum_Ofinite__2,type,
    finite_2: $tType ).

thf(ty_t_Enum_Ofinite__1,type,
    finite_1: $tType ).

thf(ty_t_Uint32_Ouint32,type,
    uint32: $tType ).

thf(ty_t_Option_Ooption,type,
    option: $tType > $tType ).

thf(ty_t_Filter_Ofilter,type,
    filter: $tType > $tType ).

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

thf(ty_t_Heap_Oarray,type,
    array: $tType > $tType ).

thf(ty_t_Word_Oword,type,
    word: $tType > $tType ).

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

thf(ty_t_List_Olist,type,
    list: $tType > $tType ).

thf(ty_t_Heap_Oref,type,
    ref: $tType > $tType ).

thf(ty_t_Set_Oset,type,
    set: $tType > $tType ).

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

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

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

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

thf(ty_t_itself,type,
    itself: $tType > $tType ).

% Explicit typings (676)
thf(sy_cl_Finite__Set_Ofinite,type,
    finite_finite: 
      !>[A: $tType] : $o ).

thf(sy_cl_Complete__Lattices_Ocomplete__linorder,type,
    comple5582772986160207858norder: 
      !>[A: $tType] : $o ).

thf(sy_cl_Enum_Oenum,type,
    enum: 
      !>[A: $tType] : $o ).

thf(sy_cl_Typerep_Otyperep,type,
    typerep: 
      !>[A: $tType] : $o ).

thf(sy_cl_HOL_Otype,type,
    type: 
      !>[A: $tType] : $o ).

thf(sy_cl_Nat_Osize,type,
    size: 
      !>[A: $tType] : $o ).

thf(sy_cl_HOL_Oequal,type,
    cl_HOL_Oequal: 
      !>[A: $tType] : $o ).

thf(sy_cl_Heap_Oheap,type,
    heap: 
      !>[A: $tType] : $o ).

thf(sy_cl_Rings_Odvd,type,
    dvd: 
      !>[A: $tType] : $o ).

thf(sy_cl_Groups_Oone,type,
    one: 
      !>[A: $tType] : $o ).

thf(sy_cl_Rings_Oidom,type,
    idom: 
      !>[A: $tType] : $o ).

thf(sy_cl_Rings_Oring,type,
    ring: 
      !>[A: $tType] : $o ).

thf(sy_cl_Groups_Oplus,type,
    plus: 
      !>[A: $tType] : $o ).

thf(sy_cl_Groups_Ozero,type,
    zero: 
      !>[A: $tType] : $o ).

thf(sy_cl_Num_Onumeral,type,
    numeral: 
      !>[A: $tType] : $o ).

thf(sy_cl_Power_Opower,type,
    power: 
      !>[A: $tType] : $o ).

thf(sy_cl_Fields_Ofield,type,
    field: 
      !>[A: $tType] : $o ).

thf(sy_cl_GCD_Oring__gcd,type,
    ring_gcd: 
      !>[A: $tType] : $o ).

thf(sy_cl_Groups_Ominus,type,
    minus: 
      !>[A: $tType] : $o ).

thf(sy_cl_Rings_Oabs__if,type,
    abs_if: 
      !>[A: $tType] : $o ).

thf(sy_cl_Rings_Oring__1,type,
    ring_1: 
      !>[A: $tType] : $o ).

thf(sy_cl_Groups_Ouminus,type,
    uminus: 
      !>[A: $tType] : $o ).

thf(sy_cl_Orderings_Oord,type,
    ord: 
      !>[A: $tType] : $o ).

thf(sy_cl_Rings_Osemiring,type,
    semiring: 
      !>[A: $tType] : $o ).

thf(sy_cl_Nat_Oring__char__0,type,
    ring_char_0: 
      !>[A: $tType] : $o ).

thf(sy_cl_Num_Oneg__numeral,type,
    neg_numeral: 
      !>[A: $tType] : $o ).

thf(sy_cl_Orderings_Oorder,type,
    order: 
      !>[A: $tType] : $o ).

thf(sy_cl_Rings_Ocomm__ring,type,
    comm_ring: 
      !>[A: $tType] : $o ).

thf(sy_cl_Rings_Omult__zero,type,
    mult_zero: 
      !>[A: $tType] : $o ).

thf(sy_cl_Type__Length_Olen,type,
    type_len: 
      !>[A: $tType] : $o ).

thf(sy_cl_GCD_Osemiring__gcd,type,
    semiring_gcd: 
      !>[A: $tType] : $o ).

thf(sy_cl_Groups_Ogroup__add,type,
    group_add: 
      !>[A: $tType] : $o ).

thf(sy_cl_Orderings_Ono__bot,type,
    no_bot: 
      !>[A: $tType] : $o ).

thf(sy_cl_Orderings_Ono__top,type,
    no_top: 
      !>[A: $tType] : $o ).

thf(sy_cl_Rings_Osemiring__0,type,
    semiring_0: 
      !>[A: $tType] : $o ).

thf(sy_cl_Rings_Osemiring__1,type,
    semiring_1: 
      !>[A: $tType] : $o ).

thf(sy_cl_Type__Length_Olen0,type,
    type_len0: 
      !>[A: $tType] : $o ).

thf(sy_cl_Groups_Omonoid__add,type,
    monoid_add: 
      !>[A: $tType] : $o ).

thf(sy_cl_Rings_Ocomm__ring__1,type,
    comm_ring_1: 
      !>[A: $tType] : $o ).

thf(sy_cl_Rings_Oidom__divide,type,
    idom_divide: 
      !>[A: $tType] : $o ).

thf(sy_cl_Rings_Oidom__modulo,type,
    idom_modulo: 
      !>[A: $tType] : $o ).

thf(sy_cl_Transcendental_Oln,type,
    ln: 
      !>[A: $tType] : $o ).

thf(sy_cl_Groups_Omonoid__mult,type,
    monoid_mult: 
      !>[A: $tType] : $o ).

thf(sy_cl_Orderings_Olinorder,type,
    linorder: 
      !>[A: $tType] : $o ).

thf(sy_cl_Orderings_Opreorder,type,
    preorder: 
      !>[A: $tType] : $o ).

thf(sy_cl_Parity_Oring__parity,type,
    ring_parity: 
      !>[A: $tType] : $o ).

thf(sy_cl_Rings_Oidom__abs__sgn,type,
    idom_abs_sgn: 
      !>[A: $tType] : $o ).

thf(sy_cl_Rings_Oordered__ring,type,
    ordered_ring: 
      !>[A: $tType] : $o ).

thf(sy_cl_Rings_Ozero__neq__one,type,
    zero_neq_one: 
      !>[A: $tType] : $o ).

thf(sy_cl_Fields_Ofield__char__0,type,
    field_char_0: 
      !>[A: $tType] : $o ).

thf(sy_cl_Groups_Oab__group__add,type,
    ab_group_add: 
      !>[A: $tType] : $o ).

thf(sy_cl_Nat_Osemiring__char__0,type,
    semiring_char_0: 
      !>[A: $tType] : $o ).

thf(sy_cl_Orderings_Oorder__bot,type,
    order_bot: 
      !>[A: $tType] : $o ).

thf(sy_cl_Orderings_Oorder__top,type,
    order_top: 
      !>[A: $tType] : $o ).

thf(sy_cl_Orderings_Owellorder,type,
    wellorder: 
      !>[A: $tType] : $o ).

thf(sy_cl_Rings_Ocomm__semiring,type,
    comm_semiring: 
      !>[A: $tType] : $o ).

thf(sy_cl_Rings_Ozero__less__one,type,
    zero_less_one: 
      !>[A: $tType] : $o ).

thf(sy_cl_Fields_Odivision__ring,type,
    division_ring: 
      !>[A: $tType] : $o ).

thf(sy_cl_Fields_Ofield__abs__sgn,type,
    field_abs_sgn: 
      !>[A: $tType] : $o ).

thf(sy_cl_Groups_Osemigroup__add,type,
    semigroup_add: 
      !>[A: $tType] : $o ).

thf(sy_cl_Num_Osemiring__numeral,type,
    semiring_numeral: 
      !>[A: $tType] : $o ).

thf(sy_cl_Rings_Osemidom__divide,type,
    semidom_divide: 
      !>[A: $tType] : $o ).

thf(sy_cl_Rings_Osemidom__modulo,type,
    semidom_modulo: 
      !>[A: $tType] : $o ).

thf(sy_cl_Groups_Osemigroup__mult,type,
    semigroup_mult: 
      !>[A: $tType] : $o ).

thf(sy_cl_Orderings_Odense__order,type,
    dense_order: 
      !>[A: $tType] : $o ).

thf(sy_cl_Rings_Ocomm__semiring__0,type,
    comm_semiring_0: 
      !>[A: $tType] : $o ).

thf(sy_cl_Rings_Ocomm__semiring__1,type,
    comm_semiring_1: 
      !>[A: $tType] : $o ).

thf(sy_cl_Rings_Olinordered__idom,type,
    linordered_idom: 
      !>[A: $tType] : $o ).

thf(sy_cl_Rings_Olinordered__ring,type,
    linordered_ring: 
      !>[A: $tType] : $o ).

thf(sy_cl_Rings_Osemiring__modulo,type,
    semiring_modulo: 
      !>[A: $tType] : $o ).

thf(sy_cl_Groups_Ocomm__monoid__add,type,
    comm_monoid_add: 
      !>[A: $tType] : $o ).

thf(sy_cl_Parity_Osemiring__parity,type,
    semiring_parity: 
      !>[A: $tType] : $o ).

thf(sy_cl_Rings_Oordered__ring__abs,type,
    ordered_ring_abs: 
      !>[A: $tType] : $o ).

thf(sy_cl_Rings_Oordered__semiring,type,
    ordered_semiring: 
      !>[A: $tType] : $o ).

thf(sy_cl_Code__Evaluation_Oterm__of,type,
    code_term_of: 
      !>[A: $tType] : $o ).

thf(sy_cl_Fields_Olinordered__field,type,
    linordered_field: 
      !>[A: $tType] : $o ).

thf(sy_cl_Generic__set__bit_Oset__bit,type,
    generic_set_set_bit: 
      !>[A: $tType] : $o ).

thf(sy_cl_Groups_Oab__semigroup__add,type,
    ab_semigroup_add: 
      !>[A: $tType] : $o ).

thf(sy_cl_Groups_Ocomm__monoid__diff,type,
    comm_monoid_diff: 
      !>[A: $tType] : $o ).

thf(sy_cl_Groups_Ocomm__monoid__mult,type,
    comm_monoid_mult: 
      !>[A: $tType] : $o ).

thf(sy_cl_Rings_Oalgebraic__semidom,type,
    algebraic_semidom: 
      !>[A: $tType] : $o ).

thf(sy_cl_Rings_Osemiring__1__cancel,type,
    semiring_1_cancel: 
      !>[A: $tType] : $o ).

thf(sy_cl_Groups_Oab__semigroup__mult,type,
    ab_semigroup_mult: 
      !>[A: $tType] : $o ).

thf(sy_cl_Orderings_Odense__linorder,type,
    dense_linorder: 
      !>[A: $tType] : $o ).

thf(sy_cl_Rings_Olinordered__semidom,type,
    linordered_semidom: 
      !>[A: $tType] : $o ).

thf(sy_cl_Rings_Oordered__semiring__0,type,
    ordered_semiring_0: 
      !>[A: $tType] : $o ).

thf(sy_cl_Least__significant__bit_Olsb,type,
    least_6119777620449941438nt_lsb: 
      !>[A: $tType] : $o ).

thf(sy_cl_Real__Vector__Spaces_Obanach,type,
    real_Vector_banach: 
      !>[A: $tType] : $o ).

thf(sy_cl_Rings_Olinordered__semiring,type,
    linordered_semiring: 
      !>[A: $tType] : $o ).

thf(sy_cl_Groups_Ocancel__semigroup__add,type,
    cancel_semigroup_add: 
      !>[A: $tType] : $o ).

thf(sy_cl_Groups_Oordered__ab__group__add,type,
    ordered_ab_group_add: 
      !>[A: $tType] : $o ).

thf(sy_cl_Rings_Olinordered__semiring__1,type,
    linord6961819062388156250ring_1: 
      !>[A: $tType] : $o ).

thf(sy_cl_Rings_Oordered__comm__semiring,type,
    ordere2520102378445227354miring: 
      !>[A: $tType] : $o ).

thf(sy_cl_Topological__Spaces_Ot2__space,type,
    topological_t2_space: 
      !>[A: $tType] : $o ).

thf(sy_cl_Bit__Operations_Osemiring__bits,type,
    bit_semiring_bits: 
      !>[A: $tType] : $o ).

thf(sy_cl_Rings_Ocomm__semiring__1__cancel,type,
    comm_s4317794764714335236cancel: 
      !>[A: $tType] : $o ).

thf(sy_cl_Rings_Olinordered__ring__strict,type,
    linord4710134922213307826strict: 
      !>[A: $tType] : $o ).

thf(sy_cl_Groups_Ocancel__comm__monoid__add,type,
    cancel1802427076303600483id_add: 
      !>[A: $tType] : $o ).

thf(sy_cl_Real__Vector__Spaces_Oreal__field,type,
    real_V7773925162809079976_field: 
      !>[A: $tType] : $o ).

thf(sy_cl_Rings_Oring__1__no__zero__divisors,type,
    ring_15535105094025558882visors: 
      !>[A: $tType] : $o ).

thf(sy_cl_Groups_Ocancel__ab__semigroup__add,type,
    cancel2418104881723323429up_add: 
      !>[A: $tType] : $o ).

thf(sy_cl_Groups_Olinordered__ab__group__add,type,
    linord5086331880401160121up_add: 
      !>[A: $tType] : $o ).

thf(sy_cl_Groups_Oordered__comm__monoid__add,type,
    ordere6911136660526730532id_add: 
      !>[A: $tType] : $o ).

thf(sy_cl_Real__Vector__Spaces_Oreal__vector,type,
    real_V4867850818363320053vector: 
      !>[A: $tType] : $o ).

thf(sy_cl_Archimedean__Field_Ofloor__ceiling,type,
    archim2362893244070406136eiling: 
      !>[A: $tType] : $o ).

thf(sy_cl_GCD_Osemiring__gcd__mult__normalize,type,
    semiri6843258321239162965malize: 
      !>[A: $tType] : $o ).

thf(sy_cl_Groups_Oordered__ab__group__add__abs,type,
    ordere166539214618696060dd_abs: 
      !>[A: $tType] : $o ).

thf(sy_cl_Groups_Oordered__ab__semigroup__add,type,
    ordere6658533253407199908up_add: 
      !>[A: $tType] : $o ).

thf(sy_cl_Limits_Otopological__ab__group__add,type,
    topolo1287966508704411220up_add: 
      !>[A: $tType] : $o ).

thf(sy_cl_Real__Vector__Spaces_Oreal__algebra,type,
    real_V6157519004096292374lgebra: 
      !>[A: $tType] : $o ).

thf(sy_cl_Rings_Osemiring__no__zero__divisors,type,
    semiri3467727345109120633visors: 
      !>[A: $tType] : $o ).

thf(sy_cl_Boolean__Algebras_Oboolean__algebra,type,
    boolea8198339166811842893lgebra: 
      !>[A: $tType] : $o ).

thf(sy_cl_Rings_Olinordered__semiring__strict,type,
    linord8928482502909563296strict: 
      !>[A: $tType] : $o ).

thf(sy_cl_Euclidean__Division_Oeuclidean__ring,type,
    euclid5891614535332579305n_ring: 
      !>[A: $tType] : $o ).

thf(sy_cl_Real__Vector__Spaces_Oreal__algebra__1,type,
    real_V2191834092415804123ebra_1: 
      !>[A: $tType] : $o ).

thf(sy_cl_Rings_Olinordered__nonzero__semiring,type,
    linord181362715937106298miring: 
      !>[A: $tType] : $o ).

thf(sy_cl_Rings_Osemiring__1__no__zero__divisors,type,
    semiri2026040879449505780visors: 
      !>[A: $tType] : $o ).

thf(sy_cl_Topological__Spaces_Oorder__topology,type,
    topolo2564578578187576103pology: 
      !>[A: $tType] : $o ).

thf(sy_cl_Bit__Operations_Oring__bit__operations,type,
    bit_ri3973907225187159222ations: 
      !>[A: $tType] : $o ).

thf(sy_cl_Groups_Olinordered__ab__semigroup__add,type,
    linord4140545234300271783up_add: 
      !>[A: $tType] : $o ).

thf(sy_cl_Limits_Otopological__comm__monoid__add,type,
    topolo5987344860129210374id_add: 
      !>[A: $tType] : $o ).

thf(sy_cl_Rings_Olinordered__semiring__1__strict,type,
    linord715952674999750819strict: 
      !>[A: $tType] : $o ).

thf(sy_cl_Archimedean__Field_Oarchimedean__field,type,
    archim462609752435547400_field: 
      !>[A: $tType] : $o ).

thf(sy_cl_Bit__Comprehension_Obit__comprehension,type,
    bit_bi6583157726757044596ension: 
      !>[A: $tType] : $o ).

thf(sy_cl_Real__Vector__Spaces_Oreal__div__algebra,type,
    real_V5047593784448816457lgebra: 
      !>[A: $tType] : $o ).

thf(sy_cl_Quickcheck__Narrowing_Opartial__term__of,type,
    quickc6926020345158392990erm_of: 
      !>[A: $tType] : $o ).

thf(sy_cl_Real__Vector__Spaces_Oreal__normed__field,type,
    real_V3459762299906320749_field: 
      !>[A: $tType] : $o ).

thf(sy_cl_Topological__Spaces_Olinorder__topology,type,
    topolo1944317154257567458pology: 
      !>[A: $tType] : $o ).

thf(sy_cl_Topological__Spaces_Otopological__space,type,
    topolo4958980785337419405_space: 
      !>[A: $tType] : $o ).

thf(sy_cl_Euclidean__Division_Oeuclidean__semiring,type,
    euclid3725896446679973847miring: 
      !>[A: $tType] : $o ).

thf(sy_cl_Groups_Ocanonically__ordered__monoid__add,type,
    canoni5634975068530333245id_add: 
      !>[A: $tType] : $o ).

thf(sy_cl_Groups_Oordered__cancel__comm__monoid__add,type,
    ordere8940638589300402666id_add: 
      !>[A: $tType] : $o ).

thf(sy_cl_Groups_Ostrict__ordered__comm__monoid__add,type,
    strict7427464778891057005id_add: 
      !>[A: $tType] : $o ).

thf(sy_cl_Real__Vector__Spaces_Oreal__normed__vector,type,
    real_V822414075346904944vector: 
      !>[A: $tType] : $o ).

thf(sy_cl_Rings_Olinordered__comm__semiring__strict,type,
    linord2810124833399127020strict: 
      !>[A: $tType] : $o ).

thf(sy_cl_Bit__Operations_Osemiring__bit__operations,type,
    bit_se359711467146920520ations: 
      !>[A: $tType] : $o ).

thf(sy_cl_Groups_Oordered__ab__semigroup__add__imp__le,type,
    ordere2412721322843649153imp_le: 
      !>[A: $tType] : $o ).

thf(sy_cl_Groups_Oordered__cancel__ab__semigroup__add,type,
    ordere580206878836729694up_add: 
      !>[A: $tType] : $o ).

thf(sy_cl_Groups_Oordered__cancel__comm__monoid__diff,type,
    ordere1170586879665033532d_diff: 
      !>[A: $tType] : $o ).

thf(sy_cl_Groups_Ostrict__ordered__ab__semigroup__add,type,
    strict9044650504122735259up_add: 
      !>[A: $tType] : $o ).

thf(sy_cl_Real__Vector__Spaces_Oordered__real__vector,type,
    real_V5355595471888546746vector: 
      !>[A: $tType] : $o ).

thf(sy_cl_Real__Vector__Spaces_Oreal__normed__algebra,type,
    real_V4412858255891104859lgebra: 
      !>[A: $tType] : $o ).

thf(sy_cl_Rings_Osemiring__no__zero__divisors__cancel,type,
    semiri6575147826004484403cancel: 
      !>[A: $tType] : $o ).

thf(sy_cl_Euclidean__Division_Oeuclidean__ring__cancel,type,
    euclid8851590272496341667cancel: 
      !>[A: $tType] : $o ).

thf(sy_cl_Real__Vector__Spaces_Oreal__normed__algebra__1,type,
    real_V2822296259951069270ebra_1: 
      !>[A: $tType] : $o ).

thf(sy_cl_Divides_Ounique__euclidean__semiring__numeral,type,
    unique1627219031080169319umeral: 
      !>[A: $tType] : $o ).

thf(sy_cl_Real__Vector__Spaces_Oreal__normed__div__algebra,type,
    real_V8999393235501362500lgebra: 
      !>[A: $tType] : $o ).

thf(sy_cl_Euclidean__Division_Oeuclidean__semiring__cancel,type,
    euclid4440199948858584721cancel: 
      !>[A: $tType] : $o ).

thf(sy_cl_Euclidean__Division_Ounique__euclidean__semiring,type,
    euclid3128863361964157862miring: 
      !>[A: $tType] : $o ).

thf(sy_cl_Groups_Oordered__ab__semigroup__monoid__add__imp__le,type,
    ordere1937475149494474687imp_le: 
      !>[A: $tType] : $o ).

thf(sy_cl_Euclidean__Division_Ounique__euclidean__ring__with__nat,type,
    euclid8789492081693882211th_nat: 
      !>[A: $tType] : $o ).

thf(sy_cl_Euclidean__Division_Ounique__euclidean__semiring__with__nat,type,
    euclid5411537665997757685th_nat: 
      !>[A: $tType] : $o ).

thf(sy_cl_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct,type,
    semiri1453513574482234551roduct: 
      !>[A: $tType] : $o ).

thf(sy_cl_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations,type,
    bit_un5681908812861735899ations: 
      !>[A: $tType] : $o ).

thf(sy_cl_Conditionally__Complete__Lattices_Oconditionally__complete__linorder,type,
    condit6923001295902523014norder: 
      !>[A: $tType] : $o ).

thf(sy_c_Archimedean__Field_Oceiling,type,
    archimedean_ceiling: 
      !>[A: $tType] : ( A > int ) ).

thf(sy_c_Archimedean__Field_Ofloor__ceiling__class_Ofloor,type,
    archim6421214686448440834_floor: 
      !>[A: $tType] : ( A > int ) ).

thf(sy_c_Archimedean__Field_Ofrac,type,
    archimedean_frac: 
      !>[A: $tType] : ( A > A ) ).

thf(sy_c_Archimedean__Field_Oround,type,
    archimedean_round: 
      !>[A: $tType] : ( A > int ) ).

thf(sy_c_Array__Time_Oalloc,type,
    array_alloc: 
      !>[A: $tType] : ( ( list @ A ) > ( heap_ext @ product_unit ) > ( product_prod @ ( array @ A ) @ ( heap_ext @ product_unit ) ) ) ).

thf(sy_c_Array__Time_Ofreeze,type,
    array_freeze: 
      !>[A: $tType] : ( ( array @ A ) > ( heap_Time_Heap @ ( list @ A ) ) ) ).

thf(sy_c_Array__Time_Olen,type,
    array_len: 
      !>[A: $tType] : ( ( array @ A ) > ( heap_Time_Heap @ nat ) ) ).

thf(sy_c_Array__Time_Olen_H,type,
    array_len2: 
      !>[B: $tType,A: $tType] : ( ( array @ B ) > ( heap_Time_Heap @ A ) ) ).

thf(sy_c_Array__Time_Omake,type,
    array_make: 
      !>[A: $tType] : ( nat > ( nat > A ) > ( heap_Time_Heap @ ( array @ A ) ) ) ).

thf(sy_c_Array__Time_Omap__entry,type,
    array_map_entry: 
      !>[A: $tType] : ( nat > ( A > A ) > ( array @ A ) > ( heap_Time_Heap @ ( array @ A ) ) ) ).

thf(sy_c_Array__Time_Onew,type,
    array_new: 
      !>[A: $tType] : ( nat > A > ( heap_Time_Heap @ ( array @ A ) ) ) ).

thf(sy_c_Array__Time_Onth,type,
    array_nth: 
      !>[A: $tType] : ( ( array @ A ) > nat > ( heap_Time_Heap @ A ) ) ).

thf(sy_c_Array__Time_Onth_H,type,
    array_nth2: 
      !>[A: $tType] : ( ( array @ A ) > code_integer > ( heap_Time_Heap @ A ) ) ).

thf(sy_c_Array__Time_Oof__list,type,
    array_of_list: 
      !>[A: $tType] : ( ( list @ A ) > ( heap_Time_Heap @ ( array @ A ) ) ) ).

thf(sy_c_Array__Time_Oswap,type,
    array_swap: 
      !>[A: $tType] : ( nat > A > ( array @ A ) > ( heap_Time_Heap @ A ) ) ).

thf(sy_c_Array__Time_Oupd,type,
    array_upd: 
      !>[A: $tType] : ( nat > A > ( array @ A ) > ( heap_Time_Heap @ ( array @ A ) ) ) ).

thf(sy_c_Assertions_Oassn_ORep__assn,type,
    rep_assn: assn > ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ).

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

thf(sy_c_Assertions_Oex__assn,type,
    ex_assn: 
      !>[A: $tType] : ( ( A > assn ) > assn ) ).

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

thf(sy_c_Assertions_Osnga__assn,type,
    snga_assn: 
      !>[A: $tType] : ( ( array @ A ) > ( list @ A ) > assn ) ).

thf(sy_c_Automation_OFI,type,
    fi: ( list @ ( product_prod @ assn @ assn ) ) > assn > assn > assn > assn > assn > $o ).

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

thf(sy_c_Automation_OFI__RESULT,type,
    fI_RESULT: ( list @ ( product_prod @ assn @ assn ) ) > assn > assn > assn > $o ).

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

thf(sy_c_Binomial_Ogbinomial,type,
    gbinomial: 
      !>[A: $tType] : ( A > nat > A ) ).

thf(sy_c_Bit__Comprehension_Obit__comprehension__class_Oset__bits,type,
    bit_bi4170147762399595738t_bits: 
      !>[A: $tType] : ( ( nat > $o ) > A ) ).

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_Oor__not__num__neg__rel,type,
    bit_or3848514188828904588eg_rel: ( product_prod @ num @ num ) > ( product_prod @ num @ num ) > $o ).

thf(sy_c_Bit__Operations_Oring__bit__operations__class_Onot,type,
    bit_ri4277139882892585799ns_not: 
      !>[A: $tType] : ( A > A ) ).

thf(sy_c_Bit__Operations_Oring__bit__operations__class_Osigned__take__bit,type,
    bit_ri4674362597316999326ke_bit: 
      !>[A: $tType] : ( nat > A > A ) ).

thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oand,type,
    bit_se5824344872417868541ns_and: 
      !>[A: $tType] : ( A > A > A ) ).

thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Odrop__bit,type,
    bit_se4197421643247451524op_bit: 
      !>[A: $tType] : ( nat > A > A ) ).

thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oflip__bit,type,
    bit_se8732182000553998342ip_bit: 
      !>[A: $tType] : ( nat > A > A ) ).

thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Omask,type,
    bit_se2239418461657761734s_mask: 
      !>[A: $tType] : ( nat > A ) ).

thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oor,type,
    bit_se1065995026697491101ons_or: 
      !>[A: $tType] : ( A > A > A ) ).

thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Opush__bit,type,
    bit_se4730199178511100633sh_bit: 
      !>[A: $tType] : ( nat > A > A ) ).

thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oset__bit,type,
    bit_se5668285175392031749et_bit: 
      !>[A: $tType] : ( nat > A > A ) ).

thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Otake__bit,type,
    bit_se2584673776208193580ke_bit: 
      !>[A: $tType] : ( nat > A > A ) ).

thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Ounset__bit,type,
    bit_se2638667681897837118et_bit: 
      !>[A: $tType] : ( nat > A > A ) ).

thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oxor,type,
    bit_se5824344971392196577ns_xor: 
      !>[A: $tType] : ( A > A > A ) ).

thf(sy_c_Bit__Operations_Osemiring__bits__class_Obit,type,
    bit_se5641148757651400278ts_bit: 
      !>[A: $tType] : ( A > nat > $o ) ).

thf(sy_c_Bit__Operations_Osemiring__bits__class_Opossible__bit,type,
    bit_se6407376104438227557le_bit: 
      !>[A: $tType] : ( ( itself @ A ) > 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,type,
    bit_Sh4282982442137083160shiftl: 
      !>[A: $tType] : ( A > nat > A ) ).

thf(sy_c_Bit__Shifts__Infix__Syntax_Osemiring__bit__operations__class_Oshiftr,type,
    bit_Sh4282982442137083166shiftr: 
      !>[A: $tType] : ( A > nat > A ) ).

thf(sy_c_Bit__Shifts__Infix__Syntax_Osshiftr,type,
    bit_Sh8784991116023147202shiftr: 
      !>[A: $tType] : ( ( word @ A ) > nat > ( word @ A ) ) ).

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_Bits__Integer_Ointeger__set__bit,type,
    bits_integer_set_bit: code_integer > code_integer > $o > code_integer ).

thf(sy_c_Boolean__Algebras_Oabstract__boolean__algebra__sym__diff,type,
    boolea3799213064322606851m_diff: 
      !>[A: $tType] : ( ( A > A > A ) > ( A > A > A ) > ( A > A ) > A > A > ( A > A > A ) > $o ) ).

thf(sy_c_Code__Numeral_Obit__cut__integer,type,
    code_bit_cut_integer: code_integer > ( product_prod @ code_integer @ $o ) ).

thf(sy_c_Code__Numeral_Odivmod__abs,type,
    code_divmod_abs: code_integer > code_integer > ( product_prod @ code_integer @ code_integer ) ).

thf(sy_c_Code__Numeral_Odivmod__integer,type,
    code_divmod_integer: code_integer > code_integer > ( product_prod @ code_integer @ code_integer ) ).

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__nat,type,
    code_integer_of_nat: nat > code_integer ).

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

thf(sy_c_Code__Numeral_Onegative,type,
    code_negative: num > code_integer ).

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

thf(sy_c_Code__Numeral_Opositive,type,
    code_positive: num > code_integer ).

thf(sy_c_Code__Target__Int_Onegative,type,
    code_Target_negative: num > int ).

thf(sy_c_Code__Target__Word__Base_Oquickcheck__narrowing__samples_Onarrowing__samples,type,
    code_T4080844693773952564amples: 
      !>[A: $tType] : ( ( code_integer > ( product_prod @ A @ A ) ) > A > code_integer > ( list @ A ) ) ).

thf(sy_c_Code__Target__Word__Base_Oquickcheck__narrowing__samples_Onarrowing__samples__rel,type,
    code_T1710151556404007877es_rel: 
      !>[A: $tType] : ( ( code_integer > ( product_prod @ A @ A ) ) > code_integer > code_integer > $o ) ).

thf(sy_c_Code__Target__Word__Base_Oquickcheck__narrowing__samples_Opartial__term__of__sample,type,
    code_T4081349890594273596sample: 
      !>[A: $tType] : ( ( code_integer > ( product_prod @ A @ A ) ) > A > code_integer > A ) ).

thf(sy_c_Code__Target__Word__Base_Oset__bits__aux,type,
    code_T2661198915054445665ts_aux: 
      !>[A: $tType] : ( ( nat > $o ) > nat > ( word @ A ) > ( word @ A ) ) ).

thf(sy_c_Complete__Partial__Order_Occpo_Oadmissible,type,
    comple1908693960933563346ssible: 
      !>[A: $tType] : ( ( ( set @ A ) > A ) > ( A > A > $o ) > ( A > $o ) > $o ) ).

thf(sy_c_Complete__Partial__Order_Occpo_Ofixp,type,
    comple187402453842119260l_fixp: 
      !>[A: $tType] : ( ( ( set @ A ) > A ) > ( A > A > $o ) > ( A > A ) > A ) ).

thf(sy_c_Complete__Partial__Order_Omonotone,type,
    comple7038119648293358887notone: 
      !>[A: $tType,B: $tType] : ( ( A > A > $o ) > ( B > B > $o ) > ( A > B ) > $o ) ).

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

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

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

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

thf(sy_c_Complex_Oimaginary__unit,type,
    imaginary_unit: complex ).

thf(sy_c_Deriv_Ohas__field__derivative,type,
    has_field_derivative: 
      !>[A: $tType] : ( ( A > A ) > A > ( filter @ A ) > $o ) ).

thf(sy_c_Divides_Oadjust__div,type,
    adjust_div: ( product_prod @ int @ int ) > int ).

thf(sy_c_Divides_Oadjust__mod,type,
    adjust_mod: int > int > int ).

thf(sy_c_Divides_Odivmod__nat,type,
    divmod_nat: nat > nat > ( product_prod @ nat @ nat ) ).

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

thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivides__aux,type,
    unique5940410009612947441es_aux: 
      !>[A: $tType] : ( ( product_prod @ A @ A ) > $o ) ).

thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod,type,
    unique8689654367752047608divmod: 
      !>[A: $tType] : ( num > num > ( product_prod @ A @ A ) ) ).

thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod__step,type,
    unique1321980374590559556d_step: 
      !>[A: $tType] : ( num > ( product_prod @ A @ A ) > ( product_prod @ A @ A ) ) ).

thf(sy_c_Factorial_Ocomm__semiring__1__class_Opochhammer,type,
    comm_s3205402744901411588hammer: 
      !>[A: $tType] : ( A > nat > A ) ).

thf(sy_c_Factorial_Osemiring__char__0__class_Ofact,type,
    semiring_char_0_fact: 
      !>[A: $tType] : ( nat > A ) ).

thf(sy_c_Fields_Oinverse__class_Oinverse,type,
    inverse_inverse: 
      !>[A: $tType] : ( A > A ) ).

thf(sy_c_Finite__Set_Ofinite,type,
    finite_finite2: 
      !>[A: $tType] : ( ( set @ A ) > $o ) ).

thf(sy_c_Finite__Set_Ofold,type,
    finite_fold: 
      !>[A: $tType,B: $tType] : ( ( A > B > B ) > B > ( set @ A ) > B ) ).

thf(sy_c_Fun_Obij__betw,type,
    bij_betw: 
      !>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > ( set @ B ) > $o ) ).

thf(sy_c_Fun_Ocomp,type,
    comp: 
      !>[B: $tType,C: $tType,A: $tType] : ( ( B > C ) > ( A > B ) > A > C ) ).

thf(sy_c_Fun_Oinj__on,type,
    inj_on: 
      !>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > $o ) ).

thf(sy_c_Fun_Othe__inv__into,type,
    the_inv_into: 
      !>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > B ) > B > A ) ).

thf(sy_c_GCD_Obezw,type,
    bezw: nat > nat > ( product_prod @ int @ int ) ).

thf(sy_c_GCD_Obezw__rel,type,
    bezw_rel: ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) > $o ).

thf(sy_c_GCD_Ogcd__class_Ogcd,type,
    gcd_gcd: 
      !>[A: $tType] : ( A > A > A ) ).

thf(sy_c_GCD_Ogcd__nat__rel,type,
    gcd_nat_rel: ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) > $o ).

thf(sy_c_GCD_Osemiring__1__class_Osemiring__char,type,
    semiri4206861660011772517g_char: 
      !>[A: $tType] : ( ( itself @ A ) > nat ) ).

thf(sy_c_Generic__set__bit_Oset__bit__class_Oset__bit,type,
    generi7602027413899671122et_bit: 
      !>[A: $tType] : ( A > nat > $o > A ) ).

thf(sy_c_Groups_Oabs__class_Oabs,type,
    abs_abs: 
      !>[A: $tType] : ( A > A ) ).

thf(sy_c_Groups_Ominus__class_Ominus,type,
    minus_minus: 
      !>[A: $tType] : ( A > A > A ) ).

thf(sy_c_Groups_Oone__class_Oone,type,
    one_one: 
      !>[A: $tType] : A ).

thf(sy_c_Groups_Oplus__class_Oplus,type,
    plus_plus: 
      !>[A: $tType] : ( A > A > A ) ).

thf(sy_c_Groups_Osgn__class_Osgn,type,
    sgn_sgn: 
      !>[A: $tType] : ( A > A ) ).

thf(sy_c_Groups_Otimes__class_Otimes,type,
    times_times: 
      !>[A: $tType] : ( A > A > A ) ).

thf(sy_c_Groups_Ouminus__class_Ouminus,type,
    uminus_uminus: 
      !>[A: $tType] : ( A > A ) ).

thf(sy_c_Groups_Ozero__class_Ozero,type,
    zero_zero: 
      !>[A: $tType] : A ).

thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum,type,
    groups7311177749621191930dd_sum: 
      !>[B: $tType,A: $tType] : ( ( B > A ) > ( set @ B ) > A ) ).

thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_H,type,
    groups1027152243600224163dd_sum: 
      !>[C: $tType,A: $tType] : ( ( C > A ) > ( set @ C ) > A ) ).

thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod,type,
    groups7121269368397514597t_prod: 
      !>[B: $tType,A: $tType] : ( ( B > A ) > ( set @ B ) > A ) ).

thf(sy_c_Groups__List_Ocomm__semiring__0__class_Ohorner__sum,type,
    groups4207007520872428315er_sum: 
      !>[B: $tType,A: $tType] : ( ( B > A ) > A > ( list @ B ) > A ) ).

thf(sy_c_HOL_OThe,type,
    the: 
      !>[A: $tType] : ( ( A > $o ) > A ) ).

thf(sy_c_HOL_Oundefined,type,
    undefined: 
      !>[A: $tType] : A ).

thf(sy_c_Hash__Instances_Ohash__code__list,type,
    hash_hash_code_list: 
      !>[A: $tType] : ( ( A > uint32 ) > ( list @ A ) > uint32 ) ).

thf(sy_c_Hash__Instances_Ohash__code__option,type,
    hash_h1887023736457453652option: 
      !>[A: $tType] : ( ( A > uint32 ) > ( option @ A ) > uint32 ) ).

thf(sy_c_Hash__Instances_Ohash__code__prod,type,
    hash_hash_code_prod: 
      !>[A: $tType,B: $tType] : ( ( A > uint32 ) > ( B > uint32 ) > ( product_prod @ A @ B ) > uint32 ) ).

thf(sy_c_Heap_Oarray_Osize__array,type,
    size_array: 
      !>[A: $tType] : ( ( A > nat ) > ( array @ A ) > nat ) ).

thf(sy_c_Heap__Time__Monad_OHeap_OHeap,type,
    heap_Time_Heap2: 
      !>[A: $tType] : ( ( ( heap_ext @ product_unit ) > ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) > ( heap_Time_Heap @ A ) ) ).

thf(sy_c_Heap__Time__Monad_OHeap_Osize__Heap,type,
    heap_Time_size_Heap: 
      !>[A: $tType] : ( ( A > nat ) > ( heap_Time_Heap @ A ) > nat ) ).

thf(sy_c_Heap__Time__Monad_OHeap__lub,type,
    heap_Time_Heap_lub: 
      !>[A: $tType] : ( ( set @ ( heap_Time_Heap @ A ) ) > ( heap_Time_Heap @ A ) ) ).

thf(sy_c_Heap__Time__Monad_OHeap__ord,type,
    heap_Time_Heap_ord: 
      !>[A: $tType] : ( ( heap_Time_Heap @ A ) > ( heap_Time_Heap @ A ) > $o ) ).

thf(sy_c_Heap__Time__Monad_Oassert,type,
    heap_Time_assert: 
      !>[A: $tType] : ( ( A > $o ) > A > ( heap_Time_Heap @ A ) ) ).

thf(sy_c_Heap__Time__Monad_Obind,type,
    heap_Time_bind: 
      !>[A: $tType,B: $tType] : ( ( heap_Time_Heap @ A ) > ( A > ( heap_Time_Heap @ B ) ) > ( heap_Time_Heap @ B ) ) ).

thf(sy_c_Heap__Time__Monad_Oeffect,type,
    heap_Time_effect: 
      !>[A: $tType] : ( ( heap_Time_Heap @ A ) > ( heap_ext @ product_unit ) > ( heap_ext @ product_unit ) > A > nat > $o ) ).

thf(sy_c_Heap__Time__Monad_Oexecute,type,
    heap_Time_execute: 
      !>[A: $tType] : ( ( heap_Time_Heap @ A ) > ( heap_ext @ product_unit ) > ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) ).

thf(sy_c_Heap__Time__Monad_Oguard,type,
    heap_Time_guard: 
      !>[A: $tType] : ( ( ( heap_ext @ product_unit ) > $o ) > ( ( heap_ext @ product_unit ) > ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) > ( heap_Time_Heap @ A ) ) ).

thf(sy_c_Heap__Time__Monad_Oheap,type,
    heap_Time_heap: 
      !>[A: $tType] : ( ( ( heap_ext @ product_unit ) > ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) > ( heap_Time_Heap @ A ) ) ).

thf(sy_c_Heap__Time__Monad_Olift,type,
    heap_Time_lift: 
      !>[A: $tType,B: $tType] : ( ( A > B ) > A > ( heap_Time_Heap @ B ) ) ).

thf(sy_c_Heap__Time__Monad_Oreturn,type,
    heap_Time_return: 
      !>[A: $tType] : ( A > ( heap_Time_Heap @ A ) ) ).

thf(sy_c_Heap__Time__Monad_Osuccess,type,
    heap_Time_success: 
      !>[A: $tType] : ( ( heap_Time_Heap @ A ) > ( heap_ext @ product_unit ) > $o ) ).

thf(sy_c_Heap__Time__Monad_Otap,type,
    heap_Time_tap: 
      !>[A: $tType] : ( ( ( heap_ext @ product_unit ) > A ) > ( heap_Time_Heap @ A ) ) ).

thf(sy_c_Heap__Time__Monad_OtimeFrame,type,
    heap_Time_timeFrame: 
      !>[A: $tType] : ( nat > ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) > ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) ).

thf(sy_c_Heap__Time__Monad_OtimeFrame__rel,type,
    heap_T5500966940807335491me_rel: 
      !>[A: $tType] : ( ( product_prod @ nat @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) > ( product_prod @ nat @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) > $o ) ).

thf(sy_c_Heap__Time__Monad_Oureturn,type,
    heap_Time_ureturn: 
      !>[A: $tType] : ( A > ( heap_Time_Heap @ A ) ) ).

thf(sy_c_Heap__Time__Monad_Owait,type,
    heap_Time_wait: nat > ( heap_Time_Heap @ product_unit ) ).

thf(sy_c_Hoare__Triple_Ohoare__triple,type,
    hoare_hoare_triple: 
      !>[A: $tType] : ( assn > ( heap_Time_Heap @ A ) > ( A > assn ) > $o ) ).

thf(sy_c_Hoare__Triple_Onew__addrs,type,
    hoare_new_addrs: ( heap_ext @ product_unit ) > ( set @ nat ) > ( heap_ext @ product_unit ) > ( set @ nat ) ).

thf(sy_c_If,type,
    if: 
      !>[A: $tType] : ( $o > A > A > A ) ).

thf(sy_c_Int_Oint__ge__less__than,type,
    int_ge_less_than: int > ( set @ ( product_prod @ int @ int ) ) ).

thf(sy_c_Int_Oint__ge__less__than2,type,
    int_ge_less_than2: int > ( set @ ( product_prod @ int @ int ) ) ).

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

thf(sy_c_Int_Oring__1__class_OInts,type,
    ring_1_Ints: 
      !>[A: $tType] : ( set @ A ) ).

thf(sy_c_Int_Oring__1__class_Oof__int,type,
    ring_1_of_int: 
      !>[A: $tType] : ( int > A ) ).

thf(sy_c_Lattices__Big_Olinorder__class_OMax,type,
    lattic643756798349783984er_Max: 
      !>[A: $tType] : ( ( set @ A ) > A ) ).

thf(sy_c_Least__significant__bit_Olsb__class_Olsb,type,
    least_8051144512741203767sb_lsb: 
      !>[A: $tType] : ( A > $o ) ).

thf(sy_c_List_OBleast,type,
    bleast: 
      !>[A: $tType] : ( ( set @ A ) > ( A > $o ) > A ) ).

thf(sy_c_List_Oabort__Bleast,type,
    abort_Bleast: 
      !>[A: $tType] : ( ( set @ A ) > ( A > $o ) > A ) ).

thf(sy_c_List_Oappend,type,
    append: 
      !>[A: $tType] : ( ( list @ A ) > ( list @ A ) > ( list @ A ) ) ).

thf(sy_c_List_Oconcat,type,
    concat: 
      !>[A: $tType] : ( ( list @ ( list @ A ) ) > ( list @ A ) ) ).

thf(sy_c_List_Ocount__list,type,
    count_list: 
      !>[A: $tType] : ( ( list @ A ) > A > nat ) ).

thf(sy_c_List_Oenumerate,type,
    enumerate: 
      !>[A: $tType] : ( nat > ( list @ A ) > ( list @ ( product_prod @ nat @ A ) ) ) ).

thf(sy_c_List_Oextract,type,
    extract: 
      !>[A: $tType] : ( ( A > $o ) > ( list @ A ) > ( option @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) ) ) ).

thf(sy_c_List_Ofilter,type,
    filter2: 
      !>[A: $tType] : ( ( A > $o ) > ( list @ A ) > ( list @ A ) ) ).

thf(sy_c_List_Ofoldr,type,
    foldr: 
      !>[A: $tType,B: $tType] : ( ( A > B > B ) > ( list @ A ) > B > B ) ).

thf(sy_c_List_Olenlex,type,
    lenlex: 
      !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) ) ).

thf(sy_c_List_Olex,type,
    lex: 
      !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) ) ).

thf(sy_c_List_Olexn,type,
    lexn: 
      !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > nat > ( set @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) ) ).

thf(sy_c_List_Olinorder_Ostable__sort__key,type,
    stable_sort_key: 
      !>[B: $tType,A: $tType] : ( ( ( B > A ) > ( list @ B ) > ( list @ B ) ) > $o ) ).

thf(sy_c_List_Olinorder__class_Osort__key,type,
    linorder_sort_key: 
      !>[B: $tType,A: $tType] : ( ( B > A ) > ( list @ B ) > ( list @ B ) ) ).

thf(sy_c_List_Olinorder__class_Osorted__list__of__set,type,
    linord4507533701916653071of_set: 
      !>[A: $tType] : ( ( set @ A ) > ( list @ A ) ) ).

thf(sy_c_List_Olist_OCons,type,
    cons: 
      !>[A: $tType] : ( A > ( list @ A ) > ( list @ A ) ) ).

thf(sy_c_List_Olist_ONil,type,
    nil: 
      !>[A: $tType] : ( list @ A ) ).

thf(sy_c_List_Olist_Ocase__list,type,
    case_list: 
      !>[B: $tType,A: $tType] : ( B > ( A > ( list @ A ) > B ) > ( list @ A ) > B ) ).

thf(sy_c_List_Olist_Ohd,type,
    hd: 
      !>[A: $tType] : ( ( list @ A ) > A ) ).

thf(sy_c_List_Olist_Omap,type,
    map: 
      !>[A: $tType,Aa: $tType] : ( ( A > Aa ) > ( list @ A ) > ( list @ Aa ) ) ).

thf(sy_c_List_Olist_Oset,type,
    set2: 
      !>[A: $tType] : ( ( list @ A ) > ( set @ A ) ) ).

thf(sy_c_List_Olist_Osize__list,type,
    size_list: 
      !>[A: $tType] : ( ( A > nat ) > ( list @ A ) > nat ) ).

thf(sy_c_List_Olist_Otl,type,
    tl: 
      !>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ).

thf(sy_c_List_Olist__update,type,
    list_update: 
      !>[A: $tType] : ( ( list @ A ) > nat > A > ( list @ A ) ) ).

thf(sy_c_List_Omap__filter,type,
    map_filter: 
      !>[A: $tType,B: $tType] : ( ( A > ( option @ B ) ) > ( list @ A ) > ( list @ B ) ) ).

thf(sy_c_List_Omin__list,type,
    min_list: 
      !>[A: $tType] : ( ( list @ A ) > A ) ).

thf(sy_c_List_Omin__list__rel,type,
    min_list_rel: 
      !>[A: $tType] : ( ( list @ A ) > ( list @ A ) > $o ) ).

thf(sy_c_List_On__lists,type,
    n_lists: 
      !>[A: $tType] : ( nat > ( list @ A ) > ( list @ ( list @ A ) ) ) ).

thf(sy_c_List_Onth,type,
    nth: 
      !>[A: $tType] : ( ( list @ A ) > nat > A ) ).

thf(sy_c_List_Oproduct,type,
    product: 
      !>[A: $tType,B: $tType] : ( ( list @ A ) > ( list @ B ) > ( list @ ( product_prod @ A @ B ) ) ) ).

thf(sy_c_List_Oproduct__lists,type,
    product_lists: 
      !>[A: $tType] : ( ( list @ ( list @ A ) ) > ( list @ ( list @ A ) ) ) ).

thf(sy_c_List_Oreplicate,type,
    replicate: 
      !>[A: $tType] : ( nat > A > ( list @ A ) ) ).

thf(sy_c_List_Oset__Cons,type,
    set_Cons: 
      !>[A: $tType] : ( ( set @ A ) > ( set @ ( list @ A ) ) > ( set @ ( list @ A ) ) ) ).

thf(sy_c_List_Osplice,type,
    splice: 
      !>[A: $tType] : ( ( list @ A ) > ( list @ A ) > ( list @ A ) ) ).

thf(sy_c_List_Osplice__rel,type,
    splice_rel: 
      !>[A: $tType] : ( ( product_prod @ ( list @ A ) @ ( list @ A ) ) > ( product_prod @ ( list @ A ) @ ( list @ A ) ) > $o ) ).

thf(sy_c_List_Osubseqs,type,
    subseqs: 
      !>[A: $tType] : ( ( list @ A ) > ( list @ ( list @ A ) ) ) ).

thf(sy_c_List_Otranspose,type,
    transpose: 
      !>[A: $tType] : ( ( list @ ( list @ A ) ) > ( list @ ( list @ A ) ) ) ).

thf(sy_c_List_Otranspose__rel,type,
    transpose_rel: 
      !>[A: $tType] : ( ( list @ ( list @ A ) ) > ( list @ ( list @ A ) ) > $o ) ).

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_Map_Oran,type,
    ran: 
      !>[A: $tType,B: $tType] : ( ( A > ( option @ B ) ) > ( set @ B ) ) ).

thf(sy_c_Misc_OEps__Opt,type,
    eps_Opt: 
      !>[A: $tType] : ( ( A > $o ) > ( option @ A ) ) ).

thf(sy_c_Misc_Ofun__of__rel,type,
    fun_of_rel: 
      !>[B: $tType,A: $tType] : ( ( set @ ( product_prod @ B @ A ) ) > B > A ) ).

thf(sy_c_Misc_Oinv__on,type,
    inv_on: 
      !>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > B > A ) ).

thf(sy_c_Misc_Omergesort,type,
    mergesort: 
      !>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ).

thf(sy_c_Misc_Omergesort__by__rel,type,
    mergesort_by_rel: 
      !>[A: $tType] : ( ( A > A > $o ) > ( list @ A ) > ( list @ A ) ) ).

thf(sy_c_Misc_Omergesort__by__rel__merge,type,
    merges9089515139780605204_merge: 
      !>[A: $tType] : ( ( A > A > $o ) > ( list @ A ) > ( list @ A ) > ( list @ A ) ) ).

thf(sy_c_Misc_Omergesort__by__rel__merge__rel,type,
    merges2244889521215249637ge_rel: 
      !>[A: $tType] : ( ( product_prod @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) > ( product_prod @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) > $o ) ).

thf(sy_c_Misc_Omergesort__by__rel__rel,type,
    mergesort_by_rel_rel: 
      !>[A: $tType] : ( ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) > ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) > $o ) ).

thf(sy_c_Misc_Omergesort__by__rel__split,type,
    merges295452479951948502_split: 
      !>[A: $tType] : ( ( product_prod @ ( list @ A ) @ ( list @ A ) ) > ( list @ A ) > ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) ).

thf(sy_c_Misc_Omergesort__by__rel__split__rel,type,
    merges7066485432131860899it_rel: 
      !>[A: $tType] : ( ( product_prod @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) ) > ( product_prod @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) ) > $o ) ).

thf(sy_c_Misc_Opairself,type,
    pairself: 
      !>[A: $tType,B: $tType] : ( ( A > B ) > ( product_prod @ A @ A ) > ( product_prod @ B @ B ) ) ).

thf(sy_c_Misc_Opairself__rel,type,
    pairself_rel: 
      !>[A: $tType,B: $tType] : ( ( product_prod @ ( A > B ) @ ( product_prod @ A @ A ) ) > ( product_prod @ ( A > B ) @ ( product_prod @ A @ A ) ) > $o ) ).

thf(sy_c_Misc_Orel__of,type,
    rel_of: 
      !>[A: $tType,B: $tType] : ( ( A > ( option @ B ) ) > ( ( product_prod @ A @ B ) > $o ) > ( set @ ( product_prod @ A @ B ) ) ) ).

thf(sy_c_Misc_Oslice,type,
    slice: 
      !>[A: $tType] : ( nat > nat > ( list @ A ) > ( list @ A ) ) ).

thf(sy_c_Most__significant__bit_Omsb__class_Omsb,type,
    most_s684356279273892711sb_msb: 
      !>[A: $tType] : ( A > $o ) ).

thf(sy_c_Multiset_Olinorder__class_Opart,type,
    linorder_part: 
      !>[B: $tType,A: $tType] : ( ( B > A ) > A > ( list @ B ) > ( product_prod @ ( list @ B ) @ ( product_prod @ ( list @ B ) @ ( list @ B ) ) ) ) ).

thf(sy_c_Multiset_Osubset__eq__mset__impl,type,
    subset_eq_mset_impl: 
      !>[A: $tType] : ( ( list @ A ) > ( list @ A ) > ( option @ $o ) ) ).

thf(sy_c_Multiset_Osubset__eq__mset__impl__rel,type,
    subset751672762298770561pl_rel: 
      !>[A: $tType] : ( ( product_prod @ ( list @ A ) @ ( list @ A ) ) > ( product_prod @ ( list @ A ) @ ( list @ A ) ) > $o ) ).

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

thf(sy_c_Nat_Onat_Ocase__nat,type,
    case_nat: 
      !>[A: $tType] : ( A > ( nat > A ) > nat > A ) ).

thf(sy_c_Nat_Osemiring__1__class_Oof__nat,type,
    semiring_1_of_nat: 
      !>[A: $tType] : ( nat > A ) ).

thf(sy_c_Nat_Osemiring__1__class_Oof__nat__aux,type,
    semiri8178284476397505188at_aux: 
      !>[A: $tType] : ( ( A > A ) > nat > A > A ) ).

thf(sy_c_Nat_Osize__class_Osize,type,
    size_size: 
      !>[A: $tType] : ( A > nat ) ).

thf(sy_c_Nat__Bijection_Oset__decode,type,
    nat_set_decode: nat > ( set @ nat ) ).

thf(sy_c_Nat__Bijection_Oset__encode,type,
    nat_set_encode: ( set @ nat ) > nat ).

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

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

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

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

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

thf(sy_c_Num_Oneg__numeral__class_Odbl,type,
    neg_numeral_dbl: 
      !>[A: $tType] : ( A > A ) ).

thf(sy_c_Num_Oneg__numeral__class_Odbl__dec,type,
    neg_numeral_dbl_dec: 
      !>[A: $tType] : ( A > A ) ).

thf(sy_c_Num_Oneg__numeral__class_Odbl__inc,type,
    neg_numeral_dbl_inc: 
      !>[A: $tType] : ( A > A ) ).

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

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

thf(sy_c_Num_Onum_OOne,type,
    one2: num ).

thf(sy_c_Num_Onum_Ocase__num,type,
    case_num: 
      !>[A: $tType] : ( A > ( num > A ) > ( num > A ) > num > A ) ).

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

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

thf(sy_c_Num_Onumeral__class_Onumeral,type,
    numeral_numeral: 
      !>[A: $tType] : ( num > A ) ).

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

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

thf(sy_c_Option_Ooption_ONone,type,
    none: 
      !>[A: $tType] : ( option @ A ) ).

thf(sy_c_Option_Ooption_OSome,type,
    some: 
      !>[A: $tType] : ( A > ( option @ A ) ) ).

thf(sy_c_Option_Ooption_Ocase__option,type,
    case_option: 
      !>[B: $tType,A: $tType] : ( B > ( A > B ) > ( option @ A ) > B ) ).

thf(sy_c_Option_Ooption_Osize__option,type,
    size_option: 
      !>[A: $tType] : ( ( A > nat ) > ( option @ A ) > nat ) ).

thf(sy_c_Option_Ooption_Othe,type,
    the2: 
      !>[A: $tType] : ( ( option @ A ) > A ) ).

thf(sy_c_Orderings_Obot__class_Obot,type,
    bot_bot: 
      !>[A: $tType] : A ).

thf(sy_c_Orderings_Oord__class_OLeast,type,
    ord_Least: 
      !>[A: $tType] : ( ( A > $o ) > A ) ).

thf(sy_c_Orderings_Oord__class_Oless,type,
    ord_less: 
      !>[A: $tType] : ( A > A > $o ) ).

thf(sy_c_Orderings_Oord__class_Oless__eq,type,
    ord_less_eq: 
      !>[A: $tType] : ( A > A > $o ) ).

thf(sy_c_Orderings_Oord__class_Omax,type,
    ord_max: 
      !>[A: $tType] : ( A > A > A ) ).

thf(sy_c_Orderings_Oord__class_Omin,type,
    ord_min: 
      !>[A: $tType] : ( A > A > A ) ).

thf(sy_c_Orderings_Oorder__class_OGreatest,type,
    order_Greatest: 
      !>[A: $tType] : ( ( A > $o ) > A ) ).

thf(sy_c_Orderings_Otop__class_Otop,type,
    top_top: 
      !>[A: $tType] : A ).

thf(sy_c_Partial__Function_Oflat__lub,type,
    partial_flat_lub: 
      !>[A: $tType] : ( A > ( set @ A ) > A ) ).

thf(sy_c_Partial__Function_Oflat__ord,type,
    partial_flat_ord: 
      !>[A: $tType] : ( A > A > A > $o ) ).

thf(sy_c_Partial__Function_Ofun__lub,type,
    partial_fun_lub: 
      !>[C: $tType,B: $tType,A: $tType] : ( ( ( set @ C ) > B ) > ( set @ ( A > C ) ) > A > B ) ).

thf(sy_c_Partial__Function_Ofun__ord,type,
    partial_fun_ord: 
      !>[A: $tType,B: $tType,C: $tType] : ( ( A > B > $o ) > ( C > A ) > ( C > B ) > $o ) ).

thf(sy_c_Partial__Function_Oimg__lub,type,
    partial_img_lub: 
      !>[A: $tType,C: $tType,D: $tType,B: $tType] : ( ( A > C ) > ( D > B ) > ( ( set @ C ) > D ) > ( set @ A ) > B ) ).

thf(sy_c_Partial__Function_Oimg__ord,type,
    partial_img_ord: 
      !>[A: $tType,C: $tType,B: $tType] : ( ( A > C ) > ( C > C > B ) > A > A > B ) ).

thf(sy_c_Partial__Function_Opartial__function__definitions,type,
    partia7178651479351089652itions: 
      !>[A: $tType] : ( ( A > A > $o ) > ( ( set @ A ) > A ) > $o ) ).

thf(sy_c_Power_Opower__class_Opower,type,
    power_power: 
      !>[A: $tType] : ( A > nat > A ) ).

thf(sy_c_Product__Type_OPair,type,
    product_Pair: 
      !>[A: $tType,B: $tType] : ( A > B > ( product_prod @ A @ B ) ) ).

thf(sy_c_Product__Type_Oapsnd,type,
    product_apsnd: 
      !>[B: $tType,C: $tType,A: $tType] : ( ( B > C ) > ( product_prod @ A @ B ) > ( product_prod @ A @ C ) ) ).

thf(sy_c_Product__Type_Ocurry,type,
    product_curry: 
      !>[A: $tType,B: $tType,C: $tType] : ( ( ( product_prod @ A @ B ) > C ) > A > B > C ) ).

thf(sy_c_Product__Type_Oold_Oprod_Orec__prod,type,
    product_rec_prod: 
      !>[A: $tType,B: $tType,T: $tType] : ( ( A > B > T ) > ( product_prod @ A @ B ) > T ) ).

thf(sy_c_Product__Type_Oold_Oprod_Orec__set__prod,type,
    product_rec_set_prod: 
      !>[A: $tType,B: $tType,T: $tType] : ( ( A > B > T ) > ( product_prod @ A @ B ) > T > $o ) ).

thf(sy_c_Product__Type_Oprod_Ocase__prod,type,
    product_case_prod: 
      !>[A: $tType,B: $tType,C: $tType] : ( ( A > B > C ) > ( product_prod @ A @ B ) > C ) ).

thf(sy_c_Product__Type_Oprod_Ofst,type,
    product_fst: 
      !>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > A ) ).

thf(sy_c_Product__Type_Oprod_Osnd,type,
    product_snd: 
      !>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > B ) ).

thf(sy_c_Product__Type_Oproduct,type,
    product_product: 
      !>[A: $tType,B: $tType] : ( ( set @ A ) > ( set @ B ) > ( set @ ( product_prod @ A @ B ) ) ) ).

thf(sy_c_Pure_Otype,type,
    type2: 
      !>[A: $tType] : ( itself @ A ) ).

thf(sy_c_Quicksort_Olinorder__class_Oquicksort,type,
    linorder_quicksort: 
      !>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ).

thf(sy_c_Quicksort_Olinorder__class_Oquicksort__rel,type,
    linord6200660962353139674rt_rel: 
      !>[A: $tType] : ( ( list @ A ) > ( list @ A ) > $o ) ).

thf(sy_c_Rat_OFrct,type,
    frct: ( product_prod @ int @ int ) > rat ).

thf(sy_c_Rat_Onormalize,type,
    normalize: ( product_prod @ int @ int ) > ( product_prod @ int @ int ) ).

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

thf(sy_c_Rat_Oquotient__of,type,
    quotient_of: rat > ( product_prod @ int @ int ) ).

thf(sy_c_Real__Vector__Spaces_Onorm__class_Onorm,type,
    real_V7770717601297561774m_norm: 
      !>[A: $tType] : ( A > real ) ).

thf(sy_c_Real__Vector__Spaces_Oof__real,type,
    real_Vector_of_real: 
      !>[A: $tType] : ( real > A ) ).

thf(sy_c_Real__Vector__Spaces_OscaleR__class_OscaleR,type,
    real_V8093663219630862766scaleR: 
      !>[A: $tType] : ( real > A > A ) ).

thf(sy_c_Refine__Imp__Hol_Oassert_H,type,
    refine_Imp_assert: $o > ( heap_Time_Heap @ product_unit ) ).

thf(sy_c_Refine__Imp__Hol_Orefines,type,
    refine_Imp_refines: 
      !>[A: $tType] : ( ( heap_Time_Heap @ A ) > ( heap_Time_Heap @ A ) > $o ) ).

thf(sy_c_Rings_Odivide__class_Odivide,type,
    divide_divide: 
      !>[A: $tType] : ( A > A > A ) ).

thf(sy_c_Rings_Odvd__class_Odvd,type,
    dvd_dvd: 
      !>[A: $tType] : ( A > A > $o ) ).

thf(sy_c_Rings_Omodulo__class_Omodulo,type,
    modulo_modulo: 
      !>[A: $tType] : ( A > A > A ) ).

thf(sy_c_Rings_Ozero__neq__one__class_Oof__bool,type,
    zero_neq_one_of_bool: 
      !>[A: $tType] : ( $o > A ) ).

thf(sy_c_Series_Osuminf,type,
    suminf: 
      !>[A: $tType] : ( ( nat > A ) > A ) ).

thf(sy_c_Series_Osummable,type,
    summable: 
      !>[A: $tType] : ( ( nat > A ) > $o ) ).

thf(sy_c_Series_Osums,type,
    sums: 
      !>[A: $tType] : ( ( nat > A ) > A > $o ) ).

thf(sy_c_Set_OBall,type,
    ball: 
      !>[A: $tType] : ( ( set @ A ) > ( A > $o ) > $o ) ).

thf(sy_c_Set_OCollect,type,
    collect: 
      !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).

thf(sy_c_Set_Ofilter,type,
    filter3: 
      !>[A: $tType] : ( ( A > $o ) > ( set @ A ) > ( set @ A ) ) ).

thf(sy_c_Set_Oimage,type,
    image: 
      !>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > ( set @ B ) ) ).

thf(sy_c_Set_Oinsert,type,
    insert: 
      !>[A: $tType] : ( A > ( set @ A ) > ( set @ A ) ) ).

thf(sy_c_Set_Othe__elem,type,
    the_elem: 
      !>[A: $tType] : ( ( set @ A ) > A ) ).

thf(sy_c_Set__Interval_Ofold__atLeastAtMost__nat,type,
    set_fo6178422350223883121st_nat: 
      !>[A: $tType] : ( ( nat > A > A ) > nat > nat > A > A ) ).

thf(sy_c_Set__Interval_Ofold__atLeastAtMost__nat__rel,type,
    set_fo1817059534552279752at_rel: 
      !>[A: $tType] : ( ( product_prod @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) ) ) > ( product_prod @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) ) ) > $o ) ).

thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost,type,
    set_or1337092689740270186AtMost: 
      !>[A: $tType] : ( A > A > ( set @ A ) ) ).

thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan,type,
    set_or7035219750837199246ssThan: 
      !>[A: $tType] : ( A > A > ( set @ A ) ) ).

thf(sy_c_Set__Interval_Oord__class_OatMost,type,
    set_ord_atMost: 
      !>[A: $tType] : ( A > ( set @ A ) ) ).

thf(sy_c_Set__Interval_Oord__class_OlessThan,type,
    set_ord_lessThan: 
      !>[A: $tType] : ( A > ( set @ A ) ) ).

thf(sy_c_Signed__Division_Osigned__division__class_Osigned__divide,type,
    signed7115095781618012415divide: 
      !>[A: $tType] : ( A > A > A ) ).

thf(sy_c_Signed__Division_Osigned__division__class_Osigned__modulo,type,
    signed6721504322012087516modulo: 
      !>[A: $tType] : ( A > A > A ) ).

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__of__integer,type,
    char_of_integer: code_integer > char ).

thf(sy_c_String_Ocomm__semiring__1__class_Oof__char,type,
    comm_s6883823935334413003f_char: 
      !>[A: $tType] : ( char > A ) ).

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,type,
    unique5772411509450598832har_of: 
      !>[A: $tType] : ( A > char ) ).

thf(sy_c_Syntax__Match_Osyntax__fo__nomatch,type,
    syntax7388354845996824322omatch: 
      !>[A: $tType,B: $tType] : ( A > B > $o ) ).

thf(sy_c_Time__Reasoning_OEQ,type,
    time_EQ: 
      !>[A: $tType] : ( A > A > $o ) ).

thf(sy_c_Time__Reasoning_OTBOUND,type,
    time_TBOUND: 
      !>[A: $tType] : ( ( heap_Time_Heap @ A ) > nat > $o ) ).

thf(sy_c_Time__Reasoning_Ofails,type,
    time_fails: 
      !>[A: $tType] : ( ( heap_Time_Heap @ A ) > ( heap_ext @ product_unit ) > $o ) ).

thf(sy_c_Time__Reasoning_Ohtt,type,
    time_htt: 
      !>[A: $tType] : ( assn > ( heap_Time_Heap @ A ) > ( A > assn ) > nat > $o ) ).

thf(sy_c_Time__Reasoning_Othe__heap,type,
    time_the_heap: 
      !>[A: $tType] : ( ( heap_Time_Heap @ A ) > ( heap_ext @ product_unit ) > ( heap_ext @ product_unit ) ) ).

thf(sy_c_Time__Reasoning_Othe__res,type,
    time_the_res: 
      !>[A: $tType] : ( ( heap_Time_Heap @ A ) > ( heap_ext @ product_unit ) > A ) ).

thf(sy_c_Time__Reasoning_Otime,type,
    time_time: 
      !>[A: $tType] : ( ( heap_Time_Heap @ A ) > ( heap_ext @ product_unit ) > nat ) ).

thf(sy_c_Topological__Spaces_Omonoseq,type,
    topological_monoseq: 
      !>[A: $tType] : ( ( nat > A ) > $o ) ).

thf(sy_c_Topological__Spaces_Otopological__space__class_Oat__within,type,
    topolo174197925503356063within: 
      !>[A: $tType] : ( A > ( set @ A ) > ( filter @ A ) ) ).

thf(sy_c_Topological__Spaces_Ouniform__space__class_OCauchy,type,
    topolo3814608138187158403Cauchy: 
      !>[A: $tType] : ( ( nat > A ) > $o ) ).

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

thf(sy_c_Transcendental_Oarcosh,type,
    arcosh: 
      !>[A: $tType] : ( A > A ) ).

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

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

thf(sy_c_Transcendental_Oarsinh,type,
    arsinh: 
      !>[A: $tType] : ( A > A ) ).

thf(sy_c_Transcendental_Oartanh,type,
    artanh: 
      !>[A: $tType] : ( A > A ) ).

thf(sy_c_Transcendental_Ocos,type,
    cos: 
      !>[A: $tType] : ( A > A ) ).

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

thf(sy_c_Transcendental_Ocosh,type,
    cosh: 
      !>[A: $tType] : ( A > A ) ).

thf(sy_c_Transcendental_Ocot,type,
    cot: 
      !>[A: $tType] : ( A > A ) ).

thf(sy_c_Transcendental_Odiffs,type,
    diffs: 
      !>[A: $tType] : ( ( nat > A ) > nat > A ) ).

thf(sy_c_Transcendental_Oexp,type,
    exp: 
      !>[A: $tType] : ( A > A ) ).

thf(sy_c_Transcendental_Oln__class_Oln,type,
    ln_ln: 
      !>[A: $tType] : ( A > A ) ).

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

thf(sy_c_Transcendental_Opi,type,
    pi: real ).

thf(sy_c_Transcendental_Opowr,type,
    powr: 
      !>[A: $tType] : ( A > A > A ) ).

thf(sy_c_Transcendental_Osin,type,
    sin: 
      !>[A: $tType] : ( A > A ) ).

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

thf(sy_c_Transcendental_Osinh,type,
    sinh: 
      !>[A: $tType] : ( A > A ) ).

thf(sy_c_Transcendental_Otan,type,
    tan: 
      !>[A: $tType] : ( A > A ) ).

thf(sy_c_Transcendental_Otanh,type,
    tanh: 
      !>[A: $tType] : ( A > A ) ).

thf(sy_c_Type__Length_Olen0__class_Olen__of,type,
    type_len0_len_of: 
      !>[A: $tType] : ( ( itself @ A ) > nat ) ).

thf(sy_c_Uint32_ORep__uint32_H,type,
    rep_uint32: uint32 > ( word @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ numeral_num1 ) ) ) ) ) ) ).

thf(sy_c_Uint32_Odiv0__uint32,type,
    div0_uint32: uint32 > uint32 ).

thf(sy_c_Uint32_Ointeger__of__uint32,type,
    integer_of_uint32: uint32 > code_integer ).

thf(sy_c_Uint32_Ointeger__of__uint32__signed,type,
    intege5370686899274169573signed: uint32 > code_integer ).

thf(sy_c_Uint32_Omod0__uint32,type,
    mod0_uint32: uint32 > uint32 ).

thf(sy_c_Uint32_Oset__bits__aux__uint32,type,
    set_bits_aux_uint32: ( nat > $o ) > nat > uint32 > uint32 ).

thf(sy_c_Uint32_Osigned__drop__bit__uint32,type,
    signed489701013188660438uint32: nat > uint32 > uint32 ).

thf(sy_c_Uint32_Ouint32_OAbs__uint32,type,
    abs_uint32: ( word @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ numeral_num1 ) ) ) ) ) ) > uint32 ).

thf(sy_c_Uint32_Ouint32_ORep__uint32,type,
    rep_uint322: uint32 > ( word @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ numeral_num1 ) ) ) ) ) ) ).

thf(sy_c_Uint32_Ouint32__div,type,
    uint32_div: uint32 > uint32 > uint32 ).

thf(sy_c_Uint32_Ouint32__divmod,type,
    uint32_divmod: uint32 > uint32 > ( product_prod @ uint32 @ uint32 ) ).

thf(sy_c_Uint32_Ouint32__sdiv,type,
    uint32_sdiv: uint32 > uint32 > uint32 ).

thf(sy_c_Uint32_Ouint32__set__bit,type,
    uint32_set_bit: uint32 > code_integer > $o > uint32 ).

thf(sy_c_Uint32_Ouint32__shiftl,type,
    uint32_shiftl: uint32 > code_integer > uint32 ).

thf(sy_c_Uint32_Ouint32__shiftr,type,
    uint32_shiftr: uint32 > code_integer > uint32 ).

thf(sy_c_Uint32_Ouint32__sshiftr,type,
    uint32_sshiftr: uint32 > code_integer > uint32 ).

thf(sy_c_Uint32_Ouint32__test__bit,type,
    uint32_test_bit: uint32 > code_integer > $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,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: ( product_prod @ vEBT_VEBT @ nat ) > ( product_prod @ vEBT_VEBT @ 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: ( product_prod @ vEBT_VEBT @ nat ) > ( product_prod @ vEBT_VEBT @ 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: ( product_prod @ vEBT_VEBT @ nat ) > ( product_prod @ vEBT_VEBT @ 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: ( product_prod @ vEBT_VEBT @ nat ) > ( product_prod @ vEBT_VEBT @ 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: ( product_prod @ vEBT_VEBT @ nat ) > ( product_prod @ vEBT_VEBT @ 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: ( product_prod @ vEBT_VEBT @ nat ) > ( product_prod @ vEBT_VEBT @ 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: ( product_prod @ vEBT_VEBT @ nat ) > ( product_prod @ vEBT_VEBT @ 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: ( product_prod @ vEBT_VEBT @ nat ) > ( product_prod @ vEBT_VEBT @ 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_OminNulli,type,
    vEBT_VEBT_minNulli: vEBT_VEBTi > ( heap_Time_Heap @ $o ) ).

thf(sy_c_VEBT__BuildupMemImp_OVEBT__internal_OminNulli__rel,type,
    vEBT_V5740978063120863272li_rel: vEBT_VEBTi > vEBT_VEBTi > $o ).

thf(sy_c_VEBT__BuildupMemImp_OVEBT__internal_Oreplicatei,type,
    vEBT_VEBT_replicatei: 
      !>[A: $tType] : ( nat > ( heap_Time_Heap @ A ) > ( heap_Time_Heap @ ( list @ A ) ) ) ).

thf(sy_c_VEBT__BuildupMemImp_OVEBT__internal_Oreplicatei__rel,type,
    vEBT_V8535202610096940344ei_rel: 
      !>[A: $tType] : ( ( product_prod @ nat @ ( heap_Time_Heap @ A ) ) > ( product_prod @ nat @ ( heap_Time_Heap @ A ) ) > $o ) ).

thf(sy_c_VEBT__BuildupMemImp_OVEBT__internal_Ovebt__buildupi_H,type,
    vEBT_V739175172307565963ildupi: nat > ( heap_Time_Heap @ vEBT_VEBTi ) ).

thf(sy_c_VEBT__BuildupMemImp_OVEBT__internal_Ovebt__buildupi_H__rel,type,
    vEBT_V254170901696579886pi_rel: nat > nat > $o ).

thf(sy_c_VEBT__BuildupMemImp_OVEBT__internal_Ovebt__inserti_H,type,
    vEBT_V3964819847710782039nserti: vEBT_VEBT > vEBT_VEBTi > nat > ( heap_Time_Heap @ vEBT_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: ( option @ ( product_prod @ nat @ nat ) ) > nat > ( array @ vEBT_VEBTi ) > vEBT_VEBTi > vEBT_VEBTi ).

thf(sy_c_VEBT__BuildupMemImp_OVEBTi_Ocase__VEBTi,type,
    vEBT_case_VEBTi: 
      !>[A: $tType] : ( ( ( option @ ( product_prod @ nat @ nat ) ) > nat > ( array @ vEBT_VEBTi ) > vEBT_VEBTi > A ) > ( $o > $o > A ) > vEBT_VEBTi > A ) ).

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: ( product_prod @ vEBT_VEBT @ vEBT_VEBTi ) > ( product_prod @ vEBT_VEBT @ vEBT_VEBTi ) > $o ).

thf(sy_c_VEBT__BuildupMemImp_Ovebt__buildupi,type,
    vEBT_vebt_buildupi: nat > ( heap_Time_Heap @ vEBT_VEBTi ) ).

thf(sy_c_VEBT__BuildupMemImp_Ovebt__buildupi__rel,type,
    vEBT_v1230518104690509829pi_rel: nat > nat > $o ).

thf(sy_c_VEBT__BuildupMemImp_Ovebt__inserti,type,
    vEBT_vebt_inserti: vEBT_VEBTi > nat > ( heap_Time_Heap @ vEBT_VEBTi ) ).

thf(sy_c_VEBT__BuildupMemImp_Ovebt__maxti,type,
    vEBT_vebt_maxti: vEBT_VEBTi > ( heap_Time_Heap @ ( option @ 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__memberi,type,
    vEBT_vebt_memberi: vEBT_VEBTi > nat > ( heap_Time_Heap @ $o ) ).

thf(sy_c_VEBT__BuildupMemImp_Ovebt__minti,type,
    vEBT_vebt_minti: vEBT_VEBTi > ( heap_Time_Heap @ ( option @ 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: ( option @ ( product_prod @ nat @ nat ) ) > nat > ( list @ vEBT_VEBT ) > vEBT_VEBT > vEBT_VEBT ).

thf(sy_c_VEBT__Definitions_OVEBT_Ocase__VEBT,type,
    vEBT_case_VEBT: 
      !>[A: $tType] : ( ( ( option @ ( product_prod @ nat @ nat ) ) > nat > ( list @ vEBT_VEBT ) > vEBT_VEBT > A ) > ( $o > $o > A ) > vEBT_VEBT > A ) ).

thf(sy_c_VEBT__Definitions_OVEBT_Ois__Node,type,
    vEBT_is_Node: vEBT_VEBT > $o ).

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: ( product_prod @ vEBT_VEBT @ nat ) > ( product_prod @ vEBT_VEBT @ 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: ( product_prod @ vEBT_VEBT @ nat ) > ( product_prod @ vEBT_VEBT @ nat ) > $o ).

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

thf(sy_c_VEBT__Definitions_OVEBT__internal_Ovalid_H__rel,type,
    vEBT_VEBT_valid_rel: ( product_prod @ vEBT_VEBT @ nat ) > ( product_prod @ 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: ( product_prod @ vEBT_VEBT @ nat ) > ( product_prod @ vEBT_VEBT @ 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: ( product_prod @ vEBT_VEBT @ nat ) > ( product_prod @ vEBT_VEBT @ 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: ( product_prod @ vEBT_VEBT @ nat ) > ( product_prod @ vEBT_VEBT @ 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: ( product_prod @ vEBT_VEBT @ nat ) > ( product_prod @ vEBT_VEBT @ nat ) > $o ).

thf(sy_c_VEBT__List__Assn_OlistI__assn,type,
    vEBT_List_listI_assn: 
      !>[A: $tType,B: $tType] : ( ( set @ nat ) > ( A > B > assn ) > ( list @ A ) > ( list @ B ) > assn ) ).

thf(sy_c_VEBT__List__Assn_Olist__assn,type,
    vEBT_List_list_assn: 
      !>[A: $tType,C: $tType] : ( ( A > C > assn ) > ( list @ A ) > ( list @ C ) > assn ) ).

thf(sy_c_VEBT__List__Assn_Olist__assn__rel,type,
    vEBT_L4249061453398456502sn_rel: 
      !>[A: $tType,C: $tType] : ( ( product_prod @ ( A > C > assn ) @ ( product_prod @ ( list @ A ) @ ( list @ C ) ) ) > ( product_prod @ ( A > C > assn ) @ ( product_prod @ ( list @ A ) @ ( list @ C ) ) ) > $o ) ).

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

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

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

thf(sy_c_VEBT__Member_OVEBT__internal_Oset__vebt_H,type,
    vEBT_VEBT_set_vebt: vEBT_VEBT > ( set @ nat ) ).

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

thf(sy_c_VEBT__Member_Ovebt__member__rel,type,
    vEBT_vebt_member_rel: ( product_prod @ vEBT_VEBT @ nat ) > ( product_prod @ vEBT_VEBT @ 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,type,
    vEBT_V2048590022279873568_shift: 
      !>[A: $tType] : ( ( A > A > A ) > ( option @ A ) > ( option @ A ) > ( option @ A ) ) ).

thf(sy_c_VEBT__MinMax_OVEBT__internal_Ooption__shift__rel,type,
    vEBT_V459564278314245337ft_rel: 
      !>[A: $tType] : ( ( product_prod @ ( A > A > A ) @ ( product_prod @ ( option @ A ) @ ( option @ A ) ) ) > ( product_prod @ ( A > A > A ) @ ( product_prod @ ( option @ A ) @ ( option @ A ) ) ) > $o ) ).

thf(sy_c_VEBT__MinMax_OVEBT__internal_Opower,type,
    vEBT_VEBT_power: ( option @ nat ) > ( option @ nat ) > ( option @ nat ) ).

thf(sy_c_VEBT__MinMax_Ovebt__maxt,type,
    vEBT_vebt_maxt: vEBT_VEBT > ( option @ nat ) ).

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

thf(sy_c_VEBT__MinMax_Ovebt__mint,type,
    vEBT_vebt_mint: vEBT_VEBT > ( option @ nat ) ).

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

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

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

thf(sy_c_VEBT__Pred_Ovebt__pred__rel,type,
    vEBT_vebt_pred_rel: ( product_prod @ vEBT_VEBT @ nat ) > ( product_prod @ vEBT_VEBT @ 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__SuccPredImperative_OVEBT__internal_Ovebt__predi_H,type,
    vEBT_VEBT_vebt_predi: vEBT_VEBT > vEBT_VEBTi > nat > ( heap_Time_Heap @ ( option @ nat ) ) ).

thf(sy_c_VEBT__SuccPredImperative_OVEBT__internal_Ovebt__succi_H,type,
    vEBT_VEBT_vebt_succi: vEBT_VEBT > vEBT_VEBTi > nat > ( heap_Time_Heap @ ( option @ nat ) ) ).

thf(sy_c_VEBT__SuccPredImperative_Ovebt__predi,type,
    vEBT_vebt_predi: vEBT_VEBTi > nat > ( heap_Time_Heap @ ( option @ nat ) ) ).

thf(sy_c_VEBT__SuccPredImperative_Ovebt__succi,type,
    vEBT_vebt_succi: vEBT_VEBTi > nat > ( heap_Time_Heap @ ( option @ nat ) ) ).

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: ( product_prod @ vEBT_VEBT @ nat ) > ( product_prod @ vEBT_VEBT @ nat ) > $o ).

thf(sy_c_Wellfounded_Oaccp,type,
    accp: 
      !>[A: $tType] : ( ( A > A > $o ) > A > $o ) ).

thf(sy_c_Wellfounded_Ofinite__psubset,type,
    finite_psubset: 
      !>[A: $tType] : ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) ).

thf(sy_c_Wellfounded_Omeasure,type,
    measure: 
      !>[A: $tType] : ( ( A > nat ) > ( set @ ( product_prod @ A @ A ) ) ) ).

thf(sy_c_Word_OWord,type,
    word2: 
      !>[A: $tType] : ( int > ( word @ A ) ) ).

thf(sy_c_Word_Oeven__word,type,
    even_word: 
      !>[A: $tType] : ( ( word @ A ) > $o ) ).

thf(sy_c_Word_Orevcast,type,
    revcast: 
      !>[A: $tType,B: $tType] : ( ( word @ A ) > ( word @ B ) ) ).

thf(sy_c_Word_Oring__1__class_Osigned,type,
    ring_1_signed: 
      !>[B: $tType,A: $tType] : ( ( word @ B ) > A ) ).

thf(sy_c_Word_Osemiring__1__class_Ounsigned,type,
    semiring_1_unsigned: 
      !>[B: $tType,A: $tType] : ( ( word @ B ) > A ) ).

thf(sy_c_Word_Osigned__cast,type,
    signed_cast: 
      !>[A: $tType,B: $tType] : ( ( word @ A ) > ( word @ B ) ) ).

thf(sy_c_Word_Osigned__drop__bit,type,
    signed_drop_bit: 
      !>[A: $tType] : ( nat > ( word @ A ) > ( word @ A ) ) ).

thf(sy_c_Word_Oslice,type,
    slice2: 
      !>[A: $tType,B: $tType] : ( nat > ( word @ A ) > ( word @ B ) ) ).

thf(sy_c_Word_Oslice1,type,
    slice1: 
      !>[A: $tType,B: $tType] : ( nat > ( word @ A ) > ( word @ B ) ) ).

thf(sy_c_Word_Othe__signed__int,type,
    the_signed_int: 
      !>[A: $tType] : ( ( word @ A ) > int ) ).

thf(sy_c_Word_Oword__cat,type,
    word_cat: 
      !>[A: $tType,B: $tType,C: $tType] : ( ( word @ A ) > ( word @ B ) > ( word @ C ) ) ).

thf(sy_c_Word_Oword__int__case,type,
    word_int_case: 
      !>[B: $tType,A: $tType] : ( ( int > B ) > ( word @ A ) > B ) ).

thf(sy_c_Word_Oword__pred,type,
    word_pred: 
      !>[A: $tType] : ( ( word @ A ) > ( word @ A ) ) ).

thf(sy_c_Word_Oword__roti,type,
    word_roti: 
      !>[A: $tType] : ( int > ( word @ A ) > ( word @ A ) ) ).

thf(sy_c_Word_Oword__rotl,type,
    word_rotl: 
      !>[A: $tType] : ( nat > ( word @ A ) > ( word @ A ) ) ).

thf(sy_c_Word_Oword__rotr,type,
    word_rotr: 
      !>[A: $tType] : ( nat > ( word @ A ) > ( word @ A ) ) ).

thf(sy_c_Word_Oword__sle,type,
    word_sle: 
      !>[A: $tType] : ( ( word @ A ) > ( word @ A ) > $o ) ).

thf(sy_c_Word_Oword__sless,type,
    word_sless: 
      !>[A: $tType] : ( ( word @ A ) > ( word @ A ) > $o ) ).

thf(sy_c_Word_Oword__split,type,
    word_split: 
      !>[A: $tType,B: $tType,C: $tType] : ( ( word @ A ) > ( product_prod @ ( word @ B ) @ ( word @ C ) ) ) ).

thf(sy_c_Word_Oword__succ,type,
    word_succ: 
      !>[A: $tType] : ( ( word @ A ) > ( word @ A ) ) ).

thf(sy_c_Word__Type__Copies_Oword__type__copy__misc,type,
    word_T1870391057261780392y_misc: 
      !>[B: $tType,A: $tType] : ( ( ( word @ B ) > A ) > ( A > ( word @ B ) ) > ( nat > A > A ) > ( nat > A ) > ( A > nat ) > ( int > A ) > ( A > int ) > ( code_integer > A ) > ( A > code_integer ) > nat > ( ( nat > $o ) > nat > A > A ) > $o ) ).

thf(sy_c_Word__Type__Copies_Oword__type__copy__misc__axioms,type,
    word_T8964210463127689547axioms: 
      !>[A: $tType,B: $tType] : ( ( A > ( word @ B ) ) > nat > ( ( nat > $o ) > nat > A > A ) > $o ) ).

thf(sy_c_fChoice,type,
    fChoice: 
      !>[A: $tType] : ( ( A > $o ) > A ) ).

thf(sy_c_member,type,
    member: 
      !>[A: $tType] : ( A > ( set @ A ) > $o ) ).

thf(sy_v_ma____,type,
    ma: nat ).

thf(sy_v_mi____,type,
    mi: nat ).

thf(sy_v_na____,type,
    na: nat ).

thf(sy_v_summary____,type,
    summary: vEBT_VEBT ).

thf(sy_v_tia____,type,
    tia: vEBT_VEBTi ).

thf(sy_v_treeList____,type,
    treeList: list @ vEBT_VEBT ).

thf(sy_v_tree__is,type,
    tree_is: list @ vEBT_VEBTi ).

thf(sy_v_va____,type,
    va: nat ).

thf(sy_v_x11______,type,
    x11: option @ ( product_prod @ nat @ nat ) ).

thf(sy_v_x12______,type,
    x12: 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 (8174)
thf(fact_0_greater__shift,axiom,
    ( ( ord_less @ nat )
    = ( ^ [Y: nat,X: nat] : ( vEBT_VEBT_greater @ ( some @ nat @ X ) @ ( some @ nat @ Y ) ) ) ) ).

% greater_shift
thf(fact_1_max__in__set__def,axiom,
    ( vEBT_VEBT_max_in_set
    = ( ^ [Xs: set @ nat,X: nat] :
          ( ( member @ nat @ X @ Xs )
          & ! [Y: nat] :
              ( ( member @ nat @ Y @ Xs )
             => ( ord_less_eq @ nat @ Y @ X ) ) ) ) ) ).

% max_in_set_def
thf(fact_2_min__in__set__def,axiom,
    ( vEBT_VEBT_min_in_set
    = ( ^ [Xs: set @ nat,X: nat] :
          ( ( member @ nat @ X @ Xs )
          & ! [Y: nat] :
              ( ( member @ nat @ Y @ Xs )
             => ( ord_less_eq @ nat @ X @ Y ) ) ) ) ) ).

% min_in_set_def
thf(fact_3_power__shift,axiom,
    ! [X2: nat,Y2: nat,Z: nat] :
      ( ( ( power_power @ nat @ X2 @ Y2 )
        = Z )
      = ( ( vEBT_VEBT_power @ ( some @ nat @ X2 ) @ ( some @ nat @ Y2 ) )
        = ( some @ nat @ Z ) ) ) ).

% power_shift
thf(fact_4_bit__split__inv,axiom,
    ! [X2: nat,D2: nat] :
      ( ( vEBT_VEBT_bit_concat @ ( vEBT_VEBT_high @ X2 @ D2 ) @ ( vEBT_VEBT_low @ X2 @ D2 ) @ D2 )
      = X2 ) ).

% bit_split_inv
thf(fact_5_mimapr,axiom,
    ord_less_eq @ nat @ mi @ ma ).

% mimapr
thf(fact_6_high__def,axiom,
    ( vEBT_VEBT_high
    = ( ^ [X: nat,N: nat] : ( divide_divide @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).

% high_def
thf(fact_7_vebt__minti__h,axiom,
    ! [T2: vEBT_VEBT,Ti: vEBT_VEBTi] :
      ( hoare_hoare_triple @ ( option @ nat ) @ ( vEBT_vebt_assn_raw @ T2 @ Ti ) @ ( vEBT_vebt_minti @ Ti )
      @ ^ [R: option @ nat] :
          ( times_times @ assn @ ( vEBT_vebt_assn_raw @ T2 @ Ti )
          @ ( pure_assn
            @ ( R
              = ( vEBT_vebt_mint @ T2 ) ) ) ) ) ).

% vebt_minti_h
thf(fact_8_vebt__maxti__h,axiom,
    ! [T2: vEBT_VEBT,Ti: vEBT_VEBTi] :
      ( hoare_hoare_triple @ ( option @ nat ) @ ( vEBT_vebt_assn_raw @ T2 @ Ti ) @ ( vEBT_vebt_maxti @ Ti )
      @ ^ [R: option @ nat] :
          ( times_times @ assn @ ( vEBT_vebt_assn_raw @ T2 @ Ti )
          @ ( pure_assn
            @ ( R
              = ( vEBT_vebt_maxt @ T2 ) ) ) ) ) ).

% vebt_maxti_h
thf(fact_9_lesseq__shift,axiom,
    ( ( ord_less_eq @ nat )
    = ( ^ [X: nat,Y: nat] : ( vEBT_VEBT_lesseq @ ( some @ nat @ X ) @ ( some @ nat @ Y ) ) ) ) ).

% lesseq_shift
thf(fact_10__C7_OIH_C_I2_J,axiom,
    ! [Xa: nat,Xb: nat,Xc: option @ nat,N2: nat,Ti: vEBT_VEBTi] :
      ( ~ ( ord_less @ nat @ ma @ xa )
     => ( ( Xa
          = ( vEBT_VEBT_low @ xa @ ( divide_divide @ nat @ ( suc @ ( suc @ va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
       => ( ( Xb
            = ( vEBT_VEBT_high @ xa @ ( divide_divide @ nat @ ( suc @ ( suc @ va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
         => ( ( ord_less @ nat @ Xb @ ( size_size @ ( list @ vEBT_VEBT ) @ treeList ) )
           => ( ( Xc
                = ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ treeList @ Xb ) ) )
             => ( ~ ( ( Xc
                     != ( none @ nat ) )
                    & ( vEBT_VEBT_greater @ ( some @ nat @ Xa ) @ Xc ) )
               => ( ( vEBT_invar_vebt @ summary @ N2 )
                 => ( hoare_hoare_triple @ ( option @ nat ) @ ( vEBT_vebt_assn_raw @ summary @ Ti ) @ ( vEBT_VEBT_vebt_predi @ summary @ Ti @ Xb )
                    @ ^ [R: option @ nat] :
                        ( times_times @ assn @ ( vEBT_vebt_assn_raw @ summary @ Ti )
                        @ ( pure_assn
                          @ ( R
                            = ( vEBT_vebt_pred @ summary @ Xb ) ) ) ) ) ) ) ) ) ) ) ) ).

% "7.IH"(2)
thf(fact_11__C7_OIH_C_I1_J,axiom,
    ! [Xa: nat,Xb: nat,Xc: option @ nat,N2: nat,Ti: vEBT_VEBTi] :
      ( ~ ( ord_less @ nat @ ma @ xa )
     => ( ( Xa
          = ( vEBT_VEBT_low @ xa @ ( divide_divide @ nat @ ( suc @ ( suc @ va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
       => ( ( Xb
            = ( vEBT_VEBT_high @ xa @ ( divide_divide @ nat @ ( suc @ ( suc @ va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
         => ( ( ord_less @ nat @ Xb @ ( size_size @ ( list @ vEBT_VEBT ) @ treeList ) )
           => ( ( Xc
                = ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ treeList @ Xb ) ) )
             => ( ( ( Xc
                   != ( none @ nat ) )
                  & ( vEBT_VEBT_greater @ ( some @ nat @ Xa ) @ Xc ) )
               => ( ( vEBT_invar_vebt @ ( nth @ vEBT_VEBT @ treeList @ Xb ) @ N2 )
                 => ( hoare_hoare_triple @ ( option @ nat ) @ ( vEBT_vebt_assn_raw @ ( nth @ vEBT_VEBT @ treeList @ Xb ) @ Ti ) @ ( vEBT_VEBT_vebt_predi @ ( nth @ vEBT_VEBT @ treeList @ Xb ) @ Ti @ Xa )
                    @ ^ [R: option @ nat] :
                        ( times_times @ assn @ ( vEBT_vebt_assn_raw @ ( nth @ vEBT_VEBT @ treeList @ Xb ) @ Ti )
                        @ ( pure_assn
                          @ ( R
                            = ( vEBT_vebt_pred @ ( nth @ vEBT_VEBT @ treeList @ Xb ) @ Xa ) ) ) ) ) ) ) ) ) ) ) ) ).

% "7.IH"(1)
thf(fact_12_pred__less__length__list,axiom,
    ! [Deg: nat,X2: nat,Ma: nat,TreeList: list @ vEBT_VEBT,Mi: nat,Summary: vEBT_VEBT] :
      ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
     => ( ( ord_less_eq @ nat @ X2 @ Ma )
       => ( ( ord_less @ nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
         => ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X2 )
            = ( if @ ( option @ nat )
              @ ( ( ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
                 != ( none @ nat ) )
                & ( vEBT_VEBT_greater @ ( some @ nat @ ( vEBT_VEBT_low @ X2 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) )
              @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( some @ nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_pred @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X2 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
              @ ( if @ ( option @ nat )
                @ ( ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
                  = ( none @ nat ) )
                @ ( if @ ( option @ nat ) @ ( ord_less @ nat @ Mi @ X2 ) @ ( some @ nat @ Mi ) @ ( none @ nat ) )
                @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% pred_less_length_list
thf(fact_13_pred__lesseq__max,axiom,
    ! [Deg: nat,X2: nat,Ma: nat,Mi: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
     => ( ( ord_less_eq @ nat @ X2 @ Ma )
       => ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X2 )
          = ( if @ ( option @ nat ) @ ( ord_less @ nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
            @ ( if @ ( option @ nat )
              @ ( ( ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
                 != ( none @ nat ) )
                & ( vEBT_VEBT_greater @ ( some @ nat @ ( vEBT_VEBT_low @ X2 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) )
              @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( some @ nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_pred @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X2 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
              @ ( if @ ( option @ nat )
                @ ( ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
                  = ( none @ nat ) )
                @ ( if @ ( option @ nat ) @ ( ord_less @ nat @ Mi @ X2 ) @ ( some @ nat @ Mi ) @ ( none @ nat ) )
                @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) )
            @ ( none @ nat ) ) ) ) ) ).

% pred_lesseq_max
thf(fact_14_vebt__maxti__hT,axiom,
    ! [T2: vEBT_VEBT,Ti: vEBT_VEBTi] :
      ( time_htt @ ( option @ nat ) @ ( vEBT_vebt_assn_raw @ T2 @ Ti ) @ ( vEBT_vebt_maxti @ Ti )
      @ ^ [R: option @ nat] :
          ( times_times @ assn @ ( vEBT_vebt_assn_raw @ T2 @ Ti )
          @ ( pure_assn
            @ ( R
              = ( vEBT_vebt_maxt @ T2 ) ) ) )
      @ ( one_one @ nat ) ) ).

% vebt_maxti_hT
thf(fact_15_vebt__minti__hT,axiom,
    ! [T2: vEBT_VEBT,Ti: vEBT_VEBTi] :
      ( time_htt @ ( option @ nat ) @ ( vEBT_vebt_assn_raw @ T2 @ Ti ) @ ( vEBT_vebt_minti @ Ti )
      @ ^ [R: option @ nat] :
          ( times_times @ assn @ ( vEBT_vebt_assn_raw @ T2 @ Ti )
          @ ( pure_assn
            @ ( R
              = ( vEBT_vebt_mint @ T2 ) ) ) )
      @ ( one_one @ nat ) ) ).

% vebt_minti_hT
thf(fact_16_div2__Suc__Suc,axiom,
    ! [M: nat] :
      ( ( divide_divide @ nat @ ( suc @ ( suc @ M ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
      = ( suc @ ( divide_divide @ nat @ M @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).

% div2_Suc_Suc
thf(fact_17_Suc__1,axiom,
    ( ( suc @ ( one_one @ nat ) )
    = ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ).

% Suc_1
thf(fact_18_power__increasing__iff,axiom,
    ! [A: $tType] :
      ( ( linordered_semidom @ A )
     => ! [B2: A,X2: nat,Y2: nat] :
          ( ( ord_less @ A @ ( one_one @ A ) @ B2 )
         => ( ( ord_less_eq @ A @ ( power_power @ A @ B2 @ X2 ) @ ( power_power @ A @ B2 @ Y2 ) )
            = ( ord_less_eq @ nat @ X2 @ Y2 ) ) ) ) ).

% power_increasing_iff
thf(fact_19_power__strict__increasing__iff,axiom,
    ! [A: $tType] :
      ( ( linordered_semidom @ A )
     => ! [B2: A,X2: nat,Y2: nat] :
          ( ( ord_less @ A @ ( one_one @ A ) @ B2 )
         => ( ( ord_less @ A @ ( power_power @ A @ B2 @ X2 ) @ ( power_power @ A @ B2 @ Y2 ) )
            = ( ord_less @ nat @ X2 @ Y2 ) ) ) ) ).

% power_strict_increasing_iff
thf(fact_20_divide__less__eq__numeral1_I1_J,axiom,
    ! [A: $tType] :
      ( ( linordered_field @ A )
     => ! [B2: A,W: num,A2: A] :
          ( ( ord_less @ A @ ( divide_divide @ A @ B2 @ ( numeral_numeral @ A @ W ) ) @ A2 )
          = ( ord_less @ A @ B2 @ ( times_times @ A @ A2 @ ( numeral_numeral @ A @ W ) ) ) ) ) ).

% divide_less_eq_numeral1(1)
thf(fact_21_less__divide__eq__numeral1_I1_J,axiom,
    ! [A: $tType] :
      ( ( linordered_field @ A )
     => ! [A2: A,B2: A,W: num] :
          ( ( ord_less @ A @ A2 @ ( divide_divide @ A @ B2 @ ( numeral_numeral @ A @ W ) ) )
          = ( ord_less @ A @ ( times_times @ A @ A2 @ ( numeral_numeral @ A @ W ) ) @ B2 ) ) ) ).

% less_divide_eq_numeral1(1)
thf(fact_22_deg__deg__n,axiom,
    ! [Info: option @ ( product_prod @ nat @ nat ),Deg: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,N2: nat] :
      ( ( vEBT_invar_vebt @ ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) @ N2 )
     => ( Deg = N2 ) ) ).

% deg_deg_n
thf(fact_23_mul__shift,axiom,
    ! [X2: nat,Y2: nat,Z: nat] :
      ( ( ( times_times @ nat @ X2 @ Y2 )
        = Z )
      = ( ( vEBT_VEBT_mul @ ( some @ nat @ X2 ) @ ( some @ nat @ Y2 ) )
        = ( some @ nat @ Z ) ) ) ).

% mul_shift
thf(fact_24_deg__SUcn__Node,axiom,
    ! [Tree: vEBT_VEBT,N2: nat] :
      ( ( vEBT_invar_vebt @ Tree @ ( suc @ ( suc @ N2 ) ) )
     => ? [Info2: option @ ( product_prod @ nat @ nat ),TreeList2: list @ vEBT_VEBT,S: vEBT_VEBT] :
          ( Tree
          = ( vEBT_Node @ Info2 @ ( suc @ ( suc @ N2 ) ) @ TreeList2 @ S ) ) ) ).

% deg_SUcn_Node
thf(fact_25_numeral__eq__iff,axiom,
    ! [A: $tType] :
      ( ( semiring_char_0 @ A )
     => ! [M: num,N2: num] :
          ( ( ( numeral_numeral @ A @ M )
            = ( numeral_numeral @ A @ N2 ) )
          = ( M = N2 ) ) ) ).

% numeral_eq_iff
thf(fact_26_numeral__le__iff,axiom,
    ! [A: $tType] :
      ( ( linord181362715937106298miring @ A )
     => ! [M: num,N2: num] :
          ( ( ord_less_eq @ A @ ( numeral_numeral @ A @ M ) @ ( numeral_numeral @ A @ N2 ) )
          = ( ord_less_eq @ num @ M @ N2 ) ) ) ).

% numeral_le_iff
thf(fact_27_numeral__less__iff,axiom,
    ! [A: $tType] :
      ( ( linord181362715937106298miring @ A )
     => ! [M: num,N2: num] :
          ( ( ord_less @ A @ ( numeral_numeral @ A @ M ) @ ( numeral_numeral @ A @ N2 ) )
          = ( ord_less @ num @ M @ N2 ) ) ) ).

% numeral_less_iff
thf(fact_28_mult__numeral__left__semiring__numeral,axiom,
    ! [A: $tType] :
      ( ( semiring_numeral @ A )
     => ! [V: num,W: num,Z: A] :
          ( ( times_times @ A @ ( numeral_numeral @ A @ V ) @ ( times_times @ A @ ( numeral_numeral @ A @ W ) @ Z ) )
          = ( times_times @ A @ ( numeral_numeral @ A @ ( times_times @ num @ V @ W ) ) @ Z ) ) ) ).

% mult_numeral_left_semiring_numeral
thf(fact_29_numeral__times__numeral,axiom,
    ! [A: $tType] :
      ( ( semiring_numeral @ A )
     => ! [M: num,N2: num] :
          ( ( times_times @ A @ ( numeral_numeral @ A @ M ) @ ( numeral_numeral @ A @ N2 ) )
          = ( numeral_numeral @ A @ ( times_times @ num @ M @ N2 ) ) ) ) ).

% numeral_times_numeral
thf(fact_30_num__double,axiom,
    ! [N2: num] :
      ( ( times_times @ num @ ( bit0 @ one2 ) @ N2 )
      = ( bit0 @ N2 ) ) ).

% num_double
thf(fact_31_misiz,axiom,
    ! [T2: vEBT_VEBT,N2: nat,M: nat] :
      ( ( vEBT_invar_vebt @ T2 @ N2 )
     => ( ( ( some @ nat @ M )
          = ( vEBT_vebt_mint @ T2 ) )
       => ( ord_less @ nat @ M @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ) ) ).

% misiz
thf(fact_32_power__one,axiom,
    ! [A: $tType] :
      ( ( monoid_mult @ A )
     => ! [N2: nat] :
          ( ( power_power @ A @ ( one_one @ A ) @ N2 )
          = ( one_one @ A ) ) ) ).

% power_one
thf(fact_33_power__mult__numeral,axiom,
    ! [A: $tType] :
      ( ( monoid_mult @ A )
     => ! [A2: A,M: num,N2: num] :
          ( ( power_power @ A @ ( power_power @ A @ A2 @ ( numeral_numeral @ nat @ M ) ) @ ( numeral_numeral @ nat @ N2 ) )
          = ( power_power @ A @ A2 @ ( numeral_numeral @ nat @ ( times_times @ num @ M @ N2 ) ) ) ) ) ).

% power_mult_numeral
thf(fact_34_helpypredd,axiom,
    ! [T2: vEBT_VEBT,N2: nat,X2: nat,Y2: nat] :
      ( ( vEBT_invar_vebt @ T2 @ N2 )
     => ( ( ( vEBT_vebt_pred @ T2 @ X2 )
          = ( some @ nat @ Y2 ) )
       => ( ord_less @ nat @ Y2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ) ) ).

% helpypredd
thf(fact_35_power__one__right,axiom,
    ! [A: $tType] :
      ( ( monoid_mult @ A )
     => ! [A2: A] :
          ( ( power_power @ A @ A2 @ ( one_one @ nat ) )
          = A2 ) ) ).

% power_one_right
thf(fact_36_mi__ma__2__deg,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,N2: nat] :
      ( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ N2 )
     => ( ( ord_less_eq @ nat @ Mi @ Ma )
        & ( ord_less @ nat @ Ma @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg ) ) ) ) ).

% mi_ma_2_deg
thf(fact_37_pred__max,axiom,
    ! [Deg: nat,Ma: nat,X2: nat,Mi: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
     => ( ( ord_less @ nat @ Ma @ X2 )
       => ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X2 )
          = ( some @ nat @ Ma ) ) ) ) ).

% pred_max
thf(fact_38_summaxma,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ Deg )
     => ( ( Mi != Ma )
       => ( ( the2 @ nat @ ( vEBT_vebt_maxt @ Summary ) )
          = ( vEBT_VEBT_high @ Ma @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ).

% summaxma
thf(fact_39_one__eq__numeral__iff,axiom,
    ! [A: $tType] :
      ( ( semiring_char_0 @ A )
     => ! [N2: num] :
          ( ( ( one_one @ A )
            = ( numeral_numeral @ A @ N2 ) )
          = ( one2 = N2 ) ) ) ).

% one_eq_numeral_iff
thf(fact_40_numeral__eq__one__iff,axiom,
    ! [A: $tType] :
      ( ( semiring_char_0 @ A )
     => ! [N2: num] :
          ( ( ( numeral_numeral @ A @ N2 )
            = ( one_one @ A ) )
          = ( N2 = one2 ) ) ) ).

% numeral_eq_one_iff
thf(fact_41_power__inject__exp,axiom,
    ! [A: $tType] :
      ( ( linordered_semidom @ A )
     => ! [A2: A,M: nat,N2: nat] :
          ( ( ord_less @ A @ ( one_one @ A ) @ A2 )
         => ( ( ( power_power @ A @ A2 @ M )
              = ( power_power @ A @ A2 @ N2 ) )
            = ( M = N2 ) ) ) ) ).

% power_inject_exp
thf(fact_42_pred__list__to__short,axiom,
    ! [Deg: nat,X2: nat,Ma: nat,TreeList: list @ vEBT_VEBT,Mi: nat,Summary: vEBT_VEBT] :
      ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
     => ( ( ord_less_eq @ nat @ X2 @ Ma )
       => ( ( ord_less_eq @ nat @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
         => ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X2 )
            = ( none @ nat ) ) ) ) ) ).

% pred_list_to_short
thf(fact_43_numeral__le__one__iff,axiom,
    ! [A: $tType] :
      ( ( linord181362715937106298miring @ A )
     => ! [N2: num] :
          ( ( ord_less_eq @ A @ ( numeral_numeral @ A @ N2 ) @ ( one_one @ A ) )
          = ( ord_less_eq @ num @ N2 @ one2 ) ) ) ).

% numeral_le_one_iff
thf(fact_44_mem__Collect__eq,axiom,
    ! [A: $tType,A2: A,P: A > $o] :
      ( ( member @ A @ A2 @ ( collect @ A @ P ) )
      = ( P @ A2 ) ) ).

% mem_Collect_eq
thf(fact_45_Collect__mem__eq,axiom,
    ! [A: $tType,A3: set @ A] :
      ( ( collect @ A
        @ ^ [X: A] : ( member @ A @ X @ A3 ) )
      = A3 ) ).

% Collect_mem_eq
thf(fact_46_Collect__cong,axiom,
    ! [A: $tType,P: A > $o,Q: A > $o] :
      ( ! [X3: A] :
          ( ( P @ X3 )
          = ( Q @ X3 ) )
     => ( ( collect @ A @ P )
        = ( collect @ A @ Q ) ) ) ).

% Collect_cong
thf(fact_47_HOL_Oext,axiom,
    ! [B: $tType,A: $tType,F: A > B,G: A > B] :
      ( ! [X3: A] :
          ( ( F @ X3 )
          = ( G @ X3 ) )
     => ( F = G ) ) ).

% HOL.ext
thf(fact_48_one__less__numeral__iff,axiom,
    ! [A: $tType] :
      ( ( linord181362715937106298miring @ A )
     => ! [N2: num] :
          ( ( ord_less @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ N2 ) )
          = ( ord_less @ num @ one2 @ N2 ) ) ) ).

% one_less_numeral_iff
thf(fact_49_le__divide__eq__numeral1_I1_J,axiom,
    ! [A: $tType] :
      ( ( linordered_field @ A )
     => ! [A2: A,B2: A,W: num] :
          ( ( ord_less_eq @ A @ A2 @ ( divide_divide @ A @ B2 @ ( numeral_numeral @ A @ W ) ) )
          = ( ord_less_eq @ A @ ( times_times @ A @ A2 @ ( numeral_numeral @ A @ W ) ) @ B2 ) ) ) ).

% le_divide_eq_numeral1(1)
thf(fact_50_divide__le__eq__numeral1_I1_J,axiom,
    ! [A: $tType] :
      ( ( linordered_field @ A )
     => ! [B2: A,W: num,A2: A] :
          ( ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ ( numeral_numeral @ A @ W ) ) @ A2 )
          = ( ord_less_eq @ A @ B2 @ ( times_times @ A @ A2 @ ( numeral_numeral @ A @ W ) ) ) ) ) ).

% divide_le_eq_numeral1(1)
thf(fact_51_mul__def,axiom,
    ( vEBT_VEBT_mul
    = ( vEBT_V2048590022279873568_shift @ nat @ ( times_times @ nat ) ) ) ).

% mul_def
thf(fact_52__C7_Oprems_C,axiom,
    vEBT_invar_vebt @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ mi @ ma ) ) @ ( suc @ ( suc @ va ) ) @ treeList @ summary ) @ na ).

% "7.prems"
thf(fact_53_local_Opower__def,axiom,
    ( vEBT_VEBT_power
    = ( vEBT_V2048590022279873568_shift @ nat @ ( power_power @ nat ) ) ) ).

% local.power_def
thf(fact_54_le__num__One__iff,axiom,
    ! [X2: num] :
      ( ( ord_less_eq @ num @ X2 @ one2 )
      = ( X2 = one2 ) ) ).

% le_num_One_iff
thf(fact_55_power__mult,axiom,
    ! [A: $tType] :
      ( ( monoid_mult @ A )
     => ! [A2: A,M: nat,N2: nat] :
          ( ( power_power @ A @ A2 @ ( times_times @ nat @ M @ N2 ) )
          = ( power_power @ A @ ( power_power @ A @ A2 @ M ) @ N2 ) ) ) ).

% power_mult
thf(fact_56_div__mult2__eq,axiom,
    ! [M: nat,N2: nat,Q2: nat] :
      ( ( divide_divide @ nat @ M @ ( times_times @ nat @ N2 @ Q2 ) )
      = ( divide_divide @ nat @ ( divide_divide @ nat @ M @ N2 ) @ Q2 ) ) ).

% div_mult2_eq
thf(fact_57_div__mult2__numeral__eq,axiom,
    ! [A: $tType] :
      ( ( euclid5411537665997757685th_nat @ A )
     => ! [A2: A,K: num,L: num] :
          ( ( divide_divide @ A @ ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ K ) ) @ ( numeral_numeral @ A @ L ) )
          = ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ ( times_times @ num @ K @ L ) ) ) ) ) ).

% div_mult2_numeral_eq
thf(fact_58_less__mult__imp__div__less,axiom,
    ! [M: nat,I: nat,N2: nat] :
      ( ( ord_less @ nat @ M @ ( times_times @ nat @ I @ N2 ) )
     => ( ord_less @ nat @ ( divide_divide @ nat @ M @ N2 ) @ I ) ) ).

% less_mult_imp_div_less
thf(fact_59_times__div__less__eq__dividend,axiom,
    ! [N2: nat,M: nat] : ( ord_less_eq @ nat @ ( times_times @ nat @ N2 @ ( divide_divide @ nat @ M @ N2 ) ) @ M ) ).

% times_div_less_eq_dividend
thf(fact_60_div__times__less__eq__dividend,axiom,
    ! [M: nat,N2: nat] : ( ord_less_eq @ nat @ ( times_times @ nat @ ( divide_divide @ nat @ M @ N2 ) @ N2 ) @ M ) ).

% div_times_less_eq_dividend
thf(fact_61_power__even__eq,axiom,
    ! [A: $tType] :
      ( ( monoid_mult @ A )
     => ! [A2: A,N2: nat] :
          ( ( power_power @ A @ A2 @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) )
          = ( power_power @ A @ ( power_power @ A @ A2 @ N2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).

% power_even_eq
thf(fact_62_div__nat__eqI,axiom,
    ! [N2: nat,Q2: nat,M: nat] :
      ( ( ord_less_eq @ nat @ ( times_times @ nat @ N2 @ Q2 ) @ M )
     => ( ( ord_less @ nat @ M @ ( times_times @ nat @ N2 @ ( suc @ Q2 ) ) )
       => ( ( divide_divide @ nat @ M @ N2 )
          = Q2 ) ) ) ).

% div_nat_eqI
thf(fact_63_le__numeral__extra_I4_J,axiom,
    ! [A: $tType] :
      ( ( linord181362715937106298miring @ A )
     => ( ord_less_eq @ A @ ( one_one @ A ) @ ( one_one @ A ) ) ) ).

% le_numeral_extra(4)
thf(fact_64_less__numeral__extra_I4_J,axiom,
    ! [A: $tType] :
      ( ( linord181362715937106298miring @ A )
     => ~ ( ord_less @ A @ ( one_one @ A ) @ ( one_one @ A ) ) ) ).

% less_numeral_extra(4)
thf(fact_65_power__commuting__commutes,axiom,
    ! [A: $tType] :
      ( ( monoid_mult @ A )
     => ! [X2: A,Y2: A,N2: nat] :
          ( ( ( times_times @ A @ X2 @ Y2 )
            = ( times_times @ A @ Y2 @ X2 ) )
         => ( ( times_times @ A @ ( power_power @ A @ X2 @ N2 ) @ Y2 )
            = ( times_times @ A @ Y2 @ ( power_power @ A @ X2 @ N2 ) ) ) ) ) ).

% power_commuting_commutes
thf(fact_66_power__mult__distrib,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_mult @ A )
     => ! [A2: A,B2: A,N2: nat] :
          ( ( power_power @ A @ ( times_times @ A @ A2 @ B2 ) @ N2 )
          = ( times_times @ A @ ( power_power @ A @ A2 @ N2 ) @ ( power_power @ A @ B2 @ N2 ) ) ) ) ).

% power_mult_distrib
thf(fact_67_power__commutes,axiom,
    ! [A: $tType] :
      ( ( monoid_mult @ A )
     => ! [A2: A,N2: nat] :
          ( ( times_times @ A @ ( power_power @ A @ A2 @ N2 ) @ A2 )
          = ( times_times @ A @ A2 @ ( power_power @ A @ A2 @ N2 ) ) ) ) ).

% power_commutes
thf(fact_68_power__divide,axiom,
    ! [A: $tType] :
      ( ( field @ A )
     => ! [A2: A,B2: A,N2: nat] :
          ( ( power_power @ A @ ( divide_divide @ A @ A2 @ B2 ) @ N2 )
          = ( divide_divide @ A @ ( power_power @ A @ A2 @ N2 ) @ ( power_power @ A @ B2 @ N2 ) ) ) ) ).

% power_divide
thf(fact_69_div__le__dividend,axiom,
    ! [M: nat,N2: nat] : ( ord_less_eq @ nat @ ( divide_divide @ nat @ M @ N2 ) @ M ) ).

% div_le_dividend
thf(fact_70_div__le__mono,axiom,
    ! [M: nat,N2: nat,K: nat] :
      ( ( ord_less_eq @ nat @ M @ N2 )
     => ( ord_less_eq @ nat @ ( divide_divide @ nat @ M @ K ) @ ( divide_divide @ nat @ N2 @ K ) ) ) ).

% div_le_mono
thf(fact_71_one__le__numeral,axiom,
    ! [A: $tType] :
      ( ( linord181362715937106298miring @ A )
     => ! [N2: num] : ( ord_less_eq @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ N2 ) ) ) ).

% one_le_numeral
thf(fact_72_not__numeral__less__one,axiom,
    ! [A: $tType] :
      ( ( linord181362715937106298miring @ A )
     => ! [N2: num] :
          ~ ( ord_less @ A @ ( numeral_numeral @ A @ N2 ) @ ( one_one @ A ) ) ) ).

% not_numeral_less_one
thf(fact_73_numeral__One,axiom,
    ! [A: $tType] :
      ( ( numeral @ A )
     => ( ( numeral_numeral @ A @ one2 )
        = ( one_one @ A ) ) ) ).

% numeral_One
thf(fact_74_mult__numeral__1__right,axiom,
    ! [A: $tType] :
      ( ( semiring_numeral @ A )
     => ! [A2: A] :
          ( ( times_times @ A @ A2 @ ( numeral_numeral @ A @ one2 ) )
          = A2 ) ) ).

% mult_numeral_1_right
thf(fact_75_mult__numeral__1,axiom,
    ! [A: $tType] :
      ( ( semiring_numeral @ A )
     => ! [A2: A] :
          ( ( times_times @ A @ ( numeral_numeral @ A @ one2 ) @ A2 )
          = A2 ) ) ).

% mult_numeral_1
thf(fact_76_one__le__power,axiom,
    ! [A: $tType] :
      ( ( linordered_semidom @ A )
     => ! [A2: A,N2: nat] :
          ( ( ord_less_eq @ A @ ( one_one @ A ) @ A2 )
         => ( ord_less_eq @ A @ ( one_one @ A ) @ ( power_power @ A @ A2 @ N2 ) ) ) ) ).

% one_le_power
thf(fact_77_divide__numeral__1,axiom,
    ! [A: $tType] :
      ( ( field @ A )
     => ! [A2: A] :
          ( ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ one2 ) )
          = A2 ) ) ).

% divide_numeral_1
thf(fact_78_left__right__inverse__power,axiom,
    ! [A: $tType] :
      ( ( monoid_mult @ A )
     => ! [X2: A,Y2: A,N2: nat] :
          ( ( ( times_times @ A @ X2 @ Y2 )
            = ( one_one @ A ) )
         => ( ( times_times @ A @ ( power_power @ A @ X2 @ N2 ) @ ( power_power @ A @ Y2 @ N2 ) )
            = ( one_one @ A ) ) ) ) ).

% left_right_inverse_power
thf(fact_79_power__Suc2,axiom,
    ! [A: $tType] :
      ( ( monoid_mult @ A )
     => ! [A2: A,N2: nat] :
          ( ( power_power @ A @ A2 @ ( suc @ N2 ) )
          = ( times_times @ A @ ( power_power @ A @ A2 @ N2 ) @ A2 ) ) ) ).

% power_Suc2
thf(fact_80_power__Suc,axiom,
    ! [A: $tType] :
      ( ( power @ A )
     => ! [A2: A,N2: nat] :
          ( ( power_power @ A @ A2 @ ( suc @ N2 ) )
          = ( times_times @ A @ A2 @ ( power_power @ A @ A2 @ N2 ) ) ) ) ).

% power_Suc
thf(fact_81_power__one__over,axiom,
    ! [A: $tType] :
      ( ( division_ring @ A )
     => ! [A2: A,N2: nat] :
          ( ( power_power @ A @ ( divide_divide @ A @ ( one_one @ A ) @ A2 ) @ N2 )
          = ( divide_divide @ A @ ( one_one @ A ) @ ( power_power @ A @ A2 @ N2 ) ) ) ) ).

% power_one_over
thf(fact_82_numerals_I1_J,axiom,
    ( ( numeral_numeral @ nat @ one2 )
    = ( one_one @ nat ) ) ).

% numerals(1)
thf(fact_83_Suc__div__le__mono,axiom,
    ! [M: nat,N2: nat] : ( ord_less_eq @ nat @ ( divide_divide @ nat @ M @ N2 ) @ ( divide_divide @ nat @ ( suc @ M ) @ N2 ) ) ).

% Suc_div_le_mono
thf(fact_84_numeral__Bit0__div__2,axiom,
    ! [A: $tType] :
      ( ( euclid5411537665997757685th_nat @ A )
     => ! [N2: num] :
          ( ( divide_divide @ A @ ( numeral_numeral @ A @ ( bit0 @ N2 ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
          = ( numeral_numeral @ A @ N2 ) ) ) ).

% numeral_Bit0_div_2
thf(fact_85_power__less__power__Suc,axiom,
    ! [A: $tType] :
      ( ( linordered_semidom @ A )
     => ! [A2: A,N2: nat] :
          ( ( ord_less @ A @ ( one_one @ A ) @ A2 )
         => ( ord_less @ A @ ( power_power @ A @ A2 @ N2 ) @ ( times_times @ A @ A2 @ ( power_power @ A @ A2 @ N2 ) ) ) ) ) ).

% power_less_power_Suc
thf(fact_86_power__gt1__lemma,axiom,
    ! [A: $tType] :
      ( ( linordered_semidom @ A )
     => ! [A2: A,N2: nat] :
          ( ( ord_less @ A @ ( one_one @ A ) @ A2 )
         => ( ord_less @ A @ ( one_one @ A ) @ ( times_times @ A @ A2 @ ( power_power @ A @ A2 @ N2 ) ) ) ) ) ).

% power_gt1_lemma
thf(fact_87_power__gt1,axiom,
    ! [A: $tType] :
      ( ( linordered_semidom @ A )
     => ! [A2: A,N2: nat] :
          ( ( ord_less @ A @ ( one_one @ A ) @ A2 )
         => ( ord_less @ A @ ( one_one @ A ) @ ( power_power @ A @ A2 @ ( suc @ N2 ) ) ) ) ) ).

% power_gt1
thf(fact_88_power__strict__increasing,axiom,
    ! [A: $tType] :
      ( ( linordered_semidom @ A )
     => ! [N2: nat,N3: nat,A2: A] :
          ( ( ord_less @ nat @ N2 @ N3 )
         => ( ( ord_less @ A @ ( one_one @ A ) @ A2 )
           => ( ord_less @ A @ ( power_power @ A @ A2 @ N2 ) @ ( power_power @ A @ A2 @ N3 ) ) ) ) ) ).

% power_strict_increasing
thf(fact_89_power__less__imp__less__exp,axiom,
    ! [A: $tType] :
      ( ( linordered_semidom @ A )
     => ! [A2: A,M: nat,N2: nat] :
          ( ( ord_less @ A @ ( one_one @ A ) @ A2 )
         => ( ( ord_less @ A @ ( power_power @ A @ A2 @ M ) @ ( power_power @ A @ A2 @ N2 ) )
           => ( ord_less @ nat @ M @ N2 ) ) ) ) ).

% power_less_imp_less_exp
thf(fact_90_power__increasing,axiom,
    ! [A: $tType] :
      ( ( linordered_semidom @ A )
     => ! [N2: nat,N3: nat,A2: A] :
          ( ( ord_less_eq @ nat @ N2 @ N3 )
         => ( ( ord_less_eq @ A @ ( one_one @ A ) @ A2 )
           => ( ord_less_eq @ A @ ( power_power @ A @ A2 @ N2 ) @ ( power_power @ A @ A2 @ N3 ) ) ) ) ) ).

% power_increasing
thf(fact_91_power__numeral__even,axiom,
    ! [A: $tType] :
      ( ( monoid_mult @ A )
     => ! [Z: A,W: num] :
          ( ( power_power @ A @ Z @ ( numeral_numeral @ nat @ ( bit0 @ W ) ) )
          = ( times_times @ A @ ( power_power @ A @ Z @ ( numeral_numeral @ nat @ W ) ) @ ( power_power @ A @ Z @ ( numeral_numeral @ nat @ W ) ) ) ) ) ).

% power_numeral_even
thf(fact_92_one__power2,axiom,
    ! [A: $tType] :
      ( ( semiring_1 @ A )
     => ( ( power_power @ A @ ( one_one @ A ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
        = ( one_one @ A ) ) ) ).

% one_power2
thf(fact_93_power2__eq__square,axiom,
    ! [A: $tType] :
      ( ( monoid_mult @ A )
     => ! [A2: A] :
          ( ( power_power @ A @ A2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
          = ( times_times @ A @ A2 @ A2 ) ) ) ).

% power2_eq_square
thf(fact_94_power4__eq__xxxx,axiom,
    ! [A: $tType] :
      ( ( monoid_mult @ A )
     => ! [X2: A] :
          ( ( power_power @ A @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) )
          = ( times_times @ A @ ( times_times @ A @ ( times_times @ A @ X2 @ X2 ) @ X2 ) @ X2 ) ) ) ).

% power4_eq_xxxx
thf(fact_95_power__le__imp__le__exp,axiom,
    ! [A: $tType] :
      ( ( linordered_semidom @ A )
     => ! [A2: A,M: nat,N2: nat] :
          ( ( ord_less @ A @ ( one_one @ A ) @ A2 )
         => ( ( ord_less_eq @ A @ ( power_power @ A @ A2 @ M ) @ ( power_power @ A @ A2 @ N2 ) )
           => ( ord_less_eq @ nat @ M @ N2 ) ) ) ) ).

% power_le_imp_le_exp
thf(fact_96_power2__nat__le__imp__le,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq @ nat @ ( power_power @ nat @ M @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ N2 )
     => ( ord_less_eq @ nat @ M @ N2 ) ) ).

% power2_nat_le_imp_le
thf(fact_97_power2__nat__le__eq__le,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq @ nat @ ( power_power @ nat @ M @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
      = ( ord_less_eq @ nat @ M @ N2 ) ) ).

% power2_nat_le_eq_le
thf(fact_98_self__le__ge2__pow,axiom,
    ! [K: nat,M: nat] :
      ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ K )
     => ( ord_less_eq @ nat @ M @ ( power_power @ nat @ K @ M ) ) ) ).

% self_le_ge2_pow
thf(fact_99_power__odd__eq,axiom,
    ! [A: $tType] :
      ( ( monoid_mult @ A )
     => ! [A2: A,N2: nat] :
          ( ( power_power @ A @ A2 @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) )
          = ( times_times @ A @ A2 @ ( power_power @ A @ ( power_power @ A @ A2 @ N2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).

% power_odd_eq
thf(fact_100_vebt__pred_Osimps_I7_J,axiom,
    ! [Ma: nat,X2: nat,Mi: nat,Va: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( ord_less @ nat @ Ma @ X2 )
       => ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X2 )
          = ( some @ nat @ Ma ) ) )
      & ( ~ ( ord_less @ nat @ Ma @ X2 )
       => ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X2 )
          = ( if @ ( option @ nat ) @ ( ord_less @ nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
            @ ( if @ ( option @ nat )
              @ ( ( ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
                 != ( none @ nat ) )
                & ( vEBT_VEBT_greater @ ( some @ nat @ ( vEBT_VEBT_low @ X2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) )
              @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( some @ nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_pred @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
              @ ( if @ ( option @ nat )
                @ ( ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
                  = ( none @ nat ) )
                @ ( if @ ( option @ nat ) @ ( ord_less @ nat @ Mi @ X2 ) @ ( some @ nat @ Mi ) @ ( none @ nat ) )
                @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) )
            @ ( none @ nat ) ) ) ) ) ).

% vebt_pred.simps(7)
thf(fact_101_tdeletemimi_H,axiom,
    ! [Deg: nat,Mi: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,X2: nat] :
      ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
     => ( ord_less_eq @ nat @ ( vEBT_V1232361888498592333_e_t_e @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Mi ) ) @ Deg @ TreeList @ Summary ) @ X2 ) @ ( one_one @ nat ) ) ) ).

% tdeletemimi'
thf(fact_102__092_060open_062_092_060And_062x22_Ax21_O_Ati_A_061_ALeafi_Ax21_Ax22_A_092_060Longrightarrow_062_A_060vebt__assn__raw_A_INode_A_ISome_A_Imi_M_Ama_J_J_A_ISuc_A_ISuc_Ava_J_J_AtreeList_Asummary_J_Ati_062_Avebt__predi_H_A_INode_A_ISome_A_Imi_M_Ama_J_J_A_ISuc_A_ISuc_Ava_J_J_AtreeList_Asummary_J_Ati_Ax_A_060_092_060lambda_062r_O_Avebt__assn__raw_A_INode_A_ISome_A_Imi_M_Ama_J_J_A_ISuc_A_ISuc_Ava_J_J_AtreeList_Asummary_J_Ati_A_K_A_092_060up_062_A_Ir_A_061_Avebt__pred_A_INode_A_ISome_A_Imi_M_Ama_J_J_A_ISuc_A_ISuc_Ava_J_J_AtreeList_Asummary_J_Ax_J_062_092_060close_062,axiom,
    ! [X21: $o,X22: $o] :
      ( ( tia
        = ( vEBT_Leafi @ X21 @ X22 ) )
     => ( hoare_hoare_triple @ ( option @ nat ) @ ( vEBT_vebt_assn_raw @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ mi @ ma ) ) @ ( suc @ ( suc @ va ) ) @ treeList @ summary ) @ tia ) @ ( vEBT_VEBT_vebt_predi @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ mi @ ma ) ) @ ( suc @ ( suc @ va ) ) @ treeList @ summary ) @ tia @ xa )
        @ ^ [R: option @ nat] :
            ( times_times @ assn @ ( vEBT_vebt_assn_raw @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ mi @ ma ) ) @ ( suc @ ( suc @ va ) ) @ treeList @ summary ) @ tia )
            @ ( pure_assn
              @ ( R
                = ( vEBT_vebt_pred @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ mi @ ma ) ) @ ( suc @ ( suc @ va ) ) @ treeList @ summary ) @ xa ) ) ) ) ) ) ).

% \<open>\<And>x22 x21. ti = Leafi x21 x22 \<Longrightarrow> <vebt_assn_raw (Node (Some (mi, ma)) (Suc (Suc va)) treeList summary) ti> vebt_predi' (Node (Some (mi, ma)) (Suc (Suc va)) treeList summary) ti x <\<lambda>r. vebt_assn_raw (Node (Some (mi, ma)) (Suc (Suc va)) treeList summary) ti * \<up> (r = vebt_pred (Node (Some (mi, ma)) (Suc (Suc va)) treeList summary) x)>\<close>
thf(fact_103_maxt__sound,axiom,
    ! [T2: vEBT_VEBT,N2: nat,X2: nat] :
      ( ( vEBT_invar_vebt @ T2 @ N2 )
     => ( ( vEBT_VEBT_max_in_set @ ( vEBT_VEBT_set_vebt @ T2 ) @ X2 )
       => ( ( vEBT_vebt_maxt @ T2 )
          = ( some @ nat @ X2 ) ) ) ) ).

% maxt_sound
thf(fact_104_maxt__corr,axiom,
    ! [T2: vEBT_VEBT,N2: nat,X2: nat] :
      ( ( vEBT_invar_vebt @ T2 @ N2 )
     => ( ( ( vEBT_vebt_maxt @ T2 )
          = ( some @ nat @ X2 ) )
       => ( vEBT_VEBT_max_in_set @ ( vEBT_VEBT_set_vebt @ T2 ) @ X2 ) ) ) ).

% maxt_corr
thf(fact_105_mint__sound,axiom,
    ! [T2: vEBT_VEBT,N2: nat,X2: nat] :
      ( ( vEBT_invar_vebt @ T2 @ N2 )
     => ( ( vEBT_VEBT_min_in_set @ ( vEBT_VEBT_set_vebt @ T2 ) @ X2 )
       => ( ( vEBT_vebt_mint @ T2 )
          = ( some @ nat @ X2 ) ) ) ) ).

% mint_sound
thf(fact_106_mint__corr,axiom,
    ! [T2: vEBT_VEBT,N2: nat,X2: nat] :
      ( ( vEBT_invar_vebt @ T2 @ N2 )
     => ( ( ( vEBT_vebt_mint @ T2 )
          = ( some @ nat @ X2 ) )
       => ( vEBT_VEBT_min_in_set @ ( vEBT_VEBT_set_vebt @ T2 ) @ X2 ) ) ) ).

% mint_corr
thf(fact_107_nested__mint,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,N2: nat,Va: nat] :
      ( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ N2 )
     => ( ( N2
          = ( suc @ ( suc @ Va ) ) )
       => ( ~ ( ord_less @ nat @ Ma @ Mi )
         => ( ( Ma != Mi )
           => ( ord_less @ nat @ ( vEBT_VEBT_high @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Va @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( suc @ ( divide_divide @ nat @ Va @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) ) ) ) ) ) ).

% nested_mint
thf(fact_108_member__inv,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,X2: nat] :
      ( ( vEBT_vebt_member @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X2 )
     => ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
        & ( ( X2 = Mi )
          | ( X2 = Ma )
          | ( ( ord_less @ nat @ X2 @ Ma )
            & ( ord_less @ nat @ Mi @ X2 )
            & ( ord_less @ nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
            & ( vEBT_vebt_member @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X2 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ).

% member_inv
thf(fact_109_succ__list__to__short,axiom,
    ! [Deg: nat,Mi: nat,X2: nat,TreeList: list @ vEBT_VEBT,Ma: nat,Summary: vEBT_VEBT] :
      ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
     => ( ( ord_less_eq @ nat @ Mi @ X2 )
       => ( ( ord_less_eq @ nat @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
         => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X2 )
            = ( none @ nat ) ) ) ) ) ).

% succ_list_to_short
thf(fact_110_lowi__hT,axiom,
    ! [X2: nat,N2: nat] :
      ( time_htt @ nat @ ( one_one @ assn ) @ ( vEBT_VEBT_lowi @ X2 @ N2 )
      @ ^ [R: nat] :
          ( pure_assn
          @ ( R
            = ( vEBT_VEBT_low @ X2 @ N2 ) ) )
      @ ( one_one @ nat ) ) ).

% lowi_hT
thf(fact_111_highi__hT,axiom,
    ! [X2: nat,N2: nat] :
      ( time_htt @ nat @ ( one_one @ assn ) @ ( vEBT_VEBT_highi @ X2 @ N2 )
      @ ^ [R: nat] :
          ( pure_assn
          @ ( R
            = ( vEBT_VEBT_high @ X2 @ N2 ) ) )
      @ ( one_one @ nat ) ) ).

% highi_hT
thf(fact_112_even__odd__cases,axiom,
    ! [X2: nat] :
      ( ! [N4: nat] :
          ( X2
         != ( plus_plus @ nat @ N4 @ N4 ) )
     => ~ ! [N4: nat] :
            ( X2
           != ( plus_plus @ nat @ N4 @ ( suc @ N4 ) ) ) ) ).

% even_odd_cases
thf(fact_113_set__vebt_H__def,axiom,
    ( vEBT_VEBT_set_vebt
    = ( ^ [T3: vEBT_VEBT] : ( collect @ nat @ ( vEBT_vebt_member @ T3 ) ) ) ) ).

% set_vebt'_def
thf(fact_114_add__shift,axiom,
    ! [X2: nat,Y2: nat,Z: nat] :
      ( ( ( plus_plus @ nat @ X2 @ Y2 )
        = Z )
      = ( ( vEBT_VEBT_add @ ( some @ nat @ X2 ) @ ( some @ nat @ Y2 ) )
        = ( some @ nat @ Z ) ) ) ).

% add_shift
thf(fact_115_add__def,axiom,
    ( vEBT_VEBT_add
    = ( vEBT_V2048590022279873568_shift @ nat @ ( plus_plus @ nat ) ) ) ).

% add_def
thf(fact_116_mint__member,axiom,
    ! [T2: vEBT_VEBT,N2: nat,Maxi: nat] :
      ( ( vEBT_invar_vebt @ T2 @ N2 )
     => ( ( ( vEBT_vebt_mint @ T2 )
          = ( some @ nat @ Maxi ) )
       => ( vEBT_vebt_member @ T2 @ Maxi ) ) ) ).

% mint_member
thf(fact_117_maxt__member,axiom,
    ! [T2: vEBT_VEBT,N2: nat,Maxi: nat] :
      ( ( vEBT_invar_vebt @ T2 @ N2 )
     => ( ( ( vEBT_vebt_maxt @ T2 )
          = ( some @ nat @ Maxi ) )
       => ( vEBT_vebt_member @ T2 @ Maxi ) ) ) ).

% maxt_member
thf(fact_118_mint__corr__help,axiom,
    ! [T2: vEBT_VEBT,N2: nat,Mini: nat,X2: nat] :
      ( ( vEBT_invar_vebt @ T2 @ N2 )
     => ( ( ( vEBT_vebt_mint @ T2 )
          = ( some @ nat @ Mini ) )
       => ( ( vEBT_vebt_member @ T2 @ X2 )
         => ( ord_less_eq @ nat @ Mini @ X2 ) ) ) ) ).

% mint_corr_help
thf(fact_119_maxt__corr__help,axiom,
    ! [T2: vEBT_VEBT,N2: nat,Maxi: nat,X2: nat] :
      ( ( vEBT_invar_vebt @ T2 @ N2 )
     => ( ( ( vEBT_vebt_maxt @ T2 )
          = ( some @ nat @ Maxi ) )
       => ( ( vEBT_vebt_member @ T2 @ X2 )
         => ( ord_less_eq @ nat @ X2 @ Maxi ) ) ) ) ).

% maxt_corr_help
thf(fact_120_pow__sum,axiom,
    ! [A2: nat,B2: nat] :
      ( ( divide_divide @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( plus_plus @ nat @ A2 @ B2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ A2 ) )
      = ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ B2 ) ) ).

% pow_sum
thf(fact_121_high__bound__aux,axiom,
    ! [Ma: nat,N2: nat,M: nat] :
      ( ( ord_less @ nat @ Ma @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( plus_plus @ nat @ N2 @ M ) ) )
     => ( ord_less @ nat @ ( vEBT_VEBT_high @ Ma @ N2 ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) ) ) ).

% high_bound_aux
thf(fact_122_member__bound,axiom,
    ! [Tree: vEBT_VEBT,X2: nat,N2: nat] :
      ( ( vEBT_vebt_member @ Tree @ X2 )
     => ( ( vEBT_invar_vebt @ Tree @ N2 )
       => ( ord_less @ nat @ X2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ) ) ).

% member_bound
thf(fact_123_geqmaxNone,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,N2: nat,X2: nat] :
      ( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ N2 )
     => ( ( ord_less_eq @ nat @ Ma @ X2 )
       => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X2 )
          = ( none @ nat ) ) ) ) ).

% geqmaxNone
thf(fact_124_high__inv,axiom,
    ! [X2: nat,N2: nat,Y2: nat] :
      ( ( ord_less @ nat @ X2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) )
     => ( ( vEBT_VEBT_high @ ( plus_plus @ nat @ ( times_times @ nat @ Y2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) @ X2 ) @ N2 )
        = Y2 ) ) ).

% high_inv
thf(fact_125_low__inv,axiom,
    ! [X2: nat,N2: nat,Y2: nat] :
      ( ( ord_less @ nat @ X2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) )
     => ( ( vEBT_VEBT_low @ ( plus_plus @ nat @ ( times_times @ nat @ Y2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) @ X2 ) @ N2 )
        = X2 ) ) ).

% low_inv
thf(fact_126_add__numeral__left,axiom,
    ! [A: $tType] :
      ( ( numeral @ A )
     => ! [V: num,W: num,Z: A] :
          ( ( plus_plus @ A @ ( numeral_numeral @ A @ V ) @ ( plus_plus @ A @ ( numeral_numeral @ A @ W ) @ Z ) )
          = ( plus_plus @ A @ ( numeral_numeral @ A @ ( plus_plus @ num @ V @ W ) ) @ Z ) ) ) ).

% add_numeral_left
thf(fact_127_numeral__plus__numeral,axiom,
    ! [A: $tType] :
      ( ( numeral @ A )
     => ! [M: num,N2: num] :
          ( ( plus_plus @ A @ ( numeral_numeral @ A @ M ) @ ( numeral_numeral @ A @ N2 ) )
          = ( numeral_numeral @ A @ ( plus_plus @ num @ M @ N2 ) ) ) ) ).

% numeral_plus_numeral
thf(fact_128_bit__concat__def,axiom,
    ( vEBT_VEBT_bit_concat
    = ( ^ [H: nat,L2: nat,D3: nat] : ( plus_plus @ nat @ ( times_times @ nat @ H @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ D3 ) ) @ L2 ) ) ) ).

% bit_concat_def
thf(fact_129_helpyd,axiom,
    ! [T2: vEBT_VEBT,N2: nat,X2: nat,Y2: nat] :
      ( ( vEBT_invar_vebt @ T2 @ N2 )
     => ( ( ( vEBT_vebt_succ @ T2 @ X2 )
          = ( some @ nat @ Y2 ) )
       => ( ord_less @ nat @ Y2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ) ) ).

% helpyd
thf(fact_130_succ__min,axiom,
    ! [Deg: nat,X2: nat,Mi: nat,Ma: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
     => ( ( ord_less @ nat @ X2 @ Mi )
       => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X2 )
          = ( some @ nat @ Mi ) ) ) ) ).

% succ_min
thf(fact_131_member__correct,axiom,
    ! [T2: vEBT_VEBT,N2: nat,X2: nat] :
      ( ( vEBT_invar_vebt @ T2 @ N2 )
     => ( ( vEBT_vebt_member @ T2 @ X2 )
        = ( member @ nat @ X2 @ ( vEBT_set_vebt @ T2 ) ) ) ) ).

% member_correct
thf(fact_132_set__vebt__set__vebt_H__valid,axiom,
    ! [T2: vEBT_VEBT,N2: nat] :
      ( ( vEBT_invar_vebt @ T2 @ N2 )
     => ( ( vEBT_set_vebt @ T2 )
        = ( vEBT_VEBT_set_vebt @ T2 ) ) ) ).

% set_vebt_set_vebt'_valid
thf(fact_133_distrib__left__numeral,axiom,
    ! [A: $tType] :
      ( ( ( numeral @ A )
        & ( semiring @ A ) )
     => ! [V: num,B2: A,C2: A] :
          ( ( times_times @ A @ ( numeral_numeral @ A @ V ) @ ( plus_plus @ A @ B2 @ C2 ) )
          = ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ V ) @ B2 ) @ ( times_times @ A @ ( numeral_numeral @ A @ V ) @ C2 ) ) ) ) ).

% distrib_left_numeral
thf(fact_134_distrib__right__numeral,axiom,
    ! [A: $tType] :
      ( ( ( numeral @ A )
        & ( semiring @ A ) )
     => ! [A2: A,B2: A,V: num] :
          ( ( times_times @ A @ ( plus_plus @ A @ A2 @ B2 ) @ ( numeral_numeral @ A @ V ) )
          = ( plus_plus @ A @ ( times_times @ A @ A2 @ ( numeral_numeral @ A @ V ) ) @ ( times_times @ A @ B2 @ ( numeral_numeral @ A @ V ) ) ) ) ) ).

% distrib_right_numeral
thf(fact_135_numeral__plus__one,axiom,
    ! [A: $tType] :
      ( ( numeral @ A )
     => ! [N2: num] :
          ( ( plus_plus @ A @ ( numeral_numeral @ A @ N2 ) @ ( one_one @ A ) )
          = ( numeral_numeral @ A @ ( plus_plus @ num @ N2 @ one2 ) ) ) ) ).

% numeral_plus_one
thf(fact_136_one__plus__numeral,axiom,
    ! [A: $tType] :
      ( ( numeral @ A )
     => ! [N2: num] :
          ( ( plus_plus @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ N2 ) )
          = ( numeral_numeral @ A @ ( plus_plus @ num @ one2 @ N2 ) ) ) ) ).

% one_plus_numeral
thf(fact_137_one__add__one,axiom,
    ! [A: $tType] :
      ( ( numeral @ A )
     => ( ( plus_plus @ A @ ( one_one @ A ) @ ( one_one @ A ) )
        = ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ).

% one_add_one
thf(fact_138_add__2__eq__Suc,axiom,
    ! [N2: nat] :
      ( ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 )
      = ( suc @ ( suc @ N2 ) ) ) ).

% add_2_eq_Suc
thf(fact_139_add__2__eq__Suc_H,axiom,
    ! [N2: nat] :
      ( ( plus_plus @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
      = ( suc @ ( suc @ N2 ) ) ) ).

% add_2_eq_Suc'
thf(fact_140_add__self__div__2,axiom,
    ! [M: nat] :
      ( ( divide_divide @ nat @ ( plus_plus @ nat @ M @ M ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
      = M ) ).

% add_self_div_2
thf(fact_141_pred__member,axiom,
    ! [T2: vEBT_VEBT,X2: nat,Y2: nat] :
      ( ( vEBT_is_pred_in_set @ ( vEBT_VEBT_set_vebt @ T2 ) @ X2 @ Y2 )
      = ( ( vEBT_vebt_member @ T2 @ Y2 )
        & ( ord_less @ nat @ Y2 @ X2 )
        & ! [Z2: nat] :
            ( ( ( vEBT_vebt_member @ T2 @ Z2 )
              & ( ord_less @ nat @ Z2 @ X2 ) )
           => ( ord_less_eq @ nat @ Z2 @ Y2 ) ) ) ) ).

% pred_member
thf(fact_142_succ__member,axiom,
    ! [T2: vEBT_VEBT,X2: nat,Y2: nat] :
      ( ( vEBT_is_succ_in_set @ ( vEBT_VEBT_set_vebt @ T2 ) @ X2 @ Y2 )
      = ( ( vEBT_vebt_member @ T2 @ Y2 )
        & ( ord_less @ nat @ X2 @ Y2 )
        & ! [Z2: nat] :
            ( ( ( vEBT_vebt_member @ T2 @ Z2 )
              & ( ord_less @ nat @ X2 @ Z2 ) )
           => ( ord_less_eq @ nat @ Y2 @ Z2 ) ) ) ) ).

% succ_member
thf(fact_143_pred__corr,axiom,
    ! [T2: vEBT_VEBT,N2: nat,X2: nat,Px: nat] :
      ( ( vEBT_invar_vebt @ T2 @ N2 )
     => ( ( ( vEBT_vebt_pred @ T2 @ X2 )
          = ( some @ nat @ Px ) )
        = ( vEBT_is_pred_in_set @ ( vEBT_VEBT_set_vebt @ T2 ) @ X2 @ Px ) ) ) ).

% pred_corr
thf(fact_144_succ__corr,axiom,
    ! [T2: vEBT_VEBT,N2: nat,X2: nat,Sx: nat] :
      ( ( vEBT_invar_vebt @ T2 @ N2 )
     => ( ( ( vEBT_vebt_succ @ T2 @ X2 )
          = ( some @ nat @ Sx ) )
        = ( vEBT_is_succ_in_set @ ( vEBT_VEBT_set_vebt @ T2 ) @ X2 @ Sx ) ) ) ).

% succ_corr
thf(fact_145_vebt__succi_H__rf__abstr,axiom,
    ! [T2: vEBT_VEBT,N2: nat,Ti: vEBT_VEBTi,X2: nat] :
      ( ( vEBT_invar_vebt @ T2 @ N2 )
     => ( hoare_hoare_triple @ ( option @ nat ) @ ( vEBT_vebt_assn_raw @ T2 @ Ti ) @ ( vEBT_VEBT_vebt_succi @ T2 @ Ti @ X2 )
        @ ^ [R: option @ nat] :
            ( times_times @ assn @ ( vEBT_vebt_assn_raw @ T2 @ Ti )
            @ ( pure_assn
              @ ( R
                = ( vEBT_vebt_succ @ T2 @ X2 ) ) ) ) ) ) ).

% vebt_succi'_rf_abstr
thf(fact_146_is__num__normalize_I1_J,axiom,
    ! [A: $tType] :
      ( ( neg_numeral @ A )
     => ! [A2: A,B2: A,C2: A] :
          ( ( plus_plus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 )
          = ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B2 @ C2 ) ) ) ) ).

% is_num_normalize(1)
thf(fact_147_numeral__Bit0,axiom,
    ! [A: $tType] :
      ( ( numeral @ A )
     => ! [N2: num] :
          ( ( numeral_numeral @ A @ ( bit0 @ N2 ) )
          = ( plus_plus @ A @ ( numeral_numeral @ A @ N2 ) @ ( numeral_numeral @ A @ N2 ) ) ) ) ).

% numeral_Bit0
thf(fact_148_one__plus__numeral__commute,axiom,
    ! [A: $tType] :
      ( ( numeral @ A )
     => ! [X2: num] :
          ( ( plus_plus @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ X2 ) )
          = ( plus_plus @ A @ ( numeral_numeral @ A @ X2 ) @ ( one_one @ A ) ) ) ) ).

% one_plus_numeral_commute
thf(fact_149_power__add,axiom,
    ! [A: $tType] :
      ( ( monoid_mult @ A )
     => ! [A2: A,M: nat,N2: nat] :
          ( ( power_power @ A @ A2 @ ( plus_plus @ nat @ M @ N2 ) )
          = ( times_times @ A @ ( power_power @ A @ A2 @ M ) @ ( power_power @ A @ A2 @ N2 ) ) ) ) ).

% power_add
thf(fact_150_numeral__code_I2_J,axiom,
    ! [A: $tType] :
      ( ( numeral @ A )
     => ! [N2: num] :
          ( ( numeral_numeral @ A @ ( bit0 @ N2 ) )
          = ( plus_plus @ A @ ( numeral_numeral @ A @ N2 ) @ ( numeral_numeral @ A @ N2 ) ) ) ) ).

% numeral_code(2)
thf(fact_151_mult__2,axiom,
    ! [A: $tType] :
      ( ( semiring_numeral @ A )
     => ! [Z: A] :
          ( ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ Z )
          = ( plus_plus @ A @ Z @ Z ) ) ) ).

% mult_2
thf(fact_152_mult__2__right,axiom,
    ! [A: $tType] :
      ( ( semiring_numeral @ A )
     => ! [Z: A] :
          ( ( times_times @ A @ Z @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
          = ( plus_plus @ A @ Z @ Z ) ) ) ).

% mult_2_right
thf(fact_153_left__add__twice,axiom,
    ! [A: $tType] :
      ( ( semiring_numeral @ A )
     => ! [A2: A,B2: A] :
          ( ( plus_plus @ A @ A2 @ ( plus_plus @ A @ A2 @ B2 ) )
          = ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) @ B2 ) ) ) ).

% left_add_twice
thf(fact_154_nat__1__add__1,axiom,
    ( ( plus_plus @ nat @ ( one_one @ nat ) @ ( one_one @ nat ) )
    = ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ).

% nat_1_add_1
thf(fact_155_power2__sum,axiom,
    ! [A: $tType] :
      ( ( comm_semiring_1 @ A )
     => ! [X2: A,Y2: A] :
          ( ( power_power @ A @ ( plus_plus @ A @ X2 @ Y2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
          = ( plus_plus @ A @ ( plus_plus @ A @ ( power_power @ A @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( times_times @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ X2 ) @ Y2 ) ) ) ) ).

% power2_sum
thf(fact_156_ex__power__ivl1,axiom,
    ! [B2: nat,K: nat] :
      ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ B2 )
     => ( ( ord_less_eq @ nat @ ( one_one @ nat ) @ K )
       => ? [N4: nat] :
            ( ( ord_less_eq @ nat @ ( power_power @ nat @ B2 @ N4 ) @ K )
            & ( ord_less @ nat @ K @ ( power_power @ nat @ B2 @ ( plus_plus @ nat @ N4 @ ( one_one @ nat ) ) ) ) ) ) ) ).

% ex_power_ivl1
thf(fact_157_ex__power__ivl2,axiom,
    ! [B2: nat,K: nat] :
      ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ B2 )
     => ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ K )
       => ? [N4: nat] :
            ( ( ord_less @ nat @ ( power_power @ nat @ B2 @ N4 ) @ K )
            & ( ord_less_eq @ nat @ K @ ( power_power @ nat @ B2 @ ( plus_plus @ nat @ N4 @ ( one_one @ nat ) ) ) ) ) ) ) ).

% ex_power_ivl2
thf(fact_158_succ__greatereq__min,axiom,
    ! [Deg: nat,Mi: nat,X2: nat,Ma: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
     => ( ( ord_less_eq @ nat @ Mi @ X2 )
       => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X2 )
          = ( if @ ( option @ nat ) @ ( ord_less @ nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
            @ ( if @ ( option @ nat )
              @ ( ( ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
                 != ( none @ nat ) )
                & ( vEBT_VEBT_less @ ( some @ nat @ ( vEBT_VEBT_low @ X2 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) )
              @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( some @ nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_succ @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X2 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
              @ ( if @ ( option @ nat )
                @ ( ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
                  = ( none @ nat ) )
                @ ( none @ nat )
                @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) )
            @ ( none @ nat ) ) ) ) ) ).

% succ_greatereq_min
thf(fact_159_succ__less__length__list,axiom,
    ! [Deg: nat,Mi: nat,X2: nat,TreeList: list @ vEBT_VEBT,Ma: nat,Summary: vEBT_VEBT] :
      ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
     => ( ( ord_less_eq @ nat @ Mi @ X2 )
       => ( ( ord_less @ nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
         => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X2 )
            = ( if @ ( option @ nat )
              @ ( ( ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
                 != ( none @ nat ) )
                & ( vEBT_VEBT_less @ ( some @ nat @ ( vEBT_VEBT_low @ X2 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) )
              @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( some @ nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_succ @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X2 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
              @ ( if @ ( option @ nat )
                @ ( ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
                  = ( none @ nat ) )
                @ ( none @ nat )
                @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% succ_less_length_list
thf(fact_160_vebt__member_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,Va: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,X2: nat] :
      ( ( vEBT_vebt_member @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X2 )
      = ( ( X2 != Mi )
       => ( ( X2 != Ma )
         => ( ~ ( ord_less @ nat @ X2 @ Mi )
            & ( ~ ( ord_less @ nat @ X2 @ Mi )
             => ( ~ ( ord_less @ nat @ Ma @ X2 )
                & ( ~ ( ord_less @ nat @ Ma @ X2 )
                 => ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
                     => ( vEBT_vebt_member @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
                    & ( ord_less @ nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.simps(5)
thf(fact_161_post__member__pre__member,axiom,
    ! [T2: vEBT_VEBT,N2: nat,X2: nat,Y2: nat] :
      ( ( vEBT_invar_vebt @ T2 @ N2 )
     => ( ( ord_less @ nat @ X2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) )
       => ( ( ord_less @ nat @ Y2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) )
         => ( ( vEBT_vebt_member @ ( vEBT_vebt_insert @ T2 @ X2 ) @ Y2 )
           => ( ( vEBT_vebt_member @ T2 @ Y2 )
              | ( X2 = Y2 ) ) ) ) ) ) ).

% post_member_pre_member
thf(fact_162_vebt__assn__raw_Osimps_I2_J,axiom,
    ! [Mmo: option @ ( product_prod @ nat @ nat ),Deg: nat,Tree_list: list @ vEBT_VEBT,Summary: vEBT_VEBT,Mmoi: option @ ( product_prod @ nat @ nat ),Degi: nat,Tree_array: array @ vEBT_VEBTi,Summaryi: vEBT_VEBTi] :
      ( ( vEBT_vebt_assn_raw @ ( vEBT_Node @ Mmo @ Deg @ Tree_list @ Summary ) @ ( vEBT_Nodei @ Mmoi @ Degi @ Tree_array @ Summaryi ) )
      = ( times_times @ assn
        @ ( times_times @ assn
          @ ( pure_assn
            @ ( ( Mmoi = Mmo )
              & ( Degi = Deg ) ) )
          @ ( vEBT_vebt_assn_raw @ Summary @ Summaryi ) )
        @ ( ex_assn @ ( list @ vEBT_VEBTi )
          @ ^ [Tree_is: list @ vEBT_VEBTi] : ( times_times @ assn @ ( snga_assn @ vEBT_VEBTi @ Tree_array @ Tree_is ) @ ( vEBT_List_list_assn @ vEBT_VEBT @ vEBT_VEBTi @ vEBT_vebt_assn_raw @ Tree_list @ Tree_is ) ) ) ) ) ).

% vebt_assn_raw.simps(2)
thf(fact_163_insert__simp__mima,axiom,
    ! [X2: nat,Mi: nat,Ma: nat,Deg: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( X2 = Mi )
        | ( X2 = Ma ) )
     => ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
       => ( ( vEBT_vebt_insert @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X2 )
          = ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) ) ) ) ).

% insert_simp_mima
thf(fact_164_mintlistlength,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,N2: nat] :
      ( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ N2 )
     => ( ( Mi != Ma )
       => ( ( ord_less @ nat @ Mi @ Ma )
          & ? [M2: nat] :
              ( ( ( some @ nat @ M2 )
                = ( vEBT_vebt_mint @ Summary ) )
              & ( ord_less @ nat @ M2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ N2 @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ).

% mintlistlength
thf(fact_165_nth__rule,axiom,
    ! [A: $tType] :
      ( ( heap @ A )
     => ! [I: nat,Xs2: list @ A,A2: array @ A] :
          ( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs2 ) )
         => ( hoare_hoare_triple @ A @ ( snga_assn @ A @ A2 @ Xs2 ) @ ( array_nth @ A @ A2 @ I )
            @ ^ [R: A] :
                ( times_times @ assn @ ( snga_assn @ A @ A2 @ Xs2 )
                @ ( pure_assn
                  @ ( R
                    = ( nth @ A @ Xs2 @ I ) ) ) ) ) ) ) ).

% nth_rule
thf(fact_166_power__2__mult__step__le,axiom,
    ! [N5: nat,N2: nat,K2: nat,K: nat] :
      ( ( ord_less_eq @ nat @ N5 @ N2 )
     => ( ( ord_less @ nat @ ( times_times @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N5 ) @ K2 ) @ ( times_times @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) @ K ) )
       => ( ord_less_eq @ nat @ ( times_times @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N5 ) @ ( plus_plus @ nat @ K2 @ ( one_one @ nat ) ) ) @ ( times_times @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) @ K ) ) ) ) ).

% power_2_mult_step_le
thf(fact_167_norm__pre__pure__iff__sng,axiom,
    ! [A: $tType,B2: $o,F: heap_Time_Heap @ A,Q: A > assn] :
      ( ( hoare_hoare_triple @ A @ ( pure_assn @ B2 ) @ F @ Q )
      = ( B2
       => ( hoare_hoare_triple @ A @ ( one_one @ assn ) @ F @ Q ) ) ) ).

% norm_pre_pure_iff_sng
thf(fact_168_both__member__options__from__chilf__to__complete__tree,axiom,
    ! [X2: nat,Deg: nat,TreeList: list @ vEBT_VEBT,Mi: nat,Ma: nat,Summary: vEBT_VEBT] :
      ( ( ord_less @ nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
     => ( ( ord_less_eq @ nat @ ( one_one @ nat ) @ Deg )
       => ( ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X2 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
         => ( vEBT_V8194947554948674370ptions @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X2 ) ) ) ) ).

% both_member_options_from_chilf_to_complete_tree
thf(fact_169_norm__pre__pure__iff,axiom,
    ! [A: $tType,P: assn,B2: $o,F: heap_Time_Heap @ A,Q: A > assn] :
      ( ( hoare_hoare_triple @ A @ ( times_times @ assn @ P @ ( pure_assn @ B2 ) ) @ F @ Q )
      = ( B2
       => ( hoare_hoare_triple @ A @ P @ F @ Q ) ) ) ).

% norm_pre_pure_iff
thf(fact_170_less__shift,axiom,
    ( ( ord_less @ nat )
    = ( ^ [X: nat,Y: nat] : ( vEBT_VEBT_less @ ( some @ nat @ X ) @ ( some @ nat @ Y ) ) ) ) ).

% less_shift
thf(fact_171_maxbmo,axiom,
    ! [T2: vEBT_VEBT,X2: nat] :
      ( ( ( vEBT_vebt_maxt @ T2 )
        = ( some @ nat @ X2 ) )
     => ( vEBT_V8194947554948674370ptions @ T2 @ X2 ) ) ).

% maxbmo
thf(fact_172_both__member__options__equiv__member,axiom,
    ! [T2: vEBT_VEBT,N2: nat,X2: nat] :
      ( ( vEBT_invar_vebt @ T2 @ N2 )
     => ( ( vEBT_V8194947554948674370ptions @ T2 @ X2 )
        = ( vEBT_vebt_member @ T2 @ X2 ) ) ) ).

% both_member_options_equiv_member
thf(fact_173_valid__member__both__member__options,axiom,
    ! [T2: vEBT_VEBT,N2: nat,X2: nat] :
      ( ( vEBT_invar_vebt @ T2 @ N2 )
     => ( ( vEBT_V8194947554948674370ptions @ T2 @ X2 )
       => ( vEBT_vebt_member @ T2 @ X2 ) ) ) ).

% valid_member_both_member_options
thf(fact_174_highi__h,axiom,
    ! [X2: nat,N2: nat] :
      ( hoare_hoare_triple @ nat @ ( one_one @ assn ) @ ( vEBT_VEBT_highi @ X2 @ N2 )
      @ ^ [R: nat] :
          ( pure_assn
          @ ( R
            = ( vEBT_VEBT_high @ X2 @ N2 ) ) ) ) ).

% highi_h
thf(fact_175_lowi__h,axiom,
    ! [X2: nat,N2: nat] :
      ( hoare_hoare_triple @ nat @ ( one_one @ assn ) @ ( vEBT_VEBT_lowi @ X2 @ N2 )
      @ ^ [R: nat] :
          ( pure_assn
          @ ( R
            = ( vEBT_VEBT_low @ X2 @ N2 ) ) ) ) ).

% lowi_h
thf(fact_176_VEBTi_Oinject_I1_J,axiom,
    ! [X11: option @ ( product_prod @ nat @ nat ),X12: nat,X13: array @ vEBT_VEBTi,X14: vEBT_VEBTi,Y11: option @ ( product_prod @ nat @ 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_177_VEBTi_Oinject_I2_J,axiom,
    ! [X212: $o,X222: $o,Y21: $o,Y22: $o] :
      ( ( ( vEBT_Leafi @ X212 @ X222 )
        = ( vEBT_Leafi @ Y21 @ Y22 ) )
      = ( ( X212 = Y21 )
        & ( X222 = Y22 ) ) ) ).

% VEBTi.inject(2)
thf(fact_178_pred__correct,axiom,
    ! [T2: vEBT_VEBT,N2: nat,X2: nat,Sx: nat] :
      ( ( vEBT_invar_vebt @ T2 @ N2 )
     => ( ( ( vEBT_vebt_pred @ T2 @ X2 )
          = ( some @ nat @ Sx ) )
        = ( vEBT_is_pred_in_set @ ( vEBT_set_vebt @ T2 ) @ X2 @ Sx ) ) ) ).

% pred_correct
thf(fact_179_succ__correct,axiom,
    ! [T2: vEBT_VEBT,N2: nat,X2: nat,Sx: nat] :
      ( ( vEBT_invar_vebt @ T2 @ N2 )
     => ( ( ( vEBT_vebt_succ @ T2 @ X2 )
          = ( some @ nat @ Sx ) )
        = ( vEBT_is_succ_in_set @ ( vEBT_set_vebt @ T2 ) @ X2 @ Sx ) ) ) ).

% succ_correct
thf(fact_180_valid__insert__both__member__options__pres,axiom,
    ! [T2: vEBT_VEBT,N2: nat,X2: nat,Y2: nat] :
      ( ( vEBT_invar_vebt @ T2 @ N2 )
     => ( ( ord_less @ nat @ X2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) )
       => ( ( ord_less @ nat @ Y2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) )
         => ( ( vEBT_V8194947554948674370ptions @ T2 @ X2 )
           => ( vEBT_V8194947554948674370ptions @ ( vEBT_vebt_insert @ T2 @ Y2 ) @ X2 ) ) ) ) ) ).

% valid_insert_both_member_options_pres
thf(fact_181_valid__insert__both__member__options__add,axiom,
    ! [T2: vEBT_VEBT,N2: nat,X2: nat] :
      ( ( vEBT_invar_vebt @ T2 @ N2 )
     => ( ( ord_less @ nat @ X2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) )
       => ( vEBT_V8194947554948674370ptions @ ( vEBT_vebt_insert @ T2 @ X2 ) @ X2 ) ) ) ).

% valid_insert_both_member_options_add
thf(fact_182_sumprop,axiom,
    vEBT_invar_vebt @ summary @ ( minus_minus @ nat @ na @ ( divide_divide @ nat @ na @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ).

% sumprop
thf(fact_183_listlength,axiom,
    ( ( size_size @ ( list @ vEBT_VEBT ) @ treeList )
    = ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ na @ ( divide_divide @ nat @ na @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).

% listlength
thf(fact_184_right__diff__distrib__numeral,axiom,
    ! [A: $tType] :
      ( ( ( numeral @ A )
        & ( ring @ A ) )
     => ! [V: num,B2: A,C2: A] :
          ( ( times_times @ A @ ( numeral_numeral @ A @ V ) @ ( minus_minus @ A @ B2 @ C2 ) )
          = ( minus_minus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ V ) @ B2 ) @ ( times_times @ A @ ( numeral_numeral @ A @ V ) @ C2 ) ) ) ) ).

% right_diff_distrib_numeral
thf(fact_185_left__diff__distrib__numeral,axiom,
    ! [A: $tType] :
      ( ( ( numeral @ A )
        & ( ring @ A ) )
     => ! [A2: A,B2: A,V: num] :
          ( ( times_times @ A @ ( minus_minus @ A @ A2 @ B2 ) @ ( numeral_numeral @ A @ V ) )
          = ( minus_minus @ A @ ( times_times @ A @ A2 @ ( numeral_numeral @ A @ V ) ) @ ( times_times @ A @ B2 @ ( numeral_numeral @ A @ V ) ) ) ) ) ).

% left_diff_distrib_numeral
thf(fact_186_vebt__mintilist,axiom,
    ! [I: nat,Ts: list @ vEBT_VEBT,Tsi: list @ vEBT_VEBTi] :
      ( ( ord_less @ nat @ I @ ( size_size @ ( list @ vEBT_VEBT ) @ Ts ) )
     => ( hoare_hoare_triple @ ( option @ nat ) @ ( vEBT_List_list_assn @ vEBT_VEBT @ vEBT_VEBTi @ vEBT_vebt_assn_raw @ Ts @ Tsi ) @ ( vEBT_vebt_minti @ ( nth @ vEBT_VEBTi @ Tsi @ I ) )
        @ ^ [R: option @ nat] :
            ( times_times @ assn
            @ ( pure_assn
              @ ( R
                = ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ Ts @ I ) ) ) )
            @ ( vEBT_List_list_assn @ vEBT_VEBT @ vEBT_VEBTi @ vEBT_vebt_assn_raw @ Ts @ Tsi ) ) ) ) ).

% vebt_mintilist
thf(fact_187_vebt__maxtilist,axiom,
    ! [I: nat,Ts: list @ vEBT_VEBT,Tsi: list @ vEBT_VEBTi] :
      ( ( ord_less @ nat @ I @ ( size_size @ ( list @ vEBT_VEBT ) @ Ts ) )
     => ( hoare_hoare_triple @ ( option @ nat ) @ ( vEBT_List_list_assn @ vEBT_VEBT @ vEBT_VEBTi @ vEBT_vebt_assn_raw @ Ts @ Tsi ) @ ( vEBT_vebt_maxti @ ( nth @ vEBT_VEBTi @ Tsi @ I ) )
        @ ^ [R: option @ nat] :
            ( times_times @ assn
            @ ( pure_assn
              @ ( R
                = ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ Ts @ I ) ) ) )
            @ ( vEBT_List_list_assn @ vEBT_VEBT @ vEBT_VEBTi @ vEBT_vebt_assn_raw @ Ts @ Tsi ) ) ) ) ).

% vebt_maxtilist
thf(fact_188_both__member__options__ding,axiom,
    ! [Info: option @ ( product_prod @ nat @ nat ),Deg: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,N2: nat,X2: nat] :
      ( ( vEBT_invar_vebt @ ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) @ N2 )
     => ( ( ord_less @ nat @ X2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg ) )
       => ( ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X2 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
         => ( vEBT_V8194947554948674370ptions @ ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) @ X2 ) ) ) ) ).

% both_member_options_ding
thf(fact_189_both__member__options__from__complete__tree__to__child,axiom,
    ! [Deg: nat,Mi: nat,Ma: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,X2: nat] :
      ( ( ord_less_eq @ nat @ ( one_one @ nat ) @ Deg )
     => ( ( vEBT_V8194947554948674370ptions @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X2 )
       => ( ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X2 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
          | ( X2 = Mi )
          | ( X2 = Ma ) ) ) ) ).

% both_member_options_from_complete_tree_to_child
thf(fact_190_Suc__numeral,axiom,
    ! [N2: num] :
      ( ( suc @ ( numeral_numeral @ nat @ N2 ) )
      = ( numeral_numeral @ nat @ ( plus_plus @ num @ N2 @ one2 ) ) ) ).

% Suc_numeral
thf(fact_191_power__add__numeral2,axiom,
    ! [A: $tType] :
      ( ( monoid_mult @ A )
     => ! [A2: A,M: num,N2: num,B2: A] :
          ( ( times_times @ A @ ( power_power @ A @ A2 @ ( numeral_numeral @ nat @ M ) ) @ ( times_times @ A @ ( power_power @ A @ A2 @ ( numeral_numeral @ nat @ N2 ) ) @ B2 ) )
          = ( times_times @ A @ ( power_power @ A @ A2 @ ( numeral_numeral @ nat @ ( plus_plus @ num @ M @ N2 ) ) ) @ B2 ) ) ) ).

% power_add_numeral2
thf(fact_192_power__add__numeral,axiom,
    ! [A: $tType] :
      ( ( monoid_mult @ A )
     => ! [A2: A,M: num,N2: num] :
          ( ( times_times @ A @ ( power_power @ A @ A2 @ ( numeral_numeral @ nat @ M ) ) @ ( power_power @ A @ A2 @ ( numeral_numeral @ nat @ N2 ) ) )
          = ( power_power @ A @ A2 @ ( numeral_numeral @ nat @ ( plus_plus @ num @ M @ N2 ) ) ) ) ) ).

% power_add_numeral
thf(fact_193_vebt__memberi_H__rf__abstr,axiom,
    ! [T2: vEBT_VEBT,Ti: vEBT_VEBTi,X2: nat] :
      ( hoare_hoare_triple @ $o @ ( vEBT_vebt_assn_raw @ T2 @ Ti ) @ ( vEBT_V854960066525838166emberi @ T2 @ Ti @ X2 )
      @ ^ [R: $o] :
          ( times_times @ assn @ ( vEBT_vebt_assn_raw @ T2 @ Ti )
          @ ( pure_assn
            @ ( R
              = ( vEBT_vebt_member @ T2 @ X2 ) ) ) ) ) ).

% vebt_memberi'_rf_abstr
thf(fact_194_vebt__inserti_H__rf__abstr,axiom,
    ! [T2: vEBT_VEBT,Ti: vEBT_VEBTi,X2: nat] : ( hoare_hoare_triple @ vEBT_VEBTi @ ( vEBT_vebt_assn_raw @ T2 @ Ti ) @ ( vEBT_V3964819847710782039nserti @ T2 @ Ti @ X2 ) @ ( vEBT_vebt_assn_raw @ ( vEBT_vebt_insert @ T2 @ X2 ) ) ) ).

% vebt_inserti'_rf_abstr
thf(fact_195_diff__diff__less,axiom,
    ! [I: nat,M: nat,N2: nat] :
      ( ( ord_less @ nat @ I @ ( minus_minus @ nat @ M @ ( minus_minus @ nat @ M @ N2 ) ) )
      = ( ( ord_less @ nat @ I @ M )
        & ( ord_less @ nat @ I @ N2 ) ) ) ).

% diff_diff_less
thf(fact_196_add__One__commute,axiom,
    ! [N2: num] :
      ( ( plus_plus @ num @ one2 @ N2 )
      = ( plus_plus @ num @ N2 @ one2 ) ) ).

% add_One_commute
thf(fact_197_if__rule,axiom,
    ! [A: $tType,B2: $o,P: assn,F: heap_Time_Heap @ A,Q: A > assn,G: heap_Time_Heap @ A] :
      ( ( B2
       => ( hoare_hoare_triple @ A @ P @ F @ Q ) )
     => ( ( ~ B2
         => ( hoare_hoare_triple @ A @ P @ G @ Q ) )
       => ( hoare_hoare_triple @ A @ P @ ( if @ ( heap_Time_Heap @ A ) @ B2 @ F @ G ) @ Q ) ) ) ).

% if_rule
thf(fact_198_power2__commute,axiom,
    ! [A: $tType] :
      ( ( comm_ring_1 @ A )
     => ! [X2: A,Y2: A] :
          ( ( power_power @ A @ ( minus_minus @ A @ X2 @ Y2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
          = ( power_power @ A @ ( minus_minus @ A @ Y2 @ X2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).

% power2_commute
thf(fact_199_Suc__nat__number__of__add,axiom,
    ! [V: num,N2: nat] :
      ( ( suc @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ V ) @ N2 ) )
      = ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( plus_plus @ num @ V @ one2 ) ) @ N2 ) ) ).

% Suc_nat_number_of_add
thf(fact_200_nat__power__less__diff,axiom,
    ! [N2: nat,Q2: nat,M: nat] :
      ( ( ord_less @ nat @ ( times_times @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) @ Q2 ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) )
     => ( ord_less @ nat @ Q2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ M @ N2 ) ) ) ) ).

% nat_power_less_diff
thf(fact_201_nat__le__power__trans,axiom,
    ! [N2: nat,M: nat,K: nat] :
      ( ( ord_less_eq @ nat @ N2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( 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 @ one2 ) ) @ K ) @ N2 ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) ) ) ) ).

% nat_le_power_trans
thf(fact_202_is__pred__in__set__def,axiom,
    ( vEBT_is_pred_in_set
    = ( ^ [Xs: set @ nat,X: nat,Y: nat] :
          ( ( member @ nat @ Y @ Xs )
          & ( ord_less @ nat @ Y @ X )
          & ! [Z2: nat] :
              ( ( member @ nat @ Z2 @ Xs )
             => ( ( ord_less @ nat @ Z2 @ X )
               => ( ord_less_eq @ nat @ Z2 @ Y ) ) ) ) ) ) ).

% is_pred_in_set_def
thf(fact_203_diff__le__diff__pow,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ K )
     => ( ord_less_eq @ nat @ ( minus_minus @ nat @ M @ N2 ) @ ( minus_minus @ nat @ ( power_power @ nat @ K @ M ) @ ( power_power @ nat @ K @ N2 ) ) ) ) ).

% diff_le_diff_pow
thf(fact_204_nat__less__power__trans,axiom,
    ! [N2: nat,M: nat,K: nat] :
      ( ( ord_less @ nat @ N2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ M @ K ) ) )
     => ( ( ord_less_eq @ nat @ K @ M )
       => ( ord_less @ nat @ ( times_times @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ K ) @ N2 ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) ) ) ) ).

% nat_less_power_trans
thf(fact_205_power2__diff,axiom,
    ! [A: $tType] :
      ( ( comm_ring_1 @ A )
     => ! [X2: A,Y2: A] :
          ( ( power_power @ A @ ( minus_minus @ A @ X2 @ Y2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
          = ( minus_minus @ A @ ( plus_plus @ A @ ( power_power @ A @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( times_times @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ X2 ) @ Y2 ) ) ) ) ).

% power2_diff
thf(fact_206_return__wp__rule,axiom,
    ! [A: $tType,Q: A > assn,X2: A] : ( hoare_hoare_triple @ A @ ( Q @ X2 ) @ ( heap_Time_return @ A @ X2 ) @ Q ) ).

% return_wp_rule
thf(fact_207_bind__rule,axiom,
    ! [A: $tType,B: $tType,P: assn,F: heap_Time_Heap @ A,R2: A > assn,G: A > ( heap_Time_Heap @ B ),Q: B > assn] :
      ( ( hoare_hoare_triple @ A @ P @ F @ R2 )
     => ( ! [X3: A] : ( hoare_hoare_triple @ B @ ( R2 @ X3 ) @ ( G @ X3 ) @ Q )
       => ( hoare_hoare_triple @ B @ P @ ( heap_Time_bind @ A @ B @ F @ G ) @ Q ) ) ) ).

% bind_rule
thf(fact_208_VEBTi_Oexhaust,axiom,
    ! [Y2: vEBT_VEBTi] :
      ( ! [X112: option @ ( product_prod @ nat @ nat ),X122: nat,X132: array @ vEBT_VEBTi,X142: vEBT_VEBTi] :
          ( Y2
         != ( vEBT_Nodei @ X112 @ X122 @ X132 @ X142 ) )
     => ~ ! [X213: $o,X223: $o] :
            ( Y2
           != ( vEBT_Leafi @ X213 @ X223 ) ) ) ).

% VEBTi.exhaust
thf(fact_209_VEBTi_Odistinct_I1_J,axiom,
    ! [X11: option @ ( product_prod @ nat @ nat ),X12: nat,X13: array @ vEBT_VEBTi,X14: vEBT_VEBTi,X212: $o,X222: $o] :
      ( ( vEBT_Nodei @ X11 @ X12 @ X13 @ X14 )
     != ( vEBT_Leafi @ X212 @ X222 ) ) ).

% VEBTi.distinct(1)
thf(fact_210_frame__rule,axiom,
    ! [A: $tType,P: assn,C2: heap_Time_Heap @ A,Q: A > assn,R2: assn] :
      ( ( hoare_hoare_triple @ A @ P @ C2 @ Q )
     => ( hoare_hoare_triple @ A @ ( times_times @ assn @ P @ R2 ) @ C2
        @ ^ [X: A] : ( times_times @ assn @ ( Q @ X ) @ R2 ) ) ) ).

% frame_rule
thf(fact_211_norm__pre__ex__rule,axiom,
    ! [A: $tType,B: $tType,P: A > assn,F: heap_Time_Heap @ B,Q: B > assn] :
      ( ! [X3: A] : ( hoare_hoare_triple @ B @ ( P @ X3 ) @ F @ Q )
     => ( hoare_hoare_triple @ B @ ( ex_assn @ A @ P ) @ F @ Q ) ) ).

% norm_pre_ex_rule
thf(fact_212_post__exI__rule,axiom,
    ! [B: $tType,A: $tType,P: assn,C2: heap_Time_Heap @ A,Q: A > B > assn,X2: B] :
      ( ( hoare_hoare_triple @ A @ P @ C2
        @ ^ [R: A] : ( Q @ R @ X2 ) )
     => ( hoare_hoare_triple @ A @ P @ C2
        @ ^ [R: A] : ( ex_assn @ B @ ( Q @ R ) ) ) ) ).

% post_exI_rule
thf(fact_213_norm__pre__pure__rule1,axiom,
    ! [A: $tType,B2: $o,P: assn,F: heap_Time_Heap @ A,Q: A > assn] :
      ( ( B2
       => ( hoare_hoare_triple @ A @ P @ F @ Q ) )
     => ( hoare_hoare_triple @ A @ ( times_times @ assn @ P @ ( pure_assn @ B2 ) ) @ F @ Q ) ) ).

% norm_pre_pure_rule1
thf(fact_214_norm__pre__pure__rule2,axiom,
    ! [A: $tType,B2: $o,F: heap_Time_Heap @ A,Q: A > assn] :
      ( ( B2
       => ( hoare_hoare_triple @ A @ ( one_one @ assn ) @ F @ Q ) )
     => ( hoare_hoare_triple @ A @ ( pure_assn @ B2 ) @ F @ Q ) ) ).

% norm_pre_pure_rule2
thf(fact_215_bind__rule_H,axiom,
    ! [A: $tType,B: $tType,C: $tType,P: assn,F: heap_Time_Heap @ B,R2: B > assn,G: B > ( heap_Time_Heap @ C ),Q: C > assn] :
      ( ! [R3: A] :
          ( hoare_hoare_triple @ B @ P @ F
          @ ^ [S2: B] : ( times_times @ assn @ P @ ( R2 @ S2 ) ) )
     => ( ! [X3: B] : ( hoare_hoare_triple @ C @ ( times_times @ assn @ ( R2 @ X3 ) @ P ) @ ( G @ X3 ) @ Q )
       => ( hoare_hoare_triple @ C @ P @ ( heap_Time_bind @ B @ C @ F @ G ) @ Q ) ) ) ).

% bind_rule'
thf(fact_216_case__option__rule,axiom,
    ! [A: $tType,B: $tType,V: option @ A,P: assn,Fn: heap_Time_Heap @ B,Q: B > assn,Fs: A > ( heap_Time_Heap @ B )] :
      ( ( ( V
          = ( none @ A ) )
       => ( hoare_hoare_triple @ B @ P @ Fn @ Q ) )
     => ( ! [X3: A] :
            ( ( V
              = ( some @ A @ X3 ) )
           => ( hoare_hoare_triple @ B @ P @ ( Fs @ X3 ) @ Q ) )
       => ( hoare_hoare_triple @ B @ P @ ( case_option @ ( heap_Time_Heap @ B ) @ A @ Fn @ Fs @ V ) @ Q ) ) ) ).

% case_option_rule
thf(fact_217_return__sp__rule,axiom,
    ! [A: $tType,P: assn,X2: A] :
      ( hoare_hoare_triple @ A @ P @ ( heap_Time_return @ A @ X2 )
      @ ^ [R: A] : ( times_times @ assn @ P @ ( pure_assn @ ( R = X2 ) ) ) ) ).

% return_sp_rule
thf(fact_218_n__less__equal__power__2,axiom,
    ! [N2: nat] : ( ord_less @ nat @ N2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ).

% n_less_equal_power_2
thf(fact_219_vebt__minti_Osimps_I3_J,axiom,
    ! [Mi: nat,Ma: nat,Ux: nat,Uy: array @ vEBT_VEBTi,Uz: vEBT_VEBTi] :
      ( ( vEBT_vebt_minti @ ( vEBT_Nodei @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Ux @ Uy @ Uz ) )
      = ( heap_Time_return @ ( option @ nat ) @ ( some @ nat @ Mi ) ) ) ).

% vebt_minti.simps(3)
thf(fact_220_vebt__maxti_Osimps_I3_J,axiom,
    ! [Mi: nat,Ma: nat,Ux: nat,Uy: array @ vEBT_VEBTi,Uz: vEBT_VEBTi] :
      ( ( vEBT_vebt_maxti @ ( vEBT_Nodei @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Ux @ Uy @ Uz ) )
      = ( heap_Time_return @ ( option @ nat ) @ ( some @ nat @ Ma ) ) ) ).

% vebt_maxti.simps(3)
thf(fact_221_nat__add__offset__less,axiom,
    ! [Y2: nat,N2: nat,X2: nat,M: nat,Sz: nat] :
      ( ( ord_less @ nat @ Y2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) )
     => ( ( ord_less @ nat @ X2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) )
       => ( ( Sz
            = ( plus_plus @ nat @ M @ N2 ) )
         => ( ord_less @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ X2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) @ Y2 ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Sz ) ) ) ) ) ).

% nat_add_offset_less
thf(fact_222_vebt__succ_Osimps_I6_J,axiom,
    ! [X2: nat,Mi: nat,Ma: nat,Va: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( ord_less @ nat @ X2 @ Mi )
       => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X2 )
          = ( some @ nat @ Mi ) ) )
      & ( ~ ( ord_less @ nat @ X2 @ Mi )
       => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X2 )
          = ( if @ ( option @ nat ) @ ( ord_less @ nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
            @ ( if @ ( option @ nat )
              @ ( ( ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
                 != ( none @ nat ) )
                & ( vEBT_VEBT_less @ ( some @ nat @ ( vEBT_VEBT_low @ X2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) )
              @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( some @ nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_succ @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
              @ ( if @ ( option @ nat )
                @ ( ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
                  = ( none @ nat ) )
                @ ( none @ nat )
                @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) )
            @ ( none @ nat ) ) ) ) ) ).

% vebt_succ.simps(6)
thf(fact_223_in__children__def,axiom,
    ( vEBT_V5917875025757280293ildren
    = ( ^ [N: nat,TreeList3: list @ vEBT_VEBT,X: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ X @ N ) ) @ ( vEBT_VEBT_low @ X @ N ) ) ) ) ).

% in_children_def
thf(fact_224_Suc__diff,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq @ nat @ M @ N2 )
     => ( ( ord_less_eq @ nat @ ( one_one @ nat ) @ M )
       => ( ( suc @ ( minus_minus @ nat @ N2 @ M ) )
          = ( minus_minus @ nat @ N2 @ ( minus_minus @ nat @ M @ ( one_one @ nat ) ) ) ) ) ) ).

% Suc_diff
thf(fact_225_diff__Suc__diff__eq2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( ( ord_less_eq @ nat @ K @ J )
     => ( ( minus_minus @ nat @ ( suc @ ( minus_minus @ nat @ J @ K ) ) @ I )
        = ( minus_minus @ nat @ ( suc @ J ) @ ( plus_plus @ nat @ K @ I ) ) ) ) ).

% diff_Suc_diff_eq2
thf(fact_226_diff__Suc__diff__eq1,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( ( ord_less_eq @ nat @ K @ J )
     => ( ( minus_minus @ nat @ I @ ( suc @ ( minus_minus @ nat @ J @ K ) ) )
        = ( minus_minus @ nat @ ( plus_plus @ nat @ I @ K ) @ ( suc @ J ) ) ) ) ).

% diff_Suc_diff_eq1
thf(fact_227_semiring__norm_I76_J,axiom,
    ! [N2: num] : ( ord_less @ num @ one2 @ ( bit0 @ N2 ) ) ).

% semiring_norm(76)
thf(fact_228_semiring__norm_I69_J,axiom,
    ! [M: num] :
      ~ ( ord_less_eq @ num @ ( bit0 @ M ) @ one2 ) ).

% semiring_norm(69)
thf(fact_229_less__two__pow__divI,axiom,
    ! [X2: nat,N2: nat,M: nat] :
      ( ( ord_less @ nat @ X2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ N2 @ M ) ) )
     => ( ( ord_less_eq @ nat @ M @ N2 )
       => ( ord_less @ nat @ X2 @ ( divide_divide @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) ) ) ) ) ).

% less_two_pow_divI
thf(fact_230_less__two__pow__divD,axiom,
    ! [X2: nat,N2: nat,M: nat] :
      ( ( ord_less @ nat @ X2 @ ( divide_divide @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) ) )
     => ( ( ord_less_eq @ nat @ M @ N2 )
        & ( ord_less @ nat @ X2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ N2 @ M ) ) ) ) ) ).

% less_two_pow_divD
thf(fact_231_sum__squares__bound,axiom,
    ! [A: $tType] :
      ( ( linordered_field @ A )
     => ! [X2: A,Y2: A] : ( ord_less_eq @ A @ ( times_times @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ X2 ) @ Y2 ) @ ( plus_plus @ A @ ( power_power @ A @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).

% sum_squares_bound
thf(fact_232_option_Ocollapse,axiom,
    ! [A: $tType,Option: option @ A] :
      ( ( Option
       != ( none @ A ) )
     => ( ( some @ A @ ( the2 @ A @ Option ) )
        = Option ) ) ).

% option.collapse
thf(fact_233_highsimp,axiom,
    ! [X2: nat,N2: nat] :
      ( ( heap_Time_return @ nat @ ( vEBT_VEBT_high @ X2 @ N2 ) )
      = ( vEBT_VEBT_highi @ X2 @ N2 ) ) ).

% highsimp
thf(fact_234_lowsimp,axiom,
    ! [X2: nat,N2: nat] :
      ( ( heap_Time_return @ nat @ ( vEBT_VEBT_low @ X2 @ N2 ) )
      = ( vEBT_VEBT_lowi @ X2 @ N2 ) ) ).

% lowsimp
thf(fact_235_semiring__norm_I87_J,axiom,
    ! [M: num,N2: num] :
      ( ( ( bit0 @ M )
        = ( bit0 @ N2 ) )
      = ( M = N2 ) ) ).

% semiring_norm(87)
thf(fact_236_nat_Oinject,axiom,
    ! [X23: nat,Y23: nat] :
      ( ( ( suc @ X23 )
        = ( suc @ Y23 ) )
      = ( X23 = Y23 ) ) ).

% nat.inject
thf(fact_237_old_Onat_Oinject,axiom,
    ! [Nat: nat,Nat2: nat] :
      ( ( ( suc @ Nat )
        = ( suc @ Nat2 ) )
      = ( Nat = Nat2 ) ) ).

% old.nat.inject
thf(fact_238_option_Oinject,axiom,
    ! [A: $tType,X23: A,Y23: A] :
      ( ( ( some @ A @ X23 )
        = ( some @ A @ Y23 ) )
      = ( X23 = Y23 ) ) ).

% option.inject
thf(fact_239_highi__def,axiom,
    ( vEBT_VEBT_highi
    = ( ^ [X: nat,N: nat] : ( heap_Time_return @ nat @ ( divide_divide @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) ) ) ).

% highi_def
thf(fact_240_semiring__norm_I85_J,axiom,
    ! [M: num] :
      ( ( bit0 @ M )
     != one2 ) ).

% semiring_norm(85)
thf(fact_241_semiring__norm_I83_J,axiom,
    ! [N2: num] :
      ( one2
     != ( bit0 @ N2 ) ) ).

% semiring_norm(83)
thf(fact_242_lessI,axiom,
    ! [N2: nat] : ( ord_less @ nat @ N2 @ ( suc @ N2 ) ) ).

% lessI
thf(fact_243_Suc__mono,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less @ nat @ M @ N2 )
     => ( ord_less @ nat @ ( suc @ M ) @ ( suc @ N2 ) ) ) ).

% Suc_mono
thf(fact_244_Suc__less__eq,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less @ nat @ ( suc @ M ) @ ( suc @ N2 ) )
      = ( ord_less @ nat @ M @ N2 ) ) ).

% Suc_less_eq
thf(fact_245_add__Suc__right,axiom,
    ! [M: nat,N2: nat] :
      ( ( plus_plus @ nat @ M @ ( suc @ N2 ) )
      = ( suc @ ( plus_plus @ nat @ M @ N2 ) ) ) ).

% add_Suc_right
thf(fact_246_Suc__le__mono,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_eq @ nat @ ( suc @ N2 ) @ ( suc @ M ) )
      = ( ord_less_eq @ nat @ N2 @ M ) ) ).

% Suc_le_mono
thf(fact_247_nat__add__left__cancel__less,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( ord_less @ nat @ ( plus_plus @ nat @ K @ M ) @ ( plus_plus @ nat @ K @ N2 ) )
      = ( ord_less @ nat @ M @ N2 ) ) ).

% nat_add_left_cancel_less
thf(fact_248_diff__Suc__Suc,axiom,
    ! [M: nat,N2: nat] :
      ( ( minus_minus @ nat @ ( suc @ M ) @ ( suc @ N2 ) )
      = ( minus_minus @ nat @ M @ N2 ) ) ).

% diff_Suc_Suc
thf(fact_249_Suc__diff__diff,axiom,
    ! [M: nat,N2: nat,K: nat] :
      ( ( minus_minus @ nat @ ( minus_minus @ nat @ ( suc @ M ) @ N2 ) @ ( suc @ K ) )
      = ( minus_minus @ nat @ ( minus_minus @ nat @ M @ N2 ) @ K ) ) ).

% Suc_diff_diff
thf(fact_250_nat__add__left__cancel__le,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( ord_less_eq @ nat @ ( plus_plus @ nat @ K @ M ) @ ( plus_plus @ nat @ K @ N2 ) )
      = ( ord_less_eq @ nat @ M @ N2 ) ) ).

% nat_add_left_cancel_le
thf(fact_251_not__Some__eq2,axiom,
    ! [B: $tType,A: $tType,V: option @ ( product_prod @ A @ B )] :
      ( ( ! [X: A,Y: B] :
            ( V
           != ( some @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y ) ) ) )
      = ( V
        = ( none @ ( product_prod @ A @ B ) ) ) ) ).

% not_Some_eq2
thf(fact_252_diff__diff__left,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( minus_minus @ nat @ ( minus_minus @ nat @ I @ J ) @ K )
      = ( minus_minus @ nat @ I @ ( plus_plus @ nat @ J @ K ) ) ) ).

% diff_diff_left
thf(fact_253_diff__diff__cancel,axiom,
    ! [I: nat,N2: nat] :
      ( ( ord_less_eq @ nat @ I @ N2 )
     => ( ( minus_minus @ nat @ N2 @ ( minus_minus @ nat @ N2 @ I ) )
        = I ) ) ).

% diff_diff_cancel
thf(fact_254_not__None__eq,axiom,
    ! [A: $tType,X2: option @ A] :
      ( ( X2
       != ( none @ A ) )
      = ( ? [Y: A] :
            ( X2
            = ( some @ A @ Y ) ) ) ) ).

% not_None_eq
thf(fact_255_not__Some__eq,axiom,
    ! [A: $tType,X2: option @ A] :
      ( ( ! [Y: A] :
            ( X2
           != ( some @ A @ Y ) ) )
      = ( X2
        = ( none @ A ) ) ) ).

% not_Some_eq
thf(fact_256_nat__1__eq__mult__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( ( one_one @ nat )
        = ( times_times @ nat @ M @ N2 ) )
      = ( ( M
          = ( one_one @ nat ) )
        & ( N2
          = ( one_one @ nat ) ) ) ) ).

% nat_1_eq_mult_iff
thf(fact_257_nat__mult__eq__1__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( ( times_times @ nat @ M @ N2 )
        = ( one_one @ nat ) )
      = ( ( M
          = ( one_one @ nat ) )
        & ( N2
          = ( one_one @ nat ) ) ) ) ).

% nat_mult_eq_1_iff
thf(fact_258_semiring__norm_I6_J,axiom,
    ! [M: num,N2: num] :
      ( ( plus_plus @ num @ ( bit0 @ M ) @ ( bit0 @ N2 ) )
      = ( bit0 @ ( plus_plus @ num @ M @ N2 ) ) ) ).

% semiring_norm(6)
thf(fact_259_semiring__norm_I13_J,axiom,
    ! [M: num,N2: num] :
      ( ( times_times @ num @ ( bit0 @ M ) @ ( bit0 @ N2 ) )
      = ( bit0 @ ( bit0 @ ( times_times @ num @ M @ N2 ) ) ) ) ).

% semiring_norm(13)
thf(fact_260_semiring__norm_I12_J,axiom,
    ! [N2: num] :
      ( ( times_times @ num @ one2 @ N2 )
      = N2 ) ).

% semiring_norm(12)
thf(fact_261_semiring__norm_I11_J,axiom,
    ! [M: num] :
      ( ( times_times @ num @ M @ one2 )
      = M ) ).

% semiring_norm(11)
thf(fact_262_semiring__norm_I71_J,axiom,
    ! [M: num,N2: num] :
      ( ( ord_less_eq @ num @ ( bit0 @ M ) @ ( bit0 @ N2 ) )
      = ( ord_less_eq @ num @ M @ N2 ) ) ).

% semiring_norm(71)
thf(fact_263_semiring__norm_I78_J,axiom,
    ! [M: num,N2: num] :
      ( ( ord_less @ num @ ( bit0 @ M ) @ ( bit0 @ N2 ) )
      = ( ord_less @ num @ M @ N2 ) ) ).

% semiring_norm(78)
thf(fact_264_semiring__norm_I68_J,axiom,
    ! [N2: num] : ( ord_less_eq @ num @ one2 @ N2 ) ).

% semiring_norm(68)
thf(fact_265_semiring__norm_I75_J,axiom,
    ! [M: num] :
      ~ ( ord_less @ num @ M @ one2 ) ).

% semiring_norm(75)
thf(fact_266_diff__Suc__1,axiom,
    ! [N2: nat] :
      ( ( minus_minus @ nat @ ( suc @ N2 ) @ ( one_one @ nat ) )
      = N2 ) ).

% diff_Suc_1
thf(fact_267_mult__Suc__right,axiom,
    ! [M: nat,N2: nat] :
      ( ( times_times @ nat @ M @ ( suc @ N2 ) )
      = ( plus_plus @ nat @ M @ ( times_times @ nat @ M @ N2 ) ) ) ).

% mult_Suc_right
thf(fact_268_Nat_Oadd__diff__assoc,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( ( ord_less_eq @ nat @ K @ J )
     => ( ( plus_plus @ nat @ I @ ( minus_minus @ nat @ J @ K ) )
        = ( minus_minus @ nat @ ( plus_plus @ nat @ I @ J ) @ K ) ) ) ).

% Nat.add_diff_assoc
thf(fact_269_Nat_Oadd__diff__assoc2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( ( ord_less_eq @ nat @ K @ J )
     => ( ( plus_plus @ nat @ ( minus_minus @ nat @ J @ K ) @ I )
        = ( minus_minus @ nat @ ( plus_plus @ nat @ J @ I ) @ K ) ) ) ).

% Nat.add_diff_assoc2
thf(fact_270_Nat_Odiff__diff__right,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( ( ord_less_eq @ nat @ K @ J )
     => ( ( minus_minus @ nat @ I @ ( minus_minus @ nat @ J @ K ) )
        = ( minus_minus @ nat @ ( plus_plus @ nat @ I @ K ) @ J ) ) ) ).

% Nat.diff_diff_right
thf(fact_271_semiring__norm_I2_J,axiom,
    ( ( plus_plus @ num @ one2 @ one2 )
    = ( bit0 @ one2 ) ) ).

% semiring_norm(2)
thf(fact_272_vebt__succ_Osimps_I3_J,axiom,
    ! [Ux: nat,Uy: list @ vEBT_VEBT,Uz: vEBT_VEBT,Va: nat] :
      ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Ux @ Uy @ Uz ) @ Va )
      = ( none @ nat ) ) ).

% vebt_succ.simps(3)
thf(fact_273_vebt__member_Osimps_I2_J,axiom,
    ! [Uu: nat,Uv: list @ vEBT_VEBT,Uw: vEBT_VEBT,X2: nat] :
      ~ ( vEBT_vebt_member @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu @ Uv @ Uw ) @ X2 ) ).

% vebt_member.simps(2)
thf(fact_274_ord__eq__le__eq__trans,axiom,
    ! [A: $tType] :
      ( ( ord @ A )
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( ( A2 = B2 )
         => ( ( ord_less_eq @ A @ B2 @ C2 )
           => ( ( C2 = D2 )
             => ( ord_less_eq @ A @ A2 @ D2 ) ) ) ) ) ).

% ord_eq_le_eq_trans
thf(fact_275_measure__induct,axiom,
    ! [B: $tType,A: $tType] :
      ( ( wellorder @ B )
     => ! [F: A > B,P: A > $o,A2: A] :
          ( ! [X3: A] :
              ( ! [Y3: A] :
                  ( ( ord_less @ B @ ( F @ Y3 ) @ ( F @ X3 ) )
                 => ( P @ Y3 ) )
             => ( P @ X3 ) )
         => ( P @ A2 ) ) ) ).

% measure_induct
thf(fact_276_measure__induct__rule,axiom,
    ! [B: $tType,A: $tType] :
      ( ( wellorder @ B )
     => ! [F: A > B,P: A > $o,A2: A] :
          ( ! [X3: A] :
              ( ! [Y3: A] :
                  ( ( ord_less @ B @ ( F @ Y3 ) @ ( F @ X3 ) )
                 => ( P @ Y3 ) )
             => ( P @ X3 ) )
         => ( P @ A2 ) ) ) ).

% measure_induct_rule
thf(fact_277_pairself_Ocases,axiom,
    ! [B: $tType,A: $tType,X2: product_prod @ ( A > B ) @ ( product_prod @ A @ A )] :
      ~ ! [F2: A > B,A4: A,B3: A] :
          ( X2
         != ( product_Pair @ ( A > B ) @ ( product_prod @ A @ A ) @ F2 @ ( product_Pair @ A @ A @ A4 @ B3 ) ) ) ).

% pairself.cases
thf(fact_278_bex2I,axiom,
    ! [A: $tType,B: $tType,A2: A,B2: B,S3: set @ ( product_prod @ A @ B ),P: A > B > $o] :
      ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B2 ) @ S3 )
     => ( ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B2 ) @ S3 )
         => ( P @ A2 @ B2 ) )
       => ? [A4: A,B3: B] :
            ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A4 @ B3 ) @ S3 )
            & ( P @ A4 @ B3 ) ) ) ) ).

% bex2I
thf(fact_279_Suc__inject,axiom,
    ! [X2: nat,Y2: nat] :
      ( ( ( suc @ X2 )
        = ( suc @ Y2 ) )
     => ( X2 = Y2 ) ) ).

% Suc_inject
thf(fact_280_n__not__Suc__n,axiom,
    ! [N2: nat] :
      ( N2
     != ( suc @ N2 ) ) ).

% n_not_Suc_n
thf(fact_281_nat__neq__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( M != N2 )
      = ( ( ord_less @ nat @ M @ N2 )
        | ( ord_less @ nat @ N2 @ M ) ) ) ).

% nat_neq_iff
thf(fact_282_less__not__refl,axiom,
    ! [N2: nat] :
      ~ ( ord_less @ nat @ N2 @ N2 ) ).

% less_not_refl
thf(fact_283_less__not__refl2,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less @ nat @ N2 @ M )
     => ( M != N2 ) ) ).

% less_not_refl2
thf(fact_284_less__not__refl3,axiom,
    ! [S4: nat,T2: nat] :
      ( ( ord_less @ nat @ S4 @ T2 )
     => ( S4 != T2 ) ) ).

% less_not_refl3
thf(fact_285_less__irrefl__nat,axiom,
    ! [N2: nat] :
      ~ ( ord_less @ nat @ N2 @ N2 ) ).

% less_irrefl_nat
thf(fact_286_nat__less__induct,axiom,
    ! [P: nat > $o,N2: nat] :
      ( ! [N4: nat] :
          ( ! [M3: nat] :
              ( ( ord_less @ nat @ M3 @ N4 )
             => ( P @ M3 ) )
         => ( P @ N4 ) )
     => ( P @ N2 ) ) ).

% nat_less_induct
thf(fact_287_infinite__descent,axiom,
    ! [P: nat > $o,N2: nat] :
      ( ! [N4: nat] :
          ( ~ ( P @ N4 )
         => ? [M3: nat] :
              ( ( ord_less @ nat @ M3 @ N4 )
              & ~ ( P @ M3 ) ) )
     => ( P @ N2 ) ) ).

% infinite_descent
thf(fact_288_linorder__neqE__nat,axiom,
    ! [X2: nat,Y2: nat] :
      ( ( X2 != Y2 )
     => ( ~ ( ord_less @ nat @ X2 @ Y2 )
       => ( ord_less @ nat @ Y2 @ X2 ) ) ) ).

% linorder_neqE_nat
thf(fact_289_infinite__descent__measure,axiom,
    ! [A: $tType,P: A > $o,V2: A > nat,X2: A] :
      ( ! [X3: A] :
          ( ~ ( P @ X3 )
         => ? [Y3: A] :
              ( ( ord_less @ nat @ ( V2 @ Y3 ) @ ( V2 @ X3 ) )
              & ~ ( P @ Y3 ) ) )
     => ( P @ X2 ) ) ).

% infinite_descent_measure
thf(fact_290_size__neq__size__imp__neq,axiom,
    ! [A: $tType] :
      ( ( size @ A )
     => ! [X2: A,Y2: A] :
          ( ( ( size_size @ A @ X2 )
           != ( size_size @ A @ Y2 ) )
         => ( X2 != Y2 ) ) ) ).

% size_neq_size_imp_neq
thf(fact_291_le__refl,axiom,
    ! [N2: nat] : ( ord_less_eq @ nat @ N2 @ N2 ) ).

% le_refl
thf(fact_292_le__trans,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( ord_less_eq @ nat @ I @ J )
     => ( ( ord_less_eq @ nat @ J @ K )
       => ( ord_less_eq @ nat @ I @ K ) ) ) ).

% le_trans
thf(fact_293_eq__imp__le,axiom,
    ! [M: nat,N2: nat] :
      ( ( M = N2 )
     => ( ord_less_eq @ nat @ M @ N2 ) ) ).

% eq_imp_le
thf(fact_294_le__antisym,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq @ nat @ M @ N2 )
     => ( ( ord_less_eq @ nat @ N2 @ M )
       => ( M = N2 ) ) ) ).

% le_antisym
thf(fact_295_nat__le__linear,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq @ nat @ M @ N2 )
      | ( ord_less_eq @ nat @ N2 @ M ) ) ).

% nat_le_linear
thf(fact_296_Nat_Oex__has__greatest__nat,axiom,
    ! [P: nat > $o,K: nat,B2: nat] :
      ( ( P @ K )
     => ( ! [Y4: nat] :
            ( ( P @ Y4 )
           => ( ord_less_eq @ nat @ Y4 @ B2 ) )
       => ? [X3: nat] :
            ( ( P @ X3 )
            & ! [Y3: nat] :
                ( ( P @ Y3 )
               => ( ord_less_eq @ nat @ Y3 @ X3 ) ) ) ) ) ).

% Nat.ex_has_greatest_nat
thf(fact_297_diff__commute,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( minus_minus @ nat @ ( minus_minus @ nat @ I @ J ) @ K )
      = ( minus_minus @ nat @ ( minus_minus @ nat @ I @ K ) @ J ) ) ).

% diff_commute
thf(fact_298_vebt__pred_Osimps_I4_J,axiom,
    ! [Uy: nat,Uz: list @ vEBT_VEBT,Va: vEBT_VEBT,Vb: nat] :
      ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uy @ Uz @ Va ) @ Vb )
      = ( none @ nat ) ) ).

% vebt_pred.simps(4)
thf(fact_299_VEBT__internal_OminNulli_Ocases,axiom,
    ! [X2: vEBT_VEBTi] :
      ( ( X2
       != ( vEBT_Leafi @ $false @ $false ) )
     => ( ! [Uv2: $o] :
            ( X2
           != ( vEBT_Leafi @ $true @ Uv2 ) )
       => ( ! [Uu2: $o] :
              ( X2
             != ( vEBT_Leafi @ Uu2 @ $true ) )
         => ( ! [Uw2: nat,Ux2: array @ vEBT_VEBTi,Uy2: vEBT_VEBTi] :
                ( X2
               != ( vEBT_Nodei @ ( none @ ( product_prod @ nat @ nat ) ) @ Uw2 @ Ux2 @ Uy2 ) )
           => ~ ! [Uz2: product_prod @ nat @ nat,Va2: nat,Vb2: array @ vEBT_VEBTi,Vc: vEBT_VEBTi] :
                  ( X2
                 != ( vEBT_Nodei @ ( some @ ( product_prod @ nat @ nat ) @ Uz2 ) @ Va2 @ Vb2 @ Vc ) ) ) ) ) ) ).

% VEBT_internal.minNulli.cases
thf(fact_300_vebt__minti_Ocases,axiom,
    ! [X2: vEBT_VEBTi] :
      ( ! [A4: $o,B3: $o] :
          ( X2
         != ( vEBT_Leafi @ A4 @ B3 ) )
     => ( ! [Uu2: nat,Uv2: array @ vEBT_VEBTi,Uw2: vEBT_VEBTi] :
            ( X2
           != ( vEBT_Nodei @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw2 ) )
       => ~ ! [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: array @ vEBT_VEBTi,Uz2: vEBT_VEBTi] :
              ( X2
             != ( vEBT_Nodei @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) ) ) ) ).

% vebt_minti.cases
thf(fact_301_vebt__maxti_Osimps_I2_J,axiom,
    ! [Uu: nat,Uv: array @ vEBT_VEBTi,Uw: vEBT_VEBTi] :
      ( ( vEBT_vebt_maxti @ ( vEBT_Nodei @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu @ Uv @ Uw ) )
      = ( heap_Time_return @ ( option @ nat ) @ ( none @ nat ) ) ) ).

% vebt_maxti.simps(2)
thf(fact_302_vebt__minti_Osimps_I2_J,axiom,
    ! [Uu: nat,Uv: array @ vEBT_VEBTi,Uw: vEBT_VEBTi] :
      ( ( vEBT_vebt_minti @ ( vEBT_Nodei @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu @ Uv @ Uw ) )
      = ( heap_Time_return @ ( option @ nat ) @ ( none @ nat ) ) ) ).

% vebt_minti.simps(2)
thf(fact_303_option_Ocase__distrib,axiom,
    ! [C: $tType,B: $tType,A: $tType,H2: B > C,F1: B,F22: A > B,Option: option @ A] :
      ( ( H2 @ ( case_option @ B @ A @ F1 @ F22 @ Option ) )
      = ( case_option @ C @ A @ ( H2 @ F1 )
        @ ^ [X: A] : ( H2 @ ( F22 @ X ) )
        @ Option ) ) ).

% option.case_distrib
thf(fact_304_z1pdiv2,axiom,
    ! [B2: int] :
      ( ( divide_divide @ int @ ( plus_plus @ int @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ B2 ) @ ( one_one @ int ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) )
      = B2 ) ).

% z1pdiv2
thf(fact_305_le__some__optE,axiom,
    ! [A: $tType] :
      ( ( preorder @ A )
     => ! [M: A,X2: option @ A] :
          ( ( ord_less_eq @ ( option @ A ) @ ( some @ A @ M ) @ X2 )
         => ~ ! [M4: A] :
                ( ( X2
                  = ( some @ A @ M4 ) )
               => ~ ( ord_less_eq @ A @ M @ M4 ) ) ) ) ).

% le_some_optE
thf(fact_306_Nat_OlessE,axiom,
    ! [I: nat,K: nat] :
      ( ( ord_less @ nat @ I @ K )
     => ( ( K
         != ( suc @ I ) )
       => ~ ! [J2: nat] :
              ( ( ord_less @ nat @ I @ J2 )
             => ( K
               != ( suc @ J2 ) ) ) ) ) ).

% Nat.lessE
thf(fact_307_Suc__lessD,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less @ nat @ ( suc @ M ) @ N2 )
     => ( ord_less @ nat @ M @ N2 ) ) ).

% Suc_lessD
thf(fact_308_Suc__lessE,axiom,
    ! [I: nat,K: nat] :
      ( ( ord_less @ nat @ ( suc @ I ) @ K )
     => ~ ! [J2: nat] :
            ( ( ord_less @ nat @ I @ J2 )
           => ( K
             != ( suc @ J2 ) ) ) ) ).

% Suc_lessE
thf(fact_309_Suc__lessI,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less @ nat @ M @ N2 )
     => ( ( ( suc @ M )
         != N2 )
       => ( ord_less @ nat @ ( suc @ M ) @ N2 ) ) ) ).

% Suc_lessI
thf(fact_310_less__SucE,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less @ nat @ M @ ( suc @ N2 ) )
     => ( ~ ( ord_less @ nat @ M @ N2 )
       => ( M = N2 ) ) ) ).

% less_SucE
thf(fact_311_less__SucI,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less @ nat @ M @ N2 )
     => ( ord_less @ nat @ M @ ( suc @ N2 ) ) ) ).

% less_SucI
thf(fact_312_Ex__less__Suc,axiom,
    ! [N2: nat,P: nat > $o] :
      ( ( ? [I2: nat] :
            ( ( ord_less @ nat @ I2 @ ( suc @ N2 ) )
            & ( P @ I2 ) ) )
      = ( ( P @ N2 )
        | ? [I2: nat] :
            ( ( ord_less @ nat @ I2 @ N2 )
            & ( P @ I2 ) ) ) ) ).

% Ex_less_Suc
thf(fact_313_less__Suc__eq,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less @ nat @ M @ ( suc @ N2 ) )
      = ( ( ord_less @ nat @ M @ N2 )
        | ( M = N2 ) ) ) ).

% less_Suc_eq
thf(fact_314_not__less__eq,axiom,
    ! [M: nat,N2: nat] :
      ( ( ~ ( ord_less @ nat @ M @ N2 ) )
      = ( ord_less @ nat @ N2 @ ( suc @ M ) ) ) ).

% not_less_eq
thf(fact_315_Nat_OAll__less__Suc,axiom,
    ! [N2: nat,P: nat > $o] :
      ( ( ! [I2: nat] :
            ( ( ord_less @ nat @ I2 @ ( suc @ N2 ) )
           => ( P @ I2 ) ) )
      = ( ( P @ N2 )
        & ! [I2: nat] :
            ( ( ord_less @ nat @ I2 @ N2 )
           => ( P @ I2 ) ) ) ) ).

% Nat.All_less_Suc
thf(fact_316_Suc__less__eq2,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less @ nat @ ( suc @ N2 ) @ M )
      = ( ? [M5: nat] :
            ( ( M
              = ( suc @ M5 ) )
            & ( ord_less @ nat @ N2 @ M5 ) ) ) ) ).

% Suc_less_eq2
thf(fact_317_less__antisym,axiom,
    ! [N2: nat,M: nat] :
      ( ~ ( ord_less @ nat @ N2 @ M )
     => ( ( ord_less @ nat @ N2 @ ( suc @ M ) )
       => ( M = N2 ) ) ) ).

% less_antisym
thf(fact_318_Suc__less__SucD,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less @ nat @ ( suc @ M ) @ ( suc @ N2 ) )
     => ( ord_less @ nat @ M @ N2 ) ) ).

% Suc_less_SucD
thf(fact_319_less__trans__Suc,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( ord_less @ nat @ I @ J )
     => ( ( ord_less @ nat @ J @ K )
       => ( ord_less @ nat @ ( suc @ I ) @ K ) ) ) ).

% less_trans_Suc
thf(fact_320_less__Suc__induct,axiom,
    ! [I: nat,J: nat,P: nat > nat > $o] :
      ( ( ord_less @ nat @ I @ J )
     => ( ! [I3: nat] : ( P @ I3 @ ( suc @ I3 ) )
       => ( ! [I3: nat,J2: nat,K3: nat] :
              ( ( ord_less @ nat @ I3 @ J2 )
             => ( ( ord_less @ nat @ J2 @ K3 )
               => ( ( P @ I3 @ J2 )
                 => ( ( P @ J2 @ K3 )
                   => ( P @ I3 @ K3 ) ) ) ) )
         => ( P @ I @ J ) ) ) ) ).

% less_Suc_induct
thf(fact_321_strict__inc__induct,axiom,
    ! [I: nat,J: nat,P: nat > $o] :
      ( ( ord_less @ nat @ I @ J )
     => ( ! [I3: nat] :
            ( ( J
              = ( suc @ I3 ) )
           => ( P @ I3 ) )
       => ( ! [I3: nat] :
              ( ( ord_less @ nat @ I3 @ J )
             => ( ( P @ ( suc @ I3 ) )
               => ( P @ I3 ) ) )
         => ( P @ I ) ) ) ) ).

% strict_inc_induct
thf(fact_322_not__less__less__Suc__eq,axiom,
    ! [N2: nat,M: nat] :
      ( ~ ( ord_less @ nat @ N2 @ M )
     => ( ( ord_less @ nat @ N2 @ ( suc @ M ) )
        = ( N2 = M ) ) ) ).

% not_less_less_Suc_eq
thf(fact_323_nat__arith_Osuc1,axiom,
    ! [A3: nat,K: nat,A2: nat] :
      ( ( A3
        = ( plus_plus @ nat @ K @ A2 ) )
     => ( ( suc @ A3 )
        = ( plus_plus @ nat @ K @ ( suc @ A2 ) ) ) ) ).

% nat_arith.suc1
thf(fact_324_add__Suc,axiom,
    ! [M: nat,N2: nat] :
      ( ( plus_plus @ nat @ ( suc @ M ) @ N2 )
      = ( suc @ ( plus_plus @ nat @ M @ N2 ) ) ) ).

% add_Suc
thf(fact_325_add__Suc__shift,axiom,
    ! [M: nat,N2: nat] :
      ( ( plus_plus @ nat @ ( suc @ M ) @ N2 )
      = ( plus_plus @ nat @ M @ ( suc @ N2 ) ) ) ).

% add_Suc_shift
thf(fact_326_Suc__leD,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq @ nat @ ( suc @ M ) @ N2 )
     => ( ord_less_eq @ nat @ M @ N2 ) ) ).

% Suc_leD
thf(fact_327_le__SucE,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq @ nat @ M @ ( suc @ N2 ) )
     => ( ~ ( ord_less_eq @ nat @ M @ N2 )
       => ( M
          = ( suc @ N2 ) ) ) ) ).

% le_SucE
thf(fact_328_le__SucI,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq @ nat @ M @ N2 )
     => ( ord_less_eq @ nat @ M @ ( suc @ N2 ) ) ) ).

% le_SucI
thf(fact_329_Suc__le__D,axiom,
    ! [N2: nat,M6: nat] :
      ( ( ord_less_eq @ nat @ ( suc @ N2 ) @ M6 )
     => ? [M2: nat] :
          ( M6
          = ( suc @ M2 ) ) ) ).

% Suc_le_D
thf(fact_330_le__Suc__eq,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq @ nat @ M @ ( suc @ N2 ) )
      = ( ( ord_less_eq @ nat @ M @ N2 )
        | ( M
          = ( suc @ N2 ) ) ) ) ).

% le_Suc_eq
thf(fact_331_Suc__n__not__le__n,axiom,
    ! [N2: nat] :
      ~ ( ord_less_eq @ nat @ ( suc @ N2 ) @ N2 ) ).

% Suc_n_not_le_n
thf(fact_332_not__less__eq__eq,axiom,
    ! [M: nat,N2: nat] :
      ( ( ~ ( ord_less_eq @ nat @ M @ N2 ) )
      = ( ord_less_eq @ nat @ ( suc @ N2 ) @ M ) ) ).

% not_less_eq_eq
thf(fact_333_full__nat__induct,axiom,
    ! [P: nat > $o,N2: nat] :
      ( ! [N4: nat] :
          ( ! [M3: nat] :
              ( ( ord_less_eq @ nat @ ( suc @ M3 ) @ N4 )
             => ( P @ M3 ) )
         => ( P @ N4 ) )
     => ( P @ N2 ) ) ).

% full_nat_induct
thf(fact_334_nat__induct__at__least,axiom,
    ! [M: nat,N2: nat,P: nat > $o] :
      ( ( ord_less_eq @ nat @ M @ N2 )
     => ( ( P @ M )
       => ( ! [N4: nat] :
              ( ( ord_less_eq @ nat @ M @ N4 )
             => ( ( P @ N4 )
               => ( P @ ( suc @ N4 ) ) ) )
         => ( P @ N2 ) ) ) ) ).

% nat_induct_at_least
thf(fact_335_transitive__stepwise__le,axiom,
    ! [M: nat,N2: nat,R2: nat > nat > $o] :
      ( ( ord_less_eq @ nat @ M @ N2 )
     => ( ! [X3: nat] : ( R2 @ X3 @ X3 )
       => ( ! [X3: nat,Y4: nat,Z3: nat] :
              ( ( R2 @ X3 @ Y4 )
             => ( ( R2 @ Y4 @ Z3 )
               => ( R2 @ X3 @ Z3 ) ) )
         => ( ! [N4: nat] : ( R2 @ N4 @ ( suc @ N4 ) )
           => ( R2 @ M @ N2 ) ) ) ) ) ).

% transitive_stepwise_le
thf(fact_336_add__lessD1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( ord_less @ nat @ ( plus_plus @ nat @ I @ J ) @ K )
     => ( ord_less @ nat @ I @ K ) ) ).

% add_lessD1
thf(fact_337_add__less__mono,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ord_less @ nat @ I @ J )
     => ( ( ord_less @ nat @ K @ L )
       => ( ord_less @ nat @ ( plus_plus @ nat @ I @ K ) @ ( plus_plus @ nat @ J @ L ) ) ) ) ).

% add_less_mono
thf(fact_338_not__add__less1,axiom,
    ! [I: nat,J: nat] :
      ~ ( ord_less @ nat @ ( plus_plus @ nat @ I @ J ) @ I ) ).

% not_add_less1
thf(fact_339_not__add__less2,axiom,
    ! [J: nat,I: nat] :
      ~ ( ord_less @ nat @ ( plus_plus @ nat @ J @ I ) @ I ) ).

% not_add_less2
thf(fact_340_add__less__mono1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( ord_less @ nat @ I @ J )
     => ( ord_less @ nat @ ( plus_plus @ nat @ I @ K ) @ ( plus_plus @ nat @ J @ K ) ) ) ).

% add_less_mono1
thf(fact_341_trans__less__add1,axiom,
    ! [I: nat,J: nat,M: nat] :
      ( ( ord_less @ nat @ I @ J )
     => ( ord_less @ nat @ I @ ( plus_plus @ nat @ J @ M ) ) ) ).

% trans_less_add1
thf(fact_342_trans__less__add2,axiom,
    ! [I: nat,J: nat,M: nat] :
      ( ( ord_less @ nat @ I @ J )
     => ( ord_less @ nat @ I @ ( plus_plus @ nat @ M @ J ) ) ) ).

% trans_less_add2
thf(fact_343_less__add__eq__less,axiom,
    ! [K: nat,L: nat,M: nat,N2: nat] :
      ( ( ord_less @ nat @ K @ L )
     => ( ( ( plus_plus @ nat @ M @ L )
          = ( plus_plus @ nat @ K @ N2 ) )
       => ( ord_less @ nat @ M @ N2 ) ) ) ).

% less_add_eq_less
thf(fact_344_exists__leI,axiom,
    ! [N2: nat,P: nat > $o] :
      ( ( ! [N6: nat] :
            ( ( ord_less @ nat @ N6 @ N2 )
           => ~ ( P @ N6 ) )
       => ( P @ N2 ) )
     => ? [N7: nat] :
          ( ( ord_less_eq @ nat @ N7 @ N2 )
          & ( P @ N7 ) ) ) ).

% exists_leI
thf(fact_345_nat__less__le,axiom,
    ( ( ord_less @ nat )
    = ( ^ [M7: nat,N: nat] :
          ( ( ord_less_eq @ nat @ M7 @ N )
          & ( M7 != N ) ) ) ) ).

% nat_less_le
thf(fact_346_less__imp__le__nat,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less @ nat @ M @ N2 )
     => ( ord_less_eq @ nat @ M @ N2 ) ) ).

% less_imp_le_nat
thf(fact_347_le__eq__less__or__eq,axiom,
    ( ( ord_less_eq @ nat )
    = ( ^ [M7: nat,N: nat] :
          ( ( ord_less @ nat @ M7 @ N )
          | ( M7 = N ) ) ) ) ).

% le_eq_less_or_eq
thf(fact_348_less__or__eq__imp__le,axiom,
    ! [M: nat,N2: nat] :
      ( ( ( ord_less @ nat @ M @ N2 )
        | ( M = N2 ) )
     => ( ord_less_eq @ nat @ M @ N2 ) ) ).

% less_or_eq_imp_le
thf(fact_349_le__neq__implies__less,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq @ nat @ M @ N2 )
     => ( ( M != N2 )
       => ( ord_less @ nat @ M @ N2 ) ) ) ).

% le_neq_implies_less
thf(fact_350_less__mono__imp__le__mono,axiom,
    ! [F: nat > nat,I: nat,J: nat] :
      ( ! [I3: nat,J2: nat] :
          ( ( ord_less @ nat @ I3 @ J2 )
         => ( ord_less @ nat @ ( F @ I3 ) @ ( F @ J2 ) ) )
     => ( ( ord_less_eq @ nat @ I @ J )
       => ( ord_less_eq @ nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).

% less_mono_imp_le_mono
thf(fact_351_zero__induct__lemma,axiom,
    ! [P: nat > $o,K: nat,I: nat] :
      ( ( P @ K )
     => ( ! [N4: nat] :
            ( ( P @ ( suc @ N4 ) )
           => ( P @ N4 ) )
       => ( P @ ( minus_minus @ nat @ K @ I ) ) ) ) ).

% zero_induct_lemma
thf(fact_352_diff__less__mono2,axiom,
    ! [M: nat,N2: nat,L: nat] :
      ( ( ord_less @ nat @ M @ N2 )
     => ( ( ord_less @ nat @ M @ L )
       => ( ord_less @ nat @ ( minus_minus @ nat @ L @ N2 ) @ ( minus_minus @ nat @ L @ M ) ) ) ) ).

% diff_less_mono2
thf(fact_353_less__imp__diff__less,axiom,
    ! [J: nat,K: nat,N2: nat] :
      ( ( ord_less @ nat @ J @ K )
     => ( ord_less @ nat @ ( minus_minus @ nat @ J @ N2 ) @ K ) ) ).

% less_imp_diff_less
thf(fact_354_Suc__mult__cancel1,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( ( times_times @ nat @ ( suc @ K ) @ M )
        = ( times_times @ nat @ ( suc @ K ) @ N2 ) )
      = ( M = N2 ) ) ).

% Suc_mult_cancel1
thf(fact_355_add__leE,axiom,
    ! [M: nat,K: nat,N2: nat] :
      ( ( ord_less_eq @ nat @ ( plus_plus @ nat @ M @ K ) @ N2 )
     => ~ ( ( ord_less_eq @ nat @ M @ N2 )
         => ~ ( ord_less_eq @ nat @ K @ N2 ) ) ) ).

% add_leE
thf(fact_356_le__add1,axiom,
    ! [N2: nat,M: nat] : ( ord_less_eq @ nat @ N2 @ ( plus_plus @ nat @ N2 @ M ) ) ).

% le_add1
thf(fact_357_le__add2,axiom,
    ! [N2: nat,M: nat] : ( ord_less_eq @ nat @ N2 @ ( plus_plus @ nat @ M @ N2 ) ) ).

% le_add2
thf(fact_358_add__leD1,axiom,
    ! [M: nat,K: nat,N2: nat] :
      ( ( ord_less_eq @ nat @ ( plus_plus @ nat @ M @ K ) @ N2 )
     => ( ord_less_eq @ nat @ M @ N2 ) ) ).

% add_leD1
thf(fact_359_add__leD2,axiom,
    ! [M: nat,K: nat,N2: nat] :
      ( ( ord_less_eq @ nat @ ( plus_plus @ nat @ M @ K ) @ N2 )
     => ( ord_less_eq @ nat @ K @ N2 ) ) ).

% add_leD2
thf(fact_360_le__Suc__ex,axiom,
    ! [K: nat,L: nat] :
      ( ( ord_less_eq @ nat @ K @ L )
     => ? [N4: nat] :
          ( L
          = ( plus_plus @ nat @ K @ N4 ) ) ) ).

% le_Suc_ex
thf(fact_361_add__le__mono,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ord_less_eq @ nat @ I @ J )
     => ( ( ord_less_eq @ nat @ K @ L )
       => ( ord_less_eq @ nat @ ( plus_plus @ nat @ I @ K ) @ ( plus_plus @ nat @ J @ L ) ) ) ) ).

% add_le_mono
thf(fact_362_add__le__mono1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( ord_less_eq @ nat @ I @ J )
     => ( ord_less_eq @ nat @ ( plus_plus @ nat @ I @ K ) @ ( plus_plus @ nat @ J @ K ) ) ) ).

% add_le_mono1
thf(fact_363_trans__le__add1,axiom,
    ! [I: nat,J: nat,M: nat] :
      ( ( ord_less_eq @ nat @ I @ J )
     => ( ord_less_eq @ nat @ I @ ( plus_plus @ nat @ J @ M ) ) ) ).

% trans_le_add1
thf(fact_364_trans__le__add2,axiom,
    ! [I: nat,J: nat,M: nat] :
      ( ( ord_less_eq @ nat @ I @ J )
     => ( ord_less_eq @ nat @ I @ ( plus_plus @ nat @ M @ J ) ) ) ).

% trans_le_add2
thf(fact_365_nat__le__iff__add,axiom,
    ( ( ord_less_eq @ nat )
    = ( ^ [M7: nat,N: nat] :
        ? [K4: nat] :
          ( N
          = ( plus_plus @ nat @ M7 @ K4 ) ) ) ) ).

% nat_le_iff_add
thf(fact_366_fstE,axiom,
    ! [B: $tType,A: $tType,X2: product_prod @ A @ B,A2: A,B2: B,P: A > $o] :
      ( ( X2
        = ( product_Pair @ A @ B @ A2 @ B2 ) )
     => ( ( P @ ( product_fst @ A @ B @ X2 ) )
       => ( P @ A2 ) ) ) ).

% fstE
thf(fact_367_diff__add__inverse2,axiom,
    ! [M: nat,N2: nat] :
      ( ( minus_minus @ nat @ ( plus_plus @ nat @ M @ N2 ) @ N2 )
      = M ) ).

% diff_add_inverse2
thf(fact_368_diff__add__inverse,axiom,
    ! [N2: nat,M: nat] :
      ( ( minus_minus @ nat @ ( plus_plus @ nat @ N2 @ M ) @ N2 )
      = M ) ).

% diff_add_inverse
thf(fact_369_diff__cancel2,axiom,
    ! [M: nat,K: nat,N2: nat] :
      ( ( minus_minus @ nat @ ( plus_plus @ nat @ M @ K ) @ ( plus_plus @ nat @ N2 @ K ) )
      = ( minus_minus @ nat @ M @ N2 ) ) ).

% diff_cancel2
thf(fact_370_Nat_Odiff__cancel,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( minus_minus @ nat @ ( plus_plus @ nat @ K @ M ) @ ( plus_plus @ nat @ K @ N2 ) )
      = ( minus_minus @ nat @ M @ N2 ) ) ).

% Nat.diff_cancel
thf(fact_371_eq__diff__iff,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( ord_less_eq @ nat @ K @ M )
     => ( ( ord_less_eq @ nat @ K @ N2 )
       => ( ( ( minus_minus @ nat @ M @ K )
            = ( minus_minus @ nat @ N2 @ K ) )
          = ( M = N2 ) ) ) ) ).

% eq_diff_iff
thf(fact_372_le__diff__iff,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( ord_less_eq @ nat @ K @ M )
     => ( ( ord_less_eq @ nat @ K @ N2 )
       => ( ( ord_less_eq @ nat @ ( minus_minus @ nat @ M @ K ) @ ( minus_minus @ nat @ N2 @ K ) )
          = ( ord_less_eq @ nat @ M @ N2 ) ) ) ) ).

% le_diff_iff
thf(fact_373_Nat_Odiff__diff__eq,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( ord_less_eq @ nat @ K @ M )
     => ( ( ord_less_eq @ nat @ K @ N2 )
       => ( ( minus_minus @ nat @ ( minus_minus @ nat @ M @ K ) @ ( minus_minus @ nat @ N2 @ K ) )
          = ( minus_minus @ nat @ M @ N2 ) ) ) ) ).

% Nat.diff_diff_eq
thf(fact_374_diff__le__mono,axiom,
    ! [M: nat,N2: nat,L: nat] :
      ( ( ord_less_eq @ nat @ M @ N2 )
     => ( ord_less_eq @ nat @ ( minus_minus @ nat @ M @ L ) @ ( minus_minus @ nat @ N2 @ L ) ) ) ).

% diff_le_mono
thf(fact_375_diff__le__self,axiom,
    ! [M: nat,N2: nat] : ( ord_less_eq @ nat @ ( minus_minus @ nat @ M @ N2 ) @ M ) ).

% diff_le_self
thf(fact_376_le__diff__iff_H,axiom,
    ! [A2: nat,C2: nat,B2: nat] :
      ( ( ord_less_eq @ nat @ A2 @ C2 )
     => ( ( ord_less_eq @ nat @ B2 @ C2 )
       => ( ( ord_less_eq @ nat @ ( minus_minus @ nat @ C2 @ A2 ) @ ( minus_minus @ nat @ C2 @ B2 ) )
          = ( ord_less_eq @ nat @ B2 @ A2 ) ) ) ) ).

% le_diff_iff'
thf(fact_377_diff__le__mono2,axiom,
    ! [M: nat,N2: nat,L: nat] :
      ( ( ord_less_eq @ nat @ M @ N2 )
     => ( ord_less_eq @ nat @ ( minus_minus @ nat @ L @ N2 ) @ ( minus_minus @ nat @ L @ M ) ) ) ).

% diff_le_mono2
thf(fact_378_option_Odistinct_I1_J,axiom,
    ! [A: $tType,X23: A] :
      ( ( none @ A )
     != ( some @ A @ X23 ) ) ).

% option.distinct(1)
thf(fact_379_option_OdiscI,axiom,
    ! [A: $tType,Option: option @ A,X23: A] :
      ( ( Option
        = ( some @ A @ X23 ) )
     => ( Option
       != ( none @ A ) ) ) ).

% option.discI
thf(fact_380_option_Oexhaust,axiom,
    ! [A: $tType,Y2: option @ A] :
      ( ( Y2
       != ( none @ A ) )
     => ~ ! [X24: A] :
            ( Y2
           != ( some @ A @ X24 ) ) ) ).

% option.exhaust
thf(fact_381_split__option__ex,axiom,
    ! [A: $tType] :
      ( ( ^ [P2: ( option @ A ) > $o] :
          ? [X4: option @ A] : ( P2 @ X4 ) )
      = ( ^ [P3: ( option @ A ) > $o] :
            ( ( P3 @ ( none @ A ) )
            | ? [X: A] : ( P3 @ ( some @ A @ X ) ) ) ) ) ).

% split_option_ex
thf(fact_382_split__option__all,axiom,
    ! [A: $tType] :
      ( ( ^ [P2: ( option @ A ) > $o] :
          ! [X4: option @ A] : ( P2 @ X4 ) )
      = ( ^ [P3: ( option @ A ) > $o] :
            ( ( P3 @ ( none @ A ) )
            & ! [X: A] : ( P3 @ ( some @ A @ X ) ) ) ) ) ).

% split_option_all
thf(fact_383_combine__options__cases,axiom,
    ! [A: $tType,B: $tType,X2: option @ A,P: ( option @ A ) > ( option @ B ) > $o,Y2: option @ B] :
      ( ( ( X2
          = ( none @ A ) )
       => ( P @ X2 @ Y2 ) )
     => ( ( ( Y2
            = ( none @ B ) )
         => ( P @ X2 @ Y2 ) )
       => ( ! [A4: A,B3: B] :
              ( ( X2
                = ( some @ A @ A4 ) )
             => ( ( Y2
                  = ( some @ B @ B3 ) )
               => ( P @ X2 @ Y2 ) ) )
         => ( P @ X2 @ Y2 ) ) ) ) ).

% combine_options_cases
thf(fact_384_sndE,axiom,
    ! [A: $tType,B: $tType,X2: product_prod @ A @ B,A2: A,B2: B,P: B > $o] :
      ( ( X2
        = ( product_Pair @ A @ B @ A2 @ B2 ) )
     => ( ( P @ ( product_snd @ A @ B @ X2 ) )
       => ( P @ B2 ) ) ) ).

% sndE
thf(fact_385_add__mult__distrib,axiom,
    ! [M: nat,N2: nat,K: nat] :
      ( ( times_times @ nat @ ( plus_plus @ nat @ M @ N2 ) @ K )
      = ( plus_plus @ nat @ ( times_times @ nat @ M @ K ) @ ( times_times @ nat @ N2 @ K ) ) ) ).

% add_mult_distrib
thf(fact_386_add__mult__distrib2,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( times_times @ nat @ K @ ( plus_plus @ nat @ M @ N2 ) )
      = ( plus_plus @ nat @ ( times_times @ nat @ K @ M ) @ ( times_times @ nat @ K @ N2 ) ) ) ).

% add_mult_distrib2
thf(fact_387_left__add__mult__distrib,axiom,
    ! [I: nat,U: nat,J: nat,K: nat] :
      ( ( plus_plus @ nat @ ( times_times @ nat @ I @ U ) @ ( plus_plus @ nat @ ( times_times @ nat @ J @ U ) @ K ) )
      = ( plus_plus @ nat @ ( times_times @ nat @ ( plus_plus @ nat @ I @ J ) @ U ) @ K ) ) ).

% left_add_mult_distrib
thf(fact_388_le__cube,axiom,
    ! [M: nat] : ( ord_less_eq @ nat @ M @ ( times_times @ nat @ M @ ( times_times @ nat @ M @ M ) ) ) ).

% le_cube
thf(fact_389_le__square,axiom,
    ! [M: nat] : ( ord_less_eq @ nat @ M @ ( times_times @ nat @ M @ M ) ) ).

% le_square
thf(fact_390_mult__le__mono,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ord_less_eq @ nat @ I @ J )
     => ( ( ord_less_eq @ nat @ K @ L )
       => ( ord_less_eq @ nat @ ( times_times @ nat @ I @ K ) @ ( times_times @ nat @ J @ L ) ) ) ) ).

% mult_le_mono
thf(fact_391_mult__le__mono1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( ord_less_eq @ nat @ I @ J )
     => ( ord_less_eq @ nat @ ( times_times @ nat @ I @ K ) @ ( times_times @ nat @ J @ K ) ) ) ).

% mult_le_mono1
thf(fact_392_mult__le__mono2,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( ord_less_eq @ nat @ I @ J )
     => ( ord_less_eq @ nat @ ( times_times @ nat @ K @ I ) @ ( times_times @ nat @ K @ J ) ) ) ).

% mult_le_mono2
thf(fact_393_nat__mult__1,axiom,
    ! [N2: nat] :
      ( ( times_times @ nat @ ( one_one @ nat ) @ N2 )
      = N2 ) ).

% nat_mult_1
thf(fact_394_nat__mult__1__right,axiom,
    ! [N2: nat] :
      ( ( times_times @ nat @ N2 @ ( one_one @ nat ) )
      = N2 ) ).

% nat_mult_1_right
thf(fact_395_diff__mult__distrib,axiom,
    ! [M: nat,N2: nat,K: nat] :
      ( ( times_times @ nat @ ( minus_minus @ nat @ M @ N2 ) @ K )
      = ( minus_minus @ nat @ ( times_times @ nat @ M @ K ) @ ( times_times @ nat @ N2 @ K ) ) ) ).

% diff_mult_distrib
thf(fact_396_diff__mult__distrib2,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( times_times @ nat @ K @ ( minus_minus @ nat @ M @ N2 ) )
      = ( minus_minus @ nat @ ( times_times @ nat @ K @ M ) @ ( times_times @ nat @ K @ N2 ) ) ) ).

% diff_mult_distrib2
thf(fact_397_option_Osel,axiom,
    ! [A: $tType,X23: A] :
      ( ( the2 @ A @ ( some @ A @ X23 ) )
      = X23 ) ).

% option.sel
thf(fact_398_Ex__prod__contract,axiom,
    ! [B: $tType,A: $tType,P: A > B > $o] :
      ( ( ? [A5: A,X5: B] : ( P @ A5 @ X5 ) )
      = ( ? [Z2: product_prod @ A @ B] : ( P @ ( product_fst @ A @ B @ Z2 ) @ ( product_snd @ A @ B @ Z2 ) ) ) ) ).

% Ex_prod_contract
thf(fact_399_All__prod__contract,axiom,
    ! [B: $tType,A: $tType,P: A > B > $o] :
      ( ( ! [A5: A,X5: B] : ( P @ A5 @ X5 ) )
      = ( ! [Z2: product_prod @ A @ B] : ( P @ ( product_fst @ A @ B @ Z2 ) @ ( product_snd @ A @ B @ Z2 ) ) ) ) ).

% All_prod_contract
thf(fact_400_option_Oexpand,axiom,
    ! [A: $tType,Option: option @ A,Option2: option @ A] :
      ( ( ( Option
          = ( none @ A ) )
        = ( Option2
          = ( none @ A ) ) )
     => ( ( ( Option
           != ( none @ A ) )
         => ( ( Option2
             != ( none @ A ) )
           => ( ( the2 @ A @ Option )
              = ( the2 @ A @ Option2 ) ) ) )
       => ( Option = Option2 ) ) ) ).

% option.expand
thf(fact_401_option_Osimps_I5_J,axiom,
    ! [B: $tType,A: $tType,F1: B,F22: A > B,X23: A] :
      ( ( case_option @ B @ A @ F1 @ F22 @ ( some @ A @ X23 ) )
      = ( F22 @ X23 ) ) ).

% option.simps(5)
thf(fact_402_option_Osimps_I4_J,axiom,
    ! [A: $tType,B: $tType,F1: B,F22: A > B] :
      ( ( case_option @ B @ A @ F1 @ F22 @ ( none @ A ) )
      = F1 ) ).

% option.simps(4)
thf(fact_403_four__x__squared,axiom,
    ! [X2: real] :
      ( ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
      = ( power_power @ real @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ X2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ).

% four_x_squared
thf(fact_404_L2__set__mult__ineq__lemma,axiom,
    ! [A2: real,C2: real,B2: real,D2: real] : ( ord_less_eq @ real @ ( times_times @ real @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( times_times @ real @ A2 @ C2 ) ) @ ( times_times @ real @ B2 @ D2 ) ) @ ( plus_plus @ real @ ( times_times @ real @ ( power_power @ real @ A2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ D2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( times_times @ real @ ( power_power @ real @ B2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ C2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).

% L2_set_mult_ineq_lemma
thf(fact_405_lift__Suc__mono__less,axiom,
    ! [A: $tType] :
      ( ( order @ A )
     => ! [F: nat > A,N2: nat,N5: nat] :
          ( ! [N4: nat] : ( ord_less @ A @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
         => ( ( ord_less @ nat @ N2 @ N5 )
           => ( ord_less @ A @ ( F @ N2 ) @ ( F @ N5 ) ) ) ) ) ).

% lift_Suc_mono_less
thf(fact_406_lift__Suc__mono__less__iff,axiom,
    ! [A: $tType] :
      ( ( order @ A )
     => ! [F: nat > A,N2: nat,M: nat] :
          ( ! [N4: nat] : ( ord_less @ A @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
         => ( ( ord_less @ A @ ( F @ N2 ) @ ( F @ M ) )
            = ( ord_less @ nat @ N2 @ M ) ) ) ) ).

% lift_Suc_mono_less_iff
thf(fact_407_lift__Suc__mono__le,axiom,
    ! [A: $tType] :
      ( ( order @ A )
     => ! [F: nat > A,N2: nat,N5: nat] :
          ( ! [N4: nat] : ( ord_less_eq @ A @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
         => ( ( ord_less_eq @ nat @ N2 @ N5 )
           => ( ord_less_eq @ A @ ( F @ N2 ) @ ( F @ N5 ) ) ) ) ) ).

% lift_Suc_mono_le
thf(fact_408_lift__Suc__antimono__le,axiom,
    ! [A: $tType] :
      ( ( order @ A )
     => ! [F: nat > A,N2: nat,N5: nat] :
          ( ! [N4: nat] : ( ord_less_eq @ A @ ( F @ ( suc @ N4 ) ) @ ( F @ N4 ) )
         => ( ( ord_less_eq @ nat @ N2 @ N5 )
           => ( ord_less_eq @ A @ ( F @ N5 ) @ ( F @ N2 ) ) ) ) ) ).

% lift_Suc_antimono_le
thf(fact_409_less__natE,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less @ nat @ M @ N2 )
     => ~ ! [Q3: nat] :
            ( N2
           != ( suc @ ( plus_plus @ nat @ M @ Q3 ) ) ) ) ).

% less_natE
thf(fact_410_less__add__Suc1,axiom,
    ! [I: nat,M: nat] : ( ord_less @ nat @ I @ ( suc @ ( plus_plus @ nat @ I @ M ) ) ) ).

% less_add_Suc1
thf(fact_411_less__add__Suc2,axiom,
    ! [I: nat,M: nat] : ( ord_less @ nat @ I @ ( suc @ ( plus_plus @ nat @ M @ I ) ) ) ).

% less_add_Suc2
thf(fact_412_less__iff__Suc__add,axiom,
    ( ( ord_less @ nat )
    = ( ^ [M7: nat,N: nat] :
        ? [K4: nat] :
          ( N
          = ( suc @ ( plus_plus @ nat @ M7 @ K4 ) ) ) ) ) ).

% less_iff_Suc_add
thf(fact_413_less__imp__Suc__add,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less @ nat @ M @ N2 )
     => ? [K3: nat] :
          ( N2
          = ( suc @ ( plus_plus @ nat @ M @ K3 ) ) ) ) ).

% less_imp_Suc_add
thf(fact_414_nat__in__between__eq_I2_J,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ( ord_less_eq @ nat @ A2 @ B2 )
        & ( ord_less @ nat @ B2 @ ( suc @ A2 ) ) )
      = ( B2 = A2 ) ) ).

% nat_in_between_eq(2)
thf(fact_415_nat__in__between__eq_I1_J,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ( ord_less @ nat @ A2 @ B2 )
        & ( ord_less_eq @ nat @ B2 @ ( suc @ A2 ) ) )
      = ( B2
        = ( suc @ A2 ) ) ) ).

% nat_in_between_eq(1)
thf(fact_416_Suc__leI,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less @ nat @ M @ N2 )
     => ( ord_less_eq @ nat @ ( suc @ M ) @ N2 ) ) ).

% Suc_leI
thf(fact_417_Suc__le__eq,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq @ nat @ ( suc @ M ) @ N2 )
      = ( ord_less @ nat @ M @ N2 ) ) ).

% Suc_le_eq
thf(fact_418_dec__induct,axiom,
    ! [I: nat,J: nat,P: nat > $o] :
      ( ( ord_less_eq @ nat @ I @ J )
     => ( ( P @ I )
       => ( ! [N4: nat] :
              ( ( ord_less_eq @ nat @ I @ N4 )
             => ( ( ord_less @ nat @ N4 @ J )
               => ( ( P @ N4 )
                 => ( P @ ( suc @ N4 ) ) ) ) )
         => ( P @ J ) ) ) ) ).

% dec_induct
thf(fact_419_inc__induct,axiom,
    ! [I: nat,J: nat,P: nat > $o] :
      ( ( ord_less_eq @ nat @ I @ J )
     => ( ( P @ J )
       => ( ! [N4: nat] :
              ( ( ord_less_eq @ nat @ I @ N4 )
             => ( ( ord_less @ nat @ N4 @ J )
               => ( ( P @ ( suc @ N4 ) )
                 => ( P @ N4 ) ) ) )
         => ( P @ I ) ) ) ) ).

% inc_induct
thf(fact_420_Suc__le__lessD,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq @ nat @ ( suc @ M ) @ N2 )
     => ( ord_less @ nat @ M @ N2 ) ) ).

% Suc_le_lessD
thf(fact_421_le__less__Suc__eq,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq @ nat @ M @ N2 )
     => ( ( ord_less @ nat @ N2 @ ( suc @ M ) )
        = ( N2 = M ) ) ) ).

% le_less_Suc_eq
thf(fact_422_less__Suc__eq__le,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less @ nat @ M @ ( suc @ N2 ) )
      = ( ord_less_eq @ nat @ M @ N2 ) ) ).

% less_Suc_eq_le
thf(fact_423_less__eq__Suc__le,axiom,
    ( ( ord_less @ nat )
    = ( ^ [N: nat] : ( ord_less_eq @ nat @ ( suc @ N ) ) ) ) ).

% less_eq_Suc_le
thf(fact_424_le__imp__less__Suc,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq @ nat @ M @ N2 )
     => ( ord_less @ nat @ M @ ( suc @ N2 ) ) ) ).

% le_imp_less_Suc
thf(fact_425_Suc__diff__Suc,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less @ nat @ N2 @ M )
     => ( ( suc @ ( minus_minus @ nat @ M @ ( suc @ N2 ) ) )
        = ( minus_minus @ nat @ M @ N2 ) ) ) ).

% Suc_diff_Suc
thf(fact_426_diff__less__Suc,axiom,
    ! [M: nat,N2: nat] : ( ord_less @ nat @ ( minus_minus @ nat @ M @ N2 ) @ ( suc @ M ) ) ).

% diff_less_Suc
thf(fact_427_Suc__eq__plus1,axiom,
    ( suc
    = ( ^ [N: nat] : ( plus_plus @ nat @ N @ ( one_one @ nat ) ) ) ) ).

% Suc_eq_plus1
thf(fact_428_plus__1__eq__Suc,axiom,
    ( ( plus_plus @ nat @ ( one_one @ nat ) )
    = suc ) ).

% plus_1_eq_Suc
thf(fact_429_Suc__eq__plus1__left,axiom,
    ( suc
    = ( plus_plus @ nat @ ( one_one @ nat ) ) ) ).

% Suc_eq_plus1_left
thf(fact_430_Suc__mult__less__cancel1,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( ord_less @ nat @ ( times_times @ nat @ ( suc @ K ) @ M ) @ ( times_times @ nat @ ( suc @ K ) @ N2 ) )
      = ( ord_less @ nat @ M @ N2 ) ) ).

% Suc_mult_less_cancel1
thf(fact_431_mono__nat__linear__lb,axiom,
    ! [F: nat > nat,M: nat,K: nat] :
      ( ! [M2: nat,N4: nat] :
          ( ( ord_less @ nat @ M2 @ N4 )
         => ( ord_less @ nat @ ( F @ M2 ) @ ( F @ N4 ) ) )
     => ( ord_less_eq @ nat @ ( plus_plus @ nat @ ( F @ M ) @ K ) @ ( F @ ( plus_plus @ nat @ M @ K ) ) ) ) ).

% mono_nat_linear_lb
thf(fact_432_Suc__diff__le,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_eq @ nat @ N2 @ M )
     => ( ( minus_minus @ nat @ ( suc @ M ) @ N2 )
        = ( suc @ ( minus_minus @ nat @ M @ N2 ) ) ) ) ).

% Suc_diff_le
thf(fact_433_diff__Suc__eq__diff__pred,axiom,
    ! [M: nat,N2: nat] :
      ( ( minus_minus @ nat @ M @ ( suc @ N2 ) )
      = ( minus_minus @ nat @ ( minus_minus @ nat @ M @ ( one_one @ nat ) ) @ N2 ) ) ).

% diff_Suc_eq_diff_pred
thf(fact_434_less__diff__conv,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( ord_less @ nat @ I @ ( minus_minus @ nat @ J @ K ) )
      = ( ord_less @ nat @ ( plus_plus @ nat @ I @ K ) @ J ) ) ).

% less_diff_conv
thf(fact_435_add__diff__inverse__nat,axiom,
    ! [M: nat,N2: nat] :
      ( ~ ( ord_less @ nat @ M @ N2 )
     => ( ( plus_plus @ nat @ N2 @ ( minus_minus @ nat @ M @ N2 ) )
        = M ) ) ).

% add_diff_inverse_nat
thf(fact_436_less__diff__iff,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( ord_less_eq @ nat @ K @ M )
     => ( ( ord_less_eq @ nat @ K @ N2 )
       => ( ( ord_less @ nat @ ( minus_minus @ nat @ M @ K ) @ ( minus_minus @ nat @ N2 @ K ) )
          = ( ord_less @ nat @ M @ N2 ) ) ) ) ).

% less_diff_iff
thf(fact_437_diff__less__mono,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( ( ord_less @ nat @ A2 @ B2 )
     => ( ( ord_less_eq @ nat @ C2 @ A2 )
       => ( ord_less @ nat @ ( minus_minus @ nat @ A2 @ C2 ) @ ( minus_minus @ nat @ B2 @ C2 ) ) ) ) ).

% diff_less_mono
thf(fact_438_mult__Suc,axiom,
    ! [M: nat,N2: nat] :
      ( ( times_times @ nat @ ( suc @ M ) @ N2 )
      = ( plus_plus @ nat @ N2 @ ( times_times @ nat @ M @ N2 ) ) ) ).

% mult_Suc
thf(fact_439_Suc__mult__le__cancel1,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( ord_less_eq @ nat @ ( times_times @ nat @ ( suc @ K ) @ M ) @ ( times_times @ nat @ ( suc @ K ) @ N2 ) )
      = ( ord_less_eq @ nat @ M @ N2 ) ) ).

% Suc_mult_le_cancel1
thf(fact_440_mlex__snd__decrI,axiom,
    ! [A2: nat,A6: nat,B2: nat,B4: nat,N3: nat] :
      ( ( A2 = A6 )
     => ( ( ord_less @ nat @ B2 @ B4 )
       => ( ord_less @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ A2 @ N3 ) @ B2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ A6 @ N3 ) @ B4 ) ) ) ) ).

% mlex_snd_decrI
thf(fact_441_mlex__fst__decrI,axiom,
    ! [A2: nat,A6: nat,B2: nat,N3: nat,B4: nat] :
      ( ( ord_less @ nat @ A2 @ A6 )
     => ( ( ord_less @ nat @ B2 @ N3 )
       => ( ( ord_less @ nat @ B4 @ N3 )
         => ( ord_less @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ A2 @ N3 ) @ B2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ A6 @ N3 ) @ B4 ) ) ) ) ) ).

% mlex_fst_decrI
thf(fact_442_mlex__bound,axiom,
    ! [A2: nat,A3: nat,B2: nat,N3: nat] :
      ( ( ord_less @ nat @ A2 @ A3 )
     => ( ( ord_less @ nat @ B2 @ N3 )
       => ( ord_less @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ A2 @ N3 ) @ B2 ) @ ( times_times @ nat @ A3 @ N3 ) ) ) ) ).

% mlex_bound
thf(fact_443_le__diff__conv,axiom,
    ! [J: nat,K: nat,I: nat] :
      ( ( ord_less_eq @ nat @ ( minus_minus @ nat @ J @ K ) @ I )
      = ( ord_less_eq @ nat @ J @ ( plus_plus @ nat @ I @ K ) ) ) ).

% le_diff_conv
thf(fact_444_Nat_Ole__diff__conv2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( ( ord_less_eq @ nat @ K @ J )
     => ( ( ord_less_eq @ nat @ I @ ( minus_minus @ nat @ J @ K ) )
        = ( ord_less_eq @ nat @ ( plus_plus @ nat @ I @ K ) @ J ) ) ) ).

% Nat.le_diff_conv2
thf(fact_445_Nat_Odiff__add__assoc,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( ( ord_less_eq @ nat @ K @ J )
     => ( ( minus_minus @ nat @ ( plus_plus @ nat @ I @ J ) @ K )
        = ( plus_plus @ nat @ I @ ( minus_minus @ nat @ J @ K ) ) ) ) ).

% Nat.diff_add_assoc
thf(fact_446_Nat_Odiff__add__assoc2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( ( ord_less_eq @ nat @ K @ J )
     => ( ( minus_minus @ nat @ ( plus_plus @ nat @ J @ I ) @ K )
        = ( plus_plus @ nat @ ( minus_minus @ nat @ J @ K ) @ I ) ) ) ).

% Nat.diff_add_assoc2
thf(fact_447_Nat_Ole__imp__diff__is__add,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( ord_less_eq @ nat @ I @ J )
     => ( ( ( minus_minus @ nat @ J @ I )
          = K )
        = ( J
          = ( plus_plus @ nat @ K @ I ) ) ) ) ).

% Nat.le_imp_diff_is_add
thf(fact_448_obtain__list__from__elements,axiom,
    ! [A: $tType,N2: nat,P: A > nat > $o] :
      ( ! [I3: nat] :
          ( ( ord_less @ nat @ I3 @ N2 )
         => ? [Li: A] : ( P @ Li @ I3 ) )
     => ~ ! [L3: list @ A] :
            ( ( ( size_size @ ( list @ A ) @ L3 )
              = N2 )
           => ~ ! [I4: nat] :
                  ( ( ord_less @ nat @ I4 @ N2 )
                 => ( P @ ( nth @ A @ L3 @ I4 ) @ I4 ) ) ) ) ).

% obtain_list_from_elements
thf(fact_449_mlex__leI,axiom,
    ! [A2: nat,A6: nat,B2: nat,B4: nat,N3: nat] :
      ( ( ord_less_eq @ nat @ A2 @ A6 )
     => ( ( ord_less_eq @ nat @ B2 @ B4 )
       => ( ord_less_eq @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ A2 @ N3 ) @ B2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ A6 @ N3 ) @ B4 ) ) ) ) ).

% mlex_leI
thf(fact_450_div__mult__le,axiom,
    ! [A2: nat,B2: nat] : ( ord_less_eq @ nat @ ( times_times @ nat @ ( divide_divide @ nat @ A2 @ B2 ) @ B2 ) @ A2 ) ).

% div_mult_le
thf(fact_451_option_Oexhaust__sel,axiom,
    ! [A: $tType,Option: option @ A] :
      ( ( Option
       != ( none @ A ) )
     => ( Option
        = ( some @ A @ ( the2 @ A @ Option ) ) ) ) ).

% option.exhaust_sel
thf(fact_452_option_Ocase__eq__if,axiom,
    ! [A: $tType,B: $tType] :
      ( ( case_option @ B @ A )
      = ( ^ [F12: B,F23: A > B,Option3: option @ A] :
            ( if @ B
            @ ( Option3
              = ( none @ A ) )
            @ F12
            @ ( F23 @ ( the2 @ A @ Option3 ) ) ) ) ) ).

% option.case_eq_if
thf(fact_453_is__succ__in__set__def,axiom,
    ( vEBT_is_succ_in_set
    = ( ^ [Xs: set @ nat,X: nat,Y: nat] :
          ( ( member @ nat @ Y @ Xs )
          & ( ord_less @ nat @ X @ Y )
          & ! [Z2: nat] :
              ( ( member @ nat @ Z2 @ Xs )
             => ( ( ord_less @ nat @ X @ Z2 )
               => ( ord_less_eq @ nat @ Y @ Z2 ) ) ) ) ) ) ).

% is_succ_in_set_def
thf(fact_454_less__diff__conv2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( ( ord_less_eq @ nat @ K @ J )
     => ( ( ord_less @ nat @ ( minus_minus @ nat @ J @ K ) @ I )
        = ( ord_less @ nat @ J @ ( plus_plus @ nat @ I @ K ) ) ) ) ).

% less_diff_conv2
thf(fact_455_nat__eq__add__iff1,axiom,
    ! [J: nat,I: nat,U: nat,M: nat,N2: nat] :
      ( ( ord_less_eq @ nat @ J @ I )
     => ( ( ( plus_plus @ nat @ ( times_times @ nat @ I @ U ) @ M )
          = ( plus_plus @ nat @ ( times_times @ nat @ J @ U ) @ N2 ) )
        = ( ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ I @ J ) @ U ) @ M )
          = N2 ) ) ) ).

% nat_eq_add_iff1
thf(fact_456_nat__eq__add__iff2,axiom,
    ! [I: nat,J: nat,U: nat,M: nat,N2: nat] :
      ( ( ord_less_eq @ nat @ I @ J )
     => ( ( ( plus_plus @ nat @ ( times_times @ nat @ I @ U ) @ M )
          = ( plus_plus @ nat @ ( times_times @ nat @ J @ U ) @ N2 ) )
        = ( M
          = ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ J @ I ) @ U ) @ N2 ) ) ) ) ).

% nat_eq_add_iff2
thf(fact_457_nat__le__add__iff1,axiom,
    ! [J: nat,I: nat,U: nat,M: nat,N2: nat] :
      ( ( ord_less_eq @ nat @ J @ I )
     => ( ( ord_less_eq @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ I @ U ) @ M ) @ ( plus_plus @ nat @ ( times_times @ nat @ J @ U ) @ N2 ) )
        = ( ord_less_eq @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ I @ J ) @ U ) @ M ) @ N2 ) ) ) ).

% nat_le_add_iff1
thf(fact_458_nat__le__add__iff2,axiom,
    ! [I: nat,J: nat,U: nat,M: nat,N2: nat] :
      ( ( ord_less_eq @ nat @ I @ J )
     => ( ( ord_less_eq @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ I @ U ) @ M ) @ ( plus_plus @ nat @ ( times_times @ nat @ J @ U ) @ N2 ) )
        = ( ord_less_eq @ nat @ M @ ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ J @ I ) @ U ) @ N2 ) ) ) ) ).

% nat_le_add_iff2
thf(fact_459_nat__diff__add__eq1,axiom,
    ! [J: nat,I: nat,U: nat,M: nat,N2: nat] :
      ( ( ord_less_eq @ nat @ J @ I )
     => ( ( minus_minus @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ I @ U ) @ M ) @ ( plus_plus @ nat @ ( times_times @ nat @ J @ U ) @ N2 ) )
        = ( minus_minus @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ I @ J ) @ U ) @ M ) @ N2 ) ) ) ).

% nat_diff_add_eq1
thf(fact_460_nat__diff__add__eq2,axiom,
    ! [I: nat,J: nat,U: nat,M: nat,N2: nat] :
      ( ( ord_less_eq @ nat @ I @ J )
     => ( ( minus_minus @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ I @ U ) @ M ) @ ( plus_plus @ nat @ ( times_times @ nat @ J @ U ) @ N2 ) )
        = ( minus_minus @ nat @ M @ ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ J @ I ) @ U ) @ N2 ) ) ) ) ).

% nat_diff_add_eq2
thf(fact_461_option_Osplit__sel,axiom,
    ! [B: $tType,A: $tType,P: B > $o,F1: B,F22: A > B,Option: option @ A] :
      ( ( P @ ( case_option @ B @ A @ F1 @ F22 @ Option ) )
      = ( ( ( Option
            = ( none @ A ) )
         => ( P @ F1 ) )
        & ( ( Option
            = ( some @ A @ ( the2 @ A @ Option ) ) )
         => ( P @ ( F22 @ ( the2 @ A @ Option ) ) ) ) ) ) ).

% option.split_sel
thf(fact_462_option_Osplit__sel__asm,axiom,
    ! [B: $tType,A: $tType,P: B > $o,F1: B,F22: A > B,Option: option @ A] :
      ( ( P @ ( case_option @ B @ A @ F1 @ F22 @ Option ) )
      = ( ~ ( ( ( Option
                = ( none @ A ) )
              & ~ ( P @ F1 ) )
            | ( ( Option
                = ( some @ A @ ( the2 @ A @ Option ) ) )
              & ~ ( P @ ( F22 @ ( the2 @ A @ Option ) ) ) ) ) ) ) ).

% option.split_sel_asm
thf(fact_463_Suc__n__minus__m__eq,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq @ nat @ M @ N2 )
     => ( ( ord_less @ nat @ ( one_one @ nat ) @ M )
       => ( ( suc @ ( minus_minus @ nat @ N2 @ M ) )
          = ( minus_minus @ nat @ N2 @ ( minus_minus @ nat @ M @ ( one_one @ nat ) ) ) ) ) ) ).

% Suc_n_minus_m_eq
thf(fact_464_nat__less__add__iff1,axiom,
    ! [J: nat,I: nat,U: nat,M: nat,N2: nat] :
      ( ( ord_less_eq @ nat @ J @ I )
     => ( ( ord_less @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ I @ U ) @ M ) @ ( plus_plus @ nat @ ( times_times @ nat @ J @ U ) @ N2 ) )
        = ( ord_less @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ I @ J ) @ U ) @ M ) @ N2 ) ) ) ).

% nat_less_add_iff1
thf(fact_465_nat__less__add__iff2,axiom,
    ! [I: nat,J: nat,U: nat,M: nat,N2: nat] :
      ( ( ord_less_eq @ nat @ I @ J )
     => ( ( ord_less @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ I @ U ) @ M ) @ ( plus_plus @ nat @ ( times_times @ nat @ J @ U ) @ N2 ) )
        = ( ord_less @ nat @ M @ ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ J @ I ) @ U ) @ N2 ) ) ) ) ).

% nat_less_add_iff2
thf(fact_466_two__pow__div__gt__le,axiom,
    ! [V: nat,N2: nat,M: nat] :
      ( ( ord_less @ nat @ V @ ( divide_divide @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) ) )
     => ( ord_less_eq @ nat @ M @ N2 ) ) ).

% two_pow_div_gt_le
thf(fact_467_power__minus__is__div,axiom,
    ! [B2: nat,A2: nat] :
      ( ( ord_less_eq @ nat @ B2 @ A2 )
     => ( ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ A2 @ B2 ) )
        = ( divide_divide @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ A2 ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ B2 ) ) ) ) ).

% power_minus_is_div
thf(fact_468_cnt__bound_H,axiom,
    ! [T2: vEBT_VEBT,N2: nat] :
      ( ( vEBT_invar_vebt @ T2 @ N2 )
     => ( ord_less_eq @ real @ ( vEBT_VEBT_cnt @ T2 ) @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( minus_minus @ real @ ( power_power @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ N2 ) @ ( one_one @ real ) ) ) ) ) ).

% cnt_bound'
thf(fact_469_prod_Ocollapse,axiom,
    ! [B: $tType,A: $tType,Prod: product_prod @ A @ B] :
      ( ( product_Pair @ A @ B @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_470_le__add__diff__inverse,axiom,
    ! [A: $tType] :
      ( ( linordered_semidom @ A )
     => ! [B2: A,A2: A] :
          ( ( ord_less_eq @ A @ B2 @ A2 )
         => ( ( plus_plus @ A @ B2 @ ( minus_minus @ A @ A2 @ B2 ) )
            = A2 ) ) ) ).

% le_add_diff_inverse
thf(fact_471_le__add__diff__inverse2,axiom,
    ! [A: $tType] :
      ( ( linordered_semidom @ A )
     => ! [B2: A,A2: A] :
          ( ( ord_less_eq @ A @ B2 @ A2 )
         => ( ( plus_plus @ A @ ( minus_minus @ A @ A2 @ B2 ) @ B2 )
            = A2 ) ) ) ).

% le_add_diff_inverse2
thf(fact_472_two__powr__height__bound__deg,axiom,
    ! [T2: vEBT_VEBT,N2: nat] :
      ( ( vEBT_invar_vebt @ T2 @ N2 )
     => ( ord_less_eq @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( vEBT_VEBT_height @ T2 ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ) ).

% two_powr_height_bound_deg
thf(fact_473_setprop,axiom,
    ! [T2: vEBT_VEBT] :
      ( ( member @ vEBT_VEBT @ T2 @ ( set2 @ vEBT_VEBT @ treeList ) )
     => ( vEBT_invar_vebt @ T2 @ ( divide_divide @ nat @ na @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).

% setprop
thf(fact_474_norm__assertion__simps_I17_J,axiom,
    ! [B: $tType,R2: assn,Q: B > assn] :
      ( ( times_times @ assn @ R2 @ ( ex_assn @ B @ Q ) )
      = ( ex_assn @ B
        @ ^ [X: B] : ( times_times @ assn @ R2 @ ( Q @ X ) ) ) ) ).

% norm_assertion_simps(17)
thf(fact_475_norm__assertion__simps_I16_J,axiom,
    ! [A: $tType,Q: A > assn,R2: assn] :
      ( ( times_times @ assn @ ( ex_assn @ A @ Q ) @ R2 )
      = ( ex_assn @ A
        @ ^ [X: A] : ( times_times @ assn @ ( Q @ X ) @ R2 ) ) ) ).

% norm_assertion_simps(16)
thf(fact_476_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Osimps_I6_J,axiom,
    ! [X2: nat,Mi: nat,Ma: nat,Va: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( ord_less @ nat @ X2 @ Mi )
       => ( ( vEBT_T_s_u_c_c2 @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X2 )
          = ( one_one @ nat ) ) )
      & ( ~ ( ord_less @ nat @ X2 @ Mi )
       => ( ( vEBT_T_s_u_c_c2 @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X2 )
          = ( if @ nat @ ( ord_less @ nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
            @ ( if @ nat
              @ ( ( ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
                 != ( none @ nat ) )
                & ( vEBT_VEBT_less @ ( some @ nat @ ( vEBT_VEBT_low @ X2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) )
              @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_T_s_u_c_c2 @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
              @ ( plus_plus @ nat @ ( vEBT_T_s_u_c_c2 @ Summary @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( one_one @ nat ) ) )
            @ ( one_one @ nat ) ) ) ) ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.simps(6)
thf(fact_477_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Osimps_I7_J,axiom,
    ! [Ma: nat,X2: nat,Mi: nat,Va: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( ord_less @ nat @ Ma @ X2 )
       => ( ( vEBT_T_p_r_e_d2 @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X2 )
          = ( one_one @ nat ) ) )
      & ( ~ ( ord_less @ nat @ Ma @ X2 )
       => ( ( vEBT_T_p_r_e_d2 @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X2 )
          = ( if @ nat @ ( ord_less @ nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
            @ ( if @ nat
              @ ( ( ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
                 != ( none @ nat ) )
                & ( vEBT_VEBT_greater @ ( some @ nat @ ( vEBT_VEBT_low @ X2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) )
              @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_T_p_r_e_d2 @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
              @ ( plus_plus @ nat @ ( vEBT_T_p_r_e_d2 @ Summary @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( one_one @ nat ) ) )
            @ ( one_one @ nat ) ) ) ) ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.simps(7)
thf(fact_478_set__n__deg__not__0,axiom,
    ! [TreeList: list @ vEBT_VEBT,N2: nat,M: nat] :
      ( ! [X3: vEBT_VEBT] :
          ( ( member @ vEBT_VEBT @ X3 @ ( set2 @ vEBT_VEBT @ TreeList ) )
         => ( vEBT_invar_vebt @ X3 @ N2 ) )
     => ( ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList )
          = ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) )
       => ( ord_less_eq @ nat @ ( one_one @ nat ) @ N2 ) ) ) ).

% set_n_deg_not_0
thf(fact_479_inthall,axiom,
    ! [A: $tType,Xs2: list @ A,P: A > $o,N2: nat] :
      ( ! [X3: A] :
          ( ( member @ A @ X3 @ ( set2 @ A @ Xs2 ) )
         => ( P @ X3 ) )
     => ( ( ord_less @ nat @ N2 @ ( size_size @ ( list @ A ) @ Xs2 ) )
       => ( P @ ( nth @ A @ Xs2 @ N2 ) ) ) ) ).

% inthall
thf(fact_480_old_Oprod_Oinject,axiom,
    ! [A: $tType,B: $tType,A2: A,B2: B,A6: A,B4: B] :
      ( ( ( product_Pair @ A @ B @ A2 @ B2 )
        = ( product_Pair @ A @ B @ A6 @ B4 ) )
      = ( ( A2 = A6 )
        & ( B2 = B4 ) ) ) ).

% old.prod.inject
thf(fact_481_prod_Oinject,axiom,
    ! [A: $tType,B: $tType,X1: A,X23: B,Y1: A,Y23: B] :
      ( ( ( product_Pair @ A @ B @ X1 @ X23 )
        = ( product_Pair @ A @ B @ Y1 @ Y23 ) )
      = ( ( X1 = Y1 )
        & ( X23 = Y23 ) ) ) ).

% prod.inject
thf(fact_482_real__divide__square__eq,axiom,
    ! [R4: real,A2: real] :
      ( ( divide_divide @ real @ ( times_times @ real @ R4 @ A2 ) @ ( times_times @ real @ R4 @ R4 ) )
      = ( divide_divide @ real @ A2 @ R4 ) ) ).

% real_divide_square_eq
thf(fact_483_height__compose__summary,axiom,
    ! [Summary: vEBT_VEBT,Info: option @ ( product_prod @ nat @ 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_484_height__compose__child,axiom,
    ! [T2: vEBT_VEBT,TreeList: list @ vEBT_VEBT,Info: option @ ( product_prod @ nat @ nat ),Deg: nat,Summary: vEBT_VEBT] :
      ( ( member @ vEBT_VEBT @ T2 @ ( set2 @ vEBT_VEBT @ TreeList ) )
     => ( ord_less_eq @ nat @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_VEBT_height @ T2 ) ) @ ( vEBT_VEBT_height @ ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) ) ) ) ).

% height_compose_child
thf(fact_485_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 @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ Deg )
     => ( ( Mi = Ma )
       => ( ! [X6: vEBT_VEBT] :
              ( ( member @ vEBT_VEBT @ X6 @ ( set2 @ vEBT_VEBT @ TreeList ) )
             => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X6 @ X_1 ) )
          & ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X_1 ) ) ) ) ).

% mi_eq_ma_no_ch
thf(fact_486_delete__bound__height_H,axiom,
    ! [T2: vEBT_VEBT,N2: nat,X2: nat] :
      ( ( vEBT_invar_vebt @ T2 @ N2 )
     => ( ord_less_eq @ nat @ ( vEBT_V1232361888498592333_e_t_e @ T2 @ X2 ) @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_VEBT_height @ T2 ) ) ) ) ).

% delete_bound_height'
thf(fact_487_div__by__1,axiom,
    ! [A: $tType] :
      ( ( semidom_divide @ A )
     => ! [A2: A] :
          ( ( divide_divide @ A @ A2 @ ( one_one @ A ) )
          = A2 ) ) ).

% div_by_1
thf(fact_488_option_Odisc__eq__case_I1_J,axiom,
    ! [A: $tType,Option: option @ A] :
      ( ( Option
        = ( none @ A ) )
      = ( case_option @ $o @ A @ $true
        @ ^ [Uu3: A] : $false
        @ Option ) ) ).

% option.disc_eq_case(1)
thf(fact_489_option_Odisc__eq__case_I2_J,axiom,
    ! [A: $tType,Option: option @ A] :
      ( ( Option
       != ( none @ A ) )
      = ( case_option @ $o @ A @ $false
        @ ^ [Uu3: A] : $true
        @ Option ) ) ).

% option.disc_eq_case(2)
thf(fact_490_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Osimps_I4_J,axiom,
    ! [Uy: nat,Uz: list @ vEBT_VEBT,Va: vEBT_VEBT,Vb: nat] :
      ( ( vEBT_T_p_r_e_d2 @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uy @ Uz @ Va ) @ Vb )
      = ( one_one @ nat ) ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.simps(4)
thf(fact_491_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Osimps_I3_J,axiom,
    ! [Ux: nat,Uy: list @ vEBT_VEBT,Uz: vEBT_VEBT,Va: nat] :
      ( ( vEBT_T_s_u_c_c2 @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Ux @ Uy @ Uz ) @ Va )
      = ( one_one @ nat ) ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.simps(3)
thf(fact_492_pred__bound__height_H,axiom,
    ! [T2: vEBT_VEBT,N2: nat,X2: nat] :
      ( ( vEBT_invar_vebt @ T2 @ N2 )
     => ( ord_less_eq @ nat @ ( vEBT_T_p_r_e_d2 @ T2 @ X2 ) @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_VEBT_height @ T2 ) ) ) ) ).

% pred_bound_height'
thf(fact_493_succ_H__bound__height,axiom,
    ! [T2: vEBT_VEBT,N2: nat,X2: nat] :
      ( ( vEBT_invar_vebt @ T2 @ N2 )
     => ( ord_less_eq @ nat @ ( vEBT_T_s_u_c_c2 @ T2 @ X2 ) @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_VEBT_height @ T2 ) ) ) ) ).

% succ'_bound_height
thf(fact_494_case__optionE,axiom,
    ! [A: $tType,P: $o,Q: A > $o,X2: option @ A] :
      ( ( case_option @ $o @ A @ P @ Q @ X2 )
     => ( ( ( X2
            = ( none @ A ) )
         => ~ P )
       => ~ ! [Y4: A] :
              ( ( X2
                = ( some @ A @ Y4 ) )
             => ~ ( Q @ Y4 ) ) ) ) ).

% case_optionE
thf(fact_495_all__set__conv__nth,axiom,
    ! [A: $tType,L: list @ A,P: A > $o] :
      ( ( ! [X: A] :
            ( ( member @ A @ X @ ( set2 @ A @ L ) )
           => ( P @ X ) ) )
      = ( ! [I2: nat] :
            ( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ L ) )
           => ( P @ ( nth @ A @ L @ I2 ) ) ) ) ) ).

% all_set_conv_nth
thf(fact_496_linorder__neqE__linordered__idom,axiom,
    ! [A: $tType] :
      ( ( linordered_idom @ A )
     => ! [X2: A,Y2: A] :
          ( ( X2 != Y2 )
         => ( ~ ( ord_less @ A @ X2 @ Y2 )
           => ( ord_less @ A @ Y2 @ X2 ) ) ) ) ).

% linorder_neqE_linordered_idom
thf(fact_497_prod__induct7,axiom,
    ! [G2: $tType,F3: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G2 ) ) ) ) ) ) > $o,X2: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G2 ) ) ) ) )] :
      ( ! [A4: A,B3: B,C3: C,D4: D,E2: E,F2: F3,G3: G2] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G2 ) ) ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G2 ) ) ) ) @ B3 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G2 ) ) ) @ C3 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G2 ) ) @ D4 @ ( product_Pair @ E @ ( product_prod @ F3 @ G2 ) @ E2 @ ( product_Pair @ F3 @ G2 @ F2 @ G3 ) ) ) ) ) ) )
     => ( P @ X2 ) ) ).

% prod_induct7
thf(fact_498_prod__induct6,axiom,
    ! [F3: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) ) ) ) > $o,X2: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) ) )] :
      ( ! [A4: A,B3: B,C3: C,D4: D,E2: E,F2: F3] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) ) @ B3 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) @ C3 @ ( product_Pair @ D @ ( product_prod @ E @ F3 ) @ D4 @ ( product_Pair @ E @ F3 @ E2 @ F2 ) ) ) ) ) )
     => ( P @ X2 ) ) ).

% prod_induct6
thf(fact_499_prod__induct5,axiom,
    ! [E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) ) > $o,X2: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) )] :
      ( ! [A4: A,B3: B,C3: C,D4: D,E2: E] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) @ B3 @ ( product_Pair @ C @ ( product_prod @ D @ E ) @ C3 @ ( product_Pair @ D @ E @ D4 @ E2 ) ) ) ) )
     => ( P @ X2 ) ) ).

% prod_induct5
thf(fact_500_prod__induct4,axiom,
    ! [D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) ) > $o,X2: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
      ( ! [A4: A,B3: B,C3: C,D4: D] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B3 @ ( product_Pair @ C @ D @ C3 @ D4 ) ) ) )
     => ( P @ X2 ) ) ).

% prod_induct4
thf(fact_501_prod__induct3,axiom,
    ! [C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ C ) ) > $o,X2: product_prod @ A @ ( product_prod @ B @ C )] :
      ( ! [A4: A,B3: B,C3: C] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ C ) @ A4 @ ( product_Pair @ B @ C @ B3 @ C3 ) ) )
     => ( P @ X2 ) ) ).

% prod_induct3
thf(fact_502_prod__cases7,axiom,
    ! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F3: $tType,G2: $tType,Y2: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G2 ) ) ) ) )] :
      ~ ! [A4: A,B3: B,C3: C,D4: D,E2: E,F2: F3,G3: G2] :
          ( Y2
         != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G2 ) ) ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G2 ) ) ) ) @ B3 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G2 ) ) ) @ C3 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G2 ) ) @ D4 @ ( product_Pair @ E @ ( product_prod @ F3 @ G2 ) @ E2 @ ( product_Pair @ F3 @ G2 @ F2 @ G3 ) ) ) ) ) ) ) ).

% prod_cases7
thf(fact_503_prod__cases6,axiom,
    ! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F3: $tType,Y2: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) ) )] :
      ~ ! [A4: A,B3: B,C3: C,D4: D,E2: E,F2: F3] :
          ( Y2
         != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) ) @ B3 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) @ C3 @ ( product_Pair @ D @ ( product_prod @ E @ F3 ) @ D4 @ ( product_Pair @ E @ F3 @ E2 @ F2 ) ) ) ) ) ) ).

% prod_cases6
thf(fact_504_prod__cases5,axiom,
    ! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,Y2: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) )] :
      ~ ! [A4: A,B3: B,C3: C,D4: D,E2: E] :
          ( Y2
         != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) @ B3 @ ( product_Pair @ C @ ( product_prod @ D @ E ) @ C3 @ ( product_Pair @ D @ E @ D4 @ E2 ) ) ) ) ) ).

% prod_cases5
thf(fact_505_prod__cases4,axiom,
    ! [A: $tType,B: $tType,C: $tType,D: $tType,Y2: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
      ~ ! [A4: A,B3: B,C3: C,D4: D] :
          ( Y2
         != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B3 @ ( product_Pair @ C @ D @ C3 @ D4 ) ) ) ) ).

% prod_cases4
thf(fact_506_prod__cases3,axiom,
    ! [A: $tType,B: $tType,C: $tType,Y2: product_prod @ A @ ( product_prod @ B @ C )] :
      ~ ! [A4: A,B3: B,C3: C] :
          ( Y2
         != ( product_Pair @ A @ ( product_prod @ B @ C ) @ A4 @ ( product_Pair @ B @ C @ B3 @ C3 ) ) ) ).

% prod_cases3
thf(fact_507_Pair__inject,axiom,
    ! [A: $tType,B: $tType,A2: A,B2: B,A6: A,B4: B] :
      ( ( ( product_Pair @ A @ B @ A2 @ B2 )
        = ( product_Pair @ A @ B @ A6 @ B4 ) )
     => ~ ( ( A2 = A6 )
         => ( B2 != B4 ) ) ) ).

% Pair_inject
thf(fact_508_prod__cases,axiom,
    ! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,P4: product_prod @ A @ B] :
      ( ! [A4: A,B3: B] : ( P @ ( product_Pair @ A @ B @ A4 @ B3 ) )
     => ( P @ P4 ) ) ).

% prod_cases
thf(fact_509_surj__pair,axiom,
    ! [A: $tType,B: $tType,P4: product_prod @ A @ B] :
    ? [X3: A,Y4: B] :
      ( P4
      = ( product_Pair @ A @ B @ X3 @ Y4 ) ) ).

% surj_pair
thf(fact_510_old_Oprod_Oexhaust,axiom,
    ! [A: $tType,B: $tType,Y2: product_prod @ A @ B] :
      ~ ! [A4: A,B3: B] :
          ( Y2
         != ( product_Pair @ A @ B @ A4 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_511_star__assoc,axiom,
    ! [A2: assn,B2: assn,C2: assn] :
      ( ( times_times @ assn @ ( times_times @ assn @ A2 @ B2 ) @ C2 )
      = ( times_times @ assn @ A2 @ ( times_times @ assn @ B2 @ C2 ) ) ) ).

% star_assoc
thf(fact_512_star__aci_I2_J,axiom,
    ( ( times_times @ assn )
    = ( ^ [A5: assn,B5: assn] : ( times_times @ assn @ B5 @ A5 ) ) ) ).

% star_aci(2)
thf(fact_513_star__aci_I3_J,axiom,
    ! [A2: assn,B2: assn,C2: assn] :
      ( ( times_times @ assn @ A2 @ ( times_times @ assn @ B2 @ C2 ) )
      = ( times_times @ assn @ B2 @ ( times_times @ assn @ A2 @ C2 ) ) ) ).

% star_aci(3)
thf(fact_514_assn__aci_I10_J,axiom,
    ! [A2: assn,B2: assn,C2: assn] :
      ( ( times_times @ assn @ ( times_times @ assn @ A2 @ B2 ) @ C2 )
      = ( times_times @ assn @ ( times_times @ assn @ A2 @ C2 ) @ B2 ) ) ).

% assn_aci(10)
thf(fact_515_is__hoare__triple,axiom,
    ! [A: $tType,P: assn,C2: heap_Time_Heap @ A,Q: A > assn] :
      ( ( hoare_hoare_triple @ A @ P @ C2 @ Q )
     => ( hoare_hoare_triple @ A @ P @ C2 @ Q ) ) ).

% is_hoare_triple
thf(fact_516_height__node,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,N2: nat] :
      ( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ N2 )
     => ( ord_less_eq @ nat @ ( one_one @ nat ) @ ( vEBT_VEBT_height @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) ) ) ) ).

% height_node
thf(fact_517_combine__common__factor,axiom,
    ! [A: $tType] :
      ( ( semiring @ A )
     => ! [A2: A,E3: A,B2: A,C2: A] :
          ( ( plus_plus @ A @ ( times_times @ A @ A2 @ E3 ) @ ( plus_plus @ A @ ( times_times @ A @ B2 @ E3 ) @ C2 ) )
          = ( plus_plus @ A @ ( times_times @ A @ ( plus_plus @ A @ A2 @ B2 ) @ E3 ) @ C2 ) ) ) ).

% combine_common_factor
thf(fact_518_distrib__right,axiom,
    ! [A: $tType] :
      ( ( semiring @ A )
     => ! [A2: A,B2: A,C2: A] :
          ( ( times_times @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 )
          = ( plus_plus @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) ) ) ) ).

% distrib_right
thf(fact_519_distrib__left,axiom,
    ! [A: $tType] :
      ( ( semiring @ A )
     => ! [A2: A,B2: A,C2: A] :
          ( ( times_times @ A @ A2 @ ( plus_plus @ A @ B2 @ C2 ) )
          = ( plus_plus @ A @ ( times_times @ A @ A2 @ B2 ) @ ( times_times @ A @ A2 @ C2 ) ) ) ) ).

% distrib_left
thf(fact_520_comm__semiring__class_Odistrib,axiom,
    ! [A: $tType] :
      ( ( comm_semiring @ A )
     => ! [A2: A,B2: A,C2: A] :
          ( ( times_times @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 )
          = ( plus_plus @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) ) ) ) ).

% comm_semiring_class.distrib
thf(fact_521_ring__class_Oring__distribs_I1_J,axiom,
    ! [A: $tType] :
      ( ( ring @ A )
     => ! [A2: A,B2: A,C2: A] :
          ( ( times_times @ A @ A2 @ ( plus_plus @ A @ B2 @ C2 ) )
          = ( plus_plus @ A @ ( times_times @ A @ A2 @ B2 ) @ ( times_times @ A @ A2 @ C2 ) ) ) ) ).

% ring_class.ring_distribs(1)
thf(fact_522_ring__class_Oring__distribs_I2_J,axiom,
    ! [A: $tType] :
      ( ( ring @ A )
     => ! [A2: A,B2: A,C2: A] :
          ( ( times_times @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 )
          = ( plus_plus @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) ) ) ) ).

% ring_class.ring_distribs(2)
thf(fact_523_right__diff__distrib_H,axiom,
    ! [A: $tType] :
      ( ( comm_s4317794764714335236cancel @ A )
     => ! [A2: A,B2: A,C2: A] :
          ( ( times_times @ A @ A2 @ ( minus_minus @ A @ B2 @ C2 ) )
          = ( minus_minus @ A @ ( times_times @ A @ A2 @ B2 ) @ ( times_times @ A @ A2 @ C2 ) ) ) ) ).

% right_diff_distrib'
thf(fact_524_left__diff__distrib_H,axiom,
    ! [A: $tType] :
      ( ( comm_s4317794764714335236cancel @ A )
     => ! [B2: A,C2: A,A2: A] :
          ( ( times_times @ A @ ( minus_minus @ A @ B2 @ C2 ) @ A2 )
          = ( minus_minus @ A @ ( times_times @ A @ B2 @ A2 ) @ ( times_times @ A @ C2 @ A2 ) ) ) ) ).

% left_diff_distrib'
thf(fact_525_right__diff__distrib,axiom,
    ! [A: $tType] :
      ( ( ring @ A )
     => ! [A2: A,B2: A,C2: A] :
          ( ( times_times @ A @ A2 @ ( minus_minus @ A @ B2 @ C2 ) )
          = ( minus_minus @ A @ ( times_times @ A @ A2 @ B2 ) @ ( times_times @ A @ A2 @ C2 ) ) ) ) ).

% right_diff_distrib
thf(fact_526_left__diff__distrib,axiom,
    ! [A: $tType] :
      ( ( ring @ A )
     => ! [A2: A,B2: A,C2: A] :
          ( ( times_times @ A @ ( minus_minus @ A @ A2 @ B2 ) @ C2 )
          = ( minus_minus @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) ) ) ) ).

% left_diff_distrib
thf(fact_527_fst__conv,axiom,
    ! [B: $tType,A: $tType,X1: A,X23: B] :
      ( ( product_fst @ A @ B @ ( product_Pair @ A @ B @ X1 @ X23 ) )
      = X1 ) ).

% fst_conv
thf(fact_528_fst__eqD,axiom,
    ! [B: $tType,A: $tType,X2: A,Y2: B,A2: A] :
      ( ( ( product_fst @ A @ B @ ( product_Pair @ A @ B @ X2 @ Y2 ) )
        = A2 )
     => ( X2 = A2 ) ) ).

% fst_eqD
thf(fact_529_snd__conv,axiom,
    ! [Aa: $tType,A: $tType,X1: Aa,X23: A] :
      ( ( product_snd @ Aa @ A @ ( product_Pair @ Aa @ A @ X1 @ X23 ) )
      = X23 ) ).

% snd_conv
thf(fact_530_snd__eqD,axiom,
    ! [B: $tType,A: $tType,X2: B,Y2: A,A2: A] :
      ( ( ( product_snd @ B @ A @ ( product_Pair @ B @ A @ X2 @ Y2 ) )
        = A2 )
     => ( Y2 = A2 ) ) ).

% snd_eqD
thf(fact_531_prod__eq__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( ( ^ [Y5: product_prod @ A @ B,Z4: product_prod @ A @ B] : Y5 = Z4 )
      = ( ^ [S2: product_prod @ A @ B,T3: product_prod @ A @ B] :
            ( ( ( product_fst @ A @ B @ S2 )
              = ( product_fst @ A @ B @ T3 ) )
            & ( ( product_snd @ A @ B @ S2 )
              = ( product_snd @ A @ B @ T3 ) ) ) ) ) ).

% prod_eq_iff
thf(fact_532_prod__eqI,axiom,
    ! [B: $tType,A: $tType,P4: product_prod @ A @ B,Q2: product_prod @ A @ B] :
      ( ( ( product_fst @ A @ B @ P4 )
        = ( product_fst @ A @ B @ Q2 ) )
     => ( ( ( product_snd @ A @ B @ P4 )
          = ( product_snd @ A @ B @ Q2 ) )
       => ( P4 = Q2 ) ) ) ).

% prod_eqI
thf(fact_533_prod_Oexpand,axiom,
    ! [B: $tType,A: $tType,Prod: product_prod @ A @ B,Prod2: product_prod @ A @ B] :
      ( ( ( ( product_fst @ A @ B @ Prod )
          = ( product_fst @ A @ B @ Prod2 ) )
        & ( ( product_snd @ A @ B @ Prod )
          = ( product_snd @ A @ B @ Prod2 ) ) )
     => ( Prod = Prod2 ) ) ).

% prod.expand
thf(fact_534_norm__assertion__simps_I1_J,axiom,
    ! [A2: assn] :
      ( ( times_times @ assn @ ( one_one @ assn ) @ A2 )
      = A2 ) ).

% norm_assertion_simps(1)
thf(fact_535_norm__assertion__simps_I2_J,axiom,
    ! [A2: assn] :
      ( ( times_times @ assn @ A2 @ ( one_one @ assn ) )
      = A2 ) ).

% norm_assertion_simps(2)
thf(fact_536_lambda__one,axiom,
    ! [A: $tType] :
      ( ( monoid_mult @ A )
     => ( ( ^ [X: A] : X )
        = ( times_times @ A @ ( one_one @ A ) ) ) ) ).

% lambda_one
thf(fact_537_frame__rule__left,axiom,
    ! [A: $tType,P: assn,C2: heap_Time_Heap @ A,Q: A > assn,R2: assn] :
      ( ( hoare_hoare_triple @ A @ P @ C2 @ Q )
     => ( hoare_hoare_triple @ A @ ( times_times @ assn @ R2 @ P ) @ C2
        @ ^ [X: A] : ( times_times @ assn @ R2 @ ( Q @ X ) ) ) ) ).

% frame_rule_left
thf(fact_538_add__mono1,axiom,
    ! [A: $tType] :
      ( ( linord181362715937106298miring @ A )
     => ! [A2: A,B2: A] :
          ( ( ord_less @ A @ A2 @ B2 )
         => ( ord_less @ A @ ( plus_plus @ A @ A2 @ ( one_one @ A ) ) @ ( plus_plus @ A @ B2 @ ( one_one @ A ) ) ) ) ) ).

% add_mono1
thf(fact_539_less__add__one,axiom,
    ! [A: $tType] :
      ( ( linordered_semidom @ A )
     => ! [A2: A] : ( ord_less @ A @ A2 @ ( plus_plus @ A @ A2 @ ( one_one @ A ) ) ) ) ).

% less_add_one
thf(fact_540_less__1__mult,axiom,
    ! [A: $tType] :
      ( ( linordered_semidom @ A )
     => ! [M: A,N2: A] :
          ( ( ord_less @ A @ ( one_one @ A ) @ M )
         => ( ( ord_less @ A @ ( one_one @ A ) @ N2 )
           => ( ord_less @ A @ ( one_one @ A ) @ ( times_times @ A @ M @ N2 ) ) ) ) ) ).

% less_1_mult
thf(fact_541_add__le__add__imp__diff__le,axiom,
    ! [A: $tType] :
      ( ( linordered_semidom @ A )
     => ! [I: A,K: A,N2: A,J: A] :
          ( ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ N2 )
         => ( ( ord_less_eq @ A @ N2 @ ( plus_plus @ A @ J @ K ) )
           => ( ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ N2 )
             => ( ( ord_less_eq @ A @ N2 @ ( plus_plus @ A @ J @ K ) )
               => ( ord_less_eq @ A @ ( minus_minus @ A @ N2 @ K ) @ J ) ) ) ) ) ) ).

% add_le_add_imp_diff_le
thf(fact_542_add__le__imp__le__diff,axiom,
    ! [A: $tType] :
      ( ( linordered_semidom @ A )
     => ! [I: A,K: A,N2: A] :
          ( ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ N2 )
         => ( ord_less_eq @ A @ I @ ( minus_minus @ A @ N2 @ K ) ) ) ) ).

% add_le_imp_le_diff
thf(fact_543_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [A: $tType] :
      ( ( linordered_semidom @ A )
     => ! [A2: A,B2: A] :
          ( ~ ( ord_less @ A @ A2 @ B2 )
         => ( ( plus_plus @ A @ B2 @ ( minus_minus @ A @ A2 @ B2 ) )
            = A2 ) ) ) ).

% linordered_semidom_class.add_diff_inverse
thf(fact_544_square__diff__square__factored,axiom,
    ! [A: $tType] :
      ( ( comm_ring @ A )
     => ! [X2: A,Y2: A] :
          ( ( minus_minus @ A @ ( times_times @ A @ X2 @ X2 ) @ ( times_times @ A @ Y2 @ Y2 ) )
          = ( times_times @ A @ ( plus_plus @ A @ X2 @ Y2 ) @ ( minus_minus @ A @ X2 @ Y2 ) ) ) ) ).

% square_diff_square_factored
thf(fact_545_eq__add__iff2,axiom,
    ! [A: $tType] :
      ( ( ring @ A )
     => ! [A2: A,E3: A,C2: A,B2: A,D2: A] :
          ( ( ( plus_plus @ A @ ( times_times @ A @ A2 @ E3 ) @ C2 )
            = ( plus_plus @ A @ ( times_times @ A @ B2 @ E3 ) @ D2 ) )
          = ( C2
            = ( plus_plus @ A @ ( times_times @ A @ ( minus_minus @ A @ B2 @ A2 ) @ E3 ) @ D2 ) ) ) ) ).

% eq_add_iff2
thf(fact_546_eq__add__iff1,axiom,
    ! [A: $tType] :
      ( ( ring @ A )
     => ! [A2: A,E3: A,C2: A,B2: A,D2: A] :
          ( ( ( plus_plus @ A @ ( times_times @ A @ A2 @ E3 ) @ C2 )
            = ( plus_plus @ A @ ( times_times @ A @ B2 @ E3 ) @ D2 ) )
          = ( ( plus_plus @ A @ ( times_times @ A @ ( minus_minus @ A @ A2 @ B2 ) @ E3 ) @ C2 )
            = D2 ) ) ) ).

% eq_add_iff1
thf(fact_547_surjective__pairing,axiom,
    ! [B: $tType,A: $tType,T2: product_prod @ A @ B] :
      ( T2
      = ( product_Pair @ A @ B @ ( product_fst @ A @ B @ T2 ) @ ( product_snd @ A @ B @ T2 ) ) ) ).

% surjective_pairing
thf(fact_548_prod_Oexhaust__sel,axiom,
    ! [B: $tType,A: $tType,Prod: product_prod @ A @ B] :
      ( Prod
      = ( product_Pair @ A @ B @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) ) ).

% prod.exhaust_sel
thf(fact_549_ordered__ring__class_Ole__add__iff2,axiom,
    ! [A: $tType] :
      ( ( ordered_ring @ A )
     => ! [A2: A,E3: A,C2: A,B2: A,D2: A] :
          ( ( ord_less_eq @ A @ ( plus_plus @ A @ ( times_times @ A @ A2 @ E3 ) @ C2 ) @ ( plus_plus @ A @ ( times_times @ A @ B2 @ E3 ) @ D2 ) )
          = ( ord_less_eq @ A @ C2 @ ( plus_plus @ A @ ( times_times @ A @ ( minus_minus @ A @ B2 @ A2 ) @ E3 ) @ D2 ) ) ) ) ).

% ordered_ring_class.le_add_iff2
thf(fact_550_ordered__ring__class_Ole__add__iff1,axiom,
    ! [A: $tType] :
      ( ( ordered_ring @ A )
     => ! [A2: A,E3: A,C2: A,B2: A,D2: A] :
          ( ( ord_less_eq @ A @ ( plus_plus @ A @ ( times_times @ A @ A2 @ E3 ) @ C2 ) @ ( plus_plus @ A @ ( times_times @ A @ B2 @ E3 ) @ D2 ) )
          = ( ord_less_eq @ A @ ( plus_plus @ A @ ( times_times @ A @ ( minus_minus @ A @ A2 @ B2 ) @ E3 ) @ C2 ) @ D2 ) ) ) ).

% ordered_ring_class.le_add_iff1
thf(fact_551_less__add__iff2,axiom,
    ! [A: $tType] :
      ( ( ordered_ring @ A )
     => ! [A2: A,E3: A,C2: A,B2: A,D2: A] :
          ( ( ord_less @ A @ ( plus_plus @ A @ ( times_times @ A @ A2 @ E3 ) @ C2 ) @ ( plus_plus @ A @ ( times_times @ A @ B2 @ E3 ) @ D2 ) )
          = ( ord_less @ A @ C2 @ ( plus_plus @ A @ ( times_times @ A @ ( minus_minus @ A @ B2 @ A2 ) @ E3 ) @ D2 ) ) ) ) ).

% less_add_iff2
thf(fact_552_less__add__iff1,axiom,
    ! [A: $tType] :
      ( ( ordered_ring @ A )
     => ! [A2: A,E3: A,C2: A,B2: A,D2: A] :
          ( ( ord_less @ A @ ( plus_plus @ A @ ( times_times @ A @ A2 @ E3 ) @ C2 ) @ ( plus_plus @ A @ ( times_times @ A @ B2 @ E3 ) @ D2 ) )
          = ( ord_less @ A @ ( plus_plus @ A @ ( times_times @ A @ ( minus_minus @ A @ A2 @ B2 ) @ E3 ) @ C2 ) @ D2 ) ) ) ).

% less_add_iff1
thf(fact_553_square__diff__one__factored,axiom,
    ! [A: $tType] :
      ( ( ring_1 @ A )
     => ! [X2: A] :
          ( ( minus_minus @ A @ ( times_times @ A @ X2 @ X2 ) @ ( one_one @ A ) )
          = ( times_times @ A @ ( plus_plus @ A @ X2 @ ( one_one @ A ) ) @ ( minus_minus @ A @ X2 @ ( one_one @ A ) ) ) ) ) ).

% square_diff_one_factored
thf(fact_554_invar__vebt_Ointros_I5_J,axiom,
    ! [TreeList: list @ vEBT_VEBT,N2: nat,Summary: vEBT_VEBT,M: nat,Deg: nat,Mi: nat,Ma: nat] :
      ( ! [X3: vEBT_VEBT] :
          ( ( member @ vEBT_VEBT @ X3 @ ( set2 @ vEBT_VEBT @ TreeList ) )
         => ( vEBT_invar_vebt @ X3 @ N2 ) )
     => ( ( vEBT_invar_vebt @ Summary @ M )
       => ( ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList )
            = ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) )
         => ( ( M
              = ( suc @ N2 ) )
           => ( ( Deg
                = ( plus_plus @ nat @ N2 @ M ) )
             => ( ! [I3: nat] :
                    ( ( ord_less @ nat @ I3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) )
                   => ( ( ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList @ I3 ) @ X5 ) )
                      = ( vEBT_V8194947554948674370ptions @ Summary @ I3 ) ) )
               => ( ( ( Mi = Ma )
                   => ! [X3: vEBT_VEBT] :
                        ( ( member @ vEBT_VEBT @ X3 @ ( set2 @ vEBT_VEBT @ TreeList ) )
                       => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X_12 ) ) )
                 => ( ( ord_less_eq @ nat @ Mi @ Ma )
                   => ( ( ord_less @ nat @ Ma @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg ) )
                     => ( ( ( Mi != Ma )
                         => ! [I3: nat] :
                              ( ( ord_less @ nat @ I3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) )
                             => ( ( ( ( vEBT_VEBT_high @ Ma @ N2 )
                                    = I3 )
                                 => ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList @ I3 ) @ ( vEBT_VEBT_low @ Ma @ N2 ) ) )
                                & ! [X3: nat] :
                                    ( ( ( ( vEBT_VEBT_high @ X3 @ N2 )
                                        = I3 )
                                      & ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList @ I3 ) @ ( vEBT_VEBT_low @ X3 @ N2 ) ) )
                                   => ( ( ord_less @ nat @ Mi @ X3 )
                                      & ( ord_less_eq @ nat @ X3 @ Ma ) ) ) ) ) )
                       => ( vEBT_invar_vebt @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ Deg ) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.intros(5)
thf(fact_555_invar__vebt_Ointros_I4_J,axiom,
    ! [TreeList: list @ vEBT_VEBT,N2: nat,Summary: vEBT_VEBT,M: nat,Deg: nat,Mi: nat,Ma: nat] :
      ( ! [X3: vEBT_VEBT] :
          ( ( member @ vEBT_VEBT @ X3 @ ( set2 @ vEBT_VEBT @ TreeList ) )
         => ( vEBT_invar_vebt @ X3 @ N2 ) )
     => ( ( vEBT_invar_vebt @ Summary @ M )
       => ( ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList )
            = ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) )
         => ( ( M = N2 )
           => ( ( Deg
                = ( plus_plus @ nat @ N2 @ M ) )
             => ( ! [I3: nat] :
                    ( ( ord_less @ nat @ I3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) )
                   => ( ( ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList @ I3 ) @ X5 ) )
                      = ( vEBT_V8194947554948674370ptions @ Summary @ I3 ) ) )
               => ( ( ( Mi = Ma )
                   => ! [X3: vEBT_VEBT] :
                        ( ( member @ vEBT_VEBT @ X3 @ ( set2 @ vEBT_VEBT @ TreeList ) )
                       => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X_12 ) ) )
                 => ( ( ord_less_eq @ nat @ Mi @ Ma )
                   => ( ( ord_less @ nat @ Ma @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg ) )
                     => ( ( ( Mi != Ma )
                         => ! [I3: nat] :
                              ( ( ord_less @ nat @ I3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) )
                             => ( ( ( ( vEBT_VEBT_high @ Ma @ N2 )
                                    = I3 )
                                 => ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList @ I3 ) @ ( vEBT_VEBT_low @ Ma @ N2 ) ) )
                                & ! [X3: nat] :
                                    ( ( ( ( vEBT_VEBT_high @ X3 @ N2 )
                                        = I3 )
                                      & ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList @ I3 ) @ ( vEBT_VEBT_low @ X3 @ N2 ) ) )
                                   => ( ( ord_less @ nat @ Mi @ X3 )
                                      & ( ord_less_eq @ nat @ X3 @ Ma ) ) ) ) ) )
                       => ( vEBT_invar_vebt @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ Deg ) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.intros(4)
thf(fact_556_real__average__minus__first,axiom,
    ! [A2: real,B2: real] :
      ( ( minus_minus @ real @ ( divide_divide @ real @ ( plus_plus @ real @ A2 @ B2 ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ A2 )
      = ( divide_divide @ real @ ( minus_minus @ real @ B2 @ A2 ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ).

% real_average_minus_first
thf(fact_557_real__average__minus__second,axiom,
    ! [B2: real,A2: real] :
      ( ( minus_minus @ real @ ( divide_divide @ real @ ( plus_plus @ real @ B2 @ A2 ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ A2 )
      = ( divide_divide @ real @ ( minus_minus @ real @ B2 @ A2 ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ).

% real_average_minus_second
thf(fact_558_invar__vebt_Ointros_I3_J,axiom,
    ! [TreeList: list @ vEBT_VEBT,N2: nat,Summary: vEBT_VEBT,M: nat,Deg: nat] :
      ( ! [X3: vEBT_VEBT] :
          ( ( member @ vEBT_VEBT @ X3 @ ( set2 @ vEBT_VEBT @ TreeList ) )
         => ( vEBT_invar_vebt @ X3 @ N2 ) )
     => ( ( vEBT_invar_vebt @ Summary @ M )
       => ( ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList )
            = ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) )
         => ( ( M
              = ( suc @ N2 ) )
           => ( ( Deg
                = ( plus_plus @ nat @ N2 @ M ) )
             => ( ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X_12 )
               => ( ! [X3: vEBT_VEBT] :
                      ( ( member @ vEBT_VEBT @ X3 @ ( set2 @ vEBT_VEBT @ TreeList ) )
                     => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X_12 ) )
                 => ( vEBT_invar_vebt @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg @ TreeList @ Summary ) @ Deg ) ) ) ) ) ) ) ) ).

% invar_vebt.intros(3)
thf(fact_559_invar__vebt_Ointros_I2_J,axiom,
    ! [TreeList: list @ vEBT_VEBT,N2: nat,Summary: vEBT_VEBT,M: nat,Deg: nat] :
      ( ! [X3: vEBT_VEBT] :
          ( ( member @ vEBT_VEBT @ X3 @ ( set2 @ vEBT_VEBT @ TreeList ) )
         => ( vEBT_invar_vebt @ X3 @ N2 ) )
     => ( ( vEBT_invar_vebt @ Summary @ M )
       => ( ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList )
            = ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) )
         => ( ( M = N2 )
           => ( ( Deg
                = ( plus_plus @ nat @ N2 @ M ) )
             => ( ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X_12 )
               => ( ! [X3: vEBT_VEBT] :
                      ( ( member @ vEBT_VEBT @ X3 @ ( set2 @ vEBT_VEBT @ TreeList ) )
                     => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X_12 ) )
                 => ( vEBT_invar_vebt @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg @ TreeList @ Summary ) @ Deg ) ) ) ) ) ) ) ) ).

% invar_vebt.intros(2)
thf(fact_560_less__eq__option__Some__None,axiom,
    ! [A: $tType] :
      ( ( preorder @ A )
     => ! [X2: A] :
          ~ ( ord_less_eq @ ( option @ A ) @ ( some @ A @ X2 ) @ ( none @ A ) ) ) ).

% less_eq_option_Some_None
thf(fact_561_count__buildup,axiom,
    ! [N2: nat] : ( ord_less_eq @ real @ ( vEBT_VEBT_cnt @ ( vEBT_vebt_buildup @ N2 ) ) @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( power_power @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ N2 ) ) ) ).

% count_buildup
thf(fact_562_less__eq__option__Some,axiom,
    ! [A: $tType] :
      ( ( preorder @ A )
     => ! [X2: A,Y2: A] :
          ( ( ord_less_eq @ ( option @ A ) @ ( some @ A @ X2 ) @ ( some @ A @ Y2 ) )
          = ( ord_less_eq @ A @ X2 @ Y2 ) ) ) ).

% less_eq_option_Some
thf(fact_563_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_564_less__option__None,axiom,
    ! [A: $tType] :
      ( ( preorder @ A )
     => ! [X2: option @ A] :
          ~ ( ord_less @ ( option @ A ) @ X2 @ ( none @ A ) ) ) ).

% less_option_None
thf(fact_565_VEBT_Oinject_I1_J,axiom,
    ! [X11: option @ ( product_prod @ nat @ nat ),X12: nat,X13: list @ vEBT_VEBT,X14: vEBT_VEBT,Y11: option @ ( product_prod @ nat @ 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_566_less__option__Some,axiom,
    ! [A: $tType] :
      ( ( preorder @ A )
     => ! [X2: A,Y2: A] :
          ( ( ord_less @ ( option @ A ) @ ( some @ A @ X2 ) @ ( some @ A @ Y2 ) )
          = ( ord_less @ A @ X2 @ Y2 ) ) ) ).

% less_option_Some
thf(fact_567_less__option__None__Some__code,axiom,
    ! [A: $tType] :
      ( ( preorder @ A )
     => ! [X2: A] : ( ord_less @ ( option @ A ) @ ( none @ A ) @ ( some @ A @ X2 ) ) ) ).

% less_option_None_Some_code
thf(fact_568_less__eq__option__None__code,axiom,
    ! [A: $tType] :
      ( ( preorder @ A )
     => ! [X2: option @ A] : ( ord_less_eq @ ( option @ A ) @ ( none @ A ) @ X2 ) ) ).

% less_eq_option_None_code
thf(fact_569_less__option__def,axiom,
    ! [A: $tType] :
      ( ( preorder @ A )
     => ( ( ord_less @ ( option @ A ) )
        = ( ^ [X: option @ A] :
              ( case_option @ $o @ A @ $false
              @ ^ [Y: A] :
                  ( case_option @ $o @ A @ $true
                  @ ^ [Z2: A] : ( ord_less @ A @ Z2 @ Y )
                  @ X ) ) ) ) ) ).

% less_option_def
thf(fact_570_mult__commute__abs,axiom,
    ! [A: $tType] :
      ( ( ab_semigroup_mult @ A )
     => ! [C2: A] :
          ( ( ^ [X: A] : ( times_times @ A @ X @ C2 ) )
          = ( times_times @ A @ C2 ) ) ) ).

% mult_commute_abs
thf(fact_571_less__option__None__Some,axiom,
    ! [A: $tType] :
      ( ( preorder @ A )
     => ! [X2: A] : ( ord_less @ ( option @ A ) @ ( none @ A ) @ ( some @ A @ X2 ) ) ) ).

% less_option_None_Some
thf(fact_572_less__option__None__is__Some,axiom,
    ! [A: $tType] :
      ( ( preorder @ A )
     => ! [X2: option @ A] :
          ( ( ord_less @ ( option @ A ) @ ( none @ A ) @ X2 )
         => ? [Z3: A] :
              ( X2
              = ( some @ A @ Z3 ) ) ) ) ).

% less_option_None_is_Some
thf(fact_573_less__eq__option__None,axiom,
    ! [A: $tType] :
      ( ( preorder @ A )
     => ! [X2: option @ A] : ( ord_less_eq @ ( option @ A ) @ ( none @ A ) @ X2 ) ) ).

% less_eq_option_None
thf(fact_574_less__eq__option__None__is__None,axiom,
    ! [A: $tType] :
      ( ( preorder @ A )
     => ! [X2: option @ A] :
          ( ( ord_less_eq @ ( option @ A ) @ X2 @ ( none @ A ) )
         => ( X2
            = ( none @ A ) ) ) ) ).

% less_eq_option_None_is_None
thf(fact_575_less__eq__option__def,axiom,
    ! [A: $tType] :
      ( ( preorder @ A )
     => ( ( ord_less_eq @ ( option @ A ) )
        = ( ^ [X: option @ A,Y: option @ A] :
              ( case_option @ $o @ A @ $true
              @ ^ [Z2: A] : ( case_option @ $o @ A @ $false @ ( ord_less_eq @ A @ Z2 ) @ Y )
              @ X ) ) ) ) ).

% less_eq_option_def
thf(fact_576_set__vebt__def,axiom,
    ( vEBT_set_vebt
    = ( ^ [T3: vEBT_VEBT] : ( collect @ nat @ ( vEBT_V8194947554948674370ptions @ T3 ) ) ) ) ).

% set_vebt_def
thf(fact_577_discrete,axiom,
    ! [A: $tType] :
      ( ( unique1627219031080169319umeral @ A )
     => ( ( ord_less @ A )
        = ( ^ [A5: A] : ( ord_less_eq @ A @ ( plus_plus @ A @ A5 @ ( one_one @ A ) ) ) ) ) ) ).

% discrete
thf(fact_578_builupi_Hcorr,axiom,
    ! [N2: nat] : ( hoare_hoare_triple @ vEBT_VEBTi @ ( one_one @ assn ) @ ( vEBT_V739175172307565963ildupi @ N2 ) @ ( vEBT_vebt_assn_raw @ ( vEBT_vebt_buildup @ N2 ) ) ) ).

% builupi'corr
thf(fact_579_builupicorr,axiom,
    ! [N2: nat] : ( hoare_hoare_triple @ vEBT_VEBTi @ ( one_one @ assn ) @ ( vEBT_vebt_buildupi @ N2 ) @ ( vEBT_vebt_assn_raw @ ( vEBT_vebt_buildup @ N2 ) ) ) ).

% builupicorr
thf(fact_580_divmod__step__eq,axiom,
    ! [A: $tType] :
      ( ( unique1627219031080169319umeral @ A )
     => ! [L: num,R4: A,Q2: A] :
          ( ( ( ord_less_eq @ A @ ( numeral_numeral @ A @ L ) @ R4 )
           => ( ( unique1321980374590559556d_step @ A @ L @ ( product_Pair @ A @ A @ Q2 @ R4 ) )
              = ( product_Pair @ A @ A @ ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ Q2 ) @ ( one_one @ A ) ) @ ( minus_minus @ A @ R4 @ ( numeral_numeral @ A @ L ) ) ) ) )
          & ( ~ ( ord_less_eq @ A @ ( numeral_numeral @ A @ L ) @ R4 )
           => ( ( unique1321980374590559556d_step @ A @ L @ ( product_Pair @ A @ A @ Q2 @ R4 ) )
              = ( product_Pair @ A @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ Q2 ) @ R4 ) ) ) ) ) ).

% divmod_step_eq
thf(fact_581_cnt__bound,axiom,
    ! [T2: vEBT_VEBT,N2: nat] :
      ( ( vEBT_invar_vebt @ T2 @ N2 )
     => ( ord_less_eq @ real @ ( vEBT_VEBT_cnt @ T2 ) @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( minus_minus @ real @ ( power_power @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ N2 ) @ ( divide_divide @ real @ ( numeral_numeral @ real @ ( bit1 @ ( bit1 @ ( bit1 @ one2 ) ) ) ) @ ( numeral_numeral @ real @ ( bit0 @ ( bit1 @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ).

% cnt_bound
thf(fact_582_TBOUND__vebt__memberi,axiom,
    ! [T2: vEBT_VEBT,Ti: vEBT_VEBTi,X2: nat] : ( time_TBOUND @ $o @ ( vEBT_V854960066525838166emberi @ T2 @ Ti @ X2 ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_VEBT_height @ T2 ) ) ) ) ).

% TBOUND_vebt_memberi
thf(fact_583_vebt__insert_Osimps_I4_J,axiom,
    ! [V: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,X2: nat] :
      ( ( vEBT_vebt_insert @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ V ) ) @ TreeList @ Summary ) @ X2 )
      = ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ X2 @ X2 ) ) @ ( suc @ ( suc @ V ) ) @ TreeList @ Summary ) ) ).

% vebt_insert.simps(4)
thf(fact_584_two__realpow__ge__one,axiom,
    ! [N2: nat] : ( ord_less_eq @ real @ ( one_one @ real ) @ ( power_power @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ N2 ) ) ).

% two_realpow_ge_one
thf(fact_585_count__buildup_H,axiom,
    ! [N2: nat] : ( ord_less_eq @ real @ ( vEBT_VEBT_cnt @ ( vEBT_vebt_buildup @ N2 ) ) @ ( semiring_1_of_nat @ real @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ) ) ).

% count_buildup'
thf(fact_586_heaphelp,axiom,
    ! [A: $tType,Xa: array @ vEBT_VEBTi,Tree_is2: list @ vEBT_VEBTi,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,Xb: vEBT_VEBTi,N2: nat,Xc: vEBT_VEBTi,H2: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
      ( ( rep_assn
        @ ( times_times @ assn
          @ ( times_times @ assn @ ( times_times @ assn @ ( times_times @ assn @ ( snga_assn @ vEBT_VEBTi @ Xa @ Tree_is2 ) @ ( vEBT_List_list_assn @ vEBT_VEBT @ vEBT_VEBTi @ vEBT_vebt_assn_raw @ TreeList @ Tree_is2 ) ) @ ( vEBT_vebt_assn_raw @ Summary @ Xb ) )
            @ ( pure_assn
              @ ( ( ( none @ A )
                  = ( none @ A ) )
                & ( N2 = N2 ) ) ) )
          @ ( pure_assn
            @ ( Xc
              = ( vEBT_Nodei @ ( none @ ( product_prod @ nat @ nat ) ) @ N2 @ Xa @ Xb ) ) ) )
        @ H2 )
     => ( rep_assn @ ( vEBT_vebt_assn_raw @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ N2 @ TreeList @ Summary ) @ Xc ) @ H2 ) ) ).

% heaphelp
thf(fact_587_div__exp__eq,axiom,
    ! [A: $tType] :
      ( ( bit_semiring_bits @ A )
     => ! [A2: A,M: nat,N2: nat] :
          ( ( divide_divide @ A @ ( divide_divide @ A @ A2 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) )
          = ( divide_divide @ A @ A2 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( plus_plus @ nat @ M @ N2 ) ) ) ) ) ).

% div_exp_eq
thf(fact_588_semiring__norm_I90_J,axiom,
    ! [M: num,N2: num] :
      ( ( ( bit1 @ M )
        = ( bit1 @ N2 ) )
      = ( M = N2 ) ) ).

% semiring_norm(90)
thf(fact_589_of__nat__eq__iff,axiom,
    ! [A: $tType] :
      ( ( semiring_char_0 @ A )
     => ! [M: nat,N2: nat] :
          ( ( ( semiring_1_of_nat @ A @ M )
            = ( semiring_1_of_nat @ A @ N2 ) )
          = ( M = N2 ) ) ) ).

% of_nat_eq_iff
thf(fact_590_two__realpow__ge__two,axiom,
    ! [N2: nat] :
      ( ( ord_less_eq @ real @ ( one_one @ real ) @ ( semiring_1_of_nat @ real @ N2 ) )
     => ( ord_less_eq @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( power_power @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ N2 ) ) ) ).

% two_realpow_ge_two
thf(fact_591_bits__div__by__1,axiom,
    ! [A: $tType] :
      ( ( bit_semiring_bits @ A )
     => ! [A2: A] :
          ( ( divide_divide @ A @ A2 @ ( one_one @ A ) )
          = A2 ) ) ).

% bits_div_by_1
thf(fact_592_semiring__norm_I88_J,axiom,
    ! [M: num,N2: num] :
      ( ( bit0 @ M )
     != ( bit1 @ N2 ) ) ).

% semiring_norm(88)
thf(fact_593_semiring__norm_I89_J,axiom,
    ! [M: num,N2: num] :
      ( ( bit1 @ M )
     != ( bit0 @ N2 ) ) ).

% semiring_norm(89)
thf(fact_594_semiring__norm_I84_J,axiom,
    ! [N2: num] :
      ( one2
     != ( bit1 @ N2 ) ) ).

% semiring_norm(84)
thf(fact_595_semiring__norm_I86_J,axiom,
    ! [M: num] :
      ( ( bit1 @ M )
     != one2 ) ).

% semiring_norm(86)
thf(fact_596_semiring__norm_I73_J,axiom,
    ! [M: num,N2: num] :
      ( ( ord_less_eq @ num @ ( bit1 @ M ) @ ( bit1 @ N2 ) )
      = ( ord_less_eq @ num @ M @ N2 ) ) ).

% semiring_norm(73)
thf(fact_597_semiring__norm_I80_J,axiom,
    ! [M: num,N2: num] :
      ( ( ord_less @ num @ ( bit1 @ M ) @ ( bit1 @ N2 ) )
      = ( ord_less @ num @ M @ N2 ) ) ).

% semiring_norm(80)
thf(fact_598_of__nat__numeral,axiom,
    ! [A: $tType] :
      ( ( semiring_1 @ A )
     => ! [N2: num] :
          ( ( semiring_1_of_nat @ A @ ( numeral_numeral @ nat @ N2 ) )
          = ( numeral_numeral @ A @ N2 ) ) ) ).

% of_nat_numeral
thf(fact_599_of__nat__less__iff,axiom,
    ! [A: $tType] :
      ( ( linord181362715937106298miring @ A )
     => ! [M: nat,N2: nat] :
          ( ( ord_less @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N2 ) )
          = ( ord_less @ nat @ M @ N2 ) ) ) ).

% of_nat_less_iff
thf(fact_600_of__nat__le__iff,axiom,
    ! [A: $tType] :
      ( ( linord181362715937106298miring @ A )
     => ! [M: nat,N2: nat] :
          ( ( ord_less_eq @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N2 ) )
          = ( ord_less_eq @ nat @ M @ N2 ) ) ) ).

% of_nat_le_iff
thf(fact_601_of__nat__add,axiom,
    ! [A: $tType] :
      ( ( semiring_1 @ A )
     => ! [M: nat,N2: nat] :
          ( ( semiring_1_of_nat @ A @ ( plus_plus @ nat @ M @ N2 ) )
          = ( plus_plus @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N2 ) ) ) ) ).

% of_nat_add
thf(fact_602_of__nat__eq__1__iff,axiom,
    ! [A: $tType] :
      ( ( semiring_char_0 @ A )
     => ! [N2: nat] :
          ( ( ( semiring_1_of_nat @ A @ N2 )
            = ( one_one @ A ) )
          = ( N2
            = ( one_one @ nat ) ) ) ) ).

% of_nat_eq_1_iff
thf(fact_603_of__nat__1__eq__iff,axiom,
    ! [A: $tType] :
      ( ( semiring_char_0 @ A )
     => ! [N2: nat] :
          ( ( ( one_one @ A )
            = ( semiring_1_of_nat @ A @ N2 ) )
          = ( N2
            = ( one_one @ nat ) ) ) ) ).

% of_nat_1_eq_iff
thf(fact_604_of__nat__1,axiom,
    ! [A: $tType] :
      ( ( semiring_1 @ A )
     => ( ( semiring_1_of_nat @ A @ ( one_one @ nat ) )
        = ( one_one @ A ) ) ) ).

% of_nat_1
thf(fact_605_of__nat__mult,axiom,
    ! [A: $tType] :
      ( ( semiring_1 @ A )
     => ! [M: nat,N2: nat] :
          ( ( semiring_1_of_nat @ A @ ( times_times @ nat @ M @ N2 ) )
          = ( times_times @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N2 ) ) ) ) ).

% of_nat_mult
thf(fact_606_of__nat__power__eq__of__nat__cancel__iff,axiom,
    ! [A: $tType] :
      ( ( semiring_char_0 @ A )
     => ! [X2: nat,B2: nat,W: nat] :
          ( ( ( semiring_1_of_nat @ A @ X2 )
            = ( power_power @ A @ ( semiring_1_of_nat @ A @ B2 ) @ W ) )
          = ( X2
            = ( power_power @ nat @ B2 @ W ) ) ) ) ).

% of_nat_power_eq_of_nat_cancel_iff
thf(fact_607_of__nat__eq__of__nat__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( ( semiring_char_0 @ A )
     => ! [B2: nat,W: nat,X2: nat] :
          ( ( ( power_power @ A @ ( semiring_1_of_nat @ A @ B2 ) @ W )
            = ( semiring_1_of_nat @ A @ X2 ) )
          = ( ( power_power @ nat @ B2 @ W )
            = X2 ) ) ) ).

% of_nat_eq_of_nat_power_cancel_iff
thf(fact_608_semiring__1__class_Oof__nat__power,axiom,
    ! [A: $tType] :
      ( ( semiring_1 @ A )
     => ! [M: nat,N2: nat] :
          ( ( semiring_1_of_nat @ A @ ( power_power @ nat @ M @ N2 ) )
          = ( power_power @ A @ ( semiring_1_of_nat @ A @ M ) @ N2 ) ) ) ).

% semiring_1_class.of_nat_power
thf(fact_609_semiring__norm_I7_J,axiom,
    ! [M: num,N2: num] :
      ( ( plus_plus @ num @ ( bit0 @ M ) @ ( bit1 @ N2 ) )
      = ( bit1 @ ( plus_plus @ num @ M @ N2 ) ) ) ).

% semiring_norm(7)
thf(fact_610_semiring__norm_I9_J,axiom,
    ! [M: num,N2: num] :
      ( ( plus_plus @ num @ ( bit1 @ M ) @ ( bit0 @ N2 ) )
      = ( bit1 @ ( plus_plus @ num @ M @ N2 ) ) ) ).

% semiring_norm(9)
thf(fact_611_semiring__norm_I15_J,axiom,
    ! [M: num,N2: num] :
      ( ( times_times @ num @ ( bit1 @ M ) @ ( bit0 @ N2 ) )
      = ( bit0 @ ( times_times @ num @ ( bit1 @ M ) @ N2 ) ) ) ).

% semiring_norm(15)
thf(fact_612_semiring__norm_I14_J,axiom,
    ! [M: num,N2: num] :
      ( ( times_times @ num @ ( bit0 @ M ) @ ( bit1 @ N2 ) )
      = ( bit0 @ ( times_times @ num @ M @ ( bit1 @ N2 ) ) ) ) ).

% semiring_norm(14)
thf(fact_613_semiring__norm_I81_J,axiom,
    ! [M: num,N2: num] :
      ( ( ord_less @ num @ ( bit1 @ M ) @ ( bit0 @ N2 ) )
      = ( ord_less @ num @ M @ N2 ) ) ).

% semiring_norm(81)
thf(fact_614_semiring__norm_I72_J,axiom,
    ! [M: num,N2: num] :
      ( ( ord_less_eq @ num @ ( bit0 @ M ) @ ( bit1 @ N2 ) )
      = ( ord_less_eq @ num @ M @ N2 ) ) ).

% semiring_norm(72)
thf(fact_615_semiring__norm_I77_J,axiom,
    ! [N2: num] : ( ord_less @ num @ one2 @ ( bit1 @ N2 ) ) ).

% semiring_norm(77)
thf(fact_616_semiring__norm_I70_J,axiom,
    ! [M: num] :
      ~ ( ord_less_eq @ num @ ( bit1 @ M ) @ one2 ) ).

% semiring_norm(70)
thf(fact_617_of__nat__Suc,axiom,
    ! [A: $tType] :
      ( ( semiring_1 @ A )
     => ! [M: nat] :
          ( ( semiring_1_of_nat @ A @ ( suc @ M ) )
          = ( plus_plus @ A @ ( one_one @ A ) @ ( semiring_1_of_nat @ A @ M ) ) ) ) ).

% of_nat_Suc
thf(fact_618_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_619_real__of__nat__less__numeral__iff,axiom,
    ! [N2: nat,W: num] :
      ( ( ord_less @ real @ ( semiring_1_of_nat @ real @ N2 ) @ ( numeral_numeral @ real @ W ) )
      = ( ord_less @ nat @ N2 @ ( numeral_numeral @ nat @ W ) ) ) ).

% real_of_nat_less_numeral_iff
thf(fact_620_numeral__less__real__of__nat__iff,axiom,
    ! [W: num,N2: nat] :
      ( ( ord_less @ real @ ( numeral_numeral @ real @ W ) @ ( semiring_1_of_nat @ real @ N2 ) )
      = ( ord_less @ nat @ ( numeral_numeral @ nat @ W ) @ N2 ) ) ).

% numeral_less_real_of_nat_iff
thf(fact_621_semiring__norm_I3_J,axiom,
    ! [N2: num] :
      ( ( plus_plus @ num @ one2 @ ( bit0 @ N2 ) )
      = ( bit1 @ N2 ) ) ).

% semiring_norm(3)
thf(fact_622_semiring__norm_I4_J,axiom,
    ! [N2: num] :
      ( ( plus_plus @ num @ one2 @ ( bit1 @ N2 ) )
      = ( bit0 @ ( plus_plus @ num @ N2 @ one2 ) ) ) ).

% semiring_norm(4)
thf(fact_623_semiring__norm_I5_J,axiom,
    ! [M: num] :
      ( ( plus_plus @ num @ ( bit0 @ M ) @ one2 )
      = ( bit1 @ M ) ) ).

% semiring_norm(5)
thf(fact_624_semiring__norm_I8_J,axiom,
    ! [M: num] :
      ( ( plus_plus @ num @ ( bit1 @ M ) @ one2 )
      = ( bit0 @ ( plus_plus @ num @ M @ one2 ) ) ) ).

% semiring_norm(8)
thf(fact_625_semiring__norm_I10_J,axiom,
    ! [M: num,N2: num] :
      ( ( plus_plus @ num @ ( bit1 @ M ) @ ( bit1 @ N2 ) )
      = ( bit0 @ ( plus_plus @ num @ ( plus_plus @ num @ M @ N2 ) @ one2 ) ) ) ).

% semiring_norm(10)
thf(fact_626_semiring__norm_I16_J,axiom,
    ! [M: num,N2: num] :
      ( ( times_times @ num @ ( bit1 @ M ) @ ( bit1 @ N2 ) )
      = ( bit1 @ ( plus_plus @ num @ ( plus_plus @ num @ M @ N2 ) @ ( bit0 @ ( times_times @ num @ M @ N2 ) ) ) ) ) ).

% semiring_norm(16)
thf(fact_627_semiring__norm_I79_J,axiom,
    ! [M: num,N2: num] :
      ( ( ord_less @ num @ ( bit0 @ M ) @ ( bit1 @ N2 ) )
      = ( ord_less_eq @ num @ M @ N2 ) ) ).

% semiring_norm(79)
thf(fact_628_semiring__norm_I74_J,axiom,
    ! [M: num,N2: num] :
      ( ( ord_less_eq @ num @ ( bit1 @ M ) @ ( bit0 @ N2 ) )
      = ( ord_less @ num @ M @ N2 ) ) ).

% semiring_norm(74)
thf(fact_629_real__of__nat__eq__numeral__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( ( semiring_char_0 @ A )
     => ! [Y2: nat,X2: num,N2: nat] :
          ( ( ( semiring_1_of_nat @ A @ Y2 )
            = ( power_power @ A @ ( numeral_numeral @ A @ X2 ) @ N2 ) )
          = ( Y2
            = ( power_power @ nat @ ( numeral_numeral @ nat @ X2 ) @ N2 ) ) ) ) ).

% real_of_nat_eq_numeral_power_cancel_iff
thf(fact_630_numeral__power__eq__of__nat__cancel__iff,axiom,
    ! [A: $tType] :
      ( ( semiring_char_0 @ A )
     => ! [X2: num,N2: nat,Y2: nat] :
          ( ( ( power_power @ A @ ( numeral_numeral @ A @ X2 ) @ N2 )
            = ( semiring_1_of_nat @ A @ Y2 ) )
          = ( ( power_power @ nat @ ( numeral_numeral @ nat @ X2 ) @ N2 )
            = Y2 ) ) ) ).

% numeral_power_eq_of_nat_cancel_iff
thf(fact_631_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [A: $tType] :
      ( ( linordered_semidom @ A )
     => ! [X2: nat,B2: nat,W: nat] :
          ( ( ord_less @ A @ ( semiring_1_of_nat @ A @ X2 ) @ ( power_power @ A @ ( semiring_1_of_nat @ A @ B2 ) @ W ) )
          = ( ord_less @ nat @ X2 @ ( power_power @ nat @ B2 @ W ) ) ) ) ).

% of_nat_power_less_of_nat_cancel_iff
thf(fact_632_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( ( linordered_semidom @ A )
     => ! [B2: nat,W: nat,X2: nat] :
          ( ( ord_less @ A @ ( power_power @ A @ ( semiring_1_of_nat @ A @ B2 ) @ W ) @ ( semiring_1_of_nat @ A @ X2 ) )
          = ( ord_less @ nat @ ( power_power @ nat @ B2 @ W ) @ X2 ) ) ) ).

% of_nat_less_of_nat_power_cancel_iff
thf(fact_633_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [A: $tType] :
      ( ( linordered_semidom @ A )
     => ! [X2: nat,B2: nat,W: nat] :
          ( ( ord_less_eq @ A @ ( semiring_1_of_nat @ A @ X2 ) @ ( power_power @ A @ ( semiring_1_of_nat @ A @ B2 ) @ W ) )
          = ( ord_less_eq @ nat @ X2 @ ( power_power @ nat @ B2 @ W ) ) ) ) ).

% of_nat_power_le_of_nat_cancel_iff
thf(fact_634_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( ( linordered_semidom @ A )
     => ! [B2: nat,W: nat,X2: nat] :
          ( ( ord_less_eq @ A @ ( power_power @ A @ ( semiring_1_of_nat @ A @ B2 ) @ W ) @ ( semiring_1_of_nat @ A @ X2 ) )
          = ( ord_less_eq @ nat @ ( power_power @ nat @ B2 @ W ) @ X2 ) ) ) ).

% of_nat_le_of_nat_power_cancel_iff
thf(fact_635_numeral__le__real__of__nat__iff,axiom,
    ! [N2: num,M: nat] :
      ( ( ord_less_eq @ real @ ( numeral_numeral @ real @ N2 ) @ ( semiring_1_of_nat @ real @ M ) )
      = ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ N2 ) @ M ) ) ).

% numeral_le_real_of_nat_iff
thf(fact_636_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 @ one2 ) ) @ M ) @ ( numeral_numeral @ nat @ V ) ) ) ).

% Suc_div_eq_add3_div_numeral
thf(fact_637_div__Suc__eq__div__add3,axiom,
    ! [M: nat,N2: nat] :
      ( ( divide_divide @ nat @ M @ ( suc @ ( suc @ ( suc @ N2 ) ) ) )
      = ( divide_divide @ nat @ M @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) @ N2 ) ) ) ).

% div_Suc_eq_div_add3
thf(fact_638_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( ( linordered_semidom @ A )
     => ! [X2: nat,I: num,N2: nat] :
          ( ( ord_less @ A @ ( semiring_1_of_nat @ A @ X2 ) @ ( power_power @ A @ ( numeral_numeral @ A @ I ) @ N2 ) )
          = ( ord_less @ nat @ X2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ I ) @ N2 ) ) ) ) ).

% of_nat_less_numeral_power_cancel_iff
thf(fact_639_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [A: $tType] :
      ( ( linordered_semidom @ A )
     => ! [I: num,N2: nat,X2: nat] :
          ( ( ord_less @ A @ ( power_power @ A @ ( numeral_numeral @ A @ I ) @ N2 ) @ ( semiring_1_of_nat @ A @ X2 ) )
          = ( ord_less @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ I ) @ N2 ) @ X2 ) ) ) ).

% numeral_power_less_of_nat_cancel_iff
thf(fact_640_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( ( linordered_semidom @ A )
     => ! [X2: nat,I: num,N2: nat] :
          ( ( ord_less_eq @ A @ ( semiring_1_of_nat @ A @ X2 ) @ ( power_power @ A @ ( numeral_numeral @ A @ I ) @ N2 ) )
          = ( ord_less_eq @ nat @ X2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ I ) @ N2 ) ) ) ) ).

% of_nat_le_numeral_power_cancel_iff
thf(fact_641_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [A: $tType] :
      ( ( linordered_semidom @ A )
     => ! [I: num,N2: nat,X2: nat] :
          ( ( ord_less_eq @ A @ ( power_power @ A @ ( numeral_numeral @ A @ I ) @ N2 ) @ ( semiring_1_of_nat @ A @ X2 ) )
          = ( ord_less_eq @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ I ) @ N2 ) @ X2 ) ) ) ).

% numeral_power_le_of_nat_cancel_iff
thf(fact_642_mult__of__nat__commute,axiom,
    ! [A: $tType] :
      ( ( semiring_1 @ A )
     => ! [X2: nat,Y2: A] :
          ( ( times_times @ A @ ( semiring_1_of_nat @ A @ X2 ) @ Y2 )
          = ( times_times @ A @ Y2 @ ( semiring_1_of_nat @ A @ X2 ) ) ) ) ).

% mult_of_nat_commute
thf(fact_643_hoare__triple__preI,axiom,
    ! [A: $tType,P: assn,C2: heap_Time_Heap @ A,Q: A > assn] :
      ( ! [H3: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
          ( ( rep_assn @ P @ H3 )
         => ( hoare_hoare_triple @ A @ P @ C2 @ Q ) )
     => ( hoare_hoare_triple @ A @ P @ C2 @ Q ) ) ).

% hoare_triple_preI
thf(fact_644_xor__num_Ocases,axiom,
    ! [X2: product_prod @ num @ num] :
      ( ( X2
       != ( product_Pair @ num @ num @ one2 @ one2 ) )
     => ( ! [N4: num] :
            ( X2
           != ( product_Pair @ num @ num @ one2 @ ( bit0 @ N4 ) ) )
       => ( ! [N4: num] :
              ( X2
             != ( product_Pair @ num @ num @ one2 @ ( bit1 @ N4 ) ) )
         => ( ! [M2: num] :
                ( X2
               != ( product_Pair @ num @ num @ ( bit0 @ M2 ) @ one2 ) )
           => ( ! [M2: num,N4: num] :
                  ( X2
                 != ( product_Pair @ num @ num @ ( bit0 @ M2 ) @ ( bit0 @ N4 ) ) )
             => ( ! [M2: num,N4: num] :
                    ( X2
                   != ( product_Pair @ num @ num @ ( bit0 @ M2 ) @ ( bit1 @ N4 ) ) )
               => ( ! [M2: num] :
                      ( X2
                     != ( product_Pair @ num @ num @ ( bit1 @ M2 ) @ one2 ) )
                 => ( ! [M2: num,N4: num] :
                        ( X2
                       != ( product_Pair @ num @ num @ ( bit1 @ M2 ) @ ( bit0 @ N4 ) ) )
                   => ~ ! [M2: num,N4: num] :
                          ( X2
                         != ( product_Pair @ num @ num @ ( bit1 @ M2 ) @ ( bit1 @ N4 ) ) ) ) ) ) ) ) ) ) ) ).

% xor_num.cases
thf(fact_645_of__nat__less__imp__less,axiom,
    ! [A: $tType] :
      ( ( linord181362715937106298miring @ A )
     => ! [M: nat,N2: nat] :
          ( ( ord_less @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N2 ) )
         => ( ord_less @ nat @ M @ N2 ) ) ) ).

% of_nat_less_imp_less
thf(fact_646_less__imp__of__nat__less,axiom,
    ! [A: $tType] :
      ( ( linord181362715937106298miring @ A )
     => ! [M: nat,N2: nat] :
          ( ( ord_less @ nat @ M @ N2 )
         => ( ord_less @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N2 ) ) ) ) ).

% less_imp_of_nat_less
thf(fact_647_div__mult2__eq_H,axiom,
    ! [A: $tType] :
      ( ( euclid5411537665997757685th_nat @ A )
     => ! [A2: A,M: nat,N2: nat] :
          ( ( divide_divide @ A @ A2 @ ( times_times @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N2 ) ) )
          = ( divide_divide @ A @ ( divide_divide @ A @ A2 @ ( semiring_1_of_nat @ A @ M ) ) @ ( semiring_1_of_nat @ A @ N2 ) ) ) ) ).

% div_mult2_eq'
thf(fact_648_of__nat__mono,axiom,
    ! [A: $tType] :
      ( ( linord181362715937106298miring @ A )
     => ! [I: nat,J: nat] :
          ( ( ord_less_eq @ nat @ I @ J )
         => ( ord_less_eq @ A @ ( semiring_1_of_nat @ A @ I ) @ ( semiring_1_of_nat @ A @ J ) ) ) ) ).

% of_nat_mono
thf(fact_649_unique__euclidean__semiring__with__nat__class_Oof__nat__div,axiom,
    ! [A: $tType] :
      ( ( euclid5411537665997757685th_nat @ A )
     => ! [M: nat,N2: nat] :
          ( ( semiring_1_of_nat @ A @ ( divide_divide @ nat @ M @ N2 ) )
          = ( divide_divide @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N2 ) ) ) ) ).

% unique_euclidean_semiring_with_nat_class.of_nat_div
thf(fact_650_num_Oexhaust,axiom,
    ! [Y2: num] :
      ( ( Y2 != one2 )
     => ( ! [X24: num] :
            ( Y2
           != ( bit0 @ X24 ) )
       => ~ ! [X32: num] :
              ( Y2
             != ( bit1 @ X32 ) ) ) ) ).

% num.exhaust
thf(fact_651_real__of__nat__div4,axiom,
    ! [N2: nat,X2: nat] : ( ord_less_eq @ real @ ( semiring_1_of_nat @ real @ ( divide_divide @ nat @ N2 @ X2 ) ) @ ( divide_divide @ real @ ( semiring_1_of_nat @ real @ N2 ) @ ( semiring_1_of_nat @ real @ X2 ) ) ) ).

% real_of_nat_div4
thf(fact_652_nat__le__real__less,axiom,
    ( ( ord_less_eq @ nat )
    = ( ^ [N: nat,M7: nat] : ( ord_less @ real @ ( semiring_1_of_nat @ real @ N ) @ ( plus_plus @ real @ ( semiring_1_of_nat @ real @ M7 ) @ ( one_one @ real ) ) ) ) ) ).

% nat_le_real_less
thf(fact_653_nat__less__real__le,axiom,
    ( ( ord_less @ nat )
    = ( ^ [N: nat,M7: nat] : ( ord_less_eq @ real @ ( plus_plus @ real @ ( semiring_1_of_nat @ real @ N ) @ ( one_one @ real ) ) @ ( semiring_1_of_nat @ real @ M7 ) ) ) ) ).

% nat_less_real_le
thf(fact_654_of__nat__diff,axiom,
    ! [A: $tType] :
      ( ( semiring_1_cancel @ A )
     => ! [N2: nat,M: nat] :
          ( ( ord_less_eq @ nat @ N2 @ M )
         => ( ( semiring_1_of_nat @ A @ ( minus_minus @ nat @ M @ N2 ) )
            = ( minus_minus @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N2 ) ) ) ) ) ).

% of_nat_diff
thf(fact_655_numeral__Bit1,axiom,
    ! [A: $tType] :
      ( ( numeral @ A )
     => ! [N2: num] :
          ( ( numeral_numeral @ A @ ( bit1 @ N2 ) )
          = ( plus_plus @ A @ ( plus_plus @ A @ ( numeral_numeral @ A @ N2 ) @ ( numeral_numeral @ A @ N2 ) ) @ ( one_one @ A ) ) ) ) ).

% numeral_Bit1
thf(fact_656_eval__nat__numeral_I3_J,axiom,
    ! [N2: num] :
      ( ( numeral_numeral @ nat @ ( bit1 @ N2 ) )
      = ( suc @ ( numeral_numeral @ nat @ ( bit0 @ N2 ) ) ) ) ).

% eval_nat_numeral(3)
thf(fact_657_complete__real,axiom,
    ! [S3: set @ real] :
      ( ? [X6: real] : ( member @ real @ X6 @ S3 )
     => ( ? [Z5: real] :
          ! [X3: real] :
            ( ( member @ real @ X3 @ S3 )
           => ( ord_less_eq @ real @ X3 @ Z5 ) )
       => ? [Y4: real] :
            ( ! [X6: real] :
                ( ( member @ real @ X6 @ S3 )
               => ( ord_less_eq @ real @ X6 @ Y4 ) )
            & ! [Z5: real] :
                ( ! [X3: real] :
                    ( ( member @ real @ X3 @ S3 )
                   => ( ord_less_eq @ real @ X3 @ Z5 ) )
               => ( ord_less_eq @ real @ Y4 @ Z5 ) ) ) ) ) ).

% complete_real
thf(fact_658_less__eq__real__def,axiom,
    ( ( ord_less_eq @ real )
    = ( ^ [X: real,Y: real] :
          ( ( ord_less @ real @ X @ Y )
          | ( X = Y ) ) ) ) ).

% less_eq_real_def
thf(fact_659_of__nat__less__two__power,axiom,
    ! [A: $tType] :
      ( ( linordered_idom @ A )
     => ! [N2: nat] : ( ord_less @ A @ ( semiring_1_of_nat @ A @ N2 ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) ) ) ).

% of_nat_less_two_power
thf(fact_660_numeral__code_I3_J,axiom,
    ! [A: $tType] :
      ( ( numeral @ A )
     => ! [N2: num] :
          ( ( numeral_numeral @ A @ ( bit1 @ N2 ) )
          = ( plus_plus @ A @ ( plus_plus @ A @ ( numeral_numeral @ A @ N2 ) @ ( numeral_numeral @ A @ N2 ) ) @ ( one_one @ A ) ) ) ) ).

% numeral_code(3)
thf(fact_661_power__numeral__odd,axiom,
    ! [A: $tType] :
      ( ( monoid_mult @ A )
     => ! [Z: A,W: num] :
          ( ( power_power @ A @ Z @ ( numeral_numeral @ nat @ ( bit1 @ W ) ) )
          = ( times_times @ A @ ( times_times @ A @ Z @ ( power_power @ A @ Z @ ( numeral_numeral @ nat @ W ) ) ) @ ( power_power @ A @ Z @ ( numeral_numeral @ nat @ W ) ) ) ) ) ).

% power_numeral_odd
thf(fact_662_real__of__nat__div3,axiom,
    ! [N2: nat,X2: nat] : ( ord_less_eq @ real @ ( minus_minus @ real @ ( divide_divide @ real @ ( semiring_1_of_nat @ real @ N2 ) @ ( semiring_1_of_nat @ real @ X2 ) ) @ ( semiring_1_of_nat @ real @ ( divide_divide @ nat @ N2 @ X2 ) ) ) @ ( one_one @ real ) ) ).

% real_of_nat_div3
thf(fact_663_numeral__Bit1__div__2,axiom,
    ! [A: $tType] :
      ( ( euclid5411537665997757685th_nat @ A )
     => ! [N2: num] :
          ( ( divide_divide @ A @ ( numeral_numeral @ A @ ( bit1 @ N2 ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
          = ( numeral_numeral @ A @ N2 ) ) ) ).

% numeral_Bit1_div_2
thf(fact_664_power3__eq__cube,axiom,
    ! [A: $tType] :
      ( ( monoid_mult @ A )
     => ! [A2: A] :
          ( ( power_power @ A @ A2 @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) )
          = ( times_times @ A @ ( times_times @ A @ A2 @ A2 ) @ A2 ) ) ) ).

% power3_eq_cube
thf(fact_665_Suc3__eq__add__3,axiom,
    ! [N2: nat] :
      ( ( suc @ ( suc @ ( suc @ N2 ) ) )
      = ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) @ N2 ) ) ).

% Suc3_eq_add_3
thf(fact_666_Suc__div__eq__add3__div,axiom,
    ! [M: nat,N2: nat] :
      ( ( divide_divide @ nat @ ( suc @ ( suc @ ( suc @ M ) ) ) @ N2 )
      = ( divide_divide @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) @ M ) @ N2 ) ) ).

% Suc_div_eq_add3_div
thf(fact_667_add__diff__add,axiom,
    ! [A: $tType] :
      ( ( ab_group_add @ A )
     => ! [A2: A,C2: A,B2: A,D2: A] :
          ( ( minus_minus @ A @ ( plus_plus @ A @ A2 @ C2 ) @ ( plus_plus @ A @ B2 @ D2 ) )
          = ( plus_plus @ A @ ( minus_minus @ A @ A2 @ B2 ) @ ( minus_minus @ A @ C2 @ D2 ) ) ) ) ).

% add_diff_add
thf(fact_668_real__arch__pow,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less @ real @ ( one_one @ real ) @ X2 )
     => ? [N4: nat] : ( ord_less @ real @ Y2 @ ( power_power @ real @ X2 @ N4 ) ) ) ).

% real_arch_pow
thf(fact_669_small__powers__of__2,axiom,
    ! [X2: nat] :
      ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) @ X2 )
     => ( ord_less @ nat @ X2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ X2 @ ( one_one @ nat ) ) ) ) ) ).

% small_powers_of_2
thf(fact_670_mult__diff__mult,axiom,
    ! [A: $tType] :
      ( ( ring @ A )
     => ! [X2: A,Y2: A,A2: A,B2: A] :
          ( ( minus_minus @ A @ ( times_times @ A @ X2 @ Y2 ) @ ( times_times @ A @ A2 @ B2 ) )
          = ( plus_plus @ A @ ( times_times @ A @ X2 @ ( minus_minus @ A @ Y2 @ B2 ) ) @ ( times_times @ A @ ( minus_minus @ A @ X2 @ A2 ) @ B2 ) ) ) ) ).

% mult_diff_mult
thf(fact_671_field__sum__of__halves,axiom,
    ! [A: $tType] :
      ( ( linordered_field @ A )
     => ! [X2: A] :
          ( ( plus_plus @ A @ ( divide_divide @ A @ X2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ A @ X2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) )
          = X2 ) ) ).

% field_sum_of_halves
thf(fact_672_field__less__half__sum,axiom,
    ! [A: $tType] :
      ( ( linordered_field @ A )
     => ! [X2: A,Y2: A] :
          ( ( ord_less @ A @ X2 @ Y2 )
         => ( ord_less @ A @ X2 @ ( divide_divide @ A @ ( plus_plus @ A @ X2 @ Y2 ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).

% field_less_half_sum
thf(fact_673_space__bound,axiom,
    ! [T2: vEBT_VEBT,N2: nat,U: nat] :
      ( ( vEBT_invar_vebt @ T2 @ N2 )
     => ( ( U
          = ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) )
       => ( ord_less_eq @ nat @ ( vEBT_VEBT_space @ T2 ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ ( bit1 @ one2 ) ) ) ) @ U ) ) ) ) ).

% space_bound
thf(fact_674_cnt__cnt__eq,axiom,
    ( vEBT_VEBT_cnt
    = ( ^ [T3: vEBT_VEBT] : ( semiring_1_of_nat @ real @ ( vEBT_VEBT_cnt2 @ T3 ) ) ) ) ).

% cnt_cnt_eq
thf(fact_675_space__2__pow__bound,axiom,
    ! [T2: vEBT_VEBT,N2: nat] :
      ( ( vEBT_invar_vebt @ T2 @ N2 )
     => ( ord_less_eq @ real @ ( semiring_1_of_nat @ real @ ( vEBT_VEBT_space2 @ T2 ) ) @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ ( bit0 @ ( bit1 @ one2 ) ) ) ) @ ( minus_minus @ real @ ( power_power @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ N2 ) @ ( one_one @ real ) ) ) ) ) ).

% space_2_pow_bound
thf(fact_676_space__cnt,axiom,
    ! [T2: vEBT_VEBT] : ( ord_less_eq @ real @ ( semiring_1_of_nat @ real @ ( vEBT_VEBT_space2 @ T2 ) ) @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ ( bit1 @ one2 ) ) ) @ ( vEBT_VEBT_cnt @ T2 ) ) ) ).

% space_cnt
thf(fact_677_space_H__bound,axiom,
    ! [T2: vEBT_VEBT,N2: nat,U: nat] :
      ( ( vEBT_invar_vebt @ T2 @ N2 )
     => ( ( U
          = ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) )
       => ( ord_less_eq @ nat @ ( vEBT_VEBT_space2 @ T2 ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ ( bit1 @ one2 ) ) ) ) @ U ) ) ) ) ).

% space'_bound
thf(fact_678_TBOUND__replicate,axiom,
    ! [A: $tType,X2: heap_Time_Heap @ A,C2: nat,N2: nat] :
      ( ( time_TBOUND @ A @ X2 @ C2 )
     => ( time_TBOUND @ ( list @ A ) @ ( vEBT_VEBT_replicatei @ A @ N2 @ X2 ) @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( times_times @ nat @ ( plus_plus @ nat @ ( one_one @ nat ) @ C2 ) @ N2 ) ) ) ) ).

% TBOUND_replicate
thf(fact_679_delete__bound__height,axiom,
    ! [T2: vEBT_VEBT,N2: nat,X2: nat] :
      ( ( vEBT_invar_vebt @ T2 @ N2 )
     => ( ord_less_eq @ nat @ ( vEBT_T_d_e_l_e_t_e @ T2 @ X2 ) @ ( times_times @ nat @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_VEBT_height @ T2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ).

% delete_bound_height
thf(fact_680_t__build__cnt,axiom,
    ! [N2: nat] : ( ord_less_eq @ real @ ( semiring_1_of_nat @ real @ ( vEBT_V8646137997579335489_i_l_d @ N2 ) ) @ ( times_times @ real @ ( vEBT_VEBT_cnt @ ( vEBT_vebt_buildup @ N2 ) ) @ ( numeral_numeral @ real @ ( bit1 @ ( bit0 @ ( bit1 @ one2 ) ) ) ) ) ) ).

% t_build_cnt
thf(fact_681_t__buildup__cnt,axiom,
    ! [N2: nat] : ( ord_less_eq @ real @ ( semiring_1_of_nat @ real @ ( vEBT_V8346862874174094_d_u_p @ N2 ) ) @ ( times_times @ real @ ( vEBT_VEBT_cnt @ ( vEBT_vebt_buildup @ N2 ) ) @ ( numeral_numeral @ real @ ( bit1 @ ( bit0 @ ( bit1 @ one2 ) ) ) ) ) ) ).

% t_buildup_cnt
thf(fact_682_tdeletemimi,axiom,
    ! [Deg: nat,Mi: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,X2: nat] :
      ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
     => ( ord_less_eq @ nat @ ( vEBT_T_d_e_l_e_t_e @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Mi ) ) @ Deg @ TreeList @ Summary ) @ X2 ) @ ( numeral_numeral @ nat @ ( bit1 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) ).

% tdeletemimi
thf(fact_683_TBOUND__highi,axiom,
    ! [X2: nat,N2: nat] : ( time_TBOUND @ nat @ ( vEBT_VEBT_highi @ X2 @ N2 ) @ ( one_one @ nat ) ) ).

% TBOUND_highi
thf(fact_684_TBOUND__lowi,axiom,
    ! [X2: nat,N2: nat] : ( time_TBOUND @ nat @ ( vEBT_VEBT_lowi @ X2 @ N2 ) @ ( one_one @ nat ) ) ).

% TBOUND_lowi
thf(fact_685_TBOUND__vebt__maxti,axiom,
    ! [T2: vEBT_VEBTi] : ( time_TBOUND @ ( option @ nat ) @ ( vEBT_vebt_maxti @ T2 ) @ ( one_one @ nat ) ) ).

% TBOUND_vebt_maxti
thf(fact_686_TBOUND__vebt__minti,axiom,
    ! [T2: vEBT_VEBTi] : ( time_TBOUND @ ( option @ nat ) @ ( vEBT_vebt_minti @ T2 ) @ ( one_one @ nat ) ) ).

% TBOUND_vebt_minti
thf(fact_687_buildup__build__time,axiom,
    ! [N2: nat] : ( ord_less @ nat @ ( vEBT_V8346862874174094_d_u_p @ N2 ) @ ( vEBT_V8646137997579335489_i_l_d @ N2 ) ) ).

% buildup_build_time
thf(fact_688_space__space_H,axiom,
    ! [T2: vEBT_VEBT] : ( ord_less @ nat @ ( vEBT_VEBT_space @ T2 ) @ ( vEBT_VEBT_space2 @ T2 ) ) ).

% space_space'
thf(fact_689_vebt__buildup__bound,axiom,
    ! [U: nat,N2: nat] :
      ( ( U
        = ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) )
     => ( ord_less_eq @ nat @ ( vEBT_V8346862874174094_d_u_p @ N2 ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ one2 ) ) ) ) ) @ U ) ) ) ).

% vebt_buildup_bound
thf(fact_690_TBOUND__vebt__succi,axiom,
    ! [T2: vEBT_VEBT,Ti: vEBT_VEBTi,X2: nat] : ( time_TBOUND @ ( option @ nat ) @ ( vEBT_VEBT_vebt_succi @ T2 @ Ti @ X2 ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit1 @ one2 ) ) ) @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_VEBT_height @ T2 ) ) ) ) ).

% TBOUND_vebt_succi
thf(fact_691_zdiv__int,axiom,
    ! [A2: nat,B2: nat] :
      ( ( semiring_1_of_nat @ int @ ( divide_divide @ nat @ A2 @ B2 ) )
      = ( divide_divide @ int @ ( semiring_1_of_nat @ int @ A2 ) @ ( semiring_1_of_nat @ int @ B2 ) ) ) ).

% zdiv_int
thf(fact_692_Tb_H__cnt,axiom,
    ! [N2: nat] : ( ord_less_eq @ nat @ ( vEBT_VEBT_Tb2 @ N2 ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit0 @ one2 ) ) ) @ ( vEBT_VEBT_cnt2 @ ( vEBT_vebt_buildup @ N2 ) ) ) ) ).

% Tb'_cnt
thf(fact_693_htt__vebt__buildupi_H__univ,axiom,
    ! [U: nat,N2: nat] :
      ( ( U
        = ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) )
     => ( time_htt @ vEBT_VEBTi @ ( one_one @ assn ) @ ( vEBT_V739175172307565963ildupi @ N2 ) @ ( vEBT_vebt_assn_raw @ ( vEBT_vebt_buildup @ N2 ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit1 @ ( bit0 @ one2 ) ) ) ) @ U ) ) ) ).

% htt_vebt_buildupi'_univ
thf(fact_694_htt__vebt__buildupi__univ,axiom,
    ! [U: nat,N2: nat] :
      ( ( U
        = ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) )
     => ( time_htt @ vEBT_VEBTi @ ( one_one @ assn ) @ ( vEBT_vebt_buildupi @ N2 ) @ ( vEBT_vebt_assn_raw @ ( vEBT_vebt_buildup @ N2 ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit1 @ ( bit0 @ one2 ) ) ) ) @ U ) ) ) ).

% htt_vebt_buildupi_univ
thf(fact_695_T__vebt__buildupi__cnt_H,axiom,
    ! [N2: nat] : ( ord_less_eq @ real @ ( semiring_1_of_nat @ real @ ( vEBT_V441764108873111860ildupi @ N2 ) ) @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit1 @ ( bit0 @ one2 ) ) ) @ ( vEBT_VEBT_cnt @ ( vEBT_vebt_buildup @ N2 ) ) ) ) ).

% T_vebt_buildupi_cnt'
thf(fact_696_T__vebt__buildupi__univ,axiom,
    ! [U: nat,N2: nat] :
      ( ( U
        = ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) )
     => ( ord_less_eq @ nat @ ( vEBT_V441764108873111860ildupi @ N2 ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit1 @ ( bit0 @ one2 ) ) ) ) @ U ) ) ) ).

% T_vebt_buildupi_univ
thf(fact_697_TBOUND__minNulli,axiom,
    ! [T2: vEBT_VEBTi] : ( time_TBOUND @ $o @ ( vEBT_VEBT_minNulli @ T2 ) @ ( one_one @ nat ) ) ).

% TBOUND_minNulli
thf(fact_698_mod__pure__star__dist,axiom,
    ! [P: assn,B2: $o,H2: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
      ( ( rep_assn @ ( times_times @ assn @ P @ ( pure_assn @ B2 ) ) @ H2 )
      = ( ( rep_assn @ P @ H2 )
        & B2 ) ) ).

% mod_pure_star_dist
thf(fact_699_TBOUND__buildupi,axiom,
    ! [N2: nat] :
      ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
     => ( time_TBOUND @ vEBT_VEBTi @ ( vEBT_vebt_buildupi @ N2 ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit1 @ ( bit0 @ one2 ) ) ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ) ) ).

% TBOUND_buildupi
thf(fact_700_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,Va: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,X2: nat] :
      ( ( vEBT_T_m_e_m_b_e_r @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X2 )
      = ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( if @ nat @ ( X2 = Mi ) @ ( one_one @ nat ) @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( if @ nat @ ( X2 = Ma ) @ ( one_one @ nat ) @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( if @ nat @ ( ord_less @ nat @ X2 @ Mi ) @ ( one_one @ nat ) @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( if @ nat @ ( ord_less @ nat @ Ma @ X2 ) @ ( one_one @ nat ) @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) @ ( if @ nat @ ( ord_less @ nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) ) @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_T_m_e_m_b_e_r @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( one_one @ nat ) ) ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.simps(5)
thf(fact_701_TBOUND__vebt__inserti,axiom,
    ! [T2: vEBT_VEBT,Ti: vEBT_VEBTi,X2: nat] : ( time_TBOUND @ vEBT_VEBTi @ ( vEBT_V3964819847710782039nserti @ T2 @ Ti @ X2 ) @ ( if @ nat @ ( vEBT_VEBT_minNull @ T2 ) @ ( one_one @ nat ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit0 @ ( bit1 @ one2 ) ) ) ) @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_VEBT_height @ T2 ) ) ) ) ) ).

% TBOUND_vebt_inserti
thf(fact_702_valid__0__not,axiom,
    ! [T2: vEBT_VEBT] :
      ~ ( vEBT_invar_vebt @ T2 @ ( zero_zero @ nat ) ) ).

% valid_0_not
thf(fact_703_valid__tree__deg__neq__0,axiom,
    ! [T2: vEBT_VEBT] :
      ~ ( vEBT_invar_vebt @ T2 @ ( zero_zero @ nat ) ) ).

% valid_tree_deg_neq_0
thf(fact_704_not__min__Null__member,axiom,
    ! [T2: vEBT_VEBT] :
      ( ~ ( vEBT_VEBT_minNull @ T2 )
     => ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ T2 @ X_12 ) ) ).

% not_min_Null_member
thf(fact_705_min__Null__member,axiom,
    ! [T2: vEBT_VEBT,X2: nat] :
      ( ( vEBT_VEBT_minNull @ T2 )
     => ~ ( vEBT_vebt_member @ T2 @ X2 ) ) ).

% min_Null_member
thf(fact_706_deg__not__0,axiom,
    ! [T2: vEBT_VEBT,N2: nat] :
      ( ( vEBT_invar_vebt @ T2 @ N2 )
     => ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) ) ).

% deg_not_0
thf(fact_707_T__vebt__buildupi__gq__0,axiom,
    ! [N2: nat] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( vEBT_V441764108873111860ildupi @ N2 ) ) ).

% T_vebt_buildupi_gq_0
thf(fact_708_minminNull,axiom,
    ! [T2: vEBT_VEBT] :
      ( ( ( vEBT_vebt_mint @ T2 )
        = ( none @ nat ) )
     => ( vEBT_VEBT_minNull @ T2 ) ) ).

% minminNull
thf(fact_709_minNullmin,axiom,
    ! [T2: vEBT_VEBT] :
      ( ( vEBT_VEBT_minNull @ T2 )
     => ( ( vEBT_vebt_mint @ T2 )
        = ( none @ nat ) ) ) ).

% minNullmin
thf(fact_710_TBOUND__vebt__buildupi,axiom,
    ! [N2: nat] : ( time_TBOUND @ vEBT_VEBTi @ ( vEBT_V739175172307565963ildupi @ N2 ) @ ( vEBT_V441764108873111860ildupi @ N2 ) ) ).

% TBOUND_vebt_buildupi
thf(fact_711_buildup__gives__valid,axiom,
    ! [N2: nat] :
      ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
     => ( vEBT_invar_vebt @ ( vEBT_vebt_buildup @ N2 ) @ N2 ) ) ).

% buildup_gives_valid
thf(fact_712_pure__assn__eq__conv,axiom,
    ! [P: $o,Q: $o] :
      ( ( ( pure_assn @ P )
        = ( pure_assn @ Q ) )
      = ( P = Q ) ) ).

% pure_assn_eq_conv
thf(fact_713_TBOUND__minNull,axiom,
    ! [T2: vEBT_VEBT,Ti: vEBT_VEBTi,X2: nat] :
      ( ( vEBT_VEBT_minNull @ T2 )
     => ( time_TBOUND @ vEBT_VEBTi @ ( vEBT_V3964819847710782039nserti @ T2 @ Ti @ X2 ) @ ( one_one @ nat ) ) ) ).

% TBOUND_minNull
thf(fact_714_ex__assn__const,axiom,
    ! [A: $tType,C2: assn] :
      ( ( ex_assn @ A
        @ ^ [X: A] : C2 )
      = C2 ) ).

% ex_assn_const
thf(fact_715_Tb__T__vebt__buildupi_H_H,axiom,
    ! [N2: nat] : ( ord_less_eq @ nat @ ( vEBT_V441764108873111860ildupi @ N2 ) @ ( minus_minus @ nat @ ( vEBT_VEBT_Tb2 @ N2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ).

% Tb_T_vebt_buildupi''
thf(fact_716_htt__vebt__buildupi,axiom,
    ! [N2: nat] : ( time_htt @ vEBT_VEBTi @ ( one_one @ assn ) @ ( vEBT_vebt_buildupi @ N2 ) @ ( vEBT_vebt_assn_raw @ ( vEBT_vebt_buildup @ N2 ) ) @ ( vEBT_V441764108873111860ildupi @ N2 ) ) ).

% htt_vebt_buildupi
thf(fact_717_htt__vebt__buildupi_H,axiom,
    ! [N2: nat] : ( time_htt @ vEBT_VEBTi @ ( one_one @ assn ) @ ( vEBT_V739175172307565963ildupi @ N2 ) @ ( vEBT_vebt_assn_raw @ ( vEBT_vebt_buildup @ N2 ) ) @ ( vEBT_V441764108873111860ildupi @ N2 ) ) ).

% htt_vebt_buildupi'
thf(fact_718_mult__cancel__right,axiom,
    ! [A: $tType] :
      ( ( semiri6575147826004484403cancel @ A )
     => ! [A2: A,C2: A,B2: A] :
          ( ( ( times_times @ A @ A2 @ C2 )
            = ( times_times @ A @ B2 @ C2 ) )
          = ( ( C2
              = ( zero_zero @ A ) )
            | ( A2 = B2 ) ) ) ) ).

% mult_cancel_right
thf(fact_719_mult__cancel__left,axiom,
    ! [A: $tType] :
      ( ( semiri6575147826004484403cancel @ A )
     => ! [C2: A,A2: A,B2: A] :
          ( ( ( times_times @ A @ C2 @ A2 )
            = ( times_times @ A @ C2 @ B2 ) )
          = ( ( C2
              = ( zero_zero @ A ) )
            | ( A2 = B2 ) ) ) ) ).

% mult_cancel_left
thf(fact_720_mult__eq__0__iff,axiom,
    ! [A: $tType] :
      ( ( semiri3467727345109120633visors @ A )
     => ! [A2: A,B2: A] :
          ( ( ( times_times @ A @ A2 @ B2 )
            = ( zero_zero @ A ) )
          = ( ( A2
              = ( zero_zero @ A ) )
            | ( B2
              = ( zero_zero @ A ) ) ) ) ) ).

% mult_eq_0_iff
thf(fact_721_mult__zero__right,axiom,
    ! [A: $tType] :
      ( ( mult_zero @ A )
     => ! [A2: A] :
          ( ( times_times @ A @ A2 @ ( zero_zero @ A ) )
          = ( zero_zero @ A ) ) ) ).

% mult_zero_right
thf(fact_722_mult__zero__left,axiom,
    ! [A: $tType] :
      ( ( mult_zero @ A )
     => ! [A2: A] :
          ( ( times_times @ A @ ( zero_zero @ A ) @ A2 )
          = ( zero_zero @ A ) ) ) ).

% mult_zero_left
thf(fact_723_bits__div__by__0,axiom,
    ! [A: $tType] :
      ( ( bit_semiring_bits @ A )
     => ! [A2: A] :
          ( ( divide_divide @ A @ A2 @ ( zero_zero @ A ) )
          = ( zero_zero @ A ) ) ) ).

% bits_div_by_0
thf(fact_724_bits__div__0,axiom,
    ! [A: $tType] :
      ( ( bit_semiring_bits @ A )
     => ! [A2: A] :
          ( ( divide_divide @ A @ ( zero_zero @ A ) @ A2 )
          = ( zero_zero @ A ) ) ) ).

% bits_div_0
thf(fact_725_div__by__0,axiom,
    ! [A: $tType] :
      ( ( semidom_divide @ A )
     => ! [A2: A] :
          ( ( divide_divide @ A @ A2 @ ( zero_zero @ A ) )
          = ( zero_zero @ A ) ) ) ).

% div_by_0
thf(fact_726_div__0,axiom,
    ! [A: $tType] :
      ( ( semidom_divide @ A )
     => ! [A2: A] :
          ( ( divide_divide @ A @ ( zero_zero @ A ) @ A2 )
          = ( zero_zero @ A ) ) ) ).

% div_0
thf(fact_727_less__nat__zero__code,axiom,
    ! [N2: nat] :
      ~ ( ord_less @ nat @ N2 @ ( zero_zero @ nat ) ) ).

% less_nat_zero_code
thf(fact_728_neq0__conv,axiom,
    ! [N2: nat] :
      ( ( N2
       != ( zero_zero @ nat ) )
      = ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) ) ).

% neq0_conv
thf(fact_729_bot__nat__0_Onot__eq__extremum,axiom,
    ! [A2: nat] :
      ( ( A2
       != ( zero_zero @ nat ) )
      = ( ord_less @ nat @ ( zero_zero @ nat ) @ A2 ) ) ).

% bot_nat_0.not_eq_extremum
thf(fact_730_le0,axiom,
    ! [N2: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ N2 ) ).

% le0
thf(fact_731_bot__nat__0_Oextremum,axiom,
    ! [A2: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ A2 ) ).

% bot_nat_0.extremum
thf(fact_732_add__is__0,axiom,
    ! [M: nat,N2: nat] :
      ( ( ( plus_plus @ nat @ M @ N2 )
        = ( zero_zero @ nat ) )
      = ( ( M
          = ( zero_zero @ nat ) )
        & ( N2
          = ( zero_zero @ nat ) ) ) ) ).

% add_is_0
thf(fact_733_Nat_Oadd__0__right,axiom,
    ! [M: nat] :
      ( ( plus_plus @ nat @ M @ ( zero_zero @ nat ) )
      = M ) ).

% Nat.add_0_right
thf(fact_734_diff__0__eq__0,axiom,
    ! [N2: nat] :
      ( ( minus_minus @ nat @ ( zero_zero @ nat ) @ N2 )
      = ( zero_zero @ nat ) ) ).

% diff_0_eq_0
thf(fact_735_diff__self__eq__0,axiom,
    ! [M: nat] :
      ( ( minus_minus @ nat @ M @ M )
      = ( zero_zero @ nat ) ) ).

% diff_self_eq_0
thf(fact_736_minNulli__rule,axiom,
    ! [T2: vEBT_VEBT,Ti: vEBT_VEBTi] :
      ( hoare_hoare_triple @ $o @ ( vEBT_vebt_assn_raw @ T2 @ Ti ) @ ( vEBT_VEBT_minNulli @ Ti )
      @ ^ [R: $o] :
          ( times_times @ assn @ ( vEBT_vebt_assn_raw @ T2 @ Ti )
          @ ( pure_assn
            @ ( R
              = ( vEBT_VEBT_minNull @ T2 ) ) ) ) ) ).

% minNulli_rule
thf(fact_737_mult__cancel2,axiom,
    ! [M: nat,K: nat,N2: nat] :
      ( ( ( times_times @ nat @ M @ K )
        = ( times_times @ nat @ N2 @ K ) )
      = ( ( M = N2 )
        | ( K
          = ( zero_zero @ nat ) ) ) ) ).

% mult_cancel2
thf(fact_738_mult__cancel1,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( ( times_times @ nat @ K @ M )
        = ( times_times @ nat @ K @ N2 ) )
      = ( ( M = N2 )
        | ( K
          = ( zero_zero @ nat ) ) ) ) ).

% mult_cancel1
thf(fact_739_mult__0__right,axiom,
    ! [M: nat] :
      ( ( times_times @ nat @ M @ ( zero_zero @ nat ) )
      = ( zero_zero @ nat ) ) ).

% mult_0_right
thf(fact_740_mult__is__0,axiom,
    ! [M: nat,N2: nat] :
      ( ( ( times_times @ nat @ M @ N2 )
        = ( zero_zero @ nat ) )
      = ( ( M
          = ( zero_zero @ nat ) )
        | ( N2
          = ( zero_zero @ nat ) ) ) ) ).

% mult_is_0
thf(fact_741_merge__pure__star,axiom,
    ! [A2: $o,B2: $o] :
      ( ( times_times @ assn @ ( pure_assn @ A2 ) @ ( pure_assn @ B2 ) )
      = ( pure_assn
        @ ( A2
          & B2 ) ) ) ).

% merge_pure_star
thf(fact_742_pure__assn__eq__emp__iff,axiom,
    ! [P: $o] :
      ( ( ( pure_assn @ P )
        = ( one_one @ assn ) )
      = P ) ).

% pure_assn_eq_emp_iff
thf(fact_743_pure__true,axiom,
    ( ( pure_assn @ $true )
    = ( one_one @ assn ) ) ).

% pure_true
thf(fact_744_mod__ex__dist,axiom,
    ! [A: $tType,P: A > assn,H2: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
      ( ( rep_assn @ ( ex_assn @ A @ P ) @ H2 )
      = ( ? [X: A] : ( rep_assn @ ( P @ X ) @ H2 ) ) ) ).

% mod_ex_dist
thf(fact_745_zero__comp__diff__simps_I1_J,axiom,
    ! [A: $tType] :
      ( ( linordered_idom @ A )
     => ! [A2: A,B2: A] :
          ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( minus_minus @ A @ A2 @ B2 ) )
          = ( ord_less_eq @ A @ B2 @ A2 ) ) ) ).

% zero_comp_diff_simps(1)
thf(fact_746_zero__comp__diff__simps_I2_J,axiom,
    ! [A: $tType] :
      ( ( linordered_idom @ A )
     => ! [A2: A,B2: A] :
          ( ( ord_less @ A @ ( zero_zero @ A ) @ ( minus_minus @ A @ A2 @ B2 ) )
          = ( ord_less @ A @ B2 @ A2 ) ) ) ).

% zero_comp_diff_simps(2)
thf(fact_747_sum__squares__eq__zero__iff,axiom,
    ! [A: $tType] :
      ( ( linord4710134922213307826strict @ A )
     => ! [X2: A,Y2: A] :
          ( ( ( plus_plus @ A @ ( times_times @ A @ X2 @ X2 ) @ ( times_times @ A @ Y2 @ Y2 ) )
            = ( zero_zero @ A ) )
          = ( ( X2
              = ( zero_zero @ A ) )
            & ( Y2
              = ( zero_zero @ A ) ) ) ) ) ).

% sum_squares_eq_zero_iff
thf(fact_748_mult__cancel__right2,axiom,
    ! [A: $tType] :
      ( ( ring_15535105094025558882visors @ A )
     => ! [A2: A,C2: A] :
          ( ( ( times_times @ A @ A2 @ C2 )
            = C2 )
          = ( ( C2
              = ( zero_zero @ A ) )
            | ( A2
              = ( one_one @ A ) ) ) ) ) ).

% mult_cancel_right2
thf(fact_749_mult__cancel__right1,axiom,
    ! [A: $tType] :
      ( ( ring_15535105094025558882visors @ A )
     => ! [C2: A,B2: A] :
          ( ( C2
            = ( times_times @ A @ B2 @ C2 ) )
          = ( ( C2
              = ( zero_zero @ A ) )
            | ( B2
              = ( one_one @ A ) ) ) ) ) ).

% mult_cancel_right1
thf(fact_750_mult__cancel__left2,axiom,
    ! [A: $tType] :
      ( ( ring_15535105094025558882visors @ A )
     => ! [C2: A,A2: A] :
          ( ( ( times_times @ A @ C2 @ A2 )
            = C2 )
          = ( ( C2
              = ( zero_zero @ A ) )
            | ( A2
              = ( one_one @ A ) ) ) ) ) ).

% mult_cancel_left2
thf(fact_751_mult__cancel__left1,axiom,
    ! [A: $tType] :
      ( ( ring_15535105094025558882visors @ A )
     => ! [C2: A,B2: A] :
          ( ( C2
            = ( times_times @ A @ C2 @ B2 ) )
          = ( ( C2
              = ( zero_zero @ A ) )
            | ( B2
              = ( one_one @ A ) ) ) ) ) ).

% mult_cancel_left1
thf(fact_752_diff__numeral__special_I9_J,axiom,
    ! [A: $tType] :
      ( ( neg_numeral @ A )
     => ( ( minus_minus @ A @ ( one_one @ A ) @ ( one_one @ A ) )
        = ( zero_zero @ A ) ) ) ).

% diff_numeral_special(9)
thf(fact_753_div__self,axiom,
    ! [A: $tType] :
      ( ( semidom_divide @ A )
     => ! [A2: A] :
          ( ( A2
           != ( zero_zero @ A ) )
         => ( ( divide_divide @ A @ A2 @ A2 )
            = ( one_one @ A ) ) ) ) ).

% div_self
thf(fact_754_nonzero__mult__div__cancel__right,axiom,
    ! [A: $tType] :
      ( ( semidom_divide @ A )
     => ! [B2: A,A2: A] :
          ( ( B2
           != ( zero_zero @ A ) )
         => ( ( divide_divide @ A @ ( times_times @ A @ A2 @ B2 ) @ B2 )
            = A2 ) ) ) ).

% nonzero_mult_div_cancel_right
thf(fact_755_nonzero__mult__div__cancel__left,axiom,
    ! [A: $tType] :
      ( ( semidom_divide @ A )
     => ! [A2: A,B2: A] :
          ( ( A2
           != ( zero_zero @ A ) )
         => ( ( divide_divide @ A @ ( times_times @ A @ A2 @ B2 ) @ A2 )
            = B2 ) ) ) ).

% nonzero_mult_div_cancel_left
thf(fact_756_div__mult__mult1__if,axiom,
    ! [A: $tType] :
      ( ( euclid4440199948858584721cancel @ A )
     => ! [C2: A,A2: A,B2: A] :
          ( ( ( C2
              = ( zero_zero @ A ) )
           => ( ( divide_divide @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B2 ) )
              = ( zero_zero @ A ) ) )
          & ( ( C2
             != ( zero_zero @ A ) )
           => ( ( divide_divide @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B2 ) )
              = ( divide_divide @ A @ A2 @ B2 ) ) ) ) ) ).

% div_mult_mult1_if
thf(fact_757_div__mult__mult2,axiom,
    ! [A: $tType] :
      ( ( euclid4440199948858584721cancel @ A )
     => ! [C2: A,A2: A,B2: A] :
          ( ( C2
           != ( zero_zero @ A ) )
         => ( ( divide_divide @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) )
            = ( divide_divide @ A @ A2 @ B2 ) ) ) ) ).

% div_mult_mult2
thf(fact_758_div__mult__mult1,axiom,
    ! [A: $tType] :
      ( ( euclid4440199948858584721cancel @ A )
     => ! [C2: A,A2: A,B2: A] :
          ( ( C2
           != ( zero_zero @ A ) )
         => ( ( divide_divide @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B2 ) )
            = ( divide_divide @ A @ A2 @ B2 ) ) ) ) ).

% div_mult_mult1
thf(fact_759_power__0__Suc,axiom,
    ! [A: $tType] :
      ( ( semiring_1 @ A )
     => ! [N2: nat] :
          ( ( power_power @ A @ ( zero_zero @ A ) @ ( suc @ N2 ) )
          = ( zero_zero @ A ) ) ) ).

% power_0_Suc
thf(fact_760_power__zero__numeral,axiom,
    ! [A: $tType] :
      ( ( semiring_1 @ A )
     => ! [K: num] :
          ( ( power_power @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ nat @ K ) )
          = ( zero_zero @ A ) ) ) ).

% power_zero_numeral
thf(fact_761_of__nat__eq__0__iff,axiom,
    ! [A: $tType] :
      ( ( semiring_char_0 @ A )
     => ! [M: nat] :
          ( ( ( semiring_1_of_nat @ A @ M )
            = ( zero_zero @ A ) )
          = ( M
            = ( zero_zero @ nat ) ) ) ) ).

% of_nat_eq_0_iff
thf(fact_762_of__nat__0__eq__iff,axiom,
    ! [A: $tType] :
      ( ( semiring_char_0 @ A )
     => ! [N2: nat] :
          ( ( ( zero_zero @ A )
            = ( semiring_1_of_nat @ A @ N2 ) )
          = ( ( zero_zero @ nat )
            = N2 ) ) ) ).

% of_nat_0_eq_iff
thf(fact_763_semiring__1__class_Oof__nat__0,axiom,
    ! [A: $tType] :
      ( ( semiring_1 @ A )
     => ( ( semiring_1_of_nat @ A @ ( zero_zero @ nat ) )
        = ( zero_zero @ A ) ) ) ).

% semiring_1_class.of_nat_0
thf(fact_764_power__Suc0__right,axiom,
    ! [A: $tType] :
      ( ( monoid_mult @ A )
     => ! [A2: A] :
          ( ( power_power @ A @ A2 @ ( suc @ ( zero_zero @ nat ) ) )
          = A2 ) ) ).

% power_Suc0_right
thf(fact_765_zero__less__Suc,axiom,
    ! [N2: nat] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( suc @ N2 ) ) ).

% zero_less_Suc
thf(fact_766_less__Suc0,axiom,
    ! [N2: nat] :
      ( ( ord_less @ nat @ N2 @ ( suc @ ( zero_zero @ nat ) ) )
      = ( N2
        = ( zero_zero @ nat ) ) ) ).

% less_Suc0
thf(fact_767_add__gr__0,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( plus_plus @ nat @ M @ N2 ) )
      = ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
        | ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ).

% add_gr_0
thf(fact_768_less__one,axiom,
    ! [N2: nat] :
      ( ( ord_less @ nat @ N2 @ ( one_one @ nat ) )
      = ( N2
        = ( zero_zero @ nat ) ) ) ).

% less_one
thf(fact_769_div__by__Suc__0,axiom,
    ! [M: nat] :
      ( ( divide_divide @ nat @ M @ ( suc @ ( zero_zero @ nat ) ) )
      = M ) ).

% div_by_Suc_0
thf(fact_770_zero__less__diff,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( minus_minus @ nat @ N2 @ M ) )
      = ( ord_less @ nat @ M @ N2 ) ) ).

% zero_less_diff
thf(fact_771_mult__eq__1__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( ( times_times @ nat @ M @ N2 )
        = ( suc @ ( zero_zero @ nat ) ) )
      = ( ( M
          = ( suc @ ( zero_zero @ nat ) ) )
        & ( N2
          = ( suc @ ( zero_zero @ nat ) ) ) ) ) ).

% mult_eq_1_iff
thf(fact_772_one__eq__mult__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( ( suc @ ( zero_zero @ nat ) )
        = ( times_times @ nat @ M @ N2 ) )
      = ( ( M
          = ( suc @ ( zero_zero @ nat ) ) )
        & ( N2
          = ( suc @ ( zero_zero @ nat ) ) ) ) ) ).

% one_eq_mult_iff
thf(fact_773_div__less,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less @ nat @ M @ N2 )
     => ( ( divide_divide @ nat @ M @ N2 )
        = ( zero_zero @ nat ) ) ) ).

% div_less
thf(fact_774_nat__power__eq__Suc__0__iff,axiom,
    ! [X2: nat,M: nat] :
      ( ( ( power_power @ nat @ X2 @ M )
        = ( suc @ ( zero_zero @ nat ) ) )
      = ( ( M
          = ( zero_zero @ nat ) )
        | ( X2
          = ( suc @ ( zero_zero @ nat ) ) ) ) ) ).

% nat_power_eq_Suc_0_iff
thf(fact_775_power__Suc__0,axiom,
    ! [N2: nat] :
      ( ( power_power @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N2 )
      = ( suc @ ( zero_zero @ nat ) ) ) ).

% power_Suc_0
thf(fact_776_mult__less__cancel2,axiom,
    ! [M: nat,K: nat,N2: nat] :
      ( ( ord_less @ nat @ ( times_times @ nat @ M @ K ) @ ( times_times @ nat @ N2 @ K ) )
      = ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
        & ( ord_less @ nat @ M @ N2 ) ) ) ).

% mult_less_cancel2
thf(fact_777_nat__0__less__mult__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( times_times @ nat @ M @ N2 ) )
      = ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
        & ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ).

% nat_0_less_mult_iff
thf(fact_778_nat__mult__less__cancel__disj,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( ord_less @ nat @ ( times_times @ nat @ K @ M ) @ ( times_times @ nat @ K @ N2 ) )
      = ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
        & ( ord_less @ nat @ M @ N2 ) ) ) ).

% nat_mult_less_cancel_disj
thf(fact_779_diff__is__0__eq,axiom,
    ! [M: nat,N2: nat] :
      ( ( ( minus_minus @ nat @ M @ N2 )
        = ( zero_zero @ nat ) )
      = ( ord_less_eq @ nat @ M @ N2 ) ) ).

% diff_is_0_eq
thf(fact_780_diff__is__0__eq_H,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq @ nat @ M @ N2 )
     => ( ( minus_minus @ nat @ M @ N2 )
        = ( zero_zero @ nat ) ) ) ).

% diff_is_0_eq'
thf(fact_781_nat__zero__less__power__iff,axiom,
    ! [X2: nat,N2: nat] :
      ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( power_power @ nat @ X2 @ N2 ) )
      = ( ( ord_less @ nat @ ( zero_zero @ nat ) @ X2 )
        | ( N2
          = ( zero_zero @ nat ) ) ) ) ).

% nat_zero_less_power_iff
thf(fact_782_nat__mult__div__cancel__disj,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( ( K
          = ( zero_zero @ nat ) )
       => ( ( divide_divide @ nat @ ( times_times @ nat @ K @ M ) @ ( times_times @ nat @ K @ N2 ) )
          = ( zero_zero @ nat ) ) )
      & ( ( K
         != ( zero_zero @ nat ) )
       => ( ( divide_divide @ nat @ ( times_times @ nat @ K @ M ) @ ( times_times @ nat @ K @ N2 ) )
          = ( divide_divide @ nat @ M @ N2 ) ) ) ) ).

% nat_mult_div_cancel_disj
thf(fact_783_vebt__buildupi__rule,axiom,
    ! [N2: nat] : ( time_htt @ vEBT_VEBTi @ ( pure_assn @ ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) ) @ ( vEBT_vebt_buildupi @ N2 ) @ ( vEBT_vebt_assn_raw @ ( vEBT_vebt_buildup @ N2 ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit1 @ ( bit0 @ one2 ) ) ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ) ).

% vebt_buildupi_rule
thf(fact_784_eq__divide__eq__numeral1_I1_J,axiom,
    ! [A: $tType] :
      ( ( division_ring @ A )
     => ! [A2: A,B2: A,W: num] :
          ( ( A2
            = ( divide_divide @ A @ B2 @ ( numeral_numeral @ A @ W ) ) )
          = ( ( ( ( numeral_numeral @ A @ W )
               != ( zero_zero @ A ) )
             => ( ( times_times @ A @ A2 @ ( numeral_numeral @ A @ W ) )
                = B2 ) )
            & ( ( ( numeral_numeral @ A @ W )
                = ( zero_zero @ A ) )
             => ( A2
                = ( zero_zero @ A ) ) ) ) ) ) ).

% eq_divide_eq_numeral1(1)
thf(fact_785_divide__eq__eq__numeral1_I1_J,axiom,
    ! [A: $tType] :
      ( ( division_ring @ A )
     => ! [B2: A,W: num,A2: A] :
          ( ( ( divide_divide @ A @ B2 @ ( numeral_numeral @ A @ W ) )
            = A2 )
          = ( ( ( ( numeral_numeral @ A @ W )
               != ( zero_zero @ A ) )
             => ( B2
                = ( times_times @ A @ A2 @ ( numeral_numeral @ A @ W ) ) ) )
            & ( ( ( numeral_numeral @ A @ W )
                = ( zero_zero @ A ) )
             => ( A2
                = ( zero_zero @ A ) ) ) ) ) ) ).

% divide_eq_eq_numeral1(1)
thf(fact_786_div__mult__self4,axiom,
    ! [A: $tType] :
      ( ( euclid4440199948858584721cancel @ A )
     => ! [B2: A,C2: A,A2: A] :
          ( ( B2
           != ( zero_zero @ A ) )
         => ( ( divide_divide @ A @ ( plus_plus @ A @ ( times_times @ A @ B2 @ C2 ) @ A2 ) @ B2 )
            = ( plus_plus @ A @ C2 @ ( divide_divide @ A @ A2 @ B2 ) ) ) ) ) ).

% div_mult_self4
thf(fact_787_div__mult__self3,axiom,
    ! [A: $tType] :
      ( ( euclid4440199948858584721cancel @ A )
     => ! [B2: A,C2: A,A2: A] :
          ( ( B2
           != ( zero_zero @ A ) )
         => ( ( divide_divide @ A @ ( plus_plus @ A @ ( times_times @ A @ C2 @ B2 ) @ A2 ) @ B2 )
            = ( plus_plus @ A @ C2 @ ( divide_divide @ A @ A2 @ B2 ) ) ) ) ) ).

% div_mult_self3
thf(fact_788_div__mult__self2,axiom,
    ! [A: $tType] :
      ( ( euclid4440199948858584721cancel @ A )
     => ! [B2: A,A2: A,C2: A] :
          ( ( B2
           != ( zero_zero @ A ) )
         => ( ( divide_divide @ A @ ( plus_plus @ A @ A2 @ ( times_times @ A @ B2 @ C2 ) ) @ B2 )
            = ( plus_plus @ A @ C2 @ ( divide_divide @ A @ A2 @ B2 ) ) ) ) ) ).

% div_mult_self2
thf(fact_789_div__mult__self1,axiom,
    ! [A: $tType] :
      ( ( euclid4440199948858584721cancel @ A )
     => ! [B2: A,A2: A,C2: A] :
          ( ( B2
           != ( zero_zero @ A ) )
         => ( ( divide_divide @ A @ ( plus_plus @ A @ A2 @ ( times_times @ A @ C2 @ B2 ) ) @ B2 )
            = ( plus_plus @ A @ C2 @ ( divide_divide @ A @ A2 @ B2 ) ) ) ) ) ).

% div_mult_self1
thf(fact_790_of__nat__le__0__iff,axiom,
    ! [A: $tType] :
      ( ( linord181362715937106298miring @ A )
     => ! [M: nat] :
          ( ( ord_less_eq @ A @ ( semiring_1_of_nat @ A @ M ) @ ( zero_zero @ A ) )
          = ( M
            = ( zero_zero @ nat ) ) ) ) ).

% of_nat_le_0_iff
thf(fact_791_power__eq__0__iff,axiom,
    ! [A: $tType] :
      ( ( semiri2026040879449505780visors @ A )
     => ! [A2: A,N2: nat] :
          ( ( ( power_power @ A @ A2 @ N2 )
            = ( zero_zero @ A ) )
          = ( ( A2
              = ( zero_zero @ A ) )
            & ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ) ).

% power_eq_0_iff
thf(fact_792_Suc__pred,axiom,
    ! [N2: nat] :
      ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
     => ( ( suc @ ( minus_minus @ nat @ N2 @ ( suc @ ( zero_zero @ nat ) ) ) )
        = N2 ) ) ).

% Suc_pred
thf(fact_793_one__le__mult__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( times_times @ nat @ M @ N2 ) )
      = ( ( ord_less_eq @ nat @ ( suc @ ( zero_zero @ nat ) ) @ M )
        & ( ord_less_eq @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N2 ) ) ) ).

% one_le_mult_iff
thf(fact_794_div__eq__dividend__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
     => ( ( ( divide_divide @ nat @ M @ N2 )
          = M )
        = ( N2
          = ( one_one @ nat ) ) ) ) ).

% div_eq_dividend_iff
thf(fact_795_nat__mult__le__cancel__disj,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( ord_less_eq @ nat @ ( times_times @ nat @ K @ M ) @ ( times_times @ nat @ K @ N2 ) )
      = ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
       => ( ord_less_eq @ nat @ M @ N2 ) ) ) ).

% nat_mult_le_cancel_disj
thf(fact_796_mult__le__cancel2,axiom,
    ! [M: nat,K: nat,N2: nat] :
      ( ( ord_less_eq @ nat @ ( times_times @ nat @ M @ K ) @ ( times_times @ nat @ N2 @ K ) )
      = ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
       => ( ord_less_eq @ nat @ M @ N2 ) ) ) ).

% mult_le_cancel2
thf(fact_797_div__mult__self1__is__m,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
     => ( ( divide_divide @ nat @ ( times_times @ nat @ N2 @ M ) @ N2 )
        = M ) ) ).

% div_mult_self1_is_m
thf(fact_798_div__mult__self__is__m,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
     => ( ( divide_divide @ nat @ ( times_times @ nat @ M @ N2 ) @ N2 )
        = M ) ) ).

% div_mult_self_is_m
thf(fact_799_zero__eq__power2,axiom,
    ! [A: $tType] :
      ( ( semiri2026040879449505780visors @ A )
     => ! [A2: A] :
          ( ( ( power_power @ A @ A2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
            = ( zero_zero @ A ) )
          = ( A2
            = ( zero_zero @ A ) ) ) ) ).

% zero_eq_power2
thf(fact_800_power__strict__decreasing__iff,axiom,
    ! [A: $tType] :
      ( ( linordered_semidom @ A )
     => ! [B2: A,M: nat,N2: nat] :
          ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
         => ( ( ord_less @ A @ B2 @ ( one_one @ A ) )
           => ( ( ord_less @ A @ ( power_power @ A @ B2 @ M ) @ ( power_power @ A @ B2 @ N2 ) )
              = ( ord_less @ nat @ N2 @ M ) ) ) ) ) ).

% power_strict_decreasing_iff
thf(fact_801_power__mono__iff,axiom,
    ! [A: $tType] :
      ( ( linordered_semidom @ A )
     => ! [A2: A,B2: A,N2: nat] :
          ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
         => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
           => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
             => ( ( ord_less_eq @ A @ ( power_power @ A @ A2 @ N2 ) @ ( power_power @ A @ B2 @ N2 ) )
                = ( ord_less_eq @ A @ A2 @ B2 ) ) ) ) ) ) ).

% power_mono_iff
thf(fact_802_of__nat__0__less__iff,axiom,
    ! [A: $tType] :
      ( ( linord181362715937106298miring @ A )
     => ! [N2: nat] :
          ( ( ord_less @ A @ ( zero_zero @ A ) @ ( semiring_1_of_nat @ A @ N2 ) )
          = ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ).

% of_nat_0_less_iff
thf(fact_803_Suc__diff__1,axiom,
    ! [N2: nat] :
      ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
     => ( ( suc @ ( minus_minus @ nat @ N2 @ ( one_one @ nat ) ) )
        = N2 ) ) ).

% Suc_diff_1
thf(fact_804_bits__1__div__2,axiom,
    ! [A: $tType] :
      ( ( bit_semiring_bits @ A )
     => ( ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
        = ( zero_zero @ A ) ) ) ).

% bits_1_div_2
thf(fact_805_one__div__two__eq__zero,axiom,
    ! [A: $tType] :
      ( ( euclid5411537665997757685th_nat @ A )
     => ( ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
        = ( zero_zero @ A ) ) ) ).

% one_div_two_eq_zero
thf(fact_806_power2__eq__iff__nonneg,axiom,
    ! [A: $tType] :
      ( ( linordered_semidom @ A )
     => ! [X2: A,Y2: A] :
          ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X2 )
         => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y2 )
           => ( ( ( power_power @ A @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
                = ( power_power @ A @ Y2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
              = ( X2 = Y2 ) ) ) ) ) ).

% power2_eq_iff_nonneg
thf(fact_807_power2__less__eq__zero__iff,axiom,
    ! [A: $tType] :
      ( ( linordered_idom @ A )
     => ! [A2: A] :
          ( ( ord_less_eq @ A @ ( power_power @ A @ A2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( zero_zero @ A ) )
          = ( A2
            = ( zero_zero @ A ) ) ) ) ).

% power2_less_eq_zero_iff
thf(fact_808_zero__less__power2,axiom,
    ! [A: $tType] :
      ( ( linordered_idom @ A )
     => ! [A2: A] :
          ( ( ord_less @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
          = ( A2
           != ( zero_zero @ A ) ) ) ) ).

% zero_less_power2
thf(fact_809_sum__power2__eq__zero__iff,axiom,
    ! [A: $tType] :
      ( ( linordered_idom @ A )
     => ! [X2: A,Y2: A] :
          ( ( ( plus_plus @ A @ ( power_power @ A @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
            = ( zero_zero @ A ) )
          = ( ( X2
              = ( zero_zero @ A ) )
            & ( Y2
              = ( zero_zero @ A ) ) ) ) ) ).

% sum_power2_eq_zero_iff
thf(fact_810_power__decreasing__iff,axiom,
    ! [A: $tType] :
      ( ( linordered_semidom @ A )
     => ! [B2: A,M: nat,N2: nat] :
          ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
         => ( ( ord_less @ A @ B2 @ ( one_one @ A ) )
           => ( ( ord_less_eq @ A @ ( power_power @ A @ B2 @ M ) @ ( power_power @ A @ B2 @ N2 ) )
              = ( ord_less_eq @ nat @ N2 @ M ) ) ) ) ) ).

% power_decreasing_iff
thf(fact_811_of__nat__zero__less__power__iff,axiom,
    ! [A: $tType] :
      ( ( linordered_semidom @ A )
     => ! [X2: nat,N2: nat] :
          ( ( ord_less @ A @ ( zero_zero @ A ) @ ( power_power @ A @ ( semiring_1_of_nat @ A @ X2 ) @ N2 ) )
          = ( ( ord_less @ nat @ ( zero_zero @ nat ) @ X2 )
            | ( N2
              = ( zero_zero @ nat ) ) ) ) ) ).

% of_nat_zero_less_power_iff
thf(fact_812_htt__vebt__inserti,axiom,
    ! [T2: vEBT_VEBT,Ti: vEBT_VEBTi,X2: nat] : ( time_htt @ vEBT_VEBTi @ ( vEBT_vebt_assn_raw @ T2 @ Ti ) @ ( vEBT_vebt_inserti @ Ti @ X2 ) @ ( vEBT_vebt_assn_raw @ ( vEBT_vebt_insert @ T2 @ X2 ) ) @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit0 @ ( bit1 @ one2 ) ) ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit0 @ ( bit1 @ one2 ) ) ) ) @ ( vEBT_VEBT_height @ T2 ) ) ) ) ).

% htt_vebt_inserti
thf(fact_813_option_Osize__neq,axiom,
    ! [A: $tType,X2: option @ A] :
      ( ( size_size @ ( option @ A ) @ X2 )
     != ( zero_zero @ nat ) ) ).

% option.size_neq
thf(fact_814_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_815_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_816_power__0__left,axiom,
    ! [A: $tType] :
      ( ( semiring_1 @ A )
     => ! [N2: nat] :
          ( ( ( N2
              = ( zero_zero @ nat ) )
           => ( ( power_power @ A @ ( zero_zero @ A ) @ N2 )
              = ( one_one @ A ) ) )
          & ( ( N2
             != ( zero_zero @ nat ) )
           => ( ( power_power @ A @ ( zero_zero @ A ) @ N2 )
              = ( zero_zero @ A ) ) ) ) ) ).

% power_0_left
thf(fact_817_zero__power,axiom,
    ! [A: $tType] :
      ( ( semiring_1 @ A )
     => ! [N2: nat] :
          ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
         => ( ( power_power @ A @ ( zero_zero @ A ) @ N2 )
            = ( zero_zero @ A ) ) ) ) ).

% zero_power
thf(fact_818_le__numeral__extra_I3_J,axiom,
    ! [A: $tType] :
      ( ( linord181362715937106298miring @ A )
     => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( zero_zero @ A ) ) ) ).

% le_numeral_extra(3)
thf(fact_819_field__lbound__gt__zero,axiom,
    ! [A: $tType] :
      ( ( linordered_field @ A )
     => ! [D1: A,D22: A] :
          ( ( ord_less @ A @ ( zero_zero @ A ) @ D1 )
         => ( ( ord_less @ A @ ( zero_zero @ A ) @ D22 )
           => ? [E2: A] :
                ( ( ord_less @ A @ ( zero_zero @ A ) @ E2 )
                & ( ord_less @ A @ E2 @ D1 )
                & ( ord_less @ A @ E2 @ D22 ) ) ) ) ) ).

% field_lbound_gt_zero
thf(fact_820_less__numeral__extra_I3_J,axiom,
    ! [A: $tType] :
      ( ( linord181362715937106298miring @ A )
     => ~ ( ord_less @ A @ ( zero_zero @ A ) @ ( zero_zero @ A ) ) ) ).

% less_numeral_extra(3)
thf(fact_821_zero__neq__numeral,axiom,
    ! [A: $tType] :
      ( ( semiring_char_0 @ A )
     => ! [N2: num] :
          ( ( zero_zero @ A )
         != ( numeral_numeral @ A @ N2 ) ) ) ).

% zero_neq_numeral
thf(fact_822_zero__neq__one,axiom,
    ! [A: $tType] :
      ( ( zero_neq_one @ A )
     => ( ( zero_zero @ A )
       != ( one_one @ A ) ) ) ).

% zero_neq_one
thf(fact_823_mult__right__cancel,axiom,
    ! [A: $tType] :
      ( ( semiri6575147826004484403cancel @ A )
     => ! [C2: A,A2: A,B2: A] :
          ( ( C2
           != ( zero_zero @ A ) )
         => ( ( ( times_times @ A @ A2 @ C2 )
              = ( times_times @ A @ B2 @ C2 ) )
            = ( A2 = B2 ) ) ) ) ).

% mult_right_cancel
thf(fact_824_mult__left__cancel,axiom,
    ! [A: $tType] :
      ( ( semiri6575147826004484403cancel @ A )
     => ! [C2: A,A2: A,B2: A] :
          ( ( C2
           != ( zero_zero @ A ) )
         => ( ( ( times_times @ A @ C2 @ A2 )
              = ( times_times @ A @ C2 @ B2 ) )
            = ( A2 = B2 ) ) ) ) ).

% mult_left_cancel
thf(fact_825_no__zero__divisors,axiom,
    ! [A: $tType] :
      ( ( semiri3467727345109120633visors @ A )
     => ! [A2: A,B2: A] :
          ( ( A2
           != ( zero_zero @ A ) )
         => ( ( B2
             != ( zero_zero @ A ) )
           => ( ( times_times @ A @ A2 @ B2 )
             != ( zero_zero @ A ) ) ) ) ) ).

% no_zero_divisors
thf(fact_826_divisors__zero,axiom,
    ! [A: $tType] :
      ( ( semiri3467727345109120633visors @ A )
     => ! [A2: A,B2: A] :
          ( ( ( times_times @ A @ A2 @ B2 )
            = ( zero_zero @ A ) )
         => ( ( A2
              = ( zero_zero @ A ) )
            | ( B2
              = ( zero_zero @ A ) ) ) ) ) ).

% divisors_zero
thf(fact_827_mult__not__zero,axiom,
    ! [A: $tType] :
      ( ( mult_zero @ A )
     => ! [A2: A,B2: A] :
          ( ( ( times_times @ A @ A2 @ B2 )
           != ( zero_zero @ A ) )
         => ( ( A2
             != ( zero_zero @ A ) )
            & ( B2
             != ( zero_zero @ A ) ) ) ) ) ).

% mult_not_zero
thf(fact_828_semiring__1__no__zero__divisors__class_Opower__not__zero,axiom,
    ! [A: $tType] :
      ( ( semiri2026040879449505780visors @ A )
     => ! [A2: A,N2: nat] :
          ( ( A2
           != ( zero_zero @ A ) )
         => ( ( power_power @ A @ A2 @ N2 )
           != ( zero_zero @ A ) ) ) ) ).

% semiring_1_no_zero_divisors_class.power_not_zero
thf(fact_829_num_Osize_I4_J,axiom,
    ( ( size_size @ num @ one2 )
    = ( zero_zero @ nat ) ) ).

% num.size(4)
thf(fact_830_VEBT__internal_Oreplicatei_Ocases,axiom,
    ! [A: $tType,X2: product_prod @ nat @ ( heap_Time_Heap @ A )] :
      ( ! [X3: heap_Time_Heap @ A] :
          ( X2
         != ( product_Pair @ nat @ ( heap_Time_Heap @ A ) @ ( zero_zero @ nat ) @ X3 ) )
     => ~ ! [N4: nat,X3: heap_Time_Heap @ A] :
            ( X2
           != ( product_Pair @ nat @ ( heap_Time_Heap @ A ) @ ( suc @ N4 ) @ X3 ) ) ) ).

% VEBT_internal.replicatei.cases
thf(fact_831_not0__implies__Suc,axiom,
    ! [N2: nat] :
      ( ( N2
       != ( zero_zero @ nat ) )
     => ? [M2: nat] :
          ( N2
          = ( suc @ M2 ) ) ) ).

% not0_implies_Suc
thf(fact_832_Zero__not__Suc,axiom,
    ! [M: nat] :
      ( ( zero_zero @ nat )
     != ( suc @ M ) ) ).

% Zero_not_Suc
thf(fact_833_Zero__neq__Suc,axiom,
    ! [M: nat] :
      ( ( zero_zero @ nat )
     != ( suc @ M ) ) ).

% Zero_neq_Suc
thf(fact_834_Suc__neq__Zero,axiom,
    ! [M: nat] :
      ( ( suc @ M )
     != ( zero_zero @ nat ) ) ).

% Suc_neq_Zero
thf(fact_835_zero__induct,axiom,
    ! [P: nat > $o,K: nat] :
      ( ( P @ K )
     => ( ! [N4: nat] :
            ( ( P @ ( suc @ N4 ) )
           => ( P @ N4 ) )
       => ( P @ ( zero_zero @ nat ) ) ) ) ).

% zero_induct
thf(fact_836_diff__induct,axiom,
    ! [P: nat > nat > $o,M: nat,N2: nat] :
      ( ! [X3: nat] : ( P @ X3 @ ( zero_zero @ nat ) )
     => ( ! [Y4: nat] : ( P @ ( zero_zero @ nat ) @ ( suc @ Y4 ) )
       => ( ! [X3: nat,Y4: nat] :
              ( ( P @ X3 @ Y4 )
             => ( P @ ( suc @ X3 ) @ ( suc @ Y4 ) ) )
         => ( P @ M @ N2 ) ) ) ) ).

% diff_induct
thf(fact_837_nat__induct,axiom,
    ! [P: nat > $o,N2: nat] :
      ( ( P @ ( zero_zero @ nat ) )
     => ( ! [N4: nat] :
            ( ( P @ N4 )
           => ( P @ ( suc @ N4 ) ) )
       => ( P @ N2 ) ) ) ).

% nat_induct
thf(fact_838_old_Onat_Oexhaust,axiom,
    ! [Y2: nat] :
      ( ( Y2
       != ( zero_zero @ nat ) )
     => ~ ! [Nat3: nat] :
            ( Y2
           != ( suc @ Nat3 ) ) ) ).

% old.nat.exhaust
thf(fact_839_nat_OdiscI,axiom,
    ! [Nat: nat,X23: nat] :
      ( ( Nat
        = ( suc @ X23 ) )
     => ( Nat
       != ( zero_zero @ nat ) ) ) ).

% nat.discI
thf(fact_840_old_Onat_Odistinct_I1_J,axiom,
    ! [Nat2: nat] :
      ( ( zero_zero @ nat )
     != ( suc @ Nat2 ) ) ).

% old.nat.distinct(1)
thf(fact_841_old_Onat_Odistinct_I2_J,axiom,
    ! [Nat4: nat] :
      ( ( suc @ Nat4 )
     != ( zero_zero @ nat ) ) ).

% old.nat.distinct(2)
thf(fact_842_nat_Odistinct_I1_J,axiom,
    ! [X23: nat] :
      ( ( zero_zero @ nat )
     != ( suc @ X23 ) ) ).

% nat.distinct(1)
thf(fact_843_infinite__descent0__measure,axiom,
    ! [A: $tType,V2: A > nat,P: A > $o,X2: A] :
      ( ! [X3: A] :
          ( ( ( V2 @ X3 )
            = ( zero_zero @ nat ) )
         => ( P @ X3 ) )
     => ( ! [X3: A] :
            ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( V2 @ X3 ) )
           => ( ~ ( P @ X3 )
             => ? [Y3: A] :
                  ( ( ord_less @ nat @ ( V2 @ Y3 ) @ ( V2 @ X3 ) )
                  & ~ ( P @ Y3 ) ) ) )
       => ( P @ X2 ) ) ) ).

% infinite_descent0_measure
thf(fact_844_infinite__descent0,axiom,
    ! [P: nat > $o,N2: nat] :
      ( ( P @ ( zero_zero @ nat ) )
     => ( ! [N4: nat] :
            ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N4 )
           => ( ~ ( P @ N4 )
             => ? [M3: nat] :
                  ( ( ord_less @ nat @ M3 @ N4 )
                  & ~ ( P @ M3 ) ) ) )
       => ( P @ N2 ) ) ) ).

% infinite_descent0
thf(fact_845_gr__implies__not0,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less @ nat @ M @ N2 )
     => ( N2
       != ( zero_zero @ nat ) ) ) ).

% gr_implies_not0
thf(fact_846_less__zeroE,axiom,
    ! [N2: nat] :
      ~ ( ord_less @ nat @ N2 @ ( zero_zero @ nat ) ) ).

% less_zeroE
thf(fact_847_not__less0,axiom,
    ! [N2: nat] :
      ~ ( ord_less @ nat @ N2 @ ( zero_zero @ nat ) ) ).

% not_less0
thf(fact_848_not__gr0,axiom,
    ! [N2: nat] :
      ( ( ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) )
      = ( N2
        = ( zero_zero @ nat ) ) ) ).

% not_gr0
thf(fact_849_gr0I,axiom,
    ! [N2: nat] :
      ( ( N2
       != ( zero_zero @ nat ) )
     => ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) ) ).

% gr0I
thf(fact_850_bot__nat__0_Oextremum__strict,axiom,
    ! [A2: nat] :
      ~ ( ord_less @ nat @ A2 @ ( zero_zero @ nat ) ) ).

% bot_nat_0.extremum_strict
thf(fact_851_le__0__eq,axiom,
    ! [N2: nat] :
      ( ( ord_less_eq @ nat @ N2 @ ( zero_zero @ nat ) )
      = ( N2
        = ( zero_zero @ nat ) ) ) ).

% le_0_eq
thf(fact_852_bot__nat__0_Oextremum__uniqueI,axiom,
    ! [A2: nat] :
      ( ( ord_less_eq @ nat @ A2 @ ( zero_zero @ nat ) )
     => ( A2
        = ( zero_zero @ nat ) ) ) ).

% bot_nat_0.extremum_uniqueI
thf(fact_853_bot__nat__0_Oextremum__unique,axiom,
    ! [A2: nat] :
      ( ( ord_less_eq @ nat @ A2 @ ( zero_zero @ nat ) )
      = ( A2
        = ( zero_zero @ nat ) ) ) ).

% bot_nat_0.extremum_unique
thf(fact_854_less__eq__nat_Osimps_I1_J,axiom,
    ! [N2: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ N2 ) ).

% less_eq_nat.simps(1)
thf(fact_855_plus__nat_Oadd__0,axiom,
    ! [N2: nat] :
      ( ( plus_plus @ nat @ ( zero_zero @ nat ) @ N2 )
      = N2 ) ).

% plus_nat.add_0
thf(fact_856_add__eq__self__zero,axiom,
    ! [M: nat,N2: nat] :
      ( ( ( plus_plus @ nat @ M @ N2 )
        = M )
     => ( N2
        = ( zero_zero @ nat ) ) ) ).

% add_eq_self_zero
thf(fact_857_minus__nat_Odiff__0,axiom,
    ! [M: nat] :
      ( ( minus_minus @ nat @ M @ ( zero_zero @ nat ) )
      = M ) ).

% minus_nat.diff_0
thf(fact_858_diffs0__imp__equal,axiom,
    ! [M: nat,N2: nat] :
      ( ( ( minus_minus @ nat @ M @ N2 )
        = ( zero_zero @ nat ) )
     => ( ( ( minus_minus @ nat @ N2 @ M )
          = ( zero_zero @ nat ) )
       => ( M = N2 ) ) ) ).

% diffs0_imp_equal
thf(fact_859_nat__mult__eq__cancel__disj,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( ( times_times @ nat @ K @ M )
        = ( times_times @ nat @ K @ N2 ) )
      = ( ( K
          = ( zero_zero @ nat ) )
        | ( M = N2 ) ) ) ).

% nat_mult_eq_cancel_disj
thf(fact_860_mult__0,axiom,
    ! [N2: nat] :
      ( ( times_times @ nat @ ( zero_zero @ nat ) @ N2 )
      = ( zero_zero @ nat ) ) ).

% mult_0
thf(fact_861_VEBTi_Osize_I4_J,axiom,
    ! [X212: $o,X222: $o] :
      ( ( size_size @ vEBT_VEBTi @ ( vEBT_Leafi @ X212 @ X222 ) )
      = ( zero_zero @ nat ) ) ).

% VEBTi.size(4)
thf(fact_862_power__eq__imp__eq__base,axiom,
    ! [A: $tType] :
      ( ( linordered_semidom @ A )
     => ! [A2: A,N2: nat,B2: A] :
          ( ( ( power_power @ A @ A2 @ N2 )
            = ( power_power @ A @ B2 @ N2 ) )
         => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
           => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
             => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
               => ( A2 = B2 ) ) ) ) ) ) ).

% power_eq_imp_eq_base
thf(fact_863_power__eq__iff__eq__base,axiom,
    ! [A: $tType] :
      ( ( linordered_semidom @ A )
     => ! [N2: nat,A2: A,B2: A] :
          ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
         => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
           => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
             => ( ( ( power_power @ A @ A2 @ N2 )
                  = ( power_power @ A @ B2 @ N2 ) )
                = ( A2 = B2 ) ) ) ) ) ) ).

% power_eq_iff_eq_base
thf(fact_864_lambda__zero,axiom,
    ! [A: $tType] :
      ( ( mult_zero @ A )
     => ( ( ^ [H: A] : ( zero_zero @ A ) )
        = ( times_times @ A @ ( zero_zero @ A ) ) ) ) ).

% lambda_zero
thf(fact_865_VEBT__internal_OTb_H_Osimps_I1_J,axiom,
    ( ( vEBT_VEBT_Tb2 @ ( zero_zero @ nat ) )
    = ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) ) ).

% VEBT_internal.Tb'.simps(1)
thf(fact_866_VEBT__internal_OminNulli_Osimps_I1_J,axiom,
    ( ( vEBT_VEBT_minNulli @ ( vEBT_Leafi @ $false @ $false ) )
    = ( heap_Time_return @ $o @ $true ) ) ).

% VEBT_internal.minNulli.simps(1)
thf(fact_867_VEBT__internal_OminNulli_Osimps_I2_J,axiom,
    ! [Uv: $o] :
      ( ( vEBT_VEBT_minNulli @ ( vEBT_Leafi @ $true @ Uv ) )
      = ( heap_Time_return @ $o @ $false ) ) ).

% VEBT_internal.minNulli.simps(2)
thf(fact_868_VEBT__internal_OminNulli_Osimps_I3_J,axiom,
    ! [Uu: $o] :
      ( ( vEBT_VEBT_minNulli @ ( vEBT_Leafi @ Uu @ $true ) )
      = ( heap_Time_return @ $o @ $false ) ) ).

% VEBT_internal.minNulli.simps(3)
thf(fact_869_power__strict__mono,axiom,
    ! [A: $tType] :
      ( ( linordered_semidom @ A )
     => ! [A2: A,B2: A,N2: nat] :
          ( ( ord_less @ A @ A2 @ B2 )
         => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
           => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
             => ( ord_less @ A @ ( power_power @ A @ A2 @ N2 ) @ ( power_power @ A @ B2 @ N2 ) ) ) ) ) ) ).

% power_strict_mono
thf(fact_870_not__numeral__le__zero,axiom,
    ! [A: $tType] :
      ( ( linord181362715937106298miring @ A )
     => ! [N2: num] :
          ~ ( ord_less_eq @ A @ ( numeral_numeral @ A @ N2 ) @ ( zero_zero @ A ) ) ) ).

% not_numeral_le_zero
thf(fact_871_zero__le__numeral,axiom,
    ! [A: $tType] :
      ( ( linord181362715937106298miring @ A )
     => ! [N2: num] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ N2 ) ) ) ).

% zero_le_numeral
thf(fact_872_VEBT__internal_OminNull_Osimps_I5_J,axiom,
    ! [Uz: product_prod @ nat @ nat,Va: nat,Vb: list @ vEBT_VEBT,Vc2: vEBT_VEBT] :
      ~ ( vEBT_VEBT_minNull @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ Uz ) @ Va @ Vb @ Vc2 ) ) ).

% VEBT_internal.minNull.simps(5)
thf(fact_873_not__numeral__less__zero,axiom,
    ! [A: $tType] :
      ( ( linord181362715937106298miring @ A )
     => ! [N2: num] :
          ~ ( ord_less @ A @ ( numeral_numeral @ A @ N2 ) @ ( zero_zero @ A ) ) ) ).

% not_numeral_less_zero
thf(fact_874_zero__less__numeral,axiom,
    ! [A: $tType] :
      ( ( linord181362715937106298miring @ A )
     => ! [N2: num] : ( ord_less @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ N2 ) ) ) ).

% zero_less_numeral
thf(fact_875_not__one__le__zero,axiom,
    ! [A: $tType] :
      ( ( linord181362715937106298miring @ A )
     => ~ ( ord_less_eq @ A @ ( one_one @ A ) @ ( zero_zero @ A ) ) ) ).

% not_one_le_zero
thf(fact_876_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
    ! [A: $tType] :
      ( ( linord181362715937106298miring @ A )
     => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( one_one @ A ) ) ) ).

% linordered_nonzero_semiring_class.zero_le_one
thf(fact_877_zero__less__one__class_Ozero__le__one,axiom,
    ! [A: $tType] :
      ( ( zero_less_one @ A )
     => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( one_one @ A ) ) ) ).

% zero_less_one_class.zero_le_one
thf(fact_878_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: $tType] :
      ( ( ordere2520102378445227354miring @ A )
     => ! [A2: A,B2: A,C2: A] :
          ( ( ord_less_eq @ A @ A2 @ B2 )
         => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
           => ( ord_less_eq @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B2 ) ) ) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_879_zero__le__mult__iff,axiom,
    ! [A: $tType] :
      ( ( linord4710134922213307826strict @ A )
     => ! [A2: A,B2: A] :
          ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A2 @ B2 ) )
          = ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
              & ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) )
            | ( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) )
              & ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) ) ) ) ) ) ).

% zero_le_mult_iff
thf(fact_880_mult__nonneg__nonpos2,axiom,
    ! [A: $tType] :
      ( ( ordered_semiring_0 @ A )
     => ! [A2: A,B2: A] :
          ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
         => ( ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) )
           => ( ord_less_eq @ A @ ( times_times @ A @ B2 @ A2 ) @ ( zero_zero @ A ) ) ) ) ) ).

% mult_nonneg_nonpos2
thf(fact_881_mult__nonpos__nonneg,axiom,
    ! [A: $tType] :
      ( ( ordered_semiring_0 @ A )
     => ! [A2: A,B2: A] :
          ( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) )
         => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
           => ( ord_less_eq @ A @ ( times_times @ A @ A2 @ B2 ) @ ( zero_zero @ A ) ) ) ) ) ).

% mult_nonpos_nonneg
thf(fact_882_mult__nonneg__nonpos,axiom,
    ! [A: $tType] :
      ( ( ordered_semiring_0 @ A )
     => ! [A2: A,B2: A] :
          ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
         => ( ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) )
           => ( ord_less_eq @ A @ ( times_times @ A @ A2 @ B2 ) @ ( zero_zero @ A ) ) ) ) ) ).

% mult_nonneg_nonpos
thf(fact_883_mult__nonneg__nonneg,axiom,
    ! [A: $tType] :
      ( ( ordered_semiring_0 @ A )
     => ! [A2: A,B2: A] :
          ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
         => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
           => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A2 @ B2 ) ) ) ) ) ).

% mult_nonneg_nonneg
thf(fact_884_split__mult__neg__le,axiom,
    ! [A: $tType] :
      ( ( ordered_semiring_0 @ A )
     => ! [A2: A,B2: A] :
          ( ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
              & ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) ) )
            | ( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) )
              & ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) ) )
         => ( ord_less_eq @ A @ ( times_times @ A @ A2 @ B2 ) @ ( zero_zero @ A ) ) ) ) ).

% split_mult_neg_le
thf(fact_885_mult__le__0__iff,axiom,
    ! [A: $tType] :
      ( ( linord4710134922213307826strict @ A )
     => ! [A2: A,B2: A] :
          ( ( ord_less_eq @ A @ ( times_times @ A @ A2 @ B2 ) @ ( zero_zero @ A ) )
          = ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
              & ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) ) )
            | ( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) )
              & ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) ) ) ) ) ).

% mult_le_0_iff
thf(fact_886_mult__right__mono,axiom,
    ! [A: $tType] :
      ( ( ordered_semiring @ A )
     => ! [A2: A,B2: A,C2: A] :
          ( ( ord_less_eq @ A @ A2 @ B2 )
         => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
           => ( ord_less_eq @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) ) ) ) ) ).

% mult_right_mono
thf(fact_887_mult__right__mono__neg,axiom,
    ! [A: $tType] :
      ( ( ordered_ring @ A )
     => ! [B2: A,A2: A,C2: A] :
          ( ( ord_less_eq @ A @ B2 @ A2 )
         => ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) )
           => ( ord_less_eq @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) ) ) ) ) ).

% mult_right_mono_neg
thf(fact_888_mult__left__mono,axiom,
    ! [A: $tType] :
      ( ( ordered_semiring @ A )
     => ! [A2: A,B2: A,C2: A] :
          ( ( ord_less_eq @ A @ A2 @ B2 )
         => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
           => ( ord_less_eq @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B2 ) ) ) ) ) ).

% mult_left_mono
thf(fact_889_mult__nonpos__nonpos,axiom,
    ! [A: $tType] :
      ( ( ordered_ring @ A )
     => ! [A2: A,B2: A] :
          ( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) )
         => ( ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) )
           => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A2 @ B2 ) ) ) ) ) ).

% mult_nonpos_nonpos
thf(fact_890_mult__left__mono__neg,axiom,
    ! [A: $tType] :
      ( ( ordered_ring @ A )
     => ! [B2: A,A2: A,C2: A] :
          ( ( ord_less_eq @ A @ B2 @ A2 )
         => ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) )
           => ( ord_less_eq @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B2 ) ) ) ) ) ).

% mult_left_mono_neg
thf(fact_891_split__mult__pos__le,axiom,
    ! [A: $tType] :
      ( ( ordered_ring @ A )
     => ! [A2: A,B2: A] :
          ( ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
              & ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) )
            | ( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) )
              & ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) ) ) )
         => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A2 @ B2 ) ) ) ) ).

% split_mult_pos_le
thf(fact_892_zero__le__square,axiom,
    ! [A: $tType] :
      ( ( linordered_ring @ A )
     => ! [A2: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A2 @ A2 ) ) ) ).

% zero_le_square
thf(fact_893_mult__mono_H,axiom,
    ! [A: $tType] :
      ( ( ordered_semiring @ A )
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( ( ord_less_eq @ A @ A2 @ B2 )
         => ( ( ord_less_eq @ A @ C2 @ D2 )
           => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
             => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
               => ( ord_less_eq @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ D2 ) ) ) ) ) ) ) ).

% mult_mono'
thf(fact_894_mult__mono,axiom,
    ! [A: $tType] :
      ( ( ordered_semiring @ A )
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( ( ord_less_eq @ A @ A2 @ B2 )
         => ( ( ord_less_eq @ A @ C2 @ D2 )
           => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
             => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
               => ( ord_less_eq @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ D2 ) ) ) ) ) ) ) ).

% mult_mono
thf(fact_895_add__less__zeroD,axiom,
    ! [A: $tType] :
      ( ( linordered_idom @ A )
     => ! [X2: A,Y2: A] :
          ( ( ord_less @ A @ ( plus_plus @ A @ X2 @ Y2 ) @ ( zero_zero @ A ) )
         => ( ( ord_less @ A @ X2 @ ( zero_zero @ A ) )
            | ( ord_less @ A @ Y2 @ ( zero_zero @ A ) ) ) ) ) ).

% add_less_zeroD
thf(fact_896_not__one__less__zero,axiom,
    ! [A: $tType] :
      ( ( linord181362715937106298miring @ A )
     => ~ ( ord_less @ A @ ( one_one @ A ) @ ( zero_zero @ A ) ) ) ).

% not_one_less_zero
thf(fact_897_zero__less__one,axiom,
    ! [A: $tType] :
      ( ( zero_less_one @ A )
     => ( ord_less @ A @ ( zero_zero @ A ) @ ( one_one @ A ) ) ) ).

% zero_less_one
thf(fact_898_less__numeral__extra_I1_J,axiom,
    ! [A: $tType] :
      ( ( linord181362715937106298miring @ A )
     => ( ord_less @ A @ ( zero_zero @ A ) @ ( one_one @ A ) ) ) ).

% less_numeral_extra(1)
thf(fact_899_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: $tType] :
      ( ( linord2810124833399127020strict @ A )
     => ! [A2: A,B2: A,C2: A] :
          ( ( ord_less @ A @ A2 @ B2 )
         => ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
           => ( ord_less @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B2 ) ) ) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_900_mult__less__cancel__right__disj,axiom,
    ! [A: $tType] :
      ( ( linord4710134922213307826strict @ A )
     => ! [A2: A,C2: A,B2: A] :
          ( ( ord_less @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) )
          = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
              & ( ord_less @ A @ A2 @ B2 ) )
            | ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
              & ( ord_less @ A @ B2 @ A2 ) ) ) ) ) ).

% mult_less_cancel_right_disj
thf(fact_901_mult__strict__right__mono,axiom,
    ! [A: $tType] :
      ( ( linord8928482502909563296strict @ A )
     => ! [A2: A,B2: A,C2: A] :
          ( ( ord_less @ A @ A2 @ B2 )
         => ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
           => ( ord_less @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) ) ) ) ) ).

% mult_strict_right_mono
thf(fact_902_mult__strict__right__mono__neg,axiom,
    ! [A: $tType] :
      ( ( linord4710134922213307826strict @ A )
     => ! [B2: A,A2: A,C2: A] :
          ( ( ord_less @ A @ B2 @ A2 )
         => ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
           => ( ord_less @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) ) ) ) ) ).

% mult_strict_right_mono_neg
thf(fact_903_mult__less__cancel__left__disj,axiom,
    ! [A: $tType] :
      ( ( linord4710134922213307826strict @ A )
     => ! [C2: A,A2: A,B2: A] :
          ( ( ord_less @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B2 ) )
          = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
              & ( ord_less @ A @ A2 @ B2 ) )
            | ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
              & ( ord_less @ A @ B2 @ A2 ) ) ) ) ) ).

% mult_less_cancel_left_disj
thf(fact_904_mult__strict__left__mono,axiom,
    ! [A: $tType] :
      ( ( linord8928482502909563296strict @ A )
     => ! [A2: A,B2: A,C2: A] :
          ( ( ord_less @ A @ A2 @ B2 )
         => ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
           => ( ord_less @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B2 ) ) ) ) ) ).

% mult_strict_left_mono
thf(fact_905_mult__strict__left__mono__neg,axiom,
    ! [A: $tType] :
      ( ( linord4710134922213307826strict @ A )
     => ! [B2: A,A2: A,C2: A] :
          ( ( ord_less @ A @ B2 @ A2 )
         => ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
           => ( ord_less @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B2 ) ) ) ) ) ).

% mult_strict_left_mono_neg
thf(fact_906_mult__less__cancel__left__pos,axiom,
    ! [A: $tType] :
      ( ( linord4710134922213307826strict @ A )
     => ! [C2: A,A2: A,B2: A] :
          ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
         => ( ( ord_less @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B2 ) )
            = ( ord_less @ A @ A2 @ B2 ) ) ) ) ).

% mult_less_cancel_left_pos
thf(fact_907_mult__less__cancel__left__neg,axiom,
    ! [A: $tType] :
      ( ( linord4710134922213307826strict @ A )
     => ! [C2: A,A2: A,B2: A] :
          ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
         => ( ( ord_less @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B2 ) )
            = ( ord_less @ A @ B2 @ A2 ) ) ) ) ).

% mult_less_cancel_left_neg
thf(fact_908_zero__less__mult__pos2,axiom,
    ! [A: $tType] :
      ( ( linord8928482502909563296strict @ A )
     => ! [B2: A,A2: A] :
          ( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ B2 @ A2 ) )
         => ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
           => ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) ) ) ) ).

% zero_less_mult_pos2
thf(fact_909_zero__less__mult__pos,axiom,
    ! [A: $tType] :
      ( ( linord8928482502909563296strict @ A )
     => ! [A2: A,B2: A] :
          ( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A2 @ B2 ) )
         => ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
           => ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) ) ) ) ).

% zero_less_mult_pos
thf(fact_910_zero__less__mult__iff,axiom,
    ! [A: $tType] :
      ( ( linord4710134922213307826strict @ A )
     => ! [A2: A,B2: A] :
          ( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A2 @ B2 ) )
          = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
              & ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) )
            | ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
              & ( ord_less @ A @ B2 @ ( zero_zero @ A ) ) ) ) ) ) ).

% zero_less_mult_iff
thf(fact_911_mult__pos__neg2,axiom,
    ! [A: $tType] :
      ( ( linord8928482502909563296strict @ A )
     => ! [A2: A,B2: A] :
          ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
         => ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
           => ( ord_less @ A @ ( times_times @ A @ B2 @ A2 ) @ ( zero_zero @ A ) ) ) ) ) ).

% mult_pos_neg2
thf(fact_912_mult__pos__pos,axiom,
    ! [A: $tType] :
      ( ( linord8928482502909563296strict @ A )
     => ! [A2: A,B2: A] :
          ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
         => ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
           => ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A2 @ B2 ) ) ) ) ) ).

% mult_pos_pos
thf(fact_913_mult__pos__neg,axiom,
    ! [A: $tType] :
      ( ( linord8928482502909563296strict @ A )
     => ! [A2: A,B2: A] :
          ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
         => ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
           => ( ord_less @ A @ ( times_times @ A @ A2 @ B2 ) @ ( zero_zero @ A ) ) ) ) ) ).

% mult_pos_neg
thf(fact_914_mult__neg__pos,axiom,
    ! [A: $tType] :
      ( ( linord8928482502909563296strict @ A )
     => ! [A2: A,B2: A] :
          ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
         => ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
           => ( ord_less @ A @ ( times_times @ A @ A2 @ B2 ) @ ( zero_zero @ A ) ) ) ) ) ).

% mult_neg_pos
thf(fact_915_mult__less__0__iff,axiom,
    ! [A: $tType] :
      ( ( linord4710134922213307826strict @ A )
     => ! [A2: A,B2: A] :
          ( ( ord_less @ A @ ( times_times @ A @ A2 @ B2 ) @ ( zero_zero @ A ) )
          = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
              & ( ord_less @ A @ B2 @ ( zero_zero @ A ) ) )
            | ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
              & ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) ) ) ) ) ).

% mult_less_0_iff
thf(fact_916_not__square__less__zero,axiom,
    ! [A: $tType] :
      ( ( linordered_ring @ A )
     => ! [A2: A] :
          ~ ( ord_less @ A @ ( times_times @ A @ A2 @ A2 ) @ ( zero_zero @ A ) ) ) ).

% not_square_less_zero
thf(fact_917_mult__neg__neg,axiom,
    ! [A: $tType] :
      ( ( linord4710134922213307826strict @ A )
     => ! [A2: A,B2: A] :
          ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
         => ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
           => ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A2 @ B2 ) ) ) ) ) ).

% mult_neg_neg
thf(fact_918_zero__le__power,axiom,
    ! [A: $tType] :
      ( ( linordered_semidom @ A )
     => ! [A2: A,N2: nat] :
          ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
         => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A2 @ N2 ) ) ) ) ).

% zero_le_power
thf(fact_919_power__mono,axiom,
    ! [A: $tType] :
      ( ( linordered_semidom @ A )
     => ! [A2: A,B2: A,N2: nat] :
          ( ( ord_less_eq @ A @ A2 @ B2 )
         => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
           => ( ord_less_eq @ A @ ( power_power @ A @ A2 @ N2 ) @ ( power_power @ A @ B2 @ N2 ) ) ) ) ) ).

% power_mono
thf(fact_920_zero__less__power,axiom,
    ! [A: $tType] :
      ( ( linordered_semidom @ A )
     => ! [A2: A,N2: nat] :
          ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
         => ( ord_less @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A2 @ N2 ) ) ) ) ).

% zero_less_power
thf(fact_921_VEBT__internal_OminNull_Osimps_I4_J,axiom,
    ! [Uw: nat,Ux: list @ vEBT_VEBT,Uy: vEBT_VEBT] : ( vEBT_VEBT_minNull @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uw @ Ux @ Uy ) ) ).

% VEBT_internal.minNull.simps(4)
thf(fact_922_of__nat__0__le__iff,axiom,
    ! [A: $tType] :
      ( ( linord181362715937106298miring @ A )
     => ! [N2: nat] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( semiring_1_of_nat @ A @ N2 ) ) ) ).

% of_nat_0_le_iff
thf(fact_923_of__nat__less__0__iff,axiom,
    ! [A: $tType] :
      ( ( linord181362715937106298miring @ A )
     => ! [M: nat] :
          ~ ( ord_less @ A @ ( semiring_1_of_nat @ A @ M ) @ ( zero_zero @ A ) ) ) ).

% of_nat_less_0_iff
thf(fact_924_semiring__char__0__class_Oof__nat__neq__0,axiom,
    ! [A: $tType] :
      ( ( semiring_char_0 @ A )
     => ! [N2: nat] :
          ( ( semiring_1_of_nat @ A @ ( suc @ N2 ) )
         != ( zero_zero @ A ) ) ) ).

% semiring_char_0_class.of_nat_neq_0
thf(fact_925_power__0,axiom,
    ! [A: $tType] :
      ( ( power @ A )
     => ! [A2: A] :
          ( ( power_power @ A @ A2 @ ( zero_zero @ nat ) )
          = ( one_one @ A ) ) ) ).

% power_0
thf(fact_926_less__Suc__eq__0__disj,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less @ nat @ M @ ( suc @ N2 ) )
      = ( ( M
          = ( zero_zero @ nat ) )
        | ? [J3: nat] :
            ( ( M
              = ( suc @ J3 ) )
            & ( ord_less @ nat @ J3 @ N2 ) ) ) ) ).

% less_Suc_eq_0_disj
thf(fact_927_gr0__implies__Suc,axiom,
    ! [N2: nat] :
      ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
     => ? [M2: nat] :
          ( N2
          = ( suc @ M2 ) ) ) ).

% gr0_implies_Suc
thf(fact_928_All__less__Suc2,axiom,
    ! [N2: nat,P: nat > $o] :
      ( ( ! [I2: nat] :
            ( ( ord_less @ nat @ I2 @ ( suc @ N2 ) )
           => ( P @ I2 ) ) )
      = ( ( P @ ( zero_zero @ nat ) )
        & ! [I2: nat] :
            ( ( ord_less @ nat @ I2 @ N2 )
           => ( P @ ( suc @ I2 ) ) ) ) ) ).

% All_less_Suc2
thf(fact_929_gr0__conv__Suc,axiom,
    ! [N2: nat] :
      ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
      = ( ? [M7: nat] :
            ( N2
            = ( suc @ M7 ) ) ) ) ).

% gr0_conv_Suc
thf(fact_930_Ex__less__Suc2,axiom,
    ! [N2: nat,P: nat > $o] :
      ( ( ? [I2: nat] :
            ( ( ord_less @ nat @ I2 @ ( suc @ N2 ) )
            & ( P @ I2 ) ) )
      = ( ( P @ ( zero_zero @ nat ) )
        | ? [I2: nat] :
            ( ( ord_less @ nat @ I2 @ N2 )
            & ( P @ ( suc @ I2 ) ) ) ) ) ).

% Ex_less_Suc2
thf(fact_931_nat__compl__induct_H,axiom,
    ! [P: nat > $o,N2: nat] :
      ( ( P @ ( zero_zero @ nat ) )
     => ( ! [N4: nat] :
            ( ! [Nn: nat] :
                ( ( ord_less_eq @ nat @ Nn @ N4 )
               => ( P @ Nn ) )
           => ( P @ ( suc @ N4 ) ) )
       => ( P @ N2 ) ) ) ).

% nat_compl_induct'
thf(fact_932_nat__compl__induct,axiom,
    ! [P: nat > $o,N2: nat] :
      ( ( P @ ( zero_zero @ nat ) )
     => ( ! [N4: nat] :
            ( ! [Nn: nat] :
                ( ( ord_less_eq @ nat @ Nn @ N4 )
               => ( P @ Nn ) )
           => ( P @ ( suc @ N4 ) ) )
       => ( P @ N2 ) ) ) ).

% nat_compl_induct
thf(fact_933_add__is__1,axiom,
    ! [M: nat,N2: nat] :
      ( ( ( plus_plus @ nat @ M @ N2 )
        = ( suc @ ( zero_zero @ nat ) ) )
      = ( ( ( M
            = ( suc @ ( zero_zero @ nat ) ) )
          & ( N2
            = ( zero_zero @ nat ) ) )
        | ( ( M
            = ( zero_zero @ nat ) )
          & ( N2
            = ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ).

% add_is_1
thf(fact_934_one__is__add,axiom,
    ! [M: nat,N2: nat] :
      ( ( ( suc @ ( zero_zero @ nat ) )
        = ( plus_plus @ nat @ M @ N2 ) )
      = ( ( ( M
            = ( suc @ ( zero_zero @ nat ) ) )
          & ( N2
            = ( zero_zero @ nat ) ) )
        | ( ( M
            = ( zero_zero @ nat ) )
          & ( N2
            = ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ).

% one_is_add
thf(fact_935_One__nat__def,axiom,
    ( ( one_one @ nat )
    = ( suc @ ( zero_zero @ nat ) ) ) ).

% One_nat_def
thf(fact_936_option_Osize_I4_J,axiom,
    ! [A: $tType,X23: A] :
      ( ( size_size @ ( option @ A ) @ ( some @ A @ X23 ) )
      = ( suc @ ( zero_zero @ nat ) ) ) ).

% option.size(4)
thf(fact_937_option_Osize_I3_J,axiom,
    ! [A: $tType] :
      ( ( size_size @ ( option @ A ) @ ( none @ A ) )
      = ( suc @ ( zero_zero @ nat ) ) ) ).

% option.size(3)
thf(fact_938_ex__least__nat__le,axiom,
    ! [P: nat > $o,N2: nat] :
      ( ( P @ N2 )
     => ( ~ ( P @ ( zero_zero @ nat ) )
       => ? [K3: nat] :
            ( ( ord_less_eq @ nat @ K3 @ N2 )
            & ! [I4: nat] :
                ( ( ord_less @ nat @ I4 @ K3 )
               => ~ ( P @ I4 ) )
            & ( P @ K3 ) ) ) ) ).

% ex_least_nat_le
thf(fact_939_less__imp__add__positive,axiom,
    ! [I: nat,J: nat] :
      ( ( ord_less @ nat @ I @ J )
     => ? [K3: nat] :
          ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K3 )
          & ( ( plus_plus @ nat @ I @ K3 )
            = J ) ) ) ).

% less_imp_add_positive
thf(fact_940_Suc__to__right,axiom,
    ! [N2: nat,M: nat] :
      ( ( ( suc @ N2 )
        = M )
     => ( N2
        = ( minus_minus @ nat @ M @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ).

% Suc_to_right
thf(fact_941_diff__less,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
     => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
       => ( ord_less @ nat @ ( minus_minus @ nat @ M @ N2 ) @ M ) ) ) ).

% diff_less
thf(fact_942_Euclidean__Division_Odiv__eq__0__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( ( divide_divide @ nat @ M @ N2 )
        = ( zero_zero @ nat ) )
      = ( ( ord_less @ nat @ M @ N2 )
        | ( N2
          = ( zero_zero @ nat ) ) ) ) ).

% Euclidean_Division.div_eq_0_iff
thf(fact_943_nat__geq__1__eq__neqz,axiom,
    ! [X2: nat] :
      ( ( ord_less_eq @ nat @ ( one_one @ nat ) @ X2 )
      = ( X2
       != ( zero_zero @ nat ) ) ) ).

% nat_geq_1_eq_neqz
thf(fact_944_VEBT__internal_OTb_H_Osimps_I2_J,axiom,
    ( ( vEBT_VEBT_Tb2 @ ( suc @ ( zero_zero @ nat ) ) )
    = ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) ) ).

% VEBT_internal.Tb'.simps(2)
thf(fact_945_mult__less__mono1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( ord_less @ nat @ I @ J )
     => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
       => ( ord_less @ nat @ ( times_times @ nat @ I @ K ) @ ( times_times @ nat @ J @ K ) ) ) ) ).

% mult_less_mono1
thf(fact_946_mult__less__mono2,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( ord_less @ nat @ I @ J )
     => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
       => ( ord_less @ nat @ ( times_times @ nat @ K @ I ) @ ( times_times @ nat @ K @ J ) ) ) ) ).

% mult_less_mono2
thf(fact_947_nat__mult__eq__cancel1,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
     => ( ( ( times_times @ nat @ K @ M )
          = ( times_times @ nat @ K @ N2 ) )
        = ( M = N2 ) ) ) ).

% nat_mult_eq_cancel1
thf(fact_948_nat__mult__less__cancel1,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
     => ( ( ord_less @ nat @ ( times_times @ nat @ K @ M ) @ ( times_times @ nat @ K @ N2 ) )
        = ( ord_less @ nat @ M @ N2 ) ) ) ).

% nat_mult_less_cancel1
thf(fact_949_diff__add__0,axiom,
    ! [N2: nat,M: nat] :
      ( ( minus_minus @ nat @ N2 @ ( plus_plus @ nat @ N2 @ M ) )
      = ( zero_zero @ nat ) ) ).

% diff_add_0
thf(fact_950_nat__power__less__imp__less,axiom,
    ! [I: nat,M: nat,N2: nat] :
      ( ( ord_less @ nat @ ( zero_zero @ nat ) @ I )
     => ( ( ord_less @ nat @ ( power_power @ nat @ I @ M ) @ ( power_power @ nat @ I @ N2 ) )
       => ( ord_less @ nat @ M @ N2 ) ) ) ).

% nat_power_less_imp_less
thf(fact_951_mult__eq__self__implies__10,axiom,
    ! [M: nat,N2: nat] :
      ( ( M
        = ( times_times @ nat @ M @ N2 ) )
     => ( ( N2
          = ( one_one @ nat ) )
        | ( M
          = ( zero_zero @ nat ) ) ) ) ).

% mult_eq_self_implies_10
thf(fact_952_vebt__insert_Osimps_I2_J,axiom,
    ! [Info: option @ ( product_prod @ nat @ nat ),Ts: list @ vEBT_VEBT,S4: vEBT_VEBT,X2: nat] :
      ( ( vEBT_vebt_insert @ ( vEBT_Node @ Info @ ( zero_zero @ nat ) @ Ts @ S4 ) @ X2 )
      = ( vEBT_Node @ Info @ ( zero_zero @ nat ) @ Ts @ S4 ) ) ).

% vebt_insert.simps(2)
thf(fact_953_vebt__buildupi_Osimps_I1_J,axiom,
    ( ( vEBT_vebt_buildupi @ ( zero_zero @ nat ) )
    = ( heap_Time_return @ vEBT_VEBTi @ ( vEBT_Leafi @ $false @ $false ) ) ) ).

% vebt_buildupi.simps(1)
thf(fact_954_VEBT__internal_Ovebt__buildupi_H_Osimps_I1_J,axiom,
    ( ( vEBT_V739175172307565963ildupi @ ( zero_zero @ nat ) )
    = ( heap_Time_return @ vEBT_VEBTi @ ( vEBT_Leafi @ $false @ $false ) ) ) ).

% VEBT_internal.vebt_buildupi'.simps(1)
thf(fact_955_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,Uy: list @ vEBT_VEBT,Uz: vEBT_VEBT,X2: nat] :
      ( ( vEBT_T_m_e_m_b_e_r @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V ) @ ( zero_zero @ nat ) @ Uy @ Uz ) @ X2 )
      = ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.simps(3)
thf(fact_956_mult__less__le__imp__less,axiom,
    ! [A: $tType] :
      ( ( linord8928482502909563296strict @ A )
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( ( ord_less @ A @ A2 @ B2 )
         => ( ( ord_less_eq @ A @ C2 @ D2 )
           => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
             => ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
               => ( ord_less @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ D2 ) ) ) ) ) ) ) ).

% mult_less_le_imp_less
thf(fact_957_mult__le__less__imp__less,axiom,
    ! [A: $tType] :
      ( ( linord8928482502909563296strict @ A )
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( ( ord_less_eq @ A @ A2 @ B2 )
         => ( ( ord_less @ A @ C2 @ D2 )
           => ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
             => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
               => ( ord_less @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ D2 ) ) ) ) ) ) ) ).

% mult_le_less_imp_less
thf(fact_958_mult__right__le__imp__le,axiom,
    ! [A: $tType] :
      ( ( linord8928482502909563296strict @ A )
     => ! [A2: A,C2: A,B2: A] :
          ( ( ord_less_eq @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) )
         => ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
           => ( ord_less_eq @ A @ A2 @ B2 ) ) ) ) ).

% mult_right_le_imp_le
thf(fact_959_mult__left__le__imp__le,axiom,
    ! [A: $tType] :
      ( ( linord8928482502909563296strict @ A )
     => ! [C2: A,A2: A,B2: A] :
          ( ( ord_less_eq @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B2 ) )
         => ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
           => ( ord_less_eq @ A @ A2 @ B2 ) ) ) ) ).

% mult_left_le_imp_le
thf(fact_960_mult__le__cancel__left__pos,axiom,
    ! [A: $tType] :
      ( ( linord4710134922213307826strict @ A )
     => ! [C2: A,A2: A,B2: A] :
          ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
         => ( ( ord_less_eq @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B2 ) )
            = ( ord_less_eq @ A @ A2 @ B2 ) ) ) ) ).

% mult_le_cancel_left_pos
thf(fact_961_mult__le__cancel__left__neg,axiom,
    ! [A: $tType] :
      ( ( linord4710134922213307826strict @ A )
     => ! [C2: A,A2: A,B2: A] :
          ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
         => ( ( ord_less_eq @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B2 ) )
            = ( ord_less_eq @ A @ B2 @ A2 ) ) ) ) ).

% mult_le_cancel_left_neg
thf(fact_962_mult__less__cancel__right,axiom,
    ! [A: $tType] :
      ( ( linord4710134922213307826strict @ A )
     => ! [A2: A,C2: A,B2: A] :
          ( ( ord_less @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) )
          = ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
             => ( ord_less @ A @ A2 @ B2 ) )
            & ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) )
             => ( ord_less @ A @ B2 @ A2 ) ) ) ) ) ).

% mult_less_cancel_right
thf(fact_963_mult__strict__mono_H,axiom,
    ! [A: $tType] :
      ( ( linord8928482502909563296strict @ A )
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( ( ord_less @ A @ A2 @ B2 )
         => ( ( ord_less @ A @ C2 @ D2 )
           => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
             => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
               => ( ord_less @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ D2 ) ) ) ) ) ) ) ).

% mult_strict_mono'
thf(fact_964_mult__right__less__imp__less,axiom,
    ! [A: $tType] :
      ( ( linordered_semiring @ A )
     => ! [A2: A,C2: A,B2: A] :
          ( ( ord_less @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) )
         => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
           => ( ord_less @ A @ A2 @ B2 ) ) ) ) ).

% mult_right_less_imp_less
thf(fact_965_mult__less__cancel__left,axiom,
    ! [A: $tType] :
      ( ( linord4710134922213307826strict @ A )
     => ! [C2: A,A2: A,B2: A] :
          ( ( ord_less @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B2 ) )
          = ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
             => ( ord_less @ A @ A2 @ B2 ) )
            & ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) )
             => ( ord_less @ A @ B2 @ A2 ) ) ) ) ) ).

% mult_less_cancel_left
thf(fact_966_mult__strict__mono,axiom,
    ! [A: $tType] :
      ( ( linord8928482502909563296strict @ A )
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( ( ord_less @ A @ A2 @ B2 )
         => ( ( ord_less @ A @ C2 @ D2 )
           => ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
             => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
               => ( ord_less @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ D2 ) ) ) ) ) ) ) ).

% mult_strict_mono
thf(fact_967_mult__left__less__imp__less,axiom,
    ! [A: $tType] :
      ( ( linordered_semiring @ A )
     => ! [C2: A,A2: A,B2: A] :
          ( ( ord_less @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B2 ) )
         => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
           => ( ord_less @ A @ A2 @ B2 ) ) ) ) ).

% mult_left_less_imp_less
thf(fact_968_mult__le__cancel__right,axiom,
    ! [A: $tType] :
      ( ( linord4710134922213307826strict @ A )
     => ! [A2: A,C2: A,B2: A] :
          ( ( ord_less_eq @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) )
          = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
             => ( ord_less_eq @ A @ A2 @ B2 ) )
            & ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
             => ( ord_less_eq @ A @ B2 @ A2 ) ) ) ) ) ).

% mult_le_cancel_right
thf(fact_969_mult__le__cancel__left,axiom,
    ! [A: $tType] :
      ( ( linord4710134922213307826strict @ A )
     => ! [C2: A,A2: A,B2: A] :
          ( ( ord_less_eq @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B2 ) )
          = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
             => ( ord_less_eq @ A @ A2 @ B2 ) )
            & ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
             => ( ord_less_eq @ A @ B2 @ A2 ) ) ) ) ) ).

% mult_le_cancel_left
thf(fact_970_sum__squares__ge__zero,axiom,
    ! [A: $tType] :
      ( ( linordered_ring @ A )
     => ! [X2: A,Y2: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ ( times_times @ A @ X2 @ X2 ) @ ( times_times @ A @ Y2 @ Y2 ) ) ) ) ).

% sum_squares_ge_zero
thf(fact_971_sum__squares__le__zero__iff,axiom,
    ! [A: $tType] :
      ( ( linord4710134922213307826strict @ A )
     => ! [X2: A,Y2: A] :
          ( ( ord_less_eq @ A @ ( plus_plus @ A @ ( times_times @ A @ X2 @ X2 ) @ ( times_times @ A @ Y2 @ Y2 ) ) @ ( zero_zero @ A ) )
          = ( ( X2
              = ( zero_zero @ A ) )
            & ( Y2
              = ( zero_zero @ A ) ) ) ) ) ).

% sum_squares_le_zero_iff
thf(fact_972_mult__left__le,axiom,
    ! [A: $tType] :
      ( ( linord181362715937106298miring @ A )
     => ! [C2: A,A2: A] :
          ( ( ord_less_eq @ A @ C2 @ ( one_one @ A ) )
         => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
           => ( ord_less_eq @ A @ ( times_times @ A @ A2 @ C2 ) @ A2 ) ) ) ) ).

% mult_left_le
thf(fact_973_mult__le__one,axiom,
    ! [A: $tType] :
      ( ( linord181362715937106298miring @ A )
     => ! [A2: A,B2: A] :
          ( ( ord_less_eq @ A @ A2 @ ( one_one @ A ) )
         => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
           => ( ( ord_less_eq @ A @ B2 @ ( one_one @ A ) )
             => ( ord_less_eq @ A @ ( times_times @ A @ A2 @ B2 ) @ ( one_one @ A ) ) ) ) ) ) ).

% mult_le_one
thf(fact_974_mult__right__le__one__le,axiom,
    ! [A: $tType] :
      ( ( linordered_idom @ A )
     => ! [X2: A,Y2: A] :
          ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X2 )
         => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y2 )
           => ( ( ord_less_eq @ A @ Y2 @ ( one_one @ A ) )
             => ( ord_less_eq @ A @ ( times_times @ A @ X2 @ Y2 ) @ X2 ) ) ) ) ) ).

% mult_right_le_one_le
thf(fact_975_mult__left__le__one__le,axiom,
    ! [A: $tType] :
      ( ( linordered_idom @ A )
     => ! [X2: A,Y2: A] :
          ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X2 )
         => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y2 )
           => ( ( ord_less_eq @ A @ Y2 @ ( one_one @ A ) )
             => ( ord_less_eq @ A @ ( times_times @ A @ Y2 @ X2 ) @ X2 ) ) ) ) ) ).

% mult_left_le_one_le
thf(fact_976_zero__less__two,axiom,
    ! [A: $tType] :
      ( ( linord181362715937106298miring @ A )
     => ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ ( one_one @ A ) @ ( one_one @ A ) ) ) ) ).

% zero_less_two
thf(fact_977_div__positive,axiom,
    ! [A: $tType] :
      ( ( unique1627219031080169319umeral @ A )
     => ! [B2: A,A2: A] :
          ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
         => ( ( ord_less_eq @ A @ B2 @ A2 )
           => ( ord_less @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ A2 @ B2 ) ) ) ) ) ).

% div_positive
thf(fact_978_unique__euclidean__semiring__numeral__class_Odiv__less,axiom,
    ! [A: $tType] :
      ( ( unique1627219031080169319umeral @ A )
     => ! [A2: A,B2: A] :
          ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
         => ( ( ord_less @ A @ A2 @ B2 )
           => ( ( divide_divide @ A @ A2 @ B2 )
              = ( zero_zero @ A ) ) ) ) ) ).

% unique_euclidean_semiring_numeral_class.div_less
thf(fact_979_not__sum__squares__lt__zero,axiom,
    ! [A: $tType] :
      ( ( linordered_ring @ A )
     => ! [X2: A,Y2: A] :
          ~ ( ord_less @ A @ ( plus_plus @ A @ ( times_times @ A @ X2 @ X2 ) @ ( times_times @ A @ Y2 @ Y2 ) ) @ ( zero_zero @ A ) ) ) ).

% not_sum_squares_lt_zero
thf(fact_980_sum__squares__gt__zero__iff,axiom,
    ! [A: $tType] :
      ( ( linord4710134922213307826strict @ A )
     => ! [X2: A,Y2: A] :
          ( ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ ( times_times @ A @ X2 @ X2 ) @ ( times_times @ A @ Y2 @ Y2 ) ) )
          = ( ( X2
             != ( zero_zero @ A ) )
            | ( Y2
             != ( zero_zero @ A ) ) ) ) ) ).

% sum_squares_gt_zero_iff
thf(fact_981_power__less__imp__less__base,axiom,
    ! [A: $tType] :
      ( ( linordered_semidom @ A )
     => ! [A2: A,N2: nat,B2: A] :
          ( ( ord_less @ A @ ( power_power @ A @ A2 @ N2 ) @ ( power_power @ A @ B2 @ N2 ) )
         => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
           => ( ord_less @ A @ A2 @ B2 ) ) ) ) ).

% power_less_imp_less_base
thf(fact_982_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
    ! [A: $tType] :
      ( ( unique1627219031080169319umeral @ A )
     => ! [C2: A,A2: A,B2: A] :
          ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
         => ( ( divide_divide @ A @ A2 @ ( times_times @ A @ B2 @ C2 ) )
            = ( divide_divide @ A @ ( divide_divide @ A @ A2 @ B2 ) @ C2 ) ) ) ) ).

% unique_euclidean_semiring_numeral_class.div_mult2_eq
thf(fact_983_power__le__one,axiom,
    ! [A: $tType] :
      ( ( linordered_semidom @ A )
     => ! [A2: A,N2: nat] :
          ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
         => ( ( ord_less_eq @ A @ A2 @ ( one_one @ A ) )
           => ( ord_less_eq @ A @ ( power_power @ A @ A2 @ N2 ) @ ( one_one @ A ) ) ) ) ) ).

% power_le_one
thf(fact_984_eq__divide__eq__numeral_I1_J,axiom,
    ! [A: $tType] :
      ( ( division_ring @ A )
     => ! [W: num,B2: A,C2: A] :
          ( ( ( numeral_numeral @ A @ W )
            = ( divide_divide @ A @ B2 @ C2 ) )
          = ( ( ( C2
               != ( zero_zero @ A ) )
             => ( ( times_times @ A @ ( numeral_numeral @ A @ W ) @ C2 )
                = B2 ) )
            & ( ( C2
                = ( zero_zero @ A ) )
             => ( ( numeral_numeral @ A @ W )
                = ( zero_zero @ A ) ) ) ) ) ) ).

% eq_divide_eq_numeral(1)
thf(fact_985_divide__eq__eq__numeral_I1_J,axiom,
    ! [A: $tType] :
      ( ( division_ring @ A )
     => ! [B2: A,C2: A,W: num] :
          ( ( ( divide_divide @ A @ B2 @ C2 )
            = ( numeral_numeral @ A @ W ) )
          = ( ( ( C2
               != ( zero_zero @ A ) )
             => ( B2
                = ( times_times @ A @ ( numeral_numeral @ A @ W ) @ C2 ) ) )
            & ( ( C2
                = ( zero_zero @ A ) )
             => ( ( numeral_numeral @ A @ W )
                = ( zero_zero @ A ) ) ) ) ) ) ).

% divide_eq_eq_numeral(1)
thf(fact_986_power__le__imp__le__base,axiom,
    ! [A: $tType] :
      ( ( linordered_semidom @ A )
     => ! [A2: A,N2: nat,B2: A] :
          ( ( ord_less_eq @ A @ ( power_power @ A @ A2 @ ( suc @ N2 ) ) @ ( power_power @ A @ B2 @ ( suc @ N2 ) ) )
         => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
           => ( ord_less_eq @ A @ A2 @ B2 ) ) ) ) ).

% power_le_imp_le_base
thf(fact_987_power__inject__base,axiom,
    ! [A: $tType] :
      ( ( linordered_semidom @ A )
     => ! [A2: A,N2: nat,B2: A] :
          ( ( ( power_power @ A @ A2 @ ( suc @ N2 ) )
            = ( power_power @ A @ B2 @ ( suc @ N2 ) ) )
         => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
           => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
             => ( A2 = B2 ) ) ) ) ) ).

% power_inject_base
thf(fact_988_div__add__self2,axiom,
    ! [A: $tType] :
      ( ( euclid4440199948858584721cancel @ A )
     => ! [B2: A,A2: A] :
          ( ( B2
           != ( zero_zero @ A ) )
         => ( ( divide_divide @ A @ ( plus_plus @ A @ A2 @ B2 ) @ B2 )
            = ( plus_plus @ A @ ( divide_divide @ A @ A2 @ B2 ) @ ( one_one @ A ) ) ) ) ) ).

% div_add_self2
thf(fact_989_div__add__self1,axiom,
    ! [A: $tType] :
      ( ( euclid4440199948858584721cancel @ A )
     => ! [B2: A,A2: A] :
          ( ( B2
           != ( zero_zero @ A ) )
         => ( ( divide_divide @ A @ ( plus_plus @ A @ B2 @ A2 ) @ B2 )
            = ( plus_plus @ A @ ( divide_divide @ A @ A2 @ B2 ) @ ( one_one @ A ) ) ) ) ) ).

% div_add_self1
thf(fact_990_VEBT__internal_OminNulli_Osimps_I5_J,axiom,
    ! [Uz: product_prod @ nat @ nat,Va: nat,Vb: array @ vEBT_VEBTi,Vc2: vEBT_VEBTi] :
      ( ( vEBT_VEBT_minNulli @ ( vEBT_Nodei @ ( some @ ( product_prod @ nat @ nat ) @ Uz ) @ Va @ Vb @ Vc2 ) )
      = ( heap_Time_return @ $o @ $false ) ) ).

% VEBT_internal.minNulli.simps(5)
thf(fact_991_numeral__1__eq__Suc__0,axiom,
    ( ( numeral_numeral @ nat @ one2 )
    = ( suc @ ( zero_zero @ nat ) ) ) ).

% numeral_1_eq_Suc_0
thf(fact_992_VEBT__internal_OminNulli_Osimps_I4_J,axiom,
    ! [Uw: nat,Ux: array @ vEBT_VEBTi,Uy: vEBT_VEBTi] :
      ( ( vEBT_VEBT_minNulli @ ( vEBT_Nodei @ ( none @ ( product_prod @ nat @ nat ) ) @ Uw @ Ux @ Uy ) )
      = ( heap_Time_return @ $o @ $true ) ) ).

% VEBT_internal.minNulli.simps(4)
thf(fact_993_num_Osize_I5_J,axiom,
    ! [X23: num] :
      ( ( size_size @ num @ ( bit0 @ X23 ) )
      = ( plus_plus @ nat @ ( size_size @ num @ X23 ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ).

% num.size(5)
thf(fact_994_ex__least__nat__less,axiom,
    ! [P: nat > $o,N2: nat] :
      ( ( P @ N2 )
     => ( ~ ( P @ ( zero_zero @ nat ) )
       => ? [K3: nat] :
            ( ( ord_less @ nat @ K3 @ N2 )
            & ! [I4: nat] :
                ( ( ord_less_eq @ nat @ I4 @ K3 )
               => ~ ( P @ I4 ) )
            & ( P @ ( suc @ K3 ) ) ) ) ) ).

% ex_least_nat_less
thf(fact_995_nat__induct__non__zero,axiom,
    ! [N2: nat,P: nat > $o] :
      ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
     => ( ( P @ ( one_one @ nat ) )
       => ( ! [N4: nat] :
              ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N4 )
             => ( ( P @ N4 )
               => ( P @ ( suc @ N4 ) ) ) )
         => ( P @ N2 ) ) ) ) ).

% nat_induct_non_zero
thf(fact_996_num_Osize_I6_J,axiom,
    ! [X33: num] :
      ( ( size_size @ num @ ( bit1 @ X33 ) )
      = ( plus_plus @ nat @ ( size_size @ num @ X33 ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ).

% num.size(6)
thf(fact_997_diff__Suc__less,axiom,
    ! [N2: nat,I: nat] :
      ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
     => ( ord_less @ nat @ ( minus_minus @ nat @ N2 @ ( suc @ I ) ) @ N2 ) ) ).

% diff_Suc_less
thf(fact_998_n__less__n__mult__m,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
     => ( ( ord_less @ nat @ ( suc @ ( zero_zero @ nat ) ) @ M )
       => ( ord_less @ nat @ N2 @ ( times_times @ nat @ N2 @ M ) ) ) ) ).

% n_less_n_mult_m
thf(fact_999_n__less__m__mult__n,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
     => ( ( ord_less @ nat @ ( suc @ ( zero_zero @ nat ) ) @ M )
       => ( ord_less @ nat @ N2 @ ( times_times @ nat @ M @ N2 ) ) ) ) ).

% n_less_m_mult_n
thf(fact_1000_one__less__mult,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N2 )
     => ( ( ord_less @ nat @ ( suc @ ( zero_zero @ nat ) ) @ M )
       => ( ord_less @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( times_times @ nat @ M @ N2 ) ) ) ) ).

% one_less_mult
thf(fact_1001_power__gt__expt,axiom,
    ! [N2: nat,K: nat] :
      ( ( ord_less @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N2 )
     => ( ord_less @ nat @ K @ ( power_power @ nat @ N2 @ K ) ) ) ).

% power_gt_expt
thf(fact_1002_nat__diff__split__asm,axiom,
    ! [P: nat > $o,A2: nat,B2: nat] :
      ( ( P @ ( minus_minus @ nat @ A2 @ B2 ) )
      = ( ~ ( ( ( ord_less @ nat @ A2 @ B2 )
              & ~ ( P @ ( zero_zero @ nat ) ) )
            | ? [D3: nat] :
                ( ( A2
                  = ( plus_plus @ nat @ B2 @ D3 ) )
                & ~ ( P @ D3 ) ) ) ) ) ).

% nat_diff_split_asm
thf(fact_1003_nat__diff__split,axiom,
    ! [P: nat > $o,A2: nat,B2: nat] :
      ( ( P @ ( minus_minus @ nat @ A2 @ B2 ) )
      = ( ( ( ord_less @ nat @ A2 @ B2 )
         => ( P @ ( zero_zero @ nat ) ) )
        & ! [D3: nat] :
            ( ( A2
              = ( plus_plus @ nat @ B2 @ D3 ) )
           => ( P @ D3 ) ) ) ) ).

% nat_diff_split
thf(fact_1004_div__greater__zero__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( divide_divide @ nat @ M @ N2 ) )
      = ( ( ord_less_eq @ nat @ N2 @ M )
        & ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ).

% div_greater_zero_iff
thf(fact_1005_div__le__mono2,axiom,
    ! [M: nat,N2: nat,K: nat] :
      ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
     => ( ( ord_less_eq @ nat @ M @ N2 )
       => ( ord_less_eq @ nat @ ( divide_divide @ nat @ K @ N2 ) @ ( divide_divide @ nat @ K @ M ) ) ) ) ).

% div_le_mono2
thf(fact_1006_div__less__dividend,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less @ nat @ ( one_one @ nat ) @ N2 )
     => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
       => ( ord_less @ nat @ ( divide_divide @ nat @ M @ N2 ) @ M ) ) ) ).

% div_less_dividend
thf(fact_1007_nat__one__le__power,axiom,
    ! [I: nat,N2: nat] :
      ( ( ord_less_eq @ nat @ ( suc @ ( zero_zero @ nat ) ) @ I )
     => ( ord_less_eq @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( power_power @ nat @ I @ N2 ) ) ) ).

% nat_one_le_power
thf(fact_1008_nat__mult__le__cancel1,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
     => ( ( ord_less_eq @ nat @ ( times_times @ nat @ K @ M ) @ ( times_times @ nat @ K @ N2 ) )
        = ( ord_less_eq @ nat @ M @ N2 ) ) ) ).

% nat_mult_le_cancel1
thf(fact_1009_div__less__iff__less__mult,axiom,
    ! [Q2: nat,M: nat,N2: nat] :
      ( ( ord_less @ nat @ ( zero_zero @ nat ) @ Q2 )
     => ( ( ord_less @ nat @ ( divide_divide @ nat @ M @ Q2 ) @ N2 )
        = ( ord_less @ nat @ M @ ( times_times @ nat @ N2 @ Q2 ) ) ) ) ).

% div_less_iff_less_mult
thf(fact_1010_nat__mult__div__cancel1,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
     => ( ( divide_divide @ nat @ ( times_times @ nat @ K @ M ) @ ( times_times @ nat @ K @ N2 ) )
        = ( divide_divide @ nat @ M @ N2 ) ) ) ).

% nat_mult_div_cancel1
thf(fact_1011_td__gal__lt,axiom,
    ! [C2: nat,A2: nat,B2: nat] :
      ( ( ord_less @ nat @ ( zero_zero @ nat ) @ C2 )
     => ( ( ord_less @ nat @ A2 @ ( times_times @ nat @ B2 @ C2 ) )
        = ( ord_less @ nat @ ( divide_divide @ nat @ A2 @ C2 ) @ B2 ) ) ) ).

% td_gal_lt
thf(fact_1012_assn__times__assoc,axiom,
    ! [P: assn,Q: assn,R2: assn] :
      ( ( times_times @ assn @ ( times_times @ assn @ P @ Q ) @ R2 )
      = ( times_times @ assn @ P @ ( times_times @ assn @ Q @ R2 ) ) ) ).

% assn_times_assoc
thf(fact_1013_assn__times__comm,axiom,
    ( ( times_times @ assn )
    = ( ^ [P3: assn,Q4: assn] : ( times_times @ assn @ Q4 @ P3 ) ) ) ).

% assn_times_comm
thf(fact_1014_vebt__insert_Osimps_I3_J,axiom,
    ! [Info: option @ ( product_prod @ nat @ nat ),Ts: list @ vEBT_VEBT,S4: vEBT_VEBT,X2: nat] :
      ( ( vEBT_vebt_insert @ ( vEBT_Node @ Info @ ( suc @ ( zero_zero @ nat ) ) @ Ts @ S4 ) @ X2 )
      = ( vEBT_Node @ Info @ ( suc @ ( zero_zero @ nat ) ) @ Ts @ S4 ) ) ).

% vebt_insert.simps(3)
thf(fact_1015_vebt__member_Osimps_I3_J,axiom,
    ! [V: product_prod @ nat @ nat,Uy: list @ vEBT_VEBT,Uz: vEBT_VEBT,X2: nat] :
      ~ ( vEBT_vebt_member @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V ) @ ( zero_zero @ nat ) @ Uy @ Uz ) @ X2 ) ).

% vebt_member.simps(3)
thf(fact_1016_vebt__buildupi_Osimps_I2_J,axiom,
    ( ( vEBT_vebt_buildupi @ ( suc @ ( zero_zero @ nat ) ) )
    = ( heap_Time_return @ vEBT_VEBTi @ ( vEBT_Leafi @ $false @ $false ) ) ) ).

% vebt_buildupi.simps(2)
thf(fact_1017_VEBT__internal_Ovebt__buildupi_H_Osimps_I2_J,axiom,
    ( ( vEBT_V739175172307565963ildupi @ ( suc @ ( zero_zero @ nat ) ) )
    = ( heap_Time_return @ vEBT_VEBTi @ ( vEBT_Leafi @ $false @ $false ) ) ) ).

% VEBT_internal.vebt_buildupi'.simps(2)
thf(fact_1018_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,Vb: list @ vEBT_VEBT,Vc2: vEBT_VEBT,X2: nat] :
      ( ( vEBT_T_m_e_m_b_e_r @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V ) @ ( suc @ ( zero_zero @ nat ) ) @ Vb @ Vc2 ) @ X2 )
      = ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.simps(4)
thf(fact_1019_mult__less__cancel__right2,axiom,
    ! [A: $tType] :
      ( ( linordered_idom @ A )
     => ! [A2: A,C2: A] :
          ( ( ord_less @ A @ ( times_times @ A @ A2 @ C2 ) @ C2 )
          = ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
             => ( ord_less @ A @ A2 @ ( one_one @ A ) ) )
            & ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) )
             => ( ord_less @ A @ ( one_one @ A ) @ A2 ) ) ) ) ) ).

% mult_less_cancel_right2
thf(fact_1020_mult__less__cancel__right1,axiom,
    ! [A: $tType] :
      ( ( linordered_idom @ A )
     => ! [C2: A,B2: A] :
          ( ( ord_less @ A @ C2 @ ( times_times @ A @ B2 @ C2 ) )
          = ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
             => ( ord_less @ A @ ( one_one @ A ) @ B2 ) )
            & ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) )
             => ( ord_less @ A @ B2 @ ( one_one @ A ) ) ) ) ) ) ).

% mult_less_cancel_right1
thf(fact_1021_mult__less__cancel__left2,axiom,
    ! [A: $tType] :
      ( ( linordered_idom @ A )
     => ! [C2: A,A2: A] :
          ( ( ord_less @ A @ ( times_times @ A @ C2 @ A2 ) @ C2 )
          = ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
             => ( ord_less @ A @ A2 @ ( one_one @ A ) ) )
            & ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) )
             => ( ord_less @ A @ ( one_one @ A ) @ A2 ) ) ) ) ) ).

% mult_less_cancel_left2
thf(fact_1022_mult__less__cancel__left1,axiom,
    ! [A: $tType] :
      ( ( linordered_idom @ A )
     => ! [C2: A,B2: A] :
          ( ( ord_less @ A @ C2 @ ( times_times @ A @ C2 @ B2 ) )
          = ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
             => ( ord_less @ A @ ( one_one @ A ) @ B2 ) )
            & ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) )
             => ( ord_less @ A @ B2 @ ( one_one @ A ) ) ) ) ) ) ).

% mult_less_cancel_left1
thf(fact_1023_mult__le__cancel__right2,axiom,
    ! [A: $tType] :
      ( ( linordered_idom @ A )
     => ! [A2: A,C2: A] :
          ( ( ord_less_eq @ A @ ( times_times @ A @ A2 @ C2 ) @ C2 )
          = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
             => ( ord_less_eq @ A @ A2 @ ( one_one @ A ) ) )
            & ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
             => ( ord_less_eq @ A @ ( one_one @ A ) @ A2 ) ) ) ) ) ).

% mult_le_cancel_right2
thf(fact_1024_mult__le__cancel__right1,axiom,
    ! [A: $tType] :
      ( ( linordered_idom @ A )
     => ! [C2: A,B2: A] :
          ( ( ord_less_eq @ A @ C2 @ ( times_times @ A @ B2 @ C2 ) )
          = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
             => ( ord_less_eq @ A @ ( one_one @ A ) @ B2 ) )
            & ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
             => ( ord_less_eq @ A @ B2 @ ( one_one @ A ) ) ) ) ) ) ).

% mult_le_cancel_right1
thf(fact_1025_mult__le__cancel__left2,axiom,
    ! [A: $tType] :
      ( ( linordered_idom @ A )
     => ! [C2: A,A2: A] :
          ( ( ord_less_eq @ A @ ( times_times @ A @ C2 @ A2 ) @ C2 )
          = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
             => ( ord_less_eq @ A @ A2 @ ( one_one @ A ) ) )
            & ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
             => ( ord_less_eq @ A @ ( one_one @ A ) @ A2 ) ) ) ) ) ).

% mult_le_cancel_left2
thf(fact_1026_mult__le__cancel__left1,axiom,
    ! [A: $tType] :
      ( ( linordered_idom @ A )
     => ! [C2: A,B2: A] :
          ( ( ord_less_eq @ A @ C2 @ ( times_times @ A @ C2 @ B2 ) )
          = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
             => ( ord_less_eq @ A @ ( one_one @ A ) @ B2 ) )
            & ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
             => ( ord_less_eq @ A @ B2 @ ( one_one @ A ) ) ) ) ) ) ).

% mult_le_cancel_left1
thf(fact_1027_convex__bound__le,axiom,
    ! [A: $tType] :
      ( ( linord6961819062388156250ring_1 @ A )
     => ! [X2: A,A2: A,Y2: A,U: A,V: A] :
          ( ( ord_less_eq @ A @ X2 @ A2 )
         => ( ( ord_less_eq @ A @ Y2 @ A2 )
           => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ U )
             => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ V )
               => ( ( ( plus_plus @ A @ U @ V )
                    = ( one_one @ A ) )
                 => ( ord_less_eq @ A @ ( plus_plus @ A @ ( times_times @ A @ U @ X2 ) @ ( times_times @ A @ V @ Y2 ) ) @ A2 ) ) ) ) ) ) ) ).

% convex_bound_le
thf(fact_1028_less__divide__eq__numeral_I1_J,axiom,
    ! [A: $tType] :
      ( ( linordered_field @ A )
     => ! [W: num,B2: A,C2: A] :
          ( ( ord_less @ A @ ( numeral_numeral @ A @ W ) @ ( divide_divide @ A @ B2 @ C2 ) )
          = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
             => ( ord_less @ A @ ( times_times @ A @ ( numeral_numeral @ A @ W ) @ C2 ) @ B2 ) )
            & ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
             => ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
                 => ( ord_less @ A @ B2 @ ( times_times @ A @ ( numeral_numeral @ A @ W ) @ C2 ) ) )
                & ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
                 => ( ord_less @ A @ ( numeral_numeral @ A @ W ) @ ( zero_zero @ A ) ) ) ) ) ) ) ) ).

% less_divide_eq_numeral(1)
thf(fact_1029_divide__less__eq__numeral_I1_J,axiom,
    ! [A: $tType] :
      ( ( linordered_field @ A )
     => ! [B2: A,C2: A,W: num] :
          ( ( ord_less @ A @ ( divide_divide @ A @ B2 @ C2 ) @ ( numeral_numeral @ A @ W ) )
          = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
             => ( ord_less @ A @ B2 @ ( times_times @ A @ ( numeral_numeral @ A @ W ) @ C2 ) ) )
            & ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
             => ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
                 => ( ord_less @ A @ ( times_times @ A @ ( numeral_numeral @ A @ W ) @ C2 ) @ B2 ) )
                & ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
                 => ( ord_less @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ W ) ) ) ) ) ) ) ) ).

% divide_less_eq_numeral(1)
thf(fact_1030_power__Suc__less,axiom,
    ! [A: $tType] :
      ( ( linordered_semidom @ A )
     => ! [A2: A,N2: nat] :
          ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
         => ( ( ord_less @ A @ A2 @ ( one_one @ A ) )
           => ( ord_less @ A @ ( times_times @ A @ A2 @ ( power_power @ A @ A2 @ N2 ) ) @ ( power_power @ A @ A2 @ N2 ) ) ) ) ) ).

% power_Suc_less
thf(fact_1031_power__Suc__le__self,axiom,
    ! [A: $tType] :
      ( ( linordered_semidom @ A )
     => ! [A2: A,N2: nat] :
          ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
         => ( ( ord_less_eq @ A @ A2 @ ( one_one @ A ) )
           => ( ord_less_eq @ A @ ( power_power @ A @ A2 @ ( suc @ N2 ) ) @ A2 ) ) ) ) ).

% power_Suc_le_self
thf(fact_1032_power__Suc__less__one,axiom,
    ! [A: $tType] :
      ( ( linordered_semidom @ A )
     => ! [A2: A,N2: nat] :
          ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
         => ( ( ord_less @ A @ A2 @ ( one_one @ A ) )
           => ( ord_less @ A @ ( power_power @ A @ A2 @ ( suc @ N2 ) ) @ ( one_one @ A ) ) ) ) ) ).

% power_Suc_less_one
thf(fact_1033_zero__power2,axiom,
    ! [A: $tType] :
      ( ( semiring_1 @ A )
     => ( ( power_power @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
        = ( zero_zero @ A ) ) ) ).

% zero_power2
thf(fact_1034_power__strict__decreasing,axiom,
    ! [A: $tType] :
      ( ( linordered_semidom @ A )
     => ! [N2: nat,N3: nat,A2: A] :
          ( ( ord_less @ nat @ N2 @ N3 )
         => ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
           => ( ( ord_less @ A @ A2 @ ( one_one @ A ) )
             => ( ord_less @ A @ ( power_power @ A @ A2 @ N3 ) @ ( power_power @ A @ A2 @ N2 ) ) ) ) ) ) ).

% power_strict_decreasing
thf(fact_1035_power__decreasing,axiom,
    ! [A: $tType] :
      ( ( linordered_semidom @ A )
     => ! [N2: nat,N3: nat,A2: A] :
          ( ( ord_less_eq @ nat @ N2 @ N3 )
         => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
           => ( ( ord_less_eq @ A @ A2 @ ( one_one @ A ) )
             => ( ord_less_eq @ A @ ( power_power @ A @ A2 @ N3 ) @ ( power_power @ A @ A2 @ N2 ) ) ) ) ) ) ).

% power_decreasing
thf(fact_1036_numeral__2__eq__2,axiom,
    ( ( numeral_numeral @ nat @ ( bit0 @ one2 ) )
    = ( suc @ ( suc @ ( zero_zero @ nat ) ) ) ) ).

% numeral_2_eq_2
thf(fact_1037_self__le__power,axiom,
    ! [A: $tType] :
      ( ( linordered_semidom @ A )
     => ! [A2: A,N2: nat] :
          ( ( ord_less_eq @ A @ ( one_one @ A ) @ A2 )
         => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
           => ( ord_less_eq @ A @ A2 @ ( power_power @ A @ A2 @ N2 ) ) ) ) ) ).

% self_le_power
thf(fact_1038_pos2,axiom,
    ord_less @ nat @ ( zero_zero @ nat ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ).

% pos2
thf(fact_1039_one__less__power,axiom,
    ! [A: $tType] :
      ( ( linordered_semidom @ A )
     => ! [A2: A,N2: nat] :
          ( ( ord_less @ A @ ( one_one @ A ) @ A2 )
         => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
           => ( ord_less @ A @ ( one_one @ A ) @ ( power_power @ A @ A2 @ N2 ) ) ) ) ) ).

% one_less_power
thf(fact_1040_power__diff,axiom,
    ! [A: $tType] :
      ( ( semidom_divide @ A )
     => ! [A2: A,N2: nat,M: nat] :
          ( ( A2
           != ( zero_zero @ A ) )
         => ( ( ord_less_eq @ nat @ N2 @ M )
           => ( ( power_power @ A @ A2 @ ( minus_minus @ nat @ M @ N2 ) )
              = ( divide_divide @ A @ ( power_power @ A @ A2 @ M ) @ ( power_power @ A @ A2 @ N2 ) ) ) ) ) ) ).

% power_diff
thf(fact_1041_numeral__3__eq__3,axiom,
    ( ( numeral_numeral @ nat @ ( bit1 @ one2 ) )
    = ( suc @ ( suc @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ).

% numeral_3_eq_3
thf(fact_1042_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_1043_Suc__diff__eq__diff__pred,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
     => ( ( minus_minus @ nat @ ( suc @ M ) @ N2 )
        = ( minus_minus @ nat @ M @ ( minus_minus @ nat @ N2 @ ( one_one @ nat ) ) ) ) ) ).

% Suc_diff_eq_diff_pred
thf(fact_1044_Suc__pred_H,axiom,
    ! [N2: nat] :
      ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
     => ( N2
        = ( suc @ ( minus_minus @ nat @ N2 @ ( one_one @ nat ) ) ) ) ) ).

% Suc_pred'
thf(fact_1045_div__if,axiom,
    ( ( divide_divide @ nat )
    = ( ^ [M7: nat,N: nat] :
          ( if @ nat
          @ ( ( ord_less @ nat @ M7 @ N )
            | ( N
              = ( zero_zero @ nat ) ) )
          @ ( zero_zero @ nat )
          @ ( suc @ ( divide_divide @ nat @ ( minus_minus @ nat @ M7 @ N ) @ N ) ) ) ) ) ).

% div_if
thf(fact_1046_div__geq,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
     => ( ~ ( ord_less @ nat @ M @ N2 )
       => ( ( divide_divide @ nat @ M @ N2 )
          = ( suc @ ( divide_divide @ nat @ ( minus_minus @ nat @ M @ N2 ) @ N2 ) ) ) ) ) ).

% div_geq
thf(fact_1047_add__eq__if,axiom,
    ( ( plus_plus @ nat )
    = ( ^ [M7: nat,N: nat] :
          ( if @ nat
          @ ( M7
            = ( zero_zero @ nat ) )
          @ N
          @ ( suc @ ( plus_plus @ nat @ ( minus_minus @ nat @ M7 @ ( one_one @ nat ) ) @ N ) ) ) ) ) ).

% add_eq_if
thf(fact_1048_msrevs_I1_J,axiom,
    ! [N2: nat,K: nat,M: nat] :
      ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
     => ( ( divide_divide @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ K @ N2 ) @ M ) @ N2 )
        = ( plus_plus @ nat @ ( divide_divide @ nat @ M @ N2 ) @ K ) ) ) ).

% msrevs(1)
thf(fact_1049_less__eq__div__iff__mult__less__eq,axiom,
    ! [Q2: nat,M: nat,N2: nat] :
      ( ( ord_less @ nat @ ( zero_zero @ nat ) @ Q2 )
     => ( ( ord_less_eq @ nat @ M @ ( divide_divide @ nat @ N2 @ Q2 ) )
        = ( ord_less_eq @ nat @ ( times_times @ nat @ M @ Q2 ) @ N2 ) ) ) ).

% less_eq_div_iff_mult_less_eq
thf(fact_1050_dividend__less__times__div,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
     => ( ord_less @ nat @ M @ ( plus_plus @ nat @ N2 @ ( times_times @ nat @ N2 @ ( divide_divide @ nat @ M @ N2 ) ) ) ) ) ).

% dividend_less_times_div
thf(fact_1051_dividend__less__div__times,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
     => ( ord_less @ nat @ M @ ( plus_plus @ nat @ N2 @ ( times_times @ nat @ ( divide_divide @ nat @ M @ N2 ) @ N2 ) ) ) ) ).

% dividend_less_div_times
thf(fact_1052_split__div,axiom,
    ! [P: nat > $o,M: nat,N2: nat] :
      ( ( P @ ( divide_divide @ nat @ M @ N2 ) )
      = ( ( ( N2
            = ( zero_zero @ nat ) )
         => ( P @ ( zero_zero @ nat ) ) )
        & ( ( N2
           != ( zero_zero @ nat ) )
         => ! [I2: nat,J3: nat] :
              ( ( ord_less @ nat @ J3 @ N2 )
             => ( ( M
                  = ( plus_plus @ nat @ ( times_times @ nat @ N2 @ I2 ) @ J3 ) )
               => ( P @ I2 ) ) ) ) ) ) ).

% split_div
thf(fact_1053_td__gal,axiom,
    ! [C2: nat,B2: nat,A2: nat] :
      ( ( ord_less @ nat @ ( zero_zero @ nat ) @ C2 )
     => ( ( ord_less_eq @ nat @ ( times_times @ nat @ B2 @ C2 ) @ A2 )
        = ( ord_less_eq @ nat @ B2 @ ( divide_divide @ nat @ A2 @ C2 ) ) ) ) ).

% td_gal
thf(fact_1054_mult__eq__if,axiom,
    ( ( times_times @ nat )
    = ( ^ [M7: nat,N: nat] :
          ( if @ nat
          @ ( M7
            = ( zero_zero @ nat ) )
          @ ( zero_zero @ nat )
          @ ( plus_plus @ nat @ N @ ( times_times @ nat @ ( minus_minus @ nat @ M7 @ ( one_one @ nat ) ) @ N ) ) ) ) ) ).

% mult_eq_if
thf(fact_1055_nat__mult__power__less__eq,axiom,
    ! [B2: nat,A2: nat,N2: nat,M: nat] :
      ( ( ord_less @ nat @ ( zero_zero @ nat ) @ B2 )
     => ( ( ord_less @ nat @ ( times_times @ nat @ A2 @ ( power_power @ nat @ B2 @ N2 ) ) @ ( power_power @ nat @ B2 @ M ) )
        = ( ord_less @ nat @ A2 @ ( power_power @ nat @ B2 @ ( minus_minus @ nat @ M @ N2 ) ) ) ) ) ).

% nat_mult_power_less_eq
thf(fact_1056_vebt__member_Osimps_I4_J,axiom,
    ! [V: product_prod @ nat @ nat,Vb: list @ vEBT_VEBT,Vc2: vEBT_VEBT,X2: nat] :
      ~ ( vEBT_vebt_member @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V ) @ ( suc @ ( zero_zero @ nat ) ) @ Vb @ Vc2 ) @ X2 ) ).

% vebt_member.simps(4)
thf(fact_1057_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 @ ( product_prod @ nat @ nat ) @ V ) @ ( zero_zero @ nat ) @ Vd @ Ve ) @ Vf )
      = ( none @ nat ) ) ).

% vebt_pred.simps(5)
thf(fact_1058_vebt__succ_Osimps_I4_J,axiom,
    ! [V: product_prod @ nat @ nat,Vc2: list @ vEBT_VEBT,Vd: vEBT_VEBT,Ve: nat] :
      ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V ) @ ( zero_zero @ nat ) @ Vc2 @ Vd ) @ Ve )
      = ( none @ nat ) ) ).

% vebt_succ.simps(4)
thf(fact_1059_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 @ ( product_prod @ nat @ nat ) @ V ) @ ( zero_zero @ nat ) @ Vd @ Ve ) @ Vf )
      = ( one_one @ nat ) ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.simps(5)
thf(fact_1060_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,Vc2: list @ vEBT_VEBT,Vd: vEBT_VEBT,Ve: nat] :
      ( ( vEBT_T_s_u_c_c2 @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V ) @ ( zero_zero @ nat ) @ Vc2 @ Vd ) @ Ve )
      = ( one_one @ nat ) ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.simps(4)
thf(fact_1061_convex__bound__lt,axiom,
    ! [A: $tType] :
      ( ( linord715952674999750819strict @ A )
     => ! [X2: A,A2: A,Y2: A,U: A,V: A] :
          ( ( ord_less @ A @ X2 @ A2 )
         => ( ( ord_less @ A @ Y2 @ A2 )
           => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ U )
             => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ V )
               => ( ( ( plus_plus @ A @ U @ V )
                    = ( one_one @ A ) )
                 => ( ord_less @ A @ ( plus_plus @ A @ ( times_times @ A @ U @ X2 ) @ ( times_times @ A @ V @ Y2 ) ) @ A2 ) ) ) ) ) ) ) ).

% convex_bound_lt
thf(fact_1062_le__divide__eq__numeral_I1_J,axiom,
    ! [A: $tType] :
      ( ( linordered_field @ A )
     => ! [W: num,B2: A,C2: A] :
          ( ( ord_less_eq @ A @ ( numeral_numeral @ A @ W ) @ ( divide_divide @ A @ B2 @ C2 ) )
          = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
             => ( ord_less_eq @ A @ ( times_times @ A @ ( numeral_numeral @ A @ W ) @ C2 ) @ B2 ) )
            & ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
             => ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
                 => ( ord_less_eq @ A @ B2 @ ( times_times @ A @ ( numeral_numeral @ A @ W ) @ C2 ) ) )
                & ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
                 => ( ord_less_eq @ A @ ( numeral_numeral @ A @ W ) @ ( zero_zero @ A ) ) ) ) ) ) ) ) ).

% le_divide_eq_numeral(1)
thf(fact_1063_divide__le__eq__numeral_I1_J,axiom,
    ! [A: $tType] :
      ( ( linordered_field @ A )
     => ! [B2: A,C2: A,W: num] :
          ( ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ C2 ) @ ( numeral_numeral @ A @ W ) )
          = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
             => ( ord_less_eq @ A @ B2 @ ( times_times @ A @ ( numeral_numeral @ A @ W ) @ C2 ) ) )
            & ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
             => ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
                 => ( ord_less_eq @ A @ ( times_times @ A @ ( numeral_numeral @ A @ W ) @ C2 ) @ B2 ) )
                & ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
                 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ W ) ) ) ) ) ) ) ) ).

% divide_le_eq_numeral(1)
thf(fact_1064_half__gt__zero__iff,axiom,
    ! [A: $tType] :
      ( ( linordered_field @ A )
     => ! [A2: A] :
          ( ( ord_less @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) )
          = ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).

% half_gt_zero_iff
thf(fact_1065_half__gt__zero,axiom,
    ! [A: $tType] :
      ( ( linordered_field @ A )
     => ! [A2: A] :
          ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
         => ( ord_less @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).

% half_gt_zero
thf(fact_1066_power2__le__imp__le,axiom,
    ! [A: $tType] :
      ( ( linordered_semidom @ A )
     => ! [X2: A,Y2: A] :
          ( ( ord_less_eq @ A @ ( power_power @ A @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
         => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y2 )
           => ( ord_less_eq @ A @ X2 @ Y2 ) ) ) ) ).

% power2_le_imp_le
thf(fact_1067_power2__eq__imp__eq,axiom,
    ! [A: $tType] :
      ( ( linordered_semidom @ A )
     => ! [X2: A,Y2: A] :
          ( ( ( power_power @ A @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
            = ( power_power @ A @ Y2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
         => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X2 )
           => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y2 )
             => ( X2 = Y2 ) ) ) ) ) ).

% power2_eq_imp_eq
thf(fact_1068_zero__le__power2,axiom,
    ! [A: $tType] :
      ( ( linordered_idom @ A )
     => ! [A2: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).

% zero_le_power2
thf(fact_1069_power2__less__0,axiom,
    ! [A: $tType] :
      ( ( linordered_idom @ A )
     => ! [A2: A] :
          ~ ( ord_less @ A @ ( power_power @ A @ A2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( zero_zero @ A ) ) ) ).

% power2_less_0
thf(fact_1070_exp__add__not__zero__imp__right,axiom,
    ! [A: $tType] :
      ( ( bit_semiring_bits @ A )
     => ! [M: nat,N2: nat] :
          ( ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( plus_plus @ nat @ M @ N2 ) )
           != ( zero_zero @ A ) )
         => ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 )
           != ( zero_zero @ A ) ) ) ) ).

% exp_add_not_zero_imp_right
thf(fact_1071_exp__add__not__zero__imp__left,axiom,
    ! [A: $tType] :
      ( ( bit_semiring_bits @ A )
     => ! [M: nat,N2: nat] :
          ( ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( plus_plus @ nat @ M @ N2 ) )
           != ( zero_zero @ A ) )
         => ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M )
           != ( zero_zero @ A ) ) ) ) ).

% exp_add_not_zero_imp_left
thf(fact_1072_exp__not__zero__imp__exp__diff__not__zero,axiom,
    ! [A: $tType] :
      ( ( bit_semiring_bits @ A )
     => ! [N2: nat,M: nat] :
          ( ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 )
           != ( zero_zero @ A ) )
         => ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ N2 @ M ) )
           != ( zero_zero @ A ) ) ) ) ).

% exp_not_zero_imp_exp_diff_not_zero
thf(fact_1073_power__diff__power__eq,axiom,
    ! [A: $tType] :
      ( ( euclid4440199948858584721cancel @ A )
     => ! [A2: A,N2: nat,M: nat] :
          ( ( A2
           != ( zero_zero @ A ) )
         => ( ( ( ord_less_eq @ nat @ N2 @ M )
             => ( ( divide_divide @ A @ ( power_power @ A @ A2 @ M ) @ ( power_power @ A @ A2 @ N2 ) )
                = ( power_power @ A @ A2 @ ( minus_minus @ nat @ M @ N2 ) ) ) )
            & ( ~ ( ord_less_eq @ nat @ N2 @ M )
             => ( ( divide_divide @ A @ ( power_power @ A @ A2 @ M ) @ ( power_power @ A @ A2 @ N2 ) )
                = ( divide_divide @ A @ ( one_one @ A ) @ ( power_power @ A @ A2 @ ( minus_minus @ nat @ N2 @ M ) ) ) ) ) ) ) ) ).

% power_diff_power_eq
thf(fact_1074_less__2__cases__iff,axiom,
    ! [N2: nat] :
      ( ( ord_less @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
      = ( ( N2
          = ( zero_zero @ nat ) )
        | ( N2
          = ( suc @ ( zero_zero @ nat ) ) ) ) ) ).

% less_2_cases_iff
thf(fact_1075_less__2__cases,axiom,
    ! [N2: nat] :
      ( ( ord_less @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
     => ( ( N2
          = ( zero_zero @ nat ) )
        | ( N2
          = ( suc @ ( zero_zero @ nat ) ) ) ) ) ).

% less_2_cases
thf(fact_1076_inverse__of__nat__le,axiom,
    ! [A: $tType] :
      ( ( linordered_field @ A )
     => ! [N2: nat,M: nat] :
          ( ( ord_less_eq @ nat @ N2 @ M )
         => ( ( N2
             != ( zero_zero @ nat ) )
           => ( ord_less_eq @ A @ ( divide_divide @ A @ ( one_one @ A ) @ ( semiring_1_of_nat @ A @ M ) ) @ ( divide_divide @ A @ ( one_one @ A ) @ ( semiring_1_of_nat @ A @ N2 ) ) ) ) ) ) ).

% inverse_of_nat_le
thf(fact_1077_power__eq__if,axiom,
    ! [A: $tType] :
      ( ( power @ A )
     => ( ( power_power @ A )
        = ( ^ [P5: A,M7: nat] :
              ( if @ A
              @ ( M7
                = ( zero_zero @ nat ) )
              @ ( one_one @ A )
              @ ( times_times @ A @ P5 @ ( power_power @ A @ P5 @ ( minus_minus @ nat @ M7 @ ( one_one @ nat ) ) ) ) ) ) ) ) ).

% power_eq_if
thf(fact_1078_power__minus__mult,axiom,
    ! [A: $tType] :
      ( ( monoid_mult @ A )
     => ! [N2: nat,A2: A] :
          ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
         => ( ( times_times @ A @ ( power_power @ A @ A2 @ ( minus_minus @ nat @ N2 @ ( one_one @ nat ) ) ) @ A2 )
            = ( power_power @ A @ A2 @ N2 ) ) ) ) ).

% power_minus_mult
thf(fact_1079_le__div__geq,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
     => ( ( ord_less_eq @ nat @ N2 @ M )
       => ( ( divide_divide @ nat @ M @ N2 )
          = ( suc @ ( divide_divide @ nat @ ( minus_minus @ nat @ M @ N2 ) @ N2 ) ) ) ) ) ).

% le_div_geq
thf(fact_1080_split__div_H,axiom,
    ! [P: nat > $o,M: nat,N2: nat] :
      ( ( P @ ( divide_divide @ nat @ M @ N2 ) )
      = ( ( ( N2
            = ( zero_zero @ nat ) )
          & ( P @ ( zero_zero @ nat ) ) )
        | ? [Q5: nat] :
            ( ( ord_less_eq @ nat @ ( times_times @ nat @ N2 @ Q5 ) @ M )
            & ( ord_less @ nat @ M @ ( times_times @ nat @ N2 @ ( suc @ Q5 ) ) )
            & ( P @ Q5 ) ) ) ) ).

% split_div'
thf(fact_1081_power__sub,axiom,
    ! [N2: nat,M: nat,A2: nat] :
      ( ( ord_less_eq @ nat @ N2 @ M )
     => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ A2 )
       => ( ( power_power @ nat @ A2 @ ( minus_minus @ nat @ M @ N2 ) )
          = ( divide_divide @ nat @ ( power_power @ nat @ A2 @ M ) @ ( power_power @ nat @ A2 @ N2 ) ) ) ) ) ).

% power_sub
thf(fact_1082_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 @ ( product_prod @ nat @ nat ) @ V ) @ ( suc @ ( zero_zero @ nat ) ) @ Vh @ Vi ) @ Vj )
      = ( none @ nat ) ) ).

% vebt_pred.simps(6)
thf(fact_1083_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 @ ( product_prod @ nat @ nat ) @ V ) @ ( suc @ ( zero_zero @ nat ) ) @ Vg @ Vh ) @ Vi )
      = ( none @ nat ) ) ).

% vebt_succ.simps(5)
thf(fact_1084_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 @ ( product_prod @ nat @ 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_1085_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 @ ( product_prod @ nat @ 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_1086_power2__less__imp__less,axiom,
    ! [A: $tType] :
      ( ( linordered_semidom @ A )
     => ! [X2: A,Y2: A] :
          ( ( ord_less @ A @ ( power_power @ A @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
         => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y2 )
           => ( ord_less @ A @ X2 @ Y2 ) ) ) ) ).

% power2_less_imp_less
thf(fact_1087_sum__power2__le__zero__iff,axiom,
    ! [A: $tType] :
      ( ( linordered_idom @ A )
     => ! [X2: A,Y2: A] :
          ( ( ord_less_eq @ A @ ( plus_plus @ A @ ( power_power @ A @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( zero_zero @ A ) )
          = ( ( X2
              = ( zero_zero @ A ) )
            & ( Y2
              = ( zero_zero @ A ) ) ) ) ) ).

% sum_power2_le_zero_iff
thf(fact_1088_sum__power2__ge__zero,axiom,
    ! [A: $tType] :
      ( ( linordered_idom @ A )
     => ! [X2: A,Y2: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ ( power_power @ A @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).

% sum_power2_ge_zero
thf(fact_1089_sum__power2__gt__zero__iff,axiom,
    ! [A: $tType] :
      ( ( linordered_idom @ A )
     => ! [X2: A,Y2: A] :
          ( ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ ( power_power @ A @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
          = ( ( X2
             != ( zero_zero @ A ) )
            | ( Y2
             != ( zero_zero @ A ) ) ) ) ) ).

% sum_power2_gt_zero_iff
thf(fact_1090_not__sum__power2__lt__zero,axiom,
    ! [A: $tType] :
      ( ( linordered_idom @ A )
     => ! [X2: A,Y2: A] :
          ~ ( ord_less @ A @ ( plus_plus @ A @ ( power_power @ A @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( zero_zero @ A ) ) ) ).

% not_sum_power2_lt_zero
thf(fact_1091_VEBT__internal_OminNulli_Oelims,axiom,
    ! [X2: vEBT_VEBTi,Y2: heap_Time_Heap @ $o] :
      ( ( ( vEBT_VEBT_minNulli @ X2 )
        = Y2 )
     => ( ( ( X2
            = ( vEBT_Leafi @ $false @ $false ) )
         => ( Y2
           != ( heap_Time_return @ $o @ $true ) ) )
       => ( ( ? [Uv2: $o] :
                ( X2
                = ( vEBT_Leafi @ $true @ Uv2 ) )
           => ( Y2
             != ( heap_Time_return @ $o @ $false ) ) )
         => ( ( ? [Uu2: $o] :
                  ( X2
                  = ( vEBT_Leafi @ Uu2 @ $true ) )
             => ( Y2
               != ( heap_Time_return @ $o @ $false ) ) )
           => ( ( ? [Uw2: nat,Ux2: array @ vEBT_VEBTi,Uy2: vEBT_VEBTi] :
                    ( X2
                    = ( vEBT_Nodei @ ( none @ ( product_prod @ nat @ nat ) ) @ Uw2 @ Ux2 @ Uy2 ) )
               => ( Y2
                 != ( heap_Time_return @ $o @ $true ) ) )
             => ~ ( ? [Uz2: product_prod @ nat @ nat,Va2: nat,Vb2: array @ vEBT_VEBTi,Vc: vEBT_VEBTi] :
                      ( X2
                      = ( vEBT_Nodei @ ( some @ ( product_prod @ nat @ nat ) @ Uz2 ) @ Va2 @ Vb2 @ Vc ) )
                 => ( Y2
                   != ( heap_Time_return @ $o @ $false ) ) ) ) ) ) ) ) ).

% VEBT_internal.minNulli.elims
thf(fact_1092_zero__le__even__power_H,axiom,
    ! [A: $tType] :
      ( ( linordered_idom @ A )
     => ! [A2: A,N2: nat] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A2 @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ) ) ).

% zero_le_even_power'
thf(fact_1093_Suc__n__div__2__gt__zero,axiom,
    ! [N2: nat] :
      ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
     => ( ord_less @ nat @ ( zero_zero @ nat ) @ ( divide_divide @ nat @ ( suc @ N2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).

% Suc_n_div_2_gt_zero
thf(fact_1094_div__2__gt__zero,axiom,
    ! [N2: nat] :
      ( ( ord_less @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N2 )
     => ( ord_less @ nat @ ( zero_zero @ nat ) @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).

% div_2_gt_zero
thf(fact_1095_nat__bit__induct,axiom,
    ! [P: nat > $o,N2: nat] :
      ( ( P @ ( zero_zero @ nat ) )
     => ( ! [N4: nat] :
            ( ( P @ N4 )
           => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N4 )
             => ( P @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N4 ) ) ) )
       => ( ! [N4: nat] :
              ( ( P @ N4 )
             => ( P @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N4 ) ) ) )
         => ( P @ N2 ) ) ) ) ).

% nat_bit_induct
thf(fact_1096_vebt__minti_Osimps_I1_J,axiom,
    ! [A2: $o,B2: $o] :
      ( ( A2
       => ( ( vEBT_vebt_minti @ ( vEBT_Leafi @ A2 @ B2 ) )
          = ( heap_Time_return @ ( option @ nat ) @ ( some @ nat @ ( zero_zero @ nat ) ) ) ) )
      & ( ~ A2
       => ( ( B2
           => ( ( vEBT_vebt_minti @ ( vEBT_Leafi @ A2 @ B2 ) )
              = ( heap_Time_return @ ( option @ nat ) @ ( some @ nat @ ( one_one @ nat ) ) ) ) )
          & ( ~ B2
           => ( ( vEBT_vebt_minti @ ( vEBT_Leafi @ A2 @ B2 ) )
              = ( heap_Time_return @ ( option @ nat ) @ ( none @ nat ) ) ) ) ) ) ) ).

% vebt_minti.simps(1)
thf(fact_1097_vebt__maxti_Osimps_I1_J,axiom,
    ! [B2: $o,A2: $o] :
      ( ( B2
       => ( ( vEBT_vebt_maxti @ ( vEBT_Leafi @ A2 @ B2 ) )
          = ( heap_Time_return @ ( option @ nat ) @ ( some @ nat @ ( one_one @ nat ) ) ) ) )
      & ( ~ B2
       => ( ( A2
           => ( ( vEBT_vebt_maxti @ ( vEBT_Leafi @ A2 @ B2 ) )
              = ( heap_Time_return @ ( option @ nat ) @ ( some @ nat @ ( zero_zero @ nat ) ) ) ) )
          & ( ~ A2
           => ( ( vEBT_vebt_maxti @ ( vEBT_Leafi @ A2 @ B2 ) )
              = ( heap_Time_return @ ( option @ nat ) @ ( none @ nat ) ) ) ) ) ) ) ).

% vebt_maxti.simps(1)
thf(fact_1098_mod__starE,axiom,
    ! [A2: assn,B2: assn,H2: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
      ( ( rep_assn @ ( times_times @ assn @ A2 @ B2 ) @ H2 )
     => ~ ( ? [X_12: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] : ( rep_assn @ A2 @ X_12 )
         => ! [H_2: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
              ~ ( rep_assn @ B2 @ H_2 ) ) ) ).

% mod_starE
thf(fact_1099_mod__starD,axiom,
    ! [A3: assn,B6: assn,H2: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
      ( ( rep_assn @ ( times_times @ assn @ A3 @ B6 ) @ H2 )
     => ? [H1: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ),H22: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
          ( ( rep_assn @ A3 @ H1 )
          & ( rep_assn @ B6 @ H22 ) ) ) ).

% mod_starD
thf(fact_1100_odd__0__le__power__imp__0__le,axiom,
    ! [A: $tType] :
      ( ( linordered_idom @ A )
     => ! [A2: A,N2: nat] :
          ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A2 @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ) )
         => ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).

% odd_0_le_power_imp_0_le
thf(fact_1101_odd__power__less__zero,axiom,
    ! [A: $tType] :
      ( ( linordered_idom @ A )
     => ! [A2: A,N2: nat] :
          ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
         => ( ord_less @ A @ ( power_power @ A @ A2 @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ) @ ( zero_zero @ A ) ) ) ) ).

% odd_power_less_zero
thf(fact_1102_assn__one__left,axiom,
    ! [P: assn] :
      ( ( times_times @ assn @ ( one_one @ assn ) @ P )
      = P ) ).

% assn_one_left
thf(fact_1103_VEBT__internal_Oexp__split__high__low_I1_J,axiom,
    ! [X2: nat,N2: nat,M: nat] :
      ( ( ord_less @ nat @ X2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( plus_plus @ nat @ N2 @ M ) ) )
     => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
       => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
         => ( ord_less @ nat @ ( vEBT_VEBT_high @ X2 @ N2 ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) ) ) ) ) ).

% VEBT_internal.exp_split_high_low(1)
thf(fact_1104_VEBT__internal_Oexp__split__high__low_I2_J,axiom,
    ! [X2: nat,N2: nat,M: nat] :
      ( ( ord_less @ nat @ X2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( plus_plus @ nat @ N2 @ M ) ) )
     => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
       => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
         => ( ord_less @ nat @ ( vEBT_VEBT_low @ X2 @ N2 ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ) ) ) ).

% VEBT_internal.exp_split_high_low(2)
thf(fact_1105_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,
    ! [Uu: nat,Uv: list @ vEBT_VEBT,Uw: vEBT_VEBT,X2: nat] :
      ( ( vEBT_T_m_e_m_b_e_r @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu @ Uv @ Uw ) @ X2 )
      = ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.simps(2)
thf(fact_1106_nat__div__eq__Suc__0__iff,axiom,
    ! [N2: nat,M: nat] :
      ( ( ( divide_divide @ nat @ N2 @ M )
        = ( suc @ ( zero_zero @ nat ) ) )
      = ( ( ord_less_eq @ nat @ M @ N2 )
        & ( ord_less @ nat @ N2 @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) ) ) ) ).

% nat_div_eq_Suc_0_iff
thf(fact_1107_ex__distrib__star,axiom,
    ! [A: $tType,P: A > assn,Q: assn] :
      ( ( ex_assn @ A
        @ ^ [X: A] : ( times_times @ assn @ ( P @ X ) @ Q ) )
      = ( times_times @ assn @ ( ex_assn @ A @ P ) @ Q ) ) ).

% ex_distrib_star
thf(fact_1108_ex__one__point__gen,axiom,
    ! [A: $tType,P: A > assn,V: A] :
      ( ! [H3: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ),X3: A] :
          ( ( rep_assn @ ( P @ X3 ) @ H3 )
         => ( X3 = V ) )
     => ( ( ex_assn @ A @ P )
        = ( P @ V ) ) ) ).

% ex_one_point_gen
thf(fact_1109_mod__exI,axiom,
    ! [A: $tType,P: A > assn,H2: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
      ( ? [X6: A] : ( rep_assn @ ( P @ X6 ) @ H2 )
     => ( rep_assn @ ( ex_assn @ A @ P ) @ H2 ) ) ).

% mod_exI
thf(fact_1110_mod__exE,axiom,
    ! [A: $tType,P: A > assn,H2: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
      ( ( rep_assn @ ( ex_assn @ A @ P ) @ H2 )
     => ~ ! [X3: A] :
            ~ ( rep_assn @ ( P @ X3 ) @ H2 ) ) ).

% mod_exE
thf(fact_1111_arith__geo__mean,axiom,
    ! [A: $tType] :
      ( ( linordered_field @ A )
     => ! [U: A,X2: A,Y2: A] :
          ( ( ( power_power @ A @ U @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
            = ( times_times @ A @ X2 @ Y2 ) )
         => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X2 )
           => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y2 )
             => ( ord_less_eq @ A @ U @ ( divide_divide @ A @ ( plus_plus @ A @ X2 @ Y2 ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).

% arith_geo_mean
thf(fact_1112_vebt__minti_Oelims,axiom,
    ! [X2: vEBT_VEBTi,Y2: heap_Time_Heap @ ( option @ nat )] :
      ( ( ( vEBT_vebt_minti @ X2 )
        = Y2 )
     => ( ! [A4: $o,B3: $o] :
            ( ( X2
              = ( vEBT_Leafi @ A4 @ B3 ) )
           => ~ ( ( A4
                 => ( Y2
                    = ( heap_Time_return @ ( option @ nat ) @ ( some @ nat @ ( zero_zero @ nat ) ) ) ) )
                & ( ~ A4
                 => ( ( B3
                     => ( Y2
                        = ( heap_Time_return @ ( option @ nat ) @ ( some @ nat @ ( one_one @ nat ) ) ) ) )
                    & ( ~ B3
                     => ( Y2
                        = ( heap_Time_return @ ( option @ nat ) @ ( none @ nat ) ) ) ) ) ) ) )
       => ( ( ? [Uu2: nat,Uv2: array @ vEBT_VEBTi,Uw2: vEBT_VEBTi] :
                ( X2
                = ( vEBT_Nodei @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw2 ) )
           => ( Y2
             != ( heap_Time_return @ ( option @ nat ) @ ( none @ nat ) ) ) )
         => ~ ! [Mi2: nat] :
                ( ? [Ma2: nat,Ux2: nat,Uy2: array @ vEBT_VEBTi,Uz2: vEBT_VEBTi] :
                    ( X2
                    = ( vEBT_Nodei @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
               => ( Y2
                 != ( heap_Time_return @ ( option @ nat ) @ ( some @ nat @ Mi2 ) ) ) ) ) ) ) ).

% vebt_minti.elims
thf(fact_1113_vebt__maxti_Oelims,axiom,
    ! [X2: vEBT_VEBTi,Y2: heap_Time_Heap @ ( option @ nat )] :
      ( ( ( vEBT_vebt_maxti @ X2 )
        = Y2 )
     => ( ! [A4: $o,B3: $o] :
            ( ( X2
              = ( vEBT_Leafi @ A4 @ B3 ) )
           => ~ ( ( B3
                 => ( Y2
                    = ( heap_Time_return @ ( option @ nat ) @ ( some @ nat @ ( one_one @ nat ) ) ) ) )
                & ( ~ B3
                 => ( ( A4
                     => ( Y2
                        = ( heap_Time_return @ ( option @ nat ) @ ( some @ nat @ ( zero_zero @ nat ) ) ) ) )
                    & ( ~ A4
                     => ( Y2
                        = ( heap_Time_return @ ( option @ nat ) @ ( none @ nat ) ) ) ) ) ) ) )
       => ( ( ? [Uu2: nat,Uv2: array @ vEBT_VEBTi,Uw2: vEBT_VEBTi] :
                ( X2
                = ( vEBT_Nodei @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw2 ) )
           => ( Y2
             != ( heap_Time_return @ ( option @ nat ) @ ( none @ nat ) ) ) )
         => ~ ! [Mi2: nat,Ma2: nat] :
                ( ? [Ux2: nat,Uy2: array @ vEBT_VEBTi,Uz2: vEBT_VEBTi] :
                    ( X2
                    = ( vEBT_Nodei @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
               => ( Y2
                 != ( heap_Time_return @ ( option @ nat ) @ ( some @ nat @ Ma2 ) ) ) ) ) ) ) ).

% vebt_maxti.elims
thf(fact_1114_member__bound__height,axiom,
    ! [T2: vEBT_VEBT,N2: nat,X2: nat] :
      ( ( vEBT_invar_vebt @ T2 @ N2 )
     => ( ord_less_eq @ nat @ ( vEBT_T_m_e_m_b_e_r @ T2 @ X2 ) @ ( times_times @ nat @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_VEBT_height @ T2 ) ) @ ( numeral_numeral @ nat @ ( bit1 @ ( bit1 @ ( bit1 @ one2 ) ) ) ) ) ) ) ).

% member_bound_height
thf(fact_1115_minNrulli__ruleT,axiom,
    ! [T2: vEBT_VEBT,Ti: vEBT_VEBTi] :
      ( time_htt @ $o @ ( vEBT_vebt_assn_raw @ T2 @ Ti ) @ ( vEBT_VEBT_minNulli @ Ti )
      @ ^ [R: $o] :
          ( times_times @ assn @ ( vEBT_vebt_assn_raw @ T2 @ Ti )
          @ ( pure_assn
            @ ( R
              = ( vEBT_VEBT_minNull @ T2 ) ) ) )
      @ ( one_one @ nat ) ) ).

% minNrulli_ruleT
thf(fact_1116_Tb__T__vebt__buildupi,axiom,
    ! [N2: nat] : ( ord_less_eq @ int @ ( semiring_1_of_nat @ int @ ( vEBT_V441764108873111860ildupi @ N2 ) ) @ ( minus_minus @ int @ ( vEBT_VEBT_Tb @ N2 ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ).

% Tb_T_vebt_buildupi
thf(fact_1117_le__divide__eq__1__pos,axiom,
    ! [A: $tType] :
      ( ( linordered_field @ A )
     => ! [A2: A,B2: A] :
          ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
         => ( ( ord_less_eq @ A @ ( one_one @ A ) @ ( divide_divide @ A @ B2 @ A2 ) )
            = ( ord_less_eq @ A @ A2 @ B2 ) ) ) ) ).

% le_divide_eq_1_pos
thf(fact_1118_le__divide__eq__1__neg,axiom,
    ! [A: $tType] :
      ( ( linordered_field @ A )
     => ! [A2: A,B2: A] :
          ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
         => ( ( ord_less_eq @ A @ ( one_one @ A ) @ ( divide_divide @ A @ B2 @ A2 ) )
            = ( ord_less_eq @ A @ B2 @ A2 ) ) ) ) ).

% le_divide_eq_1_neg
thf(fact_1119_divide__le__eq__1__pos,axiom,
    ! [A: $tType] :
      ( ( linordered_field @ A )
     => ! [A2: A,B2: A] :
          ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
         => ( ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ A2 ) @ ( one_one @ A ) )
            = ( ord_less_eq @ A @ B2 @ A2 ) ) ) ) ).

% divide_le_eq_1_pos
thf(fact_1120_divide__le__eq__1__neg,axiom,
    ! [A: $tType] :
      ( ( linordered_field @ A )
     => ! [A2: A,B2: A] :
          ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
         => ( ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ A2 ) @ ( one_one @ A ) )
            = ( ord_less_eq @ A @ A2 @ B2 ) ) ) ) ).

% divide_le_eq_1_neg
thf(fact_1121_T__vebt__buildupi,axiom,
    ! [N2: nat,H2: heap_ext @ product_unit] : ( ord_less_eq @ nat @ ( time_time @ vEBT_VEBTi @ ( vEBT_V739175172307565963ildupi @ N2 ) @ H2 ) @ ( vEBT_V441764108873111860ildupi @ N2 ) ) ).

% T_vebt_buildupi
thf(fact_1122_nonzero__divide__mult__cancel__right,axiom,
    ! [A: $tType] :
      ( ( field @ A )
     => ! [B2: A,A2: A] :
          ( ( B2
           != ( zero_zero @ A ) )
         => ( ( divide_divide @ A @ B2 @ ( times_times @ A @ A2 @ B2 ) )
            = ( divide_divide @ A @ ( one_one @ A ) @ A2 ) ) ) ) ).

% nonzero_divide_mult_cancel_right
thf(fact_1123_nonzero__divide__mult__cancel__left,axiom,
    ! [A: $tType] :
      ( ( field @ A )
     => ! [A2: A,B2: A] :
          ( ( A2
           != ( zero_zero @ A ) )
         => ( ( divide_divide @ A @ A2 @ ( times_times @ A @ A2 @ B2 ) )
            = ( divide_divide @ A @ ( one_one @ A ) @ B2 ) ) ) ) ).

% nonzero_divide_mult_cancel_left
thf(fact_1124_zero__less__divide__1__iff,axiom,
    ! [A: $tType] :
      ( ( linordered_field @ A )
     => ! [A2: A] :
          ( ( ord_less @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ ( one_one @ A ) @ A2 ) )
          = ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).

% zero_less_divide_1_iff
thf(fact_1125_cnt__non__neg,axiom,
    ! [T2: vEBT_VEBT] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( vEBT_VEBT_cnt @ T2 ) ) ).

% cnt_non_neg
thf(fact_1126_Tb__Tb_H,axiom,
    ( vEBT_VEBT_Tb
    = ( ^ [T3: nat] : ( semiring_1_of_nat @ int @ ( vEBT_VEBT_Tb2 @ T3 ) ) ) ) ).

% Tb_Tb'
thf(fact_1127_time__replicate,axiom,
    ! [A: $tType,X2: heap_Time_Heap @ A,C2: nat,N2: nat,H2: heap_ext @ product_unit] :
      ( ! [H3: heap_ext @ product_unit] : ( ord_less_eq @ nat @ ( time_time @ A @ X2 @ H3 ) @ C2 )
     => ( ord_less_eq @ nat @ ( time_time @ ( list @ A ) @ ( vEBT_VEBT_replicatei @ A @ N2 @ X2 ) @ H2 ) @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( times_times @ nat @ ( plus_plus @ nat @ ( one_one @ nat ) @ C2 ) @ N2 ) ) ) ) ).

% time_replicate
thf(fact_1128_divide__eq__0__iff,axiom,
    ! [A: $tType] :
      ( ( field @ A )
     => ! [A2: A,B2: A] :
          ( ( ( divide_divide @ A @ A2 @ B2 )
            = ( zero_zero @ A ) )
          = ( ( A2
              = ( zero_zero @ A ) )
            | ( B2
              = ( zero_zero @ A ) ) ) ) ) ).

% divide_eq_0_iff
thf(fact_1129_divide__cancel__left,axiom,
    ! [A: $tType] :
      ( ( field @ A )
     => ! [C2: A,A2: A,B2: A] :
          ( ( ( divide_divide @ A @ C2 @ A2 )
            = ( divide_divide @ A @ C2 @ B2 ) )
          = ( ( C2
              = ( zero_zero @ A ) )
            | ( A2 = B2 ) ) ) ) ).

% divide_cancel_left
thf(fact_1130_divide__cancel__right,axiom,
    ! [A: $tType] :
      ( ( field @ A )
     => ! [A2: A,C2: A,B2: A] :
          ( ( ( divide_divide @ A @ A2 @ C2 )
            = ( divide_divide @ A @ B2 @ C2 ) )
          = ( ( C2
              = ( zero_zero @ A ) )
            | ( A2 = B2 ) ) ) ) ).

% divide_cancel_right
thf(fact_1131_division__ring__divide__zero,axiom,
    ! [A: $tType] :
      ( ( division_ring @ A )
     => ! [A2: A] :
          ( ( divide_divide @ A @ A2 @ ( zero_zero @ A ) )
          = ( zero_zero @ A ) ) ) ).

% division_ring_divide_zero
thf(fact_1132_times__divide__eq__left,axiom,
    ! [A: $tType] :
      ( ( field @ A )
     => ! [B2: A,C2: A,A2: A] :
          ( ( times_times @ A @ ( divide_divide @ A @ B2 @ C2 ) @ A2 )
          = ( divide_divide @ A @ ( times_times @ A @ B2 @ A2 ) @ C2 ) ) ) ).

% times_divide_eq_left
thf(fact_1133_divide__divide__eq__left,axiom,
    ! [A: $tType] :
      ( ( field @ A )
     => ! [A2: A,B2: A,C2: A] :
          ( ( divide_divide @ A @ ( divide_divide @ A @ A2 @ B2 ) @ C2 )
          = ( divide_divide @ A @ A2 @ ( times_times @ A @ B2 @ C2 ) ) ) ) ).

% divide_divide_eq_left
thf(fact_1134_divide__divide__eq__right,axiom,
    ! [A: $tType] :
      ( ( field @ A )
     => ! [A2: A,B2: A,C2: A] :
          ( ( divide_divide @ A @ A2 @ ( divide_divide @ A @ B2 @ C2 ) )
          = ( divide_divide @ A @ ( times_times @ A @ A2 @ C2 ) @ B2 ) ) ) ).

% divide_divide_eq_right
thf(fact_1135_times__divide__eq__right,axiom,
    ! [A: $tType] :
      ( ( division_ring @ A )
     => ! [A2: A,B2: A,C2: A] :
          ( ( times_times @ A @ A2 @ ( divide_divide @ A @ B2 @ C2 ) )
          = ( divide_divide @ A @ ( times_times @ A @ A2 @ B2 ) @ C2 ) ) ) ).

% times_divide_eq_right
thf(fact_1136_not__real__square__gt__zero,axiom,
    ! [X2: real] :
      ( ( ~ ( ord_less @ real @ ( zero_zero @ real ) @ ( times_times @ real @ X2 @ X2 ) ) )
      = ( X2
        = ( zero_zero @ real ) ) ) ).

% not_real_square_gt_zero
thf(fact_1137_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_1138_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_1139_int__div__same__is__1,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less @ int @ ( zero_zero @ int ) @ A2 )
     => ( ( ( divide_divide @ int @ A2 @ B2 )
          = A2 )
        = ( B2
          = ( one_one @ int ) ) ) ) ).

% int_div_same_is_1
thf(fact_1140_divide__eq__1__iff,axiom,
    ! [A: $tType] :
      ( ( field @ A )
     => ! [A2: A,B2: A] :
          ( ( ( divide_divide @ A @ A2 @ B2 )
            = ( one_one @ A ) )
          = ( ( B2
             != ( zero_zero @ A ) )
            & ( A2 = B2 ) ) ) ) ).

% divide_eq_1_iff
thf(fact_1141_one__eq__divide__iff,axiom,
    ! [A: $tType] :
      ( ( field @ A )
     => ! [A2: A,B2: A] :
          ( ( ( one_one @ A )
            = ( divide_divide @ A @ A2 @ B2 ) )
          = ( ( B2
             != ( zero_zero @ A ) )
            & ( A2 = B2 ) ) ) ) ).

% one_eq_divide_iff
thf(fact_1142_divide__self,axiom,
    ! [A: $tType] :
      ( ( division_ring @ A )
     => ! [A2: A] :
          ( ( A2
           != ( zero_zero @ A ) )
         => ( ( divide_divide @ A @ A2 @ A2 )
            = ( one_one @ A ) ) ) ) ).

% divide_self
thf(fact_1143_divide__self__if,axiom,
    ! [A: $tType] :
      ( ( division_ring @ A )
     => ! [A2: A] :
          ( ( ( A2
              = ( zero_zero @ A ) )
           => ( ( divide_divide @ A @ A2 @ A2 )
              = ( zero_zero @ A ) ) )
          & ( ( A2
             != ( zero_zero @ A ) )
           => ( ( divide_divide @ A @ A2 @ A2 )
              = ( one_one @ A ) ) ) ) ) ).

% divide_self_if
thf(fact_1144_divide__eq__eq__1,axiom,
    ! [A: $tType] :
      ( ( linordered_field @ A )
     => ! [B2: A,A2: A] :
          ( ( ( divide_divide @ A @ B2 @ A2 )
            = ( one_one @ A ) )
          = ( ( A2
             != ( zero_zero @ A ) )
            & ( A2 = B2 ) ) ) ) ).

% divide_eq_eq_1
thf(fact_1145_eq__divide__eq__1,axiom,
    ! [A: $tType] :
      ( ( linordered_field @ A )
     => ! [B2: A,A2: A] :
          ( ( ( one_one @ A )
            = ( divide_divide @ A @ B2 @ A2 ) )
          = ( ( A2
             != ( zero_zero @ A ) )
            & ( A2 = B2 ) ) ) ) ).

% eq_divide_eq_1
thf(fact_1146_one__divide__eq__0__iff,axiom,
    ! [A: $tType] :
      ( ( linordered_field @ A )
     => ! [A2: A] :
          ( ( ( divide_divide @ A @ ( one_one @ A ) @ A2 )
            = ( zero_zero @ A ) )
          = ( A2
            = ( zero_zero @ A ) ) ) ) ).

% one_divide_eq_0_iff
thf(fact_1147_zero__eq__1__divide__iff,axiom,
    ! [A: $tType] :
      ( ( linordered_field @ A )
     => ! [A2: A] :
          ( ( ( zero_zero @ A )
            = ( divide_divide @ A @ ( one_one @ A ) @ A2 ) )
          = ( A2
            = ( zero_zero @ A ) ) ) ) ).

% zero_eq_1_divide_iff
thf(fact_1148_mult__divide__mult__cancel__left__if,axiom,
    ! [A: $tType] :
      ( ( field @ A )
     => ! [C2: A,A2: A,B2: A] :
          ( ( ( C2
              = ( zero_zero @ A ) )
           => ( ( divide_divide @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B2 ) )
              = ( zero_zero @ A ) ) )
          & ( ( C2
             != ( zero_zero @ A ) )
           => ( ( divide_divide @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B2 ) )
              = ( divide_divide @ A @ A2 @ B2 ) ) ) ) ) ).

% mult_divide_mult_cancel_left_if
thf(fact_1149_nonzero__mult__divide__mult__cancel__left,axiom,
    ! [A: $tType] :
      ( ( field @ A )
     => ! [C2: A,A2: A,B2: A] :
          ( ( C2
           != ( zero_zero @ A ) )
         => ( ( divide_divide @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B2 ) )
            = ( divide_divide @ A @ A2 @ B2 ) ) ) ) ).

% nonzero_mult_divide_mult_cancel_left
thf(fact_1150_nonzero__mult__divide__mult__cancel__left2,axiom,
    ! [A: $tType] :
      ( ( field @ A )
     => ! [C2: A,A2: A,B2: A] :
          ( ( C2
           != ( zero_zero @ A ) )
         => ( ( divide_divide @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ B2 @ C2 ) )
            = ( divide_divide @ A @ A2 @ B2 ) ) ) ) ).

% nonzero_mult_divide_mult_cancel_left2
thf(fact_1151_nonzero__mult__divide__mult__cancel__right,axiom,
    ! [A: $tType] :
      ( ( field @ A )
     => ! [C2: A,A2: A,B2: A] :
          ( ( C2
           != ( zero_zero @ A ) )
         => ( ( divide_divide @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) )
            = ( divide_divide @ A @ A2 @ B2 ) ) ) ) ).

% nonzero_mult_divide_mult_cancel_right
thf(fact_1152_nonzero__mult__divide__mult__cancel__right2,axiom,
    ! [A: $tType] :
      ( ( field @ A )
     => ! [C2: A,A2: A,B2: A] :
          ( ( C2
           != ( zero_zero @ A ) )
         => ( ( divide_divide @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ C2 @ B2 ) )
            = ( divide_divide @ A @ A2 @ B2 ) ) ) ) ).

% nonzero_mult_divide_mult_cancel_right2
thf(fact_1153_divide__le__0__1__iff,axiom,
    ! [A: $tType] :
      ( ( linordered_field @ A )
     => ! [A2: A] :
          ( ( ord_less_eq @ A @ ( divide_divide @ A @ ( one_one @ A ) @ A2 ) @ ( zero_zero @ A ) )
          = ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).

% divide_le_0_1_iff
thf(fact_1154_zero__le__divide__1__iff,axiom,
    ! [A: $tType] :
      ( ( linordered_field @ A )
     => ! [A2: A] :
          ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ ( one_one @ A ) @ A2 ) )
          = ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).

% zero_le_divide_1_iff
thf(fact_1155_divide__less__0__1__iff,axiom,
    ! [A: $tType] :
      ( ( linordered_field @ A )
     => ! [A2: A] :
          ( ( ord_less @ A @ ( divide_divide @ A @ ( one_one @ A ) @ A2 ) @ ( zero_zero @ A ) )
          = ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).

% divide_less_0_1_iff
thf(fact_1156_divide__less__eq__1__neg,axiom,
    ! [A: $tType] :
      ( ( linordered_field @ A )
     => ! [A2: A,B2: A] :
          ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
         => ( ( ord_less @ A @ ( divide_divide @ A @ B2 @ A2 ) @ ( one_one @ A ) )
            = ( ord_less @ A @ A2 @ B2 ) ) ) ) ).

% divide_less_eq_1_neg
thf(fact_1157_divide__less__eq__1__pos,axiom,
    ! [A: $tType] :
      ( ( linordered_field @ A )
     => ! [A2: A,B2: A] :
          ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
         => ( ( ord_less @ A @ ( divide_divide @ A @ B2 @ A2 ) @ ( one_one @ A ) )
            = ( ord_less @ A @ B2 @ A2 ) ) ) ) ).

% divide_less_eq_1_pos
thf(fact_1158_less__divide__eq__1__neg,axiom,
    ! [A: $tType] :
      ( ( linordered_field @ A )
     => ! [A2: A,B2: A] :
          ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
         => ( ( ord_less @ A @ ( one_one @ A ) @ ( divide_divide @ A @ B2 @ A2 ) )
            = ( ord_less @ A @ B2 @ A2 ) ) ) ) ).

% less_divide_eq_1_neg
thf(fact_1159_less__divide__eq__1__pos,axiom,
    ! [A: $tType] :
      ( ( linordered_field @ A )
     => ! [A2: A,B2: A] :
          ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
         => ( ( ord_less @ A @ ( one_one @ A ) @ ( divide_divide @ A @ B2 @ A2 ) )
            = ( ord_less @ A @ A2 @ B2 ) ) ) ) ).

% less_divide_eq_1_pos
thf(fact_1160_half__negative__int__iff,axiom,
    ! [K: int] :
      ( ( ord_less @ int @ ( divide_divide @ int @ K @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( zero_zero @ int ) )
      = ( ord_less @ int @ K @ ( zero_zero @ int ) ) ) ).

% half_negative_int_iff
thf(fact_1161_half__nonnegative__int__iff,axiom,
    ! [K: int] :
      ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( divide_divide @ int @ K @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) )
      = ( ord_less_eq @ int @ ( zero_zero @ int ) @ K ) ) ).

% half_nonnegative_int_iff
thf(fact_1162_htt__vebt__memberi,axiom,
    ! [T2: vEBT_VEBT,Ti: vEBT_VEBTi,X2: nat] :
      ( time_htt @ $o @ ( vEBT_vebt_assn_raw @ T2 @ Ti ) @ ( vEBT_vebt_memberi @ Ti @ X2 )
      @ ^ [R: $o] :
          ( times_times @ assn @ ( vEBT_vebt_assn_raw @ T2 @ Ti )
          @ ( pure_assn
            @ ( R
              = ( vEBT_vebt_member @ T2 @ X2 ) ) ) )
      @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit0 @ one2 ) ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit0 @ one2 ) ) ) @ ( vEBT_VEBT_height @ T2 ) ) ) ) ).

% htt_vebt_memberi
thf(fact_1163_nonneg1__imp__zdiv__pos__iff,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ A2 )
     => ( ( ord_less @ int @ ( zero_zero @ int ) @ ( divide_divide @ int @ A2 @ B2 ) )
        = ( ( ord_less_eq @ int @ B2 @ A2 )
          & ( ord_less @ int @ ( zero_zero @ int ) @ B2 ) ) ) ) ).

% nonneg1_imp_zdiv_pos_iff
thf(fact_1164_pos__imp__zdiv__nonneg__iff,axiom,
    ! [B2: int,A2: int] :
      ( ( ord_less @ int @ ( zero_zero @ int ) @ B2 )
     => ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( divide_divide @ int @ A2 @ B2 ) )
        = ( ord_less_eq @ int @ ( zero_zero @ int ) @ A2 ) ) ) ).

% pos_imp_zdiv_nonneg_iff
thf(fact_1165_neg__imp__zdiv__nonneg__iff,axiom,
    ! [B2: int,A2: int] :
      ( ( ord_less @ int @ B2 @ ( zero_zero @ int ) )
     => ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( divide_divide @ int @ A2 @ B2 ) )
        = ( ord_less_eq @ int @ A2 @ ( zero_zero @ int ) ) ) ) ).

% neg_imp_zdiv_nonneg_iff
thf(fact_1166_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_1167_pos__imp__zdiv__neg__iff,axiom,
    ! [B2: int,A2: int] :
      ( ( ord_less @ int @ ( zero_zero @ int ) @ B2 )
     => ( ( ord_less @ int @ ( divide_divide @ int @ A2 @ B2 ) @ ( zero_zero @ int ) )
        = ( ord_less @ int @ A2 @ ( zero_zero @ int ) ) ) ) ).

% pos_imp_zdiv_neg_iff
thf(fact_1168_neg__imp__zdiv__neg__iff,axiom,
    ! [B2: int,A2: int] :
      ( ( ord_less @ int @ B2 @ ( zero_zero @ int ) )
     => ( ( ord_less @ int @ ( divide_divide @ int @ A2 @ B2 ) @ ( zero_zero @ int ) )
        = ( ord_less @ int @ ( zero_zero @ int ) @ A2 ) ) ) ).

% neg_imp_zdiv_neg_iff
thf(fact_1169_div__nonpos__pos__le0,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_eq @ int @ A2 @ ( zero_zero @ int ) )
     => ( ( ord_less @ int @ ( zero_zero @ int ) @ B2 )
       => ( ord_less_eq @ int @ ( divide_divide @ int @ A2 @ B2 ) @ ( zero_zero @ int ) ) ) ) ).

% div_nonpos_pos_le0
thf(fact_1170_div__nonneg__neg__le0,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ A2 )
     => ( ( ord_less @ int @ B2 @ ( zero_zero @ int ) )
       => ( ord_less_eq @ int @ ( divide_divide @ int @ A2 @ B2 ) @ ( zero_zero @ int ) ) ) ) ).

% div_nonneg_neg_le0
thf(fact_1171_div__neg__pos__less0,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less @ int @ A2 @ ( zero_zero @ int ) )
     => ( ( ord_less @ int @ ( zero_zero @ int ) @ B2 )
       => ( ord_less @ int @ ( divide_divide @ int @ A2 @ B2 ) @ ( zero_zero @ int ) ) ) ) ).

% div_neg_pos_less0
thf(fact_1172_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_1173_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_1174_zdiv__mono2__neg,axiom,
    ! [A2: int,B4: int,B2: int] :
      ( ( ord_less @ int @ A2 @ ( zero_zero @ int ) )
     => ( ( ord_less @ int @ ( zero_zero @ int ) @ B4 )
       => ( ( ord_less_eq @ int @ B4 @ B2 )
         => ( ord_less_eq @ int @ ( divide_divide @ int @ A2 @ B4 ) @ ( divide_divide @ int @ A2 @ B2 ) ) ) ) ) ).

% zdiv_mono2_neg
thf(fact_1175_zdiv__mono1__neg,axiom,
    ! [A2: int,A6: int,B2: int] :
      ( ( ord_less_eq @ int @ A2 @ A6 )
     => ( ( ord_less @ int @ B2 @ ( zero_zero @ int ) )
       => ( ord_less_eq @ int @ ( divide_divide @ int @ A6 @ B2 ) @ ( divide_divide @ int @ A2 @ B2 ) ) ) ) ).

% zdiv_mono1_neg
thf(fact_1176_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_1177_zdiv__mono2,axiom,
    ! [A2: int,B4: int,B2: int] :
      ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ A2 )
     => ( ( ord_less @ int @ ( zero_zero @ int ) @ B4 )
       => ( ( ord_less_eq @ int @ B4 @ B2 )
         => ( ord_less_eq @ int @ ( divide_divide @ int @ A2 @ B2 ) @ ( divide_divide @ int @ A2 @ B4 ) ) ) ) ) ).

% zdiv_mono2
thf(fact_1178_zdiv__mono1,axiom,
    ! [A2: int,A6: int,B2: int] :
      ( ( ord_less_eq @ int @ A2 @ A6 )
     => ( ( ord_less @ int @ ( zero_zero @ int ) @ B2 )
       => ( ord_less_eq @ int @ ( divide_divide @ int @ A2 @ B2 ) @ ( divide_divide @ int @ A6 @ B2 ) ) ) ) ).

% zdiv_mono1
thf(fact_1179_zdiv__le__dividend,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ A2 )
     => ( ( ord_less @ int @ ( zero_zero @ int ) @ B2 )
       => ( ord_less_eq @ int @ ( divide_divide @ int @ A2 @ B2 ) @ A2 ) ) ) ).

% zdiv_le_dividend
thf(fact_1180_q__pos__lemma,axiom,
    ! [B4: int,Q6: int,R5: int] :
      ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( plus_plus @ int @ ( times_times @ int @ B4 @ Q6 ) @ R5 ) )
     => ( ( ord_less @ int @ R5 @ B4 )
       => ( ( ord_less @ int @ ( zero_zero @ int ) @ B4 )
         => ( ord_less_eq @ int @ ( zero_zero @ int ) @ Q6 ) ) ) ) ).

% q_pos_lemma
thf(fact_1181_zdiv__mono2__lemma,axiom,
    ! [B2: int,Q2: int,R4: int,B4: int,Q6: int,R5: int] :
      ( ( ( plus_plus @ int @ ( times_times @ int @ B2 @ Q2 ) @ R4 )
        = ( plus_plus @ int @ ( times_times @ int @ B4 @ Q6 ) @ R5 ) )
     => ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( plus_plus @ int @ ( times_times @ int @ B4 @ Q6 ) @ R5 ) )
       => ( ( ord_less @ int @ R5 @ B4 )
         => ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ R4 )
           => ( ( ord_less @ int @ ( zero_zero @ int ) @ B4 )
             => ( ( ord_less_eq @ int @ B4 @ B2 )
               => ( ord_less_eq @ int @ Q2 @ Q6 ) ) ) ) ) ) ) ).

% zdiv_mono2_lemma
thf(fact_1182_zdiv__mono2__neg__lemma,axiom,
    ! [B2: int,Q2: int,R4: int,B4: int,Q6: int,R5: int] :
      ( ( ( plus_plus @ int @ ( times_times @ int @ B2 @ Q2 ) @ R4 )
        = ( plus_plus @ int @ ( times_times @ int @ B4 @ Q6 ) @ R5 ) )
     => ( ( ord_less @ int @ ( plus_plus @ int @ ( times_times @ int @ B4 @ Q6 ) @ R5 ) @ ( zero_zero @ int ) )
       => ( ( ord_less @ int @ R4 @ B2 )
         => ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ R5 )
           => ( ( ord_less @ int @ ( zero_zero @ int ) @ B4 )
             => ( ( ord_less_eq @ int @ B4 @ B2 )
               => ( ord_less_eq @ int @ Q6 @ Q2 ) ) ) ) ) ) ) ).

% zdiv_mono2_neg_lemma
thf(fact_1183_unique__quotient__lemma,axiom,
    ! [B2: int,Q6: int,R5: int,Q2: int,R4: int] :
      ( ( ord_less_eq @ int @ ( plus_plus @ int @ ( times_times @ int @ B2 @ Q6 ) @ R5 ) @ ( plus_plus @ int @ ( times_times @ int @ B2 @ Q2 ) @ R4 ) )
     => ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ R5 )
       => ( ( ord_less @ int @ R5 @ B2 )
         => ( ( ord_less @ int @ R4 @ B2 )
           => ( ord_less_eq @ int @ Q6 @ Q2 ) ) ) ) ) ).

% unique_quotient_lemma
thf(fact_1184_unique__quotient__lemma__neg,axiom,
    ! [B2: int,Q6: int,R5: int,Q2: int,R4: int] :
      ( ( ord_less_eq @ int @ ( plus_plus @ int @ ( times_times @ int @ B2 @ Q6 ) @ R5 ) @ ( plus_plus @ int @ ( times_times @ int @ B2 @ Q2 ) @ R4 ) )
     => ( ( ord_less_eq @ int @ R4 @ ( zero_zero @ int ) )
       => ( ( ord_less @ int @ B2 @ R4 )
         => ( ( ord_less @ int @ B2 @ R5 )
           => ( ord_less_eq @ int @ Q2 @ Q6 ) ) ) ) ) ).

% unique_quotient_lemma_neg
thf(fact_1185_int__div__less__self,axiom,
    ! [X2: int,K: int] :
      ( ( ord_less @ int @ ( zero_zero @ int ) @ X2 )
     => ( ( ord_less @ int @ ( one_one @ int ) @ K )
       => ( ord_less @ int @ ( divide_divide @ int @ X2 @ K ) @ X2 ) ) ) ).

% int_div_less_self
thf(fact_1186_zdiv__zmult2__eq,axiom,
    ! [C2: int,A2: int,B2: int] :
      ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ C2 )
     => ( ( divide_divide @ int @ A2 @ ( times_times @ int @ B2 @ C2 ) )
        = ( divide_divide @ int @ ( divide_divide @ int @ A2 @ B2 ) @ C2 ) ) ) ).

% zdiv_zmult2_eq
thf(fact_1187_linordered__field__no__lb,axiom,
    ! [A: $tType] :
      ( ( linordered_field @ A )
     => ! [X6: A] :
        ? [Y4: A] : ( ord_less @ A @ Y4 @ X6 ) ) ).

% linordered_field_no_lb
thf(fact_1188_linordered__field__no__ub,axiom,
    ! [A: $tType] :
      ( ( linordered_field @ A )
     => ! [X6: A] :
        ? [X_12: A] : ( ord_less @ A @ X6 @ X_12 ) ) ).

% linordered_field_no_ub
thf(fact_1189_not__exp__less__eq__0__int,axiom,
    ! [N2: nat] :
      ~ ( ord_less_eq @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N2 ) @ ( zero_zero @ int ) ) ).

% not_exp_less_eq_0_int
thf(fact_1190_realpow__pos__nth2,axiom,
    ! [A2: real,N2: nat] :
      ( ( ord_less @ real @ ( zero_zero @ real ) @ A2 )
     => ? [R3: real] :
          ( ( ord_less @ real @ ( zero_zero @ real ) @ R3 )
          & ( ( power_power @ real @ R3 @ ( suc @ N2 ) )
            = A2 ) ) ) ).

% realpow_pos_nth2
thf(fact_1191_real__arch__pow__inv,axiom,
    ! [Y2: real,X2: real] :
      ( ( ord_less @ real @ ( zero_zero @ real ) @ Y2 )
     => ( ( ord_less @ real @ X2 @ ( one_one @ real ) )
       => ? [N4: nat] : ( ord_less @ real @ ( power_power @ real @ X2 @ N4 ) @ Y2 ) ) ) ).

% real_arch_pow_inv
thf(fact_1192_reals__Archimedean3,axiom,
    ! [X2: real] :
      ( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
     => ! [Y3: real] :
        ? [N4: nat] : ( ord_less @ real @ Y3 @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N4 ) @ X2 ) ) ) ).

% reals_Archimedean3
thf(fact_1193_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_1194_split__zdiv,axiom,
    ! [P: int > $o,N2: int,K: int] :
      ( ( P @ ( divide_divide @ int @ N2 @ 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 )
                & ( N2
                  = ( 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 ) )
                & ( N2
                  = ( plus_plus @ int @ ( times_times @ int @ K @ I2 ) @ J3 ) ) )
             => ( P @ I2 ) ) ) ) ) ).

% split_zdiv
thf(fact_1195_int__div__neg__eq,axiom,
    ! [A2: int,B2: int,Q2: int,R4: int] :
      ( ( A2
        = ( plus_plus @ int @ ( times_times @ int @ B2 @ Q2 ) @ R4 ) )
     => ( ( ord_less_eq @ int @ R4 @ ( zero_zero @ int ) )
       => ( ( ord_less @ int @ B2 @ R4 )
         => ( ( divide_divide @ int @ A2 @ B2 )
            = Q2 ) ) ) ) ).

% int_div_neg_eq
thf(fact_1196_int__div__pos__eq,axiom,
    ! [A2: int,B2: int,Q2: int,R4: int] :
      ( ( A2
        = ( plus_plus @ int @ ( times_times @ int @ B2 @ Q2 ) @ R4 ) )
     => ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ R4 )
       => ( ( ord_less @ int @ R4 @ B2 )
         => ( ( divide_divide @ int @ A2 @ B2 )
            = Q2 ) ) ) ) ).

% int_div_pos_eq
thf(fact_1197_zdiv__mult__self,axiom,
    ! [M: int,A2: int,N2: int] :
      ( ( M
       != ( zero_zero @ int ) )
     => ( ( divide_divide @ int @ ( plus_plus @ int @ A2 @ ( times_times @ int @ M @ N2 ) ) @ M )
        = ( plus_plus @ int @ ( divide_divide @ int @ A2 @ M ) @ N2 ) ) ) ).

% zdiv_mult_self
thf(fact_1198_realpow__pos__nth__unique,axiom,
    ! [N2: nat,A2: real] :
      ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
     => ( ( ord_less @ real @ ( zero_zero @ real ) @ A2 )
       => ? [X3: real] :
            ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
            & ( ( power_power @ real @ X3 @ N2 )
              = A2 )
            & ! [Y3: real] :
                ( ( ( ord_less @ real @ ( zero_zero @ real ) @ Y3 )
                  & ( ( power_power @ real @ Y3 @ N2 )
                    = A2 ) )
               => ( Y3 = X3 ) ) ) ) ) ).

% realpow_pos_nth_unique
thf(fact_1199_realpow__pos__nth,axiom,
    ! [N2: nat,A2: real] :
      ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
     => ( ( ord_less @ real @ ( zero_zero @ real ) @ A2 )
       => ? [R3: real] :
            ( ( ord_less @ real @ ( zero_zero @ real ) @ R3 )
            & ( ( power_power @ real @ R3 @ N2 )
              = A2 ) ) ) ) ).

% realpow_pos_nth
thf(fact_1200_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_1201_VEBT__internal_OTb_Osimps_I1_J,axiom,
    ( ( vEBT_VEBT_Tb @ ( zero_zero @ nat ) )
    = ( numeral_numeral @ int @ ( bit1 @ one2 ) ) ) ).

% VEBT_internal.Tb.simps(1)
thf(fact_1202_real__archimedian__rdiv__eq__0,axiom,
    ! [X2: real,C2: real] :
      ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 )
     => ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ C2 )
       => ( ! [M2: nat] :
              ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M2 )
             => ( ord_less_eq @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ M2 ) @ X2 ) @ C2 ) )
         => ( X2
            = ( zero_zero @ real ) ) ) ) ) ).

% real_archimedian_rdiv_eq_0
thf(fact_1203_real__of__nat__div2,axiom,
    ! [N2: nat,X2: nat] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( minus_minus @ real @ ( divide_divide @ real @ ( semiring_1_of_nat @ real @ N2 ) @ ( semiring_1_of_nat @ real @ X2 ) ) @ ( semiring_1_of_nat @ real @ ( divide_divide @ nat @ N2 @ X2 ) ) ) ) ).

% real_of_nat_div2
thf(fact_1204_axxdiv2,axiom,
    ! [X2: int] :
      ( ( ( divide_divide @ int @ ( plus_plus @ int @ ( plus_plus @ int @ ( one_one @ int ) @ X2 ) @ X2 ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) )
        = X2 )
      & ( ( divide_divide @ int @ ( plus_plus @ int @ ( plus_plus @ int @ ( zero_zero @ int ) @ X2 ) @ X2 ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) )
        = X2 ) ) ).

% axxdiv2
thf(fact_1205_add__divide__distrib,axiom,
    ! [A: $tType] :
      ( ( division_ring @ A )
     => ! [A2: A,B2: A,C2: A] :
          ( ( divide_divide @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 )
          = ( plus_plus @ A @ ( divide_divide @ A @ A2 @ C2 ) @ ( divide_divide @ A @ B2 @ C2 ) ) ) ) ).

% add_divide_distrib
thf(fact_1206_times__divide__times__eq,axiom,
    ! [A: $tType] :
      ( ( field @ A )
     => ! [X2: A,Y2: A,Z: A,W: A] :
          ( ( times_times @ A @ ( divide_divide @ A @ X2 @ Y2 ) @ ( divide_divide @ A @ Z @ W ) )
          = ( divide_divide @ A @ ( times_times @ A @ X2 @ Z ) @ ( times_times @ A @ Y2 @ W ) ) ) ) ).

% times_divide_times_eq
thf(fact_1207_divide__divide__times__eq,axiom,
    ! [A: $tType] :
      ( ( field @ A )
     => ! [X2: A,Y2: A,Z: A,W: A] :
          ( ( divide_divide @ A @ ( divide_divide @ A @ X2 @ Y2 ) @ ( divide_divide @ A @ Z @ W ) )
          = ( divide_divide @ A @ ( times_times @ A @ X2 @ W ) @ ( times_times @ A @ Y2 @ Z ) ) ) ) ).

% divide_divide_times_eq
thf(fact_1208_divide__divide__eq__left_H,axiom,
    ! [A: $tType] :
      ( ( division_ring @ A )
     => ! [A2: A,B2: A,C2: A] :
          ( ( divide_divide @ A @ ( divide_divide @ A @ A2 @ B2 ) @ C2 )
          = ( divide_divide @ A @ A2 @ ( times_times @ A @ C2 @ B2 ) ) ) ) ).

% divide_divide_eq_left'
thf(fact_1209_diff__divide__distrib,axiom,
    ! [A: $tType] :
      ( ( division_ring @ A )
     => ! [A2: A,B2: A,C2: A] :
          ( ( divide_divide @ A @ ( minus_minus @ A @ A2 @ B2 ) @ C2 )
          = ( minus_minus @ A @ ( divide_divide @ A @ A2 @ C2 ) @ ( divide_divide @ A @ B2 @ C2 ) ) ) ) ).

% diff_divide_distrib
thf(fact_1210_VEBT__internal_OTb_Osimps_I2_J,axiom,
    ( ( vEBT_VEBT_Tb @ ( suc @ ( zero_zero @ nat ) ) )
    = ( numeral_numeral @ int @ ( bit1 @ one2 ) ) ) ).

% VEBT_internal.Tb.simps(2)
thf(fact_1211_pos__zdiv__mult__2,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ A2 )
     => ( ( divide_divide @ int @ ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ B2 ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ A2 ) )
        = ( divide_divide @ int @ B2 @ A2 ) ) ) ).

% pos_zdiv_mult_2
thf(fact_1212_neg__zdiv__mult__2,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_eq @ int @ A2 @ ( zero_zero @ int ) )
     => ( ( divide_divide @ int @ ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ B2 ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ A2 ) )
        = ( divide_divide @ int @ ( plus_plus @ int @ B2 @ ( one_one @ int ) ) @ A2 ) ) ) ).

% neg_zdiv_mult_2
thf(fact_1213_divide__le__0__iff,axiom,
    ! [A: $tType] :
      ( ( linordered_field @ A )
     => ! [A2: A,B2: A] :
          ( ( ord_less_eq @ A @ ( divide_divide @ A @ A2 @ B2 ) @ ( zero_zero @ A ) )
          = ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
              & ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) ) )
            | ( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) )
              & ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) ) ) ) ) ).

% divide_le_0_iff
thf(fact_1214_divide__right__mono,axiom,
    ! [A: $tType] :
      ( ( linordered_field @ A )
     => ! [A2: A,B2: A,C2: A] :
          ( ( ord_less_eq @ A @ A2 @ B2 )
         => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
           => ( ord_less_eq @ A @ ( divide_divide @ A @ A2 @ C2 ) @ ( divide_divide @ A @ B2 @ C2 ) ) ) ) ) ).

% divide_right_mono
thf(fact_1215_zero__le__divide__iff,axiom,
    ! [A: $tType] :
      ( ( linordered_field @ A )
     => ! [A2: A,B2: A] :
          ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ A2 @ B2 ) )
          = ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
              & ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) )
            | ( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) )
              & ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) ) ) ) ) ) ).

% zero_le_divide_iff
thf(fact_1216_divide__nonneg__nonneg,axiom,
    ! [A: $tType] :
      ( ( linordered_field @ A )
     => ! [X2: A,Y2: A] :
          ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X2 )
         => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y2 )
           => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ X2 @ Y2 ) ) ) ) ) ).

% divide_nonneg_nonneg
thf(fact_1217_divide__nonneg__nonpos,axiom,
    ! [A: $tType] :
      ( ( linordered_field @ A )
     => ! [X2: A,Y2: A] :
          ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X2 )
         => ( ( ord_less_eq @ A @ Y2 @ ( zero_zero @ A ) )
           => ( ord_less_eq @ A @ ( divide_divide @ A @ X2 @ Y2 ) @ ( zero_zero @ A ) ) ) ) ) ).

% divide_nonneg_nonpos
thf(fact_1218_divide__nonpos__nonneg,axiom,
    ! [A: $tType] :
      ( ( linordered_field @ A )
     => ! [X2: A,Y2: A] :
          ( ( ord_less_eq @ A @ X2 @ ( zero_zero @ A ) )
         => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y2 )
           => ( ord_less_eq @ A @ ( divide_divide @ A @ X2 @ Y2 ) @ ( zero_zero @ A ) ) ) ) ) ).

% divide_nonpos_nonneg
thf(fact_1219_divide__nonpos__nonpos,axiom,
    ! [A: $tType] :
      ( ( linordered_field @ A )
     => ! [X2: A,Y2: A] :
          ( ( ord_less_eq @ A @ X2 @ ( zero_zero @ A ) )
         => ( ( ord_less_eq @ A @ Y2 @ ( zero_zero @ A ) )
           => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ X2 @ Y2 ) ) ) ) ) ).

% divide_nonpos_nonpos
thf(fact_1220_divide__right__mono__neg,axiom,
    ! [A: $tType] :
      ( ( linordered_field @ A )
     => ! [A2: A,B2: A,C2: A] :
          ( ( ord_less_eq @ A @ A2 @ B2 )
         => ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) )
           => ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ C2 ) @ ( divide_divide @ A @ A2 @ C2 ) ) ) ) ) ).

% divide_right_mono_neg
thf(fact_1221_divide__neg__neg,axiom,
    ! [A: $tType] :
      ( ( linordered_field @ A )
     => ! [X2: A,Y2: A] :
          ( ( ord_less @ A @ X2 @ ( zero_zero @ A ) )
         => ( ( ord_less @ A @ Y2 @ ( zero_zero @ A ) )
           => ( ord_less @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ X2 @ Y2 ) ) ) ) ) ).

% divide_neg_neg
thf(fact_1222_divide__neg__pos,axiom,
    ! [A: $tType] :
      ( ( linordered_field @ A )
     => ! [X2: A,Y2: A] :
          ( ( ord_less @ A @ X2 @ ( zero_zero @ A ) )
         => ( ( ord_less @ A @ ( zero_zero @ A ) @ Y2 )
           => ( ord_less @ A @ ( divide_divide @ A @ X2 @ Y2 ) @ ( zero_zero @ A ) ) ) ) ) ).

% divide_neg_pos
thf(fact_1223_divide__pos__neg,axiom,
    ! [A: $tType] :
      ( ( linordered_field @ A )
     => ! [X2: A,Y2: A] :
          ( ( ord_less @ A @ ( zero_zero @ A ) @ X2 )
         => ( ( ord_less @ A @ Y2 @ ( zero_zero @ A ) )
           => ( ord_less @ A @ ( divide_divide @ A @ X2 @ Y2 ) @ ( zero_zero @ A ) ) ) ) ) ).

% divide_pos_neg
thf(fact_1224_divide__pos__pos,axiom,
    ! [A: $tType] :
      ( ( linordered_field @ A )
     => ! [X2: A,Y2: A] :
          ( ( ord_less @ A @ ( zero_zero @ A ) @ X2 )
         => ( ( ord_less @ A @ ( zero_zero @ A ) @ Y2 )
           => ( ord_less @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ X2 @ Y2 ) ) ) ) ) ).

% divide_pos_pos
thf(fact_1225_divide__less__0__iff,axiom,
    ! [A: $tType] :
      ( ( linordered_field @ A )
     => ! [A2: A,B2: A] :
          ( ( ord_less @ A @ ( divide_divide @ A @ A2 @ B2 ) @ ( zero_zero @ A ) )
          = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
              & ( ord_less @ A @ B2 @ ( zero_zero @ A ) ) )
            | ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
              & ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) ) ) ) ) ).

% divide_less_0_iff
thf(fact_1226_divide__less__cancel,axiom,
    ! [A: $tType] :
      ( ( linordered_field @ A )
     => ! [A2: A,C2: A,B2: A] :
          ( ( ord_less @ A @ ( divide_divide @ A @ A2 @ C2 ) @ ( divide_divide @ A @ B2 @ C2 ) )
          = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
             => ( ord_less @ A @ A2 @ B2 ) )
            & ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
             => ( ord_less @ A @ B2 @ A2 ) )
            & ( C2
             != ( zero_zero @ A ) ) ) ) ) ).

% divide_less_cancel
thf(fact_1227_zero__less__divide__iff,axiom,
    ! [A: $tType] :
      ( ( linordered_field @ A )
     => ! [A2: A,B2: A] :
          ( ( ord_less @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ A2 @ B2 ) )
          = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
              & ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) )
            | ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
              & ( ord_less @ A @ B2 @ ( zero_zero @ A ) ) ) ) ) ) ).

% zero_less_divide_iff
thf(fact_1228_divide__strict__right__mono,axiom,
    ! [A: $tType] :
      ( ( linordered_field @ A )
     => ! [A2: A,B2: A,C2: A] :
          ( ( ord_less @ A @ A2 @ B2 )
         => ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
           => ( ord_less @ A @ ( divide_divide @ A @ A2 @ C2 ) @ ( divide_divide @ A @ B2 @ C2 ) ) ) ) ) ).

% divide_strict_right_mono
thf(fact_1229_divide__strict__right__mono__neg,axiom,
    ! [A: $tType] :
      ( ( linordered_field @ A )
     => ! [B2: A,A2: A,C2: A] :
          ( ( ord_less @ A @ B2 @ A2 )
         => ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
           => ( ord_less @ A @ ( divide_divide @ A @ A2 @ C2 ) @ ( divide_divide @ A @ B2 @ C2 ) ) ) ) ) ).

% divide_strict_right_mono_neg
thf(fact_1230_right__inverse__eq,axiom,
    ! [A: $tType] :
      ( ( division_ring @ A )
     => ! [B2: A,A2: A] :
          ( ( B2
           != ( zero_zero @ A ) )
         => ( ( ( divide_divide @ A @ A2 @ B2 )
              = ( one_one @ A ) )
            = ( A2 = B2 ) ) ) ) ).

% right_inverse_eq
thf(fact_1231_frac__eq__eq,axiom,
    ! [A: $tType] :
      ( ( field @ A )
     => ! [Y2: A,Z: A,X2: A,W: A] :
          ( ( Y2
           != ( zero_zero @ A ) )
         => ( ( Z
             != ( zero_zero @ A ) )
           => ( ( ( divide_divide @ A @ X2 @ Y2 )
                = ( divide_divide @ A @ W @ Z ) )
              = ( ( times_times @ A @ X2 @ Z )
                = ( times_times @ A @ W @ Y2 ) ) ) ) ) ) ).

% frac_eq_eq
thf(fact_1232_divide__eq__eq,axiom,
    ! [A: $tType] :
      ( ( division_ring @ A )
     => ! [B2: A,C2: A,A2: A] :
          ( ( ( divide_divide @ A @ B2 @ C2 )
            = A2 )
          = ( ( ( C2
               != ( zero_zero @ A ) )
             => ( B2
                = ( times_times @ A @ A2 @ C2 ) ) )
            & ( ( C2
                = ( zero_zero @ A ) )
             => ( A2
                = ( zero_zero @ A ) ) ) ) ) ) ).

% divide_eq_eq
thf(fact_1233_eq__divide__eq,axiom,
    ! [A: $tType] :
      ( ( division_ring @ A )
     => ! [A2: A,B2: A,C2: A] :
          ( ( A2
            = ( divide_divide @ A @ B2 @ C2 ) )
          = ( ( ( C2
               != ( zero_zero @ A ) )
             => ( ( times_times @ A @ A2 @ C2 )
                = B2 ) )
            & ( ( C2
                = ( zero_zero @ A ) )
             => ( A2
                = ( zero_zero @ A ) ) ) ) ) ) ).

% eq_divide_eq
thf(fact_1234_divide__eq__imp,axiom,
    ! [A: $tType] :
      ( ( division_ring @ A )
     => ! [C2: A,B2: A,A2: A] :
          ( ( C2
           != ( zero_zero @ A ) )
         => ( ( B2
              = ( times_times @ A @ A2 @ C2 ) )
           => ( ( divide_divide @ A @ B2 @ C2 )
              = A2 ) ) ) ) ).

% divide_eq_imp
thf(fact_1235_eq__divide__imp,axiom,
    ! [A: $tType] :
      ( ( division_ring @ A )
     => ! [C2: A,A2: A,B2: A] :
          ( ( C2
           != ( zero_zero @ A ) )
         => ( ( ( times_times @ A @ A2 @ C2 )
              = B2 )
           => ( A2
              = ( divide_divide @ A @ B2 @ C2 ) ) ) ) ) ).

% eq_divide_imp
thf(fact_1236_nonzero__divide__eq__eq,axiom,
    ! [A: $tType] :
      ( ( division_ring @ A )
     => ! [C2: A,B2: A,A2: A] :
          ( ( C2
           != ( zero_zero @ A ) )
         => ( ( ( divide_divide @ A @ B2 @ C2 )
              = A2 )
            = ( B2
              = ( times_times @ A @ A2 @ C2 ) ) ) ) ) ).

% nonzero_divide_eq_eq
thf(fact_1237_nonzero__eq__divide__eq,axiom,
    ! [A: $tType] :
      ( ( division_ring @ A )
     => ! [C2: A,A2: A,B2: A] :
          ( ( C2
           != ( zero_zero @ A ) )
         => ( ( A2
              = ( divide_divide @ A @ B2 @ C2 ) )
            = ( ( times_times @ A @ A2 @ C2 )
              = B2 ) ) ) ) ).

% nonzero_eq_divide_eq
thf(fact_1238_field__le__epsilon,axiom,
    ! [A: $tType] :
      ( ( linordered_field @ A )
     => ! [X2: A,Y2: A] :
          ( ! [E2: A] :
              ( ( ord_less @ A @ ( zero_zero @ A ) @ E2 )
             => ( ord_less_eq @ A @ X2 @ ( plus_plus @ A @ Y2 @ E2 ) ) )
         => ( ord_less_eq @ A @ X2 @ Y2 ) ) ) ).

% field_le_epsilon
thf(fact_1239_frac__le,axiom,
    ! [A: $tType] :
      ( ( linordered_field @ A )
     => ! [Y2: A,X2: A,W: A,Z: A] :
          ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y2 )
         => ( ( ord_less_eq @ A @ X2 @ Y2 )
           => ( ( ord_less @ A @ ( zero_zero @ A ) @ W )
             => ( ( ord_less_eq @ A @ W @ Z )
               => ( ord_less_eq @ A @ ( divide_divide @ A @ X2 @ Z ) @ ( divide_divide @ A @ Y2 @ W ) ) ) ) ) ) ) ).

% frac_le
thf(fact_1240_frac__less,axiom,
    ! [A: $tType] :
      ( ( linordered_field @ A )
     => ! [X2: A,Y2: A,W: A,Z: A] :
          ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X2 )
         => ( ( ord_less @ A @ X2 @ Y2 )
           => ( ( ord_less @ A @ ( zero_zero @ A ) @ W )
             => ( ( ord_less_eq @ A @ W @ Z )
               => ( ord_less @ A @ ( divide_divide @ A @ X2 @ Z ) @ ( divide_divide @ A @ Y2 @ W ) ) ) ) ) ) ) ).

% frac_less
thf(fact_1241_frac__less2,axiom,
    ! [A: $tType] :
      ( ( linordered_field @ A )
     => ! [X2: A,Y2: A,W: A,Z: A] :
          ( ( ord_less @ A @ ( zero_zero @ A ) @ X2 )
         => ( ( ord_less_eq @ A @ X2 @ Y2 )
           => ( ( ord_less @ A @ ( zero_zero @ A ) @ W )
             => ( ( ord_less @ A @ W @ Z )
               => ( ord_less @ A @ ( divide_divide @ A @ X2 @ Z ) @ ( divide_divide @ A @ Y2 @ W ) ) ) ) ) ) ) ).

% frac_less2
thf(fact_1242_divide__le__cancel,axiom,
    ! [A: $tType] :
      ( ( linordered_field @ A )
     => ! [A2: A,C2: A,B2: A] :
          ( ( ord_less_eq @ A @ ( divide_divide @ A @ A2 @ C2 ) @ ( divide_divide @ A @ B2 @ C2 ) )
          = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
             => ( ord_less_eq @ A @ A2 @ B2 ) )
            & ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
             => ( ord_less_eq @ A @ B2 @ A2 ) ) ) ) ) ).

% divide_le_cancel
thf(fact_1243_divide__nonneg__neg,axiom,
    ! [A: $tType] :
      ( ( linordered_field @ A )
     => ! [X2: A,Y2: A] :
          ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X2 )
         => ( ( ord_less @ A @ Y2 @ ( zero_zero @ A ) )
           => ( ord_less_eq @ A @ ( divide_divide @ A @ X2 @ Y2 ) @ ( zero_zero @ A ) ) ) ) ) ).

% divide_nonneg_neg
thf(fact_1244_divide__nonneg__pos,axiom,
    ! [A: $tType] :
      ( ( linordered_field @ A )
     => ! [X2: A,Y2: A] :
          ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X2 )
         => ( ( ord_less @ A @ ( zero_zero @ A ) @ Y2 )
           => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ X2 @ Y2 ) ) ) ) ) ).

% divide_nonneg_pos
thf(fact_1245_divide__nonpos__neg,axiom,
    ! [A: $tType] :
      ( ( linordered_field @ A )
     => ! [X2: A,Y2: A] :
          ( ( ord_less_eq @ A @ X2 @ ( zero_zero @ A ) )
         => ( ( ord_less @ A @ Y2 @ ( zero_zero @ A ) )
           => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ X2 @ Y2 ) ) ) ) ) ).

% divide_nonpos_neg
thf(fact_1246_divide__nonpos__pos,axiom,
    ! [A: $tType] :
      ( ( linordered_field @ A )
     => ! [X2: A,Y2: A] :
          ( ( ord_less_eq @ A @ X2 @ ( zero_zero @ A ) )
         => ( ( ord_less @ A @ ( zero_zero @ A ) @ Y2 )
           => ( ord_less_eq @ A @ ( divide_divide @ A @ X2 @ Y2 ) @ ( zero_zero @ A ) ) ) ) ) ).

% divide_nonpos_pos
thf(fact_1247_divide__less__eq__1,axiom,
    ! [A: $tType] :
      ( ( linordered_field @ A )
     => ! [B2: A,A2: A] :
          ( ( ord_less @ A @ ( divide_divide @ A @ B2 @ A2 ) @ ( one_one @ A ) )
          = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
              & ( ord_less @ A @ B2 @ A2 ) )
            | ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
              & ( ord_less @ A @ A2 @ B2 ) )
            | ( A2
              = ( zero_zero @ A ) ) ) ) ) ).

% divide_less_eq_1
thf(fact_1248_less__divide__eq__1,axiom,
    ! [A: $tType] :
      ( ( linordered_field @ A )
     => ! [B2: A,A2: A] :
          ( ( ord_less @ A @ ( one_one @ A ) @ ( divide_divide @ A @ B2 @ A2 ) )
          = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
              & ( ord_less @ A @ A2 @ B2 ) )
            | ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
              & ( ord_less @ A @ B2 @ A2 ) ) ) ) ) ).

% less_divide_eq_1
thf(fact_1249_divide__less__eq,axiom,
    ! [A: $tType] :
      ( ( linordered_field @ A )
     => ! [B2: A,C2: A,A2: A] :
          ( ( ord_less @ A @ ( divide_divide @ A @ B2 @ C2 ) @ A2 )
          = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
             => ( ord_less @ A @ B2 @ ( times_times @ A @ A2 @ C2 ) ) )
            & ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
             => ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
                 => ( ord_less @ A @ ( times_times @ A @ A2 @ C2 ) @ B2 ) )
                & ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
                 => ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ) ) ) ) ).

% divide_less_eq
thf(fact_1250_less__divide__eq,axiom,
    ! [A: $tType] :
      ( ( linordered_field @ A )
     => ! [A2: A,B2: A,C2: A] :
          ( ( ord_less @ A @ A2 @ ( divide_divide @ A @ B2 @ C2 ) )
          = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
             => ( ord_less @ A @ ( times_times @ A @ A2 @ C2 ) @ B2 ) )
            & ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
             => ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
                 => ( ord_less @ A @ B2 @ ( times_times @ A @ A2 @ C2 ) ) )
                & ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
                 => ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ) ) ) ) ).

% less_divide_eq
thf(fact_1251_neg__divide__less__eq,axiom,
    ! [A: $tType] :
      ( ( linordered_field @ A )
     => ! [C2: A,B2: A,A2: A] :
          ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
         => ( ( ord_less @ A @ ( divide_divide @ A @ B2 @ C2 ) @ A2 )
            = ( ord_less @ A @ ( times_times @ A @ A2 @ C2 ) @ B2 ) ) ) ) ).

% neg_divide_less_eq
thf(fact_1252_neg__less__divide__eq,axiom,
    ! [A: $tType] :
      ( ( linordered_field @ A )
     => ! [C2: A,A2: A,B2: A] :
          ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
         => ( ( ord_less @ A @ A2 @ ( divide_divide @ A @ B2 @ C2 ) )
            = ( ord_less @ A @ B2 @ ( times_times @ A @ A2 @ C2 ) ) ) ) ) ).

% neg_less_divide_eq
thf(fact_1253_pos__divide__less__eq,axiom,
    ! [A: $tType] :
      ( ( linordered_field @ A )
     => ! [C2: A,B2: A,A2: A] :
          ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
         => ( ( ord_less @ A @ ( divide_divide @ A @ B2 @ C2 ) @ A2 )
            = ( ord_less @ A @ B2 @ ( times_times @ A @ A2 @ C2 ) ) ) ) ) ).

% pos_divide_less_eq
thf(fact_1254_pos__less__divide__eq,axiom,
    ! [A: $tType] :
      ( ( linordered_field @ A )
     => ! [C2: A,A2: A,B2: A] :
          ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
         => ( ( ord_less @ A @ A2 @ ( divide_divide @ A @ B2 @ C2 ) )
            = ( ord_less @ A @ ( times_times @ A @ A2 @ C2 ) @ B2 ) ) ) ) ).

% pos_less_divide_eq
thf(fact_1255_mult__imp__div__pos__less,axiom,
    ! [A: $tType] :
      ( ( linordered_field @ A )
     => ! [Y2: A,X2: A,Z: A] :
          ( ( ord_less @ A @ ( zero_zero @ A ) @ Y2 )
         => ( ( ord_less @ A @ X2 @ ( times_times @ A @ Z @ Y2 ) )
           => ( ord_less @ A @ ( divide_divide @ A @ X2 @ Y2 ) @ Z ) ) ) ) ).

% mult_imp_div_pos_less
thf(fact_1256_mult__imp__less__div__pos,axiom,
    ! [A: $tType] :
      ( ( linordered_field @ A )
     => ! [Y2: A,Z: A,X2: A] :
          ( ( ord_less @ A @ ( zero_zero @ A ) @ Y2 )
         => ( ( ord_less @ A @ ( times_times @ A @ Z @ Y2 ) @ X2 )
           => ( ord_less @ A @ Z @ ( divide_divide @ A @ X2 @ Y2 ) ) ) ) ) ).

% mult_imp_less_div_pos
thf(fact_1257_divide__strict__left__mono,axiom,
    ! [A: $tType] :
      ( ( linordered_field @ A )
     => ! [B2: A,A2: A,C2: A] :
          ( ( ord_less @ A @ B2 @ A2 )
         => ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
           => ( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A2 @ B2 ) )
             => ( ord_less @ A @ ( divide_divide @ A @ C2 @ A2 ) @ ( divide_divide @ A @ C2 @ B2 ) ) ) ) ) ) ).

% divide_strict_left_mono
thf(fact_1258_divide__strict__left__mono__neg,axiom,
    ! [A: $tType] :
      ( ( linordered_field @ A )
     => ! [A2: A,B2: A,C2: A] :
          ( ( ord_less @ A @ A2 @ B2 )
         => ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
           => ( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A2 @ B2 ) )
             => ( ord_less @ A @ ( divide_divide @ A @ C2 @ A2 ) @ ( divide_divide @ A @ C2 @ B2 ) ) ) ) ) ) ).

% divide_strict_left_mono_neg
thf(fact_1259_add__divide__eq__if__simps_I2_J,axiom,
    ! [A: $tType] :
      ( ( division_ring @ A )
     => ! [Z: A,A2: A,B2: A] :
          ( ( ( Z
              = ( zero_zero @ A ) )
           => ( ( plus_plus @ A @ ( divide_divide @ A @ A2 @ Z ) @ B2 )
              = B2 ) )
          & ( ( Z
             != ( zero_zero @ A ) )
           => ( ( plus_plus @ A @ ( divide_divide @ A @ A2 @ Z ) @ B2 )
              = ( divide_divide @ A @ ( plus_plus @ A @ A2 @ ( times_times @ A @ B2 @ Z ) ) @ Z ) ) ) ) ) ).

% add_divide_eq_if_simps(2)
thf(fact_1260_add__divide__eq__if__simps_I1_J,axiom,
    ! [A: $tType] :
      ( ( division_ring @ A )
     => ! [Z: A,A2: A,B2: A] :
          ( ( ( Z
              = ( zero_zero @ A ) )
           => ( ( plus_plus @ A @ A2 @ ( divide_divide @ A @ B2 @ Z ) )
              = A2 ) )
          & ( ( Z
             != ( zero_zero @ A ) )
           => ( ( plus_plus @ A @ A2 @ ( divide_divide @ A @ B2 @ Z ) )
              = ( divide_divide @ A @ ( plus_plus @ A @ ( times_times @ A @ A2 @ Z ) @ B2 ) @ Z ) ) ) ) ) ).

% add_divide_eq_if_simps(1)
thf(fact_1261_add__frac__eq,axiom,
    ! [A: $tType] :
      ( ( field @ A )
     => ! [Y2: A,Z: A,X2: A,W: A] :
          ( ( Y2
           != ( zero_zero @ A ) )
         => ( ( Z
             != ( zero_zero @ A ) )
           => ( ( plus_plus @ A @ ( divide_divide @ A @ X2 @ Y2 ) @ ( divide_divide @ A @ W @ Z ) )
              = ( divide_divide @ A @ ( plus_plus @ A @ ( times_times @ A @ X2 @ Z ) @ ( times_times @ A @ W @ Y2 ) ) @ ( times_times @ A @ Y2 @ Z ) ) ) ) ) ) ).

% add_frac_eq
thf(fact_1262_add__frac__num,axiom,
    ! [A: $tType] :
      ( ( field @ A )
     => ! [Y2: A,X2: A,Z: A] :
          ( ( Y2
           != ( zero_zero @ A ) )
         => ( ( plus_plus @ A @ ( divide_divide @ A @ X2 @ Y2 ) @ Z )
            = ( divide_divide @ A @ ( plus_plus @ A @ X2 @ ( times_times @ A @ Z @ Y2 ) ) @ Y2 ) ) ) ) ).

% add_frac_num
thf(fact_1263_add__num__frac,axiom,
    ! [A: $tType] :
      ( ( field @ A )
     => ! [Y2: A,Z: A,X2: A] :
          ( ( Y2
           != ( zero_zero @ A ) )
         => ( ( plus_plus @ A @ Z @ ( divide_divide @ A @ X2 @ Y2 ) )
            = ( divide_divide @ A @ ( plus_plus @ A @ X2 @ ( times_times @ A @ Z @ Y2 ) ) @ Y2 ) ) ) ) ).

% add_num_frac
thf(fact_1264_add__divide__eq__iff,axiom,
    ! [A: $tType] :
      ( ( division_ring @ A )
     => ! [Z: A,X2: A,Y2: A] :
          ( ( Z
           != ( zero_zero @ A ) )
         => ( ( plus_plus @ A @ X2 @ ( divide_divide @ A @ Y2 @ Z ) )
            = ( divide_divide @ A @ ( plus_plus @ A @ ( times_times @ A @ X2 @ Z ) @ Y2 ) @ Z ) ) ) ) ).

% add_divide_eq_iff
thf(fact_1265_divide__add__eq__iff,axiom,
    ! [A: $tType] :
      ( ( division_ring @ A )
     => ! [Z: A,X2: A,Y2: A] :
          ( ( Z
           != ( zero_zero @ A ) )
         => ( ( plus_plus @ A @ ( divide_divide @ A @ X2 @ Z ) @ Y2 )
            = ( divide_divide @ A @ ( plus_plus @ A @ X2 @ ( times_times @ A @ Y2 @ Z ) ) @ Z ) ) ) ) ).

% divide_add_eq_iff
thf(fact_1266_gt__half__sum,axiom,
    ! [A: $tType] :
      ( ( linordered_field @ A )
     => ! [A2: A,B2: A] :
          ( ( ord_less @ A @ A2 @ B2 )
         => ( ord_less @ A @ ( divide_divide @ A @ ( plus_plus @ A @ A2 @ B2 ) @ ( plus_plus @ A @ ( one_one @ A ) @ ( one_one @ A ) ) ) @ B2 ) ) ) ).

% gt_half_sum
thf(fact_1267_less__half__sum,axiom,
    ! [A: $tType] :
      ( ( linordered_field @ A )
     => ! [A2: A,B2: A] :
          ( ( ord_less @ A @ A2 @ B2 )
         => ( ord_less @ A @ A2 @ ( divide_divide @ A @ ( plus_plus @ A @ A2 @ B2 ) @ ( plus_plus @ A @ ( one_one @ A ) @ ( one_one @ A ) ) ) ) ) ) ).

% less_half_sum
thf(fact_1268_add__divide__eq__if__simps_I4_J,axiom,
    ! [A: $tType] :
      ( ( division_ring @ A )
     => ! [Z: A,A2: A,B2: A] :
          ( ( ( Z
              = ( zero_zero @ A ) )
           => ( ( minus_minus @ A @ A2 @ ( divide_divide @ A @ B2 @ Z ) )
              = A2 ) )
          & ( ( Z
             != ( zero_zero @ A ) )
           => ( ( minus_minus @ A @ A2 @ ( divide_divide @ A @ B2 @ Z ) )
              = ( divide_divide @ A @ ( minus_minus @ A @ ( times_times @ A @ A2 @ Z ) @ B2 ) @ Z ) ) ) ) ) ).

% add_divide_eq_if_simps(4)
thf(fact_1269_diff__frac__eq,axiom,
    ! [A: $tType] :
      ( ( field @ A )
     => ! [Y2: A,Z: A,X2: A,W: A] :
          ( ( Y2
           != ( zero_zero @ A ) )
         => ( ( Z
             != ( zero_zero @ A ) )
           => ( ( minus_minus @ A @ ( divide_divide @ A @ X2 @ Y2 ) @ ( divide_divide @ A @ W @ Z ) )
              = ( divide_divide @ A @ ( minus_minus @ A @ ( times_times @ A @ X2 @ Z ) @ ( times_times @ A @ W @ Y2 ) ) @ ( times_times @ A @ Y2 @ Z ) ) ) ) ) ) ).

% diff_frac_eq
thf(fact_1270_diff__divide__eq__iff,axiom,
    ! [A: $tType] :
      ( ( division_ring @ A )
     => ! [Z: A,X2: A,Y2: A] :
          ( ( Z
           != ( zero_zero @ A ) )
         => ( ( minus_minus @ A @ X2 @ ( divide_divide @ A @ Y2 @ Z ) )
            = ( divide_divide @ A @ ( minus_minus @ A @ ( times_times @ A @ X2 @ Z ) @ Y2 ) @ Z ) ) ) ) ).

% diff_divide_eq_iff
thf(fact_1271_divide__diff__eq__iff,axiom,
    ! [A: $tType] :
      ( ( division_ring @ A )
     => ! [Z: A,X2: A,Y2: A] :
          ( ( Z
           != ( zero_zero @ A ) )
         => ( ( minus_minus @ A @ ( divide_divide @ A @ X2 @ Z ) @ Y2 )
            = ( divide_divide @ A @ ( minus_minus @ A @ X2 @ ( times_times @ A @ Y2 @ Z ) ) @ Z ) ) ) ) ).

% divide_diff_eq_iff
thf(fact_1272_field__le__mult__one__interval,axiom,
    ! [A: $tType] :
      ( ( linordered_field @ A )
     => ! [X2: A,Y2: A] :
          ( ! [Z3: A] :
              ( ( ord_less @ A @ ( zero_zero @ A ) @ Z3 )
             => ( ( ord_less @ A @ Z3 @ ( one_one @ A ) )
               => ( ord_less_eq @ A @ ( times_times @ A @ Z3 @ X2 ) @ Y2 ) ) )
         => ( ord_less_eq @ A @ X2 @ Y2 ) ) ) ).

% field_le_mult_one_interval
thf(fact_1273_divide__le__eq__1,axiom,
    ! [A: $tType] :
      ( ( linordered_field @ A )
     => ! [B2: A,A2: A] :
          ( ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ A2 ) @ ( one_one @ A ) )
          = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
              & ( ord_less_eq @ A @ B2 @ A2 ) )
            | ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
              & ( ord_less_eq @ A @ A2 @ B2 ) )
            | ( A2
              = ( zero_zero @ A ) ) ) ) ) ).

% divide_le_eq_1
thf(fact_1274_le__divide__eq__1,axiom,
    ! [A: $tType] :
      ( ( linordered_field @ A )
     => ! [B2: A,A2: A] :
          ( ( ord_less_eq @ A @ ( one_one @ A ) @ ( divide_divide @ A @ B2 @ A2 ) )
          = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
              & ( ord_less_eq @ A @ A2 @ B2 ) )
            | ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
              & ( ord_less_eq @ A @ B2 @ A2 ) ) ) ) ) ).

% le_divide_eq_1
thf(fact_1275_divide__le__eq,axiom,
    ! [A: $tType] :
      ( ( linordered_field @ A )
     => ! [B2: A,C2: A,A2: A] :
          ( ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ C2 ) @ A2 )
          = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
             => ( ord_less_eq @ A @ B2 @ ( times_times @ A @ A2 @ C2 ) ) )
            & ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
             => ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
                 => ( ord_less_eq @ A @ ( times_times @ A @ A2 @ C2 ) @ B2 ) )
                & ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
                 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) ) ) ) ) ) ) ).

% divide_le_eq
thf(fact_1276_le__divide__eq,axiom,
    ! [A: $tType] :
      ( ( linordered_field @ A )
     => ! [A2: A,B2: A,C2: A] :
          ( ( ord_less_eq @ A @ A2 @ ( divide_divide @ A @ B2 @ C2 ) )
          = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
             => ( ord_less_eq @ A @ ( times_times @ A @ A2 @ C2 ) @ B2 ) )
            & ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
             => ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
                 => ( ord_less_eq @ A @ B2 @ ( times_times @ A @ A2 @ C2 ) ) )
                & ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
                 => ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ) ) ) ) ).

% le_divide_eq
thf(fact_1277_divide__left__mono,axiom,
    ! [A: $tType] :
      ( ( linordered_field @ A )
     => ! [B2: A,A2: A,C2: A] :
          ( ( ord_less_eq @ A @ B2 @ A2 )
         => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
           => ( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A2 @ B2 ) )
             => ( ord_less_eq @ A @ ( divide_divide @ A @ C2 @ A2 ) @ ( divide_divide @ A @ C2 @ B2 ) ) ) ) ) ) ).

% divide_left_mono
thf(fact_1278_neg__divide__le__eq,axiom,
    ! [A: $tType] :
      ( ( linordered_field @ A )
     => ! [C2: A,B2: A,A2: A] :
          ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
         => ( ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ C2 ) @ A2 )
            = ( ord_less_eq @ A @ ( times_times @ A @ A2 @ C2 ) @ B2 ) ) ) ) ).

% neg_divide_le_eq
thf(fact_1279_neg__le__divide__eq,axiom,
    ! [A: $tType] :
      ( ( linordered_field @ A )
     => ! [C2: A,A2: A,B2: A] :
          ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
         => ( ( ord_less_eq @ A @ A2 @ ( divide_divide @ A @ B2 @ C2 ) )
            = ( ord_less_eq @ A @ B2 @ ( times_times @ A @ A2 @ C2 ) ) ) ) ) ).

% neg_le_divide_eq
thf(fact_1280_pos__divide__le__eq,axiom,
    ! [A: $tType] :
      ( ( linordered_field @ A )
     => ! [C2: A,B2: A,A2: A] :
          ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
         => ( ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ C2 ) @ A2 )
            = ( ord_less_eq @ A @ B2 @ ( times_times @ A @ A2 @ C2 ) ) ) ) ) ).

% pos_divide_le_eq
thf(fact_1281_pos__le__divide__eq,axiom,
    ! [A: $tType] :
      ( ( linordered_field @ A )
     => ! [C2: A,A2: A,B2: A] :
          ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
         => ( ( ord_less_eq @ A @ A2 @ ( divide_divide @ A @ B2 @ C2 ) )
            = ( ord_less_eq @ A @ ( times_times @ A @ A2 @ C2 ) @ B2 ) ) ) ) ).

% pos_le_divide_eq
thf(fact_1282_mult__imp__div__pos__le,axiom,
    ! [A: $tType] :
      ( ( linordered_field @ A )
     => ! [Y2: A,X2: A,Z: A] :
          ( ( ord_less @ A @ ( zero_zero @ A ) @ Y2 )
         => ( ( ord_less_eq @ A @ X2 @ ( times_times @ A @ Z @ Y2 ) )
           => ( ord_less_eq @ A @ ( divide_divide @ A @ X2 @ Y2 ) @ Z ) ) ) ) ).

% mult_imp_div_pos_le
thf(fact_1283_mult__imp__le__div__pos,axiom,
    ! [A: $tType] :
      ( ( linordered_field @ A )
     => ! [Y2: A,Z: A,X2: A] :
          ( ( ord_less @ A @ ( zero_zero @ A ) @ Y2 )
         => ( ( ord_less_eq @ A @ ( times_times @ A @ Z @ Y2 ) @ X2 )
           => ( ord_less_eq @ A @ Z @ ( divide_divide @ A @ X2 @ Y2 ) ) ) ) ) ).

% mult_imp_le_div_pos
thf(fact_1284_divide__left__mono__neg,axiom,
    ! [A: $tType] :
      ( ( linordered_field @ A )
     => ! [A2: A,B2: A,C2: A] :
          ( ( ord_less_eq @ A @ A2 @ B2 )
         => ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) )
           => ( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A2 @ B2 ) )
             => ( ord_less_eq @ A @ ( divide_divide @ A @ C2 @ A2 ) @ ( divide_divide @ A @ C2 @ B2 ) ) ) ) ) ) ).

% divide_left_mono_neg
thf(fact_1285_frac__le__eq,axiom,
    ! [A: $tType] :
      ( ( linordered_field @ A )
     => ! [Y2: A,Z: A,X2: A,W: A] :
          ( ( Y2
           != ( zero_zero @ A ) )
         => ( ( Z
             != ( zero_zero @ A ) )
           => ( ( ord_less_eq @ A @ ( divide_divide @ A @ X2 @ Y2 ) @ ( divide_divide @ A @ W @ Z ) )
              = ( ord_less_eq @ A @ ( divide_divide @ A @ ( minus_minus @ A @ ( times_times @ A @ X2 @ Z ) @ ( times_times @ A @ W @ Y2 ) ) @ ( times_times @ A @ Y2 @ Z ) ) @ ( zero_zero @ A ) ) ) ) ) ) ).

% frac_le_eq
thf(fact_1286_frac__less__eq,axiom,
    ! [A: $tType] :
      ( ( linordered_field @ A )
     => ! [Y2: A,Z: A,X2: A,W: A] :
          ( ( Y2
           != ( zero_zero @ A ) )
         => ( ( Z
             != ( zero_zero @ A ) )
           => ( ( ord_less @ A @ ( divide_divide @ A @ X2 @ Y2 ) @ ( divide_divide @ A @ W @ Z ) )
              = ( ord_less @ A @ ( divide_divide @ A @ ( minus_minus @ A @ ( times_times @ A @ X2 @ Z ) @ ( times_times @ A @ W @ Y2 ) ) @ ( times_times @ A @ Y2 @ Z ) ) @ ( zero_zero @ A ) ) ) ) ) ) ).

% frac_less_eq
thf(fact_1287_scaling__mono,axiom,
    ! [A: $tType] :
      ( ( linordered_field @ A )
     => ! [U: A,V: A,R4: A,S4: A] :
          ( ( ord_less_eq @ A @ U @ V )
         => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ R4 )
           => ( ( ord_less_eq @ A @ R4 @ S4 )
             => ( ord_less_eq @ A @ ( plus_plus @ A @ U @ ( divide_divide @ A @ ( times_times @ A @ R4 @ ( minus_minus @ A @ V @ U ) ) @ S4 ) ) @ V ) ) ) ) ) ).

% scaling_mono
thf(fact_1288_Tb__T__vebt__buildupi_H,axiom,
    ! [N2: nat] : ( ord_less_eq @ int @ ( vEBT_V9176841429113362141ildupi @ N2 ) @ ( minus_minus @ int @ ( vEBT_VEBT_Tb @ N2 ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ).

% Tb_T_vebt_buildupi'
thf(fact_1289_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,Va: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,X2: nat] :
      ( ( vEBT_T_m_e_m_b_e_r2 @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X2 )
      = ( plus_plus @ nat @ ( one_one @ nat )
        @ ( if @ nat @ ( X2 = Mi ) @ ( zero_zero @ nat )
          @ ( if @ nat @ ( X2 = Ma ) @ ( zero_zero @ nat )
            @ ( if @ nat @ ( ord_less @ nat @ X2 @ Mi ) @ ( zero_zero @ nat )
              @ ( if @ nat @ ( ord_less @ nat @ Ma @ X2 ) @ ( zero_zero @ nat )
                @ ( if @ nat
                  @ ( ( ord_less @ nat @ Mi @ X2 )
                    & ( ord_less @ nat @ X2 @ Ma ) )
                  @ ( if @ nat @ ( ord_less @ nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) ) @ ( vEBT_T_m_e_m_b_e_r2 @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( zero_zero @ nat ) )
                  @ ( zero_zero @ nat ) ) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.simps(5)
thf(fact_1290_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_1291_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_1292_int__eq__iff__numeral,axiom,
    ! [M: nat,V: num] :
      ( ( ( semiring_1_of_nat @ int @ M )
        = ( numeral_numeral @ int @ V ) )
      = ( M
        = ( numeral_numeral @ nat @ V ) ) ) ).

% int_eq_iff_numeral
thf(fact_1293_vebt__succ_Oelims,axiom,
    ! [X2: vEBT_VEBT,Xa: nat,Y2: option @ nat] :
      ( ( ( vEBT_vebt_succ @ X2 @ Xa )
        = Y2 )
     => ( ! [Uu2: $o,B3: $o] :
            ( ( X2
              = ( vEBT_Leaf @ Uu2 @ B3 ) )
           => ( ( Xa
                = ( zero_zero @ nat ) )
             => ~ ( ( B3
                   => ( Y2
                      = ( some @ nat @ ( one_one @ nat ) ) ) )
                  & ( ~ B3
                   => ( Y2
                      = ( none @ nat ) ) ) ) ) )
       => ( ( ? [Uv2: $o,Uw2: $o] :
                ( X2
                = ( vEBT_Leaf @ Uv2 @ Uw2 ) )
           => ( ? [N4: nat] :
                  ( Xa
                  = ( suc @ N4 ) )
             => ( Y2
               != ( none @ nat ) ) ) )
         => ( ( ? [Ux2: nat,Uy2: list @ vEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( X2
                  = ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Ux2 @ Uy2 @ Uz2 ) )
             => ( Y2
               != ( none @ nat ) ) )
           => ( ( ? [V3: product_prod @ nat @ nat,Vc: list @ vEBT_VEBT,Vd2: vEBT_VEBT] :
                    ( X2
                    = ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( zero_zero @ nat ) @ Vc @ Vd2 ) )
               => ( Y2
                 != ( none @ nat ) ) )
             => ( ( ? [V3: product_prod @ nat @ nat,Vg2: list @ vEBT_VEBT,Vh2: vEBT_VEBT] :
                      ( X2
                      = ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vg2 @ Vh2 ) )
                 => ( Y2
                   != ( none @ nat ) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X2
                        = ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
                     => ~ ( ( ( ord_less @ nat @ Xa @ Mi2 )
                           => ( Y2
                              = ( some @ nat @ Mi2 ) ) )
                          & ( ~ ( ord_less @ nat @ Xa @ Mi2 )
                           => ( Y2
                              = ( if @ ( option @ nat ) @ ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
                                @ ( if @ ( option @ nat )
                                  @ ( ( ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
                                     != ( none @ nat ) )
                                    & ( vEBT_VEBT_less @ ( some @ nat @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) )
                                  @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( some @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_succ @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
                                  @ ( if @ ( option @ nat )
                                    @ ( ( vEBT_vebt_succ @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
                                      = ( none @ nat ) )
                                    @ ( none @ nat )
                                    @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_succ @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_succ @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) )
                                @ ( none @ nat ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_succ.elims
thf(fact_1294_vebt__pred_Oelims,axiom,
    ! [X2: vEBT_VEBT,Xa: nat,Y2: option @ nat] :
      ( ( ( vEBT_vebt_pred @ X2 @ Xa )
        = Y2 )
     => ( ( ? [Uu2: $o,Uv2: $o] :
              ( X2
              = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
         => ( ( Xa
              = ( zero_zero @ nat ) )
           => ( Y2
             != ( none @ nat ) ) ) )
       => ( ! [A4: $o] :
              ( ? [Uw2: $o] :
                  ( X2
                  = ( vEBT_Leaf @ A4 @ Uw2 ) )
             => ( ( Xa
                  = ( suc @ ( zero_zero @ nat ) ) )
               => ~ ( ( A4
                     => ( Y2
                        = ( some @ nat @ ( zero_zero @ nat ) ) ) )
                    & ( ~ A4
                     => ( Y2
                        = ( none @ nat ) ) ) ) ) )
         => ( ! [A4: $o,B3: $o] :
                ( ( X2
                  = ( vEBT_Leaf @ A4 @ B3 ) )
               => ( ? [Va3: nat] :
                      ( Xa
                      = ( suc @ ( suc @ Va3 ) ) )
                 => ~ ( ( B3
                       => ( Y2
                          = ( some @ nat @ ( one_one @ nat ) ) ) )
                      & ( ~ B3
                       => ( ( A4
                           => ( Y2
                              = ( some @ nat @ ( zero_zero @ nat ) ) ) )
                          & ( ~ A4
                           => ( Y2
                              = ( none @ nat ) ) ) ) ) ) ) )
           => ( ( ? [Uy2: nat,Uz2: list @ vEBT_VEBT,Va2: vEBT_VEBT] :
                    ( X2
                    = ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uy2 @ Uz2 @ Va2 ) )
               => ( Y2
                 != ( none @ nat ) ) )
             => ( ( ? [V3: product_prod @ nat @ nat,Vd2: list @ vEBT_VEBT,Ve2: vEBT_VEBT] :
                      ( X2
                      = ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( zero_zero @ nat ) @ Vd2 @ Ve2 ) )
                 => ( Y2
                   != ( none @ nat ) ) )
               => ( ( ? [V3: product_prod @ nat @ nat,Vh2: list @ vEBT_VEBT,Vi2: vEBT_VEBT] :
                        ( X2
                        = ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vh2 @ Vi2 ) )
                   => ( Y2
                     != ( none @ nat ) ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
                        ( ( X2
                          = ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
                       => ~ ( ( ( ord_less @ nat @ Ma2 @ Xa )
                             => ( Y2
                                = ( some @ nat @ Ma2 ) ) )
                            & ( ~ ( ord_less @ nat @ Ma2 @ Xa )
                             => ( Y2
                                = ( if @ ( option @ nat ) @ ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
                                  @ ( if @ ( option @ nat )
                                    @ ( ( ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
                                       != ( none @ nat ) )
                                      & ( vEBT_VEBT_greater @ ( some @ nat @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) )
                                    @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( some @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_pred @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
                                    @ ( if @ ( option @ nat )
                                      @ ( ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
                                        = ( none @ nat ) )
                                      @ ( if @ ( option @ nat ) @ ( ord_less @ nat @ Mi2 @ Xa ) @ ( some @ nat @ Mi2 ) @ ( none @ nat ) )
                                      @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) )
                                  @ ( none @ nat ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_pred.elims
thf(fact_1295_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,X2: nat] :
      ( ( vEBT_T_d_e_l_e_t_e @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( zero_zero @ nat ) ) @ TreeList @ Summary ) @ X2 )
      = ( one_one @ nat ) ) ).

% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.simps(6)
thf(fact_1296_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,X2: nat] :
      ( ( vEBT_V1232361888498592333_e_t_e @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( zero_zero @ nat ) ) @ TreeList @ Summary ) @ X2 )
      = ( one_one @ nat ) ) ).

% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.simps(6)
thf(fact_1297_Leaf__0__not,axiom,
    ! [A2: $o,B2: $o] :
      ~ ( vEBT_invar_vebt @ ( vEBT_Leaf @ A2 @ B2 ) @ ( zero_zero @ nat ) ) ).

% Leaf_0_not
thf(fact_1298_deg__1__Leafy,axiom,
    ! [T2: vEBT_VEBT,N2: nat] :
      ( ( vEBT_invar_vebt @ T2 @ N2 )
     => ( ( N2
          = ( one_one @ nat ) )
       => ? [A4: $o,B3: $o] :
            ( T2
            = ( vEBT_Leaf @ A4 @ B3 ) ) ) ) ).

% deg_1_Leafy
thf(fact_1299_deg__1__Leaf,axiom,
    ! [T2: vEBT_VEBT] :
      ( ( vEBT_invar_vebt @ T2 @ ( one_one @ nat ) )
     => ? [A4: $o,B3: $o] :
          ( T2
          = ( vEBT_Leaf @ A4 @ B3 ) ) ) ).

% deg_1_Leaf
thf(fact_1300_deg1Leaf,axiom,
    ! [T2: vEBT_VEBT] :
      ( ( vEBT_invar_vebt @ T2 @ ( one_one @ nat ) )
      = ( ? [A5: $o,B5: $o] :
            ( T2
            = ( vEBT_Leaf @ A5 @ B5 ) ) ) ) ).

% deg1Leaf
thf(fact_1301_Tbuildupi__buildupi_H,axiom,
    ! [N2: nat] :
      ( ( semiring_1_of_nat @ int @ ( vEBT_V441764108873111860ildupi @ N2 ) )
      = ( vEBT_V9176841429113362141ildupi @ N2 ) ) ).

% Tbuildupi_buildupi'
thf(fact_1302_double__eq__0__iff,axiom,
    ! [A: $tType] :
      ( ( linord5086331880401160121up_add @ A )
     => ! [A2: A] :
          ( ( ( plus_plus @ A @ A2 @ A2 )
            = ( zero_zero @ A ) )
          = ( A2
            = ( zero_zero @ A ) ) ) ) ).

% double_eq_0_iff
thf(fact_1303_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_I1_J,axiom,
    ! [A2: $o,B2: $o,X2: nat] :
      ( ( vEBT_T_m_e_m_b_e_r2 @ ( vEBT_Leaf @ A2 @ B2 ) @ X2 )
      = ( one_one @ nat ) ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.simps(1)
thf(fact_1304_VEBT_Osize_I4_J,axiom,
    ! [X212: $o,X222: $o] :
      ( ( size_size @ vEBT_VEBT @ ( vEBT_Leaf @ X212 @ X222 ) )
      = ( zero_zero @ nat ) ) ).

% VEBT.size(4)
thf(fact_1305_VEBT__internal_Ovalid_H_Ocases,axiom,
    ! [X2: product_prod @ vEBT_VEBT @ nat] :
      ( ! [Uu2: $o,Uv2: $o,D4: nat] :
          ( X2
         != ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ D4 ) )
     => ~ ! [Mima: option @ ( product_prod @ nat @ nat ),Deg2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT,Deg3: nat] :
            ( X2
           != ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Mima @ Deg2 @ TreeList2 @ Summary2 ) @ Deg3 ) ) ) ).

% VEBT_internal.valid'.cases
thf(fact_1306_VEBT_Odistinct_I1_J,axiom,
    ! [X11: option @ ( product_prod @ nat @ nat ),X12: nat,X13: list @ vEBT_VEBT,X14: vEBT_VEBT,X212: $o,X222: $o] :
      ( ( vEBT_Node @ X11 @ X12 @ X13 @ X14 )
     != ( vEBT_Leaf @ X212 @ X222 ) ) ).

% VEBT.distinct(1)
thf(fact_1307_VEBT_Oexhaust,axiom,
    ! [Y2: vEBT_VEBT] :
      ( ! [X112: option @ ( product_prod @ nat @ nat ),X122: nat,X132: list @ vEBT_VEBT,X142: vEBT_VEBT] :
          ( Y2
         != ( vEBT_Node @ X112 @ X122 @ X132 @ X142 ) )
     => ~ ! [X213: $o,X223: $o] :
            ( Y2
           != ( vEBT_Leaf @ X213 @ X223 ) ) ) ).

% VEBT.exhaust
thf(fact_1308_VEBT__internal_OminNull_Osimps_I1_J,axiom,
    vEBT_VEBT_minNull @ ( vEBT_Leaf @ $false @ $false ) ).

% VEBT_internal.minNull.simps(1)
thf(fact_1309_VEBT__internal_OminNull_Osimps_I2_J,axiom,
    ! [Uv: $o] :
      ~ ( vEBT_VEBT_minNull @ ( vEBT_Leaf @ $true @ Uv ) ) ).

% VEBT_internal.minNull.simps(2)
thf(fact_1310_VEBT__internal_OminNull_Osimps_I3_J,axiom,
    ! [Uu: $o] :
      ~ ( vEBT_VEBT_minNull @ ( vEBT_Leaf @ Uu @ $true ) ) ).

% VEBT_internal.minNull.simps(3)
thf(fact_1311_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,
    ! [A2: $o,B2: $o,N2: nat] :
      ( ( vEBT_V1232361888498592333_e_t_e @ ( vEBT_Leaf @ A2 @ B2 ) @ ( suc @ ( suc @ N2 ) ) )
      = ( one_one @ nat ) ) ).

% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.simps(3)
thf(fact_1312_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,
    ! [A2: $o,B2: $o,N2: nat] :
      ( ( vEBT_T_d_e_l_e_t_e @ ( vEBT_Leaf @ A2 @ B2 ) @ ( suc @ ( suc @ N2 ) ) )
      = ( one_one @ nat ) ) ).

% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.simps(3)
thf(fact_1313_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,
    ! [A2: $o,B2: $o] :
      ( ( vEBT_V1232361888498592333_e_t_e @ ( vEBT_Leaf @ A2 @ B2 ) @ ( 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_1314_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,
    ! [A2: $o,B2: $o] :
      ( ( vEBT_T_d_e_l_e_t_e @ ( vEBT_Leaf @ A2 @ B2 ) @ ( zero_zero @ nat ) )
      = ( one_one @ nat ) ) ).

% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.simps(1)
thf(fact_1315_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,
    ! [A2: $o,B2: $o] :
      ( ( vEBT_V1232361888498592333_e_t_e @ ( vEBT_Leaf @ A2 @ B2 ) @ ( 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_1316_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,
    ! [A2: $o,B2: $o] :
      ( ( vEBT_T_d_e_l_e_t_e @ ( vEBT_Leaf @ A2 @ B2 ) @ ( suc @ ( zero_zero @ nat ) ) )
      = ( one_one @ nat ) ) ).

% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.simps(2)
thf(fact_1317_VEBT__internal_Onaive__member_Ocases,axiom,
    ! [X2: product_prod @ vEBT_VEBT @ nat] :
      ( ! [A4: $o,B3: $o,X3: nat] :
          ( X2
         != ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A4 @ B3 ) @ X3 ) )
     => ( ! [Uu2: option @ ( product_prod @ nat @ nat ),Uv2: list @ vEBT_VEBT,Uw2: vEBT_VEBT,Ux2: nat] :
            ( X2
           != ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Uu2 @ ( zero_zero @ nat ) @ Uv2 @ Uw2 ) @ Ux2 ) )
       => ~ ! [Uy2: option @ ( product_prod @ nat @ nat ),V3: nat,TreeList2: list @ vEBT_VEBT,S: vEBT_VEBT,X3: nat] :
              ( X2
             != ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Uy2 @ ( suc @ V3 ) @ TreeList2 @ S ) @ X3 ) ) ) ) ).

% VEBT_internal.naive_member.cases
thf(fact_1318_invar__vebt_Ointros_I1_J,axiom,
    ! [A2: $o,B2: $o] : ( vEBT_invar_vebt @ ( vEBT_Leaf @ A2 @ B2 ) @ ( suc @ ( zero_zero @ nat ) ) ) ).

% invar_vebt.intros(1)
thf(fact_1319_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_1320_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,
    ! [X2: vEBT_VEBT] :
      ( ( X2
       != ( vEBT_Leaf @ $false @ $false ) )
     => ( ! [Uv2: $o] :
            ( X2
           != ( vEBT_Leaf @ $true @ Uv2 ) )
       => ( ! [Uu2: $o] :
              ( X2
             != ( vEBT_Leaf @ Uu2 @ $true ) )
         => ( ! [Uw2: nat,Ux2: list @ vEBT_VEBT,Uy2: vEBT_VEBT] :
                ( X2
               != ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uw2 @ Ux2 @ Uy2 ) )
           => ~ ! [Uz2: product_prod @ nat @ nat,Va2: nat,Vb2: list @ vEBT_VEBT,Vc: vEBT_VEBT] :
                  ( X2
                 != ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ Uz2 ) @ Va2 @ Vb2 @ Vc ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.cases
thf(fact_1321_vebt__member_Osimps_I1_J,axiom,
    ! [A2: $o,B2: $o,X2: nat] :
      ( ( vEBT_vebt_member @ ( vEBT_Leaf @ A2 @ B2 ) @ X2 )
      = ( ( ( X2
            = ( zero_zero @ nat ) )
         => A2 )
        & ( ( X2
           != ( zero_zero @ nat ) )
         => ( ( ( X2
                = ( one_one @ nat ) )
             => B2 )
            & ( X2
              = ( one_one @ nat ) ) ) ) ) ) ).

% vebt_member.simps(1)
thf(fact_1322_vebt__buildup_Osimps_I2_J,axiom,
    ( ( vEBT_vebt_buildup @ ( suc @ ( zero_zero @ nat ) ) )
    = ( vEBT_Leaf @ $false @ $false ) ) ).

% vebt_buildup.simps(2)
thf(fact_1323_VEBT__internal_OminNull_Oelims_I3_J,axiom,
    ! [X2: vEBT_VEBT] :
      ( ~ ( vEBT_VEBT_minNull @ X2 )
     => ( ! [Uv2: $o] :
            ( X2
           != ( vEBT_Leaf @ $true @ Uv2 ) )
       => ( ! [Uu2: $o] :
              ( X2
             != ( vEBT_Leaf @ Uu2 @ $true ) )
         => ~ ! [Uz2: product_prod @ nat @ nat,Va2: nat,Vb2: list @ vEBT_VEBT,Vc: vEBT_VEBT] :
                ( X2
               != ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ Uz2 ) @ Va2 @ Vb2 @ Vc ) ) ) ) ) ).

% VEBT_internal.minNull.elims(3)
thf(fact_1324_vebt__insert_Osimps_I1_J,axiom,
    ! [X2: nat,A2: $o,B2: $o] :
      ( ( ( X2
          = ( zero_zero @ nat ) )
       => ( ( vEBT_vebt_insert @ ( vEBT_Leaf @ A2 @ B2 ) @ X2 )
          = ( vEBT_Leaf @ $true @ B2 ) ) )
      & ( ( X2
         != ( zero_zero @ nat ) )
       => ( ( ( X2
              = ( one_one @ nat ) )
           => ( ( vEBT_vebt_insert @ ( vEBT_Leaf @ A2 @ B2 ) @ X2 )
              = ( vEBT_Leaf @ A2 @ $true ) ) )
          & ( ( X2
             != ( one_one @ nat ) )
           => ( ( vEBT_vebt_insert @ ( vEBT_Leaf @ A2 @ B2 ) @ X2 )
              = ( vEBT_Leaf @ A2 @ B2 ) ) ) ) ) ) ).

% vebt_insert.simps(1)
thf(fact_1325_vebt__succ_Osimps_I2_J,axiom,
    ! [Uv: $o,Uw: $o,N2: nat] :
      ( ( vEBT_vebt_succ @ ( vEBT_Leaf @ Uv @ Uw ) @ ( suc @ N2 ) )
      = ( none @ nat ) ) ).

% vebt_succ.simps(2)
thf(fact_1326_vebt__pred_Osimps_I1_J,axiom,
    ! [Uu: $o,Uv: $o] :
      ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ Uu @ Uv ) @ ( zero_zero @ nat ) )
      = ( none @ nat ) ) ).

% vebt_pred.simps(1)
thf(fact_1327_VEBT__internal_OminNull_Oelims_I2_J,axiom,
    ! [X2: vEBT_VEBT] :
      ( ( vEBT_VEBT_minNull @ X2 )
     => ( ( X2
         != ( vEBT_Leaf @ $false @ $false ) )
       => ~ ! [Uw2: nat,Ux2: list @ vEBT_VEBT,Uy2: vEBT_VEBT] :
              ( X2
             != ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uw2 @ Ux2 @ Uy2 ) ) ) ) ).

% VEBT_internal.minNull.elims(2)
thf(fact_1328_vebt__assn__raw_Ocases,axiom,
    ! [X2: product_prod @ vEBT_VEBT @ vEBT_VEBTi] :
      ( ! [A4: $o,B3: $o,Ai: $o,Bi: $o] :
          ( X2
         != ( product_Pair @ vEBT_VEBT @ vEBT_VEBTi @ ( vEBT_Leaf @ A4 @ B3 ) @ ( vEBT_Leafi @ Ai @ Bi ) ) )
     => ( ! [Mmo2: option @ ( product_prod @ nat @ nat ),Deg2: nat,Tree_list2: list @ vEBT_VEBT,Summary2: vEBT_VEBT,Mmoi2: option @ ( product_prod @ nat @ nat ),Degi2: nat,Tree_array2: array @ vEBT_VEBTi,Summaryi2: vEBT_VEBTi] :
            ( X2
           != ( product_Pair @ vEBT_VEBT @ vEBT_VEBTi @ ( vEBT_Node @ Mmo2 @ Deg2 @ Tree_list2 @ Summary2 ) @ ( vEBT_Nodei @ Mmoi2 @ Degi2 @ Tree_array2 @ Summaryi2 ) ) )
       => ( ! [V3: option @ ( product_prod @ nat @ nat ),Va3: nat,Vb3: list @ vEBT_VEBT,Vc3: vEBT_VEBT,Vd3: $o,Ve3: $o] :
              ( X2
             != ( product_Pair @ vEBT_VEBT @ vEBT_VEBTi @ ( vEBT_Node @ V3 @ Va3 @ Vb3 @ Vc3 ) @ ( vEBT_Leafi @ Vd3 @ Ve3 ) ) )
         => ~ ! [Vd3: $o,Ve3: $o,V3: option @ ( product_prod @ nat @ nat ),Va3: nat,Vb3: array @ vEBT_VEBTi,Vc3: vEBT_VEBTi] :
                ( X2
               != ( product_Pair @ vEBT_VEBT @ vEBT_VEBTi @ ( vEBT_Leaf @ Vd3 @ Ve3 ) @ ( vEBT_Nodei @ V3 @ Va3 @ Vb3 @ Vc3 ) ) ) ) ) ) ).

% vebt_assn_raw.cases
thf(fact_1329_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Osimps_I3_J,axiom,
    ! [A2: $o,B2: $o,Va: nat] :
      ( ( vEBT_T_p_r_e_d2 @ ( vEBT_Leaf @ A2 @ B2 ) @ ( suc @ ( suc @ Va ) ) )
      = ( one_one @ nat ) ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.simps(3)
thf(fact_1330_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Osimps_I1_J,axiom,
    ! [Uu: $o,Uv: $o] :
      ( ( vEBT_T_p_r_e_d2 @ ( vEBT_Leaf @ Uu @ Uv ) @ ( zero_zero @ nat ) )
      = ( one_one @ nat ) ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.simps(1)
thf(fact_1331_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Osimps_I2_J,axiom,
    ! [Uv: $o,Uw: $o,N2: nat] :
      ( ( vEBT_T_s_u_c_c2 @ ( vEBT_Leaf @ Uv @ Uw ) @ ( suc @ N2 ) )
      = ( one_one @ nat ) ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.simps(2)
thf(fact_1332_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Osimps_I1_J,axiom,
    ! [Uu: $o,B2: $o] :
      ( ( vEBT_T_s_u_c_c2 @ ( vEBT_Leaf @ Uu @ B2 ) @ ( zero_zero @ nat ) )
      = ( one_one @ nat ) ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.simps(1)
thf(fact_1333_vebt__assn__raw_Osimps_I1_J,axiom,
    ! [A2: $o,B2: $o,Ai2: $o,Bi2: $o] :
      ( ( vEBT_vebt_assn_raw @ ( vEBT_Leaf @ A2 @ B2 ) @ ( vEBT_Leafi @ Ai2 @ Bi2 ) )
      = ( pure_assn
        @ ( ( Ai2 = A2 )
          & ( Bi2 = B2 ) ) ) ) ).

% vebt_assn_raw.simps(1)
thf(fact_1334_less__eq__int__code_I1_J,axiom,
    ord_less_eq @ int @ ( zero_zero @ int ) @ ( zero_zero @ int ) ).

% less_eq_int_code(1)
thf(fact_1335_plus__int__code_I2_J,axiom,
    ! [L: int] :
      ( ( plus_plus @ int @ ( zero_zero @ int ) @ L )
      = L ) ).

% plus_int_code(2)
thf(fact_1336_plus__int__code_I1_J,axiom,
    ! [K: int] :
      ( ( plus_plus @ int @ K @ ( zero_zero @ int ) )
      = K ) ).

% plus_int_code(1)
thf(fact_1337_times__int__code_I2_J,axiom,
    ! [L: int] :
      ( ( times_times @ int @ ( zero_zero @ int ) @ L )
      = ( zero_zero @ int ) ) ).

% times_int_code(2)
thf(fact_1338_times__int__code_I1_J,axiom,
    ! [K: int] :
      ( ( times_times @ int @ K @ ( zero_zero @ int ) )
      = ( zero_zero @ int ) ) ).

% times_int_code(1)
thf(fact_1339_zmult__zless__mono2,axiom,
    ! [I: int,J: int,K: int] :
      ( ( ord_less @ int @ I @ J )
     => ( ( ord_less @ int @ ( zero_zero @ int ) @ K )
       => ( ord_less @ int @ ( times_times @ int @ K @ I ) @ ( times_times @ int @ K @ J ) ) ) ) ).

% zmult_zless_mono2
thf(fact_1340_minus__int__code_I1_J,axiom,
    ! [K: int] :
      ( ( minus_minus @ int @ K @ ( zero_zero @ int ) )
      = K ) ).

% minus_int_code(1)
thf(fact_1341_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t_Ocases,axiom,
    ! [X2: vEBT_VEBT] :
      ( ! [A4: $o,B3: $o] :
          ( X2
         != ( vEBT_Leaf @ A4 @ B3 ) )
     => ( ! [Uu2: nat,Uv2: list @ vEBT_VEBT,Uw2: vEBT_VEBT] :
            ( X2
           != ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw2 ) )
       => ~ ! [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list @ vEBT_VEBT,Uz2: vEBT_VEBT] :
              ( X2
             != ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) ) ) ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>t.cases
thf(fact_1342_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Osimps_I2_J,axiom,
    ! [A2: $o,Uw: $o] :
      ( ( vEBT_T_p_r_e_d2 @ ( vEBT_Leaf @ A2 @ Uw ) @ ( suc @ ( zero_zero @ nat ) ) )
      = ( one_one @ nat ) ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.simps(2)
thf(fact_1343_VEBT__internal_OminNull_Oelims_I1_J,axiom,
    ! [X2: vEBT_VEBT,Y2: $o] :
      ( ( ( vEBT_VEBT_minNull @ X2 )
        = Y2 )
     => ( ( ( X2
            = ( vEBT_Leaf @ $false @ $false ) )
         => ~ Y2 )
       => ( ( ? [Uv2: $o] :
                ( X2
                = ( vEBT_Leaf @ $true @ Uv2 ) )
           => Y2 )
         => ( ( ? [Uu2: $o] :
                  ( X2
                  = ( vEBT_Leaf @ Uu2 @ $true ) )
             => Y2 )
           => ( ( ? [Uw2: nat,Ux2: list @ vEBT_VEBT,Uy2: vEBT_VEBT] :
                    ( X2
                    = ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uw2 @ Ux2 @ Uy2 ) )
               => ~ Y2 )
             => ~ ( ? [Uz2: product_prod @ nat @ nat,Va2: nat,Vb2: list @ vEBT_VEBT,Vc: vEBT_VEBT] :
                      ( X2
                      = ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ Uz2 ) @ Va2 @ Vb2 @ Vc ) )
                 => Y2 ) ) ) ) ) ) ).

% VEBT_internal.minNull.elims(1)
thf(fact_1344_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,
    ! [X2: product_prod @ vEBT_VEBT @ nat] :
      ( ! [A4: $o,B3: $o] :
          ( X2
         != ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A4 @ B3 ) @ ( zero_zero @ nat ) ) )
     => ( ! [A4: $o,B3: $o] :
            ( X2
           != ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A4 @ B3 ) @ ( suc @ ( zero_zero @ nat ) ) ) )
       => ( ! [A4: $o,B3: $o,N4: nat] :
              ( X2
             != ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A4 @ B3 ) @ ( suc @ ( suc @ N4 ) ) ) )
         => ( ! [Deg2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT,Uu2: nat] :
                ( X2
               != ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg2 @ TreeList2 @ Summary2 ) @ Uu2 ) )
           => ( ! [Mi2: nat,Ma2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT,X3: nat] :
                  ( X2
                 != ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ 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] :
                    ( X2
                   != ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( zero_zero @ nat ) ) @ TreeList2 @ Summary2 ) @ X3 ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT,X3: nat] :
                      ( X2
                     != ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) @ X3 ) ) ) ) ) ) ) ) ).

% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.cases
thf(fact_1345_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_1346_vebt__pred_Osimps_I2_J,axiom,
    ! [A2: $o,Uw: $o] :
      ( ( A2
       => ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ A2 @ Uw ) @ ( suc @ ( zero_zero @ nat ) ) )
          = ( some @ nat @ ( zero_zero @ nat ) ) ) )
      & ( ~ A2
       => ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ A2 @ Uw ) @ ( suc @ ( zero_zero @ nat ) ) )
          = ( none @ nat ) ) ) ) ).

% vebt_pred.simps(2)
thf(fact_1347_vebt__succ_Osimps_I1_J,axiom,
    ! [B2: $o,Uu: $o] :
      ( ( B2
       => ( ( vEBT_vebt_succ @ ( vEBT_Leaf @ Uu @ B2 ) @ ( zero_zero @ nat ) )
          = ( some @ nat @ ( one_one @ nat ) ) ) )
      & ( ~ B2
       => ( ( vEBT_vebt_succ @ ( vEBT_Leaf @ Uu @ B2 ) @ ( zero_zero @ nat ) )
          = ( none @ nat ) ) ) ) ).

% vebt_succ.simps(1)
thf(fact_1348_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_I2_J,axiom,
    ! [Uu: nat,Uv: list @ vEBT_VEBT,Uw: vEBT_VEBT,X2: nat] :
      ( ( vEBT_T_m_e_m_b_e_r2 @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu @ Uv @ Uw ) @ X2 )
      = ( one_one @ nat ) ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.simps(2)
thf(fact_1349_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,
    ! [X2: product_prod @ vEBT_VEBT @ nat] :
      ( ! [A4: $o,B3: $o,X3: nat] :
          ( X2
         != ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A4 @ B3 ) @ X3 ) )
     => ( ! [Uu2: nat,Uv2: list @ vEBT_VEBT,Uw2: vEBT_VEBT,X3: nat] :
            ( X2
           != ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw2 ) @ X3 ) )
       => ( ! [V3: product_prod @ nat @ nat,Uy2: list @ vEBT_VEBT,Uz2: vEBT_VEBT,X3: nat] :
              ( X2
             != ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( zero_zero @ nat ) @ Uy2 @ Uz2 ) @ X3 ) )
         => ( ! [V3: product_prod @ nat @ nat,Vb2: list @ vEBT_VEBT,Vc: vEBT_VEBT,X3: nat] :
                ( X2
               != ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vb2 @ Vc ) @ X3 ) )
           => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT,X3: nat] :
                  ( X2
                 != ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) @ X3 ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.cases
thf(fact_1350_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,
    ! [X2: product_prod @ vEBT_VEBT @ nat] :
      ( ! [A4: $o,B3: $o,X3: nat] :
          ( X2
         != ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A4 @ B3 ) @ X3 ) )
     => ( ! [Info2: option @ ( product_prod @ nat @ nat ),Ts2: list @ vEBT_VEBT,S: vEBT_VEBT,X3: nat] :
            ( X2
           != ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Info2 @ ( zero_zero @ nat ) @ Ts2 @ S ) @ X3 ) )
       => ( ! [Info2: option @ ( product_prod @ nat @ nat ),Ts2: list @ vEBT_VEBT,S: vEBT_VEBT,X3: nat] :
              ( X2
             != ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Info2 @ ( suc @ ( zero_zero @ nat ) ) @ Ts2 @ S ) @ X3 ) )
         => ( ! [V3: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT,X3: nat] :
                ( X2
               != ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ V3 ) ) @ TreeList2 @ Summary2 ) @ X3 ) )
           => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT,X3: nat] :
                  ( X2
                 != ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) @ X3 ) ) ) ) ) ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.cases
thf(fact_1351_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Ocases,axiom,
    ! [X2: product_prod @ vEBT_VEBT @ nat] :
      ( ! [Uu2: $o,B3: $o] :
          ( X2
         != ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ Uu2 @ B3 ) @ ( zero_zero @ nat ) ) )
     => ( ! [Uv2: $o,Uw2: $o,N4: nat] :
            ( X2
           != ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ Uv2 @ Uw2 ) @ ( suc @ N4 ) ) )
       => ( ! [Ux2: nat,Uy2: list @ vEBT_VEBT,Uz2: vEBT_VEBT,Va2: nat] :
              ( X2
             != ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Ux2 @ Uy2 @ Uz2 ) @ Va2 ) )
         => ( ! [V3: product_prod @ nat @ nat,Vc: list @ vEBT_VEBT,Vd2: vEBT_VEBT,Ve2: nat] :
                ( X2
               != ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( zero_zero @ nat ) @ Vc @ Vd2 ) @ Ve2 ) )
           => ( ! [V3: product_prod @ nat @ nat,Vg2: list @ vEBT_VEBT,Vh2: vEBT_VEBT,Vi2: nat] :
                  ( X2
                 != ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vg2 @ Vh2 ) @ Vi2 ) )
             => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT,X3: nat] :
                    ( X2
                   != ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) @ X3 ) ) ) ) ) ) ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.cases
thf(fact_1352_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Ocases,axiom,
    ! [X2: product_prod @ vEBT_VEBT @ nat] :
      ( ! [Uu2: $o,Uv2: $o] :
          ( X2
         != ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ ( zero_zero @ nat ) ) )
     => ( ! [A4: $o,Uw2: $o] :
            ( X2
           != ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A4 @ Uw2 ) @ ( suc @ ( zero_zero @ nat ) ) ) )
       => ( ! [A4: $o,B3: $o,Va3: nat] :
              ( X2
             != ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A4 @ B3 ) @ ( suc @ ( suc @ Va3 ) ) ) )
         => ( ! [Uy2: nat,Uz2: list @ vEBT_VEBT,Va2: vEBT_VEBT,Vb2: nat] :
                ( X2
               != ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uy2 @ Uz2 @ Va2 ) @ Vb2 ) )
           => ( ! [V3: product_prod @ nat @ nat,Vd2: list @ vEBT_VEBT,Ve2: vEBT_VEBT,Vf2: nat] :
                  ( X2
                 != ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( zero_zero @ nat ) @ Vd2 @ Ve2 ) @ Vf2 ) )
             => ( ! [V3: product_prod @ nat @ nat,Vh2: list @ vEBT_VEBT,Vi2: vEBT_VEBT,Vj2: nat] :
                    ( X2
                   != ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vh2 @ Vi2 ) @ Vj2 ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT,X3: nat] :
                      ( X2
                     != ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) @ X3 ) ) ) ) ) ) ) ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.cases
thf(fact_1353_VEBT__internal_Omembermima_Ocases,axiom,
    ! [X2: product_prod @ vEBT_VEBT @ nat] :
      ( ! [Uu2: $o,Uv2: $o,Uw2: nat] :
          ( X2
         != ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Uw2 ) )
     => ( ! [Ux2: list @ vEBT_VEBT,Uy2: vEBT_VEBT,Uz2: nat] :
            ( X2
           != ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( zero_zero @ nat ) @ Ux2 @ Uy2 ) @ Uz2 ) )
       => ( ! [Mi2: nat,Ma2: nat,Va2: list @ vEBT_VEBT,Vb2: vEBT_VEBT,X3: nat] :
              ( X2
             != ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ Va2 @ Vb2 ) @ X3 ) )
         => ( ! [Mi2: nat,Ma2: nat,V3: nat,TreeList2: list @ vEBT_VEBT,Vc: vEBT_VEBT,X3: nat] :
                ( X2
               != ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ V3 ) @ TreeList2 @ Vc ) @ X3 ) )
           => ~ ! [V3: nat,TreeList2: list @ vEBT_VEBT,Vd2: vEBT_VEBT,X3: nat] :
                  ( X2
                 != ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ V3 ) @ TreeList2 @ Vd2 ) @ X3 ) ) ) ) ) ) ).

% VEBT_internal.membermima.cases
thf(fact_1354_vebt__pred_Osimps_I3_J,axiom,
    ! [B2: $o,A2: $o,Va: nat] :
      ( ( B2
       => ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ A2 @ B2 ) @ ( suc @ ( suc @ Va ) ) )
          = ( some @ nat @ ( one_one @ nat ) ) ) )
      & ( ~ B2
       => ( ( A2
           => ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ A2 @ B2 ) @ ( suc @ ( suc @ Va ) ) )
              = ( some @ nat @ ( zero_zero @ nat ) ) ) )
          & ( ~ A2
           => ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ A2 @ B2 ) @ ( suc @ ( suc @ Va ) ) )
              = ( none @ nat ) ) ) ) ) ) ).

% vebt_pred.simps(3)
thf(fact_1355_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,
    ! [A2: $o,B2: $o,X2: nat] :
      ( ( vEBT_T_m_e_m_b_e_r @ ( vEBT_Leaf @ A2 @ B2 ) @ X2 )
      = ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) )
        @ ( if @ nat
          @ ( X2
            = ( 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_1356_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,Uy: list @ vEBT_VEBT,Uz: vEBT_VEBT,X2: nat] :
      ( ( vEBT_T_m_e_m_b_e_r2 @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V ) @ ( zero_zero @ nat ) @ Uy @ Uz ) @ X2 )
      = ( one_one @ nat ) ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.simps(3)
thf(fact_1357_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_1358_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_1359_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_1360_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_1361_odd__nonzero,axiom,
    ! [Z: int] :
      ( ( plus_plus @ int @ ( plus_plus @ int @ ( one_one @ int ) @ Z ) @ Z )
     != ( zero_zero @ int ) ) ).

% odd_nonzero
thf(fact_1362_pos__zmult__eq__1__iff,axiom,
    ! [M: int,N2: int] :
      ( ( ord_less @ int @ ( zero_zero @ int ) @ M )
     => ( ( ( times_times @ int @ M @ N2 )
          = ( one_one @ int ) )
        = ( ( M
            = ( one_one @ int ) )
          & ( N2
            = ( one_one @ int ) ) ) ) ) ).

% pos_zmult_eq_1_iff
thf(fact_1363_zero__le__imp__eq__int,axiom,
    ! [K: int] :
      ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
     => ? [N4: nat] :
          ( K
          = ( semiring_1_of_nat @ int @ N4 ) ) ) ).

% zero_le_imp_eq_int
thf(fact_1364_nonneg__int__cases,axiom,
    ! [K: int] :
      ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
     => ~ ! [N4: nat] :
            ( K
           != ( semiring_1_of_nat @ int @ N4 ) ) ) ).

% nonneg_int_cases
thf(fact_1365_int__distrib_I1_J,axiom,
    ! [Z1: int,Z22: int,W: int] :
      ( ( times_times @ int @ ( plus_plus @ int @ Z1 @ Z22 ) @ W )
      = ( plus_plus @ int @ ( times_times @ int @ Z1 @ W ) @ ( times_times @ int @ Z22 @ W ) ) ) ).

% int_distrib(1)
thf(fact_1366_int__distrib_I2_J,axiom,
    ! [W: int,Z1: int,Z22: int] :
      ( ( times_times @ int @ W @ ( plus_plus @ int @ Z1 @ Z22 ) )
      = ( plus_plus @ int @ ( times_times @ int @ W @ Z1 ) @ ( times_times @ int @ W @ Z22 ) ) ) ).

% int_distrib(2)
thf(fact_1367_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_1368_int__distrib_I4_J,axiom,
    ! [W: int,Z1: int,Z22: int] :
      ( ( times_times @ int @ W @ ( minus_minus @ int @ Z1 @ Z22 ) )
      = ( minus_minus @ int @ ( times_times @ int @ W @ Z1 ) @ ( times_times @ int @ W @ Z22 ) ) ) ).

% int_distrib(4)
thf(fact_1369_int__distrib_I3_J,axiom,
    ! [Z1: int,Z22: int,W: int] :
      ( ( times_times @ int @ ( minus_minus @ int @ Z1 @ Z22 ) @ W )
      = ( minus_minus @ int @ ( times_times @ int @ Z1 @ W ) @ ( times_times @ int @ Z22 @ W ) ) ) ).

% int_distrib(3)
thf(fact_1370_int__diff__cases,axiom,
    ! [Z: int] :
      ~ ! [M2: nat,N4: nat] :
          ( Z
         != ( minus_minus @ int @ ( semiring_1_of_nat @ int @ M2 ) @ ( semiring_1_of_nat @ int @ N4 ) ) ) ).

% int_diff_cases
thf(fact_1371_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,Vb: list @ vEBT_VEBT,Vc2: vEBT_VEBT,X2: nat] :
      ( ( vEBT_T_m_e_m_b_e_r2 @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V ) @ ( suc @ ( zero_zero @ nat ) ) @ Vb @ Vc2 ) @ X2 )
      = ( one_one @ nat ) ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.simps(4)
thf(fact_1372_member__bound__height_H,axiom,
    ! [T2: vEBT_VEBT,N2: nat,X2: nat] :
      ( ( vEBT_invar_vebt @ T2 @ N2 )
     => ( ord_less_eq @ nat @ ( vEBT_T_m_e_m_b_e_r2 @ T2 @ X2 ) @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_VEBT_height @ T2 ) ) ) ) ).

% member_bound_height'
thf(fact_1373_pos__int__cases,axiom,
    ! [K: int] :
      ( ( ord_less @ int @ ( zero_zero @ int ) @ K )
     => ~ ! [N4: nat] :
            ( ( K
              = ( semiring_1_of_nat @ int @ N4 ) )
           => ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ N4 ) ) ) ).

% pos_int_cases
thf(fact_1374_zero__less__imp__eq__int,axiom,
    ! [K: int] :
      ( ( ord_less @ int @ ( zero_zero @ int ) @ K )
     => ? [N4: nat] :
          ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N4 )
          & ( K
            = ( semiring_1_of_nat @ int @ N4 ) ) ) ) ).

% zero_less_imp_eq_int
thf(fact_1375_zless__iff__Suc__zadd,axiom,
    ( ( ord_less @ int )
    = ( ^ [W2: int,Z2: int] :
        ? [N: nat] :
          ( Z2
          = ( plus_plus @ int @ W2 @ ( semiring_1_of_nat @ int @ ( suc @ N ) ) ) ) ) ) ).

% zless_iff_Suc_zadd
thf(fact_1376_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,
    ! [X2: vEBT_VEBT,Xa: nat,Y2: nat] :
      ( ( ( vEBT_T_m_e_m_b_e_r2 @ X2 @ Xa )
        = Y2 )
     => ( ( ? [A4: $o,B3: $o] :
              ( X2
              = ( vEBT_Leaf @ A4 @ B3 ) )
         => ( Y2
           != ( one_one @ nat ) ) )
       => ( ( ? [Uu2: nat,Uv2: list @ vEBT_VEBT,Uw2: vEBT_VEBT] :
                ( X2
                = ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw2 ) )
           => ( Y2
             != ( one_one @ nat ) ) )
         => ( ( ? [V3: product_prod @ nat @ nat,Uy2: list @ vEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( X2
                  = ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( zero_zero @ nat ) @ Uy2 @ Uz2 ) )
             => ( Y2
               != ( one_one @ nat ) ) )
           => ( ( ? [V3: product_prod @ nat @ nat,Vb2: list @ vEBT_VEBT,Vc: vEBT_VEBT] :
                    ( X2
                    = ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vb2 @ Vc ) )
               => ( Y2
                 != ( one_one @ nat ) ) )
             => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list @ vEBT_VEBT] :
                    ( ? [Summary2: vEBT_VEBT] :
                        ( X2
                        = ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
                   => ( Y2
                     != ( plus_plus @ nat @ ( one_one @ nat )
                        @ ( if @ nat @ ( Xa = Mi2 ) @ ( zero_zero @ nat )
                          @ ( if @ nat @ ( Xa = Ma2 ) @ ( zero_zero @ nat )
                            @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi2 ) @ ( zero_zero @ nat )
                              @ ( if @ nat @ ( ord_less @ nat @ Ma2 @ Xa ) @ ( zero_zero @ nat )
                                @ ( if @ nat
                                  @ ( ( ord_less @ nat @ Mi2 @ Xa )
                                    & ( ord_less @ nat @ Xa @ Ma2 ) )
                                  @ ( if @ nat @ ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) @ ( vEBT_T_m_e_m_b_e_r2 @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( zero_zero @ nat ) )
                                  @ ( zero_zero @ nat ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.elims
thf(fact_1377_zle__int,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq @ int @ ( semiring_1_of_nat @ int @ M ) @ ( semiring_1_of_nat @ int @ N2 ) )
      = ( ord_less_eq @ nat @ M @ N2 ) ) ).

% zle_int
thf(fact_1378_zadd__int__left,axiom,
    ! [M: nat,N2: nat,Z: int] :
      ( ( plus_plus @ int @ ( semiring_1_of_nat @ int @ M ) @ ( plus_plus @ int @ ( semiring_1_of_nat @ int @ N2 ) @ Z ) )
      = ( plus_plus @ int @ ( semiring_1_of_nat @ int @ ( plus_plus @ nat @ M @ N2 ) ) @ Z ) ) ).

% zadd_int_left
thf(fact_1379_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_1380_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_1381_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_1382_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_1383_zle__iff__zadd,axiom,
    ( ( ord_less_eq @ int )
    = ( ^ [W2: int,Z2: int] :
        ? [N: nat] :
          ( Z2
          = ( plus_plus @ int @ W2 @ ( semiring_1_of_nat @ int @ N ) ) ) ) ) ).

% zle_iff_zadd
thf(fact_1384_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_1385_zmult__zless__mono2__lemma,axiom,
    ! [I: int,J: int,K: nat] :
      ( ( ord_less @ int @ I @ J )
     => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
       => ( ord_less @ int @ ( times_times @ int @ ( semiring_1_of_nat @ int @ K ) @ I ) @ ( times_times @ int @ ( semiring_1_of_nat @ int @ K ) @ J ) ) ) ) ).

% zmult_zless_mono2_lemma
thf(fact_1386_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_1387_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_I4_J,axiom,
    ! [Deg: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,Uu: nat] :
      ( ( vEBT_V1232361888498592333_e_t_e @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg @ TreeList @ Summary ) @ Uu )
      = ( one_one @ nat ) ) ).

% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.simps(4)
thf(fact_1388_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_I4_J,axiom,
    ! [Deg: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,Uu: nat] :
      ( ( vEBT_T_d_e_l_e_t_e @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg @ TreeList @ Summary ) @ Uu )
      = ( one_one @ nat ) ) ).

% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.simps(4)
thf(fact_1389_vebt__member_Oelims_I2_J,axiom,
    ! [X2: vEBT_VEBT,Xa: nat] :
      ( ( vEBT_vebt_member @ X2 @ Xa )
     => ( ! [A4: $o,B3: $o] :
            ( ( X2
              = ( vEBT_Leaf @ A4 @ B3 ) )
           => ~ ( ( ( Xa
                    = ( zero_zero @ nat ) )
                 => A4 )
                & ( ( Xa
                   != ( zero_zero @ nat ) )
                 => ( ( ( Xa
                        = ( one_one @ nat ) )
                     => B3 )
                    & ( Xa
                      = ( one_one @ nat ) ) ) ) ) )
       => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list @ vEBT_VEBT] :
              ( ? [Summary2: vEBT_VEBT] :
                  ( X2
                  = ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
             => ~ ( ( Xa != Mi2 )
                 => ( ( Xa != Ma2 )
                   => ( ~ ( ord_less @ nat @ Xa @ Mi2 )
                      & ( ~ ( ord_less @ nat @ Xa @ Mi2 )
                       => ( ~ ( ord_less @ nat @ Ma2 @ Xa )
                          & ( ~ ( ord_less @ nat @ Ma2 @ Xa )
                           => ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
                               => ( vEBT_vebt_member @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
                              & ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(2)
thf(fact_1390_vebt__member_Oelims_I1_J,axiom,
    ! [X2: vEBT_VEBT,Xa: nat,Y2: $o] :
      ( ( ( vEBT_vebt_member @ X2 @ Xa )
        = Y2 )
     => ( ! [A4: $o,B3: $o] :
            ( ( X2
              = ( vEBT_Leaf @ A4 @ B3 ) )
           => ( Y2
              = ( ~ ( ( ( Xa
                        = ( zero_zero @ nat ) )
                     => A4 )
                    & ( ( Xa
                       != ( zero_zero @ nat ) )
                     => ( ( ( Xa
                            = ( one_one @ nat ) )
                         => B3 )
                        & ( Xa
                          = ( one_one @ nat ) ) ) ) ) ) ) )
       => ( ( ? [Uu2: nat,Uv2: list @ vEBT_VEBT,Uw2: vEBT_VEBT] :
                ( X2
                = ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw2 ) )
           => Y2 )
         => ( ( ? [V3: product_prod @ nat @ nat,Uy2: list @ vEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( X2
                  = ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( zero_zero @ nat ) @ Uy2 @ Uz2 ) )
             => Y2 )
           => ( ( ? [V3: product_prod @ nat @ nat,Vb2: list @ vEBT_VEBT,Vc: vEBT_VEBT] :
                    ( X2
                    = ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vb2 @ Vc ) )
               => Y2 )
             => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list @ vEBT_VEBT] :
                    ( ? [Summary2: vEBT_VEBT] :
                        ( X2
                        = ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
                   => ( Y2
                      = ( ~ ( ( Xa != Mi2 )
                           => ( ( Xa != Ma2 )
                             => ( ~ ( ord_less @ nat @ Xa @ Mi2 )
                                & ( ~ ( ord_less @ nat @ Xa @ Mi2 )
                                 => ( ~ ( ord_less @ nat @ Ma2 @ Xa )
                                    & ( ~ ( ord_less @ nat @ Ma2 @ Xa )
                                     => ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
                                         => ( vEBT_vebt_member @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
                                        & ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(1)
thf(fact_1391_vebt__member_Oelims_I3_J,axiom,
    ! [X2: vEBT_VEBT,Xa: nat] :
      ( ~ ( vEBT_vebt_member @ X2 @ Xa )
     => ( ! [A4: $o,B3: $o] :
            ( ( X2
              = ( vEBT_Leaf @ A4 @ B3 ) )
           => ( ( ( Xa
                  = ( zero_zero @ nat ) )
               => A4 )
              & ( ( Xa
                 != ( zero_zero @ nat ) )
               => ( ( ( Xa
                      = ( one_one @ nat ) )
                   => B3 )
                  & ( Xa
                    = ( one_one @ nat ) ) ) ) ) )
       => ( ! [Uu2: nat,Uv2: list @ vEBT_VEBT,Uw2: vEBT_VEBT] :
              ( X2
             != ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw2 ) )
         => ( ! [V3: product_prod @ nat @ nat,Uy2: list @ vEBT_VEBT,Uz2: vEBT_VEBT] :
                ( X2
               != ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( zero_zero @ nat ) @ Uy2 @ Uz2 ) )
           => ( ! [V3: product_prod @ nat @ nat,Vb2: list @ vEBT_VEBT,Vc: vEBT_VEBT] :
                  ( X2
                 != ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vb2 @ Vc ) )
             => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list @ vEBT_VEBT] :
                    ( ? [Summary2: vEBT_VEBT] :
                        ( X2
                        = ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
                   => ( ( Xa != Mi2 )
                     => ( ( Xa != Ma2 )
                       => ( ~ ( ord_less @ nat @ Xa @ Mi2 )
                          & ( ~ ( ord_less @ nat @ Xa @ Mi2 )
                           => ( ~ ( ord_less @ nat @ Ma2 @ Xa )
                              & ( ~ ( ord_less @ nat @ Ma2 @ Xa )
                               => ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
                                   => ( vEBT_vebt_member @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
                                  & ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(3)
thf(fact_1392_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,
    ! [X2: vEBT_VEBT,Xa: nat,Y2: nat] :
      ( ( ( vEBT_T_m_e_m_b_e_r @ X2 @ Xa )
        = Y2 )
     => ( ( ? [A4: $o,B3: $o] :
              ( X2
              = ( vEBT_Leaf @ A4 @ B3 ) )
         => ( Y2
           != ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) )
              @ ( if @ nat
                @ ( Xa
                  = ( zero_zero @ nat ) )
                @ ( one_one @ nat )
                @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( one_one @ nat ) ) ) ) ) )
       => ( ( ? [Uu2: nat,Uv2: list @ vEBT_VEBT,Uw2: vEBT_VEBT] :
                ( X2
                = ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw2 ) )
           => ( Y2
             != ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
         => ( ( ? [V3: product_prod @ nat @ nat,Uy2: list @ vEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( X2
                  = ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( zero_zero @ nat ) @ Uy2 @ Uz2 ) )
             => ( Y2
               != ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
           => ( ( ? [V3: product_prod @ nat @ nat,Vb2: list @ vEBT_VEBT,Vc: vEBT_VEBT] :
                    ( X2
                    = ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vb2 @ Vc ) )
               => ( Y2
                 != ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
             => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list @ vEBT_VEBT] :
                    ( ? [Summary2: vEBT_VEBT] :
                        ( X2
                        = ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
                   => ( Y2
                     != ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( if @ nat @ ( Xa = Mi2 ) @ ( one_one @ nat ) @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( if @ nat @ ( Xa = Ma2 ) @ ( one_one @ nat ) @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi2 ) @ ( one_one @ nat ) @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( if @ nat @ ( ord_less @ nat @ Ma2 @ Xa ) @ ( one_one @ nat ) @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) @ ( if @ nat @ ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_T_m_e_m_b_e_r @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( one_one @ nat ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.elims
thf(fact_1393_invar__vebt_Osimps,axiom,
    ( vEBT_invar_vebt
    = ( ^ [A1: vEBT_VEBT,A22: nat] :
          ( ( ? [A5: $o,B5: $o] :
                ( A1
                = ( vEBT_Leaf @ A5 @ B5 ) )
            & ( A22
              = ( suc @ ( zero_zero @ nat ) ) ) )
          | ? [TreeList3: list @ vEBT_VEBT,N: nat,Summary3: vEBT_VEBT] :
              ( ( A1
                = ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ A22 @ TreeList3 @ Summary3 ) )
              & ! [X: vEBT_VEBT] :
                  ( ( member @ vEBT_VEBT @ X @ ( set2 @ vEBT_VEBT @ TreeList3 ) )
                 => ( vEBT_invar_vebt @ X @ N ) )
              & ( vEBT_invar_vebt @ Summary3 @ N )
              & ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 )
                = ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) )
              & ( A22
                = ( plus_plus @ nat @ N @ N ) )
              & ~ ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ Summary3 @ X5 )
              & ! [X: vEBT_VEBT] :
                  ( ( member @ vEBT_VEBT @ X @ ( set2 @ vEBT_VEBT @ TreeList3 ) )
                 => ~ ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ X @ X5 ) ) )
          | ? [TreeList3: list @ vEBT_VEBT,N: nat,Summary3: vEBT_VEBT] :
              ( ( A1
                = ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ A22 @ TreeList3 @ Summary3 ) )
              & ! [X: vEBT_VEBT] :
                  ( ( member @ vEBT_VEBT @ X @ ( set2 @ vEBT_VEBT @ TreeList3 ) )
                 => ( vEBT_invar_vebt @ X @ N ) )
              & ( vEBT_invar_vebt @ Summary3 @ ( suc @ N ) )
              & ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 )
                = ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ N ) ) )
              & ( A22
                = ( plus_plus @ nat @ N @ ( suc @ N ) ) )
              & ~ ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ Summary3 @ X5 )
              & ! [X: vEBT_VEBT] :
                  ( ( member @ vEBT_VEBT @ X @ ( set2 @ vEBT_VEBT @ TreeList3 ) )
                 => ~ ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ X @ X5 ) ) )
          | ? [TreeList3: list @ vEBT_VEBT,N: nat,Summary3: vEBT_VEBT,Mi3: nat,Ma3: nat] :
              ( ( A1
                = ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi3 @ Ma3 ) ) @ A22 @ TreeList3 @ Summary3 ) )
              & ! [X: vEBT_VEBT] :
                  ( ( member @ vEBT_VEBT @ X @ ( set2 @ vEBT_VEBT @ TreeList3 ) )
                 => ( vEBT_invar_vebt @ X @ N ) )
              & ( vEBT_invar_vebt @ Summary3 @ N )
              & ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 )
                = ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) )
              & ( A22
                = ( plus_plus @ nat @ N @ N ) )
              & ! [I2: nat] :
                  ( ( ord_less @ nat @ I2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) )
                 => ( ( ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList3 @ I2 ) @ X5 ) )
                    = ( vEBT_V8194947554948674370ptions @ Summary3 @ I2 ) ) )
              & ( ( Mi3 = Ma3 )
               => ! [X: vEBT_VEBT] :
                    ( ( member @ vEBT_VEBT @ X @ ( set2 @ vEBT_VEBT @ TreeList3 ) )
                   => ~ ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ X @ X5 ) ) )
              & ( ord_less_eq @ nat @ Mi3 @ Ma3 )
              & ( ord_less @ nat @ Ma3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ A22 ) )
              & ( ( Mi3 != Ma3 )
               => ! [I2: nat] :
                    ( ( ord_less @ nat @ I2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) )
                   => ( ( ( ( vEBT_VEBT_high @ Ma3 @ N )
                          = I2 )
                       => ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList3 @ I2 ) @ ( vEBT_VEBT_low @ Ma3 @ N ) ) )
                      & ! [X: nat] :
                          ( ( ( ( vEBT_VEBT_high @ X @ N )
                              = I2 )
                            & ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList3 @ I2 ) @ ( vEBT_VEBT_low @ X @ N ) ) )
                         => ( ( ord_less @ nat @ Mi3 @ X )
                            & ( ord_less_eq @ nat @ X @ Ma3 ) ) ) ) ) ) )
          | ? [TreeList3: list @ vEBT_VEBT,N: nat,Summary3: vEBT_VEBT,Mi3: nat,Ma3: nat] :
              ( ( A1
                = ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi3 @ Ma3 ) ) @ A22 @ TreeList3 @ Summary3 ) )
              & ! [X: vEBT_VEBT] :
                  ( ( member @ vEBT_VEBT @ X @ ( set2 @ vEBT_VEBT @ TreeList3 ) )
                 => ( vEBT_invar_vebt @ X @ N ) )
              & ( vEBT_invar_vebt @ Summary3 @ ( suc @ N ) )
              & ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 )
                = ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ N ) ) )
              & ( A22
                = ( plus_plus @ nat @ N @ ( suc @ N ) ) )
              & ! [I2: nat] :
                  ( ( ord_less @ nat @ I2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ N ) ) )
                 => ( ( ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList3 @ I2 ) @ X5 ) )
                    = ( vEBT_V8194947554948674370ptions @ Summary3 @ I2 ) ) )
              & ( ( Mi3 = Ma3 )
               => ! [X: vEBT_VEBT] :
                    ( ( member @ vEBT_VEBT @ X @ ( set2 @ vEBT_VEBT @ TreeList3 ) )
                   => ~ ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ X @ X5 ) ) )
              & ( ord_less_eq @ nat @ Mi3 @ Ma3 )
              & ( ord_less @ nat @ Ma3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ A22 ) )
              & ( ( Mi3 != Ma3 )
               => ! [I2: nat] :
                    ( ( ord_less @ nat @ I2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ N ) ) )
                   => ( ( ( ( vEBT_VEBT_high @ Ma3 @ N )
                          = I2 )
                       => ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList3 @ I2 ) @ ( vEBT_VEBT_low @ Ma3 @ N ) ) )
                      & ! [X: nat] :
                          ( ( ( ( vEBT_VEBT_high @ X @ N )
                              = I2 )
                            & ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList3 @ I2 ) @ ( vEBT_VEBT_low @ X @ N ) ) )
                         => ( ( ord_less @ nat @ Mi3 @ X )
                            & ( ord_less_eq @ nat @ X @ Ma3 ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.simps
thf(fact_1394_invar__vebt_Ocases,axiom,
    ! [A12: vEBT_VEBT,A23: nat] :
      ( ( vEBT_invar_vebt @ A12 @ A23 )
     => ( ( ? [A4: $o,B3: $o] :
              ( A12
              = ( vEBT_Leaf @ A4 @ B3 ) )
         => ( A23
           != ( suc @ ( zero_zero @ nat ) ) ) )
       => ( ! [TreeList2: list @ vEBT_VEBT,N4: nat,Summary2: vEBT_VEBT,M2: nat,Deg2: nat] :
              ( ( A12
                = ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg2 @ TreeList2 @ Summary2 ) )
             => ( ( A23 = Deg2 )
               => ( ! [X6: vEBT_VEBT] :
                      ( ( member @ vEBT_VEBT @ X6 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
                     => ( vEBT_invar_vebt @ X6 @ N4 ) )
                 => ( ( vEBT_invar_vebt @ Summary2 @ M2 )
                   => ( ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 )
                        = ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M2 ) )
                     => ( ( M2 = N4 )
                       => ( ( Deg2
                            = ( plus_plus @ nat @ N4 @ M2 ) )
                         => ( ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X_1 )
                           => ~ ! [X6: vEBT_VEBT] :
                                  ( ( member @ vEBT_VEBT @ X6 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
                                 => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X6 @ X_1 ) ) ) ) ) ) ) ) ) )
         => ( ! [TreeList2: list @ vEBT_VEBT,N4: nat,Summary2: vEBT_VEBT,M2: nat,Deg2: nat] :
                ( ( A12
                  = ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg2 @ TreeList2 @ Summary2 ) )
               => ( ( A23 = Deg2 )
                 => ( ! [X6: vEBT_VEBT] :
                        ( ( member @ vEBT_VEBT @ X6 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
                       => ( vEBT_invar_vebt @ X6 @ N4 ) )
                   => ( ( vEBT_invar_vebt @ Summary2 @ M2 )
                     => ( ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 )
                          = ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M2 ) )
                       => ( ( M2
                            = ( suc @ N4 ) )
                         => ( ( Deg2
                              = ( plus_plus @ nat @ N4 @ M2 ) )
                           => ( ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X_1 )
                             => ~ ! [X6: vEBT_VEBT] :
                                    ( ( member @ vEBT_VEBT @ X6 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
                                   => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X6 @ X_1 ) ) ) ) ) ) ) ) ) )
           => ( ! [TreeList2: list @ vEBT_VEBT,N4: nat,Summary2: vEBT_VEBT,M2: nat,Deg2: nat,Mi2: nat,Ma2: nat] :
                  ( ( A12
                    = ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ Deg2 @ TreeList2 @ Summary2 ) )
                 => ( ( A23 = Deg2 )
                   => ( ! [X6: vEBT_VEBT] :
                          ( ( member @ vEBT_VEBT @ X6 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
                         => ( vEBT_invar_vebt @ X6 @ N4 ) )
                     => ( ( vEBT_invar_vebt @ Summary2 @ M2 )
                       => ( ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 )
                            = ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M2 ) )
                         => ( ( M2 = N4 )
                           => ( ( Deg2
                                = ( plus_plus @ nat @ N4 @ M2 ) )
                             => ( ! [I4: nat] :
                                    ( ( ord_less @ nat @ I4 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M2 ) )
                                   => ( ( ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList2 @ I4 ) @ X5 ) )
                                      = ( vEBT_V8194947554948674370ptions @ Summary2 @ I4 ) ) )
                               => ( ( ( Mi2 = Ma2 )
                                   => ! [X6: vEBT_VEBT] :
                                        ( ( member @ vEBT_VEBT @ X6 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
                                       => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X6 @ X_1 ) ) )
                                 => ( ( ord_less_eq @ nat @ Mi2 @ Ma2 )
                                   => ( ( ord_less @ nat @ Ma2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) )
                                     => ~ ( ( Mi2 != Ma2 )
                                         => ! [I4: nat] :
                                              ( ( ord_less @ nat @ I4 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M2 ) )
                                             => ( ( ( ( vEBT_VEBT_high @ Ma2 @ N4 )
                                                    = I4 )
                                                 => ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList2 @ I4 ) @ ( vEBT_VEBT_low @ Ma2 @ N4 ) ) )
                                                & ! [X6: nat] :
                                                    ( ( ( ( vEBT_VEBT_high @ X6 @ N4 )
                                                        = I4 )
                                                      & ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList2 @ I4 ) @ ( vEBT_VEBT_low @ X6 @ N4 ) ) )
                                                   => ( ( ord_less @ nat @ Mi2 @ X6 )
                                                      & ( ord_less_eq @ nat @ X6 @ Ma2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
             => ~ ! [TreeList2: list @ vEBT_VEBT,N4: nat,Summary2: vEBT_VEBT,M2: nat,Deg2: nat,Mi2: nat,Ma2: nat] :
                    ( ( A12
                      = ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ Deg2 @ TreeList2 @ Summary2 ) )
                   => ( ( A23 = Deg2 )
                     => ( ! [X6: vEBT_VEBT] :
                            ( ( member @ vEBT_VEBT @ X6 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
                           => ( vEBT_invar_vebt @ X6 @ N4 ) )
                       => ( ( vEBT_invar_vebt @ Summary2 @ M2 )
                         => ( ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 )
                              = ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M2 ) )
                           => ( ( M2
                                = ( suc @ N4 ) )
                             => ( ( Deg2
                                  = ( plus_plus @ nat @ N4 @ M2 ) )
                               => ( ! [I4: nat] :
                                      ( ( ord_less @ nat @ I4 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M2 ) )
                                     => ( ( ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList2 @ I4 ) @ X5 ) )
                                        = ( vEBT_V8194947554948674370ptions @ Summary2 @ I4 ) ) )
                                 => ( ( ( Mi2 = Ma2 )
                                     => ! [X6: vEBT_VEBT] :
                                          ( ( member @ vEBT_VEBT @ X6 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
                                         => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X6 @ X_1 ) ) )
                                   => ( ( ord_less_eq @ nat @ Mi2 @ Ma2 )
                                     => ( ( ord_less @ nat @ Ma2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) )
                                       => ~ ( ( Mi2 != Ma2 )
                                           => ! [I4: nat] :
                                                ( ( ord_less @ nat @ I4 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M2 ) )
                                               => ( ( ( ( vEBT_VEBT_high @ Ma2 @ N4 )
                                                      = I4 )
                                                   => ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList2 @ I4 ) @ ( vEBT_VEBT_low @ Ma2 @ N4 ) ) )
                                                  & ! [X6: nat] :
                                                      ( ( ( ( vEBT_VEBT_high @ X6 @ N4 )
                                                          = I4 )
                                                        & ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList2 @ I4 ) @ ( vEBT_VEBT_low @ X6 @ N4 ) ) )
                                                     => ( ( ord_less @ nat @ Mi2 @ X6 )
                                                        & ( ord_less_eq @ nat @ X6 @ Ma2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.cases
thf(fact_1395_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,X2: nat] :
      ( ( vEBT_V1232361888498592333_e_t_e @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( zero_zero @ nat ) @ TreeList @ Summary ) @ X2 )
      = ( one_one @ nat ) ) ).

% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.simps(5)
thf(fact_1396_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,X2: nat] :
      ( ( vEBT_T_d_e_l_e_t_e @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( zero_zero @ nat ) @ TreeList @ Summary ) @ X2 )
      = ( one_one @ nat ) ) ).

% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.simps(5)
thf(fact_1397_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Oelims,axiom,
    ! [X2: vEBT_VEBT,Xa: nat,Y2: nat] :
      ( ( ( vEBT_T_p_r_e_d2 @ X2 @ Xa )
        = Y2 )
     => ( ( ? [Uu2: $o,Uv2: $o] :
              ( X2
              = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
         => ( ( Xa
              = ( zero_zero @ nat ) )
           => ( Y2
             != ( one_one @ nat ) ) ) )
       => ( ( ? [A4: $o,Uw2: $o] :
                ( X2
                = ( vEBT_Leaf @ A4 @ Uw2 ) )
           => ( ( Xa
                = ( suc @ ( zero_zero @ nat ) ) )
             => ( Y2
               != ( one_one @ nat ) ) ) )
         => ( ( ? [A4: $o,B3: $o] :
                  ( X2
                  = ( vEBT_Leaf @ A4 @ B3 ) )
             => ( ? [Va3: nat] :
                    ( Xa
                    = ( suc @ ( suc @ Va3 ) ) )
               => ( Y2
                 != ( one_one @ nat ) ) ) )
           => ( ( ? [Uy2: nat,Uz2: list @ vEBT_VEBT,Va2: vEBT_VEBT] :
                    ( X2
                    = ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uy2 @ Uz2 @ Va2 ) )
               => ( Y2
                 != ( one_one @ nat ) ) )
             => ( ( ? [V3: product_prod @ nat @ nat,Vd2: list @ vEBT_VEBT,Ve2: vEBT_VEBT] :
                      ( X2
                      = ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( zero_zero @ nat ) @ Vd2 @ Ve2 ) )
                 => ( Y2
                   != ( one_one @ nat ) ) )
               => ( ( ? [V3: product_prod @ nat @ nat,Vh2: list @ vEBT_VEBT,Vi2: vEBT_VEBT] :
                        ( X2
                        = ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vh2 @ Vi2 ) )
                   => ( Y2
                     != ( one_one @ nat ) ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
                        ( ( X2
                          = ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
                       => ~ ( ( ( ord_less @ nat @ Ma2 @ Xa )
                             => ( Y2
                                = ( one_one @ nat ) ) )
                            & ( ~ ( ord_less @ nat @ Ma2 @ Xa )
                             => ( Y2
                                = ( if @ nat @ ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
                                  @ ( if @ nat
                                    @ ( ( ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
                                       != ( none @ nat ) )
                                      & ( vEBT_VEBT_greater @ ( some @ nat @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) )
                                    @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_T_p_r_e_d2 @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
                                    @ ( plus_plus @ nat @ ( vEBT_T_p_r_e_d2 @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( one_one @ nat ) ) )
                                  @ ( one_one @ nat ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.elims
thf(fact_1398_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Oelims,axiom,
    ! [X2: vEBT_VEBT,Xa: nat,Y2: nat] :
      ( ( ( vEBT_T_s_u_c_c2 @ X2 @ Xa )
        = Y2 )
     => ( ( ? [Uu2: $o,B3: $o] :
              ( X2
              = ( vEBT_Leaf @ Uu2 @ B3 ) )
         => ( ( Xa
              = ( zero_zero @ nat ) )
           => ( Y2
             != ( one_one @ nat ) ) ) )
       => ( ( ? [Uv2: $o,Uw2: $o] :
                ( X2
                = ( vEBT_Leaf @ Uv2 @ Uw2 ) )
           => ( ? [N4: nat] :
                  ( Xa
                  = ( suc @ N4 ) )
             => ( Y2
               != ( one_one @ nat ) ) ) )
         => ( ( ? [Ux2: nat,Uy2: list @ vEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( X2
                  = ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Ux2 @ Uy2 @ Uz2 ) )
             => ( Y2
               != ( one_one @ nat ) ) )
           => ( ( ? [V3: product_prod @ nat @ nat,Vc: list @ vEBT_VEBT,Vd2: vEBT_VEBT] :
                    ( X2
                    = ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( zero_zero @ nat ) @ Vc @ Vd2 ) )
               => ( Y2
                 != ( one_one @ nat ) ) )
             => ( ( ? [V3: product_prod @ nat @ nat,Vg2: list @ vEBT_VEBT,Vh2: vEBT_VEBT] :
                      ( X2
                      = ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vg2 @ Vh2 ) )
                 => ( Y2
                   != ( one_one @ nat ) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X2
                        = ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
                     => ~ ( ( ( ord_less @ nat @ Xa @ Mi2 )
                           => ( Y2
                              = ( one_one @ nat ) ) )
                          & ( ~ ( ord_less @ nat @ Xa @ Mi2 )
                           => ( Y2
                              = ( if @ nat @ ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
                                @ ( if @ nat
                                  @ ( ( ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
                                     != ( none @ nat ) )
                                    & ( vEBT_VEBT_less @ ( some @ nat @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) )
                                  @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_T_s_u_c_c2 @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
                                  @ ( plus_plus @ nat @ ( vEBT_T_s_u_c_c2 @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( one_one @ nat ) ) )
                                @ ( one_one @ nat ) ) ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.elims
thf(fact_1399_vebt__maxt_Oelims,axiom,
    ! [X2: vEBT_VEBT,Y2: option @ nat] :
      ( ( ( vEBT_vebt_maxt @ X2 )
        = Y2 )
     => ( ! [A4: $o,B3: $o] :
            ( ( X2
              = ( vEBT_Leaf @ A4 @ B3 ) )
           => ~ ( ( B3
                 => ( Y2
                    = ( some @ nat @ ( one_one @ nat ) ) ) )
                & ( ~ B3
                 => ( ( A4
                     => ( Y2
                        = ( some @ nat @ ( zero_zero @ nat ) ) ) )
                    & ( ~ A4
                     => ( Y2
                        = ( none @ nat ) ) ) ) ) ) )
       => ( ( ? [Uu2: nat,Uv2: list @ vEBT_VEBT,Uw2: vEBT_VEBT] :
                ( X2
                = ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw2 ) )
           => ( Y2
             != ( none @ nat ) ) )
         => ~ ! [Mi2: nat,Ma2: nat] :
                ( ? [Ux2: nat,Uy2: list @ vEBT_VEBT,Uz2: vEBT_VEBT] :
                    ( X2
                    = ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
               => ( Y2
                 != ( some @ nat @ Ma2 ) ) ) ) ) ) ).

% vebt_maxt.elims
thf(fact_1400_vebt__mint_Oelims,axiom,
    ! [X2: vEBT_VEBT,Y2: option @ nat] :
      ( ( ( vEBT_vebt_mint @ X2 )
        = Y2 )
     => ( ! [A4: $o,B3: $o] :
            ( ( X2
              = ( vEBT_Leaf @ A4 @ B3 ) )
           => ~ ( ( A4
                 => ( Y2
                    = ( some @ nat @ ( zero_zero @ nat ) ) ) )
                & ( ~ A4
                 => ( ( B3
                     => ( Y2
                        = ( some @ nat @ ( one_one @ nat ) ) ) )
                    & ( ~ B3
                     => ( Y2
                        = ( none @ nat ) ) ) ) ) ) )
       => ( ( ? [Uu2: nat,Uv2: list @ vEBT_VEBT,Uw2: vEBT_VEBT] :
                ( X2
                = ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw2 ) )
           => ( Y2
             != ( none @ nat ) ) )
         => ~ ! [Mi2: nat] :
                ( ? [Ma2: nat,Ux2: nat,Uy2: list @ vEBT_VEBT,Uz2: vEBT_VEBT] :
                    ( X2
                    = ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
               => ( Y2
                 != ( some @ nat @ Mi2 ) ) ) ) ) ) ).

% vebt_mint.elims
thf(fact_1401_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,
    ! [X2: vEBT_VEBT,Xa: nat,Y2: nat] :
      ( ( ( vEBT_T_i_n_s_e_r_t2 @ X2 @ Xa )
        = Y2 )
     => ( ( ? [A4: $o,B3: $o] :
              ( X2
              = ( vEBT_Leaf @ A4 @ B3 ) )
         => ( Y2
           != ( one_one @ nat ) ) )
       => ( ( ? [Info2: option @ ( product_prod @ nat @ nat ),Ts2: list @ vEBT_VEBT,S: vEBT_VEBT] :
                ( X2
                = ( vEBT_Node @ Info2 @ ( zero_zero @ nat ) @ Ts2 @ S ) )
           => ( Y2
             != ( one_one @ nat ) ) )
         => ( ( ? [Info2: option @ ( product_prod @ nat @ nat ),Ts2: list @ vEBT_VEBT,S: vEBT_VEBT] :
                  ( X2
                  = ( vEBT_Node @ Info2 @ ( suc @ ( zero_zero @ nat ) ) @ Ts2 @ S ) )
             => ( Y2
               != ( one_one @ nat ) ) )
           => ( ( ? [V3: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
                    ( X2
                    = ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ V3 ) ) @ TreeList2 @ Summary2 ) )
               => ( Y2
                 != ( one_one @ nat ) ) )
             => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
                    ( ( X2
                      = ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
                   => ( Y2
                     != ( if @ nat
                        @ ( ( ord_less @ nat @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
                          & ~ ( ( Xa = Mi2 )
                              | ( Xa = Ma2 ) ) )
                        @ ( plus_plus @ nat @ ( vEBT_T_i_n_s_e_r_t2 @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( if @ nat @ ( vEBT_VEBT_minNull @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_T_i_n_s_e_r_t2 @ Summary2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( one_one @ nat ) ) )
                        @ ( one_one @ nat ) ) ) ) ) ) ) ) ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.elims
thf(fact_1402_linear__plus__1__le__power,axiom,
    ! [X2: real,N2: nat] :
      ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 )
     => ( ord_less_eq @ real @ ( plus_plus @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N2 ) @ X2 ) @ ( one_one @ real ) ) @ ( power_power @ real @ ( plus_plus @ real @ X2 @ ( one_one @ real ) ) @ N2 ) ) ) ).

% linear_plus_1_le_power
thf(fact_1403_diff__add__zero,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_diff @ A )
     => ! [A2: A,B2: A] :
          ( ( minus_minus @ A @ A2 @ ( plus_plus @ A @ A2 @ B2 ) )
          = ( zero_zero @ A ) ) ) ).

% diff_add_zero
thf(fact_1404_diff__gt__0__iff__gt,axiom,
    ! [A: $tType] :
      ( ( ordered_ab_group_add @ A )
     => ! [A2: A,B2: A] :
          ( ( ord_less @ A @ ( zero_zero @ A ) @ ( minus_minus @ A @ A2 @ B2 ) )
          = ( ord_less @ A @ B2 @ A2 ) ) ) ).

% diff_gt_0_iff_gt
thf(fact_1405_diff__ge__0__iff__ge,axiom,
    ! [A: $tType] :
      ( ( ordered_ab_group_add @ A )
     => ! [A2: A,B2: A] :
          ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( minus_minus @ A @ A2 @ B2 ) )
          = ( ord_less_eq @ A @ B2 @ A2 ) ) ) ).

% diff_ge_0_iff_ge
thf(fact_1406_zero__less__double__add__iff__zero__less__single__add,axiom,
    ! [A: $tType] :
      ( ( linord5086331880401160121up_add @ A )
     => ! [A2: A] :
          ( ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A2 @ A2 ) )
          = ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).

% zero_less_double_add_iff_zero_less_single_add
thf(fact_1407_add__right__cancel,axiom,
    ! [A: $tType] :
      ( ( cancel_semigroup_add @ A )
     => ! [B2: A,A2: A,C2: A] :
          ( ( ( plus_plus @ A @ B2 @ A2 )
            = ( plus_plus @ A @ C2 @ A2 ) )
          = ( B2 = C2 ) ) ) ).

% add_right_cancel
thf(fact_1408_add__left__cancel,axiom,
    ! [A: $tType] :
      ( ( cancel_semigroup_add @ A )
     => ! [A2: A,B2: A,C2: A] :
          ( ( ( plus_plus @ A @ A2 @ B2 )
            = ( plus_plus @ A @ A2 @ C2 ) )
          = ( B2 = C2 ) ) ) ).

% add_left_cancel
thf(fact_1409_le__zero__eq,axiom,
    ! [A: $tType] :
      ( ( canoni5634975068530333245id_add @ A )
     => ! [N2: A] :
          ( ( ord_less_eq @ A @ N2 @ ( zero_zero @ A ) )
          = ( N2
            = ( zero_zero @ A ) ) ) ) ).

% le_zero_eq
thf(fact_1410_not__gr__zero,axiom,
    ! [A: $tType] :
      ( ( canoni5634975068530333245id_add @ A )
     => ! [N2: A] :
          ( ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ N2 ) )
          = ( N2
            = ( zero_zero @ A ) ) ) ) ).

% not_gr_zero
thf(fact_1411_add__0,axiom,
    ! [A: $tType] :
      ( ( monoid_add @ A )
     => ! [A2: A] :
          ( ( plus_plus @ A @ ( zero_zero @ A ) @ A2 )
          = A2 ) ) ).

% add_0
thf(fact_1412_zero__eq__add__iff__both__eq__0,axiom,
    ! [A: $tType] :
      ( ( canoni5634975068530333245id_add @ A )
     => ! [X2: A,Y2: A] :
          ( ( ( zero_zero @ A )
            = ( plus_plus @ A @ X2 @ Y2 ) )
          = ( ( X2
              = ( zero_zero @ A ) )
            & ( Y2
              = ( zero_zero @ A ) ) ) ) ) ).

% zero_eq_add_iff_both_eq_0
thf(fact_1413_add__eq__0__iff__both__eq__0,axiom,
    ! [A: $tType] :
      ( ( canoni5634975068530333245id_add @ A )
     => ! [X2: A,Y2: A] :
          ( ( ( plus_plus @ A @ X2 @ Y2 )
            = ( zero_zero @ A ) )
          = ( ( X2
              = ( zero_zero @ A ) )
            & ( Y2
              = ( zero_zero @ A ) ) ) ) ) ).

% add_eq_0_iff_both_eq_0
thf(fact_1414_add__cancel__right__right,axiom,
    ! [A: $tType] :
      ( ( cancel1802427076303600483id_add @ A )
     => ! [A2: A,B2: A] :
          ( ( A2
            = ( plus_plus @ A @ A2 @ B2 ) )
          = ( B2
            = ( zero_zero @ A ) ) ) ) ).

% add_cancel_right_right
thf(fact_1415_add__cancel__right__left,axiom,
    ! [A: $tType] :
      ( ( cancel1802427076303600483id_add @ A )
     => ! [A2: A,B2: A] :
          ( ( A2
            = ( plus_plus @ A @ B2 @ A2 ) )
          = ( B2
            = ( zero_zero @ A ) ) ) ) ).

% add_cancel_right_left
thf(fact_1416_add__cancel__left__right,axiom,
    ! [A: $tType] :
      ( ( cancel1802427076303600483id_add @ A )
     => ! [A2: A,B2: A] :
          ( ( ( plus_plus @ A @ A2 @ B2 )
            = A2 )
          = ( B2
            = ( zero_zero @ A ) ) ) ) ).

% add_cancel_left_right
thf(fact_1417_add__cancel__left__left,axiom,
    ! [A: $tType] :
      ( ( cancel1802427076303600483id_add @ A )
     => ! [B2: A,A2: A] :
          ( ( ( plus_plus @ A @ B2 @ A2 )
            = A2 )
          = ( B2
            = ( zero_zero @ A ) ) ) ) ).

% add_cancel_left_left
thf(fact_1418_double__zero__sym,axiom,
    ! [A: $tType] :
      ( ( linord5086331880401160121up_add @ A )
     => ! [A2: A] :
          ( ( ( zero_zero @ A )
            = ( plus_plus @ A @ A2 @ A2 ) )
          = ( A2
            = ( zero_zero @ A ) ) ) ) ).

% double_zero_sym
thf(fact_1419_add_Oright__neutral,axiom,
    ! [A: $tType] :
      ( ( monoid_add @ A )
     => ! [A2: A] :
          ( ( plus_plus @ A @ A2 @ ( zero_zero @ A ) )
          = A2 ) ) ).

% add.right_neutral
thf(fact_1420_add__le__cancel__left,axiom,
    ! [A: $tType] :
      ( ( ordere2412721322843649153imp_le @ A )
     => ! [C2: A,A2: A,B2: A] :
          ( ( ord_less_eq @ A @ ( plus_plus @ A @ C2 @ A2 ) @ ( plus_plus @ A @ C2 @ B2 ) )
          = ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).

% add_le_cancel_left
thf(fact_1421_add__le__cancel__right,axiom,
    ! [A: $tType] :
      ( ( ordere2412721322843649153imp_le @ A )
     => ! [A2: A,C2: A,B2: A] :
          ( ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C2 ) @ ( plus_plus @ A @ B2 @ C2 ) )
          = ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).

% add_le_cancel_right
thf(fact_1422_add__less__cancel__left,axiom,
    ! [A: $tType] :
      ( ( ordere2412721322843649153imp_le @ A )
     => ! [C2: A,A2: A,B2: A] :
          ( ( ord_less @ A @ ( plus_plus @ A @ C2 @ A2 ) @ ( plus_plus @ A @ C2 @ B2 ) )
          = ( ord_less @ A @ A2 @ B2 ) ) ) ).

% add_less_cancel_left
thf(fact_1423_add__less__cancel__right,axiom,
    ! [A: $tType] :
      ( ( ordere2412721322843649153imp_le @ A )
     => ! [A2: A,C2: A,B2: A] :
          ( ( ord_less @ A @ ( plus_plus @ A @ A2 @ C2 ) @ ( plus_plus @ A @ B2 @ C2 ) )
          = ( ord_less @ A @ A2 @ B2 ) ) ) ).

% add_less_cancel_right
thf(fact_1424_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A: $tType] :
      ( ( cancel1802427076303600483id_add @ A )
     => ! [A2: A] :
          ( ( minus_minus @ A @ A2 @ A2 )
          = ( zero_zero @ A ) ) ) ).

% cancel_comm_monoid_add_class.diff_cancel
thf(fact_1425_diff__zero,axiom,
    ! [A: $tType] :
      ( ( cancel1802427076303600483id_add @ A )
     => ! [A2: A] :
          ( ( minus_minus @ A @ A2 @ ( zero_zero @ A ) )
          = A2 ) ) ).

% diff_zero
thf(fact_1426_zero__diff,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_diff @ A )
     => ! [A2: A] :
          ( ( minus_minus @ A @ ( zero_zero @ A ) @ A2 )
          = ( zero_zero @ A ) ) ) ).

% zero_diff
thf(fact_1427_diff__0__right,axiom,
    ! [A: $tType] :
      ( ( group_add @ A )
     => ! [A2: A] :
          ( ( minus_minus @ A @ A2 @ ( zero_zero @ A ) )
          = A2 ) ) ).

% diff_0_right
thf(fact_1428_diff__self,axiom,
    ! [A: $tType] :
      ( ( group_add @ A )
     => ! [A2: A] :
          ( ( minus_minus @ A @ A2 @ A2 )
          = ( zero_zero @ A ) ) ) ).

% diff_self
thf(fact_1429_mult_Oright__neutral,axiom,
    ! [A: $tType] :
      ( ( monoid_mult @ A )
     => ! [A2: A] :
          ( ( times_times @ A @ A2 @ ( one_one @ A ) )
          = A2 ) ) ).

% mult.right_neutral
thf(fact_1430_mult__1,axiom,
    ! [A: $tType] :
      ( ( monoid_mult @ A )
     => ! [A2: A] :
          ( ( times_times @ A @ ( one_one @ A ) @ A2 )
          = A2 ) ) ).

% mult_1
thf(fact_1431_add__diff__cancel__right_H,axiom,
    ! [A: $tType] :
      ( ( cancel2418104881723323429up_add @ A )
     => ! [A2: A,B2: A] :
          ( ( minus_minus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ B2 )
          = A2 ) ) ).

% add_diff_cancel_right'
thf(fact_1432_add__diff__cancel__right,axiom,
    ! [A: $tType] :
      ( ( cancel2418104881723323429up_add @ A )
     => ! [A2: A,C2: A,B2: A] :
          ( ( minus_minus @ A @ ( plus_plus @ A @ A2 @ C2 ) @ ( plus_plus @ A @ B2 @ C2 ) )
          = ( minus_minus @ A @ A2 @ B2 ) ) ) ).

% add_diff_cancel_right
thf(fact_1433_add__diff__cancel__left_H,axiom,
    ! [A: $tType] :
      ( ( cancel2418104881723323429up_add @ A )
     => ! [A2: A,B2: A] :
          ( ( minus_minus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ A2 )
          = B2 ) ) ).

% add_diff_cancel_left'
thf(fact_1434_add__diff__cancel__left,axiom,
    ! [A: $tType] :
      ( ( cancel2418104881723323429up_add @ A )
     => ! [C2: A,A2: A,B2: A] :
          ( ( minus_minus @ A @ ( plus_plus @ A @ C2 @ A2 ) @ ( plus_plus @ A @ C2 @ B2 ) )
          = ( minus_minus @ A @ A2 @ B2 ) ) ) ).

% add_diff_cancel_left
thf(fact_1435_diff__add__cancel,axiom,
    ! [A: $tType] :
      ( ( group_add @ A )
     => ! [A2: A,B2: A] :
          ( ( plus_plus @ A @ ( minus_minus @ A @ A2 @ B2 ) @ B2 )
          = A2 ) ) ).

% diff_add_cancel
thf(fact_1436_add__diff__cancel,axiom,
    ! [A: $tType] :
      ( ( group_add @ A )
     => ! [A2: A,B2: A] :
          ( ( minus_minus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ B2 )
          = A2 ) ) ).

% add_diff_cancel
thf(fact_1437_add__le__same__cancel1,axiom,
    ! [A: $tType] :
      ( ( ordere1937475149494474687imp_le @ A )
     => ! [B2: A,A2: A] :
          ( ( ord_less_eq @ A @ ( plus_plus @ A @ B2 @ A2 ) @ B2 )
          = ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).

% add_le_same_cancel1
thf(fact_1438_add__le__same__cancel2,axiom,
    ! [A: $tType] :
      ( ( ordere1937475149494474687imp_le @ A )
     => ! [A2: A,B2: A] :
          ( ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ B2 ) @ B2 )
          = ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).

% add_le_same_cancel2
thf(fact_1439_le__add__same__cancel1,axiom,
    ! [A: $tType] :
      ( ( ordere1937475149494474687imp_le @ A )
     => ! [A2: A,B2: A] :
          ( ( ord_less_eq @ A @ A2 @ ( plus_plus @ A @ A2 @ B2 ) )
          = ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) ) ) ).

% le_add_same_cancel1
thf(fact_1440_le__add__same__cancel2,axiom,
    ! [A: $tType] :
      ( ( ordere1937475149494474687imp_le @ A )
     => ! [A2: A,B2: A] :
          ( ( ord_less_eq @ A @ A2 @ ( plus_plus @ A @ B2 @ A2 ) )
          = ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) ) ) ).

% le_add_same_cancel2
thf(fact_1441_double__add__le__zero__iff__single__add__le__zero,axiom,
    ! [A: $tType] :
      ( ( linord5086331880401160121up_add @ A )
     => ! [A2: A] :
          ( ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ A2 ) @ ( zero_zero @ A ) )
          = ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).

% double_add_le_zero_iff_single_add_le_zero
thf(fact_1442_zero__le__double__add__iff__zero__le__single__add,axiom,
    ! [A: $tType] :
      ( ( linord5086331880401160121up_add @ A )
     => ! [A2: A] :
          ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A2 @ A2 ) )
          = ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).

% zero_le_double_add_iff_zero_le_single_add
thf(fact_1443_add__less__same__cancel1,axiom,
    ! [A: $tType] :
      ( ( ordere1937475149494474687imp_le @ A )
     => ! [B2: A,A2: A] :
          ( ( ord_less @ A @ ( plus_plus @ A @ B2 @ A2 ) @ B2 )
          = ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).

% add_less_same_cancel1
thf(fact_1444_add__less__same__cancel2,axiom,
    ! [A: $tType] :
      ( ( ordere1937475149494474687imp_le @ A )
     => ! [A2: A,B2: A] :
          ( ( ord_less @ A @ ( plus_plus @ A @ A2 @ B2 ) @ B2 )
          = ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).

% add_less_same_cancel2
thf(fact_1445_less__add__same__cancel1,axiom,
    ! [A: $tType] :
      ( ( ordere1937475149494474687imp_le @ A )
     => ! [A2: A,B2: A] :
          ( ( ord_less @ A @ A2 @ ( plus_plus @ A @ A2 @ B2 ) )
          = ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) ) ) ).

% less_add_same_cancel1
thf(fact_1446_less__add__same__cancel2,axiom,
    ! [A: $tType] :
      ( ( ordere1937475149494474687imp_le @ A )
     => ! [A2: A,B2: A] :
          ( ( ord_less @ A @ A2 @ ( plus_plus @ A @ B2 @ A2 ) )
          = ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) ) ) ).

% less_add_same_cancel2
thf(fact_1447_double__add__less__zero__iff__single__add__less__zero,axiom,
    ! [A: $tType] :
      ( ( linord5086331880401160121up_add @ A )
     => ! [A2: A] :
          ( ( ord_less @ A @ ( plus_plus @ A @ A2 @ A2 ) @ ( zero_zero @ A ) )
          = ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).

% double_add_less_zero_iff_single_add_less_zero
thf(fact_1448_add__right__imp__eq,axiom,
    ! [A: $tType] :
      ( ( cancel_semigroup_add @ A )
     => ! [B2: A,A2: A,C2: A] :
          ( ( ( plus_plus @ A @ B2 @ A2 )
            = ( plus_plus @ A @ C2 @ A2 ) )
         => ( B2 = C2 ) ) ) ).

% add_right_imp_eq
thf(fact_1449_add__left__imp__eq,axiom,
    ! [A: $tType] :
      ( ( cancel_semigroup_add @ A )
     => ! [A2: A,B2: A,C2: A] :
          ( ( ( plus_plus @ A @ A2 @ B2 )
            = ( plus_plus @ A @ A2 @ C2 ) )
         => ( B2 = C2 ) ) ) ).

% add_left_imp_eq
thf(fact_1450_ab__semigroup__add__class_Oadd_Oleft__commute,axiom,
    ! [A: $tType] :
      ( ( ab_semigroup_add @ A )
     => ! [B2: A,A2: A,C2: A] :
          ( ( plus_plus @ A @ B2 @ ( plus_plus @ A @ A2 @ C2 ) )
          = ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B2 @ C2 ) ) ) ) ).

% ab_semigroup_add_class.add.left_commute
thf(fact_1451_ab__semigroup__add__class_Oadd_Ocommute,axiom,
    ! [A: $tType] :
      ( ( ab_semigroup_add @ A )
     => ( ( plus_plus @ A )
        = ( ^ [A5: A,B5: A] : ( plus_plus @ A @ B5 @ A5 ) ) ) ) ).

% ab_semigroup_add_class.add.commute
thf(fact_1452_add_Oright__cancel,axiom,
    ! [A: $tType] :
      ( ( group_add @ A )
     => ! [B2: A,A2: A,C2: A] :
          ( ( ( plus_plus @ A @ B2 @ A2 )
            = ( plus_plus @ A @ C2 @ A2 ) )
          = ( B2 = C2 ) ) ) ).

% add.right_cancel
thf(fact_1453_add_Oleft__cancel,axiom,
    ! [A: $tType] :
      ( ( group_add @ A )
     => ! [A2: A,B2: A,C2: A] :
          ( ( ( plus_plus @ A @ A2 @ B2 )
            = ( plus_plus @ A @ A2 @ C2 ) )
          = ( B2 = C2 ) ) ) ).

% add.left_cancel
thf(fact_1454_add_Oassoc,axiom,
    ! [A: $tType] :
      ( ( semigroup_add @ A )
     => ! [A2: A,B2: A,C2: A] :
          ( ( plus_plus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 )
          = ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B2 @ C2 ) ) ) ) ).

% add.assoc
thf(fact_1455_group__cancel_Oadd2,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add @ A )
     => ! [B6: A,K: A,B2: A,A2: A] :
          ( ( B6
            = ( plus_plus @ A @ K @ B2 ) )
         => ( ( plus_plus @ A @ A2 @ B6 )
            = ( plus_plus @ A @ K @ ( plus_plus @ A @ A2 @ B2 ) ) ) ) ) ).

% group_cancel.add2
thf(fact_1456_group__cancel_Oadd1,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add @ A )
     => ! [A3: A,K: A,A2: A,B2: A] :
          ( ( A3
            = ( plus_plus @ A @ K @ A2 ) )
         => ( ( plus_plus @ A @ A3 @ B2 )
            = ( plus_plus @ A @ K @ ( plus_plus @ A @ A2 @ B2 ) ) ) ) ) ).

% group_cancel.add1
thf(fact_1457_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [A: $tType] :
      ( ( ordere6658533253407199908up_add @ A )
     => ! [I: A,J: A,K: A,L: A] :
          ( ( ( I = J )
            & ( K = L ) )
         => ( ( plus_plus @ A @ I @ K )
            = ( plus_plus @ A @ J @ L ) ) ) ) ).

% add_mono_thms_linordered_semiring(4)
thf(fact_1458_one__reorient,axiom,
    ! [A: $tType] :
      ( ( one @ A )
     => ! [X2: A] :
          ( ( ( one_one @ A )
            = X2 )
          = ( X2
            = ( one_one @ A ) ) ) ) ).

% one_reorient
thf(fact_1459_mult_Oassoc,axiom,
    ! [A: $tType] :
      ( ( semigroup_mult @ A )
     => ! [A2: A,B2: A,C2: A] :
          ( ( times_times @ A @ ( times_times @ A @ A2 @ B2 ) @ C2 )
          = ( times_times @ A @ A2 @ ( times_times @ A @ B2 @ C2 ) ) ) ) ).

% mult.assoc
thf(fact_1460_ab__semigroup__mult__class_Omult_Ocommute,axiom,
    ! [A: $tType] :
      ( ( ab_semigroup_mult @ A )
     => ( ( times_times @ A )
        = ( ^ [A5: A,B5: A] : ( times_times @ A @ B5 @ A5 ) ) ) ) ).

% ab_semigroup_mult_class.mult.commute
thf(fact_1461_ab__semigroup__mult__class_Omult_Oleft__commute,axiom,
    ! [A: $tType] :
      ( ( ab_semigroup_mult @ A )
     => ! [B2: A,A2: A,C2: A] :
          ( ( times_times @ A @ B2 @ ( times_times @ A @ A2 @ C2 ) )
          = ( times_times @ A @ A2 @ ( times_times @ A @ B2 @ C2 ) ) ) ) ).

% ab_semigroup_mult_class.mult.left_commute
thf(fact_1462_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
    ! [A: $tType] :
      ( ( cancel2418104881723323429up_add @ A )
     => ! [A2: A,C2: A,B2: A] :
          ( ( minus_minus @ A @ ( minus_minus @ A @ A2 @ C2 ) @ B2 )
          = ( minus_minus @ A @ ( minus_minus @ A @ A2 @ B2 ) @ C2 ) ) ) ).

% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_1463_diff__eq__diff__eq,axiom,
    ! [A: $tType] :
      ( ( group_add @ A )
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( ( ( minus_minus @ A @ A2 @ B2 )
            = ( minus_minus @ A @ C2 @ D2 ) )
         => ( ( A2 = B2 )
            = ( C2 = D2 ) ) ) ) ).

% diff_eq_diff_eq
thf(fact_1464_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_I1_J,axiom,
    ! [A2: $o,B2: $o,X2: nat] :
      ( ( vEBT_T_i_n_s_e_r_t2 @ ( vEBT_Leaf @ A2 @ B2 ) @ X2 )
      = ( one_one @ nat ) ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.simps(1)
thf(fact_1465_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: option @ ( product_prod @ nat @ nat ),Ts: list @ vEBT_VEBT,S4: vEBT_VEBT,X2: nat] :
      ( ( vEBT_T_i_n_s_e_r_t2 @ ( vEBT_Node @ Info @ ( zero_zero @ nat ) @ Ts @ S4 ) @ X2 )
      = ( one_one @ nat ) ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.simps(2)
thf(fact_1466_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: option @ ( product_prod @ nat @ nat ),Ts: list @ vEBT_VEBT,S4: vEBT_VEBT,X2: nat] :
      ( ( vEBT_T_i_n_s_e_r_t2 @ ( vEBT_Node @ Info @ ( suc @ ( zero_zero @ nat ) ) @ Ts @ S4 ) @ X2 )
      = ( one_one @ nat ) ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.simps(3)
thf(fact_1467_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,X2: nat] :
      ( ( vEBT_T_i_n_s_e_r_t2 @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ V ) ) @ TreeList @ Summary ) @ X2 )
      = ( one_one @ nat ) ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.simps(4)
thf(fact_1468_insersimp_H,axiom,
    ! [T2: vEBT_VEBT,N2: nat,Y2: nat] :
      ( ( vEBT_invar_vebt @ T2 @ N2 )
     => ( ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ T2 @ X_12 )
       => ( ord_less_eq @ nat @ ( vEBT_T_i_n_s_e_r_t2 @ T2 @ Y2 ) @ ( one_one @ nat ) ) ) ) ).

% insersimp'
thf(fact_1469_insertsimp_H,axiom,
    ! [T2: vEBT_VEBT,N2: nat,L: nat] :
      ( ( vEBT_invar_vebt @ T2 @ N2 )
     => ( ( vEBT_VEBT_minNull @ T2 )
       => ( ord_less_eq @ nat @ ( vEBT_T_i_n_s_e_r_t2 @ T2 @ L ) @ ( one_one @ nat ) ) ) ) ).

% insertsimp'
thf(fact_1470_zero__le,axiom,
    ! [A: $tType] :
      ( ( canoni5634975068530333245id_add @ A )
     => ! [X2: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ X2 ) ) ).

% zero_le
thf(fact_1471_gr__zeroI,axiom,
    ! [A: $tType] :
      ( ( canoni5634975068530333245id_add @ A )
     => ! [N2: A] :
          ( ( N2
           != ( zero_zero @ A ) )
         => ( ord_less @ A @ ( zero_zero @ A ) @ N2 ) ) ) ).

% gr_zeroI
thf(fact_1472_not__less__zero,axiom,
    ! [A: $tType] :
      ( ( canoni5634975068530333245id_add @ A )
     => ! [N2: A] :
          ~ ( ord_less @ A @ N2 @ ( zero_zero @ A ) ) ) ).

% not_less_zero
thf(fact_1473_gr__implies__not__zero,axiom,
    ! [A: $tType] :
      ( ( canoni5634975068530333245id_add @ A )
     => ! [M: A,N2: A] :
          ( ( ord_less @ A @ M @ N2 )
         => ( N2
           != ( zero_zero @ A ) ) ) ) ).

% gr_implies_not_zero
thf(fact_1474_zero__less__iff__neq__zero,axiom,
    ! [A: $tType] :
      ( ( canoni5634975068530333245id_add @ A )
     => ! [N2: A] :
          ( ( ord_less @ A @ ( zero_zero @ A ) @ N2 )
          = ( N2
           != ( zero_zero @ A ) ) ) ) ).

% zero_less_iff_neq_zero
thf(fact_1475_add_Ogroup__left__neutral,axiom,
    ! [A: $tType] :
      ( ( group_add @ A )
     => ! [A2: A] :
          ( ( plus_plus @ A @ ( zero_zero @ A ) @ A2 )
          = A2 ) ) ).

% add.group_left_neutral
thf(fact_1476_add_Ocomm__neutral,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add @ A )
     => ! [A2: A] :
          ( ( plus_plus @ A @ A2 @ ( zero_zero @ A ) )
          = A2 ) ) ).

% add.comm_neutral
thf(fact_1477_comm__monoid__add__class_Oadd__0,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add @ A )
     => ! [A2: A] :
          ( ( plus_plus @ A @ ( zero_zero @ A ) @ A2 )
          = A2 ) ) ).

% comm_monoid_add_class.add_0
thf(fact_1478_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [A: $tType] :
      ( ( ordere6658533253407199908up_add @ A )
     => ! [I: A,J: A,K: A,L: A] :
          ( ( ( ord_less_eq @ A @ I @ J )
            & ( K = L ) )
         => ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).

% add_mono_thms_linordered_semiring(3)
thf(fact_1479_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [A: $tType] :
      ( ( ordere6658533253407199908up_add @ A )
     => ! [I: A,J: A,K: A,L: A] :
          ( ( ( I = J )
            & ( ord_less_eq @ A @ K @ L ) )
         => ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).

% add_mono_thms_linordered_semiring(2)
thf(fact_1480_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [A: $tType] :
      ( ( ordere6658533253407199908up_add @ A )
     => ! [I: A,J: A,K: A,L: A] :
          ( ( ( ord_less_eq @ A @ I @ J )
            & ( ord_less_eq @ A @ K @ L ) )
         => ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).

% add_mono_thms_linordered_semiring(1)
thf(fact_1481_add__mono,axiom,
    ! [A: $tType] :
      ( ( ordere6658533253407199908up_add @ A )
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( ( ord_less_eq @ A @ A2 @ B2 )
         => ( ( ord_less_eq @ A @ C2 @ D2 )
           => ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C2 ) @ ( plus_plus @ A @ B2 @ D2 ) ) ) ) ) ).

% add_mono
thf(fact_1482_add__left__mono,axiom,
    ! [A: $tType] :
      ( ( ordere6658533253407199908up_add @ A )
     => ! [A2: A,B2: A,C2: A] :
          ( ( ord_less_eq @ A @ A2 @ B2 )
         => ( ord_less_eq @ A @ ( plus_plus @ A @ C2 @ A2 ) @ ( plus_plus @ A @ C2 @ B2 ) ) ) ) ).

% add_left_mono
thf(fact_1483_less__eqE,axiom,
    ! [A: $tType] :
      ( ( canoni5634975068530333245id_add @ A )
     => ! [A2: A,B2: A] :
          ( ( ord_less_eq @ A @ A2 @ B2 )
         => ~ ! [C3: A] :
                ( B2
               != ( plus_plus @ A @ A2 @ C3 ) ) ) ) ).

% less_eqE
thf(fact_1484_add__right__mono,axiom,
    ! [A: $tType] :
      ( ( ordere6658533253407199908up_add @ A )
     => ! [A2: A,B2: A,C2: A] :
          ( ( ord_less_eq @ A @ A2 @ B2 )
         => ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C2 ) @ ( plus_plus @ A @ B2 @ C2 ) ) ) ) ).

% add_right_mono
thf(fact_1485_le__iff__add,axiom,
    ! [A: $tType] :
      ( ( canoni5634975068530333245id_add @ A )
     => ( ( ord_less_eq @ A )
        = ( ^ [A5: A,B5: A] :
            ? [C4: A] :
              ( B5
              = ( plus_plus @ A @ A5 @ C4 ) ) ) ) ) ).

% le_iff_add
thf(fact_1486_add__le__imp__le__left,axiom,
    ! [A: $tType] :
      ( ( ordere2412721322843649153imp_le @ A )
     => ! [C2: A,A2: A,B2: A] :
          ( ( ord_less_eq @ A @ ( plus_plus @ A @ C2 @ A2 ) @ ( plus_plus @ A @ C2 @ B2 ) )
         => ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).

% add_le_imp_le_left
thf(fact_1487_add__le__imp__le__right,axiom,
    ! [A: $tType] :
      ( ( ordere2412721322843649153imp_le @ A )
     => ! [A2: A,C2: A,B2: A] :
          ( ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C2 ) @ ( plus_plus @ A @ B2 @ C2 ) )
         => ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).

% add_le_imp_le_right
thf(fact_1488_add__mono__thms__linordered__field_I5_J,axiom,
    ! [A: $tType] :
      ( ( ordere580206878836729694up_add @ A )
     => ! [I: A,J: A,K: A,L: A] :
          ( ( ( ord_less @ A @ I @ J )
            & ( ord_less @ A @ K @ L ) )
         => ( ord_less @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).

% add_mono_thms_linordered_field(5)
thf(fact_1489_add__mono__thms__linordered__field_I2_J,axiom,
    ! [A: $tType] :
      ( ( ordere580206878836729694up_add @ A )
     => ! [I: A,J: A,K: A,L: A] :
          ( ( ( I = J )
            & ( ord_less @ A @ K @ L ) )
         => ( ord_less @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).

% add_mono_thms_linordered_field(2)
thf(fact_1490_add__mono__thms__linordered__field_I1_J,axiom,
    ! [A: $tType] :
      ( ( ordere580206878836729694up_add @ A )
     => ! [I: A,J: A,K: A,L: A] :
          ( ( ( ord_less @ A @ I @ J )
            & ( K = L ) )
         => ( ord_less @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).

% add_mono_thms_linordered_field(1)
thf(fact_1491_add__strict__mono,axiom,
    ! [A: $tType] :
      ( ( strict9044650504122735259up_add @ A )
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( ( ord_less @ A @ A2 @ B2 )
         => ( ( ord_less @ A @ C2 @ D2 )
           => ( ord_less @ A @ ( plus_plus @ A @ A2 @ C2 ) @ ( plus_plus @ A @ B2 @ D2 ) ) ) ) ) ).

% add_strict_mono
thf(fact_1492_add__strict__left__mono,axiom,
    ! [A: $tType] :
      ( ( ordere580206878836729694up_add @ A )
     => ! [A2: A,B2: A,C2: A] :
          ( ( ord_less @ A @ A2 @ B2 )
         => ( ord_less @ A @ ( plus_plus @ A @ C2 @ A2 ) @ ( plus_plus @ A @ C2 @ B2 ) ) ) ) ).

% add_strict_left_mono
thf(fact_1493_add__strict__right__mono,axiom,
    ! [A: $tType] :
      ( ( ordere580206878836729694up_add @ A )
     => ! [A2: A,B2: A,C2: A] :
          ( ( ord_less @ A @ A2 @ B2 )
         => ( ord_less @ A @ ( plus_plus @ A @ A2 @ C2 ) @ ( plus_plus @ A @ B2 @ C2 ) ) ) ) ).

% add_strict_right_mono
thf(fact_1494_add__less__imp__less__left,axiom,
    ! [A: $tType] :
      ( ( ordere2412721322843649153imp_le @ A )
     => ! [C2: A,A2: A,B2: A] :
          ( ( ord_less @ A @ ( plus_plus @ A @ C2 @ A2 ) @ ( plus_plus @ A @ C2 @ B2 ) )
         => ( ord_less @ A @ A2 @ B2 ) ) ) ).

% add_less_imp_less_left
thf(fact_1495_add__less__imp__less__right,axiom,
    ! [A: $tType] :
      ( ( ordere2412721322843649153imp_le @ A )
     => ! [A2: A,C2: A,B2: A] :
          ( ( ord_less @ A @ ( plus_plus @ A @ A2 @ C2 ) @ ( plus_plus @ A @ B2 @ C2 ) )
         => ( ord_less @ A @ A2 @ B2 ) ) ) ).

% add_less_imp_less_right
thf(fact_1496_eq__iff__diff__eq__0,axiom,
    ! [A: $tType] :
      ( ( group_add @ A )
     => ( ( ^ [Y5: A,Z4: A] : Y5 = Z4 )
        = ( ^ [A5: A,B5: A] :
              ( ( minus_minus @ A @ A5 @ B5 )
              = ( zero_zero @ A ) ) ) ) ) ).

% eq_iff_diff_eq_0
thf(fact_1497_diff__mono,axiom,
    ! [A: $tType] :
      ( ( ordered_ab_group_add @ A )
     => ! [A2: A,B2: A,D2: A,C2: A] :
          ( ( ord_less_eq @ A @ A2 @ B2 )
         => ( ( ord_less_eq @ A @ D2 @ C2 )
           => ( ord_less_eq @ A @ ( minus_minus @ A @ A2 @ C2 ) @ ( minus_minus @ A @ B2 @ D2 ) ) ) ) ) ).

% diff_mono
thf(fact_1498_diff__left__mono,axiom,
    ! [A: $tType] :
      ( ( ordered_ab_group_add @ A )
     => ! [B2: A,A2: A,C2: A] :
          ( ( ord_less_eq @ A @ B2 @ A2 )
         => ( ord_less_eq @ A @ ( minus_minus @ A @ C2 @ A2 ) @ ( minus_minus @ A @ C2 @ B2 ) ) ) ) ).

% diff_left_mono
thf(fact_1499_diff__right__mono,axiom,
    ! [A: $tType] :
      ( ( ordered_ab_group_add @ A )
     => ! [A2: A,B2: A,C2: A] :
          ( ( ord_less_eq @ A @ A2 @ B2 )
         => ( ord_less_eq @ A @ ( minus_minus @ A @ A2 @ C2 ) @ ( minus_minus @ A @ B2 @ C2 ) ) ) ) ).

% diff_right_mono
thf(fact_1500_diff__eq__diff__less__eq,axiom,
    ! [A: $tType] :
      ( ( ordered_ab_group_add @ A )
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( ( ( minus_minus @ A @ A2 @ B2 )
            = ( minus_minus @ A @ C2 @ D2 ) )
         => ( ( ord_less_eq @ A @ A2 @ B2 )
            = ( ord_less_eq @ A @ C2 @ D2 ) ) ) ) ).

% diff_eq_diff_less_eq
thf(fact_1501_diff__strict__mono,axiom,
    ! [A: $tType] :
      ( ( ordered_ab_group_add @ A )
     => ! [A2: A,B2: A,D2: A,C2: A] :
          ( ( ord_less @ A @ A2 @ B2 )
         => ( ( ord_less @ A @ D2 @ C2 )
           => ( ord_less @ A @ ( minus_minus @ A @ A2 @ C2 ) @ ( minus_minus @ A @ B2 @ D2 ) ) ) ) ) ).

% diff_strict_mono
thf(fact_1502_diff__eq__diff__less,axiom,
    ! [A: $tType] :
      ( ( ordered_ab_group_add @ A )
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( ( ( minus_minus @ A @ A2 @ B2 )
            = ( minus_minus @ A @ C2 @ D2 ) )
         => ( ( ord_less @ A @ A2 @ B2 )
            = ( ord_less @ A @ C2 @ D2 ) ) ) ) ).

% diff_eq_diff_less
thf(fact_1503_diff__strict__left__mono,axiom,
    ! [A: $tType] :
      ( ( ordered_ab_group_add @ A )
     => ! [B2: A,A2: A,C2: A] :
          ( ( ord_less @ A @ B2 @ A2 )
         => ( ord_less @ A @ ( minus_minus @ A @ C2 @ A2 ) @ ( minus_minus @ A @ C2 @ B2 ) ) ) ) ).

% diff_strict_left_mono
thf(fact_1504_diff__strict__right__mono,axiom,
    ! [A: $tType] :
      ( ( ordered_ab_group_add @ A )
     => ! [A2: A,B2: A,C2: A] :
          ( ( ord_less @ A @ A2 @ B2 )
         => ( ord_less @ A @ ( minus_minus @ A @ A2 @ C2 ) @ ( minus_minus @ A @ B2 @ C2 ) ) ) ) ).

% diff_strict_right_mono
thf(fact_1505_comm__monoid__mult__class_Omult__1,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_mult @ A )
     => ! [A2: A] :
          ( ( times_times @ A @ ( one_one @ A ) @ A2 )
          = A2 ) ) ).

% comm_monoid_mult_class.mult_1
thf(fact_1506_mult_Ocomm__neutral,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_mult @ A )
     => ! [A2: A] :
          ( ( times_times @ A @ A2 @ ( one_one @ A ) )
          = A2 ) ) ).

% mult.comm_neutral
thf(fact_1507_diff__diff__eq,axiom,
    ! [A: $tType] :
      ( ( cancel2418104881723323429up_add @ A )
     => ! [A2: A,B2: A,C2: A] :
          ( ( minus_minus @ A @ ( minus_minus @ A @ A2 @ B2 ) @ C2 )
          = ( minus_minus @ A @ A2 @ ( plus_plus @ A @ B2 @ C2 ) ) ) ) ).

% diff_diff_eq
thf(fact_1508_add__implies__diff,axiom,
    ! [A: $tType] :
      ( ( cancel1802427076303600483id_add @ A )
     => ! [C2: A,B2: A,A2: A] :
          ( ( ( plus_plus @ A @ C2 @ B2 )
            = A2 )
         => ( C2
            = ( minus_minus @ A @ A2 @ B2 ) ) ) ) ).

% add_implies_diff
thf(fact_1509_diff__add__eq__diff__diff__swap,axiom,
    ! [A: $tType] :
      ( ( group_add @ A )
     => ! [A2: A,B2: A,C2: A] :
          ( ( minus_minus @ A @ A2 @ ( plus_plus @ A @ B2 @ C2 ) )
          = ( minus_minus @ A @ ( minus_minus @ A @ A2 @ C2 ) @ B2 ) ) ) ).

% diff_add_eq_diff_diff_swap
thf(fact_1510_diff__add__eq,axiom,
    ! [A: $tType] :
      ( ( ab_group_add @ A )
     => ! [A2: A,B2: A,C2: A] :
          ( ( plus_plus @ A @ ( minus_minus @ A @ A2 @ B2 ) @ C2 )
          = ( minus_minus @ A @ ( plus_plus @ A @ A2 @ C2 ) @ B2 ) ) ) ).

% diff_add_eq
thf(fact_1511_diff__diff__eq2,axiom,
    ! [A: $tType] :
      ( ( group_add @ A )
     => ! [A2: A,B2: A,C2: A] :
          ( ( minus_minus @ A @ A2 @ ( minus_minus @ A @ B2 @ C2 ) )
          = ( minus_minus @ A @ ( plus_plus @ A @ A2 @ C2 ) @ B2 ) ) ) ).

% diff_diff_eq2
thf(fact_1512_add__diff__eq,axiom,
    ! [A: $tType] :
      ( ( group_add @ A )
     => ! [A2: A,B2: A,C2: A] :
          ( ( plus_plus @ A @ A2 @ ( minus_minus @ A @ B2 @ C2 ) )
          = ( minus_minus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 ) ) ) ).

% add_diff_eq
thf(fact_1513_eq__diff__eq,axiom,
    ! [A: $tType] :
      ( ( group_add @ A )
     => ! [A2: A,C2: A,B2: A] :
          ( ( A2
            = ( minus_minus @ A @ C2 @ B2 ) )
          = ( ( plus_plus @ A @ A2 @ B2 )
            = C2 ) ) ) ).

% eq_diff_eq
thf(fact_1514_diff__eq__eq,axiom,
    ! [A: $tType] :
      ( ( group_add @ A )
     => ! [A2: A,B2: A,C2: A] :
          ( ( ( minus_minus @ A @ A2 @ B2 )
            = C2 )
          = ( A2
            = ( plus_plus @ A @ C2 @ B2 ) ) ) ) ).

% diff_eq_eq
thf(fact_1515_group__cancel_Osub1,axiom,
    ! [A: $tType] :
      ( ( ab_group_add @ A )
     => ! [A3: A,K: A,A2: A,B2: A] :
          ( ( A3
            = ( plus_plus @ A @ K @ A2 ) )
         => ( ( minus_minus @ A @ A3 @ B2 )
            = ( plus_plus @ A @ K @ ( minus_minus @ A @ A2 @ B2 ) ) ) ) ) ).

% group_cancel.sub1
thf(fact_1516_VEBT__internal_Ooption__comp__shift_Ocases,axiom,
    ! [A: $tType,X2: product_prod @ ( A > A > $o ) @ ( product_prod @ ( option @ A ) @ ( option @ A ) )] :
      ( ! [Uu2: A > A > $o,Uv2: option @ A] :
          ( X2
         != ( product_Pair @ ( A > A > $o ) @ ( product_prod @ ( option @ A ) @ ( option @ A ) ) @ Uu2 @ ( product_Pair @ ( option @ A ) @ ( option @ A ) @ ( none @ A ) @ Uv2 ) ) )
     => ( ! [Uw2: A > A > $o,V3: A] :
            ( X2
           != ( product_Pair @ ( A > A > $o ) @ ( product_prod @ ( option @ A ) @ ( option @ A ) ) @ Uw2 @ ( product_Pair @ ( option @ A ) @ ( option @ A ) @ ( some @ A @ V3 ) @ ( none @ A ) ) ) )
       => ~ ! [F2: A > A > $o,X3: A,Y4: A] :
              ( X2
             != ( product_Pair @ ( A > A > $o ) @ ( product_prod @ ( option @ A ) @ ( option @ A ) ) @ F2 @ ( product_Pair @ ( option @ A ) @ ( option @ A ) @ ( some @ A @ X3 ) @ ( some @ A @ Y4 ) ) ) ) ) ) ).

% VEBT_internal.option_comp_shift.cases
thf(fact_1517_VEBT__internal_Ooption__shift_Ocases,axiom,
    ! [A: $tType,X2: product_prod @ ( A > A > A ) @ ( product_prod @ ( option @ A ) @ ( option @ A ) )] :
      ( ! [Uu2: A > A > A,Uv2: option @ A] :
          ( X2
         != ( product_Pair @ ( A > A > A ) @ ( product_prod @ ( option @ A ) @ ( option @ A ) ) @ Uu2 @ ( product_Pair @ ( option @ A ) @ ( option @ A ) @ ( none @ A ) @ Uv2 ) ) )
     => ( ! [Uw2: A > A > A,V3: A] :
            ( X2
           != ( product_Pair @ ( A > A > A ) @ ( product_prod @ ( option @ A ) @ ( option @ A ) ) @ Uw2 @ ( product_Pair @ ( option @ A ) @ ( option @ A ) @ ( some @ A @ V3 ) @ ( none @ A ) ) ) )
       => ~ ! [F2: A > A > A,A4: A,B3: A] :
              ( X2
             != ( product_Pair @ ( A > A > A ) @ ( product_prod @ ( option @ A ) @ ( option @ A ) ) @ F2 @ ( product_Pair @ ( option @ A ) @ ( option @ A ) @ ( some @ A @ A4 ) @ ( some @ A @ B3 ) ) ) ) ) ) ).

% VEBT_internal.option_shift.cases
thf(fact_1518_insert_H__bound__height,axiom,
    ! [T2: vEBT_VEBT,N2: nat,X2: nat] :
      ( ( vEBT_invar_vebt @ T2 @ N2 )
     => ( ord_less_eq @ nat @ ( vEBT_T_i_n_s_e_r_t2 @ T2 @ X2 ) @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_VEBT_height @ T2 ) ) ) ) ).

% insert'_bound_height
thf(fact_1519_VEBT__internal_Ooption__shift_Osimps_I3_J,axiom,
    ! [A: $tType,F: A > A > A,A2: A,B2: A] :
      ( ( vEBT_V2048590022279873568_shift @ A @ F @ ( some @ A @ A2 ) @ ( some @ A @ B2 ) )
      = ( some @ A @ ( F @ A2 @ B2 ) ) ) ).

% VEBT_internal.option_shift.simps(3)
thf(fact_1520_VEBT__internal_Ooption__shift_Osimps_I1_J,axiom,
    ! [A: $tType,Uu: A > A > A,Uv: option @ A] :
      ( ( vEBT_V2048590022279873568_shift @ A @ Uu @ ( none @ A ) @ Uv )
      = ( none @ A ) ) ).

% VEBT_internal.option_shift.simps(1)
thf(fact_1521_add__decreasing,axiom,
    ! [A: $tType] :
      ( ( ordere6911136660526730532id_add @ A )
     => ! [A2: A,C2: A,B2: A] :
          ( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) )
         => ( ( ord_less_eq @ A @ C2 @ B2 )
           => ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C2 ) @ B2 ) ) ) ) ).

% add_decreasing
thf(fact_1522_add__increasing,axiom,
    ! [A: $tType] :
      ( ( ordere6911136660526730532id_add @ A )
     => ! [A2: A,B2: A,C2: A] :
          ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
         => ( ( ord_less_eq @ A @ B2 @ C2 )
           => ( ord_less_eq @ A @ B2 @ ( plus_plus @ A @ A2 @ C2 ) ) ) ) ) ).

% add_increasing
thf(fact_1523_add__decreasing2,axiom,
    ! [A: $tType] :
      ( ( ordere6911136660526730532id_add @ A )
     => ! [C2: A,A2: A,B2: A] :
          ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) )
         => ( ( ord_less_eq @ A @ A2 @ B2 )
           => ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C2 ) @ B2 ) ) ) ) ).

% add_decreasing2
thf(fact_1524_add__increasing2,axiom,
    ! [A: $tType] :
      ( ( ordere6911136660526730532id_add @ A )
     => ! [C2: A,B2: A,A2: A] :
          ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
         => ( ( ord_less_eq @ A @ B2 @ A2 )
           => ( ord_less_eq @ A @ B2 @ ( plus_plus @ A @ A2 @ C2 ) ) ) ) ) ).

% add_increasing2
thf(fact_1525_add__nonneg__nonneg,axiom,
    ! [A: $tType] :
      ( ( ordere6911136660526730532id_add @ A )
     => ! [A2: A,B2: A] :
          ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
         => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
           => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A2 @ B2 ) ) ) ) ) ).

% add_nonneg_nonneg
thf(fact_1526_add__nonpos__nonpos,axiom,
    ! [A: $tType] :
      ( ( ordere6911136660526730532id_add @ A )
     => ! [A2: A,B2: A] :
          ( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) )
         => ( ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) )
           => ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ B2 ) @ ( zero_zero @ A ) ) ) ) ) ).

% add_nonpos_nonpos
thf(fact_1527_add__nonneg__eq__0__iff,axiom,
    ! [A: $tType] :
      ( ( ordere6911136660526730532id_add @ A )
     => ! [X2: A,Y2: A] :
          ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X2 )
         => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y2 )
           => ( ( ( plus_plus @ A @ X2 @ Y2 )
                = ( zero_zero @ A ) )
              = ( ( X2
                  = ( zero_zero @ A ) )
                & ( Y2
                  = ( zero_zero @ A ) ) ) ) ) ) ) ).

% add_nonneg_eq_0_iff
thf(fact_1528_add__nonpos__eq__0__iff,axiom,
    ! [A: $tType] :
      ( ( ordere6911136660526730532id_add @ A )
     => ! [X2: A,Y2: A] :
          ( ( ord_less_eq @ A @ X2 @ ( zero_zero @ A ) )
         => ( ( ord_less_eq @ A @ Y2 @ ( zero_zero @ A ) )
           => ( ( ( plus_plus @ A @ X2 @ Y2 )
                = ( zero_zero @ A ) )
              = ( ( X2
                  = ( zero_zero @ A ) )
                & ( Y2
                  = ( zero_zero @ A ) ) ) ) ) ) ) ).

% add_nonpos_eq_0_iff
thf(fact_1529_add__neg__neg,axiom,
    ! [A: $tType] :
      ( ( ordere6911136660526730532id_add @ A )
     => ! [A2: A,B2: A] :
          ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
         => ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
           => ( ord_less @ A @ ( plus_plus @ A @ A2 @ B2 ) @ ( zero_zero @ A ) ) ) ) ) ).

% add_neg_neg
thf(fact_1530_add__pos__pos,axiom,
    ! [A: $tType] :
      ( ( ordere6911136660526730532id_add @ A )
     => ! [A2: A,B2: A] :
          ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
         => ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
           => ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A2 @ B2 ) ) ) ) ) ).

% add_pos_pos
thf(fact_1531_canonically__ordered__monoid__add__class_OlessE,axiom,
    ! [A: $tType] :
      ( ( canoni5634975068530333245id_add @ A )
     => ! [A2: A,B2: A] :
          ( ( ord_less @ A @ A2 @ B2 )
         => ~ ! [C3: A] :
                ( ( B2
                  = ( plus_plus @ A @ A2 @ C3 ) )
               => ( C3
                  = ( zero_zero @ A ) ) ) ) ) ).

% canonically_ordered_monoid_add_class.lessE
thf(fact_1532_pos__add__strict,axiom,
    ! [A: $tType] :
      ( ( strict7427464778891057005id_add @ A )
     => ! [A2: A,B2: A,C2: A] :
          ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
         => ( ( ord_less @ A @ B2 @ C2 )
           => ( ord_less @ A @ B2 @ ( plus_plus @ A @ A2 @ C2 ) ) ) ) ) ).

% pos_add_strict
thf(fact_1533_add__mono__thms__linordered__field_I4_J,axiom,
    ! [A: $tType] :
      ( ( ordere580206878836729694up_add @ A )
     => ! [I: A,J: A,K: A,L: A] :
          ( ( ( ord_less_eq @ A @ I @ J )
            & ( ord_less @ A @ K @ L ) )
         => ( ord_less @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).

% add_mono_thms_linordered_field(4)
thf(fact_1534_add__mono__thms__linordered__field_I3_J,axiom,
    ! [A: $tType] :
      ( ( ordere580206878836729694up_add @ A )
     => ! [I: A,J: A,K: A,L: A] :
          ( ( ( ord_less @ A @ I @ J )
            & ( ord_less_eq @ A @ K @ L ) )
         => ( ord_less @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).

% add_mono_thms_linordered_field(3)
thf(fact_1535_add__le__less__mono,axiom,
    ! [A: $tType] :
      ( ( ordere580206878836729694up_add @ A )
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( ( ord_less_eq @ A @ A2 @ B2 )
         => ( ( ord_less @ A @ C2 @ D2 )
           => ( ord_less @ A @ ( plus_plus @ A @ A2 @ C2 ) @ ( plus_plus @ A @ B2 @ D2 ) ) ) ) ) ).

% add_le_less_mono
thf(fact_1536_add__less__le__mono,axiom,
    ! [A: $tType] :
      ( ( ordere580206878836729694up_add @ A )
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( ( ord_less @ A @ A2 @ B2 )
         => ( ( ord_less_eq @ A @ C2 @ D2 )
           => ( ord_less @ A @ ( plus_plus @ A @ A2 @ C2 ) @ ( plus_plus @ A @ B2 @ D2 ) ) ) ) ) ).

% add_less_le_mono
thf(fact_1537_le__iff__diff__le__0,axiom,
    ! [A: $tType] :
      ( ( ordered_ab_group_add @ A )
     => ( ( ord_less_eq @ A )
        = ( ^ [A5: A,B5: A] : ( ord_less_eq @ A @ ( minus_minus @ A @ A5 @ B5 ) @ ( zero_zero @ A ) ) ) ) ) ).

% le_iff_diff_le_0
thf(fact_1538_less__iff__diff__less__0,axiom,
    ! [A: $tType] :
      ( ( ordered_ab_group_add @ A )
     => ( ( ord_less @ A )
        = ( ^ [A5: A,B5: A] : ( ord_less @ A @ ( minus_minus @ A @ A5 @ B5 ) @ ( zero_zero @ A ) ) ) ) ) ).

% less_iff_diff_less_0
thf(fact_1539_diff__le__eq,axiom,
    ! [A: $tType] :
      ( ( ordered_ab_group_add @ A )
     => ! [A2: A,B2: A,C2: A] :
          ( ( ord_less_eq @ A @ ( minus_minus @ A @ A2 @ B2 ) @ C2 )
          = ( ord_less_eq @ A @ A2 @ ( plus_plus @ A @ C2 @ B2 ) ) ) ) ).

% diff_le_eq
thf(fact_1540_le__diff__eq,axiom,
    ! [A: $tType] :
      ( ( ordered_ab_group_add @ A )
     => ! [A2: A,C2: A,B2: A] :
          ( ( ord_less_eq @ A @ A2 @ ( minus_minus @ A @ C2 @ B2 ) )
          = ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 ) ) ) ).

% le_diff_eq
thf(fact_1541_ordered__cancel__comm__monoid__diff__class_Odiff__add,axiom,
    ! [A: $tType] :
      ( ( ordere1170586879665033532d_diff @ A )
     => ! [A2: A,B2: A] :
          ( ( ord_less_eq @ A @ A2 @ B2 )
         => ( ( plus_plus @ A @ ( minus_minus @ A @ B2 @ A2 ) @ A2 )
            = B2 ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add
thf(fact_1542_le__add__diff,axiom,
    ! [A: $tType] :
      ( ( ordere1170586879665033532d_diff @ A )
     => ! [A2: A,B2: A,C2: A] :
          ( ( ord_less_eq @ A @ A2 @ B2 )
         => ( ord_less_eq @ A @ C2 @ ( minus_minus @ A @ ( plus_plus @ A @ B2 @ C2 ) @ A2 ) ) ) ) ).

% le_add_diff
thf(fact_1543_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
    ! [A: $tType] :
      ( ( ordere1170586879665033532d_diff @ A )
     => ! [A2: A,B2: A,C2: A] :
          ( ( ord_less_eq @ A @ A2 @ B2 )
         => ( ( ord_less_eq @ A @ C2 @ ( minus_minus @ A @ B2 @ A2 ) )
            = ( ord_less_eq @ A @ ( plus_plus @ A @ C2 @ A2 ) @ B2 ) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
thf(fact_1544_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
    ! [A: $tType] :
      ( ( ordere1170586879665033532d_diff @ A )
     => ! [A2: A,B2: A,C2: A] :
          ( ( ord_less_eq @ A @ A2 @ B2 )
         => ( ( plus_plus @ A @ C2 @ ( minus_minus @ A @ B2 @ A2 ) )
            = ( minus_minus @ A @ ( plus_plus @ A @ C2 @ B2 ) @ A2 ) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
thf(fact_1545_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
    ! [A: $tType] :
      ( ( ordere1170586879665033532d_diff @ A )
     => ! [A2: A,B2: A,C2: A] :
          ( ( ord_less_eq @ A @ A2 @ B2 )
         => ( ( minus_minus @ A @ ( plus_plus @ A @ C2 @ B2 ) @ A2 )
            = ( plus_plus @ A @ C2 @ ( minus_minus @ A @ B2 @ A2 ) ) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
thf(fact_1546_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
    ! [A: $tType] :
      ( ( ordere1170586879665033532d_diff @ A )
     => ! [A2: A,B2: A,C2: A] :
          ( ( ord_less_eq @ A @ A2 @ B2 )
         => ( ( plus_plus @ A @ ( minus_minus @ A @ B2 @ A2 ) @ C2 )
            = ( minus_minus @ A @ ( plus_plus @ A @ B2 @ C2 ) @ A2 ) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
thf(fact_1547_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
    ! [A: $tType] :
      ( ( ordere1170586879665033532d_diff @ A )
     => ! [A2: A,B2: A,C2: A] :
          ( ( ord_less_eq @ A @ A2 @ B2 )
         => ( ( minus_minus @ A @ ( plus_plus @ A @ B2 @ C2 ) @ A2 )
            = ( plus_plus @ A @ ( minus_minus @ A @ B2 @ A2 ) @ C2 ) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
thf(fact_1548_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
    ! [A: $tType] :
      ( ( ordere1170586879665033532d_diff @ A )
     => ! [A2: A,B2: A,C2: A] :
          ( ( ord_less_eq @ A @ A2 @ B2 )
         => ( ( minus_minus @ A @ C2 @ ( minus_minus @ A @ B2 @ A2 ) )
            = ( minus_minus @ A @ ( plus_plus @ A @ C2 @ A2 ) @ B2 ) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_diff_right
thf(fact_1549_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
    ! [A: $tType] :
      ( ( ordere1170586879665033532d_diff @ A )
     => ! [A2: A,B2: A] :
          ( ( ord_less_eq @ A @ A2 @ B2 )
         => ( ( plus_plus @ A @ A2 @ ( minus_minus @ A @ B2 @ A2 ) )
            = B2 ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_inverse
thf(fact_1550_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
    ! [A: $tType] :
      ( ( ordere1170586879665033532d_diff @ A )
     => ! [A2: A,B2: A,C2: A] :
          ( ( ord_less_eq @ A @ A2 @ B2 )
         => ( ( ord_less_eq @ A @ A2 @ B2 )
           => ( ( ( minus_minus @ A @ B2 @ A2 )
                = C2 )
              = ( B2
                = ( plus_plus @ A @ C2 @ A2 ) ) ) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
thf(fact_1551_diff__less__eq,axiom,
    ! [A: $tType] :
      ( ( ordered_ab_group_add @ A )
     => ! [A2: A,B2: A,C2: A] :
          ( ( ord_less @ A @ ( minus_minus @ A @ A2 @ B2 ) @ C2 )
          = ( ord_less @ A @ A2 @ ( plus_plus @ A @ C2 @ B2 ) ) ) ) ).

% diff_less_eq
thf(fact_1552_less__diff__eq,axiom,
    ! [A: $tType] :
      ( ( ordered_ab_group_add @ A )
     => ! [A2: A,C2: A,B2: A] :
          ( ( ord_less @ A @ A2 @ ( minus_minus @ A @ C2 @ B2 ) )
          = ( ord_less @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 ) ) ) ).

% less_diff_eq
thf(fact_1553_VEBT__internal_Ooption__shift_Osimps_I2_J,axiom,
    ! [A: $tType,Uw: A > A > A,V: A] :
      ( ( vEBT_V2048590022279873568_shift @ A @ Uw @ ( some @ A @ V ) @ ( none @ A ) )
      = ( none @ A ) ) ).

% VEBT_internal.option_shift.simps(2)
thf(fact_1554_VEBT__internal_Ooption__shift_Oelims,axiom,
    ! [A: $tType,X2: A > A > A,Xa: option @ A,Xb: option @ A,Y2: option @ A] :
      ( ( ( vEBT_V2048590022279873568_shift @ A @ X2 @ Xa @ Xb )
        = Y2 )
     => ( ( ( Xa
            = ( none @ A ) )
         => ( Y2
           != ( none @ A ) ) )
       => ( ( ? [V3: A] :
                ( Xa
                = ( some @ A @ V3 ) )
           => ( ( Xb
                = ( none @ A ) )
             => ( Y2
               != ( none @ A ) ) ) )
         => ~ ! [A4: A] :
                ( ( Xa
                  = ( some @ A @ A4 ) )
               => ! [B3: A] :
                    ( ( Xb
                      = ( some @ A @ B3 ) )
                   => ( Y2
                     != ( some @ A @ ( X2 @ A4 @ B3 ) ) ) ) ) ) ) ) ).

% VEBT_internal.option_shift.elims
thf(fact_1555_add__neg__nonpos,axiom,
    ! [A: $tType] :
      ( ( ordere6911136660526730532id_add @ A )
     => ! [A2: A,B2: A] :
          ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
         => ( ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) )
           => ( ord_less @ A @ ( plus_plus @ A @ A2 @ B2 ) @ ( zero_zero @ A ) ) ) ) ) ).

% add_neg_nonpos
thf(fact_1556_add__nonneg__pos,axiom,
    ! [A: $tType] :
      ( ( ordere6911136660526730532id_add @ A )
     => ! [A2: A,B2: A] :
          ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
         => ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
           => ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A2 @ B2 ) ) ) ) ) ).

% add_nonneg_pos
thf(fact_1557_add__nonpos__neg,axiom,
    ! [A: $tType] :
      ( ( ordere6911136660526730532id_add @ A )
     => ! [A2: A,B2: A] :
          ( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) )
         => ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
           => ( ord_less @ A @ ( plus_plus @ A @ A2 @ B2 ) @ ( zero_zero @ A ) ) ) ) ) ).

% add_nonpos_neg
thf(fact_1558_add__pos__nonneg,axiom,
    ! [A: $tType] :
      ( ( ordere6911136660526730532id_add @ A )
     => ! [A2: A,B2: A] :
          ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
         => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
           => ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A2 @ B2 ) ) ) ) ) ).

% add_pos_nonneg
thf(fact_1559_add__strict__increasing,axiom,
    ! [A: $tType] :
      ( ( ordere8940638589300402666id_add @ A )
     => ! [A2: A,B2: A,C2: A] :
          ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
         => ( ( ord_less_eq @ A @ B2 @ C2 )
           => ( ord_less @ A @ B2 @ ( plus_plus @ A @ A2 @ C2 ) ) ) ) ) ).

% add_strict_increasing
thf(fact_1560_add__strict__increasing2,axiom,
    ! [A: $tType] :
      ( ( ordere8940638589300402666id_add @ A )
     => ! [A2: A,B2: A,C2: A] :
          ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
         => ( ( ord_less @ A @ B2 @ C2 )
           => ( ord_less @ A @ B2 @ ( plus_plus @ A @ A2 @ C2 ) ) ) ) ) ).

% add_strict_increasing2
thf(fact_1561_Bolzano,axiom,
    ! [A2: real,B2: real,P: real > real > $o] :
      ( ( ord_less_eq @ real @ A2 @ B2 )
     => ( ! [A4: real,B3: real,C3: real] :
            ( ( P @ A4 @ B3 )
           => ( ( P @ B3 @ C3 )
             => ( ( ord_less_eq @ real @ A4 @ B3 )
               => ( ( ord_less_eq @ real @ B3 @ C3 )
                 => ( P @ A4 @ C3 ) ) ) ) )
       => ( ! [X3: real] :
              ( ( ord_less_eq @ real @ A2 @ X3 )
             => ( ( ord_less_eq @ real @ X3 @ B2 )
               => ? [D5: real] :
                    ( ( ord_less @ real @ ( zero_zero @ real ) @ D5 )
                    & ! [A4: real,B3: real] :
                        ( ( ( ord_less_eq @ real @ A4 @ X3 )
                          & ( ord_less_eq @ real @ X3 @ B3 )
                          & ( ord_less @ real @ ( minus_minus @ real @ B3 @ A4 ) @ D5 ) )
                       => ( P @ A4 @ B3 ) ) ) ) )
         => ( P @ A2 @ B2 ) ) ) ) ).

% Bolzano
thf(fact_1562_vebt__mint_Osimps_I2_J,axiom,
    ! [Uu: nat,Uv: list @ vEBT_VEBT,Uw: vEBT_VEBT] :
      ( ( vEBT_vebt_mint @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu @ Uv @ Uw ) )
      = ( none @ nat ) ) ).

% vebt_mint.simps(2)
thf(fact_1563_vebt__maxt_Osimps_I2_J,axiom,
    ! [Uu: nat,Uv: list @ vEBT_VEBT,Uw: vEBT_VEBT] :
      ( ( vEBT_vebt_maxt @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu @ Uv @ Uw ) )
      = ( none @ nat ) ) ).

% vebt_maxt.simps(2)
thf(fact_1564_vebt__mint_Osimps_I3_J,axiom,
    ! [Mi: nat,Ma: nat,Ux: nat,Uy: list @ vEBT_VEBT,Uz: vEBT_VEBT] :
      ( ( vEBT_vebt_mint @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Ux @ Uy @ Uz ) )
      = ( some @ nat @ Mi ) ) ).

% vebt_mint.simps(3)
thf(fact_1565_vebt__maxt_Osimps_I3_J,axiom,
    ! [Mi: nat,Ma: nat,Ux: nat,Uy: list @ vEBT_VEBT,Uz: vEBT_VEBT] :
      ( ( vEBT_vebt_maxt @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Ux @ Uy @ Uz ) )
      = ( some @ nat @ Ma ) ) ).

% vebt_maxt.simps(3)
thf(fact_1566_vebt__mint_Osimps_I1_J,axiom,
    ! [A2: $o,B2: $o] :
      ( ( A2
       => ( ( vEBT_vebt_mint @ ( vEBT_Leaf @ A2 @ B2 ) )
          = ( some @ nat @ ( zero_zero @ nat ) ) ) )
      & ( ~ A2
       => ( ( B2
           => ( ( vEBT_vebt_mint @ ( vEBT_Leaf @ A2 @ B2 ) )
              = ( some @ nat @ ( one_one @ nat ) ) ) )
          & ( ~ B2
           => ( ( vEBT_vebt_mint @ ( vEBT_Leaf @ A2 @ B2 ) )
              = ( none @ nat ) ) ) ) ) ) ).

% vebt_mint.simps(1)
thf(fact_1567_vebt__maxt_Osimps_I1_J,axiom,
    ! [B2: $o,A2: $o] :
      ( ( B2
       => ( ( vEBT_vebt_maxt @ ( vEBT_Leaf @ A2 @ B2 ) )
          = ( some @ nat @ ( one_one @ nat ) ) ) )
      & ( ~ B2
       => ( ( A2
           => ( ( vEBT_vebt_maxt @ ( vEBT_Leaf @ A2 @ B2 ) )
              = ( some @ nat @ ( zero_zero @ nat ) ) ) )
          & ( ~ A2
           => ( ( vEBT_vebt_maxt @ ( vEBT_Leaf @ A2 @ B2 ) )
              = ( none @ nat ) ) ) ) ) ) ).

% vebt_maxt.simps(1)
thf(fact_1568_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,Va: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,X2: nat] :
      ( ( vEBT_T_i_n_s_e_r_t2 @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X2 )
      = ( if @ nat
        @ ( ( ord_less @ nat @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ X2 @ Mi ) @ Mi @ X2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
          & ~ ( ( X2 = Mi )
              | ( X2 = Ma ) ) )
        @ ( plus_plus @ nat @ ( vEBT_T_i_n_s_e_r_t2 @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ X2 @ Mi ) @ Mi @ X2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( ord_less @ nat @ X2 @ Mi ) @ Mi @ X2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( if @ nat @ ( vEBT_VEBT_minNull @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ X2 @ Mi ) @ Mi @ X2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_T_i_n_s_e_r_t2 @ Summary @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ X2 @ Mi ) @ Mi @ X2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( one_one @ nat ) ) )
        @ ( one_one @ nat ) ) ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.simps(5)
thf(fact_1569_vebt__predi_Osimps,axiom,
    ( vEBT_vebt_predi
    = ( ^ [T3: vEBT_VEBTi,X: nat] :
          ( vEBT_case_VEBTi @ ( heap_Time_Heap @ ( option @ nat ) )
          @ ^ [Info3: option @ ( product_prod @ nat @ nat ),Deg4: nat,TreeArray: array @ vEBT_VEBTi,Summary3: vEBT_VEBTi] :
              ( case_option @ ( heap_Time_Heap @ ( option @ nat ) ) @ ( product_prod @ nat @ nat ) @ ( heap_Time_return @ ( option @ nat ) @ ( none @ nat ) )
              @ ^ [Mima2: product_prod @ nat @ nat] :
                  ( if @ ( heap_Time_Heap @ ( option @ nat ) ) @ ( ord_less_eq @ nat @ Deg4 @ ( one_one @ nat ) ) @ ( heap_Time_return @ ( option @ nat ) @ ( none @ nat ) )
                  @ ( if @ ( heap_Time_Heap @ ( option @ nat ) ) @ ( ord_less @ nat @ ( product_snd @ nat @ nat @ Mima2 ) @ X ) @ ( heap_Time_return @ ( option @ nat ) @ ( some @ nat @ ( product_snd @ nat @ nat @ Mima2 ) ) )
                    @ ( heap_Time_bind @ nat @ ( option @ nat ) @ ( vEBT_VEBT_lowi @ X @ ( divide_divide @ nat @ Deg4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
                      @ ^ [L2: nat] :
                          ( heap_Time_bind @ nat @ ( option @ nat ) @ ( vEBT_VEBT_highi @ X @ ( divide_divide @ nat @ Deg4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
                          @ ^ [H: nat] :
                              ( heap_Time_bind @ vEBT_VEBTi @ ( option @ nat ) @ ( array_nth @ vEBT_VEBTi @ TreeArray @ H )
                              @ ^ [Aktnode: vEBT_VEBTi] :
                                  ( heap_Time_bind @ ( option @ nat ) @ ( option @ nat ) @ ( vEBT_vebt_minti @ Aktnode )
                                  @ ^ [Minlow: option @ nat] :
                                      ( if @ ( heap_Time_Heap @ ( option @ nat ) )
                                      @ ( ( Minlow
                                         != ( none @ nat ) )
                                        & ( vEBT_VEBT_greater @ ( some @ nat @ L2 ) @ Minlow ) )
                                      @ ( heap_Time_bind @ ( option @ nat ) @ ( option @ nat ) @ ( vEBT_vebt_predi @ Aktnode @ L2 )
                                        @ ^ [Predy: option @ nat] : ( heap_Time_return @ ( option @ nat ) @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( some @ nat @ H ) ) @ Predy ) ) )
                                      @ ( heap_Time_bind @ ( option @ nat ) @ ( option @ nat ) @ ( vEBT_vebt_predi @ Summary3 @ H )
                                        @ ^ [Predsum: option @ nat] :
                                            ( if @ ( heap_Time_Heap @ ( option @ nat ) )
                                            @ ( Predsum
                                              = ( none @ nat ) )
                                            @ ( if @ ( heap_Time_Heap @ ( option @ nat ) ) @ ( ord_less @ nat @ ( product_fst @ nat @ nat @ Mima2 ) @ X ) @ ( heap_Time_return @ ( option @ nat ) @ ( some @ nat @ ( product_fst @ nat @ nat @ Mima2 ) ) ) @ ( heap_Time_return @ ( option @ nat ) @ ( none @ nat ) ) )
                                            @ ( heap_Time_bind @ vEBT_VEBTi @ ( option @ nat ) @ ( array_nth @ vEBT_VEBTi @ TreeArray @ ( the2 @ nat @ Predsum ) )
                                              @ ^ [Nextnode: vEBT_VEBTi] :
                                                  ( heap_Time_bind @ ( option @ nat ) @ ( option @ nat ) @ ( vEBT_vebt_maxti @ Nextnode )
                                                  @ ^ [Maxnext: option @ nat] : ( heap_Time_return @ ( option @ nat ) @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ Predsum ) @ Maxnext ) ) ) ) ) ) ) ) ) ) ) ) )
              @ Info3 )
          @ ^ [A5: $o,B5: $o] :
              ( if @ ( heap_Time_Heap @ ( option @ nat ) ) @ ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ X ) @ ( if @ ( heap_Time_Heap @ ( option @ nat ) ) @ B5 @ ( heap_Time_return @ ( option @ nat ) @ ( some @ nat @ ( one_one @ nat ) ) ) @ ( if @ ( heap_Time_Heap @ ( option @ nat ) ) @ A5 @ ( heap_Time_return @ ( option @ nat ) @ ( some @ nat @ ( zero_zero @ nat ) ) ) @ ( heap_Time_return @ ( option @ nat ) @ ( none @ nat ) ) ) )
              @ ( if @ ( heap_Time_Heap @ ( option @ nat ) )
                @ ( X
                  = ( one_one @ nat ) )
                @ ( if @ ( heap_Time_Heap @ ( option @ nat ) ) @ A5 @ ( heap_Time_return @ ( option @ nat ) @ ( some @ nat @ ( zero_zero @ nat ) ) ) @ ( heap_Time_return @ ( option @ nat ) @ ( none @ nat ) ) )
                @ ( heap_Time_return @ ( option @ nat ) @ ( none @ nat ) ) ) )
          @ T3 ) ) ) ).

% vebt_predi.simps
thf(fact_1570_delete__bound__size__univ,axiom,
    ! [T2: vEBT_VEBT,N2: nat,U: real,X2: nat] :
      ( ( vEBT_invar_vebt @ T2 @ N2 )
     => ( ( U
          = ( power_power @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ N2 ) )
       => ( ord_less_eq @ real @ ( semiring_1_of_nat @ real @ ( vEBT_T_d_e_l_e_t_e @ T2 @ X2 ) ) @ ( plus_plus @ real @ ( numeral_numeral @ real @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) ) ) @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) ) @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ U ) ) ) ) ) ) ) ).

% delete_bound_size_univ
thf(fact_1571_vebt__succi_Osimps,axiom,
    ( vEBT_vebt_succi
    = ( ^ [T3: vEBT_VEBTi,X: nat] :
          ( vEBT_case_VEBTi @ ( heap_Time_Heap @ ( option @ nat ) )
          @ ^ [Info3: option @ ( product_prod @ nat @ nat ),Deg4: nat,TreeArray: array @ vEBT_VEBTi,Summary3: vEBT_VEBTi] :
              ( case_option @ ( heap_Time_Heap @ ( option @ nat ) ) @ ( product_prod @ nat @ nat ) @ ( heap_Time_return @ ( option @ nat ) @ ( none @ nat ) )
              @ ^ [Mima2: product_prod @ nat @ nat] :
                  ( if @ ( heap_Time_Heap @ ( option @ nat ) ) @ ( ord_less_eq @ nat @ Deg4 @ ( one_one @ nat ) ) @ ( heap_Time_return @ ( option @ nat ) @ ( none @ nat ) )
                  @ ( if @ ( heap_Time_Heap @ ( option @ nat ) ) @ ( ord_less @ nat @ X @ ( product_fst @ nat @ nat @ Mima2 ) ) @ ( heap_Time_return @ ( option @ nat ) @ ( some @ nat @ ( product_fst @ nat @ nat @ Mima2 ) ) )
                    @ ( if @ ( heap_Time_Heap @ ( option @ nat ) ) @ ( ord_less_eq @ nat @ ( product_snd @ nat @ nat @ Mima2 ) @ X ) @ ( heap_Time_return @ ( option @ nat ) @ ( none @ nat ) )
                      @ ( heap_Time_bind @ nat @ ( option @ nat ) @ ( vEBT_VEBT_lowi @ X @ ( divide_divide @ nat @ Deg4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
                        @ ^ [L2: nat] :
                            ( heap_Time_bind @ nat @ ( option @ nat ) @ ( vEBT_VEBT_highi @ X @ ( divide_divide @ nat @ Deg4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
                            @ ^ [H: nat] :
                                ( heap_Time_bind @ vEBT_VEBTi @ ( option @ nat ) @ ( array_nth @ vEBT_VEBTi @ TreeArray @ H )
                                @ ^ [Aktnode: vEBT_VEBTi] :
                                    ( heap_Time_bind @ ( option @ nat ) @ ( option @ nat ) @ ( vEBT_vebt_maxti @ Aktnode )
                                    @ ^ [Maxlow: option @ nat] :
                                        ( if @ ( heap_Time_Heap @ ( option @ nat ) )
                                        @ ( ( Maxlow
                                           != ( none @ nat ) )
                                          & ( vEBT_VEBT_less @ ( some @ nat @ L2 ) @ Maxlow ) )
                                        @ ( heap_Time_bind @ ( option @ nat ) @ ( option @ nat ) @ ( vEBT_vebt_succi @ Aktnode @ L2 )
                                          @ ^ [Succy: option @ nat] : ( heap_Time_return @ ( option @ nat ) @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( some @ nat @ H ) ) @ Succy ) ) )
                                        @ ( heap_Time_bind @ ( option @ nat ) @ ( option @ nat ) @ ( vEBT_vebt_succi @ Summary3 @ H )
                                          @ ^ [Succsum: option @ nat] :
                                              ( if @ ( heap_Time_Heap @ ( option @ nat ) )
                                              @ ( Succsum
                                                = ( none @ nat ) )
                                              @ ( heap_Time_return @ ( option @ nat ) @ ( none @ nat ) )
                                              @ ( heap_Time_bind @ vEBT_VEBTi @ ( option @ nat ) @ ( array_nth @ vEBT_VEBTi @ TreeArray @ ( the2 @ nat @ Succsum ) )
                                                @ ^ [Nextnode: vEBT_VEBTi] :
                                                    ( heap_Time_bind @ ( option @ nat ) @ ( option @ nat ) @ ( vEBT_vebt_minti @ Nextnode )
                                                    @ ^ [Minnext: option @ nat] : ( heap_Time_return @ ( option @ nat ) @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ Succsum ) @ Minnext ) ) ) ) ) ) ) ) ) ) ) ) ) )
              @ Info3 )
          @ ^ [A5: $o,B5: $o] :
              ( if @ ( heap_Time_Heap @ ( option @ nat ) )
              @ ( X
                = ( zero_zero @ nat ) )
              @ ( if @ ( heap_Time_Heap @ ( option @ nat ) ) @ B5 @ ( heap_Time_return @ ( option @ nat ) @ ( some @ nat @ ( one_one @ nat ) ) ) @ ( heap_Time_return @ ( option @ nat ) @ ( none @ nat ) ) )
              @ ( heap_Time_return @ ( option @ nat ) @ ( none @ nat ) ) )
          @ T3 ) ) ) ).

% vebt_succi.simps
thf(fact_1572_inrange,axiom,
    ! [T2: vEBT_VEBT,N2: nat] :
      ( ( vEBT_invar_vebt @ T2 @ N2 )
     => ( ord_less_eq @ ( set @ nat ) @ ( vEBT_VEBT_set_vebt @ T2 ) @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( minus_minus @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) @ ( one_one @ nat ) ) ) ) ) ).

% inrange
thf(fact_1573_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 @ one2 ) ) ) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d\<^sub>u\<^sub>p.simps(2)
thf(fact_1574_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 @ one2 ) ) ) ) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d.simps(2)
thf(fact_1575_height__double__log__univ__size,axiom,
    ! [U: real,Deg: nat,T2: vEBT_VEBT] :
      ( ( U
        = ( power_power @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ Deg ) )
     => ( ( vEBT_invar_vebt @ T2 @ Deg )
       => ( ord_less_eq @ real @ ( semiring_1_of_nat @ real @ ( vEBT_VEBT_height @ T2 ) ) @ ( plus_plus @ real @ ( one_one @ real ) @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ U ) ) ) ) ) ) ).

% height_double_log_univ_size
thf(fact_1576_zdiff__int__split,axiom,
    ! [P: int > $o,X2: nat,Y2: nat] :
      ( ( P @ ( semiring_1_of_nat @ int @ ( minus_minus @ nat @ X2 @ Y2 ) ) )
      = ( ( ( ord_less_eq @ nat @ Y2 @ X2 )
         => ( P @ ( minus_minus @ int @ ( semiring_1_of_nat @ int @ X2 ) @ ( semiring_1_of_nat @ int @ Y2 ) ) ) )
        & ( ( ord_less @ nat @ X2 @ Y2 )
         => ( P @ ( zero_zero @ int ) ) ) ) ) ).

% zdiff_int_split
thf(fact_1577_delete__bound__size__univ_H,axiom,
    ! [T2: vEBT_VEBT,N2: nat,U: real,X2: nat] :
      ( ( vEBT_invar_vebt @ T2 @ N2 )
     => ( ( U
          = ( power_power @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ N2 ) )
       => ( ord_less_eq @ real @ ( semiring_1_of_nat @ real @ ( vEBT_V1232361888498592333_e_t_e @ T2 @ X2 ) ) @ ( plus_plus @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ U ) ) ) ) ) ) ).

% delete_bound_size_univ'
thf(fact_1578_TBOUND__VEBT__case,axiom,
    ! [A: $tType,Ti: vEBT_VEBTi,F: $o > $o > ( heap_Time_Heap @ A ),Bnd: $o > $o > nat,F4: ( option @ ( product_prod @ nat @ nat ) ) > nat > ( array @ vEBT_VEBTi ) > vEBT_VEBTi > ( heap_Time_Heap @ A ),Bnd2: ( option @ ( product_prod @ nat @ nat ) ) > nat > ( array @ vEBT_VEBTi ) > vEBT_VEBTi > nat] :
      ( ! [A4: $o,B3: $o] :
          ( ( Ti
            = ( vEBT_Leafi @ A4 @ B3 ) )
         => ( time_TBOUND @ A @ ( F @ A4 @ B3 ) @ ( Bnd @ A4 @ B3 ) ) )
     => ( ! [Info2: option @ ( product_prod @ nat @ nat ),Deg2: nat,TreeArray2: array @ vEBT_VEBTi,Summary2: vEBT_VEBTi] :
            ( ( Ti
              = ( vEBT_Nodei @ Info2 @ Deg2 @ TreeArray2 @ Summary2 ) )
           => ( time_TBOUND @ A @ ( F4 @ Info2 @ Deg2 @ TreeArray2 @ Summary2 ) @ ( Bnd2 @ Info2 @ Deg2 @ TreeArray2 @ Summary2 ) ) )
       => ( time_TBOUND @ A @ ( vEBT_case_VEBTi @ ( heap_Time_Heap @ A ) @ F4 @ F @ Ti ) @ ( vEBT_case_VEBTi @ nat @ Bnd2 @ Bnd @ Ti ) ) ) ) ).

% TBOUND_VEBT_case
thf(fact_1579_subset__Collect__conv,axiom,
    ! [A: $tType,S3: set @ A,P: A > $o] :
      ( ( ord_less_eq @ ( set @ A ) @ S3 @ ( collect @ A @ P ) )
      = ( ! [X: A] :
            ( ( member @ A @ X @ S3 )
           => ( P @ X ) ) ) ) ).

% subset_Collect_conv
thf(fact_1580_VEBTi_Ocase__distrib,axiom,
    ! [A: $tType,B: $tType,H2: A > B,F1: ( option @ ( product_prod @ nat @ nat ) ) > nat > ( array @ vEBT_VEBTi ) > vEBT_VEBTi > A,F22: $o > $o > A,VEBTi: vEBT_VEBTi] :
      ( ( H2 @ ( vEBT_case_VEBTi @ A @ F1 @ F22 @ VEBTi ) )
      = ( vEBT_case_VEBTi @ B
        @ ^ [X15: option @ ( product_prod @ nat @ nat ),X25: nat,X34: array @ vEBT_VEBTi,X42: vEBT_VEBTi] : ( H2 @ ( F1 @ X15 @ X25 @ X34 @ X42 ) )
        @ ^ [X15: $o,X25: $o] : ( H2 @ ( F22 @ X15 @ X25 ) )
        @ VEBTi ) ) ).

% VEBTi.case_distrib
thf(fact_1581_VEBTi_Osimps_I5_J,axiom,
    ! [A: $tType,F1: ( option @ ( product_prod @ nat @ nat ) ) > nat > ( array @ vEBT_VEBTi ) > vEBT_VEBTi > A,F22: $o > $o > A,X11: option @ ( product_prod @ nat @ nat ),X12: nat,X13: array @ vEBT_VEBTi,X14: vEBT_VEBTi] :
      ( ( vEBT_case_VEBTi @ A @ F1 @ F22 @ ( vEBT_Nodei @ X11 @ X12 @ X13 @ X14 ) )
      = ( F1 @ X11 @ X12 @ X13 @ X14 ) ) ).

% VEBTi.simps(5)
thf(fact_1582_VEBTi_Osimps_I6_J,axiom,
    ! [A: $tType,F1: ( option @ ( product_prod @ nat @ nat ) ) > nat > ( array @ vEBT_VEBTi ) > vEBT_VEBTi > A,F22: $o > $o > A,X212: $o,X222: $o] :
      ( ( vEBT_case_VEBTi @ A @ F1 @ F22 @ ( vEBT_Leafi @ X212 @ X222 ) )
      = ( F22 @ X212 @ X222 ) ) ).

% VEBTi.simps(6)
thf(fact_1583_pinf_I1_J,axiom,
    ! [A: $tType] :
      ( ( linorder @ A )
     => ! [P: A > $o,P6: A > $o,Q: A > $o,Q7: A > $o] :
          ( ? [Z5: A] :
            ! [X3: A] :
              ( ( ord_less @ A @ Z5 @ X3 )
             => ( ( P @ X3 )
                = ( P6 @ X3 ) ) )
         => ( ? [Z5: A] :
              ! [X3: A] :
                ( ( ord_less @ A @ Z5 @ X3 )
               => ( ( Q @ X3 )
                  = ( Q7 @ X3 ) ) )
           => ? [Z3: A] :
              ! [X6: A] :
                ( ( ord_less @ A @ Z3 @ X6 )
               => ( ( ( P @ X6 )
                    & ( Q @ X6 ) )
                  = ( ( P6 @ X6 )
                    & ( Q7 @ X6 ) ) ) ) ) ) ) ).

% pinf(1)
thf(fact_1584_pinf_I2_J,axiom,
    ! [A: $tType] :
      ( ( linorder @ A )
     => ! [P: A > $o,P6: A > $o,Q: A > $o,Q7: A > $o] :
          ( ? [Z5: A] :
            ! [X3: A] :
              ( ( ord_less @ A @ Z5 @ X3 )
             => ( ( P @ X3 )
                = ( P6 @ X3 ) ) )
         => ( ? [Z5: A] :
              ! [X3: A] :
                ( ( ord_less @ A @ Z5 @ X3 )
               => ( ( Q @ X3 )
                  = ( Q7 @ X3 ) ) )
           => ? [Z3: A] :
              ! [X6: A] :
                ( ( ord_less @ A @ Z3 @ X6 )
               => ( ( ( P @ X6 )
                    | ( Q @ X6 ) )
                  = ( ( P6 @ X6 )
                    | ( Q7 @ X6 ) ) ) ) ) ) ) ).

% pinf(2)
thf(fact_1585_pinf_I3_J,axiom,
    ! [A: $tType] :
      ( ( linorder @ A )
     => ! [T2: A] :
        ? [Z3: A] :
        ! [X6: A] :
          ( ( ord_less @ A @ Z3 @ X6 )
         => ( X6 != T2 ) ) ) ).

% pinf(3)
thf(fact_1586_pinf_I4_J,axiom,
    ! [A: $tType] :
      ( ( linorder @ A )
     => ! [T2: A] :
        ? [Z3: A] :
        ! [X6: A] :
          ( ( ord_less @ A @ Z3 @ X6 )
         => ( X6 != T2 ) ) ) ).

% pinf(4)
thf(fact_1587_pinf_I5_J,axiom,
    ! [A: $tType] :
      ( ( linorder @ A )
     => ! [T2: A] :
        ? [Z3: A] :
        ! [X6: A] :
          ( ( ord_less @ A @ Z3 @ X6 )
         => ~ ( ord_less @ A @ X6 @ T2 ) ) ) ).

% pinf(5)
thf(fact_1588_pinf_I7_J,axiom,
    ! [A: $tType] :
      ( ( linorder @ A )
     => ! [T2: A] :
        ? [Z3: A] :
        ! [X6: A] :
          ( ( ord_less @ A @ Z3 @ X6 )
         => ( ord_less @ A @ T2 @ X6 ) ) ) ).

% pinf(7)
thf(fact_1589_pinf_I11_J,axiom,
    ! [C: $tType,D: $tType] :
      ( ( ord @ C )
     => ! [F5: D] :
        ? [Z3: C] :
        ! [X6: C] :
          ( ( ord_less @ C @ Z3 @ X6 )
         => ( F5 = F5 ) ) ) ).

% pinf(11)
thf(fact_1590_minf_I1_J,axiom,
    ! [A: $tType] :
      ( ( linorder @ A )
     => ! [P: A > $o,P6: A > $o,Q: A > $o,Q7: A > $o] :
          ( ? [Z5: A] :
            ! [X3: A] :
              ( ( ord_less @ A @ X3 @ Z5 )
             => ( ( P @ X3 )
                = ( P6 @ X3 ) ) )
         => ( ? [Z5: A] :
              ! [X3: A] :
                ( ( ord_less @ A @ X3 @ Z5 )
               => ( ( Q @ X3 )
                  = ( Q7 @ X3 ) ) )
           => ? [Z3: A] :
              ! [X6: A] :
                ( ( ord_less @ A @ X6 @ Z3 )
               => ( ( ( P @ X6 )
                    & ( Q @ X6 ) )
                  = ( ( P6 @ X6 )
                    & ( Q7 @ X6 ) ) ) ) ) ) ) ).

% minf(1)
thf(fact_1591_minf_I2_J,axiom,
    ! [A: $tType] :
      ( ( linorder @ A )
     => ! [P: A > $o,P6: A > $o,Q: A > $o,Q7: A > $o] :
          ( ? [Z5: A] :
            ! [X3: A] :
              ( ( ord_less @ A @ X3 @ Z5 )
             => ( ( P @ X3 )
                = ( P6 @ X3 ) ) )
         => ( ? [Z5: A] :
              ! [X3: A] :
                ( ( ord_less @ A @ X3 @ Z5 )
               => ( ( Q @ X3 )
                  = ( Q7 @ X3 ) ) )
           => ? [Z3: A] :
              ! [X6: A] :
                ( ( ord_less @ A @ X6 @ Z3 )
               => ( ( ( P @ X6 )
                    | ( Q @ X6 ) )
                  = ( ( P6 @ X6 )
                    | ( Q7 @ X6 ) ) ) ) ) ) ) ).

% minf(2)
thf(fact_1592_minf_I3_J,axiom,
    ! [A: $tType] :
      ( ( linorder @ A )
     => ! [T2: A] :
        ? [Z3: A] :
        ! [X6: A] :
          ( ( ord_less @ A @ X6 @ Z3 )
         => ( X6 != T2 ) ) ) ).

% minf(3)
thf(fact_1593_minf_I4_J,axiom,
    ! [A: $tType] :
      ( ( linorder @ A )
     => ! [T2: A] :
        ? [Z3: A] :
        ! [X6: A] :
          ( ( ord_less @ A @ X6 @ Z3 )
         => ( X6 != T2 ) ) ) ).

% minf(4)
thf(fact_1594_minf_I5_J,axiom,
    ! [A: $tType] :
      ( ( linorder @ A )
     => ! [T2: A] :
        ? [Z3: A] :
        ! [X6: A] :
          ( ( ord_less @ A @ X6 @ Z3 )
         => ( ord_less @ A @ X6 @ T2 ) ) ) ).

% minf(5)
thf(fact_1595_minf_I7_J,axiom,
    ! [A: $tType] :
      ( ( linorder @ A )
     => ! [T2: A] :
        ? [Z3: A] :
        ! [X6: A] :
          ( ( ord_less @ A @ X6 @ Z3 )
         => ~ ( ord_less @ A @ T2 @ X6 ) ) ) ).

% minf(7)
thf(fact_1596_minf_I11_J,axiom,
    ! [C: $tType,D: $tType] :
      ( ( ord @ C )
     => ! [F5: D] :
        ? [Z3: C] :
        ! [X6: C] :
          ( ( ord_less @ C @ X6 @ Z3 )
         => ( F5 = F5 ) ) ) ).

% minf(11)
thf(fact_1597_pinf_I6_J,axiom,
    ! [A: $tType] :
      ( ( linorder @ A )
     => ! [T2: A] :
        ? [Z3: A] :
        ! [X6: A] :
          ( ( ord_less @ A @ Z3 @ X6 )
         => ~ ( ord_less_eq @ A @ X6 @ T2 ) ) ) ).

% pinf(6)
thf(fact_1598_pinf_I8_J,axiom,
    ! [A: $tType] :
      ( ( linorder @ A )
     => ! [T2: A] :
        ? [Z3: A] :
        ! [X6: A] :
          ( ( ord_less @ A @ Z3 @ X6 )
         => ( ord_less_eq @ A @ T2 @ X6 ) ) ) ).

% pinf(8)
thf(fact_1599_minf_I6_J,axiom,
    ! [A: $tType] :
      ( ( linorder @ A )
     => ! [T2: A] :
        ? [Z3: A] :
        ! [X6: A] :
          ( ( ord_less @ A @ X6 @ Z3 )
         => ( ord_less_eq @ A @ X6 @ T2 ) ) ) ).

% minf(6)
thf(fact_1600_minf_I8_J,axiom,
    ! [A: $tType] :
      ( ( linorder @ A )
     => ! [T2: A] :
        ? [Z3: A] :
        ! [X6: A] :
          ( ( ord_less @ A @ X6 @ Z3 )
         => ~ ( ord_less_eq @ A @ T2 @ X6 ) ) ) ).

% minf(8)
thf(fact_1601_inf__period_I1_J,axiom,
    ! [A: $tType] :
      ( ( ( comm_ring @ A )
        & ( dvd @ A ) )
     => ! [P: A > $o,D6: A,Q: A > $o] :
          ( ! [X3: A,K3: A] :
              ( ( P @ X3 )
              = ( P @ ( minus_minus @ A @ X3 @ ( times_times @ A @ K3 @ D6 ) ) ) )
         => ( ! [X3: A,K3: A] :
                ( ( Q @ X3 )
                = ( Q @ ( minus_minus @ A @ X3 @ ( times_times @ A @ K3 @ D6 ) ) ) )
           => ! [X6: A,K5: A] :
                ( ( ( P @ X6 )
                  & ( Q @ X6 ) )
                = ( ( P @ ( minus_minus @ A @ X6 @ ( times_times @ A @ K5 @ D6 ) ) )
                  & ( Q @ ( minus_minus @ A @ X6 @ ( times_times @ A @ K5 @ D6 ) ) ) ) ) ) ) ) ).

% inf_period(1)
thf(fact_1602_inf__period_I2_J,axiom,
    ! [A: $tType] :
      ( ( ( comm_ring @ A )
        & ( dvd @ A ) )
     => ! [P: A > $o,D6: A,Q: A > $o] :
          ( ! [X3: A,K3: A] :
              ( ( P @ X3 )
              = ( P @ ( minus_minus @ A @ X3 @ ( times_times @ A @ K3 @ D6 ) ) ) )
         => ( ! [X3: A,K3: A] :
                ( ( Q @ X3 )
                = ( Q @ ( minus_minus @ A @ X3 @ ( times_times @ A @ K3 @ D6 ) ) ) )
           => ! [X6: A,K5: A] :
                ( ( ( P @ X6 )
                  | ( Q @ X6 ) )
                = ( ( P @ ( minus_minus @ A @ X6 @ ( times_times @ A @ K5 @ D6 ) ) )
                  | ( Q @ ( minus_minus @ A @ X6 @ ( times_times @ A @ K5 @ D6 ) ) ) ) ) ) ) ) ).

% inf_period(2)
thf(fact_1603_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,
    ! [X2: nat] :
      ( ( X2
       != ( zero_zero @ nat ) )
     => ( ( X2
         != ( suc @ ( zero_zero @ nat ) ) )
       => ~ ! [Va3: nat] :
              ( X2
             != ( suc @ ( suc @ Va3 ) ) ) ) ) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d\<^sub>u\<^sub>p.cases
thf(fact_1604_imp__le__cong,axiom,
    ! [X2: int,X7: int,P: $o,P6: $o] :
      ( ( X2 = X7 )
     => ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X7 )
         => ( P = P6 ) )
       => ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X2 )
           => P )
          = ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X7 )
           => P6 ) ) ) ) ).

% imp_le_cong
thf(fact_1605_conj__le__cong,axiom,
    ! [X2: int,X7: int,P: $o,P6: $o] :
      ( ( X2 = X7 )
     => ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X7 )
         => ( P = P6 ) )
       => ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X2 )
            & P )
          = ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X7 )
            & P6 ) ) ) ) ).

% conj_le_cong
thf(fact_1606_pred__bound__size__univ_H,axiom,
    ! [T2: vEBT_VEBT,N2: nat,U: real,X2: nat] :
      ( ( vEBT_invar_vebt @ T2 @ N2 )
     => ( ( U
          = ( power_power @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ N2 ) )
       => ( ord_less_eq @ real @ ( semiring_1_of_nat @ real @ ( vEBT_T_p_r_e_d2 @ T2 @ X2 ) ) @ ( plus_plus @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ U ) ) ) ) ) ) ).

% pred_bound_size_univ'
thf(fact_1607_succ__bound__size__univ_H,axiom,
    ! [T2: vEBT_VEBT,N2: nat,U: real,X2: nat] :
      ( ( vEBT_invar_vebt @ T2 @ N2 )
     => ( ( U
          = ( power_power @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ N2 ) )
       => ( ord_less_eq @ real @ ( semiring_1_of_nat @ real @ ( vEBT_T_s_u_c_c2 @ T2 @ X2 ) ) @ ( plus_plus @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ U ) ) ) ) ) ) ).

% succ_bound_size_univ'
thf(fact_1608_member__bound__size__univ,axiom,
    ! [T2: vEBT_VEBT,N2: nat,U: real,X2: nat] :
      ( ( vEBT_invar_vebt @ T2 @ N2 )
     => ( ( U
          = ( power_power @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ N2 ) )
       => ( ord_less_eq @ real @ ( semiring_1_of_nat @ real @ ( vEBT_T_m_e_m_b_e_r @ T2 @ X2 ) ) @ ( plus_plus @ real @ ( numeral_numeral @ real @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ one2 ) ) ) ) ) @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit1 @ ( bit1 @ ( bit1 @ one2 ) ) ) ) @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ U ) ) ) ) ) ) ) ).

% member_bound_size_univ
thf(fact_1609_VEBT__internal_Ocnt_Osimps_I1_J,axiom,
    ! [A2: $o,B2: $o] :
      ( ( vEBT_VEBT_cnt @ ( vEBT_Leaf @ A2 @ B2 ) )
      = ( one_one @ real ) ) ).

% VEBT_internal.cnt.simps(1)
thf(fact_1610_VEBT__internal_Ocnt_H_Osimps_I1_J,axiom,
    ! [A2: $o,B2: $o] :
      ( ( vEBT_VEBT_cnt2 @ ( vEBT_Leaf @ A2 @ B2 ) )
      = ( one_one @ nat ) ) ).

% VEBT_internal.cnt'.simps(1)
thf(fact_1611_plusinfinity,axiom,
    ! [D2: int,P6: int > $o,P: int > $o] :
      ( ( ord_less @ int @ ( zero_zero @ int ) @ D2 )
     => ( ! [X3: int,K3: int] :
            ( ( P6 @ X3 )
            = ( P6 @ ( minus_minus @ int @ X3 @ ( times_times @ int @ K3 @ D2 ) ) ) )
       => ( ? [Z5: int] :
            ! [X3: int] :
              ( ( ord_less @ int @ Z5 @ X3 )
             => ( ( P @ X3 )
                = ( P6 @ X3 ) ) )
         => ( ? [X_1: int] : ( P6 @ X_1 )
           => ? [X_12: int] : ( P @ X_12 ) ) ) ) ) ).

% plusinfinity
thf(fact_1612_minusinfinity,axiom,
    ! [D2: int,P1: int > $o,P: int > $o] :
      ( ( ord_less @ int @ ( zero_zero @ int ) @ D2 )
     => ( ! [X3: int,K3: int] :
            ( ( P1 @ X3 )
            = ( P1 @ ( minus_minus @ int @ X3 @ ( times_times @ int @ K3 @ D2 ) ) ) )
       => ( ? [Z5: int] :
            ! [X3: int] :
              ( ( ord_less @ int @ X3 @ Z5 )
             => ( ( P @ X3 )
                = ( P1 @ X3 ) ) )
         => ( ? [X_1: int] : ( P1 @ X_1 )
           => ? [X_12: int] : ( P @ X_12 ) ) ) ) ) ).

% minusinfinity
thf(fact_1613_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 )
         => ! [X6: int] :
              ( ( P @ X6 )
             => ( P @ ( plus_plus @ int @ X6 @ ( times_times @ int @ K @ D2 ) ) ) ) ) ) ) ).

% incr_mult_lemma
thf(fact_1614_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 )
         => ! [X6: int] :
              ( ( P @ X6 )
             => ( P @ ( minus_minus @ int @ X6 @ ( times_times @ int @ K @ D2 ) ) ) ) ) ) ) ).

% decr_mult_lemma
thf(fact_1615_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 @ one2 ) ) ) ) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d.simps(1)
thf(fact_1616_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 @ one2 ) ) ) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d\<^sub>u\<^sub>p.simps(1)
thf(fact_1617_VEBT__internal_Ospace_H_Osimps_I1_J,axiom,
    ! [A2: $o,B2: $o] :
      ( ( vEBT_VEBT_space2 @ ( vEBT_Leaf @ A2 @ B2 ) )
      = ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ).

% VEBT_internal.space'.simps(1)
thf(fact_1618_VEBT__internal_Ospace_Osimps_I1_J,axiom,
    ! [A2: $o,B2: $o] :
      ( ( vEBT_VEBT_space @ ( vEBT_Leaf @ A2 @ B2 ) )
      = ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) ) ).

% VEBT_internal.space.simps(1)
thf(fact_1619_log__pow__cancel,axiom,
    ! [A2: real,B2: nat] :
      ( ( ord_less @ real @ ( zero_zero @ real ) @ A2 )
     => ( ( A2
         != ( one_one @ real ) )
       => ( ( log @ A2 @ ( power_power @ real @ A2 @ B2 ) )
          = ( semiring_1_of_nat @ real @ B2 ) ) ) ) ).

% log_pow_cancel
thf(fact_1620_log__le__cancel__iff,axiom,
    ! [A2: real,X2: real,Y2: real] :
      ( ( ord_less @ real @ ( one_one @ real ) @ A2 )
     => ( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
       => ( ( ord_less @ real @ ( zero_zero @ real ) @ Y2 )
         => ( ( ord_less_eq @ real @ ( log @ A2 @ X2 ) @ ( log @ A2 @ Y2 ) )
            = ( ord_less_eq @ real @ X2 @ Y2 ) ) ) ) ) ).

% log_le_cancel_iff
thf(fact_1621_log__le__one__cancel__iff,axiom,
    ! [A2: real,X2: real] :
      ( ( ord_less @ real @ ( one_one @ real ) @ A2 )
     => ( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
       => ( ( ord_less_eq @ real @ ( log @ A2 @ X2 ) @ ( one_one @ real ) )
          = ( ord_less_eq @ real @ X2 @ A2 ) ) ) ) ).

% log_le_one_cancel_iff
thf(fact_1622_one__le__log__cancel__iff,axiom,
    ! [A2: real,X2: real] :
      ( ( ord_less @ real @ ( one_one @ real ) @ A2 )
     => ( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
       => ( ( ord_less_eq @ real @ ( one_one @ real ) @ ( log @ A2 @ X2 ) )
          = ( ord_less_eq @ real @ A2 @ X2 ) ) ) ) ).

% one_le_log_cancel_iff
thf(fact_1623_log__le__zero__cancel__iff,axiom,
    ! [A2: real,X2: real] :
      ( ( ord_less @ real @ ( one_one @ real ) @ A2 )
     => ( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
       => ( ( ord_less_eq @ real @ ( log @ A2 @ X2 ) @ ( zero_zero @ real ) )
          = ( ord_less_eq @ real @ X2 @ ( one_one @ real ) ) ) ) ) ).

% log_le_zero_cancel_iff
thf(fact_1624_zero__le__log__cancel__iff,axiom,
    ! [A2: real,X2: real] :
      ( ( ord_less @ real @ ( one_one @ real ) @ A2 )
     => ( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
       => ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( log @ A2 @ X2 ) )
          = ( ord_less_eq @ real @ ( one_one @ real ) @ X2 ) ) ) ) ).

% zero_le_log_cancel_iff
thf(fact_1625_log2__of__power__le,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq @ nat @ M @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) )
     => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
       => ( ord_less_eq @ real @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ real @ M ) ) @ ( semiring_1_of_nat @ real @ N2 ) ) ) ) ).

% log2_of_power_le
thf(fact_1626_log__eq__one,axiom,
    ! [A2: real] :
      ( ( ord_less @ real @ ( zero_zero @ real ) @ A2 )
     => ( ( A2
         != ( one_one @ real ) )
       => ( ( log @ A2 @ A2 )
          = ( one_one @ real ) ) ) ) ).

% log_eq_one
thf(fact_1627_log__one,axiom,
    ! [A2: real] :
      ( ( log @ A2 @ ( one_one @ real ) )
      = ( zero_zero @ real ) ) ).

% log_one
thf(fact_1628_zero__less__log__cancel__iff,axiom,
    ! [A2: real,X2: real] :
      ( ( ord_less @ real @ ( one_one @ real ) @ A2 )
     => ( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
       => ( ( ord_less @ real @ ( zero_zero @ real ) @ ( log @ A2 @ X2 ) )
          = ( ord_less @ real @ ( one_one @ real ) @ X2 ) ) ) ) ).

% zero_less_log_cancel_iff
thf(fact_1629_log__less__zero__cancel__iff,axiom,
    ! [A2: real,X2: real] :
      ( ( ord_less @ real @ ( one_one @ real ) @ A2 )
     => ( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
       => ( ( ord_less @ real @ ( log @ A2 @ X2 ) @ ( zero_zero @ real ) )
          = ( ord_less @ real @ X2 @ ( one_one @ real ) ) ) ) ) ).

% log_less_zero_cancel_iff
thf(fact_1630_one__less__log__cancel__iff,axiom,
    ! [A2: real,X2: real] :
      ( ( ord_less @ real @ ( one_one @ real ) @ A2 )
     => ( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
       => ( ( ord_less @ real @ ( one_one @ real ) @ ( log @ A2 @ X2 ) )
          = ( ord_less @ real @ A2 @ X2 ) ) ) ) ).

% one_less_log_cancel_iff
thf(fact_1631_log__less__one__cancel__iff,axiom,
    ! [A2: real,X2: real] :
      ( ( ord_less @ real @ ( one_one @ real ) @ A2 )
     => ( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
       => ( ( ord_less @ real @ ( log @ A2 @ X2 ) @ ( one_one @ real ) )
          = ( ord_less @ real @ X2 @ A2 ) ) ) ) ).

% log_less_one_cancel_iff
thf(fact_1632_log__less__cancel__iff,axiom,
    ! [A2: real,X2: real,Y2: real] :
      ( ( ord_less @ real @ ( one_one @ real ) @ A2 )
     => ( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
       => ( ( ord_less @ real @ ( zero_zero @ real ) @ Y2 )
         => ( ( ord_less @ real @ ( log @ A2 @ X2 ) @ ( log @ A2 @ Y2 ) )
            = ( ord_less @ real @ X2 @ Y2 ) ) ) ) ) ).

% log_less_cancel_iff
thf(fact_1633_log__base__change,axiom,
    ! [A2: real,B2: real,X2: real] :
      ( ( ord_less @ real @ ( zero_zero @ real ) @ A2 )
     => ( ( A2
         != ( one_one @ real ) )
       => ( ( log @ B2 @ X2 )
          = ( divide_divide @ real @ ( log @ A2 @ X2 ) @ ( log @ A2 @ B2 ) ) ) ) ) ).

% log_base_change
thf(fact_1634_less__log__of__power,axiom,
    ! [B2: real,N2: nat,M: real] :
      ( ( ord_less @ real @ ( power_power @ real @ B2 @ N2 ) @ M )
     => ( ( ord_less @ real @ ( one_one @ real ) @ B2 )
       => ( ord_less @ real @ ( semiring_1_of_nat @ real @ N2 ) @ ( log @ B2 @ M ) ) ) ) ).

% less_log_of_power
thf(fact_1635_log__of__power__eq,axiom,
    ! [M: nat,B2: real,N2: nat] :
      ( ( ( semiring_1_of_nat @ real @ M )
        = ( power_power @ real @ B2 @ N2 ) )
     => ( ( ord_less @ real @ ( one_one @ real ) @ B2 )
       => ( ( semiring_1_of_nat @ real @ N2 )
          = ( log @ B2 @ ( semiring_1_of_nat @ real @ M ) ) ) ) ) ).

% log_of_power_eq
thf(fact_1636_log__mult,axiom,
    ! [A2: real,X2: real,Y2: real] :
      ( ( ord_less @ real @ ( zero_zero @ real ) @ A2 )
     => ( ( A2
         != ( one_one @ real ) )
       => ( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
         => ( ( ord_less @ real @ ( zero_zero @ real ) @ Y2 )
           => ( ( log @ A2 @ ( times_times @ real @ X2 @ Y2 ) )
              = ( plus_plus @ real @ ( log @ A2 @ X2 ) @ ( log @ A2 @ Y2 ) ) ) ) ) ) ) ).

% log_mult
thf(fact_1637_le__log__of__power,axiom,
    ! [B2: real,N2: nat,M: real] :
      ( ( ord_less_eq @ real @ ( power_power @ real @ B2 @ N2 ) @ M )
     => ( ( ord_less @ real @ ( one_one @ real ) @ B2 )
       => ( ord_less_eq @ real @ ( semiring_1_of_nat @ real @ N2 ) @ ( log @ B2 @ M ) ) ) ) ).

% le_log_of_power
thf(fact_1638_log__divide,axiom,
    ! [A2: real,X2: real,Y2: real] :
      ( ( ord_less @ real @ ( zero_zero @ real ) @ A2 )
     => ( ( A2
         != ( one_one @ real ) )
       => ( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
         => ( ( ord_less @ real @ ( zero_zero @ real ) @ Y2 )
           => ( ( log @ A2 @ ( divide_divide @ real @ X2 @ Y2 ) )
              = ( minus_minus @ real @ ( log @ A2 @ X2 ) @ ( log @ A2 @ Y2 ) ) ) ) ) ) ) ).

% log_divide
thf(fact_1639_log__base__pow,axiom,
    ! [A2: real,N2: nat,X2: real] :
      ( ( ord_less @ real @ ( zero_zero @ real ) @ A2 )
     => ( ( log @ ( power_power @ real @ A2 @ N2 ) @ X2 )
        = ( divide_divide @ real @ ( log @ A2 @ X2 ) @ ( semiring_1_of_nat @ real @ N2 ) ) ) ) ).

% log_base_pow
thf(fact_1640_log__nat__power,axiom,
    ! [X2: real,B2: real,N2: nat] :
      ( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
     => ( ( log @ B2 @ ( power_power @ real @ X2 @ N2 ) )
        = ( times_times @ real @ ( semiring_1_of_nat @ real @ N2 ) @ ( log @ B2 @ X2 ) ) ) ) ).

% log_nat_power
thf(fact_1641_log2__of__power__eq,axiom,
    ! [M: nat,N2: nat] :
      ( ( M
        = ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) )
     => ( ( semiring_1_of_nat @ real @ N2 )
        = ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ real @ M ) ) ) ) ).

% log2_of_power_eq
thf(fact_1642_log__of__power__less,axiom,
    ! [M: nat,B2: real,N2: nat] :
      ( ( ord_less @ real @ ( semiring_1_of_nat @ real @ M ) @ ( power_power @ real @ B2 @ N2 ) )
     => ( ( ord_less @ real @ ( one_one @ real ) @ B2 )
       => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
         => ( ord_less @ real @ ( log @ B2 @ ( semiring_1_of_nat @ real @ M ) ) @ ( semiring_1_of_nat @ real @ N2 ) ) ) ) ) ).

% log_of_power_less
thf(fact_1643_log__of__power__le,axiom,
    ! [M: nat,B2: real,N2: nat] :
      ( ( ord_less_eq @ real @ ( semiring_1_of_nat @ real @ M ) @ ( power_power @ real @ B2 @ N2 ) )
     => ( ( ord_less @ real @ ( one_one @ real ) @ B2 )
       => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
         => ( ord_less_eq @ real @ ( log @ B2 @ ( semiring_1_of_nat @ real @ M ) ) @ ( semiring_1_of_nat @ real @ N2 ) ) ) ) ) ).

% log_of_power_le
thf(fact_1644_less__log2__of__power,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) @ M )
     => ( ord_less @ real @ ( semiring_1_of_nat @ real @ N2 ) @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ real @ M ) ) ) ) ).

% less_log2_of_power
thf(fact_1645_le__log2__of__power,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_eq @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) @ M )
     => ( ord_less_eq @ real @ ( semiring_1_of_nat @ real @ N2 ) @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ real @ M ) ) ) ) ).

% le_log2_of_power
thf(fact_1646_log2__of__power__less,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less @ nat @ M @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) )
     => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
       => ( ord_less @ real @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ real @ M ) ) @ ( semiring_1_of_nat @ real @ N2 ) ) ) ) ).

% log2_of_power_less
thf(fact_1647_atLeastatMost__subset__iff,axiom,
    ! [A: $tType] :
      ( ( preorder @ A )
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( ( ord_less_eq @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) @ ( set_or1337092689740270186AtMost @ A @ C2 @ D2 ) )
          = ( ~ ( ord_less_eq @ A @ A2 @ B2 )
            | ( ( ord_less_eq @ A @ C2 @ A2 )
              & ( ord_less_eq @ A @ B2 @ D2 ) ) ) ) ) ).

% atLeastatMost_subset_iff
thf(fact_1648_arcosh__1,axiom,
    ! [A: $tType] :
      ( ( ln @ A )
     => ( ( arcosh @ A @ ( one_one @ A ) )
        = ( zero_zero @ A ) ) ) ).

% arcosh_1
thf(fact_1649_minNull__delete__time__bound,axiom,
    ! [T2: vEBT_VEBT,N2: nat,X2: nat] :
      ( ( vEBT_invar_vebt @ T2 @ N2 )
     => ( ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ T2 @ X2 ) )
       => ( ord_less_eq @ nat @ ( vEBT_T_d_e_l_e_t_e @ T2 @ X2 ) @ ( numeral_numeral @ nat @ ( bit1 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) ) ).

% minNull_delete_time_bound
thf(fact_1650_nat__induct2,axiom,
    ! [P: nat > $o,N2: nat] :
      ( ( P @ ( zero_zero @ nat ) )
     => ( ( P @ ( one_one @ nat ) )
       => ( ! [N4: nat] :
              ( ( P @ N4 )
             => ( P @ ( plus_plus @ nat @ N4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
         => ( P @ N2 ) ) ) ) ).

% nat_induct2
thf(fact_1651_VEBT__internal_Omembermima_Oelims_I3_J,axiom,
    ! [X2: vEBT_VEBT,Xa: nat] :
      ( ~ ( vEBT_VEBT_membermima @ X2 @ Xa )
     => ( ! [Uu2: $o,Uv2: $o] :
            ( X2
           != ( vEBT_Leaf @ Uu2 @ Uv2 ) )
       => ( ! [Ux2: list @ vEBT_VEBT,Uy2: vEBT_VEBT] :
              ( X2
             != ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( zero_zero @ nat ) @ Ux2 @ Uy2 ) )
         => ( ! [Mi2: nat,Ma2: nat] :
                ( ? [Va2: list @ vEBT_VEBT,Vb2: vEBT_VEBT] :
                    ( X2
                    = ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ Va2 @ Vb2 ) )
               => ( ( Xa = Mi2 )
                  | ( Xa = Ma2 ) ) )
           => ( ! [Mi2: nat,Ma2: nat,V3: nat,TreeList2: list @ vEBT_VEBT] :
                  ( ? [Vc: vEBT_VEBT] :
                      ( X2
                      = ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ V3 ) @ TreeList2 @ Vc ) )
                 => ( ( Xa = Mi2 )
                    | ( Xa = Ma2 )
                    | ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
                       => ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
                      & ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) )
             => ~ ! [V3: nat,TreeList2: list @ vEBT_VEBT] :
                    ( ? [Vd2: vEBT_VEBT] :
                        ( X2
                        = ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ V3 ) @ TreeList2 @ Vd2 ) )
                   => ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
                       => ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
                      & ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(3)
thf(fact_1652_VEBT__internal_Omembermima_Oelims_I1_J,axiom,
    ! [X2: vEBT_VEBT,Xa: nat,Y2: $o] :
      ( ( ( vEBT_VEBT_membermima @ X2 @ Xa )
        = Y2 )
     => ( ( ? [Uu2: $o,Uv2: $o] :
              ( X2
              = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
         => Y2 )
       => ( ( ? [Ux2: list @ vEBT_VEBT,Uy2: vEBT_VEBT] :
                ( X2
                = ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( zero_zero @ nat ) @ Ux2 @ Uy2 ) )
           => Y2 )
         => ( ! [Mi2: nat,Ma2: nat] :
                ( ? [Va2: list @ vEBT_VEBT,Vb2: vEBT_VEBT] :
                    ( X2
                    = ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ Va2 @ Vb2 ) )
               => ( Y2
                  = ( ~ ( ( Xa = Mi2 )
                        | ( Xa = Ma2 ) ) ) ) )
           => ( ! [Mi2: nat,Ma2: nat,V3: nat,TreeList2: list @ vEBT_VEBT] :
                  ( ? [Vc: vEBT_VEBT] :
                      ( X2
                      = ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ V3 ) @ TreeList2 @ Vc ) )
                 => ( Y2
                    = ( ~ ( ( Xa = Mi2 )
                          | ( Xa = Ma2 )
                          | ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
                             => ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
                            & ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) ) )
             => ~ ! [V3: nat,TreeList2: list @ vEBT_VEBT] :
                    ( ? [Vd2: vEBT_VEBT] :
                        ( X2
                        = ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ V3 ) @ TreeList2 @ Vd2 ) )
                   => ( Y2
                      = ( ~ ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
                             => ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
                            & ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(1)
thf(fact_1653_nat__approx__posE,axiom,
    ! [A: $tType] :
      ( ( archim2362893244070406136eiling @ A )
     => ! [E3: A] :
          ( ( ord_less @ A @ ( zero_zero @ A ) @ E3 )
         => ~ ! [N4: nat] :
                ~ ( ord_less @ A @ ( divide_divide @ A @ ( one_one @ A ) @ ( semiring_1_of_nat @ A @ ( suc @ N4 ) ) ) @ E3 ) ) ) ).

% nat_approx_posE
thf(fact_1654_del__single__cont,axiom,
    ! [X2: nat,Mi: nat,Ma: nat,Deg: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( X2 = Mi )
        & ( X2 = Ma ) )
     => ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
       => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X2 )
          = ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg @ TreeList @ Summary ) ) ) ) ).

% del_single_cont
thf(fact_1655_delete__pres__valid,axiom,
    ! [T2: vEBT_VEBT,N2: nat,X2: nat] :
      ( ( vEBT_invar_vebt @ T2 @ N2 )
     => ( vEBT_invar_vebt @ ( vEBT_vebt_delete @ T2 @ X2 ) @ N2 ) ) ).

% delete_pres_valid
thf(fact_1656_buildup__nothing__in__min__max,axiom,
    ! [N2: nat,X2: nat] :
      ~ ( vEBT_VEBT_membermima @ ( vEBT_vebt_buildup @ N2 ) @ X2 ) ).

% buildup_nothing_in_min_max
thf(fact_1657_dele__bmo__cont__corr,axiom,
    ! [T2: vEBT_VEBT,N2: nat,X2: nat,Y2: nat] :
      ( ( vEBT_invar_vebt @ T2 @ N2 )
     => ( ( vEBT_V8194947554948674370ptions @ ( vEBT_vebt_delete @ T2 @ X2 ) @ Y2 )
        = ( ( X2 != Y2 )
          & ( vEBT_V8194947554948674370ptions @ T2 @ Y2 ) ) ) ) ).

% dele_bmo_cont_corr
thf(fact_1658_dele__member__cont__corr,axiom,
    ! [T2: vEBT_VEBT,N2: nat,X2: nat,Y2: nat] :
      ( ( vEBT_invar_vebt @ T2 @ N2 )
     => ( ( vEBT_vebt_member @ ( vEBT_vebt_delete @ T2 @ X2 ) @ Y2 )
        = ( ( X2 != Y2 )
          & ( vEBT_vebt_member @ T2 @ Y2 ) ) ) ) ).

% dele_member_cont_corr
thf(fact_1659_minNull__delete__time__bound_H,axiom,
    ! [T2: vEBT_VEBT,N2: nat,X2: nat] :
      ( ( vEBT_invar_vebt @ T2 @ N2 )
     => ( ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ T2 @ X2 ) )
       => ( ord_less_eq @ nat @ ( vEBT_V1232361888498592333_e_t_e @ T2 @ X2 ) @ ( one_one @ nat ) ) ) ) ).

% minNull_delete_time_bound'
thf(fact_1660_Icc__eq__Icc,axiom,
    ! [A: $tType] :
      ( ( order @ A )
     => ! [L: A,H2: A,L4: A,H4: A] :
          ( ( ( set_or1337092689740270186AtMost @ A @ L @ H2 )
            = ( set_or1337092689740270186AtMost @ A @ L4 @ H4 ) )
          = ( ( ( L = L4 )
              & ( H2 = H4 ) )
            | ( ~ ( ord_less_eq @ A @ L @ H2 )
              & ~ ( ord_less_eq @ A @ L4 @ H4 ) ) ) ) ) ).

% Icc_eq_Icc
thf(fact_1661_atLeastAtMost__iff,axiom,
    ! [A: $tType] :
      ( ( ord @ A )
     => ! [I: A,L: A,U: A] :
          ( ( member @ A @ I @ ( set_or1337092689740270186AtMost @ A @ L @ U ) )
          = ( ( ord_less_eq @ A @ L @ I )
            & ( ord_less_eq @ A @ I @ U ) ) ) ) ).

% atLeastAtMost_iff
thf(fact_1662_delt__out__of__range,axiom,
    ! [X2: nat,Mi: nat,Ma: nat,Deg: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( ord_less @ nat @ X2 @ Mi )
        | ( ord_less @ nat @ Ma @ X2 ) )
     => ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
       => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X2 )
          = ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) ) ) ) ).

% delt_out_of_range
thf(fact_1663_aset_I2_J,axiom,
    ! [D6: int,A3: set @ int,P: int > $o,Q: int > $o] :
      ( ! [X3: int] :
          ( ! [Xa2: int] :
              ( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D6 ) )
             => ! [Xb2: int] :
                  ( ( member @ int @ Xb2 @ A3 )
                 => ( X3
                   != ( minus_minus @ int @ Xb2 @ Xa2 ) ) ) )
         => ( ( P @ X3 )
           => ( P @ ( plus_plus @ int @ X3 @ D6 ) ) ) )
     => ( ! [X3: int] :
            ( ! [Xa2: int] :
                ( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D6 ) )
               => ! [Xb2: int] :
                    ( ( member @ int @ Xb2 @ A3 )
                   => ( X3
                     != ( minus_minus @ int @ Xb2 @ Xa2 ) ) ) )
           => ( ( Q @ X3 )
             => ( Q @ ( plus_plus @ int @ X3 @ D6 ) ) ) )
       => ! [X6: int] :
            ( ! [Xa3: int] :
                ( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D6 ) )
               => ! [Xb3: int] :
                    ( ( member @ int @ Xb3 @ A3 )
                   => ( X6
                     != ( minus_minus @ int @ Xb3 @ Xa3 ) ) ) )
           => ( ( ( P @ X6 )
                | ( Q @ X6 ) )
             => ( ( P @ ( plus_plus @ int @ X6 @ D6 ) )
                | ( Q @ ( plus_plus @ int @ X6 @ D6 ) ) ) ) ) ) ) ).

% aset(2)
thf(fact_1664_aset_I1_J,axiom,
    ! [D6: int,A3: set @ int,P: int > $o,Q: int > $o] :
      ( ! [X3: int] :
          ( ! [Xa2: int] :
              ( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D6 ) )
             => ! [Xb2: int] :
                  ( ( member @ int @ Xb2 @ A3 )
                 => ( X3
                   != ( minus_minus @ int @ Xb2 @ Xa2 ) ) ) )
         => ( ( P @ X3 )
           => ( P @ ( plus_plus @ int @ X3 @ D6 ) ) ) )
     => ( ! [X3: int] :
            ( ! [Xa2: int] :
                ( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D6 ) )
               => ! [Xb2: int] :
                    ( ( member @ int @ Xb2 @ A3 )
                   => ( X3
                     != ( minus_minus @ int @ Xb2 @ Xa2 ) ) ) )
           => ( ( Q @ X3 )
             => ( Q @ ( plus_plus @ int @ X3 @ D6 ) ) ) )
       => ! [X6: int] :
            ( ! [Xa3: int] :
                ( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D6 ) )
               => ! [Xb3: int] :
                    ( ( member @ int @ Xb3 @ A3 )
                   => ( X6
                     != ( minus_minus @ int @ Xb3 @ Xa3 ) ) ) )
           => ( ( ( P @ X6 )
                & ( Q @ X6 ) )
             => ( ( P @ ( plus_plus @ int @ X6 @ D6 ) )
                & ( Q @ ( plus_plus @ int @ X6 @ D6 ) ) ) ) ) ) ) ).

% aset(1)
thf(fact_1665_bset_I2_J,axiom,
    ! [D6: int,B6: set @ int,P: int > $o,Q: int > $o] :
      ( ! [X3: int] :
          ( ! [Xa2: int] :
              ( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D6 ) )
             => ! [Xb2: int] :
                  ( ( member @ int @ Xb2 @ B6 )
                 => ( X3
                   != ( plus_plus @ int @ Xb2 @ Xa2 ) ) ) )
         => ( ( P @ X3 )
           => ( P @ ( minus_minus @ int @ X3 @ D6 ) ) ) )
     => ( ! [X3: int] :
            ( ! [Xa2: int] :
                ( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D6 ) )
               => ! [Xb2: int] :
                    ( ( member @ int @ Xb2 @ B6 )
                   => ( X3
                     != ( plus_plus @ int @ Xb2 @ Xa2 ) ) ) )
           => ( ( Q @ X3 )
             => ( Q @ ( minus_minus @ int @ X3 @ D6 ) ) ) )
       => ! [X6: int] :
            ( ! [Xa3: int] :
                ( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D6 ) )
               => ! [Xb3: int] :
                    ( ( member @ int @ Xb3 @ B6 )
                   => ( X6
                     != ( plus_plus @ int @ Xb3 @ Xa3 ) ) ) )
           => ( ( ( P @ X6 )
                | ( Q @ X6 ) )
             => ( ( P @ ( minus_minus @ int @ X6 @ D6 ) )
                | ( Q @ ( minus_minus @ int @ X6 @ D6 ) ) ) ) ) ) ) ).

% bset(2)
thf(fact_1666_bset_I1_J,axiom,
    ! [D6: int,B6: set @ int,P: int > $o,Q: int > $o] :
      ( ! [X3: int] :
          ( ! [Xa2: int] :
              ( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D6 ) )
             => ! [Xb2: int] :
                  ( ( member @ int @ Xb2 @ B6 )
                 => ( X3
                   != ( plus_plus @ int @ Xb2 @ Xa2 ) ) ) )
         => ( ( P @ X3 )
           => ( P @ ( minus_minus @ int @ X3 @ D6 ) ) ) )
     => ( ! [X3: int] :
            ( ! [Xa2: int] :
                ( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D6 ) )
               => ! [Xb2: int] :
                    ( ( member @ int @ Xb2 @ B6 )
                   => ( X3
                     != ( plus_plus @ int @ Xb2 @ Xa2 ) ) ) )
           => ( ( Q @ X3 )
             => ( Q @ ( minus_minus @ int @ X3 @ D6 ) ) ) )
       => ! [X6: int] :
            ( ! [Xa3: int] :
                ( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D6 ) )
               => ! [Xb3: int] :
                    ( ( member @ int @ Xb3 @ B6 )
                   => ( X6
                     != ( plus_plus @ int @ Xb3 @ Xa3 ) ) ) )
           => ( ( ( P @ X6 )
                & ( Q @ X6 ) )
             => ( ( P @ ( minus_minus @ int @ X6 @ D6 ) )
                & ( Q @ ( minus_minus @ int @ X6 @ D6 ) ) ) ) ) ) ) ).

% bset(1)
thf(fact_1667_bounded__Max__nat,axiom,
    ! [P: nat > $o,X2: nat,M8: nat] :
      ( ( P @ X2 )
     => ( ! [X3: nat] :
            ( ( P @ X3 )
           => ( ord_less_eq @ nat @ X3 @ M8 ) )
       => ~ ! [M2: nat] :
              ( ( P @ M2 )
             => ~ ! [X6: nat] :
                    ( ( P @ X6 )
                   => ( ord_less_eq @ nat @ X6 @ M2 ) ) ) ) ) ).

% bounded_Max_nat
thf(fact_1668_fold__atLeastAtMost__nat_Ocases,axiom,
    ! [A: $tType,X2: product_prod @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) )] :
      ~ ! [F2: nat > A > A,A4: nat,B3: nat,Acc: A] :
          ( X2
         != ( product_Pair @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) ) @ F2 @ ( product_Pair @ nat @ ( product_prod @ nat @ A ) @ A4 @ ( product_Pair @ nat @ A @ B3 @ Acc ) ) ) ) ).

% fold_atLeastAtMost_nat.cases
thf(fact_1669_VEBT__internal_Omembermima_Osimps_I2_J,axiom,
    ! [Ux: list @ vEBT_VEBT,Uy: vEBT_VEBT,Uz: nat] :
      ~ ( vEBT_VEBT_membermima @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( zero_zero @ nat ) @ Ux @ Uy ) @ Uz ) ).

% VEBT_internal.membermima.simps(2)
thf(fact_1670_aset_I7_J,axiom,
    ! [D6: int,A3: set @ int,T2: int] :
      ( ( ord_less @ int @ ( zero_zero @ int ) @ D6 )
     => ! [X6: int] :
          ( ! [Xa3: int] :
              ( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D6 ) )
             => ! [Xb3: int] :
                  ( ( member @ int @ Xb3 @ A3 )
                 => ( X6
                   != ( minus_minus @ int @ Xb3 @ Xa3 ) ) ) )
         => ( ( ord_less @ int @ T2 @ X6 )
           => ( ord_less @ int @ T2 @ ( plus_plus @ int @ X6 @ D6 ) ) ) ) ) ).

% aset(7)
thf(fact_1671_aset_I5_J,axiom,
    ! [D6: int,T2: int,A3: set @ int] :
      ( ( ord_less @ int @ ( zero_zero @ int ) @ D6 )
     => ( ( member @ int @ T2 @ A3 )
       => ! [X6: int] :
            ( ! [Xa3: int] :
                ( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D6 ) )
               => ! [Xb3: int] :
                    ( ( member @ int @ Xb3 @ A3 )
                   => ( X6
                     != ( minus_minus @ int @ Xb3 @ Xa3 ) ) ) )
           => ( ( ord_less @ int @ X6 @ T2 )
             => ( ord_less @ int @ ( plus_plus @ int @ X6 @ D6 ) @ T2 ) ) ) ) ) ).

% aset(5)
thf(fact_1672_aset_I4_J,axiom,
    ! [D6: int,T2: int,A3: set @ int] :
      ( ( ord_less @ int @ ( zero_zero @ int ) @ D6 )
     => ( ( member @ int @ T2 @ A3 )
       => ! [X6: int] :
            ( ! [Xa3: int] :
                ( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D6 ) )
               => ! [Xb3: int] :
                    ( ( member @ int @ Xb3 @ A3 )
                   => ( X6
                     != ( minus_minus @ int @ Xb3 @ Xa3 ) ) ) )
           => ( ( X6 != T2 )
             => ( ( plus_plus @ int @ X6 @ D6 )
               != T2 ) ) ) ) ) ).

% aset(4)
thf(fact_1673_aset_I3_J,axiom,
    ! [D6: int,T2: int,A3: set @ int] :
      ( ( ord_less @ int @ ( zero_zero @ int ) @ D6 )
     => ( ( member @ int @ ( plus_plus @ int @ T2 @ ( one_one @ int ) ) @ A3 )
       => ! [X6: int] :
            ( ! [Xa3: int] :
                ( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D6 ) )
               => ! [Xb3: int] :
                    ( ( member @ int @ Xb3 @ A3 )
                   => ( X6
                     != ( minus_minus @ int @ Xb3 @ Xa3 ) ) ) )
           => ( ( X6 = T2 )
             => ( ( plus_plus @ int @ X6 @ D6 )
                = T2 ) ) ) ) ) ).

% aset(3)
thf(fact_1674_bset_I7_J,axiom,
    ! [D6: int,T2: int,B6: set @ int] :
      ( ( ord_less @ int @ ( zero_zero @ int ) @ D6 )
     => ( ( member @ int @ T2 @ B6 )
       => ! [X6: int] :
            ( ! [Xa3: int] :
                ( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D6 ) )
               => ! [Xb3: int] :
                    ( ( member @ int @ Xb3 @ B6 )
                   => ( X6
                     != ( plus_plus @ int @ Xb3 @ Xa3 ) ) ) )
           => ( ( ord_less @ int @ T2 @ X6 )
             => ( ord_less @ int @ T2 @ ( minus_minus @ int @ X6 @ D6 ) ) ) ) ) ) ).

% bset(7)
thf(fact_1675_bset_I5_J,axiom,
    ! [D6: int,B6: set @ int,T2: int] :
      ( ( ord_less @ int @ ( zero_zero @ int ) @ D6 )
     => ! [X6: int] :
          ( ! [Xa3: int] :
              ( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D6 ) )
             => ! [Xb3: int] :
                  ( ( member @ int @ Xb3 @ B6 )
                 => ( X6
                   != ( plus_plus @ int @ Xb3 @ Xa3 ) ) ) )
         => ( ( ord_less @ int @ X6 @ T2 )
           => ( ord_less @ int @ ( minus_minus @ int @ X6 @ D6 ) @ T2 ) ) ) ) ).

% bset(5)
thf(fact_1676_bset_I4_J,axiom,
    ! [D6: int,T2: int,B6: set @ int] :
      ( ( ord_less @ int @ ( zero_zero @ int ) @ D6 )
     => ( ( member @ int @ T2 @ B6 )
       => ! [X6: int] :
            ( ! [Xa3: int] :
                ( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D6 ) )
               => ! [Xb3: int] :
                    ( ( member @ int @ Xb3 @ B6 )
                   => ( X6
                     != ( plus_plus @ int @ Xb3 @ Xa3 ) ) ) )
           => ( ( X6 != T2 )
             => ( ( minus_minus @ int @ X6 @ D6 )
               != T2 ) ) ) ) ) ).

% bset(4)
thf(fact_1677_bset_I3_J,axiom,
    ! [D6: int,T2: int,B6: set @ int] :
      ( ( ord_less @ int @ ( zero_zero @ int ) @ D6 )
     => ( ( member @ int @ ( minus_minus @ int @ T2 @ ( one_one @ int ) ) @ B6 )
       => ! [X6: int] :
            ( ! [Xa3: int] :
                ( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D6 ) )
               => ! [Xb3: int] :
                    ( ( member @ int @ Xb3 @ B6 )
                   => ( X6
                     != ( plus_plus @ int @ Xb3 @ Xa3 ) ) ) )
           => ( ( X6 = T2 )
             => ( ( minus_minus @ int @ X6 @ D6 )
                = T2 ) ) ) ) ) ).

% bset(3)
thf(fact_1678_periodic__finite__ex,axiom,
    ! [D2: int,P: int > $o] :
      ( ( ord_less @ int @ ( zero_zero @ int ) @ D2 )
     => ( ! [X3: int,K3: int] :
            ( ( P @ X3 )
            = ( P @ ( minus_minus @ int @ X3 @ ( times_times @ int @ K3 @ D2 ) ) ) )
       => ( ( ? [X5: int] : ( P @ X5 ) )
          = ( ? [X: int] :
                ( ( member @ int @ X @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D2 ) )
                & ( P @ X ) ) ) ) ) ) ).

% periodic_finite_ex
thf(fact_1679_VEBT__internal_Omembermima_Osimps_I3_J,axiom,
    ! [Mi: nat,Ma: nat,Va: list @ vEBT_VEBT,Vb: vEBT_VEBT,X2: nat] :
      ( ( vEBT_VEBT_membermima @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( zero_zero @ nat ) @ Va @ Vb ) @ X2 )
      = ( ( X2 = Mi )
        | ( X2 = Ma ) ) ) ).

% VEBT_internal.membermima.simps(3)
thf(fact_1680_aset_I8_J,axiom,
    ! [D6: int,A3: set @ int,T2: int] :
      ( ( ord_less @ int @ ( zero_zero @ int ) @ D6 )
     => ! [X6: int] :
          ( ! [Xa3: int] :
              ( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D6 ) )
             => ! [Xb3: int] :
                  ( ( member @ int @ Xb3 @ A3 )
                 => ( X6
                   != ( minus_minus @ int @ Xb3 @ Xa3 ) ) ) )
         => ( ( ord_less_eq @ int @ T2 @ X6 )
           => ( ord_less_eq @ int @ T2 @ ( plus_plus @ int @ X6 @ D6 ) ) ) ) ) ).

% aset(8)
thf(fact_1681_aset_I6_J,axiom,
    ! [D6: int,T2: int,A3: set @ int] :
      ( ( ord_less @ int @ ( zero_zero @ int ) @ D6 )
     => ( ( member @ int @ ( plus_plus @ int @ T2 @ ( one_one @ int ) ) @ A3 )
       => ! [X6: int] :
            ( ! [Xa3: int] :
                ( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D6 ) )
               => ! [Xb3: int] :
                    ( ( member @ int @ Xb3 @ A3 )
                   => ( X6
                     != ( minus_minus @ int @ Xb3 @ Xa3 ) ) ) )
           => ( ( ord_less_eq @ int @ X6 @ T2 )
             => ( ord_less_eq @ int @ ( plus_plus @ int @ X6 @ D6 ) @ T2 ) ) ) ) ) ).

% aset(6)
thf(fact_1682_bset_I8_J,axiom,
    ! [D6: int,T2: int,B6: set @ int] :
      ( ( ord_less @ int @ ( zero_zero @ int ) @ D6 )
     => ( ( member @ int @ ( minus_minus @ int @ T2 @ ( one_one @ int ) ) @ B6 )
       => ! [X6: int] :
            ( ! [Xa3: int] :
                ( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D6 ) )
               => ! [Xb3: int] :
                    ( ( member @ int @ Xb3 @ B6 )
                   => ( X6
                     != ( plus_plus @ int @ Xb3 @ Xa3 ) ) ) )
           => ( ( ord_less_eq @ int @ T2 @ X6 )
             => ( ord_less_eq @ int @ T2 @ ( minus_minus @ int @ X6 @ D6 ) ) ) ) ) ) ).

% bset(8)
thf(fact_1683_bset_I6_J,axiom,
    ! [D6: int,B6: set @ int,T2: int] :
      ( ( ord_less @ int @ ( zero_zero @ int ) @ D6 )
     => ! [X6: int] :
          ( ! [Xa3: int] :
              ( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D6 ) )
             => ! [Xb3: int] :
                  ( ( member @ int @ Xb3 @ B6 )
                 => ( X6
                   != ( plus_plus @ int @ Xb3 @ Xa3 ) ) ) )
         => ( ( ord_less_eq @ int @ X6 @ T2 )
           => ( ord_less_eq @ int @ ( minus_minus @ int @ X6 @ D6 ) @ T2 ) ) ) ) ).

% bset(6)
thf(fact_1684_cpmi,axiom,
    ! [D6: int,P: int > $o,P6: int > $o,B6: set @ int] :
      ( ( ord_less @ int @ ( zero_zero @ int ) @ D6 )
     => ( ? [Z5: int] :
          ! [X3: int] :
            ( ( ord_less @ int @ X3 @ Z5 )
           => ( ( P @ X3 )
              = ( P6 @ X3 ) ) )
       => ( ! [X3: int] :
              ( ! [Xa2: int] :
                  ( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D6 ) )
                 => ! [Xb2: int] :
                      ( ( member @ int @ Xb2 @ B6 )
                     => ( X3
                       != ( plus_plus @ int @ Xb2 @ Xa2 ) ) ) )
             => ( ( P @ X3 )
               => ( P @ ( minus_minus @ int @ X3 @ D6 ) ) ) )
         => ( ! [X3: int,K3: int] :
                ( ( P6 @ X3 )
                = ( P6 @ ( minus_minus @ int @ X3 @ ( times_times @ int @ K3 @ D6 ) ) ) )
           => ( ( ? [X5: int] : ( P @ X5 ) )
              = ( ? [X: int] :
                    ( ( member @ int @ X @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D6 ) )
                    & ( P6 @ X ) )
                | ? [X: int] :
                    ( ( member @ int @ X @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D6 ) )
                    & ? [Y: int] :
                        ( ( member @ int @ Y @ B6 )
                        & ( P @ ( plus_plus @ int @ Y @ X ) ) ) ) ) ) ) ) ) ) ).

% cpmi
thf(fact_1685_cppi,axiom,
    ! [D6: int,P: int > $o,P6: int > $o,A3: set @ int] :
      ( ( ord_less @ int @ ( zero_zero @ int ) @ D6 )
     => ( ? [Z5: int] :
          ! [X3: int] :
            ( ( ord_less @ int @ Z5 @ X3 )
           => ( ( P @ X3 )
              = ( P6 @ X3 ) ) )
       => ( ! [X3: int] :
              ( ! [Xa2: int] :
                  ( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D6 ) )
                 => ! [Xb2: int] :
                      ( ( member @ int @ Xb2 @ A3 )
                     => ( X3
                       != ( minus_minus @ int @ Xb2 @ Xa2 ) ) ) )
             => ( ( P @ X3 )
               => ( P @ ( plus_plus @ int @ X3 @ D6 ) ) ) )
         => ( ! [X3: int,K3: int] :
                ( ( P6 @ X3 )
                = ( P6 @ ( minus_minus @ int @ X3 @ ( times_times @ int @ K3 @ D6 ) ) ) )
           => ( ( ? [X5: int] : ( P @ X5 ) )
              = ( ? [X: int] :
                    ( ( member @ int @ X @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D6 ) )
                    & ( P6 @ X ) )
                | ? [X: int] :
                    ( ( member @ int @ X @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D6 ) )
                    & ? [Y: int] :
                        ( ( member @ int @ Y @ A3 )
                        & ( P @ ( minus_minus @ int @ Y @ X ) ) ) ) ) ) ) ) ) ) ).

% cppi
thf(fact_1686_exists__least__lemma,axiom,
    ! [P: nat > $o] :
      ( ~ ( P @ ( zero_zero @ nat ) )
     => ( ? [X_1: nat] : ( P @ X_1 )
       => ? [N4: nat] :
            ( ~ ( P @ N4 )
            & ( P @ ( suc @ N4 ) ) ) ) ) ).

% exists_least_lemma
thf(fact_1687_real__arch__simple,axiom,
    ! [A: $tType] :
      ( ( archim462609752435547400_field @ A )
     => ! [X2: A] :
        ? [N4: nat] : ( ord_less_eq @ A @ X2 @ ( semiring_1_of_nat @ A @ N4 ) ) ) ).

% real_arch_simple
thf(fact_1688_reals__Archimedean2,axiom,
    ! [A: $tType] :
      ( ( archim462609752435547400_field @ A )
     => ! [X2: A] :
        ? [N4: nat] : ( ord_less @ A @ X2 @ ( semiring_1_of_nat @ A @ N4 ) ) ) ).

% reals_Archimedean2
thf(fact_1689_atLeastatMost__psubset__iff,axiom,
    ! [A: $tType] :
      ( ( preorder @ A )
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( ( ord_less @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) @ ( set_or1337092689740270186AtMost @ A @ C2 @ D2 ) )
          = ( ( ~ ( ord_less_eq @ A @ A2 @ B2 )
              | ( ( ord_less_eq @ A @ C2 @ A2 )
                & ( ord_less_eq @ A @ B2 @ D2 )
                & ( ( ord_less @ A @ C2 @ A2 )
                  | ( ord_less @ A @ B2 @ D2 ) ) ) )
            & ( ord_less_eq @ A @ C2 @ D2 ) ) ) ) ).

% atLeastatMost_psubset_iff
thf(fact_1690_ex__less__of__nat__mult,axiom,
    ! [A: $tType] :
      ( ( archim462609752435547400_field @ A )
     => ! [X2: A,Y2: A] :
          ( ( ord_less @ A @ ( zero_zero @ A ) @ X2 )
         => ? [N4: nat] : ( ord_less @ A @ Y2 @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N4 ) @ X2 ) ) ) ) ).

% ex_less_of_nat_mult
thf(fact_1691_VEBT__internal_Omembermima_Osimps_I5_J,axiom,
    ! [V: nat,TreeList: list @ vEBT_VEBT,Vd: vEBT_VEBT,X2: nat] :
      ( ( vEBT_VEBT_membermima @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ V ) @ TreeList @ Vd ) @ X2 )
      = ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ ( suc @ V ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
         => ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ ( suc @ V ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X2 @ ( divide_divide @ nat @ ( suc @ V ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
        & ( ord_less @ nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ ( suc @ V ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) ) ) ) ).

% VEBT_internal.membermima.simps(5)
thf(fact_1692_VEBT__internal_Omembermima_Osimps_I4_J,axiom,
    ! [Mi: nat,Ma: nat,V: nat,TreeList: list @ vEBT_VEBT,Vc2: vEBT_VEBT,X2: nat] :
      ( ( vEBT_VEBT_membermima @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ V ) @ TreeList @ Vc2 ) @ X2 )
      = ( ( X2 = Mi )
        | ( X2 = Ma )
        | ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ ( suc @ V ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
           => ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ ( suc @ V ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X2 @ ( divide_divide @ nat @ ( suc @ V ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
          & ( ord_less @ nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ ( suc @ V ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) ) ) ) ) ).

% VEBT_internal.membermima.simps(4)
thf(fact_1693_double__not__eq__Suc__double,axiom,
    ! [M: nat,N2: nat] :
      ( ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M )
     != ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ) ).

% double_not_eq_Suc_double
thf(fact_1694_Suc__double__not__eq__double,axiom,
    ! [M: nat,N2: nat] :
      ( ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) )
     != ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ).

% Suc_double_not_eq_double
thf(fact_1695_VEBT__internal_Omembermima_Oelims_I2_J,axiom,
    ! [X2: vEBT_VEBT,Xa: nat] :
      ( ( vEBT_VEBT_membermima @ X2 @ Xa )
     => ( ! [Mi2: nat,Ma2: nat] :
            ( ? [Va2: list @ vEBT_VEBT,Vb2: vEBT_VEBT] :
                ( X2
                = ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ Va2 @ Vb2 ) )
           => ~ ( ( Xa = Mi2 )
                | ( Xa = Ma2 ) ) )
       => ( ! [Mi2: nat,Ma2: nat,V3: nat,TreeList2: list @ vEBT_VEBT] :
              ( ? [Vc: vEBT_VEBT] :
                  ( X2
                  = ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ V3 ) @ TreeList2 @ Vc ) )
             => ~ ( ( Xa = Mi2 )
                  | ( Xa = Ma2 )
                  | ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
                     => ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
                    & ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) )
         => ~ ! [V3: nat,TreeList2: list @ vEBT_VEBT] :
                ( ? [Vd2: vEBT_VEBT] :
                    ( X2
                    = ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ V3 ) @ TreeList2 @ Vd2 ) )
               => ~ ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
                     => ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
                    & ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(2)
thf(fact_1696_del__x__mia,axiom,
    ! [X2: nat,Mi: nat,Ma: nat,Deg: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( X2 = Mi )
        & ( ord_less @ nat @ X2 @ Ma ) )
     => ( ( Mi != Ma )
       => ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
         => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X2 )
            = ( if @ vEBT_VEBT @ ( ord_less @ nat @ ( vEBT_VEBT_high @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
              @ ( if @ vEBT_VEBT @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
                @ ( vEBT_Node
                  @ ( some @ ( product_prod @ nat @ nat )
                    @ ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
                      @ ( if @ nat
                        @ ( ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
                          = Ma )
                        @ ( if @ nat
                          @ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
                            = ( none @ nat ) )
                          @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
                          @ ( plus_plus @ nat @ ( times_times @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ ( list_update @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) )
                        @ Ma ) ) )
                  @ Deg
                  @ ( list_update @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
                  @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
                @ ( vEBT_Node
                  @ ( some @ ( product_prod @ nat @ nat )
                    @ ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
                      @ ( if @ nat
                        @ ( ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
                          = Ma )
                        @ ( plus_plus @ nat @ ( times_times @ nat @ ( vEBT_VEBT_high @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ ( list_update @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_VEBT_high @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) )
                        @ Ma ) ) )
                  @ Deg
                  @ ( list_update @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
                  @ Summary ) )
              @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) ) ) ) ) ) ).

% del_x_mia
thf(fact_1697_del__x__mi__lets__in__minNull,axiom,
    ! [X2: 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] :
      ( ( ( X2 = Mi )
        & ( ord_less @ nat @ X2 @ Ma ) )
     => ( ( Mi != Ma )
       => ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
         => ( ( ( vEBT_VEBT_high @ Xn @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
              = H2 )
           => ( ( Xn
                = ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) )
             => ( ( ( vEBT_VEBT_low @ Xn @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
                  = L )
               => ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xn @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
                 => ( ( Newnode
                      = ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ H2 ) @ L ) )
                   => ( ( Newlist
                        = ( list_update @ vEBT_VEBT @ TreeList @ H2 @ Newnode ) )
                     => ( ( vEBT_VEBT_minNull @ Newnode )
                       => ( ( Sn
                            = ( vEBT_vebt_delete @ Summary @ H2 ) )
                         => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X2 )
                            = ( vEBT_Node
                              @ ( some @ ( product_prod @ nat @ 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 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ Sn ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ Newlist @ ( the2 @ nat @ ( vEBT_vebt_maxt @ Sn ) ) ) ) ) ) )
                                    @ Ma ) ) )
                              @ Deg
                              @ Newlist
                              @ Sn ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% del_x_mi_lets_in_minNull
thf(fact_1698_del__x__mi__lets__in,axiom,
    ! [X2: 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] :
      ( ( ( X2 = Mi )
        & ( ord_less @ nat @ X2 @ Ma ) )
     => ( ( Mi != Ma )
       => ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
         => ( ( ( vEBT_VEBT_high @ Xn @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
              = H2 )
           => ( ( Xn
                = ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) )
             => ( ( ( vEBT_VEBT_low @ Xn @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
                  = L )
               => ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xn @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
                 => ( ( Newnode
                      = ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ H2 ) @ L ) )
                   => ( ( Newlist
                        = ( list_update @ vEBT_VEBT @ TreeList @ H2 @ Newnode ) )
                     => ( ( ( vEBT_VEBT_minNull @ Newnode )
                         => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X2 )
                            = ( vEBT_Node
                              @ ( some @ ( product_prod @ nat @ 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 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ Newlist @ ( the2 @ 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 @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X2 )
                            = ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Xn @ ( if @ nat @ ( Xn = Ma ) @ ( plus_plus @ nat @ ( times_times @ nat @ H2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ Newlist @ H2 ) ) ) ) @ Ma ) ) ) @ Deg @ Newlist @ Summary ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% del_x_mi_lets_in
thf(fact_1699_del__x__mi,axiom,
    ! [X2: nat,Mi: nat,Ma: nat,Deg: nat,Xn: nat,H2: nat,Summary: vEBT_VEBT,TreeList: list @ vEBT_VEBT,L: nat] :
      ( ( ( X2 = Mi )
        & ( ord_less @ nat @ X2 @ Ma ) )
     => ( ( Mi != Ma )
       => ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
         => ( ( ( vEBT_VEBT_high @ Xn @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
              = H2 )
           => ( ( Xn
                = ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) )
             => ( ( ( vEBT_VEBT_low @ Xn @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
                  = L )
               => ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xn @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
                 => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X2 )
                    = ( if @ vEBT_VEBT @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ H2 ) @ L ) )
                      @ ( vEBT_Node
                        @ ( some @ ( product_prod @ nat @ 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 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ ( list_update @ vEBT_VEBT @ TreeList @ H2 @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ H2 ) @ L ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) ) ) ) ) ) ) )
                              @ Ma ) ) )
                        @ Deg
                        @ ( list_update @ vEBT_VEBT @ TreeList @ H2 @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ H2 ) @ L ) )
                        @ ( vEBT_vebt_delete @ Summary @ H2 ) )
                      @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Xn @ ( if @ nat @ ( Xn = Ma ) @ ( plus_plus @ nat @ ( times_times @ nat @ H2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ ( list_update @ vEBT_VEBT @ TreeList @ H2 @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ H2 ) @ L ) ) @ H2 ) ) ) ) @ Ma ) ) ) @ Deg @ ( list_update @ vEBT_VEBT @ TreeList @ H2 @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ H2 ) @ L ) ) @ Summary ) ) ) ) ) ) ) ) ) ) ).

% del_x_mi
thf(fact_1700_del__in__range,axiom,
    ! [Mi: nat,X2: nat,Ma: nat,Deg: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( ord_less_eq @ nat @ Mi @ X2 )
        & ( ord_less_eq @ nat @ X2 @ Ma ) )
     => ( ( Mi != Ma )
       => ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
         => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X2 )
            = ( if @ vEBT_VEBT @ ( ord_less @ nat @ ( vEBT_VEBT_high @ ( if @ nat @ ( X2 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
              @ ( if @ vEBT_VEBT @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( X2 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( X2 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
                @ ( vEBT_Node
                  @ ( some @ ( product_prod @ nat @ nat )
                    @ ( product_Pair @ nat @ nat @ ( if @ nat @ ( X2 = Mi ) @ ( if @ nat @ ( X2 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ Mi )
                      @ ( if @ nat
                        @ ( ( ( X2 = Mi )
                           => ( ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
                              = Ma ) )
                          & ( ( X2 != Mi )
                           => ( X2 = Ma ) ) )
                        @ ( if @ nat
                          @ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if @ nat @ ( X2 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
                            = ( none @ nat ) )
                          @ ( if @ nat @ ( X2 = Mi ) @ ( if @ nat @ ( X2 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ Mi )
                          @ ( plus_plus @ nat @ ( times_times @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if @ nat @ ( X2 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ ( list_update @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( X2 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( X2 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( X2 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if @ nat @ ( X2 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) )
                        @ Ma ) ) )
                  @ Deg
                  @ ( list_update @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( X2 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( X2 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( X2 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
                  @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if @ nat @ ( X2 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
                @ ( vEBT_Node
                  @ ( some @ ( product_prod @ nat @ nat )
                    @ ( product_Pair @ nat @ nat @ ( if @ nat @ ( X2 = Mi ) @ ( if @ nat @ ( X2 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ Mi )
                      @ ( if @ nat
                        @ ( ( ( X2 = Mi )
                           => ( ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
                              = Ma ) )
                          & ( ( X2 != Mi )
                           => ( X2 = Ma ) ) )
                        @ ( plus_plus @ nat @ ( times_times @ nat @ ( vEBT_VEBT_high @ ( if @ nat @ ( X2 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ ( list_update @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( X2 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( X2 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( X2 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_VEBT_high @ ( if @ nat @ ( X2 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) )
                        @ Ma ) ) )
                  @ Deg
                  @ ( list_update @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( X2 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( X2 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( X2 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
                  @ Summary ) )
              @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) ) ) ) ) ) ).

% del_in_range
thf(fact_1701_del__x__not__mia,axiom,
    ! [Mi: nat,X2: nat,Ma: nat,Deg: nat,H2: nat,L: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( ord_less @ nat @ Mi @ X2 )
        & ( ord_less_eq @ nat @ X2 @ Ma ) )
     => ( ( Mi != Ma )
       => ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
         => ( ( ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
              = H2 )
           => ( ( ( vEBT_VEBT_low @ X2 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
                = L )
             => ( ( ord_less @ nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
               => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X2 )
                  = ( if @ vEBT_VEBT @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ H2 ) @ L ) )
                    @ ( vEBT_Node
                      @ ( some @ ( product_prod @ nat @ nat )
                        @ ( product_Pair @ nat @ nat @ Mi
                          @ ( if @ nat @ ( X2 = Ma )
                            @ ( if @ nat
                              @ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) )
                                = ( none @ nat ) )
                              @ Mi
                              @ ( plus_plus @ nat @ ( times_times @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ ( list_update @ vEBT_VEBT @ TreeList @ H2 @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ H2 ) @ L ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) ) ) ) ) ) ) )
                            @ Ma ) ) )
                      @ Deg
                      @ ( list_update @ vEBT_VEBT @ TreeList @ H2 @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ H2 ) @ L ) )
                      @ ( vEBT_vebt_delete @ Summary @ H2 ) )
                    @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ ( if @ nat @ ( X2 = Ma ) @ ( plus_plus @ nat @ ( times_times @ nat @ H2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ ( list_update @ vEBT_VEBT @ TreeList @ H2 @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ H2 ) @ L ) ) @ H2 ) ) ) ) @ Ma ) ) ) @ Deg @ ( list_update @ vEBT_VEBT @ TreeList @ H2 @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ H2 ) @ L ) ) @ Summary ) ) ) ) ) ) ) ) ) ).

% del_x_not_mia
thf(fact_1702_del__x__not__mi__new__node__nil,axiom,
    ! [Mi: nat,X2: 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 @ X2 )
        & ( ord_less_eq @ nat @ X2 @ Ma ) )
     => ( ( Mi != Ma )
       => ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
         => ( ( ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
              = H2 )
           => ( ( ( vEBT_VEBT_low @ X2 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
                = L )
             => ( ( Newnode
                  = ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ H2 ) @ L ) )
               => ( ( vEBT_VEBT_minNull @ Newnode )
                 => ( ( Sn
                      = ( vEBT_vebt_delete @ Summary @ H2 ) )
                   => ( ( Newlist
                        = ( list_update @ vEBT_VEBT @ TreeList @ H2 @ Newnode ) )
                     => ( ( ord_less @ nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
                       => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X2 )
                          = ( vEBT_Node
                            @ ( some @ ( product_prod @ nat @ nat )
                              @ ( product_Pair @ nat @ nat @ Mi
                                @ ( if @ nat @ ( X2 = Ma )
                                  @ ( if @ nat
                                    @ ( ( vEBT_vebt_maxt @ Sn )
                                      = ( none @ nat ) )
                                    @ Mi
                                    @ ( plus_plus @ nat @ ( times_times @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ Sn ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ Newlist @ ( the2 @ nat @ ( vEBT_vebt_maxt @ Sn ) ) ) ) ) ) )
                                  @ Ma ) ) )
                            @ Deg
                            @ Newlist
                            @ Sn ) ) ) ) ) ) ) ) ) ) ) ) ).

% del_x_not_mi_new_node_nil
thf(fact_1703_del__x__not__mi,axiom,
    ! [Mi: nat,X2: 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 @ X2 )
        & ( ord_less_eq @ nat @ X2 @ Ma ) )
     => ( ( Mi != Ma )
       => ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
         => ( ( ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
              = H2 )
           => ( ( ( vEBT_VEBT_low @ X2 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
                = L )
             => ( ( Newnode
                  = ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ H2 ) @ L ) )
               => ( ( Newlist
                    = ( list_update @ vEBT_VEBT @ TreeList @ H2 @ Newnode ) )
                 => ( ( ord_less @ nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
                   => ( ( ( vEBT_VEBT_minNull @ Newnode )
                       => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X2 )
                          = ( vEBT_Node
                            @ ( some @ ( product_prod @ nat @ nat )
                              @ ( product_Pair @ nat @ nat @ Mi
                                @ ( if @ nat @ ( X2 = Ma )
                                  @ ( if @ nat
                                    @ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) )
                                      = ( none @ nat ) )
                                    @ Mi
                                    @ ( plus_plus @ nat @ ( times_times @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ Newlist @ ( the2 @ 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 @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X2 )
                          = ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ ( if @ nat @ ( X2 = Ma ) @ ( plus_plus @ nat @ ( times_times @ nat @ H2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ Newlist @ H2 ) ) ) ) @ Ma ) ) ) @ Deg @ Newlist @ Summary ) ) ) ) ) ) ) ) ) ) ) ) ).

% del_x_not_mi
thf(fact_1704_nth__update__invalid,axiom,
    ! [A: $tType,I: nat,L: list @ A,J: nat,X2: A] :
      ( ~ ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ L ) )
     => ( ( nth @ A @ ( list_update @ A @ L @ J @ X2 ) @ I )
        = ( nth @ A @ L @ I ) ) ) ).

% nth_update_invalid
thf(fact_1705_del__x__not__mi__newnode__not__nil,axiom,
    ! [Mi: nat,X2: 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 @ X2 )
        & ( ord_less_eq @ nat @ X2 @ Ma ) )
     => ( ( Mi != Ma )
       => ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
         => ( ( ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
              = H2 )
           => ( ( ( vEBT_VEBT_low @ X2 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
                = L )
             => ( ( Newnode
                  = ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ H2 ) @ L ) )
               => ( ~ ( vEBT_VEBT_minNull @ Newnode )
                 => ( ( Newlist
                      = ( list_update @ vEBT_VEBT @ TreeList @ H2 @ Newnode ) )
                   => ( ( ord_less @ nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
                     => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X2 )
                        = ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ ( if @ nat @ ( X2 = Ma ) @ ( plus_plus @ nat @ ( times_times @ nat @ H2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ Newlist @ H2 ) ) ) ) @ Ma ) ) ) @ Deg @ Newlist @ Summary ) ) ) ) ) ) ) ) ) ) ) ).

% del_x_not_mi_newnode_not_nil
thf(fact_1706_del__x__mi__lets__in__not__minNull,axiom,
    ! [X2: 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] :
      ( ( ( X2 = Mi )
        & ( ord_less @ nat @ X2 @ Ma ) )
     => ( ( Mi != Ma )
       => ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
         => ( ( ( vEBT_VEBT_high @ Xn @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
              = H2 )
           => ( ( Xn
                = ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) )
             => ( ( ( vEBT_VEBT_low @ Xn @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
                  = L )
               => ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xn @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
                 => ( ( Newnode
                      = ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ H2 ) @ L ) )
                   => ( ( Newlist
                        = ( list_update @ vEBT_VEBT @ TreeList @ H2 @ Newnode ) )
                     => ( ~ ( vEBT_VEBT_minNull @ Newnode )
                       => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X2 )
                          = ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Xn @ ( if @ nat @ ( Xn = Ma ) @ ( plus_plus @ nat @ ( times_times @ nat @ H2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ Newlist @ H2 ) ) ) ) @ Ma ) ) ) @ Deg @ Newlist @ Summary ) ) ) ) ) ) ) ) ) ) ) ) ).

% del_x_mi_lets_in_not_minNull
thf(fact_1707_in__set__upd__eq,axiom,
    ! [A: $tType,I: nat,L: list @ A,X2: A,Y2: A] :
      ( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ L ) )
     => ( ( member @ A @ X2 @ ( set2 @ A @ ( list_update @ A @ L @ I @ Y2 ) ) )
        = ( ( X2 = Y2 )
          | ( ( member @ A @ X2 @ ( set2 @ A @ L ) )
            & ! [Y: A] : ( member @ A @ X2 @ ( set2 @ A @ ( list_update @ A @ L @ I @ Y ) ) ) ) ) ) ) ).

% in_set_upd_eq
thf(fact_1708_in__set__upd__cases,axiom,
    ! [A: $tType,X2: A,L: list @ A,I: nat,Y2: A] :
      ( ( member @ A @ X2 @ ( set2 @ A @ ( list_update @ A @ L @ I @ Y2 ) ) )
     => ( ( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ L ) )
         => ( X2 != Y2 ) )
       => ( member @ A @ X2 @ ( set2 @ A @ L ) ) ) ) ).

% in_set_upd_cases
thf(fact_1709_in__set__upd__eq__aux,axiom,
    ! [A: $tType,I: nat,L: list @ A,X2: A,Y2: A] :
      ( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ L ) )
     => ( ( member @ A @ X2 @ ( set2 @ A @ ( list_update @ A @ L @ I @ Y2 ) ) )
        = ( ( X2 = Y2 )
          | ! [Y: A] : ( member @ A @ X2 @ ( set2 @ A @ ( list_update @ A @ L @ I @ Y ) ) ) ) ) ) ).

% in_set_upd_eq_aux
thf(fact_1710_nth__list__update_H,axiom,
    ! [A: $tType,I: nat,J: nat,L: list @ A,X2: A] :
      ( ( ( ( I = J )
          & ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ L ) ) )
       => ( ( nth @ A @ ( list_update @ A @ L @ I @ X2 ) @ J )
          = X2 ) )
      & ( ~ ( ( I = J )
            & ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ L ) ) )
       => ( ( nth @ A @ ( list_update @ A @ L @ I @ X2 ) @ J )
          = ( nth @ A @ L @ J ) ) ) ) ).

% nth_list_update'
thf(fact_1711_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,
    ! [X2: nat,Mi: nat,Ma: nat,Va: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( ( ord_less @ nat @ X2 @ Mi )
          | ( ord_less @ nat @ Ma @ X2 ) )
       => ( ( vEBT_V1232361888498592333_e_t_e @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X2 )
          = ( one_one @ nat ) ) )
      & ( ~ ( ( ord_less @ nat @ X2 @ Mi )
            | ( ord_less @ nat @ Ma @ X2 ) )
       => ( ( ( ( X2 = Mi )
              & ( X2 = Ma ) )
           => ( ( vEBT_V1232361888498592333_e_t_e @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X2 )
              = ( one_one @ nat ) ) )
          & ( ~ ( ( X2 = Mi )
                & ( X2 = Ma ) )
           => ( ( vEBT_V1232361888498592333_e_t_e @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X2 )
              = ( if @ nat @ ( ord_less @ nat @ ( vEBT_VEBT_high @ ( if @ nat @ ( X2 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) ) @ ( plus_plus @ nat @ ( vEBT_V1232361888498592333_e_t_e @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( X2 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( X2 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( if @ nat @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( X2 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( X2 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_V1232361888498592333_e_t_e @ Summary @ ( vEBT_VEBT_high @ ( if @ nat @ ( X2 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( 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_1712_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,
    ! [X2: vEBT_VEBT,Xa: nat,Y2: nat] :
      ( ( ( vEBT_V1232361888498592333_e_t_e @ X2 @ Xa )
        = Y2 )
     => ( ( ? [A4: $o,B3: $o] :
              ( X2
              = ( vEBT_Leaf @ A4 @ B3 ) )
         => ( ( Xa
              = ( zero_zero @ nat ) )
           => ( Y2
             != ( one_one @ nat ) ) ) )
       => ( ( ? [A4: $o,B3: $o] :
                ( X2
                = ( vEBT_Leaf @ A4 @ B3 ) )
           => ( ( Xa
                = ( suc @ ( zero_zero @ nat ) ) )
             => ( Y2
               != ( one_one @ nat ) ) ) )
         => ( ( ? [A4: $o,B3: $o] :
                  ( X2
                  = ( vEBT_Leaf @ A4 @ B3 ) )
             => ( ? [N4: nat] :
                    ( Xa
                    = ( suc @ ( suc @ N4 ) ) )
               => ( Y2
                 != ( one_one @ nat ) ) ) )
           => ( ( ? [Deg2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
                    ( X2
                    = ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg2 @ TreeList2 @ Summary2 ) )
               => ( Y2
                 != ( one_one @ nat ) ) )
             => ( ( ? [Mi2: nat,Ma2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( X2
                      = ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ TreeList2 @ Summary2 ) )
                 => ( Y2
                   != ( one_one @ nat ) ) )
               => ( ( ? [Mi2: nat,Ma2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
                        ( X2
                        = ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( zero_zero @ nat ) ) @ TreeList2 @ Summary2 ) )
                   => ( Y2
                     != ( one_one @ nat ) ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
                        ( ( X2
                          = ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
                       => ~ ( ( ( ( ord_less @ nat @ Xa @ Mi2 )
                                | ( ord_less @ nat @ Ma2 @ Xa ) )
                             => ( Y2
                                = ( one_one @ nat ) ) )
                            & ( ~ ( ( ord_less @ nat @ Xa @ Mi2 )
                                  | ( ord_less @ nat @ Ma2 @ Xa ) )
                             => ( ( ( ( Xa = Mi2 )
                                    & ( Xa = Ma2 ) )
                                 => ( Y2
                                    = ( one_one @ nat ) ) )
                                & ( ~ ( ( Xa = Mi2 )
                                      & ( Xa = Ma2 ) )
                                 => ( Y2
                                    = ( if @ nat @ ( ord_less @ nat @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) @ ( plus_plus @ nat @ ( vEBT_V1232361888498592333_e_t_e @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( if @ nat @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_V1232361888498592333_e_t_e @ Summary2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( one_one @ nat ) ) ) @ ( one_one @ nat ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.elims
thf(fact_1713_vebt__delete_Oelims,axiom,
    ! [X2: vEBT_VEBT,Xa: nat,Y2: vEBT_VEBT] :
      ( ( ( vEBT_vebt_delete @ X2 @ Xa )
        = Y2 )
     => ( ! [A4: $o,B3: $o] :
            ( ( X2
              = ( vEBT_Leaf @ A4 @ B3 ) )
           => ( ( Xa
                = ( zero_zero @ nat ) )
             => ( Y2
               != ( vEBT_Leaf @ $false @ B3 ) ) ) )
       => ( ! [A4: $o] :
              ( ? [B3: $o] :
                  ( X2
                  = ( vEBT_Leaf @ A4 @ B3 ) )
             => ( ( Xa
                  = ( suc @ ( zero_zero @ nat ) ) )
               => ( Y2
                 != ( vEBT_Leaf @ A4 @ $false ) ) ) )
         => ( ! [A4: $o,B3: $o] :
                ( ( X2
                  = ( vEBT_Leaf @ A4 @ B3 ) )
               => ( ? [N4: nat] :
                      ( Xa
                      = ( suc @ ( suc @ N4 ) ) )
                 => ( Y2
                   != ( vEBT_Leaf @ A4 @ B3 ) ) ) )
           => ( ! [Deg2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
                  ( ( X2
                    = ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg2 @ TreeList2 @ Summary2 ) )
                 => ( Y2
                   != ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg2 @ TreeList2 @ Summary2 ) ) )
             => ( ! [Mi2: nat,Ma2: nat,TrLst: list @ vEBT_VEBT,Smry: vEBT_VEBT] :
                    ( ( X2
                      = ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ TrLst @ Smry ) )
                   => ( Y2
                     != ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ TrLst @ Smry ) ) )
               => ( ! [Mi2: nat,Ma2: nat,Tr: list @ vEBT_VEBT,Sm: vEBT_VEBT] :
                      ( ( X2
                        = ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( zero_zero @ nat ) ) @ Tr @ Sm ) )
                     => ( Y2
                       != ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( zero_zero @ nat ) ) @ Tr @ Sm ) ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
                        ( ( X2
                          = ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
                       => ~ ( ( ( ( ord_less @ nat @ Xa @ Mi2 )
                                | ( ord_less @ nat @ Ma2 @ Xa ) )
                             => ( Y2
                                = ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) ) )
                            & ( ~ ( ( ord_less @ nat @ Xa @ Mi2 )
                                  | ( ord_less @ nat @ Ma2 @ Xa ) )
                             => ( ( ( ( Xa = Mi2 )
                                    & ( Xa = Ma2 ) )
                                 => ( Y2
                                    = ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) ) )
                                & ( ~ ( ( Xa = Mi2 )
                                      & ( Xa = Ma2 ) )
                                 => ( Y2
                                    = ( if @ vEBT_VEBT @ ( ord_less @ nat @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
                                      @ ( if @ vEBT_VEBT @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
                                        @ ( vEBT_Node
                                          @ ( some @ ( product_prod @ nat @ nat )
                                            @ ( product_Pair @ nat @ nat @ ( if @ nat @ ( Xa = Mi2 ) @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ Mi2 )
                                              @ ( if @ nat
                                                @ ( ( ( Xa = Mi2 )
                                                   => ( ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) )
                                                      = Ma2 ) )
                                                  & ( ( Xa != Mi2 )
                                                   => ( Xa = Ma2 ) ) )
                                                @ ( if @ nat
                                                  @ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
                                                    = ( none @ nat ) )
                                                  @ ( if @ nat @ ( Xa = Mi2 ) @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ Mi2 )
                                                  @ ( plus_plus @ nat @ ( times_times @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ ( list_update @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) )
                                                @ Ma2 ) ) )
                                          @ ( suc @ ( suc @ Va3 ) )
                                          @ ( list_update @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
                                          @ ( vEBT_vebt_delete @ Summary2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
                                        @ ( vEBT_Node
                                          @ ( some @ ( product_prod @ nat @ nat )
                                            @ ( product_Pair @ nat @ nat @ ( if @ nat @ ( Xa = Mi2 ) @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ Mi2 )
                                              @ ( if @ nat
                                                @ ( ( ( Xa = Mi2 )
                                                   => ( ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) )
                                                      = Ma2 ) )
                                                  & ( ( Xa != Mi2 )
                                                   => ( Xa = Ma2 ) ) )
                                                @ ( plus_plus @ nat @ ( times_times @ nat @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ ( list_update @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) )
                                                @ Ma2 ) ) )
                                          @ ( suc @ ( suc @ Va3 ) )
                                          @ ( list_update @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
                                          @ Summary2 ) )
                                      @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_delete.elims
thf(fact_1714_vebt__delete_Osimps_I7_J,axiom,
    ! [X2: nat,Mi: nat,Ma: nat,Va: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( ( ord_less @ nat @ X2 @ Mi )
          | ( ord_less @ nat @ Ma @ X2 ) )
       => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X2 )
          = ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) ) )
      & ( ~ ( ( ord_less @ nat @ X2 @ Mi )
            | ( ord_less @ nat @ Ma @ X2 ) )
       => ( ( ( ( X2 = Mi )
              & ( X2 = Ma ) )
           => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X2 )
              = ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) ) )
          & ( ~ ( ( X2 = Mi )
                & ( X2 = Ma ) )
           => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X2 )
              = ( if @ vEBT_VEBT @ ( ord_less @ nat @ ( vEBT_VEBT_high @ ( if @ nat @ ( X2 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
                @ ( if @ vEBT_VEBT @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( X2 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( X2 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
                  @ ( vEBT_Node
                    @ ( some @ ( product_prod @ nat @ nat )
                      @ ( product_Pair @ nat @ nat @ ( if @ nat @ ( X2 = Mi ) @ ( if @ nat @ ( X2 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ Mi )
                        @ ( if @ nat
                          @ ( ( ( X2 = Mi )
                             => ( ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
                                = Ma ) )
                            & ( ( X2 != Mi )
                             => ( X2 = Ma ) ) )
                          @ ( if @ nat
                            @ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if @ nat @ ( X2 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
                              = ( none @ nat ) )
                            @ ( if @ nat @ ( X2 = Mi ) @ ( if @ nat @ ( X2 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ Mi )
                            @ ( plus_plus @ nat @ ( times_times @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if @ nat @ ( X2 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ ( list_update @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( X2 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( X2 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( X2 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if @ nat @ ( X2 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) )
                          @ Ma ) ) )
                    @ ( suc @ ( suc @ Va ) )
                    @ ( list_update @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( X2 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( X2 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( X2 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
                    @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if @ nat @ ( X2 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
                  @ ( vEBT_Node
                    @ ( some @ ( product_prod @ nat @ nat )
                      @ ( product_Pair @ nat @ nat @ ( if @ nat @ ( X2 = Mi ) @ ( if @ nat @ ( X2 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ Mi )
                        @ ( if @ nat
                          @ ( ( ( X2 = Mi )
                             => ( ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
                                = Ma ) )
                            & ( ( X2 != Mi )
                             => ( X2 = Ma ) ) )
                          @ ( plus_plus @ nat @ ( times_times @ nat @ ( vEBT_VEBT_high @ ( if @ nat @ ( X2 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ ( list_update @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( X2 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( X2 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( X2 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_VEBT_high @ ( if @ nat @ ( X2 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) )
                          @ Ma ) ) )
                    @ ( suc @ ( suc @ Va ) )
                    @ ( list_update @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( X2 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( X2 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( X2 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
                    @ Summary ) )
                @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) ) ) ) ) ) ) ).

% vebt_delete.simps(7)
thf(fact_1715_set__swap,axiom,
    ! [A: $tType,I: nat,Xs2: list @ A,J: nat] :
      ( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs2 ) )
     => ( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ Xs2 ) )
       => ( ( set2 @ A @ ( list_update @ A @ ( list_update @ A @ Xs2 @ I @ ( nth @ A @ Xs2 @ J ) ) @ J @ ( nth @ A @ Xs2 @ I ) ) )
          = ( set2 @ A @ Xs2 ) ) ) ) ).

% set_swap
thf(fact_1716_insert__simp__excp,axiom,
    ! [Mi: nat,Deg: nat,TreeList: list @ vEBT_VEBT,X2: nat,Ma: nat,Summary: vEBT_VEBT] :
      ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Mi @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
     => ( ( ord_less @ nat @ X2 @ Mi )
       => ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
         => ( ( X2 != Ma )
           => ( ( vEBT_vebt_insert @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X2 )
              = ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ X2 @ ( ord_max @ nat @ Mi @ Ma ) ) ) @ Deg @ ( list_update @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Mi @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_insert @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Mi @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Mi @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( if @ vEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Mi @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary @ ( vEBT_VEBT_high @ Mi @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ Summary ) ) ) ) ) ) ) ).

% insert_simp_excp
thf(fact_1717_insert__simp__norm,axiom,
    ! [X2: nat,Deg: nat,TreeList: list @ vEBT_VEBT,Mi: nat,Ma: nat,Summary: vEBT_VEBT] :
      ( ( ord_less @ nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
     => ( ( ord_less @ nat @ Mi @ X2 )
       => ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
         => ( ( X2 != Ma )
           => ( ( vEBT_vebt_insert @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X2 )
              = ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ ( ord_max @ nat @ X2 @ Ma ) ) ) @ Deg @ ( list_update @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_insert @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X2 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( if @ vEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ Summary ) ) ) ) ) ) ) ).

% insert_simp_norm
thf(fact_1718_nth__list__update__eq,axiom,
    ! [A: $tType,I: nat,Xs2: list @ A,X2: A] :
      ( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs2 ) )
     => ( ( nth @ A @ ( list_update @ A @ Xs2 @ I @ X2 ) @ I )
        = X2 ) ) ).

% nth_list_update_eq
thf(fact_1719_member__valid__both__member__options,axiom,
    ! [Tree: vEBT_VEBT,N2: nat,X2: nat] :
      ( ( vEBT_invar_vebt @ Tree @ N2 )
     => ( ( vEBT_vebt_member @ Tree @ X2 )
       => ( ( vEBT_V5719532721284313246member @ Tree @ X2 )
          | ( vEBT_VEBT_membermima @ Tree @ X2 ) ) ) ) ).

% member_valid_both_member_options
thf(fact_1720_buildup__nothing__in__leaf,axiom,
    ! [N2: nat,X2: nat] :
      ~ ( vEBT_V5719532721284313246member @ ( vEBT_vebt_buildup @ N2 ) @ X2 ) ).

% buildup_nothing_in_leaf
thf(fact_1721_both__member__options__def,axiom,
    ( vEBT_V8194947554948674370ptions
    = ( ^ [T3: vEBT_VEBT,X: nat] :
          ( ( vEBT_V5719532721284313246member @ T3 @ X )
          | ( vEBT_VEBT_membermima @ T3 @ X ) ) ) ) ).

% both_member_options_def
thf(fact_1722_length__list__update,axiom,
    ! [A: $tType,Xs2: list @ A,I: nat,X2: A] :
      ( ( size_size @ ( list @ A ) @ ( list_update @ A @ Xs2 @ I @ X2 ) )
      = ( size_size @ ( list @ A ) @ Xs2 ) ) ).

% length_list_update
thf(fact_1723_list__update__id,axiom,
    ! [A: $tType,Xs2: list @ A,I: nat] :
      ( ( list_update @ A @ Xs2 @ I @ ( nth @ A @ Xs2 @ I ) )
      = Xs2 ) ).

% list_update_id
thf(fact_1724_nth__list__update__neq,axiom,
    ! [A: $tType,I: nat,J: nat,Xs2: list @ A,X2: A] :
      ( ( I != J )
     => ( ( nth @ A @ ( list_update @ A @ Xs2 @ I @ X2 ) @ J )
        = ( nth @ A @ Xs2 @ J ) ) ) ).

% nth_list_update_neq
thf(fact_1725_max__Suc__Suc,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_max @ nat @ ( suc @ M ) @ ( suc @ N2 ) )
      = ( suc @ ( ord_max @ nat @ M @ N2 ) ) ) ).

% max_Suc_Suc
thf(fact_1726_max__0R,axiom,
    ! [N2: nat] :
      ( ( ord_max @ nat @ N2 @ ( zero_zero @ nat ) )
      = N2 ) ).

% max_0R
thf(fact_1727_max__0L,axiom,
    ! [N2: nat] :
      ( ( ord_max @ nat @ ( zero_zero @ nat ) @ N2 )
      = N2 ) ).

% max_0L
thf(fact_1728_max__nat_Oright__neutral,axiom,
    ! [A2: nat] :
      ( ( ord_max @ nat @ A2 @ ( zero_zero @ nat ) )
      = A2 ) ).

% max_nat.right_neutral
thf(fact_1729_max__nat_Oneutr__eq__iff,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ( zero_zero @ nat )
        = ( ord_max @ nat @ A2 @ B2 ) )
      = ( ( A2
          = ( zero_zero @ nat ) )
        & ( B2
          = ( zero_zero @ nat ) ) ) ) ).

% max_nat.neutr_eq_iff
thf(fact_1730_max__nat_Oleft__neutral,axiom,
    ! [A2: nat] :
      ( ( ord_max @ nat @ ( zero_zero @ nat ) @ A2 )
      = A2 ) ).

% max_nat.left_neutral
thf(fact_1731_max__nat_Oeq__neutr__iff,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ( ord_max @ nat @ A2 @ B2 )
        = ( zero_zero @ nat ) )
      = ( ( A2
          = ( zero_zero @ nat ) )
        & ( B2
          = ( zero_zero @ nat ) ) ) ) ).

% max_nat.eq_neutr_iff
thf(fact_1732_max__0__1_I4_J,axiom,
    ! [A: $tType] :
      ( ( linord181362715937106298miring @ A )
     => ! [X2: num] :
          ( ( ord_max @ A @ ( numeral_numeral @ A @ X2 ) @ ( zero_zero @ A ) )
          = ( numeral_numeral @ A @ X2 ) ) ) ).

% max_0_1(4)
thf(fact_1733_max__0__1_I3_J,axiom,
    ! [A: $tType] :
      ( ( linord181362715937106298miring @ A )
     => ! [X2: num] :
          ( ( ord_max @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ X2 ) )
          = ( numeral_numeral @ A @ X2 ) ) ) ).

% max_0_1(3)
thf(fact_1734_max__number__of_I1_J,axiom,
    ! [A: $tType] :
      ( ( ( numeral @ A )
        & ( ord @ A ) )
     => ! [U: num,V: num] :
          ( ( ( ord_less_eq @ A @ ( numeral_numeral @ A @ U ) @ ( numeral_numeral @ A @ V ) )
           => ( ( ord_max @ A @ ( numeral_numeral @ A @ U ) @ ( numeral_numeral @ A @ V ) )
              = ( numeral_numeral @ A @ V ) ) )
          & ( ~ ( ord_less_eq @ A @ ( numeral_numeral @ A @ U ) @ ( numeral_numeral @ A @ V ) )
           => ( ( ord_max @ A @ ( numeral_numeral @ A @ U ) @ ( numeral_numeral @ A @ V ) )
              = ( numeral_numeral @ A @ U ) ) ) ) ) ).

% max_number_of(1)
thf(fact_1735_max__0__1_I2_J,axiom,
    ! [A: $tType] :
      ( ( linord181362715937106298miring @ A )
     => ( ( ord_max @ A @ ( one_one @ A ) @ ( zero_zero @ A ) )
        = ( one_one @ A ) ) ) ).

% max_0_1(2)
thf(fact_1736_max__0__1_I1_J,axiom,
    ! [A: $tType] :
      ( ( linord181362715937106298miring @ A )
     => ( ( ord_max @ A @ ( zero_zero @ A ) @ ( one_one @ A ) )
        = ( one_one @ A ) ) ) ).

% max_0_1(1)
thf(fact_1737_max__0__1_I5_J,axiom,
    ! [A: $tType] :
      ( ( linord181362715937106298miring @ A )
     => ! [X2: num] :
          ( ( ord_max @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ X2 ) )
          = ( numeral_numeral @ A @ X2 ) ) ) ).

% max_0_1(5)
thf(fact_1738_max__0__1_I6_J,axiom,
    ! [A: $tType] :
      ( ( linord181362715937106298miring @ A )
     => ! [X2: num] :
          ( ( ord_max @ A @ ( numeral_numeral @ A @ X2 ) @ ( one_one @ A ) )
          = ( numeral_numeral @ A @ X2 ) ) ) ).

% max_0_1(6)
thf(fact_1739_list__update__beyond,axiom,
    ! [A: $tType,Xs2: list @ A,I: nat,X2: A] :
      ( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs2 ) @ I )
     => ( ( list_update @ A @ Xs2 @ I @ X2 )
        = Xs2 ) ) ).

% list_update_beyond
thf(fact_1740_of__nat__max,axiom,
    ! [A: $tType] :
      ( ( linord181362715937106298miring @ A )
     => ! [X2: nat,Y2: nat] :
          ( ( semiring_1_of_nat @ A @ ( ord_max @ nat @ X2 @ Y2 ) )
          = ( ord_max @ A @ ( semiring_1_of_nat @ A @ X2 ) @ ( semiring_1_of_nat @ A @ Y2 ) ) ) ) ).

% of_nat_max
thf(fact_1741_max__add__distrib__right,axiom,
    ! [A: $tType] :
      ( ( ordere2412721322843649153imp_le @ A )
     => ! [X2: A,Y2: A,Z: A] :
          ( ( plus_plus @ A @ X2 @ ( ord_max @ A @ Y2 @ Z ) )
          = ( ord_max @ A @ ( plus_plus @ A @ X2 @ Y2 ) @ ( plus_plus @ A @ X2 @ Z ) ) ) ) ).

% max_add_distrib_right
thf(fact_1742_max__add__distrib__left,axiom,
    ! [A: $tType] :
      ( ( ordere2412721322843649153imp_le @ A )
     => ! [X2: A,Y2: A,Z: A] :
          ( ( plus_plus @ A @ ( ord_max @ A @ X2 @ Y2 ) @ Z )
          = ( ord_max @ A @ ( plus_plus @ A @ X2 @ Z ) @ ( plus_plus @ A @ Y2 @ Z ) ) ) ) ).

% max_add_distrib_left
thf(fact_1743_max__diff__distrib__left,axiom,
    ! [A: $tType] :
      ( ( ordered_ab_group_add @ A )
     => ! [X2: A,Y2: A,Z: A] :
          ( ( minus_minus @ A @ ( ord_max @ A @ X2 @ Y2 ) @ Z )
          = ( ord_max @ A @ ( minus_minus @ A @ X2 @ Z ) @ ( minus_minus @ A @ Y2 @ Z ) ) ) ) ).

% max_diff_distrib_left
thf(fact_1744_nat__add__max__left,axiom,
    ! [M: nat,N2: nat,Q2: nat] :
      ( ( plus_plus @ nat @ ( ord_max @ nat @ M @ N2 ) @ Q2 )
      = ( ord_max @ nat @ ( plus_plus @ nat @ M @ Q2 ) @ ( plus_plus @ nat @ N2 @ Q2 ) ) ) ).

% nat_add_max_left
thf(fact_1745_nat__add__max__right,axiom,
    ! [M: nat,N2: nat,Q2: nat] :
      ( ( plus_plus @ nat @ M @ ( ord_max @ nat @ N2 @ Q2 ) )
      = ( ord_max @ nat @ ( plus_plus @ nat @ M @ N2 ) @ ( plus_plus @ nat @ M @ Q2 ) ) ) ).

% nat_add_max_right
thf(fact_1746_nat__mult__max__right,axiom,
    ! [M: nat,N2: nat,Q2: nat] :
      ( ( times_times @ nat @ M @ ( ord_max @ nat @ N2 @ Q2 ) )
      = ( ord_max @ nat @ ( times_times @ nat @ M @ N2 ) @ ( times_times @ nat @ M @ Q2 ) ) ) ).

% nat_mult_max_right
thf(fact_1747_nat__mult__max__left,axiom,
    ! [M: nat,N2: nat,Q2: nat] :
      ( ( times_times @ nat @ ( ord_max @ nat @ M @ N2 ) @ Q2 )
      = ( ord_max @ nat @ ( times_times @ nat @ M @ Q2 ) @ ( times_times @ nat @ N2 @ Q2 ) ) ) ).

% nat_mult_max_left
thf(fact_1748_nat__minus__add__max,axiom,
    ! [N2: nat,M: nat] :
      ( ( plus_plus @ nat @ ( minus_minus @ nat @ N2 @ M ) @ M )
      = ( ord_max @ nat @ N2 @ M ) ) ).

% nat_minus_add_max
thf(fact_1749_VEBT__internal_Onaive__member_Osimps_I2_J,axiom,
    ! [Uu: option @ ( product_prod @ nat @ nat ),Uv: list @ vEBT_VEBT,Uw: vEBT_VEBT,Ux: nat] :
      ~ ( vEBT_V5719532721284313246member @ ( vEBT_Node @ Uu @ ( zero_zero @ nat ) @ Uv @ Uw ) @ Ux ) ).

% VEBT_internal.naive_member.simps(2)
thf(fact_1750_neq__if__length__neq,axiom,
    ! [A: $tType,Xs2: list @ A,Ys: list @ A] :
      ( ( ( size_size @ ( list @ A ) @ Xs2 )
       != ( size_size @ ( list @ A ) @ Ys ) )
     => ( Xs2 != Ys ) ) ).

% neq_if_length_neq
thf(fact_1751_Ex__list__of__length,axiom,
    ! [A: $tType,N2: nat] :
    ? [Xs3: list @ A] :
      ( ( size_size @ ( list @ A ) @ Xs3 )
      = N2 ) ).

% Ex_list_of_length
thf(fact_1752_VEBT__internal_Onaive__member_Osimps_I1_J,axiom,
    ! [A2: $o,B2: $o,X2: nat] :
      ( ( vEBT_V5719532721284313246member @ ( vEBT_Leaf @ A2 @ B2 ) @ X2 )
      = ( ( ( X2
            = ( zero_zero @ nat ) )
         => A2 )
        & ( ( X2
           != ( zero_zero @ nat ) )
         => ( ( ( X2
                = ( one_one @ nat ) )
             => B2 )
            & ( X2
              = ( one_one @ nat ) ) ) ) ) ) ).

% VEBT_internal.naive_member.simps(1)
thf(fact_1753_length__induct,axiom,
    ! [A: $tType,P: ( list @ A ) > $o,Xs2: list @ A] :
      ( ! [Xs3: list @ A] :
          ( ! [Ys2: list @ A] :
              ( ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Ys2 ) @ ( size_size @ ( list @ A ) @ Xs3 ) )
             => ( P @ Ys2 ) )
         => ( P @ Xs3 ) )
     => ( P @ Xs2 ) ) ).

% length_induct
thf(fact_1754_subset__code_I1_J,axiom,
    ! [A: $tType,Xs2: list @ A,B6: set @ A] :
      ( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs2 ) @ B6 )
      = ( ! [X: A] :
            ( ( member @ A @ X @ ( set2 @ A @ Xs2 ) )
           => ( member @ A @ X @ B6 ) ) ) ) ).

% subset_code(1)
thf(fact_1755_list__eq__iff__nth__eq,axiom,
    ! [A: $tType] :
      ( ( ^ [Y5: list @ A,Z4: list @ A] : Y5 = Z4 )
      = ( ^ [Xs: list @ A,Ys3: list @ A] :
            ( ( ( size_size @ ( list @ A ) @ Xs )
              = ( size_size @ ( list @ A ) @ Ys3 ) )
            & ! [I2: nat] :
                ( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ Xs ) )
               => ( ( nth @ A @ Xs @ I2 )
                  = ( nth @ A @ Ys3 @ I2 ) ) ) ) ) ) ).

% list_eq_iff_nth_eq
thf(fact_1756_Skolem__list__nth,axiom,
    ! [A: $tType,K: nat,P: nat > A > $o] :
      ( ( ! [I2: nat] :
            ( ( ord_less @ nat @ I2 @ K )
           => ? [X5: A] : ( P @ I2 @ X5 ) ) )
      = ( ? [Xs: list @ A] :
            ( ( ( size_size @ ( list @ A ) @ Xs )
              = K )
            & ! [I2: nat] :
                ( ( ord_less @ nat @ I2 @ K )
               => ( P @ I2 @ ( nth @ A @ Xs @ I2 ) ) ) ) ) ) ).

% Skolem_list_nth
thf(fact_1757_nth__equalityI,axiom,
    ! [A: $tType,Xs2: list @ A,Ys: list @ A] :
      ( ( ( size_size @ ( list @ A ) @ Xs2 )
        = ( size_size @ ( list @ A ) @ Ys ) )
     => ( ! [I3: nat] :
            ( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ Xs2 ) )
           => ( ( nth @ A @ Xs2 @ I3 )
              = ( nth @ A @ Ys @ I3 ) ) )
       => ( Xs2 = Ys ) ) ) ).

% nth_equalityI
thf(fact_1758_all__nat__less,axiom,
    ! [N2: nat,P: nat > $o] :
      ( ( ! [M7: nat] :
            ( ( ord_less_eq @ nat @ M7 @ N2 )
           => ( P @ M7 ) ) )
      = ( ! [X: nat] :
            ( ( member @ nat @ X @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N2 ) )
           => ( P @ X ) ) ) ) ).

% all_nat_less
thf(fact_1759_ex__nat__less,axiom,
    ! [N2: nat,P: nat > $o] :
      ( ( ? [M7: nat] :
            ( ( ord_less_eq @ nat @ M7 @ N2 )
            & ( P @ M7 ) ) )
      = ( ? [X: nat] :
            ( ( member @ nat @ X @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N2 ) )
            & ( P @ X ) ) ) ) ).

% ex_nat_less
thf(fact_1760_set__update__subsetI,axiom,
    ! [A: $tType,Xs2: list @ A,A3: set @ A,X2: A,I: nat] :
      ( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs2 ) @ A3 )
     => ( ( member @ A @ X2 @ A3 )
       => ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ ( list_update @ A @ Xs2 @ I @ X2 ) ) @ A3 ) ) ) ).

% set_update_subsetI
thf(fact_1761_vebt__delete_Osimps_I3_J,axiom,
    ! [A2: $o,B2: $o,N2: nat] :
      ( ( vEBT_vebt_delete @ ( vEBT_Leaf @ A2 @ B2 ) @ ( suc @ ( suc @ N2 ) ) )
      = ( vEBT_Leaf @ A2 @ B2 ) ) ).

% vebt_delete.simps(3)
thf(fact_1762_vebt__delete_Osimps_I4_J,axiom,
    ! [Deg: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,Uu: nat] :
      ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg @ TreeList @ Summary ) @ Uu )
      = ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg @ TreeList @ Summary ) ) ).

% vebt_delete.simps(4)
thf(fact_1763_length__pos__if__in__set,axiom,
    ! [A: $tType,X2: A,Xs2: list @ A] :
      ( ( member @ A @ X2 @ ( set2 @ A @ Xs2 ) )
     => ( ord_less @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ A ) @ Xs2 ) ) ) ).

% length_pos_if_in_set
thf(fact_1764_all__set__conv__all__nth,axiom,
    ! [A: $tType,Xs2: list @ A,P: A > $o] :
      ( ( ! [X: A] :
            ( ( member @ A @ X @ ( set2 @ A @ Xs2 ) )
           => ( P @ X ) ) )
      = ( ! [I2: nat] :
            ( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ Xs2 ) )
           => ( P @ ( nth @ A @ Xs2 @ I2 ) ) ) ) ) ).

% all_set_conv_all_nth
thf(fact_1765_all__nth__imp__all__set,axiom,
    ! [A: $tType,Xs2: list @ A,P: A > $o,X2: A] :
      ( ! [I3: nat] :
          ( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ Xs2 ) )
         => ( P @ ( nth @ A @ Xs2 @ I3 ) ) )
     => ( ( member @ A @ X2 @ ( set2 @ A @ Xs2 ) )
       => ( P @ X2 ) ) ) ).

% all_nth_imp_all_set
thf(fact_1766_in__set__conv__nth,axiom,
    ! [A: $tType,X2: A,Xs2: list @ A] :
      ( ( member @ A @ X2 @ ( set2 @ A @ Xs2 ) )
      = ( ? [I2: nat] :
            ( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ Xs2 ) )
            & ( ( nth @ A @ Xs2 @ I2 )
              = X2 ) ) ) ) ).

% in_set_conv_nth
thf(fact_1767_list__ball__nth,axiom,
    ! [A: $tType,N2: nat,Xs2: list @ A,P: A > $o] :
      ( ( ord_less @ nat @ N2 @ ( size_size @ ( list @ A ) @ Xs2 ) )
     => ( ! [X3: A] :
            ( ( member @ A @ X3 @ ( set2 @ A @ Xs2 ) )
           => ( P @ X3 ) )
       => ( P @ ( nth @ A @ Xs2 @ N2 ) ) ) ) ).

% list_ball_nth
thf(fact_1768_nth__mem,axiom,
    ! [A: $tType,N2: nat,Xs2: list @ A] :
      ( ( ord_less @ nat @ N2 @ ( size_size @ ( list @ A ) @ Xs2 ) )
     => ( member @ A @ ( nth @ A @ Xs2 @ N2 ) @ ( set2 @ A @ Xs2 ) ) ) ).

% nth_mem
thf(fact_1769_set__update__memI,axiom,
    ! [A: $tType,N2: nat,Xs2: list @ A,X2: A] :
      ( ( ord_less @ nat @ N2 @ ( size_size @ ( list @ A ) @ Xs2 ) )
     => ( member @ A @ X2 @ ( set2 @ A @ ( list_update @ A @ Xs2 @ N2 @ X2 ) ) ) ) ).

% set_update_memI
thf(fact_1770_list__update__same__conv,axiom,
    ! [A: $tType,I: nat,Xs2: list @ A,X2: A] :
      ( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs2 ) )
     => ( ( ( list_update @ A @ Xs2 @ I @ X2 )
          = Xs2 )
        = ( ( nth @ A @ Xs2 @ I )
          = X2 ) ) ) ).

% list_update_same_conv
thf(fact_1771_nth__list__update,axiom,
    ! [A: $tType,I: nat,Xs2: list @ A,J: nat,X2: A] :
      ( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs2 ) )
     => ( ( ( I = J )
         => ( ( nth @ A @ ( list_update @ A @ Xs2 @ I @ X2 ) @ J )
            = X2 ) )
        & ( ( I != J )
         => ( ( nth @ A @ ( list_update @ A @ Xs2 @ I @ X2 ) @ J )
            = ( nth @ A @ Xs2 @ J ) ) ) ) ) ).

% nth_list_update
thf(fact_1772_vebt__delete_Osimps_I2_J,axiom,
    ! [A2: $o,B2: $o] :
      ( ( vEBT_vebt_delete @ ( vEBT_Leaf @ A2 @ B2 ) @ ( suc @ ( zero_zero @ nat ) ) )
      = ( vEBT_Leaf @ A2 @ $false ) ) ).

% vebt_delete.simps(2)
thf(fact_1773_vebt__insert_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,Va: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,X2: nat] :
      ( ( vEBT_vebt_insert @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X2 )
      = ( if @ vEBT_VEBT
        @ ( ( ord_less @ nat @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ X2 @ Mi ) @ Mi @ X2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
          & ~ ( ( X2 = Mi )
              | ( X2 = Ma ) ) )
        @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ ( if @ nat @ ( ord_less @ nat @ X2 @ Mi ) @ X2 @ Mi ) @ ( ord_max @ nat @ ( if @ nat @ ( ord_less @ nat @ X2 @ Mi ) @ Mi @ X2 ) @ Ma ) ) ) @ ( suc @ ( suc @ Va ) ) @ ( list_update @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ X2 @ Mi ) @ Mi @ X2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_insert @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ X2 @ Mi ) @ Mi @ X2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( ord_less @ nat @ X2 @ Mi ) @ Mi @ X2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( if @ vEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ X2 @ Mi ) @ Mi @ X2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ X2 @ Mi ) @ Mi @ X2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ Summary ) )
        @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) ) ) ).

% vebt_insert.simps(5)
thf(fact_1774_VEBT__internal_Onaive__member_Osimps_I3_J,axiom,
    ! [Uy: option @ ( product_prod @ nat @ nat ),V: nat,TreeList: list @ vEBT_VEBT,S4: vEBT_VEBT,X2: nat] :
      ( ( vEBT_V5719532721284313246member @ ( vEBT_Node @ Uy @ ( suc @ V ) @ TreeList @ S4 ) @ X2 )
      = ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ ( suc @ V ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
         => ( vEBT_V5719532721284313246member @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ ( suc @ V ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X2 @ ( divide_divide @ nat @ ( suc @ V ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
        & ( ord_less @ nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ ( suc @ V ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) ) ) ) ).

% VEBT_internal.naive_member.simps(3)
thf(fact_1775_vebt__delete_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,TrLst2: list @ vEBT_VEBT,Smry2: vEBT_VEBT,X2: nat] :
      ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( zero_zero @ nat ) @ TrLst2 @ Smry2 ) @ X2 )
      = ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( zero_zero @ nat ) @ TrLst2 @ Smry2 ) ) ).

% vebt_delete.simps(5)
thf(fact_1776_vebt__insert_Oelims,axiom,
    ! [X2: vEBT_VEBT,Xa: nat,Y2: vEBT_VEBT] :
      ( ( ( vEBT_vebt_insert @ X2 @ Xa )
        = Y2 )
     => ( ! [A4: $o,B3: $o] :
            ( ( X2
              = ( vEBT_Leaf @ A4 @ B3 ) )
           => ~ ( ( ( Xa
                    = ( zero_zero @ nat ) )
                 => ( Y2
                    = ( vEBT_Leaf @ $true @ B3 ) ) )
                & ( ( Xa
                   != ( zero_zero @ nat ) )
                 => ( ( ( Xa
                        = ( one_one @ nat ) )
                     => ( Y2
                        = ( vEBT_Leaf @ A4 @ $true ) ) )
                    & ( ( Xa
                       != ( one_one @ nat ) )
                     => ( Y2
                        = ( vEBT_Leaf @ A4 @ B3 ) ) ) ) ) ) )
       => ( ! [Info2: option @ ( product_prod @ nat @ nat ),Ts2: list @ vEBT_VEBT,S: vEBT_VEBT] :
              ( ( X2
                = ( vEBT_Node @ Info2 @ ( zero_zero @ nat ) @ Ts2 @ S ) )
             => ( Y2
               != ( vEBT_Node @ Info2 @ ( zero_zero @ nat ) @ Ts2 @ S ) ) )
         => ( ! [Info2: option @ ( product_prod @ nat @ nat ),Ts2: list @ vEBT_VEBT,S: vEBT_VEBT] :
                ( ( X2
                  = ( vEBT_Node @ Info2 @ ( suc @ ( zero_zero @ nat ) ) @ Ts2 @ S ) )
               => ( Y2
                 != ( vEBT_Node @ Info2 @ ( suc @ ( zero_zero @ nat ) ) @ Ts2 @ S ) ) )
           => ( ! [V3: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
                  ( ( X2
                    = ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ V3 ) ) @ TreeList2 @ Summary2 ) )
                 => ( Y2
                   != ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Xa @ Xa ) ) @ ( suc @ ( suc @ V3 ) ) @ TreeList2 @ Summary2 ) ) )
             => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
                    ( ( X2
                      = ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
                   => ( Y2
                     != ( if @ vEBT_VEBT
                        @ ( ( ord_less @ nat @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
                          & ~ ( ( Xa = Mi2 )
                              | ( Xa = Ma2 ) ) )
                        @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi2 ) @ Xa @ Mi2 ) @ ( ord_max @ nat @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ Ma2 ) ) ) @ ( suc @ ( suc @ Va3 ) ) @ ( list_update @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_insert @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( if @ vEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ Summary2 ) )
                        @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) ) ) ) ) ) ) ) ) ).

% vebt_insert.elims
thf(fact_1777_VEBT__internal_Onaive__member_Oelims_I1_J,axiom,
    ! [X2: vEBT_VEBT,Xa: nat,Y2: $o] :
      ( ( ( vEBT_V5719532721284313246member @ X2 @ Xa )
        = Y2 )
     => ( ! [A4: $o,B3: $o] :
            ( ( X2
              = ( vEBT_Leaf @ A4 @ B3 ) )
           => ( Y2
              = ( ~ ( ( ( Xa
                        = ( zero_zero @ nat ) )
                     => A4 )
                    & ( ( Xa
                       != ( zero_zero @ nat ) )
                     => ( ( ( Xa
                            = ( one_one @ nat ) )
                         => B3 )
                        & ( Xa
                          = ( one_one @ nat ) ) ) ) ) ) ) )
       => ( ( ? [Uu2: option @ ( product_prod @ nat @ nat ),Uv2: list @ vEBT_VEBT,Uw2: vEBT_VEBT] :
                ( X2
                = ( vEBT_Node @ Uu2 @ ( zero_zero @ nat ) @ Uv2 @ Uw2 ) )
           => Y2 )
         => ~ ! [Uy2: option @ ( product_prod @ nat @ nat ),V3: nat,TreeList2: list @ vEBT_VEBT] :
                ( ? [S: vEBT_VEBT] :
                    ( X2
                    = ( vEBT_Node @ Uy2 @ ( suc @ V3 ) @ TreeList2 @ S ) )
               => ( Y2
                  = ( ~ ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
                         => ( vEBT_V5719532721284313246member @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
                        & ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.elims(1)
thf(fact_1778_VEBT__internal_Onaive__member_Oelims_I2_J,axiom,
    ! [X2: vEBT_VEBT,Xa: nat] :
      ( ( vEBT_V5719532721284313246member @ X2 @ Xa )
     => ( ! [A4: $o,B3: $o] :
            ( ( X2
              = ( vEBT_Leaf @ A4 @ B3 ) )
           => ~ ( ( ( Xa
                    = ( zero_zero @ nat ) )
                 => A4 )
                & ( ( Xa
                   != ( zero_zero @ nat ) )
                 => ( ( ( Xa
                        = ( one_one @ nat ) )
                     => B3 )
                    & ( Xa
                      = ( one_one @ nat ) ) ) ) ) )
       => ~ ! [Uy2: option @ ( product_prod @ nat @ nat ),V3: nat,TreeList2: list @ vEBT_VEBT] :
              ( ? [S: vEBT_VEBT] :
                  ( X2
                  = ( vEBT_Node @ Uy2 @ ( suc @ V3 ) @ TreeList2 @ S ) )
             => ~ ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
                   => ( vEBT_V5719532721284313246member @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
                  & ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) ) ).

% VEBT_internal.naive_member.elims(2)
thf(fact_1779_VEBT__internal_Onaive__member_Oelims_I3_J,axiom,
    ! [X2: vEBT_VEBT,Xa: nat] :
      ( ~ ( vEBT_V5719532721284313246member @ X2 @ Xa )
     => ( ! [A4: $o,B3: $o] :
            ( ( X2
              = ( vEBT_Leaf @ A4 @ B3 ) )
           => ( ( ( Xa
                  = ( zero_zero @ nat ) )
               => A4 )
              & ( ( Xa
                 != ( zero_zero @ nat ) )
               => ( ( ( Xa
                      = ( one_one @ nat ) )
                   => B3 )
                  & ( Xa
                    = ( one_one @ nat ) ) ) ) ) )
       => ( ! [Uu2: option @ ( product_prod @ nat @ nat ),Uv2: list @ vEBT_VEBT,Uw2: vEBT_VEBT] :
              ( X2
             != ( vEBT_Node @ Uu2 @ ( zero_zero @ nat ) @ Uv2 @ Uw2 ) )
         => ~ ! [Uy2: option @ ( product_prod @ nat @ nat ),V3: nat,TreeList2: list @ vEBT_VEBT] :
                ( ? [S: vEBT_VEBT] :
                    ( X2
                    = ( vEBT_Node @ Uy2 @ ( suc @ V3 ) @ TreeList2 @ S ) )
               => ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
                   => ( vEBT_V5719532721284313246member @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
                  & ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.elims(3)
thf(fact_1780_vebt__delete_Osimps_I6_J,axiom,
    ! [Mi: nat,Ma: nat,Tr2: list @ vEBT_VEBT,Sm2: vEBT_VEBT,X2: nat] :
      ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( zero_zero @ nat ) ) @ Tr2 @ Sm2 ) @ X2 )
      = ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( zero_zero @ nat ) ) @ Tr2 @ Sm2 ) ) ).

% vebt_delete.simps(6)
thf(fact_1781_httI,axiom,
    ! [A: $tType,P: assn,C2: heap_Time_Heap @ A,Q: A > assn,T2: nat] :
      ( ( hoare_hoare_triple @ A @ P @ C2 @ Q )
     => ( ! [H3: heap_ext @ product_unit,As: set @ nat] :
            ( ( rep_assn @ P @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ As ) )
           => ( ord_less_eq @ nat @ ( time_time @ A @ C2 @ H3 ) @ T2 ) )
       => ( time_htt @ A @ P @ C2 @ Q @ T2 ) ) ) ).

% httI
thf(fact_1782_htt__def,axiom,
    ! [A: $tType] :
      ( ( time_htt @ A )
      = ( ^ [P3: assn,C4: heap_Time_Heap @ A,Q4: A > assn,T3: nat] :
            ( ( hoare_hoare_triple @ A @ P3 @ C4 @ Q4 )
            & ! [H: heap_ext @ product_unit,As2: set @ nat] :
                ( ( rep_assn @ P3 @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As2 ) )
               => ( ord_less_eq @ nat @ ( time_time @ A @ C4 @ H ) @ T3 ) ) ) ) ) ).

% htt_def
thf(fact_1783_succ__empty,axiom,
    ! [T2: vEBT_VEBT,N2: nat,X2: nat] :
      ( ( vEBT_invar_vebt @ T2 @ N2 )
     => ( ( ( vEBT_vebt_succ @ T2 @ X2 )
          = ( none @ nat ) )
        = ( ( collect @ nat
            @ ^ [Y: nat] :
                ( ( vEBT_vebt_member @ T2 @ Y )
                & ( ord_less @ nat @ X2 @ Y ) ) )
          = ( bot_bot @ ( set @ nat ) ) ) ) ) ).

% succ_empty
thf(fact_1784_pred__empty,axiom,
    ! [T2: vEBT_VEBT,N2: nat,X2: nat] :
      ( ( vEBT_invar_vebt @ T2 @ N2 )
     => ( ( ( vEBT_vebt_pred @ T2 @ X2 )
          = ( none @ nat ) )
        = ( ( collect @ nat
            @ ^ [Y: nat] :
                ( ( vEBT_vebt_member @ T2 @ Y )
                & ( ord_less @ nat @ Y @ X2 ) ) )
          = ( bot_bot @ ( set @ nat ) ) ) ) ) ).

% pred_empty
thf(fact_1785_vebt__succ_Opelims,axiom,
    ! [X2: vEBT_VEBT,Xa: nat,Y2: option @ nat] :
      ( ( ( vEBT_vebt_succ @ X2 @ Xa )
        = Y2 )
     => ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_succ_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X2 @ Xa ) )
       => ( ! [Uu2: $o,B3: $o] :
              ( ( X2
                = ( vEBT_Leaf @ Uu2 @ B3 ) )
             => ( ( Xa
                  = ( zero_zero @ nat ) )
               => ( ( ( B3
                     => ( Y2
                        = ( some @ nat @ ( one_one @ nat ) ) ) )
                    & ( ~ B3
                     => ( Y2
                        = ( none @ nat ) ) ) )
                 => ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_succ_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ Uu2 @ B3 ) @ ( zero_zero @ nat ) ) ) ) ) )
         => ( ! [Uv2: $o,Uw2: $o] :
                ( ( X2
                  = ( vEBT_Leaf @ Uv2 @ Uw2 ) )
               => ! [N4: nat] :
                    ( ( Xa
                      = ( suc @ N4 ) )
                   => ( ( Y2
                        = ( none @ nat ) )
                     => ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_succ_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ Uv2 @ Uw2 ) @ ( suc @ N4 ) ) ) ) ) )
           => ( ! [Ux2: nat,Uy2: list @ vEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( X2
                    = ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Ux2 @ Uy2 @ Uz2 ) )
                 => ( ( Y2
                      = ( none @ nat ) )
                   => ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_succ_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Ux2 @ Uy2 @ Uz2 ) @ Xa ) ) ) )
             => ( ! [V3: product_prod @ nat @ nat,Vc: list @ vEBT_VEBT,Vd2: vEBT_VEBT] :
                    ( ( X2
                      = ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( zero_zero @ nat ) @ Vc @ Vd2 ) )
                   => ( ( Y2
                        = ( none @ nat ) )
                     => ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_succ_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( zero_zero @ nat ) @ Vc @ Vd2 ) @ Xa ) ) ) )
               => ( ! [V3: product_prod @ nat @ nat,Vg2: list @ vEBT_VEBT,Vh2: vEBT_VEBT] :
                      ( ( X2
                        = ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vg2 @ Vh2 ) )
                     => ( ( Y2
                          = ( none @ nat ) )
                       => ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_succ_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vg2 @ Vh2 ) @ Xa ) ) ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
                        ( ( X2
                          = ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
                       => ( ( ( ( ord_less @ nat @ Xa @ Mi2 )
                             => ( Y2
                                = ( some @ nat @ Mi2 ) ) )
                            & ( ~ ( ord_less @ nat @ Xa @ Mi2 )
                             => ( Y2
                                = ( if @ ( option @ nat ) @ ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
                                  @ ( if @ ( option @ nat )
                                    @ ( ( ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
                                       != ( none @ nat ) )
                                      & ( vEBT_VEBT_less @ ( some @ nat @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) )
                                    @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( some @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_succ @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
                                    @ ( if @ ( option @ nat )
                                      @ ( ( vEBT_vebt_succ @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
                                        = ( none @ nat ) )
                                      @ ( none @ nat )
                                      @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_succ @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_succ @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) )
                                  @ ( none @ nat ) ) ) ) )
                         => ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_succ_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) @ Xa ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_succ.pelims
thf(fact_1786_psubsetI,axiom,
    ! [A: $tType,A3: set @ A,B6: set @ A] :
      ( ( ord_less_eq @ ( set @ A ) @ A3 @ B6 )
     => ( ( A3 != B6 )
       => ( ord_less @ ( set @ A ) @ A3 @ B6 ) ) ) ).

% psubsetI
thf(fact_1787_int__ops_I6_J,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ( ord_less @ int @ ( semiring_1_of_nat @ int @ A2 ) @ ( semiring_1_of_nat @ int @ B2 ) )
       => ( ( semiring_1_of_nat @ int @ ( minus_minus @ nat @ A2 @ B2 ) )
          = ( zero_zero @ int ) ) )
      & ( ~ ( ord_less @ int @ ( semiring_1_of_nat @ int @ A2 ) @ ( semiring_1_of_nat @ int @ B2 ) )
       => ( ( semiring_1_of_nat @ int @ ( minus_minus @ nat @ A2 @ B2 ) )
          = ( minus_minus @ int @ ( semiring_1_of_nat @ int @ A2 ) @ ( semiring_1_of_nat @ int @ B2 ) ) ) ) ) ).

% int_ops(6)
thf(fact_1788_buildup__gives__empty,axiom,
    ! [N2: nat] :
      ( ( vEBT_VEBT_set_vebt @ ( vEBT_vebt_buildup @ N2 ) )
      = ( bot_bot @ ( set @ nat ) ) ) ).

% buildup_gives_empty
thf(fact_1789_verit__eq__simplify_I8_J,axiom,
    ! [X23: num,Y23: num] :
      ( ( ( bit0 @ X23 )
        = ( bit0 @ Y23 ) )
      = ( X23 = Y23 ) ) ).

% verit_eq_simplify(8)
thf(fact_1790_Diff__cancel,axiom,
    ! [A: $tType,A3: set @ A] :
      ( ( minus_minus @ ( set @ A ) @ A3 @ A3 )
      = ( bot_bot @ ( set @ A ) ) ) ).

% Diff_cancel
thf(fact_1791_empty__Diff,axiom,
    ! [A: $tType,A3: set @ A] :
      ( ( minus_minus @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ A3 )
      = ( bot_bot @ ( set @ A ) ) ) ).

% empty_Diff
thf(fact_1792_Diff__empty,axiom,
    ! [A: $tType,A3: set @ A] :
      ( ( minus_minus @ ( set @ A ) @ A3 @ ( bot_bot @ ( set @ A ) ) )
      = A3 ) ).

% Diff_empty
thf(fact_1793_subset__antisym,axiom,
    ! [A: $tType,A3: set @ A,B6: set @ A] :
      ( ( ord_less_eq @ ( set @ A ) @ A3 @ B6 )
     => ( ( ord_less_eq @ ( set @ A ) @ B6 @ A3 )
       => ( A3 = B6 ) ) ) ).

% subset_antisym
thf(fact_1794_subsetI,axiom,
    ! [A: $tType,A3: set @ A,B6: set @ A] :
      ( ! [X3: A] :
          ( ( member @ A @ X3 @ A3 )
         => ( member @ A @ X3 @ B6 ) )
     => ( ord_less_eq @ ( set @ A ) @ A3 @ B6 ) ) ).

% subsetI
thf(fact_1795_mint__corr__help__empty,axiom,
    ! [T2: vEBT_VEBT,N2: nat] :
      ( ( vEBT_invar_vebt @ T2 @ N2 )
     => ( ( ( vEBT_vebt_mint @ T2 )
          = ( none @ nat ) )
       => ( ( vEBT_VEBT_set_vebt @ T2 )
          = ( bot_bot @ ( set @ nat ) ) ) ) ) ).

% mint_corr_help_empty
thf(fact_1796_maxt__corr__help__empty,axiom,
    ! [T2: vEBT_VEBT,N2: nat] :
      ( ( vEBT_invar_vebt @ T2 @ N2 )
     => ( ( ( vEBT_vebt_maxt @ T2 )
          = ( none @ nat ) )
       => ( ( vEBT_VEBT_set_vebt @ T2 )
          = ( bot_bot @ ( set @ nat ) ) ) ) ) ).

% maxt_corr_help_empty
thf(fact_1797_Diff__eq__empty__iff,axiom,
    ! [A: $tType,A3: set @ A,B6: set @ A] :
      ( ( ( minus_minus @ ( set @ A ) @ A3 @ B6 )
        = ( bot_bot @ ( set @ A ) ) )
      = ( ord_less_eq @ ( set @ A ) @ A3 @ B6 ) ) ).

% Diff_eq_empty_iff
thf(fact_1798_subset__empty,axiom,
    ! [A: $tType,A3: set @ A] :
      ( ( ord_less_eq @ ( set @ A ) @ A3 @ ( bot_bot @ ( set @ A ) ) )
      = ( A3
        = ( bot_bot @ ( set @ A ) ) ) ) ).

% subset_empty
thf(fact_1799_empty__subsetI,axiom,
    ! [A: $tType,A3: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ A3 ) ).

% empty_subsetI
thf(fact_1800_atLeastatMost__empty__iff2,axiom,
    ! [A: $tType] :
      ( ( preorder @ A )
     => ! [A2: A,B2: A] :
          ( ( ( bot_bot @ ( set @ A ) )
            = ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) )
          = ( ~ ( ord_less_eq @ A @ A2 @ B2 ) ) ) ) ).

% atLeastatMost_empty_iff2
thf(fact_1801_atLeastatMost__empty__iff,axiom,
    ! [A: $tType] :
      ( ( preorder @ A )
     => ! [A2: A,B2: A] :
          ( ( ( set_or1337092689740270186AtMost @ A @ A2 @ B2 )
            = ( bot_bot @ ( set @ A ) ) )
          = ( ~ ( ord_less_eq @ A @ A2 @ B2 ) ) ) ) ).

% atLeastatMost_empty_iff
thf(fact_1802_atLeastatMost__empty,axiom,
    ! [A: $tType] :
      ( ( order @ A )
     => ! [B2: A,A2: A] :
          ( ( ord_less @ A @ B2 @ A2 )
         => ( ( set_or1337092689740270186AtMost @ A @ A2 @ B2 )
            = ( bot_bot @ ( set @ A ) ) ) ) ) ).

% atLeastatMost_empty
thf(fact_1803_mod__pure,axiom,
    ! [B2: $o,H2: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
      ( ( rep_assn @ ( pure_assn @ B2 ) @ H2 )
      = ( ( ( product_snd @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 )
          = ( bot_bot @ ( set @ nat ) ) )
        & B2 ) ) ).

% mod_pure
thf(fact_1804_mod__h__bot__iff_I5_J,axiom,
    ! [P: assn,Q: assn,H2: heap_ext @ product_unit] :
      ( ( rep_assn @ ( times_times @ assn @ P @ Q ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ ( bot_bot @ ( set @ nat ) ) ) )
      = ( ( rep_assn @ P @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ ( bot_bot @ ( set @ nat ) ) ) )
        & ( rep_assn @ Q @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ ( bot_bot @ ( set @ nat ) ) ) ) ) ) ).

% mod_h_bot_iff(5)
thf(fact_1805_mod__h__bot__iff_I1_J,axiom,
    ! [B2: $o,H2: heap_ext @ product_unit] :
      ( ( rep_assn @ ( pure_assn @ B2 ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ ( bot_bot @ ( set @ nat ) ) ) )
      = B2 ) ).

% mod_h_bot_iff(1)
thf(fact_1806_mod__h__bot__iff_I4_J,axiom,
    ! [B: $tType] :
      ( ( heap @ B )
     => ! [Q2: array @ B,Y2: list @ B,H2: heap_ext @ product_unit] :
          ~ ( rep_assn @ ( snga_assn @ B @ Q2 @ Y2 ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ ( bot_bot @ ( set @ nat ) ) ) ) ) ).

% mod_h_bot_iff(4)
thf(fact_1807_mod__h__bot__iff_I8_J,axiom,
    ! [C: $tType,R2: C > assn,H2: heap_ext @ product_unit] :
      ( ( rep_assn @ ( ex_assn @ C @ R2 ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ ( bot_bot @ ( set @ nat ) ) ) )
      = ( ? [X: C] : ( rep_assn @ ( R2 @ X ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ ( bot_bot @ ( set @ nat ) ) ) ) ) ) ).

% mod_h_bot_iff(8)
thf(fact_1808_Set_Oempty__def,axiom,
    ! [A: $tType] :
      ( ( bot_bot @ ( set @ A ) )
      = ( collect @ A
        @ ^ [X: A] : $false ) ) ).

% Set.empty_def
thf(fact_1809_set__notEmptyE,axiom,
    ! [A: $tType,S3: set @ A] :
      ( ( S3
       != ( bot_bot @ ( set @ A ) ) )
     => ~ ! [X3: A] :
            ~ ( member @ A @ X3 @ S3 ) ) ).

% set_notEmptyE
thf(fact_1810_memb__imp__not__empty,axiom,
    ! [A: $tType,X2: A,S3: set @ A] :
      ( ( member @ A @ X2 @ S3 )
     => ( S3
       != ( bot_bot @ ( set @ A ) ) ) ) ).

% memb_imp_not_empty
thf(fact_1811_times__assn__raw_Ocases,axiom,
    ! [X2: product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) )] :
      ~ ! [P7: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Q8: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,H3: heap_ext @ product_unit,As: set @ nat] :
          ( X2
         != ( product_Pair @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) @ P7 @ ( product_Pair @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ Q8 @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ As ) ) ) ) ).

% times_assn_raw.cases
thf(fact_1812_sngr__assn__raw_Ocases,axiom,
    ! [A: $tType] :
      ( ( heap @ A )
     => ! [X2: product_prod @ ( ref @ A ) @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) )] :
          ~ ! [R3: ref @ A,X3: A,H3: heap_ext @ product_unit,As: set @ nat] :
              ( X2
             != ( product_Pair @ ( ref @ A ) @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) @ R3 @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ X3 @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ As ) ) ) ) ) ).

% sngr_assn_raw.cases
thf(fact_1813_snga__assn__raw_Ocases,axiom,
    ! [A: $tType] :
      ( ( heap @ A )
     => ! [X2: product_prod @ ( array @ A ) @ ( product_prod @ ( list @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) )] :
          ~ ! [R3: array @ A,X3: list @ A,H3: heap_ext @ product_unit,As: set @ nat] :
              ( X2
             != ( product_Pair @ ( array @ A ) @ ( product_prod @ ( list @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) @ R3 @ ( product_Pair @ ( list @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ X3 @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ As ) ) ) ) ) ).

% snga_assn_raw.cases
thf(fact_1814_pure__assn__raw_Ocases,axiom,
    ! [A: $tType,B: $tType,X2: product_prod @ $o @ ( product_prod @ A @ ( set @ B ) )] :
      ~ ! [B3: $o,H3: A,As: set @ B] :
          ( X2
         != ( product_Pair @ $o @ ( product_prod @ A @ ( set @ B ) ) @ B3 @ ( product_Pair @ A @ ( set @ B ) @ H3 @ As ) ) ) ).

% pure_assn_raw.cases
thf(fact_1815_one__assn__raw_Ocases,axiom,
    ! [X2: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
      ~ ! [H3: heap_ext @ product_unit,As: set @ nat] :
          ( X2
         != ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ As ) ) ).

% one_assn_raw.cases
thf(fact_1816_subset__minus__empty,axiom,
    ! [A: $tType,A3: set @ A,B6: set @ A] :
      ( ( ord_less_eq @ ( set @ A ) @ A3 @ B6 )
     => ( ( minus_minus @ ( set @ A ) @ A3 @ B6 )
        = ( bot_bot @ ( set @ A ) ) ) ) ).

% subset_minus_empty
thf(fact_1817_mod__h__bot__indep,axiom,
    ! [P: assn,H2: heap_ext @ product_unit,H4: heap_ext @ product_unit] :
      ( ( rep_assn @ P @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ ( bot_bot @ ( set @ nat ) ) ) )
      = ( rep_assn @ P @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H4 @ ( bot_bot @ ( set @ nat ) ) ) ) ) ).

% mod_h_bot_indep
thf(fact_1818_mod__emp,axiom,
    ! [H2: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
      ( ( rep_assn @ ( one_one @ assn ) @ H2 )
      = ( ( product_snd @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 )
        = ( bot_bot @ ( set @ nat ) ) ) ) ).

% mod_emp
thf(fact_1819_mod__emp__simp,axiom,
    ! [H2: heap_ext @ product_unit] : ( rep_assn @ ( one_one @ assn ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ ( bot_bot @ ( set @ nat ) ) ) ) ).

% mod_emp_simp
thf(fact_1820_verit__comp__simplify1_I2_J,axiom,
    ! [A: $tType] :
      ( ( order @ A )
     => ! [A2: A] : ( ord_less_eq @ A @ A2 @ A2 ) ) ).

% verit_comp_simplify1(2)
thf(fact_1821_verit__la__disequality,axiom,
    ! [A: $tType] :
      ( ( linorder @ A )
     => ! [A2: A,B2: A] :
          ( ( A2 = B2 )
          | ~ ( ord_less_eq @ A @ A2 @ B2 )
          | ~ ( ord_less_eq @ A @ B2 @ A2 ) ) ) ).

% verit_la_disequality
thf(fact_1822_verit__comp__simplify1_I1_J,axiom,
    ! [A: $tType] :
      ( ( order @ A )
     => ! [A2: A] :
          ~ ( ord_less @ A @ A2 @ A2 ) ) ).

% verit_comp_simplify1(1)
thf(fact_1823_Collect__mono__iff,axiom,
    ! [A: $tType,P: A > $o,Q: A > $o] :
      ( ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) )
      = ( ! [X: A] :
            ( ( P @ X )
           => ( Q @ X ) ) ) ) ).

% Collect_mono_iff
thf(fact_1824_set__eq__subset,axiom,
    ! [A: $tType] :
      ( ( ^ [Y5: set @ A,Z4: set @ A] : Y5 = Z4 )
      = ( ^ [A7: set @ A,B7: set @ A] :
            ( ( ord_less_eq @ ( set @ A ) @ A7 @ B7 )
            & ( ord_less_eq @ ( set @ A ) @ B7 @ A7 ) ) ) ) ).

% set_eq_subset
thf(fact_1825_subset__trans,axiom,
    ! [A: $tType,A3: set @ A,B6: set @ A,C5: set @ A] :
      ( ( ord_less_eq @ ( set @ A ) @ A3 @ B6 )
     => ( ( ord_less_eq @ ( set @ A ) @ B6 @ C5 )
       => ( ord_less_eq @ ( set @ A ) @ A3 @ C5 ) ) ) ).

% subset_trans
thf(fact_1826_Collect__mono,axiom,
    ! [A: $tType,P: A > $o,Q: A > $o] :
      ( ! [X3: A] :
          ( ( P @ X3 )
         => ( Q @ X3 ) )
     => ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) ) ) ).

% Collect_mono
thf(fact_1827_subset__refl,axiom,
    ! [A: $tType,A3: set @ A] : ( ord_less_eq @ ( set @ A ) @ A3 @ A3 ) ).

% subset_refl
thf(fact_1828_subset__iff,axiom,
    ! [A: $tType] :
      ( ( ord_less_eq @ ( set @ A ) )
      = ( ^ [A7: set @ A,B7: set @ A] :
          ! [T3: A] :
            ( ( member @ A @ T3 @ A7 )
           => ( member @ A @ T3 @ B7 ) ) ) ) ).

% subset_iff
thf(fact_1829_Set_OequalityD2,axiom,
    ! [A: $tType,A3: set @ A,B6: set @ A] :
      ( ( A3 = B6 )
     => ( ord_less_eq @ ( set @ A ) @ B6 @ A3 ) ) ).

% Set.equalityD2
thf(fact_1830_equalityD1,axiom,
    ! [A: $tType,A3: set @ A,B6: set @ A] :
      ( ( A3 = B6 )
     => ( ord_less_eq @ ( set @ A ) @ A3 @ B6 ) ) ).

% equalityD1
thf(fact_1831_subset__eq,axiom,
    ! [A: $tType] :
      ( ( ord_less_eq @ ( set @ A ) )
      = ( ^ [A7: set @ A,B7: set @ A] :
          ! [X: A] :
            ( ( member @ A @ X @ A7 )
           => ( member @ A @ X @ B7 ) ) ) ) ).

% subset_eq
thf(fact_1832_equalityE,axiom,
    ! [A: $tType,A3: set @ A,B6: set @ A] :
      ( ( A3 = B6 )
     => ~ ( ( ord_less_eq @ ( set @ A ) @ A3 @ B6 )
         => ~ ( ord_less_eq @ ( set @ A ) @ B6 @ A3 ) ) ) ).

% equalityE
thf(fact_1833_subsetD,axiom,
    ! [A: $tType,A3: set @ A,B6: set @ A,C2: A] :
      ( ( ord_less_eq @ ( set @ A ) @ A3 @ B6 )
     => ( ( member @ A @ C2 @ A3 )
       => ( member @ A @ C2 @ B6 ) ) ) ).

% subsetD
thf(fact_1834_in__mono,axiom,
    ! [A: $tType,A3: set @ A,B6: set @ A,X2: A] :
      ( ( ord_less_eq @ ( set @ A ) @ A3 @ B6 )
     => ( ( member @ A @ X2 @ A3 )
       => ( member @ A @ X2 @ B6 ) ) ) ).

% in_mono
thf(fact_1835_double__diff,axiom,
    ! [A: $tType,A3: set @ A,B6: set @ A,C5: set @ A] :
      ( ( ord_less_eq @ ( set @ A ) @ A3 @ B6 )
     => ( ( ord_less_eq @ ( set @ A ) @ B6 @ C5 )
       => ( ( minus_minus @ ( set @ A ) @ B6 @ ( minus_minus @ ( set @ A ) @ C5 @ A3 ) )
          = A3 ) ) ) ).

% double_diff
thf(fact_1836_Diff__subset,axiom,
    ! [A: $tType,A3: set @ A,B6: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ B6 ) @ A3 ) ).

% Diff_subset
thf(fact_1837_Diff__mono,axiom,
    ! [A: $tType,A3: set @ A,C5: set @ A,D6: set @ A,B6: set @ A] :
      ( ( ord_less_eq @ ( set @ A ) @ A3 @ C5 )
     => ( ( ord_less_eq @ ( set @ A ) @ D6 @ B6 )
       => ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ B6 ) @ ( minus_minus @ ( set @ A ) @ C5 @ D6 ) ) ) ) ).

% Diff_mono
thf(fact_1838_verit__la__generic,axiom,
    ! [A2: int,X2: int] :
      ( ( ord_less_eq @ int @ A2 @ X2 )
      | ( A2 = X2 )
      | ( ord_less_eq @ int @ X2 @ A2 ) ) ).

% verit_la_generic
thf(fact_1839_psubset__imp__ex__mem,axiom,
    ! [A: $tType,A3: set @ A,B6: set @ A] :
      ( ( ord_less @ ( set @ A ) @ A3 @ B6 )
     => ? [B3: A] : ( member @ A @ B3 @ ( minus_minus @ ( set @ A ) @ B6 @ A3 ) ) ) ).

% psubset_imp_ex_mem
thf(fact_1840_less__eq__set__def,axiom,
    ! [A: $tType] :
      ( ( ord_less_eq @ ( set @ A ) )
      = ( ^ [A7: set @ A,B7: set @ A] :
            ( ord_less_eq @ ( A > $o )
            @ ^ [X: A] : ( member @ A @ X @ A7 )
            @ ^ [X: A] : ( member @ A @ X @ B7 ) ) ) ) ).

% less_eq_set_def
thf(fact_1841_Collect__subset,axiom,
    ! [A: $tType,A3: set @ A,P: A > $o] :
      ( ord_less_eq @ ( set @ A )
      @ ( collect @ A
        @ ^ [X: A] :
            ( ( member @ A @ X @ A3 )
            & ( P @ X ) ) )
      @ A3 ) ).

% Collect_subset
thf(fact_1842_verit__comp__simplify1_I3_J,axiom,
    ! [B: $tType] :
      ( ( linorder @ B )
     => ! [B4: B,A6: B] :
          ( ( ~ ( ord_less_eq @ B @ B4 @ A6 ) )
          = ( ord_less @ B @ A6 @ B4 ) ) ) ).

% verit_comp_simplify1(3)
thf(fact_1843_verit__sum__simplify,axiom,
    ! [A: $tType] :
      ( ( cancel1802427076303600483id_add @ A )
     => ! [A2: A] :
          ( ( plus_plus @ A @ A2 @ ( zero_zero @ A ) )
          = A2 ) ) ).

% verit_sum_simplify
thf(fact_1844_verit__eq__simplify_I10_J,axiom,
    ! [X23: num] :
      ( one2
     != ( bit0 @ X23 ) ) ).

% verit_eq_simplify(10)
thf(fact_1845_verit__eq__simplify_I14_J,axiom,
    ! [X23: num,X33: num] :
      ( ( bit0 @ X23 )
     != ( bit1 @ X33 ) ) ).

% verit_eq_simplify(14)
thf(fact_1846_verit__eq__simplify_I12_J,axiom,
    ! [X33: num] :
      ( one2
     != ( bit1 @ X33 ) ) ).

% verit_eq_simplify(12)
thf(fact_1847_psubsetE,axiom,
    ! [A: $tType,A3: set @ A,B6: set @ A] :
      ( ( ord_less @ ( set @ A ) @ A3 @ B6 )
     => ~ ( ( ord_less_eq @ ( set @ A ) @ A3 @ B6 )
         => ( ord_less_eq @ ( set @ A ) @ B6 @ A3 ) ) ) ).

% psubsetE
thf(fact_1848_psubset__eq,axiom,
    ! [A: $tType] :
      ( ( ord_less @ ( set @ A ) )
      = ( ^ [A7: set @ A,B7: set @ A] :
            ( ( ord_less_eq @ ( set @ A ) @ A7 @ B7 )
            & ( A7 != B7 ) ) ) ) ).

% psubset_eq
thf(fact_1849_psubset__imp__subset,axiom,
    ! [A: $tType,A3: set @ A,B6: set @ A] :
      ( ( ord_less @ ( set @ A ) @ A3 @ B6 )
     => ( ord_less_eq @ ( set @ A ) @ A3 @ B6 ) ) ).

% psubset_imp_subset
thf(fact_1850_psubset__subset__trans,axiom,
    ! [A: $tType,A3: set @ A,B6: set @ A,C5: set @ A] :
      ( ( ord_less @ ( set @ A ) @ A3 @ B6 )
     => ( ( ord_less_eq @ ( set @ A ) @ B6 @ C5 )
       => ( ord_less @ ( set @ A ) @ A3 @ C5 ) ) ) ).

% psubset_subset_trans
thf(fact_1851_subset__not__subset__eq,axiom,
    ! [A: $tType] :
      ( ( ord_less @ ( set @ A ) )
      = ( ^ [A7: set @ A,B7: set @ A] :
            ( ( ord_less_eq @ ( set @ A ) @ A7 @ B7 )
            & ~ ( ord_less_eq @ ( set @ A ) @ B7 @ A7 ) ) ) ) ).

% subset_not_subset_eq
thf(fact_1852_subset__psubset__trans,axiom,
    ! [A: $tType,A3: set @ A,B6: set @ A,C5: set @ A] :
      ( ( ord_less_eq @ ( set @ A ) @ A3 @ B6 )
     => ( ( ord_less @ ( set @ A ) @ B6 @ C5 )
       => ( ord_less @ ( set @ A ) @ A3 @ C5 ) ) ) ).

% subset_psubset_trans
thf(fact_1853_subset__iff__psubset__eq,axiom,
    ! [A: $tType] :
      ( ( ord_less_eq @ ( set @ A ) )
      = ( ^ [A7: set @ A,B7: set @ A] :
            ( ( ord_less @ ( set @ A ) @ A7 @ B7 )
            | ( A7 = B7 ) ) ) ) ).

% subset_iff_psubset_eq
thf(fact_1854_time__assert_H,axiom,
    ! [P: $o,H2: heap_ext @ product_unit] :
      ( ( time_time @ product_unit @ ( refine_Imp_assert @ P ) @ H2 )
      = ( zero_zero @ nat ) ) ).

% time_assert'
thf(fact_1855_TBOUND__mono,axiom,
    ! [A: $tType,C2: heap_Time_Heap @ A,T2: nat,T4: nat] :
      ( ( time_TBOUND @ A @ C2 @ T2 )
     => ( ( ord_less_eq @ nat @ T2 @ T4 )
       => ( time_TBOUND @ A @ C2 @ T4 ) ) ) ).

% TBOUND_mono
thf(fact_1856_htt__htD,axiom,
    ! [A: $tType,P: assn,C2: heap_Time_Heap @ A,Q: A > assn,T2: nat] :
      ( ( time_htt @ A @ P @ C2 @ Q @ T2 )
     => ( hoare_hoare_triple @ A @ P @ C2 @ Q ) ) ).

% htt_htD
thf(fact_1857_max__def__raw,axiom,
    ! [A: $tType] :
      ( ( ord @ A )
     => ( ( ord_max @ A )
        = ( ^ [A5: A,B5: A] : ( if @ A @ ( ord_less_eq @ A @ A5 @ B5 ) @ B5 @ A5 ) ) ) ) ).

% max_def_raw
thf(fact_1858_norm__pre__ex__rule__htt,axiom,
    ! [B: $tType,A: $tType,P: A > assn,F: heap_Time_Heap @ B,Q: B > assn,T2: nat] :
      ( ! [X3: A] : ( time_htt @ B @ ( P @ X3 ) @ F @ Q @ T2 )
     => ( time_htt @ B @ ( ex_assn @ A @ P ) @ F @ Q @ T2 ) ) ).

% norm_pre_ex_rule_htt
thf(fact_1859_norm__post__ex__rule__htt,axiom,
    ! [A: $tType,B: $tType,P: assn,F: heap_Time_Heap @ A,Q: B > A > assn,X2: B,T2: nat] :
      ( ( time_htt @ A @ P @ F @ ( Q @ X2 ) @ T2 )
     => ( time_htt @ A @ P @ F
        @ ^ [R: A] :
            ( ex_assn @ B
            @ ^ [X: B] : ( Q @ X @ R ) )
        @ T2 ) ) ).

% norm_post_ex_rule_htt
thf(fact_1860_int__ops_I3_J,axiom,
    ! [N2: num] :
      ( ( semiring_1_of_nat @ int @ ( numeral_numeral @ nat @ N2 ) )
      = ( numeral_numeral @ int @ N2 ) ) ).

% int_ops(3)
thf(fact_1861_nat__int__comparison_I2_J,axiom,
    ( ( ord_less @ nat )
    = ( ^ [A5: nat,B5: nat] : ( ord_less @ int @ ( semiring_1_of_nat @ int @ A5 ) @ ( semiring_1_of_nat @ int @ B5 ) ) ) ) ).

% nat_int_comparison(2)
thf(fact_1862_nat__int__comparison_I3_J,axiom,
    ( ( ord_less_eq @ nat )
    = ( ^ [A5: nat,B5: nat] : ( ord_less_eq @ int @ ( semiring_1_of_nat @ int @ A5 ) @ ( semiring_1_of_nat @ int @ B5 ) ) ) ) ).

% nat_int_comparison(3)
thf(fact_1863_int__ops_I2_J,axiom,
    ( ( semiring_1_of_nat @ int @ ( one_one @ nat ) )
    = ( one_one @ int ) ) ).

% int_ops(2)
thf(fact_1864_int__ops_I5_J,axiom,
    ! [A2: nat,B2: nat] :
      ( ( semiring_1_of_nat @ int @ ( plus_plus @ nat @ A2 @ B2 ) )
      = ( plus_plus @ int @ ( semiring_1_of_nat @ int @ A2 ) @ ( semiring_1_of_nat @ int @ B2 ) ) ) ).

% int_ops(5)
thf(fact_1865_int__plus,axiom,
    ! [N2: nat,M: nat] :
      ( ( semiring_1_of_nat @ int @ ( plus_plus @ nat @ N2 @ M ) )
      = ( plus_plus @ int @ ( semiring_1_of_nat @ int @ N2 ) @ ( semiring_1_of_nat @ int @ M ) ) ) ).

% int_plus
thf(fact_1866_int__ops_I7_J,axiom,
    ! [A2: nat,B2: nat] :
      ( ( semiring_1_of_nat @ int @ ( times_times @ nat @ A2 @ B2 ) )
      = ( times_times @ int @ ( semiring_1_of_nat @ int @ A2 ) @ ( semiring_1_of_nat @ int @ B2 ) ) ) ).

% int_ops(7)
thf(fact_1867_TBOUND__return,axiom,
    ! [A: $tType,X2: A] : ( time_TBOUND @ A @ ( heap_Time_return @ A @ X2 ) @ ( one_one @ nat ) ) ).

% TBOUND_return
thf(fact_1868_time__return,axiom,
    ! [A: $tType,X2: A,H2: heap_ext @ product_unit] :
      ( ( time_time @ A @ ( heap_Time_return @ A @ X2 ) @ H2 )
      = ( one_one @ nat ) ) ).

% time_return
thf(fact_1869_TBOUND__nth,axiom,
    ! [A: $tType] :
      ( ( heap @ A )
     => ! [Xs2: array @ A,I: nat] : ( time_TBOUND @ A @ ( array_nth @ A @ Xs2 @ I ) @ ( one_one @ nat ) ) ) ).

% TBOUND_nth
thf(fact_1870_TBOUNDD,axiom,
    ! [A: $tType,M: heap_Time_Heap @ A,T2: nat,H2: heap_ext @ product_unit] :
      ( ( time_TBOUND @ A @ M @ T2 )
     => ( ord_less_eq @ nat @ ( time_time @ A @ M @ H2 ) @ T2 ) ) ).

% TBOUNDD
thf(fact_1871_TBOUNDI,axiom,
    ! [A: $tType,M: heap_Time_Heap @ A,T2: nat] :
      ( ! [H3: heap_ext @ product_unit] : ( ord_less_eq @ nat @ ( time_time @ A @ M @ H3 ) @ T2 )
     => ( time_TBOUND @ A @ M @ T2 ) ) ).

% TBOUNDI
thf(fact_1872_TBOUND__def,axiom,
    ! [A: $tType] :
      ( ( time_TBOUND @ A )
      = ( ^ [M7: heap_Time_Heap @ A,T3: nat] :
          ! [H: heap_ext @ product_unit] : ( ord_less_eq @ nat @ ( time_time @ A @ M7 @ H ) @ T3 ) ) ) ).

% TBOUND_def
thf(fact_1873_norm__pre__pure__iff__htt,axiom,
    ! [A: $tType,P: assn,B2: $o,F: heap_Time_Heap @ A,Q: A > assn,T2: nat] :
      ( ( time_htt @ A @ ( times_times @ assn @ P @ ( pure_assn @ B2 ) ) @ F @ Q @ T2 )
      = ( B2
       => ( time_htt @ A @ P @ F @ Q @ T2 ) ) ) ).

% norm_pre_pure_iff_htt
thf(fact_1874_norm__pre__pure__iff__htt_H,axiom,
    ! [A: $tType,B2: $o,P: assn,F: heap_Time_Heap @ A,Q: A > assn,T2: nat] :
      ( ( time_htt @ A @ ( times_times @ assn @ ( pure_assn @ B2 ) @ P ) @ F @ Q @ T2 )
      = ( B2
       => ( time_htt @ A @ P @ F @ Q @ T2 ) ) ) ).

% norm_pre_pure_iff_htt'
thf(fact_1875_nat__less__as__int,axiom,
    ( ( ord_less @ nat )
    = ( ^ [A5: nat,B5: nat] : ( ord_less @ int @ ( semiring_1_of_nat @ int @ A5 ) @ ( semiring_1_of_nat @ int @ B5 ) ) ) ) ).

% nat_less_as_int
thf(fact_1876_nat__leq__as__int,axiom,
    ( ( ord_less_eq @ nat )
    = ( ^ [A5: nat,B5: nat] : ( ord_less_eq @ int @ ( semiring_1_of_nat @ int @ A5 ) @ ( semiring_1_of_nat @ int @ B5 ) ) ) ) ).

% nat_leq_as_int
thf(fact_1877_httI__TBOUND,axiom,
    ! [A: $tType,P: assn,C2: heap_Time_Heap @ A,Q: A > assn,T2: nat] :
      ( ( hoare_hoare_triple @ A @ P @ C2 @ Q )
     => ( ( time_TBOUND @ A @ C2 @ T2 )
       => ( time_htt @ A @ P @ C2 @ Q @ T2 ) ) ) ).

% httI_TBOUND
thf(fact_1878_TBOUND__assert_H__weak,axiom,
    ! [A: $tType,P: $o,M: heap_Time_Heap @ A,T2: nat] :
      ( ( P
       => ( time_TBOUND @ A @ M @ T2 ) )
     => ( time_TBOUND @ A
        @ ( heap_Time_bind @ product_unit @ A @ ( refine_Imp_assert @ P )
          @ ^ [Uu3: product_unit] : M )
        @ T2 ) ) ).

% TBOUND_assert'_weak
thf(fact_1879_int__Suc,axiom,
    ! [N2: nat] :
      ( ( semiring_1_of_nat @ int @ ( suc @ N2 ) )
      = ( plus_plus @ int @ ( semiring_1_of_nat @ int @ N2 ) @ ( one_one @ int ) ) ) ).

% int_Suc
thf(fact_1880_int__ops_I4_J,axiom,
    ! [A2: nat] :
      ( ( semiring_1_of_nat @ int @ ( suc @ A2 ) )
      = ( plus_plus @ int @ ( semiring_1_of_nat @ int @ A2 ) @ ( one_one @ int ) ) ) ).

% int_ops(4)
thf(fact_1881_TBOUND__assert_H__bind__strong,axiom,
    ! [A: $tType,P: $o,M: heap_Time_Heap @ A,T2: nat] :
      ( ( P
       => ( time_TBOUND @ A @ M @ T2 ) )
     => ( time_TBOUND @ A
        @ ( heap_Time_bind @ product_unit @ A @ ( refine_Imp_assert @ P )
          @ ^ [Uu3: product_unit] : M )
        @ ( if @ nat @ P @ T2 @ ( zero_zero @ nat ) ) ) ) ).

% TBOUND_assert'_bind_strong
thf(fact_1882_TBOUND__option__case,axiom,
    ! [B: $tType,A: $tType,T2: option @ A,F: heap_Time_Heap @ B,Bnd: nat,F4: A > ( heap_Time_Heap @ B ),Bnd2: A > nat] :
      ( ( ( T2
          = ( none @ A ) )
       => ( time_TBOUND @ B @ F @ Bnd ) )
     => ( ! [X3: A] :
            ( ( T2
              = ( some @ A @ X3 ) )
           => ( time_TBOUND @ B @ ( F4 @ X3 ) @ ( Bnd2 @ X3 ) ) )
       => ( time_TBOUND @ B @ ( case_option @ ( heap_Time_Heap @ B ) @ A @ F @ F4 @ T2 ) @ ( case_option @ nat @ A @ Bnd @ Bnd2 @ T2 ) ) ) ) ).

% TBOUND_option_case
thf(fact_1883_vebt__pred_Opelims,axiom,
    ! [X2: vEBT_VEBT,Xa: nat,Y2: option @ nat] :
      ( ( ( vEBT_vebt_pred @ X2 @ Xa )
        = Y2 )
     => ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_pred_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X2 @ Xa ) )
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( X2
                = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
             => ( ( Xa
                  = ( zero_zero @ nat ) )
               => ( ( Y2
                    = ( none @ nat ) )
                 => ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_pred_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ ( zero_zero @ nat ) ) ) ) ) )
         => ( ! [A4: $o,Uw2: $o] :
                ( ( X2
                  = ( vEBT_Leaf @ A4 @ Uw2 ) )
               => ( ( Xa
                    = ( suc @ ( zero_zero @ nat ) ) )
                 => ( ( ( A4
                       => ( Y2
                          = ( some @ nat @ ( zero_zero @ nat ) ) ) )
                      & ( ~ A4
                       => ( Y2
                          = ( none @ nat ) ) ) )
                   => ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_pred_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A4 @ Uw2 ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) )
           => ( ! [A4: $o,B3: $o] :
                  ( ( X2
                    = ( vEBT_Leaf @ A4 @ B3 ) )
                 => ! [Va3: nat] :
                      ( ( Xa
                        = ( suc @ ( suc @ Va3 ) ) )
                     => ( ( ( B3
                           => ( Y2
                              = ( some @ nat @ ( one_one @ nat ) ) ) )
                          & ( ~ B3
                           => ( ( A4
                               => ( Y2
                                  = ( some @ nat @ ( zero_zero @ nat ) ) ) )
                              & ( ~ A4
                               => ( Y2
                                  = ( none @ nat ) ) ) ) ) )
                       => ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_pred_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A4 @ B3 ) @ ( suc @ ( suc @ Va3 ) ) ) ) ) ) )
             => ( ! [Uy2: nat,Uz2: list @ vEBT_VEBT,Va2: vEBT_VEBT] :
                    ( ( X2
                      = ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uy2 @ Uz2 @ Va2 ) )
                   => ( ( Y2
                        = ( none @ nat ) )
                     => ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_pred_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uy2 @ Uz2 @ Va2 ) @ Xa ) ) ) )
               => ( ! [V3: product_prod @ nat @ nat,Vd2: list @ vEBT_VEBT,Ve2: vEBT_VEBT] :
                      ( ( X2
                        = ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( zero_zero @ nat ) @ Vd2 @ Ve2 ) )
                     => ( ( Y2
                          = ( none @ nat ) )
                       => ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_pred_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( zero_zero @ nat ) @ Vd2 @ Ve2 ) @ Xa ) ) ) )
                 => ( ! [V3: product_prod @ nat @ nat,Vh2: list @ vEBT_VEBT,Vi2: vEBT_VEBT] :
                        ( ( X2
                          = ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vh2 @ Vi2 ) )
                       => ( ( Y2
                            = ( none @ nat ) )
                         => ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_pred_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vh2 @ Vi2 ) @ Xa ) ) ) )
                   => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
                          ( ( X2
                            = ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
                         => ( ( ( ( ord_less @ nat @ Ma2 @ Xa )
                               => ( Y2
                                  = ( some @ nat @ Ma2 ) ) )
                              & ( ~ ( ord_less @ nat @ Ma2 @ Xa )
                               => ( Y2
                                  = ( if @ ( option @ nat ) @ ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
                                    @ ( if @ ( option @ nat )
                                      @ ( ( ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
                                         != ( none @ nat ) )
                                        & ( vEBT_VEBT_greater @ ( some @ nat @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) )
                                      @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( some @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_pred @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
                                      @ ( if @ ( option @ nat )
                                        @ ( ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
                                          = ( none @ nat ) )
                                        @ ( if @ ( option @ nat ) @ ( ord_less @ nat @ Mi2 @ Xa ) @ ( some @ nat @ Mi2 ) @ ( none @ nat ) )
                                        @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) )
                                    @ ( none @ nat ) ) ) ) )
                           => ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_pred_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) @ Xa ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_pred.pelims
thf(fact_1884_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,
    ! [X2: vEBT_VEBT,Xa: nat,Y2: nat] :
      ( ( ( vEBT_V1232361888498592333_e_t_e @ X2 @ Xa )
        = Y2 )
     => ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V6368547301243506412_e_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X2 @ Xa ) )
       => ( ! [A4: $o,B3: $o] :
              ( ( X2
                = ( vEBT_Leaf @ A4 @ B3 ) )
             => ( ( Xa
                  = ( zero_zero @ nat ) )
               => ( ( Y2
                    = ( one_one @ nat ) )
                 => ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V6368547301243506412_e_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A4 @ B3 ) @ ( zero_zero @ nat ) ) ) ) ) )
         => ( ! [A4: $o,B3: $o] :
                ( ( X2
                  = ( vEBT_Leaf @ A4 @ B3 ) )
               => ( ( Xa
                    = ( suc @ ( zero_zero @ nat ) ) )
                 => ( ( Y2
                      = ( one_one @ nat ) )
                   => ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V6368547301243506412_e_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A4 @ B3 ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) )
           => ( ! [A4: $o,B3: $o] :
                  ( ( X2
                    = ( vEBT_Leaf @ A4 @ B3 ) )
                 => ! [N4: nat] :
                      ( ( Xa
                        = ( suc @ ( suc @ N4 ) ) )
                     => ( ( Y2
                          = ( one_one @ nat ) )
                       => ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V6368547301243506412_e_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A4 @ B3 ) @ ( suc @ ( suc @ N4 ) ) ) ) ) ) )
             => ( ! [Deg2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
                    ( ( X2
                      = ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg2 @ TreeList2 @ Summary2 ) )
                   => ( ( Y2
                        = ( one_one @ nat ) )
                     => ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V6368547301243506412_e_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg2 @ TreeList2 @ Summary2 ) @ Xa ) ) ) )
               => ( ! [Mi2: nat,Ma2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X2
                        = ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ TreeList2 @ Summary2 ) )
                     => ( ( Y2
                          = ( one_one @ nat ) )
                       => ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V6368547301243506412_e_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ TreeList2 @ Summary2 ) @ Xa ) ) ) )
                 => ( ! [Mi2: nat,Ma2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
                        ( ( X2
                          = ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( zero_zero @ nat ) ) @ TreeList2 @ Summary2 ) )
                       => ( ( Y2
                            = ( one_one @ nat ) )
                         => ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V6368547301243506412_e_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( zero_zero @ nat ) ) @ TreeList2 @ Summary2 ) @ Xa ) ) ) )
                   => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
                          ( ( X2
                            = ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
                         => ( ( ( ( ( ord_less @ nat @ Xa @ Mi2 )
                                  | ( ord_less @ nat @ Ma2 @ Xa ) )
                               => ( Y2
                                  = ( one_one @ nat ) ) )
                              & ( ~ ( ( ord_less @ nat @ Xa @ Mi2 )
                                    | ( ord_less @ nat @ Ma2 @ Xa ) )
                               => ( ( ( ( Xa = Mi2 )
                                      & ( Xa = Ma2 ) )
                                   => ( Y2
                                      = ( one_one @ nat ) ) )
                                  & ( ~ ( ( Xa = Mi2 )
                                        & ( Xa = Ma2 ) )
                                   => ( Y2
                                      = ( if @ nat @ ( ord_less @ nat @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) @ ( plus_plus @ nat @ ( vEBT_V1232361888498592333_e_t_e @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( if @ nat @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_V1232361888498592333_e_t_e @ Summary2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( one_one @ nat ) ) ) @ ( one_one @ nat ) ) ) ) ) ) )
                           => ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V6368547301243506412_e_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) @ Xa ) ) ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.pelims
thf(fact_1885_vebt__delete_Opelims,axiom,
    ! [X2: vEBT_VEBT,Xa: nat,Y2: vEBT_VEBT] :
      ( ( ( vEBT_vebt_delete @ X2 @ Xa )
        = Y2 )
     => ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_delete_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X2 @ Xa ) )
       => ( ! [A4: $o,B3: $o] :
              ( ( X2
                = ( vEBT_Leaf @ A4 @ B3 ) )
             => ( ( Xa
                  = ( zero_zero @ nat ) )
               => ( ( Y2
                    = ( vEBT_Leaf @ $false @ B3 ) )
                 => ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_delete_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A4 @ B3 ) @ ( zero_zero @ nat ) ) ) ) ) )
         => ( ! [A4: $o,B3: $o] :
                ( ( X2
                  = ( vEBT_Leaf @ A4 @ B3 ) )
               => ( ( Xa
                    = ( suc @ ( zero_zero @ nat ) ) )
                 => ( ( Y2
                      = ( vEBT_Leaf @ A4 @ $false ) )
                   => ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_delete_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A4 @ B3 ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) )
           => ( ! [A4: $o,B3: $o] :
                  ( ( X2
                    = ( vEBT_Leaf @ A4 @ B3 ) )
                 => ! [N4: nat] :
                      ( ( Xa
                        = ( suc @ ( suc @ N4 ) ) )
                     => ( ( Y2
                          = ( vEBT_Leaf @ A4 @ B3 ) )
                       => ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_delete_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A4 @ B3 ) @ ( suc @ ( suc @ N4 ) ) ) ) ) ) )
             => ( ! [Deg2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
                    ( ( X2
                      = ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg2 @ TreeList2 @ Summary2 ) )
                   => ( ( Y2
                        = ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg2 @ TreeList2 @ Summary2 ) )
                     => ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_delete_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg2 @ TreeList2 @ Summary2 ) @ Xa ) ) ) )
               => ( ! [Mi2: nat,Ma2: nat,TrLst: list @ vEBT_VEBT,Smry: vEBT_VEBT] :
                      ( ( X2
                        = ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ TrLst @ Smry ) )
                     => ( ( Y2
                          = ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ TrLst @ Smry ) )
                       => ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_delete_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ TrLst @ Smry ) @ Xa ) ) ) )
                 => ( ! [Mi2: nat,Ma2: nat,Tr: list @ vEBT_VEBT,Sm: vEBT_VEBT] :
                        ( ( X2
                          = ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( zero_zero @ nat ) ) @ Tr @ Sm ) )
                       => ( ( Y2
                            = ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( zero_zero @ nat ) ) @ Tr @ Sm ) )
                         => ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_delete_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( zero_zero @ nat ) ) @ Tr @ Sm ) @ Xa ) ) ) )
                   => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
                          ( ( X2
                            = ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
                         => ( ( ( ( ( ord_less @ nat @ Xa @ Mi2 )
                                  | ( ord_less @ nat @ Ma2 @ Xa ) )
                               => ( Y2
                                  = ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) ) )
                              & ( ~ ( ( ord_less @ nat @ Xa @ Mi2 )
                                    | ( ord_less @ nat @ Ma2 @ Xa ) )
                               => ( ( ( ( Xa = Mi2 )
                                      & ( Xa = Ma2 ) )
                                   => ( Y2
                                      = ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) ) )
                                  & ( ~ ( ( Xa = Mi2 )
                                        & ( Xa = Ma2 ) )
                                   => ( Y2
                                      = ( if @ vEBT_VEBT @ ( ord_less @ nat @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
                                        @ ( if @ vEBT_VEBT @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
                                          @ ( vEBT_Node
                                            @ ( some @ ( product_prod @ nat @ nat )
                                              @ ( product_Pair @ nat @ nat @ ( if @ nat @ ( Xa = Mi2 ) @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ Mi2 )
                                                @ ( if @ nat
                                                  @ ( ( ( Xa = Mi2 )
                                                     => ( ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) )
                                                        = Ma2 ) )
                                                    & ( ( Xa != Mi2 )
                                                     => ( Xa = Ma2 ) ) )
                                                  @ ( if @ nat
                                                    @ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
                                                      = ( none @ nat ) )
                                                    @ ( if @ nat @ ( Xa = Mi2 ) @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ Mi2 )
                                                    @ ( plus_plus @ nat @ ( times_times @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ ( list_update @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) )
                                                  @ Ma2 ) ) )
                                            @ ( suc @ ( suc @ Va3 ) )
                                            @ ( list_update @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
                                            @ ( vEBT_vebt_delete @ Summary2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
                                          @ ( vEBT_Node
                                            @ ( some @ ( product_prod @ nat @ nat )
                                              @ ( product_Pair @ nat @ nat @ ( if @ nat @ ( Xa = Mi2 ) @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ Mi2 )
                                                @ ( if @ nat
                                                  @ ( ( ( Xa = Mi2 )
                                                     => ( ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) )
                                                        = Ma2 ) )
                                                    & ( ( Xa != Mi2 )
                                                     => ( Xa = Ma2 ) ) )
                                                  @ ( plus_plus @ nat @ ( times_times @ nat @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ ( list_update @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) )
                                                  @ Ma2 ) ) )
                                            @ ( suc @ ( suc @ Va3 ) )
                                            @ ( list_update @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
                                            @ Summary2 ) )
                                        @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) ) ) ) ) ) )
                           => ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_delete_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) @ Xa ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_delete.pelims
thf(fact_1886_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Opelims,axiom,
    ! [X2: vEBT_VEBT,Xa: nat,Y2: nat] :
      ( ( ( vEBT_T_s_u_c_c2 @ X2 @ Xa )
        = Y2 )
     => ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T_s_u_c_c_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X2 @ Xa ) )
       => ( ! [Uu2: $o,B3: $o] :
              ( ( X2
                = ( vEBT_Leaf @ Uu2 @ B3 ) )
             => ( ( Xa
                  = ( zero_zero @ nat ) )
               => ( ( Y2
                    = ( one_one @ nat ) )
                 => ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T_s_u_c_c_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ Uu2 @ B3 ) @ ( zero_zero @ nat ) ) ) ) ) )
         => ( ! [Uv2: $o,Uw2: $o] :
                ( ( X2
                  = ( vEBT_Leaf @ Uv2 @ Uw2 ) )
               => ! [N4: nat] :
                    ( ( Xa
                      = ( suc @ N4 ) )
                   => ( ( Y2
                        = ( one_one @ nat ) )
                     => ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T_s_u_c_c_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ Uv2 @ Uw2 ) @ ( suc @ N4 ) ) ) ) ) )
           => ( ! [Ux2: nat,Uy2: list @ vEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( X2
                    = ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Ux2 @ Uy2 @ Uz2 ) )
                 => ( ( Y2
                      = ( one_one @ nat ) )
                   => ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T_s_u_c_c_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Ux2 @ Uy2 @ Uz2 ) @ Xa ) ) ) )
             => ( ! [V3: product_prod @ nat @ nat,Vc: list @ vEBT_VEBT,Vd2: vEBT_VEBT] :
                    ( ( X2
                      = ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( zero_zero @ nat ) @ Vc @ Vd2 ) )
                   => ( ( Y2
                        = ( one_one @ nat ) )
                     => ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T_s_u_c_c_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( zero_zero @ nat ) @ Vc @ Vd2 ) @ Xa ) ) ) )
               => ( ! [V3: product_prod @ nat @ nat,Vg2: list @ vEBT_VEBT,Vh2: vEBT_VEBT] :
                      ( ( X2
                        = ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vg2 @ Vh2 ) )
                     => ( ( Y2
                          = ( one_one @ nat ) )
                       => ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T_s_u_c_c_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vg2 @ Vh2 ) @ Xa ) ) ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
                        ( ( X2
                          = ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
                       => ( ( ( ( ord_less @ nat @ Xa @ Mi2 )
                             => ( Y2
                                = ( one_one @ nat ) ) )
                            & ( ~ ( ord_less @ nat @ Xa @ Mi2 )
                             => ( Y2
                                = ( if @ nat @ ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
                                  @ ( if @ nat
                                    @ ( ( ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
                                       != ( none @ nat ) )
                                      & ( vEBT_VEBT_less @ ( some @ nat @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) )
                                    @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_T_s_u_c_c2 @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
                                    @ ( plus_plus @ nat @ ( vEBT_T_s_u_c_c2 @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( one_one @ nat ) ) )
                                  @ ( one_one @ nat ) ) ) ) )
                         => ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T_s_u_c_c_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) @ Xa ) ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.pelims
thf(fact_1887_vebt__insert_Opelims,axiom,
    ! [X2: vEBT_VEBT,Xa: nat,Y2: vEBT_VEBT] :
      ( ( ( vEBT_vebt_insert @ X2 @ Xa )
        = Y2 )
     => ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_insert_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X2 @ Xa ) )
       => ( ! [A4: $o,B3: $o] :
              ( ( X2
                = ( vEBT_Leaf @ A4 @ B3 ) )
             => ( ( ( ( Xa
                      = ( zero_zero @ nat ) )
                   => ( Y2
                      = ( vEBT_Leaf @ $true @ B3 ) ) )
                  & ( ( Xa
                     != ( zero_zero @ nat ) )
                   => ( ( ( Xa
                          = ( one_one @ nat ) )
                       => ( Y2
                          = ( vEBT_Leaf @ A4 @ $true ) ) )
                      & ( ( Xa
                         != ( one_one @ nat ) )
                       => ( Y2
                          = ( vEBT_Leaf @ A4 @ B3 ) ) ) ) ) )
               => ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_insert_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A4 @ B3 ) @ Xa ) ) ) )
         => ( ! [Info2: option @ ( product_prod @ nat @ nat ),Ts2: list @ vEBT_VEBT,S: vEBT_VEBT] :
                ( ( X2
                  = ( vEBT_Node @ Info2 @ ( zero_zero @ nat ) @ Ts2 @ S ) )
               => ( ( Y2
                    = ( vEBT_Node @ Info2 @ ( zero_zero @ nat ) @ Ts2 @ S ) )
                 => ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_insert_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Info2 @ ( zero_zero @ nat ) @ Ts2 @ S ) @ Xa ) ) ) )
           => ( ! [Info2: option @ ( product_prod @ nat @ nat ),Ts2: list @ vEBT_VEBT,S: vEBT_VEBT] :
                  ( ( X2
                    = ( vEBT_Node @ Info2 @ ( suc @ ( zero_zero @ nat ) ) @ Ts2 @ S ) )
                 => ( ( Y2
                      = ( vEBT_Node @ Info2 @ ( suc @ ( zero_zero @ nat ) ) @ Ts2 @ S ) )
                   => ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_insert_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Info2 @ ( suc @ ( zero_zero @ nat ) ) @ Ts2 @ S ) @ Xa ) ) ) )
             => ( ! [V3: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
                    ( ( X2
                      = ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ V3 ) ) @ TreeList2 @ Summary2 ) )
                   => ( ( Y2
                        = ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Xa @ Xa ) ) @ ( suc @ ( suc @ V3 ) ) @ TreeList2 @ Summary2 ) )
                     => ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_insert_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ V3 ) ) @ TreeList2 @ Summary2 ) @ Xa ) ) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X2
                        = ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
                     => ( ( Y2
                          = ( if @ vEBT_VEBT
                            @ ( ( ord_less @ nat @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
                              & ~ ( ( Xa = Mi2 )
                                  | ( Xa = Ma2 ) ) )
                            @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi2 ) @ Xa @ Mi2 ) @ ( ord_max @ nat @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ Ma2 ) ) ) @ ( suc @ ( suc @ Va3 ) ) @ ( list_update @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_insert @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( if @ vEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ Summary2 ) )
                            @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) ) )
                       => ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_insert_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) @ Xa ) ) ) ) ) ) ) ) ) ) ).

% vebt_insert.pelims
thf(fact_1888_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Opelims,axiom,
    ! [X2: vEBT_VEBT,Xa: nat,Y2: nat] :
      ( ( ( vEBT_T_p_r_e_d2 @ X2 @ Xa )
        = Y2 )
     => ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T_p_r_e_d_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X2 @ Xa ) )
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( X2
                = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
             => ( ( Xa
                  = ( zero_zero @ nat ) )
               => ( ( Y2
                    = ( one_one @ nat ) )
                 => ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T_p_r_e_d_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ ( zero_zero @ nat ) ) ) ) ) )
         => ( ! [A4: $o,Uw2: $o] :
                ( ( X2
                  = ( vEBT_Leaf @ A4 @ Uw2 ) )
               => ( ( Xa
                    = ( suc @ ( zero_zero @ nat ) ) )
                 => ( ( Y2
                      = ( one_one @ nat ) )
                   => ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T_p_r_e_d_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A4 @ Uw2 ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) )
           => ( ! [A4: $o,B3: $o] :
                  ( ( X2
                    = ( vEBT_Leaf @ A4 @ B3 ) )
                 => ! [Va3: nat] :
                      ( ( Xa
                        = ( suc @ ( suc @ Va3 ) ) )
                     => ( ( Y2
                          = ( one_one @ nat ) )
                       => ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T_p_r_e_d_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A4 @ B3 ) @ ( suc @ ( suc @ Va3 ) ) ) ) ) ) )
             => ( ! [Uy2: nat,Uz2: list @ vEBT_VEBT,Va2: vEBT_VEBT] :
                    ( ( X2
                      = ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uy2 @ Uz2 @ Va2 ) )
                   => ( ( Y2
                        = ( one_one @ nat ) )
                     => ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T_p_r_e_d_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uy2 @ Uz2 @ Va2 ) @ Xa ) ) ) )
               => ( ! [V3: product_prod @ nat @ nat,Vd2: list @ vEBT_VEBT,Ve2: vEBT_VEBT] :
                      ( ( X2
                        = ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( zero_zero @ nat ) @ Vd2 @ Ve2 ) )
                     => ( ( Y2
                          = ( one_one @ nat ) )
                       => ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T_p_r_e_d_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( zero_zero @ nat ) @ Vd2 @ Ve2 ) @ Xa ) ) ) )
                 => ( ! [V3: product_prod @ nat @ nat,Vh2: list @ vEBT_VEBT,Vi2: vEBT_VEBT] :
                        ( ( X2
                          = ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vh2 @ Vi2 ) )
                       => ( ( Y2
                            = ( one_one @ nat ) )
                         => ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T_p_r_e_d_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vh2 @ Vi2 ) @ Xa ) ) ) )
                   => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
                          ( ( X2
                            = ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
                         => ( ( ( ( ord_less @ nat @ Ma2 @ Xa )
                               => ( Y2
                                  = ( one_one @ nat ) ) )
                              & ( ~ ( ord_less @ nat @ Ma2 @ Xa )
                               => ( Y2
                                  = ( if @ nat @ ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
                                    @ ( if @ nat
                                      @ ( ( ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
                                         != ( none @ nat ) )
                                        & ( vEBT_VEBT_greater @ ( some @ nat @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) )
                                      @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_T_p_r_e_d2 @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
                                      @ ( plus_plus @ nat @ ( vEBT_T_p_r_e_d2 @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( one_one @ nat ) ) )
                                    @ ( one_one @ nat ) ) ) ) )
                           => ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T_p_r_e_d_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) @ Xa ) ) ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.pelims
thf(fact_1889_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,
    ! [X2: vEBT_VEBT,Xa: nat,Y2: nat] :
      ( ( ( vEBT_T_i_n_s_e_r_t2 @ X2 @ Xa )
        = Y2 )
     => ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T5076183648494686801_t_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X2 @ Xa ) )
       => ( ! [A4: $o,B3: $o] :
              ( ( X2
                = ( vEBT_Leaf @ A4 @ B3 ) )
             => ( ( Y2
                  = ( one_one @ nat ) )
               => ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T5076183648494686801_t_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A4 @ B3 ) @ Xa ) ) ) )
         => ( ! [Info2: option @ ( product_prod @ nat @ nat ),Ts2: list @ vEBT_VEBT,S: vEBT_VEBT] :
                ( ( X2
                  = ( vEBT_Node @ Info2 @ ( zero_zero @ nat ) @ Ts2 @ S ) )
               => ( ( Y2
                    = ( one_one @ nat ) )
                 => ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T5076183648494686801_t_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Info2 @ ( zero_zero @ nat ) @ Ts2 @ S ) @ Xa ) ) ) )
           => ( ! [Info2: option @ ( product_prod @ nat @ nat ),Ts2: list @ vEBT_VEBT,S: vEBT_VEBT] :
                  ( ( X2
                    = ( vEBT_Node @ Info2 @ ( suc @ ( zero_zero @ nat ) ) @ Ts2 @ S ) )
                 => ( ( Y2
                      = ( one_one @ nat ) )
                   => ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T5076183648494686801_t_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Info2 @ ( suc @ ( zero_zero @ nat ) ) @ Ts2 @ S ) @ Xa ) ) ) )
             => ( ! [V3: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
                    ( ( X2
                      = ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ V3 ) ) @ TreeList2 @ Summary2 ) )
                   => ( ( Y2
                        = ( one_one @ nat ) )
                     => ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T5076183648494686801_t_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ V3 ) ) @ TreeList2 @ Summary2 ) @ Xa ) ) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X2
                        = ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
                     => ( ( Y2
                          = ( if @ nat
                            @ ( ( ord_less @ nat @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
                              & ~ ( ( Xa = Mi2 )
                                  | ( Xa = Ma2 ) ) )
                            @ ( plus_plus @ nat @ ( vEBT_T_i_n_s_e_r_t2 @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( if @ nat @ ( vEBT_VEBT_minNull @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_T_i_n_s_e_r_t2 @ Summary2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( one_one @ nat ) ) )
                            @ ( one_one @ nat ) ) )
                       => ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T5076183648494686801_t_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) @ Xa ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.pelims
thf(fact_1890_Diff__idemp,axiom,
    ! [A: $tType,A3: set @ A,B6: set @ A] :
      ( ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ B6 ) @ B6 )
      = ( minus_minus @ ( set @ A ) @ A3 @ B6 ) ) ).

% Diff_idemp
thf(fact_1891_Diff__iff,axiom,
    ! [A: $tType,C2: A,A3: set @ A,B6: set @ A] :
      ( ( member @ A @ C2 @ ( minus_minus @ ( set @ A ) @ A3 @ B6 ) )
      = ( ( member @ A @ C2 @ A3 )
        & ~ ( member @ A @ C2 @ B6 ) ) ) ).

% Diff_iff
thf(fact_1892_DiffI,axiom,
    ! [A: $tType,C2: A,A3: set @ A,B6: set @ A] :
      ( ( member @ A @ C2 @ A3 )
     => ( ~ ( member @ A @ C2 @ B6 )
       => ( member @ A @ C2 @ ( minus_minus @ ( set @ A ) @ A3 @ B6 ) ) ) ) ).

% DiffI
thf(fact_1893_minus__set__def,axiom,
    ! [A: $tType] :
      ( ( minus_minus @ ( set @ A ) )
      = ( ^ [A7: set @ A,B7: set @ A] :
            ( collect @ A
            @ ( minus_minus @ ( A > $o )
              @ ^ [X: A] : ( member @ A @ X @ A7 )
              @ ^ [X: A] : ( member @ A @ X @ B7 ) ) ) ) ) ).

% minus_set_def
thf(fact_1894_set__diff__eq,axiom,
    ! [A: $tType] :
      ( ( minus_minus @ ( set @ A ) )
      = ( ^ [A7: set @ A,B7: set @ A] :
            ( collect @ A
            @ ^ [X: A] :
                ( ( member @ A @ X @ A7 )
                & ~ ( member @ A @ X @ B7 ) ) ) ) ) ).

% set_diff_eq
thf(fact_1895_DiffD2,axiom,
    ! [A: $tType,C2: A,A3: set @ A,B6: set @ A] :
      ( ( member @ A @ C2 @ ( minus_minus @ ( set @ A ) @ A3 @ B6 ) )
     => ~ ( member @ A @ C2 @ B6 ) ) ).

% DiffD2
thf(fact_1896_DiffD1,axiom,
    ! [A: $tType,C2: A,A3: set @ A,B6: set @ A] :
      ( ( member @ A @ C2 @ ( minus_minus @ ( set @ A ) @ A3 @ B6 ) )
     => ( member @ A @ C2 @ A3 ) ) ).

% DiffD1
thf(fact_1897_DiffE,axiom,
    ! [A: $tType,C2: A,A3: set @ A,B6: set @ A] :
      ( ( member @ A @ C2 @ ( minus_minus @ ( set @ A ) @ A3 @ B6 ) )
     => ~ ( ( member @ A @ C2 @ A3 )
         => ( member @ A @ C2 @ B6 ) ) ) ).

% DiffE
thf(fact_1898_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,
    ! [X2: vEBT_VEBT,Xa: nat,Y2: nat] :
      ( ( ( vEBT_T_m_e_m_b_e_r @ X2 @ Xa )
        = Y2 )
     => ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T5837161174952499735_r_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X2 @ Xa ) )
       => ( ! [A4: $o,B3: $o] :
              ( ( X2
                = ( vEBT_Leaf @ A4 @ B3 ) )
             => ( ( Y2
                  = ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) )
                    @ ( if @ nat
                      @ ( Xa
                        = ( zero_zero @ nat ) )
                      @ ( one_one @ nat )
                      @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( one_one @ nat ) ) ) ) )
               => ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T5837161174952499735_r_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A4 @ B3 ) @ Xa ) ) ) )
         => ( ! [Uu2: nat,Uv2: list @ vEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( X2
                  = ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw2 ) )
               => ( ( Y2
                    = ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
                 => ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T5837161174952499735_r_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw2 ) @ Xa ) ) ) )
           => ( ! [V3: product_prod @ nat @ nat,Uy2: list @ vEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( X2
                    = ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( zero_zero @ nat ) @ Uy2 @ Uz2 ) )
                 => ( ( Y2
                      = ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
                   => ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T5837161174952499735_r_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( zero_zero @ nat ) @ Uy2 @ Uz2 ) @ Xa ) ) ) )
             => ( ! [V3: product_prod @ nat @ nat,Vb2: list @ vEBT_VEBT,Vc: vEBT_VEBT] :
                    ( ( X2
                      = ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vb2 @ Vc ) )
                   => ( ( Y2
                        = ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
                     => ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T5837161174952499735_r_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vb2 @ Vc ) @ Xa ) ) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X2
                        = ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
                     => ( ( Y2
                          = ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( if @ nat @ ( Xa = Mi2 ) @ ( one_one @ nat ) @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( if @ nat @ ( Xa = Ma2 ) @ ( one_one @ nat ) @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi2 ) @ ( one_one @ nat ) @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( if @ nat @ ( ord_less @ nat @ Ma2 @ Xa ) @ ( one_one @ nat ) @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) @ ( if @ nat @ ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_T_m_e_m_b_e_r @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( one_one @ nat ) ) ) ) ) ) ) ) ) ) ) )
                       => ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T5837161174952499735_r_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) @ Xa ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.pelims
thf(fact_1899_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,
    ! [X2: vEBT_VEBT,Xa: nat,Y2: nat] :
      ( ( ( vEBT_T_m_e_m_b_e_r2 @ X2 @ Xa )
        = Y2 )
     => ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T8099345112685741742_r_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X2 @ Xa ) )
       => ( ! [A4: $o,B3: $o] :
              ( ( X2
                = ( vEBT_Leaf @ A4 @ B3 ) )
             => ( ( Y2
                  = ( one_one @ nat ) )
               => ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T8099345112685741742_r_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A4 @ B3 ) @ Xa ) ) ) )
         => ( ! [Uu2: nat,Uv2: list @ vEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( X2
                  = ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw2 ) )
               => ( ( Y2
                    = ( one_one @ nat ) )
                 => ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T8099345112685741742_r_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw2 ) @ Xa ) ) ) )
           => ( ! [V3: product_prod @ nat @ nat,Uy2: list @ vEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( X2
                    = ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( zero_zero @ nat ) @ Uy2 @ Uz2 ) )
                 => ( ( Y2
                      = ( one_one @ nat ) )
                   => ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T8099345112685741742_r_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( zero_zero @ nat ) @ Uy2 @ Uz2 ) @ Xa ) ) ) )
             => ( ! [V3: product_prod @ nat @ nat,Vb2: list @ vEBT_VEBT,Vc: vEBT_VEBT] :
                    ( ( X2
                      = ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vb2 @ Vc ) )
                   => ( ( Y2
                        = ( one_one @ nat ) )
                     => ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T8099345112685741742_r_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vb2 @ Vc ) @ Xa ) ) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X2
                        = ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
                     => ( ( Y2
                          = ( plus_plus @ nat @ ( one_one @ nat )
                            @ ( if @ nat @ ( Xa = Mi2 ) @ ( zero_zero @ nat )
                              @ ( if @ nat @ ( Xa = Ma2 ) @ ( zero_zero @ nat )
                                @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi2 ) @ ( zero_zero @ nat )
                                  @ ( if @ nat @ ( ord_less @ nat @ Ma2 @ Xa ) @ ( zero_zero @ nat )
                                    @ ( if @ nat
                                      @ ( ( ord_less @ nat @ Mi2 @ Xa )
                                        & ( ord_less @ nat @ Xa @ Ma2 ) )
                                      @ ( if @ nat @ ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) @ ( vEBT_T_m_e_m_b_e_r2 @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( zero_zero @ nat ) )
                                      @ ( zero_zero @ nat ) ) ) ) ) ) ) )
                       => ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T8099345112685741742_r_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) @ Xa ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.pelims
thf(fact_1900_vebt__member_Opelims_I1_J,axiom,
    ! [X2: vEBT_VEBT,Xa: nat,Y2: $o] :
      ( ( ( vEBT_vebt_member @ X2 @ Xa )
        = Y2 )
     => ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X2 @ Xa ) )
       => ( ! [A4: $o,B3: $o] :
              ( ( X2
                = ( vEBT_Leaf @ A4 @ B3 ) )
             => ( ( Y2
                  = ( ( ( Xa
                        = ( zero_zero @ nat ) )
                     => A4 )
                    & ( ( Xa
                       != ( zero_zero @ nat ) )
                     => ( ( ( Xa
                            = ( one_one @ nat ) )
                         => B3 )
                        & ( Xa
                          = ( one_one @ nat ) ) ) ) ) )
               => ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A4 @ B3 ) @ Xa ) ) ) )
         => ( ! [Uu2: nat,Uv2: list @ vEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( X2
                  = ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw2 ) )
               => ( ~ Y2
                 => ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw2 ) @ Xa ) ) ) )
           => ( ! [V3: product_prod @ nat @ nat,Uy2: list @ vEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( X2
                    = ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( zero_zero @ nat ) @ Uy2 @ Uz2 ) )
                 => ( ~ Y2
                   => ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( zero_zero @ nat ) @ Uy2 @ Uz2 ) @ Xa ) ) ) )
             => ( ! [V3: product_prod @ nat @ nat,Vb2: list @ vEBT_VEBT,Vc: vEBT_VEBT] :
                    ( ( X2
                      = ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vb2 @ Vc ) )
                   => ( ~ Y2
                     => ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vb2 @ Vc ) @ Xa ) ) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X2
                        = ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
                     => ( ( Y2
                          = ( ( Xa != Mi2 )
                           => ( ( Xa != Ma2 )
                             => ( ~ ( ord_less @ nat @ Xa @ Mi2 )
                                & ( ~ ( ord_less @ nat @ Xa @ Mi2 )
                                 => ( ~ ( ord_less @ nat @ Ma2 @ Xa )
                                    & ( ~ ( ord_less @ nat @ Ma2 @ Xa )
                                     => ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
                                         => ( vEBT_vebt_member @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
                                        & ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) ) ) ) ) )
                       => ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) @ Xa ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(1)
thf(fact_1901_vebt__member_Opelims_I3_J,axiom,
    ! [X2: vEBT_VEBT,Xa: nat] :
      ( ~ ( vEBT_vebt_member @ X2 @ Xa )
     => ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X2 @ Xa ) )
       => ( ! [A4: $o,B3: $o] :
              ( ( X2
                = ( vEBT_Leaf @ A4 @ B3 ) )
             => ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A4 @ B3 ) @ Xa ) )
               => ( ( ( Xa
                      = ( zero_zero @ nat ) )
                   => A4 )
                  & ( ( Xa
                     != ( zero_zero @ nat ) )
                   => ( ( ( Xa
                          = ( one_one @ nat ) )
                       => B3 )
                      & ( Xa
                        = ( one_one @ nat ) ) ) ) ) ) )
         => ( ! [Uu2: nat,Uv2: list @ vEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( X2
                  = ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw2 ) )
               => ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw2 ) @ Xa ) ) )
           => ( ! [V3: product_prod @ nat @ nat,Uy2: list @ vEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( X2
                    = ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( zero_zero @ nat ) @ Uy2 @ Uz2 ) )
                 => ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( zero_zero @ nat ) @ Uy2 @ Uz2 ) @ Xa ) ) )
             => ( ! [V3: product_prod @ nat @ nat,Vb2: list @ vEBT_VEBT,Vc: vEBT_VEBT] :
                    ( ( X2
                      = ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vb2 @ Vc ) )
                   => ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vb2 @ Vc ) @ Xa ) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X2
                        = ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
                     => ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) @ Xa ) )
                       => ( ( Xa != Mi2 )
                         => ( ( Xa != Ma2 )
                           => ( ~ ( ord_less @ nat @ Xa @ Mi2 )
                              & ( ~ ( ord_less @ nat @ Xa @ Mi2 )
                               => ( ~ ( ord_less @ nat @ Ma2 @ Xa )
                                  & ( ~ ( ord_less @ nat @ Ma2 @ Xa )
                                   => ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
                                       => ( vEBT_vebt_member @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
                                      & ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(3)
thf(fact_1902_VEBT__internal_Onaive__member_Opelims_I1_J,axiom,
    ! [X2: vEBT_VEBT,Xa: nat,Y2: $o] :
      ( ( ( vEBT_V5719532721284313246member @ X2 @ Xa )
        = Y2 )
     => ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V5765760719290551771er_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X2 @ Xa ) )
       => ( ! [A4: $o,B3: $o] :
              ( ( X2
                = ( vEBT_Leaf @ A4 @ B3 ) )
             => ( ( Y2
                  = ( ( ( Xa
                        = ( zero_zero @ nat ) )
                     => A4 )
                    & ( ( Xa
                       != ( zero_zero @ nat ) )
                     => ( ( ( Xa
                            = ( one_one @ nat ) )
                         => B3 )
                        & ( Xa
                          = ( one_one @ nat ) ) ) ) ) )
               => ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V5765760719290551771er_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A4 @ B3 ) @ Xa ) ) ) )
         => ( ! [Uu2: option @ ( product_prod @ nat @ nat ),Uv2: list @ vEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( X2
                  = ( vEBT_Node @ Uu2 @ ( zero_zero @ nat ) @ Uv2 @ Uw2 ) )
               => ( ~ Y2
                 => ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V5765760719290551771er_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Uu2 @ ( zero_zero @ nat ) @ Uv2 @ Uw2 ) @ Xa ) ) ) )
           => ~ ! [Uy2: option @ ( product_prod @ nat @ nat ),V3: nat,TreeList2: list @ vEBT_VEBT,S: vEBT_VEBT] :
                  ( ( X2
                    = ( vEBT_Node @ Uy2 @ ( suc @ V3 ) @ TreeList2 @ S ) )
                 => ( ( Y2
                      = ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
                         => ( vEBT_V5719532721284313246member @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
                        & ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) )
                   => ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V5765760719290551771er_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Uy2 @ ( suc @ V3 ) @ TreeList2 @ S ) @ Xa ) ) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(1)
thf(fact_1903_VEBT__internal_Onaive__member_Opelims_I2_J,axiom,
    ! [X2: vEBT_VEBT,Xa: nat] :
      ( ( vEBT_V5719532721284313246member @ X2 @ Xa )
     => ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V5765760719290551771er_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X2 @ Xa ) )
       => ( ! [A4: $o,B3: $o] :
              ( ( X2
                = ( vEBT_Leaf @ A4 @ B3 ) )
             => ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V5765760719290551771er_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A4 @ B3 ) @ Xa ) )
               => ~ ( ( ( Xa
                        = ( zero_zero @ nat ) )
                     => A4 )
                    & ( ( Xa
                       != ( zero_zero @ nat ) )
                     => ( ( ( Xa
                            = ( one_one @ nat ) )
                         => B3 )
                        & ( Xa
                          = ( one_one @ nat ) ) ) ) ) ) )
         => ~ ! [Uy2: option @ ( product_prod @ nat @ nat ),V3: nat,TreeList2: list @ vEBT_VEBT,S: vEBT_VEBT] :
                ( ( X2
                  = ( vEBT_Node @ Uy2 @ ( suc @ V3 ) @ TreeList2 @ S ) )
               => ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V5765760719290551771er_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Uy2 @ ( suc @ V3 ) @ TreeList2 @ S ) @ Xa ) )
                 => ~ ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
                       => ( vEBT_V5719532721284313246member @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
                      & ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(2)
thf(fact_1904_VEBT__internal_Onaive__member_Opelims_I3_J,axiom,
    ! [X2: vEBT_VEBT,Xa: nat] :
      ( ~ ( vEBT_V5719532721284313246member @ X2 @ Xa )
     => ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V5765760719290551771er_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X2 @ Xa ) )
       => ( ! [A4: $o,B3: $o] :
              ( ( X2
                = ( vEBT_Leaf @ A4 @ B3 ) )
             => ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V5765760719290551771er_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A4 @ B3 ) @ Xa ) )
               => ( ( ( Xa
                      = ( zero_zero @ nat ) )
                   => A4 )
                  & ( ( Xa
                     != ( zero_zero @ nat ) )
                   => ( ( ( Xa
                          = ( one_one @ nat ) )
                       => B3 )
                      & ( Xa
                        = ( one_one @ nat ) ) ) ) ) ) )
         => ( ! [Uu2: option @ ( product_prod @ nat @ nat ),Uv2: list @ vEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( X2
                  = ( vEBT_Node @ Uu2 @ ( zero_zero @ nat ) @ Uv2 @ Uw2 ) )
               => ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V5765760719290551771er_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Uu2 @ ( zero_zero @ nat ) @ Uv2 @ Uw2 ) @ Xa ) ) )
           => ~ ! [Uy2: option @ ( product_prod @ nat @ nat ),V3: nat,TreeList2: list @ vEBT_VEBT,S: vEBT_VEBT] :
                  ( ( X2
                    = ( vEBT_Node @ Uy2 @ ( suc @ V3 ) @ TreeList2 @ S ) )
                 => ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V5765760719290551771er_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Uy2 @ ( suc @ V3 ) @ TreeList2 @ S ) @ Xa ) )
                   => ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
                       => ( vEBT_V5719532721284313246member @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
                      & ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(3)
thf(fact_1905_vebt__member_Opelims_I2_J,axiom,
    ! [X2: vEBT_VEBT,Xa: nat] :
      ( ( vEBT_vebt_member @ X2 @ Xa )
     => ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X2 @ Xa ) )
       => ( ! [A4: $o,B3: $o] :
              ( ( X2
                = ( vEBT_Leaf @ A4 @ B3 ) )
             => ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A4 @ B3 ) @ Xa ) )
               => ~ ( ( ( Xa
                        = ( zero_zero @ nat ) )
                     => A4 )
                    & ( ( Xa
                       != ( zero_zero @ nat ) )
                     => ( ( ( Xa
                            = ( one_one @ nat ) )
                         => B3 )
                        & ( Xa
                          = ( one_one @ nat ) ) ) ) ) ) )
         => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( X2
                  = ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
               => ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) @ Xa ) )
                 => ~ ( ( Xa != Mi2 )
                     => ( ( Xa != Ma2 )
                       => ( ~ ( ord_less @ nat @ Xa @ Mi2 )
                          & ( ~ ( ord_less @ nat @ Xa @ Mi2 )
                           => ( ~ ( ord_less @ nat @ Ma2 @ Xa )
                              & ( ~ ( ord_less @ nat @ Ma2 @ Xa )
                               => ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
                                   => ( vEBT_vebt_member @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
                                  & ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(2)
thf(fact_1906_VEBT__internal_Omembermima_Opelims_I3_J,axiom,
    ! [X2: vEBT_VEBT,Xa: nat] :
      ( ~ ( vEBT_VEBT_membermima @ X2 @ Xa )
     => ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X2 @ Xa ) )
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( X2
                = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
             => ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa ) ) )
         => ( ! [Ux2: list @ vEBT_VEBT,Uy2: vEBT_VEBT] :
                ( ( X2
                  = ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( zero_zero @ nat ) @ Ux2 @ Uy2 ) )
               => ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( zero_zero @ nat ) @ Ux2 @ Uy2 ) @ Xa ) ) )
           => ( ! [Mi2: nat,Ma2: nat,Va2: list @ vEBT_VEBT,Vb2: vEBT_VEBT] :
                  ( ( X2
                    = ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ Va2 @ Vb2 ) )
                 => ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ Va2 @ Vb2 ) @ Xa ) )
                   => ( ( Xa = Mi2 )
                      | ( Xa = Ma2 ) ) ) )
             => ( ! [Mi2: nat,Ma2: nat,V3: nat,TreeList2: list @ vEBT_VEBT,Vc: vEBT_VEBT] :
                    ( ( X2
                      = ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ V3 ) @ TreeList2 @ Vc ) )
                   => ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ V3 ) @ TreeList2 @ Vc ) @ Xa ) )
                     => ( ( Xa = Mi2 )
                        | ( Xa = Ma2 )
                        | ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
                           => ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
                          & ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) )
               => ~ ! [V3: nat,TreeList2: list @ vEBT_VEBT,Vd2: vEBT_VEBT] :
                      ( ( X2
                        = ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ V3 ) @ TreeList2 @ Vd2 ) )
                     => ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ V3 ) @ TreeList2 @ Vd2 ) @ Xa ) )
                       => ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
                           => ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
                          & ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(3)
thf(fact_1907_VEBT__internal_Omembermima_Opelims_I1_J,axiom,
    ! [X2: vEBT_VEBT,Xa: nat,Y2: $o] :
      ( ( ( vEBT_VEBT_membermima @ X2 @ Xa )
        = Y2 )
     => ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X2 @ Xa ) )
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( X2
                = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
             => ( ~ Y2
               => ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa ) ) ) )
         => ( ! [Ux2: list @ vEBT_VEBT,Uy2: vEBT_VEBT] :
                ( ( X2
                  = ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( zero_zero @ nat ) @ Ux2 @ Uy2 ) )
               => ( ~ Y2
                 => ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( zero_zero @ nat ) @ Ux2 @ Uy2 ) @ Xa ) ) ) )
           => ( ! [Mi2: nat,Ma2: nat,Va2: list @ vEBT_VEBT,Vb2: vEBT_VEBT] :
                  ( ( X2
                    = ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ Va2 @ Vb2 ) )
                 => ( ( Y2
                      = ( ( Xa = Mi2 )
                        | ( Xa = Ma2 ) ) )
                   => ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ Va2 @ Vb2 ) @ Xa ) ) ) )
             => ( ! [Mi2: nat,Ma2: nat,V3: nat,TreeList2: list @ vEBT_VEBT,Vc: vEBT_VEBT] :
                    ( ( X2
                      = ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ V3 ) @ TreeList2 @ Vc ) )
                   => ( ( Y2
                        = ( ( Xa = Mi2 )
                          | ( Xa = Ma2 )
                          | ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
                             => ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
                            & ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) )
                     => ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ V3 ) @ TreeList2 @ Vc ) @ Xa ) ) ) )
               => ~ ! [V3: nat,TreeList2: list @ vEBT_VEBT,Vd2: vEBT_VEBT] :
                      ( ( X2
                        = ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ V3 ) @ TreeList2 @ Vd2 ) )
                     => ( ( Y2
                          = ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
                             => ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
                            & ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) )
                       => ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ V3 ) @ TreeList2 @ Vd2 ) @ Xa ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(1)
thf(fact_1908_VEBT__internal_Omembermima_Opelims_I2_J,axiom,
    ! [X2: vEBT_VEBT,Xa: nat] :
      ( ( vEBT_VEBT_membermima @ X2 @ Xa )
     => ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X2 @ Xa ) )
       => ( ! [Mi2: nat,Ma2: nat,Va2: list @ vEBT_VEBT,Vb2: vEBT_VEBT] :
              ( ( X2
                = ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ Va2 @ Vb2 ) )
             => ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ Va2 @ Vb2 ) @ Xa ) )
               => ~ ( ( Xa = Mi2 )
                    | ( Xa = Ma2 ) ) ) )
         => ( ! [Mi2: nat,Ma2: nat,V3: nat,TreeList2: list @ vEBT_VEBT,Vc: vEBT_VEBT] :
                ( ( X2
                  = ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ V3 ) @ TreeList2 @ Vc ) )
               => ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ V3 ) @ TreeList2 @ Vc ) @ Xa ) )
                 => ~ ( ( Xa = Mi2 )
                      | ( Xa = Ma2 )
                      | ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
                         => ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
                        & ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) )
           => ~ ! [V3: nat,TreeList2: list @ vEBT_VEBT,Vd2: vEBT_VEBT] :
                  ( ( X2
                    = ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ V3 ) @ TreeList2 @ Vd2 ) )
                 => ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ V3 ) @ TreeList2 @ Vd2 ) @ Xa ) )
                   => ~ ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
                         => ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
                        & ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(2)
thf(fact_1909_minus__apply,axiom,
    ! [B: $tType,A: $tType] :
      ( ( minus @ B )
     => ( ( minus_minus @ ( A > B ) )
        = ( ^ [A7: A > B,B7: A > B,X: A] : ( minus_minus @ B @ ( A7 @ X ) @ ( B7 @ X ) ) ) ) ) ).

% minus_apply
thf(fact_1910_pos__mult__pos__ge,axiom,
    ! [X2: int,N2: int] :
      ( ( ord_less @ int @ ( zero_zero @ int ) @ X2 )
     => ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ N2 )
       => ( ord_less_eq @ int @ ( times_times @ int @ N2 @ ( one_one @ int ) ) @ ( times_times @ int @ N2 @ X2 ) ) ) ) ).

% pos_mult_pos_ge
thf(fact_1911_max__less__iff__conj,axiom,
    ! [A: $tType] :
      ( ( linorder @ A )
     => ! [X2: A,Y2: A,Z: A] :
          ( ( ord_less @ A @ ( ord_max @ A @ X2 @ Y2 ) @ Z )
          = ( ( ord_less @ A @ X2 @ Z )
            & ( ord_less @ A @ Y2 @ Z ) ) ) ) ).

% max_less_iff_conj
thf(fact_1912_max_Oabsorb4,axiom,
    ! [A: $tType] :
      ( ( linorder @ A )
     => ! [A2: A,B2: A] :
          ( ( ord_less @ A @ A2 @ B2 )
         => ( ( ord_max @ A @ A2 @ B2 )
            = B2 ) ) ) ).

% max.absorb4
thf(fact_1913_max_Oabsorb1,axiom,
    ! [A: $tType] :
      ( ( linorder @ A )
     => ! [B2: A,A2: A] :
          ( ( ord_less_eq @ A @ B2 @ A2 )
         => ( ( ord_max @ A @ A2 @ B2 )
            = A2 ) ) ) ).

% max.absorb1
thf(fact_1914_max_Oabsorb2,axiom,
    ! [A: $tType] :
      ( ( linorder @ A )
     => ! [A2: A,B2: A] :
          ( ( ord_less_eq @ A @ A2 @ B2 )
         => ( ( ord_max @ A @ A2 @ B2 )
            = B2 ) ) ) ).

% max.absorb2
thf(fact_1915_max_Obounded__iff,axiom,
    ! [A: $tType] :
      ( ( linorder @ A )
     => ! [B2: A,C2: A,A2: A] :
          ( ( ord_less_eq @ A @ ( ord_max @ A @ B2 @ C2 ) @ A2 )
          = ( ( ord_less_eq @ A @ B2 @ A2 )
            & ( ord_less_eq @ A @ C2 @ A2 ) ) ) ) ).

% max.bounded_iff
thf(fact_1916_max_Oabsorb3,axiom,
    ! [A: $tType] :
      ( ( linorder @ A )
     => ! [B2: A,A2: A] :
          ( ( ord_less @ A @ B2 @ A2 )
         => ( ( ord_max @ A @ A2 @ B2 )
            = A2 ) ) ) ).

% max.absorb3
thf(fact_1917_max_OcoboundedI2,axiom,
    ! [A: $tType] :
      ( ( linorder @ A )
     => ! [C2: A,B2: A,A2: A] :
          ( ( ord_less_eq @ A @ C2 @ B2 )
         => ( ord_less_eq @ A @ C2 @ ( ord_max @ A @ A2 @ B2 ) ) ) ) ).

% max.coboundedI2
thf(fact_1918_max_OcoboundedI1,axiom,
    ! [A: $tType] :
      ( ( linorder @ A )
     => ! [C2: A,A2: A,B2: A] :
          ( ( ord_less_eq @ A @ C2 @ A2 )
         => ( ord_less_eq @ A @ C2 @ ( ord_max @ A @ A2 @ B2 ) ) ) ) ).

% max.coboundedI1
thf(fact_1919_max_Oabsorb__iff2,axiom,
    ! [A: $tType] :
      ( ( linorder @ A )
     => ( ( ord_less_eq @ A )
        = ( ^ [A5: A,B5: A] :
              ( ( ord_max @ A @ A5 @ B5 )
              = B5 ) ) ) ) ).

% max.absorb_iff2
thf(fact_1920_max_Oabsorb__iff1,axiom,
    ! [A: $tType] :
      ( ( linorder @ A )
     => ( ( ord_less_eq @ A )
        = ( ^ [B5: A,A5: A] :
              ( ( ord_max @ A @ A5 @ B5 )
              = A5 ) ) ) ) ).

% max.absorb_iff1
thf(fact_1921_le__max__iff__disj,axiom,
    ! [A: $tType] :
      ( ( linorder @ A )
     => ! [Z: A,X2: A,Y2: A] :
          ( ( ord_less_eq @ A @ Z @ ( ord_max @ A @ X2 @ Y2 ) )
          = ( ( ord_less_eq @ A @ Z @ X2 )
            | ( ord_less_eq @ A @ Z @ Y2 ) ) ) ) ).

% le_max_iff_disj
thf(fact_1922_max_Ocobounded2,axiom,
    ! [A: $tType] :
      ( ( linorder @ A )
     => ! [B2: A,A2: A] : ( ord_less_eq @ A @ B2 @ ( ord_max @ A @ A2 @ B2 ) ) ) ).

% max.cobounded2
thf(fact_1923_max_Ocobounded1,axiom,
    ! [A: $tType] :
      ( ( linorder @ A )
     => ! [A2: A,B2: A] : ( ord_less_eq @ A @ A2 @ ( ord_max @ A @ A2 @ B2 ) ) ) ).

% max.cobounded1
thf(fact_1924_max_Oorder__iff,axiom,
    ! [A: $tType] :
      ( ( linorder @ A )
     => ( ( ord_less_eq @ A )
        = ( ^ [B5: A,A5: A] :
              ( A5
              = ( ord_max @ A @ A5 @ B5 ) ) ) ) ) ).

% max.order_iff
thf(fact_1925_max_OboundedI,axiom,
    ! [A: $tType] :
      ( ( linorder @ A )
     => ! [B2: A,A2: A,C2: A] :
          ( ( ord_less_eq @ A @ B2 @ A2 )
         => ( ( ord_less_eq @ A @ C2 @ A2 )
           => ( ord_less_eq @ A @ ( ord_max @ A @ B2 @ C2 ) @ A2 ) ) ) ) ).

% max.boundedI
thf(fact_1926_max_OboundedE,axiom,
    ! [A: $tType] :
      ( ( linorder @ A )
     => ! [B2: A,C2: A,A2: A] :
          ( ( ord_less_eq @ A @ ( ord_max @ A @ B2 @ C2 ) @ A2 )
         => ~ ( ( ord_less_eq @ A @ B2 @ A2 )
             => ~ ( ord_less_eq @ A @ C2 @ A2 ) ) ) ) ).

% max.boundedE
thf(fact_1927_max_OorderI,axiom,
    ! [A: $tType] :
      ( ( linorder @ A )
     => ! [A2: A,B2: A] :
          ( ( A2
            = ( ord_max @ A @ A2 @ B2 ) )
         => ( ord_less_eq @ A @ B2 @ A2 ) ) ) ).

% max.orderI
thf(fact_1928_max_OorderE,axiom,
    ! [A: $tType] :
      ( ( linorder @ A )
     => ! [B2: A,A2: A] :
          ( ( ord_less_eq @ A @ B2 @ A2 )
         => ( A2
            = ( ord_max @ A @ A2 @ B2 ) ) ) ) ).

% max.orderE
thf(fact_1929_max_Omono,axiom,
    ! [A: $tType] :
      ( ( linorder @ A )
     => ! [C2: A,A2: A,D2: A,B2: A] :
          ( ( ord_less_eq @ A @ C2 @ A2 )
         => ( ( ord_less_eq @ A @ D2 @ B2 )
           => ( ord_less_eq @ A @ ( ord_max @ A @ C2 @ D2 ) @ ( ord_max @ A @ A2 @ B2 ) ) ) ) ) ).

% max.mono
thf(fact_1930_less__max__iff__disj,axiom,
    ! [A: $tType] :
      ( ( linorder @ A )
     => ! [Z: A,X2: A,Y2: A] :
          ( ( ord_less @ A @ Z @ ( ord_max @ A @ X2 @ Y2 ) )
          = ( ( ord_less @ A @ Z @ X2 )
            | ( ord_less @ A @ Z @ Y2 ) ) ) ) ).

% less_max_iff_disj
thf(fact_1931_max_Ostrict__boundedE,axiom,
    ! [A: $tType] :
      ( ( linorder @ A )
     => ! [B2: A,C2: A,A2: A] :
          ( ( ord_less @ A @ ( ord_max @ A @ B2 @ C2 ) @ A2 )
         => ~ ( ( ord_less @ A @ B2 @ A2 )
             => ~ ( ord_less @ A @ C2 @ A2 ) ) ) ) ).

% max.strict_boundedE
thf(fact_1932_max_Ostrict__order__iff,axiom,
    ! [A: $tType] :
      ( ( linorder @ A )
     => ( ( ord_less @ A )
        = ( ^ [B5: A,A5: A] :
              ( ( A5
                = ( ord_max @ A @ A5 @ B5 ) )
              & ( A5 != B5 ) ) ) ) ) ).

% max.strict_order_iff
thf(fact_1933_max_Ostrict__coboundedI1,axiom,
    ! [A: $tType] :
      ( ( linorder @ A )
     => ! [C2: A,A2: A,B2: A] :
          ( ( ord_less @ A @ C2 @ A2 )
         => ( ord_less @ A @ C2 @ ( ord_max @ A @ A2 @ B2 ) ) ) ) ).

% max.strict_coboundedI1
thf(fact_1934_max_Ostrict__coboundedI2,axiom,
    ! [A: $tType] :
      ( ( linorder @ A )
     => ! [C2: A,B2: A,A2: A] :
          ( ( ord_less @ A @ C2 @ B2 )
         => ( ord_less @ A @ C2 @ ( ord_max @ A @ A2 @ B2 ) ) ) ) ).

% max.strict_coboundedI2
thf(fact_1935_fun__diff__def,axiom,
    ! [B: $tType,A: $tType] :
      ( ( minus @ B )
     => ( ( minus_minus @ ( A > B ) )
        = ( ^ [A7: A > B,B7: A > B,X: A] : ( minus_minus @ B @ ( A7 @ X ) @ ( B7 @ X ) ) ) ) ) ).

% fun_diff_def
thf(fact_1936_p2__eq__1,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [N2: nat] :
          ( ( ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N2 )
            = ( one_one @ ( word @ A ) ) )
          = ( N2
            = ( zero_zero @ nat ) ) ) ) ).

% p2_eq_1
thf(fact_1937_word__unat__power,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [N2: nat] :
          ( ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N2 )
          = ( semiring_1_of_nat @ ( word @ A ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ) ) ).

% word_unat_power
thf(fact_1938_word__less__two__pow__divD,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [X2: word @ A,N2: nat,M: nat] :
          ( ( ord_less @ ( word @ A ) @ X2 @ ( divide_divide @ ( word @ A ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N2 ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ M ) ) )
         => ( ( ord_less_eq @ nat @ M @ N2 )
            & ( ord_less @ ( word @ A ) @ X2 @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ N2 @ M ) ) ) ) ) ) ).

% word_less_two_pow_divD
thf(fact_1939_less__1__helper,axiom,
    ! [N2: int,M: int] :
      ( ( ord_less_eq @ int @ N2 @ M )
     => ( ord_less @ int @ ( minus_minus @ int @ N2 @ ( one_one @ int ) ) @ M ) ) ).

% less_1_helper
thf(fact_1940_enat__ord__number_I1_J,axiom,
    ! [M: num,N2: num] :
      ( ( ord_less_eq @ extended_enat @ ( numeral_numeral @ extended_enat @ M ) @ ( numeral_numeral @ extended_enat @ N2 ) )
      = ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ M ) @ ( numeral_numeral @ nat @ N2 ) ) ) ).

% enat_ord_number(1)
thf(fact_1941_heigt__uplog__rel,axiom,
    ! [T2: vEBT_VEBT,N2: nat] :
      ( ( vEBT_invar_vebt @ T2 @ N2 )
     => ( ( semiring_1_of_nat @ int @ ( vEBT_VEBT_height @ T2 ) )
        = ( archimedean_ceiling @ real @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ real @ N2 ) ) ) ) ) ).

% heigt_uplog_rel
thf(fact_1942_word__size__gt__0,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [W: word @ A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( word @ A ) @ W ) ) ) ).

% word_size_gt_0
thf(fact_1943_enat__ord__number_I2_J,axiom,
    ! [M: num,N2: num] :
      ( ( ord_less @ extended_enat @ ( numeral_numeral @ extended_enat @ M ) @ ( numeral_numeral @ extended_enat @ N2 ) )
      = ( ord_less @ nat @ ( numeral_numeral @ nat @ M ) @ ( numeral_numeral @ nat @ N2 ) ) ) ).

% enat_ord_number(2)
thf(fact_1944_set__bit__0,axiom,
    ! [A: $tType] :
      ( ( bit_se359711467146920520ations @ A )
     => ! [A2: A] :
          ( ( bit_se5668285175392031749et_bit @ A @ ( zero_zero @ nat ) @ A2 )
          = ( plus_plus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ).

% set_bit_0
thf(fact_1945_assert_H__bind__rule,axiom,
    ! [A: $tType,P: assn,Phi: $o,C2: heap_Time_Heap @ A,Q: A > assn] :
      ( ! [H3: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
          ( ( rep_assn @ P @ H3 )
         => Phi )
     => ( ( Phi
         => ( hoare_hoare_triple @ A @ P @ C2 @ Q ) )
       => ( hoare_hoare_triple @ A @ P
          @ ( heap_Time_bind @ product_unit @ A @ ( refine_Imp_assert @ Phi )
            @ ^ [Uu3: product_unit] : C2 )
          @ Q ) ) ) ).

% assert'_bind_rule
thf(fact_1946_succ__bound__size__univ,axiom,
    ! [T2: vEBT_VEBT,N2: nat,U: real,X2: nat] :
      ( ( vEBT_invar_vebt @ T2 @ N2 )
     => ( ( U
          = ( power_power @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ N2 ) )
       => ( ord_less_eq @ real @ ( semiring_1_of_nat @ real @ ( vEBT_T_s_u_c_c @ T2 @ X2 ) ) @ ( plus_plus @ real @ ( numeral_numeral @ real @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ one2 ) ) ) ) ) ) @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ one2 ) ) ) ) ) @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ U ) ) ) ) ) ) ) ).

% succ_bound_size_univ
thf(fact_1947_div__of__0__id,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [N2: word @ A] :
          ( ( divide_divide @ ( word @ A ) @ ( zero_zero @ ( word @ A ) ) @ N2 )
          = ( zero_zero @ ( word @ A ) ) ) ) ).

% div_of_0_id
thf(fact_1948_set__bit__nonnegative__int__iff,axiom,
    ! [N2: nat,K: int] :
      ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_se5668285175392031749et_bit @ int @ N2 @ K ) )
      = ( ord_less_eq @ int @ ( zero_zero @ int ) @ K ) ) ).

% set_bit_nonnegative_int_iff
thf(fact_1949_word__le__sub1__numberof,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [W: num] :
          ( ( ( numeral_numeral @ ( word @ A ) @ W )
           != ( zero_zero @ ( word @ A ) ) )
         => ( ( ord_less_eq @ ( word @ A ) @ ( one_one @ ( word @ A ) ) @ ( numeral_numeral @ ( word @ A ) @ W ) )
            = ( ord_less_eq @ ( word @ A ) @ ( zero_zero @ ( word @ A ) ) @ ( minus_minus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ W ) @ ( one_one @ ( word @ A ) ) ) ) ) ) ) ).

% word_le_sub1_numberof
thf(fact_1950_word__gt__0__no,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [Y2: num] :
          ( ( ord_less @ ( word @ A ) @ ( zero_zero @ ( word @ A ) ) @ ( numeral_numeral @ ( word @ A ) @ Y2 ) )
          = ( ( zero_zero @ ( word @ A ) )
           != ( numeral_numeral @ ( word @ A ) @ Y2 ) ) ) ) ).

% word_gt_0_no
thf(fact_1951_word__less__1,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [X2: word @ A] :
          ( ( ord_less @ ( word @ A ) @ X2 @ ( one_one @ ( word @ A ) ) )
          = ( X2
            = ( zero_zero @ ( word @ A ) ) ) ) ) ).

% word_less_1
thf(fact_1952_ceiling__numeral,axiom,
    ! [A: $tType] :
      ( ( archim2362893244070406136eiling @ A )
     => ! [V: num] :
          ( ( archimedean_ceiling @ A @ ( numeral_numeral @ A @ V ) )
          = ( numeral_numeral @ int @ V ) ) ) ).

% ceiling_numeral
thf(fact_1953_ceiling__one,axiom,
    ! [A: $tType] :
      ( ( archim2362893244070406136eiling @ A )
     => ( ( archimedean_ceiling @ A @ ( one_one @ A ) )
        = ( one_one @ int ) ) ) ).

% ceiling_one
thf(fact_1954_word__less__sub1__numberof,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [W: num] :
          ( ( ( numeral_numeral @ ( word @ A ) @ W )
           != ( zero_zero @ ( word @ A ) ) )
         => ( ( ord_less @ ( word @ A ) @ ( one_one @ ( word @ A ) ) @ ( numeral_numeral @ ( word @ A ) @ W ) )
            = ( ord_less @ ( word @ A ) @ ( zero_zero @ ( word @ A ) ) @ ( minus_minus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ W ) @ ( one_one @ ( word @ A ) ) ) ) ) ) ) ).

% word_less_sub1_numberof
thf(fact_1955_ceiling__le__zero,axiom,
    ! [A: $tType] :
      ( ( archim2362893244070406136eiling @ A )
     => ! [X2: A] :
          ( ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ X2 ) @ ( zero_zero @ int ) )
          = ( ord_less_eq @ A @ X2 @ ( zero_zero @ A ) ) ) ) ).

% ceiling_le_zero
thf(fact_1956_ceiling__le__numeral,axiom,
    ! [A: $tType] :
      ( ( archim2362893244070406136eiling @ A )
     => ! [X2: A,V: num] :
          ( ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ X2 ) @ ( numeral_numeral @ int @ V ) )
          = ( ord_less_eq @ A @ X2 @ ( numeral_numeral @ A @ V ) ) ) ) ).

% ceiling_le_numeral
thf(fact_1957_zero__less__ceiling,axiom,
    ! [A: $tType] :
      ( ( archim2362893244070406136eiling @ A )
     => ! [X2: A] :
          ( ( ord_less @ int @ ( zero_zero @ int ) @ ( archimedean_ceiling @ A @ X2 ) )
          = ( ord_less @ A @ ( zero_zero @ A ) @ X2 ) ) ) ).

% zero_less_ceiling
thf(fact_1958_numeral__less__ceiling,axiom,
    ! [A: $tType] :
      ( ( archim2362893244070406136eiling @ A )
     => ! [V: num,X2: A] :
          ( ( ord_less @ int @ ( numeral_numeral @ int @ V ) @ ( archimedean_ceiling @ A @ X2 ) )
          = ( ord_less @ A @ ( numeral_numeral @ A @ V ) @ X2 ) ) ) ).

% numeral_less_ceiling
thf(fact_1959_ceiling__less__one,axiom,
    ! [A: $tType] :
      ( ( archim2362893244070406136eiling @ A )
     => ! [X2: A] :
          ( ( ord_less @ int @ ( archimedean_ceiling @ A @ X2 ) @ ( one_one @ int ) )
          = ( ord_less_eq @ A @ X2 @ ( zero_zero @ A ) ) ) ) ).

% ceiling_less_one
thf(fact_1960_one__le__ceiling,axiom,
    ! [A: $tType] :
      ( ( archim2362893244070406136eiling @ A )
     => ! [X2: A] :
          ( ( ord_less_eq @ int @ ( one_one @ int ) @ ( archimedean_ceiling @ A @ X2 ) )
          = ( ord_less @ A @ ( zero_zero @ A ) @ X2 ) ) ) ).

% one_le_ceiling
thf(fact_1961_ceiling__add__numeral,axiom,
    ! [A: $tType] :
      ( ( archim2362893244070406136eiling @ A )
     => ! [X2: A,V: num] :
          ( ( archimedean_ceiling @ A @ ( plus_plus @ A @ X2 @ ( numeral_numeral @ A @ V ) ) )
          = ( plus_plus @ int @ ( archimedean_ceiling @ A @ X2 ) @ ( numeral_numeral @ int @ V ) ) ) ) ).

% ceiling_add_numeral
thf(fact_1962_ceiling__le__one,axiom,
    ! [A: $tType] :
      ( ( archim2362893244070406136eiling @ A )
     => ! [X2: A] :
          ( ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ X2 ) @ ( one_one @ int ) )
          = ( ord_less_eq @ A @ X2 @ ( one_one @ A ) ) ) ) ).

% ceiling_le_one
thf(fact_1963_one__less__ceiling,axiom,
    ! [A: $tType] :
      ( ( archim2362893244070406136eiling @ A )
     => ! [X2: A] :
          ( ( ord_less @ int @ ( one_one @ int ) @ ( archimedean_ceiling @ A @ X2 ) )
          = ( ord_less @ A @ ( one_one @ A ) @ X2 ) ) ) ).

% one_less_ceiling
thf(fact_1964_ceiling__add__one,axiom,
    ! [A: $tType] :
      ( ( archim2362893244070406136eiling @ A )
     => ! [X2: A] :
          ( ( archimedean_ceiling @ A @ ( plus_plus @ A @ X2 @ ( one_one @ A ) ) )
          = ( plus_plus @ int @ ( archimedean_ceiling @ A @ X2 ) @ ( one_one @ int ) ) ) ) ).

% ceiling_add_one
thf(fact_1965_ceiling__diff__numeral,axiom,
    ! [A: $tType] :
      ( ( archim2362893244070406136eiling @ A )
     => ! [X2: A,V: num] :
          ( ( archimedean_ceiling @ A @ ( minus_minus @ A @ X2 @ ( numeral_numeral @ A @ V ) ) )
          = ( minus_minus @ int @ ( archimedean_ceiling @ A @ X2 ) @ ( numeral_numeral @ int @ V ) ) ) ) ).

% ceiling_diff_numeral
thf(fact_1966_ceiling__numeral__power,axiom,
    ! [A: $tType] :
      ( ( archim2362893244070406136eiling @ A )
     => ! [X2: num,N2: nat] :
          ( ( archimedean_ceiling @ A @ ( power_power @ A @ ( numeral_numeral @ A @ X2 ) @ N2 ) )
          = ( power_power @ int @ ( numeral_numeral @ int @ X2 ) @ N2 ) ) ) ).

% ceiling_numeral_power
thf(fact_1967_ceiling__diff__one,axiom,
    ! [A: $tType] :
      ( ( archim2362893244070406136eiling @ A )
     => ! [X2: A] :
          ( ( archimedean_ceiling @ A @ ( minus_minus @ A @ X2 @ ( one_one @ A ) ) )
          = ( minus_minus @ int @ ( archimedean_ceiling @ A @ X2 ) @ ( one_one @ int ) ) ) ) ).

% ceiling_diff_one
thf(fact_1968_ceiling__less__numeral,axiom,
    ! [A: $tType] :
      ( ( archim2362893244070406136eiling @ A )
     => ! [X2: A,V: num] :
          ( ( ord_less @ int @ ( archimedean_ceiling @ A @ X2 ) @ ( numeral_numeral @ int @ V ) )
          = ( ord_less_eq @ A @ X2 @ ( minus_minus @ A @ ( numeral_numeral @ A @ V ) @ ( one_one @ A ) ) ) ) ) ).

% ceiling_less_numeral
thf(fact_1969_numeral__le__ceiling,axiom,
    ! [A: $tType] :
      ( ( archim2362893244070406136eiling @ A )
     => ! [V: num,X2: A] :
          ( ( ord_less_eq @ int @ ( numeral_numeral @ int @ V ) @ ( archimedean_ceiling @ A @ X2 ) )
          = ( ord_less @ A @ ( minus_minus @ A @ ( numeral_numeral @ A @ V ) @ ( one_one @ A ) ) @ X2 ) ) ) ).

% numeral_le_ceiling
thf(fact_1970_word__subset__less,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [X2: word @ A,R4: word @ A,Y2: word @ A,S4: word @ A] :
          ( ( ord_less_eq @ ( set @ ( word @ A ) ) @ ( set_or1337092689740270186AtMost @ ( word @ A ) @ X2 @ ( minus_minus @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ X2 @ R4 ) @ ( one_one @ ( word @ A ) ) ) ) @ ( set_or1337092689740270186AtMost @ ( word @ A ) @ Y2 @ ( minus_minus @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ Y2 @ S4 ) @ ( one_one @ ( word @ A ) ) ) ) )
         => ( ( ord_less_eq @ ( word @ A ) @ X2 @ ( minus_minus @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ X2 @ R4 ) @ ( one_one @ ( word @ A ) ) ) )
           => ( ( ord_less_eq @ ( word @ A ) @ Y2 @ ( minus_minus @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ Y2 @ S4 ) @ ( one_one @ ( word @ A ) ) ) )
             => ( ( S4
                 != ( zero_zero @ ( word @ A ) ) )
               => ( ord_less_eq @ ( word @ A ) @ R4 @ S4 ) ) ) ) ) ) ).

% word_subset_less
thf(fact_1971_gt0__iff__gem1,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [X2: word @ A] :
          ( ( ord_less @ ( word @ A ) @ ( zero_zero @ ( word @ A ) ) @ X2 )
          = ( ord_less @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ X2 @ ( one_one @ ( word @ A ) ) ) @ X2 ) ) ) ).

% gt0_iff_gem1
thf(fact_1972_word__less__sub1,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [X2: word @ A] :
          ( ( X2
           != ( zero_zero @ ( word @ A ) ) )
         => ( ( ord_less @ ( word @ A ) @ ( one_one @ ( word @ A ) ) @ X2 )
            = ( ord_less @ ( word @ A ) @ ( zero_zero @ ( word @ A ) ) @ ( minus_minus @ ( word @ A ) @ X2 @ ( one_one @ ( word @ A ) ) ) ) ) ) ) ).

% word_less_sub1
thf(fact_1973_sub__wrap__lt,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [X2: word @ A,Z: word @ A] :
          ( ( ord_less @ ( word @ A ) @ X2 @ ( minus_minus @ ( word @ A ) @ X2 @ Z ) )
          = ( ord_less @ ( word @ A ) @ X2 @ Z ) ) ) ).

% sub_wrap_lt
thf(fact_1974_word__sub__less__iff,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [X2: word @ A,Y2: word @ A] :
          ( ( ord_less @ ( word @ A ) @ X2 @ ( minus_minus @ ( word @ A ) @ X2 @ Y2 ) )
          = ( ord_less @ ( word @ A ) @ X2 @ Y2 ) ) ) ).

% word_sub_less_iff
thf(fact_1975_word__less__cases,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [X2: word @ A,Y2: word @ A] :
          ( ( ord_less @ ( word @ A ) @ X2 @ Y2 )
         => ( ( X2
              = ( minus_minus @ ( word @ A ) @ Y2 @ ( one_one @ ( word @ A ) ) ) )
            | ( ord_less @ ( word @ A ) @ X2 @ ( minus_minus @ ( word @ A ) @ Y2 @ ( one_one @ ( word @ A ) ) ) ) ) ) ) ).

% word_less_cases
thf(fact_1976_word__less__minus__mono,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [A2: word @ A,C2: word @ A,D2: word @ A,B2: word @ A] :
          ( ( ord_less @ ( word @ A ) @ A2 @ C2 )
         => ( ( ord_less @ ( word @ A ) @ D2 @ B2 )
           => ( ( ord_less @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ A2 @ B2 ) @ A2 )
             => ( ( ord_less @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ C2 @ D2 ) @ C2 )
               => ( ord_less @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ A2 @ B2 ) @ ( minus_minus @ ( word @ A ) @ C2 @ D2 ) ) ) ) ) ) ) ).

% word_less_minus_mono
thf(fact_1977_word__div__sub,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [Y2: word @ A,X2: word @ A] :
          ( ( ord_less_eq @ ( word @ A ) @ Y2 @ X2 )
         => ( ( ord_less @ ( word @ A ) @ ( zero_zero @ ( word @ A ) ) @ Y2 )
           => ( ( divide_divide @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ X2 @ Y2 ) @ Y2 )
              = ( minus_minus @ ( word @ A ) @ ( divide_divide @ ( word @ A ) @ X2 @ Y2 ) @ ( one_one @ ( word @ A ) ) ) ) ) ) ) ).

% word_div_sub
thf(fact_1978_word__leq__minus__one__le,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [Y2: word @ A,X2: word @ A] :
          ( ( Y2
           != ( zero_zero @ ( word @ A ) ) )
         => ( ( ord_less_eq @ ( word @ A ) @ X2 @ ( minus_minus @ ( word @ A ) @ Y2 @ ( one_one @ ( word @ A ) ) ) )
           => ( ord_less @ ( word @ A ) @ X2 @ Y2 ) ) ) ) ).

% word_leq_minus_one_le
thf(fact_1979_word__leq__le__minus__one,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [X2: word @ A,Y2: word @ A] :
          ( ( ord_less_eq @ ( word @ A ) @ X2 @ Y2 )
         => ( ( X2
             != ( zero_zero @ ( word @ A ) ) )
           => ( ord_less @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ X2 @ ( one_one @ ( word @ A ) ) ) @ Y2 ) ) ) ) ).

% word_leq_le_minus_one
thf(fact_1980_word__less__nowrapI_H,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [X2: word @ A,Z: word @ A,K: word @ A] :
          ( ( ord_less_eq @ ( word @ A ) @ X2 @ ( minus_minus @ ( word @ A ) @ Z @ K ) )
         => ( ( ord_less_eq @ ( word @ A ) @ K @ Z )
           => ( ( ord_less @ ( word @ A ) @ ( zero_zero @ ( word @ A ) ) @ K )
             => ( ord_less @ ( word @ A ) @ X2 @ ( plus_plus @ ( word @ A ) @ X2 @ K ) ) ) ) ) ) ).

% word_less_nowrapI'
thf(fact_1981_plus__minus__not__NULL,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [X2: word @ A,Ab: word @ A,C2: word @ A] :
          ( ( ord_less @ ( word @ A ) @ X2 @ ( minus_minus @ ( word @ A ) @ Ab @ C2 ) )
         => ( ( ord_less_eq @ ( word @ A ) @ C2 @ Ab )
           => ( ( C2
               != ( zero_zero @ ( word @ A ) ) )
             => ( ( plus_plus @ ( word @ A ) @ X2 @ C2 )
               != ( zero_zero @ ( word @ A ) ) ) ) ) ) ) ).

% plus_minus_not_NULL
thf(fact_1982_word__diff__less,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [N2: word @ A,M: word @ A] :
          ( ( ord_less @ ( word @ A ) @ ( zero_zero @ ( word @ A ) ) @ N2 )
         => ( ( ord_less @ ( word @ A ) @ ( zero_zero @ ( word @ A ) ) @ M )
           => ( ( ord_less_eq @ ( word @ A ) @ N2 @ M )
             => ( ord_less @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ M @ N2 ) @ M ) ) ) ) ) ).

% word_diff_less
thf(fact_1983_word__less__nowrapI,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [X2: word @ A,Z: word @ A,K: word @ A] :
          ( ( ord_less @ ( word @ A ) @ X2 @ ( minus_minus @ ( word @ A ) @ Z @ K ) )
         => ( ( ord_less_eq @ ( word @ A ) @ K @ Z )
           => ( ( ord_less @ ( word @ A ) @ ( zero_zero @ ( word @ A ) ) @ K )
             => ( ord_less @ ( word @ A ) @ X2 @ ( plus_plus @ ( word @ A ) @ X2 @ K ) ) ) ) ) ) ).

% word_less_nowrapI
thf(fact_1984_le__m1__iff__lt,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [X2: word @ A,Y2: word @ A] :
          ( ( ord_less @ ( word @ A ) @ ( zero_zero @ ( word @ A ) ) @ X2 )
          = ( ( ord_less_eq @ ( word @ A ) @ Y2 @ ( minus_minus @ ( word @ A ) @ X2 @ ( one_one @ ( word @ A ) ) ) )
            = ( ord_less @ ( word @ A ) @ Y2 @ X2 ) ) ) ) ).

% le_m1_iff_lt
thf(fact_1985_less__1__simp,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [N2: word @ A,M: word @ A] :
          ( ( ord_less @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ N2 @ ( one_one @ ( word @ A ) ) ) @ M )
          = ( ( ord_less_eq @ ( word @ A ) @ N2 @ M )
            & ( N2
             != ( zero_zero @ ( word @ A ) ) ) ) ) ) ).

% less_1_simp
thf(fact_1986_sub__wrap,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [X2: word @ A,Z: word @ A] :
          ( ( ord_less_eq @ ( word @ A ) @ X2 @ ( minus_minus @ ( word @ A ) @ X2 @ Z ) )
          = ( ( Z
              = ( zero_zero @ ( word @ A ) ) )
            | ( ord_less @ ( word @ A ) @ X2 @ Z ) ) ) ) ).

% sub_wrap
thf(fact_1987_word__diff__ls_I2_J,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [Y2: word @ A,W: word @ A,Xa: word @ A,X2: word @ A] :
          ( ( ord_less @ ( word @ A ) @ Y2 @ ( plus_plus @ ( word @ A ) @ W @ Xa ) )
         => ( ( ord_less_eq @ ( word @ A ) @ X2 @ Y2 )
           => ( ord_less @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ Y2 @ X2 ) @ ( minus_minus @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ W @ Xa ) @ X2 ) ) ) ) ) ).

% word_diff_ls(2)
thf(fact_1988_word__diff__ls_I1_J,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [Y2: word @ A,X2: word @ A,W: word @ A,Xa: word @ A] :
          ( ( ord_less @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ Y2 @ X2 ) @ ( minus_minus @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ W @ Xa ) @ X2 ) )
         => ( ( ord_less_eq @ ( word @ A ) @ X2 @ ( plus_plus @ ( word @ A ) @ W @ Xa ) )
           => ( ord_less @ ( word @ A ) @ Y2 @ ( plus_plus @ ( word @ A ) @ W @ Xa ) ) ) ) ) ).

% word_diff_ls(1)
thf(fact_1989_Word_Oword__l__diffs_I2_J,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [W: word @ A,Xa: word @ A,Z: word @ A,X2: word @ A] :
          ( ( ord_less @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ W @ Xa ) @ Z )
         => ( ( ord_less_eq @ ( word @ A ) @ X2 @ ( plus_plus @ ( word @ A ) @ W @ Xa ) )
           => ( ord_less @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ W @ Xa ) @ X2 ) @ ( minus_minus @ ( word @ A ) @ Z @ X2 ) ) ) ) ) ).

% Word.word_l_diffs(2)
thf(fact_1990_Word_Oword__l__diffs_I1_J,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [W: word @ A,Xa: word @ A,X2: word @ A,Z: word @ A] :
          ( ( ord_less @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ W @ Xa ) @ X2 ) @ ( minus_minus @ ( word @ A ) @ Z @ X2 ) )
         => ( ( ord_less_eq @ ( word @ A ) @ X2 @ Z )
           => ( ord_less @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ W @ Xa ) @ Z ) ) ) ) ).

% Word.word_l_diffs(1)
thf(fact_1991_word__plus__mcs_I2_J,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [V: word @ A,Xb: word @ A,W: word @ A,Xa: word @ A,X2: word @ A] :
          ( ( ord_less @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ V @ Xb ) @ ( plus_plus @ ( word @ A ) @ W @ Xa ) )
         => ( ( ord_less_eq @ ( word @ A ) @ X2 @ ( plus_plus @ ( word @ A ) @ V @ Xb ) )
           => ( ord_less @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ V @ Xb ) @ X2 ) @ ( minus_minus @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ W @ Xa ) @ X2 ) ) ) ) ) ).

% word_plus_mcs(2)
thf(fact_1992_word__plus__mcs_I1_J,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [V: word @ A,Xb: word @ A,X2: word @ A,W: word @ A,Xa: word @ A] :
          ( ( ord_less @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ V @ Xb ) @ X2 ) @ ( minus_minus @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ W @ Xa ) @ X2 ) )
         => ( ( ord_less_eq @ ( word @ A ) @ X2 @ ( plus_plus @ ( word @ A ) @ W @ Xa ) )
           => ( ord_less @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ V @ Xb ) @ ( plus_plus @ ( word @ A ) @ W @ Xa ) ) ) ) ) ).

% word_plus_mcs(1)
thf(fact_1993_More__Word_Oword__l__diffs_I2_J,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [W: word @ A,X2: word @ A,Z: word @ A] :
          ( ( ord_less @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ W @ X2 ) @ Z )
         => ( ( ord_less_eq @ ( word @ A ) @ X2 @ ( plus_plus @ ( word @ A ) @ W @ X2 ) )
           => ( ord_less @ ( word @ A ) @ W @ ( minus_minus @ ( word @ A ) @ Z @ X2 ) ) ) ) ) ).

% More_Word.word_l_diffs(2)
thf(fact_1994_More__Word_Oword__l__diffs_I1_J,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [W: word @ A,Z: word @ A,X2: word @ A] :
          ( ( ord_less @ ( word @ A ) @ W @ ( minus_minus @ ( word @ A ) @ Z @ X2 ) )
         => ( ( ord_less_eq @ ( word @ A ) @ X2 @ Z )
           => ( ord_less @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ W @ X2 ) @ Z ) ) ) ) ).

% More_Word.word_l_diffs(1)
thf(fact_1995_word__diff__ls_H_I2_J,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [Y2: word @ A,W: word @ A,X2: word @ A] :
          ( ( ord_less @ ( word @ A ) @ Y2 @ ( plus_plus @ ( word @ A ) @ W @ X2 ) )
         => ( ( ord_less_eq @ ( word @ A ) @ X2 @ Y2 )
           => ( ord_less @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ Y2 @ X2 ) @ W ) ) ) ) ).

% word_diff_ls'(2)
thf(fact_1996_word__diff__ls_H_I1_J,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [Y2: word @ A,X2: word @ A,W: word @ A] :
          ( ( ord_less @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ Y2 @ X2 ) @ W )
         => ( ( ord_less_eq @ ( word @ A ) @ X2 @ ( plus_plus @ ( word @ A ) @ W @ X2 ) )
           => ( ord_less @ ( word @ A ) @ Y2 @ ( plus_plus @ ( word @ A ) @ W @ X2 ) ) ) ) ) ).

% word_diff_ls'(1)
thf(fact_1997_word__l__diffs_H_I2_J,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [W: word @ A,X2: word @ A,Z: word @ A] :
          ( ( ord_less @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ W @ X2 ) @ Z )
         => ( ( ord_less_eq @ ( word @ A ) @ X2 @ ( plus_plus @ ( word @ A ) @ W @ X2 ) )
           => ( ord_less @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ W @ X2 ) @ X2 ) @ ( minus_minus @ ( word @ A ) @ Z @ X2 ) ) ) ) ) ).

% word_l_diffs'(2)
thf(fact_1998_word__l__diffs_H_I1_J,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [W: word @ A,X2: word @ A,Z: word @ A] :
          ( ( ord_less @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ W @ X2 ) @ X2 ) @ ( minus_minus @ ( word @ A ) @ Z @ X2 ) )
         => ( ( ord_less_eq @ ( word @ A ) @ X2 @ Z )
           => ( ord_less @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ W @ X2 ) @ Z ) ) ) ) ).

% word_l_diffs'(1)
thf(fact_1999_word__diff__ls_H_H_I2_J,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [Y2: word @ A,W: word @ A,X2: word @ A] :
          ( ( ord_less @ ( word @ A ) @ Y2 @ ( plus_plus @ ( word @ A ) @ W @ X2 ) )
         => ( ( ord_less_eq @ ( word @ A ) @ X2 @ Y2 )
           => ( ord_less @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ Y2 @ X2 ) @ ( minus_minus @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ W @ X2 ) @ X2 ) ) ) ) ) ).

% word_diff_ls''(2)
thf(fact_2000_word__diff__ls_H_H_I1_J,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [Y2: word @ A,X2: word @ A,W: word @ A] :
          ( ( ord_less @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ Y2 @ X2 ) @ ( minus_minus @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ W @ X2 ) @ X2 ) )
         => ( ( ord_less_eq @ ( word @ A ) @ X2 @ ( plus_plus @ ( word @ A ) @ W @ X2 ) )
           => ( ord_less @ ( word @ A ) @ Y2 @ ( plus_plus @ ( word @ A ) @ W @ X2 ) ) ) ) ) ).

% word_diff_ls''(1)
thf(fact_2001_word__less__add__right,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [X2: word @ A,Y2: word @ A,Z: word @ A] :
          ( ( ord_less @ ( word @ A ) @ X2 @ ( minus_minus @ ( word @ A ) @ Y2 @ Z ) )
         => ( ( ord_less_eq @ ( word @ A ) @ Z @ Y2 )
           => ( ord_less @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ X2 @ Z ) @ Y2 ) ) ) ) ).

% word_less_add_right
thf(fact_2002_word__less__sub__right,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [X2: word @ A,Y2: word @ A,Z: word @ A] :
          ( ( ord_less @ ( word @ A ) @ X2 @ ( plus_plus @ ( word @ A ) @ Y2 @ Z ) )
         => ( ( ord_less_eq @ ( word @ A ) @ Y2 @ X2 )
           => ( ord_less @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ X2 @ Y2 ) @ Z ) ) ) ) ).

% word_less_sub_right
thf(fact_2003_plus__minus__no__overflow,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [X2: word @ A,Ab: word @ A,C2: word @ A] :
          ( ( ord_less @ ( word @ A ) @ X2 @ ( minus_minus @ ( word @ A ) @ Ab @ C2 ) )
         => ( ( ord_less_eq @ ( word @ A ) @ C2 @ Ab )
           => ( ord_less_eq @ ( word @ A ) @ X2 @ ( plus_plus @ ( word @ A ) @ X2 @ C2 ) ) ) ) ) ).

% plus_minus_no_overflow
thf(fact_2004_word__less__minus__cancel,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [Y2: word @ A,X2: word @ A,Z: word @ A] :
          ( ( ord_less @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ Y2 @ X2 ) @ ( minus_minus @ ( word @ A ) @ Z @ X2 ) )
         => ( ( ord_less_eq @ ( word @ A ) @ X2 @ Z )
           => ( ord_less @ ( word @ A ) @ Y2 @ Z ) ) ) ) ).

% word_less_minus_cancel
thf(fact_2005_word__less__minus__mono__left,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [Y2: word @ A,Z: word @ A,X2: word @ A] :
          ( ( ord_less @ ( word @ A ) @ Y2 @ Z )
         => ( ( ord_less_eq @ ( word @ A ) @ X2 @ Y2 )
           => ( ord_less @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ Y2 @ X2 ) @ ( minus_minus @ ( word @ A ) @ Z @ X2 ) ) ) ) ) ).

% word_less_minus_mono_left
thf(fact_2006_word__le__minus__one__leq,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [X2: word @ A,Y2: word @ A] :
          ( ( ord_less @ ( word @ A ) @ X2 @ Y2 )
         => ( ord_less_eq @ ( word @ A ) @ X2 @ ( minus_minus @ ( word @ A ) @ Y2 @ ( one_one @ ( word @ A ) ) ) ) ) ) ).

% word_le_minus_one_leq
thf(fact_2007_word__minus__one__le__leq,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [X2: word @ A,Y2: word @ A] :
          ( ( ord_less @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ X2 @ ( one_one @ ( word @ A ) ) ) @ Y2 )
         => ( ord_less_eq @ ( word @ A ) @ X2 @ Y2 ) ) ) ).

% word_minus_one_le_leq
thf(fact_2008_word__less__imp__diff__less,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [K: word @ A,N2: word @ A,M: word @ A] :
          ( ( ord_less_eq @ ( word @ A ) @ K @ N2 )
         => ( ( ord_less @ ( word @ A ) @ N2 @ M )
           => ( ord_less @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ N2 @ K ) @ M ) ) ) ) ).

% word_less_imp_diff_less
thf(fact_2009_le__step__down__word,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [I: word @ A,N2: word @ A] :
          ( ( ord_less_eq @ ( word @ A ) @ I @ N2 )
         => ( ( I != N2 )
           => ( ord_less_eq @ ( word @ A ) @ I @ ( minus_minus @ ( word @ A ) @ N2 @ ( one_one @ ( word @ A ) ) ) ) ) ) ) ).

% le_step_down_word
thf(fact_2010_le__step__down__word__2,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [X2: word @ A,Y2: word @ A] :
          ( ( ord_less_eq @ ( word @ A ) @ X2 @ Y2 )
         => ( ( X2 != Y2 )
           => ( ord_less_eq @ ( word @ A ) @ X2 @ ( minus_minus @ ( word @ A ) @ Y2 @ ( one_one @ ( word @ A ) ) ) ) ) ) ) ).

% le_step_down_word_2
thf(fact_2011_add__diff__assoc__enat,axiom,
    ! [Z: extended_enat,Y2: extended_enat,X2: extended_enat] :
      ( ( ord_less_eq @ extended_enat @ Z @ Y2 )
     => ( ( plus_plus @ extended_enat @ X2 @ ( minus_minus @ extended_enat @ Y2 @ Z ) )
        = ( minus_minus @ extended_enat @ ( plus_plus @ extended_enat @ X2 @ Y2 ) @ Z ) ) ) ).

% add_diff_assoc_enat
thf(fact_2012_word__le__sub1,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [X2: word @ A] :
          ( ( X2
           != ( zero_zero @ ( word @ A ) ) )
         => ( ( ord_less_eq @ ( word @ A ) @ ( one_one @ ( word @ A ) ) @ X2 )
            = ( ord_less_eq @ ( word @ A ) @ ( zero_zero @ ( word @ A ) ) @ ( minus_minus @ ( word @ A ) @ X2 @ ( one_one @ ( word @ A ) ) ) ) ) ) ) ).

% word_le_sub1
thf(fact_2013_word__sub__1__le,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [X2: word @ A] :
          ( ( X2
           != ( zero_zero @ ( word @ A ) ) )
         => ( ord_less_eq @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ X2 @ ( one_one @ ( word @ A ) ) ) @ X2 ) ) ) ).

% word_sub_1_le
thf(fact_2014_word__must__wrap,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [X2: word @ A,N2: word @ A] :
          ( ( ord_less_eq @ ( word @ A ) @ X2 @ ( minus_minus @ ( word @ A ) @ N2 @ ( one_one @ ( word @ A ) ) ) )
         => ( ( ord_less_eq @ ( word @ A ) @ N2 @ X2 )
           => ( N2
              = ( zero_zero @ ( word @ A ) ) ) ) ) ) ).

% word_must_wrap
thf(fact_2015_word__sub__plus__one__nonzero,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [N5: word @ A,N2: word @ A] :
          ( ( ord_less_eq @ ( word @ A ) @ N5 @ N2 )
         => ( ( N5
             != ( zero_zero @ ( word @ A ) ) )
           => ( ( plus_plus @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ N2 @ N5 ) @ ( one_one @ ( word @ A ) ) )
             != ( zero_zero @ ( word @ A ) ) ) ) ) ) ).

% word_sub_plus_one_nonzero
thf(fact_2016_word__div__less,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [W: word @ A,V: word @ A] :
          ( ( ord_less @ ( word @ A ) @ W @ V )
         => ( ( divide_divide @ ( word @ A ) @ W @ V )
            = ( zero_zero @ ( word @ A ) ) ) ) ) ).

% word_div_less
thf(fact_2017_More__Word_Oword__div__mult,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [C2: word @ A,A2: word @ A,B2: word @ A] :
          ( ( ord_less @ ( word @ A ) @ ( zero_zero @ ( word @ A ) ) @ C2 )
         => ( ( ord_less @ ( word @ A ) @ A2 @ ( times_times @ ( word @ A ) @ B2 @ C2 ) )
           => ( ord_less @ ( word @ A ) @ ( divide_divide @ ( word @ A ) @ A2 @ C2 ) @ B2 ) ) ) ) ).

% More_Word.word_div_mult
thf(fact_2018_word__less__div,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [X2: word @ A,Y2: word @ A] :
          ( ( ( divide_divide @ ( word @ A ) @ X2 @ Y2 )
            = ( zero_zero @ ( word @ A ) ) )
         => ( ( Y2
              = ( zero_zero @ ( word @ A ) ) )
            | ( ord_less @ ( word @ A ) @ X2 @ Y2 ) ) ) ) ).

% word_less_div
thf(fact_2019_word__div__lt__eq__0,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [X2: word @ A,Y2: word @ A] :
          ( ( ord_less @ ( word @ A ) @ X2 @ Y2 )
         => ( ( divide_divide @ ( word @ A ) @ X2 @ Y2 )
            = ( zero_zero @ ( word @ A ) ) ) ) ) ).

% word_div_lt_eq_0
thf(fact_2020_div__less__dividend__word,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [X2: word @ A,N2: word @ A] :
          ( ( X2
           != ( zero_zero @ ( word @ A ) ) )
         => ( ( N2
             != ( one_one @ ( word @ A ) ) )
           => ( ord_less @ ( word @ A ) @ ( divide_divide @ ( word @ A ) @ X2 @ N2 ) @ X2 ) ) ) ) ).

% div_less_dividend_word
thf(fact_2021_word__overflow,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [X2: word @ A] :
          ( ( ord_less @ ( word @ A ) @ X2 @ ( plus_plus @ ( word @ A ) @ X2 @ ( one_one @ ( word @ A ) ) ) )
          | ( ( plus_plus @ ( word @ A ) @ X2 @ ( one_one @ ( word @ A ) ) )
            = ( zero_zero @ ( word @ A ) ) ) ) ) ).

% word_overflow
thf(fact_2022_word__gr0__conv__Suc,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [M: word @ A] :
          ( ( ord_less @ ( word @ A ) @ ( zero_zero @ ( word @ A ) ) @ M )
         => ? [N4: word @ A] :
              ( M
              = ( plus_plus @ ( word @ A ) @ N4 @ ( one_one @ ( word @ A ) ) ) ) ) ) ).

% word_gr0_conv_Suc
thf(fact_2023_less__is__non__zero__p1,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [A2: word @ A,K: word @ A] :
          ( ( ord_less @ ( word @ A ) @ A2 @ K )
         => ( ( plus_plus @ ( word @ A ) @ A2 @ ( one_one @ ( word @ A ) ) )
           != ( zero_zero @ ( word @ A ) ) ) ) ) ).

% less_is_non_zero_p1
thf(fact_2024_word__div__1,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [N2: word @ A] :
          ( ( divide_divide @ ( word @ A ) @ N2 @ ( one_one @ ( word @ A ) ) )
          = N2 ) ) ).

% word_div_1
thf(fact_2025_div__by__0__word,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [X2: word @ A] :
          ( ( divide_divide @ ( word @ A ) @ X2 @ ( zero_zero @ ( word @ A ) ) )
          = ( zero_zero @ ( word @ A ) ) ) ) ).

% div_by_0_word
thf(fact_2026_word__plus__one__nonzero,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [X2: word @ A,Y2: word @ A] :
          ( ( ord_less_eq @ ( word @ A ) @ X2 @ ( plus_plus @ ( word @ A ) @ X2 @ Y2 ) )
         => ( ( Y2
             != ( zero_zero @ ( word @ A ) ) )
           => ( ( plus_plus @ ( word @ A ) @ X2 @ ( one_one @ ( word @ A ) ) )
             != ( zero_zero @ ( word @ A ) ) ) ) ) ) ).

% word_plus_one_nonzero
thf(fact_2027_lt1__neq0,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [X2: word @ A] :
          ( ( ord_less_eq @ ( word @ A ) @ ( one_one @ ( word @ A ) ) @ X2 )
          = ( X2
           != ( zero_zero @ ( word @ A ) ) ) ) ) ).

% lt1_neq0
thf(fact_2028_div__to__mult__word__lt,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [X2: word @ A,Y2: word @ A,Z: word @ A] :
          ( ( ord_less_eq @ ( word @ A ) @ X2 @ ( divide_divide @ ( word @ A ) @ Y2 @ Z ) )
         => ( ord_less_eq @ ( word @ A ) @ ( times_times @ ( word @ A ) @ X2 @ Z ) @ Y2 ) ) ) ).

% div_to_mult_word_lt
thf(fact_2029_word__le__add,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [X2: word @ A,Y2: word @ A] :
          ( ( ord_less_eq @ ( word @ A ) @ X2 @ Y2 )
         => ? [N4: nat] :
              ( Y2
              = ( plus_plus @ ( word @ A ) @ X2 @ ( semiring_1_of_nat @ ( word @ A ) @ N4 ) ) ) ) ) ).

% word_le_add
thf(fact_2030_word__le__plus,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [A2: word @ A,B2: word @ A,C2: word @ A] :
          ( ( ord_less @ ( word @ A ) @ A2 @ ( plus_plus @ ( word @ A ) @ A2 @ B2 ) )
         => ( ( ord_less @ ( word @ A ) @ C2 @ B2 )
           => ( ord_less_eq @ ( word @ A ) @ A2 @ ( plus_plus @ ( word @ A ) @ A2 @ C2 ) ) ) ) ) ).

% word_le_plus
thf(fact_2031_word__le__plus__1,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [Y2: word @ A,N2: word @ A,A2: word @ A] :
          ( ( ord_less @ ( word @ A ) @ Y2 @ ( plus_plus @ ( word @ A ) @ Y2 @ N2 ) )
         => ( ( ord_less @ ( word @ A ) @ A2 @ N2 )
           => ( ord_less_eq @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ Y2 @ A2 ) @ ( plus_plus @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ Y2 @ A2 ) @ ( one_one @ ( word @ A ) ) ) ) ) ) ) ).

% word_le_plus_1
thf(fact_2032_plus__one__helper,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [X2: word @ A,N2: word @ A] :
          ( ( ord_less @ ( word @ A ) @ X2 @ ( plus_plus @ ( word @ A ) @ N2 @ ( one_one @ ( word @ A ) ) ) )
         => ( ord_less_eq @ ( word @ A ) @ X2 @ N2 ) ) ) ).

% plus_one_helper
thf(fact_2033_word__plus__strict__mono__right,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [Y2: word @ A,Z: word @ A,X2: word @ A] :
          ( ( ord_less @ ( word @ A ) @ Y2 @ Z )
         => ( ( ord_less_eq @ ( word @ A ) @ X2 @ ( plus_plus @ ( word @ A ) @ X2 @ Z ) )
           => ( ord_less @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ X2 @ Y2 ) @ ( plus_plus @ ( word @ A ) @ X2 @ Z ) ) ) ) ) ).

% word_plus_strict_mono_right
thf(fact_2034_word__le__not__less,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ( ( ord_less_eq @ ( word @ A ) )
        = ( ^ [B5: word @ A,A5: word @ A] :
              ~ ( ord_less @ ( word @ A ) @ A5 @ B5 ) ) ) ) ).

% word_le_not_less
thf(fact_2035_word__not__le,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [X2: word @ A,Y2: word @ A] :
          ( ( ~ ( ord_less_eq @ ( word @ A ) @ X2 @ Y2 ) )
          = ( ord_less @ ( word @ A ) @ Y2 @ X2 ) ) ) ).

% word_not_le
thf(fact_2036_plus__one__helper2,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [X2: word @ A,N2: word @ A] :
          ( ( ord_less_eq @ ( word @ A ) @ X2 @ N2 )
         => ( ( ( plus_plus @ ( word @ A ) @ N2 @ ( one_one @ ( word @ A ) ) )
             != ( zero_zero @ ( word @ A ) ) )
           => ( ord_less @ ( word @ A ) @ X2 @ ( plus_plus @ ( word @ A ) @ N2 @ ( one_one @ ( word @ A ) ) ) ) ) ) ) ).

% plus_one_helper2
thf(fact_2037_word__1__0,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [A2: word @ A,B2: word @ A,X2: nat] :
          ( ( ord_less_eq @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ A2 @ ( one_one @ ( word @ A ) ) ) @ B2 )
         => ( ( ord_less @ ( word @ A ) @ A2 @ ( semiring_1_of_nat @ ( word @ A ) @ X2 ) )
           => ( ord_less @ ( word @ A ) @ A2 @ B2 ) ) ) ) ).

% word_1_0
thf(fact_2038_div__le__mult,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [I: word @ A,K: word @ A,X2: word @ A] :
          ( ( ord_less_eq @ ( word @ A ) @ I @ ( divide_divide @ ( word @ A ) @ K @ X2 ) )
         => ( ( ord_less @ ( word @ A ) @ ( zero_zero @ ( word @ A ) ) @ X2 )
           => ( ord_less_eq @ ( word @ A ) @ ( times_times @ ( word @ A ) @ I @ X2 ) @ K ) ) ) ) ).

% div_le_mult
thf(fact_2039_word__le__less__eq,axiom,
    ! [Z6: $tType] :
      ( ( type_len @ Z6 )
     => ( ( ord_less_eq @ ( word @ Z6 ) )
        = ( ^ [X: word @ Z6,Y: word @ Z6] :
              ( ( X = Y )
              | ( ord_less @ ( word @ Z6 ) @ X @ Y ) ) ) ) ) ).

% word_le_less_eq
thf(fact_2040_plus__le__left__cancel__nowrap,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [X2: word @ A,Y6: word @ A,Y2: word @ A] :
          ( ( ord_less_eq @ ( word @ A ) @ X2 @ ( plus_plus @ ( word @ A ) @ X2 @ Y6 ) )
         => ( ( ord_less_eq @ ( word @ A ) @ X2 @ ( plus_plus @ ( word @ A ) @ X2 @ Y2 ) )
           => ( ( ord_less @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ X2 @ Y6 ) @ ( plus_plus @ ( word @ A ) @ X2 @ Y2 ) )
              = ( ord_less @ ( word @ A ) @ Y6 @ Y2 ) ) ) ) ) ).

% plus_le_left_cancel_nowrap
thf(fact_2041_inc__le,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [I: word @ A,M: word @ A] :
          ( ( ord_less @ ( word @ A ) @ I @ M )
         => ( ord_less_eq @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ I @ ( one_one @ ( word @ A ) ) ) @ M ) ) ) ).

% inc_le
thf(fact_2042_inc__i,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [I: word @ A,M: word @ A] :
          ( ( ord_less_eq @ ( word @ A ) @ ( one_one @ ( word @ A ) ) @ I )
         => ( ( ord_less @ ( word @ A ) @ I @ M )
           => ( ( ord_less_eq @ ( word @ A ) @ ( one_one @ ( word @ A ) ) @ ( plus_plus @ ( word @ A ) @ I @ ( one_one @ ( word @ A ) ) ) )
              & ( ord_less_eq @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ I @ ( one_one @ ( word @ A ) ) ) @ M ) ) ) ) ) ).

% inc_i
thf(fact_2043_word__div__mult__le,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [A2: word @ A,B2: word @ A] : ( ord_less_eq @ ( word @ A ) @ ( times_times @ ( word @ A ) @ ( divide_divide @ ( word @ A ) @ A2 @ B2 ) @ B2 ) @ A2 ) ) ).

% word_div_mult_le
thf(fact_2044_i0__lb,axiom,
    ! [N2: extended_enat] : ( ord_less_eq @ extended_enat @ ( zero_zero @ extended_enat ) @ N2 ) ).

% i0_lb
thf(fact_2045_ile0__eq,axiom,
    ! [N2: extended_enat] :
      ( ( ord_less_eq @ extended_enat @ N2 @ ( zero_zero @ extended_enat ) )
      = ( N2
        = ( zero_zero @ extended_enat ) ) ) ).

% ile0_eq
thf(fact_2046_div__word__self,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [W: word @ A] :
          ( ( W
           != ( zero_zero @ ( word @ A ) ) )
         => ( ( divide_divide @ ( word @ A ) @ W @ W )
            = ( one_one @ ( word @ A ) ) ) ) ) ).

% div_word_self
thf(fact_2047_word__induct,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [P: ( word @ A ) > $o,M: word @ A] :
          ( ( P @ ( zero_zero @ ( word @ A ) ) )
         => ( ! [N4: word @ A] :
                ( ( P @ N4 )
               => ( P @ ( plus_plus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) @ N4 ) ) )
           => ( P @ M ) ) ) ) ).

% word_induct
thf(fact_2048_word__induct2,axiom,
    ! [B: $tType] :
      ( ( type_len @ B )
     => ! [P: ( word @ B ) > $o,N2: word @ B] :
          ( ( P @ ( zero_zero @ ( word @ B ) ) )
         => ( ! [N4: word @ B] :
                ( ( ( plus_plus @ ( word @ B ) @ ( one_one @ ( word @ B ) ) @ N4 )
                 != ( zero_zero @ ( word @ B ) ) )
               => ( ( P @ N4 )
                 => ( P @ ( plus_plus @ ( word @ B ) @ ( one_one @ ( word @ B ) ) @ N4 ) ) ) )
           => ( P @ N2 ) ) ) ) ).

% word_induct2
thf(fact_2049_plus__le__left__cancel__wrap,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [X2: word @ A,Y6: word @ A,Y2: word @ A] :
          ( ( ord_less @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ X2 @ Y6 ) @ X2 )
         => ( ( ord_less @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ X2 @ Y2 ) @ X2 )
           => ( ( ord_less @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ X2 @ Y6 ) @ ( plus_plus @ ( word @ A ) @ X2 @ Y2 ) )
              = ( ord_less @ ( word @ A ) @ Y6 @ Y2 ) ) ) ) ) ).

% plus_le_left_cancel_wrap
thf(fact_2050_enat__0__less__mult__iff,axiom,
    ! [M: extended_enat,N2: extended_enat] :
      ( ( ord_less @ extended_enat @ ( zero_zero @ extended_enat ) @ ( times_times @ extended_enat @ M @ N2 ) )
      = ( ( ord_less @ extended_enat @ ( zero_zero @ extended_enat ) @ M )
        & ( ord_less @ extended_enat @ ( zero_zero @ extended_enat ) @ N2 ) ) ) ).

% enat_0_less_mult_iff
thf(fact_2051_word__induct__less,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [P: ( word @ A ) > $o,M: word @ A] :
          ( ( P @ ( zero_zero @ ( word @ A ) ) )
         => ( ! [N4: word @ A] :
                ( ( ord_less @ ( word @ A ) @ N4 @ M )
               => ( ( P @ N4 )
                 => ( P @ ( plus_plus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) @ N4 ) ) ) )
           => ( P @ M ) ) ) ) ).

% word_induct_less
thf(fact_2052_div__lt__mult,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [I: word @ A,K: word @ A,X2: word @ A] :
          ( ( ord_less @ ( word @ A ) @ I @ ( divide_divide @ ( word @ A ) @ K @ X2 ) )
         => ( ( ord_less @ ( word @ A ) @ ( zero_zero @ ( word @ A ) ) @ X2 )
           => ( ord_less @ ( word @ A ) @ ( times_times @ ( word @ A ) @ I @ X2 ) @ K ) ) ) ) ).

% div_lt_mult
thf(fact_2053_set__bit__greater__eq,axiom,
    ! [K: int,N2: nat] : ( ord_less_eq @ int @ K @ ( bit_se5668285175392031749et_bit @ int @ N2 @ K ) ) ).

% set_bit_greater_eq
thf(fact_2054_ceiling__mono,axiom,
    ! [A: $tType] :
      ( ( archim2362893244070406136eiling @ A )
     => ! [Y2: A,X2: A] :
          ( ( ord_less_eq @ A @ Y2 @ X2 )
         => ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ Y2 ) @ ( archimedean_ceiling @ A @ X2 ) ) ) ) ).

% ceiling_mono
thf(fact_2055_ceiling__less__cancel,axiom,
    ! [A: $tType] :
      ( ( archim2362893244070406136eiling @ A )
     => ! [X2: A,Y2: A] :
          ( ( ord_less @ int @ ( archimedean_ceiling @ A @ X2 ) @ ( archimedean_ceiling @ A @ Y2 ) )
         => ( ord_less @ A @ X2 @ Y2 ) ) ) ).

% ceiling_less_cancel
thf(fact_2056_ceiling__add__le,axiom,
    ! [A: $tType] :
      ( ( archim2362893244070406136eiling @ A )
     => ! [X2: A,Y2: A] : ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ ( plus_plus @ A @ X2 @ Y2 ) ) @ ( plus_plus @ int @ ( archimedean_ceiling @ A @ X2 ) @ ( archimedean_ceiling @ A @ Y2 ) ) ) ) ).

% ceiling_add_le
thf(fact_2057_le__minus,axiom,
    ! [Aa: $tType,A: $tType] :
      ( ( ( type_len @ A )
        & ( order @ Aa ) )
     => ! [Y2: Aa,X2: Aa,A2: word @ A,C2: word @ A,B2: word @ A] :
          ( ( ord_less_eq @ Aa @ Y2 @ X2 )
         => ( ( ord_less_eq @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ A2 @ C2 ) @ B2 )
           => ( ( ord_less_eq @ ( word @ A ) @ A2 @ ( plus_plus @ ( word @ A ) @ A2 @ C2 ) )
             => ( ord_less_eq @ ( word @ A ) @ C2 @ ( minus_minus @ ( word @ A ) @ B2 @ A2 ) ) ) ) ) ) ).

% le_minus
thf(fact_2058_size__0__same_H,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [W: word @ A,V: word @ A] :
          ( ( ( size_size @ ( word @ A ) @ W )
            = ( zero_zero @ nat ) )
         => ( W = V ) ) ) ).

% size_0_same'
thf(fact_2059_lens__not__0,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [W: word @ A] :
          ( ( size_size @ ( word @ A ) @ W )
         != ( zero_zero @ nat ) ) ) ).

% lens_not_0
thf(fact_2060_size__0__eq,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [W: word @ A,V: word @ A] :
          ( ( ( size_size @ ( word @ A ) @ W )
            = ( zero_zero @ nat ) )
         => ( V = W ) ) ) ).

% size_0_eq
thf(fact_2061_Abs__fnat__hom__mult,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [A2: nat,B2: nat] :
          ( ( times_times @ ( word @ A ) @ ( semiring_1_of_nat @ ( word @ A ) @ A2 ) @ ( semiring_1_of_nat @ ( word @ A ) @ B2 ) )
          = ( semiring_1_of_nat @ ( word @ A ) @ ( times_times @ nat @ A2 @ B2 ) ) ) ) ).

% Abs_fnat_hom_mult
thf(fact_2062_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Osimps_I2_J,axiom,
    ! [Uv: $o,Uw: $o,N2: nat] :
      ( ( vEBT_T_s_u_c_c @ ( vEBT_Leaf @ Uv @ Uw ) @ ( suc @ N2 ) )
      = ( one_one @ nat ) ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.simps(2)
thf(fact_2063_mult__ceiling__le,axiom,
    ! [A: $tType] :
      ( ( archim2362893244070406136eiling @ A )
     => ! [A2: A,B2: A] :
          ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
         => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
           => ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ ( times_times @ A @ A2 @ B2 ) ) @ ( times_times @ int @ ( archimedean_ceiling @ A @ A2 ) @ ( archimedean_ceiling @ A @ B2 ) ) ) ) ) ) ).

% mult_ceiling_le
thf(fact_2064_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Osimps_I3_J,axiom,
    ! [Ux: nat,Uy: list @ vEBT_VEBT,Uz: vEBT_VEBT,Va: nat] :
      ( ( vEBT_T_s_u_c_c @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Ux @ Uy @ Uz ) @ Va )
      = ( one_one @ nat ) ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.simps(3)
thf(fact_2065_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Osimps_I1_J,axiom,
    ! [Uu: $o,B2: $o] :
      ( ( vEBT_T_s_u_c_c @ ( vEBT_Leaf @ Uu @ B2 ) @ ( 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_2066_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,Vc2: list @ vEBT_VEBT,Vd: vEBT_VEBT,Ve: nat] :
      ( ( vEBT_T_s_u_c_c @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V ) @ ( zero_zero @ nat ) @ Vc2 @ Vd ) @ Ve )
      = ( one_one @ nat ) ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.simps(4)
thf(fact_2067_Abs__fnat__hom__1,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ( ( one_one @ ( word @ A ) )
        = ( semiring_1_of_nat @ ( word @ A ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ).

% Abs_fnat_hom_1
thf(fact_2068_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 @ ( product_prod @ nat @ 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_2069_assert_H__rule,axiom,
    ! [P: assn,Phi: $o] :
      ( ! [H3: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
          ( ( rep_assn @ P @ H3 )
         => Phi )
     => ( hoare_hoare_triple @ product_unit @ P @ ( refine_Imp_assert @ Phi )
        @ ^ [Uu3: product_unit] : P ) ) ).

% assert'_rule
thf(fact_2070_Abs__fnat__hom__add,axiom,
    ! [A: $tType] :
      ( ( semiring_1 @ A )
     => ! [A2: nat,B2: nat] :
          ( ( plus_plus @ A @ ( semiring_1_of_nat @ A @ A2 ) @ ( semiring_1_of_nat @ A @ B2 ) )
          = ( semiring_1_of_nat @ A @ ( plus_plus @ nat @ A2 @ B2 ) ) ) ) ).

% Abs_fnat_hom_add
thf(fact_2071_ceiling__log__nat__eq__if,axiom,
    ! [B2: nat,N2: nat,K: nat] :
      ( ( ord_less @ nat @ ( power_power @ nat @ B2 @ N2 ) @ K )
     => ( ( ord_less_eq @ nat @ K @ ( power_power @ nat @ B2 @ ( plus_plus @ nat @ N2 @ ( one_one @ nat ) ) ) )
       => ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ B2 )
         => ( ( archimedean_ceiling @ real @ ( log @ ( semiring_1_of_nat @ real @ B2 ) @ ( semiring_1_of_nat @ real @ K ) ) )
            = ( plus_plus @ int @ ( semiring_1_of_nat @ int @ N2 ) @ ( one_one @ int ) ) ) ) ) ) ).

% ceiling_log_nat_eq_if
thf(fact_2072_ceiling__log2__div2,axiom,
    ! [N2: nat] :
      ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 )
     => ( ( archimedean_ceiling @ real @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ real @ N2 ) ) )
        = ( plus_plus @ int @ ( archimedean_ceiling @ real @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ real @ ( plus_plus @ nat @ ( divide_divide @ nat @ ( minus_minus @ nat @ N2 @ ( one_one @ nat ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ nat ) ) ) ) ) @ ( one_one @ int ) ) ) ) ).

% ceiling_log2_div2
thf(fact_2073_ceiling__log__nat__eq__powr__iff,axiom,
    ! [B2: nat,K: nat,N2: nat] :
      ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ B2 )
     => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
       => ( ( ( archimedean_ceiling @ real @ ( log @ ( semiring_1_of_nat @ real @ B2 ) @ ( semiring_1_of_nat @ real @ K ) ) )
            = ( plus_plus @ int @ ( semiring_1_of_nat @ int @ N2 ) @ ( one_one @ int ) ) )
          = ( ( ord_less @ nat @ ( power_power @ nat @ B2 @ N2 ) @ K )
            & ( ord_less_eq @ nat @ K @ ( power_power @ nat @ B2 @ ( plus_plus @ nat @ N2 @ ( one_one @ nat ) ) ) ) ) ) ) ) ).

% ceiling_log_nat_eq_powr_iff
thf(fact_2074_of__nat__gt__0,axiom,
    ! [A: $tType] :
      ( ( semiring_1 @ A )
     => ! [K: nat] :
          ( ( ( semiring_1_of_nat @ A @ K )
           != ( zero_zero @ A ) )
         => ( ord_less @ nat @ ( zero_zero @ nat ) @ K ) ) ) ).

% of_nat_gt_0
thf(fact_2075_succ__bound__height,axiom,
    ! [T2: vEBT_VEBT,N2: nat,X2: nat] :
      ( ( vEBT_invar_vebt @ T2 @ N2 )
     => ( ord_less_eq @ nat @ ( vEBT_T_s_u_c_c @ T2 @ X2 ) @ ( times_times @ nat @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_VEBT_height @ T2 ) ) @ ( numeral_numeral @ nat @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ one2 ) ) ) ) ) ) ) ) ).

% succ_bound_height
thf(fact_2076_htt__vebt__succi,axiom,
    ! [T2: vEBT_VEBT,N2: nat,Ti: vEBT_VEBTi,X2: nat] :
      ( ( vEBT_invar_vebt @ T2 @ N2 )
     => ( time_htt @ ( option @ nat ) @ ( vEBT_vebt_assn_raw @ T2 @ Ti ) @ ( vEBT_vebt_succi @ Ti @ X2 )
        @ ^ [R: option @ nat] :
            ( times_times @ assn @ ( vEBT_vebt_assn_raw @ T2 @ Ti )
            @ ( pure_assn
              @ ( R
                = ( vEBT_vebt_succ @ T2 @ X2 ) ) ) )
        @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit1 @ one2 ) ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit1 @ one2 ) ) ) @ ( nat2 @ ( archimedean_ceiling @ real @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ real @ N2 ) ) ) ) ) ) ) ) ).

% htt_vebt_succi
thf(fact_2077_htt__vebt__memberi__invar__vebt,axiom,
    ! [T2: vEBT_VEBT,N2: nat,Ti: vEBT_VEBTi,X2: nat] :
      ( ( vEBT_invar_vebt @ T2 @ N2 )
     => ( time_htt @ $o @ ( vEBT_vebt_assn_raw @ T2 @ Ti ) @ ( vEBT_vebt_memberi @ Ti @ X2 )
        @ ^ [R: $o] :
            ( times_times @ assn @ ( vEBT_vebt_assn_raw @ T2 @ Ti )
            @ ( pure_assn
              @ ( R
                = ( vEBT_vebt_member @ T2 @ X2 ) ) ) )
        @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit0 @ one2 ) ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit0 @ one2 ) ) ) @ ( nat2 @ ( archimedean_ceiling @ real @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ real @ N2 ) ) ) ) ) ) ) ) ).

% htt_vebt_memberi_invar_vebt
thf(fact_2078_htt__vebt__inserti__invar__vebt,axiom,
    ! [T2: vEBT_VEBT,N2: nat,Ti: vEBT_VEBTi,X2: nat] :
      ( ( vEBT_invar_vebt @ T2 @ N2 )
     => ( time_htt @ vEBT_VEBTi @ ( vEBT_vebt_assn_raw @ T2 @ Ti ) @ ( vEBT_vebt_inserti @ Ti @ X2 ) @ ( vEBT_vebt_assn_raw @ ( vEBT_vebt_insert @ T2 @ X2 ) ) @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit0 @ ( bit1 @ one2 ) ) ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit0 @ ( bit1 @ one2 ) ) ) ) @ ( nat2 @ ( archimedean_ceiling @ real @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ real @ N2 ) ) ) ) ) ) ) ) ).

% htt_vebt_inserti_invar_vebt
thf(fact_2079_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Oelims,axiom,
    ! [X2: vEBT_VEBT,Xa: nat,Y2: nat] :
      ( ( ( vEBT_T_s_u_c_c @ X2 @ Xa )
        = Y2 )
     => ( ( ? [Uu2: $o,B3: $o] :
              ( X2
              = ( vEBT_Leaf @ Uu2 @ B3 ) )
         => ( ( Xa
              = ( zero_zero @ nat ) )
           => ( Y2
             != ( plus_plus @ nat @ ( one_one @ nat ) @ ( one_one @ nat ) ) ) ) )
       => ( ( ? [Uv2: $o,Uw2: $o] :
                ( X2
                = ( vEBT_Leaf @ Uv2 @ Uw2 ) )
           => ( ? [N4: nat] :
                  ( Xa
                  = ( suc @ N4 ) )
             => ( Y2
               != ( one_one @ nat ) ) ) )
         => ( ( ? [Ux2: nat,Uy2: list @ vEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( X2
                  = ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Ux2 @ Uy2 @ Uz2 ) )
             => ( Y2
               != ( one_one @ nat ) ) )
           => ( ( ? [V3: product_prod @ nat @ nat,Vc: list @ vEBT_VEBT,Vd2: vEBT_VEBT] :
                    ( X2
                    = ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( zero_zero @ nat ) @ Vc @ Vd2 ) )
               => ( Y2
                 != ( one_one @ nat ) ) )
             => ( ( ? [V3: product_prod @ nat @ nat,Vg2: list @ vEBT_VEBT,Vh2: vEBT_VEBT] :
                      ( X2
                      = ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vg2 @ Vh2 ) )
                 => ( Y2
                   != ( one_one @ nat ) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X2
                        = ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
                     => ( Y2
                       != ( plus_plus @ nat @ ( one_one @ nat )
                          @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi2 ) @ ( one_one @ nat )
                            @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit1 @ ( bit0 @ one2 ) ) ) )
                              @ ( if @ nat @ ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
                                @ ( plus_plus @ nat @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_T_m_a_x_t @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) )
                                  @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) )
                                    @ ( if @ nat
                                      @ ( ( ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
                                         != ( none @ nat ) )
                                        & ( vEBT_VEBT_less @ ( some @ nat @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) )
                                      @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) @ ( vEBT_T_s_u_c_c @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
                                      @ ( plus_plus @ nat @ ( plus_plus @ nat @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_T_s_u_c_c @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( one_one @ nat ) )
                                        @ ( if @ nat
                                          @ ( ( vEBT_vebt_succ @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
                                            = ( none @ nat ) )
                                          @ ( one_one @ nat )
                                          @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) @ ( vEBT_T_m_i_n_t @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_succ @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) )
                                @ ( one_one @ nat ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.elims
thf(fact_2080_log__ceil__idem,axiom,
    ! [X2: real] :
      ( ( ord_less_eq @ real @ ( one_one @ real ) @ X2 )
     => ( ( archimedean_ceiling @ real @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ X2 ) )
        = ( archimedean_ceiling @ real @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( ring_1_of_int @ real @ ( archimedean_ceiling @ real @ X2 ) ) ) ) ) ) ).

% log_ceil_idem
thf(fact_2081_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,Va: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,X2: nat] :
      ( ( vEBT_T_s_u_c_c @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X2 )
      = ( plus_plus @ nat @ ( one_one @ nat )
        @ ( if @ nat @ ( ord_less @ nat @ X2 @ Mi ) @ ( one_one @ nat )
          @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit1 @ ( bit0 @ one2 ) ) ) )
            @ ( if @ nat @ ( ord_less @ nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
              @ ( plus_plus @ nat @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_T_m_a_x_t @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) )
                @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) )
                  @ ( if @ nat
                    @ ( ( ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
                       != ( none @ nat ) )
                      & ( vEBT_VEBT_less @ ( some @ nat @ ( vEBT_VEBT_low @ X2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) )
                    @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) @ ( vEBT_T_s_u_c_c @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
                    @ ( plus_plus @ nat @ ( plus_plus @ nat @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_T_s_u_c_c @ Summary @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( one_one @ nat ) )
                      @ ( if @ nat
                        @ ( ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
                          = ( none @ nat ) )
                        @ ( one_one @ nat )
                        @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) @ ( vEBT_T_m_i_n_t @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) )
              @ ( one_one @ nat ) ) ) ) ) ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.simps(6)
thf(fact_2082_insert__bound__size__univ,axiom,
    ! [T2: vEBT_VEBT,N2: nat,U: real,X2: nat] :
      ( ( vEBT_invar_vebt @ T2 @ N2 )
     => ( ( U
          = ( power_power @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ N2 ) )
       => ( ord_less_eq @ real @ ( semiring_1_of_nat @ real @ ( vEBT_T_i_n_s_e_r_t @ T2 @ X2 ) ) @ ( plus_plus @ real @ ( numeral_numeral @ real @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ one2 ) ) ) ) ) ) @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ one2 ) ) ) ) ) @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ U ) ) ) ) ) ) ) ).

% insert_bound_size_univ
thf(fact_2083_word__coorder_Oextremum__unique,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [A2: word @ A] :
          ( ( ord_less_eq @ ( word @ A ) @ A2 @ ( zero_zero @ ( word @ A ) ) )
          = ( A2
            = ( zero_zero @ ( word @ A ) ) ) ) ) ).

% word_coorder.extremum_unique
thf(fact_2084_word__le__0__iff,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [X2: word @ A] :
          ( ( ord_less_eq @ ( word @ A ) @ X2 @ ( zero_zero @ ( word @ A ) ) )
          = ( X2
            = ( zero_zero @ ( word @ A ) ) ) ) ) ).

% word_le_0_iff
thf(fact_2085_idiff__0,axiom,
    ! [N2: extended_enat] :
      ( ( minus_minus @ extended_enat @ ( zero_zero @ extended_enat ) @ N2 )
      = ( zero_zero @ extended_enat ) ) ).

% idiff_0
thf(fact_2086_idiff__0__right,axiom,
    ! [N2: extended_enat] :
      ( ( minus_minus @ extended_enat @ N2 @ ( zero_zero @ extended_enat ) )
      = N2 ) ).

% idiff_0_right
thf(fact_2087_of__int__eq__numeral__iff,axiom,
    ! [A: $tType] :
      ( ( ring_char_0 @ A )
     => ! [Z: int,N2: num] :
          ( ( ( ring_1_of_int @ A @ Z )
            = ( numeral_numeral @ A @ N2 ) )
          = ( Z
            = ( numeral_numeral @ int @ N2 ) ) ) ) ).

% of_int_eq_numeral_iff
thf(fact_2088_of__int__numeral,axiom,
    ! [A: $tType] :
      ( ( ring_1 @ A )
     => ! [K: num] :
          ( ( ring_1_of_int @ A @ ( numeral_numeral @ int @ K ) )
          = ( numeral_numeral @ A @ K ) ) ) ).

% of_int_numeral
thf(fact_2089_of__int__le__iff,axiom,
    ! [A: $tType] :
      ( ( linordered_idom @ A )
     => ! [W: int,Z: int] :
          ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ W ) @ ( ring_1_of_int @ A @ Z ) )
          = ( ord_less_eq @ int @ W @ Z ) ) ) ).

% of_int_le_iff
thf(fact_2090_of__int__less__iff,axiom,
    ! [A: $tType] :
      ( ( linordered_idom @ A )
     => ! [W: int,Z: int] :
          ( ( ord_less @ A @ ( ring_1_of_int @ A @ W ) @ ( ring_1_of_int @ A @ Z ) )
          = ( ord_less @ int @ W @ Z ) ) ) ).

% of_int_less_iff
thf(fact_2091_of__int__eq__1__iff,axiom,
    ! [A: $tType] :
      ( ( ring_char_0 @ A )
     => ! [Z: int] :
          ( ( ( ring_1_of_int @ A @ Z )
            = ( one_one @ A ) )
          = ( Z
            = ( one_one @ int ) ) ) ) ).

% of_int_eq_1_iff
thf(fact_2092_of__int__1,axiom,
    ! [A: $tType] :
      ( ( ring_1 @ A )
     => ( ( ring_1_of_int @ A @ ( one_one @ int ) )
        = ( one_one @ A ) ) ) ).

% of_int_1
thf(fact_2093_of__int__add,axiom,
    ! [A: $tType] :
      ( ( ring_1 @ A )
     => ! [W: int,Z: int] :
          ( ( ring_1_of_int @ A @ ( plus_plus @ int @ W @ Z ) )
          = ( plus_plus @ A @ ( ring_1_of_int @ A @ W ) @ ( ring_1_of_int @ A @ Z ) ) ) ) ).

% of_int_add
thf(fact_2094_of__int__mult,axiom,
    ! [A: $tType] :
      ( ( ring_1 @ A )
     => ! [W: int,Z: int] :
          ( ( ring_1_of_int @ A @ ( times_times @ int @ W @ Z ) )
          = ( times_times @ A @ ( ring_1_of_int @ A @ W ) @ ( ring_1_of_int @ A @ Z ) ) ) ) ).

% of_int_mult
thf(fact_2095_nat__numeral,axiom,
    ! [K: num] :
      ( ( nat2 @ ( numeral_numeral @ int @ K ) )
      = ( numeral_numeral @ nat @ K ) ) ).

% nat_numeral
thf(fact_2096_of__int__diff,axiom,
    ! [A: $tType] :
      ( ( ring_1 @ A )
     => ! [W: int,Z: int] :
          ( ( ring_1_of_int @ A @ ( minus_minus @ int @ W @ Z ) )
          = ( minus_minus @ A @ ( ring_1_of_int @ A @ W ) @ ( ring_1_of_int @ A @ Z ) ) ) ) ).

% of_int_diff
thf(fact_2097_of__int__power__eq__of__int__cancel__iff,axiom,
    ! [A: $tType] :
      ( ( ring_char_0 @ A )
     => ! [X2: int,B2: int,W: nat] :
          ( ( ( ring_1_of_int @ A @ X2 )
            = ( power_power @ A @ ( ring_1_of_int @ A @ B2 ) @ W ) )
          = ( X2
            = ( power_power @ int @ B2 @ W ) ) ) ) ).

% of_int_power_eq_of_int_cancel_iff
thf(fact_2098_of__int__eq__of__int__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( ( ring_char_0 @ A )
     => ! [B2: int,W: nat,X2: int] :
          ( ( ( power_power @ A @ ( ring_1_of_int @ A @ B2 ) @ W )
            = ( ring_1_of_int @ A @ X2 ) )
          = ( ( power_power @ int @ B2 @ W )
            = X2 ) ) ) ).

% of_int_eq_of_int_power_cancel_iff
thf(fact_2099_of__int__power,axiom,
    ! [A: $tType] :
      ( ( ring_1 @ A )
     => ! [Z: int,N2: nat] :
          ( ( ring_1_of_int @ A @ ( power_power @ int @ Z @ N2 ) )
          = ( power_power @ A @ ( ring_1_of_int @ A @ Z ) @ N2 ) ) ) ).

% of_int_power
thf(fact_2100_nat__1,axiom,
    ( ( nat2 @ ( one_one @ int ) )
    = ( suc @ ( zero_zero @ nat ) ) ) ).

% nat_1
thf(fact_2101_nat__le__0,axiom,
    ! [Z: int] :
      ( ( ord_less_eq @ int @ Z @ ( zero_zero @ int ) )
     => ( ( nat2 @ Z )
        = ( zero_zero @ nat ) ) ) ).

% nat_le_0
thf(fact_2102_nat__0__iff,axiom,
    ! [I: int] :
      ( ( ( nat2 @ I )
        = ( zero_zero @ nat ) )
      = ( ord_less_eq @ int @ I @ ( zero_zero @ int ) ) ) ).

% nat_0_iff
thf(fact_2103_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_2104_int__nat__eq,axiom,
    ! [Z: int] :
      ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z )
       => ( ( semiring_1_of_nat @ int @ ( nat2 @ Z ) )
          = Z ) )
      & ( ~ ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z )
       => ( ( semiring_1_of_nat @ int @ ( nat2 @ Z ) )
          = ( zero_zero @ int ) ) ) ) ).

% int_nat_eq
thf(fact_2105_ceiling__add__of__int,axiom,
    ! [A: $tType] :
      ( ( archim2362893244070406136eiling @ A )
     => ! [X2: A,Z: int] :
          ( ( archimedean_ceiling @ A @ ( plus_plus @ A @ X2 @ ( ring_1_of_int @ A @ Z ) ) )
          = ( plus_plus @ int @ ( archimedean_ceiling @ A @ X2 ) @ Z ) ) ) ).

% ceiling_add_of_int
thf(fact_2106_ceiling__diff__of__int,axiom,
    ! [A: $tType] :
      ( ( archim2362893244070406136eiling @ A )
     => ! [X2: A,Z: int] :
          ( ( archimedean_ceiling @ A @ ( minus_minus @ A @ X2 @ ( ring_1_of_int @ A @ Z ) ) )
          = ( minus_minus @ int @ ( archimedean_ceiling @ A @ X2 ) @ Z ) ) ) ).

% ceiling_diff_of_int
thf(fact_2107_of__int__le__0__iff,axiom,
    ! [A: $tType] :
      ( ( linordered_idom @ A )
     => ! [Z: int] :
          ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ Z ) @ ( zero_zero @ A ) )
          = ( ord_less_eq @ int @ Z @ ( zero_zero @ int ) ) ) ) ).

% of_int_le_0_iff
thf(fact_2108_of__int__0__le__iff,axiom,
    ! [A: $tType] :
      ( ( linordered_idom @ A )
     => ! [Z: int] :
          ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( ring_1_of_int @ A @ Z ) )
          = ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z ) ) ) ).

% of_int_0_le_iff
thf(fact_2109_of__int__numeral__le__iff,axiom,
    ! [A: $tType] :
      ( ( linordered_idom @ A )
     => ! [N2: num,Z: int] :
          ( ( ord_less_eq @ A @ ( numeral_numeral @ A @ N2 ) @ ( ring_1_of_int @ A @ Z ) )
          = ( ord_less_eq @ int @ ( numeral_numeral @ int @ N2 ) @ Z ) ) ) ).

% of_int_numeral_le_iff
thf(fact_2110_of__int__le__numeral__iff,axiom,
    ! [A: $tType] :
      ( ( linordered_idom @ A )
     => ! [Z: int,N2: num] :
          ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ Z ) @ ( numeral_numeral @ A @ N2 ) )
          = ( ord_less_eq @ int @ Z @ ( numeral_numeral @ int @ N2 ) ) ) ) ).

% of_int_le_numeral_iff
thf(fact_2111_of__int__0__less__iff,axiom,
    ! [A: $tType] :
      ( ( linordered_idom @ A )
     => ! [Z: int] :
          ( ( ord_less @ A @ ( zero_zero @ A ) @ ( ring_1_of_int @ A @ Z ) )
          = ( ord_less @ int @ ( zero_zero @ int ) @ Z ) ) ) ).

% of_int_0_less_iff
thf(fact_2112_of__int__less__0__iff,axiom,
    ! [A: $tType] :
      ( ( linordered_idom @ A )
     => ! [Z: int] :
          ( ( ord_less @ A @ ( ring_1_of_int @ A @ Z ) @ ( zero_zero @ A ) )
          = ( ord_less @ int @ Z @ ( zero_zero @ int ) ) ) ) ).

% of_int_less_0_iff
thf(fact_2113_of__int__numeral__less__iff,axiom,
    ! [A: $tType] :
      ( ( linordered_idom @ A )
     => ! [N2: num,Z: int] :
          ( ( ord_less @ A @ ( numeral_numeral @ A @ N2 ) @ ( ring_1_of_int @ A @ Z ) )
          = ( ord_less @ int @ ( numeral_numeral @ int @ N2 ) @ Z ) ) ) ).

% of_int_numeral_less_iff
thf(fact_2114_of__int__less__numeral__iff,axiom,
    ! [A: $tType] :
      ( ( linordered_idom @ A )
     => ! [Z: int,N2: num] :
          ( ( ord_less @ A @ ( ring_1_of_int @ A @ Z ) @ ( numeral_numeral @ A @ N2 ) )
          = ( ord_less @ int @ Z @ ( numeral_numeral @ int @ N2 ) ) ) ) ).

% of_int_less_numeral_iff
thf(fact_2115_of__int__le__1__iff,axiom,
    ! [A: $tType] :
      ( ( linordered_idom @ A )
     => ! [Z: int] :
          ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ Z ) @ ( one_one @ A ) )
          = ( ord_less_eq @ int @ Z @ ( one_one @ int ) ) ) ) ).

% of_int_le_1_iff
thf(fact_2116_of__int__1__le__iff,axiom,
    ! [A: $tType] :
      ( ( linordered_idom @ A )
     => ! [Z: int] :
          ( ( ord_less_eq @ A @ ( one_one @ A ) @ ( ring_1_of_int @ A @ Z ) )
          = ( ord_less_eq @ int @ ( one_one @ int ) @ Z ) ) ) ).

% of_int_1_le_iff
thf(fact_2117_of__int__less__1__iff,axiom,
    ! [A: $tType] :
      ( ( linordered_idom @ A )
     => ! [Z: int] :
          ( ( ord_less @ A @ ( ring_1_of_int @ A @ Z ) @ ( one_one @ A ) )
          = ( ord_less @ int @ Z @ ( one_one @ int ) ) ) ) ).

% of_int_less_1_iff
thf(fact_2118_of__int__1__less__iff,axiom,
    ! [A: $tType] :
      ( ( linordered_idom @ A )
     => ! [Z: int] :
          ( ( ord_less @ A @ ( one_one @ A ) @ ( ring_1_of_int @ A @ Z ) )
          = ( ord_less @ int @ ( one_one @ int ) @ Z ) ) ) ).

% of_int_1_less_iff
thf(fact_2119_of__int__eq__numeral__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( ( ring_char_0 @ A )
     => ! [Y2: int,X2: num,N2: nat] :
          ( ( ( ring_1_of_int @ A @ Y2 )
            = ( power_power @ A @ ( numeral_numeral @ A @ X2 ) @ N2 ) )
          = ( Y2
            = ( power_power @ int @ ( numeral_numeral @ int @ X2 ) @ N2 ) ) ) ) ).

% of_int_eq_numeral_power_cancel_iff
thf(fact_2120_numeral__power__eq__of__int__cancel__iff,axiom,
    ! [A: $tType] :
      ( ( ring_char_0 @ A )
     => ! [X2: num,N2: nat,Y2: int] :
          ( ( ( power_power @ A @ ( numeral_numeral @ A @ X2 ) @ N2 )
            = ( ring_1_of_int @ A @ Y2 ) )
          = ( ( power_power @ int @ ( numeral_numeral @ int @ X2 ) @ N2 )
            = Y2 ) ) ) ).

% numeral_power_eq_of_int_cancel_iff
thf(fact_2121_of__int__power__le__of__int__cancel__iff,axiom,
    ! [A: $tType] :
      ( ( linordered_idom @ A )
     => ! [X2: int,B2: int,W: nat] :
          ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ X2 ) @ ( power_power @ A @ ( ring_1_of_int @ A @ B2 ) @ W ) )
          = ( ord_less_eq @ int @ X2 @ ( power_power @ int @ B2 @ W ) ) ) ) ).

% of_int_power_le_of_int_cancel_iff
thf(fact_2122_of__int__le__of__int__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( ( linordered_idom @ A )
     => ! [B2: int,W: nat,X2: int] :
          ( ( ord_less_eq @ A @ ( power_power @ A @ ( ring_1_of_int @ A @ B2 ) @ W ) @ ( ring_1_of_int @ A @ X2 ) )
          = ( ord_less_eq @ int @ ( power_power @ int @ B2 @ W ) @ X2 ) ) ) ).

% of_int_le_of_int_power_cancel_iff
thf(fact_2123_of__int__power__less__of__int__cancel__iff,axiom,
    ! [A: $tType] :
      ( ( linordered_idom @ A )
     => ! [X2: int,B2: int,W: nat] :
          ( ( ord_less @ A @ ( ring_1_of_int @ A @ X2 ) @ ( power_power @ A @ ( ring_1_of_int @ A @ B2 ) @ W ) )
          = ( ord_less @ int @ X2 @ ( power_power @ int @ B2 @ W ) ) ) ) ).

% of_int_power_less_of_int_cancel_iff
thf(fact_2124_of__int__less__of__int__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( ( linordered_idom @ A )
     => ! [B2: int,W: nat,X2: int] :
          ( ( ord_less @ A @ ( power_power @ A @ ( ring_1_of_int @ A @ B2 ) @ W ) @ ( ring_1_of_int @ A @ X2 ) )
          = ( ord_less @ int @ ( power_power @ int @ B2 @ W ) @ X2 ) ) ) ).

% of_int_less_of_int_power_cancel_iff
thf(fact_2125_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_2126_diff__nat__numeral,axiom,
    ! [V: num,V4: num] :
      ( ( minus_minus @ nat @ ( numeral_numeral @ nat @ V ) @ ( numeral_numeral @ nat @ V4 ) )
      = ( nat2 @ ( minus_minus @ int @ ( numeral_numeral @ int @ V ) @ ( numeral_numeral @ int @ V4 ) ) ) ) ).

% diff_nat_numeral
thf(fact_2127_of__nat__nat,axiom,
    ! [A: $tType] :
      ( ( ring_1 @ A )
     => ! [Z: int] :
          ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z )
         => ( ( semiring_1_of_nat @ A @ ( nat2 @ Z ) )
            = ( ring_1_of_int @ A @ Z ) ) ) ) ).

% of_nat_nat
thf(fact_2128_numeral__power__eq__nat__cancel__iff,axiom,
    ! [X2: num,N2: nat,Y2: int] :
      ( ( ( power_power @ nat @ ( numeral_numeral @ nat @ X2 ) @ N2 )
        = ( nat2 @ Y2 ) )
      = ( ( power_power @ int @ ( numeral_numeral @ int @ X2 ) @ N2 )
        = Y2 ) ) ).

% numeral_power_eq_nat_cancel_iff
thf(fact_2129_nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y2: int,X2: num,N2: nat] :
      ( ( ( nat2 @ Y2 )
        = ( power_power @ nat @ ( numeral_numeral @ nat @ X2 ) @ N2 ) )
      = ( Y2
        = ( power_power @ int @ ( numeral_numeral @ int @ X2 ) @ N2 ) ) ) ).

% nat_eq_numeral_power_cancel_iff
thf(fact_2130_nat__ceiling__le__eq,axiom,
    ! [X2: real,A2: nat] :
      ( ( ord_less_eq @ nat @ ( nat2 @ ( archimedean_ceiling @ real @ X2 ) ) @ A2 )
      = ( ord_less_eq @ real @ X2 @ ( semiring_1_of_nat @ real @ A2 ) ) ) ).

% nat_ceiling_le_eq
thf(fact_2131_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_2132_of__int__le__numeral__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( ( linordered_idom @ A )
     => ! [A2: int,X2: num,N2: nat] :
          ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ A2 ) @ ( power_power @ A @ ( numeral_numeral @ A @ X2 ) @ N2 ) )
          = ( ord_less_eq @ int @ A2 @ ( power_power @ int @ ( numeral_numeral @ int @ X2 ) @ N2 ) ) ) ) ).

% of_int_le_numeral_power_cancel_iff
thf(fact_2133_numeral__power__le__of__int__cancel__iff,axiom,
    ! [A: $tType] :
      ( ( linordered_idom @ A )
     => ! [X2: num,N2: nat,A2: int] :
          ( ( ord_less_eq @ A @ ( power_power @ A @ ( numeral_numeral @ A @ X2 ) @ N2 ) @ ( ring_1_of_int @ A @ A2 ) )
          = ( ord_less_eq @ int @ ( power_power @ int @ ( numeral_numeral @ int @ X2 ) @ N2 ) @ A2 ) ) ) ).

% numeral_power_le_of_int_cancel_iff
thf(fact_2134_of__int__less__numeral__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( ( linordered_idom @ A )
     => ! [A2: int,X2: num,N2: nat] :
          ( ( ord_less @ A @ ( ring_1_of_int @ A @ A2 ) @ ( power_power @ A @ ( numeral_numeral @ A @ X2 ) @ N2 ) )
          = ( ord_less @ int @ A2 @ ( power_power @ int @ ( numeral_numeral @ int @ X2 ) @ N2 ) ) ) ) ).

% of_int_less_numeral_power_cancel_iff
thf(fact_2135_numeral__power__less__of__int__cancel__iff,axiom,
    ! [A: $tType] :
      ( ( linordered_idom @ A )
     => ! [X2: num,N2: nat,A2: int] :
          ( ( ord_less @ A @ ( power_power @ A @ ( numeral_numeral @ A @ X2 ) @ N2 ) @ ( ring_1_of_int @ A @ A2 ) )
          = ( ord_less @ int @ ( power_power @ int @ ( numeral_numeral @ int @ X2 ) @ N2 ) @ A2 ) ) ) ).

% numeral_power_less_of_int_cancel_iff
thf(fact_2136_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_2137_numeral__power__less__nat__cancel__iff,axiom,
    ! [X2: num,N2: nat,A2: int] :
      ( ( ord_less @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ X2 ) @ N2 ) @ ( nat2 @ A2 ) )
      = ( ord_less @ int @ ( power_power @ int @ ( numeral_numeral @ int @ X2 ) @ N2 ) @ A2 ) ) ).

% numeral_power_less_nat_cancel_iff
thf(fact_2138_nat__less__numeral__power__cancel__iff,axiom,
    ! [A2: int,X2: num,N2: nat] :
      ( ( ord_less @ nat @ ( nat2 @ A2 ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ X2 ) @ N2 ) )
      = ( ord_less @ int @ A2 @ ( power_power @ int @ ( numeral_numeral @ int @ X2 ) @ N2 ) ) ) ).

% nat_less_numeral_power_cancel_iff
thf(fact_2139_numeral__power__le__nat__cancel__iff,axiom,
    ! [X2: num,N2: nat,A2: int] :
      ( ( ord_less_eq @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ X2 ) @ N2 ) @ ( nat2 @ A2 ) )
      = ( ord_less_eq @ int @ ( power_power @ int @ ( numeral_numeral @ int @ X2 ) @ N2 ) @ A2 ) ) ).

% numeral_power_le_nat_cancel_iff
thf(fact_2140_nat__le__numeral__power__cancel__iff,axiom,
    ! [A2: int,X2: num,N2: nat] :
      ( ( ord_less_eq @ nat @ ( nat2 @ A2 ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ X2 ) @ N2 ) )
      = ( ord_less_eq @ int @ A2 @ ( power_power @ int @ ( numeral_numeral @ int @ X2 ) @ N2 ) ) ) ).

% nat_le_numeral_power_cancel_iff
thf(fact_2141_range__subset__card,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [A2: word @ A,B2: word @ A,C2: word @ A,D2: word @ A] :
          ( ( ord_less_eq @ ( set @ ( word @ A ) ) @ ( set_or1337092689740270186AtMost @ ( word @ A ) @ A2 @ B2 ) @ ( set_or1337092689740270186AtMost @ ( word @ A ) @ C2 @ D2 ) )
         => ( ( ord_less_eq @ ( word @ A ) @ A2 @ B2 )
           => ( ( ord_less_eq @ ( word @ A ) @ C2 @ D2 )
              & ( ord_less_eq @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ B2 @ A2 ) @ ( minus_minus @ ( word @ A ) @ D2 @ C2 ) ) ) ) ) ) ).

% range_subset_card
thf(fact_2142_word__diff__ls_I4_J,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [Y2: word @ A,W: word @ A,Xa: word @ A,X2: word @ A] :
          ( ( ord_less_eq @ ( word @ A ) @ Y2 @ ( plus_plus @ ( word @ A ) @ W @ Xa ) )
         => ( ( ord_less_eq @ ( word @ A ) @ X2 @ Y2 )
           => ( ord_less_eq @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ Y2 @ X2 ) @ ( minus_minus @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ W @ Xa ) @ X2 ) ) ) ) ) ).

% word_diff_ls(4)
thf(fact_2143_word__diff__ls_I3_J,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [Y2: word @ A,X2: word @ A,W: word @ A,Xa: word @ A] :
          ( ( ord_less_eq @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ Y2 @ X2 ) @ ( minus_minus @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ W @ Xa ) @ X2 ) )
         => ( ( ord_less_eq @ ( word @ A ) @ X2 @ ( plus_plus @ ( word @ A ) @ W @ Xa ) )
           => ( ord_less_eq @ ( word @ A ) @ Y2 @ ( plus_plus @ ( word @ A ) @ W @ Xa ) ) ) ) ) ).

% word_diff_ls(3)
thf(fact_2144_Word_Oword__l__diffs_I4_J,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [W: word @ A,Xa: word @ A,Z: word @ A,X2: word @ A] :
          ( ( ord_less_eq @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ W @ Xa ) @ Z )
         => ( ( ord_less_eq @ ( word @ A ) @ X2 @ ( plus_plus @ ( word @ A ) @ W @ Xa ) )
           => ( ord_less_eq @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ W @ Xa ) @ X2 ) @ ( minus_minus @ ( word @ A ) @ Z @ X2 ) ) ) ) ) ).

% Word.word_l_diffs(4)
thf(fact_2145_Word_Oword__l__diffs_I3_J,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [W: word @ A,Xa: word @ A,X2: word @ A,Z: word @ A] :
          ( ( ord_less_eq @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ W @ Xa ) @ X2 ) @ ( minus_minus @ ( word @ A ) @ Z @ X2 ) )
         => ( ( ord_less_eq @ ( word @ A ) @ X2 @ Z )
           => ( ord_less_eq @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ W @ Xa ) @ Z ) ) ) ) ).

% Word.word_l_diffs(3)
thf(fact_2146_word__plus__mcs_I4_J,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [V: word @ A,Xb: word @ A,W: word @ A,Xa: word @ A,X2: word @ A] :
          ( ( ord_less_eq @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ V @ Xb ) @ ( plus_plus @ ( word @ A ) @ W @ Xa ) )
         => ( ( ord_less_eq @ ( word @ A ) @ X2 @ ( plus_plus @ ( word @ A ) @ V @ Xb ) )
           => ( ord_less_eq @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ V @ Xb ) @ X2 ) @ ( minus_minus @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ W @ Xa ) @ X2 ) ) ) ) ) ).

% word_plus_mcs(4)
thf(fact_2147_word__plus__mcs_I3_J,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [V: word @ A,Xb: word @ A,X2: word @ A,W: word @ A,Xa: word @ A] :
          ( ( ord_less_eq @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ V @ Xb ) @ X2 ) @ ( minus_minus @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ W @ Xa ) @ X2 ) )
         => ( ( ord_less_eq @ ( word @ A ) @ X2 @ ( plus_plus @ ( word @ A ) @ W @ Xa ) )
           => ( ord_less_eq @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ V @ Xb ) @ ( plus_plus @ ( word @ A ) @ W @ Xa ) ) ) ) ) ).

% word_plus_mcs(3)
thf(fact_2148_More__Word_Oword__l__diffs_I4_J,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [W: word @ A,X2: word @ A,Z: word @ A] :
          ( ( ord_less_eq @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ W @ X2 ) @ Z )
         => ( ( ord_less_eq @ ( word @ A ) @ X2 @ ( plus_plus @ ( word @ A ) @ W @ X2 ) )
           => ( ord_less_eq @ ( word @ A ) @ W @ ( minus_minus @ ( word @ A ) @ Z @ X2 ) ) ) ) ) ).

% More_Word.word_l_diffs(4)
thf(fact_2149_More__Word_Oword__l__diffs_I3_J,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [W: word @ A,Z: word @ A,X2: word @ A] :
          ( ( ord_less_eq @ ( word @ A ) @ W @ ( minus_minus @ ( word @ A ) @ Z @ X2 ) )
         => ( ( ord_less_eq @ ( word @ A ) @ X2 @ Z )
           => ( ord_less_eq @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ W @ X2 ) @ Z ) ) ) ) ).

% More_Word.word_l_diffs(3)
thf(fact_2150_word__diff__ls_H_I4_J,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [Y2: word @ A,W: word @ A,X2: word @ A] :
          ( ( ord_less_eq @ ( word @ A ) @ Y2 @ ( plus_plus @ ( word @ A ) @ W @ X2 ) )
         => ( ( ord_less_eq @ ( word @ A ) @ X2 @ Y2 )
           => ( ord_less_eq @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ Y2 @ X2 ) @ W ) ) ) ) ).

% word_diff_ls'(4)
thf(fact_2151_word__diff__ls_H_I3_J,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [Y2: word @ A,X2: word @ A,W: word @ A] :
          ( ( ord_less_eq @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ Y2 @ X2 ) @ W )
         => ( ( ord_less_eq @ ( word @ A ) @ X2 @ ( plus_plus @ ( word @ A ) @ W @ X2 ) )
           => ( ord_less_eq @ ( word @ A ) @ Y2 @ ( plus_plus @ ( word @ A ) @ W @ X2 ) ) ) ) ) ).

% word_diff_ls'(3)
thf(fact_2152_word__l__diffs_H_I4_J,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [W: word @ A,X2: word @ A,Z: word @ A] :
          ( ( ord_less_eq @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ W @ X2 ) @ Z )
         => ( ( ord_less_eq @ ( word @ A ) @ X2 @ ( plus_plus @ ( word @ A ) @ W @ X2 ) )
           => ( ord_less_eq @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ W @ X2 ) @ X2 ) @ ( minus_minus @ ( word @ A ) @ Z @ X2 ) ) ) ) ) ).

% word_l_diffs'(4)
thf(fact_2153_word__l__diffs_H_I3_J,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [W: word @ A,X2: word @ A,Z: word @ A] :
          ( ( ord_less_eq @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ W @ X2 ) @ X2 ) @ ( minus_minus @ ( word @ A ) @ Z @ X2 ) )
         => ( ( ord_less_eq @ ( word @ A ) @ X2 @ Z )
           => ( ord_less_eq @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ W @ X2 ) @ Z ) ) ) ) ).

% word_l_diffs'(3)
thf(fact_2154_word__diff__ls_H_H_I4_J,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [Y2: word @ A,W: word @ A,X2: word @ A] :
          ( ( ord_less_eq @ ( word @ A ) @ Y2 @ ( plus_plus @ ( word @ A ) @ W @ X2 ) )
         => ( ( ord_less_eq @ ( word @ A ) @ X2 @ Y2 )
           => ( ord_less_eq @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ Y2 @ X2 ) @ ( minus_minus @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ W @ X2 ) @ X2 ) ) ) ) ) ).

% word_diff_ls''(4)
thf(fact_2155_word__diff__ls_H_H_I3_J,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [Y2: word @ A,X2: word @ A,W: word @ A] :
          ( ( ord_less_eq @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ Y2 @ X2 ) @ ( minus_minus @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ W @ X2 ) @ X2 ) )
         => ( ( ord_less_eq @ ( word @ A ) @ X2 @ ( plus_plus @ ( word @ A ) @ W @ X2 ) )
           => ( ord_less_eq @ ( word @ A ) @ Y2 @ ( plus_plus @ ( word @ A ) @ W @ X2 ) ) ) ) ) ).

% word_diff_ls''(3)
thf(fact_2156_le__plus,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [C2: word @ A,B2: word @ A,A2: word @ A] :
          ( ( ord_less_eq @ ( word @ A ) @ C2 @ ( minus_minus @ ( word @ A ) @ B2 @ A2 ) )
         => ( ( ord_less_eq @ ( word @ A ) @ A2 @ B2 )
           => ( ord_less_eq @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ A2 @ C2 ) @ B2 ) ) ) ) ).

% le_plus
thf(fact_2157_le__plus_H,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [A2: word @ A,B2: word @ A,C2: word @ A] :
          ( ( ord_less_eq @ ( word @ A ) @ A2 @ B2 )
         => ( ( ord_less_eq @ ( word @ A ) @ C2 @ ( minus_minus @ ( word @ A ) @ B2 @ A2 ) )
           => ( ord_less_eq @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ A2 @ C2 ) @ B2 ) ) ) ) ).

% le_plus'
thf(fact_2158_le__minus_H,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [A2: word @ A,C2: word @ A,B2: word @ A] :
          ( ( ord_less_eq @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ A2 @ C2 ) @ B2 )
         => ( ( ord_less_eq @ ( word @ A ) @ A2 @ ( plus_plus @ ( word @ A ) @ A2 @ C2 ) )
           => ( ord_less_eq @ ( word @ A ) @ C2 @ ( minus_minus @ ( word @ A ) @ B2 @ A2 ) ) ) ) ) ).

% le_minus'
thf(fact_2159_word__sub__le,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [Y2: word @ A,X2: word @ A] :
          ( ( ord_less_eq @ ( word @ A ) @ Y2 @ X2 )
         => ( ord_less_eq @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ X2 @ Y2 ) @ X2 ) ) ) ).

% word_sub_le
thf(fact_2160_word__sub__le__iff,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [X2: word @ A,Y2: word @ A] :
          ( ( ord_less_eq @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ X2 @ Y2 ) @ X2 )
          = ( ord_less_eq @ ( word @ A ) @ Y2 @ X2 ) ) ) ).

% word_sub_le_iff
thf(fact_2161_word__le__minus__mono,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [A2: word @ A,C2: word @ A,D2: word @ A,B2: word @ A] :
          ( ( ord_less_eq @ ( word @ A ) @ A2 @ C2 )
         => ( ( ord_less_eq @ ( word @ A ) @ D2 @ B2 )
           => ( ( ord_less_eq @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ A2 @ B2 ) @ A2 )
             => ( ( ord_less_eq @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ C2 @ D2 ) @ C2 )
               => ( ord_less_eq @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ A2 @ B2 ) @ ( minus_minus @ ( word @ A ) @ C2 @ D2 ) ) ) ) ) ) ) ).

% word_le_minus_mono
thf(fact_2162_word__le__minus__cancel,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [Y2: word @ A,X2: word @ A,Z: word @ A] :
          ( ( ord_less_eq @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ Y2 @ X2 ) @ ( minus_minus @ ( word @ A ) @ Z @ X2 ) )
         => ( ( ord_less_eq @ ( word @ A ) @ X2 @ Z )
           => ( ord_less_eq @ ( word @ A ) @ Y2 @ Z ) ) ) ) ).

% word_le_minus_cancel
thf(fact_2163_word__le__minus__mono__left,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [Y2: word @ A,Z: word @ A,X2: word @ A] :
          ( ( ord_less_eq @ ( word @ A ) @ Y2 @ Z )
         => ( ( ord_less_eq @ ( word @ A ) @ X2 @ Y2 )
           => ( ord_less_eq @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ Y2 @ X2 ) @ ( minus_minus @ ( word @ A ) @ Z @ X2 ) ) ) ) ) ).

% word_le_minus_mono_left
thf(fact_2164_word__le__imp__diff__le,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [K: word @ A,N2: word @ A,M: word @ A] :
          ( ( ord_less_eq @ ( word @ A ) @ K @ N2 )
         => ( ( ord_less_eq @ ( word @ A ) @ N2 @ M )
           => ( ord_less_eq @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ N2 @ K ) @ M ) ) ) ) ).

% word_le_imp_diff_le
thf(fact_2165_plus__minus__no__overflow__ab,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [X2: word @ A,Ab: word @ A,C2: word @ A] :
          ( ( ord_less_eq @ ( word @ A ) @ X2 @ ( minus_minus @ ( word @ A ) @ Ab @ C2 ) )
         => ( ( ord_less_eq @ ( word @ A ) @ C2 @ Ab )
           => ( ord_less_eq @ ( word @ A ) @ X2 @ ( plus_plus @ ( word @ A ) @ X2 @ C2 ) ) ) ) ) ).

% plus_minus_no_overflow_ab
thf(fact_2166_word__le__minus__mono__right,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [Z: word @ A,Y2: word @ A,X2: word @ A] :
          ( ( ord_less_eq @ ( word @ A ) @ Z @ Y2 )
         => ( ( ord_less_eq @ ( word @ A ) @ Y2 @ X2 )
           => ( ( ord_less_eq @ ( word @ A ) @ Z @ X2 )
             => ( ord_less_eq @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ X2 @ Y2 ) @ ( minus_minus @ ( word @ A ) @ X2 @ Z ) ) ) ) ) ) ).

% word_le_minus_mono_right
thf(fact_2167_plus__minus__not__NULL__ab,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [X2: word @ A,Ab: word @ A,C2: word @ A] :
          ( ( ord_less_eq @ ( word @ A ) @ X2 @ ( minus_minus @ ( word @ A ) @ Ab @ C2 ) )
         => ( ( ord_less_eq @ ( word @ A ) @ C2 @ Ab )
           => ( ( C2
               != ( zero_zero @ ( word @ A ) ) )
             => ( ( plus_plus @ ( word @ A ) @ X2 @ C2 )
               != ( zero_zero @ ( word @ A ) ) ) ) ) ) ) ).

% plus_minus_not_NULL_ab
thf(fact_2168_neq__0__no__wrap,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [X2: word @ A,Y2: word @ A] :
          ( ( ord_less_eq @ ( word @ A ) @ X2 @ ( plus_plus @ ( word @ A ) @ X2 @ Y2 ) )
         => ( ( X2
             != ( zero_zero @ ( word @ A ) ) )
           => ( ( plus_plus @ ( word @ A ) @ X2 @ Y2 )
             != ( zero_zero @ ( word @ A ) ) ) ) ) ) ).

% neq_0_no_wrap
thf(fact_2169_word__random,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [P4: word @ A,X7: word @ A,X2: word @ A] :
          ( ( ord_less_eq @ ( word @ A ) @ P4 @ ( plus_plus @ ( word @ A ) @ P4 @ X7 ) )
         => ( ( ord_less_eq @ ( word @ A ) @ X2 @ X7 )
           => ( ord_less_eq @ ( word @ A ) @ P4 @ ( plus_plus @ ( word @ A ) @ P4 @ X2 ) ) ) ) ) ).

% word_random
thf(fact_2170_word__sub__mono2,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [A2: word @ A,B2: word @ A,C2: word @ A,D2: word @ A] :
          ( ( ord_less_eq @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ A2 @ B2 ) @ ( plus_plus @ ( word @ A ) @ C2 @ D2 ) )
         => ( ( ord_less_eq @ ( word @ A ) @ C2 @ A2 )
           => ( ( ord_less_eq @ ( word @ A ) @ B2 @ ( plus_plus @ ( word @ A ) @ A2 @ B2 ) )
             => ( ( ord_less_eq @ ( word @ A ) @ D2 @ ( plus_plus @ ( word @ A ) @ C2 @ D2 ) )
               => ( ord_less_eq @ ( word @ A ) @ B2 @ D2 ) ) ) ) ) ) ).

% word_sub_mono2
thf(fact_2171_word__plus__mcs__3,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [V: word @ A,W: word @ A,X2: word @ A] :
          ( ( ord_less_eq @ ( word @ A ) @ V @ W )
         => ( ( ord_less_eq @ ( word @ A ) @ X2 @ ( plus_plus @ ( word @ A ) @ W @ X2 ) )
           => ( ord_less_eq @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ V @ X2 ) @ ( plus_plus @ ( word @ A ) @ W @ X2 ) ) ) ) ) ).

% word_plus_mcs_3
thf(fact_2172_word__plus__mcs__4,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [V: word @ A,X2: word @ A,W: word @ A] :
          ( ( ord_less_eq @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ V @ X2 ) @ ( plus_plus @ ( word @ A ) @ W @ X2 ) )
         => ( ( ord_less_eq @ ( word @ A ) @ X2 @ ( plus_plus @ ( word @ A ) @ V @ X2 ) )
           => ( ord_less_eq @ ( word @ A ) @ V @ W ) ) ) ) ).

% word_plus_mcs_4
thf(fact_2173_word__add__le__iff2,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [I: word @ A,K: word @ A,J: word @ A] :
          ( ( ord_less_eq @ ( word @ A ) @ I @ ( plus_plus @ ( word @ A ) @ I @ K ) )
         => ( ( ord_less_eq @ ( word @ A ) @ J @ ( plus_plus @ ( word @ A ) @ J @ K ) )
           => ( ( ord_less_eq @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ I @ K ) @ ( plus_plus @ ( word @ A ) @ J @ K ) )
              = ( ord_less_eq @ ( word @ A ) @ I @ J ) ) ) ) ) ).

% word_add_le_iff2
thf(fact_2174_word__plus__mcs__4_H,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [X2: word @ A,V: word @ A,W: word @ A] :
          ( ( ord_less_eq @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ X2 @ V ) @ ( plus_plus @ ( word @ A ) @ X2 @ W ) )
         => ( ( ord_less_eq @ ( word @ A ) @ X2 @ ( plus_plus @ ( word @ A ) @ X2 @ V ) )
           => ( ord_less_eq @ ( word @ A ) @ V @ W ) ) ) ) ).

% word_plus_mcs_4'
thf(fact_2175_word__add__increasing,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [P4: word @ A,W: word @ A,X2: word @ A] :
          ( ( ord_less_eq @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ P4 @ W ) @ X2 )
         => ( ( ord_less_eq @ ( word @ A ) @ P4 @ ( plus_plus @ ( word @ A ) @ P4 @ W ) )
           => ( ord_less_eq @ ( word @ A ) @ P4 @ X2 ) ) ) ) ).

% word_add_increasing
thf(fact_2176_word__plus__mono__left,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [Y2: word @ A,Z: word @ A,X2: word @ A] :
          ( ( ord_less_eq @ ( word @ A ) @ Y2 @ Z )
         => ( ( ord_less_eq @ ( word @ A ) @ X2 @ ( plus_plus @ ( word @ A ) @ X2 @ Z ) )
           => ( ord_less_eq @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ Y2 @ X2 ) @ ( plus_plus @ ( word @ A ) @ Z @ X2 ) ) ) ) ) ).

% word_plus_mono_left
thf(fact_2177_constraint__expand,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [X2: word @ A,Lower: word @ A,Upper: word @ A] :
          ( ( member @ ( word @ A ) @ X2
            @ ( collect @ ( word @ A )
              @ ^ [Y: word @ A] :
                  ( ( ord_less_eq @ ( word @ A ) @ Lower @ Y )
                  & ( ord_less_eq @ ( word @ A ) @ Y @ Upper ) ) ) )
          = ( ( ord_less_eq @ ( word @ A ) @ Lower @ X2 )
            & ( ord_less_eq @ ( word @ A ) @ X2 @ Upper ) ) ) ) ).

% constraint_expand
thf(fact_2178_word__plus__mono__right2,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [A2: word @ A,B2: word @ A,C2: word @ A] :
          ( ( ord_less_eq @ ( word @ A ) @ A2 @ ( plus_plus @ ( word @ A ) @ A2 @ B2 ) )
         => ( ( ord_less_eq @ ( word @ A ) @ C2 @ B2 )
           => ( ord_less_eq @ ( word @ A ) @ A2 @ ( plus_plus @ ( word @ A ) @ A2 @ C2 ) ) ) ) ) ).

% word_plus_mono_right2
thf(fact_2179_word__plus__mono__right,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [Y2: word @ A,Z: word @ A,X2: word @ A] :
          ( ( ord_less_eq @ ( word @ A ) @ Y2 @ Z )
         => ( ( ord_less_eq @ ( word @ A ) @ X2 @ ( plus_plus @ ( word @ A ) @ X2 @ Z ) )
           => ( ord_less_eq @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ X2 @ Y2 ) @ ( plus_plus @ ( word @ A ) @ X2 @ Z ) ) ) ) ) ).

% word_plus_mono_right
thf(fact_2180_word__le__plus__either,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [X2: word @ A,Y2: word @ A,Z: word @ A] :
          ( ( ( ord_less_eq @ ( word @ A ) @ X2 @ Y2 )
            | ( ord_less_eq @ ( word @ A ) @ X2 @ Z ) )
         => ( ( ord_less_eq @ ( word @ A ) @ Y2 @ ( plus_plus @ ( word @ A ) @ Y2 @ Z ) )
           => ( ord_less_eq @ ( word @ A ) @ X2 @ ( plus_plus @ ( word @ A ) @ Y2 @ Z ) ) ) ) ) ).

% word_le_plus_either
thf(fact_2181_le__no__overflow,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [X2: word @ A,B2: word @ A,A2: word @ A] :
          ( ( ord_less_eq @ ( word @ A ) @ X2 @ B2 )
         => ( ( ord_less_eq @ ( word @ A ) @ A2 @ ( plus_plus @ ( word @ A ) @ A2 @ B2 ) )
           => ( ord_less_eq @ ( word @ A ) @ X2 @ ( plus_plus @ ( word @ A ) @ A2 @ B2 ) ) ) ) ) ).

% le_no_overflow
thf(fact_2182_olen__add__eqv,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [Y2: word @ A,X2: word @ A] :
          ( ( ord_less_eq @ ( word @ A ) @ Y2 @ ( plus_plus @ ( word @ A ) @ Y2 @ X2 ) )
          = ( ord_less_eq @ ( word @ A ) @ X2 @ ( plus_plus @ ( word @ A ) @ Y2 @ X2 ) ) ) ) ).

% olen_add_eqv
thf(fact_2183_word__coorder_Oextremum,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [A2: word @ A] : ( ord_less_eq @ ( word @ A ) @ ( zero_zero @ ( word @ A ) ) @ A2 ) ) ).

% word_coorder.extremum
thf(fact_2184_word__coorder_Oextremum__uniqueI,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [A2: word @ A] :
          ( ( ord_less_eq @ ( word @ A ) @ A2 @ ( zero_zero @ ( word @ A ) ) )
         => ( A2
            = ( zero_zero @ ( word @ A ) ) ) ) ) ).

% word_coorder.extremum_uniqueI
thf(fact_2185_word__zero__le,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [Y2: word @ A] : ( ord_less_eq @ ( word @ A ) @ ( zero_zero @ ( word @ A ) ) @ Y2 ) ) ).

% word_zero_le
thf(fact_2186_iadd__is__0,axiom,
    ! [M: extended_enat,N2: extended_enat] :
      ( ( ( plus_plus @ extended_enat @ M @ N2 )
        = ( zero_zero @ extended_enat ) )
      = ( ( M
          = ( zero_zero @ extended_enat ) )
        & ( N2
          = ( zero_zero @ extended_enat ) ) ) ) ).

% iadd_is_0
thf(fact_2187_imult__is__0,axiom,
    ! [M: extended_enat,N2: extended_enat] :
      ( ( ( times_times @ extended_enat @ M @ N2 )
        = ( zero_zero @ extended_enat ) )
      = ( ( M
          = ( zero_zero @ extended_enat ) )
        | ( N2
          = ( zero_zero @ extended_enat ) ) ) ) ).

% imult_is_0
thf(fact_2188_zero__one__enat__neq_I1_J,axiom,
    ( ( zero_zero @ extended_enat )
   != ( one_one @ extended_enat ) ) ).

% zero_one_enat_neq(1)
thf(fact_2189_ex__le__of__int,axiom,
    ! [A: $tType] :
      ( ( archim462609752435547400_field @ A )
     => ! [X2: A] :
        ? [Z3: int] : ( ord_less_eq @ A @ X2 @ ( ring_1_of_int @ A @ Z3 ) ) ) ).

% ex_le_of_int
thf(fact_2190_ex__of__int__less,axiom,
    ! [A: $tType] :
      ( ( archim462609752435547400_field @ A )
     => ! [X2: A] :
        ? [Z3: int] : ( ord_less @ A @ ( ring_1_of_int @ A @ Z3 ) @ X2 ) ) ).

% ex_of_int_less
thf(fact_2191_ex__less__of__int,axiom,
    ! [A: $tType] :
      ( ( archim462609752435547400_field @ A )
     => ! [X2: A] :
        ? [Z3: int] : ( ord_less @ A @ X2 @ ( ring_1_of_int @ A @ Z3 ) ) ) ).

% ex_less_of_int
thf(fact_2192_mult__of__int__commute,axiom,
    ! [A: $tType] :
      ( ( ring_1 @ A )
     => ! [X2: int,Y2: A] :
          ( ( times_times @ A @ ( ring_1_of_int @ A @ X2 ) @ Y2 )
          = ( times_times @ A @ Y2 @ ( ring_1_of_int @ A @ X2 ) ) ) ) ).

% mult_of_int_commute
thf(fact_2193_nat__numeral__as__int,axiom,
    ( ( numeral_numeral @ nat )
    = ( ^ [I2: num] : ( nat2 @ ( numeral_numeral @ int @ I2 ) ) ) ) ).

% nat_numeral_as_int
thf(fact_2194_nat__mono,axiom,
    ! [X2: int,Y2: int] :
      ( ( ord_less_eq @ int @ X2 @ Y2 )
     => ( ord_less_eq @ nat @ ( nat2 @ X2 ) @ ( nat2 @ Y2 ) ) ) ).

% nat_mono
thf(fact_2195_nat__one__as__int,axiom,
    ( ( one_one @ nat )
    = ( nat2 @ ( one_one @ int ) ) ) ).

% nat_one_as_int
thf(fact_2196_eq__nat__nat__iff,axiom,
    ! [Z: int,Z7: int] :
      ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z )
     => ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z7 )
       => ( ( ( nat2 @ Z )
            = ( nat2 @ Z7 ) )
          = ( Z = Z7 ) ) ) ) ).

% eq_nat_nat_iff
thf(fact_2197_all__nat,axiom,
    ( ( ^ [P2: nat > $o] :
        ! [X4: nat] : ( P2 @ X4 ) )
    = ( ^ [P3: nat > $o] :
        ! [X: int] :
          ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
         => ( P3 @ ( nat2 @ X ) ) ) ) ) ).

% all_nat
thf(fact_2198_ex__nat,axiom,
    ( ( ^ [P2: nat > $o] :
        ? [X4: nat] : ( P2 @ X4 ) )
    = ( ^ [P3: nat > $o] :
        ? [X: int] :
          ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
          & ( P3 @ ( nat2 @ X ) ) ) ) ) ).

% ex_nat
thf(fact_2199_le__of__int__ceiling,axiom,
    ! [A: $tType] :
      ( ( archim2362893244070406136eiling @ A )
     => ! [X2: A] : ( ord_less_eq @ A @ X2 @ ( ring_1_of_int @ A @ ( archimedean_ceiling @ A @ X2 ) ) ) ) ).

% le_of_int_ceiling
thf(fact_2200_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_2201_zless__nat__eq__int__zless,axiom,
    ! [M: nat,Z: int] :
      ( ( ord_less @ nat @ M @ ( nat2 @ Z ) )
      = ( ord_less @ int @ ( semiring_1_of_nat @ int @ M ) @ Z ) ) ).

% zless_nat_eq_int_zless
thf(fact_2202_nat__le__iff,axiom,
    ! [X2: int,N2: nat] :
      ( ( ord_less_eq @ nat @ ( nat2 @ X2 ) @ N2 )
      = ( ord_less_eq @ int @ X2 @ ( semiring_1_of_nat @ int @ N2 ) ) ) ).

% nat_le_iff
thf(fact_2203_of__nat__ceiling,axiom,
    ! [A: $tType] :
      ( ( archim2362893244070406136eiling @ A )
     => ! [R4: A] : ( ord_less_eq @ A @ R4 @ ( semiring_1_of_nat @ A @ ( nat2 @ ( archimedean_ceiling @ A @ R4 ) ) ) ) ) ).

% of_nat_ceiling
thf(fact_2204_nat__int__add,axiom,
    ! [A2: nat,B2: nat] :
      ( ( nat2 @ ( plus_plus @ int @ ( semiring_1_of_nat @ int @ A2 ) @ ( semiring_1_of_nat @ int @ B2 ) ) )
      = ( plus_plus @ nat @ A2 @ B2 ) ) ).

% nat_int_add
thf(fact_2205_nat__0__le,axiom,
    ! [Z: int] :
      ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z )
     => ( ( semiring_1_of_nat @ int @ ( nat2 @ Z ) )
        = Z ) ) ).

% nat_0_le
thf(fact_2206_int__eq__iff,axiom,
    ! [M: nat,Z: int] :
      ( ( ( semiring_1_of_nat @ int @ M )
        = Z )
      = ( ( M
          = ( nat2 @ Z ) )
        & ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z ) ) ) ).

% int_eq_iff
thf(fact_2207_ceiling__le__iff,axiom,
    ! [A: $tType] :
      ( ( archim2362893244070406136eiling @ A )
     => ! [X2: A,Z: int] :
          ( ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ X2 ) @ Z )
          = ( ord_less_eq @ A @ X2 @ ( ring_1_of_int @ A @ Z ) ) ) ) ).

% ceiling_le_iff
thf(fact_2208_ceiling__le,axiom,
    ! [A: $tType] :
      ( ( archim2362893244070406136eiling @ A )
     => ! [X2: A,A2: int] :
          ( ( ord_less_eq @ A @ X2 @ ( ring_1_of_int @ A @ A2 ) )
         => ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ X2 ) @ A2 ) ) ) ).

% ceiling_le
thf(fact_2209_less__ceiling__iff,axiom,
    ! [A: $tType] :
      ( ( archim2362893244070406136eiling @ A )
     => ! [Z: int,X2: A] :
          ( ( ord_less @ int @ Z @ ( archimedean_ceiling @ A @ X2 ) )
          = ( ord_less @ A @ ( ring_1_of_int @ A @ Z ) @ X2 ) ) ) ).

% less_ceiling_iff
thf(fact_2210_int__minus,axiom,
    ! [N2: nat,M: nat] :
      ( ( semiring_1_of_nat @ int @ ( minus_minus @ nat @ N2 @ M ) )
      = ( semiring_1_of_nat @ int @ ( nat2 @ ( minus_minus @ int @ ( semiring_1_of_nat @ int @ N2 ) @ ( semiring_1_of_nat @ int @ M ) ) ) ) ) ).

% int_minus
thf(fact_2211_real__of__int__div4,axiom,
    ! [N2: int,X2: int] : ( ord_less_eq @ real @ ( ring_1_of_int @ real @ ( divide_divide @ int @ N2 @ X2 ) ) @ ( divide_divide @ real @ ( ring_1_of_int @ real @ N2 ) @ ( ring_1_of_int @ real @ X2 ) ) ) ).

% real_of_int_div4
thf(fact_2212_real__nat__ceiling__ge,axiom,
    ! [X2: real] : ( ord_less_eq @ real @ X2 @ ( semiring_1_of_nat @ real @ ( nat2 @ ( archimedean_ceiling @ real @ X2 ) ) ) ) ).

% real_nat_ceiling_ge
thf(fact_2213_nat__plus__as__int,axiom,
    ( ( plus_plus @ nat )
    = ( ^ [A5: nat,B5: nat] : ( nat2 @ ( plus_plus @ int @ ( semiring_1_of_nat @ int @ A5 ) @ ( semiring_1_of_nat @ int @ B5 ) ) ) ) ) ).

% nat_plus_as_int
thf(fact_2214_nat__times__as__int,axiom,
    ( ( times_times @ nat )
    = ( ^ [A5: nat,B5: nat] : ( nat2 @ ( times_times @ int @ ( semiring_1_of_nat @ int @ A5 ) @ ( semiring_1_of_nat @ int @ B5 ) ) ) ) ) ).

% nat_times_as_int
thf(fact_2215_nat__minus__as__int,axiom,
    ( ( minus_minus @ nat )
    = ( ^ [A5: nat,B5: nat] : ( nat2 @ ( minus_minus @ int @ ( semiring_1_of_nat @ int @ A5 ) @ ( semiring_1_of_nat @ int @ B5 ) ) ) ) ) ).

% nat_minus_as_int
thf(fact_2216_nat__div__as__int,axiom,
    ( ( divide_divide @ nat )
    = ( ^ [A5: nat,B5: nat] : ( nat2 @ ( divide_divide @ int @ ( semiring_1_of_nat @ int @ A5 ) @ ( semiring_1_of_nat @ int @ B5 ) ) ) ) ) ).

% nat_div_as_int
thf(fact_2217_of__int__nonneg,axiom,
    ! [A: $tType] :
      ( ( linordered_idom @ A )
     => ! [Z: int] :
          ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z )
         => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( ring_1_of_int @ A @ Z ) ) ) ) ).

% of_int_nonneg
thf(fact_2218_of__int__pos,axiom,
    ! [A: $tType] :
      ( ( linordered_idom @ A )
     => ! [Z: int] :
          ( ( ord_less @ int @ ( zero_zero @ int ) @ Z )
         => ( ord_less @ A @ ( zero_zero @ A ) @ ( ring_1_of_int @ A @ Z ) ) ) ) ).

% of_int_pos
thf(fact_2219_floor__exists1,axiom,
    ! [A: $tType] :
      ( ( archim462609752435547400_field @ A )
     => ! [X2: A] :
        ? [X3: int] :
          ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ X3 ) @ X2 )
          & ( ord_less @ A @ X2 @ ( ring_1_of_int @ A @ ( plus_plus @ int @ X3 @ ( one_one @ int ) ) ) )
          & ! [Y3: int] :
              ( ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ Y3 ) @ X2 )
                & ( ord_less @ A @ X2 @ ( ring_1_of_int @ A @ ( plus_plus @ int @ Y3 @ ( one_one @ int ) ) ) ) )
             => ( Y3 = X3 ) ) ) ) ).

% floor_exists1
thf(fact_2220_floor__exists,axiom,
    ! [A: $tType] :
      ( ( archim462609752435547400_field @ A )
     => ! [X2: A] :
        ? [Z3: int] :
          ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ Z3 ) @ X2 )
          & ( ord_less @ A @ X2 @ ( ring_1_of_int @ A @ ( plus_plus @ int @ Z3 @ ( one_one @ int ) ) ) ) ) ) ).

% floor_exists
thf(fact_2221_of__int__ceiling__le__add__one,axiom,
    ! [A: $tType] :
      ( ( archim2362893244070406136eiling @ A )
     => ! [R4: A] : ( ord_less_eq @ A @ ( ring_1_of_int @ A @ ( archimedean_ceiling @ A @ R4 ) ) @ ( plus_plus @ A @ R4 @ ( one_one @ A ) ) ) ) ).

% of_int_ceiling_le_add_one
thf(fact_2222_of__int__ceiling__diff__one__le,axiom,
    ! [A: $tType] :
      ( ( archim2362893244070406136eiling @ A )
     => ! [R4: A] : ( ord_less_eq @ A @ ( minus_minus @ A @ ( ring_1_of_int @ A @ ( archimedean_ceiling @ A @ R4 ) ) @ ( one_one @ A ) ) @ R4 ) ) ).

% of_int_ceiling_diff_one_le
thf(fact_2223_of__nat__less__of__int__iff,axiom,
    ! [A: $tType] :
      ( ( linordered_idom @ A )
     => ! [N2: nat,X2: int] :
          ( ( ord_less @ A @ ( semiring_1_of_nat @ A @ N2 ) @ ( ring_1_of_int @ A @ X2 ) )
          = ( ord_less @ int @ ( semiring_1_of_nat @ int @ N2 ) @ X2 ) ) ) ).

% of_nat_less_of_int_iff
thf(fact_2224_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_2225_nat__eq__iff2,axiom,
    ! [M: nat,W: int] :
      ( ( M
        = ( nat2 @ W ) )
      = ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ W )
         => ( W
            = ( semiring_1_of_nat @ int @ M ) ) )
        & ( ~ ( ord_less_eq @ int @ ( zero_zero @ int ) @ W )
         => ( M
            = ( zero_zero @ nat ) ) ) ) ) ).

% nat_eq_iff2
thf(fact_2226_nat__eq__iff,axiom,
    ! [W: int,M: nat] :
      ( ( ( nat2 @ W )
        = M )
      = ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ W )
         => ( W
            = ( semiring_1_of_nat @ int @ M ) ) )
        & ( ~ ( ord_less_eq @ int @ ( zero_zero @ int ) @ W )
         => ( M
            = ( zero_zero @ nat ) ) ) ) ) ).

% nat_eq_iff
thf(fact_2227_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_2228_nat__add__distrib,axiom,
    ! [Z: int,Z7: int] :
      ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z )
     => ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z7 )
       => ( ( nat2 @ ( plus_plus @ int @ Z @ Z7 ) )
          = ( plus_plus @ nat @ ( nat2 @ Z ) @ ( nat2 @ Z7 ) ) ) ) ) ).

% nat_add_distrib
thf(fact_2229_le__nat__iff,axiom,
    ! [K: int,N2: nat] :
      ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
     => ( ( ord_less_eq @ nat @ N2 @ ( nat2 @ K ) )
        = ( ord_less_eq @ int @ ( semiring_1_of_nat @ int @ N2 ) @ K ) ) ) ).

% le_nat_iff
thf(fact_2230_Suc__as__int,axiom,
    ( suc
    = ( ^ [A5: nat] : ( nat2 @ ( plus_plus @ int @ ( semiring_1_of_nat @ int @ A5 ) @ ( one_one @ int ) ) ) ) ) ).

% Suc_as_int
thf(fact_2231_nat__mult__distrib,axiom,
    ! [Z: int,Z7: int] :
      ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z )
     => ( ( nat2 @ ( times_times @ int @ Z @ Z7 ) )
        = ( times_times @ nat @ ( nat2 @ Z ) @ ( nat2 @ Z7 ) ) ) ) ).

% nat_mult_distrib
thf(fact_2232_nat__diff__distrib,axiom,
    ! [Z7: int,Z: int] :
      ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z7 )
     => ( ( ord_less_eq @ int @ Z7 @ Z )
       => ( ( nat2 @ ( minus_minus @ int @ Z @ Z7 ) )
          = ( minus_minus @ nat @ ( nat2 @ Z ) @ ( nat2 @ Z7 ) ) ) ) ) ).

% nat_diff_distrib
thf(fact_2233_nat__diff__distrib_H,axiom,
    ! [X2: int,Y2: int] :
      ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X2 )
     => ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y2 )
       => ( ( nat2 @ ( minus_minus @ int @ X2 @ Y2 ) )
          = ( minus_minus @ nat @ ( nat2 @ X2 ) @ ( nat2 @ Y2 ) ) ) ) ) ).

% nat_diff_distrib'
thf(fact_2234_nat__div__distrib,axiom,
    ! [X2: int,Y2: int] :
      ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X2 )
     => ( ( nat2 @ ( divide_divide @ int @ X2 @ Y2 ) )
        = ( divide_divide @ nat @ ( nat2 @ X2 ) @ ( nat2 @ Y2 ) ) ) ) ).

% nat_div_distrib
thf(fact_2235_nat__div__distrib_H,axiom,
    ! [Y2: int,X2: int] :
      ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y2 )
     => ( ( nat2 @ ( divide_divide @ int @ X2 @ Y2 ) )
        = ( divide_divide @ nat @ ( nat2 @ X2 ) @ ( nat2 @ Y2 ) ) ) ) ).

% nat_div_distrib'
thf(fact_2236_nat__power__eq,axiom,
    ! [Z: int,N2: nat] :
      ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z )
     => ( ( nat2 @ ( power_power @ int @ Z @ N2 ) )
        = ( power_power @ nat @ ( nat2 @ Z ) @ N2 ) ) ) ).

% nat_power_eq
thf(fact_2237_int__le__real__less,axiom,
    ( ( ord_less_eq @ int )
    = ( ^ [N: int,M7: int] : ( ord_less @ real @ ( ring_1_of_int @ real @ N ) @ ( plus_plus @ real @ ( ring_1_of_int @ real @ M7 ) @ ( one_one @ real ) ) ) ) ) ).

% int_le_real_less
thf(fact_2238_int__less__real__le,axiom,
    ( ( ord_less @ int )
    = ( ^ [N: int,M7: int] : ( ord_less_eq @ real @ ( plus_plus @ real @ ( ring_1_of_int @ real @ N ) @ ( one_one @ real ) ) @ ( ring_1_of_int @ real @ M7 ) ) ) ) ).

% int_less_real_le
thf(fact_2239_T_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062x_092_060_094sub_062t_Osimps_I1_J,axiom,
    ! [A2: $o,B2: $o] :
      ( ( vEBT_T_m_a_x_t @ ( vEBT_Leaf @ A2 @ B2 ) )
      = ( plus_plus @ nat @ ( one_one @ nat ) @ ( if @ nat @ B2 @ ( 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_2240_T_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062x_092_060_094sub_062t_Osimps_I2_J,axiom,
    ! [Uu: nat,Uv: list @ vEBT_VEBT,Uw: vEBT_VEBT] :
      ( ( vEBT_T_m_a_x_t @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu @ Uv @ Uw ) )
      = ( one_one @ nat ) ) ).

% T\<^sub>m\<^sub>a\<^sub>x\<^sub>t.simps(2)
thf(fact_2241_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: option @ ( product_prod @ nat @ nat ),Ts: list @ vEBT_VEBT,S4: vEBT_VEBT,X2: nat] :
      ( ( vEBT_T_i_n_s_e_r_t @ ( vEBT_Node @ Info @ ( zero_zero @ nat ) @ Ts @ S4 ) @ X2 )
      = ( one_one @ nat ) ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.simps(2)
thf(fact_2242_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t_Osimps_I2_J,axiom,
    ! [Uu: nat,Uv: list @ vEBT_VEBT,Uw: vEBT_VEBT] :
      ( ( vEBT_T_m_i_n_t @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu @ Uv @ Uw ) )
      = ( one_one @ nat ) ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>t.simps(2)
thf(fact_2243_nat__2,axiom,
    ( ( nat2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) )
    = ( suc @ ( suc @ ( zero_zero @ nat ) ) ) ) ).

% nat_2
thf(fact_2244_ceiling__correct,axiom,
    ! [A: $tType] :
      ( ( archim2362893244070406136eiling @ A )
     => ! [X2: A] :
          ( ( ord_less @ A @ ( minus_minus @ A @ ( ring_1_of_int @ A @ ( archimedean_ceiling @ A @ X2 ) ) @ ( one_one @ A ) ) @ X2 )
          & ( ord_less_eq @ A @ X2 @ ( ring_1_of_int @ A @ ( archimedean_ceiling @ A @ X2 ) ) ) ) ) ).

% ceiling_correct
thf(fact_2245_ceiling__unique,axiom,
    ! [A: $tType] :
      ( ( archim2362893244070406136eiling @ A )
     => ! [Z: int,X2: A] :
          ( ( ord_less @ A @ ( minus_minus @ A @ ( ring_1_of_int @ A @ Z ) @ ( one_one @ A ) ) @ X2 )
         => ( ( ord_less_eq @ A @ X2 @ ( ring_1_of_int @ A @ Z ) )
           => ( ( archimedean_ceiling @ A @ X2 )
              = Z ) ) ) ) ).

% ceiling_unique
thf(fact_2246_ceiling__eq__iff,axiom,
    ! [A: $tType] :
      ( ( archim2362893244070406136eiling @ A )
     => ! [X2: A,A2: int] :
          ( ( ( archimedean_ceiling @ A @ X2 )
            = A2 )
          = ( ( ord_less @ A @ ( minus_minus @ A @ ( ring_1_of_int @ A @ A2 ) @ ( one_one @ A ) ) @ X2 )
            & ( ord_less_eq @ A @ X2 @ ( ring_1_of_int @ A @ A2 ) ) ) ) ) ).

% ceiling_eq_iff
thf(fact_2247_ceiling__split,axiom,
    ! [A: $tType] :
      ( ( archim2362893244070406136eiling @ A )
     => ! [P: int > $o,T2: A] :
          ( ( P @ ( archimedean_ceiling @ A @ T2 ) )
          = ( ! [I2: int] :
                ( ( ( ord_less @ A @ ( minus_minus @ A @ ( ring_1_of_int @ A @ I2 ) @ ( one_one @ A ) ) @ T2 )
                  & ( ord_less_eq @ A @ T2 @ ( ring_1_of_int @ A @ I2 ) ) )
               => ( P @ I2 ) ) ) ) ) ).

% ceiling_split
thf(fact_2248_ceiling__less__iff,axiom,
    ! [A: $tType] :
      ( ( archim2362893244070406136eiling @ A )
     => ! [X2: A,Z: int] :
          ( ( ord_less @ int @ ( archimedean_ceiling @ A @ X2 ) @ Z )
          = ( ord_less_eq @ A @ X2 @ ( minus_minus @ A @ ( ring_1_of_int @ A @ Z ) @ ( one_one @ A ) ) ) ) ) ).

% ceiling_less_iff
thf(fact_2249_le__ceiling__iff,axiom,
    ! [A: $tType] :
      ( ( archim2362893244070406136eiling @ A )
     => ! [Z: int,X2: A] :
          ( ( ord_less_eq @ int @ Z @ ( archimedean_ceiling @ A @ X2 ) )
          = ( ord_less @ A @ ( minus_minus @ A @ ( ring_1_of_int @ A @ Z ) @ ( one_one @ A ) ) @ X2 ) ) ) ).

% le_ceiling_iff
thf(fact_2250_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_2251_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 @ ( semiring_1_of_nat @ int @ M ) ) ) ) ).

% nat_less_iff
thf(fact_2252_real__of__int__div2,axiom,
    ! [N2: int,X2: int] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( minus_minus @ real @ ( divide_divide @ real @ ( ring_1_of_int @ real @ N2 ) @ ( ring_1_of_int @ real @ X2 ) ) @ ( ring_1_of_int @ real @ ( divide_divide @ int @ N2 @ X2 ) ) ) ) ).

% real_of_int_div2
thf(fact_2253_real__of__int__div3,axiom,
    ! [N2: int,X2: int] : ( ord_less_eq @ real @ ( minus_minus @ real @ ( divide_divide @ real @ ( ring_1_of_int @ real @ N2 ) @ ( ring_1_of_int @ real @ X2 ) ) @ ( ring_1_of_int @ real @ ( divide_divide @ int @ N2 @ X2 ) ) ) @ ( one_one @ real ) ) ).

% real_of_int_div3
thf(fact_2254_diff__nat__eq__if,axiom,
    ! [Z7: int,Z: int] :
      ( ( ( ord_less @ int @ Z7 @ ( zero_zero @ int ) )
       => ( ( minus_minus @ nat @ ( nat2 @ Z ) @ ( nat2 @ Z7 ) )
          = ( nat2 @ Z ) ) )
      & ( ~ ( ord_less @ int @ Z7 @ ( zero_zero @ int ) )
       => ( ( minus_minus @ nat @ ( nat2 @ Z ) @ ( nat2 @ Z7 ) )
          = ( if @ nat @ ( ord_less @ int @ ( minus_minus @ int @ Z @ Z7 ) @ ( zero_zero @ int ) ) @ ( zero_zero @ nat ) @ ( nat2 @ ( minus_minus @ int @ Z @ Z7 ) ) ) ) ) ) ).

% diff_nat_eq_if
thf(fact_2255_maxt__bound,axiom,
    ! [T2: vEBT_VEBT] : ( ord_less_eq @ nat @ ( vEBT_T_m_a_x_t @ T2 ) @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) ) ).

% maxt_bound
thf(fact_2256_ceiling__divide__upper,axiom,
    ! [A: $tType] :
      ( ( archim2362893244070406136eiling @ A )
     => ! [Q2: A,P4: A] :
          ( ( ord_less @ A @ ( zero_zero @ A ) @ Q2 )
         => ( ord_less_eq @ A @ P4 @ ( times_times @ A @ ( ring_1_of_int @ A @ ( archimedean_ceiling @ A @ ( divide_divide @ A @ P4 @ Q2 ) ) ) @ Q2 ) ) ) ) ).

% ceiling_divide_upper
thf(fact_2257_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,Ux: nat,Uy: list @ vEBT_VEBT,Uz: vEBT_VEBT] :
      ( ( vEBT_T_m_a_x_t @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Ux @ Uy @ Uz ) )
      = ( one_one @ nat ) ) ).

% T\<^sub>m\<^sub>a\<^sub>x\<^sub>t.simps(3)
thf(fact_2258_mint__bound,axiom,
    ! [T2: vEBT_VEBT] : ( ord_less_eq @ nat @ ( vEBT_T_m_i_n_t @ T2 ) @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) ) ).

% mint_bound
thf(fact_2259_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t_Osimps_I1_J,axiom,
    ! [A2: $o,B2: $o] :
      ( ( vEBT_T_m_i_n_t @ ( vEBT_Leaf @ A2 @ B2 ) )
      = ( plus_plus @ nat @ ( one_one @ nat ) @ ( if @ nat @ A2 @ ( 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_2260_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: option @ ( product_prod @ nat @ nat ),Ts: list @ vEBT_VEBT,S4: vEBT_VEBT,X2: nat] :
      ( ( vEBT_T_i_n_s_e_r_t @ ( vEBT_Node @ Info @ ( suc @ ( zero_zero @ nat ) ) @ Ts @ S4 ) @ X2 )
      = ( one_one @ nat ) ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.simps(3)
thf(fact_2261_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,Ux: nat,Uy: list @ vEBT_VEBT,Uz: vEBT_VEBT] :
      ( ( vEBT_T_m_i_n_t @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Ux @ Uy @ Uz ) )
      = ( one_one @ nat ) ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>t.simps(3)
thf(fact_2262_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,
    ! [A2: $o,B2: $o,X2: nat] :
      ( ( vEBT_T_i_n_s_e_r_t @ ( vEBT_Leaf @ A2 @ B2 ) @ X2 )
      = ( plus_plus @ nat @ ( one_one @ nat )
        @ ( if @ nat
          @ ( X2
            = ( 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_2263_ceiling__divide__lower,axiom,
    ! [A: $tType] :
      ( ( archim2362893244070406136eiling @ A )
     => ! [Q2: A,P4: A] :
          ( ( ord_less @ A @ ( zero_zero @ A ) @ Q2 )
         => ( ord_less @ A @ ( times_times @ A @ ( minus_minus @ A @ ( ring_1_of_int @ A @ ( archimedean_ceiling @ A @ ( divide_divide @ A @ P4 @ Q2 ) ) ) @ ( one_one @ A ) ) @ Q2 ) @ P4 ) ) ) ).

% ceiling_divide_lower
thf(fact_2264_ceiling__eq,axiom,
    ! [A: $tType] :
      ( ( archim2362893244070406136eiling @ A )
     => ! [N2: int,X2: A] :
          ( ( ord_less @ A @ ( ring_1_of_int @ A @ N2 ) @ X2 )
         => ( ( ord_less_eq @ A @ X2 @ ( plus_plus @ A @ ( ring_1_of_int @ A @ N2 ) @ ( one_one @ A ) ) )
           => ( ( archimedean_ceiling @ A @ X2 )
              = ( plus_plus @ int @ N2 @ ( one_one @ int ) ) ) ) ) ) ).

% ceiling_eq
thf(fact_2265_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,X2: nat] :
      ( ( vEBT_T_i_n_s_e_r_t @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ V ) ) @ TreeList @ Summary ) @ X2 )
      = ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.simps(4)
thf(fact_2266_insersimp,axiom,
    ! [T2: vEBT_VEBT,N2: nat,Y2: nat] :
      ( ( vEBT_invar_vebt @ T2 @ N2 )
     => ( ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ T2 @ X_12 )
       => ( ord_less_eq @ nat @ ( vEBT_T_i_n_s_e_r_t @ T2 @ Y2 ) @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) ) ) ) ).

% insersimp
thf(fact_2267_insertsimp,axiom,
    ! [T2: vEBT_VEBT,N2: nat,L: nat] :
      ( ( vEBT_invar_vebt @ T2 @ N2 )
     => ( ( vEBT_VEBT_minNull @ T2 )
       => ( ord_less_eq @ nat @ ( vEBT_T_i_n_s_e_r_t @ T2 @ L ) @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) ) ) ) ).

% insertsimp
thf(fact_2268_T_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062x_092_060_094sub_062t_Oelims,axiom,
    ! [X2: vEBT_VEBT,Y2: nat] :
      ( ( ( vEBT_T_m_a_x_t @ X2 )
        = Y2 )
     => ( ! [A4: $o,B3: $o] :
            ( ( X2
              = ( vEBT_Leaf @ A4 @ B3 ) )
           => ( Y2
             != ( plus_plus @ nat @ ( one_one @ nat ) @ ( if @ nat @ B3 @ ( one_one @ nat ) @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( one_one @ nat ) ) ) ) ) )
       => ( ( ? [Uu2: nat,Uv2: list @ vEBT_VEBT,Uw2: vEBT_VEBT] :
                ( X2
                = ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw2 ) )
           => ( Y2
             != ( one_one @ nat ) ) )
         => ~ ( ? [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list @ vEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( X2
                  = ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
             => ( Y2
               != ( one_one @ nat ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>a\<^sub>x\<^sub>t.elims
thf(fact_2269_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t_Oelims,axiom,
    ! [X2: vEBT_VEBT,Y2: nat] :
      ( ( ( vEBT_T_m_i_n_t @ X2 )
        = Y2 )
     => ( ! [A4: $o] :
            ( ? [B3: $o] :
                ( X2
                = ( vEBT_Leaf @ A4 @ B3 ) )
           => ( Y2
             != ( plus_plus @ nat @ ( one_one @ nat ) @ ( if @ nat @ A4 @ ( zero_zero @ nat ) @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( one_one @ nat ) ) ) ) ) )
       => ( ( ? [Uu2: nat,Uv2: list @ vEBT_VEBT,Uw2: vEBT_VEBT] :
                ( X2
                = ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw2 ) )
           => ( Y2
             != ( one_one @ nat ) ) )
         => ~ ( ? [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list @ vEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( X2
                  = ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
             => ( Y2
               != ( one_one @ nat ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>t.elims
thf(fact_2270_insert__bound__height,axiom,
    ! [T2: vEBT_VEBT,N2: nat,X2: nat] :
      ( ( vEBT_invar_vebt @ T2 @ N2 )
     => ( ord_less_eq @ nat @ ( vEBT_T_i_n_s_e_r_t @ T2 @ X2 ) @ ( times_times @ nat @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_VEBT_height @ T2 ) ) @ ( numeral_numeral @ nat @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).

% insert_bound_height
thf(fact_2271_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Opelims,axiom,
    ! [X2: vEBT_VEBT,Xa: nat,Y2: nat] :
      ( ( ( vEBT_T_s_u_c_c @ X2 @ Xa )
        = Y2 )
     => ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T_s_u_c_c_rel2 @ ( product_Pair @ vEBT_VEBT @ nat @ X2 @ Xa ) )
       => ( ! [Uu2: $o,B3: $o] :
              ( ( X2
                = ( vEBT_Leaf @ Uu2 @ B3 ) )
             => ( ( Xa
                  = ( zero_zero @ nat ) )
               => ( ( Y2
                    = ( plus_plus @ nat @ ( one_one @ nat ) @ ( one_one @ nat ) ) )
                 => ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T_s_u_c_c_rel2 @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ Uu2 @ B3 ) @ ( zero_zero @ nat ) ) ) ) ) )
         => ( ! [Uv2: $o,Uw2: $o] :
                ( ( X2
                  = ( vEBT_Leaf @ Uv2 @ Uw2 ) )
               => ! [N4: nat] :
                    ( ( Xa
                      = ( suc @ N4 ) )
                   => ( ( Y2
                        = ( one_one @ nat ) )
                     => ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T_s_u_c_c_rel2 @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ Uv2 @ Uw2 ) @ ( suc @ N4 ) ) ) ) ) )
           => ( ! [Ux2: nat,Uy2: list @ vEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( X2
                    = ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Ux2 @ Uy2 @ Uz2 ) )
                 => ( ( Y2
                      = ( one_one @ nat ) )
                   => ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T_s_u_c_c_rel2 @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Ux2 @ Uy2 @ Uz2 ) @ Xa ) ) ) )
             => ( ! [V3: product_prod @ nat @ nat,Vc: list @ vEBT_VEBT,Vd2: vEBT_VEBT] :
                    ( ( X2
                      = ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( zero_zero @ nat ) @ Vc @ Vd2 ) )
                   => ( ( Y2
                        = ( one_one @ nat ) )
                     => ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T_s_u_c_c_rel2 @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( zero_zero @ nat ) @ Vc @ Vd2 ) @ Xa ) ) ) )
               => ( ! [V3: product_prod @ nat @ nat,Vg2: list @ vEBT_VEBT,Vh2: vEBT_VEBT] :
                      ( ( X2
                        = ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vg2 @ Vh2 ) )
                     => ( ( Y2
                          = ( one_one @ nat ) )
                       => ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T_s_u_c_c_rel2 @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vg2 @ Vh2 ) @ Xa ) ) ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
                        ( ( X2
                          = ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
                       => ( ( Y2
                            = ( plus_plus @ nat @ ( one_one @ nat )
                              @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi2 ) @ ( one_one @ nat )
                                @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit1 @ ( bit0 @ one2 ) ) ) )
                                  @ ( if @ nat @ ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
                                    @ ( plus_plus @ nat @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_T_m_a_x_t @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) )
                                      @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) )
                                        @ ( if @ nat
                                          @ ( ( ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
                                             != ( none @ nat ) )
                                            & ( vEBT_VEBT_less @ ( some @ nat @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) )
                                          @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) @ ( vEBT_T_s_u_c_c @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
                                          @ ( plus_plus @ nat @ ( plus_plus @ nat @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_T_s_u_c_c @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( one_one @ nat ) )
                                            @ ( if @ nat
                                              @ ( ( vEBT_vebt_succ @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
                                                = ( none @ nat ) )
                                              @ ( one_one @ nat )
                                              @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) @ ( vEBT_T_m_i_n_t @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_succ @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) )
                                    @ ( one_one @ nat ) ) ) ) ) )
                         => ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T_s_u_c_c_rel2 @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) @ Xa ) ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.pelims
thf(fact_2272_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,
    ! [X2: vEBT_VEBT,Xa: nat,Y2: nat] :
      ( ( ( vEBT_T_d_e_l_e_t_e @ X2 @ Xa )
        = Y2 )
     => ( ( ? [A4: $o,B3: $o] :
              ( X2
              = ( vEBT_Leaf @ A4 @ B3 ) )
         => ( ( Xa
              = ( zero_zero @ nat ) )
           => ( Y2
             != ( one_one @ nat ) ) ) )
       => ( ( ? [A4: $o,B3: $o] :
                ( X2
                = ( vEBT_Leaf @ A4 @ B3 ) )
           => ( ( Xa
                = ( suc @ ( zero_zero @ nat ) ) )
             => ( Y2
               != ( one_one @ nat ) ) ) )
         => ( ( ? [A4: $o,B3: $o] :
                  ( X2
                  = ( vEBT_Leaf @ A4 @ B3 ) )
             => ( ? [N4: nat] :
                    ( Xa
                    = ( suc @ ( suc @ N4 ) ) )
               => ( Y2
                 != ( one_one @ nat ) ) ) )
           => ( ( ? [Deg2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
                    ( X2
                    = ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg2 @ TreeList2 @ Summary2 ) )
               => ( Y2
                 != ( one_one @ nat ) ) )
             => ( ( ? [Mi2: nat,Ma2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( X2
                      = ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ TreeList2 @ Summary2 ) )
                 => ( Y2
                   != ( one_one @ nat ) ) )
               => ( ( ? [Mi2: nat,Ma2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
                        ( X2
                        = ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( zero_zero @ nat ) ) @ TreeList2 @ Summary2 ) )
                   => ( Y2
                     != ( one_one @ nat ) ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
                        ( ( X2
                          = ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
                       => ( Y2
                         != ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) )
                            @ ( if @ nat
                              @ ( ( ord_less @ nat @ Xa @ Mi2 )
                                | ( ord_less @ nat @ Ma2 @ Xa ) )
                              @ ( one_one @ nat )
                              @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) )
                                @ ( if @ nat
                                  @ ( ( Xa = Mi2 )
                                    & ( Xa = Ma2 ) )
                                  @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) )
                                  @ ( plus_plus @ nat @ ( plus_plus @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit0 @ ( bit1 @ one2 ) ) ) ) @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( plus_plus @ nat @ ( vEBT_T_m_i_n_t @ Summary2 ) @ ( vEBT_T_m_i_n_t @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) @ ( numeral_numeral @ nat @ ( bit1 @ ( bit1 @ one2 ) ) ) ) @ ( one_one @ nat ) ) ) @ ( one_one @ nat ) )
                                    @ ( if @ nat @ ( ord_less @ nat @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
                                      @ ( plus_plus @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) @ ( vEBT_T_d_e_l_e_t_e @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
                                        @ ( 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 @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) )
                                          @ ( if @ nat @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
                                            @ ( plus_plus @ nat @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_T_d_e_l_e_t_e @ Summary2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
                                              @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) )
                                                @ ( if @ nat
                                                  @ ( ( ( Xa = Mi2 )
                                                     => ( ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) )
                                                        = Ma2 ) )
                                                    & ( ( Xa != Mi2 )
                                                     => ( Xa = 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 @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) )
                                                    @ ( plus_plus @ nat @ ( one_one @ nat )
                                                      @ ( if @ nat
                                                        @ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
                                                          = ( none @ nat ) )
                                                        @ ( one_one @ nat )
                                                        @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_T_m_a_x_t @ ( nth @ vEBT_VEBT @ ( list_update @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) )
                                                  @ ( one_one @ nat ) ) ) )
                                            @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) )
                                              @ ( if @ nat
                                                @ ( ( ( Xa = Mi2 )
                                                   => ( ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) )
                                                      = Ma2 ) )
                                                  & ( ( Xa != Mi2 )
                                                   => ( Xa = Ma2 ) ) )
                                                @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit1 @ one2 ) ) ) @ ( vEBT_T_m_a_x_t @ ( nth @ vEBT_VEBT @ ( list_update @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) )
                                                @ ( one_one @ nat ) ) ) ) ) )
                                      @ ( one_one @ nat ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.elims
thf(fact_2273_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,Va: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,X2: nat] :
      ( ( vEBT_T_d_e_l_e_t_e @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X2 )
      = ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) )
        @ ( if @ nat
          @ ( ( ord_less @ nat @ X2 @ Mi )
            | ( ord_less @ nat @ Ma @ X2 ) )
          @ ( one_one @ nat )
          @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) )
            @ ( if @ nat
              @ ( ( X2 = Mi )
                & ( X2 = Ma ) )
              @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) )
              @ ( plus_plus @ nat @ ( plus_plus @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit0 @ ( bit1 @ one2 ) ) ) ) @ ( if @ nat @ ( X2 = Mi ) @ ( plus_plus @ nat @ ( plus_plus @ nat @ ( vEBT_T_m_i_n_t @ Summary ) @ ( vEBT_T_m_i_n_t @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) @ ( numeral_numeral @ nat @ ( bit1 @ ( bit1 @ one2 ) ) ) ) @ ( one_one @ nat ) ) ) @ ( one_one @ nat ) )
                @ ( if @ nat @ ( ord_less @ nat @ ( vEBT_VEBT_high @ ( if @ nat @ ( X2 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
                  @ ( plus_plus @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) @ ( vEBT_T_d_e_l_e_t_e @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( X2 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( X2 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
                    @ ( 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 @ ( X2 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( X2 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) )
                      @ ( if @ nat @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( X2 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( X2 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
                        @ ( plus_plus @ nat @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_T_d_e_l_e_t_e @ Summary @ ( vEBT_VEBT_high @ ( if @ nat @ ( X2 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
                          @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) )
                            @ ( if @ nat
                              @ ( ( ( X2 = Mi )
                                 => ( ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
                                    = Ma ) )
                                & ( ( X2 != Mi )
                                 => ( X2 = 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 @ ( X2 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) )
                                @ ( plus_plus @ nat @ ( one_one @ nat )
                                  @ ( if @ nat
                                    @ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if @ nat @ ( X2 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
                                      = ( none @ nat ) )
                                    @ ( one_one @ nat )
                                    @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_T_m_a_x_t @ ( nth @ vEBT_VEBT @ ( list_update @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( X2 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( X2 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( X2 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if @ nat @ ( X2 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) )
                              @ ( one_one @ nat ) ) ) )
                        @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) )
                          @ ( if @ nat
                            @ ( ( ( X2 = Mi )
                               => ( ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
                                  = Ma ) )
                              & ( ( X2 != Mi )
                               => ( X2 = Ma ) ) )
                            @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit1 @ one2 ) ) ) @ ( vEBT_T_m_a_x_t @ ( nth @ vEBT_VEBT @ ( list_update @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( X2 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( X2 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( X2 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_VEBT_high @ ( if @ nat @ ( X2 = Mi ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) )
                            @ ( one_one @ nat ) ) ) ) ) )
                  @ ( one_one @ nat ) ) ) ) ) ) ) ) ).

% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.simps(7)
thf(fact_2274_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Oelims,axiom,
    ! [X2: vEBT_VEBT,Xa: nat,Y2: nat] :
      ( ( ( vEBT_T_p_r_e_d @ X2 @ Xa )
        = Y2 )
     => ( ( ? [Uu2: $o,Uv2: $o] :
              ( X2
              = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
         => ( ( Xa
              = ( zero_zero @ nat ) )
           => ( Y2
             != ( one_one @ nat ) ) ) )
       => ( ( ? [A4: $o,Uw2: $o] :
                ( X2
                = ( vEBT_Leaf @ A4 @ Uw2 ) )
           => ( ( Xa
                = ( suc @ ( zero_zero @ nat ) ) )
             => ( Y2
               != ( plus_plus @ nat @ ( one_one @ nat ) @ ( one_one @ nat ) ) ) ) )
         => ( ! [A4: $o,B3: $o] :
                ( ( X2
                  = ( vEBT_Leaf @ A4 @ B3 ) )
               => ( ? [Va3: nat] :
                      ( Xa
                      = ( suc @ ( suc @ Va3 ) ) )
                 => ( Y2
                   != ( plus_plus @ nat @ ( one_one @ nat ) @ ( if @ nat @ B3 @ ( one_one @ nat ) @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( one_one @ nat ) ) ) ) ) ) )
           => ( ( ? [Uy2: nat,Uz2: list @ vEBT_VEBT,Va2: vEBT_VEBT] :
                    ( X2
                    = ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uy2 @ Uz2 @ Va2 ) )
               => ( Y2
                 != ( one_one @ nat ) ) )
             => ( ( ? [V3: product_prod @ nat @ nat,Vd2: list @ vEBT_VEBT,Ve2: vEBT_VEBT] :
                      ( X2
                      = ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( zero_zero @ nat ) @ Vd2 @ Ve2 ) )
                 => ( Y2
                   != ( one_one @ nat ) ) )
               => ( ( ? [V3: product_prod @ nat @ nat,Vh2: list @ vEBT_VEBT,Vi2: vEBT_VEBT] :
                        ( X2
                        = ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vh2 @ Vi2 ) )
                   => ( Y2
                     != ( one_one @ nat ) ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
                        ( ( X2
                          = ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
                       => ( Y2
                         != ( plus_plus @ nat @ ( one_one @ nat )
                            @ ( if @ nat @ ( ord_less @ nat @ Ma2 @ Xa ) @ ( one_one @ nat )
                              @ ( plus_plus @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit1 @ ( bit0 @ one2 ) ) ) ) @ ( one_one @ nat ) )
                                @ ( if @ nat @ ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
                                  @ ( plus_plus @ nat @ ( plus_plus @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( vEBT_T_m_i_n_t @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) )
                                    @ ( if @ nat
                                      @ ( ( ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
                                         != ( none @ nat ) )
                                        & ( vEBT_VEBT_greater @ ( some @ nat @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) )
                                      @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) @ ( vEBT_T_p_r_e_d @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
                                      @ ( plus_plus @ nat @ ( plus_plus @ nat @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_T_p_r_e_d @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( one_one @ nat ) )
                                        @ ( if @ nat
                                          @ ( ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
                                            = ( none @ nat ) )
                                          @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( one_one @ nat ) )
                                          @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) @ ( vEBT_T_m_a_x_t @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) )
                                  @ ( one_one @ nat ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.elims
thf(fact_2275_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,Va: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,X2: nat] :
      ( ( vEBT_T_p_r_e_d @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X2 )
      = ( plus_plus @ nat @ ( one_one @ nat )
        @ ( if @ nat @ ( ord_less @ nat @ Ma @ X2 ) @ ( one_one @ nat )
          @ ( plus_plus @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit1 @ ( bit0 @ one2 ) ) ) ) @ ( one_one @ nat ) )
            @ ( if @ nat @ ( ord_less @ nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
              @ ( plus_plus @ nat @ ( plus_plus @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( vEBT_T_m_i_n_t @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) )
                @ ( if @ nat
                  @ ( ( ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
                     != ( none @ nat ) )
                    & ( vEBT_VEBT_greater @ ( some @ nat @ ( vEBT_VEBT_low @ X2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) )
                  @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) @ ( vEBT_T_p_r_e_d @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
                  @ ( plus_plus @ nat @ ( plus_plus @ nat @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_T_p_r_e_d @ Summary @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( one_one @ nat ) )
                    @ ( if @ nat
                      @ ( ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
                        = ( none @ nat ) )
                      @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( one_one @ nat ) )
                      @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) @ ( vEBT_T_m_a_x_t @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) )
              @ ( one_one @ nat ) ) ) ) ) ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.simps(7)
thf(fact_2276_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,
    ! [X2: vEBT_VEBT,Xa: nat,Y2: nat] :
      ( ( ( vEBT_T_i_n_s_e_r_t @ X2 @ Xa )
        = Y2 )
     => ( ( ? [A4: $o,B3: $o] :
              ( X2
              = ( vEBT_Leaf @ A4 @ B3 ) )
         => ( Y2
           != ( plus_plus @ nat @ ( one_one @ nat )
              @ ( if @ nat
                @ ( Xa
                  = ( zero_zero @ nat ) )
                @ ( one_one @ nat )
                @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( one_one @ nat ) ) ) ) ) )
       => ( ( ? [Info2: option @ ( product_prod @ nat @ nat ),Ts2: list @ vEBT_VEBT,S: vEBT_VEBT] :
                ( X2
                = ( vEBT_Node @ Info2 @ ( zero_zero @ nat ) @ Ts2 @ S ) )
           => ( Y2
             != ( one_one @ nat ) ) )
         => ( ( ? [Info2: option @ ( product_prod @ nat @ nat ),Ts2: list @ vEBT_VEBT,S: vEBT_VEBT] :
                  ( X2
                  = ( vEBT_Node @ Info2 @ ( suc @ ( zero_zero @ nat ) ) @ Ts2 @ S ) )
             => ( Y2
               != ( one_one @ nat ) ) )
           => ( ( ? [V3: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
                    ( X2
                    = ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ V3 ) ) @ TreeList2 @ Summary2 ) )
               => ( Y2
                 != ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
             => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
                    ( ( X2
                      = ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
                   => ( Y2
                     != ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) )
                        @ ( if @ nat
                          @ ( ( ord_less @ nat @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
                            & ~ ( ( Xa = Mi2 )
                                | ( Xa = 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 @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_T_m_i_n_N_u_l_l @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) @ ( if @ nat @ ( vEBT_VEBT_minNull @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_T_i_n_s_e_r_t @ Summary2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( one_one @ nat ) ) )
                          @ ( one_one @ nat ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.elims
thf(fact_2277_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,
    ! [X2: vEBT_VEBT,Xa: nat,Y2: nat] :
      ( ( ( vEBT_T_d_e_l_e_t_e @ X2 @ Xa )
        = Y2 )
     => ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T8441311223069195367_e_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X2 @ Xa ) )
       => ( ! [A4: $o,B3: $o] :
              ( ( X2
                = ( vEBT_Leaf @ A4 @ B3 ) )
             => ( ( Xa
                  = ( zero_zero @ nat ) )
               => ( ( Y2
                    = ( one_one @ nat ) )
                 => ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T8441311223069195367_e_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A4 @ B3 ) @ ( zero_zero @ nat ) ) ) ) ) )
         => ( ! [A4: $o,B3: $o] :
                ( ( X2
                  = ( vEBT_Leaf @ A4 @ B3 ) )
               => ( ( Xa
                    = ( suc @ ( zero_zero @ nat ) ) )
                 => ( ( Y2
                      = ( one_one @ nat ) )
                   => ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T8441311223069195367_e_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A4 @ B3 ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) )
           => ( ! [A4: $o,B3: $o] :
                  ( ( X2
                    = ( vEBT_Leaf @ A4 @ B3 ) )
                 => ! [N4: nat] :
                      ( ( Xa
                        = ( suc @ ( suc @ N4 ) ) )
                     => ( ( Y2
                          = ( one_one @ nat ) )
                       => ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T8441311223069195367_e_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A4 @ B3 ) @ ( suc @ ( suc @ N4 ) ) ) ) ) ) )
             => ( ! [Deg2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
                    ( ( X2
                      = ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg2 @ TreeList2 @ Summary2 ) )
                   => ( ( Y2
                        = ( one_one @ nat ) )
                     => ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T8441311223069195367_e_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg2 @ TreeList2 @ Summary2 ) @ Xa ) ) ) )
               => ( ! [Mi2: nat,Ma2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X2
                        = ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ TreeList2 @ Summary2 ) )
                     => ( ( Y2
                          = ( one_one @ nat ) )
                       => ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T8441311223069195367_e_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ TreeList2 @ Summary2 ) @ Xa ) ) ) )
                 => ( ! [Mi2: nat,Ma2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
                        ( ( X2
                          = ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( zero_zero @ nat ) ) @ TreeList2 @ Summary2 ) )
                       => ( ( Y2
                            = ( one_one @ nat ) )
                         => ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T8441311223069195367_e_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( zero_zero @ nat ) ) @ TreeList2 @ Summary2 ) @ Xa ) ) ) )
                   => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
                          ( ( X2
                            = ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
                         => ( ( Y2
                              = ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) )
                                @ ( if @ nat
                                  @ ( ( ord_less @ nat @ Xa @ Mi2 )
                                    | ( ord_less @ nat @ Ma2 @ Xa ) )
                                  @ ( one_one @ nat )
                                  @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) )
                                    @ ( if @ nat
                                      @ ( ( Xa = Mi2 )
                                        & ( Xa = Ma2 ) )
                                      @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) )
                                      @ ( plus_plus @ nat @ ( plus_plus @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit0 @ ( bit1 @ one2 ) ) ) ) @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( plus_plus @ nat @ ( vEBT_T_m_i_n_t @ Summary2 ) @ ( vEBT_T_m_i_n_t @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) @ ( numeral_numeral @ nat @ ( bit1 @ ( bit1 @ one2 ) ) ) ) @ ( one_one @ nat ) ) ) @ ( one_one @ nat ) )
                                        @ ( if @ nat @ ( ord_less @ nat @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
                                          @ ( plus_plus @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) @ ( vEBT_T_d_e_l_e_t_e @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
                                            @ ( 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 @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) )
                                              @ ( if @ nat @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
                                                @ ( plus_plus @ nat @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_T_d_e_l_e_t_e @ Summary2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
                                                  @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) )
                                                    @ ( if @ nat
                                                      @ ( ( ( Xa = Mi2 )
                                                         => ( ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) )
                                                            = Ma2 ) )
                                                        & ( ( Xa != Mi2 )
                                                         => ( Xa = 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 @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) )
                                                        @ ( plus_plus @ nat @ ( one_one @ nat )
                                                          @ ( if @ nat
                                                            @ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
                                                              = ( none @ nat ) )
                                                            @ ( one_one @ nat )
                                                            @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_T_m_a_x_t @ ( nth @ vEBT_VEBT @ ( list_update @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) )
                                                      @ ( one_one @ nat ) ) ) )
                                                @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) )
                                                  @ ( if @ nat
                                                    @ ( ( ( Xa = Mi2 )
                                                       => ( ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) )
                                                          = Ma2 ) )
                                                      & ( ( Xa != Mi2 )
                                                       => ( Xa = Ma2 ) ) )
                                                    @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit1 @ one2 ) ) ) @ ( vEBT_T_m_a_x_t @ ( nth @ vEBT_VEBT @ ( list_update @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) )
                                                    @ ( one_one @ nat ) ) ) ) ) )
                                          @ ( one_one @ nat ) ) ) ) ) ) ) )
                           => ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T8441311223069195367_e_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) @ Xa ) ) ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.pelims
thf(fact_2278_word__of__int__numeral,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [Bin: num] :
          ( ( ring_1_of_int @ ( word @ A ) @ ( numeral_numeral @ int @ Bin ) )
          = ( numeral_numeral @ ( word @ A ) @ Bin ) ) ) ).

% word_of_int_numeral
thf(fact_2279_word__of__int__1,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ( ( ring_1_of_int @ ( word @ A ) @ ( one_one @ int ) )
        = ( one_one @ ( word @ A ) ) ) ) ).

% word_of_int_1
thf(fact_2280_word__numeral__alt,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ( ( numeral_numeral @ ( word @ A ) )
        = ( ^ [B5: num] : ( ring_1_of_int @ ( word @ A ) @ ( numeral_numeral @ int @ B5 ) ) ) ) ) ).

% word_numeral_alt
thf(fact_2281_wi__hom__add,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [A2: int,B2: int] :
          ( ( plus_plus @ ( word @ A ) @ ( ring_1_of_int @ ( word @ A ) @ A2 ) @ ( ring_1_of_int @ ( word @ A ) @ B2 ) )
          = ( ring_1_of_int @ ( word @ A ) @ ( plus_plus @ int @ A2 @ B2 ) ) ) ) ).

% wi_hom_add
thf(fact_2282_wi__hom__sub,axiom,
    ! [B: $tType] :
      ( ( type_len @ B )
     => ! [A2: int,B2: int] :
          ( ( minus_minus @ ( word @ B ) @ ( ring_1_of_int @ ( word @ B ) @ A2 ) @ ( ring_1_of_int @ ( word @ B ) @ B2 ) )
          = ( ring_1_of_int @ ( word @ B ) @ ( minus_minus @ int @ A2 @ B2 ) ) ) ) ).

% wi_hom_sub
thf(fact_2283_word__of__int__power__hom,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [A2: int,N2: nat] :
          ( ( power_power @ ( word @ A ) @ ( ring_1_of_int @ ( word @ A ) @ A2 ) @ N2 )
          = ( ring_1_of_int @ ( word @ A ) @ ( power_power @ int @ A2 @ N2 ) ) ) ) ).

% word_of_int_power_hom
thf(fact_2284_wi__hom__mult,axiom,
    ! [C: $tType] :
      ( ( type_len @ C )
     => ! [A2: int,B2: int] :
          ( ( times_times @ ( word @ C ) @ ( ring_1_of_int @ ( word @ C ) @ A2 ) @ ( ring_1_of_int @ ( word @ C ) @ B2 ) )
          = ( ring_1_of_int @ ( word @ C ) @ ( times_times @ int @ A2 @ B2 ) ) ) ) ).

% wi_hom_mult
thf(fact_2285_minNull__bound,axiom,
    ! [T2: vEBT_VEBT] : ( ord_less_eq @ nat @ ( vEBT_T_m_i_n_N_u_l_l @ T2 ) @ ( one_one @ nat ) ) ).

% minNull_bound
thf(fact_2286_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_I1_J,axiom,
    ( ( vEBT_T_m_i_n_N_u_l_l @ ( vEBT_Leaf @ $false @ $false ) )
    = ( one_one @ nat ) ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.simps(1)
thf(fact_2287_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_I2_J,axiom,
    ! [Uv: $o] :
      ( ( vEBT_T_m_i_n_N_u_l_l @ ( vEBT_Leaf @ $true @ Uv ) )
      = ( one_one @ nat ) ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.simps(2)
thf(fact_2288_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_I3_J,axiom,
    ! [Uu: $o] :
      ( ( vEBT_T_m_i_n_N_u_l_l @ ( vEBT_Leaf @ Uu @ $true ) )
      = ( one_one @ nat ) ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.simps(3)
thf(fact_2289_word__of__int__nat,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [X2: int] :
          ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X2 )
         => ( ( ring_1_of_int @ ( word @ A ) @ X2 )
            = ( semiring_1_of_nat @ ( word @ A ) @ ( nat2 @ X2 ) ) ) ) ) ).

% word_of_int_nat
thf(fact_2290_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Osimps_I1_J,axiom,
    ! [Uu: $o,Uv: $o] :
      ( ( vEBT_T_p_r_e_d @ ( vEBT_Leaf @ Uu @ Uv ) @ ( zero_zero @ nat ) )
      = ( one_one @ nat ) ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.simps(1)
thf(fact_2291_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Osimps_I4_J,axiom,
    ! [Uy: nat,Uz: list @ vEBT_VEBT,Va: vEBT_VEBT,Vb: nat] :
      ( ( vEBT_T_p_r_e_d @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uy @ Uz @ Va ) @ Vb )
      = ( one_one @ nat ) ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.simps(4)
thf(fact_2292_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,
    ! [Uz: product_prod @ nat @ nat,Va: nat,Vb: list @ vEBT_VEBT,Vc2: vEBT_VEBT] :
      ( ( vEBT_T_m_i_n_N_u_l_l @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ Uz ) @ Va @ Vb @ Vc2 ) )
      = ( one_one @ nat ) ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.simps(5)
thf(fact_2293_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_I4_J,axiom,
    ! [Uw: nat,Ux: list @ vEBT_VEBT,Uy: vEBT_VEBT] :
      ( ( vEBT_T_m_i_n_N_u_l_l @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uw @ Ux @ Uy ) )
      = ( one_one @ nat ) ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.simps(4)
thf(fact_2294_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Osimps_I3_J,axiom,
    ! [A2: $o,B2: $o,Va: nat] :
      ( ( vEBT_T_p_r_e_d @ ( vEBT_Leaf @ A2 @ B2 ) @ ( suc @ ( suc @ Va ) ) )
      = ( plus_plus @ nat @ ( one_one @ nat ) @ ( if @ nat @ B2 @ ( 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_2295_word__of__int__2p,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [N2: nat] :
          ( ( ring_1_of_int @ ( word @ A ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N2 ) )
          = ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N2 ) ) ) ).

% word_of_int_2p
thf(fact_2296_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 @ ( product_prod @ nat @ nat ) @ V ) @ ( zero_zero @ nat ) @ Vd @ Ve ) @ Vf )
      = ( one_one @ nat ) ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.simps(5)
thf(fact_2297_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Osimps_I2_J,axiom,
    ! [A2: $o,Uw: $o] :
      ( ( vEBT_T_p_r_e_d @ ( vEBT_Leaf @ A2 @ Uw ) @ ( 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_2298_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 @ ( product_prod @ nat @ 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_2299_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,
    ! [X2: vEBT_VEBT,Y2: nat] :
      ( ( ( vEBT_T_m_i_n_N_u_l_l @ X2 )
        = Y2 )
     => ( ( ( X2
            = ( vEBT_Leaf @ $false @ $false ) )
         => ( Y2
           != ( one_one @ nat ) ) )
       => ( ( ? [Uv2: $o] :
                ( X2
                = ( vEBT_Leaf @ $true @ Uv2 ) )
           => ( Y2
             != ( one_one @ nat ) ) )
         => ( ( ? [Uu2: $o] :
                  ( X2
                  = ( vEBT_Leaf @ Uu2 @ $true ) )
             => ( Y2
               != ( one_one @ nat ) ) )
           => ( ( ? [Uw2: nat,Ux2: list @ vEBT_VEBT,Uy2: vEBT_VEBT] :
                    ( X2
                    = ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uw2 @ Ux2 @ Uy2 ) )
               => ( Y2
                 != ( one_one @ nat ) ) )
             => ~ ( ? [Uz2: product_prod @ nat @ nat,Va2: nat,Vb2: list @ vEBT_VEBT,Vc: vEBT_VEBT] :
                      ( X2
                      = ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ Uz2 ) @ Va2 @ Vb2 @ Vc ) )
                 => ( Y2
                   != ( one_one @ nat ) ) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.elims
thf(fact_2300_pred__bound__height,axiom,
    ! [T2: vEBT_VEBT,N2: nat,X2: nat] :
      ( ( vEBT_invar_vebt @ T2 @ N2 )
     => ( ord_less_eq @ nat @ ( vEBT_T_p_r_e_d @ T2 @ X2 ) @ ( times_times @ nat @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_VEBT_height @ T2 ) ) @ ( numeral_numeral @ nat @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ one2 ) ) ) ) ) ) ) ) ).

% pred_bound_height
thf(fact_2301_pred__bound__size__univ,axiom,
    ! [T2: vEBT_VEBT,N2: nat,U: real,X2: nat] :
      ( ( vEBT_invar_vebt @ T2 @ N2 )
     => ( ( U
          = ( power_power @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ N2 ) )
       => ( ord_less_eq @ real @ ( semiring_1_of_nat @ real @ ( vEBT_T_p_r_e_d @ T2 @ X2 ) ) @ ( plus_plus @ real @ ( numeral_numeral @ real @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ one2 ) ) ) ) ) ) @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ one2 ) ) ) ) ) @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ U ) ) ) ) ) ) ) ).

% pred_bound_size_univ
thf(fact_2302_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,Va: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,X2: nat] :
      ( ( vEBT_T_i_n_s_e_r_t @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X2 )
      = ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) )
        @ ( if @ nat
          @ ( ( ord_less @ nat @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ X2 @ Mi ) @ Mi @ X2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
            & ~ ( ( X2 = Mi )
                | ( X2 = 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 @ X2 @ Mi ) @ Mi @ X2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( ord_less @ nat @ X2 @ Mi ) @ Mi @ X2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_T_m_i_n_N_u_l_l @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ X2 @ Mi ) @ Mi @ X2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) @ ( if @ nat @ ( vEBT_VEBT_minNull @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ X2 @ Mi ) @ Mi @ X2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_T_i_n_s_e_r_t @ Summary @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ X2 @ Mi ) @ Mi @ X2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( one_one @ nat ) ) )
          @ ( one_one @ nat ) ) ) ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.simps(5)
thf(fact_2303_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Opelims,axiom,
    ! [X2: vEBT_VEBT,Xa: nat,Y2: nat] :
      ( ( ( vEBT_T_p_r_e_d @ X2 @ Xa )
        = Y2 )
     => ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T_p_r_e_d_rel2 @ ( product_Pair @ vEBT_VEBT @ nat @ X2 @ Xa ) )
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( X2
                = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
             => ( ( Xa
                  = ( zero_zero @ nat ) )
               => ( ( Y2
                    = ( one_one @ nat ) )
                 => ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T_p_r_e_d_rel2 @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ ( zero_zero @ nat ) ) ) ) ) )
         => ( ! [A4: $o,Uw2: $o] :
                ( ( X2
                  = ( vEBT_Leaf @ A4 @ Uw2 ) )
               => ( ( Xa
                    = ( suc @ ( zero_zero @ nat ) ) )
                 => ( ( Y2
                      = ( plus_plus @ nat @ ( one_one @ nat ) @ ( one_one @ nat ) ) )
                   => ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T_p_r_e_d_rel2 @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A4 @ Uw2 ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) )
           => ( ! [A4: $o,B3: $o] :
                  ( ( X2
                    = ( vEBT_Leaf @ A4 @ B3 ) )
                 => ! [Va3: nat] :
                      ( ( Xa
                        = ( suc @ ( suc @ Va3 ) ) )
                     => ( ( Y2
                          = ( plus_plus @ nat @ ( one_one @ nat ) @ ( if @ nat @ B3 @ ( one_one @ nat ) @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( one_one @ nat ) ) ) ) )
                       => ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T_p_r_e_d_rel2 @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A4 @ B3 ) @ ( suc @ ( suc @ Va3 ) ) ) ) ) ) )
             => ( ! [Uy2: nat,Uz2: list @ vEBT_VEBT,Va2: vEBT_VEBT] :
                    ( ( X2
                      = ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uy2 @ Uz2 @ Va2 ) )
                   => ( ( Y2
                        = ( one_one @ nat ) )
                     => ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T_p_r_e_d_rel2 @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uy2 @ Uz2 @ Va2 ) @ Xa ) ) ) )
               => ( ! [V3: product_prod @ nat @ nat,Vd2: list @ vEBT_VEBT,Ve2: vEBT_VEBT] :
                      ( ( X2
                        = ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( zero_zero @ nat ) @ Vd2 @ Ve2 ) )
                     => ( ( Y2
                          = ( one_one @ nat ) )
                       => ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T_p_r_e_d_rel2 @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( zero_zero @ nat ) @ Vd2 @ Ve2 ) @ Xa ) ) ) )
                 => ( ! [V3: product_prod @ nat @ nat,Vh2: list @ vEBT_VEBT,Vi2: vEBT_VEBT] :
                        ( ( X2
                          = ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vh2 @ Vi2 ) )
                       => ( ( Y2
                            = ( one_one @ nat ) )
                         => ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T_p_r_e_d_rel2 @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vh2 @ Vi2 ) @ Xa ) ) ) )
                   => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
                          ( ( X2
                            = ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
                         => ( ( Y2
                              = ( plus_plus @ nat @ ( one_one @ nat )
                                @ ( if @ nat @ ( ord_less @ nat @ Ma2 @ Xa ) @ ( one_one @ nat )
                                  @ ( plus_plus @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit1 @ ( bit0 @ one2 ) ) ) ) @ ( one_one @ nat ) )
                                    @ ( if @ nat @ ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
                                      @ ( plus_plus @ nat @ ( plus_plus @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( vEBT_T_m_i_n_t @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) )
                                        @ ( if @ nat
                                          @ ( ( ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
                                             != ( none @ nat ) )
                                            & ( vEBT_VEBT_greater @ ( some @ nat @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) )
                                          @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) @ ( vEBT_T_p_r_e_d @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
                                          @ ( plus_plus @ nat @ ( plus_plus @ nat @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_T_p_r_e_d @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( one_one @ nat ) )
                                            @ ( if @ nat
                                              @ ( ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
                                                = ( none @ nat ) )
                                              @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( one_one @ nat ) )
                                              @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) @ ( vEBT_T_m_a_x_t @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) )
                                      @ ( one_one @ nat ) ) ) ) ) )
                           => ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T_p_r_e_d_rel2 @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) @ Xa ) ) ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.pelims
thf(fact_2304_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,
    ! [X2: vEBT_VEBT,Xa: nat,Y2: nat] :
      ( ( ( vEBT_T_i_n_s_e_r_t @ X2 @ Xa )
        = Y2 )
     => ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T9217963907923527482_t_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X2 @ Xa ) )
       => ( ! [A4: $o,B3: $o] :
              ( ( X2
                = ( vEBT_Leaf @ A4 @ B3 ) )
             => ( ( Y2
                  = ( plus_plus @ nat @ ( one_one @ nat )
                    @ ( if @ nat
                      @ ( Xa
                        = ( zero_zero @ nat ) )
                      @ ( one_one @ nat )
                      @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( one_one @ nat ) ) ) ) )
               => ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T9217963907923527482_t_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A4 @ B3 ) @ Xa ) ) ) )
         => ( ! [Info2: option @ ( product_prod @ nat @ nat ),Ts2: list @ vEBT_VEBT,S: vEBT_VEBT] :
                ( ( X2
                  = ( vEBT_Node @ Info2 @ ( zero_zero @ nat ) @ Ts2 @ S ) )
               => ( ( Y2
                    = ( one_one @ nat ) )
                 => ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T9217963907923527482_t_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Info2 @ ( zero_zero @ nat ) @ Ts2 @ S ) @ Xa ) ) ) )
           => ( ! [Info2: option @ ( product_prod @ nat @ nat ),Ts2: list @ vEBT_VEBT,S: vEBT_VEBT] :
                  ( ( X2
                    = ( vEBT_Node @ Info2 @ ( suc @ ( zero_zero @ nat ) ) @ Ts2 @ S ) )
                 => ( ( Y2
                      = ( one_one @ nat ) )
                   => ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T9217963907923527482_t_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Info2 @ ( suc @ ( zero_zero @ nat ) ) @ Ts2 @ S ) @ Xa ) ) ) )
             => ( ! [V3: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
                    ( ( X2
                      = ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ V3 ) ) @ TreeList2 @ Summary2 ) )
                   => ( ( Y2
                        = ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
                     => ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T9217963907923527482_t_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ V3 ) ) @ TreeList2 @ Summary2 ) @ Xa ) ) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X2
                        = ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
                     => ( ( Y2
                          = ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) )
                            @ ( if @ nat
                              @ ( ( ord_less @ nat @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
                                & ~ ( ( Xa = Mi2 )
                                    | ( Xa = 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 @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_T_m_i_n_N_u_l_l @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) @ ( if @ nat @ ( vEBT_VEBT_minNull @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_T_i_n_s_e_r_t @ Summary2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( one_one @ nat ) ) )
                              @ ( one_one @ nat ) ) ) )
                       => ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T9217963907923527482_t_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) @ Xa ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.pelims
thf(fact_2305_delete__correct,axiom,
    ! [T2: vEBT_VEBT,N2: nat,X2: nat] :
      ( ( vEBT_invar_vebt @ T2 @ N2 )
     => ( ( vEBT_VEBT_set_vebt @ ( vEBT_vebt_delete @ T2 @ X2 ) )
        = ( minus_minus @ ( set @ nat ) @ ( vEBT_set_vebt @ T2 ) @ ( insert @ nat @ X2 @ ( bot_bot @ ( set @ nat ) ) ) ) ) ) ).

% delete_correct
thf(fact_2306_lemma__termdiff3,axiom,
    ! [A: $tType] :
      ( ( real_V3459762299906320749_field @ A )
     => ! [H2: A,Z: A,K6: real,N2: nat] :
          ( ( H2
           != ( zero_zero @ A ) )
         => ( ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ Z ) @ K6 )
           => ( ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( plus_plus @ A @ Z @ H2 ) ) @ K6 )
             => ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ ( divide_divide @ A @ ( minus_minus @ A @ ( power_power @ A @ ( plus_plus @ A @ Z @ H2 ) @ N2 ) @ ( power_power @ A @ Z @ N2 ) ) @ H2 ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N2 ) @ ( power_power @ A @ Z @ ( minus_minus @ nat @ N2 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) @ ( times_times @ real @ ( times_times @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N2 ) @ ( semiring_1_of_nat @ real @ ( minus_minus @ nat @ N2 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) @ ( power_power @ real @ K6 @ ( minus_minus @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( real_V7770717601297561774m_norm @ A @ H2 ) ) ) ) ) ) ) ).

% lemma_termdiff3
thf(fact_2307_lowi__def,axiom,
    ( vEBT_VEBT_lowi
    = ( ^ [X: nat,N: nat] : ( heap_Time_return @ nat @ ( modulo_modulo @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) ) ) ).

% lowi_def
thf(fact_2308_foldr__zero,axiom,
    ! [Xs2: list @ nat,D2: nat] :
      ( ! [I3: nat] :
          ( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ nat ) @ Xs2 ) )
         => ( ord_less @ nat @ ( zero_zero @ nat ) @ ( nth @ nat @ Xs2 @ I3 ) ) )
     => ( ord_less_eq @ nat @ ( size_size @ ( list @ nat ) @ Xs2 ) @ ( minus_minus @ nat @ ( foldr @ nat @ nat @ ( plus_plus @ nat ) @ Xs2 @ D2 ) @ D2 ) ) ) ).

% foldr_zero
thf(fact_2309_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_2310_mod__mod__trivial,axiom,
    ! [A: $tType] :
      ( ( euclid4440199948858584721cancel @ A )
     => ! [A2: A,B2: A] :
          ( ( modulo_modulo @ A @ ( modulo_modulo @ A @ A2 @ B2 ) @ B2 )
          = ( modulo_modulo @ A @ A2 @ B2 ) ) ) ).

% mod_mod_trivial
thf(fact_2311_foldr__same__int,axiom,
    ! [Xs2: list @ nat,Y2: nat] :
      ( ! [X3: nat,Y4: nat] :
          ( ( member @ nat @ X3 @ ( set2 @ nat @ Xs2 ) )
         => ( ( member @ nat @ Y4 @ ( set2 @ nat @ Xs2 ) )
           => ( X3 = Y4 ) ) )
     => ( ! [X3: nat] :
            ( ( member @ nat @ X3 @ ( set2 @ nat @ Xs2 ) )
           => ( X3 = Y2 ) )
       => ( ( foldr @ nat @ nat @ ( plus_plus @ nat ) @ Xs2 @ ( zero_zero @ nat ) )
          = ( times_times @ nat @ ( size_size @ ( list @ nat ) @ Xs2 ) @ Y2 ) ) ) ) ).

% foldr_same_int
thf(fact_2312_low__def,axiom,
    ( vEBT_VEBT_low
    = ( ^ [X: nat,N: nat] : ( modulo_modulo @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).

% low_def
thf(fact_2313_delete__correct_H,axiom,
    ! [T2: vEBT_VEBT,N2: nat,X2: nat] :
      ( ( vEBT_invar_vebt @ T2 @ N2 )
     => ( ( vEBT_VEBT_set_vebt @ ( vEBT_vebt_delete @ T2 @ X2 ) )
        = ( minus_minus @ ( set @ nat ) @ ( vEBT_VEBT_set_vebt @ T2 ) @ ( insert @ nat @ X2 @ ( bot_bot @ ( set @ nat ) ) ) ) ) ) ).

% delete_correct'
thf(fact_2314_foldr__mono,axiom,
    ! [Xs2: list @ nat,Ys: list @ nat,C2: nat,D2: nat] :
      ( ( ( size_size @ ( list @ nat ) @ Xs2 )
        = ( size_size @ ( list @ nat ) @ Ys ) )
     => ( ! [I3: nat] :
            ( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ nat ) @ Xs2 ) )
           => ( ord_less @ nat @ ( nth @ nat @ Xs2 @ I3 ) @ ( nth @ nat @ Ys @ I3 ) ) )
       => ( ( ord_less_eq @ nat @ C2 @ D2 )
         => ( ord_less_eq @ nat @ ( plus_plus @ nat @ ( foldr @ nat @ nat @ ( plus_plus @ nat ) @ Xs2 @ C2 ) @ ( size_size @ ( list @ nat ) @ Ys ) ) @ ( foldr @ nat @ nat @ ( plus_plus @ nat ) @ Ys @ D2 ) ) ) ) ) ).

% foldr_mono
thf(fact_2315_mod__self,axiom,
    ! [A: $tType] :
      ( ( semidom_modulo @ A )
     => ! [A2: A] :
          ( ( modulo_modulo @ A @ A2 @ A2 )
          = ( zero_zero @ A ) ) ) ).

% mod_self
thf(fact_2316_mod__by__0,axiom,
    ! [A: $tType] :
      ( ( semidom_modulo @ A )
     => ! [A2: A] :
          ( ( modulo_modulo @ A @ A2 @ ( zero_zero @ A ) )
          = A2 ) ) ).

% mod_by_0
thf(fact_2317_mod__0,axiom,
    ! [A: $tType] :
      ( ( semidom_modulo @ A )
     => ! [A2: A] :
          ( ( modulo_modulo @ A @ ( zero_zero @ A ) @ A2 )
          = ( zero_zero @ A ) ) ) ).

% mod_0
thf(fact_2318_mod__add__self2,axiom,
    ! [A: $tType] :
      ( ( euclid4440199948858584721cancel @ A )
     => ! [A2: A,B2: A] :
          ( ( modulo_modulo @ A @ ( plus_plus @ A @ A2 @ B2 ) @ B2 )
          = ( modulo_modulo @ A @ A2 @ B2 ) ) ) ).

% mod_add_self2
thf(fact_2319_mod__add__self1,axiom,
    ! [A: $tType] :
      ( ( euclid4440199948858584721cancel @ A )
     => ! [B2: A,A2: A] :
          ( ( modulo_modulo @ A @ ( plus_plus @ A @ B2 @ A2 ) @ B2 )
          = ( modulo_modulo @ A @ A2 @ B2 ) ) ) ).

% mod_add_self1
thf(fact_2320_minus__mod__self2,axiom,
    ! [A: $tType] :
      ( ( euclid8851590272496341667cancel @ A )
     => ! [A2: A,B2: A] :
          ( ( modulo_modulo @ A @ ( minus_minus @ A @ A2 @ B2 ) @ B2 )
          = ( modulo_modulo @ A @ A2 @ B2 ) ) ) ).

% minus_mod_self2
thf(fact_2321_insert__subset,axiom,
    ! [A: $tType,X2: A,A3: set @ A,B6: set @ A] :
      ( ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ X2 @ A3 ) @ B6 )
      = ( ( member @ A @ X2 @ B6 )
        & ( ord_less_eq @ ( set @ A ) @ A3 @ B6 ) ) ) ).

% insert_subset
thf(fact_2322_insert__Diff1,axiom,
    ! [A: $tType,X2: A,B6: set @ A,A3: set @ A] :
      ( ( member @ A @ X2 @ B6 )
     => ( ( minus_minus @ ( set @ A ) @ ( insert @ A @ X2 @ A3 ) @ B6 )
        = ( minus_minus @ ( set @ A ) @ A3 @ B6 ) ) ) ).

% insert_Diff1
thf(fact_2323_Diff__insert0,axiom,
    ! [A: $tType,X2: A,A3: set @ A,B6: set @ A] :
      ( ~ ( member @ A @ X2 @ A3 )
     => ( ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X2 @ B6 ) )
        = ( minus_minus @ ( set @ A ) @ A3 @ B6 ) ) ) ).

% Diff_insert0
thf(fact_2324_mod__less,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less @ nat @ M @ N2 )
     => ( ( modulo_modulo @ nat @ M @ N2 )
        = M ) ) ).

% mod_less
thf(fact_2325_nat__mod__eq_H,axiom,
    ! [A2: nat,N2: nat] :
      ( ( ord_less @ nat @ A2 @ N2 )
     => ( ( modulo_modulo @ nat @ A2 @ N2 )
        = A2 ) ) ).

% nat_mod_eq'
thf(fact_2326_singleton__conv,axiom,
    ! [A: $tType,A2: A] :
      ( ( collect @ A
        @ ^ [X: A] : X = A2 )
      = ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ).

% singleton_conv
thf(fact_2327_singleton__conv2,axiom,
    ! [A: $tType,A2: A] :
      ( ( collect @ A
        @ ( ^ [Y5: A,Z4: A] : Y5 = Z4
          @ A2 ) )
      = ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ).

% singleton_conv2
thf(fact_2328_bits__mod__by__1,axiom,
    ! [A: $tType] :
      ( ( bit_semiring_bits @ A )
     => ! [A2: A] :
          ( ( modulo_modulo @ A @ A2 @ ( one_one @ A ) )
          = ( zero_zero @ A ) ) ) ).

% bits_mod_by_1
thf(fact_2329_mod__by__1,axiom,
    ! [A: $tType] :
      ( ( semidom_modulo @ A )
     => ! [A2: A] :
          ( ( modulo_modulo @ A @ A2 @ ( one_one @ A ) )
          = ( zero_zero @ A ) ) ) ).

% mod_by_1
thf(fact_2330_mod__mult__self1__is__0,axiom,
    ! [A: $tType] :
      ( ( euclid4440199948858584721cancel @ A )
     => ! [B2: A,A2: A] :
          ( ( modulo_modulo @ A @ ( times_times @ A @ B2 @ A2 ) @ B2 )
          = ( zero_zero @ A ) ) ) ).

% mod_mult_self1_is_0
thf(fact_2331_mod__mult__self2__is__0,axiom,
    ! [A: $tType] :
      ( ( euclid4440199948858584721cancel @ A )
     => ! [A2: A,B2: A] :
          ( ( modulo_modulo @ A @ ( times_times @ A @ A2 @ B2 ) @ B2 )
          = ( zero_zero @ A ) ) ) ).

% mod_mult_self2_is_0
thf(fact_2332_bits__mod__div__trivial,axiom,
    ! [A: $tType] :
      ( ( bit_semiring_bits @ A )
     => ! [A2: A,B2: A] :
          ( ( divide_divide @ A @ ( modulo_modulo @ A @ A2 @ B2 ) @ B2 )
          = ( zero_zero @ A ) ) ) ).

% bits_mod_div_trivial
thf(fact_2333_mod__div__trivial,axiom,
    ! [A: $tType] :
      ( ( euclid4440199948858584721cancel @ A )
     => ! [A2: A,B2: A] :
          ( ( divide_divide @ A @ ( modulo_modulo @ A @ A2 @ B2 ) @ B2 )
          = ( zero_zero @ A ) ) ) ).

% mod_div_trivial
thf(fact_2334_mod__mult__self1,axiom,
    ! [A: $tType] :
      ( ( euclid4440199948858584721cancel @ A )
     => ! [A2: A,C2: A,B2: A] :
          ( ( modulo_modulo @ A @ ( plus_plus @ A @ A2 @ ( times_times @ A @ C2 @ B2 ) ) @ B2 )
          = ( modulo_modulo @ A @ A2 @ B2 ) ) ) ).

% mod_mult_self1
thf(fact_2335_mod__mult__self2,axiom,
    ! [A: $tType] :
      ( ( euclid4440199948858584721cancel @ A )
     => ! [A2: A,B2: A,C2: A] :
          ( ( modulo_modulo @ A @ ( plus_plus @ A @ A2 @ ( times_times @ A @ B2 @ C2 ) ) @ B2 )
          = ( modulo_modulo @ A @ A2 @ B2 ) ) ) ).

% mod_mult_self2
thf(fact_2336_mod__mult__self3,axiom,
    ! [A: $tType] :
      ( ( euclid4440199948858584721cancel @ A )
     => ! [C2: A,B2: A,A2: A] :
          ( ( modulo_modulo @ A @ ( plus_plus @ A @ ( times_times @ A @ C2 @ B2 ) @ A2 ) @ B2 )
          = ( modulo_modulo @ A @ A2 @ B2 ) ) ) ).

% mod_mult_self3
thf(fact_2337_mod__mult__self4,axiom,
    ! [A: $tType] :
      ( ( euclid4440199948858584721cancel @ A )
     => ! [B2: A,C2: A,A2: A] :
          ( ( modulo_modulo @ A @ ( plus_plus @ A @ ( times_times @ A @ B2 @ C2 ) @ A2 ) @ B2 )
          = ( modulo_modulo @ A @ A2 @ B2 ) ) ) ).

% mod_mult_self4
thf(fact_2338_singleton__insert__inj__eq,axiom,
    ! [A: $tType,B2: A,A2: A,A3: set @ A] :
      ( ( ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) )
        = ( insert @ A @ A2 @ A3 ) )
      = ( ( A2 = B2 )
        & ( ord_less_eq @ ( set @ A ) @ A3 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).

% singleton_insert_inj_eq
thf(fact_2339_singleton__insert__inj__eq_H,axiom,
    ! [A: $tType,A2: A,A3: set @ A,B2: A] :
      ( ( ( insert @ A @ A2 @ A3 )
        = ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) )
      = ( ( A2 = B2 )
        & ( ord_less_eq @ ( set @ A ) @ A3 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).

% singleton_insert_inj_eq'
thf(fact_2340_mod__by__Suc__0,axiom,
    ! [M: nat] :
      ( ( modulo_modulo @ nat @ M @ ( suc @ ( zero_zero @ nat ) ) )
      = ( zero_zero @ nat ) ) ).

% mod_by_Suc_0
thf(fact_2341_insert__Diff__single,axiom,
    ! [A: $tType,A2: A,A3: set @ A] :
      ( ( insert @ A @ A2 @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) )
      = ( insert @ A @ A2 @ A3 ) ) ).

% insert_Diff_single
thf(fact_2342_Suc__mod__mult__self1,axiom,
    ! [M: nat,K: nat,N2: nat] :
      ( ( modulo_modulo @ nat @ ( suc @ ( plus_plus @ nat @ M @ ( times_times @ nat @ K @ N2 ) ) ) @ N2 )
      = ( modulo_modulo @ nat @ ( suc @ M ) @ N2 ) ) ).

% Suc_mod_mult_self1
thf(fact_2343_Suc__mod__mult__self2,axiom,
    ! [M: nat,N2: nat,K: nat] :
      ( ( modulo_modulo @ nat @ ( suc @ ( plus_plus @ nat @ M @ ( times_times @ nat @ N2 @ K ) ) ) @ N2 )
      = ( modulo_modulo @ nat @ ( suc @ M ) @ N2 ) ) ).

% Suc_mod_mult_self2
thf(fact_2344_Suc__mod__mult__self3,axiom,
    ! [K: nat,N2: nat,M: nat] :
      ( ( modulo_modulo @ nat @ ( suc @ ( plus_plus @ nat @ ( times_times @ nat @ K @ N2 ) @ M ) ) @ N2 )
      = ( modulo_modulo @ nat @ ( suc @ M ) @ N2 ) ) ).

% Suc_mod_mult_self3
thf(fact_2345_Suc__mod__mult__self4,axiom,
    ! [N2: nat,K: nat,M: nat] :
      ( ( modulo_modulo @ nat @ ( suc @ ( plus_plus @ nat @ ( times_times @ nat @ N2 @ K ) @ M ) ) @ N2 )
      = ( modulo_modulo @ nat @ ( suc @ M ) @ N2 ) ) ).

% Suc_mod_mult_self4
thf(fact_2346_foldr__length,axiom,
    ! [A: $tType,L: list @ A] :
      ( ( foldr @ A @ nat
        @ ^ [X: A] : suc
        @ L
        @ ( zero_zero @ nat ) )
      = ( size_size @ ( list @ A ) @ L ) ) ).

% foldr_length
thf(fact_2347_bits__one__mod__two__eq__one,axiom,
    ! [A: $tType] :
      ( ( bit_semiring_bits @ A )
     => ( ( modulo_modulo @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
        = ( one_one @ A ) ) ) ).

% bits_one_mod_two_eq_one
thf(fact_2348_one__mod__two__eq__one,axiom,
    ! [A: $tType] :
      ( ( euclid5411537665997757685th_nat @ A )
     => ( ( modulo_modulo @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
        = ( one_one @ A ) ) ) ).

% one_mod_two_eq_one
thf(fact_2349_mod2__Suc__Suc,axiom,
    ! [M: nat] :
      ( ( modulo_modulo @ nat @ ( suc @ ( suc @ M ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
      = ( modulo_modulo @ nat @ M @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ).

% mod2_Suc_Suc
thf(fact_2350_Suc__times__numeral__mod__eq,axiom,
    ! [K: num,N2: nat] :
      ( ( ( numeral_numeral @ nat @ K )
       != ( one_one @ nat ) )
     => ( ( modulo_modulo @ nat @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ K ) @ N2 ) ) @ ( numeral_numeral @ nat @ K ) )
        = ( one_one @ nat ) ) ) ).

% Suc_times_numeral_mod_eq
thf(fact_2351_not__mod__2__eq__1__eq__0,axiom,
    ! [A: $tType] :
      ( ( semiring_parity @ A )
     => ! [A2: A] :
          ( ( ( modulo_modulo @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
           != ( one_one @ A ) )
          = ( ( modulo_modulo @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
            = ( zero_zero @ A ) ) ) ) ).

% not_mod_2_eq_1_eq_0
thf(fact_2352_not__mod__2__eq__0__eq__1,axiom,
    ! [A: $tType] :
      ( ( semiring_parity @ A )
     => ! [A2: A] :
          ( ( ( modulo_modulo @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
           != ( zero_zero @ A ) )
          = ( ( modulo_modulo @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
            = ( one_one @ A ) ) ) ) ).

% not_mod_2_eq_0_eq_1
thf(fact_2353_not__mod2__eq__Suc__0__eq__0,axiom,
    ! [N2: nat] :
      ( ( ( modulo_modulo @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
       != ( suc @ ( zero_zero @ nat ) ) )
      = ( ( modulo_modulo @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
        = ( zero_zero @ nat ) ) ) ).

% not_mod2_eq_Suc_0_eq_0
thf(fact_2354_add__self__mod__2,axiom,
    ! [M: nat] :
      ( ( modulo_modulo @ nat @ ( plus_plus @ nat @ M @ M ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
      = ( zero_zero @ nat ) ) ).

% add_self_mod_2
thf(fact_2355_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 @ one2 ) ) @ M ) @ ( numeral_numeral @ nat @ V ) ) ) ).

% Suc_mod_eq_add3_mod_numeral
thf(fact_2356_mod__Suc__eq__mod__add3,axiom,
    ! [M: nat,N2: nat] :
      ( ( modulo_modulo @ nat @ M @ ( suc @ ( suc @ ( suc @ N2 ) ) ) )
      = ( modulo_modulo @ nat @ M @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) @ N2 ) ) ) ).

% mod_Suc_eq_mod_add3
thf(fact_2357_mod2__gr__0,axiom,
    ! [M: nat] :
      ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( modulo_modulo @ nat @ M @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
      = ( ( modulo_modulo @ nat @ M @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
        = ( one_one @ nat ) ) ) ).

% mod2_gr_0
thf(fact_2358_of__nat__mod,axiom,
    ! [A: $tType] :
      ( ( euclid5411537665997757685th_nat @ A )
     => ! [M: nat,N2: nat] :
          ( ( semiring_1_of_nat @ A @ ( modulo_modulo @ nat @ M @ N2 ) )
          = ( modulo_modulo @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N2 ) ) ) ) ).

% of_nat_mod
thf(fact_2359_insert__Collect,axiom,
    ! [A: $tType,A2: A,P: A > $o] :
      ( ( insert @ A @ A2 @ ( collect @ A @ P ) )
      = ( collect @ A
        @ ^ [U2: A] :
            ( ( U2 != A2 )
           => ( P @ U2 ) ) ) ) ).

% insert_Collect
thf(fact_2360_insert__compr,axiom,
    ! [A: $tType] :
      ( ( insert @ A )
      = ( ^ [A5: A,B7: set @ A] :
            ( collect @ A
            @ ^ [X: A] :
                ( ( X = A5 )
                | ( member @ A @ X @ B7 ) ) ) ) ) ).

% insert_compr
thf(fact_2361_mod__add__right__eq,axiom,
    ! [A: $tType] :
      ( ( euclid4440199948858584721cancel @ A )
     => ! [A2: A,B2: A,C2: A] :
          ( ( modulo_modulo @ A @ ( plus_plus @ A @ A2 @ ( modulo_modulo @ A @ B2 @ C2 ) ) @ C2 )
          = ( modulo_modulo @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 ) ) ) ).

% mod_add_right_eq
thf(fact_2362_mod__add__left__eq,axiom,
    ! [A: $tType] :
      ( ( euclid4440199948858584721cancel @ A )
     => ! [A2: A,C2: A,B2: A] :
          ( ( modulo_modulo @ A @ ( plus_plus @ A @ ( modulo_modulo @ A @ A2 @ C2 ) @ B2 ) @ C2 )
          = ( modulo_modulo @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 ) ) ) ).

% mod_add_left_eq
thf(fact_2363_mod__add__cong,axiom,
    ! [A: $tType] :
      ( ( euclid4440199948858584721cancel @ A )
     => ! [A2: A,C2: A,A6: A,B2: A,B4: A] :
          ( ( ( modulo_modulo @ A @ A2 @ C2 )
            = ( modulo_modulo @ A @ A6 @ C2 ) )
         => ( ( ( modulo_modulo @ A @ B2 @ C2 )
              = ( modulo_modulo @ A @ B4 @ C2 ) )
           => ( ( modulo_modulo @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 )
              = ( modulo_modulo @ A @ ( plus_plus @ A @ A6 @ B4 ) @ C2 ) ) ) ) ) ).

% mod_add_cong
thf(fact_2364_mod__add__eq,axiom,
    ! [A: $tType] :
      ( ( euclid4440199948858584721cancel @ A )
     => ! [A2: A,C2: A,B2: A] :
          ( ( modulo_modulo @ A @ ( plus_plus @ A @ ( modulo_modulo @ A @ A2 @ C2 ) @ ( modulo_modulo @ A @ B2 @ C2 ) ) @ C2 )
          = ( modulo_modulo @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 ) ) ) ).

% mod_add_eq
thf(fact_2365_mod__mult__right__eq,axiom,
    ! [A: $tType] :
      ( ( euclid4440199948858584721cancel @ A )
     => ! [A2: A,B2: A,C2: A] :
          ( ( modulo_modulo @ A @ ( times_times @ A @ A2 @ ( modulo_modulo @ A @ B2 @ C2 ) ) @ C2 )
          = ( modulo_modulo @ A @ ( times_times @ A @ A2 @ B2 ) @ C2 ) ) ) ).

% mod_mult_right_eq
thf(fact_2366_mod__mult__left__eq,axiom,
    ! [A: $tType] :
      ( ( euclid4440199948858584721cancel @ A )
     => ! [A2: A,C2: A,B2: A] :
          ( ( modulo_modulo @ A @ ( times_times @ A @ ( modulo_modulo @ A @ A2 @ C2 ) @ B2 ) @ C2 )
          = ( modulo_modulo @ A @ ( times_times @ A @ A2 @ B2 ) @ C2 ) ) ) ).

% mod_mult_left_eq
thf(fact_2367_mult__mod__right,axiom,
    ! [A: $tType] :
      ( ( euclid4440199948858584721cancel @ A )
     => ! [C2: A,A2: A,B2: A] :
          ( ( times_times @ A @ C2 @ ( modulo_modulo @ A @ A2 @ B2 ) )
          = ( modulo_modulo @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B2 ) ) ) ) ).

% mult_mod_right
thf(fact_2368_mod__mult__mult2,axiom,
    ! [A: $tType] :
      ( ( euclid4440199948858584721cancel @ A )
     => ! [A2: A,C2: A,B2: A] :
          ( ( modulo_modulo @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) )
          = ( times_times @ A @ ( modulo_modulo @ A @ A2 @ B2 ) @ C2 ) ) ) ).

% mod_mult_mult2
thf(fact_2369_mod__mult__cong,axiom,
    ! [A: $tType] :
      ( ( euclid4440199948858584721cancel @ A )
     => ! [A2: A,C2: A,A6: A,B2: A,B4: A] :
          ( ( ( modulo_modulo @ A @ A2 @ C2 )
            = ( modulo_modulo @ A @ A6 @ C2 ) )
         => ( ( ( modulo_modulo @ A @ B2 @ C2 )
              = ( modulo_modulo @ A @ B4 @ C2 ) )
           => ( ( modulo_modulo @ A @ ( times_times @ A @ A2 @ B2 ) @ C2 )
              = ( modulo_modulo @ A @ ( times_times @ A @ A6 @ B4 ) @ C2 ) ) ) ) ) ).

% mod_mult_cong
thf(fact_2370_mod__mult__eq,axiom,
    ! [A: $tType] :
      ( ( euclid4440199948858584721cancel @ A )
     => ! [A2: A,C2: A,B2: A] :
          ( ( modulo_modulo @ A @ ( times_times @ A @ ( modulo_modulo @ A @ A2 @ C2 ) @ ( modulo_modulo @ A @ B2 @ C2 ) ) @ C2 )
          = ( modulo_modulo @ A @ ( times_times @ A @ A2 @ B2 ) @ C2 ) ) ) ).

% mod_mult_eq
thf(fact_2371_mod__diff__right__eq,axiom,
    ! [A: $tType] :
      ( ( euclid8851590272496341667cancel @ A )
     => ! [A2: A,B2: A,C2: A] :
          ( ( modulo_modulo @ A @ ( minus_minus @ A @ A2 @ ( modulo_modulo @ A @ B2 @ C2 ) ) @ C2 )
          = ( modulo_modulo @ A @ ( minus_minus @ A @ A2 @ B2 ) @ C2 ) ) ) ).

% mod_diff_right_eq
thf(fact_2372_mod__diff__left__eq,axiom,
    ! [A: $tType] :
      ( ( euclid8851590272496341667cancel @ A )
     => ! [A2: A,C2: A,B2: A] :
          ( ( modulo_modulo @ A @ ( minus_minus @ A @ ( modulo_modulo @ A @ A2 @ C2 ) @ B2 ) @ C2 )
          = ( modulo_modulo @ A @ ( minus_minus @ A @ A2 @ B2 ) @ C2 ) ) ) ).

% mod_diff_left_eq
thf(fact_2373_mod__diff__cong,axiom,
    ! [A: $tType] :
      ( ( euclid8851590272496341667cancel @ A )
     => ! [A2: A,C2: A,A6: A,B2: A,B4: A] :
          ( ( ( modulo_modulo @ A @ A2 @ C2 )
            = ( modulo_modulo @ A @ A6 @ C2 ) )
         => ( ( ( modulo_modulo @ A @ B2 @ C2 )
              = ( modulo_modulo @ A @ B4 @ C2 ) )
           => ( ( modulo_modulo @ A @ ( minus_minus @ A @ A2 @ B2 ) @ C2 )
              = ( modulo_modulo @ A @ ( minus_minus @ A @ A6 @ B4 ) @ C2 ) ) ) ) ) ).

% mod_diff_cong
thf(fact_2374_mod__diff__eq,axiom,
    ! [A: $tType] :
      ( ( euclid8851590272496341667cancel @ A )
     => ! [A2: A,C2: A,B2: A] :
          ( ( modulo_modulo @ A @ ( minus_minus @ A @ ( modulo_modulo @ A @ A2 @ C2 ) @ ( modulo_modulo @ A @ B2 @ C2 ) ) @ C2 )
          = ( modulo_modulo @ A @ ( minus_minus @ A @ A2 @ B2 ) @ C2 ) ) ) ).

% mod_diff_eq
thf(fact_2375_power__mod,axiom,
    ! [A: $tType] :
      ( ( euclid4440199948858584721cancel @ A )
     => ! [A2: A,B2: A,N2: nat] :
          ( ( modulo_modulo @ A @ ( power_power @ A @ ( modulo_modulo @ A @ A2 @ B2 ) @ N2 ) @ B2 )
          = ( modulo_modulo @ A @ ( power_power @ A @ A2 @ N2 ) @ B2 ) ) ) ).

% power_mod
thf(fact_2376_mod__Suc__Suc__eq,axiom,
    ! [M: nat,N2: nat] :
      ( ( modulo_modulo @ nat @ ( suc @ ( suc @ ( modulo_modulo @ nat @ M @ N2 ) ) ) @ N2 )
      = ( modulo_modulo @ nat @ ( suc @ ( suc @ M ) ) @ N2 ) ) ).

% mod_Suc_Suc_eq
thf(fact_2377_mod__Suc__eq,axiom,
    ! [M: nat,N2: nat] :
      ( ( modulo_modulo @ nat @ ( suc @ ( modulo_modulo @ nat @ M @ N2 ) ) @ N2 )
      = ( modulo_modulo @ nat @ ( suc @ M ) @ N2 ) ) ).

% mod_Suc_eq
thf(fact_2378_nat__mod__eq,axiom,
    ! [B2: nat,N2: nat,A2: nat] :
      ( ( ord_less @ nat @ B2 @ N2 )
     => ( ( ( modulo_modulo @ nat @ A2 @ N2 )
          = ( modulo_modulo @ nat @ B2 @ N2 ) )
       => ( ( modulo_modulo @ nat @ A2 @ N2 )
          = B2 ) ) ) ).

% nat_mod_eq
thf(fact_2379_mod__less__eq__dividend,axiom,
    ! [M: nat,N2: nat] : ( ord_less_eq @ nat @ ( modulo_modulo @ nat @ M @ N2 ) @ M ) ).

% mod_less_eq_dividend
thf(fact_2380_mod__plus__right,axiom,
    ! [A2: nat,X2: nat,M: nat,B2: nat] :
      ( ( ( modulo_modulo @ nat @ ( plus_plus @ nat @ A2 @ X2 ) @ M )
        = ( modulo_modulo @ nat @ ( plus_plus @ nat @ B2 @ X2 ) @ M ) )
      = ( ( modulo_modulo @ nat @ A2 @ M )
        = ( modulo_modulo @ nat @ B2 @ M ) ) ) ).

% mod_plus_right
thf(fact_2381_insert__mono,axiom,
    ! [A: $tType,C5: set @ A,D6: set @ A,A2: A] :
      ( ( ord_less_eq @ ( set @ A ) @ C5 @ D6 )
     => ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ A2 @ C5 ) @ ( insert @ A @ A2 @ D6 ) ) ) ).

% insert_mono
thf(fact_2382_subset__insert,axiom,
    ! [A: $tType,X2: A,A3: set @ A,B6: set @ A] :
      ( ~ ( member @ A @ X2 @ A3 )
     => ( ( ord_less_eq @ ( set @ A ) @ A3 @ ( insert @ A @ X2 @ B6 ) )
        = ( ord_less_eq @ ( set @ A ) @ A3 @ B6 ) ) ) ).

% subset_insert
thf(fact_2383_subset__insertI,axiom,
    ! [A: $tType,B6: set @ A,A2: A] : ( ord_less_eq @ ( set @ A ) @ B6 @ ( insert @ A @ A2 @ B6 ) ) ).

% subset_insertI
thf(fact_2384_subset__insertI2,axiom,
    ! [A: $tType,A3: set @ A,B6: set @ A,B2: A] :
      ( ( ord_less_eq @ ( set @ A ) @ A3 @ B6 )
     => ( ord_less_eq @ ( set @ A ) @ A3 @ ( insert @ A @ B2 @ B6 ) ) ) ).

% subset_insertI2
thf(fact_2385_insert__Diff__if,axiom,
    ! [A: $tType,X2: A,B6: set @ A,A3: set @ A] :
      ( ( ( member @ A @ X2 @ B6 )
       => ( ( minus_minus @ ( set @ A ) @ ( insert @ A @ X2 @ A3 ) @ B6 )
          = ( minus_minus @ ( set @ A ) @ A3 @ B6 ) ) )
      & ( ~ ( member @ A @ X2 @ B6 )
       => ( ( minus_minus @ ( set @ A ) @ ( insert @ A @ X2 @ A3 ) @ B6 )
          = ( insert @ A @ X2 @ ( minus_minus @ ( set @ A ) @ A3 @ B6 ) ) ) ) ) ).

% insert_Diff_if
thf(fact_2386_foldr__cong,axiom,
    ! [B: $tType,A: $tType,A2: A,B2: A,L: list @ B,K: list @ B,F: B > A > A,G: B > A > A] :
      ( ( A2 = B2 )
     => ( ( L = K )
       => ( ! [A4: A,X3: B] :
              ( ( member @ B @ X3 @ ( set2 @ B @ L ) )
             => ( ( F @ X3 @ A4 )
                = ( G @ X3 @ A4 ) ) )
         => ( ( foldr @ B @ A @ F @ L @ A2 )
            = ( foldr @ B @ A @ G @ K @ B2 ) ) ) ) ) ).

% foldr_cong
thf(fact_2387_Collect__conv__if,axiom,
    ! [A: $tType,P: A > $o,A2: A] :
      ( ( ( P @ A2 )
       => ( ( collect @ A
            @ ^ [X: A] :
                ( ( X = A2 )
                & ( P @ X ) ) )
          = ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) )
      & ( ~ ( P @ A2 )
       => ( ( collect @ A
            @ ^ [X: A] :
                ( ( X = A2 )
                & ( P @ X ) ) )
          = ( bot_bot @ ( set @ A ) ) ) ) ) ).

% Collect_conv_if
thf(fact_2388_Collect__conv__if2,axiom,
    ! [A: $tType,P: A > $o,A2: A] :
      ( ( ( P @ A2 )
       => ( ( collect @ A
            @ ^ [X: A] :
                ( ( A2 = X )
                & ( P @ X ) ) )
          = ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) )
      & ( ~ ( P @ A2 )
       => ( ( collect @ A
            @ ^ [X: A] :
                ( ( A2 = X )
                & ( P @ X ) ) )
          = ( bot_bot @ ( set @ A ) ) ) ) ) ).

% Collect_conv_if2
thf(fact_2389_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
    ! [A: $tType] :
      ( ( unique1627219031080169319umeral @ A )
     => ! [A2: A,B2: A] :
          ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
         => ( ord_less_eq @ A @ ( modulo_modulo @ A @ A2 @ B2 ) @ A2 ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
thf(fact_2390_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
    ! [A: $tType] :
      ( ( unique1627219031080169319umeral @ A )
     => ! [B2: A,A2: A] :
          ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
         => ( ord_less @ A @ ( modulo_modulo @ A @ A2 @ B2 ) @ B2 ) ) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_bound
thf(fact_2391_cong__exp__iff__simps_I9_J,axiom,
    ! [A: $tType] :
      ( ( unique1627219031080169319umeral @ A )
     => ! [M: num,Q2: num,N2: num] :
          ( ( ( modulo_modulo @ A @ ( numeral_numeral @ A @ ( bit0 @ M ) ) @ ( numeral_numeral @ A @ ( bit0 @ Q2 ) ) )
            = ( modulo_modulo @ A @ ( numeral_numeral @ A @ ( bit0 @ N2 ) ) @ ( numeral_numeral @ A @ ( bit0 @ Q2 ) ) ) )
          = ( ( modulo_modulo @ A @ ( numeral_numeral @ A @ M ) @ ( numeral_numeral @ A @ Q2 ) )
            = ( modulo_modulo @ A @ ( numeral_numeral @ A @ N2 ) @ ( numeral_numeral @ A @ Q2 ) ) ) ) ) ).

% cong_exp_iff_simps(9)
thf(fact_2392_cong__exp__iff__simps_I4_J,axiom,
    ! [A: $tType] :
      ( ( unique1627219031080169319umeral @ A )
     => ! [M: num,N2: num] :
          ( ( modulo_modulo @ A @ ( numeral_numeral @ A @ M ) @ ( numeral_numeral @ A @ one2 ) )
          = ( modulo_modulo @ A @ ( numeral_numeral @ A @ N2 ) @ ( numeral_numeral @ A @ one2 ) ) ) ) ).

% cong_exp_iff_simps(4)
thf(fact_2393_mod__eq__self__iff__div__eq__0,axiom,
    ! [A: $tType] :
      ( ( euclid3725896446679973847miring @ A )
     => ! [A2: A,B2: A] :
          ( ( ( modulo_modulo @ A @ A2 @ B2 )
            = A2 )
          = ( ( divide_divide @ A @ A2 @ B2 )
            = ( zero_zero @ A ) ) ) ) ).

% mod_eq_self_iff_div_eq_0
thf(fact_2394_mod__eqE,axiom,
    ! [A: $tType] :
      ( ( euclid8851590272496341667cancel @ A )
     => ! [A2: A,C2: A,B2: A] :
          ( ( ( modulo_modulo @ A @ A2 @ C2 )
            = ( modulo_modulo @ A @ B2 @ C2 ) )
         => ~ ! [D4: A] :
                ( B2
               != ( plus_plus @ A @ A2 @ ( times_times @ A @ C2 @ D4 ) ) ) ) ) ).

% mod_eqE
thf(fact_2395_div__add1__eq,axiom,
    ! [A: $tType] :
      ( ( euclid3128863361964157862miring @ A )
     => ! [A2: A,B2: A,C2: A] :
          ( ( divide_divide @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 )
          = ( plus_plus @ A @ ( plus_plus @ A @ ( divide_divide @ A @ A2 @ C2 ) @ ( divide_divide @ A @ B2 @ C2 ) ) @ ( divide_divide @ A @ ( plus_plus @ A @ ( modulo_modulo @ A @ A2 @ C2 ) @ ( modulo_modulo @ A @ B2 @ C2 ) ) @ C2 ) ) ) ) ).

% div_add1_eq
thf(fact_2396_mod__Suc,axiom,
    ! [M: nat,N2: nat] :
      ( ( ( ( suc @ ( modulo_modulo @ nat @ M @ N2 ) )
          = N2 )
       => ( ( modulo_modulo @ nat @ ( suc @ M ) @ N2 )
          = ( zero_zero @ nat ) ) )
      & ( ( ( suc @ ( modulo_modulo @ nat @ M @ N2 ) )
         != N2 )
       => ( ( modulo_modulo @ nat @ ( suc @ M ) @ N2 )
          = ( suc @ ( modulo_modulo @ nat @ M @ N2 ) ) ) ) ) ).

% mod_Suc
thf(fact_2397_mod__induct,axiom,
    ! [P: nat > $o,N2: nat,P4: nat,M: nat] :
      ( ( P @ N2 )
     => ( ( ord_less @ nat @ N2 @ P4 )
       => ( ( ord_less @ nat @ M @ P4 )
         => ( ! [N4: nat] :
                ( ( ord_less @ nat @ N4 @ P4 )
               => ( ( P @ N4 )
                 => ( P @ ( modulo_modulo @ nat @ ( suc @ N4 ) @ P4 ) ) ) )
           => ( P @ M ) ) ) ) ) ).

% mod_induct
thf(fact_2398_nat__mod__lem,axiom,
    ! [N2: nat,B2: nat] :
      ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
     => ( ( ord_less @ nat @ B2 @ N2 )
        = ( ( modulo_modulo @ nat @ B2 @ N2 )
          = B2 ) ) ) ).

% nat_mod_lem
thf(fact_2399_mod__less__divisor,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
     => ( ord_less @ nat @ ( modulo_modulo @ nat @ M @ N2 ) @ N2 ) ) ).

% mod_less_divisor
thf(fact_2400_mod__Suc__le__divisor,axiom,
    ! [M: nat,N2: nat] : ( ord_less_eq @ nat @ ( modulo_modulo @ nat @ M @ ( suc @ N2 ) ) @ N2 ) ).

% mod_Suc_le_divisor
thf(fact_2401_word__rot__lem,axiom,
    ! [L: nat,K: nat,D2: nat,N2: nat] :
      ( ( ( plus_plus @ nat @ L @ K )
        = ( plus_plus @ nat @ D2 @ ( modulo_modulo @ nat @ K @ L ) ) )
     => ( ( ord_less @ nat @ N2 @ L )
       => ( ( modulo_modulo @ nat @ ( plus_plus @ nat @ D2 @ N2 ) @ L )
          = N2 ) ) ) ).

% word_rot_lem
thf(fact_2402_nat__minus__mod,axiom,
    ! [N2: nat,M: nat] :
      ( ( modulo_modulo @ nat @ ( minus_minus @ nat @ N2 @ ( modulo_modulo @ nat @ N2 @ M ) ) @ M )
      = ( zero_zero @ nat ) ) ).

% nat_minus_mod
thf(fact_2403_mod__if,axiom,
    ( ( modulo_modulo @ nat )
    = ( ^ [M7: nat,N: nat] : ( if @ nat @ ( ord_less @ nat @ M7 @ N ) @ M7 @ ( modulo_modulo @ nat @ ( minus_minus @ nat @ M7 @ N ) @ N ) ) ) ) ).

% mod_if
thf(fact_2404_mod__nat__sub,axiom,
    ! [X2: nat,Z: nat,Y2: nat] :
      ( ( ord_less @ nat @ X2 @ Z )
     => ( ( modulo_modulo @ nat @ ( minus_minus @ nat @ X2 @ Y2 ) @ Z )
        = ( minus_minus @ nat @ X2 @ Y2 ) ) ) ).

% mod_nat_sub
thf(fact_2405_mod__geq,axiom,
    ! [M: nat,N2: nat] :
      ( ~ ( ord_less @ nat @ M @ N2 )
     => ( ( modulo_modulo @ nat @ M @ N2 )
        = ( modulo_modulo @ nat @ ( minus_minus @ nat @ M @ N2 ) @ N2 ) ) ) ).

% mod_geq
thf(fact_2406_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_2407_le__mod__geq,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_eq @ nat @ N2 @ M )
     => ( ( modulo_modulo @ nat @ M @ N2 )
        = ( modulo_modulo @ nat @ ( minus_minus @ nat @ M @ N2 ) @ N2 ) ) ) ).

% le_mod_geq
thf(fact_2408_nat__minus__mod__plus__right,axiom,
    ! [N2: nat,X2: nat,M: nat] :
      ( ( modulo_modulo @ nat @ ( minus_minus @ nat @ ( plus_plus @ nat @ N2 @ X2 ) @ ( modulo_modulo @ nat @ N2 @ M ) ) @ M )
      = ( modulo_modulo @ nat @ X2 @ M ) ) ).

% nat_minus_mod_plus_right
thf(fact_2409_msrevs_I2_J,axiom,
    ! [K: nat,N2: nat,M: nat] :
      ( ( modulo_modulo @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ K @ N2 ) @ M ) @ N2 )
      = ( modulo_modulo @ nat @ M @ N2 ) ) ).

% msrevs(2)
thf(fact_2410_nat__mod__eq__iff,axiom,
    ! [X2: nat,N2: nat,Y2: nat] :
      ( ( ( modulo_modulo @ nat @ X2 @ N2 )
        = ( modulo_modulo @ nat @ Y2 @ N2 ) )
      = ( ? [Q1: nat,Q22: nat] :
            ( ( plus_plus @ nat @ X2 @ ( times_times @ nat @ N2 @ Q1 ) )
            = ( plus_plus @ nat @ Y2 @ ( times_times @ nat @ N2 @ Q22 ) ) ) ) ) ).

% nat_mod_eq_iff
thf(fact_2411_subset__singletonD,axiom,
    ! [A: $tType,A3: set @ A,X2: A] :
      ( ( ord_less_eq @ ( set @ A ) @ A3 @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) )
     => ( ( A3
          = ( bot_bot @ ( set @ A ) ) )
        | ( A3
          = ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).

% subset_singletonD
thf(fact_2412_subset__singleton__iff,axiom,
    ! [A: $tType,X8: set @ A,A2: A] :
      ( ( ord_less_eq @ ( set @ A ) @ X8 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) )
      = ( ( X8
          = ( bot_bot @ ( set @ A ) ) )
        | ( X8
          = ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).

% subset_singleton_iff
thf(fact_2413_Diff__insert,axiom,
    ! [A: $tType,A3: set @ A,A2: A,B6: set @ A] :
      ( ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ A2 @ B6 ) )
      = ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ B6 ) @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).

% Diff_insert
thf(fact_2414_insert__Diff,axiom,
    ! [A: $tType,A2: A,A3: set @ A] :
      ( ( member @ A @ A2 @ A3 )
     => ( ( insert @ A @ A2 @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) )
        = A3 ) ) ).

% insert_Diff
thf(fact_2415_Diff__insert2,axiom,
    ! [A: $tType,A3: set @ A,A2: A,B6: set @ A] :
      ( ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ A2 @ B6 ) )
      = ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) @ B6 ) ) ).

% Diff_insert2
thf(fact_2416_insert__minus__eq,axiom,
    ! [A: $tType,X2: A,Y2: A,A3: set @ A] :
      ( ( X2 != Y2 )
     => ( ( minus_minus @ ( set @ A ) @ ( insert @ A @ X2 @ A3 ) @ ( insert @ A @ Y2 @ ( bot_bot @ ( set @ A ) ) ) )
        = ( insert @ A @ X2 @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ Y2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).

% insert_minus_eq
thf(fact_2417_Diff__insert__absorb,axiom,
    ! [A: $tType,X2: A,A3: set @ A] :
      ( ~ ( member @ A @ X2 @ A3 )
     => ( ( minus_minus @ ( set @ A ) @ ( insert @ A @ X2 @ A3 ) @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) )
        = A3 ) ) ).

% Diff_insert_absorb
thf(fact_2418_set__minus__singleton__eq,axiom,
    ! [A: $tType,X2: A,X8: set @ A] :
      ( ~ ( member @ A @ X2 @ X8 )
     => ( ( minus_minus @ ( set @ A ) @ X8 @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) )
        = X8 ) ) ).

% set_minus_singleton_eq
thf(fact_2419_subset__Diff__insert,axiom,
    ! [A: $tType,A3: set @ A,B6: set @ A,X2: A,C5: set @ A] :
      ( ( ord_less_eq @ ( set @ A ) @ A3 @ ( minus_minus @ ( set @ A ) @ B6 @ ( insert @ A @ X2 @ C5 ) ) )
      = ( ( ord_less_eq @ ( set @ A ) @ A3 @ ( minus_minus @ ( set @ A ) @ B6 @ C5 ) )
        & ~ ( member @ A @ X2 @ A3 ) ) ) ).

% subset_Diff_insert
thf(fact_2420_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
    ! [A: $tType] :
      ( ( unique1627219031080169319umeral @ A )
     => ! [B2: A,A2: A] :
          ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
         => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( modulo_modulo @ A @ A2 @ B2 ) ) ) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_sign
thf(fact_2421_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
    ! [A: $tType] :
      ( ( unique1627219031080169319umeral @ A )
     => ! [A2: A,B2: A] :
          ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
         => ( ( ord_less @ A @ A2 @ B2 )
           => ( ( modulo_modulo @ A @ A2 @ B2 )
              = A2 ) ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_less
thf(fact_2422_cong__exp__iff__simps_I2_J,axiom,
    ! [A: $tType] :
      ( ( unique1627219031080169319umeral @ A )
     => ! [N2: num,Q2: num] :
          ( ( ( modulo_modulo @ A @ ( numeral_numeral @ A @ ( bit0 @ N2 ) ) @ ( numeral_numeral @ A @ ( bit0 @ Q2 ) ) )
            = ( zero_zero @ A ) )
          = ( ( modulo_modulo @ A @ ( numeral_numeral @ A @ N2 ) @ ( numeral_numeral @ A @ Q2 ) )
            = ( zero_zero @ A ) ) ) ) ).

% cong_exp_iff_simps(2)
thf(fact_2423_cong__exp__iff__simps_I1_J,axiom,
    ! [A: $tType] :
      ( ( unique1627219031080169319umeral @ A )
     => ! [N2: num] :
          ( ( modulo_modulo @ A @ ( numeral_numeral @ A @ N2 ) @ ( numeral_numeral @ A @ one2 ) )
          = ( zero_zero @ A ) ) ) ).

% cong_exp_iff_simps(1)
thf(fact_2424_cong__exp__iff__simps_I6_J,axiom,
    ! [A: $tType] :
      ( ( unique1627219031080169319umeral @ A )
     => ! [Q2: num,N2: num] :
          ( ( modulo_modulo @ A @ ( numeral_numeral @ A @ one2 ) @ ( numeral_numeral @ A @ ( bit0 @ Q2 ) ) )
         != ( modulo_modulo @ A @ ( numeral_numeral @ A @ ( bit0 @ N2 ) ) @ ( numeral_numeral @ A @ ( bit0 @ Q2 ) ) ) ) ) ).

% cong_exp_iff_simps(6)
thf(fact_2425_cong__exp__iff__simps_I8_J,axiom,
    ! [A: $tType] :
      ( ( unique1627219031080169319umeral @ A )
     => ! [M: num,Q2: num] :
          ( ( modulo_modulo @ A @ ( numeral_numeral @ A @ ( bit0 @ M ) ) @ ( numeral_numeral @ A @ ( bit0 @ Q2 ) ) )
         != ( modulo_modulo @ A @ ( numeral_numeral @ A @ one2 ) @ ( numeral_numeral @ A @ ( bit0 @ Q2 ) ) ) ) ) ).

% cong_exp_iff_simps(8)
thf(fact_2426_cong__exp__iff__simps_I10_J,axiom,
    ! [A: $tType] :
      ( ( unique1627219031080169319umeral @ A )
     => ! [M: num,Q2: num,N2: num] :
          ( ( modulo_modulo @ A @ ( numeral_numeral @ A @ ( bit0 @ M ) ) @ ( numeral_numeral @ A @ ( bit0 @ Q2 ) ) )
         != ( modulo_modulo @ A @ ( numeral_numeral @ A @ ( bit1 @ N2 ) ) @ ( numeral_numeral @ A @ ( bit0 @ Q2 ) ) ) ) ) ).

% cong_exp_iff_simps(10)
thf(fact_2427_cong__exp__iff__simps_I12_J,axiom,
    ! [A: $tType] :
      ( ( unique1627219031080169319umeral @ A )
     => ! [M: num,Q2: num,N2: num] :
          ( ( modulo_modulo @ A @ ( numeral_numeral @ A @ ( bit1 @ M ) ) @ ( numeral_numeral @ A @ ( bit0 @ Q2 ) ) )
         != ( modulo_modulo @ A @ ( numeral_numeral @ A @ ( bit0 @ N2 ) ) @ ( numeral_numeral @ A @ ( bit0 @ Q2 ) ) ) ) ) ).

% cong_exp_iff_simps(12)
thf(fact_2428_cong__exp__iff__simps_I13_J,axiom,
    ! [A: $tType] :
      ( ( unique1627219031080169319umeral @ A )
     => ! [M: num,Q2: num,N2: num] :
          ( ( ( modulo_modulo @ A @ ( numeral_numeral @ A @ ( bit1 @ M ) ) @ ( numeral_numeral @ A @ ( bit0 @ Q2 ) ) )
            = ( modulo_modulo @ A @ ( numeral_numeral @ A @ ( bit1 @ N2 ) ) @ ( numeral_numeral @ A @ ( bit0 @ Q2 ) ) ) )
          = ( ( modulo_modulo @ A @ ( numeral_numeral @ A @ M ) @ ( numeral_numeral @ A @ Q2 ) )
            = ( modulo_modulo @ A @ ( numeral_numeral @ A @ N2 ) @ ( numeral_numeral @ A @ Q2 ) ) ) ) ) ).

% cong_exp_iff_simps(13)
thf(fact_2429_mult__div__mod__eq,axiom,
    ! [A: $tType] :
      ( ( semiring_modulo @ A )
     => ! [B2: A,A2: A] :
          ( ( plus_plus @ A @ ( times_times @ A @ B2 @ ( divide_divide @ A @ A2 @ B2 ) ) @ ( modulo_modulo @ A @ A2 @ B2 ) )
          = A2 ) ) ).

% mult_div_mod_eq
thf(fact_2430_mod__mult__div__eq,axiom,
    ! [A: $tType] :
      ( ( semiring_modulo @ A )
     => ! [A2: A,B2: A] :
          ( ( plus_plus @ A @ ( modulo_modulo @ A @ A2 @ B2 ) @ ( times_times @ A @ B2 @ ( divide_divide @ A @ A2 @ B2 ) ) )
          = A2 ) ) ).

% mod_mult_div_eq
thf(fact_2431_mod__div__mult__eq,axiom,
    ! [A: $tType] :
      ( ( semiring_modulo @ A )
     => ! [A2: A,B2: A] :
          ( ( plus_plus @ A @ ( modulo_modulo @ A @ A2 @ B2 ) @ ( times_times @ A @ ( divide_divide @ A @ A2 @ B2 ) @ B2 ) )
          = A2 ) ) ).

% mod_div_mult_eq
thf(fact_2432_div__mult__mod__eq,axiom,
    ! [A: $tType] :
      ( ( semiring_modulo @ A )
     => ! [A2: A,B2: A] :
          ( ( plus_plus @ A @ ( times_times @ A @ ( divide_divide @ A @ A2 @ B2 ) @ B2 ) @ ( modulo_modulo @ A @ A2 @ B2 ) )
          = A2 ) ) ).

% div_mult_mod_eq
thf(fact_2433_mod__div__decomp,axiom,
    ! [A: $tType] :
      ( ( semiring_modulo @ A )
     => ! [A2: A,B2: A] :
          ( A2
          = ( plus_plus @ A @ ( times_times @ A @ ( divide_divide @ A @ A2 @ B2 ) @ B2 ) @ ( modulo_modulo @ A @ A2 @ B2 ) ) ) ) ).

% mod_div_decomp
thf(fact_2434_cancel__div__mod__rules_I1_J,axiom,
    ! [A: $tType] :
      ( ( semidom_modulo @ A )
     => ! [A2: A,B2: A,C2: A] :
          ( ( plus_plus @ A @ ( plus_plus @ A @ ( times_times @ A @ ( divide_divide @ A @ A2 @ B2 ) @ B2 ) @ ( modulo_modulo @ A @ A2 @ B2 ) ) @ C2 )
          = ( plus_plus @ A @ A2 @ C2 ) ) ) ).

% cancel_div_mod_rules(1)
thf(fact_2435_cancel__div__mod__rules_I2_J,axiom,
    ! [A: $tType] :
      ( ( semidom_modulo @ A )
     => ! [B2: A,A2: A,C2: A] :
          ( ( plus_plus @ A @ ( plus_plus @ A @ ( times_times @ A @ B2 @ ( divide_divide @ A @ A2 @ B2 ) ) @ ( modulo_modulo @ A @ A2 @ B2 ) ) @ C2 )
          = ( plus_plus @ A @ A2 @ C2 ) ) ) ).

% cancel_div_mod_rules(2)
thf(fact_2436_div__mult1__eq,axiom,
    ! [A: $tType] :
      ( ( euclid3128863361964157862miring @ A )
     => ! [A2: A,B2: A,C2: A] :
          ( ( divide_divide @ A @ ( times_times @ A @ A2 @ B2 ) @ C2 )
          = ( plus_plus @ A @ ( times_times @ A @ A2 @ ( divide_divide @ A @ B2 @ C2 ) ) @ ( divide_divide @ A @ ( times_times @ A @ A2 @ ( modulo_modulo @ A @ B2 @ C2 ) ) @ C2 ) ) ) ) ).

% div_mult1_eq
thf(fact_2437_minus__mult__div__eq__mod,axiom,
    ! [A: $tType] :
      ( ( semiring_modulo @ A )
     => ! [A2: A,B2: A] :
          ( ( minus_minus @ A @ A2 @ ( times_times @ A @ B2 @ ( divide_divide @ A @ A2 @ B2 ) ) )
          = ( modulo_modulo @ A @ A2 @ B2 ) ) ) ).

% minus_mult_div_eq_mod
thf(fact_2438_minus__mod__eq__mult__div,axiom,
    ! [A: $tType] :
      ( ( semiring_modulo @ A )
     => ! [A2: A,B2: A] :
          ( ( minus_minus @ A @ A2 @ ( modulo_modulo @ A @ A2 @ B2 ) )
          = ( times_times @ A @ B2 @ ( divide_divide @ A @ A2 @ B2 ) ) ) ) ).

% minus_mod_eq_mult_div
thf(fact_2439_minus__mod__eq__div__mult,axiom,
    ! [A: $tType] :
      ( ( semiring_modulo @ A )
     => ! [A2: A,B2: A] :
          ( ( minus_minus @ A @ A2 @ ( modulo_modulo @ A @ A2 @ B2 ) )
          = ( times_times @ A @ ( divide_divide @ A @ A2 @ B2 ) @ B2 ) ) ) ).

% minus_mod_eq_div_mult
thf(fact_2440_minus__div__mult__eq__mod,axiom,
    ! [A: $tType] :
      ( ( semiring_modulo @ A )
     => ! [A2: A,B2: A] :
          ( ( minus_minus @ A @ A2 @ ( times_times @ A @ ( divide_divide @ A @ A2 @ B2 ) @ B2 ) )
          = ( modulo_modulo @ A @ A2 @ B2 ) ) ) ).

% minus_div_mult_eq_mod
thf(fact_2441_zmde,axiom,
    ! [A: $tType] :
      ( ( ( group_add @ A )
        & ( semiring_modulo @ A ) )
     => ! [B2: A,A2: A] :
          ( ( times_times @ A @ B2 @ ( divide_divide @ A @ A2 @ B2 ) )
          = ( minus_minus @ A @ A2 @ ( modulo_modulo @ A @ A2 @ B2 ) ) ) ) ).

% zmde
thf(fact_2442_mod__le__divisor,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
     => ( ord_less_eq @ nat @ ( modulo_modulo @ nat @ M @ N2 ) @ N2 ) ) ).

% mod_le_divisor
thf(fact_2443_div__less__mono,axiom,
    ! [A3: nat,B6: nat,N2: nat] :
      ( ( ord_less @ nat @ A3 @ B6 )
     => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
       => ( ( ( modulo_modulo @ nat @ A3 @ N2 )
            = ( zero_zero @ nat ) )
         => ( ( ( modulo_modulo @ nat @ B6 @ N2 )
              = ( zero_zero @ nat ) )
           => ( ord_less @ nat @ ( divide_divide @ nat @ A3 @ N2 ) @ ( divide_divide @ nat @ B6 @ N2 ) ) ) ) ) ) ).

% div_less_mono
thf(fact_2444_mod__nat__add,axiom,
    ! [X2: nat,Z: nat,Y2: nat] :
      ( ( ord_less @ nat @ X2 @ Z )
     => ( ( ord_less @ nat @ Y2 @ Z )
       => ( ( ( ord_less @ nat @ ( plus_plus @ nat @ X2 @ Y2 ) @ Z )
           => ( ( modulo_modulo @ nat @ ( plus_plus @ nat @ X2 @ Y2 ) @ Z )
              = ( plus_plus @ nat @ X2 @ Y2 ) ) )
          & ( ~ ( ord_less @ nat @ ( plus_plus @ nat @ X2 @ Y2 ) @ Z )
           => ( ( modulo_modulo @ nat @ ( plus_plus @ nat @ X2 @ Y2 ) @ Z )
              = ( minus_minus @ nat @ ( plus_plus @ nat @ X2 @ Y2 ) @ Z ) ) ) ) ) ) ).

% mod_nat_add
thf(fact_2445_nat__mod__eq__lemma,axiom,
    ! [X2: nat,N2: nat,Y2: nat] :
      ( ( ( modulo_modulo @ nat @ X2 @ N2 )
        = ( modulo_modulo @ nat @ Y2 @ N2 ) )
     => ( ( ord_less_eq @ nat @ Y2 @ X2 )
       => ? [Q3: nat] :
            ( X2
            = ( plus_plus @ nat @ Y2 @ ( times_times @ nat @ N2 @ Q3 ) ) ) ) ) ).

% nat_mod_eq_lemma
thf(fact_2446_mod__eq__nat2E,axiom,
    ! [M: nat,Q2: nat,N2: nat] :
      ( ( ( modulo_modulo @ nat @ M @ Q2 )
        = ( modulo_modulo @ nat @ N2 @ Q2 ) )
     => ( ( ord_less_eq @ nat @ M @ N2 )
       => ~ ! [S: nat] :
              ( N2
             != ( plus_plus @ nat @ M @ ( times_times @ nat @ Q2 @ S ) ) ) ) ) ).

% mod_eq_nat2E
thf(fact_2447_mod__eq__nat1E,axiom,
    ! [M: nat,Q2: nat,N2: nat] :
      ( ( ( modulo_modulo @ nat @ M @ Q2 )
        = ( modulo_modulo @ nat @ N2 @ Q2 ) )
     => ( ( ord_less_eq @ nat @ N2 @ M )
       => ~ ! [S: nat] :
              ( M
             != ( plus_plus @ nat @ N2 @ ( times_times @ nat @ Q2 @ S ) ) ) ) ) ).

% mod_eq_nat1E
thf(fact_2448_mod__mult2__eq,axiom,
    ! [M: nat,N2: nat,Q2: nat] :
      ( ( modulo_modulo @ nat @ M @ ( times_times @ nat @ N2 @ Q2 ) )
      = ( plus_plus @ nat @ ( times_times @ nat @ N2 @ ( modulo_modulo @ nat @ ( divide_divide @ nat @ M @ N2 ) @ Q2 ) ) @ ( modulo_modulo @ nat @ M @ N2 ) ) ) ).

% mod_mult2_eq
thf(fact_2449_div__mod__decomp,axiom,
    ! [A3: nat,N2: nat] :
      ( A3
      = ( plus_plus @ nat @ ( times_times @ nat @ ( divide_divide @ nat @ A3 @ N2 ) @ N2 ) @ ( modulo_modulo @ nat @ A3 @ N2 ) ) ) ).

% div_mod_decomp
thf(fact_2450_modulo__nat__def,axiom,
    ( ( modulo_modulo @ nat )
    = ( ^ [M7: nat,N: nat] : ( minus_minus @ nat @ M7 @ ( times_times @ nat @ ( divide_divide @ nat @ M7 @ N ) @ N ) ) ) ) ).

% modulo_nat_def
thf(fact_2451_subset__insert__iff,axiom,
    ! [A: $tType,A3: set @ A,X2: A,B6: set @ A] :
      ( ( ord_less_eq @ ( set @ A ) @ A3 @ ( insert @ A @ X2 @ B6 ) )
      = ( ( ( member @ A @ X2 @ A3 )
         => ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) @ B6 ) )
        & ( ~ ( member @ A @ X2 @ A3 )
         => ( ord_less_eq @ ( set @ A ) @ A3 @ B6 ) ) ) ) ).

% subset_insert_iff
thf(fact_2452_Diff__single__insert,axiom,
    ! [A: $tType,A3: set @ A,X2: A,B6: set @ A] :
      ( ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) @ B6 )
     => ( ord_less_eq @ ( set @ A ) @ A3 @ ( insert @ A @ X2 @ B6 ) ) ) ).

% Diff_single_insert
thf(fact_2453_atLeast0__atMost__Suc,axiom,
    ! [N2: nat] :
      ( ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( suc @ N2 ) )
      = ( insert @ nat @ ( suc @ N2 ) @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ).

% atLeast0_atMost_Suc
thf(fact_2454_Icc__eq__insert__lb__nat,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq @ nat @ M @ N2 )
     => ( ( set_or1337092689740270186AtMost @ nat @ M @ N2 )
        = ( insert @ nat @ M @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M ) @ N2 ) ) ) ) ).

% Icc_eq_insert_lb_nat
thf(fact_2455_atLeastAtMostSuc__conv,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq @ nat @ M @ ( suc @ N2 ) )
     => ( ( set_or1337092689740270186AtMost @ nat @ M @ ( suc @ N2 ) )
        = ( insert @ nat @ ( suc @ N2 ) @ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) ) ) ) ).

% atLeastAtMostSuc_conv
thf(fact_2456_atLeastAtMost__insertL,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq @ nat @ M @ N2 )
     => ( ( insert @ nat @ M @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M ) @ N2 ) )
        = ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) ) ) ).

% atLeastAtMost_insertL
thf(fact_2457_remove__subset,axiom,
    ! [A: $tType,X2: A,S3: set @ A] :
      ( ( member @ A @ X2 @ S3 )
     => ( ord_less @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ S3 @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) @ S3 ) ) ).

% remove_subset
thf(fact_2458_set__update__subset__insert,axiom,
    ! [A: $tType,Xs2: list @ A,I: nat,X2: A] : ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ ( list_update @ A @ Xs2 @ I @ X2 ) ) @ ( insert @ A @ X2 @ ( set2 @ A @ Xs2 ) ) ) ).

% set_update_subset_insert
thf(fact_2459_cong__exp__iff__simps_I3_J,axiom,
    ! [A: $tType] :
      ( ( unique1627219031080169319umeral @ A )
     => ! [N2: num,Q2: num] :
          ( ( modulo_modulo @ A @ ( numeral_numeral @ A @ ( bit1 @ N2 ) ) @ ( numeral_numeral @ A @ ( bit0 @ Q2 ) ) )
         != ( zero_zero @ A ) ) ) ).

% cong_exp_iff_simps(3)
thf(fact_2460_mod__mult2__eq_H,axiom,
    ! [A: $tType] :
      ( ( euclid5411537665997757685th_nat @ A )
     => ! [A2: A,M: nat,N2: nat] :
          ( ( modulo_modulo @ A @ A2 @ ( times_times @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N2 ) ) )
          = ( plus_plus @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ M ) @ ( modulo_modulo @ A @ ( divide_divide @ A @ A2 @ ( semiring_1_of_nat @ A @ M ) ) @ ( semiring_1_of_nat @ A @ N2 ) ) ) @ ( modulo_modulo @ A @ A2 @ ( semiring_1_of_nat @ A @ M ) ) ) ) ) ).

% mod_mult2_eq'
thf(fact_2461_field__char__0__class_Oof__nat__div,axiom,
    ! [A: $tType] :
      ( ( field_char_0 @ A )
     => ! [M: nat,N2: nat] :
          ( ( semiring_1_of_nat @ A @ ( divide_divide @ nat @ M @ N2 ) )
          = ( divide_divide @ A @ ( minus_minus @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ ( modulo_modulo @ nat @ M @ N2 ) ) ) @ ( semiring_1_of_nat @ A @ N2 ) ) ) ) ).

% field_char_0_class.of_nat_div
thf(fact_2462_split__mod,axiom,
    ! [P: nat > $o,M: nat,N2: nat] :
      ( ( P @ ( modulo_modulo @ nat @ M @ N2 ) )
      = ( ( ( N2
            = ( zero_zero @ nat ) )
         => ( P @ M ) )
        & ( ( N2
           != ( zero_zero @ nat ) )
         => ! [I2: nat,J3: nat] :
              ( ( ord_less @ nat @ J3 @ N2 )
             => ( ( M
                  = ( plus_plus @ nat @ ( times_times @ nat @ N2 @ I2 ) @ J3 ) )
               => ( P @ J3 ) ) ) ) ) ) ).

% split_mod
thf(fact_2463_mod__lemma,axiom,
    ! [C2: nat,R4: nat,B2: nat,Q2: nat] :
      ( ( ord_less @ nat @ ( zero_zero @ nat ) @ C2 )
     => ( ( ord_less @ nat @ R4 @ B2 )
       => ( ord_less @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ B2 @ ( modulo_modulo @ nat @ Q2 @ C2 ) ) @ R4 ) @ ( times_times @ nat @ B2 @ C2 ) ) ) ) ).

% mod_lemma
thf(fact_2464_foldr__length__aux,axiom,
    ! [A: $tType,L: list @ A,A2: nat] :
      ( ( foldr @ A @ nat
        @ ^ [X: A] : suc
        @ L
        @ A2 )
      = ( plus_plus @ nat @ A2 @ ( size_size @ ( list @ A ) @ L ) ) ) ).

% foldr_length_aux
thf(fact_2465_psubset__insert__iff,axiom,
    ! [A: $tType,A3: set @ A,X2: A,B6: set @ A] :
      ( ( ord_less @ ( set @ A ) @ A3 @ ( insert @ A @ X2 @ B6 ) )
      = ( ( ( member @ A @ X2 @ B6 )
         => ( ord_less @ ( set @ A ) @ A3 @ B6 ) )
        & ( ~ ( member @ A @ X2 @ B6 )
         => ( ( ( member @ A @ X2 @ A3 )
             => ( ord_less @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) @ B6 ) )
            & ( ~ ( member @ A @ X2 @ A3 )
             => ( ord_less_eq @ ( set @ A ) @ A3 @ B6 ) ) ) ) ) ) ).

% psubset_insert_iff
thf(fact_2466_real__of__nat__div__aux,axiom,
    ! [X2: nat,D2: nat] :
      ( ( divide_divide @ real @ ( semiring_1_of_nat @ real @ X2 ) @ ( semiring_1_of_nat @ real @ D2 ) )
      = ( plus_plus @ real @ ( semiring_1_of_nat @ real @ ( divide_divide @ nat @ X2 @ D2 ) ) @ ( divide_divide @ real @ ( semiring_1_of_nat @ real @ ( modulo_modulo @ nat @ X2 @ D2 ) ) @ ( semiring_1_of_nat @ real @ D2 ) ) ) ) ).

% real_of_nat_div_aux
thf(fact_2467_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
    ! [A: $tType] :
      ( ( unique1627219031080169319umeral @ A )
     => ! [C2: A,A2: A,B2: A] :
          ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
         => ( ( modulo_modulo @ A @ A2 @ ( times_times @ A @ B2 @ C2 ) )
            = ( plus_plus @ A @ ( times_times @ A @ B2 @ ( modulo_modulo @ A @ ( divide_divide @ A @ A2 @ B2 ) @ C2 ) ) @ ( modulo_modulo @ A @ A2 @ B2 ) ) ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_mult2_eq
thf(fact_2468_cong__exp__iff__simps_I7_J,axiom,
    ! [A: $tType] :
      ( ( unique1627219031080169319umeral @ A )
     => ! [Q2: num,N2: num] :
          ( ( ( modulo_modulo @ A @ ( numeral_numeral @ A @ one2 ) @ ( numeral_numeral @ A @ ( bit0 @ Q2 ) ) )
            = ( modulo_modulo @ A @ ( numeral_numeral @ A @ ( bit1 @ N2 ) ) @ ( numeral_numeral @ A @ ( bit0 @ Q2 ) ) ) )
          = ( ( modulo_modulo @ A @ ( numeral_numeral @ A @ N2 ) @ ( numeral_numeral @ A @ Q2 ) )
            = ( zero_zero @ A ) ) ) ) ).

% cong_exp_iff_simps(7)
thf(fact_2469_cong__exp__iff__simps_I11_J,axiom,
    ! [A: $tType] :
      ( ( unique1627219031080169319umeral @ A )
     => ! [M: num,Q2: num] :
          ( ( ( modulo_modulo @ A @ ( numeral_numeral @ A @ ( bit1 @ M ) ) @ ( numeral_numeral @ A @ ( bit0 @ Q2 ) ) )
            = ( modulo_modulo @ A @ ( numeral_numeral @ A @ one2 ) @ ( numeral_numeral @ A @ ( bit0 @ Q2 ) ) ) )
          = ( ( modulo_modulo @ A @ ( numeral_numeral @ A @ M ) @ ( numeral_numeral @ A @ Q2 ) )
            = ( zero_zero @ A ) ) ) ) ).

% cong_exp_iff_simps(11)
thf(fact_2470_Suc__mod__eq__add3__mod,axiom,
    ! [M: nat,N2: nat] :
      ( ( modulo_modulo @ nat @ ( suc @ ( suc @ ( suc @ M ) ) ) @ N2 )
      = ( modulo_modulo @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) @ M ) @ N2 ) ) ).

% Suc_mod_eq_add3_mod
thf(fact_2471_Suc__times__mod__eq,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less @ nat @ ( suc @ ( zero_zero @ nat ) ) @ M )
     => ( ( modulo_modulo @ nat @ ( suc @ ( times_times @ nat @ M @ N2 ) ) @ M )
        = ( one_one @ nat ) ) ) ).

% Suc_times_mod_eq
thf(fact_2472_lemma__NBseq__def2,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V822414075346904944vector @ B )
     => ! [X8: A > B] :
          ( ( ? [K7: real] :
                ( ( ord_less @ real @ ( zero_zero @ real ) @ K7 )
                & ! [N: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ B @ ( X8 @ N ) ) @ K7 ) ) )
          = ( ? [N8: nat] :
              ! [N: A] : ( ord_less @ real @ ( real_V7770717601297561774m_norm @ B @ ( X8 @ N ) ) @ ( semiring_1_of_nat @ real @ ( suc @ N8 ) ) ) ) ) ) ).

% lemma_NBseq_def2
thf(fact_2473_lemma__NBseq__def,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V822414075346904944vector @ B )
     => ! [X8: A > B] :
          ( ( ? [K7: real] :
                ( ( ord_less @ real @ ( zero_zero @ real ) @ K7 )
                & ! [N: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ B @ ( X8 @ N ) ) @ K7 ) ) )
          = ( ? [N8: nat] :
              ! [N: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ B @ ( X8 @ N ) ) @ ( semiring_1_of_nat @ real @ ( suc @ N8 ) ) ) ) ) ) ).

% lemma_NBseq_def
thf(fact_2474_insert__swap__set__eq,axiom,
    ! [A: $tType,I: nat,L: list @ A,X2: A] :
      ( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ L ) )
     => ( ( insert @ A @ ( nth @ A @ L @ I ) @ ( set2 @ A @ ( list_update @ A @ L @ I @ X2 ) ) )
        = ( insert @ A @ X2 @ ( set2 @ A @ L ) ) ) ) ).

% insert_swap_set_eq
thf(fact_2475_divmod__digit__0_I2_J,axiom,
    ! [A: $tType] :
      ( ( unique1627219031080169319umeral @ A )
     => ! [B2: A,A2: A] :
          ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
         => ( ( ord_less @ A @ ( modulo_modulo @ A @ A2 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) ) @ B2 )
           => ( ( modulo_modulo @ A @ A2 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) )
              = ( modulo_modulo @ A @ A2 @ B2 ) ) ) ) ) ).

% divmod_digit_0(2)
thf(fact_2476_bits__stable__imp__add__self,axiom,
    ! [A: $tType] :
      ( ( bit_semiring_bits @ A )
     => ! [A2: A] :
          ( ( ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
            = A2 )
         => ( ( plus_plus @ A @ A2 @ ( modulo_modulo @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) )
            = ( zero_zero @ A ) ) ) ) ).

% bits_stable_imp_add_self
thf(fact_2477_div__exp__mod__exp__eq,axiom,
    ! [A: $tType] :
      ( ( bit_semiring_bits @ A )
     => ! [A2: A,N2: nat,M: nat] :
          ( ( modulo_modulo @ A @ ( divide_divide @ A @ A2 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) )
          = ( divide_divide @ A @ ( modulo_modulo @ A @ A2 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( plus_plus @ nat @ N2 @ M ) ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) ) ) ) ).

% div_exp_mod_exp_eq
thf(fact_2478_power__mod__div,axiom,
    ! [X2: nat,N2: nat,M: nat] :
      ( ( divide_divide @ nat @ ( modulo_modulo @ nat @ X2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) )
      = ( modulo_modulo @ nat @ ( divide_divide @ nat @ X2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ N2 @ M ) ) ) ) ).

% power_mod_div
thf(fact_2479_verit__le__mono__div,axiom,
    ! [A3: nat,B6: nat,N2: nat] :
      ( ( ord_less @ nat @ A3 @ B6 )
     => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
       => ( ord_less_eq @ nat
          @ ( plus_plus @ nat @ ( divide_divide @ nat @ A3 @ N2 )
            @ ( if @ nat
              @ ( ( modulo_modulo @ nat @ B6 @ N2 )
                = ( zero_zero @ nat ) )
              @ ( one_one @ nat )
              @ ( zero_zero @ nat ) ) )
          @ ( divide_divide @ nat @ B6 @ N2 ) ) ) ) ).

% verit_le_mono_div
thf(fact_2480_divmod__digit__0_I1_J,axiom,
    ! [A: $tType] :
      ( ( unique1627219031080169319umeral @ A )
     => ! [B2: A,A2: A] :
          ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
         => ( ( ord_less @ A @ ( modulo_modulo @ A @ A2 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) ) @ B2 )
           => ( ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ A2 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) ) )
              = ( divide_divide @ A @ A2 @ B2 ) ) ) ) ) ).

% divmod_digit_0(1)
thf(fact_2481_mult__exp__mod__exp__eq,axiom,
    ! [A: $tType] :
      ( ( bit_semiring_bits @ A )
     => ! [M: nat,N2: nat,A2: A] :
          ( ( ord_less_eq @ nat @ M @ N2 )
         => ( ( modulo_modulo @ A @ ( times_times @ A @ A2 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) )
            = ( times_times @ A @ ( modulo_modulo @ A @ A2 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ N2 @ M ) ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) ) ) ) ) ).

% mult_exp_mod_exp_eq
thf(fact_2482_mod__double__modulus,axiom,
    ! [A: $tType] :
      ( ( unique1627219031080169319umeral @ A )
     => ! [M: A,X2: A] :
          ( ( ord_less @ A @ ( zero_zero @ A ) @ M )
         => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X2 )
           => ( ( ( modulo_modulo @ A @ X2 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) )
                = ( modulo_modulo @ A @ X2 @ M ) )
              | ( ( modulo_modulo @ A @ X2 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) )
                = ( plus_plus @ A @ ( modulo_modulo @ A @ X2 @ M ) @ M ) ) ) ) ) ) ).

% mod_double_modulus
thf(fact_2483_divmod__digit__1_I2_J,axiom,
    ! [A: $tType] :
      ( ( unique1627219031080169319umeral @ A )
     => ! [A2: A,B2: A] :
          ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
         => ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
           => ( ( ord_less_eq @ A @ B2 @ ( modulo_modulo @ A @ A2 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) ) )
             => ( ( minus_minus @ A @ ( modulo_modulo @ A @ A2 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) ) @ B2 )
                = ( modulo_modulo @ A @ A2 @ B2 ) ) ) ) ) ) ).

% divmod_digit_1(2)
thf(fact_2484_set__bit__Suc,axiom,
    ! [A: $tType] :
      ( ( bit_se359711467146920520ations @ A )
     => ! [N2: nat,A2: A] :
          ( ( bit_se5668285175392031749et_bit @ A @ ( suc @ N2 ) @ A2 )
          = ( plus_plus @ A @ ( modulo_modulo @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se5668285175392031749et_bit @ A @ N2 @ ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).

% set_bit_Suc
thf(fact_2485_divmod__digit__1_I1_J,axiom,
    ! [A: $tType] :
      ( ( unique1627219031080169319umeral @ A )
     => ! [A2: A,B2: A] :
          ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
         => ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
           => ( ( ord_less_eq @ A @ B2 @ ( modulo_modulo @ A @ A2 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) ) )
             => ( ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ A2 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) ) ) @ ( one_one @ A ) )
                = ( divide_divide @ A @ A2 @ B2 ) ) ) ) ) ) ).

% divmod_digit_1(1)
thf(fact_2486_norm__divide__numeral,axiom,
    ! [A: $tType] :
      ( ( real_V3459762299906320749_field @ A )
     => ! [A2: A,W: num] :
          ( ( real_V7770717601297561774m_norm @ A @ ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ W ) ) )
          = ( divide_divide @ real @ ( real_V7770717601297561774m_norm @ A @ A2 ) @ ( numeral_numeral @ real @ W ) ) ) ) ).

% norm_divide_numeral
thf(fact_2487_norm__mult__numeral2,axiom,
    ! [A: $tType] :
      ( ( real_V3459762299906320749_field @ A )
     => ! [A2: A,W: num] :
          ( ( real_V7770717601297561774m_norm @ A @ ( times_times @ A @ A2 @ ( numeral_numeral @ A @ W ) ) )
          = ( times_times @ real @ ( real_V7770717601297561774m_norm @ A @ A2 ) @ ( numeral_numeral @ real @ W ) ) ) ) ).

% norm_mult_numeral2
thf(fact_2488_norm__mult__numeral1,axiom,
    ! [A: $tType] :
      ( ( real_V3459762299906320749_field @ A )
     => ! [W: num,A2: A] :
          ( ( real_V7770717601297561774m_norm @ A @ ( times_times @ A @ ( numeral_numeral @ A @ W ) @ A2 ) )
          = ( times_times @ real @ ( numeral_numeral @ real @ W ) @ ( real_V7770717601297561774m_norm @ A @ A2 ) ) ) ) ).

% norm_mult_numeral1
thf(fact_2489_div__half__nat,axiom,
    ! [Y2: nat,X2: nat] :
      ( ( Y2
       != ( zero_zero @ nat ) )
     => ( ( product_Pair @ nat @ nat @ ( divide_divide @ nat @ X2 @ Y2 ) @ ( modulo_modulo @ nat @ X2 @ Y2 ) )
        = ( if @ ( product_prod @ nat @ nat ) @ ( ord_less_eq @ nat @ Y2 @ ( minus_minus @ nat @ X2 @ ( times_times @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( divide_divide @ nat @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ Y2 ) ) @ Y2 ) ) ) @ ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( divide_divide @ nat @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ Y2 ) ) @ ( one_one @ nat ) ) @ ( minus_minus @ nat @ ( minus_minus @ nat @ X2 @ ( times_times @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( divide_divide @ nat @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ Y2 ) ) @ Y2 ) ) @ Y2 ) ) @ ( product_Pair @ nat @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( divide_divide @ nat @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ Y2 ) ) @ ( minus_minus @ nat @ X2 @ ( times_times @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( divide_divide @ nat @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ Y2 ) ) @ Y2 ) ) ) ) ) ) ).

% div_half_nat
thf(fact_2490_norm__le__zero__iff,axiom,
    ! [A: $tType] :
      ( ( real_V822414075346904944vector @ A )
     => ! [X2: A] :
          ( ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ X2 ) @ ( zero_zero @ real ) )
          = ( X2
            = ( zero_zero @ A ) ) ) ) ).

% norm_le_zero_iff
thf(fact_2491_norm__numeral,axiom,
    ! [A: $tType] :
      ( ( real_V2822296259951069270ebra_1 @ A )
     => ! [W: num] :
          ( ( real_V7770717601297561774m_norm @ A @ ( numeral_numeral @ A @ W ) )
          = ( numeral_numeral @ real @ W ) ) ) ).

% norm_numeral
thf(fact_2492_foldr0,axiom,
    ! [Xs2: list @ real,C2: real,D2: real] :
      ( ( foldr @ real @ real @ ( plus_plus @ real ) @ Xs2 @ ( plus_plus @ real @ C2 @ D2 ) )
      = ( plus_plus @ real @ ( foldr @ real @ real @ ( plus_plus @ real ) @ Xs2 @ D2 ) @ C2 ) ) ).

% foldr0
thf(fact_2493_foldr__same,axiom,
    ! [Xs2: list @ real,Y2: real] :
      ( ! [X3: real,Y4: real] :
          ( ( member @ real @ X3 @ ( set2 @ real @ Xs2 ) )
         => ( ( member @ real @ Y4 @ ( set2 @ real @ Xs2 ) )
           => ( X3 = Y4 ) ) )
     => ( ! [X3: real] :
            ( ( member @ real @ X3 @ ( set2 @ real @ Xs2 ) )
           => ( X3 = Y2 ) )
       => ( ( foldr @ real @ real @ ( plus_plus @ real ) @ Xs2 @ ( zero_zero @ real ) )
          = ( times_times @ real @ ( semiring_1_of_nat @ real @ ( size_size @ ( list @ real ) @ Xs2 ) ) @ Y2 ) ) ) ) ).

% foldr_same
thf(fact_2494_norm__one,axiom,
    ! [A: $tType] :
      ( ( real_V2822296259951069270ebra_1 @ A )
     => ( ( real_V7770717601297561774m_norm @ A @ ( one_one @ A ) )
        = ( one_one @ real ) ) ) ).

% norm_one
thf(fact_2495_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_2496_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_2497_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 @ one2 ) ) @ ( modulo_modulo @ int @ ( numeral_numeral @ int @ V ) @ ( numeral_numeral @ int @ W ) ) ) ) ).

% zmod_numeral_Bit0
thf(fact_2498_one__mod__exp__eq__one,axiom,
    ! [N2: nat] :
      ( ( modulo_modulo @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N2 ) ) )
      = ( one_one @ int ) ) ).

% one_mod_exp_eq_one
thf(fact_2499_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 @ one2 ) ) @ ( modulo_modulo @ int @ ( numeral_numeral @ int @ V ) @ ( numeral_numeral @ int @ W ) ) ) @ ( one_one @ int ) ) ) ).

% zmod_numeral_Bit1
thf(fact_2500_mod__plus__cong,axiom,
    ! [B2: int,B4: int,X2: int,X7: int,Y2: int,Y6: int,Z7: int] :
      ( ( B2 = B4 )
     => ( ( ( modulo_modulo @ int @ X2 @ B4 )
          = ( modulo_modulo @ int @ X7 @ B4 ) )
       => ( ( ( modulo_modulo @ int @ Y2 @ B4 )
            = ( modulo_modulo @ int @ Y6 @ B4 ) )
         => ( ( ( plus_plus @ int @ X7 @ Y6 )
              = Z7 )
           => ( ( modulo_modulo @ int @ ( plus_plus @ int @ X2 @ Y2 ) @ B2 )
              = ( modulo_modulo @ int @ Z7 @ B4 ) ) ) ) ) ) ).

% mod_plus_cong
thf(fact_2501_zmod__helper,axiom,
    ! [N2: int,M: int,K: int,A2: int] :
      ( ( ( modulo_modulo @ int @ N2 @ M )
        = K )
     => ( ( modulo_modulo @ int @ ( plus_plus @ int @ N2 @ A2 ) @ M )
        = ( modulo_modulo @ int @ ( plus_plus @ int @ K @ A2 ) @ M ) ) ) ).

% zmod_helper
thf(fact_2502_Word_Omod__minus__cong,axiom,
    ! [B2: int,B4: int,X2: int,X7: int,Y2: int,Y6: int,Z7: int] :
      ( ( B2 = B4 )
     => ( ( ( modulo_modulo @ int @ X2 @ B4 )
          = ( modulo_modulo @ int @ X7 @ B4 ) )
       => ( ( ( modulo_modulo @ int @ Y2 @ B4 )
            = ( modulo_modulo @ int @ Y6 @ B4 ) )
         => ( ( ( minus_minus @ int @ X7 @ Y6 )
              = Z7 )
           => ( ( modulo_modulo @ int @ ( minus_minus @ int @ X2 @ Y2 ) @ B2 )
              = ( modulo_modulo @ int @ Z7 @ B4 ) ) ) ) ) ) ).

% Word.mod_minus_cong
thf(fact_2503_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_2504_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_2505_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_2506_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_2507_zmod__eq__0__iff,axiom,
    ! [M: int,D2: int] :
      ( ( ( modulo_modulo @ int @ M @ D2 )
        = ( zero_zero @ int ) )
      = ( ? [Q5: int] :
            ( M
            = ( times_times @ int @ D2 @ Q5 ) ) ) ) ).

% zmod_eq_0_iff
thf(fact_2508_int__mod__ge,axiom,
    ! [A2: int,N2: int] :
      ( ( ord_less @ int @ A2 @ N2 )
     => ( ( ord_less @ int @ ( zero_zero @ int ) @ N2 )
       => ( ord_less_eq @ int @ A2 @ ( modulo_modulo @ int @ A2 @ N2 ) ) ) ) ).

% int_mod_ge
thf(fact_2509_neg__mod__conj,axiom,
    ! [B2: int,A2: int] :
      ( ( ord_less @ int @ B2 @ ( zero_zero @ int ) )
     => ( ( ord_less_eq @ int @ ( modulo_modulo @ int @ A2 @ B2 ) @ ( zero_zero @ int ) )
        & ( ord_less @ int @ B2 @ ( modulo_modulo @ int @ A2 @ B2 ) ) ) ) ).

% neg_mod_conj
thf(fact_2510_pos__mod__conj,axiom,
    ! [B2: int,A2: int] :
      ( ( ord_less @ int @ ( zero_zero @ int ) @ B2 )
     => ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( modulo_modulo @ int @ A2 @ B2 ) )
        & ( ord_less @ int @ ( modulo_modulo @ int @ A2 @ B2 ) @ B2 ) ) ) ).

% pos_mod_conj
thf(fact_2511_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_2512_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_2513_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_2514_int__mod__eq,axiom,
    ! [B2: int,N2: int,A2: int] :
      ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ B2 )
     => ( ( ord_less @ int @ B2 @ N2 )
       => ( ( ( modulo_modulo @ int @ A2 @ N2 )
            = ( modulo_modulo @ int @ B2 @ N2 ) )
         => ( ( modulo_modulo @ int @ A2 @ N2 )
            = B2 ) ) ) ) ).

% int_mod_eq
thf(fact_2515_int__mod__lem,axiom,
    ! [N2: int,B2: int] :
      ( ( ord_less @ int @ ( zero_zero @ int ) @ N2 )
     => ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ B2 )
          & ( ord_less @ int @ B2 @ N2 ) )
        = ( ( modulo_modulo @ int @ B2 @ N2 )
          = B2 ) ) ) ).

% int_mod_lem
thf(fact_2516_int__mod__le_H,axiom,
    ! [B2: int,N2: int] :
      ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( minus_minus @ int @ B2 @ N2 ) )
     => ( ord_less_eq @ int @ ( modulo_modulo @ int @ B2 @ N2 ) @ ( minus_minus @ int @ B2 @ N2 ) ) ) ).

% int_mod_le'
thf(fact_2517_nonneg__mod__div,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ A2 )
     => ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ B2 )
       => ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( modulo_modulo @ int @ A2 @ B2 ) )
          & ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( divide_divide @ int @ A2 @ B2 ) ) ) ) ) ).

% nonneg_mod_div
thf(fact_2518_zdiv__mono__strict,axiom,
    ! [A3: int,B6: int,N2: int] :
      ( ( ord_less @ int @ A3 @ B6 )
     => ( ( ord_less @ int @ ( zero_zero @ int ) @ N2 )
       => ( ( ( modulo_modulo @ int @ A3 @ N2 )
            = ( zero_zero @ int ) )
         => ( ( ( modulo_modulo @ int @ B6 @ N2 )
              = ( zero_zero @ int ) )
           => ( ord_less @ int @ ( divide_divide @ int @ A3 @ N2 ) @ ( divide_divide @ int @ B6 @ N2 ) ) ) ) ) ) ).

% zdiv_mono_strict
thf(fact_2519_div__mod__decomp__int,axiom,
    ! [A3: int,N2: int] :
      ( A3
      = ( plus_plus @ int @ ( times_times @ int @ ( divide_divide @ int @ A3 @ N2 ) @ N2 ) @ ( modulo_modulo @ int @ A3 @ N2 ) ) ) ).

% div_mod_decomp_int
thf(fact_2520_mod__div__equality__div__eq,axiom,
    ! [A2: int,B2: int] :
      ( ( times_times @ int @ ( divide_divide @ int @ A2 @ B2 ) @ B2 )
      = ( minus_minus @ int @ A2 @ ( modulo_modulo @ int @ A2 @ B2 ) ) ) ).

% mod_div_equality_div_eq
thf(fact_2521_word__mod__div__equality,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [N2: word @ A,B2: word @ A] :
          ( ( plus_plus @ ( word @ A ) @ ( times_times @ ( word @ A ) @ ( divide_divide @ ( word @ A ) @ N2 @ B2 ) @ B2 ) @ ( modulo_modulo @ ( word @ A ) @ N2 @ B2 ) )
          = N2 ) ) ).

% word_mod_div_equality
thf(fact_2522_nat__mod__as__int,axiom,
    ( ( modulo_modulo @ nat )
    = ( ^ [A5: nat,B5: nat] : ( nat2 @ ( modulo_modulo @ int @ ( semiring_1_of_nat @ int @ A5 ) @ ( semiring_1_of_nat @ int @ B5 ) ) ) ) ) ).

% nat_mod_as_int
thf(fact_2523_pos__mod__bound2,axiom,
    ! [A2: int] : ( ord_less @ int @ ( modulo_modulo @ int @ A2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ).

% pos_mod_bound2
thf(fact_2524_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_2525_int__mod__ge_H,axiom,
    ! [B2: int,N2: int] :
      ( ( ord_less @ int @ B2 @ ( zero_zero @ int ) )
     => ( ( ord_less @ int @ ( zero_zero @ int ) @ N2 )
       => ( ord_less_eq @ int @ ( plus_plus @ int @ B2 @ N2 ) @ ( modulo_modulo @ int @ B2 @ N2 ) ) ) ) ).

% int_mod_ge'
thf(fact_2526_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_2527_nat__mod__distrib,axiom,
    ! [X2: int,Y2: int] :
      ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X2 )
     => ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y2 )
       => ( ( nat2 @ ( modulo_modulo @ int @ X2 @ Y2 ) )
          = ( modulo_modulo @ nat @ ( nat2 @ X2 ) @ ( nat2 @ Y2 ) ) ) ) ) ).

% nat_mod_distrib
thf(fact_2528_real__of__int__div__aux,axiom,
    ! [X2: int,D2: int] :
      ( ( divide_divide @ real @ ( ring_1_of_int @ real @ X2 ) @ ( ring_1_of_int @ real @ D2 ) )
      = ( plus_plus @ real @ ( ring_1_of_int @ real @ ( divide_divide @ int @ X2 @ D2 ) ) @ ( divide_divide @ real @ ( ring_1_of_int @ real @ ( modulo_modulo @ int @ X2 @ D2 ) ) @ ( ring_1_of_int @ real @ D2 ) ) ) ) ).

% real_of_int_div_aux
thf(fact_2529_atLeastAtMostPlus1__int__conv,axiom,
    ! [M: int,N2: int] :
      ( ( ord_less_eq @ int @ M @ ( plus_plus @ int @ ( one_one @ int ) @ N2 ) )
     => ( ( set_or1337092689740270186AtMost @ int @ M @ ( plus_plus @ int @ ( one_one @ int ) @ N2 ) )
        = ( insert @ int @ ( plus_plus @ int @ ( one_one @ int ) @ N2 ) @ ( set_or1337092689740270186AtMost @ int @ M @ N2 ) ) ) ) ).

% atLeastAtMostPlus1_int_conv
thf(fact_2530_simp__from__to,axiom,
    ( ( set_or1337092689740270186AtMost @ int )
    = ( ^ [I2: int,J3: int] : ( if @ ( set @ int ) @ ( ord_less @ int @ J3 @ I2 ) @ ( bot_bot @ ( set @ int ) ) @ ( insert @ int @ I2 @ ( set_or1337092689740270186AtMost @ int @ ( plus_plus @ int @ I2 @ ( one_one @ int ) ) @ J3 ) ) ) ) ) ).

% simp_from_to
thf(fact_2531_pos__mod__sign2,axiom,
    ! [A2: int] : ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( modulo_modulo @ int @ A2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ).

% pos_mod_sign2
thf(fact_2532_mod__2__neq__1__eq__eq__0,axiom,
    ! [K: int] :
      ( ( ( modulo_modulo @ int @ K @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) )
       != ( one_one @ int ) )
      = ( ( modulo_modulo @ int @ K @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) )
        = ( zero_zero @ int ) ) ) ).

% mod_2_neq_1_eq_eq_0
thf(fact_2533_nmod2,axiom,
    ! [N2: int] :
      ( ( ( modulo_modulo @ int @ N2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) )
        = ( zero_zero @ int ) )
      | ( ( modulo_modulo @ int @ N2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) )
        = ( one_one @ int ) ) ) ).

% nmod2
thf(fact_2534_mod__exp__less__eq__exp,axiom,
    ! [A2: int,N2: nat] : ( ord_less @ int @ ( modulo_modulo @ int @ A2 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N2 ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N2 ) ) ).

% mod_exp_less_eq_exp
thf(fact_2535_mod__power__lem,axiom,
    ! [A2: int,M: nat,N2: nat] :
      ( ( ord_less @ int @ ( one_one @ int ) @ A2 )
     => ( ( ( ord_less_eq @ nat @ M @ N2 )
         => ( ( modulo_modulo @ int @ ( power_power @ int @ A2 @ N2 ) @ ( power_power @ int @ A2 @ M ) )
            = ( zero_zero @ int ) ) )
        & ( ~ ( ord_less_eq @ nat @ M @ N2 )
         => ( ( modulo_modulo @ int @ ( power_power @ int @ A2 @ N2 ) @ ( power_power @ int @ A2 @ M ) )
            = ( power_power @ int @ A2 @ N2 ) ) ) ) ) ).

% mod_power_lem
thf(fact_2536_split__zmod,axiom,
    ! [P: int > $o,N2: int,K: int] :
      ( ( P @ ( modulo_modulo @ int @ N2 @ K ) )
      = ( ( ( K
            = ( zero_zero @ int ) )
         => ( P @ N2 ) )
        & ( ( ord_less @ int @ ( zero_zero @ int ) @ K )
         => ! [I2: int,J3: int] :
              ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ J3 )
                & ( ord_less @ int @ J3 @ K )
                & ( N2
                  = ( 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 ) )
                & ( N2
                  = ( plus_plus @ int @ ( times_times @ int @ K @ I2 ) @ J3 ) ) )
             => ( P @ J3 ) ) ) ) ) ).

% split_zmod
thf(fact_2537_int__mod__neg__eq,axiom,
    ! [A2: int,B2: int,Q2: int,R4: int] :
      ( ( A2
        = ( plus_plus @ int @ ( times_times @ int @ B2 @ Q2 ) @ R4 ) )
     => ( ( ord_less_eq @ int @ R4 @ ( zero_zero @ int ) )
       => ( ( ord_less @ int @ B2 @ R4 )
         => ( ( modulo_modulo @ int @ A2 @ B2 )
            = R4 ) ) ) ) ).

% int_mod_neg_eq
thf(fact_2538_int__mod__pos__eq,axiom,
    ! [A2: int,B2: int,Q2: int,R4: int] :
      ( ( A2
        = ( plus_plus @ int @ ( times_times @ int @ B2 @ Q2 ) @ R4 ) )
     => ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ R4 )
       => ( ( ord_less @ int @ R4 @ B2 )
         => ( ( modulo_modulo @ int @ A2 @ B2 )
            = R4 ) ) ) ) ).

% int_mod_pos_eq
thf(fact_2539_mod__sub__if__z,axiom,
    ! [X2: int,Z: int,Y2: int] :
      ( ( ord_less @ int @ X2 @ Z )
     => ( ( ord_less @ int @ Y2 @ Z )
       => ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y2 )
         => ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X2 )
           => ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z )
             => ( ( ( ord_less_eq @ int @ Y2 @ X2 )
                 => ( ( modulo_modulo @ int @ ( minus_minus @ int @ X2 @ Y2 ) @ Z )
                    = ( minus_minus @ int @ X2 @ Y2 ) ) )
                & ( ~ ( ord_less_eq @ int @ Y2 @ X2 )
                 => ( ( modulo_modulo @ int @ ( minus_minus @ int @ X2 @ Y2 ) @ Z )
                    = ( plus_plus @ int @ ( minus_minus @ int @ X2 @ Y2 ) @ Z ) ) ) ) ) ) ) ) ) ).

% mod_sub_if_z
thf(fact_2540_mod__add__if__z,axiom,
    ! [X2: int,Z: int,Y2: int] :
      ( ( ord_less @ int @ X2 @ Z )
     => ( ( ord_less @ int @ Y2 @ Z )
       => ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y2 )
         => ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X2 )
           => ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z )
             => ( ( ( ord_less @ int @ ( plus_plus @ int @ X2 @ Y2 ) @ Z )
                 => ( ( modulo_modulo @ int @ ( plus_plus @ int @ X2 @ Y2 ) @ Z )
                    = ( plus_plus @ int @ X2 @ Y2 ) ) )
                & ( ~ ( ord_less @ int @ ( plus_plus @ int @ X2 @ Y2 ) @ Z )
                 => ( ( modulo_modulo @ int @ ( plus_plus @ int @ X2 @ Y2 ) @ Z )
                    = ( minus_minus @ int @ ( plus_plus @ int @ X2 @ Y2 ) @ Z ) ) ) ) ) ) ) ) ) ).

% mod_add_if_z
thf(fact_2541_zmod__zmult2__eq,axiom,
    ! [C2: int,A2: int,B2: int] :
      ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ C2 )
     => ( ( modulo_modulo @ int @ A2 @ ( times_times @ int @ B2 @ C2 ) )
        = ( plus_plus @ int @ ( times_times @ int @ B2 @ ( modulo_modulo @ int @ ( divide_divide @ int @ A2 @ B2 ) @ C2 ) ) @ ( modulo_modulo @ int @ A2 @ B2 ) ) ) ) ).

% zmod_zmult2_eq
thf(fact_2542_axxmod2,axiom,
    ! [X2: int] :
      ( ( ( modulo_modulo @ int @ ( plus_plus @ int @ ( plus_plus @ int @ ( one_one @ int ) @ X2 ) @ X2 ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) )
        = ( one_one @ int ) )
      & ( ( modulo_modulo @ int @ ( plus_plus @ int @ ( plus_plus @ int @ ( zero_zero @ int ) @ X2 ) @ X2 ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) )
        = ( zero_zero @ int ) ) ) ).

% axxmod2
thf(fact_2543_z1pmod2,axiom,
    ! [B2: int] :
      ( ( modulo_modulo @ int @ ( plus_plus @ int @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ B2 ) @ ( one_one @ int ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) )
      = ( one_one @ int ) ) ).

% z1pmod2
thf(fact_2544_verit__le__mono__div__int,axiom,
    ! [A3: int,B6: int,N2: int] :
      ( ( ord_less @ int @ A3 @ B6 )
     => ( ( ord_less @ int @ ( zero_zero @ int ) @ N2 )
       => ( ord_less_eq @ int
          @ ( plus_plus @ int @ ( divide_divide @ int @ A3 @ N2 )
            @ ( if @ int
              @ ( ( modulo_modulo @ int @ B6 @ N2 )
                = ( zero_zero @ int ) )
              @ ( one_one @ int )
              @ ( zero_zero @ int ) ) )
          @ ( divide_divide @ int @ B6 @ N2 ) ) ) ) ).

% verit_le_mono_div_int
thf(fact_2545_split__neg__lemma,axiom,
    ! [K: int,P: int > int > $o,N2: int] :
      ( ( ord_less @ int @ K @ ( zero_zero @ int ) )
     => ( ( P @ ( divide_divide @ int @ N2 @ K ) @ ( modulo_modulo @ int @ N2 @ K ) )
        = ( ! [I2: int,J3: int] :
              ( ( ( ord_less @ int @ K @ J3 )
                & ( ord_less_eq @ int @ J3 @ ( zero_zero @ int ) )
                & ( N2
                  = ( plus_plus @ int @ ( times_times @ int @ K @ I2 ) @ J3 ) ) )
             => ( P @ I2 @ J3 ) ) ) ) ) ).

% split_neg_lemma
thf(fact_2546_split__pos__lemma,axiom,
    ! [K: int,P: int > int > $o,N2: int] :
      ( ( ord_less @ int @ ( zero_zero @ int ) @ K )
     => ( ( P @ ( divide_divide @ int @ N2 @ K ) @ ( modulo_modulo @ int @ N2 @ K ) )
        = ( ! [I2: int,J3: int] :
              ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ J3 )
                & ( ord_less @ int @ J3 @ K )
                & ( N2
                  = ( plus_plus @ int @ ( times_times @ int @ K @ I2 ) @ J3 ) ) )
             => ( P @ I2 @ J3 ) ) ) ) ) ).

% split_pos_lemma
thf(fact_2547_norm__minus__commute,axiom,
    ! [A: $tType] :
      ( ( real_V822414075346904944vector @ A )
     => ! [A2: A,B2: A] :
          ( ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ A2 @ B2 ) )
          = ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ B2 @ A2 ) ) ) ) ).

% norm_minus_commute
thf(fact_2548_p1mod22k_H,axiom,
    ! [B2: int,N2: nat] :
      ( ( modulo_modulo @ int @ ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ B2 ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N2 ) ) )
      = ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( modulo_modulo @ int @ B2 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N2 ) ) ) ) ) ).

% p1mod22k'
thf(fact_2549_p1mod22k,axiom,
    ! [B2: int,N2: nat] :
      ( ( modulo_modulo @ int @ ( plus_plus @ int @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ B2 ) @ ( one_one @ int ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N2 ) ) )
      = ( plus_plus @ int @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( modulo_modulo @ int @ B2 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N2 ) ) ) @ ( one_one @ int ) ) ) ).

% p1mod22k
thf(fact_2550_pos__zmod__mult__2,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ A2 )
     => ( ( modulo_modulo @ int @ ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ B2 ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ A2 ) )
        = ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( modulo_modulo @ int @ B2 @ A2 ) ) ) ) ) ).

% pos_zmod_mult_2
thf(fact_2551_sb__inc__lem,axiom,
    ! [A2: int,K: nat] :
      ( ( ord_less @ int @ ( plus_plus @ int @ A2 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K ) ) @ ( zero_zero @ int ) )
     => ( ord_less_eq @ int @ ( plus_plus @ int @ ( plus_plus @ int @ A2 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( suc @ K ) ) ) @ ( modulo_modulo @ int @ ( plus_plus @ int @ A2 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( suc @ K ) ) ) ) ) ).

% sb_inc_lem
thf(fact_2552_neg__zmod__mult__2,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_eq @ int @ A2 @ ( zero_zero @ int ) )
     => ( ( modulo_modulo @ int @ ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ B2 ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ A2 ) )
        = ( minus_minus @ int @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( modulo_modulo @ int @ ( plus_plus @ int @ B2 @ ( one_one @ int ) ) @ A2 ) ) @ ( one_one @ int ) ) ) ) ).

% neg_zmod_mult_2
thf(fact_2553_norm__mult,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra @ A )
     => ! [X2: A,Y2: A] :
          ( ( real_V7770717601297561774m_norm @ A @ ( times_times @ A @ X2 @ Y2 ) )
          = ( times_times @ real @ ( real_V7770717601297561774m_norm @ A @ X2 ) @ ( real_V7770717601297561774m_norm @ A @ Y2 ) ) ) ) ).

% norm_mult
thf(fact_2554_norm__ge__zero,axiom,
    ! [A: $tType] :
      ( ( real_V822414075346904944vector @ A )
     => ! [X2: A] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( real_V7770717601297561774m_norm @ A @ X2 ) ) ) ).

% norm_ge_zero
thf(fact_2555_norm__divide,axiom,
    ! [A: $tType] :
      ( ( real_V3459762299906320749_field @ A )
     => ! [A2: A,B2: A] :
          ( ( real_V7770717601297561774m_norm @ A @ ( divide_divide @ A @ A2 @ B2 ) )
          = ( divide_divide @ real @ ( real_V7770717601297561774m_norm @ A @ A2 ) @ ( real_V7770717601297561774m_norm @ A @ B2 ) ) ) ) ).

% norm_divide
thf(fact_2556_norm__power,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra @ A )
     => ! [X2: A,N2: nat] :
          ( ( real_V7770717601297561774m_norm @ A @ ( power_power @ A @ X2 @ N2 ) )
          = ( power_power @ real @ ( real_V7770717601297561774m_norm @ A @ X2 ) @ N2 ) ) ) ).

% norm_power
thf(fact_2557_power__eq__imp__eq__norm,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra @ A )
     => ! [W: A,N2: nat,Z: A] :
          ( ( ( power_power @ A @ W @ N2 )
            = ( power_power @ A @ Z @ N2 ) )
         => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
           => ( ( real_V7770717601297561774m_norm @ A @ W )
              = ( real_V7770717601297561774m_norm @ A @ Z ) ) ) ) ) ).

% power_eq_imp_eq_norm
thf(fact_2558_nonzero__norm__divide,axiom,
    ! [A: $tType] :
      ( ( real_V3459762299906320749_field @ A )
     => ! [B2: A,A2: A] :
          ( ( B2
           != ( zero_zero @ A ) )
         => ( ( real_V7770717601297561774m_norm @ A @ ( divide_divide @ A @ A2 @ B2 ) )
            = ( divide_divide @ real @ ( real_V7770717601297561774m_norm @ A @ A2 ) @ ( real_V7770717601297561774m_norm @ A @ B2 ) ) ) ) ) ).

% nonzero_norm_divide
thf(fact_2559_norm__mult__less,axiom,
    ! [A: $tType] :
      ( ( real_V4412858255891104859lgebra @ A )
     => ! [X2: A,R4: real,Y2: A,S4: real] :
          ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X2 ) @ R4 )
         => ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ Y2 ) @ S4 )
           => ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( times_times @ A @ X2 @ Y2 ) ) @ ( times_times @ real @ R4 @ S4 ) ) ) ) ) ).

% norm_mult_less
thf(fact_2560_norm__mult__ineq,axiom,
    ! [A: $tType] :
      ( ( real_V4412858255891104859lgebra @ A )
     => ! [X2: A,Y2: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( times_times @ A @ X2 @ Y2 ) ) @ ( times_times @ real @ ( real_V7770717601297561774m_norm @ A @ X2 ) @ ( real_V7770717601297561774m_norm @ A @ Y2 ) ) ) ) ).

% norm_mult_ineq
thf(fact_2561_norm__add__less,axiom,
    ! [A: $tType] :
      ( ( real_V822414075346904944vector @ A )
     => ! [X2: A,R4: real,Y2: A,S4: real] :
          ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X2 ) @ R4 )
         => ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ Y2 ) @ S4 )
           => ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( plus_plus @ A @ X2 @ Y2 ) ) @ ( plus_plus @ real @ R4 @ S4 ) ) ) ) ) ).

% norm_add_less
thf(fact_2562_norm__triangle__lt,axiom,
    ! [A: $tType] :
      ( ( real_V822414075346904944vector @ A )
     => ! [X2: A,Y2: A,E3: real] :
          ( ( ord_less @ real @ ( plus_plus @ real @ ( real_V7770717601297561774m_norm @ A @ X2 ) @ ( real_V7770717601297561774m_norm @ A @ Y2 ) ) @ E3 )
         => ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( plus_plus @ A @ X2 @ Y2 ) ) @ E3 ) ) ) ).

% norm_triangle_lt
thf(fact_2563_norm__add__leD,axiom,
    ! [A: $tType] :
      ( ( real_V822414075346904944vector @ A )
     => ! [A2: A,B2: A,C2: real] :
          ( ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( plus_plus @ A @ A2 @ B2 ) ) @ C2 )
         => ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ B2 ) @ ( plus_plus @ real @ ( real_V7770717601297561774m_norm @ A @ A2 ) @ C2 ) ) ) ) ).

% norm_add_leD
thf(fact_2564_norm__triangle__le,axiom,
    ! [A: $tType] :
      ( ( real_V822414075346904944vector @ A )
     => ! [X2: A,Y2: A,E3: real] :
          ( ( ord_less_eq @ real @ ( plus_plus @ real @ ( real_V7770717601297561774m_norm @ A @ X2 ) @ ( real_V7770717601297561774m_norm @ A @ Y2 ) ) @ E3 )
         => ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( plus_plus @ A @ X2 @ Y2 ) ) @ E3 ) ) ) ).

% norm_triangle_le
thf(fact_2565_norm__triangle__ineq,axiom,
    ! [A: $tType] :
      ( ( real_V822414075346904944vector @ A )
     => ! [X2: A,Y2: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( plus_plus @ A @ X2 @ Y2 ) ) @ ( plus_plus @ real @ ( real_V7770717601297561774m_norm @ A @ X2 ) @ ( real_V7770717601297561774m_norm @ A @ Y2 ) ) ) ) ).

% norm_triangle_ineq
thf(fact_2566_norm__triangle__mono,axiom,
    ! [A: $tType] :
      ( ( real_V822414075346904944vector @ A )
     => ! [A2: A,R4: real,B2: A,S4: real] :
          ( ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ A2 ) @ R4 )
         => ( ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ B2 ) @ S4 )
           => ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( plus_plus @ A @ A2 @ B2 ) ) @ ( plus_plus @ real @ R4 @ S4 ) ) ) ) ) ).

% norm_triangle_mono
thf(fact_2567_norm__diff__triangle__less,axiom,
    ! [A: $tType] :
      ( ( real_V822414075346904944vector @ A )
     => ! [X2: A,Y2: A,E1: real,Z: A,E22: real] :
          ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ X2 @ Y2 ) ) @ E1 )
         => ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ Y2 @ Z ) ) @ E22 )
           => ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ X2 @ Z ) ) @ ( plus_plus @ real @ E1 @ E22 ) ) ) ) ) ).

% norm_diff_triangle_less
thf(fact_2568_norm__power__ineq,axiom,
    ! [A: $tType] :
      ( ( real_V2822296259951069270ebra_1 @ A )
     => ! [X2: A,N2: nat] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( power_power @ A @ X2 @ N2 ) ) @ ( power_power @ real @ ( real_V7770717601297561774m_norm @ A @ X2 ) @ N2 ) ) ) ).

% norm_power_ineq
thf(fact_2569_norm__triangle__le__diff,axiom,
    ! [A: $tType] :
      ( ( real_V822414075346904944vector @ A )
     => ! [X2: A,Y2: A,E3: real] :
          ( ( ord_less_eq @ real @ ( plus_plus @ real @ ( real_V7770717601297561774m_norm @ A @ X2 ) @ ( real_V7770717601297561774m_norm @ A @ Y2 ) ) @ E3 )
         => ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ X2 @ Y2 ) ) @ E3 ) ) ) ).

% norm_triangle_le_diff
thf(fact_2570_norm__diff__triangle__le,axiom,
    ! [A: $tType] :
      ( ( real_V822414075346904944vector @ A )
     => ! [X2: A,Y2: A,E1: real,Z: A,E22: real] :
          ( ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ X2 @ Y2 ) ) @ E1 )
         => ( ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ Y2 @ Z ) ) @ E22 )
           => ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ X2 @ Z ) ) @ ( plus_plus @ real @ E1 @ E22 ) ) ) ) ) ).

% norm_diff_triangle_le
thf(fact_2571_norm__triangle__ineq4,axiom,
    ! [A: $tType] :
      ( ( real_V822414075346904944vector @ A )
     => ! [A2: A,B2: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ A2 @ B2 ) ) @ ( plus_plus @ real @ ( real_V7770717601297561774m_norm @ A @ A2 ) @ ( real_V7770717601297561774m_norm @ A @ B2 ) ) ) ) ).

% norm_triangle_ineq4
thf(fact_2572_norm__triangle__sub,axiom,
    ! [A: $tType] :
      ( ( real_V822414075346904944vector @ A )
     => ! [X2: A,Y2: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ X2 ) @ ( plus_plus @ real @ ( real_V7770717601297561774m_norm @ A @ Y2 ) @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ X2 @ Y2 ) ) ) ) ) ).

% norm_triangle_sub
thf(fact_2573_norm__diff__ineq,axiom,
    ! [A: $tType] :
      ( ( real_V822414075346904944vector @ A )
     => ! [A2: A,B2: A] : ( ord_less_eq @ real @ ( minus_minus @ real @ ( real_V7770717601297561774m_norm @ A @ A2 ) @ ( real_V7770717601297561774m_norm @ A @ B2 ) ) @ ( real_V7770717601297561774m_norm @ A @ ( plus_plus @ A @ A2 @ B2 ) ) ) ) ).

% norm_diff_ineq
thf(fact_2574_norm__triangle__ineq2,axiom,
    ! [A: $tType] :
      ( ( real_V822414075346904944vector @ A )
     => ! [A2: A,B2: A] : ( ord_less_eq @ real @ ( minus_minus @ real @ ( real_V7770717601297561774m_norm @ A @ A2 ) @ ( real_V7770717601297561774m_norm @ A @ B2 ) ) @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ A2 @ B2 ) ) ) ) ).

% norm_triangle_ineq2
thf(fact_2575_power__eq__1__iff,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra @ A )
     => ! [W: A,N2: nat] :
          ( ( ( power_power @ A @ W @ N2 )
            = ( one_one @ A ) )
         => ( ( ( real_V7770717601297561774m_norm @ A @ W )
              = ( one_one @ real ) )
            | ( N2
              = ( zero_zero @ nat ) ) ) ) ) ).

% power_eq_1_iff
thf(fact_2576_norm__diff__triangle__ineq,axiom,
    ! [A: $tType] :
      ( ( real_V822414075346904944vector @ A )
     => ! [A2: A,B2: A,C2: A,D2: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ ( plus_plus @ A @ C2 @ D2 ) ) ) @ ( plus_plus @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ A2 @ C2 ) ) @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ B2 @ D2 ) ) ) ) ) ).

% norm_diff_triangle_ineq
thf(fact_2577_square__norm__one,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra @ A )
     => ! [X2: A] :
          ( ( ( power_power @ A @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
            = ( one_one @ A ) )
         => ( ( real_V7770717601297561774m_norm @ A @ X2 )
            = ( one_one @ real ) ) ) ) ).

% square_norm_one
thf(fact_2578_norm__power__diff,axiom,
    ! [A: $tType] :
      ( ( ( comm_monoid_mult @ A )
        & ( real_V2822296259951069270ebra_1 @ A ) )
     => ! [Z: A,W: A,M: nat] :
          ( ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ Z ) @ ( one_one @ real ) )
         => ( ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ W ) @ ( one_one @ real ) )
           => ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ ( power_power @ A @ Z @ M ) @ ( power_power @ A @ W @ M ) ) ) @ ( times_times @ real @ ( semiring_1_of_nat @ real @ M ) @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ Z @ W ) ) ) ) ) ) ) ).

% norm_power_diff
thf(fact_2579_mod__exhaust__less__4,axiom,
    ! [M: nat] :
      ( ( ( modulo_modulo @ nat @ M @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) )
        = ( zero_zero @ nat ) )
      | ( ( modulo_modulo @ nat @ M @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) )
        = ( one_one @ nat ) )
      | ( ( modulo_modulo @ nat @ M @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) )
        = ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
      | ( ( modulo_modulo @ nat @ M @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) )
        = ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) ) ) ).

% mod_exhaust_less_4
thf(fact_2580_rel__of__empty,axiom,
    ! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o] :
      ( ( rel_of @ A @ B
        @ ^ [X: A] : ( none @ B )
        @ P )
      = ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ).

% rel_of_empty
thf(fact_2581_vebt__inserti_Osimps,axiom,
    ( vEBT_vebt_inserti
    = ( ^ [T3: vEBT_VEBTi,X: nat] :
          ( vEBT_case_VEBTi @ ( heap_Time_Heap @ vEBT_VEBTi )
          @ ^ [Info3: option @ ( product_prod @ nat @ nat ),Deg4: nat,TreeArray: array @ vEBT_VEBTi,Summary3: vEBT_VEBTi] :
              ( case_option @ ( heap_Time_Heap @ vEBT_VEBTi ) @ ( product_prod @ nat @ nat ) @ ( if @ ( heap_Time_Heap @ vEBT_VEBTi ) @ ( ord_less_eq @ nat @ Deg4 @ ( one_one @ nat ) ) @ ( heap_Time_return @ vEBT_VEBTi @ ( vEBT_Nodei @ Info3 @ Deg4 @ TreeArray @ Summary3 ) ) @ ( heap_Time_return @ vEBT_VEBTi @ ( vEBT_Nodei @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ X @ X ) ) @ Deg4 @ TreeArray @ Summary3 ) ) )
              @ ^ [Minma: product_prod @ nat @ nat] :
                  ( if @ ( heap_Time_Heap @ vEBT_VEBTi ) @ ( ord_less_eq @ nat @ Deg4 @ ( one_one @ nat ) ) @ ( heap_Time_return @ vEBT_VEBTi @ ( vEBT_Nodei @ Info3 @ Deg4 @ TreeArray @ Summary3 ) )
                  @ ( heap_Time_bind @ nat @ vEBT_VEBTi @ ( heap_Time_return @ nat @ ( product_fst @ nat @ nat @ Minma ) )
                    @ ^ [Mi3: nat] :
                        ( heap_Time_bind @ nat @ vEBT_VEBTi @ ( heap_Time_return @ nat @ ( product_snd @ nat @ nat @ Minma ) )
                        @ ^ [Ma3: nat] :
                            ( heap_Time_bind @ nat @ vEBT_VEBTi @ ( if @ ( heap_Time_Heap @ nat ) @ ( ord_less @ nat @ X @ Mi3 ) @ ( heap_Time_return @ nat @ Mi3 ) @ ( heap_Time_return @ nat @ X ) )
                            @ ^ [Xn2: nat] :
                                ( heap_Time_bind @ nat @ vEBT_VEBTi @ ( if @ ( heap_Time_Heap @ nat ) @ ( ord_less @ nat @ X @ Mi3 ) @ ( heap_Time_return @ nat @ X ) @ ( heap_Time_return @ nat @ Mi3 ) )
                                @ ^ [Minn: nat] :
                                    ( heap_Time_bind @ nat @ vEBT_VEBTi @ ( vEBT_VEBT_lowi @ Xn2 @ ( divide_divide @ nat @ Deg4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
                                    @ ^ [L2: nat] :
                                        ( heap_Time_bind @ nat @ vEBT_VEBTi @ ( vEBT_VEBT_highi @ Xn2 @ ( divide_divide @ nat @ Deg4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
                                        @ ^ [H: nat] :
                                            ( heap_Time_bind @ nat @ vEBT_VEBTi @ ( array_len @ vEBT_VEBTi @ TreeArray )
                                            @ ^ [Len: nat] :
                                                ( if @ ( heap_Time_Heap @ vEBT_VEBTi )
                                                @ ( ( ord_less @ nat @ H @ Len )
                                                  & ~ ( ( X = Mi3 )
                                                      | ( X = Ma3 ) ) )
                                                @ ( heap_Time_bind @ vEBT_VEBTi @ vEBT_VEBTi @ ( array_nth @ vEBT_VEBTi @ TreeArray @ H )
                                                  @ ^ [Node: vEBT_VEBTi] :
                                                      ( heap_Time_bind @ $o @ vEBT_VEBTi @ ( vEBT_VEBT_minNulli @ Node )
                                                      @ ^ [Empt: $o] :
                                                          ( heap_Time_bind @ vEBT_VEBTi @ vEBT_VEBTi @ ( vEBT_vebt_inserti @ Node @ L2 )
                                                          @ ^ [Newnode2: vEBT_VEBTi] :
                                                              ( heap_Time_bind @ ( array @ vEBT_VEBTi ) @ vEBT_VEBTi @ ( array_upd @ vEBT_VEBTi @ H @ Newnode2 @ TreeArray )
                                                              @ ^ [Newarray: array @ vEBT_VEBTi] :
                                                                  ( heap_Time_bind @ vEBT_VEBTi @ vEBT_VEBTi @ ( if @ ( heap_Time_Heap @ vEBT_VEBTi ) @ Empt @ ( vEBT_vebt_inserti @ Summary3 @ H ) @ ( heap_Time_return @ vEBT_VEBTi @ Summary3 ) )
                                                                  @ ^ [Newsummary: vEBT_VEBTi] :
                                                                      ( heap_Time_bind @ nat @ vEBT_VEBTi @ ( if @ ( heap_Time_Heap @ nat ) @ ( ord_less @ nat @ Ma3 @ Xn2 ) @ ( heap_Time_return @ nat @ Xn2 ) @ ( heap_Time_return @ nat @ Ma3 ) )
                                                                      @ ^ [Man: nat] : ( heap_Time_return @ vEBT_VEBTi @ ( vEBT_Nodei @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Minn @ Man ) ) @ Deg4 @ Newarray @ Newsummary ) ) ) ) ) ) ) )
                                                @ ( heap_Time_return @ vEBT_VEBTi @ ( vEBT_Nodei @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi3 @ Ma3 ) ) @ Deg4 @ TreeArray @ Summary3 ) ) ) ) ) ) ) ) ) ) )
              @ Info3 )
          @ ^ [A5: $o,B5: $o] :
              ( if @ ( heap_Time_Heap @ vEBT_VEBTi )
              @ ( X
                = ( zero_zero @ nat ) )
              @ ( heap_Time_return @ vEBT_VEBTi @ ( vEBT_Leafi @ $true @ B5 ) )
              @ ( if @ ( heap_Time_Heap @ vEBT_VEBTi )
                @ ( X
                  = ( one_one @ nat ) )
                @ ( heap_Time_return @ vEBT_VEBTi @ ( vEBT_Leafi @ A5 @ $true ) )
                @ ( heap_Time_return @ vEBT_VEBTi @ ( vEBT_Leafi @ A5 @ B5 ) ) ) )
          @ T3 ) ) ) ).

% vebt_inserti.simps
thf(fact_2582_VEBTi_Osize_I3_J,axiom,
    ! [X11: option @ ( product_prod @ nat @ 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_array @ vEBT_VEBTi @ ( size_size @ vEBT_VEBTi ) @ X13 ) @ ( size_size @ vEBT_VEBTi @ X14 ) ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ).

% VEBTi.size(3)
thf(fact_2583_list__every__elemnt__bound__sum__bound,axiom,
    ! [A: $tType,Xs2: list @ A,F: A > nat,Bound: nat,I: nat] :
      ( ! [X3: A] :
          ( ( member @ A @ X3 @ ( set2 @ A @ Xs2 ) )
         => ( ord_less_eq @ nat @ ( F @ X3 ) @ Bound ) )
     => ( ord_less_eq @ nat @ ( foldr @ nat @ nat @ ( plus_plus @ nat ) @ ( map @ A @ nat @ F @ Xs2 ) @ I ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( size_size @ ( list @ A ) @ Xs2 ) @ Bound ) @ I ) ) ) ).

% list_every_elemnt_bound_sum_bound
thf(fact_2584_map__ident,axiom,
    ! [A: $tType] :
      ( ( map @ A @ A
        @ ^ [X: A] : X )
      = ( ^ [Xs: list @ A] : Xs ) ) ).

% map_ident
thf(fact_2585_length__map,axiom,
    ! [A: $tType,B: $tType,F: B > A,Xs2: list @ B] :
      ( ( size_size @ ( list @ A ) @ ( map @ B @ A @ F @ Xs2 ) )
      = ( size_size @ ( list @ B ) @ Xs2 ) ) ).

% length_map
thf(fact_2586_map__eq__conv,axiom,
    ! [A: $tType,B: $tType,F: B > A,Xs2: list @ B,G: B > A] :
      ( ( ( map @ B @ A @ F @ Xs2 )
        = ( map @ B @ A @ G @ Xs2 ) )
      = ( ! [X: B] :
            ( ( member @ B @ X @ ( set2 @ B @ Xs2 ) )
           => ( ( F @ X )
              = ( G @ X ) ) ) ) ) ).

% map_eq_conv
thf(fact_2587_nth__map,axiom,
    ! [B: $tType,A: $tType,N2: nat,Xs2: list @ A,F: A > B] :
      ( ( ord_less @ nat @ N2 @ ( size_size @ ( list @ A ) @ Xs2 ) )
     => ( ( nth @ B @ ( map @ A @ B @ F @ Xs2 ) @ N2 )
        = ( F @ ( nth @ A @ Xs2 @ N2 ) ) ) ) ).

% nth_map
thf(fact_2588_list_Omap__ident,axiom,
    ! [A: $tType,T2: list @ A] :
      ( ( map @ A @ A
        @ ^ [X: A] : X
        @ T2 )
      = T2 ) ).

% list.map_ident
thf(fact_2589_map__eq__imp__length__eq,axiom,
    ! [A: $tType,B: $tType,C: $tType,F: B > A,Xs2: list @ B,G: C > A,Ys: list @ C] :
      ( ( ( map @ B @ A @ F @ Xs2 )
        = ( map @ C @ A @ G @ Ys ) )
     => ( ( size_size @ ( list @ B ) @ Xs2 )
        = ( size_size @ ( list @ C ) @ Ys ) ) ) ).

% map_eq_imp_length_eq
thf(fact_2590_list_Omap__cong,axiom,
    ! [B: $tType,A: $tType,X2: list @ A,Ya: list @ A,F: A > B,G: A > B] :
      ( ( X2 = Ya )
     => ( ! [Z3: A] :
            ( ( member @ A @ Z3 @ ( set2 @ A @ Ya ) )
           => ( ( F @ Z3 )
              = ( G @ Z3 ) ) )
       => ( ( map @ A @ B @ F @ X2 )
          = ( map @ A @ B @ G @ Ya ) ) ) ) ).

% list.map_cong
thf(fact_2591_list_Omap__cong0,axiom,
    ! [B: $tType,A: $tType,X2: list @ A,F: A > B,G: A > B] :
      ( ! [Z3: A] :
          ( ( member @ A @ Z3 @ ( set2 @ A @ X2 ) )
         => ( ( F @ Z3 )
            = ( G @ Z3 ) ) )
     => ( ( map @ A @ B @ F @ X2 )
        = ( map @ A @ B @ G @ X2 ) ) ) ).

% list.map_cong0
thf(fact_2592_list_Oinj__map__strong,axiom,
    ! [B: $tType,A: $tType,X2: list @ A,Xa: list @ A,F: A > B,Fa: A > B] :
      ( ! [Z3: A,Za: A] :
          ( ( member @ A @ Z3 @ ( set2 @ A @ X2 ) )
         => ( ( member @ A @ Za @ ( set2 @ A @ Xa ) )
           => ( ( ( F @ Z3 )
                = ( Fa @ Za ) )
             => ( Z3 = Za ) ) ) )
     => ( ( ( map @ A @ B @ F @ X2 )
          = ( map @ A @ B @ Fa @ Xa ) )
       => ( X2 = Xa ) ) ) ).

% list.inj_map_strong
thf(fact_2593_map__ext,axiom,
    ! [B: $tType,A: $tType,Xs2: list @ A,F: A > B,G: A > B] :
      ( ! [X3: A] :
          ( ( member @ A @ X3 @ ( set2 @ A @ Xs2 ) )
         => ( ( F @ X3 )
            = ( G @ X3 ) ) )
     => ( ( map @ A @ B @ F @ Xs2 )
        = ( map @ A @ B @ G @ Xs2 ) ) ) ).

% map_ext
thf(fact_2594_map__idI,axiom,
    ! [A: $tType,Xs2: list @ A,F: A > A] :
      ( ! [X3: A] :
          ( ( member @ A @ X3 @ ( set2 @ A @ Xs2 ) )
         => ( ( F @ X3 )
            = X3 ) )
     => ( ( map @ A @ A @ F @ Xs2 )
        = Xs2 ) ) ).

% map_idI
thf(fact_2595_map__cong,axiom,
    ! [B: $tType,A: $tType,Xs2: list @ A,Ys: list @ A,F: A > B,G: A > B] :
      ( ( Xs2 = Ys )
     => ( ! [X3: A] :
            ( ( member @ A @ X3 @ ( set2 @ A @ Ys ) )
           => ( ( F @ X3 )
              = ( G @ X3 ) ) )
       => ( ( map @ A @ B @ F @ Xs2 )
          = ( map @ A @ B @ G @ Ys ) ) ) ) ).

% map_cong
thf(fact_2596_ex__map__conv,axiom,
    ! [A: $tType,B: $tType,Ys: list @ B,F: A > B] :
      ( ( ? [Xs: list @ A] :
            ( Ys
            = ( map @ A @ B @ F @ Xs ) ) )
      = ( ! [X: B] :
            ( ( member @ B @ X @ ( set2 @ B @ Ys ) )
           => ? [Y: A] :
                ( X
                = ( F @ Y ) ) ) ) ) ).

% ex_map_conv
thf(fact_2597_map__eq__nth__eq,axiom,
    ! [A: $tType,B: $tType,F: B > A,L: list @ B,L4: list @ B,I: nat] :
      ( ( ( map @ B @ A @ F @ L )
        = ( map @ B @ A @ F @ L4 ) )
     => ( ( F @ ( nth @ B @ L @ I ) )
        = ( F @ ( nth @ B @ L4 @ I ) ) ) ) ).

% map_eq_nth_eq
thf(fact_2598_TBOUND__upd,axiom,
    ! [A: $tType] :
      ( ( heap @ A )
     => ! [Xs2: nat,I: A,X2: array @ A] : ( time_TBOUND @ ( array @ A ) @ ( array_upd @ A @ Xs2 @ I @ X2 ) @ ( one_one @ nat ) ) ) ).

% TBOUND_upd
thf(fact_2599_time__array__len,axiom,
    ! [A: $tType] :
      ( ( heap @ A )
     => ! [P4: array @ A,H2: heap_ext @ product_unit] :
          ( ( time_time @ nat @ ( array_len @ A @ P4 ) @ H2 )
          = ( one_one @ nat ) ) ) ).

% time_array_len
thf(fact_2600_map__upd__eq,axiom,
    ! [B: $tType,A: $tType,I: nat,L: list @ A,F: A > B,X2: A] :
      ( ( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ L ) )
       => ( ( F @ ( nth @ A @ L @ I ) )
          = ( F @ X2 ) ) )
     => ( ( map @ A @ B @ F @ ( list_update @ A @ L @ I @ X2 ) )
        = ( map @ A @ B @ F @ L ) ) ) ).

% map_upd_eq
thf(fact_2601_TBOUND__len,axiom,
    ! [A: $tType] :
      ( ( heap @ A )
     => ! [Xs2: array @ A] : ( time_TBOUND @ nat @ ( array_len @ A @ Xs2 ) @ ( one_one @ nat ) ) ) ).

% TBOUND_len
thf(fact_2602_VEBT__internal_Ocnt_H_Osimps_I2_J,axiom,
    ! [Info: option @ ( product_prod @ nat @ 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_2603_VEBT__internal_Ocnt_H_Oelims,axiom,
    ! [X2: vEBT_VEBT,Y2: nat] :
      ( ( ( vEBT_VEBT_cnt2 @ X2 )
        = Y2 )
     => ( ( ? [A4: $o,B3: $o] :
              ( X2
              = ( vEBT_Leaf @ A4 @ B3 ) )
         => ( Y2
           != ( one_one @ nat ) ) )
       => ~ ! [Info2: option @ ( product_prod @ nat @ nat ),Deg2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
              ( ( X2
                = ( vEBT_Node @ Info2 @ Deg2 @ TreeList2 @ Summary2 ) )
             => ( Y2
               != ( 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_2604_eq__diff__eq_H,axiom,
    ! [X2: real,Y2: real,Z: real] :
      ( ( X2
        = ( minus_minus @ real @ Y2 @ Z ) )
      = ( Y2
        = ( plus_plus @ real @ X2 @ Z ) ) ) ).

% eq_diff_eq'
thf(fact_2605_VEBT__internal_Ospace_H_Osimps_I2_J,axiom,
    ! [Info: option @ ( product_prod @ nat @ 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 @ one2 ) ) ) @ ( 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_2606_VEBT__internal_Ospace_H_Oelims,axiom,
    ! [X2: vEBT_VEBT,Y2: nat] :
      ( ( ( vEBT_VEBT_space2 @ X2 )
        = Y2 )
     => ( ( ? [A4: $o,B3: $o] :
              ( X2
              = ( vEBT_Leaf @ A4 @ B3 ) )
         => ( Y2
           != ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) ) )
       => ~ ! [Info2: option @ ( product_prod @ nat @ nat ),Deg2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
              ( ( X2
                = ( vEBT_Node @ Info2 @ Deg2 @ TreeList2 @ Summary2 ) )
             => ( Y2
               != ( plus_plus @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit1 @ one2 ) ) ) @ ( 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_2607_VEBT__internal_Ospace_Osimps_I2_J,axiom,
    ! [Info: option @ ( product_prod @ nat @ 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 @ one2 ) ) ) @ ( vEBT_VEBT_space @ Summary ) ) @ ( size_size @ ( list @ vEBT_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_2608_VEBT__internal_Ospace_Oelims,axiom,
    ! [X2: vEBT_VEBT,Y2: nat] :
      ( ( ( vEBT_VEBT_space @ X2 )
        = Y2 )
     => ( ( ? [A4: $o,B3: $o] :
              ( X2
              = ( vEBT_Leaf @ A4 @ B3 ) )
         => ( Y2
           != ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) ) )
       => ~ ! [Info2: option @ ( product_prod @ nat @ nat ),Deg2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
              ( ( X2
                = ( vEBT_Node @ Info2 @ Deg2 @ TreeList2 @ Summary2 ) )
             => ( Y2
               != ( plus_plus @ nat @ ( plus_plus @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit0 @ one2 ) ) ) @ ( vEBT_VEBT_space @ Summary2 ) ) @ ( size_size @ ( list @ vEBT_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_2609_upd__rule,axiom,
    ! [A: $tType] :
      ( ( heap @ A )
     => ! [I: nat,Xs2: list @ A,A2: array @ A,X2: A] :
          ( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs2 ) )
         => ( hoare_hoare_triple @ ( array @ A ) @ ( snga_assn @ A @ A2 @ Xs2 ) @ ( array_upd @ A @ I @ X2 @ A2 )
            @ ^ [R: array @ A] : ( times_times @ assn @ ( snga_assn @ A @ A2 @ ( list_update @ A @ Xs2 @ I @ X2 ) ) @ ( pure_assn @ ( R = A2 ) ) ) ) ) ) ).

% upd_rule
thf(fact_2610_length__rule,axiom,
    ! [A: $tType] :
      ( ( heap @ A )
     => ! [A2: array @ A,Xs2: list @ A] :
          ( hoare_hoare_triple @ nat @ ( snga_assn @ A @ A2 @ Xs2 ) @ ( array_len @ A @ A2 )
          @ ^ [R: nat] :
              ( times_times @ assn @ ( snga_assn @ A @ A2 @ Xs2 )
              @ ( pure_assn
                @ ( R
                  = ( size_size @ ( list @ A ) @ Xs2 ) ) ) ) ) ) ).

% length_rule
thf(fact_2611_VEBTi_Osize__gen_I1_J,axiom,
    ! [X11: option @ ( product_prod @ nat @ 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_array @ vEBT_VEBTi @ vEBT_size_VEBTi @ X13 ) @ ( vEBT_size_VEBTi @ X14 ) ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ).

% VEBTi.size_gen(1)
thf(fact_2612_length__corresp,axiom,
    ! [B: $tType,A: $tType] :
      ( ( heap @ A )
     => ! [Tree_array: array @ A,Tree_is2: list @ B] :
          ( ( ( ex_assn @ ( list @ A ) @ ( snga_assn @ A @ Tree_array ) )
            = ( top_top @ assn ) )
         => ( ( heap_Time_return @ nat @ ( size_size @ ( list @ B ) @ Tree_is2 ) )
            = ( array_len @ A @ Tree_array ) ) ) ) ).

% length_corresp
thf(fact_2613_list__every__elemnt__bound__sum__bound__real,axiom,
    ! [A: $tType,Xs2: list @ A,F: A > real,Bound: real,I: real] :
      ( ! [X3: A] :
          ( ( member @ A @ X3 @ ( set2 @ A @ Xs2 ) )
         => ( ord_less_eq @ real @ ( F @ X3 ) @ Bound ) )
     => ( ord_less_eq @ real @ ( foldr @ real @ real @ ( plus_plus @ real ) @ ( map @ A @ real @ F @ Xs2 ) @ I ) @ ( plus_plus @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ ( size_size @ ( list @ A ) @ Xs2 ) ) @ Bound ) @ I ) ) ) ).

% list_every_elemnt_bound_sum_bound_real
thf(fact_2614_real__nat__list,axiom,
    ! [A: $tType,F: A > nat,Xs2: list @ A,C2: nat] :
      ( ( semiring_1_of_nat @ real @ ( foldr @ nat @ nat @ ( plus_plus @ nat ) @ ( map @ A @ nat @ F @ Xs2 ) @ C2 ) )
      = ( foldr @ real @ real @ ( plus_plus @ real )
        @ ( map @ A @ real
          @ ^ [X: A] : ( semiring_1_of_nat @ real @ ( F @ X ) )
          @ Xs2 )
        @ ( semiring_1_of_nat @ real @ C2 ) ) ) ).

% real_nat_list
thf(fact_2615_f__g__map__foldr__bound,axiom,
    ! [A: $tType,Xs2: list @ A,F: A > real,C2: real,G: A > real,D2: real] :
      ( ! [X3: A] :
          ( ( member @ A @ X3 @ ( set2 @ A @ Xs2 ) )
         => ( ord_less_eq @ real @ ( F @ X3 ) @ ( times_times @ real @ C2 @ ( G @ X3 ) ) ) )
     => ( ord_less_eq @ real @ ( foldr @ real @ real @ ( plus_plus @ real ) @ ( map @ A @ real @ F @ Xs2 ) @ D2 ) @ ( plus_plus @ real @ ( times_times @ real @ C2 @ ( foldr @ real @ real @ ( plus_plus @ real ) @ ( map @ A @ real @ G @ Xs2 ) @ ( zero_zero @ real ) ) ) @ D2 ) ) ) ).

% f_g_map_foldr_bound
thf(fact_2616_vebt__memberi_Osimps,axiom,
    ( vEBT_vebt_memberi
    = ( ^ [T3: vEBT_VEBTi,X: nat] :
          ( vEBT_case_VEBTi @ ( heap_Time_Heap @ $o )
          @ ^ [Info3: option @ ( product_prod @ nat @ nat ),Deg4: nat,TreeList3: array @ vEBT_VEBTi,Summary3: vEBT_VEBTi] :
              ( case_option @ ( heap_Time_Heap @ $o ) @ ( product_prod @ nat @ nat ) @ ( heap_Time_return @ $o @ $false )
              @ ( product_case_prod @ nat @ nat @ ( heap_Time_Heap @ $o )
                @ ^ [Mi3: nat,Ma3: nat] :
                    ( if @ ( heap_Time_Heap @ $o ) @ ( ord_less_eq @ nat @ Deg4 @ ( one_one @ nat ) ) @ ( heap_Time_return @ $o @ $false )
                    @ ( if @ ( heap_Time_Heap @ $o ) @ ( X = Mi3 ) @ ( heap_Time_return @ $o @ $true )
                      @ ( if @ ( heap_Time_Heap @ $o ) @ ( X = Ma3 ) @ ( heap_Time_return @ $o @ $true )
                        @ ( if @ ( heap_Time_Heap @ $o ) @ ( ord_less @ nat @ X @ Mi3 ) @ ( heap_Time_return @ $o @ $false )
                          @ ( if @ ( heap_Time_Heap @ $o ) @ ( ord_less @ nat @ Ma3 @ X ) @ ( heap_Time_return @ $o @ $false )
                            @ ( heap_Time_bind @ nat @ $o @ ( vEBT_VEBT_highi @ X @ ( divide_divide @ nat @ Deg4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
                              @ ^ [H: nat] :
                                  ( heap_Time_bind @ nat @ $o @ ( vEBT_VEBT_lowi @ X @ ( divide_divide @ nat @ Deg4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
                                  @ ^ [L2: nat] :
                                      ( heap_Time_bind @ nat @ $o @ ( array_len @ vEBT_VEBTi @ TreeList3 )
                                      @ ^ [Len: nat] :
                                          ( if @ ( heap_Time_Heap @ $o ) @ ( ord_less @ nat @ H @ Len )
                                          @ ( heap_Time_bind @ vEBT_VEBTi @ $o @ ( array_nth @ vEBT_VEBTi @ TreeList3 @ H )
                                            @ ^ [Th: vEBT_VEBTi] : ( vEBT_vebt_memberi @ Th @ L2 ) )
                                          @ ( heap_Time_return @ $o @ $false ) ) ) ) ) ) ) ) ) ) )
              @ Info3 )
          @ ^ [A5: $o,B5: $o] :
              ( heap_Time_return @ $o
              @ ( ( ( X
                    = ( zero_zero @ nat ) )
                 => A5 )
                & ( ( X
                   != ( zero_zero @ nat ) )
                 => ( ( ( X
                        = ( one_one @ nat ) )
                     => B5 )
                    & ( X
                      = ( one_one @ nat ) ) ) ) ) )
          @ T3 ) ) ) ).

% vebt_memberi.simps
thf(fact_2617_listsum__bound,axiom,
    ! [A: $tType,Xs2: list @ A,F: A > real,Y2: real] :
      ( ! [X3: A] :
          ( ( member @ A @ X3 @ ( set2 @ A @ Xs2 ) )
         => ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( F @ X3 ) ) )
     => ( ord_less_eq @ real @ Y2 @ ( foldr @ real @ real @ ( plus_plus @ real ) @ ( map @ A @ real @ F @ Xs2 ) @ Y2 ) ) ) ).

% listsum_bound
thf(fact_2618_merge__true__star,axiom,
    ( ( times_times @ assn @ ( top_top @ assn ) @ ( top_top @ assn ) )
    = ( top_top @ assn ) ) ).

% merge_true_star
thf(fact_2619_assn__basic__inequalities_I1_J,axiom,
    ( ( top_top @ assn )
   != ( one_one @ assn ) ) ).

% assn_basic_inequalities(1)
thf(fact_2620_case__prod__conv,axiom,
    ! [B: $tType,A: $tType,C: $tType,F: B > C > A,A2: B,B2: C] :
      ( ( product_case_prod @ B @ C @ A @ F @ ( product_Pair @ B @ C @ A2 @ B2 ) )
      = ( F @ A2 @ B2 ) ) ).

% case_prod_conv
thf(fact_2621_map__fst__mk__snd,axiom,
    ! [B: $tType,A: $tType,K: B,L: list @ A] :
      ( ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B )
        @ ( map @ A @ ( product_prod @ A @ B )
          @ ^ [X: A] : ( product_Pair @ A @ B @ X @ K )
          @ L ) )
      = L ) ).

% map_fst_mk_snd
thf(fact_2622_map__snd__mk__fst,axiom,
    ! [B: $tType,A: $tType,K: B,L: list @ A] :
      ( ( map @ ( product_prod @ B @ A ) @ A @ ( product_snd @ B @ A ) @ ( map @ A @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ K ) @ L ) )
      = L ) ).

% map_snd_mk_fst
thf(fact_2623_nested__case__prod__simp,axiom,
    ! [A: $tType,D: $tType,C: $tType,B: $tType] :
      ( ( product_case_prod @ B @ C @ ( D > A ) )
      = ( ^ [F6: B > C > D > A,X: product_prod @ B @ C,Y: D] :
            ( product_case_prod @ B @ C @ A
            @ ^ [A5: B,B5: C] : ( F6 @ A5 @ B5 @ Y )
            @ X ) ) ) ).

% nested_case_prod_simp
thf(fact_2624_top__option__def,axiom,
    ! [A: $tType] :
      ( ( order_top @ A )
     => ( ( top_top @ ( option @ A ) )
        = ( some @ A @ ( top_top @ A ) ) ) ) ).

% top_option_def
thf(fact_2625_prod_Ocase__distrib,axiom,
    ! [C: $tType,D: $tType,B: $tType,A: $tType,H2: C > D,F: A > B > C,Prod: product_prod @ A @ B] :
      ( ( H2 @ ( product_case_prod @ A @ B @ C @ F @ Prod ) )
      = ( product_case_prod @ A @ B @ D
        @ ^ [X15: A,X25: B] : ( H2 @ ( F @ X15 @ X25 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_2626_old_Oprod_Ocase,axiom,
    ! [A: $tType,C: $tType,B: $tType,F: A > B > C,X1: A,X23: B] :
      ( ( product_case_prod @ A @ B @ C @ F @ ( product_Pair @ A @ B @ X1 @ X23 ) )
      = ( F @ X1 @ X23 ) ) ).

% old.prod.case
thf(fact_2627_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_2628_case__prodE2,axiom,
    ! [B: $tType,A: $tType,C: $tType,Q: A > $o,P: B > C > A,Z: product_prod @ B @ C] :
      ( ( Q @ ( product_case_prod @ B @ C @ A @ P @ Z ) )
     => ~ ! [X3: B,Y4: C] :
            ( ( Z
              = ( product_Pair @ B @ C @ X3 @ Y4 ) )
           => ~ ( Q @ ( P @ X3 @ Y4 ) ) ) ) ).

% case_prodE2
thf(fact_2629_case__prod__eta,axiom,
    ! [C: $tType,B: $tType,A: $tType,F: ( product_prod @ A @ B ) > C] :
      ( ( product_case_prod @ A @ B @ C
        @ ^ [X: A,Y: B] : ( F @ ( product_Pair @ A @ B @ X @ Y ) ) )
      = F ) ).

% case_prod_eta
thf(fact_2630_cond__case__prod__eta,axiom,
    ! [C: $tType,B: $tType,A: $tType,F: A > B > C,G: ( product_prod @ A @ B ) > C] :
      ( ! [X3: A,Y4: B] :
          ( ( F @ X3 @ Y4 )
          = ( G @ ( product_Pair @ A @ B @ X3 @ Y4 ) ) )
     => ( ( product_case_prod @ A @ B @ C @ F )
        = G ) ) ).

% cond_case_prod_eta
thf(fact_2631_fn__fst__conv,axiom,
    ! [B: $tType,C: $tType,A: $tType,F: A > C] :
      ( ( ^ [X: product_prod @ A @ B] : ( F @ ( product_fst @ A @ B @ X ) ) )
      = ( product_case_prod @ A @ B @ C
        @ ^ [A5: A,Uu3: B] : ( F @ A5 ) ) ) ).

% fn_fst_conv
thf(fact_2632_fst__def,axiom,
    ! [B: $tType,A: $tType] :
      ( ( product_fst @ A @ B )
      = ( product_case_prod @ A @ B @ A
        @ ^ [X15: A,X25: B] : X15 ) ) ).

% fst_def
thf(fact_2633_fn__snd__conv,axiom,
    ! [B: $tType,C: $tType,A: $tType,F: B > C] :
      ( ( ^ [X: product_prod @ A @ B] : ( F @ ( product_snd @ A @ B @ X ) ) )
      = ( product_case_prod @ A @ B @ C
        @ ^ [Uu3: A] : F ) ) ).

% fn_snd_conv
thf(fact_2634_snd__def,axiom,
    ! [B: $tType,A: $tType] :
      ( ( product_snd @ A @ B )
      = ( product_case_prod @ A @ B @ B
        @ ^ [X15: A,X25: B] : X25 ) ) ).

% snd_def
thf(fact_2635_pair__list__eqI,axiom,
    ! [B: $tType,A: $tType,Xs2: list @ ( product_prod @ A @ B ),Ys: list @ ( product_prod @ A @ B )] :
      ( ( ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Xs2 )
        = ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Ys ) )
     => ( ( ( map @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) @ Xs2 )
          = ( map @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) @ Ys ) )
       => ( Xs2 = Ys ) ) ) ).

% pair_list_eqI
thf(fact_2636_split__beta,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( product_case_prod @ A @ B @ C )
      = ( ^ [F6: A > B > C,Prod3: product_prod @ A @ B] : ( F6 @ ( product_fst @ A @ B @ Prod3 ) @ ( product_snd @ A @ B @ Prod3 ) ) ) ) ).

% split_beta
thf(fact_2637_case__prod__beta,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( ( product_case_prod @ B @ C @ A )
      = ( ^ [F6: B > C > A,P5: product_prod @ B @ C] : ( F6 @ ( product_fst @ B @ C @ P5 ) @ ( product_snd @ B @ C @ P5 ) ) ) ) ).

% case_prod_beta
thf(fact_2638_mod__star__trueE,axiom,
    ! [P: assn,H2: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
      ( ( rep_assn @ ( times_times @ assn @ P @ ( top_top @ assn ) ) @ H2 )
     => ~ ! [H5: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
            ~ ( rep_assn @ P @ H5 ) ) ).

% mod_star_trueE
thf(fact_2639_mod__star__trueI,axiom,
    ! [P: assn,H2: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
      ( ( rep_assn @ P @ H2 )
     => ( rep_assn @ ( times_times @ assn @ P @ ( top_top @ assn ) ) @ H2 ) ) ).

% mod_star_trueI
thf(fact_2640_case__prod__rule,axiom,
    ! [A: $tType,B: $tType,C: $tType,X2: product_prod @ A @ B,P: assn,F: A > B > ( heap_Time_Heap @ C ),Q: C > assn] :
      ( ! [A4: A,B3: B] :
          ( ( X2
            = ( product_Pair @ A @ B @ A4 @ B3 ) )
         => ( hoare_hoare_triple @ C @ P @ ( F @ A4 @ B3 ) @ Q ) )
     => ( hoare_hoare_triple @ C @ P @ ( product_case_prod @ A @ B @ ( heap_Time_Heap @ C ) @ F @ X2 ) @ Q ) ) ).

% case_prod_rule
thf(fact_2641_split__comp__eq,axiom,
    ! [A: $tType,C: $tType,B: $tType,D: $tType,F: A > B > C,G: D > A] :
      ( ( ^ [U2: product_prod @ D @ B] : ( F @ ( G @ ( product_fst @ D @ B @ U2 ) ) @ ( product_snd @ D @ B @ U2 ) ) )
      = ( product_case_prod @ D @ B @ C
        @ ^ [X: D] : ( F @ ( G @ X ) ) ) ) ).

% split_comp_eq
thf(fact_2642_case__prod__beta_H,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( product_case_prod @ A @ B @ C )
      = ( ^ [F6: A > B > C,X: product_prod @ A @ B] : ( F6 @ ( product_fst @ A @ B @ X ) @ ( product_snd @ A @ B @ X ) ) ) ) ).

% case_prod_beta'
thf(fact_2643_case__prod__unfold,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( product_case_prod @ A @ B @ C )
      = ( ^ [C4: A > B > C,P5: product_prod @ A @ B] : ( C4 @ ( product_fst @ A @ B @ P5 ) @ ( product_snd @ A @ B @ P5 ) ) ) ) ).

% case_prod_unfold
thf(fact_2644_prod_Osplit__sel,axiom,
    ! [C: $tType,B: $tType,A: $tType,P: C > $o,F: A > B > C,Prod: product_prod @ A @ B] :
      ( ( P @ ( product_case_prod @ A @ B @ C @ F @ Prod ) )
      = ( ( Prod
          = ( product_Pair @ A @ B @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) )
       => ( P @ ( F @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) ) ) ) ).

% prod.split_sel
thf(fact_2645_prod_Osplit__sel__asm,axiom,
    ! [C: $tType,B: $tType,A: $tType,P: C > $o,F: A > B > C,Prod: product_prod @ A @ B] :
      ( ( P @ ( product_case_prod @ A @ B @ C @ F @ Prod ) )
      = ( ~ ( ( Prod
              = ( product_Pair @ A @ B @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) )
            & ~ ( P @ ( F @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) ) ) ) ) ).

% prod.split_sel_asm
thf(fact_2646_mod__h__bot__iff_I2_J,axiom,
    ! [H2: heap_ext @ product_unit] : ( rep_assn @ ( top_top @ assn ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ ( bot_bot @ ( set @ nat ) ) ) ) ).

% mod_h_bot_iff(2)
thf(fact_2647_mod__star__trueE_H,axiom,
    ! [P: assn,H2: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
      ( ( rep_assn @ ( times_times @ assn @ P @ ( top_top @ assn ) ) @ H2 )
     => ~ ! [H5: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
            ( ( ( product_fst @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H5 )
              = ( product_fst @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 ) )
           => ( ( ord_less_eq @ ( set @ nat ) @ ( product_snd @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H5 ) @ ( product_snd @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 ) )
             => ~ ( rep_assn @ P @ H5 ) ) ) ) ).

% mod_star_trueE'
thf(fact_2648_VEBTi_Osize__gen_I2_J,axiom,
    ! [X212: $o,X222: $o] :
      ( ( vEBT_size_VEBTi @ ( vEBT_Leafi @ X212 @ X222 ) )
      = ( zero_zero @ nat ) ) ).

% VEBTi.size_gen(2)
thf(fact_2649_VEBT__internal_Ocnt_Osimps_I2_J,axiom,
    ! [Info: option @ ( product_prod @ nat @ 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_2650_VEBT__internal_Ocnt_Oelims,axiom,
    ! [X2: vEBT_VEBT,Y2: real] :
      ( ( ( vEBT_VEBT_cnt @ X2 )
        = Y2 )
     => ( ( ? [A4: $o,B3: $o] :
              ( X2
              = ( vEBT_Leaf @ A4 @ B3 ) )
         => ( Y2
           != ( one_one @ real ) ) )
       => ~ ! [Info2: option @ ( product_prod @ nat @ nat ),Deg2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
              ( ( X2
                = ( vEBT_Node @ Info2 @ Deg2 @ TreeList2 @ Summary2 ) )
             => ( Y2
               != ( 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_2651_vebt__memberi__refines,axiom,
    ! [Ti: vEBT_VEBTi,X2: nat,T2: vEBT_VEBT] : ( refine_Imp_refines @ $o @ ( vEBT_vebt_memberi @ Ti @ X2 ) @ ( vEBT_V854960066525838166emberi @ T2 @ Ti @ X2 ) ) ).

% vebt_memberi_refines
thf(fact_2652_of__nat__code__if,axiom,
    ! [A: $tType] :
      ( ( semiring_1 @ A )
     => ( ( semiring_1_of_nat @ A )
        = ( ^ [N: nat] :
              ( if @ A
              @ ( N
                = ( zero_zero @ nat ) )
              @ ( zero_zero @ A )
              @ ( product_case_prod @ nat @ nat @ A
                @ ^ [M7: nat,Q5: nat] :
                    ( if @ A
                    @ ( Q5
                      = ( zero_zero @ nat ) )
                    @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ A @ M7 ) )
                    @ ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ A @ M7 ) ) @ ( one_one @ A ) ) )
                @ ( divmod_nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).

% of_nat_code_if
thf(fact_2653_Suc__if__eq,axiom,
    ! [A: $tType,F: nat > A,H2: nat > A,G: A,N2: nat] :
      ( ! [N4: nat] :
          ( ( F @ ( suc @ N4 ) )
          = ( H2 @ N4 ) )
     => ( ( ( F @ ( zero_zero @ nat ) )
          = G )
       => ( ( ( N2
              = ( zero_zero @ nat ) )
           => ( ( F @ N2 )
              = G ) )
          & ( ( N2
             != ( zero_zero @ nat ) )
           => ( ( F @ N2 )
              = ( H2 @ ( minus_minus @ nat @ N2 @ ( one_one @ nat ) ) ) ) ) ) ) ) ).

% Suc_if_eq
thf(fact_2654_vebt__succi__refines,axiom,
    ! [Ti: vEBT_VEBTi,X2: nat,T2: vEBT_VEBT] : ( refine_Imp_refines @ ( option @ nat ) @ ( vEBT_vebt_succi @ Ti @ X2 ) @ ( vEBT_VEBT_vebt_succi @ T2 @ Ti @ X2 ) ) ).

% vebt_succi_refines
thf(fact_2655_round__unique,axiom,
    ! [A: $tType] :
      ( ( archim2362893244070406136eiling @ A )
     => ! [X2: A,Y2: int] :
          ( ( ord_less @ A @ ( minus_minus @ A @ X2 @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) @ ( ring_1_of_int @ A @ Y2 ) )
         => ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ Y2 ) @ ( plus_plus @ A @ X2 @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) )
           => ( ( archimedean_round @ A @ X2 )
              = Y2 ) ) ) ) ).

% round_unique
thf(fact_2656_mult__le__cancel__iff1,axiom,
    ! [A: $tType] :
      ( ( linordered_idom @ A )
     => ! [Z: A,X2: A,Y2: A] :
          ( ( ord_less @ A @ ( zero_zero @ A ) @ Z )
         => ( ( ord_less_eq @ A @ ( times_times @ A @ X2 @ Z ) @ ( times_times @ A @ Y2 @ Z ) )
            = ( ord_less_eq @ A @ X2 @ Y2 ) ) ) ) ).

% mult_le_cancel_iff1
thf(fact_2657_refines__replicate,axiom,
    ! [A: $tType,F: heap_Time_Heap @ A,F4: heap_Time_Heap @ A,N2: nat] :
      ( ( refine_Imp_refines @ A @ F @ F4 )
     => ( refine_Imp_refines @ ( list @ A ) @ ( vEBT_VEBT_replicatei @ A @ N2 @ F ) @ ( vEBT_VEBT_replicatei @ A @ N2 @ F4 ) ) ) ).

% refines_replicate
thf(fact_2658_refines__case__VEBTi,axiom,
    ! [A: $tType,Ti: vEBT_VEBTi,Ti2: vEBT_VEBTi,F1: $o > $o > ( heap_Time_Heap @ A ),F13: $o > $o > ( heap_Time_Heap @ A ),F22: ( option @ ( product_prod @ nat @ nat ) ) > nat > ( array @ vEBT_VEBTi ) > vEBT_VEBTi > ( heap_Time_Heap @ A ),F24: ( option @ ( product_prod @ nat @ nat ) ) > nat > ( array @ vEBT_VEBTi ) > vEBT_VEBTi > ( heap_Time_Heap @ A )] :
      ( ( Ti = Ti2 )
     => ( ! [A4: $o,B3: $o] : ( refine_Imp_refines @ A @ ( F1 @ A4 @ B3 ) @ ( F13 @ A4 @ B3 ) )
       => ( ! [Info2: option @ ( product_prod @ nat @ nat ),Deg2: nat,TreeArray2: array @ vEBT_VEBTi,Summary2: vEBT_VEBTi] : ( refine_Imp_refines @ A @ ( F22 @ Info2 @ Deg2 @ TreeArray2 @ Summary2 ) @ ( F24 @ Info2 @ Deg2 @ TreeArray2 @ Summary2 ) )
         => ( refine_Imp_refines @ A @ ( vEBT_case_VEBTi @ ( heap_Time_Heap @ A ) @ F22 @ F1 @ Ti ) @ ( vEBT_case_VEBTi @ ( heap_Time_Heap @ A ) @ F24 @ F13 @ Ti2 ) ) ) ) ) ).

% refines_case_VEBTi
thf(fact_2659_case__prodI2,axiom,
    ! [B: $tType,A: $tType,P4: product_prod @ A @ B,C2: A > B > $o] :
      ( ! [A4: A,B3: B] :
          ( ( P4
            = ( product_Pair @ A @ B @ A4 @ B3 ) )
         => ( C2 @ A4 @ B3 ) )
     => ( product_case_prod @ A @ B @ $o @ C2 @ P4 ) ) ).

% case_prodI2
thf(fact_2660_case__prodI,axiom,
    ! [A: $tType,B: $tType,F: A > B > $o,A2: A,B2: B] :
      ( ( F @ A2 @ B2 )
     => ( product_case_prod @ A @ B @ $o @ F @ ( product_Pair @ A @ B @ A2 @ B2 ) ) ) ).

% case_prodI
thf(fact_2661_mem__case__prodI2,axiom,
    ! [C: $tType,B: $tType,A: $tType,P4: product_prod @ A @ B,Z: C,C2: A > B > ( set @ C )] :
      ( ! [A4: A,B3: B] :
          ( ( P4
            = ( product_Pair @ A @ B @ A4 @ B3 ) )
         => ( member @ C @ Z @ ( C2 @ A4 @ B3 ) ) )
     => ( member @ C @ Z @ ( product_case_prod @ A @ B @ ( set @ C ) @ C2 @ P4 ) ) ) ).

% mem_case_prodI2
thf(fact_2662_mem__case__prodI,axiom,
    ! [A: $tType,B: $tType,C: $tType,Z: A,C2: B > C > ( set @ A ),A2: B,B2: C] :
      ( ( member @ A @ Z @ ( C2 @ A2 @ B2 ) )
     => ( member @ A @ Z @ ( product_case_prod @ B @ C @ ( set @ A ) @ C2 @ ( product_Pair @ B @ C @ A2 @ B2 ) ) ) ) ).

% mem_case_prodI
thf(fact_2663_case__prodI2_H,axiom,
    ! [A: $tType,B: $tType,C: $tType,P4: product_prod @ A @ B,C2: A > B > C > $o,X2: C] :
      ( ! [A4: A,B3: B] :
          ( ( ( product_Pair @ A @ B @ A4 @ B3 )
            = P4 )
         => ( C2 @ A4 @ B3 @ X2 ) )
     => ( product_case_prod @ A @ B @ ( C > $o ) @ C2 @ P4 @ X2 ) ) ).

% case_prodI2'
thf(fact_2664_Collect__const,axiom,
    ! [A: $tType,P: $o] :
      ( ( P
       => ( ( collect @ A
            @ ^ [S2: A] : P )
          = ( top_top @ ( set @ A ) ) ) )
      & ( ~ P
       => ( ( collect @ A
            @ ^ [S2: A] : P )
          = ( bot_bot @ ( set @ A ) ) ) ) ) ).

% Collect_const
thf(fact_2665_Collect__const__case__prod,axiom,
    ! [B: $tType,A: $tType,P: $o] :
      ( ( P
       => ( ( collect @ ( product_prod @ A @ B )
            @ ( product_case_prod @ A @ B @ $o
              @ ^ [A5: A,B5: B] : P ) )
          = ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) ) )
      & ( ~ P
       => ( ( collect @ ( product_prod @ A @ B )
            @ ( product_case_prod @ A @ B @ $o
              @ ^ [A5: A,B5: B] : P ) )
          = ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ) ) ).

% Collect_const_case_prod
thf(fact_2666_Diff__UNIV,axiom,
    ! [A: $tType,A3: set @ A] :
      ( ( minus_minus @ ( set @ A ) @ A3 @ ( top_top @ ( set @ A ) ) )
      = ( bot_bot @ ( set @ A ) ) ) ).

% Diff_UNIV
thf(fact_2667_round__numeral,axiom,
    ! [A: $tType] :
      ( ( archim2362893244070406136eiling @ A )
     => ! [N2: num] :
          ( ( archimedean_round @ A @ ( numeral_numeral @ A @ N2 ) )
          = ( numeral_numeral @ int @ N2 ) ) ) ).

% round_numeral
thf(fact_2668_round__1,axiom,
    ! [A: $tType] :
      ( ( archim2362893244070406136eiling @ A )
     => ( ( archimedean_round @ A @ ( one_one @ A ) )
        = ( one_one @ int ) ) ) ).

% round_1
thf(fact_2669_fst__divmod__nat,axiom,
    ! [M: nat,N2: nat] :
      ( ( product_fst @ nat @ nat @ ( divmod_nat @ M @ N2 ) )
      = ( divide_divide @ nat @ M @ N2 ) ) ).

% fst_divmod_nat
thf(fact_2670_snd__divmod__nat,axiom,
    ! [M: nat,N2: nat] :
      ( ( product_snd @ nat @ nat @ ( divmod_nat @ M @ N2 ) )
      = ( modulo_modulo @ nat @ M @ N2 ) ) ).

% snd_divmod_nat
thf(fact_2671_mem__case__prodE,axiom,
    ! [B: $tType,A: $tType,C: $tType,Z: A,C2: B > C > ( set @ A ),P4: product_prod @ B @ C] :
      ( ( member @ A @ Z @ ( product_case_prod @ B @ C @ ( set @ A ) @ C2 @ P4 ) )
     => ~ ! [X3: B,Y4: C] :
            ( ( P4
              = ( product_Pair @ B @ C @ X3 @ Y4 ) )
           => ~ ( member @ A @ Z @ ( C2 @ X3 @ Y4 ) ) ) ) ).

% mem_case_prodE
thf(fact_2672_subset__UNIV,axiom,
    ! [A: $tType,A3: set @ A] : ( ord_less_eq @ ( set @ A ) @ A3 @ ( top_top @ ( set @ A ) ) ) ).

% subset_UNIV
thf(fact_2673_case__prodE,axiom,
    ! [A: $tType,B: $tType,C2: A > B > $o,P4: product_prod @ A @ B] :
      ( ( product_case_prod @ A @ B @ $o @ C2 @ P4 )
     => ~ ! [X3: A,Y4: B] :
            ( ( P4
              = ( product_Pair @ A @ B @ X3 @ Y4 ) )
           => ~ ( C2 @ X3 @ Y4 ) ) ) ).

% case_prodE
thf(fact_2674_case__prodD,axiom,
    ! [A: $tType,B: $tType,F: A > B > $o,A2: A,B2: B] :
      ( ( product_case_prod @ A @ B @ $o @ F @ ( product_Pair @ A @ B @ A2 @ B2 ) )
     => ( F @ A2 @ B2 ) ) ).

% case_prodD
thf(fact_2675_case__prodE_H,axiom,
    ! [B: $tType,A: $tType,C: $tType,C2: A > B > C > $o,P4: product_prod @ A @ B,Z: C] :
      ( ( product_case_prod @ A @ B @ ( C > $o ) @ C2 @ P4 @ Z )
     => ~ ! [X3: A,Y4: B] :
            ( ( P4
              = ( product_Pair @ A @ B @ X3 @ Y4 ) )
           => ~ ( C2 @ X3 @ Y4 @ Z ) ) ) ).

% case_prodE'
thf(fact_2676_case__prodD_H,axiom,
    ! [B: $tType,A: $tType,C: $tType,R2: A > B > C > $o,A2: A,B2: B,C2: C] :
      ( ( product_case_prod @ A @ B @ ( C > $o ) @ R2 @ ( product_Pair @ A @ B @ A2 @ B2 ) @ C2 )
     => ( R2 @ A2 @ B2 @ C2 ) ) ).

% case_prodD'
thf(fact_2677_hoare__triple__refines,axiom,
    ! [A: $tType,P: assn,C2: heap_Time_Heap @ A,Q: A > assn,C6: heap_Time_Heap @ A] :
      ( ( hoare_hoare_triple @ A @ P @ C2 @ Q )
     => ( ( refine_Imp_refines @ A @ C6 @ C2 )
       => ( hoare_hoare_triple @ A @ P @ C6 @ Q ) ) ) ).

% hoare_triple_refines
thf(fact_2678_htt__refine,axiom,
    ! [A: $tType,P: assn,C2: heap_Time_Heap @ A,Q: A > assn,T2: nat,C6: heap_Time_Heap @ A] :
      ( ( time_htt @ A @ P @ C2 @ Q @ T2 )
     => ( ( refine_Imp_refines @ A @ C6 @ C2 )
       => ( time_htt @ A @ P @ C6 @ Q @ T2 ) ) ) ).

% htt_refine
thf(fact_2679_refines__bind,axiom,
    ! [B: $tType,A: $tType,M: heap_Time_Heap @ A,M6: heap_Time_Heap @ A,F: A > ( heap_Time_Heap @ B ),F4: A > ( heap_Time_Heap @ B )] :
      ( ( refine_Imp_refines @ A @ M @ M6 )
     => ( ! [X3: A] : ( refine_Imp_refines @ B @ ( F @ X3 ) @ ( F4 @ X3 ) )
       => ( refine_Imp_refines @ B @ ( heap_Time_bind @ A @ B @ M @ F ) @ ( heap_Time_bind @ A @ B @ M6 @ F4 ) ) ) ) ).

% refines_bind
thf(fact_2680_Product__Type_OCollect__case__prodD,axiom,
    ! [B: $tType,A: $tType,X2: product_prod @ A @ B,A3: A > B > $o] :
      ( ( member @ ( product_prod @ A @ B ) @ X2 @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ A3 ) ) )
     => ( A3 @ ( product_fst @ A @ B @ X2 ) @ ( product_snd @ A @ B @ X2 ) ) ) ).

% Product_Type.Collect_case_prodD
thf(fact_2681_refines__case__prod__right,axiom,
    ! [C: $tType,B: $tType,A: $tType,M: heap_Time_Heap @ C,M6: A > B > ( heap_Time_Heap @ C ),T2: product_prod @ A @ B] :
      ( ! [A4: A,B3: B] : ( refine_Imp_refines @ C @ M @ ( M6 @ A4 @ B3 ) )
     => ( refine_Imp_refines @ C @ M @ ( product_case_prod @ A @ B @ ( heap_Time_Heap @ C ) @ M6 @ T2 ) ) ) ).

% refines_case_prod_right
thf(fact_2682_prod__case__refines,axiom,
    ! [C: $tType,B: $tType,A: $tType,P4: product_prod @ A @ B,P8: product_prod @ A @ B,F: A > B > ( heap_Time_Heap @ C ),F4: A > B > ( heap_Time_Heap @ C )] :
      ( ( P4 = P8 )
     => ( ! [A4: A,B3: B] : ( refine_Imp_refines @ C @ ( F @ A4 @ B3 ) @ ( F4 @ A4 @ B3 ) )
       => ( refine_Imp_refines @ C @ ( product_case_prod @ A @ B @ ( heap_Time_Heap @ C ) @ F @ P4 ) @ ( product_case_prod @ A @ B @ ( heap_Time_Heap @ C ) @ F4 @ P8 ) ) ) ) ).

% prod_case_refines
thf(fact_2683_refines__option,axiom,
    ! [B: $tType,A: $tType,A2: option @ A,A6: option @ A,M1: heap_Time_Heap @ B,M12: heap_Time_Heap @ B,M22: A > ( heap_Time_Heap @ B ),M23: A > ( heap_Time_Heap @ B )] :
      ( ( A2 = A6 )
     => ( ( refine_Imp_refines @ B @ M1 @ M12 )
       => ( ! [X3: A] : ( refine_Imp_refines @ B @ ( M22 @ X3 ) @ ( M23 @ X3 ) )
         => ( refine_Imp_refines @ B @ ( case_option @ ( heap_Time_Heap @ B ) @ A @ M1 @ M22 @ A2 ) @ ( case_option @ ( heap_Time_Heap @ B ) @ A @ M12 @ M23 @ A6 ) ) ) ) ) ).

% refines_option
thf(fact_2684_not__UNIV__le__Icc,axiom,
    ! [A: $tType] :
      ( ( no_top @ A )
     => ! [L: A,H2: A] :
          ~ ( ord_less_eq @ ( set @ A ) @ ( top_top @ ( set @ A ) ) @ ( set_or1337092689740270186AtMost @ A @ L @ H2 ) ) ) ).

% not_UNIV_le_Icc
thf(fact_2685_round__mono,axiom,
    ! [A: $tType] :
      ( ( archim2362893244070406136eiling @ A )
     => ! [X2: A,Y2: A] :
          ( ( ord_less_eq @ A @ X2 @ Y2 )
         => ( ord_less_eq @ int @ ( archimedean_round @ A @ X2 ) @ ( archimedean_round @ A @ Y2 ) ) ) ) ).

% round_mono
thf(fact_2686_divmod__nat__if,axiom,
    ( divmod_nat
    = ( ^ [M7: nat,N: nat] :
          ( if @ ( product_prod @ nat @ nat )
          @ ( ( N
              = ( zero_zero @ nat ) )
            | ( ord_less @ nat @ M7 @ N ) )
          @ ( product_Pair @ nat @ nat @ ( zero_zero @ nat ) @ M7 )
          @ ( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
            @ ^ [Q5: nat] : ( product_Pair @ nat @ nat @ ( suc @ Q5 ) )
            @ ( divmod_nat @ ( minus_minus @ nat @ M7 @ N ) @ N ) ) ) ) ) ).

% divmod_nat_if
thf(fact_2687_ceiling__ge__round,axiom,
    ! [A: $tType] :
      ( ( archim2362893244070406136eiling @ A )
     => ! [X2: A] : ( ord_less_eq @ int @ ( archimedean_round @ A @ X2 ) @ ( archimedean_ceiling @ A @ X2 ) ) ) ).

% ceiling_ge_round
thf(fact_2688_refines__assert_H__bind,axiom,
    ! [A: $tType,P4: heap_Time_Heap @ A,Q2: heap_Time_Heap @ A,Phi: $o] :
      ( ( refine_Imp_refines @ A @ P4 @ Q2 )
     => ( refine_Imp_refines @ A @ P4
        @ ( heap_Time_bind @ product_unit @ A @ ( refine_Imp_assert @ Phi )
          @ ^ [Uu3: product_unit] : Q2 ) ) ) ).

% refines_assert'_bind
thf(fact_2689_TBOUND__prod__case,axiom,
    ! [C: $tType,B: $tType,A: $tType,T2: product_prod @ A @ B,F: A > B > ( heap_Time_Heap @ C ),Bnd: A > B > nat] :
      ( ! [A4: A,B3: B] :
          ( ( T2
            = ( product_Pair @ A @ B @ A4 @ B3 ) )
         => ( time_TBOUND @ C @ ( F @ A4 @ B3 ) @ ( Bnd @ A4 @ B3 ) ) )
     => ( time_TBOUND @ C @ ( product_case_prod @ A @ B @ ( heap_Time_Heap @ C ) @ F @ T2 ) @ ( product_case_prod @ A @ B @ nat @ Bnd @ T2 ) ) ) ).

% TBOUND_prod_case
thf(fact_2690_rel__of__def,axiom,
    ! [B: $tType,A: $tType] :
      ( ( rel_of @ A @ B )
      = ( ^ [M7: A > ( option @ B ),P3: ( product_prod @ A @ B ) > $o] :
            ( collect @ ( product_prod @ A @ B )
            @ ( product_case_prod @ A @ B @ $o
              @ ^ [K4: A,V5: B] :
                  ( ( ( M7 @ K4 )
                    = ( some @ B @ V5 ) )
                  & ( P3 @ ( product_Pair @ A @ B @ K4 @ V5 ) ) ) ) ) ) ) ).

% rel_of_def
thf(fact_2691_divmod__nat__def,axiom,
    ( divmod_nat
    = ( ^ [M7: nat,N: nat] : ( product_Pair @ nat @ nat @ ( divide_divide @ nat @ M7 @ N ) @ ( modulo_modulo @ nat @ M7 @ N ) ) ) ) ).

% divmod_nat_def
thf(fact_2692_divmod__step__int__def,axiom,
    ( ( unique1321980374590559556d_step @ int )
    = ( ^ [L2: num] :
          ( product_case_prod @ int @ int @ ( product_prod @ int @ int )
          @ ^ [Q5: int,R: int] : ( if @ ( product_prod @ int @ int ) @ ( ord_less_eq @ int @ ( numeral_numeral @ int @ L2 ) @ R ) @ ( product_Pair @ int @ int @ ( plus_plus @ int @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ Q5 ) @ ( one_one @ int ) ) @ ( minus_minus @ int @ R @ ( numeral_numeral @ int @ L2 ) ) ) @ ( product_Pair @ int @ int @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ Q5 ) @ R ) ) ) ) ) ).

% divmod_step_int_def
thf(fact_2693_divmod__step__nat__def,axiom,
    ( ( unique1321980374590559556d_step @ nat )
    = ( ^ [L2: num] :
          ( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
          @ ^ [Q5: nat,R: nat] : ( if @ ( product_prod @ nat @ nat ) @ ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ L2 ) @ R ) @ ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Q5 ) @ ( one_one @ nat ) ) @ ( minus_minus @ nat @ R @ ( numeral_numeral @ nat @ L2 ) ) ) @ ( product_Pair @ nat @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Q5 ) @ R ) ) ) ) ) ).

% divmod_step_nat_def
thf(fact_2694_divmod__step__def,axiom,
    ! [A: $tType] :
      ( ( unique1627219031080169319umeral @ A )
     => ( ( unique1321980374590559556d_step @ A )
        = ( ^ [L2: num] :
              ( product_case_prod @ A @ A @ ( product_prod @ A @ A )
              @ ^ [Q5: A,R: A] : ( if @ ( product_prod @ A @ A ) @ ( ord_less_eq @ A @ ( numeral_numeral @ A @ L2 ) @ R ) @ ( product_Pair @ A @ A @ ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ Q5 ) @ ( one_one @ A ) ) @ ( minus_minus @ A @ R @ ( numeral_numeral @ A @ L2 ) ) ) @ ( product_Pair @ A @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ Q5 ) @ R ) ) ) ) ) ) ).

% divmod_step_def
thf(fact_2695_mult__less__iff1,axiom,
    ! [A: $tType] :
      ( ( linordered_idom @ A )
     => ! [Z: A,X2: A,Y2: A] :
          ( ( ord_less @ A @ ( zero_zero @ A ) @ Z )
         => ( ( ord_less @ A @ ( times_times @ A @ X2 @ Z ) @ ( times_times @ A @ Y2 @ Z ) )
            = ( ord_less @ A @ X2 @ Y2 ) ) ) ) ).

% mult_less_iff1
thf(fact_2696_of__int__round__le,axiom,
    ! [A: $tType] :
      ( ( archim2362893244070406136eiling @ A )
     => ! [X2: A] : ( ord_less_eq @ A @ ( ring_1_of_int @ A @ ( archimedean_round @ A @ X2 ) ) @ ( plus_plus @ A @ X2 @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).

% of_int_round_le
thf(fact_2697_of__int__round__ge,axiom,
    ! [A: $tType] :
      ( ( archim2362893244070406136eiling @ A )
     => ! [X2: A] : ( ord_less_eq @ A @ ( minus_minus @ A @ X2 @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) @ ( ring_1_of_int @ A @ ( archimedean_round @ A @ X2 ) ) ) ) ).

% of_int_round_ge
thf(fact_2698_of__int__round__gt,axiom,
    ! [A: $tType] :
      ( ( archim2362893244070406136eiling @ A )
     => ! [X2: A] : ( ord_less @ A @ ( minus_minus @ A @ X2 @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) @ ( ring_1_of_int @ A @ ( archimedean_round @ A @ X2 ) ) ) ) ).

% of_int_round_gt
thf(fact_2699_mult__le__cancel__iff2,axiom,
    ! [A: $tType] :
      ( ( linordered_idom @ A )
     => ! [Z: A,X2: A,Y2: A] :
          ( ( ord_less @ A @ ( zero_zero @ A ) @ Z )
         => ( ( ord_less_eq @ A @ ( times_times @ A @ Z @ X2 ) @ ( times_times @ A @ Z @ Y2 ) )
            = ( ord_less_eq @ A @ X2 @ Y2 ) ) ) ) ).

% mult_le_cancel_iff2
thf(fact_2700_vebt__inserti_Oraw__induct,axiom,
    ! [P: vEBT_VEBTi > nat > ( heap_ext @ product_unit ) > ( heap_ext @ product_unit ) > vEBT_VEBTi > nat > $o,Xa: product_prod @ vEBT_VEBTi @ nat,H2: heap_ext @ product_unit,H4: heap_ext @ product_unit,R4: vEBT_VEBTi,N2: nat] :
      ( ! [Vebt_inserti: vEBT_VEBTi > nat > ( heap_Time_Heap @ vEBT_VEBTi )] :
          ( ! [A8: vEBT_VEBTi,B8: nat,H6: heap_ext @ product_unit,H7: heap_ext @ product_unit,R6: vEBT_VEBTi,N9: nat] :
              ( ( heap_Time_effect @ vEBT_VEBTi @ ( Vebt_inserti @ A8 @ B8 ) @ H6 @ H7 @ R6 @ N9 )
             => ( P @ A8 @ B8 @ H6 @ H7 @ R6 @ N9 ) )
         => ! [T5: vEBT_VEBTi,X3: nat,Ta: heap_ext @ product_unit,Xa3: heap_ext @ product_unit,R3: vEBT_VEBTi,N4: nat] :
              ( ( heap_Time_effect @ vEBT_VEBTi
                @ ( vEBT_case_VEBTi @ ( heap_Time_Heap @ vEBT_VEBTi )
                  @ ^ [Info3: option @ ( product_prod @ nat @ nat ),Deg4: nat,TreeArray: array @ vEBT_VEBTi,Summary3: vEBT_VEBTi] :
                      ( case_option @ ( heap_Time_Heap @ vEBT_VEBTi ) @ ( product_prod @ nat @ nat ) @ ( if @ ( heap_Time_Heap @ vEBT_VEBTi ) @ ( ord_less_eq @ nat @ Deg4 @ ( one_one @ nat ) ) @ ( heap_Time_return @ vEBT_VEBTi @ ( vEBT_Nodei @ Info3 @ Deg4 @ TreeArray @ Summary3 ) ) @ ( heap_Time_return @ vEBT_VEBTi @ ( vEBT_Nodei @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ X3 @ X3 ) ) @ Deg4 @ TreeArray @ Summary3 ) ) )
                      @ ^ [Minma: product_prod @ nat @ nat] :
                          ( if @ ( heap_Time_Heap @ vEBT_VEBTi ) @ ( ord_less_eq @ nat @ Deg4 @ ( one_one @ nat ) ) @ ( heap_Time_return @ vEBT_VEBTi @ ( vEBT_Nodei @ Info3 @ Deg4 @ TreeArray @ Summary3 ) )
                          @ ( heap_Time_bind @ nat @ vEBT_VEBTi @ ( heap_Time_return @ nat @ ( product_fst @ nat @ nat @ Minma ) )
                            @ ^ [Mi3: nat] :
                                ( heap_Time_bind @ nat @ vEBT_VEBTi @ ( heap_Time_return @ nat @ ( product_snd @ nat @ nat @ Minma ) )
                                @ ^ [Ma3: nat] :
                                    ( heap_Time_bind @ nat @ vEBT_VEBTi @ ( if @ ( heap_Time_Heap @ nat ) @ ( ord_less @ nat @ X3 @ Mi3 ) @ ( heap_Time_return @ nat @ Mi3 ) @ ( heap_Time_return @ nat @ X3 ) )
                                    @ ^ [Xn2: nat] :
                                        ( heap_Time_bind @ nat @ vEBT_VEBTi @ ( if @ ( heap_Time_Heap @ nat ) @ ( ord_less @ nat @ X3 @ Mi3 ) @ ( heap_Time_return @ nat @ X3 ) @ ( heap_Time_return @ nat @ Mi3 ) )
                                        @ ^ [Minn: nat] :
                                            ( heap_Time_bind @ nat @ vEBT_VEBTi @ ( vEBT_VEBT_lowi @ Xn2 @ ( divide_divide @ nat @ Deg4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
                                            @ ^ [L2: nat] :
                                                ( heap_Time_bind @ nat @ vEBT_VEBTi @ ( vEBT_VEBT_highi @ Xn2 @ ( divide_divide @ nat @ Deg4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
                                                @ ^ [H: nat] :
                                                    ( heap_Time_bind @ nat @ vEBT_VEBTi @ ( array_len @ vEBT_VEBTi @ TreeArray )
                                                    @ ^ [Len: nat] :
                                                        ( if @ ( heap_Time_Heap @ vEBT_VEBTi )
                                                        @ ( ( ord_less @ nat @ H @ Len )
                                                          & ~ ( ( X3 = Mi3 )
                                                              | ( X3 = Ma3 ) ) )
                                                        @ ( heap_Time_bind @ vEBT_VEBTi @ vEBT_VEBTi @ ( array_nth @ vEBT_VEBTi @ TreeArray @ H )
                                                          @ ^ [Node: vEBT_VEBTi] :
                                                              ( heap_Time_bind @ $o @ vEBT_VEBTi @ ( vEBT_VEBT_minNulli @ Node )
                                                              @ ^ [Empt: $o] :
                                                                  ( heap_Time_bind @ vEBT_VEBTi @ vEBT_VEBTi @ ( Vebt_inserti @ Node @ L2 )
                                                                  @ ^ [Newnode2: vEBT_VEBTi] :
                                                                      ( heap_Time_bind @ ( array @ vEBT_VEBTi ) @ vEBT_VEBTi @ ( array_upd @ vEBT_VEBTi @ H @ Newnode2 @ TreeArray )
                                                                      @ ^ [Newarray: array @ vEBT_VEBTi] :
                                                                          ( heap_Time_bind @ vEBT_VEBTi @ vEBT_VEBTi @ ( if @ ( heap_Time_Heap @ vEBT_VEBTi ) @ Empt @ ( Vebt_inserti @ Summary3 @ H ) @ ( heap_Time_return @ vEBT_VEBTi @ Summary3 ) )
                                                                          @ ^ [Newsummary: vEBT_VEBTi] :
                                                                              ( heap_Time_bind @ nat @ vEBT_VEBTi @ ( if @ ( heap_Time_Heap @ nat ) @ ( ord_less @ nat @ Ma3 @ Xn2 ) @ ( heap_Time_return @ nat @ Xn2 ) @ ( heap_Time_return @ nat @ Ma3 ) )
                                                                              @ ^ [Man: nat] : ( heap_Time_return @ vEBT_VEBTi @ ( vEBT_Nodei @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Minn @ Man ) ) @ Deg4 @ Newarray @ Newsummary ) ) ) ) ) ) ) )
                                                        @ ( heap_Time_return @ vEBT_VEBTi @ ( vEBT_Nodei @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi3 @ Ma3 ) ) @ Deg4 @ TreeArray @ Summary3 ) ) ) ) ) ) ) ) ) ) )
                      @ Info3 )
                  @ ^ [A5: $o,B5: $o] :
                      ( if @ ( heap_Time_Heap @ vEBT_VEBTi )
                      @ ( X3
                        = ( zero_zero @ nat ) )
                      @ ( heap_Time_return @ vEBT_VEBTi @ ( vEBT_Leafi @ $true @ B5 ) )
                      @ ( if @ ( heap_Time_Heap @ vEBT_VEBTi )
                        @ ( X3
                          = ( one_one @ nat ) )
                        @ ( heap_Time_return @ vEBT_VEBTi @ ( vEBT_Leafi @ A5 @ $true ) )
                        @ ( heap_Time_return @ vEBT_VEBTi @ ( vEBT_Leafi @ A5 @ B5 ) ) ) )
                  @ T5 )
                @ Ta
                @ Xa3
                @ R3
                @ N4 )
             => ( P @ T5 @ X3 @ Ta @ Xa3 @ R3 @ N4 ) ) )
     => ( ( heap_Time_effect @ vEBT_VEBTi @ ( product_case_prod @ vEBT_VEBTi @ nat @ ( heap_Time_Heap @ vEBT_VEBTi ) @ vEBT_vebt_inserti @ Xa ) @ H2 @ H4 @ R4 @ N2 )
       => ( product_case_prod @ vEBT_VEBTi @ nat @ ( ( heap_ext @ product_unit ) > ( heap_ext @ product_unit ) > vEBT_VEBTi > nat > $o ) @ P @ Xa @ H2 @ H4 @ R4 @ N2 ) ) ) ).

% vebt_inserti.raw_induct
thf(fact_2701_vebt__predi_Oraw__induct,axiom,
    ! [P: vEBT_VEBTi > nat > ( heap_ext @ product_unit ) > ( heap_ext @ product_unit ) > ( option @ nat ) > nat > $o,Xa: product_prod @ vEBT_VEBTi @ nat,H2: heap_ext @ product_unit,H4: heap_ext @ product_unit,R4: option @ nat,N2: nat] :
      ( ! [Vebt_predi: vEBT_VEBTi > nat > ( heap_Time_Heap @ ( option @ nat ) )] :
          ( ! [A8: vEBT_VEBTi,B8: nat,H6: heap_ext @ product_unit,H7: heap_ext @ product_unit,R6: option @ nat,N9: nat] :
              ( ( heap_Time_effect @ ( option @ nat ) @ ( Vebt_predi @ A8 @ B8 ) @ H6 @ H7 @ R6 @ N9 )
             => ( P @ A8 @ B8 @ H6 @ H7 @ R6 @ N9 ) )
         => ! [T5: vEBT_VEBTi,X3: nat,Ta: heap_ext @ product_unit,Xa3: heap_ext @ product_unit,R3: option @ nat,N4: nat] :
              ( ( heap_Time_effect @ ( option @ nat )
                @ ( vEBT_case_VEBTi @ ( heap_Time_Heap @ ( option @ nat ) )
                  @ ^ [Info3: option @ ( product_prod @ nat @ nat ),Deg4: nat,TreeArray: array @ vEBT_VEBTi,Summary3: vEBT_VEBTi] :
                      ( case_option @ ( heap_Time_Heap @ ( option @ nat ) ) @ ( product_prod @ nat @ nat ) @ ( heap_Time_return @ ( option @ nat ) @ ( none @ nat ) )
                      @ ^ [Mima2: product_prod @ nat @ nat] :
                          ( if @ ( heap_Time_Heap @ ( option @ nat ) ) @ ( ord_less_eq @ nat @ Deg4 @ ( one_one @ nat ) ) @ ( heap_Time_return @ ( option @ nat ) @ ( none @ nat ) )
                          @ ( if @ ( heap_Time_Heap @ ( option @ nat ) ) @ ( ord_less @ nat @ ( product_snd @ nat @ nat @ Mima2 ) @ X3 ) @ ( heap_Time_return @ ( option @ nat ) @ ( some @ nat @ ( product_snd @ nat @ nat @ Mima2 ) ) )
                            @ ( heap_Time_bind @ nat @ ( option @ nat ) @ ( vEBT_VEBT_lowi @ X3 @ ( divide_divide @ nat @ Deg4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
                              @ ^ [L2: nat] :
                                  ( heap_Time_bind @ nat @ ( option @ nat ) @ ( vEBT_VEBT_highi @ X3 @ ( divide_divide @ nat @ Deg4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
                                  @ ^ [H: nat] :
                                      ( heap_Time_bind @ vEBT_VEBTi @ ( option @ nat ) @ ( array_nth @ vEBT_VEBTi @ TreeArray @ H )
                                      @ ^ [Aktnode: vEBT_VEBTi] :
                                          ( heap_Time_bind @ ( option @ nat ) @ ( option @ nat ) @ ( vEBT_vebt_minti @ Aktnode )
                                          @ ^ [Minlow: option @ nat] :
                                              ( if @ ( heap_Time_Heap @ ( option @ nat ) )
                                              @ ( ( Minlow
                                                 != ( none @ nat ) )
                                                & ( vEBT_VEBT_greater @ ( some @ nat @ L2 ) @ Minlow ) )
                                              @ ( heap_Time_bind @ ( option @ nat ) @ ( option @ nat ) @ ( Vebt_predi @ Aktnode @ L2 )
                                                @ ^ [Predy: option @ nat] : ( heap_Time_return @ ( option @ nat ) @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( some @ nat @ H ) ) @ Predy ) ) )
                                              @ ( heap_Time_bind @ ( option @ nat ) @ ( option @ nat ) @ ( Vebt_predi @ Summary3 @ H )
                                                @ ^ [Predsum: option @ nat] :
                                                    ( if @ ( heap_Time_Heap @ ( option @ nat ) )
                                                    @ ( Predsum
                                                      = ( none @ nat ) )
                                                    @ ( if @ ( heap_Time_Heap @ ( option @ nat ) ) @ ( ord_less @ nat @ ( product_fst @ nat @ nat @ Mima2 ) @ X3 ) @ ( heap_Time_return @ ( option @ nat ) @ ( some @ nat @ ( product_fst @ nat @ nat @ Mima2 ) ) ) @ ( heap_Time_return @ ( option @ nat ) @ ( none @ nat ) ) )
                                                    @ ( heap_Time_bind @ vEBT_VEBTi @ ( option @ nat ) @ ( array_nth @ vEBT_VEBTi @ TreeArray @ ( the2 @ nat @ Predsum ) )
                                                      @ ^ [Nextnode: vEBT_VEBTi] :
                                                          ( heap_Time_bind @ ( option @ nat ) @ ( option @ nat ) @ ( vEBT_vebt_maxti @ Nextnode )
                                                          @ ^ [Maxnext: option @ nat] : ( heap_Time_return @ ( option @ nat ) @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ Predsum ) @ Maxnext ) ) ) ) ) ) ) ) ) ) ) ) )
                      @ Info3 )
                  @ ^ [A5: $o,B5: $o] :
                      ( if @ ( heap_Time_Heap @ ( option @ nat ) ) @ ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ X3 ) @ ( if @ ( heap_Time_Heap @ ( option @ nat ) ) @ B5 @ ( heap_Time_return @ ( option @ nat ) @ ( some @ nat @ ( one_one @ nat ) ) ) @ ( if @ ( heap_Time_Heap @ ( option @ nat ) ) @ A5 @ ( heap_Time_return @ ( option @ nat ) @ ( some @ nat @ ( zero_zero @ nat ) ) ) @ ( heap_Time_return @ ( option @ nat ) @ ( none @ nat ) ) ) )
                      @ ( if @ ( heap_Time_Heap @ ( option @ nat ) )
                        @ ( X3
                          = ( one_one @ nat ) )
                        @ ( if @ ( heap_Time_Heap @ ( option @ nat ) ) @ A5 @ ( heap_Time_return @ ( option @ nat ) @ ( some @ nat @ ( zero_zero @ nat ) ) ) @ ( heap_Time_return @ ( option @ nat ) @ ( none @ nat ) ) )
                        @ ( heap_Time_return @ ( option @ nat ) @ ( none @ nat ) ) ) )
                  @ T5 )
                @ Ta
                @ Xa3
                @ R3
                @ N4 )
             => ( P @ T5 @ X3 @ Ta @ Xa3 @ R3 @ N4 ) ) )
     => ( ( heap_Time_effect @ ( option @ nat ) @ ( product_case_prod @ vEBT_VEBTi @ nat @ ( heap_Time_Heap @ ( option @ nat ) ) @ vEBT_vebt_predi @ Xa ) @ H2 @ H4 @ R4 @ N2 )
       => ( product_case_prod @ vEBT_VEBTi @ nat @ ( ( heap_ext @ product_unit ) > ( heap_ext @ product_unit ) > ( option @ nat ) > nat > $o ) @ P @ Xa @ H2 @ H4 @ R4 @ N2 ) ) ) ).

% vebt_predi.raw_induct
thf(fact_2702_vebt__succi_Oraw__induct,axiom,
    ! [P: vEBT_VEBTi > nat > ( heap_ext @ product_unit ) > ( heap_ext @ product_unit ) > ( option @ nat ) > nat > $o,Xa: product_prod @ vEBT_VEBTi @ nat,H2: heap_ext @ product_unit,H4: heap_ext @ product_unit,R4: option @ nat,N2: nat] :
      ( ! [Vebt_succi: vEBT_VEBTi > nat > ( heap_Time_Heap @ ( option @ nat ) )] :
          ( ! [A8: vEBT_VEBTi,B8: nat,H6: heap_ext @ product_unit,H7: heap_ext @ product_unit,R6: option @ nat,N9: nat] :
              ( ( heap_Time_effect @ ( option @ nat ) @ ( Vebt_succi @ A8 @ B8 ) @ H6 @ H7 @ R6 @ N9 )
             => ( P @ A8 @ B8 @ H6 @ H7 @ R6 @ N9 ) )
         => ! [T5: vEBT_VEBTi,X3: nat,Ta: heap_ext @ product_unit,Xa3: heap_ext @ product_unit,R3: option @ nat,N4: nat] :
              ( ( heap_Time_effect @ ( option @ nat )
                @ ( vEBT_case_VEBTi @ ( heap_Time_Heap @ ( option @ nat ) )
                  @ ^ [Info3: option @ ( product_prod @ nat @ nat ),Deg4: nat,TreeArray: array @ vEBT_VEBTi,Summary3: vEBT_VEBTi] :
                      ( case_option @ ( heap_Time_Heap @ ( option @ nat ) ) @ ( product_prod @ nat @ nat ) @ ( heap_Time_return @ ( option @ nat ) @ ( none @ nat ) )
                      @ ^ [Mima2: product_prod @ nat @ nat] :
                          ( if @ ( heap_Time_Heap @ ( option @ nat ) ) @ ( ord_less_eq @ nat @ Deg4 @ ( one_one @ nat ) ) @ ( heap_Time_return @ ( option @ nat ) @ ( none @ nat ) )
                          @ ( if @ ( heap_Time_Heap @ ( option @ nat ) ) @ ( ord_less @ nat @ X3 @ ( product_fst @ nat @ nat @ Mima2 ) ) @ ( heap_Time_return @ ( option @ nat ) @ ( some @ nat @ ( product_fst @ nat @ nat @ Mima2 ) ) )
                            @ ( if @ ( heap_Time_Heap @ ( option @ nat ) ) @ ( ord_less_eq @ nat @ ( product_snd @ nat @ nat @ Mima2 ) @ X3 ) @ ( heap_Time_return @ ( option @ nat ) @ ( none @ nat ) )
                              @ ( heap_Time_bind @ nat @ ( option @ nat ) @ ( vEBT_VEBT_lowi @ X3 @ ( divide_divide @ nat @ Deg4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
                                @ ^ [L2: nat] :
                                    ( heap_Time_bind @ nat @ ( option @ nat ) @ ( vEBT_VEBT_highi @ X3 @ ( divide_divide @ nat @ Deg4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
                                    @ ^ [H: nat] :
                                        ( heap_Time_bind @ vEBT_VEBTi @ ( option @ nat ) @ ( array_nth @ vEBT_VEBTi @ TreeArray @ H )
                                        @ ^ [Aktnode: vEBT_VEBTi] :
                                            ( heap_Time_bind @ ( option @ nat ) @ ( option @ nat ) @ ( vEBT_vebt_maxti @ Aktnode )
                                            @ ^ [Maxlow: option @ nat] :
                                                ( if @ ( heap_Time_Heap @ ( option @ nat ) )
                                                @ ( ( Maxlow
                                                   != ( none @ nat ) )
                                                  & ( vEBT_VEBT_less @ ( some @ nat @ L2 ) @ Maxlow ) )
                                                @ ( heap_Time_bind @ ( option @ nat ) @ ( option @ nat ) @ ( Vebt_succi @ Aktnode @ L2 )
                                                  @ ^ [Succy: option @ nat] : ( heap_Time_return @ ( option @ nat ) @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( some @ nat @ H ) ) @ Succy ) ) )
                                                @ ( heap_Time_bind @ ( option @ nat ) @ ( option @ nat ) @ ( Vebt_succi @ Summary3 @ H )
                                                  @ ^ [Succsum: option @ nat] :
                                                      ( if @ ( heap_Time_Heap @ ( option @ nat ) )
                                                      @ ( Succsum
                                                        = ( none @ nat ) )
                                                      @ ( heap_Time_return @ ( option @ nat ) @ ( none @ nat ) )
                                                      @ ( heap_Time_bind @ vEBT_VEBTi @ ( option @ nat ) @ ( array_nth @ vEBT_VEBTi @ TreeArray @ ( the2 @ nat @ Succsum ) )
                                                        @ ^ [Nextnode: vEBT_VEBTi] :
                                                            ( heap_Time_bind @ ( option @ nat ) @ ( option @ nat ) @ ( vEBT_vebt_minti @ Nextnode )
                                                            @ ^ [Minnext: option @ nat] : ( heap_Time_return @ ( option @ nat ) @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ Succsum ) @ Minnext ) ) ) ) ) ) ) ) ) ) ) ) ) )
                      @ Info3 )
                  @ ^ [A5: $o,B5: $o] :
                      ( if @ ( heap_Time_Heap @ ( option @ nat ) )
                      @ ( X3
                        = ( zero_zero @ nat ) )
                      @ ( if @ ( heap_Time_Heap @ ( option @ nat ) ) @ B5 @ ( heap_Time_return @ ( option @ nat ) @ ( some @ nat @ ( one_one @ nat ) ) ) @ ( heap_Time_return @ ( option @ nat ) @ ( none @ nat ) ) )
                      @ ( heap_Time_return @ ( option @ nat ) @ ( none @ nat ) ) )
                  @ T5 )
                @ Ta
                @ Xa3
                @ R3
                @ N4 )
             => ( P @ T5 @ X3 @ Ta @ Xa3 @ R3 @ N4 ) ) )
     => ( ( heap_Time_effect @ ( option @ nat ) @ ( product_case_prod @ vEBT_VEBTi @ nat @ ( heap_Time_Heap @ ( option @ nat ) ) @ vEBT_vebt_succi @ Xa ) @ H2 @ H4 @ R4 @ N2 )
       => ( product_case_prod @ vEBT_VEBTi @ nat @ ( ( heap_ext @ product_unit ) > ( heap_ext @ product_unit ) > ( option @ nat ) > nat > $o ) @ P @ Xa @ H2 @ H4 @ R4 @ N2 ) ) ) ).

% vebt_succi.raw_induct
thf(fact_2703_vebt__memberi_Oraw__induct,axiom,
    ! [P: vEBT_VEBTi > nat > ( heap_ext @ product_unit ) > ( heap_ext @ product_unit ) > $o > nat > $o,Xa: product_prod @ vEBT_VEBTi @ nat,H2: heap_ext @ product_unit,H4: heap_ext @ product_unit,R4: $o,N2: nat] :
      ( ! [Vebt_memberi: vEBT_VEBTi > nat > ( heap_Time_Heap @ $o )] :
          ( ! [A8: vEBT_VEBTi,B8: nat,H6: heap_ext @ product_unit,H7: heap_ext @ product_unit,R6: $o,N9: nat] :
              ( ( heap_Time_effect @ $o @ ( Vebt_memberi @ A8 @ B8 ) @ H6 @ H7 @ R6 @ N9 )
             => ( P @ A8 @ B8 @ H6 @ H7 @ R6 @ N9 ) )
         => ! [T5: vEBT_VEBTi,X3: nat,Ta: heap_ext @ product_unit,Xa3: heap_ext @ product_unit,R3: $o,N4: nat] :
              ( ( heap_Time_effect @ $o
                @ ( vEBT_case_VEBTi @ ( heap_Time_Heap @ $o )
                  @ ^ [Info3: option @ ( product_prod @ nat @ nat ),Deg4: nat,TreeList3: array @ vEBT_VEBTi,Summary3: vEBT_VEBTi] :
                      ( case_option @ ( heap_Time_Heap @ $o ) @ ( product_prod @ nat @ nat ) @ ( heap_Time_return @ $o @ $false )
                      @ ( product_case_prod @ nat @ nat @ ( heap_Time_Heap @ $o )
                        @ ^ [Mi3: nat,Ma3: nat] :
                            ( if @ ( heap_Time_Heap @ $o ) @ ( ord_less_eq @ nat @ Deg4 @ ( one_one @ nat ) ) @ ( heap_Time_return @ $o @ $false )
                            @ ( if @ ( heap_Time_Heap @ $o ) @ ( X3 = Mi3 ) @ ( heap_Time_return @ $o @ $true )
                              @ ( if @ ( heap_Time_Heap @ $o ) @ ( X3 = Ma3 ) @ ( heap_Time_return @ $o @ $true )
                                @ ( if @ ( heap_Time_Heap @ $o ) @ ( ord_less @ nat @ X3 @ Mi3 ) @ ( heap_Time_return @ $o @ $false )
                                  @ ( if @ ( heap_Time_Heap @ $o ) @ ( ord_less @ nat @ Ma3 @ X3 ) @ ( heap_Time_return @ $o @ $false )
                                    @ ( heap_Time_bind @ nat @ $o @ ( vEBT_VEBT_highi @ X3 @ ( divide_divide @ nat @ Deg4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
                                      @ ^ [H: nat] :
                                          ( heap_Time_bind @ nat @ $o @ ( vEBT_VEBT_lowi @ X3 @ ( divide_divide @ nat @ Deg4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
                                          @ ^ [L2: nat] :
                                              ( heap_Time_bind @ nat @ $o @ ( array_len @ vEBT_VEBTi @ TreeList3 )
                                              @ ^ [Len: nat] :
                                                  ( if @ ( heap_Time_Heap @ $o ) @ ( ord_less @ nat @ H @ Len )
                                                  @ ( heap_Time_bind @ vEBT_VEBTi @ $o @ ( array_nth @ vEBT_VEBTi @ TreeList3 @ H )
                                                    @ ^ [Th: vEBT_VEBTi] : ( Vebt_memberi @ Th @ L2 ) )
                                                  @ ( heap_Time_return @ $o @ $false ) ) ) ) ) ) ) ) ) ) )
                      @ Info3 )
                  @ ^ [A5: $o,B5: $o] :
                      ( heap_Time_return @ $o
                      @ ( ( ( X3
                            = ( zero_zero @ nat ) )
                         => A5 )
                        & ( ( X3
                           != ( zero_zero @ nat ) )
                         => ( ( ( X3
                                = ( one_one @ nat ) )
                             => B5 )
                            & ( X3
                              = ( one_one @ nat ) ) ) ) ) )
                  @ T5 )
                @ Ta
                @ Xa3
                @ R3
                @ N4 )
             => ( P @ T5 @ X3 @ Ta @ Xa3 @ R3 @ N4 ) ) )
     => ( ( heap_Time_effect @ $o @ ( product_case_prod @ vEBT_VEBTi @ nat @ ( heap_Time_Heap @ $o ) @ vEBT_vebt_memberi @ Xa ) @ H2 @ H4 @ R4 @ N2 )
       => ( product_case_prod @ vEBT_VEBTi @ nat @ ( ( heap_ext @ product_unit ) > ( heap_ext @ product_unit ) > $o > nat > $o ) @ P @ Xa @ H2 @ H4 @ R4 @ N2 ) ) ) ).

% vebt_memberi.raw_induct
thf(fact_2704_divmod__step__integer__def,axiom,
    ( ( unique1321980374590559556d_step @ code_integer )
    = ( ^ [L2: num] :
          ( product_case_prod @ code_integer @ code_integer @ ( product_prod @ code_integer @ code_integer )
          @ ^ [Q5: code_integer,R: code_integer] : ( if @ ( product_prod @ code_integer @ code_integer ) @ ( ord_less_eq @ code_integer @ ( numeral_numeral @ code_integer @ L2 ) @ R ) @ ( product_Pair @ code_integer @ code_integer @ ( plus_plus @ code_integer @ ( times_times @ code_integer @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) @ Q5 ) @ ( one_one @ code_integer ) ) @ ( minus_minus @ code_integer @ R @ ( numeral_numeral @ code_integer @ L2 ) ) ) @ ( product_Pair @ code_integer @ code_integer @ ( times_times @ code_integer @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) @ Q5 ) @ R ) ) ) ) ) ).

% divmod_step_integer_def
thf(fact_2705_VEBT__internal_Ovebt__succi_H_Osimps,axiom,
    ( vEBT_VEBT_vebt_succi
    = ( ^ [T3: vEBT_VEBT,Ti3: vEBT_VEBTi,X: nat] :
          ( vEBT_case_VEBTi @ ( heap_Time_Heap @ ( option @ nat ) )
          @ ^ [Info3: option @ ( product_prod @ nat @ nat ),Deg4: nat,TreeArray: array @ vEBT_VEBTi,Summary3: vEBT_VEBTi] :
              ( heap_Time_bind @ product_unit @ ( option @ nat ) @ ( refine_Imp_assert @ ( vEBT_is_Node @ T3 ) )
              @ ^ [Uu3: product_unit] :
                  ( product_case_prod @ ( option @ ( product_prod @ nat @ nat ) ) @ ( product_prod @ nat @ ( product_prod @ ( list @ vEBT_VEBT ) @ vEBT_VEBT ) ) @ ( heap_Time_Heap @ ( option @ nat ) )
                  @ ^ [Info4: option @ ( product_prod @ nat @ nat )] :
                      ( product_case_prod @ nat @ ( product_prod @ ( list @ vEBT_VEBT ) @ vEBT_VEBT ) @ ( heap_Time_Heap @ ( option @ nat ) )
                      @ ^ [Deg5: nat] :
                          ( product_case_prod @ ( list @ vEBT_VEBT ) @ vEBT_VEBT @ ( heap_Time_Heap @ ( option @ nat ) )
                          @ ^ [TreeList3: list @ vEBT_VEBT,Summary4: vEBT_VEBT] :
                              ( heap_Time_bind @ product_unit @ ( option @ nat )
                              @ ( refine_Imp_assert
                                @ ( ( Info4 = Info3 )
                                  & ( Deg5 = Deg4 )
                                  & ( vEBT_is_Node @ T3 ) ) )
                              @ ^ [Uv3: product_unit] :
                                  ( case_option @ ( heap_Time_Heap @ ( option @ nat ) ) @ ( product_prod @ nat @ nat ) @ ( heap_Time_return @ ( option @ nat ) @ ( none @ nat ) )
                                  @ ^ [Mima2: product_prod @ nat @ nat] :
                                      ( if @ ( heap_Time_Heap @ ( option @ nat ) ) @ ( ord_less_eq @ nat @ Deg4 @ ( one_one @ nat ) ) @ ( heap_Time_return @ ( option @ nat ) @ ( none @ nat ) )
                                      @ ( if @ ( heap_Time_Heap @ ( option @ nat ) ) @ ( ord_less @ nat @ X @ ( product_fst @ nat @ nat @ Mima2 ) ) @ ( heap_Time_return @ ( option @ nat ) @ ( some @ nat @ ( product_fst @ nat @ nat @ Mima2 ) ) )
                                        @ ( if @ ( heap_Time_Heap @ ( option @ nat ) ) @ ( ord_less_eq @ nat @ ( product_snd @ nat @ nat @ Mima2 ) @ X ) @ ( heap_Time_return @ ( option @ nat ) @ ( none @ nat ) )
                                          @ ( heap_Time_bind @ nat @ ( option @ nat ) @ ( vEBT_VEBT_lowi @ X @ ( divide_divide @ nat @ Deg4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
                                            @ ^ [L2: nat] :
                                                ( heap_Time_bind @ nat @ ( option @ nat ) @ ( vEBT_VEBT_highi @ X @ ( divide_divide @ nat @ Deg4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
                                                @ ^ [H: nat] :
                                                    ( heap_Time_bind @ product_unit @ ( option @ nat )
                                                    @ ( refine_Imp_assert
                                                      @ ( L2
                                                        = ( vEBT_VEBT_low @ X @ ( divide_divide @ nat @ Deg4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
                                                    @ ^ [Uw3: product_unit] :
                                                        ( heap_Time_bind @ product_unit @ ( option @ nat )
                                                        @ ( refine_Imp_assert
                                                          @ ( H
                                                            = ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
                                                        @ ^ [Ux3: product_unit] :
                                                            ( heap_Time_bind @ product_unit @ ( option @ nat ) @ ( refine_Imp_assert @ ( ord_less @ nat @ H @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) ) )
                                                            @ ^ [Uy3: product_unit] :
                                                                ( heap_Time_bind @ vEBT_VEBTi @ ( option @ nat ) @ ( array_nth @ vEBT_VEBTi @ TreeArray @ H )
                                                                @ ^ [Aktnode: vEBT_VEBTi] :
                                                                    ( heap_Time_bind @ ( option @ nat ) @ ( option @ nat ) @ ( vEBT_vebt_maxti @ Aktnode )
                                                                    @ ^ [Maxlow: option @ nat] :
                                                                        ( heap_Time_bind @ product_unit @ ( option @ nat )
                                                                        @ ( refine_Imp_assert
                                                                          @ ( Maxlow
                                                                            = ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList3 @ H ) ) ) )
                                                                        @ ^ [Uz3: product_unit] :
                                                                            ( if @ ( heap_Time_Heap @ ( option @ nat ) )
                                                                            @ ( ( Maxlow
                                                                               != ( none @ nat ) )
                                                                              & ( vEBT_VEBT_less @ ( some @ nat @ L2 ) @ Maxlow ) )
                                                                            @ ( heap_Time_bind @ ( option @ nat ) @ ( option @ nat ) @ ( vEBT_VEBT_vebt_succi @ ( nth @ vEBT_VEBT @ TreeList3 @ H ) @ Aktnode @ L2 )
                                                                              @ ^ [Succy: option @ nat] : ( heap_Time_return @ ( option @ nat ) @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( some @ nat @ H ) ) @ Succy ) ) )
                                                                            @ ( heap_Time_bind @ ( option @ nat ) @ ( option @ nat ) @ ( vEBT_VEBT_vebt_succi @ Summary4 @ Summary3 @ H )
                                                                              @ ^ [Succsum: option @ nat] :
                                                                                  ( heap_Time_bind @ product_unit @ ( option @ nat )
                                                                                  @ ( refine_Imp_assert
                                                                                    @ ( ( Succsum
                                                                                        = ( none @ nat ) )
                                                                                      = ( ( vEBT_vebt_succ @ Summary4 @ H )
                                                                                        = ( none @ nat ) ) ) )
                                                                                  @ ^ [Va4: product_unit] :
                                                                                      ( if @ ( heap_Time_Heap @ ( option @ nat ) )
                                                                                      @ ( Succsum
                                                                                        = ( none @ nat ) )
                                                                                      @ ( heap_Time_return @ ( option @ nat ) @ ( none @ nat ) )
                                                                                      @ ( heap_Time_bind @ vEBT_VEBTi @ ( option @ nat ) @ ( array_nth @ vEBT_VEBTi @ TreeArray @ ( the2 @ nat @ Succsum ) )
                                                                                        @ ^ [Nextnode: vEBT_VEBTi] :
                                                                                            ( heap_Time_bind @ ( option @ nat ) @ ( option @ nat ) @ ( vEBT_vebt_minti @ Nextnode )
                                                                                            @ ^ [Minnext: option @ nat] : ( heap_Time_return @ ( option @ nat ) @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ Succsum ) @ Minnext ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
                                  @ Info3 ) ) ) )
                  @ ( vEBT_case_VEBT @ ( product_prod @ ( option @ ( product_prod @ nat @ nat ) ) @ ( product_prod @ nat @ ( product_prod @ ( list @ vEBT_VEBT ) @ vEBT_VEBT ) ) )
                    @ ^ [Info4: option @ ( product_prod @ nat @ nat ),Deg5: nat,TreeList3: list @ vEBT_VEBT,Summary4: vEBT_VEBT] : ( product_Pair @ ( option @ ( product_prod @ nat @ nat ) ) @ ( product_prod @ nat @ ( product_prod @ ( list @ vEBT_VEBT ) @ vEBT_VEBT ) ) @ Info4 @ ( product_Pair @ nat @ ( product_prod @ ( list @ vEBT_VEBT ) @ vEBT_VEBT ) @ Deg5 @ ( product_Pair @ ( list @ vEBT_VEBT ) @ vEBT_VEBT @ TreeList3 @ Summary4 ) ) )
                    @ ^ [A5: $o,B5: $o] : ( undefined @ ( product_prod @ ( option @ ( product_prod @ nat @ nat ) ) @ ( product_prod @ nat @ ( product_prod @ ( list @ vEBT_VEBT ) @ vEBT_VEBT ) ) ) )
                    @ T3 ) ) )
          @ ^ [A5: $o,B5: $o] :
              ( if @ ( heap_Time_Heap @ ( option @ nat ) )
              @ ( X
                = ( zero_zero @ nat ) )
              @ ( if @ ( heap_Time_Heap @ ( option @ nat ) ) @ B5 @ ( heap_Time_return @ ( option @ nat ) @ ( some @ nat @ ( one_one @ nat ) ) ) @ ( heap_Time_return @ ( option @ nat ) @ ( none @ nat ) ) )
              @ ( heap_Time_return @ ( option @ nat ) @ ( none @ nat ) ) )
          @ Ti3 ) ) ) ).

% VEBT_internal.vebt_succi'.simps
thf(fact_2706_vebt__buildupi__refines,axiom,
    ! [N2: nat] : ( refine_Imp_refines @ vEBT_VEBTi @ ( vEBT_vebt_buildupi @ N2 ) @ ( vEBT_V739175172307565963ildupi @ N2 ) ) ).

% vebt_buildupi_refines
thf(fact_2707_vebt__inserti__refines,axiom,
    ! [Ti: vEBT_VEBTi,X2: nat,T2: vEBT_VEBT] : ( refine_Imp_refines @ vEBT_VEBTi @ ( vEBT_vebt_inserti @ Ti @ X2 ) @ ( vEBT_V3964819847710782039nserti @ T2 @ Ti @ X2 ) ) ).

% vebt_inserti_refines
thf(fact_2708_split__part,axiom,
    ! [B: $tType,A: $tType,P: $o,Q: A > B > $o] :
      ( ( product_case_prod @ A @ B @ $o
        @ ^ [A5: A,B5: B] :
            ( P
            & ( Q @ A5 @ B5 ) ) )
      = ( ^ [Ab2: product_prod @ A @ B] :
            ( P
            & ( product_case_prod @ A @ B @ $o @ Q @ Ab2 ) ) ) ) ).

% split_part
thf(fact_2709_Collect__case__prod__mono,axiom,
    ! [B: $tType,A: $tType,A3: A > B > $o,B6: A > B > $o] :
      ( ( ord_less_eq @ ( A > B > $o ) @ A3 @ B6 )
     => ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ A3 ) ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ B6 ) ) ) ) ).

% Collect_case_prod_mono
thf(fact_2710_prod_Odisc__eq__case,axiom,
    ! [B: $tType,A: $tType,Prod: product_prod @ A @ B] :
      ( product_case_prod @ A @ B @ $o
      @ ^ [Uu3: A,Uv3: B] : $true
      @ Prod ) ).

% prod.disc_eq_case
thf(fact_2711_UNIV__def,axiom,
    ! [A: $tType] :
      ( ( top_top @ ( set @ A ) )
      = ( collect @ A
        @ ^ [X: A] : $true ) ) ).

% UNIV_def
thf(fact_2712_VEBT_Ocase__distrib,axiom,
    ! [A: $tType,B: $tType,H2: A > B,F1: ( option @ ( product_prod @ nat @ nat ) ) > nat > ( list @ vEBT_VEBT ) > vEBT_VEBT > A,F22: $o > $o > A,VEBT: vEBT_VEBT] :
      ( ( H2 @ ( vEBT_case_VEBT @ A @ F1 @ F22 @ VEBT ) )
      = ( vEBT_case_VEBT @ B
        @ ^ [X15: option @ ( product_prod @ nat @ nat ),X25: nat,X34: list @ vEBT_VEBT,X42: vEBT_VEBT] : ( H2 @ ( F1 @ X15 @ X25 @ X34 @ X42 ) )
        @ ^ [X15: $o,X25: $o] : ( H2 @ ( F22 @ X15 @ X25 ) )
        @ VEBT ) ) ).

% VEBT.case_distrib
thf(fact_2713_is__Node__def,axiom,
    ( vEBT_is_Node
    = ( ^ [VEBT2: vEBT_VEBT] :
        ? [X113: option @ ( product_prod @ nat @ nat ),X123: nat,X133: list @ vEBT_VEBT,X143: vEBT_VEBT] :
          ( VEBT2
          = ( vEBT_Node @ X113 @ X123 @ X133 @ X143 ) ) ) ) ).

% is_Node_def
thf(fact_2714_VEBT_OdiscI_I1_J,axiom,
    ! [VEBT: vEBT_VEBT,X11: option @ ( product_prod @ nat @ nat ),X12: nat,X13: list @ vEBT_VEBT,X14: vEBT_VEBT] :
      ( ( VEBT
        = ( vEBT_Node @ X11 @ X12 @ X13 @ X14 ) )
     => ( vEBT_is_Node @ VEBT ) ) ).

% VEBT.discI(1)
thf(fact_2715_VEBT_Odisc_I1_J,axiom,
    ! [X11: option @ ( product_prod @ nat @ nat ),X12: nat,X13: list @ vEBT_VEBT,X14: vEBT_VEBT] : ( vEBT_is_Node @ ( vEBT_Node @ X11 @ X12 @ X13 @ X14 ) ) ).

% VEBT.disc(1)
thf(fact_2716_VEBT_Osimps_I5_J,axiom,
    ! [A: $tType,F1: ( option @ ( product_prod @ nat @ nat ) ) > nat > ( list @ vEBT_VEBT ) > vEBT_VEBT > A,F22: $o > $o > A,X11: option @ ( product_prod @ nat @ nat ),X12: nat,X13: list @ vEBT_VEBT,X14: vEBT_VEBT] :
      ( ( vEBT_case_VEBT @ A @ F1 @ F22 @ ( vEBT_Node @ X11 @ X12 @ X13 @ X14 ) )
      = ( F1 @ X11 @ X12 @ X13 @ X14 ) ) ).

% VEBT.simps(5)
thf(fact_2717_option_Othe__def,axiom,
    ! [A: $tType] :
      ( ( the2 @ A )
      = ( case_option @ A @ A @ ( undefined @ A )
        @ ^ [X25: A] : X25 ) ) ).

% option.the_def
thf(fact_2718_one__natural_Orsp,axiom,
    ( ( one_one @ nat )
    = ( one_one @ nat ) ) ).

% one_natural.rsp
thf(fact_2719_one__integer_Orsp,axiom,
    ( ( one_one @ int )
    = ( one_one @ int ) ) ).

% one_integer.rsp
thf(fact_2720_less__eq__integer__code_I1_J,axiom,
    ord_less_eq @ code_integer @ ( zero_zero @ code_integer ) @ ( zero_zero @ code_integer ) ).

% less_eq_integer_code(1)
thf(fact_2721_times__integer__code_I2_J,axiom,
    ! [L: code_integer] :
      ( ( times_times @ code_integer @ ( zero_zero @ code_integer ) @ L )
      = ( zero_zero @ code_integer ) ) ).

% times_integer_code(2)
thf(fact_2722_times__integer__code_I1_J,axiom,
    ! [K: code_integer] :
      ( ( times_times @ code_integer @ K @ ( zero_zero @ code_integer ) )
      = ( zero_zero @ code_integer ) ) ).

% times_integer_code(1)
thf(fact_2723_plus__integer__code_I2_J,axiom,
    ! [L: code_integer] :
      ( ( plus_plus @ code_integer @ ( zero_zero @ code_integer ) @ L )
      = L ) ).

% plus_integer_code(2)
thf(fact_2724_plus__integer__code_I1_J,axiom,
    ! [K: code_integer] :
      ( ( plus_plus @ code_integer @ K @ ( zero_zero @ code_integer ) )
      = K ) ).

% plus_integer_code(1)
thf(fact_2725_VEBT__internal_Ovebt__memberi_H_Oraw__induct,axiom,
    ! [P: vEBT_VEBT > vEBT_VEBTi > nat > ( heap_ext @ product_unit ) > ( heap_ext @ product_unit ) > $o > nat > $o,Xa: product_prod @ ( product_prod @ vEBT_VEBT @ vEBT_VEBTi ) @ nat,H2: heap_ext @ product_unit,H4: heap_ext @ product_unit,R4: $o,N2: nat] :
      ( ! [Vebt_memberi2: vEBT_VEBT > vEBT_VEBTi > nat > ( heap_Time_Heap @ $o )] :
          ( ! [A8: vEBT_VEBT,B8: vEBT_VEBTi,Ba: nat,H6: heap_ext @ product_unit,H7: heap_ext @ product_unit,R6: $o,N9: nat] :
              ( ( heap_Time_effect @ $o @ ( Vebt_memberi2 @ A8 @ B8 @ Ba ) @ H6 @ H7 @ R6 @ N9 )
             => ( P @ A8 @ B8 @ Ba @ H6 @ H7 @ R6 @ N9 ) )
         => ! [T5: vEBT_VEBT,Ti4: vEBT_VEBTi,X3: nat,Ta: heap_ext @ product_unit,Tia: heap_ext @ product_unit,Xa3: $o,N4: nat] :
              ( ( heap_Time_effect @ $o
                @ ( vEBT_case_VEBTi @ ( heap_Time_Heap @ $o )
                  @ ^ [Info3: option @ ( product_prod @ nat @ nat ),Deg4: nat,TreeArray: array @ vEBT_VEBTi,Summary3: vEBT_VEBTi] :
                      ( heap_Time_bind @ product_unit @ $o @ ( refine_Imp_assert @ ( vEBT_is_Node @ T5 ) )
                      @ ^ [Uu3: product_unit] :
                          ( case_option @ ( heap_Time_Heap @ $o ) @ ( product_prod @ nat @ nat ) @ ( heap_Time_return @ $o @ $false )
                          @ ( product_case_prod @ nat @ nat @ ( heap_Time_Heap @ $o )
                            @ ^ [Mi3: nat,Ma3: nat] :
                                ( if @ ( heap_Time_Heap @ $o ) @ ( ord_less_eq @ nat @ Deg4 @ ( one_one @ nat ) ) @ ( heap_Time_return @ $o @ $false )
                                @ ( if @ ( heap_Time_Heap @ $o ) @ ( X3 = Mi3 ) @ ( heap_Time_return @ $o @ $true )
                                  @ ( if @ ( heap_Time_Heap @ $o ) @ ( X3 = Ma3 ) @ ( heap_Time_return @ $o @ $true )
                                    @ ( if @ ( heap_Time_Heap @ $o ) @ ( ord_less @ nat @ X3 @ Mi3 ) @ ( heap_Time_return @ $o @ $false )
                                      @ ( if @ ( heap_Time_Heap @ $o ) @ ( ord_less @ nat @ Ma3 @ X3 ) @ ( heap_Time_return @ $o @ $false )
                                        @ ( product_case_prod @ ( option @ ( product_prod @ nat @ nat ) ) @ ( product_prod @ nat @ ( product_prod @ ( list @ vEBT_VEBT ) @ vEBT_VEBT ) ) @ ( heap_Time_Heap @ $o )
                                          @ ^ [Info4: option @ ( product_prod @ nat @ nat )] :
                                              ( product_case_prod @ nat @ ( product_prod @ ( list @ vEBT_VEBT ) @ vEBT_VEBT ) @ ( heap_Time_Heap @ $o )
                                              @ ^ [Deg5: nat] :
                                                  ( product_case_prod @ ( list @ vEBT_VEBT ) @ vEBT_VEBT @ ( heap_Time_Heap @ $o )
                                                  @ ^ [TreeList3: list @ vEBT_VEBT,Summary4: vEBT_VEBT] :
                                                      ( heap_Time_bind @ product_unit @ $o
                                                      @ ( refine_Imp_assert
                                                        @ ( ( Info3 = Info4 )
                                                          & ( Deg4 = Deg5 ) ) )
                                                      @ ^ [Uv3: product_unit] :
                                                          ( heap_Time_bind @ nat @ $o @ ( vEBT_VEBT_highi @ X3 @ ( divide_divide @ nat @ Deg4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
                                                          @ ^ [H: nat] :
                                                              ( heap_Time_bind @ nat @ $o @ ( vEBT_VEBT_lowi @ X3 @ ( divide_divide @ nat @ Deg4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
                                                              @ ^ [L2: nat] :
                                                                  ( heap_Time_bind @ product_unit @ $o
                                                                  @ ( refine_Imp_assert
                                                                    @ ( ( L2
                                                                        = ( vEBT_VEBT_low @ X3 @ ( divide_divide @ nat @ Deg4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
                                                                      & ( H
                                                                        = ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ Deg4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) )
                                                                  @ ^ [Uw3: product_unit] :
                                                                      ( heap_Time_bind @ nat @ $o @ ( array_len @ vEBT_VEBTi @ TreeArray )
                                                                      @ ^ [Len: nat] :
                                                                          ( heap_Time_bind @ product_unit @ $o
                                                                          @ ( refine_Imp_assert
                                                                            @ ( Len
                                                                              = ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) ) )
                                                                          @ ^ [Ux3: product_unit] :
                                                                              ( if @ ( heap_Time_Heap @ $o ) @ ( ord_less @ nat @ H @ Len )
                                                                              @ ( heap_Time_bind @ product_unit @ $o
                                                                                @ ( refine_Imp_assert
                                                                                  @ ( ( H
                                                                                      = ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ Deg4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
                                                                                    & ( ord_less @ nat @ H @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) ) ) )
                                                                                @ ^ [Uy3: product_unit] :
                                                                                    ( heap_Time_bind @ vEBT_VEBTi @ $o @ ( array_nth @ vEBT_VEBTi @ TreeArray @ H )
                                                                                    @ ^ [Th: vEBT_VEBTi] : ( Vebt_memberi2 @ ( nth @ vEBT_VEBT @ TreeList3 @ H ) @ Th @ L2 ) ) )
                                                                              @ ( heap_Time_return @ $o @ $false ) ) ) ) ) ) ) ) ) )
                                          @ ( vEBT_case_VEBT @ ( product_prod @ ( option @ ( product_prod @ nat @ nat ) ) @ ( product_prod @ nat @ ( product_prod @ ( list @ vEBT_VEBT ) @ vEBT_VEBT ) ) )
                                            @ ^ [Info4: option @ ( product_prod @ nat @ nat ),Deg5: nat,TreeList3: list @ vEBT_VEBT,Summary4: vEBT_VEBT] : ( product_Pair @ ( option @ ( product_prod @ nat @ nat ) ) @ ( product_prod @ nat @ ( product_prod @ ( list @ vEBT_VEBT ) @ vEBT_VEBT ) ) @ Info4 @ ( product_Pair @ nat @ ( product_prod @ ( list @ vEBT_VEBT ) @ vEBT_VEBT ) @ Deg5 @ ( product_Pair @ ( list @ vEBT_VEBT ) @ vEBT_VEBT @ TreeList3 @ Summary4 ) ) )
                                            @ ^ [A5: $o,B5: $o] : ( undefined @ ( product_prod @ ( option @ ( product_prod @ nat @ nat ) ) @ ( product_prod @ nat @ ( product_prod @ ( list @ vEBT_VEBT ) @ vEBT_VEBT ) ) ) )
                                            @ T5 ) ) ) ) ) ) ) )
                          @ Info3 ) )
                  @ ^ [A5: $o,B5: $o] :
                      ( heap_Time_return @ $o
                      @ ( ( ( X3
                            = ( zero_zero @ nat ) )
                         => A5 )
                        & ( ( X3
                           != ( zero_zero @ nat ) )
                         => ( ( ( X3
                                = ( one_one @ nat ) )
                             => B5 )
                            & ( X3
                              = ( one_one @ nat ) ) ) ) ) )
                  @ Ti4 )
                @ Ta
                @ Tia
                @ Xa3
                @ N4 )
             => ( P @ T5 @ Ti4 @ X3 @ Ta @ Tia @ Xa3 @ N4 ) ) )
     => ( ( heap_Time_effect @ $o @ ( product_case_prod @ ( product_prod @ vEBT_VEBT @ vEBT_VEBTi ) @ nat @ ( heap_Time_Heap @ $o ) @ ( product_case_prod @ vEBT_VEBT @ vEBT_VEBTi @ ( nat > ( heap_Time_Heap @ $o ) ) @ vEBT_V854960066525838166emberi ) @ Xa ) @ H2 @ H4 @ R4 @ N2 )
       => ( product_case_prod @ ( product_prod @ vEBT_VEBT @ vEBT_VEBTi ) @ nat @ ( ( heap_ext @ product_unit ) > ( heap_ext @ product_unit ) > $o > nat > $o ) @ ( product_case_prod @ vEBT_VEBT @ vEBT_VEBTi @ ( nat > ( heap_ext @ product_unit ) > ( heap_ext @ product_unit ) > $o > nat > $o ) @ P ) @ Xa @ H2 @ H4 @ R4 @ N2 ) ) ) ).

% VEBT_internal.vebt_memberi'.raw_induct
thf(fact_2726_minus__integer__code_I1_J,axiom,
    ! [K: code_integer] :
      ( ( minus_minus @ code_integer @ K @ ( zero_zero @ code_integer ) )
      = K ) ).

% minus_integer_code(1)
thf(fact_2727_VEBT__internal_Ovebt__memberi_H_Osimps,axiom,
    ( vEBT_V854960066525838166emberi
    = ( ^ [T3: vEBT_VEBT,Ti3: vEBT_VEBTi,X: nat] :
          ( vEBT_case_VEBTi @ ( heap_Time_Heap @ $o )
          @ ^ [Info3: option @ ( product_prod @ nat @ nat ),Deg4: nat,TreeArray: array @ vEBT_VEBTi,Summary3: vEBT_VEBTi] :
              ( heap_Time_bind @ product_unit @ $o @ ( refine_Imp_assert @ ( vEBT_is_Node @ T3 ) )
              @ ^ [Uu3: product_unit] :
                  ( case_option @ ( heap_Time_Heap @ $o ) @ ( product_prod @ nat @ nat ) @ ( heap_Time_return @ $o @ $false )
                  @ ( product_case_prod @ nat @ nat @ ( heap_Time_Heap @ $o )
                    @ ^ [Mi3: nat,Ma3: nat] :
                        ( if @ ( heap_Time_Heap @ $o ) @ ( ord_less_eq @ nat @ Deg4 @ ( one_one @ nat ) ) @ ( heap_Time_return @ $o @ $false )
                        @ ( if @ ( heap_Time_Heap @ $o ) @ ( X = Mi3 ) @ ( heap_Time_return @ $o @ $true )
                          @ ( if @ ( heap_Time_Heap @ $o ) @ ( X = Ma3 ) @ ( heap_Time_return @ $o @ $true )
                            @ ( if @ ( heap_Time_Heap @ $o ) @ ( ord_less @ nat @ X @ Mi3 ) @ ( heap_Time_return @ $o @ $false )
                              @ ( if @ ( heap_Time_Heap @ $o ) @ ( ord_less @ nat @ Ma3 @ X ) @ ( heap_Time_return @ $o @ $false )
                                @ ( product_case_prod @ ( option @ ( product_prod @ nat @ nat ) ) @ ( product_prod @ nat @ ( product_prod @ ( list @ vEBT_VEBT ) @ vEBT_VEBT ) ) @ ( heap_Time_Heap @ $o )
                                  @ ^ [Info4: option @ ( product_prod @ nat @ nat )] :
                                      ( product_case_prod @ nat @ ( product_prod @ ( list @ vEBT_VEBT ) @ vEBT_VEBT ) @ ( heap_Time_Heap @ $o )
                                      @ ^ [Deg5: nat] :
                                          ( product_case_prod @ ( list @ vEBT_VEBT ) @ vEBT_VEBT @ ( heap_Time_Heap @ $o )
                                          @ ^ [TreeList3: list @ vEBT_VEBT,Summary4: vEBT_VEBT] :
                                              ( heap_Time_bind @ product_unit @ $o
                                              @ ( refine_Imp_assert
                                                @ ( ( Info3 = Info4 )
                                                  & ( Deg4 = Deg5 ) ) )
                                              @ ^ [Uv3: product_unit] :
                                                  ( heap_Time_bind @ nat @ $o @ ( vEBT_VEBT_highi @ X @ ( divide_divide @ nat @ Deg4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
                                                  @ ^ [H: nat] :
                                                      ( heap_Time_bind @ nat @ $o @ ( vEBT_VEBT_lowi @ X @ ( divide_divide @ nat @ Deg4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
                                                      @ ^ [L2: nat] :
                                                          ( heap_Time_bind @ product_unit @ $o
                                                          @ ( refine_Imp_assert
                                                            @ ( ( L2
                                                                = ( vEBT_VEBT_low @ X @ ( divide_divide @ nat @ Deg4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
                                                              & ( H
                                                                = ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) )
                                                          @ ^ [Uw3: product_unit] :
                                                              ( heap_Time_bind @ nat @ $o @ ( array_len @ vEBT_VEBTi @ TreeArray )
                                                              @ ^ [Len: nat] :
                                                                  ( heap_Time_bind @ product_unit @ $o
                                                                  @ ( refine_Imp_assert
                                                                    @ ( Len
                                                                      = ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) ) )
                                                                  @ ^ [Ux3: product_unit] :
                                                                      ( if @ ( heap_Time_Heap @ $o ) @ ( ord_less @ nat @ H @ Len )
                                                                      @ ( heap_Time_bind @ product_unit @ $o
                                                                        @ ( refine_Imp_assert
                                                                          @ ( ( H
                                                                              = ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
                                                                            & ( ord_less @ nat @ H @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) ) ) )
                                                                        @ ^ [Uy3: product_unit] :
                                                                            ( heap_Time_bind @ vEBT_VEBTi @ $o @ ( array_nth @ vEBT_VEBTi @ TreeArray @ H )
                                                                            @ ^ [Th: vEBT_VEBTi] : ( vEBT_V854960066525838166emberi @ ( nth @ vEBT_VEBT @ TreeList3 @ H ) @ Th @ L2 ) ) )
                                                                      @ ( heap_Time_return @ $o @ $false ) ) ) ) ) ) ) ) ) )
                                  @ ( vEBT_case_VEBT @ ( product_prod @ ( option @ ( product_prod @ nat @ nat ) ) @ ( product_prod @ nat @ ( product_prod @ ( list @ vEBT_VEBT ) @ vEBT_VEBT ) ) )
                                    @ ^ [Info4: option @ ( product_prod @ nat @ nat ),Deg5: nat,TreeList3: list @ vEBT_VEBT,Summary4: vEBT_VEBT] : ( product_Pair @ ( option @ ( product_prod @ nat @ nat ) ) @ ( product_prod @ nat @ ( product_prod @ ( list @ vEBT_VEBT ) @ vEBT_VEBT ) ) @ Info4 @ ( product_Pair @ nat @ ( product_prod @ ( list @ vEBT_VEBT ) @ vEBT_VEBT ) @ Deg5 @ ( product_Pair @ ( list @ vEBT_VEBT ) @ vEBT_VEBT @ TreeList3 @ Summary4 ) ) )
                                    @ ^ [A5: $o,B5: $o] : ( undefined @ ( product_prod @ ( option @ ( product_prod @ nat @ nat ) ) @ ( product_prod @ nat @ ( product_prod @ ( list @ vEBT_VEBT ) @ vEBT_VEBT ) ) ) )
                                    @ T3 ) ) ) ) ) ) ) )
                  @ Info3 ) )
          @ ^ [A5: $o,B5: $o] :
              ( heap_Time_return @ $o
              @ ( ( ( X
                    = ( zero_zero @ nat ) )
                 => A5 )
                & ( ( X
                   != ( zero_zero @ nat ) )
                 => ( ( ( X
                        = ( one_one @ nat ) )
                     => B5 )
                    & ( X
                      = ( one_one @ nat ) ) ) ) ) )
          @ Ti3 ) ) ) ).

% VEBT_internal.vebt_memberi'.simps
thf(fact_2728_VEBT__internal_Ovebt__inserti_H_Oraw__induct,axiom,
    ! [P: vEBT_VEBT > vEBT_VEBTi > nat > ( heap_ext @ product_unit ) > ( heap_ext @ product_unit ) > vEBT_VEBTi > nat > $o,Xa: product_prod @ ( product_prod @ vEBT_VEBT @ vEBT_VEBTi ) @ nat,H2: heap_ext @ product_unit,H4: heap_ext @ product_unit,R4: vEBT_VEBTi,N2: nat] :
      ( ! [Vebt_inserti2: vEBT_VEBT > vEBT_VEBTi > nat > ( heap_Time_Heap @ vEBT_VEBTi )] :
          ( ! [A8: vEBT_VEBT,B8: vEBT_VEBTi,Ba: nat,H6: heap_ext @ product_unit,H7: heap_ext @ product_unit,R6: vEBT_VEBTi,N9: nat] :
              ( ( heap_Time_effect @ vEBT_VEBTi @ ( Vebt_inserti2 @ A8 @ B8 @ Ba ) @ H6 @ H7 @ R6 @ N9 )
             => ( P @ A8 @ B8 @ Ba @ H6 @ H7 @ R6 @ N9 ) )
         => ! [T5: vEBT_VEBT,Ti4: vEBT_VEBTi,X3: nat,Ta: heap_ext @ product_unit,Tia: heap_ext @ product_unit,Xa3: vEBT_VEBTi,N4: nat] :
              ( ( heap_Time_effect @ vEBT_VEBTi
                @ ( vEBT_case_VEBTi @ ( heap_Time_Heap @ vEBT_VEBTi )
                  @ ^ [Info3: option @ ( product_prod @ nat @ nat ),Deg4: nat,TreeArray: array @ vEBT_VEBTi,Summary3: vEBT_VEBTi] :
                      ( case_option @ ( heap_Time_Heap @ vEBT_VEBTi ) @ ( product_prod @ nat @ nat ) @ ( if @ ( heap_Time_Heap @ vEBT_VEBTi ) @ ( ord_less_eq @ nat @ Deg4 @ ( one_one @ nat ) ) @ ( heap_Time_return @ vEBT_VEBTi @ ( vEBT_Nodei @ Info3 @ Deg4 @ TreeArray @ Summary3 ) ) @ ( heap_Time_return @ vEBT_VEBTi @ ( vEBT_Nodei @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ X3 @ X3 ) ) @ Deg4 @ TreeArray @ Summary3 ) ) )
                      @ ^ [Minma: product_prod @ nat @ nat] :
                          ( if @ ( heap_Time_Heap @ vEBT_VEBTi ) @ ( ord_less_eq @ nat @ Deg4 @ ( one_one @ nat ) ) @ ( heap_Time_return @ vEBT_VEBTi @ ( vEBT_Nodei @ Info3 @ Deg4 @ TreeArray @ Summary3 ) )
                          @ ( heap_Time_bind @ product_unit @ vEBT_VEBTi @ ( refine_Imp_assert @ ( vEBT_is_Node @ T5 ) )
                            @ ^ [Uu3: product_unit] :
                                ( product_case_prod @ ( option @ ( product_prod @ nat @ nat ) ) @ ( product_prod @ nat @ ( product_prod @ ( list @ vEBT_VEBT ) @ vEBT_VEBT ) ) @ ( heap_Time_Heap @ vEBT_VEBTi )
                                @ ^ [Info4: option @ ( product_prod @ nat @ nat )] :
                                    ( product_case_prod @ nat @ ( product_prod @ ( list @ vEBT_VEBT ) @ vEBT_VEBT ) @ ( heap_Time_Heap @ vEBT_VEBTi )
                                    @ ^ [Deg5: nat] :
                                        ( product_case_prod @ ( list @ vEBT_VEBT ) @ vEBT_VEBT @ ( heap_Time_Heap @ vEBT_VEBTi )
                                        @ ^ [TreeList3: list @ vEBT_VEBT,Summary4: vEBT_VEBT] :
                                            ( heap_Time_bind @ product_unit @ vEBT_VEBTi
                                            @ ( refine_Imp_assert
                                              @ ( ( Info3 = Info4 )
                                                & ( Deg4 = Deg5 ) ) )
                                            @ ^ [Uv3: product_unit] :
                                                ( product_case_prod @ nat @ nat @ ( heap_Time_Heap @ vEBT_VEBTi )
                                                @ ^ [Mi4: nat,Ma4: nat] :
                                                    ( heap_Time_bind @ nat @ vEBT_VEBTi @ ( heap_Time_return @ nat @ ( product_fst @ nat @ nat @ Minma ) )
                                                    @ ^ [Mi3: nat] :
                                                        ( heap_Time_bind @ nat @ vEBT_VEBTi @ ( heap_Time_return @ nat @ ( product_snd @ nat @ nat @ Minma ) )
                                                        @ ^ [Ma3: nat] :
                                                            ( heap_Time_bind @ nat @ vEBT_VEBTi @ ( if @ ( heap_Time_Heap @ nat ) @ ( ord_less @ nat @ X3 @ Mi3 ) @ ( heap_Time_return @ nat @ Mi3 ) @ ( heap_Time_return @ nat @ X3 ) )
                                                            @ ^ [Xn2: nat] :
                                                                ( heap_Time_bind @ nat @ vEBT_VEBTi @ ( if @ ( heap_Time_Heap @ nat ) @ ( ord_less @ nat @ X3 @ Mi3 ) @ ( heap_Time_return @ nat @ X3 ) @ ( heap_Time_return @ nat @ Mi3 ) )
                                                                @ ^ [Minn: nat] :
                                                                    ( heap_Time_bind @ nat @ vEBT_VEBTi @ ( vEBT_VEBT_lowi @ Xn2 @ ( divide_divide @ nat @ Deg4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
                                                                    @ ^ [L2: nat] :
                                                                        ( heap_Time_bind @ product_unit @ vEBT_VEBTi
                                                                        @ ( refine_Imp_assert
                                                                          @ ( L2
                                                                            = ( vEBT_VEBT_low @ ( if @ nat @ ( ord_less @ nat @ X3 @ Mi4 ) @ Mi4 @ X3 ) @ ( divide_divide @ nat @ Deg5 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
                                                                        @ ^ [Uw3: product_unit] :
                                                                            ( heap_Time_bind @ nat @ vEBT_VEBTi @ ( vEBT_VEBT_highi @ Xn2 @ ( divide_divide @ nat @ Deg4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
                                                                            @ ^ [H: nat] :
                                                                                ( heap_Time_bind @ nat @ vEBT_VEBTi @ ( array_len @ vEBT_VEBTi @ TreeArray )
                                                                                @ ^ [Len: nat] :
                                                                                    ( if @ ( heap_Time_Heap @ vEBT_VEBTi )
                                                                                    @ ( ( ord_less @ nat @ H @ Len )
                                                                                      & ~ ( ( X3 = Mi3 )
                                                                                          | ( X3 = Ma3 ) ) )
                                                                                    @ ( heap_Time_bind @ product_unit @ vEBT_VEBTi
                                                                                      @ ( refine_Imp_assert
                                                                                        @ ( H
                                                                                          = ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ X3 @ Mi4 ) @ Mi4 @ X3 ) @ ( divide_divide @ nat @ Deg5 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
                                                                                      @ ^ [Ux3: product_unit] :
                                                                                          ( heap_Time_bind @ product_unit @ vEBT_VEBTi @ ( refine_Imp_assert @ ( ord_less @ nat @ H @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) ) )
                                                                                          @ ^ [Uy3: product_unit] :
                                                                                              ( heap_Time_bind @ vEBT_VEBTi @ vEBT_VEBTi @ ( array_nth @ vEBT_VEBTi @ TreeArray @ H )
                                                                                              @ ^ [Node: vEBT_VEBTi] :
                                                                                                  ( heap_Time_bind @ $o @ vEBT_VEBTi @ ( vEBT_VEBT_minNulli @ Node )
                                                                                                  @ ^ [Empt: $o] :
                                                                                                      ( heap_Time_bind @ product_unit @ vEBT_VEBTi
                                                                                                      @ ( refine_Imp_assert
                                                                                                        @ ( Empt
                                                                                                          = ( vEBT_VEBT_minNull @ ( nth @ vEBT_VEBT @ TreeList3 @ H ) ) ) )
                                                                                                      @ ^ [Uz3: product_unit] :
                                                                                                          ( heap_Time_bind @ vEBT_VEBTi @ vEBT_VEBTi @ ( Vebt_inserti2 @ ( nth @ vEBT_VEBT @ TreeList3 @ H ) @ Node @ L2 )
                                                                                                          @ ^ [Newnode2: vEBT_VEBTi] :
                                                                                                              ( heap_Time_bind @ ( array @ vEBT_VEBTi ) @ vEBT_VEBTi @ ( array_upd @ vEBT_VEBTi @ H @ Newnode2 @ TreeArray )
                                                                                                              @ ^ [Newarray: array @ vEBT_VEBTi] :
                                                                                                                  ( heap_Time_bind @ vEBT_VEBTi @ vEBT_VEBTi @ ( if @ ( heap_Time_Heap @ vEBT_VEBTi ) @ Empt @ ( Vebt_inserti2 @ Summary4 @ Summary3 @ H ) @ ( heap_Time_return @ vEBT_VEBTi @ Summary3 ) )
                                                                                                                  @ ^ [Newsummary: vEBT_VEBTi] :
                                                                                                                      ( heap_Time_bind @ nat @ vEBT_VEBTi @ ( if @ ( heap_Time_Heap @ nat ) @ ( ord_less @ nat @ Ma3 @ Xn2 ) @ ( heap_Time_return @ nat @ Xn2 ) @ ( heap_Time_return @ nat @ Ma3 ) )
                                                                                                                      @ ^ [Man: nat] : ( heap_Time_return @ vEBT_VEBTi @ ( vEBT_Nodei @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Minn @ Man ) ) @ Deg4 @ Newarray @ Newsummary ) ) ) ) ) ) ) ) ) ) )
                                                                                    @ ( heap_Time_return @ vEBT_VEBTi @ ( vEBT_Nodei @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi3 @ Ma3 ) ) @ Deg4 @ TreeArray @ Summary3 ) ) ) ) ) ) ) ) ) ) )
                                                @ ( the2 @ ( product_prod @ nat @ nat ) @ Info4 ) ) ) ) )
                                @ ( vEBT_case_VEBT @ ( product_prod @ ( option @ ( product_prod @ nat @ nat ) ) @ ( product_prod @ nat @ ( product_prod @ ( list @ vEBT_VEBT ) @ vEBT_VEBT ) ) )
                                  @ ^ [Info4: option @ ( product_prod @ nat @ nat ),Deg5: nat,TreeList3: list @ vEBT_VEBT,Summary4: vEBT_VEBT] : ( product_Pair @ ( option @ ( product_prod @ nat @ nat ) ) @ ( product_prod @ nat @ ( product_prod @ ( list @ vEBT_VEBT ) @ vEBT_VEBT ) ) @ Info4 @ ( product_Pair @ nat @ ( product_prod @ ( list @ vEBT_VEBT ) @ vEBT_VEBT ) @ Deg5 @ ( product_Pair @ ( list @ vEBT_VEBT ) @ vEBT_VEBT @ TreeList3 @ Summary4 ) ) )
                                  @ ^ [A5: $o,B5: $o] : ( undefined @ ( product_prod @ ( option @ ( product_prod @ nat @ nat ) ) @ ( product_prod @ nat @ ( product_prod @ ( list @ vEBT_VEBT ) @ vEBT_VEBT ) ) ) )
                                  @ T5 ) ) ) )
                      @ Info3 )
                  @ ^ [A5: $o,B5: $o] :
                      ( if @ ( heap_Time_Heap @ vEBT_VEBTi )
                      @ ( X3
                        = ( zero_zero @ nat ) )
                      @ ( heap_Time_return @ vEBT_VEBTi @ ( vEBT_Leafi @ $true @ B5 ) )
                      @ ( if @ ( heap_Time_Heap @ vEBT_VEBTi )
                        @ ( X3
                          = ( one_one @ nat ) )
                        @ ( heap_Time_return @ vEBT_VEBTi @ ( vEBT_Leafi @ A5 @ $true ) )
                        @ ( heap_Time_return @ vEBT_VEBTi @ ( vEBT_Leafi @ A5 @ B5 ) ) ) )
                  @ Ti4 )
                @ Ta
                @ Tia
                @ Xa3
                @ N4 )
             => ( P @ T5 @ Ti4 @ X3 @ Ta @ Tia @ Xa3 @ N4 ) ) )
     => ( ( heap_Time_effect @ vEBT_VEBTi @ ( product_case_prod @ ( product_prod @ vEBT_VEBT @ vEBT_VEBTi ) @ nat @ ( heap_Time_Heap @ vEBT_VEBTi ) @ ( product_case_prod @ vEBT_VEBT @ vEBT_VEBTi @ ( nat > ( heap_Time_Heap @ vEBT_VEBTi ) ) @ vEBT_V3964819847710782039nserti ) @ Xa ) @ H2 @ H4 @ R4 @ N2 )
       => ( product_case_prod @ ( product_prod @ vEBT_VEBT @ vEBT_VEBTi ) @ nat @ ( ( heap_ext @ product_unit ) > ( heap_ext @ product_unit ) > vEBT_VEBTi > nat > $o ) @ ( product_case_prod @ vEBT_VEBT @ vEBT_VEBTi @ ( nat > ( heap_ext @ product_unit ) > ( heap_ext @ product_unit ) > vEBT_VEBTi > nat > $o ) @ P ) @ Xa @ H2 @ H4 @ R4 @ N2 ) ) ) ).

% VEBT_internal.vebt_inserti'.raw_induct
thf(fact_2729_VEBT__internal_Ovebt__predi_H_Oraw__induct,axiom,
    ! [P: vEBT_VEBT > vEBT_VEBTi > nat > ( heap_ext @ product_unit ) > ( heap_ext @ product_unit ) > ( option @ nat ) > nat > $o,Xa: product_prod @ ( product_prod @ vEBT_VEBT @ vEBT_VEBTi ) @ nat,H2: heap_ext @ product_unit,H4: heap_ext @ product_unit,R4: option @ nat,N2: nat] :
      ( ! [Vebt_predi2: vEBT_VEBT > vEBT_VEBTi > nat > ( heap_Time_Heap @ ( option @ nat ) )] :
          ( ! [A8: vEBT_VEBT,B8: vEBT_VEBTi,Ba: nat,H6: heap_ext @ product_unit,H7: heap_ext @ product_unit,R6: option @ nat,N9: nat] :
              ( ( heap_Time_effect @ ( option @ nat ) @ ( Vebt_predi2 @ A8 @ B8 @ Ba ) @ H6 @ H7 @ R6 @ N9 )
             => ( P @ A8 @ B8 @ Ba @ H6 @ H7 @ R6 @ N9 ) )
         => ! [T5: vEBT_VEBT,Ti4: vEBT_VEBTi,X3: nat,Ta: heap_ext @ product_unit,Tia: heap_ext @ product_unit,Xa3: option @ nat,N4: nat] :
              ( ( heap_Time_effect @ ( option @ nat )
                @ ( vEBT_case_VEBTi @ ( heap_Time_Heap @ ( option @ nat ) )
                  @ ^ [Info3: option @ ( product_prod @ nat @ nat ),Deg4: nat,TreeArray: array @ vEBT_VEBTi,Summary3: vEBT_VEBTi] :
                      ( heap_Time_bind @ product_unit @ ( option @ nat ) @ ( refine_Imp_assert @ ( vEBT_is_Node @ T5 ) )
                      @ ^ [Uu3: product_unit] :
                          ( product_case_prod @ ( option @ ( product_prod @ nat @ nat ) ) @ ( product_prod @ nat @ ( product_prod @ ( list @ vEBT_VEBT ) @ vEBT_VEBT ) ) @ ( heap_Time_Heap @ ( option @ nat ) )
                          @ ^ [Info4: option @ ( product_prod @ nat @ nat )] :
                              ( product_case_prod @ nat @ ( product_prod @ ( list @ vEBT_VEBT ) @ vEBT_VEBT ) @ ( heap_Time_Heap @ ( option @ nat ) )
                              @ ^ [Deg5: nat] :
                                  ( product_case_prod @ ( list @ vEBT_VEBT ) @ vEBT_VEBT @ ( heap_Time_Heap @ ( option @ nat ) )
                                  @ ^ [TreeList3: list @ vEBT_VEBT,Summary4: vEBT_VEBT] :
                                      ( heap_Time_bind @ product_unit @ ( option @ nat )
                                      @ ( refine_Imp_assert
                                        @ ( ( Info4 = Info3 )
                                          & ( Deg5 = Deg4 )
                                          & ( vEBT_is_Node @ T5 ) ) )
                                      @ ^ [Uv3: product_unit] :
                                          ( case_option @ ( heap_Time_Heap @ ( option @ nat ) ) @ ( product_prod @ nat @ nat ) @ ( heap_Time_return @ ( option @ nat ) @ ( none @ nat ) )
                                          @ ^ [Mima2: product_prod @ nat @ nat] :
                                              ( if @ ( heap_Time_Heap @ ( option @ nat ) ) @ ( ord_less_eq @ nat @ Deg4 @ ( one_one @ nat ) ) @ ( heap_Time_return @ ( option @ nat ) @ ( none @ nat ) )
                                              @ ( if @ ( heap_Time_Heap @ ( option @ nat ) ) @ ( ord_less @ nat @ ( product_snd @ nat @ nat @ Mima2 ) @ X3 ) @ ( heap_Time_return @ ( option @ nat ) @ ( some @ nat @ ( product_snd @ nat @ nat @ Mima2 ) ) )
                                                @ ( heap_Time_bind @ nat @ ( option @ nat ) @ ( vEBT_VEBT_lowi @ X3 @ ( divide_divide @ nat @ Deg4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
                                                  @ ^ [L2: nat] :
                                                      ( heap_Time_bind @ nat @ ( option @ nat ) @ ( vEBT_VEBT_highi @ X3 @ ( divide_divide @ nat @ Deg4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
                                                      @ ^ [H: nat] :
                                                          ( heap_Time_bind @ product_unit @ ( option @ nat )
                                                          @ ( refine_Imp_assert
                                                            @ ( L2
                                                              = ( vEBT_VEBT_low @ X3 @ ( divide_divide @ nat @ Deg4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
                                                          @ ^ [Uw3: product_unit] :
                                                              ( heap_Time_bind @ product_unit @ ( option @ nat )
                                                              @ ( refine_Imp_assert
                                                                @ ( H
                                                                  = ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ Deg4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
                                                              @ ^ [Ux3: product_unit] :
                                                                  ( heap_Time_bind @ product_unit @ ( option @ nat ) @ ( refine_Imp_assert @ ( ord_less @ nat @ H @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) ) )
                                                                  @ ^ [Uy3: product_unit] :
                                                                      ( heap_Time_bind @ vEBT_VEBTi @ ( option @ nat ) @ ( array_nth @ vEBT_VEBTi @ TreeArray @ H )
                                                                      @ ^ [Aktnode: vEBT_VEBTi] :
                                                                          ( heap_Time_bind @ ( option @ nat ) @ ( option @ nat ) @ ( vEBT_vebt_minti @ Aktnode )
                                                                          @ ^ [Minlow: option @ nat] :
                                                                              ( heap_Time_bind @ product_unit @ ( option @ nat )
                                                                              @ ( refine_Imp_assert
                                                                                @ ( Minlow
                                                                                  = ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList3 @ H ) ) ) )
                                                                              @ ^ [Uz3: product_unit] :
                                                                                  ( if @ ( heap_Time_Heap @ ( option @ nat ) )
                                                                                  @ ( ( Minlow
                                                                                     != ( none @ nat ) )
                                                                                    & ( vEBT_VEBT_greater @ ( some @ nat @ L2 ) @ Minlow ) )
                                                                                  @ ( heap_Time_bind @ ( option @ nat ) @ ( option @ nat ) @ ( Vebt_predi2 @ ( nth @ vEBT_VEBT @ TreeList3 @ H ) @ Aktnode @ L2 )
                                                                                    @ ^ [Predy: option @ nat] : ( heap_Time_return @ ( option @ nat ) @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( some @ nat @ H ) ) @ Predy ) ) )
                                                                                  @ ( heap_Time_bind @ ( option @ nat ) @ ( option @ nat ) @ ( Vebt_predi2 @ Summary4 @ Summary3 @ H )
                                                                                    @ ^ [Predsum: option @ nat] :
                                                                                        ( heap_Time_bind @ product_unit @ ( option @ nat )
                                                                                        @ ( refine_Imp_assert
                                                                                          @ ( ( Predsum
                                                                                              = ( none @ nat ) )
                                                                                            = ( ( vEBT_vebt_pred @ Summary4 @ H )
                                                                                              = ( none @ nat ) ) ) )
                                                                                        @ ^ [Va4: product_unit] :
                                                                                            ( if @ ( heap_Time_Heap @ ( option @ nat ) )
                                                                                            @ ( Predsum
                                                                                              = ( none @ nat ) )
                                                                                            @ ( if @ ( heap_Time_Heap @ ( option @ nat ) ) @ ( ord_less @ nat @ ( product_fst @ nat @ nat @ Mima2 ) @ X3 ) @ ( heap_Time_return @ ( option @ nat ) @ ( some @ nat @ ( product_fst @ nat @ nat @ Mima2 ) ) ) @ ( heap_Time_return @ ( option @ nat ) @ ( none @ nat ) ) )
                                                                                            @ ( heap_Time_bind @ vEBT_VEBTi @ ( option @ nat ) @ ( array_nth @ vEBT_VEBTi @ TreeArray @ ( the2 @ nat @ Predsum ) )
                                                                                              @ ^ [Nextnode: vEBT_VEBTi] :
                                                                                                  ( heap_Time_bind @ ( option @ nat ) @ ( option @ nat ) @ ( vEBT_vebt_maxti @ Nextnode )
                                                                                                  @ ^ [Maxnext: option @ nat] : ( heap_Time_return @ ( option @ nat ) @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ Predsum ) @ Maxnext ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
                                          @ Info3 ) ) ) )
                          @ ( vEBT_case_VEBT @ ( product_prod @ ( option @ ( product_prod @ nat @ nat ) ) @ ( product_prod @ nat @ ( product_prod @ ( list @ vEBT_VEBT ) @ vEBT_VEBT ) ) )
                            @ ^ [Info4: option @ ( product_prod @ nat @ nat ),Deg5: nat,TreeList3: list @ vEBT_VEBT,Summary4: vEBT_VEBT] : ( product_Pair @ ( option @ ( product_prod @ nat @ nat ) ) @ ( product_prod @ nat @ ( product_prod @ ( list @ vEBT_VEBT ) @ vEBT_VEBT ) ) @ Info4 @ ( product_Pair @ nat @ ( product_prod @ ( list @ vEBT_VEBT ) @ vEBT_VEBT ) @ Deg5 @ ( product_Pair @ ( list @ vEBT_VEBT ) @ vEBT_VEBT @ TreeList3 @ Summary4 ) ) )
                            @ ^ [A5: $o,B5: $o] : ( undefined @ ( product_prod @ ( option @ ( product_prod @ nat @ nat ) ) @ ( product_prod @ nat @ ( product_prod @ ( list @ vEBT_VEBT ) @ vEBT_VEBT ) ) ) )
                            @ T5 ) ) )
                  @ ^ [A5: $o,B5: $o] :
                      ( if @ ( heap_Time_Heap @ ( option @ nat ) ) @ ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ X3 ) @ ( if @ ( heap_Time_Heap @ ( option @ nat ) ) @ B5 @ ( heap_Time_return @ ( option @ nat ) @ ( some @ nat @ ( one_one @ nat ) ) ) @ ( if @ ( heap_Time_Heap @ ( option @ nat ) ) @ A5 @ ( heap_Time_return @ ( option @ nat ) @ ( some @ nat @ ( zero_zero @ nat ) ) ) @ ( heap_Time_return @ ( option @ nat ) @ ( none @ nat ) ) ) )
                      @ ( if @ ( heap_Time_Heap @ ( option @ nat ) )
                        @ ( X3
                          = ( one_one @ nat ) )
                        @ ( if @ ( heap_Time_Heap @ ( option @ nat ) ) @ A5 @ ( heap_Time_return @ ( option @ nat ) @ ( some @ nat @ ( zero_zero @ nat ) ) ) @ ( heap_Time_return @ ( option @ nat ) @ ( none @ nat ) ) )
                        @ ( heap_Time_return @ ( option @ nat ) @ ( none @ nat ) ) ) )
                  @ Ti4 )
                @ Ta
                @ Tia
                @ Xa3
                @ N4 )
             => ( P @ T5 @ Ti4 @ X3 @ Ta @ Tia @ Xa3 @ N4 ) ) )
     => ( ( heap_Time_effect @ ( option @ nat ) @ ( product_case_prod @ ( product_prod @ vEBT_VEBT @ vEBT_VEBTi ) @ nat @ ( heap_Time_Heap @ ( option @ nat ) ) @ ( product_case_prod @ vEBT_VEBT @ vEBT_VEBTi @ ( nat > ( heap_Time_Heap @ ( option @ nat ) ) ) @ vEBT_VEBT_vebt_predi ) @ Xa ) @ H2 @ H4 @ R4 @ N2 )
       => ( product_case_prod @ ( product_prod @ vEBT_VEBT @ vEBT_VEBTi ) @ nat @ ( ( heap_ext @ product_unit ) > ( heap_ext @ product_unit ) > ( option @ nat ) > nat > $o ) @ ( product_case_prod @ vEBT_VEBT @ vEBT_VEBTi @ ( nat > ( heap_ext @ product_unit ) > ( heap_ext @ product_unit ) > ( option @ nat ) > nat > $o ) @ P ) @ Xa @ H2 @ H4 @ R4 @ N2 ) ) ) ).

% VEBT_internal.vebt_predi'.raw_induct
thf(fact_2730_VEBT__internal_Ovebt__succi_H_Oraw__induct,axiom,
    ! [P: vEBT_VEBT > vEBT_VEBTi > nat > ( heap_ext @ product_unit ) > ( heap_ext @ product_unit ) > ( option @ nat ) > nat > $o,Xa: product_prod @ ( product_prod @ vEBT_VEBT @ vEBT_VEBTi ) @ nat,H2: heap_ext @ product_unit,H4: heap_ext @ product_unit,R4: option @ nat,N2: nat] :
      ( ! [Vebt_succi2: vEBT_VEBT > vEBT_VEBTi > nat > ( heap_Time_Heap @ ( option @ nat ) )] :
          ( ! [A8: vEBT_VEBT,B8: vEBT_VEBTi,Ba: nat,H6: heap_ext @ product_unit,H7: heap_ext @ product_unit,R6: option @ nat,N9: nat] :
              ( ( heap_Time_effect @ ( option @ nat ) @ ( Vebt_succi2 @ A8 @ B8 @ Ba ) @ H6 @ H7 @ R6 @ N9 )
             => ( P @ A8 @ B8 @ Ba @ H6 @ H7 @ R6 @ N9 ) )
         => ! [T5: vEBT_VEBT,Ti4: vEBT_VEBTi,X3: nat,Ta: heap_ext @ product_unit,Tia: heap_ext @ product_unit,Xa3: option @ nat,N4: nat] :
              ( ( heap_Time_effect @ ( option @ nat )
                @ ( vEBT_case_VEBTi @ ( heap_Time_Heap @ ( option @ nat ) )
                  @ ^ [Info3: option @ ( product_prod @ nat @ nat ),Deg4: nat,TreeArray: array @ vEBT_VEBTi,Summary3: vEBT_VEBTi] :
                      ( heap_Time_bind @ product_unit @ ( option @ nat ) @ ( refine_Imp_assert @ ( vEBT_is_Node @ T5 ) )
                      @ ^ [Uu3: product_unit] :
                          ( product_case_prod @ ( option @ ( product_prod @ nat @ nat ) ) @ ( product_prod @ nat @ ( product_prod @ ( list @ vEBT_VEBT ) @ vEBT_VEBT ) ) @ ( heap_Time_Heap @ ( option @ nat ) )
                          @ ^ [Info4: option @ ( product_prod @ nat @ nat )] :
                              ( product_case_prod @ nat @ ( product_prod @ ( list @ vEBT_VEBT ) @ vEBT_VEBT ) @ ( heap_Time_Heap @ ( option @ nat ) )
                              @ ^ [Deg5: nat] :
                                  ( product_case_prod @ ( list @ vEBT_VEBT ) @ vEBT_VEBT @ ( heap_Time_Heap @ ( option @ nat ) )
                                  @ ^ [TreeList3: list @ vEBT_VEBT,Summary4: vEBT_VEBT] :
                                      ( heap_Time_bind @ product_unit @ ( option @ nat )
                                      @ ( refine_Imp_assert
                                        @ ( ( Info4 = Info3 )
                                          & ( Deg5 = Deg4 )
                                          & ( vEBT_is_Node @ T5 ) ) )
                                      @ ^ [Uv3: product_unit] :
                                          ( case_option @ ( heap_Time_Heap @ ( option @ nat ) ) @ ( product_prod @ nat @ nat ) @ ( heap_Time_return @ ( option @ nat ) @ ( none @ nat ) )
                                          @ ^ [Mima2: product_prod @ nat @ nat] :
                                              ( if @ ( heap_Time_Heap @ ( option @ nat ) ) @ ( ord_less_eq @ nat @ Deg4 @ ( one_one @ nat ) ) @ ( heap_Time_return @ ( option @ nat ) @ ( none @ nat ) )
                                              @ ( if @ ( heap_Time_Heap @ ( option @ nat ) ) @ ( ord_less @ nat @ X3 @ ( product_fst @ nat @ nat @ Mima2 ) ) @ ( heap_Time_return @ ( option @ nat ) @ ( some @ nat @ ( product_fst @ nat @ nat @ Mima2 ) ) )
                                                @ ( if @ ( heap_Time_Heap @ ( option @ nat ) ) @ ( ord_less_eq @ nat @ ( product_snd @ nat @ nat @ Mima2 ) @ X3 ) @ ( heap_Time_return @ ( option @ nat ) @ ( none @ nat ) )
                                                  @ ( heap_Time_bind @ nat @ ( option @ nat ) @ ( vEBT_VEBT_lowi @ X3 @ ( divide_divide @ nat @ Deg4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
                                                    @ ^ [L2: nat] :
                                                        ( heap_Time_bind @ nat @ ( option @ nat ) @ ( vEBT_VEBT_highi @ X3 @ ( divide_divide @ nat @ Deg4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
                                                        @ ^ [H: nat] :
                                                            ( heap_Time_bind @ product_unit @ ( option @ nat )
                                                            @ ( refine_Imp_assert
                                                              @ ( L2
                                                                = ( vEBT_VEBT_low @ X3 @ ( divide_divide @ nat @ Deg4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
                                                            @ ^ [Uw3: product_unit] :
                                                                ( heap_Time_bind @ product_unit @ ( option @ nat )
                                                                @ ( refine_Imp_assert
                                                                  @ ( H
                                                                    = ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ Deg4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
                                                                @ ^ [Ux3: product_unit] :
                                                                    ( heap_Time_bind @ product_unit @ ( option @ nat ) @ ( refine_Imp_assert @ ( ord_less @ nat @ H @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) ) )
                                                                    @ ^ [Uy3: product_unit] :
                                                                        ( heap_Time_bind @ vEBT_VEBTi @ ( option @ nat ) @ ( array_nth @ vEBT_VEBTi @ TreeArray @ H )
                                                                        @ ^ [Aktnode: vEBT_VEBTi] :
                                                                            ( heap_Time_bind @ ( option @ nat ) @ ( option @ nat ) @ ( vEBT_vebt_maxti @ Aktnode )
                                                                            @ ^ [Maxlow: option @ nat] :
                                                                                ( heap_Time_bind @ product_unit @ ( option @ nat )
                                                                                @ ( refine_Imp_assert
                                                                                  @ ( Maxlow
                                                                                    = ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList3 @ H ) ) ) )
                                                                                @ ^ [Uz3: product_unit] :
                                                                                    ( if @ ( heap_Time_Heap @ ( option @ nat ) )
                                                                                    @ ( ( Maxlow
                                                                                       != ( none @ nat ) )
                                                                                      & ( vEBT_VEBT_less @ ( some @ nat @ L2 ) @ Maxlow ) )
                                                                                    @ ( heap_Time_bind @ ( option @ nat ) @ ( option @ nat ) @ ( Vebt_succi2 @ ( nth @ vEBT_VEBT @ TreeList3 @ H ) @ Aktnode @ L2 )
                                                                                      @ ^ [Succy: option @ nat] : ( heap_Time_return @ ( option @ nat ) @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( some @ nat @ H ) ) @ Succy ) ) )
                                                                                    @ ( heap_Time_bind @ ( option @ nat ) @ ( option @ nat ) @ ( Vebt_succi2 @ Summary4 @ Summary3 @ H )
                                                                                      @ ^ [Succsum: option @ nat] :
                                                                                          ( heap_Time_bind @ product_unit @ ( option @ nat )
                                                                                          @ ( refine_Imp_assert
                                                                                            @ ( ( Succsum
                                                                                                = ( none @ nat ) )
                                                                                              = ( ( vEBT_vebt_succ @ Summary4 @ H )
                                                                                                = ( none @ nat ) ) ) )
                                                                                          @ ^ [Va4: product_unit] :
                                                                                              ( if @ ( heap_Time_Heap @ ( option @ nat ) )
                                                                                              @ ( Succsum
                                                                                                = ( none @ nat ) )
                                                                                              @ ( heap_Time_return @ ( option @ nat ) @ ( none @ nat ) )
                                                                                              @ ( heap_Time_bind @ vEBT_VEBTi @ ( option @ nat ) @ ( array_nth @ vEBT_VEBTi @ TreeArray @ ( the2 @ nat @ Succsum ) )
                                                                                                @ ^ [Nextnode: vEBT_VEBTi] :
                                                                                                    ( heap_Time_bind @ ( option @ nat ) @ ( option @ nat ) @ ( vEBT_vebt_minti @ Nextnode )
                                                                                                    @ ^ [Minnext: option @ nat] : ( heap_Time_return @ ( option @ nat ) @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ Succsum ) @ Minnext ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
                                          @ Info3 ) ) ) )
                          @ ( vEBT_case_VEBT @ ( product_prod @ ( option @ ( product_prod @ nat @ nat ) ) @ ( product_prod @ nat @ ( product_prod @ ( list @ vEBT_VEBT ) @ vEBT_VEBT ) ) )
                            @ ^ [Info4: option @ ( product_prod @ nat @ nat ),Deg5: nat,TreeList3: list @ vEBT_VEBT,Summary4: vEBT_VEBT] : ( product_Pair @ ( option @ ( product_prod @ nat @ nat ) ) @ ( product_prod @ nat @ ( product_prod @ ( list @ vEBT_VEBT ) @ vEBT_VEBT ) ) @ Info4 @ ( product_Pair @ nat @ ( product_prod @ ( list @ vEBT_VEBT ) @ vEBT_VEBT ) @ Deg5 @ ( product_Pair @ ( list @ vEBT_VEBT ) @ vEBT_VEBT @ TreeList3 @ Summary4 ) ) )
                            @ ^ [A5: $o,B5: $o] : ( undefined @ ( product_prod @ ( option @ ( product_prod @ nat @ nat ) ) @ ( product_prod @ nat @ ( product_prod @ ( list @ vEBT_VEBT ) @ vEBT_VEBT ) ) ) )
                            @ T5 ) ) )
                  @ ^ [A5: $o,B5: $o] :
                      ( if @ ( heap_Time_Heap @ ( option @ nat ) )
                      @ ( X3
                        = ( zero_zero @ nat ) )
                      @ ( if @ ( heap_Time_Heap @ ( option @ nat ) ) @ B5 @ ( heap_Time_return @ ( option @ nat ) @ ( some @ nat @ ( one_one @ nat ) ) ) @ ( heap_Time_return @ ( option @ nat ) @ ( none @ nat ) ) )
                      @ ( heap_Time_return @ ( option @ nat ) @ ( none @ nat ) ) )
                  @ Ti4 )
                @ Ta
                @ Tia
                @ Xa3
                @ N4 )
             => ( P @ T5 @ Ti4 @ X3 @ Ta @ Tia @ Xa3 @ N4 ) ) )
     => ( ( heap_Time_effect @ ( option @ nat ) @ ( product_case_prod @ ( product_prod @ vEBT_VEBT @ vEBT_VEBTi ) @ nat @ ( heap_Time_Heap @ ( option @ nat ) ) @ ( product_case_prod @ vEBT_VEBT @ vEBT_VEBTi @ ( nat > ( heap_Time_Heap @ ( option @ nat ) ) ) @ vEBT_VEBT_vebt_succi ) @ Xa ) @ H2 @ H4 @ R4 @ N2 )
       => ( product_case_prod @ ( product_prod @ vEBT_VEBT @ vEBT_VEBTi ) @ nat @ ( ( heap_ext @ product_unit ) > ( heap_ext @ product_unit ) > ( option @ nat ) > nat > $o ) @ ( product_case_prod @ vEBT_VEBT @ vEBT_VEBTi @ ( nat > ( heap_ext @ product_unit ) > ( heap_ext @ product_unit ) > ( option @ nat ) > nat > $o ) @ P ) @ Xa @ H2 @ H4 @ R4 @ N2 ) ) ) ).

% VEBT_internal.vebt_succi'.raw_induct
thf(fact_2731_VEBT__internal_Ovebt__inserti_H_Osimps,axiom,
    ( vEBT_V3964819847710782039nserti
    = ( ^ [T3: vEBT_VEBT,Ti3: vEBT_VEBTi,X: nat] :
          ( vEBT_case_VEBTi @ ( heap_Time_Heap @ vEBT_VEBTi )
          @ ^ [Info3: option @ ( product_prod @ nat @ nat ),Deg4: nat,TreeArray: array @ vEBT_VEBTi,Summary3: vEBT_VEBTi] :
              ( case_option @ ( heap_Time_Heap @ vEBT_VEBTi ) @ ( product_prod @ nat @ nat ) @ ( if @ ( heap_Time_Heap @ vEBT_VEBTi ) @ ( ord_less_eq @ nat @ Deg4 @ ( one_one @ nat ) ) @ ( heap_Time_return @ vEBT_VEBTi @ ( vEBT_Nodei @ Info3 @ Deg4 @ TreeArray @ Summary3 ) ) @ ( heap_Time_return @ vEBT_VEBTi @ ( vEBT_Nodei @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ X @ X ) ) @ Deg4 @ TreeArray @ Summary3 ) ) )
              @ ^ [Minma: product_prod @ nat @ nat] :
                  ( if @ ( heap_Time_Heap @ vEBT_VEBTi ) @ ( ord_less_eq @ nat @ Deg4 @ ( one_one @ nat ) ) @ ( heap_Time_return @ vEBT_VEBTi @ ( vEBT_Nodei @ Info3 @ Deg4 @ TreeArray @ Summary3 ) )
                  @ ( heap_Time_bind @ product_unit @ vEBT_VEBTi @ ( refine_Imp_assert @ ( vEBT_is_Node @ T3 ) )
                    @ ^ [Uu3: product_unit] :
                        ( product_case_prod @ ( option @ ( product_prod @ nat @ nat ) ) @ ( product_prod @ nat @ ( product_prod @ ( list @ vEBT_VEBT ) @ vEBT_VEBT ) ) @ ( heap_Time_Heap @ vEBT_VEBTi )
                        @ ^ [Info4: option @ ( product_prod @ nat @ nat )] :
                            ( product_case_prod @ nat @ ( product_prod @ ( list @ vEBT_VEBT ) @ vEBT_VEBT ) @ ( heap_Time_Heap @ vEBT_VEBTi )
                            @ ^ [Deg5: nat] :
                                ( product_case_prod @ ( list @ vEBT_VEBT ) @ vEBT_VEBT @ ( heap_Time_Heap @ vEBT_VEBTi )
                                @ ^ [TreeList3: list @ vEBT_VEBT,Summary4: vEBT_VEBT] :
                                    ( heap_Time_bind @ product_unit @ vEBT_VEBTi
                                    @ ( refine_Imp_assert
                                      @ ( ( Info3 = Info4 )
                                        & ( Deg4 = Deg5 ) ) )
                                    @ ^ [Uv3: product_unit] :
                                        ( product_case_prod @ nat @ nat @ ( heap_Time_Heap @ vEBT_VEBTi )
                                        @ ^ [Mi4: nat,Ma4: nat] :
                                            ( heap_Time_bind @ nat @ vEBT_VEBTi @ ( heap_Time_return @ nat @ ( product_fst @ nat @ nat @ Minma ) )
                                            @ ^ [Mi3: nat] :
                                                ( heap_Time_bind @ nat @ vEBT_VEBTi @ ( heap_Time_return @ nat @ ( product_snd @ nat @ nat @ Minma ) )
                                                @ ^ [Ma3: nat] :
                                                    ( heap_Time_bind @ nat @ vEBT_VEBTi @ ( if @ ( heap_Time_Heap @ nat ) @ ( ord_less @ nat @ X @ Mi3 ) @ ( heap_Time_return @ nat @ Mi3 ) @ ( heap_Time_return @ nat @ X ) )
                                                    @ ^ [Xn2: nat] :
                                                        ( heap_Time_bind @ nat @ vEBT_VEBTi @ ( if @ ( heap_Time_Heap @ nat ) @ ( ord_less @ nat @ X @ Mi3 ) @ ( heap_Time_return @ nat @ X ) @ ( heap_Time_return @ nat @ Mi3 ) )
                                                        @ ^ [Minn: nat] :
                                                            ( heap_Time_bind @ nat @ vEBT_VEBTi @ ( vEBT_VEBT_lowi @ Xn2 @ ( divide_divide @ nat @ Deg4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
                                                            @ ^ [L2: nat] :
                                                                ( heap_Time_bind @ product_unit @ vEBT_VEBTi
                                                                @ ( refine_Imp_assert
                                                                  @ ( L2
                                                                    = ( vEBT_VEBT_low @ ( if @ nat @ ( ord_less @ nat @ X @ Mi4 ) @ Mi4 @ X ) @ ( divide_divide @ nat @ Deg5 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
                                                                @ ^ [Uw3: product_unit] :
                                                                    ( heap_Time_bind @ nat @ vEBT_VEBTi @ ( vEBT_VEBT_highi @ Xn2 @ ( divide_divide @ nat @ Deg4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
                                                                    @ ^ [H: nat] :
                                                                        ( heap_Time_bind @ nat @ vEBT_VEBTi @ ( array_len @ vEBT_VEBTi @ TreeArray )
                                                                        @ ^ [Len: nat] :
                                                                            ( if @ ( heap_Time_Heap @ vEBT_VEBTi )
                                                                            @ ( ( ord_less @ nat @ H @ Len )
                                                                              & ~ ( ( X = Mi3 )
                                                                                  | ( X = Ma3 ) ) )
                                                                            @ ( heap_Time_bind @ product_unit @ vEBT_VEBTi
                                                                              @ ( refine_Imp_assert
                                                                                @ ( H
                                                                                  = ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ X @ Mi4 ) @ Mi4 @ X ) @ ( divide_divide @ nat @ Deg5 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
                                                                              @ ^ [Ux3: product_unit] :
                                                                                  ( heap_Time_bind @ product_unit @ vEBT_VEBTi @ ( refine_Imp_assert @ ( ord_less @ nat @ H @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) ) )
                                                                                  @ ^ [Uy3: product_unit] :
                                                                                      ( heap_Time_bind @ vEBT_VEBTi @ vEBT_VEBTi @ ( array_nth @ vEBT_VEBTi @ TreeArray @ H )
                                                                                      @ ^ [Node: vEBT_VEBTi] :
                                                                                          ( heap_Time_bind @ $o @ vEBT_VEBTi @ ( vEBT_VEBT_minNulli @ Node )
                                                                                          @ ^ [Empt: $o] :
                                                                                              ( heap_Time_bind @ product_unit @ vEBT_VEBTi
                                                                                              @ ( refine_Imp_assert
                                                                                                @ ( Empt
                                                                                                  = ( vEBT_VEBT_minNull @ ( nth @ vEBT_VEBT @ TreeList3 @ H ) ) ) )
                                                                                              @ ^ [Uz3: product_unit] :
                                                                                                  ( heap_Time_bind @ vEBT_VEBTi @ vEBT_VEBTi @ ( vEBT_V3964819847710782039nserti @ ( nth @ vEBT_VEBT @ TreeList3 @ H ) @ Node @ L2 )
                                                                                                  @ ^ [Newnode2: vEBT_VEBTi] :
                                                                                                      ( heap_Time_bind @ ( array @ vEBT_VEBTi ) @ vEBT_VEBTi @ ( array_upd @ vEBT_VEBTi @ H @ Newnode2 @ TreeArray )
                                                                                                      @ ^ [Newarray: array @ vEBT_VEBTi] :
                                                                                                          ( heap_Time_bind @ vEBT_VEBTi @ vEBT_VEBTi @ ( if @ ( heap_Time_Heap @ vEBT_VEBTi ) @ Empt @ ( vEBT_V3964819847710782039nserti @ Summary4 @ Summary3 @ H ) @ ( heap_Time_return @ vEBT_VEBTi @ Summary3 ) )
                                                                                                          @ ^ [Newsummary: vEBT_VEBTi] :
                                                                                                              ( heap_Time_bind @ nat @ vEBT_VEBTi @ ( if @ ( heap_Time_Heap @ nat ) @ ( ord_less @ nat @ Ma3 @ Xn2 ) @ ( heap_Time_return @ nat @ Xn2 ) @ ( heap_Time_return @ nat @ Ma3 ) )
                                                                                                              @ ^ [Man: nat] : ( heap_Time_return @ vEBT_VEBTi @ ( vEBT_Nodei @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Minn @ Man ) ) @ Deg4 @ Newarray @ Newsummary ) ) ) ) ) ) ) ) ) ) )
                                                                            @ ( heap_Time_return @ vEBT_VEBTi @ ( vEBT_Nodei @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi3 @ Ma3 ) ) @ Deg4 @ TreeArray @ Summary3 ) ) ) ) ) ) ) ) ) ) )
                                        @ ( the2 @ ( product_prod @ nat @ nat ) @ Info4 ) ) ) ) )
                        @ ( vEBT_case_VEBT @ ( product_prod @ ( option @ ( product_prod @ nat @ nat ) ) @ ( product_prod @ nat @ ( product_prod @ ( list @ vEBT_VEBT ) @ vEBT_VEBT ) ) )
                          @ ^ [Info4: option @ ( product_prod @ nat @ nat ),Deg5: nat,TreeList3: list @ vEBT_VEBT,Summary4: vEBT_VEBT] : ( product_Pair @ ( option @ ( product_prod @ nat @ nat ) ) @ ( product_prod @ nat @ ( product_prod @ ( list @ vEBT_VEBT ) @ vEBT_VEBT ) ) @ Info4 @ ( product_Pair @ nat @ ( product_prod @ ( list @ vEBT_VEBT ) @ vEBT_VEBT ) @ Deg5 @ ( product_Pair @ ( list @ vEBT_VEBT ) @ vEBT_VEBT @ TreeList3 @ Summary4 ) ) )
                          @ ^ [A5: $o,B5: $o] : ( undefined @ ( product_prod @ ( option @ ( product_prod @ nat @ nat ) ) @ ( product_prod @ nat @ ( product_prod @ ( list @ vEBT_VEBT ) @ vEBT_VEBT ) ) ) )
                          @ T3 ) ) ) )
              @ Info3 )
          @ ^ [A5: $o,B5: $o] :
              ( if @ ( heap_Time_Heap @ vEBT_VEBTi )
              @ ( X
                = ( zero_zero @ nat ) )
              @ ( heap_Time_return @ vEBT_VEBTi @ ( vEBT_Leafi @ $true @ B5 ) )
              @ ( if @ ( heap_Time_Heap @ vEBT_VEBTi )
                @ ( X
                  = ( one_one @ nat ) )
                @ ( heap_Time_return @ vEBT_VEBTi @ ( vEBT_Leafi @ A5 @ $true ) )
                @ ( heap_Time_return @ vEBT_VEBTi @ ( vEBT_Leafi @ A5 @ B5 ) ) ) )
          @ Ti3 ) ) ) ).

% VEBT_internal.vebt_inserti'.simps
thf(fact_2732_VEBT__internal_Ovebt__predi_H_Osimps,axiom,
    ( vEBT_VEBT_vebt_predi
    = ( ^ [T3: vEBT_VEBT,Ti3: vEBT_VEBTi,X: nat] :
          ( vEBT_case_VEBTi @ ( heap_Time_Heap @ ( option @ nat ) )
          @ ^ [Info3: option @ ( product_prod @ nat @ nat ),Deg4: nat,TreeArray: array @ vEBT_VEBTi,Summary3: vEBT_VEBTi] :
              ( heap_Time_bind @ product_unit @ ( option @ nat ) @ ( refine_Imp_assert @ ( vEBT_is_Node @ T3 ) )
              @ ^ [Uu3: product_unit] :
                  ( product_case_prod @ ( option @ ( product_prod @ nat @ nat ) ) @ ( product_prod @ nat @ ( product_prod @ ( list @ vEBT_VEBT ) @ vEBT_VEBT ) ) @ ( heap_Time_Heap @ ( option @ nat ) )
                  @ ^ [Info4: option @ ( product_prod @ nat @ nat )] :
                      ( product_case_prod @ nat @ ( product_prod @ ( list @ vEBT_VEBT ) @ vEBT_VEBT ) @ ( heap_Time_Heap @ ( option @ nat ) )
                      @ ^ [Deg5: nat] :
                          ( product_case_prod @ ( list @ vEBT_VEBT ) @ vEBT_VEBT @ ( heap_Time_Heap @ ( option @ nat ) )
                          @ ^ [TreeList3: list @ vEBT_VEBT,Summary4: vEBT_VEBT] :
                              ( heap_Time_bind @ product_unit @ ( option @ nat )
                              @ ( refine_Imp_assert
                                @ ( ( Info4 = Info3 )
                                  & ( Deg5 = Deg4 )
                                  & ( vEBT_is_Node @ T3 ) ) )
                              @ ^ [Uv3: product_unit] :
                                  ( case_option @ ( heap_Time_Heap @ ( option @ nat ) ) @ ( product_prod @ nat @ nat ) @ ( heap_Time_return @ ( option @ nat ) @ ( none @ nat ) )
                                  @ ^ [Mima2: product_prod @ nat @ nat] :
                                      ( if @ ( heap_Time_Heap @ ( option @ nat ) ) @ ( ord_less_eq @ nat @ Deg4 @ ( one_one @ nat ) ) @ ( heap_Time_return @ ( option @ nat ) @ ( none @ nat ) )
                                      @ ( if @ ( heap_Time_Heap @ ( option @ nat ) ) @ ( ord_less @ nat @ ( product_snd @ nat @ nat @ Mima2 ) @ X ) @ ( heap_Time_return @ ( option @ nat ) @ ( some @ nat @ ( product_snd @ nat @ nat @ Mima2 ) ) )
                                        @ ( heap_Time_bind @ nat @ ( option @ nat ) @ ( vEBT_VEBT_lowi @ X @ ( divide_divide @ nat @ Deg4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
                                          @ ^ [L2: nat] :
                                              ( heap_Time_bind @ nat @ ( option @ nat ) @ ( vEBT_VEBT_highi @ X @ ( divide_divide @ nat @ Deg4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
                                              @ ^ [H: nat] :
                                                  ( heap_Time_bind @ product_unit @ ( option @ nat )
                                                  @ ( refine_Imp_assert
                                                    @ ( L2
                                                      = ( vEBT_VEBT_low @ X @ ( divide_divide @ nat @ Deg4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
                                                  @ ^ [Uw3: product_unit] :
                                                      ( heap_Time_bind @ product_unit @ ( option @ nat )
                                                      @ ( refine_Imp_assert
                                                        @ ( H
                                                          = ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
                                                      @ ^ [Ux3: product_unit] :
                                                          ( heap_Time_bind @ product_unit @ ( option @ nat ) @ ( refine_Imp_assert @ ( ord_less @ nat @ H @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) ) )
                                                          @ ^ [Uy3: product_unit] :
                                                              ( heap_Time_bind @ vEBT_VEBTi @ ( option @ nat ) @ ( array_nth @ vEBT_VEBTi @ TreeArray @ H )
                                                              @ ^ [Aktnode: vEBT_VEBTi] :
                                                                  ( heap_Time_bind @ ( option @ nat ) @ ( option @ nat ) @ ( vEBT_vebt_minti @ Aktnode )
                                                                  @ ^ [Minlow: option @ nat] :
                                                                      ( heap_Time_bind @ product_unit @ ( option @ nat )
                                                                      @ ( refine_Imp_assert
                                                                        @ ( Minlow
                                                                          = ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList3 @ H ) ) ) )
                                                                      @ ^ [Uz3: product_unit] :
                                                                          ( if @ ( heap_Time_Heap @ ( option @ nat ) )
                                                                          @ ( ( Minlow
                                                                             != ( none @ nat ) )
                                                                            & ( vEBT_VEBT_greater @ ( some @ nat @ L2 ) @ Minlow ) )
                                                                          @ ( heap_Time_bind @ ( option @ nat ) @ ( option @ nat ) @ ( vEBT_VEBT_vebt_predi @ ( nth @ vEBT_VEBT @ TreeList3 @ H ) @ Aktnode @ L2 )
                                                                            @ ^ [Predy: option @ nat] : ( heap_Time_return @ ( option @ nat ) @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( some @ nat @ H ) ) @ Predy ) ) )
                                                                          @ ( heap_Time_bind @ ( option @ nat ) @ ( option @ nat ) @ ( vEBT_VEBT_vebt_predi @ Summary4 @ Summary3 @ H )
                                                                            @ ^ [Predsum: option @ nat] :
                                                                                ( heap_Time_bind @ product_unit @ ( option @ nat )
                                                                                @ ( refine_Imp_assert
                                                                                  @ ( ( Predsum
                                                                                      = ( none @ nat ) )
                                                                                    = ( ( vEBT_vebt_pred @ Summary4 @ H )
                                                                                      = ( none @ nat ) ) ) )
                                                                                @ ^ [Va4: product_unit] :
                                                                                    ( if @ ( heap_Time_Heap @ ( option @ nat ) )
                                                                                    @ ( Predsum
                                                                                      = ( none @ nat ) )
                                                                                    @ ( if @ ( heap_Time_Heap @ ( option @ nat ) ) @ ( ord_less @ nat @ ( product_fst @ nat @ nat @ Mima2 ) @ X ) @ ( heap_Time_return @ ( option @ nat ) @ ( some @ nat @ ( product_fst @ nat @ nat @ Mima2 ) ) ) @ ( heap_Time_return @ ( option @ nat ) @ ( none @ nat ) ) )
                                                                                    @ ( heap_Time_bind @ vEBT_VEBTi @ ( option @ nat ) @ ( array_nth @ vEBT_VEBTi @ TreeArray @ ( the2 @ nat @ Predsum ) )
                                                                                      @ ^ [Nextnode: vEBT_VEBTi] :
                                                                                          ( heap_Time_bind @ ( option @ nat ) @ ( option @ nat ) @ ( vEBT_vebt_maxti @ Nextnode )
                                                                                          @ ^ [Maxnext: option @ nat] : ( heap_Time_return @ ( option @ nat ) @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ Predsum ) @ Maxnext ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
                                  @ Info3 ) ) ) )
                  @ ( vEBT_case_VEBT @ ( product_prod @ ( option @ ( product_prod @ nat @ nat ) ) @ ( product_prod @ nat @ ( product_prod @ ( list @ vEBT_VEBT ) @ vEBT_VEBT ) ) )
                    @ ^ [Info4: option @ ( product_prod @ nat @ nat ),Deg5: nat,TreeList3: list @ vEBT_VEBT,Summary4: vEBT_VEBT] : ( product_Pair @ ( option @ ( product_prod @ nat @ nat ) ) @ ( product_prod @ nat @ ( product_prod @ ( list @ vEBT_VEBT ) @ vEBT_VEBT ) ) @ Info4 @ ( product_Pair @ nat @ ( product_prod @ ( list @ vEBT_VEBT ) @ vEBT_VEBT ) @ Deg5 @ ( product_Pair @ ( list @ vEBT_VEBT ) @ vEBT_VEBT @ TreeList3 @ Summary4 ) ) )
                    @ ^ [A5: $o,B5: $o] : ( undefined @ ( product_prod @ ( option @ ( product_prod @ nat @ nat ) ) @ ( product_prod @ nat @ ( product_prod @ ( list @ vEBT_VEBT ) @ vEBT_VEBT ) ) ) )
                    @ T3 ) ) )
          @ ^ [A5: $o,B5: $o] :
              ( if @ ( heap_Time_Heap @ ( option @ nat ) ) @ ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ X ) @ ( if @ ( heap_Time_Heap @ ( option @ nat ) ) @ B5 @ ( heap_Time_return @ ( option @ nat ) @ ( some @ nat @ ( one_one @ nat ) ) ) @ ( if @ ( heap_Time_Heap @ ( option @ nat ) ) @ A5 @ ( heap_Time_return @ ( option @ nat ) @ ( some @ nat @ ( zero_zero @ nat ) ) ) @ ( heap_Time_return @ ( option @ nat ) @ ( none @ nat ) ) ) )
              @ ( if @ ( heap_Time_Heap @ ( option @ nat ) )
                @ ( X
                  = ( one_one @ nat ) )
                @ ( if @ ( heap_Time_Heap @ ( option @ nat ) ) @ A5 @ ( heap_Time_return @ ( option @ nat ) @ ( some @ nat @ ( zero_zero @ nat ) ) ) @ ( heap_Time_return @ ( option @ nat ) @ ( none @ nat ) ) )
                @ ( heap_Time_return @ ( option @ nat ) @ ( none @ nat ) ) ) )
          @ Ti3 ) ) ) ).

% VEBT_internal.vebt_predi'.simps
thf(fact_2733_quickcheck__narrowing__samples_Opartial__term__of__sample__def,axiom,
    ! [A: $tType] :
      ( ( ( code_term_of @ A )
        & ( quickc6926020345158392990erm_of @ A ) )
     => ( ( code_T4081349890594273596sample @ A )
        = ( ^ [A_of_integer: code_integer > ( product_prod @ A @ A ),Zero: A,I2: code_integer] :
              ( if @ A @ ( ord_less @ code_integer @ I2 @ ( zero_zero @ code_integer ) ) @ ( undefined @ A )
              @ ( if @ A
                @ ( I2
                  = ( zero_zero @ code_integer ) )
                @ Zero
                @ ( if @ A
                  @ ( ( modulo_modulo @ code_integer @ I2 @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) )
                    = ( zero_zero @ code_integer ) )
                  @ ( product_snd @ A @ A @ ( A_of_integer @ ( divide_divide @ code_integer @ I2 @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) ) ) )
                  @ ( product_fst @ A @ A @ ( A_of_integer @ ( plus_plus @ code_integer @ ( divide_divide @ code_integer @ I2 @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) ) @ ( one_one @ code_integer ) ) ) ) ) ) ) ) ) ) ).

% quickcheck_narrowing_samples.partial_term_of_sample_def
thf(fact_2734_mod__h__bot__normalize,axiom,
    ! [A: $tType,H2: heap_ext @ product_unit,P: assn] :
      ( ( syntax7388354845996824322omatch @ A @ ( heap_ext @ product_unit ) @ ( undefined @ A ) @ H2 )
     => ( ( rep_assn @ P @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ ( bot_bot @ ( set @ nat ) ) ) )
        = ( rep_assn @ P @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ ( undefined @ ( heap_ext @ product_unit ) ) @ ( bot_bot @ ( set @ nat ) ) ) ) ) ) ).

% mod_h_bot_normalize
thf(fact_2735_divmod__algorithm__code_I6_J,axiom,
    ! [A: $tType] :
      ( ( unique1627219031080169319umeral @ A )
     => ! [M: num,N2: num] :
          ( ( unique8689654367752047608divmod @ A @ ( bit1 @ M ) @ ( bit0 @ N2 ) )
          = ( product_case_prod @ A @ A @ ( product_prod @ A @ A )
            @ ^ [Q5: A,R: A] : ( product_Pair @ A @ A @ Q5 @ ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ R ) @ ( one_one @ A ) ) )
            @ ( unique8689654367752047608divmod @ A @ M @ N2 ) ) ) ) ).

% divmod_algorithm_code(6)
thf(fact_2736_neg__eucl__rel__int__mult__2,axiom,
    ! [B2: int,A2: int,Q2: int,R4: int] :
      ( ( ord_less_eq @ int @ B2 @ ( zero_zero @ int ) )
     => ( ( eucl_rel_int @ ( plus_plus @ int @ A2 @ ( one_one @ int ) ) @ B2 @ ( product_Pair @ int @ int @ Q2 @ R4 ) )
       => ( eucl_rel_int @ ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ A2 ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ B2 ) @ ( product_Pair @ int @ int @ Q2 @ ( minus_minus @ int @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ R4 ) @ ( one_one @ int ) ) ) ) ) ) ).

% neg_eucl_rel_int_mult_2
thf(fact_2737_divmod__algorithm__code_I7_J,axiom,
    ! [A: $tType] :
      ( ( unique1627219031080169319umeral @ A )
     => ! [M: num,N2: num] :
          ( ( ( ord_less_eq @ num @ M @ N2 )
           => ( ( unique8689654367752047608divmod @ A @ ( bit0 @ M ) @ ( bit1 @ N2 ) )
              = ( product_Pair @ A @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ ( bit0 @ M ) ) ) ) )
          & ( ~ ( ord_less_eq @ num @ M @ N2 )
           => ( ( unique8689654367752047608divmod @ A @ ( bit0 @ M ) @ ( bit1 @ N2 ) )
              = ( unique1321980374590559556d_step @ A @ ( bit1 @ N2 ) @ ( unique8689654367752047608divmod @ A @ ( bit0 @ M ) @ ( bit0 @ ( bit1 @ N2 ) ) ) ) ) ) ) ) ).

% divmod_algorithm_code(7)
thf(fact_2738_divmod__algorithm__code_I8_J,axiom,
    ! [A: $tType] :
      ( ( unique1627219031080169319umeral @ A )
     => ! [M: num,N2: num] :
          ( ( ( ord_less @ num @ M @ N2 )
           => ( ( unique8689654367752047608divmod @ A @ ( bit1 @ M ) @ ( bit1 @ N2 ) )
              = ( product_Pair @ A @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ ( bit1 @ M ) ) ) ) )
          & ( ~ ( ord_less @ num @ M @ N2 )
           => ( ( unique8689654367752047608divmod @ A @ ( bit1 @ M ) @ ( bit1 @ N2 ) )
              = ( unique1321980374590559556d_step @ A @ ( bit1 @ N2 ) @ ( unique8689654367752047608divmod @ A @ ( bit1 @ M ) @ ( bit0 @ ( bit1 @ N2 ) ) ) ) ) ) ) ) ).

% divmod_algorithm_code(8)
thf(fact_2739_numeral__div__numeral,axiom,
    ! [A: $tType] :
      ( ( unique1627219031080169319umeral @ A )
     => ! [K: num,L: num] :
          ( ( divide_divide @ A @ ( numeral_numeral @ A @ K ) @ ( numeral_numeral @ A @ L ) )
          = ( product_fst @ A @ A @ ( unique8689654367752047608divmod @ A @ K @ L ) ) ) ) ).

% numeral_div_numeral
thf(fact_2740_numeral__mod__numeral,axiom,
    ! [A: $tType] :
      ( ( unique1627219031080169319umeral @ A )
     => ! [K: num,L: num] :
          ( ( modulo_modulo @ A @ ( numeral_numeral @ A @ K ) @ ( numeral_numeral @ A @ L ) )
          = ( product_snd @ A @ A @ ( unique8689654367752047608divmod @ A @ K @ L ) ) ) ) ).

% numeral_mod_numeral
thf(fact_2741_divmod__algorithm__code_I2_J,axiom,
    ! [A: $tType] :
      ( ( unique1627219031080169319umeral @ A )
     => ! [M: num] :
          ( ( unique8689654367752047608divmod @ A @ M @ one2 )
          = ( product_Pair @ A @ A @ ( numeral_numeral @ A @ M ) @ ( zero_zero @ A ) ) ) ) ).

% divmod_algorithm_code(2)
thf(fact_2742_divmod__algorithm__code_I3_J,axiom,
    ! [A: $tType] :
      ( ( unique1627219031080169319umeral @ A )
     => ! [N2: num] :
          ( ( unique8689654367752047608divmod @ A @ one2 @ ( bit0 @ N2 ) )
          = ( product_Pair @ A @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ one2 ) ) ) ) ).

% divmod_algorithm_code(3)
thf(fact_2743_divmod__algorithm__code_I4_J,axiom,
    ! [A: $tType] :
      ( ( unique1627219031080169319umeral @ A )
     => ! [N2: num] :
          ( ( unique8689654367752047608divmod @ A @ one2 @ ( bit1 @ N2 ) )
          = ( product_Pair @ A @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ one2 ) ) ) ) ).

% divmod_algorithm_code(4)
thf(fact_2744_Suc__0__div__numeral,axiom,
    ! [K: num] :
      ( ( divide_divide @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( numeral_numeral @ nat @ K ) )
      = ( product_fst @ nat @ nat @ ( unique8689654367752047608divmod @ nat @ one2 @ K ) ) ) ).

% Suc_0_div_numeral
thf(fact_2745_one__div__numeral,axiom,
    ! [A: $tType] :
      ( ( unique1627219031080169319umeral @ A )
     => ! [N2: num] :
          ( ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ N2 ) )
          = ( product_fst @ A @ A @ ( unique8689654367752047608divmod @ A @ one2 @ N2 ) ) ) ) ).

% one_div_numeral
thf(fact_2746_one__mod__numeral,axiom,
    ! [A: $tType] :
      ( ( unique1627219031080169319umeral @ A )
     => ! [N2: num] :
          ( ( modulo_modulo @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ N2 ) )
          = ( product_snd @ A @ A @ ( unique8689654367752047608divmod @ A @ one2 @ N2 ) ) ) ) ).

% one_mod_numeral
thf(fact_2747_Suc__0__mod__numeral,axiom,
    ! [K: num] :
      ( ( modulo_modulo @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( numeral_numeral @ nat @ K ) )
      = ( product_snd @ nat @ nat @ ( unique8689654367752047608divmod @ nat @ one2 @ K ) ) ) ).

% Suc_0_mod_numeral
thf(fact_2748_divmod__algorithm__code_I5_J,axiom,
    ! [A: $tType] :
      ( ( unique1627219031080169319umeral @ A )
     => ! [M: num,N2: num] :
          ( ( unique8689654367752047608divmod @ A @ ( bit0 @ M ) @ ( bit0 @ N2 ) )
          = ( product_case_prod @ A @ A @ ( product_prod @ A @ A )
            @ ^ [Q5: A,R: A] : ( product_Pair @ A @ A @ Q5 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ R ) )
            @ ( unique8689654367752047608divmod @ A @ M @ N2 ) ) ) ) ).

% divmod_algorithm_code(5)
thf(fact_2749_unique__remainder,axiom,
    ! [A2: int,B2: int,Q2: int,R4: int,Q6: int,R5: int] :
      ( ( eucl_rel_int @ A2 @ B2 @ ( product_Pair @ int @ int @ Q2 @ R4 ) )
     => ( ( eucl_rel_int @ A2 @ B2 @ ( product_Pair @ int @ int @ Q6 @ R5 ) )
       => ( R4 = R5 ) ) ) ).

% unique_remainder
thf(fact_2750_unique__quotient,axiom,
    ! [A2: int,B2: int,Q2: int,R4: int,Q6: int,R5: int] :
      ( ( eucl_rel_int @ A2 @ B2 @ ( product_Pair @ int @ int @ Q2 @ R4 ) )
     => ( ( eucl_rel_int @ A2 @ B2 @ ( product_Pair @ int @ int @ Q6 @ R5 ) )
       => ( Q2 = Q6 ) ) ) ).

% unique_quotient
thf(fact_2751_VEBT_Odisc__eq__case_I1_J,axiom,
    ( vEBT_is_Node
    = ( vEBT_case_VEBT @ $o
      @ ^ [Uu3: option @ ( product_prod @ nat @ nat ),Uv3: nat,Uw3: list @ vEBT_VEBT,Ux3: vEBT_VEBT] : $true
      @ ^ [Uu3: $o,Uv3: $o] : $false ) ) ).

% VEBT.disc_eq_case(1)
thf(fact_2752_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_2753_mod__int__unique,axiom,
    ! [K: int,L: int,Q2: int,R4: int] :
      ( ( eucl_rel_int @ K @ L @ ( product_Pair @ int @ int @ Q2 @ R4 ) )
     => ( ( modulo_modulo @ int @ K @ L )
        = R4 ) ) ).

% mod_int_unique
thf(fact_2754_div__int__unique,axiom,
    ! [K: int,L: int,Q2: int,R4: int] :
      ( ( eucl_rel_int @ K @ L @ ( product_Pair @ int @ int @ Q2 @ R4 ) )
     => ( ( divide_divide @ int @ K @ L )
        = Q2 ) ) ).

% div_int_unique
thf(fact_2755_divmod__integer_H__def,axiom,
    ( ( unique8689654367752047608divmod @ code_integer )
    = ( ^ [M7: num,N: num] : ( product_Pair @ code_integer @ code_integer @ ( divide_divide @ code_integer @ ( numeral_numeral @ code_integer @ M7 ) @ ( numeral_numeral @ code_integer @ N ) ) @ ( modulo_modulo @ code_integer @ ( numeral_numeral @ code_integer @ M7 ) @ ( numeral_numeral @ code_integer @ N ) ) ) ) ) ).

% divmod_integer'_def
thf(fact_2756_fst__divmod,axiom,
    ! [A: $tType] :
      ( ( unique1627219031080169319umeral @ A )
     => ! [M: num,N2: num] :
          ( ( product_fst @ A @ A @ ( unique8689654367752047608divmod @ A @ M @ N2 ) )
          = ( divide_divide @ A @ ( numeral_numeral @ A @ M ) @ ( numeral_numeral @ A @ N2 ) ) ) ) ).

% fst_divmod
thf(fact_2757_snd__divmod,axiom,
    ! [A: $tType] :
      ( ( unique1627219031080169319umeral @ A )
     => ! [M: num,N2: num] :
          ( ( product_snd @ A @ A @ ( unique8689654367752047608divmod @ A @ M @ N2 ) )
          = ( modulo_modulo @ A @ ( numeral_numeral @ A @ M ) @ ( numeral_numeral @ A @ N2 ) ) ) ) ).

% snd_divmod
thf(fact_2758_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_2759_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_2760_divmod__int__def,axiom,
    ( ( unique8689654367752047608divmod @ int )
    = ( ^ [M7: num,N: num] : ( product_Pair @ int @ int @ ( divide_divide @ int @ ( numeral_numeral @ int @ M7 ) @ ( numeral_numeral @ int @ N ) ) @ ( modulo_modulo @ int @ ( numeral_numeral @ int @ M7 ) @ ( numeral_numeral @ int @ N ) ) ) ) ) ).

% divmod_int_def
thf(fact_2761_divmod__def,axiom,
    ! [A: $tType] :
      ( ( unique1627219031080169319umeral @ A )
     => ( ( unique8689654367752047608divmod @ A )
        = ( ^ [M7: num,N: num] : ( product_Pair @ A @ A @ ( divide_divide @ A @ ( numeral_numeral @ A @ M7 ) @ ( numeral_numeral @ A @ N ) ) @ ( modulo_modulo @ A @ ( numeral_numeral @ A @ M7 ) @ ( numeral_numeral @ A @ N ) ) ) ) ) ) ).

% divmod_def
thf(fact_2762_divmod_H__nat__def,axiom,
    ( ( unique8689654367752047608divmod @ nat )
    = ( ^ [M7: num,N: num] : ( product_Pair @ nat @ nat @ ( divide_divide @ nat @ ( numeral_numeral @ nat @ M7 ) @ ( numeral_numeral @ nat @ N ) ) @ ( modulo_modulo @ nat @ ( numeral_numeral @ nat @ M7 ) @ ( numeral_numeral @ nat @ N ) ) ) ) ) ).

% divmod'_nat_def
thf(fact_2763_eucl__rel__int__iff,axiom,
    ! [K: int,L: int,Q2: int,R4: int] :
      ( ( eucl_rel_int @ K @ L @ ( product_Pair @ int @ int @ Q2 @ R4 ) )
      = ( ( K
          = ( plus_plus @ int @ ( times_times @ int @ L @ Q2 ) @ R4 ) )
        & ( ( ord_less @ int @ ( zero_zero @ int ) @ L )
         => ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ R4 )
            & ( ord_less @ int @ R4 @ L ) ) )
        & ( ~ ( ord_less @ int @ ( zero_zero @ int ) @ L )
         => ( ( ( ord_less @ int @ L @ ( zero_zero @ int ) )
             => ( ( ord_less @ int @ L @ R4 )
                & ( ord_less_eq @ int @ R4 @ ( zero_zero @ int ) ) ) )
            & ( ~ ( ord_less @ int @ L @ ( zero_zero @ int ) )
             => ( Q2
                = ( zero_zero @ int ) ) ) ) ) ) ) ).

% eucl_rel_int_iff
thf(fact_2764_divmod__divmod__step,axiom,
    ! [A: $tType] :
      ( ( unique1627219031080169319umeral @ A )
     => ( ( unique8689654367752047608divmod @ A )
        = ( ^ [M7: num,N: num] : ( if @ ( product_prod @ A @ A ) @ ( ord_less @ num @ M7 @ N ) @ ( product_Pair @ A @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ M7 ) ) @ ( unique1321980374590559556d_step @ A @ N @ ( unique8689654367752047608divmod @ A @ M7 @ ( bit0 @ N ) ) ) ) ) ) ) ).

% divmod_divmod_step
thf(fact_2765_pos__eucl__rel__int__mult__2,axiom,
    ! [B2: int,A2: int,Q2: int,R4: int] :
      ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ B2 )
     => ( ( eucl_rel_int @ A2 @ B2 @ ( product_Pair @ int @ int @ Q2 @ R4 ) )
       => ( eucl_rel_int @ ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ A2 ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ B2 ) @ ( product_Pair @ int @ int @ Q2 @ ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ R4 ) ) ) ) ) ) ).

% pos_eucl_rel_int_mult_2
thf(fact_2766_VEBT__internal_Ovebt__inserti_H_Omono,axiom,
    ! [X2: product_prod @ ( product_prod @ vEBT_VEBT @ vEBT_VEBTi ) @ nat] :
      ( comple7038119648293358887notone @ ( ( product_prod @ ( product_prod @ vEBT_VEBT @ vEBT_VEBTi ) @ nat ) > ( heap_Time_Heap @ vEBT_VEBTi ) ) @ ( heap_Time_Heap @ vEBT_VEBTi ) @ ( partial_fun_ord @ ( heap_Time_Heap @ vEBT_VEBTi ) @ ( heap_Time_Heap @ vEBT_VEBTi ) @ ( product_prod @ ( product_prod @ vEBT_VEBT @ vEBT_VEBTi ) @ nat ) @ ( heap_Time_Heap_ord @ vEBT_VEBTi ) ) @ ( heap_Time_Heap_ord @ vEBT_VEBTi )
      @ ^ [Vebt_inserti3: ( product_prod @ ( product_prod @ vEBT_VEBT @ vEBT_VEBTi ) @ nat ) > ( heap_Time_Heap @ vEBT_VEBTi )] :
          ( product_case_prod @ ( product_prod @ vEBT_VEBT @ vEBT_VEBTi ) @ nat @ ( heap_Time_Heap @ vEBT_VEBTi )
          @ ( product_case_prod @ vEBT_VEBT @ vEBT_VEBTi @ ( nat > ( heap_Time_Heap @ vEBT_VEBTi ) )
            @ ^ [T3: vEBT_VEBT,Ti3: vEBT_VEBTi,X: nat] :
                ( vEBT_case_VEBTi @ ( heap_Time_Heap @ vEBT_VEBTi )
                @ ^ [Info3: option @ ( product_prod @ nat @ nat ),Deg4: nat,TreeArray: array @ vEBT_VEBTi,Summary3: vEBT_VEBTi] :
                    ( case_option @ ( heap_Time_Heap @ vEBT_VEBTi ) @ ( product_prod @ nat @ nat ) @ ( if @ ( heap_Time_Heap @ vEBT_VEBTi ) @ ( ord_less_eq @ nat @ Deg4 @ ( one_one @ nat ) ) @ ( heap_Time_return @ vEBT_VEBTi @ ( vEBT_Nodei @ Info3 @ Deg4 @ TreeArray @ Summary3 ) ) @ ( heap_Time_return @ vEBT_VEBTi @ ( vEBT_Nodei @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ X @ X ) ) @ Deg4 @ TreeArray @ Summary3 ) ) )
                    @ ^ [Minma: product_prod @ nat @ nat] :
                        ( if @ ( heap_Time_Heap @ vEBT_VEBTi ) @ ( ord_less_eq @ nat @ Deg4 @ ( one_one @ nat ) ) @ ( heap_Time_return @ vEBT_VEBTi @ ( vEBT_Nodei @ Info3 @ Deg4 @ TreeArray @ Summary3 ) )
                        @ ( heap_Time_bind @ product_unit @ vEBT_VEBTi @ ( refine_Imp_assert @ ( vEBT_is_Node @ T3 ) )
                          @ ^ [Uu3: product_unit] :
                              ( product_case_prod @ ( option @ ( product_prod @ nat @ nat ) ) @ ( product_prod @ nat @ ( product_prod @ ( list @ vEBT_VEBT ) @ vEBT_VEBT ) ) @ ( heap_Time_Heap @ vEBT_VEBTi )
                              @ ^ [Info4: option @ ( product_prod @ nat @ nat )] :
                                  ( product_case_prod @ nat @ ( product_prod @ ( list @ vEBT_VEBT ) @ vEBT_VEBT ) @ ( heap_Time_Heap @ vEBT_VEBTi )
                                  @ ^ [Deg5: nat] :
                                      ( product_case_prod @ ( list @ vEBT_VEBT ) @ vEBT_VEBT @ ( heap_Time_Heap @ vEBT_VEBTi )
                                      @ ^ [TreeList3: list @ vEBT_VEBT,Summary4: vEBT_VEBT] :
                                          ( heap_Time_bind @ product_unit @ vEBT_VEBTi
                                          @ ( refine_Imp_assert
                                            @ ( ( Info3 = Info4 )
                                              & ( Deg4 = Deg5 ) ) )
                                          @ ^ [Uv3: product_unit] :
                                              ( product_case_prod @ nat @ nat @ ( heap_Time_Heap @ vEBT_VEBTi )
                                              @ ^ [Mi4: nat,Ma4: nat] :
                                                  ( heap_Time_bind @ nat @ vEBT_VEBTi @ ( heap_Time_return @ nat @ ( product_fst @ nat @ nat @ Minma ) )
                                                  @ ^ [Mi3: nat] :
                                                      ( heap_Time_bind @ nat @ vEBT_VEBTi @ ( heap_Time_return @ nat @ ( product_snd @ nat @ nat @ Minma ) )
                                                      @ ^ [Ma3: nat] :
                                                          ( heap_Time_bind @ nat @ vEBT_VEBTi @ ( if @ ( heap_Time_Heap @ nat ) @ ( ord_less @ nat @ X @ Mi3 ) @ ( heap_Time_return @ nat @ Mi3 ) @ ( heap_Time_return @ nat @ X ) )
                                                          @ ^ [Xn2: nat] :
                                                              ( heap_Time_bind @ nat @ vEBT_VEBTi @ ( if @ ( heap_Time_Heap @ nat ) @ ( ord_less @ nat @ X @ Mi3 ) @ ( heap_Time_return @ nat @ X ) @ ( heap_Time_return @ nat @ Mi3 ) )
                                                              @ ^ [Minn: nat] :
                                                                  ( heap_Time_bind @ nat @ vEBT_VEBTi @ ( vEBT_VEBT_lowi @ Xn2 @ ( divide_divide @ nat @ Deg4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
                                                                  @ ^ [L2: nat] :
                                                                      ( heap_Time_bind @ product_unit @ vEBT_VEBTi
                                                                      @ ( refine_Imp_assert
                                                                        @ ( L2
                                                                          = ( vEBT_VEBT_low @ ( if @ nat @ ( ord_less @ nat @ X @ Mi4 ) @ Mi4 @ X ) @ ( divide_divide @ nat @ Deg5 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
                                                                      @ ^ [Uw3: product_unit] :
                                                                          ( heap_Time_bind @ nat @ vEBT_VEBTi @ ( vEBT_VEBT_highi @ Xn2 @ ( divide_divide @ nat @ Deg4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
                                                                          @ ^ [H: nat] :
                                                                              ( heap_Time_bind @ nat @ vEBT_VEBTi @ ( array_len @ vEBT_VEBTi @ TreeArray )
                                                                              @ ^ [Len: nat] :
                                                                                  ( if @ ( heap_Time_Heap @ vEBT_VEBTi )
                                                                                  @ ( ( ord_less @ nat @ H @ Len )
                                                                                    & ~ ( ( X = Mi3 )
                                                                                        | ( X = Ma3 ) ) )
                                                                                  @ ( heap_Time_bind @ product_unit @ vEBT_VEBTi
                                                                                    @ ( refine_Imp_assert
                                                                                      @ ( H
                                                                                        = ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ X @ Mi4 ) @ Mi4 @ X ) @ ( divide_divide @ nat @ Deg5 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
                                                                                    @ ^ [Ux3: product_unit] :
                                                                                        ( heap_Time_bind @ product_unit @ vEBT_VEBTi @ ( refine_Imp_assert @ ( ord_less @ nat @ H @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) ) )
                                                                                        @ ^ [Uy3: product_unit] :
                                                                                            ( heap_Time_bind @ vEBT_VEBTi @ vEBT_VEBTi @ ( array_nth @ vEBT_VEBTi @ TreeArray @ H )
                                                                                            @ ^ [Node: vEBT_VEBTi] :
                                                                                                ( heap_Time_bind @ $o @ vEBT_VEBTi @ ( vEBT_VEBT_minNulli @ Node )
                                                                                                @ ^ [Empt: $o] :
                                                                                                    ( heap_Time_bind @ product_unit @ vEBT_VEBTi
                                                                                                    @ ( refine_Imp_assert
                                                                                                      @ ( Empt
                                                                                                        = ( vEBT_VEBT_minNull @ ( nth @ vEBT_VEBT @ TreeList3 @ H ) ) ) )
                                                                                                    @ ^ [Uz3: product_unit] :
                                                                                                        ( heap_Time_bind @ vEBT_VEBTi @ vEBT_VEBTi @ ( product_curry @ vEBT_VEBT @ vEBT_VEBTi @ ( nat > ( heap_Time_Heap @ vEBT_VEBTi ) ) @ ( product_curry @ ( product_prod @ vEBT_VEBT @ vEBT_VEBTi ) @ nat @ ( heap_Time_Heap @ vEBT_VEBTi ) @ Vebt_inserti3 ) @ ( nth @ vEBT_VEBT @ TreeList3 @ H ) @ Node @ L2 )
                                                                                                        @ ^ [Newnode2: vEBT_VEBTi] :
                                                                                                            ( heap_Time_bind @ ( array @ vEBT_VEBTi ) @ vEBT_VEBTi @ ( array_upd @ vEBT_VEBTi @ H @ Newnode2 @ TreeArray )
                                                                                                            @ ^ [Newarray: array @ vEBT_VEBTi] :
                                                                                                                ( heap_Time_bind @ vEBT_VEBTi @ vEBT_VEBTi @ ( if @ ( heap_Time_Heap @ vEBT_VEBTi ) @ Empt @ ( product_curry @ vEBT_VEBT @ vEBT_VEBTi @ ( nat > ( heap_Time_Heap @ vEBT_VEBTi ) ) @ ( product_curry @ ( product_prod @ vEBT_VEBT @ vEBT_VEBTi ) @ nat @ ( heap_Time_Heap @ vEBT_VEBTi ) @ Vebt_inserti3 ) @ Summary4 @ Summary3 @ H ) @ ( heap_Time_return @ vEBT_VEBTi @ Summary3 ) )
                                                                                                                @ ^ [Newsummary: vEBT_VEBTi] :
                                                                                                                    ( heap_Time_bind @ nat @ vEBT_VEBTi @ ( if @ ( heap_Time_Heap @ nat ) @ ( ord_less @ nat @ Ma3 @ Xn2 ) @ ( heap_Time_return @ nat @ Xn2 ) @ ( heap_Time_return @ nat @ Ma3 ) )
                                                                                                                    @ ^ [Man: nat] : ( heap_Time_return @ vEBT_VEBTi @ ( vEBT_Nodei @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Minn @ Man ) ) @ Deg4 @ Newarray @ Newsummary ) ) ) ) ) ) ) ) ) ) )
                                                                                  @ ( heap_Time_return @ vEBT_VEBTi @ ( vEBT_Nodei @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi3 @ Ma3 ) ) @ Deg4 @ TreeArray @ Summary3 ) ) ) ) ) ) ) ) ) ) )
                                              @ ( the2 @ ( product_prod @ nat @ nat ) @ Info4 ) ) ) ) )
                              @ ( vEBT_case_VEBT @ ( product_prod @ ( option @ ( product_prod @ nat @ nat ) ) @ ( product_prod @ nat @ ( product_prod @ ( list @ vEBT_VEBT ) @ vEBT_VEBT ) ) )
                                @ ^ [Info4: option @ ( product_prod @ nat @ nat ),Deg5: nat,TreeList3: list @ vEBT_VEBT,Summary4: vEBT_VEBT] : ( product_Pair @ ( option @ ( product_prod @ nat @ nat ) ) @ ( product_prod @ nat @ ( product_prod @ ( list @ vEBT_VEBT ) @ vEBT_VEBT ) ) @ Info4 @ ( product_Pair @ nat @ ( product_prod @ ( list @ vEBT_VEBT ) @ vEBT_VEBT ) @ Deg5 @ ( product_Pair @ ( list @ vEBT_VEBT ) @ vEBT_VEBT @ TreeList3 @ Summary4 ) ) )
                                @ ^ [A5: $o,B5: $o] : ( undefined @ ( product_prod @ ( option @ ( product_prod @ nat @ nat ) ) @ ( product_prod @ nat @ ( product_prod @ ( list @ vEBT_VEBT ) @ vEBT_VEBT ) ) ) )
                                @ T3 ) ) ) )
                    @ Info3 )
                @ ^ [A5: $o,B5: $o] :
                    ( if @ ( heap_Time_Heap @ vEBT_VEBTi )
                    @ ( X
                      = ( zero_zero @ nat ) )
                    @ ( heap_Time_return @ vEBT_VEBTi @ ( vEBT_Leafi @ $true @ B5 ) )
                    @ ( if @ ( heap_Time_Heap @ vEBT_VEBTi )
                      @ ( X
                        = ( one_one @ nat ) )
                      @ ( heap_Time_return @ vEBT_VEBTi @ ( vEBT_Leafi @ A5 @ $true ) )
                      @ ( heap_Time_return @ vEBT_VEBTi @ ( vEBT_Leafi @ A5 @ B5 ) ) ) )
                @ Ti3 ) )
          @ X2 ) ) ).

% VEBT_internal.vebt_inserti'.mono
thf(fact_2767_divides__aux__eq,axiom,
    ! [A: $tType] :
      ( ( unique1627219031080169319umeral @ A )
     => ! [Q2: A,R4: A] :
          ( ( unique5940410009612947441es_aux @ A @ ( product_Pair @ A @ A @ Q2 @ R4 ) )
          = ( R4
            = ( zero_zero @ A ) ) ) ) ).

% divides_aux_eq
thf(fact_2768_VEBT__internal_OT__vebt__buildupi_H_Oelims,axiom,
    ! [X2: nat,Y2: int] :
      ( ( ( vEBT_V9176841429113362141ildupi @ X2 )
        = Y2 )
     => ( ( ( X2
            = ( zero_zero @ nat ) )
         => ( Y2
           != ( one_one @ int ) ) )
       => ( ( ( X2
              = ( suc @ ( zero_zero @ nat ) ) )
           => ( Y2
             != ( one_one @ int ) ) )
         => ~ ! [N4: nat] :
                ( ( X2
                  = ( suc @ ( suc @ N4 ) ) )
               => ~ ( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N4 )
                     => ( Y2
                        = ( plus_plus @ int @ ( numeral_numeral @ int @ ( bit1 @ one2 ) ) @ ( plus_plus @ int @ ( vEBT_V9176841429113362141ildupi @ ( suc @ ( divide_divide @ nat @ N4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( plus_plus @ int @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ ( bit0 @ one2 ) ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ N4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( times_times @ int @ ( vEBT_V9176841429113362141ildupi @ ( suc @ ( divide_divide @ nat @ N4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ N4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) )
                    & ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N4 )
                     => ( Y2
                        = ( plus_plus @ int @ ( numeral_numeral @ int @ ( bit1 @ one2 ) ) @ ( plus_plus @ int @ ( vEBT_V9176841429113362141ildupi @ ( suc @ ( suc @ ( divide_divide @ nat @ N4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( plus_plus @ int @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ N4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ ( bit0 @ one2 ) ) ) @ ( times_times @ int @ ( vEBT_V9176841429113362141ildupi @ ( suc @ ( divide_divide @ nat @ N4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ N4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.T_vebt_buildupi'.elims
thf(fact_2769_flip__bit__Suc,axiom,
    ! [A: $tType] :
      ( ( bit_se359711467146920520ations @ A )
     => ! [N2: nat,A2: A] :
          ( ( bit_se8732182000553998342ip_bit @ A @ ( suc @ N2 ) @ A2 )
          = ( plus_plus @ A @ ( modulo_modulo @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se8732182000553998342ip_bit @ A @ N2 @ ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).

% flip_bit_Suc
thf(fact_2770_unset__bit__0,axiom,
    ! [A: $tType] :
      ( ( bit_se359711467146920520ations @ A )
     => ! [A2: A] :
          ( ( bit_se2638667681897837118et_bit @ A @ ( zero_zero @ nat ) @ A2 )
          = ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).

% unset_bit_0
thf(fact_2771_VEBT__internal_Ovebt__succi_H_Omono,axiom,
    ! [X2: product_prod @ ( product_prod @ vEBT_VEBT @ vEBT_VEBTi ) @ nat] :
      ( comple7038119648293358887notone @ ( ( product_prod @ ( product_prod @ vEBT_VEBT @ vEBT_VEBTi ) @ nat ) > ( heap_Time_Heap @ ( option @ nat ) ) ) @ ( heap_Time_Heap @ ( option @ nat ) ) @ ( partial_fun_ord @ ( heap_Time_Heap @ ( option @ nat ) ) @ ( heap_Time_Heap @ ( option @ nat ) ) @ ( product_prod @ ( product_prod @ vEBT_VEBT @ vEBT_VEBTi ) @ nat ) @ ( heap_Time_Heap_ord @ ( option @ nat ) ) ) @ ( heap_Time_Heap_ord @ ( option @ nat ) )
      @ ^ [Vebt_succi3: ( product_prod @ ( product_prod @ vEBT_VEBT @ vEBT_VEBTi ) @ nat ) > ( heap_Time_Heap @ ( option @ nat ) )] :
          ( product_case_prod @ ( product_prod @ vEBT_VEBT @ vEBT_VEBTi ) @ nat @ ( heap_Time_Heap @ ( option @ nat ) )
          @ ( product_case_prod @ vEBT_VEBT @ vEBT_VEBTi @ ( nat > ( heap_Time_Heap @ ( option @ nat ) ) )
            @ ^ [T3: vEBT_VEBT,Ti3: vEBT_VEBTi,X: nat] :
                ( vEBT_case_VEBTi @ ( heap_Time_Heap @ ( option @ nat ) )
                @ ^ [Info3: option @ ( product_prod @ nat @ nat ),Deg4: nat,TreeArray: array @ vEBT_VEBTi,Summary3: vEBT_VEBTi] :
                    ( heap_Time_bind @ product_unit @ ( option @ nat ) @ ( refine_Imp_assert @ ( vEBT_is_Node @ T3 ) )
                    @ ^ [Uu3: product_unit] :
                        ( product_case_prod @ ( option @ ( product_prod @ nat @ nat ) ) @ ( product_prod @ nat @ ( product_prod @ ( list @ vEBT_VEBT ) @ vEBT_VEBT ) ) @ ( heap_Time_Heap @ ( option @ nat ) )
                        @ ^ [Info4: option @ ( product_prod @ nat @ nat )] :
                            ( product_case_prod @ nat @ ( product_prod @ ( list @ vEBT_VEBT ) @ vEBT_VEBT ) @ ( heap_Time_Heap @ ( option @ nat ) )
                            @ ^ [Deg5: nat] :
                                ( product_case_prod @ ( list @ vEBT_VEBT ) @ vEBT_VEBT @ ( heap_Time_Heap @ ( option @ nat ) )
                                @ ^ [TreeList3: list @ vEBT_VEBT,Summary4: vEBT_VEBT] :
                                    ( heap_Time_bind @ product_unit @ ( option @ nat )
                                    @ ( refine_Imp_assert
                                      @ ( ( Info4 = Info3 )
                                        & ( Deg5 = Deg4 )
                                        & ( vEBT_is_Node @ T3 ) ) )
                                    @ ^ [Uv3: product_unit] :
                                        ( case_option @ ( heap_Time_Heap @ ( option @ nat ) ) @ ( product_prod @ nat @ nat ) @ ( heap_Time_return @ ( option @ nat ) @ ( none @ nat ) )
                                        @ ^ [Mima2: product_prod @ nat @ nat] :
                                            ( if @ ( heap_Time_Heap @ ( option @ nat ) ) @ ( ord_less_eq @ nat @ Deg4 @ ( one_one @ nat ) ) @ ( heap_Time_return @ ( option @ nat ) @ ( none @ nat ) )
                                            @ ( if @ ( heap_Time_Heap @ ( option @ nat ) ) @ ( ord_less @ nat @ X @ ( product_fst @ nat @ nat @ Mima2 ) ) @ ( heap_Time_return @ ( option @ nat ) @ ( some @ nat @ ( product_fst @ nat @ nat @ Mima2 ) ) )
                                              @ ( if @ ( heap_Time_Heap @ ( option @ nat ) ) @ ( ord_less_eq @ nat @ ( product_snd @ nat @ nat @ Mima2 ) @ X ) @ ( heap_Time_return @ ( option @ nat ) @ ( none @ nat ) )
                                                @ ( heap_Time_bind @ nat @ ( option @ nat ) @ ( vEBT_VEBT_lowi @ X @ ( divide_divide @ nat @ Deg4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
                                                  @ ^ [L2: nat] :
                                                      ( heap_Time_bind @ nat @ ( option @ nat ) @ ( vEBT_VEBT_highi @ X @ ( divide_divide @ nat @ Deg4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
                                                      @ ^ [H: nat] :
                                                          ( heap_Time_bind @ product_unit @ ( option @ nat )
                                                          @ ( refine_Imp_assert
                                                            @ ( L2
                                                              = ( vEBT_VEBT_low @ X @ ( divide_divide @ nat @ Deg4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
                                                          @ ^ [Uw3: product_unit] :
                                                              ( heap_Time_bind @ product_unit @ ( option @ nat )
                                                              @ ( refine_Imp_assert
                                                                @ ( H
                                                                  = ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
                                                              @ ^ [Ux3: product_unit] :
                                                                  ( heap_Time_bind @ product_unit @ ( option @ nat ) @ ( refine_Imp_assert @ ( ord_less @ nat @ H @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) ) )
                                                                  @ ^ [Uy3: product_unit] :
                                                                      ( heap_Time_bind @ vEBT_VEBTi @ ( option @ nat ) @ ( array_nth @ vEBT_VEBTi @ TreeArray @ H )
                                                                      @ ^ [Aktnode: vEBT_VEBTi] :
                                                                          ( heap_Time_bind @ ( option @ nat ) @ ( option @ nat ) @ ( vEBT_vebt_maxti @ Aktnode )
                                                                          @ ^ [Maxlow: option @ nat] :
                                                                              ( heap_Time_bind @ product_unit @ ( option @ nat )
                                                                              @ ( refine_Imp_assert
                                                                                @ ( Maxlow
                                                                                  = ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList3 @ H ) ) ) )
                                                                              @ ^ [Uz3: product_unit] :
                                                                                  ( if @ ( heap_Time_Heap @ ( option @ nat ) )
                                                                                  @ ( ( Maxlow
                                                                                     != ( none @ nat ) )
                                                                                    & ( vEBT_VEBT_less @ ( some @ nat @ L2 ) @ Maxlow ) )
                                                                                  @ ( heap_Time_bind @ ( option @ nat ) @ ( option @ nat ) @ ( product_curry @ vEBT_VEBT @ vEBT_VEBTi @ ( nat > ( heap_Time_Heap @ ( option @ nat ) ) ) @ ( product_curry @ ( product_prod @ vEBT_VEBT @ vEBT_VEBTi ) @ nat @ ( heap_Time_Heap @ ( option @ nat ) ) @ Vebt_succi3 ) @ ( nth @ vEBT_VEBT @ TreeList3 @ H ) @ Aktnode @ L2 )
                                                                                    @ ^ [Succy: option @ nat] : ( heap_Time_return @ ( option @ nat ) @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( some @ nat @ H ) ) @ Succy ) ) )
                                                                                  @ ( heap_Time_bind @ ( option @ nat ) @ ( option @ nat ) @ ( product_curry @ vEBT_VEBT @ vEBT_VEBTi @ ( nat > ( heap_Time_Heap @ ( option @ nat ) ) ) @ ( product_curry @ ( product_prod @ vEBT_VEBT @ vEBT_VEBTi ) @ nat @ ( heap_Time_Heap @ ( option @ nat ) ) @ Vebt_succi3 ) @ Summary4 @ Summary3 @ H )
                                                                                    @ ^ [Succsum: option @ nat] :
                                                                                        ( heap_Time_bind @ product_unit @ ( option @ nat )
                                                                                        @ ( refine_Imp_assert
                                                                                          @ ( ( Succsum
                                                                                              = ( none @ nat ) )
                                                                                            = ( ( vEBT_vebt_succ @ Summary4 @ H )
                                                                                              = ( none @ nat ) ) ) )
                                                                                        @ ^ [Va4: product_unit] :
                                                                                            ( if @ ( heap_Time_Heap @ ( option @ nat ) )
                                                                                            @ ( Succsum
                                                                                              = ( none @ nat ) )
                                                                                            @ ( heap_Time_return @ ( option @ nat ) @ ( none @ nat ) )
                                                                                            @ ( heap_Time_bind @ vEBT_VEBTi @ ( option @ nat ) @ ( array_nth @ vEBT_VEBTi @ TreeArray @ ( the2 @ nat @ Succsum ) )
                                                                                              @ ^ [Nextnode: vEBT_VEBTi] :
                                                                                                  ( heap_Time_bind @ ( option @ nat ) @ ( option @ nat ) @ ( vEBT_vebt_minti @ Nextnode )
                                                                                                  @ ^ [Minnext: option @ nat] : ( heap_Time_return @ ( option @ nat ) @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ Succsum ) @ Minnext ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
                                        @ Info3 ) ) ) )
                        @ ( vEBT_case_VEBT @ ( product_prod @ ( option @ ( product_prod @ nat @ nat ) ) @ ( product_prod @ nat @ ( product_prod @ ( list @ vEBT_VEBT ) @ vEBT_VEBT ) ) )
                          @ ^ [Info4: option @ ( product_prod @ nat @ nat ),Deg5: nat,TreeList3: list @ vEBT_VEBT,Summary4: vEBT_VEBT] : ( product_Pair @ ( option @ ( product_prod @ nat @ nat ) ) @ ( product_prod @ nat @ ( product_prod @ ( list @ vEBT_VEBT ) @ vEBT_VEBT ) ) @ Info4 @ ( product_Pair @ nat @ ( product_prod @ ( list @ vEBT_VEBT ) @ vEBT_VEBT ) @ Deg5 @ ( product_Pair @ ( list @ vEBT_VEBT ) @ vEBT_VEBT @ TreeList3 @ Summary4 ) ) )
                          @ ^ [A5: $o,B5: $o] : ( undefined @ ( product_prod @ ( option @ ( product_prod @ nat @ nat ) ) @ ( product_prod @ nat @ ( product_prod @ ( list @ vEBT_VEBT ) @ vEBT_VEBT ) ) ) )
                          @ T3 ) ) )
                @ ^ [A5: $o,B5: $o] :
                    ( if @ ( heap_Time_Heap @ ( option @ nat ) )
                    @ ( X
                      = ( zero_zero @ nat ) )
                    @ ( if @ ( heap_Time_Heap @ ( option @ nat ) ) @ B5 @ ( heap_Time_return @ ( option @ nat ) @ ( some @ nat @ ( one_one @ nat ) ) ) @ ( heap_Time_return @ ( option @ nat ) @ ( none @ nat ) ) )
                    @ ( heap_Time_return @ ( option @ nat ) @ ( none @ nat ) ) )
                @ Ti3 ) )
          @ X2 ) ) ).

% VEBT_internal.vebt_succi'.mono
thf(fact_2772_dvd__0__right,axiom,
    ! [A: $tType] :
      ( ( comm_semiring_1 @ A )
     => ! [A2: A] : ( dvd_dvd @ A @ A2 @ ( zero_zero @ A ) ) ) ).

% dvd_0_right
thf(fact_2773_dvd__0__left__iff,axiom,
    ! [A: $tType] :
      ( ( comm_semiring_1 @ A )
     => ! [A2: A] :
          ( ( dvd_dvd @ A @ ( zero_zero @ A ) @ A2 )
          = ( A2
            = ( zero_zero @ A ) ) ) ) ).

% dvd_0_left_iff
thf(fact_2774_dvd__add__triv__right__iff,axiom,
    ! [A: $tType] :
      ( ( comm_s4317794764714335236cancel @ A )
     => ! [A2: A,B2: A] :
          ( ( dvd_dvd @ A @ A2 @ ( plus_plus @ A @ B2 @ A2 ) )
          = ( dvd_dvd @ A @ A2 @ B2 ) ) ) ).

% dvd_add_triv_right_iff
thf(fact_2775_dvd__add__triv__left__iff,axiom,
    ! [A: $tType] :
      ( ( comm_s4317794764714335236cancel @ A )
     => ! [A2: A,B2: A] :
          ( ( dvd_dvd @ A @ A2 @ ( plus_plus @ A @ A2 @ B2 ) )
          = ( dvd_dvd @ A @ A2 @ B2 ) ) ) ).

% dvd_add_triv_left_iff
thf(fact_2776_div__dvd__div,axiom,
    ! [A: $tType] :
      ( ( algebraic_semidom @ A )
     => ! [A2: A,B2: A,C2: A] :
          ( ( dvd_dvd @ A @ A2 @ B2 )
         => ( ( dvd_dvd @ A @ A2 @ C2 )
           => ( ( dvd_dvd @ A @ ( divide_divide @ A @ B2 @ A2 ) @ ( divide_divide @ A @ C2 @ A2 ) )
              = ( dvd_dvd @ A @ B2 @ C2 ) ) ) ) ) ).

% div_dvd_div
thf(fact_2777_unset__bit__nonnegative__int__iff,axiom,
    ! [N2: nat,K: int] :
      ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_se2638667681897837118et_bit @ int @ N2 @ K ) )
      = ( ord_less_eq @ int @ ( zero_zero @ int ) @ K ) ) ).

% unset_bit_nonnegative_int_iff
thf(fact_2778_flip__bit__nonnegative__int__iff,axiom,
    ! [N2: nat,K: int] :
      ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_se8732182000553998342ip_bit @ int @ N2 @ K ) )
      = ( ord_less_eq @ int @ ( zero_zero @ int ) @ K ) ) ).

% flip_bit_nonnegative_int_iff
thf(fact_2779_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_2780_curry__conv,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( ( product_curry @ B @ C @ A )
      = ( ^ [F6: ( product_prod @ B @ C ) > A,A5: B,B5: C] : ( F6 @ ( product_Pair @ B @ C @ A5 @ B5 ) ) ) ) ).

% curry_conv
thf(fact_2781_case__prod__curry,axiom,
    ! [C: $tType,B: $tType,A: $tType,F: ( product_prod @ A @ B ) > C] :
      ( ( product_case_prod @ A @ B @ C @ ( product_curry @ A @ B @ C @ F ) )
      = F ) ).

% case_prod_curry
thf(fact_2782_curry__case__prod,axiom,
    ! [C: $tType,B: $tType,A: $tType,F: A > B > C] :
      ( ( product_curry @ A @ B @ C @ ( product_case_prod @ A @ B @ C @ F ) )
      = F ) ).

% curry_case_prod
thf(fact_2783_dvd__times__right__cancel__iff,axiom,
    ! [A: $tType] :
      ( ( algebraic_semidom @ A )
     => ! [A2: A,B2: A,C2: A] :
          ( ( A2
           != ( zero_zero @ A ) )
         => ( ( dvd_dvd @ A @ ( times_times @ A @ B2 @ A2 ) @ ( times_times @ A @ C2 @ A2 ) )
            = ( dvd_dvd @ A @ B2 @ C2 ) ) ) ) ).

% dvd_times_right_cancel_iff
thf(fact_2784_dvd__times__left__cancel__iff,axiom,
    ! [A: $tType] :
      ( ( algebraic_semidom @ A )
     => ! [A2: A,B2: A,C2: A] :
          ( ( A2
           != ( zero_zero @ A ) )
         => ( ( dvd_dvd @ A @ ( times_times @ A @ A2 @ B2 ) @ ( times_times @ A @ A2 @ C2 ) )
            = ( dvd_dvd @ A @ B2 @ C2 ) ) ) ) ).

% dvd_times_left_cancel_iff
thf(fact_2785_dvd__mult__cancel__right,axiom,
    ! [A: $tType] :
      ( ( idom @ A )
     => ! [A2: A,C2: A,B2: A] :
          ( ( dvd_dvd @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) )
          = ( ( C2
              = ( zero_zero @ A ) )
            | ( dvd_dvd @ A @ A2 @ B2 ) ) ) ) ).

% dvd_mult_cancel_right
thf(fact_2786_dvd__mult__cancel__left,axiom,
    ! [A: $tType] :
      ( ( idom @ A )
     => ! [C2: A,A2: A,B2: A] :
          ( ( dvd_dvd @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B2 ) )
          = ( ( C2
              = ( zero_zero @ A ) )
            | ( dvd_dvd @ A @ A2 @ B2 ) ) ) ) ).

% dvd_mult_cancel_left
thf(fact_2787_dvd__add__times__triv__left__iff,axiom,
    ! [A: $tType] :
      ( ( comm_s4317794764714335236cancel @ A )
     => ! [A2: A,C2: A,B2: A] :
          ( ( dvd_dvd @ A @ A2 @ ( plus_plus @ A @ ( times_times @ A @ C2 @ A2 ) @ B2 ) )
          = ( dvd_dvd @ A @ A2 @ B2 ) ) ) ).

% dvd_add_times_triv_left_iff
thf(fact_2788_dvd__add__times__triv__right__iff,axiom,
    ! [A: $tType] :
      ( ( comm_s4317794764714335236cancel @ A )
     => ! [A2: A,B2: A,C2: A] :
          ( ( dvd_dvd @ A @ A2 @ ( plus_plus @ A @ B2 @ ( times_times @ A @ C2 @ A2 ) ) )
          = ( dvd_dvd @ A @ A2 @ B2 ) ) ) ).

% dvd_add_times_triv_right_iff
thf(fact_2789_unit__prod,axiom,
    ! [A: $tType] :
      ( ( algebraic_semidom @ A )
     => ! [A2: A,B2: A] :
          ( ( dvd_dvd @ A @ A2 @ ( one_one @ A ) )
         => ( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
           => ( dvd_dvd @ A @ ( times_times @ A @ A2 @ B2 ) @ ( one_one @ A ) ) ) ) ) ).

% unit_prod
thf(fact_2790_div__add,axiom,
    ! [A: $tType] :
      ( ( algebraic_semidom @ A )
     => ! [C2: A,A2: A,B2: A] :
          ( ( dvd_dvd @ A @ C2 @ A2 )
         => ( ( dvd_dvd @ A @ C2 @ B2 )
           => ( ( divide_divide @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 )
              = ( plus_plus @ A @ ( divide_divide @ A @ A2 @ C2 ) @ ( divide_divide @ A @ B2 @ C2 ) ) ) ) ) ) ).

% div_add
thf(fact_2791_unit__div,axiom,
    ! [A: $tType] :
      ( ( algebraic_semidom @ A )
     => ! [A2: A,B2: A] :
          ( ( dvd_dvd @ A @ A2 @ ( one_one @ A ) )
         => ( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
           => ( dvd_dvd @ A @ ( divide_divide @ A @ A2 @ B2 ) @ ( one_one @ A ) ) ) ) ) ).

% unit_div
thf(fact_2792_unit__div__1__unit,axiom,
    ! [A: $tType] :
      ( ( algebraic_semidom @ A )
     => ! [A2: A] :
          ( ( dvd_dvd @ A @ A2 @ ( one_one @ A ) )
         => ( dvd_dvd @ A @ ( divide_divide @ A @ ( one_one @ A ) @ A2 ) @ ( one_one @ A ) ) ) ) ).

% unit_div_1_unit
thf(fact_2793_unit__div__1__div__1,axiom,
    ! [A: $tType] :
      ( ( algebraic_semidom @ A )
     => ! [A2: A] :
          ( ( dvd_dvd @ A @ A2 @ ( one_one @ A ) )
         => ( ( divide_divide @ A @ ( one_one @ A ) @ ( divide_divide @ A @ ( one_one @ A ) @ A2 ) )
            = A2 ) ) ) ).

% unit_div_1_div_1
thf(fact_2794_dvd__div__mult__self,axiom,
    ! [A: $tType] :
      ( ( algebraic_semidom @ A )
     => ! [A2: A,B2: A] :
          ( ( dvd_dvd @ A @ A2 @ B2 )
         => ( ( times_times @ A @ ( divide_divide @ A @ B2 @ A2 ) @ A2 )
            = B2 ) ) ) ).

% dvd_div_mult_self
thf(fact_2795_dvd__mult__div__cancel,axiom,
    ! [A: $tType] :
      ( ( algebraic_semidom @ A )
     => ! [A2: A,B2: A] :
          ( ( dvd_dvd @ A @ A2 @ B2 )
         => ( ( times_times @ A @ A2 @ ( divide_divide @ A @ B2 @ A2 ) )
            = B2 ) ) ) ).

% dvd_mult_div_cancel
thf(fact_2796_div__diff,axiom,
    ! [A: $tType] :
      ( ( idom_modulo @ A )
     => ! [C2: A,A2: A,B2: A] :
          ( ( dvd_dvd @ A @ C2 @ A2 )
         => ( ( dvd_dvd @ A @ C2 @ B2 )
           => ( ( divide_divide @ A @ ( minus_minus @ A @ A2 @ B2 ) @ C2 )
              = ( minus_minus @ A @ ( divide_divide @ A @ A2 @ C2 ) @ ( divide_divide @ A @ B2 @ C2 ) ) ) ) ) ) ).

% div_diff
thf(fact_2797_dvd__imp__mod__0,axiom,
    ! [A: $tType] :
      ( ( semidom_modulo @ A )
     => ! [A2: A,B2: A] :
          ( ( dvd_dvd @ A @ A2 @ B2 )
         => ( ( modulo_modulo @ A @ B2 @ A2 )
            = ( zero_zero @ A ) ) ) ) ).

% dvd_imp_mod_0
thf(fact_2798_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_2799_dvd__1__left,axiom,
    ! [K: nat] : ( dvd_dvd @ nat @ ( suc @ ( zero_zero @ nat ) ) @ K ) ).

% dvd_1_left
thf(fact_2800_nat__mult__dvd__cancel__disj,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( dvd_dvd @ nat @ ( times_times @ nat @ K @ M ) @ ( times_times @ nat @ K @ N2 ) )
      = ( ( K
          = ( zero_zero @ nat ) )
        | ( dvd_dvd @ nat @ M @ N2 ) ) ) ).

% nat_mult_dvd_cancel_disj
thf(fact_2801_unit__mult__div__div,axiom,
    ! [A: $tType] :
      ( ( algebraic_semidom @ A )
     => ! [A2: A,B2: A] :
          ( ( dvd_dvd @ A @ A2 @ ( one_one @ A ) )
         => ( ( times_times @ A @ B2 @ ( divide_divide @ A @ ( one_one @ A ) @ A2 ) )
            = ( divide_divide @ A @ B2 @ A2 ) ) ) ) ).

% unit_mult_div_div
thf(fact_2802_unit__div__mult__self,axiom,
    ! [A: $tType] :
      ( ( algebraic_semidom @ A )
     => ! [A2: A,B2: A] :
          ( ( dvd_dvd @ A @ A2 @ ( one_one @ A ) )
         => ( ( times_times @ A @ ( divide_divide @ A @ B2 @ A2 ) @ A2 )
            = B2 ) ) ) ).

% unit_div_mult_self
thf(fact_2803_odd__add,axiom,
    ! [A: $tType] :
      ( ( semiring_parity @ A )
     => ! [A2: A,B2: A] :
          ( ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( plus_plus @ A @ A2 @ B2 ) ) )
          = ( ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) )
           != ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) ) ) ) ) ).

% odd_add
thf(fact_2804_even__add,axiom,
    ! [A: $tType] :
      ( ( semiring_parity @ A )
     => ! [A2: A,B2: A] :
          ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( plus_plus @ A @ A2 @ B2 ) )
          = ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 )
            = ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) ) ) ) ).

% even_add
thf(fact_2805_even__mult__iff,axiom,
    ! [A: $tType] :
      ( ( semiring_parity @ A )
     => ! [A2: A,B2: A] :
          ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( times_times @ A @ A2 @ B2 ) )
          = ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 )
            | ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) ) ) ) ).

% even_mult_iff
thf(fact_2806_even__mod__2__iff,axiom,
    ! [A: $tType] :
      ( ( semiring_parity @ A )
     => ! [A2: A] :
          ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( modulo_modulo @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) )
          = ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) ) ) ).

% even_mod_2_iff
thf(fact_2807_even__Suc,axiom,
    ! [N2: nat] :
      ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ N2 ) )
      = ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ) ).

% even_Suc
thf(fact_2808_even__Suc__Suc__iff,axiom,
    ! [N2: nat] :
      ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ N2 ) ) )
      = ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ).

% even_Suc_Suc_iff
thf(fact_2809_dvd__numeral__simp,axiom,
    ! [A: $tType] :
      ( ( unique1627219031080169319umeral @ A )
     => ! [M: num,N2: num] :
          ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ M ) @ ( numeral_numeral @ A @ N2 ) )
          = ( unique5940410009612947441es_aux @ A @ ( unique8689654367752047608divmod @ A @ N2 @ M ) ) ) ) ).

% dvd_numeral_simp
thf(fact_2810_even__plus__one__iff,axiom,
    ! [A: $tType] :
      ( ( semiring_parity @ A )
     => ! [A2: A] :
          ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( plus_plus @ A @ A2 @ ( one_one @ A ) ) )
          = ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) ) ) ) ).

% even_plus_one_iff
thf(fact_2811_even__diff,axiom,
    ! [A: $tType] :
      ( ( ring_parity @ A )
     => ! [A2: A,B2: A] :
          ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( minus_minus @ A @ A2 @ B2 ) )
          = ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( plus_plus @ A @ A2 @ B2 ) ) ) ) ).

% even_diff
thf(fact_2812_odd__Suc__div__two,axiom,
    ! [N2: nat] :
      ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 )
     => ( ( divide_divide @ nat @ ( suc @ N2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
        = ( suc @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).

% odd_Suc_div_two
thf(fact_2813_even__Suc__div__two,axiom,
    ! [N2: nat] :
      ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 )
     => ( ( divide_divide @ nat @ ( suc @ N2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
        = ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).

% even_Suc_div_two
thf(fact_2814_even__succ__div__2,axiom,
    ! [A: $tType] :
      ( ( bit_semiring_bits @ A )
     => ! [A2: A] :
          ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 )
         => ( ( divide_divide @ A @ ( plus_plus @ A @ ( one_one @ A ) @ A2 ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
            = ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).

% even_succ_div_2
thf(fact_2815_odd__succ__div__two,axiom,
    ! [A: $tType] :
      ( ( euclid5411537665997757685th_nat @ A )
     => ! [A2: A] :
          ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 )
         => ( ( divide_divide @ A @ ( plus_plus @ A @ A2 @ ( one_one @ A ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
            = ( plus_plus @ A @ ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( one_one @ A ) ) ) ) ) ).

% odd_succ_div_two
thf(fact_2816_even__succ__div__two,axiom,
    ! [A: $tType] :
      ( ( euclid5411537665997757685th_nat @ A )
     => ! [A2: A] :
          ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 )
         => ( ( divide_divide @ A @ ( plus_plus @ A @ A2 @ ( one_one @ A ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
            = ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).

% even_succ_div_two
thf(fact_2817_even__power,axiom,
    ! [A: $tType] :
      ( ( semiring_parity @ A )
     => ! [A2: A,N2: nat] :
          ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( power_power @ A @ A2 @ N2 ) )
          = ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 )
            & ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ) ).

% even_power
thf(fact_2818_zero__le__power__eq__numeral,axiom,
    ! [A: $tType] :
      ( ( linordered_idom @ A )
     => ! [A2: A,W: num] :
          ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A2 @ ( numeral_numeral @ nat @ W ) ) )
          = ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ W ) )
            | ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ W ) )
              & ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) ) ) ) ) ).

% zero_le_power_eq_numeral
thf(fact_2819_power__less__zero__eq,axiom,
    ! [A: $tType] :
      ( ( linordered_idom @ A )
     => ! [A2: A,N2: nat] :
          ( ( ord_less @ A @ ( power_power @ A @ A2 @ N2 ) @ ( zero_zero @ A ) )
          = ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 )
            & ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ) ).

% power_less_zero_eq
thf(fact_2820_power__less__zero__eq__numeral,axiom,
    ! [A: $tType] :
      ( ( linordered_idom @ A )
     => ! [A2: A,W: num] :
          ( ( ord_less @ A @ ( power_power @ A @ A2 @ ( numeral_numeral @ nat @ W ) ) @ ( zero_zero @ A ) )
          = ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ W ) )
            & ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ) ).

% power_less_zero_eq_numeral
thf(fact_2821_even__of__nat,axiom,
    ! [A: $tType] :
      ( ( semiring_parity @ A )
     => ! [N2: nat] :
          ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ A @ N2 ) )
          = ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ) ).

% even_of_nat
thf(fact_2822_odd__Suc__minus__one,axiom,
    ! [N2: nat] :
      ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 )
     => ( ( suc @ ( minus_minus @ nat @ N2 @ ( suc @ ( zero_zero @ nat ) ) ) )
        = N2 ) ) ).

% odd_Suc_minus_one
thf(fact_2823_even__diff__nat,axiom,
    ! [M: nat,N2: nat] :
      ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ M @ N2 ) )
      = ( ( ord_less @ nat @ M @ N2 )
        | ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( plus_plus @ nat @ M @ N2 ) ) ) ) ).

% even_diff_nat
thf(fact_2824_odd__two__times__div__two__succ,axiom,
    ! [A: $tType] :
      ( ( euclid5411537665997757685th_nat @ A )
     => ! [A2: A] :
          ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 )
         => ( ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) @ ( one_one @ A ) )
            = A2 ) ) ) ).

% odd_two_times_div_two_succ
thf(fact_2825_semiring__parity__class_Oeven__mask__iff,axiom,
    ! [A: $tType] :
      ( ( semiring_parity @ A )
     => ! [N2: nat] :
          ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) @ ( one_one @ A ) ) )
          = ( N2
            = ( zero_zero @ nat ) ) ) ) ).

% semiring_parity_class.even_mask_iff
thf(fact_2826_zero__less__power__eq__numeral,axiom,
    ! [A: $tType] :
      ( ( linordered_idom @ A )
     => ! [A2: A,W: num] :
          ( ( ord_less @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A2 @ ( numeral_numeral @ nat @ W ) ) )
          = ( ( ( numeral_numeral @ nat @ W )
              = ( zero_zero @ nat ) )
            | ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ W ) )
              & ( A2
               != ( zero_zero @ A ) ) )
            | ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ W ) )
              & ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ) ) ).

% zero_less_power_eq_numeral
thf(fact_2827_odd__two__times__div__two__nat,axiom,
    ! [N2: nat] :
      ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 )
     => ( ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
        = ( minus_minus @ nat @ N2 @ ( one_one @ nat ) ) ) ) ).

% odd_two_times_div_two_nat
thf(fact_2828_power__le__zero__eq__numeral,axiom,
    ! [A: $tType] :
      ( ( linordered_idom @ A )
     => ! [A2: A,W: num] :
          ( ( ord_less_eq @ A @ ( power_power @ A @ A2 @ ( numeral_numeral @ nat @ W ) ) @ ( zero_zero @ A ) )
          = ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( numeral_numeral @ nat @ W ) )
            & ( ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ W ) )
                & ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) )
              | ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ W ) )
                & ( A2
                  = ( zero_zero @ A ) ) ) ) ) ) ) ).

% power_le_zero_eq_numeral
thf(fact_2829_even__succ__div__exp,axiom,
    ! [A: $tType] :
      ( ( bit_semiring_bits @ A )
     => ! [A2: A,N2: nat] :
          ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 )
         => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
           => ( ( divide_divide @ A @ ( plus_plus @ A @ ( one_one @ A ) @ A2 ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) )
              = ( divide_divide @ A @ A2 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) ) ) ) ) ) ).

% even_succ_div_exp
thf(fact_2830_even__succ__mod__exp,axiom,
    ! [A: $tType] :
      ( ( bit_semiring_bits @ A )
     => ! [A2: A,N2: nat] :
          ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 )
         => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
           => ( ( modulo_modulo @ A @ ( plus_plus @ A @ ( one_one @ A ) @ A2 ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) )
              = ( plus_plus @ A @ ( one_one @ A ) @ ( modulo_modulo @ A @ A2 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) ) ) ) ) ) ) ).

% even_succ_mod_exp
thf(fact_2831_dvd__refl,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_mult @ A )
     => ! [A2: A] : ( dvd_dvd @ A @ A2 @ A2 ) ) ).

% dvd_refl
thf(fact_2832_dvd__trans,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_mult @ A )
     => ! [A2: A,B2: A,C2: A] :
          ( ( dvd_dvd @ A @ A2 @ B2 )
         => ( ( dvd_dvd @ A @ B2 @ C2 )
           => ( dvd_dvd @ A @ A2 @ C2 ) ) ) ) ).

% dvd_trans
thf(fact_2833_dvd__antisym,axiom,
    ! [M: nat,N2: nat] :
      ( ( dvd_dvd @ nat @ M @ N2 )
     => ( ( dvd_dvd @ nat @ N2 @ M )
       => ( M = N2 ) ) ) ).

% dvd_antisym
thf(fact_2834_of__nat__dvd__iff,axiom,
    ! [A: $tType] :
      ( ( euclid5411537665997757685th_nat @ A )
     => ! [M: nat,N2: nat] :
          ( ( dvd_dvd @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N2 ) )
          = ( dvd_dvd @ nat @ M @ N2 ) ) ) ).

% of_nat_dvd_iff
thf(fact_2835_curry__K,axiom,
    ! [B: $tType,C: $tType,A: $tType,C2: C] :
      ( ( product_curry @ A @ B @ C
        @ ^ [X: product_prod @ A @ B] : C2 )
      = ( ^ [X: A,Y: B] : C2 ) ) ).

% curry_K
thf(fact_2836_dvd__0__left,axiom,
    ! [A: $tType] :
      ( ( comm_semiring_1 @ A )
     => ! [A2: A] :
          ( ( dvd_dvd @ A @ ( zero_zero @ A ) @ A2 )
         => ( A2
            = ( zero_zero @ A ) ) ) ) ).

% dvd_0_left
thf(fact_2837_dvd__add__right__iff,axiom,
    ! [A: $tType] :
      ( ( comm_s4317794764714335236cancel @ A )
     => ! [A2: A,B2: A,C2: A] :
          ( ( dvd_dvd @ A @ A2 @ B2 )
         => ( ( dvd_dvd @ A @ A2 @ ( plus_plus @ A @ B2 @ C2 ) )
            = ( dvd_dvd @ A @ A2 @ C2 ) ) ) ) ).

% dvd_add_right_iff
thf(fact_2838_dvd__add__left__iff,axiom,
    ! [A: $tType] :
      ( ( comm_s4317794764714335236cancel @ A )
     => ! [A2: A,C2: A,B2: A] :
          ( ( dvd_dvd @ A @ A2 @ C2 )
         => ( ( dvd_dvd @ A @ A2 @ ( plus_plus @ A @ B2 @ C2 ) )
            = ( dvd_dvd @ A @ A2 @ B2 ) ) ) ) ).

% dvd_add_left_iff
thf(fact_2839_dvd__add,axiom,
    ! [A: $tType] :
      ( ( comm_semiring_1 @ A )
     => ! [A2: A,B2: A,C2: A] :
          ( ( dvd_dvd @ A @ A2 @ B2 )
         => ( ( dvd_dvd @ A @ A2 @ C2 )
           => ( dvd_dvd @ A @ A2 @ ( plus_plus @ A @ B2 @ C2 ) ) ) ) ) ).

% dvd_add
thf(fact_2840_one__dvd,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_mult @ A )
     => ! [A2: A] : ( dvd_dvd @ A @ ( one_one @ A ) @ A2 ) ) ).

% one_dvd
thf(fact_2841_unit__imp__dvd,axiom,
    ! [A: $tType] :
      ( ( algebraic_semidom @ A )
     => ! [B2: A,A2: A] :
          ( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
         => ( dvd_dvd @ A @ B2 @ A2 ) ) ) ).

% unit_imp_dvd
thf(fact_2842_dvd__unit__imp__unit,axiom,
    ! [A: $tType] :
      ( ( algebraic_semidom @ A )
     => ! [A2: A,B2: A] :
          ( ( dvd_dvd @ A @ A2 @ B2 )
         => ( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
           => ( dvd_dvd @ A @ A2 @ ( one_one @ A ) ) ) ) ) ).

% dvd_unit_imp_unit
thf(fact_2843_dvdE,axiom,
    ! [A: $tType] :
      ( ( dvd @ A )
     => ! [B2: A,A2: A] :
          ( ( dvd_dvd @ A @ B2 @ A2 )
         => ~ ! [K3: A] :
                ( A2
               != ( times_times @ A @ B2 @ K3 ) ) ) ) ).

% dvdE
thf(fact_2844_dvdI,axiom,
    ! [A: $tType] :
      ( ( dvd @ A )
     => ! [A2: A,B2: A,K: A] :
          ( ( A2
            = ( times_times @ A @ B2 @ K ) )
         => ( dvd_dvd @ A @ B2 @ A2 ) ) ) ).

% dvdI
thf(fact_2845_dvd__def,axiom,
    ! [A: $tType] :
      ( ( dvd @ A )
     => ( ( dvd_dvd @ A )
        = ( ^ [B5: A,A5: A] :
            ? [K4: A] :
              ( A5
              = ( times_times @ A @ B5 @ K4 ) ) ) ) ) ).

% dvd_def
thf(fact_2846_dvd__mult,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_mult @ A )
     => ! [A2: A,C2: A,B2: A] :
          ( ( dvd_dvd @ A @ A2 @ C2 )
         => ( dvd_dvd @ A @ A2 @ ( times_times @ A @ B2 @ C2 ) ) ) ) ).

% dvd_mult
thf(fact_2847_dvd__mult2,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_mult @ A )
     => ! [A2: A,B2: A,C2: A] :
          ( ( dvd_dvd @ A @ A2 @ B2 )
         => ( dvd_dvd @ A @ A2 @ ( times_times @ A @ B2 @ C2 ) ) ) ) ).

% dvd_mult2
thf(fact_2848_dvd__mult__left,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_mult @ A )
     => ! [A2: A,B2: A,C2: A] :
          ( ( dvd_dvd @ A @ ( times_times @ A @ A2 @ B2 ) @ C2 )
         => ( dvd_dvd @ A @ A2 @ C2 ) ) ) ).

% dvd_mult_left
thf(fact_2849_dvd__triv__left,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_mult @ A )
     => ! [A2: A,B2: A] : ( dvd_dvd @ A @ A2 @ ( times_times @ A @ A2 @ B2 ) ) ) ).

% dvd_triv_left
thf(fact_2850_mult__dvd__mono,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_mult @ A )
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( ( dvd_dvd @ A @ A2 @ B2 )
         => ( ( dvd_dvd @ A @ C2 @ D2 )
           => ( dvd_dvd @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ D2 ) ) ) ) ) ).

% mult_dvd_mono
thf(fact_2851_dvd__mult__right,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_mult @ A )
     => ! [A2: A,B2: A,C2: A] :
          ( ( dvd_dvd @ A @ ( times_times @ A @ A2 @ B2 ) @ C2 )
         => ( dvd_dvd @ A @ B2 @ C2 ) ) ) ).

% dvd_mult_right
thf(fact_2852_dvd__triv__right,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_mult @ A )
     => ! [A2: A,B2: A] : ( dvd_dvd @ A @ A2 @ ( times_times @ A @ B2 @ A2 ) ) ) ).

% dvd_triv_right
thf(fact_2853_dvd__diff__commute,axiom,
    ! [A: $tType] :
      ( ( euclid5891614535332579305n_ring @ A )
     => ! [A2: A,C2: A,B2: A] :
          ( ( dvd_dvd @ A @ A2 @ ( minus_minus @ A @ C2 @ B2 ) )
          = ( dvd_dvd @ A @ A2 @ ( minus_minus @ A @ B2 @ C2 ) ) ) ) ).

% dvd_diff_commute
thf(fact_2854_dvd__diff,axiom,
    ! [A: $tType] :
      ( ( comm_ring_1 @ A )
     => ! [X2: A,Y2: A,Z: A] :
          ( ( dvd_dvd @ A @ X2 @ Y2 )
         => ( ( dvd_dvd @ A @ X2 @ Z )
           => ( dvd_dvd @ A @ X2 @ ( minus_minus @ A @ Y2 @ Z ) ) ) ) ) ).

% dvd_diff
thf(fact_2855_dvd__div__eq__iff,axiom,
    ! [A: $tType] :
      ( ( semidom_divide @ A )
     => ! [C2: A,A2: A,B2: A] :
          ( ( dvd_dvd @ A @ C2 @ A2 )
         => ( ( dvd_dvd @ A @ C2 @ B2 )
           => ( ( ( divide_divide @ A @ A2 @ C2 )
                = ( divide_divide @ A @ B2 @ C2 ) )
              = ( A2 = B2 ) ) ) ) ) ).

% dvd_div_eq_iff
thf(fact_2856_dvd__div__eq__cancel,axiom,
    ! [A: $tType] :
      ( ( semidom_divide @ A )
     => ! [A2: A,C2: A,B2: A] :
          ( ( ( divide_divide @ A @ A2 @ C2 )
            = ( divide_divide @ A @ B2 @ C2 ) )
         => ( ( dvd_dvd @ A @ C2 @ A2 )
           => ( ( dvd_dvd @ A @ C2 @ B2 )
             => ( A2 = B2 ) ) ) ) ) ).

% dvd_div_eq_cancel
thf(fact_2857_div__div__div__same,axiom,
    ! [A: $tType] :
      ( ( algebraic_semidom @ A )
     => ! [D2: A,B2: A,A2: A] :
          ( ( dvd_dvd @ A @ D2 @ B2 )
         => ( ( dvd_dvd @ A @ B2 @ A2 )
           => ( ( divide_divide @ A @ ( divide_divide @ A @ A2 @ D2 ) @ ( divide_divide @ A @ B2 @ D2 ) )
              = ( divide_divide @ A @ A2 @ B2 ) ) ) ) ) ).

% div_div_div_same
thf(fact_2858_dvd__power__same,axiom,
    ! [A: $tType] :
      ( ( comm_semiring_1 @ A )
     => ! [X2: A,Y2: A,N2: nat] :
          ( ( dvd_dvd @ A @ X2 @ Y2 )
         => ( dvd_dvd @ A @ ( power_power @ A @ X2 @ N2 ) @ ( power_power @ A @ Y2 @ N2 ) ) ) ) ).

% dvd_power_same
thf(fact_2859_dvd__mod__iff,axiom,
    ! [A: $tType] :
      ( ( semidom_modulo @ A )
     => ! [C2: A,B2: A,A2: A] :
          ( ( dvd_dvd @ A @ C2 @ B2 )
         => ( ( dvd_dvd @ A @ C2 @ ( modulo_modulo @ A @ A2 @ B2 ) )
            = ( dvd_dvd @ A @ C2 @ A2 ) ) ) ) ).

% dvd_mod_iff
thf(fact_2860_dvd__mod__imp__dvd,axiom,
    ! [A: $tType] :
      ( ( semidom_modulo @ A )
     => ! [C2: A,A2: A,B2: A] :
          ( ( dvd_dvd @ A @ C2 @ ( modulo_modulo @ A @ A2 @ B2 ) )
         => ( ( dvd_dvd @ A @ C2 @ B2 )
           => ( dvd_dvd @ A @ C2 @ A2 ) ) ) ) ).

% dvd_mod_imp_dvd
thf(fact_2861_dvd__mod,axiom,
    ! [A: $tType] :
      ( ( euclid4440199948858584721cancel @ A )
     => ! [K: A,M: A,N2: A] :
          ( ( dvd_dvd @ A @ K @ M )
         => ( ( dvd_dvd @ A @ K @ N2 )
           => ( dvd_dvd @ A @ K @ ( modulo_modulo @ A @ M @ N2 ) ) ) ) ) ).

% dvd_mod
thf(fact_2862_mod__mod__cancel,axiom,
    ! [A: $tType] :
      ( ( euclid4440199948858584721cancel @ A )
     => ! [C2: A,B2: A,A2: A] :
          ( ( dvd_dvd @ A @ C2 @ B2 )
         => ( ( modulo_modulo @ A @ ( modulo_modulo @ A @ A2 @ B2 ) @ C2 )
            = ( modulo_modulo @ A @ A2 @ C2 ) ) ) ) ).

% mod_mod_cancel
thf(fact_2863_dvd__diff__nat,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( dvd_dvd @ nat @ K @ M )
     => ( ( dvd_dvd @ nat @ K @ N2 )
       => ( dvd_dvd @ nat @ K @ ( minus_minus @ nat @ M @ N2 ) ) ) ) ).

% dvd_diff_nat
thf(fact_2864_unset__bit__less__eq,axiom,
    ! [N2: nat,K: int] : ( ord_less_eq @ int @ ( bit_se2638667681897837118et_bit @ int @ N2 @ K ) @ K ) ).

% unset_bit_less_eq
thf(fact_2865_subset__divisors__dvd,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_mult @ A )
     => ! [A2: A,B2: A] :
          ( ( ord_less_eq @ ( set @ A )
            @ ( collect @ A
              @ ^ [C4: A] : ( dvd_dvd @ A @ C4 @ A2 ) )
            @ ( collect @ A
              @ ^ [C4: A] : ( dvd_dvd @ A @ C4 @ B2 ) ) )
          = ( dvd_dvd @ A @ A2 @ B2 ) ) ) ).

% subset_divisors_dvd
thf(fact_2866_strict__subset__divisors__dvd,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_mult @ A )
     => ! [A2: A,B2: A] :
          ( ( ord_less @ ( set @ A )
            @ ( collect @ A
              @ ^ [C4: A] : ( dvd_dvd @ A @ C4 @ A2 ) )
            @ ( collect @ A
              @ ^ [C4: A] : ( dvd_dvd @ A @ C4 @ B2 ) ) )
          = ( ( dvd_dvd @ A @ A2 @ B2 )
            & ~ ( dvd_dvd @ A @ B2 @ A2 ) ) ) ) ).

% strict_subset_divisors_dvd
thf(fact_2867_not__is__unit__0,axiom,
    ! [A: $tType] :
      ( ( algebraic_semidom @ A )
     => ~ ( dvd_dvd @ A @ ( zero_zero @ A ) @ ( one_one @ A ) ) ) ).

% not_is_unit_0
thf(fact_2868_minf_I10_J,axiom,
    ! [B: $tType] :
      ( ( ( plus @ B )
        & ( linorder @ B )
        & ( dvd @ B ) )
     => ! [D2: B,S4: B] :
        ? [Z3: B] :
        ! [X6: B] :
          ( ( ord_less @ B @ X6 @ Z3 )
         => ( ( ~ ( dvd_dvd @ B @ D2 @ ( plus_plus @ B @ X6 @ S4 ) ) )
            = ( ~ ( dvd_dvd @ B @ D2 @ ( plus_plus @ B @ X6 @ S4 ) ) ) ) ) ) ).

% minf(10)
thf(fact_2869_minf_I9_J,axiom,
    ! [B: $tType] :
      ( ( ( plus @ B )
        & ( linorder @ B )
        & ( dvd @ B ) )
     => ! [D2: B,S4: B] :
        ? [Z3: B] :
        ! [X6: B] :
          ( ( ord_less @ B @ X6 @ Z3 )
         => ( ( dvd_dvd @ B @ D2 @ ( plus_plus @ B @ X6 @ S4 ) )
            = ( dvd_dvd @ B @ D2 @ ( plus_plus @ B @ X6 @ S4 ) ) ) ) ) ).

% minf(9)
thf(fact_2870_pinf_I10_J,axiom,
    ! [B: $tType] :
      ( ( ( plus @ B )
        & ( linorder @ B )
        & ( dvd @ B ) )
     => ! [D2: B,S4: B] :
        ? [Z3: B] :
        ! [X6: B] :
          ( ( ord_less @ B @ Z3 @ X6 )
         => ( ( ~ ( dvd_dvd @ B @ D2 @ ( plus_plus @ B @ X6 @ S4 ) ) )
            = ( ~ ( dvd_dvd @ B @ D2 @ ( plus_plus @ B @ X6 @ S4 ) ) ) ) ) ) ).

% pinf(10)
thf(fact_2871_pinf_I9_J,axiom,
    ! [B: $tType] :
      ( ( ( plus @ B )
        & ( linorder @ B )
        & ( dvd @ B ) )
     => ! [D2: B,S4: B] :
        ? [Z3: B] :
        ! [X6: B] :
          ( ( ord_less @ B @ Z3 @ X6 )
         => ( ( dvd_dvd @ B @ D2 @ ( plus_plus @ B @ X6 @ S4 ) )
            = ( dvd_dvd @ B @ D2 @ ( plus_plus @ B @ X6 @ S4 ) ) ) ) ) ).

% pinf(9)
thf(fact_2872_dvd__div__eq__0__iff,axiom,
    ! [A: $tType] :
      ( ( semidom_divide @ A )
     => ! [B2: A,A2: A] :
          ( ( dvd_dvd @ A @ B2 @ A2 )
         => ( ( ( divide_divide @ A @ A2 @ B2 )
              = ( zero_zero @ A ) )
            = ( A2
              = ( zero_zero @ A ) ) ) ) ) ).

% dvd_div_eq_0_iff
thf(fact_2873_is__unit__mult__iff,axiom,
    ! [A: $tType] :
      ( ( algebraic_semidom @ A )
     => ! [A2: A,B2: A] :
          ( ( dvd_dvd @ A @ ( times_times @ A @ A2 @ B2 ) @ ( one_one @ A ) )
          = ( ( dvd_dvd @ A @ A2 @ ( one_one @ A ) )
            & ( dvd_dvd @ A @ B2 @ ( one_one @ A ) ) ) ) ) ).

% is_unit_mult_iff
thf(fact_2874_dvd__mult__unit__iff,axiom,
    ! [A: $tType] :
      ( ( algebraic_semidom @ A )
     => ! [B2: A,A2: A,C2: A] :
          ( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
         => ( ( dvd_dvd @ A @ A2 @ ( times_times @ A @ C2 @ B2 ) )
            = ( dvd_dvd @ A @ A2 @ C2 ) ) ) ) ).

% dvd_mult_unit_iff
thf(fact_2875_mult__unit__dvd__iff,axiom,
    ! [A: $tType] :
      ( ( algebraic_semidom @ A )
     => ! [B2: A,A2: A,C2: A] :
          ( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
         => ( ( dvd_dvd @ A @ ( times_times @ A @ A2 @ B2 ) @ C2 )
            = ( dvd_dvd @ A @ A2 @ C2 ) ) ) ) ).

% mult_unit_dvd_iff
thf(fact_2876_dvd__mult__unit__iff_H,axiom,
    ! [A: $tType] :
      ( ( algebraic_semidom @ A )
     => ! [B2: A,A2: A,C2: A] :
          ( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
         => ( ( dvd_dvd @ A @ A2 @ ( times_times @ A @ B2 @ C2 ) )
            = ( dvd_dvd @ A @ A2 @ C2 ) ) ) ) ).

% dvd_mult_unit_iff'
thf(fact_2877_mult__unit__dvd__iff_H,axiom,
    ! [A: $tType] :
      ( ( algebraic_semidom @ A )
     => ! [A2: A,B2: A,C2: A] :
          ( ( dvd_dvd @ A @ A2 @ ( one_one @ A ) )
         => ( ( dvd_dvd @ A @ ( times_times @ A @ A2 @ B2 ) @ C2 )
            = ( dvd_dvd @ A @ B2 @ C2 ) ) ) ) ).

% mult_unit_dvd_iff'
thf(fact_2878_unit__mult__left__cancel,axiom,
    ! [A: $tType] :
      ( ( algebraic_semidom @ A )
     => ! [A2: A,B2: A,C2: A] :
          ( ( dvd_dvd @ A @ A2 @ ( one_one @ A ) )
         => ( ( ( times_times @ A @ A2 @ B2 )
              = ( times_times @ A @ A2 @ C2 ) )
            = ( B2 = C2 ) ) ) ) ).

% unit_mult_left_cancel
thf(fact_2879_unit__mult__right__cancel,axiom,
    ! [A: $tType] :
      ( ( algebraic_semidom @ A )
     => ! [A2: A,B2: A,C2: A] :
          ( ( dvd_dvd @ A @ A2 @ ( one_one @ A ) )
         => ( ( ( times_times @ A @ B2 @ A2 )
              = ( times_times @ A @ C2 @ A2 ) )
            = ( B2 = C2 ) ) ) ) ).

% unit_mult_right_cancel
thf(fact_2880_div__plus__div__distrib__dvd__right,axiom,
    ! [A: $tType] :
      ( ( euclid4440199948858584721cancel @ A )
     => ! [C2: A,B2: A,A2: A] :
          ( ( dvd_dvd @ A @ C2 @ B2 )
         => ( ( divide_divide @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 )
            = ( plus_plus @ A @ ( divide_divide @ A @ A2 @ C2 ) @ ( divide_divide @ A @ B2 @ C2 ) ) ) ) ) ).

% div_plus_div_distrib_dvd_right
thf(fact_2881_div__plus__div__distrib__dvd__left,axiom,
    ! [A: $tType] :
      ( ( euclid4440199948858584721cancel @ A )
     => ! [C2: A,A2: A,B2: A] :
          ( ( dvd_dvd @ A @ C2 @ A2 )
         => ( ( divide_divide @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 )
            = ( plus_plus @ A @ ( divide_divide @ A @ A2 @ C2 ) @ ( divide_divide @ A @ B2 @ C2 ) ) ) ) ) ).

% div_plus_div_distrib_dvd_left
thf(fact_2882_unit__div__cancel,axiom,
    ! [A: $tType] :
      ( ( algebraic_semidom @ A )
     => ! [A2: A,B2: A,C2: A] :
          ( ( dvd_dvd @ A @ A2 @ ( one_one @ A ) )
         => ( ( ( divide_divide @ A @ B2 @ A2 )
              = ( divide_divide @ A @ C2 @ A2 ) )
            = ( B2 = C2 ) ) ) ) ).

% unit_div_cancel
thf(fact_2883_div__unit__dvd__iff,axiom,
    ! [A: $tType] :
      ( ( algebraic_semidom @ A )
     => ! [B2: A,A2: A,C2: A] :
          ( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
         => ( ( dvd_dvd @ A @ ( divide_divide @ A @ A2 @ B2 ) @ C2 )
            = ( dvd_dvd @ A @ A2 @ C2 ) ) ) ) ).

% div_unit_dvd_iff
thf(fact_2884_dvd__div__unit__iff,axiom,
    ! [A: $tType] :
      ( ( algebraic_semidom @ A )
     => ! [B2: A,A2: A,C2: A] :
          ( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
         => ( ( dvd_dvd @ A @ A2 @ ( divide_divide @ A @ C2 @ B2 ) )
            = ( dvd_dvd @ A @ A2 @ C2 ) ) ) ) ).

% dvd_div_unit_iff
thf(fact_2885_dvd__div__mult,axiom,
    ! [A: $tType] :
      ( ( algebraic_semidom @ A )
     => ! [C2: A,B2: A,A2: A] :
          ( ( dvd_dvd @ A @ C2 @ B2 )
         => ( ( times_times @ A @ ( divide_divide @ A @ B2 @ C2 ) @ A2 )
            = ( divide_divide @ A @ ( times_times @ A @ B2 @ A2 ) @ C2 ) ) ) ) ).

% dvd_div_mult
thf(fact_2886_div__mult__swap,axiom,
    ! [A: $tType] :
      ( ( algebraic_semidom @ A )
     => ! [C2: A,B2: A,A2: A] :
          ( ( dvd_dvd @ A @ C2 @ B2 )
         => ( ( times_times @ A @ A2 @ ( divide_divide @ A @ B2 @ C2 ) )
            = ( divide_divide @ A @ ( times_times @ A @ A2 @ B2 ) @ C2 ) ) ) ) ).

% div_mult_swap
thf(fact_2887_div__div__eq__right,axiom,
    ! [A: $tType] :
      ( ( algebraic_semidom @ A )
     => ! [C2: A,B2: A,A2: A] :
          ( ( dvd_dvd @ A @ C2 @ B2 )
         => ( ( dvd_dvd @ A @ B2 @ A2 )
           => ( ( divide_divide @ A @ A2 @ ( divide_divide @ A @ B2 @ C2 ) )
              = ( times_times @ A @ ( divide_divide @ A @ A2 @ B2 ) @ C2 ) ) ) ) ) ).

% div_div_eq_right
thf(fact_2888_dvd__div__mult2__eq,axiom,
    ! [A: $tType] :
      ( ( algebraic_semidom @ A )
     => ! [B2: A,C2: A,A2: A] :
          ( ( dvd_dvd @ A @ ( times_times @ A @ B2 @ C2 ) @ A2 )
         => ( ( divide_divide @ A @ A2 @ ( times_times @ A @ B2 @ C2 ) )
            = ( divide_divide @ A @ ( divide_divide @ A @ A2 @ B2 ) @ C2 ) ) ) ) ).

% dvd_div_mult2_eq
thf(fact_2889_dvd__mult__imp__div,axiom,
    ! [A: $tType] :
      ( ( algebraic_semidom @ A )
     => ! [A2: A,C2: A,B2: A] :
          ( ( dvd_dvd @ A @ ( times_times @ A @ A2 @ C2 ) @ B2 )
         => ( dvd_dvd @ A @ A2 @ ( divide_divide @ A @ B2 @ C2 ) ) ) ) ).

% dvd_mult_imp_div
thf(fact_2890_div__mult__div__if__dvd,axiom,
    ! [A: $tType] :
      ( ( algebraic_semidom @ A )
     => ! [B2: A,A2: A,D2: A,C2: A] :
          ( ( dvd_dvd @ A @ B2 @ A2 )
         => ( ( dvd_dvd @ A @ D2 @ C2 )
           => ( ( times_times @ A @ ( divide_divide @ A @ A2 @ B2 ) @ ( divide_divide @ A @ C2 @ D2 ) )
              = ( divide_divide @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ D2 ) ) ) ) ) ) ).

% div_mult_div_if_dvd
thf(fact_2891_div__power,axiom,
    ! [A: $tType] :
      ( ( algebraic_semidom @ A )
     => ! [B2: A,A2: A,N2: nat] :
          ( ( dvd_dvd @ A @ B2 @ A2 )
         => ( ( power_power @ A @ ( divide_divide @ A @ A2 @ B2 ) @ N2 )
            = ( divide_divide @ A @ ( power_power @ A @ A2 @ N2 ) @ ( power_power @ A @ B2 @ N2 ) ) ) ) ) ).

% div_power
thf(fact_2892_mod__0__imp__dvd,axiom,
    ! [A: $tType] :
      ( ( semiring_modulo @ A )
     => ! [A2: A,B2: A] :
          ( ( ( modulo_modulo @ A @ A2 @ B2 )
            = ( zero_zero @ A ) )
         => ( dvd_dvd @ A @ B2 @ A2 ) ) ) ).

% mod_0_imp_dvd
thf(fact_2893_dvd__eq__mod__eq__0,axiom,
    ! [A: $tType] :
      ( ( semidom_modulo @ A )
     => ( ( dvd_dvd @ A )
        = ( ^ [A5: A,B5: A] :
              ( ( modulo_modulo @ A @ B5 @ A5 )
              = ( zero_zero @ A ) ) ) ) ) ).

% dvd_eq_mod_eq_0
thf(fact_2894_mod__eq__0__iff__dvd,axiom,
    ! [A: $tType] :
      ( ( semidom_modulo @ A )
     => ! [A2: A,B2: A] :
          ( ( ( modulo_modulo @ A @ A2 @ B2 )
            = ( zero_zero @ A ) )
          = ( dvd_dvd @ A @ B2 @ A2 ) ) ) ).

% mod_eq_0_iff_dvd
thf(fact_2895_dvd__power__le,axiom,
    ! [A: $tType] :
      ( ( comm_semiring_1 @ A )
     => ! [X2: A,Y2: A,N2: nat,M: nat] :
          ( ( dvd_dvd @ A @ X2 @ Y2 )
         => ( ( ord_less_eq @ nat @ N2 @ M )
           => ( dvd_dvd @ A @ ( power_power @ A @ X2 @ N2 ) @ ( power_power @ A @ Y2 @ M ) ) ) ) ) ).

% dvd_power_le
thf(fact_2896_power__le__dvd,axiom,
    ! [A: $tType] :
      ( ( comm_semiring_1 @ A )
     => ! [A2: A,N2: nat,B2: A,M: nat] :
          ( ( dvd_dvd @ A @ ( power_power @ A @ A2 @ N2 ) @ B2 )
         => ( ( ord_less_eq @ nat @ M @ N2 )
           => ( dvd_dvd @ A @ ( power_power @ A @ A2 @ M ) @ B2 ) ) ) ) ).

% power_le_dvd
thf(fact_2897_le__imp__power__dvd,axiom,
    ! [A: $tType] :
      ( ( comm_semiring_1 @ A )
     => ! [M: nat,N2: nat,A2: A] :
          ( ( ord_less_eq @ nat @ M @ N2 )
         => ( dvd_dvd @ A @ ( power_power @ A @ A2 @ M ) @ ( power_power @ A @ A2 @ N2 ) ) ) ) ).

% le_imp_power_dvd
thf(fact_2898_mod__eq__dvd__iff,axiom,
    ! [A: $tType] :
      ( ( euclid8851590272496341667cancel @ A )
     => ! [A2: A,C2: A,B2: A] :
          ( ( ( modulo_modulo @ A @ A2 @ C2 )
            = ( modulo_modulo @ A @ B2 @ C2 ) )
          = ( dvd_dvd @ A @ C2 @ ( minus_minus @ A @ A2 @ B2 ) ) ) ) ).

% mod_eq_dvd_iff
thf(fact_2899_dvd__minus__mod,axiom,
    ! [A: $tType] :
      ( ( semidom_modulo @ A )
     => ! [B2: A,A2: A] : ( dvd_dvd @ A @ B2 @ ( minus_minus @ A @ A2 @ ( modulo_modulo @ A @ A2 @ B2 ) ) ) ) ).

% dvd_minus_mod
thf(fact_2900_nat__dvd__not__less,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
     => ( ( ord_less @ nat @ M @ N2 )
       => ~ ( dvd_dvd @ nat @ N2 @ M ) ) ) ).

% nat_dvd_not_less
thf(fact_2901_dvd__minus__self,axiom,
    ! [M: nat,N2: nat] :
      ( ( dvd_dvd @ nat @ M @ ( minus_minus @ nat @ N2 @ M ) )
      = ( ( ord_less @ nat @ N2 @ M )
        | ( dvd_dvd @ nat @ M @ N2 ) ) ) ).

% dvd_minus_self
thf(fact_2902_less__eq__dvd__minus,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq @ nat @ M @ N2 )
     => ( ( dvd_dvd @ nat @ M @ N2 )
        = ( dvd_dvd @ nat @ M @ ( minus_minus @ nat @ N2 @ M ) ) ) ) ).

% less_eq_dvd_minus
thf(fact_2903_dvd__diffD1,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( dvd_dvd @ nat @ K @ ( minus_minus @ nat @ M @ N2 ) )
     => ( ( dvd_dvd @ nat @ K @ M )
       => ( ( ord_less_eq @ nat @ N2 @ M )
         => ( dvd_dvd @ nat @ K @ N2 ) ) ) ) ).

% dvd_diffD1
thf(fact_2904_dvd__diffD,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( dvd_dvd @ nat @ K @ ( minus_minus @ nat @ M @ N2 ) )
     => ( ( dvd_dvd @ nat @ K @ N2 )
       => ( ( ord_less_eq @ nat @ N2 @ M )
         => ( dvd_dvd @ nat @ K @ M ) ) ) ) ).

% dvd_diffD
thf(fact_2905_even__unset__bit__iff,axiom,
    ! [A: $tType] :
      ( ( bit_se359711467146920520ations @ A )
     => ! [M: nat,A2: A] :
          ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se2638667681897837118et_bit @ A @ M @ A2 ) )
          = ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 )
            | ( M
              = ( zero_zero @ nat ) ) ) ) ) ).

% even_unset_bit_iff
thf(fact_2906_even__flip__bit__iff,axiom,
    ! [A: $tType] :
      ( ( bit_se359711467146920520ations @ A )
     => ! [M: nat,A2: A] :
          ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se8732182000553998342ip_bit @ A @ M @ A2 ) )
          = ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 )
           != ( M
              = ( zero_zero @ nat ) ) ) ) ) ).

% even_flip_bit_iff
thf(fact_2907_even__of__int__iff,axiom,
    ! [A: $tType] :
      ( ( bit_ri3973907225187159222ations @ A )
     => ! [K: int] :
          ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( ring_1_of_int @ A @ K ) )
          = ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K ) ) ) ).

% even_of_int_iff
thf(fact_2908_curry__def,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( product_curry @ A @ B @ C )
      = ( ^ [C4: ( product_prod @ A @ B ) > C,X: A,Y: B] : ( C4 @ ( product_Pair @ A @ B @ X @ Y ) ) ) ) ).

% curry_def
thf(fact_2909_even__numeral,axiom,
    ! [A: $tType] :
      ( ( semiring_parity @ A )
     => ! [N2: num] : ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ A @ ( bit0 @ N2 ) ) ) ) ).

% even_numeral
thf(fact_2910_unity__coeff__ex,axiom,
    ! [A: $tType] :
      ( ( ( dvd @ A )
        & ( semiring_0 @ A ) )
     => ! [P: A > $o,L: A] :
          ( ( ? [X: A] : ( P @ ( times_times @ A @ L @ X ) ) )
          = ( ? [X: A] :
                ( ( dvd_dvd @ A @ L @ ( plus_plus @ A @ X @ ( zero_zero @ A ) ) )
                & ( P @ X ) ) ) ) ) ).

% unity_coeff_ex
thf(fact_2911_unit__dvdE,axiom,
    ! [A: $tType] :
      ( ( algebraic_semidom @ A )
     => ! [A2: A,B2: A] :
          ( ( dvd_dvd @ A @ A2 @ ( one_one @ A ) )
         => ~ ( ( A2
               != ( zero_zero @ A ) )
             => ! [C3: A] :
                  ( B2
                 != ( times_times @ A @ A2 @ C3 ) ) ) ) ) ).

% unit_dvdE
thf(fact_2912_unit__div__eq__0__iff,axiom,
    ! [A: $tType] :
      ( ( algebraic_semidom @ A )
     => ! [B2: A,A2: A] :
          ( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
         => ( ( ( divide_divide @ A @ A2 @ B2 )
              = ( zero_zero @ A ) )
            = ( A2
              = ( zero_zero @ A ) ) ) ) ) ).

% unit_div_eq_0_iff
thf(fact_2913_dvd__div__eq__mult,axiom,
    ! [A: $tType] :
      ( ( algebraic_semidom @ A )
     => ! [A2: A,B2: A,C2: A] :
          ( ( A2
           != ( zero_zero @ A ) )
         => ( ( dvd_dvd @ A @ A2 @ B2 )
           => ( ( ( divide_divide @ A @ B2 @ A2 )
                = C2 )
              = ( B2
                = ( times_times @ A @ C2 @ A2 ) ) ) ) ) ) ).

% dvd_div_eq_mult
thf(fact_2914_div__dvd__iff__mult,axiom,
    ! [A: $tType] :
      ( ( algebraic_semidom @ A )
     => ! [B2: A,A2: A,C2: A] :
          ( ( B2
           != ( zero_zero @ A ) )
         => ( ( dvd_dvd @ A @ B2 @ A2 )
           => ( ( dvd_dvd @ A @ ( divide_divide @ A @ A2 @ B2 ) @ C2 )
              = ( dvd_dvd @ A @ A2 @ ( times_times @ A @ C2 @ B2 ) ) ) ) ) ) ).

% div_dvd_iff_mult
thf(fact_2915_dvd__div__iff__mult,axiom,
    ! [A: $tType] :
      ( ( algebraic_semidom @ A )
     => ! [C2: A,B2: A,A2: A] :
          ( ( C2
           != ( zero_zero @ A ) )
         => ( ( dvd_dvd @ A @ C2 @ B2 )
           => ( ( dvd_dvd @ A @ A2 @ ( divide_divide @ A @ B2 @ C2 ) )
              = ( dvd_dvd @ A @ ( times_times @ A @ A2 @ C2 ) @ B2 ) ) ) ) ) ).

% dvd_div_iff_mult
thf(fact_2916_dvd__div__div__eq__mult,axiom,
    ! [A: $tType] :
      ( ( algebraic_semidom @ A )
     => ! [A2: A,C2: A,B2: A,D2: A] :
          ( ( A2
           != ( zero_zero @ A ) )
         => ( ( C2
             != ( zero_zero @ A ) )
           => ( ( dvd_dvd @ A @ A2 @ B2 )
             => ( ( dvd_dvd @ A @ C2 @ D2 )
               => ( ( ( divide_divide @ A @ B2 @ A2 )
                    = ( divide_divide @ A @ D2 @ C2 ) )
                  = ( ( times_times @ A @ B2 @ C2 )
                    = ( times_times @ A @ A2 @ D2 ) ) ) ) ) ) ) ) ).

% dvd_div_div_eq_mult
thf(fact_2917_inf__period_I4_J,axiom,
    ! [A: $tType] :
      ( ( ( comm_ring @ A )
        & ( dvd @ A ) )
     => ! [D2: A,D6: A,T2: A] :
          ( ( dvd_dvd @ A @ D2 @ D6 )
         => ! [X6: A,K5: A] :
              ( ( ~ ( dvd_dvd @ A @ D2 @ ( plus_plus @ A @ X6 @ T2 ) ) )
              = ( ~ ( dvd_dvd @ A @ D2 @ ( plus_plus @ A @ ( minus_minus @ A @ X6 @ ( times_times @ A @ K5 @ D6 ) ) @ T2 ) ) ) ) ) ) ).

% inf_period(4)
thf(fact_2918_inf__period_I3_J,axiom,
    ! [A: $tType] :
      ( ( ( comm_ring @ A )
        & ( dvd @ A ) )
     => ! [D2: A,D6: A,T2: A] :
          ( ( dvd_dvd @ A @ D2 @ D6 )
         => ! [X6: A,K5: A] :
              ( ( dvd_dvd @ A @ D2 @ ( plus_plus @ A @ X6 @ T2 ) )
              = ( dvd_dvd @ A @ D2 @ ( plus_plus @ A @ ( minus_minus @ A @ X6 @ ( times_times @ A @ K5 @ D6 ) ) @ T2 ) ) ) ) ) ).

% inf_period(3)
thf(fact_2919_unit__eq__div1,axiom,
    ! [A: $tType] :
      ( ( algebraic_semidom @ A )
     => ! [B2: A,A2: A,C2: A] :
          ( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
         => ( ( ( divide_divide @ A @ A2 @ B2 )
              = C2 )
            = ( A2
              = ( times_times @ A @ C2 @ B2 ) ) ) ) ) ).

% unit_eq_div1
thf(fact_2920_unit__eq__div2,axiom,
    ! [A: $tType] :
      ( ( algebraic_semidom @ A )
     => ! [B2: A,A2: A,C2: A] :
          ( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
         => ( ( A2
              = ( divide_divide @ A @ C2 @ B2 ) )
            = ( ( times_times @ A @ A2 @ B2 )
              = C2 ) ) ) ) ).

% unit_eq_div2
thf(fact_2921_div__mult__unit2,axiom,
    ! [A: $tType] :
      ( ( algebraic_semidom @ A )
     => ! [C2: A,B2: A,A2: A] :
          ( ( dvd_dvd @ A @ C2 @ ( one_one @ A ) )
         => ( ( dvd_dvd @ A @ B2 @ A2 )
           => ( ( divide_divide @ A @ A2 @ ( times_times @ A @ B2 @ C2 ) )
              = ( divide_divide @ A @ ( divide_divide @ A @ A2 @ B2 ) @ C2 ) ) ) ) ) ).

% div_mult_unit2
thf(fact_2922_unit__div__commute,axiom,
    ! [A: $tType] :
      ( ( algebraic_semidom @ A )
     => ! [B2: A,A2: A,C2: A] :
          ( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
         => ( ( times_times @ A @ ( divide_divide @ A @ A2 @ B2 ) @ C2 )
            = ( divide_divide @ A @ ( times_times @ A @ A2 @ C2 ) @ B2 ) ) ) ) ).

% unit_div_commute
thf(fact_2923_unit__div__mult__swap,axiom,
    ! [A: $tType] :
      ( ( algebraic_semidom @ A )
     => ! [C2: A,A2: A,B2: A] :
          ( ( dvd_dvd @ A @ C2 @ ( one_one @ A ) )
         => ( ( times_times @ A @ A2 @ ( divide_divide @ A @ B2 @ C2 ) )
            = ( divide_divide @ A @ ( times_times @ A @ A2 @ B2 ) @ C2 ) ) ) ) ).

% unit_div_mult_swap
thf(fact_2924_is__unit__div__mult2__eq,axiom,
    ! [A: $tType] :
      ( ( algebraic_semidom @ A )
     => ! [B2: A,C2: A,A2: A] :
          ( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
         => ( ( dvd_dvd @ A @ C2 @ ( one_one @ A ) )
           => ( ( divide_divide @ A @ A2 @ ( times_times @ A @ B2 @ C2 ) )
              = ( divide_divide @ A @ ( divide_divide @ A @ A2 @ B2 ) @ C2 ) ) ) ) ) ).

% is_unit_div_mult2_eq
thf(fact_2925_unit__imp__mod__eq__0,axiom,
    ! [A: $tType] :
      ( ( euclid3725896446679973847miring @ A )
     => ! [B2: A,A2: A] :
          ( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
         => ( ( modulo_modulo @ A @ A2 @ B2 )
            = ( zero_zero @ A ) ) ) ) ).

% unit_imp_mod_eq_0
thf(fact_2926_is__unit__power__iff,axiom,
    ! [A: $tType] :
      ( ( algebraic_semidom @ A )
     => ! [A2: A,N2: nat] :
          ( ( dvd_dvd @ A @ ( power_power @ A @ A2 @ N2 ) @ ( one_one @ A ) )
          = ( ( dvd_dvd @ A @ A2 @ ( one_one @ A ) )
            | ( N2
              = ( zero_zero @ nat ) ) ) ) ) ).

% is_unit_power_iff
thf(fact_2927_dvd__imp__le,axiom,
    ! [K: nat,N2: nat] :
      ( ( dvd_dvd @ nat @ K @ N2 )
     => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
       => ( ord_less_eq @ nat @ K @ N2 ) ) ) ).

% dvd_imp_le
thf(fact_2928_dvd__mult__cancel,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( dvd_dvd @ nat @ ( times_times @ nat @ K @ M ) @ ( times_times @ nat @ K @ N2 ) )
     => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
       => ( dvd_dvd @ nat @ M @ N2 ) ) ) ).

% dvd_mult_cancel
thf(fact_2929_nat__mult__dvd__cancel1,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
     => ( ( dvd_dvd @ nat @ ( times_times @ nat @ K @ M ) @ ( times_times @ nat @ K @ N2 ) )
        = ( dvd_dvd @ nat @ M @ N2 ) ) ) ).

% nat_mult_dvd_cancel1
thf(fact_2930_mod__greater__zero__iff__not__dvd,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( modulo_modulo @ nat @ M @ N2 ) )
      = ( ~ ( dvd_dvd @ nat @ N2 @ M ) ) ) ).

% mod_greater_zero_iff_not_dvd
thf(fact_2931_mod__eq__dvd__iff__nat,axiom,
    ! [N2: nat,M: nat,Q2: nat] :
      ( ( ord_less_eq @ nat @ N2 @ M )
     => ( ( ( modulo_modulo @ nat @ M @ Q2 )
          = ( modulo_modulo @ nat @ N2 @ Q2 ) )
        = ( dvd_dvd @ nat @ Q2 @ ( minus_minus @ nat @ M @ N2 ) ) ) ) ).

% mod_eq_dvd_iff_nat
thf(fact_2932_real__of__nat__div,axiom,
    ! [D2: nat,N2: nat] :
      ( ( dvd_dvd @ nat @ D2 @ N2 )
     => ( ( semiring_1_of_nat @ real @ ( divide_divide @ nat @ N2 @ D2 ) )
        = ( divide_divide @ real @ ( semiring_1_of_nat @ real @ N2 ) @ ( semiring_1_of_nat @ real @ D2 ) ) ) ) ).

% real_of_nat_div
thf(fact_2933_even__zero,axiom,
    ! [A: $tType] :
      ( ( semiring_parity @ A )
     => ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( zero_zero @ A ) ) ) ).

% even_zero
thf(fact_2934_odd__even__add,axiom,
    ! [A: $tType] :
      ( ( semiring_parity @ A )
     => ! [A2: A,B2: A] :
          ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 )
         => ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 )
           => ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( plus_plus @ A @ A2 @ B2 ) ) ) ) ) ).

% odd_even_add
thf(fact_2935_odd__one,axiom,
    ! [A: $tType] :
      ( ( semiring_parity @ A )
     => ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( one_one @ A ) ) ) ).

% odd_one
thf(fact_2936_evenE,axiom,
    ! [A: $tType] :
      ( ( semiring_parity @ A )
     => ! [A2: A] :
          ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 )
         => ~ ! [B3: A] :
                ( A2
               != ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B3 ) ) ) ) ).

% evenE
thf(fact_2937_bit__eq__rec,axiom,
    ! [A: $tType] :
      ( ( bit_semiring_bits @ A )
     => ( ( ^ [Y5: A,Z4: A] : Y5 = Z4 )
        = ( ^ [A5: A,B5: A] :
              ( ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A5 )
                = ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B5 ) )
              & ( ( divide_divide @ A @ A5 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
                = ( divide_divide @ A @ B5 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).

% bit_eq_rec
thf(fact_2938_is__unitE,axiom,
    ! [A: $tType] :
      ( ( algebraic_semidom @ A )
     => ! [A2: A,C2: A] :
          ( ( dvd_dvd @ A @ A2 @ ( one_one @ A ) )
         => ~ ( ( A2
               != ( zero_zero @ A ) )
             => ! [B3: A] :
                  ( ( B3
                   != ( zero_zero @ A ) )
                 => ( ( dvd_dvd @ A @ B3 @ ( one_one @ A ) )
                   => ( ( ( divide_divide @ A @ ( one_one @ A ) @ A2 )
                        = B3 )
                     => ( ( ( divide_divide @ A @ ( one_one @ A ) @ B3 )
                          = A2 )
                       => ( ( ( times_times @ A @ A2 @ B3 )
                            = ( one_one @ A ) )
                         => ( ( divide_divide @ A @ C2 @ A2 )
                           != ( times_times @ A @ C2 @ B3 ) ) ) ) ) ) ) ) ) ) ).

% is_unitE
thf(fact_2939_is__unit__div__mult__cancel__left,axiom,
    ! [A: $tType] :
      ( ( algebraic_semidom @ A )
     => ! [A2: A,B2: A] :
          ( ( A2
           != ( zero_zero @ A ) )
         => ( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
           => ( ( divide_divide @ A @ A2 @ ( times_times @ A @ A2 @ B2 ) )
              = ( divide_divide @ A @ ( one_one @ A ) @ B2 ) ) ) ) ) ).

% is_unit_div_mult_cancel_left
thf(fact_2940_is__unit__div__mult__cancel__right,axiom,
    ! [A: $tType] :
      ( ( algebraic_semidom @ A )
     => ! [A2: A,B2: A] :
          ( ( A2
           != ( zero_zero @ A ) )
         => ( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
           => ( ( divide_divide @ A @ A2 @ ( times_times @ A @ B2 @ A2 ) )
              = ( divide_divide @ A @ ( one_one @ A ) @ B2 ) ) ) ) ) ).

% is_unit_div_mult_cancel_right
thf(fact_2941_odd__numeral,axiom,
    ! [A: $tType] :
      ( ( semiring_parity @ A )
     => ! [N2: num] :
          ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ A @ ( bit1 @ N2 ) ) ) ) ).

% odd_numeral
thf(fact_2942_dvd__power__iff,axiom,
    ! [A: $tType] :
      ( ( algebraic_semidom @ A )
     => ! [X2: A,M: nat,N2: nat] :
          ( ( X2
           != ( zero_zero @ A ) )
         => ( ( dvd_dvd @ A @ ( power_power @ A @ X2 @ M ) @ ( power_power @ A @ X2 @ N2 ) )
            = ( ( dvd_dvd @ A @ X2 @ ( one_one @ A ) )
              | ( ord_less_eq @ nat @ M @ N2 ) ) ) ) ) ).

% dvd_power_iff
thf(fact_2943_dvd__power,axiom,
    ! [A: $tType] :
      ( ( comm_semiring_1 @ A )
     => ! [N2: nat,X2: A] :
          ( ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
            | ( X2
              = ( one_one @ A ) ) )
         => ( dvd_dvd @ A @ X2 @ ( power_power @ A @ X2 @ N2 ) ) ) ) ).

% dvd_power
thf(fact_2944_even__even__mod__4__iff,axiom,
    ! [N2: nat] :
      ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 )
      = ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( modulo_modulo @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) ).

% even_even_mod_4_iff
thf(fact_2945_dvd__mult__cancel1,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
     => ( ( dvd_dvd @ nat @ ( times_times @ nat @ M @ N2 ) @ M )
        = ( N2
          = ( one_one @ nat ) ) ) ) ).

% dvd_mult_cancel1
thf(fact_2946_dvd__mult__cancel2,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
     => ( ( dvd_dvd @ nat @ ( times_times @ nat @ N2 @ M ) @ M )
        = ( N2
          = ( one_one @ nat ) ) ) ) ).

% dvd_mult_cancel2
thf(fact_2947_power__dvd__imp__le,axiom,
    ! [I: nat,M: nat,N2: nat] :
      ( ( dvd_dvd @ nat @ ( power_power @ nat @ I @ M ) @ ( power_power @ nat @ I @ N2 ) )
     => ( ( ord_less @ nat @ ( one_one @ nat ) @ I )
       => ( ord_less_eq @ nat @ M @ N2 ) ) ) ).

% power_dvd_imp_le
thf(fact_2948_dvd__minus__add,axiom,
    ! [Q2: nat,N2: nat,R4: nat,M: nat] :
      ( ( ord_less_eq @ nat @ Q2 @ N2 )
     => ( ( ord_less_eq @ nat @ Q2 @ ( times_times @ nat @ R4 @ M ) )
       => ( ( dvd_dvd @ nat @ M @ ( minus_minus @ nat @ N2 @ Q2 ) )
          = ( dvd_dvd @ nat @ M @ ( plus_plus @ nat @ N2 @ ( minus_minus @ nat @ ( times_times @ nat @ R4 @ M ) @ Q2 ) ) ) ) ) ) ).

% dvd_minus_add
thf(fact_2949_diff__mod__le,axiom,
    ! [A2: nat,D2: nat,B2: nat] :
      ( ( ord_less @ nat @ A2 @ D2 )
     => ( ( dvd_dvd @ nat @ B2 @ D2 )
       => ( ord_less_eq @ nat @ ( minus_minus @ nat @ A2 @ ( modulo_modulo @ nat @ A2 @ B2 ) ) @ ( minus_minus @ nat @ D2 @ B2 ) ) ) ) ).

% diff_mod_le
thf(fact_2950_mod__nat__eqI,axiom,
    ! [R4: nat,N2: nat,M: nat] :
      ( ( ord_less @ nat @ R4 @ N2 )
     => ( ( ord_less_eq @ nat @ R4 @ M )
       => ( ( dvd_dvd @ nat @ N2 @ ( minus_minus @ nat @ M @ R4 ) )
         => ( ( modulo_modulo @ nat @ M @ N2 )
            = R4 ) ) ) ) ).

% mod_nat_eqI
thf(fact_2951_even__two__times__div__two,axiom,
    ! [A: $tType] :
      ( ( euclid5411537665997757685th_nat @ A )
     => ! [A2: A] :
          ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 )
         => ( ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) )
            = A2 ) ) ) ).

% even_two_times_div_two
thf(fact_2952_even__iff__mod__2__eq__zero,axiom,
    ! [A: $tType] :
      ( ( semiring_parity @ A )
     => ! [A2: A] :
          ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 )
          = ( ( modulo_modulo @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
            = ( zero_zero @ A ) ) ) ) ).

% even_iff_mod_2_eq_zero
thf(fact_2953_odd__iff__mod__2__eq__one,axiom,
    ! [A: $tType] :
      ( ( semiring_parity @ A )
     => ! [A2: A] :
          ( ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) )
          = ( ( modulo_modulo @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
            = ( one_one @ A ) ) ) ) ).

% odd_iff_mod_2_eq_one
thf(fact_2954_power__mono__odd,axiom,
    ! [A: $tType] :
      ( ( linordered_idom @ A )
     => ! [N2: nat,A2: A,B2: A] :
          ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 )
         => ( ( ord_less_eq @ A @ A2 @ B2 )
           => ( ord_less_eq @ A @ ( power_power @ A @ A2 @ N2 ) @ ( power_power @ A @ B2 @ N2 ) ) ) ) ) ).

% power_mono_odd
thf(fact_2955_odd__pos,axiom,
    ! [N2: nat] :
      ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 )
     => ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) ) ).

% odd_pos
thf(fact_2956_even__set__bit__iff,axiom,
    ! [A: $tType] :
      ( ( bit_se359711467146920520ations @ A )
     => ! [M: nat,A2: A] :
          ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se5668285175392031749et_bit @ A @ M @ A2 ) )
          = ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 )
            & ( M
             != ( zero_zero @ nat ) ) ) ) ) ).

% even_set_bit_iff
thf(fact_2957_dvd__power__iff__le,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ K )
     => ( ( dvd_dvd @ nat @ ( power_power @ nat @ K @ M ) @ ( power_power @ nat @ K @ N2 ) )
        = ( ord_less_eq @ nat @ M @ N2 ) ) ) ).

% dvd_power_iff_le
thf(fact_2958_oddE,axiom,
    ! [A: $tType] :
      ( ( semiring_parity @ A )
     => ! [A2: A] :
          ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 )
         => ~ ! [B3: A] :
                ( A2
               != ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B3 ) @ ( one_one @ A ) ) ) ) ) ).

% oddE
thf(fact_2959_mod2__eq__if,axiom,
    ! [A: $tType] :
      ( ( semiring_parity @ A )
     => ! [A2: A] :
          ( ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 )
           => ( ( modulo_modulo @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
              = ( zero_zero @ A ) ) )
          & ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 )
           => ( ( modulo_modulo @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
              = ( one_one @ A ) ) ) ) ) ).

% mod2_eq_if
thf(fact_2960_parity__cases,axiom,
    ! [A: $tType] :
      ( ( semiring_parity @ A )
     => ! [A2: A] :
          ( ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 )
           => ( ( modulo_modulo @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
             != ( zero_zero @ A ) ) )
         => ~ ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 )
             => ( ( modulo_modulo @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
               != ( one_one @ A ) ) ) ) ) ).

% parity_cases
thf(fact_2961_zero__le__power__eq,axiom,
    ! [A: $tType] :
      ( ( linordered_idom @ A )
     => ! [A2: A,N2: nat] :
          ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A2 @ N2 ) )
          = ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 )
            | ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 )
              & ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) ) ) ) ) ).

% zero_le_power_eq
thf(fact_2962_zero__le__odd__power,axiom,
    ! [A: $tType] :
      ( ( linordered_idom @ A )
     => ! [N2: nat,A2: A] :
          ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 )
         => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A2 @ N2 ) )
            = ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) ) ) ) ).

% zero_le_odd_power
thf(fact_2963_zero__le__even__power,axiom,
    ! [A: $tType] :
      ( ( linordered_idom @ A )
     => ! [N2: nat,A2: A] :
          ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 )
         => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A2 @ N2 ) ) ) ) ).

% zero_le_even_power
thf(fact_2964_divides__aux__def,axiom,
    ! [A: $tType] :
      ( ( unique1627219031080169319umeral @ A )
     => ( ( unique5940410009612947441es_aux @ A )
        = ( ^ [Qr: product_prod @ A @ A] :
              ( ( product_snd @ A @ A @ Qr )
              = ( zero_zero @ A ) ) ) ) ) ).

% divides_aux_def
thf(fact_2965_zero__less__power__eq,axiom,
    ! [A: $tType] :
      ( ( linordered_idom @ A )
     => ! [A2: A,N2: nat] :
          ( ( ord_less @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A2 @ N2 ) )
          = ( ( N2
              = ( zero_zero @ nat ) )
            | ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 )
              & ( A2
               != ( zero_zero @ A ) ) )
            | ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 )
              & ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ) ) ).

% zero_less_power_eq
thf(fact_2966_even__mask__div__iff_H,axiom,
    ! [A: $tType] :
      ( ( euclid5411537665997757685th_nat @ A )
     => ! [M: nat,N2: nat] :
          ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) @ ( one_one @ A ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) ) )
          = ( ord_less_eq @ nat @ M @ N2 ) ) ) ).

% even_mask_div_iff'
thf(fact_2967_power__le__zero__eq,axiom,
    ! [A: $tType] :
      ( ( linordered_idom @ A )
     => ! [A2: A,N2: nat] :
          ( ( ord_less_eq @ A @ ( power_power @ A @ A2 @ N2 ) @ ( zero_zero @ A ) )
          = ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
            & ( ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 )
                & ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) )
              | ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 )
                & ( A2
                  = ( zero_zero @ A ) ) ) ) ) ) ) ).

% power_le_zero_eq
thf(fact_2968_even__mod__4__div__2,axiom,
    ! [N2: nat] :
      ( ( ( modulo_modulo @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) )
        = ( suc @ ( zero_zero @ nat ) ) )
     => ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( minus_minus @ nat @ N2 @ ( suc @ ( zero_zero @ nat ) ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).

% even_mod_4_div_2
thf(fact_2969_even__mask__div__iff,axiom,
    ! [A: $tType] :
      ( ( bit_semiring_bits @ A )
     => ! [M: nat,N2: nat] :
          ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) @ ( one_one @ A ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) ) )
          = ( ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 )
              = ( zero_zero @ A ) )
            | ( ord_less_eq @ nat @ M @ N2 ) ) ) ) ).

% even_mask_div_iff
thf(fact_2970_odd__mod__4__div__2,axiom,
    ! [N2: nat] :
      ( ( ( modulo_modulo @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) )
        = ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) )
     => ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( minus_minus @ nat @ N2 @ ( suc @ ( zero_zero @ nat ) ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).

% odd_mod_4_div_2
thf(fact_2971_Bernoulli__inequality__even,axiom,
    ! [N2: nat,X2: real] :
      ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 )
     => ( ord_less_eq @ real @ ( plus_plus @ real @ ( one_one @ real ) @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N2 ) @ X2 ) ) @ ( power_power @ real @ ( plus_plus @ real @ ( one_one @ real ) @ X2 ) @ N2 ) ) ) ).

% Bernoulli_inequality_even
thf(fact_2972_VEBT__internal_OT__vebt__buildupi_Osimps_I3_J,axiom,
    ! [N2: nat] :
      ( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 )
       => ( ( vEBT_V441764108873111860ildupi @ ( suc @ ( suc @ N2 ) ) )
          = ( suc @ ( suc @ ( suc @ ( plus_plus @ nat @ ( vEBT_V441764108873111860ildupi @ ( suc @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( times_times @ nat @ ( vEBT_V441764108873111860ildupi @ ( suc @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) )
      & ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 )
       => ( ( vEBT_V441764108873111860ildupi @ ( suc @ ( suc @ N2 ) ) )
          = ( suc @ ( suc @ ( suc @ ( plus_plus @ nat @ ( vEBT_V441764108873111860ildupi @ ( suc @ ( suc @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) @ ( times_times @ nat @ ( vEBT_V441764108873111860ildupi @ ( suc @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.T_vebt_buildupi.simps(3)
thf(fact_2973_even__mult__exp__div__exp__iff,axiom,
    ! [A: $tType] :
      ( ( bit_semiring_bits @ A )
     => ! [A2: A,M: nat,N2: nat] :
          ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ ( times_times @ A @ A2 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) ) )
          = ( ( ord_less @ nat @ N2 @ M )
            | ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 )
              = ( zero_zero @ A ) )
            | ( ( ord_less_eq @ nat @ M @ N2 )
              & ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ A2 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ N2 @ M ) ) ) ) ) ) ) ) ).

% even_mult_exp_div_exp_iff
thf(fact_2974_VEBT__internal_OT__vebt__buildupi_Oelims,axiom,
    ! [X2: nat,Y2: nat] :
      ( ( ( vEBT_V441764108873111860ildupi @ X2 )
        = Y2 )
     => ( ( ( X2
            = ( zero_zero @ nat ) )
         => ( Y2
           != ( suc @ ( zero_zero @ nat ) ) ) )
       => ( ( ( X2
              = ( suc @ ( zero_zero @ nat ) ) )
           => ( Y2
             != ( suc @ ( zero_zero @ nat ) ) ) )
         => ~ ! [N4: nat] :
                ( ( X2
                  = ( suc @ ( suc @ N4 ) ) )
               => ~ ( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N4 )
                     => ( Y2
                        = ( suc @ ( suc @ ( suc @ ( plus_plus @ nat @ ( vEBT_V441764108873111860ildupi @ ( suc @ ( divide_divide @ nat @ N4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ N4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( times_times @ nat @ ( vEBT_V441764108873111860ildupi @ ( suc @ ( divide_divide @ nat @ N4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ N4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) )
                    & ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N4 )
                     => ( Y2
                        = ( suc @ ( suc @ ( suc @ ( plus_plus @ nat @ ( vEBT_V441764108873111860ildupi @ ( suc @ ( suc @ ( divide_divide @ nat @ N4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ N4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) @ ( times_times @ nat @ ( vEBT_V441764108873111860ildupi @ ( suc @ ( divide_divide @ nat @ N4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ N4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.T_vebt_buildupi.elims
thf(fact_2975_VEBT__internal_OTb_H_Osimps_I3_J,axiom,
    ! [N2: nat] :
      ( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 )
       => ( ( vEBT_VEBT_Tb2 @ ( suc @ ( suc @ N2 ) ) )
          = ( plus_plus @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit0 @ one2 ) ) ) @ ( vEBT_VEBT_Tb2 @ ( suc @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( times_times @ nat @ ( vEBT_VEBT_Tb2 @ ( suc @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) )
      & ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 )
       => ( ( vEBT_VEBT_Tb2 @ ( suc @ ( suc @ N2 ) ) )
          = ( plus_plus @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit0 @ one2 ) ) ) @ ( vEBT_VEBT_Tb2 @ ( suc @ ( suc @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) @ ( times_times @ nat @ ( vEBT_VEBT_Tb2 @ ( suc @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.Tb'.simps(3)
thf(fact_2976_VEBT__internal_OTb_H_Oelims,axiom,
    ! [X2: nat,Y2: nat] :
      ( ( ( vEBT_VEBT_Tb2 @ X2 )
        = Y2 )
     => ( ( ( X2
            = ( zero_zero @ nat ) )
         => ( Y2
           != ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) ) )
       => ( ( ( X2
              = ( suc @ ( zero_zero @ nat ) ) )
           => ( Y2
             != ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) ) )
         => ~ ! [N4: nat] :
                ( ( X2
                  = ( suc @ ( suc @ N4 ) ) )
               => ~ ( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N4 )
                     => ( Y2
                        = ( plus_plus @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit0 @ one2 ) ) ) @ ( vEBT_VEBT_Tb2 @ ( suc @ ( divide_divide @ nat @ N4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( times_times @ nat @ ( vEBT_VEBT_Tb2 @ ( suc @ ( divide_divide @ nat @ N4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( divide_divide @ nat @ N4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) )
                    & ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N4 )
                     => ( Y2
                        = ( plus_plus @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit0 @ one2 ) ) ) @ ( vEBT_VEBT_Tb2 @ ( suc @ ( suc @ ( divide_divide @ nat @ N4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) @ ( times_times @ nat @ ( vEBT_VEBT_Tb2 @ ( suc @ ( divide_divide @ nat @ N4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ ( divide_divide @ nat @ N4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.Tb'.elims
thf(fact_2977_VEBT__internal_OTb_Osimps_I3_J,axiom,
    ! [N2: nat] :
      ( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 )
       => ( ( vEBT_VEBT_Tb @ ( suc @ ( suc @ N2 ) ) )
          = ( plus_plus @ int @ ( plus_plus @ int @ ( numeral_numeral @ int @ ( bit1 @ ( bit0 @ one2 ) ) ) @ ( vEBT_VEBT_Tb @ ( suc @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( times_times @ int @ ( vEBT_VEBT_Tb @ ( suc @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( suc @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) )
      & ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 )
       => ( ( vEBT_VEBT_Tb @ ( suc @ ( suc @ N2 ) ) )
          = ( plus_plus @ int @ ( plus_plus @ int @ ( numeral_numeral @ int @ ( bit1 @ ( bit0 @ one2 ) ) ) @ ( vEBT_VEBT_Tb @ ( suc @ ( suc @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) @ ( times_times @ int @ ( vEBT_VEBT_Tb @ ( suc @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.Tb.simps(3)
thf(fact_2978_VEBT__internal_OT__vebt__buildupi_H_Osimps_I3_J,axiom,
    ! [N2: nat] :
      ( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 )
       => ( ( vEBT_V9176841429113362141ildupi @ ( suc @ ( suc @ N2 ) ) )
          = ( plus_plus @ int @ ( numeral_numeral @ int @ ( bit1 @ one2 ) ) @ ( plus_plus @ int @ ( vEBT_V9176841429113362141ildupi @ ( suc @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( plus_plus @ int @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ ( bit0 @ one2 ) ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( times_times @ int @ ( vEBT_V9176841429113362141ildupi @ ( suc @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) )
      & ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 )
       => ( ( vEBT_V9176841429113362141ildupi @ ( suc @ ( suc @ N2 ) ) )
          = ( plus_plus @ int @ ( numeral_numeral @ int @ ( bit1 @ one2 ) ) @ ( plus_plus @ int @ ( vEBT_V9176841429113362141ildupi @ ( suc @ ( suc @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( plus_plus @ int @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ ( bit0 @ one2 ) ) ) @ ( times_times @ int @ ( vEBT_V9176841429113362141ildupi @ ( suc @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.T_vebt_buildupi'.simps(3)
thf(fact_2979_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,
    ! [Va: nat] :
      ( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ Va ) ) )
       => ( ( vEBT_V8346862874174094_d_u_p @ ( suc @ ( suc @ Va ) ) )
          = ( plus_plus @ nat @ ( one_one @ nat ) @ ( plus_plus @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_V8346862874174094_d_u_p @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( times_times @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( plus_plus @ nat @ ( vEBT_V8346862874174094_d_u_p @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( one_one @ nat ) ) ) ) ) ) )
      & ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ Va ) ) )
       => ( ( vEBT_V8346862874174094_d_u_p @ ( suc @ ( suc @ Va ) ) )
          = ( plus_plus @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit1 @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_V8346862874174094_d_u_p @ ( suc @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( times_times @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( plus_plus @ nat @ ( vEBT_V8346862874174094_d_u_p @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( one_one @ nat ) ) ) ) ) ) ) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d\<^sub>u\<^sub>p.simps(3)
thf(fact_2980_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,
    ! [Va: nat] :
      ( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ Va ) ) )
       => ( ( vEBT_V8646137997579335489_i_l_d @ ( suc @ ( suc @ Va ) ) )
          = ( plus_plus @ nat @ ( one_one @ nat ) @ ( plus_plus @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit1 @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_V8646137997579335489_i_l_d @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( times_times @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_V8646137997579335489_i_l_d @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) )
      & ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ Va ) ) )
       => ( ( vEBT_V8646137997579335489_i_l_d @ ( suc @ ( suc @ Va ) ) )
          = ( plus_plus @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ ( bit1 @ one2 ) ) ) ) @ ( vEBT_V8646137997579335489_i_l_d @ ( suc @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( times_times @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_V8646137997579335489_i_l_d @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d.simps(3)
thf(fact_2981_VEBT__internal_OTb_Oelims,axiom,
    ! [X2: nat,Y2: int] :
      ( ( ( vEBT_VEBT_Tb @ X2 )
        = Y2 )
     => ( ( ( X2
            = ( zero_zero @ nat ) )
         => ( Y2
           != ( numeral_numeral @ int @ ( bit1 @ one2 ) ) ) )
       => ( ( ( X2
              = ( suc @ ( zero_zero @ nat ) ) )
           => ( Y2
             != ( numeral_numeral @ int @ ( bit1 @ one2 ) ) ) )
         => ~ ! [N4: nat] :
                ( ( X2
                  = ( suc @ ( suc @ N4 ) ) )
               => ~ ( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N4 )
                     => ( Y2
                        = ( plus_plus @ int @ ( plus_plus @ int @ ( numeral_numeral @ int @ ( bit1 @ ( bit0 @ one2 ) ) ) @ ( vEBT_VEBT_Tb @ ( suc @ ( divide_divide @ nat @ N4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( times_times @ int @ ( vEBT_VEBT_Tb @ ( suc @ ( divide_divide @ nat @ N4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( suc @ ( divide_divide @ nat @ N4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) )
                    & ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N4 )
                     => ( Y2
                        = ( plus_plus @ int @ ( plus_plus @ int @ ( numeral_numeral @ int @ ( bit1 @ ( bit0 @ one2 ) ) ) @ ( vEBT_VEBT_Tb @ ( suc @ ( suc @ ( divide_divide @ nat @ N4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) @ ( times_times @ int @ ( vEBT_VEBT_Tb @ ( suc @ ( divide_divide @ nat @ N4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ ( divide_divide @ nat @ N4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.Tb.elims
thf(fact_2982_unset__bit__Suc,axiom,
    ! [A: $tType] :
      ( ( bit_se359711467146920520ations @ A )
     => ! [N2: nat,A2: A] :
          ( ( bit_se2638667681897837118et_bit @ A @ ( suc @ N2 ) @ A2 )
          = ( plus_plus @ A @ ( modulo_modulo @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se2638667681897837118et_bit @ A @ N2 @ ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).

% unset_bit_Suc
thf(fact_2983_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,
    ! [X2: nat,Y2: nat] :
      ( ( ( vEBT_V8346862874174094_d_u_p @ X2 )
        = Y2 )
     => ( ( ( X2
            = ( zero_zero @ nat ) )
         => ( Y2
           != ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) ) )
       => ( ( ( X2
              = ( suc @ ( zero_zero @ nat ) ) )
           => ( Y2
             != ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) ) )
         => ~ ! [Va3: nat] :
                ( ( X2
                  = ( suc @ ( suc @ Va3 ) ) )
               => ~ ( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ Va3 ) ) )
                     => ( Y2
                        = ( plus_plus @ nat @ ( one_one @ nat ) @ ( plus_plus @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_V8346862874174094_d_u_p @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( times_times @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( plus_plus @ nat @ ( vEBT_V8346862874174094_d_u_p @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( one_one @ nat ) ) ) ) ) ) )
                    & ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ Va3 ) ) )
                     => ( Y2
                        = ( plus_plus @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit1 @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_V8346862874174094_d_u_p @ ( suc @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( times_times @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( plus_plus @ nat @ ( vEBT_V8346862874174094_d_u_p @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( one_one @ nat ) ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d\<^sub>u\<^sub>p.elims
thf(fact_2984_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,
    ! [X2: nat,Y2: nat] :
      ( ( ( vEBT_V8646137997579335489_i_l_d @ X2 )
        = Y2 )
     => ( ( ( X2
            = ( zero_zero @ nat ) )
         => ( Y2
           != ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) ) )
       => ( ( ( X2
              = ( suc @ ( zero_zero @ nat ) ) )
           => ( Y2
             != ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) ) )
         => ~ ! [Va3: nat] :
                ( ( X2
                  = ( suc @ ( suc @ Va3 ) ) )
               => ~ ( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ Va3 ) ) )
                     => ( Y2
                        = ( plus_plus @ nat @ ( one_one @ nat ) @ ( plus_plus @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit1 @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_V8646137997579335489_i_l_d @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( times_times @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_V8646137997579335489_i_l_d @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) )
                    & ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ Va3 ) ) )
                     => ( Y2
                        = ( plus_plus @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ ( bit1 @ one2 ) ) ) ) @ ( vEBT_V8646137997579335489_i_l_d @ ( suc @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( times_times @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_V8646137997579335489_i_l_d @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d.elims
thf(fact_2985_VEBT__internal_Ovebt__predi_H_Omono,axiom,
    ! [X2: product_prod @ ( product_prod @ vEBT_VEBT @ vEBT_VEBTi ) @ nat] :
      ( comple7038119648293358887notone @ ( ( product_prod @ ( product_prod @ vEBT_VEBT @ vEBT_VEBTi ) @ nat ) > ( heap_Time_Heap @ ( option @ nat ) ) ) @ ( heap_Time_Heap @ ( option @ nat ) ) @ ( partial_fun_ord @ ( heap_Time_Heap @ ( option @ nat ) ) @ ( heap_Time_Heap @ ( option @ nat ) ) @ ( product_prod @ ( product_prod @ vEBT_VEBT @ vEBT_VEBTi ) @ nat ) @ ( heap_Time_Heap_ord @ ( option @ nat ) ) ) @ ( heap_Time_Heap_ord @ ( option @ nat ) )
      @ ^ [Vebt_predi3: ( product_prod @ ( product_prod @ vEBT_VEBT @ vEBT_VEBTi ) @ nat ) > ( heap_Time_Heap @ ( option @ nat ) )] :
          ( product_case_prod @ ( product_prod @ vEBT_VEBT @ vEBT_VEBTi ) @ nat @ ( heap_Time_Heap @ ( option @ nat ) )
          @ ( product_case_prod @ vEBT_VEBT @ vEBT_VEBTi @ ( nat > ( heap_Time_Heap @ ( option @ nat ) ) )
            @ ^ [T3: vEBT_VEBT,Ti3: vEBT_VEBTi,X: nat] :
                ( vEBT_case_VEBTi @ ( heap_Time_Heap @ ( option @ nat ) )
                @ ^ [Info3: option @ ( product_prod @ nat @ nat ),Deg4: nat,TreeArray: array @ vEBT_VEBTi,Summary3: vEBT_VEBTi] :
                    ( heap_Time_bind @ product_unit @ ( option @ nat ) @ ( refine_Imp_assert @ ( vEBT_is_Node @ T3 ) )
                    @ ^ [Uu3: product_unit] :
                        ( product_case_prod @ ( option @ ( product_prod @ nat @ nat ) ) @ ( product_prod @ nat @ ( product_prod @ ( list @ vEBT_VEBT ) @ vEBT_VEBT ) ) @ ( heap_Time_Heap @ ( option @ nat ) )
                        @ ^ [Info4: option @ ( product_prod @ nat @ nat )] :
                            ( product_case_prod @ nat @ ( product_prod @ ( list @ vEBT_VEBT ) @ vEBT_VEBT ) @ ( heap_Time_Heap @ ( option @ nat ) )
                            @ ^ [Deg5: nat] :
                                ( product_case_prod @ ( list @ vEBT_VEBT ) @ vEBT_VEBT @ ( heap_Time_Heap @ ( option @ nat ) )
                                @ ^ [TreeList3: list @ vEBT_VEBT,Summary4: vEBT_VEBT] :
                                    ( heap_Time_bind @ product_unit @ ( option @ nat )
                                    @ ( refine_Imp_assert
                                      @ ( ( Info4 = Info3 )
                                        & ( Deg5 = Deg4 )
                                        & ( vEBT_is_Node @ T3 ) ) )
                                    @ ^ [Uv3: product_unit] :
                                        ( case_option @ ( heap_Time_Heap @ ( option @ nat ) ) @ ( product_prod @ nat @ nat ) @ ( heap_Time_return @ ( option @ nat ) @ ( none @ nat ) )
                                        @ ^ [Mima2: product_prod @ nat @ nat] :
                                            ( if @ ( heap_Time_Heap @ ( option @ nat ) ) @ ( ord_less_eq @ nat @ Deg4 @ ( one_one @ nat ) ) @ ( heap_Time_return @ ( option @ nat ) @ ( none @ nat ) )
                                            @ ( if @ ( heap_Time_Heap @ ( option @ nat ) ) @ ( ord_less @ nat @ ( product_snd @ nat @ nat @ Mima2 ) @ X ) @ ( heap_Time_return @ ( option @ nat ) @ ( some @ nat @ ( product_snd @ nat @ nat @ Mima2 ) ) )
                                              @ ( heap_Time_bind @ nat @ ( option @ nat ) @ ( vEBT_VEBT_lowi @ X @ ( divide_divide @ nat @ Deg4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
                                                @ ^ [L2: nat] :
                                                    ( heap_Time_bind @ nat @ ( option @ nat ) @ ( vEBT_VEBT_highi @ X @ ( divide_divide @ nat @ Deg4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
                                                    @ ^ [H: nat] :
                                                        ( heap_Time_bind @ product_unit @ ( option @ nat )
                                                        @ ( refine_Imp_assert
                                                          @ ( L2
                                                            = ( vEBT_VEBT_low @ X @ ( divide_divide @ nat @ Deg4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
                                                        @ ^ [Uw3: product_unit] :
                                                            ( heap_Time_bind @ product_unit @ ( option @ nat )
                                                            @ ( refine_Imp_assert
                                                              @ ( H
                                                                = ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
                                                            @ ^ [Ux3: product_unit] :
                                                                ( heap_Time_bind @ product_unit @ ( option @ nat ) @ ( refine_Imp_assert @ ( ord_less @ nat @ H @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) ) )
                                                                @ ^ [Uy3: product_unit] :
                                                                    ( heap_Time_bind @ vEBT_VEBTi @ ( option @ nat ) @ ( array_nth @ vEBT_VEBTi @ TreeArray @ H )
                                                                    @ ^ [Aktnode: vEBT_VEBTi] :
                                                                        ( heap_Time_bind @ ( option @ nat ) @ ( option @ nat ) @ ( vEBT_vebt_minti @ Aktnode )
                                                                        @ ^ [Minlow: option @ nat] :
                                                                            ( heap_Time_bind @ product_unit @ ( option @ nat )
                                                                            @ ( refine_Imp_assert
                                                                              @ ( Minlow
                                                                                = ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList3 @ H ) ) ) )
                                                                            @ ^ [Uz3: product_unit] :
                                                                                ( if @ ( heap_Time_Heap @ ( option @ nat ) )
                                                                                @ ( ( Minlow
                                                                                   != ( none @ nat ) )
                                                                                  & ( vEBT_VEBT_greater @ ( some @ nat @ L2 ) @ Minlow ) )
                                                                                @ ( heap_Time_bind @ ( option @ nat ) @ ( option @ nat ) @ ( product_curry @ vEBT_VEBT @ vEBT_VEBTi @ ( nat > ( heap_Time_Heap @ ( option @ nat ) ) ) @ ( product_curry @ ( product_prod @ vEBT_VEBT @ vEBT_VEBTi ) @ nat @ ( heap_Time_Heap @ ( option @ nat ) ) @ Vebt_predi3 ) @ ( nth @ vEBT_VEBT @ TreeList3 @ H ) @ Aktnode @ L2 )
                                                                                  @ ^ [Predy: option @ nat] : ( heap_Time_return @ ( option @ nat ) @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( some @ nat @ H ) ) @ Predy ) ) )
                                                                                @ ( heap_Time_bind @ ( option @ nat ) @ ( option @ nat ) @ ( product_curry @ vEBT_VEBT @ vEBT_VEBTi @ ( nat > ( heap_Time_Heap @ ( option @ nat ) ) ) @ ( product_curry @ ( product_prod @ vEBT_VEBT @ vEBT_VEBTi ) @ nat @ ( heap_Time_Heap @ ( option @ nat ) ) @ Vebt_predi3 ) @ Summary4 @ Summary3 @ H )
                                                                                  @ ^ [Predsum: option @ nat] :
                                                                                      ( heap_Time_bind @ product_unit @ ( option @ nat )
                                                                                      @ ( refine_Imp_assert
                                                                                        @ ( ( Predsum
                                                                                            = ( none @ nat ) )
                                                                                          = ( ( vEBT_vebt_pred @ Summary4 @ H )
                                                                                            = ( none @ nat ) ) ) )
                                                                                      @ ^ [Va4: product_unit] :
                                                                                          ( if @ ( heap_Time_Heap @ ( option @ nat ) )
                                                                                          @ ( Predsum
                                                                                            = ( none @ nat ) )
                                                                                          @ ( if @ ( heap_Time_Heap @ ( option @ nat ) ) @ ( ord_less @ nat @ ( product_fst @ nat @ nat @ Mima2 ) @ X ) @ ( heap_Time_return @ ( option @ nat ) @ ( some @ nat @ ( product_fst @ nat @ nat @ Mima2 ) ) ) @ ( heap_Time_return @ ( option @ nat ) @ ( none @ nat ) ) )
                                                                                          @ ( heap_Time_bind @ vEBT_VEBTi @ ( option @ nat ) @ ( array_nth @ vEBT_VEBTi @ TreeArray @ ( the2 @ nat @ Predsum ) )
                                                                                            @ ^ [Nextnode: vEBT_VEBTi] :
                                                                                                ( heap_Time_bind @ ( option @ nat ) @ ( option @ nat ) @ ( vEBT_vebt_maxti @ Nextnode )
                                                                                                @ ^ [Maxnext: option @ nat] : ( heap_Time_return @ ( option @ nat ) @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ Predsum ) @ Maxnext ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
                                        @ Info3 ) ) ) )
                        @ ( vEBT_case_VEBT @ ( product_prod @ ( option @ ( product_prod @ nat @ nat ) ) @ ( product_prod @ nat @ ( product_prod @ ( list @ vEBT_VEBT ) @ vEBT_VEBT ) ) )
                          @ ^ [Info4: option @ ( product_prod @ nat @ nat ),Deg5: nat,TreeList3: list @ vEBT_VEBT,Summary4: vEBT_VEBT] : ( product_Pair @ ( option @ ( product_prod @ nat @ nat ) ) @ ( product_prod @ nat @ ( product_prod @ ( list @ vEBT_VEBT ) @ vEBT_VEBT ) ) @ Info4 @ ( product_Pair @ nat @ ( product_prod @ ( list @ vEBT_VEBT ) @ vEBT_VEBT ) @ Deg5 @ ( product_Pair @ ( list @ vEBT_VEBT ) @ vEBT_VEBT @ TreeList3 @ Summary4 ) ) )
                          @ ^ [A5: $o,B5: $o] : ( undefined @ ( product_prod @ ( option @ ( product_prod @ nat @ nat ) ) @ ( product_prod @ nat @ ( product_prod @ ( list @ vEBT_VEBT ) @ vEBT_VEBT ) ) ) )
                          @ T3 ) ) )
                @ ^ [A5: $o,B5: $o] :
                    ( if @ ( heap_Time_Heap @ ( option @ nat ) ) @ ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ X ) @ ( if @ ( heap_Time_Heap @ ( option @ nat ) ) @ B5 @ ( heap_Time_return @ ( option @ nat ) @ ( some @ nat @ ( one_one @ nat ) ) ) @ ( if @ ( heap_Time_Heap @ ( option @ nat ) ) @ A5 @ ( heap_Time_return @ ( option @ nat ) @ ( some @ nat @ ( zero_zero @ nat ) ) ) @ ( heap_Time_return @ ( option @ nat ) @ ( none @ nat ) ) ) )
                    @ ( if @ ( heap_Time_Heap @ ( option @ nat ) )
                      @ ( X
                        = ( one_one @ nat ) )
                      @ ( if @ ( heap_Time_Heap @ ( option @ nat ) ) @ A5 @ ( heap_Time_return @ ( option @ nat ) @ ( some @ nat @ ( zero_zero @ nat ) ) ) @ ( heap_Time_return @ ( option @ nat ) @ ( none @ nat ) ) )
                      @ ( heap_Time_return @ ( option @ nat ) @ ( none @ nat ) ) ) )
                @ Ti3 ) )
          @ X2 ) ) ).

% VEBT_internal.vebt_predi'.mono
thf(fact_2986_pow__divides__pow__iff,axiom,
    ! [A: $tType] :
      ( ( semiring_gcd @ A )
     => ! [N2: nat,A2: A,B2: A] :
          ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
         => ( ( dvd_dvd @ A @ ( power_power @ A @ A2 @ N2 ) @ ( power_power @ A @ B2 @ N2 ) )
            = ( dvd_dvd @ A @ A2 @ B2 ) ) ) ) ).

% pow_divides_pow_iff
thf(fact_2987_VEBT__internal_Ovebt__buildupi_H_Oelims,axiom,
    ! [X2: nat,Y2: heap_Time_Heap @ vEBT_VEBTi] :
      ( ( ( vEBT_V739175172307565963ildupi @ X2 )
        = Y2 )
     => ( ( ( X2
            = ( zero_zero @ nat ) )
         => ( Y2
           != ( heap_Time_return @ vEBT_VEBTi @ ( vEBT_Leafi @ $false @ $false ) ) ) )
       => ( ( ( X2
              = ( suc @ ( zero_zero @ nat ) ) )
           => ( Y2
             != ( heap_Time_return @ vEBT_VEBTi @ ( vEBT_Leafi @ $false @ $false ) ) ) )
         => ~ ! [Va3: nat] :
                ( ( X2
                  = ( suc @ ( suc @ Va3 ) ) )
               => ~ ( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ Va3 ) ) )
                     => ( Y2
                        = ( heap_Time_bind @ ( list @ vEBT_VEBTi ) @ vEBT_VEBTi @ ( vEBT_VEBT_replicatei @ vEBT_VEBTi @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_V739175172307565963ildupi @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
                          @ ^ [TreeList3: list @ vEBT_VEBTi] :
                              ( heap_Time_bind @ product_unit @ vEBT_VEBTi
                              @ ( refine_Imp_assert
                                @ ( ( size_size @ ( list @ vEBT_VEBTi ) @ TreeList3 )
                                  = ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
                              @ ^ [Uu3: product_unit] :
                                  ( heap_Time_bind @ ( array @ vEBT_VEBTi ) @ vEBT_VEBTi @ ( array_of_list @ vEBT_VEBTi @ TreeList3 )
                                  @ ^ [Trees: array @ vEBT_VEBTi] :
                                      ( heap_Time_bind @ vEBT_VEBTi @ vEBT_VEBTi @ ( vEBT_V739175172307565963ildupi @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
                                      @ ^ [Summary3: vEBT_VEBTi] : ( heap_Time_return @ vEBT_VEBTi @ ( vEBT_Nodei @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ Va3 ) ) @ Trees @ Summary3 ) ) ) ) ) ) ) )
                    & ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ Va3 ) ) )
                     => ( Y2
                        = ( heap_Time_bind @ ( list @ vEBT_VEBTi ) @ vEBT_VEBTi @ ( vEBT_VEBT_replicatei @ vEBT_VEBTi @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_V739175172307565963ildupi @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
                          @ ^ [TreeList3: list @ vEBT_VEBTi] :
                              ( heap_Time_bind @ product_unit @ vEBT_VEBTi
                              @ ( refine_Imp_assert
                                @ ( ( size_size @ ( list @ vEBT_VEBTi ) @ TreeList3 )
                                  = ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) )
                              @ ^ [Uu3: product_unit] :
                                  ( heap_Time_bind @ ( array @ vEBT_VEBTi ) @ vEBT_VEBTi @ ( array_of_list @ vEBT_VEBTi @ TreeList3 )
                                  @ ^ [Trees: array @ vEBT_VEBTi] :
                                      ( heap_Time_bind @ vEBT_VEBTi @ vEBT_VEBTi @ ( vEBT_V739175172307565963ildupi @ ( suc @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
                                      @ ^ [Summary3: vEBT_VEBTi] : ( heap_Time_return @ vEBT_VEBTi @ ( vEBT_Nodei @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ Va3 ) ) @ Trees @ Summary3 ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.vebt_buildupi'.elims
thf(fact_2988_VEBT__internal_Ovebt__buildupi_H_Osimps_I3_J,axiom,
    ! [Va: nat] :
      ( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ Va ) ) )
       => ( ( vEBT_V739175172307565963ildupi @ ( suc @ ( suc @ Va ) ) )
          = ( heap_Time_bind @ ( list @ vEBT_VEBTi ) @ vEBT_VEBTi @ ( vEBT_VEBT_replicatei @ vEBT_VEBTi @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_V739175172307565963ildupi @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
            @ ^ [TreeList3: list @ vEBT_VEBTi] :
                ( heap_Time_bind @ product_unit @ vEBT_VEBTi
                @ ( refine_Imp_assert
                  @ ( ( size_size @ ( list @ vEBT_VEBTi ) @ TreeList3 )
                    = ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
                @ ^ [Uu3: product_unit] :
                    ( heap_Time_bind @ ( array @ vEBT_VEBTi ) @ vEBT_VEBTi @ ( array_of_list @ vEBT_VEBTi @ TreeList3 )
                    @ ^ [Trees: array @ vEBT_VEBTi] :
                        ( heap_Time_bind @ vEBT_VEBTi @ vEBT_VEBTi @ ( vEBT_V739175172307565963ildupi @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
                        @ ^ [Summary3: vEBT_VEBTi] : ( heap_Time_return @ vEBT_VEBTi @ ( vEBT_Nodei @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ Va ) ) @ Trees @ Summary3 ) ) ) ) ) ) ) )
      & ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ Va ) ) )
       => ( ( vEBT_V739175172307565963ildupi @ ( suc @ ( suc @ Va ) ) )
          = ( heap_Time_bind @ ( list @ vEBT_VEBTi ) @ vEBT_VEBTi @ ( vEBT_VEBT_replicatei @ vEBT_VEBTi @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_V739175172307565963ildupi @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
            @ ^ [TreeList3: list @ vEBT_VEBTi] :
                ( heap_Time_bind @ product_unit @ vEBT_VEBTi
                @ ( refine_Imp_assert
                  @ ( ( size_size @ ( list @ vEBT_VEBTi ) @ TreeList3 )
                    = ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) )
                @ ^ [Uu3: product_unit] :
                    ( heap_Time_bind @ ( array @ vEBT_VEBTi ) @ vEBT_VEBTi @ ( array_of_list @ vEBT_VEBTi @ TreeList3 )
                    @ ^ [Trees: array @ vEBT_VEBTi] :
                        ( heap_Time_bind @ vEBT_VEBTi @ vEBT_VEBTi @ ( vEBT_V739175172307565963ildupi @ ( suc @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
                        @ ^ [Summary3: vEBT_VEBTi] : ( heap_Time_return @ vEBT_VEBTi @ ( vEBT_Nodei @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ Va ) ) @ Trees @ Summary3 ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.vebt_buildupi'.simps(3)
thf(fact_2989_vebt__buildupi_Oelims,axiom,
    ! [X2: nat,Y2: heap_Time_Heap @ vEBT_VEBTi] :
      ( ( ( vEBT_vebt_buildupi @ X2 )
        = Y2 )
     => ( ( ( X2
            = ( zero_zero @ nat ) )
         => ( Y2
           != ( heap_Time_return @ vEBT_VEBTi @ ( vEBT_Leafi @ $false @ $false ) ) ) )
       => ( ( ( X2
              = ( suc @ ( zero_zero @ nat ) ) )
           => ( Y2
             != ( heap_Time_return @ vEBT_VEBTi @ ( vEBT_Leafi @ $false @ $false ) ) ) )
         => ~ ! [Va3: nat] :
                ( ( X2
                  = ( suc @ ( suc @ Va3 ) ) )
               => ~ ( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ Va3 ) ) )
                     => ( Y2
                        = ( heap_Time_bind @ ( list @ vEBT_VEBTi ) @ vEBT_VEBTi @ ( vEBT_VEBT_replicatei @ vEBT_VEBTi @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_buildupi @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
                          @ ^ [TreeList3: list @ vEBT_VEBTi] :
                              ( heap_Time_bind @ ( array @ vEBT_VEBTi ) @ vEBT_VEBTi @ ( array_of_list @ vEBT_VEBTi @ TreeList3 )
                              @ ^ [Trees: array @ vEBT_VEBTi] :
                                  ( heap_Time_bind @ vEBT_VEBTi @ vEBT_VEBTi @ ( vEBT_vebt_buildupi @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
                                  @ ^ [Summary3: vEBT_VEBTi] : ( heap_Time_return @ vEBT_VEBTi @ ( vEBT_Nodei @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ Va3 ) ) @ Trees @ Summary3 ) ) ) ) ) ) )
                    & ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ Va3 ) ) )
                     => ( Y2
                        = ( heap_Time_bind @ ( list @ vEBT_VEBTi ) @ vEBT_VEBTi @ ( vEBT_VEBT_replicatei @ vEBT_VEBTi @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_buildupi @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
                          @ ^ [TreeList3: list @ vEBT_VEBTi] :
                              ( heap_Time_bind @ ( array @ vEBT_VEBTi ) @ vEBT_VEBTi @ ( array_of_list @ vEBT_VEBTi @ TreeList3 )
                              @ ^ [Trees: array @ vEBT_VEBTi] :
                                  ( heap_Time_bind @ vEBT_VEBTi @ vEBT_VEBTi @ ( vEBT_vebt_buildupi @ ( suc @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
                                  @ ^ [Summary3: vEBT_VEBTi] : ( heap_Time_return @ vEBT_VEBTi @ ( vEBT_Nodei @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ Va3 ) ) @ Trees @ Summary3 ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_buildupi.elims
thf(fact_2990_Heap__Time__Monad_Obind__mono,axiom,
    ! [C: $tType,D: $tType,B: $tType,A: $tType,B6: ( A > ( heap_Time_Heap @ B ) ) > ( heap_Time_Heap @ C ),C5: C > ( A > ( heap_Time_Heap @ B ) ) > ( heap_Time_Heap @ D )] :
      ( ( comple7038119648293358887notone @ ( A > ( heap_Time_Heap @ B ) ) @ ( heap_Time_Heap @ C ) @ ( partial_fun_ord @ ( heap_Time_Heap @ B ) @ ( heap_Time_Heap @ B ) @ A @ ( heap_Time_Heap_ord @ B ) ) @ ( heap_Time_Heap_ord @ C ) @ B6 )
     => ( ! [Y4: C] : ( comple7038119648293358887notone @ ( A > ( heap_Time_Heap @ B ) ) @ ( heap_Time_Heap @ D ) @ ( partial_fun_ord @ ( heap_Time_Heap @ B ) @ ( heap_Time_Heap @ B ) @ A @ ( heap_Time_Heap_ord @ B ) ) @ ( heap_Time_Heap_ord @ D ) @ ( C5 @ Y4 ) )
       => ( comple7038119648293358887notone @ ( A > ( heap_Time_Heap @ B ) ) @ ( heap_Time_Heap @ D ) @ ( partial_fun_ord @ ( heap_Time_Heap @ B ) @ ( heap_Time_Heap @ B ) @ A @ ( heap_Time_Heap_ord @ B ) ) @ ( heap_Time_Heap_ord @ D )
          @ ^ [F6: A > ( heap_Time_Heap @ B )] :
              ( heap_Time_bind @ C @ D @ ( B6 @ F6 )
              @ ^ [Y: C] : ( C5 @ Y @ F6 ) ) ) ) ) ).

% Heap_Time_Monad.bind_mono
thf(fact_2991_VEBT__internal_Ovebt__inserti_H_Ofixp__induct,axiom,
    ! [P: ( vEBT_VEBT > vEBT_VEBTi > nat > ( heap_Time_Heap @ vEBT_VEBTi ) ) > $o] :
      ( ( comple1908693960933563346ssible @ ( ( product_prod @ ( product_prod @ vEBT_VEBT @ vEBT_VEBTi ) @ nat ) > ( heap_Time_Heap @ vEBT_VEBTi ) ) @ ( partial_fun_lub @ ( heap_Time_Heap @ vEBT_VEBTi ) @ ( heap_Time_Heap @ vEBT_VEBTi ) @ ( product_prod @ ( product_prod @ vEBT_VEBT @ vEBT_VEBTi ) @ nat ) @ ( heap_Time_Heap_lub @ vEBT_VEBTi ) ) @ ( partial_fun_ord @ ( heap_Time_Heap @ vEBT_VEBTi ) @ ( heap_Time_Heap @ vEBT_VEBTi ) @ ( product_prod @ ( product_prod @ vEBT_VEBT @ vEBT_VEBTi ) @ nat ) @ ( heap_Time_Heap_ord @ vEBT_VEBTi ) )
        @ ^ [Vebt_inserti3: ( product_prod @ ( product_prod @ vEBT_VEBT @ vEBT_VEBTi ) @ nat ) > ( heap_Time_Heap @ vEBT_VEBTi )] : ( P @ ( product_curry @ vEBT_VEBT @ vEBT_VEBTi @ ( nat > ( heap_Time_Heap @ vEBT_VEBTi ) ) @ ( product_curry @ ( product_prod @ vEBT_VEBT @ vEBT_VEBTi ) @ nat @ ( heap_Time_Heap @ vEBT_VEBTi ) @ Vebt_inserti3 ) ) ) )
     => ( ( P
          @ ^ [Vebt_inserti3: vEBT_VEBT,T3: vEBT_VEBTi,Ti3: nat] :
              ( heap_Time_Heap2 @ vEBT_VEBTi
              @ ^ [X: heap_ext @ product_unit] : ( none @ ( product_prod @ vEBT_VEBTi @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) )
       => ( ! [F2: vEBT_VEBT > vEBT_VEBTi > nat > ( heap_Time_Heap @ vEBT_VEBTi )] :
              ( ( P @ F2 )
             => ( P
                @ ^ [X9: vEBT_VEBT,A5: vEBT_VEBTi,B5: nat] :
                    ( vEBT_case_VEBTi @ ( heap_Time_Heap @ vEBT_VEBTi )
                    @ ^ [Info3: option @ ( product_prod @ nat @ nat ),Deg4: nat,TreeArray: array @ vEBT_VEBTi,Summary3: vEBT_VEBTi] :
                        ( case_option @ ( heap_Time_Heap @ vEBT_VEBTi ) @ ( product_prod @ nat @ nat ) @ ( if @ ( heap_Time_Heap @ vEBT_VEBTi ) @ ( ord_less_eq @ nat @ Deg4 @ ( one_one @ nat ) ) @ ( heap_Time_return @ vEBT_VEBTi @ ( vEBT_Nodei @ Info3 @ Deg4 @ TreeArray @ Summary3 ) ) @ ( heap_Time_return @ vEBT_VEBTi @ ( vEBT_Nodei @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ B5 @ B5 ) ) @ Deg4 @ TreeArray @ Summary3 ) ) )
                        @ ^ [Minma: product_prod @ nat @ nat] :
                            ( if @ ( heap_Time_Heap @ vEBT_VEBTi ) @ ( ord_less_eq @ nat @ Deg4 @ ( one_one @ nat ) ) @ ( heap_Time_return @ vEBT_VEBTi @ ( vEBT_Nodei @ Info3 @ Deg4 @ TreeArray @ Summary3 ) )
                            @ ( heap_Time_bind @ product_unit @ vEBT_VEBTi @ ( refine_Imp_assert @ ( vEBT_is_Node @ X9 ) )
                              @ ^ [Uu3: product_unit] :
                                  ( product_case_prod @ ( option @ ( product_prod @ nat @ nat ) ) @ ( product_prod @ nat @ ( product_prod @ ( list @ vEBT_VEBT ) @ vEBT_VEBT ) ) @ ( heap_Time_Heap @ vEBT_VEBTi )
                                  @ ^ [Info4: option @ ( product_prod @ nat @ nat )] :
                                      ( product_case_prod @ nat @ ( product_prod @ ( list @ vEBT_VEBT ) @ vEBT_VEBT ) @ ( heap_Time_Heap @ vEBT_VEBTi )
                                      @ ^ [Deg5: nat] :
                                          ( product_case_prod @ ( list @ vEBT_VEBT ) @ vEBT_VEBT @ ( heap_Time_Heap @ vEBT_VEBTi )
                                          @ ^ [TreeList3: list @ vEBT_VEBT,Summary4: vEBT_VEBT] :
                                              ( heap_Time_bind @ product_unit @ vEBT_VEBTi
                                              @ ( refine_Imp_assert
                                                @ ( ( Info3 = Info4 )
                                                  & ( Deg4 = Deg5 ) ) )
                                              @ ^ [Uv3: product_unit] :
                                                  ( product_case_prod @ nat @ nat @ ( heap_Time_Heap @ vEBT_VEBTi )
                                                  @ ^ [Mi4: nat,Ma4: nat] :
                                                      ( heap_Time_bind @ nat @ vEBT_VEBTi @ ( heap_Time_return @ nat @ ( product_fst @ nat @ nat @ Minma ) )
                                                      @ ^ [Mi3: nat] :
                                                          ( heap_Time_bind @ nat @ vEBT_VEBTi @ ( heap_Time_return @ nat @ ( product_snd @ nat @ nat @ Minma ) )
                                                          @ ^ [Ma3: nat] :
                                                              ( heap_Time_bind @ nat @ vEBT_VEBTi @ ( if @ ( heap_Time_Heap @ nat ) @ ( ord_less @ nat @ B5 @ Mi3 ) @ ( heap_Time_return @ nat @ Mi3 ) @ ( heap_Time_return @ nat @ B5 ) )
                                                              @ ^ [Xn2: nat] :
                                                                  ( heap_Time_bind @ nat @ vEBT_VEBTi @ ( if @ ( heap_Time_Heap @ nat ) @ ( ord_less @ nat @ B5 @ Mi3 ) @ ( heap_Time_return @ nat @ B5 ) @ ( heap_Time_return @ nat @ Mi3 ) )
                                                                  @ ^ [Minn: nat] :
                                                                      ( heap_Time_bind @ nat @ vEBT_VEBTi @ ( vEBT_VEBT_lowi @ Xn2 @ ( divide_divide @ nat @ Deg4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
                                                                      @ ^ [L2: nat] :
                                                                          ( heap_Time_bind @ product_unit @ vEBT_VEBTi
                                                                          @ ( refine_Imp_assert
                                                                            @ ( L2
                                                                              = ( vEBT_VEBT_low @ ( if @ nat @ ( ord_less @ nat @ B5 @ Mi4 ) @ Mi4 @ B5 ) @ ( divide_divide @ nat @ Deg5 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
                                                                          @ ^ [Uw3: product_unit] :
                                                                              ( heap_Time_bind @ nat @ vEBT_VEBTi @ ( vEBT_VEBT_highi @ Xn2 @ ( divide_divide @ nat @ Deg4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
                                                                              @ ^ [H: nat] :
                                                                                  ( heap_Time_bind @ nat @ vEBT_VEBTi @ ( array_len @ vEBT_VEBTi @ TreeArray )
                                                                                  @ ^ [Len: nat] :
                                                                                      ( if @ ( heap_Time_Heap @ vEBT_VEBTi )
                                                                                      @ ( ( ord_less @ nat @ H @ Len )
                                                                                        & ~ ( ( B5 = Mi3 )
                                                                                            | ( B5 = Ma3 ) ) )
                                                                                      @ ( heap_Time_bind @ product_unit @ vEBT_VEBTi
                                                                                        @ ( refine_Imp_assert
                                                                                          @ ( H
                                                                                            = ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ B5 @ Mi4 ) @ Mi4 @ B5 ) @ ( divide_divide @ nat @ Deg5 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
                                                                                        @ ^ [Ux3: product_unit] :
                                                                                            ( heap_Time_bind @ product_unit @ vEBT_VEBTi @ ( refine_Imp_assert @ ( ord_less @ nat @ H @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) ) )
                                                                                            @ ^ [Uy3: product_unit] :
                                                                                                ( heap_Time_bind @ vEBT_VEBTi @ vEBT_VEBTi @ ( array_nth @ vEBT_VEBTi @ TreeArray @ H )
                                                                                                @ ^ [Node: vEBT_VEBTi] :
                                                                                                    ( heap_Time_bind @ $o @ vEBT_VEBTi @ ( vEBT_VEBT_minNulli @ Node )
                                                                                                    @ ^ [Empt: $o] :
                                                                                                        ( heap_Time_bind @ product_unit @ vEBT_VEBTi
                                                                                                        @ ( refine_Imp_assert
                                                                                                          @ ( Empt
                                                                                                            = ( vEBT_VEBT_minNull @ ( nth @ vEBT_VEBT @ TreeList3 @ H ) ) ) )
                                                                                                        @ ^ [Uz3: product_unit] :
                                                                                                            ( heap_Time_bind @ vEBT_VEBTi @ vEBT_VEBTi @ ( F2 @ ( nth @ vEBT_VEBT @ TreeList3 @ H ) @ Node @ L2 )
                                                                                                            @ ^ [Newnode2: vEBT_VEBTi] :
                                                                                                                ( heap_Time_bind @ ( array @ vEBT_VEBTi ) @ vEBT_VEBTi @ ( array_upd @ vEBT_VEBTi @ H @ Newnode2 @ TreeArray )
                                                                                                                @ ^ [Newarray: array @ vEBT_VEBTi] :
                                                                                                                    ( heap_Time_bind @ vEBT_VEBTi @ vEBT_VEBTi @ ( if @ ( heap_Time_Heap @ vEBT_VEBTi ) @ Empt @ ( F2 @ Summary4 @ Summary3 @ H ) @ ( heap_Time_return @ vEBT_VEBTi @ Summary3 ) )
                                                                                                                    @ ^ [Newsummary: vEBT_VEBTi] :
                                                                                                                        ( heap_Time_bind @ nat @ vEBT_VEBTi @ ( if @ ( heap_Time_Heap @ nat ) @ ( ord_less @ nat @ Ma3 @ Xn2 ) @ ( heap_Time_return @ nat @ Xn2 ) @ ( heap_Time_return @ nat @ Ma3 ) )
                                                                                                                        @ ^ [Man: nat] : ( heap_Time_return @ vEBT_VEBTi @ ( vEBT_Nodei @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Minn @ Man ) ) @ Deg4 @ Newarray @ Newsummary ) ) ) ) ) ) ) ) ) ) )
                                                                                      @ ( heap_Time_return @ vEBT_VEBTi @ ( vEBT_Nodei @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi3 @ Ma3 ) ) @ Deg4 @ TreeArray @ Summary3 ) ) ) ) ) ) ) ) ) ) )
                                                  @ ( the2 @ ( product_prod @ nat @ nat ) @ Info4 ) ) ) ) )
                                  @ ( vEBT_case_VEBT @ ( product_prod @ ( option @ ( product_prod @ nat @ nat ) ) @ ( product_prod @ nat @ ( product_prod @ ( list @ vEBT_VEBT ) @ vEBT_VEBT ) ) )
                                    @ ^ [Info4: option @ ( product_prod @ nat @ nat ),Deg5: nat,TreeList3: list @ vEBT_VEBT,Summary4: vEBT_VEBT] : ( product_Pair @ ( option @ ( product_prod @ nat @ nat ) ) @ ( product_prod @ nat @ ( product_prod @ ( list @ vEBT_VEBT ) @ vEBT_VEBT ) ) @ Info4 @ ( product_Pair @ nat @ ( product_prod @ ( list @ vEBT_VEBT ) @ vEBT_VEBT ) @ Deg5 @ ( product_Pair @ ( list @ vEBT_VEBT ) @ vEBT_VEBT @ TreeList3 @ Summary4 ) ) )
                                    @ ^ [C4: $o,D3: $o] : ( undefined @ ( product_prod @ ( option @ ( product_prod @ nat @ nat ) ) @ ( product_prod @ nat @ ( product_prod @ ( list @ vEBT_VEBT ) @ vEBT_VEBT ) ) ) )
                                    @ X9 ) ) ) )
                        @ Info3 )
                    @ ^ [C4: $o,D3: $o] :
                        ( if @ ( heap_Time_Heap @ vEBT_VEBTi )
                        @ ( B5
                          = ( zero_zero @ nat ) )
                        @ ( heap_Time_return @ vEBT_VEBTi @ ( vEBT_Leafi @ $true @ D3 ) )
                        @ ( if @ ( heap_Time_Heap @ vEBT_VEBTi )
                          @ ( B5
                            = ( one_one @ nat ) )
                          @ ( heap_Time_return @ vEBT_VEBTi @ ( vEBT_Leafi @ C4 @ $true ) )
                          @ ( heap_Time_return @ vEBT_VEBTi @ ( vEBT_Leafi @ C4 @ D3 ) ) ) )
                    @ A5 ) ) )
         => ( P @ vEBT_V3964819847710782039nserti ) ) ) ) ).

% VEBT_internal.vebt_inserti'.fixp_induct
thf(fact_2992_curryI,axiom,
    ! [A: $tType,B: $tType,F: ( product_prod @ A @ B ) > $o,A2: A,B2: B] :
      ( ( F @ ( product_Pair @ A @ B @ A2 @ B2 ) )
     => ( product_curry @ A @ B @ $o @ F @ A2 @ B2 ) ) ).

% curryI
thf(fact_2993_Heap__Time__Monad_Obind__bind,axiom,
    ! [C: $tType,B: $tType,A: $tType,F: heap_Time_Heap @ A,G: A > ( heap_Time_Heap @ C ),K: C > ( heap_Time_Heap @ B )] :
      ( ( heap_Time_bind @ C @ B @ ( heap_Time_bind @ A @ C @ F @ G ) @ K )
      = ( heap_Time_bind @ A @ B @ F
        @ ^ [X: A] : ( heap_Time_bind @ C @ B @ ( G @ X ) @ K ) ) ) ).

% Heap_Time_Monad.bind_bind
thf(fact_2994_Heap__lub__empty,axiom,
    ! [A: $tType] :
      ( ( heap_Time_Heap_lub @ A @ ( bot_bot @ ( set @ ( heap_Time_Heap @ A ) ) ) )
      = ( heap_Time_Heap2 @ A
        @ ^ [X: heap_ext @ product_unit] : ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) ) ).

% Heap_lub_empty
thf(fact_2995_timeFrame_Ocases,axiom,
    ! [A: $tType,X2: product_prod @ nat @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) )] :
      ( ! [N4: nat,R3: A,H3: heap_ext @ product_unit,N7: nat] :
          ( X2
         != ( product_Pair @ nat @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) @ N4 @ ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R3 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H3 @ N7 ) ) ) ) )
     => ~ ! [N4: nat] :
            ( X2
           != ( product_Pair @ nat @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) @ N4 @ ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) ) ) ).

% timeFrame.cases
thf(fact_2996_zdvd__zdiffD,axiom,
    ! [K: int,M: int,N2: int] :
      ( ( dvd_dvd @ int @ K @ ( minus_minus @ int @ M @ N2 ) )
     => ( ( dvd_dvd @ int @ K @ N2 )
       => ( dvd_dvd @ int @ K @ M ) ) ) ).

% zdvd_zdiffD
thf(fact_2997_Heap_Osize_I2_J,axiom,
    ! [A: $tType,X2: ( heap_ext @ product_unit ) > ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) )] :
      ( ( size_size @ ( heap_Time_Heap @ A ) @ ( heap_Time_Heap2 @ A @ X2 ) )
      = ( suc @ ( zero_zero @ nat ) ) ) ).

% Heap.size(2)
thf(fact_2998_curryE,axiom,
    ! [A: $tType,B: $tType,F: ( product_prod @ A @ B ) > $o,A2: A,B2: B] :
      ( ( product_curry @ A @ B @ $o @ F @ A2 @ B2 )
     => ( F @ ( product_Pair @ A @ B @ A2 @ B2 ) ) ) ).

% curryE
thf(fact_2999_curryD,axiom,
    ! [A: $tType,B: $tType,F: ( product_prod @ A @ B ) > $o,A2: A,B2: B] :
      ( ( product_curry @ A @ B @ $o @ F @ A2 @ B2 )
     => ( F @ ( product_Pair @ A @ B @ A2 @ B2 ) ) ) ).

% curryD
thf(fact_3000_TBOUND__empty,axiom,
    ! [A: $tType,T2: nat] :
      ( time_TBOUND @ A
      @ ( heap_Time_Heap2 @ A
        @ ^ [X: heap_ext @ product_unit] : ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) )
      @ T2 ) ).

% TBOUND_empty
thf(fact_3001_refines__empty,axiom,
    ! [A: $tType,M: heap_Time_Heap @ A] :
      ( refine_Imp_refines @ A @ M
      @ ( heap_Time_Heap2 @ A
        @ ^ [X: heap_ext @ product_unit] : ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) ) ).

% refines_empty
thf(fact_3002_zdvd__antisym__nonneg,axiom,
    ! [M: int,N2: int] :
      ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ M )
     => ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ N2 )
       => ( ( dvd_dvd @ int @ M @ N2 )
         => ( ( dvd_dvd @ int @ N2 @ M )
           => ( M = N2 ) ) ) ) ) ).

% zdvd_antisym_nonneg
thf(fact_3003_zdvd__mono,axiom,
    ! [K: int,M: int,T2: int] :
      ( ( K
       != ( zero_zero @ int ) )
     => ( ( dvd_dvd @ int @ M @ T2 )
        = ( dvd_dvd @ int @ ( times_times @ int @ K @ M ) @ ( times_times @ int @ K @ T2 ) ) ) ) ).

% zdvd_mono
thf(fact_3004_zdvd__mult__cancel,axiom,
    ! [K: int,M: int,N2: int] :
      ( ( dvd_dvd @ int @ ( times_times @ int @ K @ M ) @ ( times_times @ int @ K @ N2 ) )
     => ( ( K
         != ( zero_zero @ int ) )
       => ( dvd_dvd @ int @ M @ N2 ) ) ) ).

% zdvd_mult_cancel
thf(fact_3005_zdvd__reduce,axiom,
    ! [K: int,N2: int,M: int] :
      ( ( dvd_dvd @ int @ K @ ( plus_plus @ int @ N2 @ ( times_times @ int @ K @ M ) ) )
      = ( dvd_dvd @ int @ K @ N2 ) ) ).

% zdvd_reduce
thf(fact_3006_zdvd__period,axiom,
    ! [A2: int,D2: int,X2: int,T2: int,C2: int] :
      ( ( dvd_dvd @ int @ A2 @ D2 )
     => ( ( dvd_dvd @ int @ A2 @ ( plus_plus @ int @ X2 @ T2 ) )
        = ( dvd_dvd @ int @ A2 @ ( plus_plus @ int @ ( plus_plus @ int @ X2 @ ( times_times @ int @ C2 @ D2 ) ) @ T2 ) ) ) ) ).

% zdvd_period
thf(fact_3007_zdvd__imp__le,axiom,
    ! [Z: int,N2: int] :
      ( ( dvd_dvd @ int @ Z @ N2 )
     => ( ( ord_less @ int @ ( zero_zero @ int ) @ N2 )
       => ( ord_less_eq @ int @ Z @ N2 ) ) ) ).

% zdvd_imp_le
thf(fact_3008_Heap_Osize__neq,axiom,
    ! [A: $tType,X2: heap_Time_Heap @ A] :
      ( ( size_size @ ( heap_Time_Heap @ A ) @ X2 )
     != ( zero_zero @ nat ) ) ).

% Heap.size_neq
thf(fact_3009_real__of__int__div,axiom,
    ! [D2: int,N2: int] :
      ( ( dvd_dvd @ int @ D2 @ N2 )
     => ( ( ring_1_of_int @ real @ ( divide_divide @ int @ N2 @ D2 ) )
        = ( divide_divide @ real @ ( ring_1_of_int @ real @ N2 ) @ ( ring_1_of_int @ real @ D2 ) ) ) ) ).

% real_of_int_div
thf(fact_3010_fold__if__return,axiom,
    ! [A: $tType,B2: $o,C2: A,D2: A] :
      ( ( B2
       => ( ( heap_Time_return @ A @ C2 )
          = ( heap_Time_return @ A @ ( if @ A @ B2 @ C2 @ D2 ) ) ) )
      & ( ~ B2
       => ( ( heap_Time_return @ A @ D2 )
          = ( heap_Time_return @ A @ ( if @ A @ B2 @ C2 @ D2 ) ) ) ) ) ).

% fold_if_return
thf(fact_3011_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_3012_int__div__sub__1,axiom,
    ! [M: int,N2: int] :
      ( ( ord_less_eq @ int @ ( one_one @ int ) @ M )
     => ( ( ( dvd_dvd @ int @ M @ N2 )
         => ( ( divide_divide @ int @ ( minus_minus @ int @ N2 @ ( one_one @ int ) ) @ M )
            = ( minus_minus @ int @ ( divide_divide @ int @ N2 @ M ) @ ( one_one @ int ) ) ) )
        & ( ~ ( dvd_dvd @ int @ M @ N2 )
         => ( ( divide_divide @ int @ ( minus_minus @ int @ N2 @ ( one_one @ int ) ) @ M )
            = ( divide_divide @ int @ N2 @ M ) ) ) ) ) ).

% int_div_sub_1
thf(fact_3013_aset_I10_J,axiom,
    ! [D2: int,D6: int,A3: set @ int,T2: int] :
      ( ( dvd_dvd @ int @ D2 @ D6 )
     => ! [X6: int] :
          ( ! [Xa3: int] :
              ( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D6 ) )
             => ! [Xb3: int] :
                  ( ( member @ int @ Xb3 @ A3 )
                 => ( X6
                   != ( minus_minus @ int @ Xb3 @ Xa3 ) ) ) )
         => ( ~ ( dvd_dvd @ int @ D2 @ ( plus_plus @ int @ X6 @ T2 ) )
           => ~ ( dvd_dvd @ int @ D2 @ ( plus_plus @ int @ ( plus_plus @ int @ X6 @ D6 ) @ T2 ) ) ) ) ) ).

% aset(10)
thf(fact_3014_aset_I9_J,axiom,
    ! [D2: int,D6: int,A3: set @ int,T2: int] :
      ( ( dvd_dvd @ int @ D2 @ D6 )
     => ! [X6: int] :
          ( ! [Xa3: int] :
              ( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D6 ) )
             => ! [Xb3: int] :
                  ( ( member @ int @ Xb3 @ A3 )
                 => ( X6
                   != ( minus_minus @ int @ Xb3 @ Xa3 ) ) ) )
         => ( ( dvd_dvd @ int @ D2 @ ( plus_plus @ int @ X6 @ T2 ) )
           => ( dvd_dvd @ int @ D2 @ ( plus_plus @ int @ ( plus_plus @ int @ X6 @ D6 ) @ T2 ) ) ) ) ) ).

% aset(9)
thf(fact_3015_bset_I10_J,axiom,
    ! [D2: int,D6: int,B6: set @ int,T2: int] :
      ( ( dvd_dvd @ int @ D2 @ D6 )
     => ! [X6: int] :
          ( ! [Xa3: int] :
              ( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D6 ) )
             => ! [Xb3: int] :
                  ( ( member @ int @ Xb3 @ B6 )
                 => ( X6
                   != ( plus_plus @ int @ Xb3 @ Xa3 ) ) ) )
         => ( ~ ( dvd_dvd @ int @ D2 @ ( plus_plus @ int @ X6 @ T2 ) )
           => ~ ( dvd_dvd @ int @ D2 @ ( plus_plus @ int @ ( minus_minus @ int @ X6 @ D6 ) @ T2 ) ) ) ) ) ).

% bset(10)
thf(fact_3016_bset_I9_J,axiom,
    ! [D2: int,D6: int,B6: set @ int,T2: int] :
      ( ( dvd_dvd @ int @ D2 @ D6 )
     => ! [X6: int] :
          ( ! [Xa3: int] :
              ( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D6 ) )
             => ! [Xb3: int] :
                  ( ( member @ int @ Xb3 @ B6 )
                 => ( X6
                   != ( plus_plus @ int @ Xb3 @ Xa3 ) ) ) )
         => ( ( dvd_dvd @ int @ D2 @ ( plus_plus @ int @ X6 @ T2 ) )
           => ( dvd_dvd @ int @ D2 @ ( plus_plus @ int @ ( minus_minus @ int @ X6 @ D6 ) @ T2 ) ) ) ) ) ).

% bset(9)
thf(fact_3017_distrib__if__bind,axiom,
    ! [A: $tType,B: $tType,B2: $o,C2: heap_Time_Heap @ B,D2: heap_Time_Heap @ B,F: B > ( heap_Time_Heap @ A )] :
      ( ( B2
       => ( ( heap_Time_bind @ B @ A @ ( if @ ( heap_Time_Heap @ B ) @ B2 @ C2 @ D2 ) @ F )
          = ( heap_Time_bind @ B @ A @ C2 @ F ) ) )
      & ( ~ B2
       => ( ( heap_Time_bind @ B @ A @ ( if @ ( heap_Time_Heap @ B ) @ B2 @ C2 @ D2 ) @ F )
          = ( heap_Time_bind @ B @ A @ D2 @ F ) ) ) ) ).

% distrib_if_bind
thf(fact_3018_even__diff__iff,axiom,
    ! [K: int,L: int] :
      ( ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( minus_minus @ int @ K @ L ) )
      = ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( plus_plus @ int @ K @ L ) ) ) ).

% even_diff_iff
thf(fact_3019_effect__LetI,axiom,
    ! [B: $tType,A: $tType,X2: A,T2: A,F: A > ( heap_Time_Heap @ B ),H2: heap_ext @ product_unit,H4: heap_ext @ product_unit,R4: B,N2: nat] :
      ( ( X2 = T2 )
     => ( ( heap_Time_effect @ B @ ( F @ X2 ) @ H2 @ H4 @ R4 @ N2 )
       => ( heap_Time_effect @ B @ ( F @ T2 ) @ H2 @ H4 @ R4 @ N2 ) ) ) ).

% effect_LetI
thf(fact_3020_time__array__of__list,axiom,
    ! [A: $tType] :
      ( ( heap @ A )
     => ! [Xs2: list @ A,H2: heap_ext @ product_unit] :
          ( ( time_time @ ( array @ A ) @ ( array_of_list @ A @ Xs2 ) @ H2 )
          = ( plus_plus @ nat @ ( size_size @ ( list @ A ) @ Xs2 ) @ ( one_one @ nat ) ) ) ) ).

% time_array_of_list
thf(fact_3021_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 @ ( semiring_1_of_nat @ int @ M ) ) )
        & ( ~ ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z )
         => ( M
            = ( zero_zero @ nat ) ) ) ) ) ).

% nat_dvd_iff
thf(fact_3022_vebt__memberi_Omono,axiom,
    ! [X2: product_prod @ vEBT_VEBTi @ nat] :
      ( comple7038119648293358887notone @ ( ( product_prod @ vEBT_VEBTi @ nat ) > ( heap_Time_Heap @ $o ) ) @ ( heap_Time_Heap @ $o ) @ ( partial_fun_ord @ ( heap_Time_Heap @ $o ) @ ( heap_Time_Heap @ $o ) @ ( product_prod @ vEBT_VEBTi @ nat ) @ ( heap_Time_Heap_ord @ $o ) ) @ ( heap_Time_Heap_ord @ $o )
      @ ^ [Vebt_memberi3: ( product_prod @ vEBT_VEBTi @ nat ) > ( heap_Time_Heap @ $o )] :
          ( product_case_prod @ vEBT_VEBTi @ nat @ ( heap_Time_Heap @ $o )
          @ ^ [T3: vEBT_VEBTi,X: nat] :
              ( vEBT_case_VEBTi @ ( heap_Time_Heap @ $o )
              @ ^ [Info3: option @ ( product_prod @ nat @ nat ),Deg4: nat,TreeList3: array @ vEBT_VEBTi,Summary3: vEBT_VEBTi] :
                  ( case_option @ ( heap_Time_Heap @ $o ) @ ( product_prod @ nat @ nat ) @ ( heap_Time_return @ $o @ $false )
                  @ ( product_case_prod @ nat @ nat @ ( heap_Time_Heap @ $o )
                    @ ^ [Mi3: nat,Ma3: nat] :
                        ( if @ ( heap_Time_Heap @ $o ) @ ( ord_less_eq @ nat @ Deg4 @ ( one_one @ nat ) ) @ ( heap_Time_return @ $o @ $false )
                        @ ( if @ ( heap_Time_Heap @ $o ) @ ( X = Mi3 ) @ ( heap_Time_return @ $o @ $true )
                          @ ( if @ ( heap_Time_Heap @ $o ) @ ( X = Ma3 ) @ ( heap_Time_return @ $o @ $true )
                            @ ( if @ ( heap_Time_Heap @ $o ) @ ( ord_less @ nat @ X @ Mi3 ) @ ( heap_Time_return @ $o @ $false )
                              @ ( if @ ( heap_Time_Heap @ $o ) @ ( ord_less @ nat @ Ma3 @ X ) @ ( heap_Time_return @ $o @ $false )
                                @ ( heap_Time_bind @ nat @ $o @ ( vEBT_VEBT_highi @ X @ ( divide_divide @ nat @ Deg4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
                                  @ ^ [H: nat] :
                                      ( heap_Time_bind @ nat @ $o @ ( vEBT_VEBT_lowi @ X @ ( divide_divide @ nat @ Deg4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
                                      @ ^ [L2: nat] :
                                          ( heap_Time_bind @ nat @ $o @ ( array_len @ vEBT_VEBTi @ TreeList3 )
                                          @ ^ [Len: nat] :
                                              ( if @ ( heap_Time_Heap @ $o ) @ ( ord_less @ nat @ H @ Len )
                                              @ ( heap_Time_bind @ vEBT_VEBTi @ $o @ ( array_nth @ vEBT_VEBTi @ TreeList3 @ H )
                                                @ ^ [Th: vEBT_VEBTi] : ( product_curry @ vEBT_VEBTi @ nat @ ( heap_Time_Heap @ $o ) @ Vebt_memberi3 @ Th @ L2 ) )
                                              @ ( heap_Time_return @ $o @ $false ) ) ) ) ) ) ) ) ) ) )
                  @ Info3 )
              @ ^ [A5: $o,B5: $o] :
                  ( heap_Time_return @ $o
                  @ ( ( ( X
                        = ( zero_zero @ nat ) )
                     => A5 )
                    & ( ( X
                       != ( zero_zero @ nat ) )
                     => ( ( ( X
                            = ( one_one @ nat ) )
                         => B5 )
                        & ( X
                          = ( one_one @ nat ) ) ) ) ) )
              @ T3 )
          @ X2 ) ) ).

% vebt_memberi.mono
thf(fact_3023_TBOUND__of__list,axiom,
    ! [A: $tType] :
      ( ( heap @ A )
     => ! [Xs2: list @ A] : ( time_TBOUND @ ( array @ A ) @ ( array_of_list @ A @ Xs2 ) @ ( suc @ ( size_size @ ( list @ A ) @ Xs2 ) ) ) ) ).

% TBOUND_of_list
thf(fact_3024_dvd__productE,axiom,
    ! [A: $tType] :
      ( ( semiring_gcd @ A )
     => ! [P4: A,A2: A,B2: A] :
          ( ( dvd_dvd @ A @ P4 @ ( times_times @ A @ A2 @ B2 ) )
         => ~ ! [X3: A,Y4: A] :
                ( ( P4
                  = ( times_times @ A @ X3 @ Y4 ) )
               => ( ( dvd_dvd @ A @ X3 @ A2 )
                 => ~ ( dvd_dvd @ A @ Y4 @ B2 ) ) ) ) ) ).

% dvd_productE
thf(fact_3025_division__decomp,axiom,
    ! [A: $tType] :
      ( ( semiring_gcd @ A )
     => ! [A2: A,B2: A,C2: A] :
          ( ( dvd_dvd @ A @ A2 @ ( times_times @ A @ B2 @ C2 ) )
         => ? [B9: A,C7: A] :
              ( ( A2
                = ( times_times @ A @ B9 @ C7 ) )
              & ( dvd_dvd @ A @ B9 @ B2 )
              & ( dvd_dvd @ A @ C7 @ C2 ) ) ) ) ).

% division_decomp
thf(fact_3026_Euclid__induct,axiom,
    ! [P: nat > nat > $o,A2: nat,B2: nat] :
      ( ! [A4: nat,B3: nat] :
          ( ( P @ A4 @ B3 )
          = ( P @ B3 @ A4 ) )
     => ( ! [A4: nat] : ( P @ A4 @ ( zero_zero @ nat ) )
       => ( ! [A4: nat,B3: nat] :
              ( ( P @ A4 @ B3 )
             => ( P @ A4 @ ( plus_plus @ nat @ A4 @ B3 ) ) )
         => ( P @ A2 @ B2 ) ) ) ) ).

% Euclid_induct
thf(fact_3027_even__nat__iff,axiom,
    ! [K: int] :
      ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
     => ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( nat2 @ K ) )
        = ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K ) ) ) ).

% even_nat_iff
thf(fact_3028_emep1,axiom,
    ! [N2: int,D2: int] :
      ( ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N2 )
     => ( ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ D2 )
       => ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ D2 )
         => ( ( modulo_modulo @ int @ ( plus_plus @ int @ N2 @ ( one_one @ int ) ) @ D2 )
            = ( plus_plus @ int @ ( modulo_modulo @ int @ N2 @ D2 ) @ ( one_one @ int ) ) ) ) ) ) ).

% emep1
thf(fact_3029_eme1p,axiom,
    ! [N2: int,D2: int] :
      ( ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N2 )
     => ( ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ D2 )
       => ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ D2 )
         => ( ( modulo_modulo @ int @ ( plus_plus @ int @ ( one_one @ int ) @ N2 ) @ D2 )
            = ( plus_plus @ int @ ( one_one @ int ) @ ( modulo_modulo @ int @ N2 @ D2 ) ) ) ) ) ) ).

% eme1p
thf(fact_3030_of__list__rule,axiom,
    ! [A: $tType] :
      ( ( heap @ A )
     => ! [Xs2: list @ A] :
          ( hoare_hoare_triple @ ( array @ A ) @ ( one_one @ assn ) @ ( array_of_list @ A @ Xs2 )
          @ ^ [R: array @ A] : ( snga_assn @ A @ R @ Xs2 ) ) ) ).

% of_list_rule
thf(fact_3031_heap_Oconst__mono,axiom,
    ! [A: $tType,B: $tType,Ord: B > B > $o,C2: heap_Time_Heap @ A] :
      ( comple7038119648293358887notone @ B @ ( heap_Time_Heap @ A ) @ Ord @ ( heap_Time_Heap_ord @ A )
      @ ^ [F6: B] : C2 ) ).

% heap.const_mono
thf(fact_3032_dvd__pos__nat,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
     => ( ( dvd_dvd @ nat @ M @ N2 )
       => ( ord_less @ nat @ ( zero_zero @ nat ) @ M ) ) ) ).

% dvd_pos_nat
thf(fact_3033_gcd__nat__induct,axiom,
    ! [P: nat > nat > $o,M: nat,N2: nat] :
      ( ! [M2: nat] : ( P @ M2 @ ( zero_zero @ nat ) )
     => ( ! [M2: nat,N4: nat] :
            ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N4 )
           => ( ( P @ N4 @ ( modulo_modulo @ nat @ M2 @ N4 ) )
             => ( P @ M2 @ N4 ) ) )
       => ( P @ M @ N2 ) ) ) ).

% gcd_nat_induct
thf(fact_3034_VEBT__internal_Ovebt__memberi_H_Omono,axiom,
    ! [X2: product_prod @ ( product_prod @ vEBT_VEBT @ vEBT_VEBTi ) @ nat] :
      ( comple7038119648293358887notone @ ( ( product_prod @ ( product_prod @ vEBT_VEBT @ vEBT_VEBTi ) @ nat ) > ( heap_Time_Heap @ $o ) ) @ ( heap_Time_Heap @ $o ) @ ( partial_fun_ord @ ( heap_Time_Heap @ $o ) @ ( heap_Time_Heap @ $o ) @ ( product_prod @ ( product_prod @ vEBT_VEBT @ vEBT_VEBTi ) @ nat ) @ ( heap_Time_Heap_ord @ $o ) ) @ ( heap_Time_Heap_ord @ $o )
      @ ^ [Vebt_memberi4: ( product_prod @ ( product_prod @ vEBT_VEBT @ vEBT_VEBTi ) @ nat ) > ( heap_Time_Heap @ $o )] :
          ( product_case_prod @ ( product_prod @ vEBT_VEBT @ vEBT_VEBTi ) @ nat @ ( heap_Time_Heap @ $o )
          @ ( product_case_prod @ vEBT_VEBT @ vEBT_VEBTi @ ( nat > ( heap_Time_Heap @ $o ) )
            @ ^ [T3: vEBT_VEBT,Ti3: vEBT_VEBTi,X: nat] :
                ( vEBT_case_VEBTi @ ( heap_Time_Heap @ $o )
                @ ^ [Info3: option @ ( product_prod @ nat @ nat ),Deg4: nat,TreeArray: array @ vEBT_VEBTi,Summary3: vEBT_VEBTi] :
                    ( heap_Time_bind @ product_unit @ $o @ ( refine_Imp_assert @ ( vEBT_is_Node @ T3 ) )
                    @ ^ [Uu3: product_unit] :
                        ( case_option @ ( heap_Time_Heap @ $o ) @ ( product_prod @ nat @ nat ) @ ( heap_Time_return @ $o @ $false )
                        @ ( product_case_prod @ nat @ nat @ ( heap_Time_Heap @ $o )
                          @ ^ [Mi3: nat,Ma3: nat] :
                              ( if @ ( heap_Time_Heap @ $o ) @ ( ord_less_eq @ nat @ Deg4 @ ( one_one @ nat ) ) @ ( heap_Time_return @ $o @ $false )
                              @ ( if @ ( heap_Time_Heap @ $o ) @ ( X = Mi3 ) @ ( heap_Time_return @ $o @ $true )
                                @ ( if @ ( heap_Time_Heap @ $o ) @ ( X = Ma3 ) @ ( heap_Time_return @ $o @ $true )
                                  @ ( if @ ( heap_Time_Heap @ $o ) @ ( ord_less @ nat @ X @ Mi3 ) @ ( heap_Time_return @ $o @ $false )
                                    @ ( if @ ( heap_Time_Heap @ $o ) @ ( ord_less @ nat @ Ma3 @ X ) @ ( heap_Time_return @ $o @ $false )
                                      @ ( product_case_prod @ ( option @ ( product_prod @ nat @ nat ) ) @ ( product_prod @ nat @ ( product_prod @ ( list @ vEBT_VEBT ) @ vEBT_VEBT ) ) @ ( heap_Time_Heap @ $o )
                                        @ ^ [Info4: option @ ( product_prod @ nat @ nat )] :
                                            ( product_case_prod @ nat @ ( product_prod @ ( list @ vEBT_VEBT ) @ vEBT_VEBT ) @ ( heap_Time_Heap @ $o )
                                            @ ^ [Deg5: nat] :
                                                ( product_case_prod @ ( list @ vEBT_VEBT ) @ vEBT_VEBT @ ( heap_Time_Heap @ $o )
                                                @ ^ [TreeList3: list @ vEBT_VEBT,Summary4: vEBT_VEBT] :
                                                    ( heap_Time_bind @ product_unit @ $o
                                                    @ ( refine_Imp_assert
                                                      @ ( ( Info3 = Info4 )
                                                        & ( Deg4 = Deg5 ) ) )
                                                    @ ^ [Uv3: product_unit] :
                                                        ( heap_Time_bind @ nat @ $o @ ( vEBT_VEBT_highi @ X @ ( divide_divide @ nat @ Deg4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
                                                        @ ^ [H: nat] :
                                                            ( heap_Time_bind @ nat @ $o @ ( vEBT_VEBT_lowi @ X @ ( divide_divide @ nat @ Deg4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
                                                            @ ^ [L2: nat] :
                                                                ( heap_Time_bind @ product_unit @ $o
                                                                @ ( refine_Imp_assert
                                                                  @ ( ( L2
                                                                      = ( vEBT_VEBT_low @ X @ ( divide_divide @ nat @ Deg4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
                                                                    & ( H
                                                                      = ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) )
                                                                @ ^ [Uw3: product_unit] :
                                                                    ( heap_Time_bind @ nat @ $o @ ( array_len @ vEBT_VEBTi @ TreeArray )
                                                                    @ ^ [Len: nat] :
                                                                        ( heap_Time_bind @ product_unit @ $o
                                                                        @ ( refine_Imp_assert
                                                                          @ ( Len
                                                                            = ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) ) )
                                                                        @ ^ [Ux3: product_unit] :
                                                                            ( if @ ( heap_Time_Heap @ $o ) @ ( ord_less @ nat @ H @ Len )
                                                                            @ ( heap_Time_bind @ product_unit @ $o
                                                                              @ ( refine_Imp_assert
                                                                                @ ( ( H
                                                                                    = ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
                                                                                  & ( ord_less @ nat @ H @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) ) ) )
                                                                              @ ^ [Uy3: product_unit] :
                                                                                  ( heap_Time_bind @ vEBT_VEBTi @ $o @ ( array_nth @ vEBT_VEBTi @ TreeArray @ H )
                                                                                  @ ^ [Th: vEBT_VEBTi] : ( product_curry @ vEBT_VEBT @ vEBT_VEBTi @ ( nat > ( heap_Time_Heap @ $o ) ) @ ( product_curry @ ( product_prod @ vEBT_VEBT @ vEBT_VEBTi ) @ nat @ ( heap_Time_Heap @ $o ) @ Vebt_memberi4 ) @ ( nth @ vEBT_VEBT @ TreeList3 @ H ) @ Th @ L2 ) ) )
                                                                            @ ( heap_Time_return @ $o @ $false ) ) ) ) ) ) ) ) ) )
                                        @ ( vEBT_case_VEBT @ ( product_prod @ ( option @ ( product_prod @ nat @ nat ) ) @ ( product_prod @ nat @ ( product_prod @ ( list @ vEBT_VEBT ) @ vEBT_VEBT ) ) )
                                          @ ^ [Info4: option @ ( product_prod @ nat @ nat ),Deg5: nat,TreeList3: list @ vEBT_VEBT,Summary4: vEBT_VEBT] : ( product_Pair @ ( option @ ( product_prod @ nat @ nat ) ) @ ( product_prod @ nat @ ( product_prod @ ( list @ vEBT_VEBT ) @ vEBT_VEBT ) ) @ Info4 @ ( product_Pair @ nat @ ( product_prod @ ( list @ vEBT_VEBT ) @ vEBT_VEBT ) @ Deg5 @ ( product_Pair @ ( list @ vEBT_VEBT ) @ vEBT_VEBT @ TreeList3 @ Summary4 ) ) )
                                          @ ^ [A5: $o,B5: $o] : ( undefined @ ( product_prod @ ( option @ ( product_prod @ nat @ nat ) ) @ ( product_prod @ nat @ ( product_prod @ ( list @ vEBT_VEBT ) @ vEBT_VEBT ) ) ) )
                                          @ T3 ) ) ) ) ) ) ) )
                        @ Info3 ) )
                @ ^ [A5: $o,B5: $o] :
                    ( heap_Time_return @ $o
                    @ ( ( ( X
                          = ( zero_zero @ nat ) )
                       => A5 )
                      & ( ( X
                         != ( zero_zero @ nat ) )
                       => ( ( ( X
                              = ( one_one @ nat ) )
                           => B5 )
                          & ( X
                            = ( one_one @ nat ) ) ) ) ) )
                @ Ti3 ) )
          @ X2 ) ) ).

% VEBT_internal.vebt_memberi'.mono
thf(fact_3035_bezout__lemma__nat,axiom,
    ! [D2: nat,A2: nat,B2: nat,X2: nat,Y2: nat] :
      ( ( dvd_dvd @ nat @ D2 @ A2 )
     => ( ( dvd_dvd @ nat @ D2 @ B2 )
       => ( ( ( ( times_times @ nat @ A2 @ X2 )
              = ( plus_plus @ nat @ ( times_times @ nat @ B2 @ Y2 ) @ D2 ) )
            | ( ( times_times @ nat @ B2 @ X2 )
              = ( plus_plus @ nat @ ( times_times @ nat @ A2 @ Y2 ) @ D2 ) ) )
         => ? [X3: nat,Y4: nat] :
              ( ( dvd_dvd @ nat @ D2 @ A2 )
              & ( dvd_dvd @ nat @ D2 @ ( plus_plus @ nat @ A2 @ B2 ) )
              & ( ( ( times_times @ nat @ A2 @ X3 )
                  = ( plus_plus @ nat @ ( times_times @ nat @ ( plus_plus @ nat @ A2 @ B2 ) @ Y4 ) @ D2 ) )
                | ( ( times_times @ nat @ ( plus_plus @ nat @ A2 @ B2 ) @ X3 )
                  = ( plus_plus @ nat @ ( times_times @ nat @ A2 @ Y4 ) @ D2 ) ) ) ) ) ) ) ).

% bezout_lemma_nat
thf(fact_3036_bezout__add__nat,axiom,
    ! [A2: nat,B2: nat] :
    ? [D4: nat,X3: nat,Y4: nat] :
      ( ( dvd_dvd @ nat @ D4 @ A2 )
      & ( dvd_dvd @ nat @ D4 @ B2 )
      & ( ( ( times_times @ nat @ A2 @ X3 )
          = ( plus_plus @ nat @ ( times_times @ nat @ B2 @ Y4 ) @ D4 ) )
        | ( ( times_times @ nat @ B2 @ X3 )
          = ( plus_plus @ nat @ ( times_times @ nat @ A2 @ Y4 ) @ D4 ) ) ) ) ).

% bezout_add_nat
thf(fact_3037_bezout1__nat,axiom,
    ! [A2: nat,B2: nat] :
    ? [D4: nat,X3: nat,Y4: nat] :
      ( ( dvd_dvd @ nat @ D4 @ A2 )
      & ( dvd_dvd @ nat @ D4 @ B2 )
      & ( ( ( minus_minus @ nat @ ( times_times @ nat @ A2 @ X3 ) @ ( times_times @ nat @ B2 @ Y4 ) )
          = D4 )
        | ( ( minus_minus @ nat @ ( times_times @ nat @ B2 @ X3 ) @ ( times_times @ nat @ A2 @ Y4 ) )
          = D4 ) ) ) ).

% bezout1_nat
thf(fact_3038_effect__returnI,axiom,
    ! [A: $tType,H2: heap_ext @ product_unit,H4: heap_ext @ product_unit,X2: A] :
      ( ( H2 = H4 )
     => ( heap_Time_effect @ A @ ( heap_Time_return @ A @ X2 ) @ H2 @ H4 @ X2 @ ( one_one @ nat ) ) ) ).

% effect_returnI
thf(fact_3039_effect__returnE,axiom,
    ! [A: $tType,X2: A,H2: heap_ext @ product_unit,H4: heap_ext @ product_unit,R4: A,N2: nat] :
      ( ( heap_Time_effect @ A @ ( heap_Time_return @ A @ X2 ) @ H2 @ H4 @ R4 @ N2 )
     => ~ ( ( R4 = X2 )
         => ( ( H4 = H2 )
           => ( N2
             != ( one_one @ nat ) ) ) ) ) ).

% effect_returnE
thf(fact_3040_effect__bindE,axiom,
    ! [B: $tType,A: $tType,F: heap_Time_Heap @ B,G: B > ( heap_Time_Heap @ A ),H2: heap_ext @ product_unit,H8: heap_ext @ product_unit,R5: A,N2: nat] :
      ( ( heap_Time_effect @ A @ ( heap_Time_bind @ B @ A @ F @ G ) @ H2 @ H8 @ R5 @ N2 )
     => ~ ! [H5: heap_ext @ product_unit,R3: B,N1: nat] :
            ( ( heap_Time_effect @ B @ F @ H2 @ H5 @ R3 @ N1 )
           => ! [N22: nat] :
                ( ( heap_Time_effect @ A @ ( G @ R3 ) @ H5 @ H8 @ R5 @ N22 )
               => ( N2
                 != ( plus_plus @ nat @ N1 @ N22 ) ) ) ) ) ).

% effect_bindE
thf(fact_3041_effect__bindI,axiom,
    ! [B: $tType,A: $tType,F: heap_Time_Heap @ A,H2: heap_ext @ product_unit,H4: heap_ext @ product_unit,R4: A,N2: nat,G: A > ( heap_Time_Heap @ B ),H8: heap_ext @ product_unit,R5: B,N5: nat] :
      ( ( heap_Time_effect @ A @ F @ H2 @ H4 @ R4 @ N2 )
     => ( ( heap_Time_effect @ B @ ( G @ R4 ) @ H4 @ H8 @ R5 @ N5 )
       => ( heap_Time_effect @ B @ ( heap_Time_bind @ A @ B @ F @ G ) @ H2 @ H8 @ R5 @ ( plus_plus @ nat @ N2 @ N5 ) ) ) ) ).

% effect_bindI
thf(fact_3042_vebt__predi_Omono,axiom,
    ! [X2: product_prod @ vEBT_VEBTi @ nat] :
      ( comple7038119648293358887notone @ ( ( product_prod @ vEBT_VEBTi @ nat ) > ( heap_Time_Heap @ ( option @ nat ) ) ) @ ( heap_Time_Heap @ ( option @ nat ) ) @ ( partial_fun_ord @ ( heap_Time_Heap @ ( option @ nat ) ) @ ( heap_Time_Heap @ ( option @ nat ) ) @ ( product_prod @ vEBT_VEBTi @ nat ) @ ( heap_Time_Heap_ord @ ( option @ nat ) ) ) @ ( heap_Time_Heap_ord @ ( option @ nat ) )
      @ ^ [Vebt_predi4: ( product_prod @ vEBT_VEBTi @ nat ) > ( heap_Time_Heap @ ( option @ nat ) )] :
          ( product_case_prod @ vEBT_VEBTi @ nat @ ( heap_Time_Heap @ ( option @ nat ) )
          @ ^ [T3: vEBT_VEBTi,X: nat] :
              ( vEBT_case_VEBTi @ ( heap_Time_Heap @ ( option @ nat ) )
              @ ^ [Info3: option @ ( product_prod @ nat @ nat ),Deg4: nat,TreeArray: array @ vEBT_VEBTi,Summary3: vEBT_VEBTi] :
                  ( case_option @ ( heap_Time_Heap @ ( option @ nat ) ) @ ( product_prod @ nat @ nat ) @ ( heap_Time_return @ ( option @ nat ) @ ( none @ nat ) )
                  @ ^ [Mima2: product_prod @ nat @ nat] :
                      ( if @ ( heap_Time_Heap @ ( option @ nat ) ) @ ( ord_less_eq @ nat @ Deg4 @ ( one_one @ nat ) ) @ ( heap_Time_return @ ( option @ nat ) @ ( none @ nat ) )
                      @ ( if @ ( heap_Time_Heap @ ( option @ nat ) ) @ ( ord_less @ nat @ ( product_snd @ nat @ nat @ Mima2 ) @ X ) @ ( heap_Time_return @ ( option @ nat ) @ ( some @ nat @ ( product_snd @ nat @ nat @ Mima2 ) ) )
                        @ ( heap_Time_bind @ nat @ ( option @ nat ) @ ( vEBT_VEBT_lowi @ X @ ( divide_divide @ nat @ Deg4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
                          @ ^ [L2: nat] :
                              ( heap_Time_bind @ nat @ ( option @ nat ) @ ( vEBT_VEBT_highi @ X @ ( divide_divide @ nat @ Deg4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
                              @ ^ [H: nat] :
                                  ( heap_Time_bind @ vEBT_VEBTi @ ( option @ nat ) @ ( array_nth @ vEBT_VEBTi @ TreeArray @ H )
                                  @ ^ [Aktnode: vEBT_VEBTi] :
                                      ( heap_Time_bind @ ( option @ nat ) @ ( option @ nat ) @ ( vEBT_vebt_minti @ Aktnode )
                                      @ ^ [Minlow: option @ nat] :
                                          ( if @ ( heap_Time_Heap @ ( option @ nat ) )
                                          @ ( ( Minlow
                                             != ( none @ nat ) )
                                            & ( vEBT_VEBT_greater @ ( some @ nat @ L2 ) @ Minlow ) )
                                          @ ( heap_Time_bind @ ( option @ nat ) @ ( option @ nat ) @ ( product_curry @ vEBT_VEBTi @ nat @ ( heap_Time_Heap @ ( option @ nat ) ) @ Vebt_predi4 @ Aktnode @ L2 )
                                            @ ^ [Predy: option @ nat] : ( heap_Time_return @ ( option @ nat ) @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( some @ nat @ H ) ) @ Predy ) ) )
                                          @ ( heap_Time_bind @ ( option @ nat ) @ ( option @ nat ) @ ( product_curry @ vEBT_VEBTi @ nat @ ( heap_Time_Heap @ ( option @ nat ) ) @ Vebt_predi4 @ Summary3 @ H )
                                            @ ^ [Predsum: option @ nat] :
                                                ( if @ ( heap_Time_Heap @ ( option @ nat ) )
                                                @ ( Predsum
                                                  = ( none @ nat ) )
                                                @ ( if @ ( heap_Time_Heap @ ( option @ nat ) ) @ ( ord_less @ nat @ ( product_fst @ nat @ nat @ Mima2 ) @ X ) @ ( heap_Time_return @ ( option @ nat ) @ ( some @ nat @ ( product_fst @ nat @ nat @ Mima2 ) ) ) @ ( heap_Time_return @ ( option @ nat ) @ ( none @ nat ) ) )
                                                @ ( heap_Time_bind @ vEBT_VEBTi @ ( option @ nat ) @ ( array_nth @ vEBT_VEBTi @ TreeArray @ ( the2 @ nat @ Predsum ) )
                                                  @ ^ [Nextnode: vEBT_VEBTi] :
                                                      ( heap_Time_bind @ ( option @ nat ) @ ( option @ nat ) @ ( vEBT_vebt_maxti @ Nextnode )
                                                      @ ^ [Maxnext: option @ nat] : ( heap_Time_return @ ( option @ nat ) @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ Predsum ) @ Maxnext ) ) ) ) ) ) ) ) ) ) ) ) )
                  @ Info3 )
              @ ^ [A5: $o,B5: $o] :
                  ( if @ ( heap_Time_Heap @ ( option @ nat ) ) @ ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ X ) @ ( if @ ( heap_Time_Heap @ ( option @ nat ) ) @ B5 @ ( heap_Time_return @ ( option @ nat ) @ ( some @ nat @ ( one_one @ nat ) ) ) @ ( if @ ( heap_Time_Heap @ ( option @ nat ) ) @ A5 @ ( heap_Time_return @ ( option @ nat ) @ ( some @ nat @ ( zero_zero @ nat ) ) ) @ ( heap_Time_return @ ( option @ nat ) @ ( none @ nat ) ) ) )
                  @ ( if @ ( heap_Time_Heap @ ( option @ nat ) )
                    @ ( X
                      = ( one_one @ nat ) )
                    @ ( if @ ( heap_Time_Heap @ ( option @ nat ) ) @ A5 @ ( heap_Time_return @ ( option @ nat ) @ ( some @ nat @ ( zero_zero @ nat ) ) ) @ ( heap_Time_return @ ( option @ nat ) @ ( none @ nat ) ) )
                    @ ( heap_Time_return @ ( option @ nat ) @ ( none @ nat ) ) ) )
              @ T3 )
          @ X2 ) ) ).

% vebt_predi.mono
thf(fact_3043_vebt__succi_Omono,axiom,
    ! [X2: product_prod @ vEBT_VEBTi @ nat] :
      ( comple7038119648293358887notone @ ( ( product_prod @ vEBT_VEBTi @ nat ) > ( heap_Time_Heap @ ( option @ nat ) ) ) @ ( heap_Time_Heap @ ( option @ nat ) ) @ ( partial_fun_ord @ ( heap_Time_Heap @ ( option @ nat ) ) @ ( heap_Time_Heap @ ( option @ nat ) ) @ ( product_prod @ vEBT_VEBTi @ nat ) @ ( heap_Time_Heap_ord @ ( option @ nat ) ) ) @ ( heap_Time_Heap_ord @ ( option @ nat ) )
      @ ^ [Vebt_succi4: ( product_prod @ vEBT_VEBTi @ nat ) > ( heap_Time_Heap @ ( option @ nat ) )] :
          ( product_case_prod @ vEBT_VEBTi @ nat @ ( heap_Time_Heap @ ( option @ nat ) )
          @ ^ [T3: vEBT_VEBTi,X: nat] :
              ( vEBT_case_VEBTi @ ( heap_Time_Heap @ ( option @ nat ) )
              @ ^ [Info3: option @ ( product_prod @ nat @ nat ),Deg4: nat,TreeArray: array @ vEBT_VEBTi,Summary3: vEBT_VEBTi] :
                  ( case_option @ ( heap_Time_Heap @ ( option @ nat ) ) @ ( product_prod @ nat @ nat ) @ ( heap_Time_return @ ( option @ nat ) @ ( none @ nat ) )
                  @ ^ [Mima2: product_prod @ nat @ nat] :
                      ( if @ ( heap_Time_Heap @ ( option @ nat ) ) @ ( ord_less_eq @ nat @ Deg4 @ ( one_one @ nat ) ) @ ( heap_Time_return @ ( option @ nat ) @ ( none @ nat ) )
                      @ ( if @ ( heap_Time_Heap @ ( option @ nat ) ) @ ( ord_less @ nat @ X @ ( product_fst @ nat @ nat @ Mima2 ) ) @ ( heap_Time_return @ ( option @ nat ) @ ( some @ nat @ ( product_fst @ nat @ nat @ Mima2 ) ) )
                        @ ( if @ ( heap_Time_Heap @ ( option @ nat ) ) @ ( ord_less_eq @ nat @ ( product_snd @ nat @ nat @ Mima2 ) @ X ) @ ( heap_Time_return @ ( option @ nat ) @ ( none @ nat ) )
                          @ ( heap_Time_bind @ nat @ ( option @ nat ) @ ( vEBT_VEBT_lowi @ X @ ( divide_divide @ nat @ Deg4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
                            @ ^ [L2: nat] :
                                ( heap_Time_bind @ nat @ ( option @ nat ) @ ( vEBT_VEBT_highi @ X @ ( divide_divide @ nat @ Deg4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
                                @ ^ [H: nat] :
                                    ( heap_Time_bind @ vEBT_VEBTi @ ( option @ nat ) @ ( array_nth @ vEBT_VEBTi @ TreeArray @ H )
                                    @ ^ [Aktnode: vEBT_VEBTi] :
                                        ( heap_Time_bind @ ( option @ nat ) @ ( option @ nat ) @ ( vEBT_vebt_maxti @ Aktnode )
                                        @ ^ [Maxlow: option @ nat] :
                                            ( if @ ( heap_Time_Heap @ ( option @ nat ) )
                                            @ ( ( Maxlow
                                               != ( none @ nat ) )
                                              & ( vEBT_VEBT_less @ ( some @ nat @ L2 ) @ Maxlow ) )
                                            @ ( heap_Time_bind @ ( option @ nat ) @ ( option @ nat ) @ ( product_curry @ vEBT_VEBTi @ nat @ ( heap_Time_Heap @ ( option @ nat ) ) @ Vebt_succi4 @ Aktnode @ L2 )
                                              @ ^ [Succy: option @ nat] : ( heap_Time_return @ ( option @ nat ) @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( some @ nat @ H ) ) @ Succy ) ) )
                                            @ ( heap_Time_bind @ ( option @ nat ) @ ( option @ nat ) @ ( product_curry @ vEBT_VEBTi @ nat @ ( heap_Time_Heap @ ( option @ nat ) ) @ Vebt_succi4 @ Summary3 @ H )
                                              @ ^ [Succsum: option @ nat] :
                                                  ( if @ ( heap_Time_Heap @ ( option @ nat ) )
                                                  @ ( Succsum
                                                    = ( none @ nat ) )
                                                  @ ( heap_Time_return @ ( option @ nat ) @ ( none @ nat ) )
                                                  @ ( heap_Time_bind @ vEBT_VEBTi @ ( option @ nat ) @ ( array_nth @ vEBT_VEBTi @ TreeArray @ ( the2 @ nat @ Succsum ) )
                                                    @ ^ [Nextnode: vEBT_VEBTi] :
                                                        ( heap_Time_bind @ ( option @ nat ) @ ( option @ nat ) @ ( vEBT_vebt_minti @ Nextnode )
                                                        @ ^ [Minnext: option @ nat] : ( heap_Time_return @ ( option @ nat ) @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ Succsum ) @ Minnext ) ) ) ) ) ) ) ) ) ) ) ) ) )
                  @ Info3 )
              @ ^ [A5: $o,B5: $o] :
                  ( if @ ( heap_Time_Heap @ ( option @ nat ) )
                  @ ( X
                    = ( zero_zero @ nat ) )
                  @ ( if @ ( heap_Time_Heap @ ( option @ nat ) ) @ B5 @ ( heap_Time_return @ ( option @ nat ) @ ( some @ nat @ ( one_one @ nat ) ) ) @ ( heap_Time_return @ ( option @ nat ) @ ( none @ nat ) ) )
                  @ ( heap_Time_return @ ( option @ nat ) @ ( none @ nat ) ) )
              @ T3 )
          @ X2 ) ) ).

% vebt_succi.mono
thf(fact_3044_vebt__inserti_Omono,axiom,
    ! [X2: product_prod @ vEBT_VEBTi @ nat] :
      ( comple7038119648293358887notone @ ( ( product_prod @ vEBT_VEBTi @ nat ) > ( heap_Time_Heap @ vEBT_VEBTi ) ) @ ( heap_Time_Heap @ vEBT_VEBTi ) @ ( partial_fun_ord @ ( heap_Time_Heap @ vEBT_VEBTi ) @ ( heap_Time_Heap @ vEBT_VEBTi ) @ ( product_prod @ vEBT_VEBTi @ nat ) @ ( heap_Time_Heap_ord @ vEBT_VEBTi ) ) @ ( heap_Time_Heap_ord @ vEBT_VEBTi )
      @ ^ [Vebt_inserti4: ( product_prod @ vEBT_VEBTi @ nat ) > ( heap_Time_Heap @ vEBT_VEBTi )] :
          ( product_case_prod @ vEBT_VEBTi @ nat @ ( heap_Time_Heap @ vEBT_VEBTi )
          @ ^ [T3: vEBT_VEBTi,X: nat] :
              ( vEBT_case_VEBTi @ ( heap_Time_Heap @ vEBT_VEBTi )
              @ ^ [Info3: option @ ( product_prod @ nat @ nat ),Deg4: nat,TreeArray: array @ vEBT_VEBTi,Summary3: vEBT_VEBTi] :
                  ( case_option @ ( heap_Time_Heap @ vEBT_VEBTi ) @ ( product_prod @ nat @ nat ) @ ( if @ ( heap_Time_Heap @ vEBT_VEBTi ) @ ( ord_less_eq @ nat @ Deg4 @ ( one_one @ nat ) ) @ ( heap_Time_return @ vEBT_VEBTi @ ( vEBT_Nodei @ Info3 @ Deg4 @ TreeArray @ Summary3 ) ) @ ( heap_Time_return @ vEBT_VEBTi @ ( vEBT_Nodei @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ X @ X ) ) @ Deg4 @ TreeArray @ Summary3 ) ) )
                  @ ^ [Minma: product_prod @ nat @ nat] :
                      ( if @ ( heap_Time_Heap @ vEBT_VEBTi ) @ ( ord_less_eq @ nat @ Deg4 @ ( one_one @ nat ) ) @ ( heap_Time_return @ vEBT_VEBTi @ ( vEBT_Nodei @ Info3 @ Deg4 @ TreeArray @ Summary3 ) )
                      @ ( heap_Time_bind @ nat @ vEBT_VEBTi @ ( heap_Time_return @ nat @ ( product_fst @ nat @ nat @ Minma ) )
                        @ ^ [Mi3: nat] :
                            ( heap_Time_bind @ nat @ vEBT_VEBTi @ ( heap_Time_return @ nat @ ( product_snd @ nat @ nat @ Minma ) )
                            @ ^ [Ma3: nat] :
                                ( heap_Time_bind @ nat @ vEBT_VEBTi @ ( if @ ( heap_Time_Heap @ nat ) @ ( ord_less @ nat @ X @ Mi3 ) @ ( heap_Time_return @ nat @ Mi3 ) @ ( heap_Time_return @ nat @ X ) )
                                @ ^ [Xn2: nat] :
                                    ( heap_Time_bind @ nat @ vEBT_VEBTi @ ( if @ ( heap_Time_Heap @ nat ) @ ( ord_less @ nat @ X @ Mi3 ) @ ( heap_Time_return @ nat @ X ) @ ( heap_Time_return @ nat @ Mi3 ) )
                                    @ ^ [Minn: nat] :
                                        ( heap_Time_bind @ nat @ vEBT_VEBTi @ ( vEBT_VEBT_lowi @ Xn2 @ ( divide_divide @ nat @ Deg4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
                                        @ ^ [L2: nat] :
                                            ( heap_Time_bind @ nat @ vEBT_VEBTi @ ( vEBT_VEBT_highi @ Xn2 @ ( divide_divide @ nat @ Deg4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
                                            @ ^ [H: nat] :
                                                ( heap_Time_bind @ nat @ vEBT_VEBTi @ ( array_len @ vEBT_VEBTi @ TreeArray )
                                                @ ^ [Len: nat] :
                                                    ( if @ ( heap_Time_Heap @ vEBT_VEBTi )
                                                    @ ( ( ord_less @ nat @ H @ Len )
                                                      & ~ ( ( X = Mi3 )
                                                          | ( X = Ma3 ) ) )
                                                    @ ( heap_Time_bind @ vEBT_VEBTi @ vEBT_VEBTi @ ( array_nth @ vEBT_VEBTi @ TreeArray @ H )
                                                      @ ^ [Node: vEBT_VEBTi] :
                                                          ( heap_Time_bind @ $o @ vEBT_VEBTi @ ( vEBT_VEBT_minNulli @ Node )
                                                          @ ^ [Empt: $o] :
                                                              ( heap_Time_bind @ vEBT_VEBTi @ vEBT_VEBTi @ ( product_curry @ vEBT_VEBTi @ nat @ ( heap_Time_Heap @ vEBT_VEBTi ) @ Vebt_inserti4 @ Node @ L2 )
                                                              @ ^ [Newnode2: vEBT_VEBTi] :
                                                                  ( heap_Time_bind @ ( array @ vEBT_VEBTi ) @ vEBT_VEBTi @ ( array_upd @ vEBT_VEBTi @ H @ Newnode2 @ TreeArray )
                                                                  @ ^ [Newarray: array @ vEBT_VEBTi] :
                                                                      ( heap_Time_bind @ vEBT_VEBTi @ vEBT_VEBTi @ ( if @ ( heap_Time_Heap @ vEBT_VEBTi ) @ Empt @ ( product_curry @ vEBT_VEBTi @ nat @ ( heap_Time_Heap @ vEBT_VEBTi ) @ Vebt_inserti4 @ Summary3 @ H ) @ ( heap_Time_return @ vEBT_VEBTi @ Summary3 ) )
                                                                      @ ^ [Newsummary: vEBT_VEBTi] :
                                                                          ( heap_Time_bind @ nat @ vEBT_VEBTi @ ( if @ ( heap_Time_Heap @ nat ) @ ( ord_less @ nat @ Ma3 @ Xn2 ) @ ( heap_Time_return @ nat @ Xn2 ) @ ( heap_Time_return @ nat @ Ma3 ) )
                                                                          @ ^ [Man: nat] : ( heap_Time_return @ vEBT_VEBTi @ ( vEBT_Nodei @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Minn @ Man ) ) @ Deg4 @ Newarray @ Newsummary ) ) ) ) ) ) ) )
                                                    @ ( heap_Time_return @ vEBT_VEBTi @ ( vEBT_Nodei @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi3 @ Ma3 ) ) @ Deg4 @ TreeArray @ Summary3 ) ) ) ) ) ) ) ) ) ) )
                  @ Info3 )
              @ ^ [A5: $o,B5: $o] :
                  ( if @ ( heap_Time_Heap @ vEBT_VEBTi )
                  @ ( X
                    = ( zero_zero @ nat ) )
                  @ ( heap_Time_return @ vEBT_VEBTi @ ( vEBT_Leafi @ $true @ B5 ) )
                  @ ( if @ ( heap_Time_Heap @ vEBT_VEBTi )
                    @ ( X
                      = ( one_one @ nat ) )
                    @ ( heap_Time_return @ vEBT_VEBTi @ ( vEBT_Leafi @ A5 @ $true ) )
                    @ ( heap_Time_return @ vEBT_VEBTi @ ( vEBT_Leafi @ A5 @ B5 ) ) ) )
              @ T3 )
          @ X2 ) ) ).

% vebt_inserti.mono
thf(fact_3045_bezout__add__strong__nat,axiom,
    ! [A2: nat,B2: nat] :
      ( ( A2
       != ( zero_zero @ nat ) )
     => ? [D4: nat,X3: nat,Y4: nat] :
          ( ( dvd_dvd @ nat @ D4 @ A2 )
          & ( dvd_dvd @ nat @ D4 @ B2 )
          & ( ( times_times @ nat @ A2 @ X3 )
            = ( plus_plus @ nat @ ( times_times @ nat @ B2 @ Y4 ) @ D4 ) ) ) ) ).

% bezout_add_strong_nat
thf(fact_3046_vebt__buildupi_Osimps_I3_J,axiom,
    ! [Va: nat] :
      ( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ Va ) ) )
       => ( ( vEBT_vebt_buildupi @ ( suc @ ( suc @ Va ) ) )
          = ( heap_Time_bind @ ( list @ vEBT_VEBTi ) @ vEBT_VEBTi @ ( vEBT_VEBT_replicatei @ vEBT_VEBTi @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_buildupi @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
            @ ^ [TreeList3: list @ vEBT_VEBTi] :
                ( heap_Time_bind @ ( array @ vEBT_VEBTi ) @ vEBT_VEBTi @ ( array_of_list @ vEBT_VEBTi @ TreeList3 )
                @ ^ [Trees: array @ vEBT_VEBTi] :
                    ( heap_Time_bind @ vEBT_VEBTi @ vEBT_VEBTi @ ( vEBT_vebt_buildupi @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
                    @ ^ [Summary3: vEBT_VEBTi] : ( heap_Time_return @ vEBT_VEBTi @ ( vEBT_Nodei @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ Va ) ) @ Trees @ Summary3 ) ) ) ) ) ) )
      & ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ Va ) ) )
       => ( ( vEBT_vebt_buildupi @ ( suc @ ( suc @ Va ) ) )
          = ( heap_Time_bind @ ( list @ vEBT_VEBTi ) @ vEBT_VEBTi @ ( vEBT_VEBT_replicatei @ vEBT_VEBTi @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_buildupi @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
            @ ^ [TreeList3: list @ vEBT_VEBTi] :
                ( heap_Time_bind @ ( array @ vEBT_VEBTi ) @ vEBT_VEBTi @ ( array_of_list @ vEBT_VEBTi @ TreeList3 )
                @ ^ [Trees: array @ vEBT_VEBTi] :
                    ( heap_Time_bind @ vEBT_VEBTi @ vEBT_VEBTi @ ( vEBT_vebt_buildupi @ ( suc @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
                    @ ^ [Summary3: vEBT_VEBTi] : ( heap_Time_return @ vEBT_VEBTi @ ( vEBT_Nodei @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ Va ) ) @ Trees @ Summary3 ) ) ) ) ) ) ) ) ).

% vebt_buildupi.simps(3)
thf(fact_3047_vebt__inserti_Ofixp__induct,axiom,
    ! [P: ( vEBT_VEBTi > nat > ( heap_Time_Heap @ vEBT_VEBTi ) ) > $o] :
      ( ( comple1908693960933563346ssible @ ( ( product_prod @ vEBT_VEBTi @ nat ) > ( heap_Time_Heap @ vEBT_VEBTi ) ) @ ( partial_fun_lub @ ( heap_Time_Heap @ vEBT_VEBTi ) @ ( heap_Time_Heap @ vEBT_VEBTi ) @ ( product_prod @ vEBT_VEBTi @ nat ) @ ( heap_Time_Heap_lub @ vEBT_VEBTi ) ) @ ( partial_fun_ord @ ( heap_Time_Heap @ vEBT_VEBTi ) @ ( heap_Time_Heap @ vEBT_VEBTi ) @ ( product_prod @ vEBT_VEBTi @ nat ) @ ( heap_Time_Heap_ord @ vEBT_VEBTi ) )
        @ ^ [Vebt_inserti4: ( product_prod @ vEBT_VEBTi @ nat ) > ( heap_Time_Heap @ vEBT_VEBTi )] : ( P @ ( product_curry @ vEBT_VEBTi @ nat @ ( heap_Time_Heap @ vEBT_VEBTi ) @ Vebt_inserti4 ) ) )
     => ( ( P
          @ ^ [Vebt_inserti4: vEBT_VEBTi,T3: nat] :
              ( heap_Time_Heap2 @ vEBT_VEBTi
              @ ^ [X: heap_ext @ product_unit] : ( none @ ( product_prod @ vEBT_VEBTi @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) )
       => ( ! [F2: vEBT_VEBTi > nat > ( heap_Time_Heap @ vEBT_VEBTi )] :
              ( ( P @ F2 )
             => ( P
                @ ^ [X9: vEBT_VEBTi,A5: nat] :
                    ( vEBT_case_VEBTi @ ( heap_Time_Heap @ vEBT_VEBTi )
                    @ ^ [Info3: option @ ( product_prod @ nat @ nat ),Deg4: nat,TreeArray: array @ vEBT_VEBTi,Summary3: vEBT_VEBTi] :
                        ( case_option @ ( heap_Time_Heap @ vEBT_VEBTi ) @ ( product_prod @ nat @ nat ) @ ( if @ ( heap_Time_Heap @ vEBT_VEBTi ) @ ( ord_less_eq @ nat @ Deg4 @ ( one_one @ nat ) ) @ ( heap_Time_return @ vEBT_VEBTi @ ( vEBT_Nodei @ Info3 @ Deg4 @ TreeArray @ Summary3 ) ) @ ( heap_Time_return @ vEBT_VEBTi @ ( vEBT_Nodei @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ A5 @ A5 ) ) @ Deg4 @ TreeArray @ Summary3 ) ) )
                        @ ^ [Minma: product_prod @ nat @ nat] :
                            ( if @ ( heap_Time_Heap @ vEBT_VEBTi ) @ ( ord_less_eq @ nat @ Deg4 @ ( one_one @ nat ) ) @ ( heap_Time_return @ vEBT_VEBTi @ ( vEBT_Nodei @ Info3 @ Deg4 @ TreeArray @ Summary3 ) )
                            @ ( heap_Time_bind @ nat @ vEBT_VEBTi @ ( heap_Time_return @ nat @ ( product_fst @ nat @ nat @ Minma ) )
                              @ ^ [Mi3: nat] :
                                  ( heap_Time_bind @ nat @ vEBT_VEBTi @ ( heap_Time_return @ nat @ ( product_snd @ nat @ nat @ Minma ) )
                                  @ ^ [Ma3: nat] :
                                      ( heap_Time_bind @ nat @ vEBT_VEBTi @ ( if @ ( heap_Time_Heap @ nat ) @ ( ord_less @ nat @ A5 @ Mi3 ) @ ( heap_Time_return @ nat @ Mi3 ) @ ( heap_Time_return @ nat @ A5 ) )
                                      @ ^ [Xn2: nat] :
                                          ( heap_Time_bind @ nat @ vEBT_VEBTi @ ( if @ ( heap_Time_Heap @ nat ) @ ( ord_less @ nat @ A5 @ Mi3 ) @ ( heap_Time_return @ nat @ A5 ) @ ( heap_Time_return @ nat @ Mi3 ) )
                                          @ ^ [Minn: nat] :
                                              ( heap_Time_bind @ nat @ vEBT_VEBTi @ ( vEBT_VEBT_lowi @ Xn2 @ ( divide_divide @ nat @ Deg4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
                                              @ ^ [L2: nat] :
                                                  ( heap_Time_bind @ nat @ vEBT_VEBTi @ ( vEBT_VEBT_highi @ Xn2 @ ( divide_divide @ nat @ Deg4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
                                                  @ ^ [H: nat] :
                                                      ( heap_Time_bind @ nat @ vEBT_VEBTi @ ( array_len @ vEBT_VEBTi @ TreeArray )
                                                      @ ^ [Len: nat] :
                                                          ( if @ ( heap_Time_Heap @ vEBT_VEBTi )
                                                          @ ( ( ord_less @ nat @ H @ Len )
                                                            & ~ ( ( A5 = Mi3 )
                                                                | ( A5 = Ma3 ) ) )
                                                          @ ( heap_Time_bind @ vEBT_VEBTi @ vEBT_VEBTi @ ( array_nth @ vEBT_VEBTi @ TreeArray @ H )
                                                            @ ^ [Node: vEBT_VEBTi] :
                                                                ( heap_Time_bind @ $o @ vEBT_VEBTi @ ( vEBT_VEBT_minNulli @ Node )
                                                                @ ^ [Empt: $o] :
                                                                    ( heap_Time_bind @ vEBT_VEBTi @ vEBT_VEBTi @ ( F2 @ Node @ L2 )
                                                                    @ ^ [Newnode2: vEBT_VEBTi] :
                                                                        ( heap_Time_bind @ ( array @ vEBT_VEBTi ) @ vEBT_VEBTi @ ( array_upd @ vEBT_VEBTi @ H @ Newnode2 @ TreeArray )
                                                                        @ ^ [Newarray: array @ vEBT_VEBTi] :
                                                                            ( heap_Time_bind @ vEBT_VEBTi @ vEBT_VEBTi @ ( if @ ( heap_Time_Heap @ vEBT_VEBTi ) @ Empt @ ( F2 @ Summary3 @ H ) @ ( heap_Time_return @ vEBT_VEBTi @ Summary3 ) )
                                                                            @ ^ [Newsummary: vEBT_VEBTi] :
                                                                                ( heap_Time_bind @ nat @ vEBT_VEBTi @ ( if @ ( heap_Time_Heap @ nat ) @ ( ord_less @ nat @ Ma3 @ Xn2 ) @ ( heap_Time_return @ nat @ Xn2 ) @ ( heap_Time_return @ nat @ Ma3 ) )
                                                                                @ ^ [Man: nat] : ( heap_Time_return @ vEBT_VEBTi @ ( vEBT_Nodei @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Minn @ Man ) ) @ Deg4 @ Newarray @ Newsummary ) ) ) ) ) ) ) )
                                                          @ ( heap_Time_return @ vEBT_VEBTi @ ( vEBT_Nodei @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi3 @ Ma3 ) ) @ Deg4 @ TreeArray @ Summary3 ) ) ) ) ) ) ) ) ) ) )
                        @ Info3 )
                    @ ^ [B5: $o,C4: $o] :
                        ( if @ ( heap_Time_Heap @ vEBT_VEBTi )
                        @ ( A5
                          = ( zero_zero @ nat ) )
                        @ ( heap_Time_return @ vEBT_VEBTi @ ( vEBT_Leafi @ $true @ C4 ) )
                        @ ( if @ ( heap_Time_Heap @ vEBT_VEBTi )
                          @ ( A5
                            = ( one_one @ nat ) )
                          @ ( heap_Time_return @ vEBT_VEBTi @ ( vEBT_Leafi @ B5 @ $true ) )
                          @ ( heap_Time_return @ vEBT_VEBTi @ ( vEBT_Leafi @ B5 @ C4 ) ) ) )
                    @ X9 ) ) )
         => ( P @ vEBT_vebt_inserti ) ) ) ) ).

% vebt_inserti.fixp_induct
thf(fact_3048_div2__even__ext__nat,axiom,
    ! [X2: nat,Y2: nat] :
      ( ( ( divide_divide @ nat @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
        = ( divide_divide @ nat @ Y2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
     => ( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ X2 )
          = ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Y2 ) )
       => ( X2 = Y2 ) ) ) ).

% div2_even_ext_nat
thf(fact_3049_VEBT__internal_Ovebt__succi_H_Ofixp__induct,axiom,
    ! [P: ( vEBT_VEBT > vEBT_VEBTi > nat > ( heap_Time_Heap @ ( option @ nat ) ) ) > $o] :
      ( ( comple1908693960933563346ssible @ ( ( product_prod @ ( product_prod @ vEBT_VEBT @ vEBT_VEBTi ) @ nat ) > ( heap_Time_Heap @ ( option @ nat ) ) ) @ ( partial_fun_lub @ ( heap_Time_Heap @ ( option @ nat ) ) @ ( heap_Time_Heap @ ( option @ nat ) ) @ ( product_prod @ ( product_prod @ vEBT_VEBT @ vEBT_VEBTi ) @ nat ) @ ( heap_Time_Heap_lub @ ( option @ nat ) ) ) @ ( partial_fun_ord @ ( heap_Time_Heap @ ( option @ nat ) ) @ ( heap_Time_Heap @ ( option @ nat ) ) @ ( product_prod @ ( product_prod @ vEBT_VEBT @ vEBT_VEBTi ) @ nat ) @ ( heap_Time_Heap_ord @ ( option @ nat ) ) )
        @ ^ [Vebt_succi3: ( product_prod @ ( product_prod @ vEBT_VEBT @ vEBT_VEBTi ) @ nat ) > ( heap_Time_Heap @ ( option @ nat ) )] : ( P @ ( product_curry @ vEBT_VEBT @ vEBT_VEBTi @ ( nat > ( heap_Time_Heap @ ( option @ nat ) ) ) @ ( product_curry @ ( product_prod @ vEBT_VEBT @ vEBT_VEBTi ) @ nat @ ( heap_Time_Heap @ ( option @ nat ) ) @ Vebt_succi3 ) ) ) )
     => ( ( P
          @ ^ [Vebt_succi3: vEBT_VEBT,T3: vEBT_VEBTi,Ti3: nat] :
              ( heap_Time_Heap2 @ ( option @ nat )
              @ ^ [X: heap_ext @ product_unit] : ( none @ ( product_prod @ ( option @ nat ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) )
       => ( ! [F2: vEBT_VEBT > vEBT_VEBTi > nat > ( heap_Time_Heap @ ( option @ nat ) )] :
              ( ( P @ F2 )
             => ( P
                @ ^ [X9: vEBT_VEBT,A5: vEBT_VEBTi,B5: nat] :
                    ( vEBT_case_VEBTi @ ( heap_Time_Heap @ ( option @ nat ) )
                    @ ^ [Info3: option @ ( product_prod @ nat @ nat ),Deg4: nat,TreeArray: array @ vEBT_VEBTi,Summary3: vEBT_VEBTi] :
                        ( heap_Time_bind @ product_unit @ ( option @ nat ) @ ( refine_Imp_assert @ ( vEBT_is_Node @ X9 ) )
                        @ ^ [Uu3: product_unit] :
                            ( product_case_prod @ ( option @ ( product_prod @ nat @ nat ) ) @ ( product_prod @ nat @ ( product_prod @ ( list @ vEBT_VEBT ) @ vEBT_VEBT ) ) @ ( heap_Time_Heap @ ( option @ nat ) )
                            @ ^ [Info4: option @ ( product_prod @ nat @ nat )] :
                                ( product_case_prod @ nat @ ( product_prod @ ( list @ vEBT_VEBT ) @ vEBT_VEBT ) @ ( heap_Time_Heap @ ( option @ nat ) )
                                @ ^ [Deg5: nat] :
                                    ( product_case_prod @ ( list @ vEBT_VEBT ) @ vEBT_VEBT @ ( heap_Time_Heap @ ( option @ nat ) )
                                    @ ^ [TreeList3: list @ vEBT_VEBT,Summary4: vEBT_VEBT] :
                                        ( heap_Time_bind @ product_unit @ ( option @ nat )
                                        @ ( refine_Imp_assert
                                          @ ( ( Info4 = Info3 )
                                            & ( Deg5 = Deg4 )
                                            & ( vEBT_is_Node @ X9 ) ) )
                                        @ ^ [Uv3: product_unit] :
                                            ( case_option @ ( heap_Time_Heap @ ( option @ nat ) ) @ ( product_prod @ nat @ nat ) @ ( heap_Time_return @ ( option @ nat ) @ ( none @ nat ) )
                                            @ ^ [Mima2: product_prod @ nat @ nat] :
                                                ( if @ ( heap_Time_Heap @ ( option @ nat ) ) @ ( ord_less_eq @ nat @ Deg4 @ ( one_one @ nat ) ) @ ( heap_Time_return @ ( option @ nat ) @ ( none @ nat ) )
                                                @ ( if @ ( heap_Time_Heap @ ( option @ nat ) ) @ ( ord_less @ nat @ B5 @ ( product_fst @ nat @ nat @ Mima2 ) ) @ ( heap_Time_return @ ( option @ nat ) @ ( some @ nat @ ( product_fst @ nat @ nat @ Mima2 ) ) )
                                                  @ ( if @ ( heap_Time_Heap @ ( option @ nat ) ) @ ( ord_less_eq @ nat @ ( product_snd @ nat @ nat @ Mima2 ) @ B5 ) @ ( heap_Time_return @ ( option @ nat ) @ ( none @ nat ) )
                                                    @ ( heap_Time_bind @ nat @ ( option @ nat ) @ ( vEBT_VEBT_lowi @ B5 @ ( divide_divide @ nat @ Deg4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
                                                      @ ^ [L2: nat] :
                                                          ( heap_Time_bind @ nat @ ( option @ nat ) @ ( vEBT_VEBT_highi @ B5 @ ( divide_divide @ nat @ Deg4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
                                                          @ ^ [H: nat] :
                                                              ( heap_Time_bind @ product_unit @ ( option @ nat )
                                                              @ ( refine_Imp_assert
                                                                @ ( L2
                                                                  = ( vEBT_VEBT_low @ B5 @ ( divide_divide @ nat @ Deg4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
                                                              @ ^ [Uw3: product_unit] :
                                                                  ( heap_Time_bind @ product_unit @ ( option @ nat )
                                                                  @ ( refine_Imp_assert
                                                                    @ ( H
                                                                      = ( vEBT_VEBT_high @ B5 @ ( divide_divide @ nat @ Deg4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
                                                                  @ ^ [Ux3: product_unit] :
                                                                      ( heap_Time_bind @ product_unit @ ( option @ nat ) @ ( refine_Imp_assert @ ( ord_less @ nat @ H @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) ) )
                                                                      @ ^ [Uy3: product_unit] :
                                                                          ( heap_Time_bind @ vEBT_VEBTi @ ( option @ nat ) @ ( array_nth @ vEBT_VEBTi @ TreeArray @ H )
                                                                          @ ^ [Aktnode: vEBT_VEBTi] :
                                                                              ( heap_Time_bind @ ( option @ nat ) @ ( option @ nat ) @ ( vEBT_vebt_maxti @ Aktnode )
                                                                              @ ^ [Maxlow: option @ nat] :
                                                                                  ( heap_Time_bind @ product_unit @ ( option @ nat )
                                                                                  @ ( refine_Imp_assert
                                                                                    @ ( Maxlow
                                                                                      = ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList3 @ H ) ) ) )
                                                                                  @ ^ [Uz3: product_unit] :
                                                                                      ( if @ ( heap_Time_Heap @ ( option @ nat ) )
                                                                                      @ ( ( Maxlow
                                                                                         != ( none @ nat ) )
                                                                                        & ( vEBT_VEBT_less @ ( some @ nat @ L2 ) @ Maxlow ) )
                                                                                      @ ( heap_Time_bind @ ( option @ nat ) @ ( option @ nat ) @ ( F2 @ ( nth @ vEBT_VEBT @ TreeList3 @ H ) @ Aktnode @ L2 )
                                                                                        @ ^ [Succy: option @ nat] : ( heap_Time_return @ ( option @ nat ) @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( some @ nat @ H ) ) @ Succy ) ) )
                                                                                      @ ( heap_Time_bind @ ( option @ nat ) @ ( option @ nat ) @ ( F2 @ Summary4 @ Summary3 @ H )
                                                                                        @ ^ [Succsum: option @ nat] :
                                                                                            ( heap_Time_bind @ product_unit @ ( option @ nat )
                                                                                            @ ( refine_Imp_assert
                                                                                              @ ( ( Succsum
                                                                                                  = ( none @ nat ) )
                                                                                                = ( ( vEBT_vebt_succ @ Summary4 @ H )
                                                                                                  = ( none @ nat ) ) ) )
                                                                                            @ ^ [Va4: product_unit] :
                                                                                                ( if @ ( heap_Time_Heap @ ( option @ nat ) )
                                                                                                @ ( Succsum
                                                                                                  = ( none @ nat ) )
                                                                                                @ ( heap_Time_return @ ( option @ nat ) @ ( none @ nat ) )
                                                                                                @ ( heap_Time_bind @ vEBT_VEBTi @ ( option @ nat ) @ ( array_nth @ vEBT_VEBTi @ TreeArray @ ( the2 @ nat @ Succsum ) )
                                                                                                  @ ^ [Nextnode: vEBT_VEBTi] :
                                                                                                      ( heap_Time_bind @ ( option @ nat ) @ ( option @ nat ) @ ( vEBT_vebt_minti @ Nextnode )
                                                                                                      @ ^ [Minnext: option @ nat] : ( heap_Time_return @ ( option @ nat ) @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ Succsum ) @ Minnext ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
                                            @ Info3 ) ) ) )
                            @ ( vEBT_case_VEBT @ ( product_prod @ ( option @ ( product_prod @ nat @ nat ) ) @ ( product_prod @ nat @ ( product_prod @ ( list @ vEBT_VEBT ) @ vEBT_VEBT ) ) )
                              @ ^ [Info4: option @ ( product_prod @ nat @ nat ),Deg5: nat,TreeList3: list @ vEBT_VEBT,Summary4: vEBT_VEBT] : ( product_Pair @ ( option @ ( product_prod @ nat @ nat ) ) @ ( product_prod @ nat @ ( product_prod @ ( list @ vEBT_VEBT ) @ vEBT_VEBT ) ) @ Info4 @ ( product_Pair @ nat @ ( product_prod @ ( list @ vEBT_VEBT ) @ vEBT_VEBT ) @ Deg5 @ ( product_Pair @ ( list @ vEBT_VEBT ) @ vEBT_VEBT @ TreeList3 @ Summary4 ) ) )
                              @ ^ [C4: $o,D3: $o] : ( undefined @ ( product_prod @ ( option @ ( product_prod @ nat @ nat ) ) @ ( product_prod @ nat @ ( product_prod @ ( list @ vEBT_VEBT ) @ vEBT_VEBT ) ) ) )
                              @ X9 ) ) )
                    @ ^ [C4: $o,D3: $o] :
                        ( if @ ( heap_Time_Heap @ ( option @ nat ) )
                        @ ( B5
                          = ( zero_zero @ nat ) )
                        @ ( if @ ( heap_Time_Heap @ ( option @ nat ) ) @ D3 @ ( heap_Time_return @ ( option @ nat ) @ ( some @ nat @ ( one_one @ nat ) ) ) @ ( heap_Time_return @ ( option @ nat ) @ ( none @ nat ) ) )
                        @ ( heap_Time_return @ ( option @ nat ) @ ( none @ nat ) ) )
                    @ A5 ) ) )
         => ( P @ vEBT_VEBT_vebt_succi ) ) ) ) ).

% VEBT_internal.vebt_succi'.fixp_induct
thf(fact_3050_VEBT__internal_Ovebt__predi_H_Ofixp__induct,axiom,
    ! [P: ( vEBT_VEBT > vEBT_VEBTi > nat > ( heap_Time_Heap @ ( option @ nat ) ) ) > $o] :
      ( ( comple1908693960933563346ssible @ ( ( product_prod @ ( product_prod @ vEBT_VEBT @ vEBT_VEBTi ) @ nat ) > ( heap_Time_Heap @ ( option @ nat ) ) ) @ ( partial_fun_lub @ ( heap_Time_Heap @ ( option @ nat ) ) @ ( heap_Time_Heap @ ( option @ nat ) ) @ ( product_prod @ ( product_prod @ vEBT_VEBT @ vEBT_VEBTi ) @ nat ) @ ( heap_Time_Heap_lub @ ( option @ nat ) ) ) @ ( partial_fun_ord @ ( heap_Time_Heap @ ( option @ nat ) ) @ ( heap_Time_Heap @ ( option @ nat ) ) @ ( product_prod @ ( product_prod @ vEBT_VEBT @ vEBT_VEBTi ) @ nat ) @ ( heap_Time_Heap_ord @ ( option @ nat ) ) )
        @ ^ [Vebt_predi3: ( product_prod @ ( product_prod @ vEBT_VEBT @ vEBT_VEBTi ) @ nat ) > ( heap_Time_Heap @ ( option @ nat ) )] : ( P @ ( product_curry @ vEBT_VEBT @ vEBT_VEBTi @ ( nat > ( heap_Time_Heap @ ( option @ nat ) ) ) @ ( product_curry @ ( product_prod @ vEBT_VEBT @ vEBT_VEBTi ) @ nat @ ( heap_Time_Heap @ ( option @ nat ) ) @ Vebt_predi3 ) ) ) )
     => ( ( P
          @ ^ [Vebt_predi3: vEBT_VEBT,T3: vEBT_VEBTi,Ti3: nat] :
              ( heap_Time_Heap2 @ ( option @ nat )
              @ ^ [X: heap_ext @ product_unit] : ( none @ ( product_prod @ ( option @ nat ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) )
       => ( ! [F2: vEBT_VEBT > vEBT_VEBTi > nat > ( heap_Time_Heap @ ( option @ nat ) )] :
              ( ( P @ F2 )
             => ( P
                @ ^ [X9: vEBT_VEBT,A5: vEBT_VEBTi,B5: nat] :
                    ( vEBT_case_VEBTi @ ( heap_Time_Heap @ ( option @ nat ) )
                    @ ^ [Info3: option @ ( product_prod @ nat @ nat ),Deg4: nat,TreeArray: array @ vEBT_VEBTi,Summary3: vEBT_VEBTi] :
                        ( heap_Time_bind @ product_unit @ ( option @ nat ) @ ( refine_Imp_assert @ ( vEBT_is_Node @ X9 ) )
                        @ ^ [Uu3: product_unit] :
                            ( product_case_prod @ ( option @ ( product_prod @ nat @ nat ) ) @ ( product_prod @ nat @ ( product_prod @ ( list @ vEBT_VEBT ) @ vEBT_VEBT ) ) @ ( heap_Time_Heap @ ( option @ nat ) )
                            @ ^ [Info4: option @ ( product_prod @ nat @ nat )] :
                                ( product_case_prod @ nat @ ( product_prod @ ( list @ vEBT_VEBT ) @ vEBT_VEBT ) @ ( heap_Time_Heap @ ( option @ nat ) )
                                @ ^ [Deg5: nat] :
                                    ( product_case_prod @ ( list @ vEBT_VEBT ) @ vEBT_VEBT @ ( heap_Time_Heap @ ( option @ nat ) )
                                    @ ^ [TreeList3: list @ vEBT_VEBT,Summary4: vEBT_VEBT] :
                                        ( heap_Time_bind @ product_unit @ ( option @ nat )
                                        @ ( refine_Imp_assert
                                          @ ( ( Info4 = Info3 )
                                            & ( Deg5 = Deg4 )
                                            & ( vEBT_is_Node @ X9 ) ) )
                                        @ ^ [Uv3: product_unit] :
                                            ( case_option @ ( heap_Time_Heap @ ( option @ nat ) ) @ ( product_prod @ nat @ nat ) @ ( heap_Time_return @ ( option @ nat ) @ ( none @ nat ) )
                                            @ ^ [Mima2: product_prod @ nat @ nat] :
                                                ( if @ ( heap_Time_Heap @ ( option @ nat ) ) @ ( ord_less_eq @ nat @ Deg4 @ ( one_one @ nat ) ) @ ( heap_Time_return @ ( option @ nat ) @ ( none @ nat ) )
                                                @ ( if @ ( heap_Time_Heap @ ( option @ nat ) ) @ ( ord_less @ nat @ ( product_snd @ nat @ nat @ Mima2 ) @ B5 ) @ ( heap_Time_return @ ( option @ nat ) @ ( some @ nat @ ( product_snd @ nat @ nat @ Mima2 ) ) )
                                                  @ ( heap_Time_bind @ nat @ ( option @ nat ) @ ( vEBT_VEBT_lowi @ B5 @ ( divide_divide @ nat @ Deg4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
                                                    @ ^ [L2: nat] :
                                                        ( heap_Time_bind @ nat @ ( option @ nat ) @ ( vEBT_VEBT_highi @ B5 @ ( divide_divide @ nat @ Deg4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
                                                        @ ^ [H: nat] :
                                                            ( heap_Time_bind @ product_unit @ ( option @ nat )
                                                            @ ( refine_Imp_assert
                                                              @ ( L2
                                                                = ( vEBT_VEBT_low @ B5 @ ( divide_divide @ nat @ Deg4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
                                                            @ ^ [Uw3: product_unit] :
                                                                ( heap_Time_bind @ product_unit @ ( option @ nat )
                                                                @ ( refine_Imp_assert
                                                                  @ ( H
                                                                    = ( vEBT_VEBT_high @ B5 @ ( divide_divide @ nat @ Deg4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
                                                                @ ^ [Ux3: product_unit] :
                                                                    ( heap_Time_bind @ product_unit @ ( option @ nat ) @ ( refine_Imp_assert @ ( ord_less @ nat @ H @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) ) )
                                                                    @ ^ [Uy3: product_unit] :
                                                                        ( heap_Time_bind @ vEBT_VEBTi @ ( option @ nat ) @ ( array_nth @ vEBT_VEBTi @ TreeArray @ H )
                                                                        @ ^ [Aktnode: vEBT_VEBTi] :
                                                                            ( heap_Time_bind @ ( option @ nat ) @ ( option @ nat ) @ ( vEBT_vebt_minti @ Aktnode )
                                                                            @ ^ [Minlow: option @ nat] :
                                                                                ( heap_Time_bind @ product_unit @ ( option @ nat )
                                                                                @ ( refine_Imp_assert
                                                                                  @ ( Minlow
                                                                                    = ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList3 @ H ) ) ) )
                                                                                @ ^ [Uz3: product_unit] :
                                                                                    ( if @ ( heap_Time_Heap @ ( option @ nat ) )
                                                                                    @ ( ( Minlow
                                                                                       != ( none @ nat ) )
                                                                                      & ( vEBT_VEBT_greater @ ( some @ nat @ L2 ) @ Minlow ) )
                                                                                    @ ( heap_Time_bind @ ( option @ nat ) @ ( option @ nat ) @ ( F2 @ ( nth @ vEBT_VEBT @ TreeList3 @ H ) @ Aktnode @ L2 )
                                                                                      @ ^ [Predy: option @ nat] : ( heap_Time_return @ ( option @ nat ) @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( some @ nat @ H ) ) @ Predy ) ) )
                                                                                    @ ( heap_Time_bind @ ( option @ nat ) @ ( option @ nat ) @ ( F2 @ Summary4 @ Summary3 @ H )
                                                                                      @ ^ [Predsum: option @ nat] :
                                                                                          ( heap_Time_bind @ product_unit @ ( option @ nat )
                                                                                          @ ( refine_Imp_assert
                                                                                            @ ( ( Predsum
                                                                                                = ( none @ nat ) )
                                                                                              = ( ( vEBT_vebt_pred @ Summary4 @ H )
                                                                                                = ( none @ nat ) ) ) )
                                                                                          @ ^ [Va4: product_unit] :
                                                                                              ( if @ ( heap_Time_Heap @ ( option @ nat ) )
                                                                                              @ ( Predsum
                                                                                                = ( none @ nat ) )
                                                                                              @ ( if @ ( heap_Time_Heap @ ( option @ nat ) ) @ ( ord_less @ nat @ ( product_fst @ nat @ nat @ Mima2 ) @ B5 ) @ ( heap_Time_return @ ( option @ nat ) @ ( some @ nat @ ( product_fst @ nat @ nat @ Mima2 ) ) ) @ ( heap_Time_return @ ( option @ nat ) @ ( none @ nat ) ) )
                                                                                              @ ( heap_Time_bind @ vEBT_VEBTi @ ( option @ nat ) @ ( array_nth @ vEBT_VEBTi @ TreeArray @ ( the2 @ nat @ Predsum ) )
                                                                                                @ ^ [Nextnode: vEBT_VEBTi] :
                                                                                                    ( heap_Time_bind @ ( option @ nat ) @ ( option @ nat ) @ ( vEBT_vebt_maxti @ Nextnode )
                                                                                                    @ ^ [Maxnext: option @ nat] : ( heap_Time_return @ ( option @ nat ) @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ Predsum ) @ Maxnext ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
                                            @ Info3 ) ) ) )
                            @ ( vEBT_case_VEBT @ ( product_prod @ ( option @ ( product_prod @ nat @ nat ) ) @ ( product_prod @ nat @ ( product_prod @ ( list @ vEBT_VEBT ) @ vEBT_VEBT ) ) )
                              @ ^ [Info4: option @ ( product_prod @ nat @ nat ),Deg5: nat,TreeList3: list @ vEBT_VEBT,Summary4: vEBT_VEBT] : ( product_Pair @ ( option @ ( product_prod @ nat @ nat ) ) @ ( product_prod @ nat @ ( product_prod @ ( list @ vEBT_VEBT ) @ vEBT_VEBT ) ) @ Info4 @ ( product_Pair @ nat @ ( product_prod @ ( list @ vEBT_VEBT ) @ vEBT_VEBT ) @ Deg5 @ ( product_Pair @ ( list @ vEBT_VEBT ) @ vEBT_VEBT @ TreeList3 @ Summary4 ) ) )
                              @ ^ [C4: $o,D3: $o] : ( undefined @ ( product_prod @ ( option @ ( product_prod @ nat @ nat ) ) @ ( product_prod @ nat @ ( product_prod @ ( list @ vEBT_VEBT ) @ vEBT_VEBT ) ) ) )
                              @ X9 ) ) )
                    @ ^ [C4: $o,D3: $o] :
                        ( if @ ( heap_Time_Heap @ ( option @ nat ) ) @ ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ B5 ) @ ( if @ ( heap_Time_Heap @ ( option @ nat ) ) @ D3 @ ( heap_Time_return @ ( option @ nat ) @ ( some @ nat @ ( one_one @ nat ) ) ) @ ( if @ ( heap_Time_Heap @ ( option @ nat ) ) @ C4 @ ( heap_Time_return @ ( option @ nat ) @ ( some @ nat @ ( zero_zero @ nat ) ) ) @ ( heap_Time_return @ ( option @ nat ) @ ( none @ nat ) ) ) )
                        @ ( if @ ( heap_Time_Heap @ ( option @ nat ) )
                          @ ( B5
                            = ( one_one @ nat ) )
                          @ ( if @ ( heap_Time_Heap @ ( option @ nat ) ) @ C4 @ ( heap_Time_return @ ( option @ nat ) @ ( some @ nat @ ( zero_zero @ nat ) ) ) @ ( heap_Time_return @ ( option @ nat ) @ ( none @ nat ) ) )
                          @ ( heap_Time_return @ ( option @ nat ) @ ( none @ nat ) ) ) )
                    @ A5 ) ) )
         => ( P @ vEBT_VEBT_vebt_predi ) ) ) ) ).

% VEBT_internal.vebt_predi'.fixp_induct
thf(fact_3051_VEBT__internal_Ovebt__buildupi_H_Opelims,axiom,
    ! [X2: nat,Y2: heap_Time_Heap @ vEBT_VEBTi] :
      ( ( ( vEBT_V739175172307565963ildupi @ X2 )
        = Y2 )
     => ( ( accp @ nat @ vEBT_V254170901696579886pi_rel @ X2 )
       => ( ( ( X2
              = ( zero_zero @ nat ) )
           => ( ( Y2
                = ( heap_Time_return @ vEBT_VEBTi @ ( vEBT_Leafi @ $false @ $false ) ) )
             => ~ ( accp @ nat @ vEBT_V254170901696579886pi_rel @ ( zero_zero @ nat ) ) ) )
         => ( ( ( X2
                = ( suc @ ( zero_zero @ nat ) ) )
             => ( ( Y2
                  = ( heap_Time_return @ vEBT_VEBTi @ ( vEBT_Leafi @ $false @ $false ) ) )
               => ~ ( accp @ nat @ vEBT_V254170901696579886pi_rel @ ( suc @ ( zero_zero @ nat ) ) ) ) )
           => ~ ! [Va3: nat] :
                  ( ( X2
                    = ( suc @ ( suc @ Va3 ) ) )
                 => ( ( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ Va3 ) ) )
                       => ( Y2
                          = ( heap_Time_bind @ ( list @ vEBT_VEBTi ) @ vEBT_VEBTi @ ( vEBT_VEBT_replicatei @ vEBT_VEBTi @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_V739175172307565963ildupi @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
                            @ ^ [TreeList3: list @ vEBT_VEBTi] :
                                ( heap_Time_bind @ product_unit @ vEBT_VEBTi
                                @ ( refine_Imp_assert
                                  @ ( ( size_size @ ( list @ vEBT_VEBTi ) @ TreeList3 )
                                    = ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
                                @ ^ [Uu3: product_unit] :
                                    ( heap_Time_bind @ ( array @ vEBT_VEBTi ) @ vEBT_VEBTi @ ( array_of_list @ vEBT_VEBTi @ TreeList3 )
                                    @ ^ [Trees: array @ vEBT_VEBTi] :
                                        ( heap_Time_bind @ vEBT_VEBTi @ vEBT_VEBTi @ ( vEBT_V739175172307565963ildupi @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
                                        @ ^ [Summary3: vEBT_VEBTi] : ( heap_Time_return @ vEBT_VEBTi @ ( vEBT_Nodei @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ Va3 ) ) @ Trees @ Summary3 ) ) ) ) ) ) ) )
                      & ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ Va3 ) ) )
                       => ( Y2
                          = ( heap_Time_bind @ ( list @ vEBT_VEBTi ) @ vEBT_VEBTi @ ( vEBT_VEBT_replicatei @ vEBT_VEBTi @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_V739175172307565963ildupi @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
                            @ ^ [TreeList3: list @ vEBT_VEBTi] :
                                ( heap_Time_bind @ product_unit @ vEBT_VEBTi
                                @ ( refine_Imp_assert
                                  @ ( ( size_size @ ( list @ vEBT_VEBTi ) @ TreeList3 )
                                    = ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) )
                                @ ^ [Uu3: product_unit] :
                                    ( heap_Time_bind @ ( array @ vEBT_VEBTi ) @ vEBT_VEBTi @ ( array_of_list @ vEBT_VEBTi @ TreeList3 )
                                    @ ^ [Trees: array @ vEBT_VEBTi] :
                                        ( heap_Time_bind @ vEBT_VEBTi @ vEBT_VEBTi @ ( vEBT_V739175172307565963ildupi @ ( suc @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
                                        @ ^ [Summary3: vEBT_VEBTi] : ( heap_Time_return @ vEBT_VEBTi @ ( vEBT_Nodei @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ Va3 ) ) @ Trees @ Summary3 ) ) ) ) ) ) ) ) )
                   => ~ ( accp @ nat @ vEBT_V254170901696579886pi_rel @ ( suc @ ( suc @ Va3 ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.vebt_buildupi'.pelims
thf(fact_3052_prod__decode__aux_Ocases,axiom,
    ! [X2: product_prod @ nat @ nat] :
      ~ ! [K3: nat,M2: nat] :
          ( X2
         != ( product_Pair @ nat @ nat @ K3 @ M2 ) ) ).

% prod_decode_aux.cases
thf(fact_3053_list__decode_Ocases,axiom,
    ! [X2: nat] :
      ( ( X2
       != ( zero_zero @ nat ) )
     => ~ ! [N4: nat] :
            ( X2
           != ( suc @ N4 ) ) ) ).

% list_decode.cases
thf(fact_3054_vebt__buildupi_Opelims,axiom,
    ! [X2: nat,Y2: heap_Time_Heap @ vEBT_VEBTi] :
      ( ( ( vEBT_vebt_buildupi @ X2 )
        = Y2 )
     => ( ( accp @ nat @ vEBT_v1230518104690509829pi_rel @ X2 )
       => ( ( ( X2
              = ( zero_zero @ nat ) )
           => ( ( Y2
                = ( heap_Time_return @ vEBT_VEBTi @ ( vEBT_Leafi @ $false @ $false ) ) )
             => ~ ( accp @ nat @ vEBT_v1230518104690509829pi_rel @ ( zero_zero @ nat ) ) ) )
         => ( ( ( X2
                = ( suc @ ( zero_zero @ nat ) ) )
             => ( ( Y2
                  = ( heap_Time_return @ vEBT_VEBTi @ ( vEBT_Leafi @ $false @ $false ) ) )
               => ~ ( accp @ nat @ vEBT_v1230518104690509829pi_rel @ ( suc @ ( zero_zero @ nat ) ) ) ) )
           => ~ ! [Va3: nat] :
                  ( ( X2
                    = ( suc @ ( suc @ Va3 ) ) )
                 => ( ( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ Va3 ) ) )
                       => ( Y2
                          = ( heap_Time_bind @ ( list @ vEBT_VEBTi ) @ vEBT_VEBTi @ ( vEBT_VEBT_replicatei @ vEBT_VEBTi @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_buildupi @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
                            @ ^ [TreeList3: list @ vEBT_VEBTi] :
                                ( heap_Time_bind @ ( array @ vEBT_VEBTi ) @ vEBT_VEBTi @ ( array_of_list @ vEBT_VEBTi @ TreeList3 )
                                @ ^ [Trees: array @ vEBT_VEBTi] :
                                    ( heap_Time_bind @ vEBT_VEBTi @ vEBT_VEBTi @ ( vEBT_vebt_buildupi @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
                                    @ ^ [Summary3: vEBT_VEBTi] : ( heap_Time_return @ vEBT_VEBTi @ ( vEBT_Nodei @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ Va3 ) ) @ Trees @ Summary3 ) ) ) ) ) ) )
                      & ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ Va3 ) ) )
                       => ( Y2
                          = ( heap_Time_bind @ ( list @ vEBT_VEBTi ) @ vEBT_VEBTi @ ( vEBT_VEBT_replicatei @ vEBT_VEBTi @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_buildupi @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
                            @ ^ [TreeList3: list @ vEBT_VEBTi] :
                                ( heap_Time_bind @ ( array @ vEBT_VEBTi ) @ vEBT_VEBTi @ ( array_of_list @ vEBT_VEBTi @ TreeList3 )
                                @ ^ [Trees: array @ vEBT_VEBTi] :
                                    ( heap_Time_bind @ vEBT_VEBTi @ vEBT_VEBTi @ ( vEBT_vebt_buildupi @ ( suc @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
                                    @ ^ [Summary3: vEBT_VEBTi] : ( heap_Time_return @ vEBT_VEBTi @ ( vEBT_Nodei @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ Va3 ) ) @ Trees @ Summary3 ) ) ) ) ) ) ) )
                   => ~ ( accp @ nat @ vEBT_v1230518104690509829pi_rel @ ( suc @ ( suc @ Va3 ) ) ) ) ) ) ) ) ) ).

% vebt_buildupi.pelims
thf(fact_3055_VEBT__internal_OT__vebt__buildupi_H_Opelims,axiom,
    ! [X2: nat,Y2: int] :
      ( ( ( vEBT_V9176841429113362141ildupi @ X2 )
        = Y2 )
     => ( ( accp @ nat @ vEBT_V3352910403632780892pi_rel @ X2 )
       => ( ( ( X2
              = ( zero_zero @ nat ) )
           => ( ( Y2
                = ( one_one @ int ) )
             => ~ ( accp @ nat @ vEBT_V3352910403632780892pi_rel @ ( zero_zero @ nat ) ) ) )
         => ( ( ( X2
                = ( suc @ ( zero_zero @ nat ) ) )
             => ( ( Y2
                  = ( one_one @ int ) )
               => ~ ( accp @ nat @ vEBT_V3352910403632780892pi_rel @ ( suc @ ( zero_zero @ nat ) ) ) ) )
           => ~ ! [N4: nat] :
                  ( ( X2
                    = ( suc @ ( suc @ N4 ) ) )
                 => ( ( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N4 )
                       => ( Y2
                          = ( plus_plus @ int @ ( numeral_numeral @ int @ ( bit1 @ one2 ) ) @ ( plus_plus @ int @ ( vEBT_V9176841429113362141ildupi @ ( suc @ ( divide_divide @ nat @ N4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( plus_plus @ int @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ ( bit0 @ one2 ) ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ N4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( times_times @ int @ ( vEBT_V9176841429113362141ildupi @ ( suc @ ( divide_divide @ nat @ N4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ N4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) )
                      & ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N4 )
                       => ( Y2
                          = ( plus_plus @ int @ ( numeral_numeral @ int @ ( bit1 @ one2 ) ) @ ( plus_plus @ int @ ( vEBT_V9176841429113362141ildupi @ ( suc @ ( suc @ ( divide_divide @ nat @ N4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( plus_plus @ int @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ N4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ ( bit0 @ one2 ) ) ) @ ( times_times @ int @ ( vEBT_V9176841429113362141ildupi @ ( suc @ ( divide_divide @ nat @ N4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ N4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) )
                   => ~ ( accp @ nat @ vEBT_V3352910403632780892pi_rel @ ( suc @ ( suc @ N4 ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.T_vebt_buildupi'.pelims
thf(fact_3056_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,
    ! [X2: nat,Y2: nat] :
      ( ( ( vEBT_V8646137997579335489_i_l_d @ X2 )
        = Y2 )
     => ( ( accp @ nat @ vEBT_V5144397997797733112_d_rel @ X2 )
       => ( ( ( X2
              = ( zero_zero @ nat ) )
           => ( ( Y2
                = ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) )
             => ~ ( accp @ nat @ vEBT_V5144397997797733112_d_rel @ ( zero_zero @ nat ) ) ) )
         => ( ( ( X2
                = ( suc @ ( zero_zero @ nat ) ) )
             => ( ( Y2
                  = ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) )
               => ~ ( accp @ nat @ vEBT_V5144397997797733112_d_rel @ ( suc @ ( zero_zero @ nat ) ) ) ) )
           => ~ ! [Va3: nat] :
                  ( ( X2
                    = ( suc @ ( suc @ Va3 ) ) )
                 => ( ( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ Va3 ) ) )
                       => ( Y2
                          = ( plus_plus @ nat @ ( one_one @ nat ) @ ( plus_plus @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit1 @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_V8646137997579335489_i_l_d @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( times_times @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_V8646137997579335489_i_l_d @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) )
                      & ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ Va3 ) ) )
                       => ( Y2
                          = ( plus_plus @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ ( bit1 @ one2 ) ) ) ) @ ( vEBT_V8646137997579335489_i_l_d @ ( suc @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( times_times @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_V8646137997579335489_i_l_d @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) )
                   => ~ ( accp @ nat @ vEBT_V5144397997797733112_d_rel @ ( suc @ ( suc @ Va3 ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d.pelims
thf(fact_3057_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,
    ! [X2: nat,Y2: nat] :
      ( ( ( vEBT_V8346862874174094_d_u_p @ X2 )
        = Y2 )
     => ( ( accp @ nat @ vEBT_V1247956027447740395_p_rel @ X2 )
       => ( ( ( X2
              = ( zero_zero @ nat ) )
           => ( ( Y2
                = ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) )
             => ~ ( accp @ nat @ vEBT_V1247956027447740395_p_rel @ ( zero_zero @ nat ) ) ) )
         => ( ( ( X2
                = ( suc @ ( zero_zero @ nat ) ) )
             => ( ( Y2
                  = ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) )
               => ~ ( accp @ nat @ vEBT_V1247956027447740395_p_rel @ ( suc @ ( zero_zero @ nat ) ) ) ) )
           => ~ ! [Va3: nat] :
                  ( ( X2
                    = ( suc @ ( suc @ Va3 ) ) )
                 => ( ( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ Va3 ) ) )
                       => ( Y2
                          = ( plus_plus @ nat @ ( one_one @ nat ) @ ( plus_plus @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_V8346862874174094_d_u_p @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( times_times @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( plus_plus @ nat @ ( vEBT_V8346862874174094_d_u_p @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( one_one @ nat ) ) ) ) ) ) )
                      & ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ Va3 ) ) )
                       => ( Y2
                          = ( plus_plus @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit1 @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_V8346862874174094_d_u_p @ ( suc @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( times_times @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( plus_plus @ nat @ ( vEBT_V8346862874174094_d_u_p @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( one_one @ nat ) ) ) ) ) ) )
                   => ~ ( accp @ nat @ vEBT_V1247956027447740395_p_rel @ ( suc @ ( suc @ Va3 ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d\<^sub>u\<^sub>p.pelims
thf(fact_3058_VEBT__internal_OTb_Opelims,axiom,
    ! [X2: nat,Y2: int] :
      ( ( ( vEBT_VEBT_Tb @ X2 )
        = Y2 )
     => ( ( accp @ nat @ vEBT_VEBT_Tb_rel2 @ X2 )
       => ( ( ( X2
              = ( zero_zero @ nat ) )
           => ( ( Y2
                = ( numeral_numeral @ int @ ( bit1 @ one2 ) ) )
             => ~ ( accp @ nat @ vEBT_VEBT_Tb_rel2 @ ( zero_zero @ nat ) ) ) )
         => ( ( ( X2
                = ( suc @ ( zero_zero @ nat ) ) )
             => ( ( Y2
                  = ( numeral_numeral @ int @ ( bit1 @ one2 ) ) )
               => ~ ( accp @ nat @ vEBT_VEBT_Tb_rel2 @ ( suc @ ( zero_zero @ nat ) ) ) ) )
           => ~ ! [N4: nat] :
                  ( ( X2
                    = ( suc @ ( suc @ N4 ) ) )
                 => ( ( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N4 )
                       => ( Y2
                          = ( plus_plus @ int @ ( plus_plus @ int @ ( numeral_numeral @ int @ ( bit1 @ ( bit0 @ one2 ) ) ) @ ( vEBT_VEBT_Tb @ ( suc @ ( divide_divide @ nat @ N4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( times_times @ int @ ( vEBT_VEBT_Tb @ ( suc @ ( divide_divide @ nat @ N4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( suc @ ( divide_divide @ nat @ N4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) )
                      & ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N4 )
                       => ( Y2
                          = ( plus_plus @ int @ ( plus_plus @ int @ ( numeral_numeral @ int @ ( bit1 @ ( bit0 @ one2 ) ) ) @ ( vEBT_VEBT_Tb @ ( suc @ ( suc @ ( divide_divide @ nat @ N4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) @ ( times_times @ int @ ( vEBT_VEBT_Tb @ ( suc @ ( divide_divide @ nat @ N4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ ( divide_divide @ nat @ N4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) )
                   => ~ ( accp @ nat @ vEBT_VEBT_Tb_rel2 @ ( suc @ ( suc @ N4 ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.Tb.pelims
thf(fact_3059_VEBT__internal_OTb_H_Opelims,axiom,
    ! [X2: nat,Y2: nat] :
      ( ( ( vEBT_VEBT_Tb2 @ X2 )
        = Y2 )
     => ( ( accp @ nat @ vEBT_VEBT_Tb_rel @ X2 )
       => ( ( ( X2
              = ( zero_zero @ nat ) )
           => ( ( Y2
                = ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) )
             => ~ ( accp @ nat @ vEBT_VEBT_Tb_rel @ ( zero_zero @ nat ) ) ) )
         => ( ( ( X2
                = ( suc @ ( zero_zero @ nat ) ) )
             => ( ( Y2
                  = ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) )
               => ~ ( accp @ nat @ vEBT_VEBT_Tb_rel @ ( suc @ ( zero_zero @ nat ) ) ) ) )
           => ~ ! [N4: nat] :
                  ( ( X2
                    = ( suc @ ( suc @ N4 ) ) )
                 => ( ( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N4 )
                       => ( Y2
                          = ( plus_plus @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit0 @ one2 ) ) ) @ ( vEBT_VEBT_Tb2 @ ( suc @ ( divide_divide @ nat @ N4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( times_times @ nat @ ( vEBT_VEBT_Tb2 @ ( suc @ ( divide_divide @ nat @ N4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( divide_divide @ nat @ N4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) )
                      & ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N4 )
                       => ( Y2
                          = ( plus_plus @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit0 @ one2 ) ) ) @ ( vEBT_VEBT_Tb2 @ ( suc @ ( suc @ ( divide_divide @ nat @ N4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) @ ( times_times @ nat @ ( vEBT_VEBT_Tb2 @ ( suc @ ( divide_divide @ nat @ N4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ ( divide_divide @ nat @ N4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) )
                   => ~ ( accp @ nat @ vEBT_VEBT_Tb_rel @ ( suc @ ( suc @ N4 ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.Tb'.pelims
thf(fact_3060_VEBT__internal_OT__vebt__buildupi_Opelims,axiom,
    ! [X2: nat,Y2: nat] :
      ( ( ( vEBT_V441764108873111860ildupi @ X2 )
        = Y2 )
     => ( ( accp @ nat @ vEBT_V2957053500504383685pi_rel @ X2 )
       => ( ( ( X2
              = ( zero_zero @ nat ) )
           => ( ( Y2
                = ( suc @ ( zero_zero @ nat ) ) )
             => ~ ( accp @ nat @ vEBT_V2957053500504383685pi_rel @ ( zero_zero @ nat ) ) ) )
         => ( ( ( X2
                = ( suc @ ( zero_zero @ nat ) ) )
             => ( ( Y2
                  = ( suc @ ( zero_zero @ nat ) ) )
               => ~ ( accp @ nat @ vEBT_V2957053500504383685pi_rel @ ( suc @ ( zero_zero @ nat ) ) ) ) )
           => ~ ! [N4: nat] :
                  ( ( X2
                    = ( suc @ ( suc @ N4 ) ) )
                 => ( ( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N4 )
                       => ( Y2
                          = ( suc @ ( suc @ ( suc @ ( plus_plus @ nat @ ( vEBT_V441764108873111860ildupi @ ( suc @ ( divide_divide @ nat @ N4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ N4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( times_times @ nat @ ( vEBT_V441764108873111860ildupi @ ( suc @ ( divide_divide @ nat @ N4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ N4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) )
                      & ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N4 )
                       => ( Y2
                          = ( suc @ ( suc @ ( suc @ ( plus_plus @ nat @ ( vEBT_V441764108873111860ildupi @ ( suc @ ( suc @ ( divide_divide @ nat @ N4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ N4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) @ ( times_times @ nat @ ( vEBT_V441764108873111860ildupi @ ( suc @ ( divide_divide @ nat @ N4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ N4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) )
                   => ~ ( accp @ nat @ vEBT_V2957053500504383685pi_rel @ ( suc @ ( suc @ N4 ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.T_vebt_buildupi.pelims
thf(fact_3061_VEBT__internal_Ovebt__memberi_H_Ofixp__induct,axiom,
    ! [P: ( vEBT_VEBT > vEBT_VEBTi > nat > ( heap_Time_Heap @ $o ) ) > $o] :
      ( ( comple1908693960933563346ssible @ ( ( product_prod @ ( product_prod @ vEBT_VEBT @ vEBT_VEBTi ) @ nat ) > ( heap_Time_Heap @ $o ) ) @ ( partial_fun_lub @ ( heap_Time_Heap @ $o ) @ ( heap_Time_Heap @ $o ) @ ( product_prod @ ( product_prod @ vEBT_VEBT @ vEBT_VEBTi ) @ nat ) @ ( heap_Time_Heap_lub @ $o ) ) @ ( partial_fun_ord @ ( heap_Time_Heap @ $o ) @ ( heap_Time_Heap @ $o ) @ ( product_prod @ ( product_prod @ vEBT_VEBT @ vEBT_VEBTi ) @ nat ) @ ( heap_Time_Heap_ord @ $o ) )
        @ ^ [Vebt_memberi4: ( product_prod @ ( product_prod @ vEBT_VEBT @ vEBT_VEBTi ) @ nat ) > ( heap_Time_Heap @ $o )] : ( P @ ( product_curry @ vEBT_VEBT @ vEBT_VEBTi @ ( nat > ( heap_Time_Heap @ $o ) ) @ ( product_curry @ ( product_prod @ vEBT_VEBT @ vEBT_VEBTi ) @ nat @ ( heap_Time_Heap @ $o ) @ Vebt_memberi4 ) ) ) )
     => ( ( P
          @ ^ [Vebt_memberi4: vEBT_VEBT,T3: vEBT_VEBTi,Ti3: nat] :
              ( heap_Time_Heap2 @ $o
              @ ^ [X: heap_ext @ product_unit] : ( none @ ( product_prod @ $o @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) )
       => ( ! [F2: vEBT_VEBT > vEBT_VEBTi > nat > ( heap_Time_Heap @ $o )] :
              ( ( P @ F2 )
             => ( P
                @ ^ [X9: vEBT_VEBT,A5: vEBT_VEBTi,B5: nat] :
                    ( vEBT_case_VEBTi @ ( heap_Time_Heap @ $o )
                    @ ^ [Info3: option @ ( product_prod @ nat @ nat ),Deg4: nat,TreeArray: array @ vEBT_VEBTi,Summary3: vEBT_VEBTi] :
                        ( heap_Time_bind @ product_unit @ $o @ ( refine_Imp_assert @ ( vEBT_is_Node @ X9 ) )
                        @ ^ [Uu3: product_unit] :
                            ( case_option @ ( heap_Time_Heap @ $o ) @ ( product_prod @ nat @ nat ) @ ( heap_Time_return @ $o @ $false )
                            @ ( product_case_prod @ nat @ nat @ ( heap_Time_Heap @ $o )
                              @ ^ [Mi3: nat,Ma3: nat] :
                                  ( if @ ( heap_Time_Heap @ $o ) @ ( ord_less_eq @ nat @ Deg4 @ ( one_one @ nat ) ) @ ( heap_Time_return @ $o @ $false )
                                  @ ( if @ ( heap_Time_Heap @ $o ) @ ( B5 = Mi3 ) @ ( heap_Time_return @ $o @ $true )
                                    @ ( if @ ( heap_Time_Heap @ $o ) @ ( B5 = Ma3 ) @ ( heap_Time_return @ $o @ $true )
                                      @ ( if @ ( heap_Time_Heap @ $o ) @ ( ord_less @ nat @ B5 @ Mi3 ) @ ( heap_Time_return @ $o @ $false )
                                        @ ( if @ ( heap_Time_Heap @ $o ) @ ( ord_less @ nat @ Ma3 @ B5 ) @ ( heap_Time_return @ $o @ $false )
                                          @ ( product_case_prod @ ( option @ ( product_prod @ nat @ nat ) ) @ ( product_prod @ nat @ ( product_prod @ ( list @ vEBT_VEBT ) @ vEBT_VEBT ) ) @ ( heap_Time_Heap @ $o )
                                            @ ^ [Info4: option @ ( product_prod @ nat @ nat )] :
                                                ( product_case_prod @ nat @ ( product_prod @ ( list @ vEBT_VEBT ) @ vEBT_VEBT ) @ ( heap_Time_Heap @ $o )
                                                @ ^ [Deg5: nat] :
                                                    ( product_case_prod @ ( list @ vEBT_VEBT ) @ vEBT_VEBT @ ( heap_Time_Heap @ $o )
                                                    @ ^ [TreeList3: list @ vEBT_VEBT,Summary4: vEBT_VEBT] :
                                                        ( heap_Time_bind @ product_unit @ $o
                                                        @ ( refine_Imp_assert
                                                          @ ( ( Info3 = Info4 )
                                                            & ( Deg4 = Deg5 ) ) )
                                                        @ ^ [Uv3: product_unit] :
                                                            ( heap_Time_bind @ nat @ $o @ ( vEBT_VEBT_highi @ B5 @ ( divide_divide @ nat @ Deg4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
                                                            @ ^ [H: nat] :
                                                                ( heap_Time_bind @ nat @ $o @ ( vEBT_VEBT_lowi @ B5 @ ( divide_divide @ nat @ Deg4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
                                                                @ ^ [L2: nat] :
                                                                    ( heap_Time_bind @ product_unit @ $o
                                                                    @ ( refine_Imp_assert
                                                                      @ ( ( L2
                                                                          = ( vEBT_VEBT_low @ B5 @ ( divide_divide @ nat @ Deg4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
                                                                        & ( H
                                                                          = ( vEBT_VEBT_high @ B5 @ ( divide_divide @ nat @ Deg4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) )
                                                                    @ ^ [Uw3: product_unit] :
                                                                        ( heap_Time_bind @ nat @ $o @ ( array_len @ vEBT_VEBTi @ TreeArray )
                                                                        @ ^ [Len: nat] :
                                                                            ( heap_Time_bind @ product_unit @ $o
                                                                            @ ( refine_Imp_assert
                                                                              @ ( Len
                                                                                = ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) ) )
                                                                            @ ^ [Ux3: product_unit] :
                                                                                ( if @ ( heap_Time_Heap @ $o ) @ ( ord_less @ nat @ H @ Len )
                                                                                @ ( heap_Time_bind @ product_unit @ $o
                                                                                  @ ( refine_Imp_assert
                                                                                    @ ( ( H
                                                                                        = ( vEBT_VEBT_high @ B5 @ ( divide_divide @ nat @ Deg4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
                                                                                      & ( ord_less @ nat @ H @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) ) ) )
                                                                                  @ ^ [Uy3: product_unit] :
                                                                                      ( heap_Time_bind @ vEBT_VEBTi @ $o @ ( array_nth @ vEBT_VEBTi @ TreeArray @ H )
                                                                                      @ ^ [Th: vEBT_VEBTi] : ( F2 @ ( nth @ vEBT_VEBT @ TreeList3 @ H ) @ Th @ L2 ) ) )
                                                                                @ ( heap_Time_return @ $o @ $false ) ) ) ) ) ) ) ) ) )
                                            @ ( vEBT_case_VEBT @ ( product_prod @ ( option @ ( product_prod @ nat @ nat ) ) @ ( product_prod @ nat @ ( product_prod @ ( list @ vEBT_VEBT ) @ vEBT_VEBT ) ) )
                                              @ ^ [Info4: option @ ( product_prod @ nat @ nat ),Deg5: nat,TreeList3: list @ vEBT_VEBT,Summary4: vEBT_VEBT] : ( product_Pair @ ( option @ ( product_prod @ nat @ nat ) ) @ ( product_prod @ nat @ ( product_prod @ ( list @ vEBT_VEBT ) @ vEBT_VEBT ) ) @ Info4 @ ( product_Pair @ nat @ ( product_prod @ ( list @ vEBT_VEBT ) @ vEBT_VEBT ) @ Deg5 @ ( product_Pair @ ( list @ vEBT_VEBT ) @ vEBT_VEBT @ TreeList3 @ Summary4 ) ) )
                                              @ ^ [C4: $o,D3: $o] : ( undefined @ ( product_prod @ ( option @ ( product_prod @ nat @ nat ) ) @ ( product_prod @ nat @ ( product_prod @ ( list @ vEBT_VEBT ) @ vEBT_VEBT ) ) ) )
                                              @ X9 ) ) ) ) ) ) ) )
                            @ Info3 ) )
                    @ ^ [C4: $o,D3: $o] :
                        ( heap_Time_return @ $o
                        @ ( ( ( B5
                              = ( zero_zero @ nat ) )
                           => C4 )
                          & ( ( B5
                             != ( zero_zero @ nat ) )
                           => ( ( ( B5
                                  = ( one_one @ nat ) )
                               => D3 )
                              & ( B5
                                = ( one_one @ nat ) ) ) ) ) )
                    @ A5 ) ) )
         => ( P @ vEBT_V854960066525838166emberi ) ) ) ) ).

% VEBT_internal.vebt_memberi'.fixp_induct
thf(fact_3062_vebt__predi_Ofixp__induct,axiom,
    ! [P: ( vEBT_VEBTi > nat > ( heap_Time_Heap @ ( option @ nat ) ) ) > $o] :
      ( ( comple1908693960933563346ssible @ ( ( product_prod @ vEBT_VEBTi @ nat ) > ( heap_Time_Heap @ ( option @ nat ) ) ) @ ( partial_fun_lub @ ( heap_Time_Heap @ ( option @ nat ) ) @ ( heap_Time_Heap @ ( option @ nat ) ) @ ( product_prod @ vEBT_VEBTi @ nat ) @ ( heap_Time_Heap_lub @ ( option @ nat ) ) ) @ ( partial_fun_ord @ ( heap_Time_Heap @ ( option @ nat ) ) @ ( heap_Time_Heap @ ( option @ nat ) ) @ ( product_prod @ vEBT_VEBTi @ nat ) @ ( heap_Time_Heap_ord @ ( option @ nat ) ) )
        @ ^ [Vebt_predi4: ( product_prod @ vEBT_VEBTi @ nat ) > ( heap_Time_Heap @ ( option @ nat ) )] : ( P @ ( product_curry @ vEBT_VEBTi @ nat @ ( heap_Time_Heap @ ( option @ nat ) ) @ Vebt_predi4 ) ) )
     => ( ( P
          @ ^ [Vebt_predi4: vEBT_VEBTi,T3: nat] :
              ( heap_Time_Heap2 @ ( option @ nat )
              @ ^ [X: heap_ext @ product_unit] : ( none @ ( product_prod @ ( option @ nat ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) )
       => ( ! [F2: vEBT_VEBTi > nat > ( heap_Time_Heap @ ( option @ nat ) )] :
              ( ( P @ F2 )
             => ( P
                @ ^ [X9: vEBT_VEBTi,A5: nat] :
                    ( vEBT_case_VEBTi @ ( heap_Time_Heap @ ( option @ nat ) )
                    @ ^ [Info3: option @ ( product_prod @ nat @ nat ),Deg4: nat,TreeArray: array @ vEBT_VEBTi,Summary3: vEBT_VEBTi] :
                        ( case_option @ ( heap_Time_Heap @ ( option @ nat ) ) @ ( product_prod @ nat @ nat ) @ ( heap_Time_return @ ( option @ nat ) @ ( none @ nat ) )
                        @ ^ [Mima2: product_prod @ nat @ nat] :
                            ( if @ ( heap_Time_Heap @ ( option @ nat ) ) @ ( ord_less_eq @ nat @ Deg4 @ ( one_one @ nat ) ) @ ( heap_Time_return @ ( option @ nat ) @ ( none @ nat ) )
                            @ ( if @ ( heap_Time_Heap @ ( option @ nat ) ) @ ( ord_less @ nat @ ( product_snd @ nat @ nat @ Mima2 ) @ A5 ) @ ( heap_Time_return @ ( option @ nat ) @ ( some @ nat @ ( product_snd @ nat @ nat @ Mima2 ) ) )
                              @ ( heap_Time_bind @ nat @ ( option @ nat ) @ ( vEBT_VEBT_lowi @ A5 @ ( divide_divide @ nat @ Deg4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
                                @ ^ [L2: nat] :
                                    ( heap_Time_bind @ nat @ ( option @ nat ) @ ( vEBT_VEBT_highi @ A5 @ ( divide_divide @ nat @ Deg4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
                                    @ ^ [H: nat] :
                                        ( heap_Time_bind @ vEBT_VEBTi @ ( option @ nat ) @ ( array_nth @ vEBT_VEBTi @ TreeArray @ H )
                                        @ ^ [Aktnode: vEBT_VEBTi] :
                                            ( heap_Time_bind @ ( option @ nat ) @ ( option @ nat ) @ ( vEBT_vebt_minti @ Aktnode )
                                            @ ^ [Minlow: option @ nat] :
                                                ( if @ ( heap_Time_Heap @ ( option @ nat ) )
                                                @ ( ( Minlow
                                                   != ( none @ nat ) )
                                                  & ( vEBT_VEBT_greater @ ( some @ nat @ L2 ) @ Minlow ) )
                                                @ ( heap_Time_bind @ ( option @ nat ) @ ( option @ nat ) @ ( F2 @ Aktnode @ L2 )
                                                  @ ^ [Predy: option @ nat] : ( heap_Time_return @ ( option @ nat ) @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( some @ nat @ H ) ) @ Predy ) ) )
                                                @ ( heap_Time_bind @ ( option @ nat ) @ ( option @ nat ) @ ( F2 @ Summary3 @ H )
                                                  @ ^ [Predsum: option @ nat] :
                                                      ( if @ ( heap_Time_Heap @ ( option @ nat ) )
                                                      @ ( Predsum
                                                        = ( none @ nat ) )
                                                      @ ( if @ ( heap_Time_Heap @ ( option @ nat ) ) @ ( ord_less @ nat @ ( product_fst @ nat @ nat @ Mima2 ) @ A5 ) @ ( heap_Time_return @ ( option @ nat ) @ ( some @ nat @ ( product_fst @ nat @ nat @ Mima2 ) ) ) @ ( heap_Time_return @ ( option @ nat ) @ ( none @ nat ) ) )
                                                      @ ( heap_Time_bind @ vEBT_VEBTi @ ( option @ nat ) @ ( array_nth @ vEBT_VEBTi @ TreeArray @ ( the2 @ nat @ Predsum ) )
                                                        @ ^ [Nextnode: vEBT_VEBTi] :
                                                            ( heap_Time_bind @ ( option @ nat ) @ ( option @ nat ) @ ( vEBT_vebt_maxti @ Nextnode )
                                                            @ ^ [Maxnext: option @ nat] : ( heap_Time_return @ ( option @ nat ) @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ Predsum ) @ Maxnext ) ) ) ) ) ) ) ) ) ) ) ) )
                        @ Info3 )
                    @ ^ [B5: $o,C4: $o] :
                        ( if @ ( heap_Time_Heap @ ( option @ nat ) ) @ ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ A5 ) @ ( if @ ( heap_Time_Heap @ ( option @ nat ) ) @ C4 @ ( heap_Time_return @ ( option @ nat ) @ ( some @ nat @ ( one_one @ nat ) ) ) @ ( if @ ( heap_Time_Heap @ ( option @ nat ) ) @ B5 @ ( heap_Time_return @ ( option @ nat ) @ ( some @ nat @ ( zero_zero @ nat ) ) ) @ ( heap_Time_return @ ( option @ nat ) @ ( none @ nat ) ) ) )
                        @ ( if @ ( heap_Time_Heap @ ( option @ nat ) )
                          @ ( A5
                            = ( one_one @ nat ) )
                          @ ( if @ ( heap_Time_Heap @ ( option @ nat ) ) @ B5 @ ( heap_Time_return @ ( option @ nat ) @ ( some @ nat @ ( zero_zero @ nat ) ) ) @ ( heap_Time_return @ ( option @ nat ) @ ( none @ nat ) ) )
                          @ ( heap_Time_return @ ( option @ nat ) @ ( none @ nat ) ) ) )
                    @ X9 ) ) )
         => ( P @ vEBT_vebt_predi ) ) ) ) ).

% vebt_predi.fixp_induct
thf(fact_3063_vebt__succi_Ofixp__induct,axiom,
    ! [P: ( vEBT_VEBTi > nat > ( heap_Time_Heap @ ( option @ nat ) ) ) > $o] :
      ( ( comple1908693960933563346ssible @ ( ( product_prod @ vEBT_VEBTi @ nat ) > ( heap_Time_Heap @ ( option @ nat ) ) ) @ ( partial_fun_lub @ ( heap_Time_Heap @ ( option @ nat ) ) @ ( heap_Time_Heap @ ( option @ nat ) ) @ ( product_prod @ vEBT_VEBTi @ nat ) @ ( heap_Time_Heap_lub @ ( option @ nat ) ) ) @ ( partial_fun_ord @ ( heap_Time_Heap @ ( option @ nat ) ) @ ( heap_Time_Heap @ ( option @ nat ) ) @ ( product_prod @ vEBT_VEBTi @ nat ) @ ( heap_Time_Heap_ord @ ( option @ nat ) ) )
        @ ^ [Vebt_succi4: ( product_prod @ vEBT_VEBTi @ nat ) > ( heap_Time_Heap @ ( option @ nat ) )] : ( P @ ( product_curry @ vEBT_VEBTi @ nat @ ( heap_Time_Heap @ ( option @ nat ) ) @ Vebt_succi4 ) ) )
     => ( ( P
          @ ^ [Vebt_succi4: vEBT_VEBTi,T3: nat] :
              ( heap_Time_Heap2 @ ( option @ nat )
              @ ^ [X: heap_ext @ product_unit] : ( none @ ( product_prod @ ( option @ nat ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) )
       => ( ! [F2: vEBT_VEBTi > nat > ( heap_Time_Heap @ ( option @ nat ) )] :
              ( ( P @ F2 )
             => ( P
                @ ^ [X9: vEBT_VEBTi,A5: nat] :
                    ( vEBT_case_VEBTi @ ( heap_Time_Heap @ ( option @ nat ) )
                    @ ^ [Info3: option @ ( product_prod @ nat @ nat ),Deg4: nat,TreeArray: array @ vEBT_VEBTi,Summary3: vEBT_VEBTi] :
                        ( case_option @ ( heap_Time_Heap @ ( option @ nat ) ) @ ( product_prod @ nat @ nat ) @ ( heap_Time_return @ ( option @ nat ) @ ( none @ nat ) )
                        @ ^ [Mima2: product_prod @ nat @ nat] :
                            ( if @ ( heap_Time_Heap @ ( option @ nat ) ) @ ( ord_less_eq @ nat @ Deg4 @ ( one_one @ nat ) ) @ ( heap_Time_return @ ( option @ nat ) @ ( none @ nat ) )
                            @ ( if @ ( heap_Time_Heap @ ( option @ nat ) ) @ ( ord_less @ nat @ A5 @ ( product_fst @ nat @ nat @ Mima2 ) ) @ ( heap_Time_return @ ( option @ nat ) @ ( some @ nat @ ( product_fst @ nat @ nat @ Mima2 ) ) )
                              @ ( if @ ( heap_Time_Heap @ ( option @ nat ) ) @ ( ord_less_eq @ nat @ ( product_snd @ nat @ nat @ Mima2 ) @ A5 ) @ ( heap_Time_return @ ( option @ nat ) @ ( none @ nat ) )
                                @ ( heap_Time_bind @ nat @ ( option @ nat ) @ ( vEBT_VEBT_lowi @ A5 @ ( divide_divide @ nat @ Deg4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
                                  @ ^ [L2: nat] :
                                      ( heap_Time_bind @ nat @ ( option @ nat ) @ ( vEBT_VEBT_highi @ A5 @ ( divide_divide @ nat @ Deg4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
                                      @ ^ [H: nat] :
                                          ( heap_Time_bind @ vEBT_VEBTi @ ( option @ nat ) @ ( array_nth @ vEBT_VEBTi @ TreeArray @ H )
                                          @ ^ [Aktnode: vEBT_VEBTi] :
                                              ( heap_Time_bind @ ( option @ nat ) @ ( option @ nat ) @ ( vEBT_vebt_maxti @ Aktnode )
                                              @ ^ [Maxlow: option @ nat] :
                                                  ( if @ ( heap_Time_Heap @ ( option @ nat ) )
                                                  @ ( ( Maxlow
                                                     != ( none @ nat ) )
                                                    & ( vEBT_VEBT_less @ ( some @ nat @ L2 ) @ Maxlow ) )
                                                  @ ( heap_Time_bind @ ( option @ nat ) @ ( option @ nat ) @ ( F2 @ Aktnode @ L2 )
                                                    @ ^ [Succy: option @ nat] : ( heap_Time_return @ ( option @ nat ) @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( some @ nat @ H ) ) @ Succy ) ) )
                                                  @ ( heap_Time_bind @ ( option @ nat ) @ ( option @ nat ) @ ( F2 @ Summary3 @ H )
                                                    @ ^ [Succsum: option @ nat] :
                                                        ( if @ ( heap_Time_Heap @ ( option @ nat ) )
                                                        @ ( Succsum
                                                          = ( none @ nat ) )
                                                        @ ( heap_Time_return @ ( option @ nat ) @ ( none @ nat ) )
                                                        @ ( heap_Time_bind @ vEBT_VEBTi @ ( option @ nat ) @ ( array_nth @ vEBT_VEBTi @ TreeArray @ ( the2 @ nat @ Succsum ) )
                                                          @ ^ [Nextnode: vEBT_VEBTi] :
                                                              ( heap_Time_bind @ ( option @ nat ) @ ( option @ nat ) @ ( vEBT_vebt_minti @ Nextnode )
                                                              @ ^ [Minnext: option @ nat] : ( heap_Time_return @ ( option @ nat ) @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ Succsum ) @ Minnext ) ) ) ) ) ) ) ) ) ) ) ) ) )
                        @ Info3 )
                    @ ^ [B5: $o,C4: $o] :
                        ( if @ ( heap_Time_Heap @ ( option @ nat ) )
                        @ ( A5
                          = ( zero_zero @ nat ) )
                        @ ( if @ ( heap_Time_Heap @ ( option @ nat ) ) @ C4 @ ( heap_Time_return @ ( option @ nat ) @ ( some @ nat @ ( one_one @ nat ) ) ) @ ( heap_Time_return @ ( option @ nat ) @ ( none @ nat ) ) )
                        @ ( heap_Time_return @ ( option @ nat ) @ ( none @ nat ) ) )
                    @ X9 ) ) )
         => ( P @ vEBT_vebt_succi ) ) ) ) ).

% vebt_succi.fixp_induct
thf(fact_3064_vebt__memberi_Ofixp__induct,axiom,
    ! [P: ( vEBT_VEBTi > nat > ( heap_Time_Heap @ $o ) ) > $o] :
      ( ( comple1908693960933563346ssible @ ( ( product_prod @ vEBT_VEBTi @ nat ) > ( heap_Time_Heap @ $o ) ) @ ( partial_fun_lub @ ( heap_Time_Heap @ $o ) @ ( heap_Time_Heap @ $o ) @ ( product_prod @ vEBT_VEBTi @ nat ) @ ( heap_Time_Heap_lub @ $o ) ) @ ( partial_fun_ord @ ( heap_Time_Heap @ $o ) @ ( heap_Time_Heap @ $o ) @ ( product_prod @ vEBT_VEBTi @ nat ) @ ( heap_Time_Heap_ord @ $o ) )
        @ ^ [Vebt_memberi3: ( product_prod @ vEBT_VEBTi @ nat ) > ( heap_Time_Heap @ $o )] : ( P @ ( product_curry @ vEBT_VEBTi @ nat @ ( heap_Time_Heap @ $o ) @ Vebt_memberi3 ) ) )
     => ( ( P
          @ ^ [Vebt_memberi3: vEBT_VEBTi,T3: nat] :
              ( heap_Time_Heap2 @ $o
              @ ^ [X: heap_ext @ product_unit] : ( none @ ( product_prod @ $o @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) )
       => ( ! [F2: vEBT_VEBTi > nat > ( heap_Time_Heap @ $o )] :
              ( ( P @ F2 )
             => ( P
                @ ^ [X9: vEBT_VEBTi,A5: nat] :
                    ( vEBT_case_VEBTi @ ( heap_Time_Heap @ $o )
                    @ ^ [Info3: option @ ( product_prod @ nat @ nat ),Deg4: nat,TreeList3: array @ vEBT_VEBTi,Summary3: vEBT_VEBTi] :
                        ( case_option @ ( heap_Time_Heap @ $o ) @ ( product_prod @ nat @ nat ) @ ( heap_Time_return @ $o @ $false )
                        @ ( product_case_prod @ nat @ nat @ ( heap_Time_Heap @ $o )
                          @ ^ [Mi3: nat,Ma3: nat] :
                              ( if @ ( heap_Time_Heap @ $o ) @ ( ord_less_eq @ nat @ Deg4 @ ( one_one @ nat ) ) @ ( heap_Time_return @ $o @ $false )
                              @ ( if @ ( heap_Time_Heap @ $o ) @ ( A5 = Mi3 ) @ ( heap_Time_return @ $o @ $true )
                                @ ( if @ ( heap_Time_Heap @ $o ) @ ( A5 = Ma3 ) @ ( heap_Time_return @ $o @ $true )
                                  @ ( if @ ( heap_Time_Heap @ $o ) @ ( ord_less @ nat @ A5 @ Mi3 ) @ ( heap_Time_return @ $o @ $false )
                                    @ ( if @ ( heap_Time_Heap @ $o ) @ ( ord_less @ nat @ Ma3 @ A5 ) @ ( heap_Time_return @ $o @ $false )
                                      @ ( heap_Time_bind @ nat @ $o @ ( vEBT_VEBT_highi @ A5 @ ( divide_divide @ nat @ Deg4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
                                        @ ^ [H: nat] :
                                            ( heap_Time_bind @ nat @ $o @ ( vEBT_VEBT_lowi @ A5 @ ( divide_divide @ nat @ Deg4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
                                            @ ^ [L2: nat] :
                                                ( heap_Time_bind @ nat @ $o @ ( array_len @ vEBT_VEBTi @ TreeList3 )
                                                @ ^ [Len: nat] :
                                                    ( if @ ( heap_Time_Heap @ $o ) @ ( ord_less @ nat @ H @ Len )
                                                    @ ( heap_Time_bind @ vEBT_VEBTi @ $o @ ( array_nth @ vEBT_VEBTi @ TreeList3 @ H )
                                                      @ ^ [Th: vEBT_VEBTi] : ( F2 @ Th @ L2 ) )
                                                    @ ( heap_Time_return @ $o @ $false ) ) ) ) ) ) ) ) ) ) )
                        @ Info3 )
                    @ ^ [B5: $o,C4: $o] :
                        ( heap_Time_return @ $o
                        @ ( ( ( A5
                              = ( zero_zero @ nat ) )
                           => B5 )
                          & ( ( A5
                             != ( zero_zero @ nat ) )
                           => ( ( ( A5
                                  = ( one_one @ nat ) )
                               => C4 )
                              & ( A5
                                = ( one_one @ nat ) ) ) ) ) )
                    @ X9 ) ) )
         => ( P @ vEBT_vebt_memberi ) ) ) ) ).

% vebt_memberi.fixp_induct
thf(fact_3065_heap_Ofixp__induct__uc,axiom,
    ! [B: $tType,A: $tType,C: $tType,U3: C > B > ( heap_Time_Heap @ A ),F5: C > C,C5: ( B > ( heap_Time_Heap @ A ) ) > C,F: C,P: ( B > ( heap_Time_Heap @ A ) ) > $o] :
      ( ! [X3: B] :
          ( comple7038119648293358887notone @ ( B > ( heap_Time_Heap @ A ) ) @ ( heap_Time_Heap @ A ) @ ( partial_fun_ord @ ( heap_Time_Heap @ A ) @ ( heap_Time_Heap @ A ) @ B @ ( heap_Time_Heap_ord @ A ) ) @ ( heap_Time_Heap_ord @ A )
          @ ^ [F6: B > ( heap_Time_Heap @ A )] : ( U3 @ ( F5 @ ( C5 @ F6 ) ) @ X3 ) )
     => ( ( F
          = ( C5
            @ ( comple187402453842119260l_fixp @ ( B > ( heap_Time_Heap @ A ) ) @ ( partial_fun_lub @ ( heap_Time_Heap @ A ) @ ( heap_Time_Heap @ A ) @ B @ ( heap_Time_Heap_lub @ A ) ) @ ( partial_fun_ord @ ( heap_Time_Heap @ A ) @ ( heap_Time_Heap @ A ) @ B @ ( heap_Time_Heap_ord @ A ) )
              @ ^ [F6: B > ( heap_Time_Heap @ A )] : ( U3 @ ( F5 @ ( C5 @ F6 ) ) ) ) ) )
       => ( ! [F2: B > ( heap_Time_Heap @ A )] :
              ( ( U3 @ ( C5 @ F2 ) )
              = F2 )
         => ( ( comple1908693960933563346ssible @ ( B > ( heap_Time_Heap @ A ) ) @ ( partial_fun_lub @ ( heap_Time_Heap @ A ) @ ( heap_Time_Heap @ A ) @ B @ ( heap_Time_Heap_lub @ A ) ) @ ( partial_fun_ord @ ( heap_Time_Heap @ A ) @ ( heap_Time_Heap @ A ) @ B @ ( heap_Time_Heap_ord @ A ) ) @ P )
           => ( ( P
                @ ^ [Uu3: B] :
                    ( heap_Time_Heap2 @ A
                    @ ^ [X: heap_ext @ product_unit] : ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) )
             => ( ! [F2: C] :
                    ( ( P @ ( U3 @ F2 ) )
                   => ( P @ ( U3 @ ( F5 @ F2 ) ) ) )
               => ( P @ ( U3 @ F ) ) ) ) ) ) ) ) ).

% heap.fixp_induct_uc
thf(fact_3066_bezw_Oelims,axiom,
    ! [X2: nat,Xa: nat,Y2: product_prod @ int @ int] :
      ( ( ( bezw @ X2 @ Xa )
        = Y2 )
     => ( ( ( Xa
            = ( zero_zero @ nat ) )
         => ( Y2
            = ( product_Pair @ int @ int @ ( one_one @ int ) @ ( zero_zero @ int ) ) ) )
        & ( ( Xa
           != ( zero_zero @ nat ) )
         => ( Y2
            = ( product_Pair @ int @ int @ ( product_snd @ int @ int @ ( bezw @ Xa @ ( modulo_modulo @ nat @ X2 @ Xa ) ) ) @ ( minus_minus @ int @ ( product_fst @ int @ int @ ( bezw @ Xa @ ( modulo_modulo @ nat @ X2 @ Xa ) ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ ( bezw @ Xa @ ( modulo_modulo @ nat @ X2 @ Xa ) ) ) @ ( semiring_1_of_nat @ int @ ( divide_divide @ nat @ X2 @ Xa ) ) ) ) ) ) ) ) ) ).

% bezw.elims
thf(fact_3067_bezw_Osimps,axiom,
    ( bezw
    = ( ^ [X: nat,Y: nat] :
          ( if @ ( product_prod @ int @ int )
          @ ( Y
            = ( zero_zero @ nat ) )
          @ ( product_Pair @ int @ int @ ( one_one @ int ) @ ( zero_zero @ int ) )
          @ ( product_Pair @ int @ int @ ( product_snd @ int @ int @ ( bezw @ Y @ ( modulo_modulo @ nat @ X @ Y ) ) ) @ ( minus_minus @ int @ ( product_fst @ int @ int @ ( bezw @ Y @ ( modulo_modulo @ nat @ X @ Y ) ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ ( bezw @ Y @ ( modulo_modulo @ nat @ X @ Y ) ) ) @ ( semiring_1_of_nat @ int @ ( divide_divide @ nat @ X @ Y ) ) ) ) ) ) ) ) ).

% bezw.simps
thf(fact_3068_tap__def,axiom,
    ! [A: $tType] :
      ( ( heap_Time_tap @ A )
      = ( ^ [F6: ( heap_ext @ product_unit ) > A] :
            ( heap_Time_Heap2 @ A
            @ ^ [H: heap_ext @ product_unit] : ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( F6 @ H ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H @ ( one_one @ nat ) ) ) ) ) ) ) ).

% tap_def
thf(fact_3069_bezw__0,axiom,
    ! [X2: nat] :
      ( ( bezw @ X2 @ ( zero_zero @ nat ) )
      = ( product_Pair @ int @ int @ ( one_one @ int ) @ ( zero_zero @ int ) ) ) ).

% bezw_0
thf(fact_3070_effect__tapI,axiom,
    ! [A: $tType,H4: heap_ext @ product_unit,H2: heap_ext @ product_unit,R4: A,F: ( heap_ext @ product_unit ) > A] :
      ( ( H4 = H2 )
     => ( ( R4
          = ( F @ H2 ) )
       => ( heap_Time_effect @ A @ ( heap_Time_tap @ A @ F ) @ H2 @ H4 @ R4 @ ( one_one @ nat ) ) ) ) ).

% effect_tapI
thf(fact_3071_effect__tapE,axiom,
    ! [A: $tType,F: ( heap_ext @ product_unit ) > A,H2: heap_ext @ product_unit,H4: heap_ext @ product_unit,R4: A,N2: nat] :
      ( ( heap_Time_effect @ A @ ( heap_Time_tap @ A @ F ) @ H2 @ H4 @ R4 @ N2 )
     => ~ ( ( H4 = H2 )
         => ( ( R4
              = ( F @ H2 ) )
           => ( N2
             != ( one_one @ nat ) ) ) ) ) ).

% effect_tapE
thf(fact_3072_heap_Ofixp__rule__uc,axiom,
    ! [B: $tType,A: $tType,C: $tType,U3: C > B > ( heap_Time_Heap @ A ),F5: C > C,C5: ( B > ( heap_Time_Heap @ A ) ) > C,F: C] :
      ( ! [X3: B] :
          ( comple7038119648293358887notone @ ( B > ( heap_Time_Heap @ A ) ) @ ( heap_Time_Heap @ A ) @ ( partial_fun_ord @ ( heap_Time_Heap @ A ) @ ( heap_Time_Heap @ A ) @ B @ ( heap_Time_Heap_ord @ A ) ) @ ( heap_Time_Heap_ord @ A )
          @ ^ [F6: B > ( heap_Time_Heap @ A )] : ( U3 @ ( F5 @ ( C5 @ F6 ) ) @ X3 ) )
     => ( ( F
          = ( C5
            @ ( comple187402453842119260l_fixp @ ( B > ( heap_Time_Heap @ A ) ) @ ( partial_fun_lub @ ( heap_Time_Heap @ A ) @ ( heap_Time_Heap @ A ) @ B @ ( heap_Time_Heap_lub @ A ) ) @ ( partial_fun_ord @ ( heap_Time_Heap @ A ) @ ( heap_Time_Heap @ A ) @ B @ ( heap_Time_Heap_ord @ A ) )
              @ ^ [F6: B > ( heap_Time_Heap @ A )] : ( U3 @ ( F5 @ ( C5 @ F6 ) ) ) ) ) )
       => ( ! [F2: C] :
              ( ( C5 @ ( U3 @ F2 ) )
              = F2 )
         => ( F
            = ( F5 @ F ) ) ) ) ) ).

% heap.fixp_rule_uc
thf(fact_3073_heap_Omono__body__fixp,axiom,
    ! [A: $tType,B: $tType,F5: ( B > ( heap_Time_Heap @ A ) ) > B > ( heap_Time_Heap @ A )] :
      ( ! [X3: B] :
          ( comple7038119648293358887notone @ ( B > ( heap_Time_Heap @ A ) ) @ ( heap_Time_Heap @ A ) @ ( partial_fun_ord @ ( heap_Time_Heap @ A ) @ ( heap_Time_Heap @ A ) @ B @ ( heap_Time_Heap_ord @ A ) ) @ ( heap_Time_Heap_ord @ A )
          @ ^ [F6: B > ( heap_Time_Heap @ A )] : ( F5 @ F6 @ X3 ) )
     => ( ( comple187402453842119260l_fixp @ ( B > ( heap_Time_Heap @ A ) ) @ ( partial_fun_lub @ ( heap_Time_Heap @ A ) @ ( heap_Time_Heap @ A ) @ B @ ( heap_Time_Heap_lub @ A ) ) @ ( partial_fun_ord @ ( heap_Time_Heap @ A ) @ ( heap_Time_Heap @ A ) @ B @ ( heap_Time_Heap_ord @ A ) ) @ F5 )
        = ( F5 @ ( comple187402453842119260l_fixp @ ( B > ( heap_Time_Heap @ A ) ) @ ( partial_fun_lub @ ( heap_Time_Heap @ A ) @ ( heap_Time_Heap @ A ) @ B @ ( heap_Time_Heap_lub @ A ) ) @ ( partial_fun_ord @ ( heap_Time_Heap @ A ) @ ( heap_Time_Heap @ A ) @ B @ ( heap_Time_Heap_ord @ A ) ) @ F5 ) ) ) ) ).

% heap.mono_body_fixp
thf(fact_3074_fixp__induct__heap,axiom,
    ! [C: $tType,A: $tType,B: $tType,U3: C > B > ( heap_Time_Heap @ A ),F5: C > C,C5: ( B > ( heap_Time_Heap @ A ) ) > C,F: C,P: B > ( heap_ext @ product_unit ) > ( heap_ext @ product_unit ) > A > nat > $o,X2: B,H2: heap_ext @ product_unit,H4: heap_ext @ product_unit,R4: A,N2: nat] :
      ( ! [X3: B] :
          ( comple7038119648293358887notone @ ( B > ( heap_Time_Heap @ A ) ) @ ( heap_Time_Heap @ A ) @ ( partial_fun_ord @ ( heap_Time_Heap @ A ) @ ( heap_Time_Heap @ A ) @ B @ ( heap_Time_Heap_ord @ A ) ) @ ( heap_Time_Heap_ord @ A )
          @ ^ [F6: B > ( heap_Time_Heap @ A )] : ( U3 @ ( F5 @ ( C5 @ F6 ) ) @ X3 ) )
     => ( ( F
          = ( C5
            @ ( comple187402453842119260l_fixp @ ( B > ( heap_Time_Heap @ A ) ) @ ( partial_fun_lub @ ( heap_Time_Heap @ A ) @ ( heap_Time_Heap @ A ) @ B @ ( heap_Time_Heap_lub @ A ) ) @ ( partial_fun_ord @ ( heap_Time_Heap @ A ) @ ( heap_Time_Heap @ A ) @ B @ ( heap_Time_Heap_ord @ A ) )
              @ ^ [F6: B > ( heap_Time_Heap @ A )] : ( U3 @ ( F5 @ ( C5 @ F6 ) ) ) ) ) )
       => ( ! [F2: B > ( heap_Time_Heap @ A )] :
              ( ( U3 @ ( C5 @ F2 ) )
              = F2 )
         => ( ! [F2: C,X3: B,H3: heap_ext @ product_unit,H5: heap_ext @ product_unit,R3: A,N4: nat] :
                ( ! [Xa2: B,Ha: heap_ext @ product_unit,H_a: heap_ext @ product_unit,Ra: A,Na: nat] :
                    ( ( heap_Time_effect @ A @ ( U3 @ F2 @ Xa2 ) @ Ha @ H_a @ Ra @ Na )
                   => ( P @ Xa2 @ Ha @ H_a @ Ra @ Na ) )
               => ( ( heap_Time_effect @ A @ ( U3 @ ( F5 @ F2 ) @ X3 ) @ H3 @ H5 @ R3 @ N4 )
                 => ( P @ X3 @ H3 @ H5 @ R3 @ N4 ) ) )
           => ( ( heap_Time_effect @ A @ ( U3 @ F @ X2 ) @ H2 @ H4 @ R4 @ N2 )
             => ( P @ X2 @ H2 @ H4 @ R4 @ N2 ) ) ) ) ) ) ).

% fixp_induct_heap
thf(fact_3075_bezw__non__0,axiom,
    ! [Y2: nat,X2: nat] :
      ( ( ord_less @ nat @ ( zero_zero @ nat ) @ Y2 )
     => ( ( bezw @ X2 @ Y2 )
        = ( product_Pair @ int @ int @ ( product_snd @ int @ int @ ( bezw @ Y2 @ ( modulo_modulo @ nat @ X2 @ Y2 ) ) ) @ ( minus_minus @ int @ ( product_fst @ int @ int @ ( bezw @ Y2 @ ( modulo_modulo @ nat @ X2 @ Y2 ) ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ ( bezw @ Y2 @ ( modulo_modulo @ nat @ X2 @ Y2 ) ) ) @ ( semiring_1_of_nat @ int @ ( divide_divide @ nat @ X2 @ Y2 ) ) ) ) ) ) ) ).

% bezw_non_0
thf(fact_3076_product__nth,axiom,
    ! [A: $tType,B: $tType,N2: nat,Xs2: list @ A,Ys: list @ B] :
      ( ( ord_less @ nat @ N2 @ ( times_times @ nat @ ( size_size @ ( list @ A ) @ Xs2 ) @ ( size_size @ ( list @ B ) @ Ys ) ) )
     => ( ( nth @ ( product_prod @ A @ B ) @ ( product @ A @ B @ Xs2 @ Ys ) @ N2 )
        = ( product_Pair @ A @ B @ ( nth @ A @ Xs2 @ ( divide_divide @ nat @ N2 @ ( size_size @ ( list @ B ) @ Ys ) ) ) @ ( nth @ B @ Ys @ ( modulo_modulo @ nat @ N2 @ ( size_size @ ( list @ B ) @ Ys ) ) ) ) ) ) ).

% product_nth
thf(fact_3077_pred__subset__eq2,axiom,
    ! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ B ),S3: set @ ( product_prod @ A @ B )] :
      ( ( ord_less_eq @ ( A > B > $o )
        @ ^ [X: A,Y: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y ) @ R2 )
        @ ^ [X: A,Y: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y ) @ S3 ) )
      = ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R2 @ S3 ) ) ).

% pred_subset_eq2
thf(fact_3078_even__word__def,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ( ( even_word @ A )
        = ( dvd_dvd @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) ) ) ) ).

% even_word_def
thf(fact_3079_triangle__def,axiom,
    ( nat_triangle
    = ( ^ [N: nat] : ( divide_divide @ nat @ ( times_times @ nat @ N @ ( suc @ N ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).

% triangle_def
thf(fact_3080_length__product,axiom,
    ! [A: $tType,B: $tType,Xs2: list @ A,Ys: list @ B] :
      ( ( size_size @ ( list @ ( product_prod @ A @ B ) ) @ ( product @ A @ B @ Xs2 @ Ys ) )
      = ( times_times @ nat @ ( size_size @ ( list @ A ) @ Xs2 ) @ ( size_size @ ( list @ B ) @ Ys ) ) ) ).

% length_product
thf(fact_3081_triangle__Suc,axiom,
    ! [N2: nat] :
      ( ( nat_triangle @ ( suc @ N2 ) )
      = ( plus_plus @ nat @ ( nat_triangle @ N2 ) @ ( suc @ N2 ) ) ) ).

% triangle_Suc
thf(fact_3082_exhaustive__int_H_Ocases,axiom,
    ! [X2: product_prod @ ( int > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ) ) @ ( product_prod @ int @ int )] :
      ~ ! [F2: int > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ),D4: int,I3: int] :
          ( X2
         != ( product_Pair @ ( int > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ) ) @ ( product_prod @ int @ int ) @ F2 @ ( product_Pair @ int @ int @ D4 @ I3 ) ) ) ).

% exhaustive_int'.cases
thf(fact_3083_full__exhaustive__int_H_Ocases,axiom,
    ! [X2: product_prod @ ( ( product_prod @ int @ ( product_unit > code_term ) ) > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ) ) @ ( product_prod @ int @ int )] :
      ~ ! [F2: ( product_prod @ int @ ( product_unit > code_term ) ) > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ),D4: int,I3: int] :
          ( X2
         != ( product_Pair @ ( ( product_prod @ int @ ( product_unit > code_term ) ) > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ) ) @ ( product_prod @ int @ int ) @ F2 @ ( product_Pair @ int @ int @ D4 @ I3 ) ) ) ).

% full_exhaustive_int'.cases
thf(fact_3084_exhaustive__integer_H_Ocases,axiom,
    ! [X2: product_prod @ ( code_integer > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ) ) @ ( product_prod @ code_integer @ code_integer )] :
      ~ ! [F2: code_integer > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ),D4: code_integer,I3: code_integer] :
          ( X2
         != ( product_Pair @ ( code_integer > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ) ) @ ( product_prod @ code_integer @ code_integer ) @ F2 @ ( product_Pair @ code_integer @ code_integer @ D4 @ I3 ) ) ) ).

% exhaustive_integer'.cases
thf(fact_3085_full__exhaustive__integer_H_Ocases,axiom,
    ! [X2: product_prod @ ( ( product_prod @ code_integer @ ( product_unit > code_term ) ) > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ) ) @ ( product_prod @ code_integer @ code_integer )] :
      ~ ! [F2: ( product_prod @ code_integer @ ( product_unit > code_term ) ) > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ),D4: code_integer,I3: code_integer] :
          ( X2
         != ( product_Pair @ ( ( product_prod @ code_integer @ ( product_unit > code_term ) ) > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ) ) @ ( product_prod @ code_integer @ code_integer ) @ F2 @ ( product_Pair @ code_integer @ code_integer @ D4 @ I3 ) ) ) ).

% full_exhaustive_integer'.cases
thf(fact_3086_pred__equals__eq2,axiom,
    ! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ B ),S3: set @ ( product_prod @ A @ B )] :
      ( ( ( ^ [X: A,Y: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y ) @ R2 ) )
        = ( ^ [X: A,Y: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y ) @ S3 ) ) )
      = ( R2 = S3 ) ) ).

% pred_equals_eq2
thf(fact_3087_subrelI,axiom,
    ! [B: $tType,A: $tType,R4: set @ ( product_prod @ A @ B ),S4: set @ ( product_prod @ A @ B )] :
      ( ! [X3: A,Y4: B] :
          ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y4 ) @ R4 )
         => ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y4 ) @ S4 ) )
     => ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R4 @ S4 ) ) ).

% subrelI
thf(fact_3088_top__empty__eq2,axiom,
    ! [B: $tType,A: $tType] :
      ( ( top_top @ ( A > B > $o ) )
      = ( ^ [X: A,Y: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y ) @ ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) ) ) ) ).

% top_empty_eq2
thf(fact_3089_pred__subset__eq,axiom,
    ! [A: $tType,R2: set @ A,S3: set @ A] :
      ( ( ord_less_eq @ ( A > $o )
        @ ^ [X: A] : ( member @ A @ X @ R2 )
        @ ^ [X: A] : ( member @ A @ X @ S3 ) )
      = ( ord_less_eq @ ( set @ A ) @ R2 @ S3 ) ) ).

% pred_subset_eq
thf(fact_3090_exE__realizer,axiom,
    ! [C: $tType,A: $tType,B: $tType,P: A > B > $o,P4: product_prod @ B @ A,Q: C > $o,F: B > A > C] :
      ( ( P @ ( product_snd @ B @ A @ P4 ) @ ( product_fst @ B @ A @ P4 ) )
     => ( ! [X3: B,Y4: A] :
            ( ( P @ Y4 @ X3 )
           => ( Q @ ( F @ X3 @ Y4 ) ) )
       => ( Q @ ( product_case_prod @ B @ A @ C @ F @ P4 ) ) ) ) ).

% exE_realizer
thf(fact_3091_vebt__assn__raw_Oelims,axiom,
    ! [X2: vEBT_VEBT,Xa: vEBT_VEBTi,Y2: assn] :
      ( ( ( vEBT_vebt_assn_raw @ X2 @ Xa )
        = Y2 )
     => ( ! [A4: $o,B3: $o] :
            ( ( X2
              = ( vEBT_Leaf @ A4 @ B3 ) )
           => ! [Ai: $o,Bi: $o] :
                ( ( Xa
                  = ( vEBT_Leafi @ Ai @ Bi ) )
               => ( Y2
                 != ( pure_assn
                    @ ( ( Ai = A4 )
                      & ( Bi = B3 ) ) ) ) ) )
       => ( ! [Mmo2: option @ ( product_prod @ nat @ nat ),Deg2: nat,Tree_list2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
              ( ( X2
                = ( vEBT_Node @ Mmo2 @ Deg2 @ Tree_list2 @ Summary2 ) )
             => ! [Mmoi2: option @ ( product_prod @ nat @ nat ),Degi2: nat,Tree_array2: array @ vEBT_VEBTi,Summaryi2: vEBT_VEBTi] :
                  ( ( Xa
                    = ( vEBT_Nodei @ Mmoi2 @ Degi2 @ Tree_array2 @ Summaryi2 ) )
                 => ( Y2
                   != ( times_times @ assn
                      @ ( times_times @ assn
                        @ ( pure_assn
                          @ ( ( Mmoi2 = Mmo2 )
                            & ( Degi2 = Deg2 ) ) )
                        @ ( vEBT_vebt_assn_raw @ Summary2 @ Summaryi2 ) )
                      @ ( ex_assn @ ( list @ vEBT_VEBTi )
                        @ ^ [Tree_is: list @ vEBT_VEBTi] : ( times_times @ assn @ ( snga_assn @ vEBT_VEBTi @ Tree_array2 @ Tree_is ) @ ( vEBT_List_list_assn @ vEBT_VEBT @ vEBT_VEBTi @ vEBT_vebt_assn_raw @ Tree_list2 @ Tree_is ) ) ) ) ) ) )
         => ( ( ? [V3: option @ ( product_prod @ nat @ nat ),Va3: nat,Vb3: list @ vEBT_VEBT,Vc3: vEBT_VEBT] :
                  ( X2
                  = ( vEBT_Node @ V3 @ Va3 @ Vb3 @ Vc3 ) )
             => ( ? [Vd3: $o,Ve3: $o] :
                    ( Xa
                    = ( vEBT_Leafi @ Vd3 @ Ve3 ) )
               => ( Y2
                 != ( bot_bot @ assn ) ) ) )
           => ~ ( ? [Vd3: $o,Ve3: $o] :
                    ( X2
                    = ( vEBT_Leaf @ Vd3 @ Ve3 ) )
               => ( ? [V3: option @ ( product_prod @ nat @ nat ),Va3: nat,Vb3: array @ vEBT_VEBTi,Vc3: vEBT_VEBTi] :
                      ( Xa
                      = ( vEBT_Nodei @ V3 @ Va3 @ Vb3 @ Vc3 ) )
                 => ( Y2
                   != ( bot_bot @ assn ) ) ) ) ) ) ) ) ).

% vebt_assn_raw.elims
thf(fact_3092_in__set__enumerate__eq,axiom,
    ! [A: $tType,P4: product_prod @ nat @ A,N2: nat,Xs2: list @ A] :
      ( ( member @ ( product_prod @ nat @ A ) @ P4 @ ( set2 @ ( product_prod @ nat @ A ) @ ( enumerate @ A @ N2 @ Xs2 ) ) )
      = ( ( ord_less_eq @ nat @ N2 @ ( product_fst @ nat @ A @ P4 ) )
        & ( ord_less @ nat @ ( product_fst @ nat @ A @ P4 ) @ ( plus_plus @ nat @ ( size_size @ ( list @ A ) @ Xs2 ) @ N2 ) )
        & ( ( nth @ A @ Xs2 @ ( minus_minus @ nat @ ( product_fst @ nat @ A @ P4 ) @ N2 ) )
          = ( product_snd @ nat @ A @ P4 ) ) ) ) ).

% in_set_enumerate_eq
thf(fact_3093_obtain__set__pred,axiom,
    ! [Z: nat,X2: nat,A3: set @ nat] :
      ( ( ord_less @ nat @ Z @ X2 )
     => ( ( vEBT_VEBT_min_in_set @ A3 @ Z )
       => ( ( finite_finite2 @ nat @ A3 )
         => ? [X_12: nat] : ( vEBT_is_pred_in_set @ A3 @ X2 @ X_12 ) ) ) ) ).

% obtain_set_pred
thf(fact_3094_obtain__set__succ,axiom,
    ! [X2: nat,Z: nat,A3: set @ nat,B6: set @ nat] :
      ( ( ord_less @ nat @ X2 @ Z )
     => ( ( vEBT_VEBT_max_in_set @ A3 @ Z )
       => ( ( finite_finite2 @ nat @ B6 )
         => ( ( A3 = B6 )
           => ? [X_12: nat] : ( vEBT_is_succ_in_set @ A3 @ X2 @ X_12 ) ) ) ) ) ).

% obtain_set_succ
thf(fact_3095_set__vebt__finite,axiom,
    ! [T2: vEBT_VEBT,N2: nat] :
      ( ( vEBT_invar_vebt @ T2 @ N2 )
     => ( finite_finite2 @ nat @ ( vEBT_VEBT_set_vebt @ T2 ) ) ) ).

% set_vebt_finite
thf(fact_3096_succ__none__empty,axiom,
    ! [Xs2: set @ nat,A2: nat] :
      ( ~ ? [X_12: nat] : ( vEBT_is_succ_in_set @ Xs2 @ A2 @ X_12 )
     => ( ( finite_finite2 @ nat @ Xs2 )
       => ~ ? [X6: nat] :
              ( ( member @ nat @ X6 @ Xs2 )
              & ( ord_less @ nat @ A2 @ X6 ) ) ) ) ).

% succ_none_empty
thf(fact_3097_pred__none__empty,axiom,
    ! [Xs2: set @ nat,A2: nat] :
      ( ~ ? [X_12: nat] : ( vEBT_is_pred_in_set @ Xs2 @ A2 @ X_12 )
     => ( ( finite_finite2 @ nat @ Xs2 )
       => ~ ? [X6: nat] :
              ( ( member @ nat @ X6 @ Xs2 )
              & ( ord_less @ nat @ X6 @ A2 ) ) ) ) ).

% pred_none_empty
thf(fact_3098_finite__option__UNIV,axiom,
    ! [A: $tType] :
      ( ( finite_finite2 @ ( option @ A ) @ ( top_top @ ( set @ ( option @ A ) ) ) )
      = ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) ) ) ).

% finite_option_UNIV
thf(fact_3099_List_Ofinite__set,axiom,
    ! [A: $tType,Xs2: list @ A] : ( finite_finite2 @ A @ ( set2 @ A @ Xs2 ) ) ).

% List.finite_set
thf(fact_3100_star__false__right,axiom,
    ! [P: assn] :
      ( ( times_times @ assn @ P @ ( bot_bot @ assn ) )
      = ( bot_bot @ assn ) ) ).

% star_false_right
thf(fact_3101_star__false__left,axiom,
    ! [P: assn] :
      ( ( times_times @ assn @ ( bot_bot @ assn ) @ P )
      = ( bot_bot @ assn ) ) ).

% star_false_left
thf(fact_3102_false__rule,axiom,
    ! [A: $tType,C2: heap_Time_Heap @ A,Q: A > assn] : ( hoare_hoare_triple @ A @ ( bot_bot @ assn ) @ C2 @ Q ) ).

% false_rule
thf(fact_3103_length__enumerate,axiom,
    ! [A: $tType,N2: nat,Xs2: list @ A] :
      ( ( size_size @ ( list @ ( product_prod @ nat @ A ) ) @ ( enumerate @ A @ N2 @ Xs2 ) )
      = ( size_size @ ( list @ A ) @ Xs2 ) ) ).

% length_enumerate
thf(fact_3104_pure__assn__eq__false__iff,axiom,
    ! [P: $o] :
      ( ( ( pure_assn @ P )
        = ( bot_bot @ assn ) )
      = ~ P ) ).

% pure_assn_eq_false_iff
thf(fact_3105_pure__false,axiom,
    ( ( pure_assn @ $false )
    = ( bot_bot @ assn ) ) ).

% pure_false
thf(fact_3106_assn__basic__inequalities_I3_J,axiom,
    ( ( bot_bot @ assn )
   != ( one_one @ assn ) ) ).

% assn_basic_inequalities(3)
thf(fact_3107_map__snd__enumerate,axiom,
    ! [A: $tType,N2: nat,Xs2: list @ A] :
      ( ( map @ ( product_prod @ nat @ A ) @ A @ ( product_snd @ nat @ A ) @ ( enumerate @ A @ N2 @ Xs2 ) )
      = Xs2 ) ).

% map_snd_enumerate
thf(fact_3108_infinite__Icc__iff,axiom,
    ! [A: $tType] :
      ( ( dense_linorder @ A )
     => ! [A2: A,B2: A] :
          ( ( ~ ( finite_finite2 @ A @ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) ) )
          = ( ord_less @ A @ A2 @ B2 ) ) ) ).

% infinite_Icc_iff
thf(fact_3109_snga__same__false,axiom,
    ! [A: $tType] :
      ( ( heap @ A )
     => ! [P4: array @ A,X2: list @ A,Y2: list @ A] :
          ( ( times_times @ assn @ ( snga_assn @ A @ P4 @ X2 ) @ ( snga_assn @ A @ P4 @ Y2 ) )
          = ( bot_bot @ assn ) ) ) ).

% snga_same_false
thf(fact_3110_finite__if__eq__beyond__finite,axiom,
    ! [A: $tType,S3: set @ A,S5: set @ A] :
      ( ( finite_finite2 @ A @ S3 )
     => ( finite_finite2 @ ( set @ A )
        @ ( collect @ ( set @ A )
          @ ^ [S2: set @ A] :
              ( ( minus_minus @ ( set @ A ) @ S2 @ S3 )
              = ( minus_minus @ ( set @ A ) @ S5 @ S3 ) ) ) ) ) ).

% finite_if_eq_beyond_finite
thf(fact_3111_finite__nat__set__iff__bounded,axiom,
    ( ( finite_finite2 @ nat )
    = ( ^ [N8: set @ nat] :
        ? [M7: nat] :
        ! [X: nat] :
          ( ( member @ nat @ X @ N8 )
         => ( ord_less @ nat @ X @ M7 ) ) ) ) ).

% finite_nat_set_iff_bounded
thf(fact_3112_bounded__nat__set__is__finite,axiom,
    ! [N3: set @ nat,N2: nat] :
      ( ! [X3: nat] :
          ( ( member @ nat @ X3 @ N3 )
         => ( ord_less @ nat @ X3 @ N2 ) )
     => ( finite_finite2 @ nat @ N3 ) ) ).

% bounded_nat_set_is_finite
thf(fact_3113_finite__nat__set__iff__bounded__le,axiom,
    ( ( finite_finite2 @ nat )
    = ( ^ [N8: set @ nat] :
        ? [M7: nat] :
        ! [X: nat] :
          ( ( member @ nat @ X @ N8 )
         => ( ord_less_eq @ nat @ X @ M7 ) ) ) ) ).

% finite_nat_set_iff_bounded_le
thf(fact_3114_finite__list,axiom,
    ! [A: $tType,A3: set @ A] :
      ( ( finite_finite2 @ A @ A3 )
     => ? [Xs3: list @ A] :
          ( ( set2 @ A @ Xs3 )
          = A3 ) ) ).

% finite_list
thf(fact_3115_finite__M__bounded__by__nat,axiom,
    ! [P: nat > $o,I: nat] :
      ( finite_finite2 @ nat
      @ ( collect @ nat
        @ ^ [K4: nat] :
            ( ( P @ K4 )
            & ( ord_less @ nat @ K4 @ I ) ) ) ) ).

% finite_M_bounded_by_nat
thf(fact_3116_finite__less__ub,axiom,
    ! [F: nat > nat,U: nat] :
      ( ! [N4: nat] : ( ord_less_eq @ nat @ N4 @ ( F @ N4 ) )
     => ( finite_finite2 @ nat
        @ ( collect @ nat
          @ ^ [N: nat] : ( ord_less_eq @ nat @ ( F @ N ) @ U ) ) ) ) ).

% finite_less_ub
thf(fact_3117_finite__lists__length__eq,axiom,
    ! [A: $tType,A3: set @ A,N2: nat] :
      ( ( finite_finite2 @ A @ A3 )
     => ( finite_finite2 @ ( list @ A )
        @ ( collect @ ( list @ A )
          @ ^ [Xs: list @ A] :
              ( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs ) @ A3 )
              & ( ( size_size @ ( list @ A ) @ Xs )
                = N2 ) ) ) ) ) ).

% finite_lists_length_eq
thf(fact_3118_infinite__Icc,axiom,
    ! [A: $tType] :
      ( ( dense_linorder @ A )
     => ! [A2: A,B2: A] :
          ( ( ord_less @ A @ A2 @ B2 )
         => ~ ( finite_finite2 @ A @ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) ) ) ) ).

% infinite_Icc
thf(fact_3119_finite__lists__length__le,axiom,
    ! [A: $tType,A3: set @ A,N2: nat] :
      ( ( finite_finite2 @ A @ A3 )
     => ( finite_finite2 @ ( list @ A )
        @ ( collect @ ( list @ A )
          @ ^ [Xs: list @ A] :
              ( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs ) @ A3 )
              & ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ N2 ) ) ) ) ) ).

% finite_lists_length_le
thf(fact_3120_vebt__assn__raw_Osimps_I3_J,axiom,
    ! [V: option @ ( product_prod @ nat @ nat ),Va: nat,Vb: list @ vEBT_VEBT,Vc2: vEBT_VEBT,Vd: $o,Ve: $o] :
      ( ( vEBT_vebt_assn_raw @ ( vEBT_Node @ V @ Va @ Vb @ Vc2 ) @ ( vEBT_Leafi @ Vd @ Ve ) )
      = ( bot_bot @ assn ) ) ).

% vebt_assn_raw.simps(3)
thf(fact_3121_vebt__assn__raw_Osimps_I4_J,axiom,
    ! [Vd: $o,Ve: $o,V: option @ ( product_prod @ nat @ nat ),Va: nat,Vb: array @ vEBT_VEBTi,Vc2: vEBT_VEBTi] :
      ( ( vEBT_vebt_assn_raw @ ( vEBT_Leaf @ Vd @ Ve ) @ ( vEBT_Nodei @ V @ Va @ Vb @ Vc2 ) )
      = ( bot_bot @ assn ) ) ).

% vebt_assn_raw.simps(4)
thf(fact_3122_finite__divisors__nat,axiom,
    ! [M: nat] :
      ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
     => ( finite_finite2 @ nat
        @ ( collect @ nat
          @ ^ [D3: nat] : ( dvd_dvd @ nat @ D3 @ M ) ) ) ) ).

% finite_divisors_nat
thf(fact_3123_subset__eq__atLeast0__atMost__finite,axiom,
    ! [N3: set @ nat,N2: nat] :
      ( ( ord_less_eq @ ( set @ nat ) @ N3 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N2 ) )
     => ( finite_finite2 @ nat @ N3 ) ) ).

% subset_eq_atLeast0_atMost_finite
thf(fact_3124_finite__roots__unity,axiom,
    ! [A: $tType] :
      ( ( ( real_V8999393235501362500lgebra @ A )
        & ( idom @ A ) )
     => ! [N2: nat] :
          ( ( ord_less_eq @ nat @ ( one_one @ nat ) @ N2 )
         => ( finite_finite2 @ A
            @ ( collect @ A
              @ ^ [Z2: A] :
                  ( ( power_power @ A @ Z2 @ N2 )
                  = ( one_one @ A ) ) ) ) ) ) ).

% finite_roots_unity
thf(fact_3125_exE__realizer_H,axiom,
    ! [A: $tType,B: $tType,P: A > B > $o,P4: product_prod @ B @ A] :
      ( ( P @ ( product_snd @ B @ A @ P4 ) @ ( product_fst @ B @ A @ P4 ) )
     => ~ ! [X3: B,Y4: A] :
            ~ ( P @ Y4 @ X3 ) ) ).

% exE_realizer'
thf(fact_3126_nth__enumerate__eq,axiom,
    ! [A: $tType,M: nat,Xs2: list @ A,N2: nat] :
      ( ( ord_less @ nat @ M @ ( size_size @ ( list @ A ) @ Xs2 ) )
     => ( ( nth @ ( product_prod @ nat @ A ) @ ( enumerate @ A @ N2 @ Xs2 ) @ M )
        = ( product_Pair @ nat @ A @ ( plus_plus @ nat @ N2 @ M ) @ ( nth @ A @ Xs2 @ M ) ) ) ) ).

% nth_enumerate_eq
thf(fact_3127_disjE__realizer2,axiom,
    ! [B: $tType,A: $tType,P: $o,Q: A > $o,X2: option @ A,R2: B > $o,F: B,G: A > B] :
      ( ( case_option @ $o @ A @ P @ Q @ X2 )
     => ( ( P
         => ( R2 @ F ) )
       => ( ! [Q3: A] :
              ( ( Q @ Q3 )
             => ( R2 @ ( G @ Q3 ) ) )
         => ( R2 @ ( case_option @ B @ A @ F @ G @ X2 ) ) ) ) ) ).

% disjE_realizer2
thf(fact_3128_conjI__realizer,axiom,
    ! [A: $tType,B: $tType,P: A > $o,P4: A,Q: B > $o,Q2: B] :
      ( ( P @ P4 )
     => ( ( Q @ Q2 )
       => ( ( P @ ( product_fst @ A @ B @ ( product_Pair @ A @ B @ P4 @ Q2 ) ) )
          & ( Q @ ( product_snd @ A @ B @ ( product_Pair @ A @ B @ P4 @ Q2 ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_3129_exI__realizer,axiom,
    ! [B: $tType,A: $tType,P: A > B > $o,Y2: A,X2: B] :
      ( ( P @ Y2 @ X2 )
     => ( P @ ( product_snd @ B @ A @ ( product_Pair @ B @ A @ X2 @ Y2 ) ) @ ( product_fst @ B @ A @ ( product_Pair @ B @ A @ X2 @ Y2 ) ) ) ) ).

% exI_realizer
thf(fact_3130_finite__Diff__insert,axiom,
    ! [A: $tType,A3: set @ A,A2: A,B6: set @ A] :
      ( ( finite_finite2 @ A @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ A2 @ B6 ) ) )
      = ( finite_finite2 @ A @ ( minus_minus @ ( set @ A ) @ A3 @ B6 ) ) ) ).

% finite_Diff_insert
thf(fact_3131_finite__Collect__le__nat,axiom,
    ! [K: nat] :
      ( finite_finite2 @ nat
      @ ( collect @ nat
        @ ^ [N: nat] : ( ord_less_eq @ nat @ N @ K ) ) ) ).

% finite_Collect_le_nat
thf(fact_3132_finite__Collect__less__nat,axiom,
    ! [K: nat] :
      ( finite_finite2 @ nat
      @ ( collect @ nat
        @ ^ [N: nat] : ( ord_less @ nat @ N @ K ) ) ) ).

% finite_Collect_less_nat
thf(fact_3133_finite__Collect__subsets,axiom,
    ! [A: $tType,A3: set @ A] :
      ( ( finite_finite2 @ A @ A3 )
     => ( finite_finite2 @ ( set @ A )
        @ ( collect @ ( set @ A )
          @ ^ [B7: set @ A] : ( ord_less_eq @ ( set @ A ) @ B7 @ A3 ) ) ) ) ).

% finite_Collect_subsets
thf(fact_3134_finite__Collect__not,axiom,
    ! [A: $tType,P: A > $o] :
      ( ( finite_finite2 @ A @ ( collect @ A @ P ) )
     => ( ( finite_finite2 @ A
          @ ( collect @ A
            @ ^ [X: A] :
                ~ ( P @ X ) ) )
        = ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) ) ) ) ).

% finite_Collect_not
thf(fact_3135_finite__Collect__conjI,axiom,
    ! [A: $tType,P: A > $o,Q: A > $o] :
      ( ( ( finite_finite2 @ A @ ( collect @ A @ P ) )
        | ( finite_finite2 @ A @ ( collect @ A @ Q ) ) )
     => ( finite_finite2 @ A
        @ ( collect @ A
          @ ^ [X: A] :
              ( ( P @ X )
              & ( Q @ X ) ) ) ) ) ).

% finite_Collect_conjI
thf(fact_3136_finite__Collect__disjI,axiom,
    ! [A: $tType,P: A > $o,Q: A > $o] :
      ( ( finite_finite2 @ A
        @ ( collect @ A
          @ ^ [X: A] :
              ( ( P @ X )
              | ( Q @ X ) ) ) )
      = ( ( finite_finite2 @ A @ ( collect @ A @ P ) )
        & ( finite_finite2 @ A @ ( collect @ A @ Q ) ) ) ) ).

% finite_Collect_disjI
thf(fact_3137_finite__interval__int1,axiom,
    ! [A2: int,B2: int] :
      ( finite_finite2 @ int
      @ ( collect @ int
        @ ^ [I2: int] :
            ( ( ord_less_eq @ int @ A2 @ I2 )
            & ( ord_less_eq @ int @ I2 @ B2 ) ) ) ) ).

% finite_interval_int1
thf(fact_3138_finite__interval__int4,axiom,
    ! [A2: int,B2: int] :
      ( finite_finite2 @ int
      @ ( collect @ int
        @ ^ [I2: int] :
            ( ( ord_less @ int @ A2 @ I2 )
            & ( ord_less @ int @ I2 @ B2 ) ) ) ) ).

% finite_interval_int4
thf(fact_3139_finite__Diff2,axiom,
    ! [A: $tType,B6: set @ A,A3: set @ A] :
      ( ( finite_finite2 @ A @ B6 )
     => ( ( finite_finite2 @ A @ ( minus_minus @ ( set @ A ) @ A3 @ B6 ) )
        = ( finite_finite2 @ A @ A3 ) ) ) ).

% finite_Diff2
thf(fact_3140_finite__Diff,axiom,
    ! [A: $tType,A3: set @ A,B6: set @ A] :
      ( ( finite_finite2 @ A @ A3 )
     => ( finite_finite2 @ A @ ( minus_minus @ ( set @ A ) @ A3 @ B6 ) ) ) ).

% finite_Diff
thf(fact_3141_finite__interval__int2,axiom,
    ! [A2: int,B2: int] :
      ( finite_finite2 @ int
      @ ( collect @ int
        @ ^ [I2: int] :
            ( ( ord_less_eq @ int @ A2 @ I2 )
            & ( ord_less @ int @ I2 @ B2 ) ) ) ) ).

% finite_interval_int2
thf(fact_3142_finite__interval__int3,axiom,
    ! [A2: int,B2: int] :
      ( finite_finite2 @ int
      @ ( collect @ int
        @ ^ [I2: int] :
            ( ( ord_less @ int @ A2 @ I2 )
            & ( ord_less_eq @ int @ I2 @ B2 ) ) ) ) ).

% finite_interval_int3
thf(fact_3143_finite__maxlen,axiom,
    ! [A: $tType,M8: set @ ( list @ A )] :
      ( ( finite_finite2 @ ( list @ A ) @ M8 )
     => ? [N4: nat] :
        ! [X6: list @ A] :
          ( ( member @ ( list @ A ) @ X6 @ M8 )
         => ( ord_less @ nat @ ( size_size @ ( list @ A ) @ X6 ) @ N4 ) ) ) ).

% finite_maxlen
thf(fact_3144_finite__divisors__int,axiom,
    ! [I: int] :
      ( ( I
       != ( zero_zero @ int ) )
     => ( finite_finite2 @ int
        @ ( collect @ int
          @ ^ [D3: int] : ( dvd_dvd @ int @ D3 @ I ) ) ) ) ).

% finite_divisors_int
thf(fact_3145_not__finite__existsD,axiom,
    ! [A: $tType,P: A > $o] :
      ( ~ ( finite_finite2 @ A @ ( collect @ A @ P ) )
     => ? [X_12: A] : ( P @ X_12 ) ) ).

% not_finite_existsD
thf(fact_3146_pigeonhole__infinite__rel,axiom,
    ! [B: $tType,A: $tType,A3: set @ A,B6: set @ B,R2: A > B > $o] :
      ( ~ ( finite_finite2 @ A @ A3 )
     => ( ( finite_finite2 @ B @ B6 )
       => ( ! [X3: A] :
              ( ( member @ A @ X3 @ A3 )
             => ? [Xa2: B] :
                  ( ( member @ B @ Xa2 @ B6 )
                  & ( R2 @ X3 @ Xa2 ) ) )
         => ? [X3: B] :
              ( ( member @ B @ X3 @ B6 )
              & ~ ( finite_finite2 @ A
                  @ ( collect @ A
                    @ ^ [A5: A] :
                        ( ( member @ A @ A5 @ A3 )
                        & ( R2 @ A5 @ X3 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_3147_finite__has__maximal2,axiom,
    ! [A: $tType] :
      ( ( order @ A )
     => ! [A3: set @ A,A2: A] :
          ( ( finite_finite2 @ A @ A3 )
         => ( ( member @ A @ A2 @ A3 )
           => ? [X3: A] :
                ( ( member @ A @ X3 @ A3 )
                & ( ord_less_eq @ A @ A2 @ X3 )
                & ! [Xa2: A] :
                    ( ( member @ A @ Xa2 @ A3 )
                   => ( ( ord_less_eq @ A @ X3 @ Xa2 )
                     => ( X3 = Xa2 ) ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_3148_finite__has__minimal2,axiom,
    ! [A: $tType] :
      ( ( order @ A )
     => ! [A3: set @ A,A2: A] :
          ( ( finite_finite2 @ A @ A3 )
         => ( ( member @ A @ A2 @ A3 )
           => ? [X3: A] :
                ( ( member @ A @ X3 @ A3 )
                & ( ord_less_eq @ A @ X3 @ A2 )
                & ! [Xa2: A] :
                    ( ( member @ A @ Xa2 @ A3 )
                   => ( ( ord_less_eq @ A @ Xa2 @ X3 )
                     => ( X3 = Xa2 ) ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_3149_finite__subset,axiom,
    ! [A: $tType,A3: set @ A,B6: set @ A] :
      ( ( ord_less_eq @ ( set @ A ) @ A3 @ B6 )
     => ( ( finite_finite2 @ A @ B6 )
       => ( finite_finite2 @ A @ A3 ) ) ) ).

% finite_subset
thf(fact_3150_infinite__super,axiom,
    ! [A: $tType,S3: set @ A,T6: set @ A] :
      ( ( ord_less_eq @ ( set @ A ) @ S3 @ T6 )
     => ( ~ ( finite_finite2 @ A @ S3 )
       => ~ ( finite_finite2 @ A @ T6 ) ) ) ).

% infinite_super
thf(fact_3151_rev__finite__subset,axiom,
    ! [A: $tType,B6: set @ A,A3: set @ A] :
      ( ( finite_finite2 @ A @ B6 )
     => ( ( ord_less_eq @ ( set @ A ) @ A3 @ B6 )
       => ( finite_finite2 @ A @ A3 ) ) ) ).

% rev_finite_subset
thf(fact_3152_Diff__infinite__finite,axiom,
    ! [A: $tType,T6: set @ A,S3: set @ A] :
      ( ( finite_finite2 @ A @ T6 )
     => ( ~ ( finite_finite2 @ A @ S3 )
       => ~ ( finite_finite2 @ A @ ( minus_minus @ ( set @ A ) @ S3 @ T6 ) ) ) ) ).

% Diff_infinite_finite
thf(fact_3153_finite__has__minimal,axiom,
    ! [A: $tType] :
      ( ( order @ A )
     => ! [A3: set @ A] :
          ( ( finite_finite2 @ A @ A3 )
         => ( ( A3
             != ( bot_bot @ ( set @ A ) ) )
           => ? [X3: A] :
                ( ( member @ A @ X3 @ A3 )
                & ! [Xa2: A] :
                    ( ( member @ A @ Xa2 @ A3 )
                   => ( ( ord_less_eq @ A @ Xa2 @ X3 )
                     => ( X3 = Xa2 ) ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_3154_finite__has__maximal,axiom,
    ! [A: $tType] :
      ( ( order @ A )
     => ! [A3: set @ A] :
          ( ( finite_finite2 @ A @ A3 )
         => ( ( A3
             != ( bot_bot @ ( set @ A ) ) )
           => ? [X3: A] :
                ( ( member @ A @ X3 @ A3 )
                & ! [Xa2: A] :
                    ( ( member @ A @ Xa2 @ A3 )
                   => ( ( ord_less_eq @ A @ X3 @ Xa2 )
                     => ( X3 = Xa2 ) ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_3155_finite__subset__induct_H,axiom,
    ! [A: $tType,F5: set @ A,A3: set @ A,P: ( set @ A ) > $o] :
      ( ( finite_finite2 @ A @ F5 )
     => ( ( ord_less_eq @ ( set @ A ) @ F5 @ A3 )
       => ( ( P @ ( bot_bot @ ( set @ A ) ) )
         => ( ! [A4: A,F7: set @ A] :
                ( ( finite_finite2 @ A @ F7 )
               => ( ( member @ A @ A4 @ A3 )
                 => ( ( ord_less_eq @ ( set @ A ) @ F7 @ A3 )
                   => ( ~ ( member @ A @ A4 @ F7 )
                     => ( ( P @ F7 )
                       => ( P @ ( insert @ A @ A4 @ F7 ) ) ) ) ) ) )
           => ( P @ F5 ) ) ) ) ) ).

% finite_subset_induct'
thf(fact_3156_finite__subset__induct,axiom,
    ! [A: $tType,F5: set @ A,A3: set @ A,P: ( set @ A ) > $o] :
      ( ( finite_finite2 @ A @ F5 )
     => ( ( ord_less_eq @ ( set @ A ) @ F5 @ A3 )
       => ( ( P @ ( bot_bot @ ( set @ A ) ) )
         => ( ! [A4: A,F7: set @ A] :
                ( ( finite_finite2 @ A @ F7 )
               => ( ( member @ A @ A4 @ A3 )
                 => ( ~ ( member @ A @ A4 @ F7 )
                   => ( ( P @ F7 )
                     => ( P @ ( insert @ A @ A4 @ F7 ) ) ) ) ) )
           => ( P @ F5 ) ) ) ) ) ).

% finite_subset_induct
thf(fact_3157_infinite__remove,axiom,
    ! [A: $tType,S3: set @ A,A2: A] :
      ( ~ ( finite_finite2 @ A @ S3 )
     => ~ ( finite_finite2 @ A @ ( minus_minus @ ( set @ A ) @ S3 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).

% infinite_remove
thf(fact_3158_infinite__coinduct,axiom,
    ! [A: $tType,X8: ( set @ A ) > $o,A3: set @ A] :
      ( ( X8 @ A3 )
     => ( ! [A9: set @ A] :
            ( ( X8 @ A9 )
           => ? [X6: A] :
                ( ( member @ A @ X6 @ A9 )
                & ( ( X8 @ ( minus_minus @ ( set @ A ) @ A9 @ ( insert @ A @ X6 @ ( bot_bot @ ( set @ A ) ) ) ) )
                  | ~ ( finite_finite2 @ A @ ( minus_minus @ ( set @ A ) @ A9 @ ( insert @ A @ X6 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) )
       => ~ ( finite_finite2 @ A @ A3 ) ) ) ).

% infinite_coinduct
thf(fact_3159_finite__empty__induct,axiom,
    ! [A: $tType,A3: set @ A,P: ( set @ A ) > $o] :
      ( ( finite_finite2 @ A @ A3 )
     => ( ( P @ A3 )
       => ( ! [A4: A,A9: set @ A] :
              ( ( finite_finite2 @ A @ A9 )
             => ( ( member @ A @ A4 @ A9 )
               => ( ( P @ A9 )
                 => ( P @ ( minus_minus @ ( set @ A ) @ A9 @ ( insert @ A @ A4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) )
         => ( P @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).

% finite_empty_induct
thf(fact_3160_remove__induct,axiom,
    ! [A: $tType,P: ( set @ A ) > $o,B6: set @ A] :
      ( ( P @ ( bot_bot @ ( set @ A ) ) )
     => ( ( ~ ( finite_finite2 @ A @ B6 )
         => ( P @ B6 ) )
       => ( ! [A9: set @ A] :
              ( ( finite_finite2 @ A @ A9 )
             => ( ( A9
                 != ( bot_bot @ ( set @ A ) ) )
               => ( ( ord_less_eq @ ( set @ A ) @ A9 @ B6 )
                 => ( ! [X6: A] :
                        ( ( member @ A @ X6 @ A9 )
                       => ( P @ ( minus_minus @ ( set @ A ) @ A9 @ ( insert @ A @ X6 @ ( bot_bot @ ( set @ A ) ) ) ) ) )
                   => ( P @ A9 ) ) ) ) )
         => ( P @ B6 ) ) ) ) ).

% remove_induct
thf(fact_3161_finite__remove__induct,axiom,
    ! [A: $tType,B6: set @ A,P: ( set @ A ) > $o] :
      ( ( finite_finite2 @ A @ B6 )
     => ( ( P @ ( bot_bot @ ( set @ A ) ) )
       => ( ! [A9: set @ A] :
              ( ( finite_finite2 @ A @ A9 )
             => ( ( A9
                 != ( bot_bot @ ( set @ A ) ) )
               => ( ( ord_less_eq @ ( set @ A ) @ A9 @ B6 )
                 => ( ! [X6: A] :
                        ( ( member @ A @ X6 @ A9 )
                       => ( P @ ( minus_minus @ ( set @ A ) @ A9 @ ( insert @ A @ X6 @ ( bot_bot @ ( set @ A ) ) ) ) ) )
                   => ( P @ A9 ) ) ) ) )
         => ( P @ B6 ) ) ) ) ).

% finite_remove_induct
thf(fact_3162_finite__induct__select,axiom,
    ! [A: $tType,S3: set @ A,P: ( set @ A ) > $o] :
      ( ( finite_finite2 @ A @ S3 )
     => ( ( P @ ( bot_bot @ ( set @ A ) ) )
       => ( ! [T7: set @ A] :
              ( ( ord_less @ ( set @ A ) @ T7 @ S3 )
             => ( ( P @ T7 )
               => ? [X6: A] :
                    ( ( member @ A @ X6 @ ( minus_minus @ ( set @ A ) @ S3 @ T7 ) )
                    & ( P @ ( insert @ A @ X6 @ T7 ) ) ) ) )
         => ( P @ S3 ) ) ) ) ).

% finite_induct_select
thf(fact_3163_finite__nth__roots,axiom,
    ! [N2: nat,C2: complex] :
      ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
     => ( finite_finite2 @ complex
        @ ( collect @ complex
          @ ^ [Z2: complex] :
              ( ( power_power @ complex @ Z2 @ N2 )
              = C2 ) ) ) ) ).

% finite_nth_roots
thf(fact_3164_set__encode__insert,axiom,
    ! [A3: set @ nat,N2: nat] :
      ( ( finite_finite2 @ nat @ A3 )
     => ( ~ ( member @ nat @ N2 @ A3 )
       => ( ( nat_set_encode @ ( insert @ nat @ N2 @ A3 ) )
          = ( plus_plus @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) @ ( nat_set_encode @ A3 ) ) ) ) ) ).

% set_encode_insert
thf(fact_3165_diff__preserves__multiset,axiom,
    ! [A: $tType,M8: A > nat,N3: A > nat] :
      ( ( finite_finite2 @ A
        @ ( collect @ A
          @ ^ [X: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( M8 @ X ) ) ) )
     => ( finite_finite2 @ A
        @ ( collect @ A
          @ ^ [X: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( minus_minus @ nat @ ( M8 @ X ) @ ( N3 @ X ) ) ) ) ) ) ).

% diff_preserves_multiset
thf(fact_3166_add__mset__in__multiset,axiom,
    ! [A: $tType,M8: A > nat,A2: A] :
      ( ( finite_finite2 @ A
        @ ( collect @ A
          @ ^ [X: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( M8 @ X ) ) ) )
     => ( finite_finite2 @ A
        @ ( collect @ A
          @ ^ [X: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( if @ nat @ ( X = A2 ) @ ( suc @ ( M8 @ X ) ) @ ( M8 @ X ) ) ) ) ) ) ).

% add_mset_in_multiset
thf(fact_3167_finite__linorder__max__induct,axiom,
    ! [A: $tType] :
      ( ( linorder @ A )
     => ! [A3: set @ A,P: ( set @ A ) > $o] :
          ( ( finite_finite2 @ A @ A3 )
         => ( ( P @ ( bot_bot @ ( set @ A ) ) )
           => ( ! [B3: A,A9: set @ A] :
                  ( ( finite_finite2 @ A @ A9 )
                 => ( ! [X6: A] :
                        ( ( member @ A @ X6 @ A9 )
                       => ( ord_less @ A @ X6 @ B3 ) )
                   => ( ( P @ A9 )
                     => ( P @ ( insert @ A @ B3 @ A9 ) ) ) ) )
             => ( P @ A3 ) ) ) ) ) ).

% finite_linorder_max_induct
thf(fact_3168_size__empty,axiom,
    ! [A: $tType] :
      ( ( size_size @ ( multiset @ A ) @ ( zero_zero @ ( multiset @ A ) ) )
      = ( zero_zero @ nat ) ) ).

% size_empty
thf(fact_3169_size__eq__0__iff__empty,axiom,
    ! [A: $tType,M8: multiset @ A] :
      ( ( ( size_size @ ( multiset @ A ) @ M8 )
        = ( zero_zero @ nat ) )
      = ( M8
        = ( zero_zero @ ( multiset @ A ) ) ) ) ).

% size_eq_0_iff_empty
thf(fact_3170_size__union,axiom,
    ! [A: $tType,M8: multiset @ A,N3: multiset @ A] :
      ( ( size_size @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) @ M8 @ N3 ) )
      = ( plus_plus @ nat @ ( size_size @ ( multiset @ A ) @ M8 ) @ ( size_size @ ( multiset @ A ) @ N3 ) ) ) ).

% size_union
thf(fact_3171_ex__has__least__nat,axiom,
    ! [A: $tType,P: A > $o,K: A,M: A > nat] :
      ( ( P @ K )
     => ? [X3: A] :
          ( ( P @ X3 )
          & ! [Y3: A] :
              ( ( P @ Y3 )
             => ( ord_less_eq @ nat @ ( M @ X3 ) @ ( M @ Y3 ) ) ) ) ) ).

% ex_has_least_nat
thf(fact_3172_nonempty__has__size,axiom,
    ! [A: $tType,S3: multiset @ A] :
      ( ( S3
       != ( zero_zero @ ( multiset @ A ) ) )
      = ( ord_less @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( multiset @ A ) @ S3 ) ) ) ).

% nonempty_has_size
thf(fact_3173_Lattices__Big_Oex__has__greatest__nat,axiom,
    ! [A: $tType,P: A > $o,K: A,F: A > nat,B2: nat] :
      ( ( P @ K )
     => ( ! [Y4: A] :
            ( ( P @ Y4 )
           => ( ord_less @ nat @ ( F @ Y4 ) @ B2 ) )
       => ? [X3: A] :
            ( ( P @ X3 )
            & ! [Y3: A] :
                ( ( P @ Y3 )
               => ( ord_less_eq @ nat @ ( F @ Y3 ) @ ( F @ X3 ) ) ) ) ) ) ).

% Lattices_Big.ex_has_greatest_nat
thf(fact_3174_diff__size__le__size__Diff,axiom,
    ! [A: $tType,M8: multiset @ A,M9: multiset @ A] : ( ord_less_eq @ nat @ ( minus_minus @ nat @ ( size_size @ ( multiset @ A ) @ M8 ) @ ( size_size @ ( multiset @ A ) @ M9 ) ) @ ( size_size @ ( multiset @ A ) @ ( minus_minus @ ( multiset @ A ) @ M8 @ M9 ) ) ) ).

% diff_size_le_size_Diff
thf(fact_3175_complex__mod__triangle__ineq2,axiom,
    ! [B2: complex,A2: complex] : ( ord_less_eq @ real @ ( minus_minus @ real @ ( real_V7770717601297561774m_norm @ complex @ ( plus_plus @ complex @ B2 @ A2 ) ) @ ( real_V7770717601297561774m_norm @ complex @ B2 ) ) @ ( real_V7770717601297561774m_norm @ complex @ A2 ) ) ).

% complex_mod_triangle_ineq2
thf(fact_3176_even__set__encode__iff,axiom,
    ! [A3: set @ nat] :
      ( ( finite_finite2 @ nat @ A3 )
     => ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( nat_set_encode @ A3 ) )
        = ( ~ ( member @ nat @ ( zero_zero @ nat ) @ A3 ) ) ) ) ).

% even_set_encode_iff
thf(fact_3177_infinite__growing,axiom,
    ! [A: $tType] :
      ( ( linorder @ A )
     => ! [X8: set @ A] :
          ( ( X8
           != ( bot_bot @ ( set @ A ) ) )
         => ( ! [X3: A] :
                ( ( member @ A @ X3 @ X8 )
               => ? [Xa2: A] :
                    ( ( member @ A @ Xa2 @ X8 )
                    & ( ord_less @ A @ X3 @ Xa2 ) ) )
           => ~ ( finite_finite2 @ A @ X8 ) ) ) ) ).

% infinite_growing
thf(fact_3178_ex__min__if__finite,axiom,
    ! [A: $tType] :
      ( ( order @ A )
     => ! [S3: set @ A] :
          ( ( finite_finite2 @ A @ S3 )
         => ( ( S3
             != ( bot_bot @ ( set @ A ) ) )
           => ? [X3: A] :
                ( ( member @ A @ X3 @ S3 )
                & ~ ? [Xa2: A] :
                      ( ( member @ A @ Xa2 @ S3 )
                      & ( ord_less @ A @ Xa2 @ X3 ) ) ) ) ) ) ).

% ex_min_if_finite
thf(fact_3179_ex__has__greatest__nat__lemma,axiom,
    ! [A: $tType,P: A > $o,K: A,F: A > nat,N2: nat] :
      ( ( P @ K )
     => ( ! [X3: A] :
            ( ( P @ X3 )
           => ? [Y3: A] :
                ( ( P @ Y3 )
                & ~ ( ord_less_eq @ nat @ ( F @ Y3 ) @ ( F @ X3 ) ) ) )
       => ? [Y4: A] :
            ( ( P @ Y4 )
            & ~ ( ord_less @ nat @ ( F @ Y4 ) @ ( plus_plus @ nat @ ( F @ K ) @ N2 ) ) ) ) ) ).

% ex_has_greatest_nat_lemma
thf(fact_3180_filter__preserves__multiset,axiom,
    ! [A: $tType,M8: A > nat,P: A > $o] :
      ( ( finite_finite2 @ A
        @ ( collect @ A
          @ ^ [X: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( M8 @ X ) ) ) )
     => ( finite_finite2 @ A
        @ ( collect @ A
          @ ^ [X: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( if @ nat @ ( P @ X ) @ ( M8 @ X ) @ ( zero_zero @ nat ) ) ) ) ) ) ).

% filter_preserves_multiset
thf(fact_3181_finite__ranking__induct,axiom,
    ! [A: $tType,B: $tType] :
      ( ( linorder @ A )
     => ! [S3: set @ B,P: ( set @ B ) > $o,F: B > A] :
          ( ( finite_finite2 @ B @ S3 )
         => ( ( P @ ( bot_bot @ ( set @ B ) ) )
           => ( ! [X3: B,S6: set @ B] :
                  ( ( finite_finite2 @ B @ S6 )
                 => ( ! [Y3: B] :
                        ( ( member @ B @ Y3 @ S6 )
                       => ( ord_less_eq @ A @ ( F @ Y3 ) @ ( F @ X3 ) ) )
                   => ( ( P @ S6 )
                     => ( P @ ( insert @ B @ X3 @ S6 ) ) ) ) )
             => ( P @ S3 ) ) ) ) ) ).

% finite_ranking_induct
thf(fact_3182_finite__linorder__min__induct,axiom,
    ! [A: $tType] :
      ( ( linorder @ A )
     => ! [A3: set @ A,P: ( set @ A ) > $o] :
          ( ( finite_finite2 @ A @ A3 )
         => ( ( P @ ( bot_bot @ ( set @ A ) ) )
           => ( ! [B3: A,A9: set @ A] :
                  ( ( finite_finite2 @ A @ A9 )
                 => ( ! [X6: A] :
                        ( ( member @ A @ X6 @ A9 )
                       => ( ord_less @ A @ B3 @ X6 ) )
                   => ( ( P @ A9 )
                     => ( P @ ( insert @ A @ B3 @ A9 ) ) ) ) )
             => ( P @ A3 ) ) ) ) ) ).

% finite_linorder_min_induct
thf(fact_3183_bezw_Opelims,axiom,
    ! [X2: nat,Xa: nat,Y2: product_prod @ int @ int] :
      ( ( ( bezw @ X2 @ Xa )
        = Y2 )
     => ( ( accp @ ( product_prod @ nat @ nat ) @ bezw_rel @ ( product_Pair @ nat @ nat @ X2 @ Xa ) )
       => ~ ( ( ( ( Xa
                  = ( zero_zero @ nat ) )
               => ( Y2
                  = ( product_Pair @ int @ int @ ( one_one @ int ) @ ( zero_zero @ int ) ) ) )
              & ( ( Xa
                 != ( zero_zero @ nat ) )
               => ( Y2
                  = ( product_Pair @ int @ int @ ( product_snd @ int @ int @ ( bezw @ Xa @ ( modulo_modulo @ nat @ X2 @ Xa ) ) ) @ ( minus_minus @ int @ ( product_fst @ int @ int @ ( bezw @ Xa @ ( modulo_modulo @ nat @ X2 @ Xa ) ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ ( bezw @ Xa @ ( modulo_modulo @ nat @ X2 @ Xa ) ) ) @ ( semiring_1_of_nat @ int @ ( divide_divide @ nat @ X2 @ Xa ) ) ) ) ) ) ) )
           => ~ ( accp @ ( product_prod @ nat @ nat ) @ bezw_rel @ ( product_Pair @ nat @ nat @ X2 @ Xa ) ) ) ) ) ).

% bezw.pelims
thf(fact_3184_old_Oprod_Orec,axiom,
    ! [A: $tType,T: $tType,B: $tType,F1: A > B > T,A2: A,B2: B] :
      ( ( product_rec_prod @ A @ B @ T @ F1 @ ( product_Pair @ A @ B @ A2 @ B2 ) )
      = ( F1 @ A2 @ B2 ) ) ).

% old.prod.rec
thf(fact_3185_list__assn__aux__ineq__len,axiom,
    ! [B: $tType,A: $tType,L: list @ A,Li2: list @ B,A3: A > B > assn] :
      ( ( ( size_size @ ( list @ A ) @ L )
       != ( size_size @ ( list @ B ) @ Li2 ) )
     => ( ( vEBT_List_list_assn @ A @ B @ A3 @ L @ Li2 )
        = ( bot_bot @ assn ) ) ) ).

% list_assn_aux_ineq_len
thf(fact_3186_accp__subset,axiom,
    ! [A: $tType,R1: A > A > $o,R22: A > A > $o] :
      ( ( ord_less_eq @ ( A > A > $o ) @ R1 @ R22 )
     => ( ord_less_eq @ ( A > $o ) @ ( accp @ A @ R22 ) @ ( accp @ A @ R1 ) ) ) ).

% accp_subset
thf(fact_3187_vebt__buildup_Oelims,axiom,
    ! [X2: nat,Y2: vEBT_VEBT] :
      ( ( ( vEBT_vebt_buildup @ X2 )
        = Y2 )
     => ( ( ( X2
            = ( zero_zero @ nat ) )
         => ( Y2
           != ( vEBT_Leaf @ $false @ $false ) ) )
       => ( ( ( X2
              = ( suc @ ( zero_zero @ nat ) ) )
           => ( Y2
             != ( vEBT_Leaf @ $false @ $false ) ) )
         => ~ ! [Va3: nat] :
                ( ( X2
                  = ( suc @ ( suc @ Va3 ) ) )
               => ~ ( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ Va3 ) ) )
                     => ( Y2
                        = ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ Va3 ) ) @ ( replicate @ vEBT_VEBT @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) )
                    & ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ Va3 ) ) )
                     => ( Y2
                        = ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ Va3 ) ) @ ( replicate @ vEBT_VEBT @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( suc @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_buildup.elims
thf(fact_3188_intind,axiom,
    ! [A: $tType,I: nat,N2: nat,P: A > $o,X2: A] :
      ( ( ord_less @ nat @ I @ N2 )
     => ( ( P @ X2 )
       => ( P @ ( nth @ A @ ( replicate @ A @ N2 @ X2 ) @ I ) ) ) ) ).

% intind
thf(fact_3189_repli__cons__repl,axiom,
    ! [B: $tType,A: $tType,Q: assn,X2: heap_Time_Heap @ A,A3: B > A > assn,Y2: B,N2: nat] :
      ( ( hoare_hoare_triple @ A @ Q @ X2
        @ ^ [R: A] : ( times_times @ assn @ Q @ ( A3 @ Y2 @ R ) ) )
     => ( hoare_hoare_triple @ ( list @ A ) @ Q @ ( vEBT_VEBT_replicatei @ A @ N2 @ X2 )
        @ ^ [R: list @ A] : ( times_times @ assn @ Q @ ( vEBT_List_list_assn @ B @ A @ A3 @ ( replicate @ B @ N2 @ Y2 ) @ R ) ) ) ) ).

% repli_cons_repl
thf(fact_3190_repli__emp,axiom,
    ! [A: $tType,B: $tType,X2: heap_Time_Heap @ A,A3: B > A > assn,Y2: B,N2: nat] :
      ( ( hoare_hoare_triple @ A @ ( one_one @ assn ) @ X2 @ ( A3 @ Y2 ) )
     => ( hoare_hoare_triple @ ( list @ A ) @ ( one_one @ assn ) @ ( vEBT_VEBT_replicatei @ A @ N2 @ X2 ) @ ( vEBT_List_list_assn @ B @ A @ A3 @ ( replicate @ B @ N2 @ Y2 ) ) ) ) ).

% repli_emp
thf(fact_3191_union__eq__empty,axiom,
    ! [A: $tType,M8: multiset @ A,N3: multiset @ A] :
      ( ( ( plus_plus @ ( multiset @ A ) @ M8 @ N3 )
        = ( zero_zero @ ( multiset @ A ) ) )
      = ( ( M8
          = ( zero_zero @ ( multiset @ A ) ) )
        & ( N3
          = ( zero_zero @ ( multiset @ A ) ) ) ) ) ).

% union_eq_empty
thf(fact_3192_empty__eq__union,axiom,
    ! [A: $tType,M8: multiset @ A,N3: multiset @ A] :
      ( ( ( zero_zero @ ( multiset @ A ) )
        = ( plus_plus @ ( multiset @ A ) @ M8 @ N3 ) )
      = ( ( M8
          = ( zero_zero @ ( multiset @ A ) ) )
        & ( N3
          = ( zero_zero @ ( multiset @ A ) ) ) ) ) ).

% empty_eq_union
thf(fact_3193_subset__mset_Ozero__eq__add__iff__both__eq__0,axiom,
    ! [A: $tType,X2: multiset @ A,Y2: multiset @ A] :
      ( ( ( zero_zero @ ( multiset @ A ) )
        = ( plus_plus @ ( multiset @ A ) @ X2 @ Y2 ) )
      = ( ( X2
          = ( zero_zero @ ( multiset @ A ) ) )
        & ( Y2
          = ( zero_zero @ ( multiset @ A ) ) ) ) ) ).

% subset_mset.zero_eq_add_iff_both_eq_0
thf(fact_3194_subset__mset_Oadd__eq__0__iff__both__eq__0,axiom,
    ! [A: $tType,X2: multiset @ A,Y2: multiset @ A] :
      ( ( ( plus_plus @ ( multiset @ A ) @ X2 @ Y2 )
        = ( zero_zero @ ( multiset @ A ) ) )
      = ( ( X2
          = ( zero_zero @ ( multiset @ A ) ) )
        & ( Y2
          = ( zero_zero @ ( multiset @ A ) ) ) ) ) ).

% subset_mset.add_eq_0_iff_both_eq_0
thf(fact_3195_length__replicate,axiom,
    ! [A: $tType,N2: nat,X2: A] :
      ( ( size_size @ ( list @ A ) @ ( replicate @ A @ N2 @ X2 ) )
      = N2 ) ).

% length_replicate
thf(fact_3196_diff__diff__add__mset,axiom,
    ! [A: $tType,M8: multiset @ A,N3: multiset @ A,P: multiset @ A] :
      ( ( minus_minus @ ( multiset @ A ) @ ( minus_minus @ ( multiset @ A ) @ M8 @ N3 ) @ P )
      = ( minus_minus @ ( multiset @ A ) @ M8 @ ( plus_plus @ ( multiset @ A ) @ N3 @ P ) ) ) ).

% diff_diff_add_mset
thf(fact_3197_Ball__set__replicate,axiom,
    ! [A: $tType,N2: nat,A2: A,P: A > $o] :
      ( ( ! [X: A] :
            ( ( member @ A @ X @ ( set2 @ A @ ( replicate @ A @ N2 @ A2 ) ) )
           => ( P @ X ) ) )
      = ( ( P @ A2 )
        | ( N2
          = ( zero_zero @ nat ) ) ) ) ).

% Ball_set_replicate
thf(fact_3198_Bex__set__replicate,axiom,
    ! [A: $tType,N2: nat,A2: A,P: A > $o] :
      ( ( ? [X: A] :
            ( ( member @ A @ X @ ( set2 @ A @ ( replicate @ A @ N2 @ A2 ) ) )
            & ( P @ X ) ) )
      = ( ( P @ A2 )
        & ( N2
         != ( zero_zero @ nat ) ) ) ) ).

% Bex_set_replicate
thf(fact_3199_in__set__replicate,axiom,
    ! [A: $tType,X2: A,N2: nat,Y2: A] :
      ( ( member @ A @ X2 @ ( set2 @ A @ ( replicate @ A @ N2 @ Y2 ) ) )
      = ( ( X2 = Y2 )
        & ( N2
         != ( zero_zero @ nat ) ) ) ) ).

% in_set_replicate
thf(fact_3200_nth__replicate,axiom,
    ! [A: $tType,I: nat,N2: nat,X2: A] :
      ( ( ord_less @ nat @ I @ N2 )
     => ( ( nth @ A @ ( replicate @ A @ N2 @ X2 ) @ I )
        = X2 ) ) ).

% nth_replicate
thf(fact_3201_set__replicate,axiom,
    ! [A: $tType,N2: nat,X2: A] :
      ( ( N2
       != ( zero_zero @ nat ) )
     => ( ( set2 @ A @ ( replicate @ A @ N2 @ X2 ) )
        = ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).

% set_replicate
thf(fact_3202_map__fst__mk__fst,axiom,
    ! [B: $tType,A: $tType,K: A,L: list @ B] :
      ( ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ ( map @ B @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K ) @ L ) )
      = ( replicate @ A @ ( size_size @ ( list @ B ) @ L ) @ K ) ) ).

% map_fst_mk_fst
thf(fact_3203_map__snd__mk__snd,axiom,
    ! [B: $tType,A: $tType,K: A,L: list @ B] :
      ( ( map @ ( product_prod @ B @ A ) @ A @ ( product_snd @ B @ A )
        @ ( map @ B @ ( product_prod @ B @ A )
          @ ^ [X: B] : ( product_Pair @ B @ A @ X @ K )
          @ L ) )
      = ( replicate @ A @ ( size_size @ ( list @ B ) @ L ) @ K ) ) ).

% map_snd_mk_snd
thf(fact_3204_empty__neutral_I1_J,axiom,
    ! [A: $tType,X2: multiset @ A] :
      ( ( plus_plus @ ( multiset @ A ) @ ( zero_zero @ ( multiset @ A ) ) @ X2 )
      = X2 ) ).

% empty_neutral(1)
thf(fact_3205_empty__neutral_I2_J,axiom,
    ! [A: $tType,X2: multiset @ A] :
      ( ( plus_plus @ ( multiset @ A ) @ X2 @ ( zero_zero @ ( multiset @ A ) ) )
      = X2 ) ).

% empty_neutral(2)
thf(fact_3206_union__le__mono1,axiom,
    ! [A: $tType] :
      ( ( preorder @ A )
     => ! [B6: multiset @ A,D6: multiset @ A,C5: multiset @ A] :
          ( ( ord_less @ ( multiset @ A ) @ B6 @ D6 )
         => ( ord_less @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) @ B6 @ C5 ) @ ( plus_plus @ ( multiset @ A ) @ D6 @ C5 ) ) ) ) ).

% union_le_mono1
thf(fact_3207_union__le__mono2,axiom,
    ! [A: $tType] :
      ( ( preorder @ A )
     => ! [B6: multiset @ A,D6: multiset @ A,C5: multiset @ A] :
          ( ( ord_less @ ( multiset @ A ) @ B6 @ D6 )
         => ( ord_less @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) @ C5 @ B6 ) @ ( plus_plus @ ( multiset @ A ) @ C5 @ D6 ) ) ) ) ).

% union_le_mono2
thf(fact_3208_union__less__mono,axiom,
    ! [A: $tType] :
      ( ( preorder @ A )
     => ! [A3: multiset @ A,C5: multiset @ A,B6: multiset @ A,D6: multiset @ A] :
          ( ( ord_less @ ( multiset @ A ) @ A3 @ C5 )
         => ( ( ord_less @ ( multiset @ A ) @ B6 @ D6 )
           => ( ord_less @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) @ A3 @ B6 ) @ ( plus_plus @ ( multiset @ A ) @ C5 @ D6 ) ) ) ) ) ).

% union_less_mono
thf(fact_3209_Multiset_Odiff__add,axiom,
    ! [A: $tType,M8: multiset @ A,N3: multiset @ A,Q: multiset @ A] :
      ( ( minus_minus @ ( multiset @ A ) @ M8 @ ( plus_plus @ ( multiset @ A ) @ N3 @ Q ) )
      = ( minus_minus @ ( multiset @ A ) @ ( minus_minus @ ( multiset @ A ) @ M8 @ N3 ) @ Q ) ) ).

% Multiset.diff_add
thf(fact_3210_diff__union__cancelL,axiom,
    ! [A: $tType,N3: multiset @ A,M8: multiset @ A] :
      ( ( minus_minus @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) @ N3 @ M8 ) @ N3 )
      = M8 ) ).

% diff_union_cancelL
thf(fact_3211_diff__union__cancelR,axiom,
    ! [A: $tType,M8: multiset @ A,N3: multiset @ A] :
      ( ( minus_minus @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) @ M8 @ N3 ) @ N3 )
      = M8 ) ).

% diff_union_cancelR
thf(fact_3212_multi__union__self__other__eq,axiom,
    ! [A: $tType,A3: multiset @ A,X8: multiset @ A,Y7: multiset @ A] :
      ( ( ( plus_plus @ ( multiset @ A ) @ A3 @ X8 )
        = ( plus_plus @ ( multiset @ A ) @ A3 @ Y7 ) )
     => ( X8 = Y7 ) ) ).

% multi_union_self_other_eq
thf(fact_3213_union__right__cancel,axiom,
    ! [A: $tType,M8: multiset @ A,K6: multiset @ A,N3: multiset @ A] :
      ( ( ( plus_plus @ ( multiset @ A ) @ M8 @ K6 )
        = ( plus_plus @ ( multiset @ A ) @ N3 @ K6 ) )
      = ( M8 = N3 ) ) ).

% union_right_cancel
thf(fact_3214_union__left__cancel,axiom,
    ! [A: $tType,K6: multiset @ A,M8: multiset @ A,N3: multiset @ A] :
      ( ( ( plus_plus @ ( multiset @ A ) @ K6 @ M8 )
        = ( plus_plus @ ( multiset @ A ) @ K6 @ N3 ) )
      = ( M8 = N3 ) ) ).

% union_left_cancel
thf(fact_3215_union__commute,axiom,
    ! [A: $tType] :
      ( ( plus_plus @ ( multiset @ A ) )
      = ( ^ [M10: multiset @ A,N8: multiset @ A] : ( plus_plus @ ( multiset @ A ) @ N8 @ M10 ) ) ) ).

% union_commute
thf(fact_3216_union__lcomm,axiom,
    ! [A: $tType,M8: multiset @ A,N3: multiset @ A,K6: multiset @ A] :
      ( ( plus_plus @ ( multiset @ A ) @ M8 @ ( plus_plus @ ( multiset @ A ) @ N3 @ K6 ) )
      = ( plus_plus @ ( multiset @ A ) @ N3 @ ( plus_plus @ ( multiset @ A ) @ M8 @ K6 ) ) ) ).

% union_lcomm
thf(fact_3217_union__assoc,axiom,
    ! [A: $tType,M8: multiset @ A,N3: multiset @ A,K6: multiset @ A] :
      ( ( plus_plus @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) @ M8 @ N3 ) @ K6 )
      = ( plus_plus @ ( multiset @ A ) @ M8 @ ( plus_plus @ ( multiset @ A ) @ N3 @ K6 ) ) ) ).

% union_assoc
thf(fact_3218_Multiset_Odiff__right__commute,axiom,
    ! [A: $tType,M8: multiset @ A,N3: multiset @ A,Q: multiset @ A] :
      ( ( minus_minus @ ( multiset @ A ) @ ( minus_minus @ ( multiset @ A ) @ M8 @ N3 ) @ Q )
      = ( minus_minus @ ( multiset @ A ) @ ( minus_minus @ ( multiset @ A ) @ M8 @ Q ) @ N3 ) ) ).

% Multiset.diff_right_commute
thf(fact_3219_diff__empty,axiom,
    ! [A: $tType,M8: multiset @ A] :
      ( ( ( minus_minus @ ( multiset @ A ) @ M8 @ ( zero_zero @ ( multiset @ A ) ) )
        = M8 )
      & ( ( minus_minus @ ( multiset @ A ) @ ( zero_zero @ ( multiset @ A ) ) @ M8 )
        = ( zero_zero @ ( multiset @ A ) ) ) ) ).

% diff_empty
thf(fact_3220_Multiset_Odiff__cancel,axiom,
    ! [A: $tType,A3: multiset @ A] :
      ( ( minus_minus @ ( multiset @ A ) @ A3 @ A3 )
      = ( zero_zero @ ( multiset @ A ) ) ) ).

% Multiset.diff_cancel
thf(fact_3221_mset__distrib,axiom,
    ! [A: $tType,A3: multiset @ A,B6: multiset @ A,M8: multiset @ A,N3: multiset @ A] :
      ( ( ( plus_plus @ ( multiset @ A ) @ A3 @ B6 )
        = ( plus_plus @ ( multiset @ A ) @ M8 @ N3 ) )
     => ~ ! [Am: multiset @ A,An: multiset @ A] :
            ( ( A3
              = ( plus_plus @ ( multiset @ A ) @ Am @ An ) )
           => ! [Bm: multiset @ A,Bn: multiset @ A] :
                ( ( B6
                  = ( plus_plus @ ( multiset @ A ) @ Bm @ Bn ) )
               => ( ( M8
                    = ( plus_plus @ ( multiset @ A ) @ Am @ Bm ) )
                 => ( N3
                   != ( plus_plus @ ( multiset @ A ) @ An @ Bn ) ) ) ) ) ) ).

% mset_distrib
thf(fact_3222_union__diff__assoc,axiom,
    ! [A: $tType,C5: multiset @ A,B6: multiset @ A,A3: multiset @ A] :
      ( ( ( minus_minus @ ( multiset @ A ) @ C5 @ B6 )
        = ( zero_zero @ ( multiset @ A ) ) )
     => ( ( minus_minus @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) @ A3 @ B6 ) @ C5 )
        = ( plus_plus @ ( multiset @ A ) @ A3 @ ( minus_minus @ ( multiset @ A ) @ B6 @ C5 ) ) ) ) ).

% union_diff_assoc
thf(fact_3223_replicate__eqI,axiom,
    ! [A: $tType,Xs2: list @ A,N2: nat,X2: A] :
      ( ( ( size_size @ ( list @ A ) @ Xs2 )
        = N2 )
     => ( ! [Y4: A] :
            ( ( member @ A @ Y4 @ ( set2 @ A @ Xs2 ) )
           => ( Y4 = X2 ) )
       => ( Xs2
          = ( replicate @ A @ N2 @ X2 ) ) ) ) ).

% replicate_eqI
thf(fact_3224_replicate__length__same,axiom,
    ! [A: $tType,Xs2: list @ A,X2: A] :
      ( ! [X3: A] :
          ( ( member @ A @ X3 @ ( set2 @ A @ Xs2 ) )
         => ( X3 = X2 ) )
     => ( ( replicate @ A @ ( size_size @ ( list @ A ) @ Xs2 ) @ X2 )
        = Xs2 ) ) ).

% replicate_length_same
thf(fact_3225_map__replicate__const,axiom,
    ! [B: $tType,A: $tType,K: A,Lst: list @ B] :
      ( ( map @ B @ A
        @ ^ [X: B] : K
        @ Lst )
      = ( replicate @ A @ ( size_size @ ( list @ B ) @ Lst ) @ K ) ) ).

% map_replicate_const
thf(fact_3226_set__replicate__Suc,axiom,
    ! [A: $tType,N2: nat,X2: A] :
      ( ( set2 @ A @ ( replicate @ A @ ( suc @ N2 ) @ X2 ) )
      = ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ).

% set_replicate_Suc
thf(fact_3227_set__replicate__conv__if,axiom,
    ! [A: $tType,N2: nat,X2: A] :
      ( ( ( N2
          = ( zero_zero @ nat ) )
       => ( ( set2 @ A @ ( replicate @ A @ N2 @ X2 ) )
          = ( bot_bot @ ( set @ A ) ) ) )
      & ( ( N2
         != ( zero_zero @ nat ) )
       => ( ( set2 @ A @ ( replicate @ A @ N2 @ X2 ) )
          = ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).

% set_replicate_conv_if
thf(fact_3228_list__assn__cong,axiom,
    ! [A: $tType,B: $tType,Xs2: list @ A,Xs4: list @ A,Xsi: list @ B,Xsi2: list @ B,A3: A > B > assn,A10: A > B > assn] :
      ( ( Xs2 = Xs4 )
     => ( ( Xsi = Xsi2 )
       => ( ! [X3: A,Xi: B] :
              ( ( member @ A @ X3 @ ( set2 @ A @ Xs4 ) )
             => ( ( member @ B @ Xi @ ( set2 @ B @ Xsi2 ) )
               => ( ( A3 @ X3 @ Xi )
                  = ( A10 @ X3 @ Xi ) ) ) )
         => ( ( vEBT_List_list_assn @ A @ B @ A3 @ Xs2 @ Xsi )
            = ( vEBT_List_list_assn @ A @ B @ A10 @ Xs4 @ Xsi2 ) ) ) ) ) ).

% list_assn_cong
thf(fact_3229_accp__subset__induct,axiom,
    ! [A: $tType,D6: A > $o,R2: A > A > $o,X2: A,P: A > $o] :
      ( ( ord_less_eq @ ( A > $o ) @ D6 @ ( accp @ A @ R2 ) )
     => ( ! [X3: A,Z3: A] :
            ( ( D6 @ X3 )
           => ( ( R2 @ Z3 @ X3 )
             => ( D6 @ Z3 ) ) )
       => ( ( D6 @ X2 )
         => ( ! [X3: A] :
                ( ( D6 @ X3 )
               => ( ! [Z5: A] :
                      ( ( R2 @ Z5 @ X3 )
                     => ( P @ Z5 ) )
                 => ( P @ X3 ) ) )
           => ( P @ X2 ) ) ) ) ) ).

% accp_subset_induct
thf(fact_3230_vebt__buildup_Osimps_I3_J,axiom,
    ! [Va: nat] :
      ( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ Va ) ) )
       => ( ( vEBT_vebt_buildup @ ( suc @ ( suc @ Va ) ) )
          = ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ Va ) ) @ ( replicate @ vEBT_VEBT @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) )
      & ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ Va ) ) )
       => ( ( vEBT_vebt_buildup @ ( suc @ ( suc @ Va ) ) )
          = ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ Va ) ) @ ( replicate @ vEBT_VEBT @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( suc @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ).

% vebt_buildup.simps(3)
thf(fact_3231_extract__pre__list__assn__lengthD,axiom,
    ! [B: $tType,A: $tType,A3: A > B > assn,Xs2: list @ A,Xsi: list @ B,H2: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
      ( ( rep_assn @ ( vEBT_List_list_assn @ A @ B @ A3 @ Xs2 @ Xsi ) @ H2 )
     => ( ( size_size @ ( list @ B ) @ Xsi )
        = ( size_size @ ( list @ A ) @ Xs2 ) ) ) ).

% extract_pre_list_assn_lengthD
thf(fact_3232_vebt__buildup_Opelims,axiom,
    ! [X2: nat,Y2: vEBT_VEBT] :
      ( ( ( vEBT_vebt_buildup @ X2 )
        = Y2 )
     => ( ( accp @ nat @ vEBT_v4011308405150292612up_rel @ X2 )
       => ( ( ( X2
              = ( zero_zero @ nat ) )
           => ( ( Y2
                = ( vEBT_Leaf @ $false @ $false ) )
             => ~ ( accp @ nat @ vEBT_v4011308405150292612up_rel @ ( zero_zero @ nat ) ) ) )
         => ( ( ( X2
                = ( suc @ ( zero_zero @ nat ) ) )
             => ( ( Y2
                  = ( vEBT_Leaf @ $false @ $false ) )
               => ~ ( accp @ nat @ vEBT_v4011308405150292612up_rel @ ( suc @ ( zero_zero @ nat ) ) ) ) )
           => ~ ! [Va3: nat] :
                  ( ( X2
                    = ( suc @ ( suc @ Va3 ) ) )
                 => ( ( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ Va3 ) ) )
                       => ( Y2
                          = ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ Va3 ) ) @ ( replicate @ vEBT_VEBT @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) )
                      & ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ Va3 ) ) )
                       => ( Y2
                          = ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ Va3 ) ) @ ( replicate @ vEBT_VEBT @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( suc @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) )
                   => ~ ( accp @ nat @ vEBT_v4011308405150292612up_rel @ ( suc @ ( suc @ Va3 ) ) ) ) ) ) ) ) ) ).

% vebt_buildup.pelims
thf(fact_3233_VEBT__internal_Ospace_Opelims,axiom,
    ! [X2: vEBT_VEBT,Y2: nat] :
      ( ( ( vEBT_VEBT_space @ X2 )
        = Y2 )
     => ( ( accp @ vEBT_VEBT @ vEBT_VEBT_space_rel2 @ X2 )
       => ( ! [A4: $o,B3: $o] :
              ( ( X2
                = ( vEBT_Leaf @ A4 @ B3 ) )
             => ( ( Y2
                  = ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) )
               => ~ ( accp @ vEBT_VEBT @ vEBT_VEBT_space_rel2 @ ( vEBT_Leaf @ A4 @ B3 ) ) ) )
         => ~ ! [Info2: option @ ( product_prod @ nat @ nat ),Deg2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( X2
                  = ( vEBT_Node @ Info2 @ Deg2 @ TreeList2 @ Summary2 ) )
               => ( ( Y2
                    = ( plus_plus @ nat @ ( plus_plus @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit0 @ one2 ) ) ) @ ( vEBT_VEBT_space @ Summary2 ) ) @ ( size_size @ ( list @ vEBT_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_3234_VEBT__internal_Ospace_H_Opelims,axiom,
    ! [X2: vEBT_VEBT,Y2: nat] :
      ( ( ( vEBT_VEBT_space2 @ X2 )
        = Y2 )
     => ( ( accp @ vEBT_VEBT @ vEBT_VEBT_space_rel @ X2 )
       => ( ! [A4: $o,B3: $o] :
              ( ( X2
                = ( vEBT_Leaf @ A4 @ B3 ) )
             => ( ( Y2
                  = ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) )
               => ~ ( accp @ vEBT_VEBT @ vEBT_VEBT_space_rel @ ( vEBT_Leaf @ A4 @ B3 ) ) ) )
         => ~ ! [Info2: option @ ( product_prod @ nat @ nat ),Deg2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( X2
                  = ( vEBT_Node @ Info2 @ Deg2 @ TreeList2 @ Summary2 ) )
               => ( ( Y2
                    = ( plus_plus @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit1 @ one2 ) ) ) @ ( 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_3235_vebt__assn__raw_Opelims,axiom,
    ! [X2: vEBT_VEBT,Xa: vEBT_VEBTi,Y2: assn] :
      ( ( ( vEBT_vebt_assn_raw @ X2 @ Xa )
        = Y2 )
     => ( ( accp @ ( product_prod @ vEBT_VEBT @ vEBT_VEBTi ) @ vEBT_v8524038756793281170aw_rel @ ( product_Pair @ vEBT_VEBT @ vEBT_VEBTi @ X2 @ Xa ) )
       => ( ! [A4: $o,B3: $o] :
              ( ( X2
                = ( vEBT_Leaf @ A4 @ B3 ) )
             => ! [Ai: $o,Bi: $o] :
                  ( ( Xa
                    = ( vEBT_Leafi @ Ai @ Bi ) )
                 => ( ( Y2
                      = ( pure_assn
                        @ ( ( Ai = A4 )
                          & ( Bi = B3 ) ) ) )
                   => ~ ( accp @ ( product_prod @ vEBT_VEBT @ vEBT_VEBTi ) @ vEBT_v8524038756793281170aw_rel @ ( product_Pair @ vEBT_VEBT @ vEBT_VEBTi @ ( vEBT_Leaf @ A4 @ B3 ) @ ( vEBT_Leafi @ Ai @ Bi ) ) ) ) ) )
         => ( ! [Mmo2: option @ ( product_prod @ nat @ nat ),Deg2: nat,Tree_list2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( X2
                  = ( vEBT_Node @ Mmo2 @ Deg2 @ Tree_list2 @ Summary2 ) )
               => ! [Mmoi2: option @ ( product_prod @ nat @ nat ),Degi2: nat,Tree_array2: array @ vEBT_VEBTi,Summaryi2: vEBT_VEBTi] :
                    ( ( Xa
                      = ( vEBT_Nodei @ Mmoi2 @ Degi2 @ Tree_array2 @ Summaryi2 ) )
                   => ( ( Y2
                        = ( times_times @ assn
                          @ ( times_times @ assn
                            @ ( pure_assn
                              @ ( ( Mmoi2 = Mmo2 )
                                & ( Degi2 = Deg2 ) ) )
                            @ ( vEBT_vebt_assn_raw @ Summary2 @ Summaryi2 ) )
                          @ ( ex_assn @ ( list @ vEBT_VEBTi )
                            @ ^ [Tree_is: list @ vEBT_VEBTi] : ( times_times @ assn @ ( snga_assn @ vEBT_VEBTi @ Tree_array2 @ Tree_is ) @ ( vEBT_List_list_assn @ vEBT_VEBT @ vEBT_VEBTi @ vEBT_vebt_assn_raw @ Tree_list2 @ Tree_is ) ) ) ) )
                     => ~ ( accp @ ( product_prod @ vEBT_VEBT @ vEBT_VEBTi ) @ vEBT_v8524038756793281170aw_rel @ ( product_Pair @ vEBT_VEBT @ vEBT_VEBTi @ ( vEBT_Node @ Mmo2 @ Deg2 @ Tree_list2 @ Summary2 ) @ ( vEBT_Nodei @ Mmoi2 @ Degi2 @ Tree_array2 @ Summaryi2 ) ) ) ) ) )
           => ( ! [V3: option @ ( product_prod @ nat @ nat ),Va3: nat,Vb3: list @ vEBT_VEBT,Vc3: vEBT_VEBT] :
                  ( ( X2
                    = ( vEBT_Node @ V3 @ Va3 @ Vb3 @ Vc3 ) )
                 => ! [Vd3: $o,Ve3: $o] :
                      ( ( Xa
                        = ( vEBT_Leafi @ Vd3 @ Ve3 ) )
                     => ( ( Y2
                          = ( bot_bot @ assn ) )
                       => ~ ( accp @ ( product_prod @ vEBT_VEBT @ vEBT_VEBTi ) @ vEBT_v8524038756793281170aw_rel @ ( product_Pair @ vEBT_VEBT @ vEBT_VEBTi @ ( vEBT_Node @ V3 @ Va3 @ Vb3 @ Vc3 ) @ ( vEBT_Leafi @ Vd3 @ Ve3 ) ) ) ) ) )
             => ~ ! [Vd3: $o,Ve3: $o] :
                    ( ( X2
                      = ( vEBT_Leaf @ Vd3 @ Ve3 ) )
                   => ! [V3: option @ ( product_prod @ nat @ nat ),Va3: nat,Vb3: array @ vEBT_VEBTi,Vc3: vEBT_VEBTi] :
                        ( ( Xa
                          = ( vEBT_Nodei @ V3 @ Va3 @ Vb3 @ Vc3 ) )
                       => ( ( Y2
                            = ( bot_bot @ assn ) )
                         => ~ ( accp @ ( product_prod @ vEBT_VEBT @ vEBT_VEBTi ) @ vEBT_v8524038756793281170aw_rel @ ( product_Pair @ vEBT_VEBT @ vEBT_VEBTi @ ( vEBT_Leaf @ Vd3 @ Ve3 ) @ ( vEBT_Nodei @ V3 @ Va3 @ Vb3 @ Vc3 ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_assn_raw.pelims
thf(fact_3236_vebt__maxti_Opelims,axiom,
    ! [X2: vEBT_VEBTi,Y2: heap_Time_Heap @ ( option @ nat )] :
      ( ( ( vEBT_vebt_maxti @ X2 )
        = Y2 )
     => ( ( accp @ vEBT_VEBTi @ vEBT_vebt_maxti_rel @ X2 )
       => ( ! [A4: $o,B3: $o] :
              ( ( X2
                = ( vEBT_Leafi @ A4 @ B3 ) )
             => ( ( ( B3
                   => ( Y2
                      = ( heap_Time_return @ ( option @ nat ) @ ( some @ nat @ ( one_one @ nat ) ) ) ) )
                  & ( ~ B3
                   => ( ( A4
                       => ( Y2
                          = ( heap_Time_return @ ( option @ nat ) @ ( some @ nat @ ( zero_zero @ nat ) ) ) ) )
                      & ( ~ A4
                       => ( Y2
                          = ( heap_Time_return @ ( option @ nat ) @ ( none @ nat ) ) ) ) ) ) )
               => ~ ( accp @ vEBT_VEBTi @ vEBT_vebt_maxti_rel @ ( vEBT_Leafi @ A4 @ B3 ) ) ) )
         => ( ! [Uu2: nat,Uv2: array @ vEBT_VEBTi,Uw2: vEBT_VEBTi] :
                ( ( X2
                  = ( vEBT_Nodei @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw2 ) )
               => ( ( Y2
                    = ( heap_Time_return @ ( option @ nat ) @ ( none @ nat ) ) )
                 => ~ ( accp @ vEBT_VEBTi @ vEBT_vebt_maxti_rel @ ( vEBT_Nodei @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw2 ) ) ) )
           => ~ ! [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: array @ vEBT_VEBTi,Uz2: vEBT_VEBTi] :
                  ( ( X2
                    = ( vEBT_Nodei @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
                 => ( ( Y2
                      = ( heap_Time_return @ ( option @ nat ) @ ( some @ nat @ Ma2 ) ) )
                   => ~ ( accp @ vEBT_VEBTi @ vEBT_vebt_maxti_rel @ ( vEBT_Nodei @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) ) ) ) ) ) ) ) ).

% vebt_maxti.pelims
thf(fact_3237_less__eq__multiset__def,axiom,
    ! [A: $tType] :
      ( ( preorder @ A )
     => ( ( ord_less_eq @ ( multiset @ A ) )
        = ( ^ [M10: multiset @ A,N8: multiset @ A] :
              ( ( ord_less @ ( multiset @ A ) @ M10 @ N8 )
              | ( M10 = N8 ) ) ) ) ) ).

% less_eq_multiset_def
thf(fact_3238_vebt__minti_Opelims,axiom,
    ! [X2: vEBT_VEBTi,Y2: heap_Time_Heap @ ( option @ nat )] :
      ( ( ( vEBT_vebt_minti @ X2 )
        = Y2 )
     => ( ( accp @ vEBT_VEBTi @ vEBT_vebt_minti_rel @ X2 )
       => ( ! [A4: $o,B3: $o] :
              ( ( X2
                = ( vEBT_Leafi @ A4 @ B3 ) )
             => ( ( ( A4
                   => ( Y2
                      = ( heap_Time_return @ ( option @ nat ) @ ( some @ nat @ ( zero_zero @ nat ) ) ) ) )
                  & ( ~ A4
                   => ( ( B3
                       => ( Y2
                          = ( heap_Time_return @ ( option @ nat ) @ ( some @ nat @ ( one_one @ nat ) ) ) ) )
                      & ( ~ B3
                       => ( Y2
                          = ( heap_Time_return @ ( option @ nat ) @ ( none @ nat ) ) ) ) ) ) )
               => ~ ( accp @ vEBT_VEBTi @ vEBT_vebt_minti_rel @ ( vEBT_Leafi @ A4 @ B3 ) ) ) )
         => ( ! [Uu2: nat,Uv2: array @ vEBT_VEBTi,Uw2: vEBT_VEBTi] :
                ( ( X2
                  = ( vEBT_Nodei @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw2 ) )
               => ( ( Y2
                    = ( heap_Time_return @ ( option @ nat ) @ ( none @ nat ) ) )
                 => ~ ( accp @ vEBT_VEBTi @ vEBT_vebt_minti_rel @ ( vEBT_Nodei @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw2 ) ) ) )
           => ~ ! [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: array @ vEBT_VEBTi,Uz2: vEBT_VEBTi] :
                  ( ( X2
                    = ( vEBT_Nodei @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
                 => ( ( Y2
                      = ( heap_Time_return @ ( option @ nat ) @ ( some @ nat @ Mi2 ) ) )
                   => ~ ( accp @ vEBT_VEBTi @ vEBT_vebt_minti_rel @ ( vEBT_Nodei @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) ) ) ) ) ) ) ) ).

% vebt_minti.pelims
thf(fact_3239_VEBT__internal_Ocnt_Opelims,axiom,
    ! [X2: vEBT_VEBT,Y2: real] :
      ( ( ( vEBT_VEBT_cnt @ X2 )
        = Y2 )
     => ( ( accp @ vEBT_VEBT @ vEBT_VEBT_cnt_rel2 @ X2 )
       => ( ! [A4: $o,B3: $o] :
              ( ( X2
                = ( vEBT_Leaf @ A4 @ B3 ) )
             => ( ( Y2
                  = ( one_one @ real ) )
               => ~ ( accp @ vEBT_VEBT @ vEBT_VEBT_cnt_rel2 @ ( vEBT_Leaf @ A4 @ B3 ) ) ) )
         => ~ ! [Info2: option @ ( product_prod @ nat @ nat ),Deg2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( X2
                  = ( vEBT_Node @ Info2 @ Deg2 @ TreeList2 @ Summary2 ) )
               => ( ( Y2
                    = ( 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_3240_VEBT__internal_Ocnt_H_Opelims,axiom,
    ! [X2: vEBT_VEBT,Y2: nat] :
      ( ( ( vEBT_VEBT_cnt2 @ X2 )
        = Y2 )
     => ( ( accp @ vEBT_VEBT @ vEBT_VEBT_cnt_rel @ X2 )
       => ( ! [A4: $o,B3: $o] :
              ( ( X2
                = ( vEBT_Leaf @ A4 @ B3 ) )
             => ( ( Y2
                  = ( one_one @ nat ) )
               => ~ ( accp @ vEBT_VEBT @ vEBT_VEBT_cnt_rel @ ( vEBT_Leaf @ A4 @ B3 ) ) ) )
         => ~ ! [Info2: option @ ( product_prod @ nat @ nat ),Deg2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( X2
                  = ( vEBT_Node @ Info2 @ Deg2 @ TreeList2 @ Summary2 ) )
               => ( ( Y2
                    = ( 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_3241_vebt__maxt_Opelims,axiom,
    ! [X2: vEBT_VEBT,Y2: option @ nat] :
      ( ( ( vEBT_vebt_maxt @ X2 )
        = Y2 )
     => ( ( accp @ vEBT_VEBT @ vEBT_vebt_maxt_rel @ X2 )
       => ( ! [A4: $o,B3: $o] :
              ( ( X2
                = ( vEBT_Leaf @ A4 @ B3 ) )
             => ( ( ( B3
                   => ( Y2
                      = ( some @ nat @ ( one_one @ nat ) ) ) )
                  & ( ~ B3
                   => ( ( A4
                       => ( Y2
                          = ( some @ nat @ ( zero_zero @ nat ) ) ) )
                      & ( ~ A4
                       => ( Y2
                          = ( none @ nat ) ) ) ) ) )
               => ~ ( accp @ vEBT_VEBT @ vEBT_vebt_maxt_rel @ ( vEBT_Leaf @ A4 @ B3 ) ) ) )
         => ( ! [Uu2: nat,Uv2: list @ vEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( X2
                  = ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw2 ) )
               => ( ( Y2
                    = ( none @ nat ) )
                 => ~ ( accp @ vEBT_VEBT @ vEBT_vebt_maxt_rel @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw2 ) ) ) )
           => ~ ! [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list @ vEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( X2
                    = ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
                 => ( ( Y2
                      = ( some @ nat @ Ma2 ) )
                   => ~ ( accp @ vEBT_VEBT @ vEBT_vebt_maxt_rel @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) ) ) ) ) ) ) ) ).

% vebt_maxt.pelims
thf(fact_3242_vebt__mint_Opelims,axiom,
    ! [X2: vEBT_VEBT,Y2: option @ nat] :
      ( ( ( vEBT_vebt_mint @ X2 )
        = Y2 )
     => ( ( accp @ vEBT_VEBT @ vEBT_vebt_mint_rel @ X2 )
       => ( ! [A4: $o,B3: $o] :
              ( ( X2
                = ( vEBT_Leaf @ A4 @ B3 ) )
             => ( ( ( A4
                   => ( Y2
                      = ( some @ nat @ ( zero_zero @ nat ) ) ) )
                  & ( ~ A4
                   => ( ( B3
                       => ( Y2
                          = ( some @ nat @ ( one_one @ nat ) ) ) )
                      & ( ~ B3
                       => ( Y2
                          = ( none @ nat ) ) ) ) ) )
               => ~ ( accp @ vEBT_VEBT @ vEBT_vebt_mint_rel @ ( vEBT_Leaf @ A4 @ B3 ) ) ) )
         => ( ! [Uu2: nat,Uv2: list @ vEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( X2
                  = ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw2 ) )
               => ( ( Y2
                    = ( none @ nat ) )
                 => ~ ( accp @ vEBT_VEBT @ vEBT_vebt_mint_rel @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw2 ) ) ) )
           => ~ ! [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list @ vEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( X2
                    = ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
                 => ( ( Y2
                      = ( some @ nat @ Mi2 ) )
                   => ~ ( accp @ vEBT_VEBT @ vEBT_vebt_mint_rel @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) ) ) ) ) ) ) ) ).

% vebt_mint.pelims
thf(fact_3243_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t_Opelims,axiom,
    ! [X2: vEBT_VEBT,Y2: nat] :
      ( ( ( vEBT_T_m_i_n_t @ X2 )
        = Y2 )
     => ( ( accp @ vEBT_VEBT @ vEBT_T_m_i_n_t_rel @ X2 )
       => ( ! [A4: $o,B3: $o] :
              ( ( X2
                = ( vEBT_Leaf @ A4 @ B3 ) )
             => ( ( Y2
                  = ( plus_plus @ nat @ ( one_one @ nat ) @ ( if @ nat @ A4 @ ( zero_zero @ nat ) @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( one_one @ nat ) ) ) ) )
               => ~ ( accp @ vEBT_VEBT @ vEBT_T_m_i_n_t_rel @ ( vEBT_Leaf @ A4 @ B3 ) ) ) )
         => ( ! [Uu2: nat,Uv2: list @ vEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( X2
                  = ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw2 ) )
               => ( ( Y2
                    = ( one_one @ nat ) )
                 => ~ ( accp @ vEBT_VEBT @ vEBT_T_m_i_n_t_rel @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw2 ) ) ) )
           => ~ ! [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list @ vEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( X2
                    = ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
                 => ( ( Y2
                      = ( one_one @ nat ) )
                   => ~ ( accp @ vEBT_VEBT @ vEBT_T_m_i_n_t_rel @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) ) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>t.pelims
thf(fact_3244_T_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062x_092_060_094sub_062t_Opelims,axiom,
    ! [X2: vEBT_VEBT,Y2: nat] :
      ( ( ( vEBT_T_m_a_x_t @ X2 )
        = Y2 )
     => ( ( accp @ vEBT_VEBT @ vEBT_T_m_a_x_t_rel @ X2 )
       => ( ! [A4: $o,B3: $o] :
              ( ( X2
                = ( vEBT_Leaf @ A4 @ B3 ) )
             => ( ( Y2
                  = ( plus_plus @ nat @ ( one_one @ nat ) @ ( if @ nat @ B3 @ ( one_one @ nat ) @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( one_one @ nat ) ) ) ) )
               => ~ ( accp @ vEBT_VEBT @ vEBT_T_m_a_x_t_rel @ ( vEBT_Leaf @ A4 @ B3 ) ) ) )
         => ( ! [Uu2: nat,Uv2: list @ vEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( X2
                  = ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw2 ) )
               => ( ( Y2
                    = ( one_one @ nat ) )
                 => ~ ( accp @ vEBT_VEBT @ vEBT_T_m_a_x_t_rel @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw2 ) ) ) )
           => ~ ! [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list @ vEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( X2
                    = ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
                 => ( ( Y2
                      = ( one_one @ nat ) )
                   => ~ ( accp @ vEBT_VEBT @ vEBT_T_m_a_x_t_rel @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) ) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>a\<^sub>x\<^sub>t.pelims
thf(fact_3245_VEBT__internal_OminNulli_Opelims,axiom,
    ! [X2: vEBT_VEBTi,Y2: heap_Time_Heap @ $o] :
      ( ( ( vEBT_VEBT_minNulli @ X2 )
        = Y2 )
     => ( ( accp @ vEBT_VEBTi @ vEBT_V5740978063120863272li_rel @ X2 )
       => ( ( ( X2
              = ( vEBT_Leafi @ $false @ $false ) )
           => ( ( Y2
                = ( heap_Time_return @ $o @ $true ) )
             => ~ ( accp @ vEBT_VEBTi @ vEBT_V5740978063120863272li_rel @ ( vEBT_Leafi @ $false @ $false ) ) ) )
         => ( ! [Uv2: $o] :
                ( ( X2
                  = ( vEBT_Leafi @ $true @ Uv2 ) )
               => ( ( Y2
                    = ( heap_Time_return @ $o @ $false ) )
                 => ~ ( accp @ vEBT_VEBTi @ vEBT_V5740978063120863272li_rel @ ( vEBT_Leafi @ $true @ Uv2 ) ) ) )
           => ( ! [Uu2: $o] :
                  ( ( X2
                    = ( vEBT_Leafi @ Uu2 @ $true ) )
                 => ( ( Y2
                      = ( heap_Time_return @ $o @ $false ) )
                   => ~ ( accp @ vEBT_VEBTi @ vEBT_V5740978063120863272li_rel @ ( vEBT_Leafi @ Uu2 @ $true ) ) ) )
             => ( ! [Uw2: nat,Ux2: array @ vEBT_VEBTi,Uy2: vEBT_VEBTi] :
                    ( ( X2
                      = ( vEBT_Nodei @ ( none @ ( product_prod @ nat @ nat ) ) @ Uw2 @ Ux2 @ Uy2 ) )
                   => ( ( Y2
                        = ( heap_Time_return @ $o @ $true ) )
                     => ~ ( accp @ vEBT_VEBTi @ vEBT_V5740978063120863272li_rel @ ( vEBT_Nodei @ ( none @ ( product_prod @ nat @ nat ) ) @ Uw2 @ Ux2 @ Uy2 ) ) ) )
               => ~ ! [Uz2: product_prod @ nat @ nat,Va2: nat,Vb2: array @ vEBT_VEBTi,Vc: vEBT_VEBTi] :
                      ( ( X2
                        = ( vEBT_Nodei @ ( some @ ( product_prod @ nat @ nat ) @ Uz2 ) @ Va2 @ Vb2 @ Vc ) )
                     => ( ( Y2
                          = ( heap_Time_return @ $o @ $false ) )
                       => ~ ( accp @ vEBT_VEBTi @ vEBT_V5740978063120863272li_rel @ ( vEBT_Nodei @ ( some @ ( product_prod @ nat @ nat ) @ Uz2 ) @ Va2 @ Vb2 @ Vc ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.minNulli.pelims
thf(fact_3246_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,
    ! [X2: vEBT_VEBT,Y2: nat] :
      ( ( ( vEBT_T_m_i_n_N_u_l_l @ X2 )
        = Y2 )
     => ( ( accp @ vEBT_VEBT @ vEBT_T5462971552011256508_l_rel @ X2 )
       => ( ( ( X2
              = ( vEBT_Leaf @ $false @ $false ) )
           => ( ( Y2
                = ( one_one @ nat ) )
             => ~ ( accp @ vEBT_VEBT @ vEBT_T5462971552011256508_l_rel @ ( vEBT_Leaf @ $false @ $false ) ) ) )
         => ( ! [Uv2: $o] :
                ( ( X2
                  = ( vEBT_Leaf @ $true @ Uv2 ) )
               => ( ( Y2
                    = ( one_one @ nat ) )
                 => ~ ( accp @ vEBT_VEBT @ vEBT_T5462971552011256508_l_rel @ ( vEBT_Leaf @ $true @ Uv2 ) ) ) )
           => ( ! [Uu2: $o] :
                  ( ( X2
                    = ( vEBT_Leaf @ Uu2 @ $true ) )
                 => ( ( Y2
                      = ( one_one @ nat ) )
                   => ~ ( accp @ vEBT_VEBT @ vEBT_T5462971552011256508_l_rel @ ( vEBT_Leaf @ Uu2 @ $true ) ) ) )
             => ( ! [Uw2: nat,Ux2: list @ vEBT_VEBT,Uy2: vEBT_VEBT] :
                    ( ( X2
                      = ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uw2 @ Ux2 @ Uy2 ) )
                   => ( ( Y2
                        = ( one_one @ nat ) )
                     => ~ ( accp @ vEBT_VEBT @ vEBT_T5462971552011256508_l_rel @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uw2 @ Ux2 @ Uy2 ) ) ) )
               => ~ ! [Uz2: product_prod @ nat @ nat,Va2: nat,Vb2: list @ vEBT_VEBT,Vc: vEBT_VEBT] :
                      ( ( X2
                        = ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ Uz2 ) @ Va2 @ Vb2 @ Vc ) )
                     => ( ( Y2
                          = ( one_one @ nat ) )
                       => ~ ( accp @ vEBT_VEBT @ vEBT_T5462971552011256508_l_rel @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ Uz2 ) @ Va2 @ Vb2 @ Vc ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.pelims
thf(fact_3247_fold__atLeastAtMost__nat_Opinduct,axiom,
    ! [A: $tType,A0: nat > A > A,A12: nat,A23: nat,A32: A,P: ( nat > A > A ) > nat > nat > A > $o] :
      ( ( accp @ ( product_prod @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) ) ) @ ( set_fo1817059534552279752at_rel @ A ) @ ( product_Pair @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) ) @ A0 @ ( product_Pair @ nat @ ( product_prod @ nat @ A ) @ A12 @ ( product_Pair @ nat @ A @ A23 @ A32 ) ) ) )
     => ( ! [F2: nat > A > A,A4: nat,B3: nat,Acc: A] :
            ( ( accp @ ( product_prod @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) ) ) @ ( set_fo1817059534552279752at_rel @ A ) @ ( product_Pair @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) ) @ F2 @ ( product_Pair @ nat @ ( product_prod @ nat @ A ) @ A4 @ ( product_Pair @ nat @ A @ B3 @ Acc ) ) ) )
           => ( ( ~ ( ord_less @ nat @ B3 @ A4 )
               => ( P @ F2 @ ( plus_plus @ nat @ A4 @ ( one_one @ nat ) ) @ B3 @ ( F2 @ A4 @ Acc ) ) )
             => ( P @ F2 @ A4 @ B3 @ Acc ) ) )
       => ( P @ A0 @ A12 @ A23 @ A32 ) ) ) ).

% fold_atLeastAtMost_nat.pinduct
thf(fact_3248_fold__atLeastAtMost__nat_Opsimps,axiom,
    ! [A: $tType,F: nat > A > A,A2: nat,B2: nat,Acc2: A] :
      ( ( accp @ ( product_prod @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) ) ) @ ( set_fo1817059534552279752at_rel @ A ) @ ( product_Pair @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) ) @ F @ ( product_Pair @ nat @ ( product_prod @ nat @ A ) @ A2 @ ( product_Pair @ nat @ A @ B2 @ Acc2 ) ) ) )
     => ( ( ( ord_less @ nat @ B2 @ A2 )
         => ( ( set_fo6178422350223883121st_nat @ A @ F @ A2 @ B2 @ Acc2 )
            = Acc2 ) )
        & ( ~ ( ord_less @ nat @ B2 @ A2 )
         => ( ( set_fo6178422350223883121st_nat @ A @ F @ A2 @ B2 @ Acc2 )
            = ( set_fo6178422350223883121st_nat @ A @ F @ ( plus_plus @ nat @ A2 @ ( one_one @ nat ) ) @ B2 @ ( F @ A2 @ Acc2 ) ) ) ) ) ) ).

% fold_atLeastAtMost_nat.psimps
thf(fact_3249_fold__atLeastAtMost__nat_Opelims,axiom,
    ! [A: $tType,X2: nat > A > A,Xa: nat,Xb: nat,Xc: A,Y2: A] :
      ( ( ( set_fo6178422350223883121st_nat @ A @ X2 @ Xa @ Xb @ Xc )
        = Y2 )
     => ( ( accp @ ( product_prod @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) ) ) @ ( set_fo1817059534552279752at_rel @ A ) @ ( product_Pair @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) ) @ X2 @ ( product_Pair @ nat @ ( product_prod @ nat @ A ) @ Xa @ ( product_Pair @ nat @ A @ Xb @ Xc ) ) ) )
       => ~ ( ( ( ( ord_less @ nat @ Xb @ Xa )
               => ( Y2 = Xc ) )
              & ( ~ ( ord_less @ nat @ Xb @ Xa )
               => ( Y2
                  = ( set_fo6178422350223883121st_nat @ A @ X2 @ ( plus_plus @ nat @ Xa @ ( one_one @ nat ) ) @ Xb @ ( X2 @ Xa @ Xc ) ) ) ) )
           => ~ ( accp @ ( product_prod @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) ) ) @ ( set_fo1817059534552279752at_rel @ A ) @ ( product_Pair @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) ) @ X2 @ ( product_Pair @ nat @ ( product_prod @ nat @ A ) @ Xa @ ( product_Pair @ nat @ A @ Xb @ Xc ) ) ) ) ) ) ) ).

% fold_atLeastAtMost_nat.pelims
thf(fact_3250_VEBT__internal_Ooption__shift_Opelims,axiom,
    ! [A: $tType,X2: A > A > A,Xa: option @ A,Xb: option @ A,Y2: option @ A] :
      ( ( ( vEBT_V2048590022279873568_shift @ A @ X2 @ Xa @ Xb )
        = Y2 )
     => ( ( accp @ ( product_prod @ ( A > A > A ) @ ( product_prod @ ( option @ A ) @ ( option @ A ) ) ) @ ( vEBT_V459564278314245337ft_rel @ A ) @ ( product_Pair @ ( A > A > A ) @ ( product_prod @ ( option @ A ) @ ( option @ A ) ) @ X2 @ ( product_Pair @ ( option @ A ) @ ( option @ A ) @ Xa @ Xb ) ) )
       => ( ( ( Xa
              = ( none @ A ) )
           => ( ( Y2
                = ( none @ A ) )
             => ~ ( accp @ ( product_prod @ ( A > A > A ) @ ( product_prod @ ( option @ A ) @ ( option @ A ) ) ) @ ( vEBT_V459564278314245337ft_rel @ A ) @ ( product_Pair @ ( A > A > A ) @ ( product_prod @ ( option @ A ) @ ( option @ A ) ) @ X2 @ ( product_Pair @ ( option @ A ) @ ( option @ A ) @ ( none @ A ) @ Xb ) ) ) ) )
         => ( ! [V3: A] :
                ( ( Xa
                  = ( some @ A @ V3 ) )
               => ( ( Xb
                    = ( none @ A ) )
                 => ( ( Y2
                      = ( none @ A ) )
                   => ~ ( accp @ ( product_prod @ ( A > A > A ) @ ( product_prod @ ( option @ A ) @ ( option @ A ) ) ) @ ( vEBT_V459564278314245337ft_rel @ A ) @ ( product_Pair @ ( A > A > A ) @ ( product_prod @ ( option @ A ) @ ( option @ A ) ) @ X2 @ ( product_Pair @ ( option @ A ) @ ( option @ A ) @ ( some @ A @ V3 ) @ ( none @ A ) ) ) ) ) ) )
           => ~ ! [A4: A] :
                  ( ( Xa
                    = ( some @ A @ A4 ) )
                 => ! [B3: A] :
                      ( ( Xb
                        = ( some @ A @ B3 ) )
                     => ( ( Y2
                          = ( some @ A @ ( X2 @ A4 @ B3 ) ) )
                       => ~ ( accp @ ( product_prod @ ( A > A > A ) @ ( product_prod @ ( option @ A ) @ ( option @ A ) ) ) @ ( vEBT_V459564278314245337ft_rel @ A ) @ ( product_Pair @ ( A > A > A ) @ ( product_prod @ ( option @ A ) @ ( option @ A ) ) @ X2 @ ( product_Pair @ ( option @ A ) @ ( option @ A ) @ ( some @ A @ A4 ) @ ( some @ A @ B3 ) ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.option_shift.pelims
thf(fact_3251_quickcheck__narrowing__samples_Onarrowing__samples_Opinduct,axiom,
    ! [A: $tType] :
      ( ( ( code_term_of @ A )
        & ( quickc6926020345158392990erm_of @ A ) )
     => ! [A_of_integer2: code_integer > ( product_prod @ A @ A ),A0: code_integer,P: code_integer > $o] :
          ( ( accp @ code_integer @ ( code_T1710151556404007877es_rel @ A @ A_of_integer2 ) @ A0 )
         => ( ! [I3: code_integer] :
                ( ( accp @ code_integer @ ( code_T1710151556404007877es_rel @ A @ A_of_integer2 ) @ I3 )
               => ( ! [X6: product_prod @ A @ A,Xa2: A,Y3: A] :
                      ( ( ord_less @ code_integer @ ( zero_zero @ code_integer ) @ I3 )
                     => ( ( X6
                          = ( A_of_integer2 @ I3 ) )
                       => ( ( ( product_Pair @ A @ A @ Xa2 @ Y3 )
                            = X6 )
                         => ( P @ ( minus_minus @ code_integer @ I3 @ ( one_one @ code_integer ) ) ) ) ) )
                 => ( P @ I3 ) ) )
           => ( P @ A0 ) ) ) ) ).

% quickcheck_narrowing_samples.narrowing_samples.pinduct
thf(fact_3252_in__measure,axiom,
    ! [A: $tType,X2: A,Y2: A,F: A > nat] :
      ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y2 ) @ ( measure @ A @ F ) )
      = ( ord_less @ nat @ ( F @ X2 ) @ ( F @ Y2 ) ) ) ).

% in_measure
thf(fact_3253_fold__atLeastAtMost__nat_Oelims,axiom,
    ! [A: $tType,X2: nat > A > A,Xa: nat,Xb: nat,Xc: A,Y2: A] :
      ( ( ( set_fo6178422350223883121st_nat @ A @ X2 @ Xa @ Xb @ Xc )
        = Y2 )
     => ( ( ( ord_less @ nat @ Xb @ Xa )
         => ( Y2 = Xc ) )
        & ( ~ ( ord_less @ nat @ Xb @ Xa )
         => ( Y2
            = ( set_fo6178422350223883121st_nat @ A @ X2 @ ( plus_plus @ nat @ Xa @ ( one_one @ nat ) ) @ Xb @ ( X2 @ Xa @ Xc ) ) ) ) ) ) ).

% fold_atLeastAtMost_nat.elims
thf(fact_3254_fold__atLeastAtMost__nat_Osimps,axiom,
    ! [A: $tType] :
      ( ( set_fo6178422350223883121st_nat @ A )
      = ( ^ [F6: nat > A > A,A5: nat,B5: nat,Acc3: A] : ( if @ A @ ( ord_less @ nat @ B5 @ A5 ) @ Acc3 @ ( set_fo6178422350223883121st_nat @ A @ F6 @ ( plus_plus @ nat @ A5 @ ( one_one @ nat ) ) @ B5 @ ( F6 @ A5 @ Acc3 ) ) ) ) ) ).

% fold_atLeastAtMost_nat.simps
thf(fact_3255_freeze__rule,axiom,
    ! [A: $tType] :
      ( ( heap @ A )
     => ! [A2: array @ A,Xs2: list @ A] :
          ( hoare_hoare_triple @ ( list @ A ) @ ( snga_assn @ A @ A2 @ Xs2 ) @ ( array_freeze @ A @ A2 )
          @ ^ [R: list @ A] : ( times_times @ assn @ ( snga_assn @ A @ A2 @ Xs2 ) @ ( pure_assn @ ( R = Xs2 ) ) ) ) ) ).

% freeze_rule
thf(fact_3256_in__finite__psubset,axiom,
    ! [A: $tType,A3: set @ A,B6: set @ A] :
      ( ( member @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) @ ( product_Pair @ ( set @ A ) @ ( set @ A ) @ A3 @ B6 ) @ ( finite_psubset @ A ) )
      = ( ( ord_less @ ( set @ A ) @ A3 @ B6 )
        & ( finite_finite2 @ A @ B6 ) ) ) ).

% in_finite_psubset
thf(fact_3257_prod_Ofinite__Collect__op,axiom,
    ! [A: $tType,B: $tType] :
      ( ( comm_monoid_mult @ A )
     => ! [I5: set @ B,X2: B > A,Y2: B > A] :
          ( ( finite_finite2 @ B
            @ ( collect @ B
              @ ^ [I2: B] :
                  ( ( member @ B @ I2 @ I5 )
                  & ( ( X2 @ I2 )
                   != ( one_one @ A ) ) ) ) )
         => ( ( finite_finite2 @ B
              @ ( collect @ B
                @ ^ [I2: B] :
                    ( ( member @ B @ I2 @ I5 )
                    & ( ( Y2 @ I2 )
                     != ( one_one @ A ) ) ) ) )
           => ( finite_finite2 @ B
              @ ( collect @ B
                @ ^ [I2: B] :
                    ( ( member @ B @ I2 @ I5 )
                    & ( ( times_times @ A @ ( X2 @ I2 ) @ ( Y2 @ I2 ) )
                     != ( one_one @ A ) ) ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_3258_sum_Ofinite__Collect__op,axiom,
    ! [A: $tType,B: $tType] :
      ( ( comm_monoid_add @ A )
     => ! [I5: set @ B,X2: B > A,Y2: B > A] :
          ( ( finite_finite2 @ B
            @ ( collect @ B
              @ ^ [I2: B] :
                  ( ( member @ B @ I2 @ I5 )
                  & ( ( X2 @ I2 )
                   != ( zero_zero @ A ) ) ) ) )
         => ( ( finite_finite2 @ B
              @ ( collect @ B
                @ ^ [I2: B] :
                    ( ( member @ B @ I2 @ I5 )
                    & ( ( Y2 @ I2 )
                     != ( zero_zero @ A ) ) ) ) )
           => ( finite_finite2 @ B
              @ ( collect @ B
                @ ^ [I2: B] :
                    ( ( member @ B @ I2 @ I5 )
                    & ( ( plus_plus @ A @ ( X2 @ I2 ) @ ( Y2 @ I2 ) )
                     != ( zero_zero @ A ) ) ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_3259_ceiling__log__eq__powr__iff,axiom,
    ! [X2: real,B2: real,K: nat] :
      ( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
     => ( ( ord_less @ real @ ( one_one @ real ) @ B2 )
       => ( ( ( archimedean_ceiling @ real @ ( log @ B2 @ X2 ) )
            = ( plus_plus @ int @ ( semiring_1_of_nat @ int @ K ) @ ( one_one @ int ) ) )
          = ( ( ord_less @ real @ ( powr @ real @ B2 @ ( semiring_1_of_nat @ real @ K ) ) @ X2 )
            & ( ord_less_eq @ real @ X2 @ ( powr @ real @ B2 @ ( semiring_1_of_nat @ real @ ( plus_plus @ nat @ K @ ( one_one @ nat ) ) ) ) ) ) ) ) ) ).

% ceiling_log_eq_powr_iff
thf(fact_3260_powr__one__eq__one,axiom,
    ! [A: $tType] :
      ( ( ln @ A )
     => ! [A2: A] :
          ( ( powr @ A @ ( one_one @ A ) @ A2 )
          = ( one_one @ A ) ) ) ).

% powr_one_eq_one
thf(fact_3261_powr__zero__eq__one,axiom,
    ! [A: $tType] :
      ( ( ln @ A )
     => ! [X2: A] :
          ( ( ( X2
              = ( zero_zero @ A ) )
           => ( ( powr @ A @ X2 @ ( zero_zero @ A ) )
              = ( zero_zero @ A ) ) )
          & ( ( X2
             != ( zero_zero @ A ) )
           => ( ( powr @ A @ X2 @ ( zero_zero @ A ) )
              = ( one_one @ A ) ) ) ) ) ).

% powr_zero_eq_one
thf(fact_3262_powr__nonneg__iff,axiom,
    ! [A2: real,X2: real] :
      ( ( ord_less_eq @ real @ ( powr @ real @ A2 @ X2 ) @ ( zero_zero @ real ) )
      = ( A2
        = ( zero_zero @ real ) ) ) ).

% powr_nonneg_iff
thf(fact_3263_powr__less__cancel__iff,axiom,
    ! [X2: real,A2: real,B2: real] :
      ( ( ord_less @ real @ ( one_one @ real ) @ X2 )
     => ( ( ord_less @ real @ ( powr @ real @ X2 @ A2 ) @ ( powr @ real @ X2 @ B2 ) )
        = ( ord_less @ real @ A2 @ B2 ) ) ) ).

% powr_less_cancel_iff
thf(fact_3264_powr__eq__one__iff,axiom,
    ! [A2: real,X2: real] :
      ( ( ord_less @ real @ ( one_one @ real ) @ A2 )
     => ( ( ( powr @ real @ A2 @ X2 )
          = ( one_one @ real ) )
        = ( X2
          = ( zero_zero @ real ) ) ) ) ).

% powr_eq_one_iff
thf(fact_3265_powr__one,axiom,
    ! [X2: real] :
      ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 )
     => ( ( powr @ real @ X2 @ ( one_one @ real ) )
        = X2 ) ) ).

% powr_one
thf(fact_3266_powr__one__gt__zero__iff,axiom,
    ! [X2: real] :
      ( ( ( powr @ real @ X2 @ ( one_one @ real ) )
        = X2 )
      = ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 ) ) ).

% powr_one_gt_zero_iff
thf(fact_3267_powr__le__cancel__iff,axiom,
    ! [X2: real,A2: real,B2: real] :
      ( ( ord_less @ real @ ( one_one @ real ) @ X2 )
     => ( ( ord_less_eq @ real @ ( powr @ real @ X2 @ A2 ) @ ( powr @ real @ X2 @ B2 ) )
        = ( ord_less_eq @ real @ A2 @ B2 ) ) ) ).

% powr_le_cancel_iff
thf(fact_3268_numeral__powr__numeral__real,axiom,
    ! [M: num,N2: num] :
      ( ( powr @ real @ ( numeral_numeral @ real @ M ) @ ( numeral_numeral @ real @ N2 ) )
      = ( power_power @ real @ ( numeral_numeral @ real @ M ) @ ( numeral_numeral @ nat @ N2 ) ) ) ).

% numeral_powr_numeral_real
thf(fact_3269_log__powr__cancel,axiom,
    ! [A2: real,Y2: real] :
      ( ( ord_less @ real @ ( zero_zero @ real ) @ A2 )
     => ( ( A2
         != ( one_one @ real ) )
       => ( ( log @ A2 @ ( powr @ real @ A2 @ Y2 ) )
          = Y2 ) ) ) ).

% log_powr_cancel
thf(fact_3270_powr__log__cancel,axiom,
    ! [A2: real,X2: real] :
      ( ( ord_less @ real @ ( zero_zero @ real ) @ A2 )
     => ( ( A2
         != ( one_one @ real ) )
       => ( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
         => ( ( powr @ real @ A2 @ ( log @ A2 @ X2 ) )
            = X2 ) ) ) ) ).

% powr_log_cancel
thf(fact_3271_powr__numeral,axiom,
    ! [X2: real,N2: num] :
      ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 )
     => ( ( powr @ real @ X2 @ ( numeral_numeral @ real @ N2 ) )
        = ( power_power @ real @ X2 @ ( numeral_numeral @ nat @ N2 ) ) ) ) ).

% powr_numeral
thf(fact_3272_powr__powr,axiom,
    ! [X2: real,A2: real,B2: real] :
      ( ( powr @ real @ ( powr @ real @ X2 @ A2 ) @ B2 )
      = ( powr @ real @ X2 @ ( times_times @ real @ A2 @ B2 ) ) ) ).

% powr_powr
thf(fact_3273_powr__ge__pzero,axiom,
    ! [X2: real,Y2: real] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( powr @ real @ X2 @ Y2 ) ) ).

% powr_ge_pzero
thf(fact_3274_powr__mono2,axiom,
    ! [A2: real,X2: real,Y2: real] :
      ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ A2 )
     => ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 )
       => ( ( ord_less_eq @ real @ X2 @ Y2 )
         => ( ord_less_eq @ real @ ( powr @ real @ X2 @ A2 ) @ ( powr @ real @ Y2 @ A2 ) ) ) ) ) ).

% powr_mono2
thf(fact_3275_powr__less__mono,axiom,
    ! [A2: real,B2: real,X2: real] :
      ( ( ord_less @ real @ A2 @ B2 )
     => ( ( ord_less @ real @ ( one_one @ real ) @ X2 )
       => ( ord_less @ real @ ( powr @ real @ X2 @ A2 ) @ ( powr @ real @ X2 @ B2 ) ) ) ) ).

% powr_less_mono
thf(fact_3276_powr__less__cancel,axiom,
    ! [X2: real,A2: real,B2: real] :
      ( ( ord_less @ real @ ( powr @ real @ X2 @ A2 ) @ ( powr @ real @ X2 @ B2 ) )
     => ( ( ord_less @ real @ ( one_one @ real ) @ X2 )
       => ( ord_less @ real @ A2 @ B2 ) ) ) ).

% powr_less_cancel
thf(fact_3277_powr__mono,axiom,
    ! [A2: real,B2: real,X2: real] :
      ( ( ord_less_eq @ real @ A2 @ B2 )
     => ( ( ord_less_eq @ real @ ( one_one @ real ) @ X2 )
       => ( ord_less_eq @ real @ ( powr @ real @ X2 @ A2 ) @ ( powr @ real @ X2 @ B2 ) ) ) ) ).

% powr_mono
thf(fact_3278_powr__mono2_H,axiom,
    ! [A2: real,X2: real,Y2: real] :
      ( ( ord_less_eq @ real @ A2 @ ( zero_zero @ real ) )
     => ( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
       => ( ( ord_less_eq @ real @ X2 @ Y2 )
         => ( ord_less_eq @ real @ ( powr @ real @ Y2 @ A2 ) @ ( powr @ real @ X2 @ A2 ) ) ) ) ) ).

% powr_mono2'
thf(fact_3279_powr__less__mono2,axiom,
    ! [A2: real,X2: real,Y2: real] :
      ( ( ord_less @ real @ ( zero_zero @ real ) @ A2 )
     => ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 )
       => ( ( ord_less @ real @ X2 @ Y2 )
         => ( ord_less @ real @ ( powr @ real @ X2 @ A2 ) @ ( powr @ real @ Y2 @ A2 ) ) ) ) ) ).

% powr_less_mono2
thf(fact_3280_powr__inj,axiom,
    ! [A2: real,X2: real,Y2: real] :
      ( ( ord_less @ real @ ( zero_zero @ real ) @ A2 )
     => ( ( A2
         != ( one_one @ real ) )
       => ( ( ( powr @ real @ A2 @ X2 )
            = ( powr @ real @ A2 @ Y2 ) )
          = ( X2 = Y2 ) ) ) ) ).

% powr_inj
thf(fact_3281_gr__one__powr,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less @ real @ ( one_one @ real ) @ X2 )
     => ( ( ord_less @ real @ ( zero_zero @ real ) @ Y2 )
       => ( ord_less @ real @ ( one_one @ real ) @ ( powr @ real @ X2 @ Y2 ) ) ) ) ).

% gr_one_powr
thf(fact_3282_powr__le1,axiom,
    ! [A2: real,X2: real] :
      ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ A2 )
     => ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 )
       => ( ( ord_less_eq @ real @ X2 @ ( one_one @ real ) )
         => ( ord_less_eq @ real @ ( powr @ real @ X2 @ A2 ) @ ( one_one @ real ) ) ) ) ) ).

% powr_le1
thf(fact_3283_powr__mono__both,axiom,
    ! [A2: real,B2: real,X2: real,Y2: real] :
      ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ A2 )
     => ( ( ord_less_eq @ real @ A2 @ B2 )
       => ( ( ord_less_eq @ real @ ( one_one @ real ) @ X2 )
         => ( ( ord_less_eq @ real @ X2 @ Y2 )
           => ( ord_less_eq @ real @ ( powr @ real @ X2 @ A2 ) @ ( powr @ real @ Y2 @ B2 ) ) ) ) ) ) ).

% powr_mono_both
thf(fact_3284_ge__one__powr__ge__zero,axiom,
    ! [X2: real,A2: real] :
      ( ( ord_less_eq @ real @ ( one_one @ real ) @ X2 )
     => ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ A2 )
       => ( ord_less_eq @ real @ ( one_one @ real ) @ ( powr @ real @ X2 @ A2 ) ) ) ) ).

% ge_one_powr_ge_zero
thf(fact_3285_powr__divide,axiom,
    ! [X2: real,Y2: real,A2: real] :
      ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 )
     => ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y2 )
       => ( ( powr @ real @ ( divide_divide @ real @ X2 @ Y2 ) @ A2 )
          = ( divide_divide @ real @ ( powr @ real @ X2 @ A2 ) @ ( powr @ real @ Y2 @ A2 ) ) ) ) ) ).

% powr_divide
thf(fact_3286_powr__mult,axiom,
    ! [X2: real,Y2: real,A2: real] :
      ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 )
     => ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y2 )
       => ( ( powr @ real @ ( times_times @ real @ X2 @ Y2 ) @ A2 )
          = ( times_times @ real @ ( powr @ real @ X2 @ A2 ) @ ( powr @ real @ Y2 @ A2 ) ) ) ) ) ).

% powr_mult
thf(fact_3287_log__base__powr,axiom,
    ! [A2: real,B2: real,X2: real] :
      ( ( A2
       != ( zero_zero @ real ) )
     => ( ( log @ ( powr @ real @ A2 @ B2 ) @ X2 )
        = ( divide_divide @ real @ ( log @ A2 @ X2 ) @ B2 ) ) ) ).

% log_base_powr
thf(fact_3288_log__powr,axiom,
    ! [X2: real,B2: real,Y2: real] :
      ( ( X2
       != ( zero_zero @ real ) )
     => ( ( log @ B2 @ ( powr @ real @ X2 @ Y2 ) )
        = ( times_times @ real @ Y2 @ ( log @ B2 @ X2 ) ) ) ) ).

% log_powr
thf(fact_3289_powr__add,axiom,
    ! [A: $tType] :
      ( ( ( real_V3459762299906320749_field @ A )
        & ( ln @ A ) )
     => ! [X2: A,A2: A,B2: A] :
          ( ( powr @ A @ X2 @ ( plus_plus @ A @ A2 @ B2 ) )
          = ( times_times @ A @ ( powr @ A @ X2 @ A2 ) @ ( powr @ A @ X2 @ B2 ) ) ) ) ).

% powr_add
thf(fact_3290_powr__diff,axiom,
    ! [A: $tType] :
      ( ( ( real_V3459762299906320749_field @ A )
        & ( ln @ A ) )
     => ! [W: A,Z1: A,Z22: A] :
          ( ( powr @ A @ W @ ( minus_minus @ A @ Z1 @ Z22 ) )
          = ( divide_divide @ A @ ( powr @ A @ W @ Z1 ) @ ( powr @ A @ W @ Z22 ) ) ) ) ).

% powr_diff
thf(fact_3291_powr__realpow,axiom,
    ! [X2: real,N2: nat] :
      ( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
     => ( ( powr @ real @ X2 @ ( semiring_1_of_nat @ real @ N2 ) )
        = ( power_power @ real @ X2 @ N2 ) ) ) ).

% powr_realpow
thf(fact_3292_less__log__iff,axiom,
    ! [B2: real,X2: real,Y2: real] :
      ( ( ord_less @ real @ ( one_one @ real ) @ B2 )
     => ( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
       => ( ( ord_less @ real @ Y2 @ ( log @ B2 @ X2 ) )
          = ( ord_less @ real @ ( powr @ real @ B2 @ Y2 ) @ X2 ) ) ) ) ).

% less_log_iff
thf(fact_3293_log__less__iff,axiom,
    ! [B2: real,X2: real,Y2: real] :
      ( ( ord_less @ real @ ( one_one @ real ) @ B2 )
     => ( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
       => ( ( ord_less @ real @ ( log @ B2 @ X2 ) @ Y2 )
          = ( ord_less @ real @ X2 @ ( powr @ real @ B2 @ Y2 ) ) ) ) ) ).

% log_less_iff
thf(fact_3294_less__powr__iff,axiom,
    ! [B2: real,X2: real,Y2: real] :
      ( ( ord_less @ real @ ( one_one @ real ) @ B2 )
     => ( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
       => ( ( ord_less @ real @ X2 @ ( powr @ real @ B2 @ Y2 ) )
          = ( ord_less @ real @ ( log @ B2 @ X2 ) @ Y2 ) ) ) ) ).

% less_powr_iff
thf(fact_3295_powr__less__iff,axiom,
    ! [B2: real,X2: real,Y2: real] :
      ( ( ord_less @ real @ ( one_one @ real ) @ B2 )
     => ( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
       => ( ( ord_less @ real @ ( powr @ real @ B2 @ Y2 ) @ X2 )
          = ( ord_less @ real @ Y2 @ ( log @ B2 @ X2 ) ) ) ) ) ).

% powr_less_iff
thf(fact_3296_powr__mult__base,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 )
     => ( ( times_times @ real @ X2 @ ( powr @ real @ X2 @ Y2 ) )
        = ( powr @ real @ X2 @ ( plus_plus @ real @ ( one_one @ real ) @ Y2 ) ) ) ) ).

% powr_mult_base
thf(fact_3297_le__log__iff,axiom,
    ! [B2: real,X2: real,Y2: real] :
      ( ( ord_less @ real @ ( one_one @ real ) @ B2 )
     => ( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
       => ( ( ord_less_eq @ real @ Y2 @ ( log @ B2 @ X2 ) )
          = ( ord_less_eq @ real @ ( powr @ real @ B2 @ Y2 ) @ X2 ) ) ) ) ).

% le_log_iff
thf(fact_3298_log__le__iff,axiom,
    ! [B2: real,X2: real,Y2: real] :
      ( ( ord_less @ real @ ( one_one @ real ) @ B2 )
     => ( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
       => ( ( ord_less_eq @ real @ ( log @ B2 @ X2 ) @ Y2 )
          = ( ord_less_eq @ real @ X2 @ ( powr @ real @ B2 @ Y2 ) ) ) ) ) ).

% log_le_iff
thf(fact_3299_le__powr__iff,axiom,
    ! [B2: real,X2: real,Y2: real] :
      ( ( ord_less @ real @ ( one_one @ real ) @ B2 )
     => ( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
       => ( ( ord_less_eq @ real @ X2 @ ( powr @ real @ B2 @ Y2 ) )
          = ( ord_less_eq @ real @ ( log @ B2 @ X2 ) @ Y2 ) ) ) ) ).

% le_powr_iff
thf(fact_3300_powr__le__iff,axiom,
    ! [B2: real,X2: real,Y2: real] :
      ( ( ord_less @ real @ ( one_one @ real ) @ B2 )
     => ( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
       => ( ( ord_less_eq @ real @ ( powr @ real @ B2 @ Y2 ) @ X2 )
          = ( ord_less_eq @ real @ Y2 @ ( log @ B2 @ X2 ) ) ) ) ) ).

% powr_le_iff
thf(fact_3301_log__add__eq__powr,axiom,
    ! [B2: real,X2: real,Y2: real] :
      ( ( ord_less @ real @ ( zero_zero @ real ) @ B2 )
     => ( ( B2
         != ( one_one @ real ) )
       => ( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
         => ( ( plus_plus @ real @ ( log @ B2 @ X2 ) @ Y2 )
            = ( log @ B2 @ ( times_times @ real @ X2 @ ( powr @ real @ B2 @ Y2 ) ) ) ) ) ) ) ).

% log_add_eq_powr
thf(fact_3302_add__log__eq__powr,axiom,
    ! [B2: real,X2: real,Y2: real] :
      ( ( ord_less @ real @ ( zero_zero @ real ) @ B2 )
     => ( ( B2
         != ( one_one @ real ) )
       => ( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
         => ( ( plus_plus @ real @ Y2 @ ( log @ B2 @ X2 ) )
            = ( log @ B2 @ ( times_times @ real @ ( powr @ real @ B2 @ Y2 ) @ X2 ) ) ) ) ) ) ).

% add_log_eq_powr
thf(fact_3303_minus__log__eq__powr,axiom,
    ! [B2: real,X2: real,Y2: real] :
      ( ( ord_less @ real @ ( zero_zero @ real ) @ B2 )
     => ( ( B2
         != ( one_one @ real ) )
       => ( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
         => ( ( minus_minus @ real @ Y2 @ ( log @ B2 @ X2 ) )
            = ( log @ B2 @ ( divide_divide @ real @ ( powr @ real @ B2 @ Y2 ) @ X2 ) ) ) ) ) ) ).

% minus_log_eq_powr
thf(fact_3304_finite__psubset__def,axiom,
    ! [A: $tType] :
      ( ( finite_psubset @ A )
      = ( collect @ ( product_prod @ ( set @ A ) @ ( set @ A ) )
        @ ( product_case_prod @ ( set @ A ) @ ( set @ A ) @ $o
          @ ^ [A7: set @ A,B7: set @ A] :
              ( ( ord_less @ ( set @ A ) @ A7 @ B7 )
              & ( finite_finite2 @ A @ B7 ) ) ) ) ) ).

% finite_psubset_def
thf(fact_3305_sum__gp,axiom,
    ! [A: $tType] :
      ( ( ( division_ring @ A )
        & ( comm_ring @ A ) )
     => ! [N2: nat,M: nat,X2: A] :
          ( ( ( ord_less @ nat @ N2 @ M )
           => ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X2 ) @ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) )
              = ( zero_zero @ A ) ) )
          & ( ~ ( ord_less @ nat @ N2 @ M )
           => ( ( ( X2
                  = ( one_one @ A ) )
               => ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X2 ) @ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) )
                  = ( semiring_1_of_nat @ A @ ( minus_minus @ nat @ ( plus_plus @ nat @ N2 @ ( one_one @ nat ) ) @ M ) ) ) )
              & ( ( X2
                 != ( one_one @ A ) )
               => ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X2 ) @ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) )
                  = ( divide_divide @ A @ ( minus_minus @ A @ ( power_power @ A @ X2 @ M ) @ ( power_power @ A @ X2 @ ( suc @ N2 ) ) ) @ ( minus_minus @ A @ ( one_one @ A ) @ X2 ) ) ) ) ) ) ) ) ).

% sum_gp
thf(fact_3306_dbl__inc__simps_I3_J,axiom,
    ! [A: $tType] :
      ( ( neg_numeral @ A )
     => ( ( neg_numeral_dbl_inc @ A @ ( one_one @ A ) )
        = ( numeral_numeral @ A @ ( bit1 @ one2 ) ) ) ) ).

% dbl_inc_simps(3)
thf(fact_3307_geometric__deriv__sums,axiom,
    ! [A: $tType] :
      ( ( ( real_Vector_banach @ A )
        & ( real_V3459762299906320749_field @ A ) )
     => ! [Z: A] :
          ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ Z ) @ ( one_one @ real ) )
         => ( sums @ A
            @ ^ [N: nat] : ( times_times @ A @ ( semiring_1_of_nat @ A @ ( suc @ N ) ) @ ( power_power @ A @ Z @ N ) )
            @ ( divide_divide @ A @ ( one_one @ A ) @ ( power_power @ A @ ( minus_minus @ A @ ( one_one @ A ) @ Z ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ).

% geometric_deriv_sums
thf(fact_3308_VEBT__internal_Oheight_Osimps_I1_J,axiom,
    ! [A2: $o,B2: $o] :
      ( ( vEBT_VEBT_height @ ( vEBT_Leaf @ A2 @ B2 ) )
      = ( zero_zero @ nat ) ) ).

% VEBT_internal.height.simps(1)
thf(fact_3309_sum_Oneutral__const,axiom,
    ! [B: $tType,A: $tType] :
      ( ( comm_monoid_add @ A )
     => ! [A3: set @ B] :
          ( ( groups7311177749621191930dd_sum @ B @ A
            @ ^ [Uu3: B] : ( zero_zero @ A )
            @ A3 )
          = ( zero_zero @ A ) ) ) ).

% sum.neutral_const
thf(fact_3310_of__nat__sum,axiom,
    ! [A: $tType,B: $tType] :
      ( ( semiring_1 @ A )
     => ! [F: B > nat,A3: set @ B] :
          ( ( semiring_1_of_nat @ A @ ( groups7311177749621191930dd_sum @ B @ nat @ F @ A3 ) )
          = ( groups7311177749621191930dd_sum @ B @ A
            @ ^ [X: B] : ( semiring_1_of_nat @ A @ ( F @ X ) )
            @ A3 ) ) ) ).

% of_nat_sum
thf(fact_3311_of__int__sum,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ring_1 @ A )
     => ! [F: B > int,A3: set @ B] :
          ( ( ring_1_of_int @ A @ ( groups7311177749621191930dd_sum @ B @ int @ F @ A3 ) )
          = ( groups7311177749621191930dd_sum @ B @ A
            @ ^ [X: B] : ( ring_1_of_int @ A @ ( F @ X ) )
            @ A3 ) ) ) ).

% of_int_sum
thf(fact_3312_sum_Odelta_H,axiom,
    ! [B: $tType,A: $tType] :
      ( ( comm_monoid_add @ A )
     => ! [S3: set @ B,A2: B,B2: B > A] :
          ( ( finite_finite2 @ B @ S3 )
         => ( ( ( member @ B @ A2 @ S3 )
             => ( ( groups7311177749621191930dd_sum @ B @ A
                  @ ^ [K4: B] : ( if @ A @ ( A2 = K4 ) @ ( B2 @ K4 ) @ ( zero_zero @ A ) )
                  @ S3 )
                = ( B2 @ A2 ) ) )
            & ( ~ ( member @ B @ A2 @ S3 )
             => ( ( groups7311177749621191930dd_sum @ B @ A
                  @ ^ [K4: B] : ( if @ A @ ( A2 = K4 ) @ ( B2 @ K4 ) @ ( zero_zero @ A ) )
                  @ S3 )
                = ( zero_zero @ A ) ) ) ) ) ) ).

% sum.delta'
thf(fact_3313_sum_Odelta,axiom,
    ! [B: $tType,A: $tType] :
      ( ( comm_monoid_add @ A )
     => ! [S3: set @ B,A2: B,B2: B > A] :
          ( ( finite_finite2 @ B @ S3 )
         => ( ( ( member @ B @ A2 @ S3 )
             => ( ( groups7311177749621191930dd_sum @ B @ A
                  @ ^ [K4: B] : ( if @ A @ ( K4 = A2 ) @ ( B2 @ K4 ) @ ( zero_zero @ A ) )
                  @ S3 )
                = ( B2 @ A2 ) ) )
            & ( ~ ( member @ B @ A2 @ S3 )
             => ( ( groups7311177749621191930dd_sum @ B @ A
                  @ ^ [K4: B] : ( if @ A @ ( K4 = A2 ) @ ( B2 @ K4 ) @ ( zero_zero @ A ) )
                  @ S3 )
                = ( zero_zero @ A ) ) ) ) ) ) ).

% sum.delta
thf(fact_3314_dbl__inc__simps_I2_J,axiom,
    ! [A: $tType] :
      ( ( neg_numeral @ A )
     => ( ( neg_numeral_dbl_inc @ A @ ( zero_zero @ A ) )
        = ( one_one @ A ) ) ) ).

% dbl_inc_simps(2)
thf(fact_3315_dbl__inc__simps_I5_J,axiom,
    ! [A: $tType] :
      ( ( neg_numeral @ A )
     => ! [K: num] :
          ( ( neg_numeral_dbl_inc @ A @ ( numeral_numeral @ A @ K ) )
          = ( numeral_numeral @ A @ ( bit1 @ K ) ) ) ) ).

% dbl_inc_simps(5)
thf(fact_3316_sum_Oinsert,axiom,
    ! [A: $tType,B: $tType] :
      ( ( comm_monoid_add @ A )
     => ! [A3: set @ B,X2: B,G: B > A] :
          ( ( finite_finite2 @ B @ A3 )
         => ( ~ ( member @ B @ X2 @ A3 )
           => ( ( groups7311177749621191930dd_sum @ B @ A @ G @ ( insert @ B @ X2 @ A3 ) )
              = ( plus_plus @ A @ ( G @ X2 ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ A3 ) ) ) ) ) ) ).

% sum.insert
thf(fact_3317_powser__sums__zero__iff,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra @ A )
     => ! [A2: nat > A,X2: A] :
          ( ( sums @ A
            @ ^ [N: nat] : ( times_times @ A @ ( A2 @ N ) @ ( power_power @ A @ ( zero_zero @ A ) @ N ) )
            @ X2 )
          = ( ( A2 @ ( zero_zero @ nat ) )
            = X2 ) ) ) ).

% powser_sums_zero_iff
thf(fact_3318_sum_Ocl__ivl__Suc,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add @ A )
     => ! [N2: nat,M: nat,G: nat > A] :
          ( ( ( ord_less @ nat @ ( suc @ N2 ) @ M )
           => ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ ( suc @ N2 ) ) )
              = ( zero_zero @ A ) ) )
          & ( ~ ( ord_less @ nat @ ( suc @ N2 ) @ M )
           => ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ ( suc @ N2 ) ) )
              = ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ) ) ) ).

% sum.cl_ivl_Suc
thf(fact_3319_sum__zero__power,axiom,
    ! [A: $tType] :
      ( ( division_ring @ A )
     => ! [A3: set @ nat,C2: nat > A] :
          ( ( ( ( finite_finite2 @ nat @ A3 )
              & ( member @ nat @ ( zero_zero @ nat ) @ A3 ) )
           => ( ( groups7311177749621191930dd_sum @ nat @ A
                @ ^ [I2: nat] : ( times_times @ A @ ( C2 @ I2 ) @ ( power_power @ A @ ( zero_zero @ A ) @ I2 ) )
                @ A3 )
              = ( C2 @ ( zero_zero @ nat ) ) ) )
          & ( ~ ( ( finite_finite2 @ nat @ A3 )
                & ( member @ nat @ ( zero_zero @ nat ) @ A3 ) )
           => ( ( groups7311177749621191930dd_sum @ nat @ A
                @ ^ [I2: nat] : ( times_times @ A @ ( C2 @ I2 ) @ ( power_power @ A @ ( zero_zero @ A ) @ I2 ) )
                @ A3 )
              = ( zero_zero @ A ) ) ) ) ) ).

% sum_zero_power
thf(fact_3320_sum__zero__power_H,axiom,
    ! [A: $tType] :
      ( ( field @ A )
     => ! [A3: set @ nat,C2: nat > A,D2: nat > A] :
          ( ( ( ( finite_finite2 @ nat @ A3 )
              & ( member @ nat @ ( zero_zero @ nat ) @ A3 ) )
           => ( ( groups7311177749621191930dd_sum @ nat @ A
                @ ^ [I2: nat] : ( divide_divide @ A @ ( times_times @ A @ ( C2 @ I2 ) @ ( power_power @ A @ ( zero_zero @ A ) @ I2 ) ) @ ( D2 @ I2 ) )
                @ A3 )
              = ( divide_divide @ A @ ( C2 @ ( zero_zero @ nat ) ) @ ( D2 @ ( zero_zero @ nat ) ) ) ) )
          & ( ~ ( ( finite_finite2 @ nat @ A3 )
                & ( member @ nat @ ( zero_zero @ nat ) @ A3 ) )
           => ( ( groups7311177749621191930dd_sum @ nat @ A
                @ ^ [I2: nat] : ( divide_divide @ A @ ( times_times @ A @ ( C2 @ I2 ) @ ( power_power @ A @ ( zero_zero @ A ) @ I2 ) ) @ ( D2 @ I2 ) )
                @ A3 )
              = ( zero_zero @ A ) ) ) ) ) ).

% sum_zero_power'
thf(fact_3321_sum_Oswap,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( ( comm_monoid_add @ A )
     => ! [G: B > C > A,B6: set @ C,A3: set @ B] :
          ( ( groups7311177749621191930dd_sum @ B @ A
            @ ^ [I2: B] : ( groups7311177749621191930dd_sum @ C @ A @ ( G @ I2 ) @ B6 )
            @ A3 )
          = ( groups7311177749621191930dd_sum @ C @ A
            @ ^ [J3: C] :
                ( groups7311177749621191930dd_sum @ B @ A
                @ ^ [I2: B] : ( G @ I2 @ J3 )
                @ A3 )
            @ B6 ) ) ) ).

% sum.swap
thf(fact_3322_norm__sum,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V822414075346904944vector @ A )
     => ! [F: B > A,A3: set @ B] :
          ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F @ A3 ) )
          @ ( groups7311177749621191930dd_sum @ B @ real
            @ ^ [I2: B] : ( real_V7770717601297561774m_norm @ A @ ( F @ I2 ) )
            @ A3 ) ) ) ).

% norm_sum
thf(fact_3323_sum__norm__le,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V822414075346904944vector @ A )
     => ! [S3: set @ B,F: B > A,G: B > real] :
          ( ! [X3: B] :
              ( ( member @ B @ X3 @ S3 )
             => ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( F @ X3 ) ) @ ( G @ X3 ) ) )
         => ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F @ S3 ) ) @ ( groups7311177749621191930dd_sum @ B @ real @ G @ S3 ) ) ) ) ).

% sum_norm_le
thf(fact_3324_sum__mono,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ordere6911136660526730532id_add @ A )
     => ! [K6: set @ B,F: B > A,G: B > A] :
          ( ! [I3: B] :
              ( ( member @ B @ I3 @ K6 )
             => ( ord_less_eq @ A @ ( F @ I3 ) @ ( G @ I3 ) ) )
         => ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F @ K6 ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ K6 ) ) ) ) ).

% sum_mono
thf(fact_3325_sum_Odistrib,axiom,
    ! [A: $tType,B: $tType] :
      ( ( comm_monoid_add @ A )
     => ! [G: B > A,H2: B > A,A3: set @ B] :
          ( ( groups7311177749621191930dd_sum @ B @ A
            @ ^ [X: B] : ( plus_plus @ A @ ( G @ X ) @ ( H2 @ X ) )
            @ A3 )
          = ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ G @ A3 ) @ ( groups7311177749621191930dd_sum @ B @ A @ H2 @ A3 ) ) ) ) ).

% sum.distrib
thf(fact_3326_sum__distrib__left,axiom,
    ! [A: $tType,B: $tType] :
      ( ( semiring_0 @ A )
     => ! [R4: A,F: B > A,A3: set @ B] :
          ( ( times_times @ A @ R4 @ ( groups7311177749621191930dd_sum @ B @ A @ F @ A3 ) )
          = ( groups7311177749621191930dd_sum @ B @ A
            @ ^ [N: B] : ( times_times @ A @ R4 @ ( F @ N ) )
            @ A3 ) ) ) ).

% sum_distrib_left
thf(fact_3327_sum__distrib__right,axiom,
    ! [A: $tType,B: $tType] :
      ( ( semiring_0 @ A )
     => ! [F: B > A,A3: set @ B,R4: A] :
          ( ( times_times @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F @ A3 ) @ R4 )
          = ( groups7311177749621191930dd_sum @ B @ A
            @ ^ [N: B] : ( times_times @ A @ ( F @ N ) @ R4 )
            @ A3 ) ) ) ).

% sum_distrib_right
thf(fact_3328_sum__product,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( semiring_0 @ B )
     => ! [F: A > B,A3: set @ A,G: C > B,B6: set @ C] :
          ( ( times_times @ B @ ( groups7311177749621191930dd_sum @ A @ B @ F @ A3 ) @ ( groups7311177749621191930dd_sum @ C @ B @ G @ B6 ) )
          = ( groups7311177749621191930dd_sum @ A @ B
            @ ^ [I2: A] :
                ( groups7311177749621191930dd_sum @ C @ B
                @ ^ [J3: C] : ( times_times @ B @ ( F @ I2 ) @ ( G @ J3 ) )
                @ B6 )
            @ A3 ) ) ) ).

% sum_product
thf(fact_3329_sum__subtractf,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ab_group_add @ A )
     => ! [F: B > A,G: B > A,A3: set @ B] :
          ( ( groups7311177749621191930dd_sum @ B @ A
            @ ^ [X: B] : ( minus_minus @ A @ ( F @ X ) @ ( G @ X ) )
            @ A3 )
          = ( minus_minus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F @ A3 ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ A3 ) ) ) ) ).

% sum_subtractf
thf(fact_3330_sum__divide__distrib,axiom,
    ! [A: $tType,B: $tType] :
      ( ( field @ A )
     => ! [F: B > A,A3: set @ B,R4: A] :
          ( ( divide_divide @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F @ A3 ) @ R4 )
          = ( groups7311177749621191930dd_sum @ B @ A
            @ ^ [N: B] : ( divide_divide @ A @ ( F @ N ) @ R4 )
            @ A3 ) ) ) ).

% sum_divide_distrib
thf(fact_3331_sum_Oswap__restrict,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( ( comm_monoid_add @ A )
     => ! [A3: set @ B,B6: set @ C,G: B > C > A,R2: B > C > $o] :
          ( ( finite_finite2 @ B @ A3 )
         => ( ( finite_finite2 @ C @ B6 )
           => ( ( groups7311177749621191930dd_sum @ B @ A
                @ ^ [X: B] :
                    ( groups7311177749621191930dd_sum @ C @ A @ ( G @ X )
                    @ ( collect @ C
                      @ ^ [Y: C] :
                          ( ( member @ C @ Y @ B6 )
                          & ( R2 @ X @ Y ) ) ) )
                @ A3 )
              = ( groups7311177749621191930dd_sum @ C @ A
                @ ^ [Y: C] :
                    ( groups7311177749621191930dd_sum @ B @ A
                    @ ^ [X: B] : ( G @ X @ Y )
                    @ ( collect @ B
                      @ ^ [X: B] :
                          ( ( member @ B @ X @ A3 )
                          & ( R2 @ X @ Y ) ) ) )
                @ B6 ) ) ) ) ) ).

% sum.swap_restrict
thf(fact_3332_mod__sum__eq,axiom,
    ! [B: $tType,A: $tType] :
      ( ( euclid4440199948858584721cancel @ A )
     => ! [F: B > A,A2: A,A3: set @ B] :
          ( ( modulo_modulo @ A
            @ ( groups7311177749621191930dd_sum @ B @ A
              @ ^ [I2: B] : ( modulo_modulo @ A @ ( F @ I2 ) @ A2 )
              @ A3 )
            @ A2 )
          = ( modulo_modulo @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F @ A3 ) @ A2 ) ) ) ).

% mod_sum_eq
thf(fact_3333_sum__nonpos,axiom,
    ! [B: $tType,A: $tType] :
      ( ( ordere6911136660526730532id_add @ A )
     => ! [A3: set @ B,F: B > A] :
          ( ! [X3: B] :
              ( ( member @ B @ X3 @ A3 )
             => ( ord_less_eq @ A @ ( F @ X3 ) @ ( zero_zero @ A ) ) )
         => ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F @ A3 ) @ ( zero_zero @ A ) ) ) ) ).

% sum_nonpos
thf(fact_3334_sum__nonneg,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ordere6911136660526730532id_add @ A )
     => ! [A3: set @ B,F: B > A] :
          ( ! [X3: B] :
              ( ( member @ B @ X3 @ A3 )
             => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F @ X3 ) ) )
         => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( groups7311177749621191930dd_sum @ B @ A @ F @ A3 ) ) ) ) ).

% sum_nonneg
thf(fact_3335_sum__mono__inv,axiom,
    ! [A: $tType,I6: $tType] :
      ( ( ordere8940638589300402666id_add @ A )
     => ! [F: I6 > A,I5: set @ I6,G: I6 > A,I: I6] :
          ( ( ( groups7311177749621191930dd_sum @ I6 @ A @ F @ I5 )
            = ( groups7311177749621191930dd_sum @ I6 @ A @ G @ I5 ) )
         => ( ! [I3: I6] :
                ( ( member @ I6 @ I3 @ I5 )
               => ( ord_less_eq @ A @ ( F @ I3 ) @ ( G @ I3 ) ) )
           => ( ( member @ I6 @ I @ I5 )
             => ( ( finite_finite2 @ I6 @ I5 )
               => ( ( F @ I )
                  = ( G @ I ) ) ) ) ) ) ) ).

% sum_mono_inv
thf(fact_3336_sum__cong__Suc,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add @ A )
     => ! [A3: set @ nat,F: nat > A,G: nat > A] :
          ( ~ ( member @ nat @ ( zero_zero @ nat ) @ A3 )
         => ( ! [X3: nat] :
                ( ( member @ nat @ ( suc @ X3 ) @ A3 )
               => ( ( F @ ( suc @ X3 ) )
                  = ( G @ ( suc @ X3 ) ) ) )
           => ( ( groups7311177749621191930dd_sum @ nat @ A @ F @ A3 )
              = ( groups7311177749621191930dd_sum @ nat @ A @ G @ A3 ) ) ) ) ) ).

% sum_cong_Suc
thf(fact_3337_sum_Ointer__filter,axiom,
    ! [A: $tType,B: $tType] :
      ( ( comm_monoid_add @ A )
     => ! [A3: set @ B,G: B > A,P: B > $o] :
          ( ( finite_finite2 @ B @ A3 )
         => ( ( groups7311177749621191930dd_sum @ B @ A @ G
              @ ( collect @ B
                @ ^ [X: B] :
                    ( ( member @ B @ X @ A3 )
                    & ( P @ X ) ) ) )
            = ( groups7311177749621191930dd_sum @ B @ A
              @ ^ [X: B] : ( if @ A @ ( P @ X ) @ ( G @ X ) @ ( zero_zero @ A ) )
              @ A3 ) ) ) ) ).

% sum.inter_filter
thf(fact_3338_sum_Oshift__bounds__cl__Suc__ivl,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add @ A )
     => ! [G: nat > A,M: nat,N2: nat] :
          ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M ) @ ( suc @ N2 ) ) )
          = ( groups7311177749621191930dd_sum @ nat @ A
            @ ^ [I2: nat] : ( G @ ( suc @ I2 ) )
            @ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) ) ) ) ).

% sum.shift_bounds_cl_Suc_ivl
thf(fact_3339_sum_Oshift__bounds__cl__nat__ivl,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add @ A )
     => ! [G: nat > A,M: nat,K: nat,N2: nat] :
          ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ M @ K ) @ ( plus_plus @ nat @ N2 @ K ) ) )
          = ( groups7311177749621191930dd_sum @ nat @ A
            @ ^ [I2: nat] : ( G @ ( plus_plus @ nat @ I2 @ K ) )
            @ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) ) ) ) ).

% sum.shift_bounds_cl_nat_ivl
thf(fact_3340_sum__le__included,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ( ordere6911136660526730532id_add @ A )
     => ! [S4: set @ B,T2: set @ C,G: C > A,I: C > B,F: B > A] :
          ( ( finite_finite2 @ B @ S4 )
         => ( ( finite_finite2 @ C @ T2 )
           => ( ! [X3: C] :
                  ( ( member @ C @ X3 @ T2 )
                 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( G @ X3 ) ) )
             => ( ! [X3: B] :
                    ( ( member @ B @ X3 @ S4 )
                   => ? [Xa2: C] :
                        ( ( member @ C @ Xa2 @ T2 )
                        & ( ( I @ Xa2 )
                          = X3 )
                        & ( ord_less_eq @ A @ ( F @ X3 ) @ ( G @ Xa2 ) ) ) )
               => ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F @ S4 ) @ ( groups7311177749621191930dd_sum @ C @ A @ G @ T2 ) ) ) ) ) ) ) ).

% sum_le_included
thf(fact_3341_sum__nonneg__eq__0__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ordere6911136660526730532id_add @ A )
     => ! [A3: set @ B,F: B > A] :
          ( ( finite_finite2 @ B @ A3 )
         => ( ! [X3: B] :
                ( ( member @ B @ X3 @ A3 )
               => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F @ X3 ) ) )
           => ( ( ( groups7311177749621191930dd_sum @ B @ A @ F @ A3 )
                = ( zero_zero @ A ) )
              = ( ! [X: B] :
                    ( ( member @ B @ X @ A3 )
                   => ( ( F @ X )
                      = ( zero_zero @ A ) ) ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_3342_sum__strict__mono__ex1,axiom,
    ! [A: $tType,I6: $tType] :
      ( ( ordere8940638589300402666id_add @ A )
     => ! [A3: set @ I6,F: I6 > A,G: I6 > A] :
          ( ( finite_finite2 @ I6 @ A3 )
         => ( ! [X3: I6] :
                ( ( member @ I6 @ X3 @ A3 )
               => ( ord_less_eq @ A @ ( F @ X3 ) @ ( G @ X3 ) ) )
           => ( ? [X6: I6] :
                  ( ( member @ I6 @ X6 @ A3 )
                  & ( ord_less @ A @ ( F @ X6 ) @ ( G @ X6 ) ) )
             => ( ord_less @ A @ ( groups7311177749621191930dd_sum @ I6 @ A @ F @ A3 ) @ ( groups7311177749621191930dd_sum @ I6 @ A @ G @ A3 ) ) ) ) ) ) ).

% sum_strict_mono_ex1
thf(fact_3343_sum_Orelated,axiom,
    ! [A: $tType,B: $tType] :
      ( ( comm_monoid_add @ A )
     => ! [R2: A > A > $o,S3: set @ B,H2: B > A,G: B > A] :
          ( ( R2 @ ( zero_zero @ A ) @ ( zero_zero @ A ) )
         => ( ! [X16: A,Y15: A,X24: A,Y24: A] :
                ( ( ( R2 @ X16 @ X24 )
                  & ( R2 @ Y15 @ Y24 ) )
               => ( R2 @ ( plus_plus @ A @ X16 @ Y15 ) @ ( plus_plus @ A @ X24 @ Y24 ) ) )
           => ( ( finite_finite2 @ B @ S3 )
             => ( ! [X3: B] :
                    ( ( member @ B @ X3 @ S3 )
                   => ( R2 @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
               => ( R2 @ ( groups7311177749621191930dd_sum @ B @ A @ H2 @ S3 ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ S3 ) ) ) ) ) ) ) ).

% sum.related
thf(fact_3344_sum__strict__mono,axiom,
    ! [A: $tType,B: $tType] :
      ( ( strict7427464778891057005id_add @ A )
     => ! [A3: set @ B,F: B > A,G: B > A] :
          ( ( finite_finite2 @ B @ A3 )
         => ( ( A3
             != ( bot_bot @ ( set @ B ) ) )
           => ( ! [X3: B] :
                  ( ( member @ B @ X3 @ A3 )
                 => ( ord_less @ A @ ( F @ X3 ) @ ( G @ X3 ) ) )
             => ( ord_less @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F @ A3 ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ A3 ) ) ) ) ) ) ).

% sum_strict_mono
thf(fact_3345_sum_Oinsert__if,axiom,
    ! [A: $tType,B: $tType] :
      ( ( comm_monoid_add @ A )
     => ! [A3: set @ B,X2: B,G: B > A] :
          ( ( finite_finite2 @ B @ A3 )
         => ( ( ( member @ B @ X2 @ A3 )
             => ( ( groups7311177749621191930dd_sum @ B @ A @ G @ ( insert @ B @ X2 @ A3 ) )
                = ( groups7311177749621191930dd_sum @ B @ A @ G @ A3 ) ) )
            & ( ~ ( member @ B @ X2 @ A3 )
             => ( ( groups7311177749621191930dd_sum @ B @ A @ G @ ( insert @ B @ X2 @ A3 ) )
                = ( plus_plus @ A @ ( G @ X2 ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ A3 ) ) ) ) ) ) ) ).

% sum.insert_if
thf(fact_3346_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ( comm_monoid_add @ A )
     => ! [S7: set @ B,T8: set @ C,S3: set @ B,I: C > B,J: B > C,T6: set @ C,G: B > A,H2: C > A] :
          ( ( finite_finite2 @ B @ S7 )
         => ( ( finite_finite2 @ C @ T8 )
           => ( ! [A4: B] :
                  ( ( member @ B @ A4 @ ( minus_minus @ ( set @ B ) @ S3 @ S7 ) )
                 => ( ( I @ ( J @ A4 ) )
                    = A4 ) )
             => ( ! [A4: B] :
                    ( ( member @ B @ A4 @ ( minus_minus @ ( set @ B ) @ S3 @ S7 ) )
                   => ( member @ C @ ( J @ A4 ) @ ( minus_minus @ ( set @ C ) @ T6 @ T8 ) ) )
               => ( ! [B3: C] :
                      ( ( member @ C @ B3 @ ( minus_minus @ ( set @ C ) @ T6 @ T8 ) )
                     => ( ( J @ ( I @ B3 ) )
                        = B3 ) )
                 => ( ! [B3: C] :
                        ( ( member @ C @ B3 @ ( minus_minus @ ( set @ C ) @ T6 @ T8 ) )
                       => ( member @ B @ ( I @ B3 ) @ ( minus_minus @ ( set @ B ) @ S3 @ S7 ) ) )
                   => ( ! [A4: B] :
                          ( ( member @ B @ A4 @ S7 )
                         => ( ( G @ A4 )
                            = ( zero_zero @ A ) ) )
                     => ( ! [B3: C] :
                            ( ( member @ C @ B3 @ T8 )
                           => ( ( H2 @ B3 )
                              = ( zero_zero @ A ) ) )
                       => ( ! [A4: B] :
                              ( ( member @ B @ A4 @ S3 )
                             => ( ( H2 @ ( J @ A4 ) )
                                = ( G @ A4 ) ) )
                         => ( ( groups7311177749621191930dd_sum @ B @ A @ G @ S3 )
                            = ( groups7311177749621191930dd_sum @ C @ A @ H2 @ T6 ) ) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
thf(fact_3347_sum__nonneg__0,axiom,
    ! [B: $tType,A: $tType] :
      ( ( ordere6911136660526730532id_add @ A )
     => ! [S4: set @ B,F: B > A,I: B] :
          ( ( finite_finite2 @ B @ S4 )
         => ( ! [I3: B] :
                ( ( member @ B @ I3 @ S4 )
               => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F @ I3 ) ) )
           => ( ( ( groups7311177749621191930dd_sum @ B @ A @ F @ S4 )
                = ( zero_zero @ A ) )
             => ( ( member @ B @ I @ S4 )
               => ( ( F @ I )
                  = ( zero_zero @ A ) ) ) ) ) ) ) ).

% sum_nonneg_0
thf(fact_3348_sum__nonneg__leq__bound,axiom,
    ! [B: $tType,A: $tType] :
      ( ( ordere6911136660526730532id_add @ A )
     => ! [S4: set @ B,F: B > A,B6: A,I: B] :
          ( ( finite_finite2 @ B @ S4 )
         => ( ! [I3: B] :
                ( ( member @ B @ I3 @ S4 )
               => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F @ I3 ) ) )
           => ( ( ( groups7311177749621191930dd_sum @ B @ A @ F @ S4 )
                = B6 )
             => ( ( member @ B @ I @ S4 )
               => ( ord_less_eq @ A @ ( F @ I ) @ B6 ) ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_3349_sum_Osetdiff__irrelevant,axiom,
    ! [A: $tType,B: $tType] :
      ( ( comm_monoid_add @ A )
     => ! [A3: set @ B,G: B > A] :
          ( ( finite_finite2 @ B @ A3 )
         => ( ( groups7311177749621191930dd_sum @ B @ A @ G
              @ ( minus_minus @ ( set @ B ) @ A3
                @ ( collect @ B
                  @ ^ [X: B] :
                      ( ( G @ X )
                      = ( zero_zero @ A ) ) ) ) )
            = ( groups7311177749621191930dd_sum @ B @ A @ G @ A3 ) ) ) ) ).

% sum.setdiff_irrelevant
thf(fact_3350_sum__power__add,axiom,
    ! [A: $tType] :
      ( ( ( monoid_mult @ A )
        & ( comm_ring @ A ) )
     => ! [X2: A,M: nat,I5: set @ nat] :
          ( ( groups7311177749621191930dd_sum @ nat @ A
            @ ^ [I2: nat] : ( power_power @ A @ X2 @ ( plus_plus @ nat @ M @ I2 ) )
            @ I5 )
          = ( times_times @ A @ ( power_power @ A @ X2 @ M ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X2 ) @ I5 ) ) ) ) ).

% sum_power_add
thf(fact_3351_sum_OatLeastAtMost__rev,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add @ A )
     => ! [G: nat > A,N2: nat,M: nat] :
          ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ N2 @ M ) )
          = ( groups7311177749621191930dd_sum @ nat @ A
            @ ^ [I2: nat] : ( G @ ( minus_minus @ nat @ ( plus_plus @ nat @ M @ N2 ) @ I2 ) )
            @ ( set_or1337092689740270186AtMost @ nat @ N2 @ M ) ) ) ) ).

% sum.atLeastAtMost_rev
thf(fact_3352_sum__pos2,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ordere6911136660526730532id_add @ A )
     => ! [I5: set @ B,I: B,F: B > A] :
          ( ( finite_finite2 @ B @ I5 )
         => ( ( member @ B @ I @ I5 )
           => ( ( ord_less @ A @ ( zero_zero @ A ) @ ( F @ I ) )
             => ( ! [I3: B] :
                    ( ( member @ B @ I3 @ I5 )
                   => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F @ I3 ) ) )
               => ( ord_less @ A @ ( zero_zero @ A ) @ ( groups7311177749621191930dd_sum @ B @ A @ F @ I5 ) ) ) ) ) ) ) ).

% sum_pos2
thf(fact_3353_sum__pos,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ordere6911136660526730532id_add @ A )
     => ! [I5: set @ B,F: B > A] :
          ( ( finite_finite2 @ B @ I5 )
         => ( ( I5
             != ( bot_bot @ ( set @ B ) ) )
           => ( ! [I3: B] :
                  ( ( member @ B @ I3 @ I5 )
                 => ( ord_less @ A @ ( zero_zero @ A ) @ ( F @ I3 ) ) )
             => ( ord_less @ A @ ( zero_zero @ A ) @ ( groups7311177749621191930dd_sum @ B @ A @ F @ I5 ) ) ) ) ) ) ).

% sum_pos
thf(fact_3354_sum_Osame__carrier,axiom,
    ! [A: $tType,B: $tType] :
      ( ( comm_monoid_add @ A )
     => ! [C5: set @ B,A3: set @ B,B6: set @ B,G: B > A,H2: B > A] :
          ( ( finite_finite2 @ B @ C5 )
         => ( ( ord_less_eq @ ( set @ B ) @ A3 @ C5 )
           => ( ( ord_less_eq @ ( set @ B ) @ B6 @ C5 )
             => ( ! [A4: B] :
                    ( ( member @ B @ A4 @ ( minus_minus @ ( set @ B ) @ C5 @ A3 ) )
                   => ( ( G @ A4 )
                      = ( zero_zero @ A ) ) )
               => ( ! [B3: B] :
                      ( ( member @ B @ B3 @ ( minus_minus @ ( set @ B ) @ C5 @ B6 ) )
                     => ( ( H2 @ B3 )
                        = ( zero_zero @ A ) ) )
                 => ( ( ( groups7311177749621191930dd_sum @ B @ A @ G @ A3 )
                      = ( groups7311177749621191930dd_sum @ B @ A @ H2 @ B6 ) )
                    = ( ( groups7311177749621191930dd_sum @ B @ A @ G @ C5 )
                      = ( groups7311177749621191930dd_sum @ B @ A @ H2 @ C5 ) ) ) ) ) ) ) ) ) ).

% sum.same_carrier
thf(fact_3355_sum_Osame__carrierI,axiom,
    ! [A: $tType,B: $tType] :
      ( ( comm_monoid_add @ A )
     => ! [C5: set @ B,A3: set @ B,B6: set @ B,G: B > A,H2: B > A] :
          ( ( finite_finite2 @ B @ C5 )
         => ( ( ord_less_eq @ ( set @ B ) @ A3 @ C5 )
           => ( ( ord_less_eq @ ( set @ B ) @ B6 @ C5 )
             => ( ! [A4: B] :
                    ( ( member @ B @ A4 @ ( minus_minus @ ( set @ B ) @ C5 @ A3 ) )
                   => ( ( G @ A4 )
                      = ( zero_zero @ A ) ) )
               => ( ! [B3: B] :
                      ( ( member @ B @ B3 @ ( minus_minus @ ( set @ B ) @ C5 @ B6 ) )
                     => ( ( H2 @ B3 )
                        = ( zero_zero @ A ) ) )
                 => ( ( ( groups7311177749621191930dd_sum @ B @ A @ G @ C5 )
                      = ( groups7311177749621191930dd_sum @ B @ A @ H2 @ C5 ) )
                   => ( ( groups7311177749621191930dd_sum @ B @ A @ G @ A3 )
                      = ( groups7311177749621191930dd_sum @ B @ A @ H2 @ B6 ) ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
thf(fact_3356_sum_Omono__neutral__left,axiom,
    ! [A: $tType,B: $tType] :
      ( ( comm_monoid_add @ A )
     => ! [T6: set @ B,S3: set @ B,G: B > A] :
          ( ( finite_finite2 @ B @ T6 )
         => ( ( ord_less_eq @ ( set @ B ) @ S3 @ T6 )
           => ( ! [X3: B] :
                  ( ( member @ B @ X3 @ ( minus_minus @ ( set @ B ) @ T6 @ S3 ) )
                 => ( ( G @ X3 )
                    = ( zero_zero @ A ) ) )
             => ( ( groups7311177749621191930dd_sum @ B @ A @ G @ S3 )
                = ( groups7311177749621191930dd_sum @ B @ A @ G @ T6 ) ) ) ) ) ) ).

% sum.mono_neutral_left
thf(fact_3357_sum_Omono__neutral__right,axiom,
    ! [A: $tType,B: $tType] :
      ( ( comm_monoid_add @ A )
     => ! [T6: set @ B,S3: set @ B,G: B > A] :
          ( ( finite_finite2 @ B @ T6 )
         => ( ( ord_less_eq @ ( set @ B ) @ S3 @ T6 )
           => ( ! [X3: B] :
                  ( ( member @ B @ X3 @ ( minus_minus @ ( set @ B ) @ T6 @ S3 ) )
                 => ( ( G @ X3 )
                    = ( zero_zero @ A ) ) )
             => ( ( groups7311177749621191930dd_sum @ B @ A @ G @ T6 )
                = ( groups7311177749621191930dd_sum @ B @ A @ G @ S3 ) ) ) ) ) ) ).

% sum.mono_neutral_right
thf(fact_3358_sum_Omono__neutral__cong__left,axiom,
    ! [A: $tType,B: $tType] :
      ( ( comm_monoid_add @ A )
     => ! [T6: set @ B,S3: set @ B,H2: B > A,G: B > A] :
          ( ( finite_finite2 @ B @ T6 )
         => ( ( ord_less_eq @ ( set @ B ) @ S3 @ T6 )
           => ( ! [X3: B] :
                  ( ( member @ B @ X3 @ ( minus_minus @ ( set @ B ) @ T6 @ S3 ) )
                 => ( ( H2 @ X3 )
                    = ( zero_zero @ A ) ) )
             => ( ! [X3: B] :
                    ( ( member @ B @ X3 @ S3 )
                   => ( ( G @ X3 )
                      = ( H2 @ X3 ) ) )
               => ( ( groups7311177749621191930dd_sum @ B @ A @ G @ S3 )
                  = ( groups7311177749621191930dd_sum @ B @ A @ H2 @ T6 ) ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
thf(fact_3359_sum_Omono__neutral__cong__right,axiom,
    ! [A: $tType,B: $tType] :
      ( ( comm_monoid_add @ A )
     => ! [T6: set @ B,S3: set @ B,G: B > A,H2: B > A] :
          ( ( finite_finite2 @ B @ T6 )
         => ( ( ord_less_eq @ ( set @ B ) @ S3 @ T6 )
           => ( ! [X3: B] :
                  ( ( member @ B @ X3 @ ( minus_minus @ ( set @ B ) @ T6 @ S3 ) )
                 => ( ( G @ X3 )
                    = ( zero_zero @ A ) ) )
             => ( ! [X3: B] :
                    ( ( member @ B @ X3 @ S3 )
                   => ( ( G @ X3 )
                      = ( H2 @ X3 ) ) )
               => ( ( groups7311177749621191930dd_sum @ B @ A @ G @ T6 )
                  = ( groups7311177749621191930dd_sum @ B @ A @ H2 @ S3 ) ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
thf(fact_3360_sum_Osubset__diff,axiom,
    ! [A: $tType,B: $tType] :
      ( ( comm_monoid_add @ A )
     => ! [B6: set @ B,A3: set @ B,G: B > A] :
          ( ( ord_less_eq @ ( set @ B ) @ B6 @ A3 )
         => ( ( finite_finite2 @ B @ A3 )
           => ( ( groups7311177749621191930dd_sum @ B @ A @ G @ A3 )
              = ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ G @ ( minus_minus @ ( set @ B ) @ A3 @ B6 ) ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ B6 ) ) ) ) ) ) ).

% sum.subset_diff
thf(fact_3361_sum__diff,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ab_group_add @ A )
     => ! [A3: set @ B,B6: set @ B,F: B > A] :
          ( ( finite_finite2 @ B @ A3 )
         => ( ( ord_less_eq @ ( set @ B ) @ B6 @ A3 )
           => ( ( groups7311177749621191930dd_sum @ B @ A @ F @ ( minus_minus @ ( set @ B ) @ A3 @ B6 ) )
              = ( minus_minus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F @ A3 ) @ ( groups7311177749621191930dd_sum @ B @ A @ F @ B6 ) ) ) ) ) ) ).

% sum_diff
thf(fact_3362_powser__sums__zero,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra @ A )
     => ! [A2: nat > A] :
          ( sums @ A
          @ ^ [N: nat] : ( times_times @ A @ ( A2 @ N ) @ ( power_power @ A @ ( zero_zero @ A ) @ N ) )
          @ ( A2 @ ( zero_zero @ nat ) ) ) ) ).

% powser_sums_zero
thf(fact_3363_sum__shift__lb__Suc0__0,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add @ A )
     => ! [F: nat > A,K: nat] :
          ( ( ( F @ ( zero_zero @ nat ) )
            = ( zero_zero @ A ) )
         => ( ( groups7311177749621191930dd_sum @ nat @ A @ F @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ ( zero_zero @ nat ) ) @ K ) )
            = ( groups7311177749621191930dd_sum @ nat @ A @ F @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ K ) ) ) ) ) ).

% sum_shift_lb_Suc0_0
thf(fact_3364_sum_OatLeast0__atMost__Suc,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add @ A )
     => ! [G: nat > A,N2: nat] :
          ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( suc @ N2 ) ) )
          = ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ) ).

% sum.atLeast0_atMost_Suc
thf(fact_3365_sum_Onat__ivl__Suc_H,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add @ A )
     => ! [M: nat,N2: nat,G: nat > A] :
          ( ( ord_less_eq @ nat @ M @ ( suc @ N2 ) )
         => ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ ( suc @ N2 ) ) )
            = ( plus_plus @ A @ ( G @ ( suc @ N2 ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) ) ) ) ) ) ).

% sum.nat_ivl_Suc'
thf(fact_3366_sum_OatLeast__Suc__atMost,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add @ A )
     => ! [M: nat,N2: nat,G: nat > A] :
          ( ( ord_less_eq @ nat @ M @ N2 )
         => ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) )
            = ( plus_plus @ A @ ( G @ M ) @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M ) @ N2 ) ) ) ) ) ) ).

% sum.atLeast_Suc_atMost
thf(fact_3367_dbl__inc__def,axiom,
    ! [A: $tType] :
      ( ( neg_numeral @ A )
     => ( ( neg_numeral_dbl_inc @ A )
        = ( ^ [X: A] : ( plus_plus @ A @ ( plus_plus @ A @ X @ X ) @ ( one_one @ A ) ) ) ) ) ).

% dbl_inc_def
thf(fact_3368_sum_OSuc__reindex__ivl,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add @ A )
     => ! [M: nat,N2: nat,G: nat > A] :
          ( ( ord_less_eq @ nat @ M @ N2 )
         => ( ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) ) @ ( G @ ( suc @ N2 ) ) )
            = ( plus_plus @ A @ ( G @ M )
              @ ( groups7311177749621191930dd_sum @ nat @ A
                @ ^ [I2: nat] : ( G @ ( suc @ I2 ) )
                @ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) ) ) ) ) ) ).

% sum.Suc_reindex_ivl
thf(fact_3369_sum__Suc__diff,axiom,
    ! [A: $tType] :
      ( ( ab_group_add @ A )
     => ! [M: nat,N2: nat,F: nat > A] :
          ( ( ord_less_eq @ nat @ M @ ( suc @ N2 ) )
         => ( ( groups7311177749621191930dd_sum @ nat @ A
              @ ^ [I2: nat] : ( minus_minus @ A @ ( F @ ( suc @ I2 ) ) @ ( F @ I2 ) )
              @ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) )
            = ( minus_minus @ A @ ( F @ ( suc @ N2 ) ) @ ( F @ M ) ) ) ) ) ).

% sum_Suc_diff
thf(fact_3370_sum__atLeastAtMost__code,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add @ A )
     => ! [F: nat > A,A2: nat,B2: nat] :
          ( ( groups7311177749621191930dd_sum @ nat @ A @ F @ ( set_or1337092689740270186AtMost @ nat @ A2 @ B2 ) )
          = ( set_fo6178422350223883121st_nat @ A
            @ ^ [A5: nat] : ( plus_plus @ A @ ( F @ A5 ) )
            @ A2
            @ B2
            @ ( zero_zero @ A ) ) ) ) ).

% sum_atLeastAtMost_code
thf(fact_3371_sum__mono2,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ordere6911136660526730532id_add @ A )
     => ! [B6: set @ B,A3: set @ B,F: B > A] :
          ( ( finite_finite2 @ B @ B6 )
         => ( ( ord_less_eq @ ( set @ B ) @ A3 @ B6 )
           => ( ! [B3: B] :
                  ( ( member @ B @ B3 @ ( minus_minus @ ( set @ B ) @ B6 @ A3 ) )
                 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F @ B3 ) ) )
             => ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F @ A3 ) @ ( groups7311177749621191930dd_sum @ B @ A @ F @ B6 ) ) ) ) ) ) ).

% sum_mono2
thf(fact_3372_sum_Oinsert__remove,axiom,
    ! [A: $tType,B: $tType] :
      ( ( comm_monoid_add @ A )
     => ! [A3: set @ B,G: B > A,X2: B] :
          ( ( finite_finite2 @ B @ A3 )
         => ( ( groups7311177749621191930dd_sum @ B @ A @ G @ ( insert @ B @ X2 @ A3 ) )
            = ( plus_plus @ A @ ( G @ X2 ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ ( minus_minus @ ( set @ B ) @ A3 @ ( insert @ B @ X2 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ).

% sum.insert_remove
thf(fact_3373_sum_Oremove,axiom,
    ! [A: $tType,B: $tType] :
      ( ( comm_monoid_add @ A )
     => ! [A3: set @ B,X2: B,G: B > A] :
          ( ( finite_finite2 @ B @ A3 )
         => ( ( member @ B @ X2 @ A3 )
           => ( ( groups7311177749621191930dd_sum @ B @ A @ G @ A3 )
              = ( plus_plus @ A @ ( G @ X2 ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ ( minus_minus @ ( set @ B ) @ A3 @ ( insert @ B @ X2 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ) ).

% sum.remove
thf(fact_3374_sum__diff1,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ab_group_add @ A )
     => ! [A3: set @ B,A2: B,F: B > A] :
          ( ( finite_finite2 @ B @ A3 )
         => ( ( ( member @ B @ A2 @ A3 )
             => ( ( groups7311177749621191930dd_sum @ B @ A @ F @ ( minus_minus @ ( set @ B ) @ A3 @ ( insert @ B @ A2 @ ( bot_bot @ ( set @ B ) ) ) ) )
                = ( minus_minus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F @ A3 ) @ ( F @ A2 ) ) ) )
            & ( ~ ( member @ B @ A2 @ A3 )
             => ( ( groups7311177749621191930dd_sum @ B @ A @ F @ ( minus_minus @ ( set @ B ) @ A3 @ ( insert @ B @ A2 @ ( bot_bot @ ( set @ B ) ) ) ) )
                = ( groups7311177749621191930dd_sum @ B @ A @ F @ A3 ) ) ) ) ) ) ).

% sum_diff1
thf(fact_3375_sum_Oub__add__nat,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add @ A )
     => ! [M: nat,N2: nat,G: nat > A,P4: nat] :
          ( ( ord_less_eq @ nat @ M @ ( plus_plus @ nat @ N2 @ ( one_one @ nat ) ) )
         => ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ ( plus_plus @ nat @ N2 @ P4 ) ) )
            = ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ N2 @ ( one_one @ nat ) ) @ ( plus_plus @ nat @ N2 @ P4 ) ) ) ) ) ) ) ).

% sum.ub_add_nat
thf(fact_3376_sum_Odelta__remove,axiom,
    ! [A: $tType,B: $tType] :
      ( ( comm_monoid_add @ A )
     => ! [S3: set @ B,A2: B,B2: B > A,C2: B > A] :
          ( ( finite_finite2 @ B @ S3 )
         => ( ( ( member @ B @ A2 @ S3 )
             => ( ( groups7311177749621191930dd_sum @ B @ A
                  @ ^ [K4: B] : ( if @ A @ ( K4 = A2 ) @ ( B2 @ K4 ) @ ( C2 @ K4 ) )
                  @ S3 )
                = ( plus_plus @ A @ ( B2 @ A2 ) @ ( groups7311177749621191930dd_sum @ B @ A @ C2 @ ( minus_minus @ ( set @ B ) @ S3 @ ( insert @ B @ A2 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) )
            & ( ~ ( member @ B @ A2 @ S3 )
             => ( ( groups7311177749621191930dd_sum @ B @ A
                  @ ^ [K4: B] : ( if @ A @ ( K4 = A2 ) @ ( B2 @ K4 ) @ ( C2 @ K4 ) )
                  @ S3 )
                = ( groups7311177749621191930dd_sum @ B @ A @ C2 @ ( minus_minus @ ( set @ B ) @ S3 @ ( insert @ B @ A2 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ) ).

% sum.delta_remove
thf(fact_3377_set__encode__def,axiom,
    ( nat_set_encode
    = ( groups7311177749621191930dd_sum @ nat @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).

% set_encode_def
thf(fact_3378_sum__strict__mono2,axiom,
    ! [B: $tType,A: $tType] :
      ( ( ordere8940638589300402666id_add @ B )
     => ! [B6: set @ A,A3: set @ A,B2: A,F: A > B] :
          ( ( finite_finite2 @ A @ B6 )
         => ( ( ord_less_eq @ ( set @ A ) @ A3 @ B6 )
           => ( ( member @ A @ B2 @ ( minus_minus @ ( set @ A ) @ B6 @ A3 ) )
             => ( ( ord_less @ B @ ( zero_zero @ B ) @ ( F @ B2 ) )
               => ( ! [X3: A] :
                      ( ( member @ A @ X3 @ B6 )
                     => ( ord_less_eq @ B @ ( zero_zero @ B ) @ ( F @ X3 ) ) )
                 => ( ord_less @ B @ ( groups7311177749621191930dd_sum @ A @ B @ F @ A3 ) @ ( groups7311177749621191930dd_sum @ A @ B @ F @ B6 ) ) ) ) ) ) ) ) ).

% sum_strict_mono2
thf(fact_3379_member__le__sum,axiom,
    ! [B: $tType,C: $tType] :
      ( ( ( ordere6911136660526730532id_add @ B )
        & ( semiring_1 @ B ) )
     => ! [I: C,A3: set @ C,F: C > B] :
          ( ( member @ C @ I @ A3 )
         => ( ! [X3: C] :
                ( ( member @ C @ X3 @ ( minus_minus @ ( set @ C ) @ A3 @ ( insert @ C @ I @ ( bot_bot @ ( set @ C ) ) ) ) )
               => ( ord_less_eq @ B @ ( zero_zero @ B ) @ ( F @ X3 ) ) )
           => ( ( finite_finite2 @ C @ A3 )
             => ( ord_less_eq @ B @ ( F @ I ) @ ( groups7311177749621191930dd_sum @ C @ B @ F @ A3 ) ) ) ) ) ) ).

% member_le_sum
thf(fact_3380_sum__natinterval__diff,axiom,
    ! [A: $tType] :
      ( ( ab_group_add @ A )
     => ! [M: nat,N2: nat,F: nat > A] :
          ( ( ( ord_less_eq @ nat @ M @ N2 )
           => ( ( groups7311177749621191930dd_sum @ nat @ A
                @ ^ [K4: nat] : ( minus_minus @ A @ ( F @ K4 ) @ ( F @ ( plus_plus @ nat @ K4 @ ( one_one @ nat ) ) ) )
                @ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) )
              = ( minus_minus @ A @ ( F @ M ) @ ( F @ ( plus_plus @ nat @ N2 @ ( one_one @ nat ) ) ) ) ) )
          & ( ~ ( ord_less_eq @ nat @ M @ N2 )
           => ( ( groups7311177749621191930dd_sum @ nat @ A
                @ ^ [K4: nat] : ( minus_minus @ A @ ( F @ K4 ) @ ( F @ ( plus_plus @ nat @ K4 @ ( one_one @ nat ) ) ) )
                @ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) )
              = ( zero_zero @ A ) ) ) ) ) ).

% sum_natinterval_diff
thf(fact_3381_sum__telescope_H_H,axiom,
    ! [A: $tType] :
      ( ( ab_group_add @ A )
     => ! [M: nat,N2: nat,F: nat > A] :
          ( ( ord_less_eq @ nat @ M @ N2 )
         => ( ( groups7311177749621191930dd_sum @ nat @ A
              @ ^ [K4: nat] : ( minus_minus @ A @ ( F @ K4 ) @ ( F @ ( minus_minus @ nat @ K4 @ ( one_one @ nat ) ) ) )
              @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M ) @ N2 ) )
            = ( minus_minus @ A @ ( F @ N2 ) @ ( F @ M ) ) ) ) ) ).

% sum_telescope''
thf(fact_3382_mask__eq__sum__exp,axiom,
    ! [A: $tType] :
      ( ( semiring_parity @ A )
     => ! [N2: nat] :
          ( ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) @ ( one_one @ A ) )
          = ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
            @ ( collect @ nat
              @ ^ [Q5: nat] : ( ord_less @ nat @ Q5 @ N2 ) ) ) ) ) ).

% mask_eq_sum_exp
thf(fact_3383_sum__gp__multiplied,axiom,
    ! [A: $tType] :
      ( ( ( monoid_mult @ A )
        & ( comm_ring @ A ) )
     => ! [M: nat,N2: nat,X2: A] :
          ( ( ord_less_eq @ nat @ M @ N2 )
         => ( ( times_times @ A @ ( minus_minus @ A @ ( one_one @ A ) @ X2 ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X2 ) @ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) ) )
            = ( minus_minus @ A @ ( power_power @ A @ X2 @ M ) @ ( power_power @ A @ X2 @ ( suc @ N2 ) ) ) ) ) ) ).

% sum_gp_multiplied
thf(fact_3384_sum_Oin__pairs,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add @ A )
     => ! [G: nat > A,M: nat,N2: nat] :
          ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ) )
          = ( groups7311177749621191930dd_sum @ nat @ A
            @ ^ [I2: nat] : ( plus_plus @ A @ ( G @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I2 ) ) @ ( G @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I2 ) ) ) )
            @ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) ) ) ) ).

% sum.in_pairs
thf(fact_3385_VEBT__internal_Oheight_Ocases,axiom,
    ! [X2: vEBT_VEBT] :
      ( ! [A4: $o,B3: $o] :
          ( X2
         != ( vEBT_Leaf @ A4 @ B3 ) )
     => ~ ! [Uu2: option @ ( product_prod @ nat @ nat ),Deg2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
            ( X2
           != ( vEBT_Node @ Uu2 @ Deg2 @ TreeList2 @ Summary2 ) ) ) ).

% VEBT_internal.height.cases
thf(fact_3386_sums__if_H,axiom,
    ! [G: nat > real,X2: real] :
      ( ( sums @ real @ G @ X2 )
     => ( sums @ real
        @ ^ [N: nat] : ( if @ real @ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ ( zero_zero @ real ) @ ( G @ ( divide_divide @ nat @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
        @ X2 ) ) ).

% sums_if'
thf(fact_3387_sums__if,axiom,
    ! [G: nat > real,X2: real,F: nat > real,Y2: real] :
      ( ( sums @ real @ G @ X2 )
     => ( ( sums @ real @ F @ Y2 )
       => ( sums @ real
          @ ^ [N: nat] : ( if @ real @ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ ( F @ ( divide_divide @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( G @ ( divide_divide @ nat @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
          @ ( plus_plus @ real @ X2 @ Y2 ) ) ) ) ).

% sums_if
thf(fact_3388_mask__eq__sum__exp__nat,axiom,
    ! [N2: nat] :
      ( ( minus_minus @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) @ ( suc @ ( zero_zero @ nat ) ) )
      = ( groups7311177749621191930dd_sum @ nat @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
        @ ( collect @ nat
          @ ^ [Q5: nat] : ( ord_less @ nat @ Q5 @ N2 ) ) ) ) ).

% mask_eq_sum_exp_nat
thf(fact_3389_gauss__sum__nat,axiom,
    ! [N2: nat] :
      ( ( groups7311177749621191930dd_sum @ nat @ nat
        @ ^ [X: nat] : X
        @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N2 ) )
      = ( divide_divide @ nat @ ( times_times @ nat @ N2 @ ( suc @ N2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ).

% gauss_sum_nat
thf(fact_3390_double__gauss__sum,axiom,
    ! [A: $tType] :
      ( ( comm_semiring_1 @ A )
     => ! [N2: nat] :
          ( ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( semiring_1_of_nat @ A ) @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N2 ) ) )
          = ( times_times @ A @ ( semiring_1_of_nat @ A @ N2 ) @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ N2 ) @ ( one_one @ A ) ) ) ) ) ).

% double_gauss_sum
thf(fact_3391_double__arith__series,axiom,
    ! [A: $tType] :
      ( ( comm_semiring_1 @ A )
     => ! [A2: A,D2: A,N2: nat] :
          ( ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) )
            @ ( groups7311177749621191930dd_sum @ nat @ A
              @ ^ [I2: nat] : ( plus_plus @ A @ A2 @ ( times_times @ A @ ( semiring_1_of_nat @ A @ I2 ) @ D2 ) )
              @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N2 ) ) )
          = ( times_times @ A @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ N2 ) @ ( one_one @ A ) ) @ ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N2 ) @ D2 ) ) ) ) ) ).

% double_arith_series
thf(fact_3392_arith__series__nat,axiom,
    ! [A2: nat,D2: nat,N2: nat] :
      ( ( groups7311177749621191930dd_sum @ nat @ nat
        @ ^ [I2: nat] : ( plus_plus @ nat @ A2 @ ( times_times @ nat @ I2 @ D2 ) )
        @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N2 ) )
      = ( divide_divide @ nat @ ( times_times @ nat @ ( suc @ N2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ A2 ) @ ( times_times @ nat @ N2 @ D2 ) ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ).

% arith_series_nat
thf(fact_3393_Sum__Icc__nat,axiom,
    ! [M: nat,N2: nat] :
      ( ( groups7311177749621191930dd_sum @ nat @ nat
        @ ^ [X: nat] : X
        @ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) )
      = ( divide_divide @ nat @ ( minus_minus @ nat @ ( times_times @ nat @ N2 @ ( plus_plus @ nat @ N2 @ ( one_one @ nat ) ) ) @ ( times_times @ nat @ M @ ( minus_minus @ nat @ M @ ( one_one @ nat ) ) ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ).

% Sum_Icc_nat
thf(fact_3394_double__gauss__sum__from__Suc__0,axiom,
    ! [A: $tType] :
      ( ( comm_semiring_1 @ A )
     => ! [N2: nat] :
          ( ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( semiring_1_of_nat @ A ) @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N2 ) ) )
          = ( times_times @ A @ ( semiring_1_of_nat @ A @ N2 ) @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ N2 ) @ ( one_one @ A ) ) ) ) ) ).

% double_gauss_sum_from_Suc_0
thf(fact_3395_gauss__sum,axiom,
    ! [A: $tType] :
      ( ( euclid5411537665997757685th_nat @ A )
     => ! [N2: nat] :
          ( ( groups7311177749621191930dd_sum @ nat @ A @ ( semiring_1_of_nat @ A ) @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N2 ) )
          = ( divide_divide @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N2 ) @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ N2 ) @ ( one_one @ A ) ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).

% gauss_sum
thf(fact_3396_arith__series,axiom,
    ! [A: $tType] :
      ( ( euclid5411537665997757685th_nat @ A )
     => ! [A2: A,D2: A,N2: nat] :
          ( ( groups7311177749621191930dd_sum @ nat @ A
            @ ^ [I2: nat] : ( plus_plus @ A @ A2 @ ( times_times @ A @ ( semiring_1_of_nat @ A @ I2 ) @ D2 ) )
            @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N2 ) )
          = ( divide_divide @ A @ ( times_times @ A @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ N2 ) @ ( one_one @ A ) ) @ ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N2 ) @ D2 ) ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).

% arith_series
thf(fact_3397_sum__gp__offset,axiom,
    ! [A: $tType] :
      ( ( ( division_ring @ A )
        & ( comm_ring @ A ) )
     => ! [X2: A,M: nat,N2: nat] :
          ( ( ( X2
              = ( one_one @ A ) )
           => ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X2 ) @ ( set_or1337092689740270186AtMost @ nat @ M @ ( plus_plus @ nat @ M @ N2 ) ) )
              = ( plus_plus @ A @ ( semiring_1_of_nat @ A @ N2 ) @ ( one_one @ A ) ) ) )
          & ( ( X2
             != ( one_one @ A ) )
           => ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X2 ) @ ( set_or1337092689740270186AtMost @ nat @ M @ ( plus_plus @ nat @ M @ N2 ) ) )
              = ( divide_divide @ A @ ( times_times @ A @ ( power_power @ A @ X2 @ M ) @ ( minus_minus @ A @ ( one_one @ A ) @ ( power_power @ A @ X2 @ ( suc @ N2 ) ) ) ) @ ( minus_minus @ A @ ( one_one @ A ) @ X2 ) ) ) ) ) ) ).

% sum_gp_offset
thf(fact_3398_gauss__sum__from__Suc__0,axiom,
    ! [A: $tType] :
      ( ( euclid5411537665997757685th_nat @ A )
     => ! [N2: nat] :
          ( ( groups7311177749621191930dd_sum @ nat @ A @ ( semiring_1_of_nat @ A ) @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N2 ) )
          = ( divide_divide @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N2 ) @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ N2 ) @ ( one_one @ A ) ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).

% gauss_sum_from_Suc_0
thf(fact_3399_geometric__sums,axiom,
    ! [A: $tType] :
      ( ( real_V3459762299906320749_field @ A )
     => ! [C2: A] :
          ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ C2 ) @ ( one_one @ real ) )
         => ( sums @ A @ ( power_power @ A @ C2 ) @ ( divide_divide @ A @ ( one_one @ A ) @ ( minus_minus @ A @ ( one_one @ A ) @ C2 ) ) ) ) ) ).

% geometric_sums
thf(fact_3400_power__half__series,axiom,
    ( sums @ real
    @ ^ [N: nat] : ( power_power @ real @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( suc @ N ) )
    @ ( one_one @ real ) ) ).

% power_half_series
thf(fact_3401_sums__zero,axiom,
    ! [A: $tType] :
      ( ( ( comm_monoid_add @ A )
        & ( topolo4958980785337419405_space @ A ) )
     => ( sums @ A
        @ ^ [N: nat] : ( zero_zero @ A )
        @ ( zero_zero @ A ) ) ) ).

% sums_zero
thf(fact_3402_sums__If__finite__set_H,axiom,
    ! [A: $tType] :
      ( ( ( topolo1287966508704411220up_add @ A )
        & ( topological_t2_space @ A ) )
     => ! [G: nat > A,S3: A,A3: set @ nat,S7: A,F: nat > A] :
          ( ( sums @ A @ G @ S3 )
         => ( ( finite_finite2 @ nat @ A3 )
           => ( ( S7
                = ( plus_plus @ A @ S3
                  @ ( groups7311177749621191930dd_sum @ nat @ A
                    @ ^ [N: nat] : ( minus_minus @ A @ ( F @ N ) @ ( G @ N ) )
                    @ A3 ) ) )
             => ( sums @ A
                @ ^ [N: nat] : ( if @ A @ ( member @ nat @ N @ A3 ) @ ( F @ N ) @ ( G @ N ) )
                @ S7 ) ) ) ) ) ).

% sums_If_finite_set'
thf(fact_3403_powser__sums__if,axiom,
    ! [A: $tType] :
      ( ( ( ring_1 @ A )
        & ( topolo4958980785337419405_space @ A ) )
     => ! [M: nat,Z: A] :
          ( sums @ A
          @ ^ [N: nat] : ( times_times @ A @ ( if @ A @ ( N = M ) @ ( one_one @ A ) @ ( zero_zero @ A ) ) @ ( power_power @ A @ Z @ N ) )
          @ ( power_power @ A @ Z @ M ) ) ) ).

% powser_sums_if
thf(fact_3404_int__sum,axiom,
    ! [B: $tType,F: B > nat,A3: set @ B] :
      ( ( semiring_1_of_nat @ int @ ( groups7311177749621191930dd_sum @ B @ nat @ F @ A3 ) )
      = ( groups7311177749621191930dd_sum @ B @ int
        @ ^ [X: B] : ( semiring_1_of_nat @ int @ ( F @ X ) )
        @ A3 ) ) ).

% int_sum
thf(fact_3405_sum__subtractf__nat,axiom,
    ! [A: $tType,A3: set @ A,G: A > nat,F: A > nat] :
      ( ! [X3: A] :
          ( ( member @ A @ X3 @ A3 )
         => ( ord_less_eq @ nat @ ( G @ X3 ) @ ( F @ X3 ) ) )
     => ( ( groups7311177749621191930dd_sum @ A @ nat
          @ ^ [X: A] : ( minus_minus @ nat @ ( F @ X ) @ ( G @ X ) )
          @ A3 )
        = ( minus_minus @ nat @ ( groups7311177749621191930dd_sum @ A @ nat @ F @ A3 ) @ ( groups7311177749621191930dd_sum @ A @ nat @ G @ A3 ) ) ) ) ).

% sum_subtractf_nat
thf(fact_3406_sum__SucD,axiom,
    ! [A: $tType,F: A > nat,A3: set @ A,N2: nat] :
      ( ( ( groups7311177749621191930dd_sum @ A @ nat @ F @ A3 )
        = ( suc @ N2 ) )
     => ? [X3: A] :
          ( ( member @ A @ X3 @ A3 )
          & ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F @ X3 ) ) ) ) ).

% sum_SucD
thf(fact_3407_sum__eq__Suc0__iff,axiom,
    ! [A: $tType,A3: set @ A,F: A > nat] :
      ( ( finite_finite2 @ A @ A3 )
     => ( ( ( groups7311177749621191930dd_sum @ A @ nat @ F @ A3 )
          = ( suc @ ( zero_zero @ nat ) ) )
        = ( ? [X: A] :
              ( ( member @ A @ X @ A3 )
              & ( ( F @ X )
                = ( suc @ ( zero_zero @ nat ) ) )
              & ! [Y: A] :
                  ( ( member @ A @ Y @ A3 )
                 => ( ( X != Y )
                   => ( ( F @ Y )
                      = ( zero_zero @ nat ) ) ) ) ) ) ) ) ).

% sum_eq_Suc0_iff
thf(fact_3408_sum__eq__1__iff,axiom,
    ! [A: $tType,A3: set @ A,F: A > nat] :
      ( ( finite_finite2 @ A @ A3 )
     => ( ( ( groups7311177749621191930dd_sum @ A @ nat @ F @ A3 )
          = ( one_one @ nat ) )
        = ( ? [X: A] :
              ( ( member @ A @ X @ A3 )
              & ( ( F @ X )
                = ( one_one @ nat ) )
              & ! [Y: A] :
                  ( ( member @ A @ Y @ A3 )
                 => ( ( X != Y )
                   => ( ( F @ Y )
                      = ( zero_zero @ nat ) ) ) ) ) ) ) ) ).

% sum_eq_1_iff
thf(fact_3409_sum__roots__unity,axiom,
    ! [N2: nat] :
      ( ( ord_less @ nat @ ( one_one @ nat ) @ N2 )
     => ( ( groups7311177749621191930dd_sum @ complex @ complex
          @ ^ [X: complex] : X
          @ ( collect @ complex
            @ ^ [Z2: complex] :
                ( ( power_power @ complex @ Z2 @ N2 )
                = ( one_one @ complex ) ) ) )
        = ( zero_zero @ complex ) ) ) ).

% sum_roots_unity
thf(fact_3410_sum__nth__roots,axiom,
    ! [N2: nat,C2: complex] :
      ( ( ord_less @ nat @ ( one_one @ nat ) @ N2 )
     => ( ( groups7311177749621191930dd_sum @ complex @ complex
          @ ^ [X: complex] : X
          @ ( collect @ complex
            @ ^ [Z2: complex] :
                ( ( power_power @ complex @ Z2 @ N2 )
                = C2 ) ) )
        = ( zero_zero @ complex ) ) ) ).

% sum_nth_roots
thf(fact_3411_sum__diff__nat,axiom,
    ! [A: $tType,B6: set @ A,A3: set @ A,F: A > nat] :
      ( ( finite_finite2 @ A @ B6 )
     => ( ( ord_less_eq @ ( set @ A ) @ B6 @ A3 )
       => ( ( groups7311177749621191930dd_sum @ A @ nat @ F @ ( minus_minus @ ( set @ A ) @ A3 @ B6 ) )
          = ( minus_minus @ nat @ ( groups7311177749621191930dd_sum @ A @ nat @ F @ A3 ) @ ( groups7311177749621191930dd_sum @ A @ nat @ F @ B6 ) ) ) ) ) ).

% sum_diff_nat
thf(fact_3412_sum__diff1__nat,axiom,
    ! [A: $tType,A2: A,A3: set @ A,F: A > nat] :
      ( ( ( member @ A @ A2 @ A3 )
       => ( ( groups7311177749621191930dd_sum @ A @ nat @ F @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) )
          = ( minus_minus @ nat @ ( groups7311177749621191930dd_sum @ A @ nat @ F @ A3 ) @ ( F @ A2 ) ) ) )
      & ( ~ ( member @ A @ A2 @ A3 )
       => ( ( groups7311177749621191930dd_sum @ A @ nat @ F @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) )
          = ( groups7311177749621191930dd_sum @ A @ nat @ F @ A3 ) ) ) ) ).

% sum_diff1_nat
thf(fact_3413_sums__le,axiom,
    ! [A: $tType] :
      ( ( ( ordere6911136660526730532id_add @ A )
        & ( topolo1944317154257567458pology @ A ) )
     => ! [F: nat > A,G: nat > A,S4: A,T2: A] :
          ( ! [N4: nat] : ( ord_less_eq @ A @ ( F @ N4 ) @ ( G @ N4 ) )
         => ( ( sums @ A @ F @ S4 )
           => ( ( sums @ A @ G @ T2 )
             => ( ord_less_eq @ A @ S4 @ T2 ) ) ) ) ) ).

% sums_le
thf(fact_3414_sums__single,axiom,
    ! [A: $tType] :
      ( ( ( comm_monoid_add @ A )
        & ( topolo4958980785337419405_space @ A ) )
     => ! [I: nat,F: nat > A] :
          ( sums @ A
          @ ^ [R: nat] : ( if @ A @ ( R = I ) @ ( F @ R ) @ ( zero_zero @ A ) )
          @ ( F @ I ) ) ) ).

% sums_single
thf(fact_3415_sums__add,axiom,
    ! [A: $tType] :
      ( ( ( topolo5987344860129210374id_add @ A )
        & ( topological_t2_space @ A ) )
     => ! [F: nat > A,A2: A,G: nat > A,B2: A] :
          ( ( sums @ A @ F @ A2 )
         => ( ( sums @ A @ G @ B2 )
           => ( sums @ A
              @ ^ [N: nat] : ( plus_plus @ A @ ( F @ N ) @ ( G @ N ) )
              @ ( plus_plus @ A @ A2 @ B2 ) ) ) ) ) ).

% sums_add
thf(fact_3416_sums__mult2,axiom,
    ! [A: $tType] :
      ( ( real_V4412858255891104859lgebra @ A )
     => ! [F: nat > A,A2: A,C2: A] :
          ( ( sums @ A @ F @ A2 )
         => ( sums @ A
            @ ^ [N: nat] : ( times_times @ A @ ( F @ N ) @ C2 )
            @ ( times_times @ A @ A2 @ C2 ) ) ) ) ).

% sums_mult2
thf(fact_3417_sums__mult,axiom,
    ! [A: $tType] :
      ( ( real_V4412858255891104859lgebra @ A )
     => ! [F: nat > A,A2: A,C2: A] :
          ( ( sums @ A @ F @ A2 )
         => ( sums @ A
            @ ^ [N: nat] : ( times_times @ A @ C2 @ ( F @ N ) )
            @ ( times_times @ A @ C2 @ A2 ) ) ) ) ).

% sums_mult
thf(fact_3418_sums__diff,axiom,
    ! [A: $tType] :
      ( ( real_V822414075346904944vector @ A )
     => ! [F: nat > A,A2: A,G: nat > A,B2: A] :
          ( ( sums @ A @ F @ A2 )
         => ( ( sums @ A @ G @ B2 )
           => ( sums @ A
              @ ^ [N: nat] : ( minus_minus @ A @ ( F @ N ) @ ( G @ N ) )
              @ ( minus_minus @ A @ A2 @ B2 ) ) ) ) ) ).

% sums_diff
thf(fact_3419_sums__divide,axiom,
    ! [A: $tType] :
      ( ( real_V3459762299906320749_field @ A )
     => ! [F: nat > A,A2: A,C2: A] :
          ( ( sums @ A @ F @ A2 )
         => ( sums @ A
            @ ^ [N: nat] : ( divide_divide @ A @ ( F @ N ) @ C2 )
            @ ( divide_divide @ A @ A2 @ C2 ) ) ) ) ).

% sums_divide
thf(fact_3420_sums__sum,axiom,
    ! [A: $tType,I6: $tType] :
      ( ( ( topolo5987344860129210374id_add @ A )
        & ( topological_t2_space @ A ) )
     => ! [I5: set @ I6,F: I6 > nat > A,X2: I6 > A] :
          ( ! [I3: I6] :
              ( ( member @ I6 @ I3 @ I5 )
             => ( sums @ A @ ( F @ I3 ) @ ( X2 @ I3 ) ) )
         => ( sums @ A
            @ ^ [N: nat] :
                ( groups7311177749621191930dd_sum @ I6 @ A
                @ ^ [I2: I6] : ( F @ I2 @ N )
                @ I5 )
            @ ( groups7311177749621191930dd_sum @ I6 @ A @ X2 @ I5 ) ) ) ) ).

% sums_sum
thf(fact_3421_Sum__Icc__int,axiom,
    ! [M: int,N2: int] :
      ( ( ord_less_eq @ int @ M @ N2 )
     => ( ( groups7311177749621191930dd_sum @ int @ int
          @ ^ [X: int] : X
          @ ( set_or1337092689740270186AtMost @ int @ M @ N2 ) )
        = ( divide_divide @ int @ ( minus_minus @ int @ ( times_times @ int @ N2 @ ( plus_plus @ int @ N2 @ ( one_one @ int ) ) ) @ ( times_times @ int @ M @ ( minus_minus @ int @ M @ ( one_one @ int ) ) ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ).

% Sum_Icc_int
thf(fact_3422_sums__mult2__iff,axiom,
    ! [A: $tType] :
      ( ( ( field @ A )
        & ( real_V4412858255891104859lgebra @ A ) )
     => ! [C2: A,F: nat > A,D2: A] :
          ( ( C2
           != ( zero_zero @ A ) )
         => ( ( sums @ A
              @ ^ [N: nat] : ( times_times @ A @ ( F @ N ) @ C2 )
              @ ( times_times @ A @ D2 @ C2 ) )
            = ( sums @ A @ F @ D2 ) ) ) ) ).

% sums_mult2_iff
thf(fact_3423_sums__mult__iff,axiom,
    ! [A: $tType] :
      ( ( ( field @ A )
        & ( real_V4412858255891104859lgebra @ A ) )
     => ! [C2: A,F: nat > A,D2: A] :
          ( ( C2
           != ( zero_zero @ A ) )
         => ( ( sums @ A
              @ ^ [N: nat] : ( times_times @ A @ C2 @ ( F @ N ) )
              @ ( times_times @ A @ C2 @ D2 ) )
            = ( sums @ A @ F @ D2 ) ) ) ) ).

% sums_mult_iff
thf(fact_3424_sums__mult__D,axiom,
    ! [A: $tType] :
      ( ( real_V3459762299906320749_field @ A )
     => ! [C2: A,F: nat > A,A2: A] :
          ( ( sums @ A
            @ ^ [N: nat] : ( times_times @ A @ C2 @ ( F @ N ) )
            @ A2 )
         => ( ( C2
             != ( zero_zero @ A ) )
           => ( sums @ A @ F @ ( divide_divide @ A @ A2 @ C2 ) ) ) ) ) ).

% sums_mult_D
thf(fact_3425_sums__Suc__imp,axiom,
    ! [A: $tType] :
      ( ( real_V822414075346904944vector @ A )
     => ! [F: nat > A,S4: A] :
          ( ( ( F @ ( zero_zero @ nat ) )
            = ( zero_zero @ A ) )
         => ( ( sums @ A
              @ ^ [N: nat] : ( F @ ( suc @ N ) )
              @ S4 )
           => ( sums @ A @ F @ S4 ) ) ) ) ).

% sums_Suc_imp
thf(fact_3426_sums__Suc,axiom,
    ! [A: $tType] :
      ( ( ( topolo5987344860129210374id_add @ A )
        & ( topological_t2_space @ A ) )
     => ! [F: nat > A,L: A] :
          ( ( sums @ A
            @ ^ [N: nat] : ( F @ ( suc @ N ) )
            @ L )
         => ( sums @ A @ F @ ( plus_plus @ A @ L @ ( F @ ( zero_zero @ nat ) ) ) ) ) ) ).

% sums_Suc
thf(fact_3427_sums__Suc__iff,axiom,
    ! [A: $tType] :
      ( ( real_V822414075346904944vector @ A )
     => ! [F: nat > A,S4: A] :
          ( ( sums @ A
            @ ^ [N: nat] : ( F @ ( suc @ N ) )
            @ S4 )
          = ( sums @ A @ F @ ( plus_plus @ A @ S4 @ ( F @ ( zero_zero @ nat ) ) ) ) ) ) ).

% sums_Suc_iff
thf(fact_3428_sums__zero__iff__shift,axiom,
    ! [A: $tType] :
      ( ( real_V822414075346904944vector @ A )
     => ! [N2: nat,F: nat > A,S4: A] :
          ( ! [I3: nat] :
              ( ( ord_less @ nat @ I3 @ N2 )
             => ( ( F @ I3 )
                = ( zero_zero @ A ) ) )
         => ( ( sums @ A
              @ ^ [I2: nat] : ( F @ ( plus_plus @ nat @ I2 @ N2 ) )
              @ S4 )
            = ( sums @ A @ F @ S4 ) ) ) ) ).

% sums_zero_iff_shift
thf(fact_3429_sums__finite,axiom,
    ! [A: $tType] :
      ( ( ( comm_monoid_add @ A )
        & ( topolo4958980785337419405_space @ A ) )
     => ! [N3: set @ nat,F: nat > A] :
          ( ( finite_finite2 @ nat @ N3 )
         => ( ! [N4: nat] :
                ( ~ ( member @ nat @ N4 @ N3 )
               => ( ( F @ N4 )
                  = ( zero_zero @ A ) ) )
           => ( sums @ A @ F @ ( groups7311177749621191930dd_sum @ nat @ A @ F @ N3 ) ) ) ) ) ).

% sums_finite
thf(fact_3430_sums__If__finite,axiom,
    ! [A: $tType] :
      ( ( ( comm_monoid_add @ A )
        & ( topolo4958980785337419405_space @ A ) )
     => ! [P: nat > $o,F: nat > A] :
          ( ( finite_finite2 @ nat @ ( collect @ nat @ P ) )
         => ( sums @ A
            @ ^ [R: nat] : ( if @ A @ ( P @ R ) @ ( F @ R ) @ ( zero_zero @ A ) )
            @ ( groups7311177749621191930dd_sum @ nat @ A @ F @ ( collect @ nat @ P ) ) ) ) ) ).

% sums_If_finite
thf(fact_3431_sums__If__finite__set,axiom,
    ! [A: $tType] :
      ( ( ( comm_monoid_add @ A )
        & ( topolo4958980785337419405_space @ A ) )
     => ! [A3: set @ nat,F: nat > A] :
          ( ( finite_finite2 @ nat @ A3 )
         => ( sums @ A
            @ ^ [R: nat] : ( if @ A @ ( member @ nat @ R @ A3 ) @ ( F @ R ) @ ( zero_zero @ A ) )
            @ ( groups7311177749621191930dd_sum @ nat @ A @ F @ A3 ) ) ) ) ).

% sums_If_finite_set
thf(fact_3432_lemma__termdiff2,axiom,
    ! [A: $tType] :
      ( ( field @ A )
     => ! [H2: A,Z: A,N2: nat] :
          ( ( H2
           != ( zero_zero @ A ) )
         => ( ( minus_minus @ A @ ( divide_divide @ A @ ( minus_minus @ A @ ( power_power @ A @ ( plus_plus @ A @ Z @ H2 ) @ N2 ) @ ( power_power @ A @ Z @ N2 ) ) @ H2 ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N2 ) @ ( power_power @ A @ Z @ ( minus_minus @ nat @ N2 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) )
            = ( times_times @ A @ H2
              @ ( groups7311177749621191930dd_sum @ nat @ A
                @ ^ [P5: nat] :
                    ( groups7311177749621191930dd_sum @ nat @ A
                    @ ^ [Q5: nat] : ( times_times @ A @ ( power_power @ A @ ( plus_plus @ A @ Z @ H2 ) @ Q5 ) @ ( power_power @ A @ Z @ ( minus_minus @ nat @ ( minus_minus @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ Q5 ) ) )
                    @ ( set_ord_lessThan @ nat @ ( minus_minus @ nat @ ( minus_minus @ nat @ N2 @ ( suc @ ( zero_zero @ nat ) ) ) @ P5 ) ) )
                @ ( set_ord_lessThan @ nat @ ( minus_minus @ nat @ N2 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) ) ) ).

% lemma_termdiff2
thf(fact_3433_TBOUND__fi_H__adm,axiom,
    ! [D: $tType,C: $tType,B: $tType,A: $tType,Foo: A > B > C > nat] :
      ( comple1908693960933563346ssible @ ( ( product_prod @ ( product_prod @ A @ B ) @ C ) > ( heap_Time_Heap @ D ) ) @ ( partial_fun_lub @ ( heap_Time_Heap @ D ) @ ( heap_Time_Heap @ D ) @ ( product_prod @ ( product_prod @ A @ B ) @ C ) @ ( heap_Time_Heap_lub @ D ) ) @ ( partial_fun_ord @ ( heap_Time_Heap @ D ) @ ( heap_Time_Heap @ D ) @ ( product_prod @ ( product_prod @ A @ B ) @ C ) @ ( heap_Time_Heap_ord @ D ) )
      @ ^ [Fi: ( product_prod @ ( product_prod @ A @ B ) @ C ) > ( heap_Time_Heap @ D )] :
        ! [X: A,Xa4: B,Xb4: C] : ( time_TBOUND @ D @ ( product_curry @ A @ B @ ( C > ( heap_Time_Heap @ D ) ) @ ( product_curry @ ( product_prod @ A @ B ) @ C @ ( heap_Time_Heap @ D ) @ Fi ) @ X @ Xa4 @ Xb4 ) @ ( Foo @ X @ Xa4 @ Xb4 ) ) ) ).

% TBOUND_fi'_adm
thf(fact_3434_pochhammer__double,axiom,
    ! [A: $tType] :
      ( ( field_char_0 @ A )
     => ! [Z: A,N2: nat] :
          ( ( comm_s3205402744901411588hammer @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ Z ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) )
          = ( times_times @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ) @ ( comm_s3205402744901411588hammer @ A @ Z @ N2 ) ) @ ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ Z @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) @ N2 ) ) ) ) ).

% pochhammer_double
thf(fact_3435_of__nat__code,axiom,
    ! [A: $tType] :
      ( ( semiring_1 @ A )
     => ( ( semiring_1_of_nat @ A )
        = ( ^ [N: nat] :
              ( semiri8178284476397505188at_aux @ A
              @ ^ [I2: A] : ( plus_plus @ A @ I2 @ ( one_one @ A ) )
              @ N
              @ ( zero_zero @ A ) ) ) ) ) ).

% of_nat_code
thf(fact_3436_floor__log__nat__eq__powr__iff,axiom,
    ! [B2: nat,K: nat,N2: nat] :
      ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ B2 )
     => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
       => ( ( ( archim6421214686448440834_floor @ real @ ( log @ ( semiring_1_of_nat @ real @ B2 ) @ ( semiring_1_of_nat @ real @ K ) ) )
            = ( semiring_1_of_nat @ int @ N2 ) )
          = ( ( ord_less_eq @ nat @ ( power_power @ nat @ B2 @ N2 ) @ K )
            & ( ord_less @ nat @ K @ ( power_power @ nat @ B2 @ ( plus_plus @ nat @ N2 @ ( one_one @ nat ) ) ) ) ) ) ) ) ).

% floor_log_nat_eq_powr_iff
thf(fact_3437_lessThan__iff,axiom,
    ! [A: $tType] :
      ( ( ord @ A )
     => ! [I: A,K: A] :
          ( ( member @ A @ I @ ( set_ord_lessThan @ A @ K ) )
          = ( ord_less @ A @ I @ K ) ) ) ).

% lessThan_iff
thf(fact_3438_pochhammer__1,axiom,
    ! [A: $tType] :
      ( ( comm_semiring_1 @ A )
     => ! [A2: A] :
          ( ( comm_s3205402744901411588hammer @ A @ A2 @ ( one_one @ nat ) )
          = A2 ) ) ).

% pochhammer_1
thf(fact_3439_lessThan__subset__iff,axiom,
    ! [A: $tType] :
      ( ( linorder @ A )
     => ! [X2: A,Y2: A] :
          ( ( ord_less_eq @ ( set @ A ) @ ( set_ord_lessThan @ A @ X2 ) @ ( set_ord_lessThan @ A @ Y2 ) )
          = ( ord_less_eq @ A @ X2 @ Y2 ) ) ) ).

% lessThan_subset_iff
thf(fact_3440_floor__numeral,axiom,
    ! [A: $tType] :
      ( ( archim2362893244070406136eiling @ A )
     => ! [V: num] :
          ( ( archim6421214686448440834_floor @ A @ ( numeral_numeral @ A @ V ) )
          = ( numeral_numeral @ int @ V ) ) ) ).

% floor_numeral
thf(fact_3441_floor__one,axiom,
    ! [A: $tType] :
      ( ( archim2362893244070406136eiling @ A )
     => ( ( archim6421214686448440834_floor @ A @ ( one_one @ A ) )
        = ( one_one @ int ) ) ) ).

% floor_one
thf(fact_3442_pochhammer__0,axiom,
    ! [A: $tType] :
      ( ( comm_semiring_1 @ A )
     => ! [A2: A] :
          ( ( comm_s3205402744901411588hammer @ A @ A2 @ ( zero_zero @ nat ) )
          = ( one_one @ A ) ) ) ).

% pochhammer_0
thf(fact_3443_pochhammer__Suc0,axiom,
    ! [A: $tType] :
      ( ( comm_semiring_1 @ A )
     => ! [A2: A] :
          ( ( comm_s3205402744901411588hammer @ A @ A2 @ ( suc @ ( zero_zero @ nat ) ) )
          = A2 ) ) ).

% pochhammer_Suc0
thf(fact_3444_sum_OlessThan__Suc,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add @ A )
     => ! [G: nat > A,N2: nat] :
          ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_ord_lessThan @ nat @ ( suc @ N2 ) ) )
          = ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_ord_lessThan @ nat @ N2 ) ) @ ( G @ N2 ) ) ) ) ).

% sum.lessThan_Suc
thf(fact_3445_single__Diff__lessThan,axiom,
    ! [A: $tType] :
      ( ( preorder @ A )
     => ! [K: A] :
          ( ( minus_minus @ ( set @ A ) @ ( insert @ A @ K @ ( bot_bot @ ( set @ A ) ) ) @ ( set_ord_lessThan @ A @ K ) )
          = ( insert @ A @ K @ ( bot_bot @ ( set @ A ) ) ) ) ) ).

% single_Diff_lessThan
thf(fact_3446_floor__diff__of__int,axiom,
    ! [A: $tType] :
      ( ( archim2362893244070406136eiling @ A )
     => ! [X2: A,Z: int] :
          ( ( archim6421214686448440834_floor @ A @ ( minus_minus @ A @ X2 @ ( ring_1_of_int @ A @ Z ) ) )
          = ( minus_minus @ int @ ( archim6421214686448440834_floor @ A @ X2 ) @ Z ) ) ) ).

% floor_diff_of_int
thf(fact_3447_zero__le__floor,axiom,
    ! [A: $tType] :
      ( ( archim2362893244070406136eiling @ A )
     => ! [X2: A] :
          ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( archim6421214686448440834_floor @ A @ X2 ) )
          = ( ord_less_eq @ A @ ( zero_zero @ A ) @ X2 ) ) ) ).

% zero_le_floor
thf(fact_3448_numeral__le__floor,axiom,
    ! [A: $tType] :
      ( ( archim2362893244070406136eiling @ A )
     => ! [V: num,X2: A] :
          ( ( ord_less_eq @ int @ ( numeral_numeral @ int @ V ) @ ( archim6421214686448440834_floor @ A @ X2 ) )
          = ( ord_less_eq @ A @ ( numeral_numeral @ A @ V ) @ X2 ) ) ) ).

% numeral_le_floor
thf(fact_3449_floor__less__zero,axiom,
    ! [A: $tType] :
      ( ( archim2362893244070406136eiling @ A )
     => ! [X2: A] :
          ( ( ord_less @ int @ ( archim6421214686448440834_floor @ A @ X2 ) @ ( zero_zero @ int ) )
          = ( ord_less @ A @ X2 @ ( zero_zero @ A ) ) ) ) ).

% floor_less_zero
thf(fact_3450_floor__less__numeral,axiom,
    ! [A: $tType] :
      ( ( archim2362893244070406136eiling @ A )
     => ! [X2: A,V: num] :
          ( ( ord_less @ int @ ( archim6421214686448440834_floor @ A @ X2 ) @ ( numeral_numeral @ int @ V ) )
          = ( ord_less @ A @ X2 @ ( numeral_numeral @ A @ V ) ) ) ) ).

% floor_less_numeral
thf(fact_3451_zero__less__floor,axiom,
    ! [A: $tType] :
      ( ( archim2362893244070406136eiling @ A )
     => ! [X2: A] :
          ( ( ord_less @ int @ ( zero_zero @ int ) @ ( archim6421214686448440834_floor @ A @ X2 ) )
          = ( ord_less_eq @ A @ ( one_one @ A ) @ X2 ) ) ) ).

% zero_less_floor
thf(fact_3452_floor__le__zero,axiom,
    ! [A: $tType] :
      ( ( archim2362893244070406136eiling @ A )
     => ! [X2: A] :
          ( ( ord_less_eq @ int @ ( archim6421214686448440834_floor @ A @ X2 ) @ ( zero_zero @ int ) )
          = ( ord_less @ A @ X2 @ ( one_one @ A ) ) ) ) ).

% floor_le_zero
thf(fact_3453_one__le__floor,axiom,
    ! [A: $tType] :
      ( ( archim2362893244070406136eiling @ A )
     => ! [X2: A] :
          ( ( ord_less_eq @ int @ ( one_one @ int ) @ ( archim6421214686448440834_floor @ A @ X2 ) )
          = ( ord_less_eq @ A @ ( one_one @ A ) @ X2 ) ) ) ).

% one_le_floor
thf(fact_3454_floor__less__one,axiom,
    ! [A: $tType] :
      ( ( archim2362893244070406136eiling @ A )
     => ! [X2: A] :
          ( ( ord_less @ int @ ( archim6421214686448440834_floor @ A @ X2 ) @ ( one_one @ int ) )
          = ( ord_less @ A @ X2 @ ( one_one @ A ) ) ) ) ).

% floor_less_one
thf(fact_3455_floor__diff__numeral,axiom,
    ! [A: $tType] :
      ( ( archim2362893244070406136eiling @ A )
     => ! [X2: A,V: num] :
          ( ( archim6421214686448440834_floor @ A @ ( minus_minus @ A @ X2 @ ( numeral_numeral @ A @ V ) ) )
          = ( minus_minus @ int @ ( archim6421214686448440834_floor @ A @ X2 ) @ ( numeral_numeral @ int @ V ) ) ) ) ).

% floor_diff_numeral
thf(fact_3456_floor__numeral__power,axiom,
    ! [A: $tType] :
      ( ( archim2362893244070406136eiling @ A )
     => ! [X2: num,N2: nat] :
          ( ( archim6421214686448440834_floor @ A @ ( power_power @ A @ ( numeral_numeral @ A @ X2 ) @ N2 ) )
          = ( power_power @ int @ ( numeral_numeral @ int @ X2 ) @ N2 ) ) ) ).

% floor_numeral_power
thf(fact_3457_floor__diff__one,axiom,
    ! [A: $tType] :
      ( ( archim2362893244070406136eiling @ A )
     => ! [X2: A] :
          ( ( archim6421214686448440834_floor @ A @ ( minus_minus @ A @ X2 @ ( one_one @ A ) ) )
          = ( minus_minus @ int @ ( archim6421214686448440834_floor @ A @ X2 ) @ ( one_one @ int ) ) ) ) ).

% floor_diff_one
thf(fact_3458_floor__divide__eq__div__numeral,axiom,
    ! [A2: num,B2: num] :
      ( ( archim6421214686448440834_floor @ real @ ( divide_divide @ real @ ( numeral_numeral @ real @ A2 ) @ ( numeral_numeral @ real @ B2 ) ) )
      = ( divide_divide @ int @ ( numeral_numeral @ int @ A2 ) @ ( numeral_numeral @ int @ B2 ) ) ) ).

% floor_divide_eq_div_numeral
thf(fact_3459_numeral__less__floor,axiom,
    ! [A: $tType] :
      ( ( archim2362893244070406136eiling @ A )
     => ! [V: num,X2: A] :
          ( ( ord_less @ int @ ( numeral_numeral @ int @ V ) @ ( archim6421214686448440834_floor @ A @ X2 ) )
          = ( ord_less_eq @ A @ ( plus_plus @ A @ ( numeral_numeral @ A @ V ) @ ( one_one @ A ) ) @ X2 ) ) ) ).

% numeral_less_floor
thf(fact_3460_floor__le__numeral,axiom,
    ! [A: $tType] :
      ( ( archim2362893244070406136eiling @ A )
     => ! [X2: A,V: num] :
          ( ( ord_less_eq @ int @ ( archim6421214686448440834_floor @ A @ X2 ) @ ( numeral_numeral @ int @ V ) )
          = ( ord_less @ A @ X2 @ ( plus_plus @ A @ ( numeral_numeral @ A @ V ) @ ( one_one @ A ) ) ) ) ) ).

% floor_le_numeral
thf(fact_3461_one__less__floor,axiom,
    ! [A: $tType] :
      ( ( archim2362893244070406136eiling @ A )
     => ! [X2: A] :
          ( ( ord_less @ int @ ( one_one @ int ) @ ( archim6421214686448440834_floor @ A @ X2 ) )
          = ( ord_less_eq @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ X2 ) ) ) ).

% one_less_floor
thf(fact_3462_floor__le__one,axiom,
    ! [A: $tType] :
      ( ( archim2362893244070406136eiling @ A )
     => ! [X2: A] :
          ( ( ord_less_eq @ int @ ( archim6421214686448440834_floor @ A @ X2 ) @ ( one_one @ int ) )
          = ( ord_less @ A @ X2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).

% floor_le_one
thf(fact_3463_floor__one__divide__eq__div__numeral,axiom,
    ! [B2: num] :
      ( ( archim6421214686448440834_floor @ real @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ B2 ) ) )
      = ( divide_divide @ int @ ( one_one @ int ) @ ( numeral_numeral @ int @ B2 ) ) ) ).

% floor_one_divide_eq_div_numeral
thf(fact_3464_lessThan__def,axiom,
    ! [A: $tType] :
      ( ( ord @ A )
     => ( ( set_ord_lessThan @ A )
        = ( ^ [U2: A] :
              ( collect @ A
              @ ^ [X: A] : ( ord_less @ A @ X @ U2 ) ) ) ) ) ).

% lessThan_def
thf(fact_3465_finite__set__of__finite__funs,axiom,
    ! [B: $tType,A: $tType,A3: set @ A,B6: set @ B,D2: B] :
      ( ( finite_finite2 @ A @ A3 )
     => ( ( finite_finite2 @ B @ B6 )
       => ( finite_finite2 @ ( A > B )
          @ ( collect @ ( A > B )
            @ ^ [F6: A > B] :
              ! [X: A] :
                ( ( ( member @ A @ X @ A3 )
                 => ( member @ B @ ( F6 @ X ) @ B6 ) )
                & ( ~ ( member @ A @ X @ A3 )
                 => ( ( F6 @ X )
                    = D2 ) ) ) ) ) ) ) ).

% finite_set_of_finite_funs
thf(fact_3466_of__int__floor__le,axiom,
    ! [A: $tType] :
      ( ( archim2362893244070406136eiling @ A )
     => ! [X2: A] : ( ord_less_eq @ A @ ( ring_1_of_int @ A @ ( archim6421214686448440834_floor @ A @ X2 ) ) @ X2 ) ) ).

% of_int_floor_le
thf(fact_3467_floor__mono,axiom,
    ! [A: $tType] :
      ( ( archim2362893244070406136eiling @ A )
     => ! [X2: A,Y2: A] :
          ( ( ord_less_eq @ A @ X2 @ Y2 )
         => ( ord_less_eq @ int @ ( archim6421214686448440834_floor @ A @ X2 ) @ ( archim6421214686448440834_floor @ A @ Y2 ) ) ) ) ).

% floor_mono
thf(fact_3468_floor__less__cancel,axiom,
    ! [A: $tType] :
      ( ( archim2362893244070406136eiling @ A )
     => ! [X2: A,Y2: A] :
          ( ( ord_less @ int @ ( archim6421214686448440834_floor @ A @ X2 ) @ ( archim6421214686448440834_floor @ A @ Y2 ) )
         => ( ord_less @ A @ X2 @ Y2 ) ) ) ).

% floor_less_cancel
thf(fact_3469_floor__le__ceiling,axiom,
    ! [A: $tType] :
      ( ( archim2362893244070406136eiling @ A )
     => ! [X2: A] : ( ord_less_eq @ int @ ( archim6421214686448440834_floor @ A @ X2 ) @ ( archimedean_ceiling @ A @ X2 ) ) ) ).

% floor_le_ceiling
thf(fact_3470_lessThan__strict__subset__iff,axiom,
    ! [A: $tType] :
      ( ( linorder @ A )
     => ! [M: A,N2: A] :
          ( ( ord_less @ ( set @ A ) @ ( set_ord_lessThan @ A @ M ) @ ( set_ord_lessThan @ A @ N2 ) )
          = ( ord_less @ A @ M @ N2 ) ) ) ).

% lessThan_strict_subset_iff
thf(fact_3471_pochhammer__pos,axiom,
    ! [A: $tType] :
      ( ( linordered_semidom @ A )
     => ! [X2: A,N2: nat] :
          ( ( ord_less @ A @ ( zero_zero @ A ) @ X2 )
         => ( ord_less @ A @ ( zero_zero @ A ) @ ( comm_s3205402744901411588hammer @ A @ X2 @ N2 ) ) ) ) ).

% pochhammer_pos
thf(fact_3472_lessThan__Suc,axiom,
    ! [K: nat] :
      ( ( set_ord_lessThan @ nat @ ( suc @ K ) )
      = ( insert @ nat @ K @ ( set_ord_lessThan @ nat @ K ) ) ) ).

% lessThan_Suc
thf(fact_3473_pochhammer__eq__0__mono,axiom,
    ! [A: $tType] :
      ( ( field_char_0 @ A )
     => ! [A2: A,N2: nat,M: nat] :
          ( ( ( comm_s3205402744901411588hammer @ A @ A2 @ N2 )
            = ( zero_zero @ A ) )
         => ( ( ord_less_eq @ nat @ N2 @ M )
           => ( ( comm_s3205402744901411588hammer @ A @ A2 @ M )
              = ( zero_zero @ A ) ) ) ) ) ).

% pochhammer_eq_0_mono
thf(fact_3474_pochhammer__neq__0__mono,axiom,
    ! [A: $tType] :
      ( ( field_char_0 @ A )
     => ! [A2: A,M: nat,N2: nat] :
          ( ( ( comm_s3205402744901411588hammer @ A @ A2 @ M )
           != ( zero_zero @ A ) )
         => ( ( ord_less_eq @ nat @ N2 @ M )
           => ( ( comm_s3205402744901411588hammer @ A @ A2 @ N2 )
             != ( zero_zero @ A ) ) ) ) ) ).

% pochhammer_neq_0_mono
thf(fact_3475_floor__le__round,axiom,
    ! [A: $tType] :
      ( ( archim2362893244070406136eiling @ A )
     => ! [X2: A] : ( ord_less_eq @ int @ ( archim6421214686448440834_floor @ A @ X2 ) @ ( archimedean_round @ A @ X2 ) ) ) ).

% floor_le_round
thf(fact_3476_admissible__heap,axiom,
    ! [B: $tType,A: $tType,P: A > ( heap_ext @ product_unit ) > ( heap_ext @ product_unit ) > B > nat > $o] :
      ( comple1908693960933563346ssible @ ( A > ( heap_Time_Heap @ B ) ) @ ( partial_fun_lub @ ( heap_Time_Heap @ B ) @ ( heap_Time_Heap @ B ) @ A @ ( heap_Time_Heap_lub @ B ) ) @ ( partial_fun_ord @ ( heap_Time_Heap @ B ) @ ( heap_Time_Heap @ B ) @ A @ ( heap_Time_Heap_ord @ B ) )
      @ ^ [F6: A > ( heap_Time_Heap @ B )] :
        ! [X: A,H: heap_ext @ product_unit,H9: heap_ext @ product_unit,R: B,N: nat] :
          ( ( heap_Time_effect @ B @ ( F6 @ X ) @ H @ H9 @ R @ N )
         => ( P @ X @ H @ H9 @ R @ N ) ) ) ).

% admissible_heap
thf(fact_3477_le__floor__iff,axiom,
    ! [A: $tType] :
      ( ( archim2362893244070406136eiling @ A )
     => ! [Z: int,X2: A] :
          ( ( ord_less_eq @ int @ Z @ ( archim6421214686448440834_floor @ A @ X2 ) )
          = ( ord_less_eq @ A @ ( ring_1_of_int @ A @ Z ) @ X2 ) ) ) ).

% le_floor_iff
thf(fact_3478_floor__less__iff,axiom,
    ! [A: $tType] :
      ( ( archim2362893244070406136eiling @ A )
     => ! [X2: A,Z: int] :
          ( ( ord_less @ int @ ( archim6421214686448440834_floor @ A @ X2 ) @ Z )
          = ( ord_less @ A @ X2 @ ( ring_1_of_int @ A @ Z ) ) ) ) ).

% floor_less_iff
thf(fact_3479_floor__add__int,axiom,
    ! [A: $tType] :
      ( ( archim2362893244070406136eiling @ A )
     => ! [X2: A,Z: int] :
          ( ( plus_plus @ int @ ( archim6421214686448440834_floor @ A @ X2 ) @ Z )
          = ( archim6421214686448440834_floor @ A @ ( plus_plus @ A @ X2 @ ( ring_1_of_int @ A @ Z ) ) ) ) ) ).

% floor_add_int
thf(fact_3480_int__add__floor,axiom,
    ! [A: $tType] :
      ( ( archim2362893244070406136eiling @ A )
     => ! [Z: int,X2: A] :
          ( ( plus_plus @ int @ Z @ ( archim6421214686448440834_floor @ A @ X2 ) )
          = ( archim6421214686448440834_floor @ A @ ( plus_plus @ A @ ( ring_1_of_int @ A @ Z ) @ X2 ) ) ) ) ).

% int_add_floor
thf(fact_3481_le__floor__add,axiom,
    ! [A: $tType] :
      ( ( archim2362893244070406136eiling @ A )
     => ! [X2: A,Y2: A] : ( ord_less_eq @ int @ ( plus_plus @ int @ ( archim6421214686448440834_floor @ A @ X2 ) @ ( archim6421214686448440834_floor @ A @ Y2 ) ) @ ( archim6421214686448440834_floor @ A @ ( plus_plus @ A @ X2 @ Y2 ) ) ) ) ).

% le_floor_add
thf(fact_3482_floor__power,axiom,
    ! [A: $tType] :
      ( ( archim2362893244070406136eiling @ A )
     => ! [X2: A,N2: nat] :
          ( ( X2
            = ( ring_1_of_int @ A @ ( archim6421214686448440834_floor @ A @ X2 ) ) )
         => ( ( archim6421214686448440834_floor @ A @ ( power_power @ A @ X2 @ N2 ) )
            = ( power_power @ int @ ( archim6421214686448440834_floor @ A @ X2 ) @ N2 ) ) ) ) ).

% floor_power
thf(fact_3483_floor__divide__of__int__eq,axiom,
    ! [A: $tType] :
      ( ( archim2362893244070406136eiling @ A )
     => ! [K: int,L: int] :
          ( ( archim6421214686448440834_floor @ A @ ( divide_divide @ A @ ( ring_1_of_int @ A @ K ) @ ( ring_1_of_int @ A @ L ) ) )
          = ( divide_divide @ int @ K @ L ) ) ) ).

% floor_divide_of_int_eq
thf(fact_3484_pochhammer__nonneg,axiom,
    ! [A: $tType] :
      ( ( linordered_semidom @ A )
     => ! [X2: A,N2: nat] :
          ( ( ord_less @ A @ ( zero_zero @ A ) @ X2 )
         => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( comm_s3205402744901411588hammer @ A @ X2 @ N2 ) ) ) ) ).

% pochhammer_nonneg
thf(fact_3485_of__nat__aux_Osimps_I2_J,axiom,
    ! [A: $tType] :
      ( ( semiring_1 @ A )
     => ! [Inc: A > A,N2: nat,I: A] :
          ( ( semiri8178284476397505188at_aux @ A @ Inc @ ( suc @ N2 ) @ I )
          = ( semiri8178284476397505188at_aux @ A @ Inc @ N2 @ ( Inc @ I ) ) ) ) ).

% of_nat_aux.simps(2)
thf(fact_3486_of__nat__aux_Osimps_I1_J,axiom,
    ! [A: $tType] :
      ( ( semiring_1 @ A )
     => ! [Inc: A > A,I: A] :
          ( ( semiri8178284476397505188at_aux @ A @ Inc @ ( zero_zero @ nat ) @ I )
          = I ) ) ).

% of_nat_aux.simps(1)
thf(fact_3487_pochhammer__0__left,axiom,
    ! [A: $tType] :
      ( ( comm_semiring_1 @ A )
     => ! [N2: nat] :
          ( ( ( N2
              = ( zero_zero @ nat ) )
           => ( ( comm_s3205402744901411588hammer @ A @ ( zero_zero @ A ) @ N2 )
              = ( one_one @ A ) ) )
          & ( ( N2
             != ( zero_zero @ nat ) )
           => ( ( comm_s3205402744901411588hammer @ A @ ( zero_zero @ A ) @ N2 )
              = ( zero_zero @ A ) ) ) ) ) ).

% pochhammer_0_left
thf(fact_3488_sum_Onat__diff__reindex,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add @ A )
     => ! [G: nat > A,N2: nat] :
          ( ( groups7311177749621191930dd_sum @ nat @ A
            @ ^ [I2: nat] : ( G @ ( minus_minus @ nat @ N2 @ ( suc @ I2 ) ) )
            @ ( set_ord_lessThan @ nat @ N2 ) )
          = ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_ord_lessThan @ nat @ N2 ) ) ) ) ).

% sum.nat_diff_reindex
thf(fact_3489_sum__diff__distrib,axiom,
    ! [A: $tType] :
      ( ( ord @ A )
     => ! [Q: A > nat,P: A > nat,N2: A] :
          ( ! [X3: A] : ( ord_less_eq @ nat @ ( Q @ X3 ) @ ( P @ X3 ) )
         => ( ( minus_minus @ nat @ ( groups7311177749621191930dd_sum @ A @ nat @ P @ ( set_ord_lessThan @ A @ N2 ) ) @ ( groups7311177749621191930dd_sum @ A @ nat @ Q @ ( set_ord_lessThan @ A @ N2 ) ) )
            = ( groups7311177749621191930dd_sum @ A @ nat
              @ ^ [X: A] : ( minus_minus @ nat @ ( P @ X ) @ ( Q @ X ) )
              @ ( set_ord_lessThan @ A @ N2 ) ) ) ) ) ).

% sum_diff_distrib
thf(fact_3490_of__nat__floor,axiom,
    ! [A: $tType] :
      ( ( archim2362893244070406136eiling @ A )
     => ! [R4: A] :
          ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ R4 )
         => ( ord_less_eq @ A @ ( semiring_1_of_nat @ A @ ( nat2 @ ( archim6421214686448440834_floor @ A @ R4 ) ) ) @ R4 ) ) ) ).

% of_nat_floor
thf(fact_3491_one__add__floor,axiom,
    ! [A: $tType] :
      ( ( archim2362893244070406136eiling @ A )
     => ! [X2: A] :
          ( ( plus_plus @ int @ ( archim6421214686448440834_floor @ A @ X2 ) @ ( one_one @ int ) )
          = ( archim6421214686448440834_floor @ A @ ( plus_plus @ A @ X2 @ ( one_one @ A ) ) ) ) ) ).

% one_add_floor
thf(fact_3492_floor__divide__of__nat__eq,axiom,
    ! [A: $tType] :
      ( ( archim2362893244070406136eiling @ A )
     => ! [M: nat,N2: nat] :
          ( ( archim6421214686448440834_floor @ A @ ( divide_divide @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N2 ) ) )
          = ( semiring_1_of_nat @ int @ ( divide_divide @ nat @ M @ N2 ) ) ) ) ).

% floor_divide_of_nat_eq
thf(fact_3493_le__mult__nat__floor,axiom,
    ! [A: $tType] :
      ( ( archim2362893244070406136eiling @ A )
     => ! [A2: A,B2: A] : ( ord_less_eq @ nat @ ( times_times @ nat @ ( nat2 @ ( archim6421214686448440834_floor @ A @ A2 ) ) @ ( nat2 @ ( archim6421214686448440834_floor @ A @ B2 ) ) ) @ ( nat2 @ ( archim6421214686448440834_floor @ A @ ( times_times @ A @ A2 @ B2 ) ) ) ) ) ).

% le_mult_nat_floor
thf(fact_3494_nat__floor__neg,axiom,
    ! [X2: real] :
      ( ( ord_less_eq @ real @ X2 @ ( zero_zero @ real ) )
     => ( ( nat2 @ ( archim6421214686448440834_floor @ real @ X2 ) )
        = ( zero_zero @ nat ) ) ) ).

% nat_floor_neg
thf(fact_3495_floor__eq3,axiom,
    ! [N2: nat,X2: real] :
      ( ( ord_less @ real @ ( semiring_1_of_nat @ real @ N2 ) @ X2 )
     => ( ( ord_less @ real @ X2 @ ( semiring_1_of_nat @ real @ ( suc @ N2 ) ) )
       => ( ( nat2 @ ( archim6421214686448440834_floor @ real @ X2 ) )
          = N2 ) ) ) ).

% floor_eq3
thf(fact_3496_ceiling__altdef,axiom,
    ! [A: $tType] :
      ( ( archim2362893244070406136eiling @ A )
     => ( ( archimedean_ceiling @ A )
        = ( ^ [X: A] :
              ( if @ int
              @ ( X
                = ( ring_1_of_int @ A @ ( archim6421214686448440834_floor @ A @ X ) ) )
              @ ( archim6421214686448440834_floor @ A @ X )
              @ ( plus_plus @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( one_one @ int ) ) ) ) ) ) ).

% ceiling_altdef
thf(fact_3497_le__nat__floor,axiom,
    ! [X2: nat,A2: real] :
      ( ( ord_less_eq @ real @ ( semiring_1_of_nat @ real @ X2 ) @ A2 )
     => ( ord_less_eq @ nat @ X2 @ ( nat2 @ ( archim6421214686448440834_floor @ real @ A2 ) ) ) ) ).

% le_nat_floor
thf(fact_3498_ceiling__diff__floor__le__1,axiom,
    ! [A: $tType] :
      ( ( archim2362893244070406136eiling @ A )
     => ! [X2: A] : ( ord_less_eq @ int @ ( minus_minus @ int @ ( archimedean_ceiling @ A @ X2 ) @ ( archim6421214686448440834_floor @ A @ X2 ) ) @ ( one_one @ int ) ) ) ).

% ceiling_diff_floor_le_1
thf(fact_3499_floor__eq,axiom,
    ! [N2: int,X2: real] :
      ( ( ord_less @ real @ ( ring_1_of_int @ real @ N2 ) @ X2 )
     => ( ( ord_less @ real @ X2 @ ( plus_plus @ real @ ( ring_1_of_int @ real @ N2 ) @ ( one_one @ real ) ) )
       => ( ( archim6421214686448440834_floor @ real @ X2 )
          = N2 ) ) ) ).

% floor_eq
thf(fact_3500_real__of__int__floor__add__one__gt,axiom,
    ! [R4: real] : ( ord_less @ real @ R4 @ ( plus_plus @ real @ ( ring_1_of_int @ real @ ( archim6421214686448440834_floor @ real @ R4 ) ) @ ( one_one @ real ) ) ) ).

% real_of_int_floor_add_one_gt
thf(fact_3501_real__of__int__floor__add__one__ge,axiom,
    ! [R4: real] : ( ord_less_eq @ real @ R4 @ ( plus_plus @ real @ ( ring_1_of_int @ real @ ( archim6421214686448440834_floor @ real @ R4 ) ) @ ( one_one @ real ) ) ) ).

% real_of_int_floor_add_one_ge
thf(fact_3502_real__of__int__floor__gt__diff__one,axiom,
    ! [R4: real] : ( ord_less @ real @ ( minus_minus @ real @ R4 @ ( one_one @ real ) ) @ ( ring_1_of_int @ real @ ( archim6421214686448440834_floor @ real @ R4 ) ) ) ).

% real_of_int_floor_gt_diff_one
thf(fact_3503_real__of__int__floor__ge__diff__one,axiom,
    ! [R4: real] : ( ord_less_eq @ real @ ( minus_minus @ real @ R4 @ ( one_one @ real ) ) @ ( ring_1_of_int @ real @ ( archim6421214686448440834_floor @ real @ R4 ) ) ) ).

% real_of_int_floor_ge_diff_one
thf(fact_3504_pochhammer__rec,axiom,
    ! [A: $tType] :
      ( ( comm_semiring_1 @ A )
     => ! [A2: A,N2: nat] :
          ( ( comm_s3205402744901411588hammer @ A @ A2 @ ( suc @ N2 ) )
          = ( times_times @ A @ A2 @ ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ A2 @ ( one_one @ A ) ) @ N2 ) ) ) ) ).

% pochhammer_rec
thf(fact_3505_pochhammer__Suc,axiom,
    ! [A: $tType] :
      ( ( comm_semiring_1 @ A )
     => ! [A2: A,N2: nat] :
          ( ( comm_s3205402744901411588hammer @ A @ A2 @ ( suc @ N2 ) )
          = ( times_times @ A @ ( comm_s3205402744901411588hammer @ A @ A2 @ N2 ) @ ( plus_plus @ A @ A2 @ ( semiring_1_of_nat @ A @ N2 ) ) ) ) ) ).

% pochhammer_Suc
thf(fact_3506_pochhammer__rec_H,axiom,
    ! [A: $tType] :
      ( ( comm_semiring_1 @ A )
     => ! [Z: A,N2: nat] :
          ( ( comm_s3205402744901411588hammer @ A @ Z @ ( suc @ N2 ) )
          = ( times_times @ A @ ( plus_plus @ A @ Z @ ( semiring_1_of_nat @ A @ N2 ) ) @ ( comm_s3205402744901411588hammer @ A @ Z @ N2 ) ) ) ) ).

% pochhammer_rec'
thf(fact_3507_pochhammer__product_H,axiom,
    ! [A: $tType] :
      ( ( comm_semiring_1 @ A )
     => ! [Z: A,N2: nat,M: nat] :
          ( ( comm_s3205402744901411588hammer @ A @ Z @ ( plus_plus @ nat @ N2 @ M ) )
          = ( times_times @ A @ ( comm_s3205402744901411588hammer @ A @ Z @ N2 ) @ ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ Z @ ( semiring_1_of_nat @ A @ N2 ) ) @ M ) ) ) ) ).

% pochhammer_product'
thf(fact_3508_sum_OlessThan__Suc__shift,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add @ A )
     => ! [G: nat > A,N2: nat] :
          ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_ord_lessThan @ nat @ ( suc @ N2 ) ) )
          = ( plus_plus @ A @ ( G @ ( zero_zero @ nat ) )
            @ ( groups7311177749621191930dd_sum @ nat @ A
              @ ^ [I2: nat] : ( G @ ( suc @ I2 ) )
              @ ( set_ord_lessThan @ nat @ N2 ) ) ) ) ) ).

% sum.lessThan_Suc_shift
thf(fact_3509_sum__lessThan__telescope,axiom,
    ! [A: $tType] :
      ( ( ab_group_add @ A )
     => ! [F: nat > A,M: nat] :
          ( ( groups7311177749621191930dd_sum @ nat @ A
            @ ^ [N: nat] : ( minus_minus @ A @ ( F @ ( suc @ N ) ) @ ( F @ N ) )
            @ ( set_ord_lessThan @ nat @ M ) )
          = ( minus_minus @ A @ ( F @ M ) @ ( F @ ( zero_zero @ nat ) ) ) ) ) ).

% sum_lessThan_telescope
thf(fact_3510_sum__lessThan__telescope_H,axiom,
    ! [A: $tType] :
      ( ( ab_group_add @ A )
     => ! [F: nat > A,M: nat] :
          ( ( groups7311177749621191930dd_sum @ nat @ A
            @ ^ [N: nat] : ( minus_minus @ A @ ( F @ N ) @ ( F @ ( suc @ N ) ) )
            @ ( set_ord_lessThan @ nat @ M ) )
          = ( minus_minus @ A @ ( F @ ( zero_zero @ nat ) ) @ ( F @ M ) ) ) ) ).

% sum_lessThan_telescope'
thf(fact_3511_sumr__diff__mult__const2,axiom,
    ! [A: $tType] :
      ( ( ring_1 @ A )
     => ! [F: nat > A,N2: nat,R4: A] :
          ( ( minus_minus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F @ ( set_ord_lessThan @ nat @ N2 ) ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N2 ) @ R4 ) )
          = ( groups7311177749621191930dd_sum @ nat @ A
            @ ^ [I2: nat] : ( minus_minus @ A @ ( F @ I2 ) @ R4 )
            @ ( set_ord_lessThan @ nat @ N2 ) ) ) ) ).

% sumr_diff_mult_const2
thf(fact_3512_sum_OatLeast1__atMost__eq,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add @ A )
     => ! [G: nat > A,N2: nat] :
          ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N2 ) )
          = ( groups7311177749621191930dd_sum @ nat @ A
            @ ^ [K4: nat] : ( G @ ( suc @ K4 ) )
            @ ( set_ord_lessThan @ nat @ N2 ) ) ) ) ).

% sum.atLeast1_atMost_eq
thf(fact_3513_sums__iff__shift,axiom,
    ! [A: $tType] :
      ( ( real_V822414075346904944vector @ A )
     => ! [F: nat > A,N2: nat,S4: A] :
          ( ( sums @ A
            @ ^ [I2: nat] : ( F @ ( plus_plus @ nat @ I2 @ N2 ) )
            @ S4 )
          = ( sums @ A @ F @ ( plus_plus @ A @ S4 @ ( groups7311177749621191930dd_sum @ nat @ A @ F @ ( set_ord_lessThan @ nat @ N2 ) ) ) ) ) ) ).

% sums_iff_shift
thf(fact_3514_sums__split__initial__segment,axiom,
    ! [A: $tType] :
      ( ( real_V822414075346904944vector @ A )
     => ! [F: nat > A,S4: A,N2: nat] :
          ( ( sums @ A @ F @ S4 )
         => ( sums @ A
            @ ^ [I2: nat] : ( F @ ( plus_plus @ nat @ I2 @ N2 ) )
            @ ( minus_minus @ A @ S4 @ ( groups7311177749621191930dd_sum @ nat @ A @ F @ ( set_ord_lessThan @ nat @ N2 ) ) ) ) ) ) ).

% sums_split_initial_segment
thf(fact_3515_sums__iff__shift_H,axiom,
    ! [A: $tType] :
      ( ( real_V822414075346904944vector @ A )
     => ! [F: nat > A,N2: nat,S4: A] :
          ( ( sums @ A
            @ ^ [I2: nat] : ( F @ ( plus_plus @ nat @ I2 @ N2 ) )
            @ ( minus_minus @ A @ S4 @ ( groups7311177749621191930dd_sum @ nat @ A @ F @ ( set_ord_lessThan @ nat @ N2 ) ) ) )
          = ( sums @ A @ F @ S4 ) ) ) ).

% sums_iff_shift'
thf(fact_3516_floor__split,axiom,
    ! [A: $tType] :
      ( ( archim2362893244070406136eiling @ A )
     => ! [P: int > $o,T2: A] :
          ( ( P @ ( archim6421214686448440834_floor @ A @ T2 ) )
          = ( ! [I2: int] :
                ( ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ I2 ) @ T2 )
                  & ( ord_less @ A @ T2 @ ( plus_plus @ A @ ( ring_1_of_int @ A @ I2 ) @ ( one_one @ A ) ) ) )
               => ( P @ I2 ) ) ) ) ) ).

% floor_split
thf(fact_3517_floor__eq__iff,axiom,
    ! [A: $tType] :
      ( ( archim2362893244070406136eiling @ A )
     => ! [X2: A,A2: int] :
          ( ( ( archim6421214686448440834_floor @ A @ X2 )
            = A2 )
          = ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ A2 ) @ X2 )
            & ( ord_less @ A @ X2 @ ( plus_plus @ A @ ( ring_1_of_int @ A @ A2 ) @ ( one_one @ A ) ) ) ) ) ) ).

% floor_eq_iff
thf(fact_3518_floor__unique,axiom,
    ! [A: $tType] :
      ( ( archim2362893244070406136eiling @ A )
     => ! [Z: int,X2: A] :
          ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ Z ) @ X2 )
         => ( ( ord_less @ A @ X2 @ ( plus_plus @ A @ ( ring_1_of_int @ A @ Z ) @ ( one_one @ A ) ) )
           => ( ( archim6421214686448440834_floor @ A @ X2 )
              = Z ) ) ) ) ).

% floor_unique
thf(fact_3519_less__floor__iff,axiom,
    ! [A: $tType] :
      ( ( archim2362893244070406136eiling @ A )
     => ! [Z: int,X2: A] :
          ( ( ord_less @ int @ Z @ ( archim6421214686448440834_floor @ A @ X2 ) )
          = ( ord_less_eq @ A @ ( plus_plus @ A @ ( ring_1_of_int @ A @ Z ) @ ( one_one @ A ) ) @ X2 ) ) ) ).

% less_floor_iff
thf(fact_3520_floor__le__iff,axiom,
    ! [A: $tType] :
      ( ( archim2362893244070406136eiling @ A )
     => ! [X2: A,Z: int] :
          ( ( ord_less_eq @ int @ ( archim6421214686448440834_floor @ A @ X2 ) @ Z )
          = ( ord_less @ A @ X2 @ ( plus_plus @ A @ ( ring_1_of_int @ A @ Z ) @ ( one_one @ A ) ) ) ) ) ).

% floor_le_iff
thf(fact_3521_le__mult__floor,axiom,
    ! [A: $tType] :
      ( ( archim2362893244070406136eiling @ A )
     => ! [A2: A,B2: A] :
          ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
         => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
           => ( ord_less_eq @ int @ ( times_times @ int @ ( archim6421214686448440834_floor @ A @ A2 ) @ ( archim6421214686448440834_floor @ A @ B2 ) ) @ ( archim6421214686448440834_floor @ A @ ( times_times @ A @ A2 @ B2 ) ) ) ) ) ) ).

% le_mult_floor
thf(fact_3522_floor__correct,axiom,
    ! [A: $tType] :
      ( ( archim2362893244070406136eiling @ A )
     => ! [X2: A] :
          ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ ( archim6421214686448440834_floor @ A @ X2 ) ) @ X2 )
          & ( ord_less @ A @ X2 @ ( ring_1_of_int @ A @ ( plus_plus @ int @ ( archim6421214686448440834_floor @ A @ X2 ) @ ( one_one @ int ) ) ) ) ) ) ).

% floor_correct
thf(fact_3523_floor__eq4,axiom,
    ! [N2: nat,X2: real] :
      ( ( ord_less_eq @ real @ ( semiring_1_of_nat @ real @ N2 ) @ X2 )
     => ( ( ord_less @ real @ X2 @ ( semiring_1_of_nat @ real @ ( suc @ N2 ) ) )
       => ( ( nat2 @ ( archim6421214686448440834_floor @ real @ X2 ) )
          = N2 ) ) ) ).

% floor_eq4
thf(fact_3524_floor__eq2,axiom,
    ! [N2: int,X2: real] :
      ( ( ord_less_eq @ real @ ( ring_1_of_int @ real @ N2 ) @ X2 )
     => ( ( ord_less @ real @ X2 @ ( plus_plus @ real @ ( ring_1_of_int @ real @ N2 ) @ ( one_one @ real ) ) )
       => ( ( archim6421214686448440834_floor @ real @ X2 )
          = N2 ) ) ) ).

% floor_eq2
thf(fact_3525_floor__divide__real__eq__div,axiom,
    ! [B2: int,A2: real] :
      ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ B2 )
     => ( ( archim6421214686448440834_floor @ real @ ( divide_divide @ real @ A2 @ ( ring_1_of_int @ real @ B2 ) ) )
        = ( divide_divide @ int @ ( archim6421214686448440834_floor @ real @ A2 ) @ B2 ) ) ) ).

% floor_divide_real_eq_div
thf(fact_3526_power__diff__1__eq,axiom,
    ! [A: $tType] :
      ( ( ( monoid_mult @ A )
        & ( comm_ring @ A ) )
     => ! [X2: A,N2: nat] :
          ( ( minus_minus @ A @ ( power_power @ A @ X2 @ N2 ) @ ( one_one @ A ) )
          = ( times_times @ A @ ( minus_minus @ A @ X2 @ ( one_one @ A ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X2 ) @ ( set_ord_lessThan @ nat @ N2 ) ) ) ) ) ).

% power_diff_1_eq
thf(fact_3527_one__diff__power__eq,axiom,
    ! [A: $tType] :
      ( ( ( monoid_mult @ A )
        & ( comm_ring @ A ) )
     => ! [X2: A,N2: nat] :
          ( ( minus_minus @ A @ ( one_one @ A ) @ ( power_power @ A @ X2 @ N2 ) )
          = ( times_times @ A @ ( minus_minus @ A @ ( one_one @ A ) @ X2 ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X2 ) @ ( set_ord_lessThan @ nat @ N2 ) ) ) ) ) ).

% one_diff_power_eq
thf(fact_3528_geometric__sum,axiom,
    ! [A: $tType] :
      ( ( field @ A )
     => ! [X2: A,N2: nat] :
          ( ( X2
           != ( one_one @ A ) )
         => ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X2 ) @ ( set_ord_lessThan @ nat @ N2 ) )
            = ( divide_divide @ A @ ( minus_minus @ A @ ( power_power @ A @ X2 @ N2 ) @ ( one_one @ A ) ) @ ( minus_minus @ A @ X2 @ ( one_one @ A ) ) ) ) ) ) ).

% geometric_sum
thf(fact_3529_pochhammer__product,axiom,
    ! [A: $tType] :
      ( ( comm_semiring_1 @ A )
     => ! [M: nat,N2: nat,Z: A] :
          ( ( ord_less_eq @ nat @ M @ N2 )
         => ( ( comm_s3205402744901411588hammer @ A @ Z @ N2 )
            = ( times_times @ A @ ( comm_s3205402744901411588hammer @ A @ Z @ M ) @ ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ Z @ ( semiring_1_of_nat @ A @ M ) ) @ ( minus_minus @ nat @ N2 @ M ) ) ) ) ) ) ).

% pochhammer_product
thf(fact_3530_floor__divide__lower,axiom,
    ! [A: $tType] :
      ( ( archim2362893244070406136eiling @ A )
     => ! [Q2: A,P4: A] :
          ( ( ord_less @ A @ ( zero_zero @ A ) @ Q2 )
         => ( ord_less_eq @ A @ ( times_times @ A @ ( ring_1_of_int @ A @ ( archim6421214686448440834_floor @ A @ ( divide_divide @ A @ P4 @ Q2 ) ) ) @ Q2 ) @ P4 ) ) ) ).

% floor_divide_lower
thf(fact_3531_lemma__termdiff1,axiom,
    ! [A: $tType] :
      ( ( ( monoid_mult @ A )
        & ( comm_ring @ A ) )
     => ! [Z: A,H2: A,M: nat] :
          ( ( groups7311177749621191930dd_sum @ nat @ A
            @ ^ [P5: nat] : ( minus_minus @ A @ ( times_times @ A @ ( power_power @ A @ ( plus_plus @ A @ Z @ H2 ) @ ( minus_minus @ nat @ M @ P5 ) ) @ ( power_power @ A @ Z @ P5 ) ) @ ( power_power @ A @ Z @ M ) )
            @ ( set_ord_lessThan @ nat @ M ) )
          = ( groups7311177749621191930dd_sum @ nat @ A
            @ ^ [P5: nat] : ( times_times @ A @ ( power_power @ A @ Z @ P5 ) @ ( minus_minus @ A @ ( power_power @ A @ ( plus_plus @ A @ Z @ H2 ) @ ( minus_minus @ nat @ M @ P5 ) ) @ ( power_power @ A @ Z @ ( minus_minus @ nat @ M @ P5 ) ) ) )
            @ ( set_ord_lessThan @ nat @ M ) ) ) ) ).

% lemma_termdiff1
thf(fact_3532_sum__gp__strict,axiom,
    ! [A: $tType] :
      ( ( ( division_ring @ A )
        & ( comm_ring @ A ) )
     => ! [X2: A,N2: nat] :
          ( ( ( X2
              = ( one_one @ A ) )
           => ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X2 ) @ ( set_ord_lessThan @ nat @ N2 ) )
              = ( semiring_1_of_nat @ A @ N2 ) ) )
          & ( ( X2
             != ( one_one @ A ) )
           => ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X2 ) @ ( set_ord_lessThan @ nat @ N2 ) )
              = ( divide_divide @ A @ ( minus_minus @ A @ ( one_one @ A ) @ ( power_power @ A @ X2 @ N2 ) ) @ ( minus_minus @ A @ ( one_one @ A ) @ X2 ) ) ) ) ) ) ).

% sum_gp_strict
thf(fact_3533_power__diff__sumr2,axiom,
    ! [A: $tType] :
      ( ( ( monoid_mult @ A )
        & ( comm_ring @ A ) )
     => ! [X2: A,N2: nat,Y2: A] :
          ( ( minus_minus @ A @ ( power_power @ A @ X2 @ N2 ) @ ( power_power @ A @ Y2 @ N2 ) )
          = ( times_times @ A @ ( minus_minus @ A @ X2 @ Y2 )
            @ ( groups7311177749621191930dd_sum @ nat @ A
              @ ^ [I2: nat] : ( times_times @ A @ ( power_power @ A @ Y2 @ ( minus_minus @ nat @ N2 @ ( suc @ I2 ) ) ) @ ( power_power @ A @ X2 @ I2 ) )
              @ ( set_ord_lessThan @ nat @ N2 ) ) ) ) ) ).

% power_diff_sumr2
thf(fact_3534_diff__power__eq__sum,axiom,
    ! [A: $tType] :
      ( ( ( monoid_mult @ A )
        & ( comm_ring @ A ) )
     => ! [X2: A,N2: nat,Y2: A] :
          ( ( minus_minus @ A @ ( power_power @ A @ X2 @ ( suc @ N2 ) ) @ ( power_power @ A @ Y2 @ ( suc @ N2 ) ) )
          = ( times_times @ A @ ( minus_minus @ A @ X2 @ Y2 )
            @ ( groups7311177749621191930dd_sum @ nat @ A
              @ ^ [P5: nat] : ( times_times @ A @ ( power_power @ A @ X2 @ P5 ) @ ( power_power @ A @ Y2 @ ( minus_minus @ nat @ N2 @ P5 ) ) )
              @ ( set_ord_lessThan @ nat @ ( suc @ N2 ) ) ) ) ) ) ).

% diff_power_eq_sum
thf(fact_3535_floor__divide__upper,axiom,
    ! [A: $tType] :
      ( ( archim2362893244070406136eiling @ A )
     => ! [Q2: A,P4: A] :
          ( ( ord_less @ A @ ( zero_zero @ A ) @ Q2 )
         => ( ord_less @ A @ P4 @ ( times_times @ A @ ( plus_plus @ A @ ( ring_1_of_int @ A @ ( archim6421214686448440834_floor @ A @ ( divide_divide @ A @ P4 @ Q2 ) ) ) @ ( one_one @ A ) ) @ Q2 ) ) ) ) ).

% floor_divide_upper
thf(fact_3536_round__def,axiom,
    ! [A: $tType] :
      ( ( archim2362893244070406136eiling @ A )
     => ( ( archimedean_round @ A )
        = ( ^ [X: A] : ( archim6421214686448440834_floor @ A @ ( plus_plus @ A @ X @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).

% round_def
thf(fact_3537_real__sum__nat__ivl__bounded2,axiom,
    ! [A: $tType] :
      ( ( linordered_semidom @ A )
     => ! [N2: nat,F: nat > A,K6: A,K: nat] :
          ( ! [P9: nat] :
              ( ( ord_less @ nat @ P9 @ N2 )
             => ( ord_less_eq @ A @ ( F @ P9 ) @ K6 ) )
         => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ K6 )
           => ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F @ ( set_ord_lessThan @ nat @ ( minus_minus @ nat @ N2 @ K ) ) ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N2 ) @ K6 ) ) ) ) ) ).

% real_sum_nat_ivl_bounded2
thf(fact_3538_one__diff__power__eq_H,axiom,
    ! [A: $tType] :
      ( ( ( monoid_mult @ A )
        & ( comm_ring @ A ) )
     => ! [X2: A,N2: nat] :
          ( ( minus_minus @ A @ ( one_one @ A ) @ ( power_power @ A @ X2 @ N2 ) )
          = ( times_times @ A @ ( minus_minus @ A @ ( one_one @ A ) @ X2 )
            @ ( groups7311177749621191930dd_sum @ nat @ A
              @ ^ [I2: nat] : ( power_power @ A @ X2 @ ( minus_minus @ nat @ N2 @ ( suc @ I2 ) ) )
              @ ( set_ord_lessThan @ nat @ N2 ) ) ) ) ) ).

% one_diff_power_eq'
thf(fact_3539_TBOUND__adm,axiom,
    ! [B: $tType,A: $tType,T2: A > nat] :
      ( comple1908693960933563346ssible @ ( A > ( heap_Time_Heap @ B ) ) @ ( partial_fun_lub @ ( heap_Time_Heap @ B ) @ ( heap_Time_Heap @ B ) @ A @ ( heap_Time_Heap_lub @ B ) ) @ ( partial_fun_ord @ ( heap_Time_Heap @ B ) @ ( heap_Time_Heap @ B ) @ A @ ( heap_Time_Heap_ord @ B ) )
      @ ^ [F6: A > ( heap_Time_Heap @ B )] :
        ! [X: A] : ( time_TBOUND @ B @ ( F6 @ X ) @ ( T2 @ X ) ) ) ).

% TBOUND_adm
thf(fact_3540_refines__adm,axiom,
    ! [B: $tType,A: $tType,T2: A > ( heap_Time_Heap @ B )] :
      ( comple1908693960933563346ssible @ ( A > ( heap_Time_Heap @ B ) ) @ ( partial_fun_lub @ ( heap_Time_Heap @ B ) @ ( heap_Time_Heap @ B ) @ A @ ( heap_Time_Heap_lub @ B ) ) @ ( partial_fun_ord @ ( heap_Time_Heap @ B ) @ ( heap_Time_Heap @ B ) @ A @ ( heap_Time_Heap_ord @ B ) )
      @ ^ [F6: A > ( heap_Time_Heap @ B )] :
        ! [X: A] : ( refine_Imp_refines @ B @ ( T2 @ X ) @ ( F6 @ X ) ) ) ).

% refines_adm
thf(fact_3541_sum__split__even__odd,axiom,
    ! [F: nat > real,G: nat > real,N2: nat] :
      ( ( groups7311177749621191930dd_sum @ nat @ real
        @ ^ [I2: nat] : ( if @ real @ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I2 ) @ ( F @ I2 ) @ ( G @ I2 ) )
        @ ( set_ord_lessThan @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) )
      = ( plus_plus @ real
        @ ( groups7311177749621191930dd_sum @ nat @ real
          @ ^ [I2: nat] : ( F @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I2 ) )
          @ ( set_ord_lessThan @ nat @ N2 ) )
        @ ( groups7311177749621191930dd_sum @ nat @ real
          @ ^ [I2: nat] : ( G @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I2 ) @ ( one_one @ nat ) ) )
          @ ( set_ord_lessThan @ nat @ N2 ) ) ) ) ).

% sum_split_even_odd
thf(fact_3542_floor__log__eq__powr__iff,axiom,
    ! [X2: real,B2: real,K: int] :
      ( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
     => ( ( ord_less @ real @ ( one_one @ real ) @ B2 )
       => ( ( ( archim6421214686448440834_floor @ real @ ( log @ B2 @ X2 ) )
            = K )
          = ( ( ord_less_eq @ real @ ( powr @ real @ B2 @ ( ring_1_of_int @ real @ K ) ) @ X2 )
            & ( ord_less @ real @ X2 @ ( powr @ real @ B2 @ ( ring_1_of_int @ real @ ( plus_plus @ int @ K @ ( one_one @ int ) ) ) ) ) ) ) ) ) ).

% floor_log_eq_powr_iff
thf(fact_3543_pochhammer__code,axiom,
    ! [A: $tType] :
      ( ( comm_semiring_1 @ A )
     => ( ( comm_s3205402744901411588hammer @ A )
        = ( ^ [A5: A,N: nat] :
              ( if @ A
              @ ( N
                = ( zero_zero @ nat ) )
              @ ( one_one @ A )
              @ ( set_fo6178422350223883121st_nat @ A
                @ ^ [O: nat] : ( times_times @ A @ ( plus_plus @ A @ A5 @ ( semiring_1_of_nat @ A @ O ) ) )
                @ ( zero_zero @ nat )
                @ ( minus_minus @ nat @ N @ ( one_one @ nat ) )
                @ ( one_one @ A ) ) ) ) ) ) ).

% pochhammer_code
thf(fact_3544_TBOUND__fi__adm,axiom,
    ! [C: $tType,B: $tType,A: $tType,Foo: A > B > nat] :
      ( comple1908693960933563346ssible @ ( ( product_prod @ A @ B ) > ( heap_Time_Heap @ C ) ) @ ( partial_fun_lub @ ( heap_Time_Heap @ C ) @ ( heap_Time_Heap @ C ) @ ( product_prod @ A @ B ) @ ( heap_Time_Heap_lub @ C ) ) @ ( partial_fun_ord @ ( heap_Time_Heap @ C ) @ ( heap_Time_Heap @ C ) @ ( product_prod @ A @ B ) @ ( heap_Time_Heap_ord @ C ) )
      @ ^ [Fi: ( product_prod @ A @ B ) > ( heap_Time_Heap @ C )] :
        ! [X: A,Xa4: B] : ( time_TBOUND @ C @ ( product_curry @ A @ B @ ( heap_Time_Heap @ C ) @ Fi @ X @ Xa4 ) @ ( Foo @ X @ Xa4 ) ) ) ).

% TBOUND_fi_adm
thf(fact_3545_floor__log2__div2,axiom,
    ! [N2: nat] :
      ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 )
     => ( ( archim6421214686448440834_floor @ real @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ real @ N2 ) ) )
        = ( plus_plus @ int @ ( archim6421214686448440834_floor @ real @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ real @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( one_one @ int ) ) ) ) ).

% floor_log2_div2
thf(fact_3546_floor__log__nat__eq__if,axiom,
    ! [B2: nat,N2: nat,K: nat] :
      ( ( ord_less_eq @ nat @ ( power_power @ nat @ B2 @ N2 ) @ K )
     => ( ( ord_less @ nat @ K @ ( power_power @ nat @ B2 @ ( plus_plus @ nat @ N2 @ ( one_one @ nat ) ) ) )
       => ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ B2 )
         => ( ( archim6421214686448440834_floor @ real @ ( log @ ( semiring_1_of_nat @ real @ B2 ) @ ( semiring_1_of_nat @ real @ K ) ) )
            = ( semiring_1_of_nat @ int @ N2 ) ) ) ) ) ).

% floor_log_nat_eq_if
thf(fact_3547_sum__bounds__lt__plus1,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add @ A )
     => ! [F: nat > A,Mm: nat] :
          ( ( groups7311177749621191930dd_sum @ nat @ A
            @ ^ [K4: nat] : ( F @ ( suc @ K4 ) )
            @ ( set_ord_lessThan @ nat @ Mm ) )
          = ( groups7311177749621191930dd_sum @ nat @ A @ F @ ( set_or1337092689740270186AtMost @ nat @ ( one_one @ nat ) @ Mm ) ) ) ) ).

% sum_bounds_lt_plus1
thf(fact_3548_pochhammer__times__pochhammer__half,axiom,
    ! [A: $tType] :
      ( ( field_char_0 @ A )
     => ! [Z: A,N2: nat] :
          ( ( times_times @ A @ ( comm_s3205402744901411588hammer @ A @ Z @ ( suc @ N2 ) ) @ ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ Z @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) @ ( suc @ N2 ) ) )
          = ( groups7121269368397514597t_prod @ nat @ A
            @ ^ [K4: nat] : ( plus_plus @ A @ Z @ ( divide_divide @ A @ ( semiring_1_of_nat @ A @ K4 ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) )
            @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) @ ( one_one @ nat ) ) ) ) ) ) ).

% pochhammer_times_pochhammer_half
thf(fact_3549_sumr__cos__zero__one,axiom,
    ! [N2: nat] :
      ( ( groups7311177749621191930dd_sum @ nat @ real
        @ ^ [M7: nat] : ( times_times @ real @ ( cos_coeff @ M7 ) @ ( power_power @ real @ ( zero_zero @ real ) @ M7 ) )
        @ ( set_ord_lessThan @ nat @ ( suc @ N2 ) ) )
      = ( one_one @ real ) ) ).

% sumr_cos_zero_one
thf(fact_3550_finite__nat__bounded,axiom,
    ! [S3: set @ nat] :
      ( ( finite_finite2 @ nat @ S3 )
     => ? [K3: nat] : ( ord_less_eq @ ( set @ nat ) @ S3 @ ( set_ord_lessThan @ nat @ K3 ) ) ) ).

% finite_nat_bounded
thf(fact_3551_finite__nat__iff__bounded,axiom,
    ( ( finite_finite2 @ nat )
    = ( ^ [S8: set @ nat] :
        ? [K4: nat] : ( ord_less_eq @ ( set @ nat ) @ S8 @ ( set_ord_lessThan @ nat @ K4 ) ) ) ) ).

% finite_nat_iff_bounded
thf(fact_3552_prod_Oneutral__const,axiom,
    ! [B: $tType,A: $tType] :
      ( ( comm_monoid_mult @ A )
     => ! [A3: set @ B] :
          ( ( groups7121269368397514597t_prod @ B @ A
            @ ^ [Uu3: B] : ( one_one @ A )
            @ A3 )
          = ( one_one @ A ) ) ) ).

% prod.neutral_const
thf(fact_3553_of__nat__prod,axiom,
    ! [A: $tType,B: $tType] :
      ( ( comm_semiring_1 @ A )
     => ! [F: B > nat,A3: set @ B] :
          ( ( semiring_1_of_nat @ A @ ( groups7121269368397514597t_prod @ B @ nat @ F @ A3 ) )
          = ( groups7121269368397514597t_prod @ B @ A
            @ ^ [X: B] : ( semiring_1_of_nat @ A @ ( F @ X ) )
            @ A3 ) ) ) ).

% of_nat_prod
thf(fact_3554_of__int__prod,axiom,
    ! [A: $tType,B: $tType] :
      ( ( comm_ring_1 @ A )
     => ! [F: B > int,A3: set @ B] :
          ( ( ring_1_of_int @ A @ ( groups7121269368397514597t_prod @ B @ int @ F @ A3 ) )
          = ( groups7121269368397514597t_prod @ B @ A
            @ ^ [X: B] : ( ring_1_of_int @ A @ ( F @ X ) )
            @ A3 ) ) ) ).

% of_int_prod
thf(fact_3555_prod_Oempty,axiom,
    ! [B: $tType,A: $tType] :
      ( ( comm_monoid_mult @ A )
     => ! [G: B > A] :
          ( ( groups7121269368397514597t_prod @ B @ A @ G @ ( bot_bot @ ( set @ B ) ) )
          = ( one_one @ A ) ) ) ).

% prod.empty
thf(fact_3556_prod_Oinfinite,axiom,
    ! [B: $tType,A: $tType] :
      ( ( comm_monoid_mult @ A )
     => ! [A3: set @ B,G: B > A] :
          ( ~ ( finite_finite2 @ B @ A3 )
         => ( ( groups7121269368397514597t_prod @ B @ A @ G @ A3 )
            = ( one_one @ A ) ) ) ) ).

% prod.infinite
thf(fact_3557_cos__coeff__0,axiom,
    ( ( cos_coeff @ ( zero_zero @ nat ) )
    = ( one_one @ real ) ) ).

% cos_coeff_0
thf(fact_3558_prod_Odelta_H,axiom,
    ! [B: $tType,A: $tType] :
      ( ( comm_monoid_mult @ A )
     => ! [S3: set @ B,A2: B,B2: B > A] :
          ( ( finite_finite2 @ B @ S3 )
         => ( ( ( member @ B @ A2 @ S3 )
             => ( ( groups7121269368397514597t_prod @ B @ A
                  @ ^ [K4: B] : ( if @ A @ ( A2 = K4 ) @ ( B2 @ K4 ) @ ( one_one @ A ) )
                  @ S3 )
                = ( B2 @ A2 ) ) )
            & ( ~ ( member @ B @ A2 @ S3 )
             => ( ( groups7121269368397514597t_prod @ B @ A
                  @ ^ [K4: B] : ( if @ A @ ( A2 = K4 ) @ ( B2 @ K4 ) @ ( one_one @ A ) )
                  @ S3 )
                = ( one_one @ A ) ) ) ) ) ) ).

% prod.delta'
thf(fact_3559_prod_Odelta,axiom,
    ! [B: $tType,A: $tType] :
      ( ( comm_monoid_mult @ A )
     => ! [S3: set @ B,A2: B,B2: B > A] :
          ( ( finite_finite2 @ B @ S3 )
         => ( ( ( member @ B @ A2 @ S3 )
             => ( ( groups7121269368397514597t_prod @ B @ A
                  @ ^ [K4: B] : ( if @ A @ ( K4 = A2 ) @ ( B2 @ K4 ) @ ( one_one @ A ) )
                  @ S3 )
                = ( B2 @ A2 ) ) )
            & ( ~ ( member @ B @ A2 @ S3 )
             => ( ( groups7121269368397514597t_prod @ B @ A
                  @ ^ [K4: B] : ( if @ A @ ( K4 = A2 ) @ ( B2 @ K4 ) @ ( one_one @ A ) )
                  @ S3 )
                = ( one_one @ A ) ) ) ) ) ) ).

% prod.delta
thf(fact_3560_prod_Oinsert,axiom,
    ! [A: $tType,B: $tType] :
      ( ( comm_monoid_mult @ A )
     => ! [A3: set @ B,X2: B,G: B > A] :
          ( ( finite_finite2 @ B @ A3 )
         => ( ~ ( member @ B @ X2 @ A3 )
           => ( ( groups7121269368397514597t_prod @ B @ A @ G @ ( insert @ B @ X2 @ A3 ) )
              = ( times_times @ A @ ( G @ X2 ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ A3 ) ) ) ) ) ) ).

% prod.insert
thf(fact_3561_prod_OlessThan__Suc,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_mult @ A )
     => ! [G: nat > A,N2: nat] :
          ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_ord_lessThan @ nat @ ( suc @ N2 ) ) )
          = ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_ord_lessThan @ nat @ N2 ) ) @ ( G @ N2 ) ) ) ) ).

% prod.lessThan_Suc
thf(fact_3562_prod_Ocl__ivl__Suc,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_mult @ A )
     => ! [N2: nat,M: nat,G: nat > A] :
          ( ( ( ord_less @ nat @ ( suc @ N2 ) @ M )
           => ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ ( suc @ N2 ) ) )
              = ( one_one @ A ) ) )
          & ( ~ ( ord_less @ nat @ ( suc @ N2 ) @ M )
           => ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ ( suc @ N2 ) ) )
              = ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ) ) ) ).

% prod.cl_ivl_Suc
thf(fact_3563_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [B: $tType,A: $tType] :
      ( ( comm_monoid_mult @ A )
     => ! [G: B > A,A3: set @ B] :
          ( ( ( groups7121269368397514597t_prod @ B @ A @ G @ A3 )
           != ( one_one @ A ) )
         => ~ ! [A4: B] :
                ( ( member @ B @ A4 @ A3 )
               => ( ( G @ A4 )
                  = ( one_one @ A ) ) ) ) ) ).

% prod.not_neutral_contains_not_neutral
thf(fact_3564_prod_Oneutral,axiom,
    ! [B: $tType,A: $tType] :
      ( ( comm_monoid_mult @ A )
     => ! [A3: set @ B,G: B > A] :
          ( ! [X3: B] :
              ( ( member @ B @ X3 @ A3 )
             => ( ( G @ X3 )
                = ( one_one @ A ) ) )
         => ( ( groups7121269368397514597t_prod @ B @ A @ G @ A3 )
            = ( one_one @ A ) ) ) ) ).

% prod.neutral
thf(fact_3565_norm__prod__le,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ( comm_monoid_mult @ A )
        & ( real_V2822296259951069270ebra_1 @ A ) )
     => ! [F: B > A,A3: set @ B] :
          ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( groups7121269368397514597t_prod @ B @ A @ F @ A3 ) )
          @ ( groups7121269368397514597t_prod @ B @ real
            @ ^ [A5: B] : ( real_V7770717601297561774m_norm @ A @ ( F @ A5 ) )
            @ A3 ) ) ) ).

% norm_prod_le
thf(fact_3566_prod__norm,axiom,
    ! [B: $tType,A: $tType] :
      ( ( ( real_V8999393235501362500lgebra @ B )
        & ( comm_semiring_1 @ B ) )
     => ! [F: A > B,A3: set @ A] :
          ( ( groups7121269368397514597t_prod @ A @ real
            @ ^ [X: A] : ( real_V7770717601297561774m_norm @ B @ ( F @ X ) )
            @ A3 )
          = ( real_V7770717601297561774m_norm @ B @ ( groups7121269368397514597t_prod @ A @ B @ F @ A3 ) ) ) ) ).

% prod_norm
thf(fact_3567_prod_Oswap,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( ( comm_monoid_mult @ A )
     => ! [G: B > C > A,B6: set @ C,A3: set @ B] :
          ( ( groups7121269368397514597t_prod @ B @ A
            @ ^ [I2: B] : ( groups7121269368397514597t_prod @ C @ A @ ( G @ I2 ) @ B6 )
            @ A3 )
          = ( groups7121269368397514597t_prod @ C @ A
            @ ^ [J3: C] :
                ( groups7121269368397514597t_prod @ B @ A
                @ ^ [I2: B] : ( G @ I2 @ J3 )
                @ A3 )
            @ B6 ) ) ) ).

% prod.swap
thf(fact_3568_prod_Odistrib,axiom,
    ! [A: $tType,B: $tType] :
      ( ( comm_monoid_mult @ A )
     => ! [G: B > A,H2: B > A,A3: set @ B] :
          ( ( groups7121269368397514597t_prod @ B @ A
            @ ^ [X: B] : ( times_times @ A @ ( G @ X ) @ ( H2 @ X ) )
            @ A3 )
          = ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ G @ A3 ) @ ( groups7121269368397514597t_prod @ B @ A @ H2 @ A3 ) ) ) ) ).

% prod.distrib
thf(fact_3569_prod__dividef,axiom,
    ! [A: $tType,B: $tType] :
      ( ( field @ A )
     => ! [F: B > A,G: B > A,A3: set @ B] :
          ( ( groups7121269368397514597t_prod @ B @ A
            @ ^ [X: B] : ( divide_divide @ A @ ( F @ X ) @ ( G @ X ) )
            @ A3 )
          = ( divide_divide @ A @ ( groups7121269368397514597t_prod @ B @ A @ F @ A3 ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ A3 ) ) ) ) ).

% prod_dividef
thf(fact_3570_prod__power__distrib,axiom,
    ! [B: $tType,A: $tType] :
      ( ( comm_semiring_1 @ B )
     => ! [F: A > B,A3: set @ A,N2: nat] :
          ( ( power_power @ B @ ( groups7121269368397514597t_prod @ A @ B @ F @ A3 ) @ N2 )
          = ( groups7121269368397514597t_prod @ A @ B
            @ ^ [X: A] : ( power_power @ B @ ( F @ X ) @ N2 )
            @ A3 ) ) ) ).

% prod_power_distrib
thf(fact_3571_prod_Oswap__restrict,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( ( comm_monoid_mult @ A )
     => ! [A3: set @ B,B6: set @ C,G: B > C > A,R2: B > C > $o] :
          ( ( finite_finite2 @ B @ A3 )
         => ( ( finite_finite2 @ C @ B6 )
           => ( ( groups7121269368397514597t_prod @ B @ A
                @ ^ [X: B] :
                    ( groups7121269368397514597t_prod @ C @ A @ ( G @ X )
                    @ ( collect @ C
                      @ ^ [Y: C] :
                          ( ( member @ C @ Y @ B6 )
                          & ( R2 @ X @ Y ) ) ) )
                @ A3 )
              = ( groups7121269368397514597t_prod @ C @ A
                @ ^ [Y: C] :
                    ( groups7121269368397514597t_prod @ B @ A
                    @ ^ [X: B] : ( G @ X @ Y )
                    @ ( collect @ B
                      @ ^ [X: B] :
                          ( ( member @ B @ X @ A3 )
                          & ( R2 @ X @ Y ) ) ) )
                @ B6 ) ) ) ) ) ).

% prod.swap_restrict
thf(fact_3572_mod__prod__eq,axiom,
    ! [B: $tType,A: $tType] :
      ( ( euclid4440199948858584721cancel @ A )
     => ! [F: B > A,A2: A,A3: set @ B] :
          ( ( modulo_modulo @ A
            @ ( groups7121269368397514597t_prod @ B @ A
              @ ^ [I2: B] : ( modulo_modulo @ A @ ( F @ I2 ) @ A2 )
              @ A3 )
            @ A2 )
          = ( modulo_modulo @ A @ ( groups7121269368397514597t_prod @ B @ A @ F @ A3 ) @ A2 ) ) ) ).

% mod_prod_eq
thf(fact_3573_prod__mono,axiom,
    ! [A: $tType,B: $tType] :
      ( ( linordered_semidom @ A )
     => ! [A3: set @ B,F: B > A,G: B > A] :
          ( ! [I3: B] :
              ( ( member @ B @ I3 @ A3 )
             => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F @ I3 ) )
                & ( ord_less_eq @ A @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
         => ( ord_less_eq @ A @ ( groups7121269368397514597t_prod @ B @ A @ F @ A3 ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ A3 ) ) ) ) ).

% prod_mono
thf(fact_3574_prod__nonneg,axiom,
    ! [A: $tType,B: $tType] :
      ( ( linordered_semidom @ A )
     => ! [A3: set @ B,F: B > A] :
          ( ! [X3: B] :
              ( ( member @ B @ X3 @ A3 )
             => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F @ X3 ) ) )
         => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( groups7121269368397514597t_prod @ B @ A @ F @ A3 ) ) ) ) ).

% prod_nonneg
thf(fact_3575_prod__pos,axiom,
    ! [A: $tType,B: $tType] :
      ( ( linordered_semidom @ A )
     => ! [A3: set @ B,F: B > A] :
          ( ! [X3: B] :
              ( ( member @ B @ X3 @ A3 )
             => ( ord_less @ A @ ( zero_zero @ A ) @ ( F @ X3 ) ) )
         => ( ord_less @ A @ ( zero_zero @ A ) @ ( groups7121269368397514597t_prod @ B @ A @ F @ A3 ) ) ) ) ).

% prod_pos
thf(fact_3576_prod__ge__1,axiom,
    ! [A: $tType,B: $tType] :
      ( ( linord181362715937106298miring @ A )
     => ! [A3: set @ B,F: B > A] :
          ( ! [X3: B] :
              ( ( member @ B @ X3 @ A3 )
             => ( ord_less_eq @ A @ ( one_one @ A ) @ ( F @ X3 ) ) )
         => ( ord_less_eq @ A @ ( one_one @ A ) @ ( groups7121269368397514597t_prod @ B @ A @ F @ A3 ) ) ) ) ).

% prod_ge_1
thf(fact_3577_prod_Ointer__filter,axiom,
    ! [A: $tType,B: $tType] :
      ( ( comm_monoid_mult @ A )
     => ! [A3: set @ B,G: B > A,P: B > $o] :
          ( ( finite_finite2 @ B @ A3 )
         => ( ( groups7121269368397514597t_prod @ B @ A @ G
              @ ( collect @ B
                @ ^ [X: B] :
                    ( ( member @ B @ X @ A3 )
                    & ( P @ X ) ) ) )
            = ( groups7121269368397514597t_prod @ B @ A
              @ ^ [X: B] : ( if @ A @ ( P @ X ) @ ( G @ X ) @ ( one_one @ A ) )
              @ A3 ) ) ) ) ).

% prod.inter_filter
thf(fact_3578_prod_Oshift__bounds__cl__Suc__ivl,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_mult @ A )
     => ! [G: nat > A,M: nat,N2: nat] :
          ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M ) @ ( suc @ N2 ) ) )
          = ( groups7121269368397514597t_prod @ nat @ A
            @ ^ [I2: nat] : ( G @ ( suc @ I2 ) )
            @ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) ) ) ) ).

% prod.shift_bounds_cl_Suc_ivl
thf(fact_3579_power__sum,axiom,
    ! [A: $tType,B: $tType] :
      ( ( comm_monoid_mult @ A )
     => ! [C2: A,F: B > nat,A3: set @ B] :
          ( ( power_power @ A @ C2 @ ( groups7311177749621191930dd_sum @ B @ nat @ F @ A3 ) )
          = ( groups7121269368397514597t_prod @ B @ A
            @ ^ [A5: B] : ( power_power @ A @ C2 @ ( F @ A5 ) )
            @ A3 ) ) ) ).

% power_sum
thf(fact_3580_prod_Oshift__bounds__cl__nat__ivl,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_mult @ A )
     => ! [G: nat > A,M: nat,K: nat,N2: nat] :
          ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ M @ K ) @ ( plus_plus @ nat @ N2 @ K ) ) )
          = ( groups7121269368397514597t_prod @ nat @ A
            @ ^ [I2: nat] : ( G @ ( plus_plus @ nat @ I2 @ K ) )
            @ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) ) ) ) ).

% prod.shift_bounds_cl_nat_ivl
thf(fact_3581_prod__le__1,axiom,
    ! [B: $tType,A: $tType] :
      ( ( linord181362715937106298miring @ A )
     => ! [A3: set @ B,F: B > A] :
          ( ! [X3: B] :
              ( ( member @ B @ X3 @ A3 )
             => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F @ X3 ) )
                & ( ord_less_eq @ A @ ( F @ X3 ) @ ( one_one @ A ) ) ) )
         => ( ord_less_eq @ A @ ( groups7121269368397514597t_prod @ B @ A @ F @ A3 ) @ ( one_one @ A ) ) ) ) ).

% prod_le_1
thf(fact_3582_prod_Orelated,axiom,
    ! [A: $tType,B: $tType] :
      ( ( comm_monoid_mult @ A )
     => ! [R2: A > A > $o,S3: set @ B,H2: B > A,G: B > A] :
          ( ( R2 @ ( one_one @ A ) @ ( one_one @ A ) )
         => ( ! [X16: A,Y15: A,X24: A,Y24: A] :
                ( ( ( R2 @ X16 @ X24 )
                  & ( R2 @ Y15 @ Y24 ) )
               => ( R2 @ ( times_times @ A @ X16 @ Y15 ) @ ( times_times @ A @ X24 @ Y24 ) ) )
           => ( ( finite_finite2 @ B @ S3 )
             => ( ! [X3: B] :
                    ( ( member @ B @ X3 @ S3 )
                   => ( R2 @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
               => ( R2 @ ( groups7121269368397514597t_prod @ B @ A @ H2 @ S3 ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ S3 ) ) ) ) ) ) ) ).

% prod.related
thf(fact_3583_prod_Oinsert__if,axiom,
    ! [A: $tType,B: $tType] :
      ( ( comm_monoid_mult @ A )
     => ! [A3: set @ B,X2: B,G: B > A] :
          ( ( finite_finite2 @ B @ A3 )
         => ( ( ( member @ B @ X2 @ A3 )
             => ( ( groups7121269368397514597t_prod @ B @ A @ G @ ( insert @ B @ X2 @ A3 ) )
                = ( groups7121269368397514597t_prod @ B @ A @ G @ A3 ) ) )
            & ( ~ ( member @ B @ X2 @ A3 )
             => ( ( groups7121269368397514597t_prod @ B @ A @ G @ ( insert @ B @ X2 @ A3 ) )
                = ( times_times @ A @ ( G @ X2 ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ A3 ) ) ) ) ) ) ) ).

% prod.insert_if
thf(fact_3584_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ( comm_monoid_mult @ A )
     => ! [S7: set @ B,T8: set @ C,S3: set @ B,I: C > B,J: B > C,T6: set @ C,G: B > A,H2: C > A] :
          ( ( finite_finite2 @ B @ S7 )
         => ( ( finite_finite2 @ C @ T8 )
           => ( ! [A4: B] :
                  ( ( member @ B @ A4 @ ( minus_minus @ ( set @ B ) @ S3 @ S7 ) )
                 => ( ( I @ ( J @ A4 ) )
                    = A4 ) )
             => ( ! [A4: B] :
                    ( ( member @ B @ A4 @ ( minus_minus @ ( set @ B ) @ S3 @ S7 ) )
                   => ( member @ C @ ( J @ A4 ) @ ( minus_minus @ ( set @ C ) @ T6 @ T8 ) ) )
               => ( ! [B3: C] :
                      ( ( member @ C @ B3 @ ( minus_minus @ ( set @ C ) @ T6 @ T8 ) )
                     => ( ( J @ ( I @ B3 ) )
                        = B3 ) )
                 => ( ! [B3: C] :
                        ( ( member @ C @ B3 @ ( minus_minus @ ( set @ C ) @ T6 @ T8 ) )
                       => ( member @ B @ ( I @ B3 ) @ ( minus_minus @ ( set @ B ) @ S3 @ S7 ) ) )
                   => ( ! [A4: B] :
                          ( ( member @ B @ A4 @ S7 )
                         => ( ( G @ A4 )
                            = ( one_one @ A ) ) )
                     => ( ! [B3: C] :
                            ( ( member @ C @ B3 @ T8 )
                           => ( ( H2 @ B3 )
                              = ( one_one @ A ) ) )
                       => ( ! [A4: B] :
                              ( ( member @ B @ A4 @ S3 )
                             => ( ( H2 @ ( J @ A4 ) )
                                = ( G @ A4 ) ) )
                         => ( ( groups7121269368397514597t_prod @ B @ A @ G @ S3 )
                            = ( groups7121269368397514597t_prod @ C @ A @ H2 @ T6 ) ) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
thf(fact_3585_prod__dvd__prod__subset,axiom,
    ! [A: $tType,B: $tType] :
      ( ( comm_monoid_mult @ A )
     => ! [B6: set @ B,A3: set @ B,F: B > A] :
          ( ( finite_finite2 @ B @ B6 )
         => ( ( ord_less_eq @ ( set @ B ) @ A3 @ B6 )
           => ( dvd_dvd @ A @ ( groups7121269368397514597t_prod @ B @ A @ F @ A3 ) @ ( groups7121269368397514597t_prod @ B @ A @ F @ B6 ) ) ) ) ) ).

% prod_dvd_prod_subset
thf(fact_3586_prod__dvd__prod__subset2,axiom,
    ! [A: $tType,B: $tType] :
      ( ( comm_semiring_1 @ A )
     => ! [B6: set @ B,A3: set @ B,F: B > A,G: B > A] :
          ( ( finite_finite2 @ B @ B6 )
         => ( ( ord_less_eq @ ( set @ B ) @ A3 @ B6 )
           => ( ! [A4: B] :
                  ( ( member @ B @ A4 @ A3 )
                 => ( dvd_dvd @ A @ ( F @ A4 ) @ ( G @ A4 ) ) )
             => ( dvd_dvd @ A @ ( groups7121269368397514597t_prod @ B @ A @ F @ A3 ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ B6 ) ) ) ) ) ) ).

% prod_dvd_prod_subset2
thf(fact_3587_prod_Osetdiff__irrelevant,axiom,
    ! [A: $tType,B: $tType] :
      ( ( comm_monoid_mult @ A )
     => ! [A3: set @ B,G: B > A] :
          ( ( finite_finite2 @ B @ A3 )
         => ( ( groups7121269368397514597t_prod @ B @ A @ G
              @ ( minus_minus @ ( set @ B ) @ A3
                @ ( collect @ B
                  @ ^ [X: B] :
                      ( ( G @ X )
                      = ( one_one @ A ) ) ) ) )
            = ( groups7121269368397514597t_prod @ B @ A @ G @ A3 ) ) ) ) ).

% prod.setdiff_irrelevant
thf(fact_3588_prod_Onat__diff__reindex,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_mult @ A )
     => ! [G: nat > A,N2: nat] :
          ( ( groups7121269368397514597t_prod @ nat @ A
            @ ^ [I2: nat] : ( G @ ( minus_minus @ nat @ N2 @ ( suc @ I2 ) ) )
            @ ( set_ord_lessThan @ nat @ N2 ) )
          = ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_ord_lessThan @ nat @ N2 ) ) ) ) ).

% prod.nat_diff_reindex
thf(fact_3589_prod_OatLeastAtMost__rev,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_mult @ A )
     => ! [G: nat > A,N2: nat,M: nat] :
          ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ N2 @ M ) )
          = ( groups7121269368397514597t_prod @ nat @ A
            @ ^ [I2: nat] : ( G @ ( minus_minus @ nat @ ( plus_plus @ nat @ M @ N2 ) @ I2 ) )
            @ ( set_or1337092689740270186AtMost @ nat @ N2 @ M ) ) ) ) ).

% prod.atLeastAtMost_rev
thf(fact_3590_less__1__prod2,axiom,
    ! [B: $tType,A: $tType] :
      ( ( linordered_idom @ B )
     => ! [I5: set @ A,I: A,F: A > B] :
          ( ( finite_finite2 @ A @ I5 )
         => ( ( member @ A @ I @ I5 )
           => ( ( ord_less @ B @ ( one_one @ B ) @ ( F @ I ) )
             => ( ! [I3: A] :
                    ( ( member @ A @ I3 @ I5 )
                   => ( ord_less_eq @ B @ ( one_one @ B ) @ ( F @ I3 ) ) )
               => ( ord_less @ B @ ( one_one @ B ) @ ( groups7121269368397514597t_prod @ A @ B @ F @ I5 ) ) ) ) ) ) ) ).

% less_1_prod2
thf(fact_3591_less__1__prod,axiom,
    ! [B: $tType,A: $tType] :
      ( ( linordered_idom @ B )
     => ! [I5: set @ A,F: A > B] :
          ( ( finite_finite2 @ A @ I5 )
         => ( ( I5
             != ( bot_bot @ ( set @ A ) ) )
           => ( ! [I3: A] :
                  ( ( member @ A @ I3 @ I5 )
                 => ( ord_less @ B @ ( one_one @ B ) @ ( F @ I3 ) ) )
             => ( ord_less @ B @ ( one_one @ B ) @ ( groups7121269368397514597t_prod @ A @ B @ F @ I5 ) ) ) ) ) ) ).

% less_1_prod
thf(fact_3592_prod_Osame__carrier,axiom,
    ! [A: $tType,B: $tType] :
      ( ( comm_monoid_mult @ A )
     => ! [C5: set @ B,A3: set @ B,B6: set @ B,G: B > A,H2: B > A] :
          ( ( finite_finite2 @ B @ C5 )
         => ( ( ord_less_eq @ ( set @ B ) @ A3 @ C5 )
           => ( ( ord_less_eq @ ( set @ B ) @ B6 @ C5 )
             => ( ! [A4: B] :
                    ( ( member @ B @ A4 @ ( minus_minus @ ( set @ B ) @ C5 @ A3 ) )
                   => ( ( G @ A4 )
                      = ( one_one @ A ) ) )
               => ( ! [B3: B] :
                      ( ( member @ B @ B3 @ ( minus_minus @ ( set @ B ) @ C5 @ B6 ) )
                     => ( ( H2 @ B3 )
                        = ( one_one @ A ) ) )
                 => ( ( ( groups7121269368397514597t_prod @ B @ A @ G @ A3 )
                      = ( groups7121269368397514597t_prod @ B @ A @ H2 @ B6 ) )
                    = ( ( groups7121269368397514597t_prod @ B @ A @ G @ C5 )
                      = ( groups7121269368397514597t_prod @ B @ A @ H2 @ C5 ) ) ) ) ) ) ) ) ) ).

% prod.same_carrier
thf(fact_3593_prod_Osame__carrierI,axiom,
    ! [A: $tType,B: $tType] :
      ( ( comm_monoid_mult @ A )
     => ! [C5: set @ B,A3: set @ B,B6: set @ B,G: B > A,H2: B > A] :
          ( ( finite_finite2 @ B @ C5 )
         => ( ( ord_less_eq @ ( set @ B ) @ A3 @ C5 )
           => ( ( ord_less_eq @ ( set @ B ) @ B6 @ C5 )
             => ( ! [A4: B] :
                    ( ( member @ B @ A4 @ ( minus_minus @ ( set @ B ) @ C5 @ A3 ) )
                   => ( ( G @ A4 )
                      = ( one_one @ A ) ) )
               => ( ! [B3: B] :
                      ( ( member @ B @ B3 @ ( minus_minus @ ( set @ B ) @ C5 @ B6 ) )
                     => ( ( H2 @ B3 )
                        = ( one_one @ A ) ) )
                 => ( ( ( groups7121269368397514597t_prod @ B @ A @ G @ C5 )
                      = ( groups7121269368397514597t_prod @ B @ A @ H2 @ C5 ) )
                   => ( ( groups7121269368397514597t_prod @ B @ A @ G @ A3 )
                      = ( groups7121269368397514597t_prod @ B @ A @ H2 @ B6 ) ) ) ) ) ) ) ) ) ).

% prod.same_carrierI
thf(fact_3594_prod_Omono__neutral__left,axiom,
    ! [A: $tType,B: $tType] :
      ( ( comm_monoid_mult @ A )
     => ! [T6: set @ B,S3: set @ B,G: B > A] :
          ( ( finite_finite2 @ B @ T6 )
         => ( ( ord_less_eq @ ( set @ B ) @ S3 @ T6 )
           => ( ! [X3: B] :
                  ( ( member @ B @ X3 @ ( minus_minus @ ( set @ B ) @ T6 @ S3 ) )
                 => ( ( G @ X3 )
                    = ( one_one @ A ) ) )
             => ( ( groups7121269368397514597t_prod @ B @ A @ G @ S3 )
                = ( groups7121269368397514597t_prod @ B @ A @ G @ T6 ) ) ) ) ) ) ).

% prod.mono_neutral_left
thf(fact_3595_prod_Omono__neutral__right,axiom,
    ! [A: $tType,B: $tType] :
      ( ( comm_monoid_mult @ A )
     => ! [T6: set @ B,S3: set @ B,G: B > A] :
          ( ( finite_finite2 @ B @ T6 )
         => ( ( ord_less_eq @ ( set @ B ) @ S3 @ T6 )
           => ( ! [X3: B] :
                  ( ( member @ B @ X3 @ ( minus_minus @ ( set @ B ) @ T6 @ S3 ) )
                 => ( ( G @ X3 )
                    = ( one_one @ A ) ) )
             => ( ( groups7121269368397514597t_prod @ B @ A @ G @ T6 )
                = ( groups7121269368397514597t_prod @ B @ A @ G @ S3 ) ) ) ) ) ) ).

% prod.mono_neutral_right
thf(fact_3596_prod_Omono__neutral__cong__left,axiom,
    ! [A: $tType,B: $tType] :
      ( ( comm_monoid_mult @ A )
     => ! [T6: set @ B,S3: set @ B,H2: B > A,G: B > A] :
          ( ( finite_finite2 @ B @ T6 )
         => ( ( ord_less_eq @ ( set @ B ) @ S3 @ T6 )
           => ( ! [X3: B] :
                  ( ( member @ B @ X3 @ ( minus_minus @ ( set @ B ) @ T6 @ S3 ) )
                 => ( ( H2 @ X3 )
                    = ( one_one @ A ) ) )
             => ( ! [X3: B] :
                    ( ( member @ B @ X3 @ S3 )
                   => ( ( G @ X3 )
                      = ( H2 @ X3 ) ) )
               => ( ( groups7121269368397514597t_prod @ B @ A @ G @ S3 )
                  = ( groups7121269368397514597t_prod @ B @ A @ H2 @ T6 ) ) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
thf(fact_3597_prod_Omono__neutral__cong__right,axiom,
    ! [A: $tType,B: $tType] :
      ( ( comm_monoid_mult @ A )
     => ! [T6: set @ B,S3: set @ B,G: B > A,H2: B > A] :
          ( ( finite_finite2 @ B @ T6 )
         => ( ( ord_less_eq @ ( set @ B ) @ S3 @ T6 )
           => ( ! [X3: B] :
                  ( ( member @ B @ X3 @ ( minus_minus @ ( set @ B ) @ T6 @ S3 ) )
                 => ( ( G @ X3 )
                    = ( one_one @ A ) ) )
             => ( ! [X3: B] :
                    ( ( member @ B @ X3 @ S3 )
                   => ( ( G @ X3 )
                      = ( H2 @ X3 ) ) )
               => ( ( groups7121269368397514597t_prod @ B @ A @ G @ T6 )
                  = ( groups7121269368397514597t_prod @ B @ A @ H2 @ S3 ) ) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
thf(fact_3598_prod_Osubset__diff,axiom,
    ! [A: $tType,B: $tType] :
      ( ( comm_monoid_mult @ A )
     => ! [B6: set @ B,A3: set @ B,G: B > A] :
          ( ( ord_less_eq @ ( set @ B ) @ B6 @ A3 )
         => ( ( finite_finite2 @ B @ A3 )
           => ( ( groups7121269368397514597t_prod @ B @ A @ G @ A3 )
              = ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ G @ ( minus_minus @ ( set @ B ) @ A3 @ B6 ) ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ B6 ) ) ) ) ) ) ).

% prod.subset_diff
thf(fact_3599_prod_OatLeast0__atMost__Suc,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_mult @ A )
     => ! [G: nat > A,N2: nat] :
          ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( suc @ N2 ) ) )
          = ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ) ).

% prod.atLeast0_atMost_Suc
thf(fact_3600_prod_OatLeast__Suc__atMost,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_mult @ A )
     => ! [M: nat,N2: nat,G: nat > A] :
          ( ( ord_less_eq @ nat @ M @ N2 )
         => ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) )
            = ( times_times @ A @ ( G @ M ) @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M ) @ N2 ) ) ) ) ) ) ).

% prod.atLeast_Suc_atMost
thf(fact_3601_prod_Onat__ivl__Suc_H,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_mult @ A )
     => ! [M: nat,N2: nat,G: nat > A] :
          ( ( ord_less_eq @ nat @ M @ ( suc @ N2 ) )
         => ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ ( suc @ N2 ) ) )
            = ( times_times @ A @ ( G @ ( suc @ N2 ) ) @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) ) ) ) ) ) ).

% prod.nat_ivl_Suc'
thf(fact_3602_prod_OlessThan__Suc__shift,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_mult @ A )
     => ! [G: nat > A,N2: nat] :
          ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_ord_lessThan @ nat @ ( suc @ N2 ) ) )
          = ( times_times @ A @ ( G @ ( zero_zero @ nat ) )
            @ ( groups7121269368397514597t_prod @ nat @ A
              @ ^ [I2: nat] : ( G @ ( suc @ I2 ) )
              @ ( set_ord_lessThan @ nat @ N2 ) ) ) ) ) ).

% prod.lessThan_Suc_shift
thf(fact_3603_prod_OSuc__reindex__ivl,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_mult @ A )
     => ! [M: nat,N2: nat,G: nat > A] :
          ( ( ord_less_eq @ nat @ M @ N2 )
         => ( ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) ) @ ( G @ ( suc @ N2 ) ) )
            = ( times_times @ A @ ( G @ M )
              @ ( groups7121269368397514597t_prod @ nat @ A
                @ ^ [I2: nat] : ( G @ ( suc @ I2 ) )
                @ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) ) ) ) ) ) ).

% prod.Suc_reindex_ivl
thf(fact_3604_prod_OatLeast1__atMost__eq,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_mult @ A )
     => ! [G: nat > A,N2: nat] :
          ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N2 ) )
          = ( groups7121269368397514597t_prod @ nat @ A
            @ ^ [K4: nat] : ( G @ ( suc @ K4 ) )
            @ ( set_ord_lessThan @ nat @ N2 ) ) ) ) ).

% prod.atLeast1_atMost_eq
thf(fact_3605_prod__mono__strict,axiom,
    ! [A: $tType,B: $tType] :
      ( ( linordered_semidom @ A )
     => ! [A3: set @ B,F: B > A,G: B > A] :
          ( ( finite_finite2 @ B @ A3 )
         => ( ! [I3: B] :
                ( ( member @ B @ I3 @ A3 )
               => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F @ I3 ) )
                  & ( ord_less @ A @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
           => ( ( A3
               != ( bot_bot @ ( set @ B ) ) )
             => ( ord_less @ A @ ( groups7121269368397514597t_prod @ B @ A @ F @ A3 ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ A3 ) ) ) ) ) ) ).

% prod_mono_strict
thf(fact_3606_prod__atLeastAtMost__code,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_mult @ A )
     => ! [F: nat > A,A2: nat,B2: nat] :
          ( ( groups7121269368397514597t_prod @ nat @ A @ F @ ( set_or1337092689740270186AtMost @ nat @ A2 @ B2 ) )
          = ( set_fo6178422350223883121st_nat @ A
            @ ^ [A5: nat] : ( times_times @ A @ ( F @ A5 ) )
            @ A2
            @ B2
            @ ( one_one @ A ) ) ) ) ).

% prod_atLeastAtMost_code
thf(fact_3607_even__prod__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( ( semiring_parity @ A )
     => ! [A3: set @ B,F: B > A] :
          ( ( finite_finite2 @ B @ A3 )
         => ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( groups7121269368397514597t_prod @ B @ A @ F @ A3 ) )
            = ( ? [X: B] :
                  ( ( member @ B @ X @ A3 )
                  & ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( F @ X ) ) ) ) ) ) ) ).

% even_prod_iff
thf(fact_3608_prod_Oinsert__remove,axiom,
    ! [A: $tType,B: $tType] :
      ( ( comm_monoid_mult @ A )
     => ! [A3: set @ B,G: B > A,X2: B] :
          ( ( finite_finite2 @ B @ A3 )
         => ( ( groups7121269368397514597t_prod @ B @ A @ G @ ( insert @ B @ X2 @ A3 ) )
            = ( times_times @ A @ ( G @ X2 ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ ( minus_minus @ ( set @ B ) @ A3 @ ( insert @ B @ X2 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ).

% prod.insert_remove
thf(fact_3609_prod_Oremove,axiom,
    ! [A: $tType,B: $tType] :
      ( ( comm_monoid_mult @ A )
     => ! [A3: set @ B,X2: B,G: B > A] :
          ( ( finite_finite2 @ B @ A3 )
         => ( ( member @ B @ X2 @ A3 )
           => ( ( groups7121269368397514597t_prod @ B @ A @ G @ A3 )
              = ( times_times @ A @ ( G @ X2 ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ ( minus_minus @ ( set @ B ) @ A3 @ ( insert @ B @ X2 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ) ).

% prod.remove
thf(fact_3610_prod_Oub__add__nat,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_mult @ A )
     => ! [M: nat,N2: nat,G: nat > A,P4: nat] :
          ( ( ord_less_eq @ nat @ M @ ( plus_plus @ nat @ N2 @ ( one_one @ nat ) ) )
         => ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ ( plus_plus @ nat @ N2 @ P4 ) ) )
            = ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) ) @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ N2 @ ( one_one @ nat ) ) @ ( plus_plus @ nat @ N2 @ P4 ) ) ) ) ) ) ) ).

% prod.ub_add_nat
thf(fact_3611_prod_Odelta__remove,axiom,
    ! [A: $tType,B: $tType] :
      ( ( comm_monoid_mult @ A )
     => ! [S3: set @ B,A2: B,B2: B > A,C2: B > A] :
          ( ( finite_finite2 @ B @ S3 )
         => ( ( ( member @ B @ A2 @ S3 )
             => ( ( groups7121269368397514597t_prod @ B @ A
                  @ ^ [K4: B] : ( if @ A @ ( K4 = A2 ) @ ( B2 @ K4 ) @ ( C2 @ K4 ) )
                  @ S3 )
                = ( times_times @ A @ ( B2 @ A2 ) @ ( groups7121269368397514597t_prod @ B @ A @ C2 @ ( minus_minus @ ( set @ B ) @ S3 @ ( insert @ B @ A2 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) )
            & ( ~ ( member @ B @ A2 @ S3 )
             => ( ( groups7121269368397514597t_prod @ B @ A
                  @ ^ [K4: B] : ( if @ A @ ( K4 = A2 ) @ ( B2 @ K4 ) @ ( C2 @ K4 ) )
                  @ S3 )
                = ( groups7121269368397514597t_prod @ B @ A @ C2 @ ( minus_minus @ ( set @ B ) @ S3 @ ( insert @ B @ A2 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ) ).

% prod.delta_remove
thf(fact_3612_norm__prod__diff,axiom,
    ! [A: $tType,I6: $tType] :
      ( ( ( comm_monoid_mult @ A )
        & ( real_V2822296259951069270ebra_1 @ A ) )
     => ! [I5: set @ I6,Z: I6 > A,W: I6 > A] :
          ( ! [I3: I6] :
              ( ( member @ I6 @ I3 @ I5 )
             => ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( Z @ I3 ) ) @ ( one_one @ real ) ) )
         => ( ! [I3: I6] :
                ( ( member @ I6 @ I3 @ I5 )
               => ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( W @ I3 ) ) @ ( one_one @ real ) ) )
           => ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ ( groups7121269368397514597t_prod @ I6 @ A @ Z @ I5 ) @ ( groups7121269368397514597t_prod @ I6 @ A @ W @ I5 ) ) )
              @ ( groups7311177749621191930dd_sum @ I6 @ real
                @ ^ [I2: I6] : ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ ( Z @ I2 ) @ ( W @ I2 ) ) )
                @ I5 ) ) ) ) ) ).

% norm_prod_diff
thf(fact_3613_prod__mono2,axiom,
    ! [B: $tType,A: $tType] :
      ( ( linordered_idom @ B )
     => ! [B6: set @ A,A3: set @ A,F: A > B] :
          ( ( finite_finite2 @ A @ B6 )
         => ( ( ord_less_eq @ ( set @ A ) @ A3 @ B6 )
           => ( ! [B3: A] :
                  ( ( member @ A @ B3 @ ( minus_minus @ ( set @ A ) @ B6 @ A3 ) )
                 => ( ord_less_eq @ B @ ( one_one @ B ) @ ( F @ B3 ) ) )
             => ( ! [A4: A] :
                    ( ( member @ A @ A4 @ A3 )
                   => ( ord_less_eq @ B @ ( zero_zero @ B ) @ ( F @ A4 ) ) )
               => ( ord_less_eq @ B @ ( groups7121269368397514597t_prod @ A @ B @ F @ A3 ) @ ( groups7121269368397514597t_prod @ A @ B @ F @ B6 ) ) ) ) ) ) ) ).

% prod_mono2
thf(fact_3614_prod__diff1,axiom,
    ! [A: $tType,B: $tType] :
      ( ( semidom_divide @ A )
     => ! [A3: set @ B,F: B > A,A2: B] :
          ( ( finite_finite2 @ B @ A3 )
         => ( ( ( F @ A2 )
             != ( zero_zero @ A ) )
           => ( ( ( member @ B @ A2 @ A3 )
               => ( ( groups7121269368397514597t_prod @ B @ A @ F @ ( minus_minus @ ( set @ B ) @ A3 @ ( insert @ B @ A2 @ ( bot_bot @ ( set @ B ) ) ) ) )
                  = ( divide_divide @ A @ ( groups7121269368397514597t_prod @ B @ A @ F @ A3 ) @ ( F @ A2 ) ) ) )
              & ( ~ ( member @ B @ A2 @ A3 )
               => ( ( groups7121269368397514597t_prod @ B @ A @ F @ ( minus_minus @ ( set @ B ) @ A3 @ ( insert @ B @ A2 @ ( bot_bot @ ( set @ B ) ) ) ) )
                  = ( groups7121269368397514597t_prod @ B @ A @ F @ A3 ) ) ) ) ) ) ) ).

% prod_diff1
thf(fact_3615_pochhammer__Suc__prod,axiom,
    ! [A: $tType] :
      ( ( comm_semiring_1 @ A )
     => ! [A2: A,N2: nat] :
          ( ( comm_s3205402744901411588hammer @ A @ A2 @ ( suc @ N2 ) )
          = ( groups7121269368397514597t_prod @ nat @ A
            @ ^ [I2: nat] : ( plus_plus @ A @ A2 @ ( semiring_1_of_nat @ A @ I2 ) )
            @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ) ).

% pochhammer_Suc_prod
thf(fact_3616_pochhammer__prod__rev,axiom,
    ! [A: $tType] :
      ( ( comm_semiring_1 @ A )
     => ( ( comm_s3205402744901411588hammer @ A )
        = ( ^ [A5: A,N: nat] :
              ( groups7121269368397514597t_prod @ nat @ A
              @ ^ [I2: nat] : ( plus_plus @ A @ A5 @ ( semiring_1_of_nat @ A @ ( minus_minus @ nat @ N @ I2 ) ) )
              @ ( set_or1337092689740270186AtMost @ nat @ ( one_one @ nat ) @ N ) ) ) ) ) ).

% pochhammer_prod_rev
thf(fact_3617_prod_Oin__pairs,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_mult @ A )
     => ! [G: nat > A,M: nat,N2: nat] :
          ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ) )
          = ( groups7121269368397514597t_prod @ nat @ A
            @ ^ [I2: nat] : ( times_times @ A @ ( G @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I2 ) ) @ ( G @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I2 ) ) ) )
            @ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) ) ) ) ).

% prod.in_pairs
thf(fact_3618_pochhammer__Suc__prod__rev,axiom,
    ! [A: $tType] :
      ( ( comm_semiring_1 @ A )
     => ! [A2: A,N2: nat] :
          ( ( comm_s3205402744901411588hammer @ A @ A2 @ ( suc @ N2 ) )
          = ( groups7121269368397514597t_prod @ nat @ A
            @ ^ [I2: nat] : ( plus_plus @ A @ A2 @ ( semiring_1_of_nat @ A @ ( minus_minus @ nat @ N2 @ I2 ) ) )
            @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ) ).

% pochhammer_Suc_prod_rev
thf(fact_3619_unbounded__k__infinite,axiom,
    ! [K: nat,S3: set @ nat] :
      ( ! [M2: nat] :
          ( ( ord_less @ nat @ K @ M2 )
         => ? [N9: nat] :
              ( ( ord_less @ nat @ M2 @ N9 )
              & ( member @ nat @ N9 @ S3 ) ) )
     => ~ ( finite_finite2 @ nat @ S3 ) ) ).

% unbounded_k_infinite
thf(fact_3620_infinite__nat__iff__unbounded,axiom,
    ! [S3: set @ nat] :
      ( ( ~ ( finite_finite2 @ nat @ S3 ) )
      = ( ! [M7: nat] :
          ? [N: nat] :
            ( ( ord_less @ nat @ M7 @ N )
            & ( member @ nat @ N @ S3 ) ) ) ) ).

% infinite_nat_iff_unbounded
thf(fact_3621_infinite__nat__iff__unbounded__le,axiom,
    ! [S3: set @ nat] :
      ( ( ~ ( finite_finite2 @ nat @ S3 ) )
      = ( ! [M7: nat] :
          ? [N: nat] :
            ( ( ord_less_eq @ nat @ M7 @ N )
            & ( member @ nat @ N @ S3 ) ) ) ) ).

% infinite_nat_iff_unbounded_le
thf(fact_3622_round__altdef,axiom,
    ! [A: $tType] :
      ( ( archim2362893244070406136eiling @ A )
     => ( ( archimedean_round @ A )
        = ( ^ [X: A] : ( if @ int @ ( ord_less_eq @ A @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( archimedean_frac @ A @ X ) ) @ ( archimedean_ceiling @ A @ X ) @ ( archim6421214686448440834_floor @ A @ X ) ) ) ) ) ).

% round_altdef
thf(fact_3623_gchoose__row__sum__weighted,axiom,
    ! [A: $tType] :
      ( ( field_char_0 @ A )
     => ! [R4: A,M: nat] :
          ( ( groups7311177749621191930dd_sum @ nat @ A
            @ ^ [K4: nat] : ( times_times @ A @ ( gbinomial @ A @ R4 @ K4 ) @ ( minus_minus @ A @ ( divide_divide @ A @ R4 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( semiring_1_of_nat @ A @ K4 ) ) )
            @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ M ) )
          = ( times_times @ A @ ( divide_divide @ A @ ( semiring_1_of_nat @ A @ ( suc @ M ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( gbinomial @ A @ R4 @ ( suc @ M ) ) ) ) ) ).

% gchoose_row_sum_weighted
thf(fact_3624_central__binomial__lower__bound,axiom,
    ! [N2: nat] :
      ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
     => ( ord_less_eq @ real @ ( divide_divide @ real @ ( power_power @ real @ ( numeral_numeral @ real @ ( bit0 @ ( bit0 @ one2 ) ) ) @ N2 ) @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ real @ N2 ) ) ) @ ( semiring_1_of_nat @ real @ ( binomial @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) @ N2 ) ) ) ) ).

% central_binomial_lower_bound
thf(fact_3625_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [B: $tType,A: $tType,P: A > B > $o,X2: A,Y2: B,A2: product_prod @ A @ B] :
      ( ( P @ X2 @ Y2 )
     => ( ( A2
          = ( product_Pair @ A @ B @ X2 @ Y2 ) )
       => ( P @ ( product_fst @ A @ B @ A2 ) @ ( product_snd @ A @ B @ A2 ) ) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_3626_binomial__Suc__n,axiom,
    ! [N2: nat] :
      ( ( binomial @ ( suc @ N2 ) @ N2 )
      = ( suc @ N2 ) ) ).

% binomial_Suc_n
thf(fact_3627_binomial__n__n,axiom,
    ! [N2: nat] :
      ( ( binomial @ N2 @ N2 )
      = ( one_one @ nat ) ) ).

% binomial_n_n
thf(fact_3628_gbinomial__1,axiom,
    ! [A: $tType] :
      ( ( ( semiring_char_0 @ A )
        & ( semidom_divide @ A ) )
     => ! [A2: A] :
          ( ( gbinomial @ A @ A2 @ ( one_one @ nat ) )
          = A2 ) ) ).

% gbinomial_1
thf(fact_3629_binomial__0__Suc,axiom,
    ! [K: nat] :
      ( ( binomial @ ( zero_zero @ nat ) @ ( suc @ K ) )
      = ( zero_zero @ nat ) ) ).

% binomial_0_Suc
thf(fact_3630_binomial__1,axiom,
    ! [N2: nat] :
      ( ( binomial @ N2 @ ( suc @ ( zero_zero @ nat ) ) )
      = N2 ) ).

% binomial_1
thf(fact_3631_gbinomial__0_I2_J,axiom,
    ! [B: $tType] :
      ( ( ( semiring_char_0 @ B )
        & ( semidom_divide @ B ) )
     => ! [K: nat] :
          ( ( gbinomial @ B @ ( zero_zero @ B ) @ ( suc @ K ) )
          = ( zero_zero @ B ) ) ) ).

% gbinomial_0(2)
thf(fact_3632_binomial__eq__0__iff,axiom,
    ! [N2: nat,K: nat] :
      ( ( ( binomial @ N2 @ K )
        = ( zero_zero @ nat ) )
      = ( ord_less @ nat @ N2 @ K ) ) ).

% binomial_eq_0_iff
thf(fact_3633_binomial__Suc__Suc,axiom,
    ! [N2: nat,K: nat] :
      ( ( binomial @ ( suc @ N2 ) @ ( suc @ K ) )
      = ( plus_plus @ nat @ ( binomial @ N2 @ K ) @ ( binomial @ N2 @ ( suc @ K ) ) ) ) ).

% binomial_Suc_Suc
thf(fact_3634_gbinomial__0_I1_J,axiom,
    ! [A: $tType] :
      ( ( ( semiring_char_0 @ A )
        & ( semidom_divide @ A ) )
     => ! [A2: A] :
          ( ( gbinomial @ A @ A2 @ ( zero_zero @ nat ) )
          = ( one_one @ A ) ) ) ).

% gbinomial_0(1)
thf(fact_3635_binomial__n__0,axiom,
    ! [N2: nat] :
      ( ( binomial @ N2 @ ( zero_zero @ nat ) )
      = ( one_one @ nat ) ) ).

% binomial_n_0
thf(fact_3636_gbinomial__Suc0,axiom,
    ! [A: $tType] :
      ( ( ( semiring_char_0 @ A )
        & ( semidom_divide @ A ) )
     => ! [A2: A] :
          ( ( gbinomial @ A @ A2 @ ( suc @ ( zero_zero @ nat ) ) )
          = A2 ) ) ).

% gbinomial_Suc0
thf(fact_3637_prod__eq__1__iff,axiom,
    ! [A: $tType,A3: set @ A,F: A > nat] :
      ( ( finite_finite2 @ A @ A3 )
     => ( ( ( groups7121269368397514597t_prod @ A @ nat @ F @ A3 )
          = ( one_one @ nat ) )
        = ( ! [X: A] :
              ( ( member @ A @ X @ A3 )
             => ( ( F @ X )
                = ( one_one @ nat ) ) ) ) ) ) ).

% prod_eq_1_iff
thf(fact_3638_zero__less__binomial__iff,axiom,
    ! [N2: nat,K: nat] :
      ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( binomial @ N2 @ K ) )
      = ( ord_less_eq @ nat @ K @ N2 ) ) ).

% zero_less_binomial_iff
thf(fact_3639_prod__pos__nat__iff,axiom,
    ! [A: $tType,A3: set @ A,F: A > nat] :
      ( ( finite_finite2 @ A @ A3 )
     => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( groups7121269368397514597t_prod @ A @ nat @ F @ A3 ) )
        = ( ! [X: A] :
              ( ( member @ A @ X @ A3 )
             => ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F @ X ) ) ) ) ) ) ).

% prod_pos_nat_iff
thf(fact_3640_choose__one,axiom,
    ! [N2: nat] :
      ( ( binomial @ N2 @ ( one_one @ nat ) )
      = N2 ) ).

% choose_one
thf(fact_3641_int__prod,axiom,
    ! [B: $tType,F: B > nat,A3: set @ B] :
      ( ( semiring_1_of_nat @ int @ ( groups7121269368397514597t_prod @ B @ nat @ F @ A3 ) )
      = ( groups7121269368397514597t_prod @ B @ int
        @ ^ [X: B] : ( semiring_1_of_nat @ int @ ( F @ X ) )
        @ A3 ) ) ).

% int_prod
thf(fact_3642_binomial__eq__0,axiom,
    ! [N2: nat,K: nat] :
      ( ( ord_less @ nat @ N2 @ K )
     => ( ( binomial @ N2 @ K )
        = ( zero_zero @ nat ) ) ) ).

% binomial_eq_0
thf(fact_3643_Suc__times__binomial,axiom,
    ! [K: nat,N2: nat] :
      ( ( times_times @ nat @ ( suc @ K ) @ ( binomial @ ( suc @ N2 ) @ ( suc @ K ) ) )
      = ( times_times @ nat @ ( suc @ N2 ) @ ( binomial @ N2 @ K ) ) ) ).

% Suc_times_binomial
thf(fact_3644_Suc__times__binomial__eq,axiom,
    ! [N2: nat,K: nat] :
      ( ( times_times @ nat @ ( suc @ N2 ) @ ( binomial @ N2 @ K ) )
      = ( times_times @ nat @ ( binomial @ ( suc @ N2 ) @ ( suc @ K ) ) @ ( suc @ K ) ) ) ).

% Suc_times_binomial_eq
thf(fact_3645_binomial__symmetric,axiom,
    ! [K: nat,N2: nat] :
      ( ( ord_less_eq @ nat @ K @ N2 )
     => ( ( binomial @ N2 @ K )
        = ( binomial @ N2 @ ( minus_minus @ nat @ N2 @ K ) ) ) ) ).

% binomial_symmetric
thf(fact_3646_choose__mult__lemma,axiom,
    ! [M: nat,R4: nat,K: nat] :
      ( ( times_times @ nat @ ( binomial @ ( plus_plus @ nat @ ( plus_plus @ nat @ M @ R4 ) @ K ) @ ( plus_plus @ nat @ M @ K ) ) @ ( binomial @ ( plus_plus @ nat @ M @ K ) @ K ) )
      = ( times_times @ nat @ ( binomial @ ( plus_plus @ nat @ ( plus_plus @ nat @ M @ R4 ) @ K ) @ K ) @ ( binomial @ ( plus_plus @ nat @ M @ R4 ) @ M ) ) ) ).

% choose_mult_lemma
thf(fact_3647_binomial__le__pow,axiom,
    ! [R4: nat,N2: nat] :
      ( ( ord_less_eq @ nat @ R4 @ N2 )
     => ( ord_less_eq @ nat @ ( binomial @ N2 @ R4 ) @ ( power_power @ nat @ N2 @ R4 ) ) ) ).

% binomial_le_pow
thf(fact_3648_frac__ge__0,axiom,
    ! [A: $tType] :
      ( ( archim2362893244070406136eiling @ A )
     => ! [X2: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( archimedean_frac @ A @ X2 ) ) ) ).

% frac_ge_0
thf(fact_3649_frac__lt__1,axiom,
    ! [A: $tType] :
      ( ( archim2362893244070406136eiling @ A )
     => ! [X2: A] : ( ord_less @ A @ ( archimedean_frac @ A @ X2 ) @ ( one_one @ A ) ) ) ).

% frac_lt_1
thf(fact_3650_frac__1__eq,axiom,
    ! [A: $tType] :
      ( ( archim2362893244070406136eiling @ A )
     => ! [X2: A] :
          ( ( archimedean_frac @ A @ ( plus_plus @ A @ X2 @ ( one_one @ A ) ) )
          = ( archimedean_frac @ A @ X2 ) ) ) ).

% frac_1_eq
thf(fact_3651_zero__less__binomial,axiom,
    ! [K: nat,N2: nat] :
      ( ( ord_less_eq @ nat @ K @ N2 )
     => ( ord_less @ nat @ ( zero_zero @ nat ) @ ( binomial @ N2 @ K ) ) ) ).

% zero_less_binomial
thf(fact_3652_Suc__times__binomial__add,axiom,
    ! [A2: nat,B2: nat] :
      ( ( times_times @ nat @ ( suc @ A2 ) @ ( binomial @ ( suc @ ( plus_plus @ nat @ A2 @ B2 ) ) @ ( suc @ A2 ) ) )
      = ( times_times @ nat @ ( suc @ B2 ) @ ( binomial @ ( suc @ ( plus_plus @ nat @ A2 @ B2 ) ) @ A2 ) ) ) ).

% Suc_times_binomial_add
thf(fact_3653_binomial__Suc__Suc__eq__times,axiom,
    ! [N2: nat,K: nat] :
      ( ( binomial @ ( suc @ N2 ) @ ( suc @ K ) )
      = ( divide_divide @ nat @ ( times_times @ nat @ ( suc @ N2 ) @ ( binomial @ N2 @ K ) ) @ ( suc @ K ) ) ) ).

% binomial_Suc_Suc_eq_times
thf(fact_3654_choose__mult,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( ord_less_eq @ nat @ K @ M )
     => ( ( ord_less_eq @ nat @ M @ N2 )
       => ( ( times_times @ nat @ ( binomial @ N2 @ M ) @ ( binomial @ M @ K ) )
          = ( times_times @ nat @ ( binomial @ N2 @ K ) @ ( binomial @ ( minus_minus @ nat @ N2 @ K ) @ ( minus_minus @ nat @ M @ K ) ) ) ) ) ) ).

% choose_mult
thf(fact_3655_binomial__absorb__comp,axiom,
    ! [N2: nat,K: nat] :
      ( ( times_times @ nat @ ( minus_minus @ nat @ N2 @ K ) @ ( binomial @ N2 @ K ) )
      = ( times_times @ nat @ N2 @ ( binomial @ ( minus_minus @ nat @ N2 @ ( one_one @ nat ) ) @ K ) ) ) ).

% binomial_absorb_comp
thf(fact_3656_gbinomial__Suc__Suc,axiom,
    ! [A: $tType] :
      ( ( field_char_0 @ A )
     => ! [A2: A,K: nat] :
          ( ( gbinomial @ A @ ( plus_plus @ A @ A2 @ ( one_one @ A ) ) @ ( suc @ K ) )
          = ( plus_plus @ A @ ( gbinomial @ A @ A2 @ K ) @ ( gbinomial @ A @ A2 @ ( suc @ K ) ) ) ) ) ).

% gbinomial_Suc_Suc
thf(fact_3657_gbinomial__of__nat__symmetric,axiom,
    ! [A: $tType] :
      ( ( field_char_0 @ A )
     => ! [K: nat,N2: nat] :
          ( ( ord_less_eq @ nat @ K @ N2 )
         => ( ( gbinomial @ A @ ( semiring_1_of_nat @ A @ N2 ) @ K )
            = ( gbinomial @ A @ ( semiring_1_of_nat @ A @ N2 ) @ ( minus_minus @ nat @ N2 @ K ) ) ) ) ) ).

% gbinomial_of_nat_symmetric
thf(fact_3658_prod__int__eq,axiom,
    ! [I: nat,J: nat] :
      ( ( groups7121269368397514597t_prod @ nat @ int @ ( semiring_1_of_nat @ int ) @ ( set_or1337092689740270186AtMost @ nat @ I @ J ) )
      = ( groups7121269368397514597t_prod @ int @ int
        @ ^ [X: int] : X
        @ ( set_or1337092689740270186AtMost @ int @ ( semiring_1_of_nat @ int @ I ) @ ( semiring_1_of_nat @ int @ J ) ) ) ) ).

% prod_int_eq
thf(fact_3659_small__lazy_H_Ocases,axiom,
    ! [X2: product_prod @ int @ int] :
      ~ ! [D4: int,I3: int] :
          ( X2
         != ( product_Pair @ int @ int @ D4 @ I3 ) ) ).

% small_lazy'.cases
thf(fact_3660_binomial__absorption,axiom,
    ! [K: nat,N2: nat] :
      ( ( times_times @ nat @ ( suc @ K ) @ ( binomial @ N2 @ ( suc @ K ) ) )
      = ( times_times @ nat @ N2 @ ( binomial @ ( minus_minus @ nat @ N2 @ ( one_one @ nat ) ) @ K ) ) ) ).

% binomial_absorption
thf(fact_3661_gbinomial__addition__formula,axiom,
    ! [A: $tType] :
      ( ( field_char_0 @ A )
     => ! [A2: A,K: nat] :
          ( ( gbinomial @ A @ A2 @ ( suc @ K ) )
          = ( plus_plus @ A @ ( gbinomial @ A @ ( minus_minus @ A @ A2 @ ( one_one @ A ) ) @ ( suc @ K ) ) @ ( gbinomial @ A @ ( minus_minus @ A @ A2 @ ( one_one @ A ) ) @ K ) ) ) ) ).

% gbinomial_addition_formula
thf(fact_3662_gbinomial__absorb__comp,axiom,
    ! [A: $tType] :
      ( ( field_char_0 @ A )
     => ! [A2: A,K: nat] :
          ( ( times_times @ A @ ( minus_minus @ A @ A2 @ ( semiring_1_of_nat @ A @ K ) ) @ ( gbinomial @ A @ A2 @ K ) )
          = ( times_times @ A @ A2 @ ( gbinomial @ A @ ( minus_minus @ A @ A2 @ ( one_one @ A ) ) @ K ) ) ) ) ).

% gbinomial_absorb_comp
thf(fact_3663_gbinomial__mult__1,axiom,
    ! [A: $tType] :
      ( ( field_char_0 @ A )
     => ! [A2: A,K: nat] :
          ( ( times_times @ A @ A2 @ ( gbinomial @ A @ A2 @ K ) )
          = ( plus_plus @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ K ) @ ( gbinomial @ A @ A2 @ K ) ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ ( suc @ K ) ) @ ( gbinomial @ A @ A2 @ ( suc @ K ) ) ) ) ) ) ).

% gbinomial_mult_1
thf(fact_3664_gbinomial__mult__1_H,axiom,
    ! [A: $tType] :
      ( ( field_char_0 @ A )
     => ! [A2: A,K: nat] :
          ( ( times_times @ A @ ( gbinomial @ A @ A2 @ K ) @ A2 )
          = ( plus_plus @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ K ) @ ( gbinomial @ A @ A2 @ K ) ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ ( suc @ K ) ) @ ( gbinomial @ A @ A2 @ ( suc @ K ) ) ) ) ) ) ).

% gbinomial_mult_1'
thf(fact_3665_gbinomial__ge__n__over__k__pow__k,axiom,
    ! [A: $tType] :
      ( ( linordered_field @ A )
     => ! [K: nat,A2: A] :
          ( ( ord_less_eq @ A @ ( semiring_1_of_nat @ A @ K ) @ A2 )
         => ( ord_less_eq @ A @ ( power_power @ A @ ( divide_divide @ A @ A2 @ ( semiring_1_of_nat @ A @ K ) ) @ K ) @ ( gbinomial @ A @ A2 @ K ) ) ) ) ).

% gbinomial_ge_n_over_k_pow_k
thf(fact_3666_frac__def,axiom,
    ! [A: $tType] :
      ( ( archim2362893244070406136eiling @ A )
     => ( ( archimedean_frac @ A )
        = ( ^ [X: A] : ( minus_minus @ A @ X @ ( ring_1_of_int @ A @ ( archim6421214686448440834_floor @ A @ X ) ) ) ) ) ) ).

% frac_def
thf(fact_3667_prod__int__plus__eq,axiom,
    ! [I: nat,J: nat] :
      ( ( groups7121269368397514597t_prod @ nat @ int @ ( semiring_1_of_nat @ int ) @ ( set_or1337092689740270186AtMost @ nat @ I @ ( plus_plus @ nat @ I @ J ) ) )
      = ( groups7121269368397514597t_prod @ int @ int
        @ ^ [X: int] : X
        @ ( set_or1337092689740270186AtMost @ int @ ( semiring_1_of_nat @ int @ I ) @ ( semiring_1_of_nat @ int @ ( plus_plus @ nat @ I @ J ) ) ) ) ) ).

% prod_int_plus_eq
thf(fact_3668_binomial__ge__n__over__k__pow__k,axiom,
    ! [A: $tType] :
      ( ( linordered_field @ A )
     => ! [K: nat,N2: nat] :
          ( ( ord_less_eq @ nat @ K @ N2 )
         => ( ord_less_eq @ A @ ( power_power @ A @ ( divide_divide @ A @ ( semiring_1_of_nat @ A @ N2 ) @ ( semiring_1_of_nat @ A @ K ) ) @ K ) @ ( semiring_1_of_nat @ A @ ( binomial @ N2 @ K ) ) ) ) ) ).

% binomial_ge_n_over_k_pow_k
thf(fact_3669_binomial__maximum,axiom,
    ! [N2: nat,K: nat] : ( ord_less_eq @ nat @ ( binomial @ N2 @ K ) @ ( binomial @ N2 @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).

% binomial_maximum
thf(fact_3670_binomial__antimono,axiom,
    ! [K: nat,K2: nat,N2: nat] :
      ( ( ord_less_eq @ nat @ K @ K2 )
     => ( ( ord_less_eq @ nat @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ K )
       => ( ( ord_less_eq @ nat @ K2 @ N2 )
         => ( ord_less_eq @ nat @ ( binomial @ N2 @ K2 ) @ ( binomial @ N2 @ K ) ) ) ) ) ).

% binomial_antimono
thf(fact_3671_binomial__mono,axiom,
    ! [K: nat,K2: nat,N2: nat] :
      ( ( ord_less_eq @ nat @ K @ K2 )
     => ( ( ord_less_eq @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ K2 ) @ N2 )
       => ( ord_less_eq @ nat @ ( binomial @ N2 @ K ) @ ( binomial @ N2 @ K2 ) ) ) ) ).

% binomial_mono
thf(fact_3672_binomial__maximum_H,axiom,
    ! [N2: nat,K: nat] : ( ord_less_eq @ nat @ ( binomial @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) @ K ) @ ( binomial @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) @ N2 ) ) ).

% binomial_maximum'
thf(fact_3673_binomial__le__pow2,axiom,
    ! [N2: nat,K: nat] : ( ord_less_eq @ nat @ ( binomial @ N2 @ K ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ).

% binomial_le_pow2
thf(fact_3674_choose__reduce__nat,axiom,
    ! [N2: nat,K: nat] :
      ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
     => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
       => ( ( binomial @ N2 @ K )
          = ( plus_plus @ nat @ ( binomial @ ( minus_minus @ nat @ N2 @ ( one_one @ nat ) ) @ ( minus_minus @ nat @ K @ ( one_one @ nat ) ) ) @ ( binomial @ ( minus_minus @ nat @ N2 @ ( one_one @ nat ) ) @ K ) ) ) ) ) ).

% choose_reduce_nat
thf(fact_3675_times__binomial__minus1__eq,axiom,
    ! [K: nat,N2: nat] :
      ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
     => ( ( times_times @ nat @ K @ ( binomial @ N2 @ K ) )
        = ( times_times @ nat @ N2 @ ( binomial @ ( minus_minus @ nat @ N2 @ ( one_one @ nat ) ) @ ( minus_minus @ nat @ K @ ( one_one @ nat ) ) ) ) ) ) ).

% times_binomial_minus1_eq
thf(fact_3676_subset__Collect__iff,axiom,
    ! [A: $tType,B6: set @ A,A3: set @ A,P: A > $o] :
      ( ( ord_less_eq @ ( set @ A ) @ B6 @ A3 )
     => ( ( ord_less_eq @ ( set @ A ) @ B6
          @ ( collect @ A
            @ ^ [X: A] :
                ( ( member @ A @ X @ A3 )
                & ( P @ X ) ) ) )
        = ( ! [X: A] :
              ( ( member @ A @ X @ B6 )
             => ( P @ X ) ) ) ) ) ).

% subset_Collect_iff
thf(fact_3677_subset__CollectI,axiom,
    ! [A: $tType,B6: set @ A,A3: set @ A,Q: A > $o,P: A > $o] :
      ( ( ord_less_eq @ ( set @ A ) @ B6 @ A3 )
     => ( ! [X3: A] :
            ( ( member @ A @ X3 @ B6 )
           => ( ( Q @ X3 )
             => ( P @ X3 ) ) )
       => ( ord_less_eq @ ( set @ A )
          @ ( collect @ A
            @ ^ [X: A] :
                ( ( member @ A @ X @ B6 )
                & ( Q @ X ) ) )
          @ ( collect @ A
            @ ^ [X: A] :
                ( ( member @ A @ X @ A3 )
                & ( P @ X ) ) ) ) ) ) ).

% subset_CollectI
thf(fact_3678_Suc__times__gbinomial,axiom,
    ! [A: $tType] :
      ( ( field_char_0 @ A )
     => ! [K: nat,A2: A] :
          ( ( times_times @ A @ ( semiring_1_of_nat @ A @ ( suc @ K ) ) @ ( gbinomial @ A @ ( plus_plus @ A @ A2 @ ( one_one @ A ) ) @ ( suc @ K ) ) )
          = ( times_times @ A @ ( plus_plus @ A @ A2 @ ( one_one @ A ) ) @ ( gbinomial @ A @ A2 @ K ) ) ) ) ).

% Suc_times_gbinomial
thf(fact_3679_gbinomial__absorption,axiom,
    ! [A: $tType] :
      ( ( field_char_0 @ A )
     => ! [K: nat,A2: A] :
          ( ( times_times @ A @ ( semiring_1_of_nat @ A @ ( suc @ K ) ) @ ( gbinomial @ A @ A2 @ ( suc @ K ) ) )
          = ( times_times @ A @ A2 @ ( gbinomial @ A @ ( minus_minus @ A @ A2 @ ( one_one @ A ) ) @ K ) ) ) ) ).

% gbinomial_absorption
thf(fact_3680_gbinomial__trinomial__revision,axiom,
    ! [A: $tType] :
      ( ( field_char_0 @ A )
     => ! [K: nat,M: nat,A2: A] :
          ( ( ord_less_eq @ nat @ K @ M )
         => ( ( times_times @ A @ ( gbinomial @ A @ A2 @ M ) @ ( gbinomial @ A @ ( semiring_1_of_nat @ A @ M ) @ K ) )
            = ( times_times @ A @ ( gbinomial @ A @ A2 @ K ) @ ( gbinomial @ A @ ( minus_minus @ A @ A2 @ ( semiring_1_of_nat @ A @ K ) ) @ ( minus_minus @ nat @ M @ K ) ) ) ) ) ) ).

% gbinomial_trinomial_revision
thf(fact_3681_frac__eq,axiom,
    ! [A: $tType] :
      ( ( archim2362893244070406136eiling @ A )
     => ! [X2: A] :
          ( ( ( archimedean_frac @ A @ X2 )
            = X2 )
          = ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X2 )
            & ( ord_less @ A @ X2 @ ( one_one @ A ) ) ) ) ) ).

% frac_eq
thf(fact_3682_frac__add,axiom,
    ! [A: $tType] :
      ( ( archim2362893244070406136eiling @ A )
     => ! [X2: A,Y2: A] :
          ( ( ( ord_less @ A @ ( plus_plus @ A @ ( archimedean_frac @ A @ X2 ) @ ( archimedean_frac @ A @ Y2 ) ) @ ( one_one @ A ) )
           => ( ( archimedean_frac @ A @ ( plus_plus @ A @ X2 @ Y2 ) )
              = ( plus_plus @ A @ ( archimedean_frac @ A @ X2 ) @ ( archimedean_frac @ A @ Y2 ) ) ) )
          & ( ~ ( ord_less @ A @ ( plus_plus @ A @ ( archimedean_frac @ A @ X2 ) @ ( archimedean_frac @ A @ Y2 ) ) @ ( one_one @ A ) )
           => ( ( archimedean_frac @ A @ ( plus_plus @ A @ X2 @ Y2 ) )
              = ( minus_minus @ A @ ( plus_plus @ A @ ( archimedean_frac @ A @ X2 ) @ ( archimedean_frac @ A @ Y2 ) ) @ ( one_one @ A ) ) ) ) ) ) ).

% frac_add
thf(fact_3683_binomial__less__binomial__Suc,axiom,
    ! [K: nat,N2: nat] :
      ( ( ord_less @ nat @ K @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
     => ( ord_less @ nat @ ( binomial @ N2 @ K ) @ ( binomial @ N2 @ ( suc @ K ) ) ) ) ).

% binomial_less_binomial_Suc
thf(fact_3684_binomial__strict__mono,axiom,
    ! [K: nat,K2: nat,N2: nat] :
      ( ( ord_less @ nat @ K @ K2 )
     => ( ( ord_less_eq @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ K2 ) @ N2 )
       => ( ord_less @ nat @ ( binomial @ N2 @ K ) @ ( binomial @ N2 @ K2 ) ) ) ) ).

% binomial_strict_mono
thf(fact_3685_binomial__strict__antimono,axiom,
    ! [K: nat,K2: nat,N2: nat] :
      ( ( ord_less @ nat @ K @ K2 )
     => ( ( ord_less_eq @ nat @ N2 @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ K ) )
       => ( ( ord_less_eq @ nat @ K2 @ N2 )
         => ( ord_less @ nat @ ( binomial @ N2 @ K2 ) @ ( binomial @ N2 @ K ) ) ) ) ) ).

% binomial_strict_antimono
thf(fact_3686_central__binomial__odd,axiom,
    ! [N2: nat] :
      ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 )
     => ( ( binomial @ N2 @ ( suc @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
        = ( binomial @ N2 @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).

% central_binomial_odd
thf(fact_3687_binomial__addition__formula,axiom,
    ! [N2: nat,K: nat] :
      ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
     => ( ( binomial @ N2 @ ( suc @ K ) )
        = ( plus_plus @ nat @ ( binomial @ ( minus_minus @ nat @ N2 @ ( one_one @ nat ) ) @ ( suc @ K ) ) @ ( binomial @ ( minus_minus @ nat @ N2 @ ( one_one @ nat ) ) @ K ) ) ) ) ).

% binomial_addition_formula
thf(fact_3688_gbinomial__rec,axiom,
    ! [A: $tType] :
      ( ( field_char_0 @ A )
     => ! [A2: A,K: nat] :
          ( ( gbinomial @ A @ ( plus_plus @ A @ A2 @ ( one_one @ A ) ) @ ( suc @ K ) )
          = ( times_times @ A @ ( gbinomial @ A @ A2 @ K ) @ ( divide_divide @ A @ ( plus_plus @ A @ A2 @ ( one_one @ A ) ) @ ( semiring_1_of_nat @ A @ ( suc @ K ) ) ) ) ) ) ).

% gbinomial_rec
thf(fact_3689_gbinomial__factors,axiom,
    ! [A: $tType] :
      ( ( field_char_0 @ A )
     => ! [A2: A,K: nat] :
          ( ( gbinomial @ A @ ( plus_plus @ A @ A2 @ ( one_one @ A ) ) @ ( suc @ K ) )
          = ( times_times @ A @ ( divide_divide @ A @ ( plus_plus @ A @ A2 @ ( one_one @ A ) ) @ ( semiring_1_of_nat @ A @ ( suc @ K ) ) ) @ ( gbinomial @ A @ A2 @ K ) ) ) ) ).

% gbinomial_factors
thf(fact_3690_choose__two,axiom,
    ! [N2: nat] :
      ( ( binomial @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
      = ( divide_divide @ nat @ ( times_times @ nat @ N2 @ ( minus_minus @ nat @ N2 @ ( one_one @ nat ) ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ).

% choose_two
thf(fact_3691_gbinomial__reduce__nat,axiom,
    ! [A: $tType] :
      ( ( field_char_0 @ A )
     => ! [K: nat,A2: A] :
          ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
         => ( ( gbinomial @ A @ A2 @ K )
            = ( plus_plus @ A @ ( gbinomial @ A @ ( minus_minus @ A @ A2 @ ( one_one @ A ) ) @ ( minus_minus @ nat @ K @ ( one_one @ nat ) ) ) @ ( gbinomial @ A @ ( minus_minus @ A @ A2 @ ( one_one @ A ) ) @ K ) ) ) ) ) ).

% gbinomial_reduce_nat
thf(fact_3692_floor__add,axiom,
    ! [A: $tType] :
      ( ( archim2362893244070406136eiling @ A )
     => ! [X2: A,Y2: A] :
          ( ( ( ord_less @ A @ ( plus_plus @ A @ ( archimedean_frac @ A @ X2 ) @ ( archimedean_frac @ A @ Y2 ) ) @ ( one_one @ A ) )
           => ( ( archim6421214686448440834_floor @ A @ ( plus_plus @ A @ X2 @ Y2 ) )
              = ( plus_plus @ int @ ( archim6421214686448440834_floor @ A @ X2 ) @ ( archim6421214686448440834_floor @ A @ Y2 ) ) ) )
          & ( ~ ( ord_less @ A @ ( plus_plus @ A @ ( archimedean_frac @ A @ X2 ) @ ( archimedean_frac @ A @ Y2 ) ) @ ( one_one @ A ) )
           => ( ( archim6421214686448440834_floor @ A @ ( plus_plus @ A @ X2 @ Y2 ) )
              = ( plus_plus @ int @ ( plus_plus @ int @ ( archim6421214686448440834_floor @ A @ X2 ) @ ( archim6421214686448440834_floor @ A @ Y2 ) ) @ ( one_one @ int ) ) ) ) ) ) ).

% floor_add
thf(fact_3693_gbinomial__sum__up__index,axiom,
    ! [A: $tType] :
      ( ( field_char_0 @ A )
     => ! [K: nat,N2: nat] :
          ( ( groups7311177749621191930dd_sum @ nat @ A
            @ ^ [J3: nat] : ( gbinomial @ A @ ( semiring_1_of_nat @ A @ J3 ) @ K )
            @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N2 ) )
          = ( gbinomial @ A @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ N2 ) @ ( one_one @ A ) ) @ ( plus_plus @ nat @ K @ ( one_one @ nat ) ) ) ) ) ).

% gbinomial_sum_up_index
thf(fact_3694_gbinomial__absorption_H,axiom,
    ! [A: $tType] :
      ( ( field_char_0 @ A )
     => ! [K: nat,A2: A] :
          ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
         => ( ( gbinomial @ A @ A2 @ K )
            = ( times_times @ A @ ( divide_divide @ A @ A2 @ ( semiring_1_of_nat @ A @ K ) ) @ ( gbinomial @ A @ ( minus_minus @ A @ A2 @ ( one_one @ A ) ) @ ( minus_minus @ nat @ K @ ( one_one @ nat ) ) ) ) ) ) ) ).

% gbinomial_absorption'
thf(fact_3695_choose__odd__sum,axiom,
    ! [A: $tType] :
      ( ( comm_ring_1 @ A )
     => ! [N2: nat] :
          ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
         => ( ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) )
              @ ( groups7311177749621191930dd_sum @ nat @ A
                @ ^ [I2: nat] :
                    ( if @ A
                    @ ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I2 )
                    @ ( semiring_1_of_nat @ A @ ( binomial @ N2 @ I2 ) )
                    @ ( zero_zero @ A ) )
                @ ( set_ord_atMost @ nat @ N2 ) ) )
            = ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) ) ) ) ).

% choose_odd_sum
thf(fact_3696_choose__even__sum,axiom,
    ! [A: $tType] :
      ( ( comm_ring_1 @ A )
     => ! [N2: nat] :
          ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
         => ( ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) )
              @ ( groups7311177749621191930dd_sum @ nat @ A
                @ ^ [I2: nat] : ( if @ A @ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I2 ) @ ( semiring_1_of_nat @ A @ ( binomial @ N2 @ I2 ) ) @ ( zero_zero @ A ) )
                @ ( set_ord_atMost @ nat @ N2 ) ) )
            = ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) ) ) ) ).

% choose_even_sum
thf(fact_3697_gbinomial__partial__row__sum,axiom,
    ! [A: $tType] :
      ( ( field_char_0 @ A )
     => ! [A2: A,M: nat] :
          ( ( groups7311177749621191930dd_sum @ nat @ A
            @ ^ [K4: nat] : ( times_times @ A @ ( gbinomial @ A @ A2 @ K4 ) @ ( minus_minus @ A @ ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( semiring_1_of_nat @ A @ K4 ) ) )
            @ ( set_ord_atMost @ nat @ M ) )
          = ( times_times @ A @ ( divide_divide @ A @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ M ) @ ( one_one @ A ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( gbinomial @ A @ A2 @ ( plus_plus @ nat @ M @ ( one_one @ nat ) ) ) ) ) ) ).

% gbinomial_partial_row_sum
thf(fact_3698_gbinomial__r__part__sum,axiom,
    ! [A: $tType] :
      ( ( field_char_0 @ A )
     => ! [M: nat] :
          ( ( groups7311177749621191930dd_sum @ nat @ A @ ( gbinomial @ A @ ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ A @ M ) ) @ ( one_one @ A ) ) ) @ ( set_ord_atMost @ nat @ M ) )
          = ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) ) ) ) ).

% gbinomial_r_part_sum
thf(fact_3699_atMost__iff,axiom,
    ! [A: $tType] :
      ( ( ord @ A )
     => ! [I: A,K: A] :
          ( ( member @ A @ I @ ( set_ord_atMost @ A @ K ) )
          = ( ord_less_eq @ A @ I @ K ) ) ) ).

% atMost_iff
thf(fact_3700_atMost__subset__iff,axiom,
    ! [A: $tType] :
      ( ( preorder @ A )
     => ! [X2: A,Y2: A] :
          ( ( ord_less_eq @ ( set @ A ) @ ( set_ord_atMost @ A @ X2 ) @ ( set_ord_atMost @ A @ Y2 ) )
          = ( ord_less_eq @ A @ X2 @ Y2 ) ) ) ).

% atMost_subset_iff
thf(fact_3701_Icc__subset__Iic__iff,axiom,
    ! [A: $tType] :
      ( ( preorder @ A )
     => ! [L: A,H2: A,H4: A] :
          ( ( ord_less_eq @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ L @ H2 ) @ ( set_ord_atMost @ A @ H4 ) )
          = ( ~ ( ord_less_eq @ A @ L @ H2 )
            | ( ord_less_eq @ A @ H2 @ H4 ) ) ) ) ).

% Icc_subset_Iic_iff
thf(fact_3702_sum_OatMost__Suc,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add @ A )
     => ! [G: nat > A,N2: nat] :
          ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_ord_atMost @ nat @ ( suc @ N2 ) ) )
          = ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_ord_atMost @ nat @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ) ).

% sum.atMost_Suc
thf(fact_3703_prod_OatMost__Suc,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_mult @ A )
     => ! [G: nat > A,N2: nat] :
          ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_ord_atMost @ nat @ ( suc @ N2 ) ) )
          = ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_ord_atMost @ nat @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ) ).

% prod.atMost_Suc
thf(fact_3704_atMost__def,axiom,
    ! [A: $tType] :
      ( ( ord @ A )
     => ( ( set_ord_atMost @ A )
        = ( ^ [U2: A] :
              ( collect @ A
              @ ^ [X: A] : ( ord_less_eq @ A @ X @ U2 ) ) ) ) ) ).

% atMost_def
thf(fact_3705_lessThan__Suc__atMost,axiom,
    ! [K: nat] :
      ( ( set_ord_lessThan @ nat @ ( suc @ K ) )
      = ( set_ord_atMost @ nat @ K ) ) ).

% lessThan_Suc_atMost
thf(fact_3706_atMost__Suc,axiom,
    ! [K: nat] :
      ( ( set_ord_atMost @ nat @ ( suc @ K ) )
      = ( insert @ nat @ ( suc @ K ) @ ( set_ord_atMost @ nat @ K ) ) ) ).

% atMost_Suc
thf(fact_3707_not__UNIV__le__Iic,axiom,
    ! [A: $tType] :
      ( ( no_top @ A )
     => ! [H2: A] :
          ~ ( ord_less_eq @ ( set @ A ) @ ( top_top @ ( set @ A ) ) @ ( set_ord_atMost @ A @ H2 ) ) ) ).

% not_UNIV_le_Iic
thf(fact_3708_not__Iic__le__Icc,axiom,
    ! [A: $tType] :
      ( ( no_bot @ A )
     => ! [H2: A,L4: A,H4: A] :
          ~ ( ord_less_eq @ ( set @ A ) @ ( set_ord_atMost @ A @ H2 ) @ ( set_or1337092689740270186AtMost @ A @ L4 @ H4 ) ) ) ).

% not_Iic_le_Icc
thf(fact_3709_finite__nat__iff__bounded__le,axiom,
    ( ( finite_finite2 @ nat )
    = ( ^ [S8: set @ nat] :
        ? [K4: nat] : ( ord_less_eq @ ( set @ nat ) @ S8 @ ( set_ord_atMost @ nat @ K4 ) ) ) ) ).

% finite_nat_iff_bounded_le
thf(fact_3710_Iic__subset__Iio__iff,axiom,
    ! [A: $tType] :
      ( ( linorder @ A )
     => ! [A2: A,B2: A] :
          ( ( ord_less_eq @ ( set @ A ) @ ( set_ord_atMost @ A @ A2 ) @ ( set_ord_lessThan @ A @ B2 ) )
          = ( ord_less @ A @ A2 @ B2 ) ) ) ).

% Iic_subset_Iio_iff
thf(fact_3711_sum__choose__upper,axiom,
    ! [M: nat,N2: nat] :
      ( ( groups7311177749621191930dd_sum @ nat @ nat
        @ ^ [K4: nat] : ( binomial @ K4 @ M )
        @ ( set_ord_atMost @ nat @ N2 ) )
      = ( binomial @ ( suc @ N2 ) @ ( suc @ M ) ) ) ).

% sum_choose_upper
thf(fact_3712_sum_OatMost__Suc__shift,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add @ A )
     => ! [G: nat > A,N2: nat] :
          ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_ord_atMost @ nat @ ( suc @ N2 ) ) )
          = ( plus_plus @ A @ ( G @ ( zero_zero @ nat ) )
            @ ( groups7311177749621191930dd_sum @ nat @ A
              @ ^ [I2: nat] : ( G @ ( suc @ I2 ) )
              @ ( set_ord_atMost @ nat @ N2 ) ) ) ) ) ).

% sum.atMost_Suc_shift
thf(fact_3713_sum__telescope,axiom,
    ! [A: $tType] :
      ( ( ab_group_add @ A )
     => ! [F: nat > A,I: nat] :
          ( ( groups7311177749621191930dd_sum @ nat @ A
            @ ^ [I2: nat] : ( minus_minus @ A @ ( F @ I2 ) @ ( F @ ( suc @ I2 ) ) )
            @ ( set_ord_atMost @ nat @ I ) )
          = ( minus_minus @ A @ ( F @ ( zero_zero @ nat ) ) @ ( F @ ( suc @ I ) ) ) ) ) ).

% sum_telescope
thf(fact_3714_polyfun__eq__coeffs,axiom,
    ! [A: $tType] :
      ( ( ( real_V8999393235501362500lgebra @ A )
        & ( idom @ A ) )
     => ! [C2: nat > A,N2: nat,D2: nat > A] :
          ( ( ! [X: A] :
                ( ( groups7311177749621191930dd_sum @ nat @ A
                  @ ^ [I2: nat] : ( times_times @ A @ ( C2 @ I2 ) @ ( power_power @ A @ X @ I2 ) )
                  @ ( set_ord_atMost @ nat @ N2 ) )
                = ( groups7311177749621191930dd_sum @ nat @ A
                  @ ^ [I2: nat] : ( times_times @ A @ ( D2 @ I2 ) @ ( power_power @ A @ X @ I2 ) )
                  @ ( set_ord_atMost @ nat @ N2 ) ) ) )
          = ( ! [I2: nat] :
                ( ( ord_less_eq @ nat @ I2 @ N2 )
               => ( ( C2 @ I2 )
                  = ( D2 @ I2 ) ) ) ) ) ) ).

% polyfun_eq_coeffs
thf(fact_3715_prod_OatMost__Suc__shift,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_mult @ A )
     => ! [G: nat > A,N2: nat] :
          ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_ord_atMost @ nat @ ( suc @ N2 ) ) )
          = ( times_times @ A @ ( G @ ( zero_zero @ nat ) )
            @ ( groups7121269368397514597t_prod @ nat @ A
              @ ^ [I2: nat] : ( G @ ( suc @ I2 ) )
              @ ( set_ord_atMost @ nat @ N2 ) ) ) ) ) ).

% prod.atMost_Suc_shift
thf(fact_3716_sum_Onested__swap_H,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add @ A )
     => ! [A2: nat > nat > A,N2: nat] :
          ( ( groups7311177749621191930dd_sum @ nat @ A
            @ ^ [I2: nat] : ( groups7311177749621191930dd_sum @ nat @ A @ ( A2 @ I2 ) @ ( set_ord_lessThan @ nat @ I2 ) )
            @ ( set_ord_atMost @ nat @ N2 ) )
          = ( groups7311177749621191930dd_sum @ nat @ A
            @ ^ [J3: nat] :
                ( groups7311177749621191930dd_sum @ nat @ A
                @ ^ [I2: nat] : ( A2 @ I2 @ J3 )
                @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ J3 ) @ N2 ) )
            @ ( set_ord_lessThan @ nat @ N2 ) ) ) ) ).

% sum.nested_swap'
thf(fact_3717_prod_Onested__swap_H,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_mult @ A )
     => ! [A2: nat > nat > A,N2: nat] :
          ( ( groups7121269368397514597t_prod @ nat @ A
            @ ^ [I2: nat] : ( groups7121269368397514597t_prod @ nat @ A @ ( A2 @ I2 ) @ ( set_ord_lessThan @ nat @ I2 ) )
            @ ( set_ord_atMost @ nat @ N2 ) )
          = ( groups7121269368397514597t_prod @ nat @ A
            @ ^ [J3: nat] :
                ( groups7121269368397514597t_prod @ nat @ A
                @ ^ [I2: nat] : ( A2 @ I2 @ J3 )
                @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ J3 ) @ N2 ) )
            @ ( set_ord_lessThan @ nat @ N2 ) ) ) ) ).

% prod.nested_swap'
thf(fact_3718_sum__choose__lower,axiom,
    ! [R4: nat,N2: nat] :
      ( ( groups7311177749621191930dd_sum @ nat @ nat
        @ ^ [K4: nat] : ( binomial @ ( plus_plus @ nat @ R4 @ K4 ) @ K4 )
        @ ( set_ord_atMost @ nat @ N2 ) )
      = ( binomial @ ( suc @ ( plus_plus @ nat @ R4 @ N2 ) ) @ N2 ) ) ).

% sum_choose_lower
thf(fact_3719_choose__rising__sum_I2_J,axiom,
    ! [N2: nat,M: nat] :
      ( ( groups7311177749621191930dd_sum @ nat @ nat
        @ ^ [J3: nat] : ( binomial @ ( plus_plus @ nat @ N2 @ J3 ) @ N2 )
        @ ( set_ord_atMost @ nat @ M ) )
      = ( binomial @ ( plus_plus @ nat @ ( plus_plus @ nat @ N2 @ M ) @ ( one_one @ nat ) ) @ M ) ) ).

% choose_rising_sum(2)
thf(fact_3720_choose__rising__sum_I1_J,axiom,
    ! [N2: nat,M: nat] :
      ( ( groups7311177749621191930dd_sum @ nat @ nat
        @ ^ [J3: nat] : ( binomial @ ( plus_plus @ nat @ N2 @ J3 ) @ N2 )
        @ ( set_ord_atMost @ nat @ M ) )
      = ( binomial @ ( plus_plus @ nat @ ( plus_plus @ nat @ N2 @ M ) @ ( one_one @ nat ) ) @ ( plus_plus @ nat @ N2 @ ( one_one @ nat ) ) ) ) ).

% choose_rising_sum(1)
thf(fact_3721_zero__polynom__imp__zero__coeffs,axiom,
    ! [A: $tType] :
      ( ( ( ab_semigroup_mult @ A )
        & ( real_V8999393235501362500lgebra @ A ) )
     => ! [C2: nat > A,N2: nat,K: nat] :
          ( ! [W3: A] :
              ( ( groups7311177749621191930dd_sum @ nat @ A
                @ ^ [I2: nat] : ( times_times @ A @ ( C2 @ I2 ) @ ( power_power @ A @ W3 @ I2 ) )
                @ ( set_ord_atMost @ nat @ N2 ) )
              = ( zero_zero @ A ) )
         => ( ( ord_less_eq @ nat @ K @ N2 )
           => ( ( C2 @ K )
              = ( zero_zero @ A ) ) ) ) ) ).

% zero_polynom_imp_zero_coeffs
thf(fact_3722_polyfun__eq__0,axiom,
    ! [A: $tType] :
      ( ( ( real_V8999393235501362500lgebra @ A )
        & ( idom @ A ) )
     => ! [C2: nat > A,N2: nat] :
          ( ( ! [X: A] :
                ( ( groups7311177749621191930dd_sum @ nat @ A
                  @ ^ [I2: nat] : ( times_times @ A @ ( C2 @ I2 ) @ ( power_power @ A @ X @ I2 ) )
                  @ ( set_ord_atMost @ nat @ N2 ) )
                = ( zero_zero @ A ) ) )
          = ( ! [I2: nat] :
                ( ( ord_less_eq @ nat @ I2 @ N2 )
               => ( ( C2 @ I2 )
                  = ( zero_zero @ A ) ) ) ) ) ) ).

% polyfun_eq_0
thf(fact_3723_sum_OatMost__shift,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add @ A )
     => ! [G: nat > A,N2: nat] :
          ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_ord_atMost @ nat @ N2 ) )
          = ( plus_plus @ A @ ( G @ ( zero_zero @ nat ) )
            @ ( groups7311177749621191930dd_sum @ nat @ A
              @ ^ [I2: nat] : ( G @ ( suc @ I2 ) )
              @ ( set_ord_lessThan @ nat @ N2 ) ) ) ) ) ).

% sum.atMost_shift
thf(fact_3724_sum__up__index__split,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add @ A )
     => ! [F: nat > A,M: nat,N2: nat] :
          ( ( groups7311177749621191930dd_sum @ nat @ A @ F @ ( set_ord_atMost @ nat @ ( plus_plus @ nat @ M @ N2 ) ) )
          = ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F @ ( set_ord_atMost @ nat @ M ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ F @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M ) @ ( plus_plus @ nat @ M @ N2 ) ) ) ) ) ) ).

% sum_up_index_split
thf(fact_3725_prod_OatMost__shift,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_mult @ A )
     => ! [G: nat > A,N2: nat] :
          ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_ord_atMost @ nat @ N2 ) )
          = ( times_times @ A @ ( G @ ( zero_zero @ nat ) )
            @ ( groups7121269368397514597t_prod @ nat @ A
              @ ^ [I2: nat] : ( G @ ( suc @ I2 ) )
              @ ( set_ord_lessThan @ nat @ N2 ) ) ) ) ) ).

% prod.atMost_shift
thf(fact_3726_gbinomial__parallel__sum,axiom,
    ! [A: $tType] :
      ( ( field_char_0 @ A )
     => ! [A2: A,N2: nat] :
          ( ( groups7311177749621191930dd_sum @ nat @ A
            @ ^ [K4: nat] : ( gbinomial @ A @ ( plus_plus @ A @ A2 @ ( semiring_1_of_nat @ A @ K4 ) ) @ K4 )
            @ ( set_ord_atMost @ nat @ N2 ) )
          = ( gbinomial @ A @ ( plus_plus @ A @ ( plus_plus @ A @ A2 @ ( semiring_1_of_nat @ A @ N2 ) ) @ ( one_one @ A ) ) @ N2 ) ) ) ).

% gbinomial_parallel_sum
thf(fact_3727_sum_Otriangle__reindex__eq,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add @ A )
     => ! [G: nat > nat > A,N2: nat] :
          ( ( groups7311177749621191930dd_sum @ ( product_prod @ nat @ nat ) @ A @ ( product_case_prod @ nat @ nat @ A @ G )
            @ ( collect @ ( product_prod @ nat @ nat )
              @ ( product_case_prod @ nat @ nat @ $o
                @ ^ [I2: nat,J3: nat] : ( ord_less_eq @ nat @ ( plus_plus @ nat @ I2 @ J3 ) @ N2 ) ) ) )
          = ( groups7311177749621191930dd_sum @ nat @ A
            @ ^ [K4: nat] :
                ( groups7311177749621191930dd_sum @ nat @ A
                @ ^ [I2: nat] : ( G @ I2 @ ( minus_minus @ nat @ K4 @ I2 ) )
                @ ( set_ord_atMost @ nat @ K4 ) )
            @ ( set_ord_atMost @ nat @ N2 ) ) ) ) ).

% sum.triangle_reindex_eq
thf(fact_3728_prod_Otriangle__reindex__eq,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_mult @ A )
     => ! [G: nat > nat > A,N2: nat] :
          ( ( groups7121269368397514597t_prod @ ( product_prod @ nat @ nat ) @ A @ ( product_case_prod @ nat @ nat @ A @ G )
            @ ( collect @ ( product_prod @ nat @ nat )
              @ ( product_case_prod @ nat @ nat @ $o
                @ ^ [I2: nat,J3: nat] : ( ord_less_eq @ nat @ ( plus_plus @ nat @ I2 @ J3 ) @ N2 ) ) ) )
          = ( groups7121269368397514597t_prod @ nat @ A
            @ ^ [K4: nat] :
                ( groups7121269368397514597t_prod @ nat @ A
                @ ^ [I2: nat] : ( G @ I2 @ ( minus_minus @ nat @ K4 @ I2 ) )
                @ ( set_ord_atMost @ nat @ K4 ) )
            @ ( set_ord_atMost @ nat @ N2 ) ) ) ) ).

% prod.triangle_reindex_eq
thf(fact_3729_sum__choose__diagonal,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq @ nat @ M @ N2 )
     => ( ( groups7311177749621191930dd_sum @ nat @ nat
          @ ^ [K4: nat] : ( binomial @ ( minus_minus @ nat @ N2 @ K4 ) @ ( minus_minus @ nat @ M @ K4 ) )
          @ ( set_ord_atMost @ nat @ M ) )
        = ( binomial @ ( suc @ N2 ) @ M ) ) ) ).

% sum_choose_diagonal
thf(fact_3730_vandermonde,axiom,
    ! [M: nat,N2: nat,R4: nat] :
      ( ( groups7311177749621191930dd_sum @ nat @ nat
        @ ^ [K4: nat] : ( times_times @ nat @ ( binomial @ M @ K4 ) @ ( binomial @ N2 @ ( minus_minus @ nat @ R4 @ K4 ) ) )
        @ ( set_ord_atMost @ nat @ R4 ) )
      = ( binomial @ ( plus_plus @ nat @ M @ N2 ) @ R4 ) ) ).

% vandermonde
thf(fact_3731_sum__gp__basic,axiom,
    ! [A: $tType] :
      ( ( ( monoid_mult @ A )
        & ( comm_ring @ A ) )
     => ! [X2: A,N2: nat] :
          ( ( times_times @ A @ ( minus_minus @ A @ ( one_one @ A ) @ X2 ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X2 ) @ ( set_ord_atMost @ nat @ N2 ) ) )
          = ( minus_minus @ A @ ( one_one @ A ) @ ( power_power @ A @ X2 @ ( suc @ N2 ) ) ) ) ) ).

% sum_gp_basic
thf(fact_3732_polyfun__roots__finite,axiom,
    ! [A: $tType] :
      ( ( ( real_V8999393235501362500lgebra @ A )
        & ( idom @ A ) )
     => ! [C2: nat > A,K: nat,N2: nat] :
          ( ( ( C2 @ K )
           != ( zero_zero @ A ) )
         => ( ( ord_less_eq @ nat @ K @ N2 )
           => ( finite_finite2 @ A
              @ ( collect @ A
                @ ^ [Z2: A] :
                    ( ( groups7311177749621191930dd_sum @ nat @ A
                      @ ^ [I2: nat] : ( times_times @ A @ ( C2 @ I2 ) @ ( power_power @ A @ Z2 @ I2 ) )
                      @ ( set_ord_atMost @ nat @ N2 ) )
                    = ( zero_zero @ A ) ) ) ) ) ) ) ).

% polyfun_roots_finite
thf(fact_3733_polyfun__finite__roots,axiom,
    ! [A: $tType] :
      ( ( ( real_V8999393235501362500lgebra @ A )
        & ( idom @ A ) )
     => ! [C2: nat > A,N2: nat] :
          ( ( finite_finite2 @ A
            @ ( collect @ A
              @ ^ [X: A] :
                  ( ( groups7311177749621191930dd_sum @ nat @ A
                    @ ^ [I2: nat] : ( times_times @ A @ ( C2 @ I2 ) @ ( power_power @ A @ X @ I2 ) )
                    @ ( set_ord_atMost @ nat @ N2 ) )
                  = ( zero_zero @ A ) ) ) )
          = ( ? [I2: nat] :
                ( ( ord_less_eq @ nat @ I2 @ N2 )
                & ( ( C2 @ I2 )
                 != ( zero_zero @ A ) ) ) ) ) ) ).

% polyfun_finite_roots
thf(fact_3734_polyfun__linear__factor__root,axiom,
    ! [A: $tType] :
      ( ( idom @ A )
     => ! [C2: nat > A,A2: A,N2: nat] :
          ( ( ( groups7311177749621191930dd_sum @ nat @ A
              @ ^ [I2: nat] : ( times_times @ A @ ( C2 @ I2 ) @ ( power_power @ A @ A2 @ I2 ) )
              @ ( set_ord_atMost @ nat @ N2 ) )
            = ( zero_zero @ A ) )
         => ~ ! [B3: nat > A] :
                ~ ! [Z5: A] :
                    ( ( groups7311177749621191930dd_sum @ nat @ A
                      @ ^ [I2: nat] : ( times_times @ A @ ( C2 @ I2 ) @ ( power_power @ A @ Z5 @ I2 ) )
                      @ ( set_ord_atMost @ nat @ N2 ) )
                    = ( times_times @ A @ ( minus_minus @ A @ Z5 @ A2 )
                      @ ( groups7311177749621191930dd_sum @ nat @ A
                        @ ^ [I2: nat] : ( times_times @ A @ ( B3 @ I2 ) @ ( power_power @ A @ Z5 @ I2 ) )
                        @ ( set_ord_lessThan @ nat @ N2 ) ) ) ) ) ) ).

% polyfun_linear_factor_root
thf(fact_3735_polyfun__linear__factor,axiom,
    ! [A: $tType] :
      ( ( idom @ A )
     => ! [C2: nat > A,N2: nat,A2: A] :
        ? [B3: nat > A] :
        ! [Z5: A] :
          ( ( groups7311177749621191930dd_sum @ nat @ A
            @ ^ [I2: nat] : ( times_times @ A @ ( C2 @ I2 ) @ ( power_power @ A @ Z5 @ I2 ) )
            @ ( set_ord_atMost @ nat @ N2 ) )
          = ( plus_plus @ A
            @ ( times_times @ A @ ( minus_minus @ A @ Z5 @ A2 )
              @ ( groups7311177749621191930dd_sum @ nat @ A
                @ ^ [I2: nat] : ( times_times @ A @ ( B3 @ I2 ) @ ( power_power @ A @ Z5 @ I2 ) )
                @ ( set_ord_lessThan @ nat @ N2 ) ) )
            @ ( groups7311177749621191930dd_sum @ nat @ A
              @ ^ [I2: nat] : ( times_times @ A @ ( C2 @ I2 ) @ ( power_power @ A @ A2 @ I2 ) )
              @ ( set_ord_atMost @ nat @ N2 ) ) ) ) ) ).

% polyfun_linear_factor
thf(fact_3736_sum__power__shift,axiom,
    ! [A: $tType] :
      ( ( ( monoid_mult @ A )
        & ( comm_ring @ A ) )
     => ! [M: nat,N2: nat,X2: A] :
          ( ( ord_less_eq @ nat @ M @ N2 )
         => ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X2 ) @ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) )
            = ( times_times @ A @ ( power_power @ A @ X2 @ M ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X2 ) @ ( set_ord_atMost @ nat @ ( minus_minus @ nat @ N2 @ M ) ) ) ) ) ) ) ).

% sum_power_shift
thf(fact_3737_atLeast1__atMost__eq__remove0,axiom,
    ! [N2: nat] :
      ( ( set_or1337092689740270186AtMost @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N2 )
      = ( minus_minus @ ( set @ nat ) @ ( set_ord_atMost @ nat @ N2 ) @ ( insert @ nat @ ( zero_zero @ nat ) @ ( bot_bot @ ( set @ nat ) ) ) ) ) ).

% atLeast1_atMost_eq_remove0
thf(fact_3738_sum_Otriangle__reindex,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add @ A )
     => ! [G: nat > nat > A,N2: nat] :
          ( ( groups7311177749621191930dd_sum @ ( product_prod @ nat @ nat ) @ A @ ( product_case_prod @ nat @ nat @ A @ G )
            @ ( collect @ ( product_prod @ nat @ nat )
              @ ( product_case_prod @ nat @ nat @ $o
                @ ^ [I2: nat,J3: nat] : ( ord_less @ nat @ ( plus_plus @ nat @ I2 @ J3 ) @ N2 ) ) ) )
          = ( groups7311177749621191930dd_sum @ nat @ A
            @ ^ [K4: nat] :
                ( groups7311177749621191930dd_sum @ nat @ A
                @ ^ [I2: nat] : ( G @ I2 @ ( minus_minus @ nat @ K4 @ I2 ) )
                @ ( set_ord_atMost @ nat @ K4 ) )
            @ ( set_ord_lessThan @ nat @ N2 ) ) ) ) ).

% sum.triangle_reindex
thf(fact_3739_prod_Otriangle__reindex,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_mult @ A )
     => ! [G: nat > nat > A,N2: nat] :
          ( ( groups7121269368397514597t_prod @ ( product_prod @ nat @ nat ) @ A @ ( product_case_prod @ nat @ nat @ A @ G )
            @ ( collect @ ( product_prod @ nat @ nat )
              @ ( product_case_prod @ nat @ nat @ $o
                @ ^ [I2: nat,J3: nat] : ( ord_less @ nat @ ( plus_plus @ nat @ I2 @ J3 ) @ N2 ) ) ) )
          = ( groups7121269368397514597t_prod @ nat @ A
            @ ^ [K4: nat] :
                ( groups7121269368397514597t_prod @ nat @ A
                @ ^ [I2: nat] : ( G @ I2 @ ( minus_minus @ nat @ K4 @ I2 ) )
                @ ( set_ord_atMost @ nat @ K4 ) )
            @ ( set_ord_lessThan @ nat @ N2 ) ) ) ) ).

% prod.triangle_reindex
thf(fact_3740_choose__row__sum,axiom,
    ! [N2: nat] :
      ( ( groups7311177749621191930dd_sum @ nat @ nat @ ( binomial @ N2 ) @ ( set_ord_atMost @ nat @ N2 ) )
      = ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ).

% choose_row_sum
thf(fact_3741_binomial,axiom,
    ! [A2: nat,B2: nat,N2: nat] :
      ( ( power_power @ nat @ ( plus_plus @ nat @ A2 @ B2 ) @ N2 )
      = ( groups7311177749621191930dd_sum @ nat @ nat
        @ ^ [K4: nat] : ( times_times @ nat @ ( times_times @ nat @ ( semiring_1_of_nat @ nat @ ( binomial @ N2 @ K4 ) ) @ ( power_power @ nat @ A2 @ K4 ) ) @ ( power_power @ nat @ B2 @ ( minus_minus @ nat @ N2 @ K4 ) ) )
        @ ( set_ord_atMost @ nat @ N2 ) ) ) ).

% binomial
thf(fact_3742_sum_Oin__pairs__0,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add @ A )
     => ! [G: nat > A,N2: nat] :
          ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_ord_atMost @ nat @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ) )
          = ( groups7311177749621191930dd_sum @ nat @ A
            @ ^ [I2: nat] : ( plus_plus @ A @ ( G @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I2 ) ) @ ( G @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I2 ) ) ) )
            @ ( set_ord_atMost @ nat @ N2 ) ) ) ) ).

% sum.in_pairs_0
thf(fact_3743_polynomial__product,axiom,
    ! [A: $tType] :
      ( ( idom @ A )
     => ! [M: nat,A2: nat > A,N2: nat,B2: nat > A,X2: A] :
          ( ! [I3: nat] :
              ( ( ord_less @ nat @ M @ I3 )
             => ( ( A2 @ I3 )
                = ( zero_zero @ A ) ) )
         => ( ! [J2: nat] :
                ( ( ord_less @ nat @ N2 @ J2 )
               => ( ( B2 @ J2 )
                  = ( zero_zero @ A ) ) )
           => ( ( times_times @ A
                @ ( groups7311177749621191930dd_sum @ nat @ A
                  @ ^ [I2: nat] : ( times_times @ A @ ( A2 @ I2 ) @ ( power_power @ A @ X2 @ I2 ) )
                  @ ( set_ord_atMost @ nat @ M ) )
                @ ( groups7311177749621191930dd_sum @ nat @ A
                  @ ^ [J3: nat] : ( times_times @ A @ ( B2 @ J3 ) @ ( power_power @ A @ X2 @ J3 ) )
                  @ ( set_ord_atMost @ nat @ N2 ) ) )
              = ( groups7311177749621191930dd_sum @ nat @ A
                @ ^ [R: nat] :
                    ( times_times @ A
                    @ ( groups7311177749621191930dd_sum @ nat @ A
                      @ ^ [K4: nat] : ( times_times @ A @ ( A2 @ K4 ) @ ( B2 @ ( minus_minus @ nat @ R @ K4 ) ) )
                      @ ( set_ord_atMost @ nat @ R ) )
                    @ ( power_power @ A @ X2 @ R ) )
                @ ( set_ord_atMost @ nat @ ( plus_plus @ nat @ M @ N2 ) ) ) ) ) ) ) ).

% polynomial_product
thf(fact_3744_prod_Oin__pairs__0,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_mult @ A )
     => ! [G: nat > A,N2: nat] :
          ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_ord_atMost @ nat @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ) )
          = ( groups7121269368397514597t_prod @ nat @ A
            @ ^ [I2: nat] : ( times_times @ A @ ( G @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I2 ) ) @ ( G @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I2 ) ) ) )
            @ ( set_ord_atMost @ nat @ N2 ) ) ) ) ).

% prod.in_pairs_0
thf(fact_3745_polyfun__eq__const,axiom,
    ! [A: $tType] :
      ( ( ( real_V8999393235501362500lgebra @ A )
        & ( idom @ A ) )
     => ! [C2: nat > A,N2: nat,K: A] :
          ( ( ! [X: A] :
                ( ( groups7311177749621191930dd_sum @ nat @ A
                  @ ^ [I2: nat] : ( times_times @ A @ ( C2 @ I2 ) @ ( power_power @ A @ X @ I2 ) )
                  @ ( set_ord_atMost @ nat @ N2 ) )
                = K ) )
          = ( ( ( C2 @ ( zero_zero @ nat ) )
              = K )
            & ! [X: nat] :
                ( ( member @ nat @ X @ ( set_or1337092689740270186AtMost @ nat @ ( one_one @ nat ) @ N2 ) )
               => ( ( C2 @ X )
                  = ( zero_zero @ A ) ) ) ) ) ) ).

% polyfun_eq_const
thf(fact_3746_binomial__ring,axiom,
    ! [A: $tType] :
      ( ( comm_semiring_1 @ A )
     => ! [A2: A,B2: A,N2: nat] :
          ( ( power_power @ A @ ( plus_plus @ A @ A2 @ B2 ) @ N2 )
          = ( groups7311177749621191930dd_sum @ nat @ A
            @ ^ [K4: nat] : ( times_times @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ ( binomial @ N2 @ K4 ) ) @ ( power_power @ A @ A2 @ K4 ) ) @ ( power_power @ A @ B2 @ ( minus_minus @ nat @ N2 @ K4 ) ) )
            @ ( set_ord_atMost @ nat @ N2 ) ) ) ) ).

% binomial_ring
thf(fact_3747_polynomial__product__nat,axiom,
    ! [M: nat,A2: nat > nat,N2: nat,B2: nat > nat,X2: nat] :
      ( ! [I3: nat] :
          ( ( ord_less @ nat @ M @ I3 )
         => ( ( A2 @ I3 )
            = ( zero_zero @ nat ) ) )
     => ( ! [J2: nat] :
            ( ( ord_less @ nat @ N2 @ J2 )
           => ( ( B2 @ J2 )
              = ( zero_zero @ nat ) ) )
       => ( ( times_times @ nat
            @ ( groups7311177749621191930dd_sum @ nat @ nat
              @ ^ [I2: nat] : ( times_times @ nat @ ( A2 @ I2 ) @ ( power_power @ nat @ X2 @ I2 ) )
              @ ( set_ord_atMost @ nat @ M ) )
            @ ( groups7311177749621191930dd_sum @ nat @ nat
              @ ^ [J3: nat] : ( times_times @ nat @ ( B2 @ J3 ) @ ( power_power @ nat @ X2 @ J3 ) )
              @ ( set_ord_atMost @ nat @ N2 ) ) )
          = ( groups7311177749621191930dd_sum @ nat @ nat
            @ ^ [R: nat] :
                ( times_times @ nat
                @ ( groups7311177749621191930dd_sum @ nat @ nat
                  @ ^ [K4: nat] : ( times_times @ nat @ ( A2 @ K4 ) @ ( B2 @ ( minus_minus @ nat @ R @ K4 ) ) )
                  @ ( set_ord_atMost @ nat @ R ) )
                @ ( power_power @ nat @ X2 @ R ) )
            @ ( set_ord_atMost @ nat @ ( plus_plus @ nat @ M @ N2 ) ) ) ) ) ) ).

% polynomial_product_nat
thf(fact_3748_pochhammer__binomial__sum,axiom,
    ! [A: $tType] :
      ( ( comm_ring_1 @ A )
     => ! [A2: A,B2: A,N2: nat] :
          ( ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ A2 @ B2 ) @ N2 )
          = ( groups7311177749621191930dd_sum @ nat @ A
            @ ^ [K4: nat] : ( times_times @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ ( binomial @ N2 @ K4 ) ) @ ( comm_s3205402744901411588hammer @ A @ A2 @ K4 ) ) @ ( comm_s3205402744901411588hammer @ A @ B2 @ ( minus_minus @ nat @ N2 @ K4 ) ) )
            @ ( set_ord_atMost @ nat @ N2 ) ) ) ) ).

% pochhammer_binomial_sum
thf(fact_3749_choose__square__sum,axiom,
    ! [N2: nat] :
      ( ( groups7311177749621191930dd_sum @ nat @ nat
        @ ^ [K4: nat] : ( power_power @ nat @ ( binomial @ N2 @ K4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
        @ ( set_ord_atMost @ nat @ N2 ) )
      = ( binomial @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) @ N2 ) ) ).

% choose_square_sum
thf(fact_3750_sum_Ozero__middle,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add @ A )
     => ! [P4: nat,K: nat,G: nat > A,H2: nat > A] :
          ( ( ord_less_eq @ nat @ ( one_one @ nat ) @ P4 )
         => ( ( ord_less_eq @ nat @ K @ P4 )
           => ( ( groups7311177749621191930dd_sum @ nat @ A
                @ ^ [J3: nat] : ( if @ A @ ( ord_less @ nat @ J3 @ K ) @ ( G @ J3 ) @ ( if @ A @ ( J3 = K ) @ ( zero_zero @ A ) @ ( H2 @ ( minus_minus @ nat @ J3 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) )
                @ ( set_ord_atMost @ nat @ P4 ) )
              = ( groups7311177749621191930dd_sum @ nat @ A
                @ ^ [J3: nat] : ( if @ A @ ( ord_less @ nat @ J3 @ K ) @ ( G @ J3 ) @ ( H2 @ J3 ) )
                @ ( set_ord_atMost @ nat @ ( minus_minus @ nat @ P4 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) ) ) ).

% sum.zero_middle
thf(fact_3751_prod_Ozero__middle,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_mult @ A )
     => ! [P4: nat,K: nat,G: nat > A,H2: nat > A] :
          ( ( ord_less_eq @ nat @ ( one_one @ nat ) @ P4 )
         => ( ( ord_less_eq @ nat @ K @ P4 )
           => ( ( groups7121269368397514597t_prod @ nat @ A
                @ ^ [J3: nat] : ( if @ A @ ( ord_less @ nat @ J3 @ K ) @ ( G @ J3 ) @ ( if @ A @ ( J3 = K ) @ ( one_one @ A ) @ ( H2 @ ( minus_minus @ nat @ J3 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) )
                @ ( set_ord_atMost @ nat @ P4 ) )
              = ( groups7121269368397514597t_prod @ nat @ A
                @ ^ [J3: nat] : ( if @ A @ ( ord_less @ nat @ J3 @ K ) @ ( G @ J3 ) @ ( H2 @ J3 ) )
                @ ( set_ord_atMost @ nat @ ( minus_minus @ nat @ P4 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) ) ) ).

% prod.zero_middle
thf(fact_3752_sum__gp0,axiom,
    ! [A: $tType] :
      ( ( ( division_ring @ A )
        & ( comm_ring @ A ) )
     => ! [X2: A,N2: nat] :
          ( ( ( X2
              = ( one_one @ A ) )
           => ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X2 ) @ ( set_ord_atMost @ nat @ N2 ) )
              = ( semiring_1_of_nat @ A @ ( plus_plus @ nat @ N2 @ ( one_one @ nat ) ) ) ) )
          & ( ( X2
             != ( one_one @ A ) )
           => ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X2 ) @ ( set_ord_atMost @ nat @ N2 ) )
              = ( divide_divide @ A @ ( minus_minus @ A @ ( one_one @ A ) @ ( power_power @ A @ X2 @ ( suc @ N2 ) ) ) @ ( minus_minus @ A @ ( one_one @ A ) @ X2 ) ) ) ) ) ) ).

% sum_gp0
thf(fact_3753_gbinomial__sum__nat__pow2,axiom,
    ! [A: $tType] :
      ( ( field_char_0 @ A )
     => ! [M: nat] :
          ( ( groups7311177749621191930dd_sum @ nat @ A
            @ ^ [K4: nat] : ( divide_divide @ A @ ( gbinomial @ A @ ( semiring_1_of_nat @ A @ ( plus_plus @ nat @ M @ K4 ) ) @ K4 ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ K4 ) )
            @ ( set_ord_atMost @ nat @ M ) )
          = ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) ) ) ).

% gbinomial_sum_nat_pow2
thf(fact_3754_gbinomial__partial__sum__poly__xpos,axiom,
    ! [A: $tType] :
      ( ( field_char_0 @ A )
     => ! [M: nat,A2: A,X2: A,Y2: A] :
          ( ( groups7311177749621191930dd_sum @ nat @ A
            @ ^ [K4: nat] : ( times_times @ A @ ( times_times @ A @ ( gbinomial @ A @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ M ) @ A2 ) @ K4 ) @ ( power_power @ A @ X2 @ K4 ) ) @ ( power_power @ A @ Y2 @ ( minus_minus @ nat @ M @ K4 ) ) )
            @ ( set_ord_atMost @ nat @ M ) )
          = ( groups7311177749621191930dd_sum @ nat @ A
            @ ^ [K4: nat] : ( times_times @ A @ ( times_times @ A @ ( gbinomial @ A @ ( minus_minus @ A @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ K4 ) @ A2 ) @ ( one_one @ A ) ) @ K4 ) @ ( power_power @ A @ X2 @ K4 ) ) @ ( power_power @ A @ ( plus_plus @ A @ X2 @ Y2 ) @ ( minus_minus @ nat @ M @ K4 ) ) )
            @ ( set_ord_atMost @ nat @ M ) ) ) ) ).

% gbinomial_partial_sum_poly_xpos
thf(fact_3755_polyfun__diff__alt,axiom,
    ! [A: $tType] :
      ( ( idom @ A )
     => ! [N2: nat,A2: nat > A,X2: A,Y2: A] :
          ( ( ord_less_eq @ nat @ ( one_one @ nat ) @ N2 )
         => ( ( minus_minus @ A
              @ ( groups7311177749621191930dd_sum @ nat @ A
                @ ^ [I2: nat] : ( times_times @ A @ ( A2 @ I2 ) @ ( power_power @ A @ X2 @ I2 ) )
                @ ( set_ord_atMost @ nat @ N2 ) )
              @ ( groups7311177749621191930dd_sum @ nat @ A
                @ ^ [I2: nat] : ( times_times @ A @ ( A2 @ I2 ) @ ( power_power @ A @ Y2 @ I2 ) )
                @ ( set_ord_atMost @ nat @ N2 ) ) )
            = ( times_times @ A @ ( minus_minus @ A @ X2 @ Y2 )
              @ ( groups7311177749621191930dd_sum @ nat @ A
                @ ^ [J3: nat] :
                    ( groups7311177749621191930dd_sum @ nat @ A
                    @ ^ [K4: nat] : ( times_times @ A @ ( times_times @ A @ ( A2 @ ( plus_plus @ nat @ ( plus_plus @ nat @ J3 @ K4 ) @ ( one_one @ nat ) ) ) @ ( power_power @ A @ Y2 @ K4 ) ) @ ( power_power @ A @ X2 @ J3 ) )
                    @ ( set_ord_lessThan @ nat @ ( minus_minus @ nat @ N2 @ J3 ) ) )
                @ ( set_ord_lessThan @ nat @ N2 ) ) ) ) ) ) ).

% polyfun_diff_alt
thf(fact_3756_binomial__r__part__sum,axiom,
    ! [M: nat] :
      ( ( groups7311177749621191930dd_sum @ nat @ nat @ ( binomial @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) @ ( one_one @ nat ) ) ) @ ( set_ord_atMost @ nat @ M ) )
      = ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) ) ) ).

% binomial_r_part_sum
thf(fact_3757_choose__linear__sum,axiom,
    ! [N2: nat] :
      ( ( groups7311177749621191930dd_sum @ nat @ nat
        @ ^ [I2: nat] : ( times_times @ nat @ I2 @ ( binomial @ N2 @ I2 ) )
        @ ( set_ord_atMost @ nat @ N2 ) )
      = ( times_times @ nat @ N2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ N2 @ ( one_one @ nat ) ) ) ) ) ).

% choose_linear_sum
thf(fact_3758_polyfun__extremal__lemma,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra @ A )
     => ! [E3: real,C2: nat > A,N2: nat] :
          ( ( ord_less @ real @ ( zero_zero @ real ) @ E3 )
         => ? [M11: real] :
            ! [Z5: A] :
              ( ( ord_less_eq @ real @ M11 @ ( real_V7770717601297561774m_norm @ A @ Z5 ) )
             => ( ord_less_eq @ real
                @ ( real_V7770717601297561774m_norm @ A
                  @ ( groups7311177749621191930dd_sum @ nat @ A
                    @ ^ [I2: nat] : ( times_times @ A @ ( C2 @ I2 ) @ ( power_power @ A @ Z5 @ I2 ) )
                    @ ( set_ord_atMost @ nat @ N2 ) ) )
                @ ( times_times @ real @ E3 @ ( power_power @ real @ ( real_V7770717601297561774m_norm @ A @ Z5 ) @ ( suc @ N2 ) ) ) ) ) ) ) ).

% polyfun_extremal_lemma
thf(fact_3759_polyfun__diff,axiom,
    ! [A: $tType] :
      ( ( idom @ A )
     => ! [N2: nat,A2: nat > A,X2: A,Y2: A] :
          ( ( ord_less_eq @ nat @ ( one_one @ nat ) @ N2 )
         => ( ( minus_minus @ A
              @ ( groups7311177749621191930dd_sum @ nat @ A
                @ ^ [I2: nat] : ( times_times @ A @ ( A2 @ I2 ) @ ( power_power @ A @ X2 @ I2 ) )
                @ ( set_ord_atMost @ nat @ N2 ) )
              @ ( groups7311177749621191930dd_sum @ nat @ A
                @ ^ [I2: nat] : ( times_times @ A @ ( A2 @ I2 ) @ ( power_power @ A @ Y2 @ I2 ) )
                @ ( set_ord_atMost @ nat @ N2 ) ) )
            = ( times_times @ A @ ( minus_minus @ A @ X2 @ Y2 )
              @ ( groups7311177749621191930dd_sum @ nat @ A
                @ ^ [J3: nat] :
                    ( times_times @ A
                    @ ( groups7311177749621191930dd_sum @ nat @ A
                      @ ^ [I2: nat] : ( times_times @ A @ ( A2 @ I2 ) @ ( power_power @ A @ Y2 @ ( minus_minus @ nat @ ( minus_minus @ nat @ I2 @ J3 ) @ ( one_one @ nat ) ) ) )
                      @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ J3 ) @ N2 ) )
                    @ ( power_power @ A @ X2 @ J3 ) )
                @ ( set_ord_lessThan @ nat @ N2 ) ) ) ) ) ) ).

% polyfun_diff
thf(fact_3760_gbinomial__code,axiom,
    ! [A: $tType] :
      ( ( field_char_0 @ A )
     => ( ( gbinomial @ A )
        = ( ^ [A5: A,K4: nat] :
              ( if @ A
              @ ( K4
                = ( zero_zero @ nat ) )
              @ ( one_one @ A )
              @ ( divide_divide @ A
                @ ( set_fo6178422350223883121st_nat @ A
                  @ ^ [L2: nat] : ( times_times @ A @ ( minus_minus @ A @ A5 @ ( semiring_1_of_nat @ A @ L2 ) ) )
                  @ ( zero_zero @ nat )
                  @ ( minus_minus @ nat @ K4 @ ( one_one @ nat ) )
                  @ ( one_one @ A ) )
                @ ( semiring_char_0_fact @ A @ K4 ) ) ) ) ) ) ).

% gbinomial_code
thf(fact_3761_powr__int,axiom,
    ! [X2: real,I: int] :
      ( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
     => ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ I )
         => ( ( powr @ real @ X2 @ ( ring_1_of_int @ real @ I ) )
            = ( power_power @ real @ X2 @ ( nat2 @ I ) ) ) )
        & ( ~ ( ord_less_eq @ int @ ( zero_zero @ int ) @ I )
         => ( ( powr @ real @ X2 @ ( ring_1_of_int @ real @ I ) )
            = ( divide_divide @ real @ ( one_one @ real ) @ ( power_power @ real @ X2 @ ( nat2 @ ( uminus_uminus @ int @ I ) ) ) ) ) ) ) ) ).

% powr_int
thf(fact_3762_dbl__simps_I3_J,axiom,
    ! [A: $tType] :
      ( ( neg_numeral @ A )
     => ( ( neg_numeral_dbl @ A @ ( one_one @ A ) )
        = ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ).

% dbl_simps(3)
thf(fact_3763_VEBT__internal_OminNull_Opelims_I1_J,axiom,
    ! [X2: vEBT_VEBT,Y2: $o] :
      ( ( ( vEBT_VEBT_minNull @ X2 )
        = Y2 )
     => ( ( accp @ vEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ X2 )
       => ( ( ( X2
              = ( vEBT_Leaf @ $false @ $false ) )
           => ( Y2
             => ~ ( accp @ vEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ $false @ $false ) ) ) )
         => ( ! [Uv2: $o] :
                ( ( X2
                  = ( vEBT_Leaf @ $true @ Uv2 ) )
               => ( ~ Y2
                 => ~ ( accp @ vEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ $true @ Uv2 ) ) ) )
           => ( ! [Uu2: $o] :
                  ( ( X2
                    = ( vEBT_Leaf @ Uu2 @ $true ) )
                 => ( ~ Y2
                   => ~ ( accp @ vEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ Uu2 @ $true ) ) ) )
             => ( ! [Uw2: nat,Ux2: list @ vEBT_VEBT,Uy2: vEBT_VEBT] :
                    ( ( X2
                      = ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uw2 @ Ux2 @ Uy2 ) )
                   => ( Y2
                     => ~ ( accp @ vEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uw2 @ Ux2 @ Uy2 ) ) ) )
               => ~ ! [Uz2: product_prod @ nat @ nat,Va2: nat,Vb2: list @ vEBT_VEBT,Vc: vEBT_VEBT] :
                      ( ( X2
                        = ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ Uz2 ) @ Va2 @ Vb2 @ Vc ) )
                     => ( ~ Y2
                       => ~ ( accp @ vEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ Uz2 ) @ Va2 @ Vb2 @ Vc ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.minNull.pelims(1)
thf(fact_3764_of__nat__id,axiom,
    ( ( semiring_1_of_nat @ nat )
    = ( ^ [N: nat] : N ) ) ).

% of_nat_id
thf(fact_3765_neg__le__iff__le,axiom,
    ! [A: $tType] :
      ( ( ordered_ab_group_add @ A )
     => ! [B2: A,A2: A] :
          ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ ( uminus_uminus @ A @ A2 ) )
          = ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).

% neg_le_iff_le
thf(fact_3766_neg__less__iff__less,axiom,
    ! [A: $tType] :
      ( ( ordered_ab_group_add @ A )
     => ! [B2: A,A2: A] :
          ( ( ord_less @ A @ ( uminus_uminus @ A @ B2 ) @ ( uminus_uminus @ A @ A2 ) )
          = ( ord_less @ A @ A2 @ B2 ) ) ) ).

% neg_less_iff_less
thf(fact_3767_neg__numeral__eq__iff,axiom,
    ! [A: $tType] :
      ( ( ring_char_0 @ A )
     => ! [M: num,N2: num] :
          ( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) )
            = ( uminus_uminus @ A @ ( numeral_numeral @ A @ N2 ) ) )
          = ( M = N2 ) ) ) ).

% neg_numeral_eq_iff
thf(fact_3768_minus__add__distrib,axiom,
    ! [A: $tType] :
      ( ( ab_group_add @ A )
     => ! [A2: A,B2: A] :
          ( ( uminus_uminus @ A @ ( plus_plus @ A @ A2 @ B2 ) )
          = ( plus_plus @ A @ ( uminus_uminus @ A @ A2 ) @ ( uminus_uminus @ A @ B2 ) ) ) ) ).

% minus_add_distrib
thf(fact_3769_minus__add__cancel,axiom,
    ! [A: $tType] :
      ( ( group_add @ A )
     => ! [A2: A,B2: A] :
          ( ( plus_plus @ A @ ( uminus_uminus @ A @ A2 ) @ ( plus_plus @ A @ A2 @ B2 ) )
          = B2 ) ) ).

% minus_add_cancel
thf(fact_3770_add__minus__cancel,axiom,
    ! [A: $tType] :
      ( ( group_add @ A )
     => ! [A2: A,B2: A] :
          ( ( plus_plus @ A @ A2 @ ( plus_plus @ A @ ( uminus_uminus @ A @ A2 ) @ B2 ) )
          = B2 ) ) ).

% add_minus_cancel
thf(fact_3771_mult__minus__left,axiom,
    ! [A: $tType] :
      ( ( ring @ A )
     => ! [A2: A,B2: A] :
          ( ( times_times @ A @ ( uminus_uminus @ A @ A2 ) @ B2 )
          = ( uminus_uminus @ A @ ( times_times @ A @ A2 @ B2 ) ) ) ) ).

% mult_minus_left
thf(fact_3772_minus__mult__minus,axiom,
    ! [A: $tType] :
      ( ( ring @ A )
     => ! [A2: A,B2: A] :
          ( ( times_times @ A @ ( uminus_uminus @ A @ A2 ) @ ( uminus_uminus @ A @ B2 ) )
          = ( times_times @ A @ A2 @ B2 ) ) ) ).

% minus_mult_minus
thf(fact_3773_mult__minus__right,axiom,
    ! [A: $tType] :
      ( ( ring @ A )
     => ! [A2: A,B2: A] :
          ( ( times_times @ A @ A2 @ ( uminus_uminus @ A @ B2 ) )
          = ( uminus_uminus @ A @ ( times_times @ A @ A2 @ B2 ) ) ) ) ).

% mult_minus_right
thf(fact_3774_minus__diff__eq,axiom,
    ! [A: $tType] :
      ( ( group_add @ A )
     => ! [A2: A,B2: A] :
          ( ( uminus_uminus @ A @ ( minus_minus @ A @ A2 @ B2 ) )
          = ( minus_minus @ A @ B2 @ A2 ) ) ) ).

% minus_diff_eq
thf(fact_3775_div__minus__minus,axiom,
    ! [A: $tType] :
      ( ( euclid8851590272496341667cancel @ A )
     => ! [A2: A,B2: A] :
          ( ( divide_divide @ A @ ( uminus_uminus @ A @ A2 ) @ ( uminus_uminus @ A @ B2 ) )
          = ( divide_divide @ A @ A2 @ B2 ) ) ) ).

% div_minus_minus
thf(fact_3776_dvd__minus__iff,axiom,
    ! [A: $tType] :
      ( ( comm_ring_1 @ A )
     => ! [X2: A,Y2: A] :
          ( ( dvd_dvd @ A @ X2 @ ( uminus_uminus @ A @ Y2 ) )
          = ( dvd_dvd @ A @ X2 @ Y2 ) ) ) ).

% dvd_minus_iff
thf(fact_3777_minus__dvd__iff,axiom,
    ! [A: $tType] :
      ( ( comm_ring_1 @ A )
     => ! [X2: A,Y2: A] :
          ( ( dvd_dvd @ A @ ( uminus_uminus @ A @ X2 ) @ Y2 )
          = ( dvd_dvd @ A @ X2 @ Y2 ) ) ) ).

% minus_dvd_iff
thf(fact_3778_mod__minus__minus,axiom,
    ! [A: $tType] :
      ( ( euclid8851590272496341667cancel @ A )
     => ! [A2: A,B2: A] :
          ( ( modulo_modulo @ A @ ( uminus_uminus @ A @ A2 ) @ ( uminus_uminus @ A @ B2 ) )
          = ( uminus_uminus @ A @ ( modulo_modulo @ A @ A2 @ B2 ) ) ) ) ).

% mod_minus_minus
thf(fact_3779_dbl__simps_I2_J,axiom,
    ! [A: $tType] :
      ( ( neg_numeral @ A )
     => ( ( neg_numeral_dbl @ A @ ( zero_zero @ A ) )
        = ( zero_zero @ A ) ) ) ).

% dbl_simps(2)
thf(fact_3780_neg__0__le__iff__le,axiom,
    ! [A: $tType] :
      ( ( ordered_ab_group_add @ A )
     => ! [A2: A] :
          ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ A2 ) )
          = ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).

% neg_0_le_iff_le
thf(fact_3781_neg__le__0__iff__le,axiom,
    ! [A: $tType] :
      ( ( ordered_ab_group_add @ A )
     => ! [A2: A] :
          ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ A2 ) @ ( zero_zero @ A ) )
          = ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).

% neg_le_0_iff_le
thf(fact_3782_less__eq__neg__nonpos,axiom,
    ! [A: $tType] :
      ( ( linord5086331880401160121up_add @ A )
     => ! [A2: A] :
          ( ( ord_less_eq @ A @ A2 @ ( uminus_uminus @ A @ A2 ) )
          = ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).

% less_eq_neg_nonpos
thf(fact_3783_neg__less__eq__nonneg,axiom,
    ! [A: $tType] :
      ( ( linord5086331880401160121up_add @ A )
     => ! [A2: A] :
          ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ A2 ) @ A2 )
          = ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).

% neg_less_eq_nonneg
thf(fact_3784_neg__less__0__iff__less,axiom,
    ! [A: $tType] :
      ( ( ordered_ab_group_add @ A )
     => ! [A2: A] :
          ( ( ord_less @ A @ ( uminus_uminus @ A @ A2 ) @ ( zero_zero @ A ) )
          = ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).

% neg_less_0_iff_less
thf(fact_3785_neg__0__less__iff__less,axiom,
    ! [A: $tType] :
      ( ( ordered_ab_group_add @ A )
     => ! [A2: A] :
          ( ( ord_less @ A @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ A2 ) )
          = ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).

% neg_0_less_iff_less
thf(fact_3786_neg__less__pos,axiom,
    ! [A: $tType] :
      ( ( linord5086331880401160121up_add @ A )
     => ! [A2: A] :
          ( ( ord_less @ A @ ( uminus_uminus @ A @ A2 ) @ A2 )
          = ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).

% neg_less_pos
thf(fact_3787_less__neg__neg,axiom,
    ! [A: $tType] :
      ( ( linord5086331880401160121up_add @ A )
     => ! [A2: A] :
          ( ( ord_less @ A @ A2 @ ( uminus_uminus @ A @ A2 ) )
          = ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).

% less_neg_neg
thf(fact_3788_add_Oright__inverse,axiom,
    ! [A: $tType] :
      ( ( group_add @ A )
     => ! [A2: A] :
          ( ( plus_plus @ A @ A2 @ ( uminus_uminus @ A @ A2 ) )
          = ( zero_zero @ A ) ) ) ).

% add.right_inverse
thf(fact_3789_ab__left__minus,axiom,
    ! [A: $tType] :
      ( ( group_add @ A )
     => ! [A2: A] :
          ( ( plus_plus @ A @ ( uminus_uminus @ A @ A2 ) @ A2 )
          = ( zero_zero @ A ) ) ) ).

% ab_left_minus
thf(fact_3790_add__neg__numeral__simps_I3_J,axiom,
    ! [A: $tType] :
      ( ( neg_numeral @ A )
     => ! [M: num,N2: num] :
          ( ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N2 ) ) )
          = ( uminus_uminus @ A @ ( plus_plus @ A @ ( numeral_numeral @ A @ M ) @ ( numeral_numeral @ A @ N2 ) ) ) ) ) ).

% add_neg_numeral_simps(3)
thf(fact_3791_verit__minus__simplify_I3_J,axiom,
    ! [B: $tType] :
      ( ( group_add @ B )
     => ! [B2: B] :
          ( ( minus_minus @ B @ ( zero_zero @ B ) @ B2 )
          = ( uminus_uminus @ B @ B2 ) ) ) ).

% verit_minus_simplify(3)
thf(fact_3792_diff__0,axiom,
    ! [A: $tType] :
      ( ( group_add @ A )
     => ! [A2: A] :
          ( ( minus_minus @ A @ ( zero_zero @ A ) @ A2 )
          = ( uminus_uminus @ A @ A2 ) ) ) ).

% diff_0
thf(fact_3793_mult__minus1,axiom,
    ! [A: $tType] :
      ( ( ring_1 @ A )
     => ! [Z: A] :
          ( ( times_times @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ Z )
          = ( uminus_uminus @ A @ Z ) ) ) ).

% mult_minus1
thf(fact_3794_mult__minus1__right,axiom,
    ! [A: $tType] :
      ( ( ring_1 @ A )
     => ! [Z: A] :
          ( ( times_times @ A @ Z @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
          = ( uminus_uminus @ A @ Z ) ) ) ).

% mult_minus1_right
thf(fact_3795_diff__minus__eq__add,axiom,
    ! [A: $tType] :
      ( ( group_add @ A )
     => ! [A2: A,B2: A] :
          ( ( minus_minus @ A @ A2 @ ( uminus_uminus @ A @ B2 ) )
          = ( plus_plus @ A @ A2 @ B2 ) ) ) ).

% diff_minus_eq_add
thf(fact_3796_uminus__add__conv__diff,axiom,
    ! [A: $tType] :
      ( ( ab_group_add @ A )
     => ! [A2: A,B2: A] :
          ( ( plus_plus @ A @ ( uminus_uminus @ A @ A2 ) @ B2 )
          = ( minus_minus @ A @ B2 @ A2 ) ) ) ).

% uminus_add_conv_diff
thf(fact_3797_divide__minus1,axiom,
    ! [A: $tType] :
      ( ( field @ A )
     => ! [X2: A] :
          ( ( divide_divide @ A @ X2 @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
          = ( uminus_uminus @ A @ X2 ) ) ) ).

% divide_minus1
thf(fact_3798_div__minus1__right,axiom,
    ! [A: $tType] :
      ( ( euclid8851590272496341667cancel @ A )
     => ! [A2: A] :
          ( ( divide_divide @ A @ A2 @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
          = ( uminus_uminus @ A @ A2 ) ) ) ).

% div_minus1_right
thf(fact_3799_minus__mod__self1,axiom,
    ! [A: $tType] :
      ( ( euclid8851590272496341667cancel @ A )
     => ! [B2: A,A2: A] :
          ( ( modulo_modulo @ A @ ( minus_minus @ A @ B2 @ A2 ) @ B2 )
          = ( modulo_modulo @ A @ ( uminus_uminus @ A @ A2 ) @ B2 ) ) ) ).

% minus_mod_self1
thf(fact_3800_fact__0,axiom,
    ! [A: $tType] :
      ( ( semiring_char_0 @ A )
     => ( ( semiring_char_0_fact @ A @ ( zero_zero @ nat ) )
        = ( one_one @ A ) ) ) ).

% fact_0
thf(fact_3801_fact__1,axiom,
    ! [A: $tType] :
      ( ( semiring_char_0 @ A )
     => ( ( semiring_char_0_fact @ A @ ( one_one @ nat ) )
        = ( one_one @ A ) ) ) ).

% fact_1
thf(fact_3802_negative__zle,axiom,
    ! [N2: nat,M: nat] : ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ N2 ) ) @ ( semiring_1_of_nat @ int @ M ) ) ).

% negative_zle
thf(fact_3803_dbl__simps_I5_J,axiom,
    ! [A: $tType] :
      ( ( neg_numeral @ A )
     => ! [K: num] :
          ( ( neg_numeral_dbl @ A @ ( numeral_numeral @ A @ K ) )
          = ( numeral_numeral @ A @ ( bit0 @ K ) ) ) ) ).

% dbl_simps(5)
thf(fact_3804_dbl__simps_I1_J,axiom,
    ! [A: $tType] :
      ( ( neg_numeral @ A )
     => ! [K: num] :
          ( ( neg_numeral_dbl @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ K ) ) )
          = ( uminus_uminus @ A @ ( neg_numeral_dbl @ A @ ( numeral_numeral @ A @ K ) ) ) ) ) ).

% dbl_simps(1)
thf(fact_3805_dbl__inc__simps_I4_J,axiom,
    ! [A: $tType] :
      ( ( neg_numeral @ A )
     => ( ( neg_numeral_dbl_inc @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
        = ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).

% dbl_inc_simps(4)
thf(fact_3806_add__neg__numeral__special_I8_J,axiom,
    ! [A: $tType] :
      ( ( neg_numeral @ A )
     => ( ( plus_plus @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( one_one @ A ) )
        = ( zero_zero @ A ) ) ) ).

% add_neg_numeral_special(8)
thf(fact_3807_add__neg__numeral__special_I7_J,axiom,
    ! [A: $tType] :
      ( ( neg_numeral @ A )
     => ( ( plus_plus @ A @ ( one_one @ A ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
        = ( zero_zero @ A ) ) ) ).

% add_neg_numeral_special(7)
thf(fact_3808_neg__one__eq__numeral__iff,axiom,
    ! [A: $tType] :
      ( ( ring_char_0 @ A )
     => ! [N2: num] :
          ( ( ( uminus_uminus @ A @ ( one_one @ A ) )
            = ( uminus_uminus @ A @ ( numeral_numeral @ A @ N2 ) ) )
          = ( N2 = one2 ) ) ) ).

% neg_one_eq_numeral_iff
thf(fact_3809_numeral__eq__neg__one__iff,axiom,
    ! [A: $tType] :
      ( ( ring_char_0 @ A )
     => ! [N2: num] :
          ( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ N2 ) )
            = ( uminus_uminus @ A @ ( one_one @ A ) ) )
          = ( N2 = one2 ) ) ) ).

% numeral_eq_neg_one_iff
thf(fact_3810_diff__numeral__special_I12_J,axiom,
    ! [A: $tType] :
      ( ( neg_numeral @ A )
     => ( ( minus_minus @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
        = ( zero_zero @ A ) ) ) ).

% diff_numeral_special(12)
thf(fact_3811_minus__one__mult__self,axiom,
    ! [A: $tType] :
      ( ( comm_ring_1 @ A )
     => ! [N2: nat] :
          ( ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N2 ) @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N2 ) )
          = ( one_one @ A ) ) ) ).

% minus_one_mult_self
thf(fact_3812_left__minus__one__mult__self,axiom,
    ! [A: $tType] :
      ( ( comm_ring_1 @ A )
     => ! [N2: nat,A2: A] :
          ( ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N2 ) @ ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N2 ) @ A2 ) )
          = A2 ) ) ).

% left_minus_one_mult_self
thf(fact_3813_mod__minus1__right,axiom,
    ! [A: $tType] :
      ( ( euclid8851590272496341667cancel @ A )
     => ! [A2: A] :
          ( ( modulo_modulo @ A @ A2 @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
          = ( zero_zero @ A ) ) ) ).

% mod_minus1_right
thf(fact_3814_max__number__of_I4_J,axiom,
    ! [A: $tType] :
      ( ( ( uminus @ A )
        & ( numeral @ A )
        & ( ord @ A ) )
     => ! [U: num,V: num] :
          ( ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) )
           => ( ( ord_max @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) )
              = ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) ) )
          & ( ~ ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) )
           => ( ( ord_max @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) )
              = ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) ) ) ) ) ).

% max_number_of(4)
thf(fact_3815_max__number__of_I3_J,axiom,
    ! [A: $tType] :
      ( ( ( uminus @ A )
        & ( numeral @ A )
        & ( ord @ A ) )
     => ! [U: num,V: num] :
          ( ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( numeral_numeral @ A @ V ) )
           => ( ( ord_max @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( numeral_numeral @ A @ V ) )
              = ( numeral_numeral @ A @ V ) ) )
          & ( ~ ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( numeral_numeral @ A @ V ) )
           => ( ( ord_max @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( numeral_numeral @ A @ V ) )
              = ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) ) ) ) ) ).

% max_number_of(3)
thf(fact_3816_max__number__of_I2_J,axiom,
    ! [A: $tType] :
      ( ( ( uminus @ A )
        & ( numeral @ A )
        & ( ord @ A ) )
     => ! [U: num,V: num] :
          ( ( ( ord_less_eq @ A @ ( numeral_numeral @ A @ U ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) )
           => ( ( ord_max @ A @ ( numeral_numeral @ A @ U ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) )
              = ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) ) )
          & ( ~ ( ord_less_eq @ A @ ( numeral_numeral @ A @ U ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) )
           => ( ( ord_max @ A @ ( numeral_numeral @ A @ U ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) )
              = ( numeral_numeral @ A @ U ) ) ) ) ) ).

% max_number_of(2)
thf(fact_3817_fact__Suc__0,axiom,
    ! [A: $tType] :
      ( ( semiring_char_0 @ A )
     => ( ( semiring_char_0_fact @ A @ ( suc @ ( zero_zero @ nat ) ) )
        = ( one_one @ A ) ) ) ).

% fact_Suc_0
thf(fact_3818_norm__neg__numeral,axiom,
    ! [A: $tType] :
      ( ( real_V2822296259951069270ebra_1 @ A )
     => ! [W: num] :
          ( ( real_V7770717601297561774m_norm @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) )
          = ( numeral_numeral @ real @ W ) ) ) ).

% norm_neg_numeral
thf(fact_3819_fact__Suc,axiom,
    ! [A: $tType] :
      ( ( semiring_char_0 @ A )
     => ! [N2: nat] :
          ( ( semiring_char_0_fact @ A @ ( suc @ N2 ) )
          = ( times_times @ A @ ( semiring_1_of_nat @ A @ ( suc @ N2 ) ) @ ( semiring_char_0_fact @ A @ N2 ) ) ) ) ).

% fact_Suc
thf(fact_3820_semiring__norm_I168_J,axiom,
    ! [A: $tType] :
      ( ( neg_numeral @ A )
     => ! [V: num,W: num,Y2: A] :
          ( ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) @ ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) @ Y2 ) )
          = ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( plus_plus @ num @ V @ W ) ) ) @ Y2 ) ) ) ).

% semiring_norm(168)
thf(fact_3821_diff__numeral__simps_I2_J,axiom,
    ! [A: $tType] :
      ( ( neg_numeral @ A )
     => ! [M: num,N2: num] :
          ( ( minus_minus @ A @ ( numeral_numeral @ A @ M ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N2 ) ) )
          = ( numeral_numeral @ A @ ( plus_plus @ num @ M @ N2 ) ) ) ) ).

% diff_numeral_simps(2)
thf(fact_3822_diff__numeral__simps_I3_J,axiom,
    ! [A: $tType] :
      ( ( neg_numeral @ A )
     => ! [M: num,N2: num] :
          ( ( minus_minus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( numeral_numeral @ A @ N2 ) )
          = ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( plus_plus @ num @ M @ N2 ) ) ) ) ) ).

% diff_numeral_simps(3)
thf(fact_3823_nat__neg__numeral,axiom,
    ! [K: num] :
      ( ( nat2 @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K ) ) )
      = ( zero_zero @ nat ) ) ).

% nat_neg_numeral
thf(fact_3824_negative__zless,axiom,
    ! [N2: nat,M: nat] : ( ord_less @ int @ ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ ( suc @ N2 ) ) ) @ ( semiring_1_of_nat @ int @ M ) ) ).

% negative_zless
thf(fact_3825_semiring__norm_I172_J,axiom,
    ! [A: $tType] :
      ( ( ring_1 @ A )
     => ! [V: num,W: num,Y2: A] :
          ( ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) @ Y2 ) )
          = ( times_times @ A @ ( numeral_numeral @ A @ ( times_times @ num @ V @ W ) ) @ Y2 ) ) ) ).

% semiring_norm(172)
thf(fact_3826_semiring__norm_I171_J,axiom,
    ! [A: $tType] :
      ( ( ring_1 @ A )
     => ! [V: num,W: num,Y2: A] :
          ( ( times_times @ A @ ( numeral_numeral @ A @ V ) @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) @ Y2 ) )
          = ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( times_times @ num @ V @ W ) ) ) @ Y2 ) ) ) ).

% semiring_norm(171)
thf(fact_3827_semiring__norm_I170_J,axiom,
    ! [A: $tType] :
      ( ( ring_1 @ A )
     => ! [V: num,W: num,Y2: A] :
          ( ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) @ ( times_times @ A @ ( numeral_numeral @ A @ W ) @ Y2 ) )
          = ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( times_times @ num @ V @ W ) ) ) @ Y2 ) ) ) ).

% semiring_norm(170)
thf(fact_3828_mult__neg__numeral__simps_I3_J,axiom,
    ! [A: $tType] :
      ( ( ring_1 @ A )
     => ! [M: num,N2: num] :
          ( ( times_times @ A @ ( numeral_numeral @ A @ M ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N2 ) ) )
          = ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( times_times @ num @ M @ N2 ) ) ) ) ) ).

% mult_neg_numeral_simps(3)
thf(fact_3829_mult__neg__numeral__simps_I2_J,axiom,
    ! [A: $tType] :
      ( ( ring_1 @ A )
     => ! [M: num,N2: num] :
          ( ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( numeral_numeral @ A @ N2 ) )
          = ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( times_times @ num @ M @ N2 ) ) ) ) ) ).

% mult_neg_numeral_simps(2)
thf(fact_3830_mult__neg__numeral__simps_I1_J,axiom,
    ! [A: $tType] :
      ( ( ring_1 @ A )
     => ! [M: num,N2: num] :
          ( ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N2 ) ) )
          = ( numeral_numeral @ A @ ( times_times @ num @ M @ N2 ) ) ) ) ).

% mult_neg_numeral_simps(1)
thf(fact_3831_neg__numeral__le__iff,axiom,
    ! [A: $tType] :
      ( ( linordered_idom @ A )
     => ! [M: num,N2: num] :
          ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N2 ) ) )
          = ( ord_less_eq @ num @ N2 @ M ) ) ) ).

% neg_numeral_le_iff
thf(fact_3832_neg__numeral__less__iff,axiom,
    ! [A: $tType] :
      ( ( linordered_idom @ A )
     => ! [M: num,N2: num] :
          ( ( ord_less @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N2 ) ) )
          = ( ord_less @ num @ N2 @ M ) ) ) ).

% neg_numeral_less_iff
thf(fact_3833_not__neg__one__le__neg__numeral__iff,axiom,
    ! [A: $tType] :
      ( ( linordered_idom @ A )
     => ! [M: num] :
          ( ( ~ ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) ) )
          = ( M != one2 ) ) ) ).

% not_neg_one_le_neg_numeral_iff
thf(fact_3834_neg__numeral__less__neg__one__iff,axiom,
    ! [A: $tType] :
      ( ( linordered_idom @ A )
     => ! [M: num] :
          ( ( ord_less @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
          = ( M != one2 ) ) ) ).

% neg_numeral_less_neg_one_iff
thf(fact_3835_divide__eq__eq__numeral1_I2_J,axiom,
    ! [A: $tType] :
      ( ( division_ring @ A )
     => ! [B2: A,W: num,A2: A] :
          ( ( ( divide_divide @ A @ B2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) )
            = A2 )
          = ( ( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) )
               != ( zero_zero @ A ) )
             => ( B2
                = ( times_times @ A @ A2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) ) ) )
            & ( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) )
                = ( zero_zero @ A ) )
             => ( A2
                = ( zero_zero @ A ) ) ) ) ) ) ).

% divide_eq_eq_numeral1(2)
thf(fact_3836_eq__divide__eq__numeral1_I2_J,axiom,
    ! [A: $tType] :
      ( ( division_ring @ A )
     => ! [A2: A,B2: A,W: num] :
          ( ( A2
            = ( divide_divide @ A @ B2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) ) )
          = ( ( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) )
               != ( zero_zero @ A ) )
             => ( ( times_times @ A @ A2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) )
                = B2 ) )
            & ( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) )
                = ( zero_zero @ A ) )
             => ( A2
                = ( zero_zero @ A ) ) ) ) ) ) ).

% eq_divide_eq_numeral1(2)
thf(fact_3837_divide__le__eq__numeral1_I2_J,axiom,
    ! [A: $tType] :
      ( ( linordered_field @ A )
     => ! [B2: A,W: num,A2: A] :
          ( ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) ) @ A2 )
          = ( ord_less_eq @ A @ ( times_times @ A @ A2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) ) @ B2 ) ) ) ).

% divide_le_eq_numeral1(2)
thf(fact_3838_le__divide__eq__numeral1_I2_J,axiom,
    ! [A: $tType] :
      ( ( linordered_field @ A )
     => ! [A2: A,B2: A,W: num] :
          ( ( ord_less_eq @ A @ A2 @ ( divide_divide @ A @ B2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) ) )
          = ( ord_less_eq @ A @ B2 @ ( times_times @ A @ A2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) ) ) ) ) ).

% le_divide_eq_numeral1(2)
thf(fact_3839_divide__less__eq__numeral1_I2_J,axiom,
    ! [A: $tType] :
      ( ( linordered_field @ A )
     => ! [B2: A,W: num,A2: A] :
          ( ( ord_less @ A @ ( divide_divide @ A @ B2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) ) @ A2 )
          = ( ord_less @ A @ ( times_times @ A @ A2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) ) @ B2 ) ) ) ).

% divide_less_eq_numeral1(2)
thf(fact_3840_less__divide__eq__numeral1_I2_J,axiom,
    ! [A: $tType] :
      ( ( linordered_field @ A )
     => ! [A2: A,B2: A,W: num] :
          ( ( ord_less @ A @ A2 @ ( divide_divide @ A @ B2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) ) )
          = ( ord_less @ A @ B2 @ ( times_times @ A @ A2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) ) ) ) ) ).

% less_divide_eq_numeral1(2)
thf(fact_3841_power2__minus,axiom,
    ! [A: $tType] :
      ( ( ring_1 @ A )
     => ! [A2: A] :
          ( ( power_power @ A @ ( uminus_uminus @ A @ A2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
          = ( power_power @ A @ A2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).

% power2_minus
thf(fact_3842_fact__2,axiom,
    ! [A: $tType] :
      ( ( semiring_char_0 @ A )
     => ( ( semiring_char_0_fact @ A @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
        = ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ).

% fact_2
thf(fact_3843_floor__neg__numeral,axiom,
    ! [A: $tType] :
      ( ( archim2362893244070406136eiling @ A )
     => ! [V: num] :
          ( ( archim6421214686448440834_floor @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) )
          = ( uminus_uminus @ int @ ( numeral_numeral @ int @ V ) ) ) ) ).

% floor_neg_numeral
thf(fact_3844_ceiling__neg__numeral,axiom,
    ! [A: $tType] :
      ( ( archim2362893244070406136eiling @ A )
     => ! [V: num] :
          ( ( archimedean_ceiling @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) )
          = ( uminus_uminus @ int @ ( numeral_numeral @ int @ V ) ) ) ) ).

% ceiling_neg_numeral
thf(fact_3845_int__div__minus__is__minus1,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less @ int @ A2 @ ( zero_zero @ int ) )
     => ( ( ( divide_divide @ int @ A2 @ B2 )
          = ( uminus_uminus @ int @ A2 ) )
        = ( B2
          = ( uminus_uminus @ int @ ( one_one @ int ) ) ) ) ) ).

% int_div_minus_is_minus1
thf(fact_3846_round__neg__numeral,axiom,
    ! [A: $tType] :
      ( ( archim2362893244070406136eiling @ A )
     => ! [N2: num] :
          ( ( archimedean_round @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N2 ) ) )
          = ( uminus_uminus @ int @ ( numeral_numeral @ int @ N2 ) ) ) ) ).

% round_neg_numeral
thf(fact_3847_add__neg__numeral__special_I9_J,axiom,
    ! [A: $tType] :
      ( ( neg_numeral @ A )
     => ( ( plus_plus @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
        = ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).

% add_neg_numeral_special(9)
thf(fact_3848_diff__numeral__special_I11_J,axiom,
    ! [A: $tType] :
      ( ( neg_numeral @ A )
     => ( ( minus_minus @ A @ ( one_one @ A ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
        = ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ).

% diff_numeral_special(11)
thf(fact_3849_diff__numeral__special_I10_J,axiom,
    ! [A: $tType] :
      ( ( neg_numeral @ A )
     => ( ( minus_minus @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( one_one @ A ) )
        = ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).

% diff_numeral_special(10)
thf(fact_3850_minus__1__div__2__eq,axiom,
    ! [A: $tType] :
      ( ( euclid8789492081693882211th_nat @ A )
     => ( ( divide_divide @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
        = ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).

% minus_1_div_2_eq
thf(fact_3851_bits__minus__1__mod__2__eq,axiom,
    ! [A: $tType] :
      ( ( bit_ri3973907225187159222ations @ A )
     => ( ( modulo_modulo @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
        = ( one_one @ A ) ) ) ).

% bits_minus_1_mod_2_eq
thf(fact_3852_minus__1__mod__2__eq,axiom,
    ! [A: $tType] :
      ( ( euclid8789492081693882211th_nat @ A )
     => ( ( modulo_modulo @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
        = ( one_one @ A ) ) ) ).

% minus_1_mod_2_eq
thf(fact_3853_Power_Oring__1__class_Opower__minus__even,axiom,
    ! [A: $tType] :
      ( ( ring_1 @ A )
     => ! [A2: A,N2: nat] :
          ( ( power_power @ A @ ( uminus_uminus @ A @ A2 ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) )
          = ( power_power @ A @ A2 @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ) ) ).

% Power.ring_1_class.power_minus_even
thf(fact_3854_power__minus__odd,axiom,
    ! [A: $tType] :
      ( ( ring_1 @ A )
     => ! [N2: nat,A2: A] :
          ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 )
         => ( ( power_power @ A @ ( uminus_uminus @ A @ A2 ) @ N2 )
            = ( uminus_uminus @ A @ ( power_power @ A @ A2 @ N2 ) ) ) ) ) ).

% power_minus_odd
thf(fact_3855_Parity_Oring__1__class_Opower__minus__even,axiom,
    ! [A: $tType] :
      ( ( ring_1 @ A )
     => ! [N2: nat,A2: A] :
          ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 )
         => ( ( power_power @ A @ ( uminus_uminus @ A @ A2 ) @ N2 )
            = ( power_power @ A @ A2 @ N2 ) ) ) ) ).

% Parity.ring_1_class.power_minus_even
thf(fact_3856_diff__numeral__special_I4_J,axiom,
    ! [A: $tType] :
      ( ( neg_numeral @ A )
     => ! [M: num] :
          ( ( minus_minus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( one_one @ A ) )
          = ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( plus_plus @ num @ M @ one2 ) ) ) ) ) ).

% diff_numeral_special(4)
thf(fact_3857_diff__numeral__special_I3_J,axiom,
    ! [A: $tType] :
      ( ( neg_numeral @ A )
     => ! [N2: num] :
          ( ( minus_minus @ A @ ( one_one @ A ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N2 ) ) )
          = ( numeral_numeral @ A @ ( plus_plus @ num @ one2 @ N2 ) ) ) ) ).

% diff_numeral_special(3)
thf(fact_3858_ceiling__less__zero,axiom,
    ! [A: $tType] :
      ( ( archim2362893244070406136eiling @ A )
     => ! [X2: A] :
          ( ( ord_less @ int @ ( archimedean_ceiling @ A @ X2 ) @ ( zero_zero @ int ) )
          = ( ord_less_eq @ A @ X2 @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ) ).

% ceiling_less_zero
thf(fact_3859_zero__le__ceiling,axiom,
    ! [A: $tType] :
      ( ( archim2362893244070406136eiling @ A )
     => ! [X2: A] :
          ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( archimedean_ceiling @ A @ X2 ) )
          = ( ord_less @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ X2 ) ) ) ).

% zero_le_ceiling
thf(fact_3860_dbl__simps_I4_J,axiom,
    ! [A: $tType] :
      ( ( neg_numeral @ A )
     => ( ( neg_numeral_dbl @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
        = ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).

% dbl_simps(4)
thf(fact_3861_ceiling__divide__eq__div__numeral,axiom,
    ! [A2: num,B2: num] :
      ( ( archimedean_ceiling @ real @ ( divide_divide @ real @ ( numeral_numeral @ real @ A2 ) @ ( numeral_numeral @ real @ B2 ) ) )
      = ( uminus_uminus @ int @ ( divide_divide @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ A2 ) ) @ ( numeral_numeral @ int @ B2 ) ) ) ) ).

% ceiling_divide_eq_div_numeral
thf(fact_3862_power__minus1__even,axiom,
    ! [A: $tType] :
      ( ( ring_1 @ A )
     => ! [N2: nat] :
          ( ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) )
          = ( one_one @ A ) ) ) ).

% power_minus1_even
thf(fact_3863_neg__one__odd__power,axiom,
    ! [A: $tType] :
      ( ( ring_1 @ A )
     => ! [N2: nat] :
          ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 )
         => ( ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N2 )
            = ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ) ).

% neg_one_odd_power
thf(fact_3864_neg__one__even__power,axiom,
    ! [A: $tType] :
      ( ( ring_1 @ A )
     => ! [N2: nat] :
          ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 )
         => ( ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N2 )
            = ( one_one @ A ) ) ) ) ).

% neg_one_even_power
thf(fact_3865_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( ( ring_char_0 @ A )
     => ! [Y2: int,X2: num,N2: nat] :
          ( ( ( ring_1_of_int @ A @ Y2 )
            = ( power_power @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ X2 ) ) @ N2 ) )
          = ( Y2
            = ( power_power @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ X2 ) ) @ N2 ) ) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
thf(fact_3866_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [A: $tType] :
      ( ( ring_char_0 @ A )
     => ! [X2: num,N2: nat,Y2: int] :
          ( ( ( power_power @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ X2 ) ) @ N2 )
            = ( ring_1_of_int @ A @ Y2 ) )
          = ( ( power_power @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ X2 ) ) @ N2 )
            = Y2 ) ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
thf(fact_3867_neg__numeral__le__floor,axiom,
    ! [A: $tType] :
      ( ( archim2362893244070406136eiling @ A )
     => ! [V: num,X2: A] :
          ( ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ V ) ) @ ( archim6421214686448440834_floor @ A @ X2 ) )
          = ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) @ X2 ) ) ) ).

% neg_numeral_le_floor
thf(fact_3868_floor__less__neg__numeral,axiom,
    ! [A: $tType] :
      ( ( archim2362893244070406136eiling @ A )
     => ! [X2: A,V: num] :
          ( ( ord_less @ int @ ( archim6421214686448440834_floor @ A @ X2 ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ V ) ) )
          = ( ord_less @ A @ X2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) ) ) ) ).

% floor_less_neg_numeral
thf(fact_3869_ceiling__le__neg__numeral,axiom,
    ! [A: $tType] :
      ( ( archim2362893244070406136eiling @ A )
     => ! [X2: A,V: num] :
          ( ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ X2 ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ V ) ) )
          = ( ord_less_eq @ A @ X2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) ) ) ) ).

% ceiling_le_neg_numeral
thf(fact_3870_neg__numeral__less__ceiling,axiom,
    ! [A: $tType] :
      ( ( archim2362893244070406136eiling @ A )
     => ! [V: num,X2: A] :
          ( ( ord_less @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ V ) ) @ ( archimedean_ceiling @ A @ X2 ) )
          = ( ord_less @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) @ X2 ) ) ) ).

% neg_numeral_less_ceiling
thf(fact_3871_neg__numeral__less__floor,axiom,
    ! [A: $tType] :
      ( ( archim2362893244070406136eiling @ A )
     => ! [V: num,X2: A] :
          ( ( ord_less @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ V ) ) @ ( archim6421214686448440834_floor @ A @ X2 ) )
          = ( ord_less_eq @ A @ ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) @ ( one_one @ A ) ) @ X2 ) ) ) ).

% neg_numeral_less_floor
thf(fact_3872_floor__le__neg__numeral,axiom,
    ! [A: $tType] :
      ( ( archim2362893244070406136eiling @ A )
     => ! [X2: A,V: num] :
          ( ( ord_less_eq @ int @ ( archim6421214686448440834_floor @ A @ X2 ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ V ) ) )
          = ( ord_less @ A @ X2 @ ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) @ ( one_one @ A ) ) ) ) ) ).

% floor_le_neg_numeral
thf(fact_3873_ceiling__less__neg__numeral,axiom,
    ! [A: $tType] :
      ( ( archim2362893244070406136eiling @ A )
     => ! [X2: A,V: num] :
          ( ( ord_less @ int @ ( archimedean_ceiling @ A @ X2 ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ V ) ) )
          = ( ord_less_eq @ A @ X2 @ ( minus_minus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) @ ( one_one @ A ) ) ) ) ) ).

% ceiling_less_neg_numeral
thf(fact_3874_neg__numeral__le__ceiling,axiom,
    ! [A: $tType] :
      ( ( archim2362893244070406136eiling @ A )
     => ! [V: num,X2: A] :
          ( ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ V ) ) @ ( archimedean_ceiling @ A @ X2 ) )
          = ( ord_less @ A @ ( minus_minus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) @ ( one_one @ A ) ) @ X2 ) ) ) ).

% neg_numeral_le_ceiling
thf(fact_3875_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [A: $tType] :
      ( ( linordered_idom @ A )
     => ! [X2: num,N2: nat,A2: int] :
          ( ( ord_less_eq @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ X2 ) ) @ N2 ) @ ( ring_1_of_int @ A @ A2 ) )
          = ( ord_less_eq @ int @ ( power_power @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ X2 ) ) @ N2 ) @ A2 ) ) ) ).

% neg_numeral_power_le_of_int_cancel_iff
thf(fact_3876_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( ( linordered_idom @ A )
     => ! [A2: int,X2: num,N2: nat] :
          ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ A2 ) @ ( power_power @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ X2 ) ) @ N2 ) )
          = ( ord_less_eq @ int @ A2 @ ( power_power @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ X2 ) ) @ N2 ) ) ) ) ).

% of_int_le_neg_numeral_power_cancel_iff
thf(fact_3877_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [A: $tType] :
      ( ( linordered_idom @ A )
     => ! [X2: num,N2: nat,A2: int] :
          ( ( ord_less @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ X2 ) ) @ N2 ) @ ( ring_1_of_int @ A @ A2 ) )
          = ( ord_less @ int @ ( power_power @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ X2 ) ) @ N2 ) @ A2 ) ) ) ).

% neg_numeral_power_less_of_int_cancel_iff
thf(fact_3878_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( ( linordered_idom @ A )
     => ! [A2: int,X2: num,N2: nat] :
          ( ( ord_less @ A @ ( ring_1_of_int @ A @ A2 ) @ ( power_power @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ X2 ) ) @ N2 ) )
          = ( ord_less @ int @ A2 @ ( power_power @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ X2 ) ) @ N2 ) ) ) ) ).

% of_int_less_neg_numeral_power_cancel_iff
thf(fact_3879_verit__negate__coefficient_I2_J,axiom,
    ! [A: $tType] :
      ( ( ordered_ab_group_add @ A )
     => ! [A2: A,B2: A] :
          ( ( ord_less @ A @ A2 @ B2 )
         => ( ord_less @ A @ ( uminus_uminus @ A @ B2 ) @ ( uminus_uminus @ A @ A2 ) ) ) ) ).

% verit_negate_coefficient(2)
thf(fact_3880_less__minus__iff,axiom,
    ! [A: $tType] :
      ( ( ordered_ab_group_add @ A )
     => ! [A2: A,B2: A] :
          ( ( ord_less @ A @ A2 @ ( uminus_uminus @ A @ B2 ) )
          = ( ord_less @ A @ B2 @ ( uminus_uminus @ A @ A2 ) ) ) ) ).

% less_minus_iff
thf(fact_3881_minus__less__iff,axiom,
    ! [A: $tType] :
      ( ( ordered_ab_group_add @ A )
     => ! [A2: A,B2: A] :
          ( ( ord_less @ A @ ( uminus_uminus @ A @ A2 ) @ B2 )
          = ( ord_less @ A @ ( uminus_uminus @ A @ B2 ) @ A2 ) ) ) ).

% minus_less_iff
thf(fact_3882_mod__minus__right,axiom,
    ! [A: $tType] :
      ( ( euclid8851590272496341667cancel @ A )
     => ! [A2: A,B2: A] :
          ( ( modulo_modulo @ A @ A2 @ ( uminus_uminus @ A @ B2 ) )
          = ( uminus_uminus @ A @ ( modulo_modulo @ A @ ( uminus_uminus @ A @ A2 ) @ B2 ) ) ) ) ).

% mod_minus_right
thf(fact_3883_euclidean__ring__cancel__class_Omod__minus__cong,axiom,
    ! [A: $tType] :
      ( ( euclid8851590272496341667cancel @ A )
     => ! [A2: A,B2: A,A6: A] :
          ( ( ( modulo_modulo @ A @ A2 @ B2 )
            = ( modulo_modulo @ A @ A6 @ B2 ) )
         => ( ( modulo_modulo @ A @ ( uminus_uminus @ A @ A2 ) @ B2 )
            = ( modulo_modulo @ A @ ( uminus_uminus @ A @ A6 ) @ B2 ) ) ) ) ).

% euclidean_ring_cancel_class.mod_minus_cong
thf(fact_3884_mod__minus__eq,axiom,
    ! [A: $tType] :
      ( ( euclid8851590272496341667cancel @ A )
     => ! [A2: A,B2: A] :
          ( ( modulo_modulo @ A @ ( uminus_uminus @ A @ ( modulo_modulo @ A @ A2 @ B2 ) ) @ B2 )
          = ( modulo_modulo @ A @ ( uminus_uminus @ A @ A2 ) @ B2 ) ) ) ).

% mod_minus_eq
thf(fact_3885_numeral__neq__neg__numeral,axiom,
    ! [A: $tType] :
      ( ( ring_char_0 @ A )
     => ! [M: num,N2: num] :
          ( ( numeral_numeral @ A @ M )
         != ( uminus_uminus @ A @ ( numeral_numeral @ A @ N2 ) ) ) ) ).

% numeral_neq_neg_numeral
thf(fact_3886_neg__numeral__neq__numeral,axiom,
    ! [A: $tType] :
      ( ( ring_char_0 @ A )
     => ! [M: num,N2: num] :
          ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) )
         != ( numeral_numeral @ A @ N2 ) ) ) ).

% neg_numeral_neq_numeral
thf(fact_3887_div__minus__right,axiom,
    ! [A: $tType] :
      ( ( euclid8851590272496341667cancel @ A )
     => ! [A2: A,B2: A] :
          ( ( divide_divide @ A @ A2 @ ( uminus_uminus @ A @ B2 ) )
          = ( divide_divide @ A @ ( uminus_uminus @ A @ A2 ) @ B2 ) ) ) ).

% div_minus_right
thf(fact_3888_minus__divide__left,axiom,
    ! [A: $tType] :
      ( ( division_ring @ A )
     => ! [A2: A,B2: A] :
          ( ( uminus_uminus @ A @ ( divide_divide @ A @ A2 @ B2 ) )
          = ( divide_divide @ A @ ( uminus_uminus @ A @ A2 ) @ B2 ) ) ) ).

% minus_divide_left
thf(fact_3889_minus__divide__divide,axiom,
    ! [A: $tType] :
      ( ( field @ A )
     => ! [A2: A,B2: A] :
          ( ( divide_divide @ A @ ( uminus_uminus @ A @ A2 ) @ ( uminus_uminus @ A @ B2 ) )
          = ( divide_divide @ A @ A2 @ B2 ) ) ) ).

% minus_divide_divide
thf(fact_3890_minus__divide__right,axiom,
    ! [A: $tType] :
      ( ( field @ A )
     => ! [A2: A,B2: A] :
          ( ( uminus_uminus @ A @ ( divide_divide @ A @ A2 @ B2 ) )
          = ( divide_divide @ A @ A2 @ ( uminus_uminus @ A @ B2 ) ) ) ) ).

% minus_divide_right
thf(fact_3891_minus__mult__commute,axiom,
    ! [A: $tType] :
      ( ( ring @ A )
     => ! [A2: A,B2: A] :
          ( ( times_times @ A @ ( uminus_uminus @ A @ A2 ) @ B2 )
          = ( times_times @ A @ A2 @ ( uminus_uminus @ A @ B2 ) ) ) ) ).

% minus_mult_commute
thf(fact_3892_square__eq__iff,axiom,
    ! [A: $tType] :
      ( ( idom @ A )
     => ! [A2: A,B2: A] :
          ( ( ( times_times @ A @ A2 @ A2 )
            = ( times_times @ A @ B2 @ B2 ) )
          = ( ( A2 = B2 )
            | ( A2
              = ( uminus_uminus @ A @ B2 ) ) ) ) ) ).

% square_eq_iff
thf(fact_3893_group__cancel_Oneg1,axiom,
    ! [A: $tType] :
      ( ( ab_group_add @ A )
     => ! [A3: A,K: A,A2: A] :
          ( ( A3
            = ( plus_plus @ A @ K @ A2 ) )
         => ( ( uminus_uminus @ A @ A3 )
            = ( plus_plus @ A @ ( uminus_uminus @ A @ K ) @ ( uminus_uminus @ A @ A2 ) ) ) ) ) ).

% group_cancel.neg1
thf(fact_3894_add_Oinverse__distrib__swap,axiom,
    ! [A: $tType] :
      ( ( group_add @ A )
     => ! [A2: A,B2: A] :
          ( ( uminus_uminus @ A @ ( plus_plus @ A @ A2 @ B2 ) )
          = ( plus_plus @ A @ ( uminus_uminus @ A @ B2 ) @ ( uminus_uminus @ A @ A2 ) ) ) ) ).

% add.inverse_distrib_swap
thf(fact_3895_is__num__normalize_I8_J,axiom,
    ! [A: $tType] :
      ( ( neg_numeral @ A )
     => ! [A2: A,B2: A] :
          ( ( uminus_uminus @ A @ ( plus_plus @ A @ A2 @ B2 ) )
          = ( plus_plus @ A @ ( uminus_uminus @ A @ B2 ) @ ( uminus_uminus @ A @ A2 ) ) ) ) ).

% is_num_normalize(8)
thf(fact_3896_minus__diff__commute,axiom,
    ! [A: $tType] :
      ( ( ab_group_add @ A )
     => ! [B2: A,A2: A] :
          ( ( minus_minus @ A @ ( uminus_uminus @ A @ B2 ) @ A2 )
          = ( minus_minus @ A @ ( uminus_uminus @ A @ A2 ) @ B2 ) ) ) ).

% minus_diff_commute
thf(fact_3897_minus__diff__minus,axiom,
    ! [A: $tType] :
      ( ( ab_group_add @ A )
     => ! [A2: A,B2: A] :
          ( ( minus_minus @ A @ ( uminus_uminus @ A @ A2 ) @ ( uminus_uminus @ A @ B2 ) )
          = ( uminus_uminus @ A @ ( minus_minus @ A @ A2 @ B2 ) ) ) ) ).

% minus_diff_minus
thf(fact_3898_le__imp__neg__le,axiom,
    ! [A: $tType] :
      ( ( ordered_ab_group_add @ A )
     => ! [A2: A,B2: A] :
          ( ( ord_less_eq @ A @ A2 @ B2 )
         => ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ ( uminus_uminus @ A @ A2 ) ) ) ) ).

% le_imp_neg_le
thf(fact_3899_minus__le__iff,axiom,
    ! [A: $tType] :
      ( ( ordered_ab_group_add @ A )
     => ! [A2: A,B2: A] :
          ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ A2 ) @ B2 )
          = ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ A2 ) ) ) ).

% minus_le_iff
thf(fact_3900_le__minus__iff,axiom,
    ! [A: $tType] :
      ( ( ordered_ab_group_add @ A )
     => ! [A2: A,B2: A] :
          ( ( ord_less_eq @ A @ A2 @ ( uminus_uminus @ A @ B2 ) )
          = ( ord_less_eq @ A @ B2 @ ( uminus_uminus @ A @ A2 ) ) ) ) ).

% le_minus_iff
thf(fact_3901_fact__ge__self,axiom,
    ! [N2: nat] : ( ord_less_eq @ nat @ N2 @ ( semiring_char_0_fact @ nat @ N2 ) ) ).

% fact_ge_self
thf(fact_3902_fact__mono__nat,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq @ nat @ M @ N2 )
     => ( ord_less_eq @ nat @ ( semiring_char_0_fact @ nat @ M ) @ ( semiring_char_0_fact @ nat @ N2 ) ) ) ).

% fact_mono_nat
thf(fact_3903_one__neq__neg__one,axiom,
    ! [A: $tType] :
      ( ( ring_char_0 @ A )
     => ( ( one_one @ A )
       != ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).

% one_neq_neg_one
thf(fact_3904_of__int__neg__numeral,axiom,
    ! [A: $tType] :
      ( ( ring_1 @ A )
     => ! [K: num] :
          ( ( ring_1_of_int @ A @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K ) ) )
          = ( uminus_uminus @ A @ ( numeral_numeral @ A @ K ) ) ) ) ).

% of_int_neg_numeral
thf(fact_3905_sum__negf,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ab_group_add @ A )
     => ! [F: B > A,A3: set @ B] :
          ( ( groups7311177749621191930dd_sum @ B @ A
            @ ^ [X: B] : ( uminus_uminus @ A @ ( F @ X ) )
            @ A3 )
          = ( uminus_uminus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F @ A3 ) ) ) ) ).

% sum_negf
thf(fact_3906_sums__minus,axiom,
    ! [A: $tType] :
      ( ( real_V822414075346904944vector @ A )
     => ! [F: nat > A,A2: A] :
          ( ( sums @ A @ F @ A2 )
         => ( sums @ A
            @ ^ [N: nat] : ( uminus_uminus @ A @ ( F @ N ) )
            @ ( uminus_uminus @ A @ A2 ) ) ) ) ).

% sums_minus
thf(fact_3907_fact__less__mono__nat,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
     => ( ( ord_less @ nat @ M @ N2 )
       => ( ord_less @ nat @ ( semiring_char_0_fact @ nat @ M ) @ ( semiring_char_0_fact @ nat @ N2 ) ) ) ) ).

% fact_less_mono_nat
thf(fact_3908_fact__ge__zero,axiom,
    ! [A: $tType] :
      ( ( linordered_semidom @ A )
     => ! [N2: nat] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( semiring_char_0_fact @ A @ N2 ) ) ) ).

% fact_ge_zero
thf(fact_3909_fact__not__neg,axiom,
    ! [A: $tType] :
      ( ( linordered_semidom @ A )
     => ! [N2: nat] :
          ~ ( ord_less @ A @ ( semiring_char_0_fact @ A @ N2 ) @ ( zero_zero @ A ) ) ) ).

% fact_not_neg
thf(fact_3910_fact__gt__zero,axiom,
    ! [A: $tType] :
      ( ( linordered_semidom @ A )
     => ! [N2: nat] : ( ord_less @ A @ ( zero_zero @ A ) @ ( semiring_char_0_fact @ A @ N2 ) ) ) ).

% fact_gt_zero
thf(fact_3911_fact__ge__1,axiom,
    ! [A: $tType] :
      ( ( linordered_semidom @ A )
     => ! [N2: nat] : ( ord_less_eq @ A @ ( one_one @ A ) @ ( semiring_char_0_fact @ A @ N2 ) ) ) ).

% fact_ge_1
thf(fact_3912_zero__neq__neg__numeral,axiom,
    ! [A: $tType] :
      ( ( ring_char_0 @ A )
     => ! [N2: num] :
          ( ( zero_zero @ A )
         != ( uminus_uminus @ A @ ( numeral_numeral @ A @ N2 ) ) ) ) ).

% zero_neq_neg_numeral
thf(fact_3913_neg__numeral__le__numeral,axiom,
    ! [A: $tType] :
      ( ( linordered_idom @ A )
     => ! [M: num,N2: num] : ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( numeral_numeral @ A @ N2 ) ) ) ).

% neg_numeral_le_numeral
thf(fact_3914_not__numeral__le__neg__numeral,axiom,
    ! [A: $tType] :
      ( ( linordered_idom @ A )
     => ! [M: num,N2: num] :
          ~ ( ord_less_eq @ A @ ( numeral_numeral @ A @ M ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N2 ) ) ) ) ).

% not_numeral_le_neg_numeral
thf(fact_3915_neg__numeral__less__numeral,axiom,
    ! [A: $tType] :
      ( ( linordered_idom @ A )
     => ! [M: num,N2: num] : ( ord_less @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( numeral_numeral @ A @ N2 ) ) ) ).

% neg_numeral_less_numeral
thf(fact_3916_not__numeral__less__neg__numeral,axiom,
    ! [A: $tType] :
      ( ( linordered_idom @ A )
     => ! [M: num,N2: num] :
          ~ ( ord_less @ A @ ( numeral_numeral @ A @ M ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N2 ) ) ) ) ).

% not_numeral_less_neg_numeral
thf(fact_3917_add__eq__0__iff,axiom,
    ! [A: $tType] :
      ( ( group_add @ A )
     => ! [A2: A,B2: A] :
          ( ( ( plus_plus @ A @ A2 @ B2 )
            = ( zero_zero @ A ) )
          = ( B2
            = ( uminus_uminus @ A @ A2 ) ) ) ) ).

% add_eq_0_iff
thf(fact_3918_ab__group__add__class_Oab__left__minus,axiom,
    ! [A: $tType] :
      ( ( ab_group_add @ A )
     => ! [A2: A] :
          ( ( plus_plus @ A @ ( uminus_uminus @ A @ A2 ) @ A2 )
          = ( zero_zero @ A ) ) ) ).

% ab_group_add_class.ab_left_minus
thf(fact_3919_add_Oinverse__unique,axiom,
    ! [A: $tType] :
      ( ( group_add @ A )
     => ! [A2: A,B2: A] :
          ( ( ( plus_plus @ A @ A2 @ B2 )
            = ( zero_zero @ A ) )
         => ( ( uminus_uminus @ A @ A2 )
            = B2 ) ) ) ).

% add.inverse_unique
thf(fact_3920_eq__neg__iff__add__eq__0,axiom,
    ! [A: $tType] :
      ( ( group_add @ A )
     => ! [A2: A,B2: A] :
          ( ( A2
            = ( uminus_uminus @ A @ B2 ) )
          = ( ( plus_plus @ A @ A2 @ B2 )
            = ( zero_zero @ A ) ) ) ) ).

% eq_neg_iff_add_eq_0
thf(fact_3921_neg__eq__iff__add__eq__0,axiom,
    ! [A: $tType] :
      ( ( group_add @ A )
     => ! [A2: A,B2: A] :
          ( ( ( uminus_uminus @ A @ A2 )
            = B2 )
          = ( ( plus_plus @ A @ A2 @ B2 )
            = ( zero_zero @ A ) ) ) ) ).

% neg_eq_iff_add_eq_0
thf(fact_3922_zero__neq__neg__one,axiom,
    ! [A: $tType] :
      ( ( ring_char_0 @ A )
     => ( ( zero_zero @ A )
       != ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).

% zero_neq_neg_one
thf(fact_3923_le__minus__one__simps_I4_J,axiom,
    ! [A: $tType] :
      ( ( linordered_idom @ A )
     => ~ ( ord_less_eq @ A @ ( one_one @ A ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).

% le_minus_one_simps(4)
thf(fact_3924_le__minus__one__simps_I2_J,axiom,
    ! [A: $tType] :
      ( ( linordered_idom @ A )
     => ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( one_one @ A ) ) ) ).

% le_minus_one_simps(2)
thf(fact_3925_less__minus__one__simps_I4_J,axiom,
    ! [A: $tType] :
      ( ( linordered_idom @ A )
     => ~ ( ord_less @ A @ ( one_one @ A ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).

% less_minus_one_simps(4)
thf(fact_3926_less__minus__one__simps_I2_J,axiom,
    ! [A: $tType] :
      ( ( linordered_idom @ A )
     => ( ord_less @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( one_one @ A ) ) ) ).

% less_minus_one_simps(2)
thf(fact_3927_numeral__neq__neg__one,axiom,
    ! [A: $tType] :
      ( ( ring_char_0 @ A )
     => ! [N2: num] :
          ( ( numeral_numeral @ A @ N2 )
         != ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).

% numeral_neq_neg_one
thf(fact_3928_one__neq__neg__numeral,axiom,
    ! [A: $tType] :
      ( ( ring_char_0 @ A )
     => ! [N2: num] :
          ( ( one_one @ A )
         != ( uminus_uminus @ A @ ( numeral_numeral @ A @ N2 ) ) ) ) ).

% one_neq_neg_numeral
thf(fact_3929_numeral__times__minus__swap,axiom,
    ! [A: $tType] :
      ( ( comm_ring_1 @ A )
     => ! [W: num,X2: A] :
          ( ( times_times @ A @ ( numeral_numeral @ A @ W ) @ ( uminus_uminus @ A @ X2 ) )
          = ( times_times @ A @ X2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) ) ) ) ).

% numeral_times_minus_swap
thf(fact_3930_nonzero__minus__divide__right,axiom,
    ! [A: $tType] :
      ( ( division_ring @ A )
     => ! [B2: A,A2: A] :
          ( ( B2
           != ( zero_zero @ A ) )
         => ( ( uminus_uminus @ A @ ( divide_divide @ A @ A2 @ B2 ) )
            = ( divide_divide @ A @ A2 @ ( uminus_uminus @ A @ B2 ) ) ) ) ) ).

% nonzero_minus_divide_right
thf(fact_3931_nonzero__minus__divide__divide,axiom,
    ! [A: $tType] :
      ( ( division_ring @ A )
     => ! [B2: A,A2: A] :
          ( ( B2
           != ( zero_zero @ A ) )
         => ( ( divide_divide @ A @ ( uminus_uminus @ A @ A2 ) @ ( uminus_uminus @ A @ B2 ) )
            = ( divide_divide @ A @ A2 @ B2 ) ) ) ) ).

% nonzero_minus_divide_divide
thf(fact_3932_square__eq__1__iff,axiom,
    ! [A: $tType] :
      ( ( ring_15535105094025558882visors @ A )
     => ! [X2: A] :
          ( ( ( times_times @ A @ X2 @ X2 )
            = ( one_one @ A ) )
          = ( ( X2
              = ( one_one @ A ) )
            | ( X2
              = ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ) ) ).

% square_eq_1_iff
thf(fact_3933_fact__mono,axiom,
    ! [A: $tType] :
      ( ( linordered_semidom @ A )
     => ! [M: nat,N2: nat] :
          ( ( ord_less_eq @ nat @ M @ N2 )
         => ( ord_less_eq @ A @ ( semiring_char_0_fact @ A @ M ) @ ( semiring_char_0_fact @ A @ N2 ) ) ) ) ).

% fact_mono
thf(fact_3934_group__cancel_Osub2,axiom,
    ! [A: $tType] :
      ( ( ab_group_add @ A )
     => ! [B6: A,K: A,B2: A,A2: A] :
          ( ( B6
            = ( plus_plus @ A @ K @ B2 ) )
         => ( ( minus_minus @ A @ A2 @ B6 )
            = ( plus_plus @ A @ ( uminus_uminus @ A @ K ) @ ( minus_minus @ A @ A2 @ B2 ) ) ) ) ) ).

% group_cancel.sub2
thf(fact_3935_diff__conv__add__uminus,axiom,
    ! [A: $tType] :
      ( ( group_add @ A )
     => ( ( minus_minus @ A )
        = ( ^ [A5: A,B5: A] : ( plus_plus @ A @ A5 @ ( uminus_uminus @ A @ B5 ) ) ) ) ) ).

% diff_conv_add_uminus
thf(fact_3936_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ! [A: $tType] :
      ( ( ab_group_add @ A )
     => ( ( minus_minus @ A )
        = ( ^ [A5: A,B5: A] : ( plus_plus @ A @ A5 @ ( uminus_uminus @ A @ B5 ) ) ) ) ) ).

% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_3937_dvd__neg__div,axiom,
    ! [A: $tType] :
      ( ( idom_divide @ A )
     => ! [B2: A,A2: A] :
          ( ( dvd_dvd @ A @ B2 @ A2 )
         => ( ( divide_divide @ A @ ( uminus_uminus @ A @ A2 ) @ B2 )
            = ( uminus_uminus @ A @ ( divide_divide @ A @ A2 @ B2 ) ) ) ) ) ).

% dvd_neg_div
thf(fact_3938_dvd__div__neg,axiom,
    ! [A: $tType] :
      ( ( idom_divide @ A )
     => ! [B2: A,A2: A] :
          ( ( dvd_dvd @ A @ B2 @ A2 )
         => ( ( divide_divide @ A @ A2 @ ( uminus_uminus @ A @ B2 ) )
            = ( uminus_uminus @ A @ ( divide_divide @ A @ A2 @ B2 ) ) ) ) ) ).

% dvd_div_neg
thf(fact_3939_fact__dvd,axiom,
    ! [A: $tType] :
      ( ( linordered_semidom @ A )
     => ! [N2: nat,M: nat] :
          ( ( ord_less_eq @ nat @ N2 @ M )
         => ( dvd_dvd @ A @ ( semiring_char_0_fact @ A @ N2 ) @ ( semiring_char_0_fact @ A @ M ) ) ) ) ).

% fact_dvd
thf(fact_3940_int__cases,axiom,
    ! [Z: int] :
      ( ! [N4: nat] :
          ( Z
         != ( semiring_1_of_nat @ int @ N4 ) )
     => ~ ! [N4: nat] :
            ( Z
           != ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ ( suc @ N4 ) ) ) ) ) ).

% int_cases
thf(fact_3941_int__of__nat__induct,axiom,
    ! [P: int > $o,Z: int] :
      ( ! [N4: nat] : ( P @ ( semiring_1_of_nat @ int @ N4 ) )
     => ( ! [N4: nat] : ( P @ ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ ( suc @ N4 ) ) ) )
       => ( P @ Z ) ) ) ).

% int_of_nat_induct
thf(fact_3942_dbl__def,axiom,
    ! [A: $tType] :
      ( ( neg_numeral @ A )
     => ( ( neg_numeral_dbl @ A )
        = ( ^ [X: A] : ( plus_plus @ A @ X @ X ) ) ) ) ).

% dbl_def
thf(fact_3943_pos__zmult__eq__1__iff__lemma,axiom,
    ! [M: int,N2: int] :
      ( ( ( times_times @ int @ M @ N2 )
        = ( one_one @ int ) )
     => ( ( M
          = ( one_one @ int ) )
        | ( M
          = ( uminus_uminus @ int @ ( one_one @ int ) ) ) ) ) ).

% pos_zmult_eq_1_iff_lemma
thf(fact_3944_zmult__eq__1__iff,axiom,
    ! [M: int,N2: int] :
      ( ( ( times_times @ int @ M @ N2 )
        = ( one_one @ int ) )
      = ( ( ( M
            = ( one_one @ int ) )
          & ( N2
            = ( one_one @ int ) ) )
        | ( ( M
            = ( uminus_uminus @ int @ ( one_one @ int ) ) )
          & ( N2
            = ( uminus_uminus @ int @ ( one_one @ int ) ) ) ) ) ) ).

% zmult_eq_1_iff
thf(fact_3945_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_3946_pochhammer__fact,axiom,
    ! [A: $tType] :
      ( ( ( semiring_char_0 @ A )
        & ( comm_semiring_1 @ A ) )
     => ( ( semiring_char_0_fact @ A )
        = ( comm_s3205402744901411588hammer @ A @ ( one_one @ A ) ) ) ) ).

% pochhammer_fact
thf(fact_3947_pochhammer__same,axiom,
    ! [A: $tType] :
      ( ( ( semiring_char_0 @ A )
        & ( comm_ring_1 @ A )
        & ( semiri3467727345109120633visors @ A ) )
     => ! [N2: nat] :
          ( ( comm_s3205402744901411588hammer @ A @ ( uminus_uminus @ A @ ( semiring_1_of_nat @ A @ N2 ) ) @ N2 )
          = ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N2 ) @ ( semiring_char_0_fact @ A @ N2 ) ) ) ) ).

% pochhammer_same
thf(fact_3948_fact__ge__Suc__0__nat,axiom,
    ! [N2: nat] : ( ord_less_eq @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( semiring_char_0_fact @ nat @ N2 ) ) ).

% fact_ge_Suc_0_nat
thf(fact_3949_dvd__fact,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq @ nat @ ( one_one @ nat ) @ M )
     => ( ( ord_less_eq @ nat @ M @ N2 )
       => ( dvd_dvd @ nat @ M @ ( semiring_char_0_fact @ nat @ N2 ) ) ) ) ).

% dvd_fact
thf(fact_3950_neg__numeral__le__zero,axiom,
    ! [A: $tType] :
      ( ( linordered_idom @ A )
     => ! [N2: num] : ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N2 ) ) @ ( zero_zero @ A ) ) ) ).

% neg_numeral_le_zero
thf(fact_3951_not__zero__le__neg__numeral,axiom,
    ! [A: $tType] :
      ( ( linordered_idom @ A )
     => ! [N2: num] :
          ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N2 ) ) ) ) ).

% not_zero_le_neg_numeral
thf(fact_3952_neg__numeral__less__zero,axiom,
    ! [A: $tType] :
      ( ( linordered_idom @ A )
     => ! [N2: num] : ( ord_less @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N2 ) ) @ ( zero_zero @ A ) ) ) ).

% neg_numeral_less_zero
thf(fact_3953_not__zero__less__neg__numeral,axiom,
    ! [A: $tType] :
      ( ( linordered_idom @ A )
     => ! [N2: num] :
          ~ ( ord_less @ A @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N2 ) ) ) ) ).

% not_zero_less_neg_numeral
thf(fact_3954_le__minus__one__simps_I3_J,axiom,
    ! [A: $tType] :
      ( ( linordered_idom @ A )
     => ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).

% le_minus_one_simps(3)
thf(fact_3955_le__minus__one__simps_I1_J,axiom,
    ! [A: $tType] :
      ( ( linordered_idom @ A )
     => ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( zero_zero @ A ) ) ) ).

% le_minus_one_simps(1)
thf(fact_3956_less__minus__one__simps_I3_J,axiom,
    ! [A: $tType] :
      ( ( linordered_idom @ A )
     => ~ ( ord_less @ A @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).

% less_minus_one_simps(3)
thf(fact_3957_less__minus__one__simps_I1_J,axiom,
    ! [A: $tType] :
      ( ( linordered_idom @ A )
     => ( ord_less @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( zero_zero @ A ) ) ) ).

% less_minus_one_simps(1)
thf(fact_3958_neg__numeral__le__one,axiom,
    ! [A: $tType] :
      ( ( linordered_idom @ A )
     => ! [M: num] : ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( one_one @ A ) ) ) ).

% neg_numeral_le_one
thf(fact_3959_neg__one__le__numeral,axiom,
    ! [A: $tType] :
      ( ( linordered_idom @ A )
     => ! [M: num] : ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( numeral_numeral @ A @ M ) ) ) ).

% neg_one_le_numeral
thf(fact_3960_neg__numeral__le__neg__one,axiom,
    ! [A: $tType] :
      ( ( linordered_idom @ A )
     => ! [M: num] : ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).

% neg_numeral_le_neg_one
thf(fact_3961_not__numeral__le__neg__one,axiom,
    ! [A: $tType] :
      ( ( linordered_idom @ A )
     => ! [M: num] :
          ~ ( ord_less_eq @ A @ ( numeral_numeral @ A @ M ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).

% not_numeral_le_neg_one
thf(fact_3962_not__one__le__neg__numeral,axiom,
    ! [A: $tType] :
      ( ( linordered_idom @ A )
     => ! [M: num] :
          ~ ( ord_less_eq @ A @ ( one_one @ A ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) ) ) ).

% not_one_le_neg_numeral
thf(fact_3963_neg__numeral__less__one,axiom,
    ! [A: $tType] :
      ( ( linordered_idom @ A )
     => ! [M: num] : ( ord_less @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( one_one @ A ) ) ) ).

% neg_numeral_less_one
thf(fact_3964_neg__one__less__numeral,axiom,
    ! [A: $tType] :
      ( ( linordered_idom @ A )
     => ! [M: num] : ( ord_less @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( numeral_numeral @ A @ M ) ) ) ).

% neg_one_less_numeral
thf(fact_3965_not__numeral__less__neg__one,axiom,
    ! [A: $tType] :
      ( ( linordered_idom @ A )
     => ! [M: num] :
          ~ ( ord_less @ A @ ( numeral_numeral @ A @ M ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).

% not_numeral_less_neg_one
thf(fact_3966_not__one__less__neg__numeral,axiom,
    ! [A: $tType] :
      ( ( linordered_idom @ A )
     => ! [M: num] :
          ~ ( ord_less @ A @ ( one_one @ A ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) ) ) ).

% not_one_less_neg_numeral
thf(fact_3967_not__neg__one__less__neg__numeral,axiom,
    ! [A: $tType] :
      ( ( linordered_idom @ A )
     => ! [M: num] :
          ~ ( ord_less @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) ) ) ).

% not_neg_one_less_neg_numeral
thf(fact_3968_gbinomial__pochhammer,axiom,
    ! [A: $tType] :
      ( ( field_char_0 @ A )
     => ( ( gbinomial @ A )
        = ( ^ [A5: A,K4: nat] : ( divide_divide @ A @ ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ K4 ) @ ( comm_s3205402744901411588hammer @ A @ ( uminus_uminus @ A @ A5 ) @ K4 ) ) @ ( semiring_char_0_fact @ A @ K4 ) ) ) ) ) ).

% gbinomial_pochhammer
thf(fact_3969_uminus__numeral__One,axiom,
    ! [A: $tType] :
      ( ( neg_numeral @ A )
     => ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ one2 ) )
        = ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).

% uminus_numeral_One
thf(fact_3970_mult__1s__ring__1_I2_J,axiom,
    ! [A: $tType] :
      ( ( ring_1 @ A )
     => ! [B2: A] :
          ( ( times_times @ A @ B2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ one2 ) ) )
          = ( uminus_uminus @ A @ B2 ) ) ) ).

% mult_1s_ring_1(2)
thf(fact_3971_mult__1s__ring__1_I1_J,axiom,
    ! [A: $tType] :
      ( ( ring_1 @ A )
     => ! [B2: A] :
          ( ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ one2 ) ) @ B2 )
          = ( uminus_uminus @ A @ B2 ) ) ) ).

% mult_1s_ring_1(1)
thf(fact_3972_divide__eq__minus__1__iff,axiom,
    ! [A: $tType] :
      ( ( field @ A )
     => ! [A2: A,B2: A] :
          ( ( ( divide_divide @ A @ A2 @ B2 )
            = ( uminus_uminus @ A @ ( one_one @ A ) ) )
          = ( ( B2
             != ( zero_zero @ A ) )
            & ( A2
              = ( uminus_uminus @ A @ B2 ) ) ) ) ) ).

% divide_eq_minus_1_iff
thf(fact_3973_eq__minus__divide__eq,axiom,
    ! [A: $tType] :
      ( ( division_ring @ A )
     => ! [A2: A,B2: A,C2: A] :
          ( ( A2
            = ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C2 ) ) )
          = ( ( ( C2
               != ( zero_zero @ A ) )
             => ( ( times_times @ A @ A2 @ C2 )
                = ( uminus_uminus @ A @ B2 ) ) )
            & ( ( C2
                = ( zero_zero @ A ) )
             => ( A2
                = ( zero_zero @ A ) ) ) ) ) ) ).

% eq_minus_divide_eq
thf(fact_3974_minus__divide__eq__eq,axiom,
    ! [A: $tType] :
      ( ( division_ring @ A )
     => ! [B2: A,C2: A,A2: A] :
          ( ( ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C2 ) )
            = A2 )
          = ( ( ( C2
               != ( zero_zero @ A ) )
             => ( ( uminus_uminus @ A @ B2 )
                = ( times_times @ A @ A2 @ C2 ) ) )
            & ( ( C2
                = ( zero_zero @ A ) )
             => ( A2
                = ( zero_zero @ A ) ) ) ) ) ) ).

% minus_divide_eq_eq
thf(fact_3975_nonzero__neg__divide__eq__eq,axiom,
    ! [A: $tType] :
      ( ( division_ring @ A )
     => ! [B2: A,A2: A,C2: A] :
          ( ( B2
           != ( zero_zero @ A ) )
         => ( ( ( uminus_uminus @ A @ ( divide_divide @ A @ A2 @ B2 ) )
              = C2 )
            = ( ( uminus_uminus @ A @ A2 )
              = ( times_times @ A @ C2 @ B2 ) ) ) ) ) ).

% nonzero_neg_divide_eq_eq
thf(fact_3976_nonzero__neg__divide__eq__eq2,axiom,
    ! [A: $tType] :
      ( ( division_ring @ A )
     => ! [B2: A,C2: A,A2: A] :
          ( ( B2
           != ( zero_zero @ A ) )
         => ( ( C2
              = ( uminus_uminus @ A @ ( divide_divide @ A @ A2 @ B2 ) ) )
            = ( ( times_times @ A @ C2 @ B2 )
              = ( uminus_uminus @ A @ A2 ) ) ) ) ) ).

% nonzero_neg_divide_eq_eq2
thf(fact_3977_fact__less__mono,axiom,
    ! [A: $tType] :
      ( ( linordered_semidom @ A )
     => ! [M: nat,N2: nat] :
          ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
         => ( ( ord_less @ nat @ M @ N2 )
           => ( ord_less @ A @ ( semiring_char_0_fact @ A @ M ) @ ( semiring_char_0_fact @ A @ N2 ) ) ) ) ) ).

% fact_less_mono
thf(fact_3978_power__minus,axiom,
    ! [A: $tType] :
      ( ( ring_1 @ A )
     => ! [A2: A,N2: nat] :
          ( ( power_power @ A @ ( uminus_uminus @ A @ A2 ) @ N2 )
          = ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N2 ) @ ( power_power @ A @ A2 @ N2 ) ) ) ) ).

% power_minus
thf(fact_3979_power__minus__Bit0,axiom,
    ! [A: $tType] :
      ( ( ring_1 @ A )
     => ! [X2: A,K: num] :
          ( ( power_power @ A @ ( uminus_uminus @ A @ X2 ) @ ( numeral_numeral @ nat @ ( bit0 @ K ) ) )
          = ( power_power @ A @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ K ) ) ) ) ) ).

% power_minus_Bit0
thf(fact_3980_fact__mod,axiom,
    ! [A: $tType] :
      ( ( ( linordered_semidom @ A )
        & ( semidom_modulo @ A ) )
     => ! [M: nat,N2: nat] :
          ( ( ord_less_eq @ nat @ M @ N2 )
         => ( ( modulo_modulo @ A @ ( semiring_char_0_fact @ A @ N2 ) @ ( semiring_char_0_fact @ A @ M ) )
            = ( zero_zero @ A ) ) ) ) ).

% fact_mod
thf(fact_3981_fact__fact__dvd__fact,axiom,
    ! [A: $tType] :
      ( ( linordered_semidom @ A )
     => ! [K: nat,N2: nat] : ( dvd_dvd @ A @ ( times_times @ A @ ( semiring_char_0_fact @ A @ K ) @ ( semiring_char_0_fact @ A @ N2 ) ) @ ( semiring_char_0_fact @ A @ ( plus_plus @ nat @ K @ N2 ) ) ) ) ).

% fact_fact_dvd_fact
thf(fact_3982_power__minus__Bit1,axiom,
    ! [A: $tType] :
      ( ( ring_1 @ A )
     => ! [X2: A,K: num] :
          ( ( power_power @ A @ ( uminus_uminus @ A @ X2 ) @ ( numeral_numeral @ nat @ ( bit1 @ K ) ) )
          = ( uminus_uminus @ A @ ( power_power @ A @ X2 @ ( numeral_numeral @ nat @ ( bit1 @ K ) ) ) ) ) ) ).

% power_minus_Bit1
thf(fact_3983_fact__le__power,axiom,
    ! [A: $tType] :
      ( ( linordered_semidom @ A )
     => ! [N2: nat] : ( ord_less_eq @ A @ ( semiring_char_0_fact @ A @ N2 ) @ ( semiring_1_of_nat @ A @ ( power_power @ nat @ N2 @ N2 ) ) ) ) ).

% fact_le_power
thf(fact_3984_norm__uminus__minus,axiom,
    ! [A: $tType] :
      ( ( real_V822414075346904944vector @ A )
     => ! [X2: A,Y2: A] :
          ( ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ ( uminus_uminus @ A @ X2 ) @ Y2 ) )
          = ( real_V7770717601297561774m_norm @ A @ ( plus_plus @ A @ X2 @ Y2 ) ) ) ) ).

% norm_uminus_minus
thf(fact_3985_powr__minus__divide,axiom,
    ! [A: $tType] :
      ( ( ( real_V3459762299906320749_field @ A )
        & ( ln @ A ) )
     => ! [X2: A,A2: A] :
          ( ( powr @ A @ X2 @ ( uminus_uminus @ A @ A2 ) )
          = ( divide_divide @ A @ ( one_one @ A ) @ ( powr @ A @ X2 @ A2 ) ) ) ) ).

% powr_minus_divide
thf(fact_3986_int__cases4,axiom,
    ! [M: int] :
      ( ! [N4: nat] :
          ( M
         != ( semiring_1_of_nat @ int @ N4 ) )
     => ~ ! [N4: nat] :
            ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N4 )
           => ( M
             != ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ N4 ) ) ) ) ) ).

% int_cases4
thf(fact_3987_int__zle__neg,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_eq @ int @ ( semiring_1_of_nat @ int @ N2 ) @ ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ M ) ) )
      = ( ( N2
          = ( zero_zero @ nat ) )
        & ( M
          = ( zero_zero @ nat ) ) ) ) ).

% int_zle_neg
thf(fact_3988_nonpos__int__cases,axiom,
    ! [K: int] :
      ( ( ord_less_eq @ int @ K @ ( zero_zero @ int ) )
     => ~ ! [N4: nat] :
            ( K
           != ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ N4 ) ) ) ) ).

% nonpos_int_cases
thf(fact_3989_negative__zle__0,axiom,
    ! [N2: nat] : ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ N2 ) ) @ ( zero_zero @ int ) ) ).

% negative_zle_0
thf(fact_3990_zmod__zminus2__eq__if,axiom,
    ! [A2: int,B2: int] :
      ( ( ( ( modulo_modulo @ int @ A2 @ B2 )
          = ( zero_zero @ int ) )
       => ( ( modulo_modulo @ int @ A2 @ ( uminus_uminus @ int @ B2 ) )
          = ( zero_zero @ int ) ) )
      & ( ( ( modulo_modulo @ int @ A2 @ B2 )
         != ( zero_zero @ int ) )
       => ( ( modulo_modulo @ int @ A2 @ ( uminus_uminus @ int @ B2 ) )
          = ( minus_minus @ int @ ( modulo_modulo @ int @ A2 @ B2 ) @ B2 ) ) ) ) ).

% zmod_zminus2_eq_if
thf(fact_3991_zmod__zminus1__eq__if,axiom,
    ! [A2: int,B2: int] :
      ( ( ( ( modulo_modulo @ int @ A2 @ B2 )
          = ( zero_zero @ int ) )
       => ( ( modulo_modulo @ int @ ( uminus_uminus @ int @ A2 ) @ B2 )
          = ( zero_zero @ int ) ) )
      & ( ( ( modulo_modulo @ int @ A2 @ B2 )
         != ( zero_zero @ int ) )
       => ( ( modulo_modulo @ int @ ( uminus_uminus @ int @ A2 ) @ B2 )
          = ( minus_minus @ int @ B2 @ ( modulo_modulo @ int @ A2 @ B2 ) ) ) ) ) ).

% zmod_zminus1_eq_if
thf(fact_3992_fact__prod,axiom,
    ! [A: $tType] :
      ( ( semiring_char_0 @ A )
     => ( ( semiring_char_0_fact @ A )
        = ( ^ [N: nat] :
              ( semiring_1_of_nat @ A
              @ ( groups7121269368397514597t_prod @ nat @ nat
                @ ^ [X: nat] : X
                @ ( set_or1337092689740270186AtMost @ nat @ ( one_one @ nat ) @ N ) ) ) ) ) ) ).

% fact_prod
thf(fact_3993_fact__diff__Suc,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less @ nat @ N2 @ ( suc @ M ) )
     => ( ( semiring_char_0_fact @ nat @ ( minus_minus @ nat @ ( suc @ M ) @ N2 ) )
        = ( times_times @ nat @ ( minus_minus @ nat @ ( suc @ M ) @ N2 ) @ ( semiring_char_0_fact @ nat @ ( minus_minus @ nat @ M @ N2 ) ) ) ) ) ).

% fact_diff_Suc
thf(fact_3994_fact__div__fact__le__pow,axiom,
    ! [R4: nat,N2: nat] :
      ( ( ord_less_eq @ nat @ R4 @ N2 )
     => ( ord_less_eq @ nat @ ( divide_divide @ nat @ ( semiring_char_0_fact @ nat @ N2 ) @ ( semiring_char_0_fact @ nat @ ( minus_minus @ nat @ N2 @ R4 ) ) ) @ ( power_power @ nat @ N2 @ R4 ) ) ) ).

% fact_div_fact_le_pow
thf(fact_3995_binomial__fact__lemma,axiom,
    ! [K: nat,N2: nat] :
      ( ( ord_less_eq @ nat @ K @ N2 )
     => ( ( times_times @ nat @ ( times_times @ nat @ ( semiring_char_0_fact @ nat @ K ) @ ( semiring_char_0_fact @ nat @ ( minus_minus @ nat @ N2 @ K ) ) ) @ ( binomial @ N2 @ K ) )
        = ( semiring_char_0_fact @ nat @ N2 ) ) ) ).

% binomial_fact_lemma
thf(fact_3996_less__minus__divide__eq,axiom,
    ! [A: $tType] :
      ( ( linordered_field @ A )
     => ! [A2: A,B2: A,C2: A] :
          ( ( ord_less @ A @ A2 @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C2 ) ) )
          = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
             => ( ord_less @ A @ ( times_times @ A @ A2 @ C2 ) @ ( uminus_uminus @ A @ B2 ) ) )
            & ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
             => ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
                 => ( ord_less @ A @ ( uminus_uminus @ A @ B2 ) @ ( times_times @ A @ A2 @ C2 ) ) )
                & ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
                 => ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ) ) ) ) ).

% less_minus_divide_eq
thf(fact_3997_minus__divide__less__eq,axiom,
    ! [A: $tType] :
      ( ( linordered_field @ A )
     => ! [B2: A,C2: A,A2: A] :
          ( ( ord_less @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C2 ) ) @ A2 )
          = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
             => ( ord_less @ A @ ( uminus_uminus @ A @ B2 ) @ ( times_times @ A @ A2 @ C2 ) ) )
            & ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
             => ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
                 => ( ord_less @ A @ ( times_times @ A @ A2 @ C2 ) @ ( uminus_uminus @ A @ B2 ) ) )
                & ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
                 => ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ) ) ) ) ).

% minus_divide_less_eq
thf(fact_3998_neg__less__minus__divide__eq,axiom,
    ! [A: $tType] :
      ( ( linordered_field @ A )
     => ! [C2: A,A2: A,B2: A] :
          ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
         => ( ( ord_less @ A @ A2 @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C2 ) ) )
            = ( ord_less @ A @ ( uminus_uminus @ A @ B2 ) @ ( times_times @ A @ A2 @ C2 ) ) ) ) ) ).

% neg_less_minus_divide_eq
thf(fact_3999_neg__minus__divide__less__eq,axiom,
    ! [A: $tType] :
      ( ( linordered_field @ A )
     => ! [C2: A,B2: A,A2: A] :
          ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
         => ( ( ord_less @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C2 ) ) @ A2 )
            = ( ord_less @ A @ ( times_times @ A @ A2 @ C2 ) @ ( uminus_uminus @ A @ B2 ) ) ) ) ) ).

% neg_minus_divide_less_eq
thf(fact_4000_pos__less__minus__divide__eq,axiom,
    ! [A: $tType] :
      ( ( linordered_field @ A )
     => ! [C2: A,A2: A,B2: A] :
          ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
         => ( ( ord_less @ A @ A2 @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C2 ) ) )
            = ( ord_less @ A @ ( times_times @ A @ A2 @ C2 ) @ ( uminus_uminus @ A @ B2 ) ) ) ) ) ).

% pos_less_minus_divide_eq
thf(fact_4001_pos__minus__divide__less__eq,axiom,
    ! [A: $tType] :
      ( ( linordered_field @ A )
     => ! [C2: A,B2: A,A2: A] :
          ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
         => ( ( ord_less @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C2 ) ) @ A2 )
            = ( ord_less @ A @ ( uminus_uminus @ A @ B2 ) @ ( times_times @ A @ A2 @ C2 ) ) ) ) ) ).

% pos_minus_divide_less_eq
thf(fact_4002_divide__eq__eq__numeral_I2_J,axiom,
    ! [A: $tType] :
      ( ( division_ring @ A )
     => ! [B2: A,C2: A,W: num] :
          ( ( ( divide_divide @ A @ B2 @ C2 )
            = ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) )
          = ( ( ( C2
               != ( zero_zero @ A ) )
             => ( B2
                = ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) @ C2 ) ) )
            & ( ( C2
                = ( zero_zero @ A ) )
             => ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) )
                = ( zero_zero @ A ) ) ) ) ) ) ).

% divide_eq_eq_numeral(2)
thf(fact_4003_eq__divide__eq__numeral_I2_J,axiom,
    ! [A: $tType] :
      ( ( division_ring @ A )
     => ! [W: num,B2: A,C2: A] :
          ( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) )
            = ( divide_divide @ A @ B2 @ C2 ) )
          = ( ( ( C2
               != ( zero_zero @ A ) )
             => ( ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) @ C2 )
                = B2 ) )
            & ( ( C2
                = ( zero_zero @ A ) )
             => ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) )
                = ( zero_zero @ A ) ) ) ) ) ) ).

% eq_divide_eq_numeral(2)
thf(fact_4004_add__divide__eq__if__simps_I3_J,axiom,
    ! [A: $tType] :
      ( ( division_ring @ A )
     => ! [Z: A,A2: A,B2: A] :
          ( ( ( Z
              = ( zero_zero @ A ) )
           => ( ( plus_plus @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ A2 @ Z ) ) @ B2 )
              = B2 ) )
          & ( ( Z
             != ( zero_zero @ A ) )
           => ( ( plus_plus @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ A2 @ Z ) ) @ B2 )
              = ( divide_divide @ A @ ( plus_plus @ A @ ( uminus_uminus @ A @ A2 ) @ ( times_times @ A @ B2 @ Z ) ) @ Z ) ) ) ) ) ).

% add_divide_eq_if_simps(3)
thf(fact_4005_minus__divide__add__eq__iff,axiom,
    ! [A: $tType] :
      ( ( division_ring @ A )
     => ! [Z: A,X2: A,Y2: A] :
          ( ( Z
           != ( zero_zero @ A ) )
         => ( ( plus_plus @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ X2 @ Z ) ) @ Y2 )
            = ( divide_divide @ A @ ( plus_plus @ A @ ( uminus_uminus @ A @ X2 ) @ ( times_times @ A @ Y2 @ Z ) ) @ Z ) ) ) ) ).

% minus_divide_add_eq_iff
thf(fact_4006_add__divide__eq__if__simps_I6_J,axiom,
    ! [A: $tType] :
      ( ( division_ring @ A )
     => ! [Z: A,A2: A,B2: A] :
          ( ( ( Z
              = ( zero_zero @ A ) )
           => ( ( minus_minus @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ A2 @ Z ) ) @ B2 )
              = ( uminus_uminus @ A @ B2 ) ) )
          & ( ( Z
             != ( zero_zero @ A ) )
           => ( ( minus_minus @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ A2 @ Z ) ) @ B2 )
              = ( divide_divide @ A @ ( minus_minus @ A @ ( uminus_uminus @ A @ A2 ) @ ( times_times @ A @ B2 @ Z ) ) @ Z ) ) ) ) ) ).

% add_divide_eq_if_simps(6)
thf(fact_4007_add__divide__eq__if__simps_I5_J,axiom,
    ! [A: $tType] :
      ( ( division_ring @ A )
     => ! [Z: A,A2: A,B2: A] :
          ( ( ( Z
              = ( zero_zero @ A ) )
           => ( ( minus_minus @ A @ ( divide_divide @ A @ A2 @ Z ) @ B2 )
              = ( uminus_uminus @ A @ B2 ) ) )
          & ( ( Z
             != ( zero_zero @ A ) )
           => ( ( minus_minus @ A @ ( divide_divide @ A @ A2 @ Z ) @ B2 )
              = ( divide_divide @ A @ ( minus_minus @ A @ A2 @ ( times_times @ A @ B2 @ Z ) ) @ Z ) ) ) ) ) ).

% add_divide_eq_if_simps(5)
thf(fact_4008_minus__divide__diff__eq__iff,axiom,
    ! [A: $tType] :
      ( ( division_ring @ A )
     => ! [Z: A,X2: A,Y2: A] :
          ( ( Z
           != ( zero_zero @ A ) )
         => ( ( minus_minus @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ X2 @ Z ) ) @ Y2 )
            = ( divide_divide @ A @ ( minus_minus @ A @ ( uminus_uminus @ A @ X2 ) @ ( times_times @ A @ Y2 @ Z ) ) @ Z ) ) ) ) ).

% minus_divide_diff_eq_iff
thf(fact_4009_even__minus,axiom,
    ! [A: $tType] :
      ( ( ring_parity @ A )
     => ! [A2: A] :
          ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( uminus_uminus @ A @ A2 ) )
          = ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) ) ) ).

% even_minus
thf(fact_4010_power2__eq__iff,axiom,
    ! [A: $tType] :
      ( ( idom @ A )
     => ! [X2: A,Y2: A] :
          ( ( ( power_power @ A @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
            = ( power_power @ A @ Y2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
          = ( ( X2 = Y2 )
            | ( X2
              = ( uminus_uminus @ A @ Y2 ) ) ) ) ) ).

% power2_eq_iff
thf(fact_4011_choose__dvd,axiom,
    ! [A: $tType] :
      ( ( linordered_semidom @ A )
     => ! [K: nat,N2: nat] :
          ( ( ord_less_eq @ nat @ K @ N2 )
         => ( dvd_dvd @ A @ ( times_times @ A @ ( semiring_char_0_fact @ A @ K ) @ ( semiring_char_0_fact @ A @ ( minus_minus @ nat @ N2 @ K ) ) ) @ ( semiring_char_0_fact @ A @ N2 ) ) ) ) ).

% choose_dvd
thf(fact_4012_int__cases3,axiom,
    ! [K: int] :
      ( ( K
       != ( zero_zero @ int ) )
     => ( ! [N4: nat] :
            ( ( K
              = ( semiring_1_of_nat @ int @ N4 ) )
           => ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ N4 ) )
       => ~ ! [N4: nat] :
              ( ( K
                = ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ N4 ) ) )
             => ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ N4 ) ) ) ) ).

% int_cases3
thf(fact_4013_fact__eq__fact__times,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_eq @ nat @ N2 @ M )
     => ( ( semiring_char_0_fact @ nat @ M )
        = ( times_times @ nat @ ( semiring_char_0_fact @ nat @ N2 )
          @ ( groups7121269368397514597t_prod @ nat @ nat
            @ ^ [X: nat] : X
            @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ N2 ) @ M ) ) ) ) ) ).

% fact_eq_fact_times
thf(fact_4014_pochhammer__of__nat__eq__0__lemma,axiom,
    ! [A: $tType] :
      ( ( idom @ A )
     => ! [N2: nat,K: nat] :
          ( ( ord_less @ nat @ N2 @ K )
         => ( ( comm_s3205402744901411588hammer @ A @ ( uminus_uminus @ A @ ( semiring_1_of_nat @ A @ N2 ) ) @ K )
            = ( zero_zero @ A ) ) ) ) ).

% pochhammer_of_nat_eq_0_lemma
thf(fact_4015_pochhammer__of__nat__eq__0__iff,axiom,
    ! [A: $tType] :
      ( ( ( ring_char_0 @ A )
        & ( idom @ A ) )
     => ! [N2: nat,K: nat] :
          ( ( ( comm_s3205402744901411588hammer @ A @ ( uminus_uminus @ A @ ( semiring_1_of_nat @ A @ N2 ) ) @ K )
            = ( zero_zero @ A ) )
          = ( ord_less @ nat @ N2 @ K ) ) ) ).

% pochhammer_of_nat_eq_0_iff
thf(fact_4016_pochhammer__eq__0__iff,axiom,
    ! [A: $tType] :
      ( ( field_char_0 @ A )
     => ! [A2: A,N2: nat] :
          ( ( ( comm_s3205402744901411588hammer @ A @ A2 @ N2 )
            = ( zero_zero @ A ) )
          = ( ? [K4: nat] :
                ( ( ord_less @ nat @ K4 @ N2 )
                & ( A2
                  = ( uminus_uminus @ A @ ( semiring_1_of_nat @ A @ K4 ) ) ) ) ) ) ) ).

% pochhammer_eq_0_iff
thf(fact_4017_pochhammer__of__nat__eq__0__lemma_H,axiom,
    ! [A: $tType] :
      ( ( ( ring_char_0 @ A )
        & ( idom @ A ) )
     => ! [K: nat,N2: nat] :
          ( ( ord_less_eq @ nat @ K @ N2 )
         => ( ( comm_s3205402744901411588hammer @ A @ ( uminus_uminus @ A @ ( semiring_1_of_nat @ A @ N2 ) ) @ K )
           != ( zero_zero @ A ) ) ) ) ).

% pochhammer_of_nat_eq_0_lemma'
thf(fact_4018_not__zle__0__negative,axiom,
    ! [N2: nat] :
      ~ ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ ( suc @ N2 ) ) ) ) ).

% not_zle_0_negative
thf(fact_4019_negative__zless__0,axiom,
    ! [N2: nat] : ( ord_less @ int @ ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ ( suc @ N2 ) ) ) @ ( zero_zero @ int ) ) ).

% negative_zless_0
thf(fact_4020_negD,axiom,
    ! [X2: int] :
      ( ( ord_less @ int @ X2 @ ( zero_zero @ int ) )
     => ? [N4: nat] :
          ( X2
          = ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ ( suc @ N4 ) ) ) ) ) ).

% negD
thf(fact_4021_verit__less__mono__div__int2,axiom,
    ! [A3: int,B6: int,N2: int] :
      ( ( ord_less_eq @ int @ A3 @ B6 )
     => ( ( ord_less @ int @ ( zero_zero @ int ) @ ( uminus_uminus @ int @ N2 ) )
       => ( ord_less_eq @ int @ ( divide_divide @ int @ B6 @ N2 ) @ ( divide_divide @ int @ A3 @ N2 ) ) ) ) ).

% verit_less_mono_div_int2
thf(fact_4022_div__eq__minus1,axiom,
    ! [B2: int] :
      ( ( ord_less @ int @ ( zero_zero @ int ) @ B2 )
     => ( ( divide_divide @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ B2 )
        = ( uminus_uminus @ int @ ( one_one @ int ) ) ) ) ).

% div_eq_minus1
thf(fact_4023_ceiling__divide__eq__div,axiom,
    ! [A: $tType] :
      ( ( archim2362893244070406136eiling @ A )
     => ! [A2: int,B2: int] :
          ( ( archimedean_ceiling @ A @ ( divide_divide @ A @ ( ring_1_of_int @ A @ A2 ) @ ( ring_1_of_int @ A @ B2 ) ) )
          = ( uminus_uminus @ int @ ( divide_divide @ int @ ( uminus_uminus @ int @ A2 ) @ B2 ) ) ) ) ).

% ceiling_divide_eq_div
thf(fact_4024_binomial__altdef__nat,axiom,
    ! [K: nat,N2: nat] :
      ( ( ord_less_eq @ nat @ K @ N2 )
     => ( ( binomial @ N2 @ K )
        = ( divide_divide @ nat @ ( semiring_char_0_fact @ nat @ N2 ) @ ( times_times @ nat @ ( semiring_char_0_fact @ nat @ K ) @ ( semiring_char_0_fact @ nat @ ( minus_minus @ nat @ N2 @ K ) ) ) ) ) ) ).

% binomial_altdef_nat
thf(fact_4025_le__minus__divide__eq,axiom,
    ! [A: $tType] :
      ( ( linordered_field @ A )
     => ! [A2: A,B2: A,C2: A] :
          ( ( ord_less_eq @ A @ A2 @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C2 ) ) )
          = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
             => ( ord_less_eq @ A @ ( times_times @ A @ A2 @ C2 ) @ ( uminus_uminus @ A @ B2 ) ) )
            & ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
             => ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
                 => ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ ( times_times @ A @ A2 @ C2 ) ) )
                & ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
                 => ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ) ) ) ) ).

% le_minus_divide_eq
thf(fact_4026_minus__divide__le__eq,axiom,
    ! [A: $tType] :
      ( ( linordered_field @ A )
     => ! [B2: A,C2: A,A2: A] :
          ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C2 ) ) @ A2 )
          = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
             => ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ ( times_times @ A @ A2 @ C2 ) ) )
            & ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
             => ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
                 => ( ord_less_eq @ A @ ( times_times @ A @ A2 @ C2 ) @ ( uminus_uminus @ A @ B2 ) ) )
                & ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
                 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) ) ) ) ) ) ) ).

% minus_divide_le_eq
thf(fact_4027_neg__le__minus__divide__eq,axiom,
    ! [A: $tType] :
      ( ( linordered_field @ A )
     => ! [C2: A,A2: A,B2: A] :
          ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
         => ( ( ord_less_eq @ A @ A2 @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C2 ) ) )
            = ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ ( times_times @ A @ A2 @ C2 ) ) ) ) ) ).

% neg_le_minus_divide_eq
thf(fact_4028_neg__minus__divide__le__eq,axiom,
    ! [A: $tType] :
      ( ( linordered_field @ A )
     => ! [C2: A,B2: A,A2: A] :
          ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
         => ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C2 ) ) @ A2 )
            = ( ord_less_eq @ A @ ( times_times @ A @ A2 @ C2 ) @ ( uminus_uminus @ A @ B2 ) ) ) ) ) ).

% neg_minus_divide_le_eq
thf(fact_4029_pos__le__minus__divide__eq,axiom,
    ! [A: $tType] :
      ( ( linordered_field @ A )
     => ! [C2: A,A2: A,B2: A] :
          ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
         => ( ( ord_less_eq @ A @ A2 @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C2 ) ) )
            = ( ord_less_eq @ A @ ( times_times @ A @ A2 @ C2 ) @ ( uminus_uminus @ A @ B2 ) ) ) ) ) ).

% pos_le_minus_divide_eq
thf(fact_4030_pos__minus__divide__le__eq,axiom,
    ! [A: $tType] :
      ( ( linordered_field @ A )
     => ! [C2: A,B2: A,A2: A] :
          ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
         => ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C2 ) ) @ A2 )
            = ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ ( times_times @ A @ A2 @ C2 ) ) ) ) ) ).

% pos_minus_divide_le_eq
thf(fact_4031_divide__less__eq__numeral_I2_J,axiom,
    ! [A: $tType] :
      ( ( linordered_field @ A )
     => ! [B2: A,C2: A,W: num] :
          ( ( ord_less @ A @ ( divide_divide @ A @ B2 @ C2 ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) )
          = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
             => ( ord_less @ A @ B2 @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) @ C2 ) ) )
            & ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
             => ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
                 => ( ord_less @ A @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) @ C2 ) @ B2 ) )
                & ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
                 => ( ord_less @ A @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) ) ) ) ) ) ) ) ).

% divide_less_eq_numeral(2)
thf(fact_4032_less__divide__eq__numeral_I2_J,axiom,
    ! [A: $tType] :
      ( ( linordered_field @ A )
     => ! [W: num,B2: A,C2: A] :
          ( ( ord_less @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) @ ( divide_divide @ A @ B2 @ C2 ) )
          = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
             => ( ord_less @ A @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) @ C2 ) @ B2 ) )
            & ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
             => ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
                 => ( ord_less @ A @ B2 @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) @ C2 ) ) )
                & ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
                 => ( ord_less @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) @ ( zero_zero @ A ) ) ) ) ) ) ) ) ).

% less_divide_eq_numeral(2)
thf(fact_4033_power2__eq__1__iff,axiom,
    ! [A: $tType] :
      ( ( ring_15535105094025558882visors @ A )
     => ! [A2: A] :
          ( ( ( power_power @ A @ A2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
            = ( one_one @ A ) )
          = ( ( A2
              = ( one_one @ A ) )
            | ( A2
              = ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ) ) ).

% power2_eq_1_iff
thf(fact_4034_uminus__power__if,axiom,
    ! [A: $tType] :
      ( ( ring_1 @ A )
     => ! [N2: nat,A2: A] :
          ( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 )
           => ( ( power_power @ A @ ( uminus_uminus @ A @ A2 ) @ N2 )
              = ( power_power @ A @ A2 @ N2 ) ) )
          & ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 )
           => ( ( power_power @ A @ ( uminus_uminus @ A @ A2 ) @ N2 )
              = ( uminus_uminus @ A @ ( power_power @ A @ A2 @ N2 ) ) ) ) ) ) ).

% uminus_power_if
thf(fact_4035_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [A: $tType] :
      ( ( ring_1 @ A )
     => ! [K: nat,N2: nat] :
          ( ( ord_less_eq @ nat @ K @ N2 )
         => ( ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( plus_plus @ nat @ N2 @ K ) )
            = ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( minus_minus @ nat @ N2 @ K ) ) ) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
thf(fact_4036_fact__div__fact,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_eq @ nat @ N2 @ M )
     => ( ( divide_divide @ nat @ ( semiring_char_0_fact @ nat @ M ) @ ( semiring_char_0_fact @ nat @ N2 ) )
        = ( groups7121269368397514597t_prod @ nat @ nat
          @ ^ [X: nat] : X
          @ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ N2 @ ( one_one @ nat ) ) @ M ) ) ) ) ).

% fact_div_fact
thf(fact_4037_neg__int__cases,axiom,
    ! [K: int] :
      ( ( ord_less @ int @ K @ ( zero_zero @ int ) )
     => ~ ! [N4: nat] :
            ( ( K
              = ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ N4 ) ) )
           => ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ N4 ) ) ) ).

% neg_int_cases
thf(fact_4038_nat__mult__distrib__neg,axiom,
    ! [Z: int,Z7: int] :
      ( ( ord_less_eq @ int @ Z @ ( zero_zero @ int ) )
     => ( ( nat2 @ ( times_times @ int @ Z @ Z7 ) )
        = ( times_times @ nat @ ( nat2 @ ( uminus_uminus @ int @ Z ) ) @ ( nat2 @ ( uminus_uminus @ int @ Z7 ) ) ) ) ) ).

% nat_mult_distrib_neg
thf(fact_4039_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_4040_zmod__minus1,axiom,
    ! [B2: int] :
      ( ( ord_less @ int @ ( zero_zero @ int ) @ B2 )
     => ( ( modulo_modulo @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ B2 )
        = ( minus_minus @ int @ B2 @ ( one_one @ int ) ) ) ) ).

% zmod_minus1
thf(fact_4041_zdiv__zminus1__eq__if,axiom,
    ! [B2: int,A2: int] :
      ( ( B2
       != ( zero_zero @ int ) )
     => ( ( ( ( modulo_modulo @ int @ A2 @ B2 )
            = ( zero_zero @ int ) )
         => ( ( divide_divide @ int @ ( uminus_uminus @ int @ A2 ) @ B2 )
            = ( uminus_uminus @ int @ ( divide_divide @ int @ A2 @ B2 ) ) ) )
        & ( ( ( modulo_modulo @ int @ A2 @ B2 )
           != ( zero_zero @ int ) )
         => ( ( divide_divide @ int @ ( uminus_uminus @ int @ A2 ) @ B2 )
            = ( minus_minus @ int @ ( uminus_uminus @ int @ ( divide_divide @ int @ A2 @ B2 ) ) @ ( one_one @ int ) ) ) ) ) ) ).

% zdiv_zminus1_eq_if
thf(fact_4042_zdiv__zminus2__eq__if,axiom,
    ! [B2: int,A2: int] :
      ( ( B2
       != ( zero_zero @ int ) )
     => ( ( ( ( modulo_modulo @ int @ A2 @ B2 )
            = ( zero_zero @ int ) )
         => ( ( divide_divide @ int @ A2 @ ( uminus_uminus @ int @ B2 ) )
            = ( uminus_uminus @ int @ ( divide_divide @ int @ A2 @ B2 ) ) ) )
        & ( ( ( modulo_modulo @ int @ A2 @ B2 )
           != ( zero_zero @ int ) )
         => ( ( divide_divide @ int @ A2 @ ( uminus_uminus @ int @ B2 ) )
            = ( minus_minus @ int @ ( uminus_uminus @ int @ ( divide_divide @ int @ A2 @ B2 ) ) @ ( one_one @ int ) ) ) ) ) ) ).

% zdiv_zminus2_eq_if
thf(fact_4043_square__fact__le__2__fact,axiom,
    ! [N2: nat] : ( ord_less_eq @ real @ ( times_times @ real @ ( semiring_char_0_fact @ real @ N2 ) @ ( semiring_char_0_fact @ real @ N2 ) ) @ ( semiring_char_0_fact @ real @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ) ).

% square_fact_le_2_fact
thf(fact_4044_zminus1__lemma,axiom,
    ! [A2: int,B2: int,Q2: int,R4: int] :
      ( ( eucl_rel_int @ A2 @ B2 @ ( product_Pair @ int @ int @ Q2 @ R4 ) )
     => ( ( B2
         != ( zero_zero @ int ) )
       => ( eucl_rel_int @ ( uminus_uminus @ int @ A2 ) @ B2
          @ ( product_Pair @ int @ int
            @ ( if @ int
              @ ( R4
                = ( zero_zero @ int ) )
              @ ( uminus_uminus @ int @ Q2 )
              @ ( minus_minus @ int @ ( uminus_uminus @ int @ Q2 ) @ ( one_one @ int ) ) )
            @ ( if @ int
              @ ( R4
                = ( zero_zero @ int ) )
              @ ( zero_zero @ int )
              @ ( minus_minus @ int @ B2 @ R4 ) ) ) ) ) ) ).

% zminus1_lemma
thf(fact_4045_divide__le__eq__numeral_I2_J,axiom,
    ! [A: $tType] :
      ( ( linordered_field @ A )
     => ! [B2: A,C2: A,W: num] :
          ( ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ C2 ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) )
          = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
             => ( ord_less_eq @ A @ B2 @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) @ C2 ) ) )
            & ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
             => ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
                 => ( ord_less_eq @ A @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) @ C2 ) @ B2 ) )
                & ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
                 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) ) ) ) ) ) ) ) ).

% divide_le_eq_numeral(2)
thf(fact_4046_le__divide__eq__numeral_I2_J,axiom,
    ! [A: $tType] :
      ( ( linordered_field @ A )
     => ! [W: num,B2: A,C2: A] :
          ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) @ ( divide_divide @ A @ B2 @ C2 ) )
          = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
             => ( ord_less_eq @ A @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) @ C2 ) @ B2 ) )
            & ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
             => ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
                 => ( ord_less_eq @ A @ B2 @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) @ C2 ) ) )
                & ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
                 => ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) @ ( zero_zero @ A ) ) ) ) ) ) ) ) ).

% le_divide_eq_numeral(2)
thf(fact_4047_square__le__1,axiom,
    ! [A: $tType] :
      ( ( linordered_idom @ A )
     => ! [X2: A] :
          ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ X2 )
         => ( ( ord_less_eq @ A @ X2 @ ( one_one @ A ) )
           => ( ord_less_eq @ A @ ( power_power @ A @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ A ) ) ) ) ) ).

% square_le_1
thf(fact_4048_minus__power__mult__self,axiom,
    ! [A: $tType] :
      ( ( comm_ring_1 @ A )
     => ! [A2: A,N2: nat] :
          ( ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ A2 ) @ N2 ) @ ( power_power @ A @ ( uminus_uminus @ A @ A2 ) @ N2 ) )
          = ( power_power @ A @ A2 @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ) ) ).

% minus_power_mult_self
thf(fact_4049_fact__num__eq__if,axiom,
    ! [A: $tType] :
      ( ( semiring_char_0 @ A )
     => ( ( semiring_char_0_fact @ A )
        = ( ^ [M7: nat] :
              ( if @ A
              @ ( M7
                = ( zero_zero @ nat ) )
              @ ( one_one @ A )
              @ ( times_times @ A @ ( semiring_1_of_nat @ A @ M7 ) @ ( semiring_char_0_fact @ A @ ( minus_minus @ nat @ M7 @ ( one_one @ nat ) ) ) ) ) ) ) ) ).

% fact_num_eq_if
thf(fact_4050_fact__code,axiom,
    ! [A: $tType] :
      ( ( semiring_char_0 @ A )
     => ( ( semiring_char_0_fact @ A )
        = ( ^ [N: nat] : ( semiring_1_of_nat @ A @ ( set_fo6178422350223883121st_nat @ nat @ ( times_times @ nat ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N @ ( one_one @ nat ) ) ) ) ) ) ).

% fact_code
thf(fact_4051_minus__one__power__iff,axiom,
    ! [A: $tType] :
      ( ( ring_1 @ A )
     => ! [N2: nat] :
          ( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 )
           => ( ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N2 )
              = ( one_one @ A ) ) )
          & ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 )
           => ( ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N2 )
              = ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ) ) ).

% minus_one_power_iff
thf(fact_4052_fact__reduce,axiom,
    ! [A: $tType] :
      ( ( semiring_char_0 @ A )
     => ! [N2: nat] :
          ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
         => ( ( semiring_char_0_fact @ A @ N2 )
            = ( times_times @ A @ ( semiring_1_of_nat @ A @ N2 ) @ ( semiring_char_0_fact @ A @ ( minus_minus @ nat @ N2 @ ( one_one @ nat ) ) ) ) ) ) ) ).

% fact_reduce
thf(fact_4053_binomial__fact,axiom,
    ! [A: $tType] :
      ( ( field_char_0 @ A )
     => ! [K: nat,N2: nat] :
          ( ( ord_less_eq @ nat @ K @ N2 )
         => ( ( semiring_1_of_nat @ A @ ( binomial @ N2 @ K ) )
            = ( divide_divide @ A @ ( semiring_char_0_fact @ A @ N2 ) @ ( times_times @ A @ ( semiring_char_0_fact @ A @ K ) @ ( semiring_char_0_fact @ A @ ( minus_minus @ nat @ N2 @ K ) ) ) ) ) ) ) ).

% binomial_fact
thf(fact_4054_fact__binomial,axiom,
    ! [A: $tType] :
      ( ( field_char_0 @ A )
     => ! [K: nat,N2: nat] :
          ( ( ord_less_eq @ nat @ K @ N2 )
         => ( ( times_times @ A @ ( semiring_char_0_fact @ A @ K ) @ ( semiring_1_of_nat @ A @ ( binomial @ N2 @ K ) ) )
            = ( divide_divide @ A @ ( semiring_char_0_fact @ A @ N2 ) @ ( semiring_char_0_fact @ A @ ( minus_minus @ nat @ N2 @ K ) ) ) ) ) ) ).

% fact_binomial
thf(fact_4055_pochhammer__absorb__comp,axiom,
    ! [A: $tType] :
      ( ( comm_ring_1 @ A )
     => ! [R4: A,K: nat] :
          ( ( times_times @ A @ ( minus_minus @ A @ R4 @ ( semiring_1_of_nat @ A @ K ) ) @ ( comm_s3205402744901411588hammer @ A @ ( uminus_uminus @ A @ R4 ) @ K ) )
          = ( times_times @ A @ R4 @ ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ ( uminus_uminus @ A @ R4 ) @ ( one_one @ A ) ) @ K ) ) ) ) ).

% pochhammer_absorb_comp
thf(fact_4056_gbinomial__index__swap,axiom,
    ! [A: $tType] :
      ( ( field_char_0 @ A )
     => ! [K: nat,N2: nat] :
          ( ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ K ) @ ( gbinomial @ A @ ( minus_minus @ A @ ( uminus_uminus @ A @ ( semiring_1_of_nat @ A @ N2 ) ) @ ( one_one @ A ) ) @ K ) )
          = ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N2 ) @ ( gbinomial @ A @ ( minus_minus @ A @ ( uminus_uminus @ A @ ( semiring_1_of_nat @ A @ K ) ) @ ( one_one @ A ) ) @ N2 ) ) ) ) ).

% gbinomial_index_swap
thf(fact_4057_gbinomial__negated__upper,axiom,
    ! [A: $tType] :
      ( ( field_char_0 @ A )
     => ( ( gbinomial @ A )
        = ( ^ [A5: A,K4: nat] : ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ K4 ) @ ( gbinomial @ A @ ( minus_minus @ A @ ( minus_minus @ A @ ( semiring_1_of_nat @ A @ K4 ) @ A5 ) @ ( one_one @ A ) ) @ K4 ) ) ) ) ) ).

% gbinomial_negated_upper
thf(fact_4058_minus__1__div__exp__eq__int,axiom,
    ! [N2: nat] :
      ( ( divide_divide @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N2 ) )
      = ( uminus_uminus @ int @ ( one_one @ int ) ) ) ).

% minus_1_div_exp_eq_int
thf(fact_4059_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_4060_power__minus1__odd,axiom,
    ! [A: $tType] :
      ( ( ring_1 @ A )
     => ! [N2: nat] :
          ( ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) )
          = ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).

% power_minus1_odd
thf(fact_4061_gbinomial__minus,axiom,
    ! [A: $tType] :
      ( ( field_char_0 @ A )
     => ! [A2: A,K: nat] :
          ( ( gbinomial @ A @ ( uminus_uminus @ A @ A2 ) @ K )
          = ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ K ) @ ( gbinomial @ A @ ( minus_minus @ A @ ( plus_plus @ A @ A2 @ ( semiring_1_of_nat @ A @ K ) ) @ ( one_one @ A ) ) @ K ) ) ) ) ).

% gbinomial_minus
thf(fact_4062_pochhammer__minus,axiom,
    ! [A: $tType] :
      ( ( comm_ring_1 @ A )
     => ! [B2: A,K: nat] :
          ( ( comm_s3205402744901411588hammer @ A @ ( uminus_uminus @ A @ B2 ) @ K )
          = ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ K ) @ ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ ( minus_minus @ A @ B2 @ ( semiring_1_of_nat @ A @ K ) ) @ ( one_one @ A ) ) @ K ) ) ) ) ).

% pochhammer_minus
thf(fact_4063_pochhammer__minus_H,axiom,
    ! [A: $tType] :
      ( ( comm_ring_1 @ A )
     => ! [B2: A,K: nat] :
          ( ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ ( minus_minus @ A @ B2 @ ( semiring_1_of_nat @ A @ K ) ) @ ( one_one @ A ) ) @ K )
          = ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ K ) @ ( comm_s3205402744901411588hammer @ A @ ( uminus_uminus @ A @ B2 ) @ K ) ) ) ) ).

% pochhammer_minus'
thf(fact_4064_gbinomial__pochhammer_H,axiom,
    ! [A: $tType] :
      ( ( field_char_0 @ A )
     => ( ( gbinomial @ A )
        = ( ^ [A5: A,K4: nat] : ( divide_divide @ A @ ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ ( minus_minus @ A @ A5 @ ( semiring_1_of_nat @ A @ K4 ) ) @ ( one_one @ A ) ) @ K4 ) @ ( semiring_char_0_fact @ A @ K4 ) ) ) ) ) ).

% gbinomial_pochhammer'
thf(fact_4065_int__bit__induct,axiom,
    ! [P: int > $o,K: int] :
      ( ( P @ ( zero_zero @ int ) )
     => ( ( P @ ( uminus_uminus @ int @ ( one_one @ int ) ) )
       => ( ! [K3: int] :
              ( ( P @ K3 )
             => ( ( K3
                 != ( zero_zero @ int ) )
               => ( P @ ( times_times @ int @ K3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) )
         => ( ! [K3: int] :
                ( ( P @ K3 )
               => ( ( K3
                   != ( uminus_uminus @ int @ ( one_one @ int ) ) )
                 => ( P @ ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ K3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) )
           => ( P @ K ) ) ) ) ) ).

% int_bit_induct
thf(fact_4066_m1mod2k,axiom,
    ! [N2: nat] :
      ( ( modulo_modulo @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N2 ) )
      = ( minus_minus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N2 ) @ ( one_one @ int ) ) ) ).

% m1mod2k
thf(fact_4067_gbinomial__sum__lower__neg,axiom,
    ! [A: $tType] :
      ( ( field_char_0 @ A )
     => ! [A2: A,M: nat] :
          ( ( groups7311177749621191930dd_sum @ nat @ A
            @ ^ [K4: nat] : ( times_times @ A @ ( gbinomial @ A @ A2 @ K4 ) @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ K4 ) )
            @ ( set_ord_atMost @ nat @ M ) )
          = ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ M ) @ ( gbinomial @ A @ ( minus_minus @ A @ A2 @ ( one_one @ A ) ) @ M ) ) ) ) ).

% gbinomial_sum_lower_neg
thf(fact_4068_Maclaurin__zero,axiom,
    ! [A: $tType] :
      ( ( zero @ A )
     => ! [X2: real,N2: nat,Diff: nat > A > real] :
          ( ( X2
            = ( zero_zero @ real ) )
         => ( ( N2
             != ( zero_zero @ nat ) )
           => ( ( groups7311177749621191930dd_sum @ nat @ real
                @ ^ [M7: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ M7 @ ( zero_zero @ A ) ) @ ( semiring_char_0_fact @ real @ M7 ) ) @ ( power_power @ real @ X2 @ M7 ) )
                @ ( set_ord_lessThan @ nat @ N2 ) )
              = ( Diff @ ( zero_zero @ nat ) @ ( zero_zero @ A ) ) ) ) ) ) ).

% Maclaurin_zero
thf(fact_4069_Maclaurin__lemma,axiom,
    ! [H2: real,F: real > real,J: nat > real,N2: nat] :
      ( ( ord_less @ real @ ( zero_zero @ real ) @ H2 )
     => ? [B10: real] :
          ( ( F @ H2 )
          = ( plus_plus @ real
            @ ( groups7311177749621191930dd_sum @ nat @ real
              @ ^ [M7: nat] : ( times_times @ real @ ( divide_divide @ real @ ( J @ M7 ) @ ( semiring_char_0_fact @ real @ M7 ) ) @ ( power_power @ real @ H2 @ M7 ) )
              @ ( set_ord_lessThan @ nat @ N2 ) )
            @ ( times_times @ real @ B10 @ ( divide_divide @ real @ ( power_power @ real @ H2 @ N2 ) @ ( semiring_char_0_fact @ real @ N2 ) ) ) ) ) ) ).

% Maclaurin_lemma
thf(fact_4070_sb__dec__lem_H,axiom,
    ! [K: nat,A2: int] :
      ( ( ord_less_eq @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K ) @ A2 )
     => ( ord_less_eq @ int @ ( modulo_modulo @ int @ ( plus_plus @ int @ A2 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K ) ) ) @ ( plus_plus @ int @ ( uminus_uminus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K ) ) @ A2 ) ) ) ).

% sb_dec_lem'
thf(fact_4071_m1mod22k,axiom,
    ! [N2: nat] :
      ( ( modulo_modulo @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N2 ) ) )
      = ( minus_minus @ int @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N2 ) ) @ ( one_one @ int ) ) ) ).

% m1mod22k
thf(fact_4072_gbinomial__Suc,axiom,
    ! [A: $tType] :
      ( ( ( semiring_char_0 @ A )
        & ( semidom_divide @ A ) )
     => ! [A2: A,K: nat] :
          ( ( gbinomial @ A @ A2 @ ( suc @ K ) )
          = ( divide_divide @ A
            @ ( groups7121269368397514597t_prod @ nat @ A
              @ ^ [I2: nat] : ( minus_minus @ A @ A2 @ ( semiring_1_of_nat @ A @ I2 ) )
              @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ K ) )
            @ ( semiring_char_0_fact @ A @ ( suc @ K ) ) ) ) ) ).

% gbinomial_Suc
thf(fact_4073_gbinomial__partial__sum__poly,axiom,
    ! [A: $tType] :
      ( ( field_char_0 @ A )
     => ! [M: nat,A2: A,X2: A,Y2: A] :
          ( ( groups7311177749621191930dd_sum @ nat @ A
            @ ^ [K4: nat] : ( times_times @ A @ ( times_times @ A @ ( gbinomial @ A @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ M ) @ A2 ) @ K4 ) @ ( power_power @ A @ X2 @ K4 ) ) @ ( power_power @ A @ Y2 @ ( minus_minus @ nat @ M @ K4 ) ) )
            @ ( set_ord_atMost @ nat @ M ) )
          = ( groups7311177749621191930dd_sum @ nat @ A
            @ ^ [K4: nat] : ( times_times @ A @ ( times_times @ A @ ( gbinomial @ A @ ( uminus_uminus @ A @ A2 ) @ K4 ) @ ( power_power @ A @ ( uminus_uminus @ A @ X2 ) @ K4 ) ) @ ( power_power @ A @ ( plus_plus @ A @ X2 @ Y2 ) @ ( minus_minus @ nat @ M @ K4 ) ) )
            @ ( set_ord_atMost @ nat @ M ) ) ) ) ).

% gbinomial_partial_sum_poly
thf(fact_4074_VEBT__internal_OminNull_Opelims_I3_J,axiom,
    ! [X2: vEBT_VEBT] :
      ( ~ ( vEBT_VEBT_minNull @ X2 )
     => ( ( accp @ vEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ X2 )
       => ( ! [Uv2: $o] :
              ( ( X2
                = ( vEBT_Leaf @ $true @ Uv2 ) )
             => ~ ( accp @ vEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ $true @ Uv2 ) ) )
         => ( ! [Uu2: $o] :
                ( ( X2
                  = ( vEBT_Leaf @ Uu2 @ $true ) )
               => ~ ( accp @ vEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ Uu2 @ $true ) ) )
           => ~ ! [Uz2: product_prod @ nat @ nat,Va2: nat,Vb2: list @ vEBT_VEBT,Vc: vEBT_VEBT] :
                  ( ( X2
                    = ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ Uz2 ) @ Va2 @ Vb2 @ Vc ) )
                 => ~ ( accp @ vEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ Uz2 ) @ Va2 @ Vb2 @ Vc ) ) ) ) ) ) ) ).

% VEBT_internal.minNull.pelims(3)
thf(fact_4075_VEBT__internal_OminNull_Opelims_I2_J,axiom,
    ! [X2: vEBT_VEBT] :
      ( ( vEBT_VEBT_minNull @ X2 )
     => ( ( accp @ vEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ X2 )
       => ( ( ( X2
              = ( vEBT_Leaf @ $false @ $false ) )
           => ~ ( accp @ vEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ $false @ $false ) ) )
         => ~ ! [Uw2: nat,Ux2: list @ vEBT_VEBT,Uy2: vEBT_VEBT] :
                ( ( X2
                  = ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uw2 @ Ux2 @ Uy2 ) )
               => ~ ( accp @ vEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uw2 @ Ux2 @ Uy2 ) ) ) ) ) ) ).

% VEBT_internal.minNull.pelims(2)
thf(fact_4076_sb__inc__lem_H,axiom,
    ! [A2: int,K: nat] :
      ( ( ord_less @ int @ A2 @ ( uminus_uminus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K ) ) )
     => ( ord_less_eq @ int @ ( plus_plus @ int @ ( plus_plus @ int @ A2 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( suc @ K ) ) ) @ ( modulo_modulo @ int @ ( plus_plus @ int @ A2 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( suc @ K ) ) ) ) ) ).

% sb_inc_lem'
thf(fact_4077_sb__dec__lem,axiom,
    ! [K: nat,A2: int] :
      ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( plus_plus @ int @ ( uminus_uminus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K ) ) @ A2 ) )
     => ( ord_less_eq @ int @ ( modulo_modulo @ int @ ( plus_plus @ int @ A2 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K ) ) ) @ ( plus_plus @ int @ ( uminus_uminus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K ) ) @ A2 ) ) ) ).

% sb_dec_lem
thf(fact_4078_root__polyfun,axiom,
    ! [A: $tType] :
      ( ( idom @ A )
     => ! [N2: nat,Z: A,A2: A] :
          ( ( ord_less_eq @ nat @ ( one_one @ nat ) @ N2 )
         => ( ( ( power_power @ A @ Z @ N2 )
              = A2 )
            = ( ( groups7311177749621191930dd_sum @ nat @ A
                @ ^ [I2: nat] :
                    ( times_times @ A
                    @ ( if @ A
                      @ ( I2
                        = ( zero_zero @ nat ) )
                      @ ( uminus_uminus @ A @ A2 )
                      @ ( if @ A @ ( I2 = N2 ) @ ( one_one @ A ) @ ( zero_zero @ A ) ) )
                    @ ( power_power @ A @ Z @ I2 ) )
                @ ( set_ord_atMost @ nat @ N2 ) )
              = ( zero_zero @ A ) ) ) ) ) ).

% root_polyfun
thf(fact_4079_choose__alternating__linear__sum,axiom,
    ! [A: $tType] :
      ( ( comm_ring_1 @ A )
     => ! [N2: nat] :
          ( ( N2
           != ( one_one @ nat ) )
         => ( ( groups7311177749621191930dd_sum @ nat @ A
              @ ^ [I2: nat] : ( times_times @ A @ ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ I2 ) @ ( semiring_1_of_nat @ A @ I2 ) ) @ ( semiring_1_of_nat @ A @ ( binomial @ N2 @ I2 ) ) )
              @ ( set_ord_atMost @ nat @ N2 ) )
            = ( zero_zero @ A ) ) ) ) ).

% choose_alternating_linear_sum
thf(fact_4080_fact__double,axiom,
    ! [A: $tType] :
      ( ( field_char_0 @ A )
     => ! [N2: nat] :
          ( ( semiring_char_0_fact @ A @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) )
          = ( times_times @ A @ ( times_times @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) @ ( comm_s3205402744901411588hammer @ A @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ N2 ) ) @ ( semiring_char_0_fact @ A @ N2 ) ) ) ) ).

% fact_double
thf(fact_4081_binomial__code,axiom,
    ( binomial
    = ( ^ [N: nat,K4: nat] : ( if @ nat @ ( ord_less @ nat @ N @ K4 ) @ ( zero_zero @ nat ) @ ( if @ nat @ ( ord_less @ nat @ N @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ K4 ) ) @ ( binomial @ N @ ( minus_minus @ nat @ N @ K4 ) ) @ ( divide_divide @ nat @ ( set_fo6178422350223883121st_nat @ nat @ ( times_times @ nat ) @ ( plus_plus @ nat @ ( minus_minus @ nat @ N @ K4 ) @ ( one_one @ nat ) ) @ N @ ( one_one @ nat ) ) @ ( semiring_char_0_fact @ nat @ K4 ) ) ) ) ) ) ).

% binomial_code
thf(fact_4082_choose__alternating__sum,axiom,
    ! [A: $tType] :
      ( ( comm_ring_1 @ A )
     => ! [N2: nat] :
          ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
         => ( ( groups7311177749621191930dd_sum @ nat @ A
              @ ^ [I2: nat] : ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ I2 ) @ ( semiring_1_of_nat @ A @ ( binomial @ N2 @ I2 ) ) )
              @ ( set_ord_atMost @ nat @ N2 ) )
            = ( zero_zero @ A ) ) ) ) ).

% choose_alternating_sum
thf(fact_4083_of__int__code__if,axiom,
    ! [A: $tType] :
      ( ( ring_1 @ A )
     => ( ( ring_1_of_int @ A )
        = ( ^ [K4: int] :
              ( if @ A
              @ ( K4
                = ( zero_zero @ int ) )
              @ ( zero_zero @ A )
              @ ( if @ A @ ( ord_less @ int @ K4 @ ( zero_zero @ int ) ) @ ( uminus_uminus @ A @ ( ring_1_of_int @ A @ ( uminus_uminus @ int @ K4 ) ) )
                @ ( if @ A
                  @ ( ( modulo_modulo @ int @ K4 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) )
                    = ( zero_zero @ int ) )
                  @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( ring_1_of_int @ A @ ( divide_divide @ int @ K4 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) )
                  @ ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( ring_1_of_int @ A @ ( divide_divide @ int @ K4 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) @ ( one_one @ A ) ) ) ) ) ) ) ) ).

% of_int_code_if
thf(fact_4084_one__mod__minus__numeral,axiom,
    ! [N2: num] :
      ( ( modulo_modulo @ int @ ( one_one @ int ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N2 ) ) )
      = ( uminus_uminus @ int @ ( adjust_mod @ ( numeral_numeral @ int @ N2 ) @ ( product_snd @ int @ int @ ( unique8689654367752047608divmod @ int @ one2 @ N2 ) ) ) ) ) ).

% one_mod_minus_numeral
thf(fact_4085_minus__one__mod__numeral,axiom,
    ! [N2: num] :
      ( ( modulo_modulo @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( numeral_numeral @ int @ N2 ) )
      = ( adjust_mod @ ( numeral_numeral @ int @ N2 ) @ ( product_snd @ int @ int @ ( unique8689654367752047608divmod @ int @ one2 @ N2 ) ) ) ) ).

% minus_one_mod_numeral
thf(fact_4086_and__int_Opinduct,axiom,
    ! [A0: int,A12: int,P: int > int > $o] :
      ( ( accp @ ( product_prod @ int @ int ) @ bit_and_int_rel @ ( product_Pair @ int @ int @ A0 @ A12 ) )
     => ( ! [K3: int,L3: int] :
            ( ( accp @ ( product_prod @ int @ int ) @ bit_and_int_rel @ ( product_Pair @ int @ int @ K3 @ L3 ) )
           => ( ( ~ ( ( 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 ) ) ) ) ) )
               => ( P @ ( divide_divide @ int @ K3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ L3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) )
             => ( P @ K3 @ L3 ) ) )
       => ( P @ A0 @ A12 ) ) ) ).

% and_int.pinduct
thf(fact_4087_Compl__anti__mono,axiom,
    ! [A: $tType,A3: set @ A,B6: set @ A] :
      ( ( ord_less_eq @ ( set @ A ) @ A3 @ B6 )
     => ( ord_less_eq @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ B6 ) @ ( uminus_uminus @ ( set @ A ) @ A3 ) ) ) ).

% Compl_anti_mono
thf(fact_4088_Compl__subset__Compl__iff,axiom,
    ! [A: $tType,A3: set @ A,B6: set @ A] :
      ( ( ord_less_eq @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A3 ) @ ( uminus_uminus @ ( set @ A ) @ B6 ) )
      = ( ord_less_eq @ ( set @ A ) @ B6 @ A3 ) ) ).

% Compl_subset_Compl_iff
thf(fact_4089_real__add__minus__iff,axiom,
    ! [X2: real,A2: real] :
      ( ( ( plus_plus @ real @ X2 @ ( uminus_uminus @ real @ A2 ) )
        = ( zero_zero @ real ) )
      = ( X2 = A2 ) ) ).

% real_add_minus_iff
thf(fact_4090_max__word__max,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [N2: word @ A] : ( ord_less_eq @ ( word @ A ) @ N2 @ ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) ) ) ).

% max_word_max
thf(fact_4091_word__n1__ge,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [Y2: word @ A] : ( ord_less_eq @ ( word @ A ) @ Y2 @ ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) ) ) ).

% word_n1_ge
thf(fact_4092_word__order_Oextremum__unique,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [A2: word @ A] :
          ( ( ord_less_eq @ ( word @ A ) @ ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) @ A2 )
          = ( A2
            = ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) ) ) ) ).

% word_order.extremum_unique
thf(fact_4093_word__order_Oextremum,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [A2: word @ A] : ( ord_less_eq @ ( word @ A ) @ A2 @ ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) ) ) ).

% word_order.extremum
thf(fact_4094_word__minus__one__le,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [X2: word @ A] :
          ( ( ord_less_eq @ ( word @ A ) @ ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) @ X2 )
          = ( X2
            = ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) ) ) ) ).

% word_minus_one_le
thf(fact_4095_max__word__less__eq__iff,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [W: word @ A] :
          ( ( ord_less_eq @ ( word @ A ) @ ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) @ W )
          = ( W
            = ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) ) ) ) ).

% max_word_less_eq_iff
thf(fact_4096_subset__Compl__singleton,axiom,
    ! [A: $tType,A3: set @ A,B2: A] :
      ( ( ord_less_eq @ ( set @ A ) @ A3 @ ( uminus_uminus @ ( set @ A ) @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) )
      = ( ~ ( member @ A @ B2 @ A3 ) ) ) ).

% subset_Compl_singleton
thf(fact_4097_word__of__int__neg__numeral,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [Bin: num] :
          ( ( ring_1_of_int @ ( word @ A ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ Bin ) ) )
          = ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ Bin ) ) ) ) ).

% word_of_int_neg_numeral
thf(fact_4098_word__of__int__neg__1,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ( ( ring_1_of_int @ ( word @ A ) @ ( uminus_uminus @ int @ ( one_one @ int ) ) )
        = ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) ) ) ).

% word_of_int_neg_1
thf(fact_4099_floor__minus__divide__eq__div__numeral,axiom,
    ! [A2: num,B2: num] :
      ( ( archim6421214686448440834_floor @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ ( numeral_numeral @ real @ A2 ) @ ( numeral_numeral @ real @ B2 ) ) ) )
      = ( divide_divide @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ A2 ) ) @ ( numeral_numeral @ int @ B2 ) ) ) ).

% floor_minus_divide_eq_div_numeral
thf(fact_4100_ceiling__minus__divide__eq__div__numeral,axiom,
    ! [A2: num,B2: num] :
      ( ( archimedean_ceiling @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ ( numeral_numeral @ real @ A2 ) @ ( numeral_numeral @ real @ B2 ) ) ) )
      = ( uminus_uminus @ int @ ( divide_divide @ int @ ( numeral_numeral @ int @ A2 ) @ ( numeral_numeral @ int @ B2 ) ) ) ) ).

% ceiling_minus_divide_eq_div_numeral
thf(fact_4101_numeral__mod__minus__numeral,axiom,
    ! [M: num,N2: num] :
      ( ( modulo_modulo @ int @ ( numeral_numeral @ int @ M ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N2 ) ) )
      = ( uminus_uminus @ int @ ( adjust_mod @ ( numeral_numeral @ int @ N2 ) @ ( product_snd @ int @ int @ ( unique8689654367752047608divmod @ int @ M @ N2 ) ) ) ) ) ).

% numeral_mod_minus_numeral
thf(fact_4102_minus__numeral__mod__numeral,axiom,
    ! [M: num,N2: num] :
      ( ( modulo_modulo @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M ) ) @ ( numeral_numeral @ int @ N2 ) )
      = ( adjust_mod @ ( numeral_numeral @ int @ N2 ) @ ( product_snd @ int @ int @ ( unique8689654367752047608divmod @ int @ M @ N2 ) ) ) ) ).

% minus_numeral_mod_numeral
thf(fact_4103_floor__minus__one__divide__eq__div__numeral,axiom,
    ! [B2: num] :
      ( ( archim6421214686448440834_floor @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ B2 ) ) ) )
      = ( divide_divide @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( numeral_numeral @ int @ B2 ) ) ) ).

% floor_minus_one_divide_eq_div_numeral
thf(fact_4104_subset__Compl__self__eq,axiom,
    ! [A: $tType,A3: set @ A] :
      ( ( ord_less_eq @ ( set @ A ) @ A3 @ ( uminus_uminus @ ( set @ A ) @ A3 ) )
      = ( A3
        = ( bot_bot @ ( set @ A ) ) ) ) ).

% subset_Compl_self_eq
thf(fact_4105_Compl__eq__Diff__UNIV,axiom,
    ! [A: $tType] :
      ( ( uminus_uminus @ ( set @ A ) )
      = ( minus_minus @ ( set @ A ) @ ( top_top @ ( set @ A ) ) ) ) ).

% Compl_eq_Diff_UNIV
thf(fact_4106_real__minus__mult__self__le,axiom,
    ! [U: real,X2: real] : ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( times_times @ real @ U @ U ) ) @ ( times_times @ real @ X2 @ X2 ) ) ).

% real_minus_mult_self_le
thf(fact_4107_word__order_Oextremum__uniqueI,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [A2: word @ A] :
          ( ( ord_less_eq @ ( word @ A ) @ ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) @ A2 )
         => ( A2
            = ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) ) ) ) ).

% word_order.extremum_uniqueI
thf(fact_4108_max__word__not__0,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ( ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) )
       != ( zero_zero @ ( word @ A ) ) ) ) ).

% max_word_not_0
thf(fact_4109_word__lessThan__Suc__atMost,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [K: word @ A] :
          ( ( K
           != ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) )
         => ( ( set_ord_lessThan @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ K @ ( one_one @ ( word @ A ) ) ) )
            = ( set_ord_atMost @ ( word @ A ) @ K ) ) ) ) ).

% word_lessThan_Suc_atMost
thf(fact_4110_word__order_Onot__eq__extremum,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [A2: word @ A] :
          ( ( A2
           != ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) )
          = ( ord_less @ ( word @ A ) @ A2 @ ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) ) ) ) ).

% word_order.not_eq_extremum
thf(fact_4111_word__order_Oextremum__strict,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [A2: word @ A] :
          ~ ( ord_less @ ( word @ A ) @ ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) @ A2 ) ) ).

% word_order.extremum_strict
thf(fact_4112_word__not__simps_I3_J,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [Y2: word @ A] :
          ~ ( ord_less @ ( word @ A ) @ ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) @ Y2 ) ) ).

% word_not_simps(3)
thf(fact_4113_max__word__not__less,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [X2: word @ A] :
          ~ ( ord_less @ ( word @ A ) @ ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) @ X2 ) ) ).

% max_word_not_less
thf(fact_4114_minus__real__def,axiom,
    ( ( minus_minus @ real )
    = ( ^ [X: real,Y: real] : ( plus_plus @ real @ X @ ( uminus_uminus @ real @ Y ) ) ) ) ).

% minus_real_def
thf(fact_4115_complex__mod__minus__le__complex__mod,axiom,
    ! [X2: complex] : ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( real_V7770717601297561774m_norm @ complex @ X2 ) ) @ ( real_V7770717601297561774m_norm @ complex @ X2 ) ) ).

% complex_mod_minus_le_complex_mod
thf(fact_4116_minus__integer__code_I2_J,axiom,
    ! [L: code_integer] :
      ( ( minus_minus @ code_integer @ ( zero_zero @ code_integer ) @ L )
      = ( uminus_uminus @ code_integer @ L ) ) ).

% minus_integer_code(2)
thf(fact_4117_Compl__insert,axiom,
    ! [A: $tType,X2: A,A3: set @ A] :
      ( ( uminus_uminus @ ( set @ A ) @ ( insert @ A @ X2 @ A3 ) )
      = ( minus_minus @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A3 ) @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).

% Compl_insert
thf(fact_4118_real__0__less__add__iff,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less @ real @ ( zero_zero @ real ) @ ( plus_plus @ real @ X2 @ Y2 ) )
      = ( ord_less @ real @ ( uminus_uminus @ real @ X2 ) @ Y2 ) ) ).

% real_0_less_add_iff
thf(fact_4119_real__add__less__0__iff,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less @ real @ ( plus_plus @ real @ X2 @ Y2 ) @ ( zero_zero @ real ) )
      = ( ord_less @ real @ Y2 @ ( uminus_uminus @ real @ X2 ) ) ) ).

% real_add_less_0_iff
thf(fact_4120_real__add__le__0__iff,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_eq @ real @ ( plus_plus @ real @ X2 @ Y2 ) @ ( zero_zero @ real ) )
      = ( ord_less_eq @ real @ Y2 @ ( uminus_uminus @ real @ X2 ) ) ) ).

% real_add_le_0_iff
thf(fact_4121_real__0__le__add__iff,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( plus_plus @ real @ X2 @ Y2 ) )
      = ( ord_less_eq @ real @ ( uminus_uminus @ real @ X2 ) @ Y2 ) ) ).

% real_0_le_add_iff
thf(fact_4122_overflow__plus__one__self,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [P4: word @ A] :
          ( ( ord_less_eq @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) @ P4 ) @ P4 )
          = ( P4
            = ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) ) ) ) ).

% overflow_plus_one_self
thf(fact_4123_plus__1__less,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [X2: word @ A] :
          ( ( ord_less_eq @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ X2 @ ( one_one @ ( word @ A ) ) ) @ X2 )
          = ( X2
            = ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) ) ) ) ).

% plus_1_less
thf(fact_4124_max__word__wrap,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [X2: word @ A] :
          ( ( ( plus_plus @ ( word @ A ) @ X2 @ ( one_one @ ( word @ A ) ) )
            = ( zero_zero @ ( word @ A ) ) )
         => ( X2
            = ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) ) ) ) ).

% max_word_wrap
thf(fact_4125_word__add__no__overflow,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [X2: word @ A] :
          ( ( ord_less @ ( word @ A ) @ X2 @ ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) )
         => ( ord_less @ ( word @ A ) @ X2 @ ( plus_plus @ ( word @ A ) @ X2 @ ( one_one @ ( word @ A ) ) ) ) ) ) ).

% word_add_no_overflow
thf(fact_4126_less__x__plus__1,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [X2: word @ A,Y2: word @ A] :
          ( ( X2
           != ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) )
         => ( ( ord_less @ ( word @ A ) @ Y2 @ ( plus_plus @ ( word @ A ) @ X2 @ ( one_one @ ( word @ A ) ) ) )
            = ( ( ord_less @ ( word @ A ) @ Y2 @ X2 )
              | ( Y2 = X2 ) ) ) ) ) ).

% less_x_plus_1
thf(fact_4127_no__plus__overflow__neg,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [X2: word @ A,Y2: word @ A] :
          ( ( ord_less @ ( word @ A ) @ X2 @ ( uminus_uminus @ ( word @ A ) @ Y2 ) )
         => ( ord_less_eq @ ( word @ A ) @ X2 @ ( plus_plus @ ( word @ A ) @ X2 @ Y2 ) ) ) ) ).

% no_plus_overflow_neg
thf(fact_4128_divide__powr__uminus,axiom,
    ! [A2: real,B2: real,C2: real] :
      ( ( divide_divide @ real @ A2 @ ( powr @ real @ B2 @ C2 ) )
      = ( times_times @ real @ A2 @ ( powr @ real @ B2 @ ( uminus_uminus @ real @ C2 ) ) ) ) ).

% divide_powr_uminus
thf(fact_4129_word__neg__numeral__alt,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [B2: num] :
          ( ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ B2 ) )
          = ( ring_1_of_int @ ( word @ A ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ B2 ) ) ) ) ) ).

% word_neg_numeral_alt
thf(fact_4130_word__le__make__less,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [Y2: word @ A,X2: word @ A] :
          ( ( Y2
           != ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) )
         => ( ( ord_less_eq @ ( word @ A ) @ X2 @ Y2 )
            = ( ord_less @ ( word @ A ) @ X2 @ ( plus_plus @ ( word @ A ) @ Y2 @ ( one_one @ ( word @ A ) ) ) ) ) ) ) ).

% word_le_make_less
thf(fact_4131_word__Suc__leq,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [K: word @ A,X2: word @ A] :
          ( ( K
           != ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) )
         => ( ( ord_less @ ( word @ A ) @ X2 @ ( plus_plus @ ( word @ A ) @ K @ ( one_one @ ( word @ A ) ) ) )
            = ( ord_less_eq @ ( word @ A ) @ X2 @ K ) ) ) ) ).

% word_Suc_leq
thf(fact_4132_word__Suc__le,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [X2: word @ A,K: word @ A] :
          ( ( X2
           != ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) )
         => ( ( ord_less_eq @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ X2 @ ( one_one @ ( word @ A ) ) ) @ K )
            = ( ord_less @ ( word @ A ) @ X2 @ K ) ) ) ) ).

% word_Suc_le
thf(fact_4133_realpow__square__minus__le,axiom,
    ! [U: real,X2: real] : ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( power_power @ real @ U @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( power_power @ real @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ).

% realpow_square_minus_le
thf(fact_4134_powr__neg__one,axiom,
    ! [X2: real] :
      ( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
     => ( ( powr @ real @ X2 @ ( uminus_uminus @ real @ ( one_one @ real ) ) )
        = ( divide_divide @ real @ ( one_one @ real ) @ X2 ) ) ) ).

% powr_neg_one
thf(fact_4135_Divides_Oadjust__mod__def,axiom,
    ( adjust_mod
    = ( ^ [L2: int,R: int] :
          ( if @ int
          @ ( R
            = ( zero_zero @ int ) )
          @ ( zero_zero @ int )
          @ ( minus_minus @ int @ L2 @ R ) ) ) ) ).

% Divides.adjust_mod_def
thf(fact_4136_Bernoulli__inequality,axiom,
    ! [X2: real,N2: nat] :
      ( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X2 )
     => ( ord_less_eq @ real @ ( plus_plus @ real @ ( one_one @ real ) @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N2 ) @ X2 ) ) @ ( power_power @ real @ ( plus_plus @ real @ ( one_one @ real ) @ X2 ) @ N2 ) ) ) ).

% Bernoulli_inequality
thf(fact_4137_log__minus__eq__powr,axiom,
    ! [B2: real,X2: real,Y2: real] :
      ( ( ord_less @ real @ ( zero_zero @ real ) @ B2 )
     => ( ( B2
         != ( one_one @ real ) )
       => ( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
         => ( ( minus_minus @ real @ ( log @ B2 @ X2 ) @ Y2 )
            = ( log @ B2 @ ( times_times @ real @ X2 @ ( powr @ real @ B2 @ ( uminus_uminus @ real @ Y2 ) ) ) ) ) ) ) ) ).

% log_minus_eq_powr
thf(fact_4138_powr__neg__numeral,axiom,
    ! [X2: real,N2: num] :
      ( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
     => ( ( powr @ real @ X2 @ ( uminus_uminus @ real @ ( numeral_numeral @ real @ N2 ) ) )
        = ( divide_divide @ real @ ( one_one @ real ) @ ( power_power @ real @ X2 @ ( numeral_numeral @ nat @ N2 ) ) ) ) ) ).

% powr_neg_numeral
thf(fact_4139_cos__coeff__def,axiom,
    ( cos_coeff
    = ( ^ [N: nat] : ( if @ real @ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ ( divide_divide @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ ( divide_divide @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( semiring_char_0_fact @ real @ N ) ) @ ( zero_zero @ real ) ) ) ) ).

% cos_coeff_def
thf(fact_4140_minus__one__div__numeral,axiom,
    ! [N2: num] :
      ( ( divide_divide @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( numeral_numeral @ int @ N2 ) )
      = ( uminus_uminus @ int @ ( adjust_div @ ( unique8689654367752047608divmod @ int @ one2 @ N2 ) ) ) ) ).

% minus_one_div_numeral
thf(fact_4141_one__div__minus__numeral,axiom,
    ! [N2: num] :
      ( ( divide_divide @ int @ ( one_one @ int ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N2 ) ) )
      = ( uminus_uminus @ int @ ( adjust_div @ ( unique8689654367752047608divmod @ int @ one2 @ N2 ) ) ) ) ).

% one_div_minus_numeral
thf(fact_4142_sin__coeff__def,axiom,
    ( sin_coeff
    = ( ^ [N: nat] : ( if @ real @ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ ( zero_zero @ real ) @ ( divide_divide @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ ( divide_divide @ nat @ ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( semiring_char_0_fact @ real @ N ) ) ) ) ) ).

% sin_coeff_def
thf(fact_4143_compl__less__compl__iff,axiom,
    ! [A: $tType] :
      ( ( boolea8198339166811842893lgebra @ A )
     => ! [X2: A,Y2: A] :
          ( ( ord_less @ A @ ( uminus_uminus @ A @ X2 ) @ ( uminus_uminus @ A @ Y2 ) )
          = ( ord_less @ A @ Y2 @ X2 ) ) ) ).

% compl_less_compl_iff
thf(fact_4144_compl__le__compl__iff,axiom,
    ! [A: $tType] :
      ( ( boolea8198339166811842893lgebra @ A )
     => ! [X2: A,Y2: A] :
          ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ X2 ) @ ( uminus_uminus @ A @ Y2 ) )
          = ( ord_less_eq @ A @ Y2 @ X2 ) ) ) ).

% compl_le_compl_iff
thf(fact_4145_minus__numeral__div__numeral,axiom,
    ! [M: num,N2: num] :
      ( ( divide_divide @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M ) ) @ ( numeral_numeral @ int @ N2 ) )
      = ( uminus_uminus @ int @ ( adjust_div @ ( unique8689654367752047608divmod @ int @ M @ N2 ) ) ) ) ).

% minus_numeral_div_numeral
thf(fact_4146_numeral__div__minus__numeral,axiom,
    ! [M: num,N2: num] :
      ( ( divide_divide @ int @ ( numeral_numeral @ int @ M ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N2 ) ) )
      = ( uminus_uminus @ int @ ( adjust_div @ ( unique8689654367752047608divmod @ int @ M @ N2 ) ) ) ) ).

% numeral_div_minus_numeral
thf(fact_4147_uminus__set__def,axiom,
    ! [A: $tType] :
      ( ( uminus_uminus @ ( set @ A ) )
      = ( ^ [A7: set @ A] :
            ( collect @ A
            @ ( uminus_uminus @ ( A > $o )
              @ ^ [X: A] : ( member @ A @ X @ A7 ) ) ) ) ) ).

% uminus_set_def
thf(fact_4148_Collect__neg__eq,axiom,
    ! [A: $tType,P: A > $o] :
      ( ( collect @ A
        @ ^ [X: A] :
            ~ ( P @ X ) )
      = ( uminus_uminus @ ( set @ A ) @ ( collect @ A @ P ) ) ) ).

% Collect_neg_eq
thf(fact_4149_Compl__eq,axiom,
    ! [A: $tType] :
      ( ( uminus_uminus @ ( set @ A ) )
      = ( ^ [A7: set @ A] :
            ( collect @ A
            @ ^ [X: A] :
                ~ ( member @ A @ X @ A7 ) ) ) ) ).

% Compl_eq
thf(fact_4150_compl__le__swap2,axiom,
    ! [A: $tType] :
      ( ( boolea8198339166811842893lgebra @ A )
     => ! [Y2: A,X2: A] :
          ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ Y2 ) @ X2 )
         => ( ord_less_eq @ A @ ( uminus_uminus @ A @ X2 ) @ Y2 ) ) ) ).

% compl_le_swap2
thf(fact_4151_compl__le__swap1,axiom,
    ! [A: $tType] :
      ( ( boolea8198339166811842893lgebra @ A )
     => ! [Y2: A,X2: A] :
          ( ( ord_less_eq @ A @ Y2 @ ( uminus_uminus @ A @ X2 ) )
         => ( ord_less_eq @ A @ X2 @ ( uminus_uminus @ A @ Y2 ) ) ) ) ).

% compl_le_swap1
thf(fact_4152_compl__mono,axiom,
    ! [A: $tType] :
      ( ( boolea8198339166811842893lgebra @ A )
     => ! [X2: A,Y2: A] :
          ( ( ord_less_eq @ A @ X2 @ Y2 )
         => ( ord_less_eq @ A @ ( uminus_uminus @ A @ Y2 ) @ ( uminus_uminus @ A @ X2 ) ) ) ) ).

% compl_mono
thf(fact_4153_compl__less__swap1,axiom,
    ! [A: $tType] :
      ( ( boolea8198339166811842893lgebra @ A )
     => ! [Y2: A,X2: A] :
          ( ( ord_less @ A @ Y2 @ ( uminus_uminus @ A @ X2 ) )
         => ( ord_less @ A @ X2 @ ( uminus_uminus @ A @ Y2 ) ) ) ) ).

% compl_less_swap1
thf(fact_4154_compl__less__swap2,axiom,
    ! [A: $tType] :
      ( ( boolea8198339166811842893lgebra @ A )
     => ! [Y2: A,X2: A] :
          ( ( ord_less @ A @ ( uminus_uminus @ A @ Y2 ) @ X2 )
         => ( ord_less @ A @ ( uminus_uminus @ A @ X2 ) @ Y2 ) ) ) ).

% compl_less_swap2
thf(fact_4155_sin__coeff__Suc,axiom,
    ! [N2: nat] :
      ( ( sin_coeff @ ( suc @ N2 ) )
      = ( divide_divide @ real @ ( cos_coeff @ N2 ) @ ( semiring_1_of_nat @ real @ ( suc @ N2 ) ) ) ) ).

% sin_coeff_Suc
thf(fact_4156_cos__coeff__Suc,axiom,
    ! [N2: nat] :
      ( ( cos_coeff @ ( suc @ N2 ) )
      = ( divide_divide @ real @ ( uminus_uminus @ real @ ( sin_coeff @ N2 ) ) @ ( semiring_1_of_nat @ real @ ( suc @ N2 ) ) ) ) ).

% cos_coeff_Suc
thf(fact_4157_diff__shunt__var,axiom,
    ! [A: $tType] :
      ( ( boolea8198339166811842893lgebra @ A )
     => ! [X2: A,Y2: A] :
          ( ( ( minus_minus @ A @ X2 @ Y2 )
            = ( bot_bot @ A ) )
          = ( ord_less_eq @ A @ X2 @ Y2 ) ) ) ).

% diff_shunt_var
thf(fact_4158_integer__of__int__code,axiom,
    ( code_integer_of_int
    = ( ^ [K4: int] :
          ( if @ code_integer @ ( ord_less @ int @ K4 @ ( zero_zero @ int ) ) @ ( uminus_uminus @ code_integer @ ( code_integer_of_int @ ( uminus_uminus @ int @ K4 ) ) )
          @ ( if @ code_integer
            @ ( K4
              = ( zero_zero @ int ) )
            @ ( zero_zero @ code_integer )
            @ ( if @ code_integer
              @ ( ( modulo_modulo @ int @ K4 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) )
                = ( zero_zero @ int ) )
              @ ( times_times @ code_integer @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) @ ( code_integer_of_int @ ( divide_divide @ int @ K4 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) )
              @ ( plus_plus @ code_integer @ ( times_times @ code_integer @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) @ ( code_integer_of_int @ ( divide_divide @ int @ K4 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) @ ( one_one @ code_integer ) ) ) ) ) ) ) ).

% integer_of_int_code
thf(fact_4159_dbl__dec__simps_I4_J,axiom,
    ! [A: $tType] :
      ( ( neg_numeral @ A )
     => ( ( neg_numeral_dbl_dec @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
        = ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( bit1 @ one2 ) ) ) ) ) ).

% dbl_dec_simps(4)
thf(fact_4160_upto_Opinduct,axiom,
    ! [A0: int,A12: int,P: int > int > $o] :
      ( ( accp @ ( product_prod @ int @ int ) @ upto_rel @ ( product_Pair @ int @ int @ A0 @ A12 ) )
     => ( ! [I3: int,J2: int] :
            ( ( accp @ ( product_prod @ int @ int ) @ upto_rel @ ( product_Pair @ int @ int @ I3 @ J2 ) )
           => ( ( ( ord_less_eq @ int @ I3 @ J2 )
               => ( P @ ( plus_plus @ int @ I3 @ ( one_one @ int ) ) @ J2 ) )
             => ( P @ I3 @ J2 ) ) )
       => ( P @ A0 @ A12 ) ) ) ).

% upto.pinduct
thf(fact_4161_signed__take__bit__rec,axiom,
    ! [A: $tType] :
      ( ( bit_ri3973907225187159222ations @ A )
     => ( ( bit_ri4674362597316999326ke_bit @ A )
        = ( ^ [N: nat,A5: A] :
              ( if @ A
              @ ( N
                = ( zero_zero @ nat ) )
              @ ( uminus_uminus @ A @ ( modulo_modulo @ A @ A5 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) )
              @ ( plus_plus @ A @ ( modulo_modulo @ A @ A5 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_ri4674362597316999326ke_bit @ A @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ ( divide_divide @ A @ A5 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ).

% signed_take_bit_rec
thf(fact_4162_dbl__dec__simps_I3_J,axiom,
    ! [A: $tType] :
      ( ( neg_numeral @ A )
     => ( ( neg_numeral_dbl_dec @ A @ ( one_one @ A ) )
        = ( one_one @ A ) ) ) ).

% dbl_dec_simps(3)
thf(fact_4163_signed__take__bit__Suc__1,axiom,
    ! [A: $tType] :
      ( ( bit_ri3973907225187159222ations @ A )
     => ! [N2: nat] :
          ( ( bit_ri4674362597316999326ke_bit @ A @ ( suc @ N2 ) @ ( one_one @ A ) )
          = ( one_one @ A ) ) ) ).

% signed_take_bit_Suc_1
thf(fact_4164_signed__take__bit__numeral__of__1,axiom,
    ! [A: $tType] :
      ( ( bit_ri3973907225187159222ations @ A )
     => ! [K: num] :
          ( ( bit_ri4674362597316999326ke_bit @ A @ ( numeral_numeral @ nat @ K ) @ ( one_one @ A ) )
          = ( one_one @ A ) ) ) ).

% signed_take_bit_numeral_of_1
thf(fact_4165_signed__take__bit__of__minus__1,axiom,
    ! [A: $tType] :
      ( ( bit_ri3973907225187159222ations @ A )
     => ! [N2: nat] :
          ( ( bit_ri4674362597316999326ke_bit @ A @ N2 @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
          = ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).

% signed_take_bit_of_minus_1
thf(fact_4166_dbl__dec__simps_I2_J,axiom,
    ! [A: $tType] :
      ( ( neg_numeral @ A )
     => ( ( neg_numeral_dbl_dec @ A @ ( zero_zero @ A ) )
        = ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).

% dbl_dec_simps(2)
thf(fact_4167_dbl__inc__simps_I1_J,axiom,
    ! [A: $tType] :
      ( ( neg_numeral @ A )
     => ! [K: num] :
          ( ( neg_numeral_dbl_inc @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ K ) ) )
          = ( uminus_uminus @ A @ ( neg_numeral_dbl_dec @ A @ ( numeral_numeral @ A @ K ) ) ) ) ) ).

% dbl_inc_simps(1)
thf(fact_4168_dbl__dec__simps_I1_J,axiom,
    ! [A: $tType] :
      ( ( neg_numeral @ A )
     => ! [K: num] :
          ( ( neg_numeral_dbl_dec @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ K ) ) )
          = ( uminus_uminus @ A @ ( neg_numeral_dbl_inc @ A @ ( numeral_numeral @ A @ K ) ) ) ) ) ).

% dbl_dec_simps(1)
thf(fact_4169_signed__take__bit__Suc__bit0,axiom,
    ! [N2: nat,K: num] :
      ( ( bit_ri4674362597316999326ke_bit @ int @ ( suc @ N2 ) @ ( numeral_numeral @ int @ ( bit0 @ K ) ) )
      = ( times_times @ int @ ( bit_ri4674362597316999326ke_bit @ int @ N2 @ ( numeral_numeral @ int @ K ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ).

% signed_take_bit_Suc_bit0
thf(fact_4170_signed__take__bit__Suc__minus__bit0,axiom,
    ! [N2: nat,K: num] :
      ( ( bit_ri4674362597316999326ke_bit @ int @ ( suc @ N2 ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ K ) ) ) )
      = ( times_times @ int @ ( bit_ri4674362597316999326ke_bit @ int @ N2 @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K ) ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ).

% signed_take_bit_Suc_minus_bit0
thf(fact_4171_signed__take__bit__0,axiom,
    ! [A: $tType] :
      ( ( bit_ri3973907225187159222ations @ A )
     => ! [A2: A] :
          ( ( bit_ri4674362597316999326ke_bit @ A @ ( zero_zero @ nat ) @ A2 )
          = ( uminus_uminus @ A @ ( modulo_modulo @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).

% signed_take_bit_0
thf(fact_4172_signed__take__bit__Suc__bit1,axiom,
    ! [N2: nat,K: num] :
      ( ( bit_ri4674362597316999326ke_bit @ int @ ( suc @ N2 ) @ ( numeral_numeral @ int @ ( bit1 @ K ) ) )
      = ( plus_plus @ int @ ( times_times @ int @ ( bit_ri4674362597316999326ke_bit @ int @ N2 @ ( numeral_numeral @ int @ K ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( one_one @ int ) ) ) ).

% signed_take_bit_Suc_bit1
thf(fact_4173_signed__take__bit__Suc__minus__bit1,axiom,
    ! [N2: nat,K: num] :
      ( ( bit_ri4674362597316999326ke_bit @ int @ ( suc @ N2 ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ K ) ) ) )
      = ( plus_plus @ int @ ( times_times @ int @ ( bit_ri4674362597316999326ke_bit @ int @ N2 @ ( minus_minus @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K ) ) @ ( one_one @ int ) ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( one_one @ int ) ) ) ).

% signed_take_bit_Suc_minus_bit1
thf(fact_4174_bot__nat__def,axiom,
    ( ( bot_bot @ nat )
    = ( zero_zero @ nat ) ) ).

% bot_nat_def
thf(fact_4175_signed__take__bit__add,axiom,
    ! [N2: nat,K: int,L: int] :
      ( ( bit_ri4674362597316999326ke_bit @ int @ N2 @ ( plus_plus @ int @ ( bit_ri4674362597316999326ke_bit @ int @ N2 @ K ) @ ( bit_ri4674362597316999326ke_bit @ int @ N2 @ L ) ) )
      = ( bit_ri4674362597316999326ke_bit @ int @ N2 @ ( plus_plus @ int @ K @ L ) ) ) ).

% signed_take_bit_add
thf(fact_4176_signed__take__bit__diff,axiom,
    ! [N2: nat,K: int,L: int] :
      ( ( bit_ri4674362597316999326ke_bit @ int @ N2 @ ( minus_minus @ int @ ( bit_ri4674362597316999326ke_bit @ int @ N2 @ K ) @ ( bit_ri4674362597316999326ke_bit @ int @ N2 @ L ) ) )
      = ( bit_ri4674362597316999326ke_bit @ int @ N2 @ ( minus_minus @ int @ K @ L ) ) ) ).

% signed_take_bit_diff
thf(fact_4177_signed__take__bit__mult,axiom,
    ! [N2: nat,K: int,L: int] :
      ( ( bit_ri4674362597316999326ke_bit @ int @ N2 @ ( times_times @ int @ ( bit_ri4674362597316999326ke_bit @ int @ N2 @ K ) @ ( bit_ri4674362597316999326ke_bit @ int @ N2 @ L ) ) )
      = ( bit_ri4674362597316999326ke_bit @ int @ N2 @ ( times_times @ int @ K @ L ) ) ) ).

% signed_take_bit_mult
thf(fact_4178_bot__option__def,axiom,
    ! [A: $tType] :
      ( ( order @ A )
     => ( ( bot_bot @ ( option @ A ) )
        = ( none @ A ) ) ) ).

% bot_option_def
thf(fact_4179_one__integer__def,axiom,
    ( ( one_one @ code_integer )
    = ( code_integer_of_int @ ( one_one @ int ) ) ) ).

% one_integer_def
thf(fact_4180_less__eq__integer_Oabs__eq,axiom,
    ! [Xa: int,X2: int] :
      ( ( ord_less_eq @ code_integer @ ( code_integer_of_int @ Xa ) @ ( code_integer_of_int @ X2 ) )
      = ( ord_less_eq @ int @ Xa @ X2 ) ) ).

% less_eq_integer.abs_eq
thf(fact_4181_plus__integer_Oabs__eq,axiom,
    ! [Xa: int,X2: int] :
      ( ( plus_plus @ code_integer @ ( code_integer_of_int @ Xa ) @ ( code_integer_of_int @ X2 ) )
      = ( code_integer_of_int @ ( plus_plus @ int @ Xa @ X2 ) ) ) ).

% plus_integer.abs_eq
thf(fact_4182_minus__integer_Oabs__eq,axiom,
    ! [Xa: int,X2: int] :
      ( ( minus_minus @ code_integer @ ( code_integer_of_int @ Xa ) @ ( code_integer_of_int @ X2 ) )
      = ( code_integer_of_int @ ( minus_minus @ int @ Xa @ X2 ) ) ) ).

% minus_integer.abs_eq
thf(fact_4183_times__integer_Oabs__eq,axiom,
    ! [Xa: int,X2: int] :
      ( ( times_times @ code_integer @ ( code_integer_of_int @ Xa ) @ ( code_integer_of_int @ X2 ) )
      = ( code_integer_of_int @ ( times_times @ int @ Xa @ X2 ) ) ) ).

% times_integer.abs_eq
thf(fact_4184_divide__integer_Oabs__eq,axiom,
    ! [Xa: int,X2: int] :
      ( ( divide_divide @ code_integer @ ( code_integer_of_int @ Xa ) @ ( code_integer_of_int @ X2 ) )
      = ( code_integer_of_int @ ( divide_divide @ int @ Xa @ X2 ) ) ) ).

% divide_integer.abs_eq
thf(fact_4185_bot__empty__eq2,axiom,
    ! [B: $tType,A: $tType] :
      ( ( bot_bot @ ( A > B > $o ) )
      = ( ^ [X: A,Y: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y ) @ ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ) ) ).

% bot_empty_eq2
thf(fact_4186_signed__take__bit__int__less__exp,axiom,
    ! [N2: nat,K: int] : ( ord_less @ int @ ( bit_ri4674362597316999326ke_bit @ int @ N2 @ K ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N2 ) ) ).

% signed_take_bit_int_less_exp
thf(fact_4187_even__signed__take__bit__iff,axiom,
    ! [A: $tType] :
      ( ( bit_ri3973907225187159222ations @ A )
     => ! [M: nat,A2: A] :
          ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_ri4674362597316999326ke_bit @ A @ M @ A2 ) )
          = ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) ) ) ).

% even_signed_take_bit_iff
thf(fact_4188_signed__take__bit__int__less__self__iff,axiom,
    ! [N2: nat,K: int] :
      ( ( ord_less @ int @ ( bit_ri4674362597316999326ke_bit @ int @ N2 @ K ) @ K )
      = ( ord_less_eq @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N2 ) @ K ) ) ).

% signed_take_bit_int_less_self_iff
thf(fact_4189_signed__take__bit__int__greater__eq__self__iff,axiom,
    ! [K: int,N2: nat] :
      ( ( ord_less_eq @ int @ K @ ( bit_ri4674362597316999326ke_bit @ int @ N2 @ K ) )
      = ( ord_less @ int @ K @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N2 ) ) ) ).

% signed_take_bit_int_greater_eq_self_iff
thf(fact_4190_signed__take__bit__int__less__eq__self__iff,axiom,
    ! [N2: nat,K: int] :
      ( ( ord_less_eq @ int @ ( bit_ri4674362597316999326ke_bit @ int @ N2 @ K ) @ K )
      = ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N2 ) ) @ K ) ) ).

% signed_take_bit_int_less_eq_self_iff
thf(fact_4191_signed__take__bit__int__greater__eq__minus__exp,axiom,
    ! [N2: nat,K: int] : ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N2 ) ) @ ( bit_ri4674362597316999326ke_bit @ int @ N2 @ K ) ) ).

% signed_take_bit_int_greater_eq_minus_exp
thf(fact_4192_signed__take__bit__int__greater__self__iff,axiom,
    ! [K: int,N2: nat] :
      ( ( ord_less @ int @ K @ ( bit_ri4674362597316999326ke_bit @ int @ N2 @ K ) )
      = ( ord_less @ int @ K @ ( uminus_uminus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N2 ) ) ) ) ).

% signed_take_bit_int_greater_self_iff
thf(fact_4193_dbl__dec__def,axiom,
    ! [A: $tType] :
      ( ( neg_numeral @ A )
     => ( ( neg_numeral_dbl_dec @ A )
        = ( ^ [X: A] : ( minus_minus @ A @ ( plus_plus @ A @ X @ X ) @ ( one_one @ A ) ) ) ) ) ).

% dbl_dec_def
thf(fact_4194_signed__take__bit__int__less__eq,axiom,
    ! [N2: nat,K: int] :
      ( ( ord_less_eq @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N2 ) @ K )
     => ( ord_less_eq @ int @ ( bit_ri4674362597316999326ke_bit @ int @ N2 @ K ) @ ( minus_minus @ int @ K @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( suc @ N2 ) ) ) ) ) ).

% signed_take_bit_int_less_eq
thf(fact_4195_signed__take__bit__int__eq__self,axiom,
    ! [N2: nat,K: int] :
      ( ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N2 ) ) @ K )
     => ( ( ord_less @ int @ K @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N2 ) )
       => ( ( bit_ri4674362597316999326ke_bit @ int @ N2 @ K )
          = K ) ) ) ).

% signed_take_bit_int_eq_self
thf(fact_4196_signed__take__bit__int__eq__self__iff,axiom,
    ! [N2: nat,K: int] :
      ( ( ( bit_ri4674362597316999326ke_bit @ int @ N2 @ K )
        = K )
      = ( ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N2 ) ) @ K )
        & ( ord_less @ int @ K @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N2 ) ) ) ) ).

% signed_take_bit_int_eq_self_iff
thf(fact_4197_signed__take__bit__int__greater__eq,axiom,
    ! [K: int,N2: nat] :
      ( ( ord_less @ int @ K @ ( uminus_uminus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N2 ) ) )
     => ( ord_less_eq @ int @ ( plus_plus @ int @ K @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( suc @ N2 ) ) ) @ ( bit_ri4674362597316999326ke_bit @ int @ N2 @ K ) ) ) ).

% signed_take_bit_int_greater_eq
thf(fact_4198_signed__take__bit__Suc,axiom,
    ! [A: $tType] :
      ( ( bit_ri3973907225187159222ations @ A )
     => ! [N2: nat,A2: A] :
          ( ( bit_ri4674362597316999326ke_bit @ A @ ( suc @ N2 ) @ A2 )
          = ( plus_plus @ A @ ( modulo_modulo @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_ri4674362597316999326ke_bit @ A @ N2 @ ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).

% signed_take_bit_Suc
thf(fact_4199_sin__paired,axiom,
    ! [X2: real] :
      ( sums @ real
      @ ^ [N: nat] : ( times_times @ real @ ( divide_divide @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ N ) @ ( semiring_char_0_fact @ real @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ ( one_one @ nat ) ) ) ) @ ( power_power @ real @ X2 @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ ( one_one @ nat ) ) ) )
      @ ( sin @ real @ X2 ) ) ).

% sin_paired
thf(fact_4200_signed__take__bit__numeral__minus__bit1,axiom,
    ! [L: num,K: num] :
      ( ( bit_ri4674362597316999326ke_bit @ int @ ( numeral_numeral @ nat @ L ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ K ) ) ) )
      = ( plus_plus @ int @ ( times_times @ int @ ( bit_ri4674362597316999326ke_bit @ int @ ( pred_numeral @ L ) @ ( minus_minus @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K ) ) @ ( one_one @ int ) ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( one_one @ int ) ) ) ).

% signed_take_bit_numeral_minus_bit1
thf(fact_4201_cos__paired,axiom,
    ! [X2: real] :
      ( sums @ real
      @ ^ [N: nat] : ( times_times @ real @ ( divide_divide @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ N ) @ ( semiring_char_0_fact @ real @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) @ ( power_power @ real @ X2 @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) )
      @ ( cos @ real @ X2 ) ) ).

% cos_paired
thf(fact_4202_ln__one__minus__pos__lower__bound,axiom,
    ! [X2: real] :
      ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 )
     => ( ( ord_less_eq @ real @ X2 @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
       => ( ord_less_eq @ real @ ( minus_minus @ real @ ( uminus_uminus @ real @ X2 ) @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( power_power @ real @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( ln_ln @ real @ ( minus_minus @ real @ ( one_one @ real ) @ X2 ) ) ) ) ) ).

% ln_one_minus_pos_lower_bound
thf(fact_4203_cos__zero,axiom,
    ! [A: $tType] :
      ( ( ( real_Vector_banach @ A )
        & ( real_V2822296259951069270ebra_1 @ A ) )
     => ( ( cos @ A @ ( zero_zero @ A ) )
        = ( one_one @ A ) ) ) ).

% cos_zero
thf(fact_4204_ln__one,axiom,
    ! [A: $tType] :
      ( ( ln @ A )
     => ( ( ln_ln @ A @ ( one_one @ A ) )
        = ( zero_zero @ A ) ) ) ).

% ln_one
thf(fact_4205_pred__numeral__simps_I1_J,axiom,
    ( ( pred_numeral @ one2 )
    = ( zero_zero @ nat ) ) ).

% pred_numeral_simps(1)
thf(fact_4206_Suc__eq__numeral,axiom,
    ! [N2: nat,K: num] :
      ( ( ( suc @ N2 )
        = ( numeral_numeral @ nat @ K ) )
      = ( N2
        = ( pred_numeral @ K ) ) ) ).

% Suc_eq_numeral
thf(fact_4207_eq__numeral__Suc,axiom,
    ! [K: num,N2: nat] :
      ( ( ( numeral_numeral @ nat @ K )
        = ( suc @ N2 ) )
      = ( ( pred_numeral @ K )
        = N2 ) ) ).

% eq_numeral_Suc
thf(fact_4208_pred__numeral__simps_I3_J,axiom,
    ! [K: num] :
      ( ( pred_numeral @ ( bit1 @ K ) )
      = ( numeral_numeral @ nat @ ( bit0 @ K ) ) ) ).

% pred_numeral_simps(3)
thf(fact_4209_less__Suc__numeral,axiom,
    ! [N2: nat,K: num] :
      ( ( ord_less @ nat @ ( suc @ N2 ) @ ( numeral_numeral @ nat @ K ) )
      = ( ord_less @ nat @ N2 @ ( pred_numeral @ K ) ) ) ).

% less_Suc_numeral
thf(fact_4210_less__numeral__Suc,axiom,
    ! [K: num,N2: nat] :
      ( ( ord_less @ nat @ ( numeral_numeral @ nat @ K ) @ ( suc @ N2 ) )
      = ( ord_less @ nat @ ( pred_numeral @ K ) @ N2 ) ) ).

% less_numeral_Suc
thf(fact_4211_le__Suc__numeral,axiom,
    ! [N2: nat,K: num] :
      ( ( ord_less_eq @ nat @ ( suc @ N2 ) @ ( numeral_numeral @ nat @ K ) )
      = ( ord_less_eq @ nat @ N2 @ ( pred_numeral @ K ) ) ) ).

% le_Suc_numeral
thf(fact_4212_le__numeral__Suc,axiom,
    ! [K: num,N2: nat] :
      ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ K ) @ ( suc @ N2 ) )
      = ( ord_less_eq @ nat @ ( pred_numeral @ K ) @ N2 ) ) ).

% le_numeral_Suc
thf(fact_4213_diff__Suc__numeral,axiom,
    ! [N2: nat,K: num] :
      ( ( minus_minus @ nat @ ( suc @ N2 ) @ ( numeral_numeral @ nat @ K ) )
      = ( minus_minus @ nat @ N2 @ ( pred_numeral @ K ) ) ) ).

% diff_Suc_numeral
thf(fact_4214_diff__numeral__Suc,axiom,
    ! [K: num,N2: nat] :
      ( ( minus_minus @ nat @ ( numeral_numeral @ nat @ K ) @ ( suc @ N2 ) )
      = ( minus_minus @ nat @ ( pred_numeral @ K ) @ N2 ) ) ).

% diff_numeral_Suc
thf(fact_4215_ln__le__cancel__iff,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
     => ( ( ord_less @ real @ ( zero_zero @ real ) @ Y2 )
       => ( ( ord_less_eq @ real @ ( ln_ln @ real @ X2 ) @ ( ln_ln @ real @ Y2 ) )
          = ( ord_less_eq @ real @ X2 @ Y2 ) ) ) ) ).

% ln_le_cancel_iff
thf(fact_4216_ln__less__zero__iff,axiom,
    ! [X2: real] :
      ( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
     => ( ( ord_less @ real @ ( ln_ln @ real @ X2 ) @ ( zero_zero @ real ) )
        = ( ord_less @ real @ X2 @ ( one_one @ real ) ) ) ) ).

% ln_less_zero_iff
thf(fact_4217_ln__gt__zero__iff,axiom,
    ! [X2: real] :
      ( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
     => ( ( ord_less @ real @ ( zero_zero @ real ) @ ( ln_ln @ real @ X2 ) )
        = ( ord_less @ real @ ( one_one @ real ) @ X2 ) ) ) ).

% ln_gt_zero_iff
thf(fact_4218_ln__eq__zero__iff,axiom,
    ! [X2: real] :
      ( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
     => ( ( ( ln_ln @ real @ X2 )
          = ( zero_zero @ real ) )
        = ( X2
          = ( one_one @ real ) ) ) ) ).

% ln_eq_zero_iff
thf(fact_4219_max__Suc__numeral,axiom,
    ! [N2: nat,K: num] :
      ( ( ord_max @ nat @ ( suc @ N2 ) @ ( numeral_numeral @ nat @ K ) )
      = ( suc @ ( ord_max @ nat @ N2 @ ( pred_numeral @ K ) ) ) ) ).

% max_Suc_numeral
thf(fact_4220_max__numeral__Suc,axiom,
    ! [K: num,N2: nat] :
      ( ( ord_max @ nat @ ( numeral_numeral @ nat @ K ) @ ( suc @ N2 ) )
      = ( suc @ ( ord_max @ nat @ ( pred_numeral @ K ) @ N2 ) ) ) ).

% max_numeral_Suc
thf(fact_4221_sin__cos__squared__add3,axiom,
    ! [A: $tType] :
      ( ( ( real_Vector_banach @ A )
        & ( real_V3459762299906320749_field @ A ) )
     => ! [X2: A] :
          ( ( plus_plus @ A @ ( times_times @ A @ ( cos @ A @ X2 ) @ ( cos @ A @ X2 ) ) @ ( times_times @ A @ ( sin @ A @ X2 ) @ ( sin @ A @ X2 ) ) )
          = ( one_one @ A ) ) ) ).

% sin_cos_squared_add3
thf(fact_4222_ln__le__zero__iff,axiom,
    ! [X2: real] :
      ( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
     => ( ( ord_less_eq @ real @ ( ln_ln @ real @ X2 ) @ ( zero_zero @ real ) )
        = ( ord_less_eq @ real @ X2 @ ( one_one @ real ) ) ) ) ).

% ln_le_zero_iff
thf(fact_4223_ln__ge__zero__iff,axiom,
    ! [X2: real] :
      ( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
     => ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( ln_ln @ real @ X2 ) )
        = ( ord_less_eq @ real @ ( one_one @ real ) @ X2 ) ) ) ).

% ln_ge_zero_iff
thf(fact_4224_signed__take__bit__numeral__bit0,axiom,
    ! [L: num,K: num] :
      ( ( bit_ri4674362597316999326ke_bit @ int @ ( numeral_numeral @ nat @ L ) @ ( numeral_numeral @ int @ ( bit0 @ K ) ) )
      = ( times_times @ int @ ( bit_ri4674362597316999326ke_bit @ int @ ( pred_numeral @ L ) @ ( numeral_numeral @ int @ K ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ).

% signed_take_bit_numeral_bit0
thf(fact_4225_sin__cos__squared__add,axiom,
    ! [A: $tType] :
      ( ( ( real_Vector_banach @ A )
        & ( real_V3459762299906320749_field @ A ) )
     => ! [X2: A] :
          ( ( plus_plus @ A @ ( power_power @ A @ ( sin @ A @ X2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ ( cos @ A @ X2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
          = ( one_one @ A ) ) ) ).

% sin_cos_squared_add
thf(fact_4226_sin__cos__squared__add2,axiom,
    ! [A: $tType] :
      ( ( ( real_Vector_banach @ A )
        & ( real_V3459762299906320749_field @ A ) )
     => ! [X2: A] :
          ( ( plus_plus @ A @ ( power_power @ A @ ( cos @ A @ X2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ ( sin @ A @ X2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
          = ( one_one @ A ) ) ) ).

% sin_cos_squared_add2
thf(fact_4227_signed__take__bit__numeral__minus__bit0,axiom,
    ! [L: num,K: num] :
      ( ( bit_ri4674362597316999326ke_bit @ int @ ( numeral_numeral @ nat @ L ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ K ) ) ) )
      = ( times_times @ int @ ( bit_ri4674362597316999326ke_bit @ int @ ( pred_numeral @ L ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K ) ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ).

% signed_take_bit_numeral_minus_bit0
thf(fact_4228_signed__take__bit__numeral__bit1,axiom,
    ! [L: num,K: num] :
      ( ( bit_ri4674362597316999326ke_bit @ int @ ( numeral_numeral @ nat @ L ) @ ( numeral_numeral @ int @ ( bit1 @ K ) ) )
      = ( plus_plus @ int @ ( times_times @ int @ ( bit_ri4674362597316999326ke_bit @ int @ ( pred_numeral @ L ) @ ( numeral_numeral @ int @ K ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( one_one @ int ) ) ) ).

% signed_take_bit_numeral_bit1
thf(fact_4229_sin__zero__norm__cos__one,axiom,
    ! [A: $tType] :
      ( ( ( real_Vector_banach @ A )
        & ( real_V3459762299906320749_field @ A ) )
     => ! [X2: A] :
          ( ( ( sin @ A @ X2 )
            = ( zero_zero @ A ) )
         => ( ( real_V7770717601297561774m_norm @ A @ ( cos @ A @ X2 ) )
            = ( one_one @ real ) ) ) ) ).

% sin_zero_norm_cos_one
thf(fact_4230_cos__add,axiom,
    ! [A: $tType] :
      ( ( ( real_Vector_banach @ A )
        & ( real_V3459762299906320749_field @ A ) )
     => ! [X2: A,Y2: A] :
          ( ( cos @ A @ ( plus_plus @ A @ X2 @ Y2 ) )
          = ( minus_minus @ A @ ( times_times @ A @ ( cos @ A @ X2 ) @ ( cos @ A @ Y2 ) ) @ ( times_times @ A @ ( sin @ A @ X2 ) @ ( sin @ A @ Y2 ) ) ) ) ) ).

% cos_add
thf(fact_4231_cos__diff,axiom,
    ! [A: $tType] :
      ( ( ( real_Vector_banach @ A )
        & ( real_V3459762299906320749_field @ A ) )
     => ! [X2: A,Y2: A] :
          ( ( cos @ A @ ( minus_minus @ A @ X2 @ Y2 ) )
          = ( plus_plus @ A @ ( times_times @ A @ ( cos @ A @ X2 ) @ ( cos @ A @ Y2 ) ) @ ( times_times @ A @ ( sin @ A @ X2 ) @ ( sin @ A @ Y2 ) ) ) ) ) ).

% cos_diff
thf(fact_4232_polar__Ex,axiom,
    ! [X2: real,Y2: real] :
    ? [R3: real,A4: real] :
      ( ( X2
        = ( times_times @ real @ R3 @ ( cos @ real @ A4 ) ) )
      & ( Y2
        = ( times_times @ real @ R3 @ ( sin @ real @ A4 ) ) ) ) ).

% polar_Ex
thf(fact_4233_sin__diff,axiom,
    ! [A: $tType] :
      ( ( ( real_Vector_banach @ A )
        & ( real_V3459762299906320749_field @ A ) )
     => ! [X2: A,Y2: A] :
          ( ( sin @ A @ ( minus_minus @ A @ X2 @ Y2 ) )
          = ( minus_minus @ A @ ( times_times @ A @ ( sin @ A @ X2 ) @ ( cos @ A @ Y2 ) ) @ ( times_times @ A @ ( cos @ A @ X2 ) @ ( sin @ A @ Y2 ) ) ) ) ) ).

% sin_diff
thf(fact_4234_sin__add,axiom,
    ! [A: $tType] :
      ( ( ( real_Vector_banach @ A )
        & ( real_V3459762299906320749_field @ A ) )
     => ! [X2: A,Y2: A] :
          ( ( sin @ A @ ( plus_plus @ A @ X2 @ Y2 ) )
          = ( plus_plus @ A @ ( times_times @ A @ ( sin @ A @ X2 ) @ ( cos @ A @ Y2 ) ) @ ( times_times @ A @ ( cos @ A @ X2 ) @ ( sin @ A @ Y2 ) ) ) ) ) ).

% sin_add
thf(fact_4235_cos__one__sin__zero,axiom,
    ! [A: $tType] :
      ( ( ( real_Vector_banach @ A )
        & ( real_V3459762299906320749_field @ A ) )
     => ! [X2: A] :
          ( ( ( cos @ A @ X2 )
            = ( one_one @ A ) )
         => ( ( sin @ A @ X2 )
            = ( zero_zero @ A ) ) ) ) ).

% cos_one_sin_zero
thf(fact_4236_sin__double,axiom,
    ! [A: $tType] :
      ( ( ( real_Vector_banach @ A )
        & ( real_V3459762299906320749_field @ A ) )
     => ! [X2: A] :
          ( ( sin @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ X2 ) )
          = ( times_times @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( sin @ A @ X2 ) ) @ ( cos @ A @ X2 ) ) ) ) ).

% sin_double
thf(fact_4237_sin__x__le__x,axiom,
    ! [X2: real] :
      ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 )
     => ( ord_less_eq @ real @ ( sin @ real @ X2 ) @ X2 ) ) ).

% sin_x_le_x
thf(fact_4238_sin__le__one,axiom,
    ! [X2: real] : ( ord_less_eq @ real @ ( sin @ real @ X2 ) @ ( one_one @ real ) ) ).

% sin_le_one
thf(fact_4239_cos__le__one,axiom,
    ! [X2: real] : ( ord_less_eq @ real @ ( cos @ real @ X2 ) @ ( one_one @ real ) ) ).

% cos_le_one
thf(fact_4240_log__def,axiom,
    ( log
    = ( ^ [A5: real,X: real] : ( divide_divide @ real @ ( ln_ln @ real @ X ) @ ( ln_ln @ real @ A5 ) ) ) ) ).

% log_def
thf(fact_4241_numeral__eq__Suc,axiom,
    ( ( numeral_numeral @ nat )
    = ( ^ [K4: num] : ( suc @ ( pred_numeral @ K4 ) ) ) ) ).

% numeral_eq_Suc
thf(fact_4242_cos__squared__eq,axiom,
    ! [A: $tType] :
      ( ( ( real_Vector_banach @ A )
        & ( real_V3459762299906320749_field @ A ) )
     => ! [X2: A] :
          ( ( power_power @ A @ ( cos @ A @ X2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
          = ( minus_minus @ A @ ( one_one @ A ) @ ( power_power @ A @ ( sin @ A @ X2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).

% cos_squared_eq
thf(fact_4243_sin__squared__eq,axiom,
    ! [A: $tType] :
      ( ( ( real_Vector_banach @ A )
        & ( real_V3459762299906320749_field @ A ) )
     => ! [X2: A] :
          ( ( power_power @ A @ ( sin @ A @ X2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
          = ( minus_minus @ A @ ( one_one @ A ) @ ( power_power @ A @ ( cos @ A @ X2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).

% sin_squared_eq
thf(fact_4244_ln__bound,axiom,
    ! [X2: real] :
      ( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
     => ( ord_less_eq @ real @ ( ln_ln @ real @ X2 ) @ X2 ) ) ).

% ln_bound
thf(fact_4245_ln__gt__zero__imp__gt__one,axiom,
    ! [X2: real] :
      ( ( ord_less @ real @ ( zero_zero @ real ) @ ( ln_ln @ real @ X2 ) )
     => ( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
       => ( ord_less @ real @ ( one_one @ real ) @ X2 ) ) ) ).

% ln_gt_zero_imp_gt_one
thf(fact_4246_ln__less__zero,axiom,
    ! [X2: real] :
      ( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
     => ( ( ord_less @ real @ X2 @ ( one_one @ real ) )
       => ( ord_less @ real @ ( ln_ln @ real @ X2 ) @ ( zero_zero @ real ) ) ) ) ).

% ln_less_zero
thf(fact_4247_ln__gt__zero,axiom,
    ! [X2: real] :
      ( ( ord_less @ real @ ( one_one @ real ) @ X2 )
     => ( ord_less @ real @ ( zero_zero @ real ) @ ( ln_ln @ real @ X2 ) ) ) ).

% ln_gt_zero
thf(fact_4248_ln__ge__zero,axiom,
    ! [X2: real] :
      ( ( ord_less_eq @ real @ ( one_one @ real ) @ X2 )
     => ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( ln_ln @ real @ X2 ) ) ) ).

% ln_ge_zero
thf(fact_4249_ln__powr,axiom,
    ! [X2: real,Y2: real] :
      ( ( X2
       != ( zero_zero @ real ) )
     => ( ( ln_ln @ real @ ( powr @ real @ X2 @ Y2 ) )
        = ( times_times @ real @ Y2 @ ( ln_ln @ real @ X2 ) ) ) ) ).

% ln_powr
thf(fact_4250_sin__x__ge__neg__x,axiom,
    ! [X2: real] :
      ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 )
     => ( ord_less_eq @ real @ ( uminus_uminus @ real @ X2 ) @ ( sin @ real @ X2 ) ) ) ).

% sin_x_ge_neg_x
thf(fact_4251_sin__ge__minus__one,axiom,
    ! [X2: real] : ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ ( sin @ real @ X2 ) ) ).

% sin_ge_minus_one
thf(fact_4252_cos__ge__minus__one,axiom,
    ! [X2: real] : ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ ( cos @ real @ X2 ) ) ).

% cos_ge_minus_one
thf(fact_4253_cos__diff__cos,axiom,
    ! [A: $tType] :
      ( ( ( real_Vector_banach @ A )
        & ( real_V3459762299906320749_field @ A ) )
     => ! [W: A,Z: A] :
          ( ( minus_minus @ A @ ( cos @ A @ W ) @ ( cos @ A @ Z ) )
          = ( times_times @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( sin @ A @ ( divide_divide @ A @ ( plus_plus @ A @ W @ Z ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) @ ( sin @ A @ ( divide_divide @ A @ ( minus_minus @ A @ Z @ W ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ).

% cos_diff_cos
thf(fact_4254_sin__diff__sin,axiom,
    ! [A: $tType] :
      ( ( ( real_Vector_banach @ A )
        & ( real_V3459762299906320749_field @ A ) )
     => ! [W: A,Z: A] :
          ( ( minus_minus @ A @ ( sin @ A @ W ) @ ( sin @ A @ Z ) )
          = ( times_times @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( sin @ A @ ( divide_divide @ A @ ( minus_minus @ A @ W @ Z ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) @ ( cos @ A @ ( divide_divide @ A @ ( plus_plus @ A @ W @ Z ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ).

% sin_diff_sin
thf(fact_4255_sin__plus__sin,axiom,
    ! [A: $tType] :
      ( ( ( real_Vector_banach @ A )
        & ( real_V3459762299906320749_field @ A ) )
     => ! [W: A,Z: A] :
          ( ( plus_plus @ A @ ( sin @ A @ W ) @ ( sin @ A @ Z ) )
          = ( times_times @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( sin @ A @ ( divide_divide @ A @ ( plus_plus @ A @ W @ Z ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) @ ( cos @ A @ ( divide_divide @ A @ ( minus_minus @ A @ W @ Z ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ).

% sin_plus_sin
thf(fact_4256_cos__times__sin,axiom,
    ! [A: $tType] :
      ( ( ( real_Vector_banach @ A )
        & ( real_V3459762299906320749_field @ A ) )
     => ! [W: A,Z: A] :
          ( ( times_times @ A @ ( cos @ A @ W ) @ ( sin @ A @ Z ) )
          = ( divide_divide @ A @ ( minus_minus @ A @ ( sin @ A @ ( plus_plus @ A @ W @ Z ) ) @ ( sin @ A @ ( minus_minus @ A @ W @ Z ) ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).

% cos_times_sin
thf(fact_4257_sin__times__cos,axiom,
    ! [A: $tType] :
      ( ( ( real_Vector_banach @ A )
        & ( real_V3459762299906320749_field @ A ) )
     => ! [W: A,Z: A] :
          ( ( times_times @ A @ ( sin @ A @ W ) @ ( cos @ A @ Z ) )
          = ( divide_divide @ A @ ( plus_plus @ A @ ( sin @ A @ ( plus_plus @ A @ W @ Z ) ) @ ( sin @ A @ ( minus_minus @ A @ W @ Z ) ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).

% sin_times_cos
thf(fact_4258_sin__times__sin,axiom,
    ! [A: $tType] :
      ( ( ( real_Vector_banach @ A )
        & ( real_V3459762299906320749_field @ A ) )
     => ! [W: A,Z: A] :
          ( ( times_times @ A @ ( sin @ A @ W ) @ ( sin @ A @ Z ) )
          = ( divide_divide @ A @ ( minus_minus @ A @ ( cos @ A @ ( minus_minus @ A @ W @ Z ) ) @ ( cos @ A @ ( plus_plus @ A @ W @ Z ) ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).

% sin_times_sin
thf(fact_4259_cos__double,axiom,
    ! [A: $tType] :
      ( ( ( real_Vector_banach @ A )
        & ( real_V3459762299906320749_field @ A ) )
     => ! [X2: A] :
          ( ( cos @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ X2 ) )
          = ( minus_minus @ A @ ( power_power @ A @ ( cos @ A @ X2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ ( sin @ A @ X2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).

% cos_double
thf(fact_4260_pred__numeral__def,axiom,
    ( pred_numeral
    = ( ^ [K4: num] : ( minus_minus @ nat @ ( numeral_numeral @ nat @ K4 ) @ ( one_one @ nat ) ) ) ) ).

% pred_numeral_def
thf(fact_4261_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_4262_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_4263_cos__double__sin,axiom,
    ! [A: $tType] :
      ( ( ( real_Vector_banach @ A )
        & ( real_V3459762299906320749_field @ A ) )
     => ! [W: A] :
          ( ( cos @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ W ) )
          = ( minus_minus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( power_power @ A @ ( sin @ A @ W ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ).

% cos_double_sin
thf(fact_4264_ln__ge__zero__imp__ge__one,axiom,
    ! [X2: real] :
      ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( ln_ln @ real @ X2 ) )
     => ( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
       => ( ord_less_eq @ real @ ( one_one @ real ) @ X2 ) ) ) ).

% ln_ge_zero_imp_ge_one
thf(fact_4265_ln__add__one__self__le__self,axiom,
    ! [X2: real] :
      ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 )
     => ( ord_less_eq @ real @ ( ln_ln @ real @ ( plus_plus @ real @ ( one_one @ real ) @ X2 ) ) @ X2 ) ) ).

% ln_add_one_self_le_self
thf(fact_4266_ln__mult,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
     => ( ( ord_less @ real @ ( zero_zero @ real ) @ Y2 )
       => ( ( ln_ln @ real @ ( times_times @ real @ X2 @ Y2 ) )
          = ( plus_plus @ real @ ( ln_ln @ real @ X2 ) @ ( ln_ln @ real @ Y2 ) ) ) ) ) ).

% ln_mult
thf(fact_4267_ln__eq__minus__one,axiom,
    ! [X2: real] :
      ( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
     => ( ( ( ln_ln @ real @ X2 )
          = ( minus_minus @ real @ X2 @ ( one_one @ real ) ) )
       => ( X2
          = ( one_one @ real ) ) ) ) ).

% ln_eq_minus_one
thf(fact_4268_ln__div,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
     => ( ( ord_less @ real @ ( zero_zero @ real ) @ Y2 )
       => ( ( ln_ln @ real @ ( divide_divide @ real @ X2 @ Y2 ) )
          = ( minus_minus @ real @ ( ln_ln @ real @ X2 ) @ ( ln_ln @ real @ Y2 ) ) ) ) ) ).

% ln_div
thf(fact_4269_cos__two__neq__zero,axiom,
    ( ( cos @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) )
   != ( zero_zero @ real ) ) ).

% cos_two_neq_zero
thf(fact_4270_fact__numeral,axiom,
    ! [A: $tType] :
      ( ( semiring_char_0 @ A )
     => ! [K: num] :
          ( ( semiring_char_0_fact @ A @ ( numeral_numeral @ nat @ K ) )
          = ( times_times @ A @ ( numeral_numeral @ A @ K ) @ ( semiring_char_0_fact @ A @ ( pred_numeral @ K ) ) ) ) ) ).

% fact_numeral
thf(fact_4271_ln__2__less__1,axiom,
    ord_less @ real @ ( ln_ln @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( one_one @ real ) ).

% ln_2_less_1
thf(fact_4272_ln__le__minus__one,axiom,
    ! [X2: real] :
      ( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
     => ( ord_less_eq @ real @ ( ln_ln @ real @ X2 ) @ ( minus_minus @ real @ X2 @ ( one_one @ real ) ) ) ) ).

% ln_le_minus_one
thf(fact_4273_ln__diff__le,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
     => ( ( ord_less @ real @ ( zero_zero @ real ) @ Y2 )
       => ( ord_less_eq @ real @ ( minus_minus @ real @ ( ln_ln @ real @ X2 ) @ ( ln_ln @ real @ Y2 ) ) @ ( divide_divide @ real @ ( minus_minus @ real @ X2 @ Y2 ) @ Y2 ) ) ) ) ).

% ln_diff_le
thf(fact_4274_ln__add__one__self__le__self2,axiom,
    ! [X2: real] :
      ( ( ord_less @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X2 )
     => ( ord_less_eq @ real @ ( ln_ln @ real @ ( plus_plus @ real @ ( one_one @ real ) @ X2 ) ) @ X2 ) ) ).

% ln_add_one_self_le_self2
thf(fact_4275_ln__realpow,axiom,
    ! [X2: real,N2: nat] :
      ( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
     => ( ( ln_ln @ real @ ( power_power @ real @ X2 @ N2 ) )
        = ( times_times @ real @ ( semiring_1_of_nat @ real @ N2 ) @ ( ln_ln @ real @ X2 ) ) ) ) ).

% ln_realpow
thf(fact_4276_sin__gt__zero__02,axiom,
    ! [X2: real] :
      ( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
     => ( ( ord_less @ real @ X2 @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) )
       => ( ord_less @ real @ ( zero_zero @ real ) @ ( sin @ real @ X2 ) ) ) ) ).

% sin_gt_zero_02
thf(fact_4277_cos__two__less__zero,axiom,
    ord_less @ real @ ( cos @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( zero_zero @ real ) ).

% cos_two_less_zero
thf(fact_4278_cos__two__le__zero,axiom,
    ord_less_eq @ real @ ( cos @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( zero_zero @ real ) ).

% cos_two_le_zero
thf(fact_4279_cos__is__zero,axiom,
    ? [X3: real] :
      ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X3 )
      & ( ord_less_eq @ real @ X3 @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) )
      & ( ( cos @ real @ X3 )
        = ( zero_zero @ real ) )
      & ! [Y3: real] :
          ( ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y3 )
            & ( ord_less_eq @ real @ Y3 @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) )
            & ( ( cos @ real @ Y3 )
              = ( zero_zero @ real ) ) )
         => ( Y3 = X3 ) ) ) ).

% cos_is_zero
thf(fact_4280_ln__one__minus__pos__upper__bound,axiom,
    ! [X2: real] :
      ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 )
     => ( ( ord_less @ real @ X2 @ ( one_one @ real ) )
       => ( ord_less_eq @ real @ ( ln_ln @ real @ ( minus_minus @ real @ ( one_one @ real ) @ X2 ) ) @ ( uminus_uminus @ real @ X2 ) ) ) ) ).

% ln_one_minus_pos_upper_bound
thf(fact_4281_ln__powr__bound,axiom,
    ! [X2: real,A2: real] :
      ( ( ord_less_eq @ real @ ( one_one @ real ) @ X2 )
     => ( ( ord_less @ real @ ( zero_zero @ real ) @ A2 )
       => ( ord_less_eq @ real @ ( ln_ln @ real @ X2 ) @ ( divide_divide @ real @ ( powr @ real @ X2 @ A2 ) @ A2 ) ) ) ) ).

% ln_powr_bound
thf(fact_4282_ln__powr__bound2,axiom,
    ! [X2: real,A2: real] :
      ( ( ord_less @ real @ ( one_one @ real ) @ X2 )
     => ( ( ord_less @ real @ ( zero_zero @ real ) @ A2 )
       => ( ord_less_eq @ real @ ( powr @ real @ ( ln_ln @ real @ X2 ) @ A2 ) @ ( times_times @ real @ ( powr @ real @ A2 @ A2 ) @ X2 ) ) ) ) ).

% ln_powr_bound2
thf(fact_4283_ln__prod,axiom,
    ! [A: $tType,I5: set @ A,F: A > real] :
      ( ( finite_finite2 @ A @ I5 )
     => ( ! [I3: A] :
            ( ( member @ A @ I3 @ I5 )
           => ( ord_less @ real @ ( zero_zero @ real ) @ ( F @ I3 ) ) )
       => ( ( ln_ln @ real @ ( groups7121269368397514597t_prod @ A @ real @ F @ I5 ) )
          = ( groups7311177749621191930dd_sum @ A @ real
            @ ^ [X: A] : ( ln_ln @ real @ ( F @ X ) )
            @ I5 ) ) ) ) ).

% ln_prod
thf(fact_4284_log__eq__div__ln__mult__log,axiom,
    ! [A2: real,B2: real,X2: real] :
      ( ( ord_less @ real @ ( zero_zero @ real ) @ A2 )
     => ( ( A2
         != ( one_one @ real ) )
       => ( ( ord_less @ real @ ( zero_zero @ real ) @ B2 )
         => ( ( B2
             != ( one_one @ real ) )
           => ( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
             => ( ( log @ A2 @ X2 )
                = ( times_times @ real @ ( divide_divide @ real @ ( ln_ln @ real @ B2 ) @ ( ln_ln @ real @ A2 ) ) @ ( log @ B2 @ X2 ) ) ) ) ) ) ) ) ).

% log_eq_div_ln_mult_log
thf(fact_4285_cos__plus__cos,axiom,
    ! [A: $tType] :
      ( ( ( real_Vector_banach @ A )
        & ( real_V3459762299906320749_field @ A ) )
     => ! [W: A,Z: A] :
          ( ( plus_plus @ A @ ( cos @ A @ W ) @ ( cos @ A @ Z ) )
          = ( times_times @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( cos @ A @ ( divide_divide @ A @ ( plus_plus @ A @ W @ Z ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) @ ( cos @ A @ ( divide_divide @ A @ ( minus_minus @ A @ W @ Z ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ).

% cos_plus_cos
thf(fact_4286_cos__times__cos,axiom,
    ! [A: $tType] :
      ( ( ( real_Vector_banach @ A )
        & ( real_V3459762299906320749_field @ A ) )
     => ! [W: A,Z: A] :
          ( ( times_times @ A @ ( cos @ A @ W ) @ ( cos @ A @ Z ) )
          = ( divide_divide @ A @ ( plus_plus @ A @ ( cos @ A @ ( minus_minus @ A @ W @ Z ) ) @ ( cos @ A @ ( plus_plus @ A @ W @ Z ) ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).

% cos_times_cos
thf(fact_4287_cos__double__less__one,axiom,
    ! [X2: real] :
      ( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
     => ( ( ord_less @ real @ X2 @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) )
       => ( ord_less @ real @ ( cos @ real @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ X2 ) ) @ ( one_one @ real ) ) ) ) ).

% cos_double_less_one
thf(fact_4288_cos__double__cos,axiom,
    ! [A: $tType] :
      ( ( ( real_Vector_banach @ A )
        & ( real_V3459762299906320749_field @ A ) )
     => ! [W: A] :
          ( ( cos @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ W ) )
          = ( minus_minus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( power_power @ A @ ( cos @ A @ W ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( one_one @ A ) ) ) ) ).

% cos_double_cos
thf(fact_4289_cos__treble__cos,axiom,
    ! [A: $tType] :
      ( ( ( real_Vector_banach @ A )
        & ( real_V3459762299906320749_field @ A ) )
     => ! [X2: A] :
          ( ( cos @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit1 @ one2 ) ) @ X2 ) )
          = ( minus_minus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ ( cos @ A @ X2 ) @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) ) ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit1 @ one2 ) ) @ ( cos @ A @ X2 ) ) ) ) ) ).

% cos_treble_cos
thf(fact_4290_ln__one__plus__pos__lower__bound,axiom,
    ! [X2: real] :
      ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 )
     => ( ( ord_less_eq @ real @ X2 @ ( one_one @ real ) )
       => ( ord_less_eq @ real @ ( minus_minus @ real @ X2 @ ( power_power @ real @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( ln_ln @ real @ ( plus_plus @ real @ ( one_one @ real ) @ X2 ) ) ) ) ) ).

% ln_one_plus_pos_lower_bound
thf(fact_4291_artanh__def,axiom,
    ! [A: $tType] :
      ( ( ( real_V3459762299906320749_field @ A )
        & ( ln @ A ) )
     => ( ( artanh @ A )
        = ( ^ [X: A] : ( divide_divide @ A @ ( ln_ln @ A @ ( divide_divide @ A @ ( plus_plus @ A @ ( one_one @ A ) @ X ) @ ( minus_minus @ A @ ( one_one @ A ) @ X ) ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).

% artanh_def
thf(fact_4292_sin__x__sin__y,axiom,
    ! [A: $tType] :
      ( ( ( real_Vector_banach @ A )
        & ( real_V3459762299906320749_field @ A ) )
     => ! [X2: A,Y2: A] :
          ( sums @ A
          @ ^ [P5: nat] :
              ( groups7311177749621191930dd_sum @ nat @ A
              @ ^ [N: nat] :
                  ( if @ A
                  @ ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ P5 )
                    & ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) )
                  @ ( times_times @ A @ ( real_V8093663219630862766scaleR @ A @ ( uminus_uminus @ real @ ( divide_divide @ real @ ( ring_1_of_int @ real @ ( times_times @ int @ ( power_power @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( divide_divide @ nat @ P5 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( semiring_1_of_nat @ int @ ( binomial @ P5 @ N ) ) ) ) @ ( semiring_char_0_fact @ real @ P5 ) ) ) @ ( power_power @ A @ X2 @ N ) ) @ ( power_power @ A @ Y2 @ ( minus_minus @ nat @ P5 @ N ) ) )
                  @ ( zero_zero @ A ) )
              @ ( set_ord_atMost @ nat @ P5 ) )
          @ ( times_times @ A @ ( sin @ A @ X2 ) @ ( sin @ A @ Y2 ) ) ) ) ).

% sin_x_sin_y
thf(fact_4293_sums__cos__x__plus__y,axiom,
    ! [A: $tType] :
      ( ( ( real_Vector_banach @ A )
        & ( real_V3459762299906320749_field @ A ) )
     => ! [X2: A,Y2: A] :
          ( sums @ A
          @ ^ [P5: nat] :
              ( groups7311177749621191930dd_sum @ nat @ A
              @ ^ [N: nat] : ( if @ A @ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ P5 ) @ ( times_times @ A @ ( real_V8093663219630862766scaleR @ A @ ( divide_divide @ real @ ( ring_1_of_int @ real @ ( times_times @ int @ ( power_power @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( divide_divide @ nat @ P5 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( semiring_1_of_nat @ int @ ( binomial @ P5 @ N ) ) ) ) @ ( semiring_char_0_fact @ real @ P5 ) ) @ ( power_power @ A @ X2 @ N ) ) @ ( power_power @ A @ Y2 @ ( minus_minus @ nat @ P5 @ N ) ) ) @ ( zero_zero @ A ) )
              @ ( set_ord_atMost @ nat @ P5 ) )
          @ ( cos @ A @ ( plus_plus @ A @ X2 @ Y2 ) ) ) ) ).

% sums_cos_x_plus_y
thf(fact_4294_cos__x__cos__y,axiom,
    ! [A: $tType] :
      ( ( ( real_Vector_banach @ A )
        & ( real_V3459762299906320749_field @ A ) )
     => ! [X2: A,Y2: A] :
          ( sums @ A
          @ ^ [P5: nat] :
              ( groups7311177749621191930dd_sum @ nat @ A
              @ ^ [N: nat] :
                  ( if @ A
                  @ ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ P5 )
                    & ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) )
                  @ ( times_times @ A @ ( real_V8093663219630862766scaleR @ A @ ( divide_divide @ real @ ( ring_1_of_int @ real @ ( times_times @ int @ ( power_power @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( divide_divide @ nat @ P5 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( semiring_1_of_nat @ int @ ( binomial @ P5 @ N ) ) ) ) @ ( semiring_char_0_fact @ real @ P5 ) ) @ ( power_power @ A @ X2 @ N ) ) @ ( power_power @ A @ Y2 @ ( minus_minus @ nat @ P5 @ N ) ) )
                  @ ( zero_zero @ A ) )
              @ ( set_ord_atMost @ nat @ P5 ) )
          @ ( times_times @ A @ ( cos @ A @ X2 ) @ ( cos @ A @ Y2 ) ) ) ) ).

% cos_x_cos_y
thf(fact_4295_mult__scaleR__left,axiom,
    ! [A: $tType] :
      ( ( real_V6157519004096292374lgebra @ A )
     => ! [A2: real,X2: A,Y2: A] :
          ( ( times_times @ A @ ( real_V8093663219630862766scaleR @ A @ A2 @ X2 ) @ Y2 )
          = ( real_V8093663219630862766scaleR @ A @ A2 @ ( times_times @ A @ X2 @ Y2 ) ) ) ) ).

% mult_scaleR_left
thf(fact_4296_mult__scaleR__right,axiom,
    ! [A: $tType] :
      ( ( real_V6157519004096292374lgebra @ A )
     => ! [X2: A,A2: real,Y2: A] :
          ( ( times_times @ A @ X2 @ ( real_V8093663219630862766scaleR @ A @ A2 @ Y2 ) )
          = ( real_V8093663219630862766scaleR @ A @ A2 @ ( times_times @ A @ X2 @ Y2 ) ) ) ) ).

% mult_scaleR_right
thf(fact_4297_scaleR__one,axiom,
    ! [A: $tType] :
      ( ( real_V4867850818363320053vector @ A )
     => ! [X2: A] :
          ( ( real_V8093663219630862766scaleR @ A @ ( one_one @ real ) @ X2 )
          = X2 ) ) ).

% scaleR_one
thf(fact_4298_scaleR__scaleR,axiom,
    ! [A: $tType] :
      ( ( real_V4867850818363320053vector @ A )
     => ! [A2: real,B2: real,X2: A] :
          ( ( real_V8093663219630862766scaleR @ A @ A2 @ ( real_V8093663219630862766scaleR @ A @ B2 @ X2 ) )
          = ( real_V8093663219630862766scaleR @ A @ ( times_times @ real @ A2 @ B2 ) @ X2 ) ) ) ).

% scaleR_scaleR
thf(fact_4299_scaleR__eq__iff,axiom,
    ! [A: $tType] :
      ( ( real_V4867850818363320053vector @ A )
     => ! [B2: A,U: real,A2: A] :
          ( ( ( plus_plus @ A @ B2 @ ( real_V8093663219630862766scaleR @ A @ U @ A2 ) )
            = ( plus_plus @ A @ A2 @ ( real_V8093663219630862766scaleR @ A @ U @ B2 ) ) )
          = ( ( A2 = B2 )
            | ( U
              = ( one_one @ real ) ) ) ) ) ).

% scaleR_eq_iff
thf(fact_4300_scaleR__power,axiom,
    ! [A: $tType] :
      ( ( real_V2822296259951069270ebra_1 @ A )
     => ! [X2: real,Y2: A,N2: nat] :
          ( ( power_power @ A @ ( real_V8093663219630862766scaleR @ A @ X2 @ Y2 ) @ N2 )
          = ( real_V8093663219630862766scaleR @ A @ ( power_power @ real @ X2 @ N2 ) @ ( power_power @ A @ Y2 @ N2 ) ) ) ) ).

% scaleR_power
thf(fact_4301_scaleR__minus1__left,axiom,
    ! [A: $tType] :
      ( ( real_V4867850818363320053vector @ A )
     => ! [X2: A] :
          ( ( real_V8093663219630862766scaleR @ A @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X2 )
          = ( uminus_uminus @ A @ X2 ) ) ) ).

% scaleR_minus1_left
thf(fact_4302_scaleR__collapse,axiom,
    ! [A: $tType] :
      ( ( real_V4867850818363320053vector @ A )
     => ! [U: real,A2: A] :
          ( ( plus_plus @ A @ ( real_V8093663219630862766scaleR @ A @ ( minus_minus @ real @ ( one_one @ real ) @ U ) @ A2 ) @ ( real_V8093663219630862766scaleR @ A @ U @ A2 ) )
          = A2 ) ) ).

% scaleR_collapse
thf(fact_4303_scaleR__times,axiom,
    ! [A: $tType] :
      ( ( real_V2191834092415804123ebra_1 @ A )
     => ! [U: num,W: num,A2: A] :
          ( ( real_V8093663219630862766scaleR @ A @ ( numeral_numeral @ real @ U ) @ ( times_times @ A @ ( numeral_numeral @ A @ W ) @ A2 ) )
          = ( real_V8093663219630862766scaleR @ A @ ( times_times @ real @ ( numeral_numeral @ real @ U ) @ ( numeral_numeral @ real @ W ) ) @ A2 ) ) ) ).

% scaleR_times
thf(fact_4304_inverse__scaleR__times,axiom,
    ! [A: $tType] :
      ( ( real_V2191834092415804123ebra_1 @ A )
     => ! [V: num,W: num,A2: A] :
          ( ( real_V8093663219630862766scaleR @ A @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ V ) ) @ ( times_times @ A @ ( numeral_numeral @ A @ W ) @ A2 ) )
          = ( real_V8093663219630862766scaleR @ A @ ( divide_divide @ real @ ( numeral_numeral @ real @ W ) @ ( numeral_numeral @ real @ V ) ) @ A2 ) ) ) ).

% inverse_scaleR_times
thf(fact_4305_fraction__scaleR__times,axiom,
    ! [A: $tType] :
      ( ( real_V2191834092415804123ebra_1 @ A )
     => ! [U: num,V: num,W: num,A2: A] :
          ( ( real_V8093663219630862766scaleR @ A @ ( divide_divide @ real @ ( numeral_numeral @ real @ U ) @ ( numeral_numeral @ real @ V ) ) @ ( times_times @ A @ ( numeral_numeral @ A @ W ) @ A2 ) )
          = ( real_V8093663219630862766scaleR @ A @ ( divide_divide @ real @ ( times_times @ real @ ( numeral_numeral @ real @ U ) @ ( numeral_numeral @ real @ W ) ) @ ( numeral_numeral @ real @ V ) ) @ A2 ) ) ) ).

% fraction_scaleR_times
thf(fact_4306_scaleR__half__double,axiom,
    ! [A: $tType] :
      ( ( real_V4867850818363320053vector @ A )
     => ! [A2: A] :
          ( ( real_V8093663219630862766scaleR @ A @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( plus_plus @ A @ A2 @ A2 ) )
          = A2 ) ) ).

% scaleR_half_double
thf(fact_4307_scaleR__right__distrib,axiom,
    ! [A: $tType] :
      ( ( real_V4867850818363320053vector @ A )
     => ! [A2: real,X2: A,Y2: A] :
          ( ( real_V8093663219630862766scaleR @ A @ A2 @ ( plus_plus @ A @ X2 @ Y2 ) )
          = ( plus_plus @ A @ ( real_V8093663219630862766scaleR @ A @ A2 @ X2 ) @ ( real_V8093663219630862766scaleR @ A @ A2 @ Y2 ) ) ) ) ).

% scaleR_right_distrib
thf(fact_4308_scaleR__right__diff__distrib,axiom,
    ! [A: $tType] :
      ( ( real_V4867850818363320053vector @ A )
     => ! [A2: real,X2: A,Y2: A] :
          ( ( real_V8093663219630862766scaleR @ A @ A2 @ ( minus_minus @ A @ X2 @ Y2 ) )
          = ( minus_minus @ A @ ( real_V8093663219630862766scaleR @ A @ A2 @ X2 ) @ ( real_V8093663219630862766scaleR @ A @ A2 @ Y2 ) ) ) ) ).

% scaleR_right_diff_distrib
thf(fact_4309_real__scaleR__def,axiom,
    ( ( real_V8093663219630862766scaleR @ real )
    = ( times_times @ real ) ) ).

% real_scaleR_def
thf(fact_4310_scaleR__sum__right,axiom,
    ! [A: $tType,C: $tType] :
      ( ( real_V4867850818363320053vector @ A )
     => ! [A2: real,F: C > A,A3: set @ C] :
          ( ( real_V8093663219630862766scaleR @ A @ A2 @ ( groups7311177749621191930dd_sum @ C @ A @ F @ A3 ) )
          = ( groups7311177749621191930dd_sum @ C @ A
            @ ^ [X: C] : ( real_V8093663219630862766scaleR @ A @ A2 @ ( F @ X ) )
            @ A3 ) ) ) ).

% scaleR_sum_right
thf(fact_4311_scaleR__right_Osum,axiom,
    ! [A: $tType,C: $tType] :
      ( ( real_V4867850818363320053vector @ A )
     => ! [A2: real,G: C > A,A3: set @ C] :
          ( ( real_V8093663219630862766scaleR @ A @ A2 @ ( groups7311177749621191930dd_sum @ C @ A @ G @ A3 ) )
          = ( groups7311177749621191930dd_sum @ C @ A
            @ ^ [X: C] : ( real_V8093663219630862766scaleR @ A @ A2 @ ( G @ X ) )
            @ A3 ) ) ) ).

% scaleR_right.sum
thf(fact_4312_sums__scaleR__right,axiom,
    ! [B: $tType] :
      ( ( real_V822414075346904944vector @ B )
     => ! [X8: nat > B,A2: B,R4: real] :
          ( ( sums @ B @ X8 @ A2 )
         => ( sums @ B
            @ ^ [N: nat] : ( real_V8093663219630862766scaleR @ B @ R4 @ ( X8 @ N ) )
            @ ( real_V8093663219630862766scaleR @ B @ R4 @ A2 ) ) ) ) ).

% sums_scaleR_right
thf(fact_4313_scaleR__left_Oadd,axiom,
    ! [A: $tType] :
      ( ( real_V4867850818363320053vector @ A )
     => ! [X2: real,Y2: real,Xa: A] :
          ( ( real_V8093663219630862766scaleR @ A @ ( plus_plus @ real @ X2 @ Y2 ) @ Xa )
          = ( plus_plus @ A @ ( real_V8093663219630862766scaleR @ A @ X2 @ Xa ) @ ( real_V8093663219630862766scaleR @ A @ Y2 @ Xa ) ) ) ) ).

% scaleR_left.add
thf(fact_4314_scaleR__left__distrib,axiom,
    ! [A: $tType] :
      ( ( real_V4867850818363320053vector @ A )
     => ! [A2: real,B2: real,X2: A] :
          ( ( real_V8093663219630862766scaleR @ A @ ( plus_plus @ real @ A2 @ B2 ) @ X2 )
          = ( plus_plus @ A @ ( real_V8093663219630862766scaleR @ A @ A2 @ X2 ) @ ( real_V8093663219630862766scaleR @ A @ B2 @ X2 ) ) ) ) ).

% scaleR_left_distrib
thf(fact_4315_scaleR__left_Odiff,axiom,
    ! [A: $tType] :
      ( ( real_V4867850818363320053vector @ A )
     => ! [X2: real,Y2: real,Xa: A] :
          ( ( real_V8093663219630862766scaleR @ A @ ( minus_minus @ real @ X2 @ Y2 ) @ Xa )
          = ( minus_minus @ A @ ( real_V8093663219630862766scaleR @ A @ X2 @ Xa ) @ ( real_V8093663219630862766scaleR @ A @ Y2 @ Xa ) ) ) ) ).

% scaleR_left.diff
thf(fact_4316_scaleR__left__diff__distrib,axiom,
    ! [A: $tType] :
      ( ( real_V4867850818363320053vector @ A )
     => ! [A2: real,B2: real,X2: A] :
          ( ( real_V8093663219630862766scaleR @ A @ ( minus_minus @ real @ A2 @ B2 ) @ X2 )
          = ( minus_minus @ A @ ( real_V8093663219630862766scaleR @ A @ A2 @ X2 ) @ ( real_V8093663219630862766scaleR @ A @ B2 @ X2 ) ) ) ) ).

% scaleR_left_diff_distrib
thf(fact_4317_scaleR__left_Osum,axiom,
    ! [A: $tType,C: $tType] :
      ( ( real_V4867850818363320053vector @ A )
     => ! [G: C > real,A3: set @ C,X2: A] :
          ( ( real_V8093663219630862766scaleR @ A @ ( groups7311177749621191930dd_sum @ C @ real @ G @ A3 ) @ X2 )
          = ( groups7311177749621191930dd_sum @ C @ A
            @ ^ [X: C] : ( real_V8093663219630862766scaleR @ A @ ( G @ X ) @ X2 )
            @ A3 ) ) ) ).

% scaleR_left.sum
thf(fact_4318_scaleR__sum__left,axiom,
    ! [A: $tType,C: $tType] :
      ( ( real_V4867850818363320053vector @ A )
     => ! [F: C > real,A3: set @ C,X2: A] :
          ( ( real_V8093663219630862766scaleR @ A @ ( groups7311177749621191930dd_sum @ C @ real @ F @ A3 ) @ X2 )
          = ( groups7311177749621191930dd_sum @ C @ A
            @ ^ [A5: C] : ( real_V8093663219630862766scaleR @ A @ ( F @ A5 ) @ X2 )
            @ A3 ) ) ) ).

% scaleR_sum_left
thf(fact_4319_sums__scaleR__left,axiom,
    ! [B: $tType] :
      ( ( real_V822414075346904944vector @ B )
     => ! [X8: nat > real,A2: real,X2: B] :
          ( ( sums @ real @ X8 @ A2 )
         => ( sums @ B
            @ ^ [N: nat] : ( real_V8093663219630862766scaleR @ B @ ( X8 @ N ) @ X2 )
            @ ( real_V8093663219630862766scaleR @ B @ A2 @ X2 ) ) ) ) ).

% sums_scaleR_left
thf(fact_4320_scaleR__right__mono,axiom,
    ! [A: $tType] :
      ( ( real_V5355595471888546746vector @ A )
     => ! [A2: real,B2: real,X2: A] :
          ( ( ord_less_eq @ real @ A2 @ B2 )
         => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X2 )
           => ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ A2 @ X2 ) @ ( real_V8093663219630862766scaleR @ A @ B2 @ X2 ) ) ) ) ) ).

% scaleR_right_mono
thf(fact_4321_scaleR__right__mono__neg,axiom,
    ! [A: $tType] :
      ( ( real_V5355595471888546746vector @ A )
     => ! [B2: real,A2: real,C2: A] :
          ( ( ord_less_eq @ real @ B2 @ A2 )
         => ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) )
           => ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ A2 @ C2 ) @ ( real_V8093663219630862766scaleR @ A @ B2 @ C2 ) ) ) ) ) ).

% scaleR_right_mono_neg
thf(fact_4322_scaleR__le__cancel__left,axiom,
    ! [A: $tType] :
      ( ( real_V5355595471888546746vector @ A )
     => ! [C2: real,A2: A,B2: A] :
          ( ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ C2 @ A2 ) @ ( real_V8093663219630862766scaleR @ A @ C2 @ B2 ) )
          = ( ( ( ord_less @ real @ ( zero_zero @ real ) @ C2 )
             => ( ord_less_eq @ A @ A2 @ B2 ) )
            & ( ( ord_less @ real @ C2 @ ( zero_zero @ real ) )
             => ( ord_less_eq @ A @ B2 @ A2 ) ) ) ) ) ).

% scaleR_le_cancel_left
thf(fact_4323_scaleR__le__cancel__left__neg,axiom,
    ! [A: $tType] :
      ( ( real_V5355595471888546746vector @ A )
     => ! [C2: real,A2: A,B2: A] :
          ( ( ord_less @ real @ C2 @ ( zero_zero @ real ) )
         => ( ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ C2 @ A2 ) @ ( real_V8093663219630862766scaleR @ A @ C2 @ B2 ) )
            = ( ord_less_eq @ A @ B2 @ A2 ) ) ) ) ).

% scaleR_le_cancel_left_neg
thf(fact_4324_scaleR__le__cancel__left__pos,axiom,
    ! [A: $tType] :
      ( ( real_V5355595471888546746vector @ A )
     => ! [C2: real,A2: A,B2: A] :
          ( ( ord_less @ real @ ( zero_zero @ real ) @ C2 )
         => ( ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ C2 @ A2 ) @ ( real_V8093663219630862766scaleR @ A @ C2 @ B2 ) )
            = ( ord_less_eq @ A @ A2 @ B2 ) ) ) ) ).

% scaleR_le_cancel_left_pos
thf(fact_4325_scaleR__left__mono,axiom,
    ! [A: $tType] :
      ( ( real_V5355595471888546746vector @ A )
     => ! [X2: A,Y2: A,A2: real] :
          ( ( ord_less_eq @ A @ X2 @ Y2 )
         => ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ A2 )
           => ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ A2 @ X2 ) @ ( real_V8093663219630862766scaleR @ A @ A2 @ Y2 ) ) ) ) ) ).

% scaleR_left_mono
thf(fact_4326_scaleR__left__mono__neg,axiom,
    ! [A: $tType] :
      ( ( real_V5355595471888546746vector @ A )
     => ! [B2: A,A2: A,C2: real] :
          ( ( ord_less_eq @ A @ B2 @ A2 )
         => ( ( ord_less_eq @ real @ C2 @ ( zero_zero @ real ) )
           => ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ C2 @ A2 ) @ ( real_V8093663219630862766scaleR @ A @ C2 @ B2 ) ) ) ) ) ).

% scaleR_left_mono_neg
thf(fact_4327_vector__fraction__eq__iff,axiom,
    ! [A: $tType] :
      ( ( real_V4867850818363320053vector @ A )
     => ! [U: real,V: real,A2: A,X2: A] :
          ( ( ( real_V8093663219630862766scaleR @ A @ ( divide_divide @ real @ U @ V ) @ A2 )
            = X2 )
          = ( ( ( V
                = ( zero_zero @ real ) )
             => ( X2
                = ( zero_zero @ A ) ) )
            & ( ( V
               != ( zero_zero @ real ) )
             => ( ( real_V8093663219630862766scaleR @ A @ U @ A2 )
                = ( real_V8093663219630862766scaleR @ A @ V @ X2 ) ) ) ) ) ) ).

% vector_fraction_eq_iff
thf(fact_4328_eq__vector__fraction__iff,axiom,
    ! [A: $tType] :
      ( ( real_V4867850818363320053vector @ A )
     => ! [X2: A,U: real,V: real,A2: A] :
          ( ( X2
            = ( real_V8093663219630862766scaleR @ A @ ( divide_divide @ real @ U @ V ) @ A2 ) )
          = ( ( ( V
                = ( zero_zero @ real ) )
             => ( X2
                = ( zero_zero @ A ) ) )
            & ( ( V
               != ( zero_zero @ real ) )
             => ( ( real_V8093663219630862766scaleR @ A @ V @ X2 )
                = ( real_V8093663219630862766scaleR @ A @ U @ A2 ) ) ) ) ) ) ).

% eq_vector_fraction_iff
thf(fact_4329_Real__Vector__Spaces_Ole__add__iff2,axiom,
    ! [A: $tType] :
      ( ( real_V5355595471888546746vector @ A )
     => ! [A2: real,E3: A,C2: A,B2: real,D2: A] :
          ( ( ord_less_eq @ A @ ( plus_plus @ A @ ( real_V8093663219630862766scaleR @ A @ A2 @ E3 ) @ C2 ) @ ( plus_plus @ A @ ( real_V8093663219630862766scaleR @ A @ B2 @ E3 ) @ D2 ) )
          = ( ord_less_eq @ A @ C2 @ ( plus_plus @ A @ ( real_V8093663219630862766scaleR @ A @ ( minus_minus @ real @ B2 @ A2 ) @ E3 ) @ D2 ) ) ) ) ).

% Real_Vector_Spaces.le_add_iff2
thf(fact_4330_Real__Vector__Spaces_Ole__add__iff1,axiom,
    ! [A: $tType] :
      ( ( real_V5355595471888546746vector @ A )
     => ! [A2: real,E3: A,C2: A,B2: real,D2: A] :
          ( ( ord_less_eq @ A @ ( plus_plus @ A @ ( real_V8093663219630862766scaleR @ A @ A2 @ E3 ) @ C2 ) @ ( plus_plus @ A @ ( real_V8093663219630862766scaleR @ A @ B2 @ E3 ) @ D2 ) )
          = ( ord_less_eq @ A @ ( plus_plus @ A @ ( real_V8093663219630862766scaleR @ A @ ( minus_minus @ real @ A2 @ B2 ) @ E3 ) @ C2 ) @ D2 ) ) ) ).

% Real_Vector_Spaces.le_add_iff1
thf(fact_4331_zero__le__scaleR__iff,axiom,
    ! [A: $tType] :
      ( ( real_V5355595471888546746vector @ A )
     => ! [A2: real,B2: A] :
          ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( real_V8093663219630862766scaleR @ A @ A2 @ B2 ) )
          = ( ( ( ord_less @ real @ ( zero_zero @ real ) @ A2 )
              & ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) )
            | ( ( ord_less @ real @ A2 @ ( zero_zero @ real ) )
              & ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) ) )
            | ( A2
              = ( zero_zero @ real ) ) ) ) ) ).

% zero_le_scaleR_iff
thf(fact_4332_scaleR__le__0__iff,axiom,
    ! [A: $tType] :
      ( ( real_V5355595471888546746vector @ A )
     => ! [A2: real,B2: A] :
          ( ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ A2 @ B2 ) @ ( zero_zero @ A ) )
          = ( ( ( ord_less @ real @ ( zero_zero @ real ) @ A2 )
              & ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) ) )
            | ( ( ord_less @ real @ A2 @ ( zero_zero @ real ) )
              & ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) )
            | ( A2
              = ( zero_zero @ real ) ) ) ) ) ).

% scaleR_le_0_iff
thf(fact_4333_scaleR__nonpos__nonpos,axiom,
    ! [A: $tType] :
      ( ( real_V5355595471888546746vector @ A )
     => ! [A2: real,B2: A] :
          ( ( ord_less_eq @ real @ A2 @ ( zero_zero @ real ) )
         => ( ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) )
           => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( real_V8093663219630862766scaleR @ A @ A2 @ B2 ) ) ) ) ) ).

% scaleR_nonpos_nonpos
thf(fact_4334_scaleR__nonpos__nonneg,axiom,
    ! [A: $tType] :
      ( ( real_V5355595471888546746vector @ A )
     => ! [A2: real,X2: A] :
          ( ( ord_less_eq @ real @ A2 @ ( zero_zero @ real ) )
         => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X2 )
           => ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ A2 @ X2 ) @ ( zero_zero @ A ) ) ) ) ) ).

% scaleR_nonpos_nonneg
thf(fact_4335_scaleR__nonneg__nonpos,axiom,
    ! [A: $tType] :
      ( ( real_V5355595471888546746vector @ A )
     => ! [A2: real,X2: A] :
          ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ A2 )
         => ( ( ord_less_eq @ A @ X2 @ ( zero_zero @ A ) )
           => ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ A2 @ X2 ) @ ( zero_zero @ A ) ) ) ) ) ).

% scaleR_nonneg_nonpos
thf(fact_4336_scaleR__nonneg__nonneg,axiom,
    ! [A: $tType] :
      ( ( real_V5355595471888546746vector @ A )
     => ! [A2: real,X2: A] :
          ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ A2 )
         => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X2 )
           => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( real_V8093663219630862766scaleR @ A @ A2 @ X2 ) ) ) ) ) ).

% scaleR_nonneg_nonneg
thf(fact_4337_split__scaleR__pos__le,axiom,
    ! [A: $tType] :
      ( ( real_V5355595471888546746vector @ A )
     => ! [A2: real,B2: A] :
          ( ( ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ A2 )
              & ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) )
            | ( ( ord_less_eq @ real @ A2 @ ( zero_zero @ real ) )
              & ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) ) ) )
         => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( real_V8093663219630862766scaleR @ A @ A2 @ B2 ) ) ) ) ).

% split_scaleR_pos_le
thf(fact_4338_split__scaleR__neg__le,axiom,
    ! [A: $tType] :
      ( ( real_V5355595471888546746vector @ A )
     => ! [A2: real,X2: A] :
          ( ( ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ A2 )
              & ( ord_less_eq @ A @ X2 @ ( zero_zero @ A ) ) )
            | ( ( ord_less_eq @ real @ A2 @ ( zero_zero @ real ) )
              & ( ord_less_eq @ A @ ( zero_zero @ A ) @ X2 ) ) )
         => ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ A2 @ X2 ) @ ( zero_zero @ A ) ) ) ) ).

% split_scaleR_neg_le
thf(fact_4339_scaleR__mono_H,axiom,
    ! [A: $tType] :
      ( ( real_V5355595471888546746vector @ A )
     => ! [A2: real,B2: real,C2: A,D2: A] :
          ( ( ord_less_eq @ real @ A2 @ B2 )
         => ( ( ord_less_eq @ A @ C2 @ D2 )
           => ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ A2 )
             => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
               => ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ A2 @ C2 ) @ ( real_V8093663219630862766scaleR @ A @ B2 @ D2 ) ) ) ) ) ) ) ).

% scaleR_mono'
thf(fact_4340_scaleR__mono,axiom,
    ! [A: $tType] :
      ( ( real_V5355595471888546746vector @ A )
     => ! [A2: real,B2: real,X2: A,Y2: A] :
          ( ( ord_less_eq @ real @ A2 @ B2 )
         => ( ( ord_less_eq @ A @ X2 @ Y2 )
           => ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ B2 )
             => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X2 )
               => ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ A2 @ X2 ) @ ( real_V8093663219630862766scaleR @ A @ B2 @ Y2 ) ) ) ) ) ) ) ).

% scaleR_mono
thf(fact_4341_scaleR__2,axiom,
    ! [A: $tType] :
      ( ( real_V4867850818363320053vector @ A )
     => ! [X2: A] :
          ( ( real_V8093663219630862766scaleR @ A @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ X2 )
          = ( plus_plus @ A @ X2 @ X2 ) ) ) ).

% scaleR_2
thf(fact_4342_scaleR__left__le__one__le,axiom,
    ! [A: $tType] :
      ( ( real_V5355595471888546746vector @ A )
     => ! [X2: A,A2: real] :
          ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X2 )
         => ( ( ord_less_eq @ real @ A2 @ ( one_one @ real ) )
           => ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ A2 @ X2 ) @ X2 ) ) ) ) ).

% scaleR_left_le_one_le
thf(fact_4343_sin__converges,axiom,
    ! [A: $tType] :
      ( ( ( real_Vector_banach @ A )
        & ( real_V2822296259951069270ebra_1 @ A ) )
     => ! [X2: A] :
          ( sums @ A
          @ ^ [N: nat] : ( real_V8093663219630862766scaleR @ A @ ( sin_coeff @ N ) @ ( power_power @ A @ X2 @ N ) )
          @ ( sin @ A @ X2 ) ) ) ).

% sin_converges
thf(fact_4344_cos__converges,axiom,
    ! [A: $tType] :
      ( ( ( real_Vector_banach @ A )
        & ( real_V2822296259951069270ebra_1 @ A ) )
     => ! [X2: A] :
          ( sums @ A
          @ ^ [N: nat] : ( real_V8093663219630862766scaleR @ A @ ( cos_coeff @ N ) @ ( power_power @ A @ X2 @ N ) )
          @ ( cos @ A @ X2 ) ) ) ).

% cos_converges
thf(fact_4345_sin__minus__converges,axiom,
    ! [A: $tType] :
      ( ( ( real_Vector_banach @ A )
        & ( real_V2822296259951069270ebra_1 @ A ) )
     => ! [X2: A] :
          ( sums @ A
          @ ^ [N: nat] : ( uminus_uminus @ A @ ( real_V8093663219630862766scaleR @ A @ ( sin_coeff @ N ) @ ( power_power @ A @ ( uminus_uminus @ A @ X2 ) @ N ) ) )
          @ ( sin @ A @ X2 ) ) ) ).

% sin_minus_converges
thf(fact_4346_cos__minus__converges,axiom,
    ! [A: $tType] :
      ( ( ( real_Vector_banach @ A )
        & ( real_V2822296259951069270ebra_1 @ A ) )
     => ! [X2: A] :
          ( sums @ A
          @ ^ [N: nat] : ( real_V8093663219630862766scaleR @ A @ ( cos_coeff @ N ) @ ( power_power @ A @ ( uminus_uminus @ A @ X2 ) @ N ) )
          @ ( cos @ A @ X2 ) ) ) ).

% cos_minus_converges
thf(fact_4347_Maclaurin__minus__cos__expansion,axiom,
    ! [N2: nat,X2: real] :
      ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
     => ( ( ord_less @ real @ X2 @ ( zero_zero @ real ) )
       => ? [T5: real] :
            ( ( ord_less @ real @ X2 @ T5 )
            & ( ord_less @ real @ T5 @ ( zero_zero @ real ) )
            & ( ( cos @ real @ X2 )
              = ( plus_plus @ real
                @ ( groups7311177749621191930dd_sum @ nat @ real
                  @ ^ [M7: nat] : ( times_times @ real @ ( cos_coeff @ M7 ) @ ( power_power @ real @ X2 @ M7 ) )
                  @ ( set_ord_lessThan @ nat @ N2 ) )
                @ ( times_times @ real @ ( divide_divide @ real @ ( cos @ real @ ( plus_plus @ real @ T5 @ ( times_times @ real @ ( times_times @ real @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( semiring_1_of_nat @ real @ N2 ) ) @ pi ) ) ) @ ( semiring_char_0_fact @ real @ N2 ) ) @ ( power_power @ real @ X2 @ N2 ) ) ) ) ) ) ) ).

% Maclaurin_minus_cos_expansion
thf(fact_4348_Maclaurin__cos__expansion2,axiom,
    ! [X2: real,N2: nat] :
      ( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
     => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
       => ? [T5: real] :
            ( ( ord_less @ real @ ( zero_zero @ real ) @ T5 )
            & ( ord_less @ real @ T5 @ X2 )
            & ( ( cos @ real @ X2 )
              = ( plus_plus @ real
                @ ( groups7311177749621191930dd_sum @ nat @ real
                  @ ^ [M7: nat] : ( times_times @ real @ ( cos_coeff @ M7 ) @ ( power_power @ real @ X2 @ M7 ) )
                  @ ( set_ord_lessThan @ nat @ N2 ) )
                @ ( times_times @ real @ ( divide_divide @ real @ ( cos @ real @ ( plus_plus @ real @ T5 @ ( times_times @ real @ ( times_times @ real @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( semiring_1_of_nat @ real @ N2 ) ) @ pi ) ) ) @ ( semiring_char_0_fact @ real @ N2 ) ) @ ( power_power @ real @ X2 @ N2 ) ) ) ) ) ) ) ).

% Maclaurin_cos_expansion2
thf(fact_4349_Maclaurin__sin__expansion3,axiom,
    ! [N2: nat,X2: real] :
      ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
     => ( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
       => ? [T5: real] :
            ( ( ord_less @ real @ ( zero_zero @ real ) @ T5 )
            & ( ord_less @ real @ T5 @ X2 )
            & ( ( sin @ real @ X2 )
              = ( plus_plus @ real
                @ ( groups7311177749621191930dd_sum @ nat @ real
                  @ ^ [M7: nat] : ( times_times @ real @ ( sin_coeff @ M7 ) @ ( power_power @ real @ X2 @ M7 ) )
                  @ ( set_ord_lessThan @ nat @ N2 ) )
                @ ( times_times @ real @ ( divide_divide @ real @ ( sin @ real @ ( plus_plus @ real @ T5 @ ( times_times @ real @ ( times_times @ real @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( semiring_1_of_nat @ real @ N2 ) ) @ pi ) ) ) @ ( semiring_char_0_fact @ real @ N2 ) ) @ ( power_power @ real @ X2 @ N2 ) ) ) ) ) ) ) ).

% Maclaurin_sin_expansion3
thf(fact_4350_Maclaurin__sin__expansion4,axiom,
    ! [X2: real,N2: nat] :
      ( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
     => ? [T5: real] :
          ( ( ord_less @ real @ ( zero_zero @ real ) @ T5 )
          & ( ord_less_eq @ real @ T5 @ X2 )
          & ( ( sin @ real @ X2 )
            = ( plus_plus @ real
              @ ( groups7311177749621191930dd_sum @ nat @ real
                @ ^ [M7: nat] : ( times_times @ real @ ( sin_coeff @ M7 ) @ ( power_power @ real @ X2 @ M7 ) )
                @ ( set_ord_lessThan @ nat @ N2 ) )
              @ ( times_times @ real @ ( divide_divide @ real @ ( sin @ real @ ( plus_plus @ real @ T5 @ ( times_times @ real @ ( times_times @ real @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( semiring_1_of_nat @ real @ N2 ) ) @ pi ) ) ) @ ( semiring_char_0_fact @ real @ N2 ) ) @ ( power_power @ real @ X2 @ N2 ) ) ) ) ) ) ).

% Maclaurin_sin_expansion4
thf(fact_4351_sin__pi__minus,axiom,
    ! [X2: real] :
      ( ( sin @ real @ ( minus_minus @ real @ pi @ X2 ) )
      = ( sin @ real @ X2 ) ) ).

% sin_pi_minus
thf(fact_4352_cos__pi,axiom,
    ( ( cos @ real @ pi )
    = ( uminus_uminus @ real @ ( one_one @ real ) ) ) ).

% cos_pi
thf(fact_4353_cos__periodic__pi,axiom,
    ! [X2: real] :
      ( ( cos @ real @ ( plus_plus @ real @ X2 @ pi ) )
      = ( uminus_uminus @ real @ ( cos @ real @ X2 ) ) ) ).

% cos_periodic_pi
thf(fact_4354_cos__periodic__pi2,axiom,
    ! [X2: real] :
      ( ( cos @ real @ ( plus_plus @ real @ pi @ X2 ) )
      = ( uminus_uminus @ real @ ( cos @ real @ X2 ) ) ) ).

% cos_periodic_pi2
thf(fact_4355_sin__periodic__pi,axiom,
    ! [X2: real] :
      ( ( sin @ real @ ( plus_plus @ real @ X2 @ pi ) )
      = ( uminus_uminus @ real @ ( sin @ real @ X2 ) ) ) ).

% sin_periodic_pi
thf(fact_4356_sin__periodic__pi2,axiom,
    ! [X2: real] :
      ( ( sin @ real @ ( plus_plus @ real @ pi @ X2 ) )
      = ( uminus_uminus @ real @ ( sin @ real @ X2 ) ) ) ).

% sin_periodic_pi2
thf(fact_4357_cos__pi__minus,axiom,
    ! [X2: real] :
      ( ( cos @ real @ ( minus_minus @ real @ pi @ X2 ) )
      = ( uminus_uminus @ real @ ( cos @ real @ X2 ) ) ) ).

% cos_pi_minus
thf(fact_4358_cos__minus__pi,axiom,
    ! [X2: real] :
      ( ( cos @ real @ ( minus_minus @ real @ X2 @ pi ) )
      = ( uminus_uminus @ real @ ( cos @ real @ X2 ) ) ) ).

% cos_minus_pi
thf(fact_4359_sin__minus__pi,axiom,
    ! [X2: real] :
      ( ( sin @ real @ ( minus_minus @ real @ X2 @ pi ) )
      = ( uminus_uminus @ real @ ( sin @ real @ X2 ) ) ) ).

% sin_minus_pi
thf(fact_4360_sin__npi2,axiom,
    ! [N2: nat] :
      ( ( sin @ real @ ( times_times @ real @ pi @ ( semiring_1_of_nat @ real @ N2 ) ) )
      = ( zero_zero @ real ) ) ).

% sin_npi2
thf(fact_4361_sin__npi,axiom,
    ! [N2: nat] :
      ( ( sin @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N2 ) @ pi ) )
      = ( zero_zero @ real ) ) ).

% sin_npi
thf(fact_4362_sin__npi__int,axiom,
    ! [N2: int] :
      ( ( sin @ real @ ( times_times @ real @ pi @ ( ring_1_of_int @ real @ N2 ) ) )
      = ( zero_zero @ real ) ) ).

% sin_npi_int
thf(fact_4363_cos__pi__half,axiom,
    ( ( cos @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
    = ( zero_zero @ real ) ) ).

% cos_pi_half
thf(fact_4364_sin__two__pi,axiom,
    ( ( sin @ real @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) )
    = ( zero_zero @ real ) ) ).

% sin_two_pi
thf(fact_4365_cos__two__pi,axiom,
    ( ( cos @ real @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) )
    = ( one_one @ real ) ) ).

% cos_two_pi
thf(fact_4366_sin__pi__half,axiom,
    ( ( sin @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
    = ( one_one @ real ) ) ).

% sin_pi_half
thf(fact_4367_cos__periodic,axiom,
    ! [X2: real] :
      ( ( cos @ real @ ( plus_plus @ real @ X2 @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) ) )
      = ( cos @ real @ X2 ) ) ).

% cos_periodic
thf(fact_4368_sin__periodic,axiom,
    ! [X2: real] :
      ( ( sin @ real @ ( plus_plus @ real @ X2 @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) ) )
      = ( sin @ real @ X2 ) ) ).

% sin_periodic
thf(fact_4369_cos__2pi__minus,axiom,
    ! [X2: real] :
      ( ( cos @ real @ ( minus_minus @ real @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) @ X2 ) )
      = ( cos @ real @ X2 ) ) ).

% cos_2pi_minus
thf(fact_4370_cos__npi2,axiom,
    ! [N2: nat] :
      ( ( cos @ real @ ( times_times @ real @ pi @ ( semiring_1_of_nat @ real @ N2 ) ) )
      = ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ N2 ) ) ).

% cos_npi2
thf(fact_4371_cos__npi,axiom,
    ! [N2: nat] :
      ( ( cos @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N2 ) @ pi ) )
      = ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ N2 ) ) ).

% cos_npi
thf(fact_4372_sin__2npi,axiom,
    ! [N2: nat] :
      ( ( sin @ real @ ( times_times @ real @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ real @ N2 ) ) @ pi ) )
      = ( zero_zero @ real ) ) ).

% sin_2npi
thf(fact_4373_cos__2npi,axiom,
    ! [N2: nat] :
      ( ( cos @ real @ ( times_times @ real @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ real @ N2 ) ) @ pi ) )
      = ( one_one @ real ) ) ).

% cos_2npi
thf(fact_4374_sin__2pi__minus,axiom,
    ! [X2: real] :
      ( ( sin @ real @ ( minus_minus @ real @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) @ X2 ) )
      = ( uminus_uminus @ real @ ( sin @ real @ X2 ) ) ) ).

% sin_2pi_minus
thf(fact_4375_sin__int__2pin,axiom,
    ! [N2: int] :
      ( ( sin @ real @ ( times_times @ real @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) @ ( ring_1_of_int @ real @ N2 ) ) )
      = ( zero_zero @ real ) ) ).

% sin_int_2pin
thf(fact_4376_cos__int__2pin,axiom,
    ! [N2: int] :
      ( ( cos @ real @ ( times_times @ real @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) @ ( ring_1_of_int @ real @ N2 ) ) )
      = ( one_one @ real ) ) ).

% cos_int_2pin
thf(fact_4377_cos__3over2__pi,axiom,
    ( ( cos @ real @ ( times_times @ real @ ( divide_divide @ real @ ( numeral_numeral @ real @ ( bit1 @ one2 ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ pi ) )
    = ( zero_zero @ real ) ) ).

% cos_3over2_pi
thf(fact_4378_sin__3over2__pi,axiom,
    ( ( sin @ real @ ( times_times @ real @ ( divide_divide @ real @ ( numeral_numeral @ real @ ( bit1 @ one2 ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ pi ) )
    = ( uminus_uminus @ real @ ( one_one @ real ) ) ) ).

% sin_3over2_pi
thf(fact_4379_cos__npi__int,axiom,
    ! [N2: int] :
      ( ( ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N2 )
       => ( ( cos @ real @ ( times_times @ real @ pi @ ( ring_1_of_int @ real @ N2 ) ) )
          = ( one_one @ real ) ) )
      & ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N2 )
       => ( ( cos @ real @ ( times_times @ real @ pi @ ( ring_1_of_int @ real @ N2 ) ) )
          = ( uminus_uminus @ real @ ( one_one @ real ) ) ) ) ) ).

% cos_npi_int
thf(fact_4380_cos__pi__eq__zero,axiom,
    ! [M: nat] :
      ( ( cos @ real @ ( divide_divide @ real @ ( times_times @ real @ pi @ ( semiring_1_of_nat @ real @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) ) ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
      = ( zero_zero @ real ) ) ).

% cos_pi_eq_zero
thf(fact_4381_sin__cos__npi,axiom,
    ! [N2: nat] :
      ( ( sin @ real @ ( divide_divide @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ) @ pi ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
      = ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ N2 ) ) ).

% sin_cos_npi
thf(fact_4382_pi__ge__zero,axiom,
    ord_less_eq @ real @ ( zero_zero @ real ) @ pi ).

% pi_ge_zero
thf(fact_4383_cos__monotone__0__pi__le,axiom,
    ! [Y2: real,X2: real] :
      ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y2 )
     => ( ( ord_less_eq @ real @ Y2 @ X2 )
       => ( ( ord_less_eq @ real @ X2 @ pi )
         => ( ord_less_eq @ real @ ( cos @ real @ X2 ) @ ( cos @ real @ Y2 ) ) ) ) ) ).

% cos_monotone_0_pi_le
thf(fact_4384_cos__mono__le__eq,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 )
     => ( ( ord_less_eq @ real @ X2 @ pi )
       => ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y2 )
         => ( ( ord_less_eq @ real @ Y2 @ pi )
           => ( ( ord_less_eq @ real @ ( cos @ real @ X2 ) @ ( cos @ real @ Y2 ) )
              = ( ord_less_eq @ real @ Y2 @ X2 ) ) ) ) ) ) ).

% cos_mono_le_eq
thf(fact_4385_cos__inj__pi,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 )
     => ( ( ord_less_eq @ real @ X2 @ pi )
       => ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y2 )
         => ( ( ord_less_eq @ real @ Y2 @ pi )
           => ( ( ( cos @ real @ X2 )
                = ( cos @ real @ Y2 ) )
             => ( X2 = Y2 ) ) ) ) ) ) ).

% cos_inj_pi
thf(fact_4386_sin__ge__zero,axiom,
    ! [X2: real] :
      ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 )
     => ( ( ord_less_eq @ real @ X2 @ pi )
       => ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( sin @ real @ X2 ) ) ) ) ).

% sin_ge_zero
thf(fact_4387_pi__less__4,axiom,
    ord_less @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ ( bit0 @ one2 ) ) ) ).

% pi_less_4
thf(fact_4388_pi__ge__two,axiom,
    ord_less_eq @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ).

% pi_ge_two
thf(fact_4389_pi__half__neq__two,axiom,
    ( ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) )
   != ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ).

% pi_half_neq_two
thf(fact_4390_cos__mono__less__eq,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 )
     => ( ( ord_less_eq @ real @ X2 @ pi )
       => ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y2 )
         => ( ( ord_less_eq @ real @ Y2 @ pi )
           => ( ( ord_less @ real @ ( cos @ real @ X2 ) @ ( cos @ real @ Y2 ) )
              = ( ord_less @ real @ Y2 @ X2 ) ) ) ) ) ) ).

% cos_mono_less_eq
thf(fact_4391_cos__monotone__0__pi,axiom,
    ! [Y2: real,X2: real] :
      ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y2 )
     => ( ( ord_less @ real @ Y2 @ X2 )
       => ( ( ord_less_eq @ real @ X2 @ pi )
         => ( ord_less @ real @ ( cos @ real @ X2 ) @ ( cos @ real @ Y2 ) ) ) ) ) ).

% cos_monotone_0_pi
thf(fact_4392_cos__monotone__minus__pi__0_H,axiom,
    ! [Y2: real,X2: real] :
      ( ( ord_less_eq @ real @ ( uminus_uminus @ real @ pi ) @ Y2 )
     => ( ( ord_less_eq @ real @ Y2 @ X2 )
       => ( ( ord_less_eq @ real @ X2 @ ( zero_zero @ real ) )
         => ( ord_less_eq @ real @ ( cos @ real @ Y2 ) @ ( cos @ real @ X2 ) ) ) ) ) ).

% cos_monotone_minus_pi_0'
thf(fact_4393_sin__zero__iff__int2,axiom,
    ! [X2: real] :
      ( ( ( sin @ real @ X2 )
        = ( zero_zero @ real ) )
      = ( ? [I2: int] :
            ( X2
            = ( times_times @ real @ ( ring_1_of_int @ real @ I2 ) @ pi ) ) ) ) ).

% sin_zero_iff_int2
thf(fact_4394_pi__half__neq__zero,axiom,
    ( ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) )
   != ( zero_zero @ real ) ) ).

% pi_half_neq_zero
thf(fact_4395_pi__half__less__two,axiom,
    ord_less @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ).

% pi_half_less_two
thf(fact_4396_pi__half__le__two,axiom,
    ord_less_eq @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ).

% pi_half_le_two
thf(fact_4397_cos__monotone__minus__pi__0,axiom,
    ! [Y2: real,X2: real] :
      ( ( ord_less_eq @ real @ ( uminus_uminus @ real @ pi ) @ Y2 )
     => ( ( ord_less @ real @ Y2 @ X2 )
       => ( ( ord_less_eq @ real @ X2 @ ( zero_zero @ real ) )
         => ( ord_less @ real @ ( cos @ real @ Y2 ) @ ( cos @ real @ X2 ) ) ) ) ) ).

% cos_monotone_minus_pi_0
thf(fact_4398_cos__total,axiom,
    ! [Y2: real] :
      ( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ Y2 )
     => ( ( ord_less_eq @ real @ Y2 @ ( one_one @ real ) )
       => ? [X3: real] :
            ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X3 )
            & ( ord_less_eq @ real @ X3 @ pi )
            & ( ( cos @ real @ X3 )
              = Y2 )
            & ! [Y3: real] :
                ( ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y3 )
                  & ( ord_less_eq @ real @ Y3 @ pi )
                  & ( ( cos @ real @ Y3 )
                    = Y2 ) )
               => ( Y3 = X3 ) ) ) ) ) ).

% cos_total
thf(fact_4399_sincos__principal__value,axiom,
    ! [X2: real] :
    ? [Y4: real] :
      ( ( ord_less @ real @ ( uminus_uminus @ real @ pi ) @ Y4 )
      & ( ord_less_eq @ real @ Y4 @ pi )
      & ( ( sin @ real @ Y4 )
        = ( sin @ real @ X2 ) )
      & ( ( cos @ real @ Y4 )
        = ( cos @ real @ X2 ) ) ) ).

% sincos_principal_value
thf(fact_4400_pi__half__gt__zero,axiom,
    ord_less @ real @ ( zero_zero @ real ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ).

% pi_half_gt_zero
thf(fact_4401_pi__half__ge__zero,axiom,
    ord_less_eq @ real @ ( zero_zero @ real ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ).

% pi_half_ge_zero
thf(fact_4402_m2pi__less__pi,axiom,
    ord_less @ real @ ( uminus_uminus @ real @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) ) @ pi ).

% m2pi_less_pi
thf(fact_4403_sin__pi__divide__n__ge__0,axiom,
    ! [N2: nat] :
      ( ( N2
       != ( zero_zero @ nat ) )
     => ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( sin @ real @ ( divide_divide @ real @ pi @ ( semiring_1_of_nat @ real @ N2 ) ) ) ) ) ).

% sin_pi_divide_n_ge_0
thf(fact_4404_minus__pi__half__less__zero,axiom,
    ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ ( zero_zero @ real ) ).

% minus_pi_half_less_zero
thf(fact_4405_cos__gt__zero,axiom,
    ! [X2: real] :
      ( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
     => ( ( ord_less @ real @ X2 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
       => ( ord_less @ real @ ( zero_zero @ real ) @ ( cos @ real @ X2 ) ) ) ) ).

% cos_gt_zero
thf(fact_4406_sin__gt__zero2,axiom,
    ! [X2: real] :
      ( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
     => ( ( ord_less @ real @ X2 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
       => ( ord_less @ real @ ( zero_zero @ real ) @ ( sin @ real @ X2 ) ) ) ) ).

% sin_gt_zero2
thf(fact_4407_sin__lt__zero,axiom,
    ! [X2: real] :
      ( ( ord_less @ real @ pi @ X2 )
     => ( ( ord_less @ real @ X2 @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) )
       => ( ord_less @ real @ ( sin @ real @ X2 ) @ ( zero_zero @ real ) ) ) ) ).

% sin_lt_zero
thf(fact_4408_cos__60,axiom,
    ( ( cos @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit1 @ one2 ) ) ) )
    = ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ).

% cos_60
thf(fact_4409_sin__30,axiom,
    ( ( sin @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ ( bit1 @ one2 ) ) ) ) )
    = ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ).

% sin_30
thf(fact_4410_sin__monotone__2pi__le,axiom,
    ! [Y2: real,X2: real] :
      ( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ Y2 )
     => ( ( ord_less_eq @ real @ Y2 @ X2 )
       => ( ( ord_less_eq @ real @ X2 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
         => ( ord_less_eq @ real @ ( sin @ real @ Y2 ) @ ( sin @ real @ X2 ) ) ) ) ) ).

% sin_monotone_2pi_le
thf(fact_4411_sin__mono__le__eq,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X2 )
     => ( ( ord_less_eq @ real @ X2 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
       => ( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ Y2 )
         => ( ( ord_less_eq @ real @ Y2 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
           => ( ( ord_less_eq @ real @ ( sin @ real @ X2 ) @ ( sin @ real @ Y2 ) )
              = ( ord_less_eq @ real @ X2 @ Y2 ) ) ) ) ) ) ).

% sin_mono_le_eq
thf(fact_4412_sin__inj__pi,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X2 )
     => ( ( ord_less_eq @ real @ X2 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
       => ( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ Y2 )
         => ( ( ord_less_eq @ real @ Y2 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
           => ( ( ( sin @ real @ X2 )
                = ( sin @ real @ Y2 ) )
             => ( X2 = Y2 ) ) ) ) ) ) ).

% sin_inj_pi
thf(fact_4413_cos__one__2pi__int,axiom,
    ! [X2: real] :
      ( ( ( cos @ real @ X2 )
        = ( one_one @ real ) )
      = ( ? [X: int] :
            ( X2
            = ( times_times @ real @ ( times_times @ real @ ( ring_1_of_int @ real @ X ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ pi ) ) ) ) ).

% cos_one_2pi_int
thf(fact_4414_sin__le__zero,axiom,
    ! [X2: real] :
      ( ( ord_less_eq @ real @ pi @ X2 )
     => ( ( ord_less @ real @ X2 @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) )
       => ( ord_less_eq @ real @ ( sin @ real @ X2 ) @ ( zero_zero @ real ) ) ) ) ).

% sin_le_zero
thf(fact_4415_cos__gt__zero__pi,axiom,
    ! [X2: real] :
      ( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X2 )
     => ( ( ord_less @ real @ X2 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
       => ( ord_less @ real @ ( zero_zero @ real ) @ ( cos @ real @ X2 ) ) ) ) ).

% cos_gt_zero_pi
thf(fact_4416_sin__less__zero,axiom,
    ! [X2: real] :
      ( ( ord_less @ real @ ( divide_divide @ real @ ( uminus_uminus @ real @ pi ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ X2 )
     => ( ( ord_less @ real @ X2 @ ( zero_zero @ real ) )
       => ( ord_less @ real @ ( sin @ real @ X2 ) @ ( zero_zero @ real ) ) ) ) ).

% sin_less_zero
thf(fact_4417_cos__ge__zero,axiom,
    ! [X2: real] :
      ( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X2 )
     => ( ( ord_less_eq @ real @ X2 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
       => ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( cos @ real @ X2 ) ) ) ) ).

% cos_ge_zero
thf(fact_4418_sin__monotone__2pi,axiom,
    ! [Y2: real,X2: real] :
      ( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ Y2 )
     => ( ( ord_less @ real @ Y2 @ X2 )
       => ( ( ord_less_eq @ real @ X2 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
         => ( ord_less @ real @ ( sin @ real @ Y2 ) @ ( sin @ real @ X2 ) ) ) ) ) ).

% sin_monotone_2pi
thf(fact_4419_sin__mono__less__eq,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X2 )
     => ( ( ord_less_eq @ real @ X2 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
       => ( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ Y2 )
         => ( ( ord_less_eq @ real @ Y2 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
           => ( ( ord_less @ real @ ( sin @ real @ X2 ) @ ( sin @ real @ Y2 ) )
              = ( ord_less @ real @ X2 @ Y2 ) ) ) ) ) ) ).

% sin_mono_less_eq
thf(fact_4420_sin__total,axiom,
    ! [Y2: real] :
      ( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ Y2 )
     => ( ( ord_less_eq @ real @ Y2 @ ( one_one @ real ) )
       => ? [X3: real] :
            ( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X3 )
            & ( ord_less_eq @ real @ X3 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
            & ( ( sin @ real @ X3 )
              = Y2 )
            & ! [Y3: real] :
                ( ( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ Y3 )
                  & ( ord_less_eq @ real @ Y3 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
                  & ( ( sin @ real @ Y3 )
                    = Y2 ) )
               => ( Y3 = X3 ) ) ) ) ) ).

% sin_total
thf(fact_4421_cos__one__2pi,axiom,
    ! [X2: real] :
      ( ( ( cos @ real @ X2 )
        = ( one_one @ real ) )
      = ( ? [X: nat] :
            ( X2
            = ( times_times @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ X ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ pi ) )
        | ? [X: nat] :
            ( X2
            = ( uminus_uminus @ real @ ( times_times @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ X ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ pi ) ) ) ) ) ).

% cos_one_2pi
thf(fact_4422_sin__pi__divide__n__gt__0,axiom,
    ! [N2: nat] :
      ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 )
     => ( ord_less @ real @ ( zero_zero @ real ) @ ( sin @ real @ ( divide_divide @ real @ pi @ ( semiring_1_of_nat @ real @ N2 ) ) ) ) ) ).

% sin_pi_divide_n_gt_0
thf(fact_4423_sincos__total__pi,axiom,
    ! [Y2: real,X2: real] :
      ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y2 )
     => ( ( ( plus_plus @ real @ ( power_power @ real @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
          = ( one_one @ real ) )
       => ? [T5: real] :
            ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ T5 )
            & ( ord_less_eq @ real @ T5 @ pi )
            & ( X2
              = ( cos @ real @ T5 ) )
            & ( Y2
              = ( sin @ real @ T5 ) ) ) ) ) ).

% sincos_total_pi
thf(fact_4424_sin__expansion__lemma,axiom,
    ! [X2: real,M: nat] :
      ( ( sin @ real @ ( plus_plus @ real @ X2 @ ( divide_divide @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ ( suc @ M ) ) @ pi ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) )
      = ( cos @ real @ ( plus_plus @ real @ X2 @ ( divide_divide @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ M ) @ pi ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ).

% sin_expansion_lemma
thf(fact_4425_cos__zero__iff__int,axiom,
    ! [X2: real] :
      ( ( ( cos @ real @ X2 )
        = ( zero_zero @ real ) )
      = ( ? [I2: int] :
            ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ I2 )
            & ( X2
              = ( times_times @ real @ ( ring_1_of_int @ real @ I2 ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).

% cos_zero_iff_int
thf(fact_4426_sin__zero__iff__int,axiom,
    ! [X2: real] :
      ( ( ( sin @ real @ X2 )
        = ( zero_zero @ real ) )
      = ( ? [I2: int] :
            ( ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ I2 )
            & ( X2
              = ( times_times @ real @ ( ring_1_of_int @ real @ I2 ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).

% sin_zero_iff_int
thf(fact_4427_cos__zero__lemma,axiom,
    ! [X2: real] :
      ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 )
     => ( ( ( cos @ real @ X2 )
          = ( zero_zero @ real ) )
       => ? [N4: nat] :
            ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N4 )
            & ( X2
              = ( times_times @ real @ ( semiring_1_of_nat @ real @ N4 ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).

% cos_zero_lemma
thf(fact_4428_sin__zero__lemma,axiom,
    ! [X2: real] :
      ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 )
     => ( ( ( sin @ real @ X2 )
          = ( zero_zero @ real ) )
       => ? [N4: nat] :
            ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N4 )
            & ( X2
              = ( times_times @ real @ ( semiring_1_of_nat @ real @ N4 ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).

% sin_zero_lemma
thf(fact_4429_cos__zero__iff,axiom,
    ! [X2: real] :
      ( ( ( cos @ real @ X2 )
        = ( zero_zero @ real ) )
      = ( ? [N: nat] :
            ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
            & ( X2
              = ( times_times @ real @ ( semiring_1_of_nat @ real @ N ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) )
        | ? [N: nat] :
            ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
            & ( X2
              = ( uminus_uminus @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ).

% cos_zero_iff
thf(fact_4430_sin__zero__iff,axiom,
    ! [X2: real] :
      ( ( ( sin @ real @ X2 )
        = ( zero_zero @ real ) )
      = ( ? [N: nat] :
            ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
            & ( X2
              = ( times_times @ real @ ( semiring_1_of_nat @ real @ N ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) )
        | ? [N: nat] :
            ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
            & ( X2
              = ( uminus_uminus @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ).

% sin_zero_iff
thf(fact_4431_cos__expansion__lemma,axiom,
    ! [X2: real,M: nat] :
      ( ( cos @ real @ ( plus_plus @ real @ X2 @ ( divide_divide @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ ( suc @ M ) ) @ pi ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) )
      = ( uminus_uminus @ real @ ( sin @ real @ ( plus_plus @ real @ X2 @ ( divide_divide @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ M ) @ pi ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ) ).

% cos_expansion_lemma
thf(fact_4432_sincos__total__pi__half,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 )
     => ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y2 )
       => ( ( ( plus_plus @ real @ ( power_power @ real @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
            = ( one_one @ real ) )
         => ? [T5: real] :
              ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ T5 )
              & ( ord_less_eq @ real @ T5 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
              & ( X2
                = ( cos @ real @ T5 ) )
              & ( Y2
                = ( sin @ real @ T5 ) ) ) ) ) ) ).

% sincos_total_pi_half
thf(fact_4433_sincos__total__2pi__le,axiom,
    ! [X2: real,Y2: real] :
      ( ( ( plus_plus @ real @ ( power_power @ real @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
        = ( one_one @ real ) )
     => ? [T5: real] :
          ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ T5 )
          & ( ord_less_eq @ real @ T5 @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) )
          & ( X2
            = ( cos @ real @ T5 ) )
          & ( Y2
            = ( sin @ real @ T5 ) ) ) ) ).

% sincos_total_2pi_le
thf(fact_4434_sincos__total__2pi,axiom,
    ! [X2: real,Y2: real] :
      ( ( ( plus_plus @ real @ ( power_power @ real @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
        = ( one_one @ real ) )
     => ~ ! [T5: real] :
            ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ T5 )
           => ( ( ord_less @ real @ T5 @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) )
             => ( ( X2
                  = ( cos @ real @ T5 ) )
               => ( Y2
                 != ( sin @ real @ T5 ) ) ) ) ) ) ).

% sincos_total_2pi
thf(fact_4435_Maclaurin__sin__expansion,axiom,
    ! [X2: real,N2: nat] :
    ? [T5: real] :
      ( ( sin @ real @ X2 )
      = ( plus_plus @ real
        @ ( groups7311177749621191930dd_sum @ nat @ real
          @ ^ [M7: nat] : ( times_times @ real @ ( sin_coeff @ M7 ) @ ( power_power @ real @ X2 @ M7 ) )
          @ ( set_ord_lessThan @ nat @ N2 ) )
        @ ( times_times @ real @ ( divide_divide @ real @ ( sin @ real @ ( plus_plus @ real @ T5 @ ( times_times @ real @ ( times_times @ real @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( semiring_1_of_nat @ real @ N2 ) ) @ pi ) ) ) @ ( semiring_char_0_fact @ real @ N2 ) ) @ ( power_power @ real @ X2 @ N2 ) ) ) ) ).

% Maclaurin_sin_expansion
thf(fact_4436_Maclaurin__cos__expansion,axiom,
    ! [X2: real,N2: nat] :
    ? [T5: real] :
      ( ( ord_less_eq @ real @ ( abs_abs @ real @ T5 ) @ ( abs_abs @ real @ X2 ) )
      & ( ( cos @ real @ X2 )
        = ( plus_plus @ real
          @ ( groups7311177749621191930dd_sum @ nat @ real
            @ ^ [M7: nat] : ( times_times @ real @ ( cos_coeff @ M7 ) @ ( power_power @ real @ X2 @ M7 ) )
            @ ( set_ord_lessThan @ nat @ N2 ) )
          @ ( times_times @ real @ ( divide_divide @ real @ ( cos @ real @ ( plus_plus @ real @ T5 @ ( times_times @ real @ ( times_times @ real @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( semiring_1_of_nat @ real @ N2 ) ) @ pi ) ) ) @ ( semiring_char_0_fact @ real @ N2 ) ) @ ( power_power @ real @ X2 @ N2 ) ) ) ) ) ).

% Maclaurin_cos_expansion
thf(fact_4437_Maclaurin__sin__expansion2,axiom,
    ! [X2: real,N2: nat] :
    ? [T5: real] :
      ( ( ord_less_eq @ real @ ( abs_abs @ real @ T5 ) @ ( abs_abs @ real @ X2 ) )
      & ( ( sin @ real @ X2 )
        = ( plus_plus @ real
          @ ( groups7311177749621191930dd_sum @ nat @ real
            @ ^ [M7: nat] : ( times_times @ real @ ( sin_coeff @ M7 ) @ ( power_power @ real @ X2 @ M7 ) )
            @ ( set_ord_lessThan @ nat @ N2 ) )
          @ ( times_times @ real @ ( divide_divide @ real @ ( sin @ real @ ( plus_plus @ real @ T5 @ ( times_times @ real @ ( times_times @ real @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( semiring_1_of_nat @ real @ N2 ) ) @ pi ) ) ) @ ( semiring_char_0_fact @ real @ N2 ) ) @ ( power_power @ real @ X2 @ N2 ) ) ) ) ) ).

% Maclaurin_sin_expansion2
thf(fact_4438_ln__series,axiom,
    ! [X2: real] :
      ( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
     => ( ( ord_less @ real @ X2 @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) )
       => ( ( ln_ln @ real @ X2 )
          = ( suminf @ real
            @ ^ [N: nat] : ( times_times @ real @ ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ N ) @ ( divide_divide @ real @ ( one_one @ real ) @ ( semiring_1_of_nat @ real @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) ) ) ) @ ( power_power @ real @ ( minus_minus @ real @ X2 @ ( one_one @ real ) ) @ ( suc @ N ) ) ) ) ) ) ) ).

% ln_series
thf(fact_4439_abs__ln__one__plus__x__minus__x__bound__nonpos,axiom,
    ! [X2: real] :
      ( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X2 )
     => ( ( ord_less_eq @ real @ X2 @ ( zero_zero @ real ) )
       => ( ord_less_eq @ real @ ( abs_abs @ real @ ( minus_minus @ real @ ( ln_ln @ real @ ( plus_plus @ real @ ( one_one @ real ) @ X2 ) ) @ X2 ) ) @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( power_power @ real @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ).

% abs_ln_one_plus_x_minus_x_bound_nonpos
thf(fact_4440_abs__abs,axiom,
    ! [A: $tType] :
      ( ( idom_abs_sgn @ A )
     => ! [A2: A] :
          ( ( abs_abs @ A @ ( abs_abs @ A @ A2 ) )
          = ( abs_abs @ A @ A2 ) ) ) ).

% abs_abs
thf(fact_4441_abs__0,axiom,
    ! [A: $tType] :
      ( ( idom_abs_sgn @ A )
     => ( ( abs_abs @ A @ ( zero_zero @ A ) )
        = ( zero_zero @ A ) ) ) ).

% abs_0
thf(fact_4442_abs__numeral,axiom,
    ! [A: $tType] :
      ( ( linordered_idom @ A )
     => ! [N2: num] :
          ( ( abs_abs @ A @ ( numeral_numeral @ A @ N2 ) )
          = ( numeral_numeral @ A @ N2 ) ) ) ).

% abs_numeral
thf(fact_4443_abs__add__abs,axiom,
    ! [A: $tType] :
      ( ( ordere166539214618696060dd_abs @ A )
     => ! [A2: A,B2: A] :
          ( ( abs_abs @ A @ ( plus_plus @ A @ ( abs_abs @ A @ A2 ) @ ( abs_abs @ A @ B2 ) ) )
          = ( plus_plus @ A @ ( abs_abs @ A @ A2 ) @ ( abs_abs @ A @ B2 ) ) ) ) ).

% abs_add_abs
thf(fact_4444_abs__1,axiom,
    ! [A: $tType] :
      ( ( idom_abs_sgn @ A )
     => ( ( abs_abs @ A @ ( one_one @ A ) )
        = ( one_one @ A ) ) ) ).

% abs_1
thf(fact_4445_abs__mult__self__eq,axiom,
    ! [A: $tType] :
      ( ( linordered_idom @ A )
     => ! [A2: A] :
          ( ( times_times @ A @ ( abs_abs @ A @ A2 ) @ ( abs_abs @ A @ A2 ) )
          = ( times_times @ A @ A2 @ A2 ) ) ) ).

% abs_mult_self_eq
thf(fact_4446_abs__divide,axiom,
    ! [A: $tType] :
      ( ( field_abs_sgn @ A )
     => ! [A2: A,B2: A] :
          ( ( abs_abs @ A @ ( divide_divide @ A @ A2 @ B2 ) )
          = ( divide_divide @ A @ ( abs_abs @ A @ A2 ) @ ( abs_abs @ A @ B2 ) ) ) ) ).

% abs_divide
thf(fact_4447_abs__minus,axiom,
    ! [A: $tType] :
      ( ( idom_abs_sgn @ A )
     => ! [A2: A] :
          ( ( abs_abs @ A @ ( uminus_uminus @ A @ A2 ) )
          = ( abs_abs @ A @ A2 ) ) ) ).

% abs_minus
thf(fact_4448_abs__dvd__iff,axiom,
    ! [A: $tType] :
      ( ( linordered_idom @ A )
     => ! [M: A,K: A] :
          ( ( dvd_dvd @ A @ ( abs_abs @ A @ M ) @ K )
          = ( dvd_dvd @ A @ M @ K ) ) ) ).

% abs_dvd_iff
thf(fact_4449_dvd__abs__iff,axiom,
    ! [A: $tType] :
      ( ( linordered_idom @ A )
     => ! [M: A,K: A] :
          ( ( dvd_dvd @ A @ M @ ( abs_abs @ A @ K ) )
          = ( dvd_dvd @ A @ M @ K ) ) ) ).

% dvd_abs_iff
thf(fact_4450_abs__of__nat,axiom,
    ! [A: $tType] :
      ( ( linordered_idom @ A )
     => ! [N2: nat] :
          ( ( abs_abs @ A @ ( semiring_1_of_nat @ A @ N2 ) )
          = ( semiring_1_of_nat @ A @ N2 ) ) ) ).

% abs_of_nat
thf(fact_4451_abs__sum__abs,axiom,
    ! [B: $tType,A: $tType] :
      ( ( ordere166539214618696060dd_abs @ B )
     => ! [F: A > B,A3: set @ A] :
          ( ( abs_abs @ B
            @ ( groups7311177749621191930dd_sum @ A @ B
              @ ^ [A5: A] : ( abs_abs @ B @ ( F @ A5 ) )
              @ A3 ) )
          = ( groups7311177749621191930dd_sum @ A @ B
            @ ^ [A5: A] : ( abs_abs @ B @ ( F @ A5 ) )
            @ A3 ) ) ) ).

% abs_sum_abs
thf(fact_4452_suminf__zero,axiom,
    ! [A: $tType] :
      ( ( ( comm_monoid_add @ A )
        & ( topological_t2_space @ A ) )
     => ( ( suminf @ A
          @ ^ [N: nat] : ( zero_zero @ A ) )
        = ( zero_zero @ A ) ) ) ).

% suminf_zero
thf(fact_4453_abs__le__zero__iff,axiom,
    ! [A: $tType] :
      ( ( ordere166539214618696060dd_abs @ A )
     => ! [A2: A] :
          ( ( ord_less_eq @ A @ ( abs_abs @ A @ A2 ) @ ( zero_zero @ A ) )
          = ( A2
            = ( zero_zero @ A ) ) ) ) ).

% abs_le_zero_iff
thf(fact_4454_abs__le__self__iff,axiom,
    ! [A: $tType] :
      ( ( ordere166539214618696060dd_abs @ A )
     => ! [A2: A] :
          ( ( ord_less_eq @ A @ ( abs_abs @ A @ A2 ) @ A2 )
          = ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).

% abs_le_self_iff
thf(fact_4455_abs__of__nonneg,axiom,
    ! [A: $tType] :
      ( ( ordere166539214618696060dd_abs @ A )
     => ! [A2: A] :
          ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
         => ( ( abs_abs @ A @ A2 )
            = A2 ) ) ) ).

% abs_of_nonneg
thf(fact_4456_zero__less__abs__iff,axiom,
    ! [A: $tType] :
      ( ( ordere166539214618696060dd_abs @ A )
     => ! [A2: A] :
          ( ( ord_less @ A @ ( zero_zero @ A ) @ ( abs_abs @ A @ A2 ) )
          = ( A2
           != ( zero_zero @ A ) ) ) ) ).

% zero_less_abs_iff
thf(fact_4457_abs__neg__numeral,axiom,
    ! [A: $tType] :
      ( ( linordered_idom @ A )
     => ! [N2: num] :
          ( ( abs_abs @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N2 ) ) )
          = ( numeral_numeral @ A @ N2 ) ) ) ).

% abs_neg_numeral
thf(fact_4458_abs__neg__one,axiom,
    ! [A: $tType] :
      ( ( linordered_idom @ A )
     => ( ( abs_abs @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
        = ( one_one @ A ) ) ) ).

% abs_neg_one
thf(fact_4459_abs__power__minus,axiom,
    ! [A: $tType] :
      ( ( linordered_idom @ A )
     => ! [A2: A,N2: nat] :
          ( ( abs_abs @ A @ ( power_power @ A @ ( uminus_uminus @ A @ A2 ) @ N2 ) )
          = ( abs_abs @ A @ ( power_power @ A @ A2 @ N2 ) ) ) ) ).

% abs_power_minus
thf(fact_4460_sum__abs,axiom,
    ! [B: $tType,A: $tType] :
      ( ( ordere166539214618696060dd_abs @ B )
     => ! [F: A > B,A3: set @ A] :
          ( ord_less_eq @ B @ ( abs_abs @ B @ ( groups7311177749621191930dd_sum @ A @ B @ F @ A3 ) )
          @ ( groups7311177749621191930dd_sum @ A @ B
            @ ^ [I2: A] : ( abs_abs @ B @ ( F @ I2 ) )
            @ A3 ) ) ) ).

% sum_abs
thf(fact_4461_zero__le__divide__abs__iff,axiom,
    ! [A: $tType] :
      ( ( linordered_field @ A )
     => ! [A2: A,B2: A] :
          ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ A2 @ ( abs_abs @ A @ B2 ) ) )
          = ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
            | ( B2
              = ( zero_zero @ A ) ) ) ) ) ).

% zero_le_divide_abs_iff
thf(fact_4462_divide__le__0__abs__iff,axiom,
    ! [A: $tType] :
      ( ( linordered_field @ A )
     => ! [A2: A,B2: A] :
          ( ( ord_less_eq @ A @ ( divide_divide @ A @ A2 @ ( abs_abs @ A @ B2 ) ) @ ( zero_zero @ A ) )
          = ( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) )
            | ( B2
              = ( zero_zero @ A ) ) ) ) ) ).

% divide_le_0_abs_iff
thf(fact_4463_abs__of__nonpos,axiom,
    ! [A: $tType] :
      ( ( ordere166539214618696060dd_abs @ A )
     => ! [A2: A] :
          ( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) )
         => ( ( abs_abs @ A @ A2 )
            = ( uminus_uminus @ A @ A2 ) ) ) ) ).

% abs_of_nonpos
thf(fact_4464_artanh__minus__real,axiom,
    ! [X2: real] :
      ( ( ord_less @ real @ ( abs_abs @ real @ X2 ) @ ( one_one @ real ) )
     => ( ( artanh @ real @ ( uminus_uminus @ real @ X2 ) )
        = ( uminus_uminus @ real @ ( artanh @ real @ X2 ) ) ) ) ).

% artanh_minus_real
thf(fact_4465_norm__scaleR,axiom,
    ! [A: $tType] :
      ( ( real_V822414075346904944vector @ A )
     => ! [A2: real,X2: A] :
          ( ( real_V7770717601297561774m_norm @ A @ ( real_V8093663219630862766scaleR @ A @ A2 @ X2 ) )
          = ( times_times @ real @ ( abs_abs @ real @ A2 ) @ ( real_V7770717601297561774m_norm @ A @ X2 ) ) ) ) ).

% norm_scaleR
thf(fact_4466_sum__abs__ge__zero,axiom,
    ! [B: $tType,A: $tType] :
      ( ( ordere166539214618696060dd_abs @ B )
     => ! [F: A > B,A3: set @ A] :
          ( ord_less_eq @ B @ ( zero_zero @ B )
          @ ( groups7311177749621191930dd_sum @ A @ B
            @ ^ [I2: A] : ( abs_abs @ B @ ( F @ I2 ) )
            @ A3 ) ) ) ).

% sum_abs_ge_zero
thf(fact_4467_zero__less__power__abs__iff,axiom,
    ! [A: $tType] :
      ( ( linordered_idom @ A )
     => ! [A2: A,N2: nat] :
          ( ( ord_less @ A @ ( zero_zero @ A ) @ ( power_power @ A @ ( abs_abs @ A @ A2 ) @ N2 ) )
          = ( ( A2
             != ( zero_zero @ A ) )
            | ( N2
              = ( zero_zero @ nat ) ) ) ) ) ).

% zero_less_power_abs_iff
thf(fact_4468_power2__abs,axiom,
    ! [A: $tType] :
      ( ( linordered_idom @ A )
     => ! [A2: A] :
          ( ( power_power @ A @ ( abs_abs @ A @ A2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
          = ( power_power @ A @ A2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).

% power2_abs
thf(fact_4469_abs__power2,axiom,
    ! [A: $tType] :
      ( ( linordered_idom @ A )
     => ! [A2: A] :
          ( ( abs_abs @ A @ ( power_power @ A @ A2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
          = ( power_power @ A @ A2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).

% abs_power2
thf(fact_4470_powser__zero,axiom,
    ! [A: $tType] :
      ( ( real_V2822296259951069270ebra_1 @ A )
     => ! [F: nat > A] :
          ( ( suminf @ A
            @ ^ [N: nat] : ( times_times @ A @ ( F @ N ) @ ( power_power @ A @ ( zero_zero @ A ) @ N ) ) )
          = ( F @ ( zero_zero @ nat ) ) ) ) ).

% powser_zero
thf(fact_4471_power__even__abs__numeral,axiom,
    ! [A: $tType] :
      ( ( linordered_idom @ A )
     => ! [W: num,A2: A] :
          ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ W ) )
         => ( ( power_power @ A @ ( abs_abs @ A @ A2 ) @ ( numeral_numeral @ nat @ W ) )
            = ( power_power @ A @ A2 @ ( numeral_numeral @ nat @ W ) ) ) ) ) ).

% power_even_abs_numeral
thf(fact_4472_square__powr__half,axiom,
    ! [X2: real] :
      ( ( powr @ real @ ( power_power @ real @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
      = ( abs_abs @ real @ X2 ) ) ).

% square_powr_half
thf(fact_4473_abs__one,axiom,
    ! [A: $tType] :
      ( ( linordered_idom @ A )
     => ( ( abs_abs @ A @ ( one_one @ A ) )
        = ( one_one @ A ) ) ) ).

% abs_one
thf(fact_4474_abs__eq__0__iff,axiom,
    ! [A: $tType] :
      ( ( idom_abs_sgn @ A )
     => ! [A2: A] :
          ( ( ( abs_abs @ A @ A2 )
            = ( zero_zero @ A ) )
          = ( A2
            = ( zero_zero @ A ) ) ) ) ).

% abs_eq_0_iff
thf(fact_4475_abs__le__D1,axiom,
    ! [A: $tType] :
      ( ( ordere166539214618696060dd_abs @ A )
     => ! [A2: A,B2: A] :
          ( ( ord_less_eq @ A @ ( abs_abs @ A @ A2 ) @ B2 )
         => ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).

% abs_le_D1
thf(fact_4476_abs__ge__self,axiom,
    ! [A: $tType] :
      ( ( ordere166539214618696060dd_abs @ A )
     => ! [A2: A] : ( ord_less_eq @ A @ A2 @ ( abs_abs @ A @ A2 ) ) ) ).

% abs_ge_self
thf(fact_4477_abs__minus__commute,axiom,
    ! [A: $tType] :
      ( ( ordere166539214618696060dd_abs @ A )
     => ! [A2: A,B2: A] :
          ( ( abs_abs @ A @ ( minus_minus @ A @ A2 @ B2 ) )
          = ( abs_abs @ A @ ( minus_minus @ A @ B2 @ A2 ) ) ) ) ).

% abs_minus_commute
thf(fact_4478_dvd__if__abs__eq,axiom,
    ! [A: $tType] :
      ( ( linordered_idom @ A )
     => ! [L: A,K: A] :
          ( ( ( abs_abs @ A @ L )
            = ( abs_abs @ A @ K ) )
         => ( dvd_dvd @ A @ L @ K ) ) ) ).

% dvd_if_abs_eq
thf(fact_4479_abs__mult,axiom,
    ! [A: $tType] :
      ( ( idom_abs_sgn @ A )
     => ! [A2: A,B2: A] :
          ( ( abs_abs @ A @ ( times_times @ A @ A2 @ B2 ) )
          = ( times_times @ A @ ( abs_abs @ A @ A2 ) @ ( abs_abs @ A @ B2 ) ) ) ) ).

% abs_mult
thf(fact_4480_power__abs,axiom,
    ! [A: $tType] :
      ( ( linordered_idom @ A )
     => ! [A2: A,N2: nat] :
          ( ( abs_abs @ A @ ( power_power @ A @ A2 @ N2 ) )
          = ( power_power @ A @ ( abs_abs @ A @ A2 ) @ N2 ) ) ) ).

% power_abs
thf(fact_4481_abs__prod,axiom,
    ! [A: $tType,B: $tType] :
      ( ( linordered_field @ A )
     => ! [F: B > A,A3: set @ B] :
          ( ( abs_abs @ A @ ( groups7121269368397514597t_prod @ B @ A @ F @ A3 ) )
          = ( groups7121269368397514597t_prod @ B @ A
            @ ^ [X: B] : ( abs_abs @ A @ ( F @ X ) )
            @ A3 ) ) ) ).

% abs_prod
thf(fact_4482_abs__eq__iff,axiom,
    ! [A: $tType] :
      ( ( linordered_ring @ A )
     => ! [X2: A,Y2: A] :
          ( ( ( abs_abs @ A @ X2 )
            = ( abs_abs @ A @ Y2 ) )
          = ( ( X2 = Y2 )
            | ( X2
              = ( uminus_uminus @ A @ Y2 ) ) ) ) ) ).

% abs_eq_iff
thf(fact_4483_abs__ge__zero,axiom,
    ! [A: $tType] :
      ( ( ordere166539214618696060dd_abs @ A )
     => ! [A2: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( abs_abs @ A @ A2 ) ) ) ).

% abs_ge_zero
thf(fact_4484_abs__of__pos,axiom,
    ! [A: $tType] :
      ( ( ordere166539214618696060dd_abs @ A )
     => ! [A2: A] :
          ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
         => ( ( abs_abs @ A @ A2 )
            = A2 ) ) ) ).

% abs_of_pos
thf(fact_4485_abs__not__less__zero,axiom,
    ! [A: $tType] :
      ( ( ordere166539214618696060dd_abs @ A )
     => ! [A2: A] :
          ~ ( ord_less @ A @ ( abs_abs @ A @ A2 ) @ ( zero_zero @ A ) ) ) ).

% abs_not_less_zero
thf(fact_4486_abs__triangle__ineq,axiom,
    ! [A: $tType] :
      ( ( ordere166539214618696060dd_abs @ A )
     => ! [A2: A,B2: A] : ( ord_less_eq @ A @ ( abs_abs @ A @ ( plus_plus @ A @ A2 @ B2 ) ) @ ( plus_plus @ A @ ( abs_abs @ A @ A2 ) @ ( abs_abs @ A @ B2 ) ) ) ) ).

% abs_triangle_ineq
thf(fact_4487_abs__mult__less,axiom,
    ! [A: $tType] :
      ( ( linordered_idom @ A )
     => ! [A2: A,C2: A,B2: A,D2: A] :
          ( ( ord_less @ A @ ( abs_abs @ A @ A2 ) @ C2 )
         => ( ( ord_less @ A @ ( abs_abs @ A @ B2 ) @ D2 )
           => ( ord_less @ A @ ( times_times @ A @ ( abs_abs @ A @ A2 ) @ ( abs_abs @ A @ B2 ) ) @ ( times_times @ A @ C2 @ D2 ) ) ) ) ) ).

% abs_mult_less
thf(fact_4488_abs__triangle__ineq2,axiom,
    ! [A: $tType] :
      ( ( ordere166539214618696060dd_abs @ A )
     => ! [A2: A,B2: A] : ( ord_less_eq @ A @ ( minus_minus @ A @ ( abs_abs @ A @ A2 ) @ ( abs_abs @ A @ B2 ) ) @ ( abs_abs @ A @ ( minus_minus @ A @ A2 @ B2 ) ) ) ) ).

% abs_triangle_ineq2
thf(fact_4489_abs__triangle__ineq3,axiom,
    ! [A: $tType] :
      ( ( ordere166539214618696060dd_abs @ A )
     => ! [A2: A,B2: A] : ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ ( abs_abs @ A @ A2 ) @ ( abs_abs @ A @ B2 ) ) ) @ ( abs_abs @ A @ ( minus_minus @ A @ A2 @ B2 ) ) ) ) ).

% abs_triangle_ineq3
thf(fact_4490_abs__triangle__ineq2__sym,axiom,
    ! [A: $tType] :
      ( ( ordere166539214618696060dd_abs @ A )
     => ! [A2: A,B2: A] : ( ord_less_eq @ A @ ( minus_minus @ A @ ( abs_abs @ A @ A2 ) @ ( abs_abs @ A @ B2 ) ) @ ( abs_abs @ A @ ( minus_minus @ A @ B2 @ A2 ) ) ) ) ).

% abs_triangle_ineq2_sym
thf(fact_4491_nonzero__abs__divide,axiom,
    ! [A: $tType] :
      ( ( linordered_field @ A )
     => ! [B2: A,A2: A] :
          ( ( B2
           != ( zero_zero @ A ) )
         => ( ( abs_abs @ A @ ( divide_divide @ A @ A2 @ B2 ) )
            = ( divide_divide @ A @ ( abs_abs @ A @ A2 ) @ ( abs_abs @ A @ B2 ) ) ) ) ) ).

% nonzero_abs_divide
thf(fact_4492_abs__ge__minus__self,axiom,
    ! [A: $tType] :
      ( ( ordere166539214618696060dd_abs @ A )
     => ! [A2: A] : ( ord_less_eq @ A @ ( uminus_uminus @ A @ A2 ) @ ( abs_abs @ A @ A2 ) ) ) ).

% abs_ge_minus_self
thf(fact_4493_abs__le__iff,axiom,
    ! [A: $tType] :
      ( ( ordere166539214618696060dd_abs @ A )
     => ! [A2: A,B2: A] :
          ( ( ord_less_eq @ A @ ( abs_abs @ A @ A2 ) @ B2 )
          = ( ( ord_less_eq @ A @ A2 @ B2 )
            & ( ord_less_eq @ A @ ( uminus_uminus @ A @ A2 ) @ B2 ) ) ) ) ).

% abs_le_iff
thf(fact_4494_abs__le__D2,axiom,
    ! [A: $tType] :
      ( ( ordere166539214618696060dd_abs @ A )
     => ! [A2: A,B2: A] :
          ( ( ord_less_eq @ A @ ( abs_abs @ A @ A2 ) @ B2 )
         => ( ord_less_eq @ A @ ( uminus_uminus @ A @ A2 ) @ B2 ) ) ) ).

% abs_le_D2
thf(fact_4495_abs__leI,axiom,
    ! [A: $tType] :
      ( ( ordere166539214618696060dd_abs @ A )
     => ! [A2: A,B2: A] :
          ( ( ord_less_eq @ A @ A2 @ B2 )
         => ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ A2 ) @ B2 )
           => ( ord_less_eq @ A @ ( abs_abs @ A @ A2 ) @ B2 ) ) ) ) ).

% abs_leI
thf(fact_4496_abs__less__iff,axiom,
    ! [A: $tType] :
      ( ( linordered_idom @ A )
     => ! [A2: A,B2: A] :
          ( ( ord_less @ A @ ( abs_abs @ A @ A2 ) @ B2 )
          = ( ( ord_less @ A @ A2 @ B2 )
            & ( ord_less @ A @ ( uminus_uminus @ A @ A2 ) @ B2 ) ) ) ) ).

% abs_less_iff
thf(fact_4497_abs__sin__x__le__abs__x,axiom,
    ! [X2: real] : ( ord_less_eq @ real @ ( abs_abs @ real @ ( sin @ real @ X2 ) ) @ ( abs_abs @ real @ X2 ) ) ).

% abs_sin_x_le_abs_x
thf(fact_4498_dense__eq0__I,axiom,
    ! [A: $tType] :
      ( ( ( ordere166539214618696060dd_abs @ A )
        & ( dense_linorder @ A ) )
     => ! [X2: A] :
          ( ! [E2: A] :
              ( ( ord_less @ A @ ( zero_zero @ A ) @ E2 )
             => ( ord_less_eq @ A @ ( abs_abs @ A @ X2 ) @ E2 ) )
         => ( X2
            = ( zero_zero @ A ) ) ) ) ).

% dense_eq0_I
thf(fact_4499_abs__eq__mult,axiom,
    ! [A: $tType] :
      ( ( ordered_ring_abs @ A )
     => ! [A2: A,B2: A] :
          ( ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
              | ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) )
            & ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
              | ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) ) ) )
         => ( ( abs_abs @ A @ ( times_times @ A @ A2 @ B2 ) )
            = ( times_times @ A @ ( abs_abs @ A @ A2 ) @ ( abs_abs @ A @ B2 ) ) ) ) ) ).

% abs_eq_mult
thf(fact_4500_abs__mult__pos,axiom,
    ! [A: $tType] :
      ( ( linordered_idom @ A )
     => ! [X2: A,Y2: A] :
          ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X2 )
         => ( ( times_times @ A @ ( abs_abs @ A @ Y2 ) @ X2 )
            = ( abs_abs @ A @ ( times_times @ A @ Y2 @ X2 ) ) ) ) ) ).

% abs_mult_pos
thf(fact_4501_abs__minus__le__zero,axiom,
    ! [A: $tType] :
      ( ( ordere166539214618696060dd_abs @ A )
     => ! [A2: A] : ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( abs_abs @ A @ A2 ) ) @ ( zero_zero @ A ) ) ) ).

% abs_minus_le_zero
thf(fact_4502_abs__eq__iff_H,axiom,
    ! [A: $tType] :
      ( ( linordered_ring @ A )
     => ! [A2: A,B2: A] :
          ( ( ( abs_abs @ A @ A2 )
            = B2 )
          = ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
            & ( ( A2 = B2 )
              | ( A2
                = ( uminus_uminus @ A @ B2 ) ) ) ) ) ) ).

% abs_eq_iff'
thf(fact_4503_eq__abs__iff_H,axiom,
    ! [A: $tType] :
      ( ( linordered_ring @ A )
     => ! [A2: A,B2: A] :
          ( ( A2
            = ( abs_abs @ A @ B2 ) )
          = ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
            & ( ( B2 = A2 )
              | ( B2
                = ( uminus_uminus @ A @ A2 ) ) ) ) ) ) ).

% eq_abs_iff'
thf(fact_4504_abs__div__pos,axiom,
    ! [A: $tType] :
      ( ( linordered_field @ A )
     => ! [Y2: A,X2: A] :
          ( ( ord_less @ A @ ( zero_zero @ A ) @ Y2 )
         => ( ( divide_divide @ A @ ( abs_abs @ A @ X2 ) @ Y2 )
            = ( abs_abs @ A @ ( divide_divide @ A @ X2 @ Y2 ) ) ) ) ) ).

% abs_div_pos
thf(fact_4505_zero__le__power__abs,axiom,
    ! [A: $tType] :
      ( ( linordered_idom @ A )
     => ! [A2: A,N2: nat] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( power_power @ A @ ( abs_abs @ A @ A2 ) @ N2 ) ) ) ).

% zero_le_power_abs
thf(fact_4506_abs__if__raw,axiom,
    ! [A: $tType] :
      ( ( abs_if @ A )
     => ( ( abs_abs @ A )
        = ( ^ [A5: A] : ( if @ A @ ( ord_less @ A @ A5 @ ( zero_zero @ A ) ) @ ( uminus_uminus @ A @ A5 ) @ A5 ) ) ) ) ).

% abs_if_raw
thf(fact_4507_abs__of__neg,axiom,
    ! [A: $tType] :
      ( ( ordere166539214618696060dd_abs @ A )
     => ! [A2: A] :
          ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
         => ( ( abs_abs @ A @ A2 )
            = ( uminus_uminus @ A @ A2 ) ) ) ) ).

% abs_of_neg
thf(fact_4508_abs__if,axiom,
    ! [A: $tType] :
      ( ( abs_if @ A )
     => ( ( abs_abs @ A )
        = ( ^ [A5: A] : ( if @ A @ ( ord_less @ A @ A5 @ ( zero_zero @ A ) ) @ ( uminus_uminus @ A @ A5 ) @ A5 ) ) ) ) ).

% abs_if
thf(fact_4509_abs__diff__le__iff,axiom,
    ! [A: $tType] :
      ( ( linordered_idom @ A )
     => ! [X2: A,A2: A,R4: A] :
          ( ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ X2 @ A2 ) ) @ R4 )
          = ( ( ord_less_eq @ A @ ( minus_minus @ A @ A2 @ R4 ) @ X2 )
            & ( ord_less_eq @ A @ X2 @ ( plus_plus @ A @ A2 @ R4 ) ) ) ) ) ).

% abs_diff_le_iff
thf(fact_4510_abs__diff__triangle__ineq,axiom,
    ! [A: $tType] :
      ( ( ordere166539214618696060dd_abs @ A )
     => ! [A2: A,B2: A,C2: A,D2: A] : ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ ( plus_plus @ A @ C2 @ D2 ) ) ) @ ( plus_plus @ A @ ( abs_abs @ A @ ( minus_minus @ A @ A2 @ C2 ) ) @ ( abs_abs @ A @ ( minus_minus @ A @ B2 @ D2 ) ) ) ) ) ).

% abs_diff_triangle_ineq
thf(fact_4511_abs__triangle__ineq4,axiom,
    ! [A: $tType] :
      ( ( ordere166539214618696060dd_abs @ A )
     => ! [A2: A,B2: A] : ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ A2 @ B2 ) ) @ ( plus_plus @ A @ ( abs_abs @ A @ A2 ) @ ( abs_abs @ A @ B2 ) ) ) ) ).

% abs_triangle_ineq4
thf(fact_4512_abs__diff__less__iff,axiom,
    ! [A: $tType] :
      ( ( linordered_idom @ A )
     => ! [X2: A,A2: A,R4: A] :
          ( ( ord_less @ A @ ( abs_abs @ A @ ( minus_minus @ A @ X2 @ A2 ) ) @ R4 )
          = ( ( ord_less @ A @ ( minus_minus @ A @ A2 @ R4 ) @ X2 )
            & ( ord_less @ A @ X2 @ ( plus_plus @ A @ A2 @ R4 ) ) ) ) ) ).

% abs_diff_less_iff
thf(fact_4513_lemma__interval__lt,axiom,
    ! [A2: real,X2: real,B2: real] :
      ( ( ord_less @ real @ A2 @ X2 )
     => ( ( ord_less @ real @ X2 @ B2 )
       => ? [D4: real] :
            ( ( ord_less @ real @ ( zero_zero @ real ) @ D4 )
            & ! [Y3: real] :
                ( ( ord_less @ real @ ( abs_abs @ real @ ( minus_minus @ real @ X2 @ Y3 ) ) @ D4 )
               => ( ( ord_less @ real @ A2 @ Y3 )
                  & ( ord_less @ real @ Y3 @ B2 ) ) ) ) ) ) ).

% lemma_interval_lt
thf(fact_4514_abs__cos__le__one,axiom,
    ! [X2: real] : ( ord_less_eq @ real @ ( abs_abs @ real @ ( cos @ real @ X2 ) ) @ ( one_one @ real ) ) ).

% abs_cos_le_one
thf(fact_4515_abs__sin__le__one,axiom,
    ! [X2: real] : ( ord_less_eq @ real @ ( abs_abs @ real @ ( sin @ real @ X2 ) ) @ ( one_one @ real ) ) ).

% abs_sin_le_one
thf(fact_4516_sin__bound__lemma,axiom,
    ! [X2: real,Y2: real,U: real,V: real] :
      ( ( X2 = Y2 )
     => ( ( ord_less_eq @ real @ ( abs_abs @ real @ U ) @ V )
       => ( ord_less_eq @ real @ ( abs_abs @ real @ ( minus_minus @ real @ ( plus_plus @ real @ X2 @ U ) @ Y2 ) ) @ V ) ) ) ).

% sin_bound_lemma
thf(fact_4517_suminf__finite,axiom,
    ! [A: $tType] :
      ( ( ( comm_monoid_add @ A )
        & ( topological_t2_space @ A ) )
     => ! [N3: set @ nat,F: nat > A] :
          ( ( finite_finite2 @ nat @ N3 )
         => ( ! [N4: nat] :
                ( ~ ( member @ nat @ N4 @ N3 )
               => ( ( F @ N4 )
                  = ( zero_zero @ A ) ) )
           => ( ( suminf @ A @ F )
              = ( groups7311177749621191930dd_sum @ nat @ A @ F @ N3 ) ) ) ) ) ).

% suminf_finite
thf(fact_4518_abs__add__one__gt__zero,axiom,
    ! [A: $tType] :
      ( ( linordered_idom @ A )
     => ! [X2: A] : ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ ( one_one @ A ) @ ( abs_abs @ A @ X2 ) ) ) ) ).

% abs_add_one_gt_zero
thf(fact_4519_of__int__leD,axiom,
    ! [A: $tType] :
      ( ( linordered_idom @ A )
     => ! [N2: int,X2: A] :
          ( ( ord_less_eq @ A @ ( abs_abs @ A @ ( ring_1_of_int @ A @ N2 ) ) @ X2 )
         => ( ( N2
              = ( zero_zero @ int ) )
            | ( ord_less_eq @ A @ ( one_one @ A ) @ X2 ) ) ) ) ).

% of_int_leD
thf(fact_4520_of__int__lessD,axiom,
    ! [A: $tType] :
      ( ( linordered_idom @ A )
     => ! [N2: int,X2: A] :
          ( ( ord_less @ A @ ( abs_abs @ A @ ( ring_1_of_int @ A @ N2 ) ) @ X2 )
         => ( ( N2
              = ( zero_zero @ int ) )
            | ( ord_less @ A @ ( one_one @ A ) @ X2 ) ) ) ) ).

% of_int_lessD
thf(fact_4521_lemma__interval,axiom,
    ! [A2: real,X2: real,B2: real] :
      ( ( ord_less @ real @ A2 @ X2 )
     => ( ( ord_less @ real @ X2 @ B2 )
       => ? [D4: real] :
            ( ( ord_less @ real @ ( zero_zero @ real ) @ D4 )
            & ! [Y3: real] :
                ( ( ord_less @ real @ ( abs_abs @ real @ ( minus_minus @ real @ X2 @ Y3 ) ) @ D4 )
               => ( ( ord_less_eq @ real @ A2 @ Y3 )
                  & ( ord_less_eq @ real @ Y3 @ B2 ) ) ) ) ) ) ).

% lemma_interval
thf(fact_4522_round__diff__minimal,axiom,
    ! [A: $tType] :
      ( ( archim2362893244070406136eiling @ A )
     => ! [Z: A,M: int] : ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ Z @ ( ring_1_of_int @ A @ ( archimedean_round @ A @ Z ) ) ) ) @ ( abs_abs @ A @ ( minus_minus @ A @ Z @ ( ring_1_of_int @ A @ M ) ) ) ) ) ).

% round_diff_minimal
thf(fact_4523_norm__triangle__ineq3,axiom,
    ! [A: $tType] :
      ( ( real_V822414075346904944vector @ A )
     => ! [A2: A,B2: A] : ( ord_less_eq @ real @ ( abs_abs @ real @ ( minus_minus @ real @ ( real_V7770717601297561774m_norm @ A @ A2 ) @ ( real_V7770717601297561774m_norm @ A @ B2 ) ) ) @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ A2 @ B2 ) ) ) ) ).

% norm_triangle_ineq3
thf(fact_4524_sin__zero__abs__cos__one,axiom,
    ! [X2: real] :
      ( ( ( sin @ real @ X2 )
        = ( zero_zero @ real ) )
     => ( ( abs_abs @ real @ ( cos @ real @ X2 ) )
        = ( one_one @ real ) ) ) ).

% sin_zero_abs_cos_one
thf(fact_4525_sin__def,axiom,
    ! [A: $tType] :
      ( ( ( real_Vector_banach @ A )
        & ( real_V2822296259951069270ebra_1 @ A ) )
     => ( ( sin @ A )
        = ( ^ [X: A] :
              ( suminf @ A
              @ ^ [N: nat] : ( real_V8093663219630862766scaleR @ A @ ( sin_coeff @ N ) @ ( power_power @ A @ X @ N ) ) ) ) ) ) ).

% sin_def
thf(fact_4526_cos__def,axiom,
    ! [A: $tType] :
      ( ( ( real_Vector_banach @ A )
        & ( real_V2822296259951069270ebra_1 @ A ) )
     => ( ( cos @ A )
        = ( ^ [X: A] :
              ( suminf @ A
              @ ^ [N: nat] : ( real_V8093663219630862766scaleR @ A @ ( cos_coeff @ N ) @ ( power_power @ A @ X @ N ) ) ) ) ) ) ).

% cos_def
thf(fact_4527_abs__le__square__iff,axiom,
    ! [A: $tType] :
      ( ( linordered_idom @ A )
     => ! [X2: A,Y2: A] :
          ( ( ord_less_eq @ A @ ( abs_abs @ A @ X2 ) @ ( abs_abs @ A @ Y2 ) )
          = ( ord_less_eq @ A @ ( power_power @ A @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).

% abs_le_square_iff
thf(fact_4528_abs__square__eq__1,axiom,
    ! [A: $tType] :
      ( ( linordered_idom @ A )
     => ! [X2: A] :
          ( ( ( power_power @ A @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
            = ( one_one @ A ) )
          = ( ( abs_abs @ A @ X2 )
            = ( one_one @ A ) ) ) ) ).

% abs_square_eq_1
thf(fact_4529_power__even__abs,axiom,
    ! [A: $tType] :
      ( ( linordered_idom @ A )
     => ! [N2: nat,A2: A] :
          ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 )
         => ( ( power_power @ A @ ( abs_abs @ A @ A2 ) @ N2 )
            = ( power_power @ A @ A2 @ N2 ) ) ) ) ).

% power_even_abs
thf(fact_4530_abs__sqrt__wlog,axiom,
    ! [A: $tType] :
      ( ( linordered_idom @ A )
     => ! [P: A > A > $o,X2: A] :
          ( ! [X3: A] :
              ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X3 )
             => ( P @ X3 @ ( power_power @ A @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
         => ( P @ ( abs_abs @ A @ X2 ) @ ( power_power @ A @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).

% abs_sqrt_wlog
thf(fact_4531_power2__le__iff__abs__le,axiom,
    ! [A: $tType] :
      ( ( linordered_idom @ A )
     => ! [Y2: A,X2: A] :
          ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y2 )
         => ( ( ord_less_eq @ A @ ( power_power @ A @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
            = ( ord_less_eq @ A @ ( abs_abs @ A @ X2 ) @ Y2 ) ) ) ) ).

% power2_le_iff_abs_le
thf(fact_4532_abs__square__le__1,axiom,
    ! [A: $tType] :
      ( ( linordered_idom @ A )
     => ! [X2: A] :
          ( ( ord_less_eq @ A @ ( power_power @ A @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ A ) )
          = ( ord_less_eq @ A @ ( abs_abs @ A @ X2 ) @ ( one_one @ A ) ) ) ) ).

% abs_square_le_1
thf(fact_4533_abs__square__less__1,axiom,
    ! [A: $tType] :
      ( ( linordered_idom @ A )
     => ! [X2: A] :
          ( ( ord_less @ A @ ( power_power @ A @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ A ) )
          = ( ord_less @ A @ ( abs_abs @ A @ X2 ) @ ( one_one @ A ) ) ) ) ).

% abs_square_less_1
thf(fact_4534_power__mono__even,axiom,
    ! [A: $tType] :
      ( ( linordered_idom @ A )
     => ! [N2: nat,A2: A,B2: A] :
          ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 )
         => ( ( ord_less_eq @ A @ ( abs_abs @ A @ A2 ) @ ( abs_abs @ A @ B2 ) )
           => ( ord_less_eq @ A @ ( power_power @ A @ A2 @ N2 ) @ ( power_power @ A @ B2 @ N2 ) ) ) ) ) ).

% power_mono_even
thf(fact_4535_convex__sum__bound__le,axiom,
    ! [A: $tType,B: $tType] :
      ( ( linordered_idom @ B )
     => ! [I5: set @ A,X2: A > B,A2: A > B,B2: B,Delta: B] :
          ( ! [I3: A] :
              ( ( member @ A @ I3 @ I5 )
             => ( ord_less_eq @ B @ ( zero_zero @ B ) @ ( X2 @ I3 ) ) )
         => ( ( ( groups7311177749621191930dd_sum @ A @ B @ X2 @ I5 )
              = ( one_one @ B ) )
           => ( ! [I3: A] :
                  ( ( member @ A @ I3 @ I5 )
                 => ( ord_less_eq @ B @ ( abs_abs @ B @ ( minus_minus @ B @ ( A2 @ I3 ) @ B2 ) ) @ Delta ) )
             => ( ord_less_eq @ B
                @ ( abs_abs @ B
                  @ ( minus_minus @ B
                    @ ( groups7311177749621191930dd_sum @ A @ B
                      @ ^ [I2: A] : ( times_times @ B @ ( A2 @ I2 ) @ ( X2 @ I2 ) )
                      @ I5 )
                    @ B2 ) )
                @ Delta ) ) ) ) ) ).

% convex_sum_bound_le
thf(fact_4536_sin__cos__le1,axiom,
    ! [X2: real,Y2: real] : ( ord_less_eq @ real @ ( abs_abs @ real @ ( plus_plus @ real @ ( times_times @ real @ ( sin @ real @ X2 ) @ ( sin @ real @ Y2 ) ) @ ( times_times @ real @ ( cos @ real @ X2 ) @ ( cos @ real @ Y2 ) ) ) ) @ ( one_one @ real ) ) ).

% sin_cos_le1
thf(fact_4537_suminf__geometric,axiom,
    ! [A: $tType] :
      ( ( real_V3459762299906320749_field @ A )
     => ! [C2: A] :
          ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ C2 ) @ ( one_one @ real ) )
         => ( ( suminf @ A @ ( power_power @ A @ C2 ) )
            = ( divide_divide @ A @ ( one_one @ A ) @ ( minus_minus @ A @ ( one_one @ A ) @ C2 ) ) ) ) ) ).

% suminf_geometric
thf(fact_4538_of__int__round__abs__le,axiom,
    ! [A: $tType] :
      ( ( archim2362893244070406136eiling @ A )
     => ! [X2: A] : ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ ( ring_1_of_int @ A @ ( archimedean_round @ A @ X2 ) ) @ X2 ) ) @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).

% of_int_round_abs_le
thf(fact_4539_round__unique_H,axiom,
    ! [A: $tType] :
      ( ( archim2362893244070406136eiling @ A )
     => ! [X2: A,N2: int] :
          ( ( ord_less @ A @ ( abs_abs @ A @ ( minus_minus @ A @ X2 @ ( ring_1_of_int @ A @ N2 ) ) ) @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) )
         => ( ( archimedean_round @ A @ X2 )
            = N2 ) ) ) ).

% round_unique'
thf(fact_4540_abs__ln__one__plus__x__minus__x__bound__nonneg,axiom,
    ! [X2: real] :
      ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 )
     => ( ( ord_less_eq @ real @ X2 @ ( one_one @ real ) )
       => ( ord_less_eq @ real @ ( abs_abs @ real @ ( minus_minus @ real @ ( ln_ln @ real @ ( plus_plus @ real @ ( one_one @ real ) @ X2 ) ) @ X2 ) ) @ ( power_power @ real @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).

% abs_ln_one_plus_x_minus_x_bound_nonneg
thf(fact_4541_abs__ln__one__plus__x__minus__x__bound,axiom,
    ! [X2: real] :
      ( ( ord_less_eq @ real @ ( abs_abs @ real @ X2 ) @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
     => ( ord_less_eq @ real @ ( abs_abs @ real @ ( minus_minus @ real @ ( ln_ln @ real @ ( plus_plus @ real @ ( one_one @ real ) @ X2 ) ) @ X2 ) ) @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( power_power @ real @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).

% abs_ln_one_plus_x_minus_x_bound
thf(fact_4542_pi__series,axiom,
    ( ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ ( bit0 @ one2 ) ) ) )
    = ( suminf @ real
      @ ^ [K4: nat] : ( divide_divide @ real @ ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ K4 ) @ ( one_one @ real ) ) @ ( semiring_1_of_nat @ real @ ( plus_plus @ nat @ ( times_times @ nat @ K4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ nat ) ) ) ) ) ) ).

% pi_series
thf(fact_4543_monoseq__arctan__series,axiom,
    ! [X2: real] :
      ( ( ord_less_eq @ real @ ( abs_abs @ real @ X2 ) @ ( one_one @ real ) )
     => ( topological_monoseq @ real
        @ ^ [N: nat] : ( times_times @ real @ ( divide_divide @ real @ ( one_one @ real ) @ ( semiring_1_of_nat @ real @ ( plus_plus @ nat @ ( times_times @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ nat ) ) ) ) @ ( power_power @ real @ X2 @ ( plus_plus @ nat @ ( times_times @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ nat ) ) ) ) ) ) ).

% monoseq_arctan_series
thf(fact_4544_arctan__series,axiom,
    ! [X2: real] :
      ( ( ord_less_eq @ real @ ( abs_abs @ real @ X2 ) @ ( one_one @ real ) )
     => ( ( arctan @ X2 )
        = ( suminf @ real
          @ ^ [K4: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ K4 ) @ ( times_times @ real @ ( divide_divide @ real @ ( one_one @ real ) @ ( semiring_1_of_nat @ real @ ( plus_plus @ nat @ ( times_times @ nat @ K4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ nat ) ) ) ) @ ( power_power @ real @ X2 @ ( plus_plus @ nat @ ( times_times @ nat @ K4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ nat ) ) ) ) ) ) ) ) ).

% arctan_series
thf(fact_4545_summable__arctan__series,axiom,
    ! [X2: real] :
      ( ( ord_less_eq @ real @ ( abs_abs @ real @ X2 ) @ ( one_one @ real ) )
     => ( summable @ real
        @ ^ [K4: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ K4 ) @ ( times_times @ real @ ( divide_divide @ real @ ( one_one @ real ) @ ( semiring_1_of_nat @ real @ ( plus_plus @ nat @ ( times_times @ nat @ K4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ nat ) ) ) ) @ ( power_power @ real @ X2 @ ( plus_plus @ nat @ ( times_times @ nat @ K4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ nat ) ) ) ) ) ) ) ).

% summable_arctan_series
thf(fact_4546_tanh__ln__real,axiom,
    ! [X2: real] :
      ( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
     => ( ( tanh @ real @ ( ln_ln @ real @ X2 ) )
        = ( divide_divide @ real @ ( minus_minus @ real @ ( power_power @ real @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ real ) ) @ ( plus_plus @ real @ ( power_power @ real @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ real ) ) ) ) ) ).

% tanh_ln_real
thf(fact_4547_zdvd1__eq,axiom,
    ! [X2: int] :
      ( ( dvd_dvd @ int @ X2 @ ( one_one @ int ) )
      = ( ( abs_abs @ int @ X2 )
        = ( one_one @ int ) ) ) ).

% zdvd1_eq
thf(fact_4548_tanh__real__le__iff,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_eq @ real @ ( tanh @ real @ X2 ) @ ( tanh @ real @ Y2 ) )
      = ( ord_less_eq @ real @ X2 @ Y2 ) ) ).

% tanh_real_le_iff
thf(fact_4549_summable__zero,axiom,
    ! [A: $tType] :
      ( ( ( comm_monoid_add @ A )
        & ( topolo4958980785337419405_space @ A ) )
     => ( summable @ A
        @ ^ [N: nat] : ( zero_zero @ A ) ) ) ).

% summable_zero
thf(fact_4550_summable__single,axiom,
    ! [A: $tType] :
      ( ( ( comm_monoid_add @ A )
        & ( topolo4958980785337419405_space @ A ) )
     => ! [I: nat,F: nat > A] :
          ( summable @ A
          @ ^ [R: nat] : ( if @ A @ ( R = I ) @ ( F @ R ) @ ( zero_zero @ A ) ) ) ) ).

% summable_single
thf(fact_4551_summable__iff__shift,axiom,
    ! [A: $tType] :
      ( ( real_V822414075346904944vector @ A )
     => ! [F: nat > A,K: nat] :
          ( ( summable @ A
            @ ^ [N: nat] : ( F @ ( plus_plus @ nat @ N @ K ) ) )
          = ( summable @ A @ F ) ) ) ).

% summable_iff_shift
thf(fact_4552_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_4553_zero__le__arctan__iff,axiom,
    ! [X2: real] :
      ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( arctan @ X2 ) )
      = ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 ) ) ).

% zero_le_arctan_iff
thf(fact_4554_arctan__le__zero__iff,axiom,
    ! [X2: real] :
      ( ( ord_less_eq @ real @ ( arctan @ X2 ) @ ( zero_zero @ real ) )
      = ( ord_less_eq @ real @ X2 @ ( zero_zero @ real ) ) ) ).

% arctan_le_zero_iff
thf(fact_4555_tanh__real__nonpos__iff,axiom,
    ! [X2: real] :
      ( ( ord_less_eq @ real @ ( tanh @ real @ X2 ) @ ( zero_zero @ real ) )
      = ( ord_less_eq @ real @ X2 @ ( zero_zero @ real ) ) ) ).

% tanh_real_nonpos_iff
thf(fact_4556_tanh__real__nonneg__iff,axiom,
    ! [X2: real] :
      ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( tanh @ real @ X2 ) )
      = ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 ) ) ).

% tanh_real_nonneg_iff
thf(fact_4557_summable__cmult__iff,axiom,
    ! [A: $tType] :
      ( ( real_V3459762299906320749_field @ A )
     => ! [C2: A,F: nat > A] :
          ( ( summable @ A
            @ ^ [N: nat] : ( times_times @ A @ C2 @ ( F @ N ) ) )
          = ( ( C2
              = ( zero_zero @ A ) )
            | ( summable @ A @ F ) ) ) ) ).

% summable_cmult_iff
thf(fact_4558_summable__divide__iff,axiom,
    ! [A: $tType] :
      ( ( real_V3459762299906320749_field @ A )
     => ! [F: nat > A,C2: A] :
          ( ( summable @ A
            @ ^ [N: nat] : ( divide_divide @ A @ ( F @ N ) @ C2 ) )
          = ( ( C2
              = ( zero_zero @ A ) )
            | ( summable @ A @ F ) ) ) ) ).

% summable_divide_iff
thf(fact_4559_summable__If__finite__set,axiom,
    ! [A: $tType] :
      ( ( ( comm_monoid_add @ A )
        & ( topolo4958980785337419405_space @ A ) )
     => ! [A3: set @ nat,F: nat > A] :
          ( ( finite_finite2 @ nat @ A3 )
         => ( summable @ A
            @ ^ [R: nat] : ( if @ A @ ( member @ nat @ R @ A3 ) @ ( F @ R ) @ ( zero_zero @ A ) ) ) ) ) ).

% summable_If_finite_set
thf(fact_4560_summable__If__finite,axiom,
    ! [A: $tType] :
      ( ( ( comm_monoid_add @ A )
        & ( topolo4958980785337419405_space @ A ) )
     => ! [P: nat > $o,F: nat > A] :
          ( ( finite_finite2 @ nat @ ( collect @ nat @ P ) )
         => ( summable @ A
            @ ^ [R: nat] : ( if @ A @ ( P @ R ) @ ( F @ R ) @ ( zero_zero @ A ) ) ) ) ) ).

% summable_If_finite
thf(fact_4561_summable__geometric__iff,axiom,
    ! [A: $tType] :
      ( ( real_V3459762299906320749_field @ A )
     => ! [C2: A] :
          ( ( summable @ A @ ( power_power @ A @ C2 ) )
          = ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ C2 ) @ ( one_one @ real ) ) ) ) ).

% summable_geometric_iff
thf(fact_4562_summable__rabs__cancel,axiom,
    ! [F: nat > real] :
      ( ( summable @ real
        @ ^ [N: nat] : ( abs_abs @ real @ ( F @ N ) ) )
     => ( summable @ real @ F ) ) ).

% summable_rabs_cancel
thf(fact_4563_summable__minus__iff,axiom,
    ! [A: $tType] :
      ( ( real_V822414075346904944vector @ A )
     => ! [F: nat > A] :
          ( ( summable @ A
            @ ^ [N: nat] : ( uminus_uminus @ A @ ( F @ N ) ) )
          = ( summable @ A @ F ) ) ) ).

% summable_minus_iff
thf(fact_4564_summable__minus,axiom,
    ! [A: $tType] :
      ( ( real_V822414075346904944vector @ A )
     => ! [F: nat > A] :
          ( ( summable @ A @ F )
         => ( summable @ A
            @ ^ [N: nat] : ( uminus_uminus @ A @ ( F @ N ) ) ) ) ) ).

% summable_minus
thf(fact_4565_summable__comparison__test,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach @ A )
     => ! [F: nat > A,G: nat > real] :
          ( ? [N10: nat] :
            ! [N4: nat] :
              ( ( ord_less_eq @ nat @ N10 @ N4 )
             => ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( F @ N4 ) ) @ ( G @ N4 ) ) )
         => ( ( summable @ real @ G )
           => ( summable @ A @ F ) ) ) ) ).

% summable_comparison_test
thf(fact_4566_summable__comparison__test_H,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach @ A )
     => ! [G: nat > real,N3: nat,F: nat > A] :
          ( ( summable @ real @ G )
         => ( ! [N4: nat] :
                ( ( ord_less_eq @ nat @ N3 @ N4 )
               => ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( F @ N4 ) ) @ ( G @ N4 ) ) )
           => ( summable @ A @ F ) ) ) ) ).

% summable_comparison_test'
thf(fact_4567_summable__const__iff,axiom,
    ! [A: $tType] :
      ( ( real_V822414075346904944vector @ A )
     => ! [C2: A] :
          ( ( summable @ A
            @ ^ [Uu3: nat] : C2 )
          = ( C2
            = ( zero_zero @ A ) ) ) ) ).

% summable_const_iff
thf(fact_4568_summable__add,axiom,
    ! [A: $tType] :
      ( ( ( topolo5987344860129210374id_add @ A )
        & ( topological_t2_space @ A ) )
     => ! [F: nat > A,G: nat > A] :
          ( ( summable @ A @ F )
         => ( ( summable @ A @ G )
           => ( summable @ A
              @ ^ [N: nat] : ( plus_plus @ A @ ( F @ N ) @ ( G @ N ) ) ) ) ) ) ).

% summable_add
thf(fact_4569_summable__mult2,axiom,
    ! [A: $tType] :
      ( ( real_V4412858255891104859lgebra @ A )
     => ! [F: nat > A,C2: A] :
          ( ( summable @ A @ F )
         => ( summable @ A
            @ ^ [N: nat] : ( times_times @ A @ ( F @ N ) @ C2 ) ) ) ) ).

% summable_mult2
thf(fact_4570_summable__mult,axiom,
    ! [A: $tType] :
      ( ( real_V4412858255891104859lgebra @ A )
     => ! [F: nat > A,C2: A] :
          ( ( summable @ A @ F )
         => ( summable @ A
            @ ^ [N: nat] : ( times_times @ A @ C2 @ ( F @ N ) ) ) ) ) ).

% summable_mult
thf(fact_4571_summable__diff,axiom,
    ! [A: $tType] :
      ( ( real_V822414075346904944vector @ A )
     => ! [F: nat > A,G: nat > A] :
          ( ( summable @ A @ F )
         => ( ( summable @ A @ G )
           => ( summable @ A
              @ ^ [N: nat] : ( minus_minus @ A @ ( F @ N ) @ ( G @ N ) ) ) ) ) ) ).

% summable_diff
thf(fact_4572_summable__Suc__iff,axiom,
    ! [A: $tType] :
      ( ( real_V822414075346904944vector @ A )
     => ! [F: nat > A] :
          ( ( summable @ A
            @ ^ [N: nat] : ( F @ ( suc @ N ) ) )
          = ( summable @ A @ F ) ) ) ).

% summable_Suc_iff
thf(fact_4573_summable__norm__cancel,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach @ A )
     => ! [F: nat > A] :
          ( ( summable @ real
            @ ^ [N: nat] : ( real_V7770717601297561774m_norm @ A @ ( F @ N ) ) )
         => ( summable @ A @ F ) ) ) ).

% summable_norm_cancel
thf(fact_4574_arctan__le__iff,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_eq @ real @ ( arctan @ X2 ) @ ( arctan @ Y2 ) )
      = ( ord_less_eq @ real @ X2 @ Y2 ) ) ).

% arctan_le_iff
thf(fact_4575_arctan__monotone_H,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_eq @ real @ X2 @ Y2 )
     => ( ord_less_eq @ real @ ( arctan @ X2 ) @ ( arctan @ Y2 ) ) ) ).

% arctan_monotone'
thf(fact_4576_summable__ignore__initial__segment,axiom,
    ! [A: $tType] :
      ( ( real_V822414075346904944vector @ A )
     => ! [F: nat > A,K: nat] :
          ( ( summable @ A @ F )
         => ( summable @ A
            @ ^ [N: nat] : ( F @ ( plus_plus @ nat @ N @ K ) ) ) ) ) ).

% summable_ignore_initial_segment
thf(fact_4577_summable__divide,axiom,
    ! [A: $tType] :
      ( ( real_V3459762299906320749_field @ A )
     => ! [F: nat > A,C2: A] :
          ( ( summable @ A @ F )
         => ( summable @ A
            @ ^ [N: nat] : ( divide_divide @ A @ ( F @ N ) @ C2 ) ) ) ) ).

% summable_divide
thf(fact_4578_summable__sum,axiom,
    ! [I6: $tType,A: $tType] :
      ( ( ( topolo5987344860129210374id_add @ A )
        & ( topological_t2_space @ A ) )
     => ! [I5: set @ I6,F: I6 > nat > A] :
          ( ! [I3: I6] :
              ( ( member @ I6 @ I3 @ I5 )
             => ( summable @ A @ ( F @ I3 ) ) )
         => ( summable @ A
            @ ^ [N: nat] :
                ( groups7311177749621191930dd_sum @ I6 @ A
                @ ^ [I2: I6] : ( F @ I2 @ N )
                @ I5 ) ) ) ) ).

% summable_sum
thf(fact_4579_summable__scaleR__right,axiom,
    ! [B: $tType] :
      ( ( real_V822414075346904944vector @ B )
     => ! [X8: nat > B,R4: real] :
          ( ( summable @ B @ X8 )
         => ( summable @ B
            @ ^ [N: nat] : ( real_V8093663219630862766scaleR @ B @ R4 @ ( X8 @ N ) ) ) ) ) ).

% summable_scaleR_right
thf(fact_4580_summable__scaleR__left,axiom,
    ! [B: $tType] :
      ( ( real_V822414075346904944vector @ B )
     => ! [X8: nat > real,X2: B] :
          ( ( summable @ real @ X8 )
         => ( summable @ B
            @ ^ [N: nat] : ( real_V8093663219630862766scaleR @ B @ ( X8 @ N ) @ X2 ) ) ) ) ).

% summable_scaleR_left
thf(fact_4581_powser__insidea,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra @ A )
     => ! [F: nat > A,X2: A,Z: A] :
          ( ( summable @ A
            @ ^ [N: nat] : ( times_times @ A @ ( F @ N ) @ ( power_power @ A @ X2 @ N ) ) )
         => ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ Z ) @ ( real_V7770717601297561774m_norm @ A @ X2 ) )
           => ( summable @ real
              @ ^ [N: nat] : ( real_V7770717601297561774m_norm @ A @ ( times_times @ A @ ( F @ N ) @ ( power_power @ A @ Z @ N ) ) ) ) ) ) ) ).

% powser_insidea
thf(fact_4582_suminf__le,axiom,
    ! [A: $tType] :
      ( ( ( ordere6911136660526730532id_add @ A )
        & ( topolo1944317154257567458pology @ A ) )
     => ! [F: nat > A,G: nat > A] :
          ( ! [N4: nat] : ( ord_less_eq @ A @ ( F @ N4 ) @ ( G @ N4 ) )
         => ( ( summable @ A @ F )
           => ( ( summable @ A @ G )
             => ( ord_less_eq @ A @ ( suminf @ A @ F ) @ ( suminf @ A @ G ) ) ) ) ) ) ).

% suminf_le
thf(fact_4583_abs__zmult__eq__1,axiom,
    ! [M: int,N2: int] :
      ( ( ( abs_abs @ int @ ( times_times @ int @ M @ N2 ) )
        = ( one_one @ int ) )
     => ( ( abs_abs @ int @ M )
        = ( one_one @ int ) ) ) ).

% abs_zmult_eq_1
thf(fact_4584_summable__mult__D,axiom,
    ! [A: $tType] :
      ( ( real_V3459762299906320749_field @ A )
     => ! [C2: A,F: nat > A] :
          ( ( summable @ A
            @ ^ [N: nat] : ( times_times @ A @ C2 @ ( F @ N ) ) )
         => ( ( C2
             != ( zero_zero @ A ) )
           => ( summable @ A @ F ) ) ) ) ).

% summable_mult_D
thf(fact_4585_summable__zero__power,axiom,
    ! [A: $tType] :
      ( ( ( comm_ring_1 @ A )
        & ( topolo4958980785337419405_space @ A ) )
     => ( summable @ A @ ( power_power @ A @ ( zero_zero @ A ) ) ) ) ).

% summable_zero_power
thf(fact_4586_abs__div,axiom,
    ! [Y2: int,X2: int] :
      ( ( dvd_dvd @ int @ Y2 @ X2 )
     => ( ( abs_abs @ int @ ( divide_divide @ int @ X2 @ Y2 ) )
        = ( divide_divide @ int @ ( abs_abs @ int @ X2 ) @ ( abs_abs @ int @ Y2 ) ) ) ) ).

% abs_div
thf(fact_4587_suminf__add,axiom,
    ! [A: $tType] :
      ( ( ( topolo5987344860129210374id_add @ A )
        & ( topological_t2_space @ A ) )
     => ! [F: nat > A,G: nat > A] :
          ( ( summable @ A @ F )
         => ( ( summable @ A @ G )
           => ( ( plus_plus @ A @ ( suminf @ A @ F ) @ ( suminf @ A @ G ) )
              = ( suminf @ A
                @ ^ [N: nat] : ( plus_plus @ A @ ( F @ N ) @ ( G @ N ) ) ) ) ) ) ) ).

% suminf_add
thf(fact_4588_suminf__mult,axiom,
    ! [A: $tType] :
      ( ( real_V4412858255891104859lgebra @ A )
     => ! [F: nat > A,C2: A] :
          ( ( summable @ A @ F )
         => ( ( suminf @ A
              @ ^ [N: nat] : ( times_times @ A @ C2 @ ( F @ N ) ) )
            = ( times_times @ A @ C2 @ ( suminf @ A @ F ) ) ) ) ) ).

% suminf_mult
thf(fact_4589_suminf__mult2,axiom,
    ! [A: $tType] :
      ( ( real_V4412858255891104859lgebra @ A )
     => ! [F: nat > A,C2: A] :
          ( ( summable @ A @ F )
         => ( ( times_times @ A @ ( suminf @ A @ F ) @ C2 )
            = ( suminf @ A
              @ ^ [N: nat] : ( times_times @ A @ ( F @ N ) @ C2 ) ) ) ) ) ).

% suminf_mult2
thf(fact_4590_suminf__diff,axiom,
    ! [A: $tType] :
      ( ( real_V822414075346904944vector @ A )
     => ! [F: nat > A,G: nat > A] :
          ( ( summable @ A @ F )
         => ( ( summable @ A @ G )
           => ( ( minus_minus @ A @ ( suminf @ A @ F ) @ ( suminf @ A @ G ) )
              = ( suminf @ A
                @ ^ [N: nat] : ( minus_minus @ A @ ( F @ N ) @ ( G @ N ) ) ) ) ) ) ) ).

% suminf_diff
thf(fact_4591_suminf__divide,axiom,
    ! [A: $tType] :
      ( ( real_V3459762299906320749_field @ A )
     => ! [F: nat > A,C2: A] :
          ( ( summable @ A @ F )
         => ( ( suminf @ A
              @ ^ [N: nat] : ( divide_divide @ A @ ( F @ N ) @ C2 ) )
            = ( divide_divide @ A @ ( suminf @ A @ F ) @ C2 ) ) ) ) ).

% suminf_divide
thf(fact_4592_suminf__minus,axiom,
    ! [A: $tType] :
      ( ( real_V822414075346904944vector @ A )
     => ! [F: nat > A] :
          ( ( summable @ A @ F )
         => ( ( suminf @ A
              @ ^ [N: nat] : ( uminus_uminus @ A @ ( F @ N ) ) )
            = ( uminus_uminus @ A @ ( suminf @ A @ F ) ) ) ) ) ).

% suminf_minus
thf(fact_4593_suminf__sum,axiom,
    ! [A: $tType,I6: $tType] :
      ( ( ( topolo5987344860129210374id_add @ A )
        & ( topological_t2_space @ A ) )
     => ! [I5: set @ I6,F: I6 > nat > A] :
          ( ! [I3: I6] :
              ( ( member @ I6 @ I3 @ I5 )
             => ( summable @ A @ ( F @ I3 ) ) )
         => ( ( suminf @ A
              @ ^ [N: nat] :
                  ( groups7311177749621191930dd_sum @ I6 @ A
                  @ ^ [I2: I6] : ( F @ I2 @ N )
                  @ I5 ) )
            = ( groups7311177749621191930dd_sum @ I6 @ A
              @ ^ [I2: I6] : ( suminf @ A @ ( F @ I2 ) )
              @ I5 ) ) ) ) ).

% suminf_sum
thf(fact_4594_suminf__scaleR__right,axiom,
    ! [B: $tType] :
      ( ( real_V822414075346904944vector @ B )
     => ! [X8: nat > B,R4: real] :
          ( ( summable @ B @ X8 )
         => ( ( real_V8093663219630862766scaleR @ B @ R4 @ ( suminf @ B @ X8 ) )
            = ( suminf @ B
              @ ^ [N: nat] : ( real_V8093663219630862766scaleR @ B @ R4 @ ( X8 @ N ) ) ) ) ) ) ).

% suminf_scaleR_right
thf(fact_4595_summable__Cauchy__product,axiom,
    ! [A: $tType] :
      ( ( ( real_Vector_banach @ A )
        & ( real_V4412858255891104859lgebra @ A ) )
     => ! [A2: nat > A,B2: nat > A] :
          ( ( summable @ real
            @ ^ [K4: nat] : ( real_V7770717601297561774m_norm @ A @ ( A2 @ K4 ) ) )
         => ( ( summable @ real
              @ ^ [K4: nat] : ( real_V7770717601297561774m_norm @ A @ ( B2 @ K4 ) ) )
           => ( summable @ A
              @ ^ [K4: nat] :
                  ( groups7311177749621191930dd_sum @ nat @ A
                  @ ^ [I2: nat] : ( times_times @ A @ ( A2 @ I2 ) @ ( B2 @ ( minus_minus @ nat @ K4 @ I2 ) ) )
                  @ ( set_ord_atMost @ nat @ K4 ) ) ) ) ) ) ).

% summable_Cauchy_product
thf(fact_4596_infinite__int__iff__unbounded__le,axiom,
    ! [S3: set @ int] :
      ( ( ~ ( finite_finite2 @ int @ S3 ) )
      = ( ! [M7: int] :
          ? [N: int] :
            ( ( ord_less_eq @ int @ M7 @ ( abs_abs @ int @ N ) )
            & ( member @ int @ N @ S3 ) ) ) ) ).

% infinite_int_iff_unbounded_le
thf(fact_4597_tanh__real__lt__1,axiom,
    ! [X2: real] : ( ord_less @ real @ ( tanh @ real @ X2 ) @ ( one_one @ real ) ) ).

% tanh_real_lt_1
thf(fact_4598_suminf__nonneg,axiom,
    ! [A: $tType] :
      ( ( ( ordere6911136660526730532id_add @ A )
        & ( topolo1944317154257567458pology @ A ) )
     => ! [F: nat > A] :
          ( ( summable @ A @ F )
         => ( ! [N4: nat] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F @ N4 ) )
           => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( suminf @ A @ F ) ) ) ) ) ).

% suminf_nonneg
thf(fact_4599_suminf__eq__zero__iff,axiom,
    ! [A: $tType] :
      ( ( ( ordere6911136660526730532id_add @ A )
        & ( topolo1944317154257567458pology @ A ) )
     => ! [F: nat > A] :
          ( ( summable @ A @ F )
         => ( ! [N4: nat] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F @ N4 ) )
           => ( ( ( suminf @ A @ F )
                = ( zero_zero @ A ) )
              = ( ! [N: nat] :
                    ( ( F @ N )
                    = ( zero_zero @ A ) ) ) ) ) ) ) ).

% suminf_eq_zero_iff
thf(fact_4600_suminf__pos,axiom,
    ! [A: $tType] :
      ( ( ( ordere6911136660526730532id_add @ A )
        & ( topolo1944317154257567458pology @ A ) )
     => ! [F: nat > A] :
          ( ( summable @ A @ F )
         => ( ! [N4: nat] : ( ord_less @ A @ ( zero_zero @ A ) @ ( F @ N4 ) )
           => ( ord_less @ A @ ( zero_zero @ A ) @ ( suminf @ A @ F ) ) ) ) ) ).

% suminf_pos
thf(fact_4601_summable__0__powser,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra @ A )
     => ! [F: nat > A] :
          ( summable @ A
          @ ^ [N: nat] : ( times_times @ A @ ( F @ N ) @ ( power_power @ A @ ( zero_zero @ A ) @ N ) ) ) ) ).

% summable_0_powser
thf(fact_4602_summable__zero__power_H,axiom,
    ! [A: $tType] :
      ( ( ( ring_1 @ A )
        & ( topolo4958980785337419405_space @ A ) )
     => ! [F: nat > A] :
          ( summable @ A
          @ ^ [N: nat] : ( times_times @ A @ ( F @ N ) @ ( power_power @ A @ ( zero_zero @ A ) @ N ) ) ) ) ).

% summable_zero_power'
thf(fact_4603_powser__split__head_I3_J,axiom,
    ! [A: $tType] :
      ( ( ( real_Vector_banach @ A )
        & ( real_V8999393235501362500lgebra @ A ) )
     => ! [F: nat > A,Z: A] :
          ( ( summable @ A
            @ ^ [N: nat] : ( times_times @ A @ ( F @ N ) @ ( power_power @ A @ Z @ N ) ) )
         => ( summable @ A
            @ ^ [N: nat] : ( times_times @ A @ ( F @ ( suc @ N ) ) @ ( power_power @ A @ Z @ N ) ) ) ) ) ).

% powser_split_head(3)
thf(fact_4604_summable__powser__split__head,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra @ A )
     => ! [F: nat > A,Z: A] :
          ( ( summable @ A
            @ ^ [N: nat] : ( times_times @ A @ ( F @ ( suc @ N ) ) @ ( power_power @ A @ Z @ N ) ) )
          = ( summable @ A
            @ ^ [N: nat] : ( times_times @ A @ ( F @ N ) @ ( power_power @ A @ Z @ N ) ) ) ) ) ).

% summable_powser_split_head
thf(fact_4605_summable__powser__ignore__initial__segment,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra @ A )
     => ! [F: nat > A,M: nat,Z: A] :
          ( ( summable @ A
            @ ^ [N: nat] : ( times_times @ A @ ( F @ ( plus_plus @ nat @ N @ M ) ) @ ( power_power @ A @ Z @ N ) ) )
          = ( summable @ A
            @ ^ [N: nat] : ( times_times @ A @ ( F @ N ) @ ( power_power @ A @ Z @ N ) ) ) ) ) ).

% summable_powser_ignore_initial_segment
thf(fact_4606_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_4607_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_4608_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_4609_summable__norm__comparison__test,axiom,
    ! [A: $tType] :
      ( ( real_V822414075346904944vector @ A )
     => ! [F: nat > A,G: nat > real] :
          ( ? [N10: nat] :
            ! [N4: nat] :
              ( ( ord_less_eq @ nat @ N10 @ N4 )
             => ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( F @ N4 ) ) @ ( G @ N4 ) ) )
         => ( ( summable @ real @ G )
           => ( summable @ real
              @ ^ [N: nat] : ( real_V7770717601297561774m_norm @ A @ ( F @ N ) ) ) ) ) ) ).

% summable_norm_comparison_test
thf(fact_4610_summable__rabs__comparison__test,axiom,
    ! [F: nat > real,G: nat > real] :
      ( ? [N10: nat] :
        ! [N4: nat] :
          ( ( ord_less_eq @ nat @ N10 @ N4 )
         => ( ord_less_eq @ real @ ( abs_abs @ real @ ( F @ N4 ) ) @ ( G @ N4 ) ) )
     => ( ( summable @ real @ G )
       => ( summable @ real
          @ ^ [N: nat] : ( abs_abs @ real @ ( F @ N ) ) ) ) ) ).

% summable_rabs_comparison_test
thf(fact_4611_summable__rabs,axiom,
    ! [F: nat > real] :
      ( ( summable @ real
        @ ^ [N: nat] : ( abs_abs @ real @ ( F @ N ) ) )
     => ( ord_less_eq @ real @ ( abs_abs @ real @ ( suminf @ real @ F ) )
        @ ( suminf @ real
          @ ^ [N: nat] : ( abs_abs @ real @ ( F @ N ) ) ) ) ) ).

% summable_rabs
thf(fact_4612_suminf__scaleR__left,axiom,
    ! [B: $tType] :
      ( ( real_V822414075346904944vector @ B )
     => ! [X8: nat > real,X2: B] :
          ( ( summable @ real @ X8 )
         => ( ( real_V8093663219630862766scaleR @ B @ ( suminf @ real @ X8 ) @ X2 )
            = ( suminf @ B
              @ ^ [N: nat] : ( real_V8093663219630862766scaleR @ B @ ( X8 @ N ) @ X2 ) ) ) ) ) ).

% suminf_scaleR_left
thf(fact_4613_suminf__pos__iff,axiom,
    ! [A: $tType] :
      ( ( ( ordere6911136660526730532id_add @ A )
        & ( topolo1944317154257567458pology @ A ) )
     => ! [F: nat > A] :
          ( ( summable @ A @ F )
         => ( ! [N4: nat] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F @ N4 ) )
           => ( ( ord_less @ A @ ( zero_zero @ A ) @ ( suminf @ A @ F ) )
              = ( ? [I2: nat] : ( ord_less @ A @ ( zero_zero @ A ) @ ( F @ I2 ) ) ) ) ) ) ) ).

% suminf_pos_iff
thf(fact_4614_suminf__pos2,axiom,
    ! [A: $tType] :
      ( ( ( ordere6911136660526730532id_add @ A )
        & ( topolo1944317154257567458pology @ A ) )
     => ! [F: nat > A,I: nat] :
          ( ( summable @ A @ F )
         => ( ! [N4: nat] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F @ N4 ) )
           => ( ( ord_less @ A @ ( zero_zero @ A ) @ ( F @ I ) )
             => ( ord_less @ A @ ( zero_zero @ A ) @ ( suminf @ A @ F ) ) ) ) ) ) ).

% suminf_pos2
thf(fact_4615_tanh__real__gt__neg1,axiom,
    ! [X2: real] : ( ord_less @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ ( tanh @ real @ X2 ) ) ).

% tanh_real_gt_neg1
thf(fact_4616_suminf__le__const,axiom,
    ! [A: $tType] :
      ( ( ( ordere6911136660526730532id_add @ A )
        & ( topolo1944317154257567458pology @ A ) )
     => ! [F: nat > A,X2: A] :
          ( ( summable @ A @ F )
         => ( ! [N4: nat] : ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F @ ( set_ord_lessThan @ nat @ N4 ) ) @ X2 )
           => ( ord_less_eq @ A @ ( suminf @ A @ F ) @ X2 ) ) ) ) ).

% suminf_le_const
thf(fact_4617_powser__inside,axiom,
    ! [A: $tType] :
      ( ( ( real_Vector_banach @ A )
        & ( real_V8999393235501362500lgebra @ A ) )
     => ! [F: nat > A,X2: A,Z: A] :
          ( ( summable @ A
            @ ^ [N: nat] : ( times_times @ A @ ( F @ N ) @ ( power_power @ A @ X2 @ N ) ) )
         => ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ Z ) @ ( real_V7770717601297561774m_norm @ A @ X2 ) )
           => ( summable @ A
              @ ^ [N: nat] : ( times_times @ A @ ( F @ N ) @ ( power_power @ A @ Z @ N ) ) ) ) ) ) ).

% powser_inside
thf(fact_4618_suminf__split__head,axiom,
    ! [A: $tType] :
      ( ( real_V822414075346904944vector @ A )
     => ! [F: nat > A] :
          ( ( summable @ A @ F )
         => ( ( suminf @ A
              @ ^ [N: nat] : ( F @ ( suc @ N ) ) )
            = ( minus_minus @ A @ ( suminf @ A @ F ) @ ( F @ ( zero_zero @ nat ) ) ) ) ) ) ).

% suminf_split_head
thf(fact_4619_summableI__nonneg__bounded,axiom,
    ! [A: $tType] :
      ( ( ( condit6923001295902523014norder @ A )
        & ( ordere6911136660526730532id_add @ A )
        & ( topolo1944317154257567458pology @ A ) )
     => ! [F: nat > A,X2: A] :
          ( ! [N4: nat] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F @ N4 ) )
         => ( ! [N4: nat] : ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F @ ( set_ord_lessThan @ nat @ N4 ) ) @ X2 )
           => ( summable @ A @ F ) ) ) ) ).

% summableI_nonneg_bounded
thf(fact_4620_bounded__imp__summable,axiom,
    ! [A: $tType] :
      ( ( ( condit6923001295902523014norder @ A )
        & ( linord2810124833399127020strict @ A )
        & ( topolo1944317154257567458pology @ A ) )
     => ! [A2: nat > A,B6: A] :
          ( ! [N4: nat] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( A2 @ N4 ) )
         => ( ! [N4: nat] : ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ A2 @ ( set_ord_atMost @ nat @ N4 ) ) @ B6 )
           => ( summable @ A @ A2 ) ) ) ) ).

% bounded_imp_summable
thf(fact_4621_complete__algebra__summable__geometric,axiom,
    ! [A: $tType] :
      ( ( ( real_Vector_banach @ A )
        & ( real_V2822296259951069270ebra_1 @ A ) )
     => ! [X2: A] :
          ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X2 ) @ ( one_one @ real ) )
         => ( summable @ A @ ( power_power @ A @ X2 ) ) ) ) ).

% complete_algebra_summable_geometric
thf(fact_4622_summable__geometric,axiom,
    ! [A: $tType] :
      ( ( real_V3459762299906320749_field @ A )
     => ! [C2: A] :
          ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ C2 ) @ ( one_one @ real ) )
         => ( summable @ A @ ( power_power @ A @ C2 ) ) ) ) ).

% summable_geometric
thf(fact_4623_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_4624_zdvd__mult__cancel1,axiom,
    ! [M: int,N2: int] :
      ( ( M
       != ( zero_zero @ int ) )
     => ( ( dvd_dvd @ int @ ( times_times @ int @ M @ N2 ) @ M )
        = ( ( abs_abs @ int @ N2 )
          = ( one_one @ int ) ) ) ) ).

% zdvd_mult_cancel1
thf(fact_4625_div__abs__eq__div__nat,axiom,
    ! [K: int,L: int] :
      ( ( divide_divide @ int @ ( abs_abs @ int @ K ) @ ( abs_abs @ int @ L ) )
      = ( semiring_1_of_nat @ int @ ( divide_divide @ nat @ ( nat2 @ ( abs_abs @ int @ K ) ) @ ( nat2 @ ( abs_abs @ int @ L ) ) ) ) ) ).

% div_abs_eq_div_nat
thf(fact_4626_summable__norm,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach @ A )
     => ! [F: nat > A] :
          ( ( summable @ real
            @ ^ [N: nat] : ( real_V7770717601297561774m_norm @ A @ ( F @ N ) ) )
         => ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( suminf @ A @ F ) )
            @ ( suminf @ real
              @ ^ [N: nat] : ( real_V7770717601297561774m_norm @ A @ ( F @ N ) ) ) ) ) ) ).

% summable_norm
thf(fact_4627_summable__norm__sin,axiom,
    ! [A: $tType] :
      ( ( ( real_Vector_banach @ A )
        & ( real_V2822296259951069270ebra_1 @ A ) )
     => ! [X2: A] :
          ( summable @ real
          @ ^ [N: nat] : ( real_V7770717601297561774m_norm @ A @ ( real_V8093663219630862766scaleR @ A @ ( sin_coeff @ N ) @ ( power_power @ A @ X2 @ N ) ) ) ) ) ).

% summable_norm_sin
thf(fact_4628_summable__norm__cos,axiom,
    ! [A: $tType] :
      ( ( ( real_Vector_banach @ A )
        & ( real_V2822296259951069270ebra_1 @ A ) )
     => ! [X2: A] :
          ( summable @ real
          @ ^ [N: nat] : ( real_V7770717601297561774m_norm @ A @ ( real_V8093663219630862766scaleR @ A @ ( cos_coeff @ N ) @ ( power_power @ A @ X2 @ N ) ) ) ) ) ).

% summable_norm_cos
thf(fact_4629_sum__le__suminf,axiom,
    ! [A: $tType] :
      ( ( ( ordere6911136660526730532id_add @ A )
        & ( topolo1944317154257567458pology @ A ) )
     => ! [F: nat > A,I5: set @ nat] :
          ( ( summable @ A @ F )
         => ( ( finite_finite2 @ nat @ I5 )
           => ( ! [N4: nat] :
                  ( ( member @ nat @ N4 @ ( uminus_uminus @ ( set @ nat ) @ I5 ) )
                 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F @ N4 ) ) )
             => ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F @ I5 ) @ ( suminf @ A @ F ) ) ) ) ) ) ).

% sum_le_suminf
thf(fact_4630_even__abs__add__iff,axiom,
    ! [K: int,L: int] :
      ( ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( plus_plus @ int @ ( abs_abs @ int @ K ) @ L ) )
      = ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( plus_plus @ int @ K @ L ) ) ) ).

% even_abs_add_iff
thf(fact_4631_even__add__abs__iff,axiom,
    ! [K: int,L: int] :
      ( ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( plus_plus @ int @ K @ ( abs_abs @ int @ L ) ) )
      = ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( plus_plus @ int @ K @ L ) ) ) ).

% even_add_abs_iff
thf(fact_4632_suminf__split__initial__segment,axiom,
    ! [A: $tType] :
      ( ( real_V822414075346904944vector @ A )
     => ! [F: nat > A,K: nat] :
          ( ( summable @ A @ F )
         => ( ( suminf @ A @ F )
            = ( plus_plus @ A
              @ ( suminf @ A
                @ ^ [N: nat] : ( F @ ( plus_plus @ nat @ N @ K ) ) )
              @ ( groups7311177749621191930dd_sum @ nat @ A @ F @ ( set_ord_lessThan @ nat @ K ) ) ) ) ) ) ).

% suminf_split_initial_segment
thf(fact_4633_suminf__minus__initial__segment,axiom,
    ! [A: $tType] :
      ( ( real_V822414075346904944vector @ A )
     => ! [F: nat > A,K: nat] :
          ( ( summable @ A @ F )
         => ( ( suminf @ A
              @ ^ [N: nat] : ( F @ ( plus_plus @ nat @ N @ K ) ) )
            = ( minus_minus @ A @ ( suminf @ A @ F ) @ ( groups7311177749621191930dd_sum @ nat @ A @ F @ ( set_ord_lessThan @ nat @ K ) ) ) ) ) ) ).

% suminf_minus_initial_segment
thf(fact_4634_nat__abs__int__diff,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ( ord_less_eq @ nat @ A2 @ B2 )
       => ( ( nat2 @ ( abs_abs @ int @ ( minus_minus @ int @ ( semiring_1_of_nat @ int @ A2 ) @ ( semiring_1_of_nat @ int @ B2 ) ) ) )
          = ( minus_minus @ nat @ B2 @ A2 ) ) )
      & ( ~ ( ord_less_eq @ nat @ A2 @ B2 )
       => ( ( nat2 @ ( abs_abs @ int @ ( minus_minus @ int @ ( semiring_1_of_nat @ int @ A2 ) @ ( semiring_1_of_nat @ int @ B2 ) ) ) )
          = ( minus_minus @ nat @ A2 @ B2 ) ) ) ) ).

% nat_abs_int_diff
thf(fact_4635_sum__less__suminf,axiom,
    ! [A: $tType] :
      ( ( ( ordere8940638589300402666id_add @ A )
        & ( topolo1944317154257567458pology @ A ) )
     => ! [F: nat > A,N2: nat] :
          ( ( summable @ A @ F )
         => ( ! [M2: nat] :
                ( ( ord_less_eq @ nat @ N2 @ M2 )
               => ( ord_less @ A @ ( zero_zero @ A ) @ ( F @ M2 ) ) )
           => ( ord_less @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F @ ( set_ord_lessThan @ nat @ N2 ) ) @ ( suminf @ A @ F ) ) ) ) ) ).

% sum_less_suminf
thf(fact_4636_powser__split__head_I1_J,axiom,
    ! [A: $tType] :
      ( ( ( real_Vector_banach @ A )
        & ( real_V8999393235501362500lgebra @ A ) )
     => ! [F: nat > A,Z: A] :
          ( ( summable @ A
            @ ^ [N: nat] : ( times_times @ A @ ( F @ N ) @ ( power_power @ A @ Z @ N ) ) )
         => ( ( suminf @ A
              @ ^ [N: nat] : ( times_times @ A @ ( F @ N ) @ ( power_power @ A @ Z @ N ) ) )
            = ( plus_plus @ A @ ( F @ ( zero_zero @ nat ) )
              @ ( times_times @ A
                @ ( suminf @ A
                  @ ^ [N: nat] : ( times_times @ A @ ( F @ ( suc @ N ) ) @ ( power_power @ A @ Z @ N ) ) )
                @ Z ) ) ) ) ) ).

% powser_split_head(1)
thf(fact_4637_powser__split__head_I2_J,axiom,
    ! [A: $tType] :
      ( ( ( real_Vector_banach @ A )
        & ( real_V8999393235501362500lgebra @ A ) )
     => ! [F: nat > A,Z: A] :
          ( ( summable @ A
            @ ^ [N: nat] : ( times_times @ A @ ( F @ N ) @ ( power_power @ A @ Z @ N ) ) )
         => ( ( times_times @ A
              @ ( suminf @ A
                @ ^ [N: nat] : ( times_times @ A @ ( F @ ( suc @ N ) ) @ ( power_power @ A @ Z @ N ) ) )
              @ Z )
            = ( minus_minus @ A
              @ ( suminf @ A
                @ ^ [N: nat] : ( times_times @ A @ ( F @ N ) @ ( power_power @ A @ Z @ N ) ) )
              @ ( F @ ( zero_zero @ nat ) ) ) ) ) ) ).

% powser_split_head(2)
thf(fact_4638_monoseq__realpow,axiom,
    ! [X2: real] :
      ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 )
     => ( ( ord_less_eq @ real @ X2 @ ( one_one @ real ) )
       => ( topological_monoseq @ real @ ( power_power @ real @ X2 ) ) ) ) ).

% monoseq_realpow
thf(fact_4639_suminf__exist__split,axiom,
    ! [A: $tType] :
      ( ( real_V822414075346904944vector @ A )
     => ! [R4: real,F: nat > A] :
          ( ( ord_less @ real @ ( zero_zero @ real ) @ R4 )
         => ( ( summable @ A @ F )
           => ? [N11: nat] :
              ! [N9: nat] :
                ( ( ord_less_eq @ nat @ N11 @ N9 )
               => ( ord_less @ real
                  @ ( real_V7770717601297561774m_norm @ A
                    @ ( suminf @ A
                      @ ^ [I2: nat] : ( F @ ( plus_plus @ nat @ I2 @ N9 ) ) ) )
                  @ R4 ) ) ) ) ) ).

% suminf_exist_split
thf(fact_4640_summable__partial__sum__bound,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach @ A )
     => ! [F: nat > A,E3: real] :
          ( ( summable @ A @ F )
         => ( ( ord_less @ real @ ( zero_zero @ real ) @ E3 )
           => ~ ! [N11: nat] :
                  ~ ! [M3: nat] :
                      ( ( ord_less_eq @ nat @ N11 @ M3 )
                     => ! [N9: nat] : ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F @ ( set_or1337092689740270186AtMost @ nat @ M3 @ N9 ) ) ) @ E3 ) ) ) ) ) ).

% summable_partial_sum_bound
thf(fact_4641_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_4642_Abel__lemma,axiom,
    ! [A: $tType] :
      ( ( real_V822414075346904944vector @ A )
     => ! [R4: real,R0: real,A2: nat > A,M8: real] :
          ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ R4 )
         => ( ( ord_less @ real @ R4 @ R0 )
           => ( ! [N4: nat] : ( ord_less_eq @ real @ ( times_times @ real @ ( real_V7770717601297561774m_norm @ A @ ( A2 @ N4 ) ) @ ( power_power @ real @ R0 @ N4 ) ) @ M8 )
             => ( summable @ real
                @ ^ [N: nat] : ( times_times @ real @ ( real_V7770717601297561774m_norm @ A @ ( A2 @ N ) ) @ ( power_power @ real @ R4 @ N ) ) ) ) ) ) ) ).

% Abel_lemma
thf(fact_4643_nat__intermed__int__val,axiom,
    ! [M: nat,N2: nat,F: nat > int,K: int] :
      ( ! [I3: nat] :
          ( ( ( ord_less_eq @ nat @ M @ I3 )
            & ( ord_less @ nat @ I3 @ N2 ) )
         => ( ord_less_eq @ int @ ( abs_abs @ int @ ( minus_minus @ int @ ( F @ ( suc @ I3 ) ) @ ( F @ I3 ) ) ) @ ( one_one @ int ) ) )
     => ( ( ord_less_eq @ nat @ M @ N2 )
       => ( ( ord_less_eq @ int @ ( F @ M ) @ K )
         => ( ( ord_less_eq @ int @ K @ ( F @ N2 ) )
           => ? [I3: nat] :
                ( ( ord_less_eq @ nat @ M @ I3 )
                & ( ord_less_eq @ nat @ I3 @ N2 )
                & ( ( F @ I3 )
                  = K ) ) ) ) ) ) ).

% nat_intermed_int_val
thf(fact_4644_Cauchy__product,axiom,
    ! [A: $tType] :
      ( ( ( real_Vector_banach @ A )
        & ( real_V4412858255891104859lgebra @ A ) )
     => ! [A2: nat > A,B2: nat > A] :
          ( ( summable @ real
            @ ^ [K4: nat] : ( real_V7770717601297561774m_norm @ A @ ( A2 @ K4 ) ) )
         => ( ( summable @ real
              @ ^ [K4: nat] : ( real_V7770717601297561774m_norm @ A @ ( B2 @ K4 ) ) )
           => ( ( times_times @ A @ ( suminf @ A @ A2 ) @ ( suminf @ A @ B2 ) )
              = ( suminf @ A
                @ ^ [K4: nat] :
                    ( groups7311177749621191930dd_sum @ nat @ A
                    @ ^ [I2: nat] : ( times_times @ A @ ( A2 @ I2 ) @ ( B2 @ ( minus_minus @ nat @ K4 @ I2 ) ) )
                    @ ( set_ord_atMost @ nat @ K4 ) ) ) ) ) ) ) ).

% Cauchy_product
thf(fact_4645_incr__lemma,axiom,
    ! [D2: int,Z: int,X2: int] :
      ( ( ord_less @ int @ ( zero_zero @ int ) @ D2 )
     => ( ord_less @ int @ Z @ ( plus_plus @ int @ X2 @ ( times_times @ int @ ( plus_plus @ int @ ( abs_abs @ int @ ( minus_minus @ int @ X2 @ Z ) ) @ ( one_one @ int ) ) @ D2 ) ) ) ) ).

% incr_lemma
thf(fact_4646_decr__lemma,axiom,
    ! [D2: int,X2: int,Z: int] :
      ( ( ord_less @ int @ ( zero_zero @ int ) @ D2 )
     => ( ord_less @ int @ ( minus_minus @ int @ X2 @ ( times_times @ int @ ( plus_plus @ int @ ( abs_abs @ int @ ( minus_minus @ int @ X2 @ Z ) ) @ ( one_one @ int ) ) @ D2 ) ) @ Z ) ) ).

% decr_lemma
thf(fact_4647_summable__ratio__test,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach @ A )
     => ! [C2: real,N3: nat,F: nat > A] :
          ( ( ord_less @ real @ C2 @ ( one_one @ real ) )
         => ( ! [N4: nat] :
                ( ( ord_less_eq @ nat @ N3 @ N4 )
               => ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( F @ ( suc @ N4 ) ) ) @ ( times_times @ real @ C2 @ ( real_V7770717601297561774m_norm @ A @ ( F @ N4 ) ) ) ) )
           => ( summable @ A @ F ) ) ) ) ).

% summable_ratio_test
thf(fact_4648_sum__less__suminf2,axiom,
    ! [A: $tType] :
      ( ( ( ordere8940638589300402666id_add @ A )
        & ( topolo1944317154257567458pology @ A ) )
     => ! [F: nat > A,N2: nat,I: nat] :
          ( ( summable @ A @ F )
         => ( ! [M2: nat] :
                ( ( ord_less_eq @ nat @ N2 @ M2 )
               => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F @ M2 ) ) )
           => ( ( ord_less_eq @ nat @ N2 @ I )
             => ( ( ord_less @ A @ ( zero_zero @ A ) @ ( F @ I ) )
               => ( ord_less @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F @ ( set_ord_lessThan @ nat @ N2 ) ) @ ( suminf @ A @ F ) ) ) ) ) ) ) ).

% sum_less_suminf2
thf(fact_4649_arctan__ubound,axiom,
    ! [Y2: real] : ( ord_less @ real @ ( arctan @ Y2 ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ).

% arctan_ubound
thf(fact_4650_arctan__one,axiom,
    ( ( arctan @ ( one_one @ real ) )
    = ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ).

% arctan_one
thf(fact_4651_nat__ivt__aux,axiom,
    ! [N2: nat,F: nat > int,K: int] :
      ( ! [I3: nat] :
          ( ( ord_less @ nat @ I3 @ N2 )
         => ( ord_less_eq @ int @ ( abs_abs @ int @ ( minus_minus @ int @ ( F @ ( suc @ I3 ) ) @ ( F @ I3 ) ) ) @ ( one_one @ int ) ) )
     => ( ( ord_less_eq @ int @ ( F @ ( zero_zero @ nat ) ) @ K )
       => ( ( ord_less_eq @ int @ K @ ( F @ N2 ) )
         => ? [I3: nat] :
              ( ( ord_less_eq @ nat @ I3 @ N2 )
              & ( ( F @ I3 )
                = K ) ) ) ) ) ).

% nat_ivt_aux
thf(fact_4652_Cauchy__product__sums,axiom,
    ! [A: $tType] :
      ( ( ( real_Vector_banach @ A )
        & ( real_V4412858255891104859lgebra @ A ) )
     => ! [A2: nat > A,B2: nat > A] :
          ( ( summable @ real
            @ ^ [K4: nat] : ( real_V7770717601297561774m_norm @ A @ ( A2 @ K4 ) ) )
         => ( ( summable @ real
              @ ^ [K4: nat] : ( real_V7770717601297561774m_norm @ A @ ( B2 @ K4 ) ) )
           => ( sums @ A
              @ ^ [K4: nat] :
                  ( groups7311177749621191930dd_sum @ nat @ A
                  @ ^ [I2: nat] : ( times_times @ A @ ( A2 @ I2 ) @ ( B2 @ ( minus_minus @ nat @ K4 @ I2 ) ) )
                  @ ( set_ord_atMost @ nat @ K4 ) )
              @ ( times_times @ A @ ( suminf @ A @ A2 ) @ ( suminf @ A @ B2 ) ) ) ) ) ) ).

% Cauchy_product_sums
thf(fact_4653_arctan__lbound,axiom,
    ! [Y2: real] : ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ ( arctan @ Y2 ) ) ).

% arctan_lbound
thf(fact_4654_arctan__bounded,axiom,
    ! [Y2: real] :
      ( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ ( arctan @ Y2 ) )
      & ( ord_less @ real @ ( arctan @ Y2 ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ).

% arctan_bounded
thf(fact_4655_nat0__intermed__int__val,axiom,
    ! [N2: nat,F: nat > int,K: int] :
      ( ! [I3: nat] :
          ( ( ord_less @ nat @ I3 @ N2 )
         => ( ord_less_eq @ int @ ( abs_abs @ int @ ( minus_minus @ int @ ( F @ ( plus_plus @ nat @ I3 @ ( one_one @ nat ) ) ) @ ( F @ I3 ) ) ) @ ( one_one @ int ) ) )
     => ( ( ord_less_eq @ int @ ( F @ ( zero_zero @ nat ) ) @ K )
       => ( ( ord_less_eq @ int @ K @ ( F @ N2 ) )
         => ? [I3: nat] :
              ( ( ord_less_eq @ nat @ I3 @ N2 )
              & ( ( F @ I3 )
                = K ) ) ) ) ) ).

% nat0_intermed_int_val
thf(fact_4656_arctan__add,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_eq @ real @ ( abs_abs @ real @ X2 ) @ ( one_one @ real ) )
     => ( ( ord_less @ real @ ( abs_abs @ real @ Y2 ) @ ( one_one @ real ) )
       => ( ( plus_plus @ real @ ( arctan @ X2 ) @ ( arctan @ Y2 ) )
          = ( arctan @ ( divide_divide @ real @ ( plus_plus @ real @ X2 @ Y2 ) @ ( minus_minus @ real @ ( one_one @ real ) @ ( times_times @ real @ X2 @ Y2 ) ) ) ) ) ) ) ).

% arctan_add
thf(fact_4657_machin__Euler,axiom,
    ( ( plus_plus @ real @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit1 @ ( bit0 @ one2 ) ) ) @ ( arctan @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit1 @ ( bit1 @ one2 ) ) ) ) ) ) @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( arctan @ ( divide_divide @ real @ ( numeral_numeral @ real @ ( bit1 @ one2 ) ) @ ( numeral_numeral @ real @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) )
    = ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ).

% machin_Euler
thf(fact_4658_machin,axiom,
    ( ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ ( bit0 @ one2 ) ) ) )
    = ( minus_minus @ real @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ ( bit0 @ one2 ) ) ) @ ( arctan @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit1 @ ( bit0 @ one2 ) ) ) ) ) ) @ ( arctan @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ).

% machin
thf(fact_4659_sum__pos__lt__pair,axiom,
    ! [F: nat > real,K: nat] :
      ( ( summable @ real @ F )
     => ( ! [D4: nat] : ( ord_less @ real @ ( zero_zero @ real ) @ ( plus_plus @ real @ ( F @ ( plus_plus @ nat @ K @ ( times_times @ nat @ ( suc @ ( suc @ ( zero_zero @ nat ) ) ) @ D4 ) ) ) @ ( F @ ( plus_plus @ nat @ K @ ( plus_plus @ nat @ ( times_times @ nat @ ( suc @ ( suc @ ( zero_zero @ nat ) ) ) @ D4 ) @ ( one_one @ nat ) ) ) ) ) )
       => ( ord_less @ real @ ( groups7311177749621191930dd_sum @ nat @ real @ F @ ( set_ord_lessThan @ nat @ K ) ) @ ( suminf @ real @ F ) ) ) ) ).

% sum_pos_lt_pair
thf(fact_4660_arctan__double,axiom,
    ! [X2: real] :
      ( ( ord_less @ real @ ( abs_abs @ real @ X2 ) @ ( one_one @ real ) )
     => ( ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( arctan @ X2 ) )
        = ( arctan @ ( divide_divide @ real @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ X2 ) @ ( minus_minus @ real @ ( one_one @ real ) @ ( power_power @ real @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).

% arctan_double
thf(fact_4661_diffs__equiv,axiom,
    ! [A: $tType] :
      ( ( ( real_V822414075346904944vector @ A )
        & ( ring_1 @ A ) )
     => ! [C2: nat > A,X2: A] :
          ( ( summable @ A
            @ ^ [N: nat] : ( times_times @ A @ ( diffs @ A @ C2 @ N ) @ ( power_power @ A @ X2 @ N ) ) )
         => ( sums @ A
            @ ^ [N: nat] : ( times_times @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N ) @ ( C2 @ N ) ) @ ( power_power @ A @ X2 @ ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) ) )
            @ ( suminf @ A
              @ ^ [N: nat] : ( times_times @ A @ ( diffs @ A @ C2 @ N ) @ ( power_power @ A @ X2 @ N ) ) ) ) ) ) ).

% diffs_equiv
thf(fact_4662_tan__double,axiom,
    ! [A: $tType] :
      ( ( ( real_Vector_banach @ A )
        & ( real_V3459762299906320749_field @ A ) )
     => ! [X2: A] :
          ( ( ( cos @ A @ X2 )
           != ( zero_zero @ A ) )
         => ( ( ( cos @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ X2 ) )
             != ( zero_zero @ A ) )
           => ( ( tan @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ X2 ) )
              = ( divide_divide @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( tan @ A @ X2 ) ) @ ( minus_minus @ A @ ( one_one @ A ) @ ( power_power @ A @ ( tan @ A @ X2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ).

% tan_double
thf(fact_4663_arcosh__def,axiom,
    ! [A: $tType] :
      ( ( ln @ A )
     => ( ( arcosh @ A )
        = ( ^ [X: A] : ( ln_ln @ A @ ( plus_plus @ A @ X @ ( powr @ A @ ( minus_minus @ A @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ A ) ) @ ( real_Vector_of_real @ A @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ).

% arcosh_def
thf(fact_4664_summable__complex__of__real,axiom,
    ! [F: nat > real] :
      ( ( summable @ complex
        @ ^ [N: nat] : ( real_Vector_of_real @ complex @ ( F @ N ) ) )
      = ( summable @ real @ F ) ) ).

% summable_complex_of_real
thf(fact_4665_tan__periodic__pi,axiom,
    ! [X2: real] :
      ( ( tan @ real @ ( plus_plus @ real @ X2 @ pi ) )
      = ( tan @ real @ X2 ) ) ).

% tan_periodic_pi
thf(fact_4666_of__real__eq__1__iff,axiom,
    ! [A: $tType] :
      ( ( real_V2191834092415804123ebra_1 @ A )
     => ! [X2: real] :
          ( ( ( real_Vector_of_real @ A @ X2 )
            = ( one_one @ A ) )
          = ( X2
            = ( one_one @ real ) ) ) ) ).

% of_real_eq_1_iff
thf(fact_4667_of__real__1,axiom,
    ! [A: $tType] :
      ( ( real_V2191834092415804123ebra_1 @ A )
     => ( ( real_Vector_of_real @ A @ ( one_one @ real ) )
        = ( one_one @ A ) ) ) ).

% of_real_1
thf(fact_4668_of__real__numeral,axiom,
    ! [A: $tType] :
      ( ( real_V2191834092415804123ebra_1 @ A )
     => ! [W: num] :
          ( ( real_Vector_of_real @ A @ ( numeral_numeral @ real @ W ) )
          = ( numeral_numeral @ A @ W ) ) ) ).

% of_real_numeral
thf(fact_4669_of__real__mult,axiom,
    ! [A: $tType] :
      ( ( real_V2191834092415804123ebra_1 @ A )
     => ! [X2: real,Y2: real] :
          ( ( real_Vector_of_real @ A @ ( times_times @ real @ X2 @ Y2 ) )
          = ( times_times @ A @ ( real_Vector_of_real @ A @ X2 ) @ ( real_Vector_of_real @ A @ Y2 ) ) ) ) ).

% of_real_mult
thf(fact_4670_of__real__divide,axiom,
    ! [A: $tType] :
      ( ( real_V5047593784448816457lgebra @ A )
     => ! [X2: real,Y2: real] :
          ( ( real_Vector_of_real @ A @ ( divide_divide @ real @ X2 @ Y2 ) )
          = ( divide_divide @ A @ ( real_Vector_of_real @ A @ X2 ) @ ( real_Vector_of_real @ A @ Y2 ) ) ) ) ).

% of_real_divide
thf(fact_4671_of__real__add,axiom,
    ! [A: $tType] :
      ( ( real_V2191834092415804123ebra_1 @ A )
     => ! [X2: real,Y2: real] :
          ( ( real_Vector_of_real @ A @ ( plus_plus @ real @ X2 @ Y2 ) )
          = ( plus_plus @ A @ ( real_Vector_of_real @ A @ X2 ) @ ( real_Vector_of_real @ A @ Y2 ) ) ) ) ).

% of_real_add
thf(fact_4672_of__real__power,axiom,
    ! [A: $tType] :
      ( ( real_V2191834092415804123ebra_1 @ A )
     => ! [X2: real,N2: nat] :
          ( ( real_Vector_of_real @ A @ ( power_power @ real @ X2 @ N2 ) )
          = ( power_power @ A @ ( real_Vector_of_real @ A @ X2 ) @ N2 ) ) ) ).

% of_real_power
thf(fact_4673_of__real__diff,axiom,
    ! [A: $tType] :
      ( ( real_V2191834092415804123ebra_1 @ A )
     => ! [X2: real,Y2: real] :
          ( ( real_Vector_of_real @ A @ ( minus_minus @ real @ X2 @ Y2 ) )
          = ( minus_minus @ A @ ( real_Vector_of_real @ A @ X2 ) @ ( real_Vector_of_real @ A @ Y2 ) ) ) ) ).

% of_real_diff
thf(fact_4674_of__real__sum,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V2191834092415804123ebra_1 @ A )
     => ! [F: B > real,S4: set @ B] :
          ( ( real_Vector_of_real @ A @ ( groups7311177749621191930dd_sum @ B @ real @ F @ S4 ) )
          = ( groups7311177749621191930dd_sum @ B @ A
            @ ^ [X: B] : ( real_Vector_of_real @ A @ ( F @ X ) )
            @ S4 ) ) ) ).

% of_real_sum
thf(fact_4675_of__real__prod,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ( comm_monoid_mult @ A )
        & ( real_V2191834092415804123ebra_1 @ A ) )
     => ! [F: B > real,S4: set @ B] :
          ( ( real_Vector_of_real @ A @ ( groups7121269368397514597t_prod @ B @ real @ F @ S4 ) )
          = ( groups7121269368397514597t_prod @ B @ A
            @ ^ [X: B] : ( real_Vector_of_real @ A @ ( F @ X ) )
            @ S4 ) ) ) ).

% of_real_prod
thf(fact_4676_tan__npi,axiom,
    ! [N2: nat] :
      ( ( tan @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N2 ) @ pi ) )
      = ( zero_zero @ real ) ) ).

% tan_npi
thf(fact_4677_tan__periodic__n,axiom,
    ! [X2: real,N2: num] :
      ( ( tan @ real @ ( plus_plus @ real @ X2 @ ( times_times @ real @ ( numeral_numeral @ real @ N2 ) @ pi ) ) )
      = ( tan @ real @ X2 ) ) ).

% tan_periodic_n
thf(fact_4678_tan__periodic__nat,axiom,
    ! [X2: real,N2: nat] :
      ( ( tan @ real @ ( plus_plus @ real @ X2 @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N2 ) @ pi ) ) )
      = ( tan @ real @ X2 ) ) ).

% tan_periodic_nat
thf(fact_4679_tan__periodic__int,axiom,
    ! [X2: real,I: int] :
      ( ( tan @ real @ ( plus_plus @ real @ X2 @ ( times_times @ real @ ( ring_1_of_int @ real @ I ) @ pi ) ) )
      = ( tan @ real @ X2 ) ) ).

% tan_periodic_int
thf(fact_4680_of__real__neg__numeral,axiom,
    ! [A: $tType] :
      ( ( real_V2191834092415804123ebra_1 @ A )
     => ! [W: num] :
          ( ( real_Vector_of_real @ A @ ( uminus_uminus @ real @ ( numeral_numeral @ real @ W ) ) )
          = ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) ) ) ).

% of_real_neg_numeral
thf(fact_4681_cos__of__real__pi,axiom,
    ! [A: $tType] :
      ( ( ( real_Vector_banach @ A )
        & ( real_V2822296259951069270ebra_1 @ A ) )
     => ( ( cos @ A @ ( real_Vector_of_real @ A @ pi ) )
        = ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).

% cos_of_real_pi
thf(fact_4682_tan__periodic,axiom,
    ! [X2: real] :
      ( ( tan @ real @ ( plus_plus @ real @ X2 @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) ) )
      = ( tan @ real @ X2 ) ) ).

% tan_periodic
thf(fact_4683_norm__of__real__add1,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra @ A )
     => ! [X2: real] :
          ( ( real_V7770717601297561774m_norm @ A @ ( plus_plus @ A @ ( real_Vector_of_real @ A @ X2 ) @ ( one_one @ A ) ) )
          = ( abs_abs @ real @ ( plus_plus @ real @ X2 @ ( one_one @ real ) ) ) ) ) ).

% norm_of_real_add1
thf(fact_4684_norm__of__real__addn,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra @ A )
     => ! [X2: real,B2: num] :
          ( ( real_V7770717601297561774m_norm @ A @ ( plus_plus @ A @ ( real_Vector_of_real @ A @ X2 ) @ ( numeral_numeral @ A @ B2 ) ) )
          = ( abs_abs @ real @ ( plus_plus @ real @ X2 @ ( numeral_numeral @ real @ B2 ) ) ) ) ) ).

% norm_of_real_addn
thf(fact_4685_cos__of__real__pi__half,axiom,
    ! [A: $tType] :
      ( ( ( real_Vector_banach @ A )
        & ( real_V7773925162809079976_field @ A )
        & ( real_V2822296259951069270ebra_1 @ A ) )
     => ( ( cos @ A @ ( divide_divide @ A @ ( real_Vector_of_real @ A @ pi ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) )
        = ( zero_zero @ A ) ) ) ).

% cos_of_real_pi_half
thf(fact_4686_sin__of__real__pi__half,axiom,
    ! [A: $tType] :
      ( ( ( real_Vector_banach @ A )
        & ( real_V7773925162809079976_field @ A )
        & ( real_V2822296259951069270ebra_1 @ A ) )
     => ( ( sin @ A @ ( divide_divide @ A @ ( real_Vector_of_real @ A @ pi ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) )
        = ( one_one @ A ) ) ) ).

% sin_of_real_pi_half
thf(fact_4687_diffs__of__real,axiom,
    ! [A: $tType] :
      ( ( real_V2191834092415804123ebra_1 @ A )
     => ! [F: nat > real] :
          ( ( diffs @ A
            @ ^ [N: nat] : ( real_Vector_of_real @ A @ ( F @ N ) ) )
          = ( ^ [N: nat] : ( real_Vector_of_real @ A @ ( diffs @ real @ F @ N ) ) ) ) ) ).

% diffs_of_real
thf(fact_4688_of__real__def,axiom,
    ! [A: $tType] :
      ( ( real_V2191834092415804123ebra_1 @ A )
     => ( ( real_Vector_of_real @ A )
        = ( ^ [R: real] : ( real_V8093663219630862766scaleR @ A @ R @ ( one_one @ A ) ) ) ) ) ).

% of_real_def
thf(fact_4689_scaleR__conv__of__real,axiom,
    ! [A: $tType] :
      ( ( real_V2191834092415804123ebra_1 @ A )
     => ( ( real_V8093663219630862766scaleR @ A )
        = ( ^ [R: real] : ( times_times @ A @ ( real_Vector_of_real @ A @ R ) ) ) ) ) ).

% scaleR_conv_of_real
thf(fact_4690_diffs__minus,axiom,
    ! [A: $tType] :
      ( ( ring_1 @ A )
     => ! [C2: nat > A] :
          ( ( diffs @ A
            @ ^ [N: nat] : ( uminus_uminus @ A @ ( C2 @ N ) ) )
          = ( ^ [N: nat] : ( uminus_uminus @ A @ ( diffs @ A @ C2 @ N ) ) ) ) ) ).

% diffs_minus
thf(fact_4691_summable__of__real,axiom,
    ! [A: $tType] :
      ( ( ( real_V2191834092415804123ebra_1 @ A )
        & ( real_V822414075346904944vector @ A ) )
     => ! [X8: nat > real] :
          ( ( summable @ real @ X8 )
         => ( summable @ A
            @ ^ [N: nat] : ( real_Vector_of_real @ A @ ( X8 @ N ) ) ) ) ) ).

% summable_of_real
thf(fact_4692_sums__of__real,axiom,
    ! [A: $tType] :
      ( ( ( real_V2191834092415804123ebra_1 @ A )
        & ( real_V822414075346904944vector @ A ) )
     => ! [X8: nat > real,A2: real] :
          ( ( sums @ real @ X8 @ A2 )
         => ( sums @ A
            @ ^ [N: nat] : ( real_Vector_of_real @ A @ ( X8 @ N ) )
            @ ( real_Vector_of_real @ A @ A2 ) ) ) ) ).

% sums_of_real
thf(fact_4693_sums__of__real__iff,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra @ A )
     => ! [F: nat > real,C2: real] :
          ( ( sums @ A
            @ ^ [N: nat] : ( real_Vector_of_real @ A @ ( F @ N ) )
            @ ( real_Vector_of_real @ A @ C2 ) )
          = ( sums @ real @ F @ C2 ) ) ) ).

% sums_of_real_iff
thf(fact_4694_nonzero__of__real__divide,axiom,
    ! [A: $tType] :
      ( ( real_V7773925162809079976_field @ A )
     => ! [Y2: real,X2: real] :
          ( ( Y2
           != ( zero_zero @ real ) )
         => ( ( real_Vector_of_real @ A @ ( divide_divide @ real @ X2 @ Y2 ) )
            = ( divide_divide @ A @ ( real_Vector_of_real @ A @ X2 ) @ ( real_Vector_of_real @ A @ Y2 ) ) ) ) ) ).

% nonzero_of_real_divide
thf(fact_4695_suminf__of__real,axiom,
    ! [A: $tType] :
      ( ( ( real_V2191834092415804123ebra_1 @ A )
        & ( real_V822414075346904944vector @ A ) )
     => ! [X8: nat > real] :
          ( ( summable @ real @ X8 )
         => ( ( real_Vector_of_real @ A @ ( suminf @ real @ X8 ) )
            = ( suminf @ A
              @ ^ [N: nat] : ( real_Vector_of_real @ A @ ( X8 @ N ) ) ) ) ) ) ).

% suminf_of_real
thf(fact_4696_norm__less__p1,axiom,
    ! [A: $tType] :
      ( ( real_V2822296259951069270ebra_1 @ A )
     => ! [X2: A] : ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X2 ) @ ( real_V7770717601297561774m_norm @ A @ ( plus_plus @ A @ ( real_Vector_of_real @ A @ ( real_V7770717601297561774m_norm @ A @ X2 ) ) @ ( one_one @ A ) ) ) ) ) ).

% norm_less_p1
thf(fact_4697_tan__def,axiom,
    ! [A: $tType] :
      ( ( ( real_Vector_banach @ A )
        & ( real_V3459762299906320749_field @ A ) )
     => ( ( tan @ A )
        = ( ^ [X: A] : ( divide_divide @ A @ ( sin @ A @ X ) @ ( cos @ A @ X ) ) ) ) ) ).

% tan_def
thf(fact_4698_diffs__def,axiom,
    ! [A: $tType] :
      ( ( ring_1 @ A )
     => ( ( diffs @ A )
        = ( ^ [C4: nat > A,N: nat] : ( times_times @ A @ ( semiring_1_of_nat @ A @ ( suc @ N ) ) @ ( C4 @ ( suc @ N ) ) ) ) ) ) ).

% diffs_def
thf(fact_4699_termdiff__converges__all,axiom,
    ! [A: $tType] :
      ( ( ( real_Vector_banach @ A )
        & ( real_V3459762299906320749_field @ A ) )
     => ! [C2: nat > A,X2: A] :
          ( ! [X3: A] :
              ( summable @ A
              @ ^ [N: nat] : ( times_times @ A @ ( C2 @ N ) @ ( power_power @ A @ X3 @ N ) ) )
         => ( summable @ A
            @ ^ [N: nat] : ( times_times @ A @ ( diffs @ A @ C2 @ N ) @ ( power_power @ A @ X2 @ N ) ) ) ) ) ).

% termdiff_converges_all
thf(fact_4700_norm__of__real__diff,axiom,
    ! [A: $tType] :
      ( ( real_V2822296259951069270ebra_1 @ A )
     => ! [B2: real,A2: real] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ ( real_Vector_of_real @ A @ B2 ) @ ( real_Vector_of_real @ A @ A2 ) ) ) @ ( abs_abs @ real @ ( minus_minus @ real @ B2 @ A2 ) ) ) ) ).

% norm_of_real_diff
thf(fact_4701_tan__45,axiom,
    ( ( tan @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ ( bit0 @ one2 ) ) ) ) )
    = ( one_one @ real ) ) ).

% tan_45
thf(fact_4702_cos__int__times__real,axiom,
    ! [A: $tType] :
      ( ( ( real_Vector_banach @ A )
        & ( real_V2822296259951069270ebra_1 @ A ) )
     => ! [M: int,X2: real] :
          ( ( cos @ A @ ( times_times @ A @ ( ring_1_of_int @ A @ M ) @ ( real_Vector_of_real @ A @ X2 ) ) )
          = ( real_Vector_of_real @ A @ ( cos @ real @ ( times_times @ real @ ( ring_1_of_int @ real @ M ) @ X2 ) ) ) ) ) ).

% cos_int_times_real
thf(fact_4703_sin__int__times__real,axiom,
    ! [A: $tType] :
      ( ( ( real_Vector_banach @ A )
        & ( real_V2822296259951069270ebra_1 @ A ) )
     => ! [M: int,X2: real] :
          ( ( sin @ A @ ( times_times @ A @ ( ring_1_of_int @ A @ M ) @ ( real_Vector_of_real @ A @ X2 ) ) )
          = ( real_Vector_of_real @ A @ ( sin @ real @ ( times_times @ real @ ( ring_1_of_int @ real @ M ) @ X2 ) ) ) ) ) ).

% sin_int_times_real
thf(fact_4704_lemma__tan__total,axiom,
    ! [Y2: real] :
      ( ( ord_less @ real @ ( zero_zero @ real ) @ Y2 )
     => ? [X3: real] :
          ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
          & ( ord_less @ real @ X3 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
          & ( ord_less @ real @ Y2 @ ( tan @ real @ X3 ) ) ) ) ).

% lemma_tan_total
thf(fact_4705_tan__gt__zero,axiom,
    ! [X2: real] :
      ( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
     => ( ( ord_less @ real @ X2 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
       => ( ord_less @ real @ ( zero_zero @ real ) @ ( tan @ real @ X2 ) ) ) ) ).

% tan_gt_zero
thf(fact_4706_tan__total,axiom,
    ! [Y2: real] :
    ? [X3: real] :
      ( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X3 )
      & ( ord_less @ real @ X3 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
      & ( ( tan @ real @ X3 )
        = Y2 )
      & ! [Y3: real] :
          ( ( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ Y3 )
            & ( ord_less @ real @ Y3 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
            & ( ( tan @ real @ Y3 )
              = Y2 ) )
         => ( Y3 = X3 ) ) ) ).

% tan_total
thf(fact_4707_tan__monotone,axiom,
    ! [Y2: real,X2: real] :
      ( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ Y2 )
     => ( ( ord_less @ real @ Y2 @ X2 )
       => ( ( ord_less @ real @ X2 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
         => ( ord_less @ real @ ( tan @ real @ Y2 ) @ ( tan @ real @ X2 ) ) ) ) ) ).

% tan_monotone
thf(fact_4708_tan__monotone_H,axiom,
    ! [Y2: real,X2: real] :
      ( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ Y2 )
     => ( ( ord_less @ real @ Y2 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
       => ( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X2 )
         => ( ( ord_less @ real @ X2 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
           => ( ( ord_less @ real @ Y2 @ X2 )
              = ( ord_less @ real @ ( tan @ real @ Y2 ) @ ( tan @ real @ X2 ) ) ) ) ) ) ) ).

% tan_monotone'
thf(fact_4709_tan__mono__lt__eq,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X2 )
     => ( ( ord_less @ real @ X2 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
       => ( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ Y2 )
         => ( ( ord_less @ real @ Y2 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
           => ( ( ord_less @ real @ ( tan @ real @ X2 ) @ ( tan @ real @ Y2 ) )
              = ( ord_less @ real @ X2 @ Y2 ) ) ) ) ) ) ).

% tan_mono_lt_eq
thf(fact_4710_lemma__tan__total1,axiom,
    ! [Y2: real] :
    ? [X3: real] :
      ( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X3 )
      & ( ord_less @ real @ X3 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
      & ( ( tan @ real @ X3 )
        = Y2 ) ) ).

% lemma_tan_total1
thf(fact_4711_tan__minus__45,axiom,
    ( ( tan @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) )
    = ( uminus_uminus @ real @ ( one_one @ real ) ) ) ).

% tan_minus_45
thf(fact_4712_tan__inverse,axiom,
    ! [Y2: real] :
      ( ( divide_divide @ real @ ( one_one @ real ) @ ( tan @ real @ Y2 ) )
      = ( tan @ real @ ( minus_minus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ Y2 ) ) ) ).

% tan_inverse
thf(fact_4713_add__tan__eq,axiom,
    ! [A: $tType] :
      ( ( ( real_Vector_banach @ A )
        & ( real_V3459762299906320749_field @ A ) )
     => ! [X2: A,Y2: A] :
          ( ( ( cos @ A @ X2 )
           != ( zero_zero @ A ) )
         => ( ( ( cos @ A @ Y2 )
             != ( zero_zero @ A ) )
           => ( ( plus_plus @ A @ ( tan @ A @ X2 ) @ ( tan @ A @ Y2 ) )
              = ( divide_divide @ A @ ( sin @ A @ ( plus_plus @ A @ X2 @ Y2 ) ) @ ( times_times @ A @ ( cos @ A @ X2 ) @ ( cos @ A @ Y2 ) ) ) ) ) ) ) ).

% add_tan_eq
thf(fact_4714_sin__cos__eq,axiom,
    ! [A: $tType] :
      ( ( ( real_Vector_banach @ A )
        & ( real_V3459762299906320749_field @ A ) )
     => ( ( sin @ A )
        = ( ^ [X: A] : ( cos @ A @ ( minus_minus @ A @ ( divide_divide @ A @ ( real_Vector_of_real @ A @ pi ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ X ) ) ) ) ) ).

% sin_cos_eq
thf(fact_4715_cos__sin__eq,axiom,
    ! [A: $tType] :
      ( ( ( real_Vector_banach @ A )
        & ( real_V3459762299906320749_field @ A ) )
     => ( ( cos @ A )
        = ( ^ [X: A] : ( sin @ A @ ( minus_minus @ A @ ( divide_divide @ A @ ( real_Vector_of_real @ A @ pi ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ X ) ) ) ) ) ).

% cos_sin_eq
thf(fact_4716_termdiff__converges,axiom,
    ! [A: $tType] :
      ( ( ( real_Vector_banach @ A )
        & ( real_V3459762299906320749_field @ A ) )
     => ! [X2: A,K6: real,C2: nat > A] :
          ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X2 ) @ K6 )
         => ( ! [X3: A] :
                ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X3 ) @ K6 )
               => ( summable @ A
                  @ ^ [N: nat] : ( times_times @ A @ ( C2 @ N ) @ ( power_power @ A @ X3 @ N ) ) ) )
           => ( summable @ A
              @ ^ [N: nat] : ( times_times @ A @ ( diffs @ A @ C2 @ N ) @ ( power_power @ A @ X2 @ N ) ) ) ) ) ) ).

% termdiff_converges
thf(fact_4717_tan__pos__pi2__le,axiom,
    ! [X2: real] :
      ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 )
     => ( ( ord_less @ real @ X2 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
       => ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( tan @ real @ X2 ) ) ) ) ).

% tan_pos_pi2_le
thf(fact_4718_tan__total__pos,axiom,
    ! [Y2: real] :
      ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y2 )
     => ? [X3: real] :
          ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X3 )
          & ( ord_less @ real @ X3 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
          & ( ( tan @ real @ X3 )
            = Y2 ) ) ) ).

% tan_total_pos
thf(fact_4719_tan__less__zero,axiom,
    ! [X2: real] :
      ( ( ord_less @ real @ ( divide_divide @ real @ ( uminus_uminus @ real @ pi ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ X2 )
     => ( ( ord_less @ real @ X2 @ ( zero_zero @ real ) )
       => ( ord_less @ real @ ( tan @ real @ X2 ) @ ( zero_zero @ real ) ) ) ) ).

% tan_less_zero
thf(fact_4720_tan__mono__le,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X2 )
     => ( ( ord_less_eq @ real @ X2 @ Y2 )
       => ( ( ord_less @ real @ Y2 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
         => ( ord_less_eq @ real @ ( tan @ real @ X2 ) @ ( tan @ real @ Y2 ) ) ) ) ) ).

% tan_mono_le
thf(fact_4721_tan__mono__le__eq,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X2 )
     => ( ( ord_less @ real @ X2 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
       => ( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ Y2 )
         => ( ( ord_less @ real @ Y2 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
           => ( ( ord_less_eq @ real @ ( tan @ real @ X2 ) @ ( tan @ real @ Y2 ) )
              = ( ord_less_eq @ real @ X2 @ Y2 ) ) ) ) ) ) ).

% tan_mono_le_eq
thf(fact_4722_tan__bound__pi2,axiom,
    ! [X2: real] :
      ( ( ord_less @ real @ ( abs_abs @ real @ X2 ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ ( bit0 @ one2 ) ) ) ) )
     => ( ord_less @ real @ ( abs_abs @ real @ ( tan @ real @ X2 ) ) @ ( one_one @ real ) ) ) ).

% tan_bound_pi2
thf(fact_4723_arctan__unique,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X2 )
     => ( ( ord_less @ real @ X2 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
       => ( ( ( tan @ real @ X2 )
            = Y2 )
         => ( ( arctan @ Y2 )
            = X2 ) ) ) ) ).

% arctan_unique
thf(fact_4724_arctan__tan,axiom,
    ! [X2: real] :
      ( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X2 )
     => ( ( ord_less @ real @ X2 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
       => ( ( arctan @ ( tan @ real @ X2 ) )
          = X2 ) ) ) ).

% arctan_tan
thf(fact_4725_arctan,axiom,
    ! [Y2: real] :
      ( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ ( arctan @ Y2 ) )
      & ( ord_less @ real @ ( arctan @ Y2 ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
      & ( ( tan @ real @ ( arctan @ Y2 ) )
        = Y2 ) ) ).

% arctan
thf(fact_4726_lemma__tan__add1,axiom,
    ! [A: $tType] :
      ( ( ( real_Vector_banach @ A )
        & ( real_V3459762299906320749_field @ A ) )
     => ! [X2: A,Y2: A] :
          ( ( ( cos @ A @ X2 )
           != ( zero_zero @ A ) )
         => ( ( ( cos @ A @ Y2 )
             != ( zero_zero @ A ) )
           => ( ( minus_minus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( tan @ A @ X2 ) @ ( tan @ A @ Y2 ) ) )
              = ( divide_divide @ A @ ( cos @ A @ ( plus_plus @ A @ X2 @ Y2 ) ) @ ( times_times @ A @ ( cos @ A @ X2 ) @ ( cos @ A @ Y2 ) ) ) ) ) ) ) ).

% lemma_tan_add1
thf(fact_4727_tan__diff,axiom,
    ! [A: $tType] :
      ( ( ( real_Vector_banach @ A )
        & ( real_V3459762299906320749_field @ A ) )
     => ! [X2: A,Y2: A] :
          ( ( ( cos @ A @ X2 )
           != ( zero_zero @ A ) )
         => ( ( ( cos @ A @ Y2 )
             != ( zero_zero @ A ) )
           => ( ( ( cos @ A @ ( minus_minus @ A @ X2 @ Y2 ) )
               != ( zero_zero @ A ) )
             => ( ( tan @ A @ ( minus_minus @ A @ X2 @ Y2 ) )
                = ( divide_divide @ A @ ( minus_minus @ A @ ( tan @ A @ X2 ) @ ( tan @ A @ Y2 ) ) @ ( plus_plus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( tan @ A @ X2 ) @ ( tan @ A @ Y2 ) ) ) ) ) ) ) ) ) ).

% tan_diff
thf(fact_4728_tan__add,axiom,
    ! [A: $tType] :
      ( ( ( real_Vector_banach @ A )
        & ( real_V3459762299906320749_field @ A ) )
     => ! [X2: A,Y2: A] :
          ( ( ( cos @ A @ X2 )
           != ( zero_zero @ A ) )
         => ( ( ( cos @ A @ Y2 )
             != ( zero_zero @ A ) )
           => ( ( ( cos @ A @ ( plus_plus @ A @ X2 @ Y2 ) )
               != ( zero_zero @ A ) )
             => ( ( tan @ A @ ( plus_plus @ A @ X2 @ Y2 ) )
                = ( divide_divide @ A @ ( plus_plus @ A @ ( tan @ A @ X2 ) @ ( tan @ A @ Y2 ) ) @ ( minus_minus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( tan @ A @ X2 ) @ ( tan @ A @ Y2 ) ) ) ) ) ) ) ) ) ).

% tan_add
thf(fact_4729_minus__sin__cos__eq,axiom,
    ! [A: $tType] :
      ( ( ( real_Vector_banach @ A )
        & ( real_V3459762299906320749_field @ A ) )
     => ! [X2: A] :
          ( ( uminus_uminus @ A @ ( sin @ A @ X2 ) )
          = ( cos @ A @ ( plus_plus @ A @ X2 @ ( divide_divide @ A @ ( real_Vector_of_real @ A @ pi ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ).

% minus_sin_cos_eq
thf(fact_4730_tan__total__pi4,axiom,
    ! [X2: real] :
      ( ( ord_less @ real @ ( abs_abs @ real @ X2 ) @ ( one_one @ real ) )
     => ? [Z3: real] :
          ( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) @ Z3 )
          & ( ord_less @ real @ Z3 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ ( bit0 @ one2 ) ) ) ) )
          & ( ( tan @ real @ Z3 )
            = X2 ) ) ) ).

% tan_total_pi4
thf(fact_4731_tan__half,axiom,
    ! [A: $tType] :
      ( ( ( real_Vector_banach @ A )
        & ( real_V3459762299906320749_field @ A ) )
     => ( ( tan @ A )
        = ( ^ [X: A] : ( divide_divide @ A @ ( sin @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ X ) ) @ ( plus_plus @ A @ ( cos @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ X ) ) @ ( one_one @ A ) ) ) ) ) ) ).

% tan_half
thf(fact_4732_arsinh__def,axiom,
    ! [A: $tType] :
      ( ( ln @ A )
     => ( ( arsinh @ A )
        = ( ^ [X: A] : ( ln_ln @ A @ ( plus_plus @ A @ X @ ( powr @ A @ ( plus_plus @ A @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ A ) ) @ ( real_Vector_of_real @ A @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ).

% arsinh_def
thf(fact_4733_complex__unimodular__polar,axiom,
    ! [Z: complex] :
      ( ( ( real_V7770717601297561774m_norm @ complex @ Z )
        = ( one_one @ real ) )
     => ~ ! [T5: real] :
            ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ T5 )
           => ( ( ord_less @ real @ T5 @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) )
             => ( Z
               != ( complex2 @ ( cos @ real @ T5 ) @ ( sin @ real @ T5 ) ) ) ) ) ) ).

% complex_unimodular_polar
thf(fact_4734_Maclaurin__exp__lt,axiom,
    ! [X2: real,N2: nat] :
      ( ( X2
       != ( zero_zero @ real ) )
     => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
       => ? [T5: real] :
            ( ( ord_less @ real @ ( zero_zero @ real ) @ ( abs_abs @ real @ T5 ) )
            & ( ord_less @ real @ ( abs_abs @ real @ T5 ) @ ( abs_abs @ real @ X2 ) )
            & ( ( exp @ real @ X2 )
              = ( plus_plus @ real
                @ ( groups7311177749621191930dd_sum @ nat @ real
                  @ ^ [M7: nat] : ( divide_divide @ real @ ( power_power @ real @ X2 @ M7 ) @ ( semiring_char_0_fact @ real @ M7 ) )
                  @ ( set_ord_lessThan @ nat @ N2 ) )
                @ ( times_times @ real @ ( divide_divide @ real @ ( exp @ real @ T5 ) @ ( semiring_char_0_fact @ real @ N2 ) ) @ ( power_power @ real @ X2 @ N2 ) ) ) ) ) ) ) ).

% Maclaurin_exp_lt
thf(fact_4735_exp__le__cancel__iff,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_eq @ real @ ( exp @ real @ X2 ) @ ( exp @ real @ Y2 ) )
      = ( ord_less_eq @ real @ X2 @ Y2 ) ) ).

% exp_le_cancel_iff
thf(fact_4736_exp__zero,axiom,
    ! [A: $tType] :
      ( ( ( real_Vector_banach @ A )
        & ( real_V2822296259951069270ebra_1 @ A ) )
     => ( ( exp @ A @ ( zero_zero @ A ) )
        = ( one_one @ A ) ) ) ).

% exp_zero
thf(fact_4737_exp__eq__one__iff,axiom,
    ! [X2: real] :
      ( ( ( exp @ real @ X2 )
        = ( one_one @ real ) )
      = ( X2
        = ( zero_zero @ real ) ) ) ).

% exp_eq_one_iff
thf(fact_4738_one__less__exp__iff,axiom,
    ! [X2: real] :
      ( ( ord_less @ real @ ( one_one @ real ) @ ( exp @ real @ X2 ) )
      = ( ord_less @ real @ ( zero_zero @ real ) @ X2 ) ) ).

% one_less_exp_iff
thf(fact_4739_exp__less__one__iff,axiom,
    ! [X2: real] :
      ( ( ord_less @ real @ ( exp @ real @ X2 ) @ ( one_one @ real ) )
      = ( ord_less @ real @ X2 @ ( zero_zero @ real ) ) ) ).

% exp_less_one_iff
thf(fact_4740_one__le__exp__iff,axiom,
    ! [X2: real] :
      ( ( ord_less_eq @ real @ ( one_one @ real ) @ ( exp @ real @ X2 ) )
      = ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 ) ) ).

% one_le_exp_iff
thf(fact_4741_exp__le__one__iff,axiom,
    ! [X2: real] :
      ( ( ord_less_eq @ real @ ( exp @ real @ X2 ) @ ( one_one @ real ) )
      = ( ord_less_eq @ real @ X2 @ ( zero_zero @ real ) ) ) ).

% exp_le_one_iff
thf(fact_4742_norm__cos__sin,axiom,
    ! [T2: real] :
      ( ( real_V7770717601297561774m_norm @ complex @ ( complex2 @ ( cos @ real @ T2 ) @ ( sin @ real @ T2 ) ) )
      = ( one_one @ real ) ) ).

% norm_cos_sin
thf(fact_4743_Complex__mult__complex__of__real,axiom,
    ! [X2: real,Y2: real,R4: real] :
      ( ( times_times @ complex @ ( complex2 @ X2 @ Y2 ) @ ( real_Vector_of_real @ complex @ R4 ) )
      = ( complex2 @ ( times_times @ real @ X2 @ R4 ) @ ( times_times @ real @ Y2 @ R4 ) ) ) ).

% Complex_mult_complex_of_real
thf(fact_4744_complex__of__real__mult__Complex,axiom,
    ! [R4: real,X2: real,Y2: real] :
      ( ( times_times @ complex @ ( real_Vector_of_real @ complex @ R4 ) @ ( complex2 @ X2 @ Y2 ) )
      = ( complex2 @ ( times_times @ real @ R4 @ X2 ) @ ( times_times @ real @ R4 @ Y2 ) ) ) ).

% complex_of_real_mult_Complex
thf(fact_4745_complex__diff,axiom,
    ! [A2: real,B2: real,C2: real,D2: real] :
      ( ( minus_minus @ complex @ ( complex2 @ A2 @ B2 ) @ ( complex2 @ C2 @ D2 ) )
      = ( complex2 @ ( minus_minus @ real @ A2 @ C2 ) @ ( minus_minus @ real @ B2 @ D2 ) ) ) ).

% complex_diff
thf(fact_4746_exp__times__arg__commute,axiom,
    ! [A: $tType] :
      ( ( ( real_Vector_banach @ A )
        & ( real_V2822296259951069270ebra_1 @ A ) )
     => ! [A3: A] :
          ( ( times_times @ A @ ( exp @ A @ A3 ) @ A3 )
          = ( times_times @ A @ A3 @ ( exp @ A @ A3 ) ) ) ) ).

% exp_times_arg_commute
thf(fact_4747_norm__exp,axiom,
    ! [A: $tType] :
      ( ( ( real_Vector_banach @ A )
        & ( real_V2822296259951069270ebra_1 @ A ) )
     => ! [X2: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( exp @ A @ X2 ) ) @ ( exp @ real @ ( real_V7770717601297561774m_norm @ A @ X2 ) ) ) ) ).

% norm_exp
thf(fact_4748_complex__scaleR,axiom,
    ! [R4: real,A2: real,B2: real] :
      ( ( real_V8093663219630862766scaleR @ complex @ R4 @ ( complex2 @ A2 @ B2 ) )
      = ( complex2 @ ( times_times @ real @ R4 @ A2 ) @ ( times_times @ real @ R4 @ B2 ) ) ) ).

% complex_scaleR
thf(fact_4749_exp__ge__zero,axiom,
    ! [X2: real] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( exp @ real @ X2 ) ) ).

% exp_ge_zero
thf(fact_4750_not__exp__le__zero,axiom,
    ! [X2: real] :
      ~ ( ord_less_eq @ real @ ( exp @ real @ X2 ) @ ( zero_zero @ real ) ) ).

% not_exp_le_zero
thf(fact_4751_mult__exp__exp,axiom,
    ! [A: $tType] :
      ( ( ( real_Vector_banach @ A )
        & ( real_V3459762299906320749_field @ A ) )
     => ! [X2: A,Y2: A] :
          ( ( times_times @ A @ ( exp @ A @ X2 ) @ ( exp @ A @ Y2 ) )
          = ( exp @ A @ ( plus_plus @ A @ X2 @ Y2 ) ) ) ) ).

% mult_exp_exp
thf(fact_4752_exp__add__commuting,axiom,
    ! [A: $tType] :
      ( ( ( real_Vector_banach @ A )
        & ( real_V2822296259951069270ebra_1 @ A ) )
     => ! [X2: A,Y2: A] :
          ( ( ( times_times @ A @ X2 @ Y2 )
            = ( times_times @ A @ Y2 @ X2 ) )
         => ( ( exp @ A @ ( plus_plus @ A @ X2 @ Y2 ) )
            = ( times_times @ A @ ( exp @ A @ X2 ) @ ( exp @ A @ Y2 ) ) ) ) ) ).

% exp_add_commuting
thf(fact_4753_exp__diff,axiom,
    ! [A: $tType] :
      ( ( ( real_Vector_banach @ A )
        & ( real_V3459762299906320749_field @ A ) )
     => ! [X2: A,Y2: A] :
          ( ( exp @ A @ ( minus_minus @ A @ X2 @ Y2 ) )
          = ( divide_divide @ A @ ( exp @ A @ X2 ) @ ( exp @ A @ Y2 ) ) ) ) ).

% exp_diff
thf(fact_4754_Complex__eq__numeral,axiom,
    ! [A2: real,B2: real,W: num] :
      ( ( ( complex2 @ A2 @ B2 )
        = ( numeral_numeral @ complex @ W ) )
      = ( ( A2
          = ( numeral_numeral @ real @ W ) )
        & ( B2
          = ( zero_zero @ real ) ) ) ) ).

% Complex_eq_numeral
thf(fact_4755_complex__add,axiom,
    ! [A2: real,B2: real,C2: real,D2: real] :
      ( ( plus_plus @ complex @ ( complex2 @ A2 @ B2 ) @ ( complex2 @ C2 @ D2 ) )
      = ( complex2 @ ( plus_plus @ real @ A2 @ C2 ) @ ( plus_plus @ real @ B2 @ D2 ) ) ) ).

% complex_add
thf(fact_4756_Complex__add__complex__of__real,axiom,
    ! [X2: real,Y2: real,R4: real] :
      ( ( plus_plus @ complex @ ( complex2 @ X2 @ Y2 ) @ ( real_Vector_of_real @ complex @ R4 ) )
      = ( complex2 @ ( plus_plus @ real @ X2 @ R4 ) @ Y2 ) ) ).

% Complex_add_complex_of_real
thf(fact_4757_complex__of__real__add__Complex,axiom,
    ! [R4: real,X2: real,Y2: real] :
      ( ( plus_plus @ complex @ ( real_Vector_of_real @ complex @ R4 ) @ ( complex2 @ X2 @ Y2 ) )
      = ( complex2 @ ( plus_plus @ real @ R4 @ X2 ) @ Y2 ) ) ).

% complex_of_real_add_Complex
thf(fact_4758_exp__gt__one,axiom,
    ! [X2: real] :
      ( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
     => ( ord_less @ real @ ( one_one @ real ) @ ( exp @ real @ X2 ) ) ) ).

% exp_gt_one
thf(fact_4759_exp__ge__add__one__self,axiom,
    ! [X2: real] : ( ord_less_eq @ real @ ( plus_plus @ real @ ( one_one @ real ) @ X2 ) @ ( exp @ real @ X2 ) ) ).

% exp_ge_add_one_self
thf(fact_4760_exp__minus__inverse,axiom,
    ! [A: $tType] :
      ( ( ( real_Vector_banach @ A )
        & ( real_V2822296259951069270ebra_1 @ A ) )
     => ! [X2: A] :
          ( ( times_times @ A @ ( exp @ A @ X2 ) @ ( exp @ A @ ( uminus_uminus @ A @ X2 ) ) )
          = ( one_one @ A ) ) ) ).

% exp_minus_inverse
thf(fact_4761_exp__of__nat__mult,axiom,
    ! [A: $tType] :
      ( ( ( real_Vector_banach @ A )
        & ( real_V3459762299906320749_field @ A ) )
     => ! [N2: nat,X2: A] :
          ( ( exp @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N2 ) @ X2 ) )
          = ( power_power @ A @ ( exp @ A @ X2 ) @ N2 ) ) ) ).

% exp_of_nat_mult
thf(fact_4762_exp__of__nat2__mult,axiom,
    ! [A: $tType] :
      ( ( ( real_Vector_banach @ A )
        & ( real_V3459762299906320749_field @ A ) )
     => ! [X2: A,N2: nat] :
          ( ( exp @ A @ ( times_times @ A @ X2 @ ( semiring_1_of_nat @ A @ N2 ) ) )
          = ( power_power @ A @ ( exp @ A @ X2 ) @ N2 ) ) ) ).

% exp_of_nat2_mult
thf(fact_4763_Complex__eq__neg__numeral,axiom,
    ! [A2: real,B2: real,W: num] :
      ( ( ( complex2 @ A2 @ B2 )
        = ( uminus_uminus @ complex @ ( numeral_numeral @ complex @ W ) ) )
      = ( ( A2
          = ( uminus_uminus @ real @ ( numeral_numeral @ real @ W ) ) )
        & ( B2
          = ( zero_zero @ real ) ) ) ) ).

% Complex_eq_neg_numeral
thf(fact_4764_log__ln,axiom,
    ( ( ln_ln @ real )
    = ( log @ ( exp @ real @ ( one_one @ real ) ) ) ) ).

% log_ln
thf(fact_4765_complex__mult,axiom,
    ! [A2: real,B2: real,C2: real,D2: real] :
      ( ( times_times @ complex @ ( complex2 @ A2 @ B2 ) @ ( complex2 @ C2 @ D2 ) )
      = ( complex2 @ ( minus_minus @ real @ ( times_times @ real @ A2 @ C2 ) @ ( times_times @ real @ B2 @ D2 ) ) @ ( plus_plus @ real @ ( times_times @ real @ A2 @ D2 ) @ ( times_times @ real @ B2 @ C2 ) ) ) ) ).

% complex_mult
thf(fact_4766_one__complex_Ocode,axiom,
    ( ( one_one @ complex )
    = ( complex2 @ ( one_one @ real ) @ ( zero_zero @ real ) ) ) ).

% one_complex.code
thf(fact_4767_Complex__eq__1,axiom,
    ! [A2: real,B2: real] :
      ( ( ( complex2 @ A2 @ B2 )
        = ( one_one @ complex ) )
      = ( ( A2
          = ( one_one @ real ) )
        & ( B2
          = ( zero_zero @ real ) ) ) ) ).

% Complex_eq_1
thf(fact_4768_exp__sum,axiom,
    ! [B: $tType,A: $tType] :
      ( ( ( comm_monoid_mult @ B )
        & ( real_Vector_banach @ B )
        & ( real_V2822296259951069270ebra_1 @ B ) )
     => ! [I5: set @ A,F: A > B] :
          ( ( finite_finite2 @ A @ I5 )
         => ( ( exp @ B @ ( groups7311177749621191930dd_sum @ A @ B @ F @ I5 ) )
            = ( groups7121269368397514597t_prod @ A @ B
              @ ^ [X: A] : ( exp @ B @ ( F @ X ) )
              @ I5 ) ) ) ) ).

% exp_sum
thf(fact_4769_exp__ge__add__one__self__aux,axiom,
    ! [X2: real] :
      ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 )
     => ( ord_less_eq @ real @ ( plus_plus @ real @ ( one_one @ real ) @ X2 ) @ ( exp @ real @ X2 ) ) ) ).

% exp_ge_add_one_self_aux
thf(fact_4770_lemma__exp__total,axiom,
    ! [Y2: real] :
      ( ( ord_less_eq @ real @ ( one_one @ real ) @ Y2 )
     => ? [X3: real] :
          ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X3 )
          & ( ord_less_eq @ real @ X3 @ ( minus_minus @ real @ Y2 @ ( one_one @ real ) ) )
          & ( ( exp @ real @ X3 )
            = Y2 ) ) ) ).

% lemma_exp_total
thf(fact_4771_ln__ge__iff,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
     => ( ( ord_less_eq @ real @ Y2 @ ( ln_ln @ real @ X2 ) )
        = ( ord_less_eq @ real @ ( exp @ real @ Y2 ) @ X2 ) ) ) ).

% ln_ge_iff
thf(fact_4772_ln__x__over__x__mono,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_eq @ real @ ( exp @ real @ ( one_one @ real ) ) @ X2 )
     => ( ( ord_less_eq @ real @ X2 @ Y2 )
       => ( ord_less_eq @ real @ ( divide_divide @ real @ ( ln_ln @ real @ Y2 ) @ Y2 ) @ ( divide_divide @ real @ ( ln_ln @ real @ X2 ) @ X2 ) ) ) ) ).

% ln_x_over_x_mono
thf(fact_4773_Complex__sum_H,axiom,
    ! [A: $tType,F: A > real,S4: set @ A] :
      ( ( groups7311177749621191930dd_sum @ A @ complex
        @ ^ [X: A] : ( complex2 @ ( F @ X ) @ ( zero_zero @ real ) )
        @ S4 )
      = ( complex2 @ ( groups7311177749621191930dd_sum @ A @ real @ F @ S4 ) @ ( zero_zero @ real ) ) ) ).

% Complex_sum'
thf(fact_4774_powr__def,axiom,
    ! [A: $tType] :
      ( ( ln @ A )
     => ( ( powr @ A )
        = ( ^ [X: A,A5: A] :
              ( if @ A
              @ ( X
                = ( zero_zero @ A ) )
              @ ( zero_zero @ A )
              @ ( exp @ A @ ( times_times @ A @ A5 @ ( ln_ln @ A @ X ) ) ) ) ) ) ) ).

% powr_def
thf(fact_4775_Complex__eq__neg__1,axiom,
    ! [A2: real,B2: real] :
      ( ( ( complex2 @ A2 @ B2 )
        = ( uminus_uminus @ complex @ ( one_one @ complex ) ) )
      = ( ( A2
          = ( uminus_uminus @ real @ ( one_one @ real ) ) )
        & ( B2
          = ( zero_zero @ real ) ) ) ) ).

% Complex_eq_neg_1
thf(fact_4776_exp__le,axiom,
    ord_less_eq @ real @ ( exp @ real @ ( one_one @ real ) ) @ ( numeral_numeral @ real @ ( bit1 @ one2 ) ) ).

% exp_le
thf(fact_4777_exp__divide__power__eq,axiom,
    ! [A: $tType] :
      ( ( ( real_Vector_banach @ A )
        & ( real_V3459762299906320749_field @ A ) )
     => ! [N2: nat,X2: A] :
          ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
         => ( ( power_power @ A @ ( exp @ A @ ( divide_divide @ A @ X2 @ ( semiring_1_of_nat @ A @ N2 ) ) ) @ N2 )
            = ( exp @ A @ X2 ) ) ) ) ).

% exp_divide_power_eq
thf(fact_4778_tanh__altdef,axiom,
    ! [A: $tType] :
      ( ( ( real_Vector_banach @ A )
        & ( real_V3459762299906320749_field @ A ) )
     => ( ( tanh @ A )
        = ( ^ [X: A] : ( divide_divide @ A @ ( minus_minus @ A @ ( exp @ A @ X ) @ ( exp @ A @ ( uminus_uminus @ A @ X ) ) ) @ ( plus_plus @ A @ ( exp @ A @ X ) @ ( exp @ A @ ( uminus_uminus @ A @ X ) ) ) ) ) ) ) ).

% tanh_altdef
thf(fact_4779_exp__half__le2,axiom,
    ord_less_eq @ real @ ( exp @ real @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ).

% exp_half_le2
thf(fact_4780_exp__double,axiom,
    ! [A: $tType] :
      ( ( ( real_Vector_banach @ A )
        & ( real_V2822296259951069270ebra_1 @ A ) )
     => ! [Z: A] :
          ( ( exp @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ Z ) )
          = ( power_power @ A @ ( exp @ A @ Z ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).

% exp_double
thf(fact_4781_exp__bound__half,axiom,
    ! [A: $tType] :
      ( ( ( real_Vector_banach @ A )
        & ( real_V2822296259951069270ebra_1 @ A ) )
     => ! [Z: A] :
          ( ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ Z ) @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
         => ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( exp @ A @ Z ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ).

% exp_bound_half
thf(fact_4782_exp__bound,axiom,
    ! [X2: real] :
      ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 )
     => ( ( ord_less_eq @ real @ X2 @ ( one_one @ real ) )
       => ( ord_less_eq @ real @ ( exp @ real @ X2 ) @ ( plus_plus @ real @ ( plus_plus @ real @ ( one_one @ real ) @ X2 ) @ ( power_power @ real @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ).

% exp_bound
thf(fact_4783_real__exp__bound__lemma,axiom,
    ! [X2: real] :
      ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 )
     => ( ( ord_less_eq @ real @ X2 @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
       => ( ord_less_eq @ real @ ( exp @ real @ X2 ) @ ( plus_plus @ real @ ( one_one @ real ) @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ X2 ) ) ) ) ) ).

% real_exp_bound_lemma
thf(fact_4784_exp__ge__one__plus__x__over__n__power__n,axiom,
    ! [N2: nat,X2: real] :
      ( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( semiring_1_of_nat @ real @ N2 ) ) @ X2 )
     => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
       => ( ord_less_eq @ real @ ( power_power @ real @ ( plus_plus @ real @ ( one_one @ real ) @ ( divide_divide @ real @ X2 @ ( semiring_1_of_nat @ real @ N2 ) ) ) @ N2 ) @ ( exp @ real @ X2 ) ) ) ) ).

% exp_ge_one_plus_x_over_n_power_n
thf(fact_4785_exp__ge__one__minus__x__over__n__power__n,axiom,
    ! [X2: real,N2: nat] :
      ( ( ord_less_eq @ real @ X2 @ ( semiring_1_of_nat @ real @ N2 ) )
     => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
       => ( ord_less_eq @ real @ ( power_power @ real @ ( minus_minus @ real @ ( one_one @ real ) @ ( divide_divide @ real @ X2 @ ( semiring_1_of_nat @ real @ N2 ) ) ) @ N2 ) @ ( exp @ real @ ( uminus_uminus @ real @ X2 ) ) ) ) ) ).

% exp_ge_one_minus_x_over_n_power_n
thf(fact_4786_exp__bound__lemma,axiom,
    ! [A: $tType] :
      ( ( ( real_Vector_banach @ A )
        & ( real_V2822296259951069270ebra_1 @ A ) )
     => ! [Z: A] :
          ( ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ Z ) @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
         => ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( exp @ A @ Z ) ) @ ( plus_plus @ real @ ( one_one @ real ) @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( real_V7770717601297561774m_norm @ A @ Z ) ) ) ) ) ) ).

% exp_bound_lemma
thf(fact_4787_Maclaurin__exp__le,axiom,
    ! [X2: real,N2: nat] :
    ? [T5: real] :
      ( ( ord_less_eq @ real @ ( abs_abs @ real @ T5 ) @ ( abs_abs @ real @ X2 ) )
      & ( ( exp @ real @ X2 )
        = ( plus_plus @ real
          @ ( groups7311177749621191930dd_sum @ nat @ real
            @ ^ [M7: nat] : ( divide_divide @ real @ ( power_power @ real @ X2 @ M7 ) @ ( semiring_char_0_fact @ real @ M7 ) )
            @ ( set_ord_lessThan @ nat @ N2 ) )
          @ ( times_times @ real @ ( divide_divide @ real @ ( exp @ real @ T5 ) @ ( semiring_char_0_fact @ real @ N2 ) ) @ ( power_power @ real @ X2 @ N2 ) ) ) ) ) ).

% Maclaurin_exp_le
thf(fact_4788_exp__lower__Taylor__quadratic,axiom,
    ! [X2: real] :
      ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 )
     => ( ord_less_eq @ real @ ( plus_plus @ real @ ( plus_plus @ real @ ( one_one @ real ) @ X2 ) @ ( divide_divide @ real @ ( power_power @ real @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ ( exp @ real @ X2 ) ) ) ).

% exp_lower_Taylor_quadratic
thf(fact_4789_log__base__10__eq2,axiom,
    ! [X2: real] :
      ( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
     => ( ( log @ ( numeral_numeral @ real @ ( bit0 @ ( bit1 @ ( bit0 @ one2 ) ) ) ) @ X2 )
        = ( times_times @ real @ ( log @ ( numeral_numeral @ real @ ( bit0 @ ( bit1 @ ( bit0 @ one2 ) ) ) ) @ ( exp @ real @ ( one_one @ real ) ) ) @ ( ln_ln @ real @ X2 ) ) ) ) ).

% log_base_10_eq2
thf(fact_4790_tanh__real__altdef,axiom,
    ( ( tanh @ real )
    = ( ^ [X: real] : ( divide_divide @ real @ ( minus_minus @ real @ ( one_one @ real ) @ ( exp @ real @ ( times_times @ real @ ( uminus_uminus @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ X ) ) ) @ ( plus_plus @ real @ ( one_one @ real ) @ ( exp @ real @ ( times_times @ real @ ( uminus_uminus @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ X ) ) ) ) ) ) ).

% tanh_real_altdef
thf(fact_4791_log__base__10__eq1,axiom,
    ! [X2: real] :
      ( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
     => ( ( log @ ( numeral_numeral @ real @ ( bit0 @ ( bit1 @ ( bit0 @ one2 ) ) ) ) @ X2 )
        = ( times_times @ real @ ( divide_divide @ real @ ( ln_ln @ real @ ( exp @ real @ ( one_one @ real ) ) ) @ ( ln_ln @ real @ ( numeral_numeral @ real @ ( bit0 @ ( bit1 @ ( bit0 @ one2 ) ) ) ) ) ) @ ( ln_ln @ real @ X2 ) ) ) ) ).

% log_base_10_eq1
thf(fact_4792_sin__tan,axiom,
    ! [X2: real] :
      ( ( ord_less @ real @ ( abs_abs @ real @ X2 ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
     => ( ( sin @ real @ X2 )
        = ( divide_divide @ real @ ( tan @ real @ X2 ) @ ( sqrt @ ( plus_plus @ real @ ( one_one @ real ) @ ( power_power @ real @ ( tan @ real @ X2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).

% sin_tan
thf(fact_4793_cos__tan,axiom,
    ! [X2: real] :
      ( ( ord_less @ real @ ( abs_abs @ real @ X2 ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
     => ( ( cos @ real @ X2 )
        = ( divide_divide @ real @ ( one_one @ real ) @ ( sqrt @ ( plus_plus @ real @ ( one_one @ real ) @ ( power_power @ real @ ( tan @ real @ X2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).

% cos_tan
thf(fact_4794_monoI1,axiom,
    ! [A: $tType] :
      ( ( order @ A )
     => ! [X8: nat > A] :
          ( ! [M2: nat,N4: nat] :
              ( ( ord_less_eq @ nat @ M2 @ N4 )
             => ( ord_less_eq @ A @ ( X8 @ M2 ) @ ( X8 @ N4 ) ) )
         => ( topological_monoseq @ A @ X8 ) ) ) ).

% monoI1
thf(fact_4795_real__sqrt__eq__iff,axiom,
    ! [X2: real,Y2: real] :
      ( ( ( sqrt @ X2 )
        = ( sqrt @ Y2 ) )
      = ( X2 = Y2 ) ) ).

% real_sqrt_eq_iff
thf(fact_4796_real__sqrt__zero,axiom,
    ( ( sqrt @ ( zero_zero @ real ) )
    = ( zero_zero @ real ) ) ).

% real_sqrt_zero
thf(fact_4797_real__sqrt__eq__zero__cancel__iff,axiom,
    ! [X2: real] :
      ( ( ( sqrt @ X2 )
        = ( zero_zero @ real ) )
      = ( X2
        = ( zero_zero @ real ) ) ) ).

% real_sqrt_eq_zero_cancel_iff
thf(fact_4798_real__sqrt__less__iff,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less @ real @ ( sqrt @ X2 ) @ ( sqrt @ Y2 ) )
      = ( ord_less @ real @ X2 @ Y2 ) ) ).

% real_sqrt_less_iff
thf(fact_4799_real__sqrt__le__iff,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_eq @ real @ ( sqrt @ X2 ) @ ( sqrt @ Y2 ) )
      = ( ord_less_eq @ real @ X2 @ Y2 ) ) ).

% real_sqrt_le_iff
thf(fact_4800_real__sqrt__one,axiom,
    ( ( sqrt @ ( one_one @ real ) )
    = ( one_one @ real ) ) ).

% real_sqrt_one
thf(fact_4801_real__sqrt__eq__1__iff,axiom,
    ! [X2: real] :
      ( ( ( sqrt @ X2 )
        = ( one_one @ real ) )
      = ( X2
        = ( one_one @ real ) ) ) ).

% real_sqrt_eq_1_iff
thf(fact_4802_real__sqrt__gt__0__iff,axiom,
    ! [Y2: real] :
      ( ( ord_less @ real @ ( zero_zero @ real ) @ ( sqrt @ Y2 ) )
      = ( ord_less @ real @ ( zero_zero @ real ) @ Y2 ) ) ).

% real_sqrt_gt_0_iff
thf(fact_4803_real__sqrt__lt__0__iff,axiom,
    ! [X2: real] :
      ( ( ord_less @ real @ ( sqrt @ X2 ) @ ( zero_zero @ real ) )
      = ( ord_less @ real @ X2 @ ( zero_zero @ real ) ) ) ).

% real_sqrt_lt_0_iff
thf(fact_4804_real__sqrt__le__0__iff,axiom,
    ! [X2: real] :
      ( ( ord_less_eq @ real @ ( sqrt @ X2 ) @ ( zero_zero @ real ) )
      = ( ord_less_eq @ real @ X2 @ ( zero_zero @ real ) ) ) ).

% real_sqrt_le_0_iff
thf(fact_4805_real__sqrt__ge__0__iff,axiom,
    ! [Y2: real] :
      ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( sqrt @ Y2 ) )
      = ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y2 ) ) ).

% real_sqrt_ge_0_iff
thf(fact_4806_real__sqrt__lt__1__iff,axiom,
    ! [X2: real] :
      ( ( ord_less @ real @ ( sqrt @ X2 ) @ ( one_one @ real ) )
      = ( ord_less @ real @ X2 @ ( one_one @ real ) ) ) ).

% real_sqrt_lt_1_iff
thf(fact_4807_real__sqrt__gt__1__iff,axiom,
    ! [Y2: real] :
      ( ( ord_less @ real @ ( one_one @ real ) @ ( sqrt @ Y2 ) )
      = ( ord_less @ real @ ( one_one @ real ) @ Y2 ) ) ).

% real_sqrt_gt_1_iff
thf(fact_4808_real__sqrt__le__1__iff,axiom,
    ! [X2: real] :
      ( ( ord_less_eq @ real @ ( sqrt @ X2 ) @ ( one_one @ real ) )
      = ( ord_less_eq @ real @ X2 @ ( one_one @ real ) ) ) ).

% real_sqrt_le_1_iff
thf(fact_4809_real__sqrt__ge__1__iff,axiom,
    ! [Y2: real] :
      ( ( ord_less_eq @ real @ ( one_one @ real ) @ ( sqrt @ Y2 ) )
      = ( ord_less_eq @ real @ ( one_one @ real ) @ Y2 ) ) ).

% real_sqrt_ge_1_iff
thf(fact_4810_real__sqrt__abs2,axiom,
    ! [X2: real] :
      ( ( sqrt @ ( times_times @ real @ X2 @ X2 ) )
      = ( abs_abs @ real @ X2 ) ) ).

% real_sqrt_abs2
thf(fact_4811_real__sqrt__mult__self,axiom,
    ! [A2: real] :
      ( ( times_times @ real @ ( sqrt @ A2 ) @ ( sqrt @ A2 ) )
      = ( abs_abs @ real @ A2 ) ) ).

% real_sqrt_mult_self
thf(fact_4812_real__sqrt__four,axiom,
    ( ( sqrt @ ( numeral_numeral @ real @ ( bit0 @ ( bit0 @ one2 ) ) ) )
    = ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ).

% real_sqrt_four
thf(fact_4813_real__sqrt__abs,axiom,
    ! [X2: real] :
      ( ( sqrt @ ( power_power @ real @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
      = ( abs_abs @ real @ X2 ) ) ).

% real_sqrt_abs
thf(fact_4814_real__sqrt__pow2__iff,axiom,
    ! [X2: real] :
      ( ( ( power_power @ real @ ( sqrt @ X2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
        = X2 )
      = ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 ) ) ).

% real_sqrt_pow2_iff
thf(fact_4815_real__sqrt__pow2,axiom,
    ! [X2: real] :
      ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 )
     => ( ( power_power @ real @ ( sqrt @ X2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
        = X2 ) ) ).

% real_sqrt_pow2
thf(fact_4816_real__sqrt__sum__squares__mult__squared__eq,axiom,
    ! [X2: real,Y2: real,Xa: real,Ya: real] :
      ( ( power_power @ real @ ( sqrt @ ( times_times @ real @ ( plus_plus @ real @ ( power_power @ real @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( plus_plus @ real @ ( power_power @ real @ Xa @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Ya @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
      = ( times_times @ real @ ( plus_plus @ real @ ( power_power @ real @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( plus_plus @ real @ ( power_power @ real @ Xa @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Ya @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).

% real_sqrt_sum_squares_mult_squared_eq
thf(fact_4817_complex__exp__exists,axiom,
    ! [Z: complex] :
    ? [A4: complex,R3: real] :
      ( Z
      = ( times_times @ complex @ ( real_Vector_of_real @ complex @ R3 ) @ ( exp @ complex @ A4 ) ) ) ).

% complex_exp_exists
thf(fact_4818_real__sqrt__le__mono,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_eq @ real @ X2 @ Y2 )
     => ( ord_less_eq @ real @ ( sqrt @ X2 ) @ ( sqrt @ Y2 ) ) ) ).

% real_sqrt_le_mono
thf(fact_4819_real__sqrt__mult,axiom,
    ! [X2: real,Y2: real] :
      ( ( sqrt @ ( times_times @ real @ X2 @ Y2 ) )
      = ( times_times @ real @ ( sqrt @ X2 ) @ ( sqrt @ Y2 ) ) ) ).

% real_sqrt_mult
thf(fact_4820_real__sqrt__divide,axiom,
    ! [X2: real,Y2: real] :
      ( ( sqrt @ ( divide_divide @ real @ X2 @ Y2 ) )
      = ( divide_divide @ real @ ( sqrt @ X2 ) @ ( sqrt @ Y2 ) ) ) ).

% real_sqrt_divide
thf(fact_4821_real__sqrt__power,axiom,
    ! [X2: real,K: nat] :
      ( ( sqrt @ ( power_power @ real @ X2 @ K ) )
      = ( power_power @ real @ ( sqrt @ X2 ) @ K ) ) ).

% real_sqrt_power
thf(fact_4822_real__sqrt__less__mono,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less @ real @ X2 @ Y2 )
     => ( ord_less @ real @ ( sqrt @ X2 ) @ ( sqrt @ Y2 ) ) ) ).

% real_sqrt_less_mono
thf(fact_4823_real__sqrt__minus,axiom,
    ! [X2: real] :
      ( ( sqrt @ ( uminus_uminus @ real @ X2 ) )
      = ( uminus_uminus @ real @ ( sqrt @ X2 ) ) ) ).

% real_sqrt_minus
thf(fact_4824_real__sqrt__gt__zero,axiom,
    ! [X2: real] :
      ( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
     => ( ord_less @ real @ ( zero_zero @ real ) @ ( sqrt @ X2 ) ) ) ).

% real_sqrt_gt_zero
thf(fact_4825_real__sqrt__eq__zero__cancel,axiom,
    ! [X2: real] :
      ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 )
     => ( ( ( sqrt @ X2 )
          = ( zero_zero @ real ) )
       => ( X2
          = ( zero_zero @ real ) ) ) ) ).

% real_sqrt_eq_zero_cancel
thf(fact_4826_real__sqrt__ge__zero,axiom,
    ! [X2: real] :
      ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 )
     => ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( sqrt @ X2 ) ) ) ).

% real_sqrt_ge_zero
thf(fact_4827_real__sqrt__ge__one,axiom,
    ! [X2: real] :
      ( ( ord_less_eq @ real @ ( one_one @ real ) @ X2 )
     => ( ord_less_eq @ real @ ( one_one @ real ) @ ( sqrt @ X2 ) ) ) ).

% real_sqrt_ge_one
thf(fact_4828_real__div__sqrt,axiom,
    ! [X2: real] :
      ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 )
     => ( ( divide_divide @ real @ X2 @ ( sqrt @ X2 ) )
        = ( sqrt @ X2 ) ) ) ).

% real_div_sqrt
thf(fact_4829_sqrt__add__le__add__sqrt,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 )
     => ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y2 )
       => ( ord_less_eq @ real @ ( sqrt @ ( plus_plus @ real @ X2 @ Y2 ) ) @ ( plus_plus @ real @ ( sqrt @ X2 ) @ ( sqrt @ Y2 ) ) ) ) ) ).

% sqrt_add_le_add_sqrt
thf(fact_4830_le__real__sqrt__sumsq,axiom,
    ! [X2: real,Y2: real] : ( ord_less_eq @ real @ X2 @ ( sqrt @ ( plus_plus @ real @ ( times_times @ real @ X2 @ X2 ) @ ( times_times @ real @ Y2 @ Y2 ) ) ) ) ).

% le_real_sqrt_sumsq
thf(fact_4831_sqrt2__less__2,axiom,
    ord_less @ real @ ( sqrt @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ).

% sqrt2_less_2
thf(fact_4832_real__less__rsqrt,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less @ real @ ( power_power @ real @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ Y2 )
     => ( ord_less @ real @ X2 @ ( sqrt @ Y2 ) ) ) ).

% real_less_rsqrt
thf(fact_4833_real__le__rsqrt,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_eq @ real @ ( power_power @ real @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ Y2 )
     => ( ord_less_eq @ real @ X2 @ ( sqrt @ Y2 ) ) ) ).

% real_le_rsqrt
thf(fact_4834_sqrt__le__D,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_eq @ real @ ( sqrt @ X2 ) @ Y2 )
     => ( ord_less_eq @ real @ X2 @ ( power_power @ real @ Y2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).

% sqrt_le_D
thf(fact_4835_real__sqrt__unique,axiom,
    ! [Y2: real,X2: real] :
      ( ( ( power_power @ real @ Y2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
        = X2 )
     => ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y2 )
       => ( ( sqrt @ X2 )
          = Y2 ) ) ) ).

% real_sqrt_unique
thf(fact_4836_real__le__lsqrt,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 )
     => ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y2 )
       => ( ( ord_less_eq @ real @ X2 @ ( power_power @ real @ Y2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
         => ( ord_less_eq @ real @ ( sqrt @ X2 ) @ Y2 ) ) ) ) ).

% real_le_lsqrt
thf(fact_4837_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 @ one2 ) ) ) ) @ U ) ) ).

% lemma_real_divide_sqrt_less
thf(fact_4838_real__sqrt__sum__squares__eq__cancel2,axiom,
    ! [X2: real,Y2: real] :
      ( ( ( sqrt @ ( plus_plus @ real @ ( power_power @ real @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
        = Y2 )
     => ( X2
        = ( zero_zero @ real ) ) ) ).

% real_sqrt_sum_squares_eq_cancel2
thf(fact_4839_real__sqrt__sum__squares__eq__cancel,axiom,
    ! [X2: real,Y2: real] :
      ( ( ( sqrt @ ( plus_plus @ real @ ( power_power @ real @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
        = X2 )
     => ( Y2
        = ( zero_zero @ real ) ) ) ).

% real_sqrt_sum_squares_eq_cancel
thf(fact_4840_real__sqrt__sum__squares__triangle__ineq,axiom,
    ! [A2: real,C2: real,B2: real,D2: real] : ( ord_less_eq @ real @ ( sqrt @ ( plus_plus @ real @ ( power_power @ real @ ( plus_plus @ real @ A2 @ C2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ ( plus_plus @ real @ B2 @ D2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( plus_plus @ real @ ( sqrt @ ( plus_plus @ real @ ( power_power @ real @ A2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ B2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( sqrt @ ( plus_plus @ real @ ( power_power @ real @ C2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ D2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ).

% real_sqrt_sum_squares_triangle_ineq
thf(fact_4841_real__sqrt__sum__squares__ge2,axiom,
    ! [Y2: real,X2: real] : ( ord_less_eq @ real @ Y2 @ ( sqrt @ ( plus_plus @ real @ ( power_power @ real @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).

% real_sqrt_sum_squares_ge2
thf(fact_4842_real__sqrt__sum__squares__ge1,axiom,
    ! [X2: real,Y2: real] : ( ord_less_eq @ real @ X2 @ ( sqrt @ ( plus_plus @ real @ ( power_power @ real @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).

% real_sqrt_sum_squares_ge1
thf(fact_4843_sqrt__ge__absD,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_eq @ real @ ( abs_abs @ real @ X2 ) @ ( sqrt @ Y2 ) )
     => ( ord_less_eq @ real @ ( power_power @ real @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ Y2 ) ) ).

% sqrt_ge_absD
thf(fact_4844_cos__45,axiom,
    ( ( cos @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ ( bit0 @ one2 ) ) ) ) )
    = ( divide_divide @ real @ ( sqrt @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ).

% cos_45
thf(fact_4845_sin__45,axiom,
    ( ( sin @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ ( bit0 @ one2 ) ) ) ) )
    = ( divide_divide @ real @ ( sqrt @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ).

% sin_45
thf(fact_4846_tan__60,axiom,
    ( ( tan @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit1 @ one2 ) ) ) )
    = ( sqrt @ ( numeral_numeral @ real @ ( bit1 @ one2 ) ) ) ) ).

% tan_60
thf(fact_4847_real__less__lsqrt,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 )
     => ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y2 )
       => ( ( ord_less @ real @ X2 @ ( power_power @ real @ Y2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
         => ( ord_less @ real @ ( sqrt @ X2 ) @ Y2 ) ) ) ) ).

% real_less_lsqrt
thf(fact_4848_sqrt__sum__squares__le__sum,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 )
     => ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y2 )
       => ( ord_less_eq @ real @ ( sqrt @ ( plus_plus @ real @ ( power_power @ real @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( plus_plus @ real @ X2 @ Y2 ) ) ) ) ).

% sqrt_sum_squares_le_sum
thf(fact_4849_sqrt__even__pow2,axiom,
    ! [N2: nat] :
      ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 )
     => ( ( sqrt @ ( power_power @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ N2 ) )
        = ( power_power @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).

% sqrt_even_pow2
thf(fact_4850_real__sqrt__ge__abs1,axiom,
    ! [X2: real,Y2: real] : ( ord_less_eq @ real @ ( abs_abs @ real @ X2 ) @ ( sqrt @ ( plus_plus @ real @ ( power_power @ real @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).

% real_sqrt_ge_abs1
thf(fact_4851_real__sqrt__ge__abs2,axiom,
    ! [Y2: real,X2: real] : ( ord_less_eq @ real @ ( abs_abs @ real @ Y2 ) @ ( sqrt @ ( plus_plus @ real @ ( power_power @ real @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).

% real_sqrt_ge_abs2
thf(fact_4852_sqrt__sum__squares__le__sum__abs,axiom,
    ! [X2: real,Y2: real] : ( ord_less_eq @ real @ ( sqrt @ ( plus_plus @ real @ ( power_power @ real @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( plus_plus @ real @ ( abs_abs @ real @ X2 ) @ ( abs_abs @ real @ Y2 ) ) ) ).

% sqrt_sum_squares_le_sum_abs
thf(fact_4853_ln__sqrt,axiom,
    ! [X2: real] :
      ( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
     => ( ( ln_ln @ real @ ( sqrt @ X2 ) )
        = ( divide_divide @ real @ ( ln_ln @ real @ X2 ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ).

% ln_sqrt
thf(fact_4854_cos__30,axiom,
    ( ( cos @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ ( bit1 @ one2 ) ) ) ) )
    = ( divide_divide @ real @ ( sqrt @ ( numeral_numeral @ real @ ( bit1 @ one2 ) ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ).

% cos_30
thf(fact_4855_sin__60,axiom,
    ( ( sin @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit1 @ one2 ) ) ) )
    = ( divide_divide @ real @ ( sqrt @ ( numeral_numeral @ real @ ( bit1 @ one2 ) ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ).

% sin_60
thf(fact_4856_arsinh__real__def,axiom,
    ( ( arsinh @ real )
    = ( ^ [X: real] : ( ln_ln @ real @ ( plus_plus @ real @ X @ ( sqrt @ ( plus_plus @ real @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ real ) ) ) ) ) ) ) ).

% arsinh_real_def
thf(fact_4857_complex__norm,axiom,
    ! [X2: real,Y2: real] :
      ( ( real_V7770717601297561774m_norm @ complex @ ( complex2 @ X2 @ Y2 ) )
      = ( sqrt @ ( plus_plus @ real @ ( power_power @ real @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).

% complex_norm
thf(fact_4858_real__sqrt__power__even,axiom,
    ! [N2: nat,X2: real] :
      ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 )
     => ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 )
       => ( ( power_power @ real @ ( sqrt @ X2 ) @ N2 )
          = ( power_power @ real @ X2 @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ).

% real_sqrt_power_even
thf(fact_4859_arsinh__real__aux,axiom,
    ! [X2: real] : ( ord_less @ real @ ( zero_zero @ real ) @ ( plus_plus @ real @ X2 @ ( sqrt @ ( plus_plus @ real @ ( power_power @ real @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ real ) ) ) ) ) ).

% arsinh_real_aux
thf(fact_4860_real__sqrt__sum__squares__mult__ge__zero,axiom,
    ! [X2: real,Y2: real,Xa: real,Ya: real] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( sqrt @ ( times_times @ real @ ( plus_plus @ real @ ( power_power @ real @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( plus_plus @ real @ ( power_power @ real @ Xa @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Ya @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ).

% real_sqrt_sum_squares_mult_ge_zero
thf(fact_4861_arith__geo__mean__sqrt,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 )
     => ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y2 )
       => ( ord_less_eq @ real @ ( sqrt @ ( times_times @ real @ X2 @ Y2 ) ) @ ( divide_divide @ real @ ( plus_plus @ real @ X2 @ Y2 ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ).

% arith_geo_mean_sqrt
thf(fact_4862_powr__half__sqrt,axiom,
    ! [X2: real] :
      ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 )
     => ( ( powr @ real @ X2 @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
        = ( sqrt @ X2 ) ) ) ).

% powr_half_sqrt
thf(fact_4863_tan__30,axiom,
    ( ( tan @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ ( bit1 @ one2 ) ) ) ) )
    = ( divide_divide @ real @ ( one_one @ real ) @ ( sqrt @ ( numeral_numeral @ real @ ( bit1 @ one2 ) ) ) ) ) ).

% tan_30
thf(fact_4864_cos__x__y__le__one,axiom,
    ! [X2: real,Y2: real] : ( ord_less_eq @ real @ ( abs_abs @ real @ ( divide_divide @ real @ X2 @ ( sqrt @ ( plus_plus @ real @ ( power_power @ real @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) @ ( one_one @ real ) ) ).

% cos_x_y_le_one
thf(fact_4865_real__sqrt__sum__squares__less,axiom,
    ! [X2: real,U: real,Y2: real] :
      ( ( ord_less @ real @ ( abs_abs @ real @ X2 ) @ ( divide_divide @ real @ U @ ( sqrt @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) )
     => ( ( ord_less @ real @ ( abs_abs @ real @ Y2 ) @ ( divide_divide @ real @ U @ ( sqrt @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) )
       => ( ord_less @ real @ ( sqrt @ ( plus_plus @ real @ ( power_power @ real @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ U ) ) ) ).

% real_sqrt_sum_squares_less
thf(fact_4866_arcosh__real__def,axiom,
    ! [X2: real] :
      ( ( ord_less_eq @ real @ ( one_one @ real ) @ X2 )
     => ( ( arcosh @ real @ X2 )
        = ( ln_ln @ real @ ( plus_plus @ real @ X2 @ ( sqrt @ ( minus_minus @ real @ ( power_power @ real @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ real ) ) ) ) ) ) ) ).

% arcosh_real_def
thf(fact_4867_cos__arctan,axiom,
    ! [X2: real] :
      ( ( cos @ real @ ( arctan @ X2 ) )
      = ( divide_divide @ real @ ( one_one @ real ) @ ( sqrt @ ( plus_plus @ real @ ( one_one @ real ) @ ( power_power @ real @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ).

% cos_arctan
thf(fact_4868_sin__arctan,axiom,
    ! [X2: real] :
      ( ( sin @ real @ ( arctan @ X2 ) )
      = ( divide_divide @ real @ X2 @ ( sqrt @ ( plus_plus @ real @ ( one_one @ real ) @ ( power_power @ real @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ).

% sin_arctan
thf(fact_4869_monoseq__minus,axiom,
    ! [A: $tType] :
      ( ( ordered_ab_group_add @ A )
     => ! [A2: nat > A] :
          ( ( topological_monoseq @ A @ A2 )
         => ( topological_monoseq @ A
            @ ^ [N: nat] : ( uminus_uminus @ A @ ( A2 @ N ) ) ) ) ) ).

% monoseq_minus
thf(fact_4870_sqrt__sum__squares__half__less,axiom,
    ! [X2: real,U: real,Y2: real] :
      ( ( ord_less @ real @ X2 @ ( divide_divide @ real @ U @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
     => ( ( ord_less @ real @ Y2 @ ( divide_divide @ real @ U @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
       => ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 )
         => ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y2 )
           => ( ord_less @ real @ ( sqrt @ ( plus_plus @ real @ ( power_power @ real @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ U ) ) ) ) ) ).

% sqrt_sum_squares_half_less
thf(fact_4871_sin__cos__sqrt,axiom,
    ! [X2: real] :
      ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( sin @ real @ X2 ) )
     => ( ( sin @ real @ X2 )
        = ( sqrt @ ( minus_minus @ real @ ( one_one @ real ) @ ( power_power @ real @ ( cos @ real @ X2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ).

% sin_cos_sqrt
thf(fact_4872_arctan__half,axiom,
    ( arctan
    = ( ^ [X: real] : ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( arctan @ ( divide_divide @ real @ X @ ( plus_plus @ real @ ( one_one @ real ) @ ( sqrt @ ( plus_plus @ real @ ( one_one @ real ) @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ).

% arctan_half
thf(fact_4873_monoseq__Suc,axiom,
    ! [A: $tType] :
      ( ( order @ A )
     => ( ( topological_monoseq @ A )
        = ( ^ [X5: nat > A] :
              ( ! [N: nat] : ( ord_less_eq @ A @ ( X5 @ N ) @ ( X5 @ ( suc @ N ) ) )
              | ! [N: nat] : ( ord_less_eq @ A @ ( X5 @ ( suc @ N ) ) @ ( X5 @ N ) ) ) ) ) ) ).

% monoseq_Suc
thf(fact_4874_mono__SucI2,axiom,
    ! [A: $tType] :
      ( ( order @ A )
     => ! [X8: nat > A] :
          ( ! [N4: nat] : ( ord_less_eq @ A @ ( X8 @ ( suc @ N4 ) ) @ ( X8 @ N4 ) )
         => ( topological_monoseq @ A @ X8 ) ) ) ).

% mono_SucI2
thf(fact_4875_mono__SucI1,axiom,
    ! [A: $tType] :
      ( ( order @ A )
     => ! [X8: nat > A] :
          ( ! [N4: nat] : ( ord_less_eq @ A @ ( X8 @ N4 ) @ ( X8 @ ( suc @ N4 ) ) )
         => ( topological_monoseq @ A @ X8 ) ) ) ).

% mono_SucI1
thf(fact_4876_monoseq__def,axiom,
    ! [A: $tType] :
      ( ( order @ A )
     => ( ( topological_monoseq @ A )
        = ( ^ [X5: nat > A] :
              ( ! [M7: nat,N: nat] :
                  ( ( ord_less_eq @ nat @ M7 @ N )
                 => ( ord_less_eq @ A @ ( X5 @ M7 ) @ ( X5 @ N ) ) )
              | ! [M7: nat,N: nat] :
                  ( ( ord_less_eq @ nat @ M7 @ N )
                 => ( ord_less_eq @ A @ ( X5 @ N ) @ ( X5 @ M7 ) ) ) ) ) ) ) ).

% monoseq_def
thf(fact_4877_monoI2,axiom,
    ! [A: $tType] :
      ( ( order @ A )
     => ! [X8: nat > A] :
          ( ! [M2: nat,N4: nat] :
              ( ( ord_less_eq @ nat @ M2 @ N4 )
             => ( ord_less_eq @ A @ ( X8 @ N4 ) @ ( X8 @ M2 ) ) )
         => ( topological_monoseq @ A @ X8 ) ) ) ).

% monoI2
thf(fact_4878_cos__arcsin,axiom,
    ! [X2: real] :
      ( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X2 )
     => ( ( ord_less_eq @ real @ X2 @ ( one_one @ real ) )
       => ( ( cos @ real @ ( arcsin @ X2 ) )
          = ( sqrt @ ( minus_minus @ real @ ( one_one @ real ) @ ( power_power @ real @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).

% cos_arcsin
thf(fact_4879_sin__arccos__abs,axiom,
    ! [Y2: real] :
      ( ( ord_less_eq @ real @ ( abs_abs @ real @ Y2 ) @ ( one_one @ real ) )
     => ( ( sin @ real @ ( arccos @ Y2 ) )
        = ( sqrt @ ( minus_minus @ real @ ( one_one @ real ) @ ( power_power @ real @ Y2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ).

% sin_arccos_abs
thf(fact_4880_sin__arccos,axiom,
    ! [X2: real] :
      ( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X2 )
     => ( ( ord_less_eq @ real @ X2 @ ( one_one @ real ) )
       => ( ( sin @ real @ ( arccos @ X2 ) )
          = ( sqrt @ ( minus_minus @ real @ ( one_one @ real ) @ ( power_power @ real @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).

% sin_arccos
thf(fact_4881_arccos__1,axiom,
    ( ( arccos @ ( one_one @ real ) )
    = ( zero_zero @ real ) ) ).

% arccos_1
thf(fact_4882_arccos__minus__1,axiom,
    ( ( arccos @ ( uminus_uminus @ real @ ( one_one @ real ) ) )
    = pi ) ).

% arccos_minus_1
thf(fact_4883_cos__arccos,axiom,
    ! [Y2: real] :
      ( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ Y2 )
     => ( ( ord_less_eq @ real @ Y2 @ ( one_one @ real ) )
       => ( ( cos @ real @ ( arccos @ Y2 ) )
          = Y2 ) ) ) ).

% cos_arccos
thf(fact_4884_sin__arcsin,axiom,
    ! [Y2: real] :
      ( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ Y2 )
     => ( ( ord_less_eq @ real @ Y2 @ ( one_one @ real ) )
       => ( ( sin @ real @ ( arcsin @ Y2 ) )
          = Y2 ) ) ) ).

% sin_arcsin
thf(fact_4885_arccos__0,axiom,
    ( ( arccos @ ( zero_zero @ real ) )
    = ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ).

% arccos_0
thf(fact_4886_arcsin__1,axiom,
    ( ( arcsin @ ( one_one @ real ) )
    = ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ).

% arcsin_1
thf(fact_4887_arcsin__minus__1,axiom,
    ( ( arcsin @ ( uminus_uminus @ real @ ( one_one @ real ) ) )
    = ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ).

% arcsin_minus_1
thf(fact_4888_arccos__le__arccos,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X2 )
     => ( ( ord_less_eq @ real @ X2 @ Y2 )
       => ( ( ord_less_eq @ real @ Y2 @ ( one_one @ real ) )
         => ( ord_less_eq @ real @ ( arccos @ Y2 ) @ ( arccos @ X2 ) ) ) ) ) ).

% arccos_le_arccos
thf(fact_4889_arccos__eq__iff,axiom,
    ! [X2: real,Y2: real] :
      ( ( ( ord_less_eq @ real @ ( abs_abs @ real @ X2 ) @ ( one_one @ real ) )
        & ( ord_less_eq @ real @ ( abs_abs @ real @ Y2 ) @ ( one_one @ real ) ) )
     => ( ( ( arccos @ X2 )
          = ( arccos @ Y2 ) )
        = ( X2 = Y2 ) ) ) ).

% arccos_eq_iff
thf(fact_4890_arccos__le__mono,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_eq @ real @ ( abs_abs @ real @ X2 ) @ ( one_one @ real ) )
     => ( ( ord_less_eq @ real @ ( abs_abs @ real @ Y2 ) @ ( one_one @ real ) )
       => ( ( ord_less_eq @ real @ ( arccos @ X2 ) @ ( arccos @ Y2 ) )
          = ( ord_less_eq @ real @ Y2 @ X2 ) ) ) ) ).

% arccos_le_mono
thf(fact_4891_arcsin__le__arcsin,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X2 )
     => ( ( ord_less_eq @ real @ X2 @ Y2 )
       => ( ( ord_less_eq @ real @ Y2 @ ( one_one @ real ) )
         => ( ord_less_eq @ real @ ( arcsin @ X2 ) @ ( arcsin @ Y2 ) ) ) ) ) ).

% arcsin_le_arcsin
thf(fact_4892_arcsin__minus,axiom,
    ! [X2: real] :
      ( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X2 )
     => ( ( ord_less_eq @ real @ X2 @ ( one_one @ real ) )
       => ( ( arcsin @ ( uminus_uminus @ real @ X2 ) )
          = ( uminus_uminus @ real @ ( arcsin @ X2 ) ) ) ) ) ).

% arcsin_minus
thf(fact_4893_arcsin__eq__iff,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_eq @ real @ ( abs_abs @ real @ X2 ) @ ( one_one @ real ) )
     => ( ( ord_less_eq @ real @ ( abs_abs @ real @ Y2 ) @ ( one_one @ real ) )
       => ( ( ( arcsin @ X2 )
            = ( arcsin @ Y2 ) )
          = ( X2 = Y2 ) ) ) ) ).

% arcsin_eq_iff
thf(fact_4894_arcsin__le__mono,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_eq @ real @ ( abs_abs @ real @ X2 ) @ ( one_one @ real ) )
     => ( ( ord_less_eq @ real @ ( abs_abs @ real @ Y2 ) @ ( one_one @ real ) )
       => ( ( ord_less_eq @ real @ ( arcsin @ X2 ) @ ( arcsin @ Y2 ) )
          = ( ord_less_eq @ real @ X2 @ Y2 ) ) ) ) ).

% arcsin_le_mono
thf(fact_4895_arccos__lbound,axiom,
    ! [Y2: real] :
      ( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ Y2 )
     => ( ( ord_less_eq @ real @ Y2 @ ( one_one @ real ) )
       => ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( arccos @ Y2 ) ) ) ) ).

% arccos_lbound
thf(fact_4896_arccos__less__arccos,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X2 )
     => ( ( ord_less @ real @ X2 @ Y2 )
       => ( ( ord_less_eq @ real @ Y2 @ ( one_one @ real ) )
         => ( ord_less @ real @ ( arccos @ Y2 ) @ ( arccos @ X2 ) ) ) ) ) ).

% arccos_less_arccos
thf(fact_4897_arccos__less__mono,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_eq @ real @ ( abs_abs @ real @ X2 ) @ ( one_one @ real ) )
     => ( ( ord_less_eq @ real @ ( abs_abs @ real @ Y2 ) @ ( one_one @ real ) )
       => ( ( ord_less @ real @ ( arccos @ X2 ) @ ( arccos @ Y2 ) )
          = ( ord_less @ real @ Y2 @ X2 ) ) ) ) ).

% arccos_less_mono
thf(fact_4898_arccos__ubound,axiom,
    ! [Y2: real] :
      ( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ Y2 )
     => ( ( ord_less_eq @ real @ Y2 @ ( one_one @ real ) )
       => ( ord_less_eq @ real @ ( arccos @ Y2 ) @ pi ) ) ) ).

% arccos_ubound
thf(fact_4899_arccos__cos,axiom,
    ! [X2: real] :
      ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 )
     => ( ( ord_less_eq @ real @ X2 @ pi )
       => ( ( arccos @ ( cos @ real @ X2 ) )
          = X2 ) ) ) ).

% arccos_cos
thf(fact_4900_arcsin__less__arcsin,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X2 )
     => ( ( ord_less @ real @ X2 @ Y2 )
       => ( ( ord_less_eq @ real @ Y2 @ ( one_one @ real ) )
         => ( ord_less @ real @ ( arcsin @ X2 ) @ ( arcsin @ Y2 ) ) ) ) ) ).

% arcsin_less_arcsin
thf(fact_4901_arcsin__less__mono,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_eq @ real @ ( abs_abs @ real @ X2 ) @ ( one_one @ real ) )
     => ( ( ord_less_eq @ real @ ( abs_abs @ real @ Y2 ) @ ( one_one @ real ) )
       => ( ( ord_less @ real @ ( arcsin @ X2 ) @ ( arcsin @ Y2 ) )
          = ( ord_less @ real @ X2 @ Y2 ) ) ) ) ).

% arcsin_less_mono
thf(fact_4902_cos__arccos__abs,axiom,
    ! [Y2: real] :
      ( ( ord_less_eq @ real @ ( abs_abs @ real @ Y2 ) @ ( one_one @ real ) )
     => ( ( cos @ real @ ( arccos @ Y2 ) )
        = Y2 ) ) ).

% cos_arccos_abs
thf(fact_4903_arccos__cos__eq__abs,axiom,
    ! [Theta: real] :
      ( ( ord_less_eq @ real @ ( abs_abs @ real @ Theta ) @ pi )
     => ( ( arccos @ ( cos @ real @ Theta ) )
        = ( abs_abs @ real @ Theta ) ) ) ).

% arccos_cos_eq_abs
thf(fact_4904_arccos__lt__bounded,axiom,
    ! [Y2: real] :
      ( ( ord_less @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ Y2 )
     => ( ( ord_less @ real @ Y2 @ ( one_one @ real ) )
       => ( ( ord_less @ real @ ( zero_zero @ real ) @ ( arccos @ Y2 ) )
          & ( ord_less @ real @ ( arccos @ Y2 ) @ pi ) ) ) ) ).

% arccos_lt_bounded
thf(fact_4905_arccos__bounded,axiom,
    ! [Y2: real] :
      ( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ Y2 )
     => ( ( ord_less_eq @ real @ Y2 @ ( one_one @ real ) )
       => ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( arccos @ Y2 ) )
          & ( ord_less_eq @ real @ ( arccos @ Y2 ) @ pi ) ) ) ) ).

% arccos_bounded
thf(fact_4906_sin__arccos__nonzero,axiom,
    ! [X2: real] :
      ( ( ord_less @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X2 )
     => ( ( ord_less @ real @ X2 @ ( one_one @ real ) )
       => ( ( sin @ real @ ( arccos @ X2 ) )
         != ( zero_zero @ real ) ) ) ) ).

% sin_arccos_nonzero
thf(fact_4907_arccos__cos2,axiom,
    ! [X2: real] :
      ( ( ord_less_eq @ real @ X2 @ ( zero_zero @ real ) )
     => ( ( ord_less_eq @ real @ ( uminus_uminus @ real @ pi ) @ X2 )
       => ( ( arccos @ ( cos @ real @ X2 ) )
          = ( uminus_uminus @ real @ X2 ) ) ) ) ).

% arccos_cos2
thf(fact_4908_arccos__minus,axiom,
    ! [X2: real] :
      ( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X2 )
     => ( ( ord_less_eq @ real @ X2 @ ( one_one @ real ) )
       => ( ( arccos @ ( uminus_uminus @ real @ X2 ) )
          = ( minus_minus @ real @ pi @ ( arccos @ X2 ) ) ) ) ) ).

% arccos_minus
thf(fact_4909_cos__arcsin__nonzero,axiom,
    ! [X2: real] :
      ( ( ord_less @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X2 )
     => ( ( ord_less @ real @ X2 @ ( one_one @ real ) )
       => ( ( cos @ real @ ( arcsin @ X2 ) )
         != ( zero_zero @ real ) ) ) ) ).

% cos_arcsin_nonzero
thf(fact_4910_arccos,axiom,
    ! [Y2: real] :
      ( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ Y2 )
     => ( ( ord_less_eq @ real @ Y2 @ ( one_one @ real ) )
       => ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( arccos @ Y2 ) )
          & ( ord_less_eq @ real @ ( arccos @ Y2 ) @ pi )
          & ( ( cos @ real @ ( arccos @ Y2 ) )
            = Y2 ) ) ) ) ).

% arccos
thf(fact_4911_arccos__minus__abs,axiom,
    ! [X2: real] :
      ( ( ord_less_eq @ real @ ( abs_abs @ real @ X2 ) @ ( one_one @ real ) )
     => ( ( arccos @ ( uminus_uminus @ real @ X2 ) )
        = ( minus_minus @ real @ pi @ ( arccos @ X2 ) ) ) ) ).

% arccos_minus_abs
thf(fact_4912_arccos__le__pi2,axiom,
    ! [Y2: real] :
      ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y2 )
     => ( ( ord_less_eq @ real @ Y2 @ ( one_one @ real ) )
       => ( ord_less_eq @ real @ ( arccos @ Y2 ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ).

% arccos_le_pi2
thf(fact_4913_arcsin__lt__bounded,axiom,
    ! [Y2: real] :
      ( ( ord_less @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ Y2 )
     => ( ( ord_less @ real @ Y2 @ ( one_one @ real ) )
       => ( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ ( arcsin @ Y2 ) )
          & ( ord_less @ real @ ( arcsin @ Y2 ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ) ).

% arcsin_lt_bounded
thf(fact_4914_arcsin__lbound,axiom,
    ! [Y2: real] :
      ( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ Y2 )
     => ( ( ord_less_eq @ real @ Y2 @ ( one_one @ real ) )
       => ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ ( arcsin @ Y2 ) ) ) ) ).

% arcsin_lbound
thf(fact_4915_arcsin__ubound,axiom,
    ! [Y2: real] :
      ( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ Y2 )
     => ( ( ord_less_eq @ real @ Y2 @ ( one_one @ real ) )
       => ( ord_less_eq @ real @ ( arcsin @ Y2 ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ).

% arcsin_ubound
thf(fact_4916_arcsin__bounded,axiom,
    ! [Y2: real] :
      ( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ Y2 )
     => ( ( ord_less_eq @ real @ Y2 @ ( one_one @ real ) )
       => ( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ ( arcsin @ Y2 ) )
          & ( ord_less_eq @ real @ ( arcsin @ Y2 ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ) ).

% arcsin_bounded
thf(fact_4917_arcsin__sin,axiom,
    ! [X2: real] :
      ( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X2 )
     => ( ( ord_less_eq @ real @ X2 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
       => ( ( arcsin @ ( sin @ real @ X2 ) )
          = X2 ) ) ) ).

% arcsin_sin
thf(fact_4918_le__arcsin__iff,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X2 )
     => ( ( ord_less_eq @ real @ X2 @ ( one_one @ real ) )
       => ( ( ord_less_eq @ real @ ( divide_divide @ real @ ( uminus_uminus @ real @ pi ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ Y2 )
         => ( ( ord_less_eq @ real @ Y2 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
           => ( ( ord_less_eq @ real @ Y2 @ ( arcsin @ X2 ) )
              = ( ord_less_eq @ real @ ( sin @ real @ Y2 ) @ X2 ) ) ) ) ) ) ).

% le_arcsin_iff
thf(fact_4919_arcsin__le__iff,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X2 )
     => ( ( ord_less_eq @ real @ X2 @ ( one_one @ real ) )
       => ( ( ord_less_eq @ real @ ( divide_divide @ real @ ( uminus_uminus @ real @ pi ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ Y2 )
         => ( ( ord_less_eq @ real @ Y2 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
           => ( ( ord_less_eq @ real @ ( arcsin @ X2 ) @ Y2 )
              = ( ord_less_eq @ real @ X2 @ ( sin @ real @ Y2 ) ) ) ) ) ) ) ).

% arcsin_le_iff
thf(fact_4920_arcsin__pi,axiom,
    ! [Y2: real] :
      ( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ Y2 )
     => ( ( ord_less_eq @ real @ Y2 @ ( one_one @ real ) )
       => ( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ ( arcsin @ Y2 ) )
          & ( ord_less_eq @ real @ ( arcsin @ Y2 ) @ pi )
          & ( ( sin @ real @ ( arcsin @ Y2 ) )
            = Y2 ) ) ) ) ).

% arcsin_pi
thf(fact_4921_arcsin,axiom,
    ! [Y2: real] :
      ( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ Y2 )
     => ( ( ord_less_eq @ real @ Y2 @ ( one_one @ real ) )
       => ( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ ( arcsin @ Y2 ) )
          & ( ord_less_eq @ real @ ( arcsin @ Y2 ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
          & ( ( sin @ real @ ( arcsin @ Y2 ) )
            = Y2 ) ) ) ) ).

% arcsin
thf(fact_4922_arccos__cos__eq__abs__2pi,axiom,
    ! [Theta: real] :
      ~ ! [K3: int] :
          ( ( arccos @ ( cos @ real @ Theta ) )
         != ( abs_abs @ real @ ( minus_minus @ real @ Theta @ ( times_times @ real @ ( ring_1_of_int @ real @ K3 ) @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) ) ) ) ) ).

% arccos_cos_eq_abs_2pi
thf(fact_4923_exp__two__pi__i,axiom,
    ( ( exp @ complex @ ( times_times @ complex @ ( times_times @ complex @ ( numeral_numeral @ complex @ ( bit0 @ one2 ) ) @ ( real_Vector_of_real @ complex @ pi ) ) @ imaginary_unit ) )
    = ( one_one @ complex ) ) ).

% exp_two_pi_i
thf(fact_4924_exp__two__pi__i_H,axiom,
    ( ( exp @ complex @ ( times_times @ complex @ imaginary_unit @ ( times_times @ complex @ ( real_Vector_of_real @ complex @ pi ) @ ( numeral_numeral @ complex @ ( bit0 @ one2 ) ) ) ) )
    = ( one_one @ complex ) ) ).

% exp_two_pi_i'
thf(fact_4925_exp__first__two__terms,axiom,
    ! [A: $tType] :
      ( ( ( real_Vector_banach @ A )
        & ( real_V2822296259951069270ebra_1 @ A ) )
     => ( ( exp @ A )
        = ( ^ [X: A] :
              ( plus_plus @ A @ ( plus_plus @ A @ ( one_one @ A ) @ X )
              @ ( suminf @ A
                @ ^ [N: nat] : ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ ( plus_plus @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( power_power @ A @ X @ ( plus_plus @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ).

% exp_first_two_terms
thf(fact_4926_inverse__eq__1__iff,axiom,
    ! [A: $tType] :
      ( ( field @ A )
     => ! [X2: A] :
          ( ( ( inverse_inverse @ A @ X2 )
            = ( one_one @ A ) )
          = ( X2
            = ( one_one @ A ) ) ) ) ).

% inverse_eq_1_iff
thf(fact_4927_inverse__1,axiom,
    ! [A: $tType] :
      ( ( division_ring @ A )
     => ( ( inverse_inverse @ A @ ( one_one @ A ) )
        = ( one_one @ A ) ) ) ).

% inverse_1
thf(fact_4928_inverse__mult__distrib,axiom,
    ! [A: $tType] :
      ( ( field @ A )
     => ! [A2: A,B2: A] :
          ( ( inverse_inverse @ A @ ( times_times @ A @ A2 @ B2 ) )
          = ( times_times @ A @ ( inverse_inverse @ A @ A2 ) @ ( inverse_inverse @ A @ B2 ) ) ) ) ).

% inverse_mult_distrib
thf(fact_4929_inverse__divide,axiom,
    ! [A: $tType] :
      ( ( field @ A )
     => ! [A2: A,B2: A] :
          ( ( inverse_inverse @ A @ ( divide_divide @ A @ A2 @ B2 ) )
          = ( divide_divide @ A @ B2 @ A2 ) ) ) ).

% inverse_divide
thf(fact_4930_inverse__nonpositive__iff__nonpositive,axiom,
    ! [A: $tType] :
      ( ( linordered_field @ A )
     => ! [A2: A] :
          ( ( ord_less_eq @ A @ ( inverse_inverse @ A @ A2 ) @ ( zero_zero @ A ) )
          = ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).

% inverse_nonpositive_iff_nonpositive
thf(fact_4931_inverse__nonnegative__iff__nonnegative,axiom,
    ! [A: $tType] :
      ( ( linordered_field @ A )
     => ! [A2: A] :
          ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( inverse_inverse @ A @ A2 ) )
          = ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).

% inverse_nonnegative_iff_nonnegative
thf(fact_4932_inverse__less__iff__less,axiom,
    ! [A: $tType] :
      ( ( linordered_field @ A )
     => ! [A2: A,B2: A] :
          ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
         => ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
           => ( ( ord_less @ A @ ( inverse_inverse @ A @ A2 ) @ ( inverse_inverse @ A @ B2 ) )
              = ( ord_less @ A @ B2 @ A2 ) ) ) ) ) ).

% inverse_less_iff_less
thf(fact_4933_inverse__less__iff__less__neg,axiom,
    ! [A: $tType] :
      ( ( linordered_field @ A )
     => ! [A2: A,B2: A] :
          ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
         => ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
           => ( ( ord_less @ A @ ( inverse_inverse @ A @ A2 ) @ ( inverse_inverse @ A @ B2 ) )
              = ( ord_less @ A @ B2 @ A2 ) ) ) ) ) ).

% inverse_less_iff_less_neg
thf(fact_4934_inverse__negative__iff__negative,axiom,
    ! [A: $tType] :
      ( ( linordered_field @ A )
     => ! [A2: A] :
          ( ( ord_less @ A @ ( inverse_inverse @ A @ A2 ) @ ( zero_zero @ A ) )
          = ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).

% inverse_negative_iff_negative
thf(fact_4935_inverse__positive__iff__positive,axiom,
    ! [A: $tType] :
      ( ( linordered_field @ A )
     => ! [A2: A] :
          ( ( ord_less @ A @ ( zero_zero @ A ) @ ( inverse_inverse @ A @ A2 ) )
          = ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).

% inverse_positive_iff_positive
thf(fact_4936_norm__ii,axiom,
    ( ( real_V7770717601297561774m_norm @ complex @ imaginary_unit )
    = ( one_one @ real ) ) ).

% norm_ii
thf(fact_4937_complex__i__mult__minus,axiom,
    ! [X2: complex] :
      ( ( times_times @ complex @ imaginary_unit @ ( times_times @ complex @ imaginary_unit @ X2 ) )
      = ( uminus_uminus @ complex @ X2 ) ) ).

% complex_i_mult_minus
thf(fact_4938_divide__i,axiom,
    ! [X2: complex] :
      ( ( divide_divide @ complex @ X2 @ imaginary_unit )
      = ( times_times @ complex @ ( uminus_uminus @ complex @ imaginary_unit ) @ X2 ) ) ).

% divide_i
thf(fact_4939_inverse__le__iff__le__neg,axiom,
    ! [A: $tType] :
      ( ( linordered_field @ A )
     => ! [A2: A,B2: A] :
          ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
         => ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
           => ( ( ord_less_eq @ A @ ( inverse_inverse @ A @ A2 ) @ ( inverse_inverse @ A @ B2 ) )
              = ( ord_less_eq @ A @ B2 @ A2 ) ) ) ) ) ).

% inverse_le_iff_le_neg
thf(fact_4940_inverse__le__iff__le,axiom,
    ! [A: $tType] :
      ( ( linordered_field @ A )
     => ! [A2: A,B2: A] :
          ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
         => ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
           => ( ( ord_less_eq @ A @ ( inverse_inverse @ A @ A2 ) @ ( inverse_inverse @ A @ B2 ) )
              = ( ord_less_eq @ A @ B2 @ A2 ) ) ) ) ) ).

% inverse_le_iff_le
thf(fact_4941_right__inverse,axiom,
    ! [A: $tType] :
      ( ( division_ring @ A )
     => ! [A2: A] :
          ( ( A2
           != ( zero_zero @ A ) )
         => ( ( times_times @ A @ A2 @ ( inverse_inverse @ A @ A2 ) )
            = ( one_one @ A ) ) ) ) ).

% right_inverse
thf(fact_4942_left__inverse,axiom,
    ! [A: $tType] :
      ( ( division_ring @ A )
     => ! [A2: A] :
          ( ( A2
           != ( zero_zero @ A ) )
         => ( ( times_times @ A @ ( inverse_inverse @ A @ A2 ) @ A2 )
            = ( one_one @ A ) ) ) ) ).

% left_inverse
thf(fact_4943_inverse__eq__divide__numeral,axiom,
    ! [A: $tType] :
      ( ( division_ring @ A )
     => ! [W: num] :
          ( ( inverse_inverse @ A @ ( numeral_numeral @ A @ W ) )
          = ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ W ) ) ) ) ).

% inverse_eq_divide_numeral
thf(fact_4944_i__squared,axiom,
    ( ( times_times @ complex @ imaginary_unit @ imaginary_unit )
    = ( uminus_uminus @ complex @ ( one_one @ complex ) ) ) ).

% i_squared
thf(fact_4945_divide__numeral__i,axiom,
    ! [Z: complex,N2: num] :
      ( ( divide_divide @ complex @ Z @ ( times_times @ complex @ ( numeral_numeral @ complex @ N2 ) @ imaginary_unit ) )
      = ( divide_divide @ complex @ ( uminus_uminus @ complex @ ( times_times @ complex @ imaginary_unit @ Z ) ) @ ( numeral_numeral @ complex @ N2 ) ) ) ).

% divide_numeral_i
thf(fact_4946_inverse__eq__divide__neg__numeral,axiom,
    ! [A: $tType] :
      ( ( division_ring @ A )
     => ! [W: num] :
          ( ( inverse_inverse @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) )
          = ( divide_divide @ A @ ( one_one @ A ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) ) ) ) ).

% inverse_eq_divide_neg_numeral
thf(fact_4947_power2__i,axiom,
    ( ( power_power @ complex @ imaginary_unit @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
    = ( uminus_uminus @ complex @ ( one_one @ complex ) ) ) ).

% power2_i
thf(fact_4948_exp__pi__i,axiom,
    ( ( exp @ complex @ ( times_times @ complex @ ( real_Vector_of_real @ complex @ pi ) @ imaginary_unit ) )
    = ( uminus_uminus @ complex @ ( one_one @ complex ) ) ) ).

% exp_pi_i
thf(fact_4949_exp__pi__i_H,axiom,
    ( ( exp @ complex @ ( times_times @ complex @ imaginary_unit @ ( real_Vector_of_real @ complex @ pi ) ) )
    = ( uminus_uminus @ complex @ ( one_one @ complex ) ) ) ).

% exp_pi_i'
thf(fact_4950_i__even__power,axiom,
    ! [N2: nat] :
      ( ( power_power @ complex @ imaginary_unit @ ( times_times @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
      = ( power_power @ complex @ ( uminus_uminus @ complex @ ( one_one @ complex ) ) @ N2 ) ) ).

% i_even_power
thf(fact_4951_complex__i__not__one,axiom,
    ( imaginary_unit
   != ( one_one @ complex ) ) ).

% complex_i_not_one
thf(fact_4952_power__inverse,axiom,
    ! [A: $tType] :
      ( ( division_ring @ A )
     => ! [A2: A,N2: nat] :
          ( ( power_power @ A @ ( inverse_inverse @ A @ A2 ) @ N2 )
          = ( inverse_inverse @ A @ ( power_power @ A @ A2 @ N2 ) ) ) ) ).

% power_inverse
thf(fact_4953_complex__i__not__numeral,axiom,
    ! [W: num] :
      ( imaginary_unit
     != ( numeral_numeral @ complex @ W ) ) ).

% complex_i_not_numeral
thf(fact_4954_real__sqrt__inverse,axiom,
    ! [X2: real] :
      ( ( sqrt @ ( inverse_inverse @ real @ X2 ) )
      = ( inverse_inverse @ real @ ( sqrt @ X2 ) ) ) ).

% real_sqrt_inverse
thf(fact_4955_mult__commute__imp__mult__inverse__commute,axiom,
    ! [A: $tType] :
      ( ( division_ring @ A )
     => ! [Y2: A,X2: A] :
          ( ( ( times_times @ A @ Y2 @ X2 )
            = ( times_times @ A @ X2 @ Y2 ) )
         => ( ( times_times @ A @ ( inverse_inverse @ A @ Y2 ) @ X2 )
            = ( times_times @ A @ X2 @ ( inverse_inverse @ A @ Y2 ) ) ) ) ) ).

% mult_commute_imp_mult_inverse_commute
thf(fact_4956_norm__inverse__le__norm,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra @ A )
     => ! [R4: real,X2: A] :
          ( ( ord_less_eq @ real @ R4 @ ( real_V7770717601297561774m_norm @ A @ X2 ) )
         => ( ( ord_less @ real @ ( zero_zero @ real ) @ R4 )
           => ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( inverse_inverse @ A @ X2 ) ) @ ( inverse_inverse @ real @ R4 ) ) ) ) ) ).

% norm_inverse_le_norm
thf(fact_4957_inverse__less__imp__less,axiom,
    ! [A: $tType] :
      ( ( linordered_field @ A )
     => ! [A2: A,B2: A] :
          ( ( ord_less @ A @ ( inverse_inverse @ A @ A2 ) @ ( inverse_inverse @ A @ B2 ) )
         => ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
           => ( ord_less @ A @ B2 @ A2 ) ) ) ) ).

% inverse_less_imp_less
thf(fact_4958_less__imp__inverse__less,axiom,
    ! [A: $tType] :
      ( ( linordered_field @ A )
     => ! [A2: A,B2: A] :
          ( ( ord_less @ A @ A2 @ B2 )
         => ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
           => ( ord_less @ A @ ( inverse_inverse @ A @ B2 ) @ ( inverse_inverse @ A @ A2 ) ) ) ) ) ).

% less_imp_inverse_less
thf(fact_4959_inverse__less__imp__less__neg,axiom,
    ! [A: $tType] :
      ( ( linordered_field @ A )
     => ! [A2: A,B2: A] :
          ( ( ord_less @ A @ ( inverse_inverse @ A @ A2 ) @ ( inverse_inverse @ A @ B2 ) )
         => ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
           => ( ord_less @ A @ B2 @ A2 ) ) ) ) ).

% inverse_less_imp_less_neg
thf(fact_4960_less__imp__inverse__less__neg,axiom,
    ! [A: $tType] :
      ( ( linordered_field @ A )
     => ! [A2: A,B2: A] :
          ( ( ord_less @ A @ A2 @ B2 )
         => ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
           => ( ord_less @ A @ ( inverse_inverse @ A @ B2 ) @ ( inverse_inverse @ A @ A2 ) ) ) ) ) ).

% less_imp_inverse_less_neg
thf(fact_4961_inverse__negative__imp__negative,axiom,
    ! [A: $tType] :
      ( ( linordered_field @ A )
     => ! [A2: A] :
          ( ( ord_less @ A @ ( inverse_inverse @ A @ A2 ) @ ( zero_zero @ A ) )
         => ( ( A2
             != ( zero_zero @ A ) )
           => ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ) ).

% inverse_negative_imp_negative
thf(fact_4962_inverse__positive__imp__positive,axiom,
    ! [A: $tType] :
      ( ( linordered_field @ A )
     => ! [A2: A] :
          ( ( ord_less @ A @ ( zero_zero @ A ) @ ( inverse_inverse @ A @ A2 ) )
         => ( ( A2
             != ( zero_zero @ A ) )
           => ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ) ).

% inverse_positive_imp_positive
thf(fact_4963_negative__imp__inverse__negative,axiom,
    ! [A: $tType] :
      ( ( linordered_field @ A )
     => ! [A2: A] :
          ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
         => ( ord_less @ A @ ( inverse_inverse @ A @ A2 ) @ ( zero_zero @ A ) ) ) ) ).

% negative_imp_inverse_negative
thf(fact_4964_positive__imp__inverse__positive,axiom,
    ! [A: $tType] :
      ( ( linordered_field @ A )
     => ! [A2: A] :
          ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
         => ( ord_less @ A @ ( zero_zero @ A ) @ ( inverse_inverse @ A @ A2 ) ) ) ) ).

% positive_imp_inverse_positive
thf(fact_4965_nonzero__inverse__mult__distrib,axiom,
    ! [A: $tType] :
      ( ( division_ring @ A )
     => ! [A2: A,B2: A] :
          ( ( A2
           != ( zero_zero @ A ) )
         => ( ( B2
             != ( zero_zero @ A ) )
           => ( ( inverse_inverse @ A @ ( times_times @ A @ A2 @ B2 ) )
              = ( times_times @ A @ ( inverse_inverse @ A @ B2 ) @ ( inverse_inverse @ A @ A2 ) ) ) ) ) ) ).

% nonzero_inverse_mult_distrib
thf(fact_4966_inverse__numeral__1,axiom,
    ! [A: $tType] :
      ( ( division_ring @ A )
     => ( ( inverse_inverse @ A @ ( numeral_numeral @ A @ one2 ) )
        = ( numeral_numeral @ A @ one2 ) ) ) ).

% inverse_numeral_1
thf(fact_4967_inverse__unique,axiom,
    ! [A: $tType] :
      ( ( division_ring @ A )
     => ! [A2: A,B2: A] :
          ( ( ( times_times @ A @ A2 @ B2 )
            = ( one_one @ A ) )
         => ( ( inverse_inverse @ A @ A2 )
            = B2 ) ) ) ).

% inverse_unique
thf(fact_4968_inverse__eq__divide,axiom,
    ! [A: $tType] :
      ( ( division_ring @ A )
     => ( ( inverse_inverse @ A )
        = ( divide_divide @ A @ ( one_one @ A ) ) ) ) ).

% inverse_eq_divide
thf(fact_4969_divide__inverse__commute,axiom,
    ! [A: $tType] :
      ( ( field @ A )
     => ( ( divide_divide @ A )
        = ( ^ [A5: A,B5: A] : ( times_times @ A @ ( inverse_inverse @ A @ B5 ) @ A5 ) ) ) ) ).

% divide_inverse_commute
thf(fact_4970_divide__inverse,axiom,
    ! [A: $tType] :
      ( ( division_ring @ A )
     => ( ( divide_divide @ A )
        = ( ^ [A5: A,B5: A] : ( times_times @ A @ A5 @ ( inverse_inverse @ A @ B5 ) ) ) ) ) ).

% divide_inverse
thf(fact_4971_field__class_Ofield__divide__inverse,axiom,
    ! [A: $tType] :
      ( ( field @ A )
     => ( ( divide_divide @ A )
        = ( ^ [A5: A,B5: A] : ( times_times @ A @ A5 @ ( inverse_inverse @ A @ B5 ) ) ) ) ) ).

% field_class.field_divide_inverse
thf(fact_4972_power__mult__power__inverse__commute,axiom,
    ! [A: $tType] :
      ( ( division_ring @ A )
     => ! [X2: A,M: nat,N2: nat] :
          ( ( times_times @ A @ ( power_power @ A @ X2 @ M ) @ ( power_power @ A @ ( inverse_inverse @ A @ X2 ) @ N2 ) )
          = ( times_times @ A @ ( power_power @ A @ ( inverse_inverse @ A @ X2 ) @ N2 ) @ ( power_power @ A @ X2 @ M ) ) ) ) ).

% power_mult_power_inverse_commute
thf(fact_4973_power__mult__inverse__distrib,axiom,
    ! [A: $tType] :
      ( ( division_ring @ A )
     => ! [X2: A,M: nat] :
          ( ( times_times @ A @ ( power_power @ A @ X2 @ M ) @ ( inverse_inverse @ A @ X2 ) )
          = ( times_times @ A @ ( inverse_inverse @ A @ X2 ) @ ( power_power @ A @ X2 @ M ) ) ) ) ).

% power_mult_inverse_distrib
thf(fact_4974_mult__inverse__of__nat__commute,axiom,
    ! [A: $tType] :
      ( ( division_ring @ A )
     => ! [Xa: nat,X2: A] :
          ( ( times_times @ A @ ( inverse_inverse @ A @ ( semiring_1_of_nat @ A @ Xa ) ) @ X2 )
          = ( times_times @ A @ X2 @ ( inverse_inverse @ A @ ( semiring_1_of_nat @ A @ Xa ) ) ) ) ) ).

% mult_inverse_of_nat_commute
thf(fact_4975_mult__inverse__of__int__commute,axiom,
    ! [A: $tType] :
      ( ( division_ring @ A )
     => ! [Xa: int,X2: A] :
          ( ( times_times @ A @ ( inverse_inverse @ A @ ( ring_1_of_int @ A @ Xa ) ) @ X2 )
          = ( times_times @ A @ X2 @ ( inverse_inverse @ A @ ( ring_1_of_int @ A @ Xa ) ) ) ) ) ).

% mult_inverse_of_int_commute
thf(fact_4976_divide__real__def,axiom,
    ( ( divide_divide @ real )
    = ( ^ [X: real,Y: real] : ( times_times @ real @ X @ ( inverse_inverse @ real @ Y ) ) ) ) ).

% divide_real_def
thf(fact_4977_exp__fdiffs,axiom,
    ! [A: $tType] :
      ( ( ( real_Vector_banach @ A )
        & ( real_V3459762299906320749_field @ A ) )
     => ( ( diffs @ A
          @ ^ [N: nat] : ( inverse_inverse @ A @ ( semiring_char_0_fact @ A @ N ) ) )
        = ( ^ [N: nat] : ( inverse_inverse @ A @ ( semiring_char_0_fact @ A @ N ) ) ) ) ) ).

% exp_fdiffs
thf(fact_4978_le__imp__inverse__le__neg,axiom,
    ! [A: $tType] :
      ( ( linordered_field @ A )
     => ! [A2: A,B2: A] :
          ( ( ord_less_eq @ A @ A2 @ B2 )
         => ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
           => ( ord_less_eq @ A @ ( inverse_inverse @ A @ B2 ) @ ( inverse_inverse @ A @ A2 ) ) ) ) ) ).

% le_imp_inverse_le_neg
thf(fact_4979_inverse__le__imp__le__neg,axiom,
    ! [A: $tType] :
      ( ( linordered_field @ A )
     => ! [A2: A,B2: A] :
          ( ( ord_less_eq @ A @ ( inverse_inverse @ A @ A2 ) @ ( inverse_inverse @ A @ B2 ) )
         => ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
           => ( ord_less_eq @ A @ B2 @ A2 ) ) ) ) ).

% inverse_le_imp_le_neg
thf(fact_4980_le__imp__inverse__le,axiom,
    ! [A: $tType] :
      ( ( linordered_field @ A )
     => ! [A2: A,B2: A] :
          ( ( ord_less_eq @ A @ A2 @ B2 )
         => ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
           => ( ord_less_eq @ A @ ( inverse_inverse @ A @ B2 ) @ ( inverse_inverse @ A @ A2 ) ) ) ) ) ).

% le_imp_inverse_le
thf(fact_4981_inverse__le__imp__le,axiom,
    ! [A: $tType] :
      ( ( linordered_field @ A )
     => ! [A2: A,B2: A] :
          ( ( ord_less_eq @ A @ ( inverse_inverse @ A @ A2 ) @ ( inverse_inverse @ A @ B2 ) )
         => ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
           => ( ord_less_eq @ A @ B2 @ A2 ) ) ) ) ).

% inverse_le_imp_le
thf(fact_4982_i__times__eq__iff,axiom,
    ! [W: complex,Z: complex] :
      ( ( ( times_times @ complex @ imaginary_unit @ W )
        = Z )
      = ( W
        = ( uminus_uminus @ complex @ ( times_times @ complex @ imaginary_unit @ Z ) ) ) ) ).

% i_times_eq_iff
thf(fact_4983_inverse__le__1__iff,axiom,
    ! [A: $tType] :
      ( ( linordered_field @ A )
     => ! [X2: A] :
          ( ( ord_less_eq @ A @ ( inverse_inverse @ A @ X2 ) @ ( one_one @ A ) )
          = ( ( ord_less_eq @ A @ X2 @ ( zero_zero @ A ) )
            | ( ord_less_eq @ A @ ( one_one @ A ) @ X2 ) ) ) ) ).

% inverse_le_1_iff
thf(fact_4984_one__less__inverse__iff,axiom,
    ! [A: $tType] :
      ( ( linordered_field @ A )
     => ! [X2: A] :
          ( ( ord_less @ A @ ( one_one @ A ) @ ( inverse_inverse @ A @ X2 ) )
          = ( ( ord_less @ A @ ( zero_zero @ A ) @ X2 )
            & ( ord_less @ A @ X2 @ ( one_one @ A ) ) ) ) ) ).

% one_less_inverse_iff
thf(fact_4985_one__less__inverse,axiom,
    ! [A: $tType] :
      ( ( linordered_field @ A )
     => ! [A2: A] :
          ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
         => ( ( ord_less @ A @ A2 @ ( one_one @ A ) )
           => ( ord_less @ A @ ( one_one @ A ) @ ( inverse_inverse @ A @ A2 ) ) ) ) ) ).

% one_less_inverse
thf(fact_4986_division__ring__inverse__add,axiom,
    ! [A: $tType] :
      ( ( division_ring @ A )
     => ! [A2: A,B2: A] :
          ( ( A2
           != ( zero_zero @ A ) )
         => ( ( B2
             != ( zero_zero @ A ) )
           => ( ( plus_plus @ A @ ( inverse_inverse @ A @ A2 ) @ ( inverse_inverse @ A @ B2 ) )
              = ( times_times @ A @ ( times_times @ A @ ( inverse_inverse @ A @ A2 ) @ ( plus_plus @ A @ A2 @ B2 ) ) @ ( inverse_inverse @ A @ B2 ) ) ) ) ) ) ).

% division_ring_inverse_add
thf(fact_4987_inverse__add,axiom,
    ! [A: $tType] :
      ( ( field @ A )
     => ! [A2: A,B2: A] :
          ( ( A2
           != ( zero_zero @ A ) )
         => ( ( B2
             != ( zero_zero @ A ) )
           => ( ( plus_plus @ A @ ( inverse_inverse @ A @ A2 ) @ ( inverse_inverse @ A @ B2 ) )
              = ( times_times @ A @ ( times_times @ A @ ( plus_plus @ A @ A2 @ B2 ) @ ( inverse_inverse @ A @ A2 ) ) @ ( inverse_inverse @ A @ B2 ) ) ) ) ) ) ).

% inverse_add
thf(fact_4988_field__class_Ofield__inverse,axiom,
    ! [A: $tType] :
      ( ( field @ A )
     => ! [A2: A] :
          ( ( A2
           != ( zero_zero @ A ) )
         => ( ( times_times @ A @ ( inverse_inverse @ A @ A2 ) @ A2 )
            = ( one_one @ A ) ) ) ) ).

% field_class.field_inverse
thf(fact_4989_division__ring__inverse__diff,axiom,
    ! [A: $tType] :
      ( ( division_ring @ A )
     => ! [A2: A,B2: A] :
          ( ( A2
           != ( zero_zero @ A ) )
         => ( ( B2
             != ( zero_zero @ A ) )
           => ( ( minus_minus @ A @ ( inverse_inverse @ A @ A2 ) @ ( inverse_inverse @ A @ B2 ) )
              = ( times_times @ A @ ( times_times @ A @ ( inverse_inverse @ A @ A2 ) @ ( minus_minus @ A @ B2 @ A2 ) ) @ ( inverse_inverse @ A @ B2 ) ) ) ) ) ) ).

% division_ring_inverse_diff
thf(fact_4990_nonzero__inverse__eq__divide,axiom,
    ! [A: $tType] :
      ( ( division_ring @ A )
     => ! [A2: A] :
          ( ( A2
           != ( zero_zero @ A ) )
         => ( ( inverse_inverse @ A @ A2 )
            = ( divide_divide @ A @ ( one_one @ A ) @ A2 ) ) ) ) ).

% nonzero_inverse_eq_divide
thf(fact_4991_complex__i__not__neg__numeral,axiom,
    ! [W: num] :
      ( imaginary_unit
     != ( uminus_uminus @ complex @ ( numeral_numeral @ complex @ W ) ) ) ).

% complex_i_not_neg_numeral
thf(fact_4992_inverse__powr,axiom,
    ! [Y2: real,A2: real] :
      ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y2 )
     => ( ( powr @ real @ ( inverse_inverse @ real @ Y2 ) @ A2 )
        = ( inverse_inverse @ real @ ( powr @ real @ Y2 @ A2 ) ) ) ) ).

% inverse_powr
thf(fact_4993_Complex__eq__i,axiom,
    ! [X2: real,Y2: real] :
      ( ( ( complex2 @ X2 @ Y2 )
        = imaginary_unit )
      = ( ( X2
          = ( zero_zero @ real ) )
        & ( Y2
          = ( one_one @ real ) ) ) ) ).

% Complex_eq_i
thf(fact_4994_imaginary__unit_Ocode,axiom,
    ( imaginary_unit
    = ( complex2 @ ( zero_zero @ real ) @ ( one_one @ real ) ) ) ).

% imaginary_unit.code
thf(fact_4995_one__le__inverse__iff,axiom,
    ! [A: $tType] :
      ( ( linordered_field @ A )
     => ! [X2: A] :
          ( ( ord_less_eq @ A @ ( one_one @ A ) @ ( inverse_inverse @ A @ X2 ) )
          = ( ( ord_less @ A @ ( zero_zero @ A ) @ X2 )
            & ( ord_less_eq @ A @ X2 @ ( one_one @ A ) ) ) ) ) ).

% one_le_inverse_iff
thf(fact_4996_inverse__less__1__iff,axiom,
    ! [A: $tType] :
      ( ( linordered_field @ A )
     => ! [X2: A] :
          ( ( ord_less @ A @ ( inverse_inverse @ A @ X2 ) @ ( one_one @ A ) )
          = ( ( ord_less_eq @ A @ X2 @ ( zero_zero @ A ) )
            | ( ord_less @ A @ ( one_one @ A ) @ X2 ) ) ) ) ).

% inverse_less_1_iff
thf(fact_4997_one__le__inverse,axiom,
    ! [A: $tType] :
      ( ( linordered_field @ A )
     => ! [A2: A] :
          ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
         => ( ( ord_less_eq @ A @ A2 @ ( one_one @ A ) )
           => ( ord_less_eq @ A @ ( one_one @ A ) @ ( inverse_inverse @ A @ A2 ) ) ) ) ) ).

% one_le_inverse
thf(fact_4998_inverse__less__iff,axiom,
    ! [A: $tType] :
      ( ( linordered_field @ A )
     => ! [A2: A,B2: A] :
          ( ( ord_less @ A @ ( inverse_inverse @ A @ A2 ) @ ( inverse_inverse @ A @ B2 ) )
          = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A2 @ B2 ) )
             => ( ord_less @ A @ B2 @ A2 ) )
            & ( ( ord_less_eq @ A @ ( times_times @ A @ A2 @ B2 ) @ ( zero_zero @ A ) )
             => ( ord_less @ A @ A2 @ B2 ) ) ) ) ) ).

% inverse_less_iff
thf(fact_4999_inverse__le__iff,axiom,
    ! [A: $tType] :
      ( ( linordered_field @ A )
     => ! [A2: A,B2: A] :
          ( ( ord_less_eq @ A @ ( inverse_inverse @ A @ A2 ) @ ( inverse_inverse @ A @ B2 ) )
          = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A2 @ B2 ) )
             => ( ord_less_eq @ A @ B2 @ A2 ) )
            & ( ( ord_less_eq @ A @ ( times_times @ A @ A2 @ B2 ) @ ( zero_zero @ A ) )
             => ( ord_less_eq @ A @ A2 @ B2 ) ) ) ) ) ).

% inverse_le_iff
thf(fact_5000_inverse__diff__inverse,axiom,
    ! [A: $tType] :
      ( ( division_ring @ A )
     => ! [A2: A,B2: A] :
          ( ( A2
           != ( zero_zero @ A ) )
         => ( ( B2
             != ( zero_zero @ A ) )
           => ( ( minus_minus @ A @ ( inverse_inverse @ A @ A2 ) @ ( inverse_inverse @ A @ B2 ) )
              = ( uminus_uminus @ A @ ( times_times @ A @ ( times_times @ A @ ( inverse_inverse @ A @ A2 ) @ ( minus_minus @ A @ A2 @ B2 ) ) @ ( inverse_inverse @ A @ B2 ) ) ) ) ) ) ) ).

% inverse_diff_inverse
thf(fact_5001_reals__Archimedean,axiom,
    ! [A: $tType] :
      ( ( archim462609752435547400_field @ A )
     => ! [X2: A] :
          ( ( ord_less @ A @ ( zero_zero @ A ) @ X2 )
         => ? [N4: nat] : ( ord_less @ A @ ( inverse_inverse @ A @ ( semiring_1_of_nat @ A @ ( suc @ N4 ) ) ) @ X2 ) ) ) ).

% reals_Archimedean
thf(fact_5002_i__mult__Complex,axiom,
    ! [A2: real,B2: real] :
      ( ( times_times @ complex @ imaginary_unit @ ( complex2 @ A2 @ B2 ) )
      = ( complex2 @ ( uminus_uminus @ real @ B2 ) @ A2 ) ) ).

% i_mult_Complex
thf(fact_5003_Complex__mult__i,axiom,
    ! [A2: real,B2: real] :
      ( ( times_times @ complex @ ( complex2 @ A2 @ B2 ) @ imaginary_unit )
      = ( complex2 @ ( uminus_uminus @ real @ B2 ) @ A2 ) ) ).

% Complex_mult_i
thf(fact_5004_real__vector__affinity__eq,axiom,
    ! [A: $tType] :
      ( ( real_V4867850818363320053vector @ A )
     => ! [M: real,X2: A,C2: A,Y2: A] :
          ( ( M
           != ( zero_zero @ real ) )
         => ( ( ( plus_plus @ A @ ( real_V8093663219630862766scaleR @ A @ M @ X2 ) @ C2 )
              = Y2 )
            = ( X2
              = ( minus_minus @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ M ) @ Y2 ) @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ M ) @ C2 ) ) ) ) ) ) ).

% real_vector_affinity_eq
thf(fact_5005_real__vector__eq__affinity,axiom,
    ! [A: $tType] :
      ( ( real_V4867850818363320053vector @ A )
     => ! [M: real,Y2: A,X2: A,C2: A] :
          ( ( M
           != ( zero_zero @ real ) )
         => ( ( Y2
              = ( plus_plus @ A @ ( real_V8093663219630862766scaleR @ A @ M @ X2 ) @ C2 ) )
            = ( ( minus_minus @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ M ) @ Y2 ) @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ M ) @ C2 ) )
              = X2 ) ) ) ) ).

% real_vector_eq_affinity
thf(fact_5006_neg__le__divideR__eq,axiom,
    ! [A: $tType] :
      ( ( real_V5355595471888546746vector @ A )
     => ! [C2: real,A2: A,B2: A] :
          ( ( ord_less @ real @ C2 @ ( zero_zero @ real ) )
         => ( ( ord_less_eq @ A @ A2 @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C2 ) @ B2 ) )
            = ( ord_less_eq @ A @ B2 @ ( real_V8093663219630862766scaleR @ A @ C2 @ A2 ) ) ) ) ) ).

% neg_le_divideR_eq
thf(fact_5007_neg__divideR__le__eq,axiom,
    ! [A: $tType] :
      ( ( real_V5355595471888546746vector @ A )
     => ! [C2: real,B2: A,A2: A] :
          ( ( ord_less @ real @ C2 @ ( zero_zero @ real ) )
         => ( ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C2 ) @ B2 ) @ A2 )
            = ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ C2 @ A2 ) @ B2 ) ) ) ) ).

% neg_divideR_le_eq
thf(fact_5008_pos__le__divideR__eq,axiom,
    ! [A: $tType] :
      ( ( real_V5355595471888546746vector @ A )
     => ! [C2: real,A2: A,B2: A] :
          ( ( ord_less @ real @ ( zero_zero @ real ) @ C2 )
         => ( ( ord_less_eq @ A @ A2 @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C2 ) @ B2 ) )
            = ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ C2 @ A2 ) @ B2 ) ) ) ) ).

% pos_le_divideR_eq
thf(fact_5009_pos__divideR__le__eq,axiom,
    ! [A: $tType] :
      ( ( real_V5355595471888546746vector @ A )
     => ! [C2: real,B2: A,A2: A] :
          ( ( ord_less @ real @ ( zero_zero @ real ) @ C2 )
         => ( ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C2 ) @ B2 ) @ A2 )
            = ( ord_less_eq @ A @ B2 @ ( real_V8093663219630862766scaleR @ A @ C2 @ A2 ) ) ) ) ) ).

% pos_divideR_le_eq
thf(fact_5010_pos__divideR__less__eq,axiom,
    ! [A: $tType] :
      ( ( real_V5355595471888546746vector @ A )
     => ! [C2: real,B2: A,A2: A] :
          ( ( ord_less @ real @ ( zero_zero @ real ) @ C2 )
         => ( ( ord_less @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C2 ) @ B2 ) @ A2 )
            = ( ord_less @ A @ B2 @ ( real_V8093663219630862766scaleR @ A @ C2 @ A2 ) ) ) ) ) ).

% pos_divideR_less_eq
thf(fact_5011_pos__less__divideR__eq,axiom,
    ! [A: $tType] :
      ( ( real_V5355595471888546746vector @ A )
     => ! [C2: real,A2: A,B2: A] :
          ( ( ord_less @ real @ ( zero_zero @ real ) @ C2 )
         => ( ( ord_less @ A @ A2 @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C2 ) @ B2 ) )
            = ( ord_less @ A @ ( real_V8093663219630862766scaleR @ A @ C2 @ A2 ) @ B2 ) ) ) ) ).

% pos_less_divideR_eq
thf(fact_5012_neg__divideR__less__eq,axiom,
    ! [A: $tType] :
      ( ( real_V5355595471888546746vector @ A )
     => ! [C2: real,B2: A,A2: A] :
          ( ( ord_less @ real @ C2 @ ( zero_zero @ real ) )
         => ( ( ord_less @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C2 ) @ B2 ) @ A2 )
            = ( ord_less @ A @ ( real_V8093663219630862766scaleR @ A @ C2 @ A2 ) @ B2 ) ) ) ) ).

% neg_divideR_less_eq
thf(fact_5013_neg__less__divideR__eq,axiom,
    ! [A: $tType] :
      ( ( real_V5355595471888546746vector @ A )
     => ! [C2: real,A2: A,B2: A] :
          ( ( ord_less @ real @ C2 @ ( zero_zero @ real ) )
         => ( ( ord_less @ A @ A2 @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C2 ) @ B2 ) )
            = ( ord_less @ A @ B2 @ ( real_V8093663219630862766scaleR @ A @ C2 @ A2 ) ) ) ) ) ).

% neg_less_divideR_eq
thf(fact_5014_forall__pos__mono__1,axiom,
    ! [P: real > $o,E3: real] :
      ( ! [D4: real,E2: real] :
          ( ( ord_less @ real @ D4 @ E2 )
         => ( ( P @ D4 )
           => ( P @ E2 ) ) )
     => ( ! [N4: nat] : ( P @ ( inverse_inverse @ real @ ( semiring_1_of_nat @ real @ ( suc @ N4 ) ) ) )
       => ( ( ord_less @ real @ ( zero_zero @ real ) @ E3 )
         => ( P @ E3 ) ) ) ) ).

% forall_pos_mono_1
thf(fact_5015_sqrt__divide__self__eq,axiom,
    ! [X2: real] :
      ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 )
     => ( ( divide_divide @ real @ ( sqrt @ X2 ) @ X2 )
        = ( inverse_inverse @ real @ ( sqrt @ X2 ) ) ) ) ).

% sqrt_divide_self_eq
thf(fact_5016_summable__exp,axiom,
    ! [A: $tType] :
      ( ( ( real_Vector_banach @ A )
        & ( real_V3459762299906320749_field @ A ) )
     => ! [X2: A] :
          ( summable @ A
          @ ^ [N: nat] : ( times_times @ A @ ( inverse_inverse @ A @ ( semiring_char_0_fact @ A @ N ) ) @ ( power_power @ A @ X2 @ N ) ) ) ) ).

% summable_exp
thf(fact_5017_summable__exp__generic,axiom,
    ! [A: $tType] :
      ( ( ( real_Vector_banach @ A )
        & ( real_V2822296259951069270ebra_1 @ A ) )
     => ! [X2: A] :
          ( summable @ A
          @ ^ [N: nat] : ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ A @ X2 @ N ) ) ) ) ).

% summable_exp_generic
thf(fact_5018_ex__inverse__of__nat__less,axiom,
    ! [A: $tType] :
      ( ( archim462609752435547400_field @ A )
     => ! [X2: A] :
          ( ( ord_less @ A @ ( zero_zero @ A ) @ X2 )
         => ? [N4: nat] :
              ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N4 )
              & ( ord_less @ A @ ( inverse_inverse @ A @ ( semiring_1_of_nat @ A @ N4 ) ) @ X2 ) ) ) ) ).

% ex_inverse_of_nat_less
thf(fact_5019_power__diff__conv__inverse,axiom,
    ! [A: $tType] :
      ( ( division_ring @ A )
     => ! [X2: A,M: nat,N2: nat] :
          ( ( X2
           != ( zero_zero @ A ) )
         => ( ( ord_less_eq @ nat @ M @ N2 )
           => ( ( power_power @ A @ X2 @ ( minus_minus @ nat @ N2 @ M ) )
              = ( times_times @ A @ ( power_power @ A @ X2 @ N2 ) @ ( power_power @ A @ ( inverse_inverse @ A @ X2 ) @ M ) ) ) ) ) ) ).

% power_diff_conv_inverse
thf(fact_5020_pos__le__minus__divideR__eq,axiom,
    ! [A: $tType] :
      ( ( real_V5355595471888546746vector @ A )
     => ! [C2: real,A2: A,B2: A] :
          ( ( ord_less @ real @ ( zero_zero @ real ) @ C2 )
         => ( ( ord_less_eq @ A @ A2 @ ( uminus_uminus @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C2 ) @ B2 ) ) )
            = ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ C2 @ A2 ) @ ( uminus_uminus @ A @ B2 ) ) ) ) ) ).

% pos_le_minus_divideR_eq
thf(fact_5021_pos__minus__divideR__le__eq,axiom,
    ! [A: $tType] :
      ( ( real_V5355595471888546746vector @ A )
     => ! [C2: real,B2: A,A2: A] :
          ( ( ord_less @ real @ ( zero_zero @ real ) @ C2 )
         => ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C2 ) @ B2 ) ) @ A2 )
            = ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ ( real_V8093663219630862766scaleR @ A @ C2 @ A2 ) ) ) ) ) ).

% pos_minus_divideR_le_eq
thf(fact_5022_neg__le__minus__divideR__eq,axiom,
    ! [A: $tType] :
      ( ( real_V5355595471888546746vector @ A )
     => ! [C2: real,A2: A,B2: A] :
          ( ( ord_less @ real @ C2 @ ( zero_zero @ real ) )
         => ( ( ord_less_eq @ A @ A2 @ ( uminus_uminus @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C2 ) @ B2 ) ) )
            = ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ ( real_V8093663219630862766scaleR @ A @ C2 @ A2 ) ) ) ) ) ).

% neg_le_minus_divideR_eq
thf(fact_5023_neg__minus__divideR__le__eq,axiom,
    ! [A: $tType] :
      ( ( real_V5355595471888546746vector @ A )
     => ! [C2: real,B2: A,A2: A] :
          ( ( ord_less @ real @ C2 @ ( zero_zero @ real ) )
         => ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C2 ) @ B2 ) ) @ A2 )
            = ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ C2 @ A2 ) @ ( uminus_uminus @ A @ B2 ) ) ) ) ) ).

% neg_minus_divideR_le_eq
thf(fact_5024_neg__minus__divideR__less__eq,axiom,
    ! [A: $tType] :
      ( ( real_V5355595471888546746vector @ A )
     => ! [C2: real,B2: A,A2: A] :
          ( ( ord_less @ real @ C2 @ ( zero_zero @ real ) )
         => ( ( ord_less @ A @ ( uminus_uminus @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C2 ) @ B2 ) ) @ A2 )
            = ( ord_less @ A @ ( real_V8093663219630862766scaleR @ A @ C2 @ A2 ) @ ( uminus_uminus @ A @ B2 ) ) ) ) ) ).

% neg_minus_divideR_less_eq
thf(fact_5025_neg__less__minus__divideR__eq,axiom,
    ! [A: $tType] :
      ( ( real_V5355595471888546746vector @ A )
     => ! [C2: real,A2: A,B2: A] :
          ( ( ord_less @ real @ C2 @ ( zero_zero @ real ) )
         => ( ( ord_less @ A @ A2 @ ( uminus_uminus @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C2 ) @ B2 ) ) )
            = ( ord_less @ A @ ( uminus_uminus @ A @ B2 ) @ ( real_V8093663219630862766scaleR @ A @ C2 @ A2 ) ) ) ) ) ).

% neg_less_minus_divideR_eq
thf(fact_5026_pos__minus__divideR__less__eq,axiom,
    ! [A: $tType] :
      ( ( real_V5355595471888546746vector @ A )
     => ! [C2: real,B2: A,A2: A] :
          ( ( ord_less @ real @ ( zero_zero @ real ) @ C2 )
         => ( ( ord_less @ A @ ( uminus_uminus @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C2 ) @ B2 ) ) @ A2 )
            = ( ord_less @ A @ ( uminus_uminus @ A @ B2 ) @ ( real_V8093663219630862766scaleR @ A @ C2 @ A2 ) ) ) ) ) ).

% pos_minus_divideR_less_eq
thf(fact_5027_pos__less__minus__divideR__eq,axiom,
    ! [A: $tType] :
      ( ( real_V5355595471888546746vector @ A )
     => ! [C2: real,A2: A,B2: A] :
          ( ( ord_less @ real @ ( zero_zero @ real ) @ C2 )
         => ( ( ord_less @ A @ A2 @ ( uminus_uminus @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C2 ) @ B2 ) ) )
            = ( ord_less @ A @ ( real_V8093663219630862766scaleR @ A @ C2 @ A2 ) @ ( uminus_uminus @ A @ B2 ) ) ) ) ) ).

% pos_less_minus_divideR_eq
thf(fact_5028_i__complex__of__real,axiom,
    ! [R4: real] :
      ( ( times_times @ complex @ imaginary_unit @ ( real_Vector_of_real @ complex @ R4 ) )
      = ( complex2 @ ( zero_zero @ real ) @ R4 ) ) ).

% i_complex_of_real
thf(fact_5029_complex__of__real__i,axiom,
    ! [R4: real] :
      ( ( times_times @ complex @ ( real_Vector_of_real @ complex @ R4 ) @ imaginary_unit )
      = ( complex2 @ ( zero_zero @ real ) @ R4 ) ) ).

% complex_of_real_i
thf(fact_5030_log__inverse,axiom,
    ! [A2: real,X2: real] :
      ( ( ord_less @ real @ ( zero_zero @ real ) @ A2 )
     => ( ( A2
         != ( one_one @ real ) )
       => ( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
         => ( ( log @ A2 @ ( inverse_inverse @ real @ X2 ) )
            = ( uminus_uminus @ real @ ( log @ A2 @ X2 ) ) ) ) ) ) ).

% log_inverse
thf(fact_5031_Complex__eq,axiom,
    ( complex2
    = ( ^ [A5: real,B5: real] : ( plus_plus @ complex @ ( real_Vector_of_real @ complex @ A5 ) @ ( times_times @ complex @ imaginary_unit @ ( real_Vector_of_real @ complex @ B5 ) ) ) ) ) ).

% Complex_eq
thf(fact_5032_exp__converges,axiom,
    ! [A: $tType] :
      ( ( ( real_Vector_banach @ A )
        & ( real_V2822296259951069270ebra_1 @ A ) )
     => ! [X2: A] :
          ( sums @ A
          @ ^ [N: nat] : ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ A @ X2 @ N ) )
          @ ( exp @ A @ X2 ) ) ) ).

% exp_converges
thf(fact_5033_exp__def,axiom,
    ! [A: $tType] :
      ( ( ( real_Vector_banach @ A )
        & ( real_V2822296259951069270ebra_1 @ A ) )
     => ( ( exp @ A )
        = ( ^ [X: A] :
              ( suminf @ A
              @ ^ [N: nat] : ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ A @ X @ N ) ) ) ) ) ) ).

% exp_def
thf(fact_5034_summable__norm__exp,axiom,
    ! [A: $tType] :
      ( ( ( real_Vector_banach @ A )
        & ( real_V2822296259951069270ebra_1 @ A ) )
     => ! [X2: A] :
          ( summable @ real
          @ ^ [N: nat] : ( real_V7770717601297561774m_norm @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ A @ X2 @ N ) ) ) ) ) ).

% summable_norm_exp
thf(fact_5035_exp__plus__inverse__exp,axiom,
    ! [X2: real] : ( ord_less_eq @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( plus_plus @ real @ ( exp @ real @ X2 ) @ ( inverse_inverse @ real @ ( exp @ real @ X2 ) ) ) ) ).

% exp_plus_inverse_exp
thf(fact_5036_complex__split__polar,axiom,
    ! [Z: complex] :
    ? [R3: real,A4: real] :
      ( Z
      = ( times_times @ complex @ ( real_Vector_of_real @ complex @ R3 ) @ ( plus_plus @ complex @ ( real_Vector_of_real @ complex @ ( cos @ real @ A4 ) ) @ ( times_times @ complex @ imaginary_unit @ ( real_Vector_of_real @ complex @ ( sin @ real @ A4 ) ) ) ) ) ) ).

% complex_split_polar
thf(fact_5037_plus__inverse__ge__2,axiom,
    ! [X2: real] :
      ( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
     => ( ord_less_eq @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( plus_plus @ real @ X2 @ ( inverse_inverse @ real @ X2 ) ) ) ) ).

% plus_inverse_ge_2
thf(fact_5038_real__inv__sqrt__pow2,axiom,
    ! [X2: real] :
      ( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
     => ( ( power_power @ real @ ( inverse_inverse @ real @ ( sqrt @ X2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
        = ( inverse_inverse @ real @ X2 ) ) ) ).

% real_inv_sqrt_pow2
thf(fact_5039_tan__cot,axiom,
    ! [X2: real] :
      ( ( tan @ real @ ( minus_minus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ X2 ) )
      = ( inverse_inverse @ real @ ( tan @ real @ X2 ) ) ) ).

% tan_cot
thf(fact_5040_real__le__x__sinh,axiom,
    ! [X2: real] :
      ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 )
     => ( ord_less_eq @ real @ X2 @ ( divide_divide @ real @ ( minus_minus @ real @ ( exp @ real @ X2 ) @ ( inverse_inverse @ real @ ( exp @ real @ X2 ) ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ).

% real_le_x_sinh
thf(fact_5041_real__le__abs__sinh,axiom,
    ! [X2: real] : ( ord_less_eq @ real @ ( abs_abs @ real @ X2 ) @ ( abs_abs @ real @ ( divide_divide @ real @ ( minus_minus @ real @ ( exp @ real @ X2 ) @ ( inverse_inverse @ real @ ( exp @ real @ X2 ) ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ).

% real_le_abs_sinh
thf(fact_5042_exp__series__add__commuting,axiom,
    ! [A: $tType] :
      ( ( ( real_Vector_banach @ A )
        & ( real_V2822296259951069270ebra_1 @ A ) )
     => ! [X2: A,Y2: A,N2: nat] :
          ( ( ( times_times @ A @ X2 @ Y2 )
            = ( times_times @ A @ Y2 @ X2 ) )
         => ( ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ N2 ) ) @ ( power_power @ A @ ( plus_plus @ A @ X2 @ Y2 ) @ N2 ) )
            = ( groups7311177749621191930dd_sum @ nat @ A
              @ ^ [I2: nat] : ( times_times @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ I2 ) ) @ ( power_power @ A @ X2 @ I2 ) ) @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ ( minus_minus @ nat @ N2 @ I2 ) ) ) @ ( power_power @ A @ Y2 @ ( minus_minus @ nat @ N2 @ I2 ) ) ) )
              @ ( set_ord_atMost @ nat @ N2 ) ) ) ) ) ).

% exp_series_add_commuting
thf(fact_5043_exp__first__term,axiom,
    ! [A: $tType] :
      ( ( ( real_Vector_banach @ A )
        & ( real_V2822296259951069270ebra_1 @ A ) )
     => ( ( exp @ A )
        = ( ^ [X: A] :
              ( plus_plus @ A @ ( one_one @ A )
              @ ( suminf @ A
                @ ^ [N: nat] : ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ ( suc @ N ) ) ) @ ( power_power @ A @ X @ ( suc @ N ) ) ) ) ) ) ) ) ).

% exp_first_term
thf(fact_5044_tan__sec,axiom,
    ! [A: $tType] :
      ( ( ( real_Vector_banach @ A )
        & ( real_V3459762299906320749_field @ A ) )
     => ! [X2: A] :
          ( ( ( cos @ A @ X2 )
           != ( zero_zero @ A ) )
         => ( ( plus_plus @ A @ ( one_one @ A ) @ ( power_power @ A @ ( tan @ A @ X2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
            = ( power_power @ A @ ( inverse_inverse @ A @ ( cos @ A @ X2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).

% tan_sec
thf(fact_5045_cmod__unit__one,axiom,
    ! [A2: real] :
      ( ( real_V7770717601297561774m_norm @ complex @ ( plus_plus @ complex @ ( real_Vector_of_real @ complex @ ( cos @ real @ A2 ) ) @ ( times_times @ complex @ imaginary_unit @ ( real_Vector_of_real @ complex @ ( sin @ real @ A2 ) ) ) ) )
      = ( one_one @ real ) ) ).

% cmod_unit_one
thf(fact_5046_cmod__complex__polar,axiom,
    ! [R4: real,A2: real] :
      ( ( real_V7770717601297561774m_norm @ complex @ ( times_times @ complex @ ( real_Vector_of_real @ complex @ R4 ) @ ( plus_plus @ complex @ ( real_Vector_of_real @ complex @ ( cos @ real @ A2 ) ) @ ( times_times @ complex @ imaginary_unit @ ( real_Vector_of_real @ complex @ ( sin @ real @ A2 ) ) ) ) ) )
      = ( abs_abs @ real @ R4 ) ) ).

% cmod_complex_polar
thf(fact_5047_powr__real__of__int,axiom,
    ! [X2: real,N2: int] :
      ( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
     => ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ N2 )
         => ( ( powr @ real @ X2 @ ( ring_1_of_int @ real @ N2 ) )
            = ( power_power @ real @ X2 @ ( nat2 @ N2 ) ) ) )
        & ( ~ ( ord_less_eq @ int @ ( zero_zero @ int ) @ N2 )
         => ( ( powr @ real @ X2 @ ( ring_1_of_int @ real @ N2 ) )
            = ( inverse_inverse @ real @ ( power_power @ real @ X2 @ ( nat2 @ ( uminus_uminus @ int @ N2 ) ) ) ) ) ) ) ) ).

% powr_real_of_int
thf(fact_5048_exp__first__terms,axiom,
    ! [A: $tType] :
      ( ( ( real_Vector_banach @ A )
        & ( real_V2822296259951069270ebra_1 @ A ) )
     => ! [K: nat] :
          ( ( exp @ A )
          = ( ^ [X: A] :
                ( plus_plus @ A
                @ ( groups7311177749621191930dd_sum @ nat @ A
                  @ ^ [N: nat] : ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ A @ X @ N ) )
                  @ ( set_ord_lessThan @ nat @ K ) )
                @ ( suminf @ A
                  @ ^ [N: nat] : ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ ( plus_plus @ nat @ N @ K ) ) ) @ ( power_power @ A @ X @ ( plus_plus @ nat @ N @ K ) ) ) ) ) ) ) ) ).

% exp_first_terms
thf(fact_5049_Maclaurin__sin__bound,axiom,
    ! [X2: real,N2: nat] :
      ( ord_less_eq @ real
      @ ( abs_abs @ real
        @ ( minus_minus @ real @ ( sin @ real @ X2 )
          @ ( groups7311177749621191930dd_sum @ nat @ real
            @ ^ [M7: nat] : ( times_times @ real @ ( sin_coeff @ M7 ) @ ( power_power @ real @ X2 @ M7 ) )
            @ ( set_ord_lessThan @ nat @ N2 ) ) ) )
      @ ( times_times @ real @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ N2 ) ) @ ( power_power @ real @ ( abs_abs @ real @ X2 ) @ N2 ) ) ) ).

% Maclaurin_sin_bound
thf(fact_5050_Arg__minus__ii,axiom,
    ( ( arg @ ( uminus_uminus @ complex @ imaginary_unit ) )
    = ( divide_divide @ real @ ( uminus_uminus @ real @ pi ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ).

% Arg_minus_ii
thf(fact_5051_csqrt__ii,axiom,
    ( ( csqrt @ imaginary_unit )
    = ( divide_divide @ complex @ ( plus_plus @ complex @ ( one_one @ complex ) @ imaginary_unit ) @ ( real_Vector_of_real @ complex @ ( sqrt @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ).

% csqrt_ii
thf(fact_5052_Arg__ii,axiom,
    ( ( arg @ imaginary_unit )
    = ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ).

% Arg_ii
thf(fact_5053_csqrt__1,axiom,
    ( ( csqrt @ ( one_one @ complex ) )
    = ( one_one @ complex ) ) ).

% csqrt_1
thf(fact_5054_csqrt__eq__1,axiom,
    ! [Z: complex] :
      ( ( ( csqrt @ Z )
        = ( one_one @ complex ) )
      = ( Z
        = ( one_one @ complex ) ) ) ).

% csqrt_eq_1
thf(fact_5055_power2__csqrt,axiom,
    ! [Z: complex] :
      ( ( power_power @ complex @ ( csqrt @ Z ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
      = Z ) ).

% power2_csqrt
thf(fact_5056_divide__complex__def,axiom,
    ( ( divide_divide @ complex )
    = ( ^ [X: complex,Y: complex] : ( times_times @ complex @ X @ ( inverse_inverse @ complex @ Y ) ) ) ) ).

% divide_complex_def
thf(fact_5057_of__real__sqrt,axiom,
    ! [X2: real] :
      ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 )
     => ( ( real_Vector_of_real @ complex @ ( sqrt @ X2 ) )
        = ( csqrt @ ( real_Vector_of_real @ complex @ X2 ) ) ) ) ).

% of_real_sqrt
thf(fact_5058_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_5059_complex__inverse,axiom,
    ! [A2: real,B2: real] :
      ( ( inverse_inverse @ complex @ ( complex2 @ A2 @ B2 ) )
      = ( complex2 @ ( divide_divide @ real @ A2 @ ( plus_plus @ real @ ( power_power @ real @ A2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ B2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( divide_divide @ real @ ( uminus_uminus @ real @ B2 ) @ ( plus_plus @ real @ ( power_power @ real @ A2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ B2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ).

% complex_inverse
thf(fact_5060_sinh__converges,axiom,
    ! [A: $tType] :
      ( ( ( real_Vector_banach @ A )
        & ( real_V2822296259951069270ebra_1 @ A ) )
     => ! [X2: A] :
          ( sums @ A
          @ ^ [N: nat] : ( if @ A @ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ ( zero_zero @ A ) @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ A @ X2 @ N ) ) )
          @ ( sinh @ A @ X2 ) ) ) ).

% sinh_converges
thf(fact_5061_cis__minus__pi__half,axiom,
    ( ( cis @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) )
    = ( uminus_uminus @ complex @ imaginary_unit ) ) ).

% cis_minus_pi_half
thf(fact_5062_cosh__converges,axiom,
    ! [A: $tType] :
      ( ( ( real_Vector_banach @ A )
        & ( real_V2822296259951069270ebra_1 @ A ) )
     => ! [X2: A] :
          ( sums @ A
          @ ^ [N: nat] : ( if @ A @ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ A @ X2 @ N ) ) @ ( zero_zero @ A ) )
          @ ( cosh @ A @ X2 ) ) ) ).

% cosh_converges
thf(fact_5063_sinh__real__le__iff,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_eq @ real @ ( sinh @ real @ X2 ) @ ( sinh @ real @ Y2 ) )
      = ( ord_less_eq @ real @ X2 @ Y2 ) ) ).

% sinh_real_le_iff
thf(fact_5064_sinh__real__nonneg__iff,axiom,
    ! [X2: real] :
      ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( sinh @ real @ X2 ) )
      = ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 ) ) ).

% sinh_real_nonneg_iff
thf(fact_5065_sinh__real__nonpos__iff,axiom,
    ! [X2: real] :
      ( ( ord_less_eq @ real @ ( sinh @ real @ X2 ) @ ( zero_zero @ real ) )
      = ( ord_less_eq @ real @ X2 @ ( zero_zero @ real ) ) ) ).

% sinh_real_nonpos_iff
thf(fact_5066_cosh__0,axiom,
    ! [A: $tType] :
      ( ( ( real_Vector_banach @ A )
        & ( real_V2822296259951069270ebra_1 @ A ) )
     => ( ( cosh @ A @ ( zero_zero @ A ) )
        = ( one_one @ A ) ) ) ).

% cosh_0
thf(fact_5067_norm__cis,axiom,
    ! [A2: real] :
      ( ( real_V7770717601297561774m_norm @ complex @ ( cis @ A2 ) )
      = ( one_one @ real ) ) ).

% norm_cis
thf(fact_5068_cis__zero,axiom,
    ( ( cis @ ( zero_zero @ real ) )
    = ( one_one @ complex ) ) ).

% cis_zero
thf(fact_5069_cis__pi,axiom,
    ( ( cis @ pi )
    = ( uminus_uminus @ complex @ ( one_one @ complex ) ) ) ).

% cis_pi
thf(fact_5070_cis__pi__half,axiom,
    ( ( cis @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
    = imaginary_unit ) ).

% cis_pi_half
thf(fact_5071_cis__2pi,axiom,
    ( ( cis @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) )
    = ( one_one @ complex ) ) ).

% cis_2pi
thf(fact_5072_tanh__def,axiom,
    ! [A: $tType] :
      ( ( ( real_Vector_banach @ A )
        & ( real_V3459762299906320749_field @ A ) )
     => ( ( tanh @ A )
        = ( ^ [X: A] : ( divide_divide @ A @ ( sinh @ A @ X ) @ ( cosh @ A @ X ) ) ) ) ) ).

% tanh_def
thf(fact_5073_cosh__minus__sinh,axiom,
    ! [A: $tType] :
      ( ( ( real_Vector_banach @ A )
        & ( real_V2822296259951069270ebra_1 @ A ) )
     => ! [X2: A] :
          ( ( minus_minus @ A @ ( cosh @ A @ X2 ) @ ( sinh @ A @ X2 ) )
          = ( exp @ A @ ( uminus_uminus @ A @ X2 ) ) ) ) ).

% cosh_minus_sinh
thf(fact_5074_sinh__minus__cosh,axiom,
    ! [A: $tType] :
      ( ( ( real_Vector_banach @ A )
        & ( real_V2822296259951069270ebra_1 @ A ) )
     => ! [X2: A] :
          ( ( minus_minus @ A @ ( sinh @ A @ X2 ) @ ( cosh @ A @ X2 ) )
          = ( uminus_uminus @ A @ ( exp @ A @ ( uminus_uminus @ A @ X2 ) ) ) ) ) ).

% sinh_minus_cosh
thf(fact_5075_cosh__plus__sinh,axiom,
    ! [A: $tType] :
      ( ( ( real_Vector_banach @ A )
        & ( real_V2822296259951069270ebra_1 @ A ) )
     => ! [X2: A] :
          ( ( plus_plus @ A @ ( cosh @ A @ X2 ) @ ( sinh @ A @ X2 ) )
          = ( exp @ A @ X2 ) ) ) ).

% cosh_plus_sinh
thf(fact_5076_sinh__plus__cosh,axiom,
    ! [A: $tType] :
      ( ( ( real_Vector_banach @ A )
        & ( real_V2822296259951069270ebra_1 @ A ) )
     => ! [X2: A] :
          ( ( plus_plus @ A @ ( sinh @ A @ X2 ) @ ( cosh @ A @ X2 ) )
          = ( exp @ A @ X2 ) ) ) ).

% sinh_plus_cosh
thf(fact_5077_sinh__le__cosh__real,axiom,
    ! [X2: real] : ( ord_less_eq @ real @ ( sinh @ real @ X2 ) @ ( cosh @ real @ X2 ) ) ).

% sinh_le_cosh_real
thf(fact_5078_cosh__diff,axiom,
    ! [A: $tType] :
      ( ( ( real_Vector_banach @ A )
        & ( real_V3459762299906320749_field @ A ) )
     => ! [X2: A,Y2: A] :
          ( ( cosh @ A @ ( minus_minus @ A @ X2 @ Y2 ) )
          = ( minus_minus @ A @ ( times_times @ A @ ( cosh @ A @ X2 ) @ ( cosh @ A @ Y2 ) ) @ ( times_times @ A @ ( sinh @ A @ X2 ) @ ( sinh @ A @ Y2 ) ) ) ) ) ).

% cosh_diff
thf(fact_5079_sinh__diff,axiom,
    ! [A: $tType] :
      ( ( ( real_Vector_banach @ A )
        & ( real_V3459762299906320749_field @ A ) )
     => ! [X2: A,Y2: A] :
          ( ( sinh @ A @ ( minus_minus @ A @ X2 @ Y2 ) )
          = ( minus_minus @ A @ ( times_times @ A @ ( sinh @ A @ X2 ) @ ( cosh @ A @ Y2 ) ) @ ( times_times @ A @ ( cosh @ A @ X2 ) @ ( sinh @ A @ Y2 ) ) ) ) ) ).

% sinh_diff
thf(fact_5080_sinh__add,axiom,
    ! [A: $tType] :
      ( ( ( real_Vector_banach @ A )
        & ( real_V3459762299906320749_field @ A ) )
     => ! [X2: A,Y2: A] :
          ( ( sinh @ A @ ( plus_plus @ A @ X2 @ Y2 ) )
          = ( plus_plus @ A @ ( times_times @ A @ ( sinh @ A @ X2 ) @ ( cosh @ A @ Y2 ) ) @ ( times_times @ A @ ( cosh @ A @ X2 ) @ ( sinh @ A @ Y2 ) ) ) ) ) ).

% sinh_add
thf(fact_5081_cosh__add,axiom,
    ! [A: $tType] :
      ( ( ( real_Vector_banach @ A )
        & ( real_V3459762299906320749_field @ A ) )
     => ! [X2: A,Y2: A] :
          ( ( cosh @ A @ ( plus_plus @ A @ X2 @ Y2 ) )
          = ( plus_plus @ A @ ( times_times @ A @ ( cosh @ A @ X2 ) @ ( cosh @ A @ Y2 ) ) @ ( times_times @ A @ ( sinh @ A @ X2 ) @ ( sinh @ A @ Y2 ) ) ) ) ) ).

% cosh_add
thf(fact_5082_cosh__real__nonpos__le__iff,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_eq @ real @ X2 @ ( zero_zero @ real ) )
     => ( ( ord_less_eq @ real @ Y2 @ ( zero_zero @ real ) )
       => ( ( ord_less_eq @ real @ ( cosh @ real @ X2 ) @ ( cosh @ real @ Y2 ) )
          = ( ord_less_eq @ real @ Y2 @ X2 ) ) ) ) ).

% cosh_real_nonpos_le_iff
thf(fact_5083_cosh__real__nonneg__le__iff,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 )
     => ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y2 )
       => ( ( ord_less_eq @ real @ ( cosh @ real @ X2 ) @ ( cosh @ real @ Y2 ) )
          = ( ord_less_eq @ real @ X2 @ Y2 ) ) ) ) ).

% cosh_real_nonneg_le_iff
thf(fact_5084_cosh__real__nonneg,axiom,
    ! [X2: real] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( cosh @ real @ X2 ) ) ).

% cosh_real_nonneg
thf(fact_5085_sinh__double,axiom,
    ! [A: $tType] :
      ( ( ( real_Vector_banach @ A )
        & ( real_V3459762299906320749_field @ A ) )
     => ! [X2: A] :
          ( ( sinh @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ X2 ) )
          = ( times_times @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( sinh @ A @ X2 ) ) @ ( cosh @ A @ X2 ) ) ) ) ).

% sinh_double
thf(fact_5086_cosh__real__ge__1,axiom,
    ! [X2: real] : ( ord_less_eq @ real @ ( one_one @ real ) @ ( cosh @ real @ X2 ) ) ).

% cosh_real_ge_1
thf(fact_5087_cis__mult,axiom,
    ! [A2: real,B2: real] :
      ( ( times_times @ complex @ ( cis @ A2 ) @ ( cis @ B2 ) )
      = ( cis @ ( plus_plus @ real @ A2 @ B2 ) ) ) ).

% cis_mult
thf(fact_5088_cis__divide,axiom,
    ! [A2: real,B2: real] :
      ( ( divide_divide @ complex @ ( cis @ A2 ) @ ( cis @ B2 ) )
      = ( cis @ ( minus_minus @ real @ A2 @ B2 ) ) ) ).

% cis_divide
thf(fact_5089_cosh__real__strict__mono,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 )
     => ( ( ord_less @ real @ X2 @ Y2 )
       => ( ord_less @ real @ ( cosh @ real @ X2 ) @ ( cosh @ real @ Y2 ) ) ) ) ).

% cosh_real_strict_mono
thf(fact_5090_cosh__real__nonneg__less__iff,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 )
     => ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y2 )
       => ( ( ord_less @ real @ ( cosh @ real @ X2 ) @ ( cosh @ real @ Y2 ) )
          = ( ord_less @ real @ X2 @ Y2 ) ) ) ) ).

% cosh_real_nonneg_less_iff
thf(fact_5091_cosh__real__nonpos__less__iff,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_eq @ real @ X2 @ ( zero_zero @ real ) )
     => ( ( ord_less_eq @ real @ Y2 @ ( zero_zero @ real ) )
       => ( ( ord_less @ real @ ( cosh @ real @ X2 ) @ ( cosh @ real @ Y2 ) )
          = ( ord_less @ real @ Y2 @ X2 ) ) ) ) ).

% cosh_real_nonpos_less_iff
thf(fact_5092_cosh__square__eq,axiom,
    ! [A: $tType] :
      ( ( ( real_Vector_banach @ A )
        & ( real_V3459762299906320749_field @ A ) )
     => ! [X2: A] :
          ( ( power_power @ A @ ( cosh @ A @ X2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
          = ( plus_plus @ A @ ( power_power @ A @ ( sinh @ A @ X2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ A ) ) ) ) ).

% cosh_square_eq
thf(fact_5093_hyperbolic__pythagoras,axiom,
    ! [A: $tType] :
      ( ( ( real_Vector_banach @ A )
        & ( real_V3459762299906320749_field @ A ) )
     => ! [X2: A] :
          ( ( minus_minus @ A @ ( power_power @ A @ ( cosh @ A @ X2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ ( sinh @ A @ X2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
          = ( one_one @ A ) ) ) ).

% hyperbolic_pythagoras
thf(fact_5094_sinh__square__eq,axiom,
    ! [A: $tType] :
      ( ( ( real_Vector_banach @ A )
        & ( real_V3459762299906320749_field @ A ) )
     => ! [X2: A] :
          ( ( power_power @ A @ ( sinh @ A @ X2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
          = ( minus_minus @ A @ ( power_power @ A @ ( cosh @ A @ X2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ A ) ) ) ) ).

% sinh_square_eq
thf(fact_5095_arcosh__cosh__real,axiom,
    ! [X2: real] :
      ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 )
     => ( ( arcosh @ real @ ( cosh @ real @ X2 ) )
        = X2 ) ) ).

% arcosh_cosh_real
thf(fact_5096_cosh__double,axiom,
    ! [A: $tType] :
      ( ( ( real_Vector_banach @ A )
        & ( real_V3459762299906320749_field @ A ) )
     => ! [X2: A] :
          ( ( cosh @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ X2 ) )
          = ( plus_plus @ A @ ( power_power @ A @ ( cosh @ A @ X2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ ( sinh @ A @ X2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).

% cosh_double
thf(fact_5097_DeMoivre,axiom,
    ! [A2: real,N2: nat] :
      ( ( power_power @ complex @ ( cis @ A2 ) @ N2 )
      = ( cis @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N2 ) @ A2 ) ) ) ).

% DeMoivre
thf(fact_5098_cis__conv__exp,axiom,
    ( cis
    = ( ^ [B5: real] : ( exp @ complex @ ( times_times @ complex @ imaginary_unit @ ( real_Vector_of_real @ complex @ B5 ) ) ) ) ) ).

% cis_conv_exp
thf(fact_5099_tanh__add,axiom,
    ! [A: $tType] :
      ( ( ( real_Vector_banach @ A )
        & ( real_V3459762299906320749_field @ A ) )
     => ! [X2: A,Y2: A] :
          ( ( ( cosh @ A @ X2 )
           != ( zero_zero @ A ) )
         => ( ( ( cosh @ A @ Y2 )
             != ( zero_zero @ A ) )
           => ( ( tanh @ A @ ( plus_plus @ A @ X2 @ Y2 ) )
              = ( divide_divide @ A @ ( plus_plus @ A @ ( tanh @ A @ X2 ) @ ( tanh @ A @ Y2 ) ) @ ( plus_plus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( tanh @ A @ X2 ) @ ( tanh @ A @ Y2 ) ) ) ) ) ) ) ) ).

% tanh_add
thf(fact_5100_sinh__zero__iff,axiom,
    ! [A: $tType] :
      ( ( ( real_Vector_banach @ A )
        & ( real_V3459762299906320749_field @ A ) )
     => ! [X2: A] :
          ( ( ( sinh @ A @ X2 )
            = ( zero_zero @ A ) )
          = ( member @ A @ ( exp @ A @ X2 ) @ ( insert @ A @ ( one_one @ A ) @ ( insert @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).

% sinh_zero_iff
thf(fact_5101_cosh__field__def,axiom,
    ! [A: $tType] :
      ( ( ( real_Vector_banach @ A )
        & ( real_V3459762299906320749_field @ A ) )
     => ( ( cosh @ A )
        = ( ^ [Z2: A] : ( divide_divide @ A @ ( plus_plus @ A @ ( exp @ A @ Z2 ) @ ( exp @ A @ ( uminus_uminus @ A @ Z2 ) ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).

% cosh_field_def
thf(fact_5102_sinh__field__def,axiom,
    ! [A: $tType] :
      ( ( ( real_Vector_banach @ A )
        & ( real_V3459762299906320749_field @ A ) )
     => ( ( sinh @ A )
        = ( ^ [Z2: A] : ( divide_divide @ A @ ( minus_minus @ A @ ( exp @ A @ Z2 ) @ ( exp @ A @ ( uminus_uminus @ A @ Z2 ) ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).

% sinh_field_def
thf(fact_5103_cosh__zero__iff,axiom,
    ! [A: $tType] :
      ( ( ( real_Vector_banach @ A )
        & ( real_V3459762299906320749_field @ A ) )
     => ! [X2: A] :
          ( ( ( cosh @ A @ X2 )
            = ( zero_zero @ A ) )
          = ( ( power_power @ A @ ( exp @ A @ X2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
            = ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ) ).

% cosh_zero_iff
thf(fact_5104_cosh__def,axiom,
    ! [A: $tType] :
      ( ( ( real_Vector_banach @ A )
        & ( real_V2822296259951069270ebra_1 @ A ) )
     => ( ( cosh @ A )
        = ( ^ [X: A] : ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( plus_plus @ A @ ( exp @ A @ X ) @ ( exp @ A @ ( uminus_uminus @ A @ X ) ) ) ) ) ) ) ).

% cosh_def
thf(fact_5105_cosh__ln__real,axiom,
    ! [X2: real] :
      ( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
     => ( ( cosh @ real @ ( ln_ln @ real @ X2 ) )
        = ( divide_divide @ real @ ( plus_plus @ real @ X2 @ ( inverse_inverse @ real @ X2 ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ).

% cosh_ln_real
thf(fact_5106_sinh__def,axiom,
    ! [A: $tType] :
      ( ( ( real_Vector_banach @ A )
        & ( real_V2822296259951069270ebra_1 @ A ) )
     => ( ( sinh @ A )
        = ( ^ [X: A] : ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( minus_minus @ A @ ( exp @ A @ X ) @ ( exp @ A @ ( uminus_uminus @ A @ X ) ) ) ) ) ) ) ).

% sinh_def
thf(fact_5107_sinh__ln__real,axiom,
    ! [X2: real] :
      ( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
     => ( ( sinh @ real @ ( ln_ln @ real @ X2 ) )
        = ( divide_divide @ real @ ( minus_minus @ real @ X2 @ ( inverse_inverse @ real @ X2 ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ).

% sinh_ln_real
thf(fact_5108_bij__betw__roots__unity,axiom,
    ! [N2: nat] :
      ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
     => ( bij_betw @ nat @ complex
        @ ^ [K4: nat] : ( cis @ ( divide_divide @ real @ ( times_times @ real @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) @ ( semiring_1_of_nat @ real @ K4 ) ) @ ( semiring_1_of_nat @ real @ N2 ) ) )
        @ ( set_ord_lessThan @ nat @ N2 )
        @ ( collect @ complex
          @ ^ [Z2: complex] :
              ( ( power_power @ complex @ Z2 @ N2 )
              = ( one_one @ complex ) ) ) ) ) ).

% bij_betw_roots_unity
thf(fact_5109_cot__less__zero,axiom,
    ! [X2: real] :
      ( ( ord_less @ real @ ( divide_divide @ real @ ( uminus_uminus @ real @ pi ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ X2 )
     => ( ( ord_less @ real @ X2 @ ( zero_zero @ real ) )
       => ( ord_less @ real @ ( cot @ real @ X2 ) @ ( zero_zero @ real ) ) ) ) ).

% cot_less_zero
thf(fact_5110_sint__range__size,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [W: word @ A] :
          ( ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ ( size_size @ ( word @ A ) @ W ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ) @ ( ring_1_signed @ A @ int @ W ) )
          & ( ord_less @ int @ ( ring_1_signed @ A @ int @ W ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ ( size_size @ ( word @ A ) @ W ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) ).

% sint_range_size
thf(fact_5111_signed__minus__1,axiom,
    ! [B: $tType,A: $tType] :
      ( ( ( ring_1 @ A )
        & ( type_len @ B ) )
     => ( ( ring_1_signed @ B @ A @ ( uminus_uminus @ ( word @ B ) @ ( one_one @ ( word @ B ) ) ) )
        = ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).

% signed_minus_1
thf(fact_5112_sint__minus1,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [X2: word @ A] :
          ( ( ( ring_1_signed @ A @ int @ X2 )
            = ( uminus_uminus @ int @ ( one_one @ int ) ) )
          = ( X2
            = ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) ) ) ) ).

% sint_minus1
thf(fact_5113_cot__npi,axiom,
    ! [N2: nat] :
      ( ( cot @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N2 ) @ pi ) )
      = ( zero_zero @ real ) ) ).

% cot_npi
thf(fact_5114_cot__periodic,axiom,
    ! [X2: real] :
      ( ( cot @ real @ ( plus_plus @ real @ X2 @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) ) )
      = ( cot @ real @ X2 ) ) ).

% cot_periodic
thf(fact_5115_sum_Oreindex__bij__betw,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ( comm_monoid_add @ A )
     => ! [H2: B > C,S3: set @ B,T6: set @ C,G: C > A] :
          ( ( bij_betw @ B @ C @ H2 @ S3 @ T6 )
         => ( ( groups7311177749621191930dd_sum @ B @ A
              @ ^ [X: B] : ( G @ ( H2 @ X ) )
              @ S3 )
            = ( groups7311177749621191930dd_sum @ C @ A @ G @ T6 ) ) ) ) ).

% sum.reindex_bij_betw
thf(fact_5116_prod_Oreindex__bij__betw,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ( comm_monoid_mult @ A )
     => ! [H2: B > C,S3: set @ B,T6: set @ C,G: C > A] :
          ( ( bij_betw @ B @ C @ H2 @ S3 @ T6 )
         => ( ( groups7121269368397514597t_prod @ B @ A
              @ ^ [X: B] : ( G @ ( H2 @ X ) )
              @ S3 )
            = ( groups7121269368397514597t_prod @ C @ A @ G @ T6 ) ) ) ) ).

% prod.reindex_bij_betw
thf(fact_5117_sum_Oreindex__bij__betw__not__neutral,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ( comm_monoid_add @ A )
     => ! [S7: set @ B,T8: set @ C,H2: B > C,S3: set @ B,T6: set @ C,G: C > A] :
          ( ( finite_finite2 @ B @ S7 )
         => ( ( finite_finite2 @ C @ T8 )
           => ( ( bij_betw @ B @ C @ H2 @ ( minus_minus @ ( set @ B ) @ S3 @ S7 ) @ ( minus_minus @ ( set @ C ) @ T6 @ T8 ) )
             => ( ! [A4: B] :
                    ( ( member @ B @ A4 @ S7 )
                   => ( ( G @ ( H2 @ A4 ) )
                      = ( zero_zero @ A ) ) )
               => ( ! [B3: C] :
                      ( ( member @ C @ B3 @ T8 )
                     => ( ( G @ B3 )
                        = ( zero_zero @ A ) ) )
                 => ( ( groups7311177749621191930dd_sum @ B @ A
                      @ ^ [X: B] : ( G @ ( H2 @ X ) )
                      @ S3 )
                    = ( groups7311177749621191930dd_sum @ C @ A @ G @ T6 ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_betw_not_neutral
thf(fact_5118_prod_Oreindex__bij__betw__not__neutral,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ( comm_monoid_mult @ A )
     => ! [S7: set @ B,T8: set @ C,H2: B > C,S3: set @ B,T6: set @ C,G: C > A] :
          ( ( finite_finite2 @ B @ S7 )
         => ( ( finite_finite2 @ C @ T8 )
           => ( ( bij_betw @ B @ C @ H2 @ ( minus_minus @ ( set @ B ) @ S3 @ S7 ) @ ( minus_minus @ ( set @ C ) @ T6 @ T8 ) )
             => ( ! [A4: B] :
                    ( ( member @ B @ A4 @ S7 )
                   => ( ( G @ ( H2 @ A4 ) )
                      = ( one_one @ A ) ) )
               => ( ! [B3: C] :
                      ( ( member @ C @ B3 @ T8 )
                     => ( ( G @ B3 )
                        = ( one_one @ A ) ) )
                 => ( ( groups7121269368397514597t_prod @ B @ A
                      @ ^ [X: B] : ( G @ ( H2 @ X ) )
                      @ S3 )
                    = ( groups7121269368397514597t_prod @ C @ A @ G @ T6 ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_betw_not_neutral
thf(fact_5119_sint__n1,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ( ( ring_1_signed @ A @ int @ ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) )
        = ( uminus_uminus @ int @ ( one_one @ int ) ) ) ) ).

% sint_n1
thf(fact_5120_cot__def,axiom,
    ! [A: $tType] :
      ( ( ( real_Vector_banach @ A )
        & ( real_V3459762299906320749_field @ A ) )
     => ( ( cot @ A )
        = ( ^ [X: A] : ( divide_divide @ A @ ( cos @ A @ X ) @ ( sin @ A @ X ) ) ) ) ) ).

% cot_def
thf(fact_5121_sint__above__size,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [W: word @ A,X2: int] :
          ( ( ord_less_eq @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ ( size_size @ ( word @ A ) @ W ) @ ( one_one @ nat ) ) ) @ X2 )
         => ( ord_less @ int @ ( ring_1_signed @ A @ int @ W ) @ X2 ) ) ) ).

% sint_above_size
thf(fact_5122_sint__below__size,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [X2: int,W: word @ A] :
          ( ( ord_less_eq @ int @ X2 @ ( uminus_uminus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ ( size_size @ ( word @ A ) @ W ) @ ( one_one @ nat ) ) ) ) )
         => ( ord_less_eq @ int @ X2 @ ( ring_1_signed @ A @ int @ W ) ) ) ) ).

% sint_below_size
thf(fact_5123_cot__gt__zero,axiom,
    ! [X2: real] :
      ( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
     => ( ( ord_less @ real @ X2 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
       => ( ord_less @ real @ ( zero_zero @ real ) @ ( cot @ real @ X2 ) ) ) ) ).

% cot_gt_zero
thf(fact_5124_tan__cot_H,axiom,
    ! [X2: real] :
      ( ( tan @ real @ ( minus_minus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ X2 ) )
      = ( cot @ real @ X2 ) ) ).

% tan_cot'
thf(fact_5125_infinite__imp__bij__betw,axiom,
    ! [A: $tType,A3: set @ A,A2: A] :
      ( ~ ( finite_finite2 @ A @ A3 )
     => ? [H3: A > A] : ( bij_betw @ A @ A @ H3 @ A3 @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).

% infinite_imp_bij_betw
thf(fact_5126_predicate1I,axiom,
    ! [A: $tType,P: A > $o,Q: A > $o] :
      ( ! [X3: A] :
          ( ( P @ X3 )
         => ( Q @ X3 ) )
     => ( ord_less_eq @ ( A > $o ) @ P @ Q ) ) ).

% predicate1I
thf(fact_5127_divmod__BitM__2__eq,axiom,
    ! [M: num] :
      ( ( unique8689654367752047608divmod @ int @ ( bitM @ M ) @ ( bit0 @ one2 ) )
      = ( product_Pair @ int @ int @ ( minus_minus @ int @ ( numeral_numeral @ int @ M ) @ ( one_one @ int ) ) @ ( one_one @ int ) ) ) ).

% divmod_BitM_2_eq
thf(fact_5128_dual__order_Orefl,axiom,
    ! [A: $tType] :
      ( ( preorder @ A )
     => ! [A2: A] : ( ord_less_eq @ A @ A2 @ A2 ) ) ).

% dual_order.refl
thf(fact_5129_order__refl,axiom,
    ! [A: $tType] :
      ( ( preorder @ A )
     => ! [X2: A] : ( ord_less_eq @ A @ X2 @ X2 ) ) ).

% order_refl
thf(fact_5130_predicate2I,axiom,
    ! [B: $tType,A: $tType,P: A > B > $o,Q: A > B > $o] :
      ( ! [X3: A,Y4: B] :
          ( ( P @ X3 @ Y4 )
         => ( Q @ X3 @ Y4 ) )
     => ( ord_less_eq @ ( A > B > $o ) @ P @ Q ) ) ).

% predicate2I
thf(fact_5131_dbl__dec__simps_I5_J,axiom,
    ! [A: $tType] :
      ( ( neg_numeral @ A )
     => ! [K: num] :
          ( ( neg_numeral_dbl_dec @ A @ ( numeral_numeral @ A @ K ) )
          = ( numeral_numeral @ A @ ( bitM @ K ) ) ) ) ).

% dbl_dec_simps(5)
thf(fact_5132_pred__numeral__simps_I2_J,axiom,
    ! [K: num] :
      ( ( pred_numeral @ ( bit0 @ K ) )
      = ( numeral_numeral @ nat @ ( bitM @ K ) ) ) ).

% pred_numeral_simps(2)
thf(fact_5133_semiring__norm_I26_J,axiom,
    ( ( bitM @ one2 )
    = one2 ) ).

% semiring_norm(26)
thf(fact_5134_scast__n1,axiom,
    ! [B: $tType,A: $tType] :
      ( ( ( type_len @ A )
        & ( type_len @ B ) )
     => ( ( ring_1_signed @ B @ ( word @ A ) @ ( uminus_uminus @ ( word @ B ) @ ( one_one @ ( word @ B ) ) ) )
        = ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) ) ) ).

% scast_n1
thf(fact_5135_semiring__norm_I28_J,axiom,
    ! [N2: num] :
      ( ( bitM @ ( bit1 @ N2 ) )
      = ( bit1 @ ( bit0 @ N2 ) ) ) ).

% semiring_norm(28)
thf(fact_5136_semiring__norm_I27_J,axiom,
    ! [N2: num] :
      ( ( bitM @ ( bit0 @ N2 ) )
      = ( bit1 @ ( bitM @ N2 ) ) ) ).

% semiring_norm(27)
thf(fact_5137_order__antisym__conv,axiom,
    ! [A: $tType] :
      ( ( order @ A )
     => ! [Y2: A,X2: A] :
          ( ( ord_less_eq @ A @ Y2 @ X2 )
         => ( ( ord_less_eq @ A @ X2 @ Y2 )
            = ( X2 = Y2 ) ) ) ) ).

% order_antisym_conv
thf(fact_5138_linorder__le__cases,axiom,
    ! [A: $tType] :
      ( ( linorder @ A )
     => ! [X2: A,Y2: A] :
          ( ~ ( ord_less_eq @ A @ X2 @ Y2 )
         => ( ord_less_eq @ A @ Y2 @ X2 ) ) ) ).

% linorder_le_cases
thf(fact_5139_ord__le__eq__subst,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ( ord @ B )
        & ( ord @ A ) )
     => ! [A2: A,B2: A,F: A > B,C2: B] :
          ( ( ord_less_eq @ A @ A2 @ B2 )
         => ( ( ( F @ B2 )
              = C2 )
           => ( ! [X3: A,Y4: A] :
                  ( ( ord_less_eq @ A @ X3 @ Y4 )
                 => ( ord_less_eq @ B @ ( F @ X3 ) @ ( F @ Y4 ) ) )
             => ( ord_less_eq @ B @ ( F @ A2 ) @ C2 ) ) ) ) ) ).

% ord_le_eq_subst
thf(fact_5140_ord__eq__le__subst,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ( ord @ B )
        & ( ord @ A ) )
     => ! [A2: A,F: B > A,B2: B,C2: B] :
          ( ( A2
            = ( F @ B2 ) )
         => ( ( ord_less_eq @ B @ B2 @ C2 )
           => ( ! [X3: B,Y4: B] :
                  ( ( ord_less_eq @ B @ X3 @ Y4 )
                 => ( ord_less_eq @ A @ ( F @ X3 ) @ ( F @ Y4 ) ) )
             => ( ord_less_eq @ A @ A2 @ ( F @ C2 ) ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_5141_linorder__linear,axiom,
    ! [A: $tType] :
      ( ( linorder @ A )
     => ! [X2: A,Y2: A] :
          ( ( ord_less_eq @ A @ X2 @ Y2 )
          | ( ord_less_eq @ A @ Y2 @ X2 ) ) ) ).

% linorder_linear
thf(fact_5142_order__eq__refl,axiom,
    ! [A: $tType] :
      ( ( preorder @ A )
     => ! [X2: A,Y2: A] :
          ( ( X2 = Y2 )
         => ( ord_less_eq @ A @ X2 @ Y2 ) ) ) ).

% order_eq_refl
thf(fact_5143_order__subst2,axiom,
    ! [A: $tType,C: $tType] :
      ( ( ( order @ C )
        & ( order @ A ) )
     => ! [A2: A,B2: A,F: A > C,C2: C] :
          ( ( ord_less_eq @ A @ A2 @ B2 )
         => ( ( ord_less_eq @ C @ ( F @ B2 ) @ C2 )
           => ( ! [X3: A,Y4: A] :
                  ( ( ord_less_eq @ A @ X3 @ Y4 )
                 => ( ord_less_eq @ C @ ( F @ X3 ) @ ( F @ Y4 ) ) )
             => ( ord_less_eq @ C @ ( F @ A2 ) @ C2 ) ) ) ) ) ).

% order_subst2
thf(fact_5144_order__subst1,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ( order @ B )
        & ( order @ A ) )
     => ! [A2: A,F: B > A,B2: B,C2: B] :
          ( ( ord_less_eq @ A @ A2 @ ( F @ B2 ) )
         => ( ( ord_less_eq @ B @ B2 @ C2 )
           => ( ! [X3: B,Y4: B] :
                  ( ( ord_less_eq @ B @ X3 @ Y4 )
                 => ( ord_less_eq @ A @ ( F @ X3 ) @ ( F @ Y4 ) ) )
             => ( ord_less_eq @ A @ A2 @ ( F @ C2 ) ) ) ) ) ) ).

% order_subst1
thf(fact_5145_Orderings_Oorder__eq__iff,axiom,
    ! [A: $tType] :
      ( ( order @ A )
     => ( ( ^ [Y5: A,Z4: A] : Y5 = Z4 )
        = ( ^ [A5: A,B5: A] :
              ( ( ord_less_eq @ A @ A5 @ B5 )
              & ( ord_less_eq @ A @ B5 @ A5 ) ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_5146_antisym,axiom,
    ! [A: $tType] :
      ( ( order @ A )
     => ! [A2: A,B2: A] :
          ( ( ord_less_eq @ A @ A2 @ B2 )
         => ( ( ord_less_eq @ A @ B2 @ A2 )
           => ( A2 = B2 ) ) ) ) ).

% antisym
thf(fact_5147_dual__order_Otrans,axiom,
    ! [A: $tType] :
      ( ( preorder @ A )
     => ! [B2: A,A2: A,C2: A] :
          ( ( ord_less_eq @ A @ B2 @ A2 )
         => ( ( ord_less_eq @ A @ C2 @ B2 )
           => ( ord_less_eq @ A @ C2 @ A2 ) ) ) ) ).

% dual_order.trans
thf(fact_5148_dual__order_Oantisym,axiom,
    ! [A: $tType] :
      ( ( order @ A )
     => ! [B2: A,A2: A] :
          ( ( ord_less_eq @ A @ B2 @ A2 )
         => ( ( ord_less_eq @ A @ A2 @ B2 )
           => ( A2 = B2 ) ) ) ) ).

% dual_order.antisym
thf(fact_5149_dual__order_Oeq__iff,axiom,
    ! [A: $tType] :
      ( ( order @ A )
     => ( ( ^ [Y5: A,Z4: A] : Y5 = Z4 )
        = ( ^ [A5: A,B5: A] :
              ( ( ord_less_eq @ A @ B5 @ A5 )
              & ( ord_less_eq @ A @ A5 @ B5 ) ) ) ) ) ).

% dual_order.eq_iff
thf(fact_5150_linorder__wlog,axiom,
    ! [A: $tType] :
      ( ( linorder @ A )
     => ! [P: A > A > $o,A2: A,B2: A] :
          ( ! [A4: A,B3: A] :
              ( ( ord_less_eq @ A @ A4 @ B3 )
             => ( P @ A4 @ B3 ) )
         => ( ! [A4: A,B3: A] :
                ( ( P @ B3 @ A4 )
               => ( P @ A4 @ B3 ) )
           => ( P @ A2 @ B2 ) ) ) ) ).

% linorder_wlog
thf(fact_5151_order__trans,axiom,
    ! [A: $tType] :
      ( ( preorder @ A )
     => ! [X2: A,Y2: A,Z: A] :
          ( ( ord_less_eq @ A @ X2 @ Y2 )
         => ( ( ord_less_eq @ A @ Y2 @ Z )
           => ( ord_less_eq @ A @ X2 @ Z ) ) ) ) ).

% order_trans
thf(fact_5152_order_Otrans,axiom,
    ! [A: $tType] :
      ( ( preorder @ A )
     => ! [A2: A,B2: A,C2: A] :
          ( ( ord_less_eq @ A @ A2 @ B2 )
         => ( ( ord_less_eq @ A @ B2 @ C2 )
           => ( ord_less_eq @ A @ A2 @ C2 ) ) ) ) ).

% order.trans
thf(fact_5153_order__antisym,axiom,
    ! [A: $tType] :
      ( ( order @ A )
     => ! [X2: A,Y2: A] :
          ( ( ord_less_eq @ A @ X2 @ Y2 )
         => ( ( ord_less_eq @ A @ Y2 @ X2 )
           => ( X2 = Y2 ) ) ) ) ).

% order_antisym
thf(fact_5154_ord__le__eq__trans,axiom,
    ! [A: $tType] :
      ( ( ord @ A )
     => ! [A2: A,B2: A,C2: A] :
          ( ( ord_less_eq @ A @ A2 @ B2 )
         => ( ( B2 = C2 )
           => ( ord_less_eq @ A @ A2 @ C2 ) ) ) ) ).

% ord_le_eq_trans
thf(fact_5155_ord__eq__le__trans,axiom,
    ! [A: $tType] :
      ( ( ord @ A )
     => ! [A2: A,B2: A,C2: A] :
          ( ( A2 = B2 )
         => ( ( ord_less_eq @ A @ B2 @ C2 )
           => ( ord_less_eq @ A @ A2 @ C2 ) ) ) ) ).

% ord_eq_le_trans
thf(fact_5156_order__class_Oorder__eq__iff,axiom,
    ! [A: $tType] :
      ( ( order @ A )
     => ( ( ^ [Y5: A,Z4: A] : Y5 = Z4 )
        = ( ^ [X: A,Y: A] :
              ( ( ord_less_eq @ A @ X @ Y )
              & ( ord_less_eq @ A @ Y @ X ) ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_5157_le__cases3,axiom,
    ! [A: $tType] :
      ( ( linorder @ A )
     => ! [X2: A,Y2: A,Z: A] :
          ( ( ( ord_less_eq @ A @ X2 @ Y2 )
           => ~ ( ord_less_eq @ A @ Y2 @ Z ) )
         => ( ( ( ord_less_eq @ A @ Y2 @ X2 )
             => ~ ( ord_less_eq @ A @ X2 @ Z ) )
           => ( ( ( ord_less_eq @ A @ X2 @ Z )
               => ~ ( ord_less_eq @ A @ Z @ Y2 ) )
             => ( ( ( ord_less_eq @ A @ Z @ Y2 )
                 => ~ ( ord_less_eq @ A @ Y2 @ X2 ) )
               => ( ( ( ord_less_eq @ A @ Y2 @ Z )
                   => ~ ( ord_less_eq @ A @ Z @ X2 ) )
                 => ~ ( ( ord_less_eq @ A @ Z @ X2 )
                     => ~ ( ord_less_eq @ A @ X2 @ Y2 ) ) ) ) ) ) ) ) ).

% le_cases3
thf(fact_5158_nle__le,axiom,
    ! [A: $tType] :
      ( ( linorder @ A )
     => ! [A2: A,B2: A] :
          ( ( ~ ( ord_less_eq @ A @ A2 @ B2 ) )
          = ( ( ord_less_eq @ A @ B2 @ A2 )
            & ( B2 != A2 ) ) ) ) ).

% nle_le
thf(fact_5159_order__less__imp__not__less,axiom,
    ! [A: $tType] :
      ( ( preorder @ A )
     => ! [X2: A,Y2: A] :
          ( ( ord_less @ A @ X2 @ Y2 )
         => ~ ( ord_less @ A @ Y2 @ X2 ) ) ) ).

% order_less_imp_not_less
thf(fact_5160_order__less__imp__not__eq2,axiom,
    ! [A: $tType] :
      ( ( order @ A )
     => ! [X2: A,Y2: A] :
          ( ( ord_less @ A @ X2 @ Y2 )
         => ( Y2 != X2 ) ) ) ).

% order_less_imp_not_eq2
thf(fact_5161_order__less__imp__not__eq,axiom,
    ! [A: $tType] :
      ( ( order @ A )
     => ! [X2: A,Y2: A] :
          ( ( ord_less @ A @ X2 @ Y2 )
         => ( X2 != Y2 ) ) ) ).

% order_less_imp_not_eq
thf(fact_5162_linorder__less__linear,axiom,
    ! [A: $tType] :
      ( ( linorder @ A )
     => ! [X2: A,Y2: A] :
          ( ( ord_less @ A @ X2 @ Y2 )
          | ( X2 = Y2 )
          | ( ord_less @ A @ Y2 @ X2 ) ) ) ).

% linorder_less_linear
thf(fact_5163_order__less__imp__triv,axiom,
    ! [A: $tType] :
      ( ( preorder @ A )
     => ! [X2: A,Y2: A,P: $o] :
          ( ( ord_less @ A @ X2 @ Y2 )
         => ( ( ord_less @ A @ Y2 @ X2 )
           => P ) ) ) ).

% order_less_imp_triv
thf(fact_5164_order__less__not__sym,axiom,
    ! [A: $tType] :
      ( ( preorder @ A )
     => ! [X2: A,Y2: A] :
          ( ( ord_less @ A @ X2 @ Y2 )
         => ~ ( ord_less @ A @ Y2 @ X2 ) ) ) ).

% order_less_not_sym
thf(fact_5165_order__less__subst2,axiom,
    ! [A: $tType,C: $tType] :
      ( ( ( order @ C )
        & ( order @ A ) )
     => ! [A2: A,B2: A,F: A > C,C2: C] :
          ( ( ord_less @ A @ A2 @ B2 )
         => ( ( ord_less @ C @ ( F @ B2 ) @ C2 )
           => ( ! [X3: A,Y4: A] :
                  ( ( ord_less @ A @ X3 @ Y4 )
                 => ( ord_less @ C @ ( F @ X3 ) @ ( F @ Y4 ) ) )
             => ( ord_less @ C @ ( F @ A2 ) @ C2 ) ) ) ) ) ).

% order_less_subst2
thf(fact_5166_order__less__subst1,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ( order @ B )
        & ( order @ A ) )
     => ! [A2: A,F: B > A,B2: B,C2: B] :
          ( ( ord_less @ A @ A2 @ ( F @ B2 ) )
         => ( ( ord_less @ B @ B2 @ C2 )
           => ( ! [X3: B,Y4: B] :
                  ( ( ord_less @ B @ X3 @ Y4 )
                 => ( ord_less @ A @ ( F @ X3 ) @ ( F @ Y4 ) ) )
             => ( ord_less @ A @ A2 @ ( F @ C2 ) ) ) ) ) ) ).

% order_less_subst1
thf(fact_5167_order__less__irrefl,axiom,
    ! [A: $tType] :
      ( ( preorder @ A )
     => ! [X2: A] :
          ~ ( ord_less @ A @ X2 @ X2 ) ) ).

% order_less_irrefl
thf(fact_5168_ord__less__eq__subst,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ( ord @ B )
        & ( ord @ A ) )
     => ! [A2: A,B2: A,F: A > B,C2: B] :
          ( ( ord_less @ A @ A2 @ B2 )
         => ( ( ( F @ B2 )
              = C2 )
           => ( ! [X3: A,Y4: A] :
                  ( ( ord_less @ A @ X3 @ Y4 )
                 => ( ord_less @ B @ ( F @ X3 ) @ ( F @ Y4 ) ) )
             => ( ord_less @ B @ ( F @ A2 ) @ C2 ) ) ) ) ) ).

% ord_less_eq_subst
thf(fact_5169_ord__eq__less__subst,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ( ord @ B )
        & ( ord @ A ) )
     => ! [A2: A,F: B > A,B2: B,C2: B] :
          ( ( A2
            = ( F @ B2 ) )
         => ( ( ord_less @ B @ B2 @ C2 )
           => ( ! [X3: B,Y4: B] :
                  ( ( ord_less @ B @ X3 @ Y4 )
                 => ( ord_less @ A @ ( F @ X3 ) @ ( F @ Y4 ) ) )
             => ( ord_less @ A @ A2 @ ( F @ C2 ) ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_5170_order__less__trans,axiom,
    ! [A: $tType] :
      ( ( preorder @ A )
     => ! [X2: A,Y2: A,Z: A] :
          ( ( ord_less @ A @ X2 @ Y2 )
         => ( ( ord_less @ A @ Y2 @ Z )
           => ( ord_less @ A @ X2 @ Z ) ) ) ) ).

% order_less_trans
thf(fact_5171_order__less__asym_H,axiom,
    ! [A: $tType] :
      ( ( preorder @ A )
     => ! [A2: A,B2: A] :
          ( ( ord_less @ A @ A2 @ B2 )
         => ~ ( ord_less @ A @ B2 @ A2 ) ) ) ).

% order_less_asym'
thf(fact_5172_linorder__neq__iff,axiom,
    ! [A: $tType] :
      ( ( linorder @ A )
     => ! [X2: A,Y2: A] :
          ( ( X2 != Y2 )
          = ( ( ord_less @ A @ X2 @ Y2 )
            | ( ord_less @ A @ Y2 @ X2 ) ) ) ) ).

% linorder_neq_iff
thf(fact_5173_order__less__asym,axiom,
    ! [A: $tType] :
      ( ( preorder @ A )
     => ! [X2: A,Y2: A] :
          ( ( ord_less @ A @ X2 @ Y2 )
         => ~ ( ord_less @ A @ Y2 @ X2 ) ) ) ).

% order_less_asym
thf(fact_5174_linorder__neqE,axiom,
    ! [A: $tType] :
      ( ( linorder @ A )
     => ! [X2: A,Y2: A] :
          ( ( X2 != Y2 )
         => ( ~ ( ord_less @ A @ X2 @ Y2 )
           => ( ord_less @ A @ Y2 @ X2 ) ) ) ) ).

% linorder_neqE
thf(fact_5175_dual__order_Ostrict__implies__not__eq,axiom,
    ! [A: $tType] :
      ( ( order @ A )
     => ! [B2: A,A2: A] :
          ( ( ord_less @ A @ B2 @ A2 )
         => ( A2 != B2 ) ) ) ).

% dual_order.strict_implies_not_eq
thf(fact_5176_order_Ostrict__implies__not__eq,axiom,
    ! [A: $tType] :
      ( ( order @ A )
     => ! [A2: A,B2: A] :
          ( ( ord_less @ A @ A2 @ B2 )
         => ( A2 != B2 ) ) ) ).

% order.strict_implies_not_eq
thf(fact_5177_dual__order_Ostrict__trans,axiom,
    ! [A: $tType] :
      ( ( preorder @ A )
     => ! [B2: A,A2: A,C2: A] :
          ( ( ord_less @ A @ B2 @ A2 )
         => ( ( ord_less @ A @ C2 @ B2 )
           => ( ord_less @ A @ C2 @ A2 ) ) ) ) ).

% dual_order.strict_trans
thf(fact_5178_not__less__iff__gr__or__eq,axiom,
    ! [A: $tType] :
      ( ( linorder @ A )
     => ! [X2: A,Y2: A] :
          ( ( ~ ( ord_less @ A @ X2 @ Y2 ) )
          = ( ( ord_less @ A @ Y2 @ X2 )
            | ( X2 = Y2 ) ) ) ) ).

% not_less_iff_gr_or_eq
thf(fact_5179_order_Ostrict__trans,axiom,
    ! [A: $tType] :
      ( ( preorder @ A )
     => ! [A2: A,B2: A,C2: A] :
          ( ( ord_less @ A @ A2 @ B2 )
         => ( ( ord_less @ A @ B2 @ C2 )
           => ( ord_less @ A @ A2 @ C2 ) ) ) ) ).

% order.strict_trans
thf(fact_5180_linorder__less__wlog,axiom,
    ! [A: $tType] :
      ( ( linorder @ A )
     => ! [P: A > A > $o,A2: A,B2: A] :
          ( ! [A4: A,B3: A] :
              ( ( ord_less @ A @ A4 @ B3 )
             => ( P @ A4 @ B3 ) )
         => ( ! [A4: A] : ( P @ A4 @ A4 )
           => ( ! [A4: A,B3: A] :
                  ( ( P @ B3 @ A4 )
                 => ( P @ A4 @ B3 ) )
             => ( P @ A2 @ B2 ) ) ) ) ) ).

% linorder_less_wlog
thf(fact_5181_exists__least__iff,axiom,
    ! [A: $tType] :
      ( ( wellorder @ A )
     => ( ( ^ [P2: A > $o] :
            ? [X4: A] : ( P2 @ X4 ) )
        = ( ^ [P3: A > $o] :
            ? [N: A] :
              ( ( P3 @ N )
              & ! [M7: A] :
                  ( ( ord_less @ A @ M7 @ N )
                 => ~ ( P3 @ M7 ) ) ) ) ) ) ).

% exists_least_iff
thf(fact_5182_dual__order_Oirrefl,axiom,
    ! [A: $tType] :
      ( ( preorder @ A )
     => ! [A2: A] :
          ~ ( ord_less @ A @ A2 @ A2 ) ) ).

% dual_order.irrefl
thf(fact_5183_dual__order_Oasym,axiom,
    ! [A: $tType] :
      ( ( preorder @ A )
     => ! [B2: A,A2: A] :
          ( ( ord_less @ A @ B2 @ A2 )
         => ~ ( ord_less @ A @ A2 @ B2 ) ) ) ).

% dual_order.asym
thf(fact_5184_linorder__cases,axiom,
    ! [A: $tType] :
      ( ( linorder @ A )
     => ! [X2: A,Y2: A] :
          ( ~ ( ord_less @ A @ X2 @ Y2 )
         => ( ( X2 != Y2 )
           => ( ord_less @ A @ Y2 @ X2 ) ) ) ) ).

% linorder_cases
thf(fact_5185_antisym__conv3,axiom,
    ! [A: $tType] :
      ( ( linorder @ A )
     => ! [Y2: A,X2: A] :
          ( ~ ( ord_less @ A @ Y2 @ X2 )
         => ( ( ~ ( ord_less @ A @ X2 @ Y2 ) )
            = ( X2 = Y2 ) ) ) ) ).

% antisym_conv3
thf(fact_5186_less__induct,axiom,
    ! [A: $tType] :
      ( ( wellorder @ A )
     => ! [P: A > $o,A2: A] :
          ( ! [X3: A] :
              ( ! [Y3: A] :
                  ( ( ord_less @ A @ Y3 @ X3 )
                 => ( P @ Y3 ) )
             => ( P @ X3 ) )
         => ( P @ A2 ) ) ) ).

% less_induct
thf(fact_5187_ord__less__eq__trans,axiom,
    ! [A: $tType] :
      ( ( ord @ A )
     => ! [A2: A,B2: A,C2: A] :
          ( ( ord_less @ A @ A2 @ B2 )
         => ( ( B2 = C2 )
           => ( ord_less @ A @ A2 @ C2 ) ) ) ) ).

% ord_less_eq_trans
thf(fact_5188_ord__eq__less__trans,axiom,
    ! [A: $tType] :
      ( ( ord @ A )
     => ! [A2: A,B2: A,C2: A] :
          ( ( A2 = B2 )
         => ( ( ord_less @ A @ B2 @ C2 )
           => ( ord_less @ A @ A2 @ C2 ) ) ) ) ).

% ord_eq_less_trans
thf(fact_5189_order_Oasym,axiom,
    ! [A: $tType] :
      ( ( preorder @ A )
     => ! [A2: A,B2: A] :
          ( ( ord_less @ A @ A2 @ B2 )
         => ~ ( ord_less @ A @ B2 @ A2 ) ) ) ).

% order.asym
thf(fact_5190_less__imp__neq,axiom,
    ! [A: $tType] :
      ( ( order @ A )
     => ! [X2: A,Y2: A] :
          ( ( ord_less @ A @ X2 @ Y2 )
         => ( X2 != Y2 ) ) ) ).

% less_imp_neq
thf(fact_5191_dense,axiom,
    ! [A: $tType] :
      ( ( dense_order @ A )
     => ! [X2: A,Y2: A] :
          ( ( ord_less @ A @ X2 @ Y2 )
         => ? [Z3: A] :
              ( ( ord_less @ A @ X2 @ Z3 )
              & ( ord_less @ A @ Z3 @ Y2 ) ) ) ) ).

% dense
thf(fact_5192_gt__ex,axiom,
    ! [A: $tType] :
      ( ( no_top @ A )
     => ! [X2: A] :
        ? [X_12: A] : ( ord_less @ A @ X2 @ X_12 ) ) ).

% gt_ex
thf(fact_5193_lt__ex,axiom,
    ! [A: $tType] :
      ( ( no_bot @ A )
     => ! [X2: A] :
        ? [Y4: A] : ( ord_less @ A @ Y4 @ X2 ) ) ).

% lt_ex
thf(fact_5194_eval__nat__numeral_I2_J,axiom,
    ! [N2: num] :
      ( ( numeral_numeral @ nat @ ( bit0 @ N2 ) )
      = ( suc @ ( numeral_numeral @ nat @ ( bitM @ N2 ) ) ) ) ).

% eval_nat_numeral(2)
thf(fact_5195_one__plus__BitM,axiom,
    ! [N2: num] :
      ( ( plus_plus @ num @ one2 @ ( bitM @ N2 ) )
      = ( bit0 @ N2 ) ) ).

% one_plus_BitM
thf(fact_5196_BitM__plus__one,axiom,
    ! [N2: num] :
      ( ( plus_plus @ num @ ( bitM @ N2 ) @ one2 )
      = ( bit0 @ N2 ) ) ).

% BitM_plus_one
thf(fact_5197_less__fun__def,axiom,
    ! [B: $tType,A: $tType] :
      ( ( ord @ B )
     => ( ( ord_less @ ( A > B ) )
        = ( ^ [F6: A > B,G4: A > B] :
              ( ( ord_less_eq @ ( A > B ) @ F6 @ G4 )
              & ~ ( ord_less_eq @ ( A > B ) @ G4 @ F6 ) ) ) ) ) ).

% less_fun_def
thf(fact_5198_predicate2D,axiom,
    ! [A: $tType,B: $tType,P: A > B > $o,Q: A > B > $o,X2: A,Y2: B] :
      ( ( ord_less_eq @ ( A > B > $o ) @ P @ Q )
     => ( ( P @ X2 @ Y2 )
       => ( Q @ X2 @ Y2 ) ) ) ).

% predicate2D
thf(fact_5199_rev__predicate2D,axiom,
    ! [A: $tType,B: $tType,P: A > B > $o,X2: A,Y2: B,Q: A > B > $o] :
      ( ( P @ X2 @ Y2 )
     => ( ( ord_less_eq @ ( A > B > $o ) @ P @ Q )
       => ( Q @ X2 @ Y2 ) ) ) ).

% rev_predicate2D
thf(fact_5200_predicate1D,axiom,
    ! [A: $tType,P: A > $o,Q: A > $o,X2: A] :
      ( ( ord_less_eq @ ( A > $o ) @ P @ Q )
     => ( ( P @ X2 )
       => ( Q @ X2 ) ) ) ).

% predicate1D
thf(fact_5201_rev__predicate1D,axiom,
    ! [A: $tType,P: A > $o,X2: A,Q: A > $o] :
      ( ( P @ X2 )
     => ( ( ord_less_eq @ ( A > $o ) @ P @ Q )
       => ( Q @ X2 ) ) ) ).

% rev_predicate1D
thf(fact_5202_numeral__BitM,axiom,
    ! [A: $tType] :
      ( ( neg_numeral @ A )
     => ! [N2: num] :
          ( ( numeral_numeral @ A @ ( bitM @ N2 ) )
          = ( minus_minus @ A @ ( numeral_numeral @ A @ ( bit0 @ N2 ) ) @ ( one_one @ A ) ) ) ) ).

% numeral_BitM
thf(fact_5203_odd__numeral__BitM,axiom,
    ! [A: $tType] :
      ( ( semiring_parity @ A )
     => ! [W: num] :
          ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ A @ ( bitM @ W ) ) ) ) ).

% odd_numeral_BitM
thf(fact_5204_order__le__imp__less__or__eq,axiom,
    ! [A: $tType] :
      ( ( order @ A )
     => ! [X2: A,Y2: A] :
          ( ( ord_less_eq @ A @ X2 @ Y2 )
         => ( ( ord_less @ A @ X2 @ Y2 )
            | ( X2 = Y2 ) ) ) ) ).

% order_le_imp_less_or_eq
thf(fact_5205_linorder__le__less__linear,axiom,
    ! [A: $tType] :
      ( ( linorder @ A )
     => ! [X2: A,Y2: A] :
          ( ( ord_less_eq @ A @ X2 @ Y2 )
          | ( ord_less @ A @ Y2 @ X2 ) ) ) ).

% linorder_le_less_linear
thf(fact_5206_order__less__le__subst2,axiom,
    ! [A: $tType,C: $tType] :
      ( ( ( order @ C )
        & ( order @ A ) )
     => ! [A2: A,B2: A,F: A > C,C2: C] :
          ( ( ord_less @ A @ A2 @ B2 )
         => ( ( ord_less_eq @ C @ ( F @ B2 ) @ C2 )
           => ( ! [X3: A,Y4: A] :
                  ( ( ord_less @ A @ X3 @ Y4 )
                 => ( ord_less @ C @ ( F @ X3 ) @ ( F @ Y4 ) ) )
             => ( ord_less @ C @ ( F @ A2 ) @ C2 ) ) ) ) ) ).

% order_less_le_subst2
thf(fact_5207_order__less__le__subst1,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ( order @ B )
        & ( order @ A ) )
     => ! [A2: A,F: B > A,B2: B,C2: B] :
          ( ( ord_less @ A @ A2 @ ( F @ B2 ) )
         => ( ( ord_less_eq @ B @ B2 @ C2 )
           => ( ! [X3: B,Y4: B] :
                  ( ( ord_less_eq @ B @ X3 @ Y4 )
                 => ( ord_less_eq @ A @ ( F @ X3 ) @ ( F @ Y4 ) ) )
             => ( ord_less @ A @ A2 @ ( F @ C2 ) ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_5208_order__le__less__subst2,axiom,
    ! [A: $tType,C: $tType] :
      ( ( ( order @ C )
        & ( order @ A ) )
     => ! [A2: A,B2: A,F: A > C,C2: C] :
          ( ( ord_less_eq @ A @ A2 @ B2 )
         => ( ( ord_less @ C @ ( F @ B2 ) @ C2 )
           => ( ! [X3: A,Y4: A] :
                  ( ( ord_less_eq @ A @ X3 @ Y4 )
                 => ( ord_less_eq @ C @ ( F @ X3 ) @ ( F @ Y4 ) ) )
             => ( ord_less @ C @ ( F @ A2 ) @ C2 ) ) ) ) ) ).

% order_le_less_subst2
thf(fact_5209_order__le__less__subst1,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ( order @ B )
        & ( order @ A ) )
     => ! [A2: A,F: B > A,B2: B,C2: B] :
          ( ( ord_less_eq @ A @ A2 @ ( F @ B2 ) )
         => ( ( ord_less @ B @ B2 @ C2 )
           => ( ! [X3: B,Y4: B] :
                  ( ( ord_less @ B @ X3 @ Y4 )
                 => ( ord_less @ A @ ( F @ X3 ) @ ( F @ Y4 ) ) )
             => ( ord_less @ A @ A2 @ ( F @ C2 ) ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_5210_order__less__le__trans,axiom,
    ! [A: $tType] :
      ( ( preorder @ A )
     => ! [X2: A,Y2: A,Z: A] :
          ( ( ord_less @ A @ X2 @ Y2 )
         => ( ( ord_less_eq @ A @ Y2 @ Z )
           => ( ord_less @ A @ X2 @ Z ) ) ) ) ).

% order_less_le_trans
thf(fact_5211_order__le__less__trans,axiom,
    ! [A: $tType] :
      ( ( preorder @ A )
     => ! [X2: A,Y2: A,Z: A] :
          ( ( ord_less_eq @ A @ X2 @ Y2 )
         => ( ( ord_less @ A @ Y2 @ Z )
           => ( ord_less @ A @ X2 @ Z ) ) ) ) ).

% order_le_less_trans
thf(fact_5212_order__neq__le__trans,axiom,
    ! [A: $tType] :
      ( ( order @ A )
     => ! [A2: A,B2: A] :
          ( ( A2 != B2 )
         => ( ( ord_less_eq @ A @ A2 @ B2 )
           => ( ord_less @ A @ A2 @ B2 ) ) ) ) ).

% order_neq_le_trans
thf(fact_5213_order__le__neq__trans,axiom,
    ! [A: $tType] :
      ( ( order @ A )
     => ! [A2: A,B2: A] :
          ( ( ord_less_eq @ A @ A2 @ B2 )
         => ( ( A2 != B2 )
           => ( ord_less @ A @ A2 @ B2 ) ) ) ) ).

% order_le_neq_trans
thf(fact_5214_order__less__imp__le,axiom,
    ! [A: $tType] :
      ( ( preorder @ A )
     => ! [X2: A,Y2: A] :
          ( ( ord_less @ A @ X2 @ Y2 )
         => ( ord_less_eq @ A @ X2 @ Y2 ) ) ) ).

% order_less_imp_le
thf(fact_5215_linorder__not__less,axiom,
    ! [A: $tType] :
      ( ( linorder @ A )
     => ! [X2: A,Y2: A] :
          ( ( ~ ( ord_less @ A @ X2 @ Y2 ) )
          = ( ord_less_eq @ A @ Y2 @ X2 ) ) ) ).

% linorder_not_less
thf(fact_5216_linorder__not__le,axiom,
    ! [A: $tType] :
      ( ( linorder @ A )
     => ! [X2: A,Y2: A] :
          ( ( ~ ( ord_less_eq @ A @ X2 @ Y2 ) )
          = ( ord_less @ A @ Y2 @ X2 ) ) ) ).

% linorder_not_le
thf(fact_5217_order__less__le,axiom,
    ! [A: $tType] :
      ( ( order @ A )
     => ( ( ord_less @ A )
        = ( ^ [X: A,Y: A] :
              ( ( ord_less_eq @ A @ X @ Y )
              & ( X != Y ) ) ) ) ) ).

% order_less_le
thf(fact_5218_order__le__less,axiom,
    ! [A: $tType] :
      ( ( order @ A )
     => ( ( ord_less_eq @ A )
        = ( ^ [X: A,Y: A] :
              ( ( ord_less @ A @ X @ Y )
              | ( X = Y ) ) ) ) ) ).

% order_le_less
thf(fact_5219_dual__order_Ostrict__implies__order,axiom,
    ! [A: $tType] :
      ( ( preorder @ A )
     => ! [B2: A,A2: A] :
          ( ( ord_less @ A @ B2 @ A2 )
         => ( ord_less_eq @ A @ B2 @ A2 ) ) ) ).

% dual_order.strict_implies_order
thf(fact_5220_order_Ostrict__implies__order,axiom,
    ! [A: $tType] :
      ( ( preorder @ A )
     => ! [A2: A,B2: A] :
          ( ( ord_less @ A @ A2 @ B2 )
         => ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).

% order.strict_implies_order
thf(fact_5221_dual__order_Ostrict__iff__not,axiom,
    ! [A: $tType] :
      ( ( preorder @ A )
     => ( ( ord_less @ A )
        = ( ^ [B5: A,A5: A] :
              ( ( ord_less_eq @ A @ B5 @ A5 )
              & ~ ( ord_less_eq @ A @ A5 @ B5 ) ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_5222_dual__order_Ostrict__trans2,axiom,
    ! [A: $tType] :
      ( ( preorder @ A )
     => ! [B2: A,A2: A,C2: A] :
          ( ( ord_less @ A @ B2 @ A2 )
         => ( ( ord_less_eq @ A @ C2 @ B2 )
           => ( ord_less @ A @ C2 @ A2 ) ) ) ) ).

% dual_order.strict_trans2
thf(fact_5223_dual__order_Ostrict__trans1,axiom,
    ! [A: $tType] :
      ( ( preorder @ A )
     => ! [B2: A,A2: A,C2: A] :
          ( ( ord_less_eq @ A @ B2 @ A2 )
         => ( ( ord_less @ A @ C2 @ B2 )
           => ( ord_less @ A @ C2 @ A2 ) ) ) ) ).

% dual_order.strict_trans1
thf(fact_5224_dual__order_Ostrict__iff__order,axiom,
    ! [A: $tType] :
      ( ( order @ A )
     => ( ( ord_less @ A )
        = ( ^ [B5: A,A5: A] :
              ( ( ord_less_eq @ A @ B5 @ A5 )
              & ( A5 != B5 ) ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_5225_dual__order_Oorder__iff__strict,axiom,
    ! [A: $tType] :
      ( ( order @ A )
     => ( ( ord_less_eq @ A )
        = ( ^ [B5: A,A5: A] :
              ( ( ord_less @ A @ B5 @ A5 )
              | ( A5 = B5 ) ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_5226_dense__le__bounded,axiom,
    ! [A: $tType] :
      ( ( dense_linorder @ A )
     => ! [X2: A,Y2: A,Z: A] :
          ( ( ord_less @ A @ X2 @ Y2 )
         => ( ! [W3: A] :
                ( ( ord_less @ A @ X2 @ W3 )
               => ( ( ord_less @ A @ W3 @ Y2 )
                 => ( ord_less_eq @ A @ W3 @ Z ) ) )
           => ( ord_less_eq @ A @ Y2 @ Z ) ) ) ) ).

% dense_le_bounded
thf(fact_5227_dense__ge__bounded,axiom,
    ! [A: $tType] :
      ( ( dense_linorder @ A )
     => ! [Z: A,X2: A,Y2: A] :
          ( ( ord_less @ A @ Z @ X2 )
         => ( ! [W3: A] :
                ( ( ord_less @ A @ Z @ W3 )
               => ( ( ord_less @ A @ W3 @ X2 )
                 => ( ord_less_eq @ A @ Y2 @ W3 ) ) )
           => ( ord_less_eq @ A @ Y2 @ Z ) ) ) ) ).

% dense_ge_bounded
thf(fact_5228_order_Ostrict__iff__not,axiom,
    ! [A: $tType] :
      ( ( preorder @ A )
     => ( ( ord_less @ A )
        = ( ^ [A5: A,B5: A] :
              ( ( ord_less_eq @ A @ A5 @ B5 )
              & ~ ( ord_less_eq @ A @ B5 @ A5 ) ) ) ) ) ).

% order.strict_iff_not
thf(fact_5229_order_Ostrict__trans2,axiom,
    ! [A: $tType] :
      ( ( preorder @ A )
     => ! [A2: A,B2: A,C2: A] :
          ( ( ord_less @ A @ A2 @ B2 )
         => ( ( ord_less_eq @ A @ B2 @ C2 )
           => ( ord_less @ A @ A2 @ C2 ) ) ) ) ).

% order.strict_trans2
thf(fact_5230_order_Ostrict__trans1,axiom,
    ! [A: $tType] :
      ( ( preorder @ A )
     => ! [A2: A,B2: A,C2: A] :
          ( ( ord_less_eq @ A @ A2 @ B2 )
         => ( ( ord_less @ A @ B2 @ C2 )
           => ( ord_less @ A @ A2 @ C2 ) ) ) ) ).

% order.strict_trans1
thf(fact_5231_order_Ostrict__iff__order,axiom,
    ! [A: $tType] :
      ( ( order @ A )
     => ( ( ord_less @ A )
        = ( ^ [A5: A,B5: A] :
              ( ( ord_less_eq @ A @ A5 @ B5 )
              & ( A5 != B5 ) ) ) ) ) ).

% order.strict_iff_order
thf(fact_5232_order_Oorder__iff__strict,axiom,
    ! [A: $tType] :
      ( ( order @ A )
     => ( ( ord_less_eq @ A )
        = ( ^ [A5: A,B5: A] :
              ( ( ord_less @ A @ A5 @ B5 )
              | ( A5 = B5 ) ) ) ) ) ).

% order.order_iff_strict
thf(fact_5233_not__le__imp__less,axiom,
    ! [A: $tType] :
      ( ( linorder @ A )
     => ! [Y2: A,X2: A] :
          ( ~ ( ord_less_eq @ A @ Y2 @ X2 )
         => ( ord_less @ A @ X2 @ Y2 ) ) ) ).

% not_le_imp_less
thf(fact_5234_less__le__not__le,axiom,
    ! [A: $tType] :
      ( ( preorder @ A )
     => ( ( ord_less @ A )
        = ( ^ [X: A,Y: A] :
              ( ( ord_less_eq @ A @ X @ Y )
              & ~ ( ord_less_eq @ A @ Y @ X ) ) ) ) ) ).

% less_le_not_le
thf(fact_5235_dense__le,axiom,
    ! [A: $tType] :
      ( ( dense_linorder @ A )
     => ! [Y2: A,Z: A] :
          ( ! [X3: A] :
              ( ( ord_less @ A @ X3 @ Y2 )
             => ( ord_less_eq @ A @ X3 @ Z ) )
         => ( ord_less_eq @ A @ Y2 @ Z ) ) ) ).

% dense_le
thf(fact_5236_dense__ge,axiom,
    ! [A: $tType] :
      ( ( dense_linorder @ A )
     => ! [Z: A,Y2: A] :
          ( ! [X3: A] :
              ( ( ord_less @ A @ Z @ X3 )
             => ( ord_less_eq @ A @ Y2 @ X3 ) )
         => ( ord_less_eq @ A @ Y2 @ Z ) ) ) ).

% dense_ge
thf(fact_5237_antisym__conv2,axiom,
    ! [A: $tType] :
      ( ( order @ A )
     => ! [X2: A,Y2: A] :
          ( ( ord_less_eq @ A @ X2 @ Y2 )
         => ( ( ~ ( ord_less @ A @ X2 @ Y2 ) )
            = ( X2 = Y2 ) ) ) ) ).

% antisym_conv2
thf(fact_5238_antisym__conv1,axiom,
    ! [A: $tType] :
      ( ( order @ A )
     => ! [X2: A,Y2: A] :
          ( ~ ( ord_less @ A @ X2 @ Y2 )
         => ( ( ord_less_eq @ A @ X2 @ Y2 )
            = ( X2 = Y2 ) ) ) ) ).

% antisym_conv1
thf(fact_5239_nless__le,axiom,
    ! [A: $tType] :
      ( ( order @ A )
     => ! [A2: A,B2: A] :
          ( ( ~ ( ord_less @ A @ A2 @ B2 ) )
          = ( ~ ( ord_less_eq @ A @ A2 @ B2 )
            | ( A2 = B2 ) ) ) ) ).

% nless_le
thf(fact_5240_leI,axiom,
    ! [A: $tType] :
      ( ( linorder @ A )
     => ! [X2: A,Y2: A] :
          ( ~ ( ord_less @ A @ X2 @ Y2 )
         => ( ord_less_eq @ A @ Y2 @ X2 ) ) ) ).

% leI
thf(fact_5241_leD,axiom,
    ! [A: $tType] :
      ( ( order @ A )
     => ! [Y2: A,X2: A] :
          ( ( ord_less_eq @ A @ Y2 @ X2 )
         => ~ ( ord_less @ A @ X2 @ Y2 ) ) ) ).

% leD
thf(fact_5242_bot_Oextremum__uniqueI,axiom,
    ! [A: $tType] :
      ( ( order_bot @ A )
     => ! [A2: A] :
          ( ( ord_less_eq @ A @ A2 @ ( bot_bot @ A ) )
         => ( A2
            = ( bot_bot @ A ) ) ) ) ).

% bot.extremum_uniqueI
thf(fact_5243_bot_Oextremum__unique,axiom,
    ! [A: $tType] :
      ( ( order_bot @ A )
     => ! [A2: A] :
          ( ( ord_less_eq @ A @ A2 @ ( bot_bot @ A ) )
          = ( A2
            = ( bot_bot @ A ) ) ) ) ).

% bot.extremum_unique
thf(fact_5244_bot_Oextremum,axiom,
    ! [A: $tType] :
      ( ( order_bot @ A )
     => ! [A2: A] : ( ord_less_eq @ A @ ( bot_bot @ A ) @ A2 ) ) ).

% bot.extremum
thf(fact_5245_top__greatest,axiom,
    ! [A: $tType] :
      ( ( order_top @ A )
     => ! [A2: A] : ( ord_less_eq @ A @ A2 @ ( top_top @ A ) ) ) ).

% top_greatest
thf(fact_5246_top_Oextremum__unique,axiom,
    ! [A: $tType] :
      ( ( order_top @ A )
     => ! [A2: A] :
          ( ( ord_less_eq @ A @ ( top_top @ A ) @ A2 )
          = ( A2
            = ( top_top @ A ) ) ) ) ).

% top.extremum_unique
thf(fact_5247_top_Oextremum__uniqueI,axiom,
    ! [A: $tType] :
      ( ( order_top @ A )
     => ! [A2: A] :
          ( ( ord_less_eq @ A @ ( top_top @ A ) @ A2 )
         => ( A2
            = ( top_top @ A ) ) ) ) ).

% top.extremum_uniqueI
thf(fact_5248_bot_Oextremum__strict,axiom,
    ! [A: $tType] :
      ( ( order_bot @ A )
     => ! [A2: A] :
          ~ ( ord_less @ A @ A2 @ ( bot_bot @ A ) ) ) ).

% bot.extremum_strict
thf(fact_5249_bot_Onot__eq__extremum,axiom,
    ! [A: $tType] :
      ( ( order_bot @ A )
     => ! [A2: A] :
          ( ( A2
           != ( bot_bot @ A ) )
          = ( ord_less @ A @ ( bot_bot @ A ) @ A2 ) ) ) ).

% bot.not_eq_extremum
thf(fact_5250_top_Oextremum__strict,axiom,
    ! [A: $tType] :
      ( ( order_top @ A )
     => ! [A2: A] :
          ~ ( ord_less @ A @ ( top_top @ A ) @ A2 ) ) ).

% top.extremum_strict
thf(fact_5251_top_Onot__eq__extremum,axiom,
    ! [A: $tType] :
      ( ( order_top @ A )
     => ! [A2: A] :
          ( ( A2
           != ( top_top @ A ) )
          = ( ord_less @ A @ A2 @ ( top_top @ A ) ) ) ) ).

% top.not_eq_extremum
thf(fact_5252_dependent__nat__choice,axiom,
    ! [A: $tType,P: nat > A > $o,Q: nat > A > A > $o] :
      ( ? [X_1: A] : ( P @ ( zero_zero @ nat ) @ X_1 )
     => ( ! [X3: A,N4: nat] :
            ( ( P @ N4 @ X3 )
           => ? [Y3: A] :
                ( ( P @ ( suc @ N4 ) @ Y3 )
                & ( Q @ N4 @ X3 @ Y3 ) ) )
       => ? [F2: nat > A] :
          ! [N9: nat] :
            ( ( P @ N9 @ ( F2 @ N9 ) )
            & ( Q @ N9 @ ( F2 @ N9 ) @ ( F2 @ ( suc @ N9 ) ) ) ) ) ) ).

% dependent_nat_choice
thf(fact_5253_max__absorb2,axiom,
    ! [A: $tType] :
      ( ( ord @ A )
     => ! [X2: A,Y2: A] :
          ( ( ord_less_eq @ A @ X2 @ Y2 )
         => ( ( ord_max @ A @ X2 @ Y2 )
            = Y2 ) ) ) ).

% max_absorb2
thf(fact_5254_max__absorb1,axiom,
    ! [A: $tType] :
      ( ( order @ A )
     => ! [Y2: A,X2: A] :
          ( ( ord_less_eq @ A @ Y2 @ X2 )
         => ( ( ord_max @ A @ X2 @ Y2 )
            = X2 ) ) ) ).

% max_absorb1
thf(fact_5255_max__def,axiom,
    ! [A: $tType] :
      ( ( ord @ A )
     => ( ( ord_max @ A )
        = ( ^ [A5: A,B5: A] : ( if @ A @ ( ord_less_eq @ A @ A5 @ B5 ) @ B5 @ A5 ) ) ) ) ).

% max_def
thf(fact_5256_le__fun__def,axiom,
    ! [B: $tType,A: $tType] :
      ( ( ord @ B )
     => ( ( ord_less_eq @ ( A > B ) )
        = ( ^ [F6: A > B,G4: A > B] :
            ! [X: A] : ( ord_less_eq @ B @ ( F6 @ X ) @ ( G4 @ X ) ) ) ) ) ).

% le_fun_def
thf(fact_5257_le__funI,axiom,
    ! [B: $tType,A: $tType] :
      ( ( ord @ B )
     => ! [F: A > B,G: A > B] :
          ( ! [X3: A] : ( ord_less_eq @ B @ ( F @ X3 ) @ ( G @ X3 ) )
         => ( ord_less_eq @ ( A > B ) @ F @ G ) ) ) ).

% le_funI
thf(fact_5258_le__funE,axiom,
    ! [B: $tType,A: $tType] :
      ( ( ord @ B )
     => ! [F: A > B,G: A > B,X2: A] :
          ( ( ord_less_eq @ ( A > B ) @ F @ G )
         => ( ord_less_eq @ B @ ( F @ X2 ) @ ( G @ X2 ) ) ) ) ).

% le_funE
thf(fact_5259_le__funD,axiom,
    ! [B: $tType,A: $tType] :
      ( ( ord @ B )
     => ! [F: A > B,G: A > B,X2: A] :
          ( ( ord_less_eq @ ( A > B ) @ F @ G )
         => ( ord_less_eq @ B @ ( F @ X2 ) @ ( G @ X2 ) ) ) ) ).

% le_funD
thf(fact_5260_bij__betw__nth__root__unity,axiom,
    ! [C2: complex,N2: nat] :
      ( ( C2
       != ( zero_zero @ complex ) )
     => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
       => ( bij_betw @ complex @ complex @ ( times_times @ complex @ ( times_times @ complex @ ( real_Vector_of_real @ complex @ ( root @ N2 @ ( real_V7770717601297561774m_norm @ complex @ C2 ) ) ) @ ( cis @ ( divide_divide @ real @ ( arg @ C2 ) @ ( semiring_1_of_nat @ real @ N2 ) ) ) ) )
          @ ( collect @ complex
            @ ^ [Z2: complex] :
                ( ( power_power @ complex @ Z2 @ N2 )
                = ( one_one @ complex ) ) )
          @ ( collect @ complex
            @ ^ [Z2: complex] :
                ( ( power_power @ complex @ Z2 @ N2 )
                = C2 ) ) ) ) ) ).

% bij_betw_nth_root_unity
thf(fact_5261_assnle,axiom,
    ! [TreeList: list @ vEBT_VEBT,Tree_is2: list @ vEBT_VEBTi,X13: array @ vEBT_VEBTi,Summary: vEBT_VEBT,X14: vEBT_VEBTi] : ( entails @ ( times_times @ assn @ ( vEBT_List_list_assn @ vEBT_VEBT @ vEBT_VEBTi @ vEBT_vebt_assn_raw @ TreeList @ Tree_is2 ) @ ( times_times @ assn @ ( snga_assn @ vEBT_VEBTi @ X13 @ Tree_is2 ) @ ( vEBT_vebt_assn_raw @ Summary @ X14 ) ) ) @ ( times_times @ assn @ ( times_times @ assn @ ( vEBT_vebt_assn_raw @ Summary @ X14 ) @ ( snga_assn @ vEBT_VEBTi @ X13 @ Tree_is2 ) ) @ ( vEBT_List_list_assn @ vEBT_VEBT @ vEBT_VEBTi @ vEBT_vebt_assn_raw @ TreeList @ Tree_is2 ) ) ) ).

% assnle
thf(fact_5262_int__ge__less__than2__def,axiom,
    ( int_ge_less_than2
    = ( ^ [D3: int] :
          ( collect @ ( product_prod @ int @ int )
          @ ( product_case_prod @ int @ int @ $o
            @ ^ [Z8: int,Z2: int] :
                ( ( ord_less_eq @ int @ D3 @ Z2 )
                & ( ord_less @ int @ Z8 @ Z2 ) ) ) ) ) ) ).

% int_ge_less_than2_def
thf(fact_5263_real__root__zero,axiom,
    ! [N2: nat] :
      ( ( root @ N2 @ ( zero_zero @ real ) )
      = ( zero_zero @ real ) ) ).

% real_root_zero
thf(fact_5264_triv__exI,axiom,
    ! [A: $tType,Q: A > assn,X2: A] : ( entails @ ( Q @ X2 ) @ ( ex_assn @ A @ Q ) ) ).

% triv_exI
thf(fact_5265_real__root__Suc__0,axiom,
    ! [X2: real] :
      ( ( root @ ( suc @ ( zero_zero @ nat ) ) @ X2 )
      = X2 ) ).

% real_root_Suc_0
thf(fact_5266_real__root__eq__iff,axiom,
    ! [N2: nat,X2: real,Y2: real] :
      ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
     => ( ( ( root @ N2 @ X2 )
          = ( root @ N2 @ Y2 ) )
        = ( X2 = Y2 ) ) ) ).

% real_root_eq_iff
thf(fact_5267_root__0,axiom,
    ! [X2: real] :
      ( ( root @ ( zero_zero @ nat ) @ X2 )
      = ( zero_zero @ real ) ) ).

% root_0
thf(fact_5268_ent__pure__pre__iff,axiom,
    ! [P: assn,B2: $o,Q: assn] :
      ( ( entails @ ( times_times @ assn @ P @ ( pure_assn @ B2 ) ) @ Q )
      = ( B2
       => ( entails @ P @ Q ) ) ) ).

% ent_pure_pre_iff
thf(fact_5269_ent__pure__pre__iff__sng,axiom,
    ! [B2: $o,Q: assn] :
      ( ( entails @ ( pure_assn @ B2 ) @ Q )
      = ( B2
       => ( entails @ ( one_one @ assn ) @ Q ) ) ) ).

% ent_pure_pre_iff_sng
thf(fact_5270_real__root__eq__0__iff,axiom,
    ! [N2: nat,X2: real] :
      ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
     => ( ( ( root @ N2 @ X2 )
          = ( zero_zero @ real ) )
        = ( X2
          = ( zero_zero @ real ) ) ) ) ).

% real_root_eq_0_iff
thf(fact_5271_real__root__less__iff,axiom,
    ! [N2: nat,X2: real,Y2: real] :
      ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
     => ( ( ord_less @ real @ ( root @ N2 @ X2 ) @ ( root @ N2 @ Y2 ) )
        = ( ord_less @ real @ X2 @ Y2 ) ) ) ).

% real_root_less_iff
thf(fact_5272_real__root__le__iff,axiom,
    ! [N2: nat,X2: real,Y2: real] :
      ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
     => ( ( ord_less_eq @ real @ ( root @ N2 @ X2 ) @ ( root @ N2 @ Y2 ) )
        = ( ord_less_eq @ real @ X2 @ Y2 ) ) ) ).

% real_root_le_iff
thf(fact_5273_real__root__eq__1__iff,axiom,
    ! [N2: nat,X2: real] :
      ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
     => ( ( ( root @ N2 @ X2 )
          = ( one_one @ real ) )
        = ( X2
          = ( one_one @ real ) ) ) ) ).

% real_root_eq_1_iff
thf(fact_5274_real__root__one,axiom,
    ! [N2: nat] :
      ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
     => ( ( root @ N2 @ ( one_one @ real ) )
        = ( one_one @ real ) ) ) ).

% real_root_one
thf(fact_5275_ent__pure__post__iff,axiom,
    ! [P: assn,Q: assn,B2: $o] :
      ( ( entails @ P @ ( times_times @ assn @ Q @ ( pure_assn @ B2 ) ) )
      = ( ! [H: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
            ( ( rep_assn @ P @ H )
           => B2 )
        & ( entails @ P @ Q ) ) ) ).

% ent_pure_post_iff
thf(fact_5276_ent__pure__post__iff__sng,axiom,
    ! [P: assn,B2: $o] :
      ( ( entails @ P @ ( pure_assn @ B2 ) )
      = ( ! [H: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
            ( ( rep_assn @ P @ H )
           => B2 )
        & ( entails @ P @ ( one_one @ assn ) ) ) ) ).

% ent_pure_post_iff_sng
thf(fact_5277_real__root__lt__0__iff,axiom,
    ! [N2: nat,X2: real] :
      ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
     => ( ( ord_less @ real @ ( root @ N2 @ X2 ) @ ( zero_zero @ real ) )
        = ( ord_less @ real @ X2 @ ( zero_zero @ real ) ) ) ) ).

% real_root_lt_0_iff
thf(fact_5278_real__root__gt__0__iff,axiom,
    ! [N2: nat,Y2: real] :
      ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
     => ( ( ord_less @ real @ ( zero_zero @ real ) @ ( root @ N2 @ Y2 ) )
        = ( ord_less @ real @ ( zero_zero @ real ) @ Y2 ) ) ) ).

% real_root_gt_0_iff
thf(fact_5279_real__root__ge__0__iff,axiom,
    ! [N2: nat,Y2: real] :
      ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
     => ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( root @ N2 @ Y2 ) )
        = ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y2 ) ) ) ).

% real_root_ge_0_iff
thf(fact_5280_real__root__le__0__iff,axiom,
    ! [N2: nat,X2: real] :
      ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
     => ( ( ord_less_eq @ real @ ( root @ N2 @ X2 ) @ ( zero_zero @ real ) )
        = ( ord_less_eq @ real @ X2 @ ( zero_zero @ real ) ) ) ) ).

% real_root_le_0_iff
thf(fact_5281_real__root__lt__1__iff,axiom,
    ! [N2: nat,X2: real] :
      ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
     => ( ( ord_less @ real @ ( root @ N2 @ X2 ) @ ( one_one @ real ) )
        = ( ord_less @ real @ X2 @ ( one_one @ real ) ) ) ) ).

% real_root_lt_1_iff
thf(fact_5282_real__root__gt__1__iff,axiom,
    ! [N2: nat,Y2: real] :
      ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
     => ( ( ord_less @ real @ ( one_one @ real ) @ ( root @ N2 @ Y2 ) )
        = ( ord_less @ real @ ( one_one @ real ) @ Y2 ) ) ) ).

% real_root_gt_1_iff
thf(fact_5283_real__root__le__1__iff,axiom,
    ! [N2: nat,X2: real] :
      ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
     => ( ( ord_less_eq @ real @ ( root @ N2 @ X2 ) @ ( one_one @ real ) )
        = ( ord_less_eq @ real @ X2 @ ( one_one @ real ) ) ) ) ).

% real_root_le_1_iff
thf(fact_5284_real__root__ge__1__iff,axiom,
    ! [N2: nat,Y2: real] :
      ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
     => ( ( ord_less_eq @ real @ ( one_one @ real ) @ ( root @ N2 @ Y2 ) )
        = ( ord_less_eq @ real @ ( one_one @ real ) @ Y2 ) ) ) ).

% real_root_ge_1_iff
thf(fact_5285_real__root__pow__pos2,axiom,
    ! [N2: nat,X2: real] :
      ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
     => ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 )
       => ( ( power_power @ real @ ( root @ N2 @ X2 ) @ N2 )
          = X2 ) ) ) ).

% real_root_pow_pos2
thf(fact_5286_real__root__inverse,axiom,
    ! [N2: nat,X2: real] :
      ( ( root @ N2 @ ( inverse_inverse @ real @ X2 ) )
      = ( inverse_inverse @ real @ ( root @ N2 @ X2 ) ) ) ).

% real_root_inverse
thf(fact_5287_fr__rot,axiom,
    ! [A3: assn,B6: assn,C5: assn] :
      ( ( entails @ ( times_times @ assn @ A3 @ B6 ) @ C5 )
     => ( entails @ ( times_times @ assn @ B6 @ A3 ) @ C5 ) ) ).

% fr_rot
thf(fact_5288_fr__refl,axiom,
    ! [A3: assn,B6: assn,C5: assn] :
      ( ( entails @ A3 @ B6 )
     => ( entails @ ( times_times @ assn @ A3 @ C5 ) @ ( times_times @ assn @ B6 @ C5 ) ) ) ).

% fr_refl
thf(fact_5289_fr__rot__rhs,axiom,
    ! [A3: assn,B6: assn,C5: assn] :
      ( ( entails @ A3 @ ( times_times @ assn @ B6 @ C5 ) )
     => ( entails @ A3 @ ( times_times @ assn @ C5 @ B6 ) ) ) ).

% fr_rot_rhs
thf(fact_5290_ent__frame__fwd,axiom,
    ! [P: assn,R2: assn,Ps: assn,F5: assn,Q: assn] :
      ( ( entails @ P @ R2 )
     => ( ( entails @ Ps @ ( times_times @ assn @ P @ F5 ) )
       => ( ( entails @ ( times_times @ assn @ R2 @ F5 ) @ Q )
         => ( entails @ Ps @ Q ) ) ) ) ).

% ent_frame_fwd
thf(fact_5291_ent__star__mono,axiom,
    ! [P: assn,P6: assn,Q: assn,Q7: assn] :
      ( ( entails @ P @ P6 )
     => ( ( entails @ Q @ Q7 )
       => ( entails @ ( times_times @ assn @ P @ Q ) @ ( times_times @ assn @ P6 @ Q7 ) ) ) ) ).

% ent_star_mono
thf(fact_5292_real__root__mult,axiom,
    ! [N2: nat,X2: real,Y2: real] :
      ( ( root @ N2 @ ( times_times @ real @ X2 @ Y2 ) )
      = ( times_times @ real @ ( root @ N2 @ X2 ) @ ( root @ N2 @ Y2 ) ) ) ).

% real_root_mult
thf(fact_5293_real__root__mult__exp,axiom,
    ! [M: nat,N2: nat,X2: real] :
      ( ( root @ ( times_times @ nat @ M @ N2 ) @ X2 )
      = ( root @ M @ ( root @ N2 @ X2 ) ) ) ).

% real_root_mult_exp
thf(fact_5294_real__root__minus,axiom,
    ! [N2: nat,X2: real] :
      ( ( root @ N2 @ ( uminus_uminus @ real @ X2 ) )
      = ( uminus_uminus @ real @ ( root @ N2 @ X2 ) ) ) ).

% real_root_minus
thf(fact_5295_cons__pre__rule,axiom,
    ! [A: $tType,P: assn,P6: assn,C2: heap_Time_Heap @ A,Q: A > assn] :
      ( ( entails @ P @ P6 )
     => ( ( hoare_hoare_triple @ A @ P6 @ C2 @ Q )
       => ( hoare_hoare_triple @ A @ P @ C2 @ Q ) ) ) ).

% cons_pre_rule
thf(fact_5296_cons__rule,axiom,
    ! [A: $tType,P: assn,P6: assn,Q: A > assn,Q7: A > assn,C2: heap_Time_Heap @ A] :
      ( ( entails @ P @ P6 )
     => ( ! [X3: A] : ( entails @ ( Q @ X3 ) @ ( Q7 @ X3 ) )
       => ( ( hoare_hoare_triple @ A @ P6 @ C2 @ Q )
         => ( hoare_hoare_triple @ A @ P @ C2 @ Q7 ) ) ) ) ).

% cons_rule
thf(fact_5297_cons__post__rule,axiom,
    ! [A: $tType,P: assn,C2: heap_Time_Heap @ A,Q: A > assn,Q7: A > assn] :
      ( ( hoare_hoare_triple @ A @ P @ C2 @ Q )
     => ( ! [X3: A] : ( entails @ ( Q @ X3 ) @ ( Q7 @ X3 ) )
       => ( hoare_hoare_triple @ A @ P @ C2 @ Q7 ) ) ) ).

% cons_post_rule
thf(fact_5298_is__entails,axiom,
    ! [P: assn,Q: assn] :
      ( ( entails @ P @ Q )
     => ( entails @ P @ Q ) ) ).

% is_entails
thf(fact_5299_real__root__commute,axiom,
    ! [M: nat,N2: nat,X2: real] :
      ( ( root @ M @ ( root @ N2 @ X2 ) )
      = ( root @ N2 @ ( root @ M @ X2 ) ) ) ).

% real_root_commute
thf(fact_5300_list__assn__mono,axiom,
    ! [A: $tType,B: $tType,P: A > B > assn,P6: A > B > assn,L: list @ A,L4: list @ B] :
      ( ! [X3: A,X10: B] : ( entails @ ( P @ X3 @ X10 ) @ ( P6 @ X3 @ X10 ) )
     => ( entails @ ( vEBT_List_list_assn @ A @ B @ P @ L @ L4 ) @ ( vEBT_List_list_assn @ A @ B @ P6 @ L @ L4 ) ) ) ).

% list_assn_mono
thf(fact_5301_ent__ex__postI,axiom,
    ! [A: $tType,P: assn,Q: A > assn,X2: A] :
      ( ( entails @ P @ ( Q @ X2 ) )
     => ( entails @ P @ ( ex_assn @ A @ Q ) ) ) ).

% ent_ex_postI
thf(fact_5302_ent__ex__preI,axiom,
    ! [A: $tType,P: A > assn,Q: assn] :
      ( ! [X3: A] : ( entails @ ( P @ X3 ) @ Q )
     => ( entails @ ( ex_assn @ A @ P ) @ Q ) ) ).

% ent_ex_preI
thf(fact_5303_enorm__exI_H,axiom,
    ! [A: $tType,Z9: A > $o,P: assn,Q: A > assn] :
      ( ! [X3: A] :
          ( ( Z9 @ X3 )
         => ( entails @ P @ ( Q @ X3 ) ) )
     => ( ? [X_1: A] : ( Z9 @ X_1 )
       => ( entails @ P @ ( ex_assn @ A @ Q ) ) ) ) ).

% enorm_exI'
thf(fact_5304_real__root__divide,axiom,
    ! [N2: nat,X2: real,Y2: real] :
      ( ( root @ N2 @ ( divide_divide @ real @ X2 @ Y2 ) )
      = ( divide_divide @ real @ ( root @ N2 @ X2 ) @ ( root @ N2 @ Y2 ) ) ) ).

% real_root_divide
thf(fact_5305_real__root__pos__pos__le,axiom,
    ! [X2: real,N2: nat] :
      ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 )
     => ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( root @ N2 @ X2 ) ) ) ).

% real_root_pos_pos_le
thf(fact_5306_mod__frame__fwd,axiom,
    ! [Ps: assn,H2: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ),P: assn,R2: assn,F5: assn] :
      ( ( rep_assn @ Ps @ H2 )
     => ( ( entails @ P @ R2 )
       => ( ( entails @ Ps @ ( times_times @ assn @ P @ F5 ) )
         => ( rep_assn @ ( times_times @ assn @ R2 @ F5 ) @ H2 ) ) ) ) ).

% mod_frame_fwd
thf(fact_5307_ent__star__mono__true,axiom,
    ! [A3: assn,A10: assn,B6: assn,B11: assn] :
      ( ( entails @ A3 @ ( times_times @ assn @ A10 @ ( top_top @ assn ) ) )
     => ( ( entails @ B6 @ ( times_times @ assn @ B11 @ ( top_top @ assn ) ) )
       => ( entails @ ( times_times @ assn @ ( times_times @ assn @ A3 @ B6 ) @ ( top_top @ assn ) ) @ ( times_times @ assn @ ( times_times @ assn @ A10 @ B11 ) @ ( top_top @ assn ) ) ) ) ) ).

% ent_star_mono_true
thf(fact_5308_ent__refl__true,axiom,
    ! [A3: assn] : ( entails @ A3 @ ( times_times @ assn @ A3 @ ( top_top @ assn ) ) ) ).

% ent_refl_true
thf(fact_5309_ent__true__drop_I1_J,axiom,
    ! [P: assn,Q: assn,R2: assn] :
      ( ( entails @ P @ ( times_times @ assn @ Q @ ( top_top @ assn ) ) )
     => ( entails @ ( times_times @ assn @ P @ R2 ) @ ( times_times @ assn @ Q @ ( top_top @ assn ) ) ) ) ).

% ent_true_drop(1)
thf(fact_5310_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_5311_return__cons__rule,axiom,
    ! [A: $tType,P: assn,Q: A > assn,X2: A] :
      ( ( entails @ P @ ( Q @ X2 ) )
     => ( hoare_hoare_triple @ A @ P @ ( heap_Time_return @ A @ X2 ) @ Q ) ) ).

% return_cons_rule
thf(fact_5312_fi__rule,axiom,
    ! [A: $tType,P: assn,C2: heap_Time_Heap @ A,Q: A > assn,Ps: assn,F5: assn] :
      ( ( hoare_hoare_triple @ A @ P @ C2 @ Q )
     => ( ( entails @ Ps @ ( times_times @ assn @ P @ F5 ) )
       => ( hoare_hoare_triple @ A @ Ps @ C2
          @ ^ [X: A] : ( times_times @ assn @ ( Q @ X ) @ F5 ) ) ) ) ).

% fi_rule
thf(fact_5313_htt__cons__rule,axiom,
    ! [A: $tType,P6: assn,C2: heap_Time_Heap @ A,Q7: A > assn,T4: nat,P: assn,Q: A > assn,T2: nat] :
      ( ( time_htt @ A @ P6 @ C2 @ Q7 @ T4 )
     => ( ( entails @ P @ P6 )
       => ( ! [X3: A] : ( entails @ ( Q7 @ X3 ) @ ( Q @ X3 ) )
         => ( ( ord_less_eq @ nat @ T4 @ T2 )
           => ( time_htt @ A @ P @ C2 @ Q @ T2 ) ) ) ) ) ).

% htt_cons_rule
thf(fact_5314_real__root__less__mono,axiom,
    ! [N2: nat,X2: real,Y2: real] :
      ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
     => ( ( ord_less @ real @ X2 @ Y2 )
       => ( ord_less @ real @ ( root @ N2 @ X2 ) @ ( root @ N2 @ Y2 ) ) ) ) ).

% real_root_less_mono
thf(fact_5315_real__root__le__mono,axiom,
    ! [N2: nat,X2: real,Y2: real] :
      ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
     => ( ( ord_less_eq @ real @ X2 @ Y2 )
       => ( ord_less_eq @ real @ ( root @ N2 @ X2 ) @ ( root @ N2 @ Y2 ) ) ) ) ).

% real_root_le_mono
thf(fact_5316_real__root__power,axiom,
    ! [N2: nat,X2: real,K: nat] :
      ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
     => ( ( root @ N2 @ ( power_power @ real @ X2 @ K ) )
        = ( power_power @ real @ ( root @ N2 @ X2 ) @ K ) ) ) ).

% real_root_power
thf(fact_5317_real__root__abs,axiom,
    ! [N2: nat,X2: real] :
      ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
     => ( ( root @ N2 @ ( abs_abs @ real @ X2 ) )
        = ( abs_abs @ real @ ( root @ N2 @ X2 ) ) ) ) ).

% real_root_abs
thf(fact_5318_real__root__gt__zero,axiom,
    ! [N2: nat,X2: real] :
      ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
     => ( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
       => ( ord_less @ real @ ( zero_zero @ real ) @ ( root @ N2 @ X2 ) ) ) ) ).

% real_root_gt_zero
thf(fact_5319_real__root__strict__decreasing,axiom,
    ! [N2: nat,N3: nat,X2: real] :
      ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
     => ( ( ord_less @ nat @ N2 @ N3 )
       => ( ( ord_less @ real @ ( one_one @ real ) @ X2 )
         => ( ord_less @ real @ ( root @ N3 @ X2 ) @ ( root @ N2 @ X2 ) ) ) ) ) ).

% real_root_strict_decreasing
thf(fact_5320_sqrt__def,axiom,
    ( sqrt
    = ( root @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ).

% sqrt_def
thf(fact_5321_root__abs__power,axiom,
    ! [N2: nat,Y2: real] :
      ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
     => ( ( abs_abs @ real @ ( root @ N2 @ ( power_power @ real @ Y2 @ N2 ) ) )
        = ( abs_abs @ real @ Y2 ) ) ) ).

% root_abs_power
thf(fact_5322_real__root__pos__pos,axiom,
    ! [N2: nat,X2: real] :
      ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
     => ( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
       => ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( root @ N2 @ X2 ) ) ) ) ).

% real_root_pos_pos
thf(fact_5323_real__root__strict__increasing,axiom,
    ! [N2: nat,N3: nat,X2: real] :
      ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
     => ( ( ord_less @ nat @ N2 @ N3 )
       => ( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
         => ( ( ord_less @ real @ X2 @ ( one_one @ real ) )
           => ( ord_less @ real @ ( root @ N2 @ X2 ) @ ( root @ N3 @ X2 ) ) ) ) ) ) ).

% real_root_strict_increasing
thf(fact_5324_real__root__decreasing,axiom,
    ! [N2: nat,N3: nat,X2: real] :
      ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
     => ( ( ord_less_eq @ nat @ N2 @ N3 )
       => ( ( ord_less_eq @ real @ ( one_one @ real ) @ X2 )
         => ( ord_less_eq @ real @ ( root @ N3 @ X2 ) @ ( root @ N2 @ X2 ) ) ) ) ) ).

% real_root_decreasing
thf(fact_5325_odd__real__root__pow,axiom,
    ! [N2: nat,X2: real] :
      ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 )
     => ( ( power_power @ real @ ( root @ N2 @ X2 ) @ N2 )
        = X2 ) ) ).

% odd_real_root_pow
thf(fact_5326_odd__real__root__unique,axiom,
    ! [N2: nat,Y2: real,X2: real] :
      ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 )
     => ( ( ( power_power @ real @ Y2 @ N2 )
          = X2 )
       => ( ( root @ N2 @ X2 )
          = Y2 ) ) ) ).

% odd_real_root_unique
thf(fact_5327_odd__real__root__power__cancel,axiom,
    ! [N2: nat,X2: real] :
      ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 )
     => ( ( root @ N2 @ ( power_power @ real @ X2 @ N2 ) )
        = X2 ) ) ).

% odd_real_root_power_cancel
thf(fact_5328_real__root__pow__pos,axiom,
    ! [N2: nat,X2: real] :
      ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
     => ( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
       => ( ( power_power @ real @ ( root @ N2 @ X2 ) @ N2 )
          = X2 ) ) ) ).

% real_root_pow_pos
thf(fact_5329_real__root__pos__unique,axiom,
    ! [N2: nat,Y2: real,X2: real] :
      ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
     => ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y2 )
       => ( ( ( power_power @ real @ Y2 @ N2 )
            = X2 )
         => ( ( root @ N2 @ X2 )
            = Y2 ) ) ) ) ).

% real_root_pos_unique
thf(fact_5330_real__root__power__cancel,axiom,
    ! [N2: nat,X2: real] :
      ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
     => ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 )
       => ( ( root @ N2 @ ( power_power @ real @ X2 @ N2 ) )
          = X2 ) ) ) ).

% real_root_power_cancel
thf(fact_5331_real__root__increasing,axiom,
    ! [N2: nat,N3: nat,X2: real] :
      ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
     => ( ( ord_less_eq @ nat @ N2 @ N3 )
       => ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 )
         => ( ( ord_less_eq @ real @ X2 @ ( one_one @ real ) )
           => ( ord_less_eq @ real @ ( root @ N2 @ X2 ) @ ( root @ N3 @ X2 ) ) ) ) ) ) ).

% real_root_increasing
thf(fact_5332_log__root,axiom,
    ! [N2: nat,A2: real,B2: real] :
      ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
     => ( ( ord_less @ real @ ( zero_zero @ real ) @ A2 )
       => ( ( log @ B2 @ ( root @ N2 @ A2 ) )
          = ( divide_divide @ real @ ( log @ B2 @ A2 ) @ ( semiring_1_of_nat @ real @ N2 ) ) ) ) ) ).

% log_root
thf(fact_5333_log__base__root,axiom,
    ! [N2: nat,B2: real,X2: real] :
      ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
     => ( ( ord_less @ real @ ( zero_zero @ real ) @ B2 )
       => ( ( log @ ( root @ N2 @ B2 ) @ X2 )
          = ( times_times @ real @ ( semiring_1_of_nat @ real @ N2 ) @ ( log @ B2 @ X2 ) ) ) ) ) ).

% log_base_root
thf(fact_5334_ln__root,axiom,
    ! [N2: nat,B2: real] :
      ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
     => ( ( ord_less @ real @ ( zero_zero @ real ) @ B2 )
       => ( ( ln_ln @ real @ ( root @ N2 @ B2 ) )
          = ( divide_divide @ real @ ( ln_ln @ real @ B2 ) @ ( semiring_1_of_nat @ real @ N2 ) ) ) ) ) ).

% ln_root
thf(fact_5335_root__powr__inverse,axiom,
    ! [N2: nat,X2: real] :
      ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
     => ( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
       => ( ( root @ N2 @ X2 )
          = ( powr @ real @ X2 @ ( divide_divide @ real @ ( one_one @ real ) @ ( semiring_1_of_nat @ real @ N2 ) ) ) ) ) ) ).

% root_powr_inverse
thf(fact_5336_listI__assn__wrap__insert,axiom,
    ! [E: $tType,P: assn,Uu: vEBT_VEBT,Uua: nat,Xi2: vEBT_VEBTi,I5: set @ nat,I: nat,Xs2: list @ vEBT_VEBT,Xsi: list @ vEBT_VEBTi,F5: assn,C2: heap_Time_Heap @ E,Q: E > assn] :
      ( ( entails @ P @ ( times_times @ assn @ ( times_times @ assn @ ( vEBT_vebt_assn_raw @ ( vEBT_vebt_insert @ Uu @ Uua ) @ Xi2 ) @ ( vEBT_List_listI_assn @ vEBT_VEBT @ vEBT_VEBTi @ ( minus_minus @ ( set @ nat ) @ I5 @ ( insert @ nat @ I @ ( bot_bot @ ( set @ nat ) ) ) ) @ vEBT_vebt_assn_raw @ Xs2 @ Xsi ) ) @ F5 ) )
     => ( ( ord_less @ nat @ I @ ( size_size @ ( list @ vEBT_VEBT ) @ Xs2 ) )
       => ( ( member @ nat @ I @ I5 )
         => ( ( hoare_hoare_triple @ E @ ( times_times @ assn @ ( vEBT_List_listI_assn @ vEBT_VEBT @ vEBT_VEBTi @ I5 @ vEBT_vebt_assn_raw @ ( list_update @ vEBT_VEBT @ Xs2 @ I @ ( vEBT_vebt_insert @ Uu @ Uua ) ) @ ( list_update @ vEBT_VEBTi @ Xsi @ I @ Xi2 ) ) @ F5 ) @ C2 @ Q )
           => ( hoare_hoare_triple @ E @ P @ C2 @ Q ) ) ) ) ) ).

% listI_assn_wrap_insert
thf(fact_5337_int__ge__less__than__def,axiom,
    ( int_ge_less_than
    = ( ^ [D3: int] :
          ( collect @ ( product_prod @ int @ int )
          @ ( product_case_prod @ int @ int @ $o
            @ ^ [Z8: int,Z2: int] :
                ( ( ord_less_eq @ int @ D3 @ Z8 )
                & ( ord_less @ int @ Z8 @ Z2 ) ) ) ) ) ) ).

% int_ge_less_than_def
thf(fact_5338_big__assn__simp,axiom,
    ! [H2: nat,TreeList: list @ vEBT_VEBT,L: nat,X2: vEBT_VEBTi,Xaa: option @ nat,X13: array @ vEBT_VEBTi,Tree_is2: list @ vEBT_VEBTi,Summary: vEBT_VEBT,X14: vEBT_VEBTi] :
      ( ( ord_less @ nat @ H2 @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
     => ( entails
        @ ( times_times @ assn
          @ ( times_times @ assn @ ( vEBT_vebt_assn_raw @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ H2 ) @ L ) @ X2 )
            @ ( 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_update @ vEBT_VEBTi @ Tree_is2 @ H2 @ X2 ) ) @ ( vEBT_vebt_assn_raw @ Summary @ X14 ) ) @ ( vEBT_List_listI_assn @ vEBT_VEBT @ vEBT_VEBTi @ ( minus_minus @ ( set @ nat ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) ) @ ( insert @ nat @ H2 @ ( bot_bot @ ( set @ nat ) ) ) ) @ vEBT_vebt_assn_raw @ TreeList @ Tree_is2 ) ) )
        @ ( times_times @ assn
          @ ( times_times @ assn @ ( times_times @ assn @ ( snga_assn @ vEBT_VEBTi @ X13 @ ( list_update @ vEBT_VEBTi @ Tree_is2 @ H2 @ X2 ) ) @ ( vEBT_vebt_assn_raw @ Summary @ X14 ) )
            @ ( pure_assn
              @ ( Xaa
                = ( vEBT_vebt_mint @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ H2 ) @ L ) ) ) ) )
          @ ( vEBT_List_list_assn @ vEBT_VEBT @ vEBT_VEBTi @ vEBT_vebt_assn_raw @ ( list_update @ vEBT_VEBT @ TreeList @ H2 @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ H2 ) @ L ) ) @ ( list_update @ vEBT_VEBTi @ Tree_is2 @ H2 @ X2 ) ) ) ) ) ).

% big_assn_simp
thf(fact_5339_atLeastLessThan__iff,axiom,
    ! [A: $tType] :
      ( ( ord @ A )
     => ! [I: A,L: A,U: A] :
          ( ( member @ A @ I @ ( set_or7035219750837199246ssThan @ A @ L @ U ) )
          = ( ( ord_less_eq @ A @ L @ I )
            & ( ord_less @ A @ I @ U ) ) ) ) ).

% atLeastLessThan_iff
thf(fact_5340_atLeastLessThan__empty,axiom,
    ! [A: $tType] :
      ( ( order @ A )
     => ! [B2: A,A2: A] :
          ( ( ord_less_eq @ A @ B2 @ A2 )
         => ( ( set_or7035219750837199246ssThan @ A @ A2 @ B2 )
            = ( bot_bot @ ( set @ A ) ) ) ) ) ).

% atLeastLessThan_empty
thf(fact_5341_atLeastLessThan__empty__iff2,axiom,
    ! [A: $tType] :
      ( ( preorder @ A )
     => ! [A2: A,B2: A] :
          ( ( ( bot_bot @ ( set @ A ) )
            = ( set_or7035219750837199246ssThan @ A @ A2 @ B2 ) )
          = ( ~ ( ord_less @ A @ A2 @ B2 ) ) ) ) ).

% atLeastLessThan_empty_iff2
thf(fact_5342_atLeastLessThan__empty__iff,axiom,
    ! [A: $tType] :
      ( ( preorder @ A )
     => ! [A2: A,B2: A] :
          ( ( ( set_or7035219750837199246ssThan @ A @ A2 @ B2 )
            = ( bot_bot @ ( set @ A ) ) )
          = ( ~ ( ord_less @ A @ A2 @ B2 ) ) ) ) ).

% atLeastLessThan_empty_iff
thf(fact_5343_infinite__Ico__iff,axiom,
    ! [A: $tType] :
      ( ( dense_linorder @ A )
     => ! [A2: A,B2: A] :
          ( ( ~ ( finite_finite2 @ A @ ( set_or7035219750837199246ssThan @ A @ A2 @ B2 ) ) )
          = ( ord_less @ A @ A2 @ B2 ) ) ) ).

% infinite_Ico_iff
thf(fact_5344_ivl__subset,axiom,
    ! [A: $tType] :
      ( ( linorder @ A )
     => ! [I: A,J: A,M: A,N2: A] :
          ( ( ord_less_eq @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ I @ J ) @ ( set_or7035219750837199246ssThan @ A @ M @ N2 ) )
          = ( ( ord_less_eq @ A @ J @ I )
            | ( ( ord_less_eq @ A @ M @ I )
              & ( ord_less_eq @ A @ J @ N2 ) ) ) ) ) ).

% ivl_subset
thf(fact_5345_ivl__diff,axiom,
    ! [A: $tType] :
      ( ( linorder @ A )
     => ! [I: A,N2: A,M: A] :
          ( ( ord_less_eq @ A @ I @ N2 )
         => ( ( minus_minus @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ I @ M ) @ ( set_or7035219750837199246ssThan @ A @ I @ N2 ) )
            = ( set_or7035219750837199246ssThan @ A @ N2 @ M ) ) ) ) ).

% ivl_diff
thf(fact_5346_lessThan__minus__lessThan,axiom,
    ! [A: $tType] :
      ( ( linorder @ A )
     => ! [N2: A,M: A] :
          ( ( minus_minus @ ( set @ A ) @ ( set_ord_lessThan @ A @ N2 ) @ ( set_ord_lessThan @ A @ M ) )
          = ( set_or7035219750837199246ssThan @ A @ M @ N2 ) ) ) ).

% lessThan_minus_lessThan
thf(fact_5347_local_Oext,axiom,
    ! [Y2: nat,TreeList: list @ vEBT_VEBT,X13: array @ vEBT_VEBTi,Tree_is2: list @ vEBT_VEBTi,Summary: vEBT_VEBT,X14: vEBT_VEBTi] :
      ( ( ord_less @ nat @ Y2 @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
     => ( entails @ ( times_times @ assn @ ( snga_assn @ vEBT_VEBTi @ X13 @ Tree_is2 ) @ ( times_times @ assn @ ( vEBT_vebt_assn_raw @ Summary @ X14 ) @ ( times_times @ assn @ ( vEBT_vebt_assn_raw @ ( nth @ vEBT_VEBT @ TreeList @ Y2 ) @ ( nth @ vEBT_VEBTi @ Tree_is2 @ Y2 ) ) @ ( vEBT_List_listI_assn @ vEBT_VEBT @ vEBT_VEBTi @ ( minus_minus @ ( set @ nat ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) ) @ ( insert @ nat @ Y2 @ ( bot_bot @ ( set @ nat ) ) ) ) @ vEBT_vebt_assn_raw @ TreeList @ Tree_is2 ) ) ) ) @ ( times_times @ assn @ ( times_times @ assn @ ( times_times @ assn @ ( snga_assn @ vEBT_VEBTi @ X13 @ Tree_is2 ) @ ( vEBT_vebt_assn_raw @ Summary @ X14 ) ) @ ( vEBT_List_listI_assn @ vEBT_VEBT @ vEBT_VEBTi @ ( minus_minus @ ( set @ nat ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) ) @ ( insert @ nat @ Y2 @ ( bot_bot @ ( set @ nat ) ) ) ) @ vEBT_vebt_assn_raw @ TreeList @ Tree_is2 ) ) @ ( vEBT_vebt_assn_raw @ ( nth @ vEBT_VEBT @ TreeList @ Y2 ) @ ( nth @ vEBT_VEBTi @ Tree_is2 @ Y2 ) ) ) ) ) ).

% local.ext
thf(fact_5348_atLeastLessThan__singleton,axiom,
    ! [M: nat] :
      ( ( set_or7035219750837199246ssThan @ nat @ M @ ( suc @ M ) )
      = ( insert @ nat @ M @ ( bot_bot @ ( set @ nat ) ) ) ) ).

% atLeastLessThan_singleton
thf(fact_5349_txe,axiom,
    ! [Y2: nat,TreeList: list @ vEBT_VEBT,Tree_is2: list @ vEBT_VEBTi,X13: array @ vEBT_VEBTi,Summary: vEBT_VEBT,X14: vEBT_VEBTi] :
      ( ( ord_less @ nat @ Y2 @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
     => ( entails @ ( times_times @ assn @ ( times_times @ assn @ ( times_times @ assn @ ( vEBT_vebt_assn_raw @ ( nth @ vEBT_VEBT @ TreeList @ Y2 ) @ ( nth @ vEBT_VEBTi @ Tree_is2 @ Y2 ) ) @ ( snga_assn @ vEBT_VEBTi @ X13 @ Tree_is2 ) ) @ ( vEBT_vebt_assn_raw @ Summary @ X14 ) ) @ ( vEBT_List_listI_assn @ vEBT_VEBT @ vEBT_VEBTi @ ( minus_minus @ ( set @ nat ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) ) @ ( insert @ nat @ Y2 @ ( bot_bot @ ( set @ nat ) ) ) ) @ vEBT_vebt_assn_raw @ TreeList @ Tree_is2 ) ) @ ( times_times @ assn @ ( times_times @ assn @ ( vEBT_vebt_assn_raw @ Summary @ X14 ) @ ( snga_assn @ vEBT_VEBTi @ X13 @ Tree_is2 ) ) @ ( vEBT_List_list_assn @ vEBT_VEBT @ vEBT_VEBTi @ vEBT_vebt_assn_raw @ TreeList @ Tree_is2 ) ) ) ) ).

% txe
thf(fact_5350_repack,axiom,
    ! [I: nat,TreeList: list @ vEBT_VEBT,Tree_is2: list @ vEBT_VEBTi,Rest: assn,X13: array @ vEBT_VEBTi,Summary: vEBT_VEBT,X14: vEBT_VEBTi] :
      ( ( ord_less @ nat @ I @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
     => ( entails @ ( times_times @ assn @ ( times_times @ assn @ ( vEBT_vebt_assn_raw @ ( nth @ vEBT_VEBT @ TreeList @ I ) @ ( nth @ vEBT_VEBTi @ Tree_is2 @ I ) ) @ Rest ) @ ( times_times @ assn @ ( times_times @ assn @ ( snga_assn @ vEBT_VEBTi @ X13 @ Tree_is2 ) @ ( vEBT_vebt_assn_raw @ Summary @ X14 ) ) @ ( vEBT_List_listI_assn @ vEBT_VEBT @ vEBT_VEBTi @ ( minus_minus @ ( set @ nat ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) ) @ ( insert @ nat @ I @ ( bot_bot @ ( set @ nat ) ) ) ) @ vEBT_vebt_assn_raw @ TreeList @ Tree_is2 ) ) ) @ ( times_times @ assn @ ( times_times @ assn @ ( times_times @ assn @ Rest @ ( vEBT_vebt_assn_raw @ Summary @ X14 ) ) @ ( snga_assn @ vEBT_VEBTi @ X13 @ Tree_is2 ) ) @ ( vEBT_List_list_assn @ vEBT_VEBT @ vEBT_VEBTi @ vEBT_vebt_assn_raw @ TreeList @ Tree_is2 ) ) ) ) ).

% repack
thf(fact_5351_recomp,axiom,
    ! [I: nat,TreeList: list @ vEBT_VEBT,Tree_is2: list @ vEBT_VEBTi,X13: array @ vEBT_VEBTi,Summary: vEBT_VEBT,X14: vEBT_VEBTi] :
      ( ( ord_less @ nat @ I @ ( size_size @ ( list @ vEBT_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_is2 @ I ) ) @ ( vEBT_List_listI_assn @ vEBT_VEBT @ vEBT_VEBTi @ ( minus_minus @ ( set @ nat ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) ) @ ( insert @ nat @ I @ ( bot_bot @ ( set @ nat ) ) ) ) @ vEBT_vebt_assn_raw @ TreeList @ Tree_is2 ) ) @ ( snga_assn @ vEBT_VEBTi @ X13 @ Tree_is2 ) ) @ ( vEBT_vebt_assn_raw @ Summary @ X14 ) ) @ ( times_times @ assn @ ( times_times @ assn @ ( vEBT_vebt_assn_raw @ Summary @ X14 ) @ ( snga_assn @ vEBT_VEBTi @ X13 @ Tree_is2 ) ) @ ( vEBT_List_list_assn @ vEBT_VEBT @ vEBT_VEBTi @ vEBT_vebt_assn_raw @ TreeList @ Tree_is2 ) ) ) ) ).

% recomp
thf(fact_5352_tcd,axiom,
    ! [A: $tType,I: nat,TreeList: list @ vEBT_VEBT,TreeList4: list @ A,Y2: vEBT_VEBT,X2: vEBT_VEBTi,X13: array @ vEBT_VEBTi,Tree_is2: list @ vEBT_VEBTi,Summary: vEBT_VEBT,X14: vEBT_VEBTi] :
      ( ( ord_less @ nat @ I @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
     => ( ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList )
          = ( size_size @ ( list @ A ) @ TreeList4 ) )
       => ( entails @ ( times_times @ assn @ ( times_times @ assn @ ( times_times @ assn @ ( vEBT_vebt_assn_raw @ Y2 @ X2 ) @ ( snga_assn @ vEBT_VEBTi @ X13 @ ( list_update @ vEBT_VEBTi @ Tree_is2 @ I @ X2 ) ) ) @ ( vEBT_vebt_assn_raw @ Summary @ X14 ) ) @ ( vEBT_List_listI_assn @ vEBT_VEBT @ vEBT_VEBTi @ ( minus_minus @ ( set @ nat ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) ) @ ( insert @ nat @ I @ ( bot_bot @ ( set @ nat ) ) ) ) @ vEBT_vebt_assn_raw @ ( list_update @ vEBT_VEBT @ TreeList @ I @ Y2 ) @ ( list_update @ vEBT_VEBTi @ Tree_is2 @ I @ X2 ) ) ) @ ( times_times @ assn @ ( times_times @ assn @ ( snga_assn @ vEBT_VEBTi @ X13 @ ( list_update @ vEBT_VEBTi @ Tree_is2 @ I @ X2 ) ) @ ( vEBT_vebt_assn_raw @ Summary @ X14 ) ) @ ( vEBT_List_list_assn @ vEBT_VEBT @ vEBT_VEBTi @ vEBT_vebt_assn_raw @ ( list_update @ vEBT_VEBT @ TreeList @ I @ Y2 ) @ ( list_update @ vEBT_VEBTi @ Tree_is2 @ I @ X2 ) ) ) ) ) ) ).

% tcd
thf(fact_5353_big__assn__simp_H,axiom,
    ! [H2: nat,TreeList: list @ vEBT_VEBT,Xaa: vEBT_VEBT,L: nat,X2: vEBT_VEBTi,Xb: option @ nat,X13: array @ vEBT_VEBTi,Tree_is2: list @ vEBT_VEBTi,Summary: vEBT_VEBT,X14: vEBT_VEBTi] :
      ( ( ord_less @ nat @ H2 @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
     => ( ( Xaa
          = ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ H2 ) @ L ) )
       => ( entails
          @ ( times_times @ assn
            @ ( times_times @ assn @ ( vEBT_vebt_assn_raw @ Xaa @ X2 )
              @ ( pure_assn
                @ ( Xb
                  = ( vEBT_vebt_mint @ Xaa ) ) ) )
            @ ( times_times @ assn @ ( times_times @ assn @ ( snga_assn @ vEBT_VEBTi @ X13 @ ( list_update @ vEBT_VEBTi @ Tree_is2 @ H2 @ X2 ) ) @ ( vEBT_vebt_assn_raw @ Summary @ X14 ) ) @ ( vEBT_List_listI_assn @ vEBT_VEBT @ vEBT_VEBTi @ ( minus_minus @ ( set @ nat ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) ) @ ( insert @ nat @ H2 @ ( bot_bot @ ( set @ nat ) ) ) ) @ vEBT_vebt_assn_raw @ TreeList @ Tree_is2 ) ) )
          @ ( times_times @ assn
            @ ( times_times @ assn @ ( times_times @ assn @ ( snga_assn @ vEBT_VEBTi @ X13 @ ( list_update @ vEBT_VEBTi @ Tree_is2 @ H2 @ X2 ) ) @ ( vEBT_vebt_assn_raw @ Summary @ X14 ) )
              @ ( pure_assn
                @ ( Xb
                  = ( vEBT_vebt_mint @ Xaa ) ) ) )
            @ ( vEBT_List_list_assn @ vEBT_VEBT @ vEBT_VEBTi @ vEBT_vebt_assn_raw @ ( list_update @ vEBT_VEBT @ TreeList @ H2 @ Xaa ) @ ( list_update @ vEBT_VEBTi @ Tree_is2 @ H2 @ X2 ) ) ) ) ) ) ).

% big_assn_simp'
thf(fact_5354_sum_Oop__ivl__Suc,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add @ A )
     => ! [N2: nat,M: nat,G: nat > A] :
          ( ( ( ord_less @ nat @ N2 @ M )
           => ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ ( suc @ N2 ) ) )
              = ( zero_zero @ A ) ) )
          & ( ~ ( ord_less @ nat @ N2 @ M )
           => ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ ( suc @ N2 ) ) )
              = ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ N2 ) ) @ ( G @ N2 ) ) ) ) ) ) ).

% sum.op_ivl_Suc
thf(fact_5355_prod_Oop__ivl__Suc,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_mult @ A )
     => ! [N2: nat,M: nat,G: nat > A] :
          ( ( ( ord_less @ nat @ N2 @ M )
           => ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ ( suc @ N2 ) ) )
              = ( one_one @ A ) ) )
          & ( ~ ( ord_less @ nat @ N2 @ M )
           => ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ ( suc @ N2 ) ) )
              = ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ N2 ) ) @ ( G @ N2 ) ) ) ) ) ) ).

% prod.op_ivl_Suc
thf(fact_5356_atLeastLessThan__inj_I2_J,axiom,
    ! [A: $tType] :
      ( ( linorder @ A )
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( ( ( set_or7035219750837199246ssThan @ A @ A2 @ B2 )
            = ( set_or7035219750837199246ssThan @ A @ C2 @ D2 ) )
         => ( ( ord_less @ A @ A2 @ B2 )
           => ( ( ord_less @ A @ C2 @ D2 )
             => ( B2 = D2 ) ) ) ) ) ).

% atLeastLessThan_inj(2)
thf(fact_5357_atLeastLessThan__inj_I1_J,axiom,
    ! [A: $tType] :
      ( ( linorder @ A )
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( ( ( set_or7035219750837199246ssThan @ A @ A2 @ B2 )
            = ( set_or7035219750837199246ssThan @ A @ C2 @ D2 ) )
         => ( ( ord_less @ A @ A2 @ B2 )
           => ( ( ord_less @ A @ C2 @ D2 )
             => ( A2 = C2 ) ) ) ) ) ).

% atLeastLessThan_inj(1)
thf(fact_5358_atLeastLessThan__eq__iff,axiom,
    ! [A: $tType] :
      ( ( linorder @ A )
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( ( ord_less @ A @ A2 @ B2 )
         => ( ( ord_less @ A @ C2 @ D2 )
           => ( ( ( set_or7035219750837199246ssThan @ A @ A2 @ B2 )
                = ( set_or7035219750837199246ssThan @ A @ C2 @ D2 ) )
              = ( ( A2 = C2 )
                & ( B2 = D2 ) ) ) ) ) ) ).

% atLeastLessThan_eq_iff
thf(fact_5359_infinite__Ico,axiom,
    ! [A: $tType] :
      ( ( dense_linorder @ A )
     => ! [A2: A,B2: A] :
          ( ( ord_less @ A @ A2 @ B2 )
         => ~ ( finite_finite2 @ A @ ( set_or7035219750837199246ssThan @ A @ A2 @ B2 ) ) ) ) ).

% infinite_Ico
thf(fact_5360_atLeastLessThan__subset__iff,axiom,
    ! [A: $tType] :
      ( ( linorder @ A )
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( ( ord_less_eq @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ A2 @ B2 ) @ ( set_or7035219750837199246ssThan @ A @ C2 @ D2 ) )
         => ( ( ord_less_eq @ A @ B2 @ A2 )
            | ( ( ord_less_eq @ A @ C2 @ A2 )
              & ( ord_less_eq @ A @ B2 @ D2 ) ) ) ) ) ).

% atLeastLessThan_subset_iff
thf(fact_5361_list__assn__conv__idx,axiom,
    ! [B: $tType,A: $tType] :
      ( ( vEBT_List_list_assn @ A @ B )
      = ( ^ [A7: A > B > assn,Xs: list @ A] : ( vEBT_List_listI_assn @ A @ B @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ A ) @ Xs ) ) @ A7 @ Xs ) ) ) ).

% list_assn_conv_idx
thf(fact_5362_listI__assn__conv,axiom,
    ! [A: $tType,B: $tType,N2: nat,Xs2: list @ A,A3: A > B > assn,Xsi: list @ B] :
      ( ( N2
        = ( size_size @ ( list @ A ) @ Xs2 ) )
     => ( ( vEBT_List_listI_assn @ A @ B @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N2 ) @ A3 @ Xs2 @ Xsi )
        = ( vEBT_List_list_assn @ A @ B @ A3 @ Xs2 @ Xsi ) ) ) ).

% listI_assn_conv
thf(fact_5363_ex__nat__less__eq,axiom,
    ! [N2: nat,P: nat > $o] :
      ( ( ? [M7: nat] :
            ( ( ord_less @ nat @ M7 @ N2 )
            & ( P @ M7 ) ) )
      = ( ? [X: nat] :
            ( ( member @ nat @ X @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N2 ) )
            & ( P @ X ) ) ) ) ).

% ex_nat_less_eq
thf(fact_5364_all__nat__less__eq,axiom,
    ! [N2: nat,P: nat > $o] :
      ( ( ! [M7: nat] :
            ( ( ord_less @ nat @ M7 @ N2 )
           => ( P @ M7 ) ) )
      = ( ! [X: nat] :
            ( ( member @ nat @ X @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N2 ) )
           => ( P @ X ) ) ) ) ).

% all_nat_less_eq
thf(fact_5365_atLeastLessThanSuc__atLeastAtMost,axiom,
    ! [L: nat,U: nat] :
      ( ( set_or7035219750837199246ssThan @ nat @ L @ ( suc @ U ) )
      = ( set_or1337092689740270186AtMost @ nat @ L @ U ) ) ).

% atLeastLessThanSuc_atLeastAtMost
thf(fact_5366_listI__assn__conv_H,axiom,
    ! [B: $tType,A: $tType,N2: nat,Xs2: list @ A,A3: A > B > assn,Xsi: list @ B,F5: assn] :
      ( ( N2
        = ( size_size @ ( list @ A ) @ Xs2 ) )
     => ( ( times_times @ assn @ ( vEBT_List_listI_assn @ A @ B @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N2 ) @ A3 @ Xs2 @ Xsi ) @ F5 )
        = ( times_times @ assn @ ( vEBT_List_list_assn @ A @ B @ A3 @ Xs2 @ Xsi ) @ F5 ) ) ) ).

% listI_assn_conv'
thf(fact_5367_sum_Oshift__bounds__Suc__ivl,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add @ A )
     => ! [G: nat > A,M: nat,N2: nat] :
          ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M ) @ ( suc @ N2 ) ) )
          = ( groups7311177749621191930dd_sum @ nat @ A
            @ ^ [I2: nat] : ( G @ ( suc @ I2 ) )
            @ ( set_or7035219750837199246ssThan @ nat @ M @ N2 ) ) ) ) ).

% sum.shift_bounds_Suc_ivl
thf(fact_5368_sum_Oshift__bounds__nat__ivl,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add @ A )
     => ! [G: nat > A,M: nat,K: nat,N2: nat] :
          ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( plus_plus @ nat @ M @ K ) @ ( plus_plus @ nat @ N2 @ K ) ) )
          = ( groups7311177749621191930dd_sum @ nat @ A
            @ ^ [I2: nat] : ( G @ ( plus_plus @ nat @ I2 @ K ) )
            @ ( set_or7035219750837199246ssThan @ nat @ M @ N2 ) ) ) ) ).

% sum.shift_bounds_nat_ivl
thf(fact_5369_prod_Oshift__bounds__Suc__ivl,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_mult @ A )
     => ! [G: nat > A,M: nat,N2: nat] :
          ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M ) @ ( suc @ N2 ) ) )
          = ( groups7121269368397514597t_prod @ nat @ A
            @ ^ [I2: nat] : ( G @ ( suc @ I2 ) )
            @ ( set_or7035219750837199246ssThan @ nat @ M @ N2 ) ) ) ) ).

% prod.shift_bounds_Suc_ivl
thf(fact_5370_prod_Oshift__bounds__nat__ivl,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_mult @ A )
     => ! [G: nat > A,M: nat,K: nat,N2: nat] :
          ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( plus_plus @ nat @ M @ K ) @ ( plus_plus @ nat @ N2 @ K ) ) )
          = ( groups7121269368397514597t_prod @ nat @ A
            @ ^ [I2: nat] : ( G @ ( plus_plus @ nat @ I2 @ K ) )
            @ ( set_or7035219750837199246ssThan @ nat @ M @ N2 ) ) ) ) ).

% prod.shift_bounds_nat_ivl
thf(fact_5371_sum_Oivl__cong,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ( ord @ B )
        & ( comm_monoid_add @ A ) )
     => ! [A2: B,C2: B,B2: B,D2: B,G: B > A,H2: B > A] :
          ( ( A2 = C2 )
         => ( ( B2 = D2 )
           => ( ! [X3: B] :
                  ( ( ord_less_eq @ B @ C2 @ X3 )
                 => ( ( ord_less @ B @ X3 @ D2 )
                   => ( ( G @ X3 )
                      = ( H2 @ X3 ) ) ) )
             => ( ( groups7311177749621191930dd_sum @ B @ A @ G @ ( set_or7035219750837199246ssThan @ B @ A2 @ B2 ) )
                = ( groups7311177749621191930dd_sum @ B @ A @ H2 @ ( set_or7035219750837199246ssThan @ B @ C2 @ D2 ) ) ) ) ) ) ) ).

% sum.ivl_cong
thf(fact_5372_prod_Oivl__cong,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ( ord @ B )
        & ( comm_monoid_mult @ A ) )
     => ! [A2: B,C2: B,B2: B,D2: B,G: B > A,H2: B > A] :
          ( ( A2 = C2 )
         => ( ( B2 = D2 )
           => ( ! [X3: B] :
                  ( ( ord_less_eq @ B @ C2 @ X3 )
                 => ( ( ord_less @ B @ X3 @ D2 )
                   => ( ( G @ X3 )
                      = ( H2 @ X3 ) ) ) )
             => ( ( groups7121269368397514597t_prod @ B @ A @ G @ ( set_or7035219750837199246ssThan @ B @ A2 @ B2 ) )
                = ( groups7121269368397514597t_prod @ B @ A @ H2 @ ( set_or7035219750837199246ssThan @ B @ C2 @ D2 ) ) ) ) ) ) ) ).

% prod.ivl_cong
thf(fact_5373_sum_OatLeastLessThan__concat,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add @ A )
     => ! [M: nat,N2: nat,P4: nat,G: nat > A] :
          ( ( ord_less_eq @ nat @ M @ N2 )
         => ( ( ord_less_eq @ nat @ N2 @ P4 )
           => ( ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ N2 ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ N2 @ P4 ) ) )
              = ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ P4 ) ) ) ) ) ) ).

% sum.atLeastLessThan_concat
thf(fact_5374_sum__diff__nat__ivl,axiom,
    ! [A: $tType] :
      ( ( ab_group_add @ A )
     => ! [M: nat,N2: nat,P4: nat,F: nat > A] :
          ( ( ord_less_eq @ nat @ M @ N2 )
         => ( ( ord_less_eq @ nat @ N2 @ P4 )
           => ( ( minus_minus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F @ ( set_or7035219750837199246ssThan @ nat @ M @ P4 ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ F @ ( set_or7035219750837199246ssThan @ nat @ M @ N2 ) ) )
              = ( groups7311177749621191930dd_sum @ nat @ A @ F @ ( set_or7035219750837199246ssThan @ nat @ N2 @ P4 ) ) ) ) ) ) ).

% sum_diff_nat_ivl
thf(fact_5375_prod_OatLeastLessThan__concat,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_mult @ A )
     => ! [M: nat,N2: nat,P4: nat,G: nat > A] :
          ( ( ord_less_eq @ nat @ M @ N2 )
         => ( ( ord_less_eq @ nat @ N2 @ P4 )
           => ( ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ N2 ) ) @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ N2 @ P4 ) ) )
              = ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ P4 ) ) ) ) ) ) ).

% prod.atLeastLessThan_concat
thf(fact_5376_atLeast0__lessThan__Suc,axiom,
    ! [N2: nat] :
      ( ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( suc @ N2 ) )
      = ( insert @ nat @ N2 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ).

% atLeast0_lessThan_Suc
thf(fact_5377_subset__eq__atLeast0__lessThan__finite,axiom,
    ! [N3: set @ nat,N2: nat] :
      ( ( ord_less_eq @ ( set @ nat ) @ N3 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N2 ) )
     => ( finite_finite2 @ nat @ N3 ) ) ).

% subset_eq_atLeast0_lessThan_finite
thf(fact_5378_listI__assn__weak__cong,axiom,
    ! [A: $tType,B: $tType,I5: set @ nat,I7: set @ nat,A3: A > B > assn,A10: A > B > assn,Xs2: list @ A,Xs4: list @ A,Xsi: list @ B,Xsi2: list @ B] :
      ( ( I5 = I7 )
     => ( ( A3 = A10 )
       => ( ( ( size_size @ ( list @ A ) @ Xs2 )
            = ( size_size @ ( list @ A ) @ Xs4 ) )
         => ( ( ( size_size @ ( list @ B ) @ Xsi )
              = ( size_size @ ( list @ B ) @ Xsi2 ) )
           => ( ! [I3: nat] :
                  ( ( member @ nat @ I3 @ I5 )
                 => ( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ Xs2 ) )
                   => ( ( ( size_size @ ( list @ A ) @ Xs2 )
                        = ( size_size @ ( list @ B ) @ Xsi ) )
                     => ( ( ( nth @ A @ Xs2 @ I3 )
                          = ( nth @ A @ Xs4 @ I3 ) )
                        & ( ( nth @ B @ Xsi @ I3 )
                          = ( nth @ B @ Xsi2 @ I3 ) ) ) ) ) )
             => ( ( vEBT_List_listI_assn @ A @ B @ I5 @ A3 @ Xs2 @ Xsi )
                = ( vEBT_List_listI_assn @ A @ B @ I7 @ A10 @ Xs4 @ Xsi2 ) ) ) ) ) ) ) ).

% listI_assn_weak_cong
thf(fact_5379_listI__assn__cong,axiom,
    ! [A: $tType,B: $tType,I5: set @ nat,I7: set @ nat,Xs2: list @ A,Xs4: list @ A,Xsi: list @ B,Xsi2: list @ B,A3: A > B > assn,A10: A > B > assn] :
      ( ( I5 = I7 )
     => ( ( ( size_size @ ( list @ A ) @ Xs2 )
          = ( size_size @ ( list @ A ) @ Xs4 ) )
       => ( ( ( size_size @ ( list @ B ) @ Xsi )
            = ( size_size @ ( list @ B ) @ Xsi2 ) )
         => ( ! [I3: nat] :
                ( ( member @ nat @ I3 @ I5 )
               => ( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ Xs2 ) )
                 => ( ( ( size_size @ ( list @ A ) @ Xs2 )
                      = ( size_size @ ( list @ B ) @ Xsi ) )
                   => ( ( ( nth @ A @ Xs2 @ I3 )
                        = ( nth @ A @ Xs4 @ I3 ) )
                      & ( ( nth @ B @ Xsi @ I3 )
                        = ( nth @ B @ Xsi2 @ I3 ) )
                      & ( ( A3 @ ( nth @ A @ Xs2 @ I3 ) @ ( nth @ B @ Xsi @ I3 ) )
                        = ( A10 @ ( nth @ A @ Xs4 @ I3 ) @ ( nth @ B @ Xsi2 @ I3 ) ) ) ) ) ) )
           => ( ( vEBT_List_listI_assn @ A @ B @ I5 @ A3 @ Xs2 @ Xsi )
              = ( vEBT_List_listI_assn @ A @ B @ I7 @ A10 @ Xs4 @ Xsi2 ) ) ) ) ) ) ).

% listI_assn_cong
thf(fact_5380_atLeastLessThan__subseteq__atLeastAtMost__iff,axiom,
    ! [A: $tType] :
      ( ( dense_linorder @ A )
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( ( ord_less_eq @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ A2 @ B2 ) @ ( set_or1337092689740270186AtMost @ A @ C2 @ D2 ) )
          = ( ( ord_less @ A @ A2 @ B2 )
           => ( ( ord_less_eq @ A @ C2 @ A2 )
              & ( ord_less_eq @ A @ B2 @ D2 ) ) ) ) ) ).

% atLeastLessThan_subseteq_atLeastAtMost_iff
thf(fact_5381_atLeastAtMost__subseteq__atLeastLessThan__iff,axiom,
    ! [A: $tType] :
      ( ( preorder @ A )
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( ( ord_less_eq @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) @ ( set_or7035219750837199246ssThan @ A @ C2 @ D2 ) )
          = ( ( ord_less_eq @ A @ A2 @ B2 )
           => ( ( ord_less_eq @ A @ C2 @ A2 )
              & ( ord_less @ A @ B2 @ D2 ) ) ) ) ) ).

% atLeastAtMost_subseteq_atLeastLessThan_iff
thf(fact_5382_subst__not__in,axiom,
    ! [A: $tType,B: $tType,I: nat,I5: set @ nat,Xs2: list @ A,A3: A > B > assn,X1: A,Xsi: list @ B,X23: B] :
      ( ~ ( member @ nat @ I @ I5 )
     => ( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs2 ) )
       => ( ( vEBT_List_listI_assn @ A @ B @ I5 @ A3 @ ( list_update @ A @ Xs2 @ I @ X1 ) @ ( list_update @ B @ Xsi @ I @ X23 ) )
          = ( vEBT_List_listI_assn @ A @ B @ I5 @ A3 @ Xs2 @ Xsi ) ) ) ) ).

% subst_not_in
thf(fact_5383_sum__shift__lb__Suc0__0__upt,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add @ A )
     => ! [F: nat > A,K: nat] :
          ( ( ( F @ ( zero_zero @ nat ) )
            = ( zero_zero @ A ) )
         => ( ( groups7311177749621191930dd_sum @ nat @ A @ F @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ ( zero_zero @ nat ) ) @ K ) )
            = ( groups7311177749621191930dd_sum @ nat @ A @ F @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ K ) ) ) ) ) ).

% sum_shift_lb_Suc0_0_upt
thf(fact_5384_sum_OatLeast0__lessThan__Suc,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add @ A )
     => ! [G: nat > A,N2: nat] :
          ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( suc @ N2 ) ) )
          = ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N2 ) ) @ ( G @ N2 ) ) ) ) ).

% sum.atLeast0_lessThan_Suc
thf(fact_5385_sum_OatLeast__Suc__lessThan,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add @ A )
     => ! [M: nat,N2: nat,G: nat > A] :
          ( ( ord_less @ nat @ M @ N2 )
         => ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ N2 ) )
            = ( plus_plus @ A @ ( G @ M ) @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M ) @ N2 ) ) ) ) ) ) ).

% sum.atLeast_Suc_lessThan
thf(fact_5386_sum_OatLeastLessThan__Suc,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add @ A )
     => ! [A2: nat,B2: nat,G: nat > A] :
          ( ( ord_less_eq @ nat @ A2 @ B2 )
         => ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ A2 @ ( suc @ B2 ) ) )
            = ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ A2 @ B2 ) ) @ ( G @ B2 ) ) ) ) ) ).

% sum.atLeastLessThan_Suc
thf(fact_5387_atLeastLessThan__eq__atLeastAtMost__diff,axiom,
    ! [A: $tType] :
      ( ( order @ A )
     => ( ( set_or7035219750837199246ssThan @ A )
        = ( ^ [A5: A,B5: A] : ( minus_minus @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ A5 @ B5 ) @ ( insert @ A @ B5 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).

% atLeastLessThan_eq_atLeastAtMost_diff
thf(fact_5388_prod_OatLeast0__lessThan__Suc,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_mult @ A )
     => ! [G: nat > A,N2: nat] :
          ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( suc @ N2 ) ) )
          = ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N2 ) ) @ ( G @ N2 ) ) ) ) ).

% prod.atLeast0_lessThan_Suc
thf(fact_5389_prod_OatLeast__Suc__lessThan,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_mult @ A )
     => ! [M: nat,N2: nat,G: nat > A] :
          ( ( ord_less @ nat @ M @ N2 )
         => ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ N2 ) )
            = ( times_times @ A @ ( G @ M ) @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M ) @ N2 ) ) ) ) ) ) ).

% prod.atLeast_Suc_lessThan
thf(fact_5390_prod_OatLeastLessThan__Suc,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_mult @ A )
     => ! [A2: nat,B2: nat,G: nat > A] :
          ( ( ord_less_eq @ nat @ A2 @ B2 )
         => ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ A2 @ ( suc @ B2 ) ) )
            = ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ A2 @ B2 ) ) @ ( G @ B2 ) ) ) ) ) ).

% prod.atLeastLessThan_Suc
thf(fact_5391_sum_Olast__plus,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add @ A )
     => ! [M: nat,N2: nat,G: nat > A] :
          ( ( ord_less_eq @ nat @ M @ N2 )
         => ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) )
            = ( plus_plus @ A @ ( G @ N2 ) @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ N2 ) ) ) ) ) ) ).

% sum.last_plus
thf(fact_5392_prod_Olast__plus,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_mult @ A )
     => ! [M: nat,N2: nat,G: nat > A] :
          ( ( ord_less_eq @ nat @ M @ N2 )
         => ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) )
            = ( times_times @ A @ ( G @ N2 ) @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ N2 ) ) ) ) ) ) ).

% prod.last_plus
thf(fact_5393_atLeastLessThanSuc,axiom,
    ! [M: nat,N2: nat] :
      ( ( ( ord_less_eq @ nat @ M @ N2 )
       => ( ( set_or7035219750837199246ssThan @ nat @ M @ ( suc @ N2 ) )
          = ( insert @ nat @ N2 @ ( set_or7035219750837199246ssThan @ nat @ M @ N2 ) ) ) )
      & ( ~ ( ord_less_eq @ nat @ M @ N2 )
       => ( ( set_or7035219750837199246ssThan @ nat @ M @ ( suc @ N2 ) )
          = ( bot_bot @ ( set @ nat ) ) ) ) ) ).

% atLeastLessThanSuc
thf(fact_5394_sum__Suc__diff_H,axiom,
    ! [A: $tType] :
      ( ( ab_group_add @ A )
     => ! [M: nat,N2: nat,F: nat > A] :
          ( ( ord_less_eq @ nat @ M @ N2 )
         => ( ( groups7311177749621191930dd_sum @ nat @ A
              @ ^ [I2: nat] : ( minus_minus @ A @ ( F @ ( suc @ I2 ) ) @ ( F @ I2 ) )
              @ ( set_or7035219750837199246ssThan @ nat @ M @ N2 ) )
            = ( minus_minus @ A @ ( F @ N2 ) @ ( F @ M ) ) ) ) ) ).

% sum_Suc_diff'
thf(fact_5395_sum_OatLeastLessThan__rev,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add @ A )
     => ! [G: nat > A,N2: nat,M: nat] :
          ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ N2 @ M ) )
          = ( groups7311177749621191930dd_sum @ nat @ A
            @ ^ [I2: nat] : ( G @ ( minus_minus @ nat @ ( plus_plus @ nat @ M @ N2 ) @ ( suc @ I2 ) ) )
            @ ( set_or7035219750837199246ssThan @ nat @ N2 @ M ) ) ) ) ).

% sum.atLeastLessThan_rev
thf(fact_5396_sum_Onested__swap,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add @ A )
     => ! [A2: nat > nat > A,N2: nat] :
          ( ( groups7311177749621191930dd_sum @ nat @ A
            @ ^ [I2: nat] : ( groups7311177749621191930dd_sum @ nat @ A @ ( A2 @ I2 ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ I2 ) )
            @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N2 ) )
          = ( groups7311177749621191930dd_sum @ nat @ A
            @ ^ [J3: nat] :
                ( groups7311177749621191930dd_sum @ nat @ A
                @ ^ [I2: nat] : ( A2 @ I2 @ J3 )
                @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ J3 ) @ N2 ) )
            @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ) ).

% sum.nested_swap
thf(fact_5397_prod_OatLeastLessThan__rev,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_mult @ A )
     => ! [G: nat > A,N2: nat,M: nat] :
          ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ N2 @ M ) )
          = ( groups7121269368397514597t_prod @ nat @ A
            @ ^ [I2: nat] : ( G @ ( minus_minus @ nat @ ( plus_plus @ nat @ M @ N2 ) @ ( suc @ I2 ) ) )
            @ ( set_or7035219750837199246ssThan @ nat @ N2 @ M ) ) ) ) ).

% prod.atLeastLessThan_rev
thf(fact_5398_prod_Onested__swap,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_mult @ A )
     => ! [A2: nat > nat > A,N2: nat] :
          ( ( groups7121269368397514597t_prod @ nat @ A
            @ ^ [I2: nat] : ( groups7121269368397514597t_prod @ nat @ A @ ( A2 @ I2 ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ I2 ) )
            @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N2 ) )
          = ( groups7121269368397514597t_prod @ nat @ A
            @ ^ [J3: nat] :
                ( groups7121269368397514597t_prod @ nat @ A
                @ ^ [I2: nat] : ( A2 @ I2 @ J3 )
                @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ J3 ) @ N2 ) )
            @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ) ).

% prod.nested_swap
thf(fact_5399_sum_Onat__group,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add @ A )
     => ! [G: nat > A,K: nat,N2: nat] :
          ( ( groups7311177749621191930dd_sum @ nat @ A
            @ ^ [M7: nat] : ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( times_times @ nat @ M7 @ K ) @ ( plus_plus @ nat @ ( times_times @ nat @ M7 @ K ) @ K ) ) )
            @ ( set_ord_lessThan @ nat @ N2 ) )
          = ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_ord_lessThan @ nat @ ( times_times @ nat @ N2 @ K ) ) ) ) ) ).

% sum.nat_group
thf(fact_5400_prod_Onat__group,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_mult @ A )
     => ! [G: nat > A,K: nat,N2: nat] :
          ( ( groups7121269368397514597t_prod @ nat @ A
            @ ^ [M7: nat] : ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( times_times @ nat @ M7 @ K ) @ ( plus_plus @ nat @ ( times_times @ nat @ M7 @ K ) @ K ) ) )
            @ ( set_ord_lessThan @ nat @ N2 ) )
          = ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_ord_lessThan @ nat @ ( times_times @ nat @ N2 @ K ) ) ) ) ) ).

% prod.nat_group
thf(fact_5401_prod__Suc__Suc__fact,axiom,
    ! [N2: nat] :
      ( ( groups7121269368397514597t_prod @ nat @ nat @ suc @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N2 ) )
      = ( semiring_char_0_fact @ nat @ N2 ) ) ).

% prod_Suc_Suc_fact
thf(fact_5402_prod__Suc__fact,axiom,
    ! [N2: nat] :
      ( ( groups7121269368397514597t_prod @ nat @ nat @ suc @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N2 ) )
      = ( semiring_char_0_fact @ nat @ N2 ) ) ).

% prod_Suc_fact
thf(fact_5403_sum_Ohead__if,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add @ A )
     => ! [N2: nat,M: nat,G: nat > A] :
          ( ( ( ord_less @ nat @ N2 @ M )
           => ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) )
              = ( zero_zero @ A ) ) )
          & ( ~ ( ord_less @ nat @ N2 @ M )
           => ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) )
              = ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ N2 ) ) @ ( G @ N2 ) ) ) ) ) ) ).

% sum.head_if
thf(fact_5404_prod_Ohead__if,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_mult @ A )
     => ! [N2: nat,M: nat,G: nat > A] :
          ( ( ( ord_less @ nat @ N2 @ M )
           => ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) )
              = ( one_one @ A ) ) )
          & ( ~ ( ord_less @ nat @ N2 @ M )
           => ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) )
              = ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ N2 ) ) @ ( G @ N2 ) ) ) ) ) ) ).

% prod.head_if
thf(fact_5405_atLeastLessThan__nat__numeral,axiom,
    ! [M: nat,K: num] :
      ( ( ( ord_less_eq @ nat @ M @ ( pred_numeral @ K ) )
       => ( ( set_or7035219750837199246ssThan @ nat @ M @ ( numeral_numeral @ nat @ K ) )
          = ( insert @ nat @ ( pred_numeral @ K ) @ ( set_or7035219750837199246ssThan @ nat @ M @ ( pred_numeral @ K ) ) ) ) )
      & ( ~ ( ord_less_eq @ nat @ M @ ( pred_numeral @ K ) )
       => ( ( set_or7035219750837199246ssThan @ nat @ M @ ( numeral_numeral @ nat @ K ) )
          = ( bot_bot @ ( set @ nat ) ) ) ) ) ).

% atLeastLessThan_nat_numeral
thf(fact_5406_fact__prod__Suc,axiom,
    ! [A: $tType] :
      ( ( semiring_char_0 @ A )
     => ( ( semiring_char_0_fact @ A )
        = ( ^ [N: nat] : ( semiring_1_of_nat @ A @ ( groups7121269368397514597t_prod @ nat @ nat @ suc @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ) ) ).

% fact_prod_Suc
thf(fact_5407_sum_OatLeastLessThan__rev__at__least__Suc__atMost,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add @ A )
     => ! [G: nat > A,N2: nat,M: nat] :
          ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ N2 @ M ) )
          = ( groups7311177749621191930dd_sum @ nat @ A
            @ ^ [I2: nat] : ( G @ ( minus_minus @ nat @ ( plus_plus @ nat @ M @ N2 ) @ I2 ) )
            @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ N2 ) @ M ) ) ) ) ).

% sum.atLeastLessThan_rev_at_least_Suc_atMost
thf(fact_5408_prod_OatLeastLessThan__rev__at__least__Suc__atMost,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_mult @ A )
     => ! [G: nat > A,N2: nat,M: nat] :
          ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ N2 @ M ) )
          = ( groups7121269368397514597t_prod @ nat @ A
            @ ^ [I2: nat] : ( G @ ( minus_minus @ nat @ ( plus_plus @ nat @ M @ N2 ) @ I2 ) )
            @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ N2 ) @ M ) ) ) ) ).

% prod.atLeastLessThan_rev_at_least_Suc_atMost
thf(fact_5409_pochhammer__prod,axiom,
    ! [A: $tType] :
      ( ( comm_semiring_1 @ A )
     => ( ( comm_s3205402744901411588hammer @ A )
        = ( ^ [A5: A,N: nat] :
              ( groups7121269368397514597t_prod @ nat @ A
              @ ^ [I2: nat] : ( plus_plus @ A @ A5 @ ( semiring_1_of_nat @ A @ I2 ) )
              @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ) ).

% pochhammer_prod
thf(fact_5410_fact__prod__rev,axiom,
    ! [A: $tType] :
      ( ( semiring_char_0 @ A )
     => ( ( semiring_char_0_fact @ A )
        = ( ^ [N: nat] : ( semiring_1_of_nat @ A @ ( groups7121269368397514597t_prod @ nat @ nat @ ( minus_minus @ nat @ N ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ) ) ).

% fact_prod_rev
thf(fact_5411_summable__Cauchy,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach @ A )
     => ( ( summable @ A )
        = ( ^ [F6: nat > A] :
            ! [E4: real] :
              ( ( ord_less @ real @ ( zero_zero @ real ) @ E4 )
             => ? [N8: nat] :
                ! [M7: nat] :
                  ( ( ord_less_eq @ nat @ N8 @ M7 )
                 => ! [N: nat] : ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F6 @ ( set_or7035219750837199246ssThan @ nat @ M7 @ N ) ) ) @ E4 ) ) ) ) ) ) ).

% summable_Cauchy
thf(fact_5412_sums__group,axiom,
    ! [A: $tType] :
      ( ( ( comm_monoid_add @ A )
        & ( topolo4958980785337419405_space @ A ) )
     => ! [F: nat > A,S4: A,K: nat] :
          ( ( sums @ A @ F @ S4 )
         => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
           => ( sums @ A
              @ ^ [N: nat] : ( groups7311177749621191930dd_sum @ nat @ A @ F @ ( set_or7035219750837199246ssThan @ nat @ ( times_times @ nat @ N @ K ) @ ( plus_plus @ nat @ ( times_times @ nat @ N @ K ) @ K ) ) )
              @ S4 ) ) ) ) ).

% sums_group
thf(fact_5413_listI__assn__insert,axiom,
    ! [A: $tType,B: $tType,I: nat,I5: set @ nat,Xs2: list @ A,A3: A > B > assn,Xsi: list @ B] :
      ( ~ ( member @ nat @ I @ I5 )
     => ( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs2 ) )
       => ( ( vEBT_List_listI_assn @ A @ B @ ( insert @ nat @ I @ I5 ) @ A3 @ Xs2 @ Xsi )
          = ( times_times @ assn @ ( A3 @ ( nth @ A @ Xs2 @ I ) @ ( nth @ B @ Xsi @ I ) ) @ ( vEBT_List_listI_assn @ A @ B @ I5 @ A3 @ Xs2 @ Xsi ) ) ) ) ) ).

% listI_assn_insert
thf(fact_5414_atLeast1__lessThan__eq__remove0,axiom,
    ! [N2: nat] :
      ( ( set_or7035219750837199246ssThan @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N2 )
      = ( minus_minus @ ( set @ nat ) @ ( set_ord_lessThan @ nat @ N2 ) @ ( insert @ nat @ ( zero_zero @ nat ) @ ( bot_bot @ ( set @ nat ) ) ) ) ) ).

% atLeast1_lessThan_eq_remove0
thf(fact_5415_listI__assn__subst,axiom,
    ! [A: $tType,B: $tType,I: nat,I5: set @ nat,Xs2: list @ A,A3: A > B > assn,X1: A,Xsi: list @ B,X23: B] :
      ( ~ ( member @ nat @ I @ I5 )
     => ( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs2 ) )
       => ( ( vEBT_List_listI_assn @ A @ B @ ( insert @ nat @ I @ I5 ) @ A3 @ ( list_update @ A @ Xs2 @ I @ X1 ) @ ( list_update @ B @ Xsi @ I @ X23 ) )
          = ( times_times @ assn @ ( A3 @ X1 @ X23 ) @ ( vEBT_List_listI_assn @ A @ B @ I5 @ A3 @ Xs2 @ Xsi ) ) ) ) ) ).

% listI_assn_subst
thf(fact_5416_rule__at__index,axiom,
    ! [A: $tType,B: $tType,C: $tType,P: assn,A3: A > B > assn,Xs2: list @ A,Xsi: list @ B,F5: assn,I: nat,C2: heap_Time_Heap @ C,Q7: C > assn,F8: C > assn] :
      ( ( entails @ P @ ( times_times @ assn @ ( vEBT_List_list_assn @ A @ B @ A3 @ Xs2 @ Xsi ) @ F5 ) )
     => ( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs2 ) )
       => ( ( hoare_hoare_triple @ C @ ( times_times @ assn @ ( times_times @ assn @ ( A3 @ ( nth @ A @ Xs2 @ I ) @ ( nth @ B @ Xsi @ I ) ) @ ( vEBT_List_listI_assn @ A @ B @ ( minus_minus @ ( set @ nat ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ A ) @ Xs2 ) ) @ ( insert @ nat @ I @ ( bot_bot @ ( set @ nat ) ) ) ) @ A3 @ Xs2 @ Xsi ) ) @ F5 ) @ C2 @ Q7 )
         => ( ! [R3: C] : ( entails @ ( Q7 @ R3 ) @ ( times_times @ assn @ ( times_times @ assn @ ( A3 @ ( nth @ A @ Xs2 @ I ) @ ( nth @ B @ Xsi @ I ) ) @ ( vEBT_List_listI_assn @ A @ B @ ( minus_minus @ ( set @ nat ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ A ) @ Xs2 ) ) @ ( insert @ nat @ I @ ( bot_bot @ ( set @ nat ) ) ) ) @ A3 @ Xs2 @ Xsi ) ) @ ( F8 @ R3 ) ) )
           => ( hoare_hoare_triple @ C @ P @ C2
              @ ^ [R: C] : ( times_times @ assn @ ( vEBT_List_list_assn @ A @ B @ A3 @ Xs2 @ Xsi ) @ ( F8 @ R ) ) ) ) ) ) ) ).

% rule_at_index
thf(fact_5417_fact__split,axiom,
    ! [A: $tType] :
      ( ( semiring_char_0 @ A )
     => ! [K: nat,N2: nat] :
          ( ( ord_less_eq @ nat @ K @ N2 )
         => ( ( semiring_char_0_fact @ A @ N2 )
            = ( times_times @ A @ ( semiring_1_of_nat @ A @ ( groups7121269368397514597t_prod @ nat @ nat @ suc @ ( set_or7035219750837199246ssThan @ nat @ ( minus_minus @ nat @ N2 @ K ) @ N2 ) ) ) @ ( semiring_char_0_fact @ A @ ( minus_minus @ nat @ N2 @ K ) ) ) ) ) ) ).

% fact_split
thf(fact_5418_binomial__altdef__of__nat,axiom,
    ! [A: $tType] :
      ( ( field_char_0 @ A )
     => ! [K: nat,N2: nat] :
          ( ( ord_less_eq @ nat @ K @ N2 )
         => ( ( semiring_1_of_nat @ A @ ( binomial @ N2 @ K ) )
            = ( groups7121269368397514597t_prod @ nat @ A
              @ ^ [I2: nat] : ( divide_divide @ A @ ( semiring_1_of_nat @ A @ ( minus_minus @ nat @ N2 @ I2 ) ) @ ( semiring_1_of_nat @ A @ ( minus_minus @ nat @ K @ I2 ) ) )
              @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ K ) ) ) ) ) ).

% binomial_altdef_of_nat
thf(fact_5419_gbinomial__altdef__of__nat,axiom,
    ! [A: $tType] :
      ( ( field_char_0 @ A )
     => ( ( gbinomial @ A )
        = ( ^ [A5: A,K4: nat] :
              ( groups7121269368397514597t_prod @ nat @ A
              @ ^ [I2: nat] : ( divide_divide @ A @ ( minus_minus @ A @ A5 @ ( semiring_1_of_nat @ A @ I2 ) ) @ ( semiring_1_of_nat @ A @ ( minus_minus @ nat @ K4 @ I2 ) ) )
              @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ K4 ) ) ) ) ) ).

% gbinomial_altdef_of_nat
thf(fact_5420_gbinomial__mult__fact_H,axiom,
    ! [A: $tType] :
      ( ( field_char_0 @ A )
     => ! [A2: A,K: nat] :
          ( ( times_times @ A @ ( gbinomial @ A @ A2 @ K ) @ ( semiring_char_0_fact @ A @ K ) )
          = ( groups7121269368397514597t_prod @ nat @ A
            @ ^ [I2: nat] : ( minus_minus @ A @ A2 @ ( semiring_1_of_nat @ A @ I2 ) )
            @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ K ) ) ) ) ).

% gbinomial_mult_fact'
thf(fact_5421_gbinomial__mult__fact,axiom,
    ! [A: $tType] :
      ( ( field_char_0 @ A )
     => ! [K: nat,A2: A] :
          ( ( times_times @ A @ ( semiring_char_0_fact @ A @ K ) @ ( gbinomial @ A @ A2 @ K ) )
          = ( groups7121269368397514597t_prod @ nat @ A
            @ ^ [I2: nat] : ( minus_minus @ A @ A2 @ ( semiring_1_of_nat @ A @ I2 ) )
            @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ K ) ) ) ) ).

% gbinomial_mult_fact
thf(fact_5422_gbinomial__prod__rev,axiom,
    ! [A: $tType] :
      ( ( ( semiring_char_0 @ A )
        & ( semidom_divide @ A ) )
     => ( ( gbinomial @ A )
        = ( ^ [A5: A,K4: nat] :
              ( divide_divide @ A
              @ ( groups7121269368397514597t_prod @ nat @ A
                @ ^ [I2: nat] : ( minus_minus @ A @ A5 @ ( semiring_1_of_nat @ A @ I2 ) )
                @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ K4 ) )
              @ ( semiring_char_0_fact @ A @ K4 ) ) ) ) ) ).

% gbinomial_prod_rev
thf(fact_5423_sum__power2,axiom,
    ! [K: nat] :
      ( ( groups7311177749621191930dd_sum @ nat @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ K ) )
      = ( minus_minus @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ K ) @ ( one_one @ nat ) ) ) ).

% sum_power2
thf(fact_5424_Sum__Ico__nat,axiom,
    ! [M: nat,N2: nat] :
      ( ( groups7311177749621191930dd_sum @ nat @ nat
        @ ^ [X: nat] : X
        @ ( set_or7035219750837199246ssThan @ nat @ M @ N2 ) )
      = ( divide_divide @ nat @ ( minus_minus @ nat @ ( times_times @ nat @ N2 @ ( minus_minus @ nat @ N2 @ ( one_one @ nat ) ) ) @ ( times_times @ nat @ M @ ( minus_minus @ nat @ M @ ( one_one @ nat ) ) ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ).

% Sum_Ico_nat
thf(fact_5425_listI__assn__extract,axiom,
    ! [A: $tType,B: $tType,I: nat,I5: set @ nat,Xs2: list @ A,A3: A > B > assn,Xsi: list @ B] :
      ( ( member @ nat @ I @ I5 )
     => ( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs2 ) )
       => ( ( vEBT_List_listI_assn @ A @ B @ I5 @ A3 @ Xs2 @ Xsi )
          = ( times_times @ assn @ ( A3 @ ( nth @ A @ Xs2 @ I ) @ ( nth @ B @ Xsi @ I ) ) @ ( vEBT_List_listI_assn @ A @ B @ ( minus_minus @ ( set @ nat ) @ I5 @ ( insert @ nat @ I @ ( bot_bot @ ( set @ nat ) ) ) ) @ A3 @ Xs2 @ Xsi ) ) ) ) ) ).

% listI_assn_extract
thf(fact_5426_listI__assn__reinsert,axiom,
    ! [B: $tType,A: $tType,P: assn,A3: A > B > assn,Xs2: list @ A,I: nat,Xsi: list @ B,I5: set @ nat,F5: assn,Q: assn] :
      ( ( entails @ P @ ( times_times @ assn @ ( times_times @ assn @ ( A3 @ ( nth @ A @ Xs2 @ I ) @ ( nth @ B @ Xsi @ I ) ) @ ( vEBT_List_listI_assn @ A @ B @ ( minus_minus @ ( set @ nat ) @ I5 @ ( insert @ nat @ I @ ( bot_bot @ ( set @ nat ) ) ) ) @ A3 @ Xs2 @ Xsi ) ) @ F5 ) )
     => ( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs2 ) )
       => ( ( member @ nat @ I @ I5 )
         => ( ( entails @ ( times_times @ assn @ ( vEBT_List_listI_assn @ A @ B @ I5 @ A3 @ Xs2 @ Xsi ) @ F5 ) @ Q )
           => ( entails @ P @ Q ) ) ) ) ) ).

% listI_assn_reinsert
thf(fact_5427_listI__assn__reinsert__upd,axiom,
    ! [D: $tType,C: $tType,P: assn,A3: C > D > assn,X2: C,Xi2: D,I5: set @ nat,I: nat,Xs2: list @ C,Xsi: list @ D,F5: assn,Q: assn] :
      ( ( entails @ P @ ( times_times @ assn @ ( times_times @ assn @ ( A3 @ X2 @ Xi2 ) @ ( vEBT_List_listI_assn @ C @ D @ ( minus_minus @ ( set @ nat ) @ I5 @ ( insert @ nat @ I @ ( bot_bot @ ( set @ nat ) ) ) ) @ A3 @ Xs2 @ Xsi ) ) @ F5 ) )
     => ( ( ord_less @ nat @ I @ ( size_size @ ( list @ C ) @ Xs2 ) )
       => ( ( member @ nat @ I @ I5 )
         => ( ( entails @ ( times_times @ assn @ ( vEBT_List_listI_assn @ C @ D @ I5 @ A3 @ ( list_update @ C @ Xs2 @ I @ X2 ) @ ( list_update @ D @ Xsi @ I @ Xi2 ) ) @ F5 ) @ Q )
           => ( entails @ P @ Q ) ) ) ) ) ).

% listI_assn_reinsert_upd
thf(fact_5428_listI__assn__reinsert_H,axiom,
    ! [A: $tType,B: $tType,C: $tType,P: assn,A3: A > B > assn,Xs2: list @ A,I: nat,Xsi: list @ B,I5: set @ nat,F5: assn,C2: heap_Time_Heap @ C,Q: C > assn] :
      ( ( entails @ P @ ( times_times @ assn @ ( times_times @ assn @ ( A3 @ ( nth @ A @ Xs2 @ I ) @ ( nth @ B @ Xsi @ I ) ) @ ( vEBT_List_listI_assn @ A @ B @ ( minus_minus @ ( set @ nat ) @ I5 @ ( insert @ nat @ I @ ( bot_bot @ ( set @ nat ) ) ) ) @ A3 @ Xs2 @ Xsi ) ) @ F5 ) )
     => ( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs2 ) )
       => ( ( member @ nat @ I @ I5 )
         => ( ( hoare_hoare_triple @ C @ ( times_times @ assn @ ( vEBT_List_listI_assn @ A @ B @ I5 @ A3 @ Xs2 @ Xsi ) @ F5 ) @ C2 @ Q )
           => ( hoare_hoare_triple @ C @ P @ C2 @ Q ) ) ) ) ) ).

% listI_assn_reinsert'
thf(fact_5429_listI__assn__reinsert__upd_H,axiom,
    ! [C: $tType,D: $tType,E: $tType,P: assn,A3: C > D > assn,X2: C,Xi2: D,I5: set @ nat,I: nat,Xs2: list @ C,Xsi: list @ D,F5: assn,C2: heap_Time_Heap @ E,Q: E > assn] :
      ( ( entails @ P @ ( times_times @ assn @ ( times_times @ assn @ ( A3 @ X2 @ Xi2 ) @ ( vEBT_List_listI_assn @ C @ D @ ( minus_minus @ ( set @ nat ) @ I5 @ ( insert @ nat @ I @ ( bot_bot @ ( set @ nat ) ) ) ) @ A3 @ Xs2 @ Xsi ) ) @ F5 ) )
     => ( ( ord_less @ nat @ I @ ( size_size @ ( list @ C ) @ Xs2 ) )
       => ( ( member @ nat @ I @ I5 )
         => ( ( hoare_hoare_triple @ E @ ( times_times @ assn @ ( vEBT_List_listI_assn @ C @ D @ I5 @ A3 @ ( list_update @ C @ Xs2 @ I @ X2 ) @ ( list_update @ D @ Xsi @ I @ Xi2 ) ) @ F5 ) @ C2 @ Q )
           => ( hoare_hoare_triple @ E @ P @ C2 @ Q ) ) ) ) ) ).

% listI_assn_reinsert_upd'
thf(fact_5430_Chebyshev__sum__upper,axiom,
    ! [A: $tType] :
      ( ( linordered_idom @ A )
     => ! [N2: nat,A2: nat > A,B2: nat > A] :
          ( ! [I3: nat,J2: nat] :
              ( ( ord_less_eq @ nat @ I3 @ J2 )
             => ( ( ord_less @ nat @ J2 @ N2 )
               => ( ord_less_eq @ A @ ( A2 @ I3 ) @ ( A2 @ J2 ) ) ) )
         => ( ! [I3: nat,J2: nat] :
                ( ( ord_less_eq @ nat @ I3 @ J2 )
               => ( ( ord_less @ nat @ J2 @ N2 )
                 => ( ord_less_eq @ A @ ( B2 @ J2 ) @ ( B2 @ I3 ) ) ) )
           => ( ord_less_eq @ A
              @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N2 )
                @ ( groups7311177749621191930dd_sum @ nat @ A
                  @ ^ [K4: nat] : ( times_times @ A @ ( A2 @ K4 ) @ ( B2 @ K4 ) )
                  @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N2 ) ) )
              @ ( times_times @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ A2 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N2 ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ B2 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ) ) ) ) ).

% Chebyshev_sum_upper
thf(fact_5431_Chebyshev__sum__upper__nat,axiom,
    ! [N2: nat,A2: nat > nat,B2: nat > nat] :
      ( ! [I3: nat,J2: nat] :
          ( ( ord_less_eq @ nat @ I3 @ J2 )
         => ( ( ord_less @ nat @ J2 @ N2 )
           => ( ord_less_eq @ nat @ ( A2 @ I3 ) @ ( A2 @ J2 ) ) ) )
     => ( ! [I3: nat,J2: nat] :
            ( ( ord_less_eq @ nat @ I3 @ J2 )
           => ( ( ord_less @ nat @ J2 @ N2 )
             => ( ord_less_eq @ nat @ ( B2 @ J2 ) @ ( B2 @ I3 ) ) ) )
       => ( ord_less_eq @ nat
          @ ( times_times @ nat @ N2
            @ ( groups7311177749621191930dd_sum @ nat @ nat
              @ ^ [I2: nat] : ( times_times @ nat @ ( A2 @ I2 ) @ ( B2 @ I2 ) )
              @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N2 ) ) )
          @ ( times_times @ nat @ ( groups7311177749621191930dd_sum @ nat @ nat @ A2 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N2 ) ) @ ( groups7311177749621191930dd_sum @ nat @ nat @ B2 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ) ) ) ).

% Chebyshev_sum_upper_nat
thf(fact_5432_option_Osize__gen_I2_J,axiom,
    ! [A: $tType,X2: A > nat,X23: A] :
      ( ( size_option @ A @ X2 @ ( some @ A @ X23 ) )
      = ( plus_plus @ nat @ ( X2 @ X23 ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ).

% option.size_gen(2)
thf(fact_5433_finite__atLeastZeroLessThan__integer,axiom,
    ! [U: code_integer] : ( finite_finite2 @ code_integer @ ( set_or7035219750837199246ssThan @ code_integer @ ( zero_zero @ code_integer ) @ U ) ) ).

% finite_atLeastZeroLessThan_integer
thf(fact_5434_word__to__1__set,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ( ( set_or7035219750837199246ssThan @ ( word @ A ) @ ( zero_zero @ ( word @ A ) ) @ ( one_one @ ( word @ A ) ) )
        = ( insert @ ( word @ A ) @ ( zero_zero @ ( word @ A ) ) @ ( bot_bot @ ( set @ ( word @ A ) ) ) ) ) ) ).

% word_to_1_set
thf(fact_5435_atLeastLessThanPlusOne__atLeastAtMost__integer,axiom,
    ! [L: code_integer,U: code_integer] :
      ( ( set_or7035219750837199246ssThan @ code_integer @ L @ ( plus_plus @ code_integer @ U @ ( one_one @ code_integer ) ) )
      = ( set_or1337092689740270186AtMost @ code_integer @ L @ U ) ) ).

% atLeastLessThanPlusOne_atLeastAtMost_integer
thf(fact_5436_atLeastLessThanPlusOne__atLeastAtMost__int,axiom,
    ! [L: int,U: int] :
      ( ( set_or7035219750837199246ssThan @ int @ L @ ( plus_plus @ int @ U @ ( one_one @ int ) ) )
      = ( set_or1337092689740270186AtMost @ int @ L @ U ) ) ).

% atLeastLessThanPlusOne_atLeastAtMost_int
thf(fact_5437_word__atLeastLessThan__Suc__atLeastAtMost,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [U: word @ A,L: word @ A] :
          ( ( U
           != ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) )
         => ( ( set_or7035219750837199246ssThan @ ( word @ A ) @ L @ ( plus_plus @ ( word @ A ) @ U @ ( one_one @ ( word @ A ) ) ) )
            = ( set_or1337092689740270186AtMost @ ( word @ A ) @ L @ U ) ) ) ) ).

% word_atLeastLessThan_Suc_atLeastAtMost
thf(fact_5438_word__range__minus__1,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [B2: word @ A,A2: word @ A] :
          ( ( B2
           != ( zero_zero @ ( word @ A ) ) )
         => ( ( set_or1337092689740270186AtMost @ ( word @ A ) @ A2 @ ( minus_minus @ ( word @ A ) @ B2 @ ( one_one @ ( word @ A ) ) ) )
            = ( set_or7035219750837199246ssThan @ ( word @ A ) @ A2 @ B2 ) ) ) ) ).

% word_range_minus_1
thf(fact_5439_option_Osize__gen_I1_J,axiom,
    ! [A: $tType,X2: A > nat] :
      ( ( size_option @ A @ X2 @ ( none @ A ) )
      = ( suc @ ( zero_zero @ nat ) ) ) ).

% option.size_gen(1)
thf(fact_5440_set__decode__0,axiom,
    ! [X2: nat] :
      ( ( member @ nat @ ( zero_zero @ nat ) @ ( nat_set_decode @ X2 ) )
      = ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ X2 ) ) ) ).

% set_decode_0
thf(fact_5441_flip__bit__0,axiom,
    ! [A: $tType] :
      ( ( bit_se359711467146920520ations @ A )
     => ! [A2: A] :
          ( ( bit_se8732182000553998342ip_bit @ A @ ( zero_zero @ nat ) @ A2 )
          = ( plus_plus @ A @ ( zero_neq_one_of_bool @ A @ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ).

% flip_bit_0
thf(fact_5442_forall__finite_I3_J,axiom,
    ! [X2: nat,P: nat > $o] :
      ( ( ! [I2: nat] :
            ( ( ord_less @ nat @ I2 @ ( suc @ ( suc @ X2 ) ) )
           => ( P @ I2 ) ) )
      = ( ( P @ ( zero_zero @ nat ) )
        & ! [I2: nat] :
            ( ( ord_less @ nat @ I2 @ ( suc @ X2 ) )
           => ( P @ ( suc @ I2 ) ) ) ) ) ).

% forall_finite(3)
thf(fact_5443_of__bool__eq_I1_J,axiom,
    ! [A: $tType] :
      ( ( zero_neq_one @ A )
     => ( ( zero_neq_one_of_bool @ A @ $false )
        = ( zero_zero @ A ) ) ) ).

% of_bool_eq(1)
thf(fact_5444_of__bool__eq__0__iff,axiom,
    ! [A: $tType] :
      ( ( zero_neq_one @ A )
     => ! [P: $o] :
          ( ( ( zero_neq_one_of_bool @ A @ P )
            = ( zero_zero @ A ) )
          = ~ P ) ) ).

% of_bool_eq_0_iff
thf(fact_5445_of__bool__less__eq__iff,axiom,
    ! [A: $tType] :
      ( ( linord181362715937106298miring @ A )
     => ! [P: $o,Q: $o] :
          ( ( ord_less_eq @ A @ ( zero_neq_one_of_bool @ A @ P ) @ ( zero_neq_one_of_bool @ A @ Q ) )
          = ( P
           => Q ) ) ) ).

% of_bool_less_eq_iff
thf(fact_5446_of__bool__less__iff,axiom,
    ! [A: $tType] :
      ( ( linord181362715937106298miring @ A )
     => ! [P: $o,Q: $o] :
          ( ( ord_less @ A @ ( zero_neq_one_of_bool @ A @ P ) @ ( zero_neq_one_of_bool @ A @ Q ) )
          = ( ~ P
            & Q ) ) ) ).

% of_bool_less_iff
thf(fact_5447_of__bool__eq_I2_J,axiom,
    ! [A: $tType] :
      ( ( zero_neq_one @ A )
     => ( ( zero_neq_one_of_bool @ A @ $true )
        = ( one_one @ A ) ) ) ).

% of_bool_eq(2)
thf(fact_5448_of__bool__eq__1__iff,axiom,
    ! [A: $tType] :
      ( ( zero_neq_one @ A )
     => ! [P: $o] :
          ( ( ( zero_neq_one_of_bool @ A @ P )
            = ( one_one @ A ) )
          = P ) ) ).

% of_bool_eq_1_iff
thf(fact_5449_of__nat__of__bool,axiom,
    ! [A: $tType] :
      ( ( semiring_1 @ A )
     => ! [P: $o] :
          ( ( semiring_1_of_nat @ A @ ( zero_neq_one_of_bool @ nat @ P ) )
          = ( zero_neq_one_of_bool @ A @ P ) ) ) ).

% of_nat_of_bool
thf(fact_5450_abs__bool__eq,axiom,
    ! [A: $tType] :
      ( ( linordered_idom @ A )
     => ! [P: $o] :
          ( ( abs_abs @ A @ ( zero_neq_one_of_bool @ A @ P ) )
          = ( zero_neq_one_of_bool @ A @ P ) ) ) ).

% abs_bool_eq
thf(fact_5451_of__bool__or__iff,axiom,
    ! [A: $tType] :
      ( ( linordered_semidom @ A )
     => ! [P: $o,Q: $o] :
          ( ( zero_neq_one_of_bool @ A
            @ ( P
              | Q ) )
          = ( ord_max @ A @ ( zero_neq_one_of_bool @ A @ P ) @ ( zero_neq_one_of_bool @ A @ Q ) ) ) ) ).

% of_bool_or_iff
thf(fact_5452_div__word__one,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [W: word @ A] :
          ( ( divide_divide @ ( word @ A ) @ ( one_one @ ( word @ A ) ) @ W )
          = ( zero_neq_one_of_bool @ ( word @ A )
            @ ( W
              = ( one_one @ ( word @ A ) ) ) ) ) ) ).

% div_word_one
thf(fact_5453_zero__less__of__bool__iff,axiom,
    ! [A: $tType] :
      ( ( linordered_semidom @ A )
     => ! [P: $o] :
          ( ( ord_less @ A @ ( zero_zero @ A ) @ ( zero_neq_one_of_bool @ A @ P ) )
          = P ) ) ).

% zero_less_of_bool_iff
thf(fact_5454_of__bool__less__one__iff,axiom,
    ! [A: $tType] :
      ( ( linordered_semidom @ A )
     => ! [P: $o] :
          ( ( ord_less @ A @ ( zero_neq_one_of_bool @ A @ P ) @ ( one_one @ A ) )
          = ~ P ) ) ).

% of_bool_less_one_iff
thf(fact_5455_Suc__0__mod__eq,axiom,
    ! [N2: nat] :
      ( ( modulo_modulo @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N2 )
      = ( zero_neq_one_of_bool @ nat
        @ ( N2
         != ( suc @ ( zero_zero @ nat ) ) ) ) ) ).

% Suc_0_mod_eq
thf(fact_5456_of__bool__not__iff,axiom,
    ! [A: $tType] :
      ( ( ring_1 @ A )
     => ! [P: $o] :
          ( ( zero_neq_one_of_bool @ A @ ~ P )
          = ( minus_minus @ A @ ( one_one @ A ) @ ( zero_neq_one_of_bool @ A @ P ) ) ) ) ).

% of_bool_not_iff
thf(fact_5457_finite__atLeastLessThan__integer,axiom,
    ! [L: code_integer,U: code_integer] : ( finite_finite2 @ code_integer @ ( set_or7035219750837199246ssThan @ code_integer @ L @ U ) ) ).

% finite_atLeastLessThan_integer
thf(fact_5458_div__word__by__minus__1__eq,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [W: word @ A] :
          ( ( divide_divide @ ( word @ A ) @ W @ ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) )
          = ( zero_neq_one_of_bool @ ( word @ A )
            @ ( W
              = ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) ) ) ) ) ).

% div_word_by_minus_1_eq
thf(fact_5459_finite__atLeastAtMost__integer,axiom,
    ! [L: code_integer,U: code_integer] : ( finite_finite2 @ code_integer @ ( set_or1337092689740270186AtMost @ code_integer @ L @ U ) ) ).

% finite_atLeastAtMost_integer
thf(fact_5460_mod__word__one,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [W: word @ A] :
          ( ( modulo_modulo @ ( word @ A ) @ ( one_one @ ( word @ A ) ) @ W )
          = ( minus_minus @ ( word @ A ) @ ( one_one @ ( word @ A ) )
            @ ( times_times @ ( word @ A ) @ W
              @ ( zero_neq_one_of_bool @ ( word @ A )
                @ ( W
                  = ( one_one @ ( word @ A ) ) ) ) ) ) ) ) ).

% mod_word_one
thf(fact_5461_Divides_Oadjust__div__eq,axiom,
    ! [Q2: int,R4: int] :
      ( ( adjust_div @ ( product_Pair @ int @ int @ Q2 @ R4 ) )
      = ( plus_plus @ int @ Q2
        @ ( zero_neq_one_of_bool @ int
          @ ( R4
           != ( zero_zero @ int ) ) ) ) ) ).

% Divides.adjust_div_eq
thf(fact_5462_odd__of__bool__self,axiom,
    ! [A: $tType] :
      ( ( semiring_parity @ A )
     => ! [P4: $o] :
          ( ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( zero_neq_one_of_bool @ A @ P4 ) ) )
          = P4 ) ) ).

% odd_of_bool_self
thf(fact_5463_mod__word__by__minus__1__eq,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [W: word @ A] :
          ( ( modulo_modulo @ ( word @ A ) @ W @ ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) )
          = ( times_times @ ( word @ A ) @ W @ ( zero_neq_one_of_bool @ ( word @ A ) @ ( ord_less @ ( word @ A ) @ W @ ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) ) ) ) ) ) ).

% mod_word_by_minus_1_eq
thf(fact_5464_of__bool__half__eq__0,axiom,
    ! [A: $tType] :
      ( ( euclid5411537665997757685th_nat @ A )
     => ! [B2: $o] :
          ( ( divide_divide @ A @ ( zero_neq_one_of_bool @ A @ B2 ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
          = ( zero_zero @ A ) ) ) ).

% of_bool_half_eq_0
thf(fact_5465_set__decode__Suc,axiom,
    ! [N2: nat,X2: nat] :
      ( ( member @ nat @ ( suc @ N2 ) @ ( nat_set_decode @ X2 ) )
      = ( member @ nat @ N2 @ ( nat_set_decode @ ( divide_divide @ nat @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).

% set_decode_Suc
thf(fact_5466_bits__1__div__exp,axiom,
    ! [A: $tType] :
      ( ( bit_semiring_bits @ A )
     => ! [N2: nat] :
          ( ( divide_divide @ A @ ( one_one @ A ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) )
          = ( zero_neq_one_of_bool @ A
            @ ( N2
              = ( zero_zero @ nat ) ) ) ) ) ).

% bits_1_div_exp
thf(fact_5467_one__div__2__pow__eq,axiom,
    ! [A: $tType] :
      ( ( euclid5411537665997757685th_nat @ A )
     => ! [N2: nat] :
          ( ( divide_divide @ A @ ( one_one @ A ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) )
          = ( zero_neq_one_of_bool @ A
            @ ( N2
              = ( zero_zero @ nat ) ) ) ) ) ).

% one_div_2_pow_eq
thf(fact_5468_one__mod__2__pow__eq,axiom,
    ! [A: $tType] :
      ( ( euclid5411537665997757685th_nat @ A )
     => ! [N2: nat] :
          ( ( modulo_modulo @ A @ ( one_one @ A ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) )
          = ( zero_neq_one_of_bool @ A @ ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ) ).

% one_mod_2_pow_eq
thf(fact_5469_zero__less__eq__of__bool,axiom,
    ! [A: $tType] :
      ( ( linordered_semidom @ A )
     => ! [P: $o] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( zero_neq_one_of_bool @ A @ P ) ) ) ).

% zero_less_eq_of_bool
thf(fact_5470_of__bool__def,axiom,
    ! [A: $tType] :
      ( ( zero_neq_one @ A )
     => ( ( zero_neq_one_of_bool @ A )
        = ( ^ [P5: $o] : ( if @ A @ P5 @ ( one_one @ A ) @ ( zero_zero @ A ) ) ) ) ) ).

% of_bool_def
thf(fact_5471_split__of__bool,axiom,
    ! [A: $tType] :
      ( ( zero_neq_one @ A )
     => ! [P: A > $o,P4: $o] :
          ( ( P @ ( zero_neq_one_of_bool @ A @ P4 ) )
          = ( ( P4
             => ( P @ ( one_one @ A ) ) )
            & ( ~ P4
             => ( P @ ( zero_zero @ A ) ) ) ) ) ) ).

% split_of_bool
thf(fact_5472_split__of__bool__asm,axiom,
    ! [A: $tType] :
      ( ( zero_neq_one @ A )
     => ! [P: A > $o,P4: $o] :
          ( ( P @ ( zero_neq_one_of_bool @ A @ P4 ) )
          = ( ~ ( ( P4
                  & ~ ( P @ ( one_one @ A ) ) )
                | ( ~ P4
                  & ~ ( P @ ( zero_zero @ A ) ) ) ) ) ) ) ).

% split_of_bool_asm
thf(fact_5473_of__bool__less__eq__one,axiom,
    ! [A: $tType] :
      ( ( linordered_semidom @ A )
     => ! [P: $o] : ( ord_less_eq @ A @ ( zero_neq_one_of_bool @ A @ P ) @ ( one_one @ A ) ) ) ).

% of_bool_less_eq_one
thf(fact_5474_of__bool__eq__iff,axiom,
    ! [A: $tType] :
      ( ( zero_neq_one @ A )
     => ! [P4: $o,Q2: $o] :
          ( ( ( zero_neq_one_of_bool @ A @ P4 )
            = ( zero_neq_one_of_bool @ A @ Q2 ) )
          = ( P4 = Q2 ) ) ) ).

% of_bool_eq_iff
thf(fact_5475_of__bool__conj,axiom,
    ! [A: $tType] :
      ( ( semiring_1 @ A )
     => ! [P: $o,Q: $o] :
          ( ( zero_neq_one_of_bool @ A
            @ ( P
              & Q ) )
          = ( times_times @ A @ ( zero_neq_one_of_bool @ A @ P ) @ ( zero_neq_one_of_bool @ A @ Q ) ) ) ) ).

% of_bool_conj
thf(fact_5476_Divides_Oadjust__div__def,axiom,
    ( adjust_div
    = ( product_case_prod @ int @ int @ int
      @ ^ [Q5: int,R: int] :
          ( plus_plus @ int @ Q5
          @ ( zero_neq_one_of_bool @ int
            @ ( R
             != ( zero_zero @ int ) ) ) ) ) ) ).

% Divides.adjust_div_def
thf(fact_5477_subset__decode__imp__le,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq @ ( set @ nat ) @ ( nat_set_decode @ M ) @ ( nat_set_decode @ N2 ) )
     => ( ord_less_eq @ nat @ M @ N2 ) ) ).

% subset_decode_imp_le
thf(fact_5478_of__bool__odd__eq__mod__2,axiom,
    ! [A: $tType] :
      ( ( semiring_parity @ A )
     => ! [A2: A] :
          ( ( zero_neq_one_of_bool @ A
            @ ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) )
          = ( modulo_modulo @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).

% of_bool_odd_eq_mod_2
thf(fact_5479_bits__induct,axiom,
    ! [A: $tType] :
      ( ( bit_semiring_bits @ A )
     => ! [P: A > $o,A2: A] :
          ( ! [A4: A] :
              ( ( ( divide_divide @ A @ A4 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
                = A4 )
             => ( P @ A4 ) )
         => ( ! [A4: A,B3: $o] :
                ( ( P @ A4 )
               => ( ( ( divide_divide @ A @ ( plus_plus @ A @ ( zero_neq_one_of_bool @ A @ B3 ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A4 ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
                    = A4 )
                 => ( P @ ( plus_plus @ A @ ( zero_neq_one_of_bool @ A @ B3 ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A4 ) ) ) ) )
           => ( P @ A2 ) ) ) ) ).

% bits_induct
thf(fact_5480_exp__mod__exp,axiom,
    ! [A: $tType] :
      ( ( euclid5411537665997757685th_nat @ A )
     => ! [M: nat,N2: nat] :
          ( ( modulo_modulo @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) )
          = ( times_times @ A @ ( zero_neq_one_of_bool @ A @ ( ord_less @ nat @ M @ N2 ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) ) ) ) ).

% exp_mod_exp
thf(fact_5481_forall__finite_I1_J,axiom,
    ! [P: nat > $o,I4: nat] :
      ( ( ord_less @ nat @ I4 @ ( zero_zero @ nat ) )
     => ( P @ I4 ) ) ).

% forall_finite(1)
thf(fact_5482_exp__div__exp__eq,axiom,
    ! [A: $tType] :
      ( ( bit_semiring_bits @ A )
     => ! [M: nat,N2: nat] :
          ( ( divide_divide @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) )
          = ( times_times @ A
            @ ( zero_neq_one_of_bool @ A
              @ ( ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M )
                 != ( zero_zero @ A ) )
                & ( ord_less_eq @ nat @ N2 @ M ) ) )
            @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ M @ N2 ) ) ) ) ) ).

% exp_div_exp_eq
thf(fact_5483_set__decode__plus__power__2,axiom,
    ! [N2: nat,Z: nat] :
      ( ~ ( member @ nat @ N2 @ ( nat_set_decode @ Z ) )
     => ( ( nat_set_decode @ ( plus_plus @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) @ Z ) )
        = ( insert @ nat @ N2 @ ( nat_set_decode @ Z ) ) ) ) ).

% set_decode_plus_power_2
thf(fact_5484_set__decode__def,axiom,
    ( nat_set_decode
    = ( ^ [X: nat] :
          ( collect @ nat
          @ ^ [N: nat] :
              ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) ) ) ) ).

% set_decode_def
thf(fact_5485_Comparator__Generator_OAll__less__Suc,axiom,
    ! [X2: nat,P: nat > $o] :
      ( ( ! [I2: nat] :
            ( ( ord_less @ nat @ I2 @ ( suc @ X2 ) )
           => ( P @ I2 ) ) )
      = ( ( P @ ( zero_zero @ nat ) )
        & ! [I2: nat] :
            ( ( ord_less @ nat @ I2 @ X2 )
           => ( P @ ( suc @ I2 ) ) ) ) ) ).

% Comparator_Generator.All_less_Suc
thf(fact_5486_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_5487_and__int_Opsimps,axiom,
    ! [K: int,L: int] :
      ( ( accp @ ( product_prod @ int @ 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_se5824344872417868541ns_and @ int @ K @ L )
            = ( uminus_uminus @ int
              @ ( zero_neq_one_of_bool @ int
                @ ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K )
                  & ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ 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_se5824344872417868541ns_and @ int @ K @ L )
            = ( plus_plus @ int
              @ ( zero_neq_one_of_bool @ int
                @ ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K )
                  & ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ L ) ) )
              @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ int @ ( divide_divide @ int @ K @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ L @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ).

% and_int.psimps
thf(fact_5488_and__int_Opelims,axiom,
    ! [X2: int,Xa: int,Y2: int] :
      ( ( ( bit_se5824344872417868541ns_and @ int @ X2 @ Xa )
        = Y2 )
     => ( ( accp @ ( product_prod @ int @ int ) @ bit_and_int_rel @ ( product_Pair @ int @ int @ X2 @ Xa ) )
       => ~ ( ( ( ( ( member @ int @ X2 @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) )
                  & ( member @ int @ Xa @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) ) )
               => ( Y2
                  = ( uminus_uminus @ int
                    @ ( zero_neq_one_of_bool @ int
                      @ ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ X2 )
                        & ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ Xa ) ) ) ) ) )
              & ( ~ ( ( member @ int @ X2 @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) )
                    & ( member @ int @ Xa @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) ) )
               => ( Y2
                  = ( plus_plus @ int
                    @ ( zero_neq_one_of_bool @ int
                      @ ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ X2 )
                        & ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ Xa ) ) )
                    @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ int @ ( divide_divide @ int @ X2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ Xa @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) )
           => ~ ( accp @ ( product_prod @ int @ int ) @ bit_and_int_rel @ ( product_Pair @ int @ int @ X2 @ Xa ) ) ) ) ) ).

% and_int.pelims
thf(fact_5489_and__int_Osimps,axiom,
    ( ( bit_se5824344872417868541ns_and @ int )
    = ( ^ [K4: int,L2: int] :
          ( if @ int
          @ ( ( member @ int @ K4 @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) )
            & ( member @ int @ L2 @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) ) )
          @ ( uminus_uminus @ int
            @ ( zero_neq_one_of_bool @ int
              @ ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K4 )
                & ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ L2 ) ) ) )
          @ ( plus_plus @ int
            @ ( zero_neq_one_of_bool @ int
              @ ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K4 )
                & ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ L2 ) ) )
            @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ int @ ( divide_divide @ int @ K4 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ L2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ).

% and_int.simps
thf(fact_5490_bit_Oconj__one__right,axiom,
    ! [A: $tType] :
      ( ( bit_ri3973907225187159222ations @ A )
     => ! [X2: A] :
          ( ( bit_se5824344872417868541ns_and @ A @ X2 @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
          = X2 ) ) ).

% bit.conj_one_right
thf(fact_5491_and_Oright__neutral,axiom,
    ! [A: $tType] :
      ( ( bit_ri3973907225187159222ations @ A )
     => ! [A2: A] :
          ( ( bit_se5824344872417868541ns_and @ A @ A2 @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
          = A2 ) ) ).

% and.right_neutral
thf(fact_5492_and_Oleft__neutral,axiom,
    ! [A: $tType] :
      ( ( bit_ri3973907225187159222ations @ A )
     => ! [A2: A] :
          ( ( bit_se5824344872417868541ns_and @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ A2 )
          = A2 ) ) ).

% and.left_neutral
thf(fact_5493_and__nonnegative__int__iff,axiom,
    ! [K: int,L: int] :
      ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_se5824344872417868541ns_and @ 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_5494_and__numerals_I8_J,axiom,
    ! [A: $tType] :
      ( ( bit_un5681908812861735899ations @ A )
     => ! [X2: num] :
          ( ( bit_se5824344872417868541ns_and @ A @ ( numeral_numeral @ A @ ( bit1 @ X2 ) ) @ ( one_one @ A ) )
          = ( one_one @ A ) ) ) ).

% and_numerals(8)
thf(fact_5495_and__numerals_I2_J,axiom,
    ! [A: $tType] :
      ( ( bit_un5681908812861735899ations @ A )
     => ! [Y2: num] :
          ( ( bit_se5824344872417868541ns_and @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit1 @ Y2 ) ) )
          = ( one_one @ A ) ) ) ).

% and_numerals(2)
thf(fact_5496_and__numerals_I5_J,axiom,
    ! [A: $tType] :
      ( ( bit_un5681908812861735899ations @ A )
     => ! [X2: num] :
          ( ( bit_se5824344872417868541ns_and @ A @ ( numeral_numeral @ A @ ( bit0 @ X2 ) ) @ ( one_one @ A ) )
          = ( zero_zero @ A ) ) ) ).

% and_numerals(5)
thf(fact_5497_and__numerals_I1_J,axiom,
    ! [A: $tType] :
      ( ( bit_un5681908812861735899ations @ A )
     => ! [Y2: num] :
          ( ( bit_se5824344872417868541ns_and @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ Y2 ) ) )
          = ( zero_zero @ A ) ) ) ).

% and_numerals(1)
thf(fact_5498_and__numerals_I3_J,axiom,
    ! [A: $tType] :
      ( ( bit_un5681908812861735899ations @ A )
     => ! [X2: num,Y2: num] :
          ( ( bit_se5824344872417868541ns_and @ A @ ( numeral_numeral @ A @ ( bit0 @ X2 ) ) @ ( numeral_numeral @ A @ ( bit0 @ Y2 ) ) )
          = ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ A @ ( numeral_numeral @ A @ X2 ) @ ( numeral_numeral @ A @ Y2 ) ) ) ) ) ).

% and_numerals(3)
thf(fact_5499_and__minus__numerals_I6_J,axiom,
    ! [N2: num] :
      ( ( bit_se5824344872417868541ns_and @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ N2 ) ) ) @ ( one_one @ int ) )
      = ( one_one @ int ) ) ).

% and_minus_numerals(6)
thf(fact_5500_and__minus__numerals_I2_J,axiom,
    ! [N2: num] :
      ( ( bit_se5824344872417868541ns_and @ int @ ( one_one @ int ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ N2 ) ) ) )
      = ( one_one @ int ) ) ).

% and_minus_numerals(2)
thf(fact_5501_and__numerals_I6_J,axiom,
    ! [A: $tType] :
      ( ( bit_un5681908812861735899ations @ A )
     => ! [X2: num,Y2: num] :
          ( ( bit_se5824344872417868541ns_and @ A @ ( numeral_numeral @ A @ ( bit1 @ X2 ) ) @ ( numeral_numeral @ A @ ( bit0 @ Y2 ) ) )
          = ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ A @ ( numeral_numeral @ A @ X2 ) @ ( numeral_numeral @ A @ Y2 ) ) ) ) ) ).

% and_numerals(6)
thf(fact_5502_and__numerals_I4_J,axiom,
    ! [A: $tType] :
      ( ( bit_un5681908812861735899ations @ A )
     => ! [X2: num,Y2: num] :
          ( ( bit_se5824344872417868541ns_and @ A @ ( numeral_numeral @ A @ ( bit0 @ X2 ) ) @ ( numeral_numeral @ A @ ( bit1 @ Y2 ) ) )
          = ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ A @ ( numeral_numeral @ A @ X2 ) @ ( numeral_numeral @ A @ Y2 ) ) ) ) ) ).

% and_numerals(4)
thf(fact_5503_and__minus__numerals_I1_J,axiom,
    ! [N2: num] :
      ( ( bit_se5824344872417868541ns_and @ int @ ( one_one @ int ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ N2 ) ) ) )
      = ( zero_zero @ int ) ) ).

% and_minus_numerals(1)
thf(fact_5504_and__minus__numerals_I5_J,axiom,
    ! [N2: num] :
      ( ( bit_se5824344872417868541ns_and @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ N2 ) ) ) @ ( one_one @ int ) )
      = ( zero_zero @ int ) ) ).

% and_minus_numerals(5)
thf(fact_5505_and__numerals_I7_J,axiom,
    ! [A: $tType] :
      ( ( bit_un5681908812861735899ations @ A )
     => ! [X2: num,Y2: num] :
          ( ( bit_se5824344872417868541ns_and @ A @ ( numeral_numeral @ A @ ( bit1 @ X2 ) ) @ ( numeral_numeral @ A @ ( bit1 @ Y2 ) ) )
          = ( plus_plus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ A @ ( numeral_numeral @ A @ X2 ) @ ( numeral_numeral @ A @ Y2 ) ) ) ) ) ) ).

% and_numerals(7)
thf(fact_5506_and__eq__minus__1__iff,axiom,
    ! [A: $tType] :
      ( ( bit_ri3973907225187159222ations @ A )
     => ! [A2: A,B2: A] :
          ( ( ( bit_se5824344872417868541ns_and @ A @ A2 @ B2 )
            = ( uminus_uminus @ A @ ( one_one @ A ) ) )
          = ( ( A2
              = ( uminus_uminus @ A @ ( one_one @ A ) ) )
            & ( B2
              = ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ) ) ).

% and_eq_minus_1_iff
thf(fact_5507_AND__upper2_H,axiom,
    ! [Y2: int,Z: int,X2: int] :
      ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y2 )
     => ( ( ord_less_eq @ int @ Y2 @ Z )
       => ( ord_less_eq @ int @ ( bit_se5824344872417868541ns_and @ int @ X2 @ Y2 ) @ Z ) ) ) ).

% AND_upper2'
thf(fact_5508_AND__upper1_H,axiom,
    ! [Y2: int,Z: int,Ya: int] :
      ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y2 )
     => ( ( ord_less_eq @ int @ Y2 @ Z )
       => ( ord_less_eq @ int @ ( bit_se5824344872417868541ns_and @ int @ Y2 @ Ya ) @ Z ) ) ) ).

% AND_upper1'
thf(fact_5509_AND__upper2,axiom,
    ! [Y2: int,X2: int] :
      ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y2 )
     => ( ord_less_eq @ int @ ( bit_se5824344872417868541ns_and @ int @ X2 @ Y2 ) @ Y2 ) ) ).

% AND_upper2
thf(fact_5510_AND__upper1,axiom,
    ! [X2: int,Y2: int] :
      ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X2 )
     => ( ord_less_eq @ int @ ( bit_se5824344872417868541ns_and @ int @ X2 @ Y2 ) @ X2 ) ) ).

% AND_upper1
thf(fact_5511_AND__lower,axiom,
    ! [X2: int,Y2: int] :
      ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X2 )
     => ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_se5824344872417868541ns_and @ int @ X2 @ Y2 ) ) ) ).

% AND_lower
thf(fact_5512_AND__upper2_H_H,axiom,
    ! [Y2: int,Z: int,X2: int] :
      ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y2 )
     => ( ( ord_less @ int @ Y2 @ Z )
       => ( ord_less @ int @ ( bit_se5824344872417868541ns_and @ int @ X2 @ Y2 ) @ Z ) ) ) ).

% AND_upper2''
thf(fact_5513_AND__upper1_H_H,axiom,
    ! [Y2: int,Z: int,Ya: int] :
      ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y2 )
     => ( ( ord_less @ int @ Y2 @ Z )
       => ( ord_less @ int @ ( bit_se5824344872417868541ns_and @ int @ Y2 @ Ya ) @ Z ) ) ) ).

% AND_upper1''
thf(fact_5514_and__less__eq,axiom,
    ! [L: int,K: int] :
      ( ( ord_less @ int @ L @ ( zero_zero @ int ) )
     => ( ord_less_eq @ int @ ( bit_se5824344872417868541ns_and @ int @ K @ L ) @ K ) ) ).

% and_less_eq
thf(fact_5515_even__and__iff,axiom,
    ! [A: $tType] :
      ( ( bit_se359711467146920520ations @ A )
     => ! [A2: A,B2: A] :
          ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ A @ A2 @ B2 ) )
          = ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 )
            | ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) ) ) ) ).

% even_and_iff
thf(fact_5516_even__and__iff__int,axiom,
    ! [K: int,L: int] :
      ( ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ int @ K @ L ) )
      = ( ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K )
        | ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ L ) ) ) ).

% even_and_iff_int
thf(fact_5517_and__one__eq,axiom,
    ! [A: $tType] :
      ( ( bit_se359711467146920520ations @ A )
     => ! [A2: A] :
          ( ( bit_se5824344872417868541ns_and @ A @ A2 @ ( one_one @ A ) )
          = ( modulo_modulo @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).

% and_one_eq
thf(fact_5518_one__and__eq,axiom,
    ! [A: $tType] :
      ( ( bit_se359711467146920520ations @ A )
     => ! [A2: A] :
          ( ( bit_se5824344872417868541ns_and @ A @ ( one_one @ A ) @ A2 )
          = ( modulo_modulo @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).

% one_and_eq
thf(fact_5519_and__int__rec,axiom,
    ( ( bit_se5824344872417868541ns_and @ int )
    = ( ^ [K4: int,L2: int] :
          ( plus_plus @ int
          @ ( zero_neq_one_of_bool @ int
            @ ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K4 )
              & ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ L2 ) ) )
          @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ int @ ( divide_divide @ int @ K4 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ L2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).

% and_int_rec
thf(fact_5520_and__int__unfold,axiom,
    ( ( bit_se5824344872417868541ns_and @ int )
    = ( ^ [K4: int,L2: int] :
          ( if @ int
          @ ( ( K4
              = ( zero_zero @ int ) )
            | ( L2
              = ( zero_zero @ int ) ) )
          @ ( zero_zero @ int )
          @ ( if @ int
            @ ( K4
              = ( uminus_uminus @ int @ ( one_one @ int ) ) )
            @ L2
            @ ( if @ int
              @ ( L2
                = ( uminus_uminus @ int @ ( one_one @ int ) ) )
              @ K4
              @ ( plus_plus @ int @ ( times_times @ int @ ( modulo_modulo @ int @ K4 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( modulo_modulo @ int @ L2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ int @ ( divide_divide @ int @ K4 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ L2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ).

% and_int_unfold
thf(fact_5521_and__int_Oelims,axiom,
    ! [X2: int,Xa: int,Y2: int] :
      ( ( ( bit_se5824344872417868541ns_and @ int @ X2 @ Xa )
        = Y2 )
     => ( ( ( ( member @ int @ X2 @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) )
            & ( member @ int @ Xa @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) ) )
         => ( Y2
            = ( uminus_uminus @ int
              @ ( zero_neq_one_of_bool @ int
                @ ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ X2 )
                  & ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ Xa ) ) ) ) ) )
        & ( ~ ( ( member @ int @ X2 @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) )
              & ( member @ int @ Xa @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) ) )
         => ( Y2
            = ( plus_plus @ int
              @ ( zero_neq_one_of_bool @ int
                @ ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ X2 )
                  & ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ Xa ) ) )
              @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ int @ ( divide_divide @ int @ X2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ Xa @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ).

% and_int.elims
thf(fact_5522_take__bit__word__Bit1__eq,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [N2: num,M: num] :
          ( ( bit_se2584673776208193580ke_bit @ ( word @ A ) @ ( numeral_numeral @ nat @ N2 ) @ ( numeral_numeral @ ( word @ A ) @ ( bit1 @ M ) ) )
          = ( plus_plus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) @ ( times_times @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ ( bit_se2584673776208193580ke_bit @ ( word @ A ) @ ( pred_numeral @ N2 ) @ ( numeral_numeral @ ( word @ A ) @ M ) ) ) ) ) ) ).

% take_bit_word_Bit1_eq
thf(fact_5523_uint32_Osize__eq,axiom,
    ( ( size_size @ uint32 )
    = ( ^ [P5: uint32] : ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).

% uint32.size_eq
thf(fact_5524_add__scale__eq__noteq,axiom,
    ! [A: $tType] :
      ( ( semiri1453513574482234551roduct @ A )
     => ! [R4: A,A2: A,B2: A,C2: A,D2: A] :
          ( ( R4
           != ( zero_zero @ A ) )
         => ( ( ( A2 = B2 )
              & ( C2 != D2 ) )
           => ( ( plus_plus @ A @ A2 @ ( times_times @ A @ R4 @ C2 ) )
             != ( plus_plus @ A @ B2 @ ( times_times @ A @ R4 @ D2 ) ) ) ) ) ) ).

% add_scale_eq_noteq
thf(fact_5525_word__and__max,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [X2: word @ A] :
          ( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X2 @ ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) )
          = X2 ) ) ).

% word_and_max
thf(fact_5526_word__bitwise__m1__simps_I2_J,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [X2: word @ A] :
          ( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) @ X2 )
          = X2 ) ) ).

% word_bitwise_m1_simps(2)
thf(fact_5527_take__bit__Suc__1,axiom,
    ! [A: $tType] :
      ( ( bit_un5681908812861735899ations @ A )
     => ! [N2: nat] :
          ( ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N2 ) @ ( one_one @ A ) )
          = ( one_one @ A ) ) ) ).

% take_bit_Suc_1
thf(fact_5528_take__bit__numeral__1,axiom,
    ! [A: $tType] :
      ( ( bit_un5681908812861735899ations @ A )
     => ! [L: num] :
          ( ( bit_se2584673776208193580ke_bit @ A @ ( numeral_numeral @ nat @ L ) @ ( one_one @ A ) )
          = ( one_one @ A ) ) ) ).

% take_bit_numeral_1
thf(fact_5529_and__nat__numerals_I1_J,axiom,
    ! [Y2: num] :
      ( ( bit_se5824344872417868541ns_and @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( numeral_numeral @ nat @ ( bit0 @ Y2 ) ) )
      = ( zero_zero @ nat ) ) ).

% and_nat_numerals(1)
thf(fact_5530_and__nat__numerals_I3_J,axiom,
    ! [X2: num] :
      ( ( bit_se5824344872417868541ns_and @ nat @ ( numeral_numeral @ nat @ ( bit0 @ X2 ) ) @ ( suc @ ( zero_zero @ nat ) ) )
      = ( zero_zero @ nat ) ) ).

% and_nat_numerals(3)
thf(fact_5531_take__bit__of__1__eq__0__iff,axiom,
    ! [A: $tType] :
      ( ( bit_un5681908812861735899ations @ A )
     => ! [N2: nat] :
          ( ( ( bit_se2584673776208193580ke_bit @ A @ N2 @ ( one_one @ A ) )
            = ( zero_zero @ A ) )
          = ( N2
            = ( zero_zero @ nat ) ) ) ) ).

% take_bit_of_1_eq_0_iff
thf(fact_5532_word__no__log__defs_I3_J,axiom,
    ! [C: $tType] :
      ( ( type_len @ C )
     => ! [A2: num,B2: num] :
          ( ( bit_se5824344872417868541ns_and @ ( word @ C ) @ ( numeral_numeral @ ( word @ C ) @ A2 ) @ ( numeral_numeral @ ( word @ C ) @ B2 ) )
          = ( ring_1_of_int @ ( word @ C ) @ ( bit_se5824344872417868541ns_and @ int @ ( numeral_numeral @ int @ A2 ) @ ( numeral_numeral @ int @ B2 ) ) ) ) ) ).

% word_no_log_defs(3)
thf(fact_5533_and__nat__numerals_I2_J,axiom,
    ! [Y2: num] :
      ( ( bit_se5824344872417868541ns_and @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( numeral_numeral @ nat @ ( bit1 @ Y2 ) ) )
      = ( one_one @ nat ) ) ).

% and_nat_numerals(2)
thf(fact_5534_and__nat__numerals_I4_J,axiom,
    ! [X2: num] :
      ( ( bit_se5824344872417868541ns_and @ nat @ ( numeral_numeral @ nat @ ( bit1 @ X2 ) ) @ ( suc @ ( zero_zero @ nat ) ) )
      = ( one_one @ nat ) ) ).

% and_nat_numerals(4)
thf(fact_5535_take__bit__of__1,axiom,
    ! [A: $tType] :
      ( ( bit_se359711467146920520ations @ A )
     => ! [N2: nat] :
          ( ( bit_se2584673776208193580ke_bit @ A @ N2 @ ( one_one @ A ) )
          = ( zero_neq_one_of_bool @ A @ ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ) ).

% take_bit_of_1
thf(fact_5536_even__take__bit__eq,axiom,
    ! [A: $tType] :
      ( ( bit_se359711467146920520ations @ A )
     => ! [N2: nat,A2: A] :
          ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se2584673776208193580ke_bit @ A @ N2 @ A2 ) )
          = ( ( N2
              = ( zero_zero @ nat ) )
            | ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) ) ) ) ).

% even_take_bit_eq
thf(fact_5537_and__Suc__0__eq,axiom,
    ! [N2: nat] :
      ( ( bit_se5824344872417868541ns_and @ nat @ N2 @ ( suc @ ( zero_zero @ nat ) ) )
      = ( modulo_modulo @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ).

% and_Suc_0_eq
thf(fact_5538_Suc__0__and__eq,axiom,
    ! [N2: nat] :
      ( ( bit_se5824344872417868541ns_and @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N2 )
      = ( modulo_modulo @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ).

% Suc_0_and_eq
thf(fact_5539_word__bitwise__1__simps_I2_J,axiom,
    ! [B: $tType] :
      ( ( type_len @ B )
     => ! [B2: num] :
          ( ( bit_se5824344872417868541ns_and @ ( word @ B ) @ ( one_one @ ( word @ B ) ) @ ( numeral_numeral @ ( word @ B ) @ B2 ) )
          = ( ring_1_of_int @ ( word @ B ) @ ( bit_se5824344872417868541ns_and @ int @ ( one_one @ int ) @ ( numeral_numeral @ int @ B2 ) ) ) ) ) ).

% word_bitwise_1_simps(2)
thf(fact_5540_word__bitwise__1__simps_I4_J,axiom,
    ! [D: $tType] :
      ( ( type_len @ D )
     => ! [A2: num] :
          ( ( bit_se5824344872417868541ns_and @ ( word @ D ) @ ( numeral_numeral @ ( word @ D ) @ A2 ) @ ( one_one @ ( word @ D ) ) )
          = ( ring_1_of_int @ ( word @ D ) @ ( bit_se5824344872417868541ns_and @ int @ ( numeral_numeral @ int @ A2 ) @ ( one_one @ int ) ) ) ) ) ).

% word_bitwise_1_simps(4)
thf(fact_5541_word__no__log__defs_I6_J,axiom,
    ! [F3: $tType] :
      ( ( type_len @ F3 )
     => ! [A2: num,B2: num] :
          ( ( bit_se5824344872417868541ns_and @ ( word @ F3 ) @ ( uminus_uminus @ ( word @ F3 ) @ ( numeral_numeral @ ( word @ F3 ) @ A2 ) ) @ ( uminus_uminus @ ( word @ F3 ) @ ( numeral_numeral @ ( word @ F3 ) @ B2 ) ) )
          = ( ring_1_of_int @ ( word @ F3 ) @ ( bit_se5824344872417868541ns_and @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ A2 ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ B2 ) ) ) ) ) ) ).

% word_no_log_defs(6)
thf(fact_5542_word__no__log__defs_I5_J,axiom,
    ! [E: $tType] :
      ( ( type_len @ E )
     => ! [A2: num,B2: num] :
          ( ( bit_se5824344872417868541ns_and @ ( word @ E ) @ ( uminus_uminus @ ( word @ E ) @ ( numeral_numeral @ ( word @ E ) @ A2 ) ) @ ( numeral_numeral @ ( word @ E ) @ B2 ) )
          = ( ring_1_of_int @ ( word @ E ) @ ( bit_se5824344872417868541ns_and @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ A2 ) ) @ ( numeral_numeral @ int @ B2 ) ) ) ) ) ).

% word_no_log_defs(5)
thf(fact_5543_word__no__log__defs_I4_J,axiom,
    ! [D: $tType] :
      ( ( type_len @ D )
     => ! [A2: num,B2: num] :
          ( ( bit_se5824344872417868541ns_and @ ( word @ D ) @ ( numeral_numeral @ ( word @ D ) @ A2 ) @ ( uminus_uminus @ ( word @ D ) @ ( numeral_numeral @ ( word @ D ) @ B2 ) ) )
          = ( ring_1_of_int @ ( word @ D ) @ ( bit_se5824344872417868541ns_and @ int @ ( numeral_numeral @ int @ A2 ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ B2 ) ) ) ) ) ) ).

% word_no_log_defs(4)
thf(fact_5544_take__bit__Suc__0,axiom,
    ! [A: $tType] :
      ( ( bit_se359711467146920520ations @ A )
     => ! [A2: A] :
          ( ( bit_se2584673776208193580ke_bit @ A @ ( suc @ ( zero_zero @ nat ) ) @ A2 )
          = ( modulo_modulo @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).

% take_bit_Suc_0
thf(fact_5545_take__bit__word__Bit0__eq,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [N2: num,M: num] :
          ( ( bit_se2584673776208193580ke_bit @ ( word @ A ) @ ( numeral_numeral @ nat @ N2 ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ M ) ) )
          = ( times_times @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ ( bit_se2584673776208193580ke_bit @ ( word @ A ) @ ( pred_numeral @ N2 ) @ ( numeral_numeral @ ( word @ A ) @ M ) ) ) ) ) ).

% take_bit_word_Bit0_eq
thf(fact_5546_take__bit__of__exp,axiom,
    ! [A: $tType] :
      ( ( bit_un5681908812861735899ations @ A )
     => ! [M: nat,N2: nat] :
          ( ( bit_se2584673776208193580ke_bit @ A @ M @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) )
          = ( times_times @ A @ ( zero_neq_one_of_bool @ A @ ( ord_less @ nat @ N2 @ M ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) ) ) ) ).

% take_bit_of_exp
thf(fact_5547_take__bit__of__2,axiom,
    ! [A: $tType] :
      ( ( bit_un5681908812861735899ations @ A )
     => ! [N2: nat] :
          ( ( bit_se2584673776208193580ke_bit @ A @ N2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
          = ( times_times @ A @ ( zero_neq_one_of_bool @ A @ ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).

% take_bit_of_2
thf(fact_5548_word__bitwise__1__simps_I3_J,axiom,
    ! [C: $tType] :
      ( ( type_len @ C )
     => ! [B2: num] :
          ( ( bit_se5824344872417868541ns_and @ ( word @ C ) @ ( one_one @ ( word @ C ) ) @ ( uminus_uminus @ ( word @ C ) @ ( numeral_numeral @ ( word @ C ) @ B2 ) ) )
          = ( ring_1_of_int @ ( word @ C ) @ ( bit_se5824344872417868541ns_and @ int @ ( one_one @ int ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ B2 ) ) ) ) ) ) ).

% word_bitwise_1_simps(3)
thf(fact_5549_word__bitwise__1__simps_I5_J,axiom,
    ! [E: $tType] :
      ( ( type_len @ E )
     => ! [A2: num] :
          ( ( bit_se5824344872417868541ns_and @ ( word @ E ) @ ( uminus_uminus @ ( word @ E ) @ ( numeral_numeral @ ( word @ E ) @ A2 ) ) @ ( one_one @ ( word @ E ) ) )
          = ( ring_1_of_int @ ( word @ E ) @ ( bit_se5824344872417868541ns_and @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ A2 ) ) @ ( one_one @ int ) ) ) ) ) ).

% word_bitwise_1_simps(5)
thf(fact_5550_take__bit__word__minus__Bit0__eq,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [N2: num,M: num] :
          ( ( bit_se2584673776208193580ke_bit @ ( word @ A ) @ ( numeral_numeral @ nat @ N2 ) @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ M ) ) ) )
          = ( times_times @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ ( bit_se2584673776208193580ke_bit @ ( word @ A ) @ ( pred_numeral @ N2 ) @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ M ) ) ) ) ) ) ).

% take_bit_word_minus_Bit0_eq
thf(fact_5551_word__and__le1,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [Y2: word @ A,A2: word @ A] : ( ord_less_eq @ ( word @ A ) @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ Y2 @ A2 ) @ A2 ) ) ).

% word_and_le1
thf(fact_5552_word__and__le2,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [A2: word @ A,Y2: word @ A] : ( ord_less_eq @ ( word @ A ) @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ A2 @ Y2 ) @ A2 ) ) ).

% word_and_le2
thf(fact_5553_take__bit__unset__bit__eq,axiom,
    ! [A: $tType] :
      ( ( bit_se359711467146920520ations @ A )
     => ! [N2: nat,M: nat,A2: A] :
          ( ( ( ord_less_eq @ nat @ N2 @ M )
           => ( ( bit_se2584673776208193580ke_bit @ A @ N2 @ ( bit_se2638667681897837118et_bit @ A @ M @ A2 ) )
              = ( bit_se2584673776208193580ke_bit @ A @ N2 @ A2 ) ) )
          & ( ~ ( ord_less_eq @ nat @ N2 @ M )
           => ( ( bit_se2584673776208193580ke_bit @ A @ N2 @ ( bit_se2638667681897837118et_bit @ A @ M @ A2 ) )
              = ( bit_se2638667681897837118et_bit @ A @ M @ ( bit_se2584673776208193580ke_bit @ A @ N2 @ A2 ) ) ) ) ) ) ).

% take_bit_unset_bit_eq
thf(fact_5554_take__bit__set__bit__eq,axiom,
    ! [A: $tType] :
      ( ( bit_se359711467146920520ations @ A )
     => ! [N2: nat,M: nat,A2: A] :
          ( ( ( ord_less_eq @ nat @ N2 @ M )
           => ( ( bit_se2584673776208193580ke_bit @ A @ N2 @ ( bit_se5668285175392031749et_bit @ A @ M @ A2 ) )
              = ( bit_se2584673776208193580ke_bit @ A @ N2 @ A2 ) ) )
          & ( ~ ( ord_less_eq @ nat @ N2 @ M )
           => ( ( bit_se2584673776208193580ke_bit @ A @ N2 @ ( bit_se5668285175392031749et_bit @ A @ M @ A2 ) )
              = ( bit_se5668285175392031749et_bit @ A @ M @ ( bit_se2584673776208193580ke_bit @ A @ N2 @ A2 ) ) ) ) ) ) ).

% take_bit_set_bit_eq
thf(fact_5555_take__bit__flip__bit__eq,axiom,
    ! [A: $tType] :
      ( ( bit_se359711467146920520ations @ A )
     => ! [N2: nat,M: nat,A2: A] :
          ( ( ( ord_less_eq @ nat @ N2 @ M )
           => ( ( bit_se2584673776208193580ke_bit @ A @ N2 @ ( bit_se8732182000553998342ip_bit @ A @ M @ A2 ) )
              = ( bit_se2584673776208193580ke_bit @ A @ N2 @ A2 ) ) )
          & ( ~ ( ord_less_eq @ nat @ N2 @ M )
           => ( ( bit_se2584673776208193580ke_bit @ A @ N2 @ ( bit_se8732182000553998342ip_bit @ A @ M @ A2 ) )
              = ( bit_se8732182000553998342ip_bit @ A @ M @ ( bit_se2584673776208193580ke_bit @ A @ N2 @ A2 ) ) ) ) ) ) ).

% take_bit_flip_bit_eq
thf(fact_5556_take__bit__tightened,axiom,
    ! [A: $tType] :
      ( ( bit_se359711467146920520ations @ A )
     => ! [N2: nat,A2: A,B2: A,M: nat] :
          ( ( ( bit_se2584673776208193580ke_bit @ A @ N2 @ A2 )
            = ( bit_se2584673776208193580ke_bit @ A @ N2 @ B2 ) )
         => ( ( ord_less_eq @ nat @ M @ N2 )
           => ( ( bit_se2584673776208193580ke_bit @ A @ M @ A2 )
              = ( bit_se2584673776208193580ke_bit @ A @ M @ B2 ) ) ) ) ) ).

% take_bit_tightened
thf(fact_5557_signed__take__bit__eq__iff__take__bit__eq,axiom,
    ! [A: $tType] :
      ( ( bit_ri3973907225187159222ations @ A )
     => ! [N2: nat,A2: A,B2: A] :
          ( ( ( bit_ri4674362597316999326ke_bit @ A @ N2 @ A2 )
            = ( bit_ri4674362597316999326ke_bit @ A @ N2 @ B2 ) )
          = ( ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N2 ) @ A2 )
            = ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N2 ) @ B2 ) ) ) ) ).

% signed_take_bit_eq_iff_take_bit_eq
thf(fact_5558_take__bit__add,axiom,
    ! [A: $tType] :
      ( ( bit_un5681908812861735899ations @ A )
     => ! [N2: nat,A2: A,B2: A] :
          ( ( bit_se2584673776208193580ke_bit @ A @ N2 @ ( plus_plus @ A @ ( bit_se2584673776208193580ke_bit @ A @ N2 @ A2 ) @ ( bit_se2584673776208193580ke_bit @ A @ N2 @ B2 ) ) )
          = ( bit_se2584673776208193580ke_bit @ A @ N2 @ ( plus_plus @ A @ A2 @ B2 ) ) ) ) ).

% take_bit_add
thf(fact_5559_signed__take__bit__take__bit,axiom,
    ! [A: $tType] :
      ( ( bit_ri3973907225187159222ations @ A )
     => ! [M: nat,N2: nat,A2: A] :
          ( ( bit_ri4674362597316999326ke_bit @ A @ M @ ( bit_se2584673776208193580ke_bit @ A @ N2 @ A2 ) )
          = ( if @ ( A > A ) @ ( ord_less_eq @ nat @ N2 @ M ) @ ( bit_se2584673776208193580ke_bit @ A @ N2 ) @ ( bit_ri4674362597316999326ke_bit @ A @ M ) @ A2 ) ) ) ).

% signed_take_bit_take_bit
thf(fact_5560_word__and__max__word,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [X2: word @ A,A2: word @ A] :
          ( ( X2
            = ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) )
         => ( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ A2 @ X2 )
            = A2 ) ) ) ).

% word_and_max_word
thf(fact_5561_take__bit__signed__take__bit,axiom,
    ! [A: $tType] :
      ( ( bit_ri3973907225187159222ations @ A )
     => ! [M: nat,N2: nat,A2: A] :
          ( ( ord_less_eq @ nat @ M @ ( suc @ N2 ) )
         => ( ( bit_se2584673776208193580ke_bit @ A @ M @ ( bit_ri4674362597316999326ke_bit @ A @ N2 @ A2 ) )
            = ( bit_se2584673776208193580ke_bit @ A @ M @ A2 ) ) ) ) ).

% take_bit_signed_take_bit
thf(fact_5562_even__word__iff,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ( ( even_word @ A )
        = ( ^ [A5: word @ A] :
              ( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ A5 @ ( one_one @ ( word @ A ) ) )
              = ( zero_zero @ ( word @ A ) ) ) ) ) ) ).

% even_word_iff
thf(fact_5563_take__bit__Suc__bit0,axiom,
    ! [A: $tType] :
      ( ( bit_un5681908812861735899ations @ A )
     => ! [N2: nat,K: num] :
          ( ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N2 ) @ ( numeral_numeral @ A @ ( bit0 @ K ) ) )
          = ( times_times @ A @ ( bit_se2584673776208193580ke_bit @ A @ N2 @ ( numeral_numeral @ A @ K ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).

% take_bit_Suc_bit0
thf(fact_5564_take__bit__eq__mod,axiom,
    ! [A: $tType] :
      ( ( bit_se359711467146920520ations @ A )
     => ( ( bit_se2584673776208193580ke_bit @ A )
        = ( ^ [N: nat,A5: A] : ( modulo_modulo @ A @ A5 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) ) ) ) ) ).

% take_bit_eq_mod
thf(fact_5565_take__bit__eq__0__iff,axiom,
    ! [A: $tType] :
      ( ( bit_se359711467146920520ations @ A )
     => ! [N2: nat,A2: A] :
          ( ( ( bit_se2584673776208193580ke_bit @ A @ N2 @ A2 )
            = ( zero_zero @ A ) )
          = ( dvd_dvd @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) @ A2 ) ) ) ).

% take_bit_eq_0_iff
thf(fact_5566_bin__last__bintrunc,axiom,
    ! [A: $tType] :
      ( ( bit_se359711467146920520ations @ A )
     => ! [L: nat,N2: A] :
          ( ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se2584673776208193580ke_bit @ A @ L @ N2 ) ) )
          = ( ( ord_less @ nat @ ( zero_zero @ nat ) @ L )
            & ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) ) ) ) ).

% bin_last_bintrunc
thf(fact_5567_take__bit__numeral__bit0,axiom,
    ! [A: $tType] :
      ( ( bit_un5681908812861735899ations @ A )
     => ! [L: num,K: num] :
          ( ( bit_se2584673776208193580ke_bit @ A @ ( numeral_numeral @ nat @ L ) @ ( numeral_numeral @ A @ ( bit0 @ K ) ) )
          = ( times_times @ A @ ( bit_se2584673776208193580ke_bit @ A @ ( pred_numeral @ L ) @ ( numeral_numeral @ A @ K ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).

% take_bit_numeral_bit0
thf(fact_5568_add__0__iff,axiom,
    ! [A: $tType] :
      ( ( semiri1453513574482234551roduct @ A )
     => ! [B2: A,A2: A] :
          ( ( B2
            = ( plus_plus @ A @ B2 @ A2 ) )
          = ( A2
            = ( zero_zero @ A ) ) ) ) ).

% add_0_iff
thf(fact_5569_crossproduct__noteq,axiom,
    ! [A: $tType] :
      ( ( semiri1453513574482234551roduct @ A )
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( ( ( A2 != B2 )
            & ( C2 != D2 ) )
          = ( ( plus_plus @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ D2 ) )
           != ( plus_plus @ A @ ( times_times @ A @ A2 @ D2 ) @ ( times_times @ A @ B2 @ C2 ) ) ) ) ) ).

% crossproduct_noteq
thf(fact_5570_crossproduct__eq,axiom,
    ! [A: $tType] :
      ( ( semiri1453513574482234551roduct @ A )
     => ! [W: A,Y2: A,X2: A,Z: A] :
          ( ( ( plus_plus @ A @ ( times_times @ A @ W @ Y2 ) @ ( times_times @ A @ X2 @ Z ) )
            = ( plus_plus @ A @ ( times_times @ A @ W @ Z ) @ ( times_times @ A @ X2 @ Y2 ) ) )
          = ( ( W = X2 )
            | ( Y2 = Z ) ) ) ) ).

% crossproduct_eq
thf(fact_5571_take__bit__Suc__bit1,axiom,
    ! [A: $tType] :
      ( ( bit_un5681908812861735899ations @ A )
     => ! [N2: nat,K: num] :
          ( ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N2 ) @ ( numeral_numeral @ A @ ( bit1 @ K ) ) )
          = ( plus_plus @ A @ ( times_times @ A @ ( bit_se2584673776208193580ke_bit @ A @ N2 @ ( numeral_numeral @ A @ K ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( one_one @ A ) ) ) ) ).

% take_bit_Suc_bit1
thf(fact_5572_take__bit__Suc__minus__1__eq,axiom,
    ! [A: $tType] :
      ( ( bit_ri3973907225187159222ations @ A )
     => ! [N2: nat] :
          ( ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N2 ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
          = ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( suc @ N2 ) ) @ ( one_one @ A ) ) ) ) ).

% take_bit_Suc_minus_1_eq
thf(fact_5573_take__bit__numeral__minus__1__eq,axiom,
    ! [A: $tType] :
      ( ( bit_ri3973907225187159222ations @ A )
     => ! [K: num] :
          ( ( bit_se2584673776208193580ke_bit @ A @ ( numeral_numeral @ nat @ K ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
          = ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ K ) ) @ ( one_one @ A ) ) ) ) ).

% take_bit_numeral_minus_1_eq
thf(fact_5574_take__bit__Suc,axiom,
    ! [A: $tType] :
      ( ( bit_se359711467146920520ations @ A )
     => ! [N2: nat,A2: A] :
          ( ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N2 ) @ A2 )
          = ( plus_plus @ A @ ( times_times @ A @ ( bit_se2584673776208193580ke_bit @ A @ N2 @ ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( modulo_modulo @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).

% take_bit_Suc
thf(fact_5575_and__nat__unfold,axiom,
    ( ( bit_se5824344872417868541ns_and @ nat )
    = ( ^ [M7: nat,N: nat] :
          ( if @ nat
          @ ( ( M7
              = ( zero_zero @ nat ) )
            | ( N
              = ( zero_zero @ nat ) ) )
          @ ( zero_zero @ nat )
          @ ( plus_plus @ nat @ ( times_times @ nat @ ( modulo_modulo @ nat @ M7 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( modulo_modulo @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ nat @ ( divide_divide @ nat @ M7 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ).

% and_nat_unfold
thf(fact_5576_and__nat__rec,axiom,
    ( ( bit_se5824344872417868541ns_and @ nat )
    = ( ^ [M7: nat,N: nat] :
          ( plus_plus @ nat
          @ ( zero_neq_one_of_bool @ nat
            @ ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M7 )
              & ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) )
          @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ nat @ ( divide_divide @ nat @ M7 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).

% and_nat_rec
thf(fact_5577_stable__imp__take__bit__eq,axiom,
    ! [A: $tType] :
      ( ( bit_se359711467146920520ations @ A )
     => ! [A2: A,N2: nat] :
          ( ( ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
            = A2 )
         => ( ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 )
             => ( ( bit_se2584673776208193580ke_bit @ A @ N2 @ A2 )
                = ( zero_zero @ A ) ) )
            & ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 )
             => ( ( bit_se2584673776208193580ke_bit @ A @ N2 @ A2 )
                = ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) @ ( one_one @ A ) ) ) ) ) ) ) ).

% stable_imp_take_bit_eq
thf(fact_5578_take__bit__numeral__bit1,axiom,
    ! [A: $tType] :
      ( ( bit_un5681908812861735899ations @ A )
     => ! [L: num,K: num] :
          ( ( bit_se2584673776208193580ke_bit @ A @ ( numeral_numeral @ nat @ L ) @ ( numeral_numeral @ A @ ( bit1 @ K ) ) )
          = ( plus_plus @ A @ ( times_times @ A @ ( bit_se2584673776208193580ke_bit @ A @ ( pred_numeral @ L ) @ ( numeral_numeral @ A @ K ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( one_one @ A ) ) ) ) ).

% take_bit_numeral_bit1
thf(fact_5579_take__bit__rec,axiom,
    ! [A: $tType] :
      ( ( bit_se359711467146920520ations @ A )
     => ( ( bit_se2584673776208193580ke_bit @ A )
        = ( ^ [N: nat,A5: A] :
              ( if @ A
              @ ( N
                = ( zero_zero @ nat ) )
              @ ( zero_zero @ A )
              @ ( plus_plus @ A @ ( times_times @ A @ ( bit_se2584673776208193580ke_bit @ A @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ ( divide_divide @ A @ A5 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( modulo_modulo @ A @ A5 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).

% take_bit_rec
thf(fact_5580_take__bit__word__minus__Bit1__eq,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [N2: num,M: num] :
          ( ( bit_se2584673776208193580ke_bit @ ( word @ A ) @ ( numeral_numeral @ nat @ N2 ) @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit1 @ M ) ) ) )
          = ( plus_plus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) @ ( times_times @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ ( bit_se2584673776208193580ke_bit @ ( word @ A ) @ ( pred_numeral @ N2 ) @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( inc @ M ) ) ) ) ) ) ) ) ).

% take_bit_word_minus_Bit1_eq
thf(fact_5581_take__bit__numeral__minus__numeral__word,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [M: num,N2: num] :
          ( ( bit_se2584673776208193580ke_bit @ ( word @ A ) @ ( numeral_numeral @ nat @ M ) @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ N2 ) ) )
          = ( case_option @ ( word @ A ) @ num @ ( zero_zero @ ( word @ A ) )
            @ ^ [Q5: num] : ( bit_se2584673776208193580ke_bit @ ( word @ A ) @ ( numeral_numeral @ nat @ M ) @ ( minus_minus @ ( word @ A ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ M ) ) @ ( numeral_numeral @ ( word @ A ) @ Q5 ) ) )
            @ ( bit_take_bit_num @ ( numeral_numeral @ nat @ M ) @ N2 ) ) ) ) ).

% take_bit_numeral_minus_numeral_word
thf(fact_5582_slice__nth,axiom,
    ! [A: $tType,From: nat,To: nat,Xs2: list @ A,I: nat] :
      ( ( ord_less @ nat @ From @ To )
     => ( ( ord_less_eq @ nat @ To @ ( size_size @ ( list @ A ) @ Xs2 ) )
       => ( ( ord_less @ nat @ I @ ( minus_minus @ nat @ To @ From ) )
         => ( ( nth @ A @ ( slice @ A @ From @ To @ Xs2 ) @ I )
            = ( nth @ A @ Xs2 @ ( plus_plus @ nat @ From @ I ) ) ) ) ) ) ).

% slice_nth
thf(fact_5583_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_5584_take__bit__num__simps_I2_J,axiom,
    ! [N2: nat] :
      ( ( bit_take_bit_num @ ( suc @ N2 ) @ one2 )
      = ( some @ num @ one2 ) ) ).

% take_bit_num_simps(2)
thf(fact_5585_take__bit__num__simps_I5_J,axiom,
    ! [R4: num] :
      ( ( bit_take_bit_num @ ( numeral_numeral @ nat @ R4 ) @ one2 )
      = ( some @ num @ one2 ) ) ).

% take_bit_num_simps(5)
thf(fact_5586_pred__numeral__inc,axiom,
    ! [K: num] :
      ( ( pred_numeral @ ( inc @ K ) )
      = ( numeral_numeral @ nat @ K ) ) ).

% pred_numeral_inc
thf(fact_5587_slice__complete,axiom,
    ! [A: $tType,Xs2: list @ A] :
      ( ( slice @ A @ ( zero_zero @ nat ) @ ( size_size @ ( list @ A ) @ Xs2 ) @ Xs2 )
      = Xs2 ) ).

% slice_complete
thf(fact_5588_take__bit__num__simps_I3_J,axiom,
    ! [N2: nat,M: num] :
      ( ( bit_take_bit_num @ ( suc @ N2 ) @ ( bit0 @ M ) )
      = ( case_option @ ( option @ num ) @ num @ ( none @ num )
        @ ^ [Q5: num] : ( some @ num @ ( bit0 @ Q5 ) )
        @ ( bit_take_bit_num @ N2 @ M ) ) ) ).

% take_bit_num_simps(3)
thf(fact_5589_take__bit__of__Suc__0,axiom,
    ! [N2: nat] :
      ( ( bit_se2584673776208193580ke_bit @ nat @ N2 @ ( suc @ ( zero_zero @ nat ) ) )
      = ( zero_neq_one_of_bool @ nat @ ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ).

% take_bit_of_Suc_0
thf(fact_5590_slice__len,axiom,
    ! [A: $tType,From: nat,To: nat,Xs2: list @ A] :
      ( ( ord_less_eq @ nat @ From @ To )
     => ( ( ord_less_eq @ nat @ To @ ( size_size @ ( list @ A ) @ Xs2 ) )
       => ( ( size_size @ ( list @ A ) @ ( slice @ A @ From @ To @ Xs2 ) )
          = ( minus_minus @ nat @ To @ From ) ) ) ) ).

% slice_len
thf(fact_5591_take__bit__num__simps_I4_J,axiom,
    ! [N2: nat,M: num] :
      ( ( bit_take_bit_num @ ( suc @ N2 ) @ ( bit1 @ M ) )
      = ( some @ num @ ( case_option @ num @ num @ one2 @ bit1 @ ( bit_take_bit_num @ N2 @ M ) ) ) ) ).

% take_bit_num_simps(4)
thf(fact_5592_take__bit__num__simps_I6_J,axiom,
    ! [R4: num,M: num] :
      ( ( bit_take_bit_num @ ( numeral_numeral @ nat @ R4 ) @ ( bit0 @ M ) )
      = ( case_option @ ( option @ num ) @ num @ ( none @ num )
        @ ^ [Q5: num] : ( some @ num @ ( bit0 @ Q5 ) )
        @ ( bit_take_bit_num @ ( pred_numeral @ R4 ) @ M ) ) ) ).

% take_bit_num_simps(6)
thf(fact_5593_add__neg__numeral__special_I6_J,axiom,
    ! [A: $tType] :
      ( ( neg_numeral @ A )
     => ! [M: num] :
          ( ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
          = ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( inc @ M ) ) ) ) ) ).

% add_neg_numeral_special(6)
thf(fact_5594_add__neg__numeral__special_I5_J,axiom,
    ! [A: $tType] :
      ( ( neg_numeral @ A )
     => ! [N2: num] :
          ( ( plus_plus @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N2 ) ) )
          = ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( inc @ N2 ) ) ) ) ) ).

% add_neg_numeral_special(5)
thf(fact_5595_diff__numeral__special_I6_J,axiom,
    ! [A: $tType] :
      ( ( neg_numeral @ A )
     => ! [M: num] :
          ( ( minus_minus @ A @ ( numeral_numeral @ A @ M ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
          = ( numeral_numeral @ A @ ( inc @ M ) ) ) ) ).

% diff_numeral_special(6)
thf(fact_5596_diff__numeral__special_I5_J,axiom,
    ! [A: $tType] :
      ( ( neg_numeral @ A )
     => ! [N2: num] :
          ( ( minus_minus @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( numeral_numeral @ A @ N2 ) )
          = ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( inc @ N2 ) ) ) ) ) ).

% diff_numeral_special(5)
thf(fact_5597_take__bit__num__simps_I7_J,axiom,
    ! [R4: num,M: num] :
      ( ( bit_take_bit_num @ ( numeral_numeral @ nat @ R4 ) @ ( bit1 @ M ) )
      = ( some @ num @ ( case_option @ num @ num @ one2 @ bit1 @ ( bit_take_bit_num @ ( pred_numeral @ R4 ) @ M ) ) ) ) ).

% take_bit_num_simps(7)
thf(fact_5598_take__bit__numeral__numeral,axiom,
    ! [A: $tType] :
      ( ( bit_se359711467146920520ations @ A )
     => ! [M: num,N2: num] :
          ( ( bit_se2584673776208193580ke_bit @ A @ ( numeral_numeral @ nat @ M ) @ ( numeral_numeral @ A @ N2 ) )
          = ( case_option @ A @ num @ ( zero_zero @ A ) @ ( numeral_numeral @ A ) @ ( bit_take_bit_num @ ( numeral_numeral @ nat @ M ) @ N2 ) ) ) ) ).

% take_bit_numeral_numeral
thf(fact_5599_take__bit__numeral__minus__numeral__int,axiom,
    ! [M: num,N2: num] :
      ( ( bit_se2584673776208193580ke_bit @ int @ ( numeral_numeral @ nat @ M ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N2 ) ) )
      = ( case_option @ int @ num @ ( zero_zero @ int )
        @ ^ [Q5: num] : ( bit_se2584673776208193580ke_bit @ int @ ( numeral_numeral @ nat @ M ) @ ( minus_minus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ M ) ) @ ( numeral_numeral @ int @ Q5 ) ) )
        @ ( bit_take_bit_num @ ( numeral_numeral @ nat @ M ) @ N2 ) ) ) ).

% take_bit_numeral_minus_numeral_int
thf(fact_5600_take__bit__nat__eq,axiom,
    ! [K: int,N2: nat] :
      ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
     => ( ( bit_se2584673776208193580ke_bit @ nat @ N2 @ ( nat2 @ K ) )
        = ( nat2 @ ( bit_se2584673776208193580ke_bit @ int @ N2 @ K ) ) ) ) ).

% take_bit_nat_eq
thf(fact_5601_nat__take__bit__eq,axiom,
    ! [K: int,N2: nat] :
      ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
     => ( ( nat2 @ ( bit_se2584673776208193580ke_bit @ int @ N2 @ K ) )
        = ( bit_se2584673776208193580ke_bit @ nat @ N2 @ ( nat2 @ K ) ) ) ) ).

% nat_take_bit_eq
thf(fact_5602_take__bit__nat__less__eq__self,axiom,
    ! [N2: nat,M: nat] : ( ord_less_eq @ nat @ ( bit_se2584673776208193580ke_bit @ nat @ N2 @ M ) @ M ) ).

% take_bit_nat_less_eq_self
thf(fact_5603_take__bit__tightened__less__eq__nat,axiom,
    ! [M: nat,N2: nat,Q2: nat] :
      ( ( ord_less_eq @ nat @ M @ N2 )
     => ( ord_less_eq @ nat @ ( bit_se2584673776208193580ke_bit @ nat @ M @ Q2 ) @ ( bit_se2584673776208193580ke_bit @ nat @ N2 @ Q2 ) ) ) ).

% take_bit_tightened_less_eq_nat
thf(fact_5604_take__bit__mult,axiom,
    ! [N2: nat,K: int,L: int] :
      ( ( bit_se2584673776208193580ke_bit @ int @ N2 @ ( times_times @ int @ ( bit_se2584673776208193580ke_bit @ int @ N2 @ K ) @ ( bit_se2584673776208193580ke_bit @ int @ N2 @ L ) ) )
      = ( bit_se2584673776208193580ke_bit @ int @ N2 @ ( times_times @ int @ K @ L ) ) ) ).

% take_bit_mult
thf(fact_5605_take__bit__diff,axiom,
    ! [N2: nat,K: int,L: int] :
      ( ( bit_se2584673776208193580ke_bit @ int @ N2 @ ( minus_minus @ int @ ( bit_se2584673776208193580ke_bit @ int @ N2 @ K ) @ ( bit_se2584673776208193580ke_bit @ int @ N2 @ L ) ) )
      = ( bit_se2584673776208193580ke_bit @ int @ N2 @ ( minus_minus @ int @ K @ L ) ) ) ).

% take_bit_diff
thf(fact_5606_add__inc,axiom,
    ! [X2: num,Y2: num] :
      ( ( plus_plus @ num @ X2 @ ( inc @ Y2 ) )
      = ( inc @ ( plus_plus @ num @ X2 @ Y2 ) ) ) ).

% add_inc
thf(fact_5607_num__induct,axiom,
    ! [P: num > $o,X2: num] :
      ( ( P @ one2 )
     => ( ! [X3: num] :
            ( ( P @ X3 )
           => ( P @ ( inc @ X3 ) ) )
       => ( P @ X2 ) ) ) ).

% num_induct
thf(fact_5608_take__bit__tightened__less__eq__int,axiom,
    ! [M: nat,N2: nat,K: int] :
      ( ( ord_less_eq @ nat @ M @ N2 )
     => ( ord_less_eq @ int @ ( bit_se2584673776208193580ke_bit @ int @ M @ K ) @ ( bit_se2584673776208193580ke_bit @ int @ N2 @ K ) ) ) ).

% take_bit_tightened_less_eq_int
thf(fact_5609_take__bit__nonnegative,axiom,
    ! [N2: nat,K: int] : ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_se2584673776208193580ke_bit @ int @ N2 @ K ) ) ).

% take_bit_nonnegative
thf(fact_5610_take__bit__int__less__eq__self__iff,axiom,
    ! [N2: nat,K: int] :
      ( ( ord_less_eq @ int @ ( bit_se2584673776208193580ke_bit @ int @ N2 @ K ) @ K )
      = ( ord_less_eq @ int @ ( zero_zero @ int ) @ K ) ) ).

% take_bit_int_less_eq_self_iff
thf(fact_5611_inc_Osimps_I1_J,axiom,
    ( ( inc @ one2 )
    = ( bit0 @ one2 ) ) ).

% inc.simps(1)
thf(fact_5612_inc_Osimps_I2_J,axiom,
    ! [X2: num] :
      ( ( inc @ ( bit0 @ X2 ) )
      = ( bit1 @ X2 ) ) ).

% inc.simps(2)
thf(fact_5613_inc_Osimps_I3_J,axiom,
    ! [X2: num] :
      ( ( inc @ ( bit1 @ X2 ) )
      = ( bit0 @ ( inc @ X2 ) ) ) ).

% inc.simps(3)
thf(fact_5614_add__One,axiom,
    ! [X2: num] :
      ( ( plus_plus @ num @ X2 @ one2 )
      = ( inc @ X2 ) ) ).

% add_One
thf(fact_5615_inc__BitM__eq,axiom,
    ! [N2: num] :
      ( ( inc @ ( bitM @ N2 ) )
      = ( bit0 @ N2 ) ) ).

% inc_BitM_eq
thf(fact_5616_BitM__inc__eq,axiom,
    ! [N2: num] :
      ( ( bitM @ ( inc @ N2 ) )
      = ( bit1 @ N2 ) ) ).

% BitM_inc_eq
thf(fact_5617_take__bit__num__eq__Some__imp,axiom,
    ! [A: $tType] :
      ( ( bit_se359711467146920520ations @ A )
     => ! [M: nat,N2: num,Q2: num] :
          ( ( ( bit_take_bit_num @ M @ N2 )
            = ( some @ num @ Q2 ) )
         => ( ( bit_se2584673776208193580ke_bit @ A @ M @ ( numeral_numeral @ A @ N2 ) )
            = ( numeral_numeral @ A @ Q2 ) ) ) ) ).

% take_bit_num_eq_Some_imp
thf(fact_5618_mult__inc,axiom,
    ! [X2: num,Y2: num] :
      ( ( times_times @ num @ X2 @ ( inc @ Y2 ) )
      = ( plus_plus @ num @ ( times_times @ num @ X2 @ Y2 ) @ X2 ) ) ).

% mult_inc
thf(fact_5619_take__bit__decr__eq,axiom,
    ! [N2: nat,K: int] :
      ( ( ( bit_se2584673776208193580ke_bit @ int @ N2 @ K )
       != ( zero_zero @ int ) )
     => ( ( bit_se2584673776208193580ke_bit @ int @ N2 @ ( minus_minus @ int @ K @ ( one_one @ int ) ) )
        = ( minus_minus @ int @ ( bit_se2584673776208193580ke_bit @ int @ N2 @ K ) @ ( one_one @ int ) ) ) ) ).

% take_bit_decr_eq
thf(fact_5620_numeral__inc,axiom,
    ! [A: $tType] :
      ( ( numeral @ A )
     => ! [X2: num] :
          ( ( numeral_numeral @ A @ ( inc @ X2 ) )
          = ( plus_plus @ A @ ( numeral_numeral @ A @ X2 ) @ ( one_one @ A ) ) ) ) ).

% numeral_inc
thf(fact_5621_take__bit__num__eq__None__imp,axiom,
    ! [A: $tType] :
      ( ( bit_se359711467146920520ations @ A )
     => ! [M: nat,N2: num] :
          ( ( ( bit_take_bit_num @ M @ N2 )
            = ( none @ num ) )
         => ( ( bit_se2584673776208193580ke_bit @ A @ M @ ( numeral_numeral @ A @ N2 ) )
            = ( zero_zero @ A ) ) ) ) ).

% take_bit_num_eq_None_imp
thf(fact_5622_take__bit__nat__eq__self,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less @ nat @ M @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) )
     => ( ( bit_se2584673776208193580ke_bit @ nat @ N2 @ M )
        = M ) ) ).

% take_bit_nat_eq_self
thf(fact_5623_take__bit__nat__less__exp,axiom,
    ! [N2: nat,M: nat] : ( ord_less @ nat @ ( bit_se2584673776208193580ke_bit @ nat @ N2 @ M ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ).

% take_bit_nat_less_exp
thf(fact_5624_take__bit__nat__eq__self__iff,axiom,
    ! [N2: nat,M: nat] :
      ( ( ( bit_se2584673776208193580ke_bit @ nat @ N2 @ M )
        = M )
      = ( ord_less @ nat @ M @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ) ).

% take_bit_nat_eq_self_iff
thf(fact_5625_take__bit__nat__def,axiom,
    ( ( bit_se2584673776208193580ke_bit @ nat )
    = ( ^ [N: nat,M7: nat] : ( modulo_modulo @ nat @ M7 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).

% take_bit_nat_def
thf(fact_5626_take__bit__Suc__minus__bit1,axiom,
    ! [N2: nat,K: num] :
      ( ( bit_se2584673776208193580ke_bit @ int @ ( suc @ N2 ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ K ) ) ) )
      = ( plus_plus @ int @ ( times_times @ int @ ( bit_se2584673776208193580ke_bit @ int @ N2 @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( inc @ K ) ) ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( one_one @ int ) ) ) ).

% take_bit_Suc_minus_bit1
thf(fact_5627_take__bit__int__less__exp,axiom,
    ! [N2: nat,K: int] : ( ord_less @ int @ ( bit_se2584673776208193580ke_bit @ int @ N2 @ K ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N2 ) ) ).

% take_bit_int_less_exp
thf(fact_5628_take__bit__int__def,axiom,
    ( ( bit_se2584673776208193580ke_bit @ int )
    = ( ^ [N: nat,K4: int] : ( modulo_modulo @ int @ K4 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).

% take_bit_int_def
thf(fact_5629_take__bit__numeral__minus__bit1,axiom,
    ! [L: num,K: num] :
      ( ( bit_se2584673776208193580ke_bit @ int @ ( numeral_numeral @ nat @ L ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ K ) ) ) )
      = ( plus_plus @ int @ ( times_times @ int @ ( bit_se2584673776208193580ke_bit @ int @ ( pred_numeral @ L ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( inc @ K ) ) ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( one_one @ int ) ) ) ).

% take_bit_numeral_minus_bit1
thf(fact_5630_take__bit__nat__less__self__iff,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less @ nat @ ( bit_se2584673776208193580ke_bit @ nat @ N2 @ M ) @ M )
      = ( ord_less_eq @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) @ M ) ) ).

% take_bit_nat_less_self_iff
thf(fact_5631_take__bit__Suc__minus__bit0,axiom,
    ! [N2: nat,K: num] :
      ( ( bit_se2584673776208193580ke_bit @ int @ ( suc @ N2 ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ K ) ) ) )
      = ( times_times @ int @ ( bit_se2584673776208193580ke_bit @ int @ N2 @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K ) ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ).

% take_bit_Suc_minus_bit0
thf(fact_5632_take__bit__int__greater__eq__self__iff,axiom,
    ! [K: int,N2: nat] :
      ( ( ord_less_eq @ int @ K @ ( bit_se2584673776208193580ke_bit @ int @ N2 @ K ) )
      = ( ord_less @ int @ K @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N2 ) ) ) ).

% take_bit_int_greater_eq_self_iff
thf(fact_5633_take__bit__int__less__self__iff,axiom,
    ! [N2: nat,K: int] :
      ( ( ord_less @ int @ ( bit_se2584673776208193580ke_bit @ int @ N2 @ K ) @ K )
      = ( ord_less_eq @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N2 ) @ K ) ) ).

% take_bit_int_less_self_iff
thf(fact_5634_take__bit__int__eq__self__iff,axiom,
    ! [N2: nat,K: int] :
      ( ( ( bit_se2584673776208193580ke_bit @ int @ N2 @ K )
        = K )
      = ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
        & ( ord_less @ int @ K @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N2 ) ) ) ) ).

% take_bit_int_eq_self_iff
thf(fact_5635_take__bit__int__eq__self,axiom,
    ! [K: int,N2: nat] :
      ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
     => ( ( ord_less @ int @ K @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N2 ) )
       => ( ( bit_se2584673776208193580ke_bit @ int @ N2 @ K )
          = K ) ) ) ).

% take_bit_int_eq_self
thf(fact_5636_take__bit__incr__eq,axiom,
    ! [N2: nat,K: int] :
      ( ( ( bit_se2584673776208193580ke_bit @ int @ N2 @ K )
       != ( minus_minus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N2 ) @ ( one_one @ int ) ) )
     => ( ( bit_se2584673776208193580ke_bit @ int @ N2 @ ( plus_plus @ int @ K @ ( one_one @ int ) ) )
        = ( plus_plus @ int @ ( one_one @ int ) @ ( bit_se2584673776208193580ke_bit @ int @ N2 @ K ) ) ) ) ).

% take_bit_incr_eq
thf(fact_5637_take__bit__numeral__minus__bit0,axiom,
    ! [L: num,K: num] :
      ( ( bit_se2584673776208193580ke_bit @ int @ ( numeral_numeral @ nat @ L ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ K ) ) ) )
      = ( times_times @ int @ ( bit_se2584673776208193580ke_bit @ int @ ( pred_numeral @ L ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K ) ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ).

% take_bit_numeral_minus_bit0
thf(fact_5638_take__bit__int__less__eq,axiom,
    ! [N2: nat,K: int] :
      ( ( ord_less_eq @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N2 ) @ K )
     => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
       => ( ord_less_eq @ int @ ( bit_se2584673776208193580ke_bit @ int @ N2 @ K ) @ ( minus_minus @ int @ K @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N2 ) ) ) ) ) ).

% take_bit_int_less_eq
thf(fact_5639_take__bit__int__greater__eq,axiom,
    ! [K: int,N2: nat] :
      ( ( ord_less @ int @ K @ ( zero_zero @ int ) )
     => ( ord_less_eq @ int @ ( plus_plus @ int @ K @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N2 ) ) @ ( bit_se2584673776208193580ke_bit @ int @ N2 @ K ) ) ) ).

% take_bit_int_greater_eq
thf(fact_5640_signed__take__bit__eq__take__bit__shift,axiom,
    ( ( bit_ri4674362597316999326ke_bit @ int )
    = ( ^ [N: nat,K4: int] : ( minus_minus @ int @ ( bit_se2584673776208193580ke_bit @ int @ ( suc @ N ) @ ( plus_plus @ int @ K4 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).

% signed_take_bit_eq_take_bit_shift
thf(fact_5641_take__bit__minus__small__eq,axiom,
    ! [K: int,N2: nat] :
      ( ( ord_less @ int @ ( zero_zero @ int ) @ K )
     => ( ( ord_less_eq @ int @ K @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N2 ) )
       => ( ( bit_se2584673776208193580ke_bit @ int @ N2 @ ( uminus_uminus @ int @ K ) )
          = ( minus_minus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N2 ) @ K ) ) ) ) ).

% take_bit_minus_small_eq
thf(fact_5642_and__mask__arith_H,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [N2: nat,W: word @ A] :
          ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
         => ( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ W @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N2 ) )
            = ( divide_divide @ ( word @ A ) @ ( times_times @ ( word @ A ) @ W @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ ( size_size @ ( word @ A ) @ W ) @ N2 ) ) ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ ( size_size @ ( word @ A ) @ W ) @ N2 ) ) ) ) ) ) ).

% and_mask_arith'
thf(fact_5643_listI__assn__def,axiom,
    ! [A: $tType,B: $tType] :
      ( ( vEBT_List_listI_assn @ A @ B )
      = ( ^ [I8: set @ nat,A7: A > B > assn,Xs: list @ A,Xsi3: list @ B] :
            ( times_times @ assn
            @ ( pure_assn
              @ ( ( ( size_size @ ( list @ B ) @ Xsi3 )
                  = ( size_size @ ( list @ A ) @ Xs ) )
                & ( ord_less_eq @ ( set @ nat ) @ I8 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ A ) @ Xs ) ) ) ) )
            @ ( finite_fold @ nat @ assn
              @ ^ [I2: nat,A5: assn] : ( times_times @ assn @ A5 @ ( A7 @ ( nth @ A @ Xs @ I2 ) @ ( nth @ B @ Xsi3 @ I2 ) ) )
              @ ( one_one @ assn )
              @ I8 ) ) ) ) ).

% listI_assn_def
thf(fact_5644_word__2p__lem,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [N2: nat,W: word @ A] :
          ( ( ord_less @ nat @ N2 @ ( size_size @ ( word @ A ) @ W ) )
         => ( ( ord_less @ ( word @ A ) @ W @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N2 ) )
            = ( ord_less @ int @ ( semiring_1_unsigned @ A @ int @ W ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N2 ) ) ) ) ) ).

% word_2p_lem
thf(fact_5645_unsigned__1,axiom,
    ! [B: $tType,A: $tType] :
      ( ( ( semiring_1 @ A )
        & ( type_len @ B ) )
     => ( ( semiring_1_unsigned @ B @ A @ ( one_one @ ( word @ B ) ) )
        = ( one_one @ A ) ) ) ).

% unsigned_1
thf(fact_5646_uint__ge__0,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [X2: word @ A] : ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( semiring_1_unsigned @ A @ int @ X2 ) ) ) ).

% uint_ge_0
thf(fact_5647_uint__le__0__iff,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [X2: word @ A] :
          ( ( ord_less_eq @ int @ ( semiring_1_unsigned @ A @ int @ X2 ) @ ( zero_zero @ int ) )
          = ( ( semiring_1_unsigned @ A @ int @ X2 )
            = ( zero_zero @ int ) ) ) ) ).

% uint_le_0_iff
thf(fact_5648_uint__nonnegative,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [W: word @ A] : ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( semiring_1_unsigned @ A @ int @ W ) ) ) ).

% uint_nonnegative
thf(fact_5649_Word_Omask__Suc__0,axiom,
    ! [A: $tType] :
      ( ( bit_se359711467146920520ations @ A )
     => ( ( bit_se2239418461657761734s_mask @ A @ ( suc @ ( zero_zero @ nat ) ) )
        = ( one_one @ A ) ) ) ).

% Word.mask_Suc_0
thf(fact_5650_take__bit__minus__one__eq__mask,axiom,
    ! [A: $tType] :
      ( ( bit_ri3973907225187159222ations @ A )
     => ! [N2: nat] :
          ( ( bit_se2584673776208193580ke_bit @ A @ N2 @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
          = ( bit_se2239418461657761734s_mask @ A @ N2 ) ) ) ).

% take_bit_minus_one_eq_mask
thf(fact_5651_word__le__no,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [A2: num,B2: num] :
          ( ( ord_less_eq @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ A2 ) @ ( numeral_numeral @ ( word @ A ) @ B2 ) )
          = ( ord_less_eq @ int @ ( semiring_1_unsigned @ A @ int @ ( numeral_numeral @ ( word @ A ) @ A2 ) ) @ ( semiring_1_unsigned @ A @ int @ ( numeral_numeral @ ( word @ A ) @ B2 ) ) ) ) ) ).

% word_le_no
thf(fact_5652_word__less__no,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [A2: num,B2: num] :
          ( ( ord_less @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ A2 ) @ ( numeral_numeral @ ( word @ A ) @ B2 ) )
          = ( ord_less @ int @ ( semiring_1_unsigned @ A @ int @ ( numeral_numeral @ ( word @ A ) @ A2 ) ) @ ( semiring_1_unsigned @ A @ int @ ( numeral_numeral @ ( word @ A ) @ B2 ) ) ) ) ) ).

% word_less_no
thf(fact_5653_word__div__no,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [A2: num,B2: num] :
          ( ( divide_divide @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ A2 ) @ ( numeral_numeral @ ( word @ A ) @ B2 ) )
          = ( ring_1_of_int @ ( word @ A ) @ ( divide_divide @ int @ ( semiring_1_unsigned @ A @ int @ ( numeral_numeral @ ( word @ A ) @ A2 ) ) @ ( semiring_1_unsigned @ A @ int @ ( numeral_numeral @ ( word @ A ) @ B2 ) ) ) ) ) ) ).

% word_div_no
thf(fact_5654_word__mod__no,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [A2: num,B2: num] :
          ( ( modulo_modulo @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ A2 ) @ ( numeral_numeral @ ( word @ A ) @ B2 ) )
          = ( ring_1_of_int @ ( word @ A ) @ ( modulo_modulo @ int @ ( semiring_1_unsigned @ A @ int @ ( numeral_numeral @ ( word @ A ) @ A2 ) ) @ ( semiring_1_unsigned @ A @ int @ ( numeral_numeral @ ( word @ A ) @ B2 ) ) ) ) ) ) ).

% word_mod_no
thf(fact_5655_mask__eqs_I7_J,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [A2: word @ A,N2: nat,B2: word @ A] :
          ( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ A2 @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N2 ) ) @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ B2 @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N2 ) ) ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N2 ) )
          = ( bit_se5824344872417868541ns_and @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ A2 @ B2 ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N2 ) ) ) ) ).

% mask_eqs(7)
thf(fact_5656_mask__eqs_I2_J,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [A2: word @ A,B2: word @ A,N2: nat] :
          ( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ A2 @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ B2 @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N2 ) ) ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N2 ) )
          = ( bit_se5824344872417868541ns_and @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ A2 @ B2 ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N2 ) ) ) ) ).

% mask_eqs(2)
thf(fact_5657_mask__eqs_I1_J,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [A2: word @ A,N2: nat,B2: word @ A] :
          ( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ A2 @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N2 ) ) @ B2 ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N2 ) )
          = ( bit_se5824344872417868541ns_and @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ A2 @ B2 ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N2 ) ) ) ) ).

% mask_eqs(1)
thf(fact_5658_and__mask__eq__iff__le__mask,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [W: word @ A,N2: nat] :
          ( ( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ W @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N2 ) )
            = W )
          = ( ord_less_eq @ ( word @ A ) @ W @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N2 ) ) ) ) ).

% and_mask_eq_iff_le_mask
thf(fact_5659_le__mask__imp__and__mask,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [X2: word @ A,N2: nat] :
          ( ( ord_less_eq @ ( word @ A ) @ X2 @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N2 ) )
         => ( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X2 @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N2 ) )
            = X2 ) ) ) ).

% le_mask_imp_and_mask
thf(fact_5660_mask__twice2,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [N2: nat,M: nat,X2: word @ A] :
          ( ( ord_less_eq @ nat @ N2 @ M )
         => ( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X2 @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ M ) ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N2 ) )
            = ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X2 @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N2 ) ) ) ) ) ).

% mask_twice2
thf(fact_5661_ex__mask__1,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ? [X3: nat] :
          ( ( bit_se2239418461657761734s_mask @ ( word @ A ) @ X3 )
          = ( one_one @ ( word @ A ) ) ) ) ).

% ex_mask_1
thf(fact_5662_uint__1__eq,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ( ( semiring_1_unsigned @ A @ int @ ( one_one @ ( word @ A ) ) )
        = ( one_one @ int ) ) ) ).

% uint_1_eq
thf(fact_5663_word__le__def,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ( ( ord_less_eq @ ( word @ A ) )
        = ( ^ [A5: word @ A,B5: word @ A] : ( ord_less_eq @ int @ ( semiring_1_unsigned @ A @ int @ A5 ) @ ( semiring_1_unsigned @ A @ int @ B5 ) ) ) ) ) ).

% word_le_def
thf(fact_5664_mask__eqs_I3_J,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [A2: word @ A,N2: nat,B2: word @ A] :
          ( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ A2 @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N2 ) ) @ B2 ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N2 ) )
          = ( bit_se5824344872417868541ns_and @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ A2 @ B2 ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N2 ) ) ) ) ).

% mask_eqs(3)
thf(fact_5665_mask__eqs_I4_J,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [A2: word @ A,B2: word @ A,N2: nat] :
          ( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ A2 @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ B2 @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N2 ) ) ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N2 ) )
          = ( bit_se5824344872417868541ns_and @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ A2 @ B2 ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N2 ) ) ) ) ).

% mask_eqs(4)
thf(fact_5666_mask__eqs_I8_J,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [A2: word @ A,N2: nat,B2: word @ A] :
          ( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ A2 @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N2 ) ) @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ B2 @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N2 ) ) ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N2 ) )
          = ( bit_se5824344872417868541ns_and @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ A2 @ B2 ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N2 ) ) ) ) ).

% mask_eqs(8)
thf(fact_5667_mask__power__eq,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [X2: word @ A,N2: nat,K: nat] :
          ( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ ( power_power @ ( word @ A ) @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X2 @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N2 ) ) @ K ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N2 ) )
          = ( bit_se5824344872417868541ns_and @ ( word @ A ) @ ( power_power @ ( word @ A ) @ X2 @ K ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N2 ) ) ) ) ).

% mask_power_eq
thf(fact_5668_mask__eqs_I5_J,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [A2: word @ A,B2: word @ A,N2: nat] :
          ( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ ( times_times @ ( word @ A ) @ A2 @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ B2 @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N2 ) ) ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N2 ) )
          = ( bit_se5824344872417868541ns_and @ ( word @ A ) @ ( times_times @ ( word @ A ) @ A2 @ B2 ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N2 ) ) ) ) ).

% mask_eqs(5)
thf(fact_5669_mask__eqs_I6_J,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [B2: word @ A,N2: nat,A2: word @ A] :
          ( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ ( times_times @ ( word @ A ) @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ B2 @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N2 ) ) @ A2 ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N2 ) )
          = ( bit_se5824344872417868541ns_and @ ( word @ A ) @ ( times_times @ ( word @ A ) @ B2 @ A2 ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N2 ) ) ) ) ).

% mask_eqs(6)
thf(fact_5670_mask__eqs_I9_J,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [A2: word @ A,N2: nat,B2: word @ A] :
          ( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ ( times_times @ ( word @ A ) @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ A2 @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N2 ) ) @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ B2 @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N2 ) ) ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N2 ) )
          = ( bit_se5824344872417868541ns_and @ ( word @ A ) @ ( times_times @ ( word @ A ) @ A2 @ B2 ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N2 ) ) ) ) ).

% mask_eqs(9)
thf(fact_5671_unsigned__greater__eq,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ( type_len @ B )
        & ( unique1627219031080169319umeral @ A ) )
     => ! [W: word @ B] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( semiring_1_unsigned @ B @ A @ W ) ) ) ).

% unsigned_greater_eq
thf(fact_5672_less__eq__mask__iff__take__bit__eq__self,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [W: word @ A,N2: nat] :
          ( ( ord_less_eq @ ( word @ A ) @ W @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N2 ) )
          = ( ( bit_se2584673776208193580ke_bit @ ( word @ A ) @ N2 @ W )
            = W ) ) ) ).

% less_eq_mask_iff_take_bit_eq_self
thf(fact_5673_uint__div,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [X2: word @ A,Y2: word @ A] :
          ( ( semiring_1_unsigned @ A @ int @ ( divide_divide @ ( word @ A ) @ X2 @ Y2 ) )
          = ( divide_divide @ int @ ( semiring_1_unsigned @ A @ int @ X2 ) @ ( semiring_1_unsigned @ A @ int @ Y2 ) ) ) ) ).

% uint_div
thf(fact_5674_uint__div__distrib,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [V: word @ A,W: word @ A] :
          ( ( semiring_1_unsigned @ A @ int @ ( divide_divide @ ( word @ A ) @ V @ W ) )
          = ( divide_divide @ int @ ( semiring_1_unsigned @ A @ int @ V ) @ ( semiring_1_unsigned @ A @ int @ W ) ) ) ) ).

% uint_div_distrib
thf(fact_5675_word__less__eq__iff__unsigned,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ( type_len @ B )
        & ( linordered_semidom @ A ) )
     => ( ( ord_less_eq @ ( word @ B ) )
        = ( ^ [A5: word @ B,B5: word @ B] : ( ord_less_eq @ A @ ( semiring_1_unsigned @ B @ A @ A5 ) @ ( semiring_1_unsigned @ B @ A @ B5 ) ) ) ) ) ).

% word_less_eq_iff_unsigned
thf(fact_5676_word__less__iff__unsigned,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ( type_len @ B )
        & ( linordered_semidom @ A ) )
     => ( ( ord_less @ ( word @ B ) )
        = ( ^ [A5: word @ B,B5: word @ B] : ( ord_less @ A @ ( semiring_1_unsigned @ B @ A @ A5 ) @ ( semiring_1_unsigned @ B @ A @ B5 ) ) ) ) ) ).

% word_less_iff_unsigned
thf(fact_5677_mask__1,axiom,
    ! [A: $tType] :
      ( ( bit_se359711467146920520ations @ A )
     => ( ( bit_se2239418461657761734s_mask @ A @ ( one_one @ nat ) )
        = ( one_one @ A ) ) ) ).

% mask_1
thf(fact_5678_mask__eq__iff,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [W: word @ A,N2: nat] :
          ( ( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ W @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N2 ) )
            = W )
          = ( ord_less @ int @ ( semiring_1_unsigned @ A @ int @ W ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N2 ) ) ) ) ).

% mask_eq_iff
thf(fact_5679_and__mask__lt__2p,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [W: word @ A,N2: nat] : ( ord_less @ int @ ( semiring_1_unsigned @ A @ int @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ W @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N2 ) ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N2 ) ) ) ).

% and_mask_lt_2p
thf(fact_5680_More__Word_Omask__Suc__0,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ( ( bit_se2239418461657761734s_mask @ ( word @ A ) @ ( suc @ ( zero_zero @ nat ) ) )
        = ( one_one @ ( word @ A ) ) ) ) ).

% More_Word.mask_Suc_0
thf(fact_5681_mask__bin,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ( ( bit_se2239418461657761734s_mask @ ( word @ A ) )
        = ( ^ [N: nat] : ( ring_1_of_int @ ( word @ A ) @ ( bit_se2584673776208193580ke_bit @ int @ N @ ( uminus_uminus @ int @ ( one_one @ int ) ) ) ) ) ) ) ).

% mask_bin
thf(fact_5682_uint__add__ge0,axiom,
    ! [Aa: $tType,A: $tType] :
      ( ( ( type_len @ A )
        & ( type_len @ Aa ) )
     => ! [Xa: word @ Aa,X2: word @ A] : ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( plus_plus @ int @ ( semiring_1_unsigned @ Aa @ int @ Xa ) @ ( semiring_1_unsigned @ A @ int @ X2 ) ) ) ) ).

% uint_add_ge0
thf(fact_5683_uint__mult__ge0,axiom,
    ! [Aa: $tType,A: $tType] :
      ( ( ( type_len @ A )
        & ( type_len @ Aa ) )
     => ! [Xa: word @ Aa,X2: word @ A] : ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( times_times @ int @ ( semiring_1_unsigned @ Aa @ int @ Xa ) @ ( semiring_1_unsigned @ A @ int @ X2 ) ) ) ) ).

% uint_mult_ge0
thf(fact_5684_nat__uint__less__helper,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [Y2: word @ A,Z: nat,X2: word @ A] :
          ( ( ( nat2 @ ( semiring_1_unsigned @ A @ int @ Y2 ) )
            = Z )
         => ( ( ord_less @ ( word @ A ) @ X2 @ Y2 )
           => ( ord_less @ nat @ ( nat2 @ ( semiring_1_unsigned @ A @ int @ X2 ) ) @ Z ) ) ) ) ).

% nat_uint_less_helper
thf(fact_5685_uint__add__le,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [X2: word @ A,Y2: word @ A] : ( ord_less_eq @ int @ ( semiring_1_unsigned @ A @ int @ ( plus_plus @ ( word @ A ) @ X2 @ Y2 ) ) @ ( plus_plus @ int @ ( semiring_1_unsigned @ A @ int @ X2 ) @ ( semiring_1_unsigned @ A @ int @ Y2 ) ) ) ) ).

% uint_add_le
thf(fact_5686_uint__plus__simple__iff,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [X2: word @ A,Y2: word @ A] :
          ( ( ord_less_eq @ ( word @ A ) @ X2 @ ( plus_plus @ ( word @ A ) @ X2 @ Y2 ) )
          = ( ( semiring_1_unsigned @ A @ int @ ( plus_plus @ ( word @ A ) @ X2 @ Y2 ) )
            = ( plus_plus @ int @ ( semiring_1_unsigned @ A @ int @ X2 ) @ ( semiring_1_unsigned @ A @ int @ Y2 ) ) ) ) ) ).

% uint_plus_simple_iff
thf(fact_5687_uint__plus__simple,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [X2: word @ A,Y2: word @ A] :
          ( ( ord_less_eq @ ( word @ A ) @ X2 @ ( plus_plus @ ( word @ A ) @ X2 @ Y2 ) )
         => ( ( semiring_1_unsigned @ A @ int @ ( plus_plus @ ( word @ A ) @ X2 @ Y2 ) )
            = ( plus_plus @ int @ ( semiring_1_unsigned @ A @ int @ X2 ) @ ( semiring_1_unsigned @ A @ int @ Y2 ) ) ) ) ) ).

% uint_plus_simple
thf(fact_5688_no__ulen__sub,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [X2: word @ A,Y2: word @ A] :
          ( ( ord_less_eq @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ X2 @ Y2 ) @ X2 )
          = ( ord_less_eq @ int @ ( semiring_1_unsigned @ A @ int @ Y2 ) @ ( semiring_1_unsigned @ A @ int @ X2 ) ) ) ) ).

% no_ulen_sub
thf(fact_5689_word__add__def,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ( ( plus_plus @ ( word @ A ) )
        = ( ^ [A5: word @ A,B5: word @ A] : ( ring_1_of_int @ ( word @ A ) @ ( plus_plus @ int @ ( semiring_1_unsigned @ A @ int @ A5 ) @ ( semiring_1_unsigned @ A @ int @ B5 ) ) ) ) ) ) ).

% word_add_def
thf(fact_5690_uint__sub__lem,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [Y2: word @ A,X2: word @ A] :
          ( ( ord_less_eq @ int @ ( semiring_1_unsigned @ A @ int @ Y2 ) @ ( semiring_1_unsigned @ A @ int @ X2 ) )
          = ( ( semiring_1_unsigned @ A @ int @ ( minus_minus @ ( word @ A ) @ X2 @ Y2 ) )
            = ( minus_minus @ int @ ( semiring_1_unsigned @ A @ int @ X2 ) @ ( semiring_1_unsigned @ A @ int @ Y2 ) ) ) ) ) ).

% uint_sub_lem
thf(fact_5691_uint__sub__ge,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [X2: word @ A,Y2: word @ A] : ( ord_less_eq @ int @ ( minus_minus @ int @ ( semiring_1_unsigned @ A @ int @ X2 ) @ ( semiring_1_unsigned @ A @ int @ Y2 ) ) @ ( semiring_1_unsigned @ A @ int @ ( minus_minus @ ( word @ A ) @ X2 @ Y2 ) ) ) ) ).

% uint_sub_ge
thf(fact_5692_uint__minus__simple__iff,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [X2: word @ A,Y2: word @ A] :
          ( ( ord_less_eq @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ X2 @ Y2 ) @ X2 )
          = ( ( semiring_1_unsigned @ A @ int @ ( minus_minus @ ( word @ A ) @ X2 @ Y2 ) )
            = ( minus_minus @ int @ ( semiring_1_unsigned @ A @ int @ X2 ) @ ( semiring_1_unsigned @ A @ int @ Y2 ) ) ) ) ) ).

% uint_minus_simple_iff
thf(fact_5693_uint__minus__simple__alt,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ( ( ord_less_eq @ ( word @ A ) )
        = ( ^ [Y: word @ A,X: word @ A] :
              ( ( semiring_1_unsigned @ A @ int @ ( minus_minus @ ( word @ A ) @ X @ Y ) )
              = ( minus_minus @ int @ ( semiring_1_unsigned @ A @ int @ X ) @ ( semiring_1_unsigned @ A @ int @ Y ) ) ) ) ) ) ).

% uint_minus_simple_alt
thf(fact_5694_word__sub__wi,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ( ( minus_minus @ ( word @ A ) )
        = ( ^ [A5: word @ A,B5: word @ A] : ( ring_1_of_int @ ( word @ A ) @ ( minus_minus @ int @ ( semiring_1_unsigned @ A @ int @ A5 ) @ ( semiring_1_unsigned @ A @ int @ B5 ) ) ) ) ) ) ).

% word_sub_wi
thf(fact_5695_word__arith__power__alt,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ( ( power_power @ ( word @ A ) )
        = ( ^ [A5: word @ A,N: nat] : ( ring_1_of_int @ ( word @ A ) @ ( power_power @ int @ ( semiring_1_unsigned @ A @ int @ A5 ) @ N ) ) ) ) ) ).

% word_arith_power_alt
thf(fact_5696_word__mult__def,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ( ( times_times @ ( word @ A ) )
        = ( ^ [A5: word @ A,B5: word @ A] : ( ring_1_of_int @ ( word @ A ) @ ( times_times @ int @ ( semiring_1_unsigned @ A @ int @ A5 ) @ ( semiring_1_unsigned @ A @ int @ B5 ) ) ) ) ) ) ).

% word_mult_def
thf(fact_5697_word__div__def,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ( ( divide_divide @ ( word @ A ) )
        = ( ^ [A5: word @ A,B5: word @ A] : ( ring_1_of_int @ ( word @ A ) @ ( divide_divide @ int @ ( semiring_1_unsigned @ A @ int @ A5 ) @ ( semiring_1_unsigned @ A @ int @ B5 ) ) ) ) ) ) ).

% word_div_def
thf(fact_5698_and__mask__mod__2p,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [W: word @ A,N2: nat] :
          ( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ W @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N2 ) )
          = ( ring_1_of_int @ ( word @ A ) @ ( modulo_modulo @ int @ ( semiring_1_unsigned @ A @ int @ W ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N2 ) ) ) ) ) ).

% and_mask_mod_2p
thf(fact_5699_and__mask__dvd,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [N2: nat,W: word @ A] :
          ( ( dvd_dvd @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N2 ) @ ( semiring_1_unsigned @ A @ int @ W ) )
          = ( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ W @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N2 ) )
            = ( zero_zero @ ( word @ A ) ) ) ) ) ).

% and_mask_dvd
thf(fact_5700_and__mask__no,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [I: num,N2: nat] :
          ( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ I ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N2 ) )
          = ( ring_1_of_int @ ( word @ A ) @ ( bit_se2584673776208193580ke_bit @ int @ N2 @ ( numeral_numeral @ int @ I ) ) ) ) ) ).

% and_mask_no
thf(fact_5701_udvd__incr__lem0,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [Up: int,Uq: int,N2: int,K6: word @ A,N5: int] :
          ( ( ord_less @ int @ Up @ Uq )
         => ( ( Up
              = ( times_times @ int @ N2 @ ( semiring_1_unsigned @ A @ int @ K6 ) ) )
           => ( ( Uq
                = ( times_times @ int @ N5 @ ( semiring_1_unsigned @ A @ int @ K6 ) ) )
             => ( ord_less_eq @ int @ ( plus_plus @ int @ Up @ ( semiring_1_unsigned @ A @ int @ K6 ) ) @ Uq ) ) ) ) ) ).

% udvd_incr_lem0
thf(fact_5702_udvd__incr__lem,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [Up: int,Uq: int,Ua: int,N2: int,K6: word @ A,N5: int] :
          ( ( ord_less @ int @ Up @ Uq )
         => ( ( Up
              = ( plus_plus @ int @ Ua @ ( times_times @ int @ N2 @ ( semiring_1_unsigned @ A @ int @ K6 ) ) ) )
           => ( ( Uq
                = ( plus_plus @ int @ Ua @ ( times_times @ int @ N5 @ ( semiring_1_unsigned @ A @ int @ K6 ) ) ) )
             => ( ord_less_eq @ int @ ( plus_plus @ int @ Up @ ( semiring_1_unsigned @ A @ int @ K6 ) ) @ Uq ) ) ) ) ) ).

% udvd_incr_lem
thf(fact_5703_udvd__incr0,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [P4: word @ A,Q2: word @ A,N2: int,K6: word @ A,N5: int] :
          ( ( ord_less @ ( word @ A ) @ P4 @ Q2 )
         => ( ( ( semiring_1_unsigned @ A @ int @ P4 )
              = ( times_times @ int @ N2 @ ( semiring_1_unsigned @ A @ int @ K6 ) ) )
           => ( ( ( semiring_1_unsigned @ A @ int @ Q2 )
                = ( times_times @ int @ N5 @ ( semiring_1_unsigned @ A @ int @ K6 ) ) )
             => ( ord_less_eq @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ P4 @ K6 ) @ Q2 ) ) ) ) ) ).

% udvd_incr0
thf(fact_5704_udvd__decr0,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [P4: word @ A,Q2: word @ A,N2: int,K6: word @ A,N5: int] :
          ( ( ord_less @ ( word @ A ) @ P4 @ Q2 )
         => ( ( ( semiring_1_unsigned @ A @ int @ P4 )
              = ( times_times @ int @ N2 @ ( semiring_1_unsigned @ A @ int @ K6 ) ) )
           => ( ( ( semiring_1_unsigned @ A @ int @ Q2 )
                = ( times_times @ int @ N5 @ ( semiring_1_unsigned @ A @ int @ K6 ) ) )
             => ( ( ( semiring_1_unsigned @ A @ int @ Q2 )
                  = ( times_times @ int @ N5 @ ( semiring_1_unsigned @ A @ int @ K6 ) ) )
               => ( ord_less_eq @ ( word @ A ) @ P4 @ ( minus_minus @ ( word @ A ) @ Q2 @ K6 ) ) ) ) ) ) ) ).

% udvd_decr0
thf(fact_5705_word__1FF__is__mask,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ( ( numeral_numeral @ ( word @ A ) @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one2 ) ) ) ) ) ) ) ) )
        = ( bit_se2239418461657761734s_mask @ ( word @ A ) @ ( numeral_numeral @ nat @ ( bit1 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) ) ).

% word_1FF_is_mask
thf(fact_5706_word__FF__is__mask,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ( ( numeral_numeral @ ( word @ A ) @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one2 ) ) ) ) ) ) ) )
        = ( bit_se2239418461657761734s_mask @ ( word @ A ) @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) ) ).

% word_FF_is_mask
thf(fact_5707_udvd__incr_H,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [P4: word @ A,Q2: word @ A,Ua: int,N2: int,K6: word @ A,N5: int] :
          ( ( ord_less @ ( word @ A ) @ P4 @ Q2 )
         => ( ( ( semiring_1_unsigned @ A @ int @ P4 )
              = ( plus_plus @ int @ Ua @ ( times_times @ int @ N2 @ ( semiring_1_unsigned @ A @ int @ K6 ) ) ) )
           => ( ( ( semiring_1_unsigned @ A @ int @ Q2 )
                = ( plus_plus @ int @ Ua @ ( times_times @ int @ N5 @ ( semiring_1_unsigned @ A @ int @ K6 ) ) ) )
             => ( ord_less_eq @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ P4 @ K6 ) @ Q2 ) ) ) ) ) ).

% udvd_incr'
thf(fact_5708_udvd__decr_H,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [P4: word @ A,Q2: word @ A,Ua: int,N2: int,K6: word @ A,N5: int] :
          ( ( ord_less @ ( word @ A ) @ P4 @ Q2 )
         => ( ( ( semiring_1_unsigned @ A @ int @ P4 )
              = ( plus_plus @ int @ Ua @ ( times_times @ int @ N2 @ ( semiring_1_unsigned @ A @ int @ K6 ) ) ) )
           => ( ( ( semiring_1_unsigned @ A @ int @ Q2 )
                = ( plus_plus @ int @ Ua @ ( times_times @ int @ N5 @ ( semiring_1_unsigned @ A @ int @ K6 ) ) ) )
             => ( ( ( semiring_1_unsigned @ A @ int @ Q2 )
                  = ( plus_plus @ int @ Ua @ ( times_times @ int @ N5 @ ( semiring_1_unsigned @ A @ int @ K6 ) ) ) )
               => ( ord_less_eq @ ( word @ A ) @ P4 @ ( minus_minus @ ( word @ A ) @ Q2 @ K6 ) ) ) ) ) ) ) ).

% udvd_decr'
thf(fact_5709_semiring__bit__operations__class_Oeven__mask__iff,axiom,
    ! [A: $tType] :
      ( ( bit_se359711467146920520ations @ A )
     => ! [N2: nat] :
          ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se2239418461657761734s_mask @ A @ N2 ) )
          = ( N2
            = ( zero_zero @ nat ) ) ) ) ).

% semiring_bit_operations_class.even_mask_iff
thf(fact_5710_mask__plus__1,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [N2: nat] :
          ( ( plus_plus @ ( word @ A ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N2 ) @ ( one_one @ ( word @ A ) ) )
          = ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N2 ) ) ) ).

% mask_plus_1
thf(fact_5711_less__mask__eq,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [X2: word @ A,N2: nat] :
          ( ( ord_less @ ( word @ A ) @ X2 @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N2 ) )
         => ( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X2 @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N2 ) )
            = X2 ) ) ) ).

% less_mask_eq
thf(fact_5712_mask__eq__decr__exp,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ( ( bit_se2239418461657761734s_mask @ ( word @ A ) )
        = ( ^ [N: nat] : ( minus_minus @ ( word @ A ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N ) @ ( one_one @ ( word @ A ) ) ) ) ) ) ).

% mask_eq_decr_exp
thf(fact_5713_mask__eq__exp__minus__1,axiom,
    ! [A: $tType] :
      ( ( bit_se359711467146920520ations @ A )
     => ( ( bit_se2239418461657761734s_mask @ A )
        = ( ^ [N: nat] : ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) @ ( one_one @ A ) ) ) ) ) ).

% mask_eq_exp_minus_1
thf(fact_5714_mask__Suc__rec,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [N2: nat] :
          ( ( bit_se2239418461657761734s_mask @ ( word @ A ) @ ( suc @ N2 ) )
          = ( plus_plus @ ( word @ A ) @ ( times_times @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N2 ) ) @ ( one_one @ ( word @ A ) ) ) ) ) ).

% mask_Suc_rec
thf(fact_5715_is__aligned__AND__less__0,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [U: word @ A,N2: nat,V: word @ A] :
          ( ( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ U @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N2 ) )
            = ( zero_zero @ ( word @ A ) ) )
         => ( ( ord_less @ ( word @ A ) @ V @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N2 ) )
           => ( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ U @ V )
              = ( zero_zero @ ( word @ A ) ) ) ) ) ) ).

% is_aligned_AND_less_0
thf(fact_5716_add__mask__fold,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [X2: word @ A,N2: nat] :
          ( ( minus_minus @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ X2 @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N2 ) ) @ ( one_one @ ( word @ A ) ) )
          = ( plus_plus @ ( word @ A ) @ X2 @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N2 ) ) ) ) ).

% add_mask_fold
thf(fact_5717_mask__eq__iff__w2p,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [N2: nat,W: word @ A] :
          ( ( ord_less @ nat @ N2 @ ( size_size @ ( word @ A ) @ W ) )
         => ( ( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ W @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N2 ) )
              = W )
            = ( ord_less @ ( word @ A ) @ W @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N2 ) ) ) ) ) ).

% mask_eq_iff_w2p
thf(fact_5718_and__mask__less__size,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [N2: nat,X2: word @ A] :
          ( ( ord_less @ nat @ N2 @ ( size_size @ ( word @ A ) @ X2 ) )
         => ( ord_less @ ( word @ A ) @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X2 @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N2 ) ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N2 ) ) ) ) ).

% and_mask_less_size
thf(fact_5719_word__and__mask__le__2pm1,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [W: word @ A,N2: nat] : ( ord_less_eq @ ( word @ A ) @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ W @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N2 ) ) @ ( minus_minus @ ( word @ A ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N2 ) @ ( one_one @ ( word @ A ) ) ) ) ) ).

% word_and_mask_le_2pm1
thf(fact_5720_uint__range__size,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [W: word @ A] :
          ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( semiring_1_unsigned @ A @ int @ W ) )
          & ( ord_less @ int @ ( semiring_1_unsigned @ A @ int @ W ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( size_size @ ( word @ A ) @ W ) ) ) ) ) ).

% uint_range_size
thf(fact_5721_word__mod__2p__is__mask,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [N2: nat,X2: word @ A] :
          ( ( ord_less @ ( word @ A ) @ ( zero_zero @ ( word @ A ) ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N2 ) )
         => ( ( modulo_modulo @ ( word @ A ) @ X2 @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N2 ) )
            = ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X2 @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N2 ) ) ) ) ) ).

% word_mod_2p_is_mask
thf(fact_5722_uint__2p,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [N2: nat] :
          ( ( ord_less @ ( word @ A ) @ ( zero_zero @ ( word @ A ) ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N2 ) )
         => ( ( semiring_1_unsigned @ A @ int @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N2 ) )
            = ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N2 ) ) ) ) ).

% uint_2p
thf(fact_5723_no__plus__overflow__uint__size,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [X2: word @ A,Y2: word @ A] :
          ( ( ord_less_eq @ ( word @ A ) @ X2 @ ( plus_plus @ ( word @ A ) @ X2 @ Y2 ) )
          = ( ord_less @ int @ ( plus_plus @ int @ ( semiring_1_unsigned @ A @ int @ X2 ) @ ( semiring_1_unsigned @ A @ int @ Y2 ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( size_size @ ( word @ A ) @ X2 ) ) ) ) ) ).

% no_plus_overflow_uint_size
thf(fact_5724_uint__plus__if__size,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [X2: word @ A,Y2: word @ A] :
          ( ( ( ord_less @ int @ ( plus_plus @ int @ ( semiring_1_unsigned @ A @ int @ X2 ) @ ( semiring_1_unsigned @ A @ int @ Y2 ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( size_size @ ( word @ A ) @ X2 ) ) )
           => ( ( semiring_1_unsigned @ A @ int @ ( plus_plus @ ( word @ A ) @ X2 @ Y2 ) )
              = ( plus_plus @ int @ ( semiring_1_unsigned @ A @ int @ X2 ) @ ( semiring_1_unsigned @ A @ int @ Y2 ) ) ) )
          & ( ~ ( ord_less @ int @ ( plus_plus @ int @ ( semiring_1_unsigned @ A @ int @ X2 ) @ ( semiring_1_unsigned @ A @ int @ Y2 ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( size_size @ ( word @ A ) @ X2 ) ) )
           => ( ( semiring_1_unsigned @ A @ int @ ( plus_plus @ ( word @ A ) @ X2 @ Y2 ) )
              = ( minus_minus @ int @ ( plus_plus @ int @ ( semiring_1_unsigned @ A @ int @ X2 ) @ ( semiring_1_unsigned @ A @ int @ Y2 ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( size_size @ ( word @ A ) @ X2 ) ) ) ) ) ) ) ).

% uint_plus_if_size
thf(fact_5725_and__mask__arith,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [W: word @ A,N2: nat] :
          ( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ W @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N2 ) )
          = ( divide_divide @ ( word @ A ) @ ( times_times @ ( word @ A ) @ W @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ ( size_size @ ( word @ A ) @ W ) @ N2 ) ) ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ ( size_size @ ( word @ A ) @ W ) @ N2 ) ) ) ) ) ).

% and_mask_arith
thf(fact_5726_uint__sub__if__size,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [Y2: word @ A,X2: word @ A] :
          ( ( ( ord_less_eq @ int @ ( semiring_1_unsigned @ A @ int @ Y2 ) @ ( semiring_1_unsigned @ A @ int @ X2 ) )
           => ( ( semiring_1_unsigned @ A @ int @ ( minus_minus @ ( word @ A ) @ X2 @ Y2 ) )
              = ( minus_minus @ int @ ( semiring_1_unsigned @ A @ int @ X2 ) @ ( semiring_1_unsigned @ A @ int @ Y2 ) ) ) )
          & ( ~ ( ord_less_eq @ int @ ( semiring_1_unsigned @ A @ int @ Y2 ) @ ( semiring_1_unsigned @ A @ int @ X2 ) )
           => ( ( semiring_1_unsigned @ A @ int @ ( minus_minus @ ( word @ A ) @ X2 @ Y2 ) )
              = ( plus_plus @ int @ ( minus_minus @ int @ ( semiring_1_unsigned @ A @ int @ X2 ) @ ( semiring_1_unsigned @ A @ int @ Y2 ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( size_size @ ( word @ A ) @ X2 ) ) ) ) ) ) ) ).

% uint_sub_if_size
thf(fact_5727_mask__numeral,axiom,
    ! [A: $tType] :
      ( ( bit_se359711467146920520ations @ A )
     => ! [N2: num] :
          ( ( bit_se2239418461657761734s_mask @ A @ ( numeral_numeral @ nat @ N2 ) )
          = ( plus_plus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se2239418461657761734s_mask @ A @ ( pred_numeral @ N2 ) ) ) ) ) ) ).

% mask_numeral
thf(fact_5728_and__minus__numerals_I7_J,axiom,
    ! [N2: num,M: num] :
      ( ( bit_se5824344872417868541ns_and @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ N2 ) ) ) @ ( numeral_numeral @ int @ M ) )
      = ( case_option @ int @ num @ ( zero_zero @ int ) @ ( numeral_numeral @ int ) @ ( bit_and_not_num @ M @ ( bitM @ N2 ) ) ) ) ).

% and_minus_numerals(7)
thf(fact_5729_and__minus__numerals_I3_J,axiom,
    ! [M: num,N2: num] :
      ( ( bit_se5824344872417868541ns_and @ int @ ( numeral_numeral @ int @ M ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ N2 ) ) ) )
      = ( case_option @ int @ num @ ( zero_zero @ int ) @ ( numeral_numeral @ int ) @ ( bit_and_not_num @ M @ ( bitM @ N2 ) ) ) ) ).

% and_minus_numerals(3)
thf(fact_5730_and__minus__numerals_I8_J,axiom,
    ! [N2: num,M: num] :
      ( ( bit_se5824344872417868541ns_and @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ N2 ) ) ) @ ( numeral_numeral @ int @ M ) )
      = ( case_option @ int @ num @ ( zero_zero @ int ) @ ( numeral_numeral @ int ) @ ( bit_and_not_num @ M @ ( bit0 @ N2 ) ) ) ) ).

% and_minus_numerals(8)
thf(fact_5731_mask__nat__positive__iff,axiom,
    ! [N2: nat] :
      ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( bit_se2239418461657761734s_mask @ nat @ N2 ) )
      = ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) ) ).

% mask_nat_positive_iff
thf(fact_5732_Suc__unat__minus__one,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [X2: word @ A] :
          ( ( X2
           != ( zero_zero @ ( word @ A ) ) )
         => ( ( suc @ ( semiring_1_unsigned @ A @ nat @ ( minus_minus @ ( word @ A ) @ X2 @ ( one_one @ ( word @ A ) ) ) ) )
            = ( semiring_1_unsigned @ A @ nat @ X2 ) ) ) ) ).

% Suc_unat_minus_one
thf(fact_5733_and__minus__numerals_I4_J,axiom,
    ! [M: num,N2: num] :
      ( ( bit_se5824344872417868541ns_and @ int @ ( numeral_numeral @ int @ M ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ N2 ) ) ) )
      = ( case_option @ int @ num @ ( zero_zero @ int ) @ ( numeral_numeral @ int ) @ ( bit_and_not_num @ M @ ( bit0 @ N2 ) ) ) ) ).

% and_minus_numerals(4)
thf(fact_5734_word__unat__and__lt,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [X2: word @ A,N2: nat,Y2: word @ A] :
          ( ( ( ord_less @ nat @ ( semiring_1_unsigned @ A @ nat @ X2 ) @ N2 )
            | ( ord_less @ nat @ ( semiring_1_unsigned @ A @ nat @ Y2 ) @ N2 ) )
         => ( ord_less @ nat @ ( semiring_1_unsigned @ A @ nat @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X2 @ Y2 ) ) @ N2 ) ) ) ).

% word_unat_and_lt
thf(fact_5735_le__ucast__ucast__le,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ( type_len @ B )
        & ( type_len @ A ) )
     => ! [X2: word @ A,Y2: word @ B] :
          ( ( ord_less_eq @ ( word @ A ) @ X2 @ ( semiring_1_unsigned @ B @ ( word @ A ) @ Y2 ) )
         => ( ord_less_eq @ ( word @ B ) @ ( semiring_1_unsigned @ A @ ( word @ B ) @ X2 ) @ Y2 ) ) ) ).

% le_ucast_ucast_le
thf(fact_5736_less__eq__mask,axiom,
    ! [N2: nat] : ( ord_less_eq @ nat @ N2 @ ( bit_se2239418461657761734s_mask @ nat @ N2 ) ) ).

% less_eq_mask
thf(fact_5737_ucast__1,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ( type_len @ B )
        & ( type_len @ A ) )
     => ( ( semiring_1_unsigned @ A @ ( word @ B ) @ ( one_one @ ( word @ A ) ) )
        = ( one_one @ ( word @ B ) ) ) ) ).

% ucast_1
thf(fact_5738_word__unat__Rep__inject1,axiom,
    ! [B: $tType,A: $tType] :
      ( ( ( type_len @ A )
        & ( type_len @ B ) )
     => ! [X2: word @ A] :
          ( ( ( semiring_1_unsigned @ A @ nat @ X2 )
            = ( semiring_1_unsigned @ B @ nat @ ( one_one @ ( word @ B ) ) ) )
          = ( X2
            = ( one_one @ ( word @ A ) ) ) ) ) ).

% word_unat_Rep_inject1
thf(fact_5739_unat__1,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ( ( semiring_1_unsigned @ A @ nat @ ( one_one @ ( word @ A ) ) )
        = ( one_one @ nat ) ) ) ).

% unat_1
thf(fact_5740_word__le__nat__alt,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ( ( ord_less_eq @ ( word @ A ) )
        = ( ^ [A5: word @ A,B5: word @ A] : ( ord_less_eq @ nat @ ( semiring_1_unsigned @ A @ nat @ A5 ) @ ( semiring_1_unsigned @ A @ nat @ B5 ) ) ) ) ) ).

% word_le_nat_alt
thf(fact_5741_unat__mono,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [A2: word @ A,B2: word @ A] :
          ( ( ord_less @ ( word @ A ) @ A2 @ B2 )
         => ( ord_less @ nat @ ( semiring_1_unsigned @ A @ nat @ A2 ) @ ( semiring_1_unsigned @ A @ nat @ B2 ) ) ) ) ).

% unat_mono
thf(fact_5742_word__less__nat__alt,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ( ( ord_less @ ( word @ A ) )
        = ( ^ [A5: word @ A,B5: word @ A] : ( ord_less @ nat @ ( semiring_1_unsigned @ A @ nat @ A5 ) @ ( semiring_1_unsigned @ A @ nat @ B5 ) ) ) ) ) ).

% word_less_nat_alt
thf(fact_5743_le__unat__uoi,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [Y2: nat,Z: word @ A] :
          ( ( ord_less_eq @ nat @ Y2 @ ( semiring_1_unsigned @ A @ nat @ Z ) )
         => ( ( semiring_1_unsigned @ A @ nat @ ( semiring_1_of_nat @ ( word @ A ) @ Y2 ) )
            = Y2 ) ) ) ).

% le_unat_uoi
thf(fact_5744_uno__simps_I2_J,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [Z: word @ A,N2: nat] :
          ( ( semiring_1_unsigned @ A @ nat @ ( semiring_1_of_nat @ ( word @ A ) @ ( divide_divide @ nat @ ( semiring_1_unsigned @ A @ nat @ Z ) @ N2 ) ) )
          = ( divide_divide @ nat @ ( semiring_1_unsigned @ A @ nat @ Z ) @ N2 ) ) ) ).

% uno_simps(2)
thf(fact_5745_mask__nonnegative__int,axiom,
    ! [N2: nat] : ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_se2239418461657761734s_mask @ int @ N2 ) ) ).

% mask_nonnegative_int
thf(fact_5746_unat__div,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [X2: word @ A,Y2: word @ A] :
          ( ( semiring_1_unsigned @ A @ nat @ ( divide_divide @ ( word @ A ) @ X2 @ Y2 ) )
          = ( divide_divide @ nat @ ( semiring_1_unsigned @ A @ nat @ X2 ) @ ( semiring_1_unsigned @ A @ nat @ Y2 ) ) ) ) ).

% unat_div
thf(fact_5747_unat__div__distrib,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [V: word @ A,W: word @ A] :
          ( ( semiring_1_unsigned @ A @ nat @ ( divide_divide @ ( word @ A ) @ V @ W ) )
          = ( divide_divide @ nat @ ( semiring_1_unsigned @ A @ nat @ V ) @ ( semiring_1_unsigned @ A @ nat @ W ) ) ) ) ).

% unat_div_distrib
thf(fact_5748_max__lt,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [A2: word @ A,B2: word @ A,C2: word @ A] :
          ( ( semiring_1_unsigned @ A @ nat @ ( divide_divide @ ( word @ A ) @ ( ord_max @ ( word @ A ) @ A2 @ B2 ) @ C2 ) )
          = ( divide_divide @ nat @ ( semiring_1_unsigned @ A @ nat @ ( ord_max @ ( word @ A ) @ A2 @ B2 ) ) @ ( semiring_1_unsigned @ A @ nat @ C2 ) ) ) ) ).

% max_lt
thf(fact_5749_and__not__num_Osimps_I1_J,axiom,
    ( ( bit_and_not_num @ one2 @ one2 )
    = ( none @ num ) ) ).

% and_not_num.simps(1)
thf(fact_5750_unat__eq__1,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [X2: word @ A] :
          ( ( ( semiring_1_unsigned @ A @ nat @ X2 )
            = ( suc @ ( zero_zero @ nat ) ) )
          = ( X2
            = ( one_one @ ( word @ A ) ) ) ) ) ).

% unat_eq_1
thf(fact_5751_unat__gt__0,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [X2: word @ A] :
          ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( semiring_1_unsigned @ A @ nat @ X2 ) )
          = ( X2
           != ( zero_zero @ ( word @ A ) ) ) ) ) ).

% unat_gt_0
thf(fact_5752_un__ui__le,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ( type_len @ B )
        & ( type_len @ A ) )
     => ! [A2: word @ A,B2: word @ B] :
          ( ( ord_less_eq @ nat @ ( semiring_1_unsigned @ A @ nat @ A2 ) @ ( semiring_1_unsigned @ B @ nat @ B2 ) )
          = ( ord_less_eq @ int @ ( semiring_1_unsigned @ A @ int @ A2 ) @ ( semiring_1_unsigned @ B @ int @ B2 ) ) ) ) ).

% un_ui_le
thf(fact_5753_unat__plus__simple,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [X2: word @ A,Y2: word @ A] :
          ( ( ord_less_eq @ ( word @ A ) @ X2 @ ( plus_plus @ ( word @ A ) @ X2 @ Y2 ) )
          = ( ( semiring_1_unsigned @ A @ nat @ ( plus_plus @ ( word @ A ) @ X2 @ Y2 ) )
            = ( plus_plus @ nat @ ( semiring_1_unsigned @ A @ nat @ X2 ) @ ( semiring_1_unsigned @ A @ nat @ Y2 ) ) ) ) ) ).

% unat_plus_simple
thf(fact_5754_unat__sub,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [B2: word @ A,A2: word @ A] :
          ( ( ord_less_eq @ ( word @ A ) @ B2 @ A2 )
         => ( ( semiring_1_unsigned @ A @ nat @ ( minus_minus @ ( word @ A ) @ A2 @ B2 ) )
            = ( minus_minus @ nat @ ( semiring_1_unsigned @ A @ nat @ A2 ) @ ( semiring_1_unsigned @ A @ nat @ B2 ) ) ) ) ) ).

% unat_sub
thf(fact_5755_word__of__nat__less,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [N2: nat,X2: word @ A] :
          ( ( ord_less @ nat @ N2 @ ( semiring_1_unsigned @ A @ nat @ X2 ) )
         => ( ord_less @ ( word @ A ) @ ( semiring_1_of_nat @ ( word @ A ) @ N2 ) @ X2 ) ) ) ).

% word_of_nat_less
thf(fact_5756_unat__less__helper,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [X2: word @ A,N2: nat] :
          ( ( ord_less @ ( word @ A ) @ X2 @ ( semiring_1_of_nat @ ( word @ A ) @ N2 ) )
         => ( ord_less @ nat @ ( semiring_1_unsigned @ A @ nat @ X2 ) @ N2 ) ) ) ).

% unat_less_helper
thf(fact_5757_word__unat__less__le,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [A2: word @ A,B2: nat] :
          ( ( ord_less_eq @ ( word @ A ) @ A2 @ ( semiring_1_of_nat @ ( word @ A ) @ B2 ) )
         => ( ord_less_eq @ nat @ ( semiring_1_unsigned @ A @ nat @ A2 ) @ B2 ) ) ) ).

% word_unat_less_le
thf(fact_5758_word__of__nat__le,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [N2: nat,X2: word @ A] :
          ( ( ord_less_eq @ nat @ N2 @ ( semiring_1_unsigned @ A @ nat @ X2 ) )
         => ( ord_less_eq @ ( word @ A ) @ ( semiring_1_of_nat @ ( word @ A ) @ N2 ) @ X2 ) ) ) ).

% word_of_nat_le
thf(fact_5759_word__arith__nat__add,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ( ( plus_plus @ ( word @ A ) )
        = ( ^ [A5: word @ A,B5: word @ A] : ( semiring_1_of_nat @ ( word @ A ) @ ( plus_plus @ nat @ ( semiring_1_unsigned @ A @ nat @ A5 ) @ ( semiring_1_unsigned @ A @ nat @ B5 ) ) ) ) ) ) ).

% word_arith_nat_add
thf(fact_5760_less__mask,axiom,
    ! [N2: nat] :
      ( ( ord_less @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N2 )
     => ( ord_less @ nat @ N2 @ ( bit_se2239418461657761734s_mask @ nat @ N2 ) ) ) ).

% less_mask
thf(fact_5761_word__arith__nat__mult,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ( ( times_times @ ( word @ A ) )
        = ( ^ [A5: word @ A,B5: word @ A] : ( semiring_1_of_nat @ ( word @ A ) @ ( times_times @ nat @ ( semiring_1_unsigned @ A @ nat @ A5 ) @ ( semiring_1_unsigned @ A @ nat @ B5 ) ) ) ) ) ) ).

% word_arith_nat_mult
thf(fact_5762_word__arith__nat__div,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ( ( divide_divide @ ( word @ A ) )
        = ( ^ [A5: word @ A,B5: word @ A] : ( semiring_1_of_nat @ ( word @ A ) @ ( divide_divide @ nat @ ( semiring_1_unsigned @ A @ nat @ A5 ) @ ( semiring_1_unsigned @ A @ nat @ B5 ) ) ) ) ) ) ).

% word_arith_nat_div
thf(fact_5763_and__not__num_Osimps_I4_J,axiom,
    ! [M: num] :
      ( ( bit_and_not_num @ ( bit0 @ M ) @ one2 )
      = ( some @ num @ ( bit0 @ M ) ) ) ).

% and_not_num.simps(4)
thf(fact_5764_and__not__num_Osimps_I2_J,axiom,
    ! [N2: num] :
      ( ( bit_and_not_num @ one2 @ ( bit0 @ N2 ) )
      = ( some @ num @ one2 ) ) ).

% and_not_num.simps(2)
thf(fact_5765_and__not__num_Osimps_I3_J,axiom,
    ! [N2: num] :
      ( ( bit_and_not_num @ one2 @ ( bit1 @ N2 ) )
      = ( none @ num ) ) ).

% and_not_num.simps(3)
thf(fact_5766_unat__1__0,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [X2: word @ A] :
          ( ( ord_less_eq @ ( word @ A ) @ ( one_one @ ( word @ A ) ) @ X2 )
          = ( ord_less @ nat @ ( zero_zero @ nat ) @ ( semiring_1_unsigned @ A @ nat @ X2 ) ) ) ) ).

% unat_1_0
thf(fact_5767_unat__max__word__pos,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ( ord_less @ nat @ ( zero_zero @ nat ) @ ( semiring_1_unsigned @ A @ nat @ ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) ) ) ) ).

% unat_max_word_pos
thf(fact_5768_unatSuc,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [N2: word @ A] :
          ( ( ( plus_plus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) @ N2 )
           != ( zero_zero @ ( word @ A ) ) )
         => ( ( semiring_1_unsigned @ A @ nat @ ( plus_plus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) @ N2 ) )
            = ( suc @ ( semiring_1_unsigned @ A @ nat @ N2 ) ) ) ) ) ).

% unatSuc
thf(fact_5769_unatSuc2,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [N2: word @ A] :
          ( ( ( plus_plus @ ( word @ A ) @ N2 @ ( one_one @ ( word @ A ) ) )
           != ( zero_zero @ ( word @ A ) ) )
         => ( ( semiring_1_unsigned @ A @ nat @ ( plus_plus @ ( word @ A ) @ N2 @ ( one_one @ ( word @ A ) ) ) )
            = ( suc @ ( semiring_1_unsigned @ A @ nat @ N2 ) ) ) ) ) ).

% unatSuc2
thf(fact_5770_Suc__unat__diff__1,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [X2: word @ A] :
          ( ( ord_less_eq @ ( word @ A ) @ ( one_one @ ( word @ A ) ) @ X2 )
         => ( ( suc @ ( semiring_1_unsigned @ A @ nat @ ( minus_minus @ ( word @ A ) @ X2 @ ( one_one @ ( word @ A ) ) ) ) )
            = ( semiring_1_unsigned @ A @ nat @ X2 ) ) ) ) ).

% Suc_unat_diff_1
thf(fact_5771_unat__Suc2,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [N2: word @ A] :
          ( ( N2
           != ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) )
         => ( ( semiring_1_unsigned @ A @ nat @ ( plus_plus @ ( word @ A ) @ N2 @ ( one_one @ ( word @ A ) ) ) )
            = ( suc @ ( semiring_1_unsigned @ A @ nat @ N2 ) ) ) ) ) ).

% unat_Suc2
thf(fact_5772_uno__simps_I1_J,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [Z: word @ A,M: nat] :
          ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( semiring_1_unsigned @ A @ nat @ Z ) )
         => ( ( semiring_1_unsigned @ A @ nat @ ( semiring_1_of_nat @ ( word @ A ) @ ( modulo_modulo @ nat @ M @ ( semiring_1_unsigned @ A @ nat @ Z ) ) ) )
            = ( modulo_modulo @ nat @ M @ ( semiring_1_unsigned @ A @ nat @ Z ) ) ) ) ) ).

% uno_simps(1)
thf(fact_5773_measure__unat,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [P4: word @ A] :
          ( ( P4
           != ( zero_zero @ ( word @ A ) ) )
         => ( ord_less @ nat @ ( semiring_1_unsigned @ A @ nat @ ( minus_minus @ ( word @ A ) @ P4 @ ( one_one @ ( word @ A ) ) ) ) @ ( semiring_1_unsigned @ A @ nat @ P4 ) ) ) ) ).

% measure_unat
thf(fact_5774_take__bit__eq__mask__iff,axiom,
    ! [N2: nat,K: int] :
      ( ( ( bit_se2584673776208193580ke_bit @ int @ N2 @ K )
        = ( bit_se2239418461657761734s_mask @ int @ N2 ) )
      = ( ( bit_se2584673776208193580ke_bit @ int @ N2 @ ( plus_plus @ int @ K @ ( one_one @ int ) ) )
        = ( zero_zero @ int ) ) ) ).

% take_bit_eq_mask_iff
thf(fact_5775_and__not__num_Osimps_I7_J,axiom,
    ! [M: num] :
      ( ( bit_and_not_num @ ( bit1 @ M ) @ one2 )
      = ( some @ num @ ( bit0 @ M ) ) ) ).

% and_not_num.simps(7)
thf(fact_5776_word__overflow__unat,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [X2: word @ A] :
          ( ( ( semiring_1_unsigned @ A @ nat @ ( plus_plus @ ( word @ A ) @ X2 @ ( one_one @ ( word @ A ) ) ) )
            = ( plus_plus @ nat @ ( semiring_1_unsigned @ A @ nat @ X2 ) @ ( one_one @ nat ) ) )
          | ( ( plus_plus @ ( word @ A ) @ X2 @ ( one_one @ ( word @ A ) ) )
            = ( zero_zero @ ( word @ A ) ) ) ) ) ).

% word_overflow_unat
thf(fact_5777_unat__minus__one,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [W: word @ A] :
          ( ( W
           != ( zero_zero @ ( word @ A ) ) )
         => ( ( semiring_1_unsigned @ A @ nat @ ( minus_minus @ ( word @ A ) @ W @ ( one_one @ ( word @ A ) ) ) )
            = ( minus_minus @ nat @ ( semiring_1_unsigned @ A @ nat @ W ) @ ( one_one @ nat ) ) ) ) ) ).

% unat_minus_one
thf(fact_5778_Suc__mask__eq__exp,axiom,
    ! [N2: nat] :
      ( ( suc @ ( bit_se2239418461657761734s_mask @ nat @ N2 ) )
      = ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ).

% Suc_mask_eq_exp
thf(fact_5779_mask__nat__less__exp,axiom,
    ! [N2: nat] : ( ord_less @ nat @ ( bit_se2239418461657761734s_mask @ nat @ N2 ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ).

% mask_nat_less_exp
thf(fact_5780_lt__plus__1__le__word,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [N2: nat,MaxBound: word @ A,X2: word @ A] :
          ( ( ord_less @ nat @ N2 @ ( semiring_1_unsigned @ A @ nat @ MaxBound ) )
         => ( ( ord_less @ ( word @ A ) @ X2 @ ( plus_plus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) @ ( semiring_1_of_nat @ ( word @ A ) @ N2 ) ) )
            = ( ord_less_eq @ ( word @ A ) @ X2 @ ( semiring_1_of_nat @ ( word @ A ) @ N2 ) ) ) ) ) ).

% lt_plus_1_le_word
thf(fact_5781_mask__half__int,axiom,
    ! [N2: nat] :
      ( ( divide_divide @ int @ ( bit_se2239418461657761734s_mask @ int @ N2 ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) )
      = ( bit_se2239418461657761734s_mask @ int @ ( minus_minus @ nat @ N2 @ ( one_one @ nat ) ) ) ) ).

% mask_half_int
thf(fact_5782_mask__int__def,axiom,
    ( ( bit_se2239418461657761734s_mask @ int )
    = ( ^ [N: nat] : ( minus_minus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) @ ( one_one @ int ) ) ) ) ).

% mask_int_def
thf(fact_5783_mask__nat__def,axiom,
    ( ( bit_se2239418461657761734s_mask @ nat )
    = ( ^ [N: nat] : ( minus_minus @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ ( one_one @ nat ) ) ) ) ).

% mask_nat_def
thf(fact_5784_odd__word__imp__even__next,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [X2: word @ A] :
          ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( semiring_1_unsigned @ A @ nat @ X2 ) )
         => ( ( ( plus_plus @ ( word @ A ) @ X2 @ ( one_one @ ( word @ A ) ) )
              = ( zero_zero @ ( word @ A ) ) )
            | ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( semiring_1_unsigned @ A @ nat @ ( plus_plus @ ( word @ A ) @ X2 @ ( one_one @ ( word @ A ) ) ) ) ) ) ) ) ).

% odd_word_imp_even_next
thf(fact_5785_even__word__imp__odd__next,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [X2: word @ A] :
          ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( semiring_1_unsigned @ A @ nat @ X2 ) )
         => ( ( ( plus_plus @ ( word @ A ) @ X2 @ ( one_one @ ( word @ A ) ) )
              = ( zero_zero @ ( word @ A ) ) )
            | ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( semiring_1_unsigned @ A @ nat @ ( plus_plus @ ( word @ A ) @ X2 @ ( one_one @ ( word @ A ) ) ) ) ) ) ) ) ).

% even_word_imp_odd_next
thf(fact_5786_and__not__num_Osimps_I8_J,axiom,
    ! [M: num,N2: num] :
      ( ( bit_and_not_num @ ( bit1 @ M ) @ ( bit0 @ N2 ) )
      = ( case_option @ ( option @ num ) @ num @ ( some @ num @ one2 )
        @ ^ [N12: num] : ( some @ num @ ( bit1 @ N12 ) )
        @ ( bit_and_not_num @ M @ N2 ) ) ) ).

% and_not_num.simps(8)
thf(fact_5787_word__div__eq__1__iff,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [N2: word @ A,M: word @ A] :
          ( ( ( divide_divide @ ( word @ A ) @ N2 @ M )
            = ( one_one @ ( word @ A ) ) )
          = ( ( ord_less_eq @ ( word @ A ) @ M @ N2 )
            & ( ord_less @ nat @ ( semiring_1_unsigned @ A @ nat @ N2 ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( semiring_1_unsigned @ A @ nat @ M ) ) ) ) ) ) ).

% word_div_eq_1_iff
thf(fact_5788_and__mask__dvd__nat,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [N2: nat,W: word @ A] :
          ( ( dvd_dvd @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) @ ( semiring_1_unsigned @ A @ nat @ W ) )
          = ( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ W @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N2 ) )
            = ( zero_zero @ ( word @ A ) ) ) ) ) ).

% and_mask_dvd_nat
thf(fact_5789_of__nat__eq__size,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [N2: nat,W: word @ A] :
          ( ( ( semiring_1_of_nat @ ( word @ A ) @ N2 )
            = W )
          = ( ? [Q5: nat] :
                ( N2
                = ( plus_plus @ nat @ ( semiring_1_unsigned @ A @ nat @ W ) @ ( times_times @ nat @ Q5 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( size_size @ ( word @ A ) @ W ) ) ) ) ) ) ) ) ).

% of_nat_eq_size
thf(fact_5790_take__bit__eq__mask__iff__exp__dvd,axiom,
    ! [N2: nat,K: int] :
      ( ( ( bit_se2584673776208193580ke_bit @ int @ N2 @ K )
        = ( bit_se2239418461657761734s_mask @ int @ N2 ) )
      = ( dvd_dvd @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N2 ) @ ( plus_plus @ int @ K @ ( one_one @ int ) ) ) ) ).

% take_bit_eq_mask_iff_exp_dvd
thf(fact_5791_unat__plus__if__size,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [X2: word @ A,Y2: word @ A] :
          ( ( ( ord_less @ nat @ ( plus_plus @ nat @ ( semiring_1_unsigned @ A @ nat @ X2 ) @ ( semiring_1_unsigned @ A @ nat @ Y2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( size_size @ ( word @ A ) @ X2 ) ) )
           => ( ( semiring_1_unsigned @ A @ nat @ ( plus_plus @ ( word @ A ) @ X2 @ Y2 ) )
              = ( plus_plus @ nat @ ( semiring_1_unsigned @ A @ nat @ X2 ) @ ( semiring_1_unsigned @ A @ nat @ Y2 ) ) ) )
          & ( ~ ( ord_less @ nat @ ( plus_plus @ nat @ ( semiring_1_unsigned @ A @ nat @ X2 ) @ ( semiring_1_unsigned @ A @ nat @ Y2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( size_size @ ( word @ A ) @ X2 ) ) )
           => ( ( semiring_1_unsigned @ A @ nat @ ( plus_plus @ ( word @ A ) @ X2 @ Y2 ) )
              = ( minus_minus @ nat @ ( plus_plus @ nat @ ( semiring_1_unsigned @ A @ nat @ X2 ) @ ( semiring_1_unsigned @ A @ nat @ Y2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( size_size @ ( word @ A ) @ X2 ) ) ) ) ) ) ) ).

% unat_plus_if_size
thf(fact_5792_unat__sub__if__size,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [Y2: word @ A,X2: word @ A] :
          ( ( ( ord_less_eq @ nat @ ( semiring_1_unsigned @ A @ nat @ Y2 ) @ ( semiring_1_unsigned @ A @ nat @ X2 ) )
           => ( ( semiring_1_unsigned @ A @ nat @ ( minus_minus @ ( word @ A ) @ X2 @ Y2 ) )
              = ( minus_minus @ nat @ ( semiring_1_unsigned @ A @ nat @ X2 ) @ ( semiring_1_unsigned @ A @ nat @ Y2 ) ) ) )
          & ( ~ ( ord_less_eq @ nat @ ( semiring_1_unsigned @ A @ nat @ Y2 ) @ ( semiring_1_unsigned @ A @ nat @ X2 ) )
           => ( ( semiring_1_unsigned @ A @ nat @ ( minus_minus @ ( word @ A ) @ X2 @ Y2 ) )
              = ( minus_minus @ nat @ ( plus_plus @ nat @ ( semiring_1_unsigned @ A @ nat @ X2 ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( size_size @ ( word @ A ) @ X2 ) ) ) @ ( semiring_1_unsigned @ A @ nat @ Y2 ) ) ) ) ) ) ).

% unat_sub_if_size
thf(fact_5793_no__plus__overflow__unat__size,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [X2: word @ A,Y2: word @ A] :
          ( ( ord_less_eq @ ( word @ A ) @ X2 @ ( plus_plus @ ( word @ A ) @ X2 @ Y2 ) )
          = ( ord_less @ nat @ ( plus_plus @ nat @ ( semiring_1_unsigned @ A @ nat @ X2 ) @ ( semiring_1_unsigned @ A @ nat @ Y2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( size_size @ ( word @ A ) @ X2 ) ) ) ) ) ).

% no_plus_overflow_unat_size
thf(fact_5794_word__unat__mask__lt,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [M: nat,W: word @ A] :
          ( ( ord_less_eq @ nat @ M @ ( size_size @ ( word @ A ) @ W ) )
         => ( ord_less @ nat @ ( semiring_1_unsigned @ A @ nat @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ W @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ M ) ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) ) ) ) ).

% word_unat_mask_lt
thf(fact_5795_Code__Abstract__Nat_Otake__bit__num__code_I3_J,axiom,
    ! [N2: nat,M: num] :
      ( ( bit_take_bit_num @ N2 @ ( bit1 @ M ) )
      = ( case_nat @ ( option @ num ) @ ( none @ num )
        @ ^ [N: nat] : ( some @ num @ ( case_option @ num @ num @ one2 @ bit1 @ ( bit_take_bit_num @ N @ M ) ) )
        @ N2 ) ) ).

% Code_Abstract_Nat.take_bit_num_code(3)
thf(fact_5796_arctan__def,axiom,
    ( arctan
    = ( ^ [Y: real] :
          ( the @ real
          @ ^ [X: real] :
              ( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X )
              & ( ord_less @ real @ X @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
              & ( ( tan @ real @ X )
                = Y ) ) ) ) ) ).

% arctan_def
thf(fact_5797_modulo__int__def,axiom,
    ( ( modulo_modulo @ int )
    = ( ^ [K4: int,L2: int] :
          ( if @ int
          @ ( L2
            = ( zero_zero @ int ) )
          @ K4
          @ ( if @ int
            @ ( ( sgn_sgn @ int @ K4 )
              = ( sgn_sgn @ int @ L2 ) )
            @ ( times_times @ int @ ( sgn_sgn @ int @ L2 ) @ ( semiring_1_of_nat @ int @ ( modulo_modulo @ nat @ ( nat2 @ ( abs_abs @ int @ K4 ) ) @ ( nat2 @ ( abs_abs @ int @ L2 ) ) ) ) )
            @ ( times_times @ int @ ( sgn_sgn @ int @ L2 )
              @ ( minus_minus @ int
                @ ( times_times @ int @ ( abs_abs @ int @ L2 )
                  @ ( zero_neq_one_of_bool @ int
                    @ ~ ( dvd_dvd @ int @ L2 @ K4 ) ) )
                @ ( semiring_1_of_nat @ int @ ( modulo_modulo @ nat @ ( nat2 @ ( abs_abs @ int @ K4 ) ) @ ( nat2 @ ( abs_abs @ int @ L2 ) ) ) ) ) ) ) ) ) ) ).

% modulo_int_def
thf(fact_5798_sgn__sgn,axiom,
    ! [A: $tType] :
      ( ( idom_abs_sgn @ A )
     => ! [A2: A] :
          ( ( sgn_sgn @ A @ ( sgn_sgn @ A @ A2 ) )
          = ( sgn_sgn @ A @ A2 ) ) ) ).

% sgn_sgn
thf(fact_5799_sgn__0,axiom,
    ! [A: $tType] :
      ( ( idom_abs_sgn @ A )
     => ( ( sgn_sgn @ A @ ( zero_zero @ A ) )
        = ( zero_zero @ A ) ) ) ).

% sgn_0
thf(fact_5800_sgn__one,axiom,
    ! [A: $tType] :
      ( ( real_V2822296259951069270ebra_1 @ A )
     => ( ( sgn_sgn @ A @ ( one_one @ A ) )
        = ( one_one @ A ) ) ) ).

% sgn_one
thf(fact_5801_sgn__1,axiom,
    ! [A: $tType] :
      ( ( idom_abs_sgn @ A )
     => ( ( sgn_sgn @ A @ ( one_one @ A ) )
        = ( one_one @ A ) ) ) ).

% sgn_1
thf(fact_5802_sgn__divide,axiom,
    ! [A: $tType] :
      ( ( field_abs_sgn @ A )
     => ! [A2: A,B2: A] :
          ( ( sgn_sgn @ A @ ( divide_divide @ A @ A2 @ B2 ) )
          = ( divide_divide @ A @ ( sgn_sgn @ A @ A2 ) @ ( sgn_sgn @ A @ B2 ) ) ) ) ).

% sgn_divide
thf(fact_5803_idom__abs__sgn__class_Osgn__minus,axiom,
    ! [A: $tType] :
      ( ( idom_abs_sgn @ A )
     => ! [A2: A] :
          ( ( sgn_sgn @ A @ ( uminus_uminus @ A @ A2 ) )
          = ( uminus_uminus @ A @ ( sgn_sgn @ A @ A2 ) ) ) ) ).

% idom_abs_sgn_class.sgn_minus
thf(fact_5804_power__sgn,axiom,
    ! [A: $tType] :
      ( ( linordered_idom @ A )
     => ! [A2: A,N2: nat] :
          ( ( sgn_sgn @ A @ ( power_power @ A @ A2 @ N2 ) )
          = ( power_power @ A @ ( sgn_sgn @ A @ A2 ) @ N2 ) ) ) ).

% power_sgn
thf(fact_5805_sgn__less,axiom,
    ! [A: $tType] :
      ( ( linordered_idom @ A )
     => ! [A2: A] :
          ( ( ord_less @ A @ ( sgn_sgn @ A @ A2 ) @ ( zero_zero @ A ) )
          = ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).

% sgn_less
thf(fact_5806_sgn__greater,axiom,
    ! [A: $tType] :
      ( ( linordered_idom @ A )
     => ! [A2: A] :
          ( ( ord_less @ A @ ( zero_zero @ A ) @ ( sgn_sgn @ A @ A2 ) )
          = ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).

% sgn_greater
thf(fact_5807_divide__sgn,axiom,
    ! [A: $tType] :
      ( ( linordered_field @ A )
     => ! [A2: A,B2: A] :
          ( ( divide_divide @ A @ A2 @ ( sgn_sgn @ A @ B2 ) )
          = ( times_times @ A @ A2 @ ( sgn_sgn @ A @ B2 ) ) ) ) ).

% divide_sgn
thf(fact_5808_case__nat__numeral,axiom,
    ! [A: $tType,A2: A,F: nat > A,V: num] :
      ( ( case_nat @ A @ A2 @ F @ ( numeral_numeral @ nat @ V ) )
      = ( F @ ( pred_numeral @ V ) ) ) ).

% case_nat_numeral
thf(fact_5809_sgn__pos,axiom,
    ! [A: $tType] :
      ( ( linordered_idom @ A )
     => ! [A2: A] :
          ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
         => ( ( sgn_sgn @ A @ A2 )
            = ( one_one @ A ) ) ) ) ).

% sgn_pos
thf(fact_5810_abs__sgn__eq__1,axiom,
    ! [A: $tType] :
      ( ( linordered_idom @ A )
     => ! [A2: A] :
          ( ( A2
           != ( zero_zero @ A ) )
         => ( ( abs_abs @ A @ ( sgn_sgn @ A @ A2 ) )
            = ( one_one @ A ) ) ) ) ).

% abs_sgn_eq_1
thf(fact_5811_sgn__mult__self__eq,axiom,
    ! [A: $tType] :
      ( ( linordered_idom @ A )
     => ! [A2: A] :
          ( ( times_times @ A @ ( sgn_sgn @ A @ A2 ) @ ( sgn_sgn @ A @ A2 ) )
          = ( zero_neq_one_of_bool @ A
            @ ( A2
             != ( zero_zero @ A ) ) ) ) ) ).

% sgn_mult_self_eq
thf(fact_5812_sgn__abs,axiom,
    ! [A: $tType] :
      ( ( idom_abs_sgn @ A )
     => ! [A2: A] :
          ( ( abs_abs @ A @ ( sgn_sgn @ A @ A2 ) )
          = ( zero_neq_one_of_bool @ A
            @ ( A2
             != ( zero_zero @ A ) ) ) ) ) ).

% sgn_abs
thf(fact_5813_idom__abs__sgn__class_Oabs__sgn,axiom,
    ! [A: $tType] :
      ( ( idom_abs_sgn @ A )
     => ! [A2: A] :
          ( ( sgn_sgn @ A @ ( abs_abs @ A @ A2 ) )
          = ( zero_neq_one_of_bool @ A
            @ ( A2
             != ( zero_zero @ A ) ) ) ) ) ).

% idom_abs_sgn_class.abs_sgn
thf(fact_5814_sgn__mult__dvd__iff,axiom,
    ! [R4: int,L: int,K: int] :
      ( ( dvd_dvd @ int @ ( times_times @ int @ ( sgn_sgn @ int @ R4 ) @ L ) @ K )
      = ( ( dvd_dvd @ int @ L @ K )
        & ( ( R4
            = ( zero_zero @ int ) )
         => ( K
            = ( zero_zero @ int ) ) ) ) ) ).

% sgn_mult_dvd_iff
thf(fact_5815_mult__sgn__dvd__iff,axiom,
    ! [L: int,R4: int,K: int] :
      ( ( dvd_dvd @ int @ ( times_times @ int @ L @ ( sgn_sgn @ int @ R4 ) ) @ K )
      = ( ( dvd_dvd @ int @ L @ K )
        & ( ( R4
            = ( zero_zero @ int ) )
         => ( K
            = ( zero_zero @ int ) ) ) ) ) ).

% mult_sgn_dvd_iff
thf(fact_5816_dvd__sgn__mult__iff,axiom,
    ! [L: int,R4: int,K: int] :
      ( ( dvd_dvd @ int @ L @ ( times_times @ int @ ( sgn_sgn @ int @ R4 ) @ K ) )
      = ( ( dvd_dvd @ int @ L @ K )
        | ( R4
          = ( zero_zero @ int ) ) ) ) ).

% dvd_sgn_mult_iff
thf(fact_5817_dvd__mult__sgn__iff,axiom,
    ! [L: int,K: int,R4: int] :
      ( ( dvd_dvd @ int @ L @ ( times_times @ int @ K @ ( sgn_sgn @ int @ R4 ) ) )
      = ( ( dvd_dvd @ int @ L @ K )
        | ( R4
          = ( zero_zero @ int ) ) ) ) ).

% dvd_mult_sgn_iff
thf(fact_5818_case__nat__add__eq__if,axiom,
    ! [A: $tType,A2: A,F: nat > A,V: num,N2: nat] :
      ( ( case_nat @ A @ A2 @ F @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ V ) @ N2 ) )
      = ( F @ ( plus_plus @ nat @ ( pred_numeral @ V ) @ N2 ) ) ) ).

% case_nat_add_eq_if
thf(fact_5819_sgn__neg,axiom,
    ! [A: $tType] :
      ( ( linordered_idom @ A )
     => ! [A2: A] :
          ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
         => ( ( sgn_sgn @ A @ A2 )
            = ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ) ).

% sgn_neg
thf(fact_5820_sgn__of__nat,axiom,
    ! [A: $tType] :
      ( ( linordered_idom @ A )
     => ! [N2: nat] :
          ( ( sgn_sgn @ A @ ( semiring_1_of_nat @ A @ N2 ) )
          = ( zero_neq_one_of_bool @ A @ ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ) ).

% sgn_of_nat
thf(fact_5821_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_5822_int__sgnE,axiom,
    ! [K: int] :
      ~ ! [N4: nat,L3: int] :
          ( K
         != ( times_times @ int @ ( sgn_sgn @ int @ L3 ) @ ( semiring_1_of_nat @ int @ N4 ) ) ) ).

% int_sgnE
thf(fact_5823_same__sgn__abs__add,axiom,
    ! [A: $tType] :
      ( ( linordered_idom @ A )
     => ! [B2: A,A2: A] :
          ( ( ( sgn_sgn @ A @ B2 )
            = ( sgn_sgn @ A @ A2 ) )
         => ( ( abs_abs @ A @ ( plus_plus @ A @ A2 @ B2 ) )
            = ( plus_plus @ A @ ( abs_abs @ A @ A2 ) @ ( abs_abs @ A @ B2 ) ) ) ) ) ).

% same_sgn_abs_add
thf(fact_5824_linordered__idom__class_Oabs__sgn,axiom,
    ! [A: $tType] :
      ( ( linordered_idom @ A )
     => ( ( abs_abs @ A )
        = ( ^ [K4: A] : ( times_times @ A @ K4 @ ( sgn_sgn @ A @ K4 ) ) ) ) ) ).

% linordered_idom_class.abs_sgn
thf(fact_5825_abs__mult__sgn,axiom,
    ! [A: $tType] :
      ( ( idom_abs_sgn @ A )
     => ! [A2: A] :
          ( ( times_times @ A @ ( abs_abs @ A @ A2 ) @ ( sgn_sgn @ A @ A2 ) )
          = A2 ) ) ).

% abs_mult_sgn
thf(fact_5826_sgn__mult__abs,axiom,
    ! [A: $tType] :
      ( ( idom_abs_sgn @ A )
     => ! [A2: A] :
          ( ( times_times @ A @ ( sgn_sgn @ A @ A2 ) @ ( abs_abs @ A @ A2 ) )
          = A2 ) ) ).

% sgn_mult_abs
thf(fact_5827_mult__sgn__abs,axiom,
    ! [A: $tType] :
      ( ( linordered_idom @ A )
     => ! [X2: A] :
          ( ( times_times @ A @ ( sgn_sgn @ A @ X2 ) @ ( abs_abs @ A @ X2 ) )
          = X2 ) ) ).

% mult_sgn_abs
thf(fact_5828_old_Onat_Osimps_I5_J,axiom,
    ! [A: $tType,F1: A,F22: nat > A,X23: nat] :
      ( ( case_nat @ A @ F1 @ F22 @ ( suc @ X23 ) )
      = ( F22 @ X23 ) ) ).

% old.nat.simps(5)
thf(fact_5829_old_Onat_Osimps_I4_J,axiom,
    ! [A: $tType,F1: A,F22: nat > A] :
      ( ( case_nat @ A @ F1 @ F22 @ ( zero_zero @ nat ) )
      = F1 ) ).

% old.nat.simps(4)
thf(fact_5830_sgn__0__0,axiom,
    ! [A: $tType] :
      ( ( linordered_idom @ A )
     => ! [A2: A] :
          ( ( ( sgn_sgn @ A @ A2 )
            = ( zero_zero @ A ) )
          = ( A2
            = ( zero_zero @ A ) ) ) ) ).

% sgn_0_0
thf(fact_5831_sgn__eq__0__iff,axiom,
    ! [A: $tType] :
      ( ( idom_abs_sgn @ A )
     => ! [A2: A] :
          ( ( ( sgn_sgn @ A @ A2 )
            = ( zero_zero @ A ) )
          = ( A2
            = ( zero_zero @ A ) ) ) ) ).

% sgn_eq_0_iff
thf(fact_5832_same__sgn__sgn__add,axiom,
    ! [A: $tType] :
      ( ( linordered_idom @ A )
     => ! [B2: A,A2: A] :
          ( ( ( sgn_sgn @ A @ B2 )
            = ( sgn_sgn @ A @ A2 ) )
         => ( ( sgn_sgn @ A @ ( plus_plus @ A @ A2 @ B2 ) )
            = ( sgn_sgn @ A @ A2 ) ) ) ) ).

% same_sgn_sgn_add
thf(fact_5833_nat_Ocase__distrib,axiom,
    ! [B: $tType,A: $tType,H2: A > B,F1: A,F22: nat > A,Nat: nat] :
      ( ( H2 @ ( case_nat @ A @ F1 @ F22 @ Nat ) )
      = ( case_nat @ B @ ( H2 @ F1 )
        @ ^ [X: nat] : ( H2 @ ( F22 @ X ) )
        @ Nat ) ) ).

% nat.case_distrib
thf(fact_5834_sgn__mult,axiom,
    ! [A: $tType] :
      ( ( idom_abs_sgn @ A )
     => ! [A2: A,B2: A] :
          ( ( sgn_sgn @ A @ ( times_times @ A @ A2 @ B2 ) )
          = ( times_times @ A @ ( sgn_sgn @ A @ A2 ) @ ( sgn_sgn @ A @ B2 ) ) ) ) ).

% sgn_mult
thf(fact_5835_Real__Vector__Spaces_Osgn__mult,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra @ A )
     => ! [X2: A,Y2: A] :
          ( ( sgn_sgn @ A @ ( times_times @ A @ X2 @ Y2 ) )
          = ( times_times @ A @ ( sgn_sgn @ A @ X2 ) @ ( sgn_sgn @ A @ Y2 ) ) ) ) ).

% Real_Vector_Spaces.sgn_mult
thf(fact_5836_sgn__minus__1,axiom,
    ! [A: $tType] :
      ( ( idom_abs_sgn @ A )
     => ( ( sgn_sgn @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
        = ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).

% sgn_minus_1
thf(fact_5837_sgn__not__eq__imp,axiom,
    ! [A: $tType] :
      ( ( linordered_idom @ A )
     => ! [B2: A,A2: A] :
          ( ( ( sgn_sgn @ A @ B2 )
           != ( sgn_sgn @ A @ A2 ) )
         => ( ( ( sgn_sgn @ A @ A2 )
             != ( zero_zero @ A ) )
           => ( ( ( sgn_sgn @ A @ B2 )
               != ( zero_zero @ A ) )
             => ( ( sgn_sgn @ A @ A2 )
                = ( uminus_uminus @ A @ ( sgn_sgn @ A @ B2 ) ) ) ) ) ) ) ).

% sgn_not_eq_imp
thf(fact_5838_ln__real__def,axiom,
    ( ( ln_ln @ real )
    = ( ^ [X: real] :
          ( the @ real
          @ ^ [U2: real] :
              ( ( exp @ real @ U2 )
              = X ) ) ) ) ).

% ln_real_def
thf(fact_5839_suminf__def,axiom,
    ! [A: $tType] :
      ( ( ( comm_monoid_add @ A )
        & ( topolo4958980785337419405_space @ A ) )
     => ( ( suminf @ A )
        = ( ^ [F6: nat > A] : ( the @ A @ ( sums @ A @ F6 ) ) ) ) ) ).

% suminf_def
thf(fact_5840_sgn__1__pos,axiom,
    ! [A: $tType] :
      ( ( linordered_idom @ A )
     => ! [A2: A] :
          ( ( ( sgn_sgn @ A @ A2 )
            = ( one_one @ A ) )
          = ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).

% sgn_1_pos
thf(fact_5841_abs__sgn__eq,axiom,
    ! [A: $tType] :
      ( ( linordered_idom @ A )
     => ! [A2: A] :
          ( ( ( A2
              = ( zero_zero @ A ) )
           => ( ( abs_abs @ A @ ( sgn_sgn @ A @ A2 ) )
              = ( zero_zero @ A ) ) )
          & ( ( A2
             != ( zero_zero @ A ) )
           => ( ( abs_abs @ A @ ( sgn_sgn @ A @ A2 ) )
              = ( one_one @ A ) ) ) ) ) ).

% abs_sgn_eq
thf(fact_5842_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_5843_ln__neg__is__const,axiom,
    ! [X2: real] :
      ( ( ord_less_eq @ real @ X2 @ ( zero_zero @ real ) )
     => ( ( ln_ln @ real @ X2 )
        = ( the @ real
          @ ^ [X: real] : $false ) ) ) ).

% ln_neg_is_const
thf(fact_5844_sgn__if,axiom,
    ! [A: $tType] :
      ( ( linordered_idom @ A )
     => ( ( sgn_sgn @ A )
        = ( ^ [X: A] :
              ( if @ A
              @ ( X
                = ( zero_zero @ A ) )
              @ ( zero_zero @ A )
              @ ( if @ A @ ( ord_less @ A @ ( zero_zero @ A ) @ X ) @ ( one_one @ A ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ) ) ) ).

% sgn_if
thf(fact_5845_sgn__1__neg,axiom,
    ! [A: $tType] :
      ( ( linordered_idom @ A )
     => ! [A2: A] :
          ( ( ( sgn_sgn @ A @ A2 )
            = ( uminus_uminus @ A @ ( one_one @ A ) ) )
          = ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).

% sgn_1_neg
thf(fact_5846_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_5847_norm__sgn,axiom,
    ! [A: $tType] :
      ( ( real_V822414075346904944vector @ A )
     => ! [X2: A] :
          ( ( ( X2
              = ( zero_zero @ A ) )
           => ( ( real_V7770717601297561774m_norm @ A @ ( sgn_sgn @ A @ X2 ) )
              = ( zero_zero @ real ) ) )
          & ( ( X2
             != ( zero_zero @ A ) )
           => ( ( real_V7770717601297561774m_norm @ A @ ( sgn_sgn @ A @ X2 ) )
              = ( one_one @ real ) ) ) ) ) ).

% norm_sgn
thf(fact_5848_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_5849_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_5850_arccos__def,axiom,
    ( arccos
    = ( ^ [Y: real] :
          ( the @ real
          @ ^ [X: real] :
              ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
              & ( ord_less_eq @ real @ X @ pi )
              & ( ( cos @ real @ X )
                = Y ) ) ) ) ) ).

% arccos_def
thf(fact_5851_eucl__rel__int__remainderI,axiom,
    ! [R4: int,L: int,K: int,Q2: int] :
      ( ( ( sgn_sgn @ int @ R4 )
        = ( sgn_sgn @ int @ L ) )
     => ( ( ord_less @ int @ ( abs_abs @ int @ R4 ) @ ( abs_abs @ int @ L ) )
       => ( ( K
            = ( plus_plus @ int @ ( times_times @ int @ Q2 @ L ) @ R4 ) )
         => ( eucl_rel_int @ K @ L @ ( product_Pair @ int @ int @ Q2 @ R4 ) ) ) ) ) ).

% eucl_rel_int_remainderI
thf(fact_5852_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_neq_one_of_bool @ int
              @ ~ ( dvd_dvd @ int @ L @ K ) ) ) ) ) ) ).

% div_noneq_sgn_abs
thf(fact_5853_eucl__rel__int_Ocases,axiom,
    ! [A12: int,A23: int,A32: product_prod @ int @ int] :
      ( ( eucl_rel_int @ A12 @ A23 @ A32 )
     => ( ( ( A23
            = ( zero_zero @ int ) )
         => ( A32
           != ( product_Pair @ int @ int @ ( zero_zero @ int ) @ A12 ) ) )
       => ( ! [Q3: int] :
              ( ( A32
                = ( product_Pair @ int @ int @ Q3 @ ( zero_zero @ int ) ) )
             => ( ( A23
                 != ( zero_zero @ int ) )
               => ( A12
                 != ( times_times @ int @ Q3 @ A23 ) ) ) )
         => ~ ! [R3: int,Q3: int] :
                ( ( A32
                  = ( product_Pair @ int @ int @ Q3 @ R3 ) )
               => ( ( ( sgn_sgn @ int @ R3 )
                    = ( sgn_sgn @ int @ A23 ) )
                 => ( ( ord_less @ int @ ( abs_abs @ int @ R3 ) @ ( abs_abs @ int @ A23 ) )
                   => ( A12
                     != ( plus_plus @ int @ ( times_times @ int @ Q3 @ A23 ) @ R3 ) ) ) ) ) ) ) ) ).

% eucl_rel_int.cases
thf(fact_5854_eucl__rel__int_Osimps,axiom,
    ( eucl_rel_int
    = ( ^ [A1: int,A22: int,A33: product_prod @ int @ int] :
          ( ? [K4: int] :
              ( ( A1 = K4 )
              & ( A22
                = ( zero_zero @ int ) )
              & ( A33
                = ( product_Pair @ int @ int @ ( zero_zero @ int ) @ K4 ) ) )
          | ? [L2: int,K4: int,Q5: int] :
              ( ( A1 = K4 )
              & ( A22 = L2 )
              & ( A33
                = ( product_Pair @ int @ int @ Q5 @ ( zero_zero @ int ) ) )
              & ( L2
               != ( zero_zero @ int ) )
              & ( K4
                = ( times_times @ int @ Q5 @ L2 ) ) )
          | ? [R: int,L2: int,K4: int,Q5: int] :
              ( ( A1 = K4 )
              & ( A22 = L2 )
              & ( A33
                = ( product_Pair @ int @ int @ Q5 @ R ) )
              & ( ( sgn_sgn @ int @ R )
                = ( sgn_sgn @ int @ L2 ) )
              & ( ord_less @ int @ ( abs_abs @ int @ R ) @ ( abs_abs @ int @ L2 ) )
              & ( K4
                = ( plus_plus @ int @ ( times_times @ int @ Q5 @ L2 ) @ R ) ) ) ) ) ) ).

% eucl_rel_int.simps
thf(fact_5855_Code__Abstract__Nat_Otake__bit__num__code_I1_J,axiom,
    ! [N2: nat] :
      ( ( bit_take_bit_num @ N2 @ one2 )
      = ( case_nat @ ( option @ num ) @ ( none @ num )
        @ ^ [N: nat] : ( some @ num @ one2 )
        @ N2 ) ) ).

% Code_Abstract_Nat.take_bit_num_code(1)
thf(fact_5856_pi__half,axiom,
    ( ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) )
    = ( the @ real
      @ ^ [X: real] :
          ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
          & ( ord_less_eq @ real @ X @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) )
          & ( ( cos @ real @ X )
            = ( zero_zero @ real ) ) ) ) ) ).

% pi_half
thf(fact_5857_pi__def,axiom,
    ( pi
    = ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) )
      @ ( the @ real
        @ ^ [X: real] :
            ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
            & ( ord_less_eq @ real @ X @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) )
            & ( ( cos @ real @ X )
              = ( zero_zero @ real ) ) ) ) ) ) ).

% pi_def
thf(fact_5858_Code__Abstract__Nat_Otake__bit__num__code_I2_J,axiom,
    ! [N2: nat,M: num] :
      ( ( bit_take_bit_num @ N2 @ ( bit0 @ M ) )
      = ( case_nat @ ( option @ num ) @ ( none @ num )
        @ ^ [N: nat] :
            ( case_option @ ( option @ num ) @ num @ ( none @ num )
            @ ^ [Q5: num] : ( some @ num @ ( bit0 @ Q5 ) )
            @ ( bit_take_bit_num @ N @ M ) )
        @ N2 ) ) ).

% Code_Abstract_Nat.take_bit_num_code(2)
thf(fact_5859_divide__int__unfold,axiom,
    ! [L: int,K: int,N2: nat,M: nat] :
      ( ( ( ( ( sgn_sgn @ int @ L )
            = ( zero_zero @ int ) )
          | ( ( sgn_sgn @ int @ K )
            = ( zero_zero @ int ) )
          | ( N2
            = ( zero_zero @ nat ) ) )
       => ( ( divide_divide @ int @ ( times_times @ int @ ( sgn_sgn @ int @ K ) @ ( semiring_1_of_nat @ int @ M ) ) @ ( times_times @ int @ ( sgn_sgn @ int @ L ) @ ( semiring_1_of_nat @ int @ N2 ) ) )
          = ( zero_zero @ int ) ) )
      & ( ~ ( ( ( sgn_sgn @ int @ L )
              = ( zero_zero @ int ) )
            | ( ( sgn_sgn @ int @ K )
              = ( zero_zero @ int ) )
            | ( N2
              = ( zero_zero @ nat ) ) )
       => ( ( ( ( sgn_sgn @ int @ K )
              = ( sgn_sgn @ int @ L ) )
           => ( ( divide_divide @ int @ ( times_times @ int @ ( sgn_sgn @ int @ K ) @ ( semiring_1_of_nat @ int @ M ) ) @ ( times_times @ int @ ( sgn_sgn @ int @ L ) @ ( semiring_1_of_nat @ int @ N2 ) ) )
              = ( semiring_1_of_nat @ int @ ( divide_divide @ nat @ M @ N2 ) ) ) )
          & ( ( ( sgn_sgn @ int @ K )
             != ( sgn_sgn @ int @ L ) )
           => ( ( divide_divide @ int @ ( times_times @ int @ ( sgn_sgn @ int @ K ) @ ( semiring_1_of_nat @ int @ M ) ) @ ( times_times @ int @ ( sgn_sgn @ int @ L ) @ ( semiring_1_of_nat @ int @ N2 ) ) )
              = ( uminus_uminus @ int
                @ ( semiring_1_of_nat @ int
                  @ ( plus_plus @ nat @ ( divide_divide @ nat @ M @ N2 )
                    @ ( zero_neq_one_of_bool @ nat
                      @ ~ ( dvd_dvd @ nat @ N2 @ M ) ) ) ) ) ) ) ) ) ) ).

% divide_int_unfold
thf(fact_5860_modulo__int__unfold,axiom,
    ! [L: int,K: int,N2: nat,M: nat] :
      ( ( ( ( ( sgn_sgn @ int @ L )
            = ( zero_zero @ int ) )
          | ( ( sgn_sgn @ int @ K )
            = ( zero_zero @ int ) )
          | ( N2
            = ( zero_zero @ nat ) ) )
       => ( ( modulo_modulo @ int @ ( times_times @ int @ ( sgn_sgn @ int @ K ) @ ( semiring_1_of_nat @ int @ M ) ) @ ( times_times @ int @ ( sgn_sgn @ int @ L ) @ ( semiring_1_of_nat @ int @ N2 ) ) )
          = ( times_times @ int @ ( sgn_sgn @ int @ K ) @ ( semiring_1_of_nat @ int @ M ) ) ) )
      & ( ~ ( ( ( sgn_sgn @ int @ L )
              = ( zero_zero @ int ) )
            | ( ( sgn_sgn @ int @ K )
              = ( zero_zero @ int ) )
            | ( N2
              = ( zero_zero @ nat ) ) )
       => ( ( ( ( sgn_sgn @ int @ K )
              = ( sgn_sgn @ int @ L ) )
           => ( ( modulo_modulo @ int @ ( times_times @ int @ ( sgn_sgn @ int @ K ) @ ( semiring_1_of_nat @ int @ M ) ) @ ( times_times @ int @ ( sgn_sgn @ int @ L ) @ ( semiring_1_of_nat @ int @ N2 ) ) )
              = ( times_times @ int @ ( sgn_sgn @ int @ L ) @ ( semiring_1_of_nat @ int @ ( modulo_modulo @ nat @ M @ N2 ) ) ) ) )
          & ( ( ( sgn_sgn @ int @ K )
             != ( sgn_sgn @ int @ L ) )
           => ( ( modulo_modulo @ int @ ( times_times @ int @ ( sgn_sgn @ int @ K ) @ ( semiring_1_of_nat @ int @ M ) ) @ ( times_times @ int @ ( sgn_sgn @ int @ L ) @ ( semiring_1_of_nat @ int @ N2 ) ) )
              = ( times_times @ int @ ( sgn_sgn @ int @ L )
                @ ( minus_minus @ int
                  @ ( semiring_1_of_nat @ int
                    @ ( times_times @ nat @ N2
                      @ ( zero_neq_one_of_bool @ nat
                        @ ~ ( dvd_dvd @ nat @ N2 @ M ) ) ) )
                  @ ( semiring_1_of_nat @ int @ ( modulo_modulo @ nat @ M @ N2 ) ) ) ) ) ) ) ) ) ).

% modulo_int_unfold
thf(fact_5861_divide__int__def,axiom,
    ( ( divide_divide @ int )
    = ( ^ [K4: int,L2: int] :
          ( if @ int
          @ ( L2
            = ( zero_zero @ int ) )
          @ ( zero_zero @ int )
          @ ( if @ int
            @ ( ( sgn_sgn @ int @ K4 )
              = ( sgn_sgn @ int @ L2 ) )
            @ ( semiring_1_of_nat @ int @ ( divide_divide @ nat @ ( nat2 @ ( abs_abs @ int @ K4 ) ) @ ( nat2 @ ( abs_abs @ int @ L2 ) ) ) )
            @ ( uminus_uminus @ int
              @ ( semiring_1_of_nat @ int
                @ ( plus_plus @ nat @ ( divide_divide @ nat @ ( nat2 @ ( abs_abs @ int @ K4 ) ) @ ( nat2 @ ( abs_abs @ int @ L2 ) ) )
                  @ ( zero_neq_one_of_bool @ nat
                    @ ~ ( dvd_dvd @ int @ L2 @ K4 ) ) ) ) ) ) ) ) ) ).

% divide_int_def
thf(fact_5862_arcsin__def,axiom,
    ( arcsin
    = ( ^ [Y: real] :
          ( the @ real
          @ ^ [X: real] :
              ( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X )
              & ( ord_less_eq @ real @ X @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
              & ( ( sin @ real @ X )
                = Y ) ) ) ) ) ).

% arcsin_def
thf(fact_5863_old_Orec__prod__def,axiom,
    ! [T: $tType,B: $tType,A: $tType] :
      ( ( product_rec_prod @ A @ B @ T )
      = ( ^ [F12: A > B > T,X: product_prod @ A @ B] : ( the @ T @ ( product_rec_set_prod @ A @ B @ T @ F12 @ X ) ) ) ) ).

% old.rec_prod_def
thf(fact_5864_Bit__Operations_Otake__bit__num__code,axiom,
    ( bit_take_bit_num
    = ( ^ [N: nat,M7: num] :
          ( product_case_prod @ nat @ num @ ( option @ num )
          @ ^ [A5: nat,X: num] :
              ( case_nat @ ( option @ num ) @ ( none @ num )
              @ ^ [O: nat] :
                  ( case_num @ ( option @ num ) @ ( some @ num @ one2 )
                  @ ^ [P5: num] :
                      ( case_option @ ( option @ num ) @ num @ ( none @ num )
                      @ ^ [Q5: num] : ( some @ num @ ( bit0 @ Q5 ) )
                      @ ( bit_take_bit_num @ O @ P5 ) )
                  @ ^ [P5: num] : ( some @ num @ ( case_option @ num @ num @ one2 @ bit1 @ ( bit_take_bit_num @ O @ P5 ) ) )
                  @ X )
              @ A5 )
          @ ( product_Pair @ nat @ num @ N @ M7 ) ) ) ) ).

% Bit_Operations.take_bit_num_code
thf(fact_5865_sgn__div__eq__sgn__mult,axiom,
    ! [A2: int,B2: int] :
      ( ( ( divide_divide @ int @ A2 @ B2 )
       != ( zero_zero @ int ) )
     => ( ( sgn_sgn @ int @ ( divide_divide @ int @ A2 @ B2 ) )
        = ( sgn_sgn @ int @ ( times_times @ int @ A2 @ B2 ) ) ) ) ).

% sgn_div_eq_sgn_mult
thf(fact_5866_zero__le__sgn__iff,axiom,
    ! [X2: real] :
      ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( sgn_sgn @ real @ X2 ) )
      = ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 ) ) ).

% zero_le_sgn_iff
thf(fact_5867_sgn__le__0__iff,axiom,
    ! [X2: real] :
      ( ( ord_less_eq @ real @ ( sgn_sgn @ real @ X2 ) @ ( zero_zero @ real ) )
      = ( ord_less_eq @ real @ X2 @ ( zero_zero @ real ) ) ) ).

% sgn_le_0_iff
thf(fact_5868_The__split__eq,axiom,
    ! [A: $tType,B: $tType,X2: A,Y2: B] :
      ( ( the @ ( product_prod @ A @ B )
        @ ( product_case_prod @ A @ B @ $o
          @ ^ [X17: A,Y8: B] :
              ( ( X2 = X17 )
              & ( Y2 = Y8 ) ) ) )
      = ( product_Pair @ A @ B @ X2 @ Y2 ) ) ).

% The_split_eq
thf(fact_5869_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_5870_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_5871_real__sgn__eq,axiom,
    ( ( sgn_sgn @ real )
    = ( ^ [X: real] : ( divide_divide @ real @ X @ ( abs_abs @ real @ X ) ) ) ) ).

% real_sgn_eq
thf(fact_5872_num_Ocase__distrib,axiom,
    ! [B: $tType,A: $tType,H2: A > B,F1: A,F22: num > A,F32: num > A,Num: num] :
      ( ( H2 @ ( case_num @ A @ F1 @ F22 @ F32 @ Num ) )
      = ( case_num @ B @ ( H2 @ F1 )
        @ ^ [X: num] : ( H2 @ ( F22 @ X ) )
        @ ^ [X: num] : ( H2 @ ( F32 @ X ) )
        @ Num ) ) ).

% num.case_distrib
thf(fact_5873_verit__eq__simplify_I17_J,axiom,
    ! [A: $tType,F1: A,F22: num > A,F32: num > A,X23: num] :
      ( ( case_num @ A @ F1 @ F22 @ F32 @ ( bit0 @ X23 ) )
      = ( F22 @ X23 ) ) ).

% verit_eq_simplify(17)
thf(fact_5874_verit__eq__simplify_I16_J,axiom,
    ! [A: $tType,F1: A,F22: num > A,F32: num > A] :
      ( ( case_num @ A @ F1 @ F22 @ F32 @ one2 )
      = F1 ) ).

% verit_eq_simplify(16)
thf(fact_5875_sgn__root,axiom,
    ! [N2: nat,X2: real] :
      ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
     => ( ( sgn_sgn @ real @ ( root @ N2 @ X2 ) )
        = ( sgn_sgn @ real @ X2 ) ) ) ).

% sgn_root
thf(fact_5876_sgn__eq,axiom,
    ( ( sgn_sgn @ complex )
    = ( ^ [Z2: complex] : ( divide_divide @ complex @ Z2 @ ( real_Vector_of_real @ complex @ ( real_V7770717601297561774m_norm @ complex @ Z2 ) ) ) ) ) ).

% sgn_eq
thf(fact_5877_less__eq__nat_Osimps_I2_J,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq @ nat @ ( suc @ M ) @ N2 )
      = ( case_nat @ $o @ $false @ ( ord_less_eq @ nat @ M ) @ N2 ) ) ).

% less_eq_nat.simps(2)
thf(fact_5878_max__Suc2,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_max @ nat @ M @ ( suc @ N2 ) )
      = ( case_nat @ nat @ ( suc @ N2 )
        @ ^ [M5: nat] : ( suc @ ( ord_max @ nat @ M5 @ N2 ) )
        @ M ) ) ).

% max_Suc2
thf(fact_5879_max__Suc1,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_max @ nat @ ( suc @ N2 ) @ M )
      = ( case_nat @ nat @ ( suc @ N2 )
        @ ^ [M5: nat] : ( suc @ ( ord_max @ nat @ N2 @ M5 ) )
        @ M ) ) ).

% max_Suc1
thf(fact_5880_The__case__prod,axiom,
    ! [B: $tType,A: $tType,P: A > B > $o] :
      ( ( the @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ P ) )
      = ( the @ ( product_prod @ A @ B )
        @ ^ [Xy: product_prod @ A @ B] : ( P @ ( product_fst @ A @ B @ Xy ) @ ( product_snd @ A @ B @ Xy ) ) ) ) ).

% The_case_prod
thf(fact_5881_sgn__real__def,axiom,
    ( ( sgn_sgn @ real )
    = ( ^ [A5: real] :
          ( if @ real
          @ ( A5
            = ( zero_zero @ real ) )
          @ ( zero_zero @ real )
          @ ( if @ real @ ( ord_less @ real @ ( zero_zero @ real ) @ A5 ) @ ( one_one @ real ) @ ( uminus_uminus @ real @ ( one_one @ real ) ) ) ) ) ) ).

% sgn_real_def
thf(fact_5882_sgn__integer__code,axiom,
    ( ( sgn_sgn @ code_integer )
    = ( ^ [K4: code_integer] :
          ( if @ code_integer
          @ ( K4
            = ( zero_zero @ code_integer ) )
          @ ( zero_zero @ code_integer )
          @ ( if @ code_integer @ ( ord_less @ code_integer @ K4 @ ( zero_zero @ code_integer ) ) @ ( uminus_uminus @ code_integer @ ( one_one @ code_integer ) ) @ ( one_one @ code_integer ) ) ) ) ) ).

% sgn_integer_code
thf(fact_5883_diff__Suc,axiom,
    ! [M: nat,N2: nat] :
      ( ( minus_minus @ nat @ M @ ( suc @ N2 ) )
      = ( case_nat @ nat @ ( zero_zero @ nat )
        @ ^ [K4: nat] : K4
        @ ( minus_minus @ nat @ M @ N2 ) ) ) ).

% diff_Suc
thf(fact_5884_sgn__power__injE,axiom,
    ! [A2: real,N2: nat,X2: real,B2: real] :
      ( ( ( times_times @ real @ ( sgn_sgn @ real @ A2 ) @ ( power_power @ real @ ( abs_abs @ real @ A2 ) @ N2 ) )
        = X2 )
     => ( ( X2
          = ( times_times @ real @ ( sgn_sgn @ real @ B2 ) @ ( power_power @ real @ ( abs_abs @ real @ B2 ) @ N2 ) ) )
       => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
         => ( A2 = B2 ) ) ) ) ).

% sgn_power_injE
thf(fact_5885_sgn__power__root,axiom,
    ! [N2: nat,X2: real] :
      ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
     => ( ( times_times @ real @ ( sgn_sgn @ real @ ( root @ N2 @ X2 ) ) @ ( power_power @ real @ ( abs_abs @ real @ ( root @ N2 @ X2 ) ) @ N2 ) )
        = X2 ) ) ).

% sgn_power_root
thf(fact_5886_root__sgn__power,axiom,
    ! [N2: nat,Y2: real] :
      ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
     => ( ( root @ N2 @ ( times_times @ real @ ( sgn_sgn @ real @ Y2 ) @ ( power_power @ real @ ( abs_abs @ real @ Y2 ) @ N2 ) ) )
        = Y2 ) ) ).

% root_sgn_power
thf(fact_5887_cis__Arg__unique,axiom,
    ! [Z: complex,X2: real] :
      ( ( ( sgn_sgn @ complex @ Z )
        = ( cis @ X2 ) )
     => ( ( ord_less @ real @ ( uminus_uminus @ real @ pi ) @ X2 )
       => ( ( ord_less_eq @ real @ X2 @ pi )
         => ( ( arg @ Z )
            = X2 ) ) ) ) ).

% cis_Arg_unique
thf(fact_5888_split__root,axiom,
    ! [P: real > $o,N2: nat,X2: real] :
      ( ( P @ ( root @ N2 @ X2 ) )
      = ( ( ( N2
            = ( zero_zero @ nat ) )
         => ( P @ ( zero_zero @ real ) ) )
        & ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
         => ! [Y: real] :
              ( ( ( times_times @ real @ ( sgn_sgn @ real @ Y ) @ ( power_power @ real @ ( abs_abs @ real @ Y ) @ N2 ) )
                = X2 )
             => ( P @ Y ) ) ) ) ) ).

% split_root
thf(fact_5889_floor__real__def,axiom,
    ( ( archim6421214686448440834_floor @ real )
    = ( ^ [X: real] :
          ( the @ int
          @ ^ [Z2: int] :
              ( ( ord_less_eq @ real @ ( ring_1_of_int @ real @ Z2 ) @ X )
              & ( ord_less @ real @ X @ ( ring_1_of_int @ real @ ( plus_plus @ int @ Z2 @ ( one_one @ int ) ) ) ) ) ) ) ) ).

% floor_real_def
thf(fact_5890_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_5891_arctan__inverse,axiom,
    ! [X2: real] :
      ( ( X2
       != ( zero_zero @ real ) )
     => ( ( arctan @ ( divide_divide @ real @ ( one_one @ real ) @ X2 ) )
        = ( minus_minus @ real @ ( divide_divide @ real @ ( times_times @ real @ ( sgn_sgn @ real @ X2 ) @ pi ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( arctan @ X2 ) ) ) ) ).

% arctan_inverse
thf(fact_5892_Nitpick_Ocase__nat__unfold,axiom,
    ! [A: $tType] :
      ( ( case_nat @ A )
      = ( ^ [X: A,F6: nat > A,N: nat] :
            ( if @ A
            @ ( N
              = ( zero_zero @ nat ) )
            @ X
            @ ( F6 @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) ) ) ) ).

% Nitpick.case_nat_unfold
thf(fact_5893_the__equality,axiom,
    ! [A: $tType,P: A > $o,A2: A] :
      ( ( P @ A2 )
     => ( ! [X3: A] :
            ( ( P @ X3 )
           => ( X3 = A2 ) )
       => ( ( the @ A @ P )
          = A2 ) ) ) ).

% the_equality
thf(fact_5894_the__eq__trivial,axiom,
    ! [A: $tType,A2: A] :
      ( ( the @ A
        @ ^ [X: A] : X = A2 )
      = A2 ) ).

% the_eq_trivial
thf(fact_5895_the__sym__eq__trivial,axiom,
    ! [A: $tType,X2: A] :
      ( ( the @ A
        @ ( ^ [Y5: A,Z4: A] : Y5 = Z4
          @ X2 ) )
      = X2 ) ).

% the_sym_eq_trivial
thf(fact_5896_the1__equality,axiom,
    ! [A: $tType,P: A > $o,A2: A] :
      ( ? [X6: A] :
          ( ( P @ X6 )
          & ! [Y4: A] :
              ( ( P @ Y4 )
             => ( Y4 = X6 ) ) )
     => ( ( P @ A2 )
       => ( ( the @ A @ P )
          = A2 ) ) ) ).

% the1_equality
thf(fact_5897_the1I2,axiom,
    ! [A: $tType,P: A > $o,Q: A > $o] :
      ( ? [X6: A] :
          ( ( P @ X6 )
          & ! [Y4: A] :
              ( ( P @ Y4 )
             => ( Y4 = X6 ) ) )
     => ( ! [X3: A] :
            ( ( P @ X3 )
           => ( Q @ X3 ) )
       => ( Q @ ( the @ A @ P ) ) ) ) ).

% the1I2
thf(fact_5898_If__def,axiom,
    ! [A: $tType] :
      ( ( if @ A )
      = ( ^ [P3: $o,X: A,Y: A] :
            ( the @ A
            @ ^ [Z2: A] :
                ( ( P3
                 => ( Z2 = X ) )
                & ( ~ P3
                 => ( Z2 = Y ) ) ) ) ) ) ).

% If_def
thf(fact_5899_theI2,axiom,
    ! [A: $tType,P: A > $o,A2: A,Q: A > $o] :
      ( ( P @ A2 )
     => ( ! [X3: A] :
            ( ( P @ X3 )
           => ( X3 = A2 ) )
       => ( ! [X3: A] :
              ( ( P @ X3 )
             => ( Q @ X3 ) )
         => ( Q @ ( the @ A @ P ) ) ) ) ) ).

% theI2
thf(fact_5900_theI_H,axiom,
    ! [A: $tType,P: A > $o] :
      ( ? [X6: A] :
          ( ( P @ X6 )
          & ! [Y4: A] :
              ( ( P @ Y4 )
             => ( Y4 = X6 ) ) )
     => ( P @ ( the @ A @ P ) ) ) ).

% theI'
thf(fact_5901_theI,axiom,
    ! [A: $tType,P: A > $o,A2: A] :
      ( ( P @ A2 )
     => ( ! [X3: A] :
            ( ( P @ X3 )
           => ( X3 = A2 ) )
       => ( P @ ( the @ A @ P ) ) ) ) ).

% theI
thf(fact_5902_floor__rat__def,axiom,
    ( ( archim6421214686448440834_floor @ rat )
    = ( ^ [X: rat] :
          ( the @ int
          @ ^ [Z2: int] :
              ( ( ord_less_eq @ rat @ ( ring_1_of_int @ rat @ Z2 ) @ X )
              & ( ord_less @ rat @ X @ ( ring_1_of_int @ rat @ ( plus_plus @ int @ Z2 @ ( one_one @ int ) ) ) ) ) ) ) ) ).

% floor_rat_def
thf(fact_5903_root__def,axiom,
    ( root
    = ( ^ [N: nat,X: real] :
          ( if @ real
          @ ( N
            = ( zero_zero @ nat ) )
          @ ( zero_zero @ real )
          @ ( the_inv_into @ real @ real @ ( top_top @ ( set @ real ) )
            @ ^ [Y: real] : ( times_times @ real @ ( sgn_sgn @ real @ Y ) @ ( power_power @ real @ ( abs_abs @ real @ Y ) @ N ) )
            @ X ) ) ) ) ).

% root_def
thf(fact_5904_Greatest__def,axiom,
    ! [A: $tType] :
      ( ( order @ A )
     => ( ( order_Greatest @ A )
        = ( ^ [P3: A > $o] :
              ( the @ A
              @ ^ [X: A] :
                  ( ( P3 @ X )
                  & ! [Y: A] :
                      ( ( P3 @ Y )
                     => ( ord_less_eq @ A @ Y @ X ) ) ) ) ) ) ) ).

% Greatest_def
thf(fact_5905_sgn__rat__def,axiom,
    ( ( sgn_sgn @ rat )
    = ( ^ [A5: rat] :
          ( if @ rat
          @ ( A5
            = ( zero_zero @ rat ) )
          @ ( zero_zero @ rat )
          @ ( if @ rat @ ( ord_less @ rat @ ( zero_zero @ rat ) @ A5 ) @ ( one_one @ rat ) @ ( uminus_uminus @ rat @ ( one_one @ rat ) ) ) ) ) ) ).

% sgn_rat_def
thf(fact_5906_obtain__pos__sum,axiom,
    ! [R4: rat] :
      ( ( ord_less @ rat @ ( zero_zero @ rat ) @ R4 )
     => ~ ! [S: rat] :
            ( ( ord_less @ rat @ ( zero_zero @ rat ) @ S )
           => ! [T5: rat] :
                ( ( ord_less @ rat @ ( zero_zero @ rat ) @ T5 )
               => ( R4
                 != ( plus_plus @ rat @ S @ T5 ) ) ) ) ) ).

% obtain_pos_sum
thf(fact_5907_less__eq__rat__def,axiom,
    ( ( ord_less_eq @ rat )
    = ( ^ [X: rat,Y: rat] :
          ( ( ord_less @ rat @ X @ Y )
          | ( X = Y ) ) ) ) ).

% less_eq_rat_def
thf(fact_5908_GreatestI__nat,axiom,
    ! [P: nat > $o,K: nat,B2: nat] :
      ( ( P @ K )
     => ( ! [Y4: nat] :
            ( ( P @ Y4 )
           => ( ord_less_eq @ nat @ Y4 @ B2 ) )
       => ( P @ ( order_Greatest @ nat @ P ) ) ) ) ).

% GreatestI_nat
thf(fact_5909_Greatest__le__nat,axiom,
    ! [P: nat > $o,K: nat,B2: nat] :
      ( ( P @ K )
     => ( ! [Y4: nat] :
            ( ( P @ Y4 )
           => ( ord_less_eq @ nat @ Y4 @ B2 ) )
       => ( ord_less_eq @ nat @ K @ ( order_Greatest @ nat @ P ) ) ) ) ).

% Greatest_le_nat
thf(fact_5910_GreatestI__ex__nat,axiom,
    ! [P: nat > $o,B2: nat] :
      ( ? [X_1: nat] : ( P @ X_1 )
     => ( ! [Y4: nat] :
            ( ( P @ Y4 )
           => ( ord_less_eq @ nat @ Y4 @ B2 ) )
       => ( P @ ( order_Greatest @ nat @ P ) ) ) ) ).

% GreatestI_ex_nat
thf(fact_5911_Greatest__equality,axiom,
    ! [A: $tType] :
      ( ( order @ A )
     => ! [P: A > $o,X2: A] :
          ( ( P @ X2 )
         => ( ! [Y4: A] :
                ( ( P @ Y4 )
               => ( ord_less_eq @ A @ Y4 @ X2 ) )
           => ( ( order_Greatest @ A @ P )
              = X2 ) ) ) ) ).

% Greatest_equality
thf(fact_5912_GreatestI2__order,axiom,
    ! [A: $tType] :
      ( ( order @ A )
     => ! [P: A > $o,X2: A,Q: A > $o] :
          ( ( P @ X2 )
         => ( ! [Y4: A] :
                ( ( P @ Y4 )
               => ( ord_less_eq @ A @ Y4 @ X2 ) )
           => ( ! [X3: A] :
                  ( ( P @ X3 )
                 => ( ! [Y3: A] :
                        ( ( P @ Y3 )
                       => ( ord_less_eq @ A @ Y3 @ X3 ) )
                   => ( Q @ X3 ) ) )
             => ( Q @ ( order_Greatest @ A @ P ) ) ) ) ) ) ).

% GreatestI2_order
thf(fact_5913_Arg__def,axiom,
    ( arg
    = ( ^ [Z2: complex] :
          ( if @ real
          @ ( Z2
            = ( zero_zero @ complex ) )
          @ ( zero_zero @ real )
          @ ( fChoice @ real
            @ ^ [A5: real] :
                ( ( ( sgn_sgn @ complex @ Z2 )
                  = ( cis @ A5 ) )
                & ( ord_less @ real @ ( uminus_uminus @ real @ pi ) @ A5 )
                & ( ord_less_eq @ real @ A5 @ pi ) ) ) ) ) ) ).

% Arg_def
thf(fact_5914_the__inv__into__def,axiom,
    ! [B: $tType,A: $tType] :
      ( ( the_inv_into @ A @ B )
      = ( ^ [A7: set @ A,F6: A > B,X: B] :
            ( the @ A
            @ ^ [Y: A] :
                ( ( member @ A @ Y @ A7 )
                & ( ( F6 @ Y )
                  = X ) ) ) ) ) ).

% the_inv_into_def
thf(fact_5915_hoare__triple__effect,axiom,
    ! [A: $tType,P: assn,C2: heap_Time_Heap @ A,Q: A > assn,H2: heap_ext @ product_unit,As3: set @ nat] :
      ( ( hoare_hoare_triple @ A @ P @ C2 @ Q )
     => ( ( rep_assn @ P @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As3 ) )
       => ? [H5: heap_ext @ product_unit,R3: A,T5: nat] :
            ( ( heap_Time_effect @ A @ C2 @ H2 @ H5 @ R3 @ T5 )
            & ( rep_assn @ ( Q @ R3 ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H5 @ ( hoare_new_addrs @ H2 @ As3 @ H5 ) ) ) ) ) ) ).

% hoare_triple_effect
thf(fact_5916_some__equality,axiom,
    ! [A: $tType,P: A > $o,A2: A] :
      ( ( P @ A2 )
     => ( ! [X3: A] :
            ( ( P @ X3 )
           => ( X3 = A2 ) )
       => ( ( fChoice @ A @ P )
          = A2 ) ) ) ).

% some_equality
thf(fact_5917_some__eq__trivial,axiom,
    ! [A: $tType,X2: A] :
      ( ( fChoice @ A
        @ ^ [Y: A] : Y = X2 )
      = X2 ) ).

% some_eq_trivial
thf(fact_5918_some__sym__eq__trivial,axiom,
    ! [A: $tType,X2: A] :
      ( ( fChoice @ A
        @ ( ^ [Y5: A,Z4: A] : Y5 = Z4
          @ X2 ) )
      = X2 ) ).

% some_sym_eq_trivial
thf(fact_5919_new__addr__refl,axiom,
    ! [H2: heap_ext @ product_unit,As3: set @ nat] :
      ( ( hoare_new_addrs @ H2 @ As3 @ H2 )
      = As3 ) ).

% new_addr_refl
thf(fact_5920_some__insert__self,axiom,
    ! [A: $tType,S3: set @ A] :
      ( ( S3
       != ( bot_bot @ ( set @ A ) ) )
     => ( ( insert @ A
          @ ( fChoice @ A
            @ ^ [X: A] : ( member @ A @ X @ S3 ) )
          @ S3 )
        = S3 ) ) ).

% some_insert_self
thf(fact_5921_diff__rat__def,axiom,
    ( ( minus_minus @ rat )
    = ( ^ [Q5: rat,R: rat] : ( plus_plus @ rat @ Q5 @ ( uminus_uminus @ rat @ R ) ) ) ) ).

% diff_rat_def
thf(fact_5922_verit__sko__ex_H,axiom,
    ! [A: $tType,P: A > $o,A3: $o] :
      ( ( ( P @ ( fChoice @ A @ P ) )
        = A3 )
     => ( ( ? [X5: A] : ( P @ X5 ) )
        = A3 ) ) ).

% verit_sko_ex'
thf(fact_5923_verit__sko__forall,axiom,
    ! [A: $tType] :
      ( ( ^ [P2: A > $o] :
          ! [X4: A] : ( P2 @ X4 ) )
      = ( ^ [P3: A > $o] :
            ( P3
            @ ( fChoice @ A
              @ ^ [X: A] :
                  ~ ( P3 @ X ) ) ) ) ) ).

% verit_sko_forall
thf(fact_5924_verit__sko__forall_H,axiom,
    ! [A: $tType,P: A > $o,A3: $o] :
      ( ( ( P
          @ ( fChoice @ A
            @ ^ [X: A] :
                ~ ( P @ X ) ) )
        = A3 )
     => ( ( ! [X5: A] : ( P @ X5 ) )
        = A3 ) ) ).

% verit_sko_forall'
thf(fact_5925_verit__sko__forall_H_H,axiom,
    ! [A: $tType,B6: A,A3: A,P: A > $o] :
      ( ( B6 = A3 )
     => ( ( ( fChoice @ A @ P )
          = A3 )
        = ( ( fChoice @ A @ P )
          = B6 ) ) ) ).

% verit_sko_forall''
thf(fact_5926_verit__sko__ex__indirect,axiom,
    ! [A: $tType,X2: A,P: A > $o] :
      ( ( X2
        = ( fChoice @ A @ P ) )
     => ( ( ? [X5: A] : ( P @ X5 ) )
        = ( P @ X2 ) ) ) ).

% verit_sko_ex_indirect
thf(fact_5927_verit__sko__ex__indirect2,axiom,
    ! [A: $tType,X2: A,P: A > $o,P6: A > $o] :
      ( ( X2
        = ( fChoice @ A @ P ) )
     => ( ! [X3: A] :
            ( ( P @ X3 )
            = ( P6 @ X3 ) )
       => ( ( ? [X5: A] : ( P6 @ X5 ) )
          = ( P @ X2 ) ) ) ) ).

% verit_sko_ex_indirect2
thf(fact_5928_verit__sko__forall__indirect,axiom,
    ! [A: $tType,X2: A,P: A > $o] :
      ( ( X2
        = ( fChoice @ A
          @ ^ [X: A] :
              ~ ( P @ X ) ) )
     => ( ( ! [X5: A] : ( P @ X5 ) )
        = ( P @ X2 ) ) ) ).

% verit_sko_forall_indirect
thf(fact_5929_verit__sko__forall__indirect2,axiom,
    ! [A: $tType,X2: A,P: A > $o,P6: A > $o] :
      ( ( X2
        = ( fChoice @ A
          @ ^ [X: A] :
              ~ ( P @ X ) ) )
     => ( ! [X3: A] :
            ( ( P @ X3 )
            = ( P6 @ X3 ) )
       => ( ( ! [X5: A] : ( P6 @ X5 ) )
          = ( P @ X2 ) ) ) ) ).

% verit_sko_forall_indirect2
thf(fact_5930_someI2,axiom,
    ! [A: $tType,P: A > $o,A2: A,Q: A > $o] :
      ( ( P @ A2 )
     => ( ! [X3: A] :
            ( ( P @ X3 )
           => ( Q @ X3 ) )
       => ( Q @ ( fChoice @ A @ P ) ) ) ) ).

% someI2
thf(fact_5931_someI__ex,axiom,
    ! [A: $tType,P: A > $o] :
      ( ? [X_1: A] : ( P @ X_1 )
     => ( P @ ( fChoice @ A @ P ) ) ) ).

% someI_ex
thf(fact_5932_someI2__ex,axiom,
    ! [A: $tType,P: A > $o,Q: A > $o] :
      ( ? [X_1: A] : ( P @ X_1 )
     => ( ! [X3: A] :
            ( ( P @ X3 )
           => ( Q @ X3 ) )
       => ( Q @ ( fChoice @ A @ P ) ) ) ) ).

% someI2_ex
thf(fact_5933_someI2__bex,axiom,
    ! [A: $tType,A3: set @ A,P: A > $o,Q: A > $o] :
      ( ? [X6: A] :
          ( ( member @ A @ X6 @ A3 )
          & ( P @ X6 ) )
     => ( ! [X3: A] :
            ( ( ( member @ A @ X3 @ A3 )
              & ( P @ X3 ) )
           => ( Q @ X3 ) )
       => ( Q
          @ ( fChoice @ A
            @ ^ [X: A] :
                ( ( member @ A @ X @ A3 )
                & ( P @ X ) ) ) ) ) ) ).

% someI2_bex
thf(fact_5934_some__eq__ex,axiom,
    ! [A: $tType,P: A > $o] :
      ( ( P @ ( fChoice @ A @ P ) )
      = ( ? [X5: A] : ( P @ X5 ) ) ) ).

% some_eq_ex
thf(fact_5935_some1__equality,axiom,
    ! [A: $tType,P: A > $o,A2: A] :
      ( ? [X6: A] :
          ( ( P @ X6 )
          & ! [Y4: A] :
              ( ( P @ Y4 )
             => ( Y4 = X6 ) ) )
     => ( ( P @ A2 )
       => ( ( fChoice @ A @ P )
          = A2 ) ) ) ).

% some1_equality
thf(fact_5936_some__in__eq,axiom,
    ! [A: $tType,A3: set @ A] :
      ( ( member @ A
        @ ( fChoice @ A
          @ ^ [X: A] : ( member @ A @ X @ A3 ) )
        @ A3 )
      = ( A3
       != ( bot_bot @ ( set @ A ) ) ) ) ).

% some_in_eq
thf(fact_5937_some__elem,axiom,
    ! [A: $tType,S3: set @ A] :
      ( ( S3
       != ( bot_bot @ ( set @ A ) ) )
     => ( member @ A
        @ ( fChoice @ A
          @ ^ [X: A] : ( member @ A @ X @ S3 ) )
        @ S3 ) ) ).

% some_elem
thf(fact_5938_fun__of__rel__def,axiom,
    ! [A: $tType,B: $tType] :
      ( ( fun_of_rel @ B @ A )
      = ( ^ [R7: set @ ( product_prod @ B @ A ),X: B] :
            ( fChoice @ A
            @ ^ [Y: A] : ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X @ Y ) @ R7 ) ) ) ) ).

% fun_of_rel_def
thf(fact_5939_TBOUND__assert_H__bind,axiom,
    ! [A: $tType,P: $o,P6: $o,M: heap_Time_Heap @ A,T2: nat] :
      ( ( time_EQ @ $o @ P @ P6 )
     => ( ( P
         => ( time_TBOUND @ A @ M @ T2 ) )
       => ( time_TBOUND @ A
          @ ( heap_Time_bind @ product_unit @ A @ ( refine_Imp_assert @ P )
            @ ^ [Uu3: product_unit] : M )
          @ ( if @ nat @ P6 @ T2 @ ( zero_zero @ nat ) ) ) ) ) ).

% TBOUND_assert'_bind
thf(fact_5940_neg__mask__is__div_H,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [N2: nat,W: word @ A] :
          ( ( ord_less @ nat @ N2 @ ( size_size @ ( word @ A ) @ W ) )
         => ( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ W @ ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N2 ) ) )
            = ( times_times @ ( word @ A ) @ ( divide_divide @ ( word @ A ) @ W @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N2 ) ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N2 ) ) ) ) ) ).

% neg_mask_is_div'
thf(fact_5941_Eps__case__prod__eq,axiom,
    ! [A: $tType,B: $tType,X2: A,Y2: B] :
      ( ( fChoice @ ( product_prod @ A @ B )
        @ ( product_case_prod @ A @ B @ $o
          @ ^ [X17: A,Y8: B] :
              ( ( X2 = X17 )
              & ( Y2 = Y8 ) ) ) )
      = ( product_Pair @ A @ B @ X2 @ Y2 ) ) ).

% Eps_case_prod_eq
thf(fact_5942_bit_Ocompl__one,axiom,
    ! [A: $tType] :
      ( ( bit_ri3973907225187159222ations @ A )
     => ( ( bit_ri4277139882892585799ns_not @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
        = ( zero_zero @ A ) ) ) ).

% bit.compl_one
thf(fact_5943_bit_Ocompl__zero,axiom,
    ! [A: $tType] :
      ( ( bit_ri3973907225187159222ations @ A )
     => ( ( bit_ri4277139882892585799ns_not @ A @ ( zero_zero @ A ) )
        = ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).

% bit.compl_zero
thf(fact_5944_minus__not__numeral__eq,axiom,
    ! [A: $tType] :
      ( ( bit_ri3973907225187159222ations @ A )
     => ! [N2: num] :
          ( ( uminus_uminus @ A @ ( bit_ri4277139882892585799ns_not @ A @ ( numeral_numeral @ A @ N2 ) ) )
          = ( numeral_numeral @ A @ ( inc @ N2 ) ) ) ) ).

% minus_not_numeral_eq
thf(fact_5945_word__bitwise__m1__simps_I1_J,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ( ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) )
        = ( zero_zero @ ( word @ A ) ) ) ) ).

% word_bitwise_m1_simps(1)
thf(fact_5946_word__add__not,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [X2: word @ A] :
          ( ( plus_plus @ ( word @ A ) @ X2 @ ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ X2 ) )
          = ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) ) ) ).

% word_add_not
thf(fact_5947_even__not__iff,axiom,
    ! [A: $tType] :
      ( ( bit_ri3973907225187159222ations @ A )
     => ! [A2: A] :
          ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_ri4277139882892585799ns_not @ A @ A2 ) )
          = ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) ) ) ) ).

% even_not_iff
thf(fact_5948_not__one__eq,axiom,
    ! [A: $tType] :
      ( ( bit_ri3973907225187159222ations @ A )
     => ( ( bit_ri4277139882892585799ns_not @ A @ ( one_one @ A ) )
        = ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).

% not_one_eq
thf(fact_5949_compl__of__1,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ( ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ ( one_one @ ( word @ A ) ) )
        = ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) ) ) ) ).

% compl_of_1
thf(fact_5950_word__plus__and__or__coroll2,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [X2: word @ A,W: word @ A] :
          ( ( plus_plus @ ( word @ A ) @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X2 @ W ) @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X2 @ ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ W ) ) )
          = X2 ) ) ).

% word_plus_and_or_coroll2
thf(fact_5951_not__add__distrib,axiom,
    ! [A: $tType] :
      ( ( bit_ri3973907225187159222ations @ A )
     => ! [A2: A,B2: A] :
          ( ( bit_ri4277139882892585799ns_not @ A @ ( plus_plus @ A @ A2 @ B2 ) )
          = ( minus_minus @ A @ ( bit_ri4277139882892585799ns_not @ A @ A2 ) @ B2 ) ) ) ).

% not_add_distrib
thf(fact_5952_not__diff__distrib,axiom,
    ! [A: $tType] :
      ( ( bit_ri3973907225187159222ations @ A )
     => ! [A2: A,B2: A] :
          ( ( bit_ri4277139882892585799ns_not @ A @ ( minus_minus @ A @ A2 @ B2 ) )
          = ( plus_plus @ A @ ( bit_ri4277139882892585799ns_not @ A @ A2 ) @ B2 ) ) ) ).

% not_diff_distrib
thf(fact_5953_of__int__not__numeral,axiom,
    ! [A: $tType] :
      ( ( bit_ri3973907225187159222ations @ A )
     => ! [K: num] :
          ( ( ring_1_of_int @ A @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ K ) ) )
          = ( bit_ri4277139882892585799ns_not @ A @ ( numeral_numeral @ A @ K ) ) ) ) ).

% of_int_not_numeral
thf(fact_5954_split__paired__Eps,axiom,
    ! [B: $tType,A: $tType] :
      ( ( fChoice @ ( product_prod @ A @ B ) )
      = ( ^ [P3: ( product_prod @ A @ B ) > $o] :
            ( fChoice @ ( product_prod @ A @ B )
            @ ( product_case_prod @ A @ B @ $o
              @ ^ [A5: A,B5: B] : ( P3 @ ( product_Pair @ A @ B @ A5 @ B5 ) ) ) ) ) ) ).

% split_paired_Eps
thf(fact_5955_minus__eq__not__plus__1,axiom,
    ! [A: $tType] :
      ( ( bit_ri3973907225187159222ations @ A )
     => ( ( uminus_uminus @ A )
        = ( ^ [A5: A] : ( plus_plus @ A @ ( bit_ri4277139882892585799ns_not @ A @ A5 ) @ ( one_one @ A ) ) ) ) ) ).

% minus_eq_not_plus_1
thf(fact_5956_minus__eq__not__minus__1,axiom,
    ! [A: $tType] :
      ( ( bit_ri3973907225187159222ations @ A )
     => ( ( uminus_uminus @ A )
        = ( ^ [A5: A] : ( bit_ri4277139882892585799ns_not @ A @ ( minus_minus @ A @ A5 @ ( one_one @ A ) ) ) ) ) ) ).

% minus_eq_not_minus_1
thf(fact_5957_not__eq__complement,axiom,
    ! [A: $tType] :
      ( ( bit_ri3973907225187159222ations @ A )
     => ( ( bit_ri4277139882892585799ns_not @ A )
        = ( ^ [A5: A] : ( minus_minus @ A @ ( uminus_uminus @ A @ A5 ) @ ( one_one @ A ) ) ) ) ) ).

% not_eq_complement
thf(fact_5958_take__bit__not__eq__mask__diff,axiom,
    ! [A: $tType] :
      ( ( bit_ri3973907225187159222ations @ A )
     => ! [N2: nat,A2: A] :
          ( ( bit_se2584673776208193580ke_bit @ A @ N2 @ ( bit_ri4277139882892585799ns_not @ A @ A2 ) )
          = ( minus_minus @ A @ ( bit_se2239418461657761734s_mask @ A @ N2 ) @ ( bit_se2584673776208193580ke_bit @ A @ N2 @ A2 ) ) ) ) ).

% take_bit_not_eq_mask_diff
thf(fact_5959_mask__lower__twice,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [N2: nat,M: nat,X2: word @ A] :
          ( ( ord_less_eq @ nat @ N2 @ M )
         => ( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X2 @ ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N2 ) ) ) @ ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ M ) ) )
            = ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X2 @ ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ M ) ) ) ) ) ) ).

% mask_lower_twice
thf(fact_5960_mask__out__first__mask__some,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [X2: word @ A,N2: nat,Y2: word @ A,M: nat] :
          ( ( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X2 @ ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N2 ) ) )
            = Y2 )
         => ( ( ord_less_eq @ nat @ N2 @ M )
           => ( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X2 @ ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ M ) ) )
              = ( bit_se5824344872417868541ns_and @ ( word @ A ) @ Y2 @ ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ M ) ) ) ) ) ) ) ).

% mask_out_first_mask_some
thf(fact_5961_NOT__eq,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ( ( bit_ri4277139882892585799ns_not @ ( word @ A ) )
        = ( ^ [X: word @ A] : ( minus_minus @ ( word @ A ) @ ( uminus_uminus @ ( word @ A ) @ X ) @ ( one_one @ ( word @ A ) ) ) ) ) ) ).

% NOT_eq
thf(fact_5962_AND__NOT__mask__plus__AND__mask__eq,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [W: word @ A,N2: nat] :
          ( ( plus_plus @ ( word @ A ) @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ W @ ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N2 ) ) ) @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ W @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N2 ) ) )
          = W ) ) ).

% AND_NOT_mask_plus_AND_mask_eq
thf(fact_5963_minus__numeral__inc__eq,axiom,
    ! [A: $tType] :
      ( ( bit_ri3973907225187159222ations @ A )
     => ! [N2: num] :
          ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( inc @ N2 ) ) )
          = ( bit_ri4277139882892585799ns_not @ A @ ( numeral_numeral @ A @ N2 ) ) ) ) ).

% minus_numeral_inc_eq
thf(fact_5964_subtract__mask_I2_J,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [P4: word @ A,N2: nat] :
          ( ( minus_minus @ ( word @ A ) @ P4 @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ P4 @ ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N2 ) ) ) )
          = ( bit_se5824344872417868541ns_and @ ( word @ A ) @ P4 @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N2 ) ) ) ) ).

% subtract_mask(2)
thf(fact_5965_subtract__mask_I1_J,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [P4: word @ A,N2: nat] :
          ( ( minus_minus @ ( word @ A ) @ P4 @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ P4 @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N2 ) ) )
          = ( bit_se5824344872417868541ns_and @ ( word @ A ) @ P4 @ ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N2 ) ) ) ) ) ).

% subtract_mask(1)
thf(fact_5966_mask__out__sub__mask,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [X2: word @ A,N2: nat] :
          ( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X2 @ ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N2 ) ) )
          = ( minus_minus @ ( word @ A ) @ X2 @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X2 @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N2 ) ) ) ) ) ).

% mask_out_sub_mask
thf(fact_5967_Eps__case__prod,axiom,
    ! [B: $tType,A: $tType,P: A > B > $o] :
      ( ( fChoice @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ P ) )
      = ( fChoice @ ( product_prod @ A @ B )
        @ ^ [Xy: product_prod @ A @ B] : ( P @ ( product_fst @ A @ B @ Xy ) @ ( product_snd @ A @ B @ Xy ) ) ) ) ).

% Eps_case_prod
thf(fact_5968_not__numeral__Bit0__eq,axiom,
    ! [A: $tType] :
      ( ( bit_ri3973907225187159222ations @ A )
     => ! [N2: num] :
          ( ( bit_ri4277139882892585799ns_not @ A @ ( numeral_numeral @ A @ ( bit0 @ N2 ) ) )
          = ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( bit1 @ N2 ) ) ) ) ) ).

% not_numeral_Bit0_eq
thf(fact_5969_take__bit__not__mask__eq__0,axiom,
    ! [A: $tType] :
      ( ( bit_ri3973907225187159222ations @ A )
     => ! [M: nat,N2: nat] :
          ( ( ord_less_eq @ nat @ M @ N2 )
         => ( ( bit_se2584673776208193580ke_bit @ A @ M @ ( bit_ri4277139882892585799ns_not @ A @ ( bit_se2239418461657761734s_mask @ A @ N2 ) ) )
            = ( zero_zero @ A ) ) ) ) ).

% take_bit_not_mask_eq_0
thf(fact_5970_not__numeral__BitM__eq,axiom,
    ! [A: $tType] :
      ( ( bit_ri3973907225187159222ations @ A )
     => ! [N2: num] :
          ( ( bit_ri4277139882892585799ns_not @ A @ ( numeral_numeral @ A @ ( bitM @ N2 ) ) )
          = ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( bit0 @ N2 ) ) ) ) ) ).

% not_numeral_BitM_eq
thf(fact_5971_multiple__mask__trivia,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [M: nat,N2: nat,X2: word @ A] :
          ( ( ord_less_eq @ nat @ M @ N2 )
         => ( ( plus_plus @ ( word @ A ) @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X2 @ ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N2 ) ) ) @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X2 @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N2 ) @ ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ M ) ) ) ) )
            = ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X2 @ ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ M ) ) ) ) ) ) ).

% multiple_mask_trivia
thf(fact_5972_and__mask__0__iff__le__mask,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [W: word @ A,N2: nat] :
          ( ( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ W @ ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N2 ) ) )
            = ( zero_zero @ ( word @ A ) ) )
          = ( ord_less_eq @ ( word @ A ) @ W @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N2 ) ) ) ) ).

% and_mask_0_iff_le_mask
thf(fact_5973_TBOUND__Let__strong,axiom,
    ! [B: $tType,A: $tType,V: A,V4: A,F: A > ( heap_Time_Heap @ B ),Bnd: A > nat] :
      ( ( time_EQ @ A @ V @ V4 )
     => ( ( time_TBOUND @ B @ ( F @ V ) @ ( Bnd @ V ) )
       => ( time_TBOUND @ B @ ( F @ V ) @ ( Bnd @ V4 ) ) ) ) ).

% TBOUND_Let_strong
thf(fact_5974_minus__exp__eq__not__mask,axiom,
    ! [A: $tType] :
      ( ( bit_ri3973907225187159222ations @ A )
     => ! [N2: nat] :
          ( ( uminus_uminus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) )
          = ( bit_ri4277139882892585799ns_not @ A @ ( bit_se2239418461657761734s_mask @ A @ N2 ) ) ) ) ).

% minus_exp_eq_not_mask
thf(fact_5975_NOT__mask,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [N2: nat] :
          ( ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N2 ) )
          = ( uminus_uminus @ ( word @ A ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N2 ) ) ) ) ).

% NOT_mask
thf(fact_5976_neg__mask__is__div,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [W: word @ A,N2: nat] :
          ( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ W @ ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N2 ) ) )
          = ( times_times @ ( word @ A ) @ ( divide_divide @ ( word @ A ) @ W @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N2 ) ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N2 ) ) ) ) ).

% neg_mask_is_div
thf(fact_5977_signed__take__bit__eq__take__bit__minus,axiom,
    ( ( bit_ri4674362597316999326ke_bit @ int )
    = ( ^ [N: nat,K4: int] : ( minus_minus @ int @ ( bit_se2584673776208193580ke_bit @ int @ ( suc @ N ) @ K4 ) @ ( times_times @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( suc @ N ) ) @ ( zero_neq_one_of_bool @ int @ ( bit_se5641148757651400278ts_bit @ int @ K4 @ N ) ) ) ) ) ) ).

% signed_take_bit_eq_take_bit_minus
thf(fact_5978_neg__mask__add__mask,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [X2: word @ A,N2: nat] :
          ( ( plus_plus @ ( word @ A ) @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X2 @ ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N2 ) ) ) @ ( minus_minus @ ( word @ A ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N2 ) @ ( one_one @ ( word @ A ) ) ) )
          = ( bit_se1065995026697491101ons_or @ ( word @ A ) @ X2 @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N2 ) ) ) ) ).

% neg_mask_add_mask
thf(fact_5979_bit_Odisj__one__right,axiom,
    ! [A: $tType] :
      ( ( bit_ri3973907225187159222ations @ A )
     => ! [X2: A] :
          ( ( bit_se1065995026697491101ons_or @ A @ X2 @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
          = ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).

% bit.disj_one_right
thf(fact_5980_bit_Odisj__one__left,axiom,
    ! [A: $tType] :
      ( ( bit_ri3973907225187159222ations @ A )
     => ! [X2: A] :
          ( ( bit_se1065995026697491101ons_or @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ X2 )
          = ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).

% bit.disj_one_left
thf(fact_5981_word__plus__and__or,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [X2: word @ A,Y2: word @ A] :
          ( ( plus_plus @ ( word @ A ) @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X2 @ Y2 ) @ ( bit_se1065995026697491101ons_or @ ( word @ A ) @ X2 @ Y2 ) )
          = ( plus_plus @ ( word @ A ) @ X2 @ Y2 ) ) ) ).

% word_plus_and_or
thf(fact_5982_word__or__max,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [X2: word @ A] :
          ( ( bit_se1065995026697491101ons_or @ ( word @ A ) @ X2 @ ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) )
          = ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) ) ) ).

% word_or_max
thf(fact_5983_word__bitwise__m1__simps_I4_J,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [X2: word @ A] :
          ( ( bit_se1065995026697491101ons_or @ ( word @ A ) @ ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) @ X2 )
          = ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) ) ) ).

% word_bitwise_m1_simps(4)
thf(fact_5984_bit__numeral__Bit0__Suc__iff,axiom,
    ! [A: $tType] :
      ( ( bit_un5681908812861735899ations @ A )
     => ! [M: num,N2: nat] :
          ( ( bit_se5641148757651400278ts_bit @ A @ ( numeral_numeral @ A @ ( bit0 @ M ) ) @ ( suc @ N2 ) )
          = ( bit_se5641148757651400278ts_bit @ A @ ( numeral_numeral @ A @ M ) @ N2 ) ) ) ).

% bit_numeral_Bit0_Suc_iff
thf(fact_5985_bit__numeral__Bit1__Suc__iff,axiom,
    ! [A: $tType] :
      ( ( bit_un5681908812861735899ations @ A )
     => ! [M: num,N2: nat] :
          ( ( bit_se5641148757651400278ts_bit @ A @ ( numeral_numeral @ A @ ( bit1 @ M ) ) @ ( suc @ N2 ) )
          = ( bit_se5641148757651400278ts_bit @ A @ ( numeral_numeral @ A @ M ) @ N2 ) ) ) ).

% bit_numeral_Bit1_Suc_iff
thf(fact_5986_or__numerals_I2_J,axiom,
    ! [A: $tType] :
      ( ( bit_un5681908812861735899ations @ A )
     => ! [Y2: num] :
          ( ( bit_se1065995026697491101ons_or @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit1 @ Y2 ) ) )
          = ( numeral_numeral @ A @ ( bit1 @ Y2 ) ) ) ) ).

% or_numerals(2)
thf(fact_5987_or__numerals_I8_J,axiom,
    ! [A: $tType] :
      ( ( bit_un5681908812861735899ations @ A )
     => ! [X2: num] :
          ( ( bit_se1065995026697491101ons_or @ A @ ( numeral_numeral @ A @ ( bit1 @ X2 ) ) @ ( one_one @ A ) )
          = ( numeral_numeral @ A @ ( bit1 @ X2 ) ) ) ) ).

% or_numerals(8)
thf(fact_5988_bit_Odisj__cancel__right,axiom,
    ! [A: $tType] :
      ( ( bit_ri3973907225187159222ations @ A )
     => ! [X2: A] :
          ( ( bit_se1065995026697491101ons_or @ A @ X2 @ ( bit_ri4277139882892585799ns_not @ A @ X2 ) )
          = ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).

% bit.disj_cancel_right
thf(fact_5989_bit_Odisj__cancel__left,axiom,
    ! [A: $tType] :
      ( ( bit_ri3973907225187159222ations @ A )
     => ! [X2: A] :
          ( ( bit_se1065995026697491101ons_or @ A @ ( bit_ri4277139882892585799ns_not @ A @ X2 ) @ X2 )
          = ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).

% bit.disj_cancel_left
thf(fact_5990_not__negative__int__iff,axiom,
    ! [K: int] :
      ( ( ord_less @ int @ ( bit_ri4277139882892585799ns_not @ int @ K ) @ ( zero_zero @ int ) )
      = ( ord_less_eq @ int @ ( zero_zero @ int ) @ K ) ) ).

% not_negative_int_iff
thf(fact_5991_not__nonnegative__int__iff,axiom,
    ! [K: int] :
      ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_ri4277139882892585799ns_not @ int @ K ) )
      = ( ord_less @ int @ K @ ( zero_zero @ int ) ) ) ).

% not_nonnegative_int_iff
thf(fact_5992_signed__take__bit__nonnegative__iff,axiom,
    ! [N2: nat,K: int] :
      ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_ri4674362597316999326ke_bit @ int @ N2 @ K ) )
      = ( ~ ( bit_se5641148757651400278ts_bit @ int @ K @ N2 ) ) ) ).

% signed_take_bit_nonnegative_iff
thf(fact_5993_word__or__not,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [X2: word @ A] :
          ( ( bit_se1065995026697491101ons_or @ ( word @ A ) @ X2 @ ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ X2 ) )
          = ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) ) ) ).

% word_or_not
thf(fact_5994_or__numerals_I3_J,axiom,
    ! [A: $tType] :
      ( ( bit_un5681908812861735899ations @ A )
     => ! [X2: num,Y2: num] :
          ( ( bit_se1065995026697491101ons_or @ A @ ( numeral_numeral @ A @ ( bit0 @ X2 ) ) @ ( numeral_numeral @ A @ ( bit0 @ Y2 ) ) )
          = ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se1065995026697491101ons_or @ A @ ( numeral_numeral @ A @ X2 ) @ ( numeral_numeral @ A @ Y2 ) ) ) ) ) ).

% or_numerals(3)
thf(fact_5995_or__numerals_I1_J,axiom,
    ! [A: $tType] :
      ( ( bit_un5681908812861735899ations @ A )
     => ! [Y2: num] :
          ( ( bit_se1065995026697491101ons_or @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ Y2 ) ) )
          = ( numeral_numeral @ A @ ( bit1 @ Y2 ) ) ) ) ).

% or_numerals(1)
thf(fact_5996_or__numerals_I5_J,axiom,
    ! [A: $tType] :
      ( ( bit_un5681908812861735899ations @ A )
     => ! [X2: num] :
          ( ( bit_se1065995026697491101ons_or @ A @ ( numeral_numeral @ A @ ( bit0 @ X2 ) ) @ ( one_one @ A ) )
          = ( numeral_numeral @ A @ ( bit1 @ X2 ) ) ) ) ).

% or_numerals(5)
thf(fact_5997_bit__minus__numeral__Bit0__Suc__iff,axiom,
    ! [W: num,N2: nat] :
      ( ( bit_se5641148757651400278ts_bit @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ W ) ) ) @ ( suc @ N2 ) )
      = ( bit_se5641148757651400278ts_bit @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ W ) ) @ N2 ) ) ).

% bit_minus_numeral_Bit0_Suc_iff
thf(fact_5998_bit__numeral__simps_I2_J,axiom,
    ! [A: $tType] :
      ( ( bit_un5681908812861735899ations @ A )
     => ! [W: num,N2: num] :
          ( ( bit_se5641148757651400278ts_bit @ A @ ( numeral_numeral @ A @ ( bit0 @ W ) ) @ ( numeral_numeral @ nat @ N2 ) )
          = ( bit_se5641148757651400278ts_bit @ A @ ( numeral_numeral @ A @ W ) @ ( pred_numeral @ N2 ) ) ) ) ).

% bit_numeral_simps(2)
thf(fact_5999_bit__minus__numeral__Bit1__Suc__iff,axiom,
    ! [W: num,N2: nat] :
      ( ( bit_se5641148757651400278ts_bit @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ W ) ) ) @ ( suc @ N2 ) )
      = ( ~ ( bit_se5641148757651400278ts_bit @ int @ ( numeral_numeral @ int @ W ) @ N2 ) ) ) ).

% bit_minus_numeral_Bit1_Suc_iff
thf(fact_6000_bit__numeral__simps_I3_J,axiom,
    ! [A: $tType] :
      ( ( bit_un5681908812861735899ations @ A )
     => ! [W: num,N2: num] :
          ( ( bit_se5641148757651400278ts_bit @ A @ ( numeral_numeral @ A @ ( bit1 @ W ) ) @ ( numeral_numeral @ nat @ N2 ) )
          = ( bit_se5641148757651400278ts_bit @ A @ ( numeral_numeral @ A @ W ) @ ( pred_numeral @ N2 ) ) ) ) ).

% bit_numeral_simps(3)
thf(fact_6001_word__no__log__defs_I1_J,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [A2: num] :
          ( ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ A2 ) )
          = ( ring_1_of_int @ ( word @ A ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ A2 ) ) ) ) ) ).

% word_no_log_defs(1)
thf(fact_6002_bit__0,axiom,
    ! [A: $tType] :
      ( ( bit_semiring_bits @ A )
     => ! [A2: A] :
          ( ( bit_se5641148757651400278ts_bit @ A @ A2 @ ( zero_zero @ nat ) )
          = ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) ) ) ) ).

% bit_0
thf(fact_6003_bit__minus__numeral__int_I1_J,axiom,
    ! [W: num,N2: num] :
      ( ( bit_se5641148757651400278ts_bit @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ W ) ) ) @ ( numeral_numeral @ nat @ N2 ) )
      = ( bit_se5641148757651400278ts_bit @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ W ) ) @ ( pred_numeral @ N2 ) ) ) ).

% bit_minus_numeral_int(1)
thf(fact_6004_bit__minus__numeral__int_I2_J,axiom,
    ! [W: num,N2: num] :
      ( ( bit_se5641148757651400278ts_bit @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ W ) ) ) @ ( numeral_numeral @ nat @ N2 ) )
      = ( ~ ( bit_se5641148757651400278ts_bit @ int @ ( numeral_numeral @ int @ W ) @ ( pred_numeral @ N2 ) ) ) ) ).

% bit_minus_numeral_int(2)
thf(fact_6005_bit__mod__2__iff,axiom,
    ! [A: $tType] :
      ( ( bit_semiring_bits @ A )
     => ! [A2: A,N2: nat] :
          ( ( bit_se5641148757651400278ts_bit @ A @ ( modulo_modulo @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ N2 )
          = ( ( N2
              = ( zero_zero @ nat ) )
            & ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) ) ) ) ).

% bit_mod_2_iff
thf(fact_6006_bin__nth__minus__Bit0,axiom,
    ! [N2: nat,W: num] :
      ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
     => ( ( bit_se5641148757651400278ts_bit @ int @ ( numeral_numeral @ int @ ( bit0 @ W ) ) @ N2 )
        = ( bit_se5641148757651400278ts_bit @ int @ ( numeral_numeral @ int @ W ) @ ( minus_minus @ nat @ N2 @ ( one_one @ nat ) ) ) ) ) ).

% bin_nth_minus_Bit0
thf(fact_6007_bin__nth__minus__Bit1,axiom,
    ! [N2: nat,W: num] :
      ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
     => ( ( bit_se5641148757651400278ts_bit @ int @ ( numeral_numeral @ int @ ( bit1 @ W ) ) @ N2 )
        = ( bit_se5641148757651400278ts_bit @ int @ ( numeral_numeral @ int @ W ) @ ( minus_minus @ nat @ N2 @ ( one_one @ nat ) ) ) ) ) ).

% bin_nth_minus_Bit1
thf(fact_6008_word__of__int__not__numeral__eq,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [Bin: num] :
          ( ( ring_1_of_int @ ( word @ A ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ Bin ) ) )
          = ( minus_minus @ ( word @ A ) @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ Bin ) ) @ ( one_one @ ( word @ A ) ) ) ) ) ).

% word_of_int_not_numeral_eq
thf(fact_6009_word__no__log__defs_I2_J,axiom,
    ! [B: $tType] :
      ( ( type_len @ B )
     => ! [A2: num] :
          ( ( bit_ri4277139882892585799ns_not @ ( word @ B ) @ ( uminus_uminus @ ( word @ B ) @ ( numeral_numeral @ ( word @ B ) @ A2 ) ) )
          = ( ring_1_of_int @ ( word @ B ) @ ( bit_ri4277139882892585799ns_not @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ A2 ) ) ) ) ) ) ).

% word_no_log_defs(2)
thf(fact_6010_or__numerals_I4_J,axiom,
    ! [A: $tType] :
      ( ( bit_un5681908812861735899ations @ A )
     => ! [X2: num,Y2: num] :
          ( ( bit_se1065995026697491101ons_or @ A @ ( numeral_numeral @ A @ ( bit0 @ X2 ) ) @ ( numeral_numeral @ A @ ( bit1 @ Y2 ) ) )
          = ( plus_plus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se1065995026697491101ons_or @ A @ ( numeral_numeral @ A @ X2 ) @ ( numeral_numeral @ A @ Y2 ) ) ) ) ) ) ).

% or_numerals(4)
thf(fact_6011_or__numerals_I6_J,axiom,
    ! [A: $tType] :
      ( ( bit_un5681908812861735899ations @ A )
     => ! [X2: num,Y2: num] :
          ( ( bit_se1065995026697491101ons_or @ A @ ( numeral_numeral @ A @ ( bit1 @ X2 ) ) @ ( numeral_numeral @ A @ ( bit0 @ Y2 ) ) )
          = ( plus_plus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se1065995026697491101ons_or @ A @ ( numeral_numeral @ A @ X2 ) @ ( numeral_numeral @ A @ Y2 ) ) ) ) ) ) ).

% or_numerals(6)
thf(fact_6012_or__numerals_I7_J,axiom,
    ! [A: $tType] :
      ( ( bit_un5681908812861735899ations @ A )
     => ! [X2: num,Y2: num] :
          ( ( bit_se1065995026697491101ons_or @ A @ ( numeral_numeral @ A @ ( bit1 @ X2 ) ) @ ( numeral_numeral @ A @ ( bit1 @ Y2 ) ) )
          = ( plus_plus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se1065995026697491101ons_or @ A @ ( numeral_numeral @ A @ X2 ) @ ( numeral_numeral @ A @ Y2 ) ) ) ) ) ) ).

% or_numerals(7)
thf(fact_6013_bit__minus__int__iff,axiom,
    ! [K: int,N2: nat] :
      ( ( bit_se5641148757651400278ts_bit @ int @ ( uminus_uminus @ int @ K ) @ N2 )
      = ( bit_se5641148757651400278ts_bit @ int @ ( bit_ri4277139882892585799ns_not @ int @ ( minus_minus @ int @ K @ ( one_one @ int ) ) ) @ N2 ) ) ).

% bit_minus_int_iff
thf(fact_6014_bit__take__bit__iff,axiom,
    ! [A: $tType] :
      ( ( bit_se359711467146920520ations @ A )
     => ! [M: nat,A2: A,N2: nat] :
          ( ( bit_se5641148757651400278ts_bit @ A @ ( bit_se2584673776208193580ke_bit @ A @ M @ A2 ) @ N2 )
          = ( ( ord_less @ nat @ N2 @ M )
            & ( bit_se5641148757651400278ts_bit @ A @ A2 @ N2 ) ) ) ) ).

% bit_take_bit_iff
thf(fact_6015_le__word__or1,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [C2: word @ A,Y2: word @ A] : ( ord_less_eq @ ( word @ A ) @ C2 @ ( bit_se1065995026697491101ons_or @ ( word @ A ) @ Y2 @ C2 ) ) ) ).

% le_word_or1
thf(fact_6016_le__word__or2,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [X2: word @ A,Y2: word @ A] : ( ord_less_eq @ ( word @ A ) @ X2 @ ( bit_se1065995026697491101ons_or @ ( word @ A ) @ X2 @ Y2 ) ) ) ).

% le_word_or2
thf(fact_6017_bit__numeral__iff,axiom,
    ! [A: $tType] :
      ( ( bit_un5681908812861735899ations @ A )
     => ! [M: num,N2: nat] :
          ( ( bit_se5641148757651400278ts_bit @ A @ ( numeral_numeral @ A @ M ) @ N2 )
          = ( bit_se5641148757651400278ts_bit @ nat @ ( numeral_numeral @ nat @ M ) @ N2 ) ) ) ).

% bit_numeral_iff
thf(fact_6018_disjunctive__add,axiom,
    ! [A: $tType] :
      ( ( bit_se359711467146920520ations @ A )
     => ! [A2: A,B2: A] :
          ( ! [N4: nat] :
              ( ~ ( bit_se5641148757651400278ts_bit @ A @ A2 @ N4 )
              | ~ ( bit_se5641148757651400278ts_bit @ A @ B2 @ N4 ) )
         => ( ( plus_plus @ A @ A2 @ B2 )
            = ( bit_se1065995026697491101ons_or @ A @ A2 @ B2 ) ) ) ) ).

% disjunctive_add
thf(fact_6019_bit__disjunctive__add__iff,axiom,
    ! [A: $tType] :
      ( ( bit_semiring_bits @ A )
     => ! [A2: A,B2: A,N2: nat] :
          ( ! [N4: nat] :
              ( ~ ( bit_se5641148757651400278ts_bit @ A @ A2 @ N4 )
              | ~ ( bit_se5641148757651400278ts_bit @ A @ B2 @ N4 ) )
         => ( ( bit_se5641148757651400278ts_bit @ A @ ( plus_plus @ A @ A2 @ B2 ) @ N2 )
            = ( ( bit_se5641148757651400278ts_bit @ A @ A2 @ N2 )
              | ( bit_se5641148757651400278ts_bit @ A @ B2 @ N2 ) ) ) ) ) ).

% bit_disjunctive_add_iff
thf(fact_6020_bit__1__iff,axiom,
    ! [A: $tType] :
      ( ( bit_semiring_bits @ A )
     => ! [N2: nat] :
          ( ( bit_se5641148757651400278ts_bit @ A @ ( one_one @ A ) @ N2 )
          = ( N2
            = ( zero_zero @ nat ) ) ) ) ).

% bit_1_iff
thf(fact_6021_bit__numeral__simps_I1_J,axiom,
    ! [A: $tType] :
      ( ( bit_un5681908812861735899ations @ A )
     => ! [N2: num] :
          ~ ( bit_se5641148757651400278ts_bit @ A @ ( one_one @ A ) @ ( numeral_numeral @ nat @ N2 ) ) ) ).

% bit_numeral_simps(1)
thf(fact_6022_not__bit__1__Suc,axiom,
    ! [A: $tType] :
      ( ( bit_se359711467146920520ations @ A )
     => ! [N2: nat] :
          ~ ( bit_se5641148757651400278ts_bit @ A @ ( one_one @ A ) @ ( suc @ N2 ) ) ) ).

% not_bit_1_Suc
thf(fact_6023_signed__take__bit__def,axiom,
    ! [A: $tType] :
      ( ( bit_ri3973907225187159222ations @ A )
     => ( ( bit_ri4674362597316999326ke_bit @ A )
        = ( ^ [N: nat,A5: A] : ( bit_se1065995026697491101ons_or @ A @ ( bit_se2584673776208193580ke_bit @ A @ N @ A5 ) @ ( times_times @ A @ ( zero_neq_one_of_bool @ A @ ( bit_se5641148757651400278ts_bit @ A @ A5 @ N ) ) @ ( bit_ri4277139882892585799ns_not @ A @ ( bit_se2239418461657761734s_mask @ A @ N ) ) ) ) ) ) ) ).

% signed_take_bit_def
thf(fact_6024_disjunctive__diff,axiom,
    ! [A: $tType] :
      ( ( bit_ri3973907225187159222ations @ A )
     => ! [B2: A,A2: A] :
          ( ! [N4: nat] :
              ( ( bit_se5641148757651400278ts_bit @ A @ B2 @ N4 )
             => ( bit_se5641148757651400278ts_bit @ A @ A2 @ N4 ) )
         => ( ( minus_minus @ A @ A2 @ B2 )
            = ( bit_se5824344872417868541ns_and @ A @ A2 @ ( bit_ri4277139882892585799ns_not @ A @ B2 ) ) ) ) ) ).

% disjunctive_diff
thf(fact_6025_not__int__def,axiom,
    ( ( bit_ri4277139882892585799ns_not @ int )
    = ( ^ [K4: int] : ( minus_minus @ int @ ( uminus_uminus @ int @ K4 ) @ ( one_one @ int ) ) ) ) ).

% not_int_def
thf(fact_6026_and__not__numerals_I1_J,axiom,
    ( ( bit_se5824344872417868541ns_and @ int @ ( one_one @ int ) @ ( bit_ri4277139882892585799ns_not @ int @ ( one_one @ int ) ) )
    = ( zero_zero @ int ) ) ).

% and_not_numerals(1)
thf(fact_6027_word__plus__and__or__coroll,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [X2: word @ A,Y2: word @ A] :
          ( ( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X2 @ Y2 )
            = ( zero_zero @ ( word @ A ) ) )
         => ( ( plus_plus @ ( word @ A ) @ X2 @ Y2 )
            = ( bit_se1065995026697491101ons_or @ ( word @ A ) @ X2 @ Y2 ) ) ) ) ).

% word_plus_and_or_coroll
thf(fact_6028_bit__not__int__iff_H,axiom,
    ! [K: int,N2: nat] :
      ( ( bit_se5641148757651400278ts_bit @ int @ ( minus_minus @ int @ ( uminus_uminus @ int @ K ) @ ( one_one @ int ) ) @ N2 )
      = ( ~ ( bit_se5641148757651400278ts_bit @ int @ K @ N2 ) ) ) ).

% bit_not_int_iff'
thf(fact_6029_even__or__iff,axiom,
    ! [A: $tType] :
      ( ( bit_se359711467146920520ations @ A )
     => ! [A2: A,B2: A] :
          ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se1065995026697491101ons_or @ A @ A2 @ B2 ) )
          = ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 )
            & ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) ) ) ) ).

% even_or_iff
thf(fact_6030_bit_Ocomplement__unique,axiom,
    ! [A: $tType] :
      ( ( bit_ri3973907225187159222ations @ A )
     => ! [A2: A,X2: A,Y2: A] :
          ( ( ( bit_se5824344872417868541ns_and @ A @ A2 @ X2 )
            = ( zero_zero @ A ) )
         => ( ( ( bit_se1065995026697491101ons_or @ A @ A2 @ X2 )
              = ( uminus_uminus @ A @ ( one_one @ A ) ) )
           => ( ( ( bit_se5824344872417868541ns_and @ A @ A2 @ Y2 )
                = ( zero_zero @ A ) )
             => ( ( ( bit_se1065995026697491101ons_or @ A @ A2 @ Y2 )
                  = ( uminus_uminus @ A @ ( one_one @ A ) ) )
               => ( X2 = Y2 ) ) ) ) ) ) ).

% bit.complement_unique
thf(fact_6031_bit__numeral__rec_I1_J,axiom,
    ! [A: $tType] :
      ( ( bit_un5681908812861735899ations @ A )
     => ! [W: num,N2: nat] :
          ( ( bit_se5641148757651400278ts_bit @ A @ ( numeral_numeral @ A @ ( bit0 @ W ) ) @ N2 )
          = ( case_nat @ $o @ $false @ ( bit_se5641148757651400278ts_bit @ A @ ( numeral_numeral @ A @ W ) ) @ N2 ) ) ) ).

% bit_numeral_rec(1)
thf(fact_6032_bit__numeral__rec_I2_J,axiom,
    ! [A: $tType] :
      ( ( bit_un5681908812861735899ations @ A )
     => ! [W: num,N2: nat] :
          ( ( bit_se5641148757651400278ts_bit @ A @ ( numeral_numeral @ A @ ( bit1 @ W ) ) @ N2 )
          = ( case_nat @ $o @ $true @ ( bit_se5641148757651400278ts_bit @ A @ ( numeral_numeral @ A @ W ) ) @ N2 ) ) ) ).

% bit_numeral_rec(2)
thf(fact_6033_not__int__div__2,axiom,
    ! [K: int] :
      ( ( divide_divide @ int @ ( bit_ri4277139882892585799ns_not @ int @ K ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) )
      = ( bit_ri4277139882892585799ns_not @ int @ ( divide_divide @ int @ K @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ).

% not_int_div_2
thf(fact_6034_even__not__iff__int,axiom,
    ! [K: int] :
      ( ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_ri4277139882892585799ns_not @ int @ K ) )
      = ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K ) ) ) ).

% even_not_iff_int
thf(fact_6035_and__not__numerals_I4_J,axiom,
    ! [M: num] :
      ( ( bit_se5824344872417868541ns_and @ int @ ( numeral_numeral @ int @ ( bit0 @ M ) ) @ ( bit_ri4277139882892585799ns_not @ int @ ( one_one @ int ) ) )
      = ( numeral_numeral @ int @ ( bit0 @ M ) ) ) ).

% and_not_numerals(4)
thf(fact_6036_and__not__numerals_I2_J,axiom,
    ! [N2: num] :
      ( ( bit_se5824344872417868541ns_and @ int @ ( one_one @ int ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ ( bit0 @ N2 ) ) ) )
      = ( one_one @ int ) ) ).

% and_not_numerals(2)
thf(fact_6037_bit__imp__take__bit__positive,axiom,
    ! [N2: nat,M: nat,K: int] :
      ( ( ord_less @ nat @ N2 @ M )
     => ( ( bit_se5641148757651400278ts_bit @ int @ K @ N2 )
       => ( ord_less @ int @ ( zero_zero @ int ) @ ( bit_se2584673776208193580ke_bit @ int @ M @ K ) ) ) ) ).

% bit_imp_take_bit_positive
thf(fact_6038_mask__subsume,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [N2: nat,M: nat,X2: word @ A,Y2: word @ A] :
          ( ( ord_less_eq @ nat @ N2 @ M )
         => ( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ ( bit_se1065995026697491101ons_or @ ( word @ A ) @ X2 @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ Y2 @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N2 ) ) ) @ ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ M ) ) )
            = ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X2 @ ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ M ) ) ) ) ) ) ).

% mask_subsume
thf(fact_6039_and__not__num__eq__Some__iff,axiom,
    ! [M: num,N2: num,Q2: num] :
      ( ( ( bit_and_not_num @ M @ N2 )
        = ( some @ num @ Q2 ) )
      = ( ( bit_se5824344872417868541ns_and @ int @ ( numeral_numeral @ int @ M ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ N2 ) ) )
        = ( numeral_numeral @ int @ Q2 ) ) ) ).

% and_not_num_eq_Some_iff
thf(fact_6040_exp__eq__0__imp__not__bit,axiom,
    ! [A: $tType] :
      ( ( bit_semiring_bits @ A )
     => ! [N2: nat,A2: A] :
          ( ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 )
            = ( zero_zero @ A ) )
         => ~ ( bit_se5641148757651400278ts_bit @ A @ A2 @ N2 ) ) ) ).

% exp_eq_0_imp_not_bit
thf(fact_6041_bit__Suc,axiom,
    ! [A: $tType] :
      ( ( bit_semiring_bits @ A )
     => ! [A2: A,N2: nat] :
          ( ( bit_se5641148757651400278ts_bit @ A @ A2 @ ( suc @ N2 ) )
          = ( bit_se5641148757651400278ts_bit @ A @ ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ N2 ) ) ) ).

% bit_Suc
thf(fact_6042_stable__imp__bit__iff__odd,axiom,
    ! [A: $tType] :
      ( ( bit_semiring_bits @ A )
     => ! [A2: A,N2: nat] :
          ( ( ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
            = A2 )
         => ( ( bit_se5641148757651400278ts_bit @ A @ A2 @ N2 )
            = ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) ) ) ) ) ).

% stable_imp_bit_iff_odd
thf(fact_6043_bit__iff__idd__imp__stable,axiom,
    ! [A: $tType] :
      ( ( bit_semiring_bits @ A )
     => ! [A2: A] :
          ( ! [N4: nat] :
              ( ( bit_se5641148757651400278ts_bit @ A @ A2 @ N4 )
              = ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) ) )
         => ( ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
            = A2 ) ) ) ).

% bit_iff_idd_imp_stable
thf(fact_6044_bit_Ocompl__unique,axiom,
    ! [A: $tType] :
      ( ( bit_ri3973907225187159222ations @ A )
     => ! [X2: A,Y2: A] :
          ( ( ( bit_se5824344872417868541ns_and @ A @ X2 @ Y2 )
            = ( zero_zero @ A ) )
         => ( ( ( bit_se1065995026697491101ons_or @ A @ X2 @ Y2 )
              = ( uminus_uminus @ A @ ( one_one @ A ) ) )
           => ( ( bit_ri4277139882892585799ns_not @ A @ X2 )
              = Y2 ) ) ) ) ).

% bit.compl_unique
thf(fact_6045_int__bit__bound,axiom,
    ! [K: int] :
      ~ ! [N4: nat] :
          ( ! [M3: nat] :
              ( ( ord_less_eq @ nat @ N4 @ M3 )
             => ( ( bit_se5641148757651400278ts_bit @ int @ K @ M3 )
                = ( bit_se5641148757651400278ts_bit @ int @ K @ N4 ) ) )
         => ~ ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N4 )
             => ( ( bit_se5641148757651400278ts_bit @ int @ K @ ( minus_minus @ nat @ N4 @ ( one_one @ nat ) ) )
                = ( ~ ( bit_se5641148757651400278ts_bit @ int @ K @ N4 ) ) ) ) ) ).

% int_bit_bound
thf(fact_6046_and__not__numerals_I5_J,axiom,
    ! [M: num,N2: num] :
      ( ( bit_se5824344872417868541ns_and @ int @ ( numeral_numeral @ int @ ( bit0 @ M ) ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ ( bit0 @ N2 ) ) ) )
      = ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ int @ ( numeral_numeral @ int @ M ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ N2 ) ) ) ) ) ).

% and_not_numerals(5)
thf(fact_6047_and__not__numerals_I7_J,axiom,
    ! [M: num] :
      ( ( bit_se5824344872417868541ns_and @ int @ ( numeral_numeral @ int @ ( bit1 @ M ) ) @ ( bit_ri4277139882892585799ns_not @ int @ ( one_one @ int ) ) )
      = ( numeral_numeral @ int @ ( bit0 @ M ) ) ) ).

% and_not_numerals(7)
thf(fact_6048_and__not__numerals_I3_J,axiom,
    ! [N2: num] :
      ( ( bit_se5824344872417868541ns_and @ int @ ( one_one @ int ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ ( bit1 @ N2 ) ) ) )
      = ( zero_zero @ int ) ) ).

% and_not_numerals(3)
thf(fact_6049_and__not__num__eq__None__iff,axiom,
    ! [M: num,N2: num] :
      ( ( ( bit_and_not_num @ M @ N2 )
        = ( none @ num ) )
      = ( ( bit_se5824344872417868541ns_and @ int @ ( numeral_numeral @ int @ M ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ N2 ) ) )
        = ( zero_zero @ int ) ) ) ).

% and_not_num_eq_None_iff
thf(fact_6050_bit__iff__odd,axiom,
    ! [A: $tType] :
      ( ( bit_semiring_bits @ A )
     => ( ( bit_se5641148757651400278ts_bit @ A )
        = ( ^ [A5: A,N: nat] :
              ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ A5 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) ) ) ) ) ) ).

% bit_iff_odd
thf(fact_6051_and__exp__eq__0__iff__not__bit,axiom,
    ! [A: $tType] :
      ( ( bit_se359711467146920520ations @ A )
     => ! [A2: A,N2: nat] :
          ( ( ( bit_se5824344872417868541ns_and @ A @ A2 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) )
            = ( zero_zero @ A ) )
          = ( ~ ( bit_se5641148757651400278ts_bit @ A @ A2 @ N2 ) ) ) ) ).

% and_exp_eq_0_iff_not_bit
thf(fact_6052_bit__not__iff__eq,axiom,
    ! [A: $tType] :
      ( ( bit_ri3973907225187159222ations @ A )
     => ! [A2: A,N2: nat] :
          ( ( bit_se5641148757651400278ts_bit @ A @ ( bit_ri4277139882892585799ns_not @ A @ A2 ) @ N2 )
          = ( ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 )
             != ( zero_zero @ A ) )
            & ~ ( bit_se5641148757651400278ts_bit @ A @ A2 @ N2 ) ) ) ) ).

% bit_not_iff_eq
thf(fact_6053_mask__Suc__exp,axiom,
    ! [A: $tType] :
      ( ( bit_se359711467146920520ations @ A )
     => ! [N2: nat] :
          ( ( bit_se2239418461657761734s_mask @ A @ ( suc @ N2 ) )
          = ( bit_se1065995026697491101ons_or @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) @ ( bit_se2239418461657761734s_mask @ A @ N2 ) ) ) ) ).

% mask_Suc_exp
thf(fact_6054_and__not__numerals_I9_J,axiom,
    ! [M: num,N2: num] :
      ( ( bit_se5824344872417868541ns_and @ int @ ( numeral_numeral @ int @ ( bit1 @ M ) ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ ( bit1 @ N2 ) ) ) )
      = ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ int @ ( numeral_numeral @ int @ M ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ N2 ) ) ) ) ) ).

% and_not_numerals(9)
thf(fact_6055_and__not__numerals_I6_J,axiom,
    ! [M: num,N2: num] :
      ( ( bit_se5824344872417868541ns_and @ int @ ( numeral_numeral @ int @ ( bit0 @ M ) ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ ( bit1 @ N2 ) ) ) )
      = ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ int @ ( numeral_numeral @ int @ M ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ N2 ) ) ) ) ) ).

% and_not_numerals(6)
thf(fact_6056_int__numeral__not__and__num,axiom,
    ! [M: num,N2: num] :
      ( ( bit_se5824344872417868541ns_and @ int @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ M ) ) @ ( numeral_numeral @ int @ N2 ) )
      = ( case_option @ int @ num @ ( zero_zero @ int ) @ ( numeral_numeral @ int ) @ ( bit_and_not_num @ N2 @ M ) ) ) ).

% int_numeral_not_and_num
thf(fact_6057_int__numeral__and__not__num,axiom,
    ! [M: num,N2: num] :
      ( ( bit_se5824344872417868541ns_and @ int @ ( numeral_numeral @ int @ M ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ N2 ) ) )
      = ( case_option @ int @ num @ ( zero_zero @ int ) @ ( numeral_numeral @ int ) @ ( bit_and_not_num @ M @ N2 ) ) ) ).

% int_numeral_and_not_num
thf(fact_6058_bit__int__def,axiom,
    ( ( bit_se5641148757651400278ts_bit @ int )
    = ( ^ [K4: int,N: nat] :
          ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( divide_divide @ int @ K4 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) ) ) ) ).

% bit_int_def
thf(fact_6059_even__bit__succ__iff,axiom,
    ! [A: $tType] :
      ( ( bit_semiring_bits @ A )
     => ! [A2: A,N2: nat] :
          ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 )
         => ( ( bit_se5641148757651400278ts_bit @ A @ ( plus_plus @ A @ ( one_one @ A ) @ A2 ) @ N2 )
            = ( ( bit_se5641148757651400278ts_bit @ A @ A2 @ N2 )
              | ( N2
                = ( zero_zero @ nat ) ) ) ) ) ) ).

% even_bit_succ_iff
thf(fact_6060_odd__bit__iff__bit__pred,axiom,
    ! [A: $tType] :
      ( ( bit_semiring_bits @ A )
     => ! [A2: A,N2: nat] :
          ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 )
         => ( ( bit_se5641148757651400278ts_bit @ A @ A2 @ N2 )
            = ( ( bit_se5641148757651400278ts_bit @ A @ ( minus_minus @ A @ A2 @ ( one_one @ A ) ) @ N2 )
              | ( N2
                = ( zero_zero @ nat ) ) ) ) ) ) ).

% odd_bit_iff_bit_pred
thf(fact_6061_mask__Suc__double,axiom,
    ! [A: $tType] :
      ( ( bit_se359711467146920520ations @ A )
     => ! [N2: nat] :
          ( ( bit_se2239418461657761734s_mask @ A @ ( suc @ N2 ) )
          = ( bit_se1065995026697491101ons_or @ A @ ( one_one @ A ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se2239418461657761734s_mask @ A @ N2 ) ) ) ) ) ).

% mask_Suc_double
thf(fact_6062_or__one__eq,axiom,
    ! [A: $tType] :
      ( ( bit_se359711467146920520ations @ A )
     => ! [A2: A] :
          ( ( bit_se1065995026697491101ons_or @ A @ A2 @ ( one_one @ A ) )
          = ( plus_plus @ A @ A2 @ ( zero_neq_one_of_bool @ A @ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) ) ) ) ) ).

% or_one_eq
thf(fact_6063_one__or__eq,axiom,
    ! [A: $tType] :
      ( ( bit_se359711467146920520ations @ A )
     => ! [A2: A] :
          ( ( bit_se1065995026697491101ons_or @ A @ ( one_one @ A ) @ A2 )
          = ( plus_plus @ A @ A2 @ ( zero_neq_one_of_bool @ A @ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) ) ) ) ) ).

% one_or_eq
thf(fact_6064_bit__sum__mult__2__cases,axiom,
    ! [A: $tType] :
      ( ( bit_se359711467146920520ations @ A )
     => ! [A2: A,B2: A,N2: nat] :
          ( ! [J2: nat] :
              ~ ( bit_se5641148757651400278ts_bit @ A @ A2 @ ( suc @ J2 ) )
         => ( ( bit_se5641148757651400278ts_bit @ A @ ( plus_plus @ A @ A2 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) ) @ N2 )
            = ( ( ( N2
                  = ( zero_zero @ nat ) )
               => ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) )
              & ( ( N2
                 != ( zero_zero @ nat ) )
               => ( bit_se5641148757651400278ts_bit @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) @ N2 ) ) ) ) ) ) ).

% bit_sum_mult_2_cases
thf(fact_6065_bit__rec,axiom,
    ! [A: $tType] :
      ( ( bit_semiring_bits @ A )
     => ( ( bit_se5641148757651400278ts_bit @ A )
        = ( ^ [A5: A,N: nat] :
              ( ( ( N
                  = ( zero_zero @ nat ) )
               => ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A5 ) )
              & ( ( N
                 != ( zero_zero @ nat ) )
               => ( bit_se5641148757651400278ts_bit @ A @ ( divide_divide @ A @ A5 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) ) ) ) ) ) ).

% bit_rec
thf(fact_6066_and__not__numerals_I8_J,axiom,
    ! [M: num,N2: num] :
      ( ( bit_se5824344872417868541ns_and @ int @ ( numeral_numeral @ int @ ( bit1 @ M ) ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ ( bit0 @ N2 ) ) ) )
      = ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ int @ ( numeral_numeral @ int @ M ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ N2 ) ) ) ) ) ) ).

% and_not_numerals(8)
thf(fact_6067_not__int__rec,axiom,
    ( ( bit_ri4277139882892585799ns_not @ int )
    = ( ^ [K4: int] : ( plus_plus @ int @ ( zero_neq_one_of_bool @ int @ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K4 ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_ri4277139882892585799ns_not @ int @ ( divide_divide @ int @ K4 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).

% not_int_rec
thf(fact_6068_Bit__Operations_Oset__bit__eq,axiom,
    ( ( bit_se5668285175392031749et_bit @ int )
    = ( ^ [N: nat,K4: int] :
          ( plus_plus @ int @ K4
          @ ( times_times @ int
            @ ( zero_neq_one_of_bool @ int
              @ ~ ( bit_se5641148757651400278ts_bit @ int @ K4 @ N ) )
            @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) ) ) ) ).

% Bit_Operations.set_bit_eq
thf(fact_6069_unset__bit__eq,axiom,
    ( ( bit_se2638667681897837118et_bit @ int )
    = ( ^ [N: nat,K4: int] : ( minus_minus @ int @ K4 @ ( times_times @ int @ ( zero_neq_one_of_bool @ int @ ( bit_se5641148757651400278ts_bit @ int @ K4 @ N ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) ) ) ) ).

% unset_bit_eq
thf(fact_6070_take__bit__Suc__from__most,axiom,
    ! [N2: nat,K: int] :
      ( ( bit_se2584673776208193580ke_bit @ int @ ( suc @ N2 ) @ K )
      = ( plus_plus @ int @ ( times_times @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N2 ) @ ( zero_neq_one_of_bool @ int @ ( bit_se5641148757651400278ts_bit @ int @ K @ N2 ) ) ) @ ( bit_se2584673776208193580ke_bit @ int @ N2 @ K ) ) ) ).

% take_bit_Suc_from_most
thf(fact_6071_int__not__code_I1_J,axiom,
    ( ( bit_ri4277139882892585799ns_not @ int @ ( zero_zero @ int ) )
    = ( uminus_uminus @ int @ ( one_one @ int ) ) ) ).

% int_not_code(1)
thf(fact_6072_bitNOT__integer__code,axiom,
    ( ( bit_ri4277139882892585799ns_not @ code_integer )
    = ( ^ [I2: code_integer] : ( minus_minus @ code_integer @ ( uminus_uminus @ code_integer @ I2 ) @ ( one_one @ code_integer ) ) ) ) ).

% bitNOT_integer_code
thf(fact_6073_test__bit__1,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [N2: nat] :
          ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( one_one @ ( word @ A ) ) @ N2 )
          = ( N2
            = ( zero_zero @ nat ) ) ) ) ).

% test_bit_1
thf(fact_6074_or__nonnegative__int__iff,axiom,
    ! [K: int,L: int] :
      ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_se1065995026697491101ons_or @ 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_6075_map__nth__0,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [Xs2: list @ nat] :
          ( ( map @ nat @ $o @ ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( zero_zero @ ( word @ A ) ) ) @ Xs2 )
          = ( replicate @ $o @ ( size_size @ ( list @ nat ) @ Xs2 ) @ $false ) ) ) ).

% map_nth_0
thf(fact_6076_or__nat__numerals_I2_J,axiom,
    ! [Y2: num] :
      ( ( bit_se1065995026697491101ons_or @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( numeral_numeral @ nat @ ( bit1 @ Y2 ) ) )
      = ( numeral_numeral @ nat @ ( bit1 @ Y2 ) ) ) ).

% or_nat_numerals(2)
thf(fact_6077_or__nat__numerals_I4_J,axiom,
    ! [X2: num] :
      ( ( bit_se1065995026697491101ons_or @ nat @ ( numeral_numeral @ nat @ ( bit1 @ X2 ) ) @ ( suc @ ( zero_zero @ nat ) ) )
      = ( numeral_numeral @ nat @ ( bit1 @ X2 ) ) ) ).

% or_nat_numerals(4)
thf(fact_6078_or__minus__numerals_I6_J,axiom,
    ! [N2: num] :
      ( ( bit_se1065995026697491101ons_or @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ N2 ) ) ) @ ( one_one @ int ) )
      = ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ N2 ) ) ) ) ).

% or_minus_numerals(6)
thf(fact_6079_or__minus__numerals_I2_J,axiom,
    ! [N2: num] :
      ( ( bit_se1065995026697491101ons_or @ int @ ( one_one @ int ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ N2 ) ) ) )
      = ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ N2 ) ) ) ) ).

% or_minus_numerals(2)
thf(fact_6080_word__no__log__defs_I7_J,axiom,
    ! [G2: $tType] :
      ( ( type_len @ G2 )
     => ! [A2: num,B2: num] :
          ( ( bit_se1065995026697491101ons_or @ ( word @ G2 ) @ ( numeral_numeral @ ( word @ G2 ) @ A2 ) @ ( numeral_numeral @ ( word @ G2 ) @ B2 ) )
          = ( ring_1_of_int @ ( word @ G2 ) @ ( bit_se1065995026697491101ons_or @ int @ ( numeral_numeral @ int @ A2 ) @ ( numeral_numeral @ int @ B2 ) ) ) ) ) ).

% word_no_log_defs(7)
thf(fact_6081_or__nat__numerals_I3_J,axiom,
    ! [X2: num] :
      ( ( bit_se1065995026697491101ons_or @ nat @ ( numeral_numeral @ nat @ ( bit0 @ X2 ) ) @ ( suc @ ( zero_zero @ nat ) ) )
      = ( numeral_numeral @ nat @ ( bit1 @ X2 ) ) ) ).

% or_nat_numerals(3)
thf(fact_6082_or__nat__numerals_I1_J,axiom,
    ! [Y2: num] :
      ( ( bit_se1065995026697491101ons_or @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( numeral_numeral @ nat @ ( bit0 @ Y2 ) ) )
      = ( numeral_numeral @ nat @ ( bit1 @ Y2 ) ) ) ).

% or_nat_numerals(1)
thf(fact_6083_word__bitwise__1__simps_I8_J,axiom,
    ! [H10: $tType] :
      ( ( type_len @ H10 )
     => ! [A2: num] :
          ( ( bit_se1065995026697491101ons_or @ ( word @ H10 ) @ ( numeral_numeral @ ( word @ H10 ) @ A2 ) @ ( one_one @ ( word @ H10 ) ) )
          = ( ring_1_of_int @ ( word @ H10 ) @ ( bit_se1065995026697491101ons_or @ int @ ( numeral_numeral @ int @ A2 ) @ ( one_one @ int ) ) ) ) ) ).

% word_bitwise_1_simps(8)
thf(fact_6084_word__bitwise__1__simps_I6_J,axiom,
    ! [F3: $tType] :
      ( ( type_len @ F3 )
     => ! [B2: num] :
          ( ( bit_se1065995026697491101ons_or @ ( word @ F3 ) @ ( one_one @ ( word @ F3 ) ) @ ( numeral_numeral @ ( word @ F3 ) @ B2 ) )
          = ( ring_1_of_int @ ( word @ F3 ) @ ( bit_se1065995026697491101ons_or @ int @ ( one_one @ int ) @ ( numeral_numeral @ int @ B2 ) ) ) ) ) ).

% word_bitwise_1_simps(6)
thf(fact_6085_word__no__log__defs_I8_J,axiom,
    ! [H10: $tType] :
      ( ( type_len @ H10 )
     => ! [A2: num,B2: num] :
          ( ( bit_se1065995026697491101ons_or @ ( word @ H10 ) @ ( numeral_numeral @ ( word @ H10 ) @ A2 ) @ ( uminus_uminus @ ( word @ H10 ) @ ( numeral_numeral @ ( word @ H10 ) @ B2 ) ) )
          = ( ring_1_of_int @ ( word @ H10 ) @ ( bit_se1065995026697491101ons_or @ int @ ( numeral_numeral @ int @ A2 ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ B2 ) ) ) ) ) ) ).

% word_no_log_defs(8)
thf(fact_6086_word__no__log__defs_I9_J,axiom,
    ! [I6: $tType] :
      ( ( type_len @ I6 )
     => ! [A2: num,B2: num] :
          ( ( bit_se1065995026697491101ons_or @ ( word @ I6 ) @ ( uminus_uminus @ ( word @ I6 ) @ ( numeral_numeral @ ( word @ I6 ) @ A2 ) ) @ ( numeral_numeral @ ( word @ I6 ) @ B2 ) )
          = ( ring_1_of_int @ ( word @ I6 ) @ ( bit_se1065995026697491101ons_or @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ A2 ) ) @ ( numeral_numeral @ int @ B2 ) ) ) ) ) ).

% word_no_log_defs(9)
thf(fact_6087_word__no__log__defs_I10_J,axiom,
    ! [J4: $tType] :
      ( ( type_len @ J4 )
     => ! [A2: num,B2: num] :
          ( ( bit_se1065995026697491101ons_or @ ( word @ J4 ) @ ( uminus_uminus @ ( word @ J4 ) @ ( numeral_numeral @ ( word @ J4 ) @ A2 ) ) @ ( uminus_uminus @ ( word @ J4 ) @ ( numeral_numeral @ ( word @ J4 ) @ B2 ) ) )
          = ( ring_1_of_int @ ( word @ J4 ) @ ( bit_se1065995026697491101ons_or @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ A2 ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ B2 ) ) ) ) ) ) ).

% word_no_log_defs(10)
thf(fact_6088_and__minus__minus__numerals,axiom,
    ! [M: num,N2: num] :
      ( ( bit_se5824344872417868541ns_and @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N2 ) ) )
      = ( bit_ri4277139882892585799ns_not @ int @ ( bit_se1065995026697491101ons_or @ int @ ( minus_minus @ int @ ( numeral_numeral @ int @ M ) @ ( one_one @ int ) ) @ ( minus_minus @ int @ ( numeral_numeral @ int @ N2 ) @ ( one_one @ int ) ) ) ) ) ).

% and_minus_minus_numerals
thf(fact_6089_or__minus__minus__numerals,axiom,
    ! [M: num,N2: num] :
      ( ( bit_se1065995026697491101ons_or @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N2 ) ) )
      = ( bit_ri4277139882892585799ns_not @ int @ ( bit_se5824344872417868541ns_and @ int @ ( minus_minus @ int @ ( numeral_numeral @ int @ M ) @ ( one_one @ int ) ) @ ( minus_minus @ int @ ( numeral_numeral @ int @ N2 ) @ ( one_one @ int ) ) ) ) ) ).

% or_minus_minus_numerals
thf(fact_6090_word__bitwise__1__simps_I9_J,axiom,
    ! [I6: $tType] :
      ( ( type_len @ I6 )
     => ! [A2: num] :
          ( ( bit_se1065995026697491101ons_or @ ( word @ I6 ) @ ( uminus_uminus @ ( word @ I6 ) @ ( numeral_numeral @ ( word @ I6 ) @ A2 ) ) @ ( one_one @ ( word @ I6 ) ) )
          = ( ring_1_of_int @ ( word @ I6 ) @ ( bit_se1065995026697491101ons_or @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ A2 ) ) @ ( one_one @ int ) ) ) ) ) ).

% word_bitwise_1_simps(9)
thf(fact_6091_word__bitwise__1__simps_I7_J,axiom,
    ! [G2: $tType] :
      ( ( type_len @ G2 )
     => ! [B2: num] :
          ( ( bit_se1065995026697491101ons_or @ ( word @ G2 ) @ ( one_one @ ( word @ G2 ) ) @ ( uminus_uminus @ ( word @ G2 ) @ ( numeral_numeral @ ( word @ G2 ) @ B2 ) ) )
          = ( ring_1_of_int @ ( word @ G2 ) @ ( bit_se1065995026697491101ons_or @ int @ ( one_one @ int ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ B2 ) ) ) ) ) ) ).

% word_bitwise_1_simps(7)
thf(fact_6092_test__bit__size,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [W: word @ A,N2: nat] :
          ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ W @ N2 )
         => ( ord_less @ nat @ N2 @ ( size_size @ ( word @ A ) @ W ) ) ) ) ).

% test_bit_size
thf(fact_6093_word__eqI,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [U: word @ A,V: word @ A] :
          ( ! [N4: nat] :
              ( ( ord_less @ nat @ N4 @ ( size_size @ ( word @ A ) @ U ) )
             => ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ U @ N4 )
                = ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ V @ N4 ) ) )
         => ( U = V ) ) ) ).

% word_eqI
thf(fact_6094_test__bit__over,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [X2: word @ A,N2: nat] :
          ( ( ord_less_eq @ nat @ ( size_size @ ( word @ A ) @ X2 ) @ N2 )
         => ~ ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ X2 @ N2 ) ) ) ).

% test_bit_over
thf(fact_6095_finite__bit__word,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [W: word @ A] : ( finite_finite2 @ nat @ ( collect @ nat @ ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ W ) ) ) ) ).

% finite_bit_word
thf(fact_6096_not__bit__Suc__0__Suc,axiom,
    ! [N2: nat] :
      ~ ( bit_se5641148757651400278ts_bit @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( suc @ N2 ) ) ).

% not_bit_Suc_0_Suc
thf(fact_6097_bit__Suc__0__iff,axiom,
    ! [N2: nat] :
      ( ( bit_se5641148757651400278ts_bit @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N2 )
      = ( N2
        = ( zero_zero @ nat ) ) ) ).

% bit_Suc_0_iff
thf(fact_6098_plus__and__or,axiom,
    ! [X2: int,Y2: int] :
      ( ( plus_plus @ int @ ( bit_se5824344872417868541ns_and @ int @ X2 @ Y2 ) @ ( bit_se1065995026697491101ons_or @ int @ X2 @ Y2 ) )
      = ( plus_plus @ int @ X2 @ Y2 ) ) ).

% plus_and_or
thf(fact_6099_OR__lower,axiom,
    ! [X2: int,Y2: int] :
      ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X2 )
     => ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y2 )
       => ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_se1065995026697491101ons_or @ int @ X2 @ Y2 ) ) ) ) ).

% OR_lower
thf(fact_6100_or__greater__eq,axiom,
    ! [L: int,K: int] :
      ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ L )
     => ( ord_less_eq @ int @ K @ ( bit_se1065995026697491101ons_or @ int @ K @ L ) ) ) ).

% or_greater_eq
thf(fact_6101_not__bit__Suc__0__numeral,axiom,
    ! [N2: num] :
      ~ ( bit_se5641148757651400278ts_bit @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( numeral_numeral @ nat @ N2 ) ) ).

% not_bit_Suc_0_numeral
thf(fact_6102_or__not__numerals_I1_J,axiom,
    ( ( bit_se1065995026697491101ons_or @ int @ ( one_one @ int ) @ ( bit_ri4277139882892585799ns_not @ int @ ( one_one @ int ) ) )
    = ( bit_ri4277139882892585799ns_not @ int @ ( zero_zero @ int ) ) ) ).

% or_not_numerals(1)
thf(fact_6103_lsb__this__or__next,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [X2: word @ A] :
          ( ~ ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ X2 @ ( one_one @ ( word @ A ) ) ) @ ( zero_zero @ nat ) )
         => ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ X2 @ ( zero_zero @ nat ) ) ) ) ).

% lsb_this_or_next
thf(fact_6104_word__leI,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [U: word @ A,V: word @ A] :
          ( ! [N4: nat] :
              ( ( ord_less @ nat @ N4 @ ( size_size @ ( word @ A ) @ U ) )
             => ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ U @ N4 )
               => ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ V @ N4 ) ) )
         => ( ord_less_eq @ ( word @ A ) @ U @ V ) ) ) ).

% word_leI
thf(fact_6105_nth__mask,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [N2: nat,I: nat] :
          ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N2 ) @ I )
          = ( ( ord_less @ nat @ I @ N2 )
            & ( ord_less @ nat @ I @ ( size_size @ ( word @ A ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N2 ) ) ) ) ) ) ).

% nth_mask
thf(fact_6106_or__not__numerals_I2_J,axiom,
    ! [N2: num] :
      ( ( bit_se1065995026697491101ons_or @ int @ ( one_one @ int ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ ( bit0 @ N2 ) ) ) )
      = ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ ( bit0 @ N2 ) ) ) ) ).

% or_not_numerals(2)
thf(fact_6107_or__not__numerals_I4_J,axiom,
    ! [M: num] :
      ( ( bit_se1065995026697491101ons_or @ int @ ( numeral_numeral @ int @ ( bit0 @ M ) ) @ ( bit_ri4277139882892585799ns_not @ int @ ( one_one @ int ) ) )
      = ( bit_ri4277139882892585799ns_not @ int @ ( one_one @ int ) ) ) ).

% or_not_numerals(4)
thf(fact_6108_bit__nat__iff,axiom,
    ! [K: int,N2: nat] :
      ( ( bit_se5641148757651400278ts_bit @ nat @ ( nat2 @ K ) @ N2 )
      = ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
        & ( bit_se5641148757651400278ts_bit @ int @ K @ N2 ) ) ) ).

% bit_nat_iff
thf(fact_6109_overflow__imp__lsb,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [X2: word @ A] :
          ( ( ( plus_plus @ ( word @ A ) @ X2 @ ( one_one @ ( word @ A ) ) )
            = ( zero_zero @ ( word @ A ) ) )
         => ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ X2 @ ( zero_zero @ nat ) ) ) ) ).

% overflow_imp_lsb
thf(fact_6110_word__and__1,axiom,
    ! [B: $tType] :
      ( ( type_len @ B )
     => ! [N2: word @ B] :
          ( ( ( bit_se5641148757651400278ts_bit @ ( word @ B ) @ N2 @ ( zero_zero @ nat ) )
           => ( ( bit_se5824344872417868541ns_and @ ( word @ B ) @ N2 @ ( one_one @ ( word @ B ) ) )
              = ( one_one @ ( word @ B ) ) ) )
          & ( ~ ( bit_se5641148757651400278ts_bit @ ( word @ B ) @ N2 @ ( zero_zero @ nat ) )
           => ( ( bit_se5824344872417868541ns_and @ ( word @ B ) @ N2 @ ( one_one @ ( word @ B ) ) )
              = ( zero_zero @ ( word @ B ) ) ) ) ) ) ).

% word_and_1
thf(fact_6111_test__bit__bin_H,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) )
        = ( ^ [W2: word @ A,N: nat] :
              ( ( ord_less @ nat @ N @ ( size_size @ ( word @ A ) @ W2 ) )
              & ( bit_se5641148757651400278ts_bit @ int @ ( semiring_1_unsigned @ A @ int @ W2 ) @ N ) ) ) ) ) ).

% test_bit_bin'
thf(fact_6112_le__mask__high__bits,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [W: word @ A,N2: nat] :
          ( ( ord_less_eq @ ( word @ A ) @ W @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N2 ) )
          = ( ! [X: nat] :
                ( ( member @ nat @ X @ ( set_or7035219750837199246ssThan @ nat @ N2 @ ( size_size @ ( word @ A ) @ W ) ) )
               => ~ ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ W @ X ) ) ) ) ) ).

% le_mask_high_bits
thf(fact_6113_or__not__numerals_I3_J,axiom,
    ! [N2: num] :
      ( ( bit_se1065995026697491101ons_or @ int @ ( one_one @ int ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ ( bit1 @ N2 ) ) ) )
      = ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ ( bit0 @ N2 ) ) ) ) ).

% or_not_numerals(3)
thf(fact_6114_or__not__numerals_I7_J,axiom,
    ! [M: num] :
      ( ( bit_se1065995026697491101ons_or @ int @ ( numeral_numeral @ int @ ( bit1 @ M ) ) @ ( bit_ri4277139882892585799ns_not @ int @ ( one_one @ int ) ) )
      = ( bit_ri4277139882892585799ns_not @ int @ ( zero_zero @ int ) ) ) ).

% or_not_numerals(7)
thf(fact_6115_bang__is__le,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [X2: word @ A,M: nat] :
          ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ X2 @ M )
         => ( ord_less_eq @ ( word @ A ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ M ) @ X2 ) ) ) ).

% bang_is_le
thf(fact_6116_bit__nat__def,axiom,
    ( ( bit_se5641148757651400278ts_bit @ nat )
    = ( ^ [M7: nat,N: nat] :
          ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ M7 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) ) ) ).

% bit_nat_def
thf(fact_6117_or__not__numerals_I6_J,axiom,
    ! [M: num,N2: num] :
      ( ( bit_se1065995026697491101ons_or @ int @ ( numeral_numeral @ int @ ( bit0 @ M ) ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ ( bit1 @ N2 ) ) ) )
      = ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se1065995026697491101ons_or @ int @ ( numeral_numeral @ int @ M ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ N2 ) ) ) ) ) ).

% or_not_numerals(6)
thf(fact_6118_odd__iff__lsb,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [X2: word @ A] :
          ( ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( semiring_1_unsigned @ A @ nat @ X2 ) ) )
          = ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ X2 @ ( zero_zero @ nat ) ) ) ) ).

% odd_iff_lsb
thf(fact_6119_and__neq__0__is__nth,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [Y2: word @ A,N2: nat,X2: word @ A] :
          ( ( Y2
            = ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N2 ) )
         => ( ( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X2 @ Y2 )
             != ( zero_zero @ ( word @ A ) ) )
            = ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ X2 @ N2 ) ) ) ) ).

% and_neq_0_is_nth
thf(fact_6120_nth__is__and__neq__0,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) )
        = ( ^ [X: word @ A,N: nat] :
              ( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N ) )
             != ( zero_zero @ ( word @ A ) ) ) ) ) ) ).

% nth_is_and_neq_0
thf(fact_6121_OR__upper,axiom,
    ! [X2: int,N2: nat,Y2: int] :
      ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X2 )
     => ( ( ord_less @ int @ X2 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N2 ) )
       => ( ( ord_less @ int @ Y2 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N2 ) )
         => ( ord_less @ int @ ( bit_se1065995026697491101ons_or @ int @ X2 @ Y2 ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N2 ) ) ) ) ) ).

% OR_upper
thf(fact_6122_or__not__numerals_I5_J,axiom,
    ! [M: num,N2: num] :
      ( ( bit_se1065995026697491101ons_or @ int @ ( numeral_numeral @ int @ ( bit0 @ M ) ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ ( bit0 @ N2 ) ) ) )
      = ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se1065995026697491101ons_or @ int @ ( numeral_numeral @ int @ M ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ N2 ) ) ) ) ) ) ).

% or_not_numerals(5)
thf(fact_6123_Suc__0__or__eq,axiom,
    ! [N2: nat] :
      ( ( bit_se1065995026697491101ons_or @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N2 )
      = ( plus_plus @ nat @ N2 @ ( zero_neq_one_of_bool @ nat @ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ) ) ).

% Suc_0_or_eq
thf(fact_6124_or__Suc__0__eq,axiom,
    ! [N2: nat] :
      ( ( bit_se1065995026697491101ons_or @ nat @ N2 @ ( suc @ ( zero_zero @ nat ) ) )
      = ( plus_plus @ nat @ N2 @ ( zero_neq_one_of_bool @ nat @ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ) ) ).

% or_Suc_0_eq
thf(fact_6125_or__nat__rec,axiom,
    ( ( bit_se1065995026697491101ons_or @ nat )
    = ( ^ [M7: nat,N: nat] :
          ( plus_plus @ nat
          @ ( zero_neq_one_of_bool @ nat
            @ ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M7 )
              | ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) )
          @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( bit_se1065995026697491101ons_or @ nat @ ( divide_divide @ nat @ M7 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).

% or_nat_rec
thf(fact_6126_or__not__numerals_I9_J,axiom,
    ! [M: num,N2: num] :
      ( ( bit_se1065995026697491101ons_or @ int @ ( numeral_numeral @ int @ ( bit1 @ M ) ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ ( bit1 @ N2 ) ) ) )
      = ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se1065995026697491101ons_or @ int @ ( numeral_numeral @ int @ M ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ N2 ) ) ) ) ) ) ).

% or_not_numerals(9)
thf(fact_6127_or__not__numerals_I8_J,axiom,
    ! [M: num,N2: num] :
      ( ( bit_se1065995026697491101ons_or @ int @ ( numeral_numeral @ int @ ( bit1 @ M ) ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ ( bit0 @ N2 ) ) ) )
      = ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se1065995026697491101ons_or @ int @ ( numeral_numeral @ int @ M ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ N2 ) ) ) ) ) ) ).

% or_not_numerals(8)
thf(fact_6128_or__int__rec,axiom,
    ( ( bit_se1065995026697491101ons_or @ int )
    = ( ^ [K4: int,L2: int] :
          ( plus_plus @ int
          @ ( zero_neq_one_of_bool @ int
            @ ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K4 )
              | ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ L2 ) ) )
          @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se1065995026697491101ons_or @ int @ ( divide_divide @ int @ K4 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ L2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).

% or_int_rec
thf(fact_6129_or__nat__unfold,axiom,
    ( ( bit_se1065995026697491101ons_or @ nat )
    = ( ^ [M7: nat,N: nat] :
          ( if @ nat
          @ ( M7
            = ( zero_zero @ nat ) )
          @ N
          @ ( if @ nat
            @ ( N
              = ( zero_zero @ nat ) )
            @ M7
            @ ( plus_plus @ nat @ ( ord_max @ nat @ ( modulo_modulo @ nat @ M7 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( modulo_modulo @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( bit_se1065995026697491101ons_or @ nat @ ( divide_divide @ nat @ M7 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ).

% or_nat_unfold
thf(fact_6130_or__int__unfold,axiom,
    ( ( bit_se1065995026697491101ons_or @ int )
    = ( ^ [K4: int,L2: int] :
          ( if @ int
          @ ( ( K4
              = ( uminus_uminus @ int @ ( one_one @ int ) ) )
            | ( L2
              = ( uminus_uminus @ int @ ( one_one @ int ) ) ) )
          @ ( uminus_uminus @ int @ ( one_one @ int ) )
          @ ( if @ int
            @ ( K4
              = ( zero_zero @ int ) )
            @ L2
            @ ( if @ int
              @ ( L2
                = ( zero_zero @ int ) )
              @ K4
              @ ( plus_plus @ int @ ( ord_max @ int @ ( modulo_modulo @ int @ K4 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( modulo_modulo @ int @ L2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se1065995026697491101ons_or @ int @ ( divide_divide @ int @ K4 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ L2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ).

% or_int_unfold
thf(fact_6131_Bit__integer_Oabs__eq,axiom,
    ! [Xa: int,X2: $o] :
      ( ( bits_Bit_integer @ ( code_integer_of_int @ Xa ) @ X2 )
      = ( code_integer_of_int @ ( plus_plus @ int @ ( zero_neq_one_of_bool @ int @ X2 ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ Xa ) ) ) ) ).

% Bit_integer.abs_eq
thf(fact_6132_xor__int__unfold,axiom,
    ( ( bit_se5824344971392196577ns_xor @ int )
    = ( ^ [K4: int,L2: int] :
          ( if @ int
          @ ( K4
            = ( uminus_uminus @ int @ ( one_one @ int ) ) )
          @ ( bit_ri4277139882892585799ns_not @ int @ L2 )
          @ ( if @ int
            @ ( L2
              = ( uminus_uminus @ int @ ( one_one @ int ) ) )
            @ ( bit_ri4277139882892585799ns_not @ int @ K4 )
            @ ( if @ int
              @ ( K4
                = ( zero_zero @ int ) )
              @ L2
              @ ( if @ int
                @ ( L2
                  = ( zero_zero @ int ) )
                @ K4
                @ ( plus_plus @ int @ ( abs_abs @ int @ ( minus_minus @ int @ ( modulo_modulo @ int @ K4 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( modulo_modulo @ int @ L2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344971392196577ns_xor @ int @ ( divide_divide @ int @ K4 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ L2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ).

% xor_int_unfold
thf(fact_6133_xor__nonnegative__int__iff,axiom,
    ! [K: int,L: int] :
      ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_se5824344971392196577ns_xor @ 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_6134_bit_Oxor__cancel__right,axiom,
    ! [A: $tType] :
      ( ( bit_ri3973907225187159222ations @ A )
     => ! [X2: A] :
          ( ( bit_se5824344971392196577ns_xor @ A @ X2 @ ( bit_ri4277139882892585799ns_not @ A @ X2 ) )
          = ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).

% bit.xor_cancel_right
thf(fact_6135_bit_Oxor__cancel__left,axiom,
    ! [A: $tType] :
      ( ( bit_ri3973907225187159222ations @ A )
     => ! [X2: A] :
          ( ( bit_se5824344971392196577ns_xor @ A @ ( bit_ri4277139882892585799ns_not @ A @ X2 ) @ X2 )
          = ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).

% bit.xor_cancel_left
thf(fact_6136_bit_Oxor__one__right,axiom,
    ! [A: $tType] :
      ( ( bit_ri3973907225187159222ations @ A )
     => ! [X2: A] :
          ( ( bit_se5824344971392196577ns_xor @ A @ X2 @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
          = ( bit_ri4277139882892585799ns_not @ A @ X2 ) ) ) ).

% bit.xor_one_right
thf(fact_6137_bit_Oxor__one__left,axiom,
    ! [A: $tType] :
      ( ( bit_ri3973907225187159222ations @ A )
     => ! [X2: A] :
          ( ( bit_se5824344971392196577ns_xor @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ X2 )
          = ( bit_ri4277139882892585799ns_not @ A @ X2 ) ) ) ).

% bit.xor_one_left
thf(fact_6138_xor__numerals_I3_J,axiom,
    ! [A: $tType] :
      ( ( bit_un5681908812861735899ations @ A )
     => ! [X2: num,Y2: num] :
          ( ( bit_se5824344971392196577ns_xor @ A @ ( numeral_numeral @ A @ ( bit0 @ X2 ) ) @ ( numeral_numeral @ A @ ( bit0 @ Y2 ) ) )
          = ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se5824344971392196577ns_xor @ A @ ( numeral_numeral @ A @ X2 ) @ ( numeral_numeral @ A @ Y2 ) ) ) ) ) ).

% xor_numerals(3)
thf(fact_6139_xor__numerals_I1_J,axiom,
    ! [A: $tType] :
      ( ( bit_un5681908812861735899ations @ A )
     => ! [Y2: num] :
          ( ( bit_se5824344971392196577ns_xor @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ Y2 ) ) )
          = ( numeral_numeral @ A @ ( bit1 @ Y2 ) ) ) ) ).

% xor_numerals(1)
thf(fact_6140_xor__numerals_I2_J,axiom,
    ! [A: $tType] :
      ( ( bit_un5681908812861735899ations @ A )
     => ! [Y2: num] :
          ( ( bit_se5824344971392196577ns_xor @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit1 @ Y2 ) ) )
          = ( numeral_numeral @ A @ ( bit0 @ Y2 ) ) ) ) ).

% xor_numerals(2)
thf(fact_6141_xor__numerals_I5_J,axiom,
    ! [A: $tType] :
      ( ( bit_un5681908812861735899ations @ A )
     => ! [X2: num] :
          ( ( bit_se5824344971392196577ns_xor @ A @ ( numeral_numeral @ A @ ( bit0 @ X2 ) ) @ ( one_one @ A ) )
          = ( numeral_numeral @ A @ ( bit1 @ X2 ) ) ) ) ).

% xor_numerals(5)
thf(fact_6142_xor__numerals_I8_J,axiom,
    ! [A: $tType] :
      ( ( bit_un5681908812861735899ations @ A )
     => ! [X2: num] :
          ( ( bit_se5824344971392196577ns_xor @ A @ ( numeral_numeral @ A @ ( bit1 @ X2 ) ) @ ( one_one @ A ) )
          = ( numeral_numeral @ A @ ( bit0 @ X2 ) ) ) ) ).

% xor_numerals(8)
thf(fact_6143_xor__numerals_I7_J,axiom,
    ! [A: $tType] :
      ( ( bit_un5681908812861735899ations @ A )
     => ! [X2: num,Y2: num] :
          ( ( bit_se5824344971392196577ns_xor @ A @ ( numeral_numeral @ A @ ( bit1 @ X2 ) ) @ ( numeral_numeral @ A @ ( bit1 @ Y2 ) ) )
          = ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se5824344971392196577ns_xor @ A @ ( numeral_numeral @ A @ X2 ) @ ( numeral_numeral @ A @ Y2 ) ) ) ) ) ).

% xor_numerals(7)
thf(fact_6144_xor__numerals_I4_J,axiom,
    ! [A: $tType] :
      ( ( bit_un5681908812861735899ations @ A )
     => ! [X2: num,Y2: num] :
          ( ( bit_se5824344971392196577ns_xor @ A @ ( numeral_numeral @ A @ ( bit0 @ X2 ) ) @ ( numeral_numeral @ A @ ( bit1 @ Y2 ) ) )
          = ( plus_plus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se5824344971392196577ns_xor @ A @ ( numeral_numeral @ A @ X2 ) @ ( numeral_numeral @ A @ Y2 ) ) ) ) ) ) ).

% xor_numerals(4)
thf(fact_6145_xor__numerals_I6_J,axiom,
    ! [A: $tType] :
      ( ( bit_un5681908812861735899ations @ A )
     => ! [X2: num,Y2: num] :
          ( ( bit_se5824344971392196577ns_xor @ A @ ( numeral_numeral @ A @ ( bit1 @ X2 ) ) @ ( numeral_numeral @ A @ ( bit0 @ Y2 ) ) )
          = ( plus_plus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se5824344971392196577ns_xor @ A @ ( numeral_numeral @ A @ X2 ) @ ( numeral_numeral @ A @ Y2 ) ) ) ) ) ) ).

% xor_numerals(6)
thf(fact_6146_XOR__lower,axiom,
    ! [X2: int,Y2: int] :
      ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X2 )
     => ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y2 )
       => ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_se5824344971392196577ns_xor @ int @ X2 @ Y2 ) ) ) ) ).

% XOR_lower
thf(fact_6147_even__xor__iff,axiom,
    ! [A: $tType] :
      ( ( bit_se359711467146920520ations @ A )
     => ! [A2: A,B2: A] :
          ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se5824344971392196577ns_xor @ A @ A2 @ B2 ) )
          = ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 )
            = ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) ) ) ) ).

% even_xor_iff
thf(fact_6148_XOR__upper,axiom,
    ! [X2: int,N2: nat,Y2: int] :
      ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X2 )
     => ( ( ord_less @ int @ X2 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N2 ) )
       => ( ( ord_less @ int @ Y2 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N2 ) )
         => ( ord_less @ int @ ( bit_se5824344971392196577ns_xor @ int @ X2 @ Y2 ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N2 ) ) ) ) ) ).

% XOR_upper
thf(fact_6149_xor__int__rec,axiom,
    ( ( bit_se5824344971392196577ns_xor @ int )
    = ( ^ [K4: int,L2: int] :
          ( plus_plus @ int
          @ ( zero_neq_one_of_bool @ int
            @ ( ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K4 ) )
             != ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ L2 ) ) ) )
          @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344971392196577ns_xor @ int @ ( divide_divide @ int @ K4 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ L2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).

% xor_int_rec
thf(fact_6150_xor__one__eq,axiom,
    ! [A: $tType] :
      ( ( bit_se359711467146920520ations @ A )
     => ! [A2: A] :
          ( ( bit_se5824344971392196577ns_xor @ A @ A2 @ ( one_one @ A ) )
          = ( minus_minus @ A @ ( plus_plus @ A @ A2 @ ( zero_neq_one_of_bool @ A @ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) ) )
            @ ( zero_neq_one_of_bool @ A
              @ ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) ) ) ) ) ).

% xor_one_eq
thf(fact_6151_one__xor__eq,axiom,
    ! [A: $tType] :
      ( ( bit_se359711467146920520ations @ A )
     => ! [A2: A] :
          ( ( bit_se5824344971392196577ns_xor @ A @ ( one_one @ A ) @ A2 )
          = ( minus_minus @ A @ ( plus_plus @ A @ A2 @ ( zero_neq_one_of_bool @ A @ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) ) )
            @ ( zero_neq_one_of_bool @ A
              @ ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) ) ) ) ) ).

% one_xor_eq
thf(fact_6152_or__minus__numerals_I1_J,axiom,
    ! [N2: num] :
      ( ( bit_se1065995026697491101ons_or @ int @ ( one_one @ int ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ N2 ) ) ) )
      = ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit_or_not_num_neg @ one2 @ ( bitM @ N2 ) ) ) ) ) ).

% or_minus_numerals(1)
thf(fact_6153_or__minus__numerals_I5_J,axiom,
    ! [N2: num] :
      ( ( bit_se1065995026697491101ons_or @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ N2 ) ) ) @ ( one_one @ int ) )
      = ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit_or_not_num_neg @ one2 @ ( bitM @ N2 ) ) ) ) ) ).

% or_minus_numerals(5)
thf(fact_6154_word__bitwise__m1__simps_I6_J,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [X2: word @ A] :
          ( ( bit_se5824344971392196577ns_xor @ ( word @ A ) @ ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) @ X2 )
          = ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ X2 ) ) ) ).

% word_bitwise_m1_simps(6)
thf(fact_6155_word__bitwise__m1__simps_I7_J,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [X2: word @ A] :
          ( ( bit_se5824344971392196577ns_xor @ ( word @ A ) @ X2 @ ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) )
          = ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ X2 ) ) ) ).

% word_bitwise_m1_simps(7)
thf(fact_6156_word__no__log__defs_I11_J,axiom,
    ! [K8: $tType] :
      ( ( type_len @ K8 )
     => ! [A2: num,B2: num] :
          ( ( bit_se5824344971392196577ns_xor @ ( word @ K8 ) @ ( numeral_numeral @ ( word @ K8 ) @ A2 ) @ ( numeral_numeral @ ( word @ K8 ) @ B2 ) )
          = ( ring_1_of_int @ ( word @ K8 ) @ ( bit_se5824344971392196577ns_xor @ int @ ( numeral_numeral @ int @ A2 ) @ ( numeral_numeral @ int @ B2 ) ) ) ) ) ).

% word_no_log_defs(11)
thf(fact_6157_xor__nat__numerals_I4_J,axiom,
    ! [X2: num] :
      ( ( bit_se5824344971392196577ns_xor @ nat @ ( numeral_numeral @ nat @ ( bit1 @ X2 ) ) @ ( suc @ ( zero_zero @ nat ) ) )
      = ( numeral_numeral @ nat @ ( bit0 @ X2 ) ) ) ).

% xor_nat_numerals(4)
thf(fact_6158_xor__nat__numerals_I3_J,axiom,
    ! [X2: num] :
      ( ( bit_se5824344971392196577ns_xor @ nat @ ( numeral_numeral @ nat @ ( bit0 @ X2 ) ) @ ( suc @ ( zero_zero @ nat ) ) )
      = ( numeral_numeral @ nat @ ( bit1 @ X2 ) ) ) ).

% xor_nat_numerals(3)
thf(fact_6159_xor__nat__numerals_I2_J,axiom,
    ! [Y2: num] :
      ( ( bit_se5824344971392196577ns_xor @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( numeral_numeral @ nat @ ( bit1 @ Y2 ) ) )
      = ( numeral_numeral @ nat @ ( bit0 @ Y2 ) ) ) ).

% xor_nat_numerals(2)
thf(fact_6160_xor__nat__numerals_I1_J,axiom,
    ! [Y2: num] :
      ( ( bit_se5824344971392196577ns_xor @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( numeral_numeral @ nat @ ( bit0 @ Y2 ) ) )
      = ( numeral_numeral @ nat @ ( bit1 @ Y2 ) ) ) ).

% xor_nat_numerals(1)
thf(fact_6161_or__minus__numerals_I8_J,axiom,
    ! [N2: num,M: num] :
      ( ( bit_se1065995026697491101ons_or @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ N2 ) ) ) @ ( numeral_numeral @ int @ M ) )
      = ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit_or_not_num_neg @ M @ ( bit0 @ N2 ) ) ) ) ) ).

% or_minus_numerals(8)
thf(fact_6162_or__minus__numerals_I4_J,axiom,
    ! [M: num,N2: num] :
      ( ( bit_se1065995026697491101ons_or @ int @ ( numeral_numeral @ int @ M ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ N2 ) ) ) )
      = ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit_or_not_num_neg @ M @ ( bit0 @ N2 ) ) ) ) ) ).

% or_minus_numerals(4)
thf(fact_6163_word__bitwise__1__simps_I12_J,axiom,
    ! [L5: $tType] :
      ( ( type_len @ L5 )
     => ! [A2: num] :
          ( ( bit_se5824344971392196577ns_xor @ ( word @ L5 ) @ ( numeral_numeral @ ( word @ L5 ) @ A2 ) @ ( one_one @ ( word @ L5 ) ) )
          = ( ring_1_of_int @ ( word @ L5 ) @ ( bit_se5824344971392196577ns_xor @ int @ ( numeral_numeral @ int @ A2 ) @ ( one_one @ int ) ) ) ) ) ).

% word_bitwise_1_simps(12)
thf(fact_6164_word__bitwise__1__simps_I10_J,axiom,
    ! [J4: $tType] :
      ( ( type_len @ J4 )
     => ! [B2: num] :
          ( ( bit_se5824344971392196577ns_xor @ ( word @ J4 ) @ ( one_one @ ( word @ J4 ) ) @ ( numeral_numeral @ ( word @ J4 ) @ B2 ) )
          = ( ring_1_of_int @ ( word @ J4 ) @ ( bit_se5824344971392196577ns_xor @ int @ ( one_one @ int ) @ ( numeral_numeral @ int @ B2 ) ) ) ) ) ).

% word_bitwise_1_simps(10)
thf(fact_6165_word__no__log__defs_I12_J,axiom,
    ! [L5: $tType] :
      ( ( type_len @ L5 )
     => ! [A2: num,B2: num] :
          ( ( bit_se5824344971392196577ns_xor @ ( word @ L5 ) @ ( numeral_numeral @ ( word @ L5 ) @ A2 ) @ ( uminus_uminus @ ( word @ L5 ) @ ( numeral_numeral @ ( word @ L5 ) @ B2 ) ) )
          = ( ring_1_of_int @ ( word @ L5 ) @ ( bit_se5824344971392196577ns_xor @ int @ ( numeral_numeral @ int @ A2 ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ B2 ) ) ) ) ) ) ).

% word_no_log_defs(12)
thf(fact_6166_word__no__log__defs_I13_J,axiom,
    ! [M13: $tType] :
      ( ( type_len @ M13 )
     => ! [A2: num,B2: num] :
          ( ( bit_se5824344971392196577ns_xor @ ( word @ M13 ) @ ( uminus_uminus @ ( word @ M13 ) @ ( numeral_numeral @ ( word @ M13 ) @ A2 ) ) @ ( numeral_numeral @ ( word @ M13 ) @ B2 ) )
          = ( ring_1_of_int @ ( word @ M13 ) @ ( bit_se5824344971392196577ns_xor @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ A2 ) ) @ ( numeral_numeral @ int @ B2 ) ) ) ) ) ).

% word_no_log_defs(13)
thf(fact_6167_word__no__log__defs_I14_J,axiom,
    ! [N13: $tType] :
      ( ( type_len @ N13 )
     => ! [A2: num,B2: num] :
          ( ( bit_se5824344971392196577ns_xor @ ( word @ N13 ) @ ( uminus_uminus @ ( word @ N13 ) @ ( numeral_numeral @ ( word @ N13 ) @ A2 ) ) @ ( uminus_uminus @ ( word @ N13 ) @ ( numeral_numeral @ ( word @ N13 ) @ B2 ) ) )
          = ( ring_1_of_int @ ( word @ N13 ) @ ( bit_se5824344971392196577ns_xor @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ A2 ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ B2 ) ) ) ) ) ) ).

% word_no_log_defs(14)
thf(fact_6168_or__minus__numerals_I7_J,axiom,
    ! [N2: num,M: num] :
      ( ( bit_se1065995026697491101ons_or @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ N2 ) ) ) @ ( numeral_numeral @ int @ M ) )
      = ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit_or_not_num_neg @ M @ ( bitM @ N2 ) ) ) ) ) ).

% or_minus_numerals(7)
thf(fact_6169_or__minus__numerals_I3_J,axiom,
    ! [M: num,N2: num] :
      ( ( bit_se1065995026697491101ons_or @ int @ ( numeral_numeral @ int @ M ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ N2 ) ) ) )
      = ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit_or_not_num_neg @ M @ ( bitM @ N2 ) ) ) ) ) ).

% or_minus_numerals(3)
thf(fact_6170_word__bitwise__1__simps_I11_J,axiom,
    ! [K8: $tType] :
      ( ( type_len @ K8 )
     => ! [B2: num] :
          ( ( bit_se5824344971392196577ns_xor @ ( word @ K8 ) @ ( one_one @ ( word @ K8 ) ) @ ( uminus_uminus @ ( word @ K8 ) @ ( numeral_numeral @ ( word @ K8 ) @ B2 ) ) )
          = ( ring_1_of_int @ ( word @ K8 ) @ ( bit_se5824344971392196577ns_xor @ int @ ( one_one @ int ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ B2 ) ) ) ) ) ) ).

% word_bitwise_1_simps(11)
thf(fact_6171_word__bitwise__1__simps_I13_J,axiom,
    ! [M13: $tType] :
      ( ( type_len @ M13 )
     => ! [A2: num] :
          ( ( bit_se5824344971392196577ns_xor @ ( word @ M13 ) @ ( uminus_uminus @ ( word @ M13 ) @ ( numeral_numeral @ ( word @ M13 ) @ A2 ) ) @ ( one_one @ ( word @ M13 ) ) )
          = ( ring_1_of_int @ ( word @ M13 ) @ ( bit_se5824344971392196577ns_xor @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ A2 ) ) @ ( one_one @ int ) ) ) ) ) ).

% word_bitwise_1_simps(13)
thf(fact_6172_or__not__num__neg_Osimps_I1_J,axiom,
    ( ( bit_or_not_num_neg @ one2 @ one2 )
    = one2 ) ).

% or_not_num_neg.simps(1)
thf(fact_6173_or__not__num__neg_Osimps_I4_J,axiom,
    ! [N2: num] :
      ( ( bit_or_not_num_neg @ ( bit0 @ N2 ) @ one2 )
      = ( bit0 @ one2 ) ) ).

% or_not_num_neg.simps(4)
thf(fact_6174_or__not__num__neg_Osimps_I6_J,axiom,
    ! [N2: num,M: num] :
      ( ( bit_or_not_num_neg @ ( bit0 @ N2 ) @ ( bit1 @ M ) )
      = ( bit0 @ ( bit_or_not_num_neg @ N2 @ M ) ) ) ).

% or_not_num_neg.simps(6)
thf(fact_6175_or__not__num__neg_Osimps_I3_J,axiom,
    ! [M: num] :
      ( ( bit_or_not_num_neg @ one2 @ ( bit1 @ M ) )
      = ( bit1 @ M ) ) ).

% or_not_num_neg.simps(3)
thf(fact_6176_or__not__num__neg_Osimps_I7_J,axiom,
    ! [N2: num] :
      ( ( bit_or_not_num_neg @ ( bit1 @ N2 ) @ one2 )
      = one2 ) ).

% or_not_num_neg.simps(7)
thf(fact_6177_or__not__num__neg_Osimps_I5_J,axiom,
    ! [N2: num,M: num] :
      ( ( bit_or_not_num_neg @ ( bit0 @ N2 ) @ ( bit0 @ M ) )
      = ( bitM @ ( bit_or_not_num_neg @ N2 @ M ) ) ) ).

% or_not_num_neg.simps(5)
thf(fact_6178_or__not__num__neg_Osimps_I2_J,axiom,
    ! [M: num] :
      ( ( bit_or_not_num_neg @ one2 @ ( bit0 @ M ) )
      = ( bit1 @ M ) ) ).

% or_not_num_neg.simps(2)
thf(fact_6179_or__not__num__neg_Osimps_I8_J,axiom,
    ! [N2: num,M: num] :
      ( ( bit_or_not_num_neg @ ( bit1 @ N2 ) @ ( bit0 @ M ) )
      = ( bitM @ ( bit_or_not_num_neg @ N2 @ M ) ) ) ).

% or_not_num_neg.simps(8)
thf(fact_6180_or__not__num__neg_Oelims,axiom,
    ! [X2: num,Xa: num,Y2: num] :
      ( ( ( bit_or_not_num_neg @ X2 @ Xa )
        = Y2 )
     => ( ( ( X2 = one2 )
         => ( ( Xa = one2 )
           => ( Y2 != one2 ) ) )
       => ( ( ( X2 = one2 )
           => ! [M2: num] :
                ( ( Xa
                  = ( bit0 @ M2 ) )
               => ( Y2
                 != ( bit1 @ M2 ) ) ) )
         => ( ( ( X2 = one2 )
             => ! [M2: num] :
                  ( ( Xa
                    = ( bit1 @ M2 ) )
                 => ( Y2
                   != ( bit1 @ M2 ) ) ) )
           => ( ( ? [N4: num] :
                    ( X2
                    = ( bit0 @ N4 ) )
               => ( ( Xa = one2 )
                 => ( Y2
                   != ( bit0 @ one2 ) ) ) )
             => ( ! [N4: num] :
                    ( ( X2
                      = ( bit0 @ N4 ) )
                   => ! [M2: num] :
                        ( ( Xa
                          = ( bit0 @ M2 ) )
                       => ( Y2
                         != ( bitM @ ( bit_or_not_num_neg @ N4 @ M2 ) ) ) ) )
               => ( ! [N4: num] :
                      ( ( X2
                        = ( bit0 @ N4 ) )
                     => ! [M2: num] :
                          ( ( Xa
                            = ( bit1 @ M2 ) )
                         => ( Y2
                           != ( bit0 @ ( bit_or_not_num_neg @ N4 @ M2 ) ) ) ) )
                 => ( ( ? [N4: num] :
                          ( X2
                          = ( bit1 @ N4 ) )
                     => ( ( Xa = one2 )
                       => ( Y2 != one2 ) ) )
                   => ( ! [N4: num] :
                          ( ( X2
                            = ( bit1 @ N4 ) )
                         => ! [M2: num] :
                              ( ( Xa
                                = ( bit0 @ M2 ) )
                             => ( Y2
                               != ( bitM @ ( bit_or_not_num_neg @ N4 @ M2 ) ) ) ) )
                     => ~ ! [N4: num] :
                            ( ( X2
                              = ( bit1 @ N4 ) )
                           => ! [M2: num] :
                                ( ( Xa
                                  = ( bit1 @ M2 ) )
                               => ( Y2
                                 != ( bitM @ ( bit_or_not_num_neg @ N4 @ M2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% or_not_num_neg.elims
thf(fact_6181_int__numeral__or__not__num__neg,axiom,
    ! [M: num,N2: num] :
      ( ( bit_se1065995026697491101ons_or @ int @ ( numeral_numeral @ int @ M ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ N2 ) ) )
      = ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit_or_not_num_neg @ M @ N2 ) ) ) ) ).

% int_numeral_or_not_num_neg
thf(fact_6182_int__numeral__not__or__num__neg,axiom,
    ! [M: num,N2: num] :
      ( ( bit_se1065995026697491101ons_or @ int @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ M ) ) @ ( numeral_numeral @ int @ N2 ) )
      = ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit_or_not_num_neg @ N2 @ M ) ) ) ) ).

% int_numeral_not_or_num_neg
thf(fact_6183_numeral__or__not__num__eq,axiom,
    ! [M: num,N2: num] :
      ( ( numeral_numeral @ int @ ( bit_or_not_num_neg @ M @ N2 ) )
      = ( uminus_uminus @ int @ ( bit_se1065995026697491101ons_or @ int @ ( numeral_numeral @ int @ M ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ N2 ) ) ) ) ) ).

% numeral_or_not_num_eq
thf(fact_6184_word__ops__nth__size,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [N2: nat,X2: word @ A,Y2: word @ A] :
          ( ( ord_less @ nat @ N2 @ ( size_size @ ( word @ A ) @ X2 ) )
         => ( ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( bit_se1065995026697491101ons_or @ ( word @ A ) @ X2 @ Y2 ) @ N2 )
              = ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ X2 @ N2 )
                | ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ Y2 @ N2 ) ) )
            & ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X2 @ Y2 ) @ N2 )
              = ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ X2 @ N2 )
                & ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ Y2 @ N2 ) ) )
            & ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( bit_se5824344971392196577ns_xor @ ( word @ A ) @ X2 @ Y2 ) @ N2 )
              = ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ X2 @ N2 )
               != ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ Y2 @ N2 ) ) )
            & ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ X2 ) @ N2 )
              = ( ~ ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ X2 @ N2 ) ) ) ) ) ) ).

% word_ops_nth_size
thf(fact_6185_bit__twiddle__max,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [X2: word @ A,Y2: word @ A] :
          ( ( bit_se5824344971392196577ns_xor @ ( word @ A ) @ X2 @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ ( bit_se5824344971392196577ns_xor @ ( word @ A ) @ X2 @ Y2 ) @ ( if @ ( word @ A ) @ ( ord_less @ ( word @ A ) @ X2 @ Y2 ) @ ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) @ ( zero_zero @ ( word @ A ) ) ) ) )
          = ( ord_max @ ( word @ A ) @ X2 @ Y2 ) ) ) ).

% bit_twiddle_max
thf(fact_6186_xor__nat__unfold,axiom,
    ( ( bit_se5824344971392196577ns_xor @ nat )
    = ( ^ [M7: nat,N: nat] :
          ( if @ nat
          @ ( M7
            = ( zero_zero @ nat ) )
          @ N
          @ ( if @ nat
            @ ( N
              = ( zero_zero @ nat ) )
            @ M7
            @ ( plus_plus @ nat @ ( modulo_modulo @ nat @ ( plus_plus @ nat @ ( modulo_modulo @ nat @ M7 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( modulo_modulo @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( bit_se5824344971392196577ns_xor @ nat @ ( divide_divide @ nat @ M7 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ).

% xor_nat_unfold
thf(fact_6187_xor__nat__rec,axiom,
    ( ( bit_se5824344971392196577ns_xor @ nat )
    = ( ^ [M7: nat,N: nat] :
          ( plus_plus @ nat
          @ ( zero_neq_one_of_bool @ nat
            @ ( ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M7 ) )
             != ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) )
          @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( bit_se5824344971392196577ns_xor @ nat @ ( divide_divide @ nat @ M7 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).

% xor_nat_rec
thf(fact_6188_Suc__0__xor__eq,axiom,
    ! [N2: nat] :
      ( ( bit_se5824344971392196577ns_xor @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N2 )
      = ( minus_minus @ nat @ ( plus_plus @ nat @ N2 @ ( zero_neq_one_of_bool @ nat @ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) )
        @ ( zero_neq_one_of_bool @ nat
          @ ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ) ) ).

% Suc_0_xor_eq
thf(fact_6189_xor__Suc__0__eq,axiom,
    ! [N2: nat] :
      ( ( bit_se5824344971392196577ns_xor @ nat @ N2 @ ( suc @ ( zero_zero @ nat ) ) )
      = ( minus_minus @ nat @ ( plus_plus @ nat @ N2 @ ( zero_neq_one_of_bool @ nat @ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) )
        @ ( zero_neq_one_of_bool @ nat
          @ ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ) ) ).

% xor_Suc_0_eq
thf(fact_6190_or__not__num__neg_Opelims,axiom,
    ! [X2: num,Xa: num,Y2: num] :
      ( ( ( bit_or_not_num_neg @ X2 @ Xa )
        = Y2 )
     => ( ( accp @ ( product_prod @ num @ num ) @ bit_or3848514188828904588eg_rel @ ( product_Pair @ num @ num @ X2 @ Xa ) )
       => ( ( ( X2 = one2 )
           => ( ( Xa = one2 )
             => ( ( Y2 = one2 )
               => ~ ( accp @ ( product_prod @ num @ num ) @ bit_or3848514188828904588eg_rel @ ( product_Pair @ num @ num @ one2 @ one2 ) ) ) ) )
         => ( ( ( X2 = one2 )
             => ! [M2: num] :
                  ( ( Xa
                    = ( bit0 @ M2 ) )
                 => ( ( Y2
                      = ( bit1 @ M2 ) )
                   => ~ ( accp @ ( product_prod @ num @ num ) @ bit_or3848514188828904588eg_rel @ ( product_Pair @ num @ num @ one2 @ ( bit0 @ M2 ) ) ) ) ) )
           => ( ( ( X2 = one2 )
               => ! [M2: num] :
                    ( ( Xa
                      = ( bit1 @ M2 ) )
                   => ( ( Y2
                        = ( bit1 @ M2 ) )
                     => ~ ( accp @ ( product_prod @ num @ num ) @ bit_or3848514188828904588eg_rel @ ( product_Pair @ num @ num @ one2 @ ( bit1 @ M2 ) ) ) ) ) )
             => ( ! [N4: num] :
                    ( ( X2
                      = ( bit0 @ N4 ) )
                   => ( ( Xa = one2 )
                     => ( ( Y2
                          = ( bit0 @ one2 ) )
                       => ~ ( accp @ ( product_prod @ num @ num ) @ bit_or3848514188828904588eg_rel @ ( product_Pair @ num @ num @ ( bit0 @ N4 ) @ one2 ) ) ) ) )
               => ( ! [N4: num] :
                      ( ( X2
                        = ( bit0 @ N4 ) )
                     => ! [M2: num] :
                          ( ( Xa
                            = ( bit0 @ M2 ) )
                         => ( ( Y2
                              = ( bitM @ ( bit_or_not_num_neg @ N4 @ M2 ) ) )
                           => ~ ( accp @ ( product_prod @ num @ num ) @ bit_or3848514188828904588eg_rel @ ( product_Pair @ num @ num @ ( bit0 @ N4 ) @ ( bit0 @ M2 ) ) ) ) ) )
                 => ( ! [N4: num] :
                        ( ( X2
                          = ( bit0 @ N4 ) )
                       => ! [M2: num] :
                            ( ( Xa
                              = ( bit1 @ M2 ) )
                           => ( ( Y2
                                = ( bit0 @ ( bit_or_not_num_neg @ N4 @ M2 ) ) )
                             => ~ ( accp @ ( product_prod @ num @ num ) @ bit_or3848514188828904588eg_rel @ ( product_Pair @ num @ num @ ( bit0 @ N4 ) @ ( bit1 @ M2 ) ) ) ) ) )
                   => ( ! [N4: num] :
                          ( ( X2
                            = ( bit1 @ N4 ) )
                         => ( ( Xa = one2 )
                           => ( ( Y2 = one2 )
                             => ~ ( accp @ ( product_prod @ num @ num ) @ bit_or3848514188828904588eg_rel @ ( product_Pair @ num @ num @ ( bit1 @ N4 ) @ one2 ) ) ) ) )
                     => ( ! [N4: num] :
                            ( ( X2
                              = ( bit1 @ N4 ) )
                           => ! [M2: num] :
                                ( ( Xa
                                  = ( bit0 @ M2 ) )
                               => ( ( Y2
                                    = ( bitM @ ( bit_or_not_num_neg @ N4 @ M2 ) ) )
                                 => ~ ( accp @ ( product_prod @ num @ num ) @ bit_or3848514188828904588eg_rel @ ( product_Pair @ num @ num @ ( bit1 @ N4 ) @ ( bit0 @ M2 ) ) ) ) ) )
                       => ~ ! [N4: num] :
                              ( ( X2
                                = ( bit1 @ N4 ) )
                             => ! [M2: num] :
                                  ( ( Xa
                                    = ( bit1 @ M2 ) )
                                 => ( ( Y2
                                      = ( bitM @ ( bit_or_not_num_neg @ N4 @ M2 ) ) )
                                   => ~ ( accp @ ( product_prod @ num @ num ) @ bit_or3848514188828904588eg_rel @ ( product_Pair @ num @ num @ ( bit1 @ N4 ) @ ( bit1 @ M2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% or_not_num_neg.pelims
thf(fact_6191_bit_Oabstract__boolean__algebra__sym__diff__axioms,axiom,
    ! [A: $tType] :
      ( ( bit_ri3973907225187159222ations @ A )
     => ( boolea3799213064322606851m_diff @ A @ ( bit_se5824344872417868541ns_and @ A ) @ ( bit_se1065995026697491101ons_or @ A ) @ ( bit_ri4277139882892585799ns_not @ A ) @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( bit_se5824344971392196577ns_xor @ A ) ) ) ).

% bit.abstract_boolean_algebra_sym_diff_axioms
thf(fact_6192_Heap_Osize__gen,axiom,
    ! [A: $tType,Xa: A > nat,X2: ( heap_ext @ product_unit ) > ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) )] :
      ( ( heap_Time_size_Heap @ A @ Xa @ ( heap_Time_Heap2 @ A @ X2 ) )
      = ( suc @ ( zero_zero @ nat ) ) ) ).

% Heap.size_gen
thf(fact_6193_num_Osize__gen_I3_J,axiom,
    ! [X33: num] :
      ( ( size_num @ ( bit1 @ X33 ) )
      = ( plus_plus @ nat @ ( size_num @ X33 ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ).

% num.size_gen(3)
thf(fact_6194_num_Osize__gen_I1_J,axiom,
    ( ( size_num @ one2 )
    = ( zero_zero @ nat ) ) ).

% num.size_gen(1)
thf(fact_6195_num_Osize__gen_I2_J,axiom,
    ! [X23: num] :
      ( ( size_num @ ( bit0 @ X23 ) )
      = ( plus_plus @ nat @ ( size_num @ X23 ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ).

% num.size_gen(2)
thf(fact_6196_new__rule,axiom,
    ! [A: $tType] :
      ( ( heap @ A )
     => ! [N2: nat,X2: A] :
          ( hoare_hoare_triple @ ( array @ A ) @ ( one_one @ assn ) @ ( array_new @ A @ N2 @ X2 )
          @ ^ [R: array @ A] : ( snga_assn @ A @ R @ ( replicate @ A @ N2 @ X2 ) ) ) ) ).

% new_rule
thf(fact_6197_aux,axiom,
    ! [B: $tType,A: $tType,P: A > B > assn,A2: A,As3: list @ A,C2: B,Cs: list @ B] :
      ( ( finite_fold @ nat @ assn
        @ ^ [I2: nat,Aa2: assn] : ( times_times @ assn @ Aa2 @ ( P @ ( nth @ A @ ( cons @ A @ A2 @ As3 ) @ I2 ) @ ( nth @ B @ ( cons @ B @ C2 @ Cs ) @ I2 ) ) )
        @ ( one_one @ assn )
        @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( suc @ ( size_size @ ( list @ A ) @ As3 ) ) ) )
      = ( times_times @ assn @ ( P @ A2 @ C2 )
        @ ( finite_fold @ nat @ assn
          @ ^ [I2: nat,Aa2: assn] : ( times_times @ assn @ Aa2 @ ( P @ ( nth @ A @ As3 @ I2 ) @ ( nth @ B @ Cs @ I2 ) ) )
          @ ( one_one @ assn )
          @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ A ) @ As3 ) ) ) ) ) ).

% aux
thf(fact_6198_length__nth__simps_I4_J,axiom,
    ! [B: $tType,X2: B,Xs2: list @ B,N2: nat] :
      ( ( nth @ B @ ( cons @ B @ X2 @ Xs2 ) @ ( suc @ N2 ) )
      = ( nth @ B @ Xs2 @ N2 ) ) ).

% length_nth_simps(4)
thf(fact_6199_nth__Cons__Suc,axiom,
    ! [A: $tType,X2: A,Xs2: list @ A,N2: nat] :
      ( ( nth @ A @ ( cons @ A @ X2 @ Xs2 ) @ ( suc @ N2 ) )
      = ( nth @ A @ Xs2 @ N2 ) ) ).

% nth_Cons_Suc
thf(fact_6200_length__nth__simps_I3_J,axiom,
    ! [B: $tType,X2: B,Xs2: list @ B] :
      ( ( nth @ B @ ( cons @ B @ X2 @ Xs2 ) @ ( zero_zero @ nat ) )
      = X2 ) ).

% length_nth_simps(3)
thf(fact_6201_nth__Cons__0,axiom,
    ! [A: $tType,X2: A,Xs2: list @ A] :
      ( ( nth @ A @ ( cons @ A @ X2 @ Xs2 ) @ ( zero_zero @ nat ) )
      = X2 ) ).

% nth_Cons_0
thf(fact_6202_list_Osimps_I15_J,axiom,
    ! [A: $tType,X212: A,X222: list @ A] :
      ( ( set2 @ A @ ( cons @ A @ X212 @ X222 ) )
      = ( insert @ A @ X212 @ ( set2 @ A @ X222 ) ) ) ).

% list.simps(15)
thf(fact_6203_enumerate__simps_I2_J,axiom,
    ! [B: $tType,N2: nat,X2: B,Xs2: list @ B] :
      ( ( enumerate @ B @ N2 @ ( cons @ B @ X2 @ Xs2 ) )
      = ( cons @ ( product_prod @ nat @ B ) @ ( product_Pair @ nat @ B @ N2 @ X2 ) @ ( enumerate @ B @ ( suc @ N2 ) @ Xs2 ) ) ) ).

% enumerate_simps(2)
thf(fact_6204_nth__Cons__numeral,axiom,
    ! [A: $tType,X2: A,Xs2: list @ A,V: num] :
      ( ( nth @ A @ ( cons @ A @ X2 @ Xs2 ) @ ( numeral_numeral @ nat @ V ) )
      = ( nth @ A @ Xs2 @ ( minus_minus @ nat @ ( numeral_numeral @ nat @ V ) @ ( one_one @ nat ) ) ) ) ).

% nth_Cons_numeral
thf(fact_6205_nth__Cons__pos,axiom,
    ! [A: $tType,N2: nat,X2: A,Xs2: list @ A] :
      ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
     => ( ( nth @ A @ ( cons @ A @ X2 @ Xs2 ) @ N2 )
        = ( nth @ A @ Xs2 @ ( minus_minus @ nat @ N2 @ ( one_one @ nat ) ) ) ) ) ).

% nth_Cons_pos
thf(fact_6206_set__subset__Cons,axiom,
    ! [A: $tType,Xs2: list @ A,X2: A] : ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs2 ) @ ( set2 @ A @ ( cons @ A @ X2 @ Xs2 ) ) ) ).

% set_subset_Cons
thf(fact_6207_impossible__Cons,axiom,
    ! [A: $tType,Xs2: list @ A,Ys: list @ A,X2: A] :
      ( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs2 ) @ ( size_size @ ( list @ A ) @ Ys ) )
     => ( Xs2
       != ( cons @ A @ X2 @ Ys ) ) ) ).

% impossible_Cons
thf(fact_6208_list__update__code_I3_J,axiom,
    ! [A: $tType,X2: A,Xs2: list @ A,I: nat,Y2: A] :
      ( ( list_update @ A @ ( cons @ A @ X2 @ Xs2 ) @ ( suc @ I ) @ Y2 )
      = ( cons @ A @ X2 @ ( list_update @ A @ Xs2 @ I @ Y2 ) ) ) ).

% list_update_code(3)
thf(fact_6209_set__ConsD,axiom,
    ! [A: $tType,Y2: A,X2: A,Xs2: list @ A] :
      ( ( member @ A @ Y2 @ ( set2 @ A @ ( cons @ A @ X2 @ Xs2 ) ) )
     => ( ( Y2 = X2 )
        | ( member @ A @ Y2 @ ( set2 @ A @ Xs2 ) ) ) ) ).

% set_ConsD
thf(fact_6210_list_Oset__cases,axiom,
    ! [A: $tType,E3: A,A2: list @ A] :
      ( ( member @ A @ E3 @ ( set2 @ A @ A2 ) )
     => ( ! [Z23: list @ A] :
            ( A2
           != ( cons @ A @ E3 @ Z23 ) )
       => ~ ! [Z12: A,Z23: list @ A] :
              ( ( A2
                = ( cons @ A @ Z12 @ Z23 ) )
             => ~ ( member @ A @ E3 @ ( set2 @ A @ Z23 ) ) ) ) ) ).

% list.set_cases
thf(fact_6211_list_Oset__intros_I1_J,axiom,
    ! [A: $tType,X212: A,X222: list @ A] : ( member @ A @ X212 @ ( set2 @ A @ ( cons @ A @ X212 @ X222 ) ) ) ).

% list.set_intros(1)
thf(fact_6212_list_Oset__intros_I2_J,axiom,
    ! [A: $tType,Y2: A,X222: list @ A,X212: A] :
      ( ( member @ A @ Y2 @ ( set2 @ A @ X222 ) )
     => ( member @ A @ Y2 @ ( set2 @ A @ ( cons @ A @ X212 @ X222 ) ) ) ) ).

% list.set_intros(2)
thf(fact_6213_list__tail__coinc,axiom,
    ! [A: $tType,N14: A,R12: list @ A,N23: A,R23: list @ A] :
      ( ( ( cons @ A @ N14 @ R12 )
        = ( cons @ A @ N23 @ R23 ) )
     => ( ( N14 = N23 )
        & ( R12 = R23 ) ) ) ).

% list_tail_coinc
thf(fact_6214_length__Cons,axiom,
    ! [A: $tType,X2: A,Xs2: list @ A] :
      ( ( size_size @ ( list @ A ) @ ( cons @ A @ X2 @ Xs2 ) )
      = ( suc @ ( size_size @ ( list @ A ) @ Xs2 ) ) ) ).

% length_Cons
thf(fact_6215_Suc__length__conv,axiom,
    ! [A: $tType,N2: nat,Xs2: list @ A] :
      ( ( ( suc @ N2 )
        = ( size_size @ ( list @ A ) @ Xs2 ) )
      = ( ? [Y: A,Ys3: list @ A] :
            ( ( Xs2
              = ( cons @ A @ Y @ Ys3 ) )
            & ( ( size_size @ ( list @ A ) @ Ys3 )
              = N2 ) ) ) ) ).

% Suc_length_conv
thf(fact_6216_length__Suc__conv,axiom,
    ! [A: $tType,Xs2: list @ A,N2: nat] :
      ( ( ( size_size @ ( list @ A ) @ Xs2 )
        = ( suc @ N2 ) )
      = ( ? [Y: A,Ys3: list @ A] :
            ( ( Xs2
              = ( cons @ A @ Y @ Ys3 ) )
            & ( ( size_size @ ( list @ A ) @ Ys3 )
              = N2 ) ) ) ) ).

% length_Suc_conv
thf(fact_6217_length__nth__simps_I2_J,axiom,
    ! [B: $tType,X2: B,Xs2: list @ B] :
      ( ( size_size @ ( list @ B ) @ ( cons @ B @ X2 @ Xs2 ) )
      = ( suc @ ( size_size @ ( list @ B ) @ Xs2 ) ) ) ).

% length_nth_simps(2)
thf(fact_6218_list__update_Osimps_I2_J,axiom,
    ! [A: $tType,X2: A,Xs2: list @ A,I: nat,V: A] :
      ( ( list_update @ A @ ( cons @ A @ X2 @ Xs2 ) @ I @ V )
      = ( case_nat @ ( list @ A ) @ ( cons @ A @ V @ Xs2 )
        @ ^ [J3: nat] : ( cons @ A @ X2 @ ( list_update @ A @ Xs2 @ J3 @ V ) )
        @ I ) ) ).

% list_update.simps(2)
thf(fact_6219_list__assn_Osimps_I2_J,axiom,
    ! [A: $tType,C: $tType,P: A > C > assn,A2: A,As3: list @ A,C2: C,Cs: list @ C] :
      ( ( vEBT_List_list_assn @ A @ C @ P @ ( cons @ A @ A2 @ As3 ) @ ( cons @ C @ C2 @ Cs ) )
      = ( times_times @ assn @ ( P @ A2 @ C2 ) @ ( vEBT_List_list_assn @ A @ C @ P @ As3 @ Cs ) ) ) ).

% list_assn.simps(2)
thf(fact_6220_list__assn__simps_I2_J,axiom,
    ! [A: $tType,B: $tType,P: A > B > assn,A2: A,As3: list @ A,C2: B,Cs: list @ B] :
      ( ( vEBT_List_list_assn @ A @ B @ P @ ( cons @ A @ A2 @ As3 ) @ ( cons @ B @ C2 @ Cs ) )
      = ( times_times @ assn @ ( P @ A2 @ C2 ) @ ( vEBT_List_list_assn @ A @ B @ P @ As3 @ Cs ) ) ) ).

% list_assn_simps(2)
thf(fact_6221_map__eq__consE,axiom,
    ! [B: $tType,A: $tType,F: B > A,Ls: list @ B,Fa: A,Fl: list @ A] :
      ( ( ( map @ B @ A @ F @ Ls )
        = ( cons @ A @ Fa @ Fl ) )
     => ~ ! [A4: B,L3: list @ B] :
            ( ( Ls
              = ( cons @ B @ A4 @ L3 ) )
           => ( ( ( F @ A4 )
                = Fa )
             => ( ( map @ B @ A @ F @ L3 )
               != Fl ) ) ) ) ).

% map_eq_consE
thf(fact_6222_map__consI_I1_J,axiom,
    ! [A: $tType,B: $tType,W: list @ A,F: B > A,Ww: list @ B,A2: B] :
      ( ( W
        = ( map @ B @ A @ F @ Ww ) )
     => ( ( cons @ A @ ( F @ A2 ) @ W )
        = ( map @ B @ A @ F @ ( cons @ B @ A2 @ Ww ) ) ) ) ).

% map_consI(1)
thf(fact_6223_replicate__Suc,axiom,
    ! [A: $tType,N2: nat,X2: A] :
      ( ( replicate @ A @ ( suc @ N2 ) @ X2 )
      = ( cons @ A @ X2 @ ( replicate @ A @ N2 @ X2 ) ) ) ).

% replicate_Suc
thf(fact_6224_VEBT__internal_Oreplicatei_Osimps_I2_J,axiom,
    ! [A: $tType,N2: nat,X2: heap_Time_Heap @ A] :
      ( ( vEBT_VEBT_replicatei @ A @ ( suc @ N2 ) @ X2 )
      = ( heap_Time_bind @ A @ ( list @ A ) @ X2
        @ ^ [Y: A] :
            ( heap_Time_bind @ ( list @ A ) @ ( list @ A ) @ ( vEBT_VEBT_replicatei @ A @ N2 @ X2 )
            @ ^ [Ys3: list @ A] : ( heap_Time_return @ ( list @ A ) @ ( cons @ A @ Y @ Ys3 ) ) ) ) ) ).

% VEBT_internal.replicatei.simps(2)
thf(fact_6225_nth__Cons,axiom,
    ! [A: $tType,X2: A,Xs2: list @ A,N2: nat] :
      ( ( nth @ A @ ( cons @ A @ X2 @ Xs2 ) @ N2 )
      = ( case_nat @ A @ X2 @ ( nth @ A @ Xs2 ) @ N2 ) ) ).

% nth_Cons
thf(fact_6226_Suc__le__length__iff,axiom,
    ! [A: $tType,N2: nat,Xs2: list @ A] :
      ( ( ord_less_eq @ nat @ ( suc @ N2 ) @ ( size_size @ ( list @ A ) @ Xs2 ) )
      = ( ? [X: A,Ys3: list @ A] :
            ( ( Xs2
              = ( cons @ A @ X @ Ys3 ) )
            & ( ord_less_eq @ nat @ N2 @ ( size_size @ ( list @ A ) @ Ys3 ) ) ) ) ) ).

% Suc_le_length_iff
thf(fact_6227_list_Osize_I4_J,axiom,
    ! [A: $tType,X212: A,X222: list @ A] :
      ( ( size_size @ ( list @ A ) @ ( cons @ A @ X212 @ X222 ) )
      = ( plus_plus @ nat @ ( size_size @ ( list @ A ) @ X222 ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ).

% list.size(4)
thf(fact_6228_nth__Cons_H,axiom,
    ! [A: $tType,N2: nat,X2: A,Xs2: list @ A] :
      ( ( ( N2
          = ( zero_zero @ nat ) )
       => ( ( nth @ A @ ( cons @ A @ X2 @ Xs2 ) @ N2 )
          = X2 ) )
      & ( ( N2
         != ( zero_zero @ nat ) )
       => ( ( nth @ A @ ( cons @ A @ X2 @ Xs2 ) @ N2 )
          = ( nth @ A @ Xs2 @ ( minus_minus @ nat @ N2 @ ( one_one @ nat ) ) ) ) ) ) ).

% nth_Cons'
thf(fact_6229_nth__non__equal__first__eq,axiom,
    ! [A: $tType,X2: A,Y2: A,Xs2: list @ A,N2: nat] :
      ( ( X2 != Y2 )
     => ( ( ( nth @ A @ ( cons @ A @ X2 @ Xs2 ) @ N2 )
          = Y2 )
        = ( ( ( nth @ A @ Xs2 @ ( minus_minus @ nat @ N2 @ ( one_one @ nat ) ) )
            = Y2 )
          & ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ) ).

% nth_non_equal_first_eq
thf(fact_6230_nth__equal__first__eq,axiom,
    ! [A: $tType,X2: A,Xs2: list @ A,N2: nat] :
      ( ~ ( member @ A @ X2 @ ( set2 @ A @ Xs2 ) )
     => ( ( ord_less_eq @ nat @ N2 @ ( size_size @ ( list @ A ) @ Xs2 ) )
       => ( ( ( nth @ A @ ( cons @ A @ X2 @ Xs2 ) @ N2 )
            = X2 )
          = ( N2
            = ( zero_zero @ nat ) ) ) ) ) ).

% nth_equal_first_eq
thf(fact_6231_Cons__replicate__eq,axiom,
    ! [A: $tType,X2: A,Xs2: list @ A,N2: nat,Y2: A] :
      ( ( ( cons @ A @ X2 @ Xs2 )
        = ( replicate @ A @ N2 @ Y2 ) )
      = ( ( X2 = Y2 )
        & ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
        & ( Xs2
          = ( replicate @ A @ ( minus_minus @ nat @ N2 @ ( one_one @ nat ) ) @ X2 ) ) ) ) ).

% Cons_replicate_eq
thf(fact_6232_slice__Cons,axiom,
    ! [A: $tType,Begin: nat,End: nat,X2: A,Xs2: list @ A] :
      ( ( ( ( Begin
            = ( zero_zero @ nat ) )
          & ( ord_less @ nat @ ( zero_zero @ nat ) @ End ) )
       => ( ( slice @ A @ Begin @ End @ ( cons @ A @ X2 @ Xs2 ) )
          = ( cons @ A @ X2 @ ( slice @ A @ Begin @ ( minus_minus @ nat @ End @ ( one_one @ nat ) ) @ Xs2 ) ) ) )
      & ( ~ ( ( Begin
              = ( zero_zero @ nat ) )
            & ( ord_less @ nat @ ( zero_zero @ nat ) @ End ) )
       => ( ( slice @ A @ Begin @ End @ ( cons @ A @ X2 @ Xs2 ) )
          = ( slice @ A @ ( minus_minus @ nat @ Begin @ ( one_one @ nat ) ) @ ( minus_minus @ nat @ End @ ( one_one @ nat ) ) @ Xs2 ) ) ) ) ).

% slice_Cons
thf(fact_6233_TBOUND__new,axiom,
    ! [A: $tType] :
      ( ( heap @ A )
     => ! [N2: nat,X2: A] : ( time_TBOUND @ ( array @ A ) @ ( array_new @ A @ N2 @ X2 ) @ ( plus_plus @ nat @ N2 @ ( one_one @ nat ) ) ) ) ).

% TBOUND_new
thf(fact_6234_time__array__new,axiom,
    ! [A: $tType] :
      ( ( heap @ A )
     => ! [N2: nat,X2: A,H2: heap_ext @ product_unit] :
          ( ( time_time @ ( array @ A ) @ ( array_new @ A @ N2 @ X2 ) @ H2 )
          = ( plus_plus @ nat @ N2 @ ( one_one @ nat ) ) ) ) ).

% time_array_new
thf(fact_6235_hash__code__prod__simps,axiom,
    ! [A: $tType,B: $tType,H_a2: A > uint32,H_b: B > uint32,X2: A,Xa: B] :
      ( ( hash_hash_code_prod @ A @ B @ H_a2 @ H_b @ ( product_Pair @ A @ B @ X2 @ Xa ) )
      = ( plus_plus @ uint32 @ ( times_times @ uint32 @ ( H_a2 @ X2 ) @ ( numeral_numeral @ uint32 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( plus_plus @ uint32 @ ( times_times @ uint32 @ ( H_b @ Xa ) @ ( numeral_numeral @ uint32 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( numeral_numeral @ uint32 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% hash_code_prod_simps
thf(fact_6236_sum__count__set,axiom,
    ! [A: $tType,Xs2: list @ A,X8: set @ A] :
      ( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs2 ) @ X8 )
     => ( ( finite_finite2 @ A @ X8 )
       => ( ( groups7311177749621191930dd_sum @ A @ nat @ ( count_list @ A @ Xs2 ) @ X8 )
          = ( size_size @ ( list @ A ) @ Xs2 ) ) ) ) ).

% sum_count_set
thf(fact_6237_count__notin,axiom,
    ! [A: $tType,X2: A,Xs2: list @ A] :
      ( ~ ( member @ A @ X2 @ ( set2 @ A @ Xs2 ) )
     => ( ( count_list @ A @ Xs2 @ X2 )
        = ( zero_zero @ nat ) ) ) ).

% count_notin
thf(fact_6238_count__le__length,axiom,
    ! [A: $tType,Xs2: list @ A,X2: A] : ( ord_less_eq @ nat @ ( count_list @ A @ Xs2 @ X2 ) @ ( size_size @ ( list @ A ) @ Xs2 ) ) ).

% count_le_length
thf(fact_6239_count__list_Osimps_I2_J,axiom,
    ! [A: $tType,X2: A,Y2: A,Xs2: list @ A] :
      ( ( ( X2 = Y2 )
       => ( ( count_list @ A @ ( cons @ A @ X2 @ Xs2 ) @ Y2 )
          = ( plus_plus @ nat @ ( count_list @ A @ Xs2 @ Y2 ) @ ( one_one @ nat ) ) ) )
      & ( ( X2 != Y2 )
       => ( ( count_list @ A @ ( cons @ A @ X2 @ Xs2 ) @ Y2 )
          = ( count_list @ A @ Xs2 @ Y2 ) ) ) ) ).

% count_list.simps(2)
thf(fact_6240_hash__code__option__simps_I2_J,axiom,
    ! [A: $tType,H_a2: A > uint32,X2: A] :
      ( ( hash_h1887023736457453652option @ A @ H_a2 @ ( some @ A @ X2 ) )
      = ( plus_plus @ uint32 @ ( times_times @ uint32 @ ( H_a2 @ X2 ) @ ( numeral_numeral @ uint32 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( numeral_numeral @ uint32 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% hash_code_option_simps(2)
thf(fact_6241_hash__code__list__simps_I2_J,axiom,
    ! [A: $tType,H_a2: A > uint32,X2: A,Xa: list @ A] :
      ( ( hash_hash_code_list @ A @ H_a2 @ ( cons @ A @ X2 @ Xa ) )
      = ( plus_plus @ uint32 @ ( times_times @ uint32 @ ( H_a2 @ X2 ) @ ( numeral_numeral @ uint32 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( plus_plus @ uint32 @ ( times_times @ uint32 @ ( hash_hash_code_list @ A @ H_a2 @ Xa ) @ ( numeral_numeral @ uint32 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( numeral_numeral @ uint32 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% hash_code_list_simps(2)
thf(fact_6242_hash__code__option__simps_I1_J,axiom,
    ! [A: $tType,H_a2: A > uint32] :
      ( ( hash_h1887023736457453652option @ A @ H_a2 @ ( none @ A ) )
      = ( numeral_numeral @ uint32 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% hash_code_option_simps(1)
thf(fact_6243_VEBT__internal_Oreplicatei_Oelims,axiom,
    ! [A: $tType,X2: nat,Xa: heap_Time_Heap @ A,Y2: heap_Time_Heap @ ( list @ A )] :
      ( ( ( vEBT_VEBT_replicatei @ A @ X2 @ Xa )
        = Y2 )
     => ( ( ( X2
            = ( zero_zero @ nat ) )
         => ( Y2
           != ( heap_Time_return @ ( list @ A ) @ ( nil @ A ) ) ) )
       => ~ ! [N4: nat] :
              ( ( X2
                = ( suc @ N4 ) )
             => ( Y2
               != ( heap_Time_bind @ A @ ( list @ A ) @ Xa
                  @ ^ [Y: A] :
                      ( heap_Time_bind @ ( list @ A ) @ ( list @ A ) @ ( vEBT_VEBT_replicatei @ A @ N4 @ Xa )
                      @ ^ [Ys3: list @ A] : ( heap_Time_return @ ( list @ A ) @ ( cons @ A @ Y @ Ys3 ) ) ) ) ) ) ) ) ).

% VEBT_internal.replicatei.elims
thf(fact_6244_call__mono,axiom,
    ! [B: $tType,A: $tType,Ord: B > B > $o,T2: A] :
      ( comple7038119648293358887notone @ ( A > B ) @ B @ ( partial_fun_ord @ B @ B @ A @ Ord ) @ Ord
      @ ^ [F6: A > B] : ( F6 @ T2 ) ) ).

% call_mono
thf(fact_6245_slice__eq__bounds__empty,axiom,
    ! [A: $tType,I: nat,Xs2: list @ A] :
      ( ( slice @ A @ I @ I @ Xs2 )
      = ( nil @ A ) ) ).

% slice_eq_bounds_empty
thf(fact_6246_slice__Nil,axiom,
    ! [A: $tType,Begin: nat,End: nat] :
      ( ( slice @ A @ Begin @ End @ ( nil @ A ) )
      = ( nil @ A ) ) ).

% slice_Nil
thf(fact_6247_length__0__conv,axiom,
    ! [A: $tType,Xs2: list @ A] :
      ( ( ( size_size @ ( list @ A ) @ Xs2 )
        = ( zero_zero @ nat ) )
      = ( Xs2
        = ( nil @ A ) ) ) ).

% length_0_conv
thf(fact_6248_set__empty2,axiom,
    ! [A: $tType,Xs2: list @ A] :
      ( ( ( bot_bot @ ( set @ A ) )
        = ( set2 @ A @ Xs2 ) )
      = ( Xs2
        = ( nil @ A ) ) ) ).

% set_empty2
thf(fact_6249_set__empty,axiom,
    ! [A: $tType,Xs2: list @ A] :
      ( ( ( set2 @ A @ Xs2 )
        = ( bot_bot @ ( set @ A ) ) )
      = ( Xs2
        = ( nil @ A ) ) ) ).

% set_empty
thf(fact_6250_list__assn__aux__simps_I2_J,axiom,
    ! [B: $tType,A: $tType,P: A > B > assn,L: list @ A] :
      ( ( vEBT_List_list_assn @ A @ B @ P @ L @ ( nil @ B ) )
      = ( pure_assn
        @ ( L
          = ( nil @ A ) ) ) ) ).

% list_assn_aux_simps(2)
thf(fact_6251_list__assn__aux__simps_I1_J,axiom,
    ! [A: $tType,B: $tType,P: A > B > assn,L4: list @ B] :
      ( ( vEBT_List_list_assn @ A @ B @ P @ ( nil @ A ) @ L4 )
      = ( pure_assn
        @ ( L4
          = ( nil @ B ) ) ) ) ).

% list_assn_aux_simps(1)
thf(fact_6252_length__greater__0__conv,axiom,
    ! [A: $tType,Xs2: list @ A] :
      ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ A ) @ Xs2 ) )
      = ( Xs2
       != ( nil @ A ) ) ) ).

% length_greater_0_conv
thf(fact_6253_length__ge__1__conv,axiom,
    ! [A: $tType,L: list @ A] :
      ( ( ord_less_eq @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( size_size @ ( list @ A ) @ L ) )
      = ( L
       != ( nil @ A ) ) ) ).

% length_ge_1_conv
thf(fact_6254_hash__code__list__simps_I1_J,axiom,
    ! [A: $tType,H_a2: A > uint32] :
      ( ( hash_hash_code_list @ A @ H_a2 @ ( nil @ A ) )
      = ( numeral_numeral @ uint32 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% hash_code_list_simps(1)
thf(fact_6255_length__nth__simps_I1_J,axiom,
    ! [A: $tType] :
      ( ( size_size @ ( list @ A ) @ ( nil @ A ) )
      = ( zero_zero @ nat ) ) ).

% length_nth_simps(1)
thf(fact_6256_empty__set,axiom,
    ! [A: $tType] :
      ( ( bot_bot @ ( set @ A ) )
      = ( set2 @ A @ ( nil @ A ) ) ) ).

% empty_set
thf(fact_6257_len__greater__imp__nonempty,axiom,
    ! [A: $tType,X2: nat,L: list @ A] :
      ( ( ord_less @ nat @ X2 @ ( size_size @ ( list @ A ) @ L ) )
     => ( L
       != ( nil @ A ) ) ) ).

% len_greater_imp_nonempty
thf(fact_6258_list__induct4,axiom,
    ! [C: $tType,A: $tType,B: $tType,D: $tType,Xs2: list @ A,Ys: list @ B,Zs: list @ C,Ws: list @ D,P: ( list @ A ) > ( list @ B ) > ( list @ C ) > ( list @ D ) > $o] :
      ( ( ( size_size @ ( list @ A ) @ Xs2 )
        = ( size_size @ ( list @ B ) @ Ys ) )
     => ( ( ( size_size @ ( list @ B ) @ Ys )
          = ( size_size @ ( list @ C ) @ Zs ) )
       => ( ( ( size_size @ ( list @ C ) @ Zs )
            = ( size_size @ ( list @ D ) @ Ws ) )
         => ( ( P @ ( nil @ A ) @ ( nil @ B ) @ ( nil @ C ) @ ( nil @ D ) )
           => ( ! [X3: A,Xs3: list @ A,Y4: B,Ys4: list @ B,Z3: C,Zs2: list @ C,W3: D,Ws2: list @ D] :
                  ( ( ( size_size @ ( list @ A ) @ Xs3 )
                    = ( size_size @ ( list @ B ) @ Ys4 ) )
                 => ( ( ( size_size @ ( list @ B ) @ Ys4 )
                      = ( size_size @ ( list @ C ) @ Zs2 ) )
                   => ( ( ( size_size @ ( list @ C ) @ Zs2 )
                        = ( size_size @ ( list @ D ) @ Ws2 ) )
                     => ( ( P @ Xs3 @ Ys4 @ Zs2 @ Ws2 )
                       => ( P @ ( cons @ A @ X3 @ Xs3 ) @ ( cons @ B @ Y4 @ Ys4 ) @ ( cons @ C @ Z3 @ Zs2 ) @ ( cons @ D @ W3 @ Ws2 ) ) ) ) ) )
             => ( P @ Xs2 @ Ys @ Zs @ Ws ) ) ) ) ) ) ).

% list_induct4
thf(fact_6259_list__induct3,axiom,
    ! [B: $tType,A: $tType,C: $tType,Xs2: list @ A,Ys: list @ B,Zs: list @ C,P: ( list @ A ) > ( list @ B ) > ( list @ C ) > $o] :
      ( ( ( size_size @ ( list @ A ) @ Xs2 )
        = ( size_size @ ( list @ B ) @ Ys ) )
     => ( ( ( size_size @ ( list @ B ) @ Ys )
          = ( size_size @ ( list @ C ) @ Zs ) )
       => ( ( P @ ( nil @ A ) @ ( nil @ B ) @ ( nil @ C ) )
         => ( ! [X3: A,Xs3: list @ A,Y4: B,Ys4: list @ B,Z3: C,Zs2: list @ C] :
                ( ( ( size_size @ ( list @ A ) @ Xs3 )
                  = ( size_size @ ( list @ B ) @ Ys4 ) )
               => ( ( ( size_size @ ( list @ B ) @ Ys4 )
                    = ( size_size @ ( list @ C ) @ Zs2 ) )
                 => ( ( P @ Xs3 @ Ys4 @ Zs2 )
                   => ( P @ ( cons @ A @ X3 @ Xs3 ) @ ( cons @ B @ Y4 @ Ys4 ) @ ( cons @ C @ Z3 @ Zs2 ) ) ) ) )
           => ( P @ Xs2 @ Ys @ Zs ) ) ) ) ) ).

% list_induct3
thf(fact_6260_list__induct2,axiom,
    ! [A: $tType,B: $tType,Xs2: list @ A,Ys: list @ B,P: ( list @ A ) > ( list @ B ) > $o] :
      ( ( ( size_size @ ( list @ A ) @ Xs2 )
        = ( size_size @ ( list @ B ) @ Ys ) )
     => ( ( P @ ( nil @ A ) @ ( nil @ B ) )
       => ( ! [X3: A,Xs3: list @ A,Y4: B,Ys4: list @ B] :
              ( ( ( size_size @ ( list @ A ) @ Xs3 )
                = ( size_size @ ( list @ B ) @ Ys4 ) )
             => ( ( P @ Xs3 @ Ys4 )
               => ( P @ ( cons @ A @ X3 @ Xs3 ) @ ( cons @ B @ Y4 @ Ys4 ) ) ) )
         => ( P @ Xs2 @ Ys ) ) ) ) ).

% list_induct2
thf(fact_6261_Misc_Ofoldr__Cons,axiom,
    ! [A: $tType,Xs2: list @ A] :
      ( ( foldr @ A @ ( list @ A ) @ ( cons @ A ) @ Xs2 @ ( nil @ A ) )
      = Xs2 ) ).

% Misc.foldr_Cons
thf(fact_6262_neq__NilE,axiom,
    ! [A: $tType,L: list @ A] :
      ( ( L
       != ( nil @ A ) )
     => ~ ! [X3: A,Xs3: list @ A] :
            ( L
           != ( cons @ A @ X3 @ Xs3 ) ) ) ).

% neq_NilE
thf(fact_6263_list__2pre__induct,axiom,
    ! [A: $tType,B: $tType,P: ( list @ A ) > ( list @ B ) > $o,W1: list @ A,W22: list @ B] :
      ( ( P @ ( nil @ A ) @ ( nil @ B ) )
     => ( ! [E2: A,W12: list @ A,W23: list @ B] :
            ( ( P @ W12 @ W23 )
           => ( P @ ( cons @ A @ E2 @ W12 ) @ W23 ) )
       => ( ! [E2: B,W13: list @ A,W24: list @ B] :
              ( ( P @ W13 @ W24 )
             => ( P @ W13 @ ( cons @ B @ E2 @ W24 ) ) )
         => ( P @ W1 @ W22 ) ) ) ) ).

% list_2pre_induct
thf(fact_6264_list__induct__first2,axiom,
    ! [A: $tType,P: ( list @ A ) > $o,Xs2: list @ A] :
      ( ( P @ ( nil @ A ) )
     => ( ! [X3: A] : ( P @ ( cons @ A @ X3 @ ( nil @ A ) ) )
       => ( ! [X16: A,X24: A,Xs3: list @ A] :
              ( ( P @ Xs3 )
             => ( P @ ( cons @ A @ X16 @ ( cons @ A @ X24 @ Xs3 ) ) ) )
         => ( P @ Xs2 ) ) ) ) ).

% list_induct_first2
thf(fact_6265_mergesort__by__rel__merge__induct,axiom,
    ! [A: $tType,B: $tType,P: ( list @ A ) > ( list @ B ) > $o,R2: A > B > $o,Xs2: list @ A,Ys: list @ B] :
      ( ! [Xs3: list @ A] : ( P @ Xs3 @ ( nil @ B ) )
     => ( ! [X_12: list @ B] : ( P @ ( nil @ A ) @ X_12 )
       => ( ! [X3: A,Xs3: list @ A,Y4: B,Ys4: list @ B] :
              ( ( R2 @ X3 @ Y4 )
             => ( ( P @ Xs3 @ ( cons @ B @ Y4 @ Ys4 ) )
               => ( P @ ( cons @ A @ X3 @ Xs3 ) @ ( cons @ B @ Y4 @ Ys4 ) ) ) )
         => ( ! [X3: A,Xs3: list @ A,Y4: B,Ys4: list @ B] :
                ( ~ ( R2 @ X3 @ Y4 )
               => ( ( P @ ( cons @ A @ X3 @ Xs3 ) @ Ys4 )
                 => ( P @ ( cons @ A @ X3 @ Xs3 ) @ ( cons @ B @ Y4 @ Ys4 ) ) ) )
           => ( P @ Xs2 @ Ys ) ) ) ) ) ).

% mergesort_by_rel_merge_induct
thf(fact_6266_subset__eq__mset__impl_Ocases,axiom,
    ! [A: $tType,X2: product_prod @ ( list @ A ) @ ( list @ A )] :
      ( ! [Ys4: list @ A] :
          ( X2
         != ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Ys4 ) )
     => ~ ! [X3: A,Xs3: list @ A,Ys4: list @ A] :
            ( X2
           != ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X3 @ Xs3 ) @ Ys4 ) ) ) ).

% subset_eq_mset_impl.cases
thf(fact_6267_zipf_Ocases,axiom,
    ! [C: $tType,A: $tType,B: $tType,X2: product_prod @ ( A > B > C ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) )] :
      ( ! [F2: A > B > C] :
          ( X2
         != ( product_Pair @ ( A > B > C ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ F2 @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( nil @ A ) @ ( nil @ B ) ) ) )
     => ( ! [F2: A > B > C,A4: A,As: list @ A,B3: B,Bs: list @ B] :
            ( X2
           != ( product_Pair @ ( A > B > C ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ F2 @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( cons @ A @ A4 @ As ) @ ( cons @ B @ B3 @ Bs ) ) ) )
       => ( ! [A4: A > B > C,V3: A,Va3: list @ A] :
              ( X2
             != ( product_Pair @ ( A > B > C ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ A4 @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( cons @ A @ V3 @ Va3 ) @ ( nil @ B ) ) ) )
         => ~ ! [A4: A > B > C,V3: B,Va3: list @ B] :
                ( X2
               != ( product_Pair @ ( A > B > C ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ A4 @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( nil @ A ) @ ( cons @ B @ V3 @ Va3 ) ) ) ) ) ) ) ).

% zipf.cases
thf(fact_6268_merge_Ocases,axiom,
    ! [A: $tType] :
      ( ( linorder @ A )
     => ! [X2: product_prod @ ( list @ A ) @ ( list @ A )] :
          ( ! [L22: list @ A] :
              ( X2
             != ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ L22 ) )
         => ( ! [V3: A,Va3: list @ A] :
                ( X2
               != ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ V3 @ Va3 ) @ ( nil @ A ) ) )
           => ~ ! [X16: A,L1: list @ A,X24: A,L22: list @ A] :
                  ( X2
                 != ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X16 @ L1 ) @ ( cons @ A @ X24 @ L22 ) ) ) ) ) ) ).

% merge.cases
thf(fact_6269_list__all__zip_Ocases,axiom,
    ! [A: $tType,B: $tType,X2: product_prod @ ( A > B > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) )] :
      ( ! [P7: A > B > $o] :
          ( X2
         != ( product_Pair @ ( A > B > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ P7 @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( nil @ A ) @ ( nil @ B ) ) ) )
     => ( ! [P7: A > B > $o,A4: A,As: list @ A,B3: B,Bs: list @ B] :
            ( X2
           != ( product_Pair @ ( A > B > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ P7 @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( cons @ A @ A4 @ As ) @ ( cons @ B @ B3 @ Bs ) ) ) )
       => ( ! [P7: A > B > $o,V3: A,Va3: list @ A] :
              ( X2
             != ( product_Pair @ ( A > B > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ P7 @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( cons @ A @ V3 @ Va3 ) @ ( nil @ B ) ) ) )
         => ~ ! [P7: A > B > $o,V3: B,Va3: list @ B] :
                ( X2
               != ( product_Pair @ ( A > B > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ P7 @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( nil @ A ) @ ( cons @ B @ V3 @ Va3 ) ) ) ) ) ) ) ).

% list_all_zip.cases
thf(fact_6270_partition__rev_Ocases,axiom,
    ! [A: $tType,X2: product_prod @ ( A > $o ) @ ( product_prod @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) )] :
      ( ! [P7: A > $o,Yes: list @ A,No: list @ A] :
          ( X2
         != ( product_Pair @ ( A > $o ) @ ( product_prod @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) ) @ P7 @ ( product_Pair @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Yes @ No ) @ ( nil @ A ) ) ) )
     => ~ ! [P7: A > $o,Yes: list @ A,No: list @ A,X3: A,Xs3: list @ A] :
            ( X2
           != ( product_Pair @ ( A > $o ) @ ( product_prod @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) ) @ P7 @ ( product_Pair @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Yes @ No ) @ ( cons @ A @ X3 @ Xs3 ) ) ) ) ) ).

% partition_rev.cases
thf(fact_6271_quicksort__by__rel_Ocases,axiom,
    ! [A: $tType,X2: product_prod @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) )] :
      ( ! [R8: A > A > $o,Sl: list @ A] :
          ( X2
         != ( product_Pair @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ R8 @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Sl @ ( nil @ A ) ) ) )
     => ~ ! [R8: A > A > $o,Sl: list @ A,X3: A,Xs3: list @ A] :
            ( X2
           != ( product_Pair @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ R8 @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Sl @ ( cons @ A @ X3 @ Xs3 ) ) ) ) ) ).

% quicksort_by_rel.cases
thf(fact_6272_mergesort__by__rel__merge_Ocases,axiom,
    ! [A: $tType,X2: product_prod @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) )] :
      ( ! [R8: A > A > $o,X3: A,Xs3: list @ A,Y4: A,Ys4: list @ A] :
          ( X2
         != ( product_Pair @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ R8 @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X3 @ Xs3 ) @ ( cons @ A @ Y4 @ Ys4 ) ) ) )
     => ( ! [R8: A > A > $o,Xs3: list @ A] :
            ( X2
           != ( product_Pair @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ R8 @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs3 @ ( nil @ A ) ) ) )
       => ~ ! [R8: A > A > $o,V3: A,Va3: list @ A] :
              ( X2
             != ( product_Pair @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ R8 @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ ( cons @ A @ V3 @ Va3 ) ) ) ) ) ) ).

% mergesort_by_rel_merge.cases
thf(fact_6273_mergesort__by__rel__split_Ocases,axiom,
    ! [A: $tType,X2: product_prod @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A )] :
      ( ! [Xs1: list @ A,Xs22: list @ A] :
          ( X2
         != ( product_Pair @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs1 @ Xs22 ) @ ( nil @ A ) ) )
     => ( ! [Xs1: list @ A,Xs22: list @ A,X3: A] :
            ( X2
           != ( product_Pair @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs1 @ Xs22 ) @ ( cons @ A @ X3 @ ( nil @ A ) ) ) )
       => ~ ! [Xs1: list @ A,Xs22: list @ A,X16: A,X24: A,Xs3: list @ A] :
              ( X2
             != ( product_Pair @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs1 @ Xs22 ) @ ( cons @ A @ X16 @ ( cons @ A @ X24 @ Xs3 ) ) ) ) ) ) ).

% mergesort_by_rel_split.cases
thf(fact_6274_shuffles_Ocases,axiom,
    ! [A: $tType,X2: product_prod @ ( list @ A ) @ ( list @ A )] :
      ( ! [Ys4: list @ A] :
          ( X2
         != ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Ys4 ) )
     => ( ! [Xs3: list @ A] :
            ( X2
           != ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs3 @ ( nil @ A ) ) )
       => ~ ! [X3: A,Xs3: list @ A,Y4: A,Ys4: list @ A] :
              ( X2
             != ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X3 @ Xs3 ) @ ( cons @ A @ Y4 @ Ys4 ) ) ) ) ) ).

% shuffles.cases
thf(fact_6275_sorted__wrt_Ocases,axiom,
    ! [A: $tType,X2: product_prod @ ( A > A > $o ) @ ( list @ A )] :
      ( ! [P7: A > A > $o] :
          ( X2
         != ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ P7 @ ( nil @ A ) ) )
     => ~ ! [P7: A > A > $o,X3: A,Ys4: list @ A] :
            ( X2
           != ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ P7 @ ( cons @ A @ X3 @ Ys4 ) ) ) ) ).

% sorted_wrt.cases
thf(fact_6276_arg__min__list_Ocases,axiom,
    ! [B: $tType,A: $tType] :
      ( ( linorder @ B )
     => ! [X2: product_prod @ ( A > B ) @ ( list @ A )] :
          ( ! [F2: A > B,X3: A] :
              ( X2
             != ( product_Pair @ ( A > B ) @ ( list @ A ) @ F2 @ ( cons @ A @ X3 @ ( nil @ A ) ) ) )
         => ( ! [F2: A > B,X3: A,Y4: A,Zs2: list @ A] :
                ( X2
               != ( product_Pair @ ( A > B ) @ ( list @ A ) @ F2 @ ( cons @ A @ X3 @ ( cons @ A @ Y4 @ Zs2 ) ) ) )
           => ~ ! [A4: A > B] :
                  ( X2
                 != ( product_Pair @ ( A > B ) @ ( list @ A ) @ A4 @ ( nil @ A ) ) ) ) ) ) ).

% arg_min_list.cases
thf(fact_6277_successively_Ocases,axiom,
    ! [A: $tType,X2: product_prod @ ( A > A > $o ) @ ( list @ A )] :
      ( ! [P7: A > A > $o] :
          ( X2
         != ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ P7 @ ( nil @ A ) ) )
     => ( ! [P7: A > A > $o,X3: A] :
            ( X2
           != ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ P7 @ ( cons @ A @ X3 @ ( nil @ A ) ) ) )
       => ~ ! [P7: A > A > $o,X3: A,Y4: A,Xs3: list @ A] :
              ( X2
             != ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ P7 @ ( cons @ A @ X3 @ ( cons @ A @ Y4 @ Xs3 ) ) ) ) ) ) ).

% successively.cases
thf(fact_6278_map__tailrec__rev_Ocases,axiom,
    ! [A: $tType,B: $tType,X2: product_prod @ ( A > B ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) )] :
      ( ! [F2: A > B,Bs: list @ B] :
          ( X2
         != ( product_Pair @ ( A > B ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ F2 @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( nil @ A ) @ Bs ) ) )
     => ~ ! [F2: A > B,A4: A,As: list @ A,Bs: list @ B] :
            ( X2
           != ( product_Pair @ ( A > B ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ F2 @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( cons @ A @ A4 @ As ) @ Bs ) ) ) ) ).

% map_tailrec_rev.cases
thf(fact_6279_list__assn_Ocases,axiom,
    ! [A: $tType,C: $tType,X2: product_prod @ ( A > C > assn ) @ ( product_prod @ ( list @ A ) @ ( list @ C ) )] :
      ( ! [P7: A > C > assn] :
          ( X2
         != ( product_Pair @ ( A > C > assn ) @ ( product_prod @ ( list @ A ) @ ( list @ C ) ) @ P7 @ ( product_Pair @ ( list @ A ) @ ( list @ C ) @ ( nil @ A ) @ ( nil @ C ) ) ) )
     => ( ! [P7: A > C > assn,A4: A,As: list @ A,C3: C,Cs2: list @ C] :
            ( X2
           != ( product_Pair @ ( A > C > assn ) @ ( product_prod @ ( list @ A ) @ ( list @ C ) ) @ P7 @ ( product_Pair @ ( list @ A ) @ ( list @ C ) @ ( cons @ A @ A4 @ As ) @ ( cons @ C @ C3 @ Cs2 ) ) ) )
       => ( ! [Uu2: A > C > assn,V3: A,Va3: list @ A] :
              ( X2
             != ( product_Pair @ ( A > C > assn ) @ ( product_prod @ ( list @ A ) @ ( list @ C ) ) @ Uu2 @ ( product_Pair @ ( list @ A ) @ ( list @ C ) @ ( cons @ A @ V3 @ Va3 ) @ ( nil @ C ) ) ) )
         => ~ ! [Uu2: A > C > assn,V3: C,Va3: list @ C] :
                ( X2
               != ( product_Pair @ ( A > C > assn ) @ ( product_prod @ ( list @ A ) @ ( list @ C ) ) @ Uu2 @ ( product_Pair @ ( list @ A ) @ ( list @ C ) @ ( nil @ A ) @ ( cons @ C @ V3 @ Va3 ) ) ) ) ) ) ) ).

% list_assn.cases
thf(fact_6280_list__assn__simps_I1_J,axiom,
    ! [B: $tType,A: $tType,P: A > B > assn] :
      ( ( vEBT_List_list_assn @ A @ B @ P @ ( nil @ A ) @ ( nil @ B ) )
      = ( one_one @ assn ) ) ).

% list_assn_simps(1)
thf(fact_6281_list__assn_Osimps_I1_J,axiom,
    ! [C: $tType,A: $tType,P: A > C > assn] :
      ( ( vEBT_List_list_assn @ A @ C @ P @ ( nil @ A ) @ ( nil @ C ) )
      = ( one_one @ assn ) ) ).

% list_assn.simps(1)
thf(fact_6282_length__compl__induct,axiom,
    ! [A: $tType,P: ( list @ A ) > $o,L: list @ A] :
      ( ( P @ ( nil @ A ) )
     => ( ! [E2: A,L3: list @ A] :
            ( ! [Ll: list @ A] :
                ( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Ll ) @ ( size_size @ ( list @ A ) @ L3 ) )
               => ( P @ Ll ) )
           => ( P @ ( cons @ A @ E2 @ L3 ) ) )
       => ( P @ L ) ) ) ).

% length_compl_induct
thf(fact_6283_list__decomp__1,axiom,
    ! [A: $tType,L: list @ A] :
      ( ( ( size_size @ ( list @ A ) @ L )
        = ( one_one @ nat ) )
     => ? [A4: A] :
          ( L
          = ( cons @ A @ A4 @ ( nil @ A ) ) ) ) ).

% list_decomp_1
thf(fact_6284_list__assn_Osimps_I4_J,axiom,
    ! [C: $tType,A: $tType,Uu: A > C > assn,V: C,Va: list @ C] :
      ( ( vEBT_List_list_assn @ A @ C @ Uu @ ( nil @ A ) @ ( cons @ C @ V @ Va ) )
      = ( bot_bot @ assn ) ) ).

% list_assn.simps(4)
thf(fact_6285_list__assn_Osimps_I3_J,axiom,
    ! [C: $tType,A: $tType,Uu: A > C > assn,V: A,Va: list @ A] :
      ( ( vEBT_List_list_assn @ A @ C @ Uu @ ( cons @ A @ V @ Va ) @ ( nil @ C ) )
      = ( bot_bot @ assn ) ) ).

% list_assn.simps(3)
thf(fact_6286_list__assn__simps_I4_J,axiom,
    ! [B: $tType,A: $tType,P: A > B > assn,C2: B,Cs: list @ B] :
      ( ( vEBT_List_list_assn @ A @ B @ P @ ( nil @ A ) @ ( cons @ B @ C2 @ Cs ) )
      = ( bot_bot @ assn ) ) ).

% list_assn_simps(4)
thf(fact_6287_list__assn__simps_I3_J,axiom,
    ! [B: $tType,A: $tType,P: A > B > assn,A2: A,As3: list @ A] :
      ( ( vEBT_List_list_assn @ A @ B @ P @ ( cons @ A @ A2 @ As3 ) @ ( nil @ B ) )
      = ( bot_bot @ assn ) ) ).

% list_assn_simps(3)
thf(fact_6288_VEBT__internal_Oreplicatei_Osimps_I1_J,axiom,
    ! [A: $tType,X2: heap_Time_Heap @ A] :
      ( ( vEBT_VEBT_replicatei @ A @ ( zero_zero @ nat ) @ X2 )
      = ( heap_Time_return @ ( list @ A ) @ ( nil @ A ) ) ) ).

% VEBT_internal.replicatei.simps(1)
thf(fact_6289_list__decomp__2,axiom,
    ! [A: $tType,L: list @ A] :
      ( ( ( size_size @ ( list @ A ) @ L )
        = ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
     => ? [A4: A,B3: A] :
          ( L
          = ( cons @ A @ A4 @ ( cons @ A @ B3 @ ( nil @ A ) ) ) ) ) ).

% list_decomp_2
thf(fact_6290_let__mono,axiom,
    ! [A: $tType,C: $tType,B: $tType,Orda: B > B > $o,Ordb: C > C > $o,B2: B > A > C,T2: A] :
      ( ! [X3: A] :
          ( comple7038119648293358887notone @ B @ C @ Orda @ Ordb
          @ ^ [F6: B] : ( B2 @ F6 @ X3 ) )
     => ( comple7038119648293358887notone @ B @ C @ Orda @ Ordb
        @ ^ [F6: B] : ( B2 @ F6 @ T2 ) ) ) ).

% let_mono
thf(fact_6291_if__mono,axiom,
    ! [B: $tType,A: $tType,Orda: A > A > $o,Ordb: B > B > $o,F5: A > B,G5: A > B,C2: $o] :
      ( ( comple7038119648293358887notone @ A @ B @ Orda @ Ordb @ F5 )
     => ( ( comple7038119648293358887notone @ A @ B @ Orda @ Ordb @ G5 )
       => ( comple7038119648293358887notone @ A @ B @ Orda @ Ordb
          @ ^ [F6: A] : ( if @ B @ C2 @ ( F5 @ F6 ) @ ( G5 @ F6 ) ) ) ) ) ).

% if_mono
thf(fact_6292_list__assn_Oelims,axiom,
    ! [C: $tType,A: $tType,X2: A > C > assn,Xa: list @ A,Xb: list @ C,Y2: assn] :
      ( ( ( vEBT_List_list_assn @ A @ C @ X2 @ Xa @ Xb )
        = Y2 )
     => ( ( ( Xa
            = ( nil @ A ) )
         => ( ( Xb
              = ( nil @ C ) )
           => ( Y2
             != ( one_one @ assn ) ) ) )
       => ( ! [A4: A,As: list @ A] :
              ( ( Xa
                = ( cons @ A @ A4 @ As ) )
             => ! [C3: C,Cs2: list @ C] :
                  ( ( Xb
                    = ( cons @ C @ C3 @ Cs2 ) )
                 => ( Y2
                   != ( times_times @ assn @ ( X2 @ A4 @ C3 ) @ ( vEBT_List_list_assn @ A @ C @ X2 @ As @ Cs2 ) ) ) ) )
         => ( ( ? [V3: A,Va3: list @ A] :
                  ( Xa
                  = ( cons @ A @ V3 @ Va3 ) )
             => ( ( Xb
                  = ( nil @ C ) )
               => ( Y2
                 != ( bot_bot @ assn ) ) ) )
           => ~ ( ( Xa
                  = ( nil @ A ) )
               => ( ? [V3: C,Va3: list @ C] :
                      ( Xb
                      = ( cons @ C @ V3 @ Va3 ) )
                 => ( Y2
                   != ( bot_bot @ assn ) ) ) ) ) ) ) ) ).

% list_assn.elims
thf(fact_6293_VEBT__internal_Oreplicatei_Opelims,axiom,
    ! [A: $tType,X2: nat,Xa: heap_Time_Heap @ A,Y2: heap_Time_Heap @ ( list @ A )] :
      ( ( ( vEBT_VEBT_replicatei @ A @ X2 @ Xa )
        = Y2 )
     => ( ( accp @ ( product_prod @ nat @ ( heap_Time_Heap @ A ) ) @ ( vEBT_V8535202610096940344ei_rel @ A ) @ ( product_Pair @ nat @ ( heap_Time_Heap @ A ) @ X2 @ Xa ) )
       => ( ( ( X2
              = ( zero_zero @ nat ) )
           => ( ( Y2
                = ( heap_Time_return @ ( list @ A ) @ ( nil @ A ) ) )
             => ~ ( accp @ ( product_prod @ nat @ ( heap_Time_Heap @ A ) ) @ ( vEBT_V8535202610096940344ei_rel @ A ) @ ( product_Pair @ nat @ ( heap_Time_Heap @ A ) @ ( zero_zero @ nat ) @ Xa ) ) ) )
         => ~ ! [N4: nat] :
                ( ( X2
                  = ( suc @ N4 ) )
               => ( ( Y2
                    = ( heap_Time_bind @ A @ ( list @ A ) @ Xa
                      @ ^ [Y: A] :
                          ( heap_Time_bind @ ( list @ A ) @ ( list @ A ) @ ( vEBT_VEBT_replicatei @ A @ N4 @ Xa )
                          @ ^ [Ys3: list @ A] : ( heap_Time_return @ ( list @ A ) @ ( cons @ A @ Y @ Ys3 ) ) ) ) )
                 => ~ ( accp @ ( product_prod @ nat @ ( heap_Time_Heap @ A ) ) @ ( vEBT_V8535202610096940344ei_rel @ A ) @ ( product_Pair @ nat @ ( heap_Time_Heap @ A ) @ ( suc @ N4 ) @ Xa ) ) ) ) ) ) ) ).

% VEBT_internal.replicatei.pelims
thf(fact_6294_list__assn_Opelims,axiom,
    ! [A: $tType,C: $tType,X2: A > C > assn,Xa: list @ A,Xb: list @ C,Y2: assn] :
      ( ( ( vEBT_List_list_assn @ A @ C @ X2 @ Xa @ Xb )
        = Y2 )
     => ( ( accp @ ( product_prod @ ( A > C > assn ) @ ( product_prod @ ( list @ A ) @ ( list @ C ) ) ) @ ( vEBT_L4249061453398456502sn_rel @ A @ C ) @ ( product_Pair @ ( A > C > assn ) @ ( product_prod @ ( list @ A ) @ ( list @ C ) ) @ X2 @ ( product_Pair @ ( list @ A ) @ ( list @ C ) @ Xa @ Xb ) ) )
       => ( ( ( Xa
              = ( nil @ A ) )
           => ( ( Xb
                = ( nil @ C ) )
             => ( ( Y2
                  = ( one_one @ assn ) )
               => ~ ( accp @ ( product_prod @ ( A > C > assn ) @ ( product_prod @ ( list @ A ) @ ( list @ C ) ) ) @ ( vEBT_L4249061453398456502sn_rel @ A @ C ) @ ( product_Pair @ ( A > C > assn ) @ ( product_prod @ ( list @ A ) @ ( list @ C ) ) @ X2 @ ( product_Pair @ ( list @ A ) @ ( list @ C ) @ ( nil @ A ) @ ( nil @ C ) ) ) ) ) ) )
         => ( ! [A4: A,As: list @ A] :
                ( ( Xa
                  = ( cons @ A @ A4 @ As ) )
               => ! [C3: C,Cs2: list @ C] :
                    ( ( Xb
                      = ( cons @ C @ C3 @ Cs2 ) )
                   => ( ( Y2
                        = ( times_times @ assn @ ( X2 @ A4 @ C3 ) @ ( vEBT_List_list_assn @ A @ C @ X2 @ As @ Cs2 ) ) )
                     => ~ ( accp @ ( product_prod @ ( A > C > assn ) @ ( product_prod @ ( list @ A ) @ ( list @ C ) ) ) @ ( vEBT_L4249061453398456502sn_rel @ A @ C ) @ ( product_Pair @ ( A > C > assn ) @ ( product_prod @ ( list @ A ) @ ( list @ C ) ) @ X2 @ ( product_Pair @ ( list @ A ) @ ( list @ C ) @ ( cons @ A @ A4 @ As ) @ ( cons @ C @ C3 @ Cs2 ) ) ) ) ) ) )
           => ( ! [V3: A,Va3: list @ A] :
                  ( ( Xa
                    = ( cons @ A @ V3 @ Va3 ) )
                 => ( ( Xb
                      = ( nil @ C ) )
                   => ( ( Y2
                        = ( bot_bot @ assn ) )
                     => ~ ( accp @ ( product_prod @ ( A > C > assn ) @ ( product_prod @ ( list @ A ) @ ( list @ C ) ) ) @ ( vEBT_L4249061453398456502sn_rel @ A @ C ) @ ( product_Pair @ ( A > C > assn ) @ ( product_prod @ ( list @ A ) @ ( list @ C ) ) @ X2 @ ( product_Pair @ ( list @ A ) @ ( list @ C ) @ ( cons @ A @ V3 @ Va3 ) @ ( nil @ C ) ) ) ) ) ) )
             => ~ ( ( Xa
                    = ( nil @ A ) )
                 => ! [V3: C,Va3: list @ C] :
                      ( ( Xb
                        = ( cons @ C @ V3 @ Va3 ) )
                     => ( ( Y2
                          = ( bot_bot @ assn ) )
                       => ~ ( accp @ ( product_prod @ ( A > C > assn ) @ ( product_prod @ ( list @ A ) @ ( list @ C ) ) ) @ ( vEBT_L4249061453398456502sn_rel @ A @ C ) @ ( product_Pair @ ( A > C > assn ) @ ( product_prod @ ( list @ A ) @ ( list @ C ) ) @ X2 @ ( product_Pair @ ( list @ A ) @ ( list @ C ) @ ( nil @ A ) @ ( cons @ C @ V3 @ Va3 ) ) ) ) ) ) ) ) ) ) ) ) ).

% list_assn.pelims
thf(fact_6295_mergesort__by__rel_Ocases,axiom,
    ! [A: $tType,X2: product_prod @ ( A > A > $o ) @ ( list @ A )] :
      ~ ! [R8: A > A > $o,Xs3: list @ A] :
          ( X2
         != ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ R8 @ Xs3 ) ) ).

% mergesort_by_rel.cases
thf(fact_6296_partial__function__definitions_Ofixp__induct__uc,axiom,
    ! [B: $tType,A: $tType,C: $tType,Leq: A > A > $o,Lub: ( set @ A ) > A,U3: C > B > A,F5: C > C,C5: ( B > A ) > C,F: C,P: ( B > A ) > $o] :
      ( ( partia7178651479351089652itions @ A @ Leq @ Lub )
     => ( ! [X3: B] :
            ( comple7038119648293358887notone @ ( B > A ) @ A @ ( partial_fun_ord @ A @ A @ B @ Leq ) @ Leq
            @ ^ [F6: B > A] : ( U3 @ ( F5 @ ( C5 @ F6 ) ) @ X3 ) )
       => ( ( F
            = ( C5
              @ ( comple187402453842119260l_fixp @ ( B > A ) @ ( partial_fun_lub @ A @ A @ B @ Lub ) @ ( partial_fun_ord @ A @ A @ B @ Leq )
                @ ^ [F6: B > A] : ( U3 @ ( F5 @ ( C5 @ F6 ) ) ) ) ) )
         => ( ! [F2: B > A] :
                ( ( U3 @ ( C5 @ F2 ) )
                = F2 )
           => ( ( comple1908693960933563346ssible @ ( B > A ) @ ( partial_fun_lub @ A @ A @ B @ Lub ) @ ( partial_fun_ord @ A @ A @ B @ Leq ) @ P )
             => ( ( P
                  @ ^ [Uu3: B] : ( Lub @ ( bot_bot @ ( set @ A ) ) ) )
               => ( ! [F2: C] :
                      ( ( P @ ( U3 @ F2 ) )
                     => ( P @ ( U3 @ ( F5 @ F2 ) ) ) )
                 => ( P @ ( U3 @ F ) ) ) ) ) ) ) ) ) ).

% partial_function_definitions.fixp_induct_uc
thf(fact_6297_in__set__simps_I2_J,axiom,
    ! [A: $tType,X2: A,Y2: A] :
      ( ( member @ A @ X2 @ ( set2 @ A @ ( cons @ A @ Y2 @ ( nil @ A ) ) ) )
      = ( X2 = Y2 ) ) ).

% in_set_simps(2)
thf(fact_6298_partial__function__definitions_Oconst__mono,axiom,
    ! [A: $tType,B: $tType,Leq: A > A > $o,Lub: ( set @ A ) > A,Ord: B > B > $o,C2: A] :
      ( ( partia7178651479351089652itions @ A @ Leq @ Lub )
     => ( comple7038119648293358887notone @ B @ A @ Ord @ Leq
        @ ^ [F6: B] : C2 ) ) ).

% partial_function_definitions.const_mono
thf(fact_6299_in__set__simps_I1_J,axiom,
    ! [A: $tType,X2: A,Y2: A,Z: A,Ys: list @ A] :
      ( ( member @ A @ X2 @ ( set2 @ A @ ( cons @ A @ Y2 @ ( cons @ A @ Z @ Ys ) ) ) )
      = ( ( X2 = Y2 )
        | ( member @ A @ X2 @ ( set2 @ A @ ( cons @ A @ Z @ Ys ) ) ) ) ) ).

% in_set_simps(1)
thf(fact_6300_in__set__simps_I3_J,axiom,
    ! [A: $tType,X2: A] :
      ~ ( member @ A @ X2 @ ( set2 @ A @ ( nil @ A ) ) ) ).

% in_set_simps(3)
thf(fact_6301_partial__function__definitions_Ofixp__rule__uc,axiom,
    ! [B: $tType,A: $tType,C: $tType,Leq: A > A > $o,Lub: ( set @ A ) > A,U3: C > B > A,F5: C > C,C5: ( B > A ) > C,F: C] :
      ( ( partia7178651479351089652itions @ A @ Leq @ Lub )
     => ( ! [X3: B] :
            ( comple7038119648293358887notone @ ( B > A ) @ A @ ( partial_fun_ord @ A @ A @ B @ Leq ) @ Leq
            @ ^ [F6: B > A] : ( U3 @ ( F5 @ ( C5 @ F6 ) ) @ X3 ) )
       => ( ( F
            = ( C5
              @ ( comple187402453842119260l_fixp @ ( B > A ) @ ( partial_fun_lub @ A @ A @ B @ Lub ) @ ( partial_fun_ord @ A @ A @ B @ Leq )
                @ ^ [F6: B > A] : ( U3 @ ( F5 @ ( C5 @ F6 ) ) ) ) ) )
         => ( ! [F2: C] :
                ( ( C5 @ ( U3 @ F2 ) )
                = F2 )
           => ( F
              = ( F5 @ F ) ) ) ) ) ) ).

% partial_function_definitions.fixp_rule_uc
thf(fact_6302_partial__function__definitions_Omono__body__fixp,axiom,
    ! [A: $tType,B: $tType,Leq: A > A > $o,Lub: ( set @ A ) > A,F5: ( B > A ) > B > A] :
      ( ( partia7178651479351089652itions @ A @ Leq @ Lub )
     => ( ! [X3: B] :
            ( comple7038119648293358887notone @ ( B > A ) @ A @ ( partial_fun_ord @ A @ A @ B @ Leq ) @ Leq
            @ ^ [F6: B > A] : ( F5 @ F6 @ X3 ) )
       => ( ( comple187402453842119260l_fixp @ ( B > A ) @ ( partial_fun_lub @ A @ A @ B @ Lub ) @ ( partial_fun_ord @ A @ A @ B @ Leq ) @ F5 )
          = ( F5 @ ( comple187402453842119260l_fixp @ ( B > A ) @ ( partial_fun_lub @ A @ A @ B @ Lub ) @ ( partial_fun_ord @ A @ A @ B @ Leq ) @ F5 ) ) ) ) ) ).

% partial_function_definitions.mono_body_fixp
thf(fact_6303_admissible__fun,axiom,
    ! [A: $tType,B: $tType,Le: A > A > $o,Lub: ( set @ A ) > A,Q: B > A > $o] :
      ( ( partia7178651479351089652itions @ A @ Le @ Lub )
     => ( ! [X3: B] : ( comple1908693960933563346ssible @ A @ Lub @ Le @ ( Q @ X3 ) )
       => ( comple1908693960933563346ssible @ ( B > A ) @ ( partial_fun_lub @ A @ A @ B @ Lub ) @ ( partial_fun_ord @ A @ A @ B @ Le )
          @ ^ [F6: B > A] :
            ! [X: B] : ( Q @ X @ ( F6 @ X ) ) ) ) ) ).

% admissible_fun
thf(fact_6304_mergesort__by__rel__split__length,axiom,
    ! [A: $tType,Xs12: list @ A,Xs23: list @ A,Xs2: list @ A] :
      ( ( ( size_size @ ( list @ A ) @ ( product_fst @ ( list @ A ) @ ( list @ A ) @ ( merges295452479951948502_split @ A @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs12 @ Xs23 ) @ Xs2 ) ) )
        = ( plus_plus @ nat @ ( plus_plus @ nat @ ( size_size @ ( list @ A ) @ Xs12 ) @ ( divide_divide @ nat @ ( size_size @ ( list @ A ) @ Xs2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( modulo_modulo @ nat @ ( size_size @ ( list @ A ) @ Xs2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
      & ( ( size_size @ ( list @ A ) @ ( product_snd @ ( list @ A ) @ ( list @ A ) @ ( merges295452479951948502_split @ A @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs12 @ Xs23 ) @ Xs2 ) ) )
        = ( plus_plus @ nat @ ( size_size @ ( list @ A ) @ Xs23 ) @ ( divide_divide @ nat @ ( size_size @ ( list @ A ) @ Xs2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).

% mergesort_by_rel_split_length
thf(fact_6305_concat__inth,axiom,
    ! [A: $tType,Xs2: list @ A,X2: A,Ys: list @ A] :
      ( ( nth @ A @ ( append @ A @ Xs2 @ ( append @ A @ ( cons @ A @ X2 @ ( nil @ A ) ) @ Ys ) ) @ ( size_size @ ( list @ A ) @ Xs2 ) )
      = X2 ) ).

% concat_inth
thf(fact_6306_empty__append__eq__id,axiom,
    ! [A: $tType] :
      ( ( append @ A @ ( nil @ A ) )
      = ( ^ [X: list @ A] : X ) ) ).

% empty_append_eq_id
thf(fact_6307_append__eq__append__conv,axiom,
    ! [A: $tType,Xs2: list @ A,Ys: list @ A,Us: list @ A,Vs: list @ A] :
      ( ( ( ( size_size @ ( list @ A ) @ Xs2 )
          = ( size_size @ ( list @ A ) @ Ys ) )
        | ( ( size_size @ ( list @ A ) @ Us )
          = ( size_size @ ( list @ A ) @ Vs ) ) )
     => ( ( ( append @ A @ Xs2 @ Us )
          = ( append @ A @ Ys @ Vs ) )
        = ( ( Xs2 = Ys )
          & ( Us = Vs ) ) ) ) ).

% append_eq_append_conv
thf(fact_6308_list__ee__eq__leel_I1_J,axiom,
    ! [A: $tType,E1: A,E22: A,L12: list @ A,E12: A,E23: A,L23: list @ A] :
      ( ( ( cons @ A @ E1 @ ( cons @ A @ E22 @ ( nil @ A ) ) )
        = ( append @ A @ L12 @ ( cons @ A @ E12 @ ( cons @ A @ E23 @ L23 ) ) ) )
      = ( ( L12
          = ( nil @ A ) )
        & ( E1 = E12 )
        & ( E22 = E23 )
        & ( L23
          = ( nil @ A ) ) ) ) ).

% list_ee_eq_leel(1)
thf(fact_6309_list__ee__eq__leel_I2_J,axiom,
    ! [A: $tType,L12: list @ A,E12: A,E23: A,L23: list @ A,E1: A,E22: A] :
      ( ( ( append @ A @ L12 @ ( cons @ A @ E12 @ ( cons @ A @ E23 @ L23 ) ) )
        = ( cons @ A @ E1 @ ( cons @ A @ E22 @ ( nil @ A ) ) ) )
      = ( ( L12
          = ( nil @ A ) )
        & ( E1 = E12 )
        & ( E22 = E23 )
        & ( L23
          = ( nil @ A ) ) ) ) ).

% list_ee_eq_leel(2)
thf(fact_6310_list__se__match_I1_J,axiom,
    ! [A: $tType,L12: list @ A,L23: list @ A,A2: A] :
      ( ( L12
       != ( nil @ A ) )
     => ( ( ( append @ A @ L12 @ L23 )
          = ( cons @ A @ A2 @ ( nil @ A ) ) )
        = ( ( L12
            = ( cons @ A @ A2 @ ( nil @ A ) ) )
          & ( L23
            = ( nil @ A ) ) ) ) ) ).

% list_se_match(1)
thf(fact_6311_list__se__match_I2_J,axiom,
    ! [A: $tType,L23: list @ A,L12: list @ A,A2: A] :
      ( ( L23
       != ( nil @ A ) )
     => ( ( ( append @ A @ L12 @ L23 )
          = ( cons @ A @ A2 @ ( nil @ A ) ) )
        = ( ( L12
            = ( nil @ A ) )
          & ( L23
            = ( cons @ A @ A2 @ ( nil @ A ) ) ) ) ) ) ).

% list_se_match(2)
thf(fact_6312_list__se__match_I3_J,axiom,
    ! [A: $tType,L12: list @ A,A2: A,L23: list @ A] :
      ( ( L12
       != ( nil @ A ) )
     => ( ( ( cons @ A @ A2 @ ( nil @ A ) )
          = ( append @ A @ L12 @ L23 ) )
        = ( ( L12
            = ( cons @ A @ A2 @ ( nil @ A ) ) )
          & ( L23
            = ( nil @ A ) ) ) ) ) ).

% list_se_match(3)
thf(fact_6313_list__se__match_I4_J,axiom,
    ! [A: $tType,L23: list @ A,A2: A,L12: list @ A] :
      ( ( L23
       != ( nil @ A ) )
     => ( ( ( cons @ A @ A2 @ ( nil @ A ) )
          = ( append @ A @ L12 @ L23 ) )
        = ( ( L12
            = ( nil @ A ) )
          & ( L23
            = ( cons @ A @ A2 @ ( nil @ A ) ) ) ) ) ) ).

% list_se_match(4)
thf(fact_6314_list__e__eq__lel_I1_J,axiom,
    ! [A: $tType,E3: A,L12: list @ A,E5: A,L23: list @ A] :
      ( ( ( cons @ A @ E3 @ ( nil @ A ) )
        = ( append @ A @ L12 @ ( cons @ A @ E5 @ L23 ) ) )
      = ( ( L12
          = ( nil @ A ) )
        & ( E5 = E3 )
        & ( L23
          = ( nil @ A ) ) ) ) ).

% list_e_eq_lel(1)
thf(fact_6315_list__e__eq__lel_I2_J,axiom,
    ! [A: $tType,L12: list @ A,E5: A,L23: list @ A,E3: A] :
      ( ( ( append @ A @ L12 @ ( cons @ A @ E5 @ L23 ) )
        = ( cons @ A @ E3 @ ( nil @ A ) ) )
      = ( ( L12
          = ( nil @ A ) )
        & ( E5 = E3 )
        & ( L23
          = ( nil @ A ) ) ) ) ).

% list_e_eq_lel(2)
thf(fact_6316_length__append,axiom,
    ! [A: $tType,Xs2: list @ A,Ys: list @ A] :
      ( ( size_size @ ( list @ A ) @ ( append @ A @ Xs2 @ Ys ) )
      = ( plus_plus @ nat @ ( size_size @ ( list @ A ) @ Xs2 ) @ ( size_size @ ( list @ A ) @ Ys ) ) ) ).

% length_append
thf(fact_6317_nth__append__first,axiom,
    ! [A: $tType,I: nat,L: list @ A,L4: list @ A] :
      ( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ L ) )
     => ( ( nth @ A @ ( append @ A @ L @ L4 ) @ I )
        = ( nth @ A @ L @ I ) ) ) ).

% nth_append_first
thf(fact_6318_nth__append__length,axiom,
    ! [A: $tType,Xs2: list @ A,X2: A,Ys: list @ A] :
      ( ( nth @ A @ ( append @ A @ Xs2 @ ( cons @ A @ X2 @ Ys ) ) @ ( size_size @ ( list @ A ) @ Xs2 ) )
      = X2 ) ).

% nth_append_length
thf(fact_6319_nth__append__length__plus,axiom,
    ! [A: $tType,Xs2: list @ A,Ys: list @ A,N2: nat] :
      ( ( nth @ A @ ( append @ A @ Xs2 @ Ys ) @ ( plus_plus @ nat @ ( size_size @ ( list @ A ) @ Xs2 ) @ N2 ) )
      = ( nth @ A @ Ys @ N2 ) ) ).

% nth_append_length_plus
thf(fact_6320_list__update__length,axiom,
    ! [A: $tType,Xs2: list @ A,X2: A,Ys: list @ A,Y2: A] :
      ( ( list_update @ A @ ( append @ A @ Xs2 @ ( cons @ A @ X2 @ Ys ) ) @ ( size_size @ ( list @ A ) @ Xs2 ) @ Y2 )
      = ( append @ A @ Xs2 @ ( cons @ A @ Y2 @ Ys ) ) ) ).

% list_update_length
thf(fact_6321_list__assn__aux__append2,axiom,
    ! [A: $tType,B: $tType,L23: list @ A,L24: list @ B,P: A > B > assn,L12: list @ A,L13: list @ B] :
      ( ( ( size_size @ ( list @ A ) @ L23 )
        = ( size_size @ ( list @ B ) @ L24 ) )
     => ( ( vEBT_List_list_assn @ A @ B @ P @ ( append @ A @ L12 @ L23 ) @ ( append @ B @ L13 @ L24 ) )
        = ( times_times @ assn @ ( vEBT_List_list_assn @ A @ B @ P @ L12 @ L13 ) @ ( vEBT_List_list_assn @ A @ B @ P @ L23 @ L24 ) ) ) ) ).

% list_assn_aux_append2
thf(fact_6322_list__assn__aux__append,axiom,
    ! [A: $tType,B: $tType,L12: list @ A,L13: list @ B,P: A > B > assn,L23: list @ A,L24: list @ B] :
      ( ( ( size_size @ ( list @ A ) @ L12 )
        = ( size_size @ ( list @ B ) @ L13 ) )
     => ( ( vEBT_List_list_assn @ A @ B @ P @ ( append @ A @ L12 @ L23 ) @ ( append @ B @ L13 @ L24 ) )
        = ( times_times @ assn @ ( vEBT_List_list_assn @ A @ B @ P @ L12 @ L13 ) @ ( vEBT_List_list_assn @ A @ B @ P @ L23 @ L24 ) ) ) ) ).

% list_assn_aux_append
thf(fact_6323_neq__Nil__revE,axiom,
    ! [A: $tType,L: list @ A] :
      ( ( L
       != ( nil @ A ) )
     => ~ ! [Ll2: list @ A,E2: A] :
            ( L
           != ( append @ A @ Ll2 @ ( cons @ A @ E2 @ ( nil @ A ) ) ) ) ) ).

% neq_Nil_revE
thf(fact_6324_rev__induct2_H,axiom,
    ! [A: $tType,B: $tType,P: ( list @ A ) > ( list @ B ) > $o,Xs2: list @ A,Ys: list @ B] :
      ( ( P @ ( nil @ A ) @ ( nil @ B ) )
     => ( ! [X3: A,Xs3: list @ A] : ( P @ ( append @ A @ Xs3 @ ( cons @ A @ X3 @ ( nil @ A ) ) ) @ ( nil @ B ) )
       => ( ! [Y4: B,Ys4: list @ B] : ( P @ ( nil @ A ) @ ( append @ B @ Ys4 @ ( cons @ B @ Y4 @ ( nil @ B ) ) ) )
         => ( ! [X3: A,Xs3: list @ A,Y4: B,Ys4: list @ B] :
                ( ( P @ Xs3 @ Ys4 )
               => ( P @ ( append @ A @ Xs3 @ ( cons @ A @ X3 @ ( nil @ A ) ) ) @ ( append @ B @ Ys4 @ ( cons @ B @ Y4 @ ( nil @ B ) ) ) ) )
           => ( P @ Xs2 @ Ys ) ) ) ) ) ).

% rev_induct2'
thf(fact_6325_neq__Nil__rev__conv,axiom,
    ! [A: $tType,L: list @ A] :
      ( ( L
       != ( nil @ A ) )
      = ( ? [Xs: list @ A,X: A] :
            ( L
            = ( append @ A @ Xs @ ( cons @ A @ X @ ( nil @ A ) ) ) ) ) ) ).

% neq_Nil_rev_conv
thf(fact_6326_rev__nonempty__induct2_H,axiom,
    ! [A: $tType,B: $tType,Xs2: list @ A,Ys: list @ B,P: ( list @ A ) > ( list @ B ) > $o] :
      ( ( Xs2
       != ( nil @ A ) )
     => ( ( Ys
         != ( nil @ B ) )
       => ( ! [X3: A,Y4: B] : ( P @ ( cons @ A @ X3 @ ( nil @ A ) ) @ ( cons @ B @ Y4 @ ( nil @ B ) ) )
         => ( ! [X3: A,Xs3: list @ A,Y4: B] :
                ( ( Xs3
                 != ( nil @ A ) )
               => ( P @ ( append @ A @ Xs3 @ ( cons @ A @ X3 @ ( nil @ A ) ) ) @ ( cons @ B @ Y4 @ ( nil @ B ) ) ) )
           => ( ! [X3: A,Y4: B,Ys4: list @ B] :
                  ( ( Ys4
                   != ( nil @ B ) )
                 => ( P @ ( cons @ A @ X3 @ ( nil @ A ) ) @ ( append @ B @ Ys4 @ ( cons @ B @ Y4 @ ( nil @ B ) ) ) ) )
             => ( ! [X3: A,Xs3: list @ A,Y4: B,Ys4: list @ B] :
                    ( ( P @ Xs3 @ Ys4 )
                   => ( ( Xs3
                       != ( nil @ A ) )
                     => ( ( Ys4
                         != ( nil @ B ) )
                       => ( P @ ( append @ A @ Xs3 @ ( cons @ A @ X3 @ ( nil @ A ) ) ) @ ( append @ B @ Ys4 @ ( cons @ B @ Y4 @ ( nil @ B ) ) ) ) ) ) )
               => ( P @ Xs2 @ Ys ) ) ) ) ) ) ) ).

% rev_nonempty_induct2'
thf(fact_6327_list__Cons__eq__append__cases,axiom,
    ! [A: $tType,X2: A,Xs2: list @ A,Ys: list @ A,Zs: list @ A] :
      ( ( ( cons @ A @ X2 @ Xs2 )
        = ( append @ A @ Ys @ Zs ) )
     => ( ( ( Ys
            = ( nil @ A ) )
         => ( Zs
           != ( cons @ A @ X2 @ Xs2 ) ) )
       => ~ ! [Ys5: list @ A] :
              ( ( Ys
                = ( cons @ A @ X2 @ Ys5 ) )
             => ( ( append @ A @ Ys5 @ Zs )
               != Xs2 ) ) ) ) ).

% list_Cons_eq_append_cases
thf(fact_6328_list__append__eq__Cons__cases,axiom,
    ! [A: $tType,Ys: list @ A,Zs: list @ A,X2: A,Xs2: list @ A] :
      ( ( ( append @ A @ Ys @ Zs )
        = ( cons @ A @ X2 @ Xs2 ) )
     => ( ( ( Ys
            = ( nil @ A ) )
         => ( Zs
           != ( cons @ A @ X2 @ Xs2 ) ) )
       => ~ ! [Ys5: list @ A] :
              ( ( Ys
                = ( cons @ A @ X2 @ Ys5 ) )
             => ( ( append @ A @ Ys5 @ Zs )
               != Xs2 ) ) ) ) ).

% list_append_eq_Cons_cases
thf(fact_6329_list__match__lel__lel,axiom,
    ! [A: $tType,C1: list @ A,Qs: A,C22: list @ A,C12: list @ A,Qs2: A,C23: list @ A] :
      ( ( ( append @ A @ C1 @ ( cons @ A @ Qs @ C22 ) )
        = ( append @ A @ C12 @ ( cons @ A @ Qs2 @ C23 ) ) )
     => ( ! [C21: list @ A] :
            ( ( C1
              = ( append @ A @ C12 @ ( cons @ A @ Qs2 @ C21 ) ) )
           => ( C23
             != ( append @ A @ C21 @ ( cons @ A @ Qs @ C22 ) ) ) )
       => ( ( ( C12 = C1 )
           => ( ( Qs2 = Qs )
             => ( C23 != C22 ) ) )
         => ~ ! [C212: list @ A] :
                ( ( C12
                  = ( append @ A @ C1 @ ( cons @ A @ Qs @ C212 ) ) )
               => ( C22
                 != ( append @ A @ C212 @ ( cons @ A @ Qs2 @ C23 ) ) ) ) ) ) ) ).

% list_match_lel_lel
thf(fact_6330_split__list__first__prop__iff,axiom,
    ! [A: $tType,Xs2: list @ A,P: A > $o] :
      ( ( ? [X: A] :
            ( ( member @ A @ X @ ( set2 @ A @ Xs2 ) )
            & ( P @ X ) ) )
      = ( ? [Ys3: list @ A,X: A] :
            ( ? [Zs3: list @ A] :
                ( Xs2
                = ( append @ A @ Ys3 @ ( cons @ A @ X @ Zs3 ) ) )
            & ( P @ X )
            & ! [Y: A] :
                ( ( member @ A @ Y @ ( set2 @ A @ Ys3 ) )
               => ~ ( P @ Y ) ) ) ) ) ).

% split_list_first_prop_iff
thf(fact_6331_split__list__last__prop__iff,axiom,
    ! [A: $tType,Xs2: list @ A,P: A > $o] :
      ( ( ? [X: A] :
            ( ( member @ A @ X @ ( set2 @ A @ Xs2 ) )
            & ( P @ X ) ) )
      = ( ? [Ys3: list @ A,X: A,Zs3: list @ A] :
            ( ( Xs2
              = ( append @ A @ Ys3 @ ( cons @ A @ X @ Zs3 ) ) )
            & ( P @ X )
            & ! [Y: A] :
                ( ( member @ A @ Y @ ( set2 @ A @ Zs3 ) )
               => ~ ( P @ Y ) ) ) ) ) ).

% split_list_last_prop_iff
thf(fact_6332_in__set__conv__decomp__first,axiom,
    ! [A: $tType,X2: A,Xs2: list @ A] :
      ( ( member @ A @ X2 @ ( set2 @ A @ Xs2 ) )
      = ( ? [Ys3: list @ A,Zs3: list @ A] :
            ( ( Xs2
              = ( append @ A @ Ys3 @ ( cons @ A @ X2 @ Zs3 ) ) )
            & ~ ( member @ A @ X2 @ ( set2 @ A @ Ys3 ) ) ) ) ) ).

% in_set_conv_decomp_first
thf(fact_6333_in__set__conv__decomp__last,axiom,
    ! [A: $tType,X2: A,Xs2: list @ A] :
      ( ( member @ A @ X2 @ ( set2 @ A @ Xs2 ) )
      = ( ? [Ys3: list @ A,Zs3: list @ A] :
            ( ( Xs2
              = ( append @ A @ Ys3 @ ( cons @ A @ X2 @ Zs3 ) ) )
            & ~ ( member @ A @ X2 @ ( set2 @ A @ Zs3 ) ) ) ) ) ).

% in_set_conv_decomp_last
thf(fact_6334_split__list__first__propE,axiom,
    ! [A: $tType,Xs2: list @ A,P: A > $o] :
      ( ? [X6: A] :
          ( ( member @ A @ X6 @ ( set2 @ A @ Xs2 ) )
          & ( P @ X6 ) )
     => ~ ! [Ys4: list @ A,X3: A] :
            ( ? [Zs2: list @ A] :
                ( Xs2
                = ( append @ A @ Ys4 @ ( cons @ A @ X3 @ Zs2 ) ) )
           => ( ( P @ X3 )
             => ~ ! [Xa2: A] :
                    ( ( member @ A @ Xa2 @ ( set2 @ A @ Ys4 ) )
                   => ~ ( P @ Xa2 ) ) ) ) ) ).

% split_list_first_propE
thf(fact_6335_split__list__last__propE,axiom,
    ! [A: $tType,Xs2: list @ A,P: A > $o] :
      ( ? [X6: A] :
          ( ( member @ A @ X6 @ ( set2 @ A @ Xs2 ) )
          & ( P @ X6 ) )
     => ~ ! [Ys4: list @ A,X3: A,Zs2: list @ A] :
            ( ( Xs2
              = ( append @ A @ Ys4 @ ( cons @ A @ X3 @ Zs2 ) ) )
           => ( ( P @ X3 )
             => ~ ! [Xa2: A] :
                    ( ( member @ A @ Xa2 @ ( set2 @ A @ Zs2 ) )
                   => ~ ( P @ Xa2 ) ) ) ) ) ).

% split_list_last_propE
thf(fact_6336_split__list__first__prop,axiom,
    ! [A: $tType,Xs2: list @ A,P: A > $o] :
      ( ? [X6: A] :
          ( ( member @ A @ X6 @ ( set2 @ A @ Xs2 ) )
          & ( P @ X6 ) )
     => ? [Ys4: list @ A,X3: A] :
          ( ? [Zs2: list @ A] :
              ( Xs2
              = ( append @ A @ Ys4 @ ( cons @ A @ X3 @ Zs2 ) ) )
          & ( P @ X3 )
          & ! [Xa2: A] :
              ( ( member @ A @ Xa2 @ ( set2 @ A @ Ys4 ) )
             => ~ ( P @ Xa2 ) ) ) ) ).

% split_list_first_prop
thf(fact_6337_split__list__last__prop,axiom,
    ! [A: $tType,Xs2: list @ A,P: A > $o] :
      ( ? [X6: A] :
          ( ( member @ A @ X6 @ ( set2 @ A @ Xs2 ) )
          & ( P @ X6 ) )
     => ? [Ys4: list @ A,X3: A,Zs2: list @ A] :
          ( ( Xs2
            = ( append @ A @ Ys4 @ ( cons @ A @ X3 @ Zs2 ) ) )
          & ( P @ X3 )
          & ! [Xa2: A] :
              ( ( member @ A @ Xa2 @ ( set2 @ A @ Zs2 ) )
             => ~ ( P @ Xa2 ) ) ) ) ).

% split_list_last_prop
thf(fact_6338_in__set__conv__decomp,axiom,
    ! [A: $tType,X2: A,Xs2: list @ A] :
      ( ( member @ A @ X2 @ ( set2 @ A @ Xs2 ) )
      = ( ? [Ys3: list @ A,Zs3: list @ A] :
            ( Xs2
            = ( append @ A @ Ys3 @ ( cons @ A @ X2 @ Zs3 ) ) ) ) ) ).

% in_set_conv_decomp
thf(fact_6339_append__Cons__eq__iff,axiom,
    ! [A: $tType,X2: A,Xs2: list @ A,Ys: list @ A,Xs4: list @ A,Ys6: list @ A] :
      ( ~ ( member @ A @ X2 @ ( set2 @ A @ Xs2 ) )
     => ( ~ ( member @ A @ X2 @ ( set2 @ A @ Ys ) )
       => ( ( ( append @ A @ Xs2 @ ( cons @ A @ X2 @ Ys ) )
            = ( append @ A @ Xs4 @ ( cons @ A @ X2 @ Ys6 ) ) )
          = ( ( Xs2 = Xs4 )
            & ( Ys = Ys6 ) ) ) ) ) ).

% append_Cons_eq_iff
thf(fact_6340_split__list__propE,axiom,
    ! [A: $tType,Xs2: list @ A,P: A > $o] :
      ( ? [X6: A] :
          ( ( member @ A @ X6 @ ( set2 @ A @ Xs2 ) )
          & ( P @ X6 ) )
     => ~ ! [Ys4: list @ A,X3: A] :
            ( ? [Zs2: list @ A] :
                ( Xs2
                = ( append @ A @ Ys4 @ ( cons @ A @ X3 @ Zs2 ) ) )
           => ~ ( P @ X3 ) ) ) ).

% split_list_propE
thf(fact_6341_split__list__first,axiom,
    ! [A: $tType,X2: A,Xs2: list @ A] :
      ( ( member @ A @ X2 @ ( set2 @ A @ Xs2 ) )
     => ? [Ys4: list @ A,Zs2: list @ A] :
          ( ( Xs2
            = ( append @ A @ Ys4 @ ( cons @ A @ X2 @ Zs2 ) ) )
          & ~ ( member @ A @ X2 @ ( set2 @ A @ Ys4 ) ) ) ) ).

% split_list_first
thf(fact_6342_split__list__prop,axiom,
    ! [A: $tType,Xs2: list @ A,P: A > $o] :
      ( ? [X6: A] :
          ( ( member @ A @ X6 @ ( set2 @ A @ Xs2 ) )
          & ( P @ X6 ) )
     => ? [Ys4: list @ A,X3: A] :
          ( ? [Zs2: list @ A] :
              ( Xs2
              = ( append @ A @ Ys4 @ ( cons @ A @ X3 @ Zs2 ) ) )
          & ( P @ X3 ) ) ) ).

% split_list_prop
thf(fact_6343_split__list__last,axiom,
    ! [A: $tType,X2: A,Xs2: list @ A] :
      ( ( member @ A @ X2 @ ( set2 @ A @ Xs2 ) )
     => ? [Ys4: list @ A,Zs2: list @ A] :
          ( ( Xs2
            = ( append @ A @ Ys4 @ ( cons @ A @ X2 @ Zs2 ) ) )
          & ~ ( member @ A @ X2 @ ( set2 @ A @ Zs2 ) ) ) ) ).

% split_list_last
thf(fact_6344_split__list,axiom,
    ! [A: $tType,X2: A,Xs2: list @ A] :
      ( ( member @ A @ X2 @ ( set2 @ A @ Xs2 ) )
     => ? [Ys4: list @ A,Zs2: list @ A] :
          ( Xs2
          = ( append @ A @ Ys4 @ ( cons @ A @ X2 @ Zs2 ) ) ) ) ).

% split_list
thf(fact_6345_xy__in__set__cases,axiom,
    ! [A: $tType,X2: A,L: list @ A,Y2: A] :
      ( ( member @ A @ X2 @ ( set2 @ A @ L ) )
     => ( ( member @ A @ Y2 @ ( set2 @ A @ L ) )
       => ( ( ( X2 = Y2 )
           => ! [L1: list @ A,L22: list @ A] :
                ( L
               != ( append @ A @ L1 @ ( cons @ A @ Y2 @ L22 ) ) ) )
         => ( ( ( X2 != Y2 )
             => ! [L1: list @ A,L22: list @ A,L32: list @ A] :
                  ( L
                 != ( append @ A @ L1 @ ( cons @ A @ X2 @ ( append @ A @ L22 @ ( cons @ A @ Y2 @ L32 ) ) ) ) ) )
           => ~ ( ( X2 != Y2 )
               => ! [L1: list @ A,L22: list @ A,L32: list @ A] :
                    ( L
                   != ( append @ A @ L1 @ ( cons @ A @ Y2 @ ( append @ A @ L22 @ ( cons @ A @ X2 @ L32 ) ) ) ) ) ) ) ) ) ) ).

% xy_in_set_cases
thf(fact_6346_in__set__list__format,axiom,
    ! [A: $tType,E3: A,L: list @ A] :
      ( ( member @ A @ E3 @ ( set2 @ A @ L ) )
     => ~ ! [L1: list @ A,L22: list @ A] :
            ( L
           != ( append @ A @ L1 @ ( cons @ A @ E3 @ L22 ) ) ) ) ).

% in_set_list_format
thf(fact_6347_map__consI_I2_J,axiom,
    ! [B: $tType,A: $tType,W: list @ A,L: list @ A,F: B > A,Ww: list @ B,A2: B] :
      ( ( ( append @ A @ W @ L )
        = ( append @ A @ ( map @ B @ A @ F @ Ww ) @ L ) )
     => ( ( cons @ A @ ( F @ A2 ) @ ( append @ A @ W @ L ) )
        = ( append @ A @ ( map @ B @ A @ F @ ( cons @ B @ A2 @ Ww ) ) @ L ) ) ) ).

% map_consI(2)
thf(fact_6348_replicate__add,axiom,
    ! [A: $tType,N2: nat,M: nat,X2: A] :
      ( ( replicate @ A @ ( plus_plus @ nat @ N2 @ M ) @ X2 )
      = ( append @ A @ ( replicate @ A @ N2 @ X2 ) @ ( replicate @ A @ M @ X2 ) ) ) ).

% replicate_add
thf(fact_6349_enumerate__append__eq,axiom,
    ! [A: $tType,N2: nat,Xs2: list @ A,Ys: list @ A] :
      ( ( enumerate @ A @ N2 @ ( append @ A @ Xs2 @ Ys ) )
      = ( append @ ( product_prod @ nat @ A ) @ ( enumerate @ A @ N2 @ Xs2 ) @ ( enumerate @ A @ ( plus_plus @ nat @ N2 @ ( size_size @ ( list @ A ) @ Xs2 ) ) @ Ys ) ) ) ).

% enumerate_append_eq
thf(fact_6350_list__rest__coinc,axiom,
    ! [A: $tType,S22: list @ A,S1: list @ A,R12: list @ A,R23: list @ A] :
      ( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ S22 ) @ ( size_size @ ( list @ A ) @ S1 ) )
     => ( ( ( append @ A @ S1 @ R12 )
          = ( append @ A @ S22 @ R23 ) )
       => ? [R1p: list @ A] :
            ( R23
            = ( append @ A @ R1p @ R12 ) ) ) ) ).

% list_rest_coinc
thf(fact_6351_append__eq__mapE,axiom,
    ! [B: $tType,A: $tType,Fl: list @ A,Fl2: list @ A,F: B > A,Ls: list @ B] :
      ( ( ( append @ A @ Fl @ Fl2 )
        = ( map @ B @ A @ F @ Ls ) )
     => ~ ! [L3: list @ B,L6: list @ B] :
            ( ( Ls
              = ( append @ B @ L3 @ L6 ) )
           => ( ( ( map @ B @ A @ F @ L3 )
                = Fl )
             => ( ( map @ B @ A @ F @ L6 )
               != Fl2 ) ) ) ) ).

% append_eq_mapE
thf(fact_6352_map__eq__appendE,axiom,
    ! [B: $tType,A: $tType,F: B > A,Ls: list @ B,Fl: list @ A,Fl2: list @ A] :
      ( ( ( map @ B @ A @ F @ Ls )
        = ( append @ A @ Fl @ Fl2 ) )
     => ~ ! [L3: list @ B,L6: list @ B] :
            ( ( Ls
              = ( append @ B @ L3 @ L6 ) )
           => ( ( ( map @ B @ A @ F @ L3 )
                = Fl )
             => ( ( map @ B @ A @ F @ L6 )
               != Fl2 ) ) ) ) ).

% map_eq_appendE
thf(fact_6353_Misc_Oappend__eq__map__conv,axiom,
    ! [A: $tType,B: $tType,Fl: list @ A,Fl2: list @ A,F: B > A,Ls: list @ B] :
      ( ( ( append @ A @ Fl @ Fl2 )
        = ( map @ B @ A @ F @ Ls ) )
      = ( ? [L2: list @ B,L7: list @ B] :
            ( ( Ls
              = ( append @ B @ L2 @ L7 ) )
            & ( ( map @ B @ A @ F @ L2 )
              = Fl )
            & ( ( map @ B @ A @ F @ L7 )
              = Fl2 ) ) ) ) ).

% Misc.append_eq_map_conv
thf(fact_6354_Misc_Omap__eq__append__conv,axiom,
    ! [A: $tType,B: $tType,F: B > A,Ls: list @ B,Fl: list @ A,Fl2: list @ A] :
      ( ( ( map @ B @ A @ F @ Ls )
        = ( append @ A @ Fl @ Fl2 ) )
      = ( ? [L2: list @ B,L7: list @ B] :
            ( ( Ls
              = ( append @ B @ L2 @ L7 ) )
            & ( ( map @ B @ A @ F @ L2 )
              = Fl )
            & ( ( map @ B @ A @ F @ L7 )
              = Fl2 ) ) ) ) ).

% Misc.map_eq_append_conv
thf(fact_6355_mergesort__by__rel__split_Osimps_I3_J,axiom,
    ! [A: $tType,Xs12: list @ A,Xs23: list @ A,X1: A,X23: A,Xs2: list @ A] :
      ( ( merges295452479951948502_split @ A @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs12 @ Xs23 ) @ ( cons @ A @ X1 @ ( cons @ A @ X23 @ Xs2 ) ) )
      = ( merges295452479951948502_split @ A @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X1 @ Xs12 ) @ ( cons @ A @ X23 @ Xs23 ) ) @ Xs2 ) ) ).

% mergesort_by_rel_split.simps(3)
thf(fact_6356_mergesort__by__rel__split_Osimps_I1_J,axiom,
    ! [A: $tType,Xs12: list @ A,Xs23: list @ A] :
      ( ( merges295452479951948502_split @ A @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs12 @ Xs23 ) @ ( nil @ A ) )
      = ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs12 @ Xs23 ) ) ).

% mergesort_by_rel_split.simps(1)
thf(fact_6357_same__length__different,axiom,
    ! [A: $tType,Xs2: list @ A,Ys: list @ A] :
      ( ( Xs2 != Ys )
     => ( ( ( size_size @ ( list @ A ) @ Xs2 )
          = ( size_size @ ( list @ A ) @ Ys ) )
       => ? [Pre: list @ A,X3: A,Xs5: list @ A,Y4: A,Ys5: list @ A] :
            ( ( X3 != Y4 )
            & ( Xs2
              = ( append @ A @ Pre @ ( append @ A @ ( cons @ A @ X3 @ ( nil @ A ) ) @ Xs5 ) ) )
            & ( Ys
              = ( append @ A @ Pre @ ( append @ A @ ( cons @ A @ Y4 @ ( nil @ A ) ) @ Ys5 ) ) ) ) ) ) ).

% same_length_different
thf(fact_6358_list__update__append1,axiom,
    ! [A: $tType,I: nat,Xs2: list @ A,Ys: list @ A,X2: A] :
      ( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs2 ) )
     => ( ( list_update @ A @ ( append @ A @ Xs2 @ Ys ) @ I @ X2 )
        = ( append @ A @ ( list_update @ A @ Xs2 @ I @ X2 ) @ Ys ) ) ) ).

% list_update_append1
thf(fact_6359_length__Suc__conv__rev,axiom,
    ! [A: $tType,Xs2: list @ A,N2: nat] :
      ( ( ( size_size @ ( list @ A ) @ Xs2 )
        = ( suc @ N2 ) )
      = ( ? [Y: A,Ys3: list @ A] :
            ( ( Xs2
              = ( append @ A @ Ys3 @ ( cons @ A @ Y @ ( nil @ A ) ) ) )
            & ( ( size_size @ ( list @ A ) @ Ys3 )
              = N2 ) ) ) ) ).

% length_Suc_conv_rev
thf(fact_6360_length__Suc__rev__conv,axiom,
    ! [A: $tType,Xs2: list @ A,N2: nat] :
      ( ( ( size_size @ ( list @ A ) @ Xs2 )
        = ( suc @ N2 ) )
      = ( ? [Ys3: list @ A,Y: A] :
            ( ( Xs2
              = ( append @ A @ Ys3 @ ( cons @ A @ Y @ ( nil @ A ) ) ) )
            & ( ( size_size @ ( list @ A ) @ Ys3 )
              = N2 ) ) ) ) ).

% length_Suc_rev_conv
thf(fact_6361_length__append__singleton,axiom,
    ! [A: $tType,Xs2: list @ A,X2: A] :
      ( ( size_size @ ( list @ A ) @ ( append @ A @ Xs2 @ ( cons @ A @ X2 @ ( nil @ A ) ) ) )
      = ( suc @ ( size_size @ ( list @ A ) @ Xs2 ) ) ) ).

% length_append_singleton
thf(fact_6362_length__compl__rev__induct,axiom,
    ! [A: $tType,P: ( list @ A ) > $o,L: list @ A] :
      ( ( P @ ( nil @ A ) )
     => ( ! [L3: list @ A,E2: A] :
            ( ! [Ll: list @ A] :
                ( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Ll ) @ ( size_size @ ( list @ A ) @ L3 ) )
               => ( P @ Ll ) )
           => ( P @ ( append @ A @ L3 @ ( cons @ A @ E2 @ ( nil @ A ) ) ) ) )
       => ( P @ L ) ) ) ).

% length_compl_rev_induct
thf(fact_6363_nth__append,axiom,
    ! [A: $tType,N2: nat,Xs2: list @ A,Ys: list @ A] :
      ( ( ( ord_less @ nat @ N2 @ ( size_size @ ( list @ A ) @ Xs2 ) )
       => ( ( nth @ A @ ( append @ A @ Xs2 @ Ys ) @ N2 )
          = ( nth @ A @ Xs2 @ N2 ) ) )
      & ( ~ ( ord_less @ nat @ N2 @ ( size_size @ ( list @ A ) @ Xs2 ) )
       => ( ( nth @ A @ ( append @ A @ Xs2 @ Ys ) @ N2 )
          = ( nth @ A @ Ys @ ( minus_minus @ nat @ N2 @ ( size_size @ ( list @ A ) @ Xs2 ) ) ) ) ) ) ).

% nth_append
thf(fact_6364_replicate__Suc__conv__snoc,axiom,
    ! [A: $tType,N2: nat,X2: A] :
      ( ( replicate @ A @ ( suc @ N2 ) @ X2 )
      = ( append @ A @ ( replicate @ A @ N2 @ X2 ) @ ( cons @ A @ X2 @ ( nil @ A ) ) ) ) ).

% replicate_Suc_conv_snoc
thf(fact_6365_mergesort__by__rel__split_Oelims,axiom,
    ! [A: $tType,X2: product_prod @ ( list @ A ) @ ( list @ A ),Xa: list @ A,Y2: product_prod @ ( list @ A ) @ ( list @ A )] :
      ( ( ( merges295452479951948502_split @ A @ X2 @ Xa )
        = Y2 )
     => ( ! [Xs1: list @ A,Xs22: list @ A] :
            ( ( X2
              = ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs1 @ Xs22 ) )
           => ( ( Xa
                = ( nil @ A ) )
             => ( Y2
               != ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs1 @ Xs22 ) ) ) )
       => ( ! [Xs1: list @ A,Xs22: list @ A] :
              ( ( X2
                = ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs1 @ Xs22 ) )
             => ! [X3: A] :
                  ( ( Xa
                    = ( cons @ A @ X3 @ ( nil @ A ) ) )
                 => ( Y2
                   != ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X3 @ Xs1 ) @ Xs22 ) ) ) )
         => ~ ! [Xs1: list @ A,Xs22: list @ A] :
                ( ( X2
                  = ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs1 @ Xs22 ) )
               => ! [X16: A,X24: A,Xs3: list @ A] :
                    ( ( Xa
                      = ( cons @ A @ X16 @ ( cons @ A @ X24 @ Xs3 ) ) )
                   => ( Y2
                     != ( merges295452479951948502_split @ A @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X16 @ Xs1 ) @ ( cons @ A @ X24 @ Xs22 ) ) @ Xs3 ) ) ) ) ) ) ) ).

% mergesort_by_rel_split.elims
thf(fact_6366_mergesort__by__rel__split_Osimps_I2_J,axiom,
    ! [A: $tType,Xs12: list @ A,Xs23: list @ A,X2: A] :
      ( ( merges295452479951948502_split @ A @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs12 @ Xs23 ) @ ( cons @ A @ X2 @ ( nil @ A ) ) )
      = ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X2 @ Xs12 ) @ Xs23 ) ) ).

% mergesort_by_rel_split.simps(2)
thf(fact_6367_list__update__append,axiom,
    ! [A: $tType,N2: nat,Xs2: list @ A,Ys: list @ A,X2: A] :
      ( ( ( ord_less @ nat @ N2 @ ( size_size @ ( list @ A ) @ Xs2 ) )
       => ( ( list_update @ A @ ( append @ A @ Xs2 @ Ys ) @ N2 @ X2 )
          = ( append @ A @ ( list_update @ A @ Xs2 @ N2 @ X2 ) @ Ys ) ) )
      & ( ~ ( ord_less @ nat @ N2 @ ( size_size @ ( list @ A ) @ Xs2 ) )
       => ( ( list_update @ A @ ( append @ A @ Xs2 @ Ys ) @ N2 @ X2 )
          = ( append @ A @ Xs2 @ ( list_update @ A @ Ys @ ( minus_minus @ nat @ N2 @ ( size_size @ ( list @ A ) @ Xs2 ) ) @ X2 ) ) ) ) ) ).

% list_update_append
thf(fact_6368_product_Osimps_I2_J,axiom,
    ! [A: $tType,B: $tType,X2: A,Xs2: list @ A,Ys: list @ B] :
      ( ( product @ A @ B @ ( cons @ A @ X2 @ Xs2 ) @ Ys )
      = ( append @ ( product_prod @ A @ B ) @ ( map @ B @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 ) @ Ys ) @ ( product @ A @ B @ Xs2 @ Ys ) ) ) ).

% product.simps(2)
thf(fact_6369_slice__prepend,axiom,
    ! [A: $tType,I: nat,K: nat,Xs2: list @ A,Ys: list @ A] :
      ( ( ord_less_eq @ nat @ I @ K )
     => ( ( ord_less_eq @ nat @ K @ ( size_size @ ( list @ A ) @ Xs2 ) )
       => ( ( slice @ A @ I @ K @ Xs2 )
          = ( slice @ A @ ( plus_plus @ nat @ I @ ( size_size @ ( list @ A ) @ Ys ) ) @ ( plus_plus @ nat @ K @ ( size_size @ ( list @ A ) @ Ys ) ) @ ( append @ A @ Ys @ Xs2 ) ) ) ) ) ).

% slice_prepend
thf(fact_6370_mergesort__by__rel_Opinduct,axiom,
    ! [A: $tType,A0: A > A > $o,A12: list @ A,P: ( A > A > $o ) > ( list @ A ) > $o] :
      ( ( accp @ ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) @ ( mergesort_by_rel_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ A0 @ A12 ) )
     => ( ! [R8: A > A > $o,Xs3: list @ A] :
            ( ( accp @ ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) @ ( mergesort_by_rel_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ R8 @ Xs3 ) )
           => ( ( ~ ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Xs3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
               => ( P @ R8 @ ( product_fst @ ( list @ A ) @ ( list @ A ) @ ( merges295452479951948502_split @ A @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ ( nil @ A ) ) @ Xs3 ) ) ) )
             => ( ( ~ ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Xs3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
                 => ( P @ R8 @ ( product_snd @ ( list @ A ) @ ( list @ A ) @ ( merges295452479951948502_split @ A @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ ( nil @ A ) ) @ Xs3 ) ) ) )
               => ( P @ R8 @ Xs3 ) ) ) )
       => ( P @ A0 @ A12 ) ) ) ).

% mergesort_by_rel.pinduct
thf(fact_6371_quickcheck__narrowing__samples_Onarrowing__samples_Opelims,axiom,
    ! [A: $tType] :
      ( ( ( code_term_of @ A )
        & ( quickc6926020345158392990erm_of @ A ) )
     => ! [A_of_integer2: code_integer > ( product_prod @ A @ A ),Zero2: A,X2: code_integer,Y2: list @ A] :
          ( ( ( code_T4080844693773952564amples @ A @ A_of_integer2 @ Zero2 @ X2 )
            = Y2 )
         => ( ( accp @ code_integer @ ( code_T1710151556404007877es_rel @ A @ A_of_integer2 ) @ X2 )
           => ~ ( ( ( ( ord_less @ code_integer @ ( zero_zero @ code_integer ) @ X2 )
                   => ( Y2
                      = ( product_case_prod @ A @ A @ ( list @ A )
                        @ ^ [A5: A,A11: A] : ( append @ A @ ( code_T4080844693773952564amples @ A @ A_of_integer2 @ Zero2 @ ( minus_minus @ code_integer @ X2 @ ( one_one @ code_integer ) ) ) @ ( cons @ A @ A5 @ ( cons @ A @ A11 @ ( nil @ A ) ) ) )
                        @ ( A_of_integer2 @ X2 ) ) ) )
                  & ( ~ ( ord_less @ code_integer @ ( zero_zero @ code_integer ) @ X2 )
                   => ( Y2
                      = ( cons @ A @ Zero2 @ ( nil @ A ) ) ) ) )
               => ~ ( accp @ code_integer @ ( code_T1710151556404007877es_rel @ A @ A_of_integer2 ) @ X2 ) ) ) ) ) ).

% quickcheck_narrowing_samples.narrowing_samples.pelims
thf(fact_6372_quickcheck__narrowing__samples_Onarrowing__samples_Osimps,axiom,
    ! [A: $tType] :
      ( ( ( code_term_of @ A )
        & ( quickc6926020345158392990erm_of @ A ) )
     => ( ( code_T4080844693773952564amples @ A )
        = ( ^ [A_of_integer: code_integer > ( product_prod @ A @ A ),Zero: A,I2: code_integer] :
              ( if @ ( list @ A ) @ ( ord_less @ code_integer @ ( zero_zero @ code_integer ) @ I2 )
              @ ( product_case_prod @ A @ A @ ( list @ A )
                @ ^ [A5: A,A11: A] : ( append @ A @ ( code_T4080844693773952564amples @ A @ A_of_integer @ Zero @ ( minus_minus @ code_integer @ I2 @ ( one_one @ code_integer ) ) ) @ ( cons @ A @ A5 @ ( cons @ A @ A11 @ ( nil @ A ) ) ) )
                @ ( A_of_integer @ I2 ) )
              @ ( cons @ A @ Zero @ ( nil @ A ) ) ) ) ) ) ).

% quickcheck_narrowing_samples.narrowing_samples.simps
thf(fact_6373_quickcheck__narrowing__samples_Onarrowing__samples_Oelims,axiom,
    ! [A: $tType] :
      ( ( ( code_term_of @ A )
        & ( quickc6926020345158392990erm_of @ A ) )
     => ! [A_of_integer2: code_integer > ( product_prod @ A @ A ),Zero2: A,X2: code_integer,Y2: list @ A] :
          ( ( ( code_T4080844693773952564amples @ A @ A_of_integer2 @ Zero2 @ X2 )
            = Y2 )
         => ( ( ( ord_less @ code_integer @ ( zero_zero @ code_integer ) @ X2 )
             => ( Y2
                = ( product_case_prod @ A @ A @ ( list @ A )
                  @ ^ [A5: A,A11: A] : ( append @ A @ ( code_T4080844693773952564amples @ A @ A_of_integer2 @ Zero2 @ ( minus_minus @ code_integer @ X2 @ ( one_one @ code_integer ) ) ) @ ( cons @ A @ A5 @ ( cons @ A @ A11 @ ( nil @ A ) ) ) )
                  @ ( A_of_integer2 @ X2 ) ) ) )
            & ( ~ ( ord_less @ code_integer @ ( zero_zero @ code_integer ) @ X2 )
             => ( Y2
                = ( cons @ A @ Zero2 @ ( nil @ A ) ) ) ) ) ) ) ).

% quickcheck_narrowing_samples.narrowing_samples.elims
thf(fact_6374_quickcheck__narrowing__samples_Onarrowing__samples_Opsimps,axiom,
    ! [A: $tType] :
      ( ( ( code_term_of @ A )
        & ( quickc6926020345158392990erm_of @ A ) )
     => ! [A_of_integer2: code_integer > ( product_prod @ A @ A ),I: code_integer,Zero2: A] :
          ( ( accp @ code_integer @ ( code_T1710151556404007877es_rel @ A @ A_of_integer2 ) @ I )
         => ( ( ( ord_less @ code_integer @ ( zero_zero @ code_integer ) @ I )
             => ( ( code_T4080844693773952564amples @ A @ A_of_integer2 @ Zero2 @ I )
                = ( product_case_prod @ A @ A @ ( list @ A )
                  @ ^ [A5: A,A11: A] : ( append @ A @ ( code_T4080844693773952564amples @ A @ A_of_integer2 @ Zero2 @ ( minus_minus @ code_integer @ I @ ( one_one @ code_integer ) ) ) @ ( cons @ A @ A5 @ ( cons @ A @ A11 @ ( nil @ A ) ) ) )
                  @ ( A_of_integer2 @ I ) ) ) )
            & ( ~ ( ord_less @ code_integer @ ( zero_zero @ code_integer ) @ I )
             => ( ( code_T4080844693773952564amples @ A @ A_of_integer2 @ Zero2 @ I )
                = ( cons @ A @ Zero2 @ ( nil @ A ) ) ) ) ) ) ) ).

% quickcheck_narrowing_samples.narrowing_samples.psimps
thf(fact_6375_mergesort__by__rel__split_Opelims,axiom,
    ! [A: $tType,X2: product_prod @ ( list @ A ) @ ( list @ A ),Xa: list @ A,Y2: product_prod @ ( list @ A ) @ ( list @ A )] :
      ( ( ( merges295452479951948502_split @ A @ X2 @ Xa )
        = Y2 )
     => ( ( accp @ ( product_prod @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) ) @ ( merges7066485432131860899it_rel @ A ) @ ( product_Pair @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) @ X2 @ Xa ) )
       => ( ! [Xs1: list @ A,Xs22: list @ A] :
              ( ( X2
                = ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs1 @ Xs22 ) )
             => ( ( Xa
                  = ( nil @ A ) )
               => ( ( Y2
                    = ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs1 @ Xs22 ) )
                 => ~ ( accp @ ( product_prod @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) ) @ ( merges7066485432131860899it_rel @ A ) @ ( product_Pair @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs1 @ Xs22 ) @ ( nil @ A ) ) ) ) ) )
         => ( ! [Xs1: list @ A,Xs22: list @ A] :
                ( ( X2
                  = ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs1 @ Xs22 ) )
               => ! [X3: A] :
                    ( ( Xa
                      = ( cons @ A @ X3 @ ( nil @ A ) ) )
                   => ( ( Y2
                        = ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X3 @ Xs1 ) @ Xs22 ) )
                     => ~ ( accp @ ( product_prod @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) ) @ ( merges7066485432131860899it_rel @ A ) @ ( product_Pair @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs1 @ Xs22 ) @ ( cons @ A @ X3 @ ( nil @ A ) ) ) ) ) ) )
           => ~ ! [Xs1: list @ A,Xs22: list @ A] :
                  ( ( X2
                    = ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs1 @ Xs22 ) )
                 => ! [X16: A,X24: A,Xs3: list @ A] :
                      ( ( Xa
                        = ( cons @ A @ X16 @ ( cons @ A @ X24 @ Xs3 ) ) )
                     => ( ( Y2
                          = ( merges295452479951948502_split @ A @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X16 @ Xs1 ) @ ( cons @ A @ X24 @ Xs22 ) ) @ Xs3 ) )
                       => ~ ( accp @ ( product_prod @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) ) @ ( merges7066485432131860899it_rel @ A ) @ ( product_Pair @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs1 @ Xs22 ) @ ( cons @ A @ X16 @ ( cons @ A @ X24 @ Xs3 ) ) ) ) ) ) ) ) ) ) ) ).

% mergesort_by_rel_split.pelims
thf(fact_6376_mergesort__by__rel_Opsimps,axiom,
    ! [A: $tType,R2: A > A > $o,Xs2: list @ A] :
      ( ( accp @ ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) @ ( mergesort_by_rel_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ R2 @ Xs2 ) )
     => ( ( ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Xs2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
         => ( ( mergesort_by_rel @ A @ R2 @ Xs2 )
            = Xs2 ) )
        & ( ~ ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Xs2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
         => ( ( mergesort_by_rel @ A @ R2 @ Xs2 )
            = ( merges9089515139780605204_merge @ A @ R2 @ ( mergesort_by_rel @ A @ R2 @ ( product_fst @ ( list @ A ) @ ( list @ A ) @ ( merges295452479951948502_split @ A @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ ( nil @ A ) ) @ Xs2 ) ) ) @ ( mergesort_by_rel @ A @ R2 @ ( product_snd @ ( list @ A ) @ ( list @ A ) @ ( merges295452479951948502_split @ A @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ ( nil @ A ) ) @ Xs2 ) ) ) ) ) ) ) ) ).

% mergesort_by_rel.psimps
thf(fact_6377_mergesort__by__rel__simps_I1_J,axiom,
    ! [A: $tType,R2: A > A > $o] :
      ( ( mergesort_by_rel @ A @ R2 @ ( nil @ A ) )
      = ( nil @ A ) ) ).

% mergesort_by_rel_simps(1)
thf(fact_6378_set__mergesort__by__rel,axiom,
    ! [A: $tType,R2: A > A > $o,Xs2: list @ A] :
      ( ( set2 @ A @ ( mergesort_by_rel @ A @ R2 @ Xs2 ) )
      = ( set2 @ A @ Xs2 ) ) ).

% set_mergesort_by_rel
thf(fact_6379_mergesort__by__rel__merge__simps_I3_J,axiom,
    ! [A: $tType,R2: A > A > $o,Ys: list @ A] :
      ( ( merges9089515139780605204_merge @ A @ R2 @ ( nil @ A ) @ Ys )
      = Ys ) ).

% mergesort_by_rel_merge_simps(3)
thf(fact_6380_mergesort__by__rel__simps_I2_J,axiom,
    ! [A: $tType,R2: A > A > $o,X2: A] :
      ( ( mergesort_by_rel @ A @ R2 @ ( cons @ A @ X2 @ ( nil @ A ) ) )
      = ( cons @ A @ X2 @ ( nil @ A ) ) ) ).

% mergesort_by_rel_simps(2)
thf(fact_6381_mergesort__by__rel__simps_I3_J,axiom,
    ! [A: $tType,R2: A > A > $o,X1: A,X23: A,Xs2: list @ A] :
      ( ( mergesort_by_rel @ A @ R2 @ ( cons @ A @ X1 @ ( cons @ A @ X23 @ Xs2 ) ) )
      = ( product_case_prod @ ( list @ A ) @ ( list @ A ) @ ( list @ A )
        @ ^ [Xs13: list @ A,Xs24: list @ A] : ( merges9089515139780605204_merge @ A @ R2 @ ( mergesort_by_rel @ A @ R2 @ Xs13 ) @ ( mergesort_by_rel @ A @ R2 @ Xs24 ) )
        @ ( merges295452479951948502_split @ A @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X1 @ ( nil @ A ) ) @ ( cons @ A @ X23 @ ( nil @ A ) ) ) @ Xs2 ) ) ) ).

% mergesort_by_rel_simps(3)
thf(fact_6382_mergesort__by__rel__merge__simps_I2_J,axiom,
    ! [A: $tType,R2: A > A > $o,Xs2: list @ A] :
      ( ( merges9089515139780605204_merge @ A @ R2 @ Xs2 @ ( nil @ A ) )
      = Xs2 ) ).

% mergesort_by_rel_merge_simps(2)
thf(fact_6383_mergesort__by__rel__merge__simps_I1_J,axiom,
    ! [A: $tType,R2: A > A > $o,X2: A,Y2: A,Xs2: list @ A,Ys: list @ A] :
      ( ( ( R2 @ X2 @ Y2 )
       => ( ( merges9089515139780605204_merge @ A @ R2 @ ( cons @ A @ X2 @ Xs2 ) @ ( cons @ A @ Y2 @ Ys ) )
          = ( cons @ A @ X2 @ ( merges9089515139780605204_merge @ A @ R2 @ Xs2 @ ( cons @ A @ Y2 @ Ys ) ) ) ) )
      & ( ~ ( R2 @ X2 @ Y2 )
       => ( ( merges9089515139780605204_merge @ A @ R2 @ ( cons @ A @ X2 @ Xs2 ) @ ( cons @ A @ Y2 @ Ys ) )
          = ( cons @ A @ Y2 @ ( merges9089515139780605204_merge @ A @ R2 @ ( cons @ A @ X2 @ Xs2 ) @ Ys ) ) ) ) ) ).

% mergesort_by_rel_merge_simps(1)
thf(fact_6384_mergesort__by__rel__merge_Oelims,axiom,
    ! [A: $tType,X2: A > A > $o,Xa: list @ A,Xb: list @ A,Y2: list @ A] :
      ( ( ( merges9089515139780605204_merge @ A @ X2 @ Xa @ Xb )
        = Y2 )
     => ( ! [X3: A,Xs3: list @ A] :
            ( ( Xa
              = ( cons @ A @ X3 @ Xs3 ) )
           => ! [Y4: A,Ys4: list @ A] :
                ( ( Xb
                  = ( cons @ A @ Y4 @ Ys4 ) )
               => ~ ( ( ( X2 @ X3 @ Y4 )
                     => ( Y2
                        = ( cons @ A @ X3 @ ( merges9089515139780605204_merge @ A @ X2 @ Xs3 @ ( cons @ A @ Y4 @ Ys4 ) ) ) ) )
                    & ( ~ ( X2 @ X3 @ Y4 )
                     => ( Y2
                        = ( cons @ A @ Y4 @ ( merges9089515139780605204_merge @ A @ X2 @ ( cons @ A @ X3 @ Xs3 ) @ Ys4 ) ) ) ) ) ) )
       => ( ( ( Xb
              = ( nil @ A ) )
           => ( Y2 != Xa ) )
         => ~ ( ( Xa
                = ( nil @ A ) )
             => ! [V3: A,Va3: list @ A] :
                  ( ( Xb
                    = ( cons @ A @ V3 @ Va3 ) )
                 => ( Y2
                   != ( cons @ A @ V3 @ Va3 ) ) ) ) ) ) ) ).

% mergesort_by_rel_merge.elims
thf(fact_6385_mergesort__by__rel__merge_Osimps_I3_J,axiom,
    ! [A: $tType,R2: A > A > $o,V: A,Va: list @ A] :
      ( ( merges9089515139780605204_merge @ A @ R2 @ ( nil @ A ) @ ( cons @ A @ V @ Va ) )
      = ( cons @ A @ V @ Va ) ) ).

% mergesort_by_rel_merge.simps(3)
thf(fact_6386_mergesort__by__rel_Osimps,axiom,
    ! [A: $tType] :
      ( ( mergesort_by_rel @ A )
      = ( ^ [R7: A > A > $o,Xs: list @ A] : ( if @ ( list @ A ) @ ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ Xs @ ( merges9089515139780605204_merge @ A @ R7 @ ( mergesort_by_rel @ A @ R7 @ ( product_fst @ ( list @ A ) @ ( list @ A ) @ ( merges295452479951948502_split @ A @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ ( nil @ A ) ) @ Xs ) ) ) @ ( mergesort_by_rel @ A @ R7 @ ( product_snd @ ( list @ A ) @ ( list @ A ) @ ( merges295452479951948502_split @ A @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ ( nil @ A ) ) @ Xs ) ) ) ) ) ) ) ).

% mergesort_by_rel.simps
thf(fact_6387_mergesort__by__rel_Oelims,axiom,
    ! [A: $tType,X2: A > A > $o,Xa: list @ A,Y2: list @ A] :
      ( ( ( mergesort_by_rel @ A @ X2 @ Xa )
        = Y2 )
     => ( ( ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Xa ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
         => ( Y2 = Xa ) )
        & ( ~ ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Xa ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
         => ( Y2
            = ( merges9089515139780605204_merge @ A @ X2 @ ( mergesort_by_rel @ A @ X2 @ ( product_fst @ ( list @ A ) @ ( list @ A ) @ ( merges295452479951948502_split @ A @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ ( nil @ A ) ) @ Xa ) ) ) @ ( mergesort_by_rel @ A @ X2 @ ( product_snd @ ( list @ A ) @ ( list @ A ) @ ( merges295452479951948502_split @ A @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ ( nil @ A ) ) @ Xa ) ) ) ) ) ) ) ) ).

% mergesort_by_rel.elims
thf(fact_6388_mergesort__by__rel_Opelims,axiom,
    ! [A: $tType,X2: A > A > $o,Xa: list @ A,Y2: list @ A] :
      ( ( ( mergesort_by_rel @ A @ X2 @ Xa )
        = Y2 )
     => ( ( accp @ ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) @ ( mergesort_by_rel_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ X2 @ Xa ) )
       => ~ ( ( ( ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Xa ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
               => ( Y2 = Xa ) )
              & ( ~ ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Xa ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
               => ( Y2
                  = ( merges9089515139780605204_merge @ A @ X2 @ ( mergesort_by_rel @ A @ X2 @ ( product_fst @ ( list @ A ) @ ( list @ A ) @ ( merges295452479951948502_split @ A @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ ( nil @ A ) ) @ Xa ) ) ) @ ( mergesort_by_rel @ A @ X2 @ ( product_snd @ ( list @ A ) @ ( list @ A ) @ ( merges295452479951948502_split @ A @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ ( nil @ A ) ) @ Xa ) ) ) ) ) ) )
           => ~ ( accp @ ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) @ ( mergesort_by_rel_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ X2 @ Xa ) ) ) ) ) ).

% mergesort_by_rel.pelims
thf(fact_6389_mergesort__by__rel__merge_Opelims,axiom,
    ! [A: $tType,X2: A > A > $o,Xa: list @ A,Xb: list @ A,Y2: list @ A] :
      ( ( ( merges9089515139780605204_merge @ A @ X2 @ Xa @ Xb )
        = Y2 )
     => ( ( accp @ ( product_prod @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) @ ( merges2244889521215249637ge_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ X2 @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xa @ Xb ) ) )
       => ( ! [X3: A,Xs3: list @ A] :
              ( ( Xa
                = ( cons @ A @ X3 @ Xs3 ) )
             => ! [Y4: A,Ys4: list @ A] :
                  ( ( Xb
                    = ( cons @ A @ Y4 @ Ys4 ) )
                 => ( ( ( ( X2 @ X3 @ Y4 )
                       => ( Y2
                          = ( cons @ A @ X3 @ ( merges9089515139780605204_merge @ A @ X2 @ Xs3 @ ( cons @ A @ Y4 @ Ys4 ) ) ) ) )
                      & ( ~ ( X2 @ X3 @ Y4 )
                       => ( Y2
                          = ( cons @ A @ Y4 @ ( merges9089515139780605204_merge @ A @ X2 @ ( cons @ A @ X3 @ Xs3 ) @ Ys4 ) ) ) ) )
                   => ~ ( accp @ ( product_prod @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) @ ( merges2244889521215249637ge_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ X2 @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X3 @ Xs3 ) @ ( cons @ A @ Y4 @ Ys4 ) ) ) ) ) ) )
         => ( ( ( Xb
                = ( nil @ A ) )
             => ( ( Y2 = Xa )
               => ~ ( accp @ ( product_prod @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) @ ( merges2244889521215249637ge_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ X2 @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xa @ ( nil @ A ) ) ) ) ) )
           => ~ ( ( Xa
                  = ( nil @ A ) )
               => ! [V3: A,Va3: list @ A] :
                    ( ( Xb
                      = ( cons @ A @ V3 @ Va3 ) )
                   => ( ( Y2
                        = ( cons @ A @ V3 @ Va3 ) )
                     => ~ ( accp @ ( product_prod @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) @ ( merges2244889521215249637ge_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ X2 @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ ( cons @ A @ V3 @ Va3 ) ) ) ) ) ) ) ) ) ) ) ).

% mergesort_by_rel_merge.pelims
thf(fact_6390_dup__1,axiom,
    ( ( code_dup @ ( one_one @ code_integer ) )
    = ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) ) ).

% dup_1
thf(fact_6391_dup_Oabs__eq,axiom,
    ! [X2: int] :
      ( ( code_dup @ ( code_integer_of_int @ X2 ) )
      = ( code_integer_of_int @ ( plus_plus @ int @ X2 @ X2 ) ) ) ).

% dup.abs_eq
thf(fact_6392_power__numeral,axiom,
    ! [A: $tType] :
      ( ( semiring_numeral @ A )
     => ! [K: num,L: num] :
          ( ( power_power @ A @ ( numeral_numeral @ A @ K ) @ ( numeral_numeral @ nat @ L ) )
          = ( numeral_numeral @ A @ ( pow @ K @ L ) ) ) ) ).

% power_numeral
thf(fact_6393_mergesort__def,axiom,
    ! [A: $tType] :
      ( ( ord @ A )
     => ( ( mergesort @ A )
        = ( mergesort_by_rel @ A @ ( ord_less_eq @ A ) ) ) ) ).

% mergesort_def
thf(fact_6394_pow_Osimps_I1_J,axiom,
    ! [X2: num] :
      ( ( pow @ X2 @ one2 )
      = X2 ) ).

% pow.simps(1)
thf(fact_6395_timeFrame_Oelims,axiom,
    ! [A: $tType,X2: nat,Xa: option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ),Y2: option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )] :
      ( ( ( heap_Time_timeFrame @ A @ X2 @ Xa )
        = Y2 )
     => ( ! [R3: A,H3: heap_ext @ product_unit,N7: nat] :
            ( ( Xa
              = ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R3 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H3 @ N7 ) ) ) )
           => ( Y2
             != ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R3 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H3 @ ( plus_plus @ nat @ X2 @ N7 ) ) ) ) ) )
       => ~ ( ( Xa
              = ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) )
           => ( Y2
             != ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) ) ) ) ).

% timeFrame.elims
thf(fact_6396_return__def,axiom,
    ! [A: $tType] :
      ( ( heap_Time_return @ A )
      = ( ^ [X: A] :
            ( heap_Time_heap @ A
            @ ^ [H: heap_ext @ product_unit] : ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ X @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H @ ( one_one @ nat ) ) ) ) ) ) ).

% return_def
thf(fact_6397_timeFrame__assoc,axiom,
    ! [A: $tType,N2: nat,N5: nat,F: option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )] :
      ( ( heap_Time_timeFrame @ A @ N2 @ ( heap_Time_timeFrame @ A @ N5 @ F ) )
      = ( heap_Time_timeFrame @ A @ ( plus_plus @ nat @ N2 @ N5 ) @ F ) ) ).

% timeFrame_assoc
thf(fact_6398_timeFrame_Osimps_I2_J,axiom,
    ! [A: $tType,N2: nat] :
      ( ( heap_Time_timeFrame @ A @ N2 @ ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) )
      = ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) ).

% timeFrame.simps(2)
thf(fact_6399_effect__heapE,axiom,
    ! [A: $tType,F: ( heap_ext @ product_unit ) > ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ),H2: heap_ext @ product_unit,H4: heap_ext @ product_unit,R4: A,N2: nat] :
      ( ( heap_Time_effect @ A @ ( heap_Time_heap @ A @ F ) @ H2 @ H4 @ R4 @ N2 )
     => ~ ( ( H4
            = ( product_fst @ ( heap_ext @ product_unit ) @ nat @ ( product_snd @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( F @ H2 ) ) ) )
         => ( ( N2
              = ( product_snd @ ( heap_ext @ product_unit ) @ nat @ ( product_snd @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( F @ H2 ) ) ) )
           => ( R4
             != ( product_fst @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( F @ H2 ) ) ) ) ) ) ).

% effect_heapE
thf(fact_6400_effect__heapI,axiom,
    ! [A: $tType,N2: nat,F: ( heap_ext @ product_unit ) > ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ),H2: heap_ext @ product_unit,H4: heap_ext @ product_unit,R4: A] :
      ( ( N2
        = ( product_snd @ ( heap_ext @ product_unit ) @ nat @ ( product_snd @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( F @ H2 ) ) ) )
     => ( ( H4
          = ( product_fst @ ( heap_ext @ product_unit ) @ nat @ ( product_snd @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( F @ H2 ) ) ) )
       => ( ( R4
            = ( product_fst @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( F @ H2 ) ) )
         => ( heap_Time_effect @ A @ ( heap_Time_heap @ A @ F ) @ H2 @ H4 @ R4 @ N2 ) ) ) ) ).

% effect_heapI
thf(fact_6401_timeFrame_Osimps_I1_J,axiom,
    ! [A: $tType,N2: nat,R4: A,H2: heap_ext @ product_unit,N5: nat] :
      ( ( heap_Time_timeFrame @ A @ N2 @ ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R4 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H2 @ N5 ) ) ) )
      = ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R4 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H2 @ ( plus_plus @ nat @ N2 @ N5 ) ) ) ) ) ).

% timeFrame.simps(1)
thf(fact_6402_timeFrame_Opelims,axiom,
    ! [A: $tType,X2: nat,Xa: option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ),Y2: option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )] :
      ( ( ( heap_Time_timeFrame @ A @ X2 @ Xa )
        = Y2 )
     => ( ( accp @ ( product_prod @ nat @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) @ ( heap_T5500966940807335491me_rel @ A ) @ ( product_Pair @ nat @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) @ X2 @ Xa ) )
       => ( ! [R3: A,H3: heap_ext @ product_unit,N7: nat] :
              ( ( Xa
                = ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R3 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H3 @ N7 ) ) ) )
             => ( ( Y2
                  = ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R3 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H3 @ ( plus_plus @ nat @ X2 @ N7 ) ) ) ) )
               => ~ ( accp @ ( product_prod @ nat @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) @ ( heap_T5500966940807335491me_rel @ A ) @ ( product_Pair @ nat @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) @ X2 @ ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R3 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H3 @ N7 ) ) ) ) ) ) )
         => ~ ( ( Xa
                = ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) )
             => ( ( Y2
                  = ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) )
               => ~ ( accp @ ( product_prod @ nat @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) @ ( heap_T5500966940807335491me_rel @ A ) @ ( product_Pair @ nat @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) @ X2 @ ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) ) ) ) ) ) ) ).

% timeFrame.pelims
thf(fact_6403_ureturn__def,axiom,
    ! [A: $tType] :
      ( ( heap_Time_ureturn @ A )
      = ( ^ [X: A] :
            ( heap_Time_heap @ A
            @ ^ [H: heap_ext @ product_unit] : ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ X @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H @ ( zero_zero @ nat ) ) ) ) ) ) ).

% ureturn_def
thf(fact_6404_bind__ureturn,axiom,
    ! [A: $tType,F: heap_Time_Heap @ A] :
      ( ( heap_Time_bind @ A @ A @ F @ ( heap_Time_ureturn @ A ) )
      = F ) ).

% bind_ureturn
thf(fact_6405_ureturn__bind,axiom,
    ! [A: $tType,B: $tType,X2: B,F: B > ( heap_Time_Heap @ A )] :
      ( ( heap_Time_bind @ B @ A @ ( heap_Time_ureturn @ B @ X2 ) @ F )
      = ( F @ X2 ) ) ).

% ureturn_bind
thf(fact_6406_effect__guardI,axiom,
    ! [A: $tType,P: ( heap_ext @ product_unit ) > $o,H2: heap_ext @ product_unit,H4: heap_ext @ product_unit,F: ( heap_ext @ product_unit ) > ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ),N2: nat,R4: A] :
      ( ( P @ H2 )
     => ( ( H4
          = ( product_fst @ ( heap_ext @ product_unit ) @ nat @ ( product_snd @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( F @ H2 ) ) ) )
       => ( ( N2
            = ( product_snd @ ( heap_ext @ product_unit ) @ nat @ ( product_snd @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( F @ H2 ) ) ) )
         => ( ( R4
              = ( product_fst @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( F @ H2 ) ) )
           => ( heap_Time_effect @ A @ ( heap_Time_guard @ A @ P @ F ) @ H2 @ H4 @ R4 @ N2 ) ) ) ) ) ).

% effect_guardI
thf(fact_6407_effect__guardE,axiom,
    ! [A: $tType,P: ( heap_ext @ product_unit ) > $o,F: ( heap_ext @ product_unit ) > ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ),H2: heap_ext @ product_unit,H4: heap_ext @ product_unit,R4: A,N2: nat] :
      ( ( heap_Time_effect @ A @ ( heap_Time_guard @ A @ P @ F ) @ H2 @ H4 @ R4 @ N2 )
     => ~ ( ( H4
            = ( product_fst @ ( heap_ext @ product_unit ) @ nat @ ( product_snd @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( F @ H2 ) ) ) )
         => ( ( N2
              = ( product_snd @ ( heap_ext @ product_unit ) @ nat @ ( product_snd @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( F @ H2 ) ) ) )
           => ( ( R4
                = ( product_fst @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( F @ H2 ) ) )
             => ~ ( P @ H2 ) ) ) ) ) ).

% effect_guardE
thf(fact_6408_guard__def,axiom,
    ! [A: $tType] :
      ( ( heap_Time_guard @ A )
      = ( ^ [P3: ( heap_ext @ product_unit ) > $o,F6: ( heap_ext @ product_unit ) > ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )] :
            ( heap_Time_Heap2 @ A
            @ ^ [H: heap_ext @ product_unit] : ( if @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) @ ( P3 @ H ) @ ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( F6 @ H ) ) @ ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) ) ) ) ).

% guard_def
thf(fact_6409_len_H__def,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ( heap @ B )
        & ( semiring_1 @ A ) )
     => ( ( array_len2 @ B @ A )
        = ( ^ [A5: array @ B] :
              ( heap_Time_bind @ nat @ A @ ( array_len @ B @ A5 )
              @ ^ [N: nat] : ( heap_Time_ureturn @ A @ ( semiring_1_of_nat @ A @ N ) ) ) ) ) ) ).

% len'_def
thf(fact_6410_cis__multiple__2pi,axiom,
    ! [N2: real] :
      ( ( member @ real @ N2 @ ( ring_1_Ints @ real ) )
     => ( ( cis @ ( times_times @ real @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) @ N2 ) )
        = ( one_one @ complex ) ) ) ).

% cis_multiple_2pi
thf(fact_6411_floor__add2,axiom,
    ! [A: $tType] :
      ( ( archim2362893244070406136eiling @ A )
     => ! [X2: A,Y2: A] :
          ( ( ( member @ A @ X2 @ ( ring_1_Ints @ A ) )
            | ( member @ A @ Y2 @ ( ring_1_Ints @ A ) ) )
         => ( ( archim6421214686448440834_floor @ A @ ( plus_plus @ A @ X2 @ Y2 ) )
            = ( plus_plus @ int @ ( archim6421214686448440834_floor @ A @ X2 ) @ ( archim6421214686448440834_floor @ A @ Y2 ) ) ) ) ) ).

% floor_add2
thf(fact_6412_frac__gt__0__iff,axiom,
    ! [A: $tType] :
      ( ( archim2362893244070406136eiling @ A )
     => ! [X2: A] :
          ( ( ord_less @ A @ ( zero_zero @ A ) @ ( archimedean_frac @ A @ X2 ) )
          = ( ~ ( member @ A @ X2 @ ( ring_1_Ints @ A ) ) ) ) ) ).

% frac_gt_0_iff
thf(fact_6413_Ints__add,axiom,
    ! [A: $tType] :
      ( ( ring_1 @ A )
     => ! [A2: A,B2: A] :
          ( ( member @ A @ A2 @ ( ring_1_Ints @ A ) )
         => ( ( member @ A @ B2 @ ( ring_1_Ints @ A ) )
           => ( member @ A @ ( plus_plus @ A @ A2 @ B2 ) @ ( ring_1_Ints @ A ) ) ) ) ) ).

% Ints_add
thf(fact_6414_Ints__double__eq__0__iff,axiom,
    ! [A: $tType] :
      ( ( ring_char_0 @ A )
     => ! [A2: A] :
          ( ( member @ A @ A2 @ ( ring_1_Ints @ A ) )
         => ( ( ( plus_plus @ A @ A2 @ A2 )
              = ( zero_zero @ A ) )
            = ( A2
              = ( zero_zero @ A ) ) ) ) ) ).

% Ints_double_eq_0_iff
thf(fact_6415_Ints__mult,axiom,
    ! [A: $tType] :
      ( ( ring_1 @ A )
     => ! [A2: A,B2: A] :
          ( ( member @ A @ A2 @ ( ring_1_Ints @ A ) )
         => ( ( member @ A @ B2 @ ( ring_1_Ints @ A ) )
           => ( member @ A @ ( times_times @ A @ A2 @ B2 ) @ ( ring_1_Ints @ A ) ) ) ) ) ).

% Ints_mult
thf(fact_6416_Ints__numeral,axiom,
    ! [A: $tType] :
      ( ( ring_1 @ A )
     => ! [N2: num] : ( member @ A @ ( numeral_numeral @ A @ N2 ) @ ( ring_1_Ints @ A ) ) ) ).

% Ints_numeral
thf(fact_6417_Ints__diff,axiom,
    ! [A: $tType] :
      ( ( ring_1 @ A )
     => ! [A2: A,B2: A] :
          ( ( member @ A @ A2 @ ( ring_1_Ints @ A ) )
         => ( ( member @ A @ B2 @ ( ring_1_Ints @ A ) )
           => ( member @ A @ ( minus_minus @ A @ A2 @ B2 ) @ ( ring_1_Ints @ A ) ) ) ) ) ).

% Ints_diff
thf(fact_6418_Ints__power,axiom,
    ! [A: $tType] :
      ( ( ring_1 @ A )
     => ! [A2: A,N2: nat] :
          ( ( member @ A @ A2 @ ( ring_1_Ints @ A ) )
         => ( member @ A @ ( power_power @ A @ A2 @ N2 ) @ ( ring_1_Ints @ A ) ) ) ) ).

% Ints_power
thf(fact_6419_Ints__1,axiom,
    ! [A: $tType] :
      ( ( ring_1 @ A )
     => ( member @ A @ ( one_one @ A ) @ ( ring_1_Ints @ A ) ) ) ).

% Ints_1
thf(fact_6420_finite__int__segment,axiom,
    ! [A: $tType] :
      ( ( archim2362893244070406136eiling @ A )
     => ! [A2: A,B2: A] :
          ( finite_finite2 @ A
          @ ( collect @ A
            @ ^ [X: A] :
                ( ( member @ A @ X @ ( ring_1_Ints @ A ) )
                & ( ord_less_eq @ A @ A2 @ X )
                & ( ord_less_eq @ A @ X @ B2 ) ) ) ) ) ).

% finite_int_segment
thf(fact_6421_Ints__odd__nonzero,axiom,
    ! [A: $tType] :
      ( ( ring_char_0 @ A )
     => ! [A2: A] :
          ( ( member @ A @ A2 @ ( ring_1_Ints @ A ) )
         => ( ( plus_plus @ A @ ( plus_plus @ A @ ( one_one @ A ) @ A2 ) @ A2 )
           != ( zero_zero @ A ) ) ) ) ).

% Ints_odd_nonzero
thf(fact_6422_of__int__divide__in__Ints,axiom,
    ! [A: $tType] :
      ( ( idom_divide @ A )
     => ! [B2: int,A2: int] :
          ( ( dvd_dvd @ int @ B2 @ A2 )
         => ( member @ A @ ( divide_divide @ A @ ( ring_1_of_int @ A @ A2 ) @ ( ring_1_of_int @ A @ B2 ) ) @ ( ring_1_Ints @ A ) ) ) ) ).

% of_int_divide_in_Ints
thf(fact_6423_finite__abs__int__segment,axiom,
    ! [A: $tType] :
      ( ( archim2362893244070406136eiling @ A )
     => ! [A2: A] :
          ( finite_finite2 @ A
          @ ( collect @ A
            @ ^ [K4: A] :
                ( ( member @ A @ K4 @ ( ring_1_Ints @ A ) )
                & ( ord_less_eq @ A @ ( abs_abs @ A @ K4 ) @ A2 ) ) ) ) ) ).

% finite_abs_int_segment
thf(fact_6424_Ints__odd__less__0,axiom,
    ! [A: $tType] :
      ( ( linordered_idom @ A )
     => ! [A2: A] :
          ( ( member @ A @ A2 @ ( ring_1_Ints @ A ) )
         => ( ( ord_less @ A @ ( plus_plus @ A @ ( plus_plus @ A @ ( one_one @ A ) @ A2 ) @ A2 ) @ ( zero_zero @ A ) )
            = ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ) ).

% Ints_odd_less_0
thf(fact_6425_Ints__nonzero__abs__ge1,axiom,
    ! [A: $tType] :
      ( ( linordered_idom @ A )
     => ! [X2: A] :
          ( ( member @ A @ X2 @ ( ring_1_Ints @ A ) )
         => ( ( X2
             != ( zero_zero @ A ) )
           => ( ord_less_eq @ A @ ( one_one @ A ) @ ( abs_abs @ A @ X2 ) ) ) ) ) ).

% Ints_nonzero_abs_ge1
thf(fact_6426_Ints__nonzero__abs__less1,axiom,
    ! [A: $tType] :
      ( ( linordered_idom @ A )
     => ! [X2: A] :
          ( ( member @ A @ X2 @ ( ring_1_Ints @ A ) )
         => ( ( ord_less @ A @ ( abs_abs @ A @ X2 ) @ ( one_one @ A ) )
           => ( X2
              = ( zero_zero @ A ) ) ) ) ) ).

% Ints_nonzero_abs_less1
thf(fact_6427_Ints__eq__abs__less1,axiom,
    ! [A: $tType] :
      ( ( linordered_idom @ A )
     => ! [X2: A,Y2: A] :
          ( ( member @ A @ X2 @ ( ring_1_Ints @ A ) )
         => ( ( member @ A @ Y2 @ ( ring_1_Ints @ A ) )
           => ( ( X2 = Y2 )
              = ( ord_less @ A @ ( abs_abs @ A @ ( minus_minus @ A @ X2 @ Y2 ) ) @ ( one_one @ A ) ) ) ) ) ) ).

% Ints_eq_abs_less1
thf(fact_6428_sin__times__pi__eq__0,axiom,
    ! [X2: real] :
      ( ( ( sin @ real @ ( times_times @ real @ X2 @ pi ) )
        = ( zero_zero @ real ) )
      = ( member @ real @ X2 @ ( ring_1_Ints @ real ) ) ) ).

% sin_times_pi_eq_0
thf(fact_6429_frac__neg,axiom,
    ! [A: $tType] :
      ( ( archim2362893244070406136eiling @ A )
     => ! [X2: A] :
          ( ( ( member @ A @ X2 @ ( ring_1_Ints @ A ) )
           => ( ( archimedean_frac @ A @ ( uminus_uminus @ A @ X2 ) )
              = ( zero_zero @ A ) ) )
          & ( ~ ( member @ A @ X2 @ ( ring_1_Ints @ A ) )
           => ( ( archimedean_frac @ A @ ( uminus_uminus @ A @ X2 ) )
              = ( minus_minus @ A @ ( one_one @ A ) @ ( archimedean_frac @ A @ X2 ) ) ) ) ) ) ).

% frac_neg
thf(fact_6430_upd__ureturn,axiom,
    ! [A: $tType] :
      ( ( heap @ A )
     => ! [I: nat,X2: A,A2: array @ A] :
          ( ( heap_Time_bind @ ( array @ A ) @ ( array @ A ) @ ( array_upd @ A @ I @ X2 @ A2 )
            @ ^ [Uu3: array @ A] : ( heap_Time_ureturn @ ( array @ A ) @ A2 ) )
          = ( array_upd @ A @ I @ X2 @ A2 ) ) ) ).

% upd_ureturn
thf(fact_6431_le__mult__floor__Ints,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ( archim2362893244070406136eiling @ B )
        & ( linordered_idom @ A ) )
     => ! [A2: B,B2: B] :
          ( ( ord_less_eq @ B @ ( zero_zero @ B ) @ A2 )
         => ( ( member @ B @ A2 @ ( ring_1_Ints @ B ) )
           => ( ord_less_eq @ A @ ( ring_1_of_int @ A @ ( times_times @ int @ ( archim6421214686448440834_floor @ B @ A2 ) @ ( archim6421214686448440834_floor @ B @ B2 ) ) ) @ ( ring_1_of_int @ A @ ( archim6421214686448440834_floor @ B @ ( times_times @ B @ A2 @ B2 ) ) ) ) ) ) ) ).

% le_mult_floor_Ints
thf(fact_6432_mult__ceiling__le__Ints,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ( archim2362893244070406136eiling @ B )
        & ( linordered_idom @ A ) )
     => ! [A2: B,B2: B] :
          ( ( ord_less_eq @ B @ ( zero_zero @ B ) @ A2 )
         => ( ( member @ B @ A2 @ ( ring_1_Ints @ B ) )
           => ( ord_less_eq @ A @ ( ring_1_of_int @ A @ ( archimedean_ceiling @ B @ ( times_times @ B @ A2 @ B2 ) ) ) @ ( ring_1_of_int @ A @ ( times_times @ int @ ( archimedean_ceiling @ B @ A2 ) @ ( archimedean_ceiling @ B @ B2 ) ) ) ) ) ) ) ).

% mult_ceiling_le_Ints
thf(fact_6433_frac__unique__iff,axiom,
    ! [A: $tType] :
      ( ( archim2362893244070406136eiling @ A )
     => ! [X2: A,A2: A] :
          ( ( ( archimedean_frac @ A @ X2 )
            = A2 )
          = ( ( member @ A @ ( minus_minus @ A @ X2 @ A2 ) @ ( ring_1_Ints @ A ) )
            & ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
            & ( ord_less @ A @ A2 @ ( one_one @ A ) ) ) ) ) ).

% frac_unique_iff
thf(fact_6434_sin__integer__2pi,axiom,
    ! [N2: real] :
      ( ( member @ real @ N2 @ ( ring_1_Ints @ real ) )
     => ( ( sin @ real @ ( times_times @ real @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) @ N2 ) )
        = ( zero_zero @ real ) ) ) ).

% sin_integer_2pi
thf(fact_6435_cos__integer__2pi,axiom,
    ! [N2: real] :
      ( ( member @ real @ N2 @ ( ring_1_Ints @ real ) )
     => ( ( cos @ real @ ( times_times @ real @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) @ N2 ) )
        = ( one_one @ real ) ) ) ).

% cos_integer_2pi
thf(fact_6436_ran__nth__set__encoding__conv,axiom,
    ! [A: $tType,L: list @ A] :
      ( ( ran @ nat @ A
        @ ^ [I2: nat] : ( if @ ( option @ A ) @ ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ L ) ) @ ( some @ A @ ( nth @ A @ L @ I2 ) ) @ ( none @ A ) ) )
      = ( set2 @ A @ L ) ) ).

% ran_nth_set_encoding_conv
thf(fact_6437_rat__inverse__code,axiom,
    ! [P4: rat] :
      ( ( quotient_of @ ( inverse_inverse @ rat @ P4 ) )
      = ( product_case_prod @ int @ int @ ( product_prod @ int @ int )
        @ ^ [A5: int,B5: int] :
            ( if @ ( product_prod @ int @ int )
            @ ( A5
              = ( zero_zero @ int ) )
            @ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) )
            @ ( product_Pair @ int @ int @ ( times_times @ int @ ( sgn_sgn @ int @ A5 ) @ B5 ) @ ( abs_abs @ int @ A5 ) ) )
        @ ( quotient_of @ P4 ) ) ) ).

% rat_inverse_code
thf(fact_6438_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_6439_rat__one__code,axiom,
    ( ( quotient_of @ ( one_one @ rat ) )
    = ( product_Pair @ int @ int @ ( one_one @ int ) @ ( one_one @ int ) ) ) ).

% rat_one_code
thf(fact_6440_map__update__eta__repair_I2_J,axiom,
    ! [B: $tType,A: $tType,M: A > ( option @ B ),K: A,V: B] :
      ( ( ( M @ K )
        = ( none @ B ) )
     => ( ( ran @ A @ B
          @ ^ [X: A] : ( if @ ( option @ B ) @ ( X = K ) @ ( some @ B @ V ) @ ( M @ X ) ) )
        = ( insert @ B @ V @ ( ran @ A @ B @ M ) ) ) ) ).

% map_update_eta_repair(2)
thf(fact_6441_rat__zero__code,axiom,
    ( ( quotient_of @ ( zero_zero @ rat ) )
    = ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) ) ) ).

% rat_zero_code
thf(fact_6442_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_6443_quotient__of__number_I4_J,axiom,
    ( ( quotient_of @ ( uminus_uminus @ rat @ ( one_one @ rat ) ) )
    = ( product_Pair @ int @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( one_one @ int ) ) ) ).

% quotient_of_number(4)
thf(fact_6444_divide__rat__def,axiom,
    ( ( divide_divide @ rat )
    = ( ^ [Q5: rat,R: rat] : ( times_times @ rat @ Q5 @ ( inverse_inverse @ rat @ R ) ) ) ) ).

% divide_rat_def
thf(fact_6445_quotient__of__div,axiom,
    ! [R4: rat,N2: int,D2: int] :
      ( ( ( quotient_of @ R4 )
        = ( product_Pair @ int @ int @ N2 @ D2 ) )
     => ( R4
        = ( divide_divide @ rat @ ( ring_1_of_int @ rat @ N2 ) @ ( ring_1_of_int @ rat @ D2 ) ) ) ) ).

% quotient_of_div
thf(fact_6446_quotient__of__denom__pos,axiom,
    ! [R4: rat,P4: int,Q2: int] :
      ( ( ( quotient_of @ R4 )
        = ( product_Pair @ int @ int @ P4 @ Q2 ) )
     => ( ord_less @ int @ ( zero_zero @ int ) @ Q2 ) ) ).

% quotient_of_denom_pos
thf(fact_6447_quotient__of__denom__pos_H,axiom,
    ! [R4: rat] : ( ord_less @ int @ ( zero_zero @ int ) @ ( product_snd @ int @ int @ ( quotient_of @ R4 ) ) ) ).

% quotient_of_denom_pos'
thf(fact_6448_rat__abs__code,axiom,
    ! [P4: rat] :
      ( ( quotient_of @ ( abs_abs @ rat @ P4 ) )
      = ( product_case_prod @ int @ int @ ( product_prod @ int @ int )
        @ ^ [A5: int] : ( product_Pair @ int @ int @ ( abs_abs @ int @ A5 ) )
        @ ( quotient_of @ P4 ) ) ) ).

% rat_abs_code
thf(fact_6449_rat__uminus__code,axiom,
    ! [P4: rat] :
      ( ( quotient_of @ ( uminus_uminus @ rat @ P4 ) )
      = ( product_case_prod @ int @ int @ ( product_prod @ int @ int )
        @ ^ [A5: int] : ( product_Pair @ int @ int @ ( uminus_uminus @ int @ A5 ) )
        @ ( quotient_of @ P4 ) ) ) ).

% rat_uminus_code
thf(fact_6450_rat__less__code,axiom,
    ( ( ord_less @ rat )
    = ( ^ [P5: rat,Q5: rat] :
          ( product_case_prod @ int @ int @ $o
          @ ^ [A5: int,C4: int] :
              ( product_case_prod @ int @ int @ $o
              @ ^ [B5: int,D3: int] : ( ord_less @ int @ ( times_times @ int @ A5 @ D3 ) @ ( times_times @ int @ C4 @ B5 ) )
              @ ( quotient_of @ Q5 ) )
          @ ( quotient_of @ P5 ) ) ) ) ).

% rat_less_code
thf(fact_6451_rat__floor__code,axiom,
    ( ( archim6421214686448440834_floor @ rat )
    = ( ^ [P5: rat] : ( product_case_prod @ int @ int @ int @ ( divide_divide @ int ) @ ( quotient_of @ P5 ) ) ) ) ).

% rat_floor_code
thf(fact_6452_rat__less__eq__code,axiom,
    ( ( ord_less_eq @ rat )
    = ( ^ [P5: rat,Q5: rat] :
          ( product_case_prod @ int @ int @ $o
          @ ^ [A5: int,C4: int] :
              ( product_case_prod @ int @ int @ $o
              @ ^ [B5: int,D3: int] : ( ord_less_eq @ int @ ( times_times @ int @ A5 @ D3 ) @ ( times_times @ int @ C4 @ B5 ) )
              @ ( quotient_of @ Q5 ) )
          @ ( quotient_of @ P5 ) ) ) ) ).

% rat_less_eq_code
thf(fact_6453_rat__sgn__code,axiom,
    ! [P4: rat] :
      ( ( quotient_of @ ( sgn_sgn @ rat @ P4 ) )
      = ( product_Pair @ int @ int @ ( sgn_sgn @ int @ ( product_fst @ int @ int @ ( quotient_of @ P4 ) ) ) @ ( one_one @ int ) ) ) ).

% rat_sgn_code
thf(fact_6454_ran__empty,axiom,
    ! [B: $tType,A: $tType] :
      ( ( ran @ B @ A
        @ ^ [X: B] : ( none @ A ) )
      = ( bot_bot @ ( set @ A ) ) ) ).

% ran_empty
thf(fact_6455_quotient__of__int,axiom,
    ! [A2: int] :
      ( ( quotient_of @ ( of_int @ A2 ) )
      = ( product_Pair @ int @ int @ A2 @ ( one_one @ int ) ) ) ).

% quotient_of_int
thf(fact_6456_ranI,axiom,
    ! [A: $tType,B: $tType,M: B > ( option @ A ),A2: B,B2: A] :
      ( ( ( M @ A2 )
        = ( some @ A @ B2 ) )
     => ( member @ A @ B2 @ ( ran @ B @ A @ M ) ) ) ).

% ranI
thf(fact_6457_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_6458_rat__minus__code,axiom,
    ! [P4: rat,Q2: rat] :
      ( ( quotient_of @ ( minus_minus @ rat @ P4 @ Q2 ) )
      = ( product_case_prod @ int @ int @ ( product_prod @ int @ int )
        @ ^ [A5: int,C4: int] :
            ( product_case_prod @ int @ int @ ( product_prod @ int @ int )
            @ ^ [B5: int,D3: int] : ( normalize @ ( product_Pair @ int @ int @ ( minus_minus @ int @ ( times_times @ int @ A5 @ D3 ) @ ( times_times @ int @ B5 @ C4 ) ) @ ( times_times @ int @ C4 @ D3 ) ) )
            @ ( quotient_of @ Q2 ) )
        @ ( quotient_of @ P4 ) ) ) ).

% rat_minus_code
thf(fact_6459_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_6460_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_6461_normalize__denom__pos,axiom,
    ! [R4: product_prod @ int @ int,P4: int,Q2: int] :
      ( ( ( normalize @ R4 )
        = ( product_Pair @ int @ int @ P4 @ Q2 ) )
     => ( ord_less @ int @ ( zero_zero @ int ) @ Q2 ) ) ).

% normalize_denom_pos
thf(fact_6462_normalize__crossproduct,axiom,
    ! [Q2: int,S4: int,P4: int,R4: int] :
      ( ( Q2
       != ( zero_zero @ int ) )
     => ( ( S4
         != ( zero_zero @ int ) )
       => ( ( ( normalize @ ( product_Pair @ int @ int @ P4 @ Q2 ) )
            = ( normalize @ ( product_Pair @ int @ int @ R4 @ S4 ) ) )
         => ( ( times_times @ int @ P4 @ S4 )
            = ( times_times @ int @ R4 @ Q2 ) ) ) ) ) ).

% normalize_crossproduct
thf(fact_6463_Frct__code__post_I2_J,axiom,
    ! [A2: int] :
      ( ( frct @ ( product_Pair @ int @ int @ A2 @ ( zero_zero @ int ) ) )
      = ( zero_zero @ rat ) ) ).

% Frct_code_post(2)
thf(fact_6464_Frct__code__post_I1_J,axiom,
    ! [A2: int] :
      ( ( frct @ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ A2 ) )
      = ( zero_zero @ rat ) ) ).

% Frct_code_post(1)
thf(fact_6465_Frct__code__post_I7_J,axiom,
    ! [A2: int,B2: int] :
      ( ( frct @ ( product_Pair @ int @ int @ ( uminus_uminus @ int @ A2 ) @ B2 ) )
      = ( uminus_uminus @ rat @ ( frct @ ( product_Pair @ int @ int @ A2 @ B2 ) ) ) ) ).

% Frct_code_post(7)
thf(fact_6466_Frct__code__post_I8_J,axiom,
    ! [A2: int,B2: int] :
      ( ( frct @ ( product_Pair @ int @ int @ A2 @ ( uminus_uminus @ int @ B2 ) ) )
      = ( uminus_uminus @ rat @ ( frct @ ( product_Pair @ int @ int @ A2 @ B2 ) ) ) ) ).

% Frct_code_post(8)
thf(fact_6467_Frct__code__post_I3_J,axiom,
    ( ( frct @ ( product_Pair @ int @ int @ ( one_one @ int ) @ ( one_one @ int ) ) )
    = ( one_one @ rat ) ) ).

% Frct_code_post(3)
thf(fact_6468_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_6469_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_6470_rat__times__code,axiom,
    ! [P4: rat,Q2: rat] :
      ( ( quotient_of @ ( times_times @ rat @ P4 @ Q2 ) )
      = ( product_case_prod @ int @ int @ ( product_prod @ int @ int )
        @ ^ [A5: int,C4: int] :
            ( product_case_prod @ int @ int @ ( product_prod @ int @ int )
            @ ^ [B5: int,D3: int] : ( normalize @ ( product_Pair @ int @ int @ ( times_times @ int @ A5 @ B5 ) @ ( times_times @ int @ C4 @ D3 ) ) )
            @ ( quotient_of @ Q2 ) )
        @ ( quotient_of @ P4 ) ) ) ).

% rat_times_code
thf(fact_6471_rat__divide__code,axiom,
    ! [P4: rat,Q2: rat] :
      ( ( quotient_of @ ( divide_divide @ rat @ P4 @ Q2 ) )
      = ( product_case_prod @ int @ int @ ( product_prod @ int @ int )
        @ ^ [A5: int,C4: int] :
            ( product_case_prod @ int @ int @ ( product_prod @ int @ int )
            @ ^ [B5: int,D3: int] : ( normalize @ ( product_Pair @ int @ int @ ( times_times @ int @ A5 @ D3 ) @ ( times_times @ int @ C4 @ B5 ) ) )
            @ ( quotient_of @ Q2 ) )
        @ ( quotient_of @ P4 ) ) ) ).

% rat_divide_code
thf(fact_6472_rat__plus__code,axiom,
    ! [P4: rat,Q2: rat] :
      ( ( quotient_of @ ( plus_plus @ rat @ P4 @ Q2 ) )
      = ( product_case_prod @ int @ int @ ( product_prod @ int @ int )
        @ ^ [A5: int,C4: int] :
            ( product_case_prod @ int @ int @ ( product_prod @ int @ int )
            @ ^ [B5: int,D3: int] : ( normalize @ ( product_Pair @ int @ int @ ( plus_plus @ int @ ( times_times @ int @ A5 @ D3 ) @ ( times_times @ int @ B5 @ C4 ) ) @ ( times_times @ int @ C4 @ D3 ) ) )
            @ ( quotient_of @ Q2 ) )
        @ ( quotient_of @ P4 ) ) ) ).

% rat_plus_code
thf(fact_6473_normalize__def,axiom,
    ( normalize
    = ( ^ [P5: product_prod @ int @ int] :
          ( if @ ( product_prod @ int @ int ) @ ( ord_less @ int @ ( zero_zero @ int ) @ ( product_snd @ int @ int @ P5 ) ) @ ( product_Pair @ int @ int @ ( divide_divide @ int @ ( product_fst @ int @ int @ P5 ) @ ( gcd_gcd @ int @ ( product_fst @ int @ int @ P5 ) @ ( product_snd @ int @ int @ P5 ) ) ) @ ( divide_divide @ int @ ( product_snd @ int @ int @ P5 ) @ ( gcd_gcd @ int @ ( product_fst @ int @ int @ P5 ) @ ( product_snd @ int @ int @ P5 ) ) ) )
          @ ( if @ ( product_prod @ int @ int )
            @ ( ( product_snd @ int @ int @ P5 )
              = ( zero_zero @ int ) )
            @ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) )
            @ ( product_Pair @ int @ int @ ( divide_divide @ int @ ( product_fst @ int @ int @ P5 ) @ ( uminus_uminus @ int @ ( gcd_gcd @ int @ ( product_fst @ int @ int @ P5 ) @ ( product_snd @ int @ int @ P5 ) ) ) ) @ ( divide_divide @ int @ ( product_snd @ int @ int @ P5 ) @ ( uminus_uminus @ int @ ( gcd_gcd @ int @ ( product_fst @ int @ int @ P5 ) @ ( product_snd @ int @ int @ P5 ) ) ) ) ) ) ) ) ) ).

% normalize_def
thf(fact_6474_setceilmax,axiom,
    ! [S4: vEBT_VEBT,M: nat,Listy: list @ vEBT_VEBT,N2: nat] :
      ( ( vEBT_invar_vebt @ S4 @ M )
     => ( ! [X3: vEBT_VEBT] :
            ( ( member @ vEBT_VEBT @ X3 @ ( set2 @ vEBT_VEBT @ Listy ) )
           => ( vEBT_invar_vebt @ X3 @ N2 ) )
       => ( ( M
            = ( suc @ N2 ) )
         => ( ! [X3: vEBT_VEBT] :
                ( ( member @ vEBT_VEBT @ X3 @ ( set2 @ vEBT_VEBT @ Listy ) )
               => ( ( semiring_1_of_nat @ int @ ( vEBT_VEBT_height @ X3 ) )
                  = ( archimedean_ceiling @ real @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ real @ N2 ) ) ) ) )
           => ( ( ( semiring_1_of_nat @ int @ ( vEBT_VEBT_height @ S4 ) )
                = ( archimedean_ceiling @ real @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ real @ M ) ) ) )
             => ( ( semiring_1_of_nat @ int @ ( lattic643756798349783984er_Max @ nat @ ( image @ vEBT_VEBT @ nat @ vEBT_VEBT_height @ ( insert @ vEBT_VEBT @ S4 @ ( set2 @ vEBT_VEBT @ Listy ) ) ) ) )
                = ( archimedean_ceiling @ real @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ real @ M ) ) ) ) ) ) ) ) ) ).

% setceilmax
thf(fact_6475_image__ident,axiom,
    ! [A: $tType,Y7: set @ A] :
      ( ( image @ A @ A
        @ ^ [X: A] : X
        @ Y7 )
      = Y7 ) ).

% image_ident
thf(fact_6476_height__compose__list,axiom,
    ! [T2: vEBT_VEBT,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( member @ vEBT_VEBT @ T2 @ ( set2 @ vEBT_VEBT @ TreeList ) )
     => ( ord_less_eq @ nat @ ( vEBT_VEBT_height @ T2 ) @ ( lattic643756798349783984er_Max @ nat @ ( image @ vEBT_VEBT @ nat @ vEBT_VEBT_height @ ( insert @ vEBT_VEBT @ Summary @ ( set2 @ vEBT_VEBT @ TreeList ) ) ) ) ) ) ).

% height_compose_list
thf(fact_6477_gcd__add1,axiom,
    ! [A: $tType] :
      ( ( semiring_gcd @ A )
     => ! [M: A,N2: A] :
          ( ( gcd_gcd @ A @ ( plus_plus @ A @ M @ N2 ) @ N2 )
          = ( gcd_gcd @ A @ M @ N2 ) ) ) ).

% gcd_add1
thf(fact_6478_gcd__add2,axiom,
    ! [A: $tType] :
      ( ( semiring_gcd @ A )
     => ! [M: A,N2: A] :
          ( ( gcd_gcd @ A @ M @ ( plus_plus @ A @ M @ N2 ) )
          = ( gcd_gcd @ A @ M @ N2 ) ) ) ).

% gcd_add2
thf(fact_6479_gcd_Obottom__left__bottom,axiom,
    ! [A: $tType] :
      ( ( semiring_gcd @ A )
     => ! [A2: A] :
          ( ( gcd_gcd @ A @ ( one_one @ A ) @ A2 )
          = ( one_one @ A ) ) ) ).

% gcd.bottom_left_bottom
thf(fact_6480_gcd_Obottom__right__bottom,axiom,
    ! [A: $tType] :
      ( ( semiring_gcd @ A )
     => ! [A2: A] :
          ( ( gcd_gcd @ A @ A2 @ ( one_one @ A ) )
          = ( one_one @ A ) ) ) ).

% gcd.bottom_right_bottom
thf(fact_6481_gcd__exp,axiom,
    ! [A: $tType] :
      ( ( semiri6843258321239162965malize @ A )
     => ! [A2: A,N2: nat,B2: A] :
          ( ( gcd_gcd @ A @ ( power_power @ A @ A2 @ N2 ) @ ( power_power @ A @ B2 @ N2 ) )
          = ( power_power @ A @ ( gcd_gcd @ A @ A2 @ B2 ) @ N2 ) ) ) ).

% gcd_exp
thf(fact_6482_gcd__1__int,axiom,
    ! [M: int] :
      ( ( gcd_gcd @ int @ M @ ( one_one @ int ) )
      = ( one_one @ int ) ) ).

% gcd_1_int
thf(fact_6483_max__ins__scaled,axiom,
    ! [N2: nat,X14: vEBT_VEBT,M: nat,X13: list @ vEBT_VEBT] : ( ord_less_eq @ nat @ ( times_times @ nat @ N2 @ ( vEBT_VEBT_height @ X14 ) ) @ ( plus_plus @ nat @ M @ ( times_times @ nat @ N2 @ ( lattic643756798349783984er_Max @ nat @ ( insert @ nat @ ( vEBT_VEBT_height @ X14 ) @ ( image @ vEBT_VEBT @ nat @ vEBT_VEBT_height @ ( set2 @ vEBT_VEBT @ X13 ) ) ) ) ) ) ) ).

% max_ins_scaled
thf(fact_6484_height__i__max,axiom,
    ! [I: nat,X13: list @ vEBT_VEBT,Foo: nat] :
      ( ( ord_less @ nat @ I @ ( size_size @ ( list @ vEBT_VEBT ) @ X13 ) )
     => ( ord_less_eq @ nat @ ( vEBT_VEBT_height @ ( nth @ vEBT_VEBT @ X13 @ I ) ) @ ( ord_max @ nat @ Foo @ ( lattic643756798349783984er_Max @ nat @ ( image @ vEBT_VEBT @ nat @ vEBT_VEBT_height @ ( set2 @ vEBT_VEBT @ X13 ) ) ) ) ) ) ).

% height_i_max
thf(fact_6485_image__add__0,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add @ A )
     => ! [S3: set @ A] :
          ( ( image @ A @ A @ ( plus_plus @ A @ ( zero_zero @ A ) ) @ S3 )
          = S3 ) ) ).

% image_add_0
thf(fact_6486_surj__plus,axiom,
    ! [A: $tType] :
      ( ( ab_group_add @ A )
     => ! [A2: A] :
          ( ( image @ A @ A @ ( plus_plus @ A @ A2 ) @ ( top_top @ ( set @ A ) ) )
          = ( top_top @ ( set @ A ) ) ) ) ).

% surj_plus
thf(fact_6487_range__add,axiom,
    ! [A: $tType] :
      ( ( group_add @ A )
     => ! [A2: A] :
          ( ( image @ A @ A @ ( plus_plus @ A @ A2 ) @ ( top_top @ ( set @ A ) ) )
          = ( top_top @ ( set @ A ) ) ) ) ).

% range_add
thf(fact_6488_range__diff,axiom,
    ! [A: $tType] :
      ( ( group_add @ A )
     => ! [A2: A] :
          ( ( image @ A @ A @ ( minus_minus @ A @ A2 ) @ ( top_top @ ( set @ A ) ) )
          = ( top_top @ ( set @ A ) ) ) ) ).

% range_diff
thf(fact_6489_image__add__atLeastAtMost,axiom,
    ! [A: $tType] :
      ( ( linordered_semidom @ A )
     => ! [K: A,I: A,J: A] :
          ( ( image @ A @ A @ ( plus_plus @ A @ K ) @ ( set_or1337092689740270186AtMost @ A @ I @ J ) )
          = ( set_or1337092689740270186AtMost @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ K ) ) ) ) ).

% image_add_atLeastAtMost
thf(fact_6490_image__diff__atLeastAtMost,axiom,
    ! [A: $tType] :
      ( ( linordered_idom @ A )
     => ! [D2: A,A2: A,B2: A] :
          ( ( image @ A @ A @ ( minus_minus @ A @ D2 ) @ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) )
          = ( set_or1337092689740270186AtMost @ A @ ( minus_minus @ A @ D2 @ B2 ) @ ( minus_minus @ A @ D2 @ A2 ) ) ) ) ).

% image_diff_atLeastAtMost
thf(fact_6491_image__add__atLeastLessThan,axiom,
    ! [A: $tType] :
      ( ( linordered_semidom @ A )
     => ! [K: A,I: A,J: A] :
          ( ( image @ A @ A @ ( plus_plus @ A @ K ) @ ( set_or7035219750837199246ssThan @ A @ I @ J ) )
          = ( set_or7035219750837199246ssThan @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ K ) ) ) ) ).

% image_add_atLeastLessThan
thf(fact_6492_gcd__neg__numeral__1,axiom,
    ! [A: $tType] :
      ( ( ring_gcd @ A )
     => ! [N2: num,A2: A] :
          ( ( gcd_gcd @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N2 ) ) @ A2 )
          = ( gcd_gcd @ A @ ( numeral_numeral @ A @ N2 ) @ A2 ) ) ) ).

% gcd_neg_numeral_1
thf(fact_6493_gcd__neg__numeral__2,axiom,
    ! [A: $tType] :
      ( ( ring_gcd @ A )
     => ! [A2: A,N2: num] :
          ( ( gcd_gcd @ A @ A2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N2 ) ) )
          = ( gcd_gcd @ A @ A2 @ ( numeral_numeral @ A @ N2 ) ) ) ) ).

% gcd_neg_numeral_2
thf(fact_6494_list_Oset__map,axiom,
    ! [B: $tType,A: $tType,F: A > B,V: list @ A] :
      ( ( set2 @ B @ ( map @ A @ B @ F @ V ) )
      = ( image @ A @ B @ F @ ( set2 @ A @ V ) ) ) ).

% list.set_map
thf(fact_6495_is__unit__gcd__iff,axiom,
    ! [A: $tType] :
      ( ( semiring_gcd @ A )
     => ! [A2: A,B2: A] :
          ( ( dvd_dvd @ A @ ( gcd_gcd @ A @ A2 @ B2 ) @ ( one_one @ A ) )
          = ( ( gcd_gcd @ A @ A2 @ B2 )
            = ( one_one @ A ) ) ) ) ).

% is_unit_gcd_iff
thf(fact_6496_image__add__atMost,axiom,
    ! [A: $tType] :
      ( ( ordered_ab_group_add @ A )
     => ! [C2: A,A2: A] :
          ( ( image @ A @ A @ ( plus_plus @ A @ C2 ) @ ( set_ord_atMost @ A @ A2 ) )
          = ( set_ord_atMost @ A @ ( plus_plus @ A @ C2 @ A2 ) ) ) ) ).

% image_add_atMost
thf(fact_6497_bij__betw__add,axiom,
    ! [A: $tType] :
      ( ( cancel_semigroup_add @ A )
     => ! [A2: A,A3: set @ A,B6: set @ A] :
          ( ( bij_betw @ A @ A @ ( plus_plus @ A @ A2 ) @ A3 @ B6 )
          = ( ( image @ A @ A @ ( plus_plus @ A @ A2 ) @ A3 )
            = B6 ) ) ) ).

% bij_betw_add
thf(fact_6498_max__idx__list,axiom,
    ! [I: nat,X13: list @ vEBT_VEBT,N2: nat,X14: vEBT_VEBT] :
      ( ( ord_less @ nat @ I @ ( size_size @ ( list @ vEBT_VEBT ) @ X13 ) )
     => ( ord_less_eq @ nat @ ( times_times @ nat @ N2 @ ( vEBT_VEBT_height @ ( nth @ vEBT_VEBT @ X13 @ I ) ) ) @ ( suc @ ( suc @ ( times_times @ nat @ N2 @ ( ord_max @ nat @ ( vEBT_VEBT_height @ X14 ) @ ( lattic643756798349783984er_Max @ nat @ ( image @ vEBT_VEBT @ nat @ vEBT_VEBT_height @ ( set2 @ vEBT_VEBT @ X13 ) ) ) ) ) ) ) ) ) ).

% max_idx_list
thf(fact_6499_gcd__neg__numeral__2__int,axiom,
    ! [X2: int,N2: num] :
      ( ( gcd_gcd @ int @ X2 @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N2 ) ) )
      = ( gcd_gcd @ int @ X2 @ ( numeral_numeral @ int @ N2 ) ) ) ).

% gcd_neg_numeral_2_int
thf(fact_6500_gcd__neg__numeral__1__int,axiom,
    ! [N2: num,X2: int] :
      ( ( gcd_gcd @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N2 ) ) @ X2 )
      = ( gcd_gcd @ int @ ( numeral_numeral @ int @ N2 ) @ X2 ) ) ).

% gcd_neg_numeral_1_int
thf(fact_6501_surj__diff__right,axiom,
    ! [A: $tType] :
      ( ( ab_group_add @ A )
     => ! [A2: A] :
          ( ( image @ A @ A
            @ ^ [X: A] : ( minus_minus @ A @ X @ A2 )
            @ ( top_top @ ( set @ A ) ) )
          = ( top_top @ ( set @ A ) ) ) ) ).

% surj_diff_right
thf(fact_6502_image__add__atLeastAtMost_H,axiom,
    ! [A: $tType] :
      ( ( linordered_semidom @ A )
     => ! [K: A,I: A,J: A] :
          ( ( image @ A @ A
            @ ^ [N: A] : ( plus_plus @ A @ N @ K )
            @ ( set_or1337092689740270186AtMost @ A @ I @ J ) )
          = ( set_or1337092689740270186AtMost @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ K ) ) ) ) ).

% image_add_atLeastAtMost'
thf(fact_6503_image__minus__const__atLeastAtMost_H,axiom,
    ! [A: $tType] :
      ( ( linordered_idom @ A )
     => ! [D2: A,A2: A,B2: A] :
          ( ( image @ A @ A
            @ ^ [T3: A] : ( minus_minus @ A @ T3 @ D2 )
            @ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) )
          = ( set_or1337092689740270186AtMost @ A @ ( minus_minus @ A @ A2 @ D2 ) @ ( minus_minus @ A @ B2 @ D2 ) ) ) ) ).

% image_minus_const_atLeastAtMost'
thf(fact_6504_image__add__atLeastLessThan_H,axiom,
    ! [A: $tType] :
      ( ( linordered_semidom @ A )
     => ! [K: A,I: A,J: A] :
          ( ( image @ A @ A
            @ ^ [N: A] : ( plus_plus @ A @ N @ K )
            @ ( set_or7035219750837199246ssThan @ A @ I @ J ) )
          = ( set_or7035219750837199246ssThan @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ K ) ) ) ) ).

% image_add_atLeastLessThan'
thf(fact_6505_Max__divisors__self__nat,axiom,
    ! [N2: nat] :
      ( ( N2
       != ( zero_zero @ nat ) )
     => ( ( lattic643756798349783984er_Max @ nat
          @ ( collect @ nat
            @ ^ [D3: nat] : ( dvd_dvd @ nat @ D3 @ N2 ) ) )
        = N2 ) ) ).

% Max_divisors_self_nat
thf(fact_6506_Max_Obounded__iff,axiom,
    ! [A: $tType] :
      ( ( linorder @ A )
     => ! [A3: set @ A,X2: A] :
          ( ( finite_finite2 @ A @ A3 )
         => ( ( A3
             != ( bot_bot @ ( set @ A ) ) )
           => ( ( ord_less_eq @ A @ ( lattic643756798349783984er_Max @ A @ A3 ) @ X2 )
              = ( ! [X: A] :
                    ( ( member @ A @ X @ A3 )
                   => ( ord_less_eq @ A @ X @ X2 ) ) ) ) ) ) ) ).

% Max.bounded_iff
thf(fact_6507_Max__less__iff,axiom,
    ! [A: $tType] :
      ( ( linorder @ A )
     => ! [A3: set @ A,X2: A] :
          ( ( finite_finite2 @ A @ A3 )
         => ( ( A3
             != ( bot_bot @ ( set @ A ) ) )
           => ( ( ord_less @ A @ ( lattic643756798349783984er_Max @ A @ A3 ) @ X2 )
              = ( ! [X: A] :
                    ( ( member @ A @ X @ A3 )
                   => ( ord_less @ A @ X @ X2 ) ) ) ) ) ) ) ).

% Max_less_iff
thf(fact_6508_range__constant,axiom,
    ! [B: $tType,A: $tType,X2: A] :
      ( ( image @ B @ A
        @ ^ [Uu3: B] : X2
        @ ( top_top @ ( set @ B ) ) )
      = ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ).

% range_constant
thf(fact_6509_Max__const,axiom,
    ! [B: $tType,A: $tType] :
      ( ( linorder @ A )
     => ! [A3: set @ B,C2: A] :
          ( ( finite_finite2 @ B @ A3 )
         => ( ( A3
             != ( bot_bot @ ( set @ B ) ) )
           => ( ( lattic643756798349783984er_Max @ A
                @ ( image @ B @ A
                  @ ^ [Uu3: B] : C2
                  @ A3 ) )
              = C2 ) ) ) ) ).

% Max_const
thf(fact_6510_image__mult__atLeastAtMost,axiom,
    ! [A: $tType] :
      ( ( linordered_field @ A )
     => ! [D2: A,A2: A,B2: A] :
          ( ( ord_less @ A @ ( zero_zero @ A ) @ D2 )
         => ( ( image @ A @ A @ ( times_times @ A @ D2 ) @ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) )
            = ( set_or1337092689740270186AtMost @ A @ ( times_times @ A @ D2 @ A2 ) @ ( times_times @ A @ D2 @ B2 ) ) ) ) ) ).

% image_mult_atLeastAtMost
thf(fact_6511_image__divide__atLeastAtMost,axiom,
    ! [A: $tType] :
      ( ( linordered_field @ A )
     => ! [D2: A,A2: A,B2: A] :
          ( ( ord_less @ A @ ( zero_zero @ A ) @ D2 )
         => ( ( image @ A @ A
              @ ^ [C4: A] : ( divide_divide @ A @ C4 @ D2 )
              @ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) )
            = ( set_or1337092689740270186AtMost @ A @ ( divide_divide @ A @ A2 @ D2 ) @ ( divide_divide @ A @ B2 @ D2 ) ) ) ) ) ).

% image_divide_atLeastAtMost
thf(fact_6512_translation__subtract__Compl,axiom,
    ! [A: $tType] :
      ( ( ab_group_add @ A )
     => ! [A2: A,T2: set @ A] :
          ( ( image @ A @ A
            @ ^ [X: A] : ( minus_minus @ A @ X @ A2 )
            @ ( uminus_uminus @ ( set @ A ) @ T2 ) )
          = ( uminus_uminus @ ( set @ A )
            @ ( image @ A @ A
              @ ^ [X: A] : ( minus_minus @ A @ X @ A2 )
              @ T2 ) ) ) ) ).

% translation_subtract_Compl
thf(fact_6513_translation__Compl,axiom,
    ! [A: $tType] :
      ( ( ab_group_add @ A )
     => ! [A2: A,T2: set @ A] :
          ( ( image @ A @ A @ ( plus_plus @ A @ A2 ) @ ( uminus_uminus @ ( set @ A ) @ T2 ) )
          = ( uminus_uminus @ ( set @ A ) @ ( image @ A @ A @ ( plus_plus @ A @ A2 ) @ T2 ) ) ) ) ).

% translation_Compl
thf(fact_6514_Max__add__commute,axiom,
    ! [B: $tType,A: $tType] :
      ( ( linord4140545234300271783up_add @ A )
     => ! [S3: set @ B,F: B > A,K: A] :
          ( ( finite_finite2 @ B @ S3 )
         => ( ( S3
             != ( bot_bot @ ( set @ B ) ) )
           => ( ( lattic643756798349783984er_Max @ A
                @ ( image @ B @ A
                  @ ^ [X: B] : ( plus_plus @ A @ ( F @ X ) @ K )
                  @ S3 ) )
              = ( plus_plus @ A @ ( lattic643756798349783984er_Max @ A @ ( image @ B @ A @ F @ S3 ) ) @ K ) ) ) ) ) ).

% Max_add_commute
thf(fact_6515_translation__diff,axiom,
    ! [A: $tType] :
      ( ( ab_group_add @ A )
     => ! [A2: A,S4: set @ A,T2: set @ A] :
          ( ( image @ A @ A @ ( plus_plus @ A @ A2 ) @ ( minus_minus @ ( set @ A ) @ S4 @ T2 ) )
          = ( minus_minus @ ( set @ A ) @ ( image @ A @ A @ ( plus_plus @ A @ A2 ) @ S4 ) @ ( image @ A @ A @ ( plus_plus @ A @ A2 ) @ T2 ) ) ) ) ).

% translation_diff
thf(fact_6516_prod_Oimage__gen,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( ( comm_monoid_mult @ A )
     => ! [S3: set @ B,H2: B > A,G: B > C] :
          ( ( finite_finite2 @ B @ S3 )
         => ( ( groups7121269368397514597t_prod @ B @ A @ H2 @ S3 )
            = ( groups7121269368397514597t_prod @ C @ A
              @ ^ [Y: C] :
                  ( groups7121269368397514597t_prod @ B @ A @ H2
                  @ ( collect @ B
                    @ ^ [X: B] :
                        ( ( member @ B @ X @ S3 )
                        & ( ( G @ X )
                          = Y ) ) ) )
              @ ( image @ B @ C @ G @ S3 ) ) ) ) ) ).

% prod.image_gen
thf(fact_6517_gcd__diff1,axiom,
    ! [A: $tType] :
      ( ( ring_gcd @ A )
     => ! [M: A,N2: A] :
          ( ( gcd_gcd @ A @ ( minus_minus @ A @ M @ N2 ) @ N2 )
          = ( gcd_gcd @ A @ M @ N2 ) ) ) ).

% gcd_diff1
thf(fact_6518_gcd__diff2,axiom,
    ! [A: $tType] :
      ( ( ring_gcd @ A )
     => ! [N2: A,M: A] :
          ( ( gcd_gcd @ A @ ( minus_minus @ A @ N2 @ M ) @ N2 )
          = ( gcd_gcd @ A @ M @ N2 ) ) ) ).

% gcd_diff2
thf(fact_6519_image__diff__subset,axiom,
    ! [A: $tType,B: $tType,F: B > A,A3: set @ B,B6: set @ B] : ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ ( image @ B @ A @ F @ A3 ) @ ( image @ B @ A @ F @ B6 ) ) @ ( image @ B @ A @ F @ ( minus_minus @ ( set @ B ) @ A3 @ B6 ) ) ) ).

% image_diff_subset
thf(fact_6520_Max_OcoboundedI,axiom,
    ! [A: $tType] :
      ( ( linorder @ A )
     => ! [A3: set @ A,A2: A] :
          ( ( finite_finite2 @ A @ A3 )
         => ( ( member @ A @ A2 @ A3 )
           => ( ord_less_eq @ A @ A2 @ ( lattic643756798349783984er_Max @ A @ A3 ) ) ) ) ) ).

% Max.coboundedI
thf(fact_6521_Max__eq__if,axiom,
    ! [A: $tType] :
      ( ( linorder @ A )
     => ! [A3: set @ A,B6: set @ A] :
          ( ( finite_finite2 @ A @ A3 )
         => ( ( finite_finite2 @ A @ B6 )
           => ( ! [X3: A] :
                  ( ( member @ A @ X3 @ A3 )
                 => ? [Xa2: A] :
                      ( ( member @ A @ Xa2 @ B6 )
                      & ( ord_less_eq @ A @ X3 @ Xa2 ) ) )
             => ( ! [X3: A] :
                    ( ( member @ A @ X3 @ B6 )
                   => ? [Xa2: A] :
                        ( ( member @ A @ Xa2 @ A3 )
                        & ( ord_less_eq @ A @ X3 @ Xa2 ) ) )
               => ( ( lattic643756798349783984er_Max @ A @ A3 )
                  = ( lattic643756798349783984er_Max @ A @ B6 ) ) ) ) ) ) ) ).

% Max_eq_if
thf(fact_6522_Max__eqI,axiom,
    ! [A: $tType] :
      ( ( linorder @ A )
     => ! [A3: set @ A,X2: A] :
          ( ( finite_finite2 @ A @ A3 )
         => ( ! [Y4: A] :
                ( ( member @ A @ Y4 @ A3 )
               => ( ord_less_eq @ A @ Y4 @ X2 ) )
           => ( ( member @ A @ X2 @ A3 )
             => ( ( lattic643756798349783984er_Max @ A @ A3 )
                = X2 ) ) ) ) ) ).

% Max_eqI
thf(fact_6523_Max__ge,axiom,
    ! [A: $tType] :
      ( ( linorder @ A )
     => ! [A3: set @ A,X2: A] :
          ( ( finite_finite2 @ A @ A3 )
         => ( ( member @ A @ X2 @ A3 )
           => ( ord_less_eq @ A @ X2 @ ( lattic643756798349783984er_Max @ A @ A3 ) ) ) ) ) ).

% Max_ge
thf(fact_6524_finite__surj,axiom,
    ! [A: $tType,B: $tType,A3: set @ A,B6: set @ B,F: A > B] :
      ( ( finite_finite2 @ A @ A3 )
     => ( ( ord_less_eq @ ( set @ B ) @ B6 @ ( image @ A @ B @ F @ A3 ) )
       => ( finite_finite2 @ B @ B6 ) ) ) ).

% finite_surj
thf(fact_6525_finite__subset__image,axiom,
    ! [A: $tType,B: $tType,B6: set @ A,F: B > A,A3: set @ B] :
      ( ( finite_finite2 @ A @ B6 )
     => ( ( ord_less_eq @ ( set @ A ) @ B6 @ ( image @ B @ A @ F @ A3 ) )
       => ? [C8: set @ B] :
            ( ( ord_less_eq @ ( set @ B ) @ C8 @ A3 )
            & ( finite_finite2 @ B @ C8 )
            & ( B6
              = ( image @ B @ A @ F @ C8 ) ) ) ) ) ).

% finite_subset_image
thf(fact_6526_ex__finite__subset__image,axiom,
    ! [A: $tType,B: $tType,F: B > A,A3: set @ B,P: ( set @ A ) > $o] :
      ( ( ? [B7: set @ A] :
            ( ( finite_finite2 @ A @ B7 )
            & ( ord_less_eq @ ( set @ A ) @ B7 @ ( image @ B @ A @ F @ A3 ) )
            & ( P @ B7 ) ) )
      = ( ? [B7: set @ B] :
            ( ( finite_finite2 @ B @ B7 )
            & ( ord_less_eq @ ( set @ B ) @ B7 @ A3 )
            & ( P @ ( image @ B @ A @ F @ B7 ) ) ) ) ) ).

% ex_finite_subset_image
thf(fact_6527_all__finite__subset__image,axiom,
    ! [A: $tType,B: $tType,F: B > A,A3: set @ B,P: ( set @ A ) > $o] :
      ( ( ! [B7: set @ A] :
            ( ( ( finite_finite2 @ A @ B7 )
              & ( ord_less_eq @ ( set @ A ) @ B7 @ ( image @ B @ A @ F @ A3 ) ) )
           => ( P @ B7 ) ) )
      = ( ! [B7: set @ B] :
            ( ( ( finite_finite2 @ B @ B7 )
              & ( ord_less_eq @ ( set @ B ) @ B7 @ A3 ) )
           => ( P @ ( image @ B @ A @ F @ B7 ) ) ) ) ) ).

% all_finite_subset_image
thf(fact_6528_range__composition,axiom,
    ! [A: $tType,C: $tType,B: $tType,F: C > A,G: B > C] :
      ( ( image @ B @ A
        @ ^ [X: B] : ( F @ ( G @ X ) )
        @ ( top_top @ ( set @ B ) ) )
      = ( image @ C @ A @ F @ ( image @ B @ C @ G @ ( top_top @ ( set @ B ) ) ) ) ) ).

% range_composition
thf(fact_6529_rangeE,axiom,
    ! [A: $tType,B: $tType,B2: A,F: B > A] :
      ( ( member @ A @ B2 @ ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) ) )
     => ~ ! [X3: B] :
            ( B2
           != ( F @ X3 ) ) ) ).

% rangeE
thf(fact_6530_image__mono,axiom,
    ! [B: $tType,A: $tType,A3: set @ A,B6: set @ A,F: A > B] :
      ( ( ord_less_eq @ ( set @ A ) @ A3 @ B6 )
     => ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F @ A3 ) @ ( image @ A @ B @ F @ B6 ) ) ) ).

% image_mono
thf(fact_6531_image__subsetI,axiom,
    ! [A: $tType,B: $tType,A3: set @ A,F: A > B,B6: set @ B] :
      ( ! [X3: A] :
          ( ( member @ A @ X3 @ A3 )
         => ( member @ B @ ( F @ X3 ) @ B6 ) )
     => ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F @ A3 ) @ B6 ) ) ).

% image_subsetI
thf(fact_6532_subset__imageE,axiom,
    ! [A: $tType,B: $tType,B6: set @ A,F: B > A,A3: set @ B] :
      ( ( ord_less_eq @ ( set @ A ) @ B6 @ ( image @ B @ A @ F @ A3 ) )
     => ~ ! [C8: set @ B] :
            ( ( ord_less_eq @ ( set @ B ) @ C8 @ A3 )
           => ( B6
             != ( image @ B @ A @ F @ C8 ) ) ) ) ).

% subset_imageE
thf(fact_6533_image__subset__iff,axiom,
    ! [A: $tType,B: $tType,F: B > A,A3: set @ B,B6: set @ A] :
      ( ( ord_less_eq @ ( set @ A ) @ ( image @ B @ A @ F @ A3 ) @ B6 )
      = ( ! [X: B] :
            ( ( member @ B @ X @ A3 )
           => ( member @ A @ ( F @ X ) @ B6 ) ) ) ) ).

% image_subset_iff
thf(fact_6534_subset__image__iff,axiom,
    ! [A: $tType,B: $tType,B6: set @ A,F: B > A,A3: set @ B] :
      ( ( ord_less_eq @ ( set @ A ) @ B6 @ ( image @ B @ A @ F @ A3 ) )
      = ( ? [AA: set @ B] :
            ( ( ord_less_eq @ ( set @ B ) @ AA @ A3 )
            & ( B6
              = ( image @ B @ A @ F @ AA ) ) ) ) ) ).

% subset_image_iff
thf(fact_6535_all__subset__image,axiom,
    ! [A: $tType,B: $tType,F: B > A,A3: set @ B,P: ( set @ A ) > $o] :
      ( ( ! [B7: set @ A] :
            ( ( ord_less_eq @ ( set @ A ) @ B7 @ ( image @ B @ A @ F @ A3 ) )
           => ( P @ B7 ) ) )
      = ( ! [B7: set @ B] :
            ( ( ord_less_eq @ ( set @ B ) @ B7 @ A3 )
           => ( P @ ( image @ B @ A @ F @ B7 ) ) ) ) ) ).

% all_subset_image
thf(fact_6536_range__subsetD,axiom,
    ! [B: $tType,A: $tType,F: B > A,B6: set @ A,I: B] :
      ( ( ord_less_eq @ ( set @ A ) @ ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) ) @ B6 )
     => ( member @ A @ ( F @ I ) @ B6 ) ) ).

% range_subsetD
thf(fact_6537_imageE,axiom,
    ! [A: $tType,B: $tType,B2: A,F: B > A,A3: set @ B] :
      ( ( member @ A @ B2 @ ( image @ B @ A @ F @ A3 ) )
     => ~ ! [X3: B] :
            ( ( B2
              = ( F @ X3 ) )
           => ~ ( member @ B @ X3 @ A3 ) ) ) ).

% imageE
thf(fact_6538_image__image,axiom,
    ! [A: $tType,B: $tType,C: $tType,F: B > A,G: C > B,A3: set @ C] :
      ( ( image @ B @ A @ F @ ( image @ C @ B @ G @ A3 ) )
      = ( image @ C @ A
        @ ^ [X: C] : ( F @ ( G @ X ) )
        @ A3 ) ) ).

% image_image
thf(fact_6539_Compr__image__eq,axiom,
    ! [A: $tType,B: $tType,F: B > A,A3: set @ B,P: A > $o] :
      ( ( collect @ A
        @ ^ [X: A] :
            ( ( member @ A @ X @ ( image @ B @ A @ F @ A3 ) )
            & ( P @ X ) ) )
      = ( image @ B @ A @ F
        @ ( collect @ B
          @ ^ [X: B] :
              ( ( member @ B @ X @ A3 )
              & ( P @ ( F @ X ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_6540_pigeonhole__infinite,axiom,
    ! [B: $tType,A: $tType,A3: set @ A,F: A > B] :
      ( ~ ( finite_finite2 @ A @ A3 )
     => ( ( finite_finite2 @ B @ ( image @ A @ B @ F @ A3 ) )
       => ? [X3: A] :
            ( ( member @ A @ X3 @ A3 )
            & ~ ( finite_finite2 @ A
                @ ( collect @ A
                  @ ^ [A5: A] :
                      ( ( member @ A @ A5 @ A3 )
                      & ( ( F @ A5 )
                        = ( F @ X3 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite
thf(fact_6541_gcd__ge__0__int,axiom,
    ! [X2: int,Y2: int] : ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( gcd_gcd @ int @ X2 @ Y2 ) ) ).

% gcd_ge_0_int
thf(fact_6542_bij__betw__byWitness,axiom,
    ! [A: $tType,B: $tType,A3: set @ A,F4: B > A,F: A > B,A10: set @ B] :
      ( ! [X3: A] :
          ( ( member @ A @ X3 @ A3 )
         => ( ( F4 @ ( F @ X3 ) )
            = X3 ) )
     => ( ! [X3: B] :
            ( ( member @ B @ X3 @ A10 )
           => ( ( F @ ( F4 @ X3 ) )
              = X3 ) )
       => ( ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F @ A3 ) @ A10 )
         => ( ( ord_less_eq @ ( set @ A ) @ ( image @ B @ A @ F4 @ A10 ) @ A3 )
           => ( bij_betw @ A @ B @ F @ A3 @ A10 ) ) ) ) ) ).

% bij_betw_byWitness
thf(fact_6543_bij__betw__subset,axiom,
    ! [A: $tType,B: $tType,F: A > B,A3: set @ A,A10: set @ B,B6: set @ A,B11: set @ B] :
      ( ( bij_betw @ A @ B @ F @ A3 @ A10 )
     => ( ( ord_less_eq @ ( set @ A ) @ B6 @ A3 )
       => ( ( ( image @ A @ B @ F @ B6 )
            = B11 )
         => ( bij_betw @ A @ B @ F @ B6 @ B11 ) ) ) ) ).

% bij_betw_subset
thf(fact_6544_image__constant__conv,axiom,
    ! [B: $tType,A: $tType,A3: set @ B,C2: A] :
      ( ( ( A3
          = ( bot_bot @ ( set @ B ) ) )
       => ( ( image @ B @ A
            @ ^ [X: B] : C2
            @ A3 )
          = ( bot_bot @ ( set @ A ) ) ) )
      & ( ( A3
         != ( bot_bot @ ( set @ B ) ) )
       => ( ( image @ B @ A
            @ ^ [X: B] : C2
            @ A3 )
          = ( insert @ A @ C2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).

% image_constant_conv
thf(fact_6545_image__constant,axiom,
    ! [A: $tType,B: $tType,X2: A,A3: set @ A,C2: B] :
      ( ( member @ A @ X2 @ A3 )
     => ( ( image @ A @ B
          @ ^ [X: A] : C2
          @ A3 )
        = ( insert @ B @ C2 @ ( bot_bot @ ( set @ B ) ) ) ) ) ).

% image_constant
thf(fact_6546_finite__range__imageI,axiom,
    ! [C: $tType,A: $tType,B: $tType,G: B > A,F: A > C] :
      ( ( finite_finite2 @ A @ ( image @ B @ A @ G @ ( top_top @ ( set @ B ) ) ) )
     => ( finite_finite2 @ C
        @ ( image @ B @ C
          @ ^ [X: B] : ( F @ ( G @ X ) )
          @ ( top_top @ ( set @ B ) ) ) ) ) ).

% finite_range_imageI
thf(fact_6547_image__set,axiom,
    ! [A: $tType,B: $tType,F: B > A,Xs2: list @ B] :
      ( ( image @ B @ A @ F @ ( set2 @ B @ Xs2 ) )
      = ( set2 @ A @ ( map @ B @ A @ F @ Xs2 ) ) ) ).

% image_set
thf(fact_6548_translation__subtract__diff,axiom,
    ! [A: $tType] :
      ( ( ab_group_add @ A )
     => ! [A2: A,S4: set @ A,T2: set @ A] :
          ( ( image @ A @ A
            @ ^ [X: A] : ( minus_minus @ A @ X @ A2 )
            @ ( minus_minus @ ( set @ A ) @ S4 @ T2 ) )
          = ( minus_minus @ ( set @ A )
            @ ( image @ A @ A
              @ ^ [X: A] : ( minus_minus @ A @ X @ A2 )
              @ S4 )
            @ ( image @ A @ A
              @ ^ [X: A] : ( minus_minus @ A @ X @ A2 )
              @ T2 ) ) ) ) ).

% translation_subtract_diff
thf(fact_6549_sum_Oimage__gen,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( ( comm_monoid_add @ A )
     => ! [S3: set @ B,H2: B > A,G: B > C] :
          ( ( finite_finite2 @ B @ S3 )
         => ( ( groups7311177749621191930dd_sum @ B @ A @ H2 @ S3 )
            = ( groups7311177749621191930dd_sum @ C @ A
              @ ^ [Y: C] :
                  ( groups7311177749621191930dd_sum @ B @ A @ H2
                  @ ( collect @ B
                    @ ^ [X: B] :
                        ( ( member @ B @ X @ S3 )
                        & ( ( G @ X )
                          = Y ) ) ) )
              @ ( image @ B @ C @ G @ S3 ) ) ) ) ) ).

% sum.image_gen
thf(fact_6550_gcd__mult__distrib__int,axiom,
    ! [K: int,M: int,N2: int] :
      ( ( times_times @ int @ ( abs_abs @ int @ K ) @ ( gcd_gcd @ int @ M @ N2 ) )
      = ( gcd_gcd @ int @ ( times_times @ int @ K @ M ) @ ( times_times @ int @ K @ N2 ) ) ) ).

% gcd_mult_distrib_int
thf(fact_6551_gcd__dvd__prod,axiom,
    ! [A: $tType] :
      ( ( semiring_gcd @ A )
     => ! [A2: A,B2: A,K: A] : ( dvd_dvd @ A @ ( gcd_gcd @ A @ A2 @ B2 ) @ ( times_times @ A @ K @ B2 ) ) ) ).

% gcd_dvd_prod
thf(fact_6552_gcd__add__mult,axiom,
    ! [A: $tType] :
      ( ( semiring_gcd @ A )
     => ! [M: A,K: A,N2: A] :
          ( ( gcd_gcd @ A @ M @ ( plus_plus @ A @ ( times_times @ A @ K @ M ) @ N2 ) )
          = ( gcd_gcd @ A @ M @ N2 ) ) ) ).

% gcd_add_mult
thf(fact_6553_bezout__int,axiom,
    ! [X2: int,Y2: int] :
    ? [U4: int,V3: int] :
      ( ( plus_plus @ int @ ( times_times @ int @ U4 @ X2 ) @ ( times_times @ int @ V3 @ Y2 ) )
      = ( gcd_gcd @ int @ X2 @ Y2 ) ) ).

% bezout_int
thf(fact_6554_gcd__mult__unit2,axiom,
    ! [A: $tType] :
      ( ( semiring_gcd @ A )
     => ! [A2: A,B2: A,C2: A] :
          ( ( dvd_dvd @ A @ A2 @ ( one_one @ A ) )
         => ( ( gcd_gcd @ A @ B2 @ ( times_times @ A @ C2 @ A2 ) )
            = ( gcd_gcd @ A @ B2 @ C2 ) ) ) ) ).

% gcd_mult_unit2
thf(fact_6555_gcd__mult__unit1,axiom,
    ! [A: $tType] :
      ( ( semiring_gcd @ A )
     => ! [A2: A,B2: A,C2: A] :
          ( ( dvd_dvd @ A @ A2 @ ( one_one @ A ) )
         => ( ( gcd_gcd @ A @ ( times_times @ A @ B2 @ A2 ) @ C2 )
            = ( gcd_gcd @ A @ B2 @ C2 ) ) ) ) ).

% gcd_mult_unit1
thf(fact_6556_gcd__div__unit1,axiom,
    ! [A: $tType] :
      ( ( semiring_gcd @ A )
     => ! [A2: A,B2: A,C2: A] :
          ( ( dvd_dvd @ A @ A2 @ ( one_one @ A ) )
         => ( ( gcd_gcd @ A @ ( divide_divide @ A @ B2 @ A2 ) @ C2 )
            = ( gcd_gcd @ A @ B2 @ C2 ) ) ) ) ).

% gcd_div_unit1
thf(fact_6557_gcd__div__unit2,axiom,
    ! [A: $tType] :
      ( ( semiring_gcd @ A )
     => ! [A2: A,B2: A,C2: A] :
          ( ( dvd_dvd @ A @ A2 @ ( one_one @ A ) )
         => ( ( gcd_gcd @ A @ B2 @ ( divide_divide @ A @ C2 @ A2 ) )
            = ( gcd_gcd @ A @ B2 @ C2 ) ) ) ) ).

% gcd_div_unit2
thf(fact_6558_gcd__le2__int,axiom,
    ! [B2: int,A2: int] :
      ( ( ord_less @ int @ ( zero_zero @ int ) @ B2 )
     => ( ord_less_eq @ int @ ( gcd_gcd @ int @ A2 @ B2 ) @ B2 ) ) ).

% gcd_le2_int
thf(fact_6559_gcd__le1__int,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less @ int @ ( zero_zero @ int ) @ A2 )
     => ( ord_less_eq @ int @ ( gcd_gcd @ int @ A2 @ B2 ) @ A2 ) ) ).

% gcd_le1_int
thf(fact_6560_gcd__cases__int,axiom,
    ! [X2: int,Y2: int,P: int > $o] :
      ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X2 )
       => ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y2 )
         => ( P @ ( gcd_gcd @ int @ X2 @ Y2 ) ) ) )
     => ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X2 )
         => ( ( ord_less_eq @ int @ Y2 @ ( zero_zero @ int ) )
           => ( P @ ( gcd_gcd @ int @ X2 @ ( uminus_uminus @ int @ Y2 ) ) ) ) )
       => ( ( ( ord_less_eq @ int @ X2 @ ( zero_zero @ int ) )
           => ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y2 )
             => ( P @ ( gcd_gcd @ int @ ( uminus_uminus @ int @ X2 ) @ Y2 ) ) ) )
         => ( ( ( ord_less_eq @ int @ X2 @ ( zero_zero @ int ) )
             => ( ( ord_less_eq @ int @ Y2 @ ( zero_zero @ int ) )
               => ( P @ ( gcd_gcd @ int @ ( uminus_uminus @ int @ X2 ) @ ( uminus_uminus @ int @ Y2 ) ) ) ) )
           => ( P @ ( gcd_gcd @ int @ X2 @ Y2 ) ) ) ) ) ) ).

% gcd_cases_int
thf(fact_6561_gcd__unique__int,axiom,
    ! [D2: int,A2: int,B2: int] :
      ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ D2 )
        & ( dvd_dvd @ int @ D2 @ A2 )
        & ( dvd_dvd @ int @ D2 @ B2 )
        & ! [E4: int] :
            ( ( ( dvd_dvd @ int @ E4 @ A2 )
              & ( dvd_dvd @ int @ E4 @ B2 ) )
           => ( dvd_dvd @ int @ E4 @ D2 ) ) )
      = ( D2
        = ( gcd_gcd @ int @ A2 @ B2 ) ) ) ).

% gcd_unique_int
thf(fact_6562_surj__Compl__image__subset,axiom,
    ! [A: $tType,B: $tType,F: B > A,A3: set @ B] :
      ( ( ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
        = ( top_top @ ( set @ A ) ) )
     => ( ord_less_eq @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ ( image @ B @ A @ F @ A3 ) ) @ ( image @ B @ A @ F @ ( uminus_uminus @ ( set @ B ) @ A3 ) ) ) ) ).

% surj_Compl_image_subset
thf(fact_6563_Max__eq__iff,axiom,
    ! [A: $tType] :
      ( ( linorder @ A )
     => ! [A3: set @ A,M: A] :
          ( ( finite_finite2 @ A @ A3 )
         => ( ( A3
             != ( bot_bot @ ( set @ A ) ) )
           => ( ( ( lattic643756798349783984er_Max @ A @ A3 )
                = M )
              = ( ( member @ A @ M @ A3 )
                & ! [X: A] :
                    ( ( member @ A @ X @ A3 )
                   => ( ord_less_eq @ A @ X @ M ) ) ) ) ) ) ) ).

% Max_eq_iff
thf(fact_6564_Max__ge__iff,axiom,
    ! [A: $tType] :
      ( ( linorder @ A )
     => ! [A3: set @ A,X2: A] :
          ( ( finite_finite2 @ A @ A3 )
         => ( ( A3
             != ( bot_bot @ ( set @ A ) ) )
           => ( ( ord_less_eq @ A @ X2 @ ( lattic643756798349783984er_Max @ A @ A3 ) )
              = ( ? [X: A] :
                    ( ( member @ A @ X @ A3 )
                    & ( ord_less_eq @ A @ X2 @ X ) ) ) ) ) ) ) ).

% Max_ge_iff
thf(fact_6565_eq__Max__iff,axiom,
    ! [A: $tType] :
      ( ( linorder @ A )
     => ! [A3: set @ A,M: A] :
          ( ( finite_finite2 @ A @ A3 )
         => ( ( A3
             != ( bot_bot @ ( set @ A ) ) )
           => ( ( M
                = ( lattic643756798349783984er_Max @ A @ A3 ) )
              = ( ( member @ A @ M @ A3 )
                & ! [X: A] :
                    ( ( member @ A @ X @ A3 )
                   => ( ord_less_eq @ A @ X @ M ) ) ) ) ) ) ) ).

% eq_Max_iff
thf(fact_6566_Max_OboundedE,axiom,
    ! [A: $tType] :
      ( ( linorder @ A )
     => ! [A3: set @ A,X2: A] :
          ( ( finite_finite2 @ A @ A3 )
         => ( ( A3
             != ( bot_bot @ ( set @ A ) ) )
           => ( ( ord_less_eq @ A @ ( lattic643756798349783984er_Max @ A @ A3 ) @ X2 )
             => ! [A8: A] :
                  ( ( member @ A @ A8 @ A3 )
                 => ( ord_less_eq @ A @ A8 @ X2 ) ) ) ) ) ) ).

% Max.boundedE
thf(fact_6567_Max_OboundedI,axiom,
    ! [A: $tType] :
      ( ( linorder @ A )
     => ! [A3: set @ A,X2: A] :
          ( ( finite_finite2 @ A @ A3 )
         => ( ( A3
             != ( bot_bot @ ( set @ A ) ) )
           => ( ! [A4: A] :
                  ( ( member @ A @ A4 @ A3 )
                 => ( ord_less_eq @ A @ A4 @ X2 ) )
             => ( ord_less_eq @ A @ ( lattic643756798349783984er_Max @ A @ A3 ) @ X2 ) ) ) ) ) ).

% Max.boundedI
thf(fact_6568_Max__gr__iff,axiom,
    ! [A: $tType] :
      ( ( linorder @ A )
     => ! [A3: set @ A,X2: A] :
          ( ( finite_finite2 @ A @ A3 )
         => ( ( A3
             != ( bot_bot @ ( set @ A ) ) )
           => ( ( ord_less @ A @ X2 @ ( lattic643756798349783984er_Max @ A @ A3 ) )
              = ( ? [X: A] :
                    ( ( member @ A @ X @ A3 )
                    & ( ord_less @ A @ X2 @ X ) ) ) ) ) ) ) ).

% Max_gr_iff
thf(fact_6569_Max__insert2,axiom,
    ! [A: $tType] :
      ( ( linorder @ A )
     => ! [A3: set @ A,A2: A] :
          ( ( finite_finite2 @ A @ A3 )
         => ( ! [B3: A] :
                ( ( member @ A @ B3 @ A3 )
               => ( ord_less_eq @ A @ B3 @ A2 ) )
           => ( ( lattic643756798349783984er_Max @ A @ ( insert @ A @ A2 @ A3 ) )
              = A2 ) ) ) ) ).

% Max_insert2
thf(fact_6570_VEBT__internal_Oheight_Osimps_I2_J,axiom,
    ! [Uu: option @ ( product_prod @ nat @ nat ),Deg: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( vEBT_VEBT_height @ ( vEBT_Node @ Uu @ Deg @ TreeList @ Summary ) )
      = ( plus_plus @ nat @ ( one_one @ nat ) @ ( lattic643756798349783984er_Max @ nat @ ( image @ vEBT_VEBT @ nat @ vEBT_VEBT_height @ ( insert @ vEBT_VEBT @ Summary @ ( set2 @ vEBT_VEBT @ TreeList ) ) ) ) ) ) ).

% VEBT_internal.height.simps(2)
thf(fact_6571_Max_Oinfinite,axiom,
    ! [A: $tType] :
      ( ( linorder @ A )
     => ! [A3: set @ A] :
          ( ~ ( finite_finite2 @ A @ A3 )
         => ( ( lattic643756798349783984er_Max @ A @ A3 )
            = ( the2 @ A @ ( none @ A ) ) ) ) ) ).

% Max.infinite
thf(fact_6572_sum_Ogroup,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( ( comm_monoid_add @ A )
     => ! [S3: set @ B,T6: set @ C,G: B > C,H2: B > A] :
          ( ( finite_finite2 @ B @ S3 )
         => ( ( finite_finite2 @ C @ T6 )
           => ( ( ord_less_eq @ ( set @ C ) @ ( image @ B @ C @ G @ S3 ) @ T6 )
             => ( ( groups7311177749621191930dd_sum @ C @ A
                  @ ^ [Y: C] :
                      ( groups7311177749621191930dd_sum @ B @ A @ H2
                      @ ( collect @ B
                        @ ^ [X: B] :
                            ( ( member @ B @ X @ S3 )
                            & ( ( G @ X )
                              = Y ) ) ) )
                  @ T6 )
                = ( groups7311177749621191930dd_sum @ B @ A @ H2 @ S3 ) ) ) ) ) ) ).

% sum.group
thf(fact_6573_prod_Ogroup,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( ( comm_monoid_mult @ A )
     => ! [S3: set @ B,T6: set @ C,G: B > C,H2: B > A] :
          ( ( finite_finite2 @ B @ S3 )
         => ( ( finite_finite2 @ C @ T6 )
           => ( ( ord_less_eq @ ( set @ C ) @ ( image @ B @ C @ G @ S3 ) @ T6 )
             => ( ( groups7121269368397514597t_prod @ C @ A
                  @ ^ [Y: C] :
                      ( groups7121269368397514597t_prod @ B @ A @ H2
                      @ ( collect @ B
                        @ ^ [X: B] :
                            ( ( member @ B @ X @ S3 )
                            & ( ( G @ X )
                              = Y ) ) ) )
                  @ T6 )
                = ( groups7121269368397514597t_prod @ B @ A @ H2 @ S3 ) ) ) ) ) ) ).

% prod.group
thf(fact_6574_set__image__eq__pointwiseI,axiom,
    ! [B: $tType,A: $tType,L: list @ A,L4: list @ A,F: A > B] :
      ( ( ( size_size @ ( list @ A ) @ L )
        = ( size_size @ ( list @ A ) @ L4 ) )
     => ( ! [I3: nat] :
            ( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ L ) )
           => ( ( F @ ( nth @ A @ L @ I3 ) )
              = ( F @ ( nth @ A @ L4 @ I3 ) ) ) )
       => ( ( image @ A @ B @ F @ ( set2 @ A @ L ) )
          = ( image @ A @ B @ F @ ( set2 @ A @ L4 ) ) ) ) ) ).

% set_image_eq_pointwiseI
thf(fact_6575_in__set__image__conv__nth,axiom,
    ! [B: $tType,A: $tType,F: B > A,X2: B,L: list @ B] :
      ( ( member @ A @ ( F @ X2 ) @ ( image @ B @ A @ F @ ( set2 @ B @ L ) ) )
      = ( ? [I2: nat] :
            ( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ B ) @ L ) )
            & ( ( F @ ( nth @ B @ L @ I2 ) )
              = ( F @ X2 ) ) ) ) ) ).

% in_set_image_conv_nth
thf(fact_6576_Max__mono,axiom,
    ! [A: $tType] :
      ( ( linorder @ A )
     => ! [M8: set @ A,N3: set @ A] :
          ( ( ord_less_eq @ ( set @ A ) @ M8 @ N3 )
         => ( ( M8
             != ( bot_bot @ ( set @ A ) ) )
           => ( ( finite_finite2 @ A @ N3 )
             => ( ord_less_eq @ A @ ( lattic643756798349783984er_Max @ A @ M8 ) @ ( lattic643756798349783984er_Max @ A @ N3 ) ) ) ) ) ) ).

% Max_mono
thf(fact_6577_Max_Osubset__imp,axiom,
    ! [A: $tType] :
      ( ( linorder @ A )
     => ! [A3: set @ A,B6: set @ A] :
          ( ( ord_less_eq @ ( set @ A ) @ A3 @ B6 )
         => ( ( A3
             != ( bot_bot @ ( set @ A ) ) )
           => ( ( finite_finite2 @ A @ B6 )
             => ( ord_less_eq @ A @ ( lattic643756798349783984er_Max @ A @ A3 ) @ ( lattic643756798349783984er_Max @ A @ B6 ) ) ) ) ) ) ).

% Max.subset_imp
thf(fact_6578_VEBT__internal_Oheight_Oelims,axiom,
    ! [X2: vEBT_VEBT,Y2: nat] :
      ( ( ( vEBT_VEBT_height @ X2 )
        = Y2 )
     => ( ( ? [A4: $o,B3: $o] :
              ( X2
              = ( vEBT_Leaf @ A4 @ B3 ) )
         => ( Y2
           != ( zero_zero @ nat ) ) )
       => ~ ! [Uu2: option @ ( product_prod @ nat @ nat ),Deg2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
              ( ( X2
                = ( vEBT_Node @ Uu2 @ Deg2 @ TreeList2 @ Summary2 ) )
             => ( Y2
               != ( plus_plus @ nat @ ( one_one @ nat ) @ ( lattic643756798349783984er_Max @ nat @ ( image @ vEBT_VEBT @ nat @ vEBT_VEBT_height @ ( insert @ vEBT_VEBT @ Summary2 @ ( set2 @ vEBT_VEBT @ TreeList2 ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.height.elims
thf(fact_6579_Max_Osubset,axiom,
    ! [A: $tType] :
      ( ( linorder @ A )
     => ! [A3: set @ A,B6: set @ A] :
          ( ( finite_finite2 @ A @ A3 )
         => ( ( B6
             != ( bot_bot @ ( set @ A ) ) )
           => ( ( ord_less_eq @ ( set @ A ) @ B6 @ A3 )
             => ( ( ord_max @ A @ ( lattic643756798349783984er_Max @ A @ B6 ) @ ( lattic643756798349783984er_Max @ A @ A3 ) )
                = ( lattic643756798349783984er_Max @ A @ A3 ) ) ) ) ) ) ).

% Max.subset
thf(fact_6580_divide__nat__def,axiom,
    ( ( divide_divide @ nat )
    = ( ^ [M7: nat,N: nat] :
          ( if @ nat
          @ ( N
            = ( zero_zero @ nat ) )
          @ ( zero_zero @ nat )
          @ ( lattic643756798349783984er_Max @ nat
            @ ( collect @ nat
              @ ^ [K4: nat] : ( ord_less_eq @ nat @ ( times_times @ nat @ K4 @ N ) @ M7 ) ) ) ) ) ) ).

% divide_nat_def
thf(fact_6581_Max_Oremove,axiom,
    ! [A: $tType] :
      ( ( linorder @ A )
     => ! [A3: set @ A,X2: A] :
          ( ( finite_finite2 @ A @ A3 )
         => ( ( member @ A @ X2 @ A3 )
           => ( ( ( ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) )
                  = ( bot_bot @ ( set @ A ) ) )
               => ( ( lattic643756798349783984er_Max @ A @ A3 )
                  = X2 ) )
              & ( ( ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) )
                 != ( bot_bot @ ( set @ A ) ) )
               => ( ( lattic643756798349783984er_Max @ A @ A3 )
                  = ( ord_max @ A @ X2 @ ( lattic643756798349783984er_Max @ A @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ) ).

% Max.remove
thf(fact_6582_Max_Oinsert__remove,axiom,
    ! [A: $tType] :
      ( ( linorder @ A )
     => ! [A3: set @ A,X2: A] :
          ( ( finite_finite2 @ A @ A3 )
         => ( ( ( ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) )
                = ( bot_bot @ ( set @ A ) ) )
             => ( ( lattic643756798349783984er_Max @ A @ ( insert @ A @ X2 @ A3 ) )
                = X2 ) )
            & ( ( ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) )
               != ( bot_bot @ ( set @ A ) ) )
             => ( ( lattic643756798349783984er_Max @ A @ ( insert @ A @ X2 @ A3 ) )
                = ( ord_max @ A @ X2 @ ( lattic643756798349783984er_Max @ A @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ).

% Max.insert_remove
thf(fact_6583_image__mult__atLeastAtMost__if,axiom,
    ! [A: $tType] :
      ( ( linordered_field @ A )
     => ! [C2: A,X2: A,Y2: A] :
          ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
           => ( ( image @ A @ A @ ( times_times @ A @ C2 ) @ ( set_or1337092689740270186AtMost @ A @ X2 @ Y2 ) )
              = ( set_or1337092689740270186AtMost @ A @ ( times_times @ A @ C2 @ X2 ) @ ( times_times @ A @ C2 @ Y2 ) ) ) )
          & ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
           => ( ( ( ord_less_eq @ A @ X2 @ Y2 )
               => ( ( image @ A @ A @ ( times_times @ A @ C2 ) @ ( set_or1337092689740270186AtMost @ A @ X2 @ Y2 ) )
                  = ( set_or1337092689740270186AtMost @ A @ ( times_times @ A @ C2 @ Y2 ) @ ( times_times @ A @ C2 @ X2 ) ) ) )
              & ( ~ ( ord_less_eq @ A @ X2 @ Y2 )
               => ( ( image @ A @ A @ ( times_times @ A @ C2 ) @ ( set_or1337092689740270186AtMost @ A @ X2 @ Y2 ) )
                  = ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).

% image_mult_atLeastAtMost_if
thf(fact_6584_image__mult__atLeastAtMost__if_H,axiom,
    ! [A: $tType] :
      ( ( linordered_field @ A )
     => ! [X2: A,Y2: A,C2: A] :
          ( ( ( ord_less_eq @ A @ X2 @ Y2 )
           => ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
               => ( ( image @ A @ A
                    @ ^ [X: A] : ( times_times @ A @ X @ C2 )
                    @ ( set_or1337092689740270186AtMost @ A @ X2 @ Y2 ) )
                  = ( set_or1337092689740270186AtMost @ A @ ( times_times @ A @ X2 @ C2 ) @ ( times_times @ A @ Y2 @ C2 ) ) ) )
              & ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
               => ( ( image @ A @ A
                    @ ^ [X: A] : ( times_times @ A @ X @ C2 )
                    @ ( set_or1337092689740270186AtMost @ A @ X2 @ Y2 ) )
                  = ( set_or1337092689740270186AtMost @ A @ ( times_times @ A @ Y2 @ C2 ) @ ( times_times @ A @ X2 @ C2 ) ) ) ) ) )
          & ( ~ ( ord_less_eq @ A @ X2 @ Y2 )
           => ( ( image @ A @ A
                @ ^ [X: A] : ( times_times @ A @ X @ C2 )
                @ ( set_or1337092689740270186AtMost @ A @ X2 @ Y2 ) )
              = ( bot_bot @ ( set @ A ) ) ) ) ) ) ).

% image_mult_atLeastAtMost_if'
thf(fact_6585_image__affinity__atLeastAtMost,axiom,
    ! [A: $tType] :
      ( ( linordered_field @ A )
     => ! [A2: A,B2: A,M: A,C2: A] :
          ( ( ( ( set_or1337092689740270186AtMost @ A @ A2 @ B2 )
              = ( bot_bot @ ( set @ A ) ) )
           => ( ( image @ A @ A
                @ ^ [X: A] : ( plus_plus @ A @ ( times_times @ A @ M @ X ) @ C2 )
                @ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) )
              = ( bot_bot @ ( set @ A ) ) ) )
          & ( ( ( set_or1337092689740270186AtMost @ A @ A2 @ B2 )
             != ( bot_bot @ ( set @ A ) ) )
           => ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ M )
               => ( ( image @ A @ A
                    @ ^ [X: A] : ( plus_plus @ A @ ( times_times @ A @ M @ X ) @ C2 )
                    @ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) )
                  = ( set_or1337092689740270186AtMost @ A @ ( plus_plus @ A @ ( times_times @ A @ M @ A2 ) @ C2 ) @ ( plus_plus @ A @ ( times_times @ A @ M @ B2 ) @ C2 ) ) ) )
              & ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ M )
               => ( ( image @ A @ A
                    @ ^ [X: A] : ( plus_plus @ A @ ( times_times @ A @ M @ X ) @ C2 )
                    @ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) )
                  = ( set_or1337092689740270186AtMost @ A @ ( plus_plus @ A @ ( times_times @ A @ M @ B2 ) @ C2 ) @ ( plus_plus @ A @ ( times_times @ A @ M @ A2 ) @ C2 ) ) ) ) ) ) ) ) ).

% image_affinity_atLeastAtMost
thf(fact_6586_image__affinity__atLeastAtMost__diff,axiom,
    ! [A: $tType] :
      ( ( linordered_field @ A )
     => ! [A2: A,B2: A,M: A,C2: A] :
          ( ( ( ( set_or1337092689740270186AtMost @ A @ A2 @ B2 )
              = ( bot_bot @ ( set @ A ) ) )
           => ( ( image @ A @ A
                @ ^ [X: A] : ( minus_minus @ A @ ( times_times @ A @ M @ X ) @ C2 )
                @ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) )
              = ( bot_bot @ ( set @ A ) ) ) )
          & ( ( ( set_or1337092689740270186AtMost @ A @ A2 @ B2 )
             != ( bot_bot @ ( set @ A ) ) )
           => ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ M )
               => ( ( image @ A @ A
                    @ ^ [X: A] : ( minus_minus @ A @ ( times_times @ A @ M @ X ) @ C2 )
                    @ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) )
                  = ( set_or1337092689740270186AtMost @ A @ ( minus_minus @ A @ ( times_times @ A @ M @ A2 ) @ C2 ) @ ( minus_minus @ A @ ( times_times @ A @ M @ B2 ) @ C2 ) ) ) )
              & ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ M )
               => ( ( image @ A @ A
                    @ ^ [X: A] : ( minus_minus @ A @ ( times_times @ A @ M @ X ) @ C2 )
                    @ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) )
                  = ( set_or1337092689740270186AtMost @ A @ ( minus_minus @ A @ ( times_times @ A @ M @ B2 ) @ C2 ) @ ( minus_minus @ A @ ( times_times @ A @ M @ A2 ) @ C2 ) ) ) ) ) ) ) ) ).

% image_affinity_atLeastAtMost_diff
thf(fact_6587_image__affinity__atLeastAtMost__div,axiom,
    ! [A: $tType] :
      ( ( linordered_field @ A )
     => ! [A2: A,B2: A,M: A,C2: A] :
          ( ( ( ( set_or1337092689740270186AtMost @ A @ A2 @ B2 )
              = ( bot_bot @ ( set @ A ) ) )
           => ( ( image @ A @ A
                @ ^ [X: A] : ( plus_plus @ A @ ( divide_divide @ A @ X @ M ) @ C2 )
                @ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) )
              = ( bot_bot @ ( set @ A ) ) ) )
          & ( ( ( set_or1337092689740270186AtMost @ A @ A2 @ B2 )
             != ( bot_bot @ ( set @ A ) ) )
           => ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ M )
               => ( ( image @ A @ A
                    @ ^ [X: A] : ( plus_plus @ A @ ( divide_divide @ A @ X @ M ) @ C2 )
                    @ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) )
                  = ( set_or1337092689740270186AtMost @ A @ ( plus_plus @ A @ ( divide_divide @ A @ A2 @ M ) @ C2 ) @ ( plus_plus @ A @ ( divide_divide @ A @ B2 @ M ) @ C2 ) ) ) )
              & ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ M )
               => ( ( image @ A @ A
                    @ ^ [X: A] : ( plus_plus @ A @ ( divide_divide @ A @ X @ M ) @ C2 )
                    @ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) )
                  = ( set_or1337092689740270186AtMost @ A @ ( plus_plus @ A @ ( divide_divide @ A @ B2 @ M ) @ C2 ) @ ( plus_plus @ A @ ( divide_divide @ A @ A2 @ M ) @ C2 ) ) ) ) ) ) ) ) ).

% image_affinity_atLeastAtMost_div
thf(fact_6588_image__affinity__atLeastAtMost__div__diff,axiom,
    ! [A: $tType] :
      ( ( linordered_field @ A )
     => ! [A2: A,B2: A,M: A,C2: A] :
          ( ( ( ( set_or1337092689740270186AtMost @ A @ A2 @ B2 )
              = ( bot_bot @ ( set @ A ) ) )
           => ( ( image @ A @ A
                @ ^ [X: A] : ( minus_minus @ A @ ( divide_divide @ A @ X @ M ) @ C2 )
                @ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) )
              = ( bot_bot @ ( set @ A ) ) ) )
          & ( ( ( set_or1337092689740270186AtMost @ A @ A2 @ B2 )
             != ( bot_bot @ ( set @ A ) ) )
           => ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ M )
               => ( ( image @ A @ A
                    @ ^ [X: A] : ( minus_minus @ A @ ( divide_divide @ A @ X @ M ) @ C2 )
                    @ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) )
                  = ( set_or1337092689740270186AtMost @ A @ ( minus_minus @ A @ ( divide_divide @ A @ A2 @ M ) @ C2 ) @ ( minus_minus @ A @ ( divide_divide @ A @ B2 @ M ) @ C2 ) ) ) )
              & ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ M )
               => ( ( image @ A @ A
                    @ ^ [X: A] : ( minus_minus @ A @ ( divide_divide @ A @ X @ M ) @ C2 )
                    @ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) )
                  = ( set_or1337092689740270186AtMost @ A @ ( minus_minus @ A @ ( divide_divide @ A @ B2 @ M ) @ C2 ) @ ( minus_minus @ A @ ( divide_divide @ A @ A2 @ M ) @ C2 ) ) ) ) ) ) ) ) ).

% image_affinity_atLeastAtMost_div_diff
thf(fact_6589_VEBT__internal_Oheight_Opelims,axiom,
    ! [X2: vEBT_VEBT,Y2: nat] :
      ( ( ( vEBT_VEBT_height @ X2 )
        = Y2 )
     => ( ( accp @ vEBT_VEBT @ vEBT_VEBT_height_rel @ X2 )
       => ( ! [A4: $o,B3: $o] :
              ( ( X2
                = ( vEBT_Leaf @ A4 @ B3 ) )
             => ( ( Y2
                  = ( zero_zero @ nat ) )
               => ~ ( accp @ vEBT_VEBT @ vEBT_VEBT_height_rel @ ( vEBT_Leaf @ A4 @ B3 ) ) ) )
         => ~ ! [Uu2: option @ ( product_prod @ nat @ nat ),Deg2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( X2
                  = ( vEBT_Node @ Uu2 @ Deg2 @ TreeList2 @ Summary2 ) )
               => ( ( Y2
                    = ( plus_plus @ nat @ ( one_one @ nat ) @ ( lattic643756798349783984er_Max @ nat @ ( image @ vEBT_VEBT @ nat @ vEBT_VEBT_height @ ( insert @ vEBT_VEBT @ Summary2 @ ( set2 @ vEBT_VEBT @ TreeList2 ) ) ) ) ) )
                 => ~ ( accp @ vEBT_VEBT @ vEBT_VEBT_height_rel @ ( vEBT_Node @ Uu2 @ Deg2 @ TreeList2 @ Summary2 ) ) ) ) ) ) ) ).

% VEBT_internal.height.pelims
thf(fact_6590_bin__last__integer__nbe,axiom,
    ( bits_b8758750999018896077nteger
    = ( ^ [I2: code_integer] :
          ( ( modulo_modulo @ code_integer @ I2 @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) )
         != ( zero_zero @ code_integer ) ) ) ) ).

% bin_last_integer_nbe
thf(fact_6591_bij__betw__Suc,axiom,
    ! [M8: set @ nat,N3: set @ nat] :
      ( ( bij_betw @ nat @ nat @ suc @ M8 @ N3 )
      = ( ( image @ nat @ nat @ suc @ M8 )
        = N3 ) ) ).

% bij_betw_Suc
thf(fact_6592_gcd__1__nat,axiom,
    ! [M: nat] :
      ( ( gcd_gcd @ nat @ M @ ( one_one @ nat ) )
      = ( one_one @ nat ) ) ).

% gcd_1_nat
thf(fact_6593_gcd__Suc__0,axiom,
    ! [M: nat] :
      ( ( gcd_gcd @ nat @ M @ ( suc @ ( zero_zero @ nat ) ) )
      = ( suc @ ( zero_zero @ nat ) ) ) ).

% gcd_Suc_0
thf(fact_6594_gcd__pos__nat,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( gcd_gcd @ nat @ M @ N2 ) )
      = ( ( M
         != ( zero_zero @ nat ) )
        | ( N2
         != ( zero_zero @ nat ) ) ) ) ).

% gcd_pos_nat
thf(fact_6595_img__fst,axiom,
    ! [B: $tType,A: $tType,A2: A,B2: B,S3: set @ ( product_prod @ A @ B )] :
      ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B2 ) @ S3 )
     => ( member @ A @ A2 @ ( image @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ S3 ) ) ) ).

% img_fst
thf(fact_6596_img__snd,axiom,
    ! [B: $tType,A: $tType,A2: A,B2: B,S3: set @ ( product_prod @ A @ B )] :
      ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B2 ) @ S3 )
     => ( member @ B @ B2 @ ( image @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) @ S3 ) ) ) ).

% img_snd
thf(fact_6597_op__conc__empty__img__id,axiom,
    ! [A: $tType,L8: set @ ( list @ A )] :
      ( ( image @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ ( nil @ A ) ) @ L8 )
      = L8 ) ).

% op_conc_empty_img_id
thf(fact_6598_image__Suc__atLeastLessThan,axiom,
    ! [I: nat,J: nat] :
      ( ( image @ nat @ nat @ suc @ ( set_or7035219750837199246ssThan @ nat @ I @ J ) )
      = ( set_or7035219750837199246ssThan @ nat @ ( suc @ I ) @ ( suc @ J ) ) ) ).

% image_Suc_atLeastLessThan
thf(fact_6599_image__Suc__atLeastAtMost,axiom,
    ! [I: nat,J: nat] :
      ( ( image @ nat @ nat @ suc @ ( set_or1337092689740270186AtMost @ nat @ I @ J ) )
      = ( set_or1337092689740270186AtMost @ nat @ ( suc @ I ) @ ( suc @ J ) ) ) ).

% image_Suc_atLeastAtMost
thf(fact_6600_pair__imageI,axiom,
    ! [C: $tType,B: $tType,A: $tType,A2: A,B2: B,A3: set @ ( product_prod @ A @ B ),F: A > B > C] :
      ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B2 ) @ A3 )
     => ( member @ C @ ( F @ A2 @ B2 ) @ ( image @ ( product_prod @ A @ B ) @ C @ ( product_case_prod @ A @ B @ C @ F ) @ A3 ) ) ) ).

% pair_imageI
thf(fact_6601_range__mult,axiom,
    ! [A2: real] :
      ( ( ( A2
          = ( zero_zero @ real ) )
       => ( ( image @ real @ real @ ( times_times @ real @ A2 ) @ ( top_top @ ( set @ real ) ) )
          = ( insert @ real @ ( zero_zero @ real ) @ ( bot_bot @ ( set @ real ) ) ) ) )
      & ( ( A2
         != ( zero_zero @ real ) )
       => ( ( image @ real @ real @ ( times_times @ real @ A2 ) @ ( top_top @ ( set @ real ) ) )
          = ( top_top @ ( set @ real ) ) ) ) ) ).

% range_mult
thf(fact_6602_range__fst,axiom,
    ! [B: $tType,A: $tType] :
      ( ( image @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) )
      = ( top_top @ ( set @ A ) ) ) ).

% range_fst
thf(fact_6603_range__snd,axiom,
    ! [B: $tType,A: $tType] :
      ( ( image @ ( product_prod @ B @ A ) @ A @ ( product_snd @ B @ A ) @ ( top_top @ ( set @ ( product_prod @ B @ A ) ) ) )
      = ( top_top @ ( set @ A ) ) ) ).

% range_snd
thf(fact_6604_Max__divisors__self__int,axiom,
    ! [N2: int] :
      ( ( N2
       != ( zero_zero @ int ) )
     => ( ( lattic643756798349783984er_Max @ int
          @ ( collect @ int
            @ ^ [D3: int] : ( dvd_dvd @ int @ D3 @ N2 ) ) )
        = ( abs_abs @ int @ N2 ) ) ) ).

% Max_divisors_self_int
thf(fact_6605_nth__image__indices,axiom,
    ! [A: $tType,L: list @ A] :
      ( ( image @ nat @ A @ ( nth @ A @ L ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ A ) @ L ) ) )
      = ( set2 @ A @ L ) ) ).

% nth_image_indices
thf(fact_6606_gcd__diff2__nat,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq @ nat @ M @ N2 )
     => ( ( gcd_gcd @ nat @ ( minus_minus @ nat @ N2 @ M ) @ N2 )
        = ( gcd_gcd @ nat @ M @ N2 ) ) ) ).

% gcd_diff2_nat
thf(fact_6607_gcd__diff1__nat,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_eq @ nat @ N2 @ M )
     => ( ( gcd_gcd @ nat @ ( minus_minus @ nat @ M @ N2 ) @ N2 )
        = ( gcd_gcd @ nat @ M @ N2 ) ) ) ).

% gcd_diff1_nat
thf(fact_6608_None__notin__image__Some,axiom,
    ! [A: $tType,A3: set @ A] :
      ~ ( member @ ( option @ A ) @ ( none @ A ) @ ( image @ A @ ( option @ A ) @ ( some @ A ) @ A3 ) ) ).

% None_notin_image_Some
thf(fact_6609_zero__notin__Suc__image,axiom,
    ! [A3: set @ nat] :
      ~ ( member @ nat @ ( zero_zero @ nat ) @ ( image @ nat @ nat @ suc @ A3 ) ) ).

% zero_notin_Suc_image
thf(fact_6610_gcd__le1__nat,axiom,
    ! [A2: nat,B2: nat] :
      ( ( A2
       != ( zero_zero @ nat ) )
     => ( ord_less_eq @ nat @ ( gcd_gcd @ nat @ A2 @ B2 ) @ A2 ) ) ).

% gcd_le1_nat
thf(fact_6611_gcd__le2__nat,axiom,
    ! [B2: nat,A2: nat] :
      ( ( B2
       != ( zero_zero @ nat ) )
     => ( ord_less_eq @ nat @ ( gcd_gcd @ nat @ A2 @ B2 ) @ B2 ) ) ).

% gcd_le2_nat
thf(fact_6612_nat__seg__image__imp__finite,axiom,
    ! [A: $tType,A3: set @ A,F: nat > A,N2: nat] :
      ( ( A3
        = ( image @ nat @ A @ F
          @ ( collect @ nat
            @ ^ [I2: nat] : ( ord_less @ nat @ I2 @ N2 ) ) ) )
     => ( finite_finite2 @ A @ A3 ) ) ).

% nat_seg_image_imp_finite
thf(fact_6613_finite__conv__nat__seg__image,axiom,
    ! [A: $tType] :
      ( ( finite_finite2 @ A )
      = ( ^ [A7: set @ A] :
          ? [N: nat,F6: nat > A] :
            ( A7
            = ( image @ nat @ A @ F6
              @ ( collect @ nat
                @ ^ [I2: nat] : ( ord_less @ nat @ I2 @ N ) ) ) ) ) ) ).

% finite_conv_nat_seg_image
thf(fact_6614_in__fst__imageE,axiom,
    ! [B: $tType,A: $tType,X2: A,S3: set @ ( product_prod @ A @ B )] :
      ( ( member @ A @ X2 @ ( image @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ S3 ) )
     => ~ ! [Y4: B] :
            ~ ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y4 ) @ S3 ) ) ).

% in_fst_imageE
thf(fact_6615_in__snd__imageE,axiom,
    ! [A: $tType,B: $tType,Y2: A,S3: set @ ( product_prod @ B @ A )] :
      ( ( member @ A @ Y2 @ ( image @ ( product_prod @ B @ A ) @ A @ ( product_snd @ B @ A ) @ S3 ) )
     => ~ ! [X3: B] :
            ~ ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X3 @ Y2 ) @ S3 ) ) ).

% in_snd_imageE
thf(fact_6616_finite__int__iff__bounded,axiom,
    ( ( finite_finite2 @ int )
    = ( ^ [S8: set @ int] :
        ? [K4: int] : ( ord_less_eq @ ( set @ int ) @ ( image @ int @ int @ ( abs_abs @ int ) @ S8 ) @ ( set_ord_lessThan @ int @ K4 ) ) ) ) ).

% finite_int_iff_bounded
thf(fact_6617_finite__int__iff__bounded__le,axiom,
    ( ( finite_finite2 @ int )
    = ( ^ [S8: set @ int] :
        ? [K4: int] : ( ord_less_eq @ ( set @ int ) @ ( image @ int @ int @ ( abs_abs @ int ) @ S8 ) @ ( set_ord_atMost @ int @ K4 ) ) ) ) ).

% finite_int_iff_bounded_le
thf(fact_6618_gcd__mult__distrib__nat,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( times_times @ nat @ K @ ( gcd_gcd @ nat @ M @ N2 ) )
      = ( gcd_gcd @ nat @ ( times_times @ nat @ K @ M ) @ ( times_times @ nat @ K @ N2 ) ) ) ).

% gcd_mult_distrib_nat
thf(fact_6619_bitval__bin__last__integer,axiom,
    ! [I: code_integer] :
      ( ( zero_neq_one_of_bool @ code_integer @ ( bits_b8758750999018896077nteger @ I ) )
      = ( bit_se5824344872417868541ns_and @ code_integer @ I @ ( one_one @ code_integer ) ) ) ).

% bitval_bin_last_integer
thf(fact_6620_bezout__nat,axiom,
    ! [A2: nat,B2: nat] :
      ( ( A2
       != ( zero_zero @ nat ) )
     => ? [X3: nat,Y4: nat] :
          ( ( times_times @ nat @ A2 @ X3 )
          = ( plus_plus @ nat @ ( times_times @ nat @ B2 @ Y4 ) @ ( gcd_gcd @ nat @ A2 @ B2 ) ) ) ) ).

% bezout_nat
thf(fact_6621_bezout__gcd__nat_H,axiom,
    ! [B2: nat,A2: nat] :
    ? [X3: nat,Y4: nat] :
      ( ( ( ord_less_eq @ nat @ ( times_times @ nat @ B2 @ Y4 ) @ ( times_times @ nat @ A2 @ X3 ) )
        & ( ( minus_minus @ nat @ ( times_times @ nat @ A2 @ X3 ) @ ( times_times @ nat @ B2 @ Y4 ) )
          = ( gcd_gcd @ nat @ A2 @ B2 ) ) )
      | ( ( ord_less_eq @ nat @ ( times_times @ nat @ A2 @ Y4 ) @ ( times_times @ nat @ B2 @ X3 ) )
        & ( ( minus_minus @ nat @ ( times_times @ nat @ B2 @ X3 ) @ ( times_times @ nat @ A2 @ Y4 ) )
          = ( gcd_gcd @ nat @ A2 @ B2 ) ) ) ) ).

% bezout_gcd_nat'
thf(fact_6622_notin__range__Some,axiom,
    ! [A: $tType,X2: option @ A] :
      ( ( ~ ( member @ ( option @ A ) @ X2 @ ( image @ A @ ( option @ A ) @ ( some @ A ) @ ( top_top @ ( set @ A ) ) ) ) )
      = ( X2
        = ( none @ A ) ) ) ).

% notin_range_Some
thf(fact_6623_fst__image__mp,axiom,
    ! [B: $tType,A: $tType,A3: set @ ( product_prod @ A @ B ),B6: set @ A,X2: A,Y2: B] :
      ( ( ord_less_eq @ ( set @ A ) @ ( image @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ A3 ) @ B6 )
     => ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y2 ) @ A3 )
       => ( member @ A @ X2 @ B6 ) ) ) ).

% fst_image_mp
thf(fact_6624_snd__image__mp,axiom,
    ! [B: $tType,A: $tType,A3: set @ ( product_prod @ B @ A ),B6: set @ A,X2: B,Y2: A] :
      ( ( ord_less_eq @ ( set @ A ) @ ( image @ ( product_prod @ B @ A ) @ A @ ( product_snd @ B @ A ) @ A3 ) @ B6 )
     => ( ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X2 @ Y2 ) @ A3 )
       => ( member @ A @ Y2 @ B6 ) ) ) ).

% snd_image_mp
thf(fact_6625_in__image__insert__iff,axiom,
    ! [A: $tType,B6: set @ ( set @ A ),X2: A,A3: set @ A] :
      ( ! [C8: set @ A] :
          ( ( member @ ( set @ A ) @ C8 @ B6 )
         => ~ ( member @ A @ X2 @ C8 ) )
     => ( ( member @ ( set @ A ) @ A3 @ ( image @ ( set @ A ) @ ( set @ A ) @ ( insert @ A @ X2 ) @ B6 ) )
        = ( ( member @ A @ X2 @ A3 )
          & ( member @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) @ B6 ) ) ) ) ).

% in_image_insert_iff
thf(fact_6626_gcd__is__Max__divisors__int,axiom,
    ! [N2: int,M: int] :
      ( ( N2
       != ( zero_zero @ int ) )
     => ( ( gcd_gcd @ int @ M @ N2 )
        = ( lattic643756798349783984er_Max @ int
          @ ( collect @ int
            @ ^ [D3: int] :
                ( ( dvd_dvd @ int @ D3 @ M )
                & ( dvd_dvd @ int @ D3 @ N2 ) ) ) ) ) ) ).

% gcd_is_Max_divisors_int
thf(fact_6627_bin__last__integer__code,axiom,
    ( bits_b8758750999018896077nteger
    = ( ^ [I2: code_integer] :
          ( ( bit_se5824344872417868541ns_and @ code_integer @ I2 @ ( one_one @ code_integer ) )
         != ( zero_zero @ code_integer ) ) ) ) ).

% bin_last_integer_code
thf(fact_6628_image__Suc__lessThan,axiom,
    ! [N2: nat] :
      ( ( image @ nat @ nat @ suc @ ( set_ord_lessThan @ nat @ N2 ) )
      = ( set_or1337092689740270186AtMost @ nat @ ( one_one @ nat ) @ N2 ) ) ).

% image_Suc_lessThan
thf(fact_6629_image__Suc__atMost,axiom,
    ! [N2: nat] :
      ( ( image @ nat @ nat @ suc @ ( set_ord_atMost @ nat @ N2 ) )
      = ( set_or1337092689740270186AtMost @ nat @ ( one_one @ nat ) @ ( suc @ N2 ) ) ) ).

% image_Suc_atMost
thf(fact_6630_atLeast0__lessThan__Suc__eq__insert__0,axiom,
    ! [N2: nat] :
      ( ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( suc @ N2 ) )
      = ( insert @ nat @ ( zero_zero @ nat ) @ ( image @ nat @ nat @ suc @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ) ).

% atLeast0_lessThan_Suc_eq_insert_0
thf(fact_6631_atLeast0__atMost__Suc__eq__insert__0,axiom,
    ! [N2: nat] :
      ( ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( suc @ N2 ) )
      = ( insert @ nat @ ( zero_zero @ nat ) @ ( image @ nat @ nat @ suc @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ) ).

% atLeast0_atMost_Suc_eq_insert_0
thf(fact_6632_lessThan__Suc__eq__insert__0,axiom,
    ! [N2: nat] :
      ( ( set_ord_lessThan @ nat @ ( suc @ N2 ) )
      = ( insert @ nat @ ( zero_zero @ nat ) @ ( image @ nat @ nat @ suc @ ( set_ord_lessThan @ nat @ N2 ) ) ) ) ).

% lessThan_Suc_eq_insert_0
thf(fact_6633_atMost__Suc__eq__insert__0,axiom,
    ! [N2: nat] :
      ( ( set_ord_atMost @ nat @ ( suc @ N2 ) )
      = ( insert @ nat @ ( zero_zero @ nat ) @ ( image @ nat @ nat @ suc @ ( set_ord_atMost @ nat @ N2 ) ) ) ) ).

% atMost_Suc_eq_insert_0
thf(fact_6634_Collect__split__mono__strong,axiom,
    ! [B: $tType,A: $tType,X8: set @ A,A3: set @ ( product_prod @ A @ B ),Y7: set @ B,P: A > B > $o,Q: A > B > $o] :
      ( ( X8
        = ( image @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ A3 ) )
     => ( ( Y7
          = ( image @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) @ A3 ) )
       => ( ! [X3: A] :
              ( ( member @ A @ X3 @ X8 )
             => ! [Xa3: B] :
                  ( ( member @ B @ Xa3 @ Y7 )
                 => ( ( P @ X3 @ Xa3 )
                   => ( Q @ X3 @ Xa3 ) ) ) )
         => ( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ A3 @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ P ) ) )
           => ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ A3 @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ Q ) ) ) ) ) ) ) ).

% Collect_split_mono_strong
thf(fact_6635_UNIV__option__conv,axiom,
    ! [A: $tType] :
      ( ( top_top @ ( set @ ( option @ A ) ) )
      = ( insert @ ( option @ A ) @ ( none @ A ) @ ( image @ A @ ( option @ A ) @ ( some @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ).

% UNIV_option_conv
thf(fact_6636_range__mod,axiom,
    ! [N2: nat] :
      ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
     => ( ( image @ nat @ nat
          @ ^ [M7: nat] : ( modulo_modulo @ nat @ M7 @ N2 )
          @ ( top_top @ ( set @ nat ) ) )
        = ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ).

% range_mod
thf(fact_6637_finite__range__Some,axiom,
    ! [A: $tType] :
      ( ( finite_finite2 @ ( option @ A ) @ ( image @ A @ ( option @ A ) @ ( some @ A ) @ ( top_top @ ( set @ A ) ) ) )
      = ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) ) ) ).

% finite_range_Some
thf(fact_6638_gcd__is__Max__divisors__nat,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
     => ( ( gcd_gcd @ nat @ M @ N2 )
        = ( lattic643756798349783984er_Max @ nat
          @ ( collect @ nat
            @ ^ [D3: nat] :
                ( ( dvd_dvd @ nat @ D3 @ M )
                & ( dvd_dvd @ nat @ D3 @ N2 ) ) ) ) ) ) ).

% gcd_is_Max_divisors_nat
thf(fact_6639_image__add__int__atLeastLessThan,axiom,
    ! [L: int,U: int] :
      ( ( image @ int @ int
        @ ^ [X: int] : ( plus_plus @ int @ X @ L )
        @ ( set_or7035219750837199246ssThan @ int @ ( zero_zero @ int ) @ ( minus_minus @ int @ U @ L ) ) )
      = ( set_or7035219750837199246ssThan @ int @ L @ U ) ) ).

% image_add_int_atLeastLessThan
thf(fact_6640_image__add__integer__atLeastLessThan,axiom,
    ! [L: code_integer,U: code_integer] :
      ( ( image @ code_integer @ code_integer
        @ ^ [X: code_integer] : ( plus_plus @ code_integer @ X @ L )
        @ ( set_or7035219750837199246ssThan @ code_integer @ ( zero_zero @ code_integer ) @ ( minus_minus @ code_integer @ U @ L ) ) )
      = ( set_or7035219750837199246ssThan @ code_integer @ L @ U ) ) ).

% image_add_integer_atLeastLessThan
thf(fact_6641_image__atLeastZeroLessThan__int,axiom,
    ! [U: int] :
      ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ U )
     => ( ( set_or7035219750837199246ssThan @ int @ ( zero_zero @ int ) @ U )
        = ( image @ nat @ int @ ( semiring_1_of_nat @ int ) @ ( set_ord_lessThan @ nat @ ( nat2 @ U ) ) ) ) ) ).

% image_atLeastZeroLessThan_int
thf(fact_6642_bezw__aux,axiom,
    ! [X2: nat,Y2: nat] :
      ( ( semiring_1_of_nat @ int @ ( gcd_gcd @ nat @ X2 @ Y2 ) )
      = ( plus_plus @ int @ ( times_times @ int @ ( product_fst @ int @ int @ ( bezw @ X2 @ Y2 ) ) @ ( semiring_1_of_nat @ int @ X2 ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ ( bezw @ X2 @ Y2 ) ) @ ( semiring_1_of_nat @ int @ Y2 ) ) ) ) ).

% bezw_aux
thf(fact_6643_image__minus__const__atLeastLessThan__nat,axiom,
    ! [C2: nat,Y2: nat,X2: nat] :
      ( ( ( ord_less @ nat @ C2 @ Y2 )
       => ( ( image @ nat @ nat
            @ ^ [I2: nat] : ( minus_minus @ nat @ I2 @ C2 )
            @ ( set_or7035219750837199246ssThan @ nat @ X2 @ Y2 ) )
          = ( set_or7035219750837199246ssThan @ nat @ ( minus_minus @ nat @ X2 @ C2 ) @ ( minus_minus @ nat @ Y2 @ C2 ) ) ) )
      & ( ~ ( ord_less @ nat @ C2 @ Y2 )
       => ( ( ( ord_less @ nat @ X2 @ Y2 )
           => ( ( image @ nat @ nat
                @ ^ [I2: nat] : ( minus_minus @ nat @ I2 @ C2 )
                @ ( set_or7035219750837199246ssThan @ nat @ X2 @ Y2 ) )
              = ( insert @ nat @ ( zero_zero @ nat ) @ ( bot_bot @ ( set @ nat ) ) ) ) )
          & ( ~ ( ord_less @ nat @ X2 @ Y2 )
           => ( ( image @ nat @ nat
                @ ^ [I2: nat] : ( minus_minus @ nat @ I2 @ C2 )
                @ ( set_or7035219750837199246ssThan @ nat @ X2 @ Y2 ) )
              = ( bot_bot @ ( set @ nat ) ) ) ) ) ) ) ).

% image_minus_const_atLeastLessThan_nat
thf(fact_6644_bin__last__integer_Oabs__eq,axiom,
    ! [X2: int] :
      ( ( bits_b8758750999018896077nteger @ ( code_integer_of_int @ X2 ) )
      = ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ X2 ) ) ) ).

% bin_last_integer.abs_eq
thf(fact_6645_gcd__nat_Opelims,axiom,
    ! [X2: nat,Xa: nat,Y2: nat] :
      ( ( ( gcd_gcd @ nat @ X2 @ Xa )
        = Y2 )
     => ( ( accp @ ( product_prod @ nat @ nat ) @ gcd_nat_rel @ ( product_Pair @ nat @ nat @ X2 @ Xa ) )
       => ~ ( ( ( ( Xa
                  = ( zero_zero @ nat ) )
               => ( Y2 = X2 ) )
              & ( ( Xa
                 != ( zero_zero @ nat ) )
               => ( Y2
                  = ( gcd_gcd @ nat @ Xa @ ( modulo_modulo @ nat @ X2 @ Xa ) ) ) ) )
           => ~ ( accp @ ( product_prod @ nat @ nat ) @ gcd_nat_rel @ ( product_Pair @ nat @ nat @ X2 @ Xa ) ) ) ) ) ).

% gcd_nat.pelims
thf(fact_6646_bitXOR__integer__unfold,axiom,
    ( ( bit_se5824344971392196577ns_xor @ code_integer )
    = ( ^ [X: code_integer,Y: code_integer] :
          ( if @ code_integer
          @ ( X
            = ( zero_zero @ code_integer ) )
          @ Y
          @ ( if @ code_integer
            @ ( X
              = ( uminus_uminus @ code_integer @ ( one_one @ code_integer ) ) )
            @ ( bit_ri4277139882892585799ns_not @ code_integer @ Y )
            @ ( bits_Bit_integer @ ( bit_se5824344971392196577ns_xor @ code_integer @ ( bits_b2549910563261871055nteger @ X ) @ ( bits_b2549910563261871055nteger @ Y ) )
              @ ( ( ~ ( bits_b8758750999018896077nteger @ X ) )
                = ( bits_b8758750999018896077nteger @ Y ) ) ) ) ) ) ) ).

% bitXOR_integer_unfold
thf(fact_6647_bin__rest__integer__code,axiom,
    ( bits_b2549910563261871055nteger
    = ( ^ [I2: code_integer] : ( divide_divide @ code_integer @ I2 @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) ) ) ) ).

% bin_rest_integer_code
thf(fact_6648_bin__rest__integer_Oabs__eq,axiom,
    ! [X2: int] :
      ( ( bits_b2549910563261871055nteger @ ( code_integer_of_int @ X2 ) )
      = ( code_integer_of_int @ ( divide_divide @ int @ X2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ).

% bin_rest_integer.abs_eq
thf(fact_6649_bitOR__integer__unfold,axiom,
    ( ( bit_se1065995026697491101ons_or @ code_integer )
    = ( ^ [X: code_integer,Y: code_integer] :
          ( if @ code_integer
          @ ( X
            = ( zero_zero @ code_integer ) )
          @ Y
          @ ( if @ code_integer
            @ ( X
              = ( uminus_uminus @ code_integer @ ( one_one @ code_integer ) ) )
            @ ( uminus_uminus @ code_integer @ ( one_one @ code_integer ) )
            @ ( bits_Bit_integer @ ( bit_se1065995026697491101ons_or @ code_integer @ ( bits_b2549910563261871055nteger @ X ) @ ( bits_b2549910563261871055nteger @ Y ) )
              @ ( ( bits_b8758750999018896077nteger @ X )
                | ( bits_b8758750999018896077nteger @ Y ) ) ) ) ) ) ) ).

% bitOR_integer_unfold
thf(fact_6650_bitAND__integer__unfold,axiom,
    ( ( bit_se5824344872417868541ns_and @ code_integer )
    = ( ^ [X: code_integer,Y: code_integer] :
          ( if @ code_integer
          @ ( X
            = ( zero_zero @ code_integer ) )
          @ ( zero_zero @ code_integer )
          @ ( if @ code_integer
            @ ( X
              = ( uminus_uminus @ code_integer @ ( one_one @ code_integer ) ) )
            @ Y
            @ ( bits_Bit_integer @ ( bit_se5824344872417868541ns_and @ code_integer @ ( bits_b2549910563261871055nteger @ X ) @ ( bits_b2549910563261871055nteger @ Y ) )
              @ ( ( bits_b8758750999018896077nteger @ X )
                & ( bits_b8758750999018896077nteger @ Y ) ) ) ) ) ) ) ).

% bitAND_integer_unfold
thf(fact_6651_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_6652_set__Cons__sing__Nil,axiom,
    ! [A: $tType,A3: set @ A] :
      ( ( set_Cons @ A @ A3 @ ( insert @ ( list @ A ) @ ( nil @ A ) @ ( bot_bot @ ( set @ ( list @ A ) ) ) ) )
      = ( image @ A @ ( list @ A )
        @ ^ [X: A] : ( cons @ A @ X @ ( nil @ A ) )
        @ A3 ) ) ).

% set_Cons_sing_Nil
thf(fact_6653_flat__lub__def,axiom,
    ! [A: $tType] :
      ( ( partial_flat_lub @ A )
      = ( ^ [B5: A,A7: set @ A] :
            ( if @ A @ ( ord_less_eq @ ( set @ A ) @ A7 @ ( insert @ A @ B5 @ ( bot_bot @ ( set @ A ) ) ) ) @ B5
            @ ( the @ A
              @ ^ [X: A] : ( member @ A @ X @ ( minus_minus @ ( set @ A ) @ A7 @ ( insert @ A @ B5 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ).

% flat_lub_def
thf(fact_6654_bind__return,axiom,
    ! [A: $tType,F: heap_Time_Heap @ A] :
      ( ( heap_Time_bind @ A @ A @ F @ ( heap_Time_return @ A ) )
      = ( heap_Time_bind @ product_unit @ A @ ( heap_Time_wait @ ( one_one @ nat ) )
        @ ^ [Uu3: product_unit] : F ) ) ).

% bind_return
thf(fact_6655_wait__bind__decon,axiom,
    ! [A: $tType,P: assn,M: heap_Time_Heap @ A,Q: A > assn,N2: nat] :
      ( ( hoare_hoare_triple @ A @ P @ M @ Q )
     => ( hoare_hoare_triple @ A @ P
        @ ( heap_Time_bind @ product_unit @ A @ ( heap_Time_wait @ N2 )
          @ ^ [Uu3: product_unit] : M )
        @ Q ) ) ).

% wait_bind_decon
thf(fact_6656_wait__rule,axiom,
    ! [N2: nat] :
      ( hoare_hoare_triple @ product_unit @ ( one_one @ assn ) @ ( heap_Time_wait @ N2 )
      @ ^ [Uu3: product_unit] : ( one_one @ assn ) ) ).

% wait_rule
thf(fact_6657_return__bind,axiom,
    ! [B: $tType,A: $tType,X2: B,F: B > ( heap_Time_Heap @ A )] :
      ( ( heap_Time_bind @ B @ A @ ( heap_Time_return @ B @ X2 ) @ F )
      = ( heap_Time_bind @ product_unit @ A @ ( heap_Time_wait @ ( one_one @ nat ) )
        @ ^ [Uu3: product_unit] : ( F @ X2 ) ) ) ).

% return_bind
thf(fact_6658_tailrec_Ofixp__induct__uc,axiom,
    ! [B: $tType,A: $tType,C: $tType,U3: C > B > A,F5: C > C,C5: ( B > A ) > C,F: C,P: ( B > A ) > $o] :
      ( ! [X3: B] :
          ( comple7038119648293358887notone @ ( B > A ) @ A @ ( partial_fun_ord @ A @ A @ B @ ( partial_flat_ord @ A @ ( undefined @ A ) ) ) @ ( partial_flat_ord @ A @ ( undefined @ A ) )
          @ ^ [F6: B > A] : ( U3 @ ( F5 @ ( C5 @ F6 ) ) @ X3 ) )
     => ( ( F
          = ( C5
            @ ( comple187402453842119260l_fixp @ ( B > A ) @ ( partial_fun_lub @ A @ A @ B @ ( partial_flat_lub @ A @ ( undefined @ A ) ) ) @ ( partial_fun_ord @ A @ A @ B @ ( partial_flat_ord @ A @ ( undefined @ A ) ) )
              @ ^ [F6: B > A] : ( U3 @ ( F5 @ ( C5 @ F6 ) ) ) ) ) )
       => ( ! [F2: B > A] :
              ( ( U3 @ ( C5 @ F2 ) )
              = F2 )
         => ( ( comple1908693960933563346ssible @ ( B > A ) @ ( partial_fun_lub @ A @ A @ B @ ( partial_flat_lub @ A @ ( undefined @ A ) ) ) @ ( partial_fun_ord @ A @ A @ B @ ( partial_flat_ord @ A @ ( undefined @ A ) ) ) @ P )
           => ( ( P
                @ ^ [Uu3: B] : ( undefined @ A ) )
             => ( ! [F2: C] :
                    ( ( P @ ( U3 @ F2 ) )
                   => ( P @ ( U3 @ ( F5 @ F2 ) ) ) )
               => ( P @ ( U3 @ F ) ) ) ) ) ) ) ) ).

% tailrec.fixp_induct_uc
thf(fact_6659_sofl__test,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [X2: word @ A,Y2: word @ A] :
          ( ( ( plus_plus @ int @ ( ring_1_signed @ A @ int @ X2 ) @ ( ring_1_signed @ A @ int @ Y2 ) )
            = ( ring_1_signed @ A @ int @ ( plus_plus @ ( word @ A ) @ X2 @ Y2 ) ) )
          = ( ( bit_se4197421643247451524op_bit @ ( word @ A ) @ ( minus_minus @ nat @ ( size_size @ ( word @ A ) @ X2 ) @ ( one_one @ nat ) ) @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ ( bit_se5824344971392196577ns_xor @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ X2 @ Y2 ) @ X2 ) @ ( bit_se5824344971392196577ns_xor @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ X2 @ Y2 ) @ Y2 ) ) )
            = ( zero_zero @ ( word @ A ) ) ) ) ) ).

% sofl_test
thf(fact_6660_drop__bit__drop__bit,axiom,
    ! [A: $tType] :
      ( ( bit_se359711467146920520ations @ A )
     => ! [M: nat,N2: nat,A2: A] :
          ( ( bit_se4197421643247451524op_bit @ A @ M @ ( bit_se4197421643247451524op_bit @ A @ N2 @ A2 ) )
          = ( bit_se4197421643247451524op_bit @ A @ ( plus_plus @ nat @ M @ N2 ) @ A2 ) ) ) ).

% drop_bit_drop_bit
thf(fact_6661_drop__bit__Suc__bit0,axiom,
    ! [A: $tType] :
      ( ( bit_un5681908812861735899ations @ A )
     => ! [N2: nat,K: num] :
          ( ( bit_se4197421643247451524op_bit @ A @ ( suc @ N2 ) @ ( numeral_numeral @ A @ ( bit0 @ K ) ) )
          = ( bit_se4197421643247451524op_bit @ A @ N2 @ ( numeral_numeral @ A @ K ) ) ) ) ).

% drop_bit_Suc_bit0
thf(fact_6662_drop__bit__Suc__bit1,axiom,
    ! [A: $tType] :
      ( ( bit_un5681908812861735899ations @ A )
     => ! [N2: nat,K: num] :
          ( ( bit_se4197421643247451524op_bit @ A @ ( suc @ N2 ) @ ( numeral_numeral @ A @ ( bit1 @ K ) ) )
          = ( bit_se4197421643247451524op_bit @ A @ N2 @ ( numeral_numeral @ A @ K ) ) ) ) ).

% drop_bit_Suc_bit1
thf(fact_6663_drop__bit__of__1,axiom,
    ! [A: $tType] :
      ( ( bit_se359711467146920520ations @ A )
     => ! [N2: nat] :
          ( ( bit_se4197421643247451524op_bit @ A @ N2 @ ( one_one @ A ) )
          = ( zero_neq_one_of_bool @ A
            @ ( N2
              = ( zero_zero @ nat ) ) ) ) ) ).

% drop_bit_of_1
thf(fact_6664_drop__bit__numeral__bit0,axiom,
    ! [A: $tType] :
      ( ( bit_un5681908812861735899ations @ A )
     => ! [L: num,K: num] :
          ( ( bit_se4197421643247451524op_bit @ A @ ( numeral_numeral @ nat @ L ) @ ( numeral_numeral @ A @ ( bit0 @ K ) ) )
          = ( bit_se4197421643247451524op_bit @ A @ ( pred_numeral @ L ) @ ( numeral_numeral @ A @ K ) ) ) ) ).

% drop_bit_numeral_bit0
thf(fact_6665_drop__bit__numeral__bit1,axiom,
    ! [A: $tType] :
      ( ( bit_un5681908812861735899ations @ A )
     => ! [L: num,K: num] :
          ( ( bit_se4197421643247451524op_bit @ A @ ( numeral_numeral @ nat @ L ) @ ( numeral_numeral @ A @ ( bit1 @ K ) ) )
          = ( bit_se4197421643247451524op_bit @ A @ ( pred_numeral @ L ) @ ( numeral_numeral @ A @ K ) ) ) ) ).

% drop_bit_numeral_bit1
thf(fact_6666_fixp__induct__option,axiom,
    ! [C: $tType,B: $tType,A: $tType,U3: C > B > ( option @ A ),F5: C > C,C5: ( B > ( option @ A ) ) > C,F: C,P: B > A > $o,X2: B,Y2: A] :
      ( ! [X3: B] :
          ( comple7038119648293358887notone @ ( B > ( option @ A ) ) @ ( option @ A ) @ ( partial_fun_ord @ ( option @ A ) @ ( option @ A ) @ B @ ( partial_flat_ord @ ( option @ A ) @ ( none @ A ) ) ) @ ( partial_flat_ord @ ( option @ A ) @ ( none @ A ) )
          @ ^ [F6: B > ( option @ A )] : ( U3 @ ( F5 @ ( C5 @ F6 ) ) @ X3 ) )
     => ( ( F
          = ( C5
            @ ( comple187402453842119260l_fixp @ ( B > ( option @ A ) ) @ ( partial_fun_lub @ ( option @ A ) @ ( option @ A ) @ B @ ( partial_flat_lub @ ( option @ A ) @ ( none @ A ) ) ) @ ( partial_fun_ord @ ( option @ A ) @ ( option @ A ) @ B @ ( partial_flat_ord @ ( option @ A ) @ ( none @ A ) ) )
              @ ^ [F6: B > ( option @ A )] : ( U3 @ ( F5 @ ( C5 @ F6 ) ) ) ) ) )
       => ( ! [F2: B > ( option @ A )] :
              ( ( U3 @ ( C5 @ F2 ) )
              = F2 )
         => ( ! [F2: C,X3: B,Y4: A] :
                ( ! [Xa2: B,Ya2: A] :
                    ( ( ( U3 @ F2 @ Xa2 )
                      = ( some @ A @ Ya2 ) )
                   => ( P @ Xa2 @ Ya2 ) )
               => ( ( ( U3 @ ( F5 @ F2 ) @ X3 )
                    = ( some @ A @ Y4 ) )
                 => ( P @ X3 @ Y4 ) ) )
           => ( ( ( U3 @ F @ X2 )
                = ( some @ A @ Y2 ) )
             => ( P @ X2 @ Y2 ) ) ) ) ) ) ).

% fixp_induct_option
thf(fact_6667_option_Omono__body__fixp,axiom,
    ! [A: $tType,B: $tType,F5: ( B > ( option @ A ) ) > B > ( option @ A )] :
      ( ! [X3: B] :
          ( comple7038119648293358887notone @ ( B > ( option @ A ) ) @ ( option @ A ) @ ( partial_fun_ord @ ( option @ A ) @ ( option @ A ) @ B @ ( partial_flat_ord @ ( option @ A ) @ ( none @ A ) ) ) @ ( partial_flat_ord @ ( option @ A ) @ ( none @ A ) )
          @ ^ [F6: B > ( option @ A )] : ( F5 @ F6 @ X3 ) )
     => ( ( comple187402453842119260l_fixp @ ( B > ( option @ A ) ) @ ( partial_fun_lub @ ( option @ A ) @ ( option @ A ) @ B @ ( partial_flat_lub @ ( option @ A ) @ ( none @ A ) ) ) @ ( partial_fun_ord @ ( option @ A ) @ ( option @ A ) @ B @ ( partial_flat_ord @ ( option @ A ) @ ( none @ A ) ) ) @ F5 )
        = ( F5 @ ( comple187402453842119260l_fixp @ ( B > ( option @ A ) ) @ ( partial_fun_lub @ ( option @ A ) @ ( option @ A ) @ B @ ( partial_flat_lub @ ( option @ A ) @ ( none @ A ) ) ) @ ( partial_fun_ord @ ( option @ A ) @ ( option @ A ) @ B @ ( partial_flat_ord @ ( option @ A ) @ ( none @ A ) ) ) @ F5 ) ) ) ) ).

% option.mono_body_fixp
thf(fact_6668_option_Ofixp__rule__uc,axiom,
    ! [B: $tType,A: $tType,C: $tType,U3: C > B > ( option @ A ),F5: C > C,C5: ( B > ( option @ A ) ) > C,F: C] :
      ( ! [X3: B] :
          ( comple7038119648293358887notone @ ( B > ( option @ A ) ) @ ( option @ A ) @ ( partial_fun_ord @ ( option @ A ) @ ( option @ A ) @ B @ ( partial_flat_ord @ ( option @ A ) @ ( none @ A ) ) ) @ ( partial_flat_ord @ ( option @ A ) @ ( none @ A ) )
          @ ^ [F6: B > ( option @ A )] : ( U3 @ ( F5 @ ( C5 @ F6 ) ) @ X3 ) )
     => ( ( F
          = ( C5
            @ ( comple187402453842119260l_fixp @ ( B > ( option @ A ) ) @ ( partial_fun_lub @ ( option @ A ) @ ( option @ A ) @ B @ ( partial_flat_lub @ ( option @ A ) @ ( none @ A ) ) ) @ ( partial_fun_ord @ ( option @ A ) @ ( option @ A ) @ B @ ( partial_flat_ord @ ( option @ A ) @ ( none @ A ) ) )
              @ ^ [F6: B > ( option @ A )] : ( U3 @ ( F5 @ ( C5 @ F6 ) ) ) ) ) )
       => ( ! [F2: C] :
              ( ( C5 @ ( U3 @ F2 ) )
              = F2 )
         => ( F
            = ( F5 @ F ) ) ) ) ) ).

% option.fixp_rule_uc
thf(fact_6669_option_Opartial__function__definitions__axioms,axiom,
    ! [A: $tType] : ( partia7178651479351089652itions @ ( option @ A ) @ ( partial_flat_ord @ ( option @ A ) @ ( none @ A ) ) @ ( partial_flat_lub @ ( option @ A ) @ ( none @ A ) ) ) ).

% option.partial_function_definitions_axioms
thf(fact_6670_option_Ofixp__induct__uc,axiom,
    ! [B: $tType,A: $tType,C: $tType,U3: C > B > ( option @ A ),F5: C > C,C5: ( B > ( option @ A ) ) > C,F: C,P: ( B > ( option @ A ) ) > $o] :
      ( ! [X3: B] :
          ( comple7038119648293358887notone @ ( B > ( option @ A ) ) @ ( option @ A ) @ ( partial_fun_ord @ ( option @ A ) @ ( option @ A ) @ B @ ( partial_flat_ord @ ( option @ A ) @ ( none @ A ) ) ) @ ( partial_flat_ord @ ( option @ A ) @ ( none @ A ) )
          @ ^ [F6: B > ( option @ A )] : ( U3 @ ( F5 @ ( C5 @ F6 ) ) @ X3 ) )
     => ( ( F
          = ( C5
            @ ( comple187402453842119260l_fixp @ ( B > ( option @ A ) ) @ ( partial_fun_lub @ ( option @ A ) @ ( option @ A ) @ B @ ( partial_flat_lub @ ( option @ A ) @ ( none @ A ) ) ) @ ( partial_fun_ord @ ( option @ A ) @ ( option @ A ) @ B @ ( partial_flat_ord @ ( option @ A ) @ ( none @ A ) ) )
              @ ^ [F6: B > ( option @ A )] : ( U3 @ ( F5 @ ( C5 @ F6 ) ) ) ) ) )
       => ( ! [F2: B > ( option @ A )] :
              ( ( U3 @ ( C5 @ F2 ) )
              = F2 )
         => ( ( comple1908693960933563346ssible @ ( B > ( option @ A ) ) @ ( partial_fun_lub @ ( option @ A ) @ ( option @ A ) @ B @ ( partial_flat_lub @ ( option @ A ) @ ( none @ A ) ) ) @ ( partial_fun_ord @ ( option @ A ) @ ( option @ A ) @ B @ ( partial_flat_ord @ ( option @ A ) @ ( none @ A ) ) ) @ P )
           => ( ( P
                @ ^ [Uu3: B] : ( none @ A ) )
             => ( ! [F2: C] :
                    ( ( P @ ( U3 @ F2 ) )
                   => ( P @ ( U3 @ ( F5 @ F2 ) ) ) )
               => ( P @ ( U3 @ F ) ) ) ) ) ) ) ) ).

% option.fixp_induct_uc
thf(fact_6671_option_Oconst__mono,axiom,
    ! [A: $tType,B: $tType,Ord: B > B > $o,C2: option @ A] :
      ( comple7038119648293358887notone @ B @ ( option @ A ) @ Ord @ ( partial_flat_ord @ ( option @ A ) @ ( none @ A ) )
      @ ^ [F6: B] : C2 ) ).

% option.const_mono
thf(fact_6672_drop__bit__take__bit,axiom,
    ! [A: $tType] :
      ( ( bit_se359711467146920520ations @ A )
     => ! [M: nat,N2: nat,A2: A] :
          ( ( bit_se4197421643247451524op_bit @ A @ M @ ( bit_se2584673776208193580ke_bit @ A @ N2 @ A2 ) )
          = ( bit_se2584673776208193580ke_bit @ A @ ( minus_minus @ nat @ N2 @ M ) @ ( bit_se4197421643247451524op_bit @ A @ M @ A2 ) ) ) ) ).

% drop_bit_take_bit
thf(fact_6673_drop__bit__mask__eq,axiom,
    ! [A: $tType] :
      ( ( bit_un5681908812861735899ations @ A )
     => ! [M: nat,N2: nat] :
          ( ( bit_se4197421643247451524op_bit @ A @ M @ ( bit_se2239418461657761734s_mask @ A @ N2 ) )
          = ( bit_se2239418461657761734s_mask @ A @ ( minus_minus @ nat @ N2 @ M ) ) ) ) ).

% drop_bit_mask_eq
thf(fact_6674_take__bit__drop__bit,axiom,
    ! [A: $tType] :
      ( ( bit_se359711467146920520ations @ A )
     => ! [M: nat,N2: nat,A2: A] :
          ( ( bit_se2584673776208193580ke_bit @ A @ M @ ( bit_se4197421643247451524op_bit @ A @ N2 @ A2 ) )
          = ( bit_se4197421643247451524op_bit @ A @ N2 @ ( bit_se2584673776208193580ke_bit @ A @ ( plus_plus @ nat @ M @ N2 ) @ A2 ) ) ) ) ).

% take_bit_drop_bit
thf(fact_6675_option_Oleq__refl,axiom,
    ! [A: $tType,X2: option @ A] : ( partial_flat_ord @ ( option @ A ) @ ( none @ A ) @ X2 @ X2 ) ).

% option.leq_refl
thf(fact_6676_option_Oleq__trans,axiom,
    ! [A: $tType,X2: option @ A,Y2: option @ A,Z: option @ A] :
      ( ( partial_flat_ord @ ( option @ A ) @ ( none @ A ) @ X2 @ Y2 )
     => ( ( partial_flat_ord @ ( option @ A ) @ ( none @ A ) @ Y2 @ Z )
       => ( partial_flat_ord @ ( option @ A ) @ ( none @ A ) @ X2 @ Z ) ) ) ).

% option.leq_trans
thf(fact_6677_option_Oleq__antisym,axiom,
    ! [A: $tType,X2: option @ A,Y2: option @ A] :
      ( ( partial_flat_ord @ ( option @ A ) @ ( none @ A ) @ X2 @ Y2 )
     => ( ( partial_flat_ord @ ( option @ A ) @ ( none @ A ) @ Y2 @ X2 )
       => ( X2 = Y2 ) ) ) ).

% option.leq_antisym
thf(fact_6678_bit__word__iff__drop__bit__and,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) )
        = ( ^ [A5: word @ A,N: nat] :
              ( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ ( bit_se4197421643247451524op_bit @ ( word @ A ) @ N @ A5 ) @ ( one_one @ ( word @ A ) ) )
              = ( one_one @ ( word @ A ) ) ) ) ) ) ).

% bit_word_iff_drop_bit_and
thf(fact_6679_bit__iff__and__drop__bit__eq__1,axiom,
    ! [A: $tType] :
      ( ( bit_se359711467146920520ations @ A )
     => ( ( bit_se5641148757651400278ts_bit @ A )
        = ( ^ [A5: A,N: nat] :
              ( ( bit_se5824344872417868541ns_and @ A @ ( bit_se4197421643247451524op_bit @ A @ N @ A5 ) @ ( one_one @ A ) )
              = ( one_one @ A ) ) ) ) ) ).

% bit_iff_and_drop_bit_eq_1
thf(fact_6680_tailrec_Oconst__mono,axiom,
    ! [A: $tType,B: $tType,Ord: B > B > $o,C2: A] :
      ( comple7038119648293358887notone @ B @ A @ Ord @ ( partial_flat_ord @ A @ ( undefined @ A ) )
      @ ^ [F6: B] : C2 ) ).

% tailrec.const_mono
thf(fact_6681_option__admissible,axiom,
    ! [B: $tType,A: $tType,P: A > B > $o] :
      ( comple1908693960933563346ssible @ ( A > ( option @ B ) ) @ ( partial_fun_lub @ ( option @ B ) @ ( option @ B ) @ A @ ( partial_flat_lub @ ( option @ B ) @ ( none @ B ) ) ) @ ( partial_fun_ord @ ( option @ B ) @ ( option @ B ) @ A @ ( partial_flat_ord @ ( option @ B ) @ ( none @ B ) ) )
      @ ^ [F6: A > ( option @ B )] :
        ! [X: A,Y: B] :
          ( ( ( F6 @ X )
            = ( some @ B @ Y ) )
         => ( P @ X @ Y ) ) ) ).

% option_admissible
thf(fact_6682_drop__bit__half,axiom,
    ! [A: $tType] :
      ( ( bit_se359711467146920520ations @ A )
     => ! [N2: nat,A2: A] :
          ( ( bit_se4197421643247451524op_bit @ A @ N2 @ ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) )
          = ( divide_divide @ A @ ( bit_se4197421643247451524op_bit @ A @ N2 @ A2 ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).

% drop_bit_half
thf(fact_6683_stable__imp__drop__bit__eq,axiom,
    ! [A: $tType] :
      ( ( bit_se359711467146920520ations @ A )
     => ! [A2: A,N2: nat] :
          ( ( ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
            = A2 )
         => ( ( bit_se4197421643247451524op_bit @ A @ N2 @ A2 )
            = A2 ) ) ) ).

% stable_imp_drop_bit_eq
thf(fact_6684_drop__bit__Suc,axiom,
    ! [A: $tType] :
      ( ( bit_se359711467146920520ations @ A )
     => ! [N2: nat,A2: A] :
          ( ( bit_se4197421643247451524op_bit @ A @ ( suc @ N2 ) @ A2 )
          = ( bit_se4197421643247451524op_bit @ A @ N2 @ ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).

% drop_bit_Suc
thf(fact_6685_drop__bit__eq__div,axiom,
    ! [A: $tType] :
      ( ( bit_se359711467146920520ations @ A )
     => ( ( bit_se4197421643247451524op_bit @ A )
        = ( ^ [N: nat,A5: A] : ( divide_divide @ A @ A5 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) ) ) ) ) ).

% drop_bit_eq_div
thf(fact_6686_even__drop__bit__iff__not__bit,axiom,
    ! [A: $tType] :
      ( ( bit_se359711467146920520ations @ A )
     => ! [N2: nat,A2: A] :
          ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se4197421643247451524op_bit @ A @ N2 @ A2 ) )
          = ( ~ ( bit_se5641148757651400278ts_bit @ A @ A2 @ N2 ) ) ) ) ).

% even_drop_bit_iff_not_bit
thf(fact_6687_bit__iff__odd__drop__bit,axiom,
    ! [A: $tType] :
      ( ( bit_se359711467146920520ations @ A )
     => ( ( bit_se5641148757651400278ts_bit @ A )
        = ( ^ [A5: A,N: nat] :
              ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se4197421643247451524op_bit @ A @ N @ A5 ) ) ) ) ) ).

% bit_iff_odd_drop_bit
thf(fact_6688_heap__step__admissible,axiom,
    ! [E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: E > A > C > B > D > $o,X2: E] :
      ( comple1908693960933563346ssible @ ( A > ( option @ ( product_prod @ B @ ( product_prod @ C @ D ) ) ) ) @ ( partial_fun_lub @ ( option @ ( product_prod @ B @ ( product_prod @ C @ D ) ) ) @ ( option @ ( product_prod @ B @ ( product_prod @ C @ D ) ) ) @ A @ ( partial_flat_lub @ ( option @ ( product_prod @ B @ ( product_prod @ C @ D ) ) ) @ ( none @ ( product_prod @ B @ ( product_prod @ C @ D ) ) ) ) ) @ ( partial_fun_ord @ ( option @ ( product_prod @ B @ ( product_prod @ C @ D ) ) ) @ ( option @ ( product_prod @ B @ ( product_prod @ C @ D ) ) ) @ A @ ( partial_flat_ord @ ( option @ ( product_prod @ B @ ( product_prod @ C @ D ) ) ) @ ( none @ ( product_prod @ B @ ( product_prod @ C @ D ) ) ) ) )
      @ ^ [F6: A > ( option @ ( product_prod @ B @ ( product_prod @ C @ D ) ) )] :
        ! [H: A,H9: C,R: B,N: D] :
          ( ( ( F6 @ H )
            = ( some @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ R @ ( product_Pair @ C @ D @ H9 @ N ) ) ) )
         => ( P @ X2 @ H @ H9 @ R @ N ) ) ) ).

% heap_step_admissible
thf(fact_6689_drop__bit__rec,axiom,
    ! [A: $tType] :
      ( ( bit_se359711467146920520ations @ A )
     => ( ( bit_se4197421643247451524op_bit @ A )
        = ( ^ [N: nat,A5: A] :
              ( if @ A
              @ ( N
                = ( zero_zero @ nat ) )
              @ A5
              @ ( bit_se4197421643247451524op_bit @ A @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ ( divide_divide @ A @ A5 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).

% drop_bit_rec
thf(fact_6690_fixp__induct__tailrec,axiom,
    ! [C: $tType,B: $tType,A: $tType,C2: A,U3: C > B > A,F5: C > C,C5: ( B > A ) > C,F: C,P: B > A > $o,X2: B,Y2: A] :
      ( ! [X3: B] :
          ( comple7038119648293358887notone @ ( B > A ) @ A @ ( partial_fun_ord @ A @ A @ B @ ( partial_flat_ord @ A @ C2 ) ) @ ( partial_flat_ord @ A @ C2 )
          @ ^ [F6: B > A] : ( U3 @ ( F5 @ ( C5 @ F6 ) ) @ X3 ) )
     => ( ( F
          = ( C5
            @ ( comple187402453842119260l_fixp @ ( B > A ) @ ( partial_fun_lub @ A @ A @ B @ ( partial_flat_lub @ A @ C2 ) ) @ ( partial_fun_ord @ A @ A @ B @ ( partial_flat_ord @ A @ C2 ) )
              @ ^ [F6: B > A] : ( U3 @ ( F5 @ ( C5 @ F6 ) ) ) ) ) )
       => ( ! [F2: B > A] :
              ( ( U3 @ ( C5 @ F2 ) )
              = F2 )
         => ( ! [F2: C,X3: B,Y4: A] :
                ( ! [Xa2: B,Ya2: A] :
                    ( ( ( U3 @ F2 @ Xa2 )
                      = Ya2 )
                   => ( ( Ya2 != C2 )
                     => ( P @ Xa2 @ Ya2 ) ) )
               => ( ( ( U3 @ ( F5 @ F2 ) @ X3 )
                    = Y4 )
                 => ( ( Y4 != C2 )
                   => ( P @ X3 @ Y4 ) ) ) )
           => ( ( ( U3 @ F @ X2 )
                = Y2 )
             => ( ( Y2 != C2 )
               => ( P @ X2 @ Y2 ) ) ) ) ) ) ) ).

% fixp_induct_tailrec
thf(fact_6691_tailrec__admissible,axiom,
    ! [B: $tType,A: $tType,C2: B,P: A > B > $o] :
      ( comple1908693960933563346ssible @ ( A > B ) @ ( partial_fun_lub @ B @ B @ A @ ( partial_flat_lub @ B @ C2 ) ) @ ( partial_fun_ord @ B @ B @ A @ ( partial_flat_ord @ B @ C2 ) )
      @ ^ [A5: A > B] :
        ! [X: A] :
          ( ( ( A5 @ X )
           != C2 )
         => ( P @ X @ ( A5 @ X ) ) ) ) ).

% tailrec_admissible
thf(fact_6692_tailrec_Ofixp__rule__uc,axiom,
    ! [B: $tType,A: $tType,C: $tType,U3: C > B > A,F5: C > C,C5: ( B > A ) > C,F: C] :
      ( ! [X3: B] :
          ( comple7038119648293358887notone @ ( B > A ) @ A @ ( partial_fun_ord @ A @ A @ B @ ( partial_flat_ord @ A @ ( undefined @ A ) ) ) @ ( partial_flat_ord @ A @ ( undefined @ A ) )
          @ ^ [F6: B > A] : ( U3 @ ( F5 @ ( C5 @ F6 ) ) @ X3 ) )
     => ( ( F
          = ( C5
            @ ( comple187402453842119260l_fixp @ ( B > A ) @ ( partial_fun_lub @ A @ A @ B @ ( partial_flat_lub @ A @ ( undefined @ A ) ) ) @ ( partial_fun_ord @ A @ A @ B @ ( partial_flat_ord @ A @ ( undefined @ A ) ) )
              @ ^ [F6: B > A] : ( U3 @ ( F5 @ ( C5 @ F6 ) ) ) ) ) )
       => ( ! [F2: C] :
              ( ( C5 @ ( U3 @ F2 ) )
              = F2 )
         => ( F
            = ( F5 @ F ) ) ) ) ) ).

% tailrec.fixp_rule_uc
thf(fact_6693_tailrec_Omono__body__fixp,axiom,
    ! [A: $tType,B: $tType,F5: ( B > A ) > B > A] :
      ( ! [X3: B] :
          ( comple7038119648293358887notone @ ( B > A ) @ A @ ( partial_fun_ord @ A @ A @ B @ ( partial_flat_ord @ A @ ( undefined @ A ) ) ) @ ( partial_flat_ord @ A @ ( undefined @ A ) )
          @ ^ [F6: B > A] : ( F5 @ F6 @ X3 ) )
     => ( ( comple187402453842119260l_fixp @ ( B > A ) @ ( partial_fun_lub @ A @ A @ B @ ( partial_flat_lub @ A @ ( undefined @ A ) ) ) @ ( partial_fun_ord @ A @ A @ B @ ( partial_flat_ord @ A @ ( undefined @ A ) ) ) @ F5 )
        = ( F5 @ ( comple187402453842119260l_fixp @ ( B > A ) @ ( partial_fun_lub @ A @ A @ B @ ( partial_flat_lub @ A @ ( undefined @ A ) ) ) @ ( partial_fun_ord @ A @ A @ B @ ( partial_flat_ord @ A @ ( undefined @ A ) ) ) @ F5 ) ) ) ) ).

% tailrec.mono_body_fixp
thf(fact_6694_div__half__word,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [Y2: word @ A,X2: word @ A] :
          ( ( Y2
           != ( zero_zero @ ( word @ A ) ) )
         => ( ( product_Pair @ ( word @ A ) @ ( word @ A ) @ ( divide_divide @ ( word @ A ) @ X2 @ Y2 ) @ ( modulo_modulo @ ( word @ A ) @ X2 @ Y2 ) )
            = ( if @ ( product_prod @ ( word @ A ) @ ( word @ A ) ) @ ( ord_less_eq @ ( word @ A ) @ Y2 @ ( minus_minus @ ( word @ A ) @ X2 @ ( times_times @ ( word @ A ) @ ( bit_se4730199178511100633sh_bit @ ( word @ A ) @ ( one_one @ nat ) @ ( divide_divide @ ( word @ A ) @ ( bit_se4197421643247451524op_bit @ ( word @ A ) @ ( one_one @ nat ) @ X2 ) @ Y2 ) ) @ Y2 ) ) ) @ ( product_Pair @ ( word @ A ) @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ ( bit_se4730199178511100633sh_bit @ ( word @ A ) @ ( one_one @ nat ) @ ( divide_divide @ ( word @ A ) @ ( bit_se4197421643247451524op_bit @ ( word @ A ) @ ( one_one @ nat ) @ X2 ) @ Y2 ) ) @ ( one_one @ ( word @ A ) ) ) @ ( minus_minus @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ X2 @ ( times_times @ ( word @ A ) @ ( bit_se4730199178511100633sh_bit @ ( word @ A ) @ ( one_one @ nat ) @ ( divide_divide @ ( word @ A ) @ ( bit_se4197421643247451524op_bit @ ( word @ A ) @ ( one_one @ nat ) @ X2 ) @ Y2 ) ) @ Y2 ) ) @ Y2 ) ) @ ( product_Pair @ ( word @ A ) @ ( word @ A ) @ ( bit_se4730199178511100633sh_bit @ ( word @ A ) @ ( one_one @ nat ) @ ( divide_divide @ ( word @ A ) @ ( bit_se4197421643247451524op_bit @ ( word @ A ) @ ( one_one @ nat ) @ X2 ) @ Y2 ) ) @ ( minus_minus @ ( word @ A ) @ X2 @ ( times_times @ ( word @ A ) @ ( bit_se4730199178511100633sh_bit @ ( word @ A ) @ ( one_one @ nat ) @ ( divide_divide @ ( word @ A ) @ ( bit_se4197421643247451524op_bit @ ( word @ A ) @ ( one_one @ nat ) @ X2 ) @ Y2 ) ) @ Y2 ) ) ) ) ) ) ) ).

% div_half_word
thf(fact_6695_take__bit__num__def,axiom,
    ( bit_take_bit_num
    = ( ^ [N: nat,M7: num] :
          ( if @ ( option @ num )
          @ ( ( bit_se2584673776208193580ke_bit @ nat @ N @ ( numeral_numeral @ nat @ M7 ) )
            = ( zero_zero @ nat ) )
          @ ( none @ num )
          @ ( some @ num @ ( num_of_nat @ ( bit_se2584673776208193580ke_bit @ nat @ N @ ( numeral_numeral @ nat @ M7 ) ) ) ) ) ) ) ).

% take_bit_num_def
thf(fact_6696_drop__bit__nonnegative__int__iff,axiom,
    ! [N2: nat,K: int] :
      ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_se4197421643247451524op_bit @ int @ N2 @ K ) )
      = ( ord_less_eq @ int @ ( zero_zero @ int ) @ K ) ) ).

% drop_bit_nonnegative_int_iff
thf(fact_6697_drop__bit__minus__one,axiom,
    ! [N2: nat] :
      ( ( bit_se4197421643247451524op_bit @ int @ N2 @ ( uminus_uminus @ int @ ( one_one @ int ) ) )
      = ( uminus_uminus @ int @ ( one_one @ int ) ) ) ).

% drop_bit_minus_one
thf(fact_6698_push__bit__push__bit,axiom,
    ! [A: $tType] :
      ( ( bit_se359711467146920520ations @ A )
     => ! [M: nat,N2: nat,A2: A] :
          ( ( bit_se4730199178511100633sh_bit @ A @ M @ ( bit_se4730199178511100633sh_bit @ A @ N2 @ A2 ) )
          = ( bit_se4730199178511100633sh_bit @ A @ ( plus_plus @ nat @ M @ N2 ) @ A2 ) ) ) ).

% push_bit_push_bit
thf(fact_6699_num__of__nat__numeral__eq,axiom,
    ! [Q2: num] :
      ( ( num_of_nat @ ( numeral_numeral @ nat @ Q2 ) )
      = Q2 ) ).

% num_of_nat_numeral_eq
thf(fact_6700_drop__bit__of__Suc__0,axiom,
    ! [N2: nat] :
      ( ( bit_se4197421643247451524op_bit @ nat @ N2 @ ( suc @ ( zero_zero @ nat ) ) )
      = ( zero_neq_one_of_bool @ nat
        @ ( N2
          = ( zero_zero @ nat ) ) ) ) ).

% drop_bit_of_Suc_0
thf(fact_6701_push__bit__Suc__numeral,axiom,
    ! [A: $tType] :
      ( ( bit_se359711467146920520ations @ A )
     => ! [N2: nat,K: num] :
          ( ( bit_se4730199178511100633sh_bit @ A @ ( suc @ N2 ) @ ( numeral_numeral @ A @ K ) )
          = ( bit_se4730199178511100633sh_bit @ A @ N2 @ ( numeral_numeral @ A @ ( bit0 @ K ) ) ) ) ) ).

% push_bit_Suc_numeral
thf(fact_6702_drop__bit__Suc__minus__bit0,axiom,
    ! [N2: nat,K: num] :
      ( ( bit_se4197421643247451524op_bit @ int @ ( suc @ N2 ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ K ) ) ) )
      = ( bit_se4197421643247451524op_bit @ int @ N2 @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K ) ) ) ) ).

% drop_bit_Suc_minus_bit0
thf(fact_6703_push__bit__Suc__minus__numeral,axiom,
    ! [A: $tType] :
      ( ( bit_ri3973907225187159222ations @ A )
     => ! [N2: nat,K: num] :
          ( ( bit_se4730199178511100633sh_bit @ A @ ( suc @ N2 ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ K ) ) )
          = ( bit_se4730199178511100633sh_bit @ A @ N2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( bit0 @ K ) ) ) ) ) ) ).

% push_bit_Suc_minus_numeral
thf(fact_6704_push__bit__minus__one__eq__not__mask,axiom,
    ! [A: $tType] :
      ( ( bit_ri3973907225187159222ations @ A )
     => ! [N2: nat] :
          ( ( bit_se4730199178511100633sh_bit @ A @ N2 @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
          = ( bit_ri4277139882892585799ns_not @ A @ ( bit_se2239418461657761734s_mask @ A @ N2 ) ) ) ) ).

% push_bit_minus_one_eq_not_mask
thf(fact_6705_push__bit__numeral,axiom,
    ! [A: $tType] :
      ( ( bit_se359711467146920520ations @ A )
     => ! [L: num,K: num] :
          ( ( bit_se4730199178511100633sh_bit @ A @ ( numeral_numeral @ nat @ L ) @ ( numeral_numeral @ A @ K ) )
          = ( bit_se4730199178511100633sh_bit @ A @ ( pred_numeral @ L ) @ ( numeral_numeral @ A @ ( bit0 @ K ) ) ) ) ) ).

% push_bit_numeral
thf(fact_6706_drop__bit__numeral__minus__bit0,axiom,
    ! [L: num,K: num] :
      ( ( bit_se4197421643247451524op_bit @ int @ ( numeral_numeral @ nat @ L ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ K ) ) ) )
      = ( bit_se4197421643247451524op_bit @ int @ ( pred_numeral @ L ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K ) ) ) ) ).

% drop_bit_numeral_minus_bit0
thf(fact_6707_drop__bit__Suc__minus__bit1,axiom,
    ! [N2: nat,K: num] :
      ( ( bit_se4197421643247451524op_bit @ int @ ( suc @ N2 ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ K ) ) ) )
      = ( bit_se4197421643247451524op_bit @ int @ N2 @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( inc @ K ) ) ) ) ) ).

% drop_bit_Suc_minus_bit1
thf(fact_6708_push__bit__Suc,axiom,
    ! [A: $tType] :
      ( ( bit_se359711467146920520ations @ A )
     => ! [N2: nat,A2: A] :
          ( ( bit_se4730199178511100633sh_bit @ A @ ( suc @ N2 ) @ A2 )
          = ( bit_se4730199178511100633sh_bit @ A @ N2 @ ( times_times @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).

% push_bit_Suc
thf(fact_6709_push__bit__of__1,axiom,
    ! [A: $tType] :
      ( ( bit_se359711467146920520ations @ A )
     => ! [N2: nat] :
          ( ( bit_se4730199178511100633sh_bit @ A @ N2 @ ( one_one @ A ) )
          = ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) ) ) ).

% push_bit_of_1
thf(fact_6710_even__push__bit__iff,axiom,
    ! [A: $tType] :
      ( ( bit_se359711467146920520ations @ A )
     => ! [N2: nat,A2: A] :
          ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se4730199178511100633sh_bit @ A @ N2 @ A2 ) )
          = ( ( N2
             != ( zero_zero @ nat ) )
            | ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) ) ) ) ).

% even_push_bit_iff
thf(fact_6711_push__bit__minus__numeral,axiom,
    ! [A: $tType] :
      ( ( bit_ri3973907225187159222ations @ A )
     => ! [L: num,K: num] :
          ( ( bit_se4730199178511100633sh_bit @ A @ ( numeral_numeral @ nat @ L ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ K ) ) )
          = ( bit_se4730199178511100633sh_bit @ A @ ( pred_numeral @ L ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( bit0 @ K ) ) ) ) ) ) ).

% push_bit_minus_numeral
thf(fact_6712_drop__bit__numeral__minus__bit1,axiom,
    ! [L: num,K: num] :
      ( ( bit_se4197421643247451524op_bit @ int @ ( numeral_numeral @ nat @ L ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ K ) ) ) )
      = ( bit_se4197421643247451524op_bit @ int @ ( pred_numeral @ L ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( inc @ K ) ) ) ) ) ).

% drop_bit_numeral_minus_bit1
thf(fact_6713_push__bit__numeral__minus__1,axiom,
    ! [A: $tType] :
      ( ( bit_ri3973907225187159222ations @ A )
     => ! [N2: num] :
          ( ( bit_se4730199178511100633sh_bit @ A @ ( numeral_numeral @ nat @ N2 ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
          = ( uminus_uminus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ N2 ) ) ) ) ) ).

% push_bit_numeral_minus_1
thf(fact_6714_drop__bit__int__code_I2_J,axiom,
    ! [N2: nat] :
      ( ( bit_se4197421643247451524op_bit @ int @ ( suc @ N2 ) @ ( zero_zero @ int ) )
      = ( zero_zero @ int ) ) ).

% drop_bit_int_code(2)
thf(fact_6715_push__bit__add,axiom,
    ! [A: $tType] :
      ( ( bit_se359711467146920520ations @ A )
     => ! [N2: nat,A2: A,B2: A] :
          ( ( bit_se4730199178511100633sh_bit @ A @ N2 @ ( plus_plus @ A @ A2 @ B2 ) )
          = ( plus_plus @ A @ ( bit_se4730199178511100633sh_bit @ A @ N2 @ A2 ) @ ( bit_se4730199178511100633sh_bit @ A @ N2 @ B2 ) ) ) ) ).

% push_bit_add
thf(fact_6716_push__bit__take__bit,axiom,
    ! [A: $tType] :
      ( ( bit_se359711467146920520ations @ A )
     => ! [M: nat,N2: nat,A2: A] :
          ( ( bit_se4730199178511100633sh_bit @ A @ M @ ( bit_se2584673776208193580ke_bit @ A @ N2 @ A2 ) )
          = ( bit_se2584673776208193580ke_bit @ A @ ( plus_plus @ nat @ M @ N2 ) @ ( bit_se4730199178511100633sh_bit @ A @ M @ A2 ) ) ) ) ).

% push_bit_take_bit
thf(fact_6717_take__bit__push__bit,axiom,
    ! [A: $tType] :
      ( ( bit_se359711467146920520ations @ A )
     => ! [M: nat,N2: nat,A2: A] :
          ( ( bit_se2584673776208193580ke_bit @ A @ M @ ( bit_se4730199178511100633sh_bit @ A @ N2 @ A2 ) )
          = ( bit_se4730199178511100633sh_bit @ A @ N2 @ ( bit_se2584673776208193580ke_bit @ A @ ( minus_minus @ nat @ M @ N2 ) @ A2 ) ) ) ) ).

% take_bit_push_bit
thf(fact_6718_num__of__nat_Osimps_I1_J,axiom,
    ( ( num_of_nat @ ( zero_zero @ nat ) )
    = one2 ) ).

% num_of_nat.simps(1)
thf(fact_6719_div__push__bit__of__1__eq__drop__bit,axiom,
    ! [A: $tType] :
      ( ( bit_se359711467146920520ations @ A )
     => ! [A2: A,N2: nat] :
          ( ( divide_divide @ A @ A2 @ ( bit_se4730199178511100633sh_bit @ A @ N2 @ ( one_one @ A ) ) )
          = ( bit_se4197421643247451524op_bit @ A @ N2 @ A2 ) ) ) ).

% div_push_bit_of_1_eq_drop_bit
thf(fact_6720_bits__ident,axiom,
    ! [A: $tType] :
      ( ( bit_se359711467146920520ations @ A )
     => ! [N2: nat,A2: A] :
          ( ( plus_plus @ A @ ( bit_se4730199178511100633sh_bit @ A @ N2 @ ( bit_se4197421643247451524op_bit @ A @ N2 @ A2 ) ) @ ( bit_se2584673776208193580ke_bit @ A @ N2 @ A2 ) )
          = A2 ) ) ).

% bits_ident
thf(fact_6721_set__bit__eq__or,axiom,
    ! [A: $tType] :
      ( ( bit_se359711467146920520ations @ A )
     => ( ( bit_se5668285175392031749et_bit @ A )
        = ( ^ [N: nat,A5: A] : ( bit_se1065995026697491101ons_or @ A @ A5 @ ( bit_se4730199178511100633sh_bit @ A @ N @ ( one_one @ A ) ) ) ) ) ) ).

% set_bit_eq_or
thf(fact_6722_flip__bit__eq__xor,axiom,
    ! [A: $tType] :
      ( ( bit_se359711467146920520ations @ A )
     => ( ( bit_se8732182000553998342ip_bit @ A )
        = ( ^ [N: nat,A5: A] : ( bit_se5824344971392196577ns_xor @ A @ A5 @ ( bit_se4730199178511100633sh_bit @ A @ N @ ( one_one @ A ) ) ) ) ) ) ).

% flip_bit_eq_xor
thf(fact_6723_push__bit__double,axiom,
    ! [A: $tType] :
      ( ( bit_se359711467146920520ations @ A )
     => ! [N2: nat,A2: A] :
          ( ( bit_se4730199178511100633sh_bit @ A @ N2 @ ( times_times @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) )
          = ( times_times @ A @ ( bit_se4730199178511100633sh_bit @ A @ N2 @ A2 ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).

% push_bit_double
thf(fact_6724_bit__iff__and__push__bit__not__eq__0,axiom,
    ! [A: $tType] :
      ( ( bit_se359711467146920520ations @ A )
     => ( ( bit_se5641148757651400278ts_bit @ A )
        = ( ^ [A5: A,N: nat] :
              ( ( bit_se5824344872417868541ns_and @ A @ A5 @ ( bit_se4730199178511100633sh_bit @ A @ N @ ( one_one @ A ) ) )
             != ( zero_zero @ A ) ) ) ) ) ).

% bit_iff_and_push_bit_not_eq_0
thf(fact_6725_numeral__num__of__nat,axiom,
    ! [N2: nat] :
      ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
     => ( ( numeral_numeral @ nat @ ( num_of_nat @ N2 ) )
        = N2 ) ) ).

% numeral_num_of_nat
thf(fact_6726_num__of__nat__One,axiom,
    ! [N2: nat] :
      ( ( ord_less_eq @ nat @ N2 @ ( one_one @ nat ) )
     => ( ( num_of_nat @ N2 )
        = one2 ) ) ).

% num_of_nat_One
thf(fact_6727_push__bit__mask__eq,axiom,
    ! [A: $tType] :
      ( ( bit_ri3973907225187159222ations @ A )
     => ! [M: nat,N2: nat] :
          ( ( bit_se4730199178511100633sh_bit @ A @ M @ ( bit_se2239418461657761734s_mask @ A @ N2 ) )
          = ( bit_se5824344872417868541ns_and @ A @ ( bit_se2239418461657761734s_mask @ A @ ( plus_plus @ nat @ N2 @ M ) ) @ ( bit_ri4277139882892585799ns_not @ A @ ( bit_se2239418461657761734s_mask @ A @ M ) ) ) ) ) ).

% push_bit_mask_eq
thf(fact_6728_unset__bit__eq__and__not,axiom,
    ! [A: $tType] :
      ( ( bit_ri3973907225187159222ations @ A )
     => ( ( bit_se2638667681897837118et_bit @ A )
        = ( ^ [N: nat,A5: A] : ( bit_se5824344872417868541ns_and @ A @ A5 @ ( bit_ri4277139882892585799ns_not @ A @ ( bit_se4730199178511100633sh_bit @ A @ N @ ( one_one @ A ) ) ) ) ) ) ) ).

% unset_bit_eq_and_not
thf(fact_6729_shiftr__integer__conv__div__pow2,axiom,
    ( ( bit_se4197421643247451524op_bit @ code_integer )
    = ( ^ [N: nat,X: code_integer] : ( divide_divide @ code_integer @ X @ ( power_power @ code_integer @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).

% shiftr_integer_conv_div_pow2
thf(fact_6730_bin__rest__code,axiom,
    ! [I: int] :
      ( ( divide_divide @ int @ I @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) )
      = ( bit_se4197421643247451524op_bit @ int @ ( one_one @ nat ) @ I ) ) ).

% bin_rest_code
thf(fact_6731_drop__bit__int__def,axiom,
    ( ( bit_se4197421643247451524op_bit @ int )
    = ( ^ [N: nat,K4: int] : ( divide_divide @ int @ K4 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).

% drop_bit_int_def
thf(fact_6732_push__bit__eq__mult,axiom,
    ! [A: $tType] :
      ( ( bit_se359711467146920520ations @ A )
     => ( ( bit_se4730199178511100633sh_bit @ A )
        = ( ^ [N: nat,A5: A] : ( times_times @ A @ A5 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) ) ) ) ) ).

% push_bit_eq_mult
thf(fact_6733_exp__dvdE,axiom,
    ! [A: $tType] :
      ( ( bit_se359711467146920520ations @ A )
     => ! [N2: nat,A2: A] :
          ( ( dvd_dvd @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) @ A2 )
         => ~ ! [B3: A] :
                ( A2
               != ( bit_se4730199178511100633sh_bit @ A @ N2 @ B3 ) ) ) ) ).

% exp_dvdE
thf(fact_6734_drop__bit__nat__def,axiom,
    ( ( bit_se4197421643247451524op_bit @ nat )
    = ( ^ [N: nat,M7: nat] : ( divide_divide @ nat @ M7 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).

% drop_bit_nat_def
thf(fact_6735_numeral__num__of__nat__unfold,axiom,
    ! [A: $tType] :
      ( ( semiring_1 @ A )
     => ! [N2: nat] :
          ( ( ( N2
              = ( zero_zero @ nat ) )
           => ( ( numeral_numeral @ A @ ( num_of_nat @ N2 ) )
              = ( one_one @ A ) ) )
          & ( ( N2
             != ( zero_zero @ nat ) )
           => ( ( numeral_numeral @ A @ ( num_of_nat @ N2 ) )
              = ( semiring_1_of_nat @ A @ N2 ) ) ) ) ) ).

% numeral_num_of_nat_unfold
thf(fact_6736_num__of__nat__double,axiom,
    ! [N2: nat] :
      ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
     => ( ( num_of_nat @ ( plus_plus @ nat @ N2 @ N2 ) )
        = ( bit0 @ ( num_of_nat @ N2 ) ) ) ) ).

% num_of_nat_double
thf(fact_6737_num__of__nat__plus__distrib,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
     => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
       => ( ( num_of_nat @ ( plus_plus @ nat @ M @ N2 ) )
          = ( plus_plus @ num @ ( num_of_nat @ M ) @ ( num_of_nat @ N2 ) ) ) ) ) ).

% num_of_nat_plus_distrib
thf(fact_6738_slice__eq__mask,axiom,
    ! [A: $tType] :
      ( ( bit_ri3973907225187159222ations @ A )
     => ! [N2: nat,M: nat,A2: A] :
          ( ( bit_se4730199178511100633sh_bit @ A @ N2 @ ( bit_se2584673776208193580ke_bit @ A @ M @ ( bit_se4197421643247451524op_bit @ A @ N2 @ A2 ) ) )
          = ( bit_se5824344872417868541ns_and @ A @ A2 @ ( bit_se5824344872417868541ns_and @ A @ ( bit_se2239418461657761734s_mask @ A @ ( plus_plus @ nat @ M @ N2 ) ) @ ( bit_ri4277139882892585799ns_not @ A @ ( bit_se2239418461657761734s_mask @ A @ N2 ) ) ) ) ) ) ).

% slice_eq_mask
thf(fact_6739_take__bit__sum,axiom,
    ! [A: $tType] :
      ( ( bit_un5681908812861735899ations @ A )
     => ( ( bit_se2584673776208193580ke_bit @ A )
        = ( ^ [N: nat,A5: A] :
              ( groups7311177749621191930dd_sum @ nat @ A
              @ ^ [K4: nat] : ( bit_se4730199178511100633sh_bit @ A @ K4 @ ( zero_neq_one_of_bool @ A @ ( bit_se5641148757651400278ts_bit @ A @ A5 @ K4 ) ) )
              @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ) ).

% take_bit_sum
thf(fact_6740_num__of__nat_Osimps_I2_J,axiom,
    ! [N2: nat] :
      ( ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
       => ( ( num_of_nat @ ( suc @ N2 ) )
          = ( inc @ ( num_of_nat @ N2 ) ) ) )
      & ( ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
       => ( ( num_of_nat @ ( suc @ N2 ) )
          = one2 ) ) ) ).

% num_of_nat.simps(2)
thf(fact_6741_word__and__mask__or__conv__and__mask,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [N2: word @ A,Index: nat] :
          ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ N2 @ Index )
         => ( ( bit_se1065995026697491101ons_or @ ( word @ A ) @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ N2 @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ Index ) ) @ ( bit_se4730199178511100633sh_bit @ ( word @ A ) @ Index @ ( one_one @ ( word @ A ) ) ) )
            = ( bit_se5824344872417868541ns_and @ ( word @ A ) @ N2 @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ ( plus_plus @ nat @ Index @ ( one_one @ nat ) ) ) ) ) ) ) ).

% word_and_mask_or_conv_and_mask
thf(fact_6742_signed__take__bit__code,axiom,
    ! [A: $tType] :
      ( ( bit_ri3973907225187159222ations @ A )
     => ( ( bit_ri4674362597316999326ke_bit @ A )
        = ( ^ [N: nat,A5: A] : ( if @ A @ ( bit_se5641148757651400278ts_bit @ A @ ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N ) @ A5 ) @ N ) @ ( plus_plus @ A @ ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N ) @ A5 ) @ ( bit_se4730199178511100633sh_bit @ A @ ( suc @ N ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) @ ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N ) @ A5 ) ) ) ) ) ).

% signed_take_bit_code
thf(fact_6743_set__bits__aux__Suc,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [F: nat > $o,N2: nat,W: word @ A] :
          ( ( code_T2661198915054445665ts_aux @ A @ F @ ( suc @ N2 ) @ W )
          = ( code_T2661198915054445665ts_aux @ A @ F @ N2 @ ( bit_se1065995026697491101ons_or @ ( word @ A ) @ ( bit_se4730199178511100633sh_bit @ ( word @ A ) @ ( one_one @ nat ) @ W ) @ ( if @ ( word @ A ) @ ( F @ N2 ) @ ( one_one @ ( word @ A ) ) @ ( zero_zero @ ( word @ A ) ) ) ) ) ) ) ).

% set_bits_aux_Suc
thf(fact_6744_set__bits__aux__rec,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ( ( code_T2661198915054445665ts_aux @ A )
        = ( ^ [F6: nat > $o,N: nat,W2: word @ A] :
              ( if @ ( word @ A )
              @ ( N
                = ( zero_zero @ nat ) )
              @ W2
              @ ( code_T2661198915054445665ts_aux @ A @ F6 @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ ( bit_se1065995026697491101ons_or @ ( word @ A ) @ ( bit_se4730199178511100633sh_bit @ ( word @ A ) @ ( one_one @ nat ) @ W2 ) @ ( if @ ( word @ A ) @ ( F6 @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) @ ( one_one @ ( word @ A ) ) @ ( zero_zero @ ( word @ A ) ) ) ) ) ) ) ) ) ).

% set_bits_aux_rec
thf(fact_6745_push__bit__nonnegative__int__iff,axiom,
    ! [N2: nat,K: int] :
      ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_se4730199178511100633sh_bit @ int @ N2 @ K ) )
      = ( ord_less_eq @ int @ ( zero_zero @ int ) @ K ) ) ).

% push_bit_nonnegative_int_iff
thf(fact_6746_push__bit__of__Suc__0,axiom,
    ! [N2: nat] :
      ( ( bit_se4730199178511100633sh_bit @ nat @ N2 @ ( suc @ ( zero_zero @ nat ) ) )
      = ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ).

% push_bit_of_Suc_0
thf(fact_6747_drop__bit__push__bit__int,axiom,
    ! [M: nat,N2: nat,K: int] :
      ( ( bit_se4197421643247451524op_bit @ int @ M @ ( bit_se4730199178511100633sh_bit @ int @ N2 @ K ) )
      = ( bit_se4197421643247451524op_bit @ int @ ( minus_minus @ nat @ M @ N2 ) @ ( bit_se4730199178511100633sh_bit @ int @ ( minus_minus @ nat @ N2 @ M ) @ K ) ) ) ).

% drop_bit_push_bit_int
thf(fact_6748_set__bit__nat__def,axiom,
    ( ( bit_se5668285175392031749et_bit @ nat )
    = ( ^ [M7: nat,N: nat] : ( bit_se1065995026697491101ons_or @ nat @ N @ ( bit_se4730199178511100633sh_bit @ nat @ M7 @ ( one_one @ nat ) ) ) ) ) ).

% set_bit_nat_def
thf(fact_6749_Bit__integer__code_I1_J,axiom,
    ! [I: code_integer] :
      ( ( bits_Bit_integer @ I @ $false )
      = ( bit_se4730199178511100633sh_bit @ code_integer @ ( one_one @ nat ) @ I ) ) ).

% Bit_integer_code(1)
thf(fact_6750_flip__bit__nat__def,axiom,
    ( ( bit_se8732182000553998342ip_bit @ nat )
    = ( ^ [M7: nat,N: nat] : ( bit_se5824344971392196577ns_xor @ nat @ N @ ( bit_se4730199178511100633sh_bit @ nat @ M7 @ ( one_one @ nat ) ) ) ) ) ).

% flip_bit_nat_def
thf(fact_6751_bit__push__bit__iff__int,axiom,
    ! [M: nat,K: int,N2: nat] :
      ( ( bit_se5641148757651400278ts_bit @ int @ ( bit_se4730199178511100633sh_bit @ int @ M @ K ) @ N2 )
      = ( ( ord_less_eq @ nat @ M @ N2 )
        & ( bit_se5641148757651400278ts_bit @ int @ K @ ( minus_minus @ nat @ N2 @ M ) ) ) ) ).

% bit_push_bit_iff_int
thf(fact_6752_Bit__Operations_Oset__bit__int__def,axiom,
    ( ( bit_se5668285175392031749et_bit @ int )
    = ( ^ [N: nat,K4: int] : ( bit_se1065995026697491101ons_or @ int @ K4 @ ( bit_se4730199178511100633sh_bit @ int @ N @ ( one_one @ int ) ) ) ) ) ).

% Bit_Operations.set_bit_int_def
thf(fact_6753_bit__push__bit__iff__nat,axiom,
    ! [M: nat,Q2: nat,N2: nat] :
      ( ( bit_se5641148757651400278ts_bit @ nat @ ( bit_se4730199178511100633sh_bit @ nat @ M @ Q2 ) @ N2 )
      = ( ( ord_less_eq @ nat @ M @ N2 )
        & ( bit_se5641148757651400278ts_bit @ nat @ Q2 @ ( minus_minus @ nat @ N2 @ M ) ) ) ) ).

% bit_push_bit_iff_nat
thf(fact_6754_flip__bit__int__def,axiom,
    ( ( bit_se8732182000553998342ip_bit @ int )
    = ( ^ [N: nat,K4: int] : ( bit_se5824344971392196577ns_xor @ int @ K4 @ ( bit_se4730199178511100633sh_bit @ int @ N @ ( one_one @ int ) ) ) ) ) ).

% flip_bit_int_def
thf(fact_6755_shiftl__integer__conv__mult__pow2,axiom,
    ( ( bit_se4730199178511100633sh_bit @ code_integer )
    = ( ^ [N: nat,X: code_integer] : ( times_times @ code_integer @ X @ ( power_power @ code_integer @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).

% shiftl_integer_conv_mult_pow2
thf(fact_6756_unset__bit__int__def,axiom,
    ( ( bit_se2638667681897837118et_bit @ int )
    = ( ^ [N: nat,K4: int] : ( bit_se5824344872417868541ns_and @ int @ K4 @ ( bit_ri4277139882892585799ns_not @ int @ ( bit_se4730199178511100633sh_bit @ int @ N @ ( one_one @ int ) ) ) ) ) ) ).

% unset_bit_int_def
thf(fact_6757_push__bit__int__def,axiom,
    ( ( bit_se4730199178511100633sh_bit @ int )
    = ( ^ [N: nat,K4: int] : ( times_times @ int @ K4 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).

% push_bit_int_def
thf(fact_6758_Bit__integer__code_I2_J,axiom,
    ! [I: code_integer] :
      ( ( bits_Bit_integer @ I @ $true )
      = ( plus_plus @ code_integer @ ( bit_se4730199178511100633sh_bit @ code_integer @ ( one_one @ nat ) @ I ) @ ( one_one @ code_integer ) ) ) ).

% Bit_integer_code(2)
thf(fact_6759_push__bit__nat__def,axiom,
    ( ( bit_se4730199178511100633sh_bit @ nat )
    = ( ^ [N: nat,M7: nat] : ( times_times @ nat @ M7 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).

% push_bit_nat_def
thf(fact_6760_push__bit__minus__one,axiom,
    ! [N2: nat] :
      ( ( bit_se4730199178511100633sh_bit @ int @ N2 @ ( uminus_uminus @ int @ ( one_one @ int ) ) )
      = ( uminus_uminus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N2 ) ) ) ).

% push_bit_minus_one
thf(fact_6761_test__bit__split,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( ( ( type_len @ B )
        & ( type_len @ C )
        & ( type_len @ A ) )
     => ! [C2: word @ C,A2: word @ A,B2: word @ B] :
          ( ( ( word_split @ C @ A @ B @ C2 )
            = ( product_Pair @ ( word @ A ) @ ( word @ B ) @ A2 @ B2 ) )
         => ( ! [N9: nat] :
                ( ( bit_se5641148757651400278ts_bit @ ( word @ B ) @ B2 @ N9 )
                = ( ( ord_less @ nat @ N9 @ ( size_size @ ( word @ B ) @ B2 ) )
                  & ( bit_se5641148757651400278ts_bit @ ( word @ C ) @ C2 @ N9 ) ) )
            & ! [M3: nat] :
                ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ A2 @ M3 )
                = ( ( ord_less @ nat @ M3 @ ( size_size @ ( word @ A ) @ A2 ) )
                  & ( bit_se5641148757651400278ts_bit @ ( word @ C ) @ C2 @ ( plus_plus @ nat @ M3 @ ( size_size @ ( word @ B ) @ B2 ) ) ) ) ) ) ) ) ).

% test_bit_split
thf(fact_6762_test__bit__split_H,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( ( ( type_len @ B )
        & ( type_len @ C )
        & ( type_len @ A ) )
     => ! [C2: word @ C,A2: word @ A,B2: word @ B] :
          ( ( ( word_split @ C @ A @ B @ C2 )
            = ( product_Pair @ ( word @ A ) @ ( word @ B ) @ A2 @ B2 ) )
         => ! [N9: nat,M3: nat] :
              ( ( ( bit_se5641148757651400278ts_bit @ ( word @ B ) @ B2 @ N9 )
                = ( ( ord_less @ nat @ N9 @ ( size_size @ ( word @ B ) @ B2 ) )
                  & ( bit_se5641148757651400278ts_bit @ ( word @ C ) @ C2 @ N9 ) ) )
              & ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ A2 @ M3 )
                = ( ( ord_less @ nat @ M3 @ ( size_size @ ( word @ A ) @ A2 ) )
                  & ( bit_se5641148757651400278ts_bit @ ( word @ C ) @ C2 @ ( plus_plus @ nat @ M3 @ ( size_size @ ( word @ B ) @ B2 ) ) ) ) ) ) ) ) ).

% test_bit_split'
thf(fact_6763_test__bit__split__eq,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( ( type_len @ A )
        & ( type_len @ C )
        & ( type_len @ B ) )
     => ! [C2: word @ C,A2: word @ A,B2: word @ B] :
          ( ( ( word_split @ C @ A @ B @ C2 )
            = ( product_Pair @ ( word @ A ) @ ( word @ B ) @ A2 @ B2 ) )
          = ( ! [N: nat] :
                ( ( bit_se5641148757651400278ts_bit @ ( word @ B ) @ B2 @ N )
                = ( ( ord_less @ nat @ N @ ( size_size @ ( word @ B ) @ B2 ) )
                  & ( bit_se5641148757651400278ts_bit @ ( word @ C ) @ C2 @ N ) ) )
            & ! [M7: nat] :
                ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ A2 @ M7 )
                = ( ( ord_less @ nat @ M7 @ ( size_size @ ( word @ A ) @ A2 ) )
                  & ( bit_se5641148757651400278ts_bit @ ( word @ C ) @ C2 @ ( plus_plus @ nat @ M7 @ ( size_size @ ( word @ B ) @ B2 ) ) ) ) ) ) ) ) ).

% test_bit_split_eq
thf(fact_6764_set__bit__integer__conv__masks,axiom,
    ( ( generi7602027413899671122et_bit @ code_integer )
    = ( ^ [X: code_integer,I2: nat,B5: $o] : ( if @ code_integer @ B5 @ ( bit_se1065995026697491101ons_or @ code_integer @ X @ ( bit_se4730199178511100633sh_bit @ code_integer @ I2 @ ( one_one @ code_integer ) ) ) @ ( bit_se5824344872417868541ns_and @ code_integer @ X @ ( bit_ri4277139882892585799ns_not @ code_integer @ ( bit_se4730199178511100633sh_bit @ code_integer @ I2 @ ( one_one @ code_integer ) ) ) ) ) ) ) ).

% set_bit_integer_conv_masks
thf(fact_6765_admissible__all,axiom,
    ! [A: $tType,B: $tType,Lub: ( set @ B ) > B,Ord: B > B > $o,P: B > A > $o] :
      ( ! [Y4: A] :
          ( comple1908693960933563346ssible @ B @ Lub @ Ord
          @ ^ [X: B] : ( P @ X @ Y4 ) )
     => ( comple1908693960933563346ssible @ B @ Lub @ Ord
        @ ^ [X: B] :
          ! [X5: A] : ( P @ X @ X5 ) ) ) ).

% admissible_all
thf(fact_6766_admissible__True,axiom,
    ! [A: $tType,Lub: ( set @ A ) > A,Ord: A > A > $o] :
      ( comple1908693960933563346ssible @ A @ Lub @ Ord
      @ ^ [X: A] : $true ) ).

% admissible_True
thf(fact_6767_admissible__conj,axiom,
    ! [A: $tType,Lub: ( set @ A ) > A,Ord: A > A > $o,P: A > $o,Q: A > $o] :
      ( ( comple1908693960933563346ssible @ A @ Lub @ Ord @ P )
     => ( ( comple1908693960933563346ssible @ A @ Lub @ Ord @ Q )
       => ( comple1908693960933563346ssible @ A @ Lub @ Ord
          @ ^ [X: A] :
              ( ( P @ X )
              & ( Q @ X ) ) ) ) ) ).

% admissible_conj
thf(fact_6768_admissible__const,axiom,
    ! [A: $tType,Lub: ( set @ A ) > A,Ord: A > A > $o,T2: $o] :
      ( comple1908693960933563346ssible @ A @ Lub @ Ord
      @ ^ [X: A] : T2 ) ).

% admissible_const
thf(fact_6769_bit__set__bit__iff__2n,axiom,
    ! [A: $tType] :
      ( ( generic_set_set_bit @ A )
     => ! [A2: A,M: nat,B2: $o,N2: nat] :
          ( ( bit_se5641148757651400278ts_bit @ A @ ( generi7602027413899671122et_bit @ A @ A2 @ M @ B2 ) @ N2 )
          = ( ( ( M = N2 )
             => B2 )
            & ( ( M != N2 )
             => ( bit_se5641148757651400278ts_bit @ A @ A2 @ N2 ) )
            & ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 )
             != ( zero_zero @ A ) ) ) ) ) ).

% bit_set_bit_iff_2n
thf(fact_6770_Code__Numeral_Opositive__def,axiom,
    ( code_positive
    = ( numeral_numeral @ code_integer ) ) ).

% Code_Numeral.positive_def
thf(fact_6771_test__bit__set,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [W: word @ A,N2: nat,X2: $o] :
          ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( generi7602027413899671122et_bit @ ( word @ A ) @ W @ N2 @ X2 ) @ N2 )
          = ( ( ord_less @ nat @ N2 @ ( size_size @ ( word @ A ) @ W ) )
            & X2 ) ) ) ).

% test_bit_set
thf(fact_6772_test__bit__set__gen,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [W: word @ A,N2: nat,X2: $o,M: nat] :
          ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( generi7602027413899671122et_bit @ ( word @ A ) @ W @ N2 @ X2 ) @ M )
          = ( ( ( M = N2 )
             => ( ( ord_less @ nat @ N2 @ ( size_size @ ( word @ A ) @ W ) )
                & X2 ) )
            & ( ( M != N2 )
             => ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ W @ M ) ) ) ) ) ).

% test_bit_set_gen
thf(fact_6773_word__set__nth__iff,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [W: word @ A,N2: nat,B2: $o] :
          ( ( ( generi7602027413899671122et_bit @ ( word @ A ) @ W @ N2 @ B2 )
            = W )
          = ( ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ W @ N2 )
              = B2 )
            | ( ord_less_eq @ nat @ ( size_size @ ( word @ A ) @ W ) @ N2 ) ) ) ) ).

% word_set_nth_iff
thf(fact_6774_set__bit__beyond,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [X2: word @ A,N2: nat,B2: $o] :
          ( ( ord_less_eq @ nat @ ( size_size @ ( word @ A ) @ X2 ) @ N2 )
         => ( ( generi7602027413899671122et_bit @ ( word @ A ) @ X2 @ N2 @ B2 )
            = X2 ) ) ) ).

% set_bit_beyond
thf(fact_6775_word__clr__le,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [W: word @ A,N2: nat] : ( ord_less_eq @ ( word @ A ) @ ( generi7602027413899671122et_bit @ ( word @ A ) @ W @ N2 @ $false ) @ W ) ) ).

% word_clr_le
thf(fact_6776_word__set__ge,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [W: word @ A,N2: nat] : ( ord_less_eq @ ( word @ A ) @ W @ ( generi7602027413899671122et_bit @ ( word @ A ) @ W @ N2 @ $true ) ) ) ).

% word_set_ge
thf(fact_6777_one__bit__pow,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [N2: nat] :
          ( ( generi7602027413899671122et_bit @ ( word @ A ) @ ( zero_zero @ ( word @ A ) ) @ N2 @ $true )
          = ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N2 ) ) ) ).

% one_bit_pow
thf(fact_6778_int__set__bit__True__conv__OR,axiom,
    ! [I: int,N2: nat] :
      ( ( generi7602027413899671122et_bit @ int @ I @ N2 @ $true )
      = ( bit_se1065995026697491101ons_or @ int @ I @ ( bit_se4730199178511100633sh_bit @ int @ N2 @ ( one_one @ int ) ) ) ) ).

% int_set_bit_True_conv_OR
thf(fact_6779_int__set__bit__False__conv__NAND,axiom,
    ! [I: int,N2: nat] :
      ( ( generi7602027413899671122et_bit @ int @ I @ N2 @ $false )
      = ( bit_se5824344872417868541ns_and @ int @ I @ ( bit_ri4277139882892585799ns_not @ int @ ( bit_se4730199178511100633sh_bit @ int @ N2 @ ( one_one @ int ) ) ) ) ) ).

% int_set_bit_False_conv_NAND
thf(fact_6780_int__set__bit__conv__ops,axiom,
    ( ( generi7602027413899671122et_bit @ int )
    = ( ^ [I2: int,N: nat,B5: $o] : ( if @ int @ B5 @ ( bit_se1065995026697491101ons_or @ int @ I2 @ ( bit_se4730199178511100633sh_bit @ int @ N @ ( one_one @ int ) ) ) @ ( bit_se5824344872417868541ns_and @ int @ I2 @ ( bit_ri4277139882892585799ns_not @ int @ ( bit_se4730199178511100633sh_bit @ int @ N @ ( one_one @ int ) ) ) ) ) ) ) ).

% int_set_bit_conv_ops
thf(fact_6781_case__prod__app,axiom,
    ! [A: $tType,D: $tType,C: $tType,B: $tType] :
      ( ( product_case_prod @ B @ C @ ( D > A ) )
      = ( ^ [F6: B > C > D > A,X: product_prod @ B @ C,Y: D] :
            ( product_case_prod @ B @ C @ A
            @ ^ [L2: B,R: C] : ( F6 @ L2 @ R @ Y )
            @ X ) ) ) ).

% case_prod_app
thf(fact_6782_the__elem__def,axiom,
    ! [A: $tType] :
      ( ( the_elem @ A )
      = ( ^ [X5: set @ A] :
            ( the @ A
            @ ^ [X: A] :
                ( X5
                = ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).

% the_elem_def
thf(fact_6783_predicate2D__conj,axiom,
    ! [A: $tType,B: $tType,P: A > B > $o,Q: A > B > $o,R2: $o,X2: A,Y2: B] :
      ( ( ( ord_less_eq @ ( A > B > $o ) @ P @ Q )
        & R2 )
     => ( R2
        & ( ( P @ X2 @ Y2 )
         => ( Q @ X2 @ Y2 ) ) ) ) ).

% predicate2D_conj
thf(fact_6784_fun__cong__unused__0,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( ( zero @ B )
     => ! [F: ( A > B ) > C,G: C] :
          ( ( F
            = ( ^ [X: A > B] : G ) )
         => ( ( F
              @ ^ [X: A] : ( zero_zero @ B ) )
            = G ) ) ) ).

% fun_cong_unused_0
thf(fact_6785_eq__subset,axiom,
    ! [A: $tType,P: A > A > $o] :
      ( ord_less_eq @ ( A > A > $o )
      @ ^ [Y5: A,Z4: A] : Y5 = Z4
      @ ^ [A5: A,B5: A] :
          ( ( P @ A5 @ B5 )
          | ( A5 = B5 ) ) ) ).

% eq_subset
thf(fact_6786_the__elem__set,axiom,
    ! [A: $tType,X2: A] :
      ( ( the_elem @ A @ ( set2 @ A @ ( cons @ A @ X2 @ ( nil @ A ) ) ) )
      = X2 ) ).

% the_elem_set
thf(fact_6787_case__prod__Pair__iden,axiom,
    ! [B: $tType,A: $tType,P4: product_prod @ A @ B] :
      ( ( product_case_prod @ A @ B @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B ) @ P4 )
      = P4 ) ).

% case_prod_Pair_iden
thf(fact_6788_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_6789_word__lsb__neg__numeral,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [Bin: num] :
          ( ( least_8051144512741203767sb_lsb @ ( word @ A ) @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ Bin ) ) )
          = ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ Bin ) ) ) ) ) ) ).

% word_lsb_neg_numeral
thf(fact_6790_word__lsb__numeral,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [Bin: num] :
          ( ( least_8051144512741203767sb_lsb @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ Bin ) )
          = ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ int @ Bin ) ) ) ) ) ).

% word_lsb_numeral
thf(fact_6791_word__lsb__1__0,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ( type_len @ B )
        & ( type_len @ A ) )
     => ( ( least_8051144512741203767sb_lsb @ ( word @ A ) @ ( one_one @ ( word @ A ) ) )
        & ~ ( least_8051144512741203767sb_lsb @ ( word @ B ) @ ( zero_zero @ ( word @ B ) ) ) ) ) ).

% word_lsb_1_0
thf(fact_6792_lsb__word__eq,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ( ( least_8051144512741203767sb_lsb @ ( word @ A ) )
        = ( ^ [A5: word @ A] :
              ~ ( dvd_dvd @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ A5 ) ) ) ) ).

% lsb_word_eq
thf(fact_6793_word__lsb__def,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ( ( least_8051144512741203767sb_lsb @ ( word @ A ) )
        = ( ^ [A5: word @ A] :
              ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( semiring_1_unsigned @ A @ int @ A5 ) ) ) ) ) ).

% word_lsb_def
thf(fact_6794_lsb__odd,axiom,
    ! [A: $tType] :
      ( ( least_6119777620449941438nt_lsb @ A )
     => ( ( least_8051144512741203767sb_lsb @ A )
        = ( ^ [A5: A] :
              ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A5 ) ) ) ) ).

% lsb_odd
thf(fact_6795_word__lsb__nat,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ( ( least_8051144512741203767sb_lsb @ ( word @ A ) )
        = ( ^ [W2: word @ A] :
              ( ( modulo_modulo @ nat @ ( semiring_1_unsigned @ A @ nat @ W2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
              = ( one_one @ nat ) ) ) ) ) ).

% word_lsb_nat
thf(fact_6796_word__lsb__int,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ( ( least_8051144512741203767sb_lsb @ ( word @ A ) )
        = ( ^ [W2: word @ A] :
              ( ( modulo_modulo @ int @ ( semiring_1_unsigned @ A @ int @ W2 ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) )
              = ( one_one @ int ) ) ) ) ) ).

% word_lsb_int
thf(fact_6797_upto_Opsimps,axiom,
    ! [I: int,J: int] :
      ( ( accp @ ( product_prod @ int @ int ) @ upto_rel @ ( product_Pair @ int @ int @ I @ J ) )
     => ( ( ( ord_less_eq @ int @ I @ J )
         => ( ( upto @ I @ J )
            = ( cons @ int @ I @ ( upto @ ( plus_plus @ int @ I @ ( one_one @ int ) ) @ J ) ) ) )
        & ( ~ ( ord_less_eq @ int @ I @ J )
         => ( ( upto @ I @ J )
            = ( nil @ int ) ) ) ) ) ).

% upto.psimps
thf(fact_6798_upto_Opelims,axiom,
    ! [X2: int,Xa: int,Y2: list @ int] :
      ( ( ( upto @ X2 @ Xa )
        = Y2 )
     => ( ( accp @ ( product_prod @ int @ int ) @ upto_rel @ ( product_Pair @ int @ int @ X2 @ Xa ) )
       => ~ ( ( ( ( ord_less_eq @ int @ X2 @ Xa )
               => ( Y2
                  = ( cons @ int @ X2 @ ( upto @ ( plus_plus @ int @ X2 @ ( one_one @ int ) ) @ Xa ) ) ) )
              & ( ~ ( ord_less_eq @ int @ X2 @ Xa )
               => ( Y2
                  = ( nil @ int ) ) ) )
           => ~ ( accp @ ( product_prod @ int @ int ) @ upto_rel @ ( product_Pair @ int @ int @ X2 @ Xa ) ) ) ) ) ).

% upto.pelims
thf(fact_6799_int__lsb__numeral_I2_J,axiom,
    least_8051144512741203767sb_lsb @ int @ ( one_one @ int ) ).

% int_lsb_numeral(2)
thf(fact_6800_int__lsb__numeral_I6_J,axiom,
    ! [W: num] :
      ~ ( least_8051144512741203767sb_lsb @ int @ ( numeral_numeral @ int @ ( bit0 @ W ) ) ) ).

% int_lsb_numeral(6)
thf(fact_6801_int__lsb__numeral_I3_J,axiom,
    least_8051144512741203767sb_lsb @ int @ ( numeral_numeral @ int @ one2 ) ).

% int_lsb_numeral(3)
thf(fact_6802_int__lsb__numeral_I7_J,axiom,
    ! [W: num] : ( least_8051144512741203767sb_lsb @ int @ ( numeral_numeral @ int @ ( bit1 @ W ) ) ) ).

% int_lsb_numeral(7)
thf(fact_6803_int__lsb__numeral_I4_J,axiom,
    least_8051144512741203767sb_lsb @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) ).

% int_lsb_numeral(4)
thf(fact_6804_int__lsb__numeral_I8_J,axiom,
    ! [W: num] :
      ~ ( least_8051144512741203767sb_lsb @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ W ) ) ) ) ).

% int_lsb_numeral(8)
thf(fact_6805_int__lsb__numeral_I5_J,axiom,
    least_8051144512741203767sb_lsb @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ one2 ) ) ).

% int_lsb_numeral(5)
thf(fact_6806_int__lsb__numeral_I9_J,axiom,
    ! [W: num] : ( least_8051144512741203767sb_lsb @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ W ) ) ) ) ).

% int_lsb_numeral(9)
thf(fact_6807_nth__upto,axiom,
    ! [I: int,K: nat,J: int] :
      ( ( ord_less_eq @ int @ ( plus_plus @ int @ I @ ( semiring_1_of_nat @ int @ K ) ) @ J )
     => ( ( nth @ int @ ( upto @ I @ J ) @ K )
        = ( plus_plus @ int @ I @ ( semiring_1_of_nat @ int @ K ) ) ) ) ).

% nth_upto
thf(fact_6808_length__upto,axiom,
    ! [I: int,J: int] :
      ( ( size_size @ ( list @ int ) @ ( upto @ I @ J ) )
      = ( nat2 @ ( plus_plus @ int @ ( minus_minus @ int @ J @ I ) @ ( one_one @ int ) ) ) ) ).

% length_upto
thf(fact_6809_upto__rec__numeral_I1_J,axiom,
    ! [M: num,N2: num] :
      ( ( ( ord_less_eq @ int @ ( numeral_numeral @ int @ M ) @ ( numeral_numeral @ int @ N2 ) )
       => ( ( upto @ ( numeral_numeral @ int @ M ) @ ( numeral_numeral @ int @ N2 ) )
          = ( cons @ int @ ( numeral_numeral @ int @ M ) @ ( upto @ ( plus_plus @ int @ ( numeral_numeral @ int @ M ) @ ( one_one @ int ) ) @ ( numeral_numeral @ int @ N2 ) ) ) ) )
      & ( ~ ( ord_less_eq @ int @ ( numeral_numeral @ int @ M ) @ ( numeral_numeral @ int @ N2 ) )
       => ( ( upto @ ( numeral_numeral @ int @ M ) @ ( numeral_numeral @ int @ N2 ) )
          = ( nil @ int ) ) ) ) ).

% upto_rec_numeral(1)
thf(fact_6810_upto__rec__numeral_I4_J,axiom,
    ! [M: num,N2: num] :
      ( ( ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N2 ) ) )
       => ( ( upto @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N2 ) ) )
          = ( cons @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M ) ) @ ( upto @ ( plus_plus @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M ) ) @ ( one_one @ int ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N2 ) ) ) ) ) )
      & ( ~ ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N2 ) ) )
       => ( ( upto @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N2 ) ) )
          = ( nil @ int ) ) ) ) ).

% upto_rec_numeral(4)
thf(fact_6811_upto__rec__numeral_I3_J,axiom,
    ! [M: num,N2: num] :
      ( ( ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M ) ) @ ( numeral_numeral @ int @ N2 ) )
       => ( ( upto @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M ) ) @ ( numeral_numeral @ int @ N2 ) )
          = ( cons @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M ) ) @ ( upto @ ( plus_plus @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M ) ) @ ( one_one @ int ) ) @ ( numeral_numeral @ int @ N2 ) ) ) ) )
      & ( ~ ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M ) ) @ ( numeral_numeral @ int @ N2 ) )
       => ( ( upto @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M ) ) @ ( numeral_numeral @ int @ N2 ) )
          = ( nil @ int ) ) ) ) ).

% upto_rec_numeral(3)
thf(fact_6812_upto__rec__numeral_I2_J,axiom,
    ! [M: num,N2: num] :
      ( ( ( ord_less_eq @ int @ ( numeral_numeral @ int @ M ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N2 ) ) )
       => ( ( upto @ ( numeral_numeral @ int @ M ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N2 ) ) )
          = ( cons @ int @ ( numeral_numeral @ int @ M ) @ ( upto @ ( plus_plus @ int @ ( numeral_numeral @ int @ M ) @ ( one_one @ int ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N2 ) ) ) ) ) )
      & ( ~ ( ord_less_eq @ int @ ( numeral_numeral @ int @ M ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N2 ) ) )
       => ( ( upto @ ( numeral_numeral @ int @ M ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N2 ) ) )
          = ( nil @ int ) ) ) ) ).

% upto_rec_numeral(2)
thf(fact_6813_atLeastAtMost__upto,axiom,
    ( ( set_or1337092689740270186AtMost @ int )
    = ( ^ [I2: int,J3: int] : ( set2 @ int @ ( upto @ I2 @ J3 ) ) ) ) ).

% atLeastAtMost_upto
thf(fact_6814_upto__split2,axiom,
    ! [I: int,J: int,K: int] :
      ( ( ord_less_eq @ int @ I @ J )
     => ( ( ord_less_eq @ int @ J @ K )
       => ( ( upto @ I @ K )
          = ( append @ int @ ( upto @ I @ J ) @ ( upto @ ( plus_plus @ int @ J @ ( one_one @ int ) ) @ K ) ) ) ) ) ).

% upto_split2
thf(fact_6815_upto__split1,axiom,
    ! [I: int,J: int,K: int] :
      ( ( ord_less_eq @ int @ I @ J )
     => ( ( ord_less_eq @ int @ J @ K )
       => ( ( upto @ I @ K )
          = ( append @ int @ ( upto @ I @ ( minus_minus @ int @ J @ ( one_one @ int ) ) ) @ ( upto @ J @ K ) ) ) ) ) ).

% upto_split1
thf(fact_6816_atLeastLessThan__upto,axiom,
    ( ( set_or7035219750837199246ssThan @ int )
    = ( ^ [I2: int,J3: int] : ( set2 @ int @ ( upto @ I2 @ ( minus_minus @ int @ J3 @ ( one_one @ int ) ) ) ) ) ) ).

% atLeastLessThan_upto
thf(fact_6817_upto_Oelims,axiom,
    ! [X2: int,Xa: int,Y2: list @ int] :
      ( ( ( upto @ X2 @ Xa )
        = Y2 )
     => ( ( ( ord_less_eq @ int @ X2 @ Xa )
         => ( Y2
            = ( cons @ int @ X2 @ ( upto @ ( plus_plus @ int @ X2 @ ( one_one @ int ) ) @ Xa ) ) ) )
        & ( ~ ( ord_less_eq @ int @ X2 @ Xa )
         => ( Y2
            = ( nil @ int ) ) ) ) ) ).

% upto.elims
thf(fact_6818_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_6819_upto__rec1,axiom,
    ! [I: int,J: int] :
      ( ( ord_less_eq @ int @ I @ J )
     => ( ( upto @ I @ J )
        = ( cons @ int @ I @ ( upto @ ( plus_plus @ int @ I @ ( one_one @ int ) ) @ J ) ) ) ) ).

% upto_rec1
thf(fact_6820_upto__rec2,axiom,
    ! [I: int,J: int] :
      ( ( ord_less_eq @ int @ I @ J )
     => ( ( upto @ I @ J )
        = ( append @ int @ ( upto @ I @ ( minus_minus @ int @ J @ ( one_one @ int ) ) ) @ ( cons @ int @ J @ ( nil @ int ) ) ) ) ) ).

% upto_rec2
thf(fact_6821_bin__last__conv__lsb,axiom,
    ( ( ^ [A5: int] :
          ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ A5 ) )
    = ( least_8051144512741203767sb_lsb @ int ) ) ).

% bin_last_conv_lsb
thf(fact_6822_upto__split3,axiom,
    ! [I: int,J: int,K: int] :
      ( ( ord_less_eq @ int @ I @ J )
     => ( ( ord_less_eq @ int @ J @ K )
       => ( ( upto @ I @ K )
          = ( append @ int @ ( upto @ I @ ( minus_minus @ int @ J @ ( one_one @ int ) ) ) @ ( cons @ int @ J @ ( upto @ ( plus_plus @ int @ J @ ( one_one @ int ) ) @ K ) ) ) ) ) ) ).

% upto_split3
thf(fact_6823_concat__bit__Suc,axiom,
    ! [N2: nat,K: int,L: int] :
      ( ( bit_concat_bit @ ( suc @ N2 ) @ K @ L )
      = ( plus_plus @ int @ ( modulo_modulo @ int @ K @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_concat_bit @ N2 @ ( divide_divide @ int @ K @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ L ) ) ) ) ).

% concat_bit_Suc
thf(fact_6824_word__cat__split__alt,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( ( type_len @ A )
        & ( type_len @ B )
        & ( type_len @ C ) )
     => ! [W: word @ A,U: word @ B,V: word @ C] :
          ( ( ord_less_eq @ nat @ ( size_size @ ( word @ A ) @ W ) @ ( plus_plus @ nat @ ( size_size @ ( word @ B ) @ U ) @ ( size_size @ ( word @ C ) @ V ) ) )
         => ( ( ( word_split @ A @ B @ C @ W )
              = ( product_Pair @ ( word @ B ) @ ( word @ C ) @ U @ V ) )
           => ( ( word_cat @ B @ C @ A @ U @ V )
              = W ) ) ) ) ).

% word_cat_split_alt
thf(fact_6825_concat__bit__nonnegative__iff,axiom,
    ! [N2: nat,K: int,L: int] :
      ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_concat_bit @ N2 @ K @ L ) )
      = ( ord_less_eq @ int @ ( zero_zero @ int ) @ L ) ) ).

% concat_bit_nonnegative_iff
thf(fact_6826_concat__bit__assoc,axiom,
    ! [N2: nat,K: int,M: nat,L: int,R4: int] :
      ( ( bit_concat_bit @ N2 @ K @ ( bit_concat_bit @ M @ L @ R4 ) )
      = ( bit_concat_bit @ ( plus_plus @ nat @ M @ N2 ) @ ( bit_concat_bit @ N2 @ K @ L ) @ R4 ) ) ).

% concat_bit_assoc
thf(fact_6827_concat__bit__eq,axiom,
    ( bit_concat_bit
    = ( ^ [N: nat,K4: int,L2: int] : ( plus_plus @ int @ ( bit_se2584673776208193580ke_bit @ int @ N @ K4 ) @ ( bit_se4730199178511100633sh_bit @ int @ N @ L2 ) ) ) ) ).

% concat_bit_eq
thf(fact_6828_bit__concat__bit__iff,axiom,
    ! [M: nat,K: int,L: int,N2: nat] :
      ( ( bit_se5641148757651400278ts_bit @ int @ ( bit_concat_bit @ M @ K @ L ) @ N2 )
      = ( ( ( ord_less @ nat @ N2 @ M )
          & ( bit_se5641148757651400278ts_bit @ int @ K @ N2 ) )
        | ( ( ord_less_eq @ nat @ M @ N2 )
          & ( bit_se5641148757651400278ts_bit @ int @ L @ ( minus_minus @ nat @ N2 @ M ) ) ) ) ) ).

% bit_concat_bit_iff
thf(fact_6829_test__bit__cat,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( ( ( type_len @ C )
        & ( type_len @ B )
        & ( type_len @ A ) )
     => ! [A2: word @ B,B2: word @ C,N2: nat] :
          ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( word_cat @ B @ C @ A @ A2 @ B2 ) @ N2 )
          = ( ( ord_less @ nat @ N2 @ ( size_size @ ( word @ A ) @ ( word_cat @ B @ C @ A @ A2 @ B2 ) ) )
            & ( ( ord_less @ nat @ N2 @ ( size_size @ ( word @ C ) @ B2 ) )
             => ( bit_se5641148757651400278ts_bit @ ( word @ C ) @ B2 @ N2 ) )
            & ( ~ ( ord_less @ nat @ N2 @ ( size_size @ ( word @ C ) @ B2 ) )
             => ( bit_se5641148757651400278ts_bit @ ( word @ B ) @ A2 @ ( minus_minus @ nat @ N2 @ ( size_size @ ( word @ C ) @ B2 ) ) ) ) ) ) ) ).

% test_bit_cat
thf(fact_6830_word__cat__split__size,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ( ( type_len @ C )
        & ( type_len @ A )
        & ( type_len @ B ) )
     => ! [T2: word @ A,U: word @ B,V: word @ C] :
          ( ( ord_less_eq @ nat @ ( size_size @ ( word @ A ) @ T2 ) @ ( plus_plus @ nat @ ( size_size @ ( word @ B ) @ U ) @ ( size_size @ ( word @ C ) @ V ) ) )
         => ( ( ( product_Pair @ ( word @ B ) @ ( word @ C ) @ U @ V )
              = ( word_split @ A @ B @ C @ T2 ) )
           => ( T2
              = ( word_cat @ B @ C @ A @ U @ V ) ) ) ) ) ).

% word_cat_split_size
thf(fact_6831_word__split__cat__alt,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( ( ( type_len @ C )
        & ( type_len @ B )
        & ( type_len @ A ) )
     => ! [W: word @ A,U: word @ B,V: word @ C] :
          ( ( W
            = ( word_cat @ B @ C @ A @ U @ V ) )
         => ( ( ord_less_eq @ nat @ ( plus_plus @ nat @ ( size_size @ ( word @ B ) @ U ) @ ( size_size @ ( word @ C ) @ V ) ) @ ( size_size @ ( word @ A ) @ W ) )
           => ( ( word_split @ A @ B @ C @ W )
              = ( product_Pair @ ( word @ B ) @ ( word @ C ) @ U @ V ) ) ) ) ) ).

% word_split_cat_alt
thf(fact_6832_cat__slices,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( ( ( type_len @ B )
        & ( type_len @ A )
        & ( type_len @ C ) )
     => ! [A2: word @ A,N2: nat,C2: word @ B,B2: word @ C] :
          ( ( A2
            = ( slice2 @ B @ A @ N2 @ C2 ) )
         => ( ( B2
              = ( slice2 @ B @ C @ ( zero_zero @ nat ) @ C2 ) )
           => ( ( N2
                = ( size_size @ ( word @ C ) @ B2 ) )
             => ( ( ord_less_eq @ nat @ ( size_size @ ( word @ B ) @ C2 ) @ ( plus_plus @ nat @ ( size_size @ ( word @ A ) @ A2 ) @ ( size_size @ ( word @ C ) @ B2 ) ) )
               => ( ( word_cat @ A @ C @ B @ A2 @ B2 )
                  = C2 ) ) ) ) ) ) ).

% cat_slices
thf(fact_6833_mult_Osafe__commute,axiom,
    ! [A: $tType] :
      ( ( ab_semigroup_mult @ A )
     => ! [X2: A,Y2: A,A2: A,B2: A] :
          ( ( syntax7388354845996824322omatch @ A @ A @ ( times_times @ A @ X2 @ Y2 ) @ A2 )
         => ( ( times_times @ A @ A2 @ B2 )
            = ( times_times @ A @ B2 @ A2 ) ) ) ) ).

% mult.safe_commute
thf(fact_6834_assn__aci_I11_J,axiom,
    ! [X2: assn,Y2: assn,A2: assn,B2: assn] :
      ( ( syntax7388354845996824322omatch @ assn @ assn @ ( times_times @ assn @ X2 @ Y2 ) @ A2 )
     => ( ( times_times @ assn @ A2 @ B2 )
        = ( times_times @ assn @ B2 @ A2 ) ) ) ).

% assn_aci(11)
thf(fact_6835_add_Oright__commute,axiom,
    ! [A: $tType] :
      ( ( ab_semigroup_add @ A )
     => ! [A2: A,B2: A,C2: A] :
          ( ( plus_plus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 )
          = ( plus_plus @ A @ ( plus_plus @ A @ A2 @ C2 ) @ B2 ) ) ) ).

% add.right_commute
thf(fact_6836_add_Oright__assoc,axiom,
    ! [A: $tType] :
      ( ( ab_semigroup_add @ A )
     => ! [A2: A,B2: A,C2: A] :
          ( ( plus_plus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 )
          = ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B2 @ C2 ) ) ) ) ).

% add.right_assoc
thf(fact_6837_mult_Oright__assoc,axiom,
    ! [A: $tType] :
      ( ( ab_semigroup_mult @ A )
     => ! [A2: A,B2: A,C2: A] :
          ( ( times_times @ A @ ( times_times @ A @ A2 @ B2 ) @ C2 )
          = ( times_times @ A @ A2 @ ( times_times @ A @ B2 @ C2 ) ) ) ) ).

% mult.right_assoc
thf(fact_6838_mult_Oright__commute,axiom,
    ! [A: $tType] :
      ( ( ab_semigroup_mult @ A )
     => ! [A2: A,B2: A,C2: A] :
          ( ( times_times @ A @ ( times_times @ A @ A2 @ B2 ) @ C2 )
          = ( times_times @ A @ ( times_times @ A @ A2 @ C2 ) @ B2 ) ) ) ).

% mult.right_commute
thf(fact_6839_slice__cat1,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( ( ( type_len @ B )
        & ( type_len @ A )
        & ( type_len @ C ) )
     => ! [A2: word @ B,B2: word @ C] :
          ( ( ord_less_eq @ nat @ ( plus_plus @ nat @ ( size_size @ ( word @ B ) @ A2 ) @ ( size_size @ ( word @ C ) @ B2 ) ) @ ( size_size @ ( word @ A ) @ ( word_cat @ B @ C @ A @ A2 @ B2 ) ) )
         => ( ( slice2 @ A @ B @ ( size_size @ ( word @ C ) @ B2 ) @ ( word_cat @ B @ C @ A @ A2 @ B2 ) )
            = A2 ) ) ) ).

% slice_cat1
thf(fact_6840_split__slices,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( ( ( type_len @ C )
        & ( type_len @ B )
        & ( type_len @ A ) )
     => ! [W: word @ C,U: word @ A,V: word @ B] :
          ( ( ( word_split @ C @ A @ B @ W )
            = ( product_Pair @ ( word @ A ) @ ( word @ B ) @ U @ V ) )
         => ( ( U
              = ( slice2 @ C @ A @ ( size_size @ ( word @ B ) @ V ) @ W ) )
            & ( V
              = ( slice2 @ C @ B @ ( zero_zero @ nat ) @ W ) ) ) ) ) ).

% split_slices
thf(fact_6841_add_Osafe__commute,axiom,
    ! [A: $tType] :
      ( ( ab_semigroup_add @ A )
     => ! [X2: A,Y2: A,A2: A,B2: A] :
          ( ( syntax7388354845996824322omatch @ A @ A @ ( plus_plus @ A @ X2 @ Y2 ) @ A2 )
         => ( ( plus_plus @ A @ A2 @ B2 )
            = ( plus_plus @ A @ B2 @ A2 ) ) ) ) ).

% add.safe_commute
thf(fact_6842_sum__diff1_H__aux,axiom,
    ! [B: $tType,A: $tType] :
      ( ( ab_group_add @ B )
     => ! [F5: set @ A,I5: set @ A,F: A > B,I: A] :
          ( ( finite_finite2 @ A @ F5 )
         => ( ( ord_less_eq @ ( set @ A )
              @ ( collect @ A
                @ ^ [I2: A] :
                    ( ( member @ A @ I2 @ I5 )
                    & ( ( F @ I2 )
                     != ( zero_zero @ B ) ) ) )
              @ F5 )
           => ( ( ( member @ A @ I @ I5 )
               => ( ( groups1027152243600224163dd_sum @ A @ B @ F @ ( minus_minus @ ( set @ A ) @ I5 @ ( insert @ A @ I @ ( bot_bot @ ( set @ A ) ) ) ) )
                  = ( minus_minus @ B @ ( groups1027152243600224163dd_sum @ A @ B @ F @ I5 ) @ ( F @ I ) ) ) )
              & ( ~ ( member @ A @ I @ I5 )
               => ( ( groups1027152243600224163dd_sum @ A @ B @ F @ ( minus_minus @ ( set @ A ) @ I5 @ ( insert @ A @ I @ ( bot_bot @ ( set @ A ) ) ) ) )
                  = ( groups1027152243600224163dd_sum @ A @ B @ F @ I5 ) ) ) ) ) ) ) ).

% sum_diff1'_aux
thf(fact_6843_bit__horner__sum__bit__iff,axiom,
    ! [A: $tType] :
      ( ( bit_un5681908812861735899ations @ A )
     => ! [Bs2: list @ $o,N2: nat] :
          ( ( bit_se5641148757651400278ts_bit @ A @ ( groups4207007520872428315er_sum @ $o @ A @ ( zero_neq_one_of_bool @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ Bs2 ) @ N2 )
          = ( ( ord_less @ nat @ N2 @ ( size_size @ ( list @ $o ) @ Bs2 ) )
            & ( nth @ $o @ Bs2 @ N2 ) ) ) ) ).

% bit_horner_sum_bit_iff
thf(fact_6844_sum_Oinsert_H,axiom,
    ! [A: $tType,B: $tType] :
      ( ( comm_monoid_add @ A )
     => ! [I5: set @ B,P4: B > A,I: B] :
          ( ( finite_finite2 @ B
            @ ( collect @ B
              @ ^ [X: B] :
                  ( ( member @ B @ X @ I5 )
                  & ( ( P4 @ X )
                   != ( zero_zero @ A ) ) ) ) )
         => ( ( ( member @ B @ I @ I5 )
             => ( ( groups1027152243600224163dd_sum @ B @ A @ P4 @ ( insert @ B @ I @ I5 ) )
                = ( groups1027152243600224163dd_sum @ B @ A @ P4 @ I5 ) ) )
            & ( ~ ( member @ B @ I @ I5 )
             => ( ( groups1027152243600224163dd_sum @ B @ A @ P4 @ ( insert @ B @ I @ I5 ) )
                = ( plus_plus @ A @ ( P4 @ I ) @ ( groups1027152243600224163dd_sum @ B @ A @ P4 @ I5 ) ) ) ) ) ) ) ).

% sum.insert'
thf(fact_6845_sum_Onon__neutral_H,axiom,
    ! [A: $tType,B: $tType] :
      ( ( comm_monoid_add @ A )
     => ! [G: B > A,I5: set @ B] :
          ( ( groups1027152243600224163dd_sum @ B @ A @ G
            @ ( collect @ B
              @ ^ [X: B] :
                  ( ( member @ B @ X @ I5 )
                  & ( ( G @ X )
                   != ( zero_zero @ A ) ) ) ) )
          = ( groups1027152243600224163dd_sum @ B @ A @ G @ I5 ) ) ) ).

% sum.non_neutral'
thf(fact_6846_sum_Odistrib__triv_H,axiom,
    ! [A: $tType,B: $tType] :
      ( ( comm_monoid_add @ A )
     => ! [I5: set @ B,G: B > A,H2: B > A] :
          ( ( finite_finite2 @ B @ I5 )
         => ( ( groups1027152243600224163dd_sum @ B @ A
              @ ^ [I2: B] : ( plus_plus @ A @ ( G @ I2 ) @ ( H2 @ I2 ) )
              @ I5 )
            = ( plus_plus @ A @ ( groups1027152243600224163dd_sum @ B @ A @ G @ I5 ) @ ( groups1027152243600224163dd_sum @ B @ A @ H2 @ I5 ) ) ) ) ) ).

% sum.distrib_triv'
thf(fact_6847_sum_Omono__neutral__left_H,axiom,
    ! [A: $tType,B: $tType] :
      ( ( comm_monoid_add @ A )
     => ! [S3: set @ B,T6: set @ B,G: B > A] :
          ( ( ord_less_eq @ ( set @ B ) @ S3 @ T6 )
         => ( ! [X3: B] :
                ( ( member @ B @ X3 @ ( minus_minus @ ( set @ B ) @ T6 @ S3 ) )
               => ( ( G @ X3 )
                  = ( zero_zero @ A ) ) )
           => ( ( groups1027152243600224163dd_sum @ B @ A @ G @ S3 )
              = ( groups1027152243600224163dd_sum @ B @ A @ G @ T6 ) ) ) ) ) ).

% sum.mono_neutral_left'
thf(fact_6848_sum_Omono__neutral__right_H,axiom,
    ! [A: $tType,B: $tType] :
      ( ( comm_monoid_add @ A )
     => ! [S3: set @ B,T6: set @ B,G: B > A] :
          ( ( ord_less_eq @ ( set @ B ) @ S3 @ T6 )
         => ( ! [X3: B] :
                ( ( member @ B @ X3 @ ( minus_minus @ ( set @ B ) @ T6 @ S3 ) )
               => ( ( G @ X3 )
                  = ( zero_zero @ A ) ) )
           => ( ( groups1027152243600224163dd_sum @ B @ A @ G @ T6 )
              = ( groups1027152243600224163dd_sum @ B @ A @ G @ S3 ) ) ) ) ) ).

% sum.mono_neutral_right'
thf(fact_6849_sum_Omono__neutral__cong__left_H,axiom,
    ! [A: $tType,B: $tType] :
      ( ( comm_monoid_add @ A )
     => ! [S3: set @ B,T6: set @ B,H2: B > A,G: B > A] :
          ( ( ord_less_eq @ ( set @ B ) @ S3 @ T6 )
         => ( ! [I3: B] :
                ( ( member @ B @ I3 @ ( minus_minus @ ( set @ B ) @ T6 @ S3 ) )
               => ( ( H2 @ I3 )
                  = ( zero_zero @ A ) ) )
           => ( ! [X3: B] :
                  ( ( member @ B @ X3 @ S3 )
                 => ( ( G @ X3 )
                    = ( H2 @ X3 ) ) )
             => ( ( groups1027152243600224163dd_sum @ B @ A @ G @ S3 )
                = ( groups1027152243600224163dd_sum @ B @ A @ H2 @ T6 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left'
thf(fact_6850_sum_Omono__neutral__cong__right_H,axiom,
    ! [A: $tType,B: $tType] :
      ( ( comm_monoid_add @ A )
     => ! [S3: set @ B,T6: set @ B,G: B > A,H2: B > A] :
          ( ( ord_less_eq @ ( set @ B ) @ S3 @ T6 )
         => ( ! [X3: B] :
                ( ( member @ B @ X3 @ ( minus_minus @ ( set @ B ) @ T6 @ S3 ) )
               => ( ( G @ X3 )
                  = ( zero_zero @ A ) ) )
           => ( ! [X3: B] :
                  ( ( member @ B @ X3 @ S3 )
                 => ( ( G @ X3 )
                    = ( H2 @ X3 ) ) )
             => ( ( groups1027152243600224163dd_sum @ B @ A @ G @ T6 )
                = ( groups1027152243600224163dd_sum @ B @ A @ H2 @ S3 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right'
thf(fact_6851_sum_Odistrib_H,axiom,
    ! [A: $tType,B: $tType] :
      ( ( comm_monoid_add @ A )
     => ! [I5: set @ B,G: B > A,H2: B > A] :
          ( ( finite_finite2 @ B
            @ ( collect @ B
              @ ^ [X: B] :
                  ( ( member @ B @ X @ I5 )
                  & ( ( G @ X )
                   != ( zero_zero @ A ) ) ) ) )
         => ( ( finite_finite2 @ B
              @ ( collect @ B
                @ ^ [X: B] :
                    ( ( member @ B @ X @ I5 )
                    & ( ( H2 @ X )
                     != ( zero_zero @ A ) ) ) ) )
           => ( ( groups1027152243600224163dd_sum @ B @ A
                @ ^ [I2: B] : ( plus_plus @ A @ ( G @ I2 ) @ ( H2 @ I2 ) )
                @ I5 )
              = ( plus_plus @ A @ ( groups1027152243600224163dd_sum @ B @ A @ G @ I5 ) @ ( groups1027152243600224163dd_sum @ B @ A @ H2 @ I5 ) ) ) ) ) ) ).

% sum.distrib'
thf(fact_6852_sum_OG__def,axiom,
    ! [A: $tType,B: $tType] :
      ( ( comm_monoid_add @ A )
     => ( ( groups1027152243600224163dd_sum @ B @ A )
        = ( ^ [P5: B > A,I8: set @ B] :
              ( if @ A
              @ ( finite_finite2 @ B
                @ ( collect @ B
                  @ ^ [X: B] :
                      ( ( member @ B @ X @ I8 )
                      & ( ( P5 @ X )
                       != ( zero_zero @ A ) ) ) ) )
              @ ( groups7311177749621191930dd_sum @ B @ A @ P5
                @ ( collect @ B
                  @ ^ [X: B] :
                      ( ( member @ B @ X @ I8 )
                      & ( ( P5 @ X )
                       != ( zero_zero @ A ) ) ) ) )
              @ ( zero_zero @ A ) ) ) ) ) ).

% sum.G_def
thf(fact_6853_sum__diff1_H,axiom,
    ! [B: $tType,A: $tType] :
      ( ( ab_group_add @ B )
     => ! [I5: set @ A,F: A > B,I: A] :
          ( ( finite_finite2 @ A
            @ ( collect @ A
              @ ^ [I2: A] :
                  ( ( member @ A @ I2 @ I5 )
                  & ( ( F @ I2 )
                   != ( zero_zero @ B ) ) ) ) )
         => ( ( ( member @ A @ I @ I5 )
             => ( ( groups1027152243600224163dd_sum @ A @ B @ F @ ( minus_minus @ ( set @ A ) @ I5 @ ( insert @ A @ I @ ( bot_bot @ ( set @ A ) ) ) ) )
                = ( minus_minus @ B @ ( groups1027152243600224163dd_sum @ A @ B @ F @ I5 ) @ ( F @ I ) ) ) )
            & ( ~ ( member @ A @ I @ I5 )
             => ( ( groups1027152243600224163dd_sum @ A @ B @ F @ ( minus_minus @ ( set @ A ) @ I5 @ ( insert @ A @ I @ ( bot_bot @ ( set @ A ) ) ) ) )
                = ( groups1027152243600224163dd_sum @ A @ B @ F @ I5 ) ) ) ) ) ) ).

% sum_diff1'
thf(fact_6854_horner__sum__of__bool__2__less,axiom,
    ! [Bs2: list @ $o] : ( ord_less @ int @ ( groups4207007520872428315er_sum @ $o @ int @ ( zero_neq_one_of_bool @ int ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ Bs2 ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( size_size @ ( list @ $o ) @ Bs2 ) ) ) ).

% horner_sum_of_bool_2_less
thf(fact_6855_horner__sum__simps_I2_J,axiom,
    ! [A: $tType,B: $tType] :
      ( ( comm_semiring_0 @ A )
     => ! [F: B > A,A2: A,X2: B,Xs2: list @ B] :
          ( ( groups4207007520872428315er_sum @ B @ A @ F @ A2 @ ( cons @ B @ X2 @ Xs2 ) )
          = ( plus_plus @ A @ ( F @ X2 ) @ ( times_times @ A @ A2 @ ( groups4207007520872428315er_sum @ B @ A @ F @ A2 @ Xs2 ) ) ) ) ) ).

% horner_sum_simps(2)
thf(fact_6856_horner__sum__eq__sum,axiom,
    ! [A: $tType,B: $tType] :
      ( ( comm_semiring_1 @ A )
     => ( ( groups4207007520872428315er_sum @ B @ A )
        = ( ^ [F6: B > A,A5: A,Xs: list @ B] :
              ( groups7311177749621191930dd_sum @ nat @ A
              @ ^ [N: nat] : ( times_times @ A @ ( F6 @ ( nth @ B @ Xs @ N ) ) @ ( power_power @ A @ A5 @ N ) )
              @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ B ) @ Xs ) ) ) ) ) ) ).

% horner_sum_eq_sum
thf(fact_6857_horner__sum__foldr,axiom,
    ! [A: $tType,B: $tType] :
      ( ( comm_semiring_0 @ A )
     => ( ( groups4207007520872428315er_sum @ B @ A )
        = ( ^ [F6: B > A,A5: A,Xs: list @ B] :
              ( foldr @ B @ A
              @ ^ [X: B,B5: A] : ( plus_plus @ A @ ( F6 @ X ) @ ( times_times @ A @ A5 @ B5 ) )
              @ Xs
              @ ( zero_zero @ A ) ) ) ) ) ).

% horner_sum_foldr
thf(fact_6858_horner__sum__append,axiom,
    ! [A: $tType,B: $tType] :
      ( ( comm_semiring_1 @ A )
     => ! [F: B > A,A2: A,Xs2: list @ B,Ys: list @ B] :
          ( ( groups4207007520872428315er_sum @ B @ A @ F @ A2 @ ( append @ B @ Xs2 @ Ys ) )
          = ( plus_plus @ A @ ( groups4207007520872428315er_sum @ B @ A @ F @ A2 @ Xs2 ) @ ( times_times @ A @ ( power_power @ A @ A2 @ ( size_size @ ( list @ B ) @ Xs2 ) ) @ ( groups4207007520872428315er_sum @ B @ A @ F @ A2 @ Ys ) ) ) ) ) ).

% horner_sum_append
thf(fact_6859_horner__sum__bit__eq__take__bit,axiom,
    ! [A: $tType] :
      ( ( bit_se359711467146920520ations @ A )
     => ! [A2: A,N2: nat] :
          ( ( groups4207007520872428315er_sum @ $o @ A @ ( zero_neq_one_of_bool @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( map @ nat @ $o @ ( bit_se5641148757651400278ts_bit @ A @ A2 ) @ ( upt @ ( zero_zero @ nat ) @ N2 ) ) )
          = ( bit_se2584673776208193580ke_bit @ A @ N2 @ A2 ) ) ) ).

% horner_sum_bit_eq_take_bit
thf(fact_6860_sdiv__word__min,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [A2: word @ A,B2: word @ A] : ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ ( size_size @ ( word @ A ) @ A2 ) @ ( one_one @ nat ) ) ) ) @ ( signed7115095781618012415divide @ int @ ( ring_1_signed @ A @ int @ A2 ) @ ( ring_1_signed @ A @ int @ B2 ) ) ) ) ).

% sdiv_word_min
thf(fact_6861_upt__0__eq__Nil__conv,axiom,
    ! [J: nat] :
      ( ( ( upt @ ( zero_zero @ nat ) @ J )
        = ( nil @ nat ) )
      = ( J
        = ( zero_zero @ nat ) ) ) ).

% upt_0_eq_Nil_conv
thf(fact_6862_upt__conv__Nil,axiom,
    ! [J: nat,I: nat] :
      ( ( ord_less_eq @ nat @ J @ I )
     => ( ( upt @ I @ J )
        = ( nil @ nat ) ) ) ).

% upt_conv_Nil
thf(fact_6863_upt__merge,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( ( ord_less_eq @ nat @ I @ J )
        & ( ord_less_eq @ nat @ J @ K ) )
     => ( ( append @ nat @ ( upt @ I @ J ) @ ( upt @ J @ K ) )
        = ( upt @ I @ K ) ) ) ).

% upt_merge
thf(fact_6864_int__sdiv__simps_I1_J,axiom,
    ! [A2: int] :
      ( ( signed7115095781618012415divide @ int @ A2 @ ( one_one @ int ) )
      = A2 ) ).

% int_sdiv_simps(1)
thf(fact_6865_upt__eq__Nil__conv,axiom,
    ! [I: nat,J: nat] :
      ( ( ( upt @ I @ J )
        = ( nil @ nat ) )
      = ( ( J
          = ( zero_zero @ nat ) )
        | ( ord_less_eq @ nat @ J @ I ) ) ) ).

% upt_eq_Nil_conv
thf(fact_6866_int__sdiv__same__is__1,axiom,
    ! [A2: int,B2: int] :
      ( ( A2
       != ( zero_zero @ int ) )
     => ( ( ( signed7115095781618012415divide @ int @ A2 @ B2 )
          = A2 )
        = ( B2
          = ( one_one @ int ) ) ) ) ).

% int_sdiv_same_is_1
thf(fact_6867_int__sdiv__simps_I3_J,axiom,
    ! [A2: int] :
      ( ( signed7115095781618012415divide @ int @ A2 @ ( uminus_uminus @ int @ ( one_one @ int ) ) )
      = ( uminus_uminus @ int @ A2 ) ) ).

% int_sdiv_simps(3)
thf(fact_6868_sdiv__int__numeral__numeral,axiom,
    ! [M: num,N2: num] :
      ( ( signed7115095781618012415divide @ int @ ( numeral_numeral @ int @ M ) @ ( numeral_numeral @ int @ N2 ) )
      = ( divide_divide @ int @ ( numeral_numeral @ int @ M ) @ ( numeral_numeral @ int @ N2 ) ) ) ).

% sdiv_int_numeral_numeral
thf(fact_6869_length__upt,axiom,
    ! [I: nat,J: nat] :
      ( ( size_size @ ( list @ nat ) @ ( upt @ I @ J ) )
      = ( minus_minus @ nat @ J @ I ) ) ).

% length_upt
thf(fact_6870_upt__rec__numeral,axiom,
    ! [M: num,N2: num] :
      ( ( ( ord_less @ nat @ ( numeral_numeral @ nat @ M ) @ ( numeral_numeral @ nat @ N2 ) )
       => ( ( upt @ ( numeral_numeral @ nat @ M ) @ ( numeral_numeral @ nat @ N2 ) )
          = ( cons @ nat @ ( numeral_numeral @ nat @ M ) @ ( upt @ ( suc @ ( numeral_numeral @ nat @ M ) ) @ ( numeral_numeral @ nat @ N2 ) ) ) ) )
      & ( ~ ( ord_less @ nat @ ( numeral_numeral @ nat @ M ) @ ( numeral_numeral @ nat @ N2 ) )
       => ( ( upt @ ( numeral_numeral @ nat @ M ) @ ( numeral_numeral @ nat @ N2 ) )
          = ( nil @ nat ) ) ) ) ).

% upt_rec_numeral
thf(fact_6871_int__sdiv__negated__is__minus1,axiom,
    ! [A2: int,B2: int] :
      ( ( A2
       != ( zero_zero @ int ) )
     => ( ( ( signed7115095781618012415divide @ int @ A2 @ B2 )
          = ( uminus_uminus @ int @ A2 ) )
        = ( B2
          = ( uminus_uminus @ int @ ( one_one @ int ) ) ) ) ) ).

% int_sdiv_negated_is_minus1
thf(fact_6872_nth__upt,axiom,
    ! [I: nat,K: nat,J: nat] :
      ( ( ord_less @ nat @ ( plus_plus @ nat @ I @ K ) @ J )
     => ( ( nth @ nat @ ( upt @ I @ J ) @ K )
        = ( plus_plus @ nat @ I @ K ) ) ) ).

% nth_upt
thf(fact_6873_map__fst__enumerate,axiom,
    ! [A: $tType,N2: nat,Xs2: list @ A] :
      ( ( map @ ( product_prod @ nat @ A ) @ nat @ ( product_fst @ nat @ A ) @ ( enumerate @ A @ N2 @ Xs2 ) )
      = ( upt @ N2 @ ( plus_plus @ nat @ N2 @ ( size_size @ ( list @ A ) @ Xs2 ) ) ) ) ).

% map_fst_enumerate
thf(fact_6874_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_6875_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_6876_map__add__upt,axiom,
    ! [N2: nat,M: nat] :
      ( ( map @ nat @ nat
        @ ^ [I2: nat] : ( plus_plus @ nat @ I2 @ N2 )
        @ ( upt @ ( zero_zero @ nat ) @ M ) )
      = ( upt @ N2 @ ( plus_plus @ nat @ M @ N2 ) ) ) ).

% map_add_upt
thf(fact_6877_map__add__upt_H,axiom,
    ! [Ofs: nat,A2: nat,B2: nat] :
      ( ( map @ nat @ nat
        @ ^ [I2: nat] : ( plus_plus @ nat @ I2 @ Ofs )
        @ ( upt @ A2 @ B2 ) )
      = ( upt @ ( plus_plus @ nat @ A2 @ Ofs ) @ ( plus_plus @ nat @ B2 @ Ofs ) ) ) ).

% map_add_upt'
thf(fact_6878_map__Suc__upt,axiom,
    ! [M: nat,N2: nat] :
      ( ( map @ nat @ nat @ suc @ ( upt @ M @ N2 ) )
      = ( upt @ ( suc @ M ) @ ( suc @ N2 ) ) ) ).

% map_Suc_upt
thf(fact_6879_upt__conv__Cons__Cons,axiom,
    ! [M: nat,N2: nat,Ns: list @ nat,Q2: nat] :
      ( ( ( cons @ nat @ M @ ( cons @ nat @ N2 @ Ns ) )
        = ( upt @ M @ Q2 ) )
      = ( ( cons @ nat @ N2 @ Ns )
        = ( upt @ ( suc @ M ) @ Q2 ) ) ) ).

% upt_conv_Cons_Cons
thf(fact_6880_upt__conv__Cons,axiom,
    ! [I: nat,J: nat] :
      ( ( ord_less @ nat @ I @ J )
     => ( ( upt @ I @ J )
        = ( cons @ nat @ I @ ( upt @ ( suc @ I ) @ J ) ) ) ) ).

% upt_conv_Cons
thf(fact_6881_upt__add__eq__append,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( ord_less_eq @ nat @ I @ J )
     => ( ( upt @ I @ ( plus_plus @ nat @ J @ K ) )
        = ( append @ nat @ ( upt @ I @ J ) @ ( upt @ J @ ( plus_plus @ nat @ J @ K ) ) ) ) ) ).

% upt_add_eq_append
thf(fact_6882_upt__eq__append__conv,axiom,
    ! [I: nat,J: nat,Xs2: list @ nat,Ys: list @ nat] :
      ( ( ord_less_eq @ nat @ I @ J )
     => ( ( ( upt @ I @ J )
          = ( append @ nat @ Xs2 @ Ys ) )
        = ( ? [K4: nat] :
              ( ( ord_less_eq @ nat @ I @ K4 )
              & ( ord_less_eq @ nat @ K4 @ J )
              & ( ( upt @ I @ K4 )
                = Xs2 )
              & ( ( upt @ K4 @ J )
                = Ys ) ) ) ) ) ).

% upt_eq_append_conv
thf(fact_6883_atLeastLessThan__upt,axiom,
    ( ( set_or7035219750837199246ssThan @ nat )
    = ( ^ [I2: nat,J3: nat] : ( set2 @ nat @ ( upt @ I2 @ J3 ) ) ) ) ).

% atLeastLessThan_upt
thf(fact_6884_upt__append,axiom,
    ! [I: nat,J: nat] :
      ( ( ord_less @ nat @ I @ J )
     => ( ( append @ nat @ ( upt @ ( zero_zero @ nat ) @ I ) @ ( upt @ I @ J ) )
        = ( upt @ ( zero_zero @ nat ) @ J ) ) ) ).

% upt_append
thf(fact_6885_upt__Suc,axiom,
    ! [I: nat,J: nat] :
      ( ( ( ord_less_eq @ nat @ I @ J )
       => ( ( upt @ I @ ( suc @ J ) )
          = ( append @ nat @ ( upt @ I @ J ) @ ( cons @ nat @ J @ ( nil @ nat ) ) ) ) )
      & ( ~ ( ord_less_eq @ nat @ I @ J )
       => ( ( upt @ I @ ( suc @ J ) )
          = ( nil @ nat ) ) ) ) ).

% upt_Suc
thf(fact_6886_upt__Suc__append,axiom,
    ! [I: nat,J: nat] :
      ( ( ord_less_eq @ nat @ I @ J )
     => ( ( upt @ I @ ( suc @ J ) )
        = ( append @ nat @ ( upt @ I @ J ) @ ( cons @ nat @ J @ ( nil @ nat ) ) ) ) ) ).

% upt_Suc_append
thf(fact_6887_upt__eq__Cons__conv,axiom,
    ! [I: nat,J: nat,X2: nat,Xs2: list @ nat] :
      ( ( ( upt @ I @ J )
        = ( cons @ nat @ X2 @ Xs2 ) )
      = ( ( ord_less @ nat @ I @ J )
        & ( I = X2 )
        & ( ( upt @ ( plus_plus @ nat @ I @ ( one_one @ nat ) ) @ J )
          = Xs2 ) ) ) ).

% upt_eq_Cons_conv
thf(fact_6888_map__decr__upt,axiom,
    ! [M: nat,N2: nat] :
      ( ( map @ nat @ nat
        @ ^ [N: nat] : ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) )
        @ ( upt @ ( suc @ M ) @ ( suc @ N2 ) ) )
      = ( upt @ M @ N2 ) ) ).

% map_decr_upt
thf(fact_6889_atLeastAtMost__upt,axiom,
    ( ( set_or1337092689740270186AtMost @ nat )
    = ( ^ [N: nat,M7: nat] : ( set2 @ nat @ ( upt @ N @ ( suc @ M7 ) ) ) ) ) ).

% atLeastAtMost_upt
thf(fact_6890_atLeast__upt,axiom,
    ( ( set_ord_lessThan @ nat )
    = ( ^ [N: nat] : ( set2 @ nat @ ( upt @ ( zero_zero @ nat ) @ N ) ) ) ) ).

% atLeast_upt
thf(fact_6891_map__replicate__trivial,axiom,
    ! [A: $tType,X2: A,I: nat] :
      ( ( map @ nat @ A
        @ ^ [I2: nat] : X2
        @ ( upt @ ( zero_zero @ nat ) @ I ) )
      = ( replicate @ A @ I @ X2 ) ) ).

% map_replicate_trivial
thf(fact_6892_enumerate__map__upt,axiom,
    ! [A: $tType,N2: nat,F: nat > A,M: nat] :
      ( ( enumerate @ A @ N2 @ ( map @ nat @ A @ F @ ( upt @ N2 @ M ) ) )
      = ( map @ nat @ ( product_prod @ nat @ A )
        @ ^ [K4: nat] : ( product_Pair @ nat @ A @ K4 @ ( F @ K4 ) )
        @ ( upt @ N2 @ M ) ) ) ).

% enumerate_map_upt
thf(fact_6893_sgn__sdiv__eq__sgn__mult,axiom,
    ! [A2: int,B2: int] :
      ( ( ( signed7115095781618012415divide @ int @ A2 @ B2 )
       != ( zero_zero @ int ) )
     => ( ( sgn_sgn @ int @ ( signed7115095781618012415divide @ int @ A2 @ B2 ) )
        = ( sgn_sgn @ int @ ( times_times @ int @ A2 @ B2 ) ) ) ) ).

% sgn_sdiv_eq_sgn_mult
thf(fact_6894_atMost__upto,axiom,
    ( ( set_ord_atMost @ nat )
    = ( ^ [N: nat] : ( set2 @ nat @ ( upt @ ( zero_zero @ nat ) @ ( suc @ N ) ) ) ) ) ).

% atMost_upto
thf(fact_6895_map__upt__Suc,axiom,
    ! [A: $tType,F: nat > A,N2: nat] :
      ( ( map @ nat @ A @ F @ ( upt @ ( zero_zero @ nat ) @ ( suc @ N2 ) ) )
      = ( cons @ A @ ( F @ ( zero_zero @ nat ) )
        @ ( map @ nat @ A
          @ ^ [I2: nat] : ( F @ ( suc @ I2 ) )
          @ ( upt @ ( zero_zero @ nat ) @ N2 ) ) ) ) ).

% map_upt_Suc
thf(fact_6896_map__nth,axiom,
    ! [A: $tType,Xs2: list @ A] :
      ( ( map @ nat @ A @ ( nth @ A @ Xs2 ) @ ( upt @ ( zero_zero @ nat ) @ ( size_size @ ( list @ A ) @ Xs2 ) ) )
      = Xs2 ) ).

% map_nth
thf(fact_6897_enumerate__replicate__eq,axiom,
    ! [A: $tType,N2: nat,M: nat,A2: A] :
      ( ( enumerate @ A @ N2 @ ( replicate @ A @ M @ A2 ) )
      = ( map @ nat @ ( product_prod @ nat @ A )
        @ ^ [Q5: nat] : ( product_Pair @ nat @ A @ Q5 @ A2 )
        @ ( upt @ N2 @ ( plus_plus @ nat @ N2 @ M ) ) ) ) ).

% enumerate_replicate_eq
thf(fact_6898_nth__map__upt,axiom,
    ! [A: $tType,I: nat,N2: nat,M: nat,F: nat > A] :
      ( ( ord_less @ nat @ I @ ( minus_minus @ nat @ N2 @ M ) )
     => ( ( nth @ A @ ( map @ nat @ A @ F @ ( upt @ M @ N2 ) ) @ I )
        = ( F @ ( plus_plus @ nat @ M @ I ) ) ) ) ).

% nth_map_upt
thf(fact_6899_signed__divide__int__def,axiom,
    ( ( signed7115095781618012415divide @ int )
    = ( ^ [K4: int,L2: int] : ( times_times @ int @ ( times_times @ int @ ( sgn_sgn @ int @ K4 ) @ ( sgn_sgn @ int @ L2 ) ) @ ( divide_divide @ int @ ( abs_abs @ int @ K4 ) @ ( abs_abs @ int @ L2 ) ) ) ) ) ).

% signed_divide_int_def
thf(fact_6900_map__upt__eqI,axiom,
    ! [A: $tType,Xs2: list @ A,N2: nat,M: nat,F: nat > A] :
      ( ( ( size_size @ ( list @ A ) @ Xs2 )
        = ( minus_minus @ nat @ N2 @ M ) )
     => ( ! [I3: nat] :
            ( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ Xs2 ) )
           => ( ( nth @ A @ Xs2 @ I3 )
              = ( F @ ( plus_plus @ nat @ M @ I3 ) ) ) )
       => ( ( map @ nat @ A @ F @ ( upt @ M @ N2 ) )
          = Xs2 ) ) ) ).

% map_upt_eqI
thf(fact_6901_sdiv__word__max,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [A2: word @ A,B2: word @ A] : ( ord_less_eq @ int @ ( signed7115095781618012415divide @ int @ ( ring_1_signed @ A @ int @ A2 ) @ ( ring_1_signed @ A @ int @ B2 ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ ( size_size @ ( word @ A ) @ A2 ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ).

% sdiv_word_max
thf(fact_6902_make__rule,axiom,
    ! [A: $tType] :
      ( ( heap @ A )
     => ! [N2: nat,F: nat > A] :
          ( hoare_hoare_triple @ ( array @ A ) @ ( one_one @ assn ) @ ( array_make @ A @ N2 @ F )
          @ ^ [R: array @ A] : ( snga_assn @ A @ R @ ( map @ nat @ A @ F @ ( upt @ ( zero_zero @ nat ) @ N2 ) ) ) ) ) ).

% make_rule
thf(fact_6903_Cauchy__iff2,axiom,
    ( ( topolo3814608138187158403Cauchy @ real )
    = ( ^ [X5: nat > real] :
        ! [J3: nat] :
        ? [M10: nat] :
        ! [M7: nat] :
          ( ( ord_less_eq @ nat @ M10 @ M7 )
         => ! [N: nat] :
              ( ( ord_less_eq @ nat @ M10 @ N )
             => ( ord_less @ real @ ( abs_abs @ real @ ( minus_minus @ real @ ( X5 @ M7 ) @ ( X5 @ N ) ) ) @ ( inverse_inverse @ real @ ( semiring_1_of_nat @ real @ ( suc @ J3 ) ) ) ) ) ) ) ) ).

% Cauchy_iff2
thf(fact_6904_array__make,axiom,
    ! [A: $tType] :
      ( ( heap @ A )
     => ( ( array_new @ A )
        = ( ^ [N: nat,X: A] :
              ( array_make @ A @ N
              @ ^ [Uu3: nat] : X ) ) ) ) ).

% array_make
thf(fact_6905_time__array__make,axiom,
    ! [A: $tType] :
      ( ( heap @ A )
     => ! [N2: nat,F: nat > A,H2: heap_ext @ product_unit] :
          ( ( time_time @ ( array @ A ) @ ( array_make @ A @ N2 @ F ) @ H2 )
          = ( plus_plus @ nat @ N2 @ ( one_one @ nat ) ) ) ) ).

% time_array_make
thf(fact_6906_TBOUND__make,axiom,
    ! [A: $tType] :
      ( ( heap @ A )
     => ! [N2: nat,F: nat > A] : ( time_TBOUND @ ( array @ A ) @ ( array_make @ A @ N2 @ F ) @ ( plus_plus @ nat @ N2 @ ( one_one @ nat ) ) ) ) ).

% TBOUND_make
thf(fact_6907_array__of__list__make,axiom,
    ! [A: $tType] :
      ( ( heap @ A )
     => ( ( array_of_list @ A )
        = ( ^ [Xs: list @ A] : ( array_make @ A @ ( size_size @ ( list @ A ) @ Xs ) @ ( nth @ A @ Xs ) ) ) ) ) ).

% array_of_list_make
thf(fact_6908_Cauchy__iff,axiom,
    ! [A: $tType] :
      ( ( real_V822414075346904944vector @ A )
     => ( ( topolo3814608138187158403Cauchy @ A )
        = ( ^ [X5: nat > A] :
            ! [E4: real] :
              ( ( ord_less @ real @ ( zero_zero @ real ) @ E4 )
             => ? [M10: nat] :
                ! [M7: nat] :
                  ( ( ord_less_eq @ nat @ M10 @ M7 )
                 => ! [N: nat] :
                      ( ( ord_less_eq @ nat @ M10 @ N )
                     => ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ ( X5 @ M7 ) @ ( X5 @ N ) ) ) @ E4 ) ) ) ) ) ) ) ).

% Cauchy_iff
thf(fact_6909_CauchyI,axiom,
    ! [A: $tType] :
      ( ( real_V822414075346904944vector @ A )
     => ! [X8: nat > A] :
          ( ! [E2: real] :
              ( ( ord_less @ real @ ( zero_zero @ real ) @ E2 )
             => ? [M14: nat] :
                ! [M2: nat] :
                  ( ( ord_less_eq @ nat @ M14 @ M2 )
                 => ! [N4: nat] :
                      ( ( ord_less_eq @ nat @ M14 @ N4 )
                     => ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ ( X8 @ M2 ) @ ( X8 @ N4 ) ) ) @ E2 ) ) ) )
         => ( topolo3814608138187158403Cauchy @ A @ X8 ) ) ) ).

% CauchyI
thf(fact_6910_CauchyD,axiom,
    ! [A: $tType] :
      ( ( real_V822414075346904944vector @ A )
     => ! [X8: nat > A,E3: real] :
          ( ( topolo3814608138187158403Cauchy @ A @ X8 )
         => ( ( ord_less @ real @ ( zero_zero @ real ) @ E3 )
           => ? [M11: nat] :
              ! [M3: nat] :
                ( ( ord_less_eq @ nat @ M11 @ M3 )
               => ! [N9: nat] :
                    ( ( ord_less_eq @ nat @ M11 @ N9 )
                   => ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ ( X8 @ M3 ) @ ( X8 @ N9 ) ) ) @ E3 ) ) ) ) ) ) ).

% CauchyD
thf(fact_6911_Array__Time_Omake__def,axiom,
    ! [A: $tType] :
      ( ( heap @ A )
     => ( ( array_make @ A )
        = ( ^ [N: nat,F6: nat > A] :
              ( heap_Time_heap @ ( array @ A )
              @ ^ [H: heap_ext @ product_unit] :
                  ( product_case_prod @ ( array @ A ) @ ( heap_ext @ product_unit ) @ ( product_prod @ ( array @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )
                  @ ^ [R: array @ A,H9: heap_ext @ product_unit] : ( product_Pair @ ( array @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H9 @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) ) )
                  @ ( array_alloc @ A @ ( map @ nat @ A @ F6 @ ( upt @ ( zero_zero @ nat ) @ N ) ) @ H ) ) ) ) ) ) ).

% Array_Time.make_def
thf(fact_6912_sorted__list__of__set__atMost__Suc,axiom,
    ! [K: nat] :
      ( ( linord4507533701916653071of_set @ nat @ ( set_ord_atMost @ nat @ ( suc @ K ) ) )
      = ( append @ nat @ ( linord4507533701916653071of_set @ nat @ ( set_ord_atMost @ nat @ K ) ) @ ( cons @ nat @ ( suc @ K ) @ ( nil @ nat ) ) ) ) ).

% sorted_list_of_set_atMost_Suc
thf(fact_6913_sorted__list__of__set_Oset__sorted__key__list__of__set,axiom,
    ! [A: $tType] :
      ( ( linorder @ A )
     => ! [A3: set @ A] :
          ( ( finite_finite2 @ A @ A3 )
         => ( ( set2 @ A @ ( linord4507533701916653071of_set @ A @ A3 ) )
            = A3 ) ) ) ).

% sorted_list_of_set.set_sorted_key_list_of_set
thf(fact_6914_sorted__list__of__set__lessThan__Suc,axiom,
    ! [K: nat] :
      ( ( linord4507533701916653071of_set @ nat @ ( set_ord_lessThan @ nat @ ( suc @ K ) ) )
      = ( append @ nat @ ( linord4507533701916653071of_set @ nat @ ( set_ord_lessThan @ nat @ K ) ) @ ( cons @ nat @ K @ ( nil @ nat ) ) ) ) ).

% sorted_list_of_set_lessThan_Suc
thf(fact_6915_effect__of__listI,axiom,
    ! [A: $tType] :
      ( ( heap @ A )
     => ! [A2: array @ A,H4: heap_ext @ product_unit,Xs2: list @ A,H2: heap_ext @ product_unit] :
          ( ( ( product_Pair @ ( array @ A ) @ ( heap_ext @ product_unit ) @ A2 @ H4 )
            = ( array_alloc @ A @ Xs2 @ H2 ) )
         => ( heap_Time_effect @ ( array @ A ) @ ( array_of_list @ A @ Xs2 ) @ H2 @ H4 @ A2 @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( size_size @ ( list @ A ) @ Xs2 ) ) ) ) ) ).

% effect_of_listI
thf(fact_6916_effect__newI,axiom,
    ! [A: $tType] :
      ( ( heap @ A )
     => ! [A2: array @ A,H4: heap_ext @ product_unit,N2: nat,X2: A,H2: heap_ext @ product_unit] :
          ( ( ( product_Pair @ ( array @ A ) @ ( heap_ext @ product_unit ) @ A2 @ H4 )
            = ( array_alloc @ A @ ( replicate @ A @ N2 @ X2 ) @ H2 ) )
         => ( heap_Time_effect @ ( array @ A ) @ ( array_new @ A @ N2 @ X2 ) @ H2 @ H4 @ A2 @ ( plus_plus @ nat @ N2 @ ( one_one @ nat ) ) ) ) ) ).

% effect_newI
thf(fact_6917_of__list__def,axiom,
    ! [A: $tType] :
      ( ( heap @ A )
     => ( ( array_of_list @ A )
        = ( ^ [Xs: list @ A] :
              ( heap_Time_heap @ ( array @ A )
              @ ^ [H: heap_ext @ product_unit] :
                  ( product_case_prod @ ( array @ A ) @ ( heap_ext @ product_unit ) @ ( product_prod @ ( array @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )
                  @ ^ [R: array @ A,H9: heap_ext @ product_unit] : ( product_Pair @ ( array @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H9 @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( size_size @ ( list @ A ) @ Xs ) ) ) )
                  @ ( array_alloc @ A @ Xs @ H ) ) ) ) ) ) ).

% of_list_def
thf(fact_6918_new__def,axiom,
    ! [A: $tType] :
      ( ( heap @ A )
     => ( ( array_new @ A )
        = ( ^ [N: nat,X: A] :
              ( heap_Time_heap @ ( array @ A )
              @ ^ [H: heap_ext @ product_unit] :
                  ( product_case_prod @ ( array @ A ) @ ( heap_ext @ product_unit ) @ ( product_prod @ ( array @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )
                  @ ^ [R: array @ A,H9: heap_ext @ product_unit] : ( product_Pair @ ( array @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H9 @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) ) )
                  @ ( array_alloc @ A @ ( replicate @ A @ N @ X ) @ H ) ) ) ) ) ) ).

% new_def
thf(fact_6919_effect__makeI,axiom,
    ! [A: $tType] :
      ( ( heap @ A )
     => ! [A2: array @ A,H4: heap_ext @ product_unit,F: nat > A,N2: nat,H2: heap_ext @ product_unit] :
          ( ( ( product_Pair @ ( array @ A ) @ ( heap_ext @ product_unit ) @ A2 @ H4 )
            = ( array_alloc @ A @ ( map @ nat @ A @ F @ ( upt @ ( zero_zero @ nat ) @ N2 ) ) @ H2 ) )
         => ( heap_Time_effect @ ( array @ A ) @ ( array_make @ A @ N2 @ F ) @ H2 @ H4 @ A2 @ ( plus_plus @ nat @ N2 @ ( one_one @ nat ) ) ) ) ) ).

% effect_makeI
thf(fact_6920_execute__make,axiom,
    ! [A: $tType] :
      ( ( heap @ A )
     => ! [N2: nat,F: nat > A,H2: heap_ext @ product_unit] :
          ( ( heap_Time_execute @ ( array @ A ) @ ( array_make @ A @ N2 @ F ) @ H2 )
          = ( some @ ( product_prod @ ( array @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )
            @ ( product_case_prod @ ( array @ A ) @ ( heap_ext @ product_unit ) @ ( product_prod @ ( array @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )
              @ ^ [R: array @ A,H9: heap_ext @ product_unit] : ( product_Pair @ ( array @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H9 @ ( plus_plus @ nat @ N2 @ ( one_one @ nat ) ) ) )
              @ ( array_alloc @ A @ ( map @ nat @ A @ F @ ( upt @ ( zero_zero @ nat ) @ N2 ) ) @ H2 ) ) ) ) ) ).

% execute_make
thf(fact_6921_execute__new,axiom,
    ! [A: $tType] :
      ( ( heap @ A )
     => ! [N2: nat,X2: A,H2: heap_ext @ product_unit] :
          ( ( heap_Time_execute @ ( array @ A ) @ ( array_new @ A @ N2 @ X2 ) @ H2 )
          = ( some @ ( product_prod @ ( array @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )
            @ ( product_case_prod @ ( array @ A ) @ ( heap_ext @ product_unit ) @ ( product_prod @ ( array @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )
              @ ^ [R: array @ A,H9: heap_ext @ product_unit] : ( product_Pair @ ( array @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H9 @ ( plus_plus @ nat @ N2 @ ( one_one @ nat ) ) ) )
              @ ( array_alloc @ A @ ( replicate @ A @ N2 @ X2 ) @ H2 ) ) ) ) ) ).

% execute_new
thf(fact_6922_time__def,axiom,
    ! [A: $tType] :
      ( ( time_time @ A )
      = ( ^ [C4: heap_Time_Heap @ A,H: heap_ext @ product_unit] :
            ( case_option @ nat @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( zero_zero @ nat )
            @ ( product_case_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ nat
              @ ^ [X: A] :
                  ( product_case_prod @ ( heap_ext @ product_unit ) @ nat @ nat
                  @ ^ [Xa4: heap_ext @ product_unit,T3: nat] : T3 ) )
            @ ( heap_Time_execute @ A @ C4 @ H ) ) ) ) ).

% time_def
thf(fact_6923_execute__bind_I2_J,axiom,
    ! [A: $tType,B: $tType,F: heap_Time_Heap @ A,H2: heap_ext @ product_unit,G: A > ( heap_Time_Heap @ B )] :
      ( ( ( heap_Time_execute @ A @ F @ H2 )
        = ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) )
     => ( ( heap_Time_execute @ B @ ( heap_Time_bind @ A @ B @ F @ G ) @ H2 )
        = ( none @ ( product_prod @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) ) ).

% execute_bind(2)
thf(fact_6924_Heap__ordE,axiom,
    ! [A: $tType,X2: heap_Time_Heap @ A,Y2: heap_Time_Heap @ A,H2: heap_ext @ product_unit] :
      ( ( heap_Time_Heap_ord @ A @ X2 @ Y2 )
     => ( ( ( heap_Time_execute @ A @ X2 @ H2 )
         != ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) )
       => ( ( heap_Time_execute @ A @ X2 @ H2 )
          = ( heap_Time_execute @ A @ Y2 @ H2 ) ) ) ) ).

% Heap_ordE
thf(fact_6925_Heap__ordI,axiom,
    ! [A: $tType,X2: heap_Time_Heap @ A,Y2: heap_Time_Heap @ A] :
      ( ! [H3: heap_ext @ product_unit] :
          ( ( ( heap_Time_execute @ A @ X2 @ H3 )
            = ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) )
          | ( ( heap_Time_execute @ A @ X2 @ H3 )
            = ( heap_Time_execute @ A @ Y2 @ H3 ) ) )
     => ( heap_Time_Heap_ord @ A @ X2 @ Y2 ) ) ).

% Heap_ordI
thf(fact_6926_execute__guard_I2_J,axiom,
    ! [A: $tType,P: ( heap_ext @ product_unit ) > $o,H2: heap_ext @ product_unit,F: ( heap_ext @ product_unit ) > ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )] :
      ( ( P @ H2 )
     => ( ( heap_Time_execute @ A @ ( heap_Time_guard @ A @ P @ F ) @ H2 )
        = ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( F @ H2 ) ) ) ) ).

% execute_guard(2)
thf(fact_6927_execute__guard_I1_J,axiom,
    ! [A: $tType,P: ( heap_ext @ product_unit ) > $o,H2: heap_ext @ product_unit,F: ( heap_ext @ product_unit ) > ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )] :
      ( ~ ( P @ H2 )
     => ( ( heap_Time_execute @ A @ ( heap_Time_guard @ A @ P @ F ) @ H2 )
        = ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) ) ).

% execute_guard(1)
thf(fact_6928_Heap__cases,axiom,
    ! [A: $tType,F: heap_Time_Heap @ A,H2: heap_ext @ product_unit] :
      ( ! [X3: A,H5: product_prod @ ( heap_ext @ product_unit ) @ nat] :
          ( ( heap_Time_execute @ A @ F @ H2 )
         != ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ X3 @ H5 ) ) )
     => ( ( heap_Time_execute @ A @ F @ H2 )
        = ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) ) ).

% Heap_cases
thf(fact_6929_execute__bind__case,axiom,
    ! [A: $tType,B: $tType,F: heap_Time_Heap @ B,G: B > ( heap_Time_Heap @ A ),H2: heap_ext @ product_unit] :
      ( ( heap_Time_execute @ A @ ( heap_Time_bind @ B @ A @ F @ G ) @ H2 )
      = ( case_option @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) @ ( product_prod @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) )
        @ ( product_case_prod @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) )
          @ ^ [X: B] :
              ( product_case_prod @ ( heap_ext @ product_unit ) @ nat @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) )
              @ ^ [H9: heap_ext @ product_unit,N: nat] : ( heap_Time_timeFrame @ A @ N @ ( heap_Time_execute @ A @ ( G @ X ) @ H9 ) ) ) )
        @ ( heap_Time_execute @ B @ F @ H2 ) ) ) ).

% execute_bind_case
thf(fact_6930_effectI,axiom,
    ! [A: $tType,C2: heap_Time_Heap @ A,H2: heap_ext @ product_unit,R4: A,H4: heap_ext @ product_unit,N2: nat] :
      ( ( ( heap_Time_execute @ A @ C2 @ H2 )
        = ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R4 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H4 @ N2 ) ) ) )
     => ( heap_Time_effect @ A @ C2 @ H2 @ H4 @ R4 @ N2 ) ) ).

% effectI
thf(fact_6931_effect__def,axiom,
    ! [A: $tType] :
      ( ( heap_Time_effect @ A )
      = ( ^ [C4: heap_Time_Heap @ A,H: heap_ext @ product_unit,H9: heap_ext @ product_unit,R: A,N: nat] :
            ( ( heap_Time_execute @ A @ C4 @ H )
            = ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H9 @ N ) ) ) ) ) ) ).

% effect_def
thf(fact_6932_refines__def,axiom,
    ! [A: $tType] :
      ( ( refine_Imp_refines @ A )
      = ( ^ [P5: heap_Time_Heap @ A,Q5: heap_Time_Heap @ A] :
          ! [H: heap_ext @ product_unit] :
            ( case_option @ $o @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ $true
            @ ( product_case_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ $o
              @ ^ [R: A] :
                  ( product_case_prod @ ( heap_ext @ product_unit ) @ nat @ $o
                  @ ^ [H9: heap_ext @ product_unit,T3: nat] :
                      ( ( heap_Time_execute @ A @ P5 @ H )
                      = ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H9 @ T3 ) ) ) ) ) )
            @ ( heap_Time_execute @ A @ Q5 @ H ) ) ) ) ).

% refines_def
thf(fact_6933_Heap__Time__Monad_Obind__def,axiom,
    ! [B: $tType,A: $tType] :
      ( ( heap_Time_bind @ A @ B )
      = ( ^ [F6: heap_Time_Heap @ A,G4: A > ( heap_Time_Heap @ B )] :
            ( heap_Time_Heap2 @ B
            @ ^ [H: heap_ext @ product_unit] :
                ( case_option @ ( option @ ( product_prod @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( none @ ( product_prod @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) )
                @ ( product_case_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( option @ ( product_prod @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) )
                  @ ^ [R: A] :
                      ( product_case_prod @ ( heap_ext @ product_unit ) @ nat @ ( option @ ( product_prod @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) )
                      @ ^ [H9: heap_ext @ product_unit,N: nat] : ( heap_Time_timeFrame @ B @ N @ ( heap_Time_execute @ B @ ( G4 @ R ) @ H9 ) ) ) )
                @ ( heap_Time_execute @ A @ F6 @ H ) ) ) ) ) ).

% Heap_Time_Monad.bind_def
thf(fact_6934_execute__bind__eq__SomeI,axiom,
    ! [A: $tType,B: $tType,F: heap_Time_Heap @ A,H2: heap_ext @ product_unit,X2: A,H4: heap_ext @ product_unit,N2: nat,G: A > ( heap_Time_Heap @ B ),Y2: B,H8: heap_ext @ product_unit,N5: nat] :
      ( ( ( heap_Time_execute @ A @ F @ H2 )
        = ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ X2 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H4 @ N2 ) ) ) )
     => ( ( ( heap_Time_execute @ B @ ( G @ X2 ) @ H4 )
          = ( some @ ( product_prod @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ Y2 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H8 @ N5 ) ) ) )
       => ( ( heap_Time_execute @ B @ ( heap_Time_bind @ A @ B @ F @ G ) @ H2 )
          = ( some @ ( product_prod @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ Y2 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H8 @ ( plus_plus @ nat @ N2 @ N5 ) ) ) ) ) ) ) ).

% execute_bind_eq_SomeI
thf(fact_6935_execute__bind_I1_J,axiom,
    ! [A: $tType,B: $tType,F: heap_Time_Heap @ A,H2: heap_ext @ product_unit,X2: A,H4: heap_ext @ product_unit,N2: nat,G: A > ( heap_Time_Heap @ B )] :
      ( ( ( heap_Time_execute @ A @ F @ H2 )
        = ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ X2 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H4 @ N2 ) ) ) )
     => ( ( heap_Time_execute @ B @ ( heap_Time_bind @ A @ B @ F @ G ) @ H2 )
        = ( heap_Time_timeFrame @ B @ N2 @ ( heap_Time_execute @ B @ ( G @ X2 ) @ H4 ) ) ) ) ).

% execute_bind(1)
thf(fact_6936_execute__tap,axiom,
    ! [A: $tType,F: ( heap_ext @ product_unit ) > A,H2: heap_ext @ product_unit] :
      ( ( heap_Time_execute @ A @ ( heap_Time_tap @ A @ F ) @ H2 )
      = ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( F @ H2 ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H2 @ ( one_one @ nat ) ) ) ) ) ).

% execute_tap
thf(fact_6937_execute__of__list,axiom,
    ! [A: $tType] :
      ( ( heap @ A )
     => ! [Xs2: list @ A,H2: heap_ext @ product_unit] :
          ( ( heap_Time_execute @ ( array @ A ) @ ( array_of_list @ A @ Xs2 ) @ H2 )
          = ( some @ ( product_prod @ ( array @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )
            @ ( product_case_prod @ ( array @ A ) @ ( heap_ext @ product_unit ) @ ( product_prod @ ( array @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )
              @ ^ [R: array @ A,H9: heap_ext @ product_unit] : ( product_Pair @ ( array @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H9 @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( size_size @ ( list @ A ) @ Xs2 ) ) ) )
              @ ( array_alloc @ A @ Xs2 @ H2 ) ) ) ) ) ).

% execute_of_list
thf(fact_6938_execute__bind__success,axiom,
    ! [B: $tType,A: $tType,F: heap_Time_Heap @ A,H2: heap_ext @ product_unit,G: A > ( heap_Time_Heap @ B )] :
      ( ( heap_Time_success @ A @ F @ H2 )
     => ( ( heap_Time_execute @ B @ ( heap_Time_bind @ A @ B @ F @ G ) @ H2 )
        = ( heap_Time_timeFrame @ B @ ( product_snd @ ( heap_ext @ product_unit ) @ nat @ ( product_snd @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( the2 @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( heap_Time_execute @ A @ F @ H2 ) ) ) ) @ ( heap_Time_execute @ B @ ( G @ ( product_fst @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( the2 @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( heap_Time_execute @ A @ F @ H2 ) ) ) ) @ ( product_fst @ ( heap_ext @ product_unit ) @ nat @ ( product_snd @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( the2 @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( heap_Time_execute @ A @ F @ H2 ) ) ) ) ) ) ) ) ).

% execute_bind_success
thf(fact_6939_Heap__lub__def,axiom,
    ! [A: $tType] :
      ( ( heap_Time_Heap_lub @ A )
      = ( partial_img_lub @ ( heap_Time_Heap @ A ) @ ( ( heap_ext @ product_unit ) > ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) @ ( ( heap_ext @ product_unit ) > ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) @ ( heap_Time_Heap @ A ) @ ( heap_Time_execute @ A ) @ ( heap_Time_Heap2 @ A ) @ ( partial_fun_lub @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) @ ( heap_ext @ product_unit ) @ ( partial_flat_lub @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) @ ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) ) ) ) ).

% Heap_lub_def
thf(fact_6940_success__returnI,axiom,
    ! [A: $tType,X2: A,H2: heap_ext @ product_unit] : ( heap_Time_success @ A @ ( heap_Time_return @ A @ X2 ) @ H2 ) ).

% success_returnI
thf(fact_6941_success__LetI,axiom,
    ! [A: $tType,B: $tType,X2: A,T2: A,F: A > ( heap_Time_Heap @ B ),H2: heap_ext @ product_unit] :
      ( ( X2 = T2 )
     => ( ( heap_Time_success @ B @ ( F @ X2 ) @ H2 )
       => ( heap_Time_success @ B @ ( F @ T2 ) @ H2 ) ) ) ).

% success_LetI
thf(fact_6942_success__bind__effectI,axiom,
    ! [A: $tType,B: $tType,F: heap_Time_Heap @ A,H2: heap_ext @ product_unit,H4: heap_ext @ product_unit,X2: A,N2: nat,G: A > ( heap_Time_Heap @ B )] :
      ( ( heap_Time_effect @ A @ F @ H2 @ H4 @ X2 @ N2 )
     => ( ( heap_Time_success @ B @ ( G @ X2 ) @ H4 )
       => ( heap_Time_success @ B @ ( heap_Time_bind @ A @ B @ F @ G ) @ H2 ) ) ) ).

% success_bind_effectI
thf(fact_6943_success__def,axiom,
    ! [A: $tType] :
      ( ( heap_Time_success @ A )
      = ( ^ [F6: heap_Time_Heap @ A,H: heap_ext @ product_unit] :
            ( ( heap_Time_execute @ A @ F6 @ H )
           != ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) ) ) ).

% success_def
thf(fact_6944_successI,axiom,
    ! [A: $tType,F: heap_Time_Heap @ A,H2: heap_ext @ product_unit] :
      ( ( ( heap_Time_execute @ A @ F @ H2 )
       != ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) )
     => ( heap_Time_success @ A @ F @ H2 ) ) ).

% successI
thf(fact_6945_hoare__triple__success,axiom,
    ! [A: $tType,P: assn,C2: heap_Time_Heap @ A,Q: A > assn,H2: heap_ext @ product_unit,As3: set @ nat] :
      ( ( hoare_hoare_triple @ A @ P @ C2 @ Q )
     => ( ( rep_assn @ P @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As3 ) )
       => ( heap_Time_success @ A @ C2 @ H2 ) ) ) ).

% hoare_triple_success
thf(fact_6946_successE,axiom,
    ! [A: $tType,F: heap_Time_Heap @ A,H2: heap_ext @ product_unit] :
      ( ( heap_Time_success @ A @ F @ H2 )
     => ~ ! [R3: A,H5: product_prod @ ( heap_ext @ product_unit ) @ nat] :
            ( ( heap_Time_execute @ A @ F @ H2 )
           != ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R3 @ H5 ) ) ) ) ).

% successE
thf(fact_6947_success__bind__executeI,axiom,
    ! [A: $tType,B: $tType,F: heap_Time_Heap @ A,H2: heap_ext @ product_unit,X2: A,H4: heap_ext @ product_unit,N2: nat,G: A > ( heap_Time_Heap @ B )] :
      ( ( ( heap_Time_execute @ A @ F @ H2 )
        = ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ X2 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H4 @ N2 ) ) ) )
     => ( ( heap_Time_success @ B @ ( G @ X2 ) @ H4 )
       => ( heap_Time_success @ B @ ( heap_Time_bind @ A @ B @ F @ G ) @ H2 ) ) ) ).

% success_bind_executeI
thf(fact_6948_effectE,axiom,
    ! [A: $tType,C2: heap_Time_Heap @ A,H2: heap_ext @ product_unit,H4: heap_ext @ product_unit,R4: A,N2: nat] :
      ( ( heap_Time_effect @ A @ C2 @ H2 @ H4 @ R4 @ N2 )
     => ~ ( ( R4
            = ( product_fst @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( the2 @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( heap_Time_execute @ A @ C2 @ H2 ) ) ) )
         => ( ( H4
              = ( product_fst @ ( heap_ext @ product_unit ) @ nat @ ( product_snd @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( the2 @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( heap_Time_execute @ A @ C2 @ H2 ) ) ) ) )
           => ( ( N2
                = ( product_snd @ ( heap_ext @ product_unit ) @ nat @ ( product_snd @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( the2 @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( heap_Time_execute @ A @ C2 @ H2 ) ) ) ) )
             => ~ ( heap_Time_success @ A @ C2 @ H2 ) ) ) ) ) ).

% effectE
thf(fact_6949_the__res__def,axiom,
    ! [A: $tType] :
      ( ( time_the_res @ A )
      = ( ^ [M7: heap_Time_Heap @ A,H: heap_ext @ product_unit] :
            ( case_option @ A @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( undefined @ A )
            @ ( product_case_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ A
              @ ^ [R: A] :
                  ( product_case_prod @ ( heap_ext @ product_unit ) @ nat @ A
                  @ ^ [X: heap_ext @ product_unit,Xa4: nat] : R ) )
            @ ( heap_Time_execute @ A @ M7 @ H ) ) ) ) ).

% the_res_def
thf(fact_6950_execute__assert_I1_J,axiom,
    ! [A: $tType,P: A > $o,X2: A,H2: heap_ext @ product_unit] :
      ( ( P @ X2 )
     => ( ( heap_Time_execute @ A @ ( heap_Time_assert @ A @ P @ X2 ) @ H2 )
        = ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ X2 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H2 @ ( one_one @ nat ) ) ) ) ) ) ).

% execute_assert(1)
thf(fact_6951_assert__cong,axiom,
    ! [B: $tType,A: $tType,P: A > $o,P6: A > $o,F: A > ( heap_Time_Heap @ B ),F4: A > ( heap_Time_Heap @ B ),X2: A] :
      ( ( P = P6 )
     => ( ! [X3: A] :
            ( ( P6 @ X3 )
           => ( ( F @ X3 )
              = ( F4 @ X3 ) ) )
       => ( ( heap_Time_bind @ A @ B @ ( heap_Time_assert @ A @ P @ X2 ) @ F )
          = ( heap_Time_bind @ A @ B @ ( heap_Time_assert @ A @ P6 @ X2 ) @ F4 ) ) ) ) ).

% assert_cong
thf(fact_6952_effect__assertI,axiom,
    ! [A: $tType,P: A > $o,X2: A,H4: heap_ext @ product_unit,H2: heap_ext @ product_unit,R4: A,N2: nat] :
      ( ( P @ X2 )
     => ( ( H4 = H2 )
       => ( ( R4 = X2 )
         => ( ( N2
              = ( one_one @ nat ) )
           => ( heap_Time_effect @ A @ ( heap_Time_assert @ A @ P @ X2 ) @ H2 @ H4 @ R4 @ N2 ) ) ) ) ) ).

% effect_assertI
thf(fact_6953_effect__assertE,axiom,
    ! [A: $tType,P: A > $o,X2: A,H2: heap_ext @ product_unit,H4: heap_ext @ product_unit,R4: A,N2: nat] :
      ( ( heap_Time_effect @ A @ ( heap_Time_assert @ A @ P @ X2 ) @ H2 @ H4 @ R4 @ N2 )
     => ~ ( ( P @ X2 )
         => ( ( R4 = X2 )
           => ( ( H4 = H2 )
             => ( N2
               != ( one_one @ nat ) ) ) ) ) ) ).

% effect_assertE
thf(fact_6954_execute__assert_I2_J,axiom,
    ! [A: $tType,P: A > $o,X2: A,H2: heap_ext @ product_unit] :
      ( ~ ( P @ X2 )
     => ( ( heap_Time_execute @ A @ ( heap_Time_assert @ A @ P @ X2 ) @ H2 )
        = ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) ) ).

% execute_assert(2)
thf(fact_6955_the__heap__def,axiom,
    ! [A: $tType] :
      ( ( time_the_heap @ A )
      = ( ^ [M7: heap_Time_Heap @ A,H: heap_ext @ product_unit] :
            ( case_option @ ( heap_ext @ product_unit ) @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( undefined @ ( heap_ext @ product_unit ) )
            @ ( product_case_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( heap_ext @ product_unit )
              @ ^ [X: A] :
                  ( product_case_prod @ ( heap_ext @ product_unit ) @ nat @ ( heap_ext @ product_unit )
                  @ ^ [H9: heap_ext @ product_unit,Xa4: nat] : H9 ) )
            @ ( heap_Time_execute @ A @ M7 @ H ) ) ) ) ).

% the_heap_def
thf(fact_6956_Heap__ord__def,axiom,
    ! [A: $tType] :
      ( ( heap_Time_Heap_ord @ A )
      = ( partial_img_ord @ ( heap_Time_Heap @ A ) @ ( ( heap_ext @ product_unit ) > ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) @ $o @ ( heap_Time_execute @ A ) @ ( partial_fun_ord @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) @ ( heap_ext @ product_unit ) @ ( partial_flat_ord @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) @ ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) ) ) ) ).

% Heap_ord_def
thf(fact_6957_img__ord__def,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( partial_img_ord @ A @ C @ B )
      = ( ^ [F6: A > C,Ord2: C > C > B,X: A,Y: A] : ( Ord2 @ ( F6 @ X ) @ ( F6 @ Y ) ) ) ) ).

% img_ord_def
thf(fact_6958_time__bind,axiom,
    ! [A: $tType,B: $tType,M: heap_Time_Heap @ B,H2: heap_ext @ product_unit,F: B > ( heap_Time_Heap @ A )] :
      ( ( ( ~ ( time_fails @ B @ M @ H2 )
          & ~ ( time_fails @ A @ ( F @ ( time_the_res @ B @ M @ H2 ) ) @ ( time_the_heap @ B @ M @ H2 ) ) )
       => ( ( time_time @ A @ ( heap_Time_bind @ B @ A @ M @ F ) @ H2 )
          = ( plus_plus @ nat @ ( time_time @ B @ M @ H2 ) @ ( time_time @ A @ ( F @ ( time_the_res @ B @ M @ H2 ) ) @ ( time_the_heap @ B @ M @ H2 ) ) ) ) )
      & ( ~ ( ~ ( time_fails @ B @ M @ H2 )
            & ~ ( time_fails @ A @ ( F @ ( time_the_res @ B @ M @ H2 ) ) @ ( time_the_heap @ B @ M @ H2 ) ) )
       => ( ( time_time @ A @ ( heap_Time_bind @ B @ A @ M @ F ) @ H2 )
          = ( zero_zero @ nat ) ) ) ) ).

% time_bind
thf(fact_6959_filter__upt__last,axiom,
    ! [A: $tType,P: A > $o,L: list @ A,Js2: list @ nat,J: nat,I: nat] :
      ( ( ( filter2 @ nat
          @ ^ [K4: nat] : ( P @ ( nth @ A @ L @ K4 ) )
          @ ( upt @ ( zero_zero @ nat ) @ ( size_size @ ( list @ A ) @ L ) ) )
        = ( append @ nat @ Js2 @ ( cons @ nat @ J @ ( nil @ nat ) ) ) )
     => ( ( ord_less @ nat @ J @ I )
       => ( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ L ) )
         => ~ ( P @ ( nth @ A @ L @ I ) ) ) ) ) ).

% filter_upt_last
thf(fact_6960_filter__filter,axiom,
    ! [A: $tType,P: A > $o,Q: A > $o,Xs2: list @ A] :
      ( ( filter2 @ A @ P @ ( filter2 @ A @ Q @ Xs2 ) )
      = ( filter2 @ A
        @ ^ [X: A] :
            ( ( Q @ X )
            & ( P @ X ) )
        @ Xs2 ) ) ).

% filter_filter
thf(fact_6961_filter__True,axiom,
    ! [A: $tType,Xs2: list @ A,P: A > $o] :
      ( ! [X3: A] :
          ( ( member @ A @ X3 @ ( set2 @ A @ Xs2 ) )
         => ( P @ X3 ) )
     => ( ( filter2 @ A @ P @ Xs2 )
        = Xs2 ) ) ).

% filter_True
thf(fact_6962_set__filter,axiom,
    ! [A: $tType,P: A > $o,Xs2: list @ A] :
      ( ( set2 @ A @ ( filter2 @ A @ P @ Xs2 ) )
      = ( collect @ A
        @ ^ [X: A] :
            ( ( member @ A @ X @ ( set2 @ A @ Xs2 ) )
            & ( P @ X ) ) ) ) ).

% set_filter
thf(fact_6963_filter__False,axiom,
    ! [A: $tType,Xs2: list @ A,P: A > $o] :
      ( ! [X3: A] :
          ( ( member @ A @ X3 @ ( set2 @ A @ Xs2 ) )
         => ~ ( P @ X3 ) )
     => ( ( filter2 @ A @ P @ Xs2 )
        = ( nil @ A ) ) ) ).

% filter_False
thf(fact_6964_length__filter__less,axiom,
    ! [A: $tType,X2: A,Xs2: list @ A,P: A > $o] :
      ( ( member @ A @ X2 @ ( set2 @ A @ Xs2 ) )
     => ( ~ ( P @ X2 )
       => ( ord_less @ nat @ ( size_size @ ( list @ A ) @ ( filter2 @ A @ P @ Xs2 ) ) @ ( size_size @ ( list @ A ) @ Xs2 ) ) ) ) ).

% length_filter_less
thf(fact_6965_length__filter__le,axiom,
    ! [A: $tType,P: A > $o,Xs2: list @ A] : ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ ( filter2 @ A @ P @ Xs2 ) ) @ ( size_size @ ( list @ A ) @ Xs2 ) ) ).

% length_filter_le
thf(fact_6966_replicate__length__filter,axiom,
    ! [A: $tType,X2: A,Xs2: list @ A] :
      ( ( replicate @ A
        @ ( size_size @ ( list @ A )
          @ ( filter2 @ A
            @ ( ^ [Y5: A,Z4: A] : Y5 = Z4
              @ X2 )
            @ Xs2 ) )
        @ X2 )
      = ( filter2 @ A
        @ ( ^ [Y5: A,Z4: A] : Y5 = Z4
          @ X2 )
        @ Xs2 ) ) ).

% replicate_length_filter
thf(fact_6967_sum__length__filter__compl,axiom,
    ! [A: $tType,P: A > $o,Xs2: list @ A] :
      ( ( plus_plus @ nat @ ( size_size @ ( list @ A ) @ ( filter2 @ A @ P @ Xs2 ) )
        @ ( size_size @ ( list @ A )
          @ ( filter2 @ A
            @ ^ [X: A] :
                ~ ( P @ X )
            @ Xs2 ) ) )
      = ( size_size @ ( list @ A ) @ Xs2 ) ) ).

% sum_length_filter_compl
thf(fact_6968_filter__cong,axiom,
    ! [A: $tType,Xs2: list @ A,Ys: list @ A,P: A > $o,Q: A > $o] :
      ( ( Xs2 = Ys )
     => ( ! [X3: A] :
            ( ( member @ A @ X3 @ ( set2 @ A @ Ys ) )
           => ( ( P @ X3 )
              = ( Q @ X3 ) ) )
       => ( ( filter2 @ A @ P @ Xs2 )
          = ( filter2 @ A @ Q @ Ys ) ) ) ) ).

% filter_cong
thf(fact_6969_filter__id__conv,axiom,
    ! [A: $tType,P: A > $o,Xs2: list @ A] :
      ( ( ( filter2 @ A @ P @ Xs2 )
        = Xs2 )
      = ( ! [X: A] :
            ( ( member @ A @ X @ ( set2 @ A @ Xs2 ) )
           => ( P @ X ) ) ) ) ).

% filter_id_conv
thf(fact_6970_filter__is__subset,axiom,
    ! [A: $tType,P: A > $o,Xs2: list @ A] : ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ ( filter2 @ A @ P @ Xs2 ) ) @ ( set2 @ A @ Xs2 ) ) ).

% filter_is_subset
thf(fact_6971_fails__assert_H,axiom,
    ! [P: $o,H2: heap_ext @ product_unit] :
      ( ( time_fails @ product_unit @ ( refine_Imp_assert @ P ) @ H2 )
      = ~ P ) ).

% fails_assert'
thf(fact_6972_fails__return,axiom,
    ! [A: $tType,X2: A,H2: heap_ext @ product_unit] :
      ~ ( time_fails @ A @ ( heap_Time_return @ A @ X2 ) @ H2 ) ).

% fails_return
thf(fact_6973_filter__eq__Cons__iff,axiom,
    ! [A: $tType,P: A > $o,Ys: list @ A,X2: A,Xs2: list @ A] :
      ( ( ( filter2 @ A @ P @ Ys )
        = ( cons @ A @ X2 @ Xs2 ) )
      = ( ? [Us2: list @ A,Vs2: list @ A] :
            ( ( Ys
              = ( append @ A @ Us2 @ ( cons @ A @ X2 @ Vs2 ) ) )
            & ! [X: A] :
                ( ( member @ A @ X @ ( set2 @ A @ Us2 ) )
               => ~ ( P @ X ) )
            & ( P @ X2 )
            & ( Xs2
              = ( filter2 @ A @ P @ Vs2 ) ) ) ) ) ).

% filter_eq_Cons_iff
thf(fact_6974_Cons__eq__filter__iff,axiom,
    ! [A: $tType,X2: A,Xs2: list @ A,P: A > $o,Ys: list @ A] :
      ( ( ( cons @ A @ X2 @ Xs2 )
        = ( filter2 @ A @ P @ Ys ) )
      = ( ? [Us2: list @ A,Vs2: list @ A] :
            ( ( Ys
              = ( append @ A @ Us2 @ ( cons @ A @ X2 @ Vs2 ) ) )
            & ! [X: A] :
                ( ( member @ A @ X @ ( set2 @ A @ Us2 ) )
               => ~ ( P @ X ) )
            & ( P @ X2 )
            & ( Xs2
              = ( filter2 @ A @ P @ Vs2 ) ) ) ) ) ).

% Cons_eq_filter_iff
thf(fact_6975_filter__eq__ConsD,axiom,
    ! [A: $tType,P: A > $o,Ys: list @ A,X2: A,Xs2: list @ A] :
      ( ( ( filter2 @ A @ P @ Ys )
        = ( cons @ A @ X2 @ Xs2 ) )
     => ? [Us3: list @ A,Vs3: list @ A] :
          ( ( Ys
            = ( append @ A @ Us3 @ ( cons @ A @ X2 @ Vs3 ) ) )
          & ! [X6: A] :
              ( ( member @ A @ X6 @ ( set2 @ A @ Us3 ) )
             => ~ ( P @ X6 ) )
          & ( P @ X2 )
          & ( Xs2
            = ( filter2 @ A @ P @ Vs3 ) ) ) ) ).

% filter_eq_ConsD
thf(fact_6976_Cons__eq__filterD,axiom,
    ! [A: $tType,X2: A,Xs2: list @ A,P: A > $o,Ys: list @ A] :
      ( ( ( cons @ A @ X2 @ Xs2 )
        = ( filter2 @ A @ P @ Ys ) )
     => ? [Us3: list @ A,Vs3: list @ A] :
          ( ( Ys
            = ( append @ A @ Us3 @ ( cons @ A @ X2 @ Vs3 ) ) )
          & ! [X6: A] :
              ( ( member @ A @ X6 @ ( set2 @ A @ Us3 ) )
             => ~ ( P @ X6 ) )
          & ( P @ X2 )
          & ( Xs2
            = ( filter2 @ A @ P @ Vs3 ) ) ) ) ).

% Cons_eq_filterD
thf(fact_6977_empty__filter__conv,axiom,
    ! [A: $tType,P: A > $o,Xs2: list @ A] :
      ( ( ( nil @ A )
        = ( filter2 @ A @ P @ Xs2 ) )
      = ( ! [X: A] :
            ( ( member @ A @ X @ ( set2 @ A @ Xs2 ) )
           => ~ ( P @ X ) ) ) ) ).

% empty_filter_conv
thf(fact_6978_filter__empty__conv,axiom,
    ! [A: $tType,P: A > $o,Xs2: list @ A] :
      ( ( ( filter2 @ A @ P @ Xs2 )
        = ( nil @ A ) )
      = ( ! [X: A] :
            ( ( member @ A @ X @ ( set2 @ A @ Xs2 ) )
           => ~ ( P @ X ) ) ) ) ).

% filter_empty_conv
thf(fact_6979_filter__eq__snocD,axiom,
    ! [A: $tType,P: A > $o,L: list @ A,L4: list @ A,X2: A] :
      ( ( ( filter2 @ A @ P @ L )
        = ( append @ A @ L4 @ ( cons @ A @ X2 @ ( nil @ A ) ) ) )
     => ( ( member @ A @ X2 @ ( set2 @ A @ L ) )
        & ( P @ X2 ) ) ) ).

% filter_eq_snocD
thf(fact_6980_filter__conv__foldr,axiom,
    ! [A: $tType] :
      ( ( filter2 @ A )
      = ( ^ [P3: A > $o,Xs: list @ A] :
            ( foldr @ A @ ( list @ A )
            @ ^ [X: A,Xt: list @ A] : ( if @ ( list @ A ) @ ( P3 @ X ) @ ( cons @ A @ X @ Xt ) @ Xt )
            @ Xs
            @ ( nil @ A ) ) ) ) ).

% filter_conv_foldr
thf(fact_6981_fails__def,axiom,
    ! [A: $tType] :
      ( ( time_fails @ A )
      = ( ^ [M7: heap_Time_Heap @ A,H: heap_ext @ product_unit] :
            ( ( heap_Time_execute @ A @ M7 @ H )
            = ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) ) ) ).

% fails_def
thf(fact_6982_set__minus__filter__out,axiom,
    ! [A: $tType,Xs2: list @ A,Y2: A] :
      ( ( minus_minus @ ( set @ A ) @ ( set2 @ A @ Xs2 ) @ ( insert @ A @ Y2 @ ( bot_bot @ ( set @ A ) ) ) )
      = ( set2 @ A
        @ ( filter2 @ A
          @ ^ [X: A] : X != Y2
          @ Xs2 ) ) ) ).

% set_minus_filter_out
thf(fact_6983_upt__filter__extend,axiom,
    ! [U: nat,U5: nat,P: nat > $o] :
      ( ( ord_less_eq @ nat @ U @ U5 )
     => ( ! [I3: nat] :
            ( ( ( ord_less_eq @ nat @ U @ I3 )
              & ( ord_less @ nat @ I3 @ U5 ) )
           => ~ ( P @ I3 ) )
       => ( ( filter2 @ nat @ P @ ( upt @ ( zero_zero @ nat ) @ U ) )
          = ( filter2 @ nat @ P @ ( upt @ ( zero_zero @ nat ) @ U5 ) ) ) ) ) ).

% upt_filter_extend
thf(fact_6984_time__refines,axiom,
    ! [A: $tType,C2: heap_Time_Heap @ A,C6: heap_Time_Heap @ A,H2: heap_ext @ product_unit] :
      ( ( refine_Imp_refines @ A @ C2 @ C6 )
     => ( ~ ( time_fails @ A @ C6 @ H2 )
       => ( ord_less_eq @ nat @ ( time_time @ A @ C2 @ H2 ) @ ( time_time @ A @ C6 @ H2 ) ) ) ) ).

% time_refines
thf(fact_6985_TBOUND__bind__weak,axiom,
    ! [A: $tType,B: $tType,M: heap_Time_Heap @ A,T_1: nat,F: A > ( heap_Time_Heap @ B ),T_2: nat] :
      ( ( time_TBOUND @ A @ M @ T_1 )
     => ( ! [X3: A,H3: heap_ext @ product_unit] :
            ( ~ ( time_fails @ A @ M @ H3 )
           => ( time_TBOUND @ B @ ( F @ X3 ) @ T_2 ) )
       => ( time_TBOUND @ B @ ( heap_Time_bind @ A @ B @ M @ F ) @ ( plus_plus @ nat @ T_1 @ T_2 ) ) ) ) ).

% TBOUND_bind_weak
thf(fact_6986_time__array__upd,axiom,
    ! [A: $tType] :
      ( ( heap @ A )
     => ! [I: nat,X2: A,P4: array @ A,H2: heap_ext @ product_unit] :
          ( ( ( time_fails @ ( array @ A ) @ ( array_upd @ A @ I @ X2 @ P4 ) @ H2 )
           => ( ( time_time @ ( array @ A ) @ ( array_upd @ A @ I @ X2 @ P4 ) @ H2 )
              = ( zero_zero @ nat ) ) )
          & ( ~ ( time_fails @ ( array @ A ) @ ( array_upd @ A @ I @ X2 @ P4 ) @ H2 )
           => ( ( time_time @ ( array @ A ) @ ( array_upd @ A @ I @ X2 @ P4 ) @ H2 )
              = ( one_one @ nat ) ) ) ) ) ).

% time_array_upd
thf(fact_6987_time__array__nth,axiom,
    ! [A: $tType] :
      ( ( heap @ A )
     => ! [P4: array @ A,I: nat,H2: heap_ext @ product_unit] :
          ( ( ( time_fails @ A @ ( array_nth @ A @ P4 @ I ) @ H2 )
           => ( ( time_time @ A @ ( array_nth @ A @ P4 @ I ) @ H2 )
              = ( zero_zero @ nat ) ) )
          & ( ~ ( time_fails @ A @ ( array_nth @ A @ P4 @ I ) @ H2 )
           => ( ( time_time @ A @ ( array_nth @ A @ P4 @ I ) @ H2 )
              = ( one_one @ nat ) ) ) ) ) ).

% time_array_nth
thf(fact_6988_fails__bind,axiom,
    ! [A: $tType,B: $tType,M: heap_Time_Heap @ B,F: B > ( heap_Time_Heap @ A ),H2: heap_ext @ product_unit] :
      ( ( time_fails @ A @ ( heap_Time_bind @ B @ A @ M @ F ) @ H2 )
      = ( ~ ( time_fails @ B @ M @ H2 )
       => ( time_fails @ A @ ( F @ ( time_the_res @ B @ M @ H2 ) ) @ ( time_the_heap @ B @ M @ H2 ) ) ) ) ).

% fails_bind
thf(fact_6989_TBOUND__bind,axiom,
    ! [A: $tType,B: $tType,M: heap_Time_Heap @ A,T_1: nat,F: A > ( heap_Time_Heap @ B ),T_2: nat] :
      ( ( time_TBOUND @ A @ M @ T_1 )
     => ( ! [X3: A,H3: heap_ext @ product_unit] :
            ( ( X3
              = ( time_the_res @ A @ M @ H3 ) )
           => ( ~ ( time_fails @ A @ M @ H3 )
             => ( time_TBOUND @ B @ ( F @ X3 ) @ T_2 ) ) )
       => ( time_TBOUND @ B @ ( heap_Time_bind @ A @ B @ M @ F ) @ ( plus_plus @ nat @ T_1 @ T_2 ) ) ) ) ).

% TBOUND_bind
thf(fact_6990_sort__key__by__quicksort,axiom,
    ! [A: $tType,B: $tType] :
      ( ( linorder @ A )
     => ( ( linorder_sort_key @ B @ A )
        = ( ^ [F6: B > A,Xs: list @ B] :
              ( append @ B
              @ ( linorder_sort_key @ B @ A @ F6
                @ ( filter2 @ B
                  @ ^ [X: B] : ( ord_less @ A @ ( F6 @ X ) @ ( F6 @ ( nth @ B @ Xs @ ( divide_divide @ nat @ ( size_size @ ( list @ B ) @ Xs ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
                  @ Xs ) )
              @ ( append @ B
                @ ( filter2 @ B
                  @ ^ [X: B] :
                      ( ( F6 @ X )
                      = ( F6 @ ( nth @ B @ Xs @ ( divide_divide @ nat @ ( size_size @ ( list @ B ) @ Xs ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
                  @ Xs )
                @ ( linorder_sort_key @ B @ A @ F6
                  @ ( filter2 @ B
                    @ ^ [X: B] : ( ord_less @ A @ ( F6 @ ( nth @ B @ Xs @ ( divide_divide @ nat @ ( size_size @ ( list @ B ) @ Xs ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( F6 @ X ) )
                    @ Xs ) ) ) ) ) ) ) ).

% sort_key_by_quicksort
thf(fact_6991_sort__upt,axiom,
    ! [M: nat,N2: nat] :
      ( ( linorder_sort_key @ nat @ nat
        @ ^ [X: nat] : X
        @ ( upt @ M @ N2 ) )
      = ( upt @ M @ N2 ) ) ).

% sort_upt
thf(fact_6992_sort__upto,axiom,
    ! [I: int,J: int] :
      ( ( linorder_sort_key @ int @ int
        @ ^ [X: int] : X
        @ ( upto @ I @ J ) )
      = ( upto @ I @ J ) ) ).

% sort_upto
thf(fact_6993_length__sort,axiom,
    ! [A: $tType,B: $tType] :
      ( ( linorder @ A )
     => ! [F: B > A,Xs2: list @ B] :
          ( ( size_size @ ( list @ B ) @ ( linorder_sort_key @ B @ A @ F @ Xs2 ) )
          = ( size_size @ ( list @ B ) @ Xs2 ) ) ) ).

% length_sort
thf(fact_6994_set__sort,axiom,
    ! [A: $tType,B: $tType] :
      ( ( linorder @ A )
     => ! [F: B > A,Xs2: list @ B] :
          ( ( set2 @ B @ ( linorder_sort_key @ B @ A @ F @ Xs2 ) )
          = ( set2 @ B @ Xs2 ) ) ) ).

% set_sort
thf(fact_6995_sort__key__stable,axiom,
    ! [B: $tType,A: $tType] :
      ( ( linorder @ B )
     => ! [F: A > B,K: B,Xs2: list @ A] :
          ( ( filter2 @ A
            @ ^ [Y: A] :
                ( ( F @ Y )
                = K )
            @ ( linorder_sort_key @ A @ B @ F @ Xs2 ) )
          = ( filter2 @ A
            @ ^ [Y: A] :
                ( ( F @ Y )
                = K )
            @ Xs2 ) ) ) ).

% sort_key_stable
thf(fact_6996_sort__key__const,axiom,
    ! [B: $tType,A: $tType] :
      ( ( linorder @ B )
     => ! [C2: B,Xs2: list @ A] :
          ( ( linorder_sort_key @ A @ B
            @ ^ [X: A] : C2
            @ Xs2 )
          = Xs2 ) ) ).

% sort_key_const
thf(fact_6997_sort__mergesort,axiom,
    ! [A: $tType] :
      ( ( linorder @ A )
     => ( ( linorder_sort_key @ A @ A
          @ ^ [X: A] : X )
        = ( mergesort @ A ) ) ) ).

% sort_mergesort
thf(fact_6998_sort__mergesort__by__rel,axiom,
    ! [A: $tType] :
      ( ( linorder @ A )
     => ( ( linorder_sort_key @ A @ A
          @ ^ [X: A] : X )
        = ( mergesort_by_rel @ A @ ( ord_less_eq @ A ) ) ) ) ).

% sort_mergesort_by_rel
thf(fact_6999_transpose__aux__max,axiom,
    ! [A: $tType,B: $tType,Xs2: list @ A,Xss: list @ ( list @ B )] :
      ( ( ord_max @ nat @ ( suc @ ( size_size @ ( list @ A ) @ Xs2 ) )
        @ ( foldr @ ( list @ B ) @ nat
          @ ^ [Xs: list @ B] : ( ord_max @ nat @ ( size_size @ ( list @ B ) @ Xs ) )
          @ Xss
          @ ( zero_zero @ nat ) ) )
      = ( suc
        @ ( ord_max @ nat @ ( size_size @ ( list @ A ) @ Xs2 )
          @ ( foldr @ ( list @ B ) @ nat
            @ ^ [X: list @ B] : ( ord_max @ nat @ ( minus_minus @ nat @ ( size_size @ ( list @ B ) @ X ) @ ( suc @ ( zero_zero @ nat ) ) ) )
            @ ( filter2 @ ( list @ B )
              @ ^ [Ys3: list @ B] :
                  ( Ys3
                 != ( nil @ B ) )
              @ Xss )
            @ ( zero_zero @ nat ) ) ) ) ) ).

% transpose_aux_max
thf(fact_7000_sort__by__quicksort,axiom,
    ! [A: $tType] :
      ( ( linorder @ A )
     => ! [Xs2: list @ A] :
          ( ( linorder_sort_key @ A @ A
            @ ^ [X: A] : X
            @ Xs2 )
          = ( append @ A
            @ ( linorder_sort_key @ A @ A
              @ ^ [X: A] : X
              @ ( filter2 @ A
                @ ^ [X: A] : ( ord_less @ A @ X @ ( nth @ A @ Xs2 @ ( divide_divide @ nat @ ( size_size @ ( list @ A ) @ Xs2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
                @ Xs2 ) )
            @ ( append @ A
              @ ( filter2 @ A
                @ ^ [X: A] :
                    ( X
                    = ( nth @ A @ Xs2 @ ( divide_divide @ nat @ ( size_size @ ( list @ A ) @ Xs2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
                @ Xs2 )
              @ ( linorder_sort_key @ A @ A
                @ ^ [X: A] : X
                @ ( filter2 @ A @ ( ord_less @ A @ ( nth @ A @ Xs2 @ ( divide_divide @ nat @ ( size_size @ ( list @ A ) @ Xs2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ Xs2 ) ) ) ) ) ) ).

% sort_by_quicksort
thf(fact_7001_time__array__map__entry,axiom,
    ! [A: $tType] :
      ( ( heap @ A )
     => ! [I: nat,F: A > A,P4: array @ A,H2: heap_ext @ product_unit] :
          ( ( ( time_fails @ ( array @ A ) @ ( array_map_entry @ A @ I @ F @ P4 ) @ H2 )
           => ( ( time_time @ ( array @ A ) @ ( array_map_entry @ A @ I @ F @ P4 ) @ H2 )
              = ( zero_zero @ nat ) ) )
          & ( ~ ( time_fails @ ( array @ A ) @ ( array_map_entry @ A @ I @ F @ P4 ) @ H2 )
           => ( ( time_time @ ( array @ A ) @ ( array_map_entry @ A @ I @ F @ P4 ) @ H2 )
              = ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).

% time_array_map_entry
thf(fact_7002_sort__key__by__quicksort__code,axiom,
    ! [A: $tType,B: $tType] :
      ( ( linorder @ A )
     => ( ( linorder_sort_key @ B @ A )
        = ( ^ [F6: B > A,Xs: list @ B] :
              ( case_list @ ( list @ B ) @ B @ ( nil @ B )
              @ ^ [X: B] :
                  ( case_list @ ( list @ B ) @ B @ Xs
                  @ ^ [Y: B] :
                      ( case_list @ ( list @ B ) @ B @ ( if @ ( list @ B ) @ ( ord_less_eq @ A @ ( F6 @ X ) @ ( F6 @ Y ) ) @ Xs @ ( cons @ B @ Y @ ( cons @ B @ X @ ( nil @ B ) ) ) )
                      @ ^ [Ab2: B,List: list @ B] :
                          ( product_case_prod @ ( list @ B ) @ ( product_prod @ ( list @ B ) @ ( list @ B ) ) @ ( list @ B )
                          @ ^ [Lts: list @ B] :
                              ( product_case_prod @ ( list @ B ) @ ( list @ B ) @ ( list @ B )
                              @ ^ [Eqs: list @ B,Gts: list @ B] : ( append @ B @ ( linorder_sort_key @ B @ A @ F6 @ Lts ) @ ( append @ B @ Eqs @ ( linorder_sort_key @ B @ A @ F6 @ Gts ) ) ) )
                          @ ( linorder_part @ B @ A @ F6 @ ( F6 @ ( nth @ B @ Xs @ ( divide_divide @ nat @ ( size_size @ ( list @ B ) @ Xs ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ Xs ) ) ) )
              @ Xs ) ) ) ) ).

% sort_key_by_quicksort_code
thf(fact_7003_list_Ocase__distrib,axiom,
    ! [B: $tType,C: $tType,A: $tType,H2: B > C,F1: B,F22: A > ( list @ A ) > B,List2: list @ A] :
      ( ( H2 @ ( case_list @ B @ A @ F1 @ F22 @ List2 ) )
      = ( case_list @ C @ A @ ( H2 @ F1 )
        @ ^ [X15: A,X25: list @ A] : ( H2 @ ( F22 @ X15 @ X25 ) )
        @ List2 ) ) ).

% list.case_distrib
thf(fact_7004_part__code_I1_J,axiom,
    ! [A: $tType,B: $tType] :
      ( ( linorder @ A )
     => ! [F: B > A,Pivot: A] :
          ( ( linorder_part @ B @ A @ F @ Pivot @ ( nil @ B ) )
          = ( product_Pair @ ( list @ B ) @ ( product_prod @ ( list @ B ) @ ( list @ B ) ) @ ( nil @ B ) @ ( product_Pair @ ( list @ B ) @ ( list @ B ) @ ( nil @ B ) @ ( nil @ B ) ) ) ) ) ).

% part_code(1)
thf(fact_7005_TBOUND__map__entry,axiom,
    ! [A: $tType] :
      ( ( heap @ A )
     => ! [I: nat,X2: A > A,A2: array @ A] : ( time_TBOUND @ ( array @ A ) @ ( array_map_entry @ A @ I @ X2 @ A2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ).

% TBOUND_map_entry
thf(fact_7006_part__code_I2_J,axiom,
    ! [A: $tType,B: $tType] :
      ( ( linorder @ A )
     => ! [F: B > A,Pivot: A,X2: B,Xs2: list @ B] :
          ( ( linorder_part @ B @ A @ F @ Pivot @ ( cons @ B @ X2 @ Xs2 ) )
          = ( product_case_prod @ ( list @ B ) @ ( product_prod @ ( list @ B ) @ ( list @ B ) ) @ ( product_prod @ ( list @ B ) @ ( product_prod @ ( list @ B ) @ ( list @ B ) ) )
            @ ^ [Lts: list @ B] :
                ( product_case_prod @ ( list @ B ) @ ( list @ B ) @ ( product_prod @ ( list @ B ) @ ( product_prod @ ( list @ B ) @ ( list @ B ) ) )
                @ ^ [Eqs: list @ B,Gts: list @ B] : ( if @ ( product_prod @ ( list @ B ) @ ( product_prod @ ( list @ B ) @ ( list @ B ) ) ) @ ( ord_less @ A @ ( F @ X2 ) @ Pivot ) @ ( product_Pair @ ( list @ B ) @ ( product_prod @ ( list @ B ) @ ( list @ B ) ) @ ( cons @ B @ X2 @ Lts ) @ ( product_Pair @ ( list @ B ) @ ( list @ B ) @ Eqs @ Gts ) ) @ ( if @ ( product_prod @ ( list @ B ) @ ( product_prod @ ( list @ B ) @ ( list @ B ) ) ) @ ( ord_less @ A @ Pivot @ ( F @ X2 ) ) @ ( product_Pair @ ( list @ B ) @ ( product_prod @ ( list @ B ) @ ( list @ B ) ) @ Lts @ ( product_Pair @ ( list @ B ) @ ( list @ B ) @ Eqs @ ( cons @ B @ X2 @ Gts ) ) ) @ ( product_Pair @ ( list @ B ) @ ( product_prod @ ( list @ B ) @ ( list @ B ) ) @ Lts @ ( product_Pair @ ( list @ B ) @ ( list @ B ) @ ( cons @ B @ X2 @ Eqs ) @ Gts ) ) ) ) )
            @ ( linorder_part @ B @ A @ F @ Pivot @ Xs2 ) ) ) ) ).

% part_code(2)
thf(fact_7007_part__def,axiom,
    ! [A: $tType,B: $tType] :
      ( ( linorder @ A )
     => ( ( linorder_part @ B @ A )
        = ( ^ [F6: B > A,Pivot2: A,Xs: list @ B] :
              ( product_Pair @ ( list @ B ) @ ( product_prod @ ( list @ B ) @ ( list @ B ) )
              @ ( filter2 @ B
                @ ^ [X: B] : ( ord_less @ A @ ( F6 @ X ) @ Pivot2 )
                @ Xs )
              @ ( product_Pair @ ( list @ B ) @ ( list @ B )
                @ ( filter2 @ B
                  @ ^ [X: B] :
                      ( ( F6 @ X )
                      = Pivot2 )
                  @ Xs )
                @ ( filter2 @ B
                  @ ^ [X: B] : ( ord_less @ A @ Pivot2 @ ( F6 @ X ) )
                  @ Xs ) ) ) ) ) ) ).

% part_def
thf(fact_7008_time__array__swap,axiom,
    ! [A: $tType] :
      ( ( heap @ A )
     => ! [I: nat,F: A > A,P4: array @ A,H2: heap_ext @ product_unit,X2: A] :
          ( ( ( time_fails @ ( array @ A ) @ ( array_map_entry @ A @ I @ F @ P4 ) @ H2 )
           => ( ( time_time @ A @ ( array_swap @ A @ I @ X2 @ P4 ) @ H2 )
              = ( zero_zero @ nat ) ) )
          & ( ~ ( time_fails @ ( array @ A ) @ ( array_map_entry @ A @ I @ F @ P4 ) @ H2 )
           => ( ( time_time @ A @ ( array_swap @ A @ I @ X2 @ P4 ) @ H2 )
              = ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).

% time_array_swap
thf(fact_7009_signed_Ostable__sort__key__def,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len @ A )
     => ( ( stable_sort_key @ B @ ( word @ A ) )
        = ( ^ [Sk: ( B > ( word @ A ) ) > ( list @ B ) > ( list @ B )] :
            ! [F6: B > ( word @ A ),Xs: list @ B,K4: word @ A] :
              ( ( filter2 @ B
                @ ^ [Y: B] :
                    ( ( F6 @ Y )
                    = K4 )
                @ ( Sk @ F6 @ Xs ) )
              = ( filter2 @ B
                @ ^ [Y: B] :
                    ( ( F6 @ Y )
                    = K4 )
                @ Xs ) ) ) ) ) ).

% signed.stable_sort_key_def
thf(fact_7010_list_Odisc__eq__case_I2_J,axiom,
    ! [A: $tType,List2: list @ A] :
      ( ( List2
       != ( nil @ A ) )
      = ( case_list @ $o @ A @ $false
        @ ^ [Uu3: A,Uv3: list @ A] : $true
        @ List2 ) ) ).

% list.disc_eq_case(2)
thf(fact_7011_list_Odisc__eq__case_I1_J,axiom,
    ! [A: $tType,List2: list @ A] :
      ( ( List2
        = ( nil @ A ) )
      = ( case_list @ $o @ A @ $true
        @ ^ [Uu3: A,Uv3: list @ A] : $false
        @ List2 ) ) ).

% list.disc_eq_case(1)
thf(fact_7012_case__list__rule,axiom,
    ! [A: $tType,B: $tType,L: list @ A,P: assn,Fn: heap_Time_Heap @ B,Q: B > assn,Fc: A > ( list @ A ) > ( heap_Time_Heap @ B )] :
      ( ( ( L
          = ( nil @ A ) )
       => ( hoare_hoare_triple @ B @ P @ Fn @ Q ) )
     => ( ! [X3: A,Xs3: list @ A] :
            ( ( L
              = ( cons @ A @ X3 @ Xs3 ) )
           => ( hoare_hoare_triple @ B @ P @ ( Fc @ X3 @ Xs3 ) @ Q ) )
       => ( hoare_hoare_triple @ B @ P @ ( case_list @ ( heap_Time_Heap @ B ) @ A @ Fn @ Fc @ L ) @ Q ) ) ) ).

% case_list_rule
thf(fact_7013_TBOUND__swap,axiom,
    ! [A: $tType] :
      ( ( heap @ A )
     => ! [I: nat,X2: A,A2: array @ A] : ( time_TBOUND @ A @ ( array_swap @ A @ I @ X2 @ A2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ).

% TBOUND_swap
thf(fact_7014_transpose__max__length,axiom,
    ! [A: $tType,Xs2: list @ ( list @ A )] :
      ( ( foldr @ ( list @ A ) @ nat
        @ ^ [Xs: list @ A] : ( ord_max @ nat @ ( size_size @ ( list @ A ) @ Xs ) )
        @ ( transpose @ A @ Xs2 )
        @ ( zero_zero @ nat ) )
      = ( size_size @ ( list @ ( list @ A ) )
        @ ( filter2 @ ( list @ A )
          @ ^ [X: list @ A] :
              ( X
             != ( nil @ A ) )
          @ Xs2 ) ) ) ).

% transpose_max_length
thf(fact_7015_Bleast__code,axiom,
    ! [A: $tType] :
      ( ( linorder @ A )
     => ! [Xs2: list @ A,P: A > $o] :
          ( ( bleast @ A @ ( set2 @ A @ Xs2 ) @ P )
          = ( case_list @ A @ A @ ( abort_Bleast @ A @ ( set2 @ A @ Xs2 ) @ P )
            @ ^ [X: A,Xs: list @ A] : X
            @ ( filter2 @ A @ P
              @ ( linorder_sort_key @ A @ A
                @ ^ [X: A] : X
                @ Xs2 ) ) ) ) ) ).

% Bleast_code
thf(fact_7016_transpose__empty,axiom,
    ! [A: $tType,Xs2: list @ ( list @ A )] :
      ( ( ( transpose @ A @ Xs2 )
        = ( nil @ ( list @ A ) ) )
      = ( ! [X: list @ A] :
            ( ( member @ ( list @ A ) @ X @ ( set2 @ ( list @ A ) @ Xs2 ) )
           => ( X
              = ( nil @ A ) ) ) ) ) ).

% transpose_empty
thf(fact_7017_nth__transpose,axiom,
    ! [A: $tType,I: nat,Xs2: list @ ( list @ A )] :
      ( ( ord_less @ nat @ I @ ( size_size @ ( list @ ( list @ A ) ) @ ( transpose @ A @ Xs2 ) ) )
     => ( ( nth @ ( list @ A ) @ ( transpose @ A @ Xs2 ) @ I )
        = ( map @ ( list @ A ) @ A
          @ ^ [Xs: list @ A] : ( nth @ A @ Xs @ I )
          @ ( filter2 @ ( list @ A )
            @ ^ [Ys3: list @ A] : ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Ys3 ) )
            @ Xs2 ) ) ) ) ).

% nth_transpose
thf(fact_7018_length__transpose,axiom,
    ! [A: $tType,Xs2: list @ ( list @ A )] :
      ( ( size_size @ ( list @ ( list @ A ) ) @ ( transpose @ A @ Xs2 ) )
      = ( foldr @ ( list @ A ) @ nat
        @ ^ [Xs: list @ A] : ( ord_max @ nat @ ( size_size @ ( list @ A ) @ Xs ) )
        @ Xs2
        @ ( zero_zero @ nat ) ) ) ).

% length_transpose
thf(fact_7019_transpose__rectangle,axiom,
    ! [A: $tType,Xs2: list @ ( list @ A ),N2: nat] :
      ( ( ( Xs2
          = ( nil @ ( list @ A ) ) )
       => ( N2
          = ( zero_zero @ nat ) ) )
     => ( ! [I3: nat] :
            ( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ ( list @ A ) ) @ Xs2 ) )
           => ( ( size_size @ ( list @ A ) @ ( nth @ ( list @ A ) @ Xs2 @ I3 ) )
              = N2 ) )
       => ( ( transpose @ A @ Xs2 )
          = ( map @ nat @ ( list @ A )
            @ ^ [I2: nat] :
                ( map @ nat @ A
                @ ^ [J3: nat] : ( nth @ A @ ( nth @ ( list @ A ) @ Xs2 @ J3 ) @ I2 )
                @ ( upt @ ( zero_zero @ nat ) @ ( size_size @ ( list @ ( list @ A ) ) @ Xs2 ) ) )
            @ ( upt @ ( zero_zero @ nat ) @ N2 ) ) ) ) ) ).

% transpose_rectangle
thf(fact_7020_length__product__lists,axiom,
    ! [B: $tType,Xss: list @ ( list @ B )] :
      ( ( size_size @ ( list @ ( list @ B ) ) @ ( product_lists @ B @ Xss ) )
      = ( foldr @ nat @ nat @ ( times_times @ nat ) @ ( map @ ( list @ B ) @ nat @ ( size_size @ ( list @ B ) ) @ Xss ) @ ( one_one @ nat ) ) ) ).

% length_product_lists
thf(fact_7021_length__subseqs,axiom,
    ! [A: $tType,Xs2: list @ A] :
      ( ( size_size @ ( list @ ( list @ A ) ) @ ( subseqs @ A @ Xs2 ) )
      = ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( size_size @ ( list @ A ) @ Xs2 ) ) ) ).

% length_subseqs
thf(fact_7022_Cons__in__subseqsD,axiom,
    ! [A: $tType,Y2: A,Ys: list @ A,Xs2: list @ A] :
      ( ( member @ ( list @ A ) @ ( cons @ A @ Y2 @ Ys ) @ ( set2 @ ( list @ A ) @ ( subseqs @ A @ Xs2 ) ) )
     => ( member @ ( list @ A ) @ Ys @ ( set2 @ ( list @ A ) @ ( subseqs @ A @ Xs2 ) ) ) ) ).

% Cons_in_subseqsD
thf(fact_7023_subseqs_Osimps_I2_J,axiom,
    ! [A: $tType,X2: A,Xs2: list @ A] :
      ( ( subseqs @ A @ ( cons @ A @ X2 @ Xs2 ) )
      = ( append @ ( list @ A ) @ ( map @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X2 ) @ ( subseqs @ A @ Xs2 ) ) @ ( subseqs @ A @ Xs2 ) ) ) ).

% subseqs.simps(2)
thf(fact_7024_subseqs__refl,axiom,
    ! [A: $tType,Xs2: list @ A] : ( member @ ( list @ A ) @ Xs2 @ ( set2 @ ( list @ A ) @ ( subseqs @ A @ Xs2 ) ) ) ).

% subseqs_refl
thf(fact_7025_merge__list_Ocases,axiom,
    ! [A: $tType] :
      ( ( linorder @ A )
     => ! [X2: product_prod @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) )] :
          ( ( X2
           != ( product_Pair @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) @ ( nil @ ( list @ A ) ) @ ( nil @ ( list @ A ) ) ) )
         => ( ! [L3: list @ A] :
                ( X2
               != ( product_Pair @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) @ ( nil @ ( list @ A ) ) @ ( cons @ ( list @ A ) @ L3 @ ( nil @ ( list @ A ) ) ) ) )
           => ( ! [La: list @ A,Acc22: list @ ( list @ A )] :
                  ( X2
                 != ( product_Pair @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) @ ( cons @ ( list @ A ) @ La @ Acc22 ) @ ( nil @ ( list @ A ) ) ) )
             => ( ! [La: list @ A,Acc22: list @ ( list @ A ),L3: list @ A] :
                    ( X2
                   != ( product_Pair @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) @ ( cons @ ( list @ A ) @ La @ Acc22 ) @ ( cons @ ( list @ A ) @ L3 @ ( nil @ ( list @ A ) ) ) ) )
               => ~ ! [Acc22: list @ ( list @ A ),L1: list @ A,L22: list @ A,Ls2: list @ ( list @ A )] :
                      ( X2
                     != ( product_Pair @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) @ Acc22 @ ( cons @ ( list @ A ) @ L1 @ ( cons @ ( list @ A ) @ L22 @ Ls2 ) ) ) ) ) ) ) ) ) ).

% merge_list.cases
thf(fact_7026_subset__subseqs,axiom,
    ! [A: $tType,X8: set @ A,Xs2: list @ A] :
      ( ( ord_less_eq @ ( set @ A ) @ X8 @ ( set2 @ A @ Xs2 ) )
     => ( member @ ( set @ A ) @ X8 @ ( image @ ( list @ A ) @ ( set @ A ) @ ( set2 @ A ) @ ( set2 @ ( list @ A ) @ ( subseqs @ A @ Xs2 ) ) ) ) ) ).

% subset_subseqs
thf(fact_7027_in__set__product__lists__length,axiom,
    ! [A: $tType,Xs2: list @ A,Xss: list @ ( list @ A )] :
      ( ( member @ ( list @ A ) @ Xs2 @ ( set2 @ ( list @ A ) @ ( product_lists @ A @ Xss ) ) )
     => ( ( size_size @ ( list @ A ) @ Xs2 )
        = ( size_size @ ( list @ ( list @ A ) ) @ Xss ) ) ) ).

% in_set_product_lists_length
thf(fact_7028_length__mul__elem,axiom,
    ! [A: $tType,Xs2: list @ ( list @ A ),N2: nat] :
      ( ! [X3: list @ A] :
          ( ( member @ ( list @ A ) @ X3 @ ( set2 @ ( list @ A ) @ Xs2 ) )
         => ( ( size_size @ ( list @ A ) @ X3 )
            = N2 ) )
     => ( ( size_size @ ( list @ A ) @ ( concat @ A @ Xs2 ) )
        = ( times_times @ nat @ ( size_size @ ( list @ ( list @ A ) ) @ Xs2 ) @ N2 ) ) ) ).

% length_mul_elem
thf(fact_7029_set__n__lists,axiom,
    ! [A: $tType,N2: nat,Xs2: list @ A] :
      ( ( set2 @ ( list @ A ) @ ( n_lists @ A @ N2 @ Xs2 ) )
      = ( collect @ ( list @ A )
        @ ^ [Ys3: list @ A] :
            ( ( ( size_size @ ( list @ A ) @ Ys3 )
              = N2 )
            & ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Ys3 ) @ ( set2 @ A @ Xs2 ) ) ) ) ) ).

% set_n_lists
thf(fact_7030_Nil__eq__concat__conv,axiom,
    ! [A: $tType,Xss: list @ ( list @ A )] :
      ( ( ( nil @ A )
        = ( concat @ A @ Xss ) )
      = ( ! [X: list @ A] :
            ( ( member @ ( list @ A ) @ X @ ( set2 @ ( list @ A ) @ Xss ) )
           => ( X
              = ( nil @ A ) ) ) ) ) ).

% Nil_eq_concat_conv
thf(fact_7031_concat__eq__Nil__conv,axiom,
    ! [A: $tType,Xss: list @ ( list @ A )] :
      ( ( ( concat @ A @ Xss )
        = ( nil @ A ) )
      = ( ! [X: list @ A] :
            ( ( member @ ( list @ A ) @ X @ ( set2 @ ( list @ A ) @ Xss ) )
           => ( X
              = ( nil @ A ) ) ) ) ) ).

% concat_eq_Nil_conv
thf(fact_7032_concat__map__singleton,axiom,
    ! [A: $tType,B: $tType,F: B > A,Xs2: list @ B] :
      ( ( concat @ A
        @ ( map @ B @ ( list @ A )
          @ ^ [X: B] : ( cons @ A @ ( F @ X ) @ ( nil @ A ) )
          @ Xs2 ) )
      = ( map @ B @ A @ F @ Xs2 ) ) ).

% concat_map_singleton
thf(fact_7033_n__lists_Osimps_I2_J,axiom,
    ! [A: $tType,N2: nat,Xs2: list @ A] :
      ( ( n_lists @ A @ ( suc @ N2 ) @ Xs2 )
      = ( concat @ ( list @ A )
        @ ( map @ ( list @ A ) @ ( list @ ( list @ A ) )
          @ ^ [Ys3: list @ A] :
              ( map @ A @ ( list @ A )
              @ ^ [Y: A] : ( cons @ A @ Y @ Ys3 )
              @ Xs2 )
          @ ( n_lists @ A @ N2 @ Xs2 ) ) ) ) ).

% n_lists.simps(2)
thf(fact_7034_product__lists_Osimps_I2_J,axiom,
    ! [A: $tType,Xs2: list @ A,Xss: list @ ( list @ A )] :
      ( ( product_lists @ A @ ( cons @ ( list @ A ) @ Xs2 @ Xss ) )
      = ( concat @ ( list @ A )
        @ ( map @ A @ ( list @ ( list @ A ) )
          @ ^ [X: A] : ( map @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X ) @ ( product_lists @ A @ Xss ) )
          @ Xs2 ) ) ) ).

% product_lists.simps(2)
thf(fact_7035_transpose_Oelims,axiom,
    ! [A: $tType,X2: list @ ( list @ A ),Y2: list @ ( list @ A )] :
      ( ( ( transpose @ A @ X2 )
        = Y2 )
     => ( ( ( X2
            = ( nil @ ( list @ A ) ) )
         => ( Y2
           != ( nil @ ( list @ A ) ) ) )
       => ( ! [Xss2: list @ ( list @ A )] :
              ( ( X2
                = ( cons @ ( list @ A ) @ ( nil @ A ) @ Xss2 ) )
             => ( Y2
               != ( transpose @ A @ Xss2 ) ) )
         => ~ ! [X3: A,Xs3: list @ A,Xss2: list @ ( list @ A )] :
                ( ( X2
                  = ( cons @ ( list @ A ) @ ( cons @ A @ X3 @ Xs3 ) @ Xss2 ) )
               => ( Y2
                 != ( cons @ ( list @ A )
                    @ ( cons @ A @ X3
                      @ ( concat @ A
                        @ ( map @ ( list @ A ) @ ( list @ A )
                          @ ( case_list @ ( list @ A ) @ A @ ( nil @ A )
                            @ ^ [H: A,T3: list @ A] : ( cons @ A @ H @ ( nil @ A ) ) )
                          @ Xss2 ) ) )
                    @ ( transpose @ A
                      @ ( cons @ ( list @ A ) @ Xs3
                        @ ( concat @ ( list @ A )
                          @ ( map @ ( list @ A ) @ ( list @ ( list @ A ) )
                            @ ( case_list @ ( list @ ( list @ A ) ) @ A @ ( nil @ ( list @ A ) )
                              @ ^ [H: A,T3: list @ A] : ( cons @ ( list @ A ) @ T3 @ ( nil @ ( list @ A ) ) ) )
                            @ Xss2 ) ) ) ) ) ) ) ) ) ) ).

% transpose.elims
thf(fact_7036_transpose_Osimps_I3_J,axiom,
    ! [A: $tType,X2: A,Xs2: list @ A,Xss: list @ ( list @ A )] :
      ( ( transpose @ A @ ( cons @ ( list @ A ) @ ( cons @ A @ X2 @ Xs2 ) @ Xss ) )
      = ( cons @ ( list @ A )
        @ ( cons @ A @ X2
          @ ( concat @ A
            @ ( map @ ( list @ A ) @ ( list @ A )
              @ ( case_list @ ( list @ A ) @ A @ ( nil @ A )
                @ ^ [H: A,T3: list @ A] : ( cons @ A @ H @ ( nil @ A ) ) )
              @ Xss ) ) )
        @ ( transpose @ A
          @ ( cons @ ( list @ A ) @ Xs2
            @ ( concat @ ( list @ A )
              @ ( map @ ( list @ A ) @ ( list @ ( list @ A ) )
                @ ( case_list @ ( list @ ( list @ A ) ) @ A @ ( nil @ ( list @ A ) )
                  @ ^ [H: A,T3: list @ A] : ( cons @ ( list @ A ) @ T3 @ ( nil @ ( list @ A ) ) ) )
                @ Xss ) ) ) ) ) ) ).

% transpose.simps(3)
thf(fact_7037_concat__filter__neq__Nil,axiom,
    ! [A: $tType,Xs2: list @ ( list @ A )] :
      ( ( concat @ A
        @ ( filter2 @ ( list @ A )
          @ ^ [Ys3: list @ A] :
              ( Ys3
             != ( nil @ A ) )
          @ Xs2 ) )
      = ( concat @ A @ Xs2 ) ) ).

% concat_filter_neq_Nil
thf(fact_7038_product__concat__map,axiom,
    ! [B: $tType,A: $tType] :
      ( ( product @ A @ B )
      = ( ^ [Xs: list @ A,Ys3: list @ B] :
            ( concat @ ( product_prod @ A @ B )
            @ ( map @ A @ ( list @ ( product_prod @ A @ B ) )
              @ ^ [X: A] : ( map @ B @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X ) @ Ys3 )
              @ Xs ) ) ) ) ).

% product_concat_map
thf(fact_7039_comm__append__is__replicate,axiom,
    ! [A: $tType,Xs2: list @ A,Ys: list @ A] :
      ( ( Xs2
       != ( nil @ A ) )
     => ( ( Ys
         != ( nil @ A ) )
       => ( ( ( append @ A @ Xs2 @ Ys )
            = ( append @ A @ Ys @ Xs2 ) )
         => ? [N4: nat,Zs2: list @ A] :
              ( ( ord_less @ nat @ ( one_one @ nat ) @ N4 )
              & ( ( concat @ A @ ( replicate @ ( list @ A ) @ N4 @ Zs2 ) )
                = ( append @ A @ Xs2 @ Ys ) ) ) ) ) ) ).

% comm_append_is_replicate
thf(fact_7040_length__n__lists__elem,axiom,
    ! [A: $tType,Ys: list @ A,N2: nat,Xs2: list @ A] :
      ( ( member @ ( list @ A ) @ Ys @ ( set2 @ ( list @ A ) @ ( n_lists @ A @ N2 @ Xs2 ) ) )
     => ( ( size_size @ ( list @ A ) @ Ys )
        = N2 ) ) ).

% length_n_lists_elem
thf(fact_7041_length__n__lists,axiom,
    ! [A: $tType,N2: nat,Xs2: list @ A] :
      ( ( size_size @ ( list @ ( list @ A ) ) @ ( n_lists @ A @ N2 @ Xs2 ) )
      = ( power_power @ nat @ ( size_size @ ( list @ A ) @ Xs2 ) @ N2 ) ) ).

% length_n_lists
thf(fact_7042_transpose_Opelims,axiom,
    ! [A: $tType,X2: list @ ( list @ A ),Y2: list @ ( list @ A )] :
      ( ( ( transpose @ A @ X2 )
        = Y2 )
     => ( ( accp @ ( list @ ( list @ A ) ) @ ( transpose_rel @ A ) @ X2 )
       => ( ( ( X2
              = ( nil @ ( list @ A ) ) )
           => ( ( Y2
                = ( nil @ ( list @ A ) ) )
             => ~ ( accp @ ( list @ ( list @ A ) ) @ ( transpose_rel @ A ) @ ( nil @ ( list @ A ) ) ) ) )
         => ( ! [Xss2: list @ ( list @ A )] :
                ( ( X2
                  = ( cons @ ( list @ A ) @ ( nil @ A ) @ Xss2 ) )
               => ( ( Y2
                    = ( transpose @ A @ Xss2 ) )
                 => ~ ( accp @ ( list @ ( list @ A ) ) @ ( transpose_rel @ A ) @ ( cons @ ( list @ A ) @ ( nil @ A ) @ Xss2 ) ) ) )
           => ~ ! [X3: A,Xs3: list @ A,Xss2: list @ ( list @ A )] :
                  ( ( X2
                    = ( cons @ ( list @ A ) @ ( cons @ A @ X3 @ Xs3 ) @ Xss2 ) )
                 => ( ( Y2
                      = ( cons @ ( list @ A )
                        @ ( cons @ A @ X3
                          @ ( concat @ A
                            @ ( map @ ( list @ A ) @ ( list @ A )
                              @ ( case_list @ ( list @ A ) @ A @ ( nil @ A )
                                @ ^ [H: A,T3: list @ A] : ( cons @ A @ H @ ( nil @ A ) ) )
                              @ Xss2 ) ) )
                        @ ( transpose @ A
                          @ ( cons @ ( list @ A ) @ Xs3
                            @ ( concat @ ( list @ A )
                              @ ( map @ ( list @ A ) @ ( list @ ( list @ A ) )
                                @ ( case_list @ ( list @ ( list @ A ) ) @ A @ ( nil @ ( list @ A ) )
                                  @ ^ [H: A,T3: list @ A] : ( cons @ ( list @ A ) @ T3 @ ( nil @ ( list @ A ) ) ) )
                                @ Xss2 ) ) ) ) ) )
                   => ~ ( accp @ ( list @ ( list @ A ) ) @ ( transpose_rel @ A ) @ ( cons @ ( list @ A ) @ ( cons @ A @ X3 @ Xs3 ) @ Xss2 ) ) ) ) ) ) ) ) ).

% transpose.pelims
thf(fact_7043_transpose_Opsimps_I3_J,axiom,
    ! [A: $tType,X2: A,Xs2: list @ A,Xss: list @ ( list @ A )] :
      ( ( accp @ ( list @ ( list @ A ) ) @ ( transpose_rel @ A ) @ ( cons @ ( list @ A ) @ ( cons @ A @ X2 @ Xs2 ) @ Xss ) )
     => ( ( transpose @ A @ ( cons @ ( list @ A ) @ ( cons @ A @ X2 @ Xs2 ) @ Xss ) )
        = ( cons @ ( list @ A )
          @ ( cons @ A @ X2
            @ ( concat @ A
              @ ( map @ ( list @ A ) @ ( list @ A )
                @ ( case_list @ ( list @ A ) @ A @ ( nil @ A )
                  @ ^ [H: A,T3: list @ A] : ( cons @ A @ H @ ( nil @ A ) ) )
                @ Xss ) ) )
          @ ( transpose @ A
            @ ( cons @ ( list @ A ) @ Xs2
              @ ( concat @ ( list @ A )
                @ ( map @ ( list @ A ) @ ( list @ ( list @ A ) )
                  @ ( case_list @ ( list @ ( list @ A ) ) @ A @ ( nil @ ( list @ A ) )
                    @ ^ [H: A,T3: list @ A] : ( cons @ ( list @ A ) @ T3 @ ( nil @ ( list @ A ) ) ) )
                  @ Xss ) ) ) ) ) ) ) ).

% transpose.psimps(3)
thf(fact_7044_transpose_Opinduct,axiom,
    ! [A: $tType,A0: list @ ( list @ A ),P: ( list @ ( list @ A ) ) > $o] :
      ( ( accp @ ( list @ ( list @ A ) ) @ ( transpose_rel @ A ) @ A0 )
     => ( ( ( accp @ ( list @ ( list @ A ) ) @ ( transpose_rel @ A ) @ ( nil @ ( list @ A ) ) )
         => ( P @ ( nil @ ( list @ A ) ) ) )
       => ( ! [Xss2: list @ ( list @ A )] :
              ( ( accp @ ( list @ ( list @ A ) ) @ ( transpose_rel @ A ) @ ( cons @ ( list @ A ) @ ( nil @ A ) @ Xss2 ) )
             => ( ( P @ Xss2 )
               => ( P @ ( cons @ ( list @ A ) @ ( nil @ A ) @ Xss2 ) ) ) )
         => ( ! [X3: A,Xs3: list @ A,Xss2: list @ ( list @ A )] :
                ( ( accp @ ( list @ ( list @ A ) ) @ ( transpose_rel @ A ) @ ( cons @ ( list @ A ) @ ( cons @ A @ X3 @ Xs3 ) @ Xss2 ) )
               => ( ( P
                    @ ( cons @ ( list @ A ) @ Xs3
                      @ ( concat @ ( list @ A )
                        @ ( map @ ( list @ A ) @ ( list @ ( list @ A ) )
                          @ ( case_list @ ( list @ ( list @ A ) ) @ A @ ( nil @ ( list @ A ) )
                            @ ^ [H: A,T3: list @ A] : ( cons @ ( list @ A ) @ T3 @ ( nil @ ( list @ A ) ) ) )
                          @ Xss2 ) ) ) )
                 => ( P @ ( cons @ ( list @ A ) @ ( cons @ A @ X3 @ Xs3 ) @ Xss2 ) ) ) )
           => ( P @ A0 ) ) ) ) ) ).

% transpose.pinduct
thf(fact_7045_product__code,axiom,
    ! [B: $tType,A: $tType,Xs2: list @ A,Ys: list @ B] :
      ( ( product_product @ A @ B @ ( set2 @ A @ Xs2 ) @ ( set2 @ B @ Ys ) )
      = ( set2 @ ( product_prod @ A @ B )
        @ ( concat @ ( product_prod @ A @ B )
          @ ( map @ A @ ( list @ ( product_prod @ A @ B ) )
            @ ^ [X: A] : ( map @ B @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X ) @ Ys )
            @ Xs2 ) ) ) ) ).

% product_code
thf(fact_7046_transpose__aux__filter__tail,axiom,
    ! [A: $tType,Xss: list @ ( list @ A )] :
      ( ( concat @ ( list @ A )
        @ ( map @ ( list @ A ) @ ( list @ ( list @ A ) )
          @ ( case_list @ ( list @ ( list @ A ) ) @ A @ ( nil @ ( list @ A ) )
            @ ^ [H: A,T3: list @ A] : ( cons @ ( list @ A ) @ T3 @ ( nil @ ( list @ A ) ) ) )
          @ Xss ) )
      = ( map @ ( list @ A ) @ ( list @ A ) @ ( tl @ A )
        @ ( filter2 @ ( list @ A )
          @ ^ [Ys3: list @ A] :
              ( Ys3
             != ( nil @ A ) )
          @ Xss ) ) ) ).

% transpose_aux_filter_tail
thf(fact_7047_tl__upt,axiom,
    ! [M: nat,N2: nat] :
      ( ( tl @ nat @ ( upt @ M @ N2 ) )
      = ( upt @ ( suc @ M ) @ N2 ) ) ).

% tl_upt
thf(fact_7048_length__tl,axiom,
    ! [A: $tType,Xs2: list @ A] :
      ( ( size_size @ ( list @ A ) @ ( tl @ A @ Xs2 ) )
      = ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs2 ) @ ( one_one @ nat ) ) ) ).

% length_tl
thf(fact_7049_tl__replicate,axiom,
    ! [A: $tType,N2: nat,X2: A] :
      ( ( tl @ A @ ( replicate @ A @ N2 @ X2 ) )
      = ( replicate @ A @ ( minus_minus @ nat @ N2 @ ( one_one @ nat ) ) @ X2 ) ) ).

% tl_replicate
thf(fact_7050_tl__append,axiom,
    ! [A: $tType,Xs2: list @ A,Ys: list @ A] :
      ( ( tl @ A @ ( append @ A @ Xs2 @ Ys ) )
      = ( case_list @ ( list @ A ) @ A @ ( tl @ A @ Ys )
        @ ^ [Z2: A,Zs3: list @ A] : ( append @ A @ Zs3 @ Ys )
        @ Xs2 ) ) ).

% tl_append
thf(fact_7051_tl__def,axiom,
    ! [A: $tType] :
      ( ( tl @ A )
      = ( case_list @ ( list @ A ) @ A @ ( nil @ A )
        @ ^ [X214: A,X224: list @ A] : X224 ) ) ).

% tl_def
thf(fact_7052_list_Oset__sel_I2_J,axiom,
    ! [A: $tType,A2: list @ A,X2: A] :
      ( ( A2
       != ( nil @ A ) )
     => ( ( member @ A @ X2 @ ( set2 @ A @ ( tl @ A @ A2 ) ) )
       => ( member @ A @ X2 @ ( set2 @ A @ A2 ) ) ) ) ).

% list.set_sel(2)
thf(fact_7053_in__set__tlD,axiom,
    ! [A: $tType,X2: A,Xs2: list @ A] :
      ( ( member @ A @ X2 @ ( set2 @ A @ ( tl @ A @ Xs2 ) ) )
     => ( member @ A @ X2 @ ( set2 @ A @ Xs2 ) ) ) ).

% in_set_tlD
thf(fact_7054_tl__obtain__elem,axiom,
    ! [A: $tType,Xs2: list @ A] :
      ( ( Xs2
       != ( nil @ A ) )
     => ( ( ( tl @ A @ Xs2 )
          = ( nil @ A ) )
       => ~ ! [E2: A] :
              ( Xs2
             != ( cons @ A @ E2 @ ( nil @ A ) ) ) ) ) ).

% tl_obtain_elem
thf(fact_7055_Misc_Onth__tl,axiom,
    ! [A: $tType,Xs2: list @ A,N2: nat] :
      ( ( Xs2
       != ( nil @ A ) )
     => ( ( nth @ A @ ( tl @ A @ Xs2 ) @ N2 )
        = ( nth @ A @ Xs2 @ ( suc @ N2 ) ) ) ) ).

% Misc.nth_tl
thf(fact_7056_tl__subset,axiom,
    ! [A: $tType,Xs2: list @ A,A3: set @ A] :
      ( ( Xs2
       != ( nil @ A ) )
     => ( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs2 ) @ A3 )
       => ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ ( tl @ A @ Xs2 ) ) @ A3 ) ) ) ).

% tl_subset
thf(fact_7057_list__take__induct__tl2,axiom,
    ! [B: $tType,A: $tType,Xs2: list @ A,Ys: list @ B,P: B > A > $o] :
      ( ( ( size_size @ ( list @ A ) @ Xs2 )
        = ( size_size @ ( list @ B ) @ Ys ) )
     => ( ! [N4: nat] :
            ( ( ord_less @ nat @ N4 @ ( size_size @ ( list @ A ) @ Xs2 ) )
           => ( P @ ( nth @ B @ Ys @ N4 ) @ ( nth @ A @ Xs2 @ N4 ) ) )
       => ! [N9: nat] :
            ( ( ord_less @ nat @ N9 @ ( size_size @ ( list @ A ) @ ( tl @ A @ Xs2 ) ) )
           => ( P @ ( nth @ B @ ( tl @ B @ Ys ) @ N9 ) @ ( nth @ A @ ( tl @ A @ Xs2 ) @ N9 ) ) ) ) ) ).

% list_take_induct_tl2
thf(fact_7058_Nitpick_Osize__list__simp_I2_J,axiom,
    ! [A: $tType] :
      ( ( size_size @ ( list @ A ) )
      = ( ^ [Xs: list @ A] :
            ( if @ nat
            @ ( Xs
              = ( nil @ A ) )
            @ ( zero_zero @ nat )
            @ ( suc @ ( size_size @ ( list @ A ) @ ( tl @ A @ Xs ) ) ) ) ) ) ).

% Nitpick.size_list_simp(2)
thf(fact_7059_List_Onth__tl,axiom,
    ! [A: $tType,N2: nat,Xs2: list @ A] :
      ( ( ord_less @ nat @ N2 @ ( size_size @ ( list @ A ) @ ( tl @ A @ Xs2 ) ) )
     => ( ( nth @ A @ ( tl @ A @ Xs2 ) @ N2 )
        = ( nth @ A @ Xs2 @ ( suc @ N2 ) ) ) ) ).

% List.nth_tl
thf(fact_7060_extract__Some__iff,axiom,
    ! [A: $tType,P: A > $o,Xs2: list @ A,Ys: list @ A,Y2: A,Zs: list @ A] :
      ( ( ( extract @ A @ P @ Xs2 )
        = ( some @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) @ ( product_Pair @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) @ Ys @ ( product_Pair @ A @ ( list @ A ) @ Y2 @ Zs ) ) ) )
      = ( ( Xs2
          = ( append @ A @ Ys @ ( cons @ A @ Y2 @ Zs ) ) )
        & ( P @ Y2 )
        & ~ ? [X: A] :
              ( ( member @ A @ X @ ( set2 @ A @ Ys ) )
              & ( P @ X ) ) ) ) ).

% extract_Some_iff
thf(fact_7061_extract__SomeE,axiom,
    ! [A: $tType,P: A > $o,Xs2: list @ A,Ys: list @ A,Y2: A,Zs: list @ A] :
      ( ( ( extract @ A @ P @ Xs2 )
        = ( some @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) @ ( product_Pair @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) @ Ys @ ( product_Pair @ A @ ( list @ A ) @ Y2 @ Zs ) ) ) )
     => ( ( Xs2
          = ( append @ A @ Ys @ ( cons @ A @ Y2 @ Zs ) ) )
        & ( P @ Y2 )
        & ~ ? [X6: A] :
              ( ( member @ A @ X6 @ ( set2 @ A @ Ys ) )
              & ( P @ X6 ) ) ) ) ).

% extract_SomeE
thf(fact_7062_extract__None__iff,axiom,
    ! [A: $tType,P: A > $o,Xs2: list @ A] :
      ( ( ( extract @ A @ P @ Xs2 )
        = ( none @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) ) )
      = ( ~ ? [X: A] :
              ( ( member @ A @ X @ ( set2 @ A @ Xs2 ) )
              & ( P @ X ) ) ) ) ).

% extract_None_iff
thf(fact_7063_extract__Nil__code,axiom,
    ! [A: $tType,P: A > $o] :
      ( ( extract @ A @ P @ ( nil @ A ) )
      = ( none @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) ) ) ).

% extract_Nil_code
thf(fact_7064_extract__Cons__code,axiom,
    ! [A: $tType,P: A > $o,X2: A,Xs2: list @ A] :
      ( ( ( P @ X2 )
       => ( ( extract @ A @ P @ ( cons @ A @ X2 @ Xs2 ) )
          = ( some @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) @ ( product_Pair @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) @ ( nil @ A ) @ ( product_Pair @ A @ ( list @ A ) @ X2 @ Xs2 ) ) ) ) )
      & ( ~ ( P @ X2 )
       => ( ( extract @ A @ P @ ( cons @ A @ X2 @ Xs2 ) )
          = ( case_option @ ( option @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) ) @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) @ ( none @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) )
            @ ( product_case_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) @ ( option @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) )
              @ ^ [Ys3: list @ A] :
                  ( product_case_prod @ A @ ( list @ A ) @ ( option @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) )
                  @ ^ [Y: A,Zs3: list @ A] : ( some @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) @ ( product_Pair @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) @ ( cons @ A @ X2 @ Ys3 ) @ ( product_Pair @ A @ ( list @ A ) @ Y @ Zs3 ) ) ) ) )
            @ ( extract @ A @ P @ Xs2 ) ) ) ) ) ).

% extract_Cons_code
thf(fact_7065_map__filter__map__filter,axiom,
    ! [A: $tType,B: $tType,F: B > A,P: B > $o,Xs2: list @ B] :
      ( ( map @ B @ A @ F @ ( filter2 @ B @ P @ Xs2 ) )
      = ( map_filter @ B @ A
        @ ^ [X: B] : ( if @ ( option @ A ) @ ( P @ X ) @ ( some @ A @ ( F @ X ) ) @ ( none @ A ) )
        @ Xs2 ) ) ).

% map_filter_map_filter
thf(fact_7066_subset__eq__mset__impl_Oelims,axiom,
    ! [A: $tType,X2: list @ A,Xa: list @ A,Y2: option @ $o] :
      ( ( ( subset_eq_mset_impl @ A @ X2 @ Xa )
        = Y2 )
     => ( ( ( X2
            = ( nil @ A ) )
         => ( Y2
           != ( some @ $o
              @ ( Xa
               != ( nil @ A ) ) ) ) )
       => ~ ! [X3: A,Xs3: list @ A] :
              ( ( X2
                = ( cons @ A @ X3 @ Xs3 ) )
             => ( Y2
               != ( case_option @ ( option @ $o ) @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) @ ( none @ $o )
                  @ ( product_case_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) @ ( option @ $o )
                    @ ^ [Ys1: list @ A] :
                        ( product_case_prod @ A @ ( list @ A ) @ ( option @ $o )
                        @ ^ [Y: A,Ys22: list @ A] : ( subset_eq_mset_impl @ A @ Xs3 @ ( append @ A @ Ys1 @ Ys22 ) ) ) )
                  @ ( extract @ A
                    @ ( ^ [Y5: A,Z4: A] : Y5 = Z4
                      @ X3 )
                    @ Xa ) ) ) ) ) ) ).

% subset_eq_mset_impl.elims
thf(fact_7067_subset__eq__mset__impl_Osimps_I1_J,axiom,
    ! [A: $tType,Ys: list @ A] :
      ( ( subset_eq_mset_impl @ A @ ( nil @ A ) @ Ys )
      = ( some @ $o
        @ ( Ys
         != ( nil @ A ) ) ) ) ).

% subset_eq_mset_impl.simps(1)
thf(fact_7068_map__filter__simps_I1_J,axiom,
    ! [A: $tType,B: $tType,F: B > ( option @ A ),X2: B,Xs2: list @ B] :
      ( ( map_filter @ B @ A @ F @ ( cons @ B @ X2 @ Xs2 ) )
      = ( case_option @ ( list @ A ) @ A @ ( map_filter @ B @ A @ F @ Xs2 )
        @ ^ [Y: A] : ( cons @ A @ Y @ ( map_filter @ B @ A @ F @ Xs2 ) )
        @ ( F @ X2 ) ) ) ).

% map_filter_simps(1)
thf(fact_7069_subset__eq__mset__impl_Osimps_I2_J,axiom,
    ! [A: $tType,X2: A,Xs2: list @ A,Ys: list @ A] :
      ( ( subset_eq_mset_impl @ A @ ( cons @ A @ X2 @ Xs2 ) @ Ys )
      = ( case_option @ ( option @ $o ) @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) @ ( none @ $o )
        @ ( product_case_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) @ ( option @ $o )
          @ ^ [Ys1: list @ A] :
              ( product_case_prod @ A @ ( list @ A ) @ ( option @ $o )
              @ ^ [X: A,Ys22: list @ A] : ( subset_eq_mset_impl @ A @ Xs2 @ ( append @ A @ Ys1 @ Ys22 ) ) ) )
        @ ( extract @ A
          @ ( ^ [Y5: A,Z4: A] : Y5 = Z4
            @ X2 )
          @ Ys ) ) ) ).

% subset_eq_mset_impl.simps(2)
thf(fact_7070_subset__eq__mset__impl_Opelims,axiom,
    ! [A: $tType,X2: list @ A,Xa: list @ A,Y2: option @ $o] :
      ( ( ( subset_eq_mset_impl @ A @ X2 @ Xa )
        = Y2 )
     => ( ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( subset751672762298770561pl_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X2 @ Xa ) )
       => ( ( ( X2
              = ( nil @ A ) )
           => ( ( Y2
                = ( some @ $o
                  @ ( Xa
                   != ( nil @ A ) ) ) )
             => ~ ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( subset751672762298770561pl_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Xa ) ) ) )
         => ~ ! [X3: A,Xs3: list @ A] :
                ( ( X2
                  = ( cons @ A @ X3 @ Xs3 ) )
               => ( ( Y2
                    = ( case_option @ ( option @ $o ) @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) @ ( none @ $o )
                      @ ( product_case_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) @ ( option @ $o )
                        @ ^ [Ys1: list @ A] :
                            ( product_case_prod @ A @ ( list @ A ) @ ( option @ $o )
                            @ ^ [Y: A,Ys22: list @ A] : ( subset_eq_mset_impl @ A @ Xs3 @ ( append @ A @ Ys1 @ Ys22 ) ) ) )
                      @ ( extract @ A
                        @ ( ^ [Y5: A,Z4: A] : Y5 = Z4
                          @ X3 )
                        @ Xa ) ) )
                 => ~ ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( subset751672762298770561pl_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X3 @ Xs3 ) @ Xa ) ) ) ) ) ) ) ).

% subset_eq_mset_impl.pelims
thf(fact_7071_quicksort_Osimps_I2_J,axiom,
    ! [A: $tType] :
      ( ( linorder @ A )
     => ! [X2: A,Xs2: list @ A] :
          ( ( linorder_quicksort @ A @ ( cons @ A @ X2 @ Xs2 ) )
          = ( append @ A
            @ ( linorder_quicksort @ A
              @ ( filter2 @ A
                @ ^ [Y: A] :
                    ~ ( ord_less_eq @ A @ X2 @ Y )
                @ Xs2 ) )
            @ ( append @ A @ ( cons @ A @ X2 @ ( nil @ A ) ) @ ( linorder_quicksort @ A @ ( filter2 @ A @ ( ord_less_eq @ A @ X2 ) @ Xs2 ) ) ) ) ) ) ).

% quicksort.simps(2)
thf(fact_7072_set__quicksort,axiom,
    ! [A: $tType] :
      ( ( linorder @ A )
     => ! [Xs2: list @ A] :
          ( ( set2 @ A @ ( linorder_quicksort @ A @ Xs2 ) )
          = ( set2 @ A @ Xs2 ) ) ) ).

% set_quicksort
thf(fact_7073_sort__quicksort,axiom,
    ! [A: $tType] :
      ( ( linorder @ A )
     => ( ( linorder_sort_key @ A @ A
          @ ^ [X: A] : X )
        = ( linorder_quicksort @ A ) ) ) ).

% sort_quicksort
thf(fact_7074_quicksort_Oelims,axiom,
    ! [A: $tType] :
      ( ( linorder @ A )
     => ! [X2: list @ A,Y2: list @ A] :
          ( ( ( linorder_quicksort @ A @ X2 )
            = Y2 )
         => ( ( ( X2
                = ( nil @ A ) )
             => ( Y2
               != ( nil @ A ) ) )
           => ~ ! [X3: A,Xs3: list @ A] :
                  ( ( X2
                    = ( cons @ A @ X3 @ Xs3 ) )
                 => ( Y2
                   != ( append @ A
                      @ ( linorder_quicksort @ A
                        @ ( filter2 @ A
                          @ ^ [Y: A] :
                              ~ ( ord_less_eq @ A @ X3 @ Y )
                          @ Xs3 ) )
                      @ ( append @ A @ ( cons @ A @ X3 @ ( nil @ A ) ) @ ( linorder_quicksort @ A @ ( filter2 @ A @ ( ord_less_eq @ A @ X3 ) @ Xs3 ) ) ) ) ) ) ) ) ) ).

% quicksort.elims
thf(fact_7075_quicksort_Opelims,axiom,
    ! [A: $tType] :
      ( ( linorder @ A )
     => ! [X2: list @ A,Y2: list @ A] :
          ( ( ( linorder_quicksort @ A @ X2 )
            = Y2 )
         => ( ( accp @ ( list @ A ) @ ( linord6200660962353139674rt_rel @ A ) @ X2 )
           => ( ( ( X2
                  = ( nil @ A ) )
               => ( ( Y2
                    = ( nil @ A ) )
                 => ~ ( accp @ ( list @ A ) @ ( linord6200660962353139674rt_rel @ A ) @ ( nil @ A ) ) ) )
             => ~ ! [X3: A,Xs3: list @ A] :
                    ( ( X2
                      = ( cons @ A @ X3 @ Xs3 ) )
                   => ( ( Y2
                        = ( append @ A
                          @ ( linorder_quicksort @ A
                            @ ( filter2 @ A
                              @ ^ [Y: A] :
                                  ~ ( ord_less_eq @ A @ X3 @ Y )
                              @ Xs3 ) )
                          @ ( append @ A @ ( cons @ A @ X3 @ ( nil @ A ) ) @ ( linorder_quicksort @ A @ ( filter2 @ A @ ( ord_less_eq @ A @ X3 ) @ Xs3 ) ) ) ) )
                     => ~ ( accp @ ( list @ A ) @ ( linord6200660962353139674rt_rel @ A ) @ ( cons @ A @ X3 @ Xs3 ) ) ) ) ) ) ) ) ).

% quicksort.pelims
thf(fact_7076_transpose__aux__filter__head,axiom,
    ! [A: $tType,Xss: list @ ( list @ A )] :
      ( ( concat @ A
        @ ( map @ ( list @ A ) @ ( list @ A )
          @ ( case_list @ ( list @ A ) @ A @ ( nil @ A )
            @ ^ [H: A,T3: list @ A] : ( cons @ A @ H @ ( nil @ A ) ) )
          @ Xss ) )
      = ( map @ ( list @ A ) @ A @ ( hd @ A )
        @ ( filter2 @ ( list @ A )
          @ ^ [Ys3: list @ A] :
              ( Ys3
             != ( nil @ A ) )
          @ Xss ) ) ) ).

% transpose_aux_filter_head
thf(fact_7077_hd__upt,axiom,
    ! [I: nat,J: nat] :
      ( ( ord_less @ nat @ I @ J )
     => ( ( hd @ nat @ ( upt @ I @ J ) )
        = I ) ) ).

% hd_upt
thf(fact_7078_in__hd__or__tl__conv,axiom,
    ! [A: $tType,L: list @ A,X2: A] :
      ( ( L
       != ( nil @ A ) )
     => ( ( ( X2
            = ( hd @ A @ L ) )
          | ( member @ A @ X2 @ ( set2 @ A @ ( tl @ A @ L ) ) ) )
        = ( member @ A @ X2 @ ( set2 @ A @ L ) ) ) ) ).

% in_hd_or_tl_conv
thf(fact_7079_not__hd__in__tl,axiom,
    ! [A: $tType,X2: A,Xs2: list @ A] :
      ( ( X2
       != ( hd @ A @ Xs2 ) )
     => ( ( member @ A @ X2 @ ( set2 @ A @ Xs2 ) )
       => ( member @ A @ X2 @ ( set2 @ A @ ( tl @ A @ Xs2 ) ) ) ) ) ).

% not_hd_in_tl
thf(fact_7080_hd__def,axiom,
    ! [A: $tType] :
      ( ( hd @ A )
      = ( case_list @ A @ A @ ( undefined @ A )
        @ ^ [X214: A,X224: list @ A] : X214 ) ) ).

% hd_def
thf(fact_7081_list_Oset__sel_I1_J,axiom,
    ! [A: $tType,A2: list @ A] :
      ( ( A2
       != ( nil @ A ) )
     => ( member @ A @ ( hd @ A @ A2 ) @ ( set2 @ A @ A2 ) ) ) ).

% list.set_sel(1)
thf(fact_7082_hd__in__set,axiom,
    ! [A: $tType,Xs2: list @ A] :
      ( ( Xs2
       != ( nil @ A ) )
     => ( member @ A @ ( hd @ A @ Xs2 ) @ ( set2 @ A @ Xs2 ) ) ) ).

% hd_in_set
thf(fact_7083_hd__conv__nth,axiom,
    ! [A: $tType,Xs2: list @ A] :
      ( ( Xs2
       != ( nil @ A ) )
     => ( ( hd @ A @ Xs2 )
        = ( nth @ A @ Xs2 @ ( zero_zero @ nat ) ) ) ) ).

% hd_conv_nth
thf(fact_7084_slice__head,axiom,
    ! [A: $tType,From: nat,To: nat,Xs2: list @ A] :
      ( ( ord_less @ nat @ From @ To )
     => ( ( ord_less_eq @ nat @ To @ ( size_size @ ( list @ A ) @ Xs2 ) )
       => ( ( hd @ A @ ( slice @ A @ From @ To @ Xs2 ) )
          = ( nth @ A @ Xs2 @ From ) ) ) ) ).

% slice_head
thf(fact_7085_Nitpick_Osize__list__simp_I1_J,axiom,
    ! [A: $tType] :
      ( ( size_list @ A )
      = ( ^ [F6: A > nat,Xs: list @ A] :
            ( if @ nat
            @ ( Xs
              = ( nil @ A ) )
            @ ( zero_zero @ nat )
            @ ( suc @ ( plus_plus @ nat @ ( F6 @ ( hd @ A @ Xs ) ) @ ( size_list @ A @ F6 @ ( tl @ A @ Xs ) ) ) ) ) ) ) ).

% Nitpick.size_list_simp(1)
thf(fact_7086_splice_Opinduct,axiom,
    ! [A: $tType,A0: list @ A,A12: list @ A,P: ( list @ A ) > ( list @ A ) > $o] :
      ( ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( splice_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ A0 @ A12 ) )
     => ( ! [Ys4: list @ A] :
            ( ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( splice_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Ys4 ) )
           => ( P @ ( nil @ A ) @ Ys4 ) )
       => ( ! [X3: A,Xs3: list @ A,Ys4: list @ A] :
              ( ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( splice_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X3 @ Xs3 ) @ Ys4 ) )
             => ( ( P @ Ys4 @ Xs3 )
               => ( P @ ( cons @ A @ X3 @ Xs3 ) @ Ys4 ) ) )
         => ( P @ A0 @ A12 ) ) ) ) ).

% splice.pinduct
thf(fact_7087_size__list__append,axiom,
    ! [A: $tType,F: A > nat,Xs2: list @ A,Ys: list @ A] :
      ( ( size_list @ A @ F @ ( append @ A @ Xs2 @ Ys ) )
      = ( plus_plus @ nat @ ( size_list @ A @ F @ Xs2 ) @ ( size_list @ A @ F @ Ys ) ) ) ).

% size_list_append
thf(fact_7088_size__list__estimation,axiom,
    ! [A: $tType,X2: A,Xs2: list @ A,Y2: nat,F: A > nat] :
      ( ( member @ A @ X2 @ ( set2 @ A @ Xs2 ) )
     => ( ( ord_less @ nat @ Y2 @ ( F @ X2 ) )
       => ( ord_less @ nat @ Y2 @ ( size_list @ A @ F @ Xs2 ) ) ) ) ).

% size_list_estimation
thf(fact_7089_size__list__pointwise,axiom,
    ! [A: $tType,Xs2: list @ A,F: A > nat,G: A > nat] :
      ( ! [X3: A] :
          ( ( member @ A @ X3 @ ( set2 @ A @ Xs2 ) )
         => ( ord_less_eq @ nat @ ( F @ X3 ) @ ( G @ X3 ) ) )
     => ( ord_less_eq @ nat @ ( size_list @ A @ F @ Xs2 ) @ ( size_list @ A @ G @ Xs2 ) ) ) ).

% size_list_pointwise
thf(fact_7090_size__list__estimation_H,axiom,
    ! [A: $tType,X2: A,Xs2: list @ A,Y2: nat,F: A > nat] :
      ( ( member @ A @ X2 @ ( set2 @ A @ Xs2 ) )
     => ( ( ord_less_eq @ nat @ Y2 @ ( F @ X2 ) )
       => ( ord_less_eq @ nat @ Y2 @ ( size_list @ A @ F @ Xs2 ) ) ) ) ).

% size_list_estimation'
thf(fact_7091_list_Osize__gen_I2_J,axiom,
    ! [A: $tType,X2: A > nat,X212: A,X222: list @ A] :
      ( ( size_list @ A @ X2 @ ( cons @ A @ X212 @ X222 ) )
      = ( plus_plus @ nat @ ( plus_plus @ nat @ ( X2 @ X212 ) @ ( size_list @ A @ X2 @ X222 ) ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ).

% list.size_gen(2)
thf(fact_7092_VEBT_Osize_I3_J,axiom,
    ! [X11: option @ ( product_prod @ nat @ 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_7093_VEBT_Osize__gen_I1_J,axiom,
    ! [X11: option @ ( product_prod @ nat @ 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_7094_splice_Opelims,axiom,
    ! [A: $tType,X2: list @ A,Xa: list @ A,Y2: list @ A] :
      ( ( ( splice @ A @ X2 @ Xa )
        = Y2 )
     => ( ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( splice_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X2 @ Xa ) )
       => ( ( ( X2
              = ( nil @ A ) )
           => ( ( Y2 = Xa )
             => ~ ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( splice_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Xa ) ) ) )
         => ~ ! [X3: A,Xs3: list @ A] :
                ( ( X2
                  = ( cons @ A @ X3 @ Xs3 ) )
               => ( ( Y2
                    = ( cons @ A @ X3 @ ( splice @ A @ Xa @ Xs3 ) ) )
                 => ~ ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( splice_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X3 @ Xs3 ) @ Xa ) ) ) ) ) ) ) ).

% splice.pelims
thf(fact_7095_length__splice,axiom,
    ! [A: $tType,Xs2: list @ A,Ys: list @ A] :
      ( ( size_size @ ( list @ A ) @ ( splice @ A @ Xs2 @ Ys ) )
      = ( plus_plus @ nat @ ( size_size @ ( list @ A ) @ Xs2 ) @ ( size_size @ ( list @ A ) @ Ys ) ) ) ).

% length_splice
thf(fact_7096_splice__replicate,axiom,
    ! [A: $tType,M: nat,X2: A,N2: nat] :
      ( ( splice @ A @ ( replicate @ A @ M @ X2 ) @ ( replicate @ A @ N2 @ X2 ) )
      = ( replicate @ A @ ( plus_plus @ nat @ M @ N2 ) @ X2 ) ) ).

% splice_replicate
thf(fact_7097_splice_Opsimps_I2_J,axiom,
    ! [A: $tType,X2: A,Xs2: list @ A,Ys: list @ A] :
      ( ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( splice_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X2 @ Xs2 ) @ Ys ) )
     => ( ( splice @ A @ ( cons @ A @ X2 @ Xs2 ) @ Ys )
        = ( cons @ A @ X2 @ ( splice @ A @ Ys @ Xs2 ) ) ) ) ).

% splice.psimps(2)
thf(fact_7098_splice_Opsimps_I1_J,axiom,
    ! [A: $tType,Ys: list @ A] :
      ( ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( splice_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Ys ) )
     => ( ( splice @ A @ ( nil @ A ) @ Ys )
        = Ys ) ) ).

% splice.psimps(1)
thf(fact_7099_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_7100_image__Collect__subsetI,axiom,
    ! [A: $tType,B: $tType,P: A > $o,F: A > B,B6: set @ B] :
      ( ! [X3: A] :
          ( ( P @ X3 )
         => ( member @ B @ ( F @ X3 ) @ B6 ) )
     => ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F @ ( collect @ A @ P ) ) @ B6 ) ) ).

% image_Collect_subsetI
thf(fact_7101_ssubst__Pair__rhs,axiom,
    ! [B: $tType,A: $tType,R4: A,S4: B,R2: set @ ( product_prod @ A @ B ),S5: B] :
      ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R4 @ S4 ) @ R2 )
     => ( ( S5 = S4 )
       => ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R4 @ S5 ) @ R2 ) ) ) ).

% ssubst_Pair_rhs
thf(fact_7102_prop__restrict,axiom,
    ! [A: $tType,X2: A,Z9: set @ A,X8: set @ A,P: A > $o] :
      ( ( member @ A @ X2 @ Z9 )
     => ( ( ord_less_eq @ ( set @ A ) @ Z9
          @ ( collect @ A
            @ ^ [X: A] :
                ( ( member @ A @ X @ X8 )
                & ( P @ X ) ) ) )
       => ( P @ X2 ) ) ) ).

% prop_restrict
thf(fact_7103_Collect__restrict,axiom,
    ! [A: $tType,X8: set @ A,P: A > $o] :
      ( ord_less_eq @ ( set @ A )
      @ ( collect @ A
        @ ^ [X: A] :
            ( ( member @ A @ X @ X8 )
            & ( P @ X ) ) )
      @ X8 ) ).

% Collect_restrict
thf(fact_7104_FI__QUERY__def,axiom,
    ( fI_QUERY
    = ( ^ [P3: assn,Q4: assn,F9: assn] : ( entails @ P3 @ ( times_times @ assn @ Q4 @ F9 ) ) ) ) ).

% FI_QUERY_def
thf(fact_7105_frame__inference__init,axiom,
    ! [P: assn,Q: assn,F5: assn] :
      ( ( fI_QUERY @ P @ Q @ F5 )
     => ( entails @ P @ ( times_times @ assn @ Q @ F5 ) ) ) ).

% frame_inference_init
thf(fact_7106_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_7107_subset__emptyI,axiom,
    ! [A: $tType,A3: set @ A] :
      ( ! [X3: A] :
          ~ ( member @ A @ X3 @ A3 )
     => ( ord_less_eq @ ( set @ A ) @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ).

% subset_emptyI
thf(fact_7108_insert__subsetI,axiom,
    ! [A: $tType,X2: A,A3: set @ A,X8: set @ A] :
      ( ( member @ A @ X2 @ A3 )
     => ( ( ord_less_eq @ ( set @ A ) @ X8 @ A3 )
       => ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ X2 @ X8 ) @ A3 ) ) ) ).

% insert_subsetI
thf(fact_7109_smod__int__range,axiom,
    ! [B2: int,A2: int] :
      ( ( B2
       != ( zero_zero @ int ) )
     => ( member @ int @ ( signed6721504322012087516modulo @ int @ A2 @ B2 ) @ ( set_or1337092689740270186AtMost @ int @ ( plus_plus @ int @ ( uminus_uminus @ int @ ( abs_abs @ int @ B2 ) ) @ ( one_one @ int ) ) @ ( minus_minus @ int @ ( abs_abs @ int @ B2 ) @ ( one_one @ int ) ) ) ) ) ).

% smod_int_range
thf(fact_7110_mod__word__minus__1__minus__numeral,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [B2: num] :
          ( ( modulo_modulo @ ( word @ A ) @ ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ B2 ) ) )
          = ( ring_1_of_int @ ( word @ A ) @ ( modulo_modulo @ int @ ( minus_minus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) @ ( one_one @ int ) ) @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ B2 ) ) ) ) ) ) ) ).

% mod_word_minus_1_minus_numeral
thf(fact_7111_smod__int__numeral__numeral,axiom,
    ! [M: num,N2: num] :
      ( ( signed6721504322012087516modulo @ int @ ( numeral_numeral @ int @ M ) @ ( numeral_numeral @ int @ N2 ) )
      = ( modulo_modulo @ int @ ( numeral_numeral @ int @ M ) @ ( numeral_numeral @ int @ N2 ) ) ) ).

% smod_int_numeral_numeral
thf(fact_7112_drop__bit__word__beyond,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [N2: nat,W: word @ A] :
          ( ( ord_less_eq @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ N2 )
         => ( ( bit_se4197421643247451524op_bit @ ( word @ A ) @ N2 @ W )
            = ( zero_zero @ ( word @ A ) ) ) ) ) ).

% drop_bit_word_beyond
thf(fact_7113_push__bit__word__beyond,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [N2: nat,W: word @ A] :
          ( ( ord_less_eq @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ N2 )
         => ( ( bit_se4730199178511100633sh_bit @ ( word @ A ) @ N2 @ W )
            = ( zero_zero @ ( word @ A ) ) ) ) ) ).

% push_bit_word_beyond
thf(fact_7114_uint__bintrunc,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [Bin: num] :
          ( ( semiring_1_unsigned @ A @ int @ ( numeral_numeral @ ( word @ A ) @ Bin ) )
          = ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( numeral_numeral @ int @ Bin ) ) ) ) ).

% uint_bintrunc
thf(fact_7115_signed__1,axiom,
    ! [B: $tType,A: $tType] :
      ( ( ( ring_1 @ A )
        & ( type_len @ B ) )
     => ( ( ( ( type_len0_len_of @ B @ ( type2 @ B ) )
            = ( one_one @ nat ) )
         => ( ( ring_1_signed @ B @ A @ ( one_one @ ( word @ B ) ) )
            = ( uminus_uminus @ A @ ( one_one @ A ) ) ) )
        & ( ( ( type_len0_len_of @ B @ ( type2 @ B ) )
           != ( one_one @ nat ) )
         => ( ( ring_1_signed @ B @ A @ ( one_one @ ( word @ B ) ) )
            = ( one_one @ A ) ) ) ) ) ).

% signed_1
thf(fact_7116_word__exp__length__eq__0,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ( ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
        = ( zero_zero @ ( word @ A ) ) ) ) ).

% word_exp_length_eq_0
thf(fact_7117_less__eq__word__numeral__numeral,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [A2: num,B2: num] :
          ( ( ord_less_eq @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ A2 ) @ ( numeral_numeral @ ( word @ A ) @ B2 ) )
          = ( ord_less_eq @ int @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( numeral_numeral @ int @ A2 ) ) @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( numeral_numeral @ int @ B2 ) ) ) ) ) ).

% less_eq_word_numeral_numeral
thf(fact_7118_less__word__numeral__numeral,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [A2: num,B2: num] :
          ( ( ord_less @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ A2 ) @ ( numeral_numeral @ ( word @ A ) @ B2 ) )
          = ( ord_less @ int @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( numeral_numeral @ int @ A2 ) ) @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( numeral_numeral @ int @ B2 ) ) ) ) ) ).

% less_word_numeral_numeral
thf(fact_7119_unat__bintrunc,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [Bin: num] :
          ( ( semiring_1_unsigned @ A @ nat @ ( numeral_numeral @ ( word @ A ) @ Bin ) )
          = ( nat2 @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( numeral_numeral @ int @ Bin ) ) ) ) ) ).

% unat_bintrunc
thf(fact_7120_bit__numeral__word__iff,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [W: num,N2: nat] :
          ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ W ) @ N2 )
          = ( ( ord_less @ nat @ N2 @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
            & ( bit_se5641148757651400278ts_bit @ int @ ( numeral_numeral @ int @ W ) @ N2 ) ) ) ) ).

% bit_numeral_word_iff
thf(fact_7121_ucast__bintr,axiom,
    ! [B: $tType,A: $tType] :
      ( ( ( type_len @ A )
        & ( type_len @ B ) )
     => ! [W: num] :
          ( ( semiring_1_unsigned @ A @ ( word @ B ) @ ( numeral_numeral @ ( word @ A ) @ W ) )
          = ( ring_1_of_int @ ( word @ B ) @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( numeral_numeral @ int @ W ) ) ) ) ) ).

% ucast_bintr
thf(fact_7122_unsigned__numeral,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ( type_len @ B )
        & ( semiring_1 @ A ) )
     => ! [N2: num] :
          ( ( semiring_1_unsigned @ B @ A @ ( numeral_numeral @ ( word @ B ) @ N2 ) )
          = ( semiring_1_of_nat @ A @ ( bit_se2584673776208193580ke_bit @ nat @ ( type_len0_len_of @ B @ ( type2 @ B ) ) @ ( numeral_numeral @ nat @ N2 ) ) ) ) ) ).

% unsigned_numeral
thf(fact_7123_unat__lt2p,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [X2: word @ A] : ( ord_less @ nat @ ( semiring_1_unsigned @ A @ nat @ X2 ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ).

% unat_lt2p
thf(fact_7124_uint__lt2p,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [X2: word @ A] : ( ord_less @ int @ ( semiring_1_unsigned @ A @ int @ X2 ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ).

% uint_lt2p
thf(fact_7125_uint__bounded,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [W: word @ A] : ( ord_less @ int @ ( semiring_1_unsigned @ A @ int @ W ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ).

% uint_bounded
thf(fact_7126_exp__eq__zero__iff,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [N2: nat] :
          ( ( ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N2 )
            = ( zero_zero @ ( word @ A ) ) )
          = ( ord_less_eq @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ N2 ) ) ) ).

% exp_eq_zero_iff
thf(fact_7127_of__nat__2p,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ( ( semiring_1_of_nat @ ( word @ A ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
        = ( zero_zero @ ( word @ A ) ) ) ) ).

% of_nat_2p
thf(fact_7128_signed__take__bit__word__Suc__numeral,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [N2: nat,K: num] :
          ( ( bit_ri4674362597316999326ke_bit @ ( word @ A ) @ ( suc @ N2 ) @ ( numeral_numeral @ ( word @ A ) @ K ) )
          = ( ring_1_of_int @ ( word @ A ) @ ( bit_ri4674362597316999326ke_bit @ int @ ( suc @ N2 ) @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( numeral_numeral @ int @ K ) ) ) ) ) ) ).

% signed_take_bit_word_Suc_numeral
thf(fact_7129_signed__take__bit__word__numeral,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [N2: num,K: num] :
          ( ( bit_ri4674362597316999326ke_bit @ ( word @ A ) @ ( numeral_numeral @ nat @ N2 ) @ ( numeral_numeral @ ( word @ A ) @ K ) )
          = ( ring_1_of_int @ ( word @ A ) @ ( bit_ri4674362597316999326ke_bit @ int @ ( numeral_numeral @ nat @ N2 ) @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( numeral_numeral @ int @ K ) ) ) ) ) ) ).

% signed_take_bit_word_numeral
thf(fact_7130_sint__sbintrunc,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [Bin: num] :
          ( ( ring_1_signed @ A @ int @ ( numeral_numeral @ ( word @ A ) @ Bin ) )
          = ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) @ ( numeral_numeral @ int @ Bin ) ) ) ) ).

% sint_sbintrunc
thf(fact_7131_uint__bintrunc__neg,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [Bin: num] :
          ( ( semiring_1_unsigned @ A @ int @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ Bin ) ) )
          = ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ Bin ) ) ) ) ) ).

% uint_bintrunc_neg
thf(fact_7132_div__word__numeral__numeral,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [A2: num,B2: num] :
          ( ( divide_divide @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ A2 ) @ ( numeral_numeral @ ( word @ A ) @ B2 ) )
          = ( ring_1_of_int @ ( word @ A ) @ ( divide_divide @ int @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( numeral_numeral @ int @ A2 ) ) @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( numeral_numeral @ int @ B2 ) ) ) ) ) ) ).

% div_word_numeral_numeral
thf(fact_7133_mod__word__numeral__numeral,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [A2: num,B2: num] :
          ( ( modulo_modulo @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ A2 ) @ ( numeral_numeral @ ( word @ A ) @ B2 ) )
          = ( ring_1_of_int @ ( word @ A ) @ ( modulo_modulo @ int @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( numeral_numeral @ int @ A2 ) ) @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( numeral_numeral @ int @ B2 ) ) ) ) ) ) ).

% mod_word_numeral_numeral
thf(fact_7134_signed__numeral,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ( type_len @ B )
        & ( ring_1 @ A ) )
     => ! [N2: num] :
          ( ( ring_1_signed @ B @ A @ ( numeral_numeral @ ( word @ B ) @ N2 ) )
          = ( ring_1_of_int @ A @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ B @ ( type2 @ B ) ) @ ( one_one @ nat ) ) @ ( numeral_numeral @ int @ N2 ) ) ) ) ) ).

% signed_numeral
thf(fact_7135_less__eq__word__minus__numeral__numeral,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [A2: num,B2: num] :
          ( ( ord_less_eq @ ( word @ A ) @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ A2 ) ) @ ( numeral_numeral @ ( word @ A ) @ B2 ) )
          = ( ord_less_eq @ int @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ A2 ) ) ) @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( numeral_numeral @ int @ B2 ) ) ) ) ) ).

% less_eq_word_minus_numeral_numeral
thf(fact_7136_less__eq__word__numeral__minus__numeral,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [A2: num,B2: num] :
          ( ( ord_less_eq @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ A2 ) @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ B2 ) ) )
          = ( ord_less_eq @ int @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( numeral_numeral @ int @ A2 ) ) @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ B2 ) ) ) ) ) ) ).

% less_eq_word_numeral_minus_numeral
thf(fact_7137_less__eq__word__minus__numeral__minus__numeral,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [A2: num,B2: num] :
          ( ( ord_less_eq @ ( word @ A ) @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ A2 ) ) @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ B2 ) ) )
          = ( ord_less_eq @ int @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ A2 ) ) ) @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ B2 ) ) ) ) ) ) ).

% less_eq_word_minus_numeral_minus_numeral
thf(fact_7138_less__word__minus__numeral__minus__numeral,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [A2: num,B2: num] :
          ( ( ord_less @ ( word @ A ) @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ A2 ) ) @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ B2 ) ) )
          = ( ord_less @ int @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ A2 ) ) ) @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ B2 ) ) ) ) ) ) ).

% less_word_minus_numeral_minus_numeral
thf(fact_7139_less__word__numeral__minus__numeral,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [A2: num,B2: num] :
          ( ( ord_less @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ A2 ) @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ B2 ) ) )
          = ( ord_less @ int @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( numeral_numeral @ int @ A2 ) ) @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ B2 ) ) ) ) ) ) ).

% less_word_numeral_minus_numeral
thf(fact_7140_less__word__minus__numeral__numeral,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [A2: num,B2: num] :
          ( ( ord_less @ ( word @ A ) @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ A2 ) ) @ ( numeral_numeral @ ( word @ A ) @ B2 ) )
          = ( ord_less @ int @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ A2 ) ) ) @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( numeral_numeral @ int @ B2 ) ) ) ) ) ).

% less_word_minus_numeral_numeral
thf(fact_7141_unat__bintrunc__neg,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [Bin: num] :
          ( ( semiring_1_unsigned @ A @ nat @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ Bin ) ) )
          = ( nat2 @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ Bin ) ) ) ) ) ) ).

% unat_bintrunc_neg
thf(fact_7142_bit__neg__numeral__word__iff,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [W: num,N2: nat] :
          ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ W ) ) @ N2 )
          = ( ( ord_less @ nat @ N2 @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
            & ( bit_se5641148757651400278ts_bit @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ W ) ) @ N2 ) ) ) ) ).

% bit_neg_numeral_word_iff
thf(fact_7143_drop__bit__word__Suc__numeral,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [N2: nat,K: num] :
          ( ( bit_se4197421643247451524op_bit @ ( word @ A ) @ ( suc @ N2 ) @ ( numeral_numeral @ ( word @ A ) @ K ) )
          = ( ring_1_of_int @ ( word @ A ) @ ( bit_se4197421643247451524op_bit @ int @ ( suc @ N2 ) @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( numeral_numeral @ int @ K ) ) ) ) ) ) ).

% drop_bit_word_Suc_numeral
thf(fact_7144_drop__bit__word__numeral,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [N2: num,K: num] :
          ( ( bit_se4197421643247451524op_bit @ ( word @ A ) @ ( numeral_numeral @ nat @ N2 ) @ ( numeral_numeral @ ( word @ A ) @ K ) )
          = ( ring_1_of_int @ ( word @ A ) @ ( bit_se4197421643247451524op_bit @ int @ ( numeral_numeral @ nat @ N2 ) @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( numeral_numeral @ int @ K ) ) ) ) ) ) ).

% drop_bit_word_numeral
thf(fact_7145_unat__power__lower,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [N2: nat] :
          ( ( ord_less @ nat @ N2 @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
         => ( ( semiring_1_unsigned @ A @ nat @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N2 ) )
            = ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ) ) ).

% unat_power_lower
thf(fact_7146_unsigned__neg__numeral,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ( type_len @ B )
        & ( semiring_1 @ A ) )
     => ! [N2: num] :
          ( ( semiring_1_unsigned @ B @ A @ ( uminus_uminus @ ( word @ B ) @ ( numeral_numeral @ ( word @ B ) @ N2 ) ) )
          = ( semiring_1_of_nat @ A @ ( nat2 @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ B @ ( type2 @ B ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N2 ) ) ) ) ) ) ) ).

% unsigned_neg_numeral
thf(fact_7147_signed__take__bit__word__Suc__minus__numeral,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [N2: nat,K: num] :
          ( ( bit_ri4674362597316999326ke_bit @ ( word @ A ) @ ( suc @ N2 ) @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ K ) ) )
          = ( ring_1_of_int @ ( word @ A ) @ ( bit_ri4674362597316999326ke_bit @ int @ ( suc @ N2 ) @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K ) ) ) ) ) ) ) ).

% signed_take_bit_word_Suc_minus_numeral
thf(fact_7148_signed__take__bit__word__minus__numeral,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [N2: num,K: num] :
          ( ( bit_ri4674362597316999326ke_bit @ ( word @ A ) @ ( numeral_numeral @ nat @ N2 ) @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ K ) ) )
          = ( ring_1_of_int @ ( word @ A ) @ ( bit_ri4674362597316999326ke_bit @ int @ ( numeral_numeral @ nat @ N2 ) @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K ) ) ) ) ) ) ) ).

% signed_take_bit_word_minus_numeral
thf(fact_7149_sint__sbintrunc__neg,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [Bin: num] :
          ( ( ring_1_signed @ A @ int @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ Bin ) ) )
          = ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ Bin ) ) ) ) ) ).

% sint_sbintrunc_neg
thf(fact_7150_scast__sbintr,axiom,
    ! [B: $tType,A: $tType] :
      ( ( ( type_len @ A )
        & ( type_len @ B ) )
     => ! [W: num] :
          ( ( ring_1_signed @ A @ ( word @ B ) @ ( numeral_numeral @ ( word @ A ) @ W ) )
          = ( ring_1_of_int @ ( word @ B ) @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) @ ( numeral_numeral @ int @ W ) ) ) ) ) ).

% scast_sbintr
thf(fact_7151_drop__bit__numeral__bit0__1,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [K: num] :
          ( ( bit_se4197421643247451524op_bit @ ( word @ A ) @ ( suc @ ( zero_zero @ nat ) ) @ ( numeral_numeral @ ( word @ A ) @ K ) )
          = ( ring_1_of_int @ ( word @ A ) @ ( bit_se4197421643247451524op_bit @ int @ ( suc @ ( zero_zero @ nat ) ) @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( numeral_numeral @ int @ K ) ) ) ) ) ) ).

% drop_bit_numeral_bit0_1
thf(fact_7152_div__word__minus__numeral__numeral,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [A2: num,B2: num] :
          ( ( divide_divide @ ( word @ A ) @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ A2 ) ) @ ( numeral_numeral @ ( word @ A ) @ B2 ) )
          = ( ring_1_of_int @ ( word @ A ) @ ( divide_divide @ int @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ A2 ) ) ) @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( numeral_numeral @ int @ B2 ) ) ) ) ) ) ).

% div_word_minus_numeral_numeral
thf(fact_7153_div__word__numeral__minus__numeral,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [A2: num,B2: num] :
          ( ( divide_divide @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ A2 ) @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ B2 ) ) )
          = ( ring_1_of_int @ ( word @ A ) @ ( divide_divide @ int @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( numeral_numeral @ int @ A2 ) ) @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ B2 ) ) ) ) ) ) ) ).

% div_word_numeral_minus_numeral
thf(fact_7154_div__word__minus__numeral__minus__numeral,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [A2: num,B2: num] :
          ( ( divide_divide @ ( word @ A ) @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ A2 ) ) @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ B2 ) ) )
          = ( ring_1_of_int @ ( word @ A ) @ ( divide_divide @ int @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ A2 ) ) ) @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ B2 ) ) ) ) ) ) ) ).

% div_word_minus_numeral_minus_numeral
thf(fact_7155_mod__word__minus__numeral__minus__numeral,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [A2: num,B2: num] :
          ( ( modulo_modulo @ ( word @ A ) @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ A2 ) ) @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ B2 ) ) )
          = ( ring_1_of_int @ ( word @ A ) @ ( modulo_modulo @ int @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ A2 ) ) ) @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ B2 ) ) ) ) ) ) ) ).

% mod_word_minus_numeral_minus_numeral
thf(fact_7156_mod__word__numeral__minus__numeral,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [A2: num,B2: num] :
          ( ( modulo_modulo @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ A2 ) @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ B2 ) ) )
          = ( ring_1_of_int @ ( word @ A ) @ ( modulo_modulo @ int @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( numeral_numeral @ int @ A2 ) ) @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ B2 ) ) ) ) ) ) ) ).

% mod_word_numeral_minus_numeral
thf(fact_7157_mod__word__minus__numeral__numeral,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [A2: num,B2: num] :
          ( ( modulo_modulo @ ( word @ A ) @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ A2 ) ) @ ( numeral_numeral @ ( word @ A ) @ B2 ) )
          = ( ring_1_of_int @ ( word @ A ) @ ( modulo_modulo @ int @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ A2 ) ) ) @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( numeral_numeral @ int @ B2 ) ) ) ) ) ) ).

% mod_word_minus_numeral_numeral
thf(fact_7158_word__less__sub__le,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [N2: nat,X2: word @ A] :
          ( ( ord_less @ nat @ N2 @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
         => ( ( ord_less_eq @ ( word @ A ) @ X2 @ ( minus_minus @ ( word @ A ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N2 ) @ ( one_one @ ( word @ A ) ) ) )
            = ( ord_less @ ( word @ A ) @ X2 @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N2 ) ) ) ) ) ).

% word_less_sub_le
thf(fact_7159_signed__neg__numeral,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ( type_len @ B )
        & ( ring_1 @ A ) )
     => ! [N2: num] :
          ( ( ring_1_signed @ B @ A @ ( uminus_uminus @ ( word @ B ) @ ( numeral_numeral @ ( word @ B ) @ N2 ) ) )
          = ( ring_1_of_int @ A @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ B @ ( type2 @ B ) ) @ ( one_one @ nat ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N2 ) ) ) ) ) ) ).

% signed_neg_numeral
thf(fact_7160_drop__bit__word__Suc__minus__numeral,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [N2: nat,K: num] :
          ( ( bit_se4197421643247451524op_bit @ ( word @ A ) @ ( suc @ N2 ) @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ K ) ) )
          = ( ring_1_of_int @ ( word @ A ) @ ( bit_se4197421643247451524op_bit @ int @ ( suc @ N2 ) @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K ) ) ) ) ) ) ) ).

% drop_bit_word_Suc_minus_numeral
thf(fact_7161_drop__bit__word__minus__numeral,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [N2: num,K: num] :
          ( ( bit_se4197421643247451524op_bit @ ( word @ A ) @ ( numeral_numeral @ nat @ N2 ) @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ K ) ) )
          = ( ring_1_of_int @ ( word @ A ) @ ( bit_se4197421643247451524op_bit @ int @ ( numeral_numeral @ nat @ N2 ) @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K ) ) ) ) ) ) ) ).

% drop_bit_word_minus_numeral
thf(fact_7162_less__word__numeral__minus__1,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [A2: num] :
          ( ( ord_less @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ A2 ) @ ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) )
          = ( ord_less @ int @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( numeral_numeral @ int @ A2 ) ) @ ( minus_minus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) @ ( one_one @ int ) ) ) ) ) ).

% less_word_numeral_minus_1
thf(fact_7163_less__word__minus__numeral__minus__1,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [A2: num] :
          ( ( ord_less @ ( word @ A ) @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ A2 ) ) @ ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) )
          = ( ord_less @ int @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ A2 ) ) ) @ ( minus_minus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) @ ( one_one @ int ) ) ) ) ) ).

% less_word_minus_numeral_minus_1
thf(fact_7164_div__word__minus__1__numeral,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [B2: num] :
          ( ( divide_divide @ ( word @ A ) @ ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) @ ( numeral_numeral @ ( word @ A ) @ B2 ) )
          = ( ring_1_of_int @ ( word @ A ) @ ( divide_divide @ int @ ( minus_minus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) @ ( one_one @ int ) ) @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( numeral_numeral @ int @ B2 ) ) ) ) ) ) ).

% div_word_minus_1_numeral
thf(fact_7165_mod__word__minus__1__numeral,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [B2: num] :
          ( ( modulo_modulo @ ( word @ A ) @ ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) @ ( numeral_numeral @ ( word @ A ) @ B2 ) )
          = ( ring_1_of_int @ ( word @ A ) @ ( modulo_modulo @ int @ ( minus_minus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) @ ( one_one @ int ) ) @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( numeral_numeral @ int @ B2 ) ) ) ) ) ) ).

% mod_word_minus_1_numeral
thf(fact_7166_div__word__minus__1__minus__numeral,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [B2: num] :
          ( ( divide_divide @ ( word @ A ) @ ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ B2 ) ) )
          = ( ring_1_of_int @ ( word @ A ) @ ( divide_divide @ int @ ( minus_minus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) @ ( one_one @ int ) ) @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ B2 ) ) ) ) ) ) ) ).

% div_word_minus_1_minus_numeral
thf(fact_7167_word__split__bin_H,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( ( type_len @ A )
        & ( type_len @ B )
        & ( type_len @ C ) )
     => ( ( word_split @ A @ B @ C )
        = ( ^ [W2: word @ A] : ( product_Pair @ ( word @ B ) @ ( word @ C ) @ ( semiring_1_unsigned @ A @ ( word @ B ) @ ( bit_se4197421643247451524op_bit @ ( word @ A ) @ ( type_len0_len_of @ C @ ( type2 @ C ) ) @ W2 ) ) @ ( semiring_1_unsigned @ A @ ( word @ C ) @ W2 ) ) ) ) ) ).

% word_split_bin'
thf(fact_7168_bit__word__cat__iff,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( ( ( type_len @ B )
        & ( type_len @ A )
        & ( type_len @ C ) )
     => ! [V: word @ A,W: word @ B,N2: nat] :
          ( ( bit_se5641148757651400278ts_bit @ ( word @ C ) @ ( word_cat @ A @ B @ C @ V @ W ) @ N2 )
          = ( ( ord_less @ nat @ N2 @ ( type_len0_len_of @ C @ ( type2 @ C ) ) )
            & ( ( ord_less @ nat @ N2 @ ( type_len0_len_of @ B @ ( type2 @ B ) ) )
             => ( bit_se5641148757651400278ts_bit @ ( word @ B ) @ W @ N2 ) )
            & ( ~ ( ord_less @ nat @ N2 @ ( type_len0_len_of @ B @ ( type2 @ B ) ) )
             => ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ V @ ( minus_minus @ nat @ N2 @ ( type_len0_len_of @ B @ ( type2 @ B ) ) ) ) ) ) ) ) ).

% bit_word_cat_iff
thf(fact_7169_word__cat__inj,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( ( ( type_len @ B )
        & ( type_len @ A )
        & ( type_len @ C ) )
     => ! [A2: word @ A,B2: word @ B,C2: word @ A,D2: word @ B] :
          ( ( ord_less_eq @ nat @ ( plus_plus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( type_len0_len_of @ B @ ( type2 @ B ) ) ) @ ( type_len0_len_of @ C @ ( type2 @ C ) ) )
         => ( ( ( word_cat @ A @ B @ C @ A2 @ B2 )
              = ( word_cat @ A @ B @ C @ C2 @ D2 ) )
            = ( ( A2 = C2 )
              & ( B2 = D2 ) ) ) ) ) ).

% word_cat_inj
thf(fact_7170_word__of__int__bin__cat__eq__iff,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( ( type_len @ A )
        & ( type_len @ B )
        & ( type_len @ C ) )
     => ! [B2: word @ B,A2: word @ A,D2: word @ B,C2: word @ A] :
          ( ( ord_less_eq @ nat @ ( plus_plus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( type_len0_len_of @ B @ ( type2 @ B ) ) ) @ ( type_len0_len_of @ C @ ( type2 @ C ) ) )
         => ( ( ( ring_1_of_int @ ( word @ C ) @ ( bit_concat_bit @ ( type_len0_len_of @ B @ ( type2 @ B ) ) @ ( semiring_1_unsigned @ B @ int @ B2 ) @ ( semiring_1_unsigned @ A @ int @ A2 ) ) )
              = ( ring_1_of_int @ ( word @ C ) @ ( bit_concat_bit @ ( type_len0_len_of @ B @ ( type2 @ B ) ) @ ( semiring_1_unsigned @ B @ int @ D2 ) @ ( semiring_1_unsigned @ A @ int @ C2 ) ) ) )
            = ( ( B2 = D2 )
              & ( A2 = C2 ) ) ) ) ) ).

% word_of_int_bin_cat_eq_iff
thf(fact_7171_word__cat__bin_H,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( ( type_len @ A )
        & ( type_len @ C )
        & ( type_len @ B ) )
     => ( ( word_cat @ A @ B @ C )
        = ( ^ [V5: word @ A,W2: word @ B] : ( plus_plus @ ( word @ C ) @ ( bit_se4730199178511100633sh_bit @ ( word @ C ) @ ( type_len0_len_of @ B @ ( type2 @ B ) ) @ ( semiring_1_unsigned @ A @ ( word @ C ) @ V5 ) ) @ ( semiring_1_unsigned @ B @ ( word @ C ) @ W2 ) ) ) ) ) ).

% word_cat_bin'
thf(fact_7172_mask__exceed,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [N2: nat,X2: word @ A] :
          ( ( ord_less_eq @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ N2 )
         => ( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X2 @ ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N2 ) ) )
            = ( zero_zero @ ( word @ A ) ) ) ) ) ).

% mask_exceed
thf(fact_7173_ucast__ucast__add,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ( type_len @ B )
        & ( type_len @ A ) )
     => ! [X2: word @ A,Y2: word @ B] :
          ( ( ord_less_eq @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( type_len0_len_of @ B @ ( type2 @ B ) ) )
         => ( ( semiring_1_unsigned @ B @ ( word @ A ) @ ( plus_plus @ ( word @ B ) @ ( semiring_1_unsigned @ A @ ( word @ B ) @ X2 ) @ Y2 ) )
            = ( plus_plus @ ( word @ A ) @ X2 @ ( semiring_1_unsigned @ B @ ( word @ A ) @ Y2 ) ) ) ) ) ).

% ucast_ucast_add
thf(fact_7174_ucast__sub__ucast,axiom,
    ! [B: $tType,A: $tType] :
      ( ( ( type_len @ A )
        & ( type_len @ B ) )
     => ! [Y2: word @ A,X2: word @ A] :
          ( ( ord_less_eq @ ( word @ A ) @ Y2 @ X2 )
         => ( ( ord_less_eq @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( type_len0_len_of @ B @ ( type2 @ B ) ) )
           => ( ( semiring_1_unsigned @ A @ ( word @ B ) @ ( minus_minus @ ( word @ A ) @ X2 @ Y2 ) )
              = ( minus_minus @ ( word @ B ) @ ( semiring_1_unsigned @ A @ ( word @ B ) @ X2 ) @ ( semiring_1_unsigned @ A @ ( word @ B ) @ Y2 ) ) ) ) ) ) ).

% ucast_sub_ucast
thf(fact_7175_ucast__mask__drop,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ( type_len @ B )
        & ( type_len @ A ) )
     => ! [N2: nat,X2: word @ B] :
          ( ( ord_less_eq @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ N2 )
         => ( ( semiring_1_unsigned @ B @ ( word @ A ) @ ( bit_se5824344872417868541ns_and @ ( word @ B ) @ X2 @ ( bit_se2239418461657761734s_mask @ ( word @ B ) @ N2 ) ) )
            = ( semiring_1_unsigned @ B @ ( word @ A ) @ X2 ) ) ) ) ).

% ucast_mask_drop
thf(fact_7176_bintr__uint,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [N2: nat,W: word @ A] :
          ( ( ord_less_eq @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ N2 )
         => ( ( bit_se2584673776208193580ke_bit @ int @ N2 @ ( semiring_1_unsigned @ A @ int @ W ) )
            = ( semiring_1_unsigned @ A @ int @ W ) ) ) ) ).

% bintr_uint
thf(fact_7177_up__ucast__inj,axiom,
    ! [B: $tType,A: $tType] :
      ( ( ( type_len @ A )
        & ( type_len @ B ) )
     => ! [X2: word @ A,Y2: word @ A] :
          ( ( ( semiring_1_unsigned @ A @ ( word @ B ) @ X2 )
            = ( semiring_1_unsigned @ A @ ( word @ B ) @ Y2 ) )
         => ( ( ord_less_eq @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( type_len0_len_of @ B @ ( type2 @ B ) ) )
           => ( X2 = Y2 ) ) ) ) ).

% up_ucast_inj
thf(fact_7178_ucast__le__ucast,axiom,
    ! [B: $tType,A: $tType] :
      ( ( ( type_len @ A )
        & ( type_len @ B ) )
     => ! [X2: word @ A,Y2: word @ A] :
          ( ( ord_less_eq @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( type_len0_len_of @ B @ ( type2 @ B ) ) )
         => ( ( ord_less_eq @ ( word @ B ) @ ( semiring_1_unsigned @ A @ ( word @ B ) @ X2 ) @ ( semiring_1_unsigned @ A @ ( word @ B ) @ Y2 ) )
            = ( ord_less_eq @ ( word @ A ) @ X2 @ Y2 ) ) ) ) ).

% ucast_le_ucast
thf(fact_7179_ucast__ucast__eq,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( ( ( type_len @ B )
        & ( type_len @ A )
        & ( type_len @ C ) )
     => ! [X2: word @ A,Y2: word @ B] :
          ( ( ( semiring_1_unsigned @ A @ ( word @ C ) @ X2 )
            = ( semiring_1_unsigned @ A @ ( word @ C ) @ ( semiring_1_unsigned @ B @ ( word @ A ) @ Y2 ) ) )
         => ( ( ord_less_eq @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( type_len0_len_of @ B @ ( type2 @ B ) ) )
           => ( ( ord_less_eq @ nat @ ( type_len0_len_of @ B @ ( type2 @ B ) ) @ ( type_len0_len_of @ C @ ( type2 @ C ) ) )
             => ( X2
                = ( semiring_1_unsigned @ B @ ( word @ A ) @ Y2 ) ) ) ) ) ) ).

% ucast_ucast_eq
thf(fact_7180_up__ucast__inj__eq,axiom,
    ! [B: $tType,A: $tType] :
      ( ( ( type_len @ A )
        & ( type_len @ B ) )
     => ! [X2: word @ A,Y2: word @ A] :
          ( ( ord_less_eq @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( type_len0_len_of @ B @ ( type2 @ B ) ) )
         => ( ( ( semiring_1_unsigned @ A @ ( word @ B ) @ X2 )
              = ( semiring_1_unsigned @ A @ ( word @ B ) @ Y2 ) )
            = ( X2 = Y2 ) ) ) ) ).

% up_ucast_inj_eq
thf(fact_7181_ucast__up__mono__le,axiom,
    ! [B: $tType,A: $tType] :
      ( ( ( type_len @ A )
        & ( type_len @ B ) )
     => ! [X2: word @ A,Y2: word @ A] :
          ( ( ord_less_eq @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( type_len0_len_of @ B @ ( type2 @ B ) ) )
         => ( ( ord_less_eq @ ( word @ A ) @ X2 @ Y2 )
           => ( ord_less_eq @ ( word @ B ) @ ( semiring_1_unsigned @ A @ ( word @ B ) @ X2 ) @ ( semiring_1_unsigned @ A @ ( word @ B ) @ Y2 ) ) ) ) ) ).

% ucast_up_mono_le
thf(fact_7182_eq__ucast__ucast__eq,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ( type_len @ B )
        & ( type_len @ A ) )
     => ! [X2: word @ A,Y2: word @ B] :
          ( ( ord_less_eq @ nat @ ( type_len0_len_of @ B @ ( type2 @ B ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
         => ( ( X2
              = ( semiring_1_unsigned @ B @ ( word @ A ) @ Y2 ) )
           => ( ( semiring_1_unsigned @ A @ ( word @ B ) @ X2 )
              = Y2 ) ) ) ) ).

% eq_ucast_ucast_eq
thf(fact_7183_unat__ucast__up__simp,axiom,
    ! [B: $tType,A: $tType] :
      ( ( ( type_len @ A )
        & ( type_len @ B ) )
     => ! [X2: word @ A] :
          ( ( ord_less_eq @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( type_len0_len_of @ B @ ( type2 @ B ) ) )
         => ( ( semiring_1_unsigned @ B @ nat @ ( semiring_1_unsigned @ A @ ( word @ B ) @ X2 ) )
            = ( semiring_1_unsigned @ A @ nat @ X2 ) ) ) ) ).

% unat_ucast_up_simp
thf(fact_7184_ucast__up__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( ( ( type_len @ A )
        & ( type_len @ B ) )
     => ! [X2: word @ A,Y2: word @ A] :
          ( ( ord_less_eq @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( type_len0_len_of @ B @ ( type2 @ B ) ) )
         => ( ( ord_less @ ( word @ A ) @ X2 @ Y2 )
           => ( ord_less @ ( word @ B ) @ ( semiring_1_unsigned @ A @ ( word @ B ) @ X2 ) @ ( semiring_1_unsigned @ A @ ( word @ B ) @ Y2 ) ) ) ) ) ).

% ucast_up_mono
thf(fact_7185_ucast__less__ucast,axiom,
    ! [B: $tType,A: $tType] :
      ( ( ( type_len @ A )
        & ( type_len @ B ) )
     => ! [X2: word @ A,Y2: word @ A] :
          ( ( ord_less_eq @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( type_len0_len_of @ B @ ( type2 @ B ) ) )
         => ( ( ord_less @ ( word @ B ) @ ( semiring_1_unsigned @ A @ ( word @ B ) @ X2 ) @ ( semiring_1_unsigned @ A @ ( word @ B ) @ Y2 ) )
            = ( ord_less @ ( word @ A ) @ X2 @ Y2 ) ) ) ) ).

% ucast_less_ucast
thf(fact_7186_less__ucast__ucast__less,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ( type_len @ B )
        & ( type_len @ A ) )
     => ! [X2: word @ A,Y2: word @ B] :
          ( ( ord_less_eq @ nat @ ( type_len0_len_of @ B @ ( type2 @ B ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
         => ( ( ord_less @ ( word @ A ) @ X2 @ ( semiring_1_unsigned @ B @ ( word @ A ) @ Y2 ) )
           => ( ord_less @ ( word @ B ) @ ( semiring_1_unsigned @ A @ ( word @ B ) @ X2 ) @ Y2 ) ) ) ) ).

% less_ucast_ucast_less
thf(fact_7187_word__of__nat__less__eq__iff,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [M: nat,N2: nat] :
          ( ( ord_less_eq @ ( word @ A ) @ ( semiring_1_of_nat @ ( word @ A ) @ M ) @ ( semiring_1_of_nat @ ( word @ A ) @ N2 ) )
          = ( ord_less_eq @ nat @ ( bit_se2584673776208193580ke_bit @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ M ) @ ( bit_se2584673776208193580ke_bit @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ N2 ) ) ) ) ).

% word_of_nat_less_eq_iff
thf(fact_7188_take__bit__word__beyond__length__eq,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [N2: nat,W: word @ A] :
          ( ( ord_less_eq @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ N2 )
         => ( ( bit_se2584673776208193580ke_bit @ ( word @ A ) @ N2 @ W )
            = W ) ) ) ).

% take_bit_word_beyond_length_eq
thf(fact_7189_wi__bintr,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [N2: nat,W: int] :
          ( ( ord_less_eq @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ N2 )
         => ( ( ring_1_of_int @ ( word @ A ) @ ( bit_se2584673776208193580ke_bit @ int @ N2 @ W ) )
            = ( ring_1_of_int @ ( word @ A ) @ W ) ) ) ) ).

% wi_bintr
thf(fact_7190_word__of__int__less__eq__iff,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [K: int,L: int] :
          ( ( ord_less_eq @ ( word @ A ) @ ( ring_1_of_int @ ( word @ A ) @ K ) @ ( ring_1_of_int @ ( word @ A ) @ L ) )
          = ( ord_less_eq @ int @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ K ) @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ L ) ) ) ) ).

% word_of_int_less_eq_iff
thf(fact_7191_up__scast__inj__eq,axiom,
    ! [B: $tType,A: $tType] :
      ( ( ( type_len @ A )
        & ( type_len @ B ) )
     => ! [X2: word @ A,Y2: word @ A] :
          ( ( ord_less_eq @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( type_len0_len_of @ B @ ( type2 @ B ) ) )
         => ( ( ( ring_1_signed @ A @ ( word @ B ) @ X2 )
              = ( ring_1_signed @ A @ ( word @ B ) @ Y2 ) )
            = ( X2 = Y2 ) ) ) ) ).

% up_scast_inj_eq
thf(fact_7192_mask__over__length,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [N2: nat] :
          ( ( ord_less_eq @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ N2 )
         => ( ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N2 )
            = ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) ) ) ) ).

% mask_over_length
thf(fact_7193_bin__nth__sint,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [N2: nat,W: word @ A] :
          ( ( ord_less_eq @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ N2 )
         => ( ( bit_se5641148757651400278ts_bit @ int @ ( ring_1_signed @ A @ int @ W ) @ N2 )
            = ( bit_se5641148757651400278ts_bit @ int @ ( ring_1_signed @ A @ int @ W ) @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) ) ) ) ) ).

% bin_nth_sint
thf(fact_7194_sint__sbintrunc_H,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [Bin: int] :
          ( ( ring_1_signed @ A @ int @ ( ring_1_of_int @ ( word @ A ) @ Bin ) )
          = ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) @ Bin ) ) ) ).

% sint_sbintrunc'
thf(fact_7195_neg__mask__test__bit,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [N2: nat,M: nat] :
          ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N2 ) ) @ M )
          = ( ( ord_less_eq @ nat @ N2 @ M )
            & ( ord_less @ nat @ M @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ).

% neg_mask_test_bit
thf(fact_7196_uint__word__arith__bintrs_I3_J,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [A2: word @ A,B2: word @ A] :
          ( ( semiring_1_unsigned @ A @ int @ ( times_times @ ( word @ A ) @ A2 @ B2 ) )
          = ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( times_times @ int @ ( semiring_1_unsigned @ A @ int @ A2 ) @ ( semiring_1_unsigned @ A @ int @ B2 ) ) ) ) ) ).

% uint_word_arith_bintrs(3)
thf(fact_7197_num__of__bintr_H,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [A2: num,B2: num] :
          ( ( ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( numeral_numeral @ int @ A2 ) )
            = ( numeral_numeral @ int @ B2 ) )
         => ( ( numeral_numeral @ ( word @ A ) @ A2 )
            = ( numeral_numeral @ ( word @ A ) @ B2 ) ) ) ) ).

% num_of_bintr'
thf(fact_7198_uint32_Osize__eq__length,axiom,
    ( ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) )
    = ( type_len0_len_of @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ numeral_num1 ) ) ) ) ) @ ( type2 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ numeral_num1 ) ) ) ) ) ) ) ) ).

% uint32.size_eq_length
thf(fact_7199_num__abs__bintr,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ( ( numeral_numeral @ ( word @ A ) )
        = ( ^ [X: num] : ( ring_1_of_int @ ( word @ A ) @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( numeral_numeral @ int @ X ) ) ) ) ) ) ).

% num_abs_bintr
thf(fact_7200_signed__take__bit__decr__length__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ( bit_ri3973907225187159222ations @ B )
        & ( type_len @ A ) )
     => ! [K: B,L: B] :
          ( ( ( bit_ri4674362597316999326ke_bit @ B @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) @ K )
            = ( bit_ri4674362597316999326ke_bit @ B @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) @ L ) )
          = ( ( bit_se2584673776208193580ke_bit @ B @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ K )
            = ( bit_se2584673776208193580ke_bit @ B @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ L ) ) ) ) ).

% signed_take_bit_decr_length_iff
thf(fact_7201_neg__test__bit,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [X2: word @ A,N2: nat] :
          ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ X2 ) @ N2 )
          = ( ~ ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ X2 @ N2 )
            & ( ord_less @ nat @ N2 @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ).

% neg_test_bit
thf(fact_7202_bit__uint__iff,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [W: word @ A,N2: nat] :
          ( ( bit_se5641148757651400278ts_bit @ int @ ( semiring_1_unsigned @ A @ int @ W ) @ N2 )
          = ( ( ord_less @ nat @ N2 @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
            & ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ W @ N2 ) ) ) ) ).

% bit_uint_iff
thf(fact_7203_bit__ucast__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ( type_len @ B )
        & ( type_len @ A ) )
     => ! [A2: word @ B,N2: nat] :
          ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( semiring_1_unsigned @ B @ ( word @ A ) @ A2 ) @ N2 )
          = ( ( ord_less @ nat @ N2 @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
            & ( bit_se5641148757651400278ts_bit @ ( word @ B ) @ A2 @ N2 ) ) ) ) ).

% bit_ucast_iff
thf(fact_7204_bin__nth__uint__imp,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [W: word @ A,N2: nat] :
          ( ( bit_se5641148757651400278ts_bit @ int @ ( semiring_1_unsigned @ A @ int @ W ) @ N2 )
         => ( ord_less @ nat @ N2 @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ).

% bin_nth_uint_imp
thf(fact_7205_bit__word__ucast__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( ( ( type_len @ A )
        & ( type_len @ B ) )
     => ! [W: word @ A,N2: nat] :
          ( ( bit_se5641148757651400278ts_bit @ ( word @ B ) @ ( semiring_1_unsigned @ A @ ( word @ B ) @ W ) @ N2 )
          = ( ( ord_less @ nat @ N2 @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
            & ( ord_less @ nat @ N2 @ ( type_len0_len_of @ B @ ( type2 @ B ) ) )
            & ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ W @ N2 ) ) ) ) ).

% bit_word_ucast_iff
thf(fact_7206_test__bit__bin,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) )
        = ( ^ [W2: word @ A,N: nat] :
              ( ( ord_less @ nat @ N @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
              & ( bit_se5641148757651400278ts_bit @ int @ ( semiring_1_unsigned @ A @ int @ W2 ) @ N ) ) ) ) ) ).

% test_bit_bin
thf(fact_7207_nth__ucast,axiom,
    ! [B: $tType,A: $tType] :
      ( ( ( type_len @ A )
        & ( type_len @ B ) )
     => ! [W: word @ B,N2: nat] :
          ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( semiring_1_unsigned @ B @ ( word @ A ) @ W ) @ N2 )
          = ( ( bit_se5641148757651400278ts_bit @ ( word @ B ) @ W @ N2 )
            & ( ord_less @ nat @ N2 @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ).

% nth_ucast
thf(fact_7208_word__eq__iff,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ( ( ^ [Y5: word @ A,Z4: word @ A] : Y5 = Z4 )
        = ( ^ [X: word @ A,Y: word @ A] :
            ! [N: nat] :
              ( ( ord_less @ nat @ N @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
             => ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ X @ N )
                = ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ Y @ N ) ) ) ) ) ) ).

% word_eq_iff
thf(fact_7209_test__bit__conj__lt,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [X2: word @ A,M: nat] :
          ( ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ X2 @ M )
            & ( ord_less @ nat @ M @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
          = ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ X2 @ M ) ) ) ).

% test_bit_conj_lt
thf(fact_7210_bit__imp__le__length,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [W: word @ A,N2: nat] :
          ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ W @ N2 )
         => ( ord_less @ nat @ N2 @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ).

% bit_imp_le_length
thf(fact_7211_bit__word__eqI,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [A2: word @ A,B2: word @ A] :
          ( ! [N4: nat] :
              ( ( ord_less @ nat @ N4 @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
             => ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ A2 @ N4 )
                = ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ B2 @ N4 ) ) )
         => ( A2 = B2 ) ) ) ).

% bit_word_eqI
thf(fact_7212_bit__word__of__int__iff,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [K: int,N2: nat] :
          ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( ring_1_of_int @ ( word @ A ) @ K ) @ N2 )
          = ( ( ord_less @ nat @ N2 @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
            & ( bit_se5641148757651400278ts_bit @ int @ K @ N2 ) ) ) ) ).

% bit_word_of_int_iff
thf(fact_7213_test__bit__wi,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [X2: int,N2: nat] :
          ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( ring_1_of_int @ ( word @ A ) @ X2 ) @ N2 )
          = ( ( ord_less @ nat @ N2 @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
            & ( bit_se5641148757651400278ts_bit @ int @ X2 @ N2 ) ) ) ) ).

% test_bit_wi
thf(fact_7214_max__test__bit,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [N2: nat] :
          ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) @ N2 )
          = ( ord_less @ nat @ N2 @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ).

% max_test_bit
thf(fact_7215_test__bit__1_H,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [N2: nat] :
          ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( one_one @ ( word @ A ) ) @ N2 )
          = ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
            & ( N2
              = ( zero_zero @ nat ) ) ) ) ) ).

% test_bit_1'
thf(fact_7216_degenerate__word,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [X2: word @ A] :
          ( ( ( type_len0_len_of @ A @ ( type2 @ A ) )
            = ( one_one @ nat ) )
         => ( ( X2
              = ( zero_zero @ ( word @ A ) ) )
            | ( X2
              = ( one_one @ ( word @ A ) ) ) ) ) ) ).

% degenerate_word
thf(fact_7217_one__word_Orsp,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ( ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ int ) )
        = ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ int ) ) ) ) ).

% one_word.rsp
thf(fact_7218_scast__1,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ( type_len @ B )
        & ( type_len @ A ) )
     => ( ( ( ( type_len0_len_of @ A @ ( type2 @ A ) )
            = ( one_one @ nat ) )
         => ( ( ring_1_signed @ A @ ( word @ B ) @ ( one_one @ ( word @ A ) ) )
            = ( uminus_uminus @ ( word @ B ) @ ( one_one @ ( word @ B ) ) ) ) )
        & ( ( ( type_len0_len_of @ A @ ( type2 @ A ) )
           != ( one_one @ nat ) )
         => ( ( ring_1_signed @ A @ ( word @ B ) @ ( one_one @ ( word @ A ) ) )
            = ( one_one @ ( word @ B ) ) ) ) ) ) ).

% scast_1
thf(fact_7219_max__word__mask,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ( ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) )
        = ( bit_se2239418461657761734s_mask @ ( word @ A ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ).

% max_word_mask
thf(fact_7220_unsigned__minus__1__eq__mask,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ( type_len @ B )
        & ( bit_se359711467146920520ations @ A ) )
     => ( ( semiring_1_unsigned @ B @ A @ ( uminus_uminus @ ( word @ B ) @ ( one_one @ ( word @ B ) ) ) )
        = ( bit_se2239418461657761734s_mask @ A @ ( type_len0_len_of @ B @ ( type2 @ B ) ) ) ) ) ).

% unsigned_minus_1_eq_mask
thf(fact_7221_uint__word__arith__bintrs_I8_J,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ( ( semiring_1_unsigned @ A @ int @ ( one_one @ ( word @ A ) ) )
        = ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ int ) ) ) ) ).

% uint_word_arith_bintrs(8)
thf(fact_7222_signed__take__bit__eq,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ( type_len @ B )
        & ( bit_ri3973907225187159222ations @ A ) )
     => ! [N2: nat,W: word @ B] :
          ( ( ( ord_less @ nat @ N2 @ ( type_len0_len_of @ B @ ( type2 @ B ) ) )
           => ( ( ring_1_signed @ B @ A @ ( bit_se2584673776208193580ke_bit @ ( word @ B ) @ N2 @ W ) )
              = ( bit_se2584673776208193580ke_bit @ A @ N2 @ ( ring_1_signed @ B @ A @ W ) ) ) )
          & ( ~ ( ord_less @ nat @ N2 @ ( type_len0_len_of @ B @ ( type2 @ B ) ) )
           => ( ( ring_1_signed @ B @ A @ ( bit_se2584673776208193580ke_bit @ ( word @ B ) @ N2 @ W ) )
              = ( ring_1_signed @ B @ A @ W ) ) ) ) ) ).

% signed_take_bit_eq
thf(fact_7223_word__of__nat__less__iff,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [M: nat,N2: nat] :
          ( ( ord_less @ ( word @ A ) @ ( semiring_1_of_nat @ ( word @ A ) @ M ) @ ( semiring_1_of_nat @ ( word @ A ) @ N2 ) )
          = ( ord_less @ nat @ ( bit_se2584673776208193580ke_bit @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ M ) @ ( bit_se2584673776208193580ke_bit @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ N2 ) ) ) ) ).

% word_of_nat_less_iff
thf(fact_7224_ucast__ucast__id,axiom,
    ! [B: $tType,A: $tType] :
      ( ( ( type_len @ A )
        & ( type_len @ B ) )
     => ! [X2: word @ A] :
          ( ( ord_less @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( type_len0_len_of @ B @ ( type2 @ B ) ) )
         => ( ( semiring_1_unsigned @ B @ ( word @ A ) @ ( semiring_1_unsigned @ A @ ( word @ B ) @ X2 ) )
            = X2 ) ) ) ).

% ucast_ucast_id
thf(fact_7225_ucast__less__ucast__weak,axiom,
    ! [B: $tType,A: $tType] :
      ( ( ( type_len @ A )
        & ( type_len @ B ) )
     => ! [X2: word @ A,Y2: word @ A] :
          ( ( ord_less @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( type_len0_len_of @ B @ ( type2 @ B ) ) )
         => ( ( ord_less @ ( word @ B ) @ ( semiring_1_unsigned @ A @ ( word @ B ) @ X2 ) @ ( semiring_1_unsigned @ A @ ( word @ B ) @ Y2 ) )
            = ( ord_less @ ( word @ A ) @ X2 @ Y2 ) ) ) ) ).

% ucast_less_ucast_weak
thf(fact_7226_uint__word__arith__bintrs_I2_J,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [A2: word @ A,B2: word @ A] :
          ( ( semiring_1_unsigned @ A @ int @ ( minus_minus @ ( word @ A ) @ A2 @ B2 ) )
          = ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( minus_minus @ int @ ( semiring_1_unsigned @ A @ int @ A2 ) @ ( semiring_1_unsigned @ A @ int @ B2 ) ) ) ) ) ).

% uint_word_arith_bintrs(2)
thf(fact_7227_uint__word__arith__bintrs_I1_J,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [A2: word @ A,B2: word @ A] :
          ( ( semiring_1_unsigned @ A @ int @ ( plus_plus @ ( word @ A ) @ A2 @ B2 ) )
          = ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( plus_plus @ int @ ( semiring_1_unsigned @ A @ int @ A2 ) @ ( semiring_1_unsigned @ A @ int @ B2 ) ) ) ) ) ).

% uint_word_arith_bintrs(1)
thf(fact_7228_word__sint__1,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ( ( ( ( type_len0_len_of @ A @ ( type2 @ A ) )
            = ( one_one @ nat ) )
         => ( ( ring_1_signed @ A @ int @ ( one_one @ ( word @ A ) ) )
            = ( uminus_uminus @ int @ ( one_one @ int ) ) ) )
        & ( ( ( type_len0_len_of @ A @ ( type2 @ A ) )
           != ( one_one @ nat ) )
         => ( ( ring_1_signed @ A @ int @ ( one_one @ ( word @ A ) ) )
            = ( one_one @ int ) ) ) ) ) ).

% word_sint_1
thf(fact_7229_num__of__sbintr_H,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [A2: num,B2: num] :
          ( ( ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) @ ( numeral_numeral @ int @ A2 ) )
            = ( numeral_numeral @ int @ B2 ) )
         => ( ( numeral_numeral @ ( word @ A ) @ A2 )
            = ( numeral_numeral @ ( word @ A ) @ B2 ) ) ) ) ).

% num_of_sbintr'
thf(fact_7230_bit__set__bit__aux,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [F: nat > $o,N2: nat,W: word @ A,M: nat] :
          ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( code_T2661198915054445665ts_aux @ A @ F @ N2 @ W ) @ M )
          = ( ( ord_less @ nat @ M @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
            & ( ( ord_less @ nat @ M @ N2 )
             => ( F @ M ) )
            & ( ~ ( ord_less @ nat @ M @ N2 )
             => ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ W @ ( minus_minus @ nat @ M @ N2 ) ) ) ) ) ) ).

% bit_set_bit_aux
thf(fact_7231_two__power__eq,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [N2: nat,M: nat] :
          ( ( ord_less @ nat @ N2 @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
         => ( ( ord_less @ nat @ M @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
           => ( ( ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N2 )
                = ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ M ) )
              = ( N2 = M ) ) ) ) ) ).

% two_power_eq
thf(fact_7232_ucast__drop__bit__eq,axiom,
    ! [B: $tType,A: $tType] :
      ( ( ( type_len @ A )
        & ( type_len @ B ) )
     => ! [N2: nat,W: word @ A] :
          ( ( ord_less_eq @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( type_len0_len_of @ B @ ( type2 @ B ) ) )
         => ( ( semiring_1_unsigned @ A @ ( word @ B ) @ ( bit_se4197421643247451524op_bit @ ( word @ A ) @ N2 @ W ) )
            = ( bit_se4197421643247451524op_bit @ ( word @ B ) @ N2 @ ( semiring_1_unsigned @ A @ ( word @ B ) @ W ) ) ) ) ) ).

% ucast_drop_bit_eq
thf(fact_7233_bit__set__bit__word__iff,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [W: word @ A,M: nat,B2: $o,N2: nat] :
          ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( generi7602027413899671122et_bit @ ( word @ A ) @ W @ M @ B2 ) @ N2 )
          = ( ( ( M = N2 )
             => ( ( ord_less @ nat @ N2 @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
                & B2 ) )
            & ( ( M != N2 )
             => ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ W @ N2 ) ) ) ) ) ).

% bit_set_bit_word_iff
thf(fact_7234_nth__slice,axiom,
    ! [B: $tType,A: $tType] :
      ( ( ( type_len @ A )
        & ( type_len @ B ) )
     => ! [N2: nat,W: word @ B,M: nat] :
          ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( slice2 @ B @ A @ N2 @ W ) @ M )
          = ( ( bit_se5641148757651400278ts_bit @ ( word @ B ) @ W @ ( plus_plus @ nat @ M @ N2 ) )
            & ( ord_less @ nat @ M @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ).

% nth_slice
thf(fact_7235_word__size,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ( ( size_size @ ( word @ A ) )
        = ( ^ [W2: word @ A] : ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ).

% word_size
thf(fact_7236_size__word_Orep__eq,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ( ( size_size @ ( word @ A ) )
        = ( ^ [X: word @ A] : ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ).

% size_word.rep_eq
thf(fact_7237_up__scast__inj,axiom,
    ! [B: $tType,A: $tType] :
      ( ( ( type_len @ A )
        & ( type_len @ B ) )
     => ! [X2: word @ A,Y2: word @ A] :
          ( ( ( ring_1_signed @ A @ ( word @ B ) @ X2 )
            = ( ring_1_signed @ A @ ( word @ B ) @ Y2 ) )
         => ( ( ord_less_eq @ nat @ ( size_size @ ( word @ A ) @ X2 ) @ ( type_len0_len_of @ B @ ( type2 @ B ) ) )
           => ( X2 = Y2 ) ) ) ) ).

% up_scast_inj
thf(fact_7238_unsigned__less,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ( type_len @ B )
        & ( unique1627219031080169319umeral @ A ) )
     => ! [W: word @ B] : ( ord_less @ A @ ( semiring_1_unsigned @ B @ A @ W ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ B @ ( type2 @ B ) ) ) ) ) ).

% unsigned_less
thf(fact_7239_not__degenerate__imp__2__neq__0,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ( ( ord_less @ nat @ ( one_one @ nat ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
       => ( ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) )
         != ( zero_zero @ ( word @ A ) ) ) ) ) ).

% not_degenerate_imp_2_neq_0
thf(fact_7240_smod__word__max,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [A2: word @ A,B2: word @ A] : ( ord_less @ int @ ( signed6721504322012087516modulo @ int @ ( ring_1_signed @ A @ int @ A2 ) @ ( ring_1_signed @ A @ int @ B2 ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ).

% smod_word_max
thf(fact_7241_word__nchotomy,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [W4: word @ A] :
        ? [N4: nat] :
          ( ( W4
            = ( semiring_1_of_nat @ ( word @ A ) @ N4 ) )
          & ( ord_less @ nat @ N4 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ).

% word_nchotomy
thf(fact_7242_word__nat__cases,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [X2: word @ A] :
          ~ ! [N4: nat] :
              ( ( X2
                = ( semiring_1_of_nat @ ( word @ A ) @ N4 ) )
             => ~ ( ord_less @ nat @ N4 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ).

% word_nat_cases
thf(fact_7243_of__nat__inj,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [X2: nat,Y2: nat] :
          ( ( ord_less @ nat @ X2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
         => ( ( ord_less @ nat @ Y2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
           => ( ( ( semiring_1_of_nat @ ( word @ A ) @ X2 )
                = ( semiring_1_of_nat @ ( word @ A ) @ Y2 ) )
              = ( X2 = Y2 ) ) ) ) ) ).

% of_nat_inj
thf(fact_7244_word__of__nat__inj,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [X2: nat,Y2: nat] :
          ( ( ord_less @ nat @ X2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
         => ( ( ord_less @ nat @ Y2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
           => ( ( ( semiring_1_of_nat @ ( word @ A ) @ X2 )
                = ( semiring_1_of_nat @ ( word @ A ) @ Y2 ) )
             => ( X2 = Y2 ) ) ) ) ) ).

% word_of_nat_inj
thf(fact_7245_More__Word_Opower__not__zero,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [N2: nat] :
          ( ( ord_less @ nat @ N2 @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
         => ( ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N2 )
           != ( zero_zero @ ( word @ A ) ) ) ) ) ).

% More_Word.power_not_zero
thf(fact_7246_word__power__increasing,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [X2: nat,Y2: nat] :
          ( ( ord_less @ ( word @ A ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ X2 ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ Y2 ) )
         => ( ( ord_less @ nat @ X2 @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
           => ( ( ord_less @ nat @ Y2 @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
             => ( ord_less @ nat @ X2 @ Y2 ) ) ) ) ) ).

% word_power_increasing
thf(fact_7247_power__overflow,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [N2: nat] :
          ( ( ord_less_eq @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ N2 )
         => ( ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N2 )
            = ( zero_zero @ ( word @ A ) ) ) ) ) ).

% power_overflow
thf(fact_7248_nth__w2p,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [N2: nat,M: nat] :
          ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N2 ) @ M )
          = ( ( M = N2 )
            & ( ord_less @ nat @ M @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ).

% nth_w2p
thf(fact_7249_nth__w2p__same,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [N2: nat] :
          ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N2 ) @ N2 )
          = ( ord_less @ nat @ N2 @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ).

% nth_w2p_same
thf(fact_7250_uint__idem,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [W: word @ A] :
          ( ( modulo_modulo @ int @ ( semiring_1_unsigned @ A @ int @ W ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
          = ( semiring_1_unsigned @ A @ int @ W ) ) ) ).

% uint_idem
thf(fact_7251_word__of__int__2p__len,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ( ( ring_1_of_int @ ( word @ A ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
        = ( zero_zero @ ( word @ A ) ) ) ) ).

% word_of_int_2p_len
thf(fact_7252_of__nat__neq__iff__word,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [X2: nat,Y2: nat] :
          ( ( ( modulo_modulo @ nat @ X2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
           != ( modulo_modulo @ nat @ Y2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) )
         => ( ( ( semiring_1_of_nat @ ( word @ A ) @ X2 )
             != ( semiring_1_of_nat @ ( word @ A ) @ Y2 ) )
            = ( X2 != Y2 ) ) ) ) ).

% of_nat_neq_iff_word
thf(fact_7253_sint__uint,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ( ( ring_1_signed @ A @ int )
        = ( ^ [W2: word @ A] : ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) @ ( semiring_1_unsigned @ A @ int @ W2 ) ) ) ) ) ).

% sint_uint
thf(fact_7254_num__abs__sbintr,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ( ( numeral_numeral @ ( word @ A ) )
        = ( ^ [X: num] : ( ring_1_of_int @ ( word @ A ) @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) @ ( numeral_numeral @ int @ X ) ) ) ) ) ) ).

% num_abs_sbintr
thf(fact_7255_ucast__mono,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ( type_len @ B )
        & ( type_len @ A ) )
     => ! [X2: word @ B,Y2: word @ B] :
          ( ( ord_less @ ( word @ B ) @ X2 @ Y2 )
         => ( ( ord_less @ ( word @ B ) @ Y2 @ ( power_power @ ( word @ B ) @ ( numeral_numeral @ ( word @ B ) @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
           => ( ord_less @ ( word @ A ) @ ( semiring_1_unsigned @ B @ ( word @ A ) @ X2 ) @ ( semiring_1_unsigned @ B @ ( word @ A ) @ Y2 ) ) ) ) ) ).

% ucast_mono
thf(fact_7256_ucast__ucast__len,axiom,
    ! [B: $tType,A: $tType] :
      ( ( ( type_len @ A )
        & ( type_len @ B ) )
     => ! [X2: word @ A] :
          ( ( ord_less @ ( word @ A ) @ X2 @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ B @ ( type2 @ B ) ) ) )
         => ( ( semiring_1_unsigned @ B @ ( word @ A ) @ ( semiring_1_unsigned @ A @ ( word @ B ) @ X2 ) )
            = X2 ) ) ) ).

% ucast_ucast_len
thf(fact_7257_horner__sum__uint__exp__Cons__eq,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [W: word @ A,Ws: list @ ( word @ A )] :
          ( ( groups4207007520872428315er_sum @ ( word @ A ) @ int @ ( semiring_1_unsigned @ A @ int ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) @ ( cons @ ( word @ A ) @ W @ Ws ) )
          = ( bit_concat_bit @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( semiring_1_unsigned @ A @ int @ W ) @ ( groups4207007520872428315er_sum @ ( word @ A ) @ int @ ( semiring_1_unsigned @ A @ int ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) @ Ws ) ) ) ) ).

% horner_sum_uint_exp_Cons_eq
thf(fact_7258_sint__word__ariths_I7_J,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ( ( ring_1_signed @ A @ int @ ( zero_zero @ ( word @ A ) ) )
        = ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) @ ( zero_zero @ int ) ) ) ) ).

% sint_word_ariths(7)
thf(fact_7259_sint__word__ariths_I8_J,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ( ( ring_1_signed @ A @ int @ ( one_one @ ( word @ A ) ) )
        = ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) @ ( one_one @ int ) ) ) ) ).

% sint_word_ariths(8)
thf(fact_7260_sint__word__ariths_I1_J,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [A2: word @ A,B2: word @ A] :
          ( ( ring_1_signed @ A @ int @ ( plus_plus @ ( word @ A ) @ A2 @ B2 ) )
          = ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) @ ( plus_plus @ int @ ( ring_1_signed @ A @ int @ A2 ) @ ( ring_1_signed @ A @ int @ B2 ) ) ) ) ) ).

% sint_word_ariths(1)
thf(fact_7261_nth__sint,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [W: word @ A,N2: nat] :
          ( ( bit_se5641148757651400278ts_bit @ int @ ( ring_1_signed @ A @ int @ W ) @ N2 )
          = ( ( ( ord_less @ nat @ N2 @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) )
             => ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ W @ N2 ) )
            & ( ~ ( ord_less @ nat @ N2 @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) )
             => ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ W @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) ) ) ) ) ) ).

% nth_sint
thf(fact_7262_bit__sint__iff,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [W: word @ A,N2: nat] :
          ( ( bit_se5641148757651400278ts_bit @ int @ ( ring_1_signed @ A @ int @ W ) @ N2 )
          = ( ( ( ord_less_eq @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ N2 )
              & ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ W @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) ) )
            | ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ W @ N2 ) ) ) ) ).

% bit_sint_iff
thf(fact_7263_sint__word__ariths_I4_J,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [A2: word @ A] :
          ( ( ring_1_signed @ A @ int @ ( uminus_uminus @ ( word @ A ) @ A2 ) )
          = ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) @ ( uminus_uminus @ int @ ( ring_1_signed @ A @ int @ A2 ) ) ) ) ) ).

% sint_word_ariths(4)
thf(fact_7264_drop__bit__eq__zero__iff__not__bit__last,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [W: word @ A] :
          ( ( ( bit_se4197421643247451524op_bit @ ( word @ A ) @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) @ W )
            = ( zero_zero @ ( word @ A ) ) )
          = ( ~ ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ W @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) ).

% drop_bit_eq_zero_iff_not_bit_last
thf(fact_7265_sint__word__ariths_I2_J,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [A2: word @ A,B2: word @ A] :
          ( ( ring_1_signed @ A @ int @ ( minus_minus @ ( word @ A ) @ A2 @ B2 ) )
          = ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) @ ( minus_minus @ int @ ( ring_1_signed @ A @ int @ A2 ) @ ( ring_1_signed @ A @ int @ B2 ) ) ) ) ) ).

% sint_word_ariths(2)
thf(fact_7266_signed__scast__eq,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( ( ( type_len @ B )
        & ( bit_ri3973907225187159222ations @ A )
        & ( type_len @ C ) )
     => ! [W: word @ B] :
          ( ( ring_1_signed @ C @ A @ ( ring_1_signed @ B @ ( word @ C ) @ W ) )
          = ( bit_ri4674362597316999326ke_bit @ A @ ( minus_minus @ nat @ ( type_len0_len_of @ C @ ( type2 @ C ) ) @ ( suc @ ( zero_zero @ nat ) ) ) @ ( ring_1_signed @ B @ A @ W ) ) ) ) ).

% signed_scast_eq
thf(fact_7267_sint__word__ariths_I3_J,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [A2: word @ A,B2: word @ A] :
          ( ( ring_1_signed @ A @ int @ ( times_times @ ( word @ A ) @ A2 @ B2 ) )
          = ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) @ ( times_times @ int @ ( ring_1_signed @ A @ int @ A2 ) @ ( ring_1_signed @ A @ int @ B2 ) ) ) ) ) ).

% sint_word_ariths(3)
thf(fact_7268_smod__word__min,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [A2: word @ A,B2: word @ A] : ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ) @ ( signed6721504322012087516modulo @ int @ ( ring_1_signed @ A @ int @ A2 ) @ ( ring_1_signed @ A @ int @ B2 ) ) ) ) ).

% smod_word_min
thf(fact_7269_smod__int__compares_I1_J,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ A2 )
     => ( ( ord_less @ int @ ( zero_zero @ int ) @ B2 )
       => ( ord_less @ int @ ( signed6721504322012087516modulo @ int @ A2 @ B2 ) @ B2 ) ) ) ).

% smod_int_compares(1)
thf(fact_7270_smod__int__compares_I2_J,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ A2 )
     => ( ( ord_less @ int @ ( zero_zero @ int ) @ B2 )
       => ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( signed6721504322012087516modulo @ int @ A2 @ B2 ) ) ) ) ).

% smod_int_compares(2)
thf(fact_7271_smod__int__compares_I4_J,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_eq @ int @ A2 @ ( zero_zero @ int ) )
     => ( ( ord_less @ int @ ( zero_zero @ int ) @ B2 )
       => ( ord_less_eq @ int @ ( signed6721504322012087516modulo @ int @ A2 @ B2 ) @ ( zero_zero @ int ) ) ) ) ).

% smod_int_compares(4)
thf(fact_7272_smod__int__compares_I6_J,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ A2 )
     => ( ( ord_less @ int @ B2 @ ( zero_zero @ int ) )
       => ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( signed6721504322012087516modulo @ int @ A2 @ B2 ) ) ) ) ).

% smod_int_compares(6)
thf(fact_7273_smod__int__compares_I7_J,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_eq @ int @ A2 @ ( zero_zero @ int ) )
     => ( ( ord_less @ int @ B2 @ ( zero_zero @ int ) )
       => ( ord_less_eq @ int @ ( signed6721504322012087516modulo @ int @ A2 @ B2 ) @ ( zero_zero @ int ) ) ) ) ).

% smod_int_compares(7)
thf(fact_7274_smod__int__compares_I8_J,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_eq @ int @ A2 @ ( zero_zero @ int ) )
     => ( ( ord_less @ int @ B2 @ ( zero_zero @ int ) )
       => ( ord_less_eq @ int @ B2 @ ( signed6721504322012087516modulo @ int @ A2 @ B2 ) ) ) ) ).

% smod_int_compares(8)
thf(fact_7275_less__Suc__unat__less__bound,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [N2: nat,X2: word @ A] :
          ( ( ord_less @ nat @ N2 @ ( suc @ ( semiring_1_unsigned @ A @ nat @ X2 ) ) )
         => ( ord_less @ nat @ N2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ).

% less_Suc_unat_less_bound
thf(fact_7276_smod__mod__positive,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ A2 )
     => ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ B2 )
       => ( ( signed6721504322012087516modulo @ int @ A2 @ B2 )
          = ( modulo_modulo @ int @ A2 @ B2 ) ) ) ) ).

% smod_mod_positive
thf(fact_7277_uint__2__id,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
       => ( ( semiring_1_unsigned @ A @ int @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) )
          = ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ).

% uint_2_id
thf(fact_7278_lt2p__lem,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [N2: nat,W: word @ A] :
          ( ( ord_less_eq @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ N2 )
         => ( ord_less @ int @ ( semiring_1_unsigned @ A @ int @ W ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N2 ) ) ) ) ).

% lt2p_lem
thf(fact_7279_power__le__mono,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [N2: nat,M: nat] :
          ( ( ord_less_eq @ ( word @ A ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N2 ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ M ) )
         => ( ( ord_less @ nat @ N2 @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
           => ( ( ord_less @ nat @ M @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
             => ( ord_less_eq @ nat @ N2 @ M ) ) ) ) ) ).

% power_le_mono
thf(fact_7280_two__power__increasing,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [N2: nat,M: nat] :
          ( ( ord_less_eq @ nat @ N2 @ M )
         => ( ( ord_less @ nat @ M @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
           => ( ord_less_eq @ ( word @ A ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N2 ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ M ) ) ) ) ) ).

% two_power_increasing
thf(fact_7281_unat__numeral,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [B2: num] :
          ( ( semiring_1_unsigned @ A @ nat @ ( numeral_numeral @ ( word @ A ) @ B2 ) )
          = ( modulo_modulo @ nat @ ( numeral_numeral @ nat @ B2 ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ).

% unat_numeral
thf(fact_7282_of__nat__mono__maybe,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [X2: nat,Y2: nat] :
          ( ( ord_less @ nat @ X2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
         => ( ( ord_less @ nat @ Y2 @ X2 )
           => ( ord_less @ ( word @ A ) @ ( semiring_1_of_nat @ ( word @ A ) @ Y2 ) @ ( semiring_1_of_nat @ ( word @ A ) @ X2 ) ) ) ) ) ).

% of_nat_mono_maybe
thf(fact_7283_of__nat__mono__maybe_H,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [X2: nat,Y2: nat] :
          ( ( ord_less @ nat @ X2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
         => ( ( ord_less @ nat @ Y2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
           => ( ( ord_less @ nat @ Y2 @ X2 )
              = ( ord_less @ ( word @ A ) @ ( semiring_1_of_nat @ ( word @ A ) @ Y2 ) @ ( semiring_1_of_nat @ ( word @ A ) @ X2 ) ) ) ) ) ) ).

% of_nat_mono_maybe'
thf(fact_7284_unat__split,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [P: nat > $o,X2: word @ A] :
          ( ( P @ ( semiring_1_unsigned @ A @ nat @ X2 ) )
          = ( ! [N: nat] :
                ( ( ( ( semiring_1_of_nat @ ( word @ A ) @ N )
                    = X2 )
                  & ( ord_less @ nat @ N @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) )
               => ( P @ N ) ) ) ) ) ).

% unat_split
thf(fact_7285_of__nat__inverse,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [R4: nat,A2: word @ A] :
          ( ( ( semiring_1_of_nat @ ( word @ A ) @ R4 )
            = A2 )
         => ( ( ord_less @ nat @ R4 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
           => ( ( semiring_1_unsigned @ A @ nat @ A2 )
              = R4 ) ) ) ) ).

% of_nat_inverse
thf(fact_7286_unat__split__asm,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [P: nat > $o,X2: word @ A] :
          ( ( P @ ( semiring_1_unsigned @ A @ nat @ X2 ) )
          = ( ~ ? [N: nat] :
                  ( ( ( semiring_1_of_nat @ ( word @ A ) @ N )
                    = X2 )
                  & ( ord_less @ nat @ N @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
                  & ~ ( P @ N ) ) ) ) ) ).

% unat_split_asm
thf(fact_7287_unat__eq__of__nat,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [N2: nat,X2: word @ A] :
          ( ( ord_less @ nat @ N2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
         => ( ( ( semiring_1_unsigned @ A @ nat @ X2 )
              = N2 )
            = ( X2
              = ( semiring_1_of_nat @ ( word @ A ) @ N2 ) ) ) ) ) ).

% unat_eq_of_nat
thf(fact_7288_unat__of__nat__len,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [X2: nat] :
          ( ( ord_less @ nat @ X2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
         => ( ( semiring_1_unsigned @ A @ nat @ ( semiring_1_of_nat @ ( word @ A ) @ X2 ) )
            = X2 ) ) ) ).

% unat_of_nat_len
thf(fact_7289_UNIV__word__eq__word__of__nat,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ( ( top_top @ ( set @ ( word @ A ) ) )
        = ( image @ nat @ ( word @ A ) @ ( semiring_1_of_nat @ ( word @ A ) ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ) ).

% UNIV_word_eq_word_of_nat
thf(fact_7290_x__less__2__0__1_H,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [X2: word @ A] :
          ( ( ( type_len0_len_of @ A @ ( type2 @ A ) )
           != ( one_one @ nat ) )
         => ( ( ord_less @ ( word @ A ) @ X2 @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) )
           => ( ( X2
                = ( zero_zero @ ( word @ A ) ) )
              | ( X2
                = ( one_one @ ( word @ A ) ) ) ) ) ) ) ).

% x_less_2_0_1'
thf(fact_7291_test__bit__2p,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [N2: nat,M: nat] :
          ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( ring_1_of_int @ ( word @ A ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N2 ) ) @ M )
          = ( ( M = N2 )
            & ( ord_less @ nat @ M @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ).

% test_bit_2p
thf(fact_7292_Word_Oof__nat__0,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [M: nat] :
          ( ( ( semiring_1_of_nat @ ( word @ A ) @ M )
            = ( zero_zero @ ( word @ A ) ) )
          = ( ? [Q5: nat] :
                ( M
                = ( times_times @ nat @ Q5 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ) ) ).

% Word.of_nat_0
thf(fact_7293_word__1__le__power,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [N2: nat] :
          ( ( ord_less @ nat @ N2 @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
         => ( ord_less_eq @ ( word @ A ) @ ( one_one @ ( word @ A ) ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N2 ) ) ) ) ).

% word_1_le_power
thf(fact_7294_uint__sub__lt2p,axiom,
    ! [B: $tType,A: $tType] :
      ( ( ( type_len @ A )
        & ( type_len @ B ) )
     => ! [X2: word @ A,Y2: word @ B] : ( ord_less @ int @ ( minus_minus @ int @ ( semiring_1_unsigned @ A @ int @ X2 ) @ ( semiring_1_unsigned @ B @ int @ Y2 ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ).

% uint_sub_lt2p
thf(fact_7295_ucast__of__nat__small,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ( type_len @ B )
        & ( type_len @ A ) )
     => ! [X2: nat] :
          ( ( ord_less @ nat @ X2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
         => ( ( semiring_1_unsigned @ A @ ( word @ B ) @ ( semiring_1_of_nat @ ( word @ A ) @ X2 ) )
            = ( semiring_1_of_nat @ ( word @ B ) @ X2 ) ) ) ) ).

% ucast_of_nat_small
thf(fact_7296_uint__numeral,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [B2: num] :
          ( ( semiring_1_unsigned @ A @ int @ ( numeral_numeral @ ( word @ A ) @ B2 ) )
          = ( modulo_modulo @ int @ ( numeral_numeral @ int @ B2 ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ).

% uint_numeral
thf(fact_7297_p2__gt__0,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [N2: nat] :
          ( ( ord_less @ ( word @ A ) @ ( zero_zero @ ( word @ A ) ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N2 ) )
          = ( ord_less @ nat @ N2 @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ).

% p2_gt_0
thf(fact_7298_word__of__nat__eq__0__iff,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [N2: nat] :
          ( ( ( semiring_1_of_nat @ ( word @ A ) @ N2 )
            = ( zero_zero @ ( word @ A ) ) )
          = ( dvd_dvd @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) @ N2 ) ) ) ).

% word_of_nat_eq_0_iff
thf(fact_7299_word__of__int__minus,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [I: int] :
          ( ( ring_1_of_int @ ( word @ A ) @ ( minus_minus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) @ I ) )
          = ( ring_1_of_int @ ( word @ A ) @ ( uminus_uminus @ int @ I ) ) ) ) ).

% word_of_int_minus
thf(fact_7300_bit__last__iff,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [W: word @ A] :
          ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ W @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) )
          = ( ord_less @ int @ ( ring_1_signed @ A @ int @ W ) @ ( zero_zero @ int ) ) ) ) ).

% bit_last_iff
thf(fact_7301_unat__ucast,axiom,
    ! [B: $tType,A: $tType] :
      ( ( ( type_len @ A )
        & ( type_len @ B ) )
     => ! [X2: word @ B] :
          ( ( semiring_1_unsigned @ A @ nat @ ( semiring_1_unsigned @ B @ ( word @ A ) @ X2 ) )
          = ( modulo_modulo @ nat @ ( semiring_1_unsigned @ B @ nat @ X2 ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ).

% unat_ucast
thf(fact_7302_mask__lt__2pn,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [N2: nat] :
          ( ( ord_less @ nat @ N2 @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
         => ( ord_less @ ( word @ A ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N2 ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N2 ) ) ) ) ).

% mask_lt_2pn
thf(fact_7303_uint__word__of__int,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [K: int] :
          ( ( semiring_1_unsigned @ A @ int @ ( ring_1_of_int @ ( word @ A ) @ K ) )
          = ( modulo_modulo @ int @ K @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ).

% uint_word_of_int
thf(fact_7304_unat__of__nat,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [X2: nat] :
          ( ( semiring_1_unsigned @ A @ nat @ ( semiring_1_of_nat @ ( word @ A ) @ X2 ) )
          = ( modulo_modulo @ nat @ X2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ).

% unat_of_nat
thf(fact_7305_ucast__less,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ( type_len @ B )
        & ( type_len @ A ) )
     => ! [X2: word @ B] :
          ( ( ord_less @ nat @ ( type_len0_len_of @ B @ ( type2 @ B ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
         => ( ord_less @ ( word @ A ) @ ( semiring_1_unsigned @ B @ ( word @ A ) @ X2 ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ B @ ( type2 @ B ) ) ) ) ) ) ).

% ucast_less
thf(fact_7306_signed__of__int,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ( type_len @ B )
        & ( ring_1 @ A ) )
     => ! [N2: int] :
          ( ( ring_1_signed @ B @ A @ ( ring_1_of_int @ ( word @ B ) @ N2 ) )
          = ( ring_1_of_int @ A @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ B @ ( type2 @ B ) ) @ ( suc @ ( zero_zero @ nat ) ) ) @ N2 ) ) ) ) ).

% signed_of_int
thf(fact_7307_word__of__int__eq__0__iff,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [K: int] :
          ( ( ( ring_1_of_int @ ( word @ A ) @ K )
            = ( zero_zero @ ( word @ A ) ) )
          = ( dvd_dvd @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) @ K ) ) ) ).

% word_of_int_eq_0_iff
thf(fact_7308_of__nat__n__less__equal__power__2,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [N2: nat] :
          ( ( ord_less @ nat @ N2 @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
         => ( ord_less @ ( word @ A ) @ ( semiring_1_of_nat @ ( word @ A ) @ N2 ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N2 ) ) ) ) ).

% of_nat_n_less_equal_power_2
thf(fact_7309_signed__ucast__eq,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( ( ( type_len @ B )
        & ( bit_ri3973907225187159222ations @ A )
        & ( type_len @ C ) )
     => ! [W: word @ B] :
          ( ( ring_1_signed @ C @ A @ ( semiring_1_unsigned @ B @ ( word @ C ) @ W ) )
          = ( bit_ri4674362597316999326ke_bit @ A @ ( minus_minus @ nat @ ( type_len0_len_of @ C @ ( type2 @ C ) ) @ ( suc @ ( zero_zero @ nat ) ) ) @ ( semiring_1_unsigned @ B @ A @ W ) ) ) ) ).

% signed_ucast_eq
thf(fact_7310_complement__nth__w2p,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [N5: nat,N2: nat] :
          ( ( ord_less @ nat @ N5 @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
         => ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N2 ) ) @ N5 )
            = ( N5 != N2 ) ) ) ) ).

% complement_nth_w2p
thf(fact_7311_upper__trivial,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [X2: word @ A] :
          ( ( X2
           != ( minus_minus @ ( word @ A ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) @ ( one_one @ ( word @ A ) ) ) )
         => ( ord_less @ ( word @ A ) @ X2 @ ( minus_minus @ ( word @ A ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) @ ( one_one @ ( word @ A ) ) ) ) ) ) ).

% upper_trivial
thf(fact_7312_range__uint,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ( ( image @ ( word @ A ) @ int @ ( semiring_1_unsigned @ A @ int ) @ ( top_top @ ( set @ ( word @ A ) ) ) )
        = ( set_or7035219750837199246ssThan @ int @ ( zero_zero @ int ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ).

% range_uint
thf(fact_7313_bit__word__scast__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( ( ( type_len @ A )
        & ( type_len @ B ) )
     => ! [W: word @ A,N2: nat] :
          ( ( bit_se5641148757651400278ts_bit @ ( word @ B ) @ ( ring_1_signed @ A @ ( word @ B ) @ W ) @ N2 )
          = ( ( ord_less @ nat @ N2 @ ( type_len0_len_of @ B @ ( type2 @ B ) ) )
            & ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ W @ N2 )
              | ( ( ord_less_eq @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ N2 )
                & ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ W @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) ) ) ).

% bit_word_scast_iff
thf(fact_7314_minus__one__word,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ( ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) )
        = ( minus_minus @ ( word @ A ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) @ ( one_one @ ( word @ A ) ) ) ) ) ).

% minus_one_word
thf(fact_7315_UNIV__eq,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ( ( top_top @ ( set @ ( word @ A ) ) )
        = ( image @ int @ ( word @ A ) @ ( ring_1_of_int @ ( word @ A ) ) @ ( set_or7035219750837199246ssThan @ int @ ( zero_zero @ int ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ) ).

% UNIV_eq
thf(fact_7316_word__power__less__diff,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [N2: nat,Q2: word @ A,M: nat] :
          ( ( ord_less @ ( word @ A ) @ ( times_times @ ( word @ A ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N2 ) @ Q2 ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ M ) )
         => ( ( ord_less @ ( word @ A ) @ Q2 @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ N2 ) ) )
           => ( ord_less @ ( word @ A ) @ Q2 @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ M @ N2 ) ) ) ) ) ) ).

% word_power_less_diff
thf(fact_7317_ucast__mono__le,axiom,
    ! [B: $tType,A: $tType] :
      ( ( ( type_len @ A )
        & ( type_len @ B ) )
     => ! [X2: word @ A,Y2: word @ A] :
          ( ( ord_less_eq @ ( word @ A ) @ X2 @ Y2 )
         => ( ( ord_less @ ( word @ A ) @ Y2 @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ B @ ( type2 @ B ) ) ) )
           => ( ord_less_eq @ ( word @ B ) @ ( semiring_1_unsigned @ A @ ( word @ B ) @ X2 ) @ ( semiring_1_unsigned @ A @ ( word @ B ) @ Y2 ) ) ) ) ) ).

% ucast_mono_le
thf(fact_7318_unat__word__ariths_I7_J,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [A2: word @ A,B2: word @ A] :
          ( ( semiring_1_unsigned @ A @ nat @ ( modulo_modulo @ ( word @ A ) @ A2 @ B2 ) )
          = ( modulo_modulo @ nat @ ( modulo_modulo @ nat @ ( semiring_1_unsigned @ A @ nat @ A2 ) @ ( semiring_1_unsigned @ A @ nat @ B2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ).

% unat_word_ariths(7)
thf(fact_7319_take__bit__word__eq__self__iff,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [N2: nat,W: word @ A] :
          ( ( ( bit_se2584673776208193580ke_bit @ ( word @ A ) @ N2 @ W )
            = W )
          = ( ( ord_less_eq @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ N2 )
            | ( ord_less @ ( word @ A ) @ W @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N2 ) ) ) ) ) ).

% take_bit_word_eq_self_iff
thf(fact_7320_signed__modulo__int__def,axiom,
    ( ( signed6721504322012087516modulo @ int )
    = ( ^ [K4: int,L2: int] : ( minus_minus @ int @ K4 @ ( times_times @ int @ ( signed7115095781618012415divide @ int @ K4 @ L2 ) @ L2 ) ) ) ) ).

% signed_modulo_int_def
thf(fact_7321_signed__push__bit__eq,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ( type_len @ B )
        & ( bit_ri3973907225187159222ations @ A ) )
     => ! [N2: nat,W: word @ B] :
          ( ( ring_1_signed @ B @ A @ ( bit_se4730199178511100633sh_bit @ ( word @ B ) @ N2 @ W ) )
          = ( bit_ri4674362597316999326ke_bit @ A @ ( minus_minus @ nat @ ( type_len0_len_of @ B @ ( type2 @ B ) ) @ ( suc @ ( zero_zero @ nat ) ) ) @ ( bit_se4730199178511100633sh_bit @ A @ N2 @ ( ring_1_signed @ B @ A @ W ) ) ) ) ) ).

% signed_push_bit_eq
thf(fact_7322_msb0,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [X2: word @ A,Y2: word @ A,I: nat] :
          ( ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( bit_se1065995026697491101ons_or @ ( word @ A ) @ X2 @ Y2 ) @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ I ) ) )
            = ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ X2 @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ I ) ) )
              | ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ Y2 @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ I ) ) ) ) )
          & ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X2 @ Y2 ) @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ I ) ) )
            = ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ X2 @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ I ) ) )
              & ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ Y2 @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ I ) ) ) ) )
          & ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( bit_se5824344971392196577ns_xor @ ( word @ A ) @ X2 @ Y2 ) @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ I ) ) )
            = ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ X2 @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ I ) ) )
             != ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ Y2 @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ I ) ) ) ) )
          & ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ X2 ) @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ I ) ) )
            = ( ~ ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ X2 @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ I ) ) ) ) ) ) ) ).

% msb0
thf(fact_7323_ucast__range__less,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ( type_len @ B )
        & ( type_len @ A ) )
     => ( ( ord_less @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( type_len0_len_of @ B @ ( type2 @ B ) ) )
       => ( ( image @ ( word @ A ) @ ( word @ B ) @ ( semiring_1_unsigned @ A @ ( word @ B ) ) @ ( top_top @ ( set @ ( word @ A ) ) ) )
          = ( collect @ ( word @ B )
            @ ^ [X: word @ B] : ( ord_less @ ( word @ B ) @ X @ ( power_power @ ( word @ B ) @ ( numeral_numeral @ ( word @ B ) @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ) ) ).

% ucast_range_less
thf(fact_7324_unat__add__lem,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [X2: word @ A,Y2: word @ A] :
          ( ( ord_less @ nat @ ( plus_plus @ nat @ ( semiring_1_unsigned @ A @ nat @ X2 ) @ ( semiring_1_unsigned @ A @ nat @ Y2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
          = ( ( semiring_1_unsigned @ A @ nat @ ( plus_plus @ ( word @ A ) @ X2 @ Y2 ) )
            = ( plus_plus @ nat @ ( semiring_1_unsigned @ A @ nat @ X2 ) @ ( semiring_1_unsigned @ A @ nat @ Y2 ) ) ) ) ) ).

% unat_add_lem
thf(fact_7325_unat__add__lem_H,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [X2: word @ A,Y2: word @ A] :
          ( ( ord_less @ nat @ ( plus_plus @ nat @ ( semiring_1_unsigned @ A @ nat @ X2 ) @ ( semiring_1_unsigned @ A @ nat @ Y2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
         => ( ( semiring_1_unsigned @ A @ nat @ ( plus_plus @ ( word @ A ) @ X2 @ Y2 ) )
            = ( plus_plus @ nat @ ( semiring_1_unsigned @ A @ nat @ X2 ) @ ( semiring_1_unsigned @ A @ nat @ Y2 ) ) ) ) ) ).

% unat_add_lem'
thf(fact_7326_Word_Oof__nat__neq__0,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [K: nat] :
          ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
         => ( ( ord_less @ nat @ K @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
           => ( ( semiring_1_of_nat @ ( word @ A ) @ K )
             != ( zero_zero @ ( word @ A ) ) ) ) ) ) ).

% Word.of_nat_neq_0
thf(fact_7327_More__Word_Oof__nat__0,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [N2: nat] :
          ( ( ( semiring_1_of_nat @ ( word @ A ) @ N2 )
            = ( zero_zero @ ( word @ A ) ) )
         => ( ( ord_less @ nat @ N2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
           => ( N2
              = ( zero_zero @ nat ) ) ) ) ) ).

% More_Word.of_nat_0
thf(fact_7328_of__nat__mono__maybe__le,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [X2: nat,Y2: nat] :
          ( ( ord_less @ nat @ X2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
         => ( ( ord_less @ nat @ Y2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
           => ( ( ord_less_eq @ nat @ Y2 @ X2 )
              = ( ord_less_eq @ ( word @ A ) @ ( semiring_1_of_nat @ ( word @ A ) @ Y2 ) @ ( semiring_1_of_nat @ ( word @ A ) @ X2 ) ) ) ) ) ) ).

% of_nat_mono_maybe_le
thf(fact_7329_unat__word__ariths_I4_J,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ( ( semiring_1_unsigned @ A @ nat @ ( zero_zero @ ( word @ A ) ) )
        = ( modulo_modulo @ nat @ ( zero_zero @ nat ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ).

% unat_word_ariths(4)
thf(fact_7330_bool__mask_H,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [X2: word @ A] :
          ( ( ord_less @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
         => ( ( ord_less @ ( word @ A ) @ ( zero_zero @ ( word @ A ) ) @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X2 @ ( one_one @ ( word @ A ) ) ) )
            = ( ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X2 @ ( one_one @ ( word @ A ) ) )
              = ( one_one @ ( word @ A ) ) ) ) ) ) ).

% bool_mask'
thf(fact_7331_unat__word__ariths_I1_J,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [A2: word @ A,B2: word @ A] :
          ( ( semiring_1_unsigned @ A @ nat @ ( plus_plus @ ( word @ A ) @ A2 @ B2 ) )
          = ( modulo_modulo @ nat @ ( plus_plus @ nat @ ( semiring_1_unsigned @ A @ nat @ A2 ) @ ( semiring_1_unsigned @ A @ nat @ B2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ).

% unat_word_ariths(1)
thf(fact_7332_uint__range_H,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [X2: word @ A] :
          ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( semiring_1_unsigned @ A @ int @ X2 ) )
          & ( ord_less @ int @ ( semiring_1_unsigned @ A @ int @ X2 ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ).

% uint_range'
thf(fact_7333_sint__lt,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [X2: word @ A] : ( ord_less @ int @ ( ring_1_signed @ A @ int @ X2 ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) ) ) ) ).

% sint_lt
thf(fact_7334_word__of__int__inj,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [X2: int,Y2: int] :
          ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X2 )
            & ( ord_less @ int @ X2 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) )
         => ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y2 )
              & ( ord_less @ int @ Y2 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) )
           => ( ( ( ring_1_of_int @ ( word @ A ) @ X2 )
                = ( ring_1_of_int @ ( word @ A ) @ Y2 ) )
              = ( X2 = Y2 ) ) ) ) ) ).

% word_of_int_inj
thf(fact_7335_word__int__cases,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [X2: word @ A] :
          ~ ! [N4: int] :
              ( ( X2
                = ( ring_1_of_int @ ( word @ A ) @ N4 ) )
             => ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ N4 )
               => ~ ( ord_less @ int @ N4 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ) ).

% word_int_cases
thf(fact_7336_ucast__mono__le_H,axiom,
    ! [B: $tType,A: $tType] :
      ( ( ( type_len @ A )
        & ( type_len @ B ) )
     => ! [Y2: word @ A,X2: word @ A] :
          ( ( ord_less @ nat @ ( semiring_1_unsigned @ A @ nat @ Y2 ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ B @ ( type2 @ B ) ) ) )
         => ( ( ord_less @ nat @ ( type_len0_len_of @ B @ ( type2 @ B ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
           => ( ( ord_less_eq @ ( word @ A ) @ X2 @ Y2 )
             => ( ord_less_eq @ ( word @ B ) @ ( semiring_1_unsigned @ A @ ( word @ B ) @ X2 ) @ ( semiring_1_unsigned @ A @ ( word @ B ) @ Y2 ) ) ) ) ) ) ).

% ucast_mono_le'
thf(fact_7337_of__nat__eq,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [N2: nat,W: word @ A] :
          ( ( ( semiring_1_of_nat @ ( word @ A ) @ N2 )
            = W )
          = ( ? [Q5: nat] :
                ( N2
                = ( plus_plus @ nat @ ( semiring_1_unsigned @ A @ nat @ W ) @ ( times_times @ nat @ Q5 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ) ) ) ).

% of_nat_eq
thf(fact_7338_unat__mult__lem,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [X2: word @ A,Y2: word @ A] :
          ( ( ord_less @ nat @ ( times_times @ nat @ ( semiring_1_unsigned @ A @ nat @ X2 ) @ ( semiring_1_unsigned @ A @ nat @ Y2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
          = ( ( semiring_1_unsigned @ A @ nat @ ( times_times @ ( word @ A ) @ X2 @ Y2 ) )
            = ( times_times @ nat @ ( semiring_1_unsigned @ A @ nat @ X2 ) @ ( semiring_1_unsigned @ A @ nat @ Y2 ) ) ) ) ) ).

% unat_mult_lem
thf(fact_7339_unat__ucast__no__overflow__le,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ( type_len @ B )
        & ( type_len @ A ) )
     => ! [B2: word @ B,F: word @ A] :
          ( ( ord_less @ nat @ ( semiring_1_unsigned @ B @ nat @ B2 ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
         => ( ( ord_less @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( type_len0_len_of @ B @ ( type2 @ B ) ) )
           => ( ( ord_less @ ( word @ B ) @ ( semiring_1_unsigned @ A @ ( word @ B ) @ F ) @ B2 )
              = ( ord_less @ nat @ ( semiring_1_unsigned @ A @ nat @ F ) @ ( semiring_1_unsigned @ B @ nat @ B2 ) ) ) ) ) ) ).

% unat_ucast_no_overflow_le
thf(fact_7340_uint__m2p__not__non__neg,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [X2: word @ A] :
          ~ ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( minus_minus @ int @ ( semiring_1_unsigned @ A @ int @ X2 ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ).

% uint_m2p_not_non_neg
thf(fact_7341_uint__m2p__neg,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [X2: word @ A] : ( ord_less @ int @ ( minus_minus @ int @ ( semiring_1_unsigned @ A @ int @ X2 ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) @ ( zero_zero @ int ) ) ) ).

% uint_m2p_neg
thf(fact_7342_unat__ucast__less__no__overflow,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [N2: nat,F: word @ A] :
          ( ( ord_less @ nat @ N2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
         => ( ( ord_less @ nat @ ( semiring_1_unsigned @ A @ nat @ F ) @ N2 )
           => ( ord_less @ ( word @ A ) @ F @ ( semiring_1_of_nat @ ( word @ A ) @ N2 ) ) ) ) ) ).

% unat_ucast_less_no_overflow
thf(fact_7343_unat__ucast__less__no__overflow__simp,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [N2: nat,F: word @ A] :
          ( ( ord_less @ nat @ N2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
         => ( ( ord_less @ nat @ ( semiring_1_unsigned @ A @ nat @ F ) @ N2 )
            = ( ord_less @ ( word @ A ) @ F @ ( semiring_1_of_nat @ ( word @ A ) @ N2 ) ) ) ) ) ).

% unat_ucast_less_no_overflow_simp
thf(fact_7344_uint__power__lower,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [N2: nat] :
          ( ( ord_less @ nat @ N2 @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
         => ( ( semiring_1_unsigned @ A @ int @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N2 ) )
            = ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N2 ) ) ) ) ).

% uint_power_lower
thf(fact_7345_less__2p__is__upper__bits__unset,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [P4: word @ A,N2: nat] :
          ( ( ord_less @ ( word @ A ) @ P4 @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N2 ) )
          = ( ( ord_less @ nat @ N2 @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
            & ! [N12: nat] :
                ( ( ord_less_eq @ nat @ N2 @ N12 )
               => ( ( ord_less @ nat @ N12 @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
                 => ~ ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ P4 @ N12 ) ) ) ) ) ) ).

% less_2p_is_upper_bits_unset
thf(fact_7346_upper__bits__unset__is__l2p,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [N2: nat,P4: word @ A] :
          ( ( ord_less @ nat @ N2 @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
         => ( ( ! [N12: nat] :
                  ( ( ord_less_eq @ nat @ N2 @ N12 )
                 => ( ( ord_less @ nat @ N12 @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
                   => ~ ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ P4 @ N12 ) ) ) )
            = ( ord_less @ ( word @ A ) @ P4 @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N2 ) ) ) ) ) ).

% upper_bits_unset_is_l2p
thf(fact_7347_nth__bounded,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [X2: word @ A,N2: nat,M: nat] :
          ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ X2 @ N2 )
         => ( ( ord_less @ ( word @ A ) @ X2 @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ M ) )
           => ( ( ord_less_eq @ nat @ M @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
             => ( ord_less @ nat @ N2 @ M ) ) ) ) ) ).

% nth_bounded
thf(fact_7348_uint__add__lem,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [X2: word @ A,Y2: word @ A] :
          ( ( ord_less @ int @ ( plus_plus @ int @ ( semiring_1_unsigned @ A @ int @ X2 ) @ ( semiring_1_unsigned @ A @ int @ Y2 ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
          = ( ( semiring_1_unsigned @ A @ int @ ( plus_plus @ ( word @ A ) @ X2 @ Y2 ) )
            = ( plus_plus @ int @ ( semiring_1_unsigned @ A @ int @ X2 ) @ ( semiring_1_unsigned @ A @ int @ Y2 ) ) ) ) ) ).

% uint_add_lem
thf(fact_7349_uint__word__ariths_I7_J,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ( ( semiring_1_unsigned @ A @ int @ ( zero_zero @ ( word @ A ) ) )
        = ( modulo_modulo @ int @ ( zero_zero @ int ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ).

% uint_word_ariths(7)
thf(fact_7350_uint__word__ariths_I8_J,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ( ( semiring_1_unsigned @ A @ int @ ( one_one @ ( word @ A ) ) )
        = ( modulo_modulo @ int @ ( one_one @ int ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ).

% uint_word_ariths(8)
thf(fact_7351_wi__le,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [N2: int,M: int] :
          ( ( ord_less_eq @ ( word @ A ) @ ( ring_1_of_int @ ( word @ A ) @ N2 ) @ ( ring_1_of_int @ ( word @ A ) @ M ) )
          = ( ord_less_eq @ int @ ( modulo_modulo @ int @ N2 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) @ ( modulo_modulo @ int @ M @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ) ).

% wi_le
thf(fact_7352_uint__word__ariths_I1_J,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [A2: word @ A,B2: word @ A] :
          ( ( semiring_1_unsigned @ A @ int @ ( plus_plus @ ( word @ A ) @ A2 @ B2 ) )
          = ( modulo_modulo @ int @ ( plus_plus @ int @ ( semiring_1_unsigned @ A @ int @ A2 ) @ ( semiring_1_unsigned @ A @ int @ B2 ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ).

% uint_word_ariths(1)
thf(fact_7353_word__2p__mult__inc,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [N2: nat,M: nat] :
          ( ( ord_less @ ( word @ A ) @ ( times_times @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N2 ) ) @ ( times_times @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ M ) ) )
         => ( ( ord_less @ nat @ ( suc @ N2 ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
           => ( ord_less @ ( word @ A ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N2 ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ M ) ) ) ) ) ).

% word_2p_mult_inc
thf(fact_7354_wi__less,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [N2: int,M: int] :
          ( ( ord_less @ ( word @ A ) @ ( ring_1_of_int @ ( word @ A ) @ N2 ) @ ( ring_1_of_int @ ( word @ A ) @ M ) )
          = ( ord_less @ int @ ( modulo_modulo @ int @ N2 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) @ ( modulo_modulo @ int @ M @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ) ).

% wi_less
thf(fact_7355_word__power__less__1,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [Sz: nat] :
          ( ( ord_less @ nat @ Sz @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
         => ( ord_less @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ Sz ) @ ( one_one @ ( word @ A ) ) ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ Sz ) ) ) ) ).

% word_power_less_1
thf(fact_7356_power__2__ge__iff,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [N2: nat] :
          ( ( ord_less @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N2 ) @ ( one_one @ ( word @ A ) ) ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N2 ) )
          = ( ord_less @ nat @ N2 @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ).

% power_2_ge_iff
thf(fact_7357_unat__word__ariths_I2_J,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [A2: word @ A,B2: word @ A] :
          ( ( semiring_1_unsigned @ A @ nat @ ( times_times @ ( word @ A ) @ A2 @ B2 ) )
          = ( modulo_modulo @ nat @ ( times_times @ nat @ ( semiring_1_unsigned @ A @ nat @ A2 ) @ ( semiring_1_unsigned @ A @ nat @ B2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ).

% unat_word_ariths(2)
thf(fact_7358_uint__word__ariths_I4_J,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [A2: word @ A] :
          ( ( semiring_1_unsigned @ A @ int @ ( uminus_uminus @ ( word @ A ) @ A2 ) )
          = ( modulo_modulo @ int @ ( uminus_uminus @ int @ ( semiring_1_unsigned @ A @ int @ A2 ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ).

% uint_word_ariths(4)
thf(fact_7359_unat__word__ariths_I6_J,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [A2: word @ A,B2: word @ A] :
          ( ( semiring_1_unsigned @ A @ nat @ ( divide_divide @ ( word @ A ) @ A2 @ B2 ) )
          = ( modulo_modulo @ nat @ ( divide_divide @ nat @ ( semiring_1_unsigned @ A @ nat @ A2 ) @ ( semiring_1_unsigned @ A @ nat @ B2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ).

% unat_word_ariths(6)
thf(fact_7360_le__mask__iff__lt__2n,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [N2: nat,W: word @ A] :
          ( ( ord_less @ nat @ N2 @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
          = ( ( ord_less_eq @ ( word @ A ) @ W @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N2 ) )
            = ( ord_less @ ( word @ A ) @ W @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N2 ) ) ) ) ) ).

% le_mask_iff_lt_2n
thf(fact_7361_and__mask__less_H,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [N2: nat,W: word @ A] :
          ( ( ord_less @ nat @ N2 @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
         => ( ord_less @ ( word @ A ) @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ W @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N2 ) ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N2 ) ) ) ) ).

% and_mask_less'
thf(fact_7362_eq__mask__less,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [W: word @ A,N2: nat] :
          ( ( W
            = ( bit_se5824344872417868541ns_and @ ( word @ A ) @ W @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N2 ) ) )
         => ( ( ord_less @ nat @ N2 @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
           => ( ord_less @ ( word @ A ) @ W @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N2 ) ) ) ) ) ).

% eq_mask_less
thf(fact_7363_sint__1__cases,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [A2: word @ A] :
          ( ( ( ( type_len0_len_of @ A @ ( type2 @ A ) )
              = ( one_one @ nat ) )
           => ( ( A2
                = ( zero_zero @ ( word @ A ) ) )
             => ( ( ring_1_signed @ A @ int @ A2 )
               != ( zero_zero @ int ) ) ) )
         => ( ( ( ( type_len0_len_of @ A @ ( type2 @ A ) )
                = ( one_one @ nat ) )
             => ( ( A2
                  = ( one_one @ ( word @ A ) ) )
               => ( ( ring_1_signed @ A @ int @ ( one_one @ ( word @ A ) ) )
                 != ( uminus_uminus @ int @ ( one_one @ int ) ) ) ) )
           => ~ ( ( ord_less @ nat @ ( one_one @ nat ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
               => ( ( ring_1_signed @ A @ int @ ( one_one @ ( word @ A ) ) )
                 != ( one_one @ int ) ) ) ) ) ) ).

% sint_1_cases
thf(fact_7364_uint__word__ariths_I2_J,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [A2: word @ A,B2: word @ A] :
          ( ( semiring_1_unsigned @ A @ int @ ( minus_minus @ ( word @ A ) @ A2 @ B2 ) )
          = ( modulo_modulo @ int @ ( minus_minus @ int @ ( semiring_1_unsigned @ A @ int @ A2 ) @ ( semiring_1_unsigned @ A @ int @ B2 ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ).

% uint_word_ariths(2)
thf(fact_7365_uint__mult__lem,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [X2: word @ A,Y2: word @ A] :
          ( ( ord_less @ int @ ( times_times @ int @ ( semiring_1_unsigned @ A @ int @ X2 ) @ ( semiring_1_unsigned @ A @ int @ Y2 ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
          = ( ( semiring_1_unsigned @ A @ int @ ( times_times @ ( word @ A ) @ X2 @ Y2 ) )
            = ( times_times @ int @ ( semiring_1_unsigned @ A @ int @ X2 ) @ ( semiring_1_unsigned @ A @ int @ Y2 ) ) ) ) ) ).

% uint_mult_lem
thf(fact_7366_uint__word__ariths_I3_J,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [A2: word @ A,B2: word @ A] :
          ( ( semiring_1_unsigned @ A @ int @ ( times_times @ ( word @ A ) @ A2 @ B2 ) )
          = ( modulo_modulo @ int @ ( times_times @ int @ ( semiring_1_unsigned @ A @ int @ A2 ) @ ( semiring_1_unsigned @ A @ int @ B2 ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ).

% uint_word_ariths(3)
thf(fact_7367_signed__of__nat,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ( type_len @ B )
        & ( ring_1 @ A ) )
     => ! [N2: nat] :
          ( ( ring_1_signed @ B @ A @ ( semiring_1_of_nat @ ( word @ B ) @ N2 ) )
          = ( ring_1_of_int @ A @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ B @ ( type2 @ B ) ) @ ( suc @ ( zero_zero @ nat ) ) ) @ ( semiring_1_of_nat @ int @ N2 ) ) ) ) ) ).

% signed_of_nat
thf(fact_7368_word__power__mod__div,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [N2: nat,M: nat,X2: word @ A] :
          ( ( ord_less @ nat @ N2 @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
         => ( ( ord_less @ nat @ M @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
           => ( ( divide_divide @ ( word @ A ) @ ( modulo_modulo @ ( word @ A ) @ X2 @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N2 ) ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ M ) )
              = ( modulo_modulo @ ( word @ A ) @ ( divide_divide @ ( word @ A ) @ X2 @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ M ) ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ N2 @ M ) ) ) ) ) ) ) ).

% word_power_mod_div
thf(fact_7369_scast__1_H,axiom,
    ! [B: $tType,A: $tType] :
      ( ( ( type_len @ A )
        & ( type_len @ B ) )
     => ( ( ring_1_signed @ A @ ( word @ B ) @ ( one_one @ ( word @ A ) ) )
        = ( ring_1_of_int @ ( word @ B ) @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) @ ( one_one @ int ) ) ) ) ) ).

% scast_1'
thf(fact_7370_msb1,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [X2: word @ A,Y2: word @ A] :
          ( ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( bit_se1065995026697491101ons_or @ ( word @ A ) @ X2 @ Y2 ) @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) )
            = ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ X2 @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) )
              | ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ Y2 @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ) )
          & ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ X2 @ Y2 ) @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) )
            = ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ X2 @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) )
              & ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ Y2 @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ) )
          & ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( bit_se5824344971392196577ns_xor @ ( word @ A ) @ X2 @ Y2 ) @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) )
            = ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ X2 @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) )
             != ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ Y2 @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ) )
          & ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( bit_ri4277139882892585799ns_not @ ( word @ A ) @ X2 ) @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) )
            = ( ~ ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ X2 @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) ) ).

% msb1
thf(fact_7371_unat__plus__if_H,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [A2: word @ A,B2: word @ A] :
          ( ( ( ord_less @ nat @ ( plus_plus @ nat @ ( semiring_1_unsigned @ A @ nat @ A2 ) @ ( semiring_1_unsigned @ A @ nat @ B2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
           => ( ( semiring_1_unsigned @ A @ nat @ ( plus_plus @ ( word @ A ) @ A2 @ B2 ) )
              = ( plus_plus @ nat @ ( semiring_1_unsigned @ A @ nat @ A2 ) @ ( semiring_1_unsigned @ A @ nat @ B2 ) ) ) )
          & ( ~ ( ord_less @ nat @ ( plus_plus @ nat @ ( semiring_1_unsigned @ A @ nat @ A2 ) @ ( semiring_1_unsigned @ A @ nat @ B2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
           => ( ( semiring_1_unsigned @ A @ nat @ ( plus_plus @ ( word @ A ) @ A2 @ B2 ) )
              = ( minus_minus @ nat @ ( plus_plus @ nat @ ( semiring_1_unsigned @ A @ nat @ A2 ) @ ( semiring_1_unsigned @ A @ nat @ B2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ) ) ).

% unat_plus_if'
thf(fact_7372_unat__word__ariths_I5_J,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ( ( semiring_1_unsigned @ A @ nat @ ( one_one @ ( word @ A ) ) )
        = ( modulo_modulo @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ).

% unat_word_ariths(5)
thf(fact_7373_unat__sub__if_H,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [Y2: word @ A,X2: word @ A] :
          ( ( ( ord_less_eq @ nat @ ( semiring_1_unsigned @ A @ nat @ Y2 ) @ ( semiring_1_unsigned @ A @ nat @ X2 ) )
           => ( ( semiring_1_unsigned @ A @ nat @ ( minus_minus @ ( word @ A ) @ X2 @ Y2 ) )
              = ( minus_minus @ nat @ ( semiring_1_unsigned @ A @ nat @ X2 ) @ ( semiring_1_unsigned @ A @ nat @ Y2 ) ) ) )
          & ( ~ ( ord_less_eq @ nat @ ( semiring_1_unsigned @ A @ nat @ Y2 ) @ ( semiring_1_unsigned @ A @ nat @ X2 ) )
           => ( ( semiring_1_unsigned @ A @ nat @ ( minus_minus @ ( word @ A ) @ X2 @ Y2 ) )
              = ( minus_minus @ nat @ ( plus_plus @ nat @ ( semiring_1_unsigned @ A @ nat @ X2 ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) @ ( semiring_1_unsigned @ A @ nat @ Y2 ) ) ) ) ) ) ).

% unat_sub_if'
thf(fact_7374_sint__less,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [W: word @ A] : ( ord_less @ int @ ( ring_1_signed @ A @ int @ W ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ).

% sint_less
thf(fact_7375_no__olen__add__nat,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [X2: word @ A,Y2: word @ A] :
          ( ( ord_less_eq @ ( word @ A ) @ X2 @ ( plus_plus @ ( word @ A ) @ X2 @ Y2 ) )
          = ( ord_less @ nat @ ( plus_plus @ nat @ ( semiring_1_unsigned @ A @ nat @ X2 ) @ ( semiring_1_unsigned @ A @ nat @ Y2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ).

% no_olen_add_nat
thf(fact_7376_word__add__le__dest,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [I: word @ A,K: word @ A,J: word @ A] :
          ( ( ord_less_eq @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ I @ K ) @ ( plus_plus @ ( word @ A ) @ J @ K ) )
         => ( ( ord_less @ nat @ ( plus_plus @ nat @ ( semiring_1_unsigned @ A @ nat @ I ) @ ( semiring_1_unsigned @ A @ nat @ K ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
           => ( ( ord_less @ nat @ ( plus_plus @ nat @ ( semiring_1_unsigned @ A @ nat @ J ) @ ( semiring_1_unsigned @ A @ nat @ K ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
             => ( ord_less_eq @ ( word @ A ) @ I @ J ) ) ) ) ) ).

% word_add_le_dest
thf(fact_7377_word__add__le__iff,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [I: word @ A,K: word @ A,J: word @ A] :
          ( ( ord_less @ nat @ ( plus_plus @ nat @ ( semiring_1_unsigned @ A @ nat @ I ) @ ( semiring_1_unsigned @ A @ nat @ K ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
         => ( ( ord_less @ nat @ ( plus_plus @ nat @ ( semiring_1_unsigned @ A @ nat @ J ) @ ( semiring_1_unsigned @ A @ nat @ K ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
           => ( ( ord_less_eq @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ I @ K ) @ ( plus_plus @ ( word @ A ) @ J @ K ) )
              = ( ord_less_eq @ ( word @ A ) @ I @ J ) ) ) ) ) ).

% word_add_le_iff
thf(fact_7378_word__add__le__mono1,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [I: word @ A,J: word @ A,K: word @ A] :
          ( ( ord_less_eq @ ( word @ A ) @ I @ J )
         => ( ( ord_less @ nat @ ( plus_plus @ nat @ ( semiring_1_unsigned @ A @ nat @ J ) @ ( semiring_1_unsigned @ A @ nat @ K ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
           => ( ord_less_eq @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ I @ K ) @ ( plus_plus @ ( word @ A ) @ J @ K ) ) ) ) ) ).

% word_add_le_mono1
thf(fact_7379_word__add__le__mono2,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [I: word @ A,J: word @ A,K: word @ A] :
          ( ( ord_less_eq @ ( word @ A ) @ I @ J )
         => ( ( ord_less @ nat @ ( plus_plus @ nat @ ( semiring_1_unsigned @ A @ nat @ J ) @ ( semiring_1_unsigned @ A @ nat @ K ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
           => ( ord_less_eq @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ K @ I ) @ ( plus_plus @ ( word @ A ) @ K @ J ) ) ) ) ) ).

% word_add_le_mono2
thf(fact_7380_smod__int__compares_I3_J,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_eq @ int @ A2 @ ( zero_zero @ int ) )
     => ( ( ord_less @ int @ ( zero_zero @ int ) @ B2 )
       => ( ord_less @ int @ ( uminus_uminus @ int @ B2 ) @ ( signed6721504322012087516modulo @ int @ A2 @ B2 ) ) ) ) ).

% smod_int_compares(3)
thf(fact_7381_smod__int__compares_I5_J,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ A2 )
     => ( ( ord_less @ int @ B2 @ ( zero_zero @ int ) )
       => ( ord_less @ int @ ( signed6721504322012087516modulo @ int @ A2 @ B2 ) @ ( uminus_uminus @ int @ B2 ) ) ) ) ).

% smod_int_compares(5)
thf(fact_7382_word__add__less__dest,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [I: word @ A,K: word @ A,J: word @ A] :
          ( ( ord_less @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ I @ K ) @ ( plus_plus @ ( word @ A ) @ J @ K ) )
         => ( ( ord_less @ nat @ ( plus_plus @ nat @ ( semiring_1_unsigned @ A @ nat @ I ) @ ( semiring_1_unsigned @ A @ nat @ K ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
           => ( ( ord_less @ nat @ ( plus_plus @ nat @ ( semiring_1_unsigned @ A @ nat @ J ) @ ( semiring_1_unsigned @ A @ nat @ K ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
             => ( ord_less @ ( word @ A ) @ I @ J ) ) ) ) ) ).

% word_add_less_dest
thf(fact_7383_word__add__less__iff,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [I: word @ A,K: word @ A,J: word @ A] :
          ( ( ord_less @ nat @ ( plus_plus @ nat @ ( semiring_1_unsigned @ A @ nat @ I ) @ ( semiring_1_unsigned @ A @ nat @ K ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
         => ( ( ord_less @ nat @ ( plus_plus @ nat @ ( semiring_1_unsigned @ A @ nat @ J ) @ ( semiring_1_unsigned @ A @ nat @ K ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
           => ( ( ord_less @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ I @ K ) @ ( plus_plus @ ( word @ A ) @ J @ K ) )
              = ( ord_less @ ( word @ A ) @ I @ J ) ) ) ) ) ).

% word_add_less_iff
thf(fact_7384_word__add__less__mono1,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [I: word @ A,J: word @ A,K: word @ A] :
          ( ( ord_less @ ( word @ A ) @ I @ J )
         => ( ( ord_less @ nat @ ( plus_plus @ nat @ ( semiring_1_unsigned @ A @ nat @ J ) @ ( semiring_1_unsigned @ A @ nat @ K ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
           => ( ord_less @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ I @ K ) @ ( plus_plus @ ( word @ A ) @ J @ K ) ) ) ) ) ).

% word_add_less_mono1
thf(fact_7385_unat__minus__one__word,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ( ( semiring_1_unsigned @ A @ nat @ ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) )
        = ( minus_minus @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) @ ( one_one @ nat ) ) ) ) ).

% unat_minus_one_word
thf(fact_7386_sint__ge,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [X2: word @ A] : ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) ) ) @ ( ring_1_signed @ A @ int @ X2 ) ) ) ).

% sint_ge
thf(fact_7387_unat__less__power,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [Sz: nat,K: word @ A] :
          ( ( ord_less @ nat @ Sz @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
         => ( ( ord_less @ ( word @ A ) @ K @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ Sz ) )
           => ( ord_less @ nat @ ( semiring_1_unsigned @ A @ nat @ K ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Sz ) ) ) ) ) ).

% unat_less_power
thf(fact_7388_word__of__int__inverse,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [R4: int,A2: word @ A] :
          ( ( ( ring_1_of_int @ ( word @ A ) @ R4 )
            = A2 )
         => ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ R4 )
           => ( ( ord_less @ int @ R4 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
             => ( ( semiring_1_unsigned @ A @ int @ A2 )
                = R4 ) ) ) ) ) ).

% word_of_int_inverse
thf(fact_7389_uint__split__asm,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [P: int > $o,X2: word @ A] :
          ( ( P @ ( semiring_1_unsigned @ A @ int @ X2 ) )
          = ( ~ ? [I2: int] :
                  ( ( ( ring_1_of_int @ ( word @ A ) @ I2 )
                    = X2 )
                  & ( ord_less_eq @ int @ ( zero_zero @ int ) @ I2 )
                  & ( ord_less @ int @ I2 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
                  & ~ ( P @ I2 ) ) ) ) ) ).

% uint_split_asm
thf(fact_7390_uint__split,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [P: int > $o,X2: word @ A] :
          ( ( P @ ( semiring_1_unsigned @ A @ int @ X2 ) )
          = ( ! [I2: int] :
                ( ( ( ( ring_1_of_int @ ( word @ A ) @ I2 )
                    = X2 )
                  & ( ord_less_eq @ int @ ( zero_zero @ int ) @ I2 )
                  & ( ord_less @ int @ I2 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) )
               => ( P @ I2 ) ) ) ) ) ).

% uint_split
thf(fact_7391_word__mult__less__dest,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [I: word @ A,K: word @ A,J: word @ A] :
          ( ( ord_less @ ( word @ A ) @ ( times_times @ ( word @ A ) @ I @ K ) @ ( times_times @ ( word @ A ) @ J @ K ) )
         => ( ( ord_less @ nat @ ( times_times @ nat @ ( semiring_1_unsigned @ A @ nat @ I ) @ ( semiring_1_unsigned @ A @ nat @ K ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
           => ( ( ord_less @ nat @ ( times_times @ nat @ ( semiring_1_unsigned @ A @ nat @ J ) @ ( semiring_1_unsigned @ A @ nat @ K ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
             => ( ord_less @ ( word @ A ) @ I @ J ) ) ) ) ) ).

% word_mult_less_dest
thf(fact_7392_div__lt_H,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [I: word @ A,K: word @ A,X2: word @ A] :
          ( ( ord_less_eq @ ( word @ A ) @ I @ ( divide_divide @ ( word @ A ) @ K @ X2 ) )
         => ( ord_less @ nat @ ( times_times @ nat @ ( semiring_1_unsigned @ A @ nat @ I ) @ ( semiring_1_unsigned @ A @ nat @ X2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ).

% div_lt'
thf(fact_7393_div__lt_H_H,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [I: word @ A,K: word @ A,X2: word @ A] :
          ( ( ord_less @ ( word @ A ) @ I @ ( divide_divide @ ( word @ A ) @ K @ X2 ) )
         => ( ord_less @ nat @ ( times_times @ nat @ ( semiring_1_unsigned @ A @ nat @ I ) @ ( semiring_1_unsigned @ A @ nat @ X2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ).

% div_lt''
thf(fact_7394_double__eq__zero__iff,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [A2: word @ A] :
          ( ( ( times_times @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ A2 )
            = ( zero_zero @ ( word @ A ) ) )
          = ( ( A2
              = ( zero_zero @ ( word @ A ) ) )
            | ( A2
              = ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) ) ).

% double_eq_zero_iff
thf(fact_7395_no__olen__add,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [X2: word @ A,Y2: word @ A] :
          ( ( ord_less_eq @ ( word @ A ) @ X2 @ ( plus_plus @ ( word @ A ) @ X2 @ Y2 ) )
          = ( ord_less @ int @ ( plus_plus @ int @ ( semiring_1_unsigned @ A @ int @ X2 ) @ ( semiring_1_unsigned @ A @ int @ Y2 ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ).

% no_olen_add
thf(fact_7396_no__olen__add_H,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [X2: word @ A,Y2: word @ A] :
          ( ( ord_less_eq @ ( word @ A ) @ X2 @ ( plus_plus @ ( word @ A ) @ Y2 @ X2 ) )
          = ( ord_less @ int @ ( plus_plus @ int @ ( semiring_1_unsigned @ A @ int @ Y2 ) @ ( semiring_1_unsigned @ A @ int @ X2 ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ).

% no_olen_add'
thf(fact_7397_word__le__exists_H,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [X2: word @ A,Y2: word @ A] :
          ( ( ord_less_eq @ ( word @ A ) @ X2 @ Y2 )
         => ? [Z3: word @ A] :
              ( ( Y2
                = ( plus_plus @ ( word @ A ) @ X2 @ Z3 ) )
              & ( ord_less @ int @ ( plus_plus @ int @ ( semiring_1_unsigned @ A @ int @ X2 ) @ ( semiring_1_unsigned @ A @ int @ Z3 ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ) ).

% word_le_exists'
thf(fact_7398_More__Word_Oof__nat__power,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [P4: nat,X2: nat] :
          ( ( ord_less @ nat @ P4 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ X2 ) )
         => ( ( ord_less @ nat @ X2 @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
           => ( ord_less @ ( word @ A ) @ ( semiring_1_of_nat @ ( word @ A ) @ P4 ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ X2 ) ) ) ) ) ).

% More_Word.of_nat_power
thf(fact_7399_uint__plus__if_H,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [A2: word @ A,B2: word @ A] :
          ( ( ( ord_less @ int @ ( plus_plus @ int @ ( semiring_1_unsigned @ A @ int @ A2 ) @ ( semiring_1_unsigned @ A @ int @ B2 ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
           => ( ( semiring_1_unsigned @ A @ int @ ( plus_plus @ ( word @ A ) @ A2 @ B2 ) )
              = ( plus_plus @ int @ ( semiring_1_unsigned @ A @ int @ A2 ) @ ( semiring_1_unsigned @ A @ int @ B2 ) ) ) )
          & ( ~ ( ord_less @ int @ ( plus_plus @ int @ ( semiring_1_unsigned @ A @ int @ A2 ) @ ( semiring_1_unsigned @ A @ int @ B2 ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
           => ( ( semiring_1_unsigned @ A @ int @ ( plus_plus @ ( word @ A ) @ A2 @ B2 ) )
              = ( minus_minus @ int @ ( plus_plus @ int @ ( semiring_1_unsigned @ A @ int @ A2 ) @ ( semiring_1_unsigned @ A @ int @ B2 ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ) ) ).

% uint_plus_if'
thf(fact_7400_word__less__power__trans2,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [N2: word @ A,M: nat,K: nat] :
          ( ( ord_less @ ( word @ A ) @ N2 @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ M @ K ) ) )
         => ( ( ord_less_eq @ nat @ K @ M )
           => ( ( ord_less @ nat @ M @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
             => ( ord_less @ ( word @ A ) @ ( times_times @ ( word @ A ) @ N2 @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ K ) ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ M ) ) ) ) ) ) ).

% word_less_power_trans2
thf(fact_7401_word__less__power__trans,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [N2: word @ A,M: nat,K: nat] :
          ( ( ord_less @ ( word @ A ) @ N2 @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ M @ K ) ) )
         => ( ( ord_less_eq @ nat @ K @ M )
           => ( ( ord_less @ nat @ M @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
             => ( ord_less @ ( word @ A ) @ ( times_times @ ( word @ A ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ K ) @ N2 ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ M ) ) ) ) ) ) ).

% word_less_power_trans
thf(fact_7402_uint__sub__if_H,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [B2: word @ A,A2: word @ A] :
          ( ( ( ord_less_eq @ int @ ( semiring_1_unsigned @ A @ int @ B2 ) @ ( semiring_1_unsigned @ A @ int @ A2 ) )
           => ( ( semiring_1_unsigned @ A @ int @ ( minus_minus @ ( word @ A ) @ A2 @ B2 ) )
              = ( minus_minus @ int @ ( semiring_1_unsigned @ A @ int @ A2 ) @ ( semiring_1_unsigned @ A @ int @ B2 ) ) ) )
          & ( ~ ( ord_less_eq @ int @ ( semiring_1_unsigned @ A @ int @ B2 ) @ ( semiring_1_unsigned @ A @ int @ A2 ) )
           => ( ( semiring_1_unsigned @ A @ int @ ( minus_minus @ ( word @ A ) @ A2 @ B2 ) )
              = ( plus_plus @ int @ ( minus_minus @ int @ ( semiring_1_unsigned @ A @ int @ A2 ) @ ( semiring_1_unsigned @ A @ int @ B2 ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ) ) ).

% uint_sub_if'
thf(fact_7403_word__less__two__pow__divI,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [X2: word @ A,N2: nat,M: nat] :
          ( ( ord_less @ ( word @ A ) @ X2 @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ N2 @ M ) ) )
         => ( ( ord_less_eq @ nat @ M @ N2 )
           => ( ( ord_less @ nat @ N2 @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
             => ( ord_less @ ( word @ A ) @ X2 @ ( divide_divide @ ( word @ A ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N2 ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ M ) ) ) ) ) ) ) ).

% word_less_two_pow_divI
thf(fact_7404_uint__neg__numeral,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [B2: num] :
          ( ( semiring_1_unsigned @ A @ int @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ B2 ) ) )
          = ( modulo_modulo @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ B2 ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ).

% uint_neg_numeral
thf(fact_7405_word__power__nonzero,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [X2: word @ A,N2: nat] :
          ( ( ord_less @ ( word @ A ) @ X2 @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ N2 ) ) )
         => ( ( ord_less @ nat @ N2 @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
           => ( ( X2
               != ( zero_zero @ ( word @ A ) ) )
             => ( ( times_times @ ( word @ A ) @ X2 @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N2 ) )
               != ( zero_zero @ ( word @ A ) ) ) ) ) ) ) ).

% word_power_nonzero
thf(fact_7406_div__lt__uint_H,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [I: word @ A,K: word @ A,X2: word @ A] :
          ( ( ord_less_eq @ ( word @ A ) @ I @ ( divide_divide @ ( word @ A ) @ K @ X2 ) )
         => ( ord_less @ int @ ( times_times @ int @ ( semiring_1_unsigned @ A @ int @ I ) @ ( semiring_1_unsigned @ A @ int @ X2 ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ).

% div_lt_uint'
thf(fact_7407_mult__pow2__inj,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [M: nat,N2: nat,X2: word @ A,Y2: word @ A] :
          ( ( ord_less_eq @ nat @ ( plus_plus @ nat @ M @ N2 ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
         => ( ( ord_less_eq @ ( word @ A ) @ X2 @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ M ) )
           => ( ( ord_less_eq @ ( word @ A ) @ Y2 @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ M ) )
             => ( ( ( times_times @ ( word @ A ) @ X2 @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N2 ) )
                  = ( times_times @ ( word @ A ) @ Y2 @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N2 ) ) )
               => ( X2 = Y2 ) ) ) ) ) ) ).

% mult_pow2_inj
thf(fact_7408_div__lt__uint_H_H,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [I: word @ A,K: word @ A,X2: word @ A] :
          ( ( ord_less @ ( word @ A ) @ I @ ( divide_divide @ ( word @ A ) @ K @ X2 ) )
         => ( ord_less @ int @ ( times_times @ int @ ( semiring_1_unsigned @ A @ int @ I ) @ ( semiring_1_unsigned @ A @ int @ X2 ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ).

% div_lt_uint''
thf(fact_7409_push__bit__word__eq__nonzero,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [W: word @ A,M: nat,N2: nat] :
          ( ( ord_less @ ( word @ A ) @ W @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ M ) )
         => ( ( ord_less @ nat @ ( plus_plus @ nat @ M @ N2 ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
           => ( ( W
               != ( zero_zero @ ( word @ A ) ) )
             => ( ( bit_se4730199178511100633sh_bit @ ( word @ A ) @ N2 @ W )
               != ( zero_zero @ ( word @ A ) ) ) ) ) ) ) ).

% push_bit_word_eq_nonzero
thf(fact_7410_smod__int__alt__def,axiom,
    ( ( signed6721504322012087516modulo @ int )
    = ( ^ [A5: int,B5: int] : ( times_times @ int @ ( sgn_sgn @ int @ A5 ) @ ( modulo_modulo @ int @ ( abs_abs @ int @ A5 ) @ ( abs_abs @ int @ B5 ) ) ) ) ) ).

% smod_int_alt_def
thf(fact_7411_uint__and__mask__or__full,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [N2: word @ A,Mask1: word @ A,Mask2: int] :
          ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ N2 @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) )
         => ( ( Mask1
              = ( bit_se2239418461657761734s_mask @ ( word @ A ) @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) ) )
           => ( ( Mask2
                = ( bit_se4730199178511100633sh_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) @ ( one_one @ int ) ) )
             => ( ( bit_se1065995026697491101ons_or @ int @ ( semiring_1_unsigned @ A @ int @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ N2 @ Mask1 ) ) @ Mask2 )
                = ( semiring_1_unsigned @ A @ int @ N2 ) ) ) ) ) ) ).

% uint_and_mask_or_full
thf(fact_7412_sint__greater__eq,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [W: word @ A] : ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ) @ ( ring_1_signed @ A @ int @ W ) ) ) ).

% sint_greater_eq
thf(fact_7413_int__eq__sint,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [X2: nat] :
          ( ( ord_less @ nat @ X2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) ) )
         => ( ( ring_1_signed @ A @ int @ ( semiring_1_of_nat @ ( word @ A ) @ X2 ) )
            = ( semiring_1_of_nat @ int @ X2 ) ) ) ) ).

% int_eq_sint
thf(fact_7414_word__mult__less__mono1,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [I: word @ A,J: word @ A,K: word @ A] :
          ( ( ord_less @ ( word @ A ) @ I @ J )
         => ( ( ord_less @ ( word @ A ) @ ( zero_zero @ ( word @ A ) ) @ K )
           => ( ( ord_less @ nat @ ( times_times @ nat @ ( semiring_1_unsigned @ A @ nat @ J ) @ ( semiring_1_unsigned @ A @ nat @ K ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
             => ( ord_less @ ( word @ A ) @ ( times_times @ ( word @ A ) @ I @ K ) @ ( times_times @ ( word @ A ) @ J @ K ) ) ) ) ) ) ).

% word_mult_less_mono1
thf(fact_7415_word__mult__less__cancel,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [K: word @ A,I: word @ A,J: word @ A] :
          ( ( ord_less @ ( word @ A ) @ ( zero_zero @ ( word @ A ) ) @ K )
         => ( ( ord_less @ nat @ ( times_times @ nat @ ( semiring_1_unsigned @ A @ nat @ I ) @ ( semiring_1_unsigned @ A @ nat @ K ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
           => ( ( ord_less @ nat @ ( times_times @ nat @ ( semiring_1_unsigned @ A @ nat @ J ) @ ( semiring_1_unsigned @ A @ nat @ K ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
             => ( ( ord_less @ ( word @ A ) @ ( times_times @ ( word @ A ) @ I @ K ) @ ( times_times @ ( word @ A ) @ J @ K ) )
                = ( ord_less @ ( word @ A ) @ I @ J ) ) ) ) ) ) ).

% word_mult_less_cancel
thf(fact_7416_le__2p__upper__bits,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [P4: word @ A,N2: nat] :
          ( ( ord_less_eq @ ( word @ A ) @ P4 @ ( minus_minus @ ( word @ A ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N2 ) @ ( one_one @ ( word @ A ) ) ) )
         => ( ( ord_less @ nat @ N2 @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
           => ! [N6: nat] :
                ( ( ord_less_eq @ nat @ N2 @ N6 )
               => ( ( ord_less @ nat @ N6 @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
                 => ~ ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ P4 @ N6 ) ) ) ) ) ) ).

% le_2p_upper_bits
thf(fact_7417_le2p__bits__unset,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [P4: word @ A,N2: nat] :
          ( ( ord_less_eq @ ( word @ A ) @ P4 @ ( minus_minus @ ( word @ A ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N2 ) @ ( one_one @ ( word @ A ) ) ) )
         => ! [N6: nat] :
              ( ( ord_less_eq @ nat @ N2 @ N6 )
             => ( ( ord_less @ nat @ N6 @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
               => ~ ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ P4 @ N6 ) ) ) ) ) ).

% le2p_bits_unset
thf(fact_7418_word__add__offset__less,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [Y2: word @ A,N2: nat,X2: word @ A,M: nat,Sz: nat] :
          ( ( ord_less @ ( word @ A ) @ Y2 @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N2 ) )
         => ( ( ord_less @ ( word @ A ) @ X2 @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ M ) )
           => ( ( ord_less @ nat @ Sz @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
             => ( ( ord_less @ ( word @ A ) @ X2 @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ N2 ) ) )
               => ( ( Sz
                    = ( plus_plus @ nat @ M @ N2 ) )
                 => ( ord_less @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ ( times_times @ ( word @ A ) @ X2 @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N2 ) ) @ Y2 ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ Sz ) ) ) ) ) ) ) ) ).

% word_add_offset_less
thf(fact_7419_bit__horner__sum__uint__exp__iff,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [Ws: list @ ( word @ A ),N2: nat] :
          ( ( bit_se5641148757651400278ts_bit @ int @ ( groups4207007520872428315er_sum @ ( word @ A ) @ int @ ( semiring_1_unsigned @ A @ int ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) @ Ws ) @ N2 )
          = ( ( ord_less @ nat @ ( divide_divide @ nat @ N2 @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) @ ( size_size @ ( list @ ( word @ A ) ) @ Ws ) )
            & ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( nth @ ( word @ A ) @ Ws @ ( divide_divide @ nat @ N2 @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) @ ( modulo_modulo @ nat @ N2 @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ) ).

% bit_horner_sum_uint_exp_iff
thf(fact_7420_div__power__helper,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [X2: nat,Y2: nat] :
          ( ( ord_less_eq @ nat @ X2 @ Y2 )
         => ( ( ord_less @ nat @ Y2 @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
           => ( ( divide_divide @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ Y2 ) @ ( one_one @ ( word @ A ) ) ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ X2 ) )
              = ( minus_minus @ ( word @ A ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ Y2 @ X2 ) ) @ ( one_one @ ( word @ A ) ) ) ) ) ) ) ).

% div_power_helper
thf(fact_7421_even__mult__exp__div__word__iff,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [A2: word @ A,M: nat,N2: nat] :
          ( ( dvd_dvd @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ ( divide_divide @ ( word @ A ) @ ( times_times @ ( word @ A ) @ A2 @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ M ) ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N2 ) ) )
          = ( ~ ( ( ord_less_eq @ nat @ M @ N2 )
                & ( ord_less @ nat @ N2 @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
                & ~ ( dvd_dvd @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ ( divide_divide @ ( word @ A ) @ A2 @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ N2 @ M ) ) ) ) ) ) ) ) ).

% even_mult_exp_div_word_iff
thf(fact_7422_Suc__2p__unat__mask,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [N2: nat,K: nat] :
          ( ( ord_less @ nat @ N2 @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
         => ( ( suc @ ( plus_plus @ nat @ ( times_times @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) @ K ) @ ( semiring_1_unsigned @ A @ nat @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N2 ) ) ) )
            = ( times_times @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) @ ( plus_plus @ nat @ K @ ( one_one @ nat ) ) ) ) ) ) ).

% Suc_2p_unat_mask
thf(fact_7423_sint__of__nat__le,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [B2: nat,A2: nat] :
          ( ( ord_less @ nat @ B2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) ) )
         => ( ( ord_less_eq @ nat @ A2 @ B2 )
           => ( ord_less_eq @ int @ ( ring_1_signed @ A @ int @ ( semiring_1_of_nat @ ( word @ A ) @ A2 ) ) @ ( ring_1_signed @ A @ int @ ( semiring_1_of_nat @ ( word @ A ) @ B2 ) ) ) ) ) ) ).

% sint_of_nat_le
thf(fact_7424_sint__of__nat__ge__zero,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [X2: nat] :
          ( ( ord_less @ nat @ X2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) ) )
         => ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( ring_1_signed @ A @ int @ ( semiring_1_of_nat @ ( word @ A ) @ X2 ) ) ) ) ) ).

% sint_of_nat_ge_zero
thf(fact_7425_sint__int__max__plus__1,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ( ( ring_1_signed @ A @ int @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) ) )
        = ( uminus_uminus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) ).

% sint_int_max_plus_1
thf(fact_7426_sint__of__int__eq,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [X2: int] :
          ( ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) ) ) @ X2 )
         => ( ( ord_less @ int @ X2 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) ) )
           => ( ( ring_1_signed @ A @ int @ ( ring_1_of_int @ ( word @ A ) @ X2 ) )
              = X2 ) ) ) ) ).

% sint_of_int_eq
thf(fact_7427_sint__numeral,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [B2: num] :
          ( ( ring_1_signed @ A @ int @ ( numeral_numeral @ ( word @ A ) @ B2 ) )
          = ( minus_minus @ int @ ( modulo_modulo @ int @ ( plus_plus @ int @ ( numeral_numeral @ int @ B2 ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) ) ) ) ) ).

% sint_numeral
thf(fact_7428_word__mult__le__iff,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [K: word @ A,I: word @ A,J: word @ A] :
          ( ( ord_less @ ( word @ A ) @ ( zero_zero @ ( word @ A ) ) @ K )
         => ( ( ord_less @ nat @ ( times_times @ nat @ ( semiring_1_unsigned @ A @ nat @ I ) @ ( semiring_1_unsigned @ A @ nat @ K ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
           => ( ( ord_less @ nat @ ( times_times @ nat @ ( semiring_1_unsigned @ A @ nat @ J ) @ ( semiring_1_unsigned @ A @ nat @ K ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
             => ( ( ord_less_eq @ ( word @ A ) @ ( times_times @ ( word @ A ) @ I @ K ) @ ( times_times @ ( word @ A ) @ J @ K ) )
                = ( ord_less_eq @ ( word @ A ) @ I @ J ) ) ) ) ) ) ).

% word_mult_le_iff
thf(fact_7429_word__mult__le__mono1,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [I: word @ A,J: word @ A,K: word @ A] :
          ( ( ord_less_eq @ ( word @ A ) @ I @ J )
         => ( ( ord_less @ ( word @ A ) @ ( zero_zero @ ( word @ A ) ) @ K )
           => ( ( ord_less @ nat @ ( times_times @ nat @ ( semiring_1_unsigned @ A @ nat @ J ) @ ( semiring_1_unsigned @ A @ nat @ K ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
             => ( ord_less_eq @ ( word @ A ) @ ( times_times @ ( word @ A ) @ I @ K ) @ ( times_times @ ( word @ A ) @ J @ K ) ) ) ) ) ) ).

% word_mult_le_mono1
thf(fact_7430_int__word__sint,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [X2: int] :
          ( ( ring_1_signed @ A @ int @ ( ring_1_of_int @ ( word @ A ) @ X2 ) )
          = ( minus_minus @ int @ ( modulo_modulo @ int @ ( plus_plus @ int @ X2 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) ) ) ) ) ).

% int_word_sint
thf(fact_7431_Word_Oword__div__mult,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [Y2: word @ A,X2: word @ A] :
          ( ( ord_less @ ( word @ A ) @ ( zero_zero @ ( word @ A ) ) @ Y2 )
         => ( ( ord_less @ nat @ ( times_times @ nat @ ( semiring_1_unsigned @ A @ nat @ X2 ) @ ( semiring_1_unsigned @ A @ nat @ Y2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
           => ( ( divide_divide @ ( word @ A ) @ ( times_times @ ( word @ A ) @ X2 @ Y2 ) @ Y2 )
              = X2 ) ) ) ) ).

% Word.word_div_mult
thf(fact_7432_of__nat__less__two__pow__div__set,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [N2: nat,M: nat] :
          ( ( ord_less @ nat @ N2 @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
         => ( ( collect @ ( word @ A )
              @ ^ [X: word @ A] : ( ord_less @ ( word @ A ) @ X @ ( divide_divide @ ( word @ A ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N2 ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ M ) ) ) )
            = ( image @ nat @ ( word @ A ) @ ( semiring_1_of_nat @ ( word @ A ) )
              @ ( collect @ nat
                @ ^ [K4: nat] : ( ord_less @ nat @ K4 @ ( divide_divide @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) ) ) ) ) ) ) ) ).

% of_nat_less_two_pow_div_set
thf(fact_7433_sint__int__min,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ( ( ring_1_signed @ A @ int @ ( uminus_uminus @ ( word @ A ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ) )
        = ( uminus_uminus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) ).

% sint_int_min
thf(fact_7434_word__bit__induct,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [P: ( word @ A ) > $o,A2: word @ A] :
          ( ( P @ ( zero_zero @ ( word @ A ) ) )
         => ( ! [A4: word @ A] :
                ( ( P @ A4 )
               => ( ( ord_less @ ( word @ A ) @ ( zero_zero @ ( word @ A ) ) @ A4 )
                 => ( ( ord_less @ ( word @ A ) @ A4 @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) ) )
                   => ( P @ ( times_times @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ A4 ) ) ) ) )
           => ( ! [A4: word @ A] :
                  ( ( P @ A4 )
                 => ( ( ord_less @ ( word @ A ) @ A4 @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) ) )
                   => ( P @ ( plus_plus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) @ ( times_times @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ A4 ) ) ) ) )
             => ( P @ A2 ) ) ) ) ) ).

% word_bit_induct
thf(fact_7435_word__less__power__trans__ofnat,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [N2: nat,M: nat,K: nat] :
          ( ( ord_less @ nat @ N2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ M @ K ) ) )
         => ( ( ord_less_eq @ nat @ K @ M )
           => ( ( ord_less @ nat @ M @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
             => ( ord_less @ ( word @ A ) @ ( times_times @ ( word @ A ) @ ( semiring_1_of_nat @ ( word @ A ) @ N2 ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ K ) ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ M ) ) ) ) ) ) ).

% word_less_power_trans_ofnat
thf(fact_7436_unat__mult__power__lem,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [K: nat,Sz: nat] :
          ( ( ord_less @ nat @ K @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ Sz ) ) )
         => ( ( semiring_1_unsigned @ A @ nat @ ( times_times @ ( word @ A ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ Sz ) @ ( semiring_1_of_nat @ ( word @ A ) @ K ) ) )
            = ( times_times @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Sz ) @ K ) ) ) ) ).

% unat_mult_power_lem
thf(fact_7437_bit__word__half__eq,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [A2: word @ A,B2: $o] :
          ( ( ord_less @ ( word @ A ) @ A2 @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) ) )
         => ( ( divide_divide @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ ( zero_neq_one_of_bool @ ( word @ A ) @ B2 ) @ ( times_times @ ( word @ A ) @ A2 @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) ) ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) )
            = A2 ) ) ) ).

% bit_word_half_eq
thf(fact_7438_word__of__int__via__signed,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [Mask: int,Shift: int,Index: nat,Overflow: int,Least: int,I: int,Arbitrary1: int > ( word @ A ),Arbitrary2: int > ( word @ A )] :
          ( ( Mask
            = ( bit_se2239418461657761734s_mask @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
         => ( ( Shift
              = ( bit_se4730199178511100633sh_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ int ) ) )
           => ( ( Index
                = ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) )
             => ( ( Overflow
                  = ( bit_se4730199178511100633sh_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) @ ( one_one @ int ) ) )
               => ( ( Least
                    = ( uminus_uminus @ int @ Overflow ) )
                 => ( ( ring_1_of_int @ ( word @ A ) @ I )
                    = ( if @ ( word @ A ) @ ( bit_se5641148757651400278ts_bit @ int @ ( bit_se5824344872417868541ns_and @ int @ I @ Mask ) @ Index )
                      @ ( if @ ( word @ A )
                        @ ( ( ord_less @ int @ ( minus_minus @ int @ ( bit_se5824344872417868541ns_and @ int @ I @ Mask ) @ Shift ) @ Least )
                          | ( ord_less_eq @ int @ Overflow @ ( minus_minus @ int @ ( bit_se5824344872417868541ns_and @ int @ I @ Mask ) @ Shift ) ) )
                        @ ( Arbitrary1 @ ( bit_se5824344872417868541ns_and @ int @ I @ Mask ) )
                        @ ( ring_1_of_int @ ( word @ A ) @ ( minus_minus @ int @ ( bit_se5824344872417868541ns_and @ int @ I @ Mask ) @ Shift ) ) )
                      @ ( if @ ( word @ A )
                        @ ( ( ord_less @ int @ ( bit_se5824344872417868541ns_and @ int @ I @ Mask ) @ Least )
                          | ( ord_less_eq @ int @ Overflow @ ( bit_se5824344872417868541ns_and @ int @ I @ Mask ) ) )
                        @ ( Arbitrary2 @ ( bit_se5824344872417868541ns_and @ int @ I @ Mask ) )
                        @ ( ring_1_of_int @ ( word @ A ) @ ( bit_se5824344872417868541ns_and @ int @ I @ Mask ) ) ) ) ) ) ) ) ) ) ) ).

% word_of_int_via_signed
thf(fact_7439_Suc__div__unat__helper,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [Sz: nat,Us: nat] :
          ( ( ord_less @ nat @ Sz @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
         => ( ( ord_less_eq @ nat @ Us @ Sz )
           => ( ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ Sz @ Us ) )
              = ( suc @ ( semiring_1_unsigned @ A @ nat @ ( divide_divide @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ Sz ) @ ( one_one @ ( word @ A ) ) ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ Us ) ) ) ) ) ) ) ) ).

% Suc_div_unat_helper
thf(fact_7440_alignUp__div__helper,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [K: nat,N2: nat,X2: word @ A,A2: word @ A] :
          ( ( ord_less @ nat @ K @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ N2 ) ) )
         => ( ( X2
              = ( times_times @ ( word @ A ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N2 ) @ ( semiring_1_of_nat @ ( word @ A ) @ K ) ) )
           => ( ( ord_less_eq @ ( word @ A ) @ A2 @ X2 )
             => ( ( ord_less @ nat @ N2 @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
               => ( ( ( modulo_modulo @ ( word @ A ) @ A2 @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N2 ) )
                   != ( zero_zero @ ( word @ A ) ) )
                 => ( ord_less @ ( word @ A ) @ ( divide_divide @ ( word @ A ) @ A2 @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ N2 ) ) @ ( semiring_1_of_nat @ ( word @ A ) @ K ) ) ) ) ) ) ) ) ).

% alignUp_div_helper
thf(fact_7441_less__eq__decr__length__iff,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [N2: nat] :
          ( ( ord_less_eq @ nat @ N2 @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) )
          = ( ord_less @ nat @ N2 @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ).

% less_eq_decr_length_iff
thf(fact_7442_decr__length__less__iff,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [N2: nat] :
          ( ( ord_less @ nat @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) @ N2 )
          = ( ord_less_eq @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ N2 ) ) ) ).

% decr_length_less_iff
thf(fact_7443_len__gt__0,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ( ord_less @ nat @ ( zero_zero @ nat ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ).

% len_gt_0
thf(fact_7444_length__not__greater__eq__2__iff,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ( ( ~ ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
        = ( ( type_len0_len_of @ A @ ( type2 @ A ) )
          = ( one_one @ nat ) ) ) ) ).

% length_not_greater_eq_2_iff
thf(fact_7445_len__of__finite__1__def,axiom,
    ( ( type_len0_len_of @ finite_1 )
    = ( ^ [X: itself @ finite_1] : ( one_one @ nat ) ) ) ).

% len_of_finite_1_def
thf(fact_7446_len__num1,axiom,
    ( ( type_len0_len_of @ numeral_num1 )
    = ( ^ [Uu4: itself @ numeral_num1] : ( one_one @ nat ) ) ) ).

% len_num1
thf(fact_7447_len__of__finite__2__def,axiom,
    ( ( type_len0_len_of @ finite_2 )
    = ( ^ [X: itself @ finite_2] : ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ).

% len_of_finite_2_def
thf(fact_7448_len__of__finite__3__def,axiom,
    ( ( type_len0_len_of @ finite_3 )
    = ( ^ [X: itself @ finite_3] : ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ).

% len_of_finite_3_def
thf(fact_7449_len__bit0,axiom,
    ! [A: $tType] :
      ( ( type_len0 @ A )
     => ( ( type_len0_len_of @ ( numeral_bit0 @ A ) )
        = ( ^ [Uu4: itself @ ( numeral_bit0 @ A )] : ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ).

% len_bit0
thf(fact_7450_two__less__eq__exp__length,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ( type_len @ B )
        & ( linordered_idom @ A ) )
     => ( ord_less_eq @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ B @ ( type2 @ B ) ) ) ) ) ).

% two_less_eq_exp_length
thf(fact_7451_len__bit1,axiom,
    ! [A: $tType] :
      ( ( type_len0 @ A )
     => ( ( type_len0_len_of @ ( numeral_bit1 @ A ) )
        = ( ^ [Uu4: itself @ ( numeral_bit1 @ A )] : ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) @ ( one_one @ nat ) ) ) ) ) ).

% len_bit1
thf(fact_7452_divmod__via__sdivmod,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [Y2: word @ A,X2: word @ A] :
          ( ( Y2
           != ( zero_zero @ ( word @ A ) ) )
         => ( ( ( ord_less_eq @ ( word @ A ) @ ( bit_se4730199178511100633sh_bit @ ( word @ A ) @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) @ ( one_one @ ( word @ A ) ) ) @ Y2 )
             => ( ( ( ord_less @ ( word @ A ) @ X2 @ Y2 )
                 => ( ( product_Pair @ ( word @ A ) @ ( word @ A ) @ ( divide_divide @ ( word @ A ) @ X2 @ Y2 ) @ ( modulo_modulo @ ( word @ A ) @ X2 @ Y2 ) )
                    = ( product_Pair @ ( word @ A ) @ ( word @ A ) @ ( zero_zero @ ( word @ A ) ) @ X2 ) ) )
                & ( ~ ( ord_less @ ( word @ A ) @ X2 @ Y2 )
                 => ( ( product_Pair @ ( word @ A ) @ ( word @ A ) @ ( divide_divide @ ( word @ A ) @ X2 @ Y2 ) @ ( modulo_modulo @ ( word @ A ) @ X2 @ Y2 ) )
                    = ( product_Pair @ ( word @ A ) @ ( word @ A ) @ ( one_one @ ( word @ A ) ) @ ( minus_minus @ ( word @ A ) @ X2 @ Y2 ) ) ) ) ) )
            & ( ~ ( ord_less_eq @ ( word @ A ) @ ( bit_se4730199178511100633sh_bit @ ( word @ A ) @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) @ ( one_one @ ( word @ A ) ) ) @ Y2 )
             => ( ( product_Pair @ ( word @ A ) @ ( word @ A ) @ ( divide_divide @ ( word @ A ) @ X2 @ Y2 ) @ ( modulo_modulo @ ( word @ A ) @ X2 @ Y2 ) )
                = ( if @ ( product_prod @ ( word @ A ) @ ( word @ A ) ) @ ( ord_less_eq @ ( word @ A ) @ Y2 @ ( minus_minus @ ( word @ A ) @ X2 @ ( times_times @ ( word @ A ) @ ( bit_se4730199178511100633sh_bit @ ( word @ A ) @ ( one_one @ nat ) @ ( signed7115095781618012415divide @ ( word @ A ) @ ( bit_se4197421643247451524op_bit @ ( word @ A ) @ ( one_one @ nat ) @ X2 ) @ Y2 ) ) @ Y2 ) ) ) @ ( product_Pair @ ( word @ A ) @ ( word @ A ) @ ( plus_plus @ ( word @ A ) @ ( bit_se4730199178511100633sh_bit @ ( word @ A ) @ ( one_one @ nat ) @ ( signed7115095781618012415divide @ ( word @ A ) @ ( bit_se4197421643247451524op_bit @ ( word @ A ) @ ( one_one @ nat ) @ X2 ) @ Y2 ) ) @ ( one_one @ ( word @ A ) ) ) @ ( minus_minus @ ( word @ A ) @ ( minus_minus @ ( word @ A ) @ X2 @ ( times_times @ ( word @ A ) @ ( bit_se4730199178511100633sh_bit @ ( word @ A ) @ ( one_one @ nat ) @ ( signed7115095781618012415divide @ ( word @ A ) @ ( bit_se4197421643247451524op_bit @ ( word @ A ) @ ( one_one @ nat ) @ X2 ) @ Y2 ) ) @ Y2 ) ) @ Y2 ) ) @ ( product_Pair @ ( word @ A ) @ ( word @ A ) @ ( bit_se4730199178511100633sh_bit @ ( word @ A ) @ ( one_one @ nat ) @ ( signed7115095781618012415divide @ ( word @ A ) @ ( bit_se4197421643247451524op_bit @ ( word @ A ) @ ( one_one @ nat ) @ X2 ) @ Y2 ) ) @ ( minus_minus @ ( word @ A ) @ X2 @ ( times_times @ ( word @ A ) @ ( bit_se4730199178511100633sh_bit @ ( word @ A ) @ ( one_one @ nat ) @ ( signed7115095781618012415divide @ ( word @ A ) @ ( bit_se4197421643247451524op_bit @ ( word @ A ) @ ( one_one @ nat ) @ X2 ) @ Y2 ) ) @ Y2 ) ) ) ) ) ) ) ) ) ).

% divmod_via_sdivmod
thf(fact_7453_signed__drop__bit__word__minus__numeral,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [N2: num,K: num] :
          ( ( signed_drop_bit @ A @ ( numeral_numeral @ nat @ N2 ) @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ K ) ) )
          = ( ring_1_of_int @ ( word @ A ) @ ( bit_se4197421643247451524op_bit @ int @ ( numeral_numeral @ nat @ N2 ) @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K ) ) ) ) ) ) ) ).

% signed_drop_bit_word_minus_numeral
thf(fact_7454_signed__drop__bit__signed__drop__bit,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [M: nat,N2: nat,W: word @ A] :
          ( ( signed_drop_bit @ A @ M @ ( signed_drop_bit @ A @ N2 @ W ) )
          = ( signed_drop_bit @ A @ ( plus_plus @ nat @ M @ N2 ) @ W ) ) ) ).

% signed_drop_bit_signed_drop_bit
thf(fact_7455_word__sdiv__div1,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [A2: word @ A] :
          ( ( signed7115095781618012415divide @ ( word @ A ) @ A2 @ ( one_one @ ( word @ A ) ) )
          = A2 ) ) ).

% word_sdiv_div1
thf(fact_7456_signed__drop__bit__of__minus__1,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [N2: nat] :
          ( ( signed_drop_bit @ A @ N2 @ ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) )
          = ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) ) ) ).

% signed_drop_bit_of_minus_1
thf(fact_7457_word__sdiv__div__minus1,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [A2: word @ A] :
          ( ( signed7115095781618012415divide @ ( word @ A ) @ A2 @ ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) )
          = ( uminus_uminus @ ( word @ A ) @ A2 ) ) ) ).

% word_sdiv_div_minus1
thf(fact_7458_minus__one__sdiv__word__eq,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [W: word @ A] :
          ( ( signed7115095781618012415divide @ ( word @ A ) @ ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) @ W )
          = ( uminus_uminus @ ( word @ A ) @ ( signed7115095781618012415divide @ ( word @ A ) @ ( one_one @ ( word @ A ) ) @ W ) ) ) ) ).

% minus_one_sdiv_word_eq
thf(fact_7459_one__sdiv__word__eq,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [W: word @ A] :
          ( ( signed7115095781618012415divide @ ( word @ A ) @ ( one_one @ ( word @ A ) ) @ W )
          = ( times_times @ ( word @ A )
            @ ( zero_neq_one_of_bool @ ( word @ A )
              @ ( ( W
                  = ( one_one @ ( word @ A ) ) )
                | ( W
                  = ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) ) ) )
            @ W ) ) ) ).

% one_sdiv_word_eq
thf(fact_7460_signed__drop__bit__of__1,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [N2: nat] :
          ( ( signed_drop_bit @ A @ N2 @ ( one_one @ ( word @ A ) ) )
          = ( zero_neq_one_of_bool @ ( word @ A )
            @ ( ( ( type_len0_len_of @ A @ ( type2 @ A ) )
                = ( one_one @ nat ) )
              | ( N2
                = ( zero_zero @ nat ) ) ) ) ) ) ).

% signed_drop_bit_of_1
thf(fact_7461_sdiv__word__numeral__numeral,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [A2: num,B2: num] :
          ( ( signed7115095781618012415divide @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ A2 ) @ ( numeral_numeral @ ( word @ A ) @ B2 ) )
          = ( ring_1_of_int @ ( word @ A ) @ ( signed7115095781618012415divide @ int @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) @ ( numeral_numeral @ int @ A2 ) ) @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) @ ( numeral_numeral @ int @ B2 ) ) ) ) ) ) ).

% sdiv_word_numeral_numeral
thf(fact_7462_signed__drop__bit__word__Suc__numeral,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [N2: nat,K: num] :
          ( ( signed_drop_bit @ A @ ( suc @ N2 ) @ ( numeral_numeral @ ( word @ A ) @ K ) )
          = ( ring_1_of_int @ ( word @ A ) @ ( bit_se4197421643247451524op_bit @ int @ ( suc @ N2 ) @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) @ ( numeral_numeral @ int @ K ) ) ) ) ) ) ).

% signed_drop_bit_word_Suc_numeral
thf(fact_7463_signed__drop__bit__word__numeral,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [N2: num,K: num] :
          ( ( signed_drop_bit @ A @ ( numeral_numeral @ nat @ N2 ) @ ( numeral_numeral @ ( word @ A ) @ K ) )
          = ( ring_1_of_int @ ( word @ A ) @ ( bit_se4197421643247451524op_bit @ int @ ( numeral_numeral @ nat @ N2 ) @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) @ ( numeral_numeral @ int @ K ) ) ) ) ) ) ).

% signed_drop_bit_word_numeral
thf(fact_7464_sdiv__word__minus__numeral__minus__numeral,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [A2: num,B2: num] :
          ( ( signed7115095781618012415divide @ ( word @ A ) @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ A2 ) ) @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ B2 ) ) )
          = ( ring_1_of_int @ ( word @ A ) @ ( signed7115095781618012415divide @ int @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ A2 ) ) ) @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ B2 ) ) ) ) ) ) ) ).

% sdiv_word_minus_numeral_minus_numeral
thf(fact_7465_sdiv__word__numeral__minus__numeral,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [A2: num,B2: num] :
          ( ( signed7115095781618012415divide @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ A2 ) @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ B2 ) ) )
          = ( ring_1_of_int @ ( word @ A ) @ ( signed7115095781618012415divide @ int @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) @ ( numeral_numeral @ int @ A2 ) ) @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ B2 ) ) ) ) ) ) ) ).

% sdiv_word_numeral_minus_numeral
thf(fact_7466_sdiv__word__minus__numeral__numeral,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [A2: num,B2: num] :
          ( ( signed7115095781618012415divide @ ( word @ A ) @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ A2 ) ) @ ( numeral_numeral @ ( word @ A ) @ B2 ) )
          = ( ring_1_of_int @ ( word @ A ) @ ( signed7115095781618012415divide @ int @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ A2 ) ) ) @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) @ ( numeral_numeral @ int @ B2 ) ) ) ) ) ) ).

% sdiv_word_minus_numeral_numeral
thf(fact_7467_signed__drop__bit__word__Suc__minus__numeral,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [N2: nat,K: num] :
          ( ( signed_drop_bit @ A @ ( suc @ N2 ) @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ K ) ) )
          = ( ring_1_of_int @ ( word @ A ) @ ( bit_se4197421643247451524op_bit @ int @ ( suc @ N2 ) @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K ) ) ) ) ) ) ) ).

% signed_drop_bit_word_Suc_minus_numeral
thf(fact_7468_word__sdiv__numerals__lhs_I1_J,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [X2: num,W: word @ A] :
          ( ( signed7115095781618012415divide @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ X2 ) @ W )
          = ( ring_1_of_int @ ( word @ A ) @ ( signed7115095781618012415divide @ int @ ( ring_1_signed @ A @ int @ ( numeral_numeral @ ( word @ A ) @ X2 ) ) @ ( ring_1_signed @ A @ int @ W ) ) ) ) ) ).

% word_sdiv_numerals_lhs(1)
thf(fact_7469_word__sdiv__numerals_I1_J,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [X2: num,Y2: num] :
          ( ( signed7115095781618012415divide @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ X2 ) @ ( numeral_numeral @ ( word @ A ) @ Y2 ) )
          = ( ring_1_of_int @ ( word @ A ) @ ( signed7115095781618012415divide @ int @ ( ring_1_signed @ A @ int @ ( numeral_numeral @ ( word @ A ) @ X2 ) ) @ ( ring_1_signed @ A @ int @ ( numeral_numeral @ ( word @ A ) @ Y2 ) ) ) ) ) ) ).

% word_sdiv_numerals(1)
thf(fact_7470_word__sdiv__numerals__lhs_I3_J,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [W: word @ A] :
          ( ( signed7115095781618012415divide @ ( word @ A ) @ ( one_one @ ( word @ A ) ) @ W )
          = ( ring_1_of_int @ ( word @ A ) @ ( signed7115095781618012415divide @ int @ ( ring_1_signed @ A @ int @ ( one_one @ ( word @ A ) ) ) @ ( ring_1_signed @ A @ int @ W ) ) ) ) ) ).

% word_sdiv_numerals_lhs(3)
thf(fact_7471_word__sdiv__numerals_I9_J,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ( ( signed7115095781618012415divide @ ( word @ A ) @ ( one_one @ ( word @ A ) ) @ ( one_one @ ( word @ A ) ) )
        = ( ring_1_of_int @ ( word @ A ) @ ( signed7115095781618012415divide @ int @ ( ring_1_signed @ A @ int @ ( one_one @ ( word @ A ) ) ) @ ( ring_1_signed @ A @ int @ ( one_one @ ( word @ A ) ) ) ) ) ) ) ).

% word_sdiv_numerals(9)
thf(fact_7472_word__sdiv__numerals_I3_J,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [Y2: num] :
          ( ( signed7115095781618012415divide @ ( word @ A ) @ ( one_one @ ( word @ A ) ) @ ( numeral_numeral @ ( word @ A ) @ Y2 ) )
          = ( ring_1_of_int @ ( word @ A ) @ ( signed7115095781618012415divide @ int @ ( ring_1_signed @ A @ int @ ( one_one @ ( word @ A ) ) ) @ ( ring_1_signed @ A @ int @ ( numeral_numeral @ ( word @ A ) @ Y2 ) ) ) ) ) ) ).

% word_sdiv_numerals(3)
thf(fact_7473_word__sdiv__numerals_I7_J,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [X2: num] :
          ( ( signed7115095781618012415divide @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ X2 ) @ ( one_one @ ( word @ A ) ) )
          = ( ring_1_of_int @ ( word @ A ) @ ( signed7115095781618012415divide @ int @ ( ring_1_signed @ A @ int @ ( numeral_numeral @ ( word @ A ) @ X2 ) ) @ ( ring_1_signed @ A @ int @ ( one_one @ ( word @ A ) ) ) ) ) ) ) ).

% word_sdiv_numerals(7)
thf(fact_7474_word__sdiv__numerals_I4_J,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [X2: num] :
          ( ( signed7115095781618012415divide @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ X2 ) @ ( zero_zero @ ( word @ A ) ) )
          = ( ring_1_of_int @ ( word @ A ) @ ( signed7115095781618012415divide @ int @ ( ring_1_signed @ A @ int @ ( numeral_numeral @ ( word @ A ) @ X2 ) ) @ ( ring_1_signed @ A @ int @ ( zero_zero @ ( word @ A ) ) ) ) ) ) ) ).

% word_sdiv_numerals(4)
thf(fact_7475_word__sdiv__numerals_I2_J,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [Y2: num] :
          ( ( signed7115095781618012415divide @ ( word @ A ) @ ( zero_zero @ ( word @ A ) ) @ ( numeral_numeral @ ( word @ A ) @ Y2 ) )
          = ( ring_1_of_int @ ( word @ A ) @ ( signed7115095781618012415divide @ int @ ( ring_1_signed @ A @ int @ ( zero_zero @ ( word @ A ) ) ) @ ( ring_1_signed @ A @ int @ ( numeral_numeral @ ( word @ A ) @ Y2 ) ) ) ) ) ) ).

% word_sdiv_numerals(2)
thf(fact_7476_word__sdiv__numerals_I6_J,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ( ( signed7115095781618012415divide @ ( word @ A ) @ ( one_one @ ( word @ A ) ) @ ( zero_zero @ ( word @ A ) ) )
        = ( ring_1_of_int @ ( word @ A ) @ ( signed7115095781618012415divide @ int @ ( ring_1_signed @ A @ int @ ( one_one @ ( word @ A ) ) ) @ ( ring_1_signed @ A @ int @ ( zero_zero @ ( word @ A ) ) ) ) ) ) ) ).

% word_sdiv_numerals(6)
thf(fact_7477_word__sdiv__numerals_I8_J,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ( ( signed7115095781618012415divide @ ( word @ A ) @ ( zero_zero @ ( word @ A ) ) @ ( one_one @ ( word @ A ) ) )
        = ( ring_1_of_int @ ( word @ A ) @ ( signed7115095781618012415divide @ int @ ( ring_1_signed @ A @ int @ ( zero_zero @ ( word @ A ) ) ) @ ( ring_1_signed @ A @ int @ ( one_one @ ( word @ A ) ) ) ) ) ) ) ).

% word_sdiv_numerals(8)
thf(fact_7478_signed__div__arith,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [A2: word @ A,B2: word @ A] :
          ( ( ring_1_signed @ A @ int @ ( signed7115095781618012415divide @ ( word @ A ) @ A2 @ B2 ) )
          = ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) @ ( signed7115095781618012415divide @ int @ ( ring_1_signed @ A @ int @ A2 ) @ ( ring_1_signed @ A @ int @ B2 ) ) ) ) ) ).

% signed_div_arith
thf(fact_7479_bit__signed__drop__bit__iff,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [M: nat,W: word @ A,N2: nat] :
          ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( signed_drop_bit @ A @ M @ W ) @ N2 )
          = ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ W
            @ ( if @ nat
              @ ( ( ord_less_eq @ nat @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ M ) @ N2 )
                & ( ord_less @ nat @ N2 @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
              @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) )
              @ ( plus_plus @ nat @ M @ N2 ) ) ) ) ) ).

% bit_signed_drop_bit_iff
thf(fact_7480_signed__drop__bit__beyond,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [N2: nat,W: word @ A] :
          ( ( ord_less_eq @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ N2 )
         => ( ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ W @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) )
             => ( ( signed_drop_bit @ A @ N2 @ W )
                = ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) ) )
            & ( ~ ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ W @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) )
             => ( ( signed_drop_bit @ A @ N2 @ W )
                = ( zero_zero @ ( word @ A ) ) ) ) ) ) ) ).

% signed_drop_bit_beyond
thf(fact_7481_word__int__split__asm,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len @ B )
     => ! [P: A > $o,F: int > A,X2: word @ B] :
          ( ( P @ ( word_int_case @ A @ B @ F @ X2 ) )
          = ( ~ ? [N: int] :
                  ( ( X2
                    = ( ring_1_of_int @ ( word @ B ) @ N ) )
                  & ( ord_less_eq @ int @ ( zero_zero @ int ) @ N )
                  & ( ord_less @ int @ N @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ B @ ( type2 @ B ) ) ) )
                  & ~ ( P @ ( F @ N ) ) ) ) ) ) ).

% word_int_split_asm
thf(fact_7482_word__int__split,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len @ B )
     => ! [P: A > $o,F: int > A,X2: word @ B] :
          ( ( P @ ( word_int_case @ A @ B @ F @ X2 ) )
          = ( ! [I2: int] :
                ( ( ( X2
                    = ( ring_1_of_int @ ( word @ B ) @ I2 ) )
                  & ( ord_less_eq @ int @ ( zero_zero @ int ) @ I2 )
                  & ( ord_less @ int @ I2 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ B @ ( type2 @ B ) ) ) ) )
               => ( P @ ( F @ I2 ) ) ) ) ) ) ).

% word_int_split
thf(fact_7483_word__int__case__wi,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len @ B )
     => ! [F: int > A,I: int] :
          ( ( word_int_case @ A @ B @ F @ ( ring_1_of_int @ ( word @ B ) @ I ) )
          = ( F @ ( modulo_modulo @ int @ I @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ B @ ( type2 @ B ) ) ) ) ) ) ) ).

% word_int_case_wi
thf(fact_7484_smod__word__minus__numeral__minus__numeral,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [A2: num,B2: num] :
          ( ( signed6721504322012087516modulo @ ( word @ A ) @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ A2 ) ) @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ B2 ) ) )
          = ( ring_1_of_int @ ( word @ A ) @ ( signed6721504322012087516modulo @ int @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ A2 ) ) ) @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ B2 ) ) ) ) ) ) ) ).

% smod_word_minus_numeral_minus_numeral
thf(fact_7485_smod__word__numeral__minus__numeral,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [A2: num,B2: num] :
          ( ( signed6721504322012087516modulo @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ A2 ) @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ B2 ) ) )
          = ( ring_1_of_int @ ( word @ A ) @ ( signed6721504322012087516modulo @ int @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) @ ( numeral_numeral @ int @ A2 ) ) @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ B2 ) ) ) ) ) ) ) ).

% smod_word_numeral_minus_numeral
thf(fact_7486_smod__word__one,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [W: word @ A] :
          ( ( signed6721504322012087516modulo @ ( word @ A ) @ W @ ( one_one @ ( word @ A ) ) )
          = ( zero_zero @ ( word @ A ) ) ) ) ).

% smod_word_one
thf(fact_7487_minus__one__smod__word__eq,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [W: word @ A] :
          ( ( signed6721504322012087516modulo @ ( word @ A ) @ ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) @ W )
          = ( uminus_uminus @ ( word @ A ) @ ( signed6721504322012087516modulo @ ( word @ A ) @ ( one_one @ ( word @ A ) ) @ W ) ) ) ) ).

% minus_one_smod_word_eq
thf(fact_7488_smod__word__minus__one,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [W: word @ A] :
          ( ( signed6721504322012087516modulo @ ( word @ A ) @ W @ ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) )
          = ( zero_zero @ ( word @ A ) ) ) ) ).

% smod_word_minus_one
thf(fact_7489_one__smod__word__eq,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [W: word @ A] :
          ( ( signed6721504322012087516modulo @ ( word @ A ) @ ( one_one @ ( word @ A ) ) @ W )
          = ( minus_minus @ ( word @ A ) @ ( one_one @ ( word @ A ) )
            @ ( zero_neq_one_of_bool @ ( word @ A )
              @ ( ( W
                  = ( one_one @ ( word @ A ) ) )
                | ( W
                  = ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) ) ) ) ) ) ) ).

% one_smod_word_eq
thf(fact_7490_smod__word__numeral__numeral,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [A2: num,B2: num] :
          ( ( signed6721504322012087516modulo @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ A2 ) @ ( numeral_numeral @ ( word @ A ) @ B2 ) )
          = ( ring_1_of_int @ ( word @ A ) @ ( signed6721504322012087516modulo @ int @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) @ ( numeral_numeral @ int @ A2 ) ) @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) @ ( numeral_numeral @ int @ B2 ) ) ) ) ) ) ).

% smod_word_numeral_numeral
thf(fact_7491_smod__word__minus__numeral__numeral,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [A2: num,B2: num] :
          ( ( signed6721504322012087516modulo @ ( word @ A ) @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ A2 ) ) @ ( numeral_numeral @ ( word @ A ) @ B2 ) )
          = ( ring_1_of_int @ ( word @ A ) @ ( signed6721504322012087516modulo @ int @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ A2 ) ) ) @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) @ ( numeral_numeral @ int @ B2 ) ) ) ) ) ) ).

% smod_word_minus_numeral_numeral
thf(fact_7492_sdiv__smod__id,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [A2: word @ A,B2: word @ A] :
          ( ( plus_plus @ ( word @ A ) @ ( times_times @ ( word @ A ) @ ( signed7115095781618012415divide @ ( word @ A ) @ A2 @ B2 ) @ B2 ) @ ( signed6721504322012087516modulo @ ( word @ A ) @ A2 @ B2 ) )
          = A2 ) ) ).

% sdiv_smod_id
thf(fact_7493_smod__word__alt__def,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ( ( signed6721504322012087516modulo @ ( word @ A ) )
        = ( ^ [A5: word @ A,B5: word @ A] : ( minus_minus @ ( word @ A ) @ A5 @ ( times_times @ ( word @ A ) @ ( signed7115095781618012415divide @ ( word @ A ) @ A5 @ B5 ) @ B5 ) ) ) ) ) ).

% smod_word_alt_def
thf(fact_7494_word__smod__numerals_I1_J,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [X2: num,Y2: num] :
          ( ( signed6721504322012087516modulo @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ X2 ) @ ( numeral_numeral @ ( word @ A ) @ Y2 ) )
          = ( ring_1_of_int @ ( word @ A ) @ ( signed6721504322012087516modulo @ int @ ( ring_1_signed @ A @ int @ ( numeral_numeral @ ( word @ A ) @ X2 ) ) @ ( ring_1_signed @ A @ int @ ( numeral_numeral @ ( word @ A ) @ Y2 ) ) ) ) ) ) ).

% word_smod_numerals(1)
thf(fact_7495_word__smod__numerals__lhs_I1_J,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [X2: num,W: word @ A] :
          ( ( signed6721504322012087516modulo @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ X2 ) @ W )
          = ( ring_1_of_int @ ( word @ A ) @ ( signed6721504322012087516modulo @ int @ ( ring_1_signed @ A @ int @ ( numeral_numeral @ ( word @ A ) @ X2 ) ) @ ( ring_1_signed @ A @ int @ W ) ) ) ) ) ).

% word_smod_numerals_lhs(1)
thf(fact_7496_word__smod__numerals_I9_J,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ( ( signed6721504322012087516modulo @ ( word @ A ) @ ( one_one @ ( word @ A ) ) @ ( one_one @ ( word @ A ) ) )
        = ( ring_1_of_int @ ( word @ A ) @ ( signed6721504322012087516modulo @ int @ ( ring_1_signed @ A @ int @ ( one_one @ ( word @ A ) ) ) @ ( ring_1_signed @ A @ int @ ( one_one @ ( word @ A ) ) ) ) ) ) ) ).

% word_smod_numerals(9)
thf(fact_7497_word__smod__numerals__lhs_I3_J,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [W: word @ A] :
          ( ( signed6721504322012087516modulo @ ( word @ A ) @ ( one_one @ ( word @ A ) ) @ W )
          = ( ring_1_of_int @ ( word @ A ) @ ( signed6721504322012087516modulo @ int @ ( ring_1_signed @ A @ int @ ( one_one @ ( word @ A ) ) ) @ ( ring_1_signed @ A @ int @ W ) ) ) ) ) ).

% word_smod_numerals_lhs(3)
thf(fact_7498_word__smod__numerals_I7_J,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [X2: num] :
          ( ( signed6721504322012087516modulo @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ X2 ) @ ( one_one @ ( word @ A ) ) )
          = ( ring_1_of_int @ ( word @ A ) @ ( signed6721504322012087516modulo @ int @ ( ring_1_signed @ A @ int @ ( numeral_numeral @ ( word @ A ) @ X2 ) ) @ ( ring_1_signed @ A @ int @ ( one_one @ ( word @ A ) ) ) ) ) ) ) ).

% word_smod_numerals(7)
thf(fact_7499_word__smod__numerals_I3_J,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [Y2: num] :
          ( ( signed6721504322012087516modulo @ ( word @ A ) @ ( one_one @ ( word @ A ) ) @ ( numeral_numeral @ ( word @ A ) @ Y2 ) )
          = ( ring_1_of_int @ ( word @ A ) @ ( signed6721504322012087516modulo @ int @ ( ring_1_signed @ A @ int @ ( one_one @ ( word @ A ) ) ) @ ( ring_1_signed @ A @ int @ ( numeral_numeral @ ( word @ A ) @ Y2 ) ) ) ) ) ) ).

% word_smod_numerals(3)
thf(fact_7500_word__smod__numerals_I2_J,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [Y2: num] :
          ( ( signed6721504322012087516modulo @ ( word @ A ) @ ( zero_zero @ ( word @ A ) ) @ ( numeral_numeral @ ( word @ A ) @ Y2 ) )
          = ( ring_1_of_int @ ( word @ A ) @ ( signed6721504322012087516modulo @ int @ ( ring_1_signed @ A @ int @ ( zero_zero @ ( word @ A ) ) ) @ ( ring_1_signed @ A @ int @ ( numeral_numeral @ ( word @ A ) @ Y2 ) ) ) ) ) ) ).

% word_smod_numerals(2)
thf(fact_7501_word__smod__numerals_I4_J,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [X2: num] :
          ( ( signed6721504322012087516modulo @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ X2 ) @ ( zero_zero @ ( word @ A ) ) )
          = ( ring_1_of_int @ ( word @ A ) @ ( signed6721504322012087516modulo @ int @ ( ring_1_signed @ A @ int @ ( numeral_numeral @ ( word @ A ) @ X2 ) ) @ ( ring_1_signed @ A @ int @ ( zero_zero @ ( word @ A ) ) ) ) ) ) ) ).

% word_smod_numerals(4)
thf(fact_7502_word__smod__numerals_I8_J,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ( ( signed6721504322012087516modulo @ ( word @ A ) @ ( zero_zero @ ( word @ A ) ) @ ( one_one @ ( word @ A ) ) )
        = ( ring_1_of_int @ ( word @ A ) @ ( signed6721504322012087516modulo @ int @ ( ring_1_signed @ A @ int @ ( zero_zero @ ( word @ A ) ) ) @ ( ring_1_signed @ A @ int @ ( one_one @ ( word @ A ) ) ) ) ) ) ) ).

% word_smod_numerals(8)
thf(fact_7503_word__smod__numerals_I6_J,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ( ( signed6721504322012087516modulo @ ( word @ A ) @ ( one_one @ ( word @ A ) ) @ ( zero_zero @ ( word @ A ) ) )
        = ( ring_1_of_int @ ( word @ A ) @ ( signed6721504322012087516modulo @ int @ ( ring_1_signed @ A @ int @ ( one_one @ ( word @ A ) ) ) @ ( ring_1_signed @ A @ int @ ( zero_zero @ ( word @ A ) ) ) ) ) ) ) ).

% word_smod_numerals(6)
thf(fact_7504_signed__mod__arith,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [A2: word @ A,B2: word @ A] :
          ( ( ring_1_signed @ A @ int @ ( signed6721504322012087516modulo @ ( word @ A ) @ A2 @ B2 ) )
          = ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) @ ( signed6721504322012087516modulo @ int @ ( ring_1_signed @ A @ int @ A2 ) @ ( ring_1_signed @ A @ int @ B2 ) ) ) ) ) ).

% signed_mod_arith
thf(fact_7505_uint__word__ariths_I6_J,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [A2: word @ A] :
          ( ( semiring_1_unsigned @ A @ int @ ( word_pred @ A @ A2 ) )
          = ( modulo_modulo @ int @ ( minus_minus @ int @ ( semiring_1_unsigned @ A @ int @ A2 ) @ ( one_one @ int ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ).

% uint_word_ariths(6)
thf(fact_7506_slice1__def,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ( type_len @ B )
        & ( type_len @ A ) )
     => ( ( slice1 @ A @ B )
        = ( ^ [N: nat,W2: word @ A] : ( if @ ( word @ B ) @ ( ord_less @ nat @ N @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) @ ( semiring_1_unsigned @ A @ ( word @ B ) @ ( bit_se4197421643247451524op_bit @ ( word @ A ) @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ N ) @ W2 ) ) @ ( bit_se4730199178511100633sh_bit @ ( word @ B ) @ ( minus_minus @ nat @ N @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) @ ( semiring_1_unsigned @ A @ ( word @ B ) @ W2 ) ) ) ) ) ) ).

% slice1_def
thf(fact_7507_succ__pred__no_I2_J,axiom,
    ! [B: $tType] :
      ( ( type_len @ B )
     => ! [W: num] :
          ( ( word_pred @ B @ ( numeral_numeral @ ( word @ B ) @ W ) )
          = ( minus_minus @ ( word @ B ) @ ( numeral_numeral @ ( word @ B ) @ W ) @ ( one_one @ ( word @ B ) ) ) ) ) ).

% succ_pred_no(2)
thf(fact_7508_succ__pred__no_I4_J,axiom,
    ! [D: $tType] :
      ( ( type_len @ D )
     => ! [W: num] :
          ( ( word_pred @ D @ ( uminus_uminus @ ( word @ D ) @ ( numeral_numeral @ ( word @ D ) @ W ) ) )
          = ( minus_minus @ ( word @ D ) @ ( uminus_uminus @ ( word @ D ) @ ( numeral_numeral @ ( word @ D ) @ W ) ) @ ( one_one @ ( word @ D ) ) ) ) ) ).

% succ_pred_no(4)
thf(fact_7509_word__m1__ge,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [Y2: word @ A] : ( ord_less_eq @ ( word @ A ) @ Y2 @ ( word_pred @ A @ ( zero_zero @ ( word @ A ) ) ) ) ) ).

% word_m1_ge
thf(fact_7510_word__pred__m1,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ( ( word_pred @ A )
        = ( ^ [A5: word @ A] : ( minus_minus @ ( word @ A ) @ A5 @ ( one_one @ ( word @ A ) ) ) ) ) ) ).

% word_pred_m1
thf(fact_7511_ucast__slice1,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ( type_len @ B )
        & ( type_len @ A ) )
     => ( ( semiring_1_unsigned @ B @ ( word @ A ) )
        = ( ^ [W2: word @ B] : ( slice1 @ B @ A @ ( size_size @ ( word @ B ) @ W2 ) @ W2 ) ) ) ) ).

% ucast_slice1
thf(fact_7512_wi__hom__pred,axiom,
    ! [F3: $tType] :
      ( ( type_len @ F3 )
     => ! [A2: int] :
          ( ( word_pred @ F3 @ ( ring_1_of_int @ ( word @ F3 ) @ A2 ) )
          = ( ring_1_of_int @ ( word @ F3 ) @ ( minus_minus @ int @ A2 @ ( one_one @ int ) ) ) ) ) ).

% wi_hom_pred
thf(fact_7513_word__pred__0__n1,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ( ( word_pred @ A @ ( zero_zero @ ( word @ A ) ) )
        = ( ring_1_of_int @ ( word @ A ) @ ( uminus_uminus @ int @ ( one_one @ int ) ) ) ) ) ).

% word_pred_0_n1
thf(fact_7514_word__pred__alt,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ( ( word_pred @ A )
        = ( ^ [A5: word @ A] : ( ring_1_of_int @ ( word @ A ) @ ( minus_minus @ int @ ( semiring_1_unsigned @ A @ int @ A5 ) @ ( one_one @ int ) ) ) ) ) ) ).

% word_pred_alt
thf(fact_7515_Word_Oslice__def,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ( type_len @ B )
        & ( type_len @ A ) )
     => ( ( slice2 @ A @ B )
        = ( ^ [N: nat] : ( slice1 @ A @ B @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ N ) ) ) ) ) ).

% Word.slice_def
thf(fact_7516_uint__word__arith__bintrs_I6_J,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [A2: word @ A] :
          ( ( semiring_1_unsigned @ A @ int @ ( word_pred @ A @ A2 ) )
          = ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( minus_minus @ int @ ( semiring_1_unsigned @ A @ int @ A2 ) @ ( one_one @ int ) ) ) ) ) ).

% uint_word_arith_bintrs(6)
thf(fact_7517_sint__word__ariths_I6_J,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [A2: word @ A] :
          ( ( ring_1_signed @ A @ int @ ( word_pred @ A @ A2 ) )
          = ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) @ ( minus_minus @ int @ ( ring_1_signed @ A @ int @ A2 ) @ ( one_one @ int ) ) ) ) ) ).

% sint_word_ariths(6)
thf(fact_7518_uint__word__ariths_I5_J,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [A2: word @ A] :
          ( ( semiring_1_unsigned @ A @ int @ ( word_succ @ A @ A2 ) )
          = ( modulo_modulo @ int @ ( plus_plus @ int @ ( semiring_1_unsigned @ A @ int @ A2 ) @ ( one_one @ int ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ).

% uint_word_ariths(5)
thf(fact_7519_bit__word__roti__iff,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [K: int,W: word @ A,N2: nat] :
          ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( word_roti @ A @ K @ W ) @ N2 )
          = ( ( ord_less @ nat @ N2 @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
            & ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ W @ ( nat2 @ ( modulo_modulo @ int @ ( plus_plus @ int @ ( semiring_1_of_nat @ int @ N2 ) @ K ) @ ( semiring_1_of_nat @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ) ) ) ).

% bit_word_roti_iff
thf(fact_7520_succ__pred__no_I1_J,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [W: num] :
          ( ( word_succ @ A @ ( numeral_numeral @ ( word @ A ) @ W ) )
          = ( plus_plus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ W ) @ ( one_one @ ( word @ A ) ) ) ) ) ).

% succ_pred_no(1)
thf(fact_7521_succ__pred__no_I3_J,axiom,
    ! [C: $tType] :
      ( ( type_len @ C )
     => ! [W: num] :
          ( ( word_succ @ C @ ( uminus_uminus @ ( word @ C ) @ ( numeral_numeral @ ( word @ C ) @ W ) ) )
          = ( plus_plus @ ( word @ C ) @ ( uminus_uminus @ ( word @ C ) @ ( numeral_numeral @ ( word @ C ) @ W ) ) @ ( one_one @ ( word @ C ) ) ) ) ) ).

% succ_pred_no(3)
thf(fact_7522_Abs__fnat__hom__Suc,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [A2: nat] :
          ( ( word_succ @ A @ ( semiring_1_of_nat @ ( word @ A ) @ A2 ) )
          = ( semiring_1_of_nat @ ( word @ A ) @ ( suc @ A2 ) ) ) ) ).

% Abs_fnat_hom_Suc
thf(fact_7523_word__succ__p1,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ( ( word_succ @ A )
        = ( ^ [A5: word @ A] : ( plus_plus @ ( word @ A ) @ A5 @ ( one_one @ ( word @ A ) ) ) ) ) ) ).

% word_succ_p1
thf(fact_7524_word__roti__add,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [M: int,N2: int,W: word @ A] :
          ( ( word_roti @ A @ ( plus_plus @ int @ M @ N2 ) @ W )
          = ( word_roti @ A @ M @ ( word_roti @ A @ N2 @ W ) ) ) ) ).

% word_roti_add
thf(fact_7525_word__mult__succ,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [A2: word @ A,B2: word @ A] :
          ( ( times_times @ ( word @ A ) @ ( word_succ @ A @ A2 ) @ B2 )
          = ( plus_plus @ ( word @ A ) @ B2 @ ( times_times @ ( word @ A ) @ A2 @ B2 ) ) ) ) ).

% word_mult_succ
thf(fact_7526_wi__hom__succ,axiom,
    ! [E: $tType] :
      ( ( type_len @ E )
     => ! [A2: int] :
          ( ( word_succ @ E @ ( ring_1_of_int @ ( word @ E ) @ A2 ) )
          = ( ring_1_of_int @ ( word @ E ) @ ( plus_plus @ int @ A2 @ ( one_one @ int ) ) ) ) ) ).

% wi_hom_succ
thf(fact_7527_word__arith__nat__Suc,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ( ( word_succ @ A )
        = ( ^ [A5: word @ A] : ( semiring_1_of_nat @ ( word @ A ) @ ( suc @ ( semiring_1_unsigned @ A @ nat @ A5 ) ) ) ) ) ) ).

% word_arith_nat_Suc
thf(fact_7528_word__roti__conv__mod_H,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ( ( word_roti @ A )
        = ( ^ [N: int,W2: word @ A] : ( word_roti @ A @ ( modulo_modulo @ int @ N @ ( semiring_1_of_nat @ int @ ( size_size @ ( word @ A ) @ W2 ) ) ) @ W2 ) ) ) ) ).

% word_roti_conv_mod'
thf(fact_7529_word__succ__alt,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ( ( word_succ @ A )
        = ( ^ [A5: word @ A] : ( ring_1_of_int @ ( word @ A ) @ ( plus_plus @ int @ ( semiring_1_unsigned @ A @ int @ A5 ) @ ( one_one @ int ) ) ) ) ) ) ).

% word_succ_alt
thf(fact_7530_word__sp__01,axiom,
    ! [C: $tType,A: $tType,B: $tType,D: $tType] :
      ( ( ( type_len @ D )
        & ( type_len @ B )
        & ( type_len @ A )
        & ( type_len @ C ) )
     => ( ( ( word_succ @ A @ ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) )
          = ( zero_zero @ ( word @ A ) ) )
        & ( ( word_succ @ B @ ( zero_zero @ ( word @ B ) ) )
          = ( one_one @ ( word @ B ) ) )
        & ( ( word_pred @ C @ ( zero_zero @ ( word @ C ) ) )
          = ( uminus_uminus @ ( word @ C ) @ ( one_one @ ( word @ C ) ) ) )
        & ( ( word_pred @ D @ ( one_one @ ( word @ D ) ) )
          = ( zero_zero @ ( word @ D ) ) ) ) ) ).

% word_sp_01
thf(fact_7531_uint__word__arith__bintrs_I5_J,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [A2: word @ A] :
          ( ( semiring_1_unsigned @ A @ int @ ( word_succ @ A @ A2 ) )
          = ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( plus_plus @ int @ ( semiring_1_unsigned @ A @ int @ A2 ) @ ( one_one @ int ) ) ) ) ) ).

% uint_word_arith_bintrs(5)
thf(fact_7532_sint__word__ariths_I5_J,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [A2: word @ A] :
          ( ( ring_1_signed @ A @ int @ ( word_succ @ A @ A2 ) )
          = ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) @ ( plus_plus @ int @ ( ring_1_signed @ A @ int @ A2 ) @ ( one_one @ int ) ) ) ) ) ).

% sint_word_ariths(5)
thf(fact_7533_unat__word__ariths_I3_J,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [A2: word @ A] :
          ( ( semiring_1_unsigned @ A @ nat @ ( word_succ @ A @ A2 ) )
          = ( modulo_modulo @ nat @ ( suc @ ( semiring_1_unsigned @ A @ nat @ A2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ).

% unat_word_ariths(3)
thf(fact_7534_sless__eq__word__minus__numeral__minus__numeral,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [A2: num,B2: num] :
          ( ( word_sle @ A @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ A2 ) ) @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ B2 ) ) )
          = ( ord_less_eq @ int @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ A2 ) ) ) @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ B2 ) ) ) ) ) ) ).

% sless_eq_word_minus_numeral_minus_numeral
thf(fact_7535_sless__eq__word__numeral__minus__numeral,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [A2: num,B2: num] :
          ( ( word_sle @ A @ ( numeral_numeral @ ( word @ A ) @ A2 ) @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ B2 ) ) )
          = ( ord_less_eq @ int @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) @ ( numeral_numeral @ int @ A2 ) ) @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ B2 ) ) ) ) ) ) ).

% sless_eq_word_numeral_minus_numeral
thf(fact_7536_word__sle__no,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [A2: num,B2: num] :
          ( ( word_sle @ A @ ( numeral_numeral @ ( word @ A ) @ A2 ) @ ( numeral_numeral @ ( word @ A ) @ B2 ) )
          = ( ord_less_eq @ int @ ( ring_1_signed @ A @ int @ ( numeral_numeral @ ( word @ A ) @ A2 ) ) @ ( ring_1_signed @ A @ int @ ( numeral_numeral @ ( word @ A ) @ B2 ) ) ) ) ) ).

% word_sle_no
thf(fact_7537_extra__sle__sless__unfolds_I6_J,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [N2: num] :
          ( ( word_sle @ A @ ( numeral_numeral @ ( word @ A ) @ N2 ) @ ( one_one @ ( word @ A ) ) )
          = ( ord_less_eq @ int @ ( ring_1_signed @ A @ int @ ( numeral_numeral @ ( word @ A ) @ N2 ) ) @ ( ring_1_signed @ A @ int @ ( one_one @ ( word @ A ) ) ) ) ) ) ).

% extra_sle_sless_unfolds(6)
thf(fact_7538_extra__sle__sless__unfolds_I4_J,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [N2: num] :
          ( ( word_sle @ A @ ( one_one @ ( word @ A ) ) @ ( numeral_numeral @ ( word @ A ) @ N2 ) )
          = ( ord_less_eq @ int @ ( ring_1_signed @ A @ int @ ( one_one @ ( word @ A ) ) ) @ ( ring_1_signed @ A @ int @ ( numeral_numeral @ ( word @ A ) @ N2 ) ) ) ) ) ).

% extra_sle_sless_unfolds(4)
thf(fact_7539_extra__sle__sless__unfolds_I5_J,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [N2: num] :
          ( ( word_sle @ A @ ( numeral_numeral @ ( word @ A ) @ N2 ) @ ( zero_zero @ ( word @ A ) ) )
          = ( ord_less_eq @ int @ ( ring_1_signed @ A @ int @ ( numeral_numeral @ ( word @ A ) @ N2 ) ) @ ( ring_1_signed @ A @ int @ ( zero_zero @ ( word @ A ) ) ) ) ) ) ).

% extra_sle_sless_unfolds(5)
thf(fact_7540_extra__sle__sless__unfolds_I2_J,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [N2: num] :
          ( ( word_sle @ A @ ( zero_zero @ ( word @ A ) ) @ ( numeral_numeral @ ( word @ A ) @ N2 ) )
          = ( ord_less_eq @ int @ ( ring_1_signed @ A @ int @ ( zero_zero @ ( word @ A ) ) ) @ ( ring_1_signed @ A @ int @ ( numeral_numeral @ ( word @ A ) @ N2 ) ) ) ) ) ).

% extra_sle_sless_unfolds(2)
thf(fact_7541_extra__sle__sless__unfolds_I1_J,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ( ( word_sle @ A @ ( zero_zero @ ( word @ A ) ) @ ( one_one @ ( word @ A ) ) )
        = ( ord_less_eq @ int @ ( ring_1_signed @ A @ int @ ( zero_zero @ ( word @ A ) ) ) @ ( ring_1_signed @ A @ int @ ( one_one @ ( word @ A ) ) ) ) ) ) ).

% extra_sle_sless_unfolds(1)
thf(fact_7542_extra__sle__sless__unfolds_I3_J,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ( ( word_sle @ A @ ( one_one @ ( word @ A ) ) @ ( zero_zero @ ( word @ A ) ) )
        = ( ord_less_eq @ int @ ( ring_1_signed @ A @ int @ ( one_one @ ( word @ A ) ) ) @ ( ring_1_signed @ A @ int @ ( zero_zero @ ( word @ A ) ) ) ) ) ) ).

% extra_sle_sless_unfolds(3)
thf(fact_7543_sless__eq__word__numeral__numeral,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [A2: num,B2: num] :
          ( ( word_sle @ A @ ( numeral_numeral @ ( word @ A ) @ A2 ) @ ( numeral_numeral @ ( word @ A ) @ B2 ) )
          = ( ord_less_eq @ int @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) @ ( numeral_numeral @ int @ A2 ) ) @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) @ ( numeral_numeral @ int @ B2 ) ) ) ) ) ).

% sless_eq_word_numeral_numeral
thf(fact_7544_sless__eq__word__minus__numeral__numeral,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [A2: num,B2: num] :
          ( ( word_sle @ A @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ A2 ) ) @ ( numeral_numeral @ ( word @ A ) @ B2 ) )
          = ( ord_less_eq @ int @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ A2 ) ) ) @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) @ ( numeral_numeral @ int @ B2 ) ) ) ) ) ).

% sless_eq_word_minus_numeral_numeral
thf(fact_7545_word__sle__eq,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ( ( word_sle @ A )
        = ( ^ [A5: word @ A,B5: word @ A] : ( ord_less_eq @ int @ ( ring_1_signed @ A @ int @ A5 ) @ ( ring_1_signed @ A @ int @ B5 ) ) ) ) ) ).

% word_sle_eq
thf(fact_7546_signed_Olift__Suc__mono__le,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [F: nat > ( word @ A ),N2: nat,N5: nat] :
          ( ! [N4: nat] : ( word_sle @ A @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
         => ( ( ord_less_eq @ nat @ N2 @ N5 )
           => ( word_sle @ A @ ( F @ N2 ) @ ( F @ N5 ) ) ) ) ) ).

% signed.lift_Suc_mono_le
thf(fact_7547_signed_Olift__Suc__antimono__le,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [F: nat > ( word @ A ),N2: nat,N5: nat] :
          ( ! [N4: nat] : ( word_sle @ A @ ( F @ ( suc @ N4 ) ) @ ( F @ N4 ) )
         => ( ( ord_less_eq @ nat @ N2 @ N5 )
           => ( word_sle @ A @ ( F @ N5 ) @ ( F @ N2 ) ) ) ) ) ).

% signed.lift_Suc_antimono_le
thf(fact_7548_word__0__sle__from__less,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [X2: word @ A] :
          ( ( ord_less @ ( word @ A ) @ X2 @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) ) )
         => ( word_sle @ A @ ( zero_zero @ ( word @ A ) ) @ X2 ) ) ) ).

% word_0_sle_from_less
thf(fact_7549_sless__word__minus__numeral__minus__numeral,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [A2: num,B2: num] :
          ( ( word_sless @ A @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ A2 ) ) @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ B2 ) ) )
          = ( ord_less @ int @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ A2 ) ) ) @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ B2 ) ) ) ) ) ) ).

% sless_word_minus_numeral_minus_numeral
thf(fact_7550_sless__word__numeral__minus__numeral,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [A2: num,B2: num] :
          ( ( word_sless @ A @ ( numeral_numeral @ ( word @ A ) @ A2 ) @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ B2 ) ) )
          = ( ord_less @ int @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) @ ( numeral_numeral @ int @ A2 ) ) @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ B2 ) ) ) ) ) ) ).

% sless_word_numeral_minus_numeral
thf(fact_7551_word__sless__no,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [A2: num,B2: num] :
          ( ( word_sless @ A @ ( numeral_numeral @ ( word @ A ) @ A2 ) @ ( numeral_numeral @ ( word @ A ) @ B2 ) )
          = ( ( word_sle @ A @ ( numeral_numeral @ ( word @ A ) @ A2 ) @ ( numeral_numeral @ ( word @ A ) @ B2 ) )
            & ( ( numeral_numeral @ ( word @ A ) @ A2 )
             != ( numeral_numeral @ ( word @ A ) @ B2 ) ) ) ) ) ).

% word_sless_no
thf(fact_7552_extra__sle__sless__unfolds_I10_J,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [N2: num] :
          ( ( word_sless @ A @ ( one_one @ ( word @ A ) ) @ ( numeral_numeral @ ( word @ A ) @ N2 ) )
          = ( ord_less @ int @ ( ring_1_signed @ A @ int @ ( one_one @ ( word @ A ) ) ) @ ( ring_1_signed @ A @ int @ ( numeral_numeral @ ( word @ A ) @ N2 ) ) ) ) ) ).

% extra_sle_sless_unfolds(10)
thf(fact_7553_extra__sle__sless__unfolds_I12_J,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [N2: num] :
          ( ( word_sless @ A @ ( numeral_numeral @ ( word @ A ) @ N2 ) @ ( one_one @ ( word @ A ) ) )
          = ( ord_less @ int @ ( ring_1_signed @ A @ int @ ( numeral_numeral @ ( word @ A ) @ N2 ) ) @ ( ring_1_signed @ A @ int @ ( one_one @ ( word @ A ) ) ) ) ) ) ).

% extra_sle_sless_unfolds(12)
thf(fact_7554_extra__sle__sless__unfolds_I8_J,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [N2: num] :
          ( ( word_sless @ A @ ( zero_zero @ ( word @ A ) ) @ ( numeral_numeral @ ( word @ A ) @ N2 ) )
          = ( ord_less @ int @ ( ring_1_signed @ A @ int @ ( zero_zero @ ( word @ A ) ) ) @ ( ring_1_signed @ A @ int @ ( numeral_numeral @ ( word @ A ) @ N2 ) ) ) ) ) ).

% extra_sle_sless_unfolds(8)
thf(fact_7555_extra__sle__sless__unfolds_I11_J,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [N2: num] :
          ( ( word_sless @ A @ ( numeral_numeral @ ( word @ A ) @ N2 ) @ ( zero_zero @ ( word @ A ) ) )
          = ( ord_less @ int @ ( ring_1_signed @ A @ int @ ( numeral_numeral @ ( word @ A ) @ N2 ) ) @ ( ring_1_signed @ A @ int @ ( zero_zero @ ( word @ A ) ) ) ) ) ) ).

% extra_sle_sless_unfolds(11)
thf(fact_7556_extra__sle__sless__unfolds_I9_J,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ( ( word_sless @ A @ ( one_one @ ( word @ A ) ) @ ( zero_zero @ ( word @ A ) ) )
        = ( ord_less @ int @ ( ring_1_signed @ A @ int @ ( one_one @ ( word @ A ) ) ) @ ( ring_1_signed @ A @ int @ ( zero_zero @ ( word @ A ) ) ) ) ) ) ).

% extra_sle_sless_unfolds(9)
thf(fact_7557_extra__sle__sless__unfolds_I7_J,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ( ( word_sless @ A @ ( zero_zero @ ( word @ A ) ) @ ( one_one @ ( word @ A ) ) )
        = ( ord_less @ int @ ( ring_1_signed @ A @ int @ ( zero_zero @ ( word @ A ) ) ) @ ( ring_1_signed @ A @ int @ ( one_one @ ( word @ A ) ) ) ) ) ) ).

% extra_sle_sless_unfolds(7)
thf(fact_7558_sless__word__numeral__numeral,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [A2: num,B2: num] :
          ( ( word_sless @ A @ ( numeral_numeral @ ( word @ A ) @ A2 ) @ ( numeral_numeral @ ( word @ A ) @ B2 ) )
          = ( ord_less @ int @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) @ ( numeral_numeral @ int @ A2 ) ) @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) @ ( numeral_numeral @ int @ B2 ) ) ) ) ) ).

% sless_word_numeral_numeral
thf(fact_7559_sless__word__minus__numeral__numeral,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [A2: num,B2: num] :
          ( ( word_sless @ A @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ A2 ) ) @ ( numeral_numeral @ ( word @ A ) @ B2 ) )
          = ( ord_less @ int @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ A2 ) ) ) @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) @ ( numeral_numeral @ int @ B2 ) ) ) ) ) ).

% sless_word_minus_numeral_numeral
thf(fact_7560_signed_Olift__Suc__mono__less,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [F: nat > ( word @ A ),N2: nat,N5: nat] :
          ( ! [N4: nat] : ( word_sless @ A @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
         => ( ( ord_less @ nat @ N2 @ N5 )
           => ( word_sless @ A @ ( F @ N2 ) @ ( F @ N5 ) ) ) ) ) ).

% signed.lift_Suc_mono_less
thf(fact_7561_signed_Olift__Suc__mono__less__iff,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [F: nat > ( word @ A ),N2: nat,M: nat] :
          ( ! [N4: nat] : ( word_sless @ A @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
         => ( ( word_sless @ A @ ( F @ N2 ) @ ( F @ M ) )
            = ( ord_less @ nat @ N2 @ M ) ) ) ) ).

% signed.lift_Suc_mono_less_iff
thf(fact_7562_word__sless__sint__le,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [X2: word @ A,Y2: word @ A] :
          ( ( word_sless @ A @ X2 @ Y2 )
         => ( ord_less_eq @ int @ ( ring_1_signed @ A @ int @ X2 ) @ ( minus_minus @ int @ ( ring_1_signed @ A @ int @ Y2 ) @ ( one_one @ int ) ) ) ) ) ).

% word_sless_sint_le
thf(fact_7563_take__bit__word__minus__numeral,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [N2: num,K: num] :
          ( ( bit_se2584673776208193580ke_bit @ ( word @ A ) @ ( numeral_numeral @ nat @ N2 ) @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ K ) ) )
          = ( ring_1_of_int @ ( word @ A ) @ ( bit_se2584673776208193580ke_bit @ int @ ( ord_min @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( numeral_numeral @ nat @ N2 ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K ) ) ) ) ) ) ).

% take_bit_word_minus_numeral
thf(fact_7564_take__bit__word__Suc__minus__numeral,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [N2: nat,K: num] :
          ( ( bit_se2584673776208193580ke_bit @ ( word @ A ) @ ( suc @ N2 ) @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ K ) ) )
          = ( ring_1_of_int @ ( word @ A ) @ ( bit_se2584673776208193580ke_bit @ int @ ( ord_min @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ N2 ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K ) ) ) ) ) ) ).

% take_bit_word_Suc_minus_numeral
thf(fact_7565_min_Obounded__iff,axiom,
    ! [A: $tType] :
      ( ( linorder @ A )
     => ! [A2: A,B2: A,C2: A] :
          ( ( ord_less_eq @ A @ A2 @ ( ord_min @ A @ B2 @ C2 ) )
          = ( ( ord_less_eq @ A @ A2 @ B2 )
            & ( ord_less_eq @ A @ A2 @ C2 ) ) ) ) ).

% min.bounded_iff
thf(fact_7566_min_Oabsorb2,axiom,
    ! [A: $tType] :
      ( ( linorder @ A )
     => ! [B2: A,A2: A] :
          ( ( ord_less_eq @ A @ B2 @ A2 )
         => ( ( ord_min @ A @ A2 @ B2 )
            = B2 ) ) ) ).

% min.absorb2
thf(fact_7567_min_Oabsorb1,axiom,
    ! [A: $tType] :
      ( ( linorder @ A )
     => ! [A2: A,B2: A] :
          ( ( ord_less_eq @ A @ A2 @ B2 )
         => ( ( ord_min @ A @ A2 @ B2 )
            = A2 ) ) ) ).

% min.absorb1
thf(fact_7568_min__eq__arg_I1_J,axiom,
    ! [A: $tType] :
      ( ( linorder @ A )
     => ! [M: A,N2: A] :
          ( ( ( ord_min @ A @ M @ N2 )
            = M )
          = ( ord_less_eq @ A @ M @ N2 ) ) ) ).

% min_eq_arg(1)
thf(fact_7569_min__eq__arg_I2_J,axiom,
    ! [A: $tType] :
      ( ( linorder @ A )
     => ! [M: A,N2: A] :
          ( ( ( ord_min @ A @ M @ N2 )
            = N2 )
          = ( ord_less_eq @ A @ N2 @ M ) ) ) ).

% min_eq_arg(2)
thf(fact_7570_min__arg__le_I1_J,axiom,
    ! [A: $tType] :
      ( ( order @ A )
     => ! [N2: A,M: A] :
          ( ( ord_less_eq @ A @ N2 @ ( ord_min @ A @ M @ N2 ) )
          = ( ( ord_min @ A @ M @ N2 )
            = N2 ) ) ) ).

% min_arg_le(1)
thf(fact_7571_min__arg__le_I2_J,axiom,
    ! [A: $tType] :
      ( ( order @ A )
     => ! [M: A,N2: A] :
          ( ( ord_less_eq @ A @ M @ ( ord_min @ A @ M @ N2 ) )
          = ( ( ord_min @ A @ M @ N2 )
            = M ) ) ) ).

% min_arg_le(2)
thf(fact_7572_min__simps_I2_J,axiom,
    ! [A: $tType] :
      ( ( order @ A )
     => ! [B2: A,A2: A] :
          ( ( ord_less @ A @ B2 @ A2 )
         => ( ( ord_min @ A @ A2 @ B2 )
            = B2 ) ) ) ).

% min_simps(2)
thf(fact_7573_min__simps_I1_J,axiom,
    ! [A: $tType] :
      ( ( order @ A )
     => ! [A2: A,B2: A] :
          ( ( ord_less @ A @ A2 @ B2 )
         => ( ( ord_min @ A @ A2 @ B2 )
            = A2 ) ) ) ).

% min_simps(1)
thf(fact_7574_min__less__self__conv_I2_J,axiom,
    ! [A: $tType] :
      ( ( linorder @ A )
     => ! [A2: A,B2: A] :
          ( ( ord_less @ A @ ( ord_min @ A @ A2 @ B2 ) @ B2 )
          = ( ord_less @ A @ A2 @ B2 ) ) ) ).

% min_less_self_conv(2)
thf(fact_7575_min__less__self__conv_I1_J,axiom,
    ! [A: $tType] :
      ( ( linorder @ A )
     => ! [A2: A,B2: A] :
          ( ( ord_less @ A @ ( ord_min @ A @ A2 @ B2 ) @ A2 )
          = ( ord_less @ A @ B2 @ A2 ) ) ) ).

% min_less_self_conv(1)
thf(fact_7576_min__arg__not__ge_I2_J,axiom,
    ! [A: $tType] :
      ( ( linorder @ A )
     => ! [M: A,N2: A] :
          ( ( ~ ( ord_less @ A @ ( ord_min @ A @ M @ N2 ) @ N2 ) )
          = ( ( ord_min @ A @ M @ N2 )
            = N2 ) ) ) ).

% min_arg_not_ge(2)
thf(fact_7577_min__arg__not__ge_I1_J,axiom,
    ! [A: $tType] :
      ( ( linorder @ A )
     => ! [M: A,N2: A] :
          ( ( ~ ( ord_less @ A @ ( ord_min @ A @ M @ N2 ) @ M ) )
          = ( ( ord_min @ A @ M @ N2 )
            = M ) ) ) ).

% min_arg_not_ge(1)
thf(fact_7578_min__less__iff__conj,axiom,
    ! [A: $tType] :
      ( ( linorder @ A )
     => ! [Z: A,X2: A,Y2: A] :
          ( ( ord_less @ A @ Z @ ( ord_min @ A @ X2 @ Y2 ) )
          = ( ( ord_less @ A @ Z @ X2 )
            & ( ord_less @ A @ Z @ Y2 ) ) ) ) ).

% min_less_iff_conj
thf(fact_7579_min_Oabsorb4,axiom,
    ! [A: $tType] :
      ( ( linorder @ A )
     => ! [B2: A,A2: A] :
          ( ( ord_less @ A @ B2 @ A2 )
         => ( ( ord_min @ A @ A2 @ B2 )
            = B2 ) ) ) ).

% min.absorb4
thf(fact_7580_min_Oabsorb3,axiom,
    ! [A: $tType] :
      ( ( linorder @ A )
     => ! [A2: A,B2: A] :
          ( ( ord_less @ A @ A2 @ B2 )
         => ( ( ord_min @ A @ A2 @ B2 )
            = A2 ) ) ) ).

% min.absorb3
thf(fact_7581_min__Suc__Suc,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_min @ nat @ ( suc @ M ) @ ( suc @ N2 ) )
      = ( suc @ ( ord_min @ nat @ M @ N2 ) ) ) ).

% min_Suc_Suc
thf(fact_7582_min__0R,axiom,
    ! [N2: nat] :
      ( ( ord_min @ nat @ N2 @ ( zero_zero @ nat ) )
      = ( zero_zero @ nat ) ) ).

% min_0R
thf(fact_7583_min__0L,axiom,
    ! [N2: nat] :
      ( ( ord_min @ nat @ ( zero_zero @ nat ) @ N2 )
      = ( zero_zero @ nat ) ) ).

% min_0L
thf(fact_7584_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_7585_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_7586_min__0__1_I4_J,axiom,
    ! [A: $tType] :
      ( ( linord181362715937106298miring @ A )
     => ! [X2: num] :
          ( ( ord_min @ A @ ( numeral_numeral @ A @ X2 ) @ ( zero_zero @ A ) )
          = ( zero_zero @ A ) ) ) ).

% min_0_1(4)
thf(fact_7587_min__0__1_I3_J,axiom,
    ! [A: $tType] :
      ( ( linord181362715937106298miring @ A )
     => ! [X2: num] :
          ( ( ord_min @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ X2 ) )
          = ( zero_zero @ A ) ) ) ).

% min_0_1(3)
thf(fact_7588_min__number__of_I1_J,axiom,
    ! [A: $tType] :
      ( ( ( numeral @ A )
        & ( ord @ A ) )
     => ! [U: num,V: num] :
          ( ( ( ord_less_eq @ A @ ( numeral_numeral @ A @ U ) @ ( numeral_numeral @ A @ V ) )
           => ( ( ord_min @ A @ ( numeral_numeral @ A @ U ) @ ( numeral_numeral @ A @ V ) )
              = ( numeral_numeral @ A @ U ) ) )
          & ( ~ ( ord_less_eq @ A @ ( numeral_numeral @ A @ U ) @ ( numeral_numeral @ A @ V ) )
           => ( ( ord_min @ A @ ( numeral_numeral @ A @ U ) @ ( numeral_numeral @ A @ V ) )
              = ( numeral_numeral @ A @ V ) ) ) ) ) ).

% min_number_of(1)
thf(fact_7589_min__0__1_I2_J,axiom,
    ! [A: $tType] :
      ( ( linord181362715937106298miring @ A )
     => ( ( ord_min @ A @ ( one_one @ A ) @ ( zero_zero @ A ) )
        = ( zero_zero @ A ) ) ) ).

% min_0_1(2)
thf(fact_7590_min__0__1_I1_J,axiom,
    ! [A: $tType] :
      ( ( linord181362715937106298miring @ A )
     => ( ( ord_min @ A @ ( zero_zero @ A ) @ ( one_one @ A ) )
        = ( zero_zero @ A ) ) ) ).

% min_0_1(1)
thf(fact_7591_min__0__1_I6_J,axiom,
    ! [A: $tType] :
      ( ( linord181362715937106298miring @ A )
     => ! [X2: num] :
          ( ( ord_min @ A @ ( numeral_numeral @ A @ X2 ) @ ( one_one @ A ) )
          = ( one_one @ A ) ) ) ).

% min_0_1(6)
thf(fact_7592_min__0__1_I5_J,axiom,
    ! [A: $tType] :
      ( ( linord181362715937106298miring @ A )
     => ! [X2: num] :
          ( ( ord_min @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ X2 ) )
          = ( one_one @ A ) ) ) ).

% min_0_1(5)
thf(fact_7593_min__Suc__gt_I2_J,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less @ nat @ A2 @ B2 )
     => ( ( ord_min @ nat @ B2 @ ( suc @ A2 ) )
        = ( suc @ A2 ) ) ) ).

% min_Suc_gt(2)
thf(fact_7594_min__Suc__gt_I1_J,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less @ nat @ A2 @ B2 )
     => ( ( ord_min @ nat @ ( suc @ A2 ) @ B2 )
        = ( suc @ A2 ) ) ) ).

% min_Suc_gt(1)
thf(fact_7595_min__pm,axiom,
    ! [A2: nat,B2: nat] :
      ( ( plus_plus @ nat @ ( ord_min @ nat @ A2 @ B2 ) @ ( minus_minus @ nat @ A2 @ B2 ) )
      = A2 ) ).

% min_pm
thf(fact_7596_min__pm1,axiom,
    ! [A2: nat,B2: nat] :
      ( ( plus_plus @ nat @ ( minus_minus @ nat @ A2 @ B2 ) @ ( ord_min @ nat @ A2 @ B2 ) )
      = A2 ) ).

% min_pm1
thf(fact_7597_rev__min__pm,axiom,
    ! [B2: nat,A2: nat] :
      ( ( plus_plus @ nat @ ( ord_min @ nat @ B2 @ A2 ) @ ( minus_minus @ nat @ A2 @ B2 ) )
      = A2 ) ).

% rev_min_pm
thf(fact_7598_rev__min__pm1,axiom,
    ! [A2: nat,B2: nat] :
      ( ( plus_plus @ nat @ ( minus_minus @ nat @ A2 @ B2 ) @ ( ord_min @ nat @ B2 @ A2 ) )
      = A2 ) ).

% rev_min_pm1
thf(fact_7599_min__number__of_I2_J,axiom,
    ! [A: $tType] :
      ( ( ( uminus @ A )
        & ( numeral @ A )
        & ( ord @ A ) )
     => ! [U: num,V: num] :
          ( ( ( ord_less_eq @ A @ ( numeral_numeral @ A @ U ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) )
           => ( ( ord_min @ A @ ( numeral_numeral @ A @ U ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) )
              = ( numeral_numeral @ A @ U ) ) )
          & ( ~ ( ord_less_eq @ A @ ( numeral_numeral @ A @ U ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) )
           => ( ( ord_min @ A @ ( numeral_numeral @ A @ U ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) )
              = ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) ) ) ) ) ).

% min_number_of(2)
thf(fact_7600_min__number__of_I3_J,axiom,
    ! [A: $tType] :
      ( ( ( uminus @ A )
        & ( numeral @ A )
        & ( ord @ A ) )
     => ! [U: num,V: num] :
          ( ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( numeral_numeral @ A @ V ) )
           => ( ( ord_min @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( numeral_numeral @ A @ V ) )
              = ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) ) )
          & ( ~ ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( numeral_numeral @ A @ V ) )
           => ( ( ord_min @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( numeral_numeral @ A @ V ) )
              = ( numeral_numeral @ A @ V ) ) ) ) ) ).

% min_number_of(3)
thf(fact_7601_min__number__of_I4_J,axiom,
    ! [A: $tType] :
      ( ( ( uminus @ A )
        & ( numeral @ A )
        & ( ord @ A ) )
     => ! [U: num,V: num] :
          ( ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) )
           => ( ( ord_min @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) )
              = ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) ) )
          & ( ~ ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) )
           => ( ( ord_min @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) )
              = ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) ) ) ) ) ).

% min_number_of(4)
thf(fact_7602_min__Suc__numeral,axiom,
    ! [N2: nat,K: num] :
      ( ( ord_min @ nat @ ( suc @ N2 ) @ ( numeral_numeral @ nat @ K ) )
      = ( suc @ ( ord_min @ nat @ N2 @ ( pred_numeral @ K ) ) ) ) ).

% min_Suc_numeral
thf(fact_7603_min__numeral__Suc,axiom,
    ! [K: num,N2: nat] :
      ( ( ord_min @ nat @ ( numeral_numeral @ nat @ K ) @ ( suc @ N2 ) )
      = ( suc @ ( ord_min @ nat @ ( pred_numeral @ K ) @ N2 ) ) ) ).

% min_numeral_Suc
thf(fact_7604_take__bit__word__Suc__numeral,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [N2: nat,K: num] :
          ( ( bit_se2584673776208193580ke_bit @ ( word @ A ) @ ( suc @ N2 ) @ ( numeral_numeral @ ( word @ A ) @ K ) )
          = ( ring_1_of_int @ ( word @ A ) @ ( bit_se2584673776208193580ke_bit @ int @ ( ord_min @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ N2 ) ) @ ( numeral_numeral @ int @ K ) ) ) ) ) ).

% take_bit_word_Suc_numeral
thf(fact_7605_take__bit__word__numeral,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [N2: num,K: num] :
          ( ( bit_se2584673776208193580ke_bit @ ( word @ A ) @ ( numeral_numeral @ nat @ N2 ) @ ( numeral_numeral @ ( word @ A ) @ K ) )
          = ( ring_1_of_int @ ( word @ A ) @ ( bit_se2584673776208193580ke_bit @ int @ ( ord_min @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( numeral_numeral @ nat @ N2 ) ) @ ( numeral_numeral @ int @ K ) ) ) ) ) ).

% take_bit_word_numeral
thf(fact_7606_min__add__distrib__right,axiom,
    ! [A: $tType] :
      ( ( ordere2412721322843649153imp_le @ A )
     => ! [X2: A,Y2: A,Z: A] :
          ( ( plus_plus @ A @ X2 @ ( ord_min @ A @ Y2 @ Z ) )
          = ( ord_min @ A @ ( plus_plus @ A @ X2 @ Y2 ) @ ( plus_plus @ A @ X2 @ Z ) ) ) ) ).

% min_add_distrib_right
thf(fact_7607_min__add__distrib__left,axiom,
    ! [A: $tType] :
      ( ( ordere2412721322843649153imp_le @ A )
     => ! [X2: A,Y2: A,Z: A] :
          ( ( plus_plus @ A @ ( ord_min @ A @ X2 @ Y2 ) @ Z )
          = ( ord_min @ A @ ( plus_plus @ A @ X2 @ Z ) @ ( plus_plus @ A @ Y2 @ Z ) ) ) ) ).

% min_add_distrib_left
thf(fact_7608_min__diff,axiom,
    ! [M: nat,I: nat,N2: nat] :
      ( ( ord_min @ nat @ ( minus_minus @ nat @ M @ I ) @ ( minus_minus @ nat @ N2 @ I ) )
      = ( minus_minus @ nat @ ( ord_min @ nat @ M @ N2 ) @ I ) ) ).

% min_diff
thf(fact_7609_min__diff__distrib__left,axiom,
    ! [A: $tType] :
      ( ( ordered_ab_group_add @ A )
     => ! [X2: A,Y2: A,Z: A] :
          ( ( minus_minus @ A @ ( ord_min @ A @ X2 @ Y2 ) @ Z )
          = ( ord_min @ A @ ( minus_minus @ A @ X2 @ Z ) @ ( minus_minus @ A @ Y2 @ Z ) ) ) ) ).

% min_diff_distrib_left
thf(fact_7610_min__less__iff__disj,axiom,
    ! [A: $tType] :
      ( ( linorder @ A )
     => ! [X2: A,Y2: A,Z: A] :
          ( ( ord_less @ A @ ( ord_min @ A @ X2 @ Y2 ) @ Z )
          = ( ( ord_less @ A @ X2 @ Z )
            | ( ord_less @ A @ Y2 @ Z ) ) ) ) ).

% min_less_iff_disj
thf(fact_7611_min_Ostrict__boundedE,axiom,
    ! [A: $tType] :
      ( ( linorder @ A )
     => ! [A2: A,B2: A,C2: A] :
          ( ( ord_less @ A @ A2 @ ( ord_min @ A @ B2 @ C2 ) )
         => ~ ( ( ord_less @ A @ A2 @ B2 )
             => ~ ( ord_less @ A @ A2 @ C2 ) ) ) ) ).

% min.strict_boundedE
thf(fact_7612_min_Ostrict__order__iff,axiom,
    ! [A: $tType] :
      ( ( linorder @ A )
     => ( ( ord_less @ A )
        = ( ^ [A5: A,B5: A] :
              ( ( A5
                = ( ord_min @ A @ A5 @ B5 ) )
              & ( A5 != B5 ) ) ) ) ) ).

% min.strict_order_iff
thf(fact_7613_min_Ostrict__coboundedI1,axiom,
    ! [A: $tType] :
      ( ( linorder @ A )
     => ! [A2: A,C2: A,B2: A] :
          ( ( ord_less @ A @ A2 @ C2 )
         => ( ord_less @ A @ ( ord_min @ A @ A2 @ B2 ) @ C2 ) ) ) ).

% min.strict_coboundedI1
thf(fact_7614_min_Ostrict__coboundedI2,axiom,
    ! [A: $tType] :
      ( ( linorder @ A )
     => ! [B2: A,C2: A,A2: A] :
          ( ( ord_less @ A @ B2 @ C2 )
         => ( ord_less @ A @ ( ord_min @ A @ A2 @ B2 ) @ C2 ) ) ) ).

% min.strict_coboundedI2
thf(fact_7615_of__nat__min,axiom,
    ! [A: $tType] :
      ( ( linord181362715937106298miring @ A )
     => ! [X2: nat,Y2: nat] :
          ( ( semiring_1_of_nat @ A @ ( ord_min @ nat @ X2 @ Y2 ) )
          = ( ord_min @ A @ ( semiring_1_of_nat @ A @ X2 ) @ ( semiring_1_of_nat @ A @ Y2 ) ) ) ) ).

% of_nat_min
thf(fact_7616_nat__mult__min__right,axiom,
    ! [M: nat,N2: nat,Q2: nat] :
      ( ( times_times @ nat @ M @ ( ord_min @ nat @ N2 @ Q2 ) )
      = ( ord_min @ nat @ ( times_times @ nat @ M @ N2 ) @ ( times_times @ nat @ M @ Q2 ) ) ) ).

% nat_mult_min_right
thf(fact_7617_nat__mult__min__left,axiom,
    ! [M: nat,N2: nat,Q2: nat] :
      ( ( times_times @ nat @ ( ord_min @ nat @ M @ N2 ) @ Q2 )
      = ( ord_min @ nat @ ( times_times @ nat @ M @ Q2 ) @ ( times_times @ nat @ N2 @ Q2 ) ) ) ).

% nat_mult_min_left
thf(fact_7618_min__absorb2,axiom,
    ! [A: $tType] :
      ( ( order @ A )
     => ! [Y2: A,X2: A] :
          ( ( ord_less_eq @ A @ Y2 @ X2 )
         => ( ( ord_min @ A @ X2 @ Y2 )
            = Y2 ) ) ) ).

% min_absorb2
thf(fact_7619_min__absorb1,axiom,
    ! [A: $tType] :
      ( ( ord @ A )
     => ! [X2: A,Y2: A] :
          ( ( ord_less_eq @ A @ X2 @ Y2 )
         => ( ( ord_min @ A @ X2 @ Y2 )
            = X2 ) ) ) ).

% min_absorb1
thf(fact_7620_min__def,axiom,
    ! [A: $tType] :
      ( ( ord @ A )
     => ( ( ord_min @ A )
        = ( ^ [A5: A,B5: A] : ( if @ A @ ( ord_less_eq @ A @ A5 @ B5 ) @ A5 @ B5 ) ) ) ) ).

% min_def
thf(fact_7621_min_Omono,axiom,
    ! [A: $tType] :
      ( ( linorder @ A )
     => ! [A2: A,C2: A,B2: A,D2: A] :
          ( ( ord_less_eq @ A @ A2 @ C2 )
         => ( ( ord_less_eq @ A @ B2 @ D2 )
           => ( ord_less_eq @ A @ ( ord_min @ A @ A2 @ B2 ) @ ( ord_min @ A @ C2 @ D2 ) ) ) ) ) ).

% min.mono
thf(fact_7622_min_OorderE,axiom,
    ! [A: $tType] :
      ( ( linorder @ A )
     => ! [A2: A,B2: A] :
          ( ( ord_less_eq @ A @ A2 @ B2 )
         => ( A2
            = ( ord_min @ A @ A2 @ B2 ) ) ) ) ).

% min.orderE
thf(fact_7623_min_OorderI,axiom,
    ! [A: $tType] :
      ( ( linorder @ A )
     => ! [A2: A,B2: A] :
          ( ( A2
            = ( ord_min @ A @ A2 @ B2 ) )
         => ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).

% min.orderI
thf(fact_7624_min_OboundedE,axiom,
    ! [A: $tType] :
      ( ( linorder @ A )
     => ! [A2: A,B2: A,C2: A] :
          ( ( ord_less_eq @ A @ A2 @ ( ord_min @ A @ B2 @ C2 ) )
         => ~ ( ( ord_less_eq @ A @ A2 @ B2 )
             => ~ ( ord_less_eq @ A @ A2 @ C2 ) ) ) ) ).

% min.boundedE
thf(fact_7625_min_OboundedI,axiom,
    ! [A: $tType] :
      ( ( linorder @ A )
     => ! [A2: A,B2: A,C2: A] :
          ( ( ord_less_eq @ A @ A2 @ B2 )
         => ( ( ord_less_eq @ A @ A2 @ C2 )
           => ( ord_less_eq @ A @ A2 @ ( ord_min @ A @ B2 @ C2 ) ) ) ) ) ).

% min.boundedI
thf(fact_7626_min_Oorder__iff,axiom,
    ! [A: $tType] :
      ( ( linorder @ A )
     => ( ( ord_less_eq @ A )
        = ( ^ [A5: A,B5: A] :
              ( A5
              = ( ord_min @ A @ A5 @ B5 ) ) ) ) ) ).

% min.order_iff
thf(fact_7627_min_Ocobounded1,axiom,
    ! [A: $tType] :
      ( ( linorder @ A )
     => ! [A2: A,B2: A] : ( ord_less_eq @ A @ ( ord_min @ A @ A2 @ B2 ) @ A2 ) ) ).

% min.cobounded1
thf(fact_7628_min_Ocobounded2,axiom,
    ! [A: $tType] :
      ( ( linorder @ A )
     => ! [A2: A,B2: A] : ( ord_less_eq @ A @ ( ord_min @ A @ A2 @ B2 ) @ B2 ) ) ).

% min.cobounded2
thf(fact_7629_min_Oabsorb__iff1,axiom,
    ! [A: $tType] :
      ( ( linorder @ A )
     => ( ( ord_less_eq @ A )
        = ( ^ [A5: A,B5: A] :
              ( ( ord_min @ A @ A5 @ B5 )
              = A5 ) ) ) ) ).

% min.absorb_iff1
thf(fact_7630_min_Oabsorb__iff2,axiom,
    ! [A: $tType] :
      ( ( linorder @ A )
     => ( ( ord_less_eq @ A )
        = ( ^ [B5: A,A5: A] :
              ( ( ord_min @ A @ A5 @ B5 )
              = B5 ) ) ) ) ).

% min.absorb_iff2
thf(fact_7631_min_OcoboundedI1,axiom,
    ! [A: $tType] :
      ( ( linorder @ A )
     => ! [A2: A,C2: A,B2: A] :
          ( ( ord_less_eq @ A @ A2 @ C2 )
         => ( ord_less_eq @ A @ ( ord_min @ A @ A2 @ B2 ) @ C2 ) ) ) ).

% min.coboundedI1
thf(fact_7632_min_OcoboundedI2,axiom,
    ! [A: $tType] :
      ( ( linorder @ A )
     => ! [B2: A,C2: A,A2: A] :
          ( ( ord_less_eq @ A @ B2 @ C2 )
         => ( ord_less_eq @ A @ ( ord_min @ A @ A2 @ B2 ) @ C2 ) ) ) ).

% min.coboundedI2
thf(fact_7633_min__le__iff__disj,axiom,
    ! [A: $tType] :
      ( ( linorder @ A )
     => ! [X2: A,Y2: A,Z: A] :
          ( ( ord_less_eq @ A @ ( ord_min @ A @ X2 @ Y2 ) @ Z )
          = ( ( ord_less_eq @ A @ X2 @ Z )
            | ( ord_less_eq @ A @ Y2 @ Z ) ) ) ) ).

% min_le_iff_disj
thf(fact_7634_min__def__raw,axiom,
    ! [A: $tType] :
      ( ( ord @ A )
     => ( ( ord_min @ A )
        = ( ^ [A5: A,B5: A] : ( if @ A @ ( ord_less_eq @ A @ A5 @ B5 ) @ A5 @ B5 ) ) ) ) ).

% min_def_raw
thf(fact_7635_concat__bit__assoc__sym,axiom,
    ! [M: nat,N2: nat,K: int,L: int,R4: int] :
      ( ( bit_concat_bit @ M @ ( bit_concat_bit @ N2 @ K @ L ) @ R4 )
      = ( bit_concat_bit @ ( ord_min @ nat @ M @ N2 ) @ K @ ( bit_concat_bit @ ( minus_minus @ nat @ M @ N2 ) @ L @ R4 ) ) ) ).

% concat_bit_assoc_sym
thf(fact_7636_take__bit__concat__bit__eq,axiom,
    ! [M: nat,N2: nat,K: int,L: int] :
      ( ( bit_se2584673776208193580ke_bit @ int @ M @ ( bit_concat_bit @ N2 @ K @ L ) )
      = ( bit_concat_bit @ ( ord_min @ nat @ M @ N2 ) @ K @ ( bit_se2584673776208193580ke_bit @ int @ ( minus_minus @ nat @ M @ N2 ) @ L ) ) ) ).

% take_bit_concat_bit_eq
thf(fact_7637_min__mult__distrib__right,axiom,
    ! [A: $tType] :
      ( ( linordered_idom @ A )
     => ! [P4: A,X2: A,Y2: A] :
          ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ P4 )
           => ( ( times_times @ A @ ( ord_min @ A @ X2 @ Y2 ) @ P4 )
              = ( ord_min @ A @ ( times_times @ A @ X2 @ P4 ) @ ( times_times @ A @ Y2 @ P4 ) ) ) )
          & ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ P4 )
           => ( ( times_times @ A @ ( ord_min @ A @ X2 @ Y2 ) @ P4 )
              = ( ord_max @ A @ ( times_times @ A @ X2 @ P4 ) @ ( times_times @ A @ Y2 @ P4 ) ) ) ) ) ) ).

% min_mult_distrib_right
thf(fact_7638_max__mult__distrib__right,axiom,
    ! [A: $tType] :
      ( ( linordered_idom @ A )
     => ! [P4: A,X2: A,Y2: A] :
          ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ P4 )
           => ( ( times_times @ A @ ( ord_max @ A @ X2 @ Y2 ) @ P4 )
              = ( ord_max @ A @ ( times_times @ A @ X2 @ P4 ) @ ( times_times @ A @ Y2 @ P4 ) ) ) )
          & ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ P4 )
           => ( ( times_times @ A @ ( ord_max @ A @ X2 @ Y2 ) @ P4 )
              = ( ord_min @ A @ ( times_times @ A @ X2 @ P4 ) @ ( times_times @ A @ Y2 @ P4 ) ) ) ) ) ) ).

% max_mult_distrib_right
thf(fact_7639_min__mult__distrib__left,axiom,
    ! [A: $tType] :
      ( ( linordered_idom @ A )
     => ! [P4: A,X2: A,Y2: A] :
          ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ P4 )
           => ( ( times_times @ A @ P4 @ ( ord_min @ A @ X2 @ Y2 ) )
              = ( ord_min @ A @ ( times_times @ A @ P4 @ X2 ) @ ( times_times @ A @ P4 @ Y2 ) ) ) )
          & ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ P4 )
           => ( ( times_times @ A @ P4 @ ( ord_min @ A @ X2 @ Y2 ) )
              = ( ord_max @ A @ ( times_times @ A @ P4 @ X2 ) @ ( times_times @ A @ P4 @ Y2 ) ) ) ) ) ) ).

% min_mult_distrib_left
thf(fact_7640_max__mult__distrib__left,axiom,
    ! [A: $tType] :
      ( ( linordered_idom @ A )
     => ! [P4: A,X2: A,Y2: A] :
          ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ P4 )
           => ( ( times_times @ A @ P4 @ ( ord_max @ A @ X2 @ Y2 ) )
              = ( ord_max @ A @ ( times_times @ A @ P4 @ X2 ) @ ( times_times @ A @ P4 @ Y2 ) ) ) )
          & ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ P4 )
           => ( ( times_times @ A @ P4 @ ( ord_max @ A @ X2 @ Y2 ) )
              = ( ord_min @ A @ ( times_times @ A @ P4 @ X2 ) @ ( times_times @ A @ P4 @ Y2 ) ) ) ) ) ) ).

% max_mult_distrib_left
thf(fact_7641_max__divide__distrib__right,axiom,
    ! [A: $tType] :
      ( ( linordered_field @ A )
     => ! [P4: A,X2: A,Y2: A] :
          ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ P4 )
           => ( ( divide_divide @ A @ ( ord_max @ A @ X2 @ Y2 ) @ P4 )
              = ( ord_max @ A @ ( divide_divide @ A @ X2 @ P4 ) @ ( divide_divide @ A @ Y2 @ P4 ) ) ) )
          & ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ P4 )
           => ( ( divide_divide @ A @ ( ord_max @ A @ X2 @ Y2 ) @ P4 )
              = ( ord_min @ A @ ( divide_divide @ A @ X2 @ P4 ) @ ( divide_divide @ A @ Y2 @ P4 ) ) ) ) ) ) ).

% max_divide_distrib_right
thf(fact_7642_min__divide__distrib__right,axiom,
    ! [A: $tType] :
      ( ( linordered_field @ A )
     => ! [P4: A,X2: A,Y2: A] :
          ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ P4 )
           => ( ( divide_divide @ A @ ( ord_min @ A @ X2 @ Y2 ) @ P4 )
              = ( ord_min @ A @ ( divide_divide @ A @ X2 @ P4 ) @ ( divide_divide @ A @ Y2 @ P4 ) ) ) )
          & ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ P4 )
           => ( ( divide_divide @ A @ ( ord_min @ A @ X2 @ Y2 ) @ P4 )
              = ( ord_max @ A @ ( divide_divide @ A @ X2 @ P4 ) @ ( divide_divide @ A @ Y2 @ P4 ) ) ) ) ) ) ).

% min_divide_distrib_right
thf(fact_7643_min__Suc2,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_min @ nat @ M @ ( suc @ N2 ) )
      = ( case_nat @ nat @ ( zero_zero @ nat )
        @ ^ [M5: nat] : ( suc @ ( ord_min @ nat @ M5 @ N2 ) )
        @ M ) ) ).

% min_Suc2
thf(fact_7644_min__Suc1,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_min @ nat @ ( suc @ N2 ) @ M )
      = ( case_nat @ nat @ ( zero_zero @ nat )
        @ ^ [M5: nat] : ( suc @ ( ord_min @ nat @ N2 @ M5 ) )
        @ M ) ) ).

% min_Suc1
thf(fact_7645_mod__exp__eq,axiom,
    ! [A: $tType] :
      ( ( bit_semiring_bits @ A )
     => ! [A2: A,M: nat,N2: nat] :
          ( ( modulo_modulo @ A @ ( modulo_modulo @ A @ A2 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) )
          = ( modulo_modulo @ A @ A2 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( ord_min @ nat @ M @ N2 ) ) ) ) ) ).

% mod_exp_eq
thf(fact_7646_mod__mod__power,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( modulo_modulo @ nat @ ( modulo_modulo @ nat @ K @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) )
      = ( modulo_modulo @ nat @ K @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( ord_min @ nat @ M @ N2 ) ) ) ) ).

% mod_mod_power
thf(fact_7647_Word_Obit__mask__iff,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [M: nat,N2: nat] :
          ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ M ) @ N2 )
          = ( ord_less @ nat @ N2 @ ( ord_min @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ M ) ) ) ) ).

% Word.bit_mask_iff
thf(fact_7648_mask__mod__exp,axiom,
    ! [A: $tType] :
      ( ( euclid5411537665997757685th_nat @ A )
     => ! [N2: nat,M: nat] :
          ( ( modulo_modulo @ A @ ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) @ ( one_one @ A ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) )
          = ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( ord_min @ nat @ M @ N2 ) ) @ ( one_one @ A ) ) ) ) ).

% mask_mod_exp
thf(fact_7649_bit__slice__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( ( ( type_len @ A )
        & ( type_len @ B ) )
     => ! [M: nat,W: word @ A,N2: nat] :
          ( ( bit_se5641148757651400278ts_bit @ ( word @ B ) @ ( slice2 @ A @ B @ M @ W ) @ N2 )
          = ( ( ord_less @ nat @ N2 @ ( ord_min @ nat @ ( type_len0_len_of @ B @ ( type2 @ B ) ) @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ M ) ) )
            & ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ W @ ( minus_minus @ nat @ ( plus_plus @ nat @ N2 @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ M ) ) ) ) ) ) ).

% bit_slice_iff
thf(fact_7650_bit__slice1__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( ( ( type_len @ A )
        & ( type_len @ B ) )
     => ! [M: nat,W: word @ A,N2: nat] :
          ( ( bit_se5641148757651400278ts_bit @ ( word @ B ) @ ( slice1 @ A @ B @ M @ W ) @ N2 )
          = ( ( ord_less_eq @ nat @ ( minus_minus @ nat @ M @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) @ N2 )
            & ( ord_less @ nat @ N2 @ ( ord_min @ nat @ ( type_len0_len_of @ B @ ( type2 @ B ) ) @ M ) )
            & ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ W @ ( minus_minus @ nat @ ( plus_plus @ nat @ N2 @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ M ) ) @ ( minus_minus @ nat @ M @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ) ) ).

% bit_slice1_iff
thf(fact_7651_unat__mask,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [N2: nat] :
          ( ( semiring_1_unsigned @ A @ nat @ ( bit_se2239418461657761734s_mask @ ( word @ A ) @ N2 ) )
          = ( minus_minus @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( ord_min @ nat @ N2 @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) @ ( one_one @ nat ) ) ) ) ).

% unat_mask
thf(fact_7652_bit__horner__sum__bit__word__iff,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [Bs2: list @ $o,N2: nat] :
          ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( groups4207007520872428315er_sum @ $o @ ( word @ A ) @ ( zero_neq_one_of_bool @ ( word @ A ) ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ Bs2 ) @ N2 )
          = ( ( ord_less @ nat @ N2 @ ( ord_min @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( size_size @ ( list @ $o ) @ Bs2 ) ) )
            & ( nth @ $o @ Bs2 @ N2 ) ) ) ) ).

% bit_horner_sum_bit_word_iff
thf(fact_7653_min__list_Oelims,axiom,
    ! [A: $tType] :
      ( ( ord @ A )
     => ! [X2: list @ A,Y2: A] :
          ( ( ( min_list @ A @ X2 )
            = Y2 )
         => ( ! [X3: A,Xs3: list @ A] :
                ( ( X2
                  = ( cons @ A @ X3 @ Xs3 ) )
               => ( Y2
                 != ( case_list @ A @ A @ X3
                    @ ^ [A5: A,List: list @ A] : ( ord_min @ A @ X3 @ ( min_list @ A @ Xs3 ) )
                    @ Xs3 ) ) )
           => ~ ( ( X2
                  = ( nil @ A ) )
               => ( Y2
                 != ( undefined @ A ) ) ) ) ) ) ).

% min_list.elims
thf(fact_7654_drop__bit__exp__eq,axiom,
    ! [A: $tType] :
      ( ( bit_se359711467146920520ations @ A )
     => ! [M: nat,N2: nat] :
          ( ( bit_se4197421643247451524op_bit @ A @ M @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) )
          = ( times_times @ A
            @ ( zero_neq_one_of_bool @ A
              @ ( ( ord_less_eq @ nat @ M @ N2 )
                & ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N2 ) ) )
            @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ N2 @ M ) ) ) ) ) ).

% drop_bit_exp_eq
thf(fact_7655_possible__bit__word,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [M: nat] :
          ( ( bit_se6407376104438227557le_bit @ ( word @ A ) @ ( type2 @ ( word @ A ) ) @ M )
          = ( ord_less @ nat @ M @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ).

% possible_bit_word
thf(fact_7656_bit__minus__1__iff,axiom,
    ! [A: $tType] :
      ( ( bit_ri3973907225187159222ations @ A )
     => ! [N2: nat] :
          ( ( bit_se5641148757651400278ts_bit @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N2 )
          = ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N2 ) ) ) ).

% bit_minus_1_iff
thf(fact_7657_bit__minus__2__iff,axiom,
    ! [A: $tType] :
      ( ( bit_ri3973907225187159222ations @ A )
     => ! [N2: nat] :
          ( ( bit_se5641148757651400278ts_bit @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ N2 )
          = ( ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N2 )
            & ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ) ).

% bit_minus_2_iff
thf(fact_7658_possible__bit__less__imp,axiom,
    ! [A: $tType] :
      ( ( bit_semiring_bits @ A )
     => ! [Tyrep: itself @ A,I: nat,J: nat] :
          ( ( bit_se6407376104438227557le_bit @ A @ Tyrep @ I )
         => ( ( ord_less_eq @ nat @ J @ I )
           => ( bit_se6407376104438227557le_bit @ A @ Tyrep @ J ) ) ) ) ).

% possible_bit_less_imp
thf(fact_7659_bit__mask__iff,axiom,
    ! [A: $tType] :
      ( ( bit_se359711467146920520ations @ A )
     => ! [M: nat,N2: nat] :
          ( ( bit_se5641148757651400278ts_bit @ A @ ( bit_se2239418461657761734s_mask @ A @ M ) @ N2 )
          = ( ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N2 )
            & ( ord_less @ nat @ N2 @ M ) ) ) ) ).

% bit_mask_iff
thf(fact_7660_possible__bit__def,axiom,
    ! [A: $tType] :
      ( ( bit_semiring_bits @ A )
     => ( ( bit_se6407376104438227557le_bit @ A )
        = ( ^ [Tyrep2: itself @ A,N: nat] :
              ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N )
             != ( zero_zero @ A ) ) ) ) ) ).

% possible_bit_def
thf(fact_7661_bit__minus__iff,axiom,
    ! [A: $tType] :
      ( ( bit_ri3973907225187159222ations @ A )
     => ! [A2: A,N2: nat] :
          ( ( bit_se5641148757651400278ts_bit @ A @ ( uminus_uminus @ A @ A2 ) @ N2 )
          = ( ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N2 )
            & ~ ( bit_se5641148757651400278ts_bit @ A @ ( minus_minus @ A @ A2 @ ( one_one @ A ) ) @ N2 ) ) ) ) ).

% bit_minus_iff
thf(fact_7662_bit__twiddle__min,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [Y2: word @ A,X2: word @ A] :
          ( ( bit_se5824344971392196577ns_xor @ ( word @ A ) @ Y2 @ ( bit_se5824344872417868541ns_and @ ( word @ A ) @ ( bit_se5824344971392196577ns_xor @ ( word @ A ) @ X2 @ Y2 ) @ ( if @ ( word @ A ) @ ( ord_less @ ( word @ A ) @ X2 @ Y2 ) @ ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) @ ( zero_zero @ ( word @ A ) ) ) ) )
          = ( ord_min @ ( word @ A ) @ X2 @ Y2 ) ) ) ).

% bit_twiddle_min
thf(fact_7663_bit__push__bit__iff,axiom,
    ! [A: $tType] :
      ( ( bit_se359711467146920520ations @ A )
     => ! [M: nat,A2: A,N2: nat] :
          ( ( bit_se5641148757651400278ts_bit @ A @ ( bit_se4730199178511100633sh_bit @ A @ M @ A2 ) @ N2 )
          = ( ( ord_less_eq @ nat @ M @ N2 )
            & ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N2 )
            & ( bit_se5641148757651400278ts_bit @ A @ A2 @ ( minus_minus @ nat @ N2 @ M ) ) ) ) ) ).

% bit_push_bit_iff
thf(fact_7664_fold__possible__bit,axiom,
    ! [A: $tType] :
      ( ( bit_semiring_bits @ A )
     => ! [N2: nat] :
          ( ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 )
            = ( zero_zero @ A ) )
          = ( ~ ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N2 ) ) ) ) ).

% fold_possible_bit
thf(fact_7665_bit__exp__iff,axiom,
    ! [A: $tType] :
      ( ( bit_semiring_bits @ A )
     => ! [M: nat,N2: nat] :
          ( ( bit_se5641148757651400278ts_bit @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) @ N2 )
          = ( ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N2 )
            & ( M = N2 ) ) ) ) ).

% bit_exp_iff
thf(fact_7666_bit__2__iff,axiom,
    ! [A: $tType] :
      ( ( bit_semiring_bits @ A )
     => ! [N2: nat] :
          ( ( bit_se5641148757651400278ts_bit @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 )
          = ( ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ ( one_one @ nat ) )
            & ( N2
              = ( one_one @ nat ) ) ) ) ) ).

% bit_2_iff
thf(fact_7667_min__list_Osimps,axiom,
    ! [A: $tType] :
      ( ( ord @ A )
     => ! [X2: A,Xs2: list @ A] :
          ( ( min_list @ A @ ( cons @ A @ X2 @ Xs2 ) )
          = ( case_list @ A @ A @ X2
            @ ^ [A5: A,List: list @ A] : ( ord_min @ A @ X2 @ ( min_list @ A @ Xs2 ) )
            @ Xs2 ) ) ) ).

% min_list.simps
thf(fact_7668_bit__not__exp__iff,axiom,
    ! [A: $tType] :
      ( ( bit_ri3973907225187159222ations @ A )
     => ! [M: nat,N2: nat] :
          ( ( bit_se5641148757651400278ts_bit @ A @ ( bit_ri4277139882892585799ns_not @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) ) @ N2 )
          = ( ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N2 )
            & ( N2 != M ) ) ) ) ).

% bit_not_exp_iff
thf(fact_7669_bit__minus__exp__iff,axiom,
    ! [A: $tType] :
      ( ( bit_ri3973907225187159222ations @ A )
     => ! [M: nat,N2: nat] :
          ( ( bit_se5641148757651400278ts_bit @ A @ ( uminus_uminus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) ) @ N2 )
          = ( ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N2 )
            & ( ord_less_eq @ nat @ M @ N2 ) ) ) ) ).

% bit_minus_exp_iff
thf(fact_7670_bit__mask__sub__iff,axiom,
    ! [A: $tType] :
      ( ( bit_semiring_bits @ A )
     => ! [M: nat,N2: nat] :
          ( ( bit_se5641148757651400278ts_bit @ A @ ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) @ ( one_one @ A ) ) @ N2 )
          = ( ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N2 )
            & ( ord_less @ nat @ N2 @ M ) ) ) ) ).

% bit_mask_sub_iff
thf(fact_7671_bit__double__iff,axiom,
    ! [A: $tType] :
      ( ( bit_semiring_bits @ A )
     => ! [A2: A,N2: nat] :
          ( ( bit_se5641148757651400278ts_bit @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) @ N2 )
          = ( ( bit_se5641148757651400278ts_bit @ A @ A2 @ ( minus_minus @ nat @ N2 @ ( one_one @ nat ) ) )
            & ( N2
             != ( zero_zero @ nat ) )
            & ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N2 ) ) ) ) ).

% bit_double_iff
thf(fact_7672_bit__signed__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ( type_len @ B )
        & ( bit_ri3973907225187159222ations @ A ) )
     => ! [W: word @ B,N2: nat] :
          ( ( bit_se5641148757651400278ts_bit @ A @ ( ring_1_signed @ B @ A @ W ) @ N2 )
          = ( ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N2 )
            & ( bit_se5641148757651400278ts_bit @ ( word @ B ) @ W @ ( ord_min @ nat @ ( minus_minus @ nat @ ( type_len0_len_of @ B @ ( type2 @ B ) ) @ ( suc @ ( zero_zero @ nat ) ) ) @ N2 ) ) ) ) ) ).

% bit_signed_iff
thf(fact_7673_min__list_Opelims,axiom,
    ! [A: $tType] :
      ( ( ord @ A )
     => ! [X2: list @ A,Y2: A] :
          ( ( ( min_list @ A @ X2 )
            = Y2 )
         => ( ( accp @ ( list @ A ) @ ( min_list_rel @ A ) @ X2 )
           => ( ! [X3: A,Xs3: list @ A] :
                  ( ( X2
                    = ( cons @ A @ X3 @ Xs3 ) )
                 => ( ( Y2
                      = ( case_list @ A @ A @ X3
                        @ ^ [A5: A,List: list @ A] : ( ord_min @ A @ X3 @ ( min_list @ A @ Xs3 ) )
                        @ Xs3 ) )
                   => ~ ( accp @ ( list @ A ) @ ( min_list_rel @ A ) @ ( cons @ A @ X3 @ Xs3 ) ) ) )
             => ~ ( ( X2
                    = ( nil @ A ) )
                 => ( ( Y2
                      = ( undefined @ A ) )
                   => ~ ( accp @ ( list @ A ) @ ( min_list_rel @ A ) @ ( nil @ A ) ) ) ) ) ) ) ) ).

% min_list.pelims
thf(fact_7674_word__set__bits__def,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ( ( bit_bi4170147762399595738t_bits @ ( word @ A ) )
        = ( ^ [P3: nat > $o] : ( groups4207007520872428315er_sum @ $o @ ( word @ A ) @ ( zero_neq_one_of_bool @ ( word @ A ) ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ ( map @ nat @ $o @ P3 @ ( upt @ ( zero_zero @ nat ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ) ) ).

% word_set_bits_def
thf(fact_7675_set__bits__False__eq,axiom,
    ! [A: $tType] :
      ( ( bit_bi6583157726757044596ension @ A )
     => ( ( bit_bi4170147762399595738t_bits @ A
          @ ^ [Uu3: nat] : $false )
        = ( zero_zero @ A ) ) ) ).

% set_bits_False_eq
thf(fact_7676_set__bits__K__False,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ( ( bit_bi4170147762399595738t_bits @ ( word @ A )
          @ ^ [Uu3: nat] : $false )
        = ( zero_zero @ ( word @ A ) ) ) ) ).

% set_bits_K_False
thf(fact_7677_bit__set__bits__word__iff,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [P: nat > $o,N2: nat] :
          ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( bit_bi4170147762399595738t_bits @ ( word @ A ) @ P ) @ N2 )
          = ( ( ord_less @ nat @ N2 @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
            & ( P @ N2 ) ) ) ) ).

% bit_set_bits_word_iff
thf(fact_7678_word__of__int__conv__set__bits,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ( ( ring_1_of_int @ ( word @ A ) )
        = ( ^ [I2: int] : ( bit_bi4170147762399595738t_bits @ ( word @ A ) @ ( bit_se5641148757651400278ts_bit @ int @ I2 ) ) ) ) ) ).

% word_of_int_conv_set_bits
thf(fact_7679_word__test__bit__set__bits,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [F: nat > $o,N2: nat] :
          ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( bit_bi4170147762399595738t_bits @ ( word @ A ) @ F ) @ N2 )
          = ( ( ord_less @ nat @ N2 @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
            & ( F @ N2 ) ) ) ) ).

% word_test_bit_set_bits
thf(fact_7680_uint__word__rotr__eq,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [N2: nat,W: word @ A] :
          ( ( semiring_1_unsigned @ A @ int @ ( word_rotr @ A @ N2 @ W ) )
          = ( bit_concat_bit @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( modulo_modulo @ nat @ N2 @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) @ ( bit_se4197421643247451524op_bit @ int @ ( modulo_modulo @ nat @ N2 @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) @ ( semiring_1_unsigned @ A @ int @ W ) ) @ ( semiring_1_unsigned @ A @ int @ ( bit_se2584673776208193580ke_bit @ ( word @ A ) @ ( modulo_modulo @ nat @ N2 @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) @ W ) ) ) ) ) ).

% uint_word_rotr_eq
thf(fact_7681_word__roti_Oabs__eq,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [Xa: int,X2: int] :
          ( ( word_roti @ A @ Xa @ ( word2 @ A @ X2 ) )
          = ( word2 @ A @ ( bit_concat_bit @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( nat2 @ ( modulo_modulo @ int @ Xa @ ( semiring_1_of_nat @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) @ ( bit_se4197421643247451524op_bit @ int @ ( nat2 @ ( modulo_modulo @ int @ Xa @ ( semiring_1_of_nat @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ X2 ) ) @ ( bit_se2584673776208193580ke_bit @ int @ ( nat2 @ ( modulo_modulo @ int @ Xa @ ( semiring_1_of_nat @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) @ X2 ) ) ) ) ) ).

% word_roti.abs_eq
thf(fact_7682_int__set__bits__K__False,axiom,
    ( ( bit_bi4170147762399595738t_bits @ int
      @ ^ [Uu3: nat] : $false )
    = ( zero_zero @ int ) ) ).

% int_set_bits_K_False
thf(fact_7683_int__set__bits__K__True,axiom,
    ( ( bit_bi4170147762399595738t_bits @ int
      @ ^ [Uu3: nat] : $true )
    = ( uminus_uminus @ int @ ( one_one @ int ) ) ) ).

% int_set_bits_K_True
thf(fact_7684_size__word_Oabs__eq,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [X2: int] :
          ( ( size_size @ ( word @ A ) @ ( word2 @ A @ X2 ) )
          = ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ).

% size_word.abs_eq
thf(fact_7685_word__pred_Oabs__eq,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [X2: int] :
          ( ( word_pred @ A @ ( word2 @ A @ X2 ) )
          = ( word2 @ A @ ( minus_minus @ int @ X2 @ ( one_one @ int ) ) ) ) ) ).

% word_pred.abs_eq
thf(fact_7686_times__word_Oabs__eq,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [Xa: int,X2: int] :
          ( ( times_times @ ( word @ A ) @ ( word2 @ A @ Xa ) @ ( word2 @ A @ X2 ) )
          = ( word2 @ A @ ( times_times @ int @ Xa @ X2 ) ) ) ) ).

% times_word.abs_eq
thf(fact_7687_one__word__def,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ( ( one_one @ ( word @ A ) )
        = ( word2 @ A @ ( one_one @ int ) ) ) ) ).

% one_word_def
thf(fact_7688_minus__word_Oabs__eq,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [Xa: int,X2: int] :
          ( ( minus_minus @ ( word @ A ) @ ( word2 @ A @ Xa ) @ ( word2 @ A @ X2 ) )
          = ( word2 @ A @ ( minus_minus @ int @ Xa @ X2 ) ) ) ) ).

% minus_word.abs_eq
thf(fact_7689_word__rotr__word__rotr__eq,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [M: nat,N2: nat,W: word @ A] :
          ( ( word_rotr @ A @ M @ ( word_rotr @ A @ N2 @ W ) )
          = ( word_rotr @ A @ ( plus_plus @ nat @ M @ N2 ) @ W ) ) ) ).

% word_rotr_word_rotr_eq
thf(fact_7690_plus__word_Oabs__eq,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [Xa: int,X2: int] :
          ( ( plus_plus @ ( word @ A ) @ ( word2 @ A @ Xa ) @ ( word2 @ A @ X2 ) )
          = ( word2 @ A @ ( plus_plus @ int @ Xa @ X2 ) ) ) ) ).

% plus_word.abs_eq
thf(fact_7691_word__succ_Oabs__eq,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [X2: int] :
          ( ( word_succ @ A @ ( word2 @ A @ X2 ) )
          = ( word2 @ A @ ( plus_plus @ int @ X2 @ ( one_one @ int ) ) ) ) ) ).

% word_succ.abs_eq
thf(fact_7692_word__rotr_Oabs__eq,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [Xa: nat,X2: int] :
          ( ( word_rotr @ A @ Xa @ ( word2 @ A @ X2 ) )
          = ( word2 @ A @ ( bit_concat_bit @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( modulo_modulo @ nat @ Xa @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) @ ( bit_se4197421643247451524op_bit @ int @ ( modulo_modulo @ nat @ Xa @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ X2 ) ) @ ( bit_se2584673776208193580ke_bit @ int @ ( modulo_modulo @ nat @ Xa @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) @ X2 ) ) ) ) ) ).

% word_rotr.abs_eq
thf(fact_7693_less__eq__word_Oabs__eq,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [Xa: int,X2: int] :
          ( ( ord_less_eq @ ( word @ A ) @ ( word2 @ A @ Xa ) @ ( word2 @ A @ X2 ) )
          = ( ord_less_eq @ int @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ Xa ) @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ X2 ) ) ) ) ).

% less_eq_word.abs_eq
thf(fact_7694_bit__word_Oabs__eq,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [X2: int] :
          ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( word2 @ A @ X2 ) )
          = ( ^ [N: nat] :
                ( ( ord_less @ nat @ N @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
                & ( bit_se5641148757651400278ts_bit @ int @ X2 @ N ) ) ) ) ) ).

% bit_word.abs_eq
thf(fact_7695_divide__word_Oabs__eq,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [Xa: int,X2: int] :
          ( ( divide_divide @ ( word @ A ) @ ( word2 @ A @ Xa ) @ ( word2 @ A @ X2 ) )
          = ( word2 @ A @ ( divide_divide @ int @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ Xa ) @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ X2 ) ) ) ) ) ).

% divide_word.abs_eq
thf(fact_7696_word__sle_Oabs__eq,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [Xa: int,X2: int] :
          ( ( word_sle @ A @ ( word2 @ A @ Xa ) @ ( word2 @ A @ X2 ) )
          = ( ord_less_eq @ int @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) @ Xa ) @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) @ X2 ) ) ) ) ).

% word_sle.abs_eq
thf(fact_7697_word__sless_Oabs__eq,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [Xa: int,X2: int] :
          ( ( word_sless @ A @ ( word2 @ A @ Xa ) @ ( word2 @ A @ X2 ) )
          = ( ord_less @ int @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) @ Xa ) @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) @ X2 ) ) ) ) ).

% word_sless.abs_eq
thf(fact_7698_signed_Oabs__eq,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ( type_len @ B )
        & ( ring_1 @ A ) )
     => ! [X2: int] :
          ( ( ring_1_signed @ B @ A @ ( word2 @ B @ X2 ) )
          = ( ring_1_of_int @ A @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ B @ ( type2 @ B ) ) @ ( suc @ ( zero_zero @ nat ) ) ) @ X2 ) ) ) ) ).

% signed.abs_eq
thf(fact_7699_signed__drop__bit_Oabs__eq,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [Xa: nat,X2: int] :
          ( ( signed_drop_bit @ A @ Xa @ ( word2 @ A @ X2 ) )
          = ( word2 @ A @ ( bit_se4197421643247451524op_bit @ int @ Xa @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) @ X2 ) ) ) ) ) ).

% signed_drop_bit.abs_eq
thf(fact_7700_bit__word__rotr__iff,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [M: nat,W: word @ A,N2: nat] :
          ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( word_rotr @ A @ M @ W ) @ N2 )
          = ( ( ord_less @ nat @ N2 @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
            & ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ W @ ( modulo_modulo @ nat @ ( plus_plus @ nat @ N2 @ M ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ) ).

% bit_word_rotr_iff
thf(fact_7701_signed__modulo__word_Oabs__eq,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [Xa: int,X2: int] :
          ( ( signed6721504322012087516modulo @ ( word @ A ) @ ( word2 @ A @ Xa ) @ ( word2 @ A @ X2 ) )
          = ( word2 @ A @ ( signed6721504322012087516modulo @ int @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) @ Xa ) @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) @ X2 ) ) ) ) ) ).

% signed_modulo_word.abs_eq
thf(fact_7702_signed__divide__word_Oabs__eq,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [Xa: int,X2: int] :
          ( ( signed7115095781618012415divide @ ( word @ A ) @ ( word2 @ A @ Xa ) @ ( word2 @ A @ X2 ) )
          = ( word2 @ A @ ( signed7115095781618012415divide @ int @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) @ Xa ) @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) @ X2 ) ) ) ) ) ).

% signed_divide_word.abs_eq
thf(fact_7703_signed__cast_Oabs__eq,axiom,
    ! [B: $tType,A: $tType] :
      ( ( ( type_len @ A )
        & ( type_len @ B ) )
     => ! [X2: int] :
          ( ( signed_cast @ A @ B @ ( word2 @ A @ X2 ) )
          = ( word2 @ B @ ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) @ X2 ) ) ) ) ).

% signed_cast.abs_eq
thf(fact_7704_the__signed__int_Oabs__eq,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [X2: int] :
          ( ( the_signed_int @ A @ ( word2 @ A @ X2 ) )
          = ( bit_ri4674362597316999326ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) @ X2 ) ) ) ).

% the_signed_int.abs_eq
thf(fact_7705_bin__last__set__bits,axiom,
    ! [F: nat > $o] :
      ( ( bit_wf_set_bits_int @ F )
     => ( ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_bi4170147762399595738t_bits @ int @ F ) ) )
        = ( F @ ( zero_zero @ nat ) ) ) ) ).

% bin_last_set_bits
thf(fact_7706_word__roti__eq__word__rotr__word__rotl,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ( ( word_roti @ A )
        = ( ^ [I2: int,W2: word @ A] : ( if @ ( word @ A ) @ ( ord_less_eq @ int @ ( zero_zero @ int ) @ I2 ) @ ( word_rotr @ A @ ( nat2 @ I2 ) @ W2 ) @ ( word_rotl @ A @ ( nat2 @ ( uminus_uminus @ int @ I2 ) ) @ W2 ) ) ) ) ) ).

% word_roti_eq_word_rotr_word_rotl
thf(fact_7707_wf__set__bits__int__Suc,axiom,
    ! [F: nat > $o] :
      ( ( bit_wf_set_bits_int
        @ ^ [N: nat] : ( F @ ( suc @ N ) ) )
      = ( bit_wf_set_bits_int @ F ) ) ).

% wf_set_bits_int_Suc
thf(fact_7708_wf__set__bits__int__const,axiom,
    ! [B2: $o] :
      ( bit_wf_set_bits_int
      @ ^ [Uu3: nat] : B2 ) ).

% wf_set_bits_int_const
thf(fact_7709_ones,axiom,
    ! [N2: nat,F: nat > $o] :
      ( ! [N7: nat] :
          ( ( ord_less_eq @ nat @ N2 @ N7 )
         => ( F @ N7 ) )
     => ( bit_wf_set_bits_int @ F ) ) ).

% ones
thf(fact_7710_wf__set__bits__int_Ocases,axiom,
    ! [F: nat > $o] :
      ( ( bit_wf_set_bits_int @ F )
     => ( ! [N4: nat] :
            ~ ! [N6: nat] :
                ( ( ord_less_eq @ nat @ N4 @ N6 )
               => ~ ( F @ N6 ) )
       => ~ ! [N4: nat] :
              ~ ! [N6: nat] :
                  ( ( ord_less_eq @ nat @ N4 @ N6 )
                 => ( F @ N6 ) ) ) ) ).

% wf_set_bits_int.cases
thf(fact_7711_wf__set__bits__int_Osimps,axiom,
    ( bit_wf_set_bits_int
    = ( ^ [F6: nat > $o] :
          ( ? [N: nat] :
            ! [N12: nat] :
              ( ( ord_less_eq @ nat @ N @ N12 )
             => ~ ( F6 @ N12 ) )
          | ? [N: nat] :
            ! [N12: nat] :
              ( ( ord_less_eq @ nat @ N @ N12 )
             => ( F6 @ N12 ) ) ) ) ) ).

% wf_set_bits_int.simps
thf(fact_7712_zeros,axiom,
    ! [N2: nat,F: nat > $o] :
      ( ! [N7: nat] :
          ( ( ord_less_eq @ nat @ N2 @ N7 )
         => ~ ( F @ N7 ) )
     => ( bit_wf_set_bits_int @ F ) ) ).

% zeros
thf(fact_7713_wf__set__bits__int__simps,axiom,
    ( bit_wf_set_bits_int
    = ( ^ [F6: nat > $o] :
        ? [N: nat] :
          ( ! [N12: nat] :
              ( ( ord_less_eq @ nat @ N @ N12 )
             => ~ ( F6 @ N12 ) )
          | ! [N12: nat] :
              ( ( ord_less_eq @ nat @ N @ N12 )
             => ( F6 @ N12 ) ) ) ) ) ).

% wf_set_bits_int_simps
thf(fact_7714_word__rotl__eq__word__rotr,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ( ( word_rotl @ A )
        = ( ^ [N: nat] : ( word_rotr @ A @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( modulo_modulo @ nat @ N @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ) ) ).

% word_rotl_eq_word_rotr
thf(fact_7715_bit__word__rotl__iff,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [M: nat,W: word @ A,N2: nat] :
          ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( word_rotl @ A @ M @ W ) @ N2 )
          = ( ( ord_less @ nat @ N2 @ ( type_len0_len_of @ A @ ( type2 @ A ) ) )
            & ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ W @ ( modulo_modulo @ nat @ ( plus_plus @ nat @ N2 @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( modulo_modulo @ nat @ M @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ) ).

% bit_word_rotl_iff
thf(fact_7716_word__rotl_Oabs__eq,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [Xa: nat,X2: int] :
          ( ( word_rotl @ A @ Xa @ ( word2 @ A @ X2 ) )
          = ( word2 @ A @ ( bit_concat_bit @ ( modulo_modulo @ nat @ Xa @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) @ ( bit_se4197421643247451524op_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( modulo_modulo @ nat @ Xa @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) @ ( bit_se2584673776208193580ke_bit @ int @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ X2 ) ) @ ( bit_se2584673776208193580ke_bit @ int @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( modulo_modulo @ nat @ Xa @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) @ X2 ) ) ) ) ) ).

% word_rotl.abs_eq
thf(fact_7717_int__set__bits__unfold__BIT,axiom,
    ! [F: nat > $o] :
      ( ( bit_wf_set_bits_int @ F )
     => ( ( bit_bi4170147762399595738t_bits @ int @ F )
        = ( plus_plus @ int @ ( zero_neq_one_of_bool @ int @ ( F @ ( zero_zero @ nat ) ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_bi4170147762399595738t_bits @ int @ ( comp @ nat @ $o @ nat @ F @ suc ) ) ) ) ) ) ).

% int_set_bits_unfold_BIT
thf(fact_7718_CHAR__eq0__iff,axiom,
    ! [A: $tType] :
      ( ( semiring_1 @ A )
     => ( ( ( semiri4206861660011772517g_char @ A @ ( type2 @ A ) )
          = ( zero_zero @ nat ) )
        = ( ! [N: nat] :
              ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
             => ( ( semiring_1_of_nat @ A @ N )
               != ( zero_zero @ A ) ) ) ) ) ) ).

% CHAR_eq0_iff
thf(fact_7719_length__filter__map,axiom,
    ! [A: $tType,B: $tType,P: A > $o,F: B > A,Xs2: list @ B] :
      ( ( size_size @ ( list @ A ) @ ( filter2 @ A @ P @ ( map @ B @ A @ F @ Xs2 ) ) )
      = ( size_size @ ( list @ B ) @ ( filter2 @ B @ ( comp @ A @ $o @ B @ P @ F ) @ Xs2 ) ) ) ).

% length_filter_map
thf(fact_7720_bin__rest__set__bits,axiom,
    ! [F: nat > $o] :
      ( ( bit_wf_set_bits_int @ F )
     => ( ( divide_divide @ int @ ( bit_bi4170147762399595738t_bits @ int @ F ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) )
        = ( bit_bi4170147762399595738t_bits @ int @ ( comp @ nat @ $o @ nat @ F @ suc ) ) ) ) ).

% bin_rest_set_bits
thf(fact_7721_sum__comp__morphism,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( ( ( comm_monoid_add @ B )
        & ( comm_monoid_add @ A ) )
     => ! [H2: B > A,G: C > B,A3: set @ C] :
          ( ( ( H2 @ ( zero_zero @ B ) )
            = ( zero_zero @ A ) )
         => ( ! [X3: B,Y4: B] :
                ( ( H2 @ ( plus_plus @ B @ X3 @ Y4 ) )
                = ( plus_plus @ A @ ( H2 @ X3 ) @ ( H2 @ Y4 ) ) )
           => ( ( groups7311177749621191930dd_sum @ C @ A @ ( comp @ B @ A @ C @ H2 @ G ) @ A3 )
              = ( H2 @ ( groups7311177749621191930dd_sum @ C @ B @ G @ A3 ) ) ) ) ) ) ).

% sum_comp_morphism
thf(fact_7722_fun__comp__eq__conv,axiom,
    ! [B: $tType,C: $tType,A: $tType,F: C > B,G: A > C,Fg: A > B] :
      ( ( ( comp @ C @ B @ A @ F @ G )
        = Fg )
      = ( ! [X: A] :
            ( ( F @ ( G @ X ) )
            = ( Fg @ X ) ) ) ) ).

% fun_comp_eq_conv
thf(fact_7723_comp__cong__right,axiom,
    ! [C: $tType,B: $tType,A: $tType,X2: A > B,Y2: A > B,F: B > C] :
      ( ( X2 = Y2 )
     => ( ( comp @ B @ C @ A @ F @ X2 )
        = ( comp @ B @ C @ A @ F @ Y2 ) ) ) ).

% comp_cong_right
thf(fact_7724_comp__cong__left,axiom,
    ! [B: $tType,A: $tType,C: $tType,X2: A > B,Y2: A > B,F: C > A] :
      ( ( X2 = Y2 )
     => ( ( comp @ A @ B @ C @ X2 @ F )
        = ( comp @ A @ B @ C @ Y2 @ F ) ) ) ).

% comp_cong_left
thf(fact_7725_prod_OatLeastAtMost__shift__bounds,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_mult @ A )
     => ! [G: nat > A,M: nat,K: nat,N2: nat] :
          ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ M @ K ) @ ( plus_plus @ nat @ N2 @ K ) ) )
          = ( groups7121269368397514597t_prod @ nat @ A @ ( comp @ nat @ A @ nat @ G @ ( plus_plus @ nat @ K ) ) @ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) ) ) ) ).

% prod.atLeastAtMost_shift_bounds
thf(fact_7726_prod_OatLeastLessThan__shift__bounds,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_mult @ A )
     => ! [G: nat > A,M: nat,K: nat,N2: nat] :
          ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( plus_plus @ nat @ M @ K ) @ ( plus_plus @ nat @ N2 @ K ) ) )
          = ( groups7121269368397514597t_prod @ nat @ A @ ( comp @ nat @ A @ nat @ G @ ( plus_plus @ nat @ K ) ) @ ( set_or7035219750837199246ssThan @ nat @ M @ N2 ) ) ) ) ).

% prod.atLeastLessThan_shift_bounds
thf(fact_7727_prod_OatLeast__Suc__atMost__Suc__shift,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_mult @ A )
     => ! [G: nat > A,M: nat,N2: nat] :
          ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M ) @ ( suc @ N2 ) ) )
          = ( groups7121269368397514597t_prod @ nat @ A @ ( comp @ nat @ A @ nat @ G @ suc ) @ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) ) ) ) ).

% prod.atLeast_Suc_atMost_Suc_shift
thf(fact_7728_prod_OatLeast__Suc__lessThan__Suc__shift,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_mult @ A )
     => ! [G: nat > A,M: nat,N2: nat] :
          ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M ) @ ( suc @ N2 ) ) )
          = ( groups7121269368397514597t_prod @ nat @ A @ ( comp @ nat @ A @ nat @ G @ suc ) @ ( set_or7035219750837199246ssThan @ nat @ M @ N2 ) ) ) ) ).

% prod.atLeast_Suc_lessThan_Suc_shift
thf(fact_7729_sum_OatLeastAtMost__shift__bounds,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add @ A )
     => ! [G: nat > A,M: nat,K: nat,N2: nat] :
          ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ M @ K ) @ ( plus_plus @ nat @ N2 @ K ) ) )
          = ( groups7311177749621191930dd_sum @ nat @ A @ ( comp @ nat @ A @ nat @ G @ ( plus_plus @ nat @ K ) ) @ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) ) ) ) ).

% sum.atLeastAtMost_shift_bounds
thf(fact_7730_sum_OatLeastLessThan__shift__bounds,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add @ A )
     => ! [G: nat > A,M: nat,K: nat,N2: nat] :
          ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( plus_plus @ nat @ M @ K ) @ ( plus_plus @ nat @ N2 @ K ) ) )
          = ( groups7311177749621191930dd_sum @ nat @ A @ ( comp @ nat @ A @ nat @ G @ ( plus_plus @ nat @ K ) ) @ ( set_or7035219750837199246ssThan @ nat @ M @ N2 ) ) ) ) ).

% sum.atLeastLessThan_shift_bounds
thf(fact_7731_bit__drop__bit__eq,axiom,
    ! [A: $tType] :
      ( ( bit_se359711467146920520ations @ A )
     => ! [N2: nat,A2: A] :
          ( ( bit_se5641148757651400278ts_bit @ A @ ( bit_se4197421643247451524op_bit @ A @ N2 @ A2 ) )
          = ( comp @ nat @ $o @ nat @ ( bit_se5641148757651400278ts_bit @ A @ A2 ) @ ( plus_plus @ nat @ N2 ) ) ) ) ).

% bit_drop_bit_eq
thf(fact_7732_sum_OatLeast__Suc__atMost__Suc__shift,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add @ A )
     => ! [G: nat > A,M: nat,N2: nat] :
          ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M ) @ ( suc @ N2 ) ) )
          = ( groups7311177749621191930dd_sum @ nat @ A @ ( comp @ nat @ A @ nat @ G @ suc ) @ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) ) ) ) ).

% sum.atLeast_Suc_atMost_Suc_shift
thf(fact_7733_sum_OatLeast__Suc__lessThan__Suc__shift,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add @ A )
     => ! [G: nat > A,M: nat,N2: nat] :
          ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M ) @ ( suc @ N2 ) ) )
          = ( groups7311177749621191930dd_sum @ nat @ A @ ( comp @ nat @ A @ nat @ G @ suc ) @ ( set_or7035219750837199246ssThan @ nat @ M @ N2 ) ) ) ) ).

% sum.atLeast_Suc_lessThan_Suc_shift
thf(fact_7734_bij__betw__comp__iff2,axiom,
    ! [C: $tType,A: $tType,B: $tType,F4: A > B,A10: set @ A,A13: set @ B,F: C > A,A3: set @ C] :
      ( ( bij_betw @ A @ B @ F4 @ A10 @ A13 )
     => ( ( ord_less_eq @ ( set @ A ) @ ( image @ C @ A @ F @ A3 ) @ A10 )
       => ( ( bij_betw @ C @ A @ F @ A3 @ A10 )
          = ( bij_betw @ C @ B @ ( comp @ A @ B @ C @ F4 @ F ) @ A3 @ A13 ) ) ) ) ).

% bij_betw_comp_iff2
thf(fact_7735_summable__inverse__divide,axiom,
    ! [A: $tType] :
      ( ( real_V3459762299906320749_field @ A )
     => ! [F: nat > A,C2: A] :
          ( ( summable @ A @ ( comp @ A @ A @ nat @ ( inverse_inverse @ A ) @ F ) )
         => ( summable @ A
            @ ^ [N: nat] : ( divide_divide @ A @ C2 @ ( F @ N ) ) ) ) ) ).

% summable_inverse_divide
thf(fact_7736_prod_Oreindex__nontrivial,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( ( comm_monoid_mult @ A )
     => ! [A3: set @ B,H2: B > C,G: C > A] :
          ( ( finite_finite2 @ B @ A3 )
         => ( ! [X3: B,Y4: B] :
                ( ( member @ B @ X3 @ A3 )
               => ( ( member @ B @ Y4 @ A3 )
                 => ( ( X3 != Y4 )
                   => ( ( ( H2 @ X3 )
                        = ( H2 @ Y4 ) )
                     => ( ( G @ ( H2 @ X3 ) )
                        = ( one_one @ A ) ) ) ) ) )
           => ( ( groups7121269368397514597t_prod @ C @ A @ G @ ( image @ B @ C @ H2 @ A3 ) )
              = ( groups7121269368397514597t_prod @ B @ A @ ( comp @ C @ A @ B @ G @ H2 ) @ A3 ) ) ) ) ) ).

% prod.reindex_nontrivial
thf(fact_7737_sum__image__le,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( ( ordere6911136660526730532id_add @ B )
     => ! [I5: set @ C,G: A > B,F: C > A] :
          ( ( finite_finite2 @ C @ I5 )
         => ( ! [I3: C] :
                ( ( member @ C @ I3 @ I5 )
               => ( ord_less_eq @ B @ ( zero_zero @ B ) @ ( G @ ( F @ I3 ) ) ) )
           => ( ord_less_eq @ B @ ( groups7311177749621191930dd_sum @ A @ B @ G @ ( image @ C @ A @ F @ I5 ) ) @ ( groups7311177749621191930dd_sum @ C @ B @ ( comp @ A @ B @ C @ G @ F ) @ I5 ) ) ) ) ) ).

% sum_image_le
thf(fact_7738_sum_OatLeast0__lessThan__Suc__shift,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add @ A )
     => ! [G: nat > A,N2: nat] :
          ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( suc @ N2 ) ) )
          = ( plus_plus @ A @ ( G @ ( zero_zero @ nat ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( comp @ nat @ A @ nat @ G @ suc ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ) ) ).

% sum.atLeast0_lessThan_Suc_shift
thf(fact_7739_sum_OatLeast0__atMost__Suc__shift,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add @ A )
     => ! [G: nat > A,N2: nat] :
          ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( suc @ N2 ) ) )
          = ( plus_plus @ A @ ( G @ ( zero_zero @ nat ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( comp @ nat @ A @ nat @ G @ suc ) @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ) ) ).

% sum.atLeast0_atMost_Suc_shift
thf(fact_7740_prod_OatLeast0__lessThan__Suc__shift,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_mult @ A )
     => ! [G: nat > A,N2: nat] :
          ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( suc @ N2 ) ) )
          = ( times_times @ A @ ( G @ ( zero_zero @ nat ) ) @ ( groups7121269368397514597t_prod @ nat @ A @ ( comp @ nat @ A @ nat @ G @ suc ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ) ) ).

% prod.atLeast0_lessThan_Suc_shift
thf(fact_7741_prod_OatLeast0__atMost__Suc__shift,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_mult @ A )
     => ! [G: nat > A,N2: nat] :
          ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( suc @ N2 ) ) )
          = ( times_times @ A @ ( G @ ( zero_zero @ nat ) ) @ ( groups7121269368397514597t_prod @ nat @ A @ ( comp @ nat @ A @ nat @ G @ suc ) @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ) ) ).

% prod.atLeast0_atMost_Suc_shift
thf(fact_7742_sum_OatLeastLessThan__shift__0,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add @ A )
     => ! [G: nat > A,M: nat,N2: nat] :
          ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ N2 ) )
          = ( groups7311177749621191930dd_sum @ nat @ A @ ( comp @ nat @ A @ nat @ G @ ( plus_plus @ nat @ M ) ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( minus_minus @ nat @ N2 @ M ) ) ) ) ) ).

% sum.atLeastLessThan_shift_0
thf(fact_7743_prod_OatLeastLessThan__shift__0,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_mult @ A )
     => ! [G: nat > A,M: nat,N2: nat] :
          ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ N2 ) )
          = ( groups7121269368397514597t_prod @ nat @ A @ ( comp @ nat @ A @ nat @ G @ ( plus_plus @ nat @ M ) ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( minus_minus @ nat @ N2 @ M ) ) ) ) ) ).

% prod.atLeastLessThan_shift_0
thf(fact_7744_sum_OatLeast__lessThan__pred__shift,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add @ A )
     => ! [G: nat > A,M: nat,N2: nat] :
          ( ( groups7311177749621191930dd_sum @ nat @ A
            @ ( comp @ nat @ A @ nat @ G
              @ ^ [N: nat] : ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) )
            @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M ) @ ( suc @ N2 ) ) )
          = ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ N2 ) ) ) ) ).

% sum.atLeast_lessThan_pred_shift
thf(fact_7745_sum_OatLeast__atMost__pred__shift,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add @ A )
     => ! [G: nat > A,M: nat,N2: nat] :
          ( ( groups7311177749621191930dd_sum @ nat @ A
            @ ( comp @ nat @ A @ nat @ G
              @ ^ [N: nat] : ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) )
            @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M ) @ ( suc @ N2 ) ) )
          = ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) ) ) ) ).

% sum.atLeast_atMost_pred_shift
thf(fact_7746_prod_OatLeast__lessThan__pred__shift,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_mult @ A )
     => ! [G: nat > A,M: nat,N2: nat] :
          ( ( groups7121269368397514597t_prod @ nat @ A
            @ ( comp @ nat @ A @ nat @ G
              @ ^ [N: nat] : ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) )
            @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M ) @ ( suc @ N2 ) ) )
          = ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ N2 ) ) ) ) ).

% prod.atLeast_lessThan_pred_shift
thf(fact_7747_prod_OatLeast__atMost__pred__shift,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_mult @ A )
     => ! [G: nat > A,M: nat,N2: nat] :
          ( ( groups7121269368397514597t_prod @ nat @ A
            @ ( comp @ nat @ A @ nat @ G
              @ ^ [N: nat] : ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) )
            @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M ) @ ( suc @ N2 ) ) )
          = ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) ) ) ) ).

% prod.atLeast_atMost_pred_shift
thf(fact_7748_sum_OatLeastAtMost__shift__0,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add @ A )
     => ! [M: nat,N2: nat,G: nat > A] :
          ( ( ord_less_eq @ nat @ M @ N2 )
         => ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) )
            = ( groups7311177749621191930dd_sum @ nat @ A @ ( comp @ nat @ A @ nat @ G @ ( plus_plus @ nat @ M ) ) @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( minus_minus @ nat @ N2 @ M ) ) ) ) ) ) ).

% sum.atLeastAtMost_shift_0
thf(fact_7749_prod_OatLeastAtMost__shift__0,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_mult @ A )
     => ! [M: nat,N2: nat,G: nat > A] :
          ( ( ord_less_eq @ nat @ M @ N2 )
         => ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) )
            = ( groups7121269368397514597t_prod @ nat @ A @ ( comp @ nat @ A @ nat @ G @ ( plus_plus @ nat @ M ) ) @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( minus_minus @ nat @ N2 @ M ) ) ) ) ) ) ).

% prod.atLeastAtMost_shift_0
thf(fact_7750_CHAR__pos__iff,axiom,
    ! [A: $tType] :
      ( ( semiring_1 @ A )
     => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( semiri4206861660011772517g_char @ A @ ( type2 @ A ) ) )
        = ( ? [N: nat] :
              ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
              & ( ( semiring_1_of_nat @ A @ N )
                = ( zero_zero @ A ) ) ) ) ) ) ).

% CHAR_pos_iff
thf(fact_7751_CHAR__eq__posI,axiom,
    ! [A: $tType] :
      ( ( semiring_1 @ A )
     => ! [C2: nat] :
          ( ( ord_less @ nat @ ( zero_zero @ nat ) @ C2 )
         => ( ( ( semiring_1_of_nat @ A @ C2 )
              = ( zero_zero @ A ) )
           => ( ! [X3: nat] :
                  ( ( ord_less @ nat @ ( zero_zero @ nat ) @ X3 )
                 => ( ( ord_less @ nat @ X3 @ C2 )
                   => ( ( semiring_1_of_nat @ A @ X3 )
                     != ( zero_zero @ A ) ) ) )
             => ( ( semiri4206861660011772517g_char @ A @ ( type2 @ A ) )
                = C2 ) ) ) ) ) ).

% CHAR_eq_posI
thf(fact_7752_bit__sshiftr__iff,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [W: word @ A,M: nat,N2: nat] :
          ( ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ ( bit_Sh8784991116023147202shiftr @ A @ W @ M ) @ N2 )
          = ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ W
            @ ( if @ nat
              @ ( ( ord_less_eq @ nat @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ M ) @ N2 )
                & ( ord_less @ nat @ N2 @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) )
              @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) )
              @ ( plus_plus @ nat @ M @ N2 ) ) ) ) ) ).

% bit_sshiftr_iff
thf(fact_7753_sshiftr__1,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [N2: nat] :
          ( ( bit_Sh8784991116023147202shiftr @ A @ ( one_one @ ( word @ A ) ) @ N2 )
          = ( zero_neq_one_of_bool @ ( word @ A )
            @ ( ( ( type_len0_len_of @ A @ ( type2 @ A ) )
                = ( one_one @ nat ) )
              | ( N2
                = ( zero_zero @ nat ) ) ) ) ) ) ).

% sshiftr_1
thf(fact_7754_sshiftr__numeral__Suc,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [M: num,N2: nat] :
          ( ( bit_Sh8784991116023147202shiftr @ A @ ( numeral_numeral @ ( word @ A ) @ M ) @ ( suc @ N2 ) )
          = ( signed_drop_bit @ A @ ( suc @ N2 ) @ ( numeral_numeral @ ( word @ A ) @ M ) ) ) ) ).

% sshiftr_numeral_Suc
thf(fact_7755_sshiftr__numeral__numeral,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [M: num,N2: num] :
          ( ( bit_Sh8784991116023147202shiftr @ A @ ( numeral_numeral @ ( word @ A ) @ M ) @ ( numeral_numeral @ nat @ N2 ) )
          = ( signed_drop_bit @ A @ ( numeral_numeral @ nat @ N2 ) @ ( numeral_numeral @ ( word @ A ) @ M ) ) ) ) ).

% sshiftr_numeral_numeral
thf(fact_7756_sshiftr__minus__numeral__Suc,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [M: num,N2: nat] :
          ( ( bit_Sh8784991116023147202shiftr @ A @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ M ) ) @ ( suc @ N2 ) )
          = ( signed_drop_bit @ A @ ( suc @ N2 ) @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ M ) ) ) ) ) ).

% sshiftr_minus_numeral_Suc
thf(fact_7757_sshiftr__minus__numeral__numeral,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [M: num,N2: num] :
          ( ( bit_Sh8784991116023147202shiftr @ A @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ M ) ) @ ( numeral_numeral @ nat @ N2 ) )
          = ( signed_drop_bit @ A @ ( numeral_numeral @ nat @ N2 ) @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ M ) ) ) ) ) ).

% sshiftr_minus_numeral_numeral
thf(fact_7758_card_Ocomp__fun__commute__on,axiom,
    ( ( comp @ nat @ nat @ nat @ suc @ suc )
    = ( comp @ nat @ nat @ nat @ suc @ suc ) ) ).

% card.comp_fun_commute_on
thf(fact_7759_fst__diag__snd,axiom,
    ! [B: $tType,A: $tType] :
      ( ( comp @ ( product_prod @ B @ B ) @ B @ ( product_prod @ A @ B ) @ ( product_fst @ B @ B )
        @ ( comp @ B @ ( product_prod @ B @ B ) @ ( product_prod @ A @ B )
          @ ^ [X: B] : ( product_Pair @ B @ B @ X @ X )
          @ ( product_snd @ A @ B ) ) )
      = ( product_snd @ A @ B ) ) ).

% fst_diag_snd
thf(fact_7760_snd__diag__fst,axiom,
    ! [B: $tType,A: $tType] :
      ( ( comp @ ( product_prod @ A @ A ) @ A @ ( product_prod @ A @ B ) @ ( product_snd @ A @ A )
        @ ( comp @ A @ ( product_prod @ A @ A ) @ ( product_prod @ A @ B )
          @ ^ [X: A] : ( product_Pair @ A @ A @ X @ X )
          @ ( product_fst @ A @ B ) ) )
      = ( product_fst @ A @ B ) ) ).

% snd_diag_fst
thf(fact_7761_fst__diag__fst,axiom,
    ! [B: $tType,A: $tType] :
      ( ( comp @ ( product_prod @ A @ A ) @ A @ ( product_prod @ A @ B ) @ ( product_fst @ A @ A )
        @ ( comp @ A @ ( product_prod @ A @ A ) @ ( product_prod @ A @ B )
          @ ^ [X: A] : ( product_Pair @ A @ A @ X @ X )
          @ ( product_fst @ A @ B ) ) )
      = ( product_fst @ A @ B ) ) ).

% fst_diag_fst
thf(fact_7762_snd__diag__snd,axiom,
    ! [B: $tType,A: $tType] :
      ( ( comp @ ( product_prod @ B @ B ) @ B @ ( product_prod @ A @ B ) @ ( product_snd @ B @ B )
        @ ( comp @ B @ ( product_prod @ B @ B ) @ ( product_prod @ A @ B )
          @ ^ [X: B] : ( product_Pair @ B @ B @ X @ X )
          @ ( product_snd @ A @ B ) ) )
      = ( product_snd @ A @ B ) ) ).

% snd_diag_snd
thf(fact_7763_set__oo__map__alt,axiom,
    ! [B: $tType,A: $tType,F: A > B] :
      ( ( comp @ ( list @ B ) @ ( set @ B ) @ ( list @ A ) @ ( set2 @ B ) @ ( map @ A @ B @ F ) )
      = ( ^ [L2: list @ A] : ( image @ A @ B @ F @ ( set2 @ A @ L2 ) ) ) ) ).

% set_oo_map_alt
thf(fact_7764_case__prod__comp,axiom,
    ! [D: $tType,A: $tType,C: $tType,B: $tType,F: D > C > A,G: B > D,X2: product_prod @ B @ C] :
      ( ( product_case_prod @ B @ C @ A @ ( comp @ D @ ( C > A ) @ B @ F @ G ) @ X2 )
      = ( F @ ( G @ ( product_fst @ B @ C @ X2 ) ) @ ( product_snd @ B @ C @ X2 ) ) ) ).

% case_prod_comp
thf(fact_7765_execute__heap,axiom,
    ! [A: $tType,F: ( heap_ext @ product_unit ) > ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )] :
      ( ( heap_Time_execute @ A @ ( heap_Time_heap @ A @ F ) )
      = ( comp @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) @ ( heap_ext @ product_unit ) @ ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) @ F ) ) ).

% execute_heap
thf(fact_7766_heap__def,axiom,
    ! [A: $tType] :
      ( ( heap_Time_heap @ A )
      = ( ^ [F6: ( heap_ext @ product_unit ) > ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )] : ( heap_Time_Heap2 @ A @ ( comp @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) @ ( heap_ext @ product_unit ) @ ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) @ F6 ) ) ) ) ).

% heap_def
thf(fact_7767_sum_Oeq__fold,axiom,
    ! [A: $tType,B: $tType] :
      ( ( comm_monoid_add @ A )
     => ( ( groups7311177749621191930dd_sum @ B @ A )
        = ( ^ [G4: B > A] : ( finite_fold @ B @ A @ ( comp @ A @ ( A > A ) @ B @ ( plus_plus @ A ) @ G4 ) @ ( zero_zero @ A ) ) ) ) ) ).

% sum.eq_fold
thf(fact_7768_prod_Oeq__fold,axiom,
    ! [A: $tType,B: $tType] :
      ( ( comm_monoid_mult @ A )
     => ( ( groups7121269368397514597t_prod @ B @ A )
        = ( ^ [G4: B > A] : ( finite_fold @ B @ A @ ( comp @ A @ ( A > A ) @ B @ ( times_times @ A ) @ G4 ) @ ( one_one @ A ) ) ) ) ) ).

% prod.eq_fold
thf(fact_7769_finite__range__prod,axiom,
    ! [A: $tType,C: $tType,B: $tType,F: B > ( product_prod @ A @ C )] :
      ( ( finite_finite2 @ A @ ( image @ B @ A @ ( comp @ ( product_prod @ A @ C ) @ A @ B @ ( product_fst @ A @ C ) @ F ) @ ( top_top @ ( set @ B ) ) ) )
     => ( ( finite_finite2 @ C @ ( image @ B @ C @ ( comp @ ( product_prod @ A @ C ) @ C @ B @ ( product_snd @ A @ C ) @ F ) @ ( top_top @ ( set @ B ) ) ) )
       => ( finite_finite2 @ ( product_prod @ A @ C ) @ ( image @ B @ ( product_prod @ A @ C ) @ F @ ( top_top @ ( set @ B ) ) ) ) ) ) ).

% finite_range_prod
thf(fact_7770_map__filter__def,axiom,
    ! [B: $tType,A: $tType] :
      ( ( map_filter @ A @ B )
      = ( ^ [F6: A > ( option @ B ),Xs: list @ A] :
            ( map @ A @ B @ ( comp @ ( option @ B ) @ B @ A @ ( the2 @ B ) @ F6 )
            @ ( filter2 @ A
              @ ^ [X: A] :
                  ( ( F6 @ X )
                 != ( none @ B ) )
              @ Xs ) ) ) ) ).

% map_filter_def
thf(fact_7771_execute__return,axiom,
    ! [A: $tType,X2: A] :
      ( ( heap_Time_execute @ A @ ( heap_Time_return @ A @ X2 ) )
      = ( comp @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) @ ( heap_ext @ product_unit ) @ ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) )
        @ ^ [H: heap_ext @ product_unit] : ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ X2 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H @ ( one_one @ nat ) ) ) ) ) ).

% execute_return
thf(fact_7772_execute__ureturn,axiom,
    ! [A: $tType,X2: A] :
      ( ( heap_Time_execute @ A @ ( heap_Time_ureturn @ A @ X2 ) )
      = ( comp @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) @ ( heap_ext @ product_unit ) @ ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) )
        @ ^ [H: heap_ext @ product_unit] : ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ X2 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H @ ( zero_zero @ nat ) ) ) ) ) ).

% execute_ureturn
thf(fact_7773_Code__Numeral_Onegative__def,axiom,
    ( code_negative
    = ( comp @ code_integer @ code_integer @ num @ ( uminus_uminus @ code_integer ) @ ( numeral_numeral @ code_integer ) ) ) ).

% Code_Numeral.negative_def
thf(fact_7774_fst__snd__flip,axiom,
    ! [B: $tType,A: $tType] :
      ( ( product_fst @ A @ B )
      = ( comp @ ( product_prod @ B @ A ) @ A @ ( product_prod @ A @ B ) @ ( product_snd @ B @ A )
        @ ( product_case_prod @ A @ B @ ( product_prod @ B @ A )
          @ ^ [X: A,Y: B] : ( product_Pair @ B @ A @ Y @ X ) ) ) ) ).

% fst_snd_flip
thf(fact_7775_ge__eq__refl,axiom,
    ! [A: $tType,R2: A > A > $o,X2: A] :
      ( ( ord_less_eq @ ( A > A > $o )
        @ ^ [Y5: A,Z4: A] : Y5 = Z4
        @ R2 )
     => ( R2 @ X2 @ X2 ) ) ).

% ge_eq_refl
thf(fact_7776_refl__ge__eq,axiom,
    ! [A: $tType,R2: A > A > $o] :
      ( ! [X3: A] : ( R2 @ X3 @ X3 )
     => ( ord_less_eq @ ( A > A > $o )
        @ ^ [Y5: A,Z4: A] : Y5 = Z4
        @ R2 ) ) ).

% refl_ge_eq
thf(fact_7777_image__fold__insert,axiom,
    ! [B: $tType,A: $tType,A3: set @ A,F: A > B] :
      ( ( finite_finite2 @ A @ A3 )
     => ( ( image @ A @ B @ F @ A3 )
        = ( finite_fold @ A @ ( set @ B )
          @ ^ [K4: A] : ( insert @ B @ ( F @ K4 ) )
          @ ( bot_bot @ ( set @ B ) )
          @ A3 ) ) ) ).

% image_fold_insert
thf(fact_7778_Max_Oeq__fold_H,axiom,
    ! [A: $tType] :
      ( ( linorder @ A )
     => ( ( lattic643756798349783984er_Max @ A )
        = ( ^ [A7: set @ A] :
              ( the2 @ A
              @ ( finite_fold @ A @ ( option @ A )
                @ ^ [X: A,Y: option @ A] : ( some @ A @ ( case_option @ A @ A @ X @ ( ord_max @ A @ X ) @ Y ) )
                @ ( none @ A )
                @ A7 ) ) ) ) ) ).

% Max.eq_fold'
thf(fact_7779_fstI,axiom,
    ! [B: $tType,A: $tType,X2: product_prod @ A @ B,Y2: A,Z: B] :
      ( ( X2
        = ( product_Pair @ A @ B @ Y2 @ Z ) )
     => ( ( product_fst @ A @ B @ X2 )
        = Y2 ) ) ).

% fstI
thf(fact_7780_sndI,axiom,
    ! [A: $tType,B: $tType,X2: product_prod @ A @ B,Y2: A,Z: B] :
      ( ( X2
        = ( product_Pair @ A @ B @ Y2 @ Z ) )
     => ( ( product_snd @ A @ B @ X2 )
        = Z ) ) ).

% sndI
thf(fact_7781_snd__fst__flip,axiom,
    ! [A: $tType,B: $tType] :
      ( ( product_snd @ B @ A )
      = ( comp @ ( product_prod @ A @ B ) @ A @ ( product_prod @ B @ A ) @ ( product_fst @ A @ B )
        @ ( product_case_prod @ B @ A @ ( product_prod @ A @ B )
          @ ^ [X: B,Y: A] : ( product_Pair @ A @ B @ Y @ X ) ) ) ) ).

% snd_fst_flip
thf(fact_7782_Code__Target__Int_Onegative__def,axiom,
    ( code_Target_negative
    = ( comp @ int @ int @ num @ ( uminus_uminus @ int ) @ ( numeral_numeral @ int ) ) ) ).

% Code_Target_Int.negative_def
thf(fact_7783_shiftl__Suc__0,axiom,
    ! [N2: nat] :
      ( ( bit_Sh4282982442137083160shiftl @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N2 )
      = ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ).

% shiftl_Suc_0
thf(fact_7784_shiftl__numeral__Suc,axiom,
    ! [A: $tType] :
      ( ( bit_se359711467146920520ations @ A )
     => ! [M: num,N2: nat] :
          ( ( bit_Sh4282982442137083160shiftl @ A @ ( numeral_numeral @ A @ M ) @ ( suc @ N2 ) )
          = ( bit_se4730199178511100633sh_bit @ A @ ( suc @ N2 ) @ ( numeral_numeral @ A @ M ) ) ) ) ).

% shiftl_numeral_Suc
thf(fact_7785_shiftl__numeral__numeral,axiom,
    ! [A: $tType] :
      ( ( bit_se359711467146920520ations @ A )
     => ! [M: num,N2: num] :
          ( ( bit_Sh4282982442137083160shiftl @ A @ ( numeral_numeral @ A @ M ) @ ( numeral_numeral @ nat @ N2 ) )
          = ( bit_se4730199178511100633sh_bit @ A @ ( numeral_numeral @ nat @ N2 ) @ ( numeral_numeral @ A @ M ) ) ) ) ).

% shiftl_numeral_numeral
thf(fact_7786_shiftl__minus__numeral__Suc,axiom,
    ! [A: $tType] :
      ( ( bit_ri3973907225187159222ations @ A )
     => ! [M: num,N2: nat] :
          ( ( bit_Sh4282982442137083160shiftl @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( suc @ N2 ) )
          = ( bit_se4730199178511100633sh_bit @ A @ ( suc @ N2 ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) ) ) ) ).

% shiftl_minus_numeral_Suc
thf(fact_7787_shiftl__minus__numeral__numeral,axiom,
    ! [A: $tType] :
      ( ( bit_ri3973907225187159222ations @ A )
     => ! [M: num,N2: num] :
          ( ( bit_Sh4282982442137083160shiftl @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( numeral_numeral @ nat @ N2 ) )
          = ( bit_se4730199178511100633sh_bit @ A @ ( numeral_numeral @ nat @ N2 ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) ) ) ) ).

% shiftl_minus_numeral_numeral
thf(fact_7788_shiftl__of__Suc,axiom,
    ! [A: $tType] :
      ( ( bit_se359711467146920520ations @ A )
     => ! [A2: A,N2: nat] :
          ( ( bit_Sh4282982442137083160shiftl @ A @ A2 @ ( suc @ N2 ) )
          = ( bit_Sh4282982442137083160shiftl @ A @ ( times_times @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ N2 ) ) ) ).

% shiftl_of_Suc
thf(fact_7789_shiftl__1,axiom,
    ! [A: $tType] :
      ( ( bit_se359711467146920520ations @ A )
     => ! [N2: nat] :
          ( ( bit_Sh4282982442137083160shiftl @ A @ ( one_one @ A ) @ N2 )
          = ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) ) ) ).

% shiftl_1
thf(fact_7790_shiftl__minus__1__numeral,axiom,
    ! [A: $tType] :
      ( ( bit_ri3973907225187159222ations @ A )
     => ! [N2: num] :
          ( ( bit_Sh4282982442137083160shiftl @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( numeral_numeral @ nat @ N2 ) )
          = ( bit_ri4277139882892585799ns_not @ A @ ( bit_se2239418461657761734s_mask @ A @ ( numeral_numeral @ nat @ N2 ) ) ) ) ) ).

% shiftl_minus_1_numeral
thf(fact_7791_shiftl__eq__mult,axiom,
    ! [A: $tType] :
      ( ( bit_se359711467146920520ations @ A )
     => ( ( bit_Sh4282982442137083160shiftl @ A )
        = ( ^ [X: A,N: nat] : ( times_times @ A @ X @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) ) ) ) ) ).

% shiftl_eq_mult
thf(fact_7792_bit__shiftl__iff,axiom,
    ! [A: $tType] :
      ( ( bit_se359711467146920520ations @ A )
     => ! [A2: A,M: nat,N2: nat] :
          ( ( bit_se5641148757651400278ts_bit @ A @ ( bit_Sh4282982442137083160shiftl @ A @ A2 @ M ) @ N2 )
          = ( ( ord_less_eq @ nat @ M @ N2 )
            & ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N2 )
            & ( bit_se5641148757651400278ts_bit @ A @ A2 @ ( minus_minus @ nat @ N2 @ M ) ) ) ) ) ).

% bit_shiftl_iff
thf(fact_7793_empty__natural,axiom,
    ! [C: $tType,B: $tType,D: $tType,A: $tType,F: A > C,G: D > B] :
      ( ( comp @ C @ ( set @ B ) @ A
        @ ^ [Uu3: C] : ( bot_bot @ ( set @ B ) )
        @ F )
      = ( comp @ ( set @ D ) @ ( set @ B ) @ A @ ( image @ D @ B @ G )
        @ ^ [Uu3: A] : ( bot_bot @ ( set @ D ) ) ) ) ).

% empty_natural
thf(fact_7794_shiftr__minus__numeral__numeral,axiom,
    ! [A: $tType] :
      ( ( bit_ri3973907225187159222ations @ A )
     => ! [M: num,N2: num] :
          ( ( bit_Sh4282982442137083166shiftr @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( numeral_numeral @ nat @ N2 ) )
          = ( bit_se4197421643247451524op_bit @ A @ ( numeral_numeral @ nat @ N2 ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) ) ) ) ).

% shiftr_minus_numeral_numeral
thf(fact_7795_shiftr__Suc__0,axiom,
    ! [N2: nat] :
      ( ( bit_Sh4282982442137083166shiftr @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N2 )
      = ( zero_neq_one_of_bool @ nat
        @ ( N2
          = ( zero_zero @ nat ) ) ) ) ).

% shiftr_Suc_0
thf(fact_7796_shiftr__numeral__Suc,axiom,
    ! [A: $tType] :
      ( ( bit_se359711467146920520ations @ A )
     => ! [M: num,N2: nat] :
          ( ( bit_Sh4282982442137083166shiftr @ A @ ( numeral_numeral @ A @ M ) @ ( suc @ N2 ) )
          = ( bit_se4197421643247451524op_bit @ A @ ( suc @ N2 ) @ ( numeral_numeral @ A @ M ) ) ) ) ).

% shiftr_numeral_Suc
thf(fact_7797_shiftr__numeral__numeral,axiom,
    ! [A: $tType] :
      ( ( bit_se359711467146920520ations @ A )
     => ! [M: num,N2: num] :
          ( ( bit_Sh4282982442137083166shiftr @ A @ ( numeral_numeral @ A @ M ) @ ( numeral_numeral @ nat @ N2 ) )
          = ( bit_se4197421643247451524op_bit @ A @ ( numeral_numeral @ nat @ N2 ) @ ( numeral_numeral @ A @ M ) ) ) ) ).

% shiftr_numeral_numeral
thf(fact_7798_shiftr__1,axiom,
    ! [A: $tType] :
      ( ( bit_se359711467146920520ations @ A )
     => ! [N2: nat] :
          ( ( bit_Sh4282982442137083166shiftr @ A @ ( one_one @ A ) @ N2 )
          = ( zero_neq_one_of_bool @ A
            @ ( N2
              = ( zero_zero @ nat ) ) ) ) ) ).

% shiftr_1
thf(fact_7799_shiftr__minus__numeral__Suc,axiom,
    ! [A: $tType] :
      ( ( bit_ri3973907225187159222ations @ A )
     => ! [M: num,N2: nat] :
          ( ( bit_Sh4282982442137083166shiftr @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( suc @ N2 ) )
          = ( bit_se4197421643247451524op_bit @ A @ ( suc @ N2 ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) ) ) ) ).

% shiftr_minus_numeral_Suc
thf(fact_7800_conj__subset__def,axiom,
    ! [A: $tType,A3: set @ A,P: A > $o,Q: A > $o] :
      ( ( ord_less_eq @ ( set @ A ) @ A3
        @ ( collect @ A
          @ ^ [X: A] :
              ( ( P @ X )
              & ( Q @ X ) ) ) )
      = ( ( ord_less_eq @ ( set @ A ) @ A3 @ ( collect @ A @ P ) )
        & ( ord_less_eq @ ( set @ A ) @ A3 @ ( collect @ A @ Q ) ) ) ) ).

% conj_subset_def
thf(fact_7801_one__bit__shiftl,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [N2: nat] :
          ( ( generi7602027413899671122et_bit @ ( word @ A ) @ ( zero_zero @ ( word @ A ) ) @ N2 @ $true )
          = ( bit_Sh4282982442137083160shiftl @ ( word @ A ) @ ( one_one @ ( word @ A ) ) @ N2 ) ) ) ).

% one_bit_shiftl
thf(fact_7802_conj__comp__iff,axiom,
    ! [B: $tType,A: $tType,P: B > $o,Q: B > $o,G: A > B] :
      ( ( comp @ B @ $o @ A
        @ ^ [X: B] :
            ( ( P @ X )
            & ( Q @ X ) )
        @ G )
      = ( ^ [X: A] :
            ( ( comp @ B @ $o @ A @ P @ G @ X )
            & ( comp @ B @ $o @ A @ Q @ G @ X ) ) ) ) ).

% conj_comp_iff
thf(fact_7803_bit__shiftr__eq,axiom,
    ! [A: $tType] :
      ( ( bit_se359711467146920520ations @ A )
     => ! [A2: A,N2: nat] :
          ( ( bit_se5641148757651400278ts_bit @ A @ ( bit_Sh4282982442137083166shiftr @ A @ A2 @ N2 ) )
          = ( comp @ nat @ $o @ nat @ ( bit_se5641148757651400278ts_bit @ A @ A2 ) @ ( plus_plus @ nat @ N2 ) ) ) ) ).

% bit_shiftr_eq
thf(fact_7804_shiftr__eq__div,axiom,
    ! [A: $tType] :
      ( ( bit_se359711467146920520ations @ A )
     => ( ( bit_Sh4282982442137083166shiftr @ A )
        = ( ^ [X: A,N: nat] : ( divide_divide @ A @ X @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) ) ) ) ) ).

% shiftr_eq_div
thf(fact_7805_bit__revcast__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( ( ( type_len @ A )
        & ( type_len @ B ) )
     => ! [W: word @ A,N2: nat] :
          ( ( bit_se5641148757651400278ts_bit @ ( word @ B ) @ ( revcast @ A @ B @ W ) @ N2 )
          = ( ( ord_less_eq @ nat @ ( minus_minus @ nat @ ( type_len0_len_of @ B @ ( type2 @ B ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) @ N2 )
            & ( ord_less @ nat @ N2 @ ( type_len0_len_of @ B @ ( type2 @ B ) ) )
            & ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ W @ ( minus_minus @ nat @ ( plus_plus @ nat @ N2 @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( type_len0_len_of @ B @ ( type2 @ B ) ) ) ) @ ( minus_minus @ nat @ ( type_len0_len_of @ B @ ( type2 @ B ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ) ) ).

% bit_revcast_iff
thf(fact_7806_Set__filter__fold,axiom,
    ! [A: $tType,A3: set @ A,P: A > $o] :
      ( ( finite_finite2 @ A @ A3 )
     => ( ( filter3 @ A @ P @ A3 )
        = ( finite_fold @ A @ ( set @ A )
          @ ^ [X: A,A14: set @ A] : ( if @ ( set @ A ) @ ( P @ X ) @ ( insert @ A @ X @ A14 ) @ A14 )
          @ ( bot_bot @ ( set @ A ) )
          @ A3 ) ) ) ).

% Set_filter_fold
thf(fact_7807_Set_Ofilter__def,axiom,
    ! [A: $tType] :
      ( ( filter3 @ A )
      = ( ^ [P3: A > $o,A7: set @ A] :
            ( collect @ A
            @ ^ [A5: A] :
                ( ( member @ A @ A5 @ A7 )
                & ( P3 @ A5 ) ) ) ) ) ).

% Set.filter_def
thf(fact_7808_revcast__slice1,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ( type_len @ B )
        & ( type_len @ A ) )
     => ! [W: word @ B] :
          ( ( slice1 @ B @ A @ ( size_size @ ( word @ A ) @ ( revcast @ B @ A @ W ) ) @ W )
          = ( revcast @ B @ A @ W ) ) ) ).

% revcast_slice1
thf(fact_7809_filter__set,axiom,
    ! [A: $tType,P: A > $o,Xs2: list @ A] :
      ( ( filter3 @ A @ P @ ( set2 @ A @ Xs2 ) )
      = ( set2 @ A @ ( filter2 @ A @ P @ Xs2 ) ) ) ).

% filter_set
thf(fact_7810_word__msb__neg__numeral,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [W: num] :
          ( ( most_s684356279273892711sb_msb @ ( word @ A ) @ ( uminus_uminus @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ W ) ) )
          = ( bit_se5641148757651400278ts_bit @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ W ) ) @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) ) ) ) ).

% word_msb_neg_numeral
thf(fact_7811_inj__on__word__of__int,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ( inj_on @ int @ ( word @ A ) @ ( ring_1_of_int @ ( word @ A ) ) @ ( set_or7035219750837199246ssThan @ int @ ( zero_zero @ int ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ).

% inj_on_word_of_int
thf(fact_7812_inj__mult__left,axiom,
    ! [A: $tType] :
      ( ( idom @ A )
     => ! [A2: A] :
          ( ( inj_on @ A @ A @ ( times_times @ A @ A2 ) @ ( top_top @ ( set @ A ) ) )
          = ( A2
           != ( zero_zero @ A ) ) ) ) ).

% inj_mult_left
thf(fact_7813_inj__divide__right,axiom,
    ! [A: $tType] :
      ( ( field @ A )
     => ! [A2: A] :
          ( ( inj_on @ A @ A
            @ ^ [B5: A] : ( divide_divide @ A @ B5 @ A2 )
            @ ( top_top @ ( set @ A ) ) )
          = ( A2
           != ( zero_zero @ A ) ) ) ) ).

% inj_divide_right
thf(fact_7814_inj__on__insert,axiom,
    ! [B: $tType,A: $tType,F: A > B,A2: A,A3: set @ A] :
      ( ( inj_on @ A @ B @ F @ ( insert @ A @ A2 @ A3 ) )
      = ( ( inj_on @ A @ B @ F @ A3 )
        & ~ ( member @ B @ ( F @ A2 ) @ ( image @ A @ B @ F @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).

% inj_on_insert
thf(fact_7815_word__msb__1,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ( ( most_s684356279273892711sb_msb @ ( word @ A ) @ ( one_one @ ( word @ A ) ) )
        = ( ( type_len0_len_of @ A @ ( type2 @ A ) )
          = ( one_one @ nat ) ) ) ) ).

% word_msb_1
thf(fact_7816_word__msb__numeral,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [W: num] :
          ( ( most_s684356279273892711sb_msb @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ W ) )
          = ( bit_se5641148757651400278ts_bit @ int @ ( numeral_numeral @ int @ W ) @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) ) ) ) ).

% word_msb_numeral
thf(fact_7817_inj__on__filter__key__eq,axiom,
    ! [B: $tType,A: $tType,F: A > B,Y2: A,Xs2: list @ A] :
      ( ( inj_on @ A @ B @ F @ ( insert @ A @ Y2 @ ( set2 @ A @ Xs2 ) ) )
     => ( ( filter2 @ A
          @ ^ [X: A] :
              ( ( F @ Y2 )
              = ( F @ X ) )
          @ Xs2 )
        = ( filter2 @ A
          @ ( ^ [Y5: A,Z4: A] : Y5 = Z4
            @ Y2 )
          @ Xs2 ) ) ) ).

% inj_on_filter_key_eq
thf(fact_7818_inj__on__add,axiom,
    ! [A: $tType] :
      ( ( cancel_semigroup_add @ A )
     => ! [A2: A,A3: set @ A] : ( inj_on @ A @ A @ ( plus_plus @ A @ A2 ) @ A3 ) ) ).

% inj_on_add
thf(fact_7819_inj__on__add_H,axiom,
    ! [A: $tType] :
      ( ( cancel_semigroup_add @ A )
     => ! [A2: A,A3: set @ A] :
          ( inj_on @ A @ A
          @ ^ [B5: A] : ( plus_plus @ A @ B5 @ A2 )
          @ A3 ) ) ).

% inj_on_add'
thf(fact_7820_inj__on__diff,axiom,
    ! [B: $tType,A: $tType,F: A > B,A3: set @ A,B6: set @ A] :
      ( ( inj_on @ A @ B @ F @ A3 )
     => ( inj_on @ A @ B @ F @ ( minus_minus @ ( set @ A ) @ A3 @ B6 ) ) ) ).

% inj_on_diff
thf(fact_7821_word__msb__n1,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ( most_s684356279273892711sb_msb @ ( word @ A ) @ ( uminus_uminus @ ( word @ A ) @ ( one_one @ ( word @ A ) ) ) ) ) ).

% word_msb_n1
thf(fact_7822_inj__on__id2,axiom,
    ! [A: $tType,A3: set @ A] :
      ( inj_on @ A @ A
      @ ^ [X: A] : X
      @ A3 ) ).

% inj_on_id2
thf(fact_7823_inj__on__mult,axiom,
    ! [A: $tType] :
      ( ( semidom_divide @ A )
     => ! [A2: A,A3: set @ A] :
          ( ( A2
           != ( zero_zero @ A ) )
         => ( inj_on @ A @ A @ ( times_times @ A @ A2 ) @ A3 ) ) ) ).

% inj_on_mult
thf(fact_7824_linorder__inj__onI,axiom,
    ! [B: $tType,A: $tType] :
      ( ( order @ A )
     => ! [A3: set @ A,F: A > B] :
          ( ! [X3: A,Y4: A] :
              ( ( ord_less @ A @ X3 @ Y4 )
             => ( ( member @ A @ X3 @ A3 )
               => ( ( member @ A @ Y4 @ A3 )
                 => ( ( F @ X3 )
                   != ( F @ Y4 ) ) ) ) )
         => ( ! [X3: A,Y4: A] :
                ( ( member @ A @ X3 @ A3 )
               => ( ( member @ A @ Y4 @ A3 )
                 => ( ( ord_less_eq @ A @ X3 @ Y4 )
                    | ( ord_less_eq @ A @ Y4 @ X3 ) ) ) )
           => ( inj_on @ A @ B @ F @ A3 ) ) ) ) ).

% linorder_inj_onI
thf(fact_7825_inj__on__subset,axiom,
    ! [B: $tType,A: $tType,F: A > B,A3: set @ A,B6: set @ A] :
      ( ( inj_on @ A @ B @ F @ A3 )
     => ( ( ord_less_eq @ ( set @ A ) @ B6 @ A3 )
       => ( inj_on @ A @ B @ F @ B6 ) ) ) ).

% inj_on_subset
thf(fact_7826_subset__inj__on,axiom,
    ! [B: $tType,A: $tType,F: A > B,B6: set @ A,A3: set @ A] :
      ( ( inj_on @ A @ B @ F @ B6 )
     => ( ( ord_less_eq @ ( set @ A ) @ A3 @ B6 )
       => ( inj_on @ A @ B @ F @ A3 ) ) ) ).

% subset_inj_on
thf(fact_7827_inj__add__left,axiom,
    ! [A: $tType] :
      ( ( cancel_semigroup_add @ A )
     => ! [A2: A] : ( inj_on @ A @ A @ ( plus_plus @ A @ A2 ) @ ( top_top @ ( set @ A ) ) ) ) ).

% inj_add_left
thf(fact_7828_inj__diff__right,axiom,
    ! [A: $tType] :
      ( ( ab_group_add @ A )
     => ! [A2: A] :
          ( inj_on @ A @ A
          @ ^ [B5: A] : ( minus_minus @ A @ B5 @ A2 )
          @ ( top_top @ ( set @ A ) ) ) ) ).

% inj_diff_right
thf(fact_7829_sorted__list__of__set_Oinj__on,axiom,
    ! [A: $tType] :
      ( ( linorder @ A )
     => ( inj_on @ A @ A
        @ ^ [X: A] : X
        @ ( top_top @ ( set @ A ) ) ) ) ).

% sorted_list_of_set.inj_on
thf(fact_7830_linorder__injI,axiom,
    ! [B: $tType,A: $tType] :
      ( ( linorder @ A )
     => ! [F: A > B] :
          ( ! [X3: A,Y4: A] :
              ( ( ord_less @ A @ X3 @ Y4 )
             => ( ( F @ X3 )
               != ( F @ Y4 ) ) )
         => ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) ) ) ) ).

% linorder_injI
thf(fact_7831_inj__fun,axiom,
    ! [B: $tType,C: $tType,A: $tType,F: A > B] :
      ( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
     => ( inj_on @ A @ ( C > B )
        @ ^ [X: A,Y: C] : ( F @ X )
        @ ( top_top @ ( set @ A ) ) ) ) ).

% inj_fun
thf(fact_7832_word__sle__msb__le,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ( ( word_sle @ A )
        = ( ^ [X: word @ A,Y: word @ A] :
              ( ( ( most_s684356279273892711sb_msb @ ( word @ A ) @ Y )
               => ( most_s684356279273892711sb_msb @ ( word @ A ) @ X ) )
              & ( ( ( most_s684356279273892711sb_msb @ ( word @ A ) @ X )
                  & ~ ( most_s684356279273892711sb_msb @ ( word @ A ) @ Y ) )
                | ( ord_less_eq @ ( word @ A ) @ X @ Y ) ) ) ) ) ) ).

% word_sle_msb_le
thf(fact_7833_injective__scaleR,axiom,
    ! [A: $tType] :
      ( ( real_V4867850818363320053vector @ A )
     => ! [C2: real] :
          ( ( C2
           != ( zero_zero @ real ) )
         => ( inj_on @ A @ A @ ( real_V8093663219630862766scaleR @ A @ C2 ) @ ( top_top @ ( set @ A ) ) ) ) ) ).

% injective_scaleR
thf(fact_7834_inj__on__image__eq__iff,axiom,
    ! [B: $tType,A: $tType,F: A > B,C5: set @ A,A3: set @ A,B6: set @ A] :
      ( ( inj_on @ A @ B @ F @ C5 )
     => ( ( ord_less_eq @ ( set @ A ) @ A3 @ C5 )
       => ( ( ord_less_eq @ ( set @ A ) @ B6 @ C5 )
         => ( ( ( image @ A @ B @ F @ A3 )
              = ( image @ A @ B @ F @ B6 ) )
            = ( A3 = B6 ) ) ) ) ) ).

% inj_on_image_eq_iff
thf(fact_7835_inj__on__image__mem__iff,axiom,
    ! [B: $tType,A: $tType,F: A > B,B6: set @ A,A2: A,A3: set @ A] :
      ( ( inj_on @ A @ B @ F @ B6 )
     => ( ( member @ A @ A2 @ B6 )
       => ( ( ord_less_eq @ ( set @ A ) @ A3 @ B6 )
         => ( ( member @ B @ ( F @ A2 ) @ ( image @ A @ B @ F @ A3 ) )
            = ( member @ A @ A2 @ A3 ) ) ) ) ) ).

% inj_on_image_mem_iff
thf(fact_7836_sum_Oimage__eq,axiom,
    ! [A: $tType,B: $tType] :
      ( ( comm_monoid_add @ A )
     => ! [G: B > A,A3: set @ B] :
          ( ( inj_on @ B @ A @ G @ A3 )
         => ( ( groups7311177749621191930dd_sum @ A @ A
              @ ^ [X: A] : X
              @ ( image @ B @ A @ G @ A3 ) )
            = ( groups7311177749621191930dd_sum @ B @ A @ G @ A3 ) ) ) ) ).

% sum.image_eq
thf(fact_7837_prod_Oimage__eq,axiom,
    ! [A: $tType,B: $tType] :
      ( ( comm_monoid_mult @ A )
     => ! [G: B > A,A3: set @ B] :
          ( ( inj_on @ B @ A @ G @ A3 )
         => ( ( groups7121269368397514597t_prod @ A @ A
              @ ^ [X: A] : X
              @ ( image @ B @ A @ G @ A3 ) )
            = ( groups7121269368397514597t_prod @ B @ A @ G @ A3 ) ) ) ) ).

% prod.image_eq
thf(fact_7838_subset__image__inj,axiom,
    ! [A: $tType,B: $tType,S3: set @ A,F: B > A,T6: set @ B] :
      ( ( ord_less_eq @ ( set @ A ) @ S3 @ ( image @ B @ A @ F @ T6 ) )
      = ( ? [U6: set @ B] :
            ( ( ord_less_eq @ ( set @ B ) @ U6 @ T6 )
            & ( inj_on @ B @ A @ F @ U6 )
            & ( S3
              = ( image @ B @ A @ F @ U6 ) ) ) ) ) ).

% subset_image_inj
thf(fact_7839_finite__surj__inj,axiom,
    ! [A: $tType,A3: set @ A,F: A > A] :
      ( ( finite_finite2 @ A @ A3 )
     => ( ( ord_less_eq @ ( set @ A ) @ A3 @ ( image @ A @ A @ F @ A3 ) )
       => ( inj_on @ A @ A @ F @ A3 ) ) ) ).

% finite_surj_inj
thf(fact_7840_inj__on__finite,axiom,
    ! [B: $tType,A: $tType,F: A > B,A3: set @ A,B6: set @ B] :
      ( ( inj_on @ A @ B @ F @ A3 )
     => ( ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F @ A3 ) @ B6 )
       => ( ( finite_finite2 @ B @ B6 )
         => ( finite_finite2 @ A @ A3 ) ) ) ) ).

% inj_on_finite
thf(fact_7841_endo__inj__surj,axiom,
    ! [A: $tType,A3: set @ A,F: A > A] :
      ( ( finite_finite2 @ A @ A3 )
     => ( ( ord_less_eq @ ( set @ A ) @ ( image @ A @ A @ F @ A3 ) @ A3 )
       => ( ( inj_on @ A @ A @ F @ A3 )
         => ( ( image @ A @ A @ F @ A3 )
            = A3 ) ) ) ) ).

% endo_inj_surj
thf(fact_7842_inj__image__subset__iff,axiom,
    ! [B: $tType,A: $tType,F: A > B,A3: set @ A,B6: set @ A] :
      ( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
     => ( ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F @ A3 ) @ ( image @ A @ B @ F @ B6 ) )
        = ( ord_less_eq @ ( set @ A ) @ A3 @ B6 ) ) ) ).

% inj_image_subset_iff
thf(fact_7843_inj__on__iff__surj,axiom,
    ! [A: $tType,B: $tType,A3: set @ A,A10: set @ B] :
      ( ( A3
       != ( bot_bot @ ( set @ A ) ) )
     => ( ( ? [F6: A > B] :
              ( ( inj_on @ A @ B @ F6 @ A3 )
              & ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F6 @ A3 ) @ A10 ) ) )
        = ( ? [G4: B > A] :
              ( ( image @ B @ A @ G4 @ A10 )
              = A3 ) ) ) ) ).

% inj_on_iff_surj
thf(fact_7844_image__set__diff,axiom,
    ! [B: $tType,A: $tType,F: A > B,A3: set @ A,B6: set @ A] :
      ( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
     => ( ( image @ A @ B @ F @ ( minus_minus @ ( set @ A ) @ A3 @ B6 ) )
        = ( minus_minus @ ( set @ B ) @ ( image @ A @ B @ F @ A3 ) @ ( image @ A @ B @ F @ B6 ) ) ) ) ).

% image_set_diff
thf(fact_7845_inj__on__image__set__diff,axiom,
    ! [B: $tType,A: $tType,F: A > B,C5: set @ A,A3: set @ A,B6: set @ A] :
      ( ( inj_on @ A @ B @ F @ C5 )
     => ( ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ B6 ) @ C5 )
       => ( ( ord_less_eq @ ( set @ A ) @ B6 @ C5 )
         => ( ( image @ A @ B @ F @ ( minus_minus @ ( set @ A ) @ A3 @ B6 ) )
            = ( minus_minus @ ( set @ B ) @ ( image @ A @ B @ F @ A3 ) @ ( image @ A @ B @ F @ B6 ) ) ) ) ) ) ).

% inj_on_image_set_diff
thf(fact_7846_finite__inverse__image,axiom,
    ! [A: $tType,B: $tType,A3: set @ A,F: B > A] :
      ( ( finite_finite2 @ A @ A3 )
     => ( ( inj_on @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
       => ( finite_finite2 @ B
          @ ( collect @ B
            @ ^ [J3: B] : ( member @ A @ ( F @ J3 ) @ A3 ) ) ) ) ) ).

% finite_inverse_image
thf(fact_7847_finite__Collect,axiom,
    ! [A: $tType,B: $tType,S3: set @ A,F: B > A] :
      ( ( finite_finite2 @ A @ S3 )
     => ( ( inj_on @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
       => ( finite_finite2 @ B
          @ ( collect @ B
            @ ^ [A5: B] : ( member @ A @ ( F @ A5 ) @ S3 ) ) ) ) ) ).

% finite_Collect
thf(fact_7848_the__inv__into__into,axiom,
    ! [B: $tType,A: $tType,F: A > B,A3: set @ A,X2: B,B6: set @ A] :
      ( ( inj_on @ A @ B @ F @ A3 )
     => ( ( member @ B @ X2 @ ( image @ A @ B @ F @ A3 ) )
       => ( ( ord_less_eq @ ( set @ A ) @ A3 @ B6 )
         => ( member @ A @ ( the_inv_into @ A @ B @ A3 @ F @ X2 ) @ B6 ) ) ) ) ).

% the_inv_into_into
thf(fact_7849_finite__inverse__image__gen,axiom,
    ! [A: $tType,B: $tType,A3: set @ A,F: B > A,D6: set @ B] :
      ( ( finite_finite2 @ A @ A3 )
     => ( ( inj_on @ B @ A @ F @ D6 )
       => ( finite_finite2 @ B
          @ ( collect @ B
            @ ^ [J3: B] :
                ( ( member @ B @ J3 @ D6 )
                & ( member @ A @ ( F @ J3 ) @ A3 ) ) ) ) ) ) ).

% finite_inverse_image_gen
thf(fact_7850_inj__image__Compl__subset,axiom,
    ! [B: $tType,A: $tType,F: A > B,A3: set @ A] :
      ( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
     => ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F @ ( uminus_uminus @ ( set @ A ) @ A3 ) ) @ ( uminus_uminus @ ( set @ B ) @ ( image @ A @ B @ F @ A3 ) ) ) ) ).

% inj_image_Compl_subset
thf(fact_7851_msb__shift,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ( ( most_s684356279273892711sb_msb @ ( word @ A ) )
        = ( ^ [W2: word @ A] :
              ( ( bit_Sh4282982442137083166shiftr @ ( word @ A ) @ W2 @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) )
             != ( zero_zero @ ( word @ A ) ) ) ) ) ) ).

% msb_shift
thf(fact_7852_msb__word__eq,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ( ( most_s684356279273892711sb_msb @ ( word @ A ) )
        = ( ^ [W2: word @ A] : ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ W2 @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) ) ) ) ) ).

% msb_word_eq
thf(fact_7853_msb__word__iff__bit,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ( ( most_s684356279273892711sb_msb @ ( word @ A ) )
        = ( ^ [W2: word @ A] : ( bit_se5641148757651400278ts_bit @ ( word @ A ) @ W2 @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) ).

% msb_word_iff_bit
thf(fact_7854_word__msb__nth,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ( ( most_s684356279273892711sb_msb @ ( word @ A ) )
        = ( ^ [W2: word @ A] : ( bit_se5641148757651400278ts_bit @ int @ ( semiring_1_unsigned @ A @ int @ W2 ) @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) ) ) ) ) ).

% word_msb_nth
thf(fact_7855_msb__word__of__int,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [X2: int] :
          ( ( most_s684356279273892711sb_msb @ ( word @ A ) @ ( ring_1_of_int @ ( word @ A ) @ X2 ) )
          = ( bit_se5641148757651400278ts_bit @ int @ X2 @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) ) ) ) ).

% msb_word_of_int
thf(fact_7856_not__msb__from__less,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ! [V: word @ A] :
          ( ( ord_less @ ( word @ A ) @ V @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( one_one @ nat ) ) ) )
         => ~ ( most_s684356279273892711sb_msb @ ( word @ A ) @ V ) ) ) ).

% not_msb_from_less
thf(fact_7857_word__sint__msb__eq,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ( ( ring_1_signed @ A @ int )
        = ( ^ [X: word @ A] : ( minus_minus @ int @ ( semiring_1_unsigned @ A @ int @ X ) @ ( if @ int @ ( most_s684356279273892711sb_msb @ ( word @ A ) @ X ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( size_size @ ( word @ A ) @ X ) ) @ ( zero_zero @ int ) ) ) ) ) ) ).

% word_sint_msb_eq
thf(fact_7858_msb__big,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ( ( most_s684356279273892711sb_msb @ ( word @ A ) )
        = ( ord_less_eq @ ( word @ A ) @ ( power_power @ ( word @ A ) @ ( numeral_numeral @ ( word @ A ) @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ ( type_len0_len_of @ A @ ( type2 @ A ) ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) ).

% msb_big
thf(fact_7859_word__type__copy__misc__axioms_Ointro,axiom,
    ! [B: $tType,A: $tType] :
      ( ( ( bit_ri3973907225187159222ations @ A )
        & ( cl_HOL_Oequal @ A )
        & ( linorder @ A )
        & ( type_len @ B ) )
     => ! [Size: nat,Word_of: A > ( word @ B ),Set_bits_aux: ( nat > $o ) > nat > A > A] :
          ( ( Size
            = ( type_len0_len_of @ B @ ( type2 @ B ) ) )
         => ( ! [P9: A] :
                ( ( most_s684356279273892711sb_msb @ A @ P9 )
                = ( most_s684356279273892711sb_msb @ ( word @ B ) @ ( Word_of @ P9 ) ) )
           => ( ! [P9: A] :
                  ( ( least_8051144512741203767sb_lsb @ A @ P9 )
                  = ( least_8051144512741203767sb_lsb @ ( word @ B ) @ ( Word_of @ P9 ) ) )
             => ( ! [P9: A] :
                    ( ( size_size @ A @ P9 )
                    = ( size_size @ ( word @ B ) @ ( Word_of @ P9 ) ) )
               => ( ! [P9: A,N4: nat,B3: $o] :
                      ( ( Word_of @ ( generi7602027413899671122et_bit @ A @ P9 @ N4 @ B3 ) )
                      = ( generi7602027413899671122et_bit @ ( word @ B ) @ ( Word_of @ P9 ) @ N4 @ B3 ) )
                 => ( ! [P7: nat > $o] :
                        ( ( Word_of @ ( bit_bi4170147762399595738t_bits @ A @ P7 ) )
                        = ( bit_bi4170147762399595738t_bits @ ( word @ B ) @ P7 ) )
                   => ( ! [P7: nat > $o,N4: nat,P9: A] :
                          ( ( Word_of @ ( Set_bits_aux @ P7 @ N4 @ P9 ) )
                          = ( code_T2661198915054445665ts_aux @ B @ P7 @ N4 @ ( Word_of @ P9 ) ) )
                     => ( word_T8964210463127689547axioms @ A @ B @ Word_of @ Size @ Set_bits_aux ) ) ) ) ) ) ) ) ) ).

% word_type_copy_misc_axioms.intro
thf(fact_7860_word__type__copy__misc__axioms__def,axiom,
    ! [B: $tType,A: $tType] :
      ( ( ( bit_ri3973907225187159222ations @ A )
        & ( cl_HOL_Oequal @ A )
        & ( linorder @ A )
        & ( type_len @ B ) )
     => ( ( word_T8964210463127689547axioms @ A @ B )
        = ( ^ [Word_of2: A > ( word @ B ),Size2: nat,Set_bits_aux2: ( nat > $o ) > nat > A > A] :
              ( ( Size2
                = ( type_len0_len_of @ B @ ( type2 @ B ) ) )
              & ! [P5: A] :
                  ( ( most_s684356279273892711sb_msb @ A @ P5 )
                  = ( most_s684356279273892711sb_msb @ ( word @ B ) @ ( Word_of2 @ P5 ) ) )
              & ! [P5: A] :
                  ( ( least_8051144512741203767sb_lsb @ A @ P5 )
                  = ( least_8051144512741203767sb_lsb @ ( word @ B ) @ ( Word_of2 @ P5 ) ) )
              & ! [P5: A] :
                  ( ( size_size @ A @ P5 )
                  = ( size_size @ ( word @ B ) @ ( Word_of2 @ P5 ) ) )
              & ! [P5: A,N: nat,B5: $o] :
                  ( ( Word_of2 @ ( generi7602027413899671122et_bit @ A @ P5 @ N @ B5 ) )
                  = ( generi7602027413899671122et_bit @ ( word @ B ) @ ( Word_of2 @ P5 ) @ N @ B5 ) )
              & ! [P3: nat > $o] :
                  ( ( Word_of2 @ ( bit_bi4170147762399595738t_bits @ A @ P3 ) )
                  = ( bit_bi4170147762399595738t_bits @ ( word @ B ) @ P3 ) )
              & ! [P3: nat > $o,N: nat,P5: A] :
                  ( ( Word_of2 @ ( Set_bits_aux2 @ P3 @ N @ P5 ) )
                  = ( code_T2661198915054445665ts_aux @ B @ P3 @ N @ ( Word_of2 @ P5 ) ) ) ) ) ) ) ).

% word_type_copy_misc_axioms_def
thf(fact_7861_msb__numeral_I1_J,axiom,
    ! [N2: num] :
      ~ ( most_s684356279273892711sb_msb @ int @ ( numeral_numeral @ int @ N2 ) ) ).

% msb_numeral(1)
thf(fact_7862_msb__1,axiom,
    ~ ( most_s684356279273892711sb_msb @ int @ ( one_one @ int ) ) ).

% msb_1
thf(fact_7863_msb__numeral_I2_J,axiom,
    ! [N2: num] : ( most_s684356279273892711sb_msb @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N2 ) ) ) ).

% msb_numeral(2)
thf(fact_7864_msb__bin__rest,axiom,
    ! [X2: int] :
      ( ( most_s684356279273892711sb_msb @ int @ ( divide_divide @ int @ X2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) )
      = ( most_s684356279273892711sb_msb @ int @ X2 ) ) ).

% msb_bin_rest
thf(fact_7865_swap__inj__on,axiom,
    ! [B: $tType,A: $tType,A3: set @ ( product_prod @ A @ B )] :
      ( inj_on @ ( product_prod @ A @ B ) @ ( product_prod @ B @ A )
      @ ( product_case_prod @ A @ B @ ( product_prod @ B @ A )
        @ ^ [I2: A,J3: B] : ( product_Pair @ B @ A @ J3 @ I2 ) )
      @ A3 ) ).

% swap_inj_on
thf(fact_7866_inj__Some,axiom,
    ! [A: $tType,A3: set @ A] : ( inj_on @ A @ ( option @ A ) @ ( some @ A ) @ A3 ) ).

% inj_Some
thf(fact_7867_inj__singleton,axiom,
    ! [A: $tType,A3: set @ A] :
      ( inj_on @ A @ ( set @ A )
      @ ^ [X: A] : ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) )
      @ A3 ) ).

% inj_singleton
thf(fact_7868_inj__Suc,axiom,
    ! [N3: set @ nat] : ( inj_on @ nat @ nat @ suc @ N3 ) ).

% inj_Suc
thf(fact_7869_inj__on__diff__nat,axiom,
    ! [N3: set @ nat,K: nat] :
      ( ! [N4: nat] :
          ( ( member @ nat @ N4 @ N3 )
         => ( ord_less_eq @ nat @ K @ N4 ) )
     => ( inj_on @ nat @ nat
        @ ^ [N: nat] : ( minus_minus @ nat @ N @ K )
        @ N3 ) ) ).

% inj_on_diff_nat
thf(fact_7870_inj__Pair_I2_J,axiom,
    ! [B: $tType,A: $tType,C2: A > B,S3: set @ A] :
      ( inj_on @ A @ ( product_prod @ B @ A )
      @ ^ [X: A] : ( product_Pair @ B @ A @ ( C2 @ X ) @ X )
      @ S3 ) ).

% inj_Pair(2)
thf(fact_7871_inj__Pair_I1_J,axiom,
    ! [B: $tType,A: $tType,C2: A > B,S3: set @ A] :
      ( inj_on @ A @ ( product_prod @ A @ B )
      @ ^ [X: A] : ( product_Pair @ A @ B @ X @ ( C2 @ X ) )
      @ S3 ) ).

% inj_Pair(1)
thf(fact_7872_inj__on__convol__ident,axiom,
    ! [B: $tType,A: $tType,F: A > B,X8: set @ A] :
      ( inj_on @ A @ ( product_prod @ A @ B )
      @ ^ [X: A] : ( product_Pair @ A @ B @ X @ ( F @ X ) )
      @ X8 ) ).

% inj_on_convol_ident
thf(fact_7873_signed_Osorted__list__of__set_Oinj__on,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ( inj_on @ ( word @ A ) @ ( word @ A )
        @ ^ [X: word @ A] : X
        @ ( top_top @ ( set @ ( word @ A ) ) ) ) ) ).

% signed.sorted_list_of_set.inj_on
thf(fact_7874_inj__of__nat,axiom,
    ! [A: $tType] :
      ( ( semiring_char_0 @ A )
     => ( inj_on @ nat @ A @ ( semiring_1_of_nat @ A ) @ ( top_top @ ( set @ nat ) ) ) ) ).

% inj_of_nat
thf(fact_7875_inj__split__Cons,axiom,
    ! [A: $tType,X8: set @ ( product_prod @ ( list @ A ) @ A )] :
      ( inj_on @ ( product_prod @ ( list @ A ) @ A ) @ ( list @ A )
      @ ( product_case_prod @ ( list @ A ) @ A @ ( list @ A )
        @ ^ [Xs: list @ A,N: A] : ( cons @ A @ N @ Xs ) )
      @ X8 ) ).

% inj_split_Cons
thf(fact_7876_inj__graph,axiom,
    ! [B: $tType,A: $tType] :
      ( inj_on @ ( A > B ) @ ( set @ ( product_prod @ A @ B ) )
      @ ^ [F6: A > B] :
          ( collect @ ( product_prod @ A @ B )
          @ ( product_case_prod @ A @ B @ $o
            @ ^ [X: A,Y: B] :
                ( Y
                = ( F6 @ X ) ) ) )
      @ ( top_top @ ( set @ ( A > B ) ) ) ) ).

% inj_graph
thf(fact_7877_finite__imp__nat__seg__image__inj__on,axiom,
    ! [A: $tType,A3: set @ A] :
      ( ( finite_finite2 @ A @ A3 )
     => ? [N4: nat,F2: nat > A] :
          ( ( A3
            = ( image @ nat @ A @ F2
              @ ( collect @ nat
                @ ^ [I2: nat] : ( ord_less @ nat @ I2 @ N4 ) ) ) )
          & ( inj_on @ nat @ A @ F2
            @ ( collect @ nat
              @ ^ [I2: nat] : ( ord_less @ nat @ I2 @ N4 ) ) ) ) ) ).

% finite_imp_nat_seg_image_inj_on
thf(fact_7878_finite__imp__inj__to__nat__seg,axiom,
    ! [A: $tType,A3: set @ A] :
      ( ( finite_finite2 @ A @ A3 )
     => ? [F2: A > nat,N4: nat] :
          ( ( ( image @ A @ nat @ F2 @ A3 )
            = ( collect @ nat
              @ ^ [I2: nat] : ( ord_less @ nat @ I2 @ N4 ) ) )
          & ( inj_on @ A @ nat @ F2 @ A3 ) ) ) ).

% finite_imp_inj_to_nat_seg
thf(fact_7879_finite__imp__inj__to__nat__seg_H,axiom,
    ! [A: $tType,A3: set @ A] :
      ( ( finite_finite2 @ A @ A3 )
     => ~ ! [F2: A > nat] :
            ( ? [N4: nat] :
                ( ( image @ A @ nat @ F2 @ A3 )
                = ( collect @ nat
                  @ ^ [I2: nat] : ( ord_less @ nat @ I2 @ N4 ) ) )
           => ~ ( inj_on @ A @ nat @ F2 @ A3 ) ) ) ).

% finite_imp_inj_to_nat_seg'
thf(fact_7880_infinite__iff__countable__subset,axiom,
    ! [A: $tType,S3: set @ A] :
      ( ( ~ ( finite_finite2 @ A @ S3 ) )
      = ( ? [F6: nat > A] :
            ( ( inj_on @ nat @ A @ F6 @ ( top_top @ ( set @ nat ) ) )
            & ( ord_less_eq @ ( set @ A ) @ ( image @ nat @ A @ F6 @ ( top_top @ ( set @ nat ) ) ) @ S3 ) ) ) ) ).

% infinite_iff_countable_subset
thf(fact_7881_infinite__countable__subset,axiom,
    ! [A: $tType,S3: set @ A] :
      ( ~ ( finite_finite2 @ A @ S3 )
     => ? [F2: nat > A] :
          ( ( inj_on @ nat @ A @ F2 @ ( top_top @ ( set @ nat ) ) )
          & ( ord_less_eq @ ( set @ A ) @ ( image @ nat @ A @ F2 @ ( top_top @ ( set @ nat ) ) ) @ S3 ) ) ) ).

% infinite_countable_subset
thf(fact_7882_summable__reindex,axiom,
    ! [F: nat > real,G: nat > nat] :
      ( ( summable @ real @ F )
     => ( ( inj_on @ nat @ nat @ G @ ( top_top @ ( set @ nat ) ) )
       => ( ! [X3: nat] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( F @ X3 ) )
         => ( summable @ real @ ( comp @ nat @ real @ nat @ F @ G ) ) ) ) ) ).

% summable_reindex
thf(fact_7883_suminf__reindex__mono,axiom,
    ! [F: nat > real,G: nat > nat] :
      ( ( summable @ real @ F )
     => ( ( inj_on @ nat @ nat @ G @ ( top_top @ ( set @ nat ) ) )
       => ( ! [X3: nat] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( F @ X3 ) )
         => ( ord_less_eq @ real @ ( suminf @ real @ ( comp @ nat @ real @ nat @ F @ G ) ) @ ( suminf @ real @ F ) ) ) ) ) ).

% suminf_reindex_mono
thf(fact_7884_uint32__msb__test__bit,axiom,
    ( ( most_s684356279273892711sb_msb @ uint32 )
    = ( ^ [X: uint32] : ( bit_se5641148757651400278ts_bit @ uint32 @ X @ ( numeral_numeral @ nat @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one2 ) ) ) ) ) ) ) ) ).

% uint32_msb_test_bit
thf(fact_7885_suminf__reindex,axiom,
    ! [F: nat > real,G: nat > nat] :
      ( ( summable @ real @ F )
     => ( ( inj_on @ nat @ nat @ G @ ( top_top @ ( set @ nat ) ) )
       => ( ! [X3: nat] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( F @ X3 ) )
         => ( ! [X3: nat] :
                ( ~ ( member @ nat @ X3 @ ( image @ nat @ nat @ G @ ( top_top @ ( set @ nat ) ) ) )
               => ( ( F @ X3 )
                  = ( zero_zero @ real ) ) )
           => ( ( suminf @ real @ ( comp @ nat @ real @ nat @ F @ G ) )
              = ( suminf @ real @ F ) ) ) ) ) ) ).

% suminf_reindex
thf(fact_7886_inj__sgn__power,axiom,
    ! [N2: nat] :
      ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
     => ( inj_on @ real @ real
        @ ^ [Y: real] : ( times_times @ real @ ( sgn_sgn @ real @ Y ) @ ( power_power @ real @ ( abs_abs @ real @ Y ) @ N2 ) )
        @ ( top_top @ ( set @ real ) ) ) ) ).

% inj_sgn_power
thf(fact_7887_inj__on__word__of__nat,axiom,
    ! [A: $tType] :
      ( ( type_len @ A )
     => ( inj_on @ nat @ ( word @ A ) @ ( semiring_1_of_nat @ ( word @ A ) ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( type_len0_len_of @ A @ ( type2 @ A ) ) ) ) ) ) ).

% inj_on_word_of_nat
thf(fact_7888_inj__on__map__inv__f,axiom,
    ! [B: $tType,A: $tType,L: list @ A,A3: set @ A,F: A > B] :
      ( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ L ) @ A3 )
     => ( ( inj_on @ A @ B @ F @ A3 )
       => ( ( map @ B @ A @ ( inv_on @ A @ B @ F @ A3 ) @ ( map @ A @ B @ F @ L ) )
          = L ) ) ) ).

% inj_on_map_inv_f
thf(fact_7889_Schroeder__Bernstein,axiom,
    ! [A: $tType,B: $tType,F: A > B,A3: set @ A,B6: set @ B,G: B > A] :
      ( ( inj_on @ A @ B @ F @ A3 )
     => ( ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F @ A3 ) @ B6 )
       => ( ( inj_on @ B @ A @ G @ B6 )
         => ( ( ord_less_eq @ ( set @ A ) @ ( image @ B @ A @ G @ B6 ) @ A3 )
           => ? [H3: A > B] : ( bij_betw @ A @ B @ H3 @ A3 @ B6 ) ) ) ) ) ).

% Schroeder_Bernstein
thf(fact_7890_inv__on__f__f,axiom,
    ! [B: $tType,A: $tType,F: A > B,A3: set @ A,X2: A] :
      ( ( inj_on @ A @ B @ F @ A3 )
     => ( ( member @ A @ X2 @ A3 )
       => ( ( inv_on @ A @ B @ F @ A3 @ ( F @ X2 ) )
          = X2 ) ) ) ).

% inv_on_f_f
thf(fact_7891_le__rel__bool__arg__iff,axiom,
    ! [A: $tType] :
      ( ( ord @ A )
     => ( ( ord_less_eq @ ( $o > A ) )
        = ( ^ [X5: $o > A,Y9: $o > A] :
              ( ( ord_less_eq @ A @ ( X5 @ $false ) @ ( Y9 @ $false ) )
              & ( ord_less_eq @ A @ ( X5 @ $true ) @ ( Y9 @ $true ) ) ) ) ) ) ).

% le_rel_bool_arg_iff
thf(fact_7892_inv__on__f__range,axiom,
    ! [A: $tType,B: $tType,Y2: A,F: B > A,A3: set @ B] :
      ( ( member @ A @ Y2 @ ( image @ B @ A @ F @ A3 ) )
     => ( member @ B @ ( inv_on @ B @ A @ F @ A3 @ Y2 ) @ A3 ) ) ).

% inv_on_f_range
thf(fact_7893_f__inv__on__f,axiom,
    ! [B: $tType,A: $tType,Y2: A,F: B > A,A3: set @ B] :
      ( ( member @ A @ Y2 @ ( image @ B @ A @ F @ A3 ) )
     => ( ( F @ ( inv_on @ B @ A @ F @ A3 @ Y2 ) )
        = Y2 ) ) ).

% f_inv_on_f
thf(fact_7894_inv__on__def,axiom,
    ! [B: $tType,A: $tType] :
      ( ( inv_on @ A @ B )
      = ( ^ [F6: A > B,A7: set @ A,X: B] :
            ( fChoice @ A
            @ ^ [Y: A] :
                ( ( member @ A @ Y @ A7 )
                & ( ( F6 @ Y )
                  = X ) ) ) ) ) ).

% inv_on_def
thf(fact_7895_msb__uint32__code,axiom,
    ( ( most_s684356279273892711sb_msb @ uint32 )
    = ( ^ [X: uint32] : ( uint32_test_bit @ X @ ( numeral_numeral @ code_integer @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one2 ) ) ) ) ) ) ) ) ).

% msb_uint32_code
thf(fact_7896_all__subset__image__inj,axiom,
    ! [A: $tType,B: $tType,F: B > A,S3: set @ B,P: ( set @ A ) > $o] :
      ( ( ! [T9: set @ A] :
            ( ( ord_less_eq @ ( set @ A ) @ T9 @ ( image @ B @ A @ F @ S3 ) )
           => ( P @ T9 ) ) )
      = ( ! [T9: set @ B] :
            ( ( ( ord_less_eq @ ( set @ B ) @ T9 @ S3 )
              & ( inj_on @ B @ A @ F @ T9 ) )
           => ( P @ ( image @ B @ A @ F @ T9 ) ) ) ) ) ).

% all_subset_image_inj
thf(fact_7897_ex__subset__image__inj,axiom,
    ! [A: $tType,B: $tType,F: B > A,S3: set @ B,P: ( set @ A ) > $o] :
      ( ( ? [T9: set @ A] :
            ( ( ord_less_eq @ ( set @ A ) @ T9 @ ( image @ B @ A @ F @ S3 ) )
            & ( P @ T9 ) ) )
      = ( ? [T9: set @ B] :
            ( ( ord_less_eq @ ( set @ B ) @ T9 @ S3 )
            & ( inj_on @ B @ A @ F @ T9 )
            & ( P @ ( image @ B @ A @ F @ T9 ) ) ) ) ) ).

% ex_subset_image_inj
thf(fact_7898_word__type__copy__misc_Oset__bits__aux__code,axiom,
    ! [B: $tType,A: $tType] :
      ( ( ( bit_ri3973907225187159222ations @ A )
        & ( cl_HOL_Oequal @ A )
        & ( linorder @ A )
        & ( type_len @ B ) )
     => ! [Of_word: ( word @ B ) > A,Word_of: A > ( word @ B ),Signed_drop_bit: nat > A > A,Of_nat: nat > A,Nat_of: A > nat,Of_int: int > A,Int_of: A > int,Of_integer: code_integer > A,Integer_of: A > code_integer,Size: nat,Set_bits_aux: ( nat > $o ) > nat > A > A,N2: nat,F: nat > $o,W: A] :
          ( ( word_T1870391057261780392y_misc @ B @ A @ Of_word @ Word_of @ Signed_drop_bit @ Of_nat @ Nat_of @ Of_int @ Int_of @ Of_integer @ Integer_of @ Size @ Set_bits_aux )
         => ( ( ( N2
                = ( zero_zero @ nat ) )
             => ( ( Set_bits_aux @ F @ N2 @ W )
                = W ) )
            & ( ( N2
               != ( zero_zero @ nat ) )
             => ( ( Set_bits_aux @ F @ N2 @ W )
                = ( Set_bits_aux @ F @ ( minus_minus @ nat @ N2 @ ( one_one @ nat ) ) @ ( bit_se1065995026697491101ons_or @ A @ ( bit_se4730199178511100633sh_bit @ A @ ( one_one @ nat ) @ W ) @ ( if @ A @ ( F @ ( minus_minus @ nat @ N2 @ ( one_one @ nat ) ) ) @ ( one_one @ A ) @ ( zero_zero @ A ) ) ) ) ) ) ) ) ) ).

% word_type_copy_misc.set_bits_aux_code
thf(fact_7899_uint32__test__bit__def,axiom,
    ( uint32_test_bit
    = ( ^ [X: uint32,N: code_integer] :
          ( ( ( ( ord_less @ code_integer @ N @ ( zero_zero @ code_integer ) )
              | ( ord_less @ code_integer @ ( numeral_numeral @ code_integer @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one2 ) ) ) ) ) @ N ) )
           => ( undefined @ ( ( uint32 > nat > $o ) > uint32 > code_integer > $o ) @ ( bit_se5641148757651400278ts_bit @ uint32 ) @ X @ N ) )
          & ( ~ ( ( ord_less @ code_integer @ N @ ( zero_zero @ code_integer ) )
                | ( ord_less @ code_integer @ ( numeral_numeral @ code_integer @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one2 ) ) ) ) ) @ N ) )
           => ( bit_se5641148757651400278ts_bit @ uint32 @ X @ ( code_nat_of_integer @ N ) ) ) ) ) ) ).

% uint32_test_bit_def
thf(fact_7900_nat__of__integer__numeral,axiom,
    ! [N2: num] :
      ( ( code_nat_of_integer @ ( numeral_numeral @ code_integer @ N2 ) )
      = ( numeral_numeral @ nat @ N2 ) ) ).

% nat_of_integer_numeral
thf(fact_7901_nat__of__integer__code__post_I3_J,axiom,
    ! [K: num] :
      ( ( code_nat_of_integer @ ( numeral_numeral @ code_integer @ K ) )
      = ( numeral_numeral @ nat @ K ) ) ).

% nat_of_integer_code_post(3)
thf(fact_7902_nat__of__integer__1,axiom,
    ( ( code_nat_of_integer @ ( one_one @ code_integer ) )
    = ( one_one @ nat ) ) ).

% nat_of_integer_1
thf(fact_7903_nat__of__integer__non__positive,axiom,
    ! [K: code_integer] :
      ( ( ord_less_eq @ code_integer @ K @ ( zero_zero @ code_integer ) )
     => ( ( code_nat_of_integer @ K )
        = ( zero_zero @ nat ) ) ) ).

% nat_of_integer_non_positive
thf(fact_7904_word__type__copy__misc_Osize__eq__word__of,axiom,
    ! [B: $tType,A: $tType] :
      ( ( ( bit_ri3973907225187159222ations @ A )
        & ( cl_HOL_Oequal @ A )
        & ( linorder @ A )
        & ( type_len @ B ) )
     => ! [Of_word: ( word @ B ) > A,Word_of: A > ( word @ B ),Signed_drop_bit: nat > A > A,Of_nat: nat > A,Nat_of: A > nat,Of_int: int > A,Int_of: A > int,Of_integer: code_integer > A,Integer_of: A > code_integer,Size: nat,Set_bits_aux: ( nat > $o ) > nat > A > A,P4: A] :
          ( ( word_T1870391057261780392y_misc @ B @ A @ Of_word @ Word_of @ Signed_drop_bit @ Of_nat @ Nat_of @ Of_int @ Int_of @ Of_integer @ Integer_of @ Size @ Set_bits_aux )
         => ( ( size_size @ A @ P4 )
            = ( size_size @ ( word @ B ) @ ( Word_of @ P4 ) ) ) ) ) ).

% word_type_copy_misc.size_eq_word_of
thf(fact_7905_word__type__copy__misc_Osize__eq,axiom,
    ! [B: $tType,A: $tType] :
      ( ( ( bit_ri3973907225187159222ations @ A )
        & ( cl_HOL_Oequal @ A )
        & ( linorder @ A )
        & ( type_len @ B ) )
     => ! [Of_word: ( word @ B ) > A,Word_of: A > ( word @ B ),Signed_drop_bit: nat > A > A,Of_nat: nat > A,Nat_of: A > nat,Of_int: int > A,Int_of: A > int,Of_integer: code_integer > A,Integer_of: A > code_integer,Size: nat,Set_bits_aux: ( nat > $o ) > nat > A > A,P4: A] :
          ( ( word_T1870391057261780392y_misc @ B @ A @ Of_word @ Word_of @ Signed_drop_bit @ Of_nat @ Nat_of @ Of_int @ Int_of @ Of_integer @ Integer_of @ Size @ Set_bits_aux )
         => ( ( size_size @ A @ P4 )
            = Size ) ) ) ).

% word_type_copy_misc.size_eq
thf(fact_7906_nat__of__integer__less__iff,axiom,
    ! [X2: code_integer,Y2: code_integer] :
      ( ( ord_less_eq @ code_integer @ ( zero_zero @ code_integer ) @ X2 )
     => ( ( ord_less_eq @ code_integer @ ( zero_zero @ code_integer ) @ Y2 )
       => ( ( ord_less @ nat @ ( code_nat_of_integer @ X2 ) @ ( code_nat_of_integer @ Y2 ) )
          = ( ord_less @ code_integer @ X2 @ Y2 ) ) ) ) ).

% nat_of_integer_less_iff
thf(fact_7907_image__atLeastZeroLessThan__integer,axiom,
    ! [U: code_integer] :
      ( ( ord_less_eq @ code_integer @ ( zero_zero @ code_integer ) @ U )
     => ( ( set_or7035219750837199246ssThan @ code_integer @ ( zero_zero @ code_integer ) @ U )
        = ( image @ nat @ code_integer @ ( semiring_1_of_nat @ code_integer ) @ ( set_ord_lessThan @ nat @ ( code_nat_of_integer @ U ) ) ) ) ) ).

% image_atLeastZeroLessThan_integer
thf(fact_7908_integer__set__bit__code,axiom,
    ( bits_integer_set_bit
    = ( ^ [X: code_integer,N: code_integer,B5: $o] : ( if @ code_integer @ ( ord_less @ code_integer @ N @ ( zero_zero @ code_integer ) ) @ ( undefined @ ( code_integer > code_integer > $o > code_integer ) @ X @ N @ B5 ) @ ( if @ code_integer @ B5 @ ( bit_se1065995026697491101ons_or @ code_integer @ X @ ( bit_se4730199178511100633sh_bit @ code_integer @ ( code_nat_of_integer @ N ) @ ( one_one @ code_integer ) ) ) @ ( bit_se5824344872417868541ns_and @ code_integer @ X @ ( bit_ri4277139882892585799ns_not @ code_integer @ ( bit_se4730199178511100633sh_bit @ code_integer @ ( code_nat_of_integer @ N ) @ ( one_one @ code_integer ) ) ) ) ) ) ) ) ).

% integer_set_bit_code
thf(fact_7909_uint32__shiftr__def,axiom,
    ( uint32_shiftr
    = ( ^ [X: uint32,N: code_integer] :
          ( if @ uint32
          @ ( ( ord_less @ code_integer @ N @ ( zero_zero @ code_integer ) )
            | ( ord_less_eq @ code_integer @ ( numeral_numeral @ code_integer @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) @ N ) )
          @ ( undefined @ ( ( nat > uint32 > uint32 ) > uint32 > code_integer > uint32 ) @ ( bit_se4197421643247451524op_bit @ uint32 ) @ X @ N )
          @ ( bit_se4197421643247451524op_bit @ uint32 @ ( code_nat_of_integer @ N ) @ X ) ) ) ) ).

% uint32_shiftr_def
thf(fact_7910_uint32__shiftl__def,axiom,
    ( uint32_shiftl
    = ( ^ [X: uint32,N: code_integer] :
          ( if @ uint32
          @ ( ( ord_less @ code_integer @ N @ ( zero_zero @ code_integer ) )
            | ( ord_less_eq @ code_integer @ ( numeral_numeral @ code_integer @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) @ N ) )
          @ ( undefined @ ( ( nat > uint32 > uint32 ) > uint32 > code_integer > uint32 ) @ ( bit_se4730199178511100633sh_bit @ uint32 ) @ X @ N )
          @ ( bit_se4730199178511100633sh_bit @ uint32 @ ( code_nat_of_integer @ N ) @ X ) ) ) ) ).

% uint32_shiftl_def
thf(fact_7911_uint32__set__bit__def,axiom,
    ( uint32_set_bit
    = ( ^ [X: uint32,N: code_integer,B5: $o] :
          ( if @ uint32
          @ ( ( ord_less @ code_integer @ N @ ( zero_zero @ code_integer ) )
            | ( ord_less @ code_integer @ ( numeral_numeral @ code_integer @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one2 ) ) ) ) ) @ N ) )
          @ ( undefined @ ( ( uint32 > nat > $o > uint32 ) > uint32 > code_integer > $o > uint32 ) @ ( generi7602027413899671122et_bit @ uint32 ) @ X @ N @ B5 )
          @ ( generi7602027413899671122et_bit @ uint32 @ X @ ( code_nat_of_integer @ N ) @ B5 ) ) ) ) ).

% uint32_set_bit_def
thf(fact_7912_nat__of__integer__code,axiom,
    ( code_nat_of_integer
    = ( ^ [K4: code_integer] :
          ( if @ nat @ ( ord_less_eq @ code_integer @ K4 @ ( zero_zero @ code_integer ) ) @ ( zero_zero @ nat )
          @ ( product_case_prod @ code_integer @ code_integer @ nat
            @ ^ [L2: code_integer,J3: code_integer] :
                ( if @ nat
                @ ( J3
                  = ( zero_zero @ code_integer ) )
                @ ( plus_plus @ nat @ ( code_nat_of_integer @ L2 ) @ ( code_nat_of_integer @ L2 ) )
                @ ( plus_plus @ nat @ ( plus_plus @ nat @ ( code_nat_of_integer @ L2 ) @ ( code_nat_of_integer @ L2 ) ) @ ( one_one @ nat ) ) )
            @ ( code_divmod_integer @ K4 @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) ) ) ) ) ) ).

% nat_of_integer_code
thf(fact_7913_nth_H__def,axiom,
    ! [A: $tType] :
      ( ( heap @ A )
     => ( ( array_nth2 @ A )
        = ( ^ [A5: array @ A] : ( comp @ nat @ ( heap_Time_Heap @ A ) @ code_integer @ ( array_nth @ A @ A5 ) @ code_nat_of_integer ) ) ) ) ).

% nth'_def
thf(fact_7914_fst__divmod__integer,axiom,
    ! [K: code_integer,L: code_integer] :
      ( ( product_fst @ code_integer @ code_integer @ ( code_divmod_integer @ K @ L ) )
      = ( divide_divide @ code_integer @ K @ L ) ) ).

% fst_divmod_integer
thf(fact_7915_snd__divmod__integer,axiom,
    ! [K: code_integer,L: code_integer] :
      ( ( product_snd @ code_integer @ code_integer @ ( code_divmod_integer @ K @ L ) )
      = ( modulo_modulo @ code_integer @ K @ L ) ) ).

% snd_divmod_integer
thf(fact_7916_divmod__integer__def,axiom,
    ( code_divmod_integer
    = ( ^ [K4: code_integer,L2: code_integer] : ( product_Pair @ code_integer @ code_integer @ ( divide_divide @ code_integer @ K4 @ L2 ) @ ( modulo_modulo @ code_integer @ K4 @ L2 ) ) ) ) ).

% divmod_integer_def
thf(fact_7917_num__of__integer__code,axiom,
    ( code_num_of_integer
    = ( ^ [K4: code_integer] :
          ( if @ num @ ( ord_less_eq @ code_integer @ K4 @ ( one_one @ code_integer ) ) @ one2
          @ ( product_case_prod @ code_integer @ code_integer @ num
            @ ^ [L2: code_integer,J3: code_integer] :
                ( if @ num
                @ ( J3
                  = ( zero_zero @ code_integer ) )
                @ ( plus_plus @ num @ ( code_num_of_integer @ L2 ) @ ( code_num_of_integer @ L2 ) )
                @ ( plus_plus @ num @ ( plus_plus @ num @ ( code_num_of_integer @ L2 ) @ ( code_num_of_integer @ L2 ) ) @ one2 ) )
            @ ( code_divmod_integer @ K4 @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) ) ) ) ) ) ).

% num_of_integer_code
thf(fact_7918_int__of__integer__code,axiom,
    ( code_int_of_integer
    = ( ^ [K4: code_integer] :
          ( if @ int @ ( ord_less @ code_integer @ K4 @ ( zero_zero @ code_integer ) ) @ ( uminus_uminus @ int @ ( code_int_of_integer @ ( uminus_uminus @ code_integer @ K4 ) ) )
          @ ( if @ int
            @ ( K4
              = ( zero_zero @ code_integer ) )
            @ ( zero_zero @ int )
            @ ( product_case_prod @ code_integer @ code_integer @ int
              @ ^ [L2: code_integer,J3: code_integer] :
                  ( if @ int
                  @ ( J3
                    = ( zero_zero @ code_integer ) )
                  @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( code_int_of_integer @ L2 ) )
                  @ ( plus_plus @ int @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( code_int_of_integer @ L2 ) ) @ ( one_one @ int ) ) )
              @ ( code_divmod_integer @ K4 @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).

% int_of_integer_code
thf(fact_7919_int__of__integer__numeral,axiom,
    ! [K: num] :
      ( ( code_int_of_integer @ ( numeral_numeral @ code_integer @ K ) )
      = ( numeral_numeral @ int @ K ) ) ).

% int_of_integer_numeral
thf(fact_7920_one__integer_Orep__eq,axiom,
    ( ( code_int_of_integer @ ( one_one @ code_integer ) )
    = ( one_one @ int ) ) ).

% one_integer.rep_eq
thf(fact_7921_plus__integer_Orep__eq,axiom,
    ! [X2: code_integer,Xa: code_integer] :
      ( ( code_int_of_integer @ ( plus_plus @ code_integer @ X2 @ Xa ) )
      = ( plus_plus @ int @ ( code_int_of_integer @ X2 ) @ ( code_int_of_integer @ Xa ) ) ) ).

% plus_integer.rep_eq
thf(fact_7922_minus__integer_Orep__eq,axiom,
    ! [X2: code_integer,Xa: code_integer] :
      ( ( code_int_of_integer @ ( minus_minus @ code_integer @ X2 @ Xa ) )
      = ( minus_minus @ int @ ( code_int_of_integer @ X2 ) @ ( code_int_of_integer @ Xa ) ) ) ).

% minus_integer.rep_eq
thf(fact_7923_times__integer_Orep__eq,axiom,
    ! [X2: code_integer,Xa: code_integer] :
      ( ( code_int_of_integer @ ( times_times @ code_integer @ X2 @ Xa ) )
      = ( times_times @ int @ ( code_int_of_integer @ X2 ) @ ( code_int_of_integer @ Xa ) ) ) ).

% times_integer.rep_eq
thf(fact_7924_divide__integer_Orep__eq,axiom,
    ! [X2: code_integer,Xa: code_integer] :
      ( ( code_int_of_integer @ ( divide_divide @ code_integer @ X2 @ Xa ) )
      = ( divide_divide @ int @ ( code_int_of_integer @ X2 ) @ ( code_int_of_integer @ Xa ) ) ) ).

% divide_integer.rep_eq
thf(fact_7925_integer__less__eq__iff,axiom,
    ( ( ord_less_eq @ code_integer )
    = ( ^ [K4: code_integer,L2: code_integer] : ( ord_less_eq @ int @ ( code_int_of_integer @ K4 ) @ ( code_int_of_integer @ L2 ) ) ) ) ).

% integer_less_eq_iff
thf(fact_7926_less__eq__integer_Orep__eq,axiom,
    ( ( ord_less_eq @ code_integer )
    = ( ^ [X: code_integer,Xa4: code_integer] : ( ord_less_eq @ int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Xa4 ) ) ) ) ).

% less_eq_integer.rep_eq
thf(fact_7927_int__of__integer__pow,axiom,
    ! [X2: code_integer,N2: nat] :
      ( ( code_int_of_integer @ ( power_power @ code_integer @ X2 @ N2 ) )
      = ( power_power @ int @ ( code_int_of_integer @ X2 ) @ N2 ) ) ).

% int_of_integer_pow
thf(fact_7928_dup_Orep__eq,axiom,
    ! [X2: code_integer] :
      ( ( code_int_of_integer @ ( code_dup @ X2 ) )
      = ( plus_plus @ int @ ( code_int_of_integer @ X2 ) @ ( code_int_of_integer @ X2 ) ) ) ).

% dup.rep_eq
thf(fact_7929_int__of__integer__less__iff,axiom,
    ! [X2: code_integer,Y2: code_integer] :
      ( ( ord_less @ int @ ( code_int_of_integer @ X2 ) @ ( code_int_of_integer @ Y2 ) )
      = ( ord_less @ code_integer @ X2 @ Y2 ) ) ).

% int_of_integer_less_iff
thf(fact_7930_bin__last__integer_Orep__eq,axiom,
    ( bits_b8758750999018896077nteger
    = ( ^ [X: code_integer] :
          ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( code_int_of_integer @ X ) ) ) ) ).

% bin_last_integer.rep_eq
thf(fact_7931_bin__rest__integer_Orep__eq,axiom,
    ! [X2: code_integer] :
      ( ( code_int_of_integer @ ( bits_b2549910563261871055nteger @ X2 ) )
      = ( divide_divide @ int @ ( code_int_of_integer @ X2 ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ).

% bin_rest_integer.rep_eq
thf(fact_7932_Bit__integer_Orep__eq,axiom,
    ! [X2: code_integer,Xa: $o] :
      ( ( code_int_of_integer @ ( bits_Bit_integer @ X2 @ Xa ) )
      = ( plus_plus @ int @ ( zero_neq_one_of_bool @ int @ Xa ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( code_int_of_integer @ X2 ) ) ) ) ).

% Bit_integer.rep_eq
thf(fact_7933_uint32__shiftr__code,axiom,
    ! [N2: code_integer,W: uint32] :
      ( ( ( ( ord_less @ code_integer @ N2 @ ( zero_zero @ code_integer ) )
          | ( ord_less_eq @ code_integer @ ( numeral_numeral @ code_integer @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) @ N2 ) )
       => ( ( rep_uint322 @ ( uint32_shiftr @ W @ N2 ) )
          = ( rep_uint322 @ ( undefined @ ( ( nat > uint32 > uint32 ) > uint32 > code_integer > uint32 ) @ ( bit_se4197421643247451524op_bit @ uint32 ) @ W @ N2 ) ) ) )
      & ( ~ ( ( ord_less @ code_integer @ N2 @ ( zero_zero @ code_integer ) )
            | ( ord_less_eq @ code_integer @ ( numeral_numeral @ code_integer @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) @ N2 ) )
       => ( ( rep_uint322 @ ( uint32_shiftr @ W @ N2 ) )
          = ( bit_se4197421643247451524op_bit @ ( word @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ numeral_num1 ) ) ) ) ) ) @ ( code_nat_of_integer @ N2 ) @ ( rep_uint322 @ W ) ) ) ) ) ).

% uint32_shiftr_code
thf(fact_7934_uint32__shiftl__code,axiom,
    ! [N2: code_integer,W: uint32] :
      ( ( ( ( ord_less @ code_integer @ N2 @ ( zero_zero @ code_integer ) )
          | ( ord_less_eq @ code_integer @ ( numeral_numeral @ code_integer @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) @ N2 ) )
       => ( ( rep_uint322 @ ( uint32_shiftl @ W @ N2 ) )
          = ( rep_uint322 @ ( undefined @ ( ( nat > uint32 > uint32 ) > uint32 > code_integer > uint32 ) @ ( bit_se4730199178511100633sh_bit @ uint32 ) @ W @ N2 ) ) ) )
      & ( ~ ( ( ord_less @ code_integer @ N2 @ ( zero_zero @ code_integer ) )
            | ( ord_less_eq @ code_integer @ ( numeral_numeral @ code_integer @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) @ N2 ) )
       => ( ( rep_uint322 @ ( uint32_shiftl @ W @ N2 ) )
          = ( bit_se4730199178511100633sh_bit @ ( word @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ numeral_num1 ) ) ) ) ) ) @ ( code_nat_of_integer @ N2 ) @ ( rep_uint322 @ W ) ) ) ) ) ).

% uint32_shiftl_code
thf(fact_7935_uint32_Oword__of__numeral,axiom,
    ! [N2: num] :
      ( ( rep_uint322 @ ( numeral_numeral @ uint32 @ N2 ) )
      = ( numeral_numeral @ ( word @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ numeral_num1 ) ) ) ) ) ) @ N2 ) ) ).

% uint32.word_of_numeral
thf(fact_7936_size__uint32_Orep__eq,axiom,
    ( ( size_size @ uint32 )
    = ( ^ [X: uint32] : ( size_size @ ( word @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ numeral_num1 ) ) ) ) ) ) @ ( rep_uint322 @ X ) ) ) ) ).

% size_uint32.rep_eq
thf(fact_7937_uint32_Osize__eq__word__of,axiom,
    ( ( size_size @ uint32 )
    = ( ^ [P5: uint32] : ( size_size @ ( word @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ numeral_num1 ) ) ) ) ) ) @ ( rep_uint322 @ P5 ) ) ) ) ).

% uint32.size_eq_word_of
thf(fact_7938_uint32_Oeven__iff__word__of,axiom,
    ! [P4: uint32] :
      ( ( dvd_dvd @ uint32 @ ( numeral_numeral @ uint32 @ ( bit0 @ one2 ) ) @ P4 )
      = ( dvd_dvd @ ( word @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ numeral_num1 ) ) ) ) ) ) @ ( numeral_numeral @ ( word @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ numeral_num1 ) ) ) ) ) ) @ ( bit0 @ one2 ) ) @ ( rep_uint322 @ P4 ) ) ) ).

% uint32.even_iff_word_of
thf(fact_7939_times__uint32_Orep__eq,axiom,
    ! [X2: uint32,Xa: uint32] :
      ( ( rep_uint322 @ ( times_times @ uint32 @ X2 @ Xa ) )
      = ( times_times @ ( word @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ numeral_num1 ) ) ) ) ) ) @ ( rep_uint322 @ X2 ) @ ( rep_uint322 @ Xa ) ) ) ).

% times_uint32.rep_eq
thf(fact_7940_uint32_Oword__of__mult,axiom,
    ! [P4: uint32,Q2: uint32] :
      ( ( rep_uint322 @ ( times_times @ uint32 @ P4 @ Q2 ) )
      = ( times_times @ ( word @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ numeral_num1 ) ) ) ) ) ) @ ( rep_uint322 @ P4 ) @ ( rep_uint322 @ Q2 ) ) ) ).

% uint32.word_of_mult
thf(fact_7941_divide__uint32_Orep__eq,axiom,
    ! [X2: uint32,Xa: uint32] :
      ( ( rep_uint322 @ ( divide_divide @ uint32 @ X2 @ Xa ) )
      = ( divide_divide @ ( word @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ numeral_num1 ) ) ) ) ) ) @ ( rep_uint322 @ X2 ) @ ( rep_uint322 @ Xa ) ) ) ).

% divide_uint32.rep_eq
thf(fact_7942_uint32_Oword__of__div,axiom,
    ! [P4: uint32,Q2: uint32] :
      ( ( rep_uint322 @ ( divide_divide @ uint32 @ P4 @ Q2 ) )
      = ( divide_divide @ ( word @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ numeral_num1 ) ) ) ) ) ) @ ( rep_uint322 @ P4 ) @ ( rep_uint322 @ Q2 ) ) ) ).

% uint32.word_of_div
thf(fact_7943_one__uint32_Orep__eq,axiom,
    ( ( rep_uint322 @ ( one_one @ uint32 ) )
    = ( one_one @ ( word @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ numeral_num1 ) ) ) ) ) ) ) ) ).

% one_uint32.rep_eq
thf(fact_7944_uint32_Oword__of__diff,axiom,
    ! [P4: uint32,Q2: uint32] :
      ( ( rep_uint322 @ ( minus_minus @ uint32 @ P4 @ Q2 ) )
      = ( minus_minus @ ( word @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ numeral_num1 ) ) ) ) ) ) @ ( rep_uint322 @ P4 ) @ ( rep_uint322 @ Q2 ) ) ) ).

% uint32.word_of_diff
thf(fact_7945_minus__uint32_Orep__eq,axiom,
    ! [X2: uint32,Xa: uint32] :
      ( ( rep_uint322 @ ( minus_minus @ uint32 @ X2 @ Xa ) )
      = ( minus_minus @ ( word @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ numeral_num1 ) ) ) ) ) ) @ ( rep_uint322 @ X2 ) @ ( rep_uint322 @ Xa ) ) ) ).

% minus_uint32.rep_eq
thf(fact_7946_plus__uint32_Orep__eq,axiom,
    ! [X2: uint32,Xa: uint32] :
      ( ( rep_uint322 @ ( plus_plus @ uint32 @ X2 @ Xa ) )
      = ( plus_plus @ ( word @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ numeral_num1 ) ) ) ) ) ) @ ( rep_uint322 @ X2 ) @ ( rep_uint322 @ Xa ) ) ) ).

% plus_uint32.rep_eq
thf(fact_7947_uint32_Oword__of__add,axiom,
    ! [P4: uint32,Q2: uint32] :
      ( ( rep_uint322 @ ( plus_plus @ uint32 @ P4 @ Q2 ) )
      = ( plus_plus @ ( word @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ numeral_num1 ) ) ) ) ) ) @ ( rep_uint322 @ P4 ) @ ( rep_uint322 @ Q2 ) ) ) ).

% uint32.word_of_add
thf(fact_7948_uint32_Oword__of__power,axiom,
    ! [P4: uint32,N2: nat] :
      ( ( rep_uint322 @ ( power_power @ uint32 @ P4 @ N2 ) )
      = ( power_power @ ( word @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ numeral_num1 ) ) ) ) ) ) @ ( rep_uint322 @ P4 ) @ N2 ) ) ).

% uint32.word_of_power
thf(fact_7949_less__eq__uint32_Orep__eq,axiom,
    ( ( ord_less_eq @ uint32 )
    = ( ^ [X: uint32,Xa4: uint32] : ( ord_less_eq @ ( word @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ numeral_num1 ) ) ) ) ) ) @ ( rep_uint322 @ X ) @ ( rep_uint322 @ Xa4 ) ) ) ) ).

% less_eq_uint32.rep_eq
thf(fact_7950_uint32_Oless__eq__iff__word__of,axiom,
    ( ( ord_less_eq @ uint32 )
    = ( ^ [P5: uint32,Q5: uint32] : ( ord_less_eq @ ( word @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ numeral_num1 ) ) ) ) ) ) @ ( rep_uint322 @ P5 ) @ ( rep_uint322 @ Q5 ) ) ) ) ).

% uint32.less_eq_iff_word_of
thf(fact_7951_uint32__test__bit__code,axiom,
    ( uint32_test_bit
    = ( ^ [W2: uint32,N: code_integer] :
          ( ( ( ( ord_less @ code_integer @ N @ ( zero_zero @ code_integer ) )
              | ( ord_less @ code_integer @ ( numeral_numeral @ code_integer @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one2 ) ) ) ) ) @ N ) )
           => ( undefined @ ( ( uint32 > nat > $o ) > uint32 > code_integer > $o ) @ ( bit_se5641148757651400278ts_bit @ uint32 ) @ W2 @ N ) )
          & ( ~ ( ( ord_less @ code_integer @ N @ ( zero_zero @ code_integer ) )
                | ( ord_less @ code_integer @ ( numeral_numeral @ code_integer @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one2 ) ) ) ) ) @ N ) )
           => ( bit_se5641148757651400278ts_bit @ ( word @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ numeral_num1 ) ) ) ) ) ) @ ( rep_uint322 @ W2 ) @ ( code_nat_of_integer @ N ) ) ) ) ) ) ).

% uint32_test_bit_code
thf(fact_7952_uint32__set__bit__code,axiom,
    ! [N2: code_integer,W: uint32,B2: $o] :
      ( ( ( ( ord_less @ code_integer @ N2 @ ( zero_zero @ code_integer ) )
          | ( ord_less @ code_integer @ ( numeral_numeral @ code_integer @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one2 ) ) ) ) ) @ N2 ) )
       => ( ( rep_uint322 @ ( uint32_set_bit @ W @ N2 @ B2 ) )
          = ( rep_uint322 @ ( undefined @ ( ( uint32 > nat > $o > uint32 ) > uint32 > code_integer > $o > uint32 ) @ ( generi7602027413899671122et_bit @ uint32 ) @ W @ N2 @ B2 ) ) ) )
      & ( ~ ( ( ord_less @ code_integer @ N2 @ ( zero_zero @ code_integer ) )
            | ( ord_less @ code_integer @ ( numeral_numeral @ code_integer @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one2 ) ) ) ) ) @ N2 ) )
       => ( ( rep_uint322 @ ( uint32_set_bit @ W @ N2 @ B2 ) )
          = ( generi7602027413899671122et_bit @ ( word @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ numeral_num1 ) ) ) ) ) ) @ ( rep_uint322 @ W ) @ ( code_nat_of_integer @ N2 ) @ B2 ) ) ) ) ).

% uint32_set_bit_code
thf(fact_7953_lift__def,axiom,
    ! [A: $tType,B: $tType] :
      ( ( heap_Time_lift @ A @ B )
      = ( comp @ B @ ( heap_Time_Heap @ B ) @ A @ ( heap_Time_return @ B ) ) ) ).

% lift_def
thf(fact_7954_set__bit__uint32__code,axiom,
    ( ( generi7602027413899671122et_bit @ uint32 )
    = ( ^ [X: uint32,N: nat,B5: $o] : ( if @ uint32 @ ( ord_less @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) ) @ ( uint32_set_bit @ X @ ( code_integer_of_nat @ N ) @ B5 ) @ X ) ) ) ).

% set_bit_uint32_code
thf(fact_7955_lift__collapse,axiom,
    ! [A: $tType,B: $tType] :
      ( ( heap_Time_lift @ B @ A )
      = ( ^ [F6: B > A,X: B] : ( heap_Time_return @ A @ ( F6 @ X ) ) ) ) ).

% lift_collapse
thf(fact_7956_integer__of__nat__1,axiom,
    ( ( code_integer_of_nat @ ( one_one @ nat ) )
    = ( one_one @ code_integer ) ) ).

% integer_of_nat_1
thf(fact_7957_one__uint32_Orsp,axiom,
    ( ( one_one @ ( word @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ numeral_num1 ) ) ) ) ) ) )
    = ( one_one @ ( word @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ numeral_num1 ) ) ) ) ) ) ) ) ).

% one_uint32.rsp
thf(fact_7958_integer__of__nat__numeral,axiom,
    ! [N2: num] :
      ( ( code_integer_of_nat @ ( numeral_numeral @ nat @ N2 ) )
      = ( numeral_numeral @ code_integer @ N2 ) ) ).

% integer_of_nat_numeral
thf(fact_7959_bind__lift,axiom,
    ! [A: $tType,B: $tType,F: heap_Time_Heap @ B,G: B > A] :
      ( ( heap_Time_bind @ B @ A @ F @ ( heap_Time_lift @ B @ A @ G ) )
      = ( heap_Time_bind @ B @ A @ F
        @ ^ [X: B] : ( heap_Time_return @ A @ ( G @ X ) ) ) ) ).

% bind_lift
thf(fact_7960_test__bit__uint32__code,axiom,
    ( ( bit_se5641148757651400278ts_bit @ uint32 )
    = ( ^ [X: uint32,N: nat] :
          ( ( ord_less @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) )
          & ( uint32_test_bit @ X @ ( code_integer_of_nat @ N ) ) ) ) ) ).

% test_bit_uint32_code
thf(fact_7961_shiftl__uint32__code,axiom,
    ( ( bit_se4730199178511100633sh_bit @ uint32 )
    = ( ^ [N: nat,X: uint32] : ( if @ uint32 @ ( ord_less @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) ) @ ( uint32_shiftl @ X @ ( code_integer_of_nat @ N ) ) @ ( zero_zero @ uint32 ) ) ) ) ).

% shiftl_uint32_code
thf(fact_7962_shiftr__uint32__code,axiom,
    ( ( bit_se4197421643247451524op_bit @ uint32 )
    = ( ^ [N: nat,X: uint32] : ( if @ uint32 @ ( ord_less @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) ) @ ( uint32_shiftr @ X @ ( code_integer_of_nat @ N ) ) @ ( zero_zero @ uint32 ) ) ) ) ).

% shiftr_uint32_code
thf(fact_7963_uint32__divmod__code,axiom,
    ( uint32_divmod
    = ( ^ [X: uint32,Y: uint32] :
          ( if @ ( product_prod @ uint32 @ uint32 ) @ ( ord_less_eq @ uint32 @ ( numeral_numeral @ uint32 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ Y ) @ ( if @ ( product_prod @ uint32 @ uint32 ) @ ( ord_less @ uint32 @ X @ Y ) @ ( product_Pair @ uint32 @ uint32 @ ( zero_zero @ uint32 ) @ X ) @ ( product_Pair @ uint32 @ uint32 @ ( one_one @ uint32 ) @ ( minus_minus @ uint32 @ X @ Y ) ) )
          @ ( if @ ( product_prod @ uint32 @ uint32 )
            @ ( Y
              = ( zero_zero @ uint32 ) )
            @ ( product_Pair @ uint32 @ uint32 @ ( div0_uint32 @ X ) @ ( mod0_uint32 @ X ) )
            @ ( if @ ( product_prod @ uint32 @ uint32 ) @ ( ord_less_eq @ uint32 @ Y @ ( minus_minus @ uint32 @ X @ ( times_times @ uint32 @ ( bit_se4730199178511100633sh_bit @ uint32 @ ( one_one @ nat ) @ ( uint32_sdiv @ ( bit_se4197421643247451524op_bit @ uint32 @ ( one_one @ nat ) @ X ) @ Y ) ) @ Y ) ) ) @ ( product_Pair @ uint32 @ uint32 @ ( plus_plus @ uint32 @ ( bit_se4730199178511100633sh_bit @ uint32 @ ( one_one @ nat ) @ ( uint32_sdiv @ ( bit_se4197421643247451524op_bit @ uint32 @ ( one_one @ nat ) @ X ) @ Y ) ) @ ( one_one @ uint32 ) ) @ ( minus_minus @ uint32 @ ( minus_minus @ uint32 @ X @ ( times_times @ uint32 @ ( bit_se4730199178511100633sh_bit @ uint32 @ ( one_one @ nat ) @ ( uint32_sdiv @ ( bit_se4197421643247451524op_bit @ uint32 @ ( one_one @ nat ) @ X ) @ Y ) ) @ Y ) ) @ Y ) ) @ ( product_Pair @ uint32 @ uint32 @ ( bit_se4730199178511100633sh_bit @ uint32 @ ( one_one @ nat ) @ ( uint32_sdiv @ ( bit_se4197421643247451524op_bit @ uint32 @ ( one_one @ nat ) @ X ) @ Y ) ) @ ( minus_minus @ uint32 @ X @ ( times_times @ uint32 @ ( bit_se4730199178511100633sh_bit @ uint32 @ ( one_one @ nat ) @ ( uint32_sdiv @ ( bit_se4197421643247451524op_bit @ uint32 @ ( one_one @ nat ) @ X ) @ Y ) ) @ Y ) ) ) ) ) ) ) ) ).

% uint32_divmod_code
thf(fact_7964_uint32_Oset__bits__aux__code,axiom,
    ( set_bits_aux_uint32
    = ( ^ [F6: nat > $o,N: nat,W2: uint32] :
          ( if @ uint32
          @ ( N
            = ( zero_zero @ nat ) )
          @ W2
          @ ( set_bits_aux_uint32 @ F6 @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ ( bit_se1065995026697491101ons_or @ uint32 @ ( bit_se4730199178511100633sh_bit @ uint32 @ ( one_one @ nat ) @ W2 ) @ ( if @ uint32 @ ( F6 @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) @ ( one_one @ uint32 ) @ ( zero_zero @ uint32 ) ) ) ) ) ) ) ).

% uint32.set_bits_aux_code
thf(fact_7965_uint32__divmod__def,axiom,
    ( uint32_divmod
    = ( ^ [X: uint32,Y: uint32] :
          ( if @ ( product_prod @ uint32 @ uint32 )
          @ ( Y
            = ( zero_zero @ uint32 ) )
          @ ( product_Pair @ uint32 @ uint32 @ ( undefined @ ( ( uint32 > uint32 > uint32 ) > uint32 > uint32 > uint32 ) @ ( divide_divide @ uint32 ) @ X @ ( zero_zero @ uint32 ) ) @ ( undefined @ ( ( uint32 > uint32 > uint32 ) > uint32 > uint32 > uint32 ) @ ( modulo_modulo @ uint32 ) @ X @ ( zero_zero @ uint32 ) ) )
          @ ( product_Pair @ uint32 @ uint32 @ ( divide_divide @ uint32 @ X @ Y ) @ ( modulo_modulo @ uint32 @ X @ Y ) ) ) ) ) ).

% uint32_divmod_def
thf(fact_7966_uint32_Oset__bits__code,axiom,
    ( ( bit_bi4170147762399595738t_bits @ uint32 )
    = ( ^ [P3: nat > $o] : ( set_bits_aux_uint32 @ P3 @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) @ ( zero_zero @ uint32 ) ) ) ) ).

% uint32.set_bits_code
thf(fact_7967_div0__uint32__def,axiom,
    ( div0_uint32
    = ( ^ [X: uint32] : ( undefined @ ( ( uint32 > uint32 > uint32 ) > uint32 > uint32 > uint32 ) @ ( divide_divide @ uint32 ) @ X @ ( zero_zero @ uint32 ) ) ) ) ).

% div0_uint32_def
thf(fact_7968_uint32__sdiv__code,axiom,
    ! [Y2: uint32,X2: uint32] :
      ( ( ( Y2
          = ( zero_zero @ uint32 ) )
       => ( ( rep_uint322 @ ( uint32_sdiv @ X2 @ Y2 ) )
          = ( rep_uint322 @ ( undefined @ ( ( uint32 > uint32 > uint32 ) > uint32 > uint32 > uint32 ) @ ( divide_divide @ uint32 ) @ X2 @ ( zero_zero @ uint32 ) ) ) ) )
      & ( ( Y2
         != ( zero_zero @ uint32 ) )
       => ( ( rep_uint322 @ ( uint32_sdiv @ X2 @ Y2 ) )
          = ( signed7115095781618012415divide @ ( word @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ numeral_num1 ) ) ) ) ) ) @ ( rep_uint322 @ X2 ) @ ( rep_uint322 @ Y2 ) ) ) ) ) ).

% uint32_sdiv_code
thf(fact_7969_div__uint32__code,axiom,
    ( ( divide_divide @ uint32 )
    = ( ^ [X: uint32,Y: uint32] :
          ( if @ uint32
          @ ( Y
            = ( zero_zero @ uint32 ) )
          @ ( zero_zero @ uint32 )
          @ ( uint32_div @ X @ Y ) ) ) ) ).

% div_uint32_code
thf(fact_7970_sshiftr__uint32__code,axiom,
    ( signed489701013188660438uint32
    = ( ^ [N: nat,X: uint32] : ( if @ uint32 @ ( ord_less @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) ) @ ( uint32_sshiftr @ X @ ( code_integer_of_nat @ N ) ) @ ( if @ uint32 @ ( bit_se5641148757651400278ts_bit @ uint32 @ X @ ( numeral_numeral @ nat @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one2 ) ) ) ) ) ) @ ( uminus_uminus @ uint32 @ ( one_one @ uint32 ) ) @ ( zero_zero @ uint32 ) ) ) ) ) ).

% sshiftr_uint32_code
thf(fact_7971_uint32__div__def,axiom,
    ( uint32_div
    = ( ^ [X: uint32,Y: uint32] : ( product_fst @ uint32 @ uint32 @ ( uint32_divmod @ X @ Y ) ) ) ) ).

% uint32_div_def
thf(fact_7972_uint32__sshiftr__def,axiom,
    ( uint32_sshiftr
    = ( ^ [X: uint32,N: code_integer] :
          ( if @ uint32
          @ ( ( ord_less @ code_integer @ N @ ( zero_zero @ code_integer ) )
            | ( ord_less_eq @ code_integer @ ( numeral_numeral @ code_integer @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) @ N ) )
          @ ( undefined @ ( ( nat > uint32 > uint32 ) > code_integer > uint32 > uint32 ) @ signed489701013188660438uint32 @ N @ X )
          @ ( signed489701013188660438uint32 @ ( code_nat_of_integer @ N ) @ X ) ) ) ) ).

% uint32_sshiftr_def
thf(fact_7973_uint32__sshiftr__code,axiom,
    ! [N2: code_integer,W: uint32] :
      ( ( ( ( ord_less @ code_integer @ N2 @ ( zero_zero @ code_integer ) )
          | ( ord_less_eq @ code_integer @ ( numeral_numeral @ code_integer @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) @ N2 ) )
       => ( ( rep_uint322 @ ( uint32_sshiftr @ W @ N2 ) )
          = ( rep_uint322 @ ( undefined @ ( ( nat > uint32 > uint32 ) > code_integer > uint32 > uint32 ) @ signed489701013188660438uint32 @ N2 @ W ) ) ) )
      & ( ~ ( ( ord_less @ code_integer @ N2 @ ( zero_zero @ code_integer ) )
            | ( ord_less_eq @ code_integer @ ( numeral_numeral @ code_integer @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) @ N2 ) )
       => ( ( rep_uint322 @ ( uint32_sshiftr @ W @ N2 ) )
          = ( signed_drop_bit @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ numeral_num1 ) ) ) ) ) @ ( code_nat_of_integer @ N2 ) @ ( rep_uint322 @ W ) ) ) ) ) ).

% uint32_sshiftr_code
thf(fact_7974_integer__of__uint32__code,axiom,
    ( integer_of_uint32
    = ( ^ [N: uint32] : ( if @ code_integer @ ( bit_se5641148757651400278ts_bit @ uint32 @ N @ ( numeral_numeral @ nat @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one2 ) ) ) ) ) ) @ ( bit_se1065995026697491101ons_or @ code_integer @ ( intege5370686899274169573signed @ ( bit_se5824344872417868541ns_and @ uint32 @ N @ ( numeral_numeral @ uint32 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( numeral_numeral @ code_integer @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( intege5370686899274169573signed @ N ) ) ) ) ).

% integer_of_uint32_code
thf(fact_7975_uint32__sdiv__def,axiom,
    ( uint32_sdiv
    = ( ^ [X: uint32,Y: uint32] :
          ( if @ uint32
          @ ( Y
            = ( zero_zero @ uint32 ) )
          @ ( undefined @ ( ( uint32 > uint32 > uint32 ) > uint32 > uint32 > uint32 ) @ ( divide_divide @ uint32 ) @ X @ ( zero_zero @ uint32 ) )
          @ ( abs_uint32 @ ( signed7115095781618012415divide @ ( word @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ numeral_num1 ) ) ) ) ) ) @ ( rep_uint322 @ X ) @ ( rep_uint322 @ Y ) ) ) ) ) ) ).

% uint32_sdiv_def
thf(fact_7976_less__eq__uint32_Oabs__eq,axiom,
    ! [Xa: word @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ numeral_num1 ) ) ) ) ),X2: word @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ numeral_num1 ) ) ) ) )] :
      ( ( ord_less_eq @ uint32 @ ( abs_uint32 @ Xa ) @ ( abs_uint32 @ X2 ) )
      = ( ord_less_eq @ ( word @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ numeral_num1 ) ) ) ) ) ) @ Xa @ X2 ) ) ).

% less_eq_uint32.abs_eq
thf(fact_7977_plus__uint32_Oabs__eq,axiom,
    ! [Xa: word @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ numeral_num1 ) ) ) ) ),X2: word @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ numeral_num1 ) ) ) ) )] :
      ( ( plus_plus @ uint32 @ ( abs_uint32 @ Xa ) @ ( abs_uint32 @ X2 ) )
      = ( abs_uint32 @ ( plus_plus @ ( word @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ numeral_num1 ) ) ) ) ) ) @ Xa @ X2 ) ) ) ).

% plus_uint32.abs_eq
thf(fact_7978_minus__uint32_Oabs__eq,axiom,
    ! [Xa: word @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ numeral_num1 ) ) ) ) ),X2: word @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ numeral_num1 ) ) ) ) )] :
      ( ( minus_minus @ uint32 @ ( abs_uint32 @ Xa ) @ ( abs_uint32 @ X2 ) )
      = ( abs_uint32 @ ( minus_minus @ ( word @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ numeral_num1 ) ) ) ) ) ) @ Xa @ X2 ) ) ) ).

% minus_uint32.abs_eq
thf(fact_7979_one__uint32__def,axiom,
    ( ( one_one @ uint32 )
    = ( abs_uint32 @ ( one_one @ ( word @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ numeral_num1 ) ) ) ) ) ) ) ) ) ).

% one_uint32_def
thf(fact_7980_divide__uint32_Oabs__eq,axiom,
    ! [Xa: word @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ numeral_num1 ) ) ) ) ),X2: word @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ numeral_num1 ) ) ) ) )] :
      ( ( divide_divide @ uint32 @ ( abs_uint32 @ Xa ) @ ( abs_uint32 @ X2 ) )
      = ( abs_uint32 @ ( divide_divide @ ( word @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ numeral_num1 ) ) ) ) ) ) @ Xa @ X2 ) ) ) ).

% divide_uint32.abs_eq
thf(fact_7981_times__uint32_Oabs__eq,axiom,
    ! [Xa: word @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ numeral_num1 ) ) ) ) ),X2: word @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ numeral_num1 ) ) ) ) )] :
      ( ( times_times @ uint32 @ ( abs_uint32 @ Xa ) @ ( abs_uint32 @ X2 ) )
      = ( abs_uint32 @ ( times_times @ ( word @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ numeral_num1 ) ) ) ) ) ) @ Xa @ X2 ) ) ) ).

% times_uint32.abs_eq
thf(fact_7982_uint32_Oof__word__numeral,axiom,
    ! [N2: num] :
      ( ( abs_uint32 @ ( numeral_numeral @ ( word @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ numeral_num1 ) ) ) ) ) ) @ N2 ) )
      = ( numeral_numeral @ uint32 @ N2 ) ) ).

% uint32.of_word_numeral
thf(fact_7983_size__uint32_Oabs__eq,axiom,
    ! [X2: word @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ numeral_num1 ) ) ) ) )] :
      ( ( size_size @ uint32 @ ( abs_uint32 @ X2 ) )
      = ( size_size @ ( word @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ numeral_num1 ) ) ) ) ) ) @ X2 ) ) ).

% size_uint32.abs_eq
thf(fact_7984_integer__of__uint32__signed__def,axiom,
    ( intege5370686899274169573signed
    = ( ^ [N: uint32] : ( if @ code_integer @ ( bit_se5641148757651400278ts_bit @ uint32 @ N @ ( numeral_numeral @ nat @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one2 ) ) ) ) ) ) @ ( undefined @ ( ( uint32 > code_integer ) > uint32 > code_integer ) @ integer_of_uint32 @ N ) @ ( integer_of_uint32 @ N ) ) ) ) ).

% integer_of_uint32_signed_def
thf(fact_7985_integer__of__uint32__signed__code,axiom,
    ( intege5370686899274169573signed
    = ( ^ [N: uint32] : ( if @ code_integer @ ( bit_se5641148757651400278ts_bit @ uint32 @ N @ ( numeral_numeral @ nat @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one2 ) ) ) ) ) ) @ ( undefined @ ( ( uint32 > code_integer ) > uint32 > code_integer ) @ integer_of_uint32 @ N ) @ ( code_integer_of_int @ ( semiring_1_unsigned @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ numeral_num1 ) ) ) ) ) @ int @ ( rep_uint32 @ N ) ) ) ) ) ) ).

% integer_of_uint32_signed_code
thf(fact_7986_divmod__integer__eq__cases,axiom,
    ( code_divmod_integer
    = ( ^ [K4: code_integer,L2: code_integer] :
          ( if @ ( product_prod @ code_integer @ code_integer )
          @ ( K4
            = ( zero_zero @ code_integer ) )
          @ ( product_Pair @ code_integer @ code_integer @ ( zero_zero @ code_integer ) @ ( zero_zero @ code_integer ) )
          @ ( if @ ( product_prod @ code_integer @ code_integer )
            @ ( L2
              = ( zero_zero @ code_integer ) )
            @ ( product_Pair @ code_integer @ code_integer @ ( zero_zero @ code_integer ) @ K4 )
            @ ( comp @ code_integer @ ( ( product_prod @ code_integer @ code_integer ) > ( product_prod @ code_integer @ code_integer ) ) @ code_integer @ ( comp @ ( code_integer > code_integer ) @ ( ( product_prod @ code_integer @ code_integer ) > ( product_prod @ code_integer @ code_integer ) ) @ code_integer @ ( product_apsnd @ code_integer @ code_integer @ code_integer ) @ ( times_times @ code_integer ) ) @ ( sgn_sgn @ code_integer ) @ L2
              @ ( if @ ( product_prod @ code_integer @ code_integer )
                @ ( ( sgn_sgn @ code_integer @ K4 )
                  = ( sgn_sgn @ code_integer @ L2 ) )
                @ ( code_divmod_abs @ K4 @ L2 )
                @ ( product_case_prod @ code_integer @ code_integer @ ( product_prod @ code_integer @ code_integer )
                  @ ^ [R: code_integer,S2: code_integer] :
                      ( if @ ( product_prod @ code_integer @ code_integer )
                      @ ( S2
                        = ( zero_zero @ code_integer ) )
                      @ ( product_Pair @ code_integer @ code_integer @ ( uminus_uminus @ code_integer @ R ) @ ( zero_zero @ code_integer ) )
                      @ ( product_Pair @ code_integer @ code_integer @ ( minus_minus @ code_integer @ ( uminus_uminus @ code_integer @ R ) @ ( one_one @ code_integer ) ) @ ( minus_minus @ code_integer @ ( abs_abs @ code_integer @ L2 ) @ S2 ) ) )
                  @ ( code_divmod_abs @ K4 @ L2 ) ) ) ) ) ) ) ) ).

% divmod_integer_eq_cases
thf(fact_7987_apsnd__conv,axiom,
    ! [A: $tType,B: $tType,C: $tType,F: C > B,X2: A,Y2: C] :
      ( ( product_apsnd @ C @ B @ A @ F @ ( product_Pair @ A @ C @ X2 @ Y2 ) )
      = ( product_Pair @ A @ B @ X2 @ ( F @ Y2 ) ) ) ).

% apsnd_conv
thf(fact_7988_fst__apsnd,axiom,
    ! [B: $tType,C: $tType,A: $tType,F: C > B,X2: product_prod @ A @ C] :
      ( ( product_fst @ A @ B @ ( product_apsnd @ C @ B @ A @ F @ X2 ) )
      = ( product_fst @ A @ C @ X2 ) ) ).

% fst_apsnd
thf(fact_7989_apsnd__eq__conv,axiom,
    ! [B: $tType,C: $tType,A: $tType,F: C > B,X2: product_prod @ A @ C,G: C > B] :
      ( ( ( product_apsnd @ C @ B @ A @ F @ X2 )
        = ( product_apsnd @ C @ B @ A @ G @ X2 ) )
      = ( ( F @ ( product_snd @ A @ C @ X2 ) )
        = ( G @ ( product_snd @ A @ C @ X2 ) ) ) ) ).

% apsnd_eq_conv
thf(fact_7990_snd__apsnd,axiom,
    ! [A: $tType,C: $tType,B: $tType,F: C > A,X2: product_prod @ B @ C] :
      ( ( product_snd @ B @ A @ ( product_apsnd @ C @ A @ B @ F @ X2 ) )
      = ( F @ ( product_snd @ B @ C @ X2 ) ) ) ).

% snd_apsnd
thf(fact_7991_fst__comp__apsnd,axiom,
    ! [C: $tType,B: $tType,A: $tType,F: B > C] :
      ( ( comp @ ( product_prod @ A @ C ) @ A @ ( product_prod @ A @ B ) @ ( product_fst @ A @ C ) @ ( product_apsnd @ B @ C @ A @ F ) )
      = ( product_fst @ A @ B ) ) ).

% fst_comp_apsnd
thf(fact_7992_snd__comp__apsnd,axiom,
    ! [C: $tType,B: $tType,A: $tType,F: B > C] :
      ( ( comp @ ( product_prod @ A @ C ) @ C @ ( product_prod @ A @ B ) @ ( product_snd @ A @ C ) @ ( product_apsnd @ B @ C @ A @ F ) )
      = ( comp @ B @ C @ ( product_prod @ A @ B ) @ F @ ( product_snd @ A @ B ) ) ) ).

% snd_comp_apsnd
thf(fact_7993_inj__apsnd,axiom,
    ! [A: $tType,C: $tType,B: $tType,F: B > C] :
      ( ( inj_on @ ( product_prod @ A @ B ) @ ( product_prod @ A @ C ) @ ( product_apsnd @ B @ C @ A @ F ) @ ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) )
      = ( inj_on @ B @ C @ F @ ( top_top @ ( set @ B ) ) ) ) ).

% inj_apsnd
thf(fact_7994_fst__divmod__abs,axiom,
    ! [K: code_integer,L: code_integer] :
      ( ( product_fst @ code_integer @ code_integer @ ( code_divmod_abs @ K @ L ) )
      = ( divide_divide @ code_integer @ ( abs_abs @ code_integer @ K ) @ ( abs_abs @ code_integer @ L ) ) ) ).

% fst_divmod_abs
thf(fact_7995_snd__divmod__abs,axiom,
    ! [K: code_integer,L: code_integer] :
      ( ( product_snd @ code_integer @ code_integer @ ( code_divmod_abs @ K @ L ) )
      = ( modulo_modulo @ code_integer @ ( abs_abs @ code_integer @ K ) @ ( abs_abs @ code_integer @ L ) ) ) ).

% snd_divmod_abs
thf(fact_7996_apsnd__compose,axiom,
    ! [C: $tType,B: $tType,D: $tType,A: $tType,F: C > B,G: D > C,X2: product_prod @ A @ D] :
      ( ( product_apsnd @ C @ B @ A @ F @ ( product_apsnd @ D @ C @ A @ G @ X2 ) )
      = ( product_apsnd @ D @ B @ A @ ( comp @ C @ B @ D @ F @ G ) @ X2 ) ) ).

% apsnd_compose
thf(fact_7997_divmod__abs__code_I6_J,axiom,
    ! [J: code_integer] :
      ( ( code_divmod_abs @ ( zero_zero @ code_integer ) @ J )
      = ( product_Pair @ code_integer @ code_integer @ ( zero_zero @ code_integer ) @ ( zero_zero @ code_integer ) ) ) ).

% divmod_abs_code(6)
thf(fact_7998_divmod__abs__code_I5_J,axiom,
    ! [J: code_integer] :
      ( ( code_divmod_abs @ J @ ( zero_zero @ code_integer ) )
      = ( product_Pair @ code_integer @ code_integer @ ( zero_zero @ code_integer ) @ ( abs_abs @ code_integer @ J ) ) ) ).

% divmod_abs_code(5)
thf(fact_7999_Rep__uint32_H__code,axiom,
    ( rep_uint32
    = ( ^ [X: uint32] : ( bit_bi4170147762399595738t_bits @ ( word @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ ( numeral_bit0 @ numeral_num1 ) ) ) ) ) ) @ ( bit_se5641148757651400278ts_bit @ uint32 @ X ) ) ) ) ).

% Rep_uint32'_code
thf(fact_8000_divmod__abs__def,axiom,
    ( code_divmod_abs
    = ( ^ [K4: code_integer,L2: code_integer] : ( product_Pair @ code_integer @ code_integer @ ( divide_divide @ code_integer @ ( abs_abs @ code_integer @ K4 ) @ ( abs_abs @ code_integer @ L2 ) ) @ ( modulo_modulo @ code_integer @ ( abs_abs @ code_integer @ K4 ) @ ( abs_abs @ code_integer @ L2 ) ) ) ) ) ).

% divmod_abs_def
thf(fact_8001_divmod__integer__code,axiom,
    ( code_divmod_integer
    = ( ^ [K4: code_integer,L2: code_integer] :
          ( if @ ( product_prod @ code_integer @ code_integer )
          @ ( K4
            = ( zero_zero @ code_integer ) )
          @ ( product_Pair @ code_integer @ code_integer @ ( zero_zero @ code_integer ) @ ( zero_zero @ code_integer ) )
          @ ( if @ ( product_prod @ code_integer @ code_integer ) @ ( ord_less @ code_integer @ ( zero_zero @ code_integer ) @ L2 )
            @ ( if @ ( product_prod @ code_integer @ code_integer ) @ ( ord_less @ code_integer @ ( zero_zero @ code_integer ) @ K4 ) @ ( code_divmod_abs @ K4 @ L2 )
              @ ( product_case_prod @ code_integer @ code_integer @ ( product_prod @ code_integer @ code_integer )
                @ ^ [R: code_integer,S2: code_integer] :
                    ( if @ ( product_prod @ code_integer @ code_integer )
                    @ ( S2
                      = ( zero_zero @ code_integer ) )
                    @ ( product_Pair @ code_integer @ code_integer @ ( uminus_uminus @ code_integer @ R ) @ ( zero_zero @ code_integer ) )
                    @ ( product_Pair @ code_integer @ code_integer @ ( minus_minus @ code_integer @ ( uminus_uminus @ code_integer @ R ) @ ( one_one @ code_integer ) ) @ ( minus_minus @ code_integer @ L2 @ S2 ) ) )
                @ ( code_divmod_abs @ K4 @ L2 ) ) )
            @ ( if @ ( product_prod @ code_integer @ code_integer )
              @ ( L2
                = ( zero_zero @ code_integer ) )
              @ ( product_Pair @ code_integer @ code_integer @ ( zero_zero @ code_integer ) @ K4 )
              @ ( product_apsnd @ code_integer @ code_integer @ code_integer @ ( uminus_uminus @ code_integer )
                @ ( if @ ( product_prod @ code_integer @ code_integer ) @ ( ord_less @ code_integer @ K4 @ ( zero_zero @ code_integer ) ) @ ( code_divmod_abs @ K4 @ L2 )
                  @ ( product_case_prod @ code_integer @ code_integer @ ( product_prod @ code_integer @ code_integer )
                    @ ^ [R: code_integer,S2: code_integer] :
                        ( if @ ( product_prod @ code_integer @ code_integer )
                        @ ( S2
                          = ( zero_zero @ code_integer ) )
                        @ ( product_Pair @ code_integer @ code_integer @ ( uminus_uminus @ code_integer @ R ) @ ( zero_zero @ code_integer ) )
                        @ ( product_Pair @ code_integer @ code_integer @ ( minus_minus @ code_integer @ ( uminus_uminus @ code_integer @ R ) @ ( one_one @ code_integer ) ) @ ( minus_minus @ code_integer @ ( uminus_uminus @ code_integer @ L2 ) @ S2 ) ) )
                    @ ( code_divmod_abs @ K4 @ L2 ) ) ) ) ) ) ) ) ) ).

% divmod_integer_code
thf(fact_8002_bit__cut__integer__code,axiom,
    ( code_bit_cut_integer
    = ( ^ [K4: code_integer] :
          ( if @ ( product_prod @ code_integer @ $o )
          @ ( K4
            = ( zero_zero @ code_integer ) )
          @ ( product_Pair @ code_integer @ $o @ ( zero_zero @ code_integer ) @ $false )
          @ ( product_case_prod @ code_integer @ code_integer @ ( product_prod @ code_integer @ $o )
            @ ^ [R: code_integer,S2: code_integer] :
                ( product_Pair @ code_integer @ $o @ ( if @ code_integer @ ( ord_less @ code_integer @ ( zero_zero @ code_integer ) @ K4 ) @ R @ ( minus_minus @ code_integer @ ( uminus_uminus @ code_integer @ R ) @ S2 ) )
                @ ( S2
                  = ( one_one @ code_integer ) ) )
            @ ( code_divmod_abs @ K4 @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) ) ) ) ) ) ).

% bit_cut_integer_code
thf(fact_8003_set__bits__int__def,axiom,
    ( ( bit_bi4170147762399595738t_bits @ int )
    = ( ^ [F6: nat > $o] :
          ( if @ int
          @ ? [N: nat] :
            ! [M7: nat] :
              ( ( ord_less_eq @ nat @ N @ M7 )
             => ( ( F6 @ M7 )
                = ( F6 @ N ) ) )
          @ ( bit_ri4674362597316999326ke_bit @ int
            @ ( ord_Least @ nat
              @ ^ [N: nat] :
                ! [M7: nat] :
                  ( ( ord_less_eq @ nat @ N @ M7 )
                 => ( ( F6 @ M7 )
                    = ( F6 @ N ) ) ) )
            @ ( groups4207007520872428315er_sum @ $o @ int @ ( zero_neq_one_of_bool @ int ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) )
              @ ( map @ nat @ $o @ F6
                @ ( upt @ ( zero_zero @ nat )
                  @ ( suc
                    @ ( ord_Least @ nat
                      @ ^ [N: nat] :
                        ! [M7: nat] :
                          ( ( ord_less_eq @ nat @ N @ M7 )
                         => ( ( F6 @ M7 )
                            = ( F6 @ N ) ) ) ) ) ) ) ) )
          @ ( zero_zero @ int ) ) ) ) ).

% set_bits_int_def
thf(fact_8004_finite__Collect__bounded__ex,axiom,
    ! [B: $tType,A: $tType,P: A > $o,Q: B > A > $o] :
      ( ( finite_finite2 @ A @ ( collect @ A @ P ) )
     => ( ( finite_finite2 @ B
          @ ( collect @ B
            @ ^ [X: B] :
              ? [Y: A] :
                ( ( P @ Y )
                & ( Q @ X @ Y ) ) ) )
        = ( ! [Y: A] :
              ( ( P @ Y )
             => ( finite_finite2 @ B
                @ ( collect @ B
                  @ ^ [X: B] : ( Q @ X @ Y ) ) ) ) ) ) ) ).

% finite_Collect_bounded_ex
thf(fact_8005_eq__or__mem__image__simp,axiom,
    ! [B: $tType,A: $tType,F: B > A,A2: B,B6: set @ B] :
      ( ( collect @ A
        @ ^ [Uu3: A] :
          ? [L2: B] :
            ( ( Uu3
              = ( F @ L2 ) )
            & ( ( L2 = A2 )
              | ( member @ B @ L2 @ B6 ) ) ) )
      = ( insert @ A @ ( F @ A2 )
        @ ( collect @ A
          @ ^ [Uu3: A] :
            ? [L2: B] :
              ( ( Uu3
                = ( F @ L2 ) )
              & ( member @ B @ L2 @ B6 ) ) ) ) ) ).

% eq_or_mem_image_simp
thf(fact_8006_Least__eq__0,axiom,
    ! [P: nat > $o] :
      ( ( P @ ( zero_zero @ nat ) )
     => ( ( ord_Least @ nat @ P )
        = ( zero_zero @ nat ) ) ) ).

% Least_eq_0
thf(fact_8007_Least__le,axiom,
    ! [A: $tType] :
      ( ( wellorder @ A )
     => ! [P: A > $o,K: A] :
          ( ( P @ K )
         => ( ord_less_eq @ A @ ( ord_Least @ A @ P ) @ K ) ) ) ).

% Least_le
thf(fact_8008_LeastI2__wellorder__ex,axiom,
    ! [A: $tType] :
      ( ( wellorder @ A )
     => ! [P: A > $o,Q: A > $o] :
          ( ? [X_1: A] : ( P @ X_1 )
         => ( ! [A4: A] :
                ( ( P @ A4 )
               => ( ! [B8: A] :
                      ( ( P @ B8 )
                     => ( ord_less_eq @ A @ A4 @ B8 ) )
                 => ( Q @ A4 ) ) )
           => ( Q @ ( ord_Least @ A @ P ) ) ) ) ) ).

% LeastI2_wellorder_ex
thf(fact_8009_LeastI2__wellorder,axiom,
    ! [A: $tType] :
      ( ( wellorder @ A )
     => ! [P: A > $o,A2: A,Q: A > $o] :
          ( ( P @ A2 )
         => ( ! [A4: A] :
                ( ( P @ A4 )
               => ( ! [B8: A] :
                      ( ( P @ B8 )
                     => ( ord_less_eq @ A @ A4 @ B8 ) )
                 => ( Q @ A4 ) ) )
           => ( Q @ ( ord_Least @ A @ P ) ) ) ) ) ).

% LeastI2_wellorder
thf(fact_8010_Least__equality,axiom,
    ! [A: $tType] :
      ( ( order @ A )
     => ! [P: A > $o,X2: A] :
          ( ( P @ X2 )
         => ( ! [Y4: A] :
                ( ( P @ Y4 )
               => ( ord_less_eq @ A @ X2 @ Y4 ) )
           => ( ( ord_Least @ A @ P )
              = X2 ) ) ) ) ).

% Least_equality
thf(fact_8011_LeastI2__order,axiom,
    ! [A: $tType] :
      ( ( order @ A )
     => ! [P: A > $o,X2: A,Q: A > $o] :
          ( ( P @ X2 )
         => ( ! [Y4: A] :
                ( ( P @ Y4 )
               => ( ord_less_eq @ A @ X2 @ Y4 ) )
           => ( ! [X3: A] :
                  ( ( P @ X3 )
                 => ( ! [Y3: A] :
                        ( ( P @ Y3 )
                       => ( ord_less_eq @ A @ X3 @ Y3 ) )
                   => ( Q @ X3 ) ) )
             => ( Q @ ( ord_Least @ A @ P ) ) ) ) ) ) ).

% LeastI2_order
thf(fact_8012_Least1__le,axiom,
    ! [A: $tType] :
      ( ( order @ A )
     => ! [P: A > $o,Z: A] :
          ( ? [X6: A] :
              ( ( P @ X6 )
              & ! [Y4: A] :
                  ( ( P @ Y4 )
                 => ( ord_less_eq @ A @ X6 @ Y4 ) )
              & ! [Y4: A] :
                  ( ( ( P @ Y4 )
                    & ! [Ya2: A] :
                        ( ( P @ Ya2 )
                       => ( ord_less_eq @ A @ Y4 @ Ya2 ) ) )
                 => ( Y4 = X6 ) ) )
         => ( ( P @ Z )
           => ( ord_less_eq @ A @ ( ord_Least @ A @ P ) @ Z ) ) ) ) ).

% Least1_le
thf(fact_8013_Least1I,axiom,
    ! [A: $tType] :
      ( ( order @ A )
     => ! [P: A > $o] :
          ( ? [X6: A] :
              ( ( P @ X6 )
              & ! [Y4: A] :
                  ( ( P @ Y4 )
                 => ( ord_less_eq @ A @ X6 @ Y4 ) )
              & ! [Y4: A] :
                  ( ( ( P @ Y4 )
                    & ! [Ya2: A] :
                        ( ( P @ Ya2 )
                       => ( ord_less_eq @ A @ Y4 @ Ya2 ) ) )
                 => ( Y4 = X6 ) ) )
         => ( P @ ( ord_Least @ A @ P ) ) ) ) ).

% Least1I
thf(fact_8014_finite__image__set,axiom,
    ! [A: $tType,B: $tType,P: A > $o,F: A > B] :
      ( ( finite_finite2 @ A @ ( collect @ A @ P ) )
     => ( finite_finite2 @ B
        @ ( collect @ B
          @ ^ [Uu3: B] :
            ? [X: A] :
              ( ( Uu3
                = ( F @ X ) )
              & ( P @ X ) ) ) ) ) ).

% finite_image_set
thf(fact_8015_finite__image__set2,axiom,
    ! [A: $tType,B: $tType,C: $tType,P: A > $o,Q: B > $o,F: A > B > C] :
      ( ( finite_finite2 @ A @ ( collect @ A @ P ) )
     => ( ( finite_finite2 @ B @ ( collect @ B @ Q ) )
       => ( finite_finite2 @ C
          @ ( collect @ C
            @ ^ [Uu3: C] :
              ? [X: A,Y: B] :
                ( ( Uu3
                  = ( F @ X @ Y ) )
                & ( P @ X )
                & ( Q @ Y ) ) ) ) ) ) ).

% finite_image_set2
thf(fact_8016_set__Cons__def,axiom,
    ! [A: $tType] :
      ( ( set_Cons @ A )
      = ( ^ [A7: set @ A,XS: set @ ( list @ A )] :
            ( collect @ ( list @ A )
            @ ^ [Z2: list @ A] :
              ? [X: A,Xs: list @ A] :
                ( ( Z2
                  = ( cons @ A @ X @ Xs ) )
                & ( member @ A @ X @ A7 )
                & ( member @ ( list @ A ) @ Xs @ XS ) ) ) ) ) ).

% set_Cons_def
thf(fact_8017_LeastI,axiom,
    ! [A: $tType] :
      ( ( wellorder @ A )
     => ! [P: A > $o,K: A] :
          ( ( P @ K )
         => ( P @ ( ord_Least @ A @ P ) ) ) ) ).

% LeastI
thf(fact_8018_some__theI,axiom,
    ! [A: $tType,B: $tType,P: A > B > $o] :
      ( ? [A8: A,X_1: B] : ( P @ A8 @ X_1 )
     => ( ! [B1: B,B22: B] :
            ( ? [A4: A] : ( P @ A4 @ B1 )
           => ( ? [A4: A] : ( P @ A4 @ B22 )
             => ( B1 = B22 ) ) )
       => ( P
          @ ( fChoice @ A
            @ ^ [A5: A] :
              ? [X5: B] : ( P @ A5 @ X5 ) )
          @ ( the @ B
            @ ^ [B5: B] :
              ? [A5: A] : ( P @ A5 @ B5 ) ) ) ) ) ).

% some_theI
thf(fact_8019_ran__def,axiom,
    ! [B: $tType,A: $tType] :
      ( ( ran @ A @ B )
      = ( ^ [M7: A > ( option @ B )] :
            ( collect @ B
            @ ^ [B5: B] :
              ? [A5: A] :
                ( ( M7 @ A5 )
                = ( some @ B @ B5 ) ) ) ) ) ).

% ran_def
thf(fact_8020_full__SetCompr__eq,axiom,
    ! [A: $tType,B: $tType,F: B > A] :
      ( ( collect @ A
        @ ^ [U2: A] :
          ? [X: B] :
            ( U2
            = ( F @ X ) ) )
      = ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) ) ) ).

% full_SetCompr_eq
thf(fact_8021_Setcompr__eq__image,axiom,
    ! [A: $tType,B: $tType,F: B > A,A3: set @ B] :
      ( ( collect @ A
        @ ^ [Uu3: A] :
          ? [X: B] :
            ( ( Uu3
              = ( F @ X ) )
            & ( member @ B @ X @ A3 ) ) )
      = ( image @ B @ A @ F @ A3 ) ) ).

% Setcompr_eq_image
thf(fact_8022_setcompr__eq__image,axiom,
    ! [A: $tType,B: $tType,F: B > A,P: B > $o] :
      ( ( collect @ A
        @ ^ [Uu3: A] :
          ? [X: B] :
            ( ( Uu3
              = ( F @ X ) )
            & ( P @ X ) ) )
      = ( image @ B @ A @ F @ ( collect @ B @ P ) ) ) ).

% setcompr_eq_image
thf(fact_8023_Bleast__def,axiom,
    ! [A: $tType] :
      ( ( ord @ A )
     => ( ( bleast @ A )
        = ( ^ [S8: set @ A,P3: A > $o] :
              ( ord_Least @ A
              @ ^ [X: A] :
                  ( ( member @ A @ X @ S8 )
                  & ( P3 @ X ) ) ) ) ) ) ).

% Bleast_def
thf(fact_8024_abort__Bleast__def,axiom,
    ! [A: $tType] :
      ( ( ord @ A )
     => ( ( abort_Bleast @ A )
        = ( ^ [S8: set @ A,P3: A > $o] :
              ( ord_Least @ A
              @ ^ [X: A] :
                  ( ( member @ A @ X @ S8 )
                  & ( P3 @ X ) ) ) ) ) ) ).

% abort_Bleast_def
thf(fact_8025_Least__Suc,axiom,
    ! [P: nat > $o,N2: nat] :
      ( ( P @ N2 )
     => ( ~ ( P @ ( zero_zero @ nat ) )
       => ( ( ord_Least @ nat @ P )
          = ( suc
            @ ( ord_Least @ nat
              @ ^ [M7: nat] : ( P @ ( suc @ M7 ) ) ) ) ) ) ) ).

% Least_Suc
thf(fact_8026_not__less__Least,axiom,
    ! [A: $tType] :
      ( ( wellorder @ A )
     => ! [K: A,P: A > $o] :
          ( ( ord_less @ A @ K @ ( ord_Least @ A @ P ) )
         => ~ ( P @ K ) ) ) ).

% not_less_Least
thf(fact_8027_Least__Suc2,axiom,
    ! [P: nat > $o,N2: nat,Q: nat > $o,M: nat] :
      ( ( P @ N2 )
     => ( ( Q @ M )
       => ( ~ ( P @ ( zero_zero @ nat ) )
         => ( ! [K3: nat] :
                ( ( P @ ( suc @ K3 ) )
                = ( Q @ K3 ) )
           => ( ( ord_Least @ nat @ P )
              = ( suc @ ( ord_Least @ nat @ Q ) ) ) ) ) ) ) ).

% Least_Suc2
thf(fact_8028_fs__contract,axiom,
    ! [B: $tType,C: $tType,A: $tType,F: A > B > C,S3: set @ C] :
      ( ( image @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B )
        @ ( collect @ ( product_prod @ A @ B )
          @ ^ [Uu3: product_prod @ A @ B] :
            ? [P5: product_prod @ A @ B] :
              ( ( Uu3 = P5 )
              & ( member @ C @ ( F @ ( product_fst @ A @ B @ P5 ) @ ( product_snd @ A @ B @ P5 ) ) @ S3 ) ) ) )
      = ( collect @ A
        @ ^ [A5: A] :
          ? [B5: B] : ( member @ C @ ( F @ A5 @ B5 ) @ S3 ) ) ) ).

% fs_contract
thf(fact_8029_set__map__filter,axiom,
    ! [B: $tType,A: $tType,G: B > ( option @ A ),Xs2: list @ B] :
      ( ( set2 @ A @ ( map_filter @ B @ A @ G @ Xs2 ) )
      = ( collect @ A
        @ ^ [Y: A] :
          ? [X: B] :
            ( ( member @ B @ X @ ( set2 @ B @ Xs2 ) )
            & ( ( G @ X )
              = ( some @ A @ Y ) ) ) ) ) ).

% set_map_filter
thf(fact_8030_set__conv__nth,axiom,
    ! [A: $tType] :
      ( ( set2 @ A )
      = ( ^ [Xs: list @ A] :
            ( collect @ A
            @ ^ [Uu3: A] :
              ? [I2: nat] :
                ( ( Uu3
                  = ( nth @ A @ Xs @ I2 ) )
                & ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ Xs ) ) ) ) ) ) ).

% set_conv_nth
thf(fact_8031_Least__def,axiom,
    ! [A: $tType] :
      ( ( ord @ A )
     => ( ( ord_Least @ A )
        = ( ^ [P3: A > $o] :
              ( the @ A
              @ ^ [X: A] :
                  ( ( P3 @ X )
                  & ! [Y: A] :
                      ( ( P3 @ Y )
                     => ( ord_less_eq @ A @ X @ Y ) ) ) ) ) ) ) ).

% Least_def
thf(fact_8032_set__bits__int__unfold_H,axiom,
    ( ( bit_bi4170147762399595738t_bits @ int )
    = ( ^ [F6: nat > $o] :
          ( if @ int
          @ ? [N: nat] :
            ! [N12: nat] :
              ( ( ord_less_eq @ nat @ N @ N12 )
             => ~ ( F6 @ N12 ) )
          @ ( groups4207007520872428315er_sum @ $o @ int @ ( zero_neq_one_of_bool @ int ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) )
            @ ( map @ nat @ $o @ F6
              @ ( upt @ ( zero_zero @ nat )
                @ ( ord_Least @ nat
                  @ ^ [N: nat] :
                    ! [N12: nat] :
                      ( ( ord_less_eq @ nat @ N @ N12 )
                     => ~ ( F6 @ N12 ) ) ) ) ) )
          @ ( if @ int
            @ ? [N: nat] :
              ! [N12: nat] :
                ( ( ord_less_eq @ nat @ N @ N12 )
               => ( F6 @ N12 ) )
            @ ( bit_ri4674362597316999326ke_bit @ int
              @ ( ord_Least @ nat
                @ ^ [N: nat] :
                  ! [N12: nat] :
                    ( ( ord_less_eq @ nat @ N @ N12 )
                   => ( F6 @ N12 ) ) )
              @ ( groups4207007520872428315er_sum @ $o @ int @ ( zero_neq_one_of_bool @ int ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) )
                @ ( append @ $o
                  @ ( map @ nat @ $o @ F6
                    @ ( upt @ ( zero_zero @ nat )
                      @ ( ord_Least @ nat
                        @ ^ [N: nat] :
                          ! [N12: nat] :
                            ( ( ord_less_eq @ nat @ N @ N12 )
                           => ( F6 @ N12 ) ) ) ) )
                  @ ( cons @ $o @ $true @ ( nil @ $o ) ) ) ) )
            @ ( zero_zero @ int ) ) ) ) ) ).

% set_bits_int_unfold'
thf(fact_8033_bit__cut__integer__def,axiom,
    ( code_bit_cut_integer
    = ( ^ [K4: code_integer] :
          ( product_Pair @ code_integer @ $o @ ( divide_divide @ code_integer @ K4 @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) )
          @ ~ ( dvd_dvd @ code_integer @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) @ K4 ) ) ) ) ).

% bit_cut_integer_def
thf(fact_8034_pairself__image__eq,axiom,
    ! [B: $tType,A: $tType,F: B > A,P: B > B > $o] :
      ( ( image @ ( product_prod @ B @ B ) @ ( product_prod @ A @ A ) @ ( pairself @ B @ A @ F ) @ ( collect @ ( product_prod @ B @ B ) @ ( product_case_prod @ B @ B @ $o @ P ) ) )
      = ( collect @ ( product_prod @ A @ A )
        @ ^ [Uu3: product_prod @ A @ A] :
          ? [A5: B,B5: B] :
            ( ( Uu3
              = ( product_Pair @ A @ A @ ( F @ A5 ) @ ( F @ B5 ) ) )
            & ( P @ A5 @ B5 ) ) ) ) ).

% pairself_image_eq
thf(fact_8035_VEBT__internal_Ovalid_H_Oelims_I3_J,axiom,
    ! [X2: vEBT_VEBT,Xa: nat] :
      ( ~ ( vEBT_VEBT_valid @ X2 @ Xa )
     => ( ( ? [Uu2: $o,Uv2: $o] :
              ( X2
              = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
         => ( Xa
            = ( one_one @ nat ) ) )
       => ~ ! [Mima: option @ ( product_prod @ nat @ nat ),Deg2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
              ( ( X2
                = ( vEBT_Node @ Mima @ Deg2 @ TreeList2 @ Summary2 ) )
             => ( ( Deg2 = Xa )
                & ! [X3: vEBT_VEBT] :
                    ( ( member @ vEBT_VEBT @ X3 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
                   => ( vEBT_VEBT_valid @ X3 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
                & ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
                & ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 )
                  = ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
                & ( case_option @ $o @ ( product_prod @ nat @ nat )
                  @ ( ~ ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X5 )
                    & ! [X: vEBT_VEBT] :
                        ( ( member @ vEBT_VEBT @ X @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
                       => ~ ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ X @ X5 ) ) )
                  @ ( product_case_prod @ nat @ nat @ $o
                    @ ^ [Mi3: nat,Ma3: nat] :
                        ( ( ord_less_eq @ nat @ Mi3 @ Ma3 )
                        & ( ord_less @ nat @ Ma3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) )
                        & ! [I2: nat] :
                            ( ( ord_less @ nat @ I2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
                           => ( ( ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList2 @ I2 ) @ X5 ) )
                              = ( vEBT_V8194947554948674370ptions @ Summary2 @ I2 ) ) )
                        & ( ( Mi3 = Ma3 )
                         => ! [X: vEBT_VEBT] :
                              ( ( member @ vEBT_VEBT @ X @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
                             => ~ ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ X @ X5 ) ) )
                        & ( ( Mi3 != Ma3 )
                         => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ TreeList2 @ Ma3 )
                            & ! [X: nat] :
                                ( ( ord_less @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) )
                               => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ TreeList2 @ X )
                                 => ( ( ord_less @ nat @ Mi3 @ X )
                                    & ( ord_less_eq @ nat @ X @ Ma3 ) ) ) ) ) ) ) )
                  @ Mima ) ) ) ) ) ).

% VEBT_internal.valid'.elims(3)
thf(fact_8036_valid__eq,axiom,
    vEBT_VEBT_valid = vEBT_invar_vebt ).

% valid_eq
thf(fact_8037_valid__eq1,axiom,
    ! [T2: vEBT_VEBT,D2: nat] :
      ( ( vEBT_invar_vebt @ T2 @ D2 )
     => ( vEBT_VEBT_valid @ T2 @ D2 ) ) ).

% valid_eq1
thf(fact_8038_valid__eq2,axiom,
    ! [T2: vEBT_VEBT,D2: nat] :
      ( ( vEBT_VEBT_valid @ T2 @ D2 )
     => ( vEBT_invar_vebt @ T2 @ D2 ) ) ).

% valid_eq2
thf(fact_8039_Ball__comp__iff,axiom,
    ! [C: $tType,B: $tType,A: $tType,A3: B > ( set @ C ),F: C > $o,G: A > B] :
      ( ( comp @ B @ $o @ A
        @ ^ [X: B] :
          ! [Y: C] :
            ( ( member @ C @ Y @ ( A3 @ X ) )
           => ( F @ Y ) )
        @ G )
      = ( ^ [X: A] :
          ! [Y: C] :
            ( ( member @ C @ Y @ ( comp @ B @ ( set @ C ) @ A @ A3 @ G @ X ) )
           => ( F @ Y ) ) ) ) ).

% Ball_comp_iff
thf(fact_8040_pairself_Oelims,axiom,
    ! [B: $tType,A: $tType,X2: A > B,Xa: product_prod @ A @ A,Y2: product_prod @ B @ B] :
      ( ( ( pairself @ A @ B @ X2 @ Xa )
        = Y2 )
     => ~ ! [A4: A,B3: A] :
            ( ( Xa
              = ( product_Pair @ A @ A @ A4 @ B3 ) )
           => ( Y2
             != ( product_Pair @ B @ B @ ( X2 @ A4 ) @ ( X2 @ B3 ) ) ) ) ) ).

% pairself.elims
thf(fact_8041_pairself_Osimps,axiom,
    ! [B: $tType,A: $tType,F: A > B,A2: A,B2: A] :
      ( ( pairself @ A @ B @ F @ ( product_Pair @ A @ A @ A2 @ B2 ) )
      = ( product_Pair @ B @ B @ ( F @ A2 ) @ ( F @ B2 ) ) ) ).

% pairself.simps
thf(fact_8042_Ball__fold,axiom,
    ! [A: $tType,A3: set @ A,P: A > $o] :
      ( ( finite_finite2 @ A @ A3 )
     => ( ( ! [X: A] :
              ( ( member @ A @ X @ A3 )
             => ( P @ X ) ) )
        = ( finite_fold @ A @ $o
          @ ^ [K4: A,S2: $o] :
              ( S2
              & ( P @ K4 ) )
          @ $true
          @ A3 ) ) ) ).

% Ball_fold
thf(fact_8043_in__set__simps_I4_J,axiom,
    ! [A: $tType,P: A > $o,X6: A] :
      ( ( member @ A @ X6 @ ( set2 @ A @ ( nil @ A ) ) )
     => ( P @ X6 ) ) ).

% in_set_simps(4)
thf(fact_8044_admissible__ball,axiom,
    ! [A: $tType,B: $tType,A3: set @ A,Lub: ( set @ B ) > B,Ord: B > B > $o,P: B > A > $o] :
      ( ! [Y4: A] :
          ( ( member @ A @ Y4 @ A3 )
         => ( comple1908693960933563346ssible @ B @ Lub @ Ord
            @ ^ [X: B] : ( P @ X @ Y4 ) ) )
     => ( comple1908693960933563346ssible @ B @ Lub @ Ord
        @ ^ [X: B] :
          ! [Y: A] :
            ( ( member @ A @ Y @ A3 )
           => ( P @ X @ Y ) ) ) ) ).

% admissible_ball
thf(fact_8045_Ball__def__raw,axiom,
    ! [A: $tType] :
      ( ( ball @ A )
      = ( ^ [A7: set @ A,P3: A > $o] :
          ! [X: A] :
            ( ( member @ A @ X @ A7 )
           => ( P3 @ X ) ) ) ) ).

% Ball_def_raw
thf(fact_8046_in__set__simps_I6_J,axiom,
    ! [A: $tType,X2: A,Y2: A,Zs: list @ A,P: A > $o] :
      ( ( ! [X: A] :
            ( ( member @ A @ X @ ( set2 @ A @ ( cons @ A @ X2 @ ( cons @ A @ Y2 @ Zs ) ) ) )
           => ( P @ X ) ) )
      = ( ( P @ X2 )
        & ! [X: A] :
            ( ( member @ A @ X @ ( set2 @ A @ ( cons @ A @ Y2 @ Zs ) ) )
           => ( P @ X ) ) ) ) ).

% in_set_simps(6)
thf(fact_8047_in__set__simps_I5_J,axiom,
    ! [A: $tType,X2: A,P: A > $o] :
      ( ( ! [X: A] :
            ( ( member @ A @ X @ ( set2 @ A @ ( cons @ A @ X2 @ ( nil @ A ) ) ) )
           => ( P @ X ) ) )
      = ( P @ X2 ) ) ).

% in_set_simps(5)
thf(fact_8048_VEBT__internal_Ovalid_H_Osimps_I1_J,axiom,
    ! [Uu: $o,Uv: $o,D2: nat] :
      ( ( vEBT_VEBT_valid @ ( vEBT_Leaf @ Uu @ Uv ) @ D2 )
      = ( D2
        = ( one_one @ nat ) ) ) ).

% VEBT_internal.valid'.simps(1)
thf(fact_8049_Ball__Collect,axiom,
    ! [A: $tType] :
      ( ( ball @ A )
      = ( ^ [A7: set @ A,P3: A > $o] : ( ord_less_eq @ ( set @ A ) @ A7 @ ( collect @ A @ P3 ) ) ) ) ).

% Ball_Collect
thf(fact_8050_VEBT__internal_Ovalid_H_Osimps_I2_J,axiom,
    ! [Mima3: option @ ( product_prod @ nat @ nat ),Deg: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,Deg6: nat] :
      ( ( vEBT_VEBT_valid @ ( vEBT_Node @ Mima3 @ Deg @ TreeList @ Summary ) @ Deg6 )
      = ( ( Deg = Deg6 )
        & ! [X: vEBT_VEBT] :
            ( ( member @ vEBT_VEBT @ X @ ( set2 @ vEBT_VEBT @ TreeList ) )
           => ( vEBT_VEBT_valid @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
        & ( vEBT_VEBT_valid @ Summary @ ( minus_minus @ nat @ Deg @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
        & ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList )
          = ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ Deg @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
        & ( case_option @ $o @ ( product_prod @ nat @ nat )
          @ ( ~ ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X5 )
            & ! [X: vEBT_VEBT] :
                ( ( member @ vEBT_VEBT @ X @ ( set2 @ vEBT_VEBT @ TreeList ) )
               => ~ ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ X @ X5 ) ) )
          @ ( product_case_prod @ nat @ nat @ $o
            @ ^ [Mi3: nat,Ma3: nat] :
                ( ( ord_less_eq @ nat @ Mi3 @ Ma3 )
                & ( ord_less @ nat @ Ma3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg ) )
                & ! [I2: nat] :
                    ( ( ord_less @ nat @ I2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ Deg @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
                   => ( ( ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList @ I2 ) @ X5 ) )
                      = ( vEBT_V8194947554948674370ptions @ Summary @ I2 ) ) )
                & ( ( Mi3 = Ma3 )
                 => ! [X: vEBT_VEBT] :
                      ( ( member @ vEBT_VEBT @ X @ ( set2 @ vEBT_VEBT @ TreeList ) )
                     => ~ ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ X @ X5 ) ) )
                & ( ( Mi3 != Ma3 )
                 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ TreeList @ Ma3 )
                    & ! [X: nat] :
                        ( ( ord_less @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg ) )
                       => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ TreeList @ X )
                         => ( ( ord_less @ nat @ Mi3 @ X )
                            & ( ord_less_eq @ nat @ X @ Ma3 ) ) ) ) ) ) ) )
          @ Mima3 ) ) ) ).

% VEBT_internal.valid'.simps(2)
thf(fact_8051_VEBT__internal_Ovalid_H_Oelims_I1_J,axiom,
    ! [X2: vEBT_VEBT,Xa: nat,Y2: $o] :
      ( ( ( vEBT_VEBT_valid @ X2 @ Xa )
        = Y2 )
     => ( ( ? [Uu2: $o,Uv2: $o] :
              ( X2
              = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
         => ( Y2
            = ( Xa
             != ( one_one @ nat ) ) ) )
       => ~ ! [Mima: option @ ( product_prod @ nat @ nat ),Deg2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
              ( ( X2
                = ( vEBT_Node @ Mima @ Deg2 @ TreeList2 @ Summary2 ) )
             => ( Y2
                = ( ~ ( ( Deg2 = Xa )
                      & ! [X: vEBT_VEBT] :
                          ( ( member @ vEBT_VEBT @ X @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
                         => ( vEBT_VEBT_valid @ X @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
                      & ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
                      & ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 )
                        = ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
                      & ( case_option @ $o @ ( product_prod @ nat @ nat )
                        @ ( ~ ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X5 )
                          & ! [X: vEBT_VEBT] :
                              ( ( member @ vEBT_VEBT @ X @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
                             => ~ ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ X @ X5 ) ) )
                        @ ( product_case_prod @ nat @ nat @ $o
                          @ ^ [Mi3: nat,Ma3: nat] :
                              ( ( ord_less_eq @ nat @ Mi3 @ Ma3 )
                              & ( ord_less @ nat @ Ma3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) )
                              & ! [I2: nat] :
                                  ( ( ord_less @ nat @ I2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
                                 => ( ( ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList2 @ I2 ) @ X5 ) )
                                    = ( vEBT_V8194947554948674370ptions @ Summary2 @ I2 ) ) )
                              & ( ( Mi3 = Ma3 )
                               => ! [X: vEBT_VEBT] :
                                    ( ( member @ vEBT_VEBT @ X @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
                                   => ~ ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ X @ X5 ) ) )
                              & ( ( Mi3 != Ma3 )
                               => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ TreeList2 @ Ma3 )
                                  & ! [X: nat] :
                                      ( ( ord_less @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) )
                                     => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ TreeList2 @ X )
                                       => ( ( ord_less @ nat @ Mi3 @ X )
                                          & ( ord_less_eq @ nat @ X @ Ma3 ) ) ) ) ) ) ) )
                        @ Mima ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.elims(1)
thf(fact_8052_VEBT__internal_Ovalid_H_Oelims_I2_J,axiom,
    ! [X2: vEBT_VEBT,Xa: nat] :
      ( ( vEBT_VEBT_valid @ X2 @ Xa )
     => ( ( ? [Uu2: $o,Uv2: $o] :
              ( X2
              = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
         => ( Xa
           != ( one_one @ nat ) ) )
       => ~ ! [Mima: option @ ( product_prod @ nat @ nat ),Deg2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
              ( ( X2
                = ( vEBT_Node @ Mima @ Deg2 @ TreeList2 @ Summary2 ) )
             => ~ ( ( Deg2 = Xa )
                  & ! [X6: vEBT_VEBT] :
                      ( ( member @ vEBT_VEBT @ X6 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
                     => ( vEBT_VEBT_valid @ X6 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
                  & ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
                  & ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 )
                    = ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
                  & ( case_option @ $o @ ( product_prod @ nat @ nat )
                    @ ( ~ ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X5 )
                      & ! [X: vEBT_VEBT] :
                          ( ( member @ vEBT_VEBT @ X @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
                         => ~ ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ X @ X5 ) ) )
                    @ ( product_case_prod @ nat @ nat @ $o
                      @ ^ [Mi3: nat,Ma3: nat] :
                          ( ( ord_less_eq @ nat @ Mi3 @ Ma3 )
                          & ( ord_less @ nat @ Ma3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) )
                          & ! [I2: nat] :
                              ( ( ord_less @ nat @ I2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
                             => ( ( ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList2 @ I2 ) @ X5 ) )
                                = ( vEBT_V8194947554948674370ptions @ Summary2 @ I2 ) ) )
                          & ( ( Mi3 = Ma3 )
                           => ! [X: vEBT_VEBT] :
                                ( ( member @ vEBT_VEBT @ X @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
                               => ~ ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ X @ X5 ) ) )
                          & ( ( Mi3 != Ma3 )
                           => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ TreeList2 @ Ma3 )
                              & ! [X: nat] :
                                  ( ( ord_less @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) )
                                 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ TreeList2 @ X )
                                   => ( ( ord_less @ nat @ Mi3 @ X )
                                      & ( ord_less_eq @ nat @ X @ Ma3 ) ) ) ) ) ) ) )
                    @ Mima ) ) ) ) ) ).

% VEBT_internal.valid'.elims(2)
thf(fact_8053_VEBT__internal_Ovalid_H_Opelims_I1_J,axiom,
    ! [X2: vEBT_VEBT,Xa: nat,Y2: $o] :
      ( ( ( vEBT_VEBT_valid @ X2 @ Xa )
        = Y2 )
     => ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_VEBT_valid_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X2 @ Xa ) )
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( X2
                = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
             => ( ( Y2
                  = ( Xa
                    = ( one_one @ nat ) ) )
               => ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_VEBT_valid_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa ) ) ) )
         => ~ ! [Mima: option @ ( product_prod @ nat @ nat ),Deg2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( X2
                  = ( vEBT_Node @ Mima @ Deg2 @ TreeList2 @ Summary2 ) )
               => ( ( Y2
                    = ( ( Deg2 = Xa )
                      & ! [X: vEBT_VEBT] :
                          ( ( member @ vEBT_VEBT @ X @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
                         => ( vEBT_VEBT_valid @ X @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
                      & ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
                      & ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 )
                        = ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
                      & ( case_option @ $o @ ( product_prod @ nat @ nat )
                        @ ( ~ ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X5 )
                          & ! [X: vEBT_VEBT] :
                              ( ( member @ vEBT_VEBT @ X @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
                             => ~ ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ X @ X5 ) ) )
                        @ ( product_case_prod @ nat @ nat @ $o
                          @ ^ [Mi3: nat,Ma3: nat] :
                              ( ( ord_less_eq @ nat @ Mi3 @ Ma3 )
                              & ( ord_less @ nat @ Ma3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) )
                              & ! [I2: nat] :
                                  ( ( ord_less @ nat @ I2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
                                 => ( ( ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList2 @ I2 ) @ X5 ) )
                                    = ( vEBT_V8194947554948674370ptions @ Summary2 @ I2 ) ) )
                              & ( ( Mi3 = Ma3 )
                               => ! [X: vEBT_VEBT] :
                                    ( ( member @ vEBT_VEBT @ X @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
                                   => ~ ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ X @ X5 ) ) )
                              & ( ( Mi3 != Ma3 )
                               => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ TreeList2 @ Ma3 )
                                  & ! [X: nat] :
                                      ( ( ord_less @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) )
                                     => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ TreeList2 @ X )
                                       => ( ( ord_less @ nat @ Mi3 @ X )
                                          & ( ord_less_eq @ nat @ X @ Ma3 ) ) ) ) ) ) ) )
                        @ Mima ) ) )
                 => ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_VEBT_valid_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Mima @ Deg2 @ TreeList2 @ Summary2 ) @ Xa ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.pelims(1)
thf(fact_8054_VEBT__internal_Ovalid_H_Opelims_I2_J,axiom,
    ! [X2: vEBT_VEBT,Xa: nat] :
      ( ( vEBT_VEBT_valid @ X2 @ Xa )
     => ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_VEBT_valid_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X2 @ Xa ) )
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( X2
                = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
             => ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_VEBT_valid_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa ) )
               => ( Xa
                 != ( one_one @ nat ) ) ) )
         => ~ ! [Mima: option @ ( product_prod @ nat @ nat ),Deg2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( X2
                  = ( vEBT_Node @ Mima @ Deg2 @ TreeList2 @ Summary2 ) )
               => ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_VEBT_valid_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Mima @ Deg2 @ TreeList2 @ Summary2 ) @ Xa ) )
                 => ~ ( ( Deg2 = Xa )
                      & ! [X6: vEBT_VEBT] :
                          ( ( member @ vEBT_VEBT @ X6 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
                         => ( vEBT_VEBT_valid @ X6 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
                      & ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
                      & ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 )
                        = ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
                      & ( case_option @ $o @ ( product_prod @ nat @ nat )
                        @ ( ~ ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X5 )
                          & ! [X: vEBT_VEBT] :
                              ( ( member @ vEBT_VEBT @ X @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
                             => ~ ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ X @ X5 ) ) )
                        @ ( product_case_prod @ nat @ nat @ $o
                          @ ^ [Mi3: nat,Ma3: nat] :
                              ( ( ord_less_eq @ nat @ Mi3 @ Ma3 )
                              & ( ord_less @ nat @ Ma3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) )
                              & ! [I2: nat] :
                                  ( ( ord_less @ nat @ I2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
                                 => ( ( ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList2 @ I2 ) @ X5 ) )
                                    = ( vEBT_V8194947554948674370ptions @ Summary2 @ I2 ) ) )
                              & ( ( Mi3 = Ma3 )
                               => ! [X: vEBT_VEBT] :
                                    ( ( member @ vEBT_VEBT @ X @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
                                   => ~ ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ X @ X5 ) ) )
                              & ( ( Mi3 != Ma3 )
                               => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ TreeList2 @ Ma3 )
                                  & ! [X: nat] :
                                      ( ( ord_less @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) )
                                     => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ TreeList2 @ X )
                                       => ( ( ord_less @ nat @ Mi3 @ X )
                                          & ( ord_less_eq @ nat @ X @ Ma3 ) ) ) ) ) ) ) )
                        @ Mima ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.pelims(2)
thf(fact_8055_fi__match__entails,axiom,
    ! [M: list @ ( product_prod @ assn @ assn )] :
      ( ! [X3: product_prod @ assn @ assn] :
          ( ( member @ ( product_prod @ assn @ assn ) @ X3 @ ( set2 @ ( product_prod @ assn @ assn ) @ M ) )
         => ( product_case_prod @ assn @ assn @ $o @ entails @ X3 ) )
     => ( entails @ ( foldr @ assn @ assn @ ( times_times @ assn ) @ ( map @ ( product_prod @ assn @ assn ) @ assn @ ( product_fst @ assn @ assn ) @ M ) @ ( one_one @ assn ) ) @ ( foldr @ assn @ assn @ ( times_times @ assn ) @ ( map @ ( product_prod @ assn @ assn ) @ assn @ ( product_snd @ assn @ assn ) @ M ) @ ( one_one @ assn ) ) ) ) ).

% fi_match_entails
thf(fact_8056_VEBT__internal_Ovalid_H_Opelims_I3_J,axiom,
    ! [X2: vEBT_VEBT,Xa: nat] :
      ( ~ ( vEBT_VEBT_valid @ X2 @ Xa )
     => ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_VEBT_valid_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X2 @ Xa ) )
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( X2
                = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
             => ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_VEBT_valid_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa ) )
               => ( Xa
                  = ( one_one @ nat ) ) ) )
         => ~ ! [Mima: option @ ( product_prod @ nat @ nat ),Deg2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( X2
                  = ( vEBT_Node @ Mima @ Deg2 @ TreeList2 @ Summary2 ) )
               => ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_VEBT_valid_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Mima @ Deg2 @ TreeList2 @ Summary2 ) @ Xa ) )
                 => ( ( Deg2 = Xa )
                    & ! [X3: vEBT_VEBT] :
                        ( ( member @ vEBT_VEBT @ X3 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
                       => ( vEBT_VEBT_valid @ X3 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
                    & ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
                    & ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 )
                      = ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
                    & ( case_option @ $o @ ( product_prod @ nat @ nat )
                      @ ( ~ ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X5 )
                        & ! [X: vEBT_VEBT] :
                            ( ( member @ vEBT_VEBT @ X @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
                           => ~ ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ X @ X5 ) ) )
                      @ ( product_case_prod @ nat @ nat @ $o
                        @ ^ [Mi3: nat,Ma3: nat] :
                            ( ( ord_less_eq @ nat @ Mi3 @ Ma3 )
                            & ( ord_less @ nat @ Ma3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) )
                            & ! [I2: nat] :
                                ( ( ord_less @ nat @ I2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
                               => ( ( ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList2 @ I2 ) @ X5 ) )
                                  = ( vEBT_V8194947554948674370ptions @ Summary2 @ I2 ) ) )
                            & ( ( Mi3 = Ma3 )
                             => ! [X: vEBT_VEBT] :
                                  ( ( member @ vEBT_VEBT @ X @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
                                 => ~ ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ X @ X5 ) ) )
                            & ( ( Mi3 != Ma3 )
                             => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ TreeList2 @ Ma3 )
                                & ! [X: nat] :
                                    ( ( ord_less @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) )
                                   => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ TreeList2 @ X )
                                     => ( ( ord_less @ nat @ Mi3 @ X )
                                        & ( ord_less_eq @ nat @ X @ Ma3 ) ) ) ) ) ) ) )
                      @ Mima ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.pelims(3)
thf(fact_8057_pairself_Opelims,axiom,
    ! [B: $tType,A: $tType,X2: A > B,Xa: product_prod @ A @ A,Y2: product_prod @ B @ B] :
      ( ( ( pairself @ A @ B @ X2 @ Xa )
        = Y2 )
     => ( ( accp @ ( product_prod @ ( A > B ) @ ( product_prod @ A @ A ) ) @ ( pairself_rel @ A @ B ) @ ( product_Pair @ ( A > B ) @ ( product_prod @ A @ A ) @ X2 @ Xa ) )
       => ~ ! [A4: A,B3: A] :
              ( ( Xa
                = ( product_Pair @ A @ A @ A4 @ B3 ) )
             => ( ( Y2
                  = ( product_Pair @ B @ B @ ( X2 @ A4 ) @ ( X2 @ B3 ) ) )
               => ~ ( accp @ ( product_prod @ ( A > B ) @ ( product_prod @ A @ A ) ) @ ( pairself_rel @ A @ B ) @ ( product_Pair @ ( A > B ) @ ( product_prod @ A @ A ) @ X2 @ ( product_Pair @ A @ A @ A4 @ B3 ) ) ) ) ) ) ) ).

% pairself.pelims
thf(fact_8058_lexn__conv,axiom,
    ! [A: $tType] :
      ( ( lexn @ A )
      = ( ^ [R: set @ ( product_prod @ A @ A ),N: nat] :
            ( collect @ ( product_prod @ ( list @ A ) @ ( list @ A ) )
            @ ( product_case_prod @ ( list @ A ) @ ( list @ A ) @ $o
              @ ^ [Xs: list @ A,Ys3: list @ A] :
                  ( ( ( size_size @ ( list @ A ) @ Xs )
                    = N )
                  & ( ( size_size @ ( list @ A ) @ Ys3 )
                    = N )
                  & ? [Xys: list @ A,X: A,Y: A,Xs6: list @ A,Ys7: list @ A] :
                      ( ( Xs
                        = ( append @ A @ Xys @ ( cons @ A @ X @ Xs6 ) ) )
                      & ( Ys3
                        = ( append @ A @ Xys @ ( cons @ A @ Y @ Ys7 ) ) )
                      & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R ) ) ) ) ) ) ) ).

% lexn_conv
thf(fact_8059_lexn__length,axiom,
    ! [A: $tType,Xs2: list @ A,Ys: list @ A,R4: set @ ( product_prod @ A @ A ),N2: nat] :
      ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs2 @ Ys ) @ ( lexn @ A @ R4 @ N2 ) )
     => ( ( ( size_size @ ( list @ A ) @ Xs2 )
          = N2 )
        & ( ( size_size @ ( list @ A ) @ Ys )
          = N2 ) ) ) ).

% lexn_length
thf(fact_8060_FI__RESULT__def,axiom,
    ( fI_RESULT
    = ( ^ [M10: list @ ( product_prod @ assn @ assn ),UP: assn,UQ: assn,F9: assn] :
          ( ! [X: product_prod @ assn @ assn] :
              ( ( member @ ( product_prod @ assn @ assn ) @ X @ ( set2 @ ( product_prod @ assn @ assn ) @ M10 ) )
             => ( product_case_prod @ assn @ assn @ $o @ entails @ X ) )
         => ( entails @ ( times_times @ assn @ ( foldr @ assn @ assn @ ( times_times @ assn ) @ ( map @ ( product_prod @ assn @ assn ) @ assn @ ( product_fst @ assn @ assn ) @ M10 ) @ ( one_one @ assn ) ) @ UP ) @ ( times_times @ assn @ ( times_times @ assn @ ( foldr @ assn @ assn @ ( times_times @ assn ) @ ( map @ ( product_prod @ assn @ assn ) @ assn @ ( product_snd @ assn @ assn ) @ M10 ) @ ( one_one @ assn ) ) @ UQ ) @ F9 ) ) ) ) ) ).

% FI_RESULT_def
thf(fact_8061_FI__def,axiom,
    ( fi
    = ( ^ [M7: list @ ( product_prod @ assn @ assn ),P5: assn,Q5: assn,Up2: assn,Uq2: assn,F6: assn] :
          ( ! [X: product_prod @ assn @ assn] :
              ( ( member @ ( product_prod @ assn @ assn ) @ X @ ( set2 @ ( product_prod @ assn @ assn ) @ M7 ) )
             => ( product_case_prod @ assn @ assn @ $o @ entails @ X ) )
         => ( entails @ ( times_times @ assn @ ( times_times @ assn @ ( foldr @ assn @ assn @ ( times_times @ assn ) @ ( map @ ( product_prod @ assn @ assn ) @ assn @ ( product_fst @ assn @ assn ) @ M7 ) @ ( one_one @ assn ) ) @ P5 ) @ Up2 ) @ ( times_times @ assn @ ( times_times @ assn @ ( times_times @ assn @ ( foldr @ assn @ assn @ ( times_times @ assn ) @ ( map @ ( product_prod @ assn @ assn ) @ assn @ ( product_snd @ assn @ assn ) @ M7 ) @ ( one_one @ assn ) ) @ Q5 ) @ Uq2 ) @ F6 ) ) ) ) ) ).

% FI_def
thf(fact_8062_entails__solve__finalize_I2_J,axiom,
    ! [M8: list @ ( product_prod @ assn @ assn )] : ( fI_RESULT @ M8 @ ( one_one @ assn ) @ ( one_one @ assn ) @ ( one_one @ assn ) ) ).

% entails_solve_finalize(2)
thf(fact_8063_frame__inference__finalize,axiom,
    ! [M8: list @ ( product_prod @ assn @ assn ),F5: assn] : ( fI_RESULT @ M8 @ F5 @ ( one_one @ assn ) @ F5 ) ).

% frame_inference_finalize
thf(fact_8064_FI__p__nomatch,axiom,
    ! [M: list @ ( product_prod @ assn @ assn ),Ps2: assn,Qs: assn,Q2: assn,P4: assn,Up: assn,Uq: assn,F: assn] :
      ( ( fi @ M @ Ps2 @ ( times_times @ assn @ Qs @ Q2 ) @ ( times_times @ assn @ P4 @ Up ) @ Uq @ F )
     => ( fi @ M @ ( times_times @ assn @ Ps2 @ P4 ) @ ( times_times @ assn @ Qs @ Q2 ) @ Up @ Uq @ F ) ) ).

% FI_p_nomatch
thf(fact_8065_FI__finalize,axiom,
    ! [M: list @ ( product_prod @ assn @ assn ),P4: assn,Up: assn,Q2: assn,Uq: assn,F: assn] :
      ( ( fI_RESULT @ M @ ( times_times @ assn @ P4 @ Up ) @ ( times_times @ assn @ Q2 @ Uq ) @ F )
     => ( fi @ M @ P4 @ Q2 @ Up @ Uq @ F ) ) ).

% FI_finalize
thf(fact_8066_entails__solve__finalize_I1_J,axiom,
    ! [M8: list @ ( product_prod @ assn @ assn ),P: assn] : ( fI_RESULT @ M8 @ P @ ( one_one @ assn ) @ ( top_top @ assn ) ) ).

% entails_solve_finalize(1)
thf(fact_8067_lex__conv,axiom,
    ! [A: $tType] :
      ( ( lex @ A )
      = ( ^ [R: set @ ( product_prod @ A @ A )] :
            ( collect @ ( product_prod @ ( list @ A ) @ ( list @ A ) )
            @ ( product_case_prod @ ( list @ A ) @ ( list @ A ) @ $o
              @ ^ [Xs: list @ A,Ys3: list @ A] :
                  ( ( ( size_size @ ( list @ A ) @ Xs )
                    = ( size_size @ ( list @ A ) @ Ys3 ) )
                  & ? [Xys: list @ A,X: A,Y: A,Xs6: list @ A,Ys7: list @ A] :
                      ( ( Xs
                        = ( append @ A @ Xys @ ( cons @ A @ X @ Xs6 ) ) )
                      & ( Ys3
                        = ( append @ A @ Xys @ ( cons @ A @ Y @ Ys7 ) ) )
                      & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R ) ) ) ) ) ) ) ).

% lex_conv
thf(fact_8068_less__eq__char__simp,axiom,
    ! [B0: $o,B12: $o,B23: $o,B32: $o,B42: $o,B52: $o,B62: $o,B72: $o,C0: $o,C1: $o,C22: $o,C32: $o,C42: $o,C52: $o,C62: $o,C72: $o] :
      ( ( ord_less_eq @ char @ ( char2 @ B0 @ B12 @ B23 @ B32 @ B42 @ B52 @ B62 @ B72 ) @ ( char2 @ C0 @ C1 @ C22 @ C32 @ C42 @ C52 @ C62 @ C72 ) )
      = ( ord_less_eq @ nat
        @ ( foldr @ $o @ nat
          @ ^ [B5: $o,K4: nat] : ( plus_plus @ nat @ ( zero_neq_one_of_bool @ nat @ B5 ) @ ( times_times @ nat @ K4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
          @ ( cons @ $o @ B0 @ ( cons @ $o @ B12 @ ( cons @ $o @ B23 @ ( cons @ $o @ B32 @ ( cons @ $o @ B42 @ ( cons @ $o @ B52 @ ( cons @ $o @ B62 @ ( cons @ $o @ B72 @ ( nil @ $o ) ) ) ) ) ) ) ) )
          @ ( zero_zero @ nat ) )
        @ ( foldr @ $o @ nat
          @ ^ [B5: $o,K4: nat] : ( plus_plus @ nat @ ( zero_neq_one_of_bool @ nat @ B5 ) @ ( times_times @ nat @ K4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
          @ ( cons @ $o @ C0 @ ( cons @ $o @ C1 @ ( cons @ $o @ C22 @ ( cons @ $o @ C32 @ ( cons @ $o @ C42 @ ( cons @ $o @ C52 @ ( cons @ $o @ C62 @ ( cons @ $o @ C72 @ ( nil @ $o ) ) ) ) ) ) ) ) )
          @ ( zero_zero @ nat ) ) ) ) ).

% less_eq_char_simp
thf(fact_8069_Cons__in__lex,axiom,
    ! [A: $tType,X2: A,Xs2: list @ A,Y2: A,Ys: list @ A,R4: set @ ( product_prod @ A @ A )] :
      ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X2 @ Xs2 ) @ ( cons @ A @ Y2 @ Ys ) ) @ ( lex @ A @ R4 ) )
      = ( ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y2 ) @ R4 )
          & ( ( size_size @ ( list @ A ) @ Xs2 )
            = ( size_size @ ( list @ A ) @ Ys ) ) )
        | ( ( X2 = Y2 )
          & ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs2 @ Ys ) @ ( lex @ A @ R4 ) ) ) ) ) ).

% Cons_in_lex
thf(fact_8070_less__char__simp,axiom,
    ! [B0: $o,B12: $o,B23: $o,B32: $o,B42: $o,B52: $o,B62: $o,B72: $o,C0: $o,C1: $o,C22: $o,C32: $o,C42: $o,C52: $o,C62: $o,C72: $o] :
      ( ( ord_less @ char @ ( char2 @ B0 @ B12 @ B23 @ B32 @ B42 @ B52 @ B62 @ B72 ) @ ( char2 @ C0 @ C1 @ C22 @ C32 @ C42 @ C52 @ C62 @ C72 ) )
      = ( ord_less @ nat
        @ ( foldr @ $o @ nat
          @ ^ [B5: $o,K4: nat] : ( plus_plus @ nat @ ( zero_neq_one_of_bool @ nat @ B5 ) @ ( times_times @ nat @ K4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
          @ ( cons @ $o @ B0 @ ( cons @ $o @ B12 @ ( cons @ $o @ B23 @ ( cons @ $o @ B32 @ ( cons @ $o @ B42 @ ( cons @ $o @ B52 @ ( cons @ $o @ B62 @ ( cons @ $o @ B72 @ ( nil @ $o ) ) ) ) ) ) ) ) )
          @ ( zero_zero @ nat ) )
        @ ( foldr @ $o @ nat
          @ ^ [B5: $o,K4: nat] : ( plus_plus @ nat @ ( zero_neq_one_of_bool @ nat @ B5 ) @ ( times_times @ nat @ K4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
          @ ( cons @ $o @ C0 @ ( cons @ $o @ C1 @ ( cons @ $o @ C22 @ ( cons @ $o @ C32 @ ( cons @ $o @ C42 @ ( cons @ $o @ C52 @ ( cons @ $o @ C62 @ ( cons @ $o @ C72 @ ( nil @ $o ) ) ) ) ) ) ) ) )
          @ ( zero_zero @ nat ) ) ) ) ).

% less_char_simp
thf(fact_8071_lex__append__leftI,axiom,
    ! [A: $tType,Ys: list @ A,Zs: list @ A,R4: set @ ( product_prod @ A @ A ),Xs2: list @ A] :
      ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Ys @ Zs ) @ ( lex @ A @ R4 ) )
     => ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Xs2 @ Ys ) @ ( append @ A @ Xs2 @ Zs ) ) @ ( lex @ A @ R4 ) ) ) ).

% lex_append_leftI
thf(fact_8072_Nil__notin__lex,axiom,
    ! [A: $tType,Ys: list @ A,R4: set @ ( product_prod @ A @ A )] :
      ~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Ys ) @ ( lex @ A @ R4 ) ) ).

% Nil_notin_lex
thf(fact_8073_Nil2__notin__lex,axiom,
    ! [A: $tType,Xs2: list @ A,R4: set @ ( product_prod @ A @ A )] :
      ~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs2 @ ( nil @ A ) ) @ ( lex @ A @ R4 ) ) ).

% Nil2_notin_lex
thf(fact_8074_lex__append__leftD,axiom,
    ! [A: $tType,R4: set @ ( product_prod @ A @ A ),Xs2: list @ A,Ys: list @ A,Zs: list @ A] :
      ( ! [X3: A] :
          ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ X3 ) @ R4 )
     => ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Xs2 @ Ys ) @ ( append @ A @ Xs2 @ Zs ) ) @ ( lex @ A @ R4 ) )
       => ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Ys @ Zs ) @ ( lex @ A @ R4 ) ) ) ) ).

% lex_append_leftD
thf(fact_8075_lex__append__left__iff,axiom,
    ! [A: $tType,R4: set @ ( product_prod @ A @ A ),Xs2: list @ A,Ys: list @ A,Zs: list @ A] :
      ( ! [X3: A] :
          ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ X3 ) @ R4 )
     => ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Xs2 @ Ys ) @ ( append @ A @ Xs2 @ Zs ) ) @ ( lex @ A @ R4 ) )
        = ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Ys @ Zs ) @ ( lex @ A @ R4 ) ) ) ) ).

% lex_append_left_iff
thf(fact_8076_lex__append__rightI,axiom,
    ! [A: $tType,Xs2: list @ A,Ys: list @ A,R4: set @ ( product_prod @ A @ A ),Vs: list @ A,Us: list @ A] :
      ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs2 @ Ys ) @ ( lex @ A @ R4 ) )
     => ( ( ( size_size @ ( list @ A ) @ Vs )
          = ( size_size @ ( list @ A ) @ Us ) )
       => ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Xs2 @ Us ) @ ( append @ A @ Ys @ Vs ) ) @ ( lex @ A @ R4 ) ) ) ) ).

% lex_append_rightI
thf(fact_8077_integer__of__char__code,axiom,
    ! [B0: $o,B12: $o,B23: $o,B32: $o,B42: $o,B52: $o,B62: $o,B72: $o] :
      ( ( integer_of_char @ ( char2 @ B0 @ B12 @ B23 @ B32 @ B42 @ B52 @ B62 @ B72 ) )
      = ( plus_plus @ code_integer @ ( times_times @ code_integer @ ( plus_plus @ code_integer @ ( times_times @ code_integer @ ( plus_plus @ code_integer @ ( times_times @ code_integer @ ( plus_plus @ code_integer @ ( times_times @ code_integer @ ( plus_plus @ code_integer @ ( times_times @ code_integer @ ( plus_plus @ code_integer @ ( times_times @ code_integer @ ( plus_plus @ code_integer @ ( times_times @ code_integer @ ( zero_neq_one_of_bool @ code_integer @ B72 ) @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) ) @ ( zero_neq_one_of_bool @ code_integer @ B62 ) ) @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) ) @ ( zero_neq_one_of_bool @ code_integer @ B52 ) ) @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) ) @ ( zero_neq_one_of_bool @ code_integer @ B42 ) ) @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) ) @ ( zero_neq_one_of_bool @ code_integer @ B32 ) ) @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) ) @ ( zero_neq_one_of_bool @ code_integer @ B23 ) ) @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) ) @ ( zero_neq_one_of_bool @ code_integer @ B12 ) ) @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) ) @ ( zero_neq_one_of_bool @ code_integer @ B0 ) ) ) ).

% integer_of_char_code
thf(fact_8078_of__char__Char,axiom,
    ! [A: $tType] :
      ( ( comm_semiring_1 @ A )
     => ! [B0: $o,B12: $o,B23: $o,B32: $o,B42: $o,B52: $o,B62: $o,B72: $o] :
          ( ( comm_s6883823935334413003f_char @ A @ ( char2 @ B0 @ B12 @ B23 @ B32 @ B42 @ B52 @ B62 @ B72 ) )
          = ( groups4207007520872428315er_sum @ $o @ A @ ( zero_neq_one_of_bool @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( cons @ $o @ B0 @ ( cons @ $o @ B12 @ ( cons @ $o @ B23 @ ( cons @ $o @ B32 @ ( cons @ $o @ B42 @ ( cons @ $o @ B52 @ ( cons @ $o @ B62 @ ( cons @ $o @ B72 @ ( nil @ $o ) ) ) ) ) ) ) ) ) ) ) ) ).

% of_char_Char
thf(fact_8079_of__char__mod__256,axiom,
    ! [A: $tType] :
      ( ( bit_un5681908812861735899ations @ A )
     => ! [C2: char] :
          ( ( modulo_modulo @ A @ ( comm_s6883823935334413003f_char @ A @ C2 ) @ ( numeral_numeral @ A @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) )
          = ( comm_s6883823935334413003f_char @ A @ C2 ) ) ) ).

% of_char_mod_256
thf(fact_8080_less__char__def,axiom,
    ( ( ord_less @ char )
    = ( ^ [C13: char,C24: char] : ( ord_less @ nat @ ( comm_s6883823935334413003f_char @ nat @ C13 ) @ ( comm_s6883823935334413003f_char @ nat @ C24 ) ) ) ) ).

% less_char_def
thf(fact_8081_nat__of__char__less__256,axiom,
    ! [C2: char] : ( ord_less @ nat @ ( comm_s6883823935334413003f_char @ nat @ C2 ) @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ).

% nat_of_char_less_256
thf(fact_8082_less__eq__char__def,axiom,
    ( ( ord_less_eq @ char )
    = ( ^ [C13: char,C24: char] : ( ord_less_eq @ nat @ ( comm_s6883823935334413003f_char @ nat @ C13 ) @ ( comm_s6883823935334413003f_char @ nat @ C24 ) ) ) ) ).

% less_eq_char_def
thf(fact_8083_range__nat__of__char,axiom,
    ( ( image @ char @ nat @ ( comm_s6883823935334413003f_char @ nat ) @ ( top_top @ ( set @ char ) ) )
    = ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ).

% range_nat_of_char
thf(fact_8084_char_Osize_I2_J,axiom,
    ! [X1: $o,X23: $o,X33: $o,X43: $o,X52: $o,X62: $o,X72: $o,X82: $o] :
      ( ( size_size @ char @ ( char2 @ X1 @ X23 @ X33 @ X43 @ X52 @ X62 @ X72 @ X82 ) )
      = ( zero_zero @ nat ) ) ).

% char.size(2)
thf(fact_8085_char__of__def,axiom,
    ! [A: $tType] :
      ( ( bit_un5681908812861735899ations @ A )
     => ( ( unique5772411509450598832har_of @ A )
        = ( ^ [N: A] :
              ( char2
              @ ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N )
              @ ( bit_se5641148757651400278ts_bit @ A @ N @ ( one_one @ nat ) )
              @ ( bit_se5641148757651400278ts_bit @ A @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
              @ ( bit_se5641148757651400278ts_bit @ A @ N @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) )
              @ ( bit_se5641148757651400278ts_bit @ A @ N @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) )
              @ ( bit_se5641148757651400278ts_bit @ A @ N @ ( numeral_numeral @ nat @ ( bit1 @ ( bit0 @ one2 ) ) ) )
              @ ( bit_se5641148757651400278ts_bit @ A @ N @ ( numeral_numeral @ nat @ ( bit0 @ ( bit1 @ one2 ) ) ) )
              @ ( bit_se5641148757651400278ts_bit @ A @ N @ ( numeral_numeral @ nat @ ( bit1 @ ( bit1 @ one2 ) ) ) ) ) ) ) ) ).

% char_of_def
thf(fact_8086_char__of__integer__code,axiom,
    ( char_of_integer
    = ( ^ [K4: code_integer] :
          ( product_case_prod @ code_integer @ $o @ char
          @ ^ [Q0: code_integer,B02: $o] :
              ( product_case_prod @ code_integer @ $o @ char
              @ ^ [Q1: code_integer,B13: $o] :
                  ( product_case_prod @ code_integer @ $o @ char
                  @ ^ [Q22: code_integer,B24: $o] :
                      ( product_case_prod @ code_integer @ $o @ char
                      @ ^ [Q32: code_integer,B33: $o] :
                          ( product_case_prod @ code_integer @ $o @ char
                          @ ^ [Q42: code_integer,B43: $o] :
                              ( product_case_prod @ code_integer @ $o @ char
                              @ ^ [Q52: code_integer,B53: $o] :
                                  ( product_case_prod @ code_integer @ $o @ char
                                  @ ^ [Q62: code_integer,B63: $o] :
                                      ( product_case_prod @ code_integer @ $o @ char
                                      @ ^ [Uu3: code_integer] : ( char2 @ B02 @ B13 @ B24 @ B33 @ B43 @ B53 @ B63 )
                                      @ ( code_bit_cut_integer @ Q62 ) )
                                  @ ( code_bit_cut_integer @ Q52 ) )
                              @ ( code_bit_cut_integer @ Q42 ) )
                          @ ( code_bit_cut_integer @ Q32 ) )
                      @ ( code_bit_cut_integer @ Q22 ) )
                  @ ( code_bit_cut_integer @ Q1 ) )
              @ ( code_bit_cut_integer @ Q0 ) )
          @ ( code_bit_cut_integer @ K4 ) ) ) ) ).

% char_of_integer_code
thf(fact_8087_char__of__quasi__inj,axiom,
    ! [A: $tType] :
      ( ( bit_un5681908812861735899ations @ A )
     => ! [M: A,N2: A] :
          ( ( ( unique5772411509450598832har_of @ A @ M )
            = ( unique5772411509450598832har_of @ A @ N2 ) )
          = ( ( modulo_modulo @ A @ M @ ( numeral_numeral @ A @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) )
            = ( modulo_modulo @ A @ N2 @ ( numeral_numeral @ A @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% char_of_quasi_inj
thf(fact_8088_char__of__mod__256,axiom,
    ! [A: $tType] :
      ( ( bit_un5681908812861735899ations @ A )
     => ! [N2: A] :
          ( ( unique5772411509450598832har_of @ A @ ( modulo_modulo @ A @ N2 @ ( numeral_numeral @ A @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) )
          = ( unique5772411509450598832har_of @ A @ N2 ) ) ) ).

% char_of_mod_256
thf(fact_8089_of__char__of,axiom,
    ! [A: $tType] :
      ( ( bit_un5681908812861735899ations @ A )
     => ! [A2: A] :
          ( ( comm_s6883823935334413003f_char @ A @ ( unique5772411509450598832har_of @ A @ A2 ) )
          = ( modulo_modulo @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ).

% of_char_of
thf(fact_8090_UNIV__char__of__nat,axiom,
    ( ( top_top @ ( set @ char ) )
    = ( image @ nat @ char @ ( unique5772411509450598832har_of @ nat ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ).

% UNIV_char_of_nat
thf(fact_8091_inj__on__char__of__nat,axiom,
    inj_on @ nat @ char @ ( unique5772411509450598832har_of @ nat ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ).

% inj_on_char_of_nat
thf(fact_8092_char__of__take__bit__eq,axiom,
    ! [A: $tType] :
      ( ( bit_un5681908812861735899ations @ A )
     => ! [N2: nat,M: A] :
          ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) @ N2 )
         => ( ( unique5772411509450598832har_of @ A @ ( bit_se2584673776208193580ke_bit @ A @ N2 @ M ) )
            = ( unique5772411509450598832har_of @ A @ M ) ) ) ) ).

% char_of_take_bit_eq
thf(fact_8093_char__of__eq__iff,axiom,
    ! [A: $tType] :
      ( ( bit_un5681908812861735899ations @ A )
     => ! [N2: A,C2: char] :
          ( ( ( unique5772411509450598832har_of @ A @ N2 )
            = C2 )
          = ( ( bit_se2584673776208193580ke_bit @ A @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) @ N2 )
            = ( comm_s6883823935334413003f_char @ A @ C2 ) ) ) ) ).

% char_of_eq_iff
thf(fact_8094_String_Ochar__of__ascii__of,axiom,
    ! [C2: char] :
      ( ( comm_s6883823935334413003f_char @ nat @ ( ascii_of @ C2 ) )
      = ( bit_se2584673776208193580ke_bit @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit1 @ one2 ) ) ) @ ( comm_s6883823935334413003f_char @ nat @ C2 ) ) ) ).

% String.char_of_ascii_of
thf(fact_8095_lenlex__conv,axiom,
    ! [A: $tType] :
      ( ( lenlex @ A )
      = ( ^ [R: set @ ( product_prod @ A @ A )] :
            ( collect @ ( product_prod @ ( list @ A ) @ ( list @ A ) )
            @ ( product_case_prod @ ( list @ A ) @ ( list @ A ) @ $o
              @ ^ [Xs: list @ A,Ys3: list @ A] :
                  ( ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( size_size @ ( list @ A ) @ Ys3 ) )
                  | ( ( ( size_size @ ( list @ A ) @ Xs )
                      = ( size_size @ ( list @ A ) @ Ys3 ) )
                    & ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys3 ) @ ( lex @ A @ R ) ) ) ) ) ) ) ) ).

% lenlex_conv
thf(fact_8096_Nil__lenlex__iff1,axiom,
    ! [A: $tType,Ns: list @ A,R4: set @ ( product_prod @ A @ A )] :
      ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Ns ) @ ( lenlex @ A @ R4 ) )
      = ( Ns
       != ( nil @ A ) ) ) ).

% Nil_lenlex_iff1
thf(fact_8097_Nil__lenlex__iff2,axiom,
    ! [A: $tType,Ns: list @ A,R4: set @ ( product_prod @ A @ A )] :
      ~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Ns @ ( nil @ A ) ) @ ( lenlex @ A @ R4 ) ) ).

% Nil_lenlex_iff2
thf(fact_8098_lenlex__irreflexive,axiom,
    ! [A: $tType,R4: set @ ( product_prod @ A @ A ),Xs2: list @ A] :
      ( ! [X3: A] :
          ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ X3 ) @ R4 )
     => ~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs2 @ Xs2 ) @ ( lenlex @ A @ R4 ) ) ) ).

% lenlex_irreflexive
thf(fact_8099_lenlex__length,axiom,
    ! [A: $tType,Ms: list @ A,Ns: list @ A,R4: set @ ( product_prod @ A @ A )] :
      ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Ms @ Ns ) @ ( lenlex @ A @ R4 ) )
     => ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Ms ) @ ( size_size @ ( list @ A ) @ Ns ) ) ) ).

% lenlex_length
thf(fact_8100_lenlex__append1,axiom,
    ! [A: $tType,Us: list @ A,Xs2: list @ A,R2: set @ ( product_prod @ A @ A ),Vs: list @ A,Ys: list @ A] :
      ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Us @ Xs2 ) @ ( lenlex @ A @ R2 ) )
     => ( ( ( size_size @ ( list @ A ) @ Vs )
          = ( size_size @ ( list @ A ) @ Ys ) )
       => ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Us @ Vs ) @ ( append @ A @ Xs2 @ Ys ) ) @ ( lenlex @ A @ R2 ) ) ) ) ).

% lenlex_append1
thf(fact_8101_Cons__lenlex__iff,axiom,
    ! [A: $tType,M: A,Ms: list @ A,N2: A,Ns: list @ A,R4: set @ ( product_prod @ A @ A )] :
      ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ M @ Ms ) @ ( cons @ A @ N2 @ Ns ) ) @ ( lenlex @ A @ R4 ) )
      = ( ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Ms ) @ ( size_size @ ( list @ A ) @ Ns ) )
        | ( ( ( size_size @ ( list @ A ) @ Ms )
            = ( size_size @ ( list @ A ) @ Ns ) )
          & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ M @ N2 ) @ R4 ) )
        | ( ( M = N2 )
          & ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Ms @ Ns ) @ ( lenlex @ A @ R4 ) ) ) ) ) ).

% Cons_lenlex_iff
thf(fact_8102_Eps__Opt__def,axiom,
    ! [A: $tType] :
      ( ( eps_Opt @ A )
      = ( ^ [P3: A > $o] :
            ( if @ ( option @ A )
            @ ? [X5: A] : ( P3 @ X5 )
            @ ( some @ A @ ( fChoice @ A @ P3 ) )
            @ ( none @ A ) ) ) ) ).

% Eps_Opt_def
thf(fact_8103_DERIV__even__real__root,axiom,
    ! [N2: nat,X2: real] :
      ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
     => ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 )
       => ( ( ord_less @ real @ X2 @ ( zero_zero @ real ) )
         => ( has_field_derivative @ real @ ( root @ N2 ) @ ( inverse_inverse @ real @ ( times_times @ real @ ( uminus_uminus @ real @ ( semiring_1_of_nat @ real @ N2 ) ) @ ( power_power @ real @ ( root @ N2 @ X2 ) @ ( minus_minus @ nat @ N2 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) ) ) ) ) ).

% DERIV_even_real_root
thf(fact_8104_some__opt__sym__eq__trivial,axiom,
    ! [A: $tType,X2: A] :
      ( ( eps_Opt @ A
        @ ( ^ [Y5: A,Z4: A] : Y5 = Z4
          @ X2 ) )
      = ( some @ A @ X2 ) ) ).

% some_opt_sym_eq_trivial
thf(fact_8105_some__opt__eq__trivial,axiom,
    ! [A: $tType,X2: A] :
      ( ( eps_Opt @ A
        @ ^ [Y: A] : Y = X2 )
      = ( some @ A @ X2 ) ) ).

% some_opt_eq_trivial
thf(fact_8106_some__opt__false__trivial,axiom,
    ! [A: $tType] :
      ( ( eps_Opt @ A
        @ ^ [Uu3: A] : $false )
      = ( none @ A ) ) ).

% some_opt_false_trivial
thf(fact_8107_Eps__Opt__eq__None,axiom,
    ! [A: $tType,P: A > $o] :
      ( ( ( eps_Opt @ A @ P )
        = ( none @ A ) )
      = ( ~ ? [X5: A] : ( P @ X5 ) ) ) ).

% Eps_Opt_eq_None
thf(fact_8108_DERIV__at__within__shift,axiom,
    ! [A: $tType] :
      ( ( real_V3459762299906320749_field @ A )
     => ! [F: A > A,Y2: A,Z: A,X2: A,S3: set @ A] :
          ( ( has_field_derivative @ A @ F @ Y2 @ ( topolo174197925503356063within @ A @ ( plus_plus @ A @ Z @ X2 ) @ ( image @ A @ A @ ( plus_plus @ A @ Z ) @ S3 ) ) )
          = ( has_field_derivative @ A
            @ ^ [X: A] : ( F @ ( plus_plus @ A @ Z @ X ) )
            @ Y2
            @ ( topolo174197925503356063within @ A @ X2 @ S3 ) ) ) ) ).

% DERIV_at_within_shift
thf(fact_8109_DERIV__at__within__shift__lemma,axiom,
    ! [A: $tType] :
      ( ( real_V3459762299906320749_field @ A )
     => ! [F: A > A,Y2: A,Z: A,X2: A,S3: set @ A] :
          ( ( has_field_derivative @ A @ F @ Y2 @ ( topolo174197925503356063within @ A @ ( plus_plus @ A @ Z @ X2 ) @ ( image @ A @ A @ ( plus_plus @ A @ Z ) @ S3 ) ) )
         => ( has_field_derivative @ A @ ( comp @ A @ A @ A @ F @ ( plus_plus @ A @ Z ) ) @ Y2 @ ( topolo174197925503356063within @ A @ X2 @ S3 ) ) ) ) ).

% DERIV_at_within_shift_lemma
thf(fact_8110_DERIV__image__chain,axiom,
    ! [A: $tType] :
      ( ( real_V3459762299906320749_field @ A )
     => ! [F: A > A,Da: A,G: A > A,X2: A,S4: set @ A,Db: A] :
          ( ( has_field_derivative @ A @ F @ Da @ ( topolo174197925503356063within @ A @ ( G @ X2 ) @ ( image @ A @ A @ G @ S4 ) ) )
         => ( ( has_field_derivative @ A @ G @ Db @ ( topolo174197925503356063within @ A @ X2 @ S4 ) )
           => ( has_field_derivative @ A @ ( comp @ A @ A @ A @ F @ G ) @ ( times_times @ A @ Da @ Db ) @ ( topolo174197925503356063within @ A @ X2 @ S4 ) ) ) ) ) ).

% DERIV_image_chain
thf(fact_8111_at__within__Icc__at__left,axiom,
    ! [A: $tType] :
      ( ( topolo2564578578187576103pology @ A )
     => ! [A2: A,B2: A] :
          ( ( ord_less @ A @ A2 @ B2 )
         => ( ( topolo174197925503356063within @ A @ B2 @ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) )
            = ( topolo174197925503356063within @ A @ B2 @ ( set_ord_lessThan @ A @ B2 ) ) ) ) ) ).

% at_within_Icc_at_left
thf(fact_8112_has__real__derivative__neg__dec__left,axiom,
    ! [F: real > real,L: real,X2: real,S3: set @ real] :
      ( ( has_field_derivative @ real @ F @ L @ ( topolo174197925503356063within @ real @ X2 @ S3 ) )
     => ( ( ord_less @ real @ L @ ( zero_zero @ real ) )
       => ? [D4: real] :
            ( ( ord_less @ real @ ( zero_zero @ real ) @ D4 )
            & ! [H6: real] :
                ( ( ord_less @ real @ ( zero_zero @ real ) @ H6 )
               => ( ( member @ real @ ( minus_minus @ real @ X2 @ H6 ) @ S3 )
                 => ( ( ord_less @ real @ H6 @ D4 )
                   => ( ord_less @ real @ ( F @ X2 ) @ ( F @ ( minus_minus @ real @ X2 @ H6 ) ) ) ) ) ) ) ) ) ).

% has_real_derivative_neg_dec_left
thf(fact_8113_has__real__derivative__pos__inc__left,axiom,
    ! [F: real > real,L: real,X2: real,S3: set @ real] :
      ( ( has_field_derivative @ real @ F @ L @ ( topolo174197925503356063within @ real @ X2 @ S3 ) )
     => ( ( ord_less @ real @ ( zero_zero @ real ) @ L )
       => ? [D4: real] :
            ( ( ord_less @ real @ ( zero_zero @ real ) @ D4 )
            & ! [H6: real] :
                ( ( ord_less @ real @ ( zero_zero @ real ) @ H6 )
               => ( ( member @ real @ ( minus_minus @ real @ X2 @ H6 ) @ S3 )
                 => ( ( ord_less @ real @ H6 @ D4 )
                   => ( ord_less @ real @ ( F @ ( minus_minus @ real @ X2 @ H6 ) ) @ ( F @ X2 ) ) ) ) ) ) ) ) ).

% has_real_derivative_pos_inc_left
thf(fact_8114_has__real__derivative__neg__dec__right,axiom,
    ! [F: real > real,L: real,X2: real,S3: set @ real] :
      ( ( has_field_derivative @ real @ F @ L @ ( topolo174197925503356063within @ real @ X2 @ S3 ) )
     => ( ( ord_less @ real @ L @ ( zero_zero @ real ) )
       => ? [D4: real] :
            ( ( ord_less @ real @ ( zero_zero @ real ) @ D4 )
            & ! [H6: real] :
                ( ( ord_less @ real @ ( zero_zero @ real ) @ H6 )
               => ( ( member @ real @ ( plus_plus @ real @ X2 @ H6 ) @ S3 )
                 => ( ( ord_less @ real @ H6 @ D4 )
                   => ( ord_less @ real @ ( F @ ( plus_plus @ real @ X2 @ H6 ) ) @ ( F @ X2 ) ) ) ) ) ) ) ) ).

% has_real_derivative_neg_dec_right
thf(fact_8115_has__real__derivative__pos__inc__right,axiom,
    ! [F: real > real,L: real,X2: real,S3: set @ real] :
      ( ( has_field_derivative @ real @ F @ L @ ( topolo174197925503356063within @ real @ X2 @ S3 ) )
     => ( ( ord_less @ real @ ( zero_zero @ real ) @ L )
       => ? [D4: real] :
            ( ( ord_less @ real @ ( zero_zero @ real ) @ D4 )
            & ! [H6: real] :
                ( ( ord_less @ real @ ( zero_zero @ real ) @ H6 )
               => ( ( member @ real @ ( plus_plus @ real @ X2 @ H6 ) @ S3 )
                 => ( ( ord_less @ real @ H6 @ D4 )
                   => ( ord_less @ real @ ( F @ X2 ) @ ( F @ ( plus_plus @ real @ X2 @ H6 ) ) ) ) ) ) ) ) ) ).

% has_real_derivative_pos_inc_right
thf(fact_8116_DERIV__pow,axiom,
    ! [N2: nat,X2: real,S4: set @ real] :
      ( has_field_derivative @ real
      @ ^ [X: real] : ( power_power @ real @ X @ N2 )
      @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N2 ) @ ( power_power @ real @ X2 @ ( minus_minus @ nat @ N2 @ ( suc @ ( zero_zero @ nat ) ) ) ) )
      @ ( topolo174197925503356063within @ real @ X2 @ S4 ) ) ).

% DERIV_pow
thf(fact_8117_DERIV__power,axiom,
    ! [A: $tType] :
      ( ( real_V3459762299906320749_field @ A )
     => ! [F: A > A,D6: A,X2: A,S4: set @ A,N2: nat] :
          ( ( has_field_derivative @ A @ F @ D6 @ ( topolo174197925503356063within @ A @ X2 @ S4 ) )
         => ( has_field_derivative @ A
            @ ^ [X: A] : ( power_power @ A @ ( F @ X ) @ N2 )
            @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N2 ) @ ( times_times @ A @ D6 @ ( power_power @ A @ ( F @ X2 ) @ ( minus_minus @ nat @ N2 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) )
            @ ( topolo174197925503356063within @ A @ X2 @ S4 ) ) ) ) ).

% DERIV_power
thf(fact_8118_DERIV__quotient,axiom,
    ! [A: $tType] :
      ( ( real_V3459762299906320749_field @ A )
     => ! [F: A > A,D2: A,X2: A,S4: set @ A,G: A > A,E3: A] :
          ( ( has_field_derivative @ A @ F @ D2 @ ( topolo174197925503356063within @ A @ X2 @ S4 ) )
         => ( ( has_field_derivative @ A @ G @ E3 @ ( topolo174197925503356063within @ A @ X2 @ S4 ) )
           => ( ( ( G @ X2 )
               != ( zero_zero @ A ) )
             => ( has_field_derivative @ A
                @ ^ [Y: A] : ( divide_divide @ A @ ( F @ Y ) @ ( G @ Y ) )
                @ ( divide_divide @ A @ ( minus_minus @ A @ ( times_times @ A @ D2 @ ( G @ X2 ) ) @ ( times_times @ A @ E3 @ ( F @ X2 ) ) ) @ ( power_power @ A @ ( G @ X2 ) @ ( suc @ ( suc @ ( zero_zero @ nat ) ) ) ) )
                @ ( topolo174197925503356063within @ A @ X2 @ S4 ) ) ) ) ) ) ).

% DERIV_quotient
thf(fact_8119_DERIV__divide,axiom,
    ! [A: $tType] :
      ( ( real_V3459762299906320749_field @ A )
     => ! [F: A > A,D6: A,X2: A,S4: set @ A,G: A > A,E6: A] :
          ( ( has_field_derivative @ A @ F @ D6 @ ( topolo174197925503356063within @ A @ X2 @ S4 ) )
         => ( ( has_field_derivative @ A @ G @ E6 @ ( topolo174197925503356063within @ A @ X2 @ S4 ) )
           => ( ( ( G @ X2 )
               != ( zero_zero @ A ) )
             => ( has_field_derivative @ A
                @ ^ [X: A] : ( divide_divide @ A @ ( F @ X ) @ ( G @ X ) )
                @ ( divide_divide @ A @ ( minus_minus @ A @ ( times_times @ A @ D6 @ ( G @ X2 ) ) @ ( times_times @ A @ ( F @ X2 ) @ E6 ) ) @ ( times_times @ A @ ( G @ X2 ) @ ( G @ X2 ) ) )
                @ ( topolo174197925503356063within @ A @ X2 @ S4 ) ) ) ) ) ) ).

% DERIV_divide
thf(fact_8120_DERIV__const,axiom,
    ! [A: $tType] :
      ( ( real_V3459762299906320749_field @ A )
     => ! [K: A,F5: filter @ A] :
          ( has_field_derivative @ A
          @ ^ [X: A] : K
          @ ( zero_zero @ A )
          @ F5 ) ) ).

% DERIV_const
thf(fact_8121_Eps__Opt__eq__Some,axiom,
    ! [A: $tType,P: A > $o,X2: A] :
      ( ! [X10: A] :
          ( ( P @ X2 )
         => ( ( P @ X10 )
           => ( X10 = X2 ) ) )
     => ( ( ( eps_Opt @ A @ P )
          = ( some @ A @ X2 ) )
        = ( P @ X2 ) ) ) ).

% Eps_Opt_eq_Some
thf(fact_8122_Eps__Opt__eq__Some__implies,axiom,
    ! [A: $tType,P: A > $o,X2: A] :
      ( ( ( eps_Opt @ A @ P )
        = ( some @ A @ X2 ) )
     => ( P @ X2 ) ) ).

% Eps_Opt_eq_Some_implies
thf(fact_8123_DERIV__sum,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( ( real_V3459762299906320749_field @ B )
     => ! [S3: set @ A,F: B > A > B,F4: C > A > B,X2: C,F5: filter @ B] :
          ( ! [N4: A] :
              ( ( member @ A @ N4 @ S3 )
             => ( has_field_derivative @ B
                @ ^ [X: B] : ( F @ X @ N4 )
                @ ( F4 @ X2 @ N4 )
                @ F5 ) )
         => ( has_field_derivative @ B
            @ ^ [X: B] : ( groups7311177749621191930dd_sum @ A @ B @ ( F @ X ) @ S3 )
            @ ( groups7311177749621191930dd_sum @ A @ B @ ( F4 @ X2 ) @ S3 )
            @ F5 ) ) ) ).

% DERIV_sum
thf(fact_8124_has__field__derivative__scaleR__right,axiom,
    ! [A: $tType] :
      ( ( real_V3459762299906320749_field @ A )
     => ! [F: A > A,D6: A,F5: filter @ A,C2: real] :
          ( ( has_field_derivative @ A @ F @ D6 @ F5 )
         => ( has_field_derivative @ A
            @ ^ [X: A] : ( real_V8093663219630862766scaleR @ A @ C2 @ ( F @ X ) )
            @ ( real_V8093663219630862766scaleR @ A @ C2 @ D6 )
            @ F5 ) ) ) ).

% has_field_derivative_scaleR_right
thf(fact_8125_DERIV__mult_H,axiom,
    ! [A: $tType] :
      ( ( real_V3459762299906320749_field @ A )
     => ! [F: A > A,D6: A,X2: A,S4: set @ A,G: A > A,E6: A] :
          ( ( has_field_derivative @ A @ F @ D6 @ ( topolo174197925503356063within @ A @ X2 @ S4 ) )
         => ( ( has_field_derivative @ A @ G @ E6 @ ( topolo174197925503356063within @ A @ X2 @ S4 ) )
           => ( has_field_derivative @ A
              @ ^ [X: A] : ( times_times @ A @ ( F @ X ) @ ( G @ X ) )
              @ ( plus_plus @ A @ ( times_times @ A @ ( F @ X2 ) @ E6 ) @ ( times_times @ A @ D6 @ ( G @ X2 ) ) )
              @ ( topolo174197925503356063within @ A @ X2 @ S4 ) ) ) ) ) ).

% DERIV_mult'
thf(fact_8126_DERIV__mult,axiom,
    ! [A: $tType] :
      ( ( real_V3459762299906320749_field @ A )
     => ! [F: A > A,Da: A,X2: A,S4: set @ A,G: A > A,Db: A] :
          ( ( has_field_derivative @ A @ F @ Da @ ( topolo174197925503356063within @ A @ X2 @ S4 ) )
         => ( ( has_field_derivative @ A @ G @ Db @ ( topolo174197925503356063within @ A @ X2 @ S4 ) )
           => ( has_field_derivative @ A
              @ ^ [X: A] : ( times_times @ A @ ( F @ X ) @ ( G @ X ) )
              @ ( plus_plus @ A @ ( times_times @ A @ Da @ ( G @ X2 ) ) @ ( times_times @ A @ Db @ ( F @ X2 ) ) )
              @ ( topolo174197925503356063within @ A @ X2 @ S4 ) ) ) ) ) ).

% DERIV_mult
thf(fact_8127_DERIV__cmult__right,axiom,
    ! [A: $tType] :
      ( ( real_V3459762299906320749_field @ A )
     => ! [F: A > A,D6: A,X2: A,S4: set @ A,C2: A] :
          ( ( has_field_derivative @ A @ F @ D6 @ ( topolo174197925503356063within @ A @ X2 @ S4 ) )
         => ( has_field_derivative @ A
            @ ^ [X: A] : ( times_times @ A @ ( F @ X ) @ C2 )
            @ ( times_times @ A @ D6 @ C2 )
            @ ( topolo174197925503356063within @ A @ X2 @ S4 ) ) ) ) ).

% DERIV_cmult_right
thf(fact_8128_DERIV__cmult,axiom,
    ! [A: $tType] :
      ( ( real_V3459762299906320749_field @ A )
     => ! [F: A > A,D6: A,X2: A,S4: set @ A,C2: A] :
          ( ( has_field_derivative @ A @ F @ D6 @ ( topolo174197925503356063within @ A @ X2 @ S4 ) )
         => ( has_field_derivative @ A
            @ ^ [X: A] : ( times_times @ A @ C2 @ ( F @ X ) )
            @ ( times_times @ A @ C2 @ D6 )
            @ ( topolo174197925503356063within @ A @ X2 @ S4 ) ) ) ) ).

% DERIV_cmult
thf(fact_8129_DERIV__cmult__Id,axiom,
    ! [A: $tType] :
      ( ( real_V3459762299906320749_field @ A )
     => ! [C2: A,X2: A,S4: set @ A] : ( has_field_derivative @ A @ ( times_times @ A @ C2 ) @ C2 @ ( topolo174197925503356063within @ A @ X2 @ S4 ) ) ) ).

% DERIV_cmult_Id
thf(fact_8130_has__field__derivative__sinh,axiom,
    ! [A15: $tType] :
      ( ( ( real_Vector_banach @ A15 )
        & ( real_V3459762299906320749_field @ A15 ) )
     => ! [G: A15 > A15,Db: A15,X2: A15,S4: set @ A15] :
          ( ( has_field_derivative @ A15 @ G @ Db @ ( topolo174197925503356063within @ A15 @ X2 @ S4 ) )
         => ( has_field_derivative @ A15
            @ ^ [X: A15] : ( sinh @ A15 @ ( G @ X ) )
            @ ( times_times @ A15 @ ( cosh @ A15 @ ( G @ X2 ) ) @ Db )
            @ ( topolo174197925503356063within @ A15 @ X2 @ S4 ) ) ) ) ).

% has_field_derivative_sinh
thf(fact_8131_has__field__derivative__cosh,axiom,
    ! [A15: $tType] :
      ( ( ( real_Vector_banach @ A15 )
        & ( real_V3459762299906320749_field @ A15 ) )
     => ! [G: A15 > A15,Db: A15,X2: A15,S4: set @ A15] :
          ( ( has_field_derivative @ A15 @ G @ Db @ ( topolo174197925503356063within @ A15 @ X2 @ S4 ) )
         => ( has_field_derivative @ A15
            @ ^ [X: A15] : ( cosh @ A15 @ ( G @ X ) )
            @ ( times_times @ A15 @ ( sinh @ A15 @ ( G @ X2 ) ) @ Db )
            @ ( topolo174197925503356063within @ A15 @ X2 @ S4 ) ) ) ) ).

% has_field_derivative_cosh
thf(fact_8132_DERIV__cdivide,axiom,
    ! [A: $tType] :
      ( ( real_V3459762299906320749_field @ A )
     => ! [F: A > A,D6: A,X2: A,S4: set @ A,C2: A] :
          ( ( has_field_derivative @ A @ F @ D6 @ ( topolo174197925503356063within @ A @ X2 @ S4 ) )
         => ( has_field_derivative @ A
            @ ^ [X: A] : ( divide_divide @ A @ ( F @ X ) @ C2 )
            @ ( divide_divide @ A @ D6 @ C2 )
            @ ( topolo174197925503356063within @ A @ X2 @ S4 ) ) ) ) ).

% DERIV_cdivide
thf(fact_8133_DERIV__ident,axiom,
    ! [A: $tType] :
      ( ( real_V3459762299906320749_field @ A )
     => ! [F5: filter @ A] :
          ( has_field_derivative @ A
          @ ^ [X: A] : X
          @ ( one_one @ A )
          @ F5 ) ) ).

% DERIV_ident
thf(fact_8134_field__differentiable__diff,axiom,
    ! [A: $tType] :
      ( ( real_V3459762299906320749_field @ A )
     => ! [F: A > A,F4: A,F5: filter @ A,G: A > A,G6: A] :
          ( ( has_field_derivative @ A @ F @ F4 @ F5 )
         => ( ( has_field_derivative @ A @ G @ G6 @ F5 )
           => ( has_field_derivative @ A
              @ ^ [Z2: A] : ( minus_minus @ A @ ( F @ Z2 ) @ ( G @ Z2 ) )
              @ ( minus_minus @ A @ F4 @ G6 )
              @ F5 ) ) ) ) ).

% field_differentiable_diff
thf(fact_8135_DERIV__diff,axiom,
    ! [A: $tType] :
      ( ( real_V3459762299906320749_field @ A )
     => ! [F: A > A,D6: A,X2: A,S4: set @ A,G: A > A,E6: A] :
          ( ( has_field_derivative @ A @ F @ D6 @ ( topolo174197925503356063within @ A @ X2 @ S4 ) )
         => ( ( has_field_derivative @ A @ G @ E6 @ ( topolo174197925503356063within @ A @ X2 @ S4 ) )
           => ( has_field_derivative @ A
              @ ^ [X: A] : ( minus_minus @ A @ ( F @ X ) @ ( G @ X ) )
              @ ( minus_minus @ A @ D6 @ E6 )
              @ ( topolo174197925503356063within @ A @ X2 @ S4 ) ) ) ) ) ).

% DERIV_diff
thf(fact_8136_field__differentiable__add,axiom,
    ! [A: $tType] :
      ( ( real_V3459762299906320749_field @ A )
     => ! [F: A > A,F4: A,F5: filter @ A,G: A > A,G6: A] :
          ( ( has_field_derivative @ A @ F @ F4 @ F5 )
         => ( ( has_field_derivative @ A @ G @ G6 @ F5 )
           => ( has_field_derivative @ A
              @ ^ [Z2: A] : ( plus_plus @ A @ ( F @ Z2 ) @ ( G @ Z2 ) )
              @ ( plus_plus @ A @ F4 @ G6 )
              @ F5 ) ) ) ) ).

% field_differentiable_add
thf(fact_8137_DERIV__add,axiom,
    ! [A: $tType] :
      ( ( real_V3459762299906320749_field @ A )
     => ! [F: A > A,D6: A,X2: A,S4: set @ A,G: A > A,E6: A] :
          ( ( has_field_derivative @ A @ F @ D6 @ ( topolo174197925503356063within @ A @ X2 @ S4 ) )
         => ( ( has_field_derivative @ A @ G @ E6 @ ( topolo174197925503356063within @ A @ X2 @ S4 ) )
           => ( has_field_derivative @ A
              @ ^ [X: A] : ( plus_plus @ A @ ( F @ X ) @ ( G @ X ) )
              @ ( plus_plus @ A @ D6 @ E6 )
              @ ( topolo174197925503356063within @ A @ X2 @ S4 ) ) ) ) ) ).

% DERIV_add
thf(fact_8138_DERIV__power__Suc,axiom,
    ! [A: $tType] :
      ( ( real_V3459762299906320749_field @ A )
     => ! [F: A > A,D6: A,X2: A,S4: set @ A,N2: nat] :
          ( ( has_field_derivative @ A @ F @ D6 @ ( topolo174197925503356063within @ A @ X2 @ S4 ) )
         => ( has_field_derivative @ A
            @ ^ [X: A] : ( power_power @ A @ ( F @ X ) @ ( suc @ N2 ) )
            @ ( times_times @ A @ ( plus_plus @ A @ ( one_one @ A ) @ ( semiring_1_of_nat @ A @ N2 ) ) @ ( times_times @ A @ D6 @ ( power_power @ A @ ( F @ X2 ) @ N2 ) ) )
            @ ( topolo174197925503356063within @ A @ X2 @ S4 ) ) ) ) ).

% DERIV_power_Suc
thf(fact_8139_DERIV__minus,axiom,
    ! [A: $tType] :
      ( ( real_V3459762299906320749_field @ A )
     => ! [F: A > A,D6: A,X2: A,S4: set @ A] :
          ( ( has_field_derivative @ A @ F @ D6 @ ( topolo174197925503356063within @ A @ X2 @ S4 ) )
         => ( has_field_derivative @ A
            @ ^ [X: A] : ( uminus_uminus @ A @ ( F @ X ) )
            @ ( uminus_uminus @ A @ D6 )
            @ ( topolo174197925503356063within @ A @ X2 @ S4 ) ) ) ) ).

% DERIV_minus
thf(fact_8140_field__differentiable__minus,axiom,
    ! [A: $tType] :
      ( ( real_V3459762299906320749_field @ A )
     => ! [F: A > A,F4: A,F5: filter @ A] :
          ( ( has_field_derivative @ A @ F @ F4 @ F5 )
         => ( has_field_derivative @ A
            @ ^ [Z2: A] : ( uminus_uminus @ A @ ( F @ Z2 ) )
            @ ( uminus_uminus @ A @ F4 )
            @ F5 ) ) ) ).

% field_differentiable_minus
thf(fact_8141_DERIV__inverse__fun,axiom,
    ! [A: $tType] :
      ( ( real_V3459762299906320749_field @ A )
     => ! [F: A > A,D2: A,X2: A,S4: set @ A] :
          ( ( has_field_derivative @ A @ F @ D2 @ ( topolo174197925503356063within @ A @ X2 @ S4 ) )
         => ( ( ( F @ X2 )
             != ( zero_zero @ A ) )
           => ( has_field_derivative @ A
              @ ^ [X: A] : ( inverse_inverse @ A @ ( F @ X ) )
              @ ( uminus_uminus @ A @ ( times_times @ A @ D2 @ ( inverse_inverse @ A @ ( power_power @ A @ ( F @ X2 ) @ ( suc @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) )
              @ ( topolo174197925503356063within @ A @ X2 @ S4 ) ) ) ) ) ).

% DERIV_inverse_fun
thf(fact_8142_DERIV__inverse_H,axiom,
    ! [A: $tType] :
      ( ( real_V3459762299906320749_field @ A )
     => ! [F: A > A,D6: A,X2: A,S4: set @ A] :
          ( ( has_field_derivative @ A @ F @ D6 @ ( topolo174197925503356063within @ A @ X2 @ S4 ) )
         => ( ( ( F @ X2 )
             != ( zero_zero @ A ) )
           => ( has_field_derivative @ A
              @ ^ [X: A] : ( inverse_inverse @ A @ ( F @ X ) )
              @ ( uminus_uminus @ A @ ( times_times @ A @ ( times_times @ A @ ( inverse_inverse @ A @ ( F @ X2 ) ) @ D6 ) @ ( inverse_inverse @ A @ ( F @ X2 ) ) ) )
              @ ( topolo174197925503356063within @ A @ X2 @ S4 ) ) ) ) ) ).

% DERIV_inverse'
thf(fact_8143_DERIV__inverse,axiom,
    ! [A: $tType] :
      ( ( real_V3459762299906320749_field @ A )
     => ! [X2: A,S4: set @ A] :
          ( ( X2
           != ( zero_zero @ A ) )
         => ( has_field_derivative @ A @ ( inverse_inverse @ A ) @ ( uminus_uminus @ A @ ( power_power @ A @ ( inverse_inverse @ A @ X2 ) @ ( suc @ ( suc @ ( zero_zero @ nat ) ) ) ) ) @ ( topolo174197925503356063within @ A @ X2 @ S4 ) ) ) ) ).

% DERIV_inverse
thf(fact_8144_has__field__derivative__subset,axiom,
    ! [A: $tType] :
      ( ( real_V3459762299906320749_field @ A )
     => ! [F: A > A,Y2: A,X2: A,S4: set @ A,T2: set @ A] :
          ( ( has_field_derivative @ A @ F @ Y2 @ ( topolo174197925503356063within @ A @ X2 @ S4 ) )
         => ( ( ord_less_eq @ ( set @ A ) @ T2 @ S4 )
           => ( has_field_derivative @ A @ F @ Y2 @ ( topolo174197925503356063within @ A @ X2 @ T2 ) ) ) ) ) ).

% has_field_derivative_subset
thf(fact_8145_DERIV__subset,axiom,
    ! [A: $tType] :
      ( ( real_V3459762299906320749_field @ A )
     => ! [F: A > A,F4: A,X2: A,S4: set @ A,T2: set @ A] :
          ( ( has_field_derivative @ A @ F @ F4 @ ( topolo174197925503356063within @ A @ X2 @ S4 ) )
         => ( ( ord_less_eq @ ( set @ A ) @ T2 @ S4 )
           => ( has_field_derivative @ A @ F @ F4 @ ( topolo174197925503356063within @ A @ X2 @ T2 ) ) ) ) ) ).

% DERIV_subset
thf(fact_8146_at__le,axiom,
    ! [A: $tType] :
      ( ( topolo4958980785337419405_space @ A )
     => ! [S4: set @ A,T2: set @ A,X2: A] :
          ( ( ord_less_eq @ ( set @ A ) @ S4 @ T2 )
         => ( ord_less_eq @ ( filter @ A ) @ ( topolo174197925503356063within @ A @ X2 @ S4 ) @ ( topolo174197925503356063within @ A @ X2 @ T2 ) ) ) ) ).

% at_le
thf(fact_8147_DERIV__nonpos__imp__nonincreasing,axiom,
    ! [A2: real,B2: real,F: real > real] :
      ( ( ord_less_eq @ real @ A2 @ B2 )
     => ( ! [X3: real] :
            ( ( ord_less_eq @ real @ A2 @ X3 )
           => ( ( ord_less_eq @ real @ X3 @ B2 )
             => ? [Y3: real] :
                  ( ( has_field_derivative @ real @ F @ Y3 @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) )
                  & ( ord_less_eq @ real @ Y3 @ ( zero_zero @ real ) ) ) ) )
       => ( ord_less_eq @ real @ ( F @ B2 ) @ ( F @ A2 ) ) ) ) ).

% DERIV_nonpos_imp_nonincreasing
thf(fact_8148_DERIV__nonneg__imp__nondecreasing,axiom,
    ! [A2: real,B2: real,F: real > real] :
      ( ( ord_less_eq @ real @ A2 @ B2 )
     => ( ! [X3: real] :
            ( ( ord_less_eq @ real @ A2 @ X3 )
           => ( ( ord_less_eq @ real @ X3 @ B2 )
             => ? [Y3: real] :
                  ( ( has_field_derivative @ real @ F @ Y3 @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) )
                  & ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y3 ) ) ) )
       => ( ord_less_eq @ real @ ( F @ A2 ) @ ( F @ B2 ) ) ) ) ).

% DERIV_nonneg_imp_nondecreasing
thf(fact_8149_deriv__nonneg__imp__mono,axiom,
    ! [A2: real,B2: real,G: real > real,G6: real > real] :
      ( ! [X3: real] :
          ( ( member @ real @ X3 @ ( set_or1337092689740270186AtMost @ real @ A2 @ B2 ) )
         => ( has_field_derivative @ real @ G @ ( G6 @ X3 ) @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) ) )
     => ( ! [X3: real] :
            ( ( member @ real @ X3 @ ( set_or1337092689740270186AtMost @ real @ A2 @ B2 ) )
           => ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( G6 @ X3 ) ) )
       => ( ( ord_less_eq @ real @ A2 @ B2 )
         => ( ord_less_eq @ real @ ( G @ A2 ) @ ( G @ B2 ) ) ) ) ) ).

% deriv_nonneg_imp_mono
thf(fact_8150_DERIV__mirror,axiom,
    ! [F: real > real,Y2: real,X2: real] :
      ( ( has_field_derivative @ real @ F @ Y2 @ ( topolo174197925503356063within @ real @ ( uminus_uminus @ real @ X2 ) @ ( top_top @ ( set @ real ) ) ) )
      = ( has_field_derivative @ real
        @ ^ [X: real] : ( F @ ( uminus_uminus @ real @ X ) )
        @ ( uminus_uminus @ real @ Y2 )
        @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) ) ) ).

% DERIV_mirror
thf(fact_8151_DERIV__cos__add,axiom,
    ! [A: $tType] :
      ( ( ( real_Vector_banach @ A )
        & ( real_V3459762299906320749_field @ A ) )
     => ! [K: A,Xa: A] :
          ( has_field_derivative @ A
          @ ^ [X: A] : ( cos @ A @ ( plus_plus @ A @ X @ K ) )
          @ ( uminus_uminus @ A @ ( sin @ A @ ( plus_plus @ A @ Xa @ K ) ) )
          @ ( topolo174197925503356063within @ A @ Xa @ ( top_top @ ( set @ A ) ) ) ) ) ).

% DERIV_cos_add
thf(fact_8152_DERIV__fun__cos,axiom,
    ! [A: $tType] :
      ( ( ( real_Vector_banach @ A )
        & ( real_V3459762299906320749_field @ A ) )
     => ! [G: A > A,M: A,X2: A] :
          ( ( has_field_derivative @ A @ G @ M @ ( topolo174197925503356063within @ A @ X2 @ ( top_top @ ( set @ A ) ) ) )
         => ( has_field_derivative @ A
            @ ^ [X: A] : ( cos @ A @ ( G @ X ) )
            @ ( times_times @ A @ ( uminus_uminus @ A @ ( sin @ A @ ( G @ X2 ) ) ) @ M )
            @ ( topolo174197925503356063within @ A @ X2 @ ( top_top @ ( set @ A ) ) ) ) ) ) ).

% DERIV_fun_cos
thf(fact_8153_DERIV__fun__pow,axiom,
    ! [G: real > real,M: real,X2: real,N2: nat] :
      ( ( has_field_derivative @ real @ G @ M @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) )
     => ( has_field_derivative @ real
        @ ^ [X: real] : ( power_power @ real @ ( G @ X ) @ N2 )
        @ ( times_times @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N2 ) @ ( power_power @ real @ ( G @ X2 ) @ ( minus_minus @ nat @ N2 @ ( one_one @ nat ) ) ) ) @ M )
        @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) ) ) ).

% DERIV_fun_pow
thf(fact_8154_DERIV__const__average,axiom,
    ! [A2: real,B2: real,V: real > real,K: real] :
      ( ( A2 != B2 )
     => ( ! [X3: real] : ( has_field_derivative @ real @ V @ K @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) )
       => ( ( V @ ( divide_divide @ real @ ( plus_plus @ real @ A2 @ B2 ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
          = ( divide_divide @ real @ ( plus_plus @ real @ ( V @ A2 ) @ ( V @ B2 ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ).

% DERIV_const_average
thf(fact_8155_DERIV__fun__sin,axiom,
    ! [A: $tType] :
      ( ( ( real_Vector_banach @ A )
        & ( real_V3459762299906320749_field @ A ) )
     => ! [G: A > A,M: A,X2: A] :
          ( ( has_field_derivative @ A @ G @ M @ ( topolo174197925503356063within @ A @ X2 @ ( top_top @ ( set @ A ) ) ) )
         => ( has_field_derivative @ A
            @ ^ [X: A] : ( sin @ A @ ( G @ X ) )
            @ ( times_times @ A @ ( cos @ A @ ( G @ X2 ) ) @ M )
            @ ( topolo174197925503356063within @ A @ X2 @ ( top_top @ ( set @ A ) ) ) ) ) ) ).

% DERIV_fun_sin
thf(fact_8156_DERIV__fun__exp,axiom,
    ! [A: $tType] :
      ( ( ( real_Vector_banach @ A )
        & ( real_V3459762299906320749_field @ A ) )
     => ! [G: A > A,M: A,X2: A] :
          ( ( has_field_derivative @ A @ G @ M @ ( topolo174197925503356063within @ A @ X2 @ ( top_top @ ( set @ A ) ) ) )
         => ( has_field_derivative @ A
            @ ^ [X: A] : ( exp @ A @ ( G @ X ) )
            @ ( times_times @ A @ ( exp @ A @ ( G @ X2 ) ) @ M )
            @ ( topolo174197925503356063within @ A @ X2 @ ( top_top @ ( set @ A ) ) ) ) ) ) ).

% DERIV_fun_exp
thf(fact_8157_DERIV__chain__s,axiom,
    ! [A: $tType] :
      ( ( real_V3459762299906320749_field @ A )
     => ! [S4: set @ A,G: A > A,G6: A > A,F: A > A,F4: A,X2: A] :
          ( ! [X3: A] :
              ( ( member @ A @ X3 @ S4 )
             => ( has_field_derivative @ A @ G @ ( G6 @ X3 ) @ ( topolo174197925503356063within @ A @ X3 @ ( top_top @ ( set @ A ) ) ) ) )
         => ( ( has_field_derivative @ A @ F @ F4 @ ( topolo174197925503356063within @ A @ X2 @ ( top_top @ ( set @ A ) ) ) )
           => ( ( member @ A @ ( F @ X2 ) @ S4 )
             => ( has_field_derivative @ A
                @ ^ [X: A] : ( G @ ( F @ X ) )
                @ ( times_times @ A @ F4 @ ( G6 @ ( F @ X2 ) ) )
                @ ( topolo174197925503356063within @ A @ X2 @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ) ).

% DERIV_chain_s
thf(fact_8158_DERIV__chain3,axiom,
    ! [A: $tType] :
      ( ( real_V3459762299906320749_field @ A )
     => ! [G: A > A,G6: A > A,F: A > A,F4: A,X2: A] :
          ( ! [X3: A] : ( has_field_derivative @ A @ G @ ( G6 @ X3 ) @ ( topolo174197925503356063within @ A @ X3 @ ( top_top @ ( set @ A ) ) ) )
         => ( ( has_field_derivative @ A @ F @ F4 @ ( topolo174197925503356063within @ A @ X2 @ ( top_top @ ( set @ A ) ) ) )
           => ( has_field_derivative @ A
              @ ^ [X: A] : ( G @ ( F @ X ) )
              @ ( times_times @ A @ F4 @ ( G6 @ ( F @ X2 ) ) )
              @ ( topolo174197925503356063within @ A @ X2 @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ).

% DERIV_chain3
thf(fact_8159_DERIV__chain2,axiom,
    ! [A: $tType] :
      ( ( real_V3459762299906320749_field @ A )
     => ! [F: A > A,Da: A,G: A > A,X2: A,Db: A,S4: set @ A] :
          ( ( has_field_derivative @ A @ F @ Da @ ( topolo174197925503356063within @ A @ ( G @ X2 ) @ ( top_top @ ( set @ A ) ) ) )
         => ( ( has_field_derivative @ A @ G @ Db @ ( topolo174197925503356063within @ A @ X2 @ S4 ) )
           => ( has_field_derivative @ A
              @ ^ [X: A] : ( F @ ( G @ X ) )
              @ ( times_times @ A @ Da @ Db )
              @ ( topolo174197925503356063within @ A @ X2 @ S4 ) ) ) ) ) ).

% DERIV_chain2
thf(fact_8160_DERIV__chain_H,axiom,
    ! [A: $tType] :
      ( ( real_V3459762299906320749_field @ A )
     => ! [F: A > A,D6: A,X2: A,S4: set @ A,G: A > A,E6: A] :
          ( ( has_field_derivative @ A @ F @ D6 @ ( topolo174197925503356063within @ A @ X2 @ S4 ) )
         => ( ( has_field_derivative @ A @ G @ E6 @ ( topolo174197925503356063within @ A @ ( F @ X2 ) @ ( top_top @ ( set @ A ) ) ) )
           => ( has_field_derivative @ A
              @ ^ [X: A] : ( G @ ( F @ X ) )
              @ ( times_times @ A @ E6 @ D6 )
              @ ( topolo174197925503356063within @ A @ X2 @ S4 ) ) ) ) ) ).

% DERIV_chain'
thf(fact_8161_DERIV__const__ratio__const,axiom,
    ! [A2: real,B2: real,F: real > real,K: real] :
      ( ( A2 != B2 )
     => ( ! [X3: real] : ( has_field_derivative @ real @ F @ K @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) )
       => ( ( minus_minus @ real @ ( F @ B2 ) @ ( F @ A2 ) )
          = ( times_times @ real @ ( minus_minus @ real @ B2 @ A2 ) @ K ) ) ) ) ).

% DERIV_const_ratio_const
thf(fact_8162_DERIV__const__ratio__const2,axiom,
    ! [A2: real,B2: real,F: real > real,K: real] :
      ( ( A2 != B2 )
     => ( ! [X3: real] : ( has_field_derivative @ real @ F @ K @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) )
       => ( ( divide_divide @ real @ ( minus_minus @ real @ ( F @ B2 ) @ ( F @ A2 ) ) @ ( minus_minus @ real @ B2 @ A2 ) )
          = K ) ) ) ).

% DERIV_const_ratio_const2
thf(fact_8163_DERIV__shift,axiom,
    ! [A: $tType] :
      ( ( real_V3459762299906320749_field @ A )
     => ! [F: A > A,Y2: A,X2: A,Z: A] :
          ( ( has_field_derivative @ A @ F @ Y2 @ ( topolo174197925503356063within @ A @ ( plus_plus @ A @ X2 @ Z ) @ ( top_top @ ( set @ A ) ) ) )
          = ( has_field_derivative @ A
            @ ^ [X: A] : ( F @ ( plus_plus @ A @ X @ Z ) )
            @ Y2
            @ ( topolo174197925503356063within @ A @ X2 @ ( top_top @ ( set @ A ) ) ) ) ) ) ).

% DERIV_shift
thf(fact_8164_termdiffs__strong__converges__everywhere,axiom,
    ! [A: $tType] :
      ( ( ( real_Vector_banach @ A )
        & ( real_V3459762299906320749_field @ A ) )
     => ! [C2: nat > A,X2: A] :
          ( ! [Y4: A] :
              ( summable @ A
              @ ^ [N: nat] : ( times_times @ A @ ( C2 @ N ) @ ( power_power @ A @ Y4 @ N ) ) )
         => ( has_field_derivative @ A
            @ ^ [X: A] :
                ( suminf @ A
                @ ^ [N: nat] : ( times_times @ A @ ( C2 @ N ) @ ( power_power @ A @ X @ N ) ) )
            @ ( suminf @ A
              @ ^ [N: nat] : ( times_times @ A @ ( diffs @ A @ C2 @ N ) @ ( power_power @ A @ X2 @ N ) ) )
            @ ( topolo174197925503356063within @ A @ X2 @ ( top_top @ ( set @ A ) ) ) ) ) ) ).

% termdiffs_strong_converges_everywhere
thf(fact_8165_DERIV__chain,axiom,
    ! [A: $tType] :
      ( ( real_V3459762299906320749_field @ A )
     => ! [F: A > A,Da: A,G: A > A,X2: A,Db: A,S4: set @ A] :
          ( ( has_field_derivative @ A @ F @ Da @ ( topolo174197925503356063within @ A @ ( G @ X2 ) @ ( top_top @ ( set @ A ) ) ) )
         => ( ( has_field_derivative @ A @ G @ Db @ ( topolo174197925503356063within @ A @ X2 @ S4 ) )
           => ( has_field_derivative @ A @ ( comp @ A @ A @ A @ F @ G ) @ ( times_times @ A @ Da @ Db ) @ ( topolo174197925503356063within @ A @ X2 @ S4 ) ) ) ) ) ).

% DERIV_chain
thf(fact_8166_termdiffs__sums__strong,axiom,
    ! [A: $tType] :
      ( ( ( real_Vector_banach @ A )
        & ( real_V3459762299906320749_field @ A ) )
     => ! [K6: real,C2: nat > A,F: A > A,F4: A,Z: A] :
          ( ! [Z3: A] :
              ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ Z3 ) @ K6 )
             => ( sums @ A
                @ ^ [N: nat] : ( times_times @ A @ ( C2 @ N ) @ ( power_power @ A @ Z3 @ N ) )
                @ ( F @ Z3 ) ) )
         => ( ( has_field_derivative @ A @ F @ F4 @ ( topolo174197925503356063within @ A @ Z @ ( top_top @ ( set @ A ) ) ) )
           => ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ Z ) @ K6 )
             => ( sums @ A
                @ ^ [N: nat] : ( times_times @ A @ ( diffs @ A @ C2 @ N ) @ ( power_power @ A @ Z @ N ) )
                @ F4 ) ) ) ) ) ).

% termdiffs_sums_strong
thf(fact_8167_DERIV__ln__divide,axiom,
    ! [X2: real] :
      ( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
     => ( has_field_derivative @ real @ ( ln_ln @ real ) @ ( divide_divide @ real @ ( one_one @ real ) @ X2 ) @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) ) ) ).

% DERIV_ln_divide
thf(fact_8168_DERIV__neg__dec__left,axiom,
    ! [F: real > real,L: real,X2: real] :
      ( ( has_field_derivative @ real @ F @ L @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) )
     => ( ( ord_less @ real @ L @ ( zero_zero @ real ) )
       => ? [D4: real] :
            ( ( ord_less @ real @ ( zero_zero @ real ) @ D4 )
            & ! [H6: real] :
                ( ( ord_less @ real @ ( zero_zero @ real ) @ H6 )
               => ( ( ord_less @ real @ H6 @ D4 )
                 => ( ord_less @ real @ ( F @ X2 ) @ ( F @ ( minus_minus @ real @ X2 @ H6 ) ) ) ) ) ) ) ) ).

% DERIV_neg_dec_left
thf(fact_8169_DERIV__pos__inc__left,axiom,
    ! [F: real > real,L: real,X2: real] :
      ( ( has_field_derivative @ real @ F @ L @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) )
     => ( ( ord_less @ real @ ( zero_zero @ real ) @ L )
       => ? [D4: real] :
            ( ( ord_less @ real @ ( zero_zero @ real ) @ D4 )
            & ! [H6: real] :
                ( ( ord_less @ real @ ( zero_zero @ real ) @ H6 )
               => ( ( ord_less @ real @ H6 @ D4 )
                 => ( ord_less @ real @ ( F @ ( minus_minus @ real @ X2 @ H6 ) ) @ ( F @ X2 ) ) ) ) ) ) ) ).

% DERIV_pos_inc_left
thf(fact_8170_DERIV__neg__dec__right,axiom,
    ! [F: real > real,L: real,X2: real] :
      ( ( has_field_derivative @ real @ F @ L @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) )
     => ( ( ord_less @ real @ L @ ( zero_zero @ real ) )
       => ? [D4: real] :
            ( ( ord_less @ real @ ( zero_zero @ real ) @ D4 )
            & ! [H6: real] :
                ( ( ord_less @ real @ ( zero_zero @ real ) @ H6 )
               => ( ( ord_less @ real @ H6 @ D4 )
                 => ( ord_less @ real @ ( F @ ( plus_plus @ real @ X2 @ H6 ) ) @ ( F @ X2 ) ) ) ) ) ) ) ).

% DERIV_neg_dec_right
thf(fact_8171_DERIV__pos__inc__right,axiom,
    ! [F: real > real,L: real,X2: real] :
      ( ( has_field_derivative @ real @ F @ L @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) )
     => ( ( ord_less @ real @ ( zero_zero @ real ) @ L )
       => ? [D4: real] :
            ( ( ord_less @ real @ ( zero_zero @ real ) @ D4 )
            & ! [H6: real] :
                ( ( ord_less @ real @ ( zero_zero @ real ) @ H6 )
               => ( ( ord_less @ real @ H6 @ D4 )
                 => ( ord_less @ real @ ( F @ X2 ) @ ( F @ ( plus_plus @ real @ X2 @ H6 ) ) ) ) ) ) ) ) ).

% DERIV_pos_inc_right
thf(fact_8172_at__within__Icc__at,axiom,
    ! [A: $tType] :
      ( ( topolo2564578578187576103pology @ A )
     => ! [A2: A,X2: A,B2: A] :
          ( ( ord_less @ A @ A2 @ X2 )
         => ( ( ord_less @ A @ X2 @ B2 )
           => ( ( topolo174197925503356063within @ A @ X2 @ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) )
              = ( topolo174197925503356063within @ A @ X2 @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ).

% at_within_Icc_at
thf(fact_8173_DERIV__local__const,axiom,
    ! [F: real > real,L: real,X2: real,D2: real] :
      ( ( has_field_derivative @ real @ F @ L @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) )
     => ( ( ord_less @ real @ ( zero_zero @ real ) @ D2 )
       => ( ! [Y4: real] :
              ( ( ord_less @ real @ ( abs_abs @ real @ ( minus_minus @ real @ X2 @ Y4 ) ) @ D2 )
             => ( ( F @ X2 )
                = ( F @ Y4 ) ) )
         => ( L
            = ( zero_zero @ real ) ) ) ) ) ).

% DERIV_local_const

% Type constructors (1316)
thf(tcon_VEBT__BuildupMemImp_OVEBTi___Typerep_Otyperep,axiom,
    typerep @ vEBT_VEBTi ).

thf(tcon_VEBT__Definitions_OVEBT___Typerep_Otyperep_1,axiom,
    typerep @ vEBT_VEBT ).

thf(tcon_Heap__Time__Monad_OHeap___Typerep_Otyperep_2,axiom,
    ! [A15: $tType] :
      ( ( typerep @ A15 )
     => ( typerep @ ( heap_Time_Heap @ A15 ) ) ) ).

thf(tcon_Code__Numeral_Ointeger___Typerep_Otyperep_3,axiom,
    typerep @ code_integer ).

thf(tcon_Code__Evaluation_Oterm___Typerep_Otyperep_4,axiom,
    typerep @ code_term ).

thf(tcon_Heap_Oheap_Oheap__ext___Typerep_Otyperep_5,axiom,
    ! [A15: $tType] :
      ( ( typerep @ A15 )
     => ( typerep @ ( heap_ext @ A15 ) ) ) ).

thf(tcon_Product__Type_Ounit___Typerep_Otyperep_6,axiom,
    typerep @ product_unit ).

thf(tcon_Product__Type_Ounit___Enum_Oenum,axiom,
    enum @ product_unit ).

thf(tcon_Product__Type_Ounit___Complete__Lattices_Ocomplete__linorder,axiom,
    comple5582772986160207858norder @ product_unit ).

thf(tcon_Product__Type_Ounit___Finite__Set_Ofinite,axiom,
    finite_finite @ product_unit ).

thf(tcon_Product__Type_Oprod___Typerep_Otyperep_7,axiom,
    ! [A15: $tType,A16: $tType] :
      ( ( ( typerep @ A15 )
        & ( typerep @ A16 ) )
     => ( typerep @ ( product_prod @ A15 @ A16 ) ) ) ).

thf(tcon_Product__Type_Oprod___Enum_Oenum_8,axiom,
    ! [A15: $tType,A16: $tType] :
      ( ( ( enum @ A15 )
        & ( enum @ A16 ) )
     => ( enum @ ( product_prod @ A15 @ A16 ) ) ) ).

thf(tcon_Product__Type_Oprod___Finite__Set_Ofinite_9,axiom,
    ! [A15: $tType,A16: $tType] :
      ( ( ( finite_finite @ A15 )
        & ( finite_finite @ A16 ) )
     => ( finite_finite @ ( product_prod @ A15 @ A16 ) ) ) ).

thf(tcon_Numeral__Type_Onum1___Typerep_Otyperep_10,axiom,
    typerep @ numeral_num1 ).

thf(tcon_Numeral__Type_Onum1___Enum_Oenum_11,axiom,
    enum @ numeral_num1 ).

thf(tcon_Numeral__Type_Onum1___Finite__Set_Ofinite_12,axiom,
    finite_finite @ numeral_num1 ).

thf(tcon_Numeral__Type_Obit1___Typerep_Otyperep_13,axiom,
    ! [A15: $tType] :
      ( ( typerep @ A15 )
     => ( typerep @ ( numeral_bit1 @ A15 ) ) ) ).

thf(tcon_Numeral__Type_Obit1___Enum_Oenum_14,axiom,
    ! [A15: $tType] :
      ( ( finite_finite @ A15 )
     => ( enum @ ( numeral_bit1 @ A15 ) ) ) ).

thf(tcon_Numeral__Type_Obit1___Finite__Set_Ofinite_15,axiom,
    ! [A15: $tType] :
      ( ( finite_finite @ A15 )
     => ( finite_finite @ ( numeral_bit1 @ A15 ) ) ) ).

thf(tcon_Numeral__Type_Obit0___Typerep_Otyperep_16,axiom,
    ! [A15: $tType] :
      ( ( typerep @ A15 )
     => ( typerep @ ( numeral_bit0 @ A15 ) ) ) ).

thf(tcon_Numeral__Type_Obit0___Enum_Oenum_17,axiom,
    ! [A15: $tType] :
      ( ( finite_finite @ A15 )
     => ( enum @ ( numeral_bit0 @ A15 ) ) ) ).

thf(tcon_Numeral__Type_Obit0___Finite__Set_Ofinite_18,axiom,
    ! [A15: $tType] :
      ( ( finite_finite @ A15 )
     => ( finite_finite @ ( numeral_bit0 @ A15 ) ) ) ).

thf(tcon_Multiset_Omultiset___Typerep_Otyperep_19,axiom,
    ! [A15: $tType] :
      ( ( typerep @ A15 )
     => ( typerep @ ( multiset @ A15 ) ) ) ).

thf(tcon_Extended__Nat_Oenat___Typerep_Otyperep_20,axiom,
    typerep @ extended_enat ).

thf(tcon_Extended__Nat_Oenat___Complete__Lattices_Ocomplete__linorder_21,axiom,
    comple5582772986160207858norder @ extended_enat ).

thf(tcon_Complex_Ocomplex___Typerep_Otyperep_22,axiom,
    typerep @ complex ).

thf(tcon_Assertions_Oassn___Typerep_Otyperep_23,axiom,
    typerep @ assn ).

thf(tcon_Uint32_Ouint32___Typerep_Otyperep_24,axiom,
    typerep @ uint32 ).

thf(tcon_Option_Ooption___Typerep_Otyperep_25,axiom,
    ! [A15: $tType] :
      ( ( typerep @ A15 )
     => ( typerep @ ( option @ A15 ) ) ) ).

thf(tcon_Option_Ooption___Enum_Oenum_26,axiom,
    ! [A15: $tType] :
      ( ( enum @ A15 )
     => ( enum @ ( option @ A15 ) ) ) ).

thf(tcon_Option_Ooption___Complete__Lattices_Ocomplete__linorder_27,axiom,
    ! [A15: $tType] :
      ( ( comple5582772986160207858norder @ A15 )
     => ( comple5582772986160207858norder @ ( option @ A15 ) ) ) ).

thf(tcon_Option_Ooption___Finite__Set_Ofinite_28,axiom,
    ! [A15: $tType] :
      ( ( finite_finite @ A15 )
     => ( finite_finite @ ( option @ A15 ) ) ) ).

thf(tcon_Filter_Ofilter___Typerep_Otyperep_29,axiom,
    ! [A15: $tType] :
      ( ( typerep @ A15 )
     => ( typerep @ ( filter @ A15 ) ) ) ).

thf(tcon_Enum_Ofinite__3___Typerep_Otyperep_30,axiom,
    typerep @ finite_3 ).

thf(tcon_Enum_Ofinite__3___Enum_Oenum_31,axiom,
    enum @ finite_3 ).

thf(tcon_Enum_Ofinite__3___Complete__Lattices_Ocomplete__linorder_32,axiom,
    comple5582772986160207858norder @ finite_3 ).

thf(tcon_Enum_Ofinite__3___Finite__Set_Ofinite_33,axiom,
    finite_finite @ finite_3 ).

thf(tcon_Enum_Ofinite__2___Typerep_Otyperep_34,axiom,
    typerep @ finite_2 ).

thf(tcon_Enum_Ofinite__2___Enum_Oenum_35,axiom,
    enum @ finite_2 ).

thf(tcon_Enum_Ofinite__2___Complete__Lattices_Ocomplete__linorder_36,axiom,
    comple5582772986160207858norder @ finite_2 ).

thf(tcon_Enum_Ofinite__2___Finite__Set_Ofinite_37,axiom,
    finite_finite @ finite_2 ).

thf(tcon_Enum_Ofinite__1___Typerep_Otyperep_38,axiom,
    typerep @ finite_1 ).

thf(tcon_Enum_Ofinite__1___Enum_Oenum_39,axiom,
    enum @ finite_1 ).

thf(tcon_Enum_Ofinite__1___Complete__Lattices_Ocomplete__linorder_40,axiom,
    comple5582772986160207858norder @ finite_1 ).

thf(tcon_Enum_Ofinite__1___Finite__Set_Ofinite_41,axiom,
    finite_finite @ finite_1 ).

thf(tcon_String_Ochar___Typerep_Otyperep_42,axiom,
    typerep @ char ).

thf(tcon_String_Ochar___Enum_Oenum_43,axiom,
    enum @ char ).

thf(tcon_String_Ochar___Finite__Set_Ofinite_44,axiom,
    finite_finite @ char ).

thf(tcon_Heap_Oarray___Typerep_Otyperep_45,axiom,
    ! [A15: $tType] : ( typerep @ ( array @ A15 ) ) ).

thf(tcon_Word_Oword___Typerep_Otyperep_46,axiom,
    ! [A15: $tType] :
      ( ( typerep @ A15 )
     => ( typerep @ ( word @ A15 ) ) ) ).

thf(tcon_Word_Oword___Enum_Oenum_47,axiom,
    ! [A15: $tType] :
      ( ( type_len @ A15 )
     => ( enum @ ( word @ A15 ) ) ) ).

thf(tcon_Word_Oword___Finite__Set_Ofinite_48,axiom,
    ! [A15: $tType] :
      ( ( type_len @ A15 )
     => ( finite_finite @ ( word @ A15 ) ) ) ).

thf(tcon_Real_Oreal___Typerep_Otyperep_49,axiom,
    typerep @ real ).

thf(tcon_List_Olist___Typerep_Otyperep_50,axiom,
    ! [A15: $tType] :
      ( ( typerep @ A15 )
     => ( typerep @ ( list @ A15 ) ) ) ).

thf(tcon_Heap_Oref___Typerep_Otyperep_51,axiom,
    ! [A15: $tType] : ( typerep @ ( ref @ A15 ) ) ).

thf(tcon_HOL_Obool___Typerep_Otyperep_52,axiom,
    typerep @ $o ).

thf(tcon_HOL_Obool___Enum_Oenum_53,axiom,
    enum @ $o ).

thf(tcon_HOL_Obool___Finite__Set_Ofinite_54,axiom,
    finite_finite @ $o ).

thf(tcon_Set_Oset___Typerep_Otyperep_55,axiom,
    ! [A15: $tType] :
      ( ( typerep @ A15 )
     => ( typerep @ ( set @ A15 ) ) ) ).

thf(tcon_Set_Oset___Enum_Oenum_56,axiom,
    ! [A15: $tType] :
      ( ( enum @ A15 )
     => ( enum @ ( set @ A15 ) ) ) ).

thf(tcon_Set_Oset___Finite__Set_Ofinite_57,axiom,
    ! [A15: $tType] :
      ( ( finite_finite @ A15 )
     => ( finite_finite @ ( set @ A15 ) ) ) ).

thf(tcon_Rat_Orat___Typerep_Otyperep_58,axiom,
    typerep @ rat ).

thf(tcon_Num_Onum___Typerep_Otyperep_59,axiom,
    typerep @ num ).

thf(tcon_Nat_Onat___Typerep_Otyperep_60,axiom,
    typerep @ nat ).

thf(tcon_Int_Oint___Typerep_Otyperep_61,axiom,
    typerep @ int ).

thf(tcon_fun___Typerep_Otyperep_62,axiom,
    ! [A15: $tType,A16: $tType] :
      ( ( ( typerep @ A15 )
        & ( typerep @ A16 ) )
     => ( typerep @ ( A15 > A16 ) ) ) ).

thf(tcon_fun___Enum_Oenum_63,axiom,
    ! [A15: $tType,A16: $tType] :
      ( ( ( enum @ A15 )
        & ( enum @ A16 ) )
     => ( enum @ ( A15 > A16 ) ) ) ).

thf(tcon_fun___Finite__Set_Ofinite_64,axiom,
    ! [A15: $tType,A16: $tType] :
      ( ( ( finite_finite @ A15 )
        & ( finite_finite @ A16 ) )
     => ( finite_finite @ ( A15 > A16 ) ) ) ).

thf(tcon_fun___Boolean__Algebras_Oboolean__algebra,axiom,
    ! [A15: $tType,A16: $tType] :
      ( ( boolea8198339166811842893lgebra @ A16 )
     => ( boolea8198339166811842893lgebra @ ( A15 > A16 ) ) ) ).

thf(tcon_fun___Code__Evaluation_Oterm__of,axiom,
    ! [A15: $tType,A16: $tType] :
      ( ( ( typerep @ A15 )
        & ( typerep @ A16 ) )
     => ( code_term_of @ ( A15 > A16 ) ) ) ).

thf(tcon_fun___Orderings_Oorder__top,axiom,
    ! [A15: $tType,A16: $tType] :
      ( ( order_top @ A16 )
     => ( order_top @ ( A15 > A16 ) ) ) ).

thf(tcon_fun___Orderings_Oorder__bot,axiom,
    ! [A15: $tType,A16: $tType] :
      ( ( order_bot @ A16 )
     => ( order_bot @ ( A15 > A16 ) ) ) ).

thf(tcon_fun___Orderings_Opreorder,axiom,
    ! [A15: $tType,A16: $tType] :
      ( ( preorder @ A16 )
     => ( preorder @ ( A15 > A16 ) ) ) ).

thf(tcon_fun___Orderings_Oorder,axiom,
    ! [A15: $tType,A16: $tType] :
      ( ( order @ A16 )
     => ( order @ ( A15 > A16 ) ) ) ).

thf(tcon_fun___Orderings_Oord,axiom,
    ! [A15: $tType,A16: $tType] :
      ( ( ord @ A16 )
     => ( ord @ ( A15 > A16 ) ) ) ).

thf(tcon_fun___Groups_Ouminus,axiom,
    ! [A15: $tType,A16: $tType] :
      ( ( uminus @ A16 )
     => ( uminus @ ( A15 > A16 ) ) ) ).

thf(tcon_fun___Groups_Ominus,axiom,
    ! [A15: $tType,A16: $tType] :
      ( ( minus @ A16 )
     => ( minus @ ( A15 > A16 ) ) ) ).

thf(tcon_fun___HOL_Oequal,axiom,
    ! [A15: $tType,A16: $tType] :
      ( ( ( enum @ A15 )
        & ( cl_HOL_Oequal @ A16 ) )
     => ( cl_HOL_Oequal @ ( A15 > A16 ) ) ) ).

thf(tcon_Int_Oint___Conditionally__Complete__Lattices_Oconditionally__complete__linorder,axiom,
    condit6923001295902523014norder @ int ).

thf(tcon_Int_Oint___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations,axiom,
    bit_un5681908812861735899ations @ int ).

thf(tcon_Int_Oint___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct,axiom,
    semiri1453513574482234551roduct @ int ).

thf(tcon_Int_Oint___Euclidean__Division_Ounique__euclidean__semiring__with__nat,axiom,
    euclid5411537665997757685th_nat @ int ).

thf(tcon_Int_Oint___Euclidean__Division_Ounique__euclidean__ring__with__nat,axiom,
    euclid8789492081693882211th_nat @ int ).

thf(tcon_Int_Oint___Groups_Oordered__ab__semigroup__monoid__add__imp__le,axiom,
    ordere1937475149494474687imp_le @ int ).

thf(tcon_Int_Oint___Euclidean__Division_Ounique__euclidean__semiring,axiom,
    euclid3128863361964157862miring @ int ).

thf(tcon_Int_Oint___Euclidean__Division_Oeuclidean__semiring__cancel,axiom,
    euclid4440199948858584721cancel @ int ).

thf(tcon_Int_Oint___Divides_Ounique__euclidean__semiring__numeral,axiom,
    unique1627219031080169319umeral @ int ).

thf(tcon_Int_Oint___Euclidean__Division_Oeuclidean__ring__cancel,axiom,
    euclid8851590272496341667cancel @ int ).

thf(tcon_Int_Oint___Rings_Osemiring__no__zero__divisors__cancel,axiom,
    semiri6575147826004484403cancel @ int ).

thf(tcon_Int_Oint___Groups_Ostrict__ordered__ab__semigroup__add,axiom,
    strict9044650504122735259up_add @ int ).

thf(tcon_Int_Oint___Groups_Oordered__cancel__ab__semigroup__add,axiom,
    ordere580206878836729694up_add @ int ).

thf(tcon_Int_Oint___Groups_Oordered__ab__semigroup__add__imp__le,axiom,
    ordere2412721322843649153imp_le @ int ).

thf(tcon_Int_Oint___Bit__Operations_Osemiring__bit__operations,axiom,
    bit_se359711467146920520ations @ int ).

thf(tcon_Int_Oint___Rings_Olinordered__comm__semiring__strict,axiom,
    linord2810124833399127020strict @ int ).

thf(tcon_Int_Oint___Groups_Ostrict__ordered__comm__monoid__add,axiom,
    strict7427464778891057005id_add @ int ).

thf(tcon_Int_Oint___Groups_Oordered__cancel__comm__monoid__add,axiom,
    ordere8940638589300402666id_add @ int ).

thf(tcon_Int_Oint___Euclidean__Division_Oeuclidean__semiring,axiom,
    euclid3725896446679973847miring @ int ).

thf(tcon_Int_Oint___Topological__Spaces_Otopological__space,axiom,
    topolo4958980785337419405_space @ int ).

thf(tcon_Int_Oint___Topological__Spaces_Olinorder__topology,axiom,
    topolo1944317154257567458pology @ int ).

thf(tcon_Int_Oint___Quickcheck__Narrowing_Opartial__term__of,axiom,
    quickc6926020345158392990erm_of @ int ).

thf(tcon_Int_Oint___Bit__Comprehension_Obit__comprehension,axiom,
    bit_bi6583157726757044596ension @ int ).

thf(tcon_Int_Oint___Rings_Olinordered__semiring__1__strict,axiom,
    linord715952674999750819strict @ int ).

thf(tcon_Int_Oint___Limits_Otopological__comm__monoid__add,axiom,
    topolo5987344860129210374id_add @ int ).

thf(tcon_Int_Oint___Groups_Olinordered__ab__semigroup__add,axiom,
    linord4140545234300271783up_add @ int ).

thf(tcon_Int_Oint___Bit__Operations_Oring__bit__operations,axiom,
    bit_ri3973907225187159222ations @ int ).

thf(tcon_Int_Oint___Topological__Spaces_Oorder__topology,axiom,
    topolo2564578578187576103pology @ int ).

thf(tcon_Int_Oint___Rings_Osemiring__1__no__zero__divisors,axiom,
    semiri2026040879449505780visors @ int ).

thf(tcon_Int_Oint___Rings_Olinordered__nonzero__semiring,axiom,
    linord181362715937106298miring @ int ).

thf(tcon_Int_Oint___Euclidean__Division_Oeuclidean__ring,axiom,
    euclid5891614535332579305n_ring @ int ).

thf(tcon_Int_Oint___Rings_Olinordered__semiring__strict,axiom,
    linord8928482502909563296strict @ int ).

thf(tcon_Int_Oint___Rings_Osemiring__no__zero__divisors,axiom,
    semiri3467727345109120633visors @ int ).

thf(tcon_Int_Oint___Groups_Oordered__ab__semigroup__add,axiom,
    ordere6658533253407199908up_add @ int ).

thf(tcon_Int_Oint___Groups_Oordered__ab__group__add__abs,axiom,
    ordere166539214618696060dd_abs @ int ).

thf(tcon_Int_Oint___GCD_Osemiring__gcd__mult__normalize,axiom,
    semiri6843258321239162965malize @ int ).

thf(tcon_Int_Oint___Groups_Oordered__comm__monoid__add,axiom,
    ordere6911136660526730532id_add @ int ).

thf(tcon_Int_Oint___Groups_Olinordered__ab__group__add,axiom,
    linord5086331880401160121up_add @ int ).

thf(tcon_Int_Oint___Groups_Ocancel__ab__semigroup__add,axiom,
    cancel2418104881723323429up_add @ int ).

thf(tcon_Int_Oint___Rings_Oring__1__no__zero__divisors,axiom,
    ring_15535105094025558882visors @ int ).

thf(tcon_Int_Oint___Groups_Ocancel__comm__monoid__add,axiom,
    cancel1802427076303600483id_add @ int ).

thf(tcon_Int_Oint___Rings_Olinordered__ring__strict,axiom,
    linord4710134922213307826strict @ int ).

thf(tcon_Int_Oint___Rings_Ocomm__semiring__1__cancel,axiom,
    comm_s4317794764714335236cancel @ int ).

thf(tcon_Int_Oint___Bit__Operations_Osemiring__bits,axiom,
    bit_semiring_bits @ int ).

thf(tcon_Int_Oint___Topological__Spaces_Ot2__space,axiom,
    topological_t2_space @ int ).

thf(tcon_Int_Oint___Rings_Oordered__comm__semiring,axiom,
    ordere2520102378445227354miring @ int ).

thf(tcon_Int_Oint___Rings_Olinordered__semiring__1,axiom,
    linord6961819062388156250ring_1 @ int ).

thf(tcon_Int_Oint___Groups_Oordered__ab__group__add,axiom,
    ordered_ab_group_add @ int ).

thf(tcon_Int_Oint___Groups_Ocancel__semigroup__add,axiom,
    cancel_semigroup_add @ int ).

thf(tcon_Int_Oint___Rings_Olinordered__semiring,axiom,
    linordered_semiring @ int ).

thf(tcon_Int_Oint___Least__significant__bit_Olsb,axiom,
    least_6119777620449941438nt_lsb @ int ).

thf(tcon_Int_Oint___Rings_Oordered__semiring__0,axiom,
    ordered_semiring_0 @ int ).

thf(tcon_Int_Oint___Rings_Olinordered__semidom,axiom,
    linordered_semidom @ int ).

thf(tcon_Int_Oint___Groups_Oab__semigroup__mult,axiom,
    ab_semigroup_mult @ int ).

thf(tcon_Int_Oint___Rings_Osemiring__1__cancel,axiom,
    semiring_1_cancel @ int ).

thf(tcon_Int_Oint___Rings_Oalgebraic__semidom,axiom,
    algebraic_semidom @ int ).

thf(tcon_Int_Oint___Groups_Ocomm__monoid__mult,axiom,
    comm_monoid_mult @ int ).

thf(tcon_Int_Oint___Groups_Oab__semigroup__add,axiom,
    ab_semigroup_add @ int ).

thf(tcon_Int_Oint___Generic__set__bit_Oset__bit,axiom,
    generic_set_set_bit @ int ).

thf(tcon_Int_Oint___Code__Evaluation_Oterm__of_65,axiom,
    code_term_of @ int ).

thf(tcon_Int_Oint___Rings_Oordered__semiring,axiom,
    ordered_semiring @ int ).

thf(tcon_Int_Oint___Rings_Oordered__ring__abs,axiom,
    ordered_ring_abs @ int ).

thf(tcon_Int_Oint___Parity_Osemiring__parity,axiom,
    semiring_parity @ int ).

thf(tcon_Int_Oint___Groups_Ocomm__monoid__add,axiom,
    comm_monoid_add @ int ).

thf(tcon_Int_Oint___Rings_Osemiring__modulo,axiom,
    semiring_modulo @ int ).

thf(tcon_Int_Oint___Rings_Olinordered__ring,axiom,
    linordered_ring @ int ).

thf(tcon_Int_Oint___Rings_Olinordered__idom,axiom,
    linordered_idom @ int ).

thf(tcon_Int_Oint___Rings_Ocomm__semiring__1,axiom,
    comm_semiring_1 @ int ).

thf(tcon_Int_Oint___Rings_Ocomm__semiring__0,axiom,
    comm_semiring_0 @ int ).

thf(tcon_Int_Oint___Groups_Osemigroup__mult,axiom,
    semigroup_mult @ int ).

thf(tcon_Int_Oint___Rings_Osemidom__modulo,axiom,
    semidom_modulo @ int ).

thf(tcon_Int_Oint___Rings_Osemidom__divide,axiom,
    semidom_divide @ int ).

thf(tcon_Int_Oint___Num_Osemiring__numeral,axiom,
    semiring_numeral @ int ).

thf(tcon_Int_Oint___Groups_Osemigroup__add,axiom,
    semigroup_add @ int ).

thf(tcon_Int_Oint___Rings_Ozero__less__one,axiom,
    zero_less_one @ int ).

thf(tcon_Int_Oint___Rings_Ocomm__semiring,axiom,
    comm_semiring @ int ).

thf(tcon_Int_Oint___Nat_Osemiring__char__0,axiom,
    semiring_char_0 @ int ).

thf(tcon_Int_Oint___Groups_Oab__group__add,axiom,
    ab_group_add @ int ).

thf(tcon_Int_Oint___Rings_Ozero__neq__one,axiom,
    zero_neq_one @ int ).

thf(tcon_Int_Oint___Rings_Oordered__ring,axiom,
    ordered_ring @ int ).

thf(tcon_Int_Oint___Rings_Oidom__abs__sgn,axiom,
    idom_abs_sgn @ int ).

thf(tcon_Int_Oint___Parity_Oring__parity,axiom,
    ring_parity @ int ).

thf(tcon_Int_Oint___Orderings_Opreorder_66,axiom,
    preorder @ int ).

thf(tcon_Int_Oint___Orderings_Olinorder,axiom,
    linorder @ int ).

thf(tcon_Int_Oint___Groups_Omonoid__mult,axiom,
    monoid_mult @ int ).

thf(tcon_Int_Oint___Rings_Oidom__modulo,axiom,
    idom_modulo @ int ).

thf(tcon_Int_Oint___Rings_Oidom__divide,axiom,
    idom_divide @ int ).

thf(tcon_Int_Oint___Rings_Ocomm__ring__1,axiom,
    comm_ring_1 @ int ).

thf(tcon_Int_Oint___Groups_Omonoid__add,axiom,
    monoid_add @ int ).

thf(tcon_Int_Oint___Rings_Osemiring__1,axiom,
    semiring_1 @ int ).

thf(tcon_Int_Oint___Rings_Osemiring__0,axiom,
    semiring_0 @ int ).

thf(tcon_Int_Oint___Orderings_Ono__top,axiom,
    no_top @ int ).

thf(tcon_Int_Oint___Orderings_Ono__bot,axiom,
    no_bot @ int ).

thf(tcon_Int_Oint___Groups_Ogroup__add,axiom,
    group_add @ int ).

thf(tcon_Int_Oint___GCD_Osemiring__gcd,axiom,
    semiring_gcd @ int ).

thf(tcon_Int_Oint___Rings_Omult__zero,axiom,
    mult_zero @ int ).

thf(tcon_Int_Oint___Rings_Ocomm__ring,axiom,
    comm_ring @ int ).

thf(tcon_Int_Oint___Orderings_Oorder_67,axiom,
    order @ int ).

thf(tcon_Int_Oint___Num_Oneg__numeral,axiom,
    neg_numeral @ int ).

thf(tcon_Int_Oint___Nat_Oring__char__0,axiom,
    ring_char_0 @ int ).

thf(tcon_Int_Oint___Rings_Osemiring,axiom,
    semiring @ int ).

thf(tcon_Int_Oint___Orderings_Oord_68,axiom,
    ord @ int ).

thf(tcon_Int_Oint___Groups_Ouminus_69,axiom,
    uminus @ int ).

thf(tcon_Int_Oint___Rings_Oring__1,axiom,
    ring_1 @ int ).

thf(tcon_Int_Oint___Rings_Oabs__if,axiom,
    abs_if @ int ).

thf(tcon_Int_Oint___Groups_Ominus_70,axiom,
    minus @ int ).

thf(tcon_Int_Oint___GCD_Oring__gcd,axiom,
    ring_gcd @ int ).

thf(tcon_Int_Oint___Power_Opower,axiom,
    power @ int ).

thf(tcon_Int_Oint___Num_Onumeral,axiom,
    numeral @ int ).

thf(tcon_Int_Oint___Groups_Ozero,axiom,
    zero @ int ).

thf(tcon_Int_Oint___Groups_Oplus,axiom,
    plus @ int ).

thf(tcon_Int_Oint___Rings_Oring,axiom,
    ring @ int ).

thf(tcon_Int_Oint___Rings_Oidom,axiom,
    idom @ int ).

thf(tcon_Int_Oint___Groups_Oone,axiom,
    one @ int ).

thf(tcon_Int_Oint___Rings_Odvd,axiom,
    dvd @ int ).

thf(tcon_Int_Oint___Heap_Oheap,axiom,
    heap @ int ).

thf(tcon_Int_Oint___HOL_Oequal_71,axiom,
    cl_HOL_Oequal @ int ).

thf(tcon_Nat_Onat___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_72,axiom,
    condit6923001295902523014norder @ nat ).

thf(tcon_Nat_Onat___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_73,axiom,
    bit_un5681908812861735899ations @ nat ).

thf(tcon_Nat_Onat___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_74,axiom,
    semiri1453513574482234551roduct @ nat ).

thf(tcon_Nat_Onat___Euclidean__Division_Ounique__euclidean__semiring__with__nat_75,axiom,
    euclid5411537665997757685th_nat @ nat ).

thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__monoid__add__imp__le_76,axiom,
    ordere1937475149494474687imp_le @ nat ).

thf(tcon_Nat_Onat___Euclidean__Division_Ounique__euclidean__semiring_77,axiom,
    euclid3128863361964157862miring @ nat ).

thf(tcon_Nat_Onat___Euclidean__Division_Oeuclidean__semiring__cancel_78,axiom,
    euclid4440199948858584721cancel @ nat ).

thf(tcon_Nat_Onat___Divides_Ounique__euclidean__semiring__numeral_79,axiom,
    unique1627219031080169319umeral @ nat ).

thf(tcon_Nat_Onat___Rings_Osemiring__no__zero__divisors__cancel_80,axiom,
    semiri6575147826004484403cancel @ nat ).

thf(tcon_Nat_Onat___Groups_Ostrict__ordered__ab__semigroup__add_81,axiom,
    strict9044650504122735259up_add @ nat ).

thf(tcon_Nat_Onat___Groups_Oordered__cancel__comm__monoid__diff,axiom,
    ordere1170586879665033532d_diff @ nat ).

thf(tcon_Nat_Onat___Groups_Oordered__cancel__ab__semigroup__add_82,axiom,
    ordere580206878836729694up_add @ nat ).

thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add__imp__le_83,axiom,
    ordere2412721322843649153imp_le @ nat ).

thf(tcon_Nat_Onat___Bit__Operations_Osemiring__bit__operations_84,axiom,
    bit_se359711467146920520ations @ nat ).

thf(tcon_Nat_Onat___Rings_Olinordered__comm__semiring__strict_85,axiom,
    linord2810124833399127020strict @ nat ).

thf(tcon_Nat_Onat___Groups_Ostrict__ordered__comm__monoid__add_86,axiom,
    strict7427464778891057005id_add @ nat ).

thf(tcon_Nat_Onat___Groups_Oordered__cancel__comm__monoid__add_87,axiom,
    ordere8940638589300402666id_add @ nat ).

thf(tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add,axiom,
    canoni5634975068530333245id_add @ nat ).

thf(tcon_Nat_Onat___Euclidean__Division_Oeuclidean__semiring_88,axiom,
    euclid3725896446679973847miring @ nat ).

thf(tcon_Nat_Onat___Topological__Spaces_Otopological__space_89,axiom,
    topolo4958980785337419405_space @ nat ).

thf(tcon_Nat_Onat___Topological__Spaces_Olinorder__topology_90,axiom,
    topolo1944317154257567458pology @ nat ).

thf(tcon_Nat_Onat___Quickcheck__Narrowing_Opartial__term__of_91,axiom,
    quickc6926020345158392990erm_of @ nat ).

thf(tcon_Nat_Onat___Limits_Otopological__comm__monoid__add_92,axiom,
    topolo5987344860129210374id_add @ nat ).

thf(tcon_Nat_Onat___Groups_Olinordered__ab__semigroup__add_93,axiom,
    linord4140545234300271783up_add @ nat ).

thf(tcon_Nat_Onat___Topological__Spaces_Oorder__topology_94,axiom,
    topolo2564578578187576103pology @ nat ).

thf(tcon_Nat_Onat___Rings_Osemiring__1__no__zero__divisors_95,axiom,
    semiri2026040879449505780visors @ nat ).

thf(tcon_Nat_Onat___Rings_Olinordered__nonzero__semiring_96,axiom,
    linord181362715937106298miring @ nat ).

thf(tcon_Nat_Onat___Rings_Olinordered__semiring__strict_97,axiom,
    linord8928482502909563296strict @ nat ).

thf(tcon_Nat_Onat___Rings_Osemiring__no__zero__divisors_98,axiom,
    semiri3467727345109120633visors @ nat ).

thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add_99,axiom,
    ordere6658533253407199908up_add @ nat ).

thf(tcon_Nat_Onat___GCD_Osemiring__gcd__mult__normalize_100,axiom,
    semiri6843258321239162965malize @ nat ).

thf(tcon_Nat_Onat___Groups_Oordered__comm__monoid__add_101,axiom,
    ordere6911136660526730532id_add @ nat ).

thf(tcon_Nat_Onat___Groups_Ocancel__ab__semigroup__add_102,axiom,
    cancel2418104881723323429up_add @ nat ).

thf(tcon_Nat_Onat___Groups_Ocancel__comm__monoid__add_103,axiom,
    cancel1802427076303600483id_add @ nat ).

thf(tcon_Nat_Onat___Rings_Ocomm__semiring__1__cancel_104,axiom,
    comm_s4317794764714335236cancel @ nat ).

thf(tcon_Nat_Onat___Bit__Operations_Osemiring__bits_105,axiom,
    bit_semiring_bits @ nat ).

thf(tcon_Nat_Onat___Topological__Spaces_Ot2__space_106,axiom,
    topological_t2_space @ nat ).

thf(tcon_Nat_Onat___Rings_Oordered__comm__semiring_107,axiom,
    ordere2520102378445227354miring @ nat ).

thf(tcon_Nat_Onat___Groups_Ocancel__semigroup__add_108,axiom,
    cancel_semigroup_add @ nat ).

thf(tcon_Nat_Onat___Rings_Olinordered__semiring_109,axiom,
    linordered_semiring @ nat ).

thf(tcon_Nat_Onat___Rings_Oordered__semiring__0_110,axiom,
    ordered_semiring_0 @ nat ).

thf(tcon_Nat_Onat___Rings_Olinordered__semidom_111,axiom,
    linordered_semidom @ nat ).

thf(tcon_Nat_Onat___Groups_Oab__semigroup__mult_112,axiom,
    ab_semigroup_mult @ nat ).

thf(tcon_Nat_Onat___Rings_Osemiring__1__cancel_113,axiom,
    semiring_1_cancel @ nat ).

thf(tcon_Nat_Onat___Rings_Oalgebraic__semidom_114,axiom,
    algebraic_semidom @ nat ).

thf(tcon_Nat_Onat___Groups_Ocomm__monoid__mult_115,axiom,
    comm_monoid_mult @ nat ).

thf(tcon_Nat_Onat___Groups_Ocomm__monoid__diff,axiom,
    comm_monoid_diff @ nat ).

thf(tcon_Nat_Onat___Groups_Oab__semigroup__add_116,axiom,
    ab_semigroup_add @ nat ).

thf(tcon_Nat_Onat___Code__Evaluation_Oterm__of_117,axiom,
    code_term_of @ nat ).

thf(tcon_Nat_Onat___Rings_Oordered__semiring_118,axiom,
    ordered_semiring @ nat ).

thf(tcon_Nat_Onat___Parity_Osemiring__parity_119,axiom,
    semiring_parity @ nat ).

thf(tcon_Nat_Onat___Groups_Ocomm__monoid__add_120,axiom,
    comm_monoid_add @ nat ).

thf(tcon_Nat_Onat___Rings_Osemiring__modulo_121,axiom,
    semiring_modulo @ nat ).

thf(tcon_Nat_Onat___Rings_Ocomm__semiring__1_122,axiom,
    comm_semiring_1 @ nat ).

thf(tcon_Nat_Onat___Rings_Ocomm__semiring__0_123,axiom,
    comm_semiring_0 @ nat ).

thf(tcon_Nat_Onat___Groups_Osemigroup__mult_124,axiom,
    semigroup_mult @ nat ).

thf(tcon_Nat_Onat___Rings_Osemidom__modulo_125,axiom,
    semidom_modulo @ nat ).

thf(tcon_Nat_Onat___Rings_Osemidom__divide_126,axiom,
    semidom_divide @ nat ).

thf(tcon_Nat_Onat___Num_Osemiring__numeral_127,axiom,
    semiring_numeral @ nat ).

thf(tcon_Nat_Onat___Groups_Osemigroup__add_128,axiom,
    semigroup_add @ nat ).

thf(tcon_Nat_Onat___Rings_Ozero__less__one_129,axiom,
    zero_less_one @ nat ).

thf(tcon_Nat_Onat___Rings_Ocomm__semiring_130,axiom,
    comm_semiring @ nat ).

thf(tcon_Nat_Onat___Orderings_Owellorder,axiom,
    wellorder @ nat ).

thf(tcon_Nat_Onat___Orderings_Oorder__bot_131,axiom,
    order_bot @ nat ).

thf(tcon_Nat_Onat___Nat_Osemiring__char__0_132,axiom,
    semiring_char_0 @ nat ).

thf(tcon_Nat_Onat___Rings_Ozero__neq__one_133,axiom,
    zero_neq_one @ nat ).

thf(tcon_Nat_Onat___Orderings_Opreorder_134,axiom,
    preorder @ nat ).

thf(tcon_Nat_Onat___Orderings_Olinorder_135,axiom,
    linorder @ nat ).

thf(tcon_Nat_Onat___Groups_Omonoid__mult_136,axiom,
    monoid_mult @ nat ).

thf(tcon_Nat_Onat___Groups_Omonoid__add_137,axiom,
    monoid_add @ nat ).

thf(tcon_Nat_Onat___Rings_Osemiring__1_138,axiom,
    semiring_1 @ nat ).

thf(tcon_Nat_Onat___Rings_Osemiring__0_139,axiom,
    semiring_0 @ nat ).

thf(tcon_Nat_Onat___Orderings_Ono__top_140,axiom,
    no_top @ nat ).

thf(tcon_Nat_Onat___GCD_Osemiring__gcd_141,axiom,
    semiring_gcd @ nat ).

thf(tcon_Nat_Onat___Rings_Omult__zero_142,axiom,
    mult_zero @ nat ).

thf(tcon_Nat_Onat___Orderings_Oorder_143,axiom,
    order @ nat ).

thf(tcon_Nat_Onat___Rings_Osemiring_144,axiom,
    semiring @ nat ).

thf(tcon_Nat_Onat___Orderings_Oord_145,axiom,
    ord @ nat ).

thf(tcon_Nat_Onat___Groups_Ominus_146,axiom,
    minus @ nat ).

thf(tcon_Nat_Onat___Power_Opower_147,axiom,
    power @ nat ).

thf(tcon_Nat_Onat___Num_Onumeral_148,axiom,
    numeral @ nat ).

thf(tcon_Nat_Onat___Groups_Ozero_149,axiom,
    zero @ nat ).

thf(tcon_Nat_Onat___Groups_Oplus_150,axiom,
    plus @ nat ).

thf(tcon_Nat_Onat___Groups_Oone_151,axiom,
    one @ nat ).

thf(tcon_Nat_Onat___Rings_Odvd_152,axiom,
    dvd @ nat ).

thf(tcon_Nat_Onat___Heap_Oheap_153,axiom,
    heap @ nat ).

thf(tcon_Nat_Onat___HOL_Oequal_154,axiom,
    cl_HOL_Oequal @ nat ).

thf(tcon_Nat_Onat___Nat_Osize,axiom,
    size @ nat ).

thf(tcon_Num_Onum___Quickcheck__Narrowing_Opartial__term__of_155,axiom,
    quickc6926020345158392990erm_of @ num ).

thf(tcon_Num_Onum___Code__Evaluation_Oterm__of_156,axiom,
    code_term_of @ num ).

thf(tcon_Num_Onum___Orderings_Opreorder_157,axiom,
    preorder @ num ).

thf(tcon_Num_Onum___Orderings_Olinorder_158,axiom,
    linorder @ num ).

thf(tcon_Num_Onum___Orderings_Oorder_159,axiom,
    order @ num ).

thf(tcon_Num_Onum___Orderings_Oord_160,axiom,
    ord @ num ).

thf(tcon_Num_Onum___Groups_Oplus_161,axiom,
    plus @ num ).

thf(tcon_Num_Onum___HOL_Oequal_162,axiom,
    cl_HOL_Oequal @ num ).

thf(tcon_Num_Onum___Nat_Osize_163,axiom,
    size @ num ).

thf(tcon_Rat_Orat___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_164,axiom,
    semiri1453513574482234551roduct @ rat ).

thf(tcon_Rat_Orat___Groups_Oordered__ab__semigroup__monoid__add__imp__le_165,axiom,
    ordere1937475149494474687imp_le @ rat ).

thf(tcon_Rat_Orat___Rings_Osemiring__no__zero__divisors__cancel_166,axiom,
    semiri6575147826004484403cancel @ rat ).

thf(tcon_Rat_Orat___Groups_Ostrict__ordered__ab__semigroup__add_167,axiom,
    strict9044650504122735259up_add @ rat ).

thf(tcon_Rat_Orat___Groups_Oordered__cancel__ab__semigroup__add_168,axiom,
    ordere580206878836729694up_add @ rat ).

thf(tcon_Rat_Orat___Groups_Oordered__ab__semigroup__add__imp__le_169,axiom,
    ordere2412721322843649153imp_le @ rat ).

thf(tcon_Rat_Orat___Rings_Olinordered__comm__semiring__strict_170,axiom,
    linord2810124833399127020strict @ rat ).

thf(tcon_Rat_Orat___Groups_Ostrict__ordered__comm__monoid__add_171,axiom,
    strict7427464778891057005id_add @ rat ).

thf(tcon_Rat_Orat___Groups_Oordered__cancel__comm__monoid__add_172,axiom,
    ordere8940638589300402666id_add @ rat ).

thf(tcon_Rat_Orat___Quickcheck__Narrowing_Opartial__term__of_173,axiom,
    quickc6926020345158392990erm_of @ rat ).

thf(tcon_Rat_Orat___Archimedean__Field_Oarchimedean__field,axiom,
    archim462609752435547400_field @ rat ).

thf(tcon_Rat_Orat___Rings_Olinordered__semiring__1__strict_174,axiom,
    linord715952674999750819strict @ rat ).

thf(tcon_Rat_Orat___Groups_Olinordered__ab__semigroup__add_175,axiom,
    linord4140545234300271783up_add @ rat ).

thf(tcon_Rat_Orat___Rings_Osemiring__1__no__zero__divisors_176,axiom,
    semiri2026040879449505780visors @ rat ).

thf(tcon_Rat_Orat___Rings_Olinordered__nonzero__semiring_177,axiom,
    linord181362715937106298miring @ rat ).

thf(tcon_Rat_Orat___Rings_Olinordered__semiring__strict_178,axiom,
    linord8928482502909563296strict @ rat ).

thf(tcon_Rat_Orat___Rings_Osemiring__no__zero__divisors_179,axiom,
    semiri3467727345109120633visors @ rat ).

thf(tcon_Rat_Orat___Groups_Oordered__ab__semigroup__add_180,axiom,
    ordere6658533253407199908up_add @ rat ).

thf(tcon_Rat_Orat___Groups_Oordered__ab__group__add__abs_181,axiom,
    ordere166539214618696060dd_abs @ rat ).

thf(tcon_Rat_Orat___Archimedean__Field_Ofloor__ceiling,axiom,
    archim2362893244070406136eiling @ rat ).

thf(tcon_Rat_Orat___Groups_Oordered__comm__monoid__add_182,axiom,
    ordere6911136660526730532id_add @ rat ).

thf(tcon_Rat_Orat___Groups_Olinordered__ab__group__add_183,axiom,
    linord5086331880401160121up_add @ rat ).

thf(tcon_Rat_Orat___Groups_Ocancel__ab__semigroup__add_184,axiom,
    cancel2418104881723323429up_add @ rat ).

thf(tcon_Rat_Orat___Rings_Oring__1__no__zero__divisors_185,axiom,
    ring_15535105094025558882visors @ rat ).

thf(tcon_Rat_Orat___Groups_Ocancel__comm__monoid__add_186,axiom,
    cancel1802427076303600483id_add @ rat ).

thf(tcon_Rat_Orat___Rings_Olinordered__ring__strict_187,axiom,
    linord4710134922213307826strict @ rat ).

thf(tcon_Rat_Orat___Rings_Ocomm__semiring__1__cancel_188,axiom,
    comm_s4317794764714335236cancel @ rat ).

thf(tcon_Rat_Orat___Rings_Oordered__comm__semiring_189,axiom,
    ordere2520102378445227354miring @ rat ).

thf(tcon_Rat_Orat___Rings_Olinordered__semiring__1_190,axiom,
    linord6961819062388156250ring_1 @ rat ).

thf(tcon_Rat_Orat___Groups_Oordered__ab__group__add_191,axiom,
    ordered_ab_group_add @ rat ).

thf(tcon_Rat_Orat___Groups_Ocancel__semigroup__add_192,axiom,
    cancel_semigroup_add @ rat ).

thf(tcon_Rat_Orat___Rings_Olinordered__semiring_193,axiom,
    linordered_semiring @ rat ).

thf(tcon_Rat_Orat___Rings_Oordered__semiring__0_194,axiom,
    ordered_semiring_0 @ rat ).

thf(tcon_Rat_Orat___Rings_Olinordered__semidom_195,axiom,
    linordered_semidom @ rat ).

thf(tcon_Rat_Orat___Orderings_Odense__linorder,axiom,
    dense_linorder @ rat ).

thf(tcon_Rat_Orat___Groups_Oab__semigroup__mult_196,axiom,
    ab_semigroup_mult @ rat ).

thf(tcon_Rat_Orat___Rings_Osemiring__1__cancel_197,axiom,
    semiring_1_cancel @ rat ).

thf(tcon_Rat_Orat___Groups_Ocomm__monoid__mult_198,axiom,
    comm_monoid_mult @ rat ).

thf(tcon_Rat_Orat___Groups_Oab__semigroup__add_199,axiom,
    ab_semigroup_add @ rat ).

thf(tcon_Rat_Orat___Fields_Olinordered__field,axiom,
    linordered_field @ rat ).

thf(tcon_Rat_Orat___Code__Evaluation_Oterm__of_200,axiom,
    code_term_of @ rat ).

thf(tcon_Rat_Orat___Rings_Oordered__semiring_201,axiom,
    ordered_semiring @ rat ).

thf(tcon_Rat_Orat___Rings_Oordered__ring__abs_202,axiom,
    ordered_ring_abs @ rat ).

thf(tcon_Rat_Orat___Groups_Ocomm__monoid__add_203,axiom,
    comm_monoid_add @ rat ).

thf(tcon_Rat_Orat___Rings_Olinordered__ring_204,axiom,
    linordered_ring @ rat ).

thf(tcon_Rat_Orat___Rings_Olinordered__idom_205,axiom,
    linordered_idom @ rat ).

thf(tcon_Rat_Orat___Rings_Ocomm__semiring__1_206,axiom,
    comm_semiring_1 @ rat ).

thf(tcon_Rat_Orat___Rings_Ocomm__semiring__0_207,axiom,
    comm_semiring_0 @ rat ).

thf(tcon_Rat_Orat___Orderings_Odense__order,axiom,
    dense_order @ rat ).

thf(tcon_Rat_Orat___Groups_Osemigroup__mult_208,axiom,
    semigroup_mult @ rat ).

thf(tcon_Rat_Orat___Rings_Osemidom__divide_209,axiom,
    semidom_divide @ rat ).

thf(tcon_Rat_Orat___Num_Osemiring__numeral_210,axiom,
    semiring_numeral @ rat ).

thf(tcon_Rat_Orat___Groups_Osemigroup__add_211,axiom,
    semigroup_add @ rat ).

thf(tcon_Rat_Orat___Fields_Ofield__abs__sgn,axiom,
    field_abs_sgn @ rat ).

thf(tcon_Rat_Orat___Fields_Odivision__ring,axiom,
    division_ring @ rat ).

thf(tcon_Rat_Orat___Rings_Ozero__less__one_212,axiom,
    zero_less_one @ rat ).

thf(tcon_Rat_Orat___Rings_Ocomm__semiring_213,axiom,
    comm_semiring @ rat ).

thf(tcon_Rat_Orat___Nat_Osemiring__char__0_214,axiom,
    semiring_char_0 @ rat ).

thf(tcon_Rat_Orat___Groups_Oab__group__add_215,axiom,
    ab_group_add @ rat ).

thf(tcon_Rat_Orat___Fields_Ofield__char__0,axiom,
    field_char_0 @ rat ).

thf(tcon_Rat_Orat___Rings_Ozero__neq__one_216,axiom,
    zero_neq_one @ rat ).

thf(tcon_Rat_Orat___Rings_Oordered__ring_217,axiom,
    ordered_ring @ rat ).

thf(tcon_Rat_Orat___Rings_Oidom__abs__sgn_218,axiom,
    idom_abs_sgn @ rat ).

thf(tcon_Rat_Orat___Orderings_Opreorder_219,axiom,
    preorder @ rat ).

thf(tcon_Rat_Orat___Orderings_Olinorder_220,axiom,
    linorder @ rat ).

thf(tcon_Rat_Orat___Groups_Omonoid__mult_221,axiom,
    monoid_mult @ rat ).

thf(tcon_Rat_Orat___Rings_Oidom__divide_222,axiom,
    idom_divide @ rat ).

thf(tcon_Rat_Orat___Rings_Ocomm__ring__1_223,axiom,
    comm_ring_1 @ rat ).

thf(tcon_Rat_Orat___Groups_Omonoid__add_224,axiom,
    monoid_add @ rat ).

thf(tcon_Rat_Orat___Rings_Osemiring__1_225,axiom,
    semiring_1 @ rat ).

thf(tcon_Rat_Orat___Rings_Osemiring__0_226,axiom,
    semiring_0 @ rat ).

thf(tcon_Rat_Orat___Orderings_Ono__top_227,axiom,
    no_top @ rat ).

thf(tcon_Rat_Orat___Orderings_Ono__bot_228,axiom,
    no_bot @ rat ).

thf(tcon_Rat_Orat___Groups_Ogroup__add_229,axiom,
    group_add @ rat ).

thf(tcon_Rat_Orat___Rings_Omult__zero_230,axiom,
    mult_zero @ rat ).

thf(tcon_Rat_Orat___Rings_Ocomm__ring_231,axiom,
    comm_ring @ rat ).

thf(tcon_Rat_Orat___Orderings_Oorder_232,axiom,
    order @ rat ).

thf(tcon_Rat_Orat___Num_Oneg__numeral_233,axiom,
    neg_numeral @ rat ).

thf(tcon_Rat_Orat___Nat_Oring__char__0_234,axiom,
    ring_char_0 @ rat ).

thf(tcon_Rat_Orat___Rings_Osemiring_235,axiom,
    semiring @ rat ).

thf(tcon_Rat_Orat___Orderings_Oord_236,axiom,
    ord @ rat ).

thf(tcon_Rat_Orat___Groups_Ouminus_237,axiom,
    uminus @ rat ).

thf(tcon_Rat_Orat___Rings_Oring__1_238,axiom,
    ring_1 @ rat ).

thf(tcon_Rat_Orat___Rings_Oabs__if_239,axiom,
    abs_if @ rat ).

thf(tcon_Rat_Orat___Groups_Ominus_240,axiom,
    minus @ rat ).

thf(tcon_Rat_Orat___Fields_Ofield,axiom,
    field @ rat ).

thf(tcon_Rat_Orat___Power_Opower_241,axiom,
    power @ rat ).

thf(tcon_Rat_Orat___Num_Onumeral_242,axiom,
    numeral @ rat ).

thf(tcon_Rat_Orat___Groups_Ozero_243,axiom,
    zero @ rat ).

thf(tcon_Rat_Orat___Groups_Oplus_244,axiom,
    plus @ rat ).

thf(tcon_Rat_Orat___Rings_Oring_245,axiom,
    ring @ rat ).

thf(tcon_Rat_Orat___Rings_Oidom_246,axiom,
    idom @ rat ).

thf(tcon_Rat_Orat___Groups_Oone_247,axiom,
    one @ rat ).

thf(tcon_Rat_Orat___Rings_Odvd_248,axiom,
    dvd @ rat ).

thf(tcon_Rat_Orat___HOL_Oequal_249,axiom,
    cl_HOL_Oequal @ rat ).

thf(tcon_Set_Oset___Quickcheck__Narrowing_Opartial__term__of_250,axiom,
    ! [A15: $tType] :
      ( ( typerep @ A15 )
     => ( quickc6926020345158392990erm_of @ ( set @ A15 ) ) ) ).

thf(tcon_Set_Oset___Boolean__Algebras_Oboolean__algebra_251,axiom,
    ! [A15: $tType] : ( boolea8198339166811842893lgebra @ ( set @ A15 ) ) ).

thf(tcon_Set_Oset___Code__Evaluation_Oterm__of_252,axiom,
    ! [A15: $tType] :
      ( ( typerep @ A15 )
     => ( code_term_of @ ( set @ A15 ) ) ) ).

thf(tcon_Set_Oset___Orderings_Oorder__top_253,axiom,
    ! [A15: $tType] : ( order_top @ ( set @ A15 ) ) ).

thf(tcon_Set_Oset___Orderings_Oorder__bot_254,axiom,
    ! [A15: $tType] : ( order_bot @ ( set @ A15 ) ) ).

thf(tcon_Set_Oset___Orderings_Opreorder_255,axiom,
    ! [A15: $tType] : ( preorder @ ( set @ A15 ) ) ).

thf(tcon_Set_Oset___Orderings_Oorder_256,axiom,
    ! [A15: $tType] : ( order @ ( set @ A15 ) ) ).

thf(tcon_Set_Oset___Orderings_Oord_257,axiom,
    ! [A15: $tType] : ( ord @ ( set @ A15 ) ) ).

thf(tcon_Set_Oset___Groups_Ouminus_258,axiom,
    ! [A15: $tType] : ( uminus @ ( set @ A15 ) ) ).

thf(tcon_Set_Oset___Groups_Ominus_259,axiom,
    ! [A15: $tType] : ( minus @ ( set @ A15 ) ) ).

thf(tcon_Set_Oset___HOL_Oequal_260,axiom,
    ! [A15: $tType] :
      ( ( cl_HOL_Oequal @ A15 )
     => ( cl_HOL_Oequal @ ( set @ A15 ) ) ) ).

thf(tcon_HOL_Obool___Topological__Spaces_Otopological__space_261,axiom,
    topolo4958980785337419405_space @ $o ).

thf(tcon_HOL_Obool___Topological__Spaces_Olinorder__topology_262,axiom,
    topolo1944317154257567458pology @ $o ).

thf(tcon_HOL_Obool___Quickcheck__Narrowing_Opartial__term__of_263,axiom,
    quickc6926020345158392990erm_of @ $o ).

thf(tcon_HOL_Obool___Topological__Spaces_Oorder__topology_264,axiom,
    topolo2564578578187576103pology @ $o ).

thf(tcon_HOL_Obool___Boolean__Algebras_Oboolean__algebra_265,axiom,
    boolea8198339166811842893lgebra @ $o ).

thf(tcon_HOL_Obool___Topological__Spaces_Ot2__space_266,axiom,
    topological_t2_space @ $o ).

thf(tcon_HOL_Obool___Code__Evaluation_Oterm__of_267,axiom,
    code_term_of @ $o ).

thf(tcon_HOL_Obool___Orderings_Oorder__top_268,axiom,
    order_top @ $o ).

thf(tcon_HOL_Obool___Orderings_Oorder__bot_269,axiom,
    order_bot @ $o ).

thf(tcon_HOL_Obool___Orderings_Opreorder_270,axiom,
    preorder @ $o ).

thf(tcon_HOL_Obool___Orderings_Olinorder_271,axiom,
    linorder @ $o ).

thf(tcon_HOL_Obool___Orderings_Oorder_272,axiom,
    order @ $o ).

thf(tcon_HOL_Obool___Orderings_Oord_273,axiom,
    ord @ $o ).

thf(tcon_HOL_Obool___Groups_Ouminus_274,axiom,
    uminus @ $o ).

thf(tcon_HOL_Obool___Groups_Ominus_275,axiom,
    minus @ $o ).

thf(tcon_HOL_Obool___Heap_Oheap_276,axiom,
    heap @ $o ).

thf(tcon_HOL_Obool___HOL_Oequal_277,axiom,
    cl_HOL_Oequal @ $o ).

thf(tcon_Heap_Oref___Quickcheck__Narrowing_Opartial__term__of_278,axiom,
    ! [A15: $tType] :
      ( ( typerep @ A15 )
     => ( quickc6926020345158392990erm_of @ ( ref @ A15 ) ) ) ).

thf(tcon_Heap_Oref___Code__Evaluation_Oterm__of_279,axiom,
    ! [A15: $tType] :
      ( ( typerep @ A15 )
     => ( code_term_of @ ( ref @ A15 ) ) ) ).

thf(tcon_Heap_Oref___Heap_Oheap_280,axiom,
    ! [A15: $tType] : ( heap @ ( ref @ A15 ) ) ).

thf(tcon_Heap_Oref___HOL_Oequal_281,axiom,
    ! [A15: $tType] : ( cl_HOL_Oequal @ ( ref @ A15 ) ) ).

thf(tcon_Heap_Oref___Nat_Osize_282,axiom,
    ! [A15: $tType] : ( size @ ( ref @ A15 ) ) ).

thf(tcon_List_Olist___Quickcheck__Narrowing_Opartial__term__of_283,axiom,
    ! [A15: $tType] :
      ( ( typerep @ A15 )
     => ( quickc6926020345158392990erm_of @ ( list @ A15 ) ) ) ).

thf(tcon_List_Olist___Code__Evaluation_Oterm__of_284,axiom,
    ! [A15: $tType] :
      ( ( typerep @ A15 )
     => ( code_term_of @ ( list @ A15 ) ) ) ).

thf(tcon_List_Olist___Heap_Oheap_285,axiom,
    ! [A15: $tType] :
      ( ( heap @ A15 )
     => ( heap @ ( list @ A15 ) ) ) ).

thf(tcon_List_Olist___HOL_Oequal_286,axiom,
    ! [A15: $tType] : ( cl_HOL_Oequal @ ( list @ A15 ) ) ).

thf(tcon_List_Olist___Nat_Osize_287,axiom,
    ! [A15: $tType] : ( size @ ( list @ A15 ) ) ).

thf(tcon_Real_Oreal___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_288,axiom,
    condit6923001295902523014norder @ real ).

thf(tcon_Real_Oreal___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_289,axiom,
    semiri1453513574482234551roduct @ real ).

thf(tcon_Real_Oreal___Groups_Oordered__ab__semigroup__monoid__add__imp__le_290,axiom,
    ordere1937475149494474687imp_le @ real ).

thf(tcon_Real_Oreal___Real__Vector__Spaces_Oreal__normed__div__algebra,axiom,
    real_V8999393235501362500lgebra @ real ).

thf(tcon_Real_Oreal___Real__Vector__Spaces_Oreal__normed__algebra__1,axiom,
    real_V2822296259951069270ebra_1 @ real ).

thf(tcon_Real_Oreal___Rings_Osemiring__no__zero__divisors__cancel_291,axiom,
    semiri6575147826004484403cancel @ real ).

thf(tcon_Real_Oreal___Real__Vector__Spaces_Oreal__normed__algebra,axiom,
    real_V4412858255891104859lgebra @ real ).

thf(tcon_Real_Oreal___Real__Vector__Spaces_Oordered__real__vector,axiom,
    real_V5355595471888546746vector @ real ).

thf(tcon_Real_Oreal___Groups_Ostrict__ordered__ab__semigroup__add_292,axiom,
    strict9044650504122735259up_add @ real ).

thf(tcon_Real_Oreal___Groups_Oordered__cancel__ab__semigroup__add_293,axiom,
    ordere580206878836729694up_add @ real ).

thf(tcon_Real_Oreal___Groups_Oordered__ab__semigroup__add__imp__le_294,axiom,
    ordere2412721322843649153imp_le @ real ).

thf(tcon_Real_Oreal___Rings_Olinordered__comm__semiring__strict_295,axiom,
    linord2810124833399127020strict @ real ).

thf(tcon_Real_Oreal___Real__Vector__Spaces_Oreal__normed__vector,axiom,
    real_V822414075346904944vector @ real ).

thf(tcon_Real_Oreal___Groups_Ostrict__ordered__comm__monoid__add_296,axiom,
    strict7427464778891057005id_add @ real ).

thf(tcon_Real_Oreal___Groups_Oordered__cancel__comm__monoid__add_297,axiom,
    ordere8940638589300402666id_add @ real ).

thf(tcon_Real_Oreal___Topological__Spaces_Otopological__space_298,axiom,
    topolo4958980785337419405_space @ real ).

thf(tcon_Real_Oreal___Topological__Spaces_Olinorder__topology_299,axiom,
    topolo1944317154257567458pology @ real ).

thf(tcon_Real_Oreal___Real__Vector__Spaces_Oreal__normed__field,axiom,
    real_V3459762299906320749_field @ real ).

thf(tcon_Real_Oreal___Quickcheck__Narrowing_Opartial__term__of_300,axiom,
    quickc6926020345158392990erm_of @ real ).

thf(tcon_Real_Oreal___Real__Vector__Spaces_Oreal__div__algebra,axiom,
    real_V5047593784448816457lgebra @ real ).

thf(tcon_Real_Oreal___Archimedean__Field_Oarchimedean__field_301,axiom,
    archim462609752435547400_field @ real ).

thf(tcon_Real_Oreal___Rings_Olinordered__semiring__1__strict_302,axiom,
    linord715952674999750819strict @ real ).

thf(tcon_Real_Oreal___Limits_Otopological__comm__monoid__add_303,axiom,
    topolo5987344860129210374id_add @ real ).

thf(tcon_Real_Oreal___Groups_Olinordered__ab__semigroup__add_304,axiom,
    linord4140545234300271783up_add @ real ).

thf(tcon_Real_Oreal___Topological__Spaces_Oorder__topology_305,axiom,
    topolo2564578578187576103pology @ real ).

thf(tcon_Real_Oreal___Rings_Osemiring__1__no__zero__divisors_306,axiom,
    semiri2026040879449505780visors @ real ).

thf(tcon_Real_Oreal___Rings_Olinordered__nonzero__semiring_307,axiom,
    linord181362715937106298miring @ real ).

thf(tcon_Real_Oreal___Real__Vector__Spaces_Oreal__algebra__1,axiom,
    real_V2191834092415804123ebra_1 @ real ).

thf(tcon_Real_Oreal___Rings_Olinordered__semiring__strict_308,axiom,
    linord8928482502909563296strict @ real ).

thf(tcon_Real_Oreal___Rings_Osemiring__no__zero__divisors_309,axiom,
    semiri3467727345109120633visors @ real ).

thf(tcon_Real_Oreal___Real__Vector__Spaces_Oreal__algebra,axiom,
    real_V6157519004096292374lgebra @ real ).

thf(tcon_Real_Oreal___Limits_Otopological__ab__group__add,axiom,
    topolo1287966508704411220up_add @ real ).

thf(tcon_Real_Oreal___Groups_Oordered__ab__semigroup__add_310,axiom,
    ordere6658533253407199908up_add @ real ).

thf(tcon_Real_Oreal___Groups_Oordered__ab__group__add__abs_311,axiom,
    ordere166539214618696060dd_abs @ real ).

thf(tcon_Real_Oreal___Archimedean__Field_Ofloor__ceiling_312,axiom,
    archim2362893244070406136eiling @ real ).

thf(tcon_Real_Oreal___Real__Vector__Spaces_Oreal__vector,axiom,
    real_V4867850818363320053vector @ real ).

thf(tcon_Real_Oreal___Groups_Oordered__comm__monoid__add_313,axiom,
    ordere6911136660526730532id_add @ real ).

thf(tcon_Real_Oreal___Groups_Olinordered__ab__group__add_314,axiom,
    linord5086331880401160121up_add @ real ).

thf(tcon_Real_Oreal___Groups_Ocancel__ab__semigroup__add_315,axiom,
    cancel2418104881723323429up_add @ real ).

thf(tcon_Real_Oreal___Rings_Oring__1__no__zero__divisors_316,axiom,
    ring_15535105094025558882visors @ real ).

thf(tcon_Real_Oreal___Real__Vector__Spaces_Oreal__field,axiom,
    real_V7773925162809079976_field @ real ).

thf(tcon_Real_Oreal___Groups_Ocancel__comm__monoid__add_317,axiom,
    cancel1802427076303600483id_add @ real ).

thf(tcon_Real_Oreal___Rings_Olinordered__ring__strict_318,axiom,
    linord4710134922213307826strict @ real ).

thf(tcon_Real_Oreal___Rings_Ocomm__semiring__1__cancel_319,axiom,
    comm_s4317794764714335236cancel @ real ).

thf(tcon_Real_Oreal___Topological__Spaces_Ot2__space_320,axiom,
    topological_t2_space @ real ).

thf(tcon_Real_Oreal___Rings_Oordered__comm__semiring_321,axiom,
    ordere2520102378445227354miring @ real ).

thf(tcon_Real_Oreal___Rings_Olinordered__semiring__1_322,axiom,
    linord6961819062388156250ring_1 @ real ).

thf(tcon_Real_Oreal___Groups_Oordered__ab__group__add_323,axiom,
    ordered_ab_group_add @ real ).

thf(tcon_Real_Oreal___Groups_Ocancel__semigroup__add_324,axiom,
    cancel_semigroup_add @ real ).

thf(tcon_Real_Oreal___Rings_Olinordered__semiring_325,axiom,
    linordered_semiring @ real ).

thf(tcon_Real_Oreal___Real__Vector__Spaces_Obanach,axiom,
    real_Vector_banach @ real ).

thf(tcon_Real_Oreal___Rings_Oordered__semiring__0_326,axiom,
    ordered_semiring_0 @ real ).

thf(tcon_Real_Oreal___Rings_Olinordered__semidom_327,axiom,
    linordered_semidom @ real ).

thf(tcon_Real_Oreal___Orderings_Odense__linorder_328,axiom,
    dense_linorder @ real ).

thf(tcon_Real_Oreal___Groups_Oab__semigroup__mult_329,axiom,
    ab_semigroup_mult @ real ).

thf(tcon_Real_Oreal___Rings_Osemiring__1__cancel_330,axiom,
    semiring_1_cancel @ real ).

thf(tcon_Real_Oreal___Groups_Ocomm__monoid__mult_331,axiom,
    comm_monoid_mult @ real ).

thf(tcon_Real_Oreal___Groups_Oab__semigroup__add_332,axiom,
    ab_semigroup_add @ real ).

thf(tcon_Real_Oreal___Fields_Olinordered__field_333,axiom,
    linordered_field @ real ).

thf(tcon_Real_Oreal___Code__Evaluation_Oterm__of_334,axiom,
    code_term_of @ real ).

thf(tcon_Real_Oreal___Rings_Oordered__semiring_335,axiom,
    ordered_semiring @ real ).

thf(tcon_Real_Oreal___Rings_Oordered__ring__abs_336,axiom,
    ordered_ring_abs @ real ).

thf(tcon_Real_Oreal___Groups_Ocomm__monoid__add_337,axiom,
    comm_monoid_add @ real ).

thf(tcon_Real_Oreal___Rings_Olinordered__ring_338,axiom,
    linordered_ring @ real ).

thf(tcon_Real_Oreal___Rings_Olinordered__idom_339,axiom,
    linordered_idom @ real ).

thf(tcon_Real_Oreal___Rings_Ocomm__semiring__1_340,axiom,
    comm_semiring_1 @ real ).

thf(tcon_Real_Oreal___Rings_Ocomm__semiring__0_341,axiom,
    comm_semiring_0 @ real ).

thf(tcon_Real_Oreal___Orderings_Odense__order_342,axiom,
    dense_order @ real ).

thf(tcon_Real_Oreal___Groups_Osemigroup__mult_343,axiom,
    semigroup_mult @ real ).

thf(tcon_Real_Oreal___Rings_Osemidom__divide_344,axiom,
    semidom_divide @ real ).

thf(tcon_Real_Oreal___Num_Osemiring__numeral_345,axiom,
    semiring_numeral @ real ).

thf(tcon_Real_Oreal___Groups_Osemigroup__add_346,axiom,
    semigroup_add @ real ).

thf(tcon_Real_Oreal___Fields_Ofield__abs__sgn_347,axiom,
    field_abs_sgn @ real ).

thf(tcon_Real_Oreal___Fields_Odivision__ring_348,axiom,
    division_ring @ real ).

thf(tcon_Real_Oreal___Rings_Ozero__less__one_349,axiom,
    zero_less_one @ real ).

thf(tcon_Real_Oreal___Rings_Ocomm__semiring_350,axiom,
    comm_semiring @ real ).

thf(tcon_Real_Oreal___Nat_Osemiring__char__0_351,axiom,
    semiring_char_0 @ real ).

thf(tcon_Real_Oreal___Groups_Oab__group__add_352,axiom,
    ab_group_add @ real ).

thf(tcon_Real_Oreal___Fields_Ofield__char__0_353,axiom,
    field_char_0 @ real ).

thf(tcon_Real_Oreal___Rings_Ozero__neq__one_354,axiom,
    zero_neq_one @ real ).

thf(tcon_Real_Oreal___Rings_Oordered__ring_355,axiom,
    ordered_ring @ real ).

thf(tcon_Real_Oreal___Rings_Oidom__abs__sgn_356,axiom,
    idom_abs_sgn @ real ).

thf(tcon_Real_Oreal___Orderings_Opreorder_357,axiom,
    preorder @ real ).

thf(tcon_Real_Oreal___Orderings_Olinorder_358,axiom,
    linorder @ real ).

thf(tcon_Real_Oreal___Groups_Omonoid__mult_359,axiom,
    monoid_mult @ real ).

thf(tcon_Real_Oreal___Transcendental_Oln,axiom,
    ln @ real ).

thf(tcon_Real_Oreal___Rings_Oidom__divide_360,axiom,
    idom_divide @ real ).

thf(tcon_Real_Oreal___Rings_Ocomm__ring__1_361,axiom,
    comm_ring_1 @ real ).

thf(tcon_Real_Oreal___Groups_Omonoid__add_362,axiom,
    monoid_add @ real ).

thf(tcon_Real_Oreal___Rings_Osemiring__1_363,axiom,
    semiring_1 @ real ).

thf(tcon_Real_Oreal___Rings_Osemiring__0_364,axiom,
    semiring_0 @ real ).

thf(tcon_Real_Oreal___Orderings_Ono__top_365,axiom,
    no_top @ real ).

thf(tcon_Real_Oreal___Orderings_Ono__bot_366,axiom,
    no_bot @ real ).

thf(tcon_Real_Oreal___Groups_Ogroup__add_367,axiom,
    group_add @ real ).

thf(tcon_Real_Oreal___Rings_Omult__zero_368,axiom,
    mult_zero @ real ).

thf(tcon_Real_Oreal___Rings_Ocomm__ring_369,axiom,
    comm_ring @ real ).

thf(tcon_Real_Oreal___Orderings_Oorder_370,axiom,
    order @ real ).

thf(tcon_Real_Oreal___Num_Oneg__numeral_371,axiom,
    neg_numeral @ real ).

thf(tcon_Real_Oreal___Nat_Oring__char__0_372,axiom,
    ring_char_0 @ real ).

thf(tcon_Real_Oreal___Rings_Osemiring_373,axiom,
    semiring @ real ).

thf(tcon_Real_Oreal___Orderings_Oord_374,axiom,
    ord @ real ).

thf(tcon_Real_Oreal___Groups_Ouminus_375,axiom,
    uminus @ real ).

thf(tcon_Real_Oreal___Rings_Oring__1_376,axiom,
    ring_1 @ real ).

thf(tcon_Real_Oreal___Rings_Oabs__if_377,axiom,
    abs_if @ real ).

thf(tcon_Real_Oreal___Groups_Ominus_378,axiom,
    minus @ real ).

thf(tcon_Real_Oreal___Fields_Ofield_379,axiom,
    field @ real ).

thf(tcon_Real_Oreal___Power_Opower_380,axiom,
    power @ real ).

thf(tcon_Real_Oreal___Num_Onumeral_381,axiom,
    numeral @ real ).

thf(tcon_Real_Oreal___Groups_Ozero_382,axiom,
    zero @ real ).

thf(tcon_Real_Oreal___Groups_Oplus_383,axiom,
    plus @ real ).

thf(tcon_Real_Oreal___Rings_Oring_384,axiom,
    ring @ real ).

thf(tcon_Real_Oreal___Rings_Oidom_385,axiom,
    idom @ real ).

thf(tcon_Real_Oreal___Groups_Oone_386,axiom,
    one @ real ).

thf(tcon_Real_Oreal___Rings_Odvd_387,axiom,
    dvd @ real ).

thf(tcon_Real_Oreal___HOL_Oequal_388,axiom,
    cl_HOL_Oequal @ real ).

thf(tcon_Word_Oword___Bit__Operations_Osemiring__bit__operations_389,axiom,
    ! [A15: $tType] :
      ( ( type_len @ A15 )
     => ( bit_se359711467146920520ations @ ( word @ A15 ) ) ) ).

thf(tcon_Word_Oword___Quickcheck__Narrowing_Opartial__term__of_390,axiom,
    ! [A15: $tType] :
      ( ( typerep @ A15 )
     => ( quickc6926020345158392990erm_of @ ( word @ A15 ) ) ) ).

thf(tcon_Word_Oword___Bit__Comprehension_Obit__comprehension_391,axiom,
    ! [A15: $tType] :
      ( ( type_len @ A15 )
     => ( bit_bi6583157726757044596ension @ ( word @ A15 ) ) ) ).

thf(tcon_Word_Oword___Bit__Operations_Oring__bit__operations_392,axiom,
    ! [A15: $tType] :
      ( ( type_len @ A15 )
     => ( bit_ri3973907225187159222ations @ ( word @ A15 ) ) ) ).

thf(tcon_Word_Oword___Groups_Ocancel__ab__semigroup__add_393,axiom,
    ! [A15: $tType] :
      ( ( type_len @ A15 )
     => ( cancel2418104881723323429up_add @ ( word @ A15 ) ) ) ).

thf(tcon_Word_Oword___Groups_Ocancel__comm__monoid__add_394,axiom,
    ! [A15: $tType] :
      ( ( type_len @ A15 )
     => ( cancel1802427076303600483id_add @ ( word @ A15 ) ) ) ).

thf(tcon_Word_Oword___Rings_Ocomm__semiring__1__cancel_395,axiom,
    ! [A15: $tType] :
      ( ( type_len @ A15 )
     => ( comm_s4317794764714335236cancel @ ( word @ A15 ) ) ) ).

thf(tcon_Word_Oword___Bit__Operations_Osemiring__bits_396,axiom,
    ! [A15: $tType] :
      ( ( type_len @ A15 )
     => ( bit_semiring_bits @ ( word @ A15 ) ) ) ).

thf(tcon_Word_Oword___Groups_Ocancel__semigroup__add_397,axiom,
    ! [A15: $tType] :
      ( ( type_len @ A15 )
     => ( cancel_semigroup_add @ ( word @ A15 ) ) ) ).

thf(tcon_Word_Oword___Least__significant__bit_Olsb_398,axiom,
    ! [A15: $tType] :
      ( ( type_len @ A15 )
     => ( least_6119777620449941438nt_lsb @ ( word @ A15 ) ) ) ).

thf(tcon_Word_Oword___Groups_Oab__semigroup__mult_399,axiom,
    ! [A15: $tType] :
      ( ( type_len @ A15 )
     => ( ab_semigroup_mult @ ( word @ A15 ) ) ) ).

thf(tcon_Word_Oword___Rings_Osemiring__1__cancel_400,axiom,
    ! [A15: $tType] :
      ( ( type_len @ A15 )
     => ( semiring_1_cancel @ ( word @ A15 ) ) ) ).

thf(tcon_Word_Oword___Groups_Ocomm__monoid__mult_401,axiom,
    ! [A15: $tType] :
      ( ( type_len @ A15 )
     => ( comm_monoid_mult @ ( word @ A15 ) ) ) ).

thf(tcon_Word_Oword___Groups_Oab__semigroup__add_402,axiom,
    ! [A15: $tType] :
      ( ( type_len @ A15 )
     => ( ab_semigroup_add @ ( word @ A15 ) ) ) ).

thf(tcon_Word_Oword___Generic__set__bit_Oset__bit_403,axiom,
    ! [A15: $tType] :
      ( ( type_len @ A15 )
     => ( generic_set_set_bit @ ( word @ A15 ) ) ) ).

thf(tcon_Word_Oword___Code__Evaluation_Oterm__of_404,axiom,
    ! [A15: $tType] :
      ( ( typerep @ A15 )
     => ( code_term_of @ ( word @ A15 ) ) ) ).

thf(tcon_Word_Oword___Parity_Osemiring__parity_405,axiom,
    ! [A15: $tType] :
      ( ( type_len @ A15 )
     => ( semiring_parity @ ( word @ A15 ) ) ) ).

thf(tcon_Word_Oword___Groups_Ocomm__monoid__add_406,axiom,
    ! [A15: $tType] :
      ( ( type_len @ A15 )
     => ( comm_monoid_add @ ( word @ A15 ) ) ) ).

thf(tcon_Word_Oword___Rings_Osemiring__modulo_407,axiom,
    ! [A15: $tType] :
      ( ( type_len @ A15 )
     => ( semiring_modulo @ ( word @ A15 ) ) ) ).

thf(tcon_Word_Oword___Rings_Ocomm__semiring__1_408,axiom,
    ! [A15: $tType] :
      ( ( type_len @ A15 )
     => ( comm_semiring_1 @ ( word @ A15 ) ) ) ).

thf(tcon_Word_Oword___Rings_Ocomm__semiring__0_409,axiom,
    ! [A15: $tType] :
      ( ( type_len @ A15 )
     => ( comm_semiring_0 @ ( word @ A15 ) ) ) ).

thf(tcon_Word_Oword___Groups_Osemigroup__mult_410,axiom,
    ! [A15: $tType] :
      ( ( type_len @ A15 )
     => ( semigroup_mult @ ( word @ A15 ) ) ) ).

thf(tcon_Word_Oword___Num_Osemiring__numeral_411,axiom,
    ! [A15: $tType] :
      ( ( type_len @ A15 )
     => ( semiring_numeral @ ( word @ A15 ) ) ) ).

thf(tcon_Word_Oword___Groups_Osemigroup__add_412,axiom,
    ! [A15: $tType] :
      ( ( type_len @ A15 )
     => ( semigroup_add @ ( word @ A15 ) ) ) ).

thf(tcon_Word_Oword___Rings_Ocomm__semiring_413,axiom,
    ! [A15: $tType] :
      ( ( type_len @ A15 )
     => ( comm_semiring @ ( word @ A15 ) ) ) ).

thf(tcon_Word_Oword___Orderings_Owellorder_414,axiom,
    ! [A15: $tType] :
      ( ( type_len @ A15 )
     => ( wellorder @ ( word @ A15 ) ) ) ).

thf(tcon_Word_Oword___Groups_Oab__group__add_415,axiom,
    ! [A15: $tType] :
      ( ( type_len @ A15 )
     => ( ab_group_add @ ( word @ A15 ) ) ) ).

thf(tcon_Word_Oword___Rings_Ozero__neq__one_416,axiom,
    ! [A15: $tType] :
      ( ( type_len @ A15 )
     => ( zero_neq_one @ ( word @ A15 ) ) ) ).

thf(tcon_Word_Oword___Parity_Oring__parity_417,axiom,
    ! [A15: $tType] :
      ( ( type_len @ A15 )
     => ( ring_parity @ ( word @ A15 ) ) ) ).

thf(tcon_Word_Oword___Orderings_Opreorder_418,axiom,
    ! [A15: $tType] :
      ( ( type_len @ A15 )
     => ( preorder @ ( word @ A15 ) ) ) ).

thf(tcon_Word_Oword___Orderings_Olinorder_419,axiom,
    ! [A15: $tType] :
      ( ( type_len @ A15 )
     => ( linorder @ ( word @ A15 ) ) ) ).

thf(tcon_Word_Oword___Groups_Omonoid__mult_420,axiom,
    ! [A15: $tType] :
      ( ( type_len @ A15 )
     => ( monoid_mult @ ( word @ A15 ) ) ) ).

thf(tcon_Word_Oword___Rings_Ocomm__ring__1_421,axiom,
    ! [A15: $tType] :
      ( ( type_len @ A15 )
     => ( comm_ring_1 @ ( word @ A15 ) ) ) ).

thf(tcon_Word_Oword___Groups_Omonoid__add_422,axiom,
    ! [A15: $tType] :
      ( ( type_len @ A15 )
     => ( monoid_add @ ( word @ A15 ) ) ) ).

thf(tcon_Word_Oword___Rings_Osemiring__1_423,axiom,
    ! [A15: $tType] :
      ( ( type_len @ A15 )
     => ( semiring_1 @ ( word @ A15 ) ) ) ).

thf(tcon_Word_Oword___Rings_Osemiring__0_424,axiom,
    ! [A15: $tType] :
      ( ( type_len @ A15 )
     => ( semiring_0 @ ( word @ A15 ) ) ) ).

thf(tcon_Word_Oword___Groups_Ogroup__add_425,axiom,
    ! [A15: $tType] :
      ( ( type_len @ A15 )
     => ( group_add @ ( word @ A15 ) ) ) ).

thf(tcon_Word_Oword___Rings_Omult__zero_426,axiom,
    ! [A15: $tType] :
      ( ( type_len @ A15 )
     => ( mult_zero @ ( word @ A15 ) ) ) ).

thf(tcon_Word_Oword___Rings_Ocomm__ring_427,axiom,
    ! [A15: $tType] :
      ( ( type_len @ A15 )
     => ( comm_ring @ ( word @ A15 ) ) ) ).

thf(tcon_Word_Oword___Orderings_Oorder_428,axiom,
    ! [A15: $tType] :
      ( ( type_len @ A15 )
     => ( order @ ( word @ A15 ) ) ) ).

thf(tcon_Word_Oword___Num_Oneg__numeral_429,axiom,
    ! [A15: $tType] :
      ( ( type_len @ A15 )
     => ( neg_numeral @ ( word @ A15 ) ) ) ).

thf(tcon_Word_Oword___Rings_Osemiring_430,axiom,
    ! [A15: $tType] :
      ( ( type_len @ A15 )
     => ( semiring @ ( word @ A15 ) ) ) ).

thf(tcon_Word_Oword___Orderings_Oord_431,axiom,
    ! [A15: $tType] :
      ( ( type_len @ A15 )
     => ( ord @ ( word @ A15 ) ) ) ).

thf(tcon_Word_Oword___Groups_Ouminus_432,axiom,
    ! [A15: $tType] :
      ( ( type_len @ A15 )
     => ( uminus @ ( word @ A15 ) ) ) ).

thf(tcon_Word_Oword___Rings_Oring__1_433,axiom,
    ! [A15: $tType] :
      ( ( type_len @ A15 )
     => ( ring_1 @ ( word @ A15 ) ) ) ).

thf(tcon_Word_Oword___Groups_Ominus_434,axiom,
    ! [A15: $tType] :
      ( ( type_len @ A15 )
     => ( minus @ ( word @ A15 ) ) ) ).

thf(tcon_Word_Oword___Power_Opower_435,axiom,
    ! [A15: $tType] :
      ( ( type_len @ A15 )
     => ( power @ ( word @ A15 ) ) ) ).

thf(tcon_Word_Oword___Num_Onumeral_436,axiom,
    ! [A15: $tType] :
      ( ( type_len @ A15 )
     => ( numeral @ ( word @ A15 ) ) ) ).

thf(tcon_Word_Oword___Groups_Ozero_437,axiom,
    ! [A15: $tType] :
      ( ( type_len @ A15 )
     => ( zero @ ( word @ A15 ) ) ) ).

thf(tcon_Word_Oword___Groups_Oplus_438,axiom,
    ! [A15: $tType] :
      ( ( type_len @ A15 )
     => ( plus @ ( word @ A15 ) ) ) ).

thf(tcon_Word_Oword___Rings_Oring_439,axiom,
    ! [A15: $tType] :
      ( ( type_len @ A15 )
     => ( ring @ ( word @ A15 ) ) ) ).

thf(tcon_Word_Oword___Groups_Oone_440,axiom,
    ! [A15: $tType] :
      ( ( type_len @ A15 )
     => ( one @ ( word @ A15 ) ) ) ).

thf(tcon_Word_Oword___Rings_Odvd_441,axiom,
    ! [A15: $tType] :
      ( ( type_len @ A15 )
     => ( dvd @ ( word @ A15 ) ) ) ).

thf(tcon_Word_Oword___HOL_Oequal_442,axiom,
    ! [A15: $tType] :
      ( ( type_len @ A15 )
     => ( cl_HOL_Oequal @ ( word @ A15 ) ) ) ).

thf(tcon_Word_Oword___Nat_Osize_443,axiom,
    ! [A15: $tType] :
      ( ( type_len @ A15 )
     => ( size @ ( word @ A15 ) ) ) ).

thf(tcon_Heap_Oarray___Quickcheck__Narrowing_Opartial__term__of_444,axiom,
    ! [A15: $tType] :
      ( ( typerep @ A15 )
     => ( quickc6926020345158392990erm_of @ ( array @ A15 ) ) ) ).

thf(tcon_Heap_Oarray___Code__Evaluation_Oterm__of_445,axiom,
    ! [A15: $tType] :
      ( ( typerep @ A15 )
     => ( code_term_of @ ( array @ A15 ) ) ) ).

thf(tcon_Heap_Oarray___Heap_Oheap_446,axiom,
    ! [A15: $tType] : ( heap @ ( array @ A15 ) ) ).

thf(tcon_Heap_Oarray___HOL_Oequal_447,axiom,
    ! [A15: $tType] : ( cl_HOL_Oequal @ ( array @ A15 ) ) ).

thf(tcon_Heap_Oarray___Nat_Osize_448,axiom,
    ! [A15: $tType] : ( size @ ( array @ A15 ) ) ).

thf(tcon_String_Ochar___Quickcheck__Narrowing_Opartial__term__of_449,axiom,
    quickc6926020345158392990erm_of @ char ).

thf(tcon_String_Ochar___Code__Evaluation_Oterm__of_450,axiom,
    code_term_of @ char ).

thf(tcon_String_Ochar___Orderings_Opreorder_451,axiom,
    preorder @ char ).

thf(tcon_String_Ochar___Orderings_Olinorder_452,axiom,
    linorder @ char ).

thf(tcon_String_Ochar___Orderings_Oorder_453,axiom,
    order @ char ).

thf(tcon_String_Ochar___Orderings_Oord_454,axiom,
    ord @ char ).

thf(tcon_String_Ochar___Heap_Oheap_455,axiom,
    heap @ char ).

thf(tcon_String_Ochar___HOL_Oequal_456,axiom,
    cl_HOL_Oequal @ char ).

thf(tcon_String_Ochar___Nat_Osize_457,axiom,
    size @ char ).

thf(tcon_Enum_Ofinite__1___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_458,axiom,
    condit6923001295902523014norder @ finite_1 ).

thf(tcon_Enum_Ofinite__1___Groups_Oordered__ab__semigroup__monoid__add__imp__le_459,axiom,
    ordere1937475149494474687imp_le @ finite_1 ).

thf(tcon_Enum_Ofinite__1___Rings_Osemiring__no__zero__divisors__cancel_460,axiom,
    semiri6575147826004484403cancel @ finite_1 ).

thf(tcon_Enum_Ofinite__1___Groups_Ostrict__ordered__ab__semigroup__add_461,axiom,
    strict9044650504122735259up_add @ finite_1 ).

thf(tcon_Enum_Ofinite__1___Groups_Oordered__cancel__comm__monoid__diff_462,axiom,
    ordere1170586879665033532d_diff @ finite_1 ).

thf(tcon_Enum_Ofinite__1___Groups_Oordered__cancel__ab__semigroup__add_463,axiom,
    ordere580206878836729694up_add @ finite_1 ).

thf(tcon_Enum_Ofinite__1___Groups_Oordered__ab__semigroup__add__imp__le_464,axiom,
    ordere2412721322843649153imp_le @ finite_1 ).

thf(tcon_Enum_Ofinite__1___Rings_Olinordered__comm__semiring__strict_465,axiom,
    linord2810124833399127020strict @ finite_1 ).

thf(tcon_Enum_Ofinite__1___Groups_Ostrict__ordered__comm__monoid__add_466,axiom,
    strict7427464778891057005id_add @ finite_1 ).

thf(tcon_Enum_Ofinite__1___Groups_Oordered__cancel__comm__monoid__add_467,axiom,
    ordere8940638589300402666id_add @ finite_1 ).

thf(tcon_Enum_Ofinite__1___Groups_Ocanonically__ordered__monoid__add_468,axiom,
    canoni5634975068530333245id_add @ finite_1 ).

thf(tcon_Enum_Ofinite__1___Quickcheck__Narrowing_Opartial__term__of_469,axiom,
    quickc6926020345158392990erm_of @ finite_1 ).

thf(tcon_Enum_Ofinite__1___Groups_Olinordered__ab__semigroup__add_470,axiom,
    linord4140545234300271783up_add @ finite_1 ).

thf(tcon_Enum_Ofinite__1___Rings_Olinordered__semiring__strict_471,axiom,
    linord8928482502909563296strict @ finite_1 ).

thf(tcon_Enum_Ofinite__1___Boolean__Algebras_Oboolean__algebra_472,axiom,
    boolea8198339166811842893lgebra @ finite_1 ).

thf(tcon_Enum_Ofinite__1___Rings_Osemiring__no__zero__divisors_473,axiom,
    semiri3467727345109120633visors @ finite_1 ).

thf(tcon_Enum_Ofinite__1___Groups_Oordered__ab__semigroup__add_474,axiom,
    ordere6658533253407199908up_add @ finite_1 ).

thf(tcon_Enum_Ofinite__1___Groups_Oordered__ab__group__add__abs_475,axiom,
    ordere166539214618696060dd_abs @ finite_1 ).

thf(tcon_Enum_Ofinite__1___Groups_Oordered__comm__monoid__add_476,axiom,
    ordere6911136660526730532id_add @ finite_1 ).

thf(tcon_Enum_Ofinite__1___Groups_Olinordered__ab__group__add_477,axiom,
    linord5086331880401160121up_add @ finite_1 ).

thf(tcon_Enum_Ofinite__1___Groups_Ocancel__ab__semigroup__add_478,axiom,
    cancel2418104881723323429up_add @ finite_1 ).

thf(tcon_Enum_Ofinite__1___Groups_Ocancel__comm__monoid__add_479,axiom,
    cancel1802427076303600483id_add @ finite_1 ).

thf(tcon_Enum_Ofinite__1___Rings_Olinordered__ring__strict_480,axiom,
    linord4710134922213307826strict @ finite_1 ).

thf(tcon_Enum_Ofinite__1___Rings_Oordered__comm__semiring_481,axiom,
    ordere2520102378445227354miring @ finite_1 ).

thf(tcon_Enum_Ofinite__1___Groups_Oordered__ab__group__add_482,axiom,
    ordered_ab_group_add @ finite_1 ).

thf(tcon_Enum_Ofinite__1___Groups_Ocancel__semigroup__add_483,axiom,
    cancel_semigroup_add @ finite_1 ).

thf(tcon_Enum_Ofinite__1___Rings_Olinordered__semiring_484,axiom,
    linordered_semiring @ finite_1 ).

thf(tcon_Enum_Ofinite__1___Rings_Oordered__semiring__0_485,axiom,
    ordered_semiring_0 @ finite_1 ).

thf(tcon_Enum_Ofinite__1___Orderings_Odense__linorder_486,axiom,
    dense_linorder @ finite_1 ).

thf(tcon_Enum_Ofinite__1___Groups_Oab__semigroup__mult_487,axiom,
    ab_semigroup_mult @ finite_1 ).

thf(tcon_Enum_Ofinite__1___Groups_Ocomm__monoid__mult_488,axiom,
    comm_monoid_mult @ finite_1 ).

thf(tcon_Enum_Ofinite__1___Groups_Ocomm__monoid__diff_489,axiom,
    comm_monoid_diff @ finite_1 ).

thf(tcon_Enum_Ofinite__1___Groups_Oab__semigroup__add_490,axiom,
    ab_semigroup_add @ finite_1 ).

thf(tcon_Enum_Ofinite__1___Code__Evaluation_Oterm__of_491,axiom,
    code_term_of @ finite_1 ).

thf(tcon_Enum_Ofinite__1___Rings_Oordered__semiring_492,axiom,
    ordered_semiring @ finite_1 ).

thf(tcon_Enum_Ofinite__1___Rings_Oordered__ring__abs_493,axiom,
    ordered_ring_abs @ finite_1 ).

thf(tcon_Enum_Ofinite__1___Groups_Ocomm__monoid__add_494,axiom,
    comm_monoid_add @ finite_1 ).

thf(tcon_Enum_Ofinite__1___Rings_Olinordered__ring_495,axiom,
    linordered_ring @ finite_1 ).

thf(tcon_Enum_Ofinite__1___Rings_Ocomm__semiring__0_496,axiom,
    comm_semiring_0 @ finite_1 ).

thf(tcon_Enum_Ofinite__1___Orderings_Odense__order_497,axiom,
    dense_order @ finite_1 ).

thf(tcon_Enum_Ofinite__1___Groups_Osemigroup__mult_498,axiom,
    semigroup_mult @ finite_1 ).

thf(tcon_Enum_Ofinite__1___Groups_Osemigroup__add_499,axiom,
    semigroup_add @ finite_1 ).

thf(tcon_Enum_Ofinite__1___Rings_Ocomm__semiring_500,axiom,
    comm_semiring @ finite_1 ).

thf(tcon_Enum_Ofinite__1___Orderings_Owellorder_501,axiom,
    wellorder @ finite_1 ).

thf(tcon_Enum_Ofinite__1___Orderings_Oorder__top_502,axiom,
    order_top @ finite_1 ).

thf(tcon_Enum_Ofinite__1___Orderings_Oorder__bot_503,axiom,
    order_bot @ finite_1 ).

thf(tcon_Enum_Ofinite__1___Groups_Oab__group__add_504,axiom,
    ab_group_add @ finite_1 ).

thf(tcon_Enum_Ofinite__1___Rings_Oordered__ring_505,axiom,
    ordered_ring @ finite_1 ).

thf(tcon_Enum_Ofinite__1___Orderings_Opreorder_506,axiom,
    preorder @ finite_1 ).

thf(tcon_Enum_Ofinite__1___Orderings_Olinorder_507,axiom,
    linorder @ finite_1 ).

thf(tcon_Enum_Ofinite__1___Groups_Omonoid__mult_508,axiom,
    monoid_mult @ finite_1 ).

thf(tcon_Enum_Ofinite__1___Groups_Omonoid__add_509,axiom,
    monoid_add @ finite_1 ).

thf(tcon_Enum_Ofinite__1___Type__Length_Olen0,axiom,
    type_len0 @ finite_1 ).

thf(tcon_Enum_Ofinite__1___Rings_Osemiring__0_510,axiom,
    semiring_0 @ finite_1 ).

thf(tcon_Enum_Ofinite__1___Groups_Ogroup__add_511,axiom,
    group_add @ finite_1 ).

thf(tcon_Enum_Ofinite__1___Type__Length_Olen,axiom,
    type_len @ finite_1 ).

thf(tcon_Enum_Ofinite__1___Rings_Omult__zero_512,axiom,
    mult_zero @ finite_1 ).

thf(tcon_Enum_Ofinite__1___Rings_Ocomm__ring_513,axiom,
    comm_ring @ finite_1 ).

thf(tcon_Enum_Ofinite__1___Orderings_Oorder_514,axiom,
    order @ finite_1 ).

thf(tcon_Enum_Ofinite__1___Rings_Osemiring_515,axiom,
    semiring @ finite_1 ).

thf(tcon_Enum_Ofinite__1___Orderings_Oord_516,axiom,
    ord @ finite_1 ).

thf(tcon_Enum_Ofinite__1___Groups_Ouminus_517,axiom,
    uminus @ finite_1 ).

thf(tcon_Enum_Ofinite__1___Rings_Oabs__if_518,axiom,
    abs_if @ finite_1 ).

thf(tcon_Enum_Ofinite__1___Groups_Ominus_519,axiom,
    minus @ finite_1 ).

thf(tcon_Enum_Ofinite__1___Power_Opower_520,axiom,
    power @ finite_1 ).

thf(tcon_Enum_Ofinite__1___Groups_Ozero_521,axiom,
    zero @ finite_1 ).

thf(tcon_Enum_Ofinite__1___Groups_Oplus_522,axiom,
    plus @ finite_1 ).

thf(tcon_Enum_Ofinite__1___Rings_Oring_523,axiom,
    ring @ finite_1 ).

thf(tcon_Enum_Ofinite__1___Groups_Oone_524,axiom,
    one @ finite_1 ).

thf(tcon_Enum_Ofinite__1___Rings_Odvd_525,axiom,
    dvd @ finite_1 ).

thf(tcon_Enum_Ofinite__1___HOL_Oequal_526,axiom,
    cl_HOL_Oequal @ finite_1 ).

thf(tcon_Enum_Ofinite__2___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_527,axiom,
    condit6923001295902523014norder @ finite_2 ).

thf(tcon_Enum_Ofinite__2___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_528,axiom,
    semiri1453513574482234551roduct @ finite_2 ).

thf(tcon_Enum_Ofinite__2___Euclidean__Division_Ounique__euclidean__semiring_529,axiom,
    euclid3128863361964157862miring @ finite_2 ).

thf(tcon_Enum_Ofinite__2___Euclidean__Division_Oeuclidean__semiring__cancel_530,axiom,
    euclid4440199948858584721cancel @ finite_2 ).

thf(tcon_Enum_Ofinite__2___Rings_Osemiring__no__zero__divisors__cancel_531,axiom,
    semiri6575147826004484403cancel @ finite_2 ).

thf(tcon_Enum_Ofinite__2___Euclidean__Division_Oeuclidean__semiring_532,axiom,
    euclid3725896446679973847miring @ finite_2 ).

thf(tcon_Enum_Ofinite__2___Quickcheck__Narrowing_Opartial__term__of_533,axiom,
    quickc6926020345158392990erm_of @ finite_2 ).

thf(tcon_Enum_Ofinite__2___Rings_Osemiring__1__no__zero__divisors_534,axiom,
    semiri2026040879449505780visors @ finite_2 ).

thf(tcon_Enum_Ofinite__2___Rings_Osemiring__no__zero__divisors_535,axiom,
    semiri3467727345109120633visors @ finite_2 ).

thf(tcon_Enum_Ofinite__2___Groups_Ocancel__ab__semigroup__add_536,axiom,
    cancel2418104881723323429up_add @ finite_2 ).

thf(tcon_Enum_Ofinite__2___Rings_Oring__1__no__zero__divisors_537,axiom,
    ring_15535105094025558882visors @ finite_2 ).

thf(tcon_Enum_Ofinite__2___Groups_Ocancel__comm__monoid__add_538,axiom,
    cancel1802427076303600483id_add @ finite_2 ).

thf(tcon_Enum_Ofinite__2___Rings_Ocomm__semiring__1__cancel_539,axiom,
    comm_s4317794764714335236cancel @ finite_2 ).

thf(tcon_Enum_Ofinite__2___Groups_Ocancel__semigroup__add_540,axiom,
    cancel_semigroup_add @ finite_2 ).

thf(tcon_Enum_Ofinite__2___Groups_Oab__semigroup__mult_541,axiom,
    ab_semigroup_mult @ finite_2 ).

thf(tcon_Enum_Ofinite__2___Rings_Osemiring__1__cancel_542,axiom,
    semiring_1_cancel @ finite_2 ).

thf(tcon_Enum_Ofinite__2___Rings_Oalgebraic__semidom_543,axiom,
    algebraic_semidom @ finite_2 ).

thf(tcon_Enum_Ofinite__2___Groups_Ocomm__monoid__mult_544,axiom,
    comm_monoid_mult @ finite_2 ).

thf(tcon_Enum_Ofinite__2___Groups_Oab__semigroup__add_545,axiom,
    ab_semigroup_add @ finite_2 ).

thf(tcon_Enum_Ofinite__2___Code__Evaluation_Oterm__of_546,axiom,
    code_term_of @ finite_2 ).

thf(tcon_Enum_Ofinite__2___Groups_Ocomm__monoid__add_547,axiom,
    comm_monoid_add @ finite_2 ).

thf(tcon_Enum_Ofinite__2___Rings_Osemiring__modulo_548,axiom,
    semiring_modulo @ finite_2 ).

thf(tcon_Enum_Ofinite__2___Rings_Ocomm__semiring__1_549,axiom,
    comm_semiring_1 @ finite_2 ).

thf(tcon_Enum_Ofinite__2___Rings_Ocomm__semiring__0_550,axiom,
    comm_semiring_0 @ finite_2 ).

thf(tcon_Enum_Ofinite__2___Groups_Osemigroup__mult_551,axiom,
    semigroup_mult @ finite_2 ).

thf(tcon_Enum_Ofinite__2___Rings_Osemidom__modulo_552,axiom,
    semidom_modulo @ finite_2 ).

thf(tcon_Enum_Ofinite__2___Rings_Osemidom__divide_553,axiom,
    semidom_divide @ finite_2 ).

thf(tcon_Enum_Ofinite__2___Num_Osemiring__numeral_554,axiom,
    semiring_numeral @ finite_2 ).

thf(tcon_Enum_Ofinite__2___Groups_Osemigroup__add_555,axiom,
    semigroup_add @ finite_2 ).

thf(tcon_Enum_Ofinite__2___Fields_Odivision__ring_556,axiom,
    division_ring @ finite_2 ).

thf(tcon_Enum_Ofinite__2___Rings_Ocomm__semiring_557,axiom,
    comm_semiring @ finite_2 ).

thf(tcon_Enum_Ofinite__2___Orderings_Owellorder_558,axiom,
    wellorder @ finite_2 ).

thf(tcon_Enum_Ofinite__2___Orderings_Oorder__top_559,axiom,
    order_top @ finite_2 ).

thf(tcon_Enum_Ofinite__2___Orderings_Oorder__bot_560,axiom,
    order_bot @ finite_2 ).

thf(tcon_Enum_Ofinite__2___Groups_Oab__group__add_561,axiom,
    ab_group_add @ finite_2 ).

thf(tcon_Enum_Ofinite__2___Rings_Ozero__neq__one_562,axiom,
    zero_neq_one @ finite_2 ).

thf(tcon_Enum_Ofinite__2___Rings_Oidom__abs__sgn_563,axiom,
    idom_abs_sgn @ finite_2 ).

thf(tcon_Enum_Ofinite__2___Orderings_Opreorder_564,axiom,
    preorder @ finite_2 ).

thf(tcon_Enum_Ofinite__2___Orderings_Olinorder_565,axiom,
    linorder @ finite_2 ).

thf(tcon_Enum_Ofinite__2___Groups_Omonoid__mult_566,axiom,
    monoid_mult @ finite_2 ).

thf(tcon_Enum_Ofinite__2___Rings_Oidom__modulo_567,axiom,
    idom_modulo @ finite_2 ).

thf(tcon_Enum_Ofinite__2___Rings_Oidom__divide_568,axiom,
    idom_divide @ finite_2 ).

thf(tcon_Enum_Ofinite__2___Rings_Ocomm__ring__1_569,axiom,
    comm_ring_1 @ finite_2 ).

thf(tcon_Enum_Ofinite__2___Groups_Omonoid__add_570,axiom,
    monoid_add @ finite_2 ).

thf(tcon_Enum_Ofinite__2___Type__Length_Olen0_571,axiom,
    type_len0 @ finite_2 ).

thf(tcon_Enum_Ofinite__2___Rings_Osemiring__1_572,axiom,
    semiring_1 @ finite_2 ).

thf(tcon_Enum_Ofinite__2___Rings_Osemiring__0_573,axiom,
    semiring_0 @ finite_2 ).

thf(tcon_Enum_Ofinite__2___Groups_Ogroup__add_574,axiom,
    group_add @ finite_2 ).

thf(tcon_Enum_Ofinite__2___Type__Length_Olen_575,axiom,
    type_len @ finite_2 ).

thf(tcon_Enum_Ofinite__2___Rings_Omult__zero_576,axiom,
    mult_zero @ finite_2 ).

thf(tcon_Enum_Ofinite__2___Rings_Ocomm__ring_577,axiom,
    comm_ring @ finite_2 ).

thf(tcon_Enum_Ofinite__2___Orderings_Oorder_578,axiom,
    order @ finite_2 ).

thf(tcon_Enum_Ofinite__2___Num_Oneg__numeral_579,axiom,
    neg_numeral @ finite_2 ).

thf(tcon_Enum_Ofinite__2___Rings_Osemiring_580,axiom,
    semiring @ finite_2 ).

thf(tcon_Enum_Ofinite__2___Orderings_Oord_581,axiom,
    ord @ finite_2 ).

thf(tcon_Enum_Ofinite__2___Groups_Ouminus_582,axiom,
    uminus @ finite_2 ).

thf(tcon_Enum_Ofinite__2___Rings_Oring__1_583,axiom,
    ring_1 @ finite_2 ).

thf(tcon_Enum_Ofinite__2___Groups_Ominus_584,axiom,
    minus @ finite_2 ).

thf(tcon_Enum_Ofinite__2___Fields_Ofield_585,axiom,
    field @ finite_2 ).

thf(tcon_Enum_Ofinite__2___Power_Opower_586,axiom,
    power @ finite_2 ).

thf(tcon_Enum_Ofinite__2___Num_Onumeral_587,axiom,
    numeral @ finite_2 ).

thf(tcon_Enum_Ofinite__2___Groups_Ozero_588,axiom,
    zero @ finite_2 ).

thf(tcon_Enum_Ofinite__2___Groups_Oplus_589,axiom,
    plus @ finite_2 ).

thf(tcon_Enum_Ofinite__2___Rings_Oring_590,axiom,
    ring @ finite_2 ).

thf(tcon_Enum_Ofinite__2___Rings_Oidom_591,axiom,
    idom @ finite_2 ).

thf(tcon_Enum_Ofinite__2___Groups_Oone_592,axiom,
    one @ finite_2 ).

thf(tcon_Enum_Ofinite__2___Rings_Odvd_593,axiom,
    dvd @ finite_2 ).

thf(tcon_Enum_Ofinite__2___HOL_Oequal_594,axiom,
    cl_HOL_Oequal @ finite_2 ).

thf(tcon_Enum_Ofinite__3___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_595,axiom,
    condit6923001295902523014norder @ finite_3 ).

thf(tcon_Enum_Ofinite__3___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_596,axiom,
    semiri1453513574482234551roduct @ finite_3 ).

thf(tcon_Enum_Ofinite__3___Euclidean__Division_Ounique__euclidean__semiring_597,axiom,
    euclid3128863361964157862miring @ finite_3 ).

thf(tcon_Enum_Ofinite__3___Euclidean__Division_Oeuclidean__semiring__cancel_598,axiom,
    euclid4440199948858584721cancel @ finite_3 ).

thf(tcon_Enum_Ofinite__3___Rings_Osemiring__no__zero__divisors__cancel_599,axiom,
    semiri6575147826004484403cancel @ finite_3 ).

thf(tcon_Enum_Ofinite__3___Euclidean__Division_Oeuclidean__semiring_600,axiom,
    euclid3725896446679973847miring @ finite_3 ).

thf(tcon_Enum_Ofinite__3___Quickcheck__Narrowing_Opartial__term__of_601,axiom,
    quickc6926020345158392990erm_of @ finite_3 ).

thf(tcon_Enum_Ofinite__3___Rings_Osemiring__1__no__zero__divisors_602,axiom,
    semiri2026040879449505780visors @ finite_3 ).

thf(tcon_Enum_Ofinite__3___Rings_Osemiring__no__zero__divisors_603,axiom,
    semiri3467727345109120633visors @ finite_3 ).

thf(tcon_Enum_Ofinite__3___Groups_Ocancel__ab__semigroup__add_604,axiom,
    cancel2418104881723323429up_add @ finite_3 ).

thf(tcon_Enum_Ofinite__3___Rings_Oring__1__no__zero__divisors_605,axiom,
    ring_15535105094025558882visors @ finite_3 ).

thf(tcon_Enum_Ofinite__3___Groups_Ocancel__comm__monoid__add_606,axiom,
    cancel1802427076303600483id_add @ finite_3 ).

thf(tcon_Enum_Ofinite__3___Rings_Ocomm__semiring__1__cancel_607,axiom,
    comm_s4317794764714335236cancel @ finite_3 ).

thf(tcon_Enum_Ofinite__3___Groups_Ocancel__semigroup__add_608,axiom,
    cancel_semigroup_add @ finite_3 ).

thf(tcon_Enum_Ofinite__3___Groups_Oab__semigroup__mult_609,axiom,
    ab_semigroup_mult @ finite_3 ).

thf(tcon_Enum_Ofinite__3___Rings_Osemiring__1__cancel_610,axiom,
    semiring_1_cancel @ finite_3 ).

thf(tcon_Enum_Ofinite__3___Rings_Oalgebraic__semidom_611,axiom,
    algebraic_semidom @ finite_3 ).

thf(tcon_Enum_Ofinite__3___Groups_Ocomm__monoid__mult_612,axiom,
    comm_monoid_mult @ finite_3 ).

thf(tcon_Enum_Ofinite__3___Groups_Oab__semigroup__add_613,axiom,
    ab_semigroup_add @ finite_3 ).

thf(tcon_Enum_Ofinite__3___Code__Evaluation_Oterm__of_614,axiom,
    code_term_of @ finite_3 ).

thf(tcon_Enum_Ofinite__3___Groups_Ocomm__monoid__add_615,axiom,
    comm_monoid_add @ finite_3 ).

thf(tcon_Enum_Ofinite__3___Rings_Osemiring__modulo_616,axiom,
    semiring_modulo @ finite_3 ).

thf(tcon_Enum_Ofinite__3___Rings_Ocomm__semiring__1_617,axiom,
    comm_semiring_1 @ finite_3 ).

thf(tcon_Enum_Ofinite__3___Rings_Ocomm__semiring__0_618,axiom,
    comm_semiring_0 @ finite_3 ).

thf(tcon_Enum_Ofinite__3___Groups_Osemigroup__mult_619,axiom,
    semigroup_mult @ finite_3 ).

thf(tcon_Enum_Ofinite__3___Rings_Osemidom__modulo_620,axiom,
    semidom_modulo @ finite_3 ).

thf(tcon_Enum_Ofinite__3___Rings_Osemidom__divide_621,axiom,
    semidom_divide @ finite_3 ).

thf(tcon_Enum_Ofinite__3___Num_Osemiring__numeral_622,axiom,
    semiring_numeral @ finite_3 ).

thf(tcon_Enum_Ofinite__3___Groups_Osemigroup__add_623,axiom,
    semigroup_add @ finite_3 ).

thf(tcon_Enum_Ofinite__3___Fields_Odivision__ring_624,axiom,
    division_ring @ finite_3 ).

thf(tcon_Enum_Ofinite__3___Rings_Ocomm__semiring_625,axiom,
    comm_semiring @ finite_3 ).

thf(tcon_Enum_Ofinite__3___Orderings_Owellorder_626,axiom,
    wellorder @ finite_3 ).

thf(tcon_Enum_Ofinite__3___Orderings_Oorder__top_627,axiom,
    order_top @ finite_3 ).

thf(tcon_Enum_Ofinite__3___Orderings_Oorder__bot_628,axiom,
    order_bot @ finite_3 ).

thf(tcon_Enum_Ofinite__3___Groups_Oab__group__add_629,axiom,
    ab_group_add @ finite_3 ).

thf(tcon_Enum_Ofinite__3___Rings_Ozero__neq__one_630,axiom,
    zero_neq_one @ finite_3 ).

thf(tcon_Enum_Ofinite__3___Rings_Oidom__abs__sgn_631,axiom,
    idom_abs_sgn @ finite_3 ).

thf(tcon_Enum_Ofinite__3___Orderings_Opreorder_632,axiom,
    preorder @ finite_3 ).

thf(tcon_Enum_Ofinite__3___Orderings_Olinorder_633,axiom,
    linorder @ finite_3 ).

thf(tcon_Enum_Ofinite__3___Groups_Omonoid__mult_634,axiom,
    monoid_mult @ finite_3 ).

thf(tcon_Enum_Ofinite__3___Rings_Oidom__modulo_635,axiom,
    idom_modulo @ finite_3 ).

thf(tcon_Enum_Ofinite__3___Rings_Oidom__divide_636,axiom,
    idom_divide @ finite_3 ).

thf(tcon_Enum_Ofinite__3___Rings_Ocomm__ring__1_637,axiom,
    comm_ring_1 @ finite_3 ).

thf(tcon_Enum_Ofinite__3___Groups_Omonoid__add_638,axiom,
    monoid_add @ finite_3 ).

thf(tcon_Enum_Ofinite__3___Type__Length_Olen0_639,axiom,
    type_len0 @ finite_3 ).

thf(tcon_Enum_Ofinite__3___Rings_Osemiring__1_640,axiom,
    semiring_1 @ finite_3 ).

thf(tcon_Enum_Ofinite__3___Rings_Osemiring__0_641,axiom,
    semiring_0 @ finite_3 ).

thf(tcon_Enum_Ofinite__3___Groups_Ogroup__add_642,axiom,
    group_add @ finite_3 ).

thf(tcon_Enum_Ofinite__3___Type__Length_Olen_643,axiom,
    type_len @ finite_3 ).

thf(tcon_Enum_Ofinite__3___Rings_Omult__zero_644,axiom,
    mult_zero @ finite_3 ).

thf(tcon_Enum_Ofinite__3___Rings_Ocomm__ring_645,axiom,
    comm_ring @ finite_3 ).

thf(tcon_Enum_Ofinite__3___Orderings_Oorder_646,axiom,
    order @ finite_3 ).

thf(tcon_Enum_Ofinite__3___Num_Oneg__numeral_647,axiom,
    neg_numeral @ finite_3 ).

thf(tcon_Enum_Ofinite__3___Rings_Osemiring_648,axiom,
    semiring @ finite_3 ).

thf(tcon_Enum_Ofinite__3___Orderings_Oord_649,axiom,
    ord @ finite_3 ).

thf(tcon_Enum_Ofinite__3___Groups_Ouminus_650,axiom,
    uminus @ finite_3 ).

thf(tcon_Enum_Ofinite__3___Rings_Oring__1_651,axiom,
    ring_1 @ finite_3 ).

thf(tcon_Enum_Ofinite__3___Groups_Ominus_652,axiom,
    minus @ finite_3 ).

thf(tcon_Enum_Ofinite__3___Fields_Ofield_653,axiom,
    field @ finite_3 ).

thf(tcon_Enum_Ofinite__3___Power_Opower_654,axiom,
    power @ finite_3 ).

thf(tcon_Enum_Ofinite__3___Num_Onumeral_655,axiom,
    numeral @ finite_3 ).

thf(tcon_Enum_Ofinite__3___Groups_Ozero_656,axiom,
    zero @ finite_3 ).

thf(tcon_Enum_Ofinite__3___Groups_Oplus_657,axiom,
    plus @ finite_3 ).

thf(tcon_Enum_Ofinite__3___Rings_Oring_658,axiom,
    ring @ finite_3 ).

thf(tcon_Enum_Ofinite__3___Rings_Oidom_659,axiom,
    idom @ finite_3 ).

thf(tcon_Enum_Ofinite__3___Groups_Oone_660,axiom,
    one @ finite_3 ).

thf(tcon_Enum_Ofinite__3___Rings_Odvd_661,axiom,
    dvd @ finite_3 ).

thf(tcon_Enum_Ofinite__3___HOL_Oequal_662,axiom,
    cl_HOL_Oequal @ finite_3 ).

thf(tcon_Filter_Ofilter___Quickcheck__Narrowing_Opartial__term__of_663,axiom,
    ! [A15: $tType] :
      ( ( typerep @ A15 )
     => ( quickc6926020345158392990erm_of @ ( filter @ A15 ) ) ) ).

thf(tcon_Filter_Ofilter___Code__Evaluation_Oterm__of_664,axiom,
    ! [A15: $tType] :
      ( ( typerep @ A15 )
     => ( code_term_of @ ( filter @ A15 ) ) ) ).

thf(tcon_Filter_Ofilter___Orderings_Oorder__top_665,axiom,
    ! [A15: $tType] : ( order_top @ ( filter @ A15 ) ) ).

thf(tcon_Filter_Ofilter___Orderings_Oorder__bot_666,axiom,
    ! [A15: $tType] : ( order_bot @ ( filter @ A15 ) ) ).

thf(tcon_Filter_Ofilter___Orderings_Opreorder_667,axiom,
    ! [A15: $tType] : ( preorder @ ( filter @ A15 ) ) ).

thf(tcon_Filter_Ofilter___Orderings_Oorder_668,axiom,
    ! [A15: $tType] : ( order @ ( filter @ A15 ) ) ).

thf(tcon_Filter_Ofilter___Orderings_Oord_669,axiom,
    ! [A15: $tType] : ( ord @ ( filter @ A15 ) ) ).

thf(tcon_Filter_Ofilter___HOL_Oequal_670,axiom,
    ! [A15: $tType] :
      ( ( cl_HOL_Oequal @ A15 )
     => ( cl_HOL_Oequal @ ( filter @ A15 ) ) ) ).

thf(tcon_Option_Ooption___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_671,axiom,
    ! [A15: $tType] :
      ( ( comple5582772986160207858norder @ A15 )
     => ( condit6923001295902523014norder @ ( option @ A15 ) ) ) ).

thf(tcon_Option_Ooption___Quickcheck__Narrowing_Opartial__term__of_672,axiom,
    ! [A15: $tType] :
      ( ( typerep @ A15 )
     => ( quickc6926020345158392990erm_of @ ( option @ A15 ) ) ) ).

thf(tcon_Option_Ooption___Code__Evaluation_Oterm__of_673,axiom,
    ! [A15: $tType] :
      ( ( typerep @ A15 )
     => ( code_term_of @ ( option @ A15 ) ) ) ).

thf(tcon_Option_Ooption___Orderings_Owellorder_674,axiom,
    ! [A15: $tType] :
      ( ( wellorder @ A15 )
     => ( wellorder @ ( option @ A15 ) ) ) ).

thf(tcon_Option_Ooption___Orderings_Oorder__top_675,axiom,
    ! [A15: $tType] :
      ( ( order_top @ A15 )
     => ( order_top @ ( option @ A15 ) ) ) ).

thf(tcon_Option_Ooption___Orderings_Oorder__bot_676,axiom,
    ! [A15: $tType] :
      ( ( order @ A15 )
     => ( order_bot @ ( option @ A15 ) ) ) ).

thf(tcon_Option_Ooption___Orderings_Opreorder_677,axiom,
    ! [A15: $tType] :
      ( ( preorder @ A15 )
     => ( preorder @ ( option @ A15 ) ) ) ).

thf(tcon_Option_Ooption___Orderings_Olinorder_678,axiom,
    ! [A15: $tType] :
      ( ( linorder @ A15 )
     => ( linorder @ ( option @ A15 ) ) ) ).

thf(tcon_Option_Ooption___Orderings_Oorder_679,axiom,
    ! [A15: $tType] :
      ( ( order @ A15 )
     => ( order @ ( option @ A15 ) ) ) ).

thf(tcon_Option_Ooption___Orderings_Oord_680,axiom,
    ! [A15: $tType] :
      ( ( preorder @ A15 )
     => ( ord @ ( option @ A15 ) ) ) ).

thf(tcon_Option_Ooption___Heap_Oheap_681,axiom,
    ! [A15: $tType] :
      ( ( heap @ A15 )
     => ( heap @ ( option @ A15 ) ) ) ).

thf(tcon_Option_Ooption___HOL_Oequal_682,axiom,
    ! [A15: $tType] : ( cl_HOL_Oequal @ ( option @ A15 ) ) ).

thf(tcon_Option_Ooption___Nat_Osize_683,axiom,
    ! [A15: $tType] : ( size @ ( option @ A15 ) ) ).

thf(tcon_Uint32_Ouint32___Bit__Operations_Osemiring__bit__operations_684,axiom,
    bit_se359711467146920520ations @ uint32 ).

thf(tcon_Uint32_Ouint32___Quickcheck__Narrowing_Opartial__term__of_685,axiom,
    quickc6926020345158392990erm_of @ uint32 ).

thf(tcon_Uint32_Ouint32___Bit__Comprehension_Obit__comprehension_686,axiom,
    bit_bi6583157726757044596ension @ uint32 ).

thf(tcon_Uint32_Ouint32___Bit__Operations_Oring__bit__operations_687,axiom,
    bit_ri3973907225187159222ations @ uint32 ).

thf(tcon_Uint32_Ouint32___Groups_Ocancel__ab__semigroup__add_688,axiom,
    cancel2418104881723323429up_add @ uint32 ).

thf(tcon_Uint32_Ouint32___Groups_Ocancel__comm__monoid__add_689,axiom,
    cancel1802427076303600483id_add @ uint32 ).

thf(tcon_Uint32_Ouint32___Rings_Ocomm__semiring__1__cancel_690,axiom,
    comm_s4317794764714335236cancel @ uint32 ).

thf(tcon_Uint32_Ouint32___Bit__Operations_Osemiring__bits_691,axiom,
    bit_semiring_bits @ uint32 ).

thf(tcon_Uint32_Ouint32___Groups_Ocancel__semigroup__add_692,axiom,
    cancel_semigroup_add @ uint32 ).

thf(tcon_Uint32_Ouint32___Least__significant__bit_Olsb_693,axiom,
    least_6119777620449941438nt_lsb @ uint32 ).

thf(tcon_Uint32_Ouint32___Groups_Oab__semigroup__mult_694,axiom,
    ab_semigroup_mult @ uint32 ).

thf(tcon_Uint32_Ouint32___Rings_Osemiring__1__cancel_695,axiom,
    semiring_1_cancel @ uint32 ).

thf(tcon_Uint32_Ouint32___Groups_Ocomm__monoid__mult_696,axiom,
    comm_monoid_mult @ uint32 ).

thf(tcon_Uint32_Ouint32___Groups_Oab__semigroup__add_697,axiom,
    ab_semigroup_add @ uint32 ).

thf(tcon_Uint32_Ouint32___Generic__set__bit_Oset__bit_698,axiom,
    generic_set_set_bit @ uint32 ).

thf(tcon_Uint32_Ouint32___Code__Evaluation_Oterm__of_699,axiom,
    code_term_of @ uint32 ).

thf(tcon_Uint32_Ouint32___Parity_Osemiring__parity_700,axiom,
    semiring_parity @ uint32 ).

thf(tcon_Uint32_Ouint32___Groups_Ocomm__monoid__add_701,axiom,
    comm_monoid_add @ uint32 ).

thf(tcon_Uint32_Ouint32___Rings_Osemiring__modulo_702,axiom,
    semiring_modulo @ uint32 ).

thf(tcon_Uint32_Ouint32___Rings_Ocomm__semiring__1_703,axiom,
    comm_semiring_1 @ uint32 ).

thf(tcon_Uint32_Ouint32___Rings_Ocomm__semiring__0_704,axiom,
    comm_semiring_0 @ uint32 ).

thf(tcon_Uint32_Ouint32___Groups_Osemigroup__mult_705,axiom,
    semigroup_mult @ uint32 ).

thf(tcon_Uint32_Ouint32___Num_Osemiring__numeral_706,axiom,
    semiring_numeral @ uint32 ).

thf(tcon_Uint32_Ouint32___Groups_Osemigroup__add_707,axiom,
    semigroup_add @ uint32 ).

thf(tcon_Uint32_Ouint32___Rings_Ocomm__semiring_708,axiom,
    comm_semiring @ uint32 ).

thf(tcon_Uint32_Ouint32___Groups_Oab__group__add_709,axiom,
    ab_group_add @ uint32 ).

thf(tcon_Uint32_Ouint32___Rings_Ozero__neq__one_710,axiom,
    zero_neq_one @ uint32 ).

thf(tcon_Uint32_Ouint32___Parity_Oring__parity_711,axiom,
    ring_parity @ uint32 ).

thf(tcon_Uint32_Ouint32___Orderings_Opreorder_712,axiom,
    preorder @ uint32 ).

thf(tcon_Uint32_Ouint32___Orderings_Olinorder_713,axiom,
    linorder @ uint32 ).

thf(tcon_Uint32_Ouint32___Groups_Omonoid__mult_714,axiom,
    monoid_mult @ uint32 ).

thf(tcon_Uint32_Ouint32___Rings_Ocomm__ring__1_715,axiom,
    comm_ring_1 @ uint32 ).

thf(tcon_Uint32_Ouint32___Groups_Omonoid__add_716,axiom,
    monoid_add @ uint32 ).

thf(tcon_Uint32_Ouint32___Rings_Osemiring__1_717,axiom,
    semiring_1 @ uint32 ).

thf(tcon_Uint32_Ouint32___Rings_Osemiring__0_718,axiom,
    semiring_0 @ uint32 ).

thf(tcon_Uint32_Ouint32___Groups_Ogroup__add_719,axiom,
    group_add @ uint32 ).

thf(tcon_Uint32_Ouint32___Rings_Omult__zero_720,axiom,
    mult_zero @ uint32 ).

thf(tcon_Uint32_Ouint32___Rings_Ocomm__ring_721,axiom,
    comm_ring @ uint32 ).

thf(tcon_Uint32_Ouint32___Orderings_Oorder_722,axiom,
    order @ uint32 ).

thf(tcon_Uint32_Ouint32___Num_Oneg__numeral_723,axiom,
    neg_numeral @ uint32 ).

thf(tcon_Uint32_Ouint32___Rings_Osemiring_724,axiom,
    semiring @ uint32 ).

thf(tcon_Uint32_Ouint32___Orderings_Oord_725,axiom,
    ord @ uint32 ).

thf(tcon_Uint32_Ouint32___Groups_Ouminus_726,axiom,
    uminus @ uint32 ).

thf(tcon_Uint32_Ouint32___Rings_Oring__1_727,axiom,
    ring_1 @ uint32 ).

thf(tcon_Uint32_Ouint32___Groups_Ominus_728,axiom,
    minus @ uint32 ).

thf(tcon_Uint32_Ouint32___Power_Opower_729,axiom,
    power @ uint32 ).

thf(tcon_Uint32_Ouint32___Num_Onumeral_730,axiom,
    numeral @ uint32 ).

thf(tcon_Uint32_Ouint32___Groups_Ozero_731,axiom,
    zero @ uint32 ).

thf(tcon_Uint32_Ouint32___Groups_Oplus_732,axiom,
    plus @ uint32 ).

thf(tcon_Uint32_Ouint32___Rings_Oring_733,axiom,
    ring @ uint32 ).

thf(tcon_Uint32_Ouint32___Groups_Oone_734,axiom,
    one @ uint32 ).

thf(tcon_Uint32_Ouint32___Rings_Odvd_735,axiom,
    dvd @ uint32 ).

thf(tcon_Uint32_Ouint32___HOL_Oequal_736,axiom,
    cl_HOL_Oequal @ uint32 ).

thf(tcon_Uint32_Ouint32___Nat_Osize_737,axiom,
    size @ uint32 ).

thf(tcon_Assertions_Oassn___Boolean__Algebras_Oboolean__algebra_738,axiom,
    boolea8198339166811842893lgebra @ assn ).

thf(tcon_Assertions_Oassn___Groups_Oab__semigroup__mult_739,axiom,
    ab_semigroup_mult @ assn ).

thf(tcon_Assertions_Oassn___Groups_Ocomm__monoid__mult_740,axiom,
    comm_monoid_mult @ assn ).

thf(tcon_Assertions_Oassn___Groups_Osemigroup__mult_741,axiom,
    semigroup_mult @ assn ).

thf(tcon_Assertions_Oassn___Orderings_Oorder__top_742,axiom,
    order_top @ assn ).

thf(tcon_Assertions_Oassn___Orderings_Oorder__bot_743,axiom,
    order_bot @ assn ).

thf(tcon_Assertions_Oassn___Orderings_Opreorder_744,axiom,
    preorder @ assn ).

thf(tcon_Assertions_Oassn___Groups_Omonoid__mult_745,axiom,
    monoid_mult @ assn ).

thf(tcon_Assertions_Oassn___Orderings_Oorder_746,axiom,
    order @ assn ).

thf(tcon_Assertions_Oassn___Orderings_Oord_747,axiom,
    ord @ assn ).

thf(tcon_Assertions_Oassn___Groups_Ouminus_748,axiom,
    uminus @ assn ).

thf(tcon_Assertions_Oassn___Groups_Ominus_749,axiom,
    minus @ assn ).

thf(tcon_Assertions_Oassn___Power_Opower_750,axiom,
    power @ assn ).

thf(tcon_Assertions_Oassn___Groups_Oone_751,axiom,
    one @ assn ).

thf(tcon_Assertions_Oassn___Rings_Odvd_752,axiom,
    dvd @ assn ).

thf(tcon_Complex_Ocomplex___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_753,axiom,
    semiri1453513574482234551roduct @ complex ).

thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__div__algebra_754,axiom,
    real_V8999393235501362500lgebra @ complex ).

thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__algebra__1_755,axiom,
    real_V2822296259951069270ebra_1 @ complex ).

thf(tcon_Complex_Ocomplex___Rings_Osemiring__no__zero__divisors__cancel_756,axiom,
    semiri6575147826004484403cancel @ complex ).

thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__algebra_757,axiom,
    real_V4412858255891104859lgebra @ complex ).

thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__vector_758,axiom,
    real_V822414075346904944vector @ complex ).

thf(tcon_Complex_Ocomplex___Topological__Spaces_Otopological__space_759,axiom,
    topolo4958980785337419405_space @ complex ).

thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__field_760,axiom,
    real_V3459762299906320749_field @ complex ).

thf(tcon_Complex_Ocomplex___Quickcheck__Narrowing_Opartial__term__of_761,axiom,
    quickc6926020345158392990erm_of @ complex ).

thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__div__algebra_762,axiom,
    real_V5047593784448816457lgebra @ complex ).

thf(tcon_Complex_Ocomplex___Limits_Otopological__comm__monoid__add_763,axiom,
    topolo5987344860129210374id_add @ complex ).

thf(tcon_Complex_Ocomplex___Rings_Osemiring__1__no__zero__divisors_764,axiom,
    semiri2026040879449505780visors @ complex ).

thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__algebra__1_765,axiom,
    real_V2191834092415804123ebra_1 @ complex ).

thf(tcon_Complex_Ocomplex___Rings_Osemiring__no__zero__divisors_766,axiom,
    semiri3467727345109120633visors @ complex ).

thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__algebra_767,axiom,
    real_V6157519004096292374lgebra @ complex ).

thf(tcon_Complex_Ocomplex___Limits_Otopological__ab__group__add_768,axiom,
    topolo1287966508704411220up_add @ complex ).

thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__vector_769,axiom,
    real_V4867850818363320053vector @ complex ).

thf(tcon_Complex_Ocomplex___Groups_Ocancel__ab__semigroup__add_770,axiom,
    cancel2418104881723323429up_add @ complex ).

thf(tcon_Complex_Ocomplex___Rings_Oring__1__no__zero__divisors_771,axiom,
    ring_15535105094025558882visors @ complex ).

thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__field_772,axiom,
    real_V7773925162809079976_field @ complex ).

thf(tcon_Complex_Ocomplex___Groups_Ocancel__comm__monoid__add_773,axiom,
    cancel1802427076303600483id_add @ complex ).

thf(tcon_Complex_Ocomplex___Rings_Ocomm__semiring__1__cancel_774,axiom,
    comm_s4317794764714335236cancel @ complex ).

thf(tcon_Complex_Ocomplex___Topological__Spaces_Ot2__space_775,axiom,
    topological_t2_space @ complex ).

thf(tcon_Complex_Ocomplex___Groups_Ocancel__semigroup__add_776,axiom,
    cancel_semigroup_add @ complex ).

thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Obanach_777,axiom,
    real_Vector_banach @ complex ).

thf(tcon_Complex_Ocomplex___Groups_Oab__semigroup__mult_778,axiom,
    ab_semigroup_mult @ complex ).

thf(tcon_Complex_Ocomplex___Rings_Osemiring__1__cancel_779,axiom,
    semiring_1_cancel @ complex ).

thf(tcon_Complex_Ocomplex___Groups_Ocomm__monoid__mult_780,axiom,
    comm_monoid_mult @ complex ).

thf(tcon_Complex_Ocomplex___Groups_Oab__semigroup__add_781,axiom,
    ab_semigroup_add @ complex ).

thf(tcon_Complex_Ocomplex___Code__Evaluation_Oterm__of_782,axiom,
    code_term_of @ complex ).

thf(tcon_Complex_Ocomplex___Groups_Ocomm__monoid__add_783,axiom,
    comm_monoid_add @ complex ).

thf(tcon_Complex_Ocomplex___Rings_Ocomm__semiring__1_784,axiom,
    comm_semiring_1 @ complex ).

thf(tcon_Complex_Ocomplex___Rings_Ocomm__semiring__0_785,axiom,
    comm_semiring_0 @ complex ).

thf(tcon_Complex_Ocomplex___Groups_Osemigroup__mult_786,axiom,
    semigroup_mult @ complex ).

thf(tcon_Complex_Ocomplex___Rings_Osemidom__divide_787,axiom,
    semidom_divide @ complex ).

thf(tcon_Complex_Ocomplex___Num_Osemiring__numeral_788,axiom,
    semiring_numeral @ complex ).

thf(tcon_Complex_Ocomplex___Groups_Osemigroup__add_789,axiom,
    semigroup_add @ complex ).

thf(tcon_Complex_Ocomplex___Fields_Ofield__abs__sgn_790,axiom,
    field_abs_sgn @ complex ).

thf(tcon_Complex_Ocomplex___Fields_Odivision__ring_791,axiom,
    division_ring @ complex ).

thf(tcon_Complex_Ocomplex___Rings_Ocomm__semiring_792,axiom,
    comm_semiring @ complex ).

thf(tcon_Complex_Ocomplex___Nat_Osemiring__char__0_793,axiom,
    semiring_char_0 @ complex ).

thf(tcon_Complex_Ocomplex___Groups_Oab__group__add_794,axiom,
    ab_group_add @ complex ).

thf(tcon_Complex_Ocomplex___Fields_Ofield__char__0_795,axiom,
    field_char_0 @ complex ).

thf(tcon_Complex_Ocomplex___Rings_Ozero__neq__one_796,axiom,
    zero_neq_one @ complex ).

thf(tcon_Complex_Ocomplex___Rings_Oidom__abs__sgn_797,axiom,
    idom_abs_sgn @ complex ).

thf(tcon_Complex_Ocomplex___Groups_Omonoid__mult_798,axiom,
    monoid_mult @ complex ).

thf(tcon_Complex_Ocomplex___Rings_Oidom__divide_799,axiom,
    idom_divide @ complex ).

thf(tcon_Complex_Ocomplex___Rings_Ocomm__ring__1_800,axiom,
    comm_ring_1 @ complex ).

thf(tcon_Complex_Ocomplex___Groups_Omonoid__add_801,axiom,
    monoid_add @ complex ).

thf(tcon_Complex_Ocomplex___Rings_Osemiring__1_802,axiom,
    semiring_1 @ complex ).

thf(tcon_Complex_Ocomplex___Rings_Osemiring__0_803,axiom,
    semiring_0 @ complex ).

thf(tcon_Complex_Ocomplex___Groups_Ogroup__add_804,axiom,
    group_add @ complex ).

thf(tcon_Complex_Ocomplex___Rings_Omult__zero_805,axiom,
    mult_zero @ complex ).

thf(tcon_Complex_Ocomplex___Rings_Ocomm__ring_806,axiom,
    comm_ring @ complex ).

thf(tcon_Complex_Ocomplex___Num_Oneg__numeral_807,axiom,
    neg_numeral @ complex ).

thf(tcon_Complex_Ocomplex___Nat_Oring__char__0_808,axiom,
    ring_char_0 @ complex ).

thf(tcon_Complex_Ocomplex___Rings_Osemiring_809,axiom,
    semiring @ complex ).

thf(tcon_Complex_Ocomplex___Groups_Ouminus_810,axiom,
    uminus @ complex ).

thf(tcon_Complex_Ocomplex___Rings_Oring__1_811,axiom,
    ring_1 @ complex ).

thf(tcon_Complex_Ocomplex___Groups_Ominus_812,axiom,
    minus @ complex ).

thf(tcon_Complex_Ocomplex___Fields_Ofield_813,axiom,
    field @ complex ).

thf(tcon_Complex_Ocomplex___Power_Opower_814,axiom,
    power @ complex ).

thf(tcon_Complex_Ocomplex___Num_Onumeral_815,axiom,
    numeral @ complex ).

thf(tcon_Complex_Ocomplex___Groups_Ozero_816,axiom,
    zero @ complex ).

thf(tcon_Complex_Ocomplex___Groups_Oplus_817,axiom,
    plus @ complex ).

thf(tcon_Complex_Ocomplex___Rings_Oring_818,axiom,
    ring @ complex ).

thf(tcon_Complex_Ocomplex___Rings_Oidom_819,axiom,
    idom @ complex ).

thf(tcon_Complex_Ocomplex___Groups_Oone_820,axiom,
    one @ complex ).

thf(tcon_Complex_Ocomplex___Rings_Odvd_821,axiom,
    dvd @ complex ).

thf(tcon_Complex_Ocomplex___HOL_Oequal_822,axiom,
    cl_HOL_Oequal @ complex ).

thf(tcon_Extended__Nat_Oenat___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_823,axiom,
    condit6923001295902523014norder @ extended_enat ).

thf(tcon_Extended__Nat_Oenat___Groups_Ostrict__ordered__ab__semigroup__add_824,axiom,
    strict9044650504122735259up_add @ extended_enat ).

thf(tcon_Extended__Nat_Oenat___Groups_Ostrict__ordered__comm__monoid__add_825,axiom,
    strict7427464778891057005id_add @ extended_enat ).

thf(tcon_Extended__Nat_Oenat___Groups_Ocanonically__ordered__monoid__add_826,axiom,
    canoni5634975068530333245id_add @ extended_enat ).

thf(tcon_Extended__Nat_Oenat___Quickcheck__Narrowing_Opartial__term__of_827,axiom,
    quickc6926020345158392990erm_of @ extended_enat ).

thf(tcon_Extended__Nat_Oenat___Groups_Olinordered__ab__semigroup__add_828,axiom,
    linord4140545234300271783up_add @ extended_enat ).

thf(tcon_Extended__Nat_Oenat___Rings_Olinordered__nonzero__semiring_829,axiom,
    linord181362715937106298miring @ extended_enat ).

thf(tcon_Extended__Nat_Oenat___Rings_Osemiring__no__zero__divisors_830,axiom,
    semiri3467727345109120633visors @ extended_enat ).

thf(tcon_Extended__Nat_Oenat___Groups_Oordered__ab__semigroup__add_831,axiom,
    ordere6658533253407199908up_add @ extended_enat ).

thf(tcon_Extended__Nat_Oenat___Groups_Oordered__comm__monoid__add_832,axiom,
    ordere6911136660526730532id_add @ extended_enat ).

thf(tcon_Extended__Nat_Oenat___Rings_Oordered__comm__semiring_833,axiom,
    ordere2520102378445227354miring @ extended_enat ).

thf(tcon_Extended__Nat_Oenat___Groups_Oab__semigroup__mult_834,axiom,
    ab_semigroup_mult @ extended_enat ).

thf(tcon_Extended__Nat_Oenat___Groups_Ocomm__monoid__mult_835,axiom,
    comm_monoid_mult @ extended_enat ).

thf(tcon_Extended__Nat_Oenat___Groups_Oab__semigroup__add_836,axiom,
    ab_semigroup_add @ extended_enat ).

thf(tcon_Extended__Nat_Oenat___Code__Evaluation_Oterm__of_837,axiom,
    code_term_of @ extended_enat ).

thf(tcon_Extended__Nat_Oenat___Rings_Oordered__semiring_838,axiom,
    ordered_semiring @ extended_enat ).

thf(tcon_Extended__Nat_Oenat___Groups_Ocomm__monoid__add_839,axiom,
    comm_monoid_add @ extended_enat ).

thf(tcon_Extended__Nat_Oenat___Rings_Ocomm__semiring__1_840,axiom,
    comm_semiring_1 @ extended_enat ).

thf(tcon_Extended__Nat_Oenat___Rings_Ocomm__semiring__0_841,axiom,
    comm_semiring_0 @ extended_enat ).

thf(tcon_Extended__Nat_Oenat___Groups_Osemigroup__mult_842,axiom,
    semigroup_mult @ extended_enat ).

thf(tcon_Extended__Nat_Oenat___Num_Osemiring__numeral_843,axiom,
    semiring_numeral @ extended_enat ).

thf(tcon_Extended__Nat_Oenat___Groups_Osemigroup__add_844,axiom,
    semigroup_add @ extended_enat ).

thf(tcon_Extended__Nat_Oenat___Rings_Ozero__less__one_845,axiom,
    zero_less_one @ extended_enat ).

thf(tcon_Extended__Nat_Oenat___Rings_Ocomm__semiring_846,axiom,
    comm_semiring @ extended_enat ).

thf(tcon_Extended__Nat_Oenat___Orderings_Owellorder_847,axiom,
    wellorder @ extended_enat ).

thf(tcon_Extended__Nat_Oenat___Orderings_Oorder__top_848,axiom,
    order_top @ extended_enat ).

thf(tcon_Extended__Nat_Oenat___Orderings_Oorder__bot_849,axiom,
    order_bot @ extended_enat ).

thf(tcon_Extended__Nat_Oenat___Nat_Osemiring__char__0_850,axiom,
    semiring_char_0 @ extended_enat ).

thf(tcon_Extended__Nat_Oenat___Rings_Ozero__neq__one_851,axiom,
    zero_neq_one @ extended_enat ).

thf(tcon_Extended__Nat_Oenat___Orderings_Opreorder_852,axiom,
    preorder @ extended_enat ).

thf(tcon_Extended__Nat_Oenat___Orderings_Olinorder_853,axiom,
    linorder @ extended_enat ).

thf(tcon_Extended__Nat_Oenat___Groups_Omonoid__mult_854,axiom,
    monoid_mult @ extended_enat ).

thf(tcon_Extended__Nat_Oenat___Groups_Omonoid__add_855,axiom,
    monoid_add @ extended_enat ).

thf(tcon_Extended__Nat_Oenat___Rings_Osemiring__1_856,axiom,
    semiring_1 @ extended_enat ).

thf(tcon_Extended__Nat_Oenat___Rings_Osemiring__0_857,axiom,
    semiring_0 @ extended_enat ).

thf(tcon_Extended__Nat_Oenat___Rings_Omult__zero_858,axiom,
    mult_zero @ extended_enat ).

thf(tcon_Extended__Nat_Oenat___Orderings_Oorder_859,axiom,
    order @ extended_enat ).

thf(tcon_Extended__Nat_Oenat___Rings_Osemiring_860,axiom,
    semiring @ extended_enat ).

thf(tcon_Extended__Nat_Oenat___Orderings_Oord_861,axiom,
    ord @ extended_enat ).

thf(tcon_Extended__Nat_Oenat___Groups_Ominus_862,axiom,
    minus @ extended_enat ).

thf(tcon_Extended__Nat_Oenat___Power_Opower_863,axiom,
    power @ extended_enat ).

thf(tcon_Extended__Nat_Oenat___Num_Onumeral_864,axiom,
    numeral @ extended_enat ).

thf(tcon_Extended__Nat_Oenat___Groups_Ozero_865,axiom,
    zero @ extended_enat ).

thf(tcon_Extended__Nat_Oenat___Groups_Oplus_866,axiom,
    plus @ extended_enat ).

thf(tcon_Extended__Nat_Oenat___Groups_Oone_867,axiom,
    one @ extended_enat ).

thf(tcon_Extended__Nat_Oenat___Rings_Odvd_868,axiom,
    dvd @ extended_enat ).

thf(tcon_Extended__Nat_Oenat___HOL_Oequal_869,axiom,
    cl_HOL_Oequal @ extended_enat ).

thf(tcon_Multiset_Omultiset___Quickcheck__Narrowing_Opartial__term__of_870,axiom,
    ! [A15: $tType] :
      ( ( typerep @ A15 )
     => ( quickc6926020345158392990erm_of @ ( multiset @ A15 ) ) ) ).

thf(tcon_Multiset_Omultiset___Groups_Oordered__ab__semigroup__add_871,axiom,
    ! [A15: $tType] :
      ( ( preorder @ A15 )
     => ( ordere6658533253407199908up_add @ ( multiset @ A15 ) ) ) ).

thf(tcon_Multiset_Omultiset___Groups_Ocancel__ab__semigroup__add_872,axiom,
    ! [A15: $tType] : ( cancel2418104881723323429up_add @ ( multiset @ A15 ) ) ).

thf(tcon_Multiset_Omultiset___Groups_Ocancel__comm__monoid__add_873,axiom,
    ! [A15: $tType] : ( cancel1802427076303600483id_add @ ( multiset @ A15 ) ) ).

thf(tcon_Multiset_Omultiset___Groups_Ocancel__semigroup__add_874,axiom,
    ! [A15: $tType] : ( cancel_semigroup_add @ ( multiset @ A15 ) ) ).

thf(tcon_Multiset_Omultiset___Groups_Ocomm__monoid__diff_875,axiom,
    ! [A15: $tType] : ( comm_monoid_diff @ ( multiset @ A15 ) ) ).

thf(tcon_Multiset_Omultiset___Groups_Oab__semigroup__add_876,axiom,
    ! [A15: $tType] : ( ab_semigroup_add @ ( multiset @ A15 ) ) ).

thf(tcon_Multiset_Omultiset___Code__Evaluation_Oterm__of_877,axiom,
    ! [A15: $tType] :
      ( ( typerep @ A15 )
     => ( code_term_of @ ( multiset @ A15 ) ) ) ).

thf(tcon_Multiset_Omultiset___Groups_Ocomm__monoid__add_878,axiom,
    ! [A15: $tType] : ( comm_monoid_add @ ( multiset @ A15 ) ) ).

thf(tcon_Multiset_Omultiset___Groups_Osemigroup__add_879,axiom,
    ! [A15: $tType] : ( semigroup_add @ ( multiset @ A15 ) ) ).

thf(tcon_Multiset_Omultiset___Orderings_Opreorder_880,axiom,
    ! [A15: $tType] :
      ( ( preorder @ A15 )
     => ( preorder @ ( multiset @ A15 ) ) ) ).

thf(tcon_Multiset_Omultiset___Groups_Omonoid__add_881,axiom,
    ! [A15: $tType] : ( monoid_add @ ( multiset @ A15 ) ) ).

thf(tcon_Multiset_Omultiset___Orderings_Oorder_882,axiom,
    ! [A15: $tType] :
      ( ( preorder @ A15 )
     => ( order @ ( multiset @ A15 ) ) ) ).

thf(tcon_Multiset_Omultiset___Orderings_Oord_883,axiom,
    ! [A15: $tType] :
      ( ( preorder @ A15 )
     => ( ord @ ( multiset @ A15 ) ) ) ).

thf(tcon_Multiset_Omultiset___Groups_Ominus_884,axiom,
    ! [A15: $tType] : ( minus @ ( multiset @ A15 ) ) ).

thf(tcon_Multiset_Omultiset___Groups_Ozero_885,axiom,
    ! [A15: $tType] : ( zero @ ( multiset @ A15 ) ) ).

thf(tcon_Multiset_Omultiset___Groups_Oplus_886,axiom,
    ! [A15: $tType] : ( plus @ ( multiset @ A15 ) ) ).

thf(tcon_Multiset_Omultiset___HOL_Oequal_887,axiom,
    ! [A15: $tType] :
      ( ( cl_HOL_Oequal @ A15 )
     => ( cl_HOL_Oequal @ ( multiset @ A15 ) ) ) ).

thf(tcon_Multiset_Omultiset___Nat_Osize_888,axiom,
    ! [A15: $tType] : ( size @ ( multiset @ A15 ) ) ).

thf(tcon_Numeral__Type_Obit0___Groups_Ocancel__ab__semigroup__add_889,axiom,
    ! [A15: $tType] :
      ( ( finite_finite @ A15 )
     => ( cancel2418104881723323429up_add @ ( numeral_bit0 @ A15 ) ) ) ).

thf(tcon_Numeral__Type_Obit0___Groups_Ocancel__comm__monoid__add_890,axiom,
    ! [A15: $tType] :
      ( ( finite_finite @ A15 )
     => ( cancel1802427076303600483id_add @ ( numeral_bit0 @ A15 ) ) ) ).

thf(tcon_Numeral__Type_Obit0___Rings_Ocomm__semiring__1__cancel_891,axiom,
    ! [A15: $tType] :
      ( ( finite_finite @ A15 )
     => ( comm_s4317794764714335236cancel @ ( numeral_bit0 @ A15 ) ) ) ).

thf(tcon_Numeral__Type_Obit0___Groups_Ocancel__semigroup__add_892,axiom,
    ! [A15: $tType] :
      ( ( finite_finite @ A15 )
     => ( cancel_semigroup_add @ ( numeral_bit0 @ A15 ) ) ) ).

thf(tcon_Numeral__Type_Obit0___Groups_Oab__semigroup__mult_893,axiom,
    ! [A15: $tType] :
      ( ( finite_finite @ A15 )
     => ( ab_semigroup_mult @ ( numeral_bit0 @ A15 ) ) ) ).

thf(tcon_Numeral__Type_Obit0___Rings_Osemiring__1__cancel_894,axiom,
    ! [A15: $tType] :
      ( ( finite_finite @ A15 )
     => ( semiring_1_cancel @ ( numeral_bit0 @ A15 ) ) ) ).

thf(tcon_Numeral__Type_Obit0___Groups_Ocomm__monoid__mult_895,axiom,
    ! [A15: $tType] :
      ( ( finite_finite @ A15 )
     => ( comm_monoid_mult @ ( numeral_bit0 @ A15 ) ) ) ).

thf(tcon_Numeral__Type_Obit0___Groups_Oab__semigroup__add_896,axiom,
    ! [A15: $tType] :
      ( ( finite_finite @ A15 )
     => ( ab_semigroup_add @ ( numeral_bit0 @ A15 ) ) ) ).

thf(tcon_Numeral__Type_Obit0___Code__Evaluation_Oterm__of_897,axiom,
    ! [A15: $tType] :
      ( ( typerep @ A15 )
     => ( code_term_of @ ( numeral_bit0 @ A15 ) ) ) ).

thf(tcon_Numeral__Type_Obit0___Groups_Ocomm__monoid__add_898,axiom,
    ! [A15: $tType] :
      ( ( finite_finite @ A15 )
     => ( comm_monoid_add @ ( numeral_bit0 @ A15 ) ) ) ).

thf(tcon_Numeral__Type_Obit0___Rings_Ocomm__semiring__1_899,axiom,
    ! [A15: $tType] :
      ( ( finite_finite @ A15 )
     => ( comm_semiring_1 @ ( numeral_bit0 @ A15 ) ) ) ).

thf(tcon_Numeral__Type_Obit0___Rings_Ocomm__semiring__0_900,axiom,
    ! [A15: $tType] :
      ( ( finite_finite @ A15 )
     => ( comm_semiring_0 @ ( numeral_bit0 @ A15 ) ) ) ).

thf(tcon_Numeral__Type_Obit0___Groups_Osemigroup__mult_901,axiom,
    ! [A15: $tType] :
      ( ( finite_finite @ A15 )
     => ( semigroup_mult @ ( numeral_bit0 @ A15 ) ) ) ).

thf(tcon_Numeral__Type_Obit0___Num_Osemiring__numeral_902,axiom,
    ! [A15: $tType] :
      ( ( finite_finite @ A15 )
     => ( semiring_numeral @ ( numeral_bit0 @ A15 ) ) ) ).

thf(tcon_Numeral__Type_Obit0___Groups_Osemigroup__add_903,axiom,
    ! [A15: $tType] :
      ( ( finite_finite @ A15 )
     => ( semigroup_add @ ( numeral_bit0 @ A15 ) ) ) ).

thf(tcon_Numeral__Type_Obit0___Rings_Ocomm__semiring_904,axiom,
    ! [A15: $tType] :
      ( ( finite_finite @ A15 )
     => ( comm_semiring @ ( numeral_bit0 @ A15 ) ) ) ).

thf(tcon_Numeral__Type_Obit0___Orderings_Owellorder_905,axiom,
    ! [A15: $tType] :
      ( ( finite_finite @ A15 )
     => ( wellorder @ ( numeral_bit0 @ A15 ) ) ) ).

thf(tcon_Numeral__Type_Obit0___Groups_Oab__group__add_906,axiom,
    ! [A15: $tType] :
      ( ( finite_finite @ A15 )
     => ( ab_group_add @ ( numeral_bit0 @ A15 ) ) ) ).

thf(tcon_Numeral__Type_Obit0___Rings_Ozero__neq__one_907,axiom,
    ! [A15: $tType] :
      ( ( finite_finite @ A15 )
     => ( zero_neq_one @ ( numeral_bit0 @ A15 ) ) ) ).

thf(tcon_Numeral__Type_Obit0___Orderings_Opreorder_908,axiom,
    ! [A15: $tType] :
      ( ( finite_finite @ A15 )
     => ( preorder @ ( numeral_bit0 @ A15 ) ) ) ).

thf(tcon_Numeral__Type_Obit0___Orderings_Olinorder_909,axiom,
    ! [A15: $tType] :
      ( ( finite_finite @ A15 )
     => ( linorder @ ( numeral_bit0 @ A15 ) ) ) ).

thf(tcon_Numeral__Type_Obit0___Groups_Omonoid__mult_910,axiom,
    ! [A15: $tType] :
      ( ( finite_finite @ A15 )
     => ( monoid_mult @ ( numeral_bit0 @ A15 ) ) ) ).

thf(tcon_Numeral__Type_Obit0___Rings_Ocomm__ring__1_911,axiom,
    ! [A15: $tType] :
      ( ( finite_finite @ A15 )
     => ( comm_ring_1 @ ( numeral_bit0 @ A15 ) ) ) ).

thf(tcon_Numeral__Type_Obit0___Groups_Omonoid__add_912,axiom,
    ! [A15: $tType] :
      ( ( finite_finite @ A15 )
     => ( monoid_add @ ( numeral_bit0 @ A15 ) ) ) ).

thf(tcon_Numeral__Type_Obit0___Type__Length_Olen0_913,axiom,
    ! [A15: $tType] :
      ( ( type_len0 @ A15 )
     => ( type_len0 @ ( numeral_bit0 @ A15 ) ) ) ).

thf(tcon_Numeral__Type_Obit0___Rings_Osemiring__1_914,axiom,
    ! [A15: $tType] :
      ( ( finite_finite @ A15 )
     => ( semiring_1 @ ( numeral_bit0 @ A15 ) ) ) ).

thf(tcon_Numeral__Type_Obit0___Rings_Osemiring__0_915,axiom,
    ! [A15: $tType] :
      ( ( finite_finite @ A15 )
     => ( semiring_0 @ ( numeral_bit0 @ A15 ) ) ) ).

thf(tcon_Numeral__Type_Obit0___Groups_Ogroup__add_916,axiom,
    ! [A15: $tType] :
      ( ( finite_finite @ A15 )
     => ( group_add @ ( numeral_bit0 @ A15 ) ) ) ).

thf(tcon_Numeral__Type_Obit0___Type__Length_Olen_917,axiom,
    ! [A15: $tType] :
      ( ( type_len @ A15 )
     => ( type_len @ ( numeral_bit0 @ A15 ) ) ) ).

thf(tcon_Numeral__Type_Obit0___Rings_Omult__zero_918,axiom,
    ! [A15: $tType] :
      ( ( finite_finite @ A15 )
     => ( mult_zero @ ( numeral_bit0 @ A15 ) ) ) ).

thf(tcon_Numeral__Type_Obit0___Rings_Ocomm__ring_919,axiom,
    ! [A15: $tType] :
      ( ( finite_finite @ A15 )
     => ( comm_ring @ ( numeral_bit0 @ A15 ) ) ) ).

thf(tcon_Numeral__Type_Obit0___Orderings_Oorder_920,axiom,
    ! [A15: $tType] :
      ( ( finite_finite @ A15 )
     => ( order @ ( numeral_bit0 @ A15 ) ) ) ).

thf(tcon_Numeral__Type_Obit0___Num_Oneg__numeral_921,axiom,
    ! [A15: $tType] :
      ( ( finite_finite @ A15 )
     => ( neg_numeral @ ( numeral_bit0 @ A15 ) ) ) ).

thf(tcon_Numeral__Type_Obit0___Rings_Osemiring_922,axiom,
    ! [A15: $tType] :
      ( ( finite_finite @ A15 )
     => ( semiring @ ( numeral_bit0 @ A15 ) ) ) ).

thf(tcon_Numeral__Type_Obit0___Orderings_Oord_923,axiom,
    ! [A15: $tType] :
      ( ( finite_finite @ A15 )
     => ( ord @ ( numeral_bit0 @ A15 ) ) ) ).

thf(tcon_Numeral__Type_Obit0___Groups_Ouminus_924,axiom,
    ! [A15: $tType] :
      ( ( finite_finite @ A15 )
     => ( uminus @ ( numeral_bit0 @ A15 ) ) ) ).

thf(tcon_Numeral__Type_Obit0___Rings_Oring__1_925,axiom,
    ! [A15: $tType] :
      ( ( finite_finite @ A15 )
     => ( ring_1 @ ( numeral_bit0 @ A15 ) ) ) ).

thf(tcon_Numeral__Type_Obit0___Groups_Ominus_926,axiom,
    ! [A15: $tType] :
      ( ( finite_finite @ A15 )
     => ( minus @ ( numeral_bit0 @ A15 ) ) ) ).

thf(tcon_Numeral__Type_Obit0___Power_Opower_927,axiom,
    ! [A15: $tType] :
      ( ( finite_finite @ A15 )
     => ( power @ ( numeral_bit0 @ A15 ) ) ) ).

thf(tcon_Numeral__Type_Obit0___Num_Onumeral_928,axiom,
    ! [A15: $tType] :
      ( ( finite_finite @ A15 )
     => ( numeral @ ( numeral_bit0 @ A15 ) ) ) ).

thf(tcon_Numeral__Type_Obit0___Groups_Ozero_929,axiom,
    ! [A15: $tType] :
      ( ( finite_finite @ A15 )
     => ( zero @ ( numeral_bit0 @ A15 ) ) ) ).

thf(tcon_Numeral__Type_Obit0___Groups_Oplus_930,axiom,
    ! [A15: $tType] :
      ( ( finite_finite @ A15 )
     => ( plus @ ( numeral_bit0 @ A15 ) ) ) ).

thf(tcon_Numeral__Type_Obit0___Rings_Oring_931,axiom,
    ! [A15: $tType] :
      ( ( finite_finite @ A15 )
     => ( ring @ ( numeral_bit0 @ A15 ) ) ) ).

thf(tcon_Numeral__Type_Obit0___Groups_Oone_932,axiom,
    ! [A15: $tType] :
      ( ( finite_finite @ A15 )
     => ( one @ ( numeral_bit0 @ A15 ) ) ) ).

thf(tcon_Numeral__Type_Obit0___Rings_Odvd_933,axiom,
    ! [A15: $tType] :
      ( ( finite_finite @ A15 )
     => ( dvd @ ( numeral_bit0 @ A15 ) ) ) ).

thf(tcon_Numeral__Type_Obit0___HOL_Oequal_934,axiom,
    ! [A15: $tType] :
      ( ( finite_finite @ A15 )
     => ( cl_HOL_Oequal @ ( numeral_bit0 @ A15 ) ) ) ).

thf(tcon_Numeral__Type_Obit1___Groups_Ocancel__ab__semigroup__add_935,axiom,
    ! [A15: $tType] :
      ( ( finite_finite @ A15 )
     => ( cancel2418104881723323429up_add @ ( numeral_bit1 @ A15 ) ) ) ).

thf(tcon_Numeral__Type_Obit1___Groups_Ocancel__comm__monoid__add_936,axiom,
    ! [A15: $tType] :
      ( ( finite_finite @ A15 )
     => ( cancel1802427076303600483id_add @ ( numeral_bit1 @ A15 ) ) ) ).

thf(tcon_Numeral__Type_Obit1___Rings_Ocomm__semiring__1__cancel_937,axiom,
    ! [A15: $tType] :
      ( ( finite_finite @ A15 )
     => ( comm_s4317794764714335236cancel @ ( numeral_bit1 @ A15 ) ) ) ).

thf(tcon_Numeral__Type_Obit1___Groups_Ocancel__semigroup__add_938,axiom,
    ! [A15: $tType] :
      ( ( finite_finite @ A15 )
     => ( cancel_semigroup_add @ ( numeral_bit1 @ A15 ) ) ) ).

thf(tcon_Numeral__Type_Obit1___Groups_Oab__semigroup__mult_939,axiom,
    ! [A15: $tType] :
      ( ( finite_finite @ A15 )
     => ( ab_semigroup_mult @ ( numeral_bit1 @ A15 ) ) ) ).

thf(tcon_Numeral__Type_Obit1___Rings_Osemiring__1__cancel_940,axiom,
    ! [A15: $tType] :
      ( ( finite_finite @ A15 )
     => ( semiring_1_cancel @ ( numeral_bit1 @ A15 ) ) ) ).

thf(tcon_Numeral__Type_Obit1___Groups_Ocomm__monoid__mult_941,axiom,
    ! [A15: $tType] :
      ( ( finite_finite @ A15 )
     => ( comm_monoid_mult @ ( numeral_bit1 @ A15 ) ) ) ).

thf(tcon_Numeral__Type_Obit1___Groups_Oab__semigroup__add_942,axiom,
    ! [A15: $tType] :
      ( ( finite_finite @ A15 )
     => ( ab_semigroup_add @ ( numeral_bit1 @ A15 ) ) ) ).

thf(tcon_Numeral__Type_Obit1___Code__Evaluation_Oterm__of_943,axiom,
    ! [A15: $tType] :
      ( ( typerep @ A15 )
     => ( code_term_of @ ( numeral_bit1 @ A15 ) ) ) ).

thf(tcon_Numeral__Type_Obit1___Groups_Ocomm__monoid__add_944,axiom,
    ! [A15: $tType] :
      ( ( finite_finite @ A15 )
     => ( comm_monoid_add @ ( numeral_bit1 @ A15 ) ) ) ).

thf(tcon_Numeral__Type_Obit1___Rings_Ocomm__semiring__1_945,axiom,
    ! [A15: $tType] :
      ( ( finite_finite @ A15 )
     => ( comm_semiring_1 @ ( numeral_bit1 @ A15 ) ) ) ).

thf(tcon_Numeral__Type_Obit1___Rings_Ocomm__semiring__0_946,axiom,
    ! [A15: $tType] :
      ( ( finite_finite @ A15 )
     => ( comm_semiring_0 @ ( numeral_bit1 @ A15 ) ) ) ).

thf(tcon_Numeral__Type_Obit1___Groups_Osemigroup__mult_947,axiom,
    ! [A15: $tType] :
      ( ( finite_finite @ A15 )
     => ( semigroup_mult @ ( numeral_bit1 @ A15 ) ) ) ).

thf(tcon_Numeral__Type_Obit1___Num_Osemiring__numeral_948,axiom,
    ! [A15: $tType] :
      ( ( finite_finite @ A15 )
     => ( semiring_numeral @ ( numeral_bit1 @ A15 ) ) ) ).

thf(tcon_Numeral__Type_Obit1___Groups_Osemigroup__add_949,axiom,
    ! [A15: $tType] :
      ( ( finite_finite @ A15 )
     => ( semigroup_add @ ( numeral_bit1 @ A15 ) ) ) ).

thf(tcon_Numeral__Type_Obit1___Rings_Ocomm__semiring_950,axiom,
    ! [A15: $tType] :
      ( ( finite_finite @ A15 )
     => ( comm_semiring @ ( numeral_bit1 @ A15 ) ) ) ).

thf(tcon_Numeral__Type_Obit1___Orderings_Owellorder_951,axiom,
    ! [A15: $tType] :
      ( ( finite_finite @ A15 )
     => ( wellorder @ ( numeral_bit1 @ A15 ) ) ) ).

thf(tcon_Numeral__Type_Obit1___Groups_Oab__group__add_952,axiom,
    ! [A15: $tType] :
      ( ( finite_finite @ A15 )
     => ( ab_group_add @ ( numeral_bit1 @ A15 ) ) ) ).

thf(tcon_Numeral__Type_Obit1___Rings_Ozero__neq__one_953,axiom,
    ! [A15: $tType] :
      ( ( finite_finite @ A15 )
     => ( zero_neq_one @ ( numeral_bit1 @ A15 ) ) ) ).

thf(tcon_Numeral__Type_Obit1___Orderings_Opreorder_954,axiom,
    ! [A15: $tType] :
      ( ( finite_finite @ A15 )
     => ( preorder @ ( numeral_bit1 @ A15 ) ) ) ).

thf(tcon_Numeral__Type_Obit1___Orderings_Olinorder_955,axiom,
    ! [A15: $tType] :
      ( ( finite_finite @ A15 )
     => ( linorder @ ( numeral_bit1 @ A15 ) ) ) ).

thf(tcon_Numeral__Type_Obit1___Groups_Omonoid__mult_956,axiom,
    ! [A15: $tType] :
      ( ( finite_finite @ A15 )
     => ( monoid_mult @ ( numeral_bit1 @ A15 ) ) ) ).

thf(tcon_Numeral__Type_Obit1___Rings_Ocomm__ring__1_957,axiom,
    ! [A15: $tType] :
      ( ( finite_finite @ A15 )
     => ( comm_ring_1 @ ( numeral_bit1 @ A15 ) ) ) ).

thf(tcon_Numeral__Type_Obit1___Groups_Omonoid__add_958,axiom,
    ! [A15: $tType] :
      ( ( finite_finite @ A15 )
     => ( monoid_add @ ( numeral_bit1 @ A15 ) ) ) ).

thf(tcon_Numeral__Type_Obit1___Type__Length_Olen0_959,axiom,
    ! [A15: $tType] :
      ( ( type_len0 @ A15 )
     => ( type_len0 @ ( numeral_bit1 @ A15 ) ) ) ).

thf(tcon_Numeral__Type_Obit1___Rings_Osemiring__1_960,axiom,
    ! [A15: $tType] :
      ( ( finite_finite @ A15 )
     => ( semiring_1 @ ( numeral_bit1 @ A15 ) ) ) ).

thf(tcon_Numeral__Type_Obit1___Rings_Osemiring__0_961,axiom,
    ! [A15: $tType] :
      ( ( finite_finite @ A15 )
     => ( semiring_0 @ ( numeral_bit1 @ A15 ) ) ) ).

thf(tcon_Numeral__Type_Obit1___Groups_Ogroup__add_962,axiom,
    ! [A15: $tType] :
      ( ( finite_finite @ A15 )
     => ( group_add @ ( numeral_bit1 @ A15 ) ) ) ).

thf(tcon_Numeral__Type_Obit1___Type__Length_Olen_963,axiom,
    ! [A15: $tType] :
      ( ( type_len0 @ A15 )
     => ( type_len @ ( numeral_bit1 @ A15 ) ) ) ).

thf(tcon_Numeral__Type_Obit1___Rings_Omult__zero_964,axiom,
    ! [A15: $tType] :
      ( ( finite_finite @ A15 )
     => ( mult_zero @ ( numeral_bit1 @ A15 ) ) ) ).

thf(tcon_Numeral__Type_Obit1___Rings_Ocomm__ring_965,axiom,
    ! [A15: $tType] :
      ( ( finite_finite @ A15 )
     => ( comm_ring @ ( numeral_bit1 @ A15 ) ) ) ).

thf(tcon_Numeral__Type_Obit1___Orderings_Oorder_966,axiom,
    ! [A15: $tType] :
      ( ( finite_finite @ A15 )
     => ( order @ ( numeral_bit1 @ A15 ) ) ) ).

thf(tcon_Numeral__Type_Obit1___Num_Oneg__numeral_967,axiom,
    ! [A15: $tType] :
      ( ( finite_finite @ A15 )
     => ( neg_numeral @ ( numeral_bit1 @ A15 ) ) ) ).

thf(tcon_Numeral__Type_Obit1___Rings_Osemiring_968,axiom,
    ! [A15: $tType] :
      ( ( finite_finite @ A15 )
     => ( semiring @ ( numeral_bit1 @ A15 ) ) ) ).

thf(tcon_Numeral__Type_Obit1___Orderings_Oord_969,axiom,
    ! [A15: $tType] :
      ( ( finite_finite @ A15 )
     => ( ord @ ( numeral_bit1 @ A15 ) ) ) ).

thf(tcon_Numeral__Type_Obit1___Groups_Ouminus_970,axiom,
    ! [A15: $tType] :
      ( ( finite_finite @ A15 )
     => ( uminus @ ( numeral_bit1 @ A15 ) ) ) ).

thf(tcon_Numeral__Type_Obit1___Rings_Oring__1_971,axiom,
    ! [A15: $tType] :
      ( ( finite_finite @ A15 )
     => ( ring_1 @ ( numeral_bit1 @ A15 ) ) ) ).

thf(tcon_Numeral__Type_Obit1___Groups_Ominus_972,axiom,
    ! [A15: $tType] :
      ( ( finite_finite @ A15 )
     => ( minus @ ( numeral_bit1 @ A15 ) ) ) ).

thf(tcon_Numeral__Type_Obit1___Power_Opower_973,axiom,
    ! [A15: $tType] :
      ( ( finite_finite @ A15 )
     => ( power @ ( numeral_bit1 @ A15 ) ) ) ).

thf(tcon_Numeral__Type_Obit1___Num_Onumeral_974,axiom,
    ! [A15: $tType] :
      ( ( finite_finite @ A15 )
     => ( numeral @ ( numeral_bit1 @ A15 ) ) ) ).

thf(tcon_Numeral__Type_Obit1___Groups_Ozero_975,axiom,
    ! [A15: $tType] :
      ( ( finite_finite @ A15 )
     => ( zero @ ( numeral_bit1 @ A15 ) ) ) ).

thf(tcon_Numeral__Type_Obit1___Groups_Oplus_976,axiom,
    ! [A15: $tType] :
      ( ( finite_finite @ A15 )
     => ( plus @ ( numeral_bit1 @ A15 ) ) ) ).

thf(tcon_Numeral__Type_Obit1___Rings_Oring_977,axiom,
    ! [A15: $tType] :
      ( ( finite_finite @ A15 )
     => ( ring @ ( numeral_bit1 @ A15 ) ) ) ).

thf(tcon_Numeral__Type_Obit1___Groups_Oone_978,axiom,
    ! [A15: $tType] :
      ( ( finite_finite @ A15 )
     => ( one @ ( numeral_bit1 @ A15 ) ) ) ).

thf(tcon_Numeral__Type_Obit1___Rings_Odvd_979,axiom,
    ! [A15: $tType] :
      ( ( finite_finite @ A15 )
     => ( dvd @ ( numeral_bit1 @ A15 ) ) ) ).

thf(tcon_Numeral__Type_Obit1___HOL_Oequal_980,axiom,
    ! [A15: $tType] :
      ( ( finite_finite @ A15 )
     => ( cl_HOL_Oequal @ ( numeral_bit1 @ A15 ) ) ) ).

thf(tcon_Numeral__Type_Onum1___Quickcheck__Narrowing_Opartial__term__of_981,axiom,
    quickc6926020345158392990erm_of @ numeral_num1 ).

thf(tcon_Numeral__Type_Onum1___Groups_Ocancel__ab__semigroup__add_982,axiom,
    cancel2418104881723323429up_add @ numeral_num1 ).

thf(tcon_Numeral__Type_Onum1___Groups_Ocancel__comm__monoid__add_983,axiom,
    cancel1802427076303600483id_add @ numeral_num1 ).

thf(tcon_Numeral__Type_Onum1___Groups_Ocancel__semigroup__add_984,axiom,
    cancel_semigroup_add @ numeral_num1 ).

thf(tcon_Numeral__Type_Onum1___Groups_Oab__semigroup__mult_985,axiom,
    ab_semigroup_mult @ numeral_num1 ).

thf(tcon_Numeral__Type_Onum1___Groups_Ocomm__monoid__mult_986,axiom,
    comm_monoid_mult @ numeral_num1 ).

thf(tcon_Numeral__Type_Onum1___Groups_Oab__semigroup__add_987,axiom,
    ab_semigroup_add @ numeral_num1 ).

thf(tcon_Numeral__Type_Onum1___Code__Evaluation_Oterm__of_988,axiom,
    code_term_of @ numeral_num1 ).

thf(tcon_Numeral__Type_Onum1___Groups_Ocomm__monoid__add_989,axiom,
    comm_monoid_add @ numeral_num1 ).

thf(tcon_Numeral__Type_Onum1___Rings_Ocomm__semiring__0_990,axiom,
    comm_semiring_0 @ numeral_num1 ).

thf(tcon_Numeral__Type_Onum1___Groups_Osemigroup__mult_991,axiom,
    semigroup_mult @ numeral_num1 ).

thf(tcon_Numeral__Type_Onum1___Groups_Osemigroup__add_992,axiom,
    semigroup_add @ numeral_num1 ).

thf(tcon_Numeral__Type_Onum1___Rings_Ocomm__semiring_993,axiom,
    comm_semiring @ numeral_num1 ).

thf(tcon_Numeral__Type_Onum1___Orderings_Owellorder_994,axiom,
    wellorder @ numeral_num1 ).

thf(tcon_Numeral__Type_Onum1___Groups_Oab__group__add_995,axiom,
    ab_group_add @ numeral_num1 ).

thf(tcon_Numeral__Type_Onum1___Orderings_Opreorder_996,axiom,
    preorder @ numeral_num1 ).

thf(tcon_Numeral__Type_Onum1___Orderings_Olinorder_997,axiom,
    linorder @ numeral_num1 ).

thf(tcon_Numeral__Type_Onum1___Groups_Omonoid__mult_998,axiom,
    monoid_mult @ numeral_num1 ).

thf(tcon_Numeral__Type_Onum1___Groups_Omonoid__add_999,axiom,
    monoid_add @ numeral_num1 ).

thf(tcon_Numeral__Type_Onum1___Type__Length_Olen0_1000,axiom,
    type_len0 @ numeral_num1 ).

thf(tcon_Numeral__Type_Onum1___Rings_Osemiring__0_1001,axiom,
    semiring_0 @ numeral_num1 ).

thf(tcon_Numeral__Type_Onum1___Groups_Ogroup__add_1002,axiom,
    group_add @ numeral_num1 ).

thf(tcon_Numeral__Type_Onum1___Type__Length_Olen_1003,axiom,
    type_len @ numeral_num1 ).

thf(tcon_Numeral__Type_Onum1___Rings_Omult__zero_1004,axiom,
    mult_zero @ numeral_num1 ).

thf(tcon_Numeral__Type_Onum1___Rings_Ocomm__ring_1005,axiom,
    comm_ring @ numeral_num1 ).

thf(tcon_Numeral__Type_Onum1___Orderings_Oorder_1006,axiom,
    order @ numeral_num1 ).

thf(tcon_Numeral__Type_Onum1___Rings_Osemiring_1007,axiom,
    semiring @ numeral_num1 ).

thf(tcon_Numeral__Type_Onum1___Orderings_Oord_1008,axiom,
    ord @ numeral_num1 ).

thf(tcon_Numeral__Type_Onum1___Groups_Ouminus_1009,axiom,
    uminus @ numeral_num1 ).

thf(tcon_Numeral__Type_Onum1___Groups_Ominus_1010,axiom,
    minus @ numeral_num1 ).

thf(tcon_Numeral__Type_Onum1___Power_Opower_1011,axiom,
    power @ numeral_num1 ).

thf(tcon_Numeral__Type_Onum1___Num_Onumeral_1012,axiom,
    numeral @ numeral_num1 ).

thf(tcon_Numeral__Type_Onum1___Groups_Ozero_1013,axiom,
    zero @ numeral_num1 ).

thf(tcon_Numeral__Type_Onum1___Groups_Oplus_1014,axiom,
    plus @ numeral_num1 ).

thf(tcon_Numeral__Type_Onum1___Rings_Oring_1015,axiom,
    ring @ numeral_num1 ).

thf(tcon_Numeral__Type_Onum1___Groups_Oone_1016,axiom,
    one @ numeral_num1 ).

thf(tcon_Numeral__Type_Onum1___Rings_Odvd_1017,axiom,
    dvd @ numeral_num1 ).

thf(tcon_Numeral__Type_Onum1___HOL_Oequal_1018,axiom,
    cl_HOL_Oequal @ numeral_num1 ).

thf(tcon_Product__Type_Oprod___Topological__Spaces_Otopological__space_1019,axiom,
    ! [A15: $tType,A16: $tType] :
      ( ( ( topolo4958980785337419405_space @ A15 )
        & ( topolo4958980785337419405_space @ A16 ) )
     => ( topolo4958980785337419405_space @ ( product_prod @ A15 @ A16 ) ) ) ).

thf(tcon_Product__Type_Oprod___Quickcheck__Narrowing_Opartial__term__of_1020,axiom,
    ! [A15: $tType,A16: $tType] :
      ( ( ( typerep @ A15 )
        & ( typerep @ A16 ) )
     => ( quickc6926020345158392990erm_of @ ( product_prod @ A15 @ A16 ) ) ) ).

thf(tcon_Product__Type_Oprod___Topological__Spaces_Ot2__space_1021,axiom,
    ! [A15: $tType,A16: $tType] :
      ( ( ( topological_t2_space @ A15 )
        & ( topological_t2_space @ A16 ) )
     => ( topological_t2_space @ ( product_prod @ A15 @ A16 ) ) ) ).

thf(tcon_Product__Type_Oprod___Code__Evaluation_Oterm__of_1022,axiom,
    ! [A15: $tType,A16: $tType] :
      ( ( ( typerep @ A15 )
        & ( typerep @ A16 ) )
     => ( code_term_of @ ( product_prod @ A15 @ A16 ) ) ) ).

thf(tcon_Product__Type_Oprod___Heap_Oheap_1023,axiom,
    ! [A15: $tType,A16: $tType] :
      ( ( ( heap @ A15 )
        & ( heap @ A16 ) )
     => ( heap @ ( product_prod @ A15 @ A16 ) ) ) ).

thf(tcon_Product__Type_Oprod___HOL_Oequal_1024,axiom,
    ! [A15: $tType,A16: $tType] : ( cl_HOL_Oequal @ ( product_prod @ A15 @ A16 ) ) ).

thf(tcon_Product__Type_Oprod___Nat_Osize_1025,axiom,
    ! [A15: $tType,A16: $tType] : ( size @ ( product_prod @ A15 @ A16 ) ) ).

thf(tcon_Product__Type_Ounit___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_1026,axiom,
    condit6923001295902523014norder @ product_unit ).

thf(tcon_Product__Type_Ounit___Quickcheck__Narrowing_Opartial__term__of_1027,axiom,
    quickc6926020345158392990erm_of @ product_unit ).

thf(tcon_Product__Type_Ounit___Boolean__Algebras_Oboolean__algebra_1028,axiom,
    boolea8198339166811842893lgebra @ product_unit ).

thf(tcon_Product__Type_Ounit___Code__Evaluation_Oterm__of_1029,axiom,
    code_term_of @ product_unit ).

thf(tcon_Product__Type_Ounit___Orderings_Owellorder_1030,axiom,
    wellorder @ product_unit ).

thf(tcon_Product__Type_Ounit___Orderings_Oorder__top_1031,axiom,
    order_top @ product_unit ).

thf(tcon_Product__Type_Ounit___Orderings_Oorder__bot_1032,axiom,
    order_bot @ product_unit ).

thf(tcon_Product__Type_Ounit___Orderings_Opreorder_1033,axiom,
    preorder @ product_unit ).

thf(tcon_Product__Type_Ounit___Orderings_Olinorder_1034,axiom,
    linorder @ product_unit ).

thf(tcon_Product__Type_Ounit___Orderings_Oorder_1035,axiom,
    order @ product_unit ).

thf(tcon_Product__Type_Ounit___Orderings_Oord_1036,axiom,
    ord @ product_unit ).

thf(tcon_Product__Type_Ounit___Groups_Ouminus_1037,axiom,
    uminus @ product_unit ).

thf(tcon_Product__Type_Ounit___Groups_Ominus_1038,axiom,
    minus @ product_unit ).

thf(tcon_Product__Type_Ounit___Heap_Oheap_1039,axiom,
    heap @ product_unit ).

thf(tcon_Product__Type_Ounit___HOL_Oequal_1040,axiom,
    cl_HOL_Oequal @ product_unit ).

thf(tcon_Heap_Oheap_Oheap__ext___Quickcheck__Narrowing_Opartial__term__of_1041,axiom,
    ! [A15: $tType] :
      ( ( typerep @ A15 )
     => ( quickc6926020345158392990erm_of @ ( heap_ext @ A15 ) ) ) ).

thf(tcon_Heap_Oheap_Oheap__ext___Code__Evaluation_Oterm__of_1042,axiom,
    ! [A15: $tType] :
      ( ( typerep @ A15 )
     => ( code_term_of @ ( heap_ext @ A15 ) ) ) ).

thf(tcon_Heap_Oheap_Oheap__ext___HOL_Oequal_1043,axiom,
    ! [A15: $tType] : ( cl_HOL_Oequal @ ( heap_ext @ A15 ) ) ).

thf(tcon_Code__Evaluation_Oterm___Quickcheck__Narrowing_Opartial__term__of_1044,axiom,
    quickc6926020345158392990erm_of @ code_term ).

thf(tcon_Code__Evaluation_Oterm___Code__Evaluation_Oterm__of_1045,axiom,
    code_term_of @ code_term ).

thf(tcon_Code__Numeral_Ointeger___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_1046,axiom,
    bit_un5681908812861735899ations @ code_integer ).

thf(tcon_Code__Numeral_Ointeger___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_1047,axiom,
    semiri1453513574482234551roduct @ code_integer ).

thf(tcon_Code__Numeral_Ointeger___Euclidean__Division_Ounique__euclidean__semiring__with__nat_1048,axiom,
    euclid5411537665997757685th_nat @ code_integer ).

thf(tcon_Code__Numeral_Ointeger___Euclidean__Division_Ounique__euclidean__ring__with__nat_1049,axiom,
    euclid8789492081693882211th_nat @ code_integer ).

thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__semigroup__monoid__add__imp__le_1050,axiom,
    ordere1937475149494474687imp_le @ code_integer ).

thf(tcon_Code__Numeral_Ointeger___Euclidean__Division_Ounique__euclidean__semiring_1051,axiom,
    euclid3128863361964157862miring @ code_integer ).

thf(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__semiring__cancel_1052,axiom,
    euclid4440199948858584721cancel @ code_integer ).

thf(tcon_Code__Numeral_Ointeger___Divides_Ounique__euclidean__semiring__numeral_1053,axiom,
    unique1627219031080169319umeral @ code_integer ).

thf(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__ring__cancel_1054,axiom,
    euclid8851590272496341667cancel @ code_integer ).

thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__no__zero__divisors__cancel_1055,axiom,
    semiri6575147826004484403cancel @ code_integer ).

thf(tcon_Code__Numeral_Ointeger___Groups_Ostrict__ordered__ab__semigroup__add_1056,axiom,
    strict9044650504122735259up_add @ code_integer ).

thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__cancel__ab__semigroup__add_1057,axiom,
    ordere580206878836729694up_add @ code_integer ).

thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__semigroup__add__imp__le_1058,axiom,
    ordere2412721322843649153imp_le @ code_integer ).

thf(tcon_Code__Numeral_Ointeger___Bit__Operations_Osemiring__bit__operations_1059,axiom,
    bit_se359711467146920520ations @ code_integer ).

thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__comm__semiring__strict_1060,axiom,
    linord2810124833399127020strict @ code_integer ).

thf(tcon_Code__Numeral_Ointeger___Groups_Ostrict__ordered__comm__monoid__add_1061,axiom,
    strict7427464778891057005id_add @ code_integer ).

thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__cancel__comm__monoid__add_1062,axiom,
    ordere8940638589300402666id_add @ code_integer ).

thf(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__semiring_1063,axiom,
    euclid3725896446679973847miring @ code_integer ).

thf(tcon_Code__Numeral_Ointeger___Quickcheck__Narrowing_Opartial__term__of_1064,axiom,
    quickc6926020345158392990erm_of @ code_integer ).

thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring__1__strict_1065,axiom,
    linord715952674999750819strict @ code_integer ).

thf(tcon_Code__Numeral_Ointeger___Groups_Olinordered__ab__semigroup__add_1066,axiom,
    linord4140545234300271783up_add @ code_integer ).

thf(tcon_Code__Numeral_Ointeger___Bit__Operations_Oring__bit__operations_1067,axiom,
    bit_ri3973907225187159222ations @ code_integer ).

thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__1__no__zero__divisors_1068,axiom,
    semiri2026040879449505780visors @ code_integer ).

thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__nonzero__semiring_1069,axiom,
    linord181362715937106298miring @ code_integer ).

thf(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__ring_1070,axiom,
    euclid5891614535332579305n_ring @ code_integer ).

thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring__strict_1071,axiom,
    linord8928482502909563296strict @ code_integer ).

thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__no__zero__divisors_1072,axiom,
    semiri3467727345109120633visors @ code_integer ).

thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__semigroup__add_1073,axiom,
    ordere6658533253407199908up_add @ code_integer ).

thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__group__add__abs_1074,axiom,
    ordere166539214618696060dd_abs @ code_integer ).

thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__comm__monoid__add_1075,axiom,
    ordere6911136660526730532id_add @ code_integer ).

thf(tcon_Code__Numeral_Ointeger___Groups_Olinordered__ab__group__add_1076,axiom,
    linord5086331880401160121up_add @ code_integer ).

thf(tcon_Code__Numeral_Ointeger___Groups_Ocancel__ab__semigroup__add_1077,axiom,
    cancel2418104881723323429up_add @ code_integer ).

thf(tcon_Code__Numeral_Ointeger___Rings_Oring__1__no__zero__divisors_1078,axiom,
    ring_15535105094025558882visors @ code_integer ).

thf(tcon_Code__Numeral_Ointeger___Groups_Ocancel__comm__monoid__add_1079,axiom,
    cancel1802427076303600483id_add @ code_integer ).

thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__ring__strict_1080,axiom,
    linord4710134922213307826strict @ code_integer ).

thf(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring__1__cancel_1081,axiom,
    comm_s4317794764714335236cancel @ code_integer ).

thf(tcon_Code__Numeral_Ointeger___Bit__Operations_Osemiring__bits_1082,axiom,
    bit_semiring_bits @ code_integer ).

thf(tcon_Code__Numeral_Ointeger___Rings_Oordered__comm__semiring_1083,axiom,
    ordere2520102378445227354miring @ code_integer ).

thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring__1_1084,axiom,
    linord6961819062388156250ring_1 @ code_integer ).

thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__group__add_1085,axiom,
    ordered_ab_group_add @ code_integer ).

thf(tcon_Code__Numeral_Ointeger___Groups_Ocancel__semigroup__add_1086,axiom,
    cancel_semigroup_add @ code_integer ).

thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring_1087,axiom,
    linordered_semiring @ code_integer ).

thf(tcon_Code__Numeral_Ointeger___Least__significant__bit_Olsb_1088,axiom,
    least_6119777620449941438nt_lsb @ code_integer ).

thf(tcon_Code__Numeral_Ointeger___Rings_Oordered__semiring__0_1089,axiom,
    ordered_semiring_0 @ code_integer ).

thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semidom_1090,axiom,
    linordered_semidom @ code_integer ).

thf(tcon_Code__Numeral_Ointeger___Groups_Oab__semigroup__mult_1091,axiom,
    ab_semigroup_mult @ code_integer ).

thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__1__cancel_1092,axiom,
    semiring_1_cancel @ code_integer ).

thf(tcon_Code__Numeral_Ointeger___Rings_Oalgebraic__semidom_1093,axiom,
    algebraic_semidom @ code_integer ).

thf(tcon_Code__Numeral_Ointeger___Groups_Ocomm__monoid__mult_1094,axiom,
    comm_monoid_mult @ code_integer ).

thf(tcon_Code__Numeral_Ointeger___Groups_Oab__semigroup__add_1095,axiom,
    ab_semigroup_add @ code_integer ).

thf(tcon_Code__Numeral_Ointeger___Generic__set__bit_Oset__bit_1096,axiom,
    generic_set_set_bit @ code_integer ).

thf(tcon_Code__Numeral_Ointeger___Code__Evaluation_Oterm__of_1097,axiom,
    code_term_of @ code_integer ).

thf(tcon_Code__Numeral_Ointeger___Rings_Oordered__semiring_1098,axiom,
    ordered_semiring @ code_integer ).

thf(tcon_Code__Numeral_Ointeger___Rings_Oordered__ring__abs_1099,axiom,
    ordered_ring_abs @ code_integer ).

thf(tcon_Code__Numeral_Ointeger___Parity_Osemiring__parity_1100,axiom,
    semiring_parity @ code_integer ).

thf(tcon_Code__Numeral_Ointeger___Groups_Ocomm__monoid__add_1101,axiom,
    comm_monoid_add @ code_integer ).

thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__modulo_1102,axiom,
    semiring_modulo @ code_integer ).

thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__ring_1103,axiom,
    linordered_ring @ code_integer ).

thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__idom_1104,axiom,
    linordered_idom @ code_integer ).

thf(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring__1_1105,axiom,
    comm_semiring_1 @ code_integer ).

thf(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring__0_1106,axiom,
    comm_semiring_0 @ code_integer ).

thf(tcon_Code__Numeral_Ointeger___Groups_Osemigroup__mult_1107,axiom,
    semigroup_mult @ code_integer ).

thf(tcon_Code__Numeral_Ointeger___Rings_Osemidom__modulo_1108,axiom,
    semidom_modulo @ code_integer ).

thf(tcon_Code__Numeral_Ointeger___Rings_Osemidom__divide_1109,axiom,
    semidom_divide @ code_integer ).

thf(tcon_Code__Numeral_Ointeger___Num_Osemiring__numeral_1110,axiom,
    semiring_numeral @ code_integer ).

thf(tcon_Code__Numeral_Ointeger___Groups_Osemigroup__add_1111,axiom,
    semigroup_add @ code_integer ).

thf(tcon_Code__Numeral_Ointeger___Rings_Ozero__less__one_1112,axiom,
    zero_less_one @ code_integer ).

thf(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring_1113,axiom,
    comm_semiring @ code_integer ).

thf(tcon_Code__Numeral_Ointeger___Nat_Osemiring__char__0_1114,axiom,
    semiring_char_0 @ code_integer ).

thf(tcon_Code__Numeral_Ointeger___Groups_Oab__group__add_1115,axiom,
    ab_group_add @ code_integer ).

thf(tcon_Code__Numeral_Ointeger___Rings_Ozero__neq__one_1116,axiom,
    zero_neq_one @ code_integer ).

thf(tcon_Code__Numeral_Ointeger___Rings_Oordered__ring_1117,axiom,
    ordered_ring @ code_integer ).

thf(tcon_Code__Numeral_Ointeger___Rings_Oidom__abs__sgn_1118,axiom,
    idom_abs_sgn @ code_integer ).

thf(tcon_Code__Numeral_Ointeger___Parity_Oring__parity_1119,axiom,
    ring_parity @ code_integer ).

thf(tcon_Code__Numeral_Ointeger___Orderings_Opreorder_1120,axiom,
    preorder @ code_integer ).

thf(tcon_Code__Numeral_Ointeger___Orderings_Olinorder_1121,axiom,
    linorder @ code_integer ).

thf(tcon_Code__Numeral_Ointeger___Groups_Omonoid__mult_1122,axiom,
    monoid_mult @ code_integer ).

thf(tcon_Code__Numeral_Ointeger___Rings_Oidom__modulo_1123,axiom,
    idom_modulo @ code_integer ).

thf(tcon_Code__Numeral_Ointeger___Rings_Oidom__divide_1124,axiom,
    idom_divide @ code_integer ).

thf(tcon_Code__Numeral_Ointeger___Rings_Ocomm__ring__1_1125,axiom,
    comm_ring_1 @ code_integer ).

thf(tcon_Code__Numeral_Ointeger___Groups_Omonoid__add_1126,axiom,
    monoid_add @ code_integer ).

thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__1_1127,axiom,
    semiring_1 @ code_integer ).

thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__0_1128,axiom,
    semiring_0 @ code_integer ).

thf(tcon_Code__Numeral_Ointeger___Groups_Ogroup__add_1129,axiom,
    group_add @ code_integer ).

thf(tcon_Code__Numeral_Ointeger___Rings_Omult__zero_1130,axiom,
    mult_zero @ code_integer ).

thf(tcon_Code__Numeral_Ointeger___Rings_Ocomm__ring_1131,axiom,
    comm_ring @ code_integer ).

thf(tcon_Code__Numeral_Ointeger___Orderings_Oorder_1132,axiom,
    order @ code_integer ).

thf(tcon_Code__Numeral_Ointeger___Num_Oneg__numeral_1133,axiom,
    neg_numeral @ code_integer ).

thf(tcon_Code__Numeral_Ointeger___Nat_Oring__char__0_1134,axiom,
    ring_char_0 @ code_integer ).

thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring_1135,axiom,
    semiring @ code_integer ).

thf(tcon_Code__Numeral_Ointeger___Orderings_Oord_1136,axiom,
    ord @ code_integer ).

thf(tcon_Code__Numeral_Ointeger___Groups_Ouminus_1137,axiom,
    uminus @ code_integer ).

thf(tcon_Code__Numeral_Ointeger___Rings_Oring__1_1138,axiom,
    ring_1 @ code_integer ).

thf(tcon_Code__Numeral_Ointeger___Rings_Oabs__if_1139,axiom,
    abs_if @ code_integer ).

thf(tcon_Code__Numeral_Ointeger___Groups_Ominus_1140,axiom,
    minus @ code_integer ).

thf(tcon_Code__Numeral_Ointeger___Power_Opower_1141,axiom,
    power @ code_integer ).

thf(tcon_Code__Numeral_Ointeger___Num_Onumeral_1142,axiom,
    numeral @ code_integer ).

thf(tcon_Code__Numeral_Ointeger___Groups_Ozero_1143,axiom,
    zero @ code_integer ).

thf(tcon_Code__Numeral_Ointeger___Groups_Oplus_1144,axiom,
    plus @ code_integer ).

thf(tcon_Code__Numeral_Ointeger___Rings_Oring_1145,axiom,
    ring @ code_integer ).

thf(tcon_Code__Numeral_Ointeger___Rings_Oidom_1146,axiom,
    idom @ code_integer ).

thf(tcon_Code__Numeral_Ointeger___Groups_Oone_1147,axiom,
    one @ code_integer ).

thf(tcon_Code__Numeral_Ointeger___Rings_Odvd_1148,axiom,
    dvd @ code_integer ).

thf(tcon_Code__Numeral_Ointeger___HOL_Oequal_1149,axiom,
    cl_HOL_Oequal @ code_integer ).

thf(tcon_Heap__Time__Monad_OHeap___Quickcheck__Narrowing_Opartial__term__of_1150,axiom,
    ! [A15: $tType] :
      ( ( typerep @ A15 )
     => ( quickc6926020345158392990erm_of @ ( heap_Time_Heap @ A15 ) ) ) ).

thf(tcon_Heap__Time__Monad_OHeap___Code__Evaluation_Oterm__of_1151,axiom,
    ! [A15: $tType] :
      ( ( typerep @ A15 )
     => ( code_term_of @ ( heap_Time_Heap @ A15 ) ) ) ).

thf(tcon_Heap__Time__Monad_OHeap___HOL_Oequal_1152,axiom,
    ! [A15: $tType] : ( cl_HOL_Oequal @ ( heap_Time_Heap @ A15 ) ) ).

thf(tcon_Heap__Time__Monad_OHeap___Nat_Osize_1153,axiom,
    ! [A15: $tType] : ( size @ ( heap_Time_Heap @ A15 ) ) ).

thf(tcon_VEBT__Definitions_OVEBT___Quickcheck__Narrowing_Opartial__term__of_1154,axiom,
    quickc6926020345158392990erm_of @ vEBT_VEBT ).

thf(tcon_VEBT__Definitions_OVEBT___Code__Evaluation_Oterm__of_1155,axiom,
    code_term_of @ vEBT_VEBT ).

thf(tcon_VEBT__Definitions_OVEBT___HOL_Oequal_1156,axiom,
    cl_HOL_Oequal @ vEBT_VEBT ).

thf(tcon_VEBT__Definitions_OVEBT___Nat_Osize_1157,axiom,
    size @ vEBT_VEBT ).

thf(tcon_VEBT__BuildupMemImp_OVEBTi___Quickcheck__Narrowing_Opartial__term__of_1158,axiom,
    quickc6926020345158392990erm_of @ vEBT_VEBTi ).

thf(tcon_VEBT__BuildupMemImp_OVEBTi___Code__Evaluation_Oterm__of_1159,axiom,
    code_term_of @ vEBT_VEBTi ).

thf(tcon_VEBT__BuildupMemImp_OVEBTi___Heap_Oheap_1160,axiom,
    heap @ vEBT_VEBTi ).

thf(tcon_VEBT__BuildupMemImp_OVEBTi___HOL_Oequal_1161,axiom,
    cl_HOL_Oequal @ vEBT_VEBTi ).

thf(tcon_VEBT__BuildupMemImp_OVEBTi___Nat_Osize_1162,axiom,
    size @ vEBT_VEBTi ).

% Helper facts (4)
thf(help_If_3_1_T,axiom,
    ! [P: $o] :
      ( ( P = $true )
      | ( P = $false ) ) ).

thf(help_If_2_1_T,axiom,
    ! [A: $tType,X2: A,Y2: A] :
      ( ( if @ A @ $false @ X2 @ Y2 )
      = Y2 ) ).

thf(help_If_1_1_T,axiom,
    ! [A: $tType,X2: A,Y2: A] :
      ( ( if @ A @ $true @ X2 @ Y2 )
      = X2 ) ).

thf(help_fChoice_1_1_T,axiom,
    ! [A: $tType,P: A > $o] :
      ( ( P @ ( fChoice @ A @ P ) )
      = ( ? [X5: A] : ( P @ X5 ) ) ) ).

% Conjectures (3)
thf(conj_0,hypothesis,
    ( tia
    = ( vEBT_Nodei @ x11 @ x12 @ x13 @ x14 ) ) ).

thf(conj_1,hypothesis,
    ( ( x11
      = ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ mi @ ma ) ) )
    & ( x12
      = ( suc @ ( suc @ va ) ) ) ) ).

thf(conj_2,conjecture,
    ( hoare_hoare_triple @ ( option @ nat ) @ ( times_times @ assn @ ( times_times @ assn @ ( vEBT_vebt_assn_raw @ summary @ x14 ) @ ( snga_assn @ vEBT_VEBTi @ x13 @ tree_is ) ) @ ( vEBT_List_list_assn @ vEBT_VEBT @ vEBT_VEBTi @ vEBT_vebt_assn_raw @ treeList @ tree_is ) )
    @ ( heap_Time_bind @ product_unit @ ( option @ nat ) @ ( refine_Imp_assert @ $true )
      @ ^ [Uu3: product_unit] :
          ( heap_Time_bind @ product_unit @ ( option @ nat )
          @ ( refine_Imp_assert
            @ ( ( ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ mi @ ma ) )
                = x11 )
              & ( ( suc @ ( suc @ va ) )
                = x12 ) ) )
          @ ^ [Uv3: product_unit] :
              ( case_option @ ( heap_Time_Heap @ ( option @ nat ) ) @ ( product_prod @ nat @ nat ) @ ( heap_Time_return @ ( option @ nat ) @ ( none @ nat ) )
              @ ^ [Mima2: product_prod @ nat @ nat] :
                  ( if @ ( heap_Time_Heap @ ( option @ nat ) ) @ ( ord_less_eq @ nat @ x12 @ ( one_one @ nat ) ) @ ( heap_Time_return @ ( option @ nat ) @ ( none @ nat ) )
                  @ ( if @ ( heap_Time_Heap @ ( option @ nat ) ) @ ( ord_less @ nat @ ( product_snd @ nat @ nat @ Mima2 ) @ xa ) @ ( heap_Time_return @ ( option @ nat ) @ ( some @ nat @ ( product_snd @ nat @ nat @ Mima2 ) ) )
                    @ ( heap_Time_bind @ nat @ ( option @ nat ) @ ( vEBT_VEBT_lowi @ xa @ ( divide_divide @ nat @ x12 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
                      @ ^ [L2: nat] :
                          ( heap_Time_bind @ nat @ ( option @ nat ) @ ( vEBT_VEBT_highi @ xa @ ( divide_divide @ nat @ x12 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
                          @ ^ [H: nat] :
                              ( heap_Time_bind @ product_unit @ ( option @ nat )
                              @ ( refine_Imp_assert
                                @ ( L2
                                  = ( vEBT_VEBT_low @ xa @ ( divide_divide @ nat @ x12 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
                              @ ^ [Uw3: product_unit] :
                                  ( heap_Time_bind @ product_unit @ ( option @ nat )
                                  @ ( refine_Imp_assert
                                    @ ( H
                                      = ( vEBT_VEBT_high @ xa @ ( divide_divide @ nat @ x12 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
                                  @ ^ [Ux3: product_unit] :
                                      ( heap_Time_bind @ product_unit @ ( option @ nat ) @ ( refine_Imp_assert @ ( ord_less @ nat @ H @ ( size_size @ ( list @ vEBT_VEBT ) @ treeList ) ) )
                                      @ ^ [Uy3: product_unit] :
                                          ( heap_Time_bind @ vEBT_VEBTi @ ( option @ nat ) @ ( array_nth @ vEBT_VEBTi @ x13 @ H )
                                          @ ^ [Aktnode: vEBT_VEBTi] :
                                              ( heap_Time_bind @ ( option @ nat ) @ ( option @ nat ) @ ( vEBT_vebt_minti @ Aktnode )
                                              @ ^ [Minlow: option @ nat] :
                                                  ( heap_Time_bind @ product_unit @ ( option @ nat )
                                                  @ ( refine_Imp_assert
                                                    @ ( Minlow
                                                      = ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ treeList @ H ) ) ) )
                                                  @ ^ [Uz3: product_unit] :
                                                      ( if @ ( heap_Time_Heap @ ( option @ nat ) )
                                                      @ ( ( Minlow
                                                         != ( none @ nat ) )
                                                        & ( vEBT_VEBT_greater @ ( some @ nat @ L2 ) @ Minlow ) )
                                                      @ ( heap_Time_bind @ ( option @ nat ) @ ( option @ nat ) @ ( vEBT_VEBT_vebt_predi @ ( nth @ vEBT_VEBT @ treeList @ H ) @ Aktnode @ L2 )
                                                        @ ^ [Predy: option @ nat] : ( heap_Time_return @ ( option @ nat ) @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ x12 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( some @ nat @ H ) ) @ Predy ) ) )
                                                      @ ( heap_Time_bind @ ( option @ nat ) @ ( option @ nat ) @ ( vEBT_VEBT_vebt_predi @ summary @ x14 @ H )
                                                        @ ^ [Predsum: option @ nat] :
                                                            ( heap_Time_bind @ product_unit @ ( option @ nat )
                                                            @ ( refine_Imp_assert
                                                              @ ( ( Predsum
                                                                  = ( none @ nat ) )
                                                                = ( ( vEBT_vebt_pred @ summary @ H )
                                                                  = ( none @ nat ) ) ) )
                                                            @ ^ [Va4: product_unit] :
                                                                ( if @ ( heap_Time_Heap @ ( option @ nat ) )
                                                                @ ( Predsum
                                                                  = ( none @ nat ) )
                                                                @ ( if @ ( heap_Time_Heap @ ( option @ nat ) ) @ ( ord_less @ nat @ ( product_fst @ nat @ nat @ Mima2 ) @ xa ) @ ( heap_Time_return @ ( option @ nat ) @ ( some @ nat @ ( product_fst @ nat @ nat @ Mima2 ) ) ) @ ( heap_Time_return @ ( option @ nat ) @ ( none @ nat ) ) )
                                                                @ ( heap_Time_bind @ vEBT_VEBTi @ ( option @ nat ) @ ( array_nth @ vEBT_VEBTi @ x13 @ ( the2 @ nat @ Predsum ) )
                                                                  @ ^ [Nextnode: vEBT_VEBTi] :
                                                                      ( heap_Time_bind @ ( option @ nat ) @ ( option @ nat ) @ ( vEBT_vebt_maxti @ Nextnode )
                                                                      @ ^ [Maxnext: option @ nat] : ( heap_Time_return @ ( option @ nat ) @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ x12 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ Predsum ) @ Maxnext ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
              @ x11 ) ) )
    @ ^ [R: option @ nat] :
        ( ex_assn @ ( list @ vEBT_VEBTi )
        @ ^ [Tree_is: list @ vEBT_VEBTi] :
            ( times_times @ assn @ ( times_times @ assn @ ( times_times @ assn @ ( vEBT_vebt_assn_raw @ summary @ x14 ) @ ( snga_assn @ vEBT_VEBTi @ x13 @ Tree_is ) ) @ ( vEBT_List_list_assn @ vEBT_VEBT @ vEBT_VEBTi @ vEBT_vebt_assn_raw @ treeList @ Tree_is ) )
            @ ( pure_assn
              @ ( ( x11
                  = ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ mi @ ma ) ) )
                & ( x12
                  = ( suc @ ( suc @ va ) ) )
                & ( ( ord_less @ nat @ ma @ xa )
                 => ( R
                    = ( some @ nat @ ma ) ) )
                & ( ~ ( ord_less @ nat @ ma @ xa )
                 => ( R
                    = ( if @ ( option @ nat ) @ ( ord_less @ nat @ ( vEBT_VEBT_high @ xa @ ( divide_divide @ nat @ ( suc @ ( suc @ va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ treeList ) )
                      @ ( if @ ( option @ nat )
                        @ ( ( ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ ( divide_divide @ nat @ ( suc @ ( suc @ va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
                           != ( none @ nat ) )
                          & ( vEBT_VEBT_greater @ ( some @ nat @ ( vEBT_VEBT_low @ xa @ ( divide_divide @ nat @ ( suc @ ( suc @ va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ ( divide_divide @ nat @ ( suc @ ( suc @ va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) )
                        @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( some @ nat @ ( vEBT_VEBT_high @ xa @ ( divide_divide @ nat @ ( suc @ ( suc @ va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_pred @ ( nth @ vEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ ( divide_divide @ nat @ ( suc @ ( suc @ va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ xa @ ( divide_divide @ nat @ ( suc @ ( suc @ va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
                        @ ( if @ ( option @ nat )
                          @ ( ( vEBT_vebt_pred @ summary @ ( vEBT_VEBT_high @ xa @ ( divide_divide @ nat @ ( suc @ ( suc @ va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
                            = ( none @ nat ) )
                          @ ( if @ ( option @ nat ) @ ( ord_less @ nat @ mi @ xa ) @ ( some @ nat @ mi ) @ ( none @ nat ) )
                          @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_pred @ summary @ ( vEBT_VEBT_high @ xa @ ( divide_divide @ nat @ ( suc @ ( suc @ va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ treeList @ ( the2 @ nat @ ( vEBT_vebt_pred @ summary @ ( vEBT_VEBT_high @ xa @ ( divide_divide @ nat @ ( suc @ ( suc @ va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) )
                      @ ( none @ nat ) ) ) ) ) ) ) ) ) ).

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