%------------------------------------------------------------------------------
% File     : ITP283_2 : TPTP v8.2.0. Released v8.0.0.
% Domain   : Interactive Theorem Proving
% Problem  : Sledgehammer problem VEBT_BuildupMemImp 00582_027779
% Version  : [Des22] axioms.
% English  :

% Refs     : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
%          : [Des22] Desharnais (2022), Email to Geoff Sutcliffe
% Source   : [Des22]
% Names    : 0093_VEBT_BuildupMemImp_00582_027779 [Des22]

% Status   : Theorem
% Rating   : 1.00 v8.1.0
% Syntax   : Number of formulae    : 12082 (2640 unt;1608 typ;   0 def)
%            Number of atoms       : 30025 (8670 equ)
%            Maximal formula atoms :   73 (   2 avg)
%            Number of connectives : 21863 (2312   ~; 326   |;2632   &)
%                                         (2097 <=>;14496  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   35 (   6 avg)
%            Maximal term depth    :   38 (   2 avg)
%            Number of types       :   18 (  17 usr)
%            Number of type conns  : 1403 (1226   >; 177   *;   0   +;   0  <<)
%            Number of predicates  :  206 ( 203 usr;   2 prp; 0-7 aty)
%            Number of functors    : 1388 (1388 usr;  66 con; 0-8 aty)
%            Number of variables   : 32933 (29644   !; 792   ?;32933   :)
%                                         (2497  !>;   0  ?*;   0  @-;   0  @+)
% SPC      : TF1_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 18:42:25.020
%------------------------------------------------------------------------------
% Could-be-implicit typings (29)
tff(ty_t_VEBT__Definitions_OVEBT,type,
    vEBT_VEBT: $tType ).

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

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

tff(ty_t_Product__Type_Oprod,type,
    product_prod: ( $tType * $tType ) > $tType ).

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

tff(ty_t_Numeral__Type_Onum0,type,
    numeral_num0: $tType ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

tff(ty_t_HOL_Obool,type,
    bool: $tType ).

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

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

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

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

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

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

tff(ty_t_fun,type,
    fun: ( $tType * $tType ) > $tType ).

% Explicit typings (1579)
tff(sy_cl_Complete__Lattices_Ocomplete__linorder,type,
    comple5582772986160207858norder: 
      !>[A: $tType] : $o ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

tff(sy_cl_Rings_Osemidom,type,
    semidom: 
      !>[A: $tType] : $o ).

tff(sy_cl_Fields_Oinverse,type,
    inverse: 
      !>[A: $tType] : $o ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

tff(sy_cl_Finite__Set_Ofinite,type,
    finite_finite: 
      !>[A: $tType] : $o ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

tff(sy_cl_Topological__Spaces_Ot1__space,type,
    topological_t1_space: 
      !>[A: $tType] : $o ).

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

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

tff(sy_cl_Limits_Otopological__group__add,type,
    topolo1633459387980952147up_add: 
      !>[A: $tType] : $o ).

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

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

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

tff(sy_cl_Limits_Otopological__monoid__add,type,
    topolo6943815403480290642id_add: 
      !>[A: $tType] : $o ).

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

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

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

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

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

tff(sy_cl_Limits_Otopological__monoid__mult,type,
    topolo1898628316856586783d_mult: 
      !>[A: $tType] : $o ).

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

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

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

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

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

tff(sy_cl_Real__Vector__Spaces_Ometric__space,type,
    real_V7819770556892013058_space: 
      !>[A: $tType] : $o ).

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

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

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

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

tff(sy_cl_Topological__Spaces_Operfect__space,type,
    topolo8386298272705272623_space: 
      !>[A: $tType] : $o ).

tff(sy_cl_Topological__Spaces_Ouniform__space,type,
    topolo7287701948861334536_space: 
      !>[A: $tType] : $o ).

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

tff(sy_cl_Limits_Otopological__semigroup__mult,type,
    topolo4211221413907600880p_mult: 
      !>[A: $tType] : $o ).

tff(sy_cl_Real__Vector__Spaces_Ocomplete__space,type,
    real_V8037385150606011577_space: 
      !>[A: $tType] : $o ).

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

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

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

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

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

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

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

tff(sy_cl_Orderings_Ounbounded__dense__linorder,type,
    unboun7993243217541854897norder: 
      !>[A: $tType] : $o ).

tff(sy_cl_Real__Vector__Spaces_Ouniformity__dist,type,
    real_V768167426530841204y_dist: 
      !>[A: $tType] : $o ).

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

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

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

tff(sy_cl_Limits_Otopological__comm__monoid__mult,type,
    topolo4987421752381908075d_mult: 
      !>[A: $tType] : $o ).

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

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

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

tff(sy_cl_Topological__Spaces_Odiscrete__topology,type,
    topolo8865339358273720382pology: 
      !>[A: $tType] : $o ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

tff(sy_cl_Topological__Spaces_Ofirst__countable__topology,type,
    topolo3112930676232923870pology: 
      !>[A: $tType] : $o ).

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

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

tff(sy_cl_Topological__Spaces_Olinear__continuum__topology,type,
    topolo8458572112393995274pology: 
      !>[A: $tType] : $o ).

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

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

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

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

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

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

tff(sy_c_ATP_058Lamp__a____,type,
    aTP_Lamp_a: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aa____,type,
    aTP_Lamp_aa: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aaa____,type,
    aTP_Lamp_aaa: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aab____,type,
    aTP_Lamp_aab: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aac____,type,
    aTP_Lamp_aac: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aad____,type,
    aTP_Lamp_aad: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aae____,type,
    aTP_Lamp_aae: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aaf____,type,
    aTP_Lamp_aaf: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aag____,type,
    aTP_Lamp_aag: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aah____,type,
    aTP_Lamp_aah: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aai____,type,
    aTP_Lamp_aai: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aaj____,type,
    aTP_Lamp_aaj: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aak____,type,
    aTP_Lamp_aak: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aal____,type,
    aTP_Lamp_aal: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aam____,type,
    aTP_Lamp_aam: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aan____,type,
    aTP_Lamp_aan: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aao____,type,
    aTP_Lamp_aao: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aap____,type,
    aTP_Lamp_aap: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aaq____,type,
    aTP_Lamp_aaq: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aar____,type,
    aTP_Lamp_aar: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aas____,type,
    aTP_Lamp_aas: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aat____,type,
    aTP_Lamp_aat: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__aau____,type,
    aTP_Lamp_aau: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aav____,type,
    aTP_Lamp_aav: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aaw____,type,
    aTP_Lamp_aaw: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__aax____,type,
    aTP_Lamp_aax: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aay____,type,
    aTP_Lamp_aay: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aaz____,type,
    aTP_Lamp_aaz: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ab____,type,
    aTP_Lamp_ab: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__aba____,type,
    aTP_Lamp_aba: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__abb____,type,
    aTP_Lamp_abb: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__abc____,type,
    aTP_Lamp_abc: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__abd____,type,
    aTP_Lamp_abd: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__abe____,type,
    aTP_Lamp_abe: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__abf____,type,
    aTP_Lamp_abf: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__abg____,type,
    aTP_Lamp_abg: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__abh____,type,
    aTP_Lamp_abh: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__abi____,type,
    aTP_Lamp_abi: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__abj____,type,
    aTP_Lamp_abj: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__abk____,type,
    aTP_Lamp_abk: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__abl____,type,
    aTP_Lamp_abl: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__abm____,type,
    aTP_Lamp_abm: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__abn____,type,
    aTP_Lamp_abn: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__abo____,type,
    aTP_Lamp_abo: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__abp____,type,
    aTP_Lamp_abp: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__abq____,type,
    aTP_Lamp_abq: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__abr____,type,
    aTP_Lamp_abr: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__abs____,type,
    aTP_Lamp_abs: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__abt____,type,
    aTP_Lamp_abt: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__abu____,type,
    aTP_Lamp_abu: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__abv____,type,
    aTP_Lamp_abv: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__abw____,type,
    aTP_Lamp_abw: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__abx____,type,
    aTP_Lamp_abx: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aby____,type,
    aTP_Lamp_aby: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__abz____,type,
    aTP_Lamp_abz: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ac____,type,
    aTP_Lamp_ac: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__aca____,type,
    aTP_Lamp_aca: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__acb____,type,
    aTP_Lamp_acb: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__acc____,type,
    aTP_Lamp_acc: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__acd____,type,
    aTP_Lamp_acd: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ace____,type,
    aTP_Lamp_ace: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__acf____,type,
    aTP_Lamp_acf: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__acg____,type,
    aTP_Lamp_acg: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ach____,type,
    aTP_Lamp_ach: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aci____,type,
    aTP_Lamp_aci: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__acj____,type,
    aTP_Lamp_acj: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ack____,type,
    aTP_Lamp_ack: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__acl____,type,
    aTP_Lamp_acl: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__acm____,type,
    aTP_Lamp_acm: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__acn____,type,
    aTP_Lamp_acn: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aco____,type,
    aTP_Lamp_aco: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__acp____,type,
    aTP_Lamp_acp: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__acq____,type,
    aTP_Lamp_acq: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__acr____,type,
    aTP_Lamp_acr: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__acs____,type,
    aTP_Lamp_acs: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__act____,type,
    aTP_Lamp_act: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__acu____,type,
    aTP_Lamp_acu: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__acv____,type,
    aTP_Lamp_acv: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__acw____,type,
    aTP_Lamp_acw: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__acx____,type,
    aTP_Lamp_acx: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__acy____,type,
    aTP_Lamp_acy: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__acz____,type,
    aTP_Lamp_acz: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ad____,type,
    aTP_Lamp_ad: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ada____,type,
    aTP_Lamp_ada: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__adb____,type,
    aTP_Lamp_adb: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__adc____,type,
    aTP_Lamp_adc: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__add____,type,
    aTP_Lamp_add: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ade____,type,
    aTP_Lamp_ade: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__adf____,type,
    aTP_Lamp_adf: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__adg____,type,
    aTP_Lamp_adg: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__adh____,type,
    aTP_Lamp_adh: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__adi____,type,
    aTP_Lamp_adi: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__adj____,type,
    aTP_Lamp_adj: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__adk____,type,
    aTP_Lamp_adk: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__adl____,type,
    aTP_Lamp_adl: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__adm____,type,
    aTP_Lamp_adm: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__adn____,type,
    aTP_Lamp_adn: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ado____,type,
    aTP_Lamp_ado: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__adp____,type,
    aTP_Lamp_adp: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__adq____,type,
    aTP_Lamp_adq: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__adr____,type,
    aTP_Lamp_adr: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ads____,type,
    aTP_Lamp_ads: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__adt____,type,
    aTP_Lamp_adt: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__adu____,type,
    aTP_Lamp_adu: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__adv____,type,
    aTP_Lamp_adv: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__adw____,type,
    aTP_Lamp_adw: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__adx____,type,
    aTP_Lamp_adx: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ady____,type,
    aTP_Lamp_ady: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__adz____,type,
    aTP_Lamp_adz: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ae____,type,
    aTP_Lamp_ae: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aea____,type,
    aTP_Lamp_aea: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aeb____,type,
    aTP_Lamp_aeb: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aec____,type,
    aTP_Lamp_aec: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aed____,type,
    aTP_Lamp_aed: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aee____,type,
    aTP_Lamp_aee: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aef____,type,
    aTP_Lamp_aef: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__aeg____,type,
    aTP_Lamp_aeg: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aeh____,type,
    aTP_Lamp_aeh: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aei____,type,
    aTP_Lamp_aei: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aej____,type,
    aTP_Lamp_aej: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aek____,type,
    aTP_Lamp_aek: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ael____,type,
    aTP_Lamp_ael: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aem____,type,
    aTP_Lamp_aem: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aen____,type,
    aTP_Lamp_aen: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aeo____,type,
    aTP_Lamp_aeo: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aep____,type,
    aTP_Lamp_aep: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aeq____,type,
    aTP_Lamp_aeq: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aer____,type,
    aTP_Lamp_aer: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aes____,type,
    aTP_Lamp_aes: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aet____,type,
    aTP_Lamp_aet: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aeu____,type,
    aTP_Lamp_aeu: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aev____,type,
    aTP_Lamp_aev: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aew____,type,
    aTP_Lamp_aew: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aex____,type,
    aTP_Lamp_aex: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aey____,type,
    aTP_Lamp_aey: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aez____,type,
    aTP_Lamp_aez: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__af____,type,
    aTP_Lamp_af: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__afa____,type,
    aTP_Lamp_afa: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__afb____,type,
    aTP_Lamp_afb: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__afc____,type,
    aTP_Lamp_afc: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__afd____,type,
    aTP_Lamp_afd: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__afe____,type,
    aTP_Lamp_afe: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aff____,type,
    aTP_Lamp_aff: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__afg____,type,
    aTP_Lamp_afg: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__afh____,type,
    aTP_Lamp_afh: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__afi____,type,
    aTP_Lamp_afi: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__afj____,type,
    aTP_Lamp_afj: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__afk____,type,
    aTP_Lamp_afk: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__afl____,type,
    aTP_Lamp_afl: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__afm____,type,
    aTP_Lamp_afm: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__afn____,type,
    aTP_Lamp_afn: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__afo____,type,
    aTP_Lamp_afo: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__afp____,type,
    aTP_Lamp_afp: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__afq____,type,
    aTP_Lamp_afq: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__afr____,type,
    aTP_Lamp_afr: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__afs____,type,
    aTP_Lamp_afs: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aft____,type,
    aTP_Lamp_aft: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__afu____,type,
    aTP_Lamp_afu: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__afv____,type,
    aTP_Lamp_afv: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__afw____,type,
    aTP_Lamp_afw: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__afx____,type,
    aTP_Lamp_afx: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__afy____,type,
    aTP_Lamp_afy: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__afz____,type,
    aTP_Lamp_afz: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ag____,type,
    aTP_Lamp_ag: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__aga____,type,
    aTP_Lamp_aga: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__agb____,type,
    aTP_Lamp_agb: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__agc____,type,
    aTP_Lamp_agc: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__agd____,type,
    aTP_Lamp_agd: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__age____,type,
    aTP_Lamp_age: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__agf____,type,
    aTP_Lamp_agf: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__agg____,type,
    aTP_Lamp_agg: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__agh____,type,
    aTP_Lamp_agh: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__agi____,type,
    aTP_Lamp_agi: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__agj____,type,
    aTP_Lamp_agj: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__agk____,type,
    aTP_Lamp_agk: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__agl____,type,
    aTP_Lamp_agl: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__agm____,type,
    aTP_Lamp_agm: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__agn____,type,
    aTP_Lamp_agn: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ago____,type,
    aTP_Lamp_ago: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__agp____,type,
    aTP_Lamp_agp: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__agq____,type,
    aTP_Lamp_agq: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__agr____,type,
    aTP_Lamp_agr: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ags____,type,
    aTP_Lamp_ags: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__agt____,type,
    aTP_Lamp_agt: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__agu____,type,
    aTP_Lamp_agu: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__agv____,type,
    aTP_Lamp_agv: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__agw____,type,
    aTP_Lamp_agw: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__agx____,type,
    aTP_Lamp_agx: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__agy____,type,
    aTP_Lamp_agy: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__agz____,type,
    aTP_Lamp_agz: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ah____,type,
    aTP_Lamp_ah: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aha____,type,
    aTP_Lamp_aha: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ahb____,type,
    aTP_Lamp_ahb: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ahc____,type,
    aTP_Lamp_ahc: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ahd____,type,
    aTP_Lamp_ahd: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ahe____,type,
    aTP_Lamp_ahe: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ahf____,type,
    aTP_Lamp_ahf: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ahg____,type,
    aTP_Lamp_ahg: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ahh____,type,
    aTP_Lamp_ahh: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ahi____,type,
    aTP_Lamp_ahi: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ahj____,type,
    aTP_Lamp_ahj: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ahk____,type,
    aTP_Lamp_ahk: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ahl____,type,
    aTP_Lamp_ahl: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ahm____,type,
    aTP_Lamp_ahm: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ahn____,type,
    aTP_Lamp_ahn: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aho____,type,
    aTP_Lamp_aho: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ahp____,type,
    aTP_Lamp_ahp: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ahq____,type,
    aTP_Lamp_ahq: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ahr____,type,
    aTP_Lamp_ahr: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ahs____,type,
    aTP_Lamp_ahs: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aht____,type,
    aTP_Lamp_aht: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ahu____,type,
    aTP_Lamp_ahu: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ahv____,type,
    aTP_Lamp_ahv: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ahw____,type,
    aTP_Lamp_ahw: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ahx____,type,
    aTP_Lamp_ahx: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ahy____,type,
    aTP_Lamp_ahy: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ahz____,type,
    aTP_Lamp_ahz: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ai____,type,
    aTP_Lamp_ai: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aia____,type,
    aTP_Lamp_aia: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aib____,type,
    aTP_Lamp_aib: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aic____,type,
    aTP_Lamp_aic: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aid____,type,
    aTP_Lamp_aid: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aie____,type,
    aTP_Lamp_aie: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aif____,type,
    aTP_Lamp_aif: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aig____,type,
    aTP_Lamp_aig: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aih____,type,
    aTP_Lamp_aih: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aii____,type,
    aTP_Lamp_aii: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aij____,type,
    aTP_Lamp_aij: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aik____,type,
    aTP_Lamp_aik: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ail____,type,
    aTP_Lamp_ail: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aim____,type,
    aTP_Lamp_aim: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ain____,type,
    aTP_Lamp_ain: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aio____,type,
    aTP_Lamp_aio: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aip____,type,
    aTP_Lamp_aip: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aiq____,type,
    aTP_Lamp_aiq: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__air____,type,
    aTP_Lamp_air: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ais____,type,
    aTP_Lamp_ais: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ait____,type,
    aTP_Lamp_ait: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aiu____,type,
    aTP_Lamp_aiu: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aiv____,type,
    aTP_Lamp_aiv: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aiw____,type,
    aTP_Lamp_aiw: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aix____,type,
    aTP_Lamp_aix: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aiy____,type,
    aTP_Lamp_aiy: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aiz____,type,
    aTP_Lamp_aiz: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aj____,type,
    aTP_Lamp_aj: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aja____,type,
    aTP_Lamp_aja: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ajb____,type,
    aTP_Lamp_ajb: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ajc____,type,
    aTP_Lamp_ajc: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ajd____,type,
    aTP_Lamp_ajd: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aje____,type,
    aTP_Lamp_aje: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ajf____,type,
    aTP_Lamp_ajf: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ajg____,type,
    aTP_Lamp_ajg: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ajh____,type,
    aTP_Lamp_ajh: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aji____,type,
    aTP_Lamp_aji: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ajj____,type,
    aTP_Lamp_ajj: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ajk____,type,
    aTP_Lamp_ajk: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ajl____,type,
    aTP_Lamp_ajl: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ajm____,type,
    aTP_Lamp_ajm: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ajn____,type,
    aTP_Lamp_ajn: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ajo____,type,
    aTP_Lamp_ajo: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ajp____,type,
    aTP_Lamp_ajp: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ajq____,type,
    aTP_Lamp_ajq: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ajr____,type,
    aTP_Lamp_ajr: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ajs____,type,
    aTP_Lamp_ajs: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ajt____,type,
    aTP_Lamp_ajt: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aju____,type,
    aTP_Lamp_aju: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ajv____,type,
    aTP_Lamp_ajv: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ajw____,type,
    aTP_Lamp_ajw: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ajx____,type,
    aTP_Lamp_ajx: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ajy____,type,
    aTP_Lamp_ajy: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ajz____,type,
    aTP_Lamp_ajz: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ak____,type,
    aTP_Lamp_ak: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aka____,type,
    aTP_Lamp_aka: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__akb____,type,
    aTP_Lamp_akb: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__akc____,type,
    aTP_Lamp_akc: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__akd____,type,
    aTP_Lamp_akd: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ake____,type,
    aTP_Lamp_ake: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__akf____,type,
    aTP_Lamp_akf: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__akg____,type,
    aTP_Lamp_akg: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__akh____,type,
    aTP_Lamp_akh: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aki____,type,
    aTP_Lamp_aki: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__akj____,type,
    aTP_Lamp_akj: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__akk____,type,
    aTP_Lamp_akk: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__akl____,type,
    aTP_Lamp_akl: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__akm____,type,
    aTP_Lamp_akm: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__akn____,type,
    aTP_Lamp_akn: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__ako____,type,
    aTP_Lamp_ako: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__akp____,type,
    aTP_Lamp_akp: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__akq____,type,
    aTP_Lamp_akq: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__akr____,type,
    aTP_Lamp_akr: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aks____,type,
    aTP_Lamp_aks: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__akt____,type,
    aTP_Lamp_akt: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aku____,type,
    aTP_Lamp_aku: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__akv____,type,
    aTP_Lamp_akv: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__akw____,type,
    aTP_Lamp_akw: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__akx____,type,
    aTP_Lamp_akx: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aky____,type,
    aTP_Lamp_aky: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__akz____,type,
    aTP_Lamp_akz: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__al____,type,
    aTP_Lamp_al: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ala____,type,
    aTP_Lamp_ala: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__alb____,type,
    aTP_Lamp_alb: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__alc____,type,
    aTP_Lamp_alc: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ald____,type,
    aTP_Lamp_ald: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__ale____,type,
    aTP_Lamp_ale: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__alf____,type,
    aTP_Lamp_alf: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__alg____,type,
    aTP_Lamp_alg: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__alh____,type,
    aTP_Lamp_alh: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ali____,type,
    aTP_Lamp_ali: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__alj____,type,
    aTP_Lamp_alj: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__alk____,type,
    aTP_Lamp_alk: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__all____,type,
    aTP_Lamp_all: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__alm____,type,
    aTP_Lamp_alm: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aln____,type,
    aTP_Lamp_aln: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__alo____,type,
    aTP_Lamp_alo: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__alp____,type,
    aTP_Lamp_alp: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__alq____,type,
    aTP_Lamp_alq: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__alr____,type,
    aTP_Lamp_alr: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__als____,type,
    aTP_Lamp_als: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__alt____,type,
    aTP_Lamp_alt: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__alu____,type,
    aTP_Lamp_alu: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__alv____,type,
    aTP_Lamp_alv: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__alw____,type,
    aTP_Lamp_alw: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__alx____,type,
    aTP_Lamp_alx: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aly____,type,
    aTP_Lamp_aly: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__alz____,type,
    aTP_Lamp_alz: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__am____,type,
    aTP_Lamp_am: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ama____,type,
    aTP_Lamp_ama: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__amb____,type,
    aTP_Lamp_amb: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__amc____,type,
    aTP_Lamp_amc: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__amd____,type,
    aTP_Lamp_amd: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__ame____,type,
    aTP_Lamp_ame: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__amf____,type,
    aTP_Lamp_amf: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__amg____,type,
    aTP_Lamp_amg: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__amh____,type,
    aTP_Lamp_amh: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ami____,type,
    aTP_Lamp_ami: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__amj____,type,
    aTP_Lamp_amj: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__amk____,type,
    aTP_Lamp_amk: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aml____,type,
    aTP_Lamp_aml: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__amm____,type,
    aTP_Lamp_amm: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__amn____,type,
    aTP_Lamp_amn: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__amo____,type,
    aTP_Lamp_amo: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__amp____,type,
    aTP_Lamp_amp: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__amq____,type,
    aTP_Lamp_amq: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__amr____,type,
    aTP_Lamp_amr: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ams____,type,
    aTP_Lamp_ams: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__amt____,type,
    aTP_Lamp_amt: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__amu____,type,
    aTP_Lamp_amu: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__amv____,type,
    aTP_Lamp_amv: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__amw____,type,
    aTP_Lamp_amw: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__amx____,type,
    aTP_Lamp_amx: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__amy____,type,
    aTP_Lamp_amy: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__amz____,type,
    aTP_Lamp_amz: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__an____,type,
    aTP_Lamp_an: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ana____,type,
    aTP_Lamp_ana: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__anb____,type,
    aTP_Lamp_anb: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__anc____,type,
    aTP_Lamp_anc: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__and____,type,
    aTP_Lamp_and: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ane____,type,
    aTP_Lamp_ane: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__anf____,type,
    aTP_Lamp_anf: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ang____,type,
    aTP_Lamp_ang: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__anh____,type,
    aTP_Lamp_anh: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ani____,type,
    aTP_Lamp_ani: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__anj____,type,
    aTP_Lamp_anj: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ank____,type,
    aTP_Lamp_ank: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__anl____,type,
    aTP_Lamp_anl: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ao____,type,
    aTP_Lamp_ao: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ap____,type,
    aTP_Lamp_ap: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aq____,type,
    aTP_Lamp_aq: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ar____,type,
    aTP_Lamp_ar: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__as____,type,
    aTP_Lamp_as: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__at____,type,
    aTP_Lamp_at: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__au____,type,
    aTP_Lamp_au: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__av____,type,
    aTP_Lamp_av: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aw____,type,
    aTP_Lamp_aw: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ax____,type,
    aTP_Lamp_ax: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ay____,type,
    aTP_Lamp_ay: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__az____,type,
    aTP_Lamp_az: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ba____,type,
    aTP_Lamp_ba: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__bb____,type,
    aTP_Lamp_bb: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__bc____,type,
    aTP_Lamp_bc: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__bd____,type,
    aTP_Lamp_bd: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__be____,type,
    aTP_Lamp_be: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__bf____,type,
    aTP_Lamp_bf: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__bg____,type,
    aTP_Lamp_bg: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__bh____,type,
    aTP_Lamp_bh: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__bi____,type,
    aTP_Lamp_bi: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__bj____,type,
    aTP_Lamp_bj: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__bk____,type,
    aTP_Lamp_bk: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__bl____,type,
    aTP_Lamp_bl: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__bm____,type,
    aTP_Lamp_bm: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__bn____,type,
    aTP_Lamp_bn: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__bo____,type,
    aTP_Lamp_bo: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__bp____,type,
    aTP_Lamp_bp: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__bq____,type,
    aTP_Lamp_bq: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__br____,type,
    aTP_Lamp_br: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__bs____,type,
    aTP_Lamp_bs: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__bt____,type,
    aTP_Lamp_bt: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__bu____,type,
    aTP_Lamp_bu: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__bv____,type,
    aTP_Lamp_bv: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__bw____,type,
    aTP_Lamp_bw: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__bx____,type,
    aTP_Lamp_bx: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__by____,type,
    aTP_Lamp_by: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__bz____,type,
    aTP_Lamp_bz: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ca____,type,
    aTP_Lamp_ca: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__cb____,type,
    aTP_Lamp_cb: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__cc____,type,
    aTP_Lamp_cc: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__cd____,type,
    aTP_Lamp_cd: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ce____,type,
    aTP_Lamp_ce: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__cf____,type,
    aTP_Lamp_cf: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__cg____,type,
    aTP_Lamp_cg: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ch____,type,
    aTP_Lamp_ch: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ci____,type,
    aTP_Lamp_ci: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__cj____,type,
    aTP_Lamp_cj: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ck____,type,
    aTP_Lamp_ck: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__cl____,type,
    aTP_Lamp_cl: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__cm____,type,
    aTP_Lamp_cm: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__cn____,type,
    aTP_Lamp_cn: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__co____,type,
    aTP_Lamp_co: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__cp____,type,
    aTP_Lamp_cp: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__cq____,type,
    aTP_Lamp_cq: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__cr____,type,
    aTP_Lamp_cr: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__cs____,type,
    aTP_Lamp_cs: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ct____,type,
    aTP_Lamp_ct: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__cu____,type,
    aTP_Lamp_cu: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__cv____,type,
    aTP_Lamp_cv: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__cw____,type,
    aTP_Lamp_cw: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__cx____,type,
    aTP_Lamp_cx: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__cy____,type,
    aTP_Lamp_cy: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__cz____,type,
    aTP_Lamp_cz: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__da____,type,
    aTP_Lamp_da: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__db____,type,
    aTP_Lamp_db: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__dc____,type,
    aTP_Lamp_dc: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__dd____,type,
    aTP_Lamp_dd: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__de____,type,
    aTP_Lamp_de: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__df____,type,
    aTP_Lamp_df: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__dg____,type,
    aTP_Lamp_dg: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__dh____,type,
    aTP_Lamp_dh: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__di____,type,
    aTP_Lamp_di: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__dj____,type,
    aTP_Lamp_dj: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__dk____,type,
    aTP_Lamp_dk: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__dl____,type,
    aTP_Lamp_dl: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__dm____,type,
    aTP_Lamp_dm: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__dn____,type,
    aTP_Lamp_dn: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__do____,type,
    aTP_Lamp_do: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__dp____,type,
    aTP_Lamp_dp: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__dq____,type,
    aTP_Lamp_dq: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__dr____,type,
    aTP_Lamp_dr: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ds____,type,
    aTP_Lamp_ds: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__dt____,type,
    aTP_Lamp_dt: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__du____,type,
    aTP_Lamp_du: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__dv____,type,
    aTP_Lamp_dv: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__dw____,type,
    aTP_Lamp_dw: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__dx____,type,
    aTP_Lamp_dx: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__dy____,type,
    aTP_Lamp_dy: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__dz____,type,
    aTP_Lamp_dz: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ea____,type,
    aTP_Lamp_ea: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__eb____,type,
    aTP_Lamp_eb: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ec____,type,
    aTP_Lamp_ec: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ed____,type,
    aTP_Lamp_ed: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ee____,type,
    aTP_Lamp_ee: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ef____,type,
    aTP_Lamp_ef: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__eg____,type,
    aTP_Lamp_eg: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__eh____,type,
    aTP_Lamp_eh: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ei____,type,
    aTP_Lamp_ei: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ej____,type,
    aTP_Lamp_ej: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ek____,type,
    aTP_Lamp_ek: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__el____,type,
    aTP_Lamp_el: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__em____,type,
    aTP_Lamp_em: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__en____,type,
    aTP_Lamp_en: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__eo____,type,
    aTP_Lamp_eo: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ep____,type,
    aTP_Lamp_ep: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__eq____,type,
    aTP_Lamp_eq: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__er____,type,
    aTP_Lamp_er: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__es____,type,
    aTP_Lamp_es: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__et____,type,
    aTP_Lamp_et: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__eu____,type,
    aTP_Lamp_eu: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ev____,type,
    aTP_Lamp_ev: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ew____,type,
    aTP_Lamp_ew: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ex____,type,
    aTP_Lamp_ex: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ey____,type,
    aTP_Lamp_ey: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ez____,type,
    aTP_Lamp_ez: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__fa____,type,
    aTP_Lamp_fa: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__fb____,type,
    aTP_Lamp_fb: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__fc____,type,
    aTP_Lamp_fc: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__fd____,type,
    aTP_Lamp_fd: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__fe____,type,
    aTP_Lamp_fe: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ff____,type,
    aTP_Lamp_ff: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__fg____,type,
    aTP_Lamp_fg: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__fh____,type,
    aTP_Lamp_fh: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__fi____,type,
    aTP_Lamp_fi: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__fj____,type,
    aTP_Lamp_fj: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__fk____,type,
    aTP_Lamp_fk: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__fl____,type,
    aTP_Lamp_fl: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__fm____,type,
    aTP_Lamp_fm: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__fn____,type,
    aTP_Lamp_fn: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__fo____,type,
    aTP_Lamp_fo: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__fp____,type,
    aTP_Lamp_fp: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__fq____,type,
    aTP_Lamp_fq: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__fr____,type,
    aTP_Lamp_fr: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__fs____,type,
    aTP_Lamp_fs: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ft____,type,
    aTP_Lamp_ft: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__fu____,type,
    aTP_Lamp_fu: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__fv____,type,
    aTP_Lamp_fv: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__fw____,type,
    aTP_Lamp_fw: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__fx____,type,
    aTP_Lamp_fx: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__fy____,type,
    aTP_Lamp_fy: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__fz____,type,
    aTP_Lamp_fz: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ga____,type,
    aTP_Lamp_ga: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__gb____,type,
    aTP_Lamp_gb: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__gc____,type,
    aTP_Lamp_gc: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__gd____,type,
    aTP_Lamp_gd: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ge____,type,
    aTP_Lamp_ge: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__gf____,type,
    aTP_Lamp_gf: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__gg____,type,
    aTP_Lamp_gg: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__gh____,type,
    aTP_Lamp_gh: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__gi____,type,
    aTP_Lamp_gi: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__gj____,type,
    aTP_Lamp_gj: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__gk____,type,
    aTP_Lamp_gk: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__gl____,type,
    aTP_Lamp_gl: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__gm____,type,
    aTP_Lamp_gm: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__gn____,type,
    aTP_Lamp_gn: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__go____,type,
    aTP_Lamp_go: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__gp____,type,
    aTP_Lamp_gp: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__gq____,type,
    aTP_Lamp_gq: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__gr____,type,
    aTP_Lamp_gr: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__gs____,type,
    aTP_Lamp_gs: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__gt____,type,
    aTP_Lamp_gt: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__gu____,type,
    aTP_Lamp_gu: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__gv____,type,
    aTP_Lamp_gv: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__gw____,type,
    aTP_Lamp_gw: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__gx____,type,
    aTP_Lamp_gx: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__gy____,type,
    aTP_Lamp_gy: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__gz____,type,
    aTP_Lamp_gz: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ha____,type,
    aTP_Lamp_ha: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__hb____,type,
    aTP_Lamp_hb: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__hc____,type,
    aTP_Lamp_hc: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__hd____,type,
    aTP_Lamp_hd: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__he____,type,
    aTP_Lamp_he: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__hf____,type,
    aTP_Lamp_hf: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__hg____,type,
    aTP_Lamp_hg: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__hh____,type,
    aTP_Lamp_hh: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__hi____,type,
    aTP_Lamp_hi: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__hj____,type,
    aTP_Lamp_hj: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__hk____,type,
    aTP_Lamp_hk: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__hl____,type,
    aTP_Lamp_hl: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__hm____,type,
    aTP_Lamp_hm: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__hn____,type,
    aTP_Lamp_hn: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ho____,type,
    aTP_Lamp_ho: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__hp____,type,
    aTP_Lamp_hp: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__hq____,type,
    aTP_Lamp_hq: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__hr____,type,
    aTP_Lamp_hr: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__hs____,type,
    aTP_Lamp_hs: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ht____,type,
    aTP_Lamp_ht: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__hu____,type,
    aTP_Lamp_hu: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__hv____,type,
    aTP_Lamp_hv: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__hw____,type,
    aTP_Lamp_hw: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__hx____,type,
    aTP_Lamp_hx: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__hy____,type,
    aTP_Lamp_hy: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__hz____,type,
    aTP_Lamp_hz: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ia____,type,
    aTP_Lamp_ia: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ib____,type,
    aTP_Lamp_ib: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ic____,type,
    aTP_Lamp_ic: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__id____,type,
    aTP_Lamp_id: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ie____,type,
    aTP_Lamp_ie: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__if____,type,
    aTP_Lamp_if: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ig____,type,
    aTP_Lamp_ig: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ih____,type,
    aTP_Lamp_ih: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ii____,type,
    aTP_Lamp_ii: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ij____,type,
    aTP_Lamp_ij: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ik____,type,
    aTP_Lamp_ik: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__il____,type,
    aTP_Lamp_il: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__im____,type,
    aTP_Lamp_im: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__in____,type,
    aTP_Lamp_in: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__io____,type,
    aTP_Lamp_io: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ip____,type,
    aTP_Lamp_ip: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__iq____,type,
    aTP_Lamp_iq: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ir____,type,
    aTP_Lamp_ir: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__is____,type,
    aTP_Lamp_is: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__it____,type,
    aTP_Lamp_it: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__iu____,type,
    aTP_Lamp_iu: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__iv____,type,
    aTP_Lamp_iv: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__iw____,type,
    aTP_Lamp_iw: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ix____,type,
    aTP_Lamp_ix: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__iy____,type,
    aTP_Lamp_iy: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__iz____,type,
    aTP_Lamp_iz: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ja____,type,
    aTP_Lamp_ja: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__jb____,type,
    aTP_Lamp_jb: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__jc____,type,
    aTP_Lamp_jc: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__jd____,type,
    aTP_Lamp_jd: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__je____,type,
    aTP_Lamp_je: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__jf____,type,
    aTP_Lamp_jf: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__jg____,type,
    aTP_Lamp_jg: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__jh____,type,
    aTP_Lamp_jh: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ji____,type,
    aTP_Lamp_ji: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__jj____,type,
    aTP_Lamp_jj: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__jk____,type,
    aTP_Lamp_jk: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__jl____,type,
    aTP_Lamp_jl: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__jm____,type,
    aTP_Lamp_jm: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__jn____,type,
    aTP_Lamp_jn: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__jo____,type,
    aTP_Lamp_jo: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__jp____,type,
    aTP_Lamp_jp: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__jq____,type,
    aTP_Lamp_jq: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__jr____,type,
    aTP_Lamp_jr: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__js____,type,
    aTP_Lamp_js: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__jt____,type,
    aTP_Lamp_jt: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ju____,type,
    aTP_Lamp_ju: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__jv____,type,
    aTP_Lamp_jv: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__jw____,type,
    aTP_Lamp_jw: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__jx____,type,
    aTP_Lamp_jx: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__jy____,type,
    aTP_Lamp_jy: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__jz____,type,
    aTP_Lamp_jz: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ka____,type,
    aTP_Lamp_ka: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__kb____,type,
    aTP_Lamp_kb: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__kc____,type,
    aTP_Lamp_kc: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__kd____,type,
    aTP_Lamp_kd: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ke____,type,
    aTP_Lamp_ke: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__kf____,type,
    aTP_Lamp_kf: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__kg____,type,
    aTP_Lamp_kg: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__kh____,type,
    aTP_Lamp_kh: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ki____,type,
    aTP_Lamp_ki: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__kj____,type,
    aTP_Lamp_kj: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__kk____,type,
    aTP_Lamp_kk: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__kl____,type,
    aTP_Lamp_kl: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__km____,type,
    aTP_Lamp_km: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__kn____,type,
    aTP_Lamp_kn: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ko____,type,
    aTP_Lamp_ko: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__kp____,type,
    aTP_Lamp_kp: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__kq____,type,
    aTP_Lamp_kq: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__kr____,type,
    aTP_Lamp_kr: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ks____,type,
    aTP_Lamp_ks: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__kt____,type,
    aTP_Lamp_kt: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ku____,type,
    aTP_Lamp_ku: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__kv____,type,
    aTP_Lamp_kv: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__kw____,type,
    aTP_Lamp_kw: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__kx____,type,
    aTP_Lamp_kx: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ky____,type,
    aTP_Lamp_ky: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__kz____,type,
    aTP_Lamp_kz: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__la____,type,
    aTP_Lamp_la: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__lb____,type,
    aTP_Lamp_lb: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__lc____,type,
    aTP_Lamp_lc: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ld____,type,
    aTP_Lamp_ld: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__le____,type,
    aTP_Lamp_le: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__lf____,type,
    aTP_Lamp_lf: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__lg____,type,
    aTP_Lamp_lg: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__lh____,type,
    aTP_Lamp_lh: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__li____,type,
    aTP_Lamp_li: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__lj____,type,
    aTP_Lamp_lj: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__lk____,type,
    aTP_Lamp_lk: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ll____,type,
    aTP_Lamp_ll: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__lm____,type,
    aTP_Lamp_lm: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ln____,type,
    aTP_Lamp_ln: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__lo____,type,
    aTP_Lamp_lo: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__lp____,type,
    aTP_Lamp_lp: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__lq____,type,
    aTP_Lamp_lq: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__lr____,type,
    aTP_Lamp_lr: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ls____,type,
    aTP_Lamp_ls: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__lt____,type,
    aTP_Lamp_lt: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__lu____,type,
    aTP_Lamp_lu: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__lv____,type,
    aTP_Lamp_lv: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__lw____,type,
    aTP_Lamp_lw: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__lx____,type,
    aTP_Lamp_lx: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__ly____,type,
    aTP_Lamp_ly: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__lz____,type,
    aTP_Lamp_lz: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ma____,type,
    aTP_Lamp_ma: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__mb____,type,
    aTP_Lamp_mb: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__mc____,type,
    aTP_Lamp_mc: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__md____,type,
    aTP_Lamp_md: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__me____,type,
    aTP_Lamp_me: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__mf____,type,
    aTP_Lamp_mf: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__mg____,type,
    aTP_Lamp_mg: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__mh____,type,
    aTP_Lamp_mh: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__mi____,type,
    aTP_Lamp_mi: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__mj____,type,
    aTP_Lamp_mj: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__mk____,type,
    aTP_Lamp_mk: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ml____,type,
    aTP_Lamp_ml: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__mm____,type,
    aTP_Lamp_mm: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__mn____,type,
    aTP_Lamp_mn: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__mo____,type,
    aTP_Lamp_mo: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__mp____,type,
    aTP_Lamp_mp: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__mq____,type,
    aTP_Lamp_mq: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__mr____,type,
    aTP_Lamp_mr: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ms____,type,
    aTP_Lamp_ms: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__mt____,type,
    aTP_Lamp_mt: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__mu____,type,
    aTP_Lamp_mu: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__mv____,type,
    aTP_Lamp_mv: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__mw____,type,
    aTP_Lamp_mw: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__mx____,type,
    aTP_Lamp_mx: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__my____,type,
    aTP_Lamp_my: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__mz____,type,
    aTP_Lamp_mz: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__na____,type,
    aTP_Lamp_na: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__nb____,type,
    aTP_Lamp_nb: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__nc____,type,
    aTP_Lamp_nc: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__nd____,type,
    aTP_Lamp_nd: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ne____,type,
    aTP_Lamp_ne: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__nf____,type,
    aTP_Lamp_nf: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ng____,type,
    aTP_Lamp_ng: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__nh____,type,
    aTP_Lamp_nh: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__ni____,type,
    aTP_Lamp_ni: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__nj____,type,
    aTP_Lamp_nj: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__nk____,type,
    aTP_Lamp_nk: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__nl____,type,
    aTP_Lamp_nl: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__nm____,type,
    aTP_Lamp_nm: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__nn____,type,
    aTP_Lamp_nn: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__no____,type,
    aTP_Lamp_no: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__np____,type,
    aTP_Lamp_np: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__nq____,type,
    aTP_Lamp_nq: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__nr____,type,
    aTP_Lamp_nr: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ns____,type,
    aTP_Lamp_ns: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__nt____,type,
    aTP_Lamp_nt: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__nu____,type,
    aTP_Lamp_nu: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__nv____,type,
    aTP_Lamp_nv: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__nw____,type,
    aTP_Lamp_nw: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__nx____,type,
    aTP_Lamp_nx: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ny____,type,
    aTP_Lamp_ny: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__nz____,type,
    aTP_Lamp_nz: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__oa____,type,
    aTP_Lamp_oa: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ob____,type,
    aTP_Lamp_ob: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__oc____,type,
    aTP_Lamp_oc: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__od____,type,
    aTP_Lamp_od: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__oe____,type,
    aTP_Lamp_oe: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__of____,type,
    aTP_Lamp_of: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__og____,type,
    aTP_Lamp_og: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__oh____,type,
    aTP_Lamp_oh: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__oi____,type,
    aTP_Lamp_oi: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__oj____,type,
    aTP_Lamp_oj: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ok____,type,
    aTP_Lamp_ok: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ol____,type,
    aTP_Lamp_ol: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__om____,type,
    aTP_Lamp_om: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__on____,type,
    aTP_Lamp_on: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__oo____,type,
    aTP_Lamp_oo: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__op____,type,
    aTP_Lamp_op: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__oq____,type,
    aTP_Lamp_oq: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__or____,type,
    aTP_Lamp_or: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__os____,type,
    aTP_Lamp_os: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ot____,type,
    aTP_Lamp_ot: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__ou____,type,
    aTP_Lamp_ou: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ov____,type,
    aTP_Lamp_ov: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ow____,type,
    aTP_Lamp_ow: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ox____,type,
    aTP_Lamp_ox: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__oy____,type,
    aTP_Lamp_oy: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__oz____,type,
    aTP_Lamp_oz: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__pa____,type,
    aTP_Lamp_pa: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__pb____,type,
    aTP_Lamp_pb: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__pc____,type,
    aTP_Lamp_pc: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__pd____,type,
    aTP_Lamp_pd: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__pe____,type,
    aTP_Lamp_pe: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__pf____,type,
    aTP_Lamp_pf: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__pg____,type,
    aTP_Lamp_pg: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ph____,type,
    aTP_Lamp_ph: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__pi____,type,
    aTP_Lamp_pi: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__pj____,type,
    aTP_Lamp_pj: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__pk____,type,
    aTP_Lamp_pk: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__pl____,type,
    aTP_Lamp_pl: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__pm____,type,
    aTP_Lamp_pm: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__pn____,type,
    aTP_Lamp_pn: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__po____,type,
    aTP_Lamp_po: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__pp____,type,
    aTP_Lamp_pp: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__pq____,type,
    aTP_Lamp_pq: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__pr____,type,
    aTP_Lamp_pr: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__ps____,type,
    aTP_Lamp_ps: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__pt____,type,
    aTP_Lamp_pt: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__pu____,type,
    aTP_Lamp_pu: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__pv____,type,
    aTP_Lamp_pv: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__pw____,type,
    aTP_Lamp_pw: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__px____,type,
    aTP_Lamp_px: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__py____,type,
    aTP_Lamp_py: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__pz____,type,
    aTP_Lamp_pz: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__qa____,type,
    aTP_Lamp_qa: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__qb____,type,
    aTP_Lamp_qb: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__qc____,type,
    aTP_Lamp_qc: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__qd____,type,
    aTP_Lamp_qd: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__qe____,type,
    aTP_Lamp_qe: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__qf____,type,
    aTP_Lamp_qf: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__qg____,type,
    aTP_Lamp_qg: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__qh____,type,
    aTP_Lamp_qh: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__qi____,type,
    aTP_Lamp_qi: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__qj____,type,
    aTP_Lamp_qj: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__qk____,type,
    aTP_Lamp_qk: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ql____,type,
    aTP_Lamp_ql: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__qm____,type,
    aTP_Lamp_qm: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__qn____,type,
    aTP_Lamp_qn: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__qo____,type,
    aTP_Lamp_qo: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__qp____,type,
    aTP_Lamp_qp: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__qq____,type,
    aTP_Lamp_qq: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__qr____,type,
    aTP_Lamp_qr: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__qs____,type,
    aTP_Lamp_qs: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__qt____,type,
    aTP_Lamp_qt: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__qu____,type,
    aTP_Lamp_qu: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__qv____,type,
    aTP_Lamp_qv: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__qw____,type,
    aTP_Lamp_qw: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__qx____,type,
    aTP_Lamp_qx: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__qy____,type,
    aTP_Lamp_qy: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__qz____,type,
    aTP_Lamp_qz: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__ra____,type,
    aTP_Lamp_ra: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__rb____,type,
    aTP_Lamp_rb: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__rc____,type,
    aTP_Lamp_rc: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__rd____,type,
    aTP_Lamp_rd: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__re____,type,
    aTP_Lamp_re: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__rf____,type,
    aTP_Lamp_rf: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__rg____,type,
    aTP_Lamp_rg: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__rh____,type,
    aTP_Lamp_rh: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ri____,type,
    aTP_Lamp_ri: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__rj____,type,
    aTP_Lamp_rj: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__rk____,type,
    aTP_Lamp_rk: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__rl____,type,
    aTP_Lamp_rl: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__rm____,type,
    aTP_Lamp_rm: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__rn____,type,
    aTP_Lamp_rn: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ro____,type,
    aTP_Lamp_ro: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__rp____,type,
    aTP_Lamp_rp: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__rq____,type,
    aTP_Lamp_rq: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__rr____,type,
    aTP_Lamp_rr: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__rs____,type,
    aTP_Lamp_rs: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__rt____,type,
    aTP_Lamp_rt: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ru____,type,
    aTP_Lamp_ru: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__rv____,type,
    aTP_Lamp_rv: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__rw____,type,
    aTP_Lamp_rw: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__rx____,type,
    aTP_Lamp_rx: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ry____,type,
    aTP_Lamp_ry: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__rz____,type,
    aTP_Lamp_rz: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__sa____,type,
    aTP_Lamp_sa: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__sb____,type,
    aTP_Lamp_sb: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__sc____,type,
    aTP_Lamp_sc: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__sd____,type,
    aTP_Lamp_sd: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__se____,type,
    aTP_Lamp_se: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__sf____,type,
    aTP_Lamp_sf: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__sg____,type,
    aTP_Lamp_sg: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__sh____,type,
    aTP_Lamp_sh: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__si____,type,
    aTP_Lamp_si: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__sj____,type,
    aTP_Lamp_sj: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__sk____,type,
    aTP_Lamp_sk: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__sl____,type,
    aTP_Lamp_sl: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__sm____,type,
    aTP_Lamp_sm: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__sn____,type,
    aTP_Lamp_sn: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__so____,type,
    aTP_Lamp_so: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__sp____,type,
    aTP_Lamp_sp: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__sq____,type,
    aTP_Lamp_sq: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__sr____,type,
    aTP_Lamp_sr: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__ss____,type,
    aTP_Lamp_ss: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__st____,type,
    aTP_Lamp_st: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__su____,type,
    aTP_Lamp_su: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__sv____,type,
    aTP_Lamp_sv: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__sw____,type,
    aTP_Lamp_sw: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__sx____,type,
    aTP_Lamp_sx: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__sy____,type,
    aTP_Lamp_sy: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__sz____,type,
    aTP_Lamp_sz: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ta____,type,
    aTP_Lamp_ta: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__tb____,type,
    aTP_Lamp_tb: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__tc____,type,
    aTP_Lamp_tc: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__td____,type,
    aTP_Lamp_td: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__te____,type,
    aTP_Lamp_te: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__tf____,type,
    aTP_Lamp_tf: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__tg____,type,
    aTP_Lamp_tg: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__th____,type,
    aTP_Lamp_th: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__ti____,type,
    aTP_Lamp_ti: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__tj____,type,
    aTP_Lamp_tj: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__tk____,type,
    aTP_Lamp_tk: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__tl____,type,
    aTP_Lamp_tl: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__tm____,type,
    aTP_Lamp_tm: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__tn____,type,
    aTP_Lamp_tn: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__to____,type,
    aTP_Lamp_to: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__tp____,type,
    aTP_Lamp_tp: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__tq____,type,
    aTP_Lamp_tq: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__tr____,type,
    aTP_Lamp_tr: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ts____,type,
    aTP_Lamp_ts: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__tt____,type,
    aTP_Lamp_tt: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__tu____,type,
    aTP_Lamp_tu: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__tv____,type,
    aTP_Lamp_tv: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__tw____,type,
    aTP_Lamp_tw: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__tx____,type,
    aTP_Lamp_tx: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ty____,type,
    aTP_Lamp_ty: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__tz____,type,
    aTP_Lamp_tz: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ua____,type,
    aTP_Lamp_ua: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ub____,type,
    aTP_Lamp_ub: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__uc____,type,
    aTP_Lamp_uc: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ud____,type,
    aTP_Lamp_ud: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ue____,type,
    aTP_Lamp_ue: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__uf____,type,
    aTP_Lamp_uf: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ug____,type,
    aTP_Lamp_ug: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__uh____,type,
    aTP_Lamp_uh: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ui____,type,
    aTP_Lamp_ui: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__uj____,type,
    aTP_Lamp_uj: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__uk____,type,
    aTP_Lamp_uk: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ul____,type,
    aTP_Lamp_ul: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__um____,type,
    aTP_Lamp_um: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__un____,type,
    aTP_Lamp_un: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__uo____,type,
    aTP_Lamp_uo: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__up____,type,
    aTP_Lamp_up: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__uq____,type,
    aTP_Lamp_uq: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ur____,type,
    aTP_Lamp_ur: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__us____,type,
    aTP_Lamp_us: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ut____,type,
    aTP_Lamp_ut: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__uu____,type,
    aTP_Lamp_uu: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__uv____,type,
    aTP_Lamp_uv: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__uw____,type,
    aTP_Lamp_uw: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ux____,type,
    aTP_Lamp_ux: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__uy____,type,
    aTP_Lamp_uy: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__uz____,type,
    aTP_Lamp_uz: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__va____,type,
    aTP_Lamp_va: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__vb____,type,
    aTP_Lamp_vb: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__vc____,type,
    aTP_Lamp_vc: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__vd____,type,
    aTP_Lamp_vd: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ve____,type,
    aTP_Lamp_ve: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__vf____,type,
    aTP_Lamp_vf: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__vg____,type,
    aTP_Lamp_vg: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__vh____,type,
    aTP_Lamp_vh: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__vi____,type,
    aTP_Lamp_vi: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__vj____,type,
    aTP_Lamp_vj: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__vk____,type,
    aTP_Lamp_vk: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__vl____,type,
    aTP_Lamp_vl: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__vm____,type,
    aTP_Lamp_vm: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__vn____,type,
    aTP_Lamp_vn: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__vo____,type,
    aTP_Lamp_vo: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__vp____,type,
    aTP_Lamp_vp: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__vq____,type,
    aTP_Lamp_vq: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__vr____,type,
    aTP_Lamp_vr: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__vs____,type,
    aTP_Lamp_vs: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__vt____,type,
    aTP_Lamp_vt: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__vu____,type,
    aTP_Lamp_vu: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__vv____,type,
    aTP_Lamp_vv: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__vw____,type,
    aTP_Lamp_vw: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__vx____,type,
    aTP_Lamp_vx: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__vy____,type,
    aTP_Lamp_vy: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__vz____,type,
    aTP_Lamp_vz: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__wa____,type,
    aTP_Lamp_wa: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__wb____,type,
    aTP_Lamp_wb: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__wc____,type,
    aTP_Lamp_wc: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__wd____,type,
    aTP_Lamp_wd: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__we____,type,
    aTP_Lamp_we: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__wf____,type,
    aTP_Lamp_wf: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__wg____,type,
    aTP_Lamp_wg: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__wh____,type,
    aTP_Lamp_wh: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__wi____,type,
    aTP_Lamp_wi: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__wj____,type,
    aTP_Lamp_wj: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__wk____,type,
    aTP_Lamp_wk: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__wl____,type,
    aTP_Lamp_wl: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__wm____,type,
    aTP_Lamp_wm: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__wn____,type,
    aTP_Lamp_wn: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__wo____,type,
    aTP_Lamp_wo: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__wp____,type,
    aTP_Lamp_wp: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__wq____,type,
    aTP_Lamp_wq: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__wr____,type,
    aTP_Lamp_wr: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ws____,type,
    aTP_Lamp_ws: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__wt____,type,
    aTP_Lamp_wt: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__wu____,type,
    aTP_Lamp_wu: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__wv____,type,
    aTP_Lamp_wv: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ww____,type,
    aTP_Lamp_ww: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__wx____,type,
    aTP_Lamp_wx: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__wy____,type,
    aTP_Lamp_wy: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__wz____,type,
    aTP_Lamp_wz: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__xa____,type,
    aTP_Lamp_xa: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__xb____,type,
    aTP_Lamp_xb: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__xc____,type,
    aTP_Lamp_xc: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__xd____,type,
    aTP_Lamp_xd: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__xe____,type,
    aTP_Lamp_xe: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__xf____,type,
    aTP_Lamp_xf: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__xg____,type,
    aTP_Lamp_xg: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__xh____,type,
    aTP_Lamp_xh: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__xi____,type,
    aTP_Lamp_xi: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__xj____,type,
    aTP_Lamp_xj: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__xk____,type,
    aTP_Lamp_xk: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__xl____,type,
    aTP_Lamp_xl: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__xm____,type,
    aTP_Lamp_xm: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__xn____,type,
    aTP_Lamp_xn: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__xo____,type,
    aTP_Lamp_xo: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__xp____,type,
    aTP_Lamp_xp: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__xq____,type,
    aTP_Lamp_xq: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__xr____,type,
    aTP_Lamp_xr: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__xs____,type,
    aTP_Lamp_xs: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__xt____,type,
    aTP_Lamp_xt: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__xu____,type,
    aTP_Lamp_xu: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__xv____,type,
    aTP_Lamp_xv: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__xw____,type,
    aTP_Lamp_xw: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__xx____,type,
    aTP_Lamp_xx: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__xy____,type,
    aTP_Lamp_xy: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__xz____,type,
    aTP_Lamp_xz: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ya____,type,
    aTP_Lamp_ya: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__yb____,type,
    aTP_Lamp_yb: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__yc____,type,
    aTP_Lamp_yc: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__yd____,type,
    aTP_Lamp_yd: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ye____,type,
    aTP_Lamp_ye: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__yf____,type,
    aTP_Lamp_yf: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__yg____,type,
    aTP_Lamp_yg: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__yh____,type,
    aTP_Lamp_yh: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__yi____,type,
    aTP_Lamp_yi: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__yj____,type,
    aTP_Lamp_yj: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__yk____,type,
    aTP_Lamp_yk: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__yl____,type,
    aTP_Lamp_yl: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ym____,type,
    aTP_Lamp_ym: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__yn____,type,
    aTP_Lamp_yn: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__yo____,type,
    aTP_Lamp_yo: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__yp____,type,
    aTP_Lamp_yp: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__yq____,type,
    aTP_Lamp_yq: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__yr____,type,
    aTP_Lamp_yr: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ys____,type,
    aTP_Lamp_ys: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__yt____,type,
    aTP_Lamp_yt: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__yu____,type,
    aTP_Lamp_yu: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__yv____,type,
    aTP_Lamp_yv: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__yw____,type,
    aTP_Lamp_yw: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__yx____,type,
    aTP_Lamp_yx: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__yy____,type,
    aTP_Lamp_yy: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__yz____,type,
    aTP_Lamp_yz: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__za____,type,
    aTP_Lamp_za: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zb____,type,
    aTP_Lamp_zb: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zc____,type,
    aTP_Lamp_zc: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zd____,type,
    aTP_Lamp_zd: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ze____,type,
    aTP_Lamp_ze: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zf____,type,
    aTP_Lamp_zf: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zg____,type,
    aTP_Lamp_zg: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zh____,type,
    aTP_Lamp_zh: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zi____,type,
    aTP_Lamp_zi: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zj____,type,
    aTP_Lamp_zj: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zk____,type,
    aTP_Lamp_zk: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zl____,type,
    aTP_Lamp_zl: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zm____,type,
    aTP_Lamp_zm: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zn____,type,
    aTP_Lamp_zn: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zo____,type,
    aTP_Lamp_zo: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zp____,type,
    aTP_Lamp_zp: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zq____,type,
    aTP_Lamp_zq: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zr____,type,
    aTP_Lamp_zr: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__zs____,type,
    aTP_Lamp_zs: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zt____,type,
    aTP_Lamp_zt: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zu____,type,
    aTP_Lamp_zu: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zv____,type,
    aTP_Lamp_zv: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__zw____,type,
    aTP_Lamp_zw: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zx____,type,
    aTP_Lamp_zx: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zy____,type,
    aTP_Lamp_zy: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zz____,type,
    aTP_Lamp_zz: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

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

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

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

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

tff(sy_c_Binomial_Obinomial,type,
    binomial: nat > fun(nat,nat) ).

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

tff(sy_c_Bit__Comprehension_Obit__comprehension__class_Oset__bits,type,
    bit_bi4170147762399595738t_bits: 
      !>[A: $tType] : ( fun(nat,bool) > A ) ).

tff(sy_c_Bit__Comprehension_Owf__set__bits__int,type,
    bit_wf_set_bits_int: fun(nat,bool) > $o ).

tff(sy_c_Bit__Operations_Oand__int__rel,type,
    bit_and_int_rel: fun(product_prod(int,int),fun(product_prod(int,int),bool)) ).

tff(sy_c_Bit__Operations_Oand__not__num,type,
    bit_and_not_num: ( num * num ) > option(num) ).

tff(sy_c_Bit__Operations_Oconcat__bit,type,
    bit_concat_bit: ( nat * int ) > fun(int,int) ).

tff(sy_c_Bit__Operations_Oor__not__num__neg,type,
    bit_or_not_num_neg: ( num * num ) > num ).

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

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

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

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

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

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

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

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

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

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

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

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

tff(sy_c_Bit__Operations_Osemiring__bits__class_Obit,type,
    bit_se5641148757651400278ts_bit: 
      !>[A: $tType] : ( A > fun(nat,bool) ) ).

tff(sy_c_Bit__Operations_Osemiring__bits__class_Opossible__bit,type,
    bit_se6407376104438227557le_bit: 
      !>[A: $tType] : ( ( itself(A) * nat ) > bool ) ).

tff(sy_c_Bit__Operations_Otake__bit__num,type,
    bit_take_bit_num: ( nat * num ) > option(num) ).

tff(sy_c_Bit__Shifts__Infix__Syntax_Osemiring__bit__operations__class_Oshiftl,type,
    bit_Sh4282982442137083160shiftl: 
      !>[A: $tType] : ( ( A * nat ) > A ) ).

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

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

tff(sy_c_Bits__Integer_OBit__integer,type,
    bits_Bit_integer: ( code_integer * bool ) > code_integer ).

tff(sy_c_Bits__Integer_Obin__last__integer,type,
    bits_b8758750999018896077nteger: code_integer > bool ).

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

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

tff(sy_c_COMBB,type,
    combb: 
      !>[B: $tType,C: $tType,A: $tType] : ( ( fun(B,C) * fun(A,B) ) > fun(A,C) ) ).

tff(sy_c_COMBC,type,
    combc: 
      !>[A: $tType,B: $tType,C: $tType] : ( ( fun(A,fun(B,C)) * B ) > fun(A,C) ) ).

tff(sy_c_COMBS,type,
    combs: 
      !>[A: $tType,B: $tType,C: $tType] : ( ( fun(A,fun(B,C)) * fun(A,B) ) > fun(A,C) ) ).

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

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

tff(sy_c_Code__Target__Int_Onegative,type,
    code_Target_negative: fun(num,int) ).

tff(sy_c_Code__Target__Nat_Oint__of__nat,type,
    code_T6385005292777649522of_nat: fun(nat,int) ).

tff(sy_c_Code__Target__Word__Base_Oquickcheck__narrowing__samples_Onarrowing__samples,type,
    code_T4080844693773952564amples: 
      !>[A: $tType] : ( ( fun(code_integer,product_prod(A,A)) * A * code_integer ) > list(A) ) ).

tff(sy_c_Code__Target__Word__Base_Oset__bits__aux,type,
    code_T2661198915054445665ts_aux: 
      !>[A: $tType] : ( ( fun(nat,bool) * nat * word(A) ) > word(A) ) ).

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

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

tff(sy_c_Complex_Ocomplex_OComplex,type,
    complex2: ( real * real ) > complex ).

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

tff(sy_c_Complex_Oimaginary__unit,type,
    imaginary_unit: complex ).

tff(sy_c_Deriv_Odifferentiable,type,
    differentiable: 
      !>[A: $tType,B: $tType] : ( ( fun(A,B) * filter(A) ) > $o ) ).

tff(sy_c_Deriv_Ohas__derivative,type,
    has_derivative: 
      !>[A: $tType,B: $tType] : ( ( fun(A,B) * fun(A,B) * filter(A) ) > $o ) ).

tff(sy_c_Deriv_Ohas__field__derivative,type,
    has_field_derivative: 
      !>[A: $tType] : ( ( fun(A,A) * A * filter(A) ) > $o ) ).

tff(sy_c_Divides_Oadjust__div,type,
    adjust_div: product_prod(int,int) > int ).

tff(sy_c_Divides_Odivmod__nat,type,
    divmod_nat: ( nat * nat ) > product_prod(nat,nat) ).

tff(sy_c_Divides_Oeucl__rel__int,type,
    eucl_rel_int: ( int * int * product_prod(int,int) ) > $o ).

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

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

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

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

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

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

tff(sy_c_Filter_Oat__bot,type,
    at_bot: 
      !>[A: $tType] : filter(A) ).

tff(sy_c_Filter_Oat__top,type,
    at_top: 
      !>[A: $tType] : filter(A) ).

tff(sy_c_Filter_Oeventually,type,
    eventually: 
      !>[A: $tType] : ( ( fun(A,bool) * filter(A) ) > $o ) ).

tff(sy_c_Filter_Ofilterlim,type,
    filterlim: 
      !>[A: $tType,B: $tType] : ( ( fun(A,B) * filter(B) * filter(A) ) > $o ) ).

tff(sy_c_Filter_Omap__filter__on,type,
    map_filter_on: 
      !>[A: $tType,B: $tType] : ( ( set(A) * fun(A,B) * filter(A) ) > filter(B) ) ).

tff(sy_c_Finite__Set_Ofinite,type,
    finite_finite2: 
      !>[A: $tType] : fun(set(A),bool) ).

tff(sy_c_Fun_Obij__betw,type,
    bij_betw: 
      !>[A: $tType,B: $tType] : ( ( fun(A,B) * set(A) * set(B) ) > $o ) ).

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

tff(sy_c_Fun_Oinj__on,type,
    inj_on: 
      !>[A: $tType,B: $tType] : ( ( fun(A,B) * set(A) ) > $o ) ).

tff(sy_c_Fun_Othe__inv__into,type,
    the_inv_into: 
      !>[A: $tType,B: $tType] : ( ( set(A) * fun(A,B) * B ) > A ) ).

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

tff(sy_c_Generic__set__bit_Oset__bit__class_Oset__bit,type,
    generi7602027413899671122et_bit: 
      !>[A: $tType] : ( ( A * nat * bool ) > A ) ).

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

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

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

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

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

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

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

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

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

tff(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_H,type,
    groups1027152243600224163dd_sum: 
      !>[C: $tType,A: $tType] : ( ( fun(C,A) * set(C) ) > A ) ).

tff(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod,type,
    groups7121269368397514597t_prod: 
      !>[B: $tType,A: $tType] : ( ( fun(B,A) * set(B) ) > A ) ).

tff(sy_c_Groups__List_Ocomm__semiring__0__class_Ohorner__sum,type,
    groups4207007520872428315er_sum: 
      !>[B: $tType,A: $tType] : ( ( fun(B,A) * A * list(B) ) > A ) ).

tff(sy_c_HOL_ONO__MATCH,type,
    nO_MATCH: 
      !>[A: $tType,B: $tType] : ( ( A * B ) > $o ) ).

tff(sy_c_HOL_OThe,type,
    the: 
      !>[A: $tType] : ( fun(A,bool) > A ) ).

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

tff(sy_c_Hash__Instances_Ohash__code__list,type,
    hash_hash_code_list: 
      !>[A: $tType] : ( ( fun(A,uint32) * list(A) ) > uint32 ) ).

tff(sy_c_Hash__Instances_Ohash__code__option,type,
    hash_h1887023736457453652option: 
      !>[A: $tType] : ( ( fun(A,uint32) * option(A) ) > uint32 ) ).

tff(sy_c_Hash__Instances_Ohash__code__prod,type,
    hash_hash_code_prod: 
      !>[A: $tType,B: $tType] : ( ( fun(A,uint32) * fun(B,uint32) * product_prod(A,B) ) > uint32 ) ).

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

tff(sy_c_If,type,
    if: 
      !>[A: $tType] : ( ( bool * A * A ) > A ) ).

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

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

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

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

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

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

tff(sy_c_Least__significant__bit_Olsb__class_Olsb,type,
    least_8051144512741203767sb_lsb: 
      !>[A: $tType] : fun(A,bool) ).

tff(sy_c_Limits_OBfun,type,
    bfun: 
      !>[A: $tType,B: $tType] : ( ( fun(A,B) * filter(A) ) > $o ) ).

tff(sy_c_Limits_Oat__infinity,type,
    at_infinity: 
      !>[A: $tType] : filter(A) ).

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

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

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

tff(sy_c_List_Ofilter,type,
    filter2: 
      !>[A: $tType] : ( ( fun(A,bool) * list(A) ) > list(A) ) ).

tff(sy_c_List_Ofoldr,type,
    foldr: 
      !>[A: $tType,B: $tType] : ( ( fun(A,fun(B,B)) * list(A) * B ) > B ) ).

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

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

tff(sy_c_List_Olinorder__class_Osort__key,type,
    linorder_sort_key: 
      !>[B: $tType,A: $tType] : ( ( fun(B,A) * list(B) ) > list(B) ) ).

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

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

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

tff(sy_c_List_Olist_Ocase__list,type,
    case_list: 
      !>[B: $tType,A: $tType] : ( ( B * fun(A,fun(list(A),B)) ) > fun(list(A),B) ) ).

tff(sy_c_List_Olist_Omap,type,
    map: 
      !>[A: $tType,Aa: $tType] : ( ( fun(A,Aa) * list(A) ) > list(Aa) ) ).

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

tff(sy_c_List_Olist_Osize__list,type,
    size_list: 
      !>[A: $tType] : ( ( fun(A,nat) * list(A) ) > nat ) ).

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

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

tff(sy_c_List_Omin__list__rel,type,
    min_list_rel: 
      !>[A: $tType] : fun(list(A),fun(list(A),bool)) ).

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

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

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

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

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

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

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

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

tff(sy_c_List_Otranspose__rel,type,
    transpose_rel: 
      !>[A: $tType] : fun(list(list(A)),fun(list(list(A)),bool)) ).

tff(sy_c_List_Oupt,type,
    upt: ( nat * nat ) > list(nat) ).

tff(sy_c_List_Oupto,type,
    upto: ( int * int ) > list(int) ).

tff(sy_c_List_Oupto__aux,type,
    upto_aux: ( int * int * list(int) ) > list(int) ).

tff(sy_c_List_Oupto__rel,type,
    upto_rel: fun(product_prod(int,int),fun(product_prod(int,int),bool)) ).

tff(sy_c_Map_Oran,type,
    ran: 
      !>[A: $tType,B: $tType] : ( fun(A,option(B)) > set(B) ) ).

tff(sy_c_Misc_OEps__Opt,type,
    eps_Opt: 
      !>[A: $tType] : ( fun(A,bool) > option(A) ) ).

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

tff(sy_c_Misc_Oinv__on,type,
    inv_on: 
      !>[A: $tType,B: $tType] : ( ( fun(A,B) * set(A) ) > fun(B,A) ) ).

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

tff(sy_c_Misc_Orel__of,type,
    rel_of: 
      !>[A: $tType,B: $tType] : ( ( fun(A,option(B)) * fun(product_prod(A,B),bool) ) > set(product_prod(A,B)) ) ).

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

tff(sy_c_Most__significant__bit_Omsb__class_Omsb,type,
    most_s684356279273892711sb_msb: 
      !>[A: $tType] : ( A > bool ) ).

tff(sy_c_Multiset_Ois__empty,type,
    is_empty: 
      !>[A: $tType] : ( multiset(A) > $o ) ).

tff(sy_c_Nat_OSuc,type,
    suc: fun(nat,nat) ).

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

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

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

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

tff(sy_c_Nat__Bijection_Oset__decode,type,
    nat_set_decode: nat > set(nat) ).

tff(sy_c_Nat__Bijection_Oset__encode,type,
    nat_set_encode: fun(set(nat),nat) ).

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

tff(sy_c_NthRoot_Oroot,type,
    root: nat > fun(real,real) ).

tff(sy_c_NthRoot_Osqrt,type,
    sqrt: fun(real,real) ).

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

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

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

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

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

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

tff(sy_c_Num_Onum_OBit1,type,
    bit1: fun(num,num) ).

tff(sy_c_Num_Onum_OOne,type,
    one2: num ).

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

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

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

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

tff(sy_c_Num_Opow,type,
    pow: ( num * num ) > num ).

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

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

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

tff(sy_c_Option_Ooption_Ocase__option,type,
    case_option: 
      !>[B: $tType,A: $tType] : ( ( B * fun(A,B) * option(A) ) > B ) ).

tff(sy_c_Option_Ooption_Osize__option,type,
    size_option: 
      !>[A: $tType] : ( ( fun(A,nat) * option(A) ) > nat ) ).

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

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

tff(sy_c_Orderings_Oord__class_OLeast,type,
    ord_Least: 
      !>[A: $tType] : ( fun(A,bool) > A ) ).

tff(sy_c_Orderings_Oord__class_Oless,type,
    ord_less: 
      !>[A: $tType] : ( A > fun(A,bool) ) ).

tff(sy_c_Orderings_Oord__class_Oless__eq,type,
    ord_less_eq: 
      !>[A: $tType] : ( A > fun(A,bool) ) ).

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

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

tff(sy_c_Orderings_Oorder__class_OGreatest,type,
    order_Greatest: 
      !>[A: $tType] : ( fun(A,bool) > A ) ).

tff(sy_c_Orderings_Oorder__class_Oantimono,type,
    order_antimono: 
      !>[A: $tType,B: $tType] : ( fun(A,B) > $o ) ).

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

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

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

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

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

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

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

tff(sy_c_Rat_Onormalize,type,
    normalize: product_prod(int,int) > product_prod(int,int) ).

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

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

tff(sy_c_Real__Vector__Spaces_Obounded__linear,type,
    real_V3181309239436604168linear: 
      !>[A: $tType,B: $tType] : ( fun(A,B) > $o ) ).

tff(sy_c_Real__Vector__Spaces_Odist__class_Odist,type,
    real_V557655796197034286t_dist: 
      !>[A: $tType] : ( ( A * A ) > real ) ).

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

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

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

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

tff(sy_c_Rings_Odvd__class_Odvd,type,
    dvd_dvd: 
      !>[A: $tType] : ( A > fun(A,bool) ) ).

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

tff(sy_c_Rings_Ozero__neq__one__class_Oof__bool,type,
    zero_neq_one_of_bool: 
      !>[A: $tType] : fun(bool,A) ).

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

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

tff(sy_c_Series_Osums,type,
    sums: 
      !>[A: $tType] : ( fun(nat,A) > fun(A,bool) ) ).

tff(sy_c_Set_OCollect,type,
    collect: 
      !>[A: $tType] : ( fun(A,bool) > set(A) ) ).

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

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

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

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

tff(sy_c_Set__Interval_Oord__class_OatLeast,type,
    set_ord_atLeast: 
      !>[A: $tType] : ( A > set(A) ) ).

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

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

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

tff(sy_c_Set__Interval_Oord__class_OgreaterThan,type,
    set_ord_greaterThan: 
      !>[A: $tType] : ( A > set(A) ) ).

tff(sy_c_Set__Interval_Oord__class_OgreaterThanAtMost,type,
    set_or3652927894154168847AtMost: 
      !>[A: $tType] : ( ( A * A ) > set(A) ) ).

tff(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan,type,
    set_or5935395276787703475ssThan: 
      !>[A: $tType] : ( ( A * A ) > set(A) ) ).

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

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

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

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

tff(sy_c_String_Ochar_OChar,type,
    char2: ( bool * bool * bool * bool * bool * bool * bool * bool ) > char ).

tff(sy_c_String_Ochar_Osize__char,type,
    size_char: char > nat ).

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

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

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

tff(sy_c_Topological__Spaces_Ocontinuous,type,
    topolo3448309680560233919inuous: 
      !>[A: $tType,B: $tType] : ( ( filter(A) * fun(A,B) ) > $o ) ).

tff(sy_c_Topological__Spaces_Ocontinuous__on,type,
    topolo81223032696312382ous_on: 
      !>[A: $tType,B: $tType] : ( ( set(A) * fun(A,B) ) > $o ) ).

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

tff(sy_c_Topological__Spaces_Oopen__class_Oopen,type,
    topolo1002775350975398744n_open: 
      !>[A: $tType] : ( set(A) > $o ) ).

tff(sy_c_Topological__Spaces_Ot2__space__class_OLim,type,
    topolo3827282254853284352ce_Lim: 
      !>[F: $tType,A: $tType] : ( ( filter(F) * fun(F,A) ) > A ) ).

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

tff(sy_c_Topological__Spaces_Otopological__space__class_Onhds,type,
    topolo7230453075368039082e_nhds: 
      !>[A: $tType] : ( A > filter(A) ) ).

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

tff(sy_c_Topological__Spaces_Ouniform__space__class_Ocauchy__filter,type,
    topolo6773858410816713723filter: 
      !>[A: $tType] : ( filter(A) > $o ) ).

tff(sy_c_Transcendental_Oarccos,type,
    arccos: fun(real,real) ).

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

tff(sy_c_Transcendental_Oarcsin,type,
    arcsin: fun(real,real) ).

tff(sy_c_Transcendental_Oarctan,type,
    arctan: fun(real,real) ).

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

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

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

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

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

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

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

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

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

tff(sy_c_Transcendental_Olog,type,
    log: real > fun(real,real) ).

tff(sy_c_Transcendental_Opi,type,
    pi: real ).

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

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

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

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

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

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

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

tff(sy_c_Uint32_OUint32,type,
    uint322: fun(code_integer,uint32) ).

tff(sy_c_Uint32_OUint32__signed,type,
    uint32_signed: code_integer > uint32 ).

tff(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 ).

tff(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 ).

tff(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: fun(product_prod(vEBT_VEBT,nat),fun(product_prod(vEBT_VEBT,nat),bool)) ).

tff(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: fun(product_prod(vEBT_VEBT,nat),fun(product_prod(vEBT_VEBT,nat),bool)) ).

tff(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 ).

tff(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: fun(vEBT_VEBT,fun(vEBT_VEBT,bool)) ).

tff(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 ).

tff(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 ).

tff(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: fun(product_prod(vEBT_VEBT,nat),fun(product_prod(vEBT_VEBT,nat),bool)) ).

tff(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: fun(product_prod(vEBT_VEBT,nat),fun(product_prod(vEBT_VEBT,nat),bool)) ).

tff(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 ).

tff(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: fun(vEBT_VEBT,fun(vEBT_VEBT,bool)) ).

tff(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 ).

tff(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: fun(vEBT_VEBT,fun(vEBT_VEBT,bool)) ).

tff(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 ).

tff(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 ).

tff(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: fun(product_prod(vEBT_VEBT,nat),fun(product_prod(vEBT_VEBT,nat),bool)) ).

tff(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: fun(product_prod(vEBT_VEBT,nat),fun(product_prod(vEBT_VEBT,nat),bool)) ).

tff(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 ).

tff(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 ).

tff(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: fun(product_prod(vEBT_VEBT,nat),fun(product_prod(vEBT_VEBT,nat),bool)) ).

tff(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: fun(product_prod(vEBT_VEBT,nat),fun(product_prod(vEBT_VEBT,nat),bool)) ).

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

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

tff(sy_c_VEBT__BuildupMemImp_OVEBT__internal_OT__vebt__buildupi_H__rel,type,
    vEBT_V3352910403632780892pi_rel: fun(nat,fun(nat,bool)) ).

tff(sy_c_VEBT__BuildupMemImp_OVEBT__internal_OT__vebt__buildupi__rel,type,
    vEBT_V2957053500504383685pi_rel: fun(nat,fun(nat,bool)) ).

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

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

tff(sy_c_VEBT__BuildupMemImp_OVEBT__internal_OTb_H__rel,type,
    vEBT_VEBT_Tb_rel: fun(nat,fun(nat,bool)) ).

tff(sy_c_VEBT__BuildupMemImp_OVEBT__internal_OTb__rel,type,
    vEBT_VEBT_Tb_rel2: fun(nat,fun(nat,bool)) ).

tff(sy_c_VEBT__BuildupMemImp_OVEBT__internal_Ohighi,type,
    vEBT_VEBT_highi: ( nat * nat ) > heap_Time_Heap(nat) ).

tff(sy_c_VEBT__BuildupMemImp_OVEBT__internal_Olowi,type,
    vEBT_VEBT_lowi: ( nat * nat ) > heap_Time_Heap(nat) ).

tff(sy_c_VEBT__Definitions_OVEBT_OLeaf,type,
    vEBT_Leaf: ( bool * bool ) > vEBT_VEBT ).

tff(sy_c_VEBT__Definitions_OVEBT_ONode,type,
    vEBT_Node: ( option(product_prod(nat,nat)) * nat * list(vEBT_VEBT) * vEBT_VEBT ) > vEBT_VEBT ).

tff(sy_c_VEBT__Definitions_OVEBT_Osize__VEBT,type,
    vEBT_size_VEBT: fun(vEBT_VEBT,nat) ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Oboth__member__options,type,
    vEBT_V8194947554948674370ptions: fun(vEBT_VEBT,fun(nat,bool)) ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Ohigh,type,
    vEBT_VEBT_high: ( nat * nat ) > nat ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Oin__children,type,
    vEBT_V5917875025757280293ildren: ( nat * list(vEBT_VEBT) * nat ) > $o ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Olow,type,
    vEBT_VEBT_low: ( nat * nat ) > nat ).

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

tff(sy_c_VEBT__Definitions_OVEBT__internal_Omembermima__rel,type,
    vEBT_V4351362008482014158ma_rel: fun(product_prod(vEBT_VEBT,nat),fun(product_prod(vEBT_VEBT,nat),bool)) ).

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

tff(sy_c_VEBT__Definitions_OVEBT__internal_Onaive__member__rel,type,
    vEBT_V5765760719290551771er_rel: fun(product_prod(vEBT_VEBT,nat),fun(product_prod(vEBT_VEBT,nat),bool)) ).

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

tff(sy_c_VEBT__Definitions_OVEBT__internal_Ovalid_H__rel,type,
    vEBT_VEBT_valid_rel: fun(product_prod(vEBT_VEBT,nat),fun(product_prod(vEBT_VEBT,nat),bool)) ).

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

tff(sy_c_VEBT__Definitions_Oset__vebt,type,
    vEBT_set_vebt: vEBT_VEBT > set(nat) ).

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

tff(sy_c_VEBT__Definitions_Ovebt__buildup__rel,type,
    vEBT_v4011308405150292612up_rel: fun(nat,fun(nat,bool)) ).

tff(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 ).

tff(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: fun(product_prod(vEBT_VEBT,nat),fun(product_prod(vEBT_VEBT,nat),bool)) ).

tff(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 ).

tff(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: fun(product_prod(vEBT_VEBT,nat),fun(product_prod(vEBT_VEBT,nat),bool)) ).

tff(sy_c_VEBT__Delete_Ovebt__delete,type,
    vEBT_vebt_delete: ( vEBT_VEBT * nat ) > vEBT_VEBT ).

tff(sy_c_VEBT__Delete_Ovebt__delete__rel,type,
    vEBT_vebt_delete_rel: fun(product_prod(vEBT_VEBT,nat),fun(product_prod(vEBT_VEBT,nat),bool)) ).

tff(sy_c_VEBT__Height_OVEBT__internal_Oheight,type,
    vEBT_VEBT_height: fun(vEBT_VEBT,nat) ).

tff(sy_c_VEBT__Height_OVEBT__internal_Oheight__rel,type,
    vEBT_VEBT_height_rel: fun(vEBT_VEBT,fun(vEBT_VEBT,bool)) ).

tff(sy_c_VEBT__Insert_Ovebt__insert,type,
    vEBT_vebt_insert: ( vEBT_VEBT * nat ) > vEBT_VEBT ).

tff(sy_c_VEBT__Insert_Ovebt__insert__rel,type,
    vEBT_vebt_insert_rel: fun(product_prod(vEBT_VEBT,nat),fun(product_prod(vEBT_VEBT,nat),bool)) ).

tff(sy_c_VEBT__Member_OVEBT__internal_Obit__concat,type,
    vEBT_VEBT_bit_concat: ( nat * nat * nat ) > nat ).

tff(sy_c_VEBT__Member_OVEBT__internal_OminNull,type,
    vEBT_VEBT_minNull: vEBT_VEBT > bool ).

tff(sy_c_VEBT__Member_OVEBT__internal_OminNull__rel,type,
    vEBT_V6963167321098673237ll_rel: fun(vEBT_VEBT,fun(vEBT_VEBT,bool)) ).

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

tff(sy_c_VEBT__Member_Ovebt__member,type,
    vEBT_vebt_member: vEBT_VEBT > fun(nat,bool) ).

tff(sy_c_VEBT__Member_Ovebt__member__rel,type,
    vEBT_vebt_member_rel: fun(product_prod(vEBT_VEBT,nat),fun(product_prod(vEBT_VEBT,nat),bool)) ).

tff(sy_c_VEBT__MinMax_OVEBT__internal_Oadd,type,
    vEBT_VEBT_add: fun(option(nat),fun(option(nat),option(nat))) ).

tff(sy_c_VEBT__MinMax_OVEBT__internal_Ogreater,type,
    vEBT_VEBT_greater: ( option(nat) * option(nat) ) > bool ).

tff(sy_c_VEBT__MinMax_OVEBT__internal_Oless,type,
    vEBT_VEBT_less: ( option(nat) * option(nat) ) > bool ).

tff(sy_c_VEBT__MinMax_OVEBT__internal_Olesseq,type,
    vEBT_VEBT_lesseq: ( option(nat) * option(nat) ) > $o ).

tff(sy_c_VEBT__MinMax_OVEBT__internal_Omax__in__set,type,
    vEBT_VEBT_max_in_set: ( set(nat) * nat ) > $o ).

tff(sy_c_VEBT__MinMax_OVEBT__internal_Omin__in__set,type,
    vEBT_VEBT_min_in_set: ( set(nat) * nat ) > $o ).

tff(sy_c_VEBT__MinMax_OVEBT__internal_Omul,type,
    vEBT_VEBT_mul: fun(option(nat),fun(option(nat),option(nat))) ).

tff(sy_c_VEBT__MinMax_OVEBT__internal_Ooption__shift,type,
    vEBT_V2048590022279873568_shift: 
      !>[A: $tType] : ( fun(A,fun(A,A)) > fun(option(A),fun(option(A),option(A))) ) ).

tff(sy_c_VEBT__MinMax_OVEBT__internal_Opower,type,
    vEBT_VEBT_power: fun(option(nat),fun(option(nat),option(nat))) ).

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

tff(sy_c_VEBT__MinMax_Ovebt__maxt__rel,type,
    vEBT_vebt_maxt_rel: fun(vEBT_VEBT,fun(vEBT_VEBT,bool)) ).

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

tff(sy_c_VEBT__MinMax_Ovebt__mint__rel,type,
    vEBT_vebt_mint_rel: fun(vEBT_VEBT,fun(vEBT_VEBT,bool)) ).

tff(sy_c_VEBT__Pred_Ois__pred__in__set,type,
    vEBT_is_pred_in_set: ( set(nat) * nat * nat ) > $o ).

tff(sy_c_VEBT__Pred_Ovebt__pred,type,
    vEBT_vebt_pred: ( vEBT_VEBT * nat ) > option(nat) ).

tff(sy_c_VEBT__Pred_Ovebt__pred__rel,type,
    vEBT_vebt_pred_rel: fun(product_prod(vEBT_VEBT,nat),fun(product_prod(vEBT_VEBT,nat),bool)) ).

tff(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 ).

tff(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 ).

tff(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: fun(nat,fun(nat,bool)) ).

tff(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: fun(nat,fun(nat,bool)) ).

tff(sy_c_VEBT__Space_OVEBT__internal_Ocnt,type,
    vEBT_VEBT_cnt: fun(vEBT_VEBT,real) ).

tff(sy_c_VEBT__Space_OVEBT__internal_Ocnt_H,type,
    vEBT_VEBT_cnt2: fun(vEBT_VEBT,nat) ).

tff(sy_c_VEBT__Space_OVEBT__internal_Ocnt_H__rel,type,
    vEBT_VEBT_cnt_rel: fun(vEBT_VEBT,fun(vEBT_VEBT,bool)) ).

tff(sy_c_VEBT__Space_OVEBT__internal_Ocnt__rel,type,
    vEBT_VEBT_cnt_rel2: fun(vEBT_VEBT,fun(vEBT_VEBT,bool)) ).

tff(sy_c_VEBT__Space_OVEBT__internal_Ospace,type,
    vEBT_VEBT_space: fun(vEBT_VEBT,nat) ).

tff(sy_c_VEBT__Space_OVEBT__internal_Ospace_H,type,
    vEBT_VEBT_space2: fun(vEBT_VEBT,nat) ).

tff(sy_c_VEBT__Space_OVEBT__internal_Ospace_H__rel,type,
    vEBT_VEBT_space_rel: fun(vEBT_VEBT,fun(vEBT_VEBT,bool)) ).

tff(sy_c_VEBT__Space_OVEBT__internal_Ospace__rel,type,
    vEBT_VEBT_space_rel2: fun(vEBT_VEBT,fun(vEBT_VEBT,bool)) ).

tff(sy_c_VEBT__Succ_Ois__succ__in__set,type,
    vEBT_is_succ_in_set: ( set(nat) * nat * nat ) > $o ).

tff(sy_c_VEBT__Succ_Ovebt__succ,type,
    vEBT_vebt_succ: ( vEBT_VEBT * nat ) > option(nat) ).

tff(sy_c_VEBT__Succ_Ovebt__succ__rel,type,
    vEBT_vebt_succ_rel: fun(product_prod(vEBT_VEBT,nat),fun(product_prod(vEBT_VEBT,nat),bool)) ).

tff(sy_c_Wellfounded_Oaccp,type,
    accp: 
      !>[A: $tType] : ( ( fun(A,fun(A,bool)) * A ) > $o ) ).

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

tff(sy_c_Word_Oeven__word,type,
    even_word: 
      !>[A: $tType] : fun(word(A),bool) ).

tff(sy_c_Word_Oof__nat,type,
    of_nat: 
      !>[A: $tType] : ( nat > word(A) ) ).

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

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

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

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

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

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

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

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

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

tff(sy_c_Word_Oword__int__case,type,
    word_int_case: 
      !>[B: $tType,A: $tType] : ( ( fun(int,B) * word(A) ) > B ) ).

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

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

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

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

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

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

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

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

tff(sy_c_aa,type,
    aa: 
      !>[A: $tType,B: $tType] : ( ( fun(A,B) * A ) > B ) ).

tff(sy_c_fAll,type,
    fAll: 
      !>[A: $tType] : ( fun(A,bool) > bool ) ).

tff(sy_c_fChoice,type,
    fChoice: 
      !>[A: $tType] : ( fun(A,bool) > A ) ).

tff(sy_c_fEx,type,
    fEx: 
      !>[A: $tType] : fun(fun(A,bool),bool) ).

tff(sy_c_fFalse,type,
    fFalse: bool ).

tff(sy_c_fNot,type,
    fNot: fun(bool,bool) ).

tff(sy_c_fTrue,type,
    fTrue: bool ).

tff(sy_c_fconj,type,
    fconj: ( bool * bool ) > bool ).

tff(sy_c_fdisj,type,
    fdisj: ( bool * bool ) > bool ).

tff(sy_c_fequal,type,
    fequal: 
      !>[A: $tType] : ( A > fun(A,bool) ) ).

tff(sy_c_fimplies,type,
    fimplies: fun(bool,fun(bool,bool)) ).

tff(sy_c_member,type,
    member: 
      !>[A: $tType] : fun(A,fun(set(A),bool)) ).

tff(sy_c_pp,type,
    pp: bool > $o ).

tff(sy_v_na____,type,
    na: nat ).

% Relevant facts (9206)
tff(fact_0_False,axiom,
    ~ pp(aa(nat,bool,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),na)) ).

% False
tff(fact_1_max__in__set__def,axiom,
    ! [Xs: set(nat),X: nat] :
      ( vEBT_VEBT_max_in_set(Xs,X)
    <=> ( pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),X),Xs))
        & ! [X2: nat] :
            ( pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),X2),Xs))
           => pp(aa(nat,bool,ord_less_eq(nat,X2),X)) ) ) ) ).

% max_in_set_def
tff(fact_2_min__in__set__def,axiom,
    ! [Xs: set(nat),X: nat] :
      ( vEBT_VEBT_min_in_set(Xs,X)
    <=> ( pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),X),Xs))
        & ! [X2: nat] :
            ( pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),X2),Xs))
           => pp(aa(nat,bool,ord_less_eq(nat,X),X2)) ) ) ) ).

% min_in_set_def
tff(fact_3__C3_OIH_C_I4_J,axiom,
    ! [X: nat] :
      ( ~ pp(aa(nat,bool,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,aa(nat,nat,suc,na))))
     => ( ( X = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,na))),aa(num,nat,numeral_numeral(nat),bit0(one2))) )
       => pp(aa(nat,bool,ord_less_eq(nat,vEBT_VEBT_Tb2(aa(nat,nat,suc,X))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,bit0(one2)))),aa(vEBT_VEBT,nat,vEBT_VEBT_cnt2,vEBT_vebt_buildup(aa(nat,nat,suc,X)))))) ) ) ).

% "3.IH"(4)
tff(fact_4__C3_OIH_C_I3_J,axiom,
    ! [X: nat] :
      ( ~ pp(aa(nat,bool,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,aa(nat,nat,suc,na))))
     => ( ( X = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,na))),aa(num,nat,numeral_numeral(nat),bit0(one2))) )
       => pp(aa(nat,bool,ord_less_eq(nat,vEBT_VEBT_Tb2(X)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,bit0(one2)))),aa(vEBT_VEBT,nat,vEBT_VEBT_cnt2,vEBT_vebt_buildup(X))))) ) ) ).

% "3.IH"(3)
tff(fact_5__C3_OIH_C_I1_J,axiom,
    ! [X: nat] :
      ( pp(aa(nat,bool,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,aa(nat,nat,suc,na))))
     => ( ( X = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,na))),aa(num,nat,numeral_numeral(nat),bit0(one2))) )
       => pp(aa(nat,bool,ord_less_eq(nat,vEBT_VEBT_Tb2(X)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,bit0(one2)))),aa(vEBT_VEBT,nat,vEBT_VEBT_cnt2,vEBT_vebt_buildup(X))))) ) ) ).

% "3.IH"(1)
tff(fact_6_vebt__buildup__bound,axiom,
    ! [U: nat,N: nat] :
      ( ( U = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N) )
     => pp(aa(nat,bool,ord_less_eq(nat,vEBT_V8346862874174094_d_u_p(N)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(aa(num,num,bit1,bit0(aa(num,num,bit1,one2)))))),U))) ) ).

% vebt_buildup_bound
tff(fact_7_odd__Suc__div__two,axiom,
    ! [N: nat] :
      ( ~ pp(aa(nat,bool,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N))
     => ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,N)),aa(num,nat,numeral_numeral(nat),bit0(one2))) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ) ) ).

% odd_Suc_div_two
tff(fact_8_even__Suc__div__two,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N))
     => ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,N)),aa(num,nat,numeral_numeral(nat),bit0(one2))) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2))) ) ) ).

% even_Suc_div_two
tff(fact_9__C0_C,axiom,
    pp(aa(nat,bool,ord_less_eq(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,bit0(one2)))),vEBT_VEBT_Tb2(aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),na),aa(num,nat,numeral_numeral(nat),bit0(one2)))))))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),vEBT_VEBT_Tb2(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),na),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),na),aa(num,nat,numeral_numeral(nat),bit0(one2))))))))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,bit0(one2)))),aa(vEBT_VEBT,nat,vEBT_VEBT_cnt2,vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,na)),replicate(vEBT_VEBT,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,na))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_vebt_buildup(aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,na))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_vebt_buildup(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,na))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))))))) ).

% "0"
tff(fact_10_even__Suc,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,N)))
    <=> ~ pp(aa(nat,bool,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N)) ) ).

% even_Suc
tff(fact_11_even__Suc__Suc__iff,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,aa(nat,nat,suc,N))))
    <=> pp(aa(nat,bool,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N)) ) ).

% even_Suc_Suc_iff
tff(fact_12_div2__Suc__Suc,axiom,
    ! [M: nat] : aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,M))),aa(num,nat,numeral_numeral(nat),bit0(one2))) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ).

% div2_Suc_Suc
tff(fact_13_vebt__buildup_Osimps_I3_J,axiom,
    ! [Va: nat] :
      ( ( pp(aa(nat,bool,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,aa(nat,nat,suc,Va))))
       => ( vEBT_vebt_buildup(aa(nat,nat,suc,aa(nat,nat,suc,Va))) = vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),replicate(vEBT_VEBT,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),vEBT_vebt_buildup(aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_vebt_buildup(aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) )
      & ( ~ pp(aa(nat,bool,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,aa(nat,nat,suc,Va))))
       => ( vEBT_vebt_buildup(aa(nat,nat,suc,aa(nat,nat,suc,Va))) = vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),replicate(vEBT_VEBT,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_vebt_buildup(aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_vebt_buildup(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))) ) ) ) ).

% vebt_buildup.simps(3)
tff(fact_14_even__mult__iff,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)))
        <=> ( pp(aa(A,bool,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A2))
            | pp(aa(A,bool,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),B2)) ) ) ) ).

% even_mult_iff
tff(fact_15_divide__le__eq__numeral1_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,W: num,A2: A] :
          ( pp(aa(A,bool,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),aa(num,A,numeral_numeral(A),W))),A2))
        <=> pp(aa(A,bool,ord_less_eq(A,B2),aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(num,A,numeral_numeral(A),W)))) ) ) ).

% divide_le_eq_numeral1(1)
tff(fact_16_le__divide__eq__numeral1_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A,W: num] :
          ( pp(aa(A,bool,ord_less_eq(A,A2),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),aa(num,A,numeral_numeral(A),W))))
        <=> pp(aa(A,bool,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(num,A,numeral_numeral(A),W))),B2)) ) ) ).

% le_divide_eq_numeral1(1)
tff(fact_17_dvd__div__mult__self,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,dvd_dvd(A,A2),B2))
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A2)),A2) = B2 ) ) ) ).

% dvd_div_mult_self
tff(fact_18_even__odd__cases,axiom,
    ! [X: nat] :
      ( ! [N2: nat] : X != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N2),N2)
     => ~ ! [N2: nat] : X != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N2),aa(nat,nat,suc,N2)) ) ).

% even_odd_cases
tff(fact_19_numeral__eq__iff,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [M: num,N: num] :
          ( ( aa(num,A,numeral_numeral(A),M) = aa(num,A,numeral_numeral(A),N) )
        <=> ( M = N ) ) ) ).

% numeral_eq_iff
tff(fact_20_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)
tff(fact_21_pow__sum,axiom,
    ! [A2: nat,B2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A2),B2))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),A2)) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),B2) ).

% pow_sum
tff(fact_22_numeral__le__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [M: num,N: num] :
          ( pp(aa(A,bool,ord_less_eq(A,aa(num,A,numeral_numeral(A),M)),aa(num,A,numeral_numeral(A),N)))
        <=> pp(aa(num,bool,ord_less_eq(num,M),N)) ) ) ).

% numeral_le_iff
tff(fact_23_numeral__plus__numeral,axiom,
    ! [A: $tType] :
      ( numeral(A)
     => ! [M: num,N: num] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),M)),aa(num,A,numeral_numeral(A),N)) = aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),plus_plus(num),M),N)) ) ).

% numeral_plus_numeral
tff(fact_24_add__numeral__left,axiom,
    ! [A: $tType] :
      ( numeral(A)
     => ! [V: num,W: num,Z: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),V)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),W)),Z)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),plus_plus(num),V),W))),Z) ) ).

% add_numeral_left
tff(fact_25_mult__numeral__left__semiring__numeral,axiom,
    ! [A: $tType] :
      ( semiring_numeral(A)
     => ! [V: num,W: num,Z: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),V)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W)),Z)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),times_times(num),V),W))),Z) ) ).

% mult_numeral_left_semiring_numeral
tff(fact_26_numeral__times__numeral,axiom,
    ! [A: $tType] :
      ( semiring_numeral(A)
     => ! [M: num,N: num] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),M)),aa(num,A,numeral_numeral(A),N)) = aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),times_times(num),M),N)) ) ).

% numeral_times_numeral
tff(fact_27_num__double,axiom,
    ! [N: num] : aa(num,num,aa(num,fun(num,num),times_times(num),bit0(one2)),N) = bit0(N) ).

% num_double
tff(fact_28_dvd__add__triv__right__iff,axiom,
    ! [A: $tType] :
      ( comm_s4317794764714335236cancel(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,dvd_dvd(A,A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A2)))
        <=> pp(aa(A,bool,dvd_dvd(A,A2),B2)) ) ) ).

% dvd_add_triv_right_iff
tff(fact_29_dvd__add__triv__left__iff,axiom,
    ! [A: $tType] :
      ( comm_s4317794764714335236cancel(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,dvd_dvd(A,A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)))
        <=> pp(aa(A,bool,dvd_dvd(A,A2),B2)) ) ) ).

% dvd_add_triv_left_iff
tff(fact_30_div__dvd__div,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,dvd_dvd(A,A2),B2))
         => ( pp(aa(A,bool,dvd_dvd(A,A2),C2))
           => ( pp(aa(A,bool,dvd_dvd(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),A2)))
            <=> pp(aa(A,bool,dvd_dvd(A,B2),C2)) ) ) ) ) ).

% div_dvd_div
tff(fact_31_bit__concat__def,axiom,
    ! [H: nat,L: nat,D: nat] : vEBT_VEBT_bit_concat(H,L,D) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),H),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),D))),L) ).

% bit_concat_def
tff(fact_32_distrib__right__numeral,axiom,
    ! [A: $tType] :
      ( ( numeral(A)
        & semiring(A) )
     => ! [A2: A,B2: A,V: num] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),aa(num,A,numeral_numeral(A),V)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(num,A,numeral_numeral(A),V))),aa(A,A,aa(A,fun(A,A),times_times(A),B2),aa(num,A,numeral_numeral(A),V))) ) ).

% distrib_right_numeral
tff(fact_33_distrib__left__numeral,axiom,
    ! [A: $tType] :
      ( ( numeral(A)
        & semiring(A) )
     => ! [V: num,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),V)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),V)),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),V)),C2)) ) ).

% distrib_left_numeral
tff(fact_34_dvd__add__times__triv__right__iff,axiom,
    ! [A: $tType] :
      ( comm_s4317794764714335236cancel(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,dvd_dvd(A,A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2))))
        <=> pp(aa(A,bool,dvd_dvd(A,A2),B2)) ) ) ).

% dvd_add_times_triv_right_iff
tff(fact_35_dvd__add__times__triv__left__iff,axiom,
    ! [A: $tType] :
      ( comm_s4317794764714335236cancel(A)
     => ! [A2: A,C2: A,B2: A] :
          ( pp(aa(A,bool,dvd_dvd(A,A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2)),B2)))
        <=> pp(aa(A,bool,dvd_dvd(A,A2),B2)) ) ) ).

% dvd_add_times_triv_left_iff
tff(fact_36_div__add,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [C2: A,A2: A,B2: A] :
          ( pp(aa(A,bool,dvd_dvd(A,C2),A2))
         => ( pp(aa(A,bool,dvd_dvd(A,C2),B2))
           => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),C2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)) ) ) ) ) ).

% div_add
tff(fact_37_dvd__mult__div__cancel,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,dvd_dvd(A,A2),B2))
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A2)) = B2 ) ) ) ).

% dvd_mult_div_cancel
tff(fact_38_even__add,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)))
        <=> ( pp(aa(A,bool,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A2))
          <=> pp(aa(A,bool,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),B2)) ) ) ) ).

% even_add
tff(fact_39_odd__add,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A2: A,B2: A] :
          ( ~ pp(aa(A,bool,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)))
        <=> ~ ( ~ pp(aa(A,bool,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A2))
            <=> ~ pp(aa(A,bool,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),B2)) ) ) ) ).

% odd_add
tff(fact_40_add__2__eq__Suc_H,axiom,
    ! [N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2))) = aa(nat,nat,suc,aa(nat,nat,suc,N)) ).

% add_2_eq_Suc'
tff(fact_41_add__2__eq__Suc,axiom,
    ! [N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N) = aa(nat,nat,suc,aa(nat,nat,suc,N)) ).

% add_2_eq_Suc
tff(fact_42_add__self__div__2,axiom,
    ! [M: nat] : aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),M)),aa(num,nat,numeral_numeral(nat),bit0(one2))) = M ).

% add_self_div_2
tff(fact_43_Suc__div__eq__add3__div__numeral,axiom,
    ! [M: nat,V: num] : aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,suc,M)))),aa(num,nat,numeral_numeral(nat),V)) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2))),M)),aa(num,nat,numeral_numeral(nat),V)) ).

% Suc_div_eq_add3_div_numeral
tff(fact_44_mem__Collect__eq,axiom,
    ! [A: $tType,A2: A,P: fun(A,bool)] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),collect(A,P)))
    <=> pp(aa(A,bool,P,A2)) ) ).

% mem_Collect_eq
tff(fact_45_Collect__mem__eq,axiom,
    ! [A: $tType,A3: set(A)] : collect(A,aTP_Lamp_a(set(A),fun(A,bool),A3)) = A3 ).

% Collect_mem_eq
tff(fact_46_Collect__cong,axiom,
    ! [A: $tType,P: fun(A,bool),Q: fun(A,bool)] :
      ( ! [X3: A] :
          ( pp(aa(A,bool,P,X3))
        <=> pp(aa(A,bool,Q,X3)) )
     => ( collect(A,P) = collect(A,Q) ) ) ).

% Collect_cong
tff(fact_47_HOL_Oext,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),G: fun(A,B)] :
      ( ! [X3: A] : aa(A,B,F2,X3) = aa(A,B,G,X3)
     => ( F2 = G ) ) ).

% HOL.ext
tff(fact_48_div__Suc__eq__div__add3,axiom,
    ! [M: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,suc,N)))) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2))),N)) ).

% div_Suc_eq_div_add3
tff(fact_49_is__num__normalize_I1_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [A2: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2)) ) ).

% is_num_normalize(1)
tff(fact_50_combine__common__factor,axiom,
    ! [A: $tType] :
      ( semiring(A)
     => ! [A2: A,E: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),E)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),E)),C2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),E)),C2) ) ).

% combine_common_factor
tff(fact_51_distrib__right,axiom,
    ! [A: $tType] :
      ( semiring(A)
     => ! [A2: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) ) ).

% distrib_right
tff(fact_52_distrib__left,axiom,
    ! [A: $tType] :
      ( semiring(A)
     => ! [A2: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)) ) ).

% distrib_left
tff(fact_53_comm__semiring__class_Odistrib,axiom,
    ! [A: $tType] :
      ( comm_semiring(A)
     => ! [A2: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) ) ).

% comm_semiring_class.distrib
tff(fact_54_ring__class_Oring__distribs_I1_J,axiom,
    ! [A: $tType] :
      ( ring(A)
     => ! [A2: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)) ) ).

% ring_class.ring_distribs(1)
tff(fact_55_ring__class_Oring__distribs_I2_J,axiom,
    ! [A: $tType] :
      ( ring(A)
     => ! [A2: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) ) ).

% ring_class.ring_distribs(2)
tff(fact_56_le__num__One__iff,axiom,
    ! [X: num] :
      ( pp(aa(num,bool,ord_less_eq(num,X),one2))
    <=> ( X = one2 ) ) ).

% le_num_One_iff
tff(fact_57_dvd__add__right__iff,axiom,
    ! [A: $tType] :
      ( comm_s4317794764714335236cancel(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,dvd_dvd(A,A2),B2))
         => ( pp(aa(A,bool,dvd_dvd(A,A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2)))
          <=> pp(aa(A,bool,dvd_dvd(A,A2),C2)) ) ) ) ).

% dvd_add_right_iff
tff(fact_58_dvd__add__left__iff,axiom,
    ! [A: $tType] :
      ( comm_s4317794764714335236cancel(A)
     => ! [A2: A,C2: A,B2: A] :
          ( pp(aa(A,bool,dvd_dvd(A,A2),C2))
         => ( pp(aa(A,bool,dvd_dvd(A,A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2)))
          <=> pp(aa(A,bool,dvd_dvd(A,A2),B2)) ) ) ) ).

% dvd_add_left_iff
tff(fact_59_dvd__add,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,dvd_dvd(A,A2),B2))
         => ( pp(aa(A,bool,dvd_dvd(A,A2),C2))
           => pp(aa(A,bool,dvd_dvd(A,A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2))) ) ) ) ).

% dvd_add
tff(fact_60_subset__divisors__dvd,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(set(A),bool,ord_less_eq(set(A),collect(A,aTP_Lamp_aa(A,fun(A,bool),A2))),collect(A,aTP_Lamp_aa(A,fun(A,bool),B2))))
        <=> pp(aa(A,bool,dvd_dvd(A,A2),B2)) ) ) ).

% subset_divisors_dvd
tff(fact_61_numeral__Bit0,axiom,
    ! [A: $tType] :
      ( numeral(A)
     => ! [N: num] : aa(num,A,numeral_numeral(A),bit0(N)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),N)),aa(num,A,numeral_numeral(A),N)) ) ).

% numeral_Bit0
tff(fact_62_div__mult2__numeral__eq,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [A2: A,K: num,L: num] : aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(num,A,numeral_numeral(A),K))),aa(num,A,numeral_numeral(A),L)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),times_times(num),K),L))) ) ).

% div_mult2_numeral_eq
tff(fact_63_div__plus__div__distrib__dvd__right,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [C2: A,B2: A,A2: A] :
          ( pp(aa(A,bool,dvd_dvd(A,C2),B2))
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),C2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)) ) ) ) ).

% div_plus_div_distrib_dvd_right
tff(fact_64_div__plus__div__distrib__dvd__left,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [C2: A,A2: A,B2: A] :
          ( pp(aa(A,bool,dvd_dvd(A,C2),A2))
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),C2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)) ) ) ) ).

% div_plus_div_distrib_dvd_left
tff(fact_65_numeral__code_I2_J,axiom,
    ! [A: $tType] :
      ( numeral(A)
     => ! [N: num] : aa(num,A,numeral_numeral(A),bit0(N)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),N)),aa(num,A,numeral_numeral(A),N)) ) ).

% numeral_code(2)
tff(fact_66_left__add__twice,axiom,
    ! [A: $tType] :
      ( semiring_numeral(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),A2)),B2) ) ).

% left_add_twice
tff(fact_67_mult__2__right,axiom,
    ! [A: $tType] :
      ( semiring_numeral(A)
     => ! [Z: A] : aa(A,A,aa(A,fun(A,A),times_times(A),Z),aa(num,A,numeral_numeral(A),bit0(one2))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Z),Z) ) ).

% mult_2_right
tff(fact_68_mult__2,axiom,
    ! [A: $tType] :
      ( semiring_numeral(A)
     => ! [Z: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),Z) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Z),Z) ) ).

% mult_2
tff(fact_69_odd__even__add,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A2: A,B2: A] :
          ( ~ pp(aa(A,bool,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A2))
         => ( ~ pp(aa(A,bool,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),B2))
           => pp(aa(A,bool,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2))) ) ) ) ).

% odd_even_add
tff(fact_70_Suc3__eq__add__3,axiom,
    ! [N: nat] : aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,suc,N))) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2))),N) ).

% Suc3_eq_add_3
tff(fact_71_dvd__trans,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,dvd_dvd(A,A2),B2))
         => ( pp(aa(A,bool,dvd_dvd(A,B2),C2))
           => pp(aa(A,bool,dvd_dvd(A,A2),C2)) ) ) ) ).

% dvd_trans
tff(fact_72_dvd__refl,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A2: A] : pp(aa(A,bool,dvd_dvd(A,A2),A2)) ) ).

% dvd_refl
tff(fact_73_Suc__div__eq__add3__div,axiom,
    ! [M: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,suc,M)))),N) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2))),M)),N) ).

% Suc_div_eq_add3_div
tff(fact_74_dvd__triv__right,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A2: A,B2: A] : pp(aa(A,bool,dvd_dvd(A,A2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),A2))) ) ).

% dvd_triv_right
tff(fact_75_dvd__mult__right,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),C2))
         => pp(aa(A,bool,dvd_dvd(A,B2),C2)) ) ) ).

% dvd_mult_right
tff(fact_76_mult__dvd__mono,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A2: A,B2: A,C2: A,D: A] :
          ( pp(aa(A,bool,dvd_dvd(A,A2),B2))
         => ( pp(aa(A,bool,dvd_dvd(A,C2),D))
           => pp(aa(A,bool,dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),D))) ) ) ) ).

% mult_dvd_mono
tff(fact_77_dvd__triv__left,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A2: A,B2: A] : pp(aa(A,bool,dvd_dvd(A,A2),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2))) ) ).

% dvd_triv_left
tff(fact_78_dvd__mult__left,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),C2))
         => pp(aa(A,bool,dvd_dvd(A,A2),C2)) ) ) ).

% dvd_mult_left
tff(fact_79_dvd__mult2,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,dvd_dvd(A,A2),B2))
         => pp(aa(A,bool,dvd_dvd(A,A2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2))) ) ) ).

% dvd_mult2
tff(fact_80_dvd__mult,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A2: A,C2: A,B2: A] :
          ( pp(aa(A,bool,dvd_dvd(A,A2),C2))
         => pp(aa(A,bool,dvd_dvd(A,A2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2))) ) ) ).

% dvd_mult
tff(fact_81_dvd__def,axiom,
    ! [A: $tType] :
      ( dvd(A)
     => ! [B2: A,A2: A] :
          ( pp(aa(A,bool,dvd_dvd(A,B2),A2))
        <=> ? [K2: A] : A2 = aa(A,A,aa(A,fun(A,A),times_times(A),B2),K2) ) ) ).

% dvd_def
tff(fact_82_dvdI,axiom,
    ! [A: $tType] :
      ( dvd(A)
     => ! [A2: A,B2: A,K: A] :
          ( ( A2 = aa(A,A,aa(A,fun(A,A),times_times(A),B2),K) )
         => pp(aa(A,bool,dvd_dvd(A,B2),A2)) ) ) ).

% dvdI
tff(fact_83_dvdE,axiom,
    ! [A: $tType] :
      ( dvd(A)
     => ! [B2: A,A2: A] :
          ( pp(aa(A,bool,dvd_dvd(A,B2),A2))
         => ~ ! [K3: A] : A2 != aa(A,A,aa(A,fun(A,A),times_times(A),B2),K3) ) ) ).

% dvdE
tff(fact_84_div__div__div__same,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [D: A,B2: A,A2: A] :
          ( pp(aa(A,bool,dvd_dvd(A,D),B2))
         => ( pp(aa(A,bool,dvd_dvd(A,B2),A2))
           => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),D)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),D)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2) ) ) ) ) ).

% div_div_div_same
tff(fact_85_dvd__div__eq__cancel,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [A2: A,C2: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),C2) = aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2) )
         => ( pp(aa(A,bool,dvd_dvd(A,C2),A2))
           => ( pp(aa(A,bool,dvd_dvd(A,C2),B2))
             => ( A2 = B2 ) ) ) ) ) ).

% dvd_div_eq_cancel
tff(fact_86_dvd__div__eq__iff,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [C2: A,A2: A,B2: A] :
          ( pp(aa(A,bool,dvd_dvd(A,C2),A2))
         => ( pp(aa(A,bool,dvd_dvd(A,C2),B2))
           => ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),C2) = aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2) )
            <=> ( A2 = B2 ) ) ) ) ) ).

% dvd_div_eq_iff
tff(fact_87_div__le__dividend,axiom,
    ! [M: nat,N: nat] : pp(aa(nat,bool,ord_less_eq(nat,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),N)),M)) ).

% div_le_dividend
tff(fact_88_div__le__mono,axiom,
    ! [M: nat,N: nat,K: nat] :
      ( pp(aa(nat,bool,ord_less_eq(nat,M),N))
     => pp(aa(nat,bool,ord_less_eq(nat,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),K)),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),K))) ) ).

% div_le_mono
tff(fact_89_div__mult2__eq,axiom,
    ! [M: nat,N: nat,Q2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),Q2)) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),N)),Q2) ).

% div_mult2_eq
tff(fact_90_mult__numeral__1__right,axiom,
    ! [A: $tType] :
      ( semiring_numeral(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(num,A,numeral_numeral(A),one2)) = A2 ) ).

% mult_numeral_1_right
tff(fact_91_mult__numeral__1,axiom,
    ! [A: $tType] :
      ( semiring_numeral(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),one2)),A2) = A2 ) ).

% mult_numeral_1
tff(fact_92_divide__numeral__1,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(num,A,numeral_numeral(A),one2)) = A2 ) ).

% divide_numeral_1
tff(fact_93_num_Oexhaust,axiom,
    ! [Y: num] :
      ( ( Y != one2 )
     => ( ! [X22: num] : Y != bit0(X22)
       => ~ ! [X32: num] : Y != aa(num,num,bit1,X32) ) ) ).

% num.exhaust
tff(fact_94_div__mult__div__if__dvd,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A2: A,D: A,C2: A] :
          ( pp(aa(A,bool,dvd_dvd(A,B2),A2))
         => ( pp(aa(A,bool,dvd_dvd(A,D),C2))
           => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),D)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),D)) ) ) ) ) ).

% div_mult_div_if_dvd
tff(fact_95_dvd__mult__imp__div,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,C2: A,B2: A] :
          ( pp(aa(A,bool,dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),B2))
         => pp(aa(A,bool,dvd_dvd(A,A2),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2))) ) ) ).

% dvd_mult_imp_div
tff(fact_96_dvd__div__mult2__eq,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,C2: A,A2: A] :
          ( pp(aa(A,bool,dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)),A2))
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)),C2) ) ) ) ).

% dvd_div_mult2_eq
tff(fact_97_div__div__eq__right,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [C2: A,B2: A,A2: A] :
          ( pp(aa(A,bool,dvd_dvd(A,C2),B2))
         => ( pp(aa(A,bool,dvd_dvd(A,B2),A2))
           => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)),C2) ) ) ) ) ).

% div_div_eq_right
tff(fact_98_div__mult__swap,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [C2: A,B2: A,A2: A] :
          ( pp(aa(A,bool,dvd_dvd(A,C2),B2))
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),C2) ) ) ) ).

% div_mult_swap
tff(fact_99_dvd__div__mult,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [C2: A,B2: A,A2: A] :
          ( pp(aa(A,bool,dvd_dvd(A,C2),B2))
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)),A2) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),A2)),C2) ) ) ) ).

% dvd_div_mult
tff(fact_100_Suc__div__le__mono,axiom,
    ! [M: nat,N: nat] : pp(aa(nat,bool,ord_less_eq(nat,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),N)),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,M)),N))) ).

% Suc_div_le_mono
tff(fact_101_VEBT__internal_OTb_H_Osimps_I3_J,axiom,
    ! [N: nat] :
      ( ( pp(aa(nat,bool,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N))
       => ( vEBT_VEBT_Tb2(aa(nat,nat,suc,aa(nat,nat,suc,N))) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,bit0(one2)))),vEBT_VEBT_Tb2(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),vEBT_VEBT_Tb2(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2))))))) ) )
      & ( ~ pp(aa(nat,bool,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N))
       => ( vEBT_VEBT_Tb2(aa(nat,nat,suc,aa(nat,nat,suc,N))) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,bit0(one2)))),vEBT_VEBT_Tb2(aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2)))))))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),vEBT_VEBT_Tb2(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2)))))))) ) ) ) ).

% VEBT_internal.Tb'.simps(3)
tff(fact_102_times__div__less__eq__dividend,axiom,
    ! [N: nat,M: nat] : pp(aa(nat,bool,ord_less_eq(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),N))),M)) ).

% times_div_less_eq_dividend
tff(fact_103_div__times__less__eq__dividend,axiom,
    ! [M: nat,N: nat] : pp(aa(nat,bool,ord_less_eq(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),N)),N)),M)) ).

% div_times_less_eq_dividend
tff(fact_104_numeral__Bit0__div__2,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [N: num] : aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(num,A,numeral_numeral(A),bit0(N))),aa(num,A,numeral_numeral(A),bit0(one2))) = aa(num,A,numeral_numeral(A),N) ) ).

% numeral_Bit0_div_2
tff(fact_105_even__numeral,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [N: num] : pp(aa(A,bool,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),aa(num,A,numeral_numeral(A),bit0(N)))) ) ).

% even_numeral
tff(fact_106_eval__nat__numeral_I3_J,axiom,
    ! [N: num] : aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,N)) = aa(nat,nat,suc,aa(num,nat,numeral_numeral(nat),bit0(N))) ).

% eval_nat_numeral(3)
tff(fact_107_evenE,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A2: A] :
          ( pp(aa(A,bool,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A2))
         => ~ ! [B3: A] : A2 != aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),B3) ) ) ).

% evenE
tff(fact_108_numeral__Bit1__div__2,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [N: num] : aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(num,A,numeral_numeral(A),aa(num,num,bit1,N))),aa(num,A,numeral_numeral(A),bit0(one2))) = aa(num,A,numeral_numeral(A),N) ) ).

% numeral_Bit1_div_2
tff(fact_109_odd__numeral,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [N: num] : ~ pp(aa(A,bool,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),aa(num,A,numeral_numeral(A),aa(num,num,bit1,N)))) ) ).

% odd_numeral
tff(fact_110_double__not__eq__Suc__double,axiom,
    ! [M: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M) != aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)) ).

% double_not_eq_Suc_double
tff(fact_111_Suc__double__not__eq__double,axiom,
    ! [M: nat,N: nat] : aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M)) != aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N) ).

% Suc_double_not_eq_double
tff(fact_112_even__two__times__div__two,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [A2: A] :
          ( pp(aa(A,bool,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A2))
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(num,A,numeral_numeral(A),bit0(one2)))) = A2 ) ) ) ).

% even_two_times_div_two
tff(fact_113_power__mono__odd,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [N: nat,A2: A,B2: A] :
          ( ~ pp(aa(nat,bool,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N))
         => ( pp(aa(A,bool,ord_less_eq(A,A2),B2))
           => pp(aa(A,bool,ord_less_eq(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),N)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),N))) ) ) ) ).

% power_mono_odd
tff(fact_114_sum__squares__bound,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y: A] : pp(aa(A,bool,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),X)),Y)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),aa(num,nat,numeral_numeral(nat),bit0(one2)))))) ) ).

% sum_squares_bound
tff(fact_115_mult__Suc__right,axiom,
    ! [M: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),aa(nat,nat,suc,N)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N)) ).

% mult_Suc_right
tff(fact_116_power__odd__eq,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [A2: A,N: nat] : aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N))) = aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),N)),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ) ).

% power_odd_eq
tff(fact_117_power2__sum,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [X: A,Y: A] : aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y)),aa(num,nat,numeral_numeral(nat),bit0(one2))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),X)),Y)) ) ).

% power2_sum
tff(fact_118_dvd__power__iff__le,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa(nat,bool,ord_less_eq(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),K))
     => ( pp(aa(nat,bool,dvd_dvd(nat,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),K),M)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),K),N)))
      <=> pp(aa(nat,bool,ord_less_eq(nat,M),N)) ) ) ).

% dvd_power_iff_le
tff(fact_119_div__exp__eq,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A,M: nat,N: nat] : aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),M))),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N))) ) ).

% div_exp_eq
tff(fact_120_nat__add__left__cancel__le,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa(nat,bool,ord_less_eq(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),M)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),N)))
    <=> pp(aa(nat,bool,ord_less_eq(nat,M),N)) ) ).

% nat_add_left_cancel_le
tff(fact_121_div2__even__ext__nat,axiom,
    ! [X: nat,Y: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),X),aa(num,nat,numeral_numeral(nat),bit0(one2))) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Y),aa(num,nat,numeral_numeral(nat),bit0(one2))) )
     => ( ( pp(aa(nat,bool,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),X))
        <=> pp(aa(nat,bool,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Y)) )
       => ( X = Y ) ) ) ).

% div2_even_ext_nat
tff(fact_122_enat__ord__number_I1_J,axiom,
    ! [M: num,N: num] :
      ( pp(aa(extended_enat,bool,ord_less_eq(extended_enat,aa(num,extended_enat,numeral_numeral(extended_enat),M)),aa(num,extended_enat,numeral_numeral(extended_enat),N)))
    <=> pp(aa(nat,bool,ord_less_eq(nat,aa(num,nat,numeral_numeral(nat),M)),aa(num,nat,numeral_numeral(nat),N))) ) ).

% enat_ord_number(1)
tff(fact_123_power2__nat__le__imp__le,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,ord_less_eq(nat,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),M),aa(num,nat,numeral_numeral(nat),bit0(one2)))),N))
     => pp(aa(nat,bool,ord_less_eq(nat,M),N)) ) ).

% power2_nat_le_imp_le
tff(fact_124_power2__nat__le__eq__le,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,ord_less_eq(nat,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),M),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2)))))
    <=> pp(aa(nat,bool,ord_less_eq(nat,M),N)) ) ).

% power2_nat_le_eq_le
tff(fact_125_self__le__ge2__pow,axiom,
    ! [K: nat,M: nat] :
      ( pp(aa(nat,bool,ord_less_eq(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),K))
     => pp(aa(nat,bool,ord_less_eq(nat,M),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),K),M))) ) ).

% self_le_ge2_pow
tff(fact_126_nat_Oinject,axiom,
    ! [X23: nat,Y2: nat] :
      ( ( aa(nat,nat,suc,X23) = aa(nat,nat,suc,Y2) )
    <=> ( X23 = Y2 ) ) ).

% nat.inject
tff(fact_127_old_Onat_Oinject,axiom,
    ! [Nat: nat,Nat2: nat] :
      ( ( aa(nat,nat,suc,Nat) = aa(nat,nat,suc,Nat2) )
    <=> ( Nat = Nat2 ) ) ).

% old.nat.inject
tff(fact_128_add__Suc__right,axiom,
    ! [M: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),aa(nat,nat,suc,N)) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N)) ).

% add_Suc_right
tff(fact_129_Suc__le__mono,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa(nat,bool,ord_less_eq(nat,aa(nat,nat,suc,N)),aa(nat,nat,suc,M)))
    <=> pp(aa(nat,bool,ord_less_eq(nat,N),M)) ) ).

% Suc_le_mono
tff(fact_130_power__mult__numeral,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [A2: A,M: num,N: num] : aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(num,nat,numeral_numeral(nat),M))),aa(num,nat,numeral_numeral(nat),N)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(num,nat,numeral_numeral(nat),aa(num,num,aa(num,fun(num,num),times_times(num),M),N))) ) ).

% power_mult_numeral
tff(fact_131_Suc__numeral,axiom,
    ! [N: num] : aa(nat,nat,suc,aa(num,nat,numeral_numeral(nat),N)) = aa(num,nat,numeral_numeral(nat),aa(num,num,aa(num,fun(num,num),plus_plus(num),N),one2)) ).

% Suc_numeral
tff(fact_132_power__add__numeral,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [A2: A,M: num,N: num] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(num,nat,numeral_numeral(nat),M))),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(num,nat,numeral_numeral(nat),N))) = aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(num,nat,numeral_numeral(nat),aa(num,num,aa(num,fun(num,num),plus_plus(num),M),N))) ) ).

% power_add_numeral
tff(fact_133_power__add__numeral2,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [A2: A,M: num,N: num,B2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(num,nat,numeral_numeral(nat),M))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(num,nat,numeral_numeral(nat),N))),B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(num,nat,numeral_numeral(nat),aa(num,num,aa(num,fun(num,num),plus_plus(num),M),N)))),B2) ) ).

% power_add_numeral2
tff(fact_134_L2__set__mult__ineq__lemma,axiom,
    ! [A2: real,C2: real,B2: real,D: real] : pp(aa(real,bool,ord_less_eq(real,aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),aa(real,real,aa(real,fun(real,real),times_times(real),A2),C2))),aa(real,real,aa(real,fun(real,real),times_times(real),B2),D))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),A2),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),D),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),B2),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),C2),aa(num,nat,numeral_numeral(nat),bit0(one2))))))) ).

% L2_set_mult_ineq_lemma
tff(fact_135_add__One__commute,axiom,
    ! [N: num] : aa(num,num,aa(num,fun(num,num),plus_plus(num),one2),N) = aa(num,num,aa(num,fun(num,num),plus_plus(num),N),one2) ).

% add_One_commute
tff(fact_136_four__x__squared,axiom,
    ! [X: real] : aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(bit0(one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),X),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),X)),aa(num,nat,numeral_numeral(nat),bit0(one2))) ).

% four_x_squared
tff(fact_137_Suc__inject,axiom,
    ! [X: nat,Y: nat] :
      ( ( aa(nat,nat,suc,X) = aa(nat,nat,suc,Y) )
     => ( X = Y ) ) ).

% Suc_inject
tff(fact_138_n__not__Suc__n,axiom,
    ! [N: nat] : N != aa(nat,nat,suc,N) ).

% n_not_Suc_n
tff(fact_139_le__refl,axiom,
    ! [N: nat] : pp(aa(nat,bool,ord_less_eq(nat,N),N)) ).

% le_refl
tff(fact_140_le__trans,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( pp(aa(nat,bool,ord_less_eq(nat,I),J))
     => ( pp(aa(nat,bool,ord_less_eq(nat,J),K))
       => pp(aa(nat,bool,ord_less_eq(nat,I),K)) ) ) ).

% le_trans
tff(fact_141_eq__imp__le,axiom,
    ! [M: nat,N: nat] :
      ( ( M = N )
     => pp(aa(nat,bool,ord_less_eq(nat,M),N)) ) ).

% eq_imp_le
tff(fact_142_le__antisym,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,ord_less_eq(nat,M),N))
     => ( pp(aa(nat,bool,ord_less_eq(nat,N),M))
       => ( M = N ) ) ) ).

% le_antisym
tff(fact_143_nat__le__linear,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,ord_less_eq(nat,M),N))
      | pp(aa(nat,bool,ord_less_eq(nat,N),M)) ) ).

% nat_le_linear
tff(fact_144_Nat_Oex__has__greatest__nat,axiom,
    ! [P: fun(nat,bool),K: nat,B2: nat] :
      ( pp(aa(nat,bool,P,K))
     => ( ! [Y3: nat] :
            ( pp(aa(nat,bool,P,Y3))
           => pp(aa(nat,bool,ord_less_eq(nat,Y3),B2)) )
       => ? [X3: nat] :
            ( pp(aa(nat,bool,P,X3))
            & ! [Y4: nat] :
                ( pp(aa(nat,bool,P,Y4))
               => pp(aa(nat,bool,ord_less_eq(nat,Y4),X3)) ) ) ) ) ).

% Nat.ex_has_greatest_nat
tff(fact_145_dvd__antisym,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,dvd_dvd(nat,M),N))
     => ( pp(aa(nat,bool,dvd_dvd(nat,N),M))
       => ( M = N ) ) ) ).

% dvd_antisym
tff(fact_146_Suc__nat__number__of__add,axiom,
    ! [V: num,N: nat] : aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),V)),N)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,aa(num,fun(num,num),plus_plus(num),V),one2))),N) ).

% Suc_nat_number_of_add
tff(fact_147_power__commutes,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [A2: A,N: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),N)),A2) = aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),N)) ) ).

% power_commutes
tff(fact_148_power__mult__distrib,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A2: A,B2: A,N: nat] : aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),N) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),N)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),N)) ) ).

% power_mult_distrib
tff(fact_149_power__commuting__commutes,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [X: A,Y: A,N: nat] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),X),Y) = aa(A,A,aa(A,fun(A,A),times_times(A),Y),X) )
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),N)),Y) = aa(A,A,aa(A,fun(A,A),times_times(A),Y),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),N)) ) ) ) ).

% power_commuting_commutes
tff(fact_150_power__divide,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A2: A,B2: A,N: nat] : aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)),N) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),N)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),N)) ) ).

% power_divide
tff(fact_151_nat__arith_Osuc1,axiom,
    ! [A3: nat,K: nat,A2: nat] :
      ( ( A3 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),A2) )
     => ( aa(nat,nat,suc,A3) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),aa(nat,nat,suc,A2)) ) ) ).

% nat_arith.suc1
tff(fact_152_add__Suc,axiom,
    ! [M: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,suc,M)),N) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N)) ).

% add_Suc
tff(fact_153_add__Suc__shift,axiom,
    ! [M: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,suc,M)),N) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),aa(nat,nat,suc,N)) ).

% add_Suc_shift
tff(fact_154_Suc__leD,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,ord_less_eq(nat,aa(nat,nat,suc,M)),N))
     => pp(aa(nat,bool,ord_less_eq(nat,M),N)) ) ).

% Suc_leD
tff(fact_155_le__SucE,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,ord_less_eq(nat,M),aa(nat,nat,suc,N)))
     => ( ~ pp(aa(nat,bool,ord_less_eq(nat,M),N))
       => ( M = aa(nat,nat,suc,N) ) ) ) ).

% le_SucE
tff(fact_156_le__SucI,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,ord_less_eq(nat,M),N))
     => pp(aa(nat,bool,ord_less_eq(nat,M),aa(nat,nat,suc,N))) ) ).

% le_SucI
tff(fact_157_Suc__le__D,axiom,
    ! [N: nat,M2: nat] :
      ( pp(aa(nat,bool,ord_less_eq(nat,aa(nat,nat,suc,N)),M2))
     => ? [M3: nat] : M2 = aa(nat,nat,suc,M3) ) ).

% Suc_le_D
tff(fact_158_le__Suc__eq,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,ord_less_eq(nat,M),aa(nat,nat,suc,N)))
    <=> ( pp(aa(nat,bool,ord_less_eq(nat,M),N))
        | ( M = aa(nat,nat,suc,N) ) ) ) ).

% le_Suc_eq
tff(fact_159_Suc__n__not__le__n,axiom,
    ! [N: nat] : ~ pp(aa(nat,bool,ord_less_eq(nat,aa(nat,nat,suc,N)),N)) ).

% Suc_n_not_le_n
tff(fact_160_not__less__eq__eq,axiom,
    ! [M: nat,N: nat] :
      ( ~ pp(aa(nat,bool,ord_less_eq(nat,M),N))
    <=> pp(aa(nat,bool,ord_less_eq(nat,aa(nat,nat,suc,N)),M)) ) ).

% not_less_eq_eq
tff(fact_161_full__nat__induct,axiom,
    ! [P: fun(nat,bool),N: nat] :
      ( ! [N2: nat] :
          ( ! [M4: nat] :
              ( pp(aa(nat,bool,ord_less_eq(nat,aa(nat,nat,suc,M4)),N2))
             => pp(aa(nat,bool,P,M4)) )
         => pp(aa(nat,bool,P,N2)) )
     => pp(aa(nat,bool,P,N)) ) ).

% full_nat_induct
tff(fact_162_nat__induct__at__least,axiom,
    ! [M: nat,N: nat,P: fun(nat,bool)] :
      ( pp(aa(nat,bool,ord_less_eq(nat,M),N))
     => ( pp(aa(nat,bool,P,M))
       => ( ! [N2: nat] :
              ( pp(aa(nat,bool,ord_less_eq(nat,M),N2))
             => ( pp(aa(nat,bool,P,N2))
               => pp(aa(nat,bool,P,aa(nat,nat,suc,N2))) ) )
         => pp(aa(nat,bool,P,N)) ) ) ) ).

% nat_induct_at_least
tff(fact_163_transitive__stepwise__le,axiom,
    ! [M: nat,N: nat,R: fun(nat,fun(nat,bool))] :
      ( pp(aa(nat,bool,ord_less_eq(nat,M),N))
     => ( ! [X3: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),R,X3),X3))
       => ( ! [X3: nat,Y3: nat,Z2: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),R,X3),Y3))
             => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),R,Y3),Z2))
               => pp(aa(nat,bool,aa(nat,fun(nat,bool),R,X3),Z2)) ) )
         => ( ! [N2: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),R,N2),aa(nat,nat,suc,N2)))
           => pp(aa(nat,bool,aa(nat,fun(nat,bool),R,M),N)) ) ) ) ) ).

% transitive_stepwise_le
tff(fact_164_dvd__power__same,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [X: A,Y: A,N: nat] :
          ( pp(aa(A,bool,dvd_dvd(A,X),Y))
         => pp(aa(A,bool,dvd_dvd(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),X),N)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),N))) ) ) ).

% dvd_power_same
tff(fact_165_Suc__mult__cancel1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,K)),M) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,K)),N) )
    <=> ( M = N ) ) ).

% Suc_mult_cancel1
tff(fact_166_add__leE,axiom,
    ! [M: nat,K: nat,N: nat] :
      ( pp(aa(nat,bool,ord_less_eq(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),K)),N))
     => ~ ( pp(aa(nat,bool,ord_less_eq(nat,M),N))
         => ~ pp(aa(nat,bool,ord_less_eq(nat,K),N)) ) ) ).

% add_leE
tff(fact_167_le__add1,axiom,
    ! [N: nat,M: nat] : pp(aa(nat,bool,ord_less_eq(nat,N),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),M))) ).

% le_add1
tff(fact_168_le__add2,axiom,
    ! [N: nat,M: nat] : pp(aa(nat,bool,ord_less_eq(nat,N),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N))) ).

% le_add2
tff(fact_169_add__leD1,axiom,
    ! [M: nat,K: nat,N: nat] :
      ( pp(aa(nat,bool,ord_less_eq(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),K)),N))
     => pp(aa(nat,bool,ord_less_eq(nat,M),N)) ) ).

% add_leD1
tff(fact_170_add__leD2,axiom,
    ! [M: nat,K: nat,N: nat] :
      ( pp(aa(nat,bool,ord_less_eq(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),K)),N))
     => pp(aa(nat,bool,ord_less_eq(nat,K),N)) ) ).

% add_leD2
tff(fact_171_le__Suc__ex,axiom,
    ! [K: nat,L: nat] :
      ( pp(aa(nat,bool,ord_less_eq(nat,K),L))
     => ? [N2: nat] : L = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),N2) ) ).

% le_Suc_ex
tff(fact_172_add__le__mono,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( pp(aa(nat,bool,ord_less_eq(nat,I),J))
     => ( pp(aa(nat,bool,ord_less_eq(nat,K),L))
       => pp(aa(nat,bool,ord_less_eq(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),K)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),L))) ) ) ).

% add_le_mono
tff(fact_173_add__le__mono1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( pp(aa(nat,bool,ord_less_eq(nat,I),J))
     => pp(aa(nat,bool,ord_less_eq(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),K)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),K))) ) ).

% add_le_mono1
tff(fact_174_trans__le__add1,axiom,
    ! [I: nat,J: nat,M: nat] :
      ( pp(aa(nat,bool,ord_less_eq(nat,I),J))
     => pp(aa(nat,bool,ord_less_eq(nat,I),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),M))) ) ).

% trans_le_add1
tff(fact_175_trans__le__add2,axiom,
    ! [I: nat,J: nat,M: nat] :
      ( pp(aa(nat,bool,ord_less_eq(nat,I),J))
     => pp(aa(nat,bool,ord_less_eq(nat,I),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),J))) ) ).

% trans_le_add2
tff(fact_176_nat__le__iff__add,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,ord_less_eq(nat,M),N))
    <=> ? [K2: nat] : N = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),K2) ) ).

% nat_le_iff_add
tff(fact_177_power__mult,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [A2: A,M: nat,N: nat] : aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),M)),N) ) ).

% power_mult
tff(fact_178_add__mult__distrib,axiom,
    ! [M: nat,N: nat,K: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N)),K) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),K)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),K)) ).

% add_mult_distrib
tff(fact_179_add__mult__distrib2,axiom,
    ! [K: nat,M: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),M)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),N)) ).

% add_mult_distrib2
tff(fact_180_le__cube,axiom,
    ! [M: nat] : pp(aa(nat,bool,ord_less_eq(nat,M),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),M)))) ).

% le_cube
tff(fact_181_le__square,axiom,
    ! [M: nat] : pp(aa(nat,bool,ord_less_eq(nat,M),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),M))) ).

% le_square
tff(fact_182_mult__le__mono,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( pp(aa(nat,bool,ord_less_eq(nat,I),J))
     => ( pp(aa(nat,bool,ord_less_eq(nat,K),L))
       => pp(aa(nat,bool,ord_less_eq(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I),K)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),L))) ) ) ).

% mult_le_mono
tff(fact_183_mult__le__mono1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( pp(aa(nat,bool,ord_less_eq(nat,I),J))
     => pp(aa(nat,bool,ord_less_eq(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I),K)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),K))) ) ).

% mult_le_mono1
tff(fact_184_mult__le__mono2,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( pp(aa(nat,bool,ord_less_eq(nat,I),J))
     => pp(aa(nat,bool,ord_less_eq(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),I)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),J))) ) ).

% mult_le_mono2
tff(fact_185_power__Suc,axiom,
    ! [A: $tType] :
      ( power(A)
     => ! [A2: A,N: nat] : aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(nat,nat,suc,N)) = aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),N)) ) ).

% power_Suc
tff(fact_186_power__Suc2,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [A2: A,N: nat] : aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(nat,nat,suc,N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),N)),A2) ) ).

% power_Suc2
tff(fact_187_lift__Suc__mono__le,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [F2: fun(nat,A),N: nat,N3: nat] :
          ( ! [N2: nat] : pp(aa(A,bool,ord_less_eq(A,aa(nat,A,F2,N2)),aa(nat,A,F2,aa(nat,nat,suc,N2))))
         => ( pp(aa(nat,bool,ord_less_eq(nat,N),N3))
           => pp(aa(A,bool,ord_less_eq(A,aa(nat,A,F2,N)),aa(nat,A,F2,N3))) ) ) ) ).

% lift_Suc_mono_le
tff(fact_188_lift__Suc__antimono__le,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [F2: fun(nat,A),N: nat,N3: nat] :
          ( ! [N2: nat] : pp(aa(A,bool,ord_less_eq(A,aa(nat,A,F2,aa(nat,nat,suc,N2))),aa(nat,A,F2,N2)))
         => ( pp(aa(nat,bool,ord_less_eq(nat,N),N3))
           => pp(aa(A,bool,ord_less_eq(A,aa(nat,A,F2,N3)),aa(nat,A,F2,N))) ) ) ) ).

% lift_Suc_antimono_le
tff(fact_189_power__add,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [A2: A,M: nat,N: nat] : aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),M)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),N)) ) ).

% power_add
tff(fact_190_div__power,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A2: A,N: nat] :
          ( pp(aa(A,bool,dvd_dvd(A,B2),A2))
         => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)),N) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),N)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),N)) ) ) ) ).

% div_power
tff(fact_191_dvd__power__le,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [X: A,Y: A,N: nat,M: nat] :
          ( pp(aa(A,bool,dvd_dvd(A,X),Y))
         => ( pp(aa(nat,bool,ord_less_eq(nat,N),M))
           => pp(aa(A,bool,dvd_dvd(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),X),N)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),M))) ) ) ) ).

% dvd_power_le
tff(fact_192_power__le__dvd,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A2: A,N: nat,B2: A,M: nat] :
          ( pp(aa(A,bool,dvd_dvd(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),N)),B2))
         => ( pp(aa(nat,bool,ord_less_eq(nat,M),N))
           => pp(aa(A,bool,dvd_dvd(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),M)),B2)) ) ) ) ).

% power_le_dvd
tff(fact_193_le__imp__power__dvd,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [M: nat,N: nat,A2: A] :
          ( pp(aa(nat,bool,ord_less_eq(nat,M),N))
         => pp(aa(A,bool,dvd_dvd(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),M)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),N))) ) ) ).

% le_imp_power_dvd
tff(fact_194_mult__Suc,axiom,
    ! [M: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,M)),N) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N)) ).

% mult_Suc
tff(fact_195_Suc__mult__le__cancel1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa(nat,bool,ord_less_eq(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,K)),M)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,K)),N)))
    <=> pp(aa(nat,bool,ord_less_eq(nat,M),N)) ) ).

% Suc_mult_le_cancel1
tff(fact_196_power__numeral__even,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [Z: A,W: num] : aa(nat,A,aa(A,fun(nat,A),power_power(A),Z),aa(num,nat,numeral_numeral(nat),bit0(W))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),Z),aa(num,nat,numeral_numeral(nat),W))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Z),aa(num,nat,numeral_numeral(nat),W))) ) ).

% power_numeral_even
tff(fact_197_power__numeral__odd,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [Z: A,W: num] : aa(nat,A,aa(A,fun(nat,A),power_power(A),Z),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,W))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),Z),aa(nat,A,aa(A,fun(nat,A),power_power(A),Z),aa(num,nat,numeral_numeral(nat),W)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Z),aa(num,nat,numeral_numeral(nat),W))) ) ).

% power_numeral_odd
tff(fact_198_power4__eq__xxxx,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [X: A] : aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(num,nat,numeral_numeral(nat),bit0(bit0(one2)))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),X)),X)),X) ) ).

% power4_eq_xxxx
tff(fact_199_power2__eq__square,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [A2: A] : aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(num,nat,numeral_numeral(nat),bit0(one2))) = aa(A,A,aa(A,fun(A,A),times_times(A),A2),A2) ) ).

% power2_eq_square
tff(fact_200_bit__eq__rec,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A,B2: A] :
          ( ( A2 = B2 )
        <=> ( ( pp(aa(A,bool,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A2))
            <=> pp(aa(A,bool,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),B2)) )
            & ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(num,A,numeral_numeral(A),bit0(one2))) = aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),aa(num,A,numeral_numeral(A),bit0(one2))) ) ) ) ) ).

% bit_eq_rec
tff(fact_201_power3__eq__cube,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [A2: A] : aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),A2)),A2) ) ).

% power3_eq_cube
tff(fact_202_power__even__eq,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [A2: A,N: nat] : aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),N)),aa(num,nat,numeral_numeral(nat),bit0(one2))) ) ).

% power_even_eq
tff(fact_203_semiring__norm_I16_J,axiom,
    ! [M: num,N: num] : aa(num,num,aa(num,fun(num,num),times_times(num),aa(num,num,bit1,M)),aa(num,num,bit1,N)) = aa(num,num,bit1,aa(num,num,aa(num,fun(num,num),plus_plus(num),aa(num,num,aa(num,fun(num,num),plus_plus(num),M),N)),bit0(aa(num,num,aa(num,fun(num,num),times_times(num),M),N)))) ).

% semiring_norm(16)
tff(fact_204_semiring__norm_I10_J,axiom,
    ! [M: num,N: num] : aa(num,num,aa(num,fun(num,num),plus_plus(num),aa(num,num,bit1,M)),aa(num,num,bit1,N)) = bit0(aa(num,num,aa(num,fun(num,num),plus_plus(num),aa(num,num,aa(num,fun(num,num),plus_plus(num),M),N)),one2)) ).

% semiring_norm(10)
tff(fact_205_semiring__norm_I8_J,axiom,
    ! [M: num] : aa(num,num,aa(num,fun(num,num),plus_plus(num),aa(num,num,bit1,M)),one2) = bit0(aa(num,num,aa(num,fun(num,num),plus_plus(num),M),one2)) ).

% semiring_norm(8)
tff(fact_206_semiring__norm_I5_J,axiom,
    ! [M: num] : aa(num,num,aa(num,fun(num,num),plus_plus(num),bit0(M)),one2) = aa(num,num,bit1,M) ).

% semiring_norm(5)
tff(fact_207_semiring__norm_I4_J,axiom,
    ! [N: num] : aa(num,num,aa(num,fun(num,num),plus_plus(num),one2),aa(num,num,bit1,N)) = bit0(aa(num,num,aa(num,fun(num,num),plus_plus(num),N),one2)) ).

% semiring_norm(4)
tff(fact_208_semiring__norm_I3_J,axiom,
    ! [N: num] : aa(num,num,aa(num,fun(num,num),plus_plus(num),one2),bit0(N)) = aa(num,num,bit1,N) ).

% semiring_norm(3)
tff(fact_209_count__buildup,axiom,
    ! [N: nat] : pp(aa(real,bool,ord_less_eq(real,aa(vEBT_VEBT,real,vEBT_VEBT_cnt,vEBT_vebt_buildup(N))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(num,real,numeral_numeral(real),bit0(one2))),N)))) ).

% count_buildup
tff(fact_210_semiring__norm_I70_J,axiom,
    ! [M: num] : ~ pp(aa(num,bool,ord_less_eq(num,aa(num,num,bit1,M)),one2)) ).

% semiring_norm(70)
tff(fact_211_semiring__norm_I72_J,axiom,
    ! [M: num,N: num] :
      ( pp(aa(num,bool,ord_less_eq(num,bit0(M)),aa(num,num,bit1,N)))
    <=> pp(aa(num,bool,ord_less_eq(num,M),N)) ) ).

% semiring_norm(72)
tff(fact_212_semiring__norm_I69_J,axiom,
    ! [M: num] : ~ pp(aa(num,bool,ord_less_eq(num,bit0(M)),one2)) ).

% semiring_norm(69)
tff(fact_213_semiring__norm_I15_J,axiom,
    ! [M: num,N: num] : aa(num,num,aa(num,fun(num,num),times_times(num),aa(num,num,bit1,M)),bit0(N)) = bit0(aa(num,num,aa(num,fun(num,num),times_times(num),aa(num,num,bit1,M)),N)) ).

% semiring_norm(15)
tff(fact_214_semiring__norm_I14_J,axiom,
    ! [M: num,N: num] : aa(num,num,aa(num,fun(num,num),times_times(num),bit0(M)),aa(num,num,bit1,N)) = bit0(aa(num,num,aa(num,fun(num,num),times_times(num),M),aa(num,num,bit1,N))) ).

% semiring_norm(14)
tff(fact_215_semiring__norm_I87_J,axiom,
    ! [M: num,N: num] :
      ( ( bit0(M) = bit0(N) )
    <=> ( M = N ) ) ).

% semiring_norm(87)
tff(fact_216_semiring__norm_I90_J,axiom,
    ! [M: num,N: num] :
      ( ( aa(num,num,bit1,M) = aa(num,num,bit1,N) )
    <=> ( M = N ) ) ).

% semiring_norm(90)
tff(fact_217_real__divide__square__eq,axiom,
    ! [R2: real,A2: real] : aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),times_times(real),R2),A2)),aa(real,real,aa(real,fun(real,real),times_times(real),R2),R2)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),A2),R2) ).

% real_divide_square_eq
tff(fact_218_semiring__norm_I83_J,axiom,
    ! [N: num] : one2 != bit0(N) ).

% semiring_norm(83)
tff(fact_219_semiring__norm_I85_J,axiom,
    ! [M: num] : bit0(M) != one2 ).

% semiring_norm(85)
tff(fact_220_semiring__norm_I88_J,axiom,
    ! [M: num,N: num] : bit0(M) != aa(num,num,bit1,N) ).

% semiring_norm(88)
tff(fact_221_semiring__norm_I89_J,axiom,
    ! [M: num,N: num] : aa(num,num,bit1,M) != bit0(N) ).

% semiring_norm(89)
tff(fact_222_semiring__norm_I84_J,axiom,
    ! [N: num] : one2 != aa(num,num,bit1,N) ).

% semiring_norm(84)
tff(fact_223_semiring__norm_I86_J,axiom,
    ! [M: num] : aa(num,num,bit1,M) != one2 ).

% semiring_norm(86)
tff(fact_224_semiring__norm_I6_J,axiom,
    ! [M: num,N: num] : aa(num,num,aa(num,fun(num,num),plus_plus(num),bit0(M)),bit0(N)) = bit0(aa(num,num,aa(num,fun(num,num),plus_plus(num),M),N)) ).

% semiring_norm(6)
tff(fact_225_semiring__norm_I13_J,axiom,
    ! [M: num,N: num] : aa(num,num,aa(num,fun(num,num),times_times(num),bit0(M)),bit0(N)) = bit0(bit0(aa(num,num,aa(num,fun(num,num),times_times(num),M),N))) ).

% semiring_norm(13)
tff(fact_226_semiring__norm_I11_J,axiom,
    ! [M: num] : aa(num,num,aa(num,fun(num,num),times_times(num),M),one2) = M ).

% semiring_norm(11)
tff(fact_227_semiring__norm_I12_J,axiom,
    ! [N: num] : aa(num,num,aa(num,fun(num,num),times_times(num),one2),N) = N ).

% semiring_norm(12)
tff(fact_228_semiring__norm_I71_J,axiom,
    ! [M: num,N: num] :
      ( pp(aa(num,bool,ord_less_eq(num,bit0(M)),bit0(N)))
    <=> pp(aa(num,bool,ord_less_eq(num,M),N)) ) ).

% semiring_norm(71)
tff(fact_229_semiring__norm_I68_J,axiom,
    ! [N: num] : pp(aa(num,bool,ord_less_eq(num,one2),N)) ).

% semiring_norm(68)
tff(fact_230_semiring__norm_I73_J,axiom,
    ! [M: num,N: num] :
      ( pp(aa(num,bool,ord_less_eq(num,aa(num,num,bit1,M)),aa(num,num,bit1,N)))
    <=> pp(aa(num,bool,ord_less_eq(num,M),N)) ) ).

% semiring_norm(73)
tff(fact_231_semiring__norm_I2_J,axiom,
    aa(num,num,aa(num,fun(num,num),plus_plus(num),one2),one2) = bit0(one2) ).

% semiring_norm(2)
tff(fact_232_semiring__norm_I7_J,axiom,
    ! [M: num,N: num] : aa(num,num,aa(num,fun(num,num),plus_plus(num),bit0(M)),aa(num,num,bit1,N)) = aa(num,num,bit1,aa(num,num,aa(num,fun(num,num),plus_plus(num),M),N)) ).

% semiring_norm(7)
tff(fact_233_semiring__norm_I9_J,axiom,
    ! [M: num,N: num] : aa(num,num,aa(num,fun(num,num),plus_plus(num),aa(num,num,bit1,M)),bit0(N)) = aa(num,num,bit1,aa(num,num,aa(num,fun(num,num),plus_plus(num),M),N)) ).

% semiring_norm(9)
tff(fact_234_left__add__mult__distrib,axiom,
    ! [I: nat,U: nat,J: nat,K: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I),U)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),U)),K)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),J)),U)),K) ).

% left_add_mult_distrib
tff(fact_235_t__buildup__cnt,axiom,
    ! [N: nat] : pp(aa(real,bool,ord_less_eq(real,aa(nat,real,semiring_1_of_nat(real),vEBT_V8346862874174094_d_u_p(N))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(vEBT_VEBT,real,vEBT_VEBT_cnt,vEBT_vebt_buildup(N))),aa(num,real,numeral_numeral(real),aa(num,num,bit1,bit0(aa(num,num,bit1,one2))))))) ).

% t_buildup_cnt
tff(fact_236_count__buildup_H,axiom,
    ! [N: nat] : pp(aa(real,bool,ord_less_eq(real,aa(vEBT_VEBT,real,vEBT_VEBT_cnt,vEBT_vebt_buildup(N))),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N))))) ).

% count_buildup'
tff(fact_237_zdiv__numeral__Bit1,axiom,
    ! [V: num,W: num] : aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,V))),aa(num,int,numeral_numeral(int),bit0(W))) = aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(num,int,numeral_numeral(int),V)),aa(num,int,numeral_numeral(int),W)) ).

% zdiv_numeral_Bit1
tff(fact_238_times__divide__eq__right,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),C2) ) ).

% times_divide_eq_right
tff(fact_239_divide__divide__eq__right,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A2: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),B2) ) ).

% divide_divide_eq_right
tff(fact_240_divide__divide__eq__left,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A2: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)),C2) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) ) ).

% divide_divide_eq_left
tff(fact_241_times__divide__eq__left,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [B2: A,C2: A,A2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)),A2) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),A2)),C2) ) ).

% times_divide_eq_left
tff(fact_242_field__sum__of__halves,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),aa(num,A,numeral_numeral(A),bit0(one2)))),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),aa(num,A,numeral_numeral(A),bit0(one2)))) = X ) ).

% field_sum_of_halves
tff(fact_243_add__le__cancel__right,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [A2: A,C2: A,B2: A] :
          ( pp(aa(A,bool,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2)))
        <=> pp(aa(A,bool,ord_less_eq(A,A2),B2)) ) ) ).

% add_le_cancel_right
tff(fact_244_add__le__cancel__left,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [C2: A,A2: A,B2: A] :
          ( pp(aa(A,bool,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),A2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),B2)))
        <=> pp(aa(A,bool,ord_less_eq(A,A2),B2)) ) ) ).

% add_le_cancel_left
tff(fact_245_buildup__nothing__in__leaf,axiom,
    ! [N: nat,X: nat] : ~ vEBT_V5719532721284313246member(vEBT_vebt_buildup(N),X) ).

% buildup_nothing_in_leaf
tff(fact_246_cnt__cnt__eq,axiom,
    ! [T: vEBT_VEBT] : aa(vEBT_VEBT,real,vEBT_VEBT_cnt,T) = aa(nat,real,semiring_1_of_nat(real),aa(vEBT_VEBT,nat,vEBT_VEBT_cnt2,T)) ).

% cnt_cnt_eq
tff(fact_247_add__right__cancel,axiom,
    ! [A: $tType] :
      ( cancel_semigroup_add(A)
     => ! [B2: A,A2: A,C2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),A2) )
        <=> ( B2 = C2 ) ) ) ).

% add_right_cancel
tff(fact_248_add__left__cancel,axiom,
    ! [A: $tType] :
      ( cancel_semigroup_add(A)
     => ! [A2: A,B2: A,C2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C2) )
        <=> ( B2 = C2 ) ) ) ).

% add_left_cancel
tff(fact_249_of__nat__eq__iff,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [M: nat,N: nat] :
          ( ( aa(nat,A,semiring_1_of_nat(A),M) = aa(nat,A,semiring_1_of_nat(A),N) )
        <=> ( M = N ) ) ) ).

% of_nat_eq_iff
tff(fact_250_of__nat__numeral,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [N: num] : aa(nat,A,semiring_1_of_nat(A),aa(num,nat,numeral_numeral(nat),N)) = aa(num,A,numeral_numeral(A),N) ) ).

% of_nat_numeral
tff(fact_251_of__nat__le__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [M: nat,N: nat] :
          ( pp(aa(A,bool,ord_less_eq(A,aa(nat,A,semiring_1_of_nat(A),M)),aa(nat,A,semiring_1_of_nat(A),N)))
        <=> pp(aa(nat,bool,ord_less_eq(nat,M),N)) ) ) ).

% of_nat_le_iff
tff(fact_252_of__nat__add,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [M: nat,N: nat] : aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),M)),aa(nat,A,semiring_1_of_nat(A),N)) ) ).

% of_nat_add
tff(fact_253_of__nat__mult,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [M: nat,N: nat] : aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),M)),aa(nat,A,semiring_1_of_nat(A),N)) ) ).

% of_nat_mult
tff(fact_254_semiring__1__class_Oof__nat__power,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [M: nat,N: nat] : aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),M),N)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(nat,A,semiring_1_of_nat(A),M)),N) ) ).

% semiring_1_class.of_nat_power
tff(fact_255_of__nat__eq__of__nat__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [B2: nat,W: nat,X: nat] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(nat,A,semiring_1_of_nat(A),B2)),W) = aa(nat,A,semiring_1_of_nat(A),X) )
        <=> ( aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),W) = X ) ) ) ).

% of_nat_eq_of_nat_power_cancel_iff
tff(fact_256_of__nat__power__eq__of__nat__cancel__iff,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [X: nat,B2: nat,W: nat] :
          ( ( aa(nat,A,semiring_1_of_nat(A),X) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(nat,A,semiring_1_of_nat(A),B2)),W) )
        <=> ( X = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),W) ) ) ) ).

% of_nat_power_eq_of_nat_cancel_iff
tff(fact_257_zdiv__numeral__Bit0,axiom,
    ! [V: num,W: num] : aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(num,int,numeral_numeral(int),bit0(V))),aa(num,int,numeral_numeral(int),bit0(W))) = aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(num,int,numeral_numeral(int),V)),aa(num,int,numeral_numeral(int),W)) ).

% zdiv_numeral_Bit0
tff(fact_258_numeral__power__eq__of__nat__cancel__iff,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [X: num,N: nat,Y: nat] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),X)),N) = aa(nat,A,semiring_1_of_nat(A),Y) )
        <=> ( aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),X)),N) = Y ) ) ) ).

% numeral_power_eq_of_nat_cancel_iff
tff(fact_259_real__of__nat__eq__numeral__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [Y: nat,X: num,N: nat] :
          ( ( aa(nat,A,semiring_1_of_nat(A),Y) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),X)),N) )
        <=> ( Y = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),X)),N) ) ) ) ).

% real_of_nat_eq_numeral_power_cancel_iff
tff(fact_260_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [B2: nat,W: nat,X: nat] :
          ( pp(aa(A,bool,ord_less_eq(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(nat,A,semiring_1_of_nat(A),B2)),W)),aa(nat,A,semiring_1_of_nat(A),X)))
        <=> pp(aa(nat,bool,ord_less_eq(nat,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),W)),X)) ) ) ).

% of_nat_le_of_nat_power_cancel_iff
tff(fact_261_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [X: nat,B2: nat,W: nat] :
          ( pp(aa(A,bool,ord_less_eq(A,aa(nat,A,semiring_1_of_nat(A),X)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(nat,A,semiring_1_of_nat(A),B2)),W)))
        <=> pp(aa(nat,bool,ord_less_eq(nat,X),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),W))) ) ) ).

% of_nat_power_le_of_nat_cancel_iff
tff(fact_262_numeral__le__real__of__nat__iff,axiom,
    ! [N: num,M: nat] :
      ( pp(aa(real,bool,ord_less_eq(real,aa(num,real,numeral_numeral(real),N)),aa(nat,real,semiring_1_of_nat(real),M)))
    <=> pp(aa(nat,bool,ord_less_eq(nat,aa(num,nat,numeral_numeral(nat),N)),M)) ) ).

% numeral_le_real_of_nat_iff
tff(fact_263_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [I: num,N: nat,X: nat] :
          ( pp(aa(A,bool,ord_less_eq(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),I)),N)),aa(nat,A,semiring_1_of_nat(A),X)))
        <=> pp(aa(nat,bool,ord_less_eq(nat,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),I)),N)),X)) ) ) ).

% numeral_power_le_of_nat_cancel_iff
tff(fact_264_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [X: nat,I: num,N: nat] :
          ( pp(aa(A,bool,ord_less_eq(A,aa(nat,A,semiring_1_of_nat(A),X)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),I)),N)))
        <=> pp(aa(nat,bool,ord_less_eq(nat,X),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),I)),N))) ) ) ).

% of_nat_le_numeral_power_cancel_iff
tff(fact_265_even__of__nat,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [N: nat] :
          ( pp(aa(A,bool,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),aa(nat,A,semiring_1_of_nat(A),N)))
        <=> pp(aa(nat,bool,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N)) ) ) ).

% even_of_nat
tff(fact_266_mult__of__nat__commute,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [X: nat,Y: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),X)),Y) = aa(A,A,aa(A,fun(A,A),times_times(A),Y),aa(nat,A,semiring_1_of_nat(A),X)) ) ).

% mult_of_nat_commute
tff(fact_267_real__of__nat__div4,axiom,
    ! [N: nat,X: nat] : pp(aa(real,bool,ord_less_eq(real,aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),X))),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(nat,real,semiring_1_of_nat(real),N)),aa(nat,real,semiring_1_of_nat(real),X)))) ).

% real_of_nat_div4
tff(fact_268_real__of__nat__div,axiom,
    ! [D: nat,N: nat] :
      ( pp(aa(nat,bool,dvd_dvd(nat,D),N))
     => ( aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),D)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(nat,real,semiring_1_of_nat(real),N)),aa(nat,real,semiring_1_of_nat(real),D)) ) ) ).

% real_of_nat_div
tff(fact_269_div__mult2__eq_H,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [A2: A,M: nat,N: nat] : aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),M)),aa(nat,A,semiring_1_of_nat(A),N))) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(nat,A,semiring_1_of_nat(A),M))),aa(nat,A,semiring_1_of_nat(A),N)) ) ).

% div_mult2_eq'
tff(fact_270_of__nat__mono,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [I: nat,J: nat] :
          ( pp(aa(nat,bool,ord_less_eq(nat,I),J))
         => pp(aa(A,bool,ord_less_eq(A,aa(nat,A,semiring_1_of_nat(A),I)),aa(nat,A,semiring_1_of_nat(A),J))) ) ) ).

% of_nat_mono
tff(fact_271_unique__euclidean__semiring__with__nat__class_Oof__nat__div,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [M: nat,N: nat] : aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),N)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,semiring_1_of_nat(A),M)),aa(nat,A,semiring_1_of_nat(A),N)) ) ).

% unique_euclidean_semiring_with_nat_class.of_nat_div
tff(fact_272_of__nat__dvd__iff,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [M: nat,N: nat] :
          ( pp(aa(A,bool,dvd_dvd(A,aa(nat,A,semiring_1_of_nat(A),M)),aa(nat,A,semiring_1_of_nat(A),N)))
        <=> pp(aa(nat,bool,dvd_dvd(nat,M),N)) ) ) ).

% of_nat_dvd_iff
tff(fact_273_add__right__imp__eq,axiom,
    ! [A: $tType] :
      ( cancel_semigroup_add(A)
     => ! [B2: A,A2: A,C2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),A2) )
         => ( B2 = C2 ) ) ) ).

% add_right_imp_eq
tff(fact_274_add__left__imp__eq,axiom,
    ! [A: $tType] :
      ( cancel_semigroup_add(A)
     => ! [A2: A,B2: A,C2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C2) )
         => ( B2 = C2 ) ) ) ).

% add_left_imp_eq
tff(fact_275_ab__semigroup__add__class_Oadd_Oleft__commute,axiom,
    ! [A: $tType] :
      ( ab_semigroup_add(A)
     => ! [B2: A,A2: A,C2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2)) ) ).

% ab_semigroup_add_class.add.left_commute
tff(fact_276_ab__semigroup__add__class_Oadd_Ocommute,axiom,
    ! [A: $tType] :
      ( ab_semigroup_add(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A2) ) ).

% ab_semigroup_add_class.add.commute
tff(fact_277_add_Oright__cancel,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [B2: A,A2: A,C2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),A2) )
        <=> ( B2 = C2 ) ) ) ).

% add.right_cancel
tff(fact_278_add_Oleft__cancel,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A,B2: A,C2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C2) )
        <=> ( B2 = C2 ) ) ) ).

% add.left_cancel
tff(fact_279_add_Oassoc,axiom,
    ! [A: $tType] :
      ( semigroup_add(A)
     => ! [A2: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2)) ) ).

% add.assoc
tff(fact_280_group__cancel_Oadd2,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [B4: A,K: A,B2: A,A2: A] :
          ( ( B4 = aa(A,A,aa(A,fun(A,A),plus_plus(A),K),B2) )
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B4) = aa(A,A,aa(A,fun(A,A),plus_plus(A),K),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)) ) ) ) ).

% group_cancel.add2
tff(fact_281_group__cancel_Oadd1,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [A3: A,K: A,A2: A,B2: A] :
          ( ( A3 = aa(A,A,aa(A,fun(A,A),plus_plus(A),K),A2) )
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),K),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)) ) ) ) ).

% group_cancel.add1
tff(fact_282_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 ) )
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K) = aa(A,A,aa(A,fun(A,A),plus_plus(A),J),L) ) ) ) ).

% add_mono_thms_linordered_semiring(4)
tff(fact_283_mult_Oassoc,axiom,
    ! [A: $tType] :
      ( semigroup_mult(A)
     => ! [A2: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),C2) = aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) ) ).

% mult.assoc
tff(fact_284_ab__semigroup__mult__class_Omult_Ocommute,axiom,
    ! [A: $tType] :
      ( ab_semigroup_mult(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2) = aa(A,A,aa(A,fun(A,A),times_times(A),B2),A2) ) ).

% ab_semigroup_mult_class.mult.commute
tff(fact_285_ab__semigroup__mult__class_Omult_Oleft__commute,axiom,
    ! [A: $tType] :
      ( ab_semigroup_mult(A)
     => ! [B2: A,A2: A,C2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)) = aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) ) ).

% ab_semigroup_mult_class.mult.left_commute
tff(fact_286_complete__real,axiom,
    ! [S: set(real)] :
      ( ? [X4: real] : pp(aa(set(real),bool,aa(real,fun(set(real),bool),member(real),X4),S))
     => ( ? [Z3: real] :
          ! [X3: real] :
            ( pp(aa(set(real),bool,aa(real,fun(set(real),bool),member(real),X3),S))
           => pp(aa(real,bool,ord_less_eq(real,X3),Z3)) )
       => ? [Y3: real] :
            ( ! [X4: real] :
                ( pp(aa(set(real),bool,aa(real,fun(set(real),bool),member(real),X4),S))
               => pp(aa(real,bool,ord_less_eq(real,X4),Y3)) )
            & ! [Z3: real] :
                ( ! [X3: real] :
                    ( pp(aa(set(real),bool,aa(real,fun(set(real),bool),member(real),X3),S))
                   => pp(aa(real,bool,ord_less_eq(real,X3),Z3)) )
               => pp(aa(real,bool,ord_less_eq(real,Y3),Z3)) ) ) ) ) ).

% complete_real
tff(fact_287_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [A: $tType] :
      ( ordere6658533253407199908up_add(A)
     => ! [I: A,J: A,K: A,L: A] :
          ( ( pp(aa(A,bool,ord_less_eq(A,I),J))
            & ( K = L ) )
         => pp(aa(A,bool,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),L))) ) ) ).

% add_mono_thms_linordered_semiring(3)
tff(fact_288_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [A: $tType] :
      ( ordere6658533253407199908up_add(A)
     => ! [I: A,J: A,K: A,L: A] :
          ( ( ( I = J )
            & pp(aa(A,bool,ord_less_eq(A,K),L)) )
         => pp(aa(A,bool,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),L))) ) ) ).

% add_mono_thms_linordered_semiring(2)
tff(fact_289_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [A: $tType] :
      ( ordere6658533253407199908up_add(A)
     => ! [I: A,J: A,K: A,L: A] :
          ( ( pp(aa(A,bool,ord_less_eq(A,I),J))
            & pp(aa(A,bool,ord_less_eq(A,K),L)) )
         => pp(aa(A,bool,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),L))) ) ) ).

% add_mono_thms_linordered_semiring(1)
tff(fact_290_add__mono,axiom,
    ! [A: $tType] :
      ( ordere6658533253407199908up_add(A)
     => ! [A2: A,B2: A,C2: A,D: A] :
          ( pp(aa(A,bool,ord_less_eq(A,A2),B2))
         => ( pp(aa(A,bool,ord_less_eq(A,C2),D))
           => pp(aa(A,bool,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),D))) ) ) ) ).

% add_mono
tff(fact_291_add__left__mono,axiom,
    ! [A: $tType] :
      ( ordere6658533253407199908up_add(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,ord_less_eq(A,A2),B2))
         => pp(aa(A,bool,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),A2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),B2))) ) ) ).

% add_left_mono
tff(fact_292_less__eqE,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,ord_less_eq(A,A2),B2))
         => ~ ! [C3: A] : B2 != aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C3) ) ) ).

% less_eqE
tff(fact_293_add__right__mono,axiom,
    ! [A: $tType] :
      ( ordere6658533253407199908up_add(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,ord_less_eq(A,A2),B2))
         => pp(aa(A,bool,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2))) ) ) ).

% add_right_mono
tff(fact_294_le__iff__add,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,ord_less_eq(A,A2),B2))
        <=> ? [C4: A] : B2 = aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C4) ) ) ).

% le_iff_add
tff(fact_295_add__le__imp__le__left,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [C2: A,A2: A,B2: A] :
          ( pp(aa(A,bool,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),A2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),B2)))
         => pp(aa(A,bool,ord_less_eq(A,A2),B2)) ) ) ).

% add_le_imp_le_left
tff(fact_296_add__le__imp__le__right,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [A2: A,C2: A,B2: A] :
          ( pp(aa(A,bool,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2)))
         => pp(aa(A,bool,ord_less_eq(A,A2),B2)) ) ) ).

% add_le_imp_le_right
tff(fact_297_add__divide__distrib,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),C2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)) ) ).

% add_divide_distrib
tff(fact_298_times__divide__times__eq,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [X: A,Y: A,Z: A,W: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y)),aa(A,A,aa(A,fun(A,A),divide_divide(A),Z),W)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),Z)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),W)) ) ).

% times_divide_times_eq
tff(fact_299_divide__divide__times__eq,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [X: A,Y: A,Z: A,W: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y)),aa(A,A,aa(A,fun(A,A),divide_divide(A),Z),W)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),W)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Z)) ) ).

% divide_divide_times_eq
tff(fact_300_divide__divide__eq__left_H,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)),C2) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)) ) ).

% divide_divide_eq_left'
tff(fact_301_space__cnt,axiom,
    ! [T: vEBT_VEBT] : pp(aa(real,bool,ord_less_eq(real,aa(nat,real,semiring_1_of_nat(real),aa(vEBT_VEBT,nat,vEBT_VEBT_space2,T))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(aa(num,num,bit1,one2)))),aa(vEBT_VEBT,real,vEBT_VEBT_cnt,T)))) ).

% space_cnt
tff(fact_302_t__build__cnt,axiom,
    ! [N: nat] : pp(aa(real,bool,ord_less_eq(real,aa(nat,real,semiring_1_of_nat(real),vEBT_V8646137997579335489_i_l_d(N))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(vEBT_VEBT,real,vEBT_VEBT_cnt,vEBT_vebt_buildup(N))),aa(num,real,numeral_numeral(real),aa(num,num,bit1,bit0(aa(num,num,bit1,one2))))))) ).

% t_build_cnt
tff(fact_303_two__realpow__ge__two,axiom,
    ! [N: nat] :
      ( pp(aa(real,bool,ord_less_eq(real,one_one(real)),aa(nat,real,semiring_1_of_nat(real),N)))
     => pp(aa(real,bool,ord_less_eq(real,aa(num,real,numeral_numeral(real),bit0(one2))),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(num,real,numeral_numeral(real),bit0(one2))),N))) ) ).

% two_realpow_ge_two
tff(fact_304_VEBT__internal_OTb_H_Oelims,axiom,
    ! [X: nat,Y: nat] :
      ( ( vEBT_VEBT_Tb2(X) = Y )
     => ( ( ( X = zero_zero(nat) )
         => ( Y != aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2)) ) )
       => ( ( ( X = aa(nat,nat,suc,zero_zero(nat)) )
           => ( Y != aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2)) ) )
         => ~ ! [N2: nat] :
                ( ( X = aa(nat,nat,suc,aa(nat,nat,suc,N2)) )
               => ~ ( ( pp(aa(nat,bool,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N2))
                     => ( Y = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,bit0(one2)))),vEBT_VEBT_Tb2(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N2),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),vEBT_VEBT_Tb2(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N2),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N2),aa(num,nat,numeral_numeral(nat),bit0(one2))))))) ) )
                    & ( ~ pp(aa(nat,bool,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N2))
                     => ( Y = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,bit0(one2)))),vEBT_VEBT_Tb2(aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N2),aa(num,nat,numeral_numeral(nat),bit0(one2)))))))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),vEBT_VEBT_Tb2(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N2),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N2),aa(num,nat,numeral_numeral(nat),bit0(one2)))))))) ) ) ) ) ) ) ) ).

% VEBT_internal.Tb'.elims
tff(fact_305_triangle__def,axiom,
    ! [N: nat] : nat_triangle(N) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),aa(nat,nat,suc,N))),aa(num,nat,numeral_numeral(nat),bit0(one2))) ).

% triangle_def
tff(fact_306_VEBT__internal_OT__vebt__buildupi_Osimps_I3_J,axiom,
    ! [N: nat] :
      ( ( pp(aa(nat,bool,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N))
       => ( vEBT_V441764108873111860ildupi(aa(nat,nat,suc,aa(nat,nat,suc,N))) = aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_V441764108873111860ildupi(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(one2)))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),vEBT_V441764108873111860ildupi(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2))))))))))) ) )
      & ( ~ pp(aa(nat,bool,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N))
       => ( vEBT_V441764108873111860ildupi(aa(nat,nat,suc,aa(nat,nat,suc,N))) = aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_V441764108873111860ildupi(aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(bit0(one2))))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(one2)))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),vEBT_V441764108873111860ildupi(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2))))))))))) ) ) ) ).

% VEBT_internal.T_vebt_buildupi.simps(3)
tff(fact_307_VEBT__internal_OT__vebt__buildupi_H_Osimps_I3_J,axiom,
    ! [N: nat] :
      ( ( pp(aa(nat,bool,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N))
       => ( vEBT_V9176841429113362141ildupi(aa(nat,nat,suc,aa(nat,nat,suc,N))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,one2))),aa(int,int,aa(int,fun(int,int),plus_plus(int),vEBT_V9176841429113362141ildupi(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(bit0(one2)))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(int,int,aa(int,fun(int,int),times_times(int),vEBT_V9176841429113362141ildupi(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2))))))))) ) )
      & ( ~ pp(aa(nat,bool,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N))
       => ( vEBT_V9176841429113362141ildupi(aa(nat,nat,suc,aa(nat,nat,suc,N))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,one2))),aa(int,int,aa(int,fun(int,int),plus_plus(int),vEBT_V9176841429113362141ildupi(aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(bit0(bit0(one2))))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(bit0(one2)))),aa(int,int,aa(int,fun(int,int),times_times(int),vEBT_V9176841429113362141ildupi(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2))))))))) ) ) ) ).

% VEBT_internal.T_vebt_buildupi'.simps(3)
tff(fact_308_high__def,axiom,
    ! [X: nat,N: nat] : vEBT_VEBT_high(X,N) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),X),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)) ).

% high_def
tff(fact_309_VEBT__internal_OTb_Osimps_I3_J,axiom,
    ! [N: nat] :
      ( ( pp(aa(nat,bool,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N))
       => ( vEBT_VEBT_Tb(aa(nat,nat,suc,aa(nat,nat,suc,N))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,bit0(one2)))),vEBT_VEBT_Tb(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),aa(int,int,aa(int,fun(int,int),times_times(int),vEBT_VEBT_Tb(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2))))))) ) )
      & ( ~ pp(aa(nat,bool,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N))
       => ( vEBT_VEBT_Tb(aa(nat,nat,suc,aa(nat,nat,suc,N))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,bit0(one2)))),vEBT_VEBT_Tb(aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2)))))))),aa(int,int,aa(int,fun(int,int),times_times(int),vEBT_VEBT_Tb(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2)))))))) ) ) ) ).

% VEBT_internal.Tb.simps(3)
tff(fact_310_bezout__add__nat,axiom,
    ! [A2: nat,B2: nat] :
    ? [D2: nat,X3: nat,Y3: nat] :
      ( pp(aa(nat,bool,dvd_dvd(nat,D2),A2))
      & pp(aa(nat,bool,dvd_dvd(nat,D2),B2))
      & ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A2),X3) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),Y3)),D2) )
        | ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),X3) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A2),Y3)),D2) ) ) ) ).

% bezout_add_nat
tff(fact_311_bezout__lemma__nat,axiom,
    ! [D: nat,A2: nat,B2: nat,X: nat,Y: nat] :
      ( pp(aa(nat,bool,dvd_dvd(nat,D),A2))
     => ( pp(aa(nat,bool,dvd_dvd(nat,D),B2))
       => ( ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A2),X) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),Y)),D) )
            | ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),X) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A2),Y)),D) ) )
         => ? [X3: nat,Y3: nat] :
              ( pp(aa(nat,bool,dvd_dvd(nat,D),A2))
              & pp(aa(nat,bool,dvd_dvd(nat,D),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A2),B2)))
              & ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A2),X3) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A2),B2)),Y3)),D) )
                | ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A2),B2)),X3) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A2),Y3)),D) ) ) ) ) ) ) ).

% bezout_lemma_nat
tff(fact_312_Tb__Tb_H,axiom,
    ! [T: nat] : vEBT_VEBT_Tb(T) = aa(nat,int,semiring_1_of_nat(int),vEBT_VEBT_Tb2(T)) ).

% Tb_Tb'
tff(fact_313_Tbuildupi__buildupi_H,axiom,
    ! [N: nat] : aa(nat,int,semiring_1_of_nat(int),vEBT_V441764108873111860ildupi(N)) = vEBT_V9176841429113362141ildupi(N) ).

% Tbuildupi_buildupi'
tff(fact_314_le__zero__eq,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [N: A] :
          ( pp(aa(A,bool,ord_less_eq(A,N),zero_zero(A)))
        <=> ( N = zero_zero(A) ) ) ) ).

% le_zero_eq
tff(fact_315_add_Oright__neutral,axiom,
    ! [A: $tType] :
      ( monoid_add(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),zero_zero(A)) = A2 ) ).

% add.right_neutral
tff(fact_316_double__zero__sym,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A2: A] :
          ( ( zero_zero(A) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),A2) )
        <=> ( A2 = zero_zero(A) ) ) ) ).

% double_zero_sym
tff(fact_317_add__cancel__left__left,axiom,
    ! [A: $tType] :
      ( cancel1802427076303600483id_add(A)
     => ! [B2: A,A2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A2) = A2 )
        <=> ( B2 = zero_zero(A) ) ) ) ).

% add_cancel_left_left
tff(fact_318_add__cancel__left__right,axiom,
    ! [A: $tType] :
      ( cancel1802427076303600483id_add(A)
     => ! [A2: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2) = A2 )
        <=> ( B2 = zero_zero(A) ) ) ) ).

% add_cancel_left_right
tff(fact_319_add__cancel__right__left,axiom,
    ! [A: $tType] :
      ( cancel1802427076303600483id_add(A)
     => ! [A2: A,B2: A] :
          ( ( A2 = aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A2) )
        <=> ( B2 = zero_zero(A) ) ) ) ).

% add_cancel_right_left
tff(fact_320_add__cancel__right__right,axiom,
    ! [A: $tType] :
      ( cancel1802427076303600483id_add(A)
     => ! [A2: A,B2: A] :
          ( ( A2 = aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2) )
        <=> ( B2 = zero_zero(A) ) ) ) ).

% add_cancel_right_right
tff(fact_321_add__eq__0__iff__both__eq__0,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [X: A,Y: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y) = zero_zero(A) )
        <=> ( ( X = zero_zero(A) )
            & ( Y = zero_zero(A) ) ) ) ) ).

% add_eq_0_iff_both_eq_0
tff(fact_322_zero__eq__add__iff__both__eq__0,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [X: A,Y: A] :
          ( ( zero_zero(A) = aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y) )
        <=> ( ( X = zero_zero(A) )
            & ( Y = zero_zero(A) ) ) ) ) ).

% zero_eq_add_iff_both_eq_0
tff(fact_323_add__0,axiom,
    ! [A: $tType] :
      ( monoid_add(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),zero_zero(A)),A2) = A2 ) ).

% add_0
tff(fact_324_mult__cancel__right,axiom,
    ! [A: $tType] :
      ( semiri6575147826004484403cancel(A)
     => ! [A2: A,C2: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2) = aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2) )
        <=> ( ( C2 = zero_zero(A) )
            | ( A2 = B2 ) ) ) ) ).

% mult_cancel_right
tff(fact_325_mult__cancel__left,axiom,
    ! [A: $tType] :
      ( semiri6575147826004484403cancel(A)
     => ! [C2: A,A2: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2) = aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2) )
        <=> ( ( C2 = zero_zero(A) )
            | ( A2 = B2 ) ) ) ) ).

% mult_cancel_left
tff(fact_326_mult__eq__0__iff,axiom,
    ! [A: $tType] :
      ( semiri3467727345109120633visors(A)
     => ! [A2: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2) = zero_zero(A) )
        <=> ( ( A2 = zero_zero(A) )
            | ( B2 = zero_zero(A) ) ) ) ) ).

% mult_eq_0_iff
tff(fact_327_mult__zero__right,axiom,
    ! [A: $tType] :
      ( mult_zero(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),A2),zero_zero(A)) = zero_zero(A) ) ).

% mult_zero_right
tff(fact_328_mult__zero__left,axiom,
    ! [A: $tType] :
      ( mult_zero(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),zero_zero(A)),A2) = zero_zero(A) ) ).

% mult_zero_left
tff(fact_329_div__by__0,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),zero_zero(A)) = zero_zero(A) ) ).

% div_by_0
tff(fact_330_div__0,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),zero_zero(A)),A2) = zero_zero(A) ) ).

% div_0
tff(fact_331_bits__div__by__0,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),zero_zero(A)) = zero_zero(A) ) ).

% bits_div_by_0
tff(fact_332_bits__div__0,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),zero_zero(A)),A2) = zero_zero(A) ) ).

% bits_div_0
tff(fact_333_division__ring__divide__zero,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),zero_zero(A)) = zero_zero(A) ) ).

% division_ring_divide_zero
tff(fact_334_divide__cancel__right,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A2: A,C2: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),C2) = aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2) )
        <=> ( ( C2 = zero_zero(A) )
            | ( A2 = B2 ) ) ) ) ).

% divide_cancel_right
tff(fact_335_divide__cancel__left,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [C2: A,A2: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),A2) = aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),B2) )
        <=> ( ( C2 = zero_zero(A) )
            | ( A2 = B2 ) ) ) ) ).

% divide_cancel_left
tff(fact_336_divide__eq__0__iff,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A2: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2) = zero_zero(A) )
        <=> ( ( A2 = zero_zero(A) )
            | ( B2 = zero_zero(A) ) ) ) ) ).

% divide_eq_0_iff
tff(fact_337_mult__1,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),one_one(A)),A2) = A2 ) ).

% mult_1
tff(fact_338_mult_Oright__neutral,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),A2),one_one(A)) = A2 ) ).

% mult.right_neutral
tff(fact_339_div__by__1,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),one_one(A)) = A2 ) ).

% div_by_1
tff(fact_340_bits__div__by__1,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),one_one(A)) = A2 ) ).

% bits_div_by_1
tff(fact_341_power__one,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [N: nat] : aa(nat,A,aa(A,fun(nat,A),power_power(A),one_one(A)),N) = one_one(A) ) ).

% power_one
tff(fact_342_dvd__0__left__iff,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A2: A] :
          ( pp(aa(A,bool,dvd_dvd(A,zero_zero(A)),A2))
        <=> ( A2 = zero_zero(A) ) ) ) ).

% dvd_0_left_iff
tff(fact_343_dvd__0__right,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A2: A] : pp(aa(A,bool,dvd_dvd(A,A2),zero_zero(A))) ) ).

% dvd_0_right
tff(fact_344_of__nat__eq__1__iff,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [N: nat] :
          ( ( aa(nat,A,semiring_1_of_nat(A),N) = one_one(A) )
        <=> ( N = one_one(nat) ) ) ) ).

% of_nat_eq_1_iff
tff(fact_345_of__nat__1__eq__iff,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [N: nat] :
          ( ( one_one(A) = aa(nat,A,semiring_1_of_nat(A),N) )
        <=> ( N = one_one(nat) ) ) ) ).

% of_nat_1_eq_iff
tff(fact_346_of__nat__1,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ( aa(nat,A,semiring_1_of_nat(A),one_one(nat)) = one_one(A) ) ) ).

% of_nat_1
tff(fact_347_add__is__0,axiom,
    ! [M: nat,N: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N) = zero_zero(nat) )
    <=> ( ( M = zero_zero(nat) )
        & ( N = zero_zero(nat) ) ) ) ).

% add_is_0
tff(fact_348_Nat_Oadd__0__right,axiom,
    ! [M: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),zero_zero(nat)) = M ).

% Nat.add_0_right
tff(fact_349_le0,axiom,
    ! [N: nat] : pp(aa(nat,bool,ord_less_eq(nat,zero_zero(nat)),N)) ).

% le0
tff(fact_350_bot__nat__0_Oextremum,axiom,
    ! [A2: nat] : pp(aa(nat,bool,ord_less_eq(nat,zero_zero(nat)),A2)) ).

% bot_nat_0.extremum
tff(fact_351_mult__cancel2,axiom,
    ! [M: nat,K: nat,N: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),K) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),K) )
    <=> ( ( M = N )
        | ( K = zero_zero(nat) ) ) ) ).

% mult_cancel2
tff(fact_352_mult__cancel1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),M) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),N) )
    <=> ( ( M = N )
        | ( K = zero_zero(nat) ) ) ) ).

% mult_cancel1
tff(fact_353_mult__0__right,axiom,
    ! [M: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),zero_zero(nat)) = zero_zero(nat) ).

% mult_0_right
tff(fact_354_mult__is__0,axiom,
    ! [M: nat,N: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N) = zero_zero(nat) )
    <=> ( ( M = zero_zero(nat) )
        | ( N = zero_zero(nat) ) ) ) ).

% mult_is_0
tff(fact_355_triangle__0,axiom,
    nat_triangle(zero_zero(nat)) = zero_zero(nat) ).

% triangle_0
tff(fact_356_add__le__same__cancel1,axiom,
    ! [A: $tType] :
      ( ordere1937475149494474687imp_le(A)
     => ! [B2: A,A2: A] :
          ( pp(aa(A,bool,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A2)),B2))
        <=> pp(aa(A,bool,ord_less_eq(A,A2),zero_zero(A))) ) ) ).

% add_le_same_cancel1
tff(fact_357_add__le__same__cancel2,axiom,
    ! [A: $tType] :
      ( ordere1937475149494474687imp_le(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),B2))
        <=> pp(aa(A,bool,ord_less_eq(A,A2),zero_zero(A))) ) ) ).

% add_le_same_cancel2
tff(fact_358_le__add__same__cancel1,axiom,
    ! [A: $tType] :
      ( ordere1937475149494474687imp_le(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,ord_less_eq(A,A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)))
        <=> pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),B2)) ) ) ).

% le_add_same_cancel1
tff(fact_359_le__add__same__cancel2,axiom,
    ! [A: $tType] :
      ( ordere1937475149494474687imp_le(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,ord_less_eq(A,A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A2)))
        <=> pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),B2)) ) ) ).

% le_add_same_cancel2
tff(fact_360_double__add__le__zero__iff__single__add__le__zero,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A2: A] :
          ( pp(aa(A,bool,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),A2)),zero_zero(A)))
        <=> pp(aa(A,bool,ord_less_eq(A,A2),zero_zero(A))) ) ) ).

% double_add_le_zero_iff_single_add_le_zero
tff(fact_361_zero__le__double__add__iff__zero__le__single__add,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A2: A] :
          ( pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),A2)))
        <=> pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),A2)) ) ) ).

% zero_le_double_add_iff_zero_le_single_add
tff(fact_362_sum__squares__eq__zero__iff,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [X: A,Y: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),X)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Y)) = zero_zero(A) )
        <=> ( ( X = zero_zero(A) )
            & ( Y = zero_zero(A) ) ) ) ) ).

% sum_squares_eq_zero_iff
tff(fact_363_mult__cancel__left1,axiom,
    ! [A: $tType] :
      ( ring_15535105094025558882visors(A)
     => ! [C2: A,B2: A] :
          ( ( C2 = aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2) )
        <=> ( ( C2 = zero_zero(A) )
            | ( B2 = one_one(A) ) ) ) ) ).

% mult_cancel_left1
tff(fact_364_mult__cancel__left2,axiom,
    ! [A: $tType] :
      ( ring_15535105094025558882visors(A)
     => ! [C2: A,A2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2) = C2 )
        <=> ( ( C2 = zero_zero(A) )
            | ( A2 = one_one(A) ) ) ) ) ).

% mult_cancel_left2
tff(fact_365_mult__cancel__right1,axiom,
    ! [A: $tType] :
      ( ring_15535105094025558882visors(A)
     => ! [C2: A,B2: A] :
          ( ( C2 = aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2) )
        <=> ( ( C2 = zero_zero(A) )
            | ( B2 = one_one(A) ) ) ) ) ).

% mult_cancel_right1
tff(fact_366_mult__cancel__right2,axiom,
    ! [A: $tType] :
      ( ring_15535105094025558882visors(A)
     => ! [A2: A,C2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2) = C2 )
        <=> ( ( C2 = zero_zero(A) )
            | ( A2 = one_one(A) ) ) ) ) ).

% mult_cancel_right2
tff(fact_367_one__eq__numeral__iff,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [N: num] :
          ( ( one_one(A) = aa(num,A,numeral_numeral(A),N) )
        <=> ( one2 = N ) ) ) ).

% one_eq_numeral_iff
tff(fact_368_numeral__eq__one__iff,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [N: num] :
          ( ( aa(num,A,numeral_numeral(A),N) = one_one(A) )
        <=> ( N = one2 ) ) ) ).

% numeral_eq_one_iff
tff(fact_369_div__self,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [A2: A] :
          ( ( A2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),A2) = one_one(A) ) ) ) ).

% div_self
tff(fact_370_zero__eq__1__divide__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( ( zero_zero(A) = aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),A2) )
        <=> ( A2 = zero_zero(A) ) ) ) ).

% zero_eq_1_divide_iff
tff(fact_371_one__divide__eq__0__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),A2) = zero_zero(A) )
        <=> ( A2 = zero_zero(A) ) ) ) ).

% one_divide_eq_0_iff
tff(fact_372_eq__divide__eq__1,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,A2: A] :
          ( ( one_one(A) = aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A2) )
        <=> ( ( A2 != zero_zero(A) )
            & ( A2 = B2 ) ) ) ) ).

% eq_divide_eq_1
tff(fact_373_divide__eq__eq__1,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,A2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A2) = one_one(A) )
        <=> ( ( A2 != zero_zero(A) )
            & ( A2 = B2 ) ) ) ) ).

% divide_eq_eq_1
tff(fact_374_divide__self__if,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A] :
          ( ( ( A2 = zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),A2) = zero_zero(A) ) )
          & ( ( A2 != zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),A2) = one_one(A) ) ) ) ) ).

% divide_self_if
tff(fact_375_divide__self,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A] :
          ( ( A2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),A2) = one_one(A) ) ) ) ).

% divide_self
tff(fact_376_one__eq__divide__iff,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A2: A,B2: A] :
          ( ( one_one(A) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2) )
        <=> ( ( B2 != zero_zero(A) )
            & ( A2 = B2 ) ) ) ) ).

% one_eq_divide_iff
tff(fact_377_divide__eq__1__iff,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A2: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2) = one_one(A) )
        <=> ( ( B2 != zero_zero(A) )
            & ( A2 = B2 ) ) ) ) ).

% divide_eq_1_iff
tff(fact_378_div__mult__mult1__if,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [C2: A,A2: A,B2: A] :
          ( ( ( C2 = zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)) = zero_zero(A) ) )
          & ( ( C2 != zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2) ) ) ) ) ).

% div_mult_mult1_if
tff(fact_379_div__mult__mult2,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [C2: A,A2: A,B2: A] :
          ( ( C2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2) ) ) ) ).

% div_mult_mult2
tff(fact_380_div__mult__mult1,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [C2: A,A2: A,B2: A] :
          ( ( C2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2) ) ) ) ).

% div_mult_mult1
tff(fact_381_nonzero__mult__div__cancel__right,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [B2: A,A2: A] :
          ( ( B2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),B2) = A2 ) ) ) ).

% nonzero_mult_div_cancel_right
tff(fact_382_nonzero__mult__div__cancel__left,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [A2: A,B2: A] :
          ( ( A2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),A2) = B2 ) ) ) ).

% nonzero_mult_div_cancel_left
tff(fact_383_nonzero__mult__divide__mult__cancel__right2,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [C2: A,A2: A,B2: A] :
          ( ( C2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2) ) ) ) ).

% nonzero_mult_divide_mult_cancel_right2
tff(fact_384_nonzero__mult__divide__mult__cancel__right,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [C2: A,A2: A,B2: A] :
          ( ( C2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2) ) ) ) ).

% nonzero_mult_divide_mult_cancel_right
tff(fact_385_nonzero__mult__divide__mult__cancel__left2,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [C2: A,A2: A,B2: A] :
          ( ( C2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2) ) ) ) ).

% nonzero_mult_divide_mult_cancel_left2
tff(fact_386_nonzero__mult__divide__mult__cancel__left,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [C2: A,A2: A,B2: A] :
          ( ( C2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2) ) ) ) ).

% nonzero_mult_divide_mult_cancel_left
tff(fact_387_mult__divide__mult__cancel__left__if,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [C2: A,A2: A,B2: A] :
          ( ( ( C2 = zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)) = zero_zero(A) ) )
          & ( ( C2 != zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2) ) ) ) ) ).

% mult_divide_mult_cancel_left_if
tff(fact_388_power__0__Suc,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [N: nat] : aa(nat,A,aa(A,fun(nat,A),power_power(A),zero_zero(A)),aa(nat,nat,suc,N)) = zero_zero(A) ) ).

% power_0_Suc
tff(fact_389_power__zero__numeral,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [K: num] : aa(nat,A,aa(A,fun(nat,A),power_power(A),zero_zero(A)),aa(num,nat,numeral_numeral(nat),K)) = zero_zero(A) ) ).

% power_zero_numeral
tff(fact_390_dvd__times__right__cancel__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A,C2: A] :
          ( ( A2 != zero_zero(A) )
         => ( pp(aa(A,bool,dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),B2),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2)))
          <=> pp(aa(A,bool,dvd_dvd(A,B2),C2)) ) ) ) ).

% dvd_times_right_cancel_iff
tff(fact_391_dvd__times__left__cancel__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A,C2: A] :
          ( ( A2 != zero_zero(A) )
         => ( pp(aa(A,bool,dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)))
          <=> pp(aa(A,bool,dvd_dvd(A,B2),C2)) ) ) ) ).

% dvd_times_left_cancel_iff
tff(fact_392_dvd__mult__cancel__right,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [A2: A,C2: A,B2: A] :
          ( pp(aa(A,bool,dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)))
        <=> ( ( C2 = zero_zero(A) )
            | pp(aa(A,bool,dvd_dvd(A,A2),B2)) ) ) ) ).

% dvd_mult_cancel_right
tff(fact_393_dvd__mult__cancel__left,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [C2: A,A2: A,B2: A] :
          ( pp(aa(A,bool,dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)))
        <=> ( ( C2 = zero_zero(A) )
            | pp(aa(A,bool,dvd_dvd(A,A2),B2)) ) ) ) ).

% dvd_mult_cancel_left
tff(fact_394_of__nat__eq__0__iff,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [M: nat] :
          ( ( aa(nat,A,semiring_1_of_nat(A),M) = zero_zero(A) )
        <=> ( M = zero_zero(nat) ) ) ) ).

% of_nat_eq_0_iff
tff(fact_395_of__nat__0__eq__iff,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [N: nat] :
          ( ( zero_zero(A) = aa(nat,A,semiring_1_of_nat(A),N) )
        <=> ( zero_zero(nat) = N ) ) ) ).

% of_nat_0_eq_iff
tff(fact_396_semiring__1__class_Oof__nat__0,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ( aa(nat,A,semiring_1_of_nat(A),zero_zero(nat)) = zero_zero(A) ) ) ).

% semiring_1_class.of_nat_0
tff(fact_397_unit__prod,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,dvd_dvd(A,A2),one_one(A)))
         => ( pp(aa(A,bool,dvd_dvd(A,B2),one_one(A)))
           => pp(aa(A,bool,dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),one_one(A))) ) ) ) ).

% unit_prod
tff(fact_398_power__Suc0__right,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [A2: A] : aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(nat,nat,suc,zero_zero(nat))) = A2 ) ).

% power_Suc0_right
tff(fact_399_unit__div__1__div__1,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A] :
          ( pp(aa(A,bool,dvd_dvd(A,A2),one_one(A)))
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),A2)) = A2 ) ) ) ).

% unit_div_1_div_1
tff(fact_400_unit__div__1__unit,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A] :
          ( pp(aa(A,bool,dvd_dvd(A,A2),one_one(A)))
         => pp(aa(A,bool,dvd_dvd(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),A2)),one_one(A))) ) ) ).

% unit_div_1_unit
tff(fact_401_unit__div,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,dvd_dvd(A,A2),one_one(A)))
         => ( pp(aa(A,bool,dvd_dvd(A,B2),one_one(A)))
           => pp(aa(A,bool,dvd_dvd(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)),one_one(A))) ) ) ) ).

% unit_div
tff(fact_402_div__by__Suc__0,axiom,
    ! [M: nat] : aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),aa(nat,nat,suc,zero_zero(nat))) = M ).

% div_by_Suc_0
tff(fact_403_one__eq__mult__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( aa(nat,nat,suc,zero_zero(nat)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N) )
    <=> ( ( M = aa(nat,nat,suc,zero_zero(nat)) )
        & ( N = aa(nat,nat,suc,zero_zero(nat)) ) ) ) ).

% one_eq_mult_iff
tff(fact_404_mult__eq__1__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N) = aa(nat,nat,suc,zero_zero(nat)) )
    <=> ( ( M = aa(nat,nat,suc,zero_zero(nat)) )
        & ( N = aa(nat,nat,suc,zero_zero(nat)) ) ) ) ).

% mult_eq_1_iff
tff(fact_405_nat__power__eq__Suc__0__iff,axiom,
    ! [X: nat,M: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),X),M) = aa(nat,nat,suc,zero_zero(nat)) )
    <=> ( ( M = zero_zero(nat) )
        | ( X = aa(nat,nat,suc,zero_zero(nat)) ) ) ) ).

% nat_power_eq_Suc_0_iff
tff(fact_406_power__Suc__0,axiom,
    ! [N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(nat,nat,suc,zero_zero(nat))),N) = aa(nat,nat,suc,zero_zero(nat)) ).

% power_Suc_0
tff(fact_407_dvd__1__iff__1,axiom,
    ! [M: nat] :
      ( pp(aa(nat,bool,dvd_dvd(nat,M),aa(nat,nat,suc,zero_zero(nat))))
    <=> ( M = aa(nat,nat,suc,zero_zero(nat)) ) ) ).

% dvd_1_iff_1
tff(fact_408_dvd__1__left,axiom,
    ! [K: nat] : pp(aa(nat,bool,dvd_dvd(nat,aa(nat,nat,suc,zero_zero(nat))),K)) ).

% dvd_1_left
tff(fact_409_nat__mult__div__cancel__disj,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ( K = zero_zero(nat) )
       => ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),M)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),N)) = zero_zero(nat) ) )
      & ( ( K != zero_zero(nat) )
       => ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),M)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),N)) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),N) ) ) ) ).

% nat_mult_div_cancel_disj
tff(fact_410_nat__mult__dvd__cancel__disj,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa(nat,bool,dvd_dvd(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),M)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),N)))
    <=> ( ( K = zero_zero(nat) )
        | pp(aa(nat,bool,dvd_dvd(nat,M),N)) ) ) ).

% nat_mult_dvd_cancel_disj
tff(fact_411_triangle__Suc,axiom,
    ! [N: nat] : nat_triangle(aa(nat,nat,suc,N)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),nat_triangle(N)),aa(nat,nat,suc,N)) ).

% triangle_Suc
tff(fact_412_zero__le__divide__1__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),A2)))
        <=> pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),A2)) ) ) ).

% zero_le_divide_1_iff
tff(fact_413_divide__le__0__1__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( pp(aa(A,bool,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),A2)),zero_zero(A)))
        <=> pp(aa(A,bool,ord_less_eq(A,A2),zero_zero(A))) ) ) ).

% divide_le_0_1_iff
tff(fact_414_eq__divide__eq__numeral1_I1_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,B2: A,W: num] :
          ( ( A2 = aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),aa(num,A,numeral_numeral(A),W)) )
        <=> ( ( ( aa(num,A,numeral_numeral(A),W) != zero_zero(A) )
             => ( aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(num,A,numeral_numeral(A),W)) = B2 ) )
            & ( ( aa(num,A,numeral_numeral(A),W) = zero_zero(A) )
             => ( A2 = zero_zero(A) ) ) ) ) ) ).

% eq_divide_eq_numeral1(1)
tff(fact_415_divide__eq__eq__numeral1_I1_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B2: A,W: num,A2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),aa(num,A,numeral_numeral(A),W)) = A2 )
        <=> ( ( ( aa(num,A,numeral_numeral(A),W) != zero_zero(A) )
             => ( B2 = aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(num,A,numeral_numeral(A),W)) ) )
            & ( ( aa(num,A,numeral_numeral(A),W) = zero_zero(A) )
             => ( A2 = zero_zero(A) ) ) ) ) ) ).

% divide_eq_eq_numeral1(1)
tff(fact_416_div__mult__self4,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [B2: A,C2: A,A2: A] :
          ( ( B2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)),A2)),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)) ) ) ) ).

% div_mult_self4
tff(fact_417_div__mult__self3,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [B2: A,C2: A,A2: A] :
          ( ( B2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)),A2)),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)) ) ) ) ).

% div_mult_self3
tff(fact_418_div__mult__self2,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [B2: A,A2: A,C2: A] :
          ( ( B2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2))),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)) ) ) ) ).

% div_mult_self2
tff(fact_419_div__mult__self1,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [B2: A,A2: A,C2: A] :
          ( ( B2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2))),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)) ) ) ) ).

% div_mult_self1
tff(fact_420_nonzero__divide__mult__cancel__right,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [B2: A,A2: A] :
          ( ( B2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),A2) ) ) ) ).

% nonzero_divide_mult_cancel_right
tff(fact_421_nonzero__divide__mult__cancel__left,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A2: A,B2: A] :
          ( ( A2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),B2) ) ) ) ).

% nonzero_divide_mult_cancel_left
tff(fact_422_of__nat__le__0__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [M: nat] :
          ( pp(aa(A,bool,ord_less_eq(A,aa(nat,A,semiring_1_of_nat(A),M)),zero_zero(A)))
        <=> ( M = zero_zero(nat) ) ) ) ).

% of_nat_le_0_iff
tff(fact_423_of__nat__Suc,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [M: nat] : aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,M)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(nat,A,semiring_1_of_nat(A),M)) ) ).

% of_nat_Suc
tff(fact_424_unit__div__mult__self,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,dvd_dvd(A,A2),one_one(A)))
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A2)),A2) = B2 ) ) ) ).

% unit_div_mult_self
tff(fact_425_unit__mult__div__div,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,dvd_dvd(A,A2),one_one(A)))
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),B2),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),A2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A2) ) ) ) ).

% unit_mult_div_div
tff(fact_426_one__le__mult__iff,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,ord_less_eq(nat,aa(nat,nat,suc,zero_zero(nat))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N)))
    <=> ( pp(aa(nat,bool,ord_less_eq(nat,aa(nat,nat,suc,zero_zero(nat))),M))
        & pp(aa(nat,bool,ord_less_eq(nat,aa(nat,nat,suc,zero_zero(nat))),N)) ) ) ).

% one_le_mult_iff
tff(fact_427_one__add__one,axiom,
    ! [A: $tType] :
      ( numeral(A)
     => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),one_one(A)) = aa(num,A,numeral_numeral(A),bit0(one2)) ) ) ).

% one_add_one
tff(fact_428_zero__eq__power2,axiom,
    ! [A: $tType] :
      ( semiri2026040879449505780visors(A)
     => ! [A2: A] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(num,nat,numeral_numeral(nat),bit0(one2))) = zero_zero(A) )
        <=> ( A2 = zero_zero(A) ) ) ) ).

% zero_eq_power2
tff(fact_429_one__plus__numeral,axiom,
    ! [A: $tType] :
      ( numeral(A)
     => ! [N: num] : aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(num,A,numeral_numeral(A),N)) = aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),plus_plus(num),one2),N)) ) ).

% one_plus_numeral
tff(fact_430_numeral__plus__one,axiom,
    ! [A: $tType] :
      ( numeral(A)
     => ! [N: num] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),N)),one_one(A)) = aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),plus_plus(num),N),one2)) ) ).

% numeral_plus_one
tff(fact_431_numeral__le__one__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [N: num] :
          ( pp(aa(A,bool,ord_less_eq(A,aa(num,A,numeral_numeral(A),N)),one_one(A)))
        <=> pp(aa(num,bool,ord_less_eq(num,N),one2)) ) ) ).

% numeral_le_one_iff
tff(fact_432_one__div__two__eq__zero,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),bit0(one2))) = zero_zero(A) ) ) ).

% one_div_two_eq_zero
tff(fact_433_bits__1__div__2,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),bit0(one2))) = zero_zero(A) ) ) ).

% bits_1_div_2
tff(fact_434_power2__eq__iff__nonneg,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),X))
         => ( pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),Y))
           => ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(num,nat,numeral_numeral(nat),bit0(one2))) = aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),aa(num,nat,numeral_numeral(nat),bit0(one2))) )
            <=> ( X = Y ) ) ) ) ) ).

% power2_eq_iff_nonneg
tff(fact_435_power2__less__eq__zero__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A] :
          ( pp(aa(A,bool,ord_less_eq(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(num,nat,numeral_numeral(nat),bit0(one2)))),zero_zero(A)))
        <=> ( A2 = zero_zero(A) ) ) ) ).

% power2_less_eq_zero_iff
tff(fact_436_sum__power2__eq__zero__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A,Y: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = zero_zero(A) )
        <=> ( ( X = zero_zero(A) )
            & ( Y = zero_zero(A) ) ) ) ) ).

% sum_power2_eq_zero_iff
tff(fact_437_even__plus__one__iff,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A2: A] :
          ( pp(aa(A,bool,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),one_one(A))))
        <=> ~ pp(aa(A,bool,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A2)) ) ) ).

% even_plus_one_iff
tff(fact_438_even__succ__div__two,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [A2: A] :
          ( pp(aa(A,bool,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A2))
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),one_one(A))),aa(num,A,numeral_numeral(A),bit0(one2))) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(num,A,numeral_numeral(A),bit0(one2))) ) ) ) ).

% even_succ_div_two
tff(fact_439_odd__succ__div__two,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [A2: A] :
          ( ~ pp(aa(A,bool,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A2))
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),one_one(A))),aa(num,A,numeral_numeral(A),bit0(one2))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(num,A,numeral_numeral(A),bit0(one2)))),one_one(A)) ) ) ) ).

% odd_succ_div_two
tff(fact_440_even__succ__div__2,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A] :
          ( pp(aa(A,bool,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A2))
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),A2)),aa(num,A,numeral_numeral(A),bit0(one2))) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(num,A,numeral_numeral(A),bit0(one2))) ) ) ) ).

% even_succ_div_2
tff(fact_441_zero__le__power__eq__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,W: num] :
          ( pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(num,nat,numeral_numeral(nat),W))))
        <=> ( pp(aa(nat,bool,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(num,nat,numeral_numeral(nat),W)))
            | ( ~ pp(aa(nat,bool,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(num,nat,numeral_numeral(nat),W)))
              & pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),A2)) ) ) ) ) ).

% zero_le_power_eq_numeral
tff(fact_442_odd__two__times__div__two__succ,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [A2: A] :
          ( ~ pp(aa(A,bool,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A2))
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(num,A,numeral_numeral(A),bit0(one2))))),one_one(A)) = A2 ) ) ) ).

% odd_two_times_div_two_succ
tff(fact_443_zero__reorient,axiom,
    ! [A: $tType] :
      ( zero(A)
     => ! [X: A] :
          ( ( zero_zero(A) = X )
        <=> ( X = zero_zero(A) ) ) ) ).

% zero_reorient
tff(fact_444_one__reorient,axiom,
    ! [A: $tType] :
      ( one(A)
     => ! [X: A] :
          ( ( one_one(A) = X )
        <=> ( X = one_one(A) ) ) ) ).

% one_reorient
tff(fact_445_zero__neq__one,axiom,
    ! [A: $tType] :
      ( zero_neq_one(A)
     => ( zero_zero(A) != one_one(A) ) ) ).

% zero_neq_one
tff(fact_446_power__0__left,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [N: nat] :
          ( ( ( N = zero_zero(nat) )
           => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),zero_zero(A)),N) = one_one(A) ) )
          & ( ( N != zero_zero(nat) )
           => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),zero_zero(A)),N) = zero_zero(A) ) ) ) ) ).

% power_0_left
tff(fact_447_zero__less__one__class_Ozero__le__one,axiom,
    ! [A: $tType] :
      ( zero_less_one(A)
     => pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),one_one(A))) ) ).

% zero_less_one_class.zero_le_one
tff(fact_448_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),one_one(A))) ) ).

% linordered_nonzero_semiring_class.zero_le_one
tff(fact_449_not__one__le__zero,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ~ pp(aa(A,bool,ord_less_eq(A,one_one(A)),zero_zero(A))) ) ).

% not_one_le_zero
tff(fact_450_right__inverse__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B2: A,A2: A] :
          ( ( B2 != zero_zero(A) )
         => ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2) = one_one(A) )
          <=> ( A2 = B2 ) ) ) ) ).

% right_inverse_eq
tff(fact_451_power__0,axiom,
    ! [A: $tType] :
      ( power(A)
     => ! [A2: A] : aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),zero_zero(nat)) = one_one(A) ) ).

% power_0
tff(fact_452_not__is__unit__0,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ~ pp(aa(A,bool,dvd_dvd(A,zero_zero(A)),one_one(A))) ) ).

% not_is_unit_0
tff(fact_453_VEBT__internal_OT__vebt__buildupi_Osimps_I1_J,axiom,
    vEBT_V441764108873111860ildupi(zero_zero(nat)) = aa(nat,nat,suc,zero_zero(nat)) ).

% VEBT_internal.T_vebt_buildupi.simps(1)
tff(fact_454_VEBT__internal_OT__vebt__buildupi_Osimps_I2_J,axiom,
    vEBT_V441764108873111860ildupi(aa(nat,nat,suc,zero_zero(nat))) = aa(nat,nat,suc,zero_zero(nat)) ).

% VEBT_internal.T_vebt_buildupi.simps(2)
tff(fact_455_Euclid__induct,axiom,
    ! [P: fun(nat,fun(nat,bool)),A2: nat,B2: nat] :
      ( ! [A4: nat,B3: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),P,A4),B3))
        <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),P,B3),A4)) )
     => ( ! [A4: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),P,A4),zero_zero(nat)))
       => ( ! [A4: nat,B3: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),P,A4),B3))
             => pp(aa(nat,bool,aa(nat,fun(nat,bool),P,A4),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A4),B3))) )
         => pp(aa(nat,bool,aa(nat,fun(nat,bool),P,A2),B2)) ) ) ) ).

% Euclid_induct
tff(fact_456_gcd__nat_Oextremum__uniqueI,axiom,
    ! [A2: nat] :
      ( pp(aa(nat,bool,dvd_dvd(nat,zero_zero(nat)),A2))
     => ( A2 = zero_zero(nat) ) ) ).

% gcd_nat.extremum_uniqueI
tff(fact_457_gcd__nat_Onot__eq__extremum,axiom,
    ! [A2: nat] :
      ( ( A2 != zero_zero(nat) )
    <=> ( pp(aa(nat,bool,dvd_dvd(nat,A2),zero_zero(nat)))
        & ( A2 != zero_zero(nat) ) ) ) ).

% gcd_nat.not_eq_extremum
tff(fact_458_gcd__nat_Oextremum__unique,axiom,
    ! [A2: nat] :
      ( pp(aa(nat,bool,dvd_dvd(nat,zero_zero(nat)),A2))
    <=> ( A2 = zero_zero(nat) ) ) ).

% gcd_nat.extremum_unique
tff(fact_459_gcd__nat_Oextremum__strict,axiom,
    ! [A2: nat] :
      ~ ( pp(aa(nat,bool,dvd_dvd(nat,zero_zero(nat)),A2))
        & ( zero_zero(nat) != A2 ) ) ).

% gcd_nat.extremum_strict
tff(fact_460_gcd__nat_Oextremum,axiom,
    ! [A2: nat] : pp(aa(nat,bool,dvd_dvd(nat,A2),zero_zero(nat))) ).

% gcd_nat.extremum
tff(fact_461_mult__left__le,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [C2: A,A2: A] :
          ( pp(aa(A,bool,ord_less_eq(A,C2),one_one(A)))
         => ( pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),A2))
           => pp(aa(A,bool,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),A2)) ) ) ) ).

% mult_left_le
tff(fact_462_mult__le__one,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,ord_less_eq(A,A2),one_one(A)))
         => ( pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),B2))
           => ( pp(aa(A,bool,ord_less_eq(A,B2),one_one(A)))
             => pp(aa(A,bool,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),one_one(A))) ) ) ) ) ).

% mult_le_one
tff(fact_463_mult__right__le__one__le,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),X))
         => ( pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),Y))
           => ( pp(aa(A,bool,ord_less_eq(A,Y),one_one(A)))
             => pp(aa(A,bool,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),times_times(A),X),Y)),X)) ) ) ) ) ).

% mult_right_le_one_le
tff(fact_464_mult__left__le__one__le,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),X))
         => ( pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),Y))
           => ( pp(aa(A,bool,ord_less_eq(A,Y),one_one(A)))
             => pp(aa(A,bool,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),times_times(A),Y),X)),X)) ) ) ) ) ).

% mult_left_le_one_le
tff(fact_465_power__le__one,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,N: nat] :
          ( pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),A2))
         => ( pp(aa(A,bool,ord_less_eq(A,A2),one_one(A)))
           => pp(aa(A,bool,ord_less_eq(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),N)),one_one(A))) ) ) ) ).

% power_le_one
tff(fact_466_div__add__self1,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [B2: A,A2: A] :
          ( ( B2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A2)),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)),one_one(A)) ) ) ) ).

% div_add_self1
tff(fact_467_div__add__self2,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [B2: A,A2: A] :
          ( ( B2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)),one_one(A)) ) ) ) ).

% div_add_self2
tff(fact_468_unit__dvdE,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,dvd_dvd(A,A2),one_one(A)))
         => ~ ( ( A2 != zero_zero(A) )
             => ! [C3: A] : B2 != aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3) ) ) ) ).

% unit_dvdE
tff(fact_469_unit__div__eq__0__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A2: A] :
          ( pp(aa(A,bool,dvd_dvd(A,B2),one_one(A)))
         => ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2) = zero_zero(A) )
          <=> ( A2 = zero_zero(A) ) ) ) ) ).

% unit_div_eq_0_iff
tff(fact_470_is__unit__power__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,N: nat] :
          ( pp(aa(A,bool,dvd_dvd(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),N)),one_one(A)))
        <=> ( pp(aa(A,bool,dvd_dvd(A,A2),one_one(A)))
            | ( N = zero_zero(nat) ) ) ) ) ).

% is_unit_power_iff
tff(fact_471_le__numeral__extra_I4_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => pp(aa(A,bool,ord_less_eq(A,one_one(A)),one_one(A))) ) ).

% le_numeral_extra(4)
tff(fact_472_mult_Ocomm__neutral,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),A2),one_one(A)) = A2 ) ).

% mult.comm_neutral
tff(fact_473_comm__monoid__mult__class_Omult__1,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),one_one(A)),A2) = A2 ) ).

% comm_monoid_mult_class.mult_1
tff(fact_474_le__numeral__extra_I3_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),zero_zero(A))) ) ).

% le_numeral_extra(3)
tff(fact_475_zero__le,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [X: A] : pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),X)) ) ).

% zero_le
tff(fact_476_zero__neq__numeral,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [N: num] : zero_zero(A) != aa(num,A,numeral_numeral(A),N) ) ).

% zero_neq_numeral
tff(fact_477_comm__monoid__add__class_Oadd__0,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),zero_zero(A)),A2) = A2 ) ).

% comm_monoid_add_class.add_0
tff(fact_478_add_Ocomm__neutral,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),zero_zero(A)) = A2 ) ).

% add.comm_neutral
tff(fact_479_add_Ogroup__left__neutral,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),zero_zero(A)),A2) = A2 ) ).

% add.group_left_neutral
tff(fact_480_mult__right__cancel,axiom,
    ! [A: $tType] :
      ( semiri6575147826004484403cancel(A)
     => ! [C2: A,A2: A,B2: A] :
          ( ( C2 != zero_zero(A) )
         => ( ( aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2) = aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2) )
          <=> ( A2 = B2 ) ) ) ) ).

% mult_right_cancel
tff(fact_481_mult__left__cancel,axiom,
    ! [A: $tType] :
      ( semiri6575147826004484403cancel(A)
     => ! [C2: A,A2: A,B2: A] :
          ( ( C2 != zero_zero(A) )
         => ( ( aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2) = aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2) )
          <=> ( A2 = B2 ) ) ) ) ).

% mult_left_cancel
tff(fact_482_no__zero__divisors,axiom,
    ! [A: $tType] :
      ( semiri3467727345109120633visors(A)
     => ! [A2: A,B2: A] :
          ( ( A2 != zero_zero(A) )
         => ( ( B2 != zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2) != zero_zero(A) ) ) ) ) ).

% no_zero_divisors
tff(fact_483_divisors__zero,axiom,
    ! [A: $tType] :
      ( semiri3467727345109120633visors(A)
     => ! [A2: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2) = zero_zero(A) )
         => ( ( A2 = zero_zero(A) )
            | ( B2 = zero_zero(A) ) ) ) ) ).

% divisors_zero
tff(fact_484_mult__not__zero,axiom,
    ! [A: $tType] :
      ( mult_zero(A)
     => ! [A2: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2) != zero_zero(A) )
         => ( ( A2 != zero_zero(A) )
            & ( B2 != zero_zero(A) ) ) ) ) ).

% mult_not_zero
tff(fact_485_dvd__unit__imp__unit,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,dvd_dvd(A,A2),B2))
         => ( pp(aa(A,bool,dvd_dvd(A,B2),one_one(A)))
           => pp(aa(A,bool,dvd_dvd(A,A2),one_one(A))) ) ) ) ).

% dvd_unit_imp_unit
tff(fact_486_unit__imp__dvd,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A2: A] :
          ( pp(aa(A,bool,dvd_dvd(A,B2),one_one(A)))
         => pp(aa(A,bool,dvd_dvd(A,B2),A2)) ) ) ).

% unit_imp_dvd
tff(fact_487_one__dvd,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A2: A] : pp(aa(A,bool,dvd_dvd(A,one_one(A)),A2)) ) ).

% one_dvd
tff(fact_488_semiring__1__no__zero__divisors__class_Opower__not__zero,axiom,
    ! [A: $tType] :
      ( semiri2026040879449505780visors(A)
     => ! [A2: A,N: nat] :
          ( ( A2 != zero_zero(A) )
         => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),N) != zero_zero(A) ) ) ) ).

% semiring_1_no_zero_divisors_class.power_not_zero
tff(fact_489_not0__implies__Suc,axiom,
    ! [N: nat] :
      ( ( N != zero_zero(nat) )
     => ? [M3: nat] : N = aa(nat,nat,suc,M3) ) ).

% not0_implies_Suc
tff(fact_490_Zero__not__Suc,axiom,
    ! [M: nat] : zero_zero(nat) != aa(nat,nat,suc,M) ).

% Zero_not_Suc
tff(fact_491_Zero__neq__Suc,axiom,
    ! [M: nat] : zero_zero(nat) != aa(nat,nat,suc,M) ).

% Zero_neq_Suc
tff(fact_492_Suc__neq__Zero,axiom,
    ! [M: nat] : aa(nat,nat,suc,M) != zero_zero(nat) ).

% Suc_neq_Zero
tff(fact_493_zero__induct,axiom,
    ! [P: fun(nat,bool),K: nat] :
      ( pp(aa(nat,bool,P,K))
     => ( ! [N2: nat] :
            ( pp(aa(nat,bool,P,aa(nat,nat,suc,N2)))
           => pp(aa(nat,bool,P,N2)) )
       => pp(aa(nat,bool,P,zero_zero(nat))) ) ) ).

% zero_induct
tff(fact_494_diff__induct,axiom,
    ! [P: fun(nat,fun(nat,bool)),M: nat,N: nat] :
      ( ! [X3: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),P,X3),zero_zero(nat)))
     => ( ! [Y3: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),P,zero_zero(nat)),aa(nat,nat,suc,Y3)))
       => ( ! [X3: nat,Y3: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),P,X3),Y3))
             => pp(aa(nat,bool,aa(nat,fun(nat,bool),P,aa(nat,nat,suc,X3)),aa(nat,nat,suc,Y3))) )
         => pp(aa(nat,bool,aa(nat,fun(nat,bool),P,M),N)) ) ) ) ).

% diff_induct
tff(fact_495_nat__induct,axiom,
    ! [P: fun(nat,bool),N: nat] :
      ( pp(aa(nat,bool,P,zero_zero(nat)))
     => ( ! [N2: nat] :
            ( pp(aa(nat,bool,P,N2))
           => pp(aa(nat,bool,P,aa(nat,nat,suc,N2))) )
       => pp(aa(nat,bool,P,N)) ) ) ).

% nat_induct
tff(fact_496_list__decode_Ocases,axiom,
    ! [X: nat] :
      ( ( X != zero_zero(nat) )
     => ~ ! [N2: nat] : X != aa(nat,nat,suc,N2) ) ).

% list_decode.cases
tff(fact_497_old_Onat_Oexhaust,axiom,
    ! [Y: nat] :
      ( ( Y != zero_zero(nat) )
     => ~ ! [Nat3: nat] : Y != aa(nat,nat,suc,Nat3) ) ).

% old.nat.exhaust
tff(fact_498_nat_OdiscI,axiom,
    ! [Nat: nat,X23: nat] :
      ( ( Nat = aa(nat,nat,suc,X23) )
     => ( Nat != zero_zero(nat) ) ) ).

% nat.discI
tff(fact_499_old_Onat_Odistinct_I1_J,axiom,
    ! [Nat2: nat] : zero_zero(nat) != aa(nat,nat,suc,Nat2) ).

% old.nat.distinct(1)
tff(fact_500_old_Onat_Odistinct_I2_J,axiom,
    ! [Nat2: nat] : aa(nat,nat,suc,Nat2) != zero_zero(nat) ).

% old.nat.distinct(2)
tff(fact_501_nat_Odistinct_I1_J,axiom,
    ! [X23: nat] : zero_zero(nat) != aa(nat,nat,suc,X23) ).

% nat.distinct(1)
tff(fact_502_dvd__0__left,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A2: A] :
          ( pp(aa(A,bool,dvd_dvd(A,zero_zero(A)),A2))
         => ( A2 = zero_zero(A) ) ) ) ).

% dvd_0_left
tff(fact_503_dvd__field__iff,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,dvd_dvd(A,A2),B2))
        <=> ( ( A2 = zero_zero(A) )
           => ( B2 = zero_zero(A) ) ) ) ) ).

% dvd_field_iff
tff(fact_504_plus__nat_Oadd__0,axiom,
    ! [N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),zero_zero(nat)),N) = N ).

% plus_nat.add_0
tff(fact_505_add__eq__self__zero,axiom,
    ! [M: nat,N: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N) = M )
     => ( N = zero_zero(nat) ) ) ).

% add_eq_self_zero
tff(fact_506_le__0__eq,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,ord_less_eq(nat,N),zero_zero(nat)))
    <=> ( N = zero_zero(nat) ) ) ).

% le_0_eq
tff(fact_507_bot__nat__0_Oextremum__uniqueI,axiom,
    ! [A2: nat] :
      ( pp(aa(nat,bool,ord_less_eq(nat,A2),zero_zero(nat)))
     => ( A2 = zero_zero(nat) ) ) ).

% bot_nat_0.extremum_uniqueI
tff(fact_508_bot__nat__0_Oextremum__unique,axiom,
    ! [A2: nat] :
      ( pp(aa(nat,bool,ord_less_eq(nat,A2),zero_zero(nat)))
    <=> ( A2 = zero_zero(nat) ) ) ).

% bot_nat_0.extremum_unique
tff(fact_509_less__eq__nat_Osimps_I1_J,axiom,
    ! [N: nat] : pp(aa(nat,bool,ord_less_eq(nat,zero_zero(nat)),N)) ).

% less_eq_nat.simps(1)
tff(fact_510_nat__mult__eq__cancel__disj,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),M) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),N) )
    <=> ( ( K = zero_zero(nat) )
        | ( M = N ) ) ) ).

% nat_mult_eq_cancel_disj
tff(fact_511_mult__0,axiom,
    ! [N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),zero_zero(nat)),N) = zero_zero(nat) ).

% mult_0
tff(fact_512_convex__bound__le,axiom,
    ! [A: $tType] :
      ( linord6961819062388156250ring_1(A)
     => ! [X: A,A2: A,Y: A,U: A,V: A] :
          ( pp(aa(A,bool,ord_less_eq(A,X),A2))
         => ( pp(aa(A,bool,ord_less_eq(A,Y),A2))
           => ( pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),U))
             => ( pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),V))
               => ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),U),V) = one_one(A) )
                 => pp(aa(A,bool,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),U),X)),aa(A,A,aa(A,fun(A,A),times_times(A),V),Y))),A2)) ) ) ) ) ) ) ).

% convex_bound_le
tff(fact_513_power__Suc__le__self,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,N: nat] :
          ( pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),A2))
         => ( pp(aa(A,bool,ord_less_eq(A,A2),one_one(A)))
           => pp(aa(A,bool,ord_less_eq(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(nat,nat,suc,N))),A2)) ) ) ) ).

% power_Suc_le_self
tff(fact_514_power__decreasing,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [N: nat,N4: nat,A2: A] :
          ( pp(aa(nat,bool,ord_less_eq(nat,N),N4))
         => ( pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),A2))
           => ( pp(aa(A,bool,ord_less_eq(A,A2),one_one(A)))
             => pp(aa(A,bool,ord_less_eq(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),N4)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),N))) ) ) ) ) ).

% power_decreasing
tff(fact_515_is__unit__div__mult__cancel__right,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A] :
          ( ( A2 != zero_zero(A) )
         => ( pp(aa(A,bool,dvd_dvd(A,B2),one_one(A)))
           => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),A2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),B2) ) ) ) ) ).

% is_unit_div_mult_cancel_right
tff(fact_516_is__unit__div__mult__cancel__left,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A] :
          ( ( A2 != zero_zero(A) )
         => ( pp(aa(A,bool,dvd_dvd(A,B2),one_one(A)))
           => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),B2) ) ) ) ) ).

% is_unit_div_mult_cancel_left
tff(fact_517_is__unitE,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,C2: A] :
          ( pp(aa(A,bool,dvd_dvd(A,A2),one_one(A)))
         => ~ ( ( A2 != zero_zero(A) )
             => ! [B3: A] :
                  ( ( B3 != zero_zero(A) )
                 => ( pp(aa(A,bool,dvd_dvd(A,B3),one_one(A)))
                   => ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),A2) = B3 )
                     => ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),B3) = A2 )
                       => ( ( aa(A,A,aa(A,fun(A,A),times_times(A),A2),B3) = one_one(A) )
                         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),A2) != aa(A,A,aa(A,fun(A,A),times_times(A),C2),B3) ) ) ) ) ) ) ) ) ) ).

% is_unitE
tff(fact_518_dvd__power__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [X: A,M: nat,N: nat] :
          ( ( X != zero_zero(A) )
         => ( pp(aa(A,bool,dvd_dvd(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),X),M)),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),N)))
          <=> ( pp(aa(A,bool,dvd_dvd(A,X),one_one(A)))
              | pp(aa(nat,bool,ord_less_eq(nat,M),N)) ) ) ) ) ).

% dvd_power_iff
tff(fact_519_lambda__one,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ( aTP_Lamp_ab(A,A) = aa(A,fun(A,A),times_times(A),one_one(A)) ) ) ).

% lambda_one
tff(fact_520_lambda__zero,axiom,
    ! [A: $tType] :
      ( mult_zero(A)
     => ( aTP_Lamp_ac(A,A) = aa(A,fun(A,A),times_times(A),zero_zero(A)) ) ) ).

% lambda_zero
tff(fact_521_VEBT__internal_OTb_Osimps_I1_J,axiom,
    vEBT_VEBT_Tb(zero_zero(nat)) = aa(num,int,numeral_numeral(int),aa(num,num,bit1,one2)) ).

% VEBT_internal.Tb.simps(1)
tff(fact_522_inverse__of__nat__le,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [N: nat,M: nat] :
          ( pp(aa(nat,bool,ord_less_eq(nat,N),M))
         => ( ( N != zero_zero(nat) )
           => pp(aa(A,bool,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(nat,A,semiring_1_of_nat(A),M))),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(nat,A,semiring_1_of_nat(A),N)))) ) ) ) ).

% inverse_of_nat_le
tff(fact_523_one__le__numeral,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [N: num] : pp(aa(A,bool,ord_less_eq(A,one_one(A)),aa(num,A,numeral_numeral(A),N))) ) ).

% one_le_numeral
tff(fact_524_one__plus__numeral__commute,axiom,
    ! [A: $tType] :
      ( numeral(A)
     => ! [X: num] : aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(num,A,numeral_numeral(A),X)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),X)),one_one(A)) ) ).

% one_plus_numeral_commute
tff(fact_525_numeral__One,axiom,
    ! [A: $tType] :
      ( numeral(A)
     => ( aa(num,A,numeral_numeral(A),one2) = one_one(A) ) ) ).

% numeral_One
tff(fact_526_one__le__power,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,N: nat] :
          ( pp(aa(A,bool,ord_less_eq(A,one_one(A)),A2))
         => pp(aa(A,bool,ord_less_eq(A,one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),N))) ) ) ).

% one_le_power
tff(fact_527_zdiv__int,axiom,
    ! [A2: nat,B2: nat] : aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),A2),B2)) = aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(nat,int,semiring_1_of_nat(int),A2)),aa(nat,int,semiring_1_of_nat(int),B2)) ).

% zdiv_int
tff(fact_528_left__right__inverse__power,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [X: A,Y: A,N: nat] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),X),Y) = one_one(A) )
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),N)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),N)) = one_one(A) ) ) ) ).

% left_right_inverse_power
tff(fact_529_not__numeral__le__zero,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [N: num] : ~ pp(aa(A,bool,ord_less_eq(A,aa(num,A,numeral_numeral(A),N)),zero_zero(A))) ) ).

% not_numeral_le_zero
tff(fact_530_zero__le__numeral,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [N: num] : pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),aa(num,A,numeral_numeral(A),N))) ) ).

% zero_le_numeral
tff(fact_531_power__one__over,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,N: nat] : aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),A2)),N) = aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),N)) ) ).

% power_one_over
tff(fact_532_add__decreasing,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A2: A,C2: A,B2: A] :
          ( pp(aa(A,bool,ord_less_eq(A,A2),zero_zero(A)))
         => ( pp(aa(A,bool,ord_less_eq(A,C2),B2))
           => pp(aa(A,bool,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C2)),B2)) ) ) ) ).

% add_decreasing
tff(fact_533_add__increasing,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),A2))
         => ( pp(aa(A,bool,ord_less_eq(A,B2),C2))
           => pp(aa(A,bool,ord_less_eq(A,B2),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C2))) ) ) ) ).

% add_increasing
tff(fact_534_add__decreasing2,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [C2: A,A2: A,B2: A] :
          ( pp(aa(A,bool,ord_less_eq(A,C2),zero_zero(A)))
         => ( pp(aa(A,bool,ord_less_eq(A,A2),B2))
           => pp(aa(A,bool,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C2)),B2)) ) ) ) ).

% add_decreasing2
tff(fact_535_add__increasing2,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [C2: A,B2: A,A2: A] :
          ( pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),C2))
         => ( pp(aa(A,bool,ord_less_eq(A,B2),A2))
           => pp(aa(A,bool,ord_less_eq(A,B2),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C2))) ) ) ) ).

% add_increasing2
tff(fact_536_add__nonneg__nonneg,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),A2))
         => ( pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),B2))
           => pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2))) ) ) ) ).

% add_nonneg_nonneg
tff(fact_537_add__nonpos__nonpos,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,ord_less_eq(A,A2),zero_zero(A)))
         => ( pp(aa(A,bool,ord_less_eq(A,B2),zero_zero(A)))
           => pp(aa(A,bool,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),zero_zero(A))) ) ) ) ).

% add_nonpos_nonpos
tff(fact_538_add__nonneg__eq__0__iff,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),X))
         => ( pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),Y))
           => ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y) = zero_zero(A) )
            <=> ( ( X = zero_zero(A) )
                & ( Y = zero_zero(A) ) ) ) ) ) ) ).

% add_nonneg_eq_0_iff
tff(fact_539_add__nonpos__eq__0__iff,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,ord_less_eq(A,X),zero_zero(A)))
         => ( pp(aa(A,bool,ord_less_eq(A,Y),zero_zero(A)))
           => ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y) = zero_zero(A) )
            <=> ( ( X = zero_zero(A) )
                & ( Y = zero_zero(A) ) ) ) ) ) ) ).

% add_nonpos_eq_0_iff
tff(fact_540_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: $tType] :
      ( ordere2520102378445227354miring(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,ord_less_eq(A,A2),B2))
         => ( pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),C2))
           => pp(aa(A,bool,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2))) ) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
tff(fact_541_zero__le__mult__iff,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)))
        <=> ( ( pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),A2))
              & pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),B2)) )
            | ( pp(aa(A,bool,ord_less_eq(A,A2),zero_zero(A)))
              & pp(aa(A,bool,ord_less_eq(A,B2),zero_zero(A))) ) ) ) ) ).

% zero_le_mult_iff
tff(fact_542_mult__nonneg__nonpos2,axiom,
    ! [A: $tType] :
      ( ordered_semiring_0(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),A2))
         => ( pp(aa(A,bool,ord_less_eq(A,B2),zero_zero(A)))
           => pp(aa(A,bool,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),times_times(A),B2),A2)),zero_zero(A))) ) ) ) ).

% mult_nonneg_nonpos2
tff(fact_543_mult__nonpos__nonneg,axiom,
    ! [A: $tType] :
      ( ordered_semiring_0(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,ord_less_eq(A,A2),zero_zero(A)))
         => ( pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),B2))
           => pp(aa(A,bool,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),zero_zero(A))) ) ) ) ).

% mult_nonpos_nonneg
tff(fact_544_mult__nonneg__nonpos,axiom,
    ! [A: $tType] :
      ( ordered_semiring_0(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),A2))
         => ( pp(aa(A,bool,ord_less_eq(A,B2),zero_zero(A)))
           => pp(aa(A,bool,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),zero_zero(A))) ) ) ) ).

% mult_nonneg_nonpos
tff(fact_545_mult__nonneg__nonneg,axiom,
    ! [A: $tType] :
      ( ordered_semiring_0(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),A2))
         => ( pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),B2))
           => pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2))) ) ) ) ).

% mult_nonneg_nonneg
tff(fact_546_split__mult__neg__le,axiom,
    ! [A: $tType] :
      ( ordered_semiring_0(A)
     => ! [A2: A,B2: A] :
          ( ( ( pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),A2))
              & pp(aa(A,bool,ord_less_eq(A,B2),zero_zero(A))) )
            | ( pp(aa(A,bool,ord_less_eq(A,A2),zero_zero(A)))
              & pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),B2)) ) )
         => pp(aa(A,bool,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),zero_zero(A))) ) ) ).

% split_mult_neg_le
tff(fact_547_mult__le__0__iff,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),zero_zero(A)))
        <=> ( ( pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),A2))
              & pp(aa(A,bool,ord_less_eq(A,B2),zero_zero(A))) )
            | ( pp(aa(A,bool,ord_less_eq(A,A2),zero_zero(A)))
              & pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),B2)) ) ) ) ) ).

% mult_le_0_iff
tff(fact_548_mult__right__mono,axiom,
    ! [A: $tType] :
      ( ordered_semiring(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,ord_less_eq(A,A2),B2))
         => ( pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),C2))
           => pp(aa(A,bool,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2))) ) ) ) ).

% mult_right_mono
tff(fact_549_mult__right__mono__neg,axiom,
    ! [A: $tType] :
      ( ordered_ring(A)
     => ! [B2: A,A2: A,C2: A] :
          ( pp(aa(A,bool,ord_less_eq(A,B2),A2))
         => ( pp(aa(A,bool,ord_less_eq(A,C2),zero_zero(A)))
           => pp(aa(A,bool,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2))) ) ) ) ).

% mult_right_mono_neg
tff(fact_550_mult__left__mono,axiom,
    ! [A: $tType] :
      ( ordered_semiring(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,ord_less_eq(A,A2),B2))
         => ( pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),C2))
           => pp(aa(A,bool,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2))) ) ) ) ).

% mult_left_mono
tff(fact_551_mult__nonpos__nonpos,axiom,
    ! [A: $tType] :
      ( ordered_ring(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,ord_less_eq(A,A2),zero_zero(A)))
         => ( pp(aa(A,bool,ord_less_eq(A,B2),zero_zero(A)))
           => pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2))) ) ) ) ).

% mult_nonpos_nonpos
tff(fact_552_mult__left__mono__neg,axiom,
    ! [A: $tType] :
      ( ordered_ring(A)
     => ! [B2: A,A2: A,C2: A] :
          ( pp(aa(A,bool,ord_less_eq(A,B2),A2))
         => ( pp(aa(A,bool,ord_less_eq(A,C2),zero_zero(A)))
           => pp(aa(A,bool,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2))) ) ) ) ).

% mult_left_mono_neg
tff(fact_553_split__mult__pos__le,axiom,
    ! [A: $tType] :
      ( ordered_ring(A)
     => ! [A2: A,B2: A] :
          ( ( ( pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),A2))
              & pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),B2)) )
            | ( pp(aa(A,bool,ord_less_eq(A,A2),zero_zero(A)))
              & pp(aa(A,bool,ord_less_eq(A,B2),zero_zero(A))) ) )
         => pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2))) ) ) ).

% split_mult_pos_le
tff(fact_554_zero__le__square,axiom,
    ! [A: $tType] :
      ( linordered_ring(A)
     => ! [A2: A] : pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),A2))) ) ).

% zero_le_square
tff(fact_555_mult__mono_H,axiom,
    ! [A: $tType] :
      ( ordered_semiring(A)
     => ! [A2: A,B2: A,C2: A,D: A] :
          ( pp(aa(A,bool,ord_less_eq(A,A2),B2))
         => ( pp(aa(A,bool,ord_less_eq(A,C2),D))
           => ( pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),A2))
             => ( pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),C2))
               => pp(aa(A,bool,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),D))) ) ) ) ) ) ).

% mult_mono'
tff(fact_556_mult__mono,axiom,
    ! [A: $tType] :
      ( ordered_semiring(A)
     => ! [A2: A,B2: A,C2: A,D: A] :
          ( pp(aa(A,bool,ord_less_eq(A,A2),B2))
         => ( pp(aa(A,bool,ord_less_eq(A,C2),D))
           => ( pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),B2))
             => ( pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),C2))
               => pp(aa(A,bool,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),D))) ) ) ) ) ) ).

% mult_mono
tff(fact_557_unit__mult__right__cancel,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,dvd_dvd(A,A2),one_one(A)))
         => ( ( aa(A,A,aa(A,fun(A,A),times_times(A),B2),A2) = aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2) )
          <=> ( B2 = C2 ) ) ) ) ).

% unit_mult_right_cancel
tff(fact_558_unit__mult__left__cancel,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,dvd_dvd(A,A2),one_one(A)))
         => ( ( aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2) = aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2) )
          <=> ( B2 = C2 ) ) ) ) ).

% unit_mult_left_cancel
tff(fact_559_mult__unit__dvd__iff_H,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,dvd_dvd(A,A2),one_one(A)))
         => ( pp(aa(A,bool,dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),C2))
          <=> pp(aa(A,bool,dvd_dvd(A,B2),C2)) ) ) ) ).

% mult_unit_dvd_iff'
tff(fact_560_dvd__mult__unit__iff_H,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A2: A,C2: A] :
          ( pp(aa(A,bool,dvd_dvd(A,B2),one_one(A)))
         => ( pp(aa(A,bool,dvd_dvd(A,A2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)))
          <=> pp(aa(A,bool,dvd_dvd(A,A2),C2)) ) ) ) ).

% dvd_mult_unit_iff'
tff(fact_561_mult__unit__dvd__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A2: A,C2: A] :
          ( pp(aa(A,bool,dvd_dvd(A,B2),one_one(A)))
         => ( pp(aa(A,bool,dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),C2))
          <=> pp(aa(A,bool,dvd_dvd(A,A2),C2)) ) ) ) ).

% mult_unit_dvd_iff
tff(fact_562_dvd__mult__unit__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A2: A,C2: A] :
          ( pp(aa(A,bool,dvd_dvd(A,B2),one_one(A)))
         => ( pp(aa(A,bool,dvd_dvd(A,A2),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)))
          <=> pp(aa(A,bool,dvd_dvd(A,A2),C2)) ) ) ) ).

% dvd_mult_unit_iff
tff(fact_563_is__unit__mult__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),one_one(A)))
        <=> ( pp(aa(A,bool,dvd_dvd(A,A2),one_one(A)))
            & pp(aa(A,bool,dvd_dvd(A,B2),one_one(A))) ) ) ) ).

% is_unit_mult_iff
tff(fact_564_divide__right__mono__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,ord_less_eq(A,A2),B2))
         => ( pp(aa(A,bool,ord_less_eq(A,C2),zero_zero(A)))
           => pp(aa(A,bool,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),C2))) ) ) ) ).

% divide_right_mono_neg
tff(fact_565_divide__nonpos__nonpos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,ord_less_eq(A,X),zero_zero(A)))
         => ( pp(aa(A,bool,ord_less_eq(A,Y),zero_zero(A)))
           => pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y))) ) ) ) ).

% divide_nonpos_nonpos
tff(fact_566_divide__nonpos__nonneg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,ord_less_eq(A,X),zero_zero(A)))
         => ( pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),Y))
           => pp(aa(A,bool,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y)),zero_zero(A))) ) ) ) ).

% divide_nonpos_nonneg
tff(fact_567_divide__nonneg__nonpos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),X))
         => ( pp(aa(A,bool,ord_less_eq(A,Y),zero_zero(A)))
           => pp(aa(A,bool,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y)),zero_zero(A))) ) ) ) ).

% divide_nonneg_nonpos
tff(fact_568_divide__nonneg__nonneg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),X))
         => ( pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),Y))
           => pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y))) ) ) ) ).

% divide_nonneg_nonneg
tff(fact_569_zero__le__divide__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)))
        <=> ( ( pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),A2))
              & pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),B2)) )
            | ( pp(aa(A,bool,ord_less_eq(A,A2),zero_zero(A)))
              & pp(aa(A,bool,ord_less_eq(A,B2),zero_zero(A))) ) ) ) ) ).

% zero_le_divide_iff
tff(fact_570_divide__right__mono,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,ord_less_eq(A,A2),B2))
         => ( pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),C2))
           => pp(aa(A,bool,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),C2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2))) ) ) ) ).

% divide_right_mono
tff(fact_571_divide__le__0__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)),zero_zero(A)))
        <=> ( ( pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),A2))
              & pp(aa(A,bool,ord_less_eq(A,B2),zero_zero(A))) )
            | ( pp(aa(A,bool,ord_less_eq(A,A2),zero_zero(A)))
              & pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),B2)) ) ) ) ) ).

% divide_le_0_iff
tff(fact_572_dvd__div__unit__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A2: A,C2: A] :
          ( pp(aa(A,bool,dvd_dvd(A,B2),one_one(A)))
         => ( pp(aa(A,bool,dvd_dvd(A,A2),aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),B2)))
          <=> pp(aa(A,bool,dvd_dvd(A,A2),C2)) ) ) ) ).

% dvd_div_unit_iff
tff(fact_573_div__unit__dvd__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A2: A,C2: A] :
          ( pp(aa(A,bool,dvd_dvd(A,B2),one_one(A)))
         => ( pp(aa(A,bool,dvd_dvd(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)),C2))
          <=> pp(aa(A,bool,dvd_dvd(A,A2),C2)) ) ) ) ).

% div_unit_dvd_iff
tff(fact_574_unit__div__cancel,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,dvd_dvd(A,A2),one_one(A)))
         => ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A2) = aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),A2) )
          <=> ( B2 = C2 ) ) ) ) ).

% unit_div_cancel
tff(fact_575_zero__le__power,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,N: nat] :
          ( pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),A2))
         => pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),N))) ) ) ).

% zero_le_power
tff(fact_576_power__mono,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,B2: A,N: nat] :
          ( pp(aa(A,bool,ord_less_eq(A,A2),B2))
         => ( pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),A2))
           => pp(aa(A,bool,ord_less_eq(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),N)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),N))) ) ) ) ).

% power_mono
tff(fact_577_nonzero__eq__divide__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [C2: A,A2: A,B2: A] :
          ( ( C2 != zero_zero(A) )
         => ( ( A2 = aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2) )
          <=> ( aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2) = B2 ) ) ) ) ).

% nonzero_eq_divide_eq
tff(fact_578_nonzero__divide__eq__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [C2: A,B2: A,A2: A] :
          ( ( C2 != zero_zero(A) )
         => ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2) = A2 )
          <=> ( B2 = aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2) ) ) ) ) ).

% nonzero_divide_eq_eq
tff(fact_579_eq__divide__imp,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [C2: A,A2: A,B2: A] :
          ( ( C2 != zero_zero(A) )
         => ( ( aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2) = B2 )
           => ( A2 = aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2) ) ) ) ) ).

% eq_divide_imp
tff(fact_580_divide__eq__imp,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [C2: A,B2: A,A2: A] :
          ( ( C2 != zero_zero(A) )
         => ( ( B2 = aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2) )
           => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2) = A2 ) ) ) ) ).

% divide_eq_imp
tff(fact_581_eq__divide__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,B2: A,C2: A] :
          ( ( A2 = aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2) )
        <=> ( ( ( C2 != zero_zero(A) )
             => ( aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2) = B2 ) )
            & ( ( C2 = zero_zero(A) )
             => ( A2 = zero_zero(A) ) ) ) ) ) ).

% eq_divide_eq
tff(fact_582_divide__eq__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B2: A,C2: A,A2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2) = A2 )
        <=> ( ( ( C2 != zero_zero(A) )
             => ( B2 = aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2) ) )
            & ( ( C2 = zero_zero(A) )
             => ( A2 = zero_zero(A) ) ) ) ) ) ).

% divide_eq_eq
tff(fact_583_frac__eq__eq,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Y: A,Z: A,X: A,W: A] :
          ( ( Y != zero_zero(A) )
         => ( ( Z != zero_zero(A) )
           => ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y) = aa(A,A,aa(A,fun(A,A),divide_divide(A),W),Z) )
            <=> ( aa(A,A,aa(A,fun(A,A),times_times(A),X),Z) = aa(A,A,aa(A,fun(A,A),times_times(A),W),Y) ) ) ) ) ) ).

% frac_eq_eq
tff(fact_584_of__nat__0__le__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [N: nat] : pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),aa(nat,A,semiring_1_of_nat(A),N))) ) ).

% of_nat_0_le_iff
tff(fact_585_semiring__char__0__class_Oof__nat__neq__0,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [N: nat] : aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,N)) != zero_zero(A) ) ).

% semiring_char_0_class.of_nat_neq_0
tff(fact_586_dvd__div__eq__0__iff,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [B2: A,A2: A] :
          ( pp(aa(A,bool,dvd_dvd(A,B2),A2))
         => ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2) = zero_zero(A) )
          <=> ( A2 = zero_zero(A) ) ) ) ) ).

% dvd_div_eq_0_iff
tff(fact_587_bezout__add__strong__nat,axiom,
    ! [A2: nat,B2: nat] :
      ( ( A2 != zero_zero(nat) )
     => ? [D2: nat,X3: nat,Y3: nat] :
          ( pp(aa(nat,bool,dvd_dvd(nat,D2),A2))
          & pp(aa(nat,bool,dvd_dvd(nat,D2),B2))
          & ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A2),X3) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),Y3)),D2) ) ) ) ).

% bezout_add_strong_nat
tff(fact_588_add__is__1,axiom,
    ! [M: nat,N: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N) = aa(nat,nat,suc,zero_zero(nat)) )
    <=> ( ( ( M = aa(nat,nat,suc,zero_zero(nat)) )
          & ( N = zero_zero(nat) ) )
        | ( ( M = zero_zero(nat) )
          & ( N = aa(nat,nat,suc,zero_zero(nat)) ) ) ) ) ).

% add_is_1
tff(fact_589_one__is__add,axiom,
    ! [M: nat,N: nat] :
      ( ( aa(nat,nat,suc,zero_zero(nat)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N) )
    <=> ( ( ( M = aa(nat,nat,suc,zero_zero(nat)) )
          & ( N = zero_zero(nat) ) )
        | ( ( M = zero_zero(nat) )
          & ( N = aa(nat,nat,suc,zero_zero(nat)) ) ) ) ) ).

% one_is_add
tff(fact_590_VEBT__internal_OTb_Osimps_I2_J,axiom,
    vEBT_VEBT_Tb(aa(nat,nat,suc,zero_zero(nat))) = aa(num,int,numeral_numeral(int),aa(num,num,bit1,one2)) ).

% VEBT_internal.Tb.simps(2)
tff(fact_591_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)
tff(fact_592_numeral__Bit1,axiom,
    ! [A: $tType] :
      ( numeral(A)
     => ! [N: num] : aa(num,A,numeral_numeral(A),aa(num,num,bit1,N)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),N)),aa(num,A,numeral_numeral(A),N))),one_one(A)) ) ).

% numeral_Bit1
tff(fact_593_power__increasing,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [N: nat,N4: nat,A2: A] :
          ( pp(aa(nat,bool,ord_less_eq(nat,N),N4))
         => ( pp(aa(A,bool,ord_less_eq(A,one_one(A)),A2))
           => pp(aa(A,bool,ord_less_eq(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),N)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),N4))) ) ) ) ).

% power_increasing
tff(fact_594_is__unit__div__mult2__eq,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,C2: A,A2: A] :
          ( pp(aa(A,bool,dvd_dvd(A,B2),one_one(A)))
         => ( pp(aa(A,bool,dvd_dvd(A,C2),one_one(A)))
           => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)),C2) ) ) ) ) ).

% is_unit_div_mult2_eq
tff(fact_595_unit__div__mult__swap,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [C2: A,A2: A,B2: A] :
          ( pp(aa(A,bool,dvd_dvd(A,C2),one_one(A)))
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),C2) ) ) ) ).

% unit_div_mult_swap
tff(fact_596_unit__div__commute,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A2: A,C2: A] :
          ( pp(aa(A,bool,dvd_dvd(A,B2),one_one(A)))
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)),C2) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),B2) ) ) ) ).

% unit_div_commute
tff(fact_597_div__mult__unit2,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [C2: A,B2: A,A2: A] :
          ( pp(aa(A,bool,dvd_dvd(A,C2),one_one(A)))
         => ( pp(aa(A,bool,dvd_dvd(A,B2),A2))
           => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)),C2) ) ) ) ) ).

% div_mult_unit2
tff(fact_598_unit__eq__div2,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A2: A,C2: A] :
          ( pp(aa(A,bool,dvd_dvd(A,B2),one_one(A)))
         => ( ( A2 = aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),B2) )
          <=> ( aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2) = C2 ) ) ) ) ).

% unit_eq_div2
tff(fact_599_unit__eq__div1,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A2: A,C2: A] :
          ( pp(aa(A,bool,dvd_dvd(A,B2),one_one(A)))
         => ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2) = C2 )
          <=> ( A2 = aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2) ) ) ) ) ).

% unit_eq_div1
tff(fact_600_sum__squares__ge__zero,axiom,
    ! [A: $tType] :
      ( linordered_ring(A)
     => ! [X: A,Y: A] : pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),X)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Y)))) ) ).

% sum_squares_ge_zero
tff(fact_601_sum__squares__le__zero__iff,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),X)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Y))),zero_zero(A)))
        <=> ( ( X = zero_zero(A) )
            & ( Y = zero_zero(A) ) ) ) ) ).

% sum_squares_le_zero_iff
tff(fact_602_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [C2: A,A2: A,B2: A] :
          ( pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),C2))
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)),C2) ) ) ) ).

% unique_euclidean_semiring_numeral_class.div_mult2_eq
tff(fact_603_eq__divide__eq__numeral_I1_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [W: num,B2: A,C2: A] :
          ( ( aa(num,A,numeral_numeral(A),W) = aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2) )
        <=> ( ( ( C2 != zero_zero(A) )
             => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W)),C2) = B2 ) )
            & ( ( C2 = zero_zero(A) )
             => ( aa(num,A,numeral_numeral(A),W) = zero_zero(A) ) ) ) ) ) ).

% eq_divide_eq_numeral(1)
tff(fact_604_divide__eq__eq__numeral_I1_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B2: A,C2: A,W: num] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2) = aa(num,A,numeral_numeral(A),W) )
        <=> ( ( ( C2 != zero_zero(A) )
             => ( B2 = aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W)),C2) ) )
            & ( ( C2 = zero_zero(A) )
             => ( aa(num,A,numeral_numeral(A),W) = zero_zero(A) ) ) ) ) ) ).

% divide_eq_eq_numeral(1)
tff(fact_605_power__le__imp__le__base,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,N: nat,B2: A] :
          ( pp(aa(A,bool,ord_less_eq(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(nat,nat,suc,N))),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),aa(nat,nat,suc,N))))
         => ( pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),B2))
           => pp(aa(A,bool,ord_less_eq(A,A2),B2)) ) ) ) ).

% power_le_imp_le_base
tff(fact_606_power__inject__base,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,N: nat,B2: A] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(nat,nat,suc,N)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),aa(nat,nat,suc,N)) )
         => ( pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),A2))
           => ( pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),B2))
             => ( A2 = B2 ) ) ) ) ) ).

% power_inject_base
tff(fact_607_divide__add__eq__iff,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Z: A,X: A,Y: A] :
          ( ( Z != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Z)),Y) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Z))),Z) ) ) ) ).

% divide_add_eq_iff
tff(fact_608_add__divide__eq__iff,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Z: A,X: A,Y: A] :
          ( ( Z != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),X),aa(A,A,aa(A,fun(A,A),divide_divide(A),Y),Z)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),Z)),Y)),Z) ) ) ) ).

% add_divide_eq_iff
tff(fact_609_add__num__frac,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Y: A,Z: A,X: A] :
          ( ( Y != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),Z),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),aa(A,A,aa(A,fun(A,A),times_times(A),Z),Y))),Y) ) ) ) ).

% add_num_frac
tff(fact_610_add__frac__num,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Y: A,X: A,Z: A] :
          ( ( Y != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y)),Z) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),aa(A,A,aa(A,fun(A,A),times_times(A),Z),Y))),Y) ) ) ) ).

% add_frac_num
tff(fact_611_add__frac__eq,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Y: A,Z: A,X: A,W: A] :
          ( ( Y != zero_zero(A) )
         => ( ( Z != zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y)),aa(A,A,aa(A,fun(A,A),divide_divide(A),W),Z)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),Z)),aa(A,A,aa(A,fun(A,A),times_times(A),W),Y))),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Z)) ) ) ) ) ).

% add_frac_eq
tff(fact_612_add__divide__eq__if__simps_I1_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Z: A,A2: A,B2: A] :
          ( ( ( Z = zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),Z)) = A2 ) )
          & ( ( Z != zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),Z)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),Z)),B2)),Z) ) ) ) ) ).

% add_divide_eq_if_simps(1)
tff(fact_613_add__divide__eq__if__simps_I2_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Z: A,A2: A,B2: A] :
          ( ( ( Z = zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),Z)),B2) = B2 ) )
          & ( ( Z != zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),Z)),B2) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),Z))),Z) ) ) ) ) ).

% add_divide_eq_if_simps(2)
tff(fact_614_dvd__div__div__eq__mult,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,C2: A,B2: A,D: A] :
          ( ( A2 != zero_zero(A) )
         => ( ( C2 != zero_zero(A) )
           => ( pp(aa(A,bool,dvd_dvd(A,A2),B2))
             => ( pp(aa(A,bool,dvd_dvd(A,C2),D))
               => ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A2) = aa(A,A,aa(A,fun(A,A),divide_divide(A),D),C2) )
                <=> ( aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2) = aa(A,A,aa(A,fun(A,A),times_times(A),A2),D) ) ) ) ) ) ) ) ).

% dvd_div_div_eq_mult
tff(fact_615_dvd__div__iff__mult,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [C2: A,B2: A,A2: A] :
          ( ( C2 != zero_zero(A) )
         => ( pp(aa(A,bool,dvd_dvd(A,C2),B2))
           => ( pp(aa(A,bool,dvd_dvd(A,A2),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)))
            <=> pp(aa(A,bool,dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),B2)) ) ) ) ) ).

% dvd_div_iff_mult
tff(fact_616_div__dvd__iff__mult,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A2: A,C2: A] :
          ( ( B2 != zero_zero(A) )
         => ( pp(aa(A,bool,dvd_dvd(A,B2),A2))
           => ( pp(aa(A,bool,dvd_dvd(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)),C2))
            <=> pp(aa(A,bool,dvd_dvd(A,A2),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2))) ) ) ) ) ).

% div_dvd_iff_mult
tff(fact_617_dvd__div__eq__mult,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A,C2: A] :
          ( ( A2 != zero_zero(A) )
         => ( pp(aa(A,bool,dvd_dvd(A,A2),B2))
           => ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A2) = C2 )
            <=> ( B2 = aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2) ) ) ) ) ) ).

% dvd_div_eq_mult
tff(fact_618_numeral__1__eq__Suc__0,axiom,
    aa(num,nat,numeral_numeral(nat),one2) = aa(nat,nat,suc,zero_zero(nat)) ).

% numeral_1_eq_Suc_0
tff(fact_619_nat__one__le__power,axiom,
    ! [I: nat,N: nat] :
      ( pp(aa(nat,bool,ord_less_eq(nat,aa(nat,nat,suc,zero_zero(nat))),I))
     => pp(aa(nat,bool,ord_less_eq(nat,aa(nat,nat,suc,zero_zero(nat))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),I),N))) ) ).

% nat_one_le_power
tff(fact_620_gcd__nat_Onot__eq__order__implies__strict,axiom,
    ! [A2: nat,B2: nat] :
      ( ( A2 != B2 )
     => ( pp(aa(nat,bool,dvd_dvd(nat,A2),B2))
       => ( pp(aa(nat,bool,dvd_dvd(nat,A2),B2))
          & ( A2 != B2 ) ) ) ) ).

% gcd_nat.not_eq_order_implies_strict
tff(fact_621_gcd__nat_Ostrict__implies__not__eq,axiom,
    ! [A2: nat,B2: nat] :
      ( ( pp(aa(nat,bool,dvd_dvd(nat,A2),B2))
        & ( A2 != B2 ) )
     => ( A2 != B2 ) ) ).

% gcd_nat.strict_implies_not_eq
tff(fact_622_gcd__nat_Ostrict__implies__order,axiom,
    ! [A2: nat,B2: nat] :
      ( ( pp(aa(nat,bool,dvd_dvd(nat,A2),B2))
        & ( A2 != B2 ) )
     => pp(aa(nat,bool,dvd_dvd(nat,A2),B2)) ) ).

% gcd_nat.strict_implies_order
tff(fact_623_gcd__nat_Ostrict__iff__order,axiom,
    ! [A2: nat,B2: nat] :
      ( ( pp(aa(nat,bool,dvd_dvd(nat,A2),B2))
        & ( A2 != B2 ) )
    <=> ( pp(aa(nat,bool,dvd_dvd(nat,A2),B2))
        & ( A2 != B2 ) ) ) ).

% gcd_nat.strict_iff_order
tff(fact_624_gcd__nat_Oorder__iff__strict,axiom,
    ! [A2: nat,B2: nat] :
      ( pp(aa(nat,bool,dvd_dvd(nat,A2),B2))
    <=> ( ( pp(aa(nat,bool,dvd_dvd(nat,A2),B2))
          & ( A2 != B2 ) )
        | ( A2 = B2 ) ) ) ).

% gcd_nat.order_iff_strict
tff(fact_625_gcd__nat_Ostrict__iff__not,axiom,
    ! [A2: nat,B2: nat] :
      ( ( pp(aa(nat,bool,dvd_dvd(nat,A2),B2))
        & ( A2 != B2 ) )
    <=> ( pp(aa(nat,bool,dvd_dvd(nat,A2),B2))
        & ~ pp(aa(nat,bool,dvd_dvd(nat,B2),A2)) ) ) ).

% gcd_nat.strict_iff_not
tff(fact_626_gcd__nat_Ostrict__trans2,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( ( pp(aa(nat,bool,dvd_dvd(nat,A2),B2))
        & ( A2 != B2 ) )
     => ( pp(aa(nat,bool,dvd_dvd(nat,B2),C2))
       => ( pp(aa(nat,bool,dvd_dvd(nat,A2),C2))
          & ( A2 != C2 ) ) ) ) ).

% gcd_nat.strict_trans2
tff(fact_627_gcd__nat_Ostrict__trans1,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( pp(aa(nat,bool,dvd_dvd(nat,A2),B2))
     => ( ( pp(aa(nat,bool,dvd_dvd(nat,B2),C2))
          & ( B2 != C2 ) )
       => ( pp(aa(nat,bool,dvd_dvd(nat,A2),C2))
          & ( A2 != C2 ) ) ) ) ).

% gcd_nat.strict_trans1
tff(fact_628_gcd__nat_Ostrict__trans,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( ( pp(aa(nat,bool,dvd_dvd(nat,A2),B2))
        & ( A2 != B2 ) )
     => ( ( pp(aa(nat,bool,dvd_dvd(nat,B2),C2))
          & ( B2 != C2 ) )
       => ( pp(aa(nat,bool,dvd_dvd(nat,A2),C2))
          & ( A2 != C2 ) ) ) ) ).

% gcd_nat.strict_trans
tff(fact_629_gcd__nat_Oantisym,axiom,
    ! [A2: nat,B2: nat] :
      ( pp(aa(nat,bool,dvd_dvd(nat,A2),B2))
     => ( pp(aa(nat,bool,dvd_dvd(nat,B2),A2))
       => ( A2 = B2 ) ) ) ).

% gcd_nat.antisym
tff(fact_630_gcd__nat_Oirrefl,axiom,
    ! [A2: nat] :
      ~ ( pp(aa(nat,bool,dvd_dvd(nat,A2),A2))
        & ( A2 != A2 ) ) ).

% gcd_nat.irrefl
tff(fact_631_gcd__nat_Oeq__iff,axiom,
    ! [A2: nat,B2: nat] :
      ( ( A2 = B2 )
    <=> ( pp(aa(nat,bool,dvd_dvd(nat,A2),B2))
        & pp(aa(nat,bool,dvd_dvd(nat,B2),A2)) ) ) ).

% gcd_nat.eq_iff
tff(fact_632_gcd__nat_Otrans,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( pp(aa(nat,bool,dvd_dvd(nat,A2),B2))
     => ( pp(aa(nat,bool,dvd_dvd(nat,B2),C2))
       => pp(aa(nat,bool,dvd_dvd(nat,A2),C2)) ) ) ).

% gcd_nat.trans
tff(fact_633_gcd__nat_Orefl,axiom,
    ! [A2: nat] : pp(aa(nat,bool,dvd_dvd(nat,A2),A2)) ).

% gcd_nat.refl
tff(fact_634_gcd__nat_Oasym,axiom,
    ! [A2: nat,B2: nat] :
      ( ( pp(aa(nat,bool,dvd_dvd(nat,A2),B2))
        & ( A2 != B2 ) )
     => ~ ( pp(aa(nat,bool,dvd_dvd(nat,B2),A2))
          & ( B2 != A2 ) ) ) ).

% gcd_nat.asym
tff(fact_635_numeral__code_I3_J,axiom,
    ! [A: $tType] :
      ( numeral(A)
     => ! [N: num] : aa(num,A,numeral_numeral(A),aa(num,num,bit1,N)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),N)),aa(num,A,numeral_numeral(A),N))),one_one(A)) ) ).

% numeral_code(3)
tff(fact_636_odd__one,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ~ pp(aa(A,bool,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),one_one(A))) ) ).

% odd_one
tff(fact_637_one__power2,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),one_one(A)),aa(num,nat,numeral_numeral(nat),bit0(one2))) = one_one(A) ) ) ).

% one_power2
tff(fact_638_even__zero,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => pp(aa(A,bool,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),zero_zero(A))) ) ).

% even_zero
tff(fact_639_zero__power2,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),zero_zero(A)),aa(num,nat,numeral_numeral(nat),bit0(one2))) = zero_zero(A) ) ) ).

% zero_power2
tff(fact_640_numeral__2__eq__2,axiom,
    aa(num,nat,numeral_numeral(nat),bit0(one2)) = aa(nat,nat,suc,aa(nat,nat,suc,zero_zero(nat))) ).

% numeral_2_eq_2
tff(fact_641_numeral__3__eq__3,axiom,
    aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2)) = aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,suc,zero_zero(nat)))) ).

% numeral_3_eq_3
tff(fact_642_VEBT__internal_OTb_H_Osimps_I1_J,axiom,
    vEBT_VEBT_Tb2(zero_zero(nat)) = aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2)) ).

% VEBT_internal.Tb'.simps(1)
tff(fact_643_VEBT__internal_OTb_Oelims,axiom,
    ! [X: nat,Y: int] :
      ( ( vEBT_VEBT_Tb(X) = Y )
     => ( ( ( X = zero_zero(nat) )
         => ( Y != aa(num,int,numeral_numeral(int),aa(num,num,bit1,one2)) ) )
       => ( ( ( X = aa(nat,nat,suc,zero_zero(nat)) )
           => ( Y != aa(num,int,numeral_numeral(int),aa(num,num,bit1,one2)) ) )
         => ~ ! [N2: nat] :
                ( ( X = aa(nat,nat,suc,aa(nat,nat,suc,N2)) )
               => ~ ( ( pp(aa(nat,bool,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N2))
                     => ( Y = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,bit0(one2)))),vEBT_VEBT_Tb(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N2),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),aa(int,int,aa(int,fun(int,int),times_times(int),vEBT_VEBT_Tb(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N2),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N2),aa(num,nat,numeral_numeral(nat),bit0(one2))))))) ) )
                    & ( ~ pp(aa(nat,bool,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N2))
                     => ( Y = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,bit0(one2)))),vEBT_VEBT_Tb(aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N2),aa(num,nat,numeral_numeral(nat),bit0(one2)))))))),aa(int,int,aa(int,fun(int,int),times_times(int),vEBT_VEBT_Tb(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N2),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N2),aa(num,nat,numeral_numeral(nat),bit0(one2)))))))) ) ) ) ) ) ) ) ).

% VEBT_internal.Tb.elims
tff(fact_644_VEBT__internal_OT__vebt__buildupi_Oelims,axiom,
    ! [X: nat,Y: nat] :
      ( ( vEBT_V441764108873111860ildupi(X) = Y )
     => ( ( ( X = zero_zero(nat) )
         => ( Y != aa(nat,nat,suc,zero_zero(nat)) ) )
       => ( ( ( X = aa(nat,nat,suc,zero_zero(nat)) )
           => ( Y != aa(nat,nat,suc,zero_zero(nat)) ) )
         => ~ ! [N2: nat] :
                ( ( X = aa(nat,nat,suc,aa(nat,nat,suc,N2)) )
               => ~ ( ( pp(aa(nat,bool,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N2))
                     => ( Y = aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_V441764108873111860ildupi(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N2),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(one2)))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N2),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),vEBT_V441764108873111860ildupi(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N2),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N2),aa(num,nat,numeral_numeral(nat),bit0(one2))))))))))) ) )
                    & ( ~ pp(aa(nat,bool,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N2))
                     => ( Y = aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_V441764108873111860ildupi(aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N2),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(bit0(one2))))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N2),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(one2)))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),vEBT_V441764108873111860ildupi(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N2),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N2),aa(num,nat,numeral_numeral(nat),bit0(one2))))))))))) ) ) ) ) ) ) ) ).

% VEBT_internal.T_vebt_buildupi.elims
tff(fact_645_power2__le__imp__le,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,ord_less_eq(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),aa(num,nat,numeral_numeral(nat),bit0(one2)))))
         => ( pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),Y))
           => pp(aa(A,bool,ord_less_eq(A,X),Y)) ) ) ) ).

% power2_le_imp_le
tff(fact_646_power2__eq__imp__eq,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [X: A,Y: A] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(num,nat,numeral_numeral(nat),bit0(one2))) = aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),aa(num,nat,numeral_numeral(nat),bit0(one2))) )
         => ( pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),X))
           => ( pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),Y))
             => ( X = Y ) ) ) ) ) ).

% power2_eq_imp_eq
tff(fact_647_zero__le__power2,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A] : pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) ).

% zero_le_power2
tff(fact_648_exp__add__not__zero__imp__right,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [M: nat,N: nat] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N)) != zero_zero(A) )
         => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N) != zero_zero(A) ) ) ) ).

% exp_add_not_zero_imp_right
tff(fact_649_exp__add__not__zero__imp__left,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [M: nat,N: nat] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N)) != zero_zero(A) )
         => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),M) != zero_zero(A) ) ) ) ).

% exp_add_not_zero_imp_left
tff(fact_650_two__realpow__ge__one,axiom,
    ! [N: nat] : pp(aa(real,bool,ord_less_eq(real,one_one(real)),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(num,real,numeral_numeral(real),bit0(one2))),N))) ).

% two_realpow_ge_one
tff(fact_651_VEBT__internal_OTb_H_Osimps_I2_J,axiom,
    vEBT_VEBT_Tb2(aa(nat,nat,suc,zero_zero(nat))) = aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2)) ).

% VEBT_internal.Tb'.simps(2)
tff(fact_652_oddE,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A2: A] :
          ( ~ pp(aa(A,bool,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A2))
         => ~ ! [B3: A] : A2 != aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),B3)),one_one(A)) ) ) ).

% oddE
tff(fact_653_sum__power2__le__zero__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),aa(num,nat,numeral_numeral(nat),bit0(one2))))),zero_zero(A)))
        <=> ( ( X = zero_zero(A) )
            & ( Y = zero_zero(A) ) ) ) ) ).

% sum_power2_le_zero_iff
tff(fact_654_sum__power2__ge__zero,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A,Y: A] : pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),aa(num,nat,numeral_numeral(nat),bit0(one2)))))) ) ).

% sum_power2_ge_zero
tff(fact_655_zero__le__even__power_H,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,N: nat] : pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)))) ) ).

% zero_le_even_power'
tff(fact_656_zero__le__even__power,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [N: nat,A2: A] :
          ( pp(aa(nat,bool,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N))
         => pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),N))) ) ) ).

% zero_le_even_power
tff(fact_657_zero__le__odd__power,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [N: nat,A2: A] :
          ( ~ pp(aa(nat,bool,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N))
         => ( pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),N)))
          <=> pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),A2)) ) ) ) ).

% zero_le_odd_power
tff(fact_658_zero__le__power__eq,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,N: nat] :
          ( pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),N)))
        <=> ( pp(aa(nat,bool,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N))
            | ( ~ pp(aa(nat,bool,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N))
              & pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),A2)) ) ) ) ) ).

% zero_le_power_eq
tff(fact_659_division__decomp,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,dvd_dvd(A,A2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)))
         => ? [B5: A,C5: A] :
              ( ( A2 = aa(A,A,aa(A,fun(A,A),times_times(A),B5),C5) )
              & pp(aa(A,bool,dvd_dvd(A,B5),B2))
              & pp(aa(A,bool,dvd_dvd(A,C5),C2)) ) ) ) ).

% division_decomp
tff(fact_660_dvd__productE,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [P2: A,A2: A,B2: A] :
          ( pp(aa(A,bool,dvd_dvd(A,P2),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)))
         => ~ ! [X3: A,Y3: A] :
                ( ( P2 = aa(A,A,aa(A,fun(A,A),times_times(A),X3),Y3) )
               => ( pp(aa(A,bool,dvd_dvd(A,X3),A2))
                 => ~ pp(aa(A,bool,dvd_dvd(A,Y3),B2)) ) ) ) ) ).

% dvd_productE
tff(fact_661_odd__0__le__power__imp__0__le,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,N: nat] :
          ( pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)))))
         => pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),A2)) ) ) ).

% odd_0_le_power_imp_0_le
tff(fact_662_arith__geo__mean,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [U: A,X: A,Y: A] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),U),aa(num,nat,numeral_numeral(nat),bit0(one2))) = aa(A,A,aa(A,fun(A,A),times_times(A),X),Y) )
         => ( pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),X))
           => ( pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),Y))
             => pp(aa(A,bool,ord_less_eq(A,U),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y)),aa(num,A,numeral_numeral(A),bit0(one2))))) ) ) ) ) ).

% arith_geo_mean
tff(fact_663_Bernoulli__inequality__even,axiom,
    ! [N: nat,X: real] :
      ( pp(aa(nat,bool,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N))
     => pp(aa(real,bool,ord_less_eq(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),N)),X))),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),X)),N))) ) ).

% Bernoulli_inequality_even
tff(fact_664_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(aa(nat,nat,suc,zero_zero(nat))) = aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2)) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d\<^sub>u\<^sub>p.simps(2)
tff(fact_665_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(aa(nat,nat,suc,zero_zero(nat))) = aa(num,nat,numeral_numeral(nat),bit0(bit0(one2))) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d.simps(2)
tff(fact_666_high__inv,axiom,
    ! [X: nat,N: nat,Y: nat] :
      ( pp(aa(nat,bool,ord_less(nat,X),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)))
     => ( vEBT_VEBT_high(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Y),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N))),X),N) = Y ) ) ).

% high_inv
tff(fact_667_replicate__eq__replicate,axiom,
    ! [A: $tType,M: nat,X: A,N: nat,Y: A] :
      ( ( replicate(A,M,X) = replicate(A,N,Y) )
    <=> ( ( M = N )
        & ( ( M != zero_zero(nat) )
         => ( X = Y ) ) ) ) ).

% replicate_eq_replicate
tff(fact_668_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)) = aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2)) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d\<^sub>u\<^sub>p.simps(1)
tff(fact_669_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)) = aa(num,nat,numeral_numeral(nat),bit0(bit0(one2))) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d.simps(1)
tff(fact_670_Tb__T__vebt__buildupi_H_H,axiom,
    ! [N: nat] : pp(aa(nat,bool,ord_less_eq(nat,vEBT_V441764108873111860ildupi(N)),aa(nat,nat,minus_minus(nat,vEBT_VEBT_Tb2(N)),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ).

% Tb_T_vebt_buildupi''
tff(fact_671_space_H__bound,axiom,
    ! [T: vEBT_VEBT,N: nat,U: nat] :
      ( vEBT_invar_vebt(T,N)
     => ( ( U = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N) )
       => pp(aa(nat,bool,ord_less_eq(nat,aa(vEBT_VEBT,nat,vEBT_VEBT_space2,T)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(aa(num,num,bit1,one2))))),U))) ) ) ).

% space'_bound
tff(fact_672_high__bound__aux,axiom,
    ! [Ma: nat,N: nat,M: nat] :
      ( pp(aa(nat,bool,ord_less(nat,Ma),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),M))))
     => pp(aa(nat,bool,ord_less(nat,vEBT_VEBT_high(Ma,N)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M))) ) ).

% high_bound_aux
tff(fact_673_int__eq__iff__numeral,axiom,
    ! [M: nat,V: num] :
      ( ( aa(nat,int,semiring_1_of_nat(int),M) = aa(num,int,numeral_numeral(int),V) )
    <=> ( M = aa(num,nat,numeral_numeral(nat),V) ) ) ).

% int_eq_iff_numeral
tff(fact_674_valid__tree__deg__neq__0,axiom,
    ! [T: vEBT_VEBT] : ~ vEBT_invar_vebt(T,zero_zero(nat)) ).

% valid_tree_deg_neq_0
tff(fact_675_valid__0__not,axiom,
    ! [T: vEBT_VEBT] : ~ vEBT_invar_vebt(T,zero_zero(nat)) ).

% valid_0_not
tff(fact_676_deg__deg__n,axiom,
    ! [Info: option(product_prod(nat,nat)),Deg: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,N: nat] :
      ( vEBT_invar_vebt(vEBT_Node(Info,Deg,TreeList,Summary),N)
     => ( Deg = N ) ) ).

% deg_deg_n
tff(fact_677_cnt__non__neg,axiom,
    ! [T: vEBT_VEBT] : pp(aa(real,bool,ord_less_eq(real,zero_zero(real)),aa(vEBT_VEBT,real,vEBT_VEBT_cnt,T))) ).

% cnt_non_neg
tff(fact_678_deg__not__0,axiom,
    ! [T: vEBT_VEBT,N: nat] :
      ( vEBT_invar_vebt(T,N)
     => pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),N)) ) ).

% deg_not_0
tff(fact_679_deg__SUcn__Node,axiom,
    ! [Tree: vEBT_VEBT,N: nat] :
      ( vEBT_invar_vebt(Tree,aa(nat,nat,suc,aa(nat,nat,suc,N)))
     => ? [Info2: option(product_prod(nat,nat)),TreeList2: list(vEBT_VEBT),S2: vEBT_VEBT] : Tree = vEBT_Node(Info2,aa(nat,nat,suc,aa(nat,nat,suc,N)),TreeList2,S2) ) ).

% deg_SUcn_Node
tff(fact_680_T__vebt__buildupi__gq__0,axiom,
    ! [N: nat] : pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),vEBT_V441764108873111860ildupi(N))) ).

% T_vebt_buildupi_gq_0
tff(fact_681_buildup__build__time,axiom,
    ! [N: nat] : pp(aa(nat,bool,ord_less(nat,vEBT_V8346862874174094_d_u_p(N)),vEBT_V8646137997579335489_i_l_d(N))) ).

% buildup_build_time
tff(fact_682_buildup__gives__valid,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),N))
     => vEBT_invar_vebt(vEBT_vebt_buildup(N),N) ) ).

% buildup_gives_valid
tff(fact_683_not__gr__zero,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [N: A] :
          ( ~ pp(aa(A,bool,ord_less(A,zero_zero(A)),N))
        <=> ( N = zero_zero(A) ) ) ) ).

% not_gr_zero
tff(fact_684_numeral__less__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [M: num,N: num] :
          ( pp(aa(A,bool,ord_less(A,aa(num,A,numeral_numeral(A),M)),aa(num,A,numeral_numeral(A),N)))
        <=> pp(aa(num,bool,ord_less(num,M),N)) ) ) ).

% numeral_less_iff
tff(fact_685_double__eq__0__iff,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),A2) = zero_zero(A) )
        <=> ( A2 = zero_zero(A) ) ) ) ).

% double_eq_0_iff
tff(fact_686_add__less__cancel__right,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [A2: A,C2: A,B2: A] :
          ( pp(aa(A,bool,ord_less(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2)))
        <=> pp(aa(A,bool,ord_less(A,A2),B2)) ) ) ).

% add_less_cancel_right
tff(fact_687_add__less__cancel__left,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [C2: A,A2: A,B2: A] :
          ( pp(aa(A,bool,ord_less(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),A2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),B2)))
        <=> pp(aa(A,bool,ord_less(A,A2),B2)) ) ) ).

% add_less_cancel_left
tff(fact_688_diff__self,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A] : aa(A,A,minus_minus(A,A2),A2) = zero_zero(A) ) ).

% diff_self
tff(fact_689_diff__0__right,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A] : aa(A,A,minus_minus(A,A2),zero_zero(A)) = A2 ) ).

% diff_0_right
tff(fact_690_zero__diff,axiom,
    ! [A: $tType] :
      ( comm_monoid_diff(A)
     => ! [A2: A] : aa(A,A,minus_minus(A,zero_zero(A)),A2) = zero_zero(A) ) ).

% zero_diff
tff(fact_691_diff__zero,axiom,
    ! [A: $tType] :
      ( cancel1802427076303600483id_add(A)
     => ! [A2: A] : aa(A,A,minus_minus(A,A2),zero_zero(A)) = A2 ) ).

% diff_zero
tff(fact_692_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A: $tType] :
      ( cancel1802427076303600483id_add(A)
     => ! [A2: A] : aa(A,A,minus_minus(A,A2),A2) = zero_zero(A) ) ).

% cancel_comm_monoid_add_class.diff_cancel
tff(fact_693_add__diff__cancel__right_H,axiom,
    ! [A: $tType] :
      ( cancel2418104881723323429up_add(A)
     => ! [A2: A,B2: A] : aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),B2) = A2 ) ).

% add_diff_cancel_right'
tff(fact_694_add__diff__cancel__right,axiom,
    ! [A: $tType] :
      ( cancel2418104881723323429up_add(A)
     => ! [A2: A,C2: A,B2: A] : aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2)) = aa(A,A,minus_minus(A,A2),B2) ) ).

% add_diff_cancel_right
tff(fact_695_add__diff__cancel__left_H,axiom,
    ! [A: $tType] :
      ( cancel2418104881723323429up_add(A)
     => ! [A2: A,B2: A] : aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),A2) = B2 ) ).

% add_diff_cancel_left'
tff(fact_696_add__diff__cancel__left,axiom,
    ! [A: $tType] :
      ( cancel2418104881723323429up_add(A)
     => ! [C2: A,A2: A,B2: A] : aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),A2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),B2)) = aa(A,A,minus_minus(A,A2),B2) ) ).

% add_diff_cancel_left
tff(fact_697_diff__add__cancel,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,minus_minus(A,A2),B2)),B2) = A2 ) ).

% diff_add_cancel
tff(fact_698_add__diff__cancel,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A,B2: A] : aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),B2) = A2 ) ).

% add_diff_cancel
tff(fact_699_lessI,axiom,
    ! [N: nat] : pp(aa(nat,bool,ord_less(nat,N),aa(nat,nat,suc,N))) ).

% lessI
tff(fact_700_Suc__mono,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,ord_less(nat,M),N))
     => pp(aa(nat,bool,ord_less(nat,aa(nat,nat,suc,M)),aa(nat,nat,suc,N))) ) ).

% Suc_mono
tff(fact_701_Suc__less__eq,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,ord_less(nat,aa(nat,nat,suc,M)),aa(nat,nat,suc,N)))
    <=> pp(aa(nat,bool,ord_less(nat,M),N)) ) ).

% Suc_less_eq
tff(fact_702_bot__nat__0_Onot__eq__extremum,axiom,
    ! [A2: nat] :
      ( ( A2 != zero_zero(nat) )
    <=> pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),A2)) ) ).

% bot_nat_0.not_eq_extremum
tff(fact_703_neq0__conv,axiom,
    ! [N: nat] :
      ( ( N != zero_zero(nat) )
    <=> pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),N)) ) ).

% neq0_conv
tff(fact_704_less__nat__zero__code,axiom,
    ! [N: nat] : ~ pp(aa(nat,bool,ord_less(nat,N),zero_zero(nat))) ).

% less_nat_zero_code
tff(fact_705_diff__Suc__Suc,axiom,
    ! [M: nat,N: nat] : aa(nat,nat,minus_minus(nat,aa(nat,nat,suc,M)),aa(nat,nat,suc,N)) = aa(nat,nat,minus_minus(nat,M),N) ).

% diff_Suc_Suc
tff(fact_706_Suc__diff__diff,axiom,
    ! [M: nat,N: nat,K: nat] : aa(nat,nat,minus_minus(nat,aa(nat,nat,minus_minus(nat,aa(nat,nat,suc,M)),N)),aa(nat,nat,suc,K)) = aa(nat,nat,minus_minus(nat,aa(nat,nat,minus_minus(nat,M),N)),K) ).

% Suc_diff_diff
tff(fact_707_nat__add__left__cancel__less,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa(nat,bool,ord_less(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),M)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),N)))
    <=> pp(aa(nat,bool,ord_less(nat,M),N)) ) ).

% nat_add_left_cancel_less
tff(fact_708_power__one__right,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [A2: A] : aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),one_one(nat)) = A2 ) ).

% power_one_right
tff(fact_709_diff__0__eq__0,axiom,
    ! [N: nat] : aa(nat,nat,minus_minus(nat,zero_zero(nat)),N) = zero_zero(nat) ).

% diff_0_eq_0
tff(fact_710_diff__self__eq__0,axiom,
    ! [M: nat] : aa(nat,nat,minus_minus(nat,M),M) = zero_zero(nat) ).

% diff_self_eq_0
tff(fact_711_diff__diff__left,axiom,
    ! [I: nat,J: nat,K: nat] : aa(nat,nat,minus_minus(nat,aa(nat,nat,minus_minus(nat,I),J)),K) = aa(nat,nat,minus_minus(nat,I),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),K)) ).

% diff_diff_left
tff(fact_712_diff__diff__cancel,axiom,
    ! [I: nat,N: nat] :
      ( pp(aa(nat,bool,ord_less_eq(nat,I),N))
     => ( aa(nat,nat,minus_minus(nat,N),aa(nat,nat,minus_minus(nat,N),I)) = I ) ) ).

% diff_diff_cancel
tff(fact_713_nat__1__eq__mult__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( one_one(nat) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N) )
    <=> ( ( M = one_one(nat) )
        & ( N = one_one(nat) ) ) ) ).

% nat_1_eq_mult_iff
tff(fact_714_nat__mult__eq__1__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N) = one_one(nat) )
    <=> ( ( M = one_one(nat) )
        & ( N = one_one(nat) ) ) ) ).

% nat_mult_eq_1_iff
tff(fact_715_nat__dvd__1__iff__1,axiom,
    ! [M: nat] :
      ( pp(aa(nat,bool,dvd_dvd(nat,M),one_one(nat)))
    <=> ( M = one_one(nat) ) ) ).

% nat_dvd_1_iff_1
tff(fact_716_int__dvd__int__iff,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(int,bool,dvd_dvd(int,aa(nat,int,semiring_1_of_nat(int),M)),aa(nat,int,semiring_1_of_nat(int),N)))
    <=> pp(aa(nat,bool,dvd_dvd(nat,M),N)) ) ).

% int_dvd_int_iff
tff(fact_717_add__less__same__cancel1,axiom,
    ! [A: $tType] :
      ( ordere1937475149494474687imp_le(A)
     => ! [B2: A,A2: A] :
          ( pp(aa(A,bool,ord_less(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A2)),B2))
        <=> pp(aa(A,bool,ord_less(A,A2),zero_zero(A))) ) ) ).

% add_less_same_cancel1
tff(fact_718_add__less__same__cancel2,axiom,
    ! [A: $tType] :
      ( ordere1937475149494474687imp_le(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,ord_less(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),B2))
        <=> pp(aa(A,bool,ord_less(A,A2),zero_zero(A))) ) ) ).

% add_less_same_cancel2
tff(fact_719_less__add__same__cancel1,axiom,
    ! [A: $tType] :
      ( ordere1937475149494474687imp_le(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,ord_less(A,A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)))
        <=> pp(aa(A,bool,ord_less(A,zero_zero(A)),B2)) ) ) ).

% less_add_same_cancel1
tff(fact_720_less__add__same__cancel2,axiom,
    ! [A: $tType] :
      ( ordere1937475149494474687imp_le(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,ord_less(A,A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A2)))
        <=> pp(aa(A,bool,ord_less(A,zero_zero(A)),B2)) ) ) ).

% less_add_same_cancel2
tff(fact_721_double__add__less__zero__iff__single__add__less__zero,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A2: A] :
          ( pp(aa(A,bool,ord_less(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),A2)),zero_zero(A)))
        <=> pp(aa(A,bool,ord_less(A,A2),zero_zero(A))) ) ) ).

% double_add_less_zero_iff_single_add_less_zero
tff(fact_722_zero__less__double__add__iff__zero__less__single__add,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A2: A] :
          ( pp(aa(A,bool,ord_less(A,zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),A2)))
        <=> pp(aa(A,bool,ord_less(A,zero_zero(A)),A2)) ) ) ).

% zero_less_double_add_iff_zero_less_single_add
tff(fact_723_diff__ge__0__iff__ge,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),aa(A,A,minus_minus(A,A2),B2)))
        <=> pp(aa(A,bool,ord_less_eq(A,B2),A2)) ) ) ).

% diff_ge_0_iff_ge
tff(fact_724_diff__gt__0__iff__gt,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,ord_less(A,zero_zero(A)),aa(A,A,minus_minus(A,A2),B2)))
        <=> pp(aa(A,bool,ord_less(A,B2),A2)) ) ) ).

% diff_gt_0_iff_gt
tff(fact_725_diff__add__zero,axiom,
    ! [A: $tType] :
      ( comm_monoid_diff(A)
     => ! [A2: A,B2: A] : aa(A,A,minus_minus(A,A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)) = zero_zero(A) ) ).

% diff_add_zero
tff(fact_726_diff__numeral__special_I9_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ( aa(A,A,minus_minus(A,one_one(A)),one_one(A)) = zero_zero(A) ) ) ).

% diff_numeral_special(9)
tff(fact_727_le__add__diff__inverse2,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [B2: A,A2: A] :
          ( pp(aa(A,bool,ord_less_eq(A,B2),A2))
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,minus_minus(A,A2),B2)),B2) = A2 ) ) ) ).

% le_add_diff_inverse2
tff(fact_728_le__add__diff__inverse,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [B2: A,A2: A] :
          ( pp(aa(A,bool,ord_less_eq(A,B2),A2))
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),aa(A,A,minus_minus(A,A2),B2)) = A2 ) ) ) ).

% le_add_diff_inverse
tff(fact_729_left__diff__distrib__numeral,axiom,
    ! [A: $tType] :
      ( ( numeral(A)
        & ring(A) )
     => ! [A2: A,B2: A,V: num] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,minus_minus(A,A2),B2)),aa(num,A,numeral_numeral(A),V)) = aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(num,A,numeral_numeral(A),V))),aa(A,A,aa(A,fun(A,A),times_times(A),B2),aa(num,A,numeral_numeral(A),V))) ) ).

% left_diff_distrib_numeral
tff(fact_730_right__diff__distrib__numeral,axiom,
    ! [A: $tType] :
      ( ( numeral(A)
        & ring(A) )
     => ! [V: num,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),V)),aa(A,A,minus_minus(A,B2),C2)) = aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),V)),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),V)),C2)) ) ).

% right_diff_distrib_numeral
tff(fact_731_power__inject__exp,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,M: nat,N: nat] :
          ( pp(aa(A,bool,ord_less(A,one_one(A)),A2))
         => ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),M) = aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),N) )
          <=> ( M = N ) ) ) ) ).

% power_inject_exp
tff(fact_732_zero__less__Suc,axiom,
    ! [N: nat] : pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),aa(nat,nat,suc,N))) ).

% zero_less_Suc
tff(fact_733_less__Suc0,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,ord_less(nat,N),aa(nat,nat,suc,zero_zero(nat))))
    <=> ( N = zero_zero(nat) ) ) ).

% less_Suc0
tff(fact_734_of__nat__less__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [M: nat,N: nat] :
          ( pp(aa(A,bool,ord_less(A,aa(nat,A,semiring_1_of_nat(A),M)),aa(nat,A,semiring_1_of_nat(A),N)))
        <=> pp(aa(nat,bool,ord_less(nat,M),N)) ) ) ).

% of_nat_less_iff
tff(fact_735_div__diff,axiom,
    ! [A: $tType] :
      ( idom_modulo(A)
     => ! [C2: A,A2: A,B2: A] :
          ( pp(aa(A,bool,dvd_dvd(A,C2),A2))
         => ( pp(aa(A,bool,dvd_dvd(A,C2),B2))
           => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,minus_minus(A,A2),B2)),C2) = aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),C2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)) ) ) ) ) ).

% div_diff
tff(fact_736_add__gr__0,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N)))
    <=> ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),M))
        | pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),N)) ) ) ).

% add_gr_0
tff(fact_737_less__one,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,ord_less(nat,N),one_one(nat)))
    <=> ( N = zero_zero(nat) ) ) ).

% less_one
tff(fact_738_zero__less__diff,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),aa(nat,nat,minus_minus(nat,N),M)))
    <=> pp(aa(nat,bool,ord_less(nat,M),N)) ) ).

% zero_less_diff
tff(fact_739_diff__Suc__1,axiom,
    ! [N: nat] : aa(nat,nat,minus_minus(nat,aa(nat,nat,suc,N)),one_one(nat)) = N ).

% diff_Suc_1
tff(fact_740_div__less,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,ord_less(nat,M),N))
     => ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),N) = zero_zero(nat) ) ) ).

% div_less
tff(fact_741_mult__less__cancel2,axiom,
    ! [M: nat,K: nat,N: nat] :
      ( pp(aa(nat,bool,ord_less(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),K)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),K)))
    <=> ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),K))
        & pp(aa(nat,bool,ord_less(nat,M),N)) ) ) ).

% mult_less_cancel2
tff(fact_742_nat__0__less__mult__iff,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N)))
    <=> ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),M))
        & pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),N)) ) ) ).

% nat_0_less_mult_iff
tff(fact_743_nat__mult__less__cancel__disj,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa(nat,bool,ord_less(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),M)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),N)))
    <=> ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),K))
        & pp(aa(nat,bool,ord_less(nat,M),N)) ) ) ).

% nat_mult_less_cancel_disj
tff(fact_744_diff__is__0__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( aa(nat,nat,minus_minus(nat,M),N) = zero_zero(nat) )
    <=> pp(aa(nat,bool,ord_less_eq(nat,M),N)) ) ).

% diff_is_0_eq
tff(fact_745_diff__is__0__eq_H,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,ord_less_eq(nat,M),N))
     => ( aa(nat,nat,minus_minus(nat,M),N) = zero_zero(nat) ) ) ).

% diff_is_0_eq'
tff(fact_746_nat__zero__less__power__iff,axiom,
    ! [X: nat,N: nat] :
      ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),X),N)))
    <=> ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),X))
        | ( N = zero_zero(nat) ) ) ) ).

% nat_zero_less_power_iff
tff(fact_747_Nat_Odiff__diff__right,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( pp(aa(nat,bool,ord_less_eq(nat,K),J))
     => ( aa(nat,nat,minus_minus(nat,I),aa(nat,nat,minus_minus(nat,J),K)) = aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),K)),J) ) ) ).

% Nat.diff_diff_right
tff(fact_748_Nat_Oadd__diff__assoc2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( pp(aa(nat,bool,ord_less_eq(nat,K),J))
     => ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,minus_minus(nat,J),K)),I) = aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),I)),K) ) ) ).

% Nat.add_diff_assoc2
tff(fact_749_Nat_Oadd__diff__assoc,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( pp(aa(nat,bool,ord_less_eq(nat,K),J))
     => ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),aa(nat,nat,minus_minus(nat,J),K)) = aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),J)),K) ) ) ).

% Nat.add_diff_assoc
tff(fact_750_one__less__numeral__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [N: num] :
          ( pp(aa(A,bool,ord_less(A,one_one(A)),aa(num,A,numeral_numeral(A),N)))
        <=> pp(aa(num,bool,ord_less(num,one2),N)) ) ) ).

% one_less_numeral_iff
tff(fact_751_zero__less__divide__1__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( pp(aa(A,bool,ord_less(A,zero_zero(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),A2)))
        <=> pp(aa(A,bool,ord_less(A,zero_zero(A)),A2)) ) ) ).

% zero_less_divide_1_iff
tff(fact_752_less__divide__eq__1__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,ord_less(A,zero_zero(A)),A2))
         => ( pp(aa(A,bool,ord_less(A,one_one(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A2)))
          <=> pp(aa(A,bool,ord_less(A,A2),B2)) ) ) ) ).

% less_divide_eq_1_pos
tff(fact_753_less__divide__eq__1__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,ord_less(A,A2),zero_zero(A)))
         => ( pp(aa(A,bool,ord_less(A,one_one(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A2)))
          <=> pp(aa(A,bool,ord_less(A,B2),A2)) ) ) ) ).

% less_divide_eq_1_neg
tff(fact_754_divide__less__eq__1__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,ord_less(A,zero_zero(A)),A2))
         => ( pp(aa(A,bool,ord_less(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A2)),one_one(A)))
          <=> pp(aa(A,bool,ord_less(A,B2),A2)) ) ) ) ).

% divide_less_eq_1_pos
tff(fact_755_divide__less__eq__1__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,ord_less(A,A2),zero_zero(A)))
         => ( pp(aa(A,bool,ord_less(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A2)),one_one(A)))
          <=> pp(aa(A,bool,ord_less(A,A2),B2)) ) ) ) ).

% divide_less_eq_1_neg
tff(fact_756_divide__less__0__1__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( pp(aa(A,bool,ord_less(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),A2)),zero_zero(A)))
        <=> pp(aa(A,bool,ord_less(A,A2),zero_zero(A))) ) ) ).

% divide_less_0_1_iff
tff(fact_757_divide__less__eq__numeral1_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,W: num,A2: A] :
          ( pp(aa(A,bool,ord_less(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),aa(num,A,numeral_numeral(A),W))),A2))
        <=> pp(aa(A,bool,ord_less(A,B2),aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(num,A,numeral_numeral(A),W)))) ) ) ).

% divide_less_eq_numeral1(1)
tff(fact_758_less__divide__eq__numeral1_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A,W: num] :
          ( pp(aa(A,bool,ord_less(A,A2),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),aa(num,A,numeral_numeral(A),W))))
        <=> pp(aa(A,bool,ord_less(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(num,A,numeral_numeral(A),W))),B2)) ) ) ).

% less_divide_eq_numeral1(1)
tff(fact_759_power__strict__increasing__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [B2: A,X: nat,Y: nat] :
          ( pp(aa(A,bool,ord_less(A,one_one(A)),B2))
         => ( pp(aa(A,bool,ord_less(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),X)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),Y)))
          <=> pp(aa(nat,bool,ord_less(nat,X),Y)) ) ) ) ).

% power_strict_increasing_iff
tff(fact_760_power__eq__0__iff,axiom,
    ! [A: $tType] :
      ( semiri2026040879449505780visors(A)
     => ! [A2: A,N: nat] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),N) = zero_zero(A) )
        <=> ( ( A2 = zero_zero(A) )
            & pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),N)) ) ) ) ).

% power_eq_0_iff
tff(fact_761_Suc__pred,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),N))
     => ( aa(nat,nat,suc,aa(nat,nat,minus_minus(nat,N),aa(nat,nat,suc,zero_zero(nat)))) = N ) ) ).

% Suc_pred
tff(fact_762_pow__divides__pow__iff,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [N: nat,A2: A,B2: A] :
          ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),N))
         => ( pp(aa(A,bool,dvd_dvd(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),N)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),N)))
          <=> pp(aa(A,bool,dvd_dvd(A,A2),B2)) ) ) ) ).

% pow_divides_pow_iff
tff(fact_763_diff__Suc__diff__eq1,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( pp(aa(nat,bool,ord_less_eq(nat,K),J))
     => ( aa(nat,nat,minus_minus(nat,I),aa(nat,nat,suc,aa(nat,nat,minus_minus(nat,J),K))) = aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),K)),aa(nat,nat,suc,J)) ) ) ).

% diff_Suc_diff_eq1
tff(fact_764_diff__Suc__diff__eq2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( pp(aa(nat,bool,ord_less_eq(nat,K),J))
     => ( aa(nat,nat,minus_minus(nat,aa(nat,nat,suc,aa(nat,nat,minus_minus(nat,J),K))),I) = aa(nat,nat,minus_minus(nat,aa(nat,nat,suc,J)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),I)) ) ) ).

% diff_Suc_diff_eq2
tff(fact_765_div__eq__dividend__iff,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),M))
     => ( ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),N) = M )
      <=> ( N = one_one(nat) ) ) ) ).

% div_eq_dividend_iff
tff(fact_766_mult__le__cancel2,axiom,
    ! [M: nat,K: nat,N: nat] :
      ( pp(aa(nat,bool,ord_less_eq(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),K)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),K)))
    <=> ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),K))
       => pp(aa(nat,bool,ord_less_eq(nat,M),N)) ) ) ).

% mult_le_cancel2
tff(fact_767_nat__mult__le__cancel__disj,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa(nat,bool,ord_less_eq(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),M)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),N)))
    <=> ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),K))
       => pp(aa(nat,bool,ord_less_eq(nat,M),N)) ) ) ).

% nat_mult_le_cancel_disj
tff(fact_768_div__mult__self__is__m,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),N))
     => ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N)),N) = M ) ) ).

% div_mult_self_is_m
tff(fact_769_div__mult__self1__is__m,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),N))
     => ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),M)),N) = M ) ) ).

% div_mult_self1_is_m
tff(fact_770_half__nonnegative__int__iff,axiom,
    ! [K: int] :
      ( pp(aa(int,bool,ord_less_eq(int,zero_zero(int)),aa(int,int,aa(int,fun(int,int),divide_divide(int),K),aa(num,int,numeral_numeral(int),bit0(one2)))))
    <=> pp(aa(int,bool,ord_less_eq(int,zero_zero(int)),K)) ) ).

% half_nonnegative_int_iff
tff(fact_771_divide__le__eq__1__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,ord_less(A,A2),zero_zero(A)))
         => ( pp(aa(A,bool,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A2)),one_one(A)))
          <=> pp(aa(A,bool,ord_less_eq(A,A2),B2)) ) ) ) ).

% divide_le_eq_1_neg
tff(fact_772_divide__le__eq__1__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,ord_less(A,zero_zero(A)),A2))
         => ( pp(aa(A,bool,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A2)),one_one(A)))
          <=> pp(aa(A,bool,ord_less_eq(A,B2),A2)) ) ) ) ).

% divide_le_eq_1_pos
tff(fact_773_le__divide__eq__1__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,ord_less(A,A2),zero_zero(A)))
         => ( pp(aa(A,bool,ord_less_eq(A,one_one(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A2)))
          <=> pp(aa(A,bool,ord_less_eq(A,B2),A2)) ) ) ) ).

% le_divide_eq_1_neg
tff(fact_774_le__divide__eq__1__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,ord_less(A,zero_zero(A)),A2))
         => ( pp(aa(A,bool,ord_less_eq(A,one_one(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A2)))
          <=> pp(aa(A,bool,ord_less_eq(A,A2),B2)) ) ) ) ).

% le_divide_eq_1_pos
tff(fact_775_power__strict__decreasing__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [B2: A,M: nat,N: nat] :
          ( pp(aa(A,bool,ord_less(A,zero_zero(A)),B2))
         => ( pp(aa(A,bool,ord_less(A,B2),one_one(A)))
           => ( pp(aa(A,bool,ord_less(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),M)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),N)))
            <=> pp(aa(nat,bool,ord_less(nat,N),M)) ) ) ) ) ).

% power_strict_decreasing_iff
tff(fact_776_power__mono__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,B2: A,N: nat] :
          ( pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),A2))
         => ( pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),B2))
           => ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),N))
             => ( pp(aa(A,bool,ord_less_eq(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),N)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),N)))
              <=> pp(aa(A,bool,ord_less_eq(A,A2),B2)) ) ) ) ) ) ).

% power_mono_iff
tff(fact_777_power__increasing__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [B2: A,X: nat,Y: nat] :
          ( pp(aa(A,bool,ord_less(A,one_one(A)),B2))
         => ( pp(aa(A,bool,ord_less_eq(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),X)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),Y)))
          <=> pp(aa(nat,bool,ord_less_eq(nat,X),Y)) ) ) ) ).

% power_increasing_iff
tff(fact_778_of__nat__0__less__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [N: nat] :
          ( pp(aa(A,bool,ord_less(A,zero_zero(A)),aa(nat,A,semiring_1_of_nat(A),N)))
        <=> pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),N)) ) ) ).

% of_nat_0_less_iff
tff(fact_779_Suc__1,axiom,
    aa(nat,nat,suc,one_one(nat)) = aa(num,nat,numeral_numeral(nat),bit0(one2)) ).

% Suc_1
tff(fact_780_Suc__diff__1,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),N))
     => ( aa(nat,nat,suc,aa(nat,nat,minus_minus(nat,N),one_one(nat))) = N ) ) ).

% Suc_diff_1
tff(fact_781_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [B2: nat,W: nat,X: nat] :
          ( pp(aa(A,bool,ord_less(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(nat,A,semiring_1_of_nat(A),B2)),W)),aa(nat,A,semiring_1_of_nat(A),X)))
        <=> pp(aa(nat,bool,ord_less(nat,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),W)),X)) ) ) ).

% of_nat_less_of_nat_power_cancel_iff
tff(fact_782_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [X: nat,B2: nat,W: nat] :
          ( pp(aa(A,bool,ord_less(A,aa(nat,A,semiring_1_of_nat(A),X)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(nat,A,semiring_1_of_nat(A),B2)),W)))
        <=> pp(aa(nat,bool,ord_less(nat,X),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),W))) ) ) ).

% of_nat_power_less_of_nat_cancel_iff
tff(fact_783_zero__less__power2,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A] :
          ( pp(aa(A,bool,ord_less(A,zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(num,nat,numeral_numeral(nat),bit0(one2)))))
        <=> ( A2 != zero_zero(A) ) ) ) ).

% zero_less_power2
tff(fact_784_power__decreasing__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [B2: A,M: nat,N: nat] :
          ( pp(aa(A,bool,ord_less(A,zero_zero(A)),B2))
         => ( pp(aa(A,bool,ord_less(A,B2),one_one(A)))
           => ( pp(aa(A,bool,ord_less_eq(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),M)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),N)))
            <=> pp(aa(nat,bool,ord_less_eq(nat,N),M)) ) ) ) ) ).

% power_decreasing_iff
tff(fact_785_even__diff,axiom,
    ! [A: $tType] :
      ( ring_parity(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,minus_minus(A,A2),B2)))
        <=> pp(aa(A,bool,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2))) ) ) ).

% even_diff
tff(fact_786_of__nat__zero__less__power__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [X: nat,N: nat] :
          ( pp(aa(A,bool,ord_less(A,zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(nat,A,semiring_1_of_nat(A),X)),N)))
        <=> ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),X))
            | ( N = zero_zero(nat) ) ) ) ) ).

% of_nat_zero_less_power_iff
tff(fact_787_even__power,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A2: A,N: nat] :
          ( pp(aa(A,bool,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),N)))
        <=> ( pp(aa(A,bool,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A2))
            & pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),N)) ) ) ) ).

% even_power
tff(fact_788_power__less__zero__eq,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,N: nat] :
          ( pp(aa(A,bool,ord_less(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),N)),zero_zero(A)))
        <=> ( ~ pp(aa(nat,bool,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N))
            & pp(aa(A,bool,ord_less(A,A2),zero_zero(A))) ) ) ) ).

% power_less_zero_eq
tff(fact_789_power__less__zero__eq__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,W: num] :
          ( pp(aa(A,bool,ord_less(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(num,nat,numeral_numeral(nat),W))),zero_zero(A)))
        <=> ( ~ pp(aa(nat,bool,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(num,nat,numeral_numeral(nat),W)))
            & pp(aa(A,bool,ord_less(A,A2),zero_zero(A))) ) ) ) ).

% power_less_zero_eq_numeral
tff(fact_790_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [I: num,N: nat,X: nat] :
          ( pp(aa(A,bool,ord_less(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),I)),N)),aa(nat,A,semiring_1_of_nat(A),X)))
        <=> pp(aa(nat,bool,ord_less(nat,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),I)),N)),X)) ) ) ).

% numeral_power_less_of_nat_cancel_iff
tff(fact_791_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [X: nat,I: num,N: nat] :
          ( pp(aa(A,bool,ord_less(A,aa(nat,A,semiring_1_of_nat(A),X)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),I)),N)))
        <=> pp(aa(nat,bool,ord_less(nat,X),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),I)),N))) ) ) ).

% of_nat_less_numeral_power_cancel_iff
tff(fact_792_odd__Suc__minus__one,axiom,
    ! [N: nat] :
      ( ~ pp(aa(nat,bool,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N))
     => ( aa(nat,nat,suc,aa(nat,nat,minus_minus(nat,N),aa(nat,nat,suc,zero_zero(nat)))) = N ) ) ).

% odd_Suc_minus_one
tff(fact_793_even__diff__nat,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,minus_minus(nat,M),N)))
    <=> ( pp(aa(nat,bool,ord_less(nat,M),N))
        | pp(aa(nat,bool,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N))) ) ) ).

% even_diff_nat
tff(fact_794_semiring__parity__class_Oeven__mask__iff,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [N: nat] :
          ( pp(aa(A,bool,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,minus_minus(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N)),one_one(A))))
        <=> ( N = zero_zero(nat) ) ) ) ).

% semiring_parity_class.even_mask_iff
tff(fact_795_zero__less__power__eq__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,W: num] :
          ( pp(aa(A,bool,ord_less(A,zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(num,nat,numeral_numeral(nat),W))))
        <=> ( ( aa(num,nat,numeral_numeral(nat),W) = zero_zero(nat) )
            | ( pp(aa(nat,bool,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(num,nat,numeral_numeral(nat),W)))
              & ( A2 != zero_zero(A) ) )
            | ( ~ pp(aa(nat,bool,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(num,nat,numeral_numeral(nat),W)))
              & pp(aa(A,bool,ord_less(A,zero_zero(A)),A2)) ) ) ) ) ).

% zero_less_power_eq_numeral
tff(fact_796_odd__two__times__div__two__nat,axiom,
    ! [N: nat] :
      ( ~ pp(aa(nat,bool,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N))
     => ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = aa(nat,nat,minus_minus(nat,N),one_one(nat)) ) ) ).

% odd_two_times_div_two_nat
tff(fact_797_power__le__zero__eq__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,W: num] :
          ( pp(aa(A,bool,ord_less_eq(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(num,nat,numeral_numeral(nat),W))),zero_zero(A)))
        <=> ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),aa(num,nat,numeral_numeral(nat),W)))
            & ( ( ~ pp(aa(nat,bool,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(num,nat,numeral_numeral(nat),W)))
                & pp(aa(A,bool,ord_less_eq(A,A2),zero_zero(A))) )
              | ( pp(aa(nat,bool,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(num,nat,numeral_numeral(nat),W)))
                & ( A2 = zero_zero(A) ) ) ) ) ) ) ).

% power_le_zero_eq_numeral
tff(fact_798_even__succ__div__exp,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A,N: nat] :
          ( pp(aa(A,bool,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A2))
         => ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),N))
           => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),A2)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N)) ) ) ) ) ).

% even_succ_div_exp
tff(fact_799_nonneg__int__cases,axiom,
    ! [K: int] :
      ( pp(aa(int,bool,ord_less_eq(int,zero_zero(int)),K))
     => ~ ! [N2: nat] : K != aa(nat,int,semiring_1_of_nat(int),N2) ) ).

% nonneg_int_cases
tff(fact_800_zero__le__imp__eq__int,axiom,
    ! [K: int] :
      ( pp(aa(int,bool,ord_less_eq(int,zero_zero(int)),K))
     => ? [N2: nat] : K = aa(nat,int,semiring_1_of_nat(int),N2) ) ).

% zero_le_imp_eq_int
tff(fact_801_int__int__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( aa(nat,int,semiring_1_of_nat(int),M) = aa(nat,int,semiring_1_of_nat(int),N) )
    <=> ( M = N ) ) ).

% int_int_eq
tff(fact_802_odd__nonzero,axiom,
    ! [Z: int] : aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),Z)),Z) != zero_zero(int) ).

% odd_nonzero
tff(fact_803_linorder__neqE__linordered__idom,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A,Y: A] :
          ( ( X != Y )
         => ( ~ pp(aa(A,bool,ord_less(A,X),Y))
           => pp(aa(A,bool,ord_less(A,Y),X)) ) ) ) ).

% linorder_neqE_linordered_idom
tff(fact_804_nat__neq__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( M != N )
    <=> ( pp(aa(nat,bool,ord_less(nat,M),N))
        | pp(aa(nat,bool,ord_less(nat,N),M)) ) ) ).

% nat_neq_iff
tff(fact_805_diff__commute,axiom,
    ! [I: nat,J: nat,K: nat] : aa(nat,nat,minus_minus(nat,aa(nat,nat,minus_minus(nat,I),J)),K) = aa(nat,nat,minus_minus(nat,aa(nat,nat,minus_minus(nat,I),K)),J) ).

% diff_commute
tff(fact_806_less__not__refl,axiom,
    ! [N: nat] : ~ pp(aa(nat,bool,ord_less(nat,N),N)) ).

% less_not_refl
tff(fact_807_less__not__refl2,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa(nat,bool,ord_less(nat,N),M))
     => ( M != N ) ) ).

% less_not_refl2
tff(fact_808_less__not__refl3,axiom,
    ! [S3: nat,T: nat] :
      ( pp(aa(nat,bool,ord_less(nat,S3),T))
     => ( S3 != T ) ) ).

% less_not_refl3
tff(fact_809_measure__induct,axiom,
    ! [B: $tType,A: $tType] :
      ( wellorder(B)
     => ! [F2: fun(A,B),P: fun(A,bool),A2: A] :
          ( ! [X3: A] :
              ( ! [Y4: A] :
                  ( pp(aa(B,bool,ord_less(B,aa(A,B,F2,Y4)),aa(A,B,F2,X3)))
                 => pp(aa(A,bool,P,Y4)) )
             => pp(aa(A,bool,P,X3)) )
         => pp(aa(A,bool,P,A2)) ) ) ).

% measure_induct
tff(fact_810_diff__less__mono2,axiom,
    ! [M: nat,N: nat,L: nat] :
      ( pp(aa(nat,bool,ord_less(nat,M),N))
     => ( pp(aa(nat,bool,ord_less(nat,M),L))
       => pp(aa(nat,bool,ord_less(nat,aa(nat,nat,minus_minus(nat,L),N)),aa(nat,nat,minus_minus(nat,L),M))) ) ) ).

% diff_less_mono2
tff(fact_811_less__irrefl__nat,axiom,
    ! [N: nat] : ~ pp(aa(nat,bool,ord_less(nat,N),N)) ).

% less_irrefl_nat
tff(fact_812_nat__less__induct,axiom,
    ! [P: fun(nat,bool),N: nat] :
      ( ! [N2: nat] :
          ( ! [M4: nat] :
              ( pp(aa(nat,bool,ord_less(nat,M4),N2))
             => pp(aa(nat,bool,P,M4)) )
         => pp(aa(nat,bool,P,N2)) )
     => pp(aa(nat,bool,P,N)) ) ).

% nat_less_induct
tff(fact_813_infinite__descent,axiom,
    ! [P: fun(nat,bool),N: nat] :
      ( ! [N2: nat] :
          ( ~ pp(aa(nat,bool,P,N2))
         => ? [M4: nat] :
              ( pp(aa(nat,bool,ord_less(nat,M4),N2))
              & ~ pp(aa(nat,bool,P,M4)) ) )
     => pp(aa(nat,bool,P,N)) ) ).

% infinite_descent
tff(fact_814_linorder__neqE__nat,axiom,
    ! [X: nat,Y: nat] :
      ( ( X != Y )
     => ( ~ pp(aa(nat,bool,ord_less(nat,X),Y))
       => pp(aa(nat,bool,ord_less(nat,Y),X)) ) ) ).

% linorder_neqE_nat
tff(fact_815_less__imp__diff__less,axiom,
    ! [J: nat,K: nat,N: nat] :
      ( pp(aa(nat,bool,ord_less(nat,J),K))
     => pp(aa(nat,bool,ord_less(nat,aa(nat,nat,minus_minus(nat,J),N)),K)) ) ).

% less_imp_diff_less
tff(fact_816_measure__induct__rule,axiom,
    ! [B: $tType,A: $tType] :
      ( wellorder(B)
     => ! [F2: fun(A,B),P: fun(A,bool),A2: A] :
          ( ! [X3: A] :
              ( ! [Y4: A] :
                  ( pp(aa(B,bool,ord_less(B,aa(A,B,F2,Y4)),aa(A,B,F2,X3)))
                 => pp(aa(A,bool,P,Y4)) )
             => pp(aa(A,bool,P,X3)) )
         => pp(aa(A,bool,P,A2)) ) ) ).

% measure_induct_rule
tff(fact_817_infinite__descent__measure,axiom,
    ! [A: $tType,P: fun(A,bool),V2: fun(A,nat),X: A] :
      ( ! [X3: A] :
          ( ~ pp(aa(A,bool,P,X3))
         => ? [Y4: A] :
              ( pp(aa(nat,bool,ord_less(nat,aa(A,nat,V2,Y4)),aa(A,nat,V2,X3)))
              & ~ pp(aa(A,bool,P,Y4)) ) )
     => pp(aa(A,bool,P,X)) ) ).

% infinite_descent_measure
tff(fact_818_diff__eq__diff__eq,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A,B2: A,C2: A,D: A] :
          ( ( aa(A,A,minus_minus(A,A2),B2) = aa(A,A,minus_minus(A,C2),D) )
         => ( ( A2 = B2 )
          <=> ( C2 = D ) ) ) ) ).

% diff_eq_diff_eq
tff(fact_819_diff__strict__mono,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A,D: A,C2: A] :
          ( pp(aa(A,bool,ord_less(A,A2),B2))
         => ( pp(aa(A,bool,ord_less(A,D),C2))
           => pp(aa(A,bool,ord_less(A,aa(A,A,minus_minus(A,A2),C2)),aa(A,A,minus_minus(A,B2),D))) ) ) ) ).

% diff_strict_mono
tff(fact_820_diff__eq__diff__less,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A,C2: A,D: A] :
          ( ( aa(A,A,minus_minus(A,A2),B2) = aa(A,A,minus_minus(A,C2),D) )
         => ( pp(aa(A,bool,ord_less(A,A2),B2))
          <=> pp(aa(A,bool,ord_less(A,C2),D)) ) ) ) ).

% diff_eq_diff_less
tff(fact_821_linordered__field__no__lb,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X4: A] :
        ? [Y3: A] : pp(aa(A,bool,ord_less(A,Y3),X4)) ) ).

% linordered_field_no_lb
tff(fact_822_linordered__field__no__ub,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X4: A] :
        ? [X_1: A] : pp(aa(A,bool,ord_less(A,X4),X_1)) ) ).

% linordered_field_no_ub
tff(fact_823_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
    ! [A: $tType] :
      ( cancel2418104881723323429up_add(A)
     => ! [A2: A,C2: A,B2: A] : aa(A,A,minus_minus(A,aa(A,A,minus_minus(A,A2),C2)),B2) = aa(A,A,minus_minus(A,aa(A,A,minus_minus(A,A2),B2)),C2) ) ).

% cancel_ab_semigroup_add_class.diff_right_commute
tff(fact_824_diff__strict__left__mono,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [B2: A,A2: A,C2: A] :
          ( pp(aa(A,bool,ord_less(A,B2),A2))
         => pp(aa(A,bool,ord_less(A,aa(A,A,minus_minus(A,C2),A2)),aa(A,A,minus_minus(A,C2),B2))) ) ) ).

% diff_strict_left_mono
tff(fact_825_diff__strict__right__mono,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,ord_less(A,A2),B2))
         => pp(aa(A,bool,ord_less(A,aa(A,A,minus_minus(A,A2),C2)),aa(A,A,minus_minus(A,B2),C2))) ) ) ).

% diff_strict_right_mono
tff(fact_826_less__iff__diff__less__0,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,ord_less(A,A2),B2))
        <=> pp(aa(A,bool,ord_less(A,aa(A,A,minus_minus(A,A2),B2)),zero_zero(A))) ) ) ).

% less_iff_diff_less_0
tff(fact_827_diff__less__eq,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,ord_less(A,aa(A,A,minus_minus(A,A2),B2)),C2))
        <=> pp(aa(A,bool,ord_less(A,A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),B2))) ) ) ).

% diff_less_eq
tff(fact_828_less__diff__eq,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,C2: A,B2: A] :
          ( pp(aa(A,bool,ord_less(A,A2),aa(A,A,minus_minus(A,C2),B2)))
        <=> pp(aa(A,bool,ord_less(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),C2)) ) ) ).

% less_diff_eq
tff(fact_829_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,B2: A] :
          ( ~ pp(aa(A,bool,ord_less(A,A2),B2))
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),aa(A,A,minus_minus(A,A2),B2)) = A2 ) ) ) ).

% linordered_semidom_class.add_diff_inverse
tff(fact_830_lift__Suc__mono__less__iff,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [F2: fun(nat,A),N: nat,M: nat] :
          ( ! [N2: nat] : pp(aa(A,bool,ord_less(A,aa(nat,A,F2,N2)),aa(nat,A,F2,aa(nat,nat,suc,N2))))
         => ( pp(aa(A,bool,ord_less(A,aa(nat,A,F2,N)),aa(nat,A,F2,M)))
          <=> pp(aa(nat,bool,ord_less(nat,N),M)) ) ) ) ).

% lift_Suc_mono_less_iff
tff(fact_831_lift__Suc__mono__less,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [F2: fun(nat,A),N: nat,N3: nat] :
          ( ! [N2: nat] : pp(aa(A,bool,ord_less(A,aa(nat,A,F2,N2)),aa(nat,A,F2,aa(nat,nat,suc,N2))))
         => ( pp(aa(nat,bool,ord_less(nat,N),N3))
           => pp(aa(A,bool,ord_less(A,aa(nat,A,F2,N)),aa(nat,A,F2,N3))) ) ) ) ).

% lift_Suc_mono_less
tff(fact_832_diff__less__Suc,axiom,
    ! [M: nat,N: nat] : pp(aa(nat,bool,ord_less(nat,aa(nat,nat,minus_minus(nat,M),N)),aa(nat,nat,suc,M))) ).

% diff_less_Suc
tff(fact_833_Suc__diff__Suc,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa(nat,bool,ord_less(nat,N),M))
     => ( aa(nat,nat,suc,aa(nat,nat,minus_minus(nat,M),aa(nat,nat,suc,N))) = aa(nat,nat,minus_minus(nat,M),N) ) ) ).

% Suc_diff_Suc
tff(fact_834_diff__less,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),N))
     => ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),M))
       => pp(aa(nat,bool,ord_less(nat,aa(nat,nat,minus_minus(nat,M),N)),M)) ) ) ).

% diff_less
tff(fact_835_less__diff__conv,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( pp(aa(nat,bool,ord_less(nat,I),aa(nat,nat,minus_minus(nat,J),K)))
    <=> pp(aa(nat,bool,ord_less(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),K)),J)) ) ).

% less_diff_conv
tff(fact_836_add__diff__inverse__nat,axiom,
    ! [M: nat,N: nat] :
      ( ~ pp(aa(nat,bool,ord_less(nat,M),N))
     => ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),aa(nat,nat,minus_minus(nat,M),N)) = M ) ) ).

% add_diff_inverse_nat
tff(fact_837_diff__Suc__eq__diff__pred,axiom,
    ! [M: nat,N: nat] : aa(nat,nat,minus_minus(nat,M),aa(nat,nat,suc,N)) = aa(nat,nat,minus_minus(nat,aa(nat,nat,minus_minus(nat,M),one_one(nat))),N) ).

% diff_Suc_eq_diff_pred
tff(fact_838_less__diff__iff,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa(nat,bool,ord_less_eq(nat,K),M))
     => ( pp(aa(nat,bool,ord_less_eq(nat,K),N))
       => ( pp(aa(nat,bool,ord_less(nat,aa(nat,nat,minus_minus(nat,M),K)),aa(nat,nat,minus_minus(nat,N),K)))
        <=> pp(aa(nat,bool,ord_less(nat,M),N)) ) ) ) ).

% less_diff_iff
tff(fact_839_diff__less__mono,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( pp(aa(nat,bool,ord_less(nat,A2),B2))
     => ( pp(aa(nat,bool,ord_less_eq(nat,C2),A2))
       => pp(aa(nat,bool,ord_less(nat,aa(nat,nat,minus_minus(nat,A2),C2)),aa(nat,nat,minus_minus(nat,B2),C2))) ) ) ).

% diff_less_mono
tff(fact_840_less__imp__of__nat__less,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [M: nat,N: nat] :
          ( pp(aa(nat,bool,ord_less(nat,M),N))
         => pp(aa(A,bool,ord_less(A,aa(nat,A,semiring_1_of_nat(A),M)),aa(nat,A,semiring_1_of_nat(A),N))) ) ) ).

% less_imp_of_nat_less
tff(fact_841_of__nat__less__imp__less,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [M: nat,N: nat] :
          ( pp(aa(A,bool,ord_less(A,aa(nat,A,semiring_1_of_nat(A),M)),aa(nat,A,semiring_1_of_nat(A),N)))
         => pp(aa(nat,bool,ord_less(nat,M),N)) ) ) ).

% of_nat_less_imp_less
tff(fact_842_dvd__minus__self,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,dvd_dvd(nat,M),aa(nat,nat,minus_minus(nat,N),M)))
    <=> ( pp(aa(nat,bool,ord_less(nat,N),M))
        | pp(aa(nat,bool,dvd_dvd(nat,M),N)) ) ) ).

% dvd_minus_self
tff(fact_843_Suc__pred_H,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),N))
     => ( N = aa(nat,nat,suc,aa(nat,nat,minus_minus(nat,N),one_one(nat))) ) ) ).

% Suc_pred'
tff(fact_844_Suc__diff__eq__diff__pred,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),N))
     => ( aa(nat,nat,minus_minus(nat,aa(nat,nat,suc,M)),N) = aa(nat,nat,minus_minus(nat,M),aa(nat,nat,minus_minus(nat,N),one_one(nat))) ) ) ).

% Suc_diff_eq_diff_pred
tff(fact_845_int__ge__induct,axiom,
    ! [K: int,I: int,P: fun(int,bool)] :
      ( pp(aa(int,bool,ord_less_eq(int,K),I))
     => ( pp(aa(int,bool,P,K))
       => ( ! [I2: int] :
              ( pp(aa(int,bool,ord_less_eq(int,K),I2))
             => ( pp(aa(int,bool,P,I2))
               => pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),I2),one_one(int)))) ) )
         => pp(aa(int,bool,P,I)) ) ) ) ).

% int_ge_induct
tff(fact_846_plus__int__code_I2_J,axiom,
    ! [L: int] : aa(int,int,aa(int,fun(int,int),plus_plus(int),zero_zero(int)),L) = L ).

% plus_int_code(2)
tff(fact_847_plus__int__code_I1_J,axiom,
    ! [K: int] : aa(int,int,aa(int,fun(int,int),plus_plus(int),K),zero_zero(int)) = K ).

% plus_int_code(1)
tff(fact_848_times__int__code_I2_J,axiom,
    ! [L: int] : aa(int,int,aa(int,fun(int,int),times_times(int),zero_zero(int)),L) = zero_zero(int) ).

% times_int_code(2)
tff(fact_849_times__int__code_I1_J,axiom,
    ! [K: int] : aa(int,int,aa(int,fun(int,int),times_times(int),K),zero_zero(int)) = zero_zero(int) ).

% times_int_code(1)
tff(fact_850_zdvd__mult__cancel,axiom,
    ! [K: int,M: int,N: int] :
      ( pp(aa(int,bool,dvd_dvd(int,aa(int,int,aa(int,fun(int,int),times_times(int),K),M)),aa(int,int,aa(int,fun(int,int),times_times(int),K),N)))
     => ( ( K != zero_zero(int) )
       => pp(aa(int,bool,dvd_dvd(int,M),N)) ) ) ).

% zdvd_mult_cancel
tff(fact_851_power__minus__mult,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [N: nat,A2: A] :
          ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),N))
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(nat,nat,minus_minus(nat,N),one_one(nat)))),A2) = aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),N) ) ) ) ).

% power_minus_mult
tff(fact_852_less__add__iff2,axiom,
    ! [A: $tType] :
      ( ordered_ring(A)
     => ! [A2: A,E: A,C2: A,B2: A,D: A] :
          ( pp(aa(A,bool,ord_less(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),E)),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),E)),D)))
        <=> pp(aa(A,bool,ord_less(A,C2),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,minus_minus(A,B2),A2)),E)),D))) ) ) ).

% less_add_iff2
tff(fact_853_less__add__iff1,axiom,
    ! [A: $tType] :
      ( ordered_ring(A)
     => ! [A2: A,E: A,C2: A,B2: A,D: A] :
          ( pp(aa(A,bool,ord_less(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),E)),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),E)),D)))
        <=> pp(aa(A,bool,ord_less(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,minus_minus(A,A2),B2)),E)),C2)),D)) ) ) ).

% less_add_iff1
tff(fact_854_of__nat__diff,axiom,
    ! [A: $tType] :
      ( semiring_1_cancel(A)
     => ! [N: nat,M: nat] :
          ( pp(aa(nat,bool,ord_less_eq(nat,N),M))
         => ( aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,minus_minus(nat,M),N)) = aa(A,A,minus_minus(A,aa(nat,A,semiring_1_of_nat(A),M)),aa(nat,A,semiring_1_of_nat(A),N)) ) ) ) ).

% of_nat_diff
tff(fact_855_power__strict__increasing,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [N: nat,N4: nat,A2: A] :
          ( pp(aa(nat,bool,ord_less(nat,N),N4))
         => ( pp(aa(A,bool,ord_less(A,one_one(A)),A2))
           => pp(aa(A,bool,ord_less(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),N)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),N4))) ) ) ) ).

% power_strict_increasing
tff(fact_856_power__less__imp__less__exp,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,M: nat,N: nat] :
          ( pp(aa(A,bool,ord_less(A,one_one(A)),A2))
         => ( pp(aa(A,bool,ord_less(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),M)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),N)))
           => pp(aa(nat,bool,ord_less(nat,M),N)) ) ) ) ).

% power_less_imp_less_exp
tff(fact_857_diff__Suc__less,axiom,
    ! [N: nat,I: nat] :
      ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),N))
     => pp(aa(nat,bool,ord_less(nat,aa(nat,nat,minus_minus(nat,N),aa(nat,nat,suc,I))),N)) ) ).

% diff_Suc_less
tff(fact_858_nat__diff__split,axiom,
    ! [P: fun(nat,bool),A2: nat,B2: nat] :
      ( pp(aa(nat,bool,P,aa(nat,nat,minus_minus(nat,A2),B2)))
    <=> ( ( pp(aa(nat,bool,ord_less(nat,A2),B2))
         => pp(aa(nat,bool,P,zero_zero(nat))) )
        & ! [D3: nat] :
            ( ( A2 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),B2),D3) )
           => pp(aa(nat,bool,P,D3)) ) ) ) ).

% nat_diff_split
tff(fact_859_nat__diff__split__asm,axiom,
    ! [P: fun(nat,bool),A2: nat,B2: nat] :
      ( pp(aa(nat,bool,P,aa(nat,nat,minus_minus(nat,A2),B2)))
    <=> ~ ( ( pp(aa(nat,bool,ord_less(nat,A2),B2))
            & ~ pp(aa(nat,bool,P,zero_zero(nat))) )
          | ? [D3: nat] :
              ( ( A2 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),B2),D3) )
              & ~ pp(aa(nat,bool,P,D3)) ) ) ) ).

% nat_diff_split_asm
tff(fact_860_less__diff__conv2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( pp(aa(nat,bool,ord_less_eq(nat,K),J))
     => ( pp(aa(nat,bool,ord_less(nat,aa(nat,nat,minus_minus(nat,J),K)),I))
      <=> pp(aa(nat,bool,ord_less(nat,J),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),K))) ) ) ).

% less_diff_conv2
tff(fact_861_nat__induct__non__zero,axiom,
    ! [N: nat,P: fun(nat,bool)] :
      ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),N))
     => ( pp(aa(nat,bool,P,one_one(nat)))
       => ( ! [N2: nat] :
              ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),N2))
             => ( pp(aa(nat,bool,P,N2))
               => pp(aa(nat,bool,P,aa(nat,nat,suc,N2))) ) )
         => pp(aa(nat,bool,P,N)) ) ) ) ).

% nat_induct_non_zero
tff(fact_862_div__less__dividend,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa(nat,bool,ord_less(nat,one_one(nat)),N))
     => ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),M))
       => pp(aa(nat,bool,ord_less(nat,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),N)),M)) ) ) ).

% div_less_dividend
tff(fact_863_eq__iff__diff__eq__0,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A,B2: A] :
          ( ( A2 = B2 )
        <=> ( aa(A,A,minus_minus(A,A2),B2) = zero_zero(A) ) ) ) ).

% eq_iff_diff_eq_0
tff(fact_864_diff__mono,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A,D: A,C2: A] :
          ( pp(aa(A,bool,ord_less_eq(A,A2),B2))
         => ( pp(aa(A,bool,ord_less_eq(A,D),C2))
           => pp(aa(A,bool,ord_less_eq(A,aa(A,A,minus_minus(A,A2),C2)),aa(A,A,minus_minus(A,B2),D))) ) ) ) ).

% diff_mono
tff(fact_865_diff__left__mono,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [B2: A,A2: A,C2: A] :
          ( pp(aa(A,bool,ord_less_eq(A,B2),A2))
         => pp(aa(A,bool,ord_less_eq(A,aa(A,A,minus_minus(A,C2),A2)),aa(A,A,minus_minus(A,C2),B2))) ) ) ).

% diff_left_mono
tff(fact_866_diff__right__mono,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,ord_less_eq(A,A2),B2))
         => pp(aa(A,bool,ord_less_eq(A,aa(A,A,minus_minus(A,A2),C2)),aa(A,A,minus_minus(A,B2),C2))) ) ) ).

% diff_right_mono
tff(fact_867_diff__eq__diff__less__eq,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A,C2: A,D: A] :
          ( ( aa(A,A,minus_minus(A,A2),B2) = aa(A,A,minus_minus(A,C2),D) )
         => ( pp(aa(A,bool,ord_less_eq(A,A2),B2))
          <=> pp(aa(A,bool,ord_less_eq(A,C2),D)) ) ) ) ).

% diff_eq_diff_less_eq
tff(fact_868_diff__diff__eq,axiom,
    ! [A: $tType] :
      ( cancel2418104881723323429up_add(A)
     => ! [A2: A,B2: A,C2: A] : aa(A,A,minus_minus(A,aa(A,A,minus_minus(A,A2),B2)),C2) = aa(A,A,minus_minus(A,A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2)) ) ).

% diff_diff_eq
tff(fact_869_add__diff__add,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A2: A,C2: A,B2: A,D: A] : aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),D)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,minus_minus(A,A2),B2)),aa(A,A,minus_minus(A,C2),D)) ) ).

% add_diff_add
tff(fact_870_add__implies__diff,axiom,
    ! [A: $tType] :
      ( cancel1802427076303600483id_add(A)
     => ! [C2: A,B2: A,A2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),B2) = A2 )
         => ( C2 = aa(A,A,minus_minus(A,A2),B2) ) ) ) ).

% add_implies_diff
tff(fact_871_diff__add__eq__diff__diff__swap,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A,B2: A,C2: A] : aa(A,A,minus_minus(A,A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2)) = aa(A,A,minus_minus(A,aa(A,A,minus_minus(A,A2),C2)),B2) ) ).

% diff_add_eq_diff_diff_swap
tff(fact_872_diff__add__eq,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A2: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,minus_minus(A,A2),B2)),C2) = aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C2)),B2) ) ).

% diff_add_eq
tff(fact_873_diff__diff__eq2,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A,B2: A,C2: A] : aa(A,A,minus_minus(A,A2),aa(A,A,minus_minus(A,B2),C2)) = aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C2)),B2) ) ).

% diff_diff_eq2
tff(fact_874_add__diff__eq,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa(A,A,minus_minus(A,B2),C2)) = aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),C2) ) ).

% add_diff_eq
tff(fact_875_eq__diff__eq,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A,C2: A,B2: A] :
          ( ( A2 = aa(A,A,minus_minus(A,C2),B2) )
        <=> ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2) = C2 ) ) ) ).

% eq_diff_eq
tff(fact_876_diff__eq__eq,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A,B2: A,C2: A] :
          ( ( aa(A,A,minus_minus(A,A2),B2) = C2 )
        <=> ( A2 = aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),B2) ) ) ) ).

% diff_eq_eq
tff(fact_877_group__cancel_Osub1,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A3: A,K: A,A2: A,B2: A] :
          ( ( A3 = aa(A,A,aa(A,fun(A,A),plus_plus(A),K),A2) )
         => ( aa(A,A,minus_minus(A,A3),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),K),aa(A,A,minus_minus(A,A2),B2)) ) ) ) ).

% group_cancel.sub1
tff(fact_878_left__diff__distrib,axiom,
    ! [A: $tType] :
      ( ring(A)
     => ! [A2: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,minus_minus(A,A2),B2)),C2) = aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) ) ).

% left_diff_distrib
tff(fact_879_right__diff__distrib,axiom,
    ! [A: $tType] :
      ( ring(A)
     => ! [A2: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(A,A,minus_minus(A,B2),C2)) = aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)) ) ).

% right_diff_distrib
tff(fact_880_left__diff__distrib_H,axiom,
    ! [A: $tType] :
      ( comm_s4317794764714335236cancel(A)
     => ! [B2: A,C2: A,A2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,minus_minus(A,B2),C2)),A2) = aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),B2),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2)) ) ).

% left_diff_distrib'
tff(fact_881_right__diff__distrib_H,axiom,
    ! [A: $tType] :
      ( comm_s4317794764714335236cancel(A)
     => ! [A2: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(A,A,minus_minus(A,B2),C2)) = aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)) ) ).

% right_diff_distrib'
tff(fact_882_diff__divide__distrib,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,minus_minus(A,A2),B2)),C2) = aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),C2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)) ) ).

% diff_divide_distrib
tff(fact_883_less__numeral__extra_I3_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ~ pp(aa(A,bool,ord_less(A,zero_zero(A)),zero_zero(A))) ) ).

% less_numeral_extra(3)
tff(fact_884_gr__zeroI,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [N: A] :
          ( ( N != zero_zero(A) )
         => pp(aa(A,bool,ord_less(A,zero_zero(A)),N)) ) ) ).

% gr_zeroI
tff(fact_885_not__less__zero,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [N: A] : ~ pp(aa(A,bool,ord_less(A,N),zero_zero(A))) ) ).

% not_less_zero
tff(fact_886_gr__implies__not__zero,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [M: A,N: A] :
          ( pp(aa(A,bool,ord_less(A,M),N))
         => ( N != zero_zero(A) ) ) ) ).

% gr_implies_not_zero
tff(fact_887_zero__less__iff__neq__zero,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [N: A] :
          ( pp(aa(A,bool,ord_less(A,zero_zero(A)),N))
        <=> ( N != zero_zero(A) ) ) ) ).

% zero_less_iff_neq_zero
tff(fact_888_field__lbound__gt__zero,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [D1: A,D22: A] :
          ( pp(aa(A,bool,ord_less(A,zero_zero(A)),D1))
         => ( pp(aa(A,bool,ord_less(A,zero_zero(A)),D22))
           => ? [E2: A] :
                ( pp(aa(A,bool,ord_less(A,zero_zero(A)),E2))
                & pp(aa(A,bool,ord_less(A,E2),D1))
                & pp(aa(A,bool,ord_less(A,E2),D22)) ) ) ) ) ).

% field_lbound_gt_zero
tff(fact_889_add__less__imp__less__right,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [A2: A,C2: A,B2: A] :
          ( pp(aa(A,bool,ord_less(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2)))
         => pp(aa(A,bool,ord_less(A,A2),B2)) ) ) ).

% add_less_imp_less_right
tff(fact_890_add__less__imp__less__left,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [C2: A,A2: A,B2: A] :
          ( pp(aa(A,bool,ord_less(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),A2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),B2)))
         => pp(aa(A,bool,ord_less(A,A2),B2)) ) ) ).

% add_less_imp_less_left
tff(fact_891_add__strict__right__mono,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,ord_less(A,A2),B2))
         => pp(aa(A,bool,ord_less(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2))) ) ) ).

% add_strict_right_mono
tff(fact_892_add__strict__left__mono,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,ord_less(A,A2),B2))
         => pp(aa(A,bool,ord_less(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),A2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),B2))) ) ) ).

% add_strict_left_mono
tff(fact_893_add__strict__mono,axiom,
    ! [A: $tType] :
      ( strict9044650504122735259up_add(A)
     => ! [A2: A,B2: A,C2: A,D: A] :
          ( pp(aa(A,bool,ord_less(A,A2),B2))
         => ( pp(aa(A,bool,ord_less(A,C2),D))
           => pp(aa(A,bool,ord_less(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),D))) ) ) ) ).

% add_strict_mono
tff(fact_894_add__mono__thms__linordered__field_I1_J,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [I: A,J: A,K: A,L: A] :
          ( ( pp(aa(A,bool,ord_less(A,I),J))
            & ( K = L ) )
         => pp(aa(A,bool,ord_less(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),L))) ) ) ).

% add_mono_thms_linordered_field(1)
tff(fact_895_add__mono__thms__linordered__field_I2_J,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [I: A,J: A,K: A,L: A] :
          ( ( ( I = J )
            & pp(aa(A,bool,ord_less(A,K),L)) )
         => pp(aa(A,bool,ord_less(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),L))) ) ) ).

% add_mono_thms_linordered_field(2)
tff(fact_896_add__mono__thms__linordered__field_I5_J,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [I: A,J: A,K: A,L: A] :
          ( ( pp(aa(A,bool,ord_less(A,I),J))
            & pp(aa(A,bool,ord_less(A,K),L)) )
         => pp(aa(A,bool,ord_less(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),L))) ) ) ).

% add_mono_thms_linordered_field(5)
tff(fact_897_dvd__diff,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [X: A,Y: A,Z: A] :
          ( pp(aa(A,bool,dvd_dvd(A,X),Y))
         => ( pp(aa(A,bool,dvd_dvd(A,X),Z))
           => pp(aa(A,bool,dvd_dvd(A,X),aa(A,A,minus_minus(A,Y),Z))) ) ) ) ).

% dvd_diff
tff(fact_898_dvd__diff__commute,axiom,
    ! [A: $tType] :
      ( euclid5891614535332579305n_ring(A)
     => ! [A2: A,C2: A,B2: A] :
          ( pp(aa(A,bool,dvd_dvd(A,A2),aa(A,A,minus_minus(A,C2),B2)))
        <=> pp(aa(A,bool,dvd_dvd(A,A2),aa(A,A,minus_minus(A,B2),C2))) ) ) ).

% dvd_diff_commute
tff(fact_899_less__numeral__extra_I4_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ~ pp(aa(A,bool,ord_less(A,one_one(A)),one_one(A))) ) ).

% less_numeral_extra(4)
tff(fact_900_zero__induct__lemma,axiom,
    ! [P: fun(nat,bool),K: nat,I: nat] :
      ( pp(aa(nat,bool,P,K))
     => ( ! [N2: nat] :
            ( pp(aa(nat,bool,P,aa(nat,nat,suc,N2)))
           => pp(aa(nat,bool,P,N2)) )
       => pp(aa(nat,bool,P,aa(nat,nat,minus_minus(nat,K),I))) ) ) ).

% zero_induct_lemma
tff(fact_901_minus__nat_Odiff__0,axiom,
    ! [M: nat] : aa(nat,nat,minus_minus(nat,M),zero_zero(nat)) = M ).

% minus_nat.diff_0
tff(fact_902_diffs0__imp__equal,axiom,
    ! [M: nat,N: nat] :
      ( ( aa(nat,nat,minus_minus(nat,M),N) = zero_zero(nat) )
     => ( ( aa(nat,nat,minus_minus(nat,N),M) = zero_zero(nat) )
       => ( M = N ) ) ) ).

% diffs0_imp_equal
tff(fact_903_diff__add__inverse2,axiom,
    ! [M: nat,N: nat] : aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N)),N) = M ).

% diff_add_inverse2
tff(fact_904_diff__add__inverse,axiom,
    ! [N: nat,M: nat] : aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),M)),N) = M ).

% diff_add_inverse
tff(fact_905_diff__cancel2,axiom,
    ! [M: nat,K: nat,N: nat] : aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),K)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),K)) = aa(nat,nat,minus_minus(nat,M),N) ).

% diff_cancel2
tff(fact_906_Nat_Odiff__cancel,axiom,
    ! [K: nat,M: nat,N: nat] : aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),M)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),N)) = aa(nat,nat,minus_minus(nat,M),N) ).

% Nat.diff_cancel
tff(fact_907_eq__diff__iff,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa(nat,bool,ord_less_eq(nat,K),M))
     => ( pp(aa(nat,bool,ord_less_eq(nat,K),N))
       => ( ( aa(nat,nat,minus_minus(nat,M),K) = aa(nat,nat,minus_minus(nat,N),K) )
        <=> ( M = N ) ) ) ) ).

% eq_diff_iff
tff(fact_908_le__diff__iff,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa(nat,bool,ord_less_eq(nat,K),M))
     => ( pp(aa(nat,bool,ord_less_eq(nat,K),N))
       => ( pp(aa(nat,bool,ord_less_eq(nat,aa(nat,nat,minus_minus(nat,M),K)),aa(nat,nat,minus_minus(nat,N),K)))
        <=> pp(aa(nat,bool,ord_less_eq(nat,M),N)) ) ) ) ).

% le_diff_iff
tff(fact_909_Nat_Odiff__diff__eq,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa(nat,bool,ord_less_eq(nat,K),M))
     => ( pp(aa(nat,bool,ord_less_eq(nat,K),N))
       => ( aa(nat,nat,minus_minus(nat,aa(nat,nat,minus_minus(nat,M),K)),aa(nat,nat,minus_minus(nat,N),K)) = aa(nat,nat,minus_minus(nat,M),N) ) ) ) ).

% Nat.diff_diff_eq
tff(fact_910_diff__le__mono,axiom,
    ! [M: nat,N: nat,L: nat] :
      ( pp(aa(nat,bool,ord_less_eq(nat,M),N))
     => pp(aa(nat,bool,ord_less_eq(nat,aa(nat,nat,minus_minus(nat,M),L)),aa(nat,nat,minus_minus(nat,N),L))) ) ).

% diff_le_mono
tff(fact_911_diff__le__self,axiom,
    ! [M: nat,N: nat] : pp(aa(nat,bool,ord_less_eq(nat,aa(nat,nat,minus_minus(nat,M),N)),M)) ).

% diff_le_self
tff(fact_912_le__diff__iff_H,axiom,
    ! [A2: nat,C2: nat,B2: nat] :
      ( pp(aa(nat,bool,ord_less_eq(nat,A2),C2))
     => ( pp(aa(nat,bool,ord_less_eq(nat,B2),C2))
       => ( pp(aa(nat,bool,ord_less_eq(nat,aa(nat,nat,minus_minus(nat,C2),A2)),aa(nat,nat,minus_minus(nat,C2),B2)))
        <=> pp(aa(nat,bool,ord_less_eq(nat,B2),A2)) ) ) ) ).

% le_diff_iff'
tff(fact_913_diff__le__mono2,axiom,
    ! [M: nat,N: nat,L: nat] :
      ( pp(aa(nat,bool,ord_less_eq(nat,M),N))
     => pp(aa(nat,bool,ord_less_eq(nat,aa(nat,nat,minus_minus(nat,L),N)),aa(nat,nat,minus_minus(nat,L),M))) ) ).

% diff_le_mono2
tff(fact_914_Nat_OlessE,axiom,
    ! [I: nat,K: nat] :
      ( pp(aa(nat,bool,ord_less(nat,I),K))
     => ( ( K != aa(nat,nat,suc,I) )
       => ~ ! [J2: nat] :
              ( pp(aa(nat,bool,ord_less(nat,I),J2))
             => ( K != aa(nat,nat,suc,J2) ) ) ) ) ).

% Nat.lessE
tff(fact_915_Suc__lessD,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,ord_less(nat,aa(nat,nat,suc,M)),N))
     => pp(aa(nat,bool,ord_less(nat,M),N)) ) ).

% Suc_lessD
tff(fact_916_Suc__lessE,axiom,
    ! [I: nat,K: nat] :
      ( pp(aa(nat,bool,ord_less(nat,aa(nat,nat,suc,I)),K))
     => ~ ! [J2: nat] :
            ( pp(aa(nat,bool,ord_less(nat,I),J2))
           => ( K != aa(nat,nat,suc,J2) ) ) ) ).

% Suc_lessE
tff(fact_917_Suc__lessI,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,ord_less(nat,M),N))
     => ( ( aa(nat,nat,suc,M) != N )
       => pp(aa(nat,bool,ord_less(nat,aa(nat,nat,suc,M)),N)) ) ) ).

% Suc_lessI
tff(fact_918_less__SucE,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,ord_less(nat,M),aa(nat,nat,suc,N)))
     => ( ~ pp(aa(nat,bool,ord_less(nat,M),N))
       => ( M = N ) ) ) ).

% less_SucE
tff(fact_919_less__SucI,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,ord_less(nat,M),N))
     => pp(aa(nat,bool,ord_less(nat,M),aa(nat,nat,suc,N))) ) ).

% less_SucI
tff(fact_920_Ex__less__Suc,axiom,
    ! [N: nat,P: fun(nat,bool)] :
      ( ? [I3: nat] :
          ( pp(aa(nat,bool,ord_less(nat,I3),aa(nat,nat,suc,N)))
          & pp(aa(nat,bool,P,I3)) )
    <=> ( pp(aa(nat,bool,P,N))
        | ? [I3: nat] :
            ( pp(aa(nat,bool,ord_less(nat,I3),N))
            & pp(aa(nat,bool,P,I3)) ) ) ) ).

% Ex_less_Suc
tff(fact_921_less__Suc__eq,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,ord_less(nat,M),aa(nat,nat,suc,N)))
    <=> ( pp(aa(nat,bool,ord_less(nat,M),N))
        | ( M = N ) ) ) ).

% less_Suc_eq
tff(fact_922_not__less__eq,axiom,
    ! [M: nat,N: nat] :
      ( ~ pp(aa(nat,bool,ord_less(nat,M),N))
    <=> pp(aa(nat,bool,ord_less(nat,N),aa(nat,nat,suc,M))) ) ).

% not_less_eq
tff(fact_923_Nat_OAll__less__Suc,axiom,
    ! [N: nat,P: fun(nat,bool)] :
      ( ! [I3: nat] :
          ( pp(aa(nat,bool,ord_less(nat,I3),aa(nat,nat,suc,N)))
         => pp(aa(nat,bool,P,I3)) )
    <=> ( pp(aa(nat,bool,P,N))
        & ! [I3: nat] :
            ( pp(aa(nat,bool,ord_less(nat,I3),N))
           => pp(aa(nat,bool,P,I3)) ) ) ) ).

% Nat.All_less_Suc
tff(fact_924_Suc__less__eq2,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa(nat,bool,ord_less(nat,aa(nat,nat,suc,N)),M))
    <=> ? [M5: nat] :
          ( ( M = aa(nat,nat,suc,M5) )
          & pp(aa(nat,bool,ord_less(nat,N),M5)) ) ) ).

% Suc_less_eq2
tff(fact_925_less__antisym,axiom,
    ! [N: nat,M: nat] :
      ( ~ pp(aa(nat,bool,ord_less(nat,N),M))
     => ( pp(aa(nat,bool,ord_less(nat,N),aa(nat,nat,suc,M)))
       => ( M = N ) ) ) ).

% less_antisym
tff(fact_926_Suc__less__SucD,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,ord_less(nat,aa(nat,nat,suc,M)),aa(nat,nat,suc,N)))
     => pp(aa(nat,bool,ord_less(nat,M),N)) ) ).

% Suc_less_SucD
tff(fact_927_less__trans__Suc,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( pp(aa(nat,bool,ord_less(nat,I),J))
     => ( pp(aa(nat,bool,ord_less(nat,J),K))
       => pp(aa(nat,bool,ord_less(nat,aa(nat,nat,suc,I)),K)) ) ) ).

% less_trans_Suc
tff(fact_928_less__Suc__induct,axiom,
    ! [I: nat,J: nat,P: fun(nat,fun(nat,bool))] :
      ( pp(aa(nat,bool,ord_less(nat,I),J))
     => ( ! [I2: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),P,I2),aa(nat,nat,suc,I2)))
       => ( ! [I2: nat,J2: nat,K3: nat] :
              ( pp(aa(nat,bool,ord_less(nat,I2),J2))
             => ( pp(aa(nat,bool,ord_less(nat,J2),K3))
               => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),P,I2),J2))
                 => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),P,J2),K3))
                   => pp(aa(nat,bool,aa(nat,fun(nat,bool),P,I2),K3)) ) ) ) )
         => pp(aa(nat,bool,aa(nat,fun(nat,bool),P,I),J)) ) ) ) ).

% less_Suc_induct
tff(fact_929_strict__inc__induct,axiom,
    ! [I: nat,J: nat,P: fun(nat,bool)] :
      ( pp(aa(nat,bool,ord_less(nat,I),J))
     => ( ! [I2: nat] :
            ( ( J = aa(nat,nat,suc,I2) )
           => pp(aa(nat,bool,P,I2)) )
       => ( ! [I2: nat] :
              ( pp(aa(nat,bool,ord_less(nat,I2),J))
             => ( pp(aa(nat,bool,P,aa(nat,nat,suc,I2)))
               => pp(aa(nat,bool,P,I2)) ) )
         => pp(aa(nat,bool,P,I)) ) ) ) ).

% strict_inc_induct
tff(fact_930_not__less__less__Suc__eq,axiom,
    ! [N: nat,M: nat] :
      ( ~ pp(aa(nat,bool,ord_less(nat,N),M))
     => ( pp(aa(nat,bool,ord_less(nat,N),aa(nat,nat,suc,M)))
      <=> ( N = M ) ) ) ).

% not_less_less_Suc_eq
tff(fact_931_bot__nat__0_Oextremum__strict,axiom,
    ! [A2: nat] : ~ pp(aa(nat,bool,ord_less(nat,A2),zero_zero(nat))) ).

% bot_nat_0.extremum_strict
tff(fact_932_gr0I,axiom,
    ! [N: nat] :
      ( ( N != zero_zero(nat) )
     => pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),N)) ) ).

% gr0I
tff(fact_933_not__gr0,axiom,
    ! [N: nat] :
      ( ~ pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),N))
    <=> ( N = zero_zero(nat) ) ) ).

% not_gr0
tff(fact_934_not__less0,axiom,
    ! [N: nat] : ~ pp(aa(nat,bool,ord_less(nat,N),zero_zero(nat))) ).

% not_less0
tff(fact_935_less__zeroE,axiom,
    ! [N: nat] : ~ pp(aa(nat,bool,ord_less(nat,N),zero_zero(nat))) ).

% less_zeroE
tff(fact_936_gr__implies__not0,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,ord_less(nat,M),N))
     => ( N != zero_zero(nat) ) ) ).

% gr_implies_not0
tff(fact_937_infinite__descent0,axiom,
    ! [P: fun(nat,bool),N: nat] :
      ( pp(aa(nat,bool,P,zero_zero(nat)))
     => ( ! [N2: nat] :
            ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),N2))
           => ( ~ pp(aa(nat,bool,P,N2))
             => ? [M4: nat] :
                  ( pp(aa(nat,bool,ord_less(nat,M4),N2))
                  & ~ pp(aa(nat,bool,P,M4)) ) ) )
       => pp(aa(nat,bool,P,N)) ) ) ).

% infinite_descent0
tff(fact_938_infinite__descent0__measure,axiom,
    ! [A: $tType,V2: fun(A,nat),P: fun(A,bool),X: A] :
      ( ! [X3: A] :
          ( ( aa(A,nat,V2,X3) = zero_zero(nat) )
         => pp(aa(A,bool,P,X3)) )
     => ( ! [X3: A] :
            ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),aa(A,nat,V2,X3)))
           => ( ~ pp(aa(A,bool,P,X3))
             => ? [Y4: A] :
                  ( pp(aa(nat,bool,ord_less(nat,aa(A,nat,V2,Y4)),aa(A,nat,V2,X3)))
                  & ~ pp(aa(A,bool,P,Y4)) ) ) )
       => pp(aa(A,bool,P,X)) ) ) ).

% infinite_descent0_measure
tff(fact_939_diff__mult__distrib,axiom,
    ! [M: nat,N: nat,K: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,minus_minus(nat,M),N)),K) = aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),K)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),K)) ).

% diff_mult_distrib
tff(fact_940_diff__mult__distrib2,axiom,
    ! [K: nat,M: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),aa(nat,nat,minus_minus(nat,M),N)) = aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),M)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),N)) ).

% diff_mult_distrib2
tff(fact_941_less__add__eq__less,axiom,
    ! [K: nat,L: nat,M: nat,N: nat] :
      ( pp(aa(nat,bool,ord_less(nat,K),L))
     => ( ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),L) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),N) )
       => pp(aa(nat,bool,ord_less(nat,M),N)) ) ) ).

% less_add_eq_less
tff(fact_942_trans__less__add2,axiom,
    ! [I: nat,J: nat,M: nat] :
      ( pp(aa(nat,bool,ord_less(nat,I),J))
     => pp(aa(nat,bool,ord_less(nat,I),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),J))) ) ).

% trans_less_add2
tff(fact_943_trans__less__add1,axiom,
    ! [I: nat,J: nat,M: nat] :
      ( pp(aa(nat,bool,ord_less(nat,I),J))
     => pp(aa(nat,bool,ord_less(nat,I),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),M))) ) ).

% trans_less_add1
tff(fact_944_add__less__mono1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( pp(aa(nat,bool,ord_less(nat,I),J))
     => pp(aa(nat,bool,ord_less(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),K)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),K))) ) ).

% add_less_mono1
tff(fact_945_not__add__less2,axiom,
    ! [J: nat,I: nat] : ~ pp(aa(nat,bool,ord_less(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),I)),I)) ).

% not_add_less2
tff(fact_946_not__add__less1,axiom,
    ! [I: nat,J: nat] : ~ pp(aa(nat,bool,ord_less(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),J)),I)) ).

% not_add_less1
tff(fact_947_add__less__mono,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( pp(aa(nat,bool,ord_less(nat,I),J))
     => ( pp(aa(nat,bool,ord_less(nat,K),L))
       => pp(aa(nat,bool,ord_less(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),K)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),L))) ) ) ).

% add_less_mono
tff(fact_948_add__lessD1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( pp(aa(nat,bool,ord_less(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),J)),K))
     => pp(aa(nat,bool,ord_less(nat,I),K)) ) ).

% add_lessD1
tff(fact_949_nat__less__le,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,ord_less(nat,M),N))
    <=> ( pp(aa(nat,bool,ord_less_eq(nat,M),N))
        & ( M != N ) ) ) ).

% nat_less_le
tff(fact_950_less__imp__le__nat,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,ord_less(nat,M),N))
     => pp(aa(nat,bool,ord_less_eq(nat,M),N)) ) ).

% less_imp_le_nat
tff(fact_951_le__eq__less__or__eq,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,ord_less_eq(nat,M),N))
    <=> ( pp(aa(nat,bool,ord_less(nat,M),N))
        | ( M = N ) ) ) ).

% le_eq_less_or_eq
tff(fact_952_less__or__eq__imp__le,axiom,
    ! [M: nat,N: nat] :
      ( ( pp(aa(nat,bool,ord_less(nat,M),N))
        | ( M = N ) )
     => pp(aa(nat,bool,ord_less_eq(nat,M),N)) ) ).

% less_or_eq_imp_le
tff(fact_953_le__neq__implies__less,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,ord_less_eq(nat,M),N))
     => ( ( M != N )
       => pp(aa(nat,bool,ord_less(nat,M),N)) ) ) ).

% le_neq_implies_less
tff(fact_954_less__mono__imp__le__mono,axiom,
    ! [F2: fun(nat,nat),I: nat,J: nat] :
      ( ! [I2: nat,J2: nat] :
          ( pp(aa(nat,bool,ord_less(nat,I2),J2))
         => pp(aa(nat,bool,ord_less(nat,aa(nat,nat,F2,I2)),aa(nat,nat,F2,J2))) )
     => ( pp(aa(nat,bool,ord_less_eq(nat,I),J))
       => pp(aa(nat,bool,ord_less_eq(nat,aa(nat,nat,F2,I)),aa(nat,nat,F2,J))) ) ) ).

% less_mono_imp_le_mono
tff(fact_955_dvd__diff__nat,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa(nat,bool,dvd_dvd(nat,K),M))
     => ( pp(aa(nat,bool,dvd_dvd(nat,K),N))
       => pp(aa(nat,bool,dvd_dvd(nat,K),aa(nat,nat,minus_minus(nat,M),N))) ) ) ).

% dvd_diff_nat
tff(fact_956_nat__mult__1,axiom,
    ! [N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),one_one(nat)),N) = N ).

% nat_mult_1
tff(fact_957_nat__mult__1__right,axiom,
    ! [N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),one_one(nat)) = N ).

% nat_mult_1_right
tff(fact_958_frac__less__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Y: A,Z: A,X: A,W: A] :
          ( ( Y != zero_zero(A) )
         => ( ( Z != zero_zero(A) )
           => ( pp(aa(A,bool,ord_less(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y)),aa(A,A,aa(A,fun(A,A),divide_divide(A),W),Z)))
            <=> pp(aa(A,bool,ord_less(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),X),Z)),aa(A,A,aa(A,fun(A,A),times_times(A),W),Y))),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Z))),zero_zero(A))) ) ) ) ) ).

% frac_less_eq
tff(fact_959_power__strict__decreasing,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [N: nat,N4: nat,A2: A] :
          ( pp(aa(nat,bool,ord_less(nat,N),N4))
         => ( pp(aa(A,bool,ord_less(A,zero_zero(A)),A2))
           => ( pp(aa(A,bool,ord_less(A,A2),one_one(A)))
             => pp(aa(A,bool,ord_less(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),N4)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),N))) ) ) ) ) ).

% power_strict_decreasing
tff(fact_960_one__less__power,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,N: nat] :
          ( pp(aa(A,bool,ord_less(A,one_one(A)),A2))
         => ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),N))
           => pp(aa(A,bool,ord_less(A,one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),N))) ) ) ) ).

% one_less_power
tff(fact_961_div__geq,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),N))
     => ( ~ pp(aa(nat,bool,ord_less(nat,M),N))
       => ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),N) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,minus_minus(nat,M),N)),N)) ) ) ) ).

% div_geq
tff(fact_962_div__if,axiom,
    ! [M: nat,N: nat] :
      ( ( ( pp(aa(nat,bool,ord_less(nat,M),N))
          | ( N = zero_zero(nat) ) )
       => ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),N) = zero_zero(nat) ) )
      & ( ~ ( pp(aa(nat,bool,ord_less(nat,M),N))
            | ( N = zero_zero(nat) ) )
       => ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),N) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,minus_minus(nat,M),N)),N)) ) ) ) ).

% div_if
tff(fact_963_add__eq__if,axiom,
    ! [M: nat,N: nat] :
      ( ( ( M = zero_zero(nat) )
       => ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N) = N ) )
      & ( ( M != zero_zero(nat) )
       => ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,minus_minus(nat,M),one_one(nat))),N)) ) ) ) ).

% add_eq_if
tff(fact_964_nat__less__add__iff1,axiom,
    ! [J: nat,I: nat,U: nat,M: nat,N: nat] :
      ( pp(aa(nat,bool,ord_less_eq(nat,J),I))
     => ( pp(aa(nat,bool,ord_less(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I),U)),M)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),U)),N)))
      <=> pp(aa(nat,bool,ord_less(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,minus_minus(nat,I),J)),U)),M)),N)) ) ) ).

% nat_less_add_iff1
tff(fact_965_nat__less__add__iff2,axiom,
    ! [I: nat,J: nat,U: nat,M: nat,N: nat] :
      ( pp(aa(nat,bool,ord_less_eq(nat,I),J))
     => ( pp(aa(nat,bool,ord_less(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I),U)),M)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),U)),N)))
      <=> pp(aa(nat,bool,ord_less(nat,M),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,minus_minus(nat,J),I)),U)),N))) ) ) ).

% nat_less_add_iff2
tff(fact_966_mult__eq__if,axiom,
    ! [M: nat,N: nat] :
      ( ( ( M = zero_zero(nat) )
       => ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N) = zero_zero(nat) ) )
      & ( ( M != zero_zero(nat) )
       => ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,minus_minus(nat,M),one_one(nat))),N)) ) ) ) ).

% mult_eq_if
tff(fact_967_i0__lb,axiom,
    ! [N: extended_enat] : pp(aa(extended_enat,bool,ord_less_eq(extended_enat,zero_zero(extended_enat)),N)) ).

% i0_lb
tff(fact_968_ile0__eq,axiom,
    ! [N: extended_enat] :
      ( pp(aa(extended_enat,bool,ord_less_eq(extended_enat,N),zero_zero(extended_enat)))
    <=> ( N = zero_zero(extended_enat) ) ) ).

% ile0_eq
tff(fact_969_dvd__mult__cancel1,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),M))
     => ( pp(aa(nat,bool,dvd_dvd(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N)),M))
      <=> ( N = one_one(nat) ) ) ) ).

% dvd_mult_cancel1
tff(fact_970_dvd__mult__cancel2,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),M))
     => ( pp(aa(nat,bool,dvd_dvd(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),M)),M))
      <=> ( N = one_one(nat) ) ) ) ).

% dvd_mult_cancel2
tff(fact_971_power__dvd__imp__le,axiom,
    ! [I: nat,M: nat,N: nat] :
      ( pp(aa(nat,bool,dvd_dvd(nat,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),I),M)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),I),N)))
     => ( pp(aa(nat,bool,ord_less(nat,one_one(nat)),I))
       => pp(aa(nat,bool,ord_less_eq(nat,M),N)) ) ) ).

% power_dvd_imp_le
tff(fact_972_power__strict__mono,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,B2: A,N: nat] :
          ( pp(aa(A,bool,ord_less(A,A2),B2))
         => ( pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),A2))
           => ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),N))
             => pp(aa(A,bool,ord_less(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),N)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),N))) ) ) ) ) ).

% power_strict_mono
tff(fact_973_le__div__geq,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),N))
     => ( pp(aa(nat,bool,ord_less_eq(nat,N),M))
       => ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),N) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,minus_minus(nat,M),N)),N)) ) ) ) ).

% le_div_geq
tff(fact_974_le__iff__diff__le__0,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,ord_less_eq(A,A2),B2))
        <=> pp(aa(A,bool,ord_less_eq(A,aa(A,A,minus_minus(A,A2),B2)),zero_zero(A))) ) ) ).

% le_iff_diff_le_0
tff(fact_975_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,ord_less_eq(A,A2),B2))
         => ( pp(aa(A,bool,ord_less_eq(A,A2),B2))
           => ( ( aa(A,A,minus_minus(A,B2),A2) = C2 )
            <=> ( B2 = aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),A2) ) ) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
tff(fact_976_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,ord_less_eq(A,A2),B2))
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa(A,A,minus_minus(A,B2),A2)) = B2 ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_inverse
tff(fact_977_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,ord_less_eq(A,A2),B2))
         => ( aa(A,A,minus_minus(A,C2),aa(A,A,minus_minus(A,B2),A2)) = aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),A2)),B2) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_diff_right
tff(fact_978_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,ord_less_eq(A,A2),B2))
         => ( aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2)),A2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,minus_minus(A,B2),A2)),C2) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
tff(fact_979_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,ord_less_eq(A,A2),B2))
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,minus_minus(A,B2),A2)),C2) = aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2)),A2) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
tff(fact_980_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,ord_less_eq(A,A2),B2))
         => ( aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),B2)),A2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),aa(A,A,minus_minus(A,B2),A2)) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
tff(fact_981_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,ord_less_eq(A,A2),B2))
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),aa(A,A,minus_minus(A,B2),A2)) = aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),B2)),A2) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
tff(fact_982_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,ord_less_eq(A,A2),B2))
         => ( pp(aa(A,bool,ord_less_eq(A,C2),aa(A,A,minus_minus(A,B2),A2)))
          <=> pp(aa(A,bool,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),A2)),B2)) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
tff(fact_983_le__add__diff,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,ord_less_eq(A,A2),B2))
         => pp(aa(A,bool,ord_less_eq(A,C2),aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2)),A2))) ) ) ).

% le_add_diff
tff(fact_984_ordered__cancel__comm__monoid__diff__class_Odiff__add,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,ord_less_eq(A,A2),B2))
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,minus_minus(A,B2),A2)),A2) = B2 ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add
tff(fact_985_le__diff__eq,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,C2: A,B2: A] :
          ( pp(aa(A,bool,ord_less_eq(A,A2),aa(A,A,minus_minus(A,C2),B2)))
        <=> pp(aa(A,bool,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),C2)) ) ) ).

% le_diff_eq
tff(fact_986_diff__le__eq,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,ord_less_eq(A,aa(A,A,minus_minus(A,A2),B2)),C2))
        <=> pp(aa(A,bool,ord_less_eq(A,A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),B2))) ) ) ).

% diff_le_eq
tff(fact_987_add__le__add__imp__diff__le,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [I: A,K: A,N: A,J: A] :
          ( pp(aa(A,bool,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K)),N))
         => ( pp(aa(A,bool,ord_less_eq(A,N),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),K)))
           => ( pp(aa(A,bool,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K)),N))
             => ( pp(aa(A,bool,ord_less_eq(A,N),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),K)))
               => pp(aa(A,bool,ord_less_eq(A,aa(A,A,minus_minus(A,N),K)),J)) ) ) ) ) ) ).

% add_le_add_imp_diff_le
tff(fact_988_add__le__imp__le__diff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [I: A,K: A,N: A] :
          ( pp(aa(A,bool,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K)),N))
         => pp(aa(A,bool,ord_less_eq(A,I),aa(A,A,minus_minus(A,N),K))) ) ) ).

% add_le_imp_le_diff
tff(fact_989_mult__diff__mult,axiom,
    ! [A: $tType] :
      ( ring(A)
     => ! [X: A,Y: A,A2: A,B2: A] : aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),X),Y)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),aa(A,A,minus_minus(A,Y),B2))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,minus_minus(A,X),A2)),B2)) ) ).

% mult_diff_mult
tff(fact_990_square__diff__square__factored,axiom,
    ! [A: $tType] :
      ( comm_ring(A)
     => ! [X: A,Y: A] : aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),X),X)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Y)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y)),aa(A,A,minus_minus(A,X),Y)) ) ).

% square_diff_square_factored
tff(fact_991_eq__add__iff2,axiom,
    ! [A: $tType] :
      ( ring(A)
     => ! [A2: A,E: A,C2: A,B2: A,D: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),E)),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),E)),D) )
        <=> ( C2 = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,minus_minus(A,B2),A2)),E)),D) ) ) ) ).

% eq_add_iff2
tff(fact_992_eq__add__iff1,axiom,
    ! [A: $tType] :
      ( ring(A)
     => ! [A2: A,E: A,C2: A,B2: A,D: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),E)),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),E)),D) )
        <=> ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,minus_minus(A,A2),B2)),E)),C2) = D ) ) ) ).

% eq_add_iff1
tff(fact_993_not__numeral__less__zero,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [N: num] : ~ pp(aa(A,bool,ord_less(A,aa(num,A,numeral_numeral(A),N)),zero_zero(A))) ) ).

% not_numeral_less_zero
tff(fact_994_zero__less__numeral,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [N: num] : pp(aa(A,bool,ord_less(A,zero_zero(A)),aa(num,A,numeral_numeral(A),N))) ) ).

% zero_less_numeral
tff(fact_995_add__neg__neg,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,ord_less(A,A2),zero_zero(A)))
         => ( pp(aa(A,bool,ord_less(A,B2),zero_zero(A)))
           => pp(aa(A,bool,ord_less(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),zero_zero(A))) ) ) ) ).

% add_neg_neg
tff(fact_996_add__pos__pos,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,ord_less(A,zero_zero(A)),A2))
         => ( pp(aa(A,bool,ord_less(A,zero_zero(A)),B2))
           => pp(aa(A,bool,ord_less(A,zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2))) ) ) ) ).

% add_pos_pos
tff(fact_997_canonically__ordered__monoid__add__class_OlessE,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,ord_less(A,A2),B2))
         => ~ ! [C3: A] :
                ( ( B2 = aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C3) )
               => ( C3 = zero_zero(A) ) ) ) ) ).

% canonically_ordered_monoid_add_class.lessE
tff(fact_998_pos__add__strict,axiom,
    ! [A: $tType] :
      ( strict7427464778891057005id_add(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,ord_less(A,zero_zero(A)),A2))
         => ( pp(aa(A,bool,ord_less(A,B2),C2))
           => pp(aa(A,bool,ord_less(A,B2),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C2))) ) ) ) ).

% pos_add_strict
tff(fact_999_add__less__zeroD,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,ord_less(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y)),zero_zero(A)))
         => ( pp(aa(A,bool,ord_less(A,X),zero_zero(A)))
            | pp(aa(A,bool,ord_less(A,Y),zero_zero(A))) ) ) ) ).

% add_less_zeroD
tff(fact_1000_less__numeral__extra_I1_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => pp(aa(A,bool,ord_less(A,zero_zero(A)),one_one(A))) ) ).

% less_numeral_extra(1)
tff(fact_1001_zero__less__one,axiom,
    ! [A: $tType] :
      ( zero_less_one(A)
     => pp(aa(A,bool,ord_less(A,zero_zero(A)),one_one(A))) ) ).

% zero_less_one
tff(fact_1002_not__one__less__zero,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ~ pp(aa(A,bool,ord_less(A,one_one(A)),zero_zero(A))) ) ).

% not_one_less_zero
tff(fact_1003_add__less__le__mono,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [A2: A,B2: A,C2: A,D: A] :
          ( pp(aa(A,bool,ord_less(A,A2),B2))
         => ( pp(aa(A,bool,ord_less_eq(A,C2),D))
           => pp(aa(A,bool,ord_less(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),D))) ) ) ) ).

% add_less_le_mono
tff(fact_1004_add__le__less__mono,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [A2: A,B2: A,C2: A,D: A] :
          ( pp(aa(A,bool,ord_less_eq(A,A2),B2))
         => ( pp(aa(A,bool,ord_less(A,C2),D))
           => pp(aa(A,bool,ord_less(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),D))) ) ) ) ).

% add_le_less_mono
tff(fact_1005_add__mono__thms__linordered__field_I3_J,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [I: A,J: A,K: A,L: A] :
          ( ( pp(aa(A,bool,ord_less(A,I),J))
            & pp(aa(A,bool,ord_less_eq(A,K),L)) )
         => pp(aa(A,bool,ord_less(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),L))) ) ) ).

% add_mono_thms_linordered_field(3)
tff(fact_1006_add__mono__thms__linordered__field_I4_J,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [I: A,J: A,K: A,L: A] :
          ( ( pp(aa(A,bool,ord_less_eq(A,I),J))
            & pp(aa(A,bool,ord_less(A,K),L)) )
         => pp(aa(A,bool,ord_less(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),L))) ) ) ).

% add_mono_thms_linordered_field(4)
tff(fact_1007_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: $tType] :
      ( linord2810124833399127020strict(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,ord_less(A,A2),B2))
         => ( pp(aa(A,bool,ord_less(A,zero_zero(A)),C2))
           => pp(aa(A,bool,ord_less(A,aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2))) ) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
tff(fact_1008_mult__less__cancel__right__disj,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [A2: A,C2: A,B2: A] :
          ( pp(aa(A,bool,ord_less(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)))
        <=> ( ( pp(aa(A,bool,ord_less(A,zero_zero(A)),C2))
              & pp(aa(A,bool,ord_less(A,A2),B2)) )
            | ( pp(aa(A,bool,ord_less(A,C2),zero_zero(A)))
              & pp(aa(A,bool,ord_less(A,B2),A2)) ) ) ) ) ).

% mult_less_cancel_right_disj
tff(fact_1009_mult__strict__right__mono,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,ord_less(A,A2),B2))
         => ( pp(aa(A,bool,ord_less(A,zero_zero(A)),C2))
           => pp(aa(A,bool,ord_less(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2))) ) ) ) ).

% mult_strict_right_mono
tff(fact_1010_mult__strict__right__mono__neg,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [B2: A,A2: A,C2: A] :
          ( pp(aa(A,bool,ord_less(A,B2),A2))
         => ( pp(aa(A,bool,ord_less(A,C2),zero_zero(A)))
           => pp(aa(A,bool,ord_less(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2))) ) ) ) ).

% mult_strict_right_mono_neg
tff(fact_1011_mult__less__cancel__left__disj,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [C2: A,A2: A,B2: A] :
          ( pp(aa(A,bool,ord_less(A,aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)))
        <=> ( ( pp(aa(A,bool,ord_less(A,zero_zero(A)),C2))
              & pp(aa(A,bool,ord_less(A,A2),B2)) )
            | ( pp(aa(A,bool,ord_less(A,C2),zero_zero(A)))
              & pp(aa(A,bool,ord_less(A,B2),A2)) ) ) ) ) ).

% mult_less_cancel_left_disj
tff(fact_1012_mult__strict__left__mono,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,ord_less(A,A2),B2))
         => ( pp(aa(A,bool,ord_less(A,zero_zero(A)),C2))
           => pp(aa(A,bool,ord_less(A,aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2))) ) ) ) ).

% mult_strict_left_mono
tff(fact_1013_mult__strict__left__mono__neg,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [B2: A,A2: A,C2: A] :
          ( pp(aa(A,bool,ord_less(A,B2),A2))
         => ( pp(aa(A,bool,ord_less(A,C2),zero_zero(A)))
           => pp(aa(A,bool,ord_less(A,aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2))) ) ) ) ).

% mult_strict_left_mono_neg
tff(fact_1014_mult__less__cancel__left__pos,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [C2: A,A2: A,B2: A] :
          ( pp(aa(A,bool,ord_less(A,zero_zero(A)),C2))
         => ( pp(aa(A,bool,ord_less(A,aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)))
          <=> pp(aa(A,bool,ord_less(A,A2),B2)) ) ) ) ).

% mult_less_cancel_left_pos
tff(fact_1015_mult__less__cancel__left__neg,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [C2: A,A2: A,B2: A] :
          ( pp(aa(A,bool,ord_less(A,C2),zero_zero(A)))
         => ( pp(aa(A,bool,ord_less(A,aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)))
          <=> pp(aa(A,bool,ord_less(A,B2),A2)) ) ) ) ).

% mult_less_cancel_left_neg
tff(fact_1016_zero__less__mult__pos2,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [B2: A,A2: A] :
          ( pp(aa(A,bool,ord_less(A,zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),A2)))
         => ( pp(aa(A,bool,ord_less(A,zero_zero(A)),A2))
           => pp(aa(A,bool,ord_less(A,zero_zero(A)),B2)) ) ) ) ).

% zero_less_mult_pos2
tff(fact_1017_zero__less__mult__pos,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,ord_less(A,zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)))
         => ( pp(aa(A,bool,ord_less(A,zero_zero(A)),A2))
           => pp(aa(A,bool,ord_less(A,zero_zero(A)),B2)) ) ) ) ).

% zero_less_mult_pos
tff(fact_1018_zero__less__mult__iff,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,ord_less(A,zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)))
        <=> ( ( pp(aa(A,bool,ord_less(A,zero_zero(A)),A2))
              & pp(aa(A,bool,ord_less(A,zero_zero(A)),B2)) )
            | ( pp(aa(A,bool,ord_less(A,A2),zero_zero(A)))
              & pp(aa(A,bool,ord_less(A,B2),zero_zero(A))) ) ) ) ) ).

% zero_less_mult_iff
tff(fact_1019_mult__pos__neg2,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,ord_less(A,zero_zero(A)),A2))
         => ( pp(aa(A,bool,ord_less(A,B2),zero_zero(A)))
           => pp(aa(A,bool,ord_less(A,aa(A,A,aa(A,fun(A,A),times_times(A),B2),A2)),zero_zero(A))) ) ) ) ).

% mult_pos_neg2
tff(fact_1020_mult__pos__pos,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,ord_less(A,zero_zero(A)),A2))
         => ( pp(aa(A,bool,ord_less(A,zero_zero(A)),B2))
           => pp(aa(A,bool,ord_less(A,zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2))) ) ) ) ).

% mult_pos_pos
tff(fact_1021_mult__pos__neg,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,ord_less(A,zero_zero(A)),A2))
         => ( pp(aa(A,bool,ord_less(A,B2),zero_zero(A)))
           => pp(aa(A,bool,ord_less(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),zero_zero(A))) ) ) ) ).

% mult_pos_neg
tff(fact_1022_mult__neg__pos,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,ord_less(A,A2),zero_zero(A)))
         => ( pp(aa(A,bool,ord_less(A,zero_zero(A)),B2))
           => pp(aa(A,bool,ord_less(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),zero_zero(A))) ) ) ) ).

% mult_neg_pos
tff(fact_1023_mult__less__0__iff,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,ord_less(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),zero_zero(A)))
        <=> ( ( pp(aa(A,bool,ord_less(A,zero_zero(A)),A2))
              & pp(aa(A,bool,ord_less(A,B2),zero_zero(A))) )
            | ( pp(aa(A,bool,ord_less(A,A2),zero_zero(A)))
              & pp(aa(A,bool,ord_less(A,zero_zero(A)),B2)) ) ) ) ) ).

% mult_less_0_iff
tff(fact_1024_not__square__less__zero,axiom,
    ! [A: $tType] :
      ( linordered_ring(A)
     => ! [A2: A] : ~ pp(aa(A,bool,ord_less(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),A2)),zero_zero(A))) ) ).

% not_square_less_zero
tff(fact_1025_mult__neg__neg,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,ord_less(A,A2),zero_zero(A)))
         => ( pp(aa(A,bool,ord_less(A,B2),zero_zero(A)))
           => pp(aa(A,bool,ord_less(A,zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2))) ) ) ) ).

% mult_neg_neg
tff(fact_1026_not__numeral__less__one,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [N: num] : ~ pp(aa(A,bool,ord_less(A,aa(num,A,numeral_numeral(A),N)),one_one(A))) ) ).

% not_numeral_less_one
tff(fact_1027_divide__neg__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,ord_less(A,X),zero_zero(A)))
         => ( pp(aa(A,bool,ord_less(A,Y),zero_zero(A)))
           => pp(aa(A,bool,ord_less(A,zero_zero(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y))) ) ) ) ).

% divide_neg_neg
tff(fact_1028_divide__neg__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,ord_less(A,X),zero_zero(A)))
         => ( pp(aa(A,bool,ord_less(A,zero_zero(A)),Y))
           => pp(aa(A,bool,ord_less(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y)),zero_zero(A))) ) ) ) ).

% divide_neg_pos
tff(fact_1029_divide__pos__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,ord_less(A,zero_zero(A)),X))
         => ( pp(aa(A,bool,ord_less(A,Y),zero_zero(A)))
           => pp(aa(A,bool,ord_less(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y)),zero_zero(A))) ) ) ) ).

% divide_pos_neg
tff(fact_1030_divide__pos__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,ord_less(A,zero_zero(A)),X))
         => ( pp(aa(A,bool,ord_less(A,zero_zero(A)),Y))
           => pp(aa(A,bool,ord_less(A,zero_zero(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y))) ) ) ) ).

% divide_pos_pos
tff(fact_1031_divide__less__0__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,ord_less(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)),zero_zero(A)))
        <=> ( ( pp(aa(A,bool,ord_less(A,zero_zero(A)),A2))
              & pp(aa(A,bool,ord_less(A,B2),zero_zero(A))) )
            | ( pp(aa(A,bool,ord_less(A,A2),zero_zero(A)))
              & pp(aa(A,bool,ord_less(A,zero_zero(A)),B2)) ) ) ) ) ).

% divide_less_0_iff
tff(fact_1032_divide__less__cancel,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,C2: A,B2: A] :
          ( pp(aa(A,bool,ord_less(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),C2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)))
        <=> ( ( pp(aa(A,bool,ord_less(A,zero_zero(A)),C2))
             => pp(aa(A,bool,ord_less(A,A2),B2)) )
            & ( pp(aa(A,bool,ord_less(A,C2),zero_zero(A)))
             => pp(aa(A,bool,ord_less(A,B2),A2)) )
            & ( C2 != zero_zero(A) ) ) ) ) ).

% divide_less_cancel
tff(fact_1033_zero__less__divide__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,ord_less(A,zero_zero(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)))
        <=> ( ( pp(aa(A,bool,ord_less(A,zero_zero(A)),A2))
              & pp(aa(A,bool,ord_less(A,zero_zero(A)),B2)) )
            | ( pp(aa(A,bool,ord_less(A,A2),zero_zero(A)))
              & pp(aa(A,bool,ord_less(A,B2),zero_zero(A))) ) ) ) ) ).

% zero_less_divide_iff
tff(fact_1034_divide__strict__right__mono,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,ord_less(A,A2),B2))
         => ( pp(aa(A,bool,ord_less(A,zero_zero(A)),C2))
           => pp(aa(A,bool,ord_less(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),C2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2))) ) ) ) ).

% divide_strict_right_mono
tff(fact_1035_divide__strict__right__mono__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,A2: A,C2: A] :
          ( pp(aa(A,bool,ord_less(A,B2),A2))
         => ( pp(aa(A,bool,ord_less(A,C2),zero_zero(A)))
           => pp(aa(A,bool,ord_less(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),C2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2))) ) ) ) ).

% divide_strict_right_mono_neg
tff(fact_1036_add__mono1,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,ord_less(A,A2),B2))
         => pp(aa(A,bool,ord_less(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),one_one(A))),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),one_one(A)))) ) ) ).

% add_mono1
tff(fact_1037_less__add__one,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A] : pp(aa(A,bool,ord_less(A,A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),one_one(A)))) ) ).

% less_add_one
tff(fact_1038_less__1__mult,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [M: A,N: A] :
          ( pp(aa(A,bool,ord_less(A,one_one(A)),M))
         => ( pp(aa(A,bool,ord_less(A,one_one(A)),N))
           => pp(aa(A,bool,ord_less(A,one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),M),N))) ) ) ) ).

% less_1_mult
tff(fact_1039_zero__less__power,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,N: nat] :
          ( pp(aa(A,bool,ord_less(A,zero_zero(A)),A2))
         => pp(aa(A,bool,ord_less(A,zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),N))) ) ) ).

% zero_less_power
tff(fact_1040_real__archimedian__rdiv__eq__0,axiom,
    ! [X: real,C2: real] :
      ( pp(aa(real,bool,ord_less_eq(real,zero_zero(real)),X))
     => ( pp(aa(real,bool,ord_less_eq(real,zero_zero(real)),C2))
       => ( ! [M3: nat] :
              ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),M3))
             => pp(aa(real,bool,ord_less_eq(real,aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),M3)),X)),C2)) )
         => ( X = zero_zero(real) ) ) ) ) ).

% real_archimedian_rdiv_eq_0
tff(fact_1041_of__nat__less__0__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [M: nat] : ~ pp(aa(A,bool,ord_less(A,aa(nat,A,semiring_1_of_nat(A),M)),zero_zero(A))) ) ).

% of_nat_less_0_iff
tff(fact_1042_Suc__diff__le,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa(nat,bool,ord_less_eq(nat,N),M))
     => ( aa(nat,nat,minus_minus(nat,aa(nat,nat,suc,M)),N) = aa(nat,nat,suc,aa(nat,nat,minus_minus(nat,M),N)) ) ) ).

% Suc_diff_le
tff(fact_1043_diff__add__0,axiom,
    ! [N: nat,M: nat] : aa(nat,nat,minus_minus(nat,N),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),M)) = zero_zero(nat) ).

% diff_add_0
tff(fact_1044_Nat_Ole__imp__diff__is__add,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( pp(aa(nat,bool,ord_less_eq(nat,I),J))
     => ( ( aa(nat,nat,minus_minus(nat,J),I) = K )
      <=> ( J = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),I) ) ) ) ).

% Nat.le_imp_diff_is_add
tff(fact_1045_Nat_Odiff__add__assoc2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( pp(aa(nat,bool,ord_less_eq(nat,K),J))
     => ( aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),I)),K) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,minus_minus(nat,J),K)),I) ) ) ).

% Nat.diff_add_assoc2
tff(fact_1046_Nat_Odiff__add__assoc,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( pp(aa(nat,bool,ord_less_eq(nat,K),J))
     => ( aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),J)),K) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),aa(nat,nat,minus_minus(nat,J),K)) ) ) ).

% Nat.diff_add_assoc
tff(fact_1047_Nat_Ole__diff__conv2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( pp(aa(nat,bool,ord_less_eq(nat,K),J))
     => ( pp(aa(nat,bool,ord_less_eq(nat,I),aa(nat,nat,minus_minus(nat,J),K)))
      <=> pp(aa(nat,bool,ord_less_eq(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),K)),J)) ) ) ).

% Nat.le_diff_conv2
tff(fact_1048_le__diff__conv,axiom,
    ! [J: nat,K: nat,I: nat] :
      ( pp(aa(nat,bool,ord_less_eq(nat,aa(nat,nat,minus_minus(nat,J),K)),I))
    <=> pp(aa(nat,bool,ord_less_eq(nat,J),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),K))) ) ).

% le_diff_conv
tff(fact_1049_less__Suc__eq__0__disj,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,ord_less(nat,M),aa(nat,nat,suc,N)))
    <=> ( ( M = zero_zero(nat) )
        | ? [J3: nat] :
            ( ( M = aa(nat,nat,suc,J3) )
            & pp(aa(nat,bool,ord_less(nat,J3),N)) ) ) ) ).

% less_Suc_eq_0_disj
tff(fact_1050_gr0__implies__Suc,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),N))
     => ? [M3: nat] : N = aa(nat,nat,suc,M3) ) ).

% gr0_implies_Suc
tff(fact_1051_All__less__Suc2,axiom,
    ! [N: nat,P: fun(nat,bool)] :
      ( ! [I3: nat] :
          ( pp(aa(nat,bool,ord_less(nat,I3),aa(nat,nat,suc,N)))
         => pp(aa(nat,bool,P,I3)) )
    <=> ( pp(aa(nat,bool,P,zero_zero(nat)))
        & ! [I3: nat] :
            ( pp(aa(nat,bool,ord_less(nat,I3),N))
           => pp(aa(nat,bool,P,aa(nat,nat,suc,I3))) ) ) ) ).

% All_less_Suc2
tff(fact_1052_gr0__conv__Suc,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),N))
    <=> ? [M6: nat] : N = aa(nat,nat,suc,M6) ) ).

% gr0_conv_Suc
tff(fact_1053_Ex__less__Suc2,axiom,
    ! [N: nat,P: fun(nat,bool)] :
      ( ? [I3: nat] :
          ( pp(aa(nat,bool,ord_less(nat,I3),aa(nat,nat,suc,N)))
          & pp(aa(nat,bool,P,I3)) )
    <=> ( pp(aa(nat,bool,P,zero_zero(nat)))
        | ? [I3: nat] :
            ( pp(aa(nat,bool,ord_less(nat,I3),N))
            & pp(aa(nat,bool,P,aa(nat,nat,suc,I3))) ) ) ) ).

% Ex_less_Suc2
tff(fact_1054_less__imp__Suc__add,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,ord_less(nat,M),N))
     => ? [K3: nat] : N = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),K3)) ) ).

% less_imp_Suc_add
tff(fact_1055_less__iff__Suc__add,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,ord_less(nat,M),N))
    <=> ? [K2: nat] : N = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),K2)) ) ).

% less_iff_Suc_add
tff(fact_1056_less__add__Suc2,axiom,
    ! [I: nat,M: nat] : pp(aa(nat,bool,ord_less(nat,I),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),I)))) ).

% less_add_Suc2
tff(fact_1057_less__add__Suc1,axiom,
    ! [I: nat,M: nat] : pp(aa(nat,bool,ord_less(nat,I),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),M)))) ).

% less_add_Suc1
tff(fact_1058_less__natE,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,ord_less(nat,M),N))
     => ~ ! [Q3: nat] : N != aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Q3)) ) ).

% less_natE
tff(fact_1059_Suc__leI,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,ord_less(nat,M),N))
     => pp(aa(nat,bool,ord_less_eq(nat,aa(nat,nat,suc,M)),N)) ) ).

% Suc_leI
tff(fact_1060_Suc__le__eq,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,ord_less_eq(nat,aa(nat,nat,suc,M)),N))
    <=> pp(aa(nat,bool,ord_less(nat,M),N)) ) ).

% Suc_le_eq
tff(fact_1061_dec__induct,axiom,
    ! [I: nat,J: nat,P: fun(nat,bool)] :
      ( pp(aa(nat,bool,ord_less_eq(nat,I),J))
     => ( pp(aa(nat,bool,P,I))
       => ( ! [N2: nat] :
              ( pp(aa(nat,bool,ord_less_eq(nat,I),N2))
             => ( pp(aa(nat,bool,ord_less(nat,N2),J))
               => ( pp(aa(nat,bool,P,N2))
                 => pp(aa(nat,bool,P,aa(nat,nat,suc,N2))) ) ) )
         => pp(aa(nat,bool,P,J)) ) ) ) ).

% dec_induct
tff(fact_1062_inc__induct,axiom,
    ! [I: nat,J: nat,P: fun(nat,bool)] :
      ( pp(aa(nat,bool,ord_less_eq(nat,I),J))
     => ( pp(aa(nat,bool,P,J))
       => ( ! [N2: nat] :
              ( pp(aa(nat,bool,ord_less_eq(nat,I),N2))
             => ( pp(aa(nat,bool,ord_less(nat,N2),J))
               => ( pp(aa(nat,bool,P,aa(nat,nat,suc,N2)))
                 => pp(aa(nat,bool,P,N2)) ) ) )
         => pp(aa(nat,bool,P,I)) ) ) ) ).

% inc_induct
tff(fact_1063_Suc__le__lessD,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,ord_less_eq(nat,aa(nat,nat,suc,M)),N))
     => pp(aa(nat,bool,ord_less(nat,M),N)) ) ).

% Suc_le_lessD
tff(fact_1064_le__less__Suc__eq,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,ord_less_eq(nat,M),N))
     => ( pp(aa(nat,bool,ord_less(nat,N),aa(nat,nat,suc,M)))
      <=> ( N = M ) ) ) ).

% le_less_Suc_eq
tff(fact_1065_less__Suc__eq__le,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,ord_less(nat,M),aa(nat,nat,suc,N)))
    <=> pp(aa(nat,bool,ord_less_eq(nat,M),N)) ) ).

% less_Suc_eq_le
tff(fact_1066_less__eq__Suc__le,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa(nat,bool,ord_less(nat,N),M))
    <=> pp(aa(nat,bool,ord_less_eq(nat,aa(nat,nat,suc,N)),M)) ) ).

% less_eq_Suc_le
tff(fact_1067_le__imp__less__Suc,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,ord_less_eq(nat,M),N))
     => pp(aa(nat,bool,ord_less(nat,M),aa(nat,nat,suc,N))) ) ).

% le_imp_less_Suc
tff(fact_1068_less__imp__add__positive,axiom,
    ! [I: nat,J: nat] :
      ( pp(aa(nat,bool,ord_less(nat,I),J))
     => ? [K3: nat] :
          ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),K3))
          & ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),K3) = J ) ) ) ).

% less_imp_add_positive
tff(fact_1069_ex__least__nat__le,axiom,
    ! [P: fun(nat,bool),N: nat] :
      ( pp(aa(nat,bool,P,N))
     => ( ~ pp(aa(nat,bool,P,zero_zero(nat)))
       => ? [K3: nat] :
            ( pp(aa(nat,bool,ord_less_eq(nat,K3),N))
            & ! [I4: nat] :
                ( pp(aa(nat,bool,ord_less(nat,I4),K3))
               => ~ pp(aa(nat,bool,P,I4)) )
            & pp(aa(nat,bool,P,K3)) ) ) ) ).

% ex_least_nat_le
tff(fact_1070_Suc__mult__less__cancel1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa(nat,bool,ord_less(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,K)),M)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,K)),N)))
    <=> pp(aa(nat,bool,ord_less(nat,M),N)) ) ).

% Suc_mult_less_cancel1
tff(fact_1071_dvd__diffD,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa(nat,bool,dvd_dvd(nat,K),aa(nat,nat,minus_minus(nat,M),N)))
     => ( pp(aa(nat,bool,dvd_dvd(nat,K),N))
       => ( pp(aa(nat,bool,ord_less_eq(nat,N),M))
         => pp(aa(nat,bool,dvd_dvd(nat,K),M)) ) ) ) ).

% dvd_diffD
tff(fact_1072_dvd__diffD1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa(nat,bool,dvd_dvd(nat,K),aa(nat,nat,minus_minus(nat,M),N)))
     => ( pp(aa(nat,bool,dvd_dvd(nat,K),M))
       => ( pp(aa(nat,bool,ord_less_eq(nat,N),M))
         => pp(aa(nat,bool,dvd_dvd(nat,K),N)) ) ) ) ).

% dvd_diffD1
tff(fact_1073_less__eq__dvd__minus,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,ord_less_eq(nat,M),N))
     => ( pp(aa(nat,bool,dvd_dvd(nat,M),N))
      <=> pp(aa(nat,bool,dvd_dvd(nat,M),aa(nat,nat,minus_minus(nat,N),M))) ) ) ).

% less_eq_dvd_minus
tff(fact_1074_mono__nat__linear__lb,axiom,
    ! [F2: fun(nat,nat),M: nat,K: nat] :
      ( ! [M3: nat,N2: nat] :
          ( pp(aa(nat,bool,ord_less(nat,M3),N2))
         => pp(aa(nat,bool,ord_less(nat,aa(nat,nat,F2,M3)),aa(nat,nat,F2,N2))) )
     => pp(aa(nat,bool,ord_less_eq(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,F2,M)),K)),aa(nat,nat,F2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),K)))) ) ).

% mono_nat_linear_lb
tff(fact_1075_Euclidean__Division_Odiv__eq__0__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),N) = zero_zero(nat) )
    <=> ( pp(aa(nat,bool,ord_less(nat,M),N))
        | ( N = zero_zero(nat) ) ) ) ).

% Euclidean_Division.div_eq_0_iff
tff(fact_1076_mult__less__mono1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( pp(aa(nat,bool,ord_less(nat,I),J))
     => ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),K))
       => pp(aa(nat,bool,ord_less(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I),K)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),K))) ) ) ).

% mult_less_mono1
tff(fact_1077_mult__less__mono2,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( pp(aa(nat,bool,ord_less(nat,I),J))
     => ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),K))
       => pp(aa(nat,bool,ord_less(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),I)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),J))) ) ) ).

% mult_less_mono2
tff(fact_1078_nat__mult__less__cancel1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),K))
     => ( pp(aa(nat,bool,ord_less(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),M)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),N)))
      <=> pp(aa(nat,bool,ord_less(nat,M),N)) ) ) ).

% nat_mult_less_cancel1
tff(fact_1079_nat__mult__eq__cancel1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),K))
     => ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),M) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),N) )
      <=> ( M = N ) ) ) ).

% nat_mult_eq_cancel1
tff(fact_1080_nat__power__less__imp__less,axiom,
    ! [I: nat,M: nat,N: nat] :
      ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),I))
     => ( pp(aa(nat,bool,ord_less(nat,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),I),M)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),I),N)))
       => pp(aa(nat,bool,ord_less(nat,M),N)) ) ) ).

% nat_power_less_imp_less
tff(fact_1081_bezout1__nat,axiom,
    ! [A2: nat,B2: nat] :
    ? [D2: nat,X3: nat,Y3: nat] :
      ( pp(aa(nat,bool,dvd_dvd(nat,D2),A2))
      & pp(aa(nat,bool,dvd_dvd(nat,D2),B2))
      & ( ( aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A2),X3)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),Y3)) = D2 )
        | ( aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),X3)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A2),Y3)) = D2 ) ) ) ).

% bezout1_nat
tff(fact_1082_nat__dvd__not__less,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),M))
     => ( pp(aa(nat,bool,ord_less(nat,M),N))
       => ~ pp(aa(nat,bool,dvd_dvd(nat,N),M)) ) ) ).

% nat_dvd_not_less
tff(fact_1083_dvd__pos__nat,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),N))
     => ( pp(aa(nat,bool,dvd_dvd(nat,M),N))
       => pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),M)) ) ) ).

% dvd_pos_nat
tff(fact_1084_numerals_I1_J,axiom,
    aa(num,nat,numeral_numeral(nat),one2) = one_one(nat) ).

% numerals(1)
tff(fact_1085_less__mult__imp__div__less,axiom,
    ! [M: nat,I: nat,N: nat] :
      ( pp(aa(nat,bool,ord_less(nat,M),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I),N)))
     => pp(aa(nat,bool,ord_less(nat,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),N)),I)) ) ).

% less_mult_imp_div_less
tff(fact_1086_One__nat__def,axiom,
    one_one(nat) = aa(nat,nat,suc,zero_zero(nat)) ).

% One_nat_def
tff(fact_1087_Suc__eq__plus1__left,axiom,
    ! [N: nat] : aa(nat,nat,suc,N) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),N) ).

% Suc_eq_plus1_left
tff(fact_1088_plus__1__eq__Suc,axiom,
    aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)) = suc ).

% plus_1_eq_Suc
tff(fact_1089_Suc__eq__plus1,axiom,
    ! [N: nat] : aa(nat,nat,suc,N) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat)) ).

% Suc_eq_plus1
tff(fact_1090_mult__eq__self__implies__10,axiom,
    ! [M: nat,N: nat] :
      ( ( M = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N) )
     => ( ( N = one_one(nat) )
        | ( M = zero_zero(nat) ) ) ) ).

% mult_eq_self_implies_10
tff(fact_1091_zdiv__zmult2__eq,axiom,
    ! [C2: int,A2: int,B2: int] :
      ( pp(aa(int,bool,ord_less_eq(int,zero_zero(int)),C2))
     => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),aa(int,int,aa(int,fun(int,int),times_times(int),B2),C2)) = aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),B2)),C2) ) ) ).

% zdiv_zmult2_eq
tff(fact_1092_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)
tff(fact_1093_ordered__ring__class_Ole__add__iff1,axiom,
    ! [A: $tType] :
      ( ordered_ring(A)
     => ! [A2: A,E: A,C2: A,B2: A,D: A] :
          ( pp(aa(A,bool,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),E)),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),E)),D)))
        <=> pp(aa(A,bool,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,minus_minus(A,A2),B2)),E)),C2)),D)) ) ) ).

% ordered_ring_class.le_add_iff1
tff(fact_1094_ordered__ring__class_Ole__add__iff2,axiom,
    ! [A: $tType] :
      ( ordered_ring(A)
     => ! [A2: A,E: A,C2: A,B2: A,D: A] :
          ( pp(aa(A,bool,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),E)),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),E)),D)))
        <=> pp(aa(A,bool,ord_less_eq(A,C2),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,minus_minus(A,B2),A2)),E)),D))) ) ) ).

% ordered_ring_class.le_add_iff2
tff(fact_1095_add__divide__eq__if__simps_I4_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Z: A,A2: A,B2: A] :
          ( ( ( Z = zero_zero(A) )
           => ( aa(A,A,minus_minus(A,A2),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),Z)) = A2 ) )
          & ( ( Z != zero_zero(A) )
           => ( aa(A,A,minus_minus(A,A2),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),Z)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),Z)),B2)),Z) ) ) ) ) ).

% add_divide_eq_if_simps(4)
tff(fact_1096_diff__frac__eq,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Y: A,Z: A,X: A,W: A] :
          ( ( Y != zero_zero(A) )
         => ( ( Z != zero_zero(A) )
           => ( aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y)),aa(A,A,aa(A,fun(A,A),divide_divide(A),W),Z)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),X),Z)),aa(A,A,aa(A,fun(A,A),times_times(A),W),Y))),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Z)) ) ) ) ) ).

% diff_frac_eq
tff(fact_1097_diff__divide__eq__iff,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Z: A,X: A,Y: A] :
          ( ( Z != zero_zero(A) )
         => ( aa(A,A,minus_minus(A,X),aa(A,A,aa(A,fun(A,A),divide_divide(A),Y),Z)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),X),Z)),Y)),Z) ) ) ) ).

% diff_divide_eq_iff
tff(fact_1098_divide__diff__eq__iff,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Z: A,X: A,Y: A] :
          ( ( Z != zero_zero(A) )
         => ( aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Z)),Y) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,minus_minus(A,X),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Z))),Z) ) ) ) ).

% divide_diff_eq_iff
tff(fact_1099_square__diff__one__factored,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [X: A] : aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),X),X)),one_one(A)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),one_one(A))),aa(A,A,minus_minus(A,X),one_one(A))) ) ).

% square_diff_one_factored
tff(fact_1100_add__strict__increasing2,axiom,
    ! [A: $tType] :
      ( ordere8940638589300402666id_add(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),A2))
         => ( pp(aa(A,bool,ord_less(A,B2),C2))
           => pp(aa(A,bool,ord_less(A,B2),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C2))) ) ) ) ).

% add_strict_increasing2
tff(fact_1101_add__strict__increasing,axiom,
    ! [A: $tType] :
      ( ordere8940638589300402666id_add(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,ord_less(A,zero_zero(A)),A2))
         => ( pp(aa(A,bool,ord_less_eq(A,B2),C2))
           => pp(aa(A,bool,ord_less(A,B2),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C2))) ) ) ) ).

% add_strict_increasing
tff(fact_1102_add__pos__nonneg,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,ord_less(A,zero_zero(A)),A2))
         => ( pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),B2))
           => pp(aa(A,bool,ord_less(A,zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2))) ) ) ) ).

% add_pos_nonneg
tff(fact_1103_add__nonpos__neg,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,ord_less_eq(A,A2),zero_zero(A)))
         => ( pp(aa(A,bool,ord_less(A,B2),zero_zero(A)))
           => pp(aa(A,bool,ord_less(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),zero_zero(A))) ) ) ) ).

% add_nonpos_neg
tff(fact_1104_add__nonneg__pos,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),A2))
         => ( pp(aa(A,bool,ord_less(A,zero_zero(A)),B2))
           => pp(aa(A,bool,ord_less(A,zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2))) ) ) ) ).

% add_nonneg_pos
tff(fact_1105_add__neg__nonpos,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,ord_less(A,A2),zero_zero(A)))
         => ( pp(aa(A,bool,ord_less_eq(A,B2),zero_zero(A)))
           => pp(aa(A,bool,ord_less(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),zero_zero(A))) ) ) ) ).

% add_neg_nonpos
tff(fact_1106_field__le__epsilon,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y: A] :
          ( ! [E2: A] :
              ( pp(aa(A,bool,ord_less(A,zero_zero(A)),E2))
             => pp(aa(A,bool,ord_less_eq(A,X),aa(A,A,aa(A,fun(A,A),plus_plus(A),Y),E2))) )
         => pp(aa(A,bool,ord_less_eq(A,X),Y)) ) ) ).

% field_le_epsilon
tff(fact_1107_mult__le__cancel__left,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [C2: A,A2: A,B2: A] :
          ( pp(aa(A,bool,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)))
        <=> ( ( pp(aa(A,bool,ord_less(A,zero_zero(A)),C2))
             => pp(aa(A,bool,ord_less_eq(A,A2),B2)) )
            & ( pp(aa(A,bool,ord_less(A,C2),zero_zero(A)))
             => pp(aa(A,bool,ord_less_eq(A,B2),A2)) ) ) ) ) ).

% mult_le_cancel_left
tff(fact_1108_mult__le__cancel__right,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [A2: A,C2: A,B2: A] :
          ( pp(aa(A,bool,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)))
        <=> ( ( pp(aa(A,bool,ord_less(A,zero_zero(A)),C2))
             => pp(aa(A,bool,ord_less_eq(A,A2),B2)) )
            & ( pp(aa(A,bool,ord_less(A,C2),zero_zero(A)))
             => pp(aa(A,bool,ord_less_eq(A,B2),A2)) ) ) ) ) ).

% mult_le_cancel_right
tff(fact_1109_mult__left__less__imp__less,axiom,
    ! [A: $tType] :
      ( linordered_semiring(A)
     => ! [C2: A,A2: A,B2: A] :
          ( pp(aa(A,bool,ord_less(A,aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)))
         => ( pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),C2))
           => pp(aa(A,bool,ord_less(A,A2),B2)) ) ) ) ).

% mult_left_less_imp_less
tff(fact_1110_mult__strict__mono,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A2: A,B2: A,C2: A,D: A] :
          ( pp(aa(A,bool,ord_less(A,A2),B2))
         => ( pp(aa(A,bool,ord_less(A,C2),D))
           => ( pp(aa(A,bool,ord_less(A,zero_zero(A)),B2))
             => ( pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),C2))
               => pp(aa(A,bool,ord_less(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),D))) ) ) ) ) ) ).

% mult_strict_mono
tff(fact_1111_mult__less__cancel__left,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [C2: A,A2: A,B2: A] :
          ( pp(aa(A,bool,ord_less(A,aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)))
        <=> ( ( pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),C2))
             => pp(aa(A,bool,ord_less(A,A2),B2)) )
            & ( pp(aa(A,bool,ord_less_eq(A,C2),zero_zero(A)))
             => pp(aa(A,bool,ord_less(A,B2),A2)) ) ) ) ) ).

% mult_less_cancel_left
tff(fact_1112_mult__right__less__imp__less,axiom,
    ! [A: $tType] :
      ( linordered_semiring(A)
     => ! [A2: A,C2: A,B2: A] :
          ( pp(aa(A,bool,ord_less(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)))
         => ( pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),C2))
           => pp(aa(A,bool,ord_less(A,A2),B2)) ) ) ) ).

% mult_right_less_imp_less
tff(fact_1113_mult__strict__mono_H,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A2: A,B2: A,C2: A,D: A] :
          ( pp(aa(A,bool,ord_less(A,A2),B2))
         => ( pp(aa(A,bool,ord_less(A,C2),D))
           => ( pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),A2))
             => ( pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),C2))
               => pp(aa(A,bool,ord_less(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),D))) ) ) ) ) ) ).

% mult_strict_mono'
tff(fact_1114_mult__less__cancel__right,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [A2: A,C2: A,B2: A] :
          ( pp(aa(A,bool,ord_less(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)))
        <=> ( ( pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),C2))
             => pp(aa(A,bool,ord_less(A,A2),B2)) )
            & ( pp(aa(A,bool,ord_less_eq(A,C2),zero_zero(A)))
             => pp(aa(A,bool,ord_less(A,B2),A2)) ) ) ) ) ).

% mult_less_cancel_right
tff(fact_1115_mult__le__cancel__left__neg,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [C2: A,A2: A,B2: A] :
          ( pp(aa(A,bool,ord_less(A,C2),zero_zero(A)))
         => ( pp(aa(A,bool,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)))
          <=> pp(aa(A,bool,ord_less_eq(A,B2),A2)) ) ) ) ).

% mult_le_cancel_left_neg
tff(fact_1116_mult__le__cancel__left__pos,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [C2: A,A2: A,B2: A] :
          ( pp(aa(A,bool,ord_less(A,zero_zero(A)),C2))
         => ( pp(aa(A,bool,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)))
          <=> pp(aa(A,bool,ord_less_eq(A,A2),B2)) ) ) ) ).

% mult_le_cancel_left_pos
tff(fact_1117_mult__left__le__imp__le,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [C2: A,A2: A,B2: A] :
          ( pp(aa(A,bool,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)))
         => ( pp(aa(A,bool,ord_less(A,zero_zero(A)),C2))
           => pp(aa(A,bool,ord_less_eq(A,A2),B2)) ) ) ) ).

% mult_left_le_imp_le
tff(fact_1118_mult__right__le__imp__le,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A2: A,C2: A,B2: A] :
          ( pp(aa(A,bool,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)))
         => ( pp(aa(A,bool,ord_less(A,zero_zero(A)),C2))
           => pp(aa(A,bool,ord_less_eq(A,A2),B2)) ) ) ) ).

% mult_right_le_imp_le
tff(fact_1119_mult__le__less__imp__less,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A2: A,B2: A,C2: A,D: A] :
          ( pp(aa(A,bool,ord_less_eq(A,A2),B2))
         => ( pp(aa(A,bool,ord_less(A,C2),D))
           => ( pp(aa(A,bool,ord_less(A,zero_zero(A)),A2))
             => ( pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),C2))
               => pp(aa(A,bool,ord_less(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),D))) ) ) ) ) ) ).

% mult_le_less_imp_less
tff(fact_1120_mult__less__le__imp__less,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A2: A,B2: A,C2: A,D: A] :
          ( pp(aa(A,bool,ord_less(A,A2),B2))
         => ( pp(aa(A,bool,ord_less_eq(A,C2),D))
           => ( pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),A2))
             => ( pp(aa(A,bool,ord_less(A,zero_zero(A)),C2))
               => pp(aa(A,bool,ord_less(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),D))) ) ) ) ) ) ).

% mult_less_le_imp_less
tff(fact_1121_zero__less__two,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => pp(aa(A,bool,ord_less(A,zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),one_one(A)))) ) ).

% zero_less_two
tff(fact_1122_sum__squares__gt__zero__iff,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,ord_less(A,zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),X)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Y))))
        <=> ( ( X != zero_zero(A) )
            | ( Y != zero_zero(A) ) ) ) ) ).

% sum_squares_gt_zero_iff
tff(fact_1123_not__sum__squares__lt__zero,axiom,
    ! [A: $tType] :
      ( linordered_ring(A)
     => ! [X: A,Y: A] : ~ pp(aa(A,bool,ord_less(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),X)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Y))),zero_zero(A))) ) ).

% not_sum_squares_lt_zero
tff(fact_1124_frac__le,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Y: A,X: A,W: A,Z: A] :
          ( pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),Y))
         => ( pp(aa(A,bool,ord_less_eq(A,X),Y))
           => ( pp(aa(A,bool,ord_less(A,zero_zero(A)),W))
             => ( pp(aa(A,bool,ord_less_eq(A,W),Z))
               => pp(aa(A,bool,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Z)),aa(A,A,aa(A,fun(A,A),divide_divide(A),Y),W))) ) ) ) ) ) ).

% frac_le
tff(fact_1125_frac__less,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y: A,W: A,Z: A] :
          ( pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),X))
         => ( pp(aa(A,bool,ord_less(A,X),Y))
           => ( pp(aa(A,bool,ord_less(A,zero_zero(A)),W))
             => ( pp(aa(A,bool,ord_less_eq(A,W),Z))
               => pp(aa(A,bool,ord_less(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Z)),aa(A,A,aa(A,fun(A,A),divide_divide(A),Y),W))) ) ) ) ) ) ).

% frac_less
tff(fact_1126_frac__less2,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y: A,W: A,Z: A] :
          ( pp(aa(A,bool,ord_less(A,zero_zero(A)),X))
         => ( pp(aa(A,bool,ord_less_eq(A,X),Y))
           => ( pp(aa(A,bool,ord_less(A,zero_zero(A)),W))
             => ( pp(aa(A,bool,ord_less(A,W),Z))
               => pp(aa(A,bool,ord_less(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Z)),aa(A,A,aa(A,fun(A,A),divide_divide(A),Y),W))) ) ) ) ) ) ).

% frac_less2
tff(fact_1127_divide__le__cancel,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,C2: A,B2: A] :
          ( pp(aa(A,bool,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),C2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)))
        <=> ( ( pp(aa(A,bool,ord_less(A,zero_zero(A)),C2))
             => pp(aa(A,bool,ord_less_eq(A,A2),B2)) )
            & ( pp(aa(A,bool,ord_less(A,C2),zero_zero(A)))
             => pp(aa(A,bool,ord_less_eq(A,B2),A2)) ) ) ) ) ).

% divide_le_cancel
tff(fact_1128_divide__nonneg__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),X))
         => ( pp(aa(A,bool,ord_less(A,Y),zero_zero(A)))
           => pp(aa(A,bool,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y)),zero_zero(A))) ) ) ) ).

% divide_nonneg_neg
tff(fact_1129_divide__nonneg__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),X))
         => ( pp(aa(A,bool,ord_less(A,zero_zero(A)),Y))
           => pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y))) ) ) ) ).

% divide_nonneg_pos
tff(fact_1130_divide__nonpos__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,ord_less_eq(A,X),zero_zero(A)))
         => ( pp(aa(A,bool,ord_less(A,Y),zero_zero(A)))
           => pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y))) ) ) ) ).

% divide_nonpos_neg
tff(fact_1131_divide__nonpos__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,ord_less_eq(A,X),zero_zero(A)))
         => ( pp(aa(A,bool,ord_less(A,zero_zero(A)),Y))
           => pp(aa(A,bool,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y)),zero_zero(A))) ) ) ) ).

% divide_nonpos_pos
tff(fact_1132_unique__euclidean__semiring__numeral__class_Odiv__less,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),A2))
         => ( pp(aa(A,bool,ord_less(A,A2),B2))
           => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2) = zero_zero(A) ) ) ) ) ).

% unique_euclidean_semiring_numeral_class.div_less
tff(fact_1133_div__positive,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [B2: A,A2: A] :
          ( pp(aa(A,bool,ord_less(A,zero_zero(A)),B2))
         => ( pp(aa(A,bool,ord_less_eq(A,B2),A2))
           => pp(aa(A,bool,ord_less(A,zero_zero(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2))) ) ) ) ).

% div_positive
tff(fact_1134_discrete,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,ord_less(A,A2),B2))
        <=> pp(aa(A,bool,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),one_one(A))),B2)) ) ) ).

% discrete
tff(fact_1135_power__less__imp__less__base,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,N: nat,B2: A] :
          ( pp(aa(A,bool,ord_less(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),N)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),N)))
         => ( pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),B2))
           => pp(aa(A,bool,ord_less(A,A2),B2)) ) ) ) ).

% power_less_imp_less_base
tff(fact_1136_less__divide__eq__1,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,A2: A] :
          ( pp(aa(A,bool,ord_less(A,one_one(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A2)))
        <=> ( ( pp(aa(A,bool,ord_less(A,zero_zero(A)),A2))
              & pp(aa(A,bool,ord_less(A,A2),B2)) )
            | ( pp(aa(A,bool,ord_less(A,A2),zero_zero(A)))
              & pp(aa(A,bool,ord_less(A,B2),A2)) ) ) ) ) ).

% less_divide_eq_1
tff(fact_1137_divide__less__eq__1,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,A2: A] :
          ( pp(aa(A,bool,ord_less(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A2)),one_one(A)))
        <=> ( ( pp(aa(A,bool,ord_less(A,zero_zero(A)),A2))
              & pp(aa(A,bool,ord_less(A,B2),A2)) )
            | ( pp(aa(A,bool,ord_less(A,A2),zero_zero(A)))
              & pp(aa(A,bool,ord_less(A,A2),B2)) )
            | ( A2 = zero_zero(A) ) ) ) ) ).

% divide_less_eq_1
tff(fact_1138_ex__power__ivl1,axiom,
    ! [B2: nat,K: nat] :
      ( pp(aa(nat,bool,ord_less_eq(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),B2))
     => ( pp(aa(nat,bool,ord_less_eq(nat,one_one(nat)),K))
       => ? [N2: nat] :
            ( pp(aa(nat,bool,ord_less_eq(nat,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),N2)),K))
            & pp(aa(nat,bool,ord_less(nat,K),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N2),one_one(nat))))) ) ) ) ).

% ex_power_ivl1
tff(fact_1139_ex__power__ivl2,axiom,
    ! [B2: nat,K: nat] :
      ( pp(aa(nat,bool,ord_less_eq(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),B2))
     => ( pp(aa(nat,bool,ord_less_eq(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),K))
       => ? [N2: nat] :
            ( pp(aa(nat,bool,ord_less(nat,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),N2)),K))
            & pp(aa(nat,bool,ord_less_eq(nat,K),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N2),one_one(nat))))) ) ) ) ).

% ex_power_ivl2
tff(fact_1140_divide__less__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,C2: A,A2: A] :
          ( pp(aa(A,bool,ord_less(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)),A2))
        <=> ( ( pp(aa(A,bool,ord_less(A,zero_zero(A)),C2))
             => pp(aa(A,bool,ord_less(A,B2),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2))) )
            & ( ~ pp(aa(A,bool,ord_less(A,zero_zero(A)),C2))
             => ( ( pp(aa(A,bool,ord_less(A,C2),zero_zero(A)))
                 => pp(aa(A,bool,ord_less(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),B2)) )
                & ( ~ pp(aa(A,bool,ord_less(A,C2),zero_zero(A)))
                 => pp(aa(A,bool,ord_less(A,zero_zero(A)),A2)) ) ) ) ) ) ) ).

% divide_less_eq
tff(fact_1141_less__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,ord_less(A,A2),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)))
        <=> ( ( pp(aa(A,bool,ord_less(A,zero_zero(A)),C2))
             => pp(aa(A,bool,ord_less(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),B2)) )
            & ( ~ pp(aa(A,bool,ord_less(A,zero_zero(A)),C2))
             => ( ( pp(aa(A,bool,ord_less(A,C2),zero_zero(A)))
                 => pp(aa(A,bool,ord_less(A,B2),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2))) )
                & ( ~ pp(aa(A,bool,ord_less(A,C2),zero_zero(A)))
                 => pp(aa(A,bool,ord_less(A,A2),zero_zero(A))) ) ) ) ) ) ) ).

% less_divide_eq
tff(fact_1142_neg__divide__less__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,B2: A,A2: A] :
          ( pp(aa(A,bool,ord_less(A,C2),zero_zero(A)))
         => ( pp(aa(A,bool,ord_less(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)),A2))
          <=> pp(aa(A,bool,ord_less(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),B2)) ) ) ) ).

% neg_divide_less_eq
tff(fact_1143_neg__less__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,A2: A,B2: A] :
          ( pp(aa(A,bool,ord_less(A,C2),zero_zero(A)))
         => ( pp(aa(A,bool,ord_less(A,A2),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)))
          <=> pp(aa(A,bool,ord_less(A,B2),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2))) ) ) ) ).

% neg_less_divide_eq
tff(fact_1144_pos__divide__less__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,B2: A,A2: A] :
          ( pp(aa(A,bool,ord_less(A,zero_zero(A)),C2))
         => ( pp(aa(A,bool,ord_less(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)),A2))
          <=> pp(aa(A,bool,ord_less(A,B2),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2))) ) ) ) ).

% pos_divide_less_eq
tff(fact_1145_pos__less__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,A2: A,B2: A] :
          ( pp(aa(A,bool,ord_less(A,zero_zero(A)),C2))
         => ( pp(aa(A,bool,ord_less(A,A2),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)))
          <=> pp(aa(A,bool,ord_less(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),B2)) ) ) ) ).

% pos_less_divide_eq
tff(fact_1146_mult__imp__div__pos__less,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Y: A,X: A,Z: A] :
          ( pp(aa(A,bool,ord_less(A,zero_zero(A)),Y))
         => ( pp(aa(A,bool,ord_less(A,X),aa(A,A,aa(A,fun(A,A),times_times(A),Z),Y)))
           => pp(aa(A,bool,ord_less(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y)),Z)) ) ) ) ).

% mult_imp_div_pos_less
tff(fact_1147_mult__imp__less__div__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Y: A,Z: A,X: A] :
          ( pp(aa(A,bool,ord_less(A,zero_zero(A)),Y))
         => ( pp(aa(A,bool,ord_less(A,aa(A,A,aa(A,fun(A,A),times_times(A),Z),Y)),X))
           => pp(aa(A,bool,ord_less(A,Z),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y))) ) ) ) ).

% mult_imp_less_div_pos
tff(fact_1148_divide__strict__left__mono,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,A2: A,C2: A] :
          ( pp(aa(A,bool,ord_less(A,B2),A2))
         => ( pp(aa(A,bool,ord_less(A,zero_zero(A)),C2))
           => ( pp(aa(A,bool,ord_less(A,zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)))
             => pp(aa(A,bool,ord_less(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),A2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),B2))) ) ) ) ) ).

% divide_strict_left_mono
tff(fact_1149_divide__strict__left__mono__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,ord_less(A,A2),B2))
         => ( pp(aa(A,bool,ord_less(A,C2),zero_zero(A)))
           => ( pp(aa(A,bool,ord_less(A,zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)))
             => pp(aa(A,bool,ord_less(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),A2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),B2))) ) ) ) ) ).

% divide_strict_left_mono_neg
tff(fact_1150_less__half__sum,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,ord_less(A,A2),B2))
         => pp(aa(A,bool,ord_less(A,A2),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),one_one(A))))) ) ) ).

% less_half_sum
tff(fact_1151_gt__half__sum,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,ord_less(A,A2),B2))
         => pp(aa(A,bool,ord_less(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),one_one(A)))),B2)) ) ) ).

% gt_half_sum
tff(fact_1152_power__gt1__lemma,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,N: nat] :
          ( pp(aa(A,bool,ord_less(A,one_one(A)),A2))
         => pp(aa(A,bool,ord_less(A,one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),N)))) ) ) ).

% power_gt1_lemma
tff(fact_1153_power__less__power__Suc,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,N: nat] :
          ( pp(aa(A,bool,ord_less(A,one_one(A)),A2))
         => pp(aa(A,bool,ord_less(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),N)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),N)))) ) ) ).

% power_less_power_Suc
tff(fact_1154_power__gt1,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,N: nat] :
          ( pp(aa(A,bool,ord_less(A,one_one(A)),A2))
         => pp(aa(A,bool,ord_less(A,one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(nat,nat,suc,N)))) ) ) ).

% power_gt1
tff(fact_1155_zero__power,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [N: nat] :
          ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),N))
         => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),zero_zero(A)),N) = zero_zero(A) ) ) ) ).

% zero_power
tff(fact_1156_nat__eq__add__iff1,axiom,
    ! [J: nat,I: nat,U: nat,M: nat,N: nat] :
      ( pp(aa(nat,bool,ord_less_eq(nat,J),I))
     => ( ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I),U)),M) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),U)),N) )
      <=> ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,minus_minus(nat,I),J)),U)),M) = N ) ) ) ).

% nat_eq_add_iff1
tff(fact_1157_nat__eq__add__iff2,axiom,
    ! [I: nat,J: nat,U: nat,M: nat,N: nat] :
      ( pp(aa(nat,bool,ord_less_eq(nat,I),J))
     => ( ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I),U)),M) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),U)),N) )
      <=> ( M = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,minus_minus(nat,J),I)),U)),N) ) ) ) ).

% nat_eq_add_iff2
tff(fact_1158_nat__le__add__iff1,axiom,
    ! [J: nat,I: nat,U: nat,M: nat,N: nat] :
      ( pp(aa(nat,bool,ord_less_eq(nat,J),I))
     => ( pp(aa(nat,bool,ord_less_eq(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I),U)),M)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),U)),N)))
      <=> pp(aa(nat,bool,ord_less_eq(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,minus_minus(nat,I),J)),U)),M)),N)) ) ) ).

% nat_le_add_iff1
tff(fact_1159_nat__le__add__iff2,axiom,
    ! [I: nat,J: nat,U: nat,M: nat,N: nat] :
      ( pp(aa(nat,bool,ord_less_eq(nat,I),J))
     => ( pp(aa(nat,bool,ord_less_eq(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I),U)),M)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),U)),N)))
      <=> pp(aa(nat,bool,ord_less_eq(nat,M),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,minus_minus(nat,J),I)),U)),N))) ) ) ).

% nat_le_add_iff2
tff(fact_1160_nat__diff__add__eq1,axiom,
    ! [J: nat,I: nat,U: nat,M: nat,N: nat] :
      ( pp(aa(nat,bool,ord_less_eq(nat,J),I))
     => ( aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I),U)),M)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),U)),N)) = aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,minus_minus(nat,I),J)),U)),M)),N) ) ) ).

% nat_diff_add_eq1
tff(fact_1161_nat__diff__add__eq2,axiom,
    ! [I: nat,J: nat,U: nat,M: nat,N: nat] :
      ( pp(aa(nat,bool,ord_less_eq(nat,I),J))
     => ( aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I),U)),M)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),U)),N)) = aa(nat,nat,minus_minus(nat,M),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,minus_minus(nat,J),I)),U)),N)) ) ) ).

% nat_diff_add_eq2
tff(fact_1162_ex__least__nat__less,axiom,
    ! [P: fun(nat,bool),N: nat] :
      ( pp(aa(nat,bool,P,N))
     => ( ~ pp(aa(nat,bool,P,zero_zero(nat)))
       => ? [K3: nat] :
            ( pp(aa(nat,bool,ord_less(nat,K3),N))
            & ! [I4: nat] :
                ( pp(aa(nat,bool,ord_less_eq(nat,I4),K3))
               => ~ pp(aa(nat,bool,P,I4)) )
            & pp(aa(nat,bool,P,aa(nat,nat,suc,K3))) ) ) ) ).

% ex_least_nat_less
tff(fact_1163_pos__zdiv__mult__2,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa(int,bool,ord_less_eq(int,zero_zero(int)),A2))
     => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),B2))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),A2)) = aa(int,int,aa(int,fun(int,int),divide_divide(int),B2),A2) ) ) ).

% pos_zdiv_mult_2
tff(fact_1164_neg__zdiv__mult__2,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa(int,bool,ord_less_eq(int,A2),zero_zero(int)))
     => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),B2))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),A2)) = aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),B2),one_one(int))),A2) ) ) ).

% neg_zdiv_mult_2
tff(fact_1165_one__less__mult,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa(nat,bool,ord_less(nat,aa(nat,nat,suc,zero_zero(nat))),N))
     => ( pp(aa(nat,bool,ord_less(nat,aa(nat,nat,suc,zero_zero(nat))),M))
       => pp(aa(nat,bool,ord_less(nat,aa(nat,nat,suc,zero_zero(nat))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N))) ) ) ).

% one_less_mult
tff(fact_1166_n__less__m__mult__n,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),N))
     => ( pp(aa(nat,bool,ord_less(nat,aa(nat,nat,suc,zero_zero(nat))),M))
       => pp(aa(nat,bool,ord_less(nat,N),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N))) ) ) ).

% n_less_m_mult_n
tff(fact_1167_n__less__n__mult__m,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),N))
     => ( pp(aa(nat,bool,ord_less(nat,aa(nat,nat,suc,zero_zero(nat))),M))
       => pp(aa(nat,bool,ord_less(nat,N),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),M))) ) ) ).

% n_less_n_mult_m
tff(fact_1168_power__gt__expt,axiom,
    ! [N: nat,K: nat] :
      ( pp(aa(nat,bool,ord_less(nat,aa(nat,nat,suc,zero_zero(nat))),N))
     => pp(aa(nat,bool,ord_less(nat,K),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),N),K))) ) ).

% power_gt_expt
tff(fact_1169_div__le__mono2,axiom,
    ! [M: nat,N: nat,K: nat] :
      ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),M))
     => ( pp(aa(nat,bool,ord_less_eq(nat,M),N))
       => pp(aa(nat,bool,ord_less_eq(nat,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),K),N)),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),K),M))) ) ) ).

% div_le_mono2
tff(fact_1170_div__greater__zero__iff,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),N)))
    <=> ( pp(aa(nat,bool,ord_less_eq(nat,N),M))
        & pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),N)) ) ) ).

% div_greater_zero_iff
tff(fact_1171_nat__mult__le__cancel1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),K))
     => ( pp(aa(nat,bool,ord_less_eq(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),M)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),N)))
      <=> pp(aa(nat,bool,ord_less_eq(nat,M),N)) ) ) ).

% nat_mult_le_cancel1
tff(fact_1172_div__less__iff__less__mult,axiom,
    ! [Q2: nat,M: nat,N: nat] :
      ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),Q2))
     => ( pp(aa(nat,bool,ord_less(nat,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),Q2)),N))
      <=> pp(aa(nat,bool,ord_less(nat,M),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),Q2))) ) ) ).

% div_less_iff_less_mult
tff(fact_1173_nat__mult__div__cancel1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),K))
     => ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),M)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),N)) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),N) ) ) ).

% nat_mult_div_cancel1
tff(fact_1174_dvd__imp__le,axiom,
    ! [K: nat,N: nat] :
      ( pp(aa(nat,bool,dvd_dvd(nat,K),N))
     => ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),N))
       => pp(aa(nat,bool,ord_less_eq(nat,K),N)) ) ) ).

% dvd_imp_le
tff(fact_1175_dvd__mult__cancel,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa(nat,bool,dvd_dvd(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),M)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),N)))
     => ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),K))
       => pp(aa(nat,bool,dvd_dvd(nat,M),N)) ) ) ).

% dvd_mult_cancel
tff(fact_1176_nat__mult__dvd__cancel1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),K))
     => ( pp(aa(nat,bool,dvd_dvd(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),M)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),N)))
      <=> pp(aa(nat,bool,dvd_dvd(nat,M),N)) ) ) ).

% nat_mult_dvd_cancel1
tff(fact_1177_zle__int,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(int,bool,ord_less_eq(int,aa(nat,int,semiring_1_of_nat(int),M)),aa(nat,int,semiring_1_of_nat(int),N)))
    <=> pp(aa(nat,bool,ord_less_eq(nat,M),N)) ) ).

% zle_int
tff(fact_1178_zle__iff__zadd,axiom,
    ! [W: int,Z: int] :
      ( pp(aa(int,bool,ord_less_eq(int,W),Z))
    <=> ? [N5: nat] : Z = aa(int,int,aa(int,fun(int,int),plus_plus(int),W),aa(nat,int,semiring_1_of_nat(int),N5)) ) ).

% zle_iff_zadd
tff(fact_1179_VEBT__internal_OT__vebt__buildupi_H_Osimps_I2_J,axiom,
    vEBT_V9176841429113362141ildupi(aa(nat,nat,suc,zero_zero(nat))) = one_one(int) ).

% VEBT_internal.T_vebt_buildupi'.simps(2)
tff(fact_1180_frac__le__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Y: A,Z: A,X: A,W: A] :
          ( ( Y != zero_zero(A) )
         => ( ( Z != zero_zero(A) )
           => ( pp(aa(A,bool,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y)),aa(A,A,aa(A,fun(A,A),divide_divide(A),W),Z)))
            <=> pp(aa(A,bool,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),X),Z)),aa(A,A,aa(A,fun(A,A),times_times(A),W),Y))),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Z))),zero_zero(A))) ) ) ) ) ).

% frac_le_eq
tff(fact_1181_power2__commute,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [X: A,Y: A] : aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,minus_minus(A,X),Y)),aa(num,nat,numeral_numeral(nat),bit0(one2))) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,minus_minus(A,Y),X)),aa(num,nat,numeral_numeral(nat),bit0(one2))) ) ).

% power2_commute
tff(fact_1182_field__le__mult__one__interval,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y: A] :
          ( ! [Z2: A] :
              ( pp(aa(A,bool,ord_less(A,zero_zero(A)),Z2))
             => ( pp(aa(A,bool,ord_less(A,Z2),one_one(A)))
               => pp(aa(A,bool,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),times_times(A),Z2),X)),Y)) ) )
         => pp(aa(A,bool,ord_less_eq(A,X),Y)) ) ) ).

% field_le_mult_one_interval
tff(fact_1183_mult__le__cancel__left1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [C2: A,B2: A] :
          ( pp(aa(A,bool,ord_less_eq(A,C2),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)))
        <=> ( ( pp(aa(A,bool,ord_less(A,zero_zero(A)),C2))
             => pp(aa(A,bool,ord_less_eq(A,one_one(A)),B2)) )
            & ( pp(aa(A,bool,ord_less(A,C2),zero_zero(A)))
             => pp(aa(A,bool,ord_less_eq(A,B2),one_one(A))) ) ) ) ) ).

% mult_le_cancel_left1
tff(fact_1184_mult__le__cancel__left2,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [C2: A,A2: A] :
          ( pp(aa(A,bool,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2)),C2))
        <=> ( ( pp(aa(A,bool,ord_less(A,zero_zero(A)),C2))
             => pp(aa(A,bool,ord_less_eq(A,A2),one_one(A))) )
            & ( pp(aa(A,bool,ord_less(A,C2),zero_zero(A)))
             => pp(aa(A,bool,ord_less_eq(A,one_one(A)),A2)) ) ) ) ) ).

% mult_le_cancel_left2
tff(fact_1185_mult__le__cancel__right1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [C2: A,B2: A] :
          ( pp(aa(A,bool,ord_less_eq(A,C2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)))
        <=> ( ( pp(aa(A,bool,ord_less(A,zero_zero(A)),C2))
             => pp(aa(A,bool,ord_less_eq(A,one_one(A)),B2)) )
            & ( pp(aa(A,bool,ord_less(A,C2),zero_zero(A)))
             => pp(aa(A,bool,ord_less_eq(A,B2),one_one(A))) ) ) ) ) ).

% mult_le_cancel_right1
tff(fact_1186_mult__le__cancel__right2,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,C2: A] :
          ( pp(aa(A,bool,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),C2))
        <=> ( ( pp(aa(A,bool,ord_less(A,zero_zero(A)),C2))
             => pp(aa(A,bool,ord_less_eq(A,A2),one_one(A))) )
            & ( pp(aa(A,bool,ord_less(A,C2),zero_zero(A)))
             => pp(aa(A,bool,ord_less_eq(A,one_one(A)),A2)) ) ) ) ) ).

% mult_le_cancel_right2
tff(fact_1187_mult__less__cancel__left1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [C2: A,B2: A] :
          ( pp(aa(A,bool,ord_less(A,C2),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)))
        <=> ( ( pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),C2))
             => pp(aa(A,bool,ord_less(A,one_one(A)),B2)) )
            & ( pp(aa(A,bool,ord_less_eq(A,C2),zero_zero(A)))
             => pp(aa(A,bool,ord_less(A,B2),one_one(A))) ) ) ) ) ).

% mult_less_cancel_left1
tff(fact_1188_mult__less__cancel__left2,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [C2: A,A2: A] :
          ( pp(aa(A,bool,ord_less(A,aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2)),C2))
        <=> ( ( pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),C2))
             => pp(aa(A,bool,ord_less(A,A2),one_one(A))) )
            & ( pp(aa(A,bool,ord_less_eq(A,C2),zero_zero(A)))
             => pp(aa(A,bool,ord_less(A,one_one(A)),A2)) ) ) ) ) ).

% mult_less_cancel_left2
tff(fact_1189_mult__less__cancel__right1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [C2: A,B2: A] :
          ( pp(aa(A,bool,ord_less(A,C2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)))
        <=> ( ( pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),C2))
             => pp(aa(A,bool,ord_less(A,one_one(A)),B2)) )
            & ( pp(aa(A,bool,ord_less_eq(A,C2),zero_zero(A)))
             => pp(aa(A,bool,ord_less(A,B2),one_one(A))) ) ) ) ) ).

% mult_less_cancel_right1
tff(fact_1190_mult__less__cancel__right2,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,C2: A] :
          ( pp(aa(A,bool,ord_less(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),C2))
        <=> ( ( pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),C2))
             => pp(aa(A,bool,ord_less(A,A2),one_one(A))) )
            & ( pp(aa(A,bool,ord_less_eq(A,C2),zero_zero(A)))
             => pp(aa(A,bool,ord_less(A,one_one(A)),A2)) ) ) ) ) ).

% mult_less_cancel_right2
tff(fact_1191_le__divide__eq__1,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,A2: A] :
          ( pp(aa(A,bool,ord_less_eq(A,one_one(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A2)))
        <=> ( ( pp(aa(A,bool,ord_less(A,zero_zero(A)),A2))
              & pp(aa(A,bool,ord_less_eq(A,A2),B2)) )
            | ( pp(aa(A,bool,ord_less(A,A2),zero_zero(A)))
              & pp(aa(A,bool,ord_less_eq(A,B2),A2)) ) ) ) ) ).

% le_divide_eq_1
tff(fact_1192_divide__le__eq__1,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,A2: A] :
          ( pp(aa(A,bool,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A2)),one_one(A)))
        <=> ( ( pp(aa(A,bool,ord_less(A,zero_zero(A)),A2))
              & pp(aa(A,bool,ord_less_eq(A,B2),A2)) )
            | ( pp(aa(A,bool,ord_less(A,A2),zero_zero(A)))
              & pp(aa(A,bool,ord_less_eq(A,A2),B2)) )
            | ( A2 = zero_zero(A) ) ) ) ) ).

% divide_le_eq_1
tff(fact_1193_divide__le__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,C2: A,A2: A] :
          ( pp(aa(A,bool,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)),A2))
        <=> ( ( pp(aa(A,bool,ord_less(A,zero_zero(A)),C2))
             => pp(aa(A,bool,ord_less_eq(A,B2),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2))) )
            & ( ~ pp(aa(A,bool,ord_less(A,zero_zero(A)),C2))
             => ( ( pp(aa(A,bool,ord_less(A,C2),zero_zero(A)))
                 => pp(aa(A,bool,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),B2)) )
                & ( ~ pp(aa(A,bool,ord_less(A,C2),zero_zero(A)))
                 => pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),A2)) ) ) ) ) ) ) ).

% divide_le_eq
tff(fact_1194_le__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,ord_less_eq(A,A2),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)))
        <=> ( ( pp(aa(A,bool,ord_less(A,zero_zero(A)),C2))
             => pp(aa(A,bool,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),B2)) )
            & ( ~ pp(aa(A,bool,ord_less(A,zero_zero(A)),C2))
             => ( ( pp(aa(A,bool,ord_less(A,C2),zero_zero(A)))
                 => pp(aa(A,bool,ord_less_eq(A,B2),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2))) )
                & ( ~ pp(aa(A,bool,ord_less(A,C2),zero_zero(A)))
                 => pp(aa(A,bool,ord_less_eq(A,A2),zero_zero(A))) ) ) ) ) ) ) ).

% le_divide_eq
tff(fact_1195_divide__left__mono,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,A2: A,C2: A] :
          ( pp(aa(A,bool,ord_less_eq(A,B2),A2))
         => ( pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),C2))
           => ( pp(aa(A,bool,ord_less(A,zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)))
             => pp(aa(A,bool,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),A2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),B2))) ) ) ) ) ).

% divide_left_mono
tff(fact_1196_neg__divide__le__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,B2: A,A2: A] :
          ( pp(aa(A,bool,ord_less(A,C2),zero_zero(A)))
         => ( pp(aa(A,bool,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)),A2))
          <=> pp(aa(A,bool,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),B2)) ) ) ) ).

% neg_divide_le_eq
tff(fact_1197_neg__le__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,A2: A,B2: A] :
          ( pp(aa(A,bool,ord_less(A,C2),zero_zero(A)))
         => ( pp(aa(A,bool,ord_less_eq(A,A2),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)))
          <=> pp(aa(A,bool,ord_less_eq(A,B2),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2))) ) ) ) ).

% neg_le_divide_eq
tff(fact_1198_pos__divide__le__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,B2: A,A2: A] :
          ( pp(aa(A,bool,ord_less(A,zero_zero(A)),C2))
         => ( pp(aa(A,bool,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)),A2))
          <=> pp(aa(A,bool,ord_less_eq(A,B2),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2))) ) ) ) ).

% pos_divide_le_eq
tff(fact_1199_pos__le__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,A2: A,B2: A] :
          ( pp(aa(A,bool,ord_less(A,zero_zero(A)),C2))
         => ( pp(aa(A,bool,ord_less_eq(A,A2),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)))
          <=> pp(aa(A,bool,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),B2)) ) ) ) ).

% pos_le_divide_eq
tff(fact_1200_mult__imp__div__pos__le,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Y: A,X: A,Z: A] :
          ( pp(aa(A,bool,ord_less(A,zero_zero(A)),Y))
         => ( pp(aa(A,bool,ord_less_eq(A,X),aa(A,A,aa(A,fun(A,A),times_times(A),Z),Y)))
           => pp(aa(A,bool,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y)),Z)) ) ) ) ).

% mult_imp_div_pos_le
tff(fact_1201_mult__imp__le__div__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Y: A,Z: A,X: A] :
          ( pp(aa(A,bool,ord_less(A,zero_zero(A)),Y))
         => ( pp(aa(A,bool,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),times_times(A),Z),Y)),X))
           => pp(aa(A,bool,ord_less_eq(A,Z),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y))) ) ) ) ).

% mult_imp_le_div_pos
tff(fact_1202_divide__left__mono__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,ord_less_eq(A,A2),B2))
         => ( pp(aa(A,bool,ord_less_eq(A,C2),zero_zero(A)))
           => ( pp(aa(A,bool,ord_less(A,zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)))
             => pp(aa(A,bool,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),A2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),B2))) ) ) ) ) ).

% divide_left_mono_neg
tff(fact_1203_divide__less__eq__numeral_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,C2: A,W: num] :
          ( pp(aa(A,bool,ord_less(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)),aa(num,A,numeral_numeral(A),W)))
        <=> ( ( pp(aa(A,bool,ord_less(A,zero_zero(A)),C2))
             => pp(aa(A,bool,ord_less(A,B2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W)),C2))) )
            & ( ~ pp(aa(A,bool,ord_less(A,zero_zero(A)),C2))
             => ( ( pp(aa(A,bool,ord_less(A,C2),zero_zero(A)))
                 => pp(aa(A,bool,ord_less(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W)),C2)),B2)) )
                & ( ~ pp(aa(A,bool,ord_less(A,C2),zero_zero(A)))
                 => pp(aa(A,bool,ord_less(A,zero_zero(A)),aa(num,A,numeral_numeral(A),W))) ) ) ) ) ) ) ).

% divide_less_eq_numeral(1)
tff(fact_1204_less__divide__eq__numeral_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [W: num,B2: A,C2: A] :
          ( pp(aa(A,bool,ord_less(A,aa(num,A,numeral_numeral(A),W)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)))
        <=> ( ( pp(aa(A,bool,ord_less(A,zero_zero(A)),C2))
             => pp(aa(A,bool,ord_less(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W)),C2)),B2)) )
            & ( ~ pp(aa(A,bool,ord_less(A,zero_zero(A)),C2))
             => ( ( pp(aa(A,bool,ord_less(A,C2),zero_zero(A)))
                 => pp(aa(A,bool,ord_less(A,B2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W)),C2))) )
                & ( ~ pp(aa(A,bool,ord_less(A,C2),zero_zero(A)))
                 => pp(aa(A,bool,ord_less(A,aa(num,A,numeral_numeral(A),W)),zero_zero(A))) ) ) ) ) ) ) ).

% less_divide_eq_numeral(1)
tff(fact_1205_power__Suc__less,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,N: nat] :
          ( pp(aa(A,bool,ord_less(A,zero_zero(A)),A2))
         => ( pp(aa(A,bool,ord_less(A,A2),one_one(A)))
           => pp(aa(A,bool,ord_less(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),N))),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),N))) ) ) ) ).

% power_Suc_less
tff(fact_1206_power__Suc__less__one,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,N: nat] :
          ( pp(aa(A,bool,ord_less(A,zero_zero(A)),A2))
         => ( pp(aa(A,bool,ord_less(A,A2),one_one(A)))
           => pp(aa(A,bool,ord_less(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(nat,nat,suc,N))),one_one(A))) ) ) ) ).

% power_Suc_less_one
tff(fact_1207_power__eq__if,axiom,
    ! [A: $tType] :
      ( power(A)
     => ! [M: nat,P2: A] :
          ( ( ( M = zero_zero(nat) )
           => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),P2),M) = one_one(A) ) )
          & ( ( M != zero_zero(nat) )
           => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),P2),M) = aa(A,A,aa(A,fun(A,A),times_times(A),P2),aa(nat,A,aa(A,fun(nat,A),power_power(A),P2),aa(nat,nat,minus_minus(nat,M),one_one(nat)))) ) ) ) ) ).

% power_eq_if
tff(fact_1208_power__le__imp__le__exp,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,M: nat,N: nat] :
          ( pp(aa(A,bool,ord_less(A,one_one(A)),A2))
         => ( pp(aa(A,bool,ord_less_eq(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),M)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),N)))
           => pp(aa(nat,bool,ord_less_eq(nat,M),N)) ) ) ) ).

% power_le_imp_le_exp
tff(fact_1209_power__diff,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [A2: A,N: nat,M: nat] :
          ( ( A2 != zero_zero(A) )
         => ( pp(aa(nat,bool,ord_less_eq(nat,N),M))
           => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(nat,nat,minus_minus(nat,M),N)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),M)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),N)) ) ) ) ) ).

% power_diff
tff(fact_1210_power__eq__iff__eq__base,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [N: nat,A2: A,B2: A] :
          ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),N))
         => ( pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),A2))
           => ( pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),B2))
             => ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),N) = aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),N) )
              <=> ( A2 = B2 ) ) ) ) ) ) ).

% power_eq_iff_eq_base
tff(fact_1211_power__eq__imp__eq__base,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,N: nat,B2: A] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),N) = aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),N) )
         => ( pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),A2))
           => ( pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),B2))
             => ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),N))
               => ( A2 = B2 ) ) ) ) ) ) ).

% power_eq_imp_eq_base
tff(fact_1212_self__le__power,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,N: nat] :
          ( pp(aa(A,bool,ord_less_eq(A,one_one(A)),A2))
         => ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),N))
           => pp(aa(A,bool,ord_less_eq(A,A2),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),N))) ) ) ) ).

% self_le_power
tff(fact_1213_pos2,axiom,
    pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ).

% pos2
tff(fact_1214_dvd__power,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [N: nat,X: A] :
          ( ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),N))
            | ( X = one_one(A) ) )
         => pp(aa(A,bool,dvd_dvd(A,X),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),N))) ) ) ).

% dvd_power
tff(fact_1215_dvd__minus__add,axiom,
    ! [Q2: nat,N: nat,R2: nat,M: nat] :
      ( pp(aa(nat,bool,ord_less_eq(nat,Q2),N))
     => ( pp(aa(nat,bool,ord_less_eq(nat,Q2),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),R2),M)))
       => ( pp(aa(nat,bool,dvd_dvd(nat,M),aa(nat,nat,minus_minus(nat,N),Q2)))
        <=> pp(aa(nat,bool,dvd_dvd(nat,M),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),R2),M)),Q2)))) ) ) ) ).

% dvd_minus_add
tff(fact_1216_div__nat__eqI,axiom,
    ! [N: nat,Q2: nat,M: nat] :
      ( pp(aa(nat,bool,ord_less_eq(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),Q2)),M))
     => ( pp(aa(nat,bool,ord_less(nat,M),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),aa(nat,nat,suc,Q2))))
       => ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),N) = Q2 ) ) ) ).

% div_nat_eqI
tff(fact_1217_split__div,axiom,
    ! [P: fun(nat,bool),M: nat,N: nat] :
      ( pp(aa(nat,bool,P,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),N)))
    <=> ( ( ( N = zero_zero(nat) )
         => pp(aa(nat,bool,P,zero_zero(nat))) )
        & ( ( N != zero_zero(nat) )
         => ! [I3: nat,J3: nat] :
              ( pp(aa(nat,bool,ord_less(nat,J3),N))
             => ( ( M = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),I3)),J3) )
               => pp(aa(nat,bool,P,I3)) ) ) ) ) ) ).

% split_div
tff(fact_1218_dividend__less__div__times,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),N))
     => pp(aa(nat,bool,ord_less(nat,M),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),N)),N)))) ) ).

% dividend_less_div_times
tff(fact_1219_dividend__less__times__div,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),N))
     => pp(aa(nat,bool,ord_less(nat,M),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),N))))) ) ).

% dividend_less_times_div
tff(fact_1220_less__eq__div__iff__mult__less__eq,axiom,
    ! [Q2: nat,M: nat,N: nat] :
      ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),Q2))
     => ( pp(aa(nat,bool,ord_less_eq(nat,M),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),Q2)))
      <=> pp(aa(nat,bool,ord_less_eq(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),Q2)),N)) ) ) ).

% less_eq_div_iff_mult_less_eq
tff(fact_1221_nat__1__add__1,axiom,
    aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),one_one(nat)) = aa(num,nat,numeral_numeral(nat),bit0(one2)) ).

% nat_1_add_1
tff(fact_1222_even__mult__exp__div__exp__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A,M: nat,N: nat] :
          ( pp(aa(A,bool,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),M))),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N))))
        <=> ( pp(aa(nat,bool,ord_less(nat,N),M))
            | ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N) = zero_zero(A) )
            | ( pp(aa(nat,bool,ord_less_eq(nat,M),N))
              & pp(aa(A,bool,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(nat,nat,minus_minus(nat,N),M))))) ) ) ) ) ).

% even_mult_exp_div_exp_iff
tff(fact_1223_nat__less__real__le,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa(nat,bool,ord_less(nat,N),M))
    <=> pp(aa(real,bool,ord_less_eq(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,semiring_1_of_nat(real),N)),one_one(real))),aa(nat,real,semiring_1_of_nat(real),M))) ) ).

% nat_less_real_le
tff(fact_1224_not__exp__less__eq__0__int,axiom,
    ! [N: nat] : ~ pp(aa(int,bool,ord_less_eq(int,aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),N)),zero_zero(int))) ).

% not_exp_less_eq_0_int
tff(fact_1225_scaling__mono,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [U: A,V: A,R2: A,S3: A] :
          ( pp(aa(A,bool,ord_less_eq(A,U),V))
         => ( pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),R2))
           => ( pp(aa(A,bool,ord_less_eq(A,R2),S3))
             => pp(aa(A,bool,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),U),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),R2),aa(A,A,minus_minus(A,V),U))),S3))),V)) ) ) ) ) ).

% scaling_mono
tff(fact_1226_convex__bound__lt,axiom,
    ! [A: $tType] :
      ( linord715952674999750819strict(A)
     => ! [X: A,A2: A,Y: A,U: A,V: A] :
          ( pp(aa(A,bool,ord_less(A,X),A2))
         => ( pp(aa(A,bool,ord_less(A,Y),A2))
           => ( pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),U))
             => ( pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),V))
               => ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),U),V) = one_one(A) )
                 => pp(aa(A,bool,ord_less(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),U),X)),aa(A,A,aa(A,fun(A,A),times_times(A),V),Y))),A2)) ) ) ) ) ) ) ).

% convex_bound_lt
tff(fact_1227_divide__le__eq__numeral_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,C2: A,W: num] :
          ( pp(aa(A,bool,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)),aa(num,A,numeral_numeral(A),W)))
        <=> ( ( pp(aa(A,bool,ord_less(A,zero_zero(A)),C2))
             => pp(aa(A,bool,ord_less_eq(A,B2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W)),C2))) )
            & ( ~ pp(aa(A,bool,ord_less(A,zero_zero(A)),C2))
             => ( ( pp(aa(A,bool,ord_less(A,C2),zero_zero(A)))
                 => pp(aa(A,bool,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W)),C2)),B2)) )
                & ( ~ pp(aa(A,bool,ord_less(A,C2),zero_zero(A)))
                 => pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),aa(num,A,numeral_numeral(A),W))) ) ) ) ) ) ) ).

% divide_le_eq_numeral(1)
tff(fact_1228_le__divide__eq__numeral_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [W: num,B2: A,C2: A] :
          ( pp(aa(A,bool,ord_less_eq(A,aa(num,A,numeral_numeral(A),W)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)))
        <=> ( ( pp(aa(A,bool,ord_less(A,zero_zero(A)),C2))
             => pp(aa(A,bool,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W)),C2)),B2)) )
            & ( ~ pp(aa(A,bool,ord_less(A,zero_zero(A)),C2))
             => ( ( pp(aa(A,bool,ord_less(A,C2),zero_zero(A)))
                 => pp(aa(A,bool,ord_less_eq(A,B2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W)),C2))) )
                & ( ~ pp(aa(A,bool,ord_less(A,C2),zero_zero(A)))
                 => pp(aa(A,bool,ord_less_eq(A,aa(num,A,numeral_numeral(A),W)),zero_zero(A))) ) ) ) ) ) ) ).

% le_divide_eq_numeral(1)
tff(fact_1229_half__gt__zero,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( pp(aa(A,bool,ord_less(A,zero_zero(A)),A2))
         => pp(aa(A,bool,ord_less(A,zero_zero(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(num,A,numeral_numeral(A),bit0(one2))))) ) ) ).

% half_gt_zero
tff(fact_1230_half__gt__zero__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( pp(aa(A,bool,ord_less(A,zero_zero(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(num,A,numeral_numeral(A),bit0(one2)))))
        <=> pp(aa(A,bool,ord_less(A,zero_zero(A)),A2)) ) ) ).

% half_gt_zero_iff
tff(fact_1231_exp__not__zero__imp__exp__diff__not__zero,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [N: nat,M: nat] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N) != zero_zero(A) )
         => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(nat,nat,minus_minus(nat,N),M)) != zero_zero(A) ) ) ) ).

% exp_not_zero_imp_exp_diff_not_zero
tff(fact_1232_field__less__half__sum,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,ord_less(A,X),Y))
         => pp(aa(A,bool,ord_less(A,X),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y)),aa(num,A,numeral_numeral(A),bit0(one2))))) ) ) ).

% field_less_half_sum
tff(fact_1233_power2__less__0,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A] : ~ pp(aa(A,bool,ord_less(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(num,nat,numeral_numeral(nat),bit0(one2)))),zero_zero(A))) ) ).

% power2_less_0
tff(fact_1234_of__nat__less__two__power,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [N: nat] : pp(aa(A,bool,ord_less(A,aa(nat,A,semiring_1_of_nat(A),N)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N))) ) ).

% of_nat_less_two_power
tff(fact_1235_power__diff__power__eq,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A2: A,N: nat,M: nat] :
          ( ( A2 != zero_zero(A) )
         => ( ( pp(aa(nat,bool,ord_less_eq(nat,N),M))
             => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),M)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),N)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(nat,nat,minus_minus(nat,M),N)) ) )
            & ( ~ pp(aa(nat,bool,ord_less_eq(nat,N),M))
             => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),M)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),N)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(nat,nat,minus_minus(nat,N),M))) ) ) ) ) ) ).

% power_diff_power_eq
tff(fact_1236_diff__le__diff__pow,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa(nat,bool,ord_less_eq(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),K))
     => pp(aa(nat,bool,ord_less_eq(nat,aa(nat,nat,minus_minus(nat,M),N)),aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),K),M)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),K),N)))) ) ).

% diff_le_diff_pow
tff(fact_1237_less__2__cases,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,ord_less(nat,N),aa(num,nat,numeral_numeral(nat),bit0(one2))))
     => ( ( N = zero_zero(nat) )
        | ( N = aa(nat,nat,suc,zero_zero(nat)) ) ) ) ).

% less_2_cases
tff(fact_1238_less__2__cases__iff,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,ord_less(nat,N),aa(num,nat,numeral_numeral(nat),bit0(one2))))
    <=> ( ( N = zero_zero(nat) )
        | ( N = aa(nat,nat,suc,zero_zero(nat)) ) ) ) ).

% less_2_cases_iff
tff(fact_1239_odd__pos,axiom,
    ! [N: nat] :
      ( ~ pp(aa(nat,bool,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N))
     => pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),N)) ) ).

% odd_pos
tff(fact_1240_split__div_H,axiom,
    ! [P: fun(nat,bool),M: nat,N: nat] :
      ( pp(aa(nat,bool,P,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),N)))
    <=> ( ( ( N = zero_zero(nat) )
          & pp(aa(nat,bool,P,zero_zero(nat))) )
        | ? [Q4: nat] :
            ( pp(aa(nat,bool,ord_less_eq(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),Q4)),M))
            & pp(aa(nat,bool,ord_less(nat,M),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),aa(nat,nat,suc,Q4))))
            & pp(aa(nat,bool,P,Q4)) ) ) ) ).

% split_div'
tff(fact_1241_nat__induct2,axiom,
    ! [P: fun(nat,bool),N: nat] :
      ( pp(aa(nat,bool,P,zero_zero(nat)))
     => ( pp(aa(nat,bool,P,one_one(nat)))
       => ( ! [N2: nat] :
              ( pp(aa(nat,bool,P,N2))
             => pp(aa(nat,bool,P,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N2),aa(num,nat,numeral_numeral(nat),bit0(one2))))) )
         => pp(aa(nat,bool,P,N)) ) ) ) ).

% nat_induct2
tff(fact_1242_power2__less__imp__less,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,ord_less(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),aa(num,nat,numeral_numeral(nat),bit0(one2)))))
         => ( pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),Y))
           => pp(aa(A,bool,ord_less(A,X),Y)) ) ) ) ).

% power2_less_imp_less
tff(fact_1243_not__sum__power2__lt__zero,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A,Y: A] : ~ pp(aa(A,bool,ord_less(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),aa(num,nat,numeral_numeral(nat),bit0(one2))))),zero_zero(A))) ) ).

% not_sum_power2_lt_zero
tff(fact_1244_sum__power2__gt__zero__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,ord_less(A,zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),aa(num,nat,numeral_numeral(nat),bit0(one2))))))
        <=> ( ( X != zero_zero(A) )
            | ( Y != zero_zero(A) ) ) ) ) ).

% sum_power2_gt_zero_iff
tff(fact_1245_div__2__gt__zero,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,ord_less(nat,aa(nat,nat,suc,zero_zero(nat))),N))
     => pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) ).

% div_2_gt_zero
tff(fact_1246_Suc__n__div__2__gt__zero,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),N))
     => pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,N)),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) ).

% Suc_n_div_2_gt_zero
tff(fact_1247_nat__bit__induct,axiom,
    ! [P: fun(nat,bool),N: nat] :
      ( pp(aa(nat,bool,P,zero_zero(nat)))
     => ( ! [N2: nat] :
            ( pp(aa(nat,bool,P,N2))
           => ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),N2))
             => pp(aa(nat,bool,P,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N2))) ) )
       => ( ! [N2: nat] :
              ( pp(aa(nat,bool,P,N2))
             => pp(aa(nat,bool,P,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N2)))) )
         => pp(aa(nat,bool,P,N)) ) ) ) ).

% nat_bit_induct
tff(fact_1248_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_092_060_094sub_062u_092_060_094sub_062p_Ocases,axiom,
    ! [X: nat] :
      ( ( X != zero_zero(nat) )
     => ( ( X != aa(nat,nat,suc,zero_zero(nat)) )
       => ~ ! [Va2: nat] : X != aa(nat,nat,suc,aa(nat,nat,suc,Va2)) ) ) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d\<^sub>u\<^sub>p.cases
tff(fact_1249_power2__diff,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [X: A,Y: A] : aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,minus_minus(A,X),Y)),aa(num,nat,numeral_numeral(nat),bit0(one2))) = aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),X)),Y)) ) ).

% power2_diff
tff(fact_1250_zadd__int__left,axiom,
    ! [M: nat,N: nat,Z: int] : aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(nat,int,semiring_1_of_nat(int),M)),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(nat,int,semiring_1_of_nat(int),N)),Z)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N))),Z) ).

% zadd_int_left
tff(fact_1251_odd__power__less__zero,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,N: nat] :
          ( pp(aa(A,bool,ord_less(A,A2),zero_zero(A)))
         => pp(aa(A,bool,ord_less(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)))),zero_zero(A))) ) ) ).

% odd_power_less_zero
tff(fact_1252_zero__less__power__eq,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,N: nat] :
          ( pp(aa(A,bool,ord_less(A,zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),N)))
        <=> ( ( N = zero_zero(nat) )
            | ( pp(aa(nat,bool,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N))
              & ( A2 != zero_zero(A) ) )
            | ( ~ pp(aa(nat,bool,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N))
              & pp(aa(A,bool,ord_less(A,zero_zero(A)),A2)) ) ) ) ) ).

% zero_less_power_eq
tff(fact_1253_int__distrib_I2_J,axiom,
    ! [W: int,Z1: int,Z22: int] : aa(int,int,aa(int,fun(int,int),times_times(int),W),aa(int,int,aa(int,fun(int,int),plus_plus(int),Z1),Z22)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),W),Z1)),aa(int,int,aa(int,fun(int,int),times_times(int),W),Z22)) ).

% int_distrib(2)
tff(fact_1254_int__distrib_I1_J,axiom,
    ! [Z1: int,Z22: int,W: int] : aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),Z1),Z22)),W) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),Z1),W)),aa(int,int,aa(int,fun(int,int),times_times(int),Z22),W)) ).

% int_distrib(1)
tff(fact_1255_zdvd__period,axiom,
    ! [A2: int,D: int,X: int,T: int,C2: int] :
      ( pp(aa(int,bool,dvd_dvd(int,A2),D))
     => ( pp(aa(int,bool,dvd_dvd(int,A2),aa(int,int,aa(int,fun(int,int),plus_plus(int),X),T)))
      <=> pp(aa(int,bool,dvd_dvd(int,A2),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),X),aa(int,int,aa(int,fun(int,int),times_times(int),C2),D))),T))) ) ) ).

% zdvd_period
tff(fact_1256_zdvd__reduce,axiom,
    ! [K: int,N: int,M: int] :
      ( pp(aa(int,bool,dvd_dvd(int,K),aa(int,int,aa(int,fun(int,int),plus_plus(int),N),aa(int,int,aa(int,fun(int,int),times_times(int),K),M))))
    <=> pp(aa(int,bool,dvd_dvd(int,K),N)) ) ).

% zdvd_reduce
tff(fact_1257_VEBT__internal_Oexp__split__high__low_I1_J,axiom,
    ! [X: nat,N: nat,M: nat] :
      ( pp(aa(nat,bool,ord_less(nat,X),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),M))))
     => ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),N))
       => ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),M))
         => pp(aa(nat,bool,ord_less(nat,vEBT_VEBT_high(X,N)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M))) ) ) ) ).

% VEBT_internal.exp_split_high_low(1)
tff(fact_1258_even__mask__div__iff_H,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [M: nat,N: nat] :
          ( pp(aa(A,bool,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,minus_minus(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),M)),one_one(A))),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N))))
        <=> pp(aa(nat,bool,ord_less_eq(nat,M),N)) ) ) ).

% even_mask_div_iff'
tff(fact_1259_power__le__zero__eq,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,N: nat] :
          ( pp(aa(A,bool,ord_less_eq(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),N)),zero_zero(A)))
        <=> ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),N))
            & ( ( ~ pp(aa(nat,bool,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N))
                & pp(aa(A,bool,ord_less_eq(A,A2),zero_zero(A))) )
              | ( pp(aa(nat,bool,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N))
                & ( A2 = zero_zero(A) ) ) ) ) ) ) ).

% power_le_zero_eq
tff(fact_1260_even__mask__div__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [M: nat,N: nat] :
          ( pp(aa(A,bool,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,minus_minus(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),M)),one_one(A))),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N))))
        <=> ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N) = zero_zero(A) )
            | pp(aa(nat,bool,ord_less_eq(nat,M),N)) ) ) ) ).

% even_mask_div_iff
tff(fact_1261_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] :
      ( ( pp(aa(nat,bool,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,aa(nat,nat,suc,Va))))
       => ( vEBT_V8346862874174094_d_u_p(aa(nat,nat,suc,aa(nat,nat,suc,Va))) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,bit0(bit0(one2))))),vEBT_V8346862874174094_d_u_p(aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_V8346862874174094_d_u_p(aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),one_one(nat))))) ) )
      & ( ~ pp(aa(nat,bool,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,aa(nat,nat,suc,Va))))
       => ( vEBT_V8346862874174094_d_u_p(aa(nat,nat,suc,aa(nat,nat,suc,Va))) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,aa(num,num,bit1,bit0(one2))))),vEBT_V8346862874174094_d_u_p(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_V8346862874174094_d_u_p(aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,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)
tff(fact_1262_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] :
      ( ( pp(aa(nat,bool,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,aa(nat,nat,suc,Va))))
       => ( vEBT_V8646137997579335489_i_l_d(aa(nat,nat,suc,aa(nat,nat,suc,Va))) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(aa(num,num,bit1,bit0(one2))))),vEBT_V8646137997579335489_i_l_d(aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_V8646137997579335489_i_l_d(aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2))))))) ) )
      & ( ~ pp(aa(nat,bool,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,aa(nat,nat,suc,Va))))
       => ( vEBT_V8646137997579335489_i_l_d(aa(nat,nat,suc,aa(nat,nat,suc,Va))) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(aa(num,num,bit1,one2))))),vEBT_V8646137997579335489_i_l_d(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),vEBT_V8646137997579335489_i_l_d(aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))) ) ) ) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d.simps(3)
tff(fact_1263_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_092_060_094sub_062u_092_060_094sub_062p_Oelims,axiom,
    ! [X: nat,Y: nat] :
      ( ( vEBT_V8346862874174094_d_u_p(X) = Y )
     => ( ( ( X = zero_zero(nat) )
         => ( Y != aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2)) ) )
       => ( ( ( X = aa(nat,nat,suc,zero_zero(nat)) )
           => ( Y != aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2)) ) )
         => ~ ! [Va2: nat] :
                ( ( X = aa(nat,nat,suc,aa(nat,nat,suc,Va2)) )
               => ~ ( ( pp(aa(nat,bool,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,aa(nat,nat,suc,Va2))))
                     => ( Y = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,bit0(bit0(one2))))),vEBT_V8346862874174094_d_u_p(aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_V8346862874174094_d_u_p(aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),one_one(nat))))) ) )
                    & ( ~ pp(aa(nat,bool,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,aa(nat,nat,suc,Va2))))
                     => ( Y = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,aa(num,num,bit1,bit0(one2))))),vEBT_V8346862874174094_d_u_p(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_V8346862874174094_d_u_p(aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,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
tff(fact_1264_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_Oelims,axiom,
    ! [X: nat,Y: nat] :
      ( ( vEBT_V8646137997579335489_i_l_d(X) = Y )
     => ( ( ( X = zero_zero(nat) )
         => ( Y != aa(num,nat,numeral_numeral(nat),bit0(bit0(one2))) ) )
       => ( ( ( X = aa(nat,nat,suc,zero_zero(nat)) )
           => ( Y != aa(num,nat,numeral_numeral(nat),bit0(bit0(one2))) ) )
         => ~ ! [Va2: nat] :
                ( ( X = aa(nat,nat,suc,aa(nat,nat,suc,Va2)) )
               => ~ ( ( pp(aa(nat,bool,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,aa(nat,nat,suc,Va2))))
                     => ( Y = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(aa(num,num,bit1,bit0(one2))))),vEBT_V8646137997579335489_i_l_d(aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_V8646137997579335489_i_l_d(aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))))))) ) )
                    & ( ~ pp(aa(nat,bool,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,aa(nat,nat,suc,Va2))))
                     => ( Y = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(aa(num,num,bit1,one2))))),vEBT_V8646137997579335489_i_l_d(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),vEBT_V8646137997579335489_i_l_d(aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))) ) ) ) ) ) ) ) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d.elims
tff(fact_1265_VEBT__internal_OT__vebt__buildupi_H_Oelims,axiom,
    ! [X: nat,Y: int] :
      ( ( vEBT_V9176841429113362141ildupi(X) = Y )
     => ( ( ( X = zero_zero(nat) )
         => ( Y != one_one(int) ) )
       => ( ( ( X = aa(nat,nat,suc,zero_zero(nat)) )
           => ( Y != one_one(int) ) )
         => ~ ! [N2: nat] :
                ( ( X = aa(nat,nat,suc,aa(nat,nat,suc,N2)) )
               => ~ ( ( pp(aa(nat,bool,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N2))
                     => ( Y = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,one2))),aa(int,int,aa(int,fun(int,int),plus_plus(int),vEBT_V9176841429113362141ildupi(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N2),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(bit0(one2)))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N2),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(int,int,aa(int,fun(int,int),times_times(int),vEBT_V9176841429113362141ildupi(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N2),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N2),aa(num,nat,numeral_numeral(nat),bit0(one2))))))))) ) )
                    & ( ~ pp(aa(nat,bool,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N2))
                     => ( Y = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,one2))),aa(int,int,aa(int,fun(int,int),plus_plus(int),vEBT_V9176841429113362141ildupi(aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N2),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(bit0(bit0(one2))))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N2),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(bit0(one2)))),aa(int,int,aa(int,fun(int,int),times_times(int),vEBT_V9176841429113362141ildupi(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N2),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N2),aa(num,nat,numeral_numeral(nat),bit0(one2))))))))) ) ) ) ) ) ) ) ).

% VEBT_internal.T_vebt_buildupi'.elims
tff(fact_1266_space__bound,axiom,
    ! [T: vEBT_VEBT,N: nat,U: nat] :
      ( vEBT_invar_vebt(T,N)
     => ( ( U = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N) )
       => pp(aa(nat,bool,ord_less_eq(nat,aa(vEBT_VEBT,nat,vEBT_VEBT_space,T)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(aa(num,num,bit1,one2))))),U))) ) ) ).

% space_bound
tff(fact_1267_space__space_H,axiom,
    ! [T: vEBT_VEBT] : pp(aa(nat,bool,ord_less(nat,aa(vEBT_VEBT,nat,vEBT_VEBT_space,T)),aa(vEBT_VEBT,nat,vEBT_VEBT_space2,T))) ).

% space_space'
tff(fact_1268_Suc__diff,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,ord_less_eq(nat,M),N))
     => ( pp(aa(nat,bool,ord_less_eq(nat,one_one(nat)),M))
       => ( aa(nat,nat,suc,aa(nat,nat,minus_minus(nat,N),M)) = aa(nat,nat,minus_minus(nat,N),aa(nat,nat,minus_minus(nat,M),one_one(nat))) ) ) ) ).

% Suc_diff
tff(fact_1269_power__2__mult__step__le,axiom,
    ! [N3: nat,N: nat,K4: nat,K: nat] :
      ( pp(aa(nat,bool,ord_less_eq(nat,N3),N))
     => ( pp(aa(nat,bool,ord_less(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N3)),K4)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)),K)))
       => pp(aa(nat,bool,ord_less_eq(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N3)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K4),one_one(nat)))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)),K))) ) ) ).

% power_2_mult_step_le
tff(fact_1270_small__powers__of__2,axiom,
    ! [X: nat] :
      ( pp(aa(nat,bool,ord_less_eq(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2))),X))
     => pp(aa(nat,bool,ord_less(nat,X),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,minus_minus(nat,X),one_one(nat))))) ) ).

% small_powers_of_2
tff(fact_1271_nat__div__eq__Suc__0__iff,axiom,
    ! [N: nat,M: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),M) = aa(nat,nat,suc,zero_zero(nat)) )
    <=> ( pp(aa(nat,bool,ord_less_eq(nat,M),N))
        & pp(aa(nat,bool,ord_less(nat,N),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M))) ) ) ).

% nat_div_eq_Suc_0_iff
tff(fact_1272_space__2__pow__bound,axiom,
    ! [T: vEBT_VEBT,N: nat] :
      ( vEBT_invar_vebt(T,N)
     => pp(aa(real,bool,ord_less_eq(real,aa(nat,real,semiring_1_of_nat(real),aa(vEBT_VEBT,nat,vEBT_VEBT_space2,T))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(bit0(aa(num,num,bit1,one2))))),aa(real,real,minus_minus(real,aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(num,real,numeral_numeral(real),bit0(one2))),N)),one_one(real))))) ) ).

% space_2_pow_bound
tff(fact_1273_nat__less__power__trans,axiom,
    ! [N: nat,M: nat,K: nat] :
      ( pp(aa(nat,bool,ord_less(nat,N),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,minus_minus(nat,M),K))))
     => ( pp(aa(nat,bool,ord_less_eq(nat,K),M))
       => pp(aa(nat,bool,ord_less(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),K)),N)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M))) ) ) ).

% nat_less_power_trans
tff(fact_1274_less__two__pow__divD,axiom,
    ! [X: nat,N: nat,M: nat] :
      ( pp(aa(nat,bool,ord_less(nat,X),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M))))
     => ( pp(aa(nat,bool,ord_less_eq(nat,M),N))
        & pp(aa(nat,bool,ord_less(nat,X),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,minus_minus(nat,N),M)))) ) ) ).

% less_two_pow_divD
tff(fact_1275_less__two__pow__divI,axiom,
    ! [X: nat,N: nat,M: nat] :
      ( pp(aa(nat,bool,ord_less(nat,X),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,minus_minus(nat,N),M))))
     => ( pp(aa(nat,bool,ord_less_eq(nat,M),N))
       => pp(aa(nat,bool,ord_less(nat,X),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M)))) ) ) ).

% less_two_pow_divI
tff(fact_1276_idiff__0__right,axiom,
    ! [N: extended_enat] : aa(extended_enat,extended_enat,minus_minus(extended_enat,N),zero_zero(extended_enat)) = N ).

% idiff_0_right
tff(fact_1277_idiff__0,axiom,
    ! [N: extended_enat] : aa(extended_enat,extended_enat,minus_minus(extended_enat,zero_zero(extended_enat)),N) = zero_zero(extended_enat) ).

% idiff_0
tff(fact_1278_i0__less,axiom,
    ! [N: extended_enat] :
      ( pp(aa(extended_enat,bool,ord_less(extended_enat,zero_zero(extended_enat)),N))
    <=> ( N != zero_zero(extended_enat) ) ) ).

% i0_less
tff(fact_1279_Tb__T__vebt__buildupi_H,axiom,
    ! [N: nat] : pp(aa(int,bool,ord_less_eq(int,vEBT_V9176841429113362141ildupi(N)),aa(int,int,minus_minus(int,vEBT_VEBT_Tb(N)),aa(num,int,numeral_numeral(int),bit0(one2))))) ).

% Tb_T_vebt_buildupi'
tff(fact_1280_Tb__T__vebt__buildupi,axiom,
    ! [N: nat] : pp(aa(int,bool,ord_less_eq(int,aa(nat,int,semiring_1_of_nat(int),vEBT_V441764108873111860ildupi(N))),aa(int,int,minus_minus(int,vEBT_VEBT_Tb(N)),aa(num,int,numeral_numeral(int),bit0(one2))))) ).

% Tb_T_vebt_buildupi
tff(fact_1281_not__real__square__gt__zero,axiom,
    ! [X: real] :
      ( ~ pp(aa(real,bool,ord_less(real,zero_zero(real)),aa(real,real,aa(real,fun(real,real),times_times(real),X),X)))
    <=> ( X = zero_zero(real) ) ) ).

% not_real_square_gt_zero
tff(fact_1282_zle__diff1__eq,axiom,
    ! [W: int,Z: int] :
      ( pp(aa(int,bool,ord_less_eq(int,W),aa(int,int,minus_minus(int,Z),one_one(int))))
    <=> pp(aa(int,bool,ord_less(int,W),Z)) ) ).

% zle_diff1_eq
tff(fact_1283_semiring__norm_I78_J,axiom,
    ! [M: num,N: num] :
      ( pp(aa(num,bool,ord_less(num,bit0(M)),bit0(N)))
    <=> pp(aa(num,bool,ord_less(num,M),N)) ) ).

% semiring_norm(78)
tff(fact_1284_semiring__norm_I75_J,axiom,
    ! [M: num] : ~ pp(aa(num,bool,ord_less(num,M),one2)) ).

% semiring_norm(75)
tff(fact_1285_semiring__norm_I80_J,axiom,
    ! [M: num,N: num] :
      ( pp(aa(num,bool,ord_less(num,aa(num,num,bit1,M)),aa(num,num,bit1,N)))
    <=> pp(aa(num,bool,ord_less(num,M),N)) ) ).

% semiring_norm(80)
tff(fact_1286_cnt__bound_H,axiom,
    ! [T: vEBT_VEBT,N: nat] :
      ( vEBT_invar_vebt(T,N)
     => pp(aa(real,bool,ord_less_eq(real,aa(vEBT_VEBT,real,vEBT_VEBT_cnt,T)),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),aa(real,real,minus_minus(real,aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(num,real,numeral_numeral(real),bit0(one2))),N)),one_one(real))))) ) ).

% cnt_bound'
tff(fact_1287_cnt__bound,axiom,
    ! [T: vEBT_VEBT,N: nat] :
      ( vEBT_invar_vebt(T,N)
     => pp(aa(real,bool,ord_less_eq(real,aa(vEBT_VEBT,real,vEBT_VEBT_cnt,T)),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),aa(real,real,minus_minus(real,aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(num,real,numeral_numeral(real),bit0(one2))),N)),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(num,real,numeral_numeral(real),aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,one2))))),aa(num,real,numeral_numeral(real),bit0(aa(num,num,bit1,bit0(one2))))))))) ) ).

% cnt_bound
tff(fact_1288_zero__comp__diff__simps_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),aa(A,A,minus_minus(A,A2),B2)))
        <=> pp(aa(A,bool,ord_less_eq(A,B2),A2)) ) ) ).

% zero_comp_diff_simps(1)
tff(fact_1289_zero__comp__diff__simps_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,ord_less(A,zero_zero(A)),aa(A,A,minus_minus(A,A2),B2)))
        <=> pp(aa(A,bool,ord_less(A,B2),A2)) ) ) ).

% zero_comp_diff_simps(2)
tff(fact_1290_zle__add1__eq__le,axiom,
    ! [W: int,Z: int] :
      ( pp(aa(int,bool,ord_less(int,W),aa(int,int,aa(int,fun(int,int),plus_plus(int),Z),one_one(int))))
    <=> pp(aa(int,bool,ord_less_eq(int,W),Z)) ) ).

% zle_add1_eq_le
tff(fact_1291_div__pos__pos__trivial,axiom,
    ! [K: int,L: int] :
      ( pp(aa(int,bool,ord_less_eq(int,zero_zero(int)),K))
     => ( pp(aa(int,bool,ord_less(int,K),L))
       => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),K),L) = zero_zero(int) ) ) ) ).

% div_pos_pos_trivial
tff(fact_1292_div__neg__neg__trivial,axiom,
    ! [K: int,L: int] :
      ( pp(aa(int,bool,ord_less_eq(int,K),zero_zero(int)))
     => ( pp(aa(int,bool,ord_less(int,L),K))
       => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),K),L) = zero_zero(int) ) ) ) ).

% div_neg_neg_trivial
tff(fact_1293_int__div__same__is__1,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa(int,bool,ord_less(int,zero_zero(int)),A2))
     => ( ( aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),B2) = A2 )
      <=> ( B2 = one_one(int) ) ) ) ).

% int_div_same_is_1
tff(fact_1294_semiring__norm_I76_J,axiom,
    ! [N: num] : pp(aa(num,bool,ord_less(num,one2),bit0(N))) ).

% semiring_norm(76)
tff(fact_1295_semiring__norm_I81_J,axiom,
    ! [M: num,N: num] :
      ( pp(aa(num,bool,ord_less(num,aa(num,num,bit1,M)),bit0(N)))
    <=> pp(aa(num,bool,ord_less(num,M),N)) ) ).

% semiring_norm(81)
tff(fact_1296_semiring__norm_I77_J,axiom,
    ! [N: num] : pp(aa(num,bool,ord_less(num,one2),aa(num,num,bit1,N))) ).

% semiring_norm(77)
tff(fact_1297_enat__ord__number_I2_J,axiom,
    ! [M: num,N: num] :
      ( pp(aa(extended_enat,bool,ord_less(extended_enat,aa(num,extended_enat,numeral_numeral(extended_enat),M)),aa(num,extended_enat,numeral_numeral(extended_enat),N)))
    <=> pp(aa(nat,bool,ord_less(nat,aa(num,nat,numeral_numeral(nat),M)),aa(num,nat,numeral_numeral(nat),N))) ) ).

% enat_ord_number(2)
tff(fact_1298_numeral__less__real__of__nat__iff,axiom,
    ! [W: num,N: nat] :
      ( pp(aa(real,bool,ord_less(real,aa(num,real,numeral_numeral(real),W)),aa(nat,real,semiring_1_of_nat(real),N)))
    <=> pp(aa(nat,bool,ord_less(nat,aa(num,nat,numeral_numeral(nat),W)),N)) ) ).

% numeral_less_real_of_nat_iff
tff(fact_1299_real__of__nat__less__numeral__iff,axiom,
    ! [N: nat,W: num] :
      ( pp(aa(real,bool,ord_less(real,aa(nat,real,semiring_1_of_nat(real),N)),aa(num,real,numeral_numeral(real),W)))
    <=> pp(aa(nat,bool,ord_less(nat,N),aa(num,nat,numeral_numeral(nat),W))) ) ).

% real_of_nat_less_numeral_iff
tff(fact_1300_semiring__norm_I74_J,axiom,
    ! [M: num,N: num] :
      ( pp(aa(num,bool,ord_less_eq(num,aa(num,num,bit1,M)),bit0(N)))
    <=> pp(aa(num,bool,ord_less(num,M),N)) ) ).

% semiring_norm(74)
tff(fact_1301_semiring__norm_I79_J,axiom,
    ! [M: num,N: num] :
      ( pp(aa(num,bool,ord_less(num,bit0(M)),aa(num,num,bit1,N)))
    <=> pp(aa(num,bool,ord_less_eq(num,M),N)) ) ).

% semiring_norm(79)
tff(fact_1302_half__negative__int__iff,axiom,
    ! [K: int] :
      ( pp(aa(int,bool,ord_less(int,aa(int,int,aa(int,fun(int,int),divide_divide(int),K),aa(num,int,numeral_numeral(int),bit0(one2)))),zero_zero(int)))
    <=> pp(aa(int,bool,ord_less(int,K),zero_zero(int))) ) ).

% half_negative_int_iff
tff(fact_1303_not__iless0,axiom,
    ! [N: extended_enat] : ~ pp(aa(extended_enat,bool,ord_less(extended_enat,N),zero_zero(extended_enat))) ).

% not_iless0
tff(fact_1304_zero__one__enat__neq_I1_J,axiom,
    zero_zero(extended_enat) != one_one(extended_enat) ).

% zero_one_enat_neq(1)
tff(fact_1305_less__int__code_I1_J,axiom,
    ~ pp(aa(int,bool,ord_less(int,zero_zero(int)),zero_zero(int))) ).

% less_int_code(1)
tff(fact_1306_minus__int__code_I1_J,axiom,
    ! [K: int] : aa(int,int,minus_minus(int,K),zero_zero(int)) = K ).

% minus_int_code(1)
tff(fact_1307_int__less__induct,axiom,
    ! [I: int,K: int,P: fun(int,bool)] :
      ( pp(aa(int,bool,ord_less(int,I),K))
     => ( pp(aa(int,bool,P,aa(int,int,minus_minus(int,K),one_one(int))))
       => ( ! [I2: int] :
              ( pp(aa(int,bool,ord_less(int,I2),K))
             => ( pp(aa(int,bool,P,I2))
               => pp(aa(int,bool,P,aa(int,int,minus_minus(int,I2),one_one(int)))) ) )
         => pp(aa(int,bool,P,I)) ) ) ) ).

% int_less_induct
tff(fact_1308_zdvd__not__zless,axiom,
    ! [M: int,N: int] :
      ( pp(aa(int,bool,ord_less(int,zero_zero(int)),M))
     => ( pp(aa(int,bool,ord_less(int,M),N))
       => ~ pp(aa(int,bool,dvd_dvd(int,N),M)) ) ) ).

% zdvd_not_zless
tff(fact_1309_enat__0__less__mult__iff,axiom,
    ! [M: extended_enat,N: extended_enat] :
      ( pp(aa(extended_enat,bool,ord_less(extended_enat,zero_zero(extended_enat)),aa(extended_enat,extended_enat,aa(extended_enat,fun(extended_enat,extended_enat),times_times(extended_enat),M),N)))
    <=> ( pp(aa(extended_enat,bool,ord_less(extended_enat,zero_zero(extended_enat)),M))
        & pp(aa(extended_enat,bool,ord_less(extended_enat,zero_zero(extended_enat)),N)) ) ) ).

% enat_0_less_mult_iff
tff(fact_1310_int__one__le__iff__zero__less,axiom,
    ! [Z: int] :
      ( pp(aa(int,bool,ord_less_eq(int,one_one(int)),Z))
    <=> pp(aa(int,bool,ord_less(int,zero_zero(int)),Z)) ) ).

% int_one_le_iff_zero_less
tff(fact_1311_zdvd__antisym__nonneg,axiom,
    ! [M: int,N: int] :
      ( pp(aa(int,bool,ord_less_eq(int,zero_zero(int)),M))
     => ( pp(aa(int,bool,ord_less_eq(int,zero_zero(int)),N))
       => ( pp(aa(int,bool,dvd_dvd(int,M),N))
         => ( pp(aa(int,bool,dvd_dvd(int,N),M))
           => ( M = N ) ) ) ) ) ).

% zdvd_antisym_nonneg
tff(fact_1312_imult__is__0,axiom,
    ! [M: extended_enat,N: extended_enat] :
      ( ( aa(extended_enat,extended_enat,aa(extended_enat,fun(extended_enat,extended_enat),times_times(extended_enat),M),N) = zero_zero(extended_enat) )
    <=> ( ( M = zero_zero(extended_enat) )
        | ( N = zero_zero(extended_enat) ) ) ) ).

% imult_is_0
tff(fact_1313_iadd__is__0,axiom,
    ! [M: extended_enat,N: extended_enat] :
      ( ( aa(extended_enat,extended_enat,aa(extended_enat,fun(extended_enat,extended_enat),plus_plus(extended_enat),M),N) = zero_zero(extended_enat) )
    <=> ( ( M = zero_zero(extended_enat) )
        & ( N = zero_zero(extended_enat) ) ) ) ).

% iadd_is_0
tff(fact_1314_int__le__induct,axiom,
    ! [I: int,K: int,P: fun(int,bool)] :
      ( pp(aa(int,bool,ord_less_eq(int,I),K))
     => ( pp(aa(int,bool,P,K))
       => ( ! [I2: int] :
              ( pp(aa(int,bool,ord_less_eq(int,I2),K))
             => ( pp(aa(int,bool,P,I2))
               => pp(aa(int,bool,P,aa(int,int,minus_minus(int,I2),one_one(int)))) ) )
         => pp(aa(int,bool,P,I)) ) ) ) ).

% int_le_induct
tff(fact_1315_zdvd__imp__le,axiom,
    ! [Z: int,N: int] :
      ( pp(aa(int,bool,dvd_dvd(int,Z),N))
     => ( pp(aa(int,bool,ord_less(int,zero_zero(int)),N))
       => pp(aa(int,bool,ord_less_eq(int,Z),N)) ) ) ).

% zdvd_imp_le
tff(fact_1316_less__eq__int__code_I1_J,axiom,
    pp(aa(int,bool,ord_less_eq(int,zero_zero(int)),zero_zero(int))) ).

% less_eq_int_code(1)
tff(fact_1317_int__div__sub__1,axiom,
    ! [M: int,N: int] :
      ( pp(aa(int,bool,ord_less_eq(int,one_one(int)),M))
     => ( ( pp(aa(int,bool,dvd_dvd(int,M),N))
         => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,minus_minus(int,N),one_one(int))),M) = aa(int,int,minus_minus(int,aa(int,int,aa(int,fun(int,int),divide_divide(int),N),M)),one_one(int)) ) )
        & ( ~ pp(aa(int,bool,dvd_dvd(int,M),N))
         => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,minus_minus(int,N),one_one(int))),M) = aa(int,int,aa(int,fun(int,int),divide_divide(int),N),M) ) ) ) ) ).

% int_div_sub_1
tff(fact_1318_zdiv__le__dividend,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa(int,bool,ord_less_eq(int,zero_zero(int)),A2))
     => ( pp(aa(int,bool,ord_less(int,zero_zero(int)),B2))
       => pp(aa(int,bool,ord_less_eq(int,aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),B2)),A2)) ) ) ).

% zdiv_le_dividend
tff(fact_1319_div__pos__geq,axiom,
    ! [L: int,K: int] :
      ( pp(aa(int,bool,ord_less(int,zero_zero(int)),L))
     => ( pp(aa(int,bool,ord_less_eq(int,L),K))
       => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),K),L) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,minus_minus(int,K),L)),L)),one_one(int)) ) ) ) ).

% div_pos_geq
tff(fact_1320_int__induct,axiom,
    ! [P: fun(int,bool),K: int,I: int] :
      ( pp(aa(int,bool,P,K))
     => ( ! [I2: int] :
            ( pp(aa(int,bool,ord_less_eq(int,K),I2))
           => ( pp(aa(int,bool,P,I2))
             => pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),I2),one_one(int)))) ) )
       => ( ! [I2: int] :
              ( pp(aa(int,bool,ord_less_eq(int,I2),K))
             => ( pp(aa(int,bool,P,I2))
               => pp(aa(int,bool,P,aa(int,int,minus_minus(int,I2),one_one(int)))) ) )
         => pp(aa(int,bool,P,I)) ) ) ) ).

% int_induct
tff(fact_1321_int__distrib_I3_J,axiom,
    ! [Z1: int,Z22: int,W: int] : aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,minus_minus(int,Z1),Z22)),W) = aa(int,int,minus_minus(int,aa(int,int,aa(int,fun(int,int),times_times(int),Z1),W)),aa(int,int,aa(int,fun(int,int),times_times(int),Z22),W)) ).

% int_distrib(3)
tff(fact_1322_int__distrib_I4_J,axiom,
    ! [W: int,Z1: int,Z22: int] : aa(int,int,aa(int,fun(int,int),times_times(int),W),aa(int,int,minus_minus(int,Z1),Z22)) = aa(int,int,minus_minus(int,aa(int,int,aa(int,fun(int,int),times_times(int),W),Z1)),aa(int,int,aa(int,fun(int,int),times_times(int),W),Z22)) ).

% int_distrib(4)
tff(fact_1323_int__diff__cases,axiom,
    ! [Z: int] :
      ~ ! [M3: nat,N2: nat] : Z != aa(int,int,minus_minus(int,aa(nat,int,semiring_1_of_nat(int),M3)),aa(nat,int,semiring_1_of_nat(int),N2)) ).

% int_diff_cases
tff(fact_1324_less__eq__real__def,axiom,
    ! [X: real,Y: real] :
      ( pp(aa(real,bool,ord_less_eq(real,X),Y))
    <=> ( pp(aa(real,bool,ord_less(real,X),Y))
        | ( X = Y ) ) ) ).

% less_eq_real_def
tff(fact_1325_add1__zle__eq,axiom,
    ! [W: int,Z: int] :
      ( pp(aa(int,bool,ord_less_eq(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),W),one_one(int))),Z))
    <=> pp(aa(int,bool,ord_less(int,W),Z)) ) ).

% add1_zle_eq
tff(fact_1326_int__gr__induct,axiom,
    ! [K: int,I: int,P: fun(int,bool)] :
      ( pp(aa(int,bool,ord_less(int,K),I))
     => ( pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),K),one_one(int))))
       => ( ! [I2: int] :
              ( pp(aa(int,bool,ord_less(int,K),I2))
             => ( pp(aa(int,bool,P,I2))
               => pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),I2),one_one(int)))) ) )
         => pp(aa(int,bool,P,I)) ) ) ) ).

% int_gr_induct
tff(fact_1327_le__imp__0__less,axiom,
    ! [Z: int] :
      ( pp(aa(int,bool,ord_less_eq(int,zero_zero(int)),Z))
     => pp(aa(int,bool,ord_less(int,zero_zero(int)),aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),Z))) ) ).

% le_imp_0_less
tff(fact_1328_zless__add1__eq,axiom,
    ! [W: int,Z: int] :
      ( pp(aa(int,bool,ord_less(int,W),aa(int,int,aa(int,fun(int,int),plus_plus(int),Z),one_one(int))))
    <=> ( pp(aa(int,bool,ord_less(int,W),Z))
        | ( W = Z ) ) ) ).

% zless_add1_eq
tff(fact_1329_odd__less__0__iff,axiom,
    ! [Z: int] :
      ( pp(aa(int,bool,ord_less(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),Z)),Z)),zero_zero(int)))
    <=> pp(aa(int,bool,ord_less(int,Z),zero_zero(int))) ) ).

% odd_less_0_iff
tff(fact_1330_zless__imp__add1__zle,axiom,
    ! [W: int,Z: int] :
      ( pp(aa(int,bool,ord_less(int,W),Z))
     => pp(aa(int,bool,ord_less_eq(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),W),one_one(int))),Z)) ) ).

% zless_imp_add1_zle
tff(fact_1331_zmult__zless__mono2,axiom,
    ! [I: int,J: int,K: int] :
      ( pp(aa(int,bool,ord_less(int,I),J))
     => ( pp(aa(int,bool,ord_less(int,zero_zero(int)),K))
       => pp(aa(int,bool,ord_less(int,aa(int,int,aa(int,fun(int,int),times_times(int),K),I)),aa(int,int,aa(int,fun(int,int),times_times(int),K),J))) ) ) ).

% zmult_zless_mono2
tff(fact_1332_pos__zmult__eq__1__iff,axiom,
    ! [M: int,N: int] :
      ( pp(aa(int,bool,ord_less(int,zero_zero(int)),M))
     => ( ( aa(int,int,aa(int,fun(int,int),times_times(int),M),N) = one_one(int) )
      <=> ( ( M = one_one(int) )
          & ( N = one_one(int) ) ) ) ) ).

% pos_zmult_eq_1_iff
tff(fact_1333_pos__imp__zdiv__neg__iff,axiom,
    ! [B2: int,A2: int] :
      ( pp(aa(int,bool,ord_less(int,zero_zero(int)),B2))
     => ( pp(aa(int,bool,ord_less(int,aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),B2)),zero_zero(int)))
      <=> pp(aa(int,bool,ord_less(int,A2),zero_zero(int))) ) ) ).

% pos_imp_zdiv_neg_iff
tff(fact_1334_neg__imp__zdiv__neg__iff,axiom,
    ! [B2: int,A2: int] :
      ( pp(aa(int,bool,ord_less(int,B2),zero_zero(int)))
     => ( pp(aa(int,bool,ord_less(int,aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),B2)),zero_zero(int)))
      <=> pp(aa(int,bool,ord_less(int,zero_zero(int)),A2)) ) ) ).

% neg_imp_zdiv_neg_iff
tff(fact_1335_div__neg__pos__less0,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa(int,bool,ord_less(int,A2),zero_zero(int)))
     => ( pp(aa(int,bool,ord_less(int,zero_zero(int)),B2))
       => pp(aa(int,bool,ord_less(int,aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),B2)),zero_zero(int))) ) ) ).

% div_neg_pos_less0
tff(fact_1336_nonneg1__imp__zdiv__pos__iff,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa(int,bool,ord_less_eq(int,zero_zero(int)),A2))
     => ( pp(aa(int,bool,ord_less(int,zero_zero(int)),aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),B2)))
      <=> ( pp(aa(int,bool,ord_less_eq(int,B2),A2))
          & pp(aa(int,bool,ord_less(int,zero_zero(int)),B2)) ) ) ) ).

% nonneg1_imp_zdiv_pos_iff
tff(fact_1337_pos__imp__zdiv__nonneg__iff,axiom,
    ! [B2: int,A2: int] :
      ( pp(aa(int,bool,ord_less(int,zero_zero(int)),B2))
     => ( pp(aa(int,bool,ord_less_eq(int,zero_zero(int)),aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),B2)))
      <=> pp(aa(int,bool,ord_less_eq(int,zero_zero(int)),A2)) ) ) ).

% pos_imp_zdiv_nonneg_iff
tff(fact_1338_neg__imp__zdiv__nonneg__iff,axiom,
    ! [B2: int,A2: int] :
      ( pp(aa(int,bool,ord_less(int,B2),zero_zero(int)))
     => ( pp(aa(int,bool,ord_less_eq(int,zero_zero(int)),aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),B2)))
      <=> pp(aa(int,bool,ord_less_eq(int,A2),zero_zero(int))) ) ) ).

% neg_imp_zdiv_nonneg_iff
tff(fact_1339_pos__imp__zdiv__pos__iff,axiom,
    ! [K: int,I: int] :
      ( pp(aa(int,bool,ord_less(int,zero_zero(int)),K))
     => ( pp(aa(int,bool,ord_less(int,zero_zero(int)),aa(int,int,aa(int,fun(int,int),divide_divide(int),I),K)))
      <=> pp(aa(int,bool,ord_less_eq(int,K),I)) ) ) ).

% pos_imp_zdiv_pos_iff
tff(fact_1340_div__nonpos__pos__le0,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa(int,bool,ord_less_eq(int,A2),zero_zero(int)))
     => ( pp(aa(int,bool,ord_less(int,zero_zero(int)),B2))
       => pp(aa(int,bool,ord_less_eq(int,aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),B2)),zero_zero(int))) ) ) ).

% div_nonpos_pos_le0
tff(fact_1341_div__nonneg__neg__le0,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa(int,bool,ord_less_eq(int,zero_zero(int)),A2))
     => ( pp(aa(int,bool,ord_less(int,B2),zero_zero(int)))
       => pp(aa(int,bool,ord_less_eq(int,aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),B2)),zero_zero(int))) ) ) ).

% div_nonneg_neg_le0
tff(fact_1342_div__positive__int,axiom,
    ! [L: int,K: int] :
      ( pp(aa(int,bool,ord_less_eq(int,L),K))
     => ( pp(aa(int,bool,ord_less(int,zero_zero(int)),L))
       => pp(aa(int,bool,ord_less(int,zero_zero(int)),aa(int,int,aa(int,fun(int,int),divide_divide(int),K),L))) ) ) ).

% div_positive_int
tff(fact_1343_div__int__pos__iff,axiom,
    ! [K: int,L: int] :
      ( pp(aa(int,bool,ord_less_eq(int,zero_zero(int)),aa(int,int,aa(int,fun(int,int),divide_divide(int),K),L)))
    <=> ( ( K = zero_zero(int) )
        | ( L = zero_zero(int) )
        | ( pp(aa(int,bool,ord_less_eq(int,zero_zero(int)),K))
          & pp(aa(int,bool,ord_less_eq(int,zero_zero(int)),L)) )
        | ( pp(aa(int,bool,ord_less(int,K),zero_zero(int)))
          & pp(aa(int,bool,ord_less(int,L),zero_zero(int))) ) ) ) ).

% div_int_pos_iff
tff(fact_1344_zdiv__mono2__neg,axiom,
    ! [A2: int,B6: int,B2: int] :
      ( pp(aa(int,bool,ord_less(int,A2),zero_zero(int)))
     => ( pp(aa(int,bool,ord_less(int,zero_zero(int)),B6))
       => ( pp(aa(int,bool,ord_less_eq(int,B6),B2))
         => pp(aa(int,bool,ord_less_eq(int,aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),B6)),aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),B2))) ) ) ) ).

% zdiv_mono2_neg
tff(fact_1345_zdiv__mono1__neg,axiom,
    ! [A2: int,A5: int,B2: int] :
      ( pp(aa(int,bool,ord_less_eq(int,A2),A5))
     => ( pp(aa(int,bool,ord_less(int,B2),zero_zero(int)))
       => pp(aa(int,bool,ord_less_eq(int,aa(int,int,aa(int,fun(int,int),divide_divide(int),A5),B2)),aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),B2))) ) ) ).

% zdiv_mono1_neg
tff(fact_1346_zdiv__eq__0__iff,axiom,
    ! [I: int,K: int] :
      ( ( aa(int,int,aa(int,fun(int,int),divide_divide(int),I),K) = zero_zero(int) )
    <=> ( ( K = zero_zero(int) )
        | ( pp(aa(int,bool,ord_less_eq(int,zero_zero(int)),I))
          & pp(aa(int,bool,ord_less(int,I),K)) )
        | ( pp(aa(int,bool,ord_less_eq(int,I),zero_zero(int)))
          & pp(aa(int,bool,ord_less(int,K),I)) ) ) ) ).

% zdiv_eq_0_iff
tff(fact_1347_zdiv__mono2,axiom,
    ! [A2: int,B6: int,B2: int] :
      ( pp(aa(int,bool,ord_less_eq(int,zero_zero(int)),A2))
     => ( pp(aa(int,bool,ord_less(int,zero_zero(int)),B6))
       => ( pp(aa(int,bool,ord_less_eq(int,B6),B2))
         => pp(aa(int,bool,ord_less_eq(int,aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),B2)),aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),B6))) ) ) ) ).

% zdiv_mono2
tff(fact_1348_zdiv__mono1,axiom,
    ! [A2: int,A5: int,B2: int] :
      ( pp(aa(int,bool,ord_less_eq(int,A2),A5))
     => ( pp(aa(int,bool,ord_less(int,zero_zero(int)),B2))
       => pp(aa(int,bool,ord_less_eq(int,aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),B2)),aa(int,int,aa(int,fun(int,int),divide_divide(int),A5),B2))) ) ) ).

% zdiv_mono1
tff(fact_1349_int__div__less__self,axiom,
    ! [X: int,K: int] :
      ( pp(aa(int,bool,ord_less(int,zero_zero(int)),X))
     => ( pp(aa(int,bool,ord_less(int,one_one(int)),K))
       => pp(aa(int,bool,ord_less(int,aa(int,int,aa(int,fun(int,int),divide_divide(int),X),K)),X)) ) ) ).

% int_div_less_self
tff(fact_1350_add__diff__assoc__enat,axiom,
    ! [Z: extended_enat,Y: extended_enat,X: extended_enat] :
      ( pp(aa(extended_enat,bool,ord_less_eq(extended_enat,Z),Y))
     => ( aa(extended_enat,extended_enat,aa(extended_enat,fun(extended_enat,extended_enat),plus_plus(extended_enat),X),aa(extended_enat,extended_enat,minus_minus(extended_enat,Y),Z)) = aa(extended_enat,extended_enat,minus_minus(extended_enat,aa(extended_enat,extended_enat,aa(extended_enat,fun(extended_enat,extended_enat),plus_plus(extended_enat),X),Y)),Z) ) ) ).

% add_diff_assoc_enat
tff(fact_1351_strict__subset__divisors__dvd,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(set(A),bool,ord_less(set(A),collect(A,aTP_Lamp_aa(A,fun(A,bool),A2))),collect(A,aTP_Lamp_aa(A,fun(A,bool),B2))))
        <=> ( pp(aa(A,bool,dvd_dvd(A,A2),B2))
            & ~ pp(aa(A,bool,dvd_dvd(A,B2),A2)) ) ) ) ).

% strict_subset_divisors_dvd
tff(fact_1352_realpow__pos__nth2,axiom,
    ! [A2: real,N: nat] :
      ( pp(aa(real,bool,ord_less(real,zero_zero(real)),A2))
     => ? [R3: real] :
          ( pp(aa(real,bool,ord_less(real,zero_zero(real)),R3))
          & ( aa(nat,real,aa(real,fun(nat,real),power_power(real),R3),aa(nat,nat,suc,N)) = A2 ) ) ) ).

% realpow_pos_nth2
tff(fact_1353_q__pos__lemma,axiom,
    ! [B6: int,Q5: int,R4: int] :
      ( pp(aa(int,bool,ord_less_eq(int,zero_zero(int)),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B6),Q5)),R4)))
     => ( pp(aa(int,bool,ord_less(int,R4),B6))
       => ( pp(aa(int,bool,ord_less(int,zero_zero(int)),B6))
         => pp(aa(int,bool,ord_less_eq(int,zero_zero(int)),Q5)) ) ) ) ).

% q_pos_lemma
tff(fact_1354_zdiv__mono2__lemma,axiom,
    ! [B2: int,Q2: int,R2: int,B6: int,Q5: int,R4: int] :
      ( ( aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B2),Q2)),R2) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B6),Q5)),R4) )
     => ( pp(aa(int,bool,ord_less_eq(int,zero_zero(int)),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B6),Q5)),R4)))
       => ( pp(aa(int,bool,ord_less(int,R4),B6))
         => ( pp(aa(int,bool,ord_less_eq(int,zero_zero(int)),R2))
           => ( pp(aa(int,bool,ord_less(int,zero_zero(int)),B6))
             => ( pp(aa(int,bool,ord_less_eq(int,B6),B2))
               => pp(aa(int,bool,ord_less_eq(int,Q2),Q5)) ) ) ) ) ) ) ).

% zdiv_mono2_lemma
tff(fact_1355_zdiv__mono2__neg__lemma,axiom,
    ! [B2: int,Q2: int,R2: int,B6: int,Q5: int,R4: int] :
      ( ( aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B2),Q2)),R2) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B6),Q5)),R4) )
     => ( pp(aa(int,bool,ord_less(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B6),Q5)),R4)),zero_zero(int)))
       => ( pp(aa(int,bool,ord_less(int,R2),B2))
         => ( pp(aa(int,bool,ord_less_eq(int,zero_zero(int)),R4))
           => ( pp(aa(int,bool,ord_less(int,zero_zero(int)),B6))
             => ( pp(aa(int,bool,ord_less_eq(int,B6),B2))
               => pp(aa(int,bool,ord_less_eq(int,Q5),Q2)) ) ) ) ) ) ) ).

% zdiv_mono2_neg_lemma
tff(fact_1356_unique__quotient__lemma,axiom,
    ! [B2: int,Q5: int,R4: int,Q2: int,R2: int] :
      ( pp(aa(int,bool,ord_less_eq(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B2),Q5)),R4)),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B2),Q2)),R2)))
     => ( pp(aa(int,bool,ord_less_eq(int,zero_zero(int)),R4))
       => ( pp(aa(int,bool,ord_less(int,R4),B2))
         => ( pp(aa(int,bool,ord_less(int,R2),B2))
           => pp(aa(int,bool,ord_less_eq(int,Q5),Q2)) ) ) ) ) ).

% unique_quotient_lemma
tff(fact_1357_unique__quotient__lemma__neg,axiom,
    ! [B2: int,Q5: int,R4: int,Q2: int,R2: int] :
      ( pp(aa(int,bool,ord_less_eq(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B2),Q5)),R4)),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B2),Q2)),R2)))
     => ( pp(aa(int,bool,ord_less_eq(int,R2),zero_zero(int)))
       => ( pp(aa(int,bool,ord_less(int,B2),R2))
         => ( pp(aa(int,bool,ord_less(int,B2),R4))
           => pp(aa(int,bool,ord_less_eq(int,Q2),Q5)) ) ) ) ) ).

% unique_quotient_lemma_neg
tff(fact_1358_real__arch__pow,axiom,
    ! [X: real,Y: real] :
      ( pp(aa(real,bool,ord_less(real,one_one(real)),X))
     => ? [N2: nat] : pp(aa(real,bool,ord_less(real,Y),aa(nat,real,aa(real,fun(nat,real),power_power(real),X),N2))) ) ).

% real_arch_pow
tff(fact_1359_real__arch__pow__inv,axiom,
    ! [Y: real,X: real] :
      ( pp(aa(real,bool,ord_less(real,zero_zero(real)),Y))
     => ( pp(aa(real,bool,ord_less(real,X),one_one(real)))
       => ? [N2: nat] : pp(aa(real,bool,ord_less(real,aa(nat,real,aa(real,fun(nat,real),power_power(real),X),N2)),Y)) ) ) ).

% real_arch_pow_inv
tff(fact_1360_reals__Archimedean3,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,ord_less(real,zero_zero(real)),X))
     => ! [Y4: real] :
        ? [N2: nat] : pp(aa(real,bool,ord_less(real,Y4),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),N2)),X))) ) ).

% reals_Archimedean3
tff(fact_1361_ord__eq__le__eq__trans,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [A2: A,B2: A,C2: A,D: A] :
          ( ( A2 = B2 )
         => ( pp(aa(A,bool,ord_less_eq(A,B2),C2))
           => ( ( C2 = D )
             => pp(aa(A,bool,ord_less_eq(A,A2),D)) ) ) ) ) ).

% ord_eq_le_eq_trans
tff(fact_1362_pos__int__cases,axiom,
    ! [K: int] :
      ( pp(aa(int,bool,ord_less(int,zero_zero(int)),K))
     => ~ ! [N2: nat] :
            ( ( K = aa(nat,int,semiring_1_of_nat(int),N2) )
           => ~ pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),N2)) ) ) ).

% pos_int_cases
tff(fact_1363_zero__less__imp__eq__int,axiom,
    ! [K: int] :
      ( pp(aa(int,bool,ord_less(int,zero_zero(int)),K))
     => ? [N2: nat] :
          ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),N2))
          & ( K = aa(nat,int,semiring_1_of_nat(int),N2) ) ) ) ).

% zero_less_imp_eq_int
tff(fact_1364_realpow__pos__nth,axiom,
    ! [N: nat,A2: real] :
      ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),N))
     => ( pp(aa(real,bool,ord_less(real,zero_zero(real)),A2))
       => ? [R3: real] :
            ( pp(aa(real,bool,ord_less(real,zero_zero(real)),R3))
            & ( aa(nat,real,aa(real,fun(nat,real),power_power(real),R3),N) = A2 ) ) ) ) ).

% realpow_pos_nth
tff(fact_1365_realpow__pos__nth__unique,axiom,
    ! [N: nat,A2: real] :
      ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),N))
     => ( pp(aa(real,bool,ord_less(real,zero_zero(real)),A2))
       => ? [X3: real] :
            ( pp(aa(real,bool,ord_less(real,zero_zero(real)),X3))
            & ( aa(nat,real,aa(real,fun(nat,real),power_power(real),X3),N) = A2 )
            & ! [Y4: real] :
                ( ( pp(aa(real,bool,ord_less(real,zero_zero(real)),Y4))
                  & ( aa(nat,real,aa(real,fun(nat,real),power_power(real),Y4),N) = A2 ) )
               => ( Y4 = X3 ) ) ) ) ) ).

% realpow_pos_nth_unique
tff(fact_1366_zless__iff__Suc__zadd,axiom,
    ! [W: int,Z: int] :
      ( pp(aa(int,bool,ord_less(int,W),Z))
    <=> ? [N5: nat] : Z = aa(int,int,aa(int,fun(int,int),plus_plus(int),W),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,suc,N5))) ) ).

% zless_iff_Suc_zadd
tff(fact_1367_subset__Collect__conv,axiom,
    ! [A: $tType,S: set(A),P: fun(A,bool)] :
      ( pp(aa(set(A),bool,ord_less_eq(set(A),S),collect(A,P)))
    <=> ! [X2: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X2),S))
         => pp(aa(A,bool,P,X2)) ) ) ).

% subset_Collect_conv
tff(fact_1368_int__div__pos__eq,axiom,
    ! [A2: int,B2: int,Q2: int,R2: int] :
      ( ( A2 = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B2),Q2)),R2) )
     => ( pp(aa(int,bool,ord_less_eq(int,zero_zero(int)),R2))
       => ( pp(aa(int,bool,ord_less(int,R2),B2))
         => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),B2) = Q2 ) ) ) ) ).

% int_div_pos_eq
tff(fact_1369_int__div__neg__eq,axiom,
    ! [A2: int,B2: int,Q2: int,R2: int] :
      ( ( A2 = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B2),Q2)),R2) )
     => ( pp(aa(int,bool,ord_less_eq(int,R2),zero_zero(int)))
       => ( pp(aa(int,bool,ord_less(int,B2),R2))
         => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),B2) = Q2 ) ) ) ) ).

% int_div_neg_eq
tff(fact_1370_split__zdiv,axiom,
    ! [P: fun(int,bool),N: int,K: int] :
      ( pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),divide_divide(int),N),K)))
    <=> ( ( ( K = zero_zero(int) )
         => pp(aa(int,bool,P,zero_zero(int))) )
        & ( pp(aa(int,bool,ord_less(int,zero_zero(int)),K))
         => ! [I3: int,J3: int] :
              ( ( pp(aa(int,bool,ord_less_eq(int,zero_zero(int)),J3))
                & pp(aa(int,bool,ord_less(int,J3),K))
                & ( N = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),K),I3)),J3) ) )
             => pp(aa(int,bool,P,I3)) ) )
        & ( pp(aa(int,bool,ord_less(int,K),zero_zero(int)))
         => ! [I3: int,J3: int] :
              ( ( pp(aa(int,bool,ord_less(int,K),J3))
                & pp(aa(int,bool,ord_less_eq(int,J3),zero_zero(int)))
                & ( N = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),K),I3)),J3) ) )
             => pp(aa(int,bool,P,I3)) ) ) ) ) ).

% split_zdiv
tff(fact_1371_even__diff__iff,axiom,
    ! [K: int,L: int] :
      ( pp(aa(int,bool,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),aa(int,int,minus_minus(int,K),L)))
    <=> pp(aa(int,bool,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),aa(int,int,aa(int,fun(int,int),plus_plus(int),K),L))) ) ).

% even_diff_iff
tff(fact_1372_real__of__nat__div2,axiom,
    ! [N: nat,X: nat] : pp(aa(real,bool,ord_less_eq(real,zero_zero(real)),aa(real,real,minus_minus(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(nat,real,semiring_1_of_nat(real),N)),aa(nat,real,semiring_1_of_nat(real),X))),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),X))))) ).

% real_of_nat_div2
tff(fact_1373_zmult__zless__mono2__lemma,axiom,
    ! [I: int,J: int,K: nat] :
      ( pp(aa(int,bool,ord_less(int,I),J))
     => ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),K))
       => pp(aa(int,bool,ord_less(int,aa(int,int,aa(int,fun(int,int),times_times(int),aa(nat,int,semiring_1_of_nat(int),K)),I)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(nat,int,semiring_1_of_nat(int),K)),J))) ) ) ).

% zmult_zless_mono2_lemma
tff(fact_1374_nat__le__real__less,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa(nat,bool,ord_less_eq(nat,N),M))
    <=> pp(aa(real,bool,ord_less(real,aa(nat,real,semiring_1_of_nat(real),N)),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,semiring_1_of_nat(real),M)),one_one(real)))) ) ).

% nat_le_real_less
tff(fact_1375_real__of__nat__div3,axiom,
    ! [N: nat,X: nat] : pp(aa(real,bool,ord_less_eq(real,aa(real,real,minus_minus(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(nat,real,semiring_1_of_nat(real),N)),aa(nat,real,semiring_1_of_nat(real),X))),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),X)))),one_one(real))) ).

% real_of_nat_div3
tff(fact_1376_int__power__div__base,axiom,
    ! [M: nat,K: int] :
      ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),M))
     => ( pp(aa(int,bool,ord_less(int,zero_zero(int)),K))
       => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),K),M)),K) = aa(nat,int,aa(int,fun(nat,int),power_power(int),K),aa(nat,nat,minus_minus(nat,M),aa(nat,nat,suc,zero_zero(nat)))) ) ) ) ).

% int_power_div_base
tff(fact_1377_exists__leI,axiom,
    ! [N: nat,P: fun(nat,bool)] :
      ( ( ! [N6: nat] :
            ( pp(aa(nat,bool,ord_less(nat,N6),N))
           => ~ pp(aa(nat,bool,P,N6)) )
       => pp(aa(nat,bool,P,N)) )
     => ? [N7: nat] :
          ( pp(aa(nat,bool,ord_less_eq(nat,N7),N))
          & pp(aa(nat,bool,P,N7)) ) ) ).

% exists_leI
tff(fact_1378_nat__compl__induct_H,axiom,
    ! [P: fun(nat,bool),N: nat] :
      ( pp(aa(nat,bool,P,zero_zero(nat)))
     => ( ! [N2: nat] :
            ( ! [Nn: nat] :
                ( pp(aa(nat,bool,ord_less_eq(nat,Nn),N2))
               => pp(aa(nat,bool,P,Nn)) )
           => pp(aa(nat,bool,P,aa(nat,nat,suc,N2))) )
       => pp(aa(nat,bool,P,N)) ) ) ).

% nat_compl_induct'
tff(fact_1379_nat__compl__induct,axiom,
    ! [P: fun(nat,bool),N: nat] :
      ( pp(aa(nat,bool,P,zero_zero(nat)))
     => ( ! [N2: nat] :
            ( ! [Nn: nat] :
                ( pp(aa(nat,bool,ord_less_eq(nat,Nn),N2))
               => pp(aa(nat,bool,P,Nn)) )
           => pp(aa(nat,bool,P,aa(nat,nat,suc,N2))) )
       => pp(aa(nat,bool,P,N)) ) ) ).

% nat_compl_induct
tff(fact_1380_nat__in__between__eq_I1_J,axiom,
    ! [A2: nat,B2: nat] :
      ( ( pp(aa(nat,bool,ord_less(nat,A2),B2))
        & pp(aa(nat,bool,ord_less_eq(nat,B2),aa(nat,nat,suc,A2))) )
    <=> ( B2 = aa(nat,nat,suc,A2) ) ) ).

% nat_in_between_eq(1)
tff(fact_1381_nat__in__between__eq_I2_J,axiom,
    ! [A2: nat,B2: nat] :
      ( ( pp(aa(nat,bool,ord_less_eq(nat,A2),B2))
        & pp(aa(nat,bool,ord_less(nat,B2),aa(nat,nat,suc,A2))) )
    <=> ( B2 = A2 ) ) ).

% nat_in_between_eq(2)
tff(fact_1382_Suc__to__right,axiom,
    ! [N: nat,M: nat] :
      ( ( aa(nat,nat,suc,N) = M )
     => ( N = aa(nat,nat,minus_minus(nat,M),aa(nat,nat,suc,zero_zero(nat))) ) ) ).

% Suc_to_right
tff(fact_1383_nat__geq__1__eq__neqz,axiom,
    ! [X: nat] :
      ( pp(aa(nat,bool,ord_less_eq(nat,one_one(nat)),X))
    <=> ( X != zero_zero(nat) ) ) ).

% nat_geq_1_eq_neqz
tff(fact_1384_mlex__bound,axiom,
    ! [A2: nat,A3: nat,B2: nat,N4: nat] :
      ( pp(aa(nat,bool,ord_less(nat,A2),A3))
     => ( pp(aa(nat,bool,ord_less(nat,B2),N4))
       => pp(aa(nat,bool,ord_less(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A2),N4)),B2)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A3),N4))) ) ) ).

% mlex_bound
tff(fact_1385_mlex__fst__decrI,axiom,
    ! [A2: nat,A5: nat,B2: nat,N4: nat,B6: nat] :
      ( pp(aa(nat,bool,ord_less(nat,A2),A5))
     => ( pp(aa(nat,bool,ord_less(nat,B2),N4))
       => ( pp(aa(nat,bool,ord_less(nat,B6),N4))
         => pp(aa(nat,bool,ord_less(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A2),N4)),B2)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A5),N4)),B6))) ) ) ) ).

% mlex_fst_decrI
tff(fact_1386_mlex__snd__decrI,axiom,
    ! [A2: nat,A5: nat,B2: nat,B6: nat,N4: nat] :
      ( ( A2 = A5 )
     => ( pp(aa(nat,bool,ord_less(nat,B2),B6))
       => pp(aa(nat,bool,ord_less(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A2),N4)),B2)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A5),N4)),B6))) ) ) ).

% mlex_snd_decrI
tff(fact_1387_mlex__leI,axiom,
    ! [A2: nat,A5: nat,B2: nat,B6: nat,N4: nat] :
      ( pp(aa(nat,bool,ord_less_eq(nat,A2),A5))
     => ( pp(aa(nat,bool,ord_less_eq(nat,B2),B6))
       => pp(aa(nat,bool,ord_less_eq(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A2),N4)),B2)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A5),N4)),B6))) ) ) ).

% mlex_leI
tff(fact_1388_div__mult__le,axiom,
    ! [A2: nat,B2: nat] : pp(aa(nat,bool,ord_less_eq(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),A2),B2)),B2)),A2)) ).

% div_mult_le
tff(fact_1389_zdiv__mult__self,axiom,
    ! [M: int,A2: int,N: int] :
      ( ( M != zero_zero(int) )
     => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),A2),aa(int,int,aa(int,fun(int,int),times_times(int),M),N))),M) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),M)),N) ) ) ).

% zdiv_mult_self
tff(fact_1390_td__gal__lt,axiom,
    ! [C2: nat,A2: nat,B2: nat] :
      ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),C2))
     => ( pp(aa(nat,bool,ord_less(nat,A2),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),C2)))
      <=> pp(aa(nat,bool,ord_less(nat,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),A2),C2)),B2)) ) ) ).

% td_gal_lt
tff(fact_1391_n__less__equal__power__2,axiom,
    ! [N: nat] : pp(aa(nat,bool,ord_less(nat,N),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N))) ).

% n_less_equal_power_2
tff(fact_1392_nz__le__conv__less,axiom,
    ! [K: nat,M: nat] :
      ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),K))
     => ( pp(aa(nat,bool,ord_less_eq(nat,K),M))
       => pp(aa(nat,bool,ord_less(nat,aa(nat,nat,minus_minus(nat,K),aa(nat,nat,suc,zero_zero(nat)))),M)) ) ) ).

% nz_le_conv_less
tff(fact_1393_Suc__n__minus__m__eq,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,ord_less_eq(nat,M),N))
     => ( pp(aa(nat,bool,ord_less(nat,one_one(nat)),M))
       => ( aa(nat,nat,suc,aa(nat,nat,minus_minus(nat,N),M)) = aa(nat,nat,minus_minus(nat,N),aa(nat,nat,minus_minus(nat,M),one_one(nat))) ) ) ) ).

% Suc_n_minus_m_eq
tff(fact_1394_msrevs_I1_J,axiom,
    ! [N: nat,K: nat,M: nat] :
      ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),N))
     => ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),N)),M)),N) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),N)),K) ) ) ).

% msrevs(1)
tff(fact_1395_td__gal,axiom,
    ! [C2: nat,B2: nat,A2: nat] :
      ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),C2))
     => ( pp(aa(nat,bool,ord_less_eq(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),C2)),A2))
      <=> pp(aa(nat,bool,ord_less_eq(nat,B2),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),A2),C2))) ) ) ).

% td_gal
tff(fact_1396_nat__mult__power__less__eq,axiom,
    ! [B2: nat,A2: nat,N: nat,M: nat] :
      ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),B2))
     => ( pp(aa(nat,bool,ord_less(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A2),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),N))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),M)))
      <=> pp(aa(nat,bool,ord_less(nat,A2),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),aa(nat,nat,minus_minus(nat,M),N)))) ) ) ).

% nat_mult_power_less_eq
tff(fact_1397_axxdiv2,axiom,
    ! [X: int] :
      ( ( aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),X)),X)),aa(num,int,numeral_numeral(int),bit0(one2))) = X )
      & ( aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),zero_zero(int)),X)),X)),aa(num,int,numeral_numeral(int),bit0(one2))) = X ) ) ).

% axxdiv2
tff(fact_1398_power__sub,axiom,
    ! [N: nat,M: nat,A2: nat] :
      ( pp(aa(nat,bool,ord_less_eq(nat,N),M))
     => ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),A2))
       => ( aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),A2),aa(nat,nat,minus_minus(nat,M),N)) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),A2),M)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),A2),N)) ) ) ) ).

% power_sub
tff(fact_1399_z1pdiv2,axiom,
    ! [B2: int] : aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),B2)),one_one(int))),aa(num,int,numeral_numeral(int),bit0(one2))) = B2 ).

% z1pdiv2
tff(fact_1400_two__pow__div__gt__le,axiom,
    ! [V: nat,N: nat,M: nat] :
      ( pp(aa(nat,bool,ord_less(nat,V),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M))))
     => pp(aa(nat,bool,ord_less_eq(nat,M),N)) ) ).

% two_pow_div_gt_le
tff(fact_1401_nat__add__offset__less,axiom,
    ! [Y: nat,N: nat,X: nat,M: nat,Sz: nat] :
      ( pp(aa(nat,bool,ord_less(nat,Y),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)))
     => ( pp(aa(nat,bool,ord_less(nat,X),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M)))
       => ( ( Sz = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N) )
         => pp(aa(nat,bool,ord_less(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),X),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N))),Y)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Sz))) ) ) ) ).

% nat_add_offset_less
tff(fact_1402_nat__power__less__diff,axiom,
    ! [N: nat,Q2: nat,M: nat] :
      ( pp(aa(nat,bool,ord_less(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)),Q2)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M)))
     => pp(aa(nat,bool,ord_less(nat,Q2),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,minus_minus(nat,M),N)))) ) ).

% nat_power_less_diff
tff(fact_1403_power__minus__is__div,axiom,
    ! [B2: nat,A2: nat] :
      ( pp(aa(nat,bool,ord_less_eq(nat,B2),A2))
     => ( aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,minus_minus(nat,A2),B2)) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),A2)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),B2)) ) ) ).

% power_minus_is_div
tff(fact_1404_nat__le__power__trans,axiom,
    ! [N: nat,M: nat,K: nat] :
      ( pp(aa(nat,bool,ord_less_eq(nat,N),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,minus_minus(nat,M),K))))
     => ( pp(aa(nat,bool,ord_less_eq(nat,K),M))
       => pp(aa(nat,bool,ord_less_eq(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),K)),N)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M))) ) ) ).

% nat_le_power_trans
tff(fact_1405_real__average__minus__first,axiom,
    ! [A2: real,B2: real] : aa(real,real,minus_minus(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),A2),B2)),aa(num,real,numeral_numeral(real),bit0(one2)))),A2) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,minus_minus(real,B2),A2)),aa(num,real,numeral_numeral(real),bit0(one2))) ).

% real_average_minus_first
tff(fact_1406_real__average__minus__second,axiom,
    ! [B2: real,A2: real] : aa(real,real,minus_minus(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),B2),A2)),aa(num,real,numeral_numeral(real),bit0(one2)))),A2) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,minus_minus(real,B2),A2)),aa(num,real,numeral_numeral(real),bit0(one2))) ).

% real_average_minus_second
tff(fact_1407_two__powr__height__bound__deg,axiom,
    ! [T: vEBT_VEBT,N: nat] :
      ( vEBT_invar_vebt(T,N)
     => pp(aa(nat,bool,ord_less_eq(nat,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(vEBT_VEBT,nat,vEBT_VEBT_height,T))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N))) ) ).

% two_powr_height_bound_deg
tff(fact_1408_linear__plus__1__le__power,axiom,
    ! [X: real,N: nat] :
      ( pp(aa(real,bool,ord_less_eq(real,zero_zero(real)),X))
     => pp(aa(real,bool,ord_less_eq(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),N)),X)),one_one(real))),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),X),one_one(real))),N))) ) ).

% linear_plus_1_le_power
tff(fact_1409_member__bound,axiom,
    ! [Tree: vEBT_VEBT,X: nat,N: nat] :
      ( pp(aa(nat,bool,vEBT_vebt_member(Tree),X))
     => ( vEBT_invar_vebt(Tree,N)
       => pp(aa(nat,bool,ord_less(nat,X),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N))) ) ) ).

% member_bound
tff(fact_1410_nat__approx__posE,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [E: A] :
          ( pp(aa(A,bool,ord_less(A,zero_zero(A)),E))
         => ~ ! [N2: nat] : ~ pp(aa(A,bool,ord_less(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,N2)))),E)) ) ) ).

% nat_approx_posE
tff(fact_1411_zdiff__int__split,axiom,
    ! [P: fun(int,bool),X: nat,Y: nat] :
      ( pp(aa(int,bool,P,aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,minus_minus(nat,X),Y))))
    <=> ( ( pp(aa(nat,bool,ord_less_eq(nat,Y),X))
         => pp(aa(int,bool,P,aa(int,int,minus_minus(int,aa(nat,int,semiring_1_of_nat(int),X)),aa(nat,int,semiring_1_of_nat(int),Y)))) )
        & ( pp(aa(nat,bool,ord_less(nat,X),Y))
         => pp(aa(int,bool,P,zero_zero(int))) ) ) ) ).

% zdiff_int_split
tff(fact_1412_set__bit__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A] : aa(A,A,aa(nat,fun(A,A),bit_se5668285175392031749et_bit(A),zero_zero(nat)),A2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(num,A,numeral_numeral(A),bit0(one2))))) ) ).

% set_bit_0
tff(fact_1413_height__compose__summary,axiom,
    ! [Summary: vEBT_VEBT,Info: option(product_prod(nat,nat)),Deg: nat,TreeList: list(vEBT_VEBT)] : pp(aa(nat,bool,ord_less_eq(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),aa(vEBT_VEBT,nat,vEBT_VEBT_height,Summary))),aa(vEBT_VEBT,nat,vEBT_VEBT_height,vEBT_Node(Info,Deg,TreeList,Summary)))) ).

% height_compose_summary
tff(fact_1414_set__bit__nonnegative__int__iff,axiom,
    ! [N: nat,K: int] :
      ( pp(aa(int,bool,ord_less_eq(int,zero_zero(int)),aa(int,int,aa(nat,fun(int,int),bit_se5668285175392031749et_bit(int),N),K)))
    <=> pp(aa(int,bool,ord_less_eq(int,zero_zero(int)),K)) ) ).

% set_bit_nonnegative_int_iff
tff(fact_1415_set__bit__negative__int__iff,axiom,
    ! [N: nat,K: int] :
      ( pp(aa(int,bool,ord_less(int,aa(int,int,aa(nat,fun(int,int),bit_se5668285175392031749et_bit(int),N),K)),zero_zero(int)))
    <=> pp(aa(int,bool,ord_less(int,K),zero_zero(int))) ) ).

% set_bit_negative_int_iff
tff(fact_1416_member__correct,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( vEBT_invar_vebt(T,N)
     => ( pp(aa(nat,bool,vEBT_vebt_member(T),X))
      <=> pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),X),vEBT_set_vebt(T))) ) ) ).

% member_correct
tff(fact_1417_enat__less__induct,axiom,
    ! [P: fun(extended_enat,bool),N: extended_enat] :
      ( ! [N2: extended_enat] :
          ( ! [M4: extended_enat] :
              ( pp(aa(extended_enat,bool,ord_less(extended_enat,M4),N2))
             => pp(aa(extended_enat,bool,P,M4)) )
         => pp(aa(extended_enat,bool,P,N2)) )
     => pp(aa(extended_enat,bool,P,N)) ) ).

% enat_less_induct
tff(fact_1418_set__bit__greater__eq,axiom,
    ! [K: int,N: nat] : pp(aa(int,bool,ord_less_eq(int,K),aa(int,int,aa(nat,fun(int,int),bit_se5668285175392031749et_bit(int),N),K))) ).

% set_bit_greater_eq
tff(fact_1419_mult__commute__abs,axiom,
    ! [A: $tType] :
      ( ab_semigroup_mult(A)
     => ! [C2: A] : aTP_Lamp_ad(A,fun(A,A),C2) = aa(A,fun(A,A),times_times(A),C2) ) ).

% mult_commute_abs
tff(fact_1420_minf_I8_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [T: A] :
        ? [Z2: A] :
        ! [X4: A] :
          ( pp(aa(A,bool,ord_less(A,X4),Z2))
         => ~ pp(aa(A,bool,ord_less_eq(A,T),X4)) ) ) ).

% minf(8)
tff(fact_1421_minf_I6_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [T: A] :
        ? [Z2: A] :
        ! [X4: A] :
          ( pp(aa(A,bool,ord_less(A,X4),Z2))
         => pp(aa(A,bool,ord_less_eq(A,X4),T)) ) ) ).

% minf(6)
tff(fact_1422_pinf_I8_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [T: A] :
        ? [Z2: A] :
        ! [X4: A] :
          ( pp(aa(A,bool,ord_less(A,Z2),X4))
         => pp(aa(A,bool,ord_less_eq(A,T),X4)) ) ) ).

% pinf(8)
tff(fact_1423_pinf_I6_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [T: A] :
        ? [Z2: A] :
        ! [X4: A] :
          ( pp(aa(A,bool,ord_less(A,Z2),X4))
         => ~ pp(aa(A,bool,ord_less_eq(A,X4),T)) ) ) ).

% pinf(6)
tff(fact_1424_inf__period_I2_J,axiom,
    ! [A: $tType] :
      ( ( comm_ring(A)
        & dvd(A) )
     => ! [P: fun(A,bool),D4: A,Q: fun(A,bool)] :
          ( ! [X3: A,K3: A] :
              ( pp(aa(A,bool,P,X3))
            <=> pp(aa(A,bool,P,aa(A,A,minus_minus(A,X3),aa(A,A,aa(A,fun(A,A),times_times(A),K3),D4)))) )
         => ( ! [X3: A,K3: A] :
                ( pp(aa(A,bool,Q,X3))
              <=> pp(aa(A,bool,Q,aa(A,A,minus_minus(A,X3),aa(A,A,aa(A,fun(A,A),times_times(A),K3),D4)))) )
           => ! [X4: A,K5: A] :
                ( ( pp(aa(A,bool,P,X4))
                  | pp(aa(A,bool,Q,X4)) )
              <=> ( pp(aa(A,bool,P,aa(A,A,minus_minus(A,X4),aa(A,A,aa(A,fun(A,A),times_times(A),K5),D4))))
                  | pp(aa(A,bool,Q,aa(A,A,minus_minus(A,X4),aa(A,A,aa(A,fun(A,A),times_times(A),K5),D4)))) ) ) ) ) ) ).

% inf_period(2)
tff(fact_1425_inf__period_I1_J,axiom,
    ! [A: $tType] :
      ( ( comm_ring(A)
        & dvd(A) )
     => ! [P: fun(A,bool),D4: A,Q: fun(A,bool)] :
          ( ! [X3: A,K3: A] :
              ( pp(aa(A,bool,P,X3))
            <=> pp(aa(A,bool,P,aa(A,A,minus_minus(A,X3),aa(A,A,aa(A,fun(A,A),times_times(A),K3),D4)))) )
         => ( ! [X3: A,K3: A] :
                ( pp(aa(A,bool,Q,X3))
              <=> pp(aa(A,bool,Q,aa(A,A,minus_minus(A,X3),aa(A,A,aa(A,fun(A,A),times_times(A),K3),D4)))) )
           => ! [X4: A,K5: A] :
                ( ( pp(aa(A,bool,P,X4))
                  & pp(aa(A,bool,Q,X4)) )
              <=> ( pp(aa(A,bool,P,aa(A,A,minus_minus(A,X4),aa(A,A,aa(A,fun(A,A),times_times(A),K5),D4))))
                  & pp(aa(A,bool,Q,aa(A,A,minus_minus(A,X4),aa(A,A,aa(A,fun(A,A),times_times(A),K5),D4)))) ) ) ) ) ) ).

% inf_period(1)
tff(fact_1426_real__arch__simple,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [X: A] :
        ? [N2: nat] : pp(aa(A,bool,ord_less_eq(A,X),aa(nat,A,semiring_1_of_nat(A),N2))) ) ).

% real_arch_simple
tff(fact_1427_exists__least__lemma,axiom,
    ! [P: fun(nat,bool)] :
      ( ~ pp(aa(nat,bool,P,zero_zero(nat)))
     => ( ? [X_12: nat] : pp(aa(nat,bool,P,X_12))
       => ? [N2: nat] :
            ( ~ pp(aa(nat,bool,P,N2))
            & pp(aa(nat,bool,P,aa(nat,nat,suc,N2))) ) ) ) ).

% exists_least_lemma
tff(fact_1428_reals__Archimedean2,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [X: A] :
        ? [N2: nat] : pp(aa(A,bool,ord_less(A,X),aa(nat,A,semiring_1_of_nat(A),N2))) ) ).

% reals_Archimedean2
tff(fact_1429_imp__le__cong,axiom,
    ! [X: int,X5: int,P: bool,P3: bool] :
      ( ( X = X5 )
     => ( ( pp(aa(int,bool,ord_less_eq(int,zero_zero(int)),X5))
         => ( pp(P)
          <=> pp(P3) ) )
       => ( ( pp(aa(int,bool,ord_less_eq(int,zero_zero(int)),X))
           => pp(P) )
        <=> ( pp(aa(int,bool,ord_less_eq(int,zero_zero(int)),X5))
           => pp(P3) ) ) ) ) ).

% imp_le_cong
tff(fact_1430_conj__le__cong,axiom,
    ! [X: int,X5: int,P: bool,P3: bool] :
      ( ( X = X5 )
     => ( ( pp(aa(int,bool,ord_less_eq(int,zero_zero(int)),X5))
         => ( pp(P)
          <=> pp(P3) ) )
       => ( ( pp(aa(int,bool,ord_less_eq(int,zero_zero(int)),X))
            & pp(P) )
        <=> ( pp(aa(int,bool,ord_less_eq(int,zero_zero(int)),X5))
            & pp(P3) ) ) ) ) ).

% conj_le_cong
tff(fact_1431_plusinfinity,axiom,
    ! [D: int,P3: fun(int,bool),P: fun(int,bool)] :
      ( pp(aa(int,bool,ord_less(int,zero_zero(int)),D))
     => ( ! [X3: int,K3: int] :
            ( pp(aa(int,bool,P3,X3))
          <=> pp(aa(int,bool,P3,aa(int,int,minus_minus(int,X3),aa(int,int,aa(int,fun(int,int),times_times(int),K3),D)))) )
       => ( ? [Z3: int] :
            ! [X3: int] :
              ( pp(aa(int,bool,ord_less(int,Z3),X3))
             => ( pp(aa(int,bool,P,X3))
              <=> pp(aa(int,bool,P3,X3)) ) )
         => ( ? [X_12: int] : pp(aa(int,bool,P3,X_12))
           => ? [X_1: int] : pp(aa(int,bool,P,X_1)) ) ) ) ) ).

% plusinfinity
tff(fact_1432_minusinfinity,axiom,
    ! [D: int,P1: fun(int,bool),P: fun(int,bool)] :
      ( pp(aa(int,bool,ord_less(int,zero_zero(int)),D))
     => ( ! [X3: int,K3: int] :
            ( pp(aa(int,bool,P1,X3))
          <=> pp(aa(int,bool,P1,aa(int,int,minus_minus(int,X3),aa(int,int,aa(int,fun(int,int),times_times(int),K3),D)))) )
       => ( ? [Z3: int] :
            ! [X3: int] :
              ( pp(aa(int,bool,ord_less(int,X3),Z3))
             => ( pp(aa(int,bool,P,X3))
              <=> pp(aa(int,bool,P1,X3)) ) )
         => ( ? [X_12: int] : pp(aa(int,bool,P1,X_12))
           => ? [X_1: int] : pp(aa(int,bool,P,X_1)) ) ) ) ) ).

% minusinfinity
tff(fact_1433_even__set__bit__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,A2: A] :
          ( pp(aa(A,bool,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(nat,fun(A,A),bit_se5668285175392031749et_bit(A),M),A2)))
        <=> ( pp(aa(A,bool,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A2))
            & ( M != zero_zero(nat) ) ) ) ) ).

% even_set_bit_iff
tff(fact_1434_pinf_I9_J,axiom,
    ! [B: $tType] :
      ( ( plus(B)
        & linorder(B)
        & dvd(B) )
     => ! [D: B,S3: B] :
        ? [Z2: B] :
        ! [X4: B] :
          ( pp(aa(B,bool,ord_less(B,Z2),X4))
         => ( pp(aa(B,bool,dvd_dvd(B,D),aa(B,B,aa(B,fun(B,B),plus_plus(B),X4),S3)))
          <=> pp(aa(B,bool,dvd_dvd(B,D),aa(B,B,aa(B,fun(B,B),plus_plus(B),X4),S3))) ) ) ) ).

% pinf(9)
tff(fact_1435_pinf_I10_J,axiom,
    ! [B: $tType] :
      ( ( plus(B)
        & linorder(B)
        & dvd(B) )
     => ! [D: B,S3: B] :
        ? [Z2: B] :
        ! [X4: B] :
          ( pp(aa(B,bool,ord_less(B,Z2),X4))
         => ( ~ pp(aa(B,bool,dvd_dvd(B,D),aa(B,B,aa(B,fun(B,B),plus_plus(B),X4),S3)))
          <=> ~ pp(aa(B,bool,dvd_dvd(B,D),aa(B,B,aa(B,fun(B,B),plus_plus(B),X4),S3))) ) ) ) ).

% pinf(10)
tff(fact_1436_minf_I9_J,axiom,
    ! [B: $tType] :
      ( ( plus(B)
        & linorder(B)
        & dvd(B) )
     => ! [D: B,S3: B] :
        ? [Z2: B] :
        ! [X4: B] :
          ( pp(aa(B,bool,ord_less(B,X4),Z2))
         => ( pp(aa(B,bool,dvd_dvd(B,D),aa(B,B,aa(B,fun(B,B),plus_plus(B),X4),S3)))
          <=> pp(aa(B,bool,dvd_dvd(B,D),aa(B,B,aa(B,fun(B,B),plus_plus(B),X4),S3))) ) ) ) ).

% minf(9)
tff(fact_1437_minf_I10_J,axiom,
    ! [B: $tType] :
      ( ( plus(B)
        & linorder(B)
        & dvd(B) )
     => ! [D: B,S3: B] :
        ? [Z2: B] :
        ! [X4: B] :
          ( pp(aa(B,bool,ord_less(B,X4),Z2))
         => ( ~ pp(aa(B,bool,dvd_dvd(B,D),aa(B,B,aa(B,fun(B,B),plus_plus(B),X4),S3)))
          <=> ~ pp(aa(B,bool,dvd_dvd(B,D),aa(B,B,aa(B,fun(B,B),plus_plus(B),X4),S3))) ) ) ) ).

% minf(10)
tff(fact_1438_decr__mult__lemma,axiom,
    ! [D: int,P: fun(int,bool),K: int] :
      ( pp(aa(int,bool,ord_less(int,zero_zero(int)),D))
     => ( ! [X3: int] :
            ( pp(aa(int,bool,P,X3))
           => pp(aa(int,bool,P,aa(int,int,minus_minus(int,X3),D))) )
       => ( pp(aa(int,bool,ord_less_eq(int,zero_zero(int)),K))
         => ! [X4: int] :
              ( pp(aa(int,bool,P,X4))
             => pp(aa(int,bool,P,aa(int,int,minus_minus(int,X4),aa(int,int,aa(int,fun(int,int),times_times(int),K),D)))) ) ) ) ) ).

% decr_mult_lemma
tff(fact_1439_Bolzano,axiom,
    ! [A2: real,B2: real,P: fun(real,fun(real,bool))] :
      ( pp(aa(real,bool,ord_less_eq(real,A2),B2))
     => ( ! [A4: real,B3: real,C3: real] :
            ( pp(aa(real,bool,aa(real,fun(real,bool),P,A4),B3))
           => ( pp(aa(real,bool,aa(real,fun(real,bool),P,B3),C3))
             => ( pp(aa(real,bool,ord_less_eq(real,A4),B3))
               => ( pp(aa(real,bool,ord_less_eq(real,B3),C3))
                 => pp(aa(real,bool,aa(real,fun(real,bool),P,A4),C3)) ) ) ) )
       => ( ! [X3: real] :
              ( pp(aa(real,bool,ord_less_eq(real,A2),X3))
             => ( pp(aa(real,bool,ord_less_eq(real,X3),B2))
               => ? [D5: real] :
                    ( pp(aa(real,bool,ord_less(real,zero_zero(real)),D5))
                    & ! [A4: real,B3: real] :
                        ( ( pp(aa(real,bool,ord_less_eq(real,A4),X3))
                          & pp(aa(real,bool,ord_less_eq(real,X3),B3))
                          & pp(aa(real,bool,ord_less(real,aa(real,real,minus_minus(real,B3),A4)),D5)) )
                       => pp(aa(real,bool,aa(real,fun(real,bool),P,A4),B3)) ) ) ) )
         => pp(aa(real,bool,aa(real,fun(real,bool),P,A2),B2)) ) ) ) ).

% Bolzano
tff(fact_1440_zdvd__mono,axiom,
    ! [K: int,M: int,T: int] :
      ( ( K != zero_zero(int) )
     => ( pp(aa(int,bool,dvd_dvd(int,M),T))
      <=> pp(aa(int,bool,dvd_dvd(int,aa(int,int,aa(int,fun(int,int),times_times(int),K),M)),aa(int,int,aa(int,fun(int,int),times_times(int),K),T))) ) ) ).

% zdvd_mono
tff(fact_1441_ex__less__of__nat__mult,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,ord_less(A,zero_zero(A)),X))
         => ? [N2: nat] : pp(aa(A,bool,ord_less(A,Y),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),N2)),X))) ) ) ).

% ex_less_of_nat_mult
tff(fact_1442_unity__coeff__ex,axiom,
    ! [A: $tType] :
      ( ( dvd(A)
        & semiring_0(A) )
     => ! [P: fun(A,bool),L: A] :
          ( ? [X2: A] : pp(aa(A,bool,P,aa(A,A,aa(A,fun(A,A),times_times(A),L),X2)))
        <=> ? [X2: A] :
              ( pp(aa(A,bool,dvd_dvd(A,L),aa(A,A,aa(A,fun(A,A),plus_plus(A),X2),zero_zero(A))))
              & pp(aa(A,bool,P,X2)) ) ) ) ).

% unity_coeff_ex
tff(fact_1443_inf__period_I4_J,axiom,
    ! [A: $tType] :
      ( ( comm_ring(A)
        & dvd(A) )
     => ! [D: A,D4: A,T: A] :
          ( pp(aa(A,bool,dvd_dvd(A,D),D4))
         => ! [X4: A,K5: A] :
              ( ~ pp(aa(A,bool,dvd_dvd(A,D),aa(A,A,aa(A,fun(A,A),plus_plus(A),X4),T)))
            <=> ~ pp(aa(A,bool,dvd_dvd(A,D),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,minus_minus(A,X4),aa(A,A,aa(A,fun(A,A),times_times(A),K5),D4))),T))) ) ) ) ).

% inf_period(4)
tff(fact_1444_inf__period_I3_J,axiom,
    ! [A: $tType] :
      ( ( comm_ring(A)
        & dvd(A) )
     => ! [D: A,D4: A,T: A] :
          ( pp(aa(A,bool,dvd_dvd(A,D),D4))
         => ! [X4: A,K5: A] :
              ( pp(aa(A,bool,dvd_dvd(A,D),aa(A,A,aa(A,fun(A,A),plus_plus(A),X4),T)))
            <=> pp(aa(A,bool,dvd_dvd(A,D),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,minus_minus(A,X4),aa(A,A,aa(A,fun(A,A),times_times(A),K5),D4))),T))) ) ) ) ).

% inf_period(3)
tff(fact_1445_incr__mult__lemma,axiom,
    ! [D: int,P: fun(int,bool),K: int] :
      ( pp(aa(int,bool,ord_less(int,zero_zero(int)),D))
     => ( ! [X3: int] :
            ( pp(aa(int,bool,P,X3))
           => pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),X3),D))) )
       => ( pp(aa(int,bool,ord_less_eq(int,zero_zero(int)),K))
         => ! [X4: int] :
              ( pp(aa(int,bool,P,X4))
             => pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),X4),aa(int,int,aa(int,fun(int,int),times_times(int),K),D)))) ) ) ) ) ).

% incr_mult_lemma
tff(fact_1446_delete__bound__height,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( vEBT_invar_vebt(T,N)
     => pp(aa(nat,bool,ord_less_eq(nat,vEBT_T_d_e_l_e_t_e(T,X)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),aa(vEBT_VEBT,nat,vEBT_VEBT_height,T))),aa(num,nat,numeral_numeral(nat),bit0(aa(num,num,bit1,aa(num,num,bit1,bit0(bit0(bit0(one2)))))))))) ) ).

% delete_bound_height
tff(fact_1447_delete__bound__height_H,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( vEBT_invar_vebt(T,N)
     => pp(aa(nat,bool,ord_less_eq(nat,vEBT_V1232361888498592333_e_t_e(T,X)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),aa(vEBT_VEBT,nat,vEBT_VEBT_height,T)))) ) ).

% delete_bound_height'
tff(fact_1448_post__member__pre__member,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat,Y: nat] :
      ( vEBT_invar_vebt(T,N)
     => ( pp(aa(nat,bool,ord_less(nat,X),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)))
       => ( pp(aa(nat,bool,ord_less(nat,Y),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)))
         => ( pp(aa(nat,bool,vEBT_vebt_member(vEBT_vebt_insert(T,X)),Y))
           => ( pp(aa(nat,bool,vEBT_vebt_member(T),Y))
              | ( X = Y ) ) ) ) ) ) ).

% post_member_pre_member
tff(fact_1449_height__double__log__univ__size,axiom,
    ! [U: real,Deg: nat,T: vEBT_VEBT] :
      ( ( U = aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(num,real,numeral_numeral(real),bit0(one2))),Deg) )
     => ( vEBT_invar_vebt(T,Deg)
       => pp(aa(real,bool,ord_less_eq(real,aa(nat,real,semiring_1_of_nat(real),aa(vEBT_VEBT,nat,vEBT_VEBT_height,T))),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),U))))) ) ) ).

% height_double_log_univ_size
tff(fact_1450_set__vebt_H__def,axiom,
    ! [T: vEBT_VEBT] : vEBT_VEBT_set_vebt(T) = collect(nat,vEBT_vebt_member(T)) ).

% set_vebt'_def
tff(fact_1451_member__valid__both__member__options,axiom,
    ! [Tree: vEBT_VEBT,N: nat,X: nat] :
      ( vEBT_invar_vebt(Tree,N)
     => ( pp(aa(nat,bool,vEBT_vebt_member(Tree),X))
       => ( vEBT_V5719532721284313246member(Tree,X)
          | vEBT_VEBT_membermima(Tree,X) ) ) ) ).

% member_valid_both_member_options
tff(fact_1452_psubsetI,axiom,
    ! [A: $tType,A3: set(A),B4: set(A)] :
      ( pp(aa(set(A),bool,ord_less_eq(set(A),A3),B4))
     => ( ( A3 != B4 )
       => pp(aa(set(A),bool,ord_less(set(A),A3),B4)) ) ) ).

% psubsetI
tff(fact_1453_int__ops_I6_J,axiom,
    ! [A2: nat,B2: nat] :
      ( ( pp(aa(int,bool,ord_less(int,aa(nat,int,semiring_1_of_nat(int),A2)),aa(nat,int,semiring_1_of_nat(int),B2)))
       => ( aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,minus_minus(nat,A2),B2)) = zero_zero(int) ) )
      & ( ~ pp(aa(int,bool,ord_less(int,aa(nat,int,semiring_1_of_nat(int),A2)),aa(nat,int,semiring_1_of_nat(int),B2)))
       => ( aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,minus_minus(nat,A2),B2)) = aa(int,int,minus_minus(int,aa(nat,int,semiring_1_of_nat(int),A2)),aa(nat,int,semiring_1_of_nat(int),B2)) ) ) ) ).

% int_ops(6)
tff(fact_1454_buildup__nothing__in__min__max,axiom,
    ! [N: nat,X: nat] : ~ vEBT_VEBT_membermima(vEBT_vebt_buildup(N),X) ).

% buildup_nothing_in_min_max
tff(fact_1455_set__vebt__set__vebt_H__valid,axiom,
    ! [T: vEBT_VEBT,N: nat] :
      ( vEBT_invar_vebt(T,N)
     => ( vEBT_set_vebt(T) = vEBT_VEBT_set_vebt(T) ) ) ).

% set_vebt_set_vebt'_valid
tff(fact_1456_verit__eq__simplify_I8_J,axiom,
    ! [X23: num,Y2: num] :
      ( ( bit0(X23) = bit0(Y2) )
    <=> ( X23 = Y2 ) ) ).

% verit_eq_simplify(8)
tff(fact_1457_verit__eq__simplify_I9_J,axiom,
    ! [X33: num,Y32: num] :
      ( ( aa(num,num,bit1,X33) = aa(num,num,bit1,Y32) )
    <=> ( X33 = Y32 ) ) ).

% verit_eq_simplify(9)
tff(fact_1458_subset__antisym,axiom,
    ! [A: $tType,A3: set(A),B4: set(A)] :
      ( pp(aa(set(A),bool,ord_less_eq(set(A),A3),B4))
     => ( pp(aa(set(A),bool,ord_less_eq(set(A),B4),A3))
       => ( A3 = B4 ) ) ) ).

% subset_antisym
tff(fact_1459_subsetI,axiom,
    ! [A: $tType,A3: set(A),B4: set(A)] :
      ( ! [X3: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),A3))
         => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),B4)) )
     => pp(aa(set(A),bool,ord_less_eq(set(A),A3),B4)) ) ).

% subsetI
tff(fact_1460_delete__bound__size__univ_H,axiom,
    ! [T: vEBT_VEBT,N: nat,U: real,X: nat] :
      ( vEBT_invar_vebt(T,N)
     => ( ( U = aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(num,real,numeral_numeral(real),bit0(one2))),N) )
       => pp(aa(real,bool,ord_less_eq(real,aa(nat,real,semiring_1_of_nat(real),vEBT_V1232361888498592333_e_t_e(T,X))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(num,real,numeral_numeral(real),bit0(one2))),aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),U))))) ) ) ).

% delete_bound_size_univ'
tff(fact_1461_delete__bound__size__univ,axiom,
    ! [T: vEBT_VEBT,N: nat,U: real,X: nat] :
      ( vEBT_invar_vebt(T,N)
     => ( ( U = aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(num,real,numeral_numeral(real),bit0(one2))),N) )
       => pp(aa(real,bool,ord_less_eq(real,aa(nat,real,semiring_1_of_nat(real),vEBT_T_d_e_l_e_t_e(T,X))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(num,real,numeral_numeral(real),bit0(bit0(aa(num,num,bit1,aa(num,num,bit1,bit0(bit0(bit0(one2))))))))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(aa(num,num,bit1,aa(num,num,bit1,bit0(bit0(bit0(one2)))))))),aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),U)))))) ) ) ).

% delete_bound_size_univ
tff(fact_1462_succ__member,axiom,
    ! [T: vEBT_VEBT,X: nat,Y: nat] :
      ( vEBT_is_succ_in_set(vEBT_VEBT_set_vebt(T),X,Y)
    <=> ( pp(aa(nat,bool,vEBT_vebt_member(T),Y))
        & pp(aa(nat,bool,ord_less(nat,X),Y))
        & ! [Z4: nat] :
            ( ( pp(aa(nat,bool,vEBT_vebt_member(T),Z4))
              & pp(aa(nat,bool,ord_less(nat,X),Z4)) )
           => pp(aa(nat,bool,ord_less_eq(nat,Y),Z4)) ) ) ) ).

% succ_member
tff(fact_1463_pred__member,axiom,
    ! [T: vEBT_VEBT,X: nat,Y: nat] :
      ( vEBT_is_pred_in_set(vEBT_VEBT_set_vebt(T),X,Y)
    <=> ( pp(aa(nat,bool,vEBT_vebt_member(T),Y))
        & pp(aa(nat,bool,ord_less(nat,Y),X))
        & ! [Z4: nat] :
            ( ( pp(aa(nat,bool,vEBT_vebt_member(T),Z4))
              & pp(aa(nat,bool,ord_less(nat,Z4),X)) )
           => pp(aa(nat,bool,ord_less_eq(nat,Z4),Y)) ) ) ) ).

% pred_member
tff(fact_1464_verit__la__disequality,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A] :
          ( ( A2 = B2 )
          | ~ pp(aa(A,bool,ord_less_eq(A,A2),B2))
          | ~ pp(aa(A,bool,ord_less_eq(A,B2),A2)) ) ) ).

% verit_la_disequality
tff(fact_1465_verit__comp__simplify1_I2_J,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A2: A] : pp(aa(A,bool,ord_less_eq(A,A2),A2)) ) ).

% verit_comp_simplify1(2)
tff(fact_1466_double__diff,axiom,
    ! [A: $tType,A3: set(A),B4: set(A),C6: set(A)] :
      ( pp(aa(set(A),bool,ord_less_eq(set(A),A3),B4))
     => ( pp(aa(set(A),bool,ord_less_eq(set(A),B4),C6))
       => ( aa(set(A),set(A),minus_minus(set(A),B4),aa(set(A),set(A),minus_minus(set(A),C6),A3)) = A3 ) ) ) ).

% double_diff
tff(fact_1467_Diff__subset,axiom,
    ! [A: $tType,A3: set(A),B4: set(A)] : pp(aa(set(A),bool,ord_less_eq(set(A),aa(set(A),set(A),minus_minus(set(A),A3),B4)),A3)) ).

% Diff_subset
tff(fact_1468_Diff__mono,axiom,
    ! [A: $tType,A3: set(A),C6: set(A),D4: set(A),B4: set(A)] :
      ( pp(aa(set(A),bool,ord_less_eq(set(A),A3),C6))
     => ( pp(aa(set(A),bool,ord_less_eq(set(A),D4),B4))
       => pp(aa(set(A),bool,ord_less_eq(set(A),aa(set(A),set(A),minus_minus(set(A),A3),B4)),aa(set(A),set(A),minus_minus(set(A),C6),D4))) ) ) ).

% Diff_mono
tff(fact_1469_Collect__mono__iff,axiom,
    ! [A: $tType,P: fun(A,bool),Q: fun(A,bool)] :
      ( pp(aa(set(A),bool,ord_less_eq(set(A),collect(A,P)),collect(A,Q)))
    <=> ! [X2: A] :
          ( pp(aa(A,bool,P,X2))
         => pp(aa(A,bool,Q,X2)) ) ) ).

% Collect_mono_iff
tff(fact_1470_set__eq__subset,axiom,
    ! [A: $tType,A3: set(A),B4: set(A)] :
      ( ( A3 = B4 )
    <=> ( pp(aa(set(A),bool,ord_less_eq(set(A),A3),B4))
        & pp(aa(set(A),bool,ord_less_eq(set(A),B4),A3)) ) ) ).

% set_eq_subset
tff(fact_1471_subset__trans,axiom,
    ! [A: $tType,A3: set(A),B4: set(A),C6: set(A)] :
      ( pp(aa(set(A),bool,ord_less_eq(set(A),A3),B4))
     => ( pp(aa(set(A),bool,ord_less_eq(set(A),B4),C6))
       => pp(aa(set(A),bool,ord_less_eq(set(A),A3),C6)) ) ) ).

% subset_trans
tff(fact_1472_Collect__mono,axiom,
    ! [A: $tType,P: fun(A,bool),Q: fun(A,bool)] :
      ( ! [X3: A] :
          ( pp(aa(A,bool,P,X3))
         => pp(aa(A,bool,Q,X3)) )
     => pp(aa(set(A),bool,ord_less_eq(set(A),collect(A,P)),collect(A,Q))) ) ).

% Collect_mono
tff(fact_1473_subset__refl,axiom,
    ! [A: $tType,A3: set(A)] : pp(aa(set(A),bool,ord_less_eq(set(A),A3),A3)) ).

% subset_refl
tff(fact_1474_subset__iff,axiom,
    ! [A: $tType,A3: set(A),B4: set(A)] :
      ( pp(aa(set(A),bool,ord_less_eq(set(A),A3),B4))
    <=> ! [T2: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),T2),A3))
         => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),T2),B4)) ) ) ).

% subset_iff
tff(fact_1475_Set_OequalityD2,axiom,
    ! [A: $tType,A3: set(A),B4: set(A)] :
      ( ( A3 = B4 )
     => pp(aa(set(A),bool,ord_less_eq(set(A),B4),A3)) ) ).

% Set.equalityD2
tff(fact_1476_equalityD1,axiom,
    ! [A: $tType,A3: set(A),B4: set(A)] :
      ( ( A3 = B4 )
     => pp(aa(set(A),bool,ord_less_eq(set(A),A3),B4)) ) ).

% equalityD1
tff(fact_1477_subset__eq,axiom,
    ! [A: $tType,A3: set(A),B4: set(A)] :
      ( pp(aa(set(A),bool,ord_less_eq(set(A),A3),B4))
    <=> ! [X2: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X2),A3))
         => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X2),B4)) ) ) ).

% subset_eq
tff(fact_1478_equalityE,axiom,
    ! [A: $tType,A3: set(A),B4: set(A)] :
      ( ( A3 = B4 )
     => ~ ( pp(aa(set(A),bool,ord_less_eq(set(A),A3),B4))
         => ~ pp(aa(set(A),bool,ord_less_eq(set(A),B4),A3)) ) ) ).

% equalityE
tff(fact_1479_subsetD,axiom,
    ! [A: $tType,A3: set(A),B4: set(A),C2: A] :
      ( pp(aa(set(A),bool,ord_less_eq(set(A),A3),B4))
     => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),C2),A3))
       => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),C2),B4)) ) ) ).

% subsetD
tff(fact_1480_in__mono,axiom,
    ! [A: $tType,A3: set(A),B4: set(A),X: A] :
      ( pp(aa(set(A),bool,ord_less_eq(set(A),A3),B4))
     => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A3))
       => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),B4)) ) ) ).

% in_mono
tff(fact_1481_nat__int__comparison_I1_J,axiom,
    ! [A2: nat,B2: nat] :
      ( ( A2 = B2 )
    <=> ( aa(nat,int,semiring_1_of_nat(int),A2) = aa(nat,int,semiring_1_of_nat(int),B2) ) ) ).

% nat_int_comparison(1)
tff(fact_1482_int__if,axiom,
    ! [P: bool,A2: nat,B2: nat] :
      ( ( pp(P)
       => ( aa(nat,int,semiring_1_of_nat(int),if(nat,P,A2,B2)) = aa(nat,int,semiring_1_of_nat(int),A2) ) )
      & ( ~ pp(P)
       => ( aa(nat,int,semiring_1_of_nat(int),if(nat,P,A2,B2)) = aa(nat,int,semiring_1_of_nat(int),B2) ) ) ) ).

% int_if
tff(fact_1483_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)
tff(fact_1484_less__eq__set__def,axiom,
    ! [A: $tType,A3: set(A),B4: set(A)] :
      ( pp(aa(set(A),bool,ord_less_eq(set(A),A3),B4))
    <=> pp(aa(fun(A,bool),bool,ord_less_eq(fun(A,bool),aTP_Lamp_a(set(A),fun(A,bool),A3)),aTP_Lamp_a(set(A),fun(A,bool),B4))) ) ).

% less_eq_set_def
tff(fact_1485_Collect__subset,axiom,
    ! [A: $tType,A3: set(A),P: fun(A,bool)] : pp(aa(set(A),bool,ord_less_eq(set(A),collect(A,aa(fun(A,bool),fun(A,bool),aTP_Lamp_ae(set(A),fun(fun(A,bool),fun(A,bool)),A3),P))),A3)) ).

% Collect_subset
tff(fact_1486_less__set__def,axiom,
    ! [A: $tType,A3: set(A),B4: set(A)] :
      ( pp(aa(set(A),bool,ord_less(set(A),A3),B4))
    <=> pp(aa(fun(A,bool),bool,ord_less(fun(A,bool),aTP_Lamp_a(set(A),fun(A,bool),A3)),aTP_Lamp_a(set(A),fun(A,bool),B4))) ) ).

% less_set_def
tff(fact_1487_verit__comp__simplify1_I3_J,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [B6: B,A5: B] :
          ( ~ pp(aa(B,bool,ord_less_eq(B,B6),A5))
        <=> pp(aa(B,bool,ord_less(B,A5),B6)) ) ) ).

% verit_comp_simplify1(3)
tff(fact_1488_verit__sum__simplify,axiom,
    ! [A: $tType] :
      ( cancel1802427076303600483id_add(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),zero_zero(A)) = A2 ) ).

% verit_sum_simplify
tff(fact_1489_verit__eq__simplify_I10_J,axiom,
    ! [X23: num] : one2 != bit0(X23) ).

% verit_eq_simplify(10)
tff(fact_1490_verit__eq__simplify_I14_J,axiom,
    ! [X23: num,X33: num] : bit0(X23) != aa(num,num,bit1,X33) ).

% verit_eq_simplify(14)
tff(fact_1491_verit__eq__simplify_I12_J,axiom,
    ! [X33: num] : one2 != aa(num,num,bit1,X33) ).

% verit_eq_simplify(12)
tff(fact_1492_subset__iff__psubset__eq,axiom,
    ! [A: $tType,A3: set(A),B4: set(A)] :
      ( pp(aa(set(A),bool,ord_less_eq(set(A),A3),B4))
    <=> ( pp(aa(set(A),bool,ord_less(set(A),A3),B4))
        | ( A3 = B4 ) ) ) ).

% subset_iff_psubset_eq
tff(fact_1493_subset__psubset__trans,axiom,
    ! [A: $tType,A3: set(A),B4: set(A),C6: set(A)] :
      ( pp(aa(set(A),bool,ord_less_eq(set(A),A3),B4))
     => ( pp(aa(set(A),bool,ord_less(set(A),B4),C6))
       => pp(aa(set(A),bool,ord_less(set(A),A3),C6)) ) ) ).

% subset_psubset_trans
tff(fact_1494_subset__not__subset__eq,axiom,
    ! [A: $tType,A3: set(A),B4: set(A)] :
      ( pp(aa(set(A),bool,ord_less(set(A),A3),B4))
    <=> ( pp(aa(set(A),bool,ord_less_eq(set(A),A3),B4))
        & ~ pp(aa(set(A),bool,ord_less_eq(set(A),B4),A3)) ) ) ).

% subset_not_subset_eq
tff(fact_1495_psubset__subset__trans,axiom,
    ! [A: $tType,A3: set(A),B4: set(A),C6: set(A)] :
      ( pp(aa(set(A),bool,ord_less(set(A),A3),B4))
     => ( pp(aa(set(A),bool,ord_less_eq(set(A),B4),C6))
       => pp(aa(set(A),bool,ord_less(set(A),A3),C6)) ) ) ).

% psubset_subset_trans
tff(fact_1496_psubset__imp__subset,axiom,
    ! [A: $tType,A3: set(A),B4: set(A)] :
      ( pp(aa(set(A),bool,ord_less(set(A),A3),B4))
     => pp(aa(set(A),bool,ord_less_eq(set(A),A3),B4)) ) ).

% psubset_imp_subset
tff(fact_1497_psubset__eq,axiom,
    ! [A: $tType,A3: set(A),B4: set(A)] :
      ( pp(aa(set(A),bool,ord_less(set(A),A3),B4))
    <=> ( pp(aa(set(A),bool,ord_less_eq(set(A),A3),B4))
        & ( A3 != B4 ) ) ) ).

% psubset_eq
tff(fact_1498_psubsetE,axiom,
    ! [A: $tType,A3: set(A),B4: set(A)] :
      ( pp(aa(set(A),bool,ord_less(set(A),A3),B4))
     => ~ ( pp(aa(set(A),bool,ord_less_eq(set(A),A3),B4))
         => pp(aa(set(A),bool,ord_less_eq(set(A),B4),A3)) ) ) ).

% psubsetE
tff(fact_1499_int__ops_I3_J,axiom,
    ! [N: num] : aa(nat,int,semiring_1_of_nat(int),aa(num,nat,numeral_numeral(nat),N)) = aa(num,int,numeral_numeral(int),N) ).

% int_ops(3)
tff(fact_1500_int__ops_I1_J,axiom,
    aa(nat,int,semiring_1_of_nat(int),zero_zero(nat)) = zero_zero(int) ).

% int_ops(1)
tff(fact_1501_nat__int__comparison_I2_J,axiom,
    ! [A2: nat,B2: nat] :
      ( pp(aa(nat,bool,ord_less(nat,A2),B2))
    <=> pp(aa(int,bool,ord_less(int,aa(nat,int,semiring_1_of_nat(int),A2)),aa(nat,int,semiring_1_of_nat(int),B2))) ) ).

% nat_int_comparison(2)
tff(fact_1502_nat__int__comparison_I3_J,axiom,
    ! [A2: nat,B2: nat] :
      ( pp(aa(nat,bool,ord_less_eq(nat,A2),B2))
    <=> pp(aa(int,bool,ord_less_eq(int,aa(nat,int,semiring_1_of_nat(int),A2)),aa(nat,int,semiring_1_of_nat(int),B2))) ) ).

% nat_int_comparison(3)
tff(fact_1503_int__ops_I2_J,axiom,
    aa(nat,int,semiring_1_of_nat(int),one_one(nat)) = one_one(int) ).

% int_ops(2)
tff(fact_1504_int__plus,axiom,
    ! [N: nat,M: nat] : aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),M)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(nat,int,semiring_1_of_nat(int),N)),aa(nat,int,semiring_1_of_nat(int),M)) ).

% int_plus
tff(fact_1505_int__ops_I5_J,axiom,
    ! [A2: nat,B2: nat] : aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A2),B2)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(nat,int,semiring_1_of_nat(int),A2)),aa(nat,int,semiring_1_of_nat(int),B2)) ).

% int_ops(5)
tff(fact_1506_int__ops_I7_J,axiom,
    ! [A2: nat,B2: nat] : aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A2),B2)) = aa(int,int,aa(int,fun(int,int),times_times(int),aa(nat,int,semiring_1_of_nat(int),A2)),aa(nat,int,semiring_1_of_nat(int),B2)) ).

% int_ops(7)
tff(fact_1507_nat__less__as__int,axiom,
    ! [X4: nat,Xa: nat] :
      ( pp(aa(nat,bool,ord_less(nat,X4),Xa))
    <=> pp(aa(int,bool,ord_less(int,aa(nat,int,semiring_1_of_nat(int),X4)),aa(nat,int,semiring_1_of_nat(int),Xa))) ) ).

% nat_less_as_int
tff(fact_1508_nat__leq__as__int,axiom,
    ! [X4: nat,Xa: nat] :
      ( pp(aa(nat,bool,ord_less_eq(nat,X4),Xa))
    <=> pp(aa(int,bool,ord_less_eq(int,aa(nat,int,semiring_1_of_nat(int),X4)),aa(nat,int,semiring_1_of_nat(int),Xa))) ) ).

% nat_leq_as_int
tff(fact_1509_int__ops_I4_J,axiom,
    ! [A2: nat] : aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,suc,A2)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(nat,int,semiring_1_of_nat(int),A2)),one_one(int)) ).

% int_ops(4)
tff(fact_1510_int__Suc,axiom,
    ! [N: nat] : aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,suc,N)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(nat,int,semiring_1_of_nat(int),N)),one_one(int)) ).

% int_Suc
tff(fact_1511_log__pow__cancel,axiom,
    ! [A2: real,B2: nat] :
      ( pp(aa(real,bool,ord_less(real,zero_zero(real)),A2))
     => ( ( A2 != one_one(real) )
       => ( aa(real,real,log(A2),aa(nat,real,aa(real,fun(nat,real),power_power(real),A2),B2)) = aa(nat,real,semiring_1_of_nat(real),B2) ) ) ) ).

% log_pow_cancel
tff(fact_1512_zero__le__log__cancel__iff,axiom,
    ! [A2: real,X: real] :
      ( pp(aa(real,bool,ord_less(real,one_one(real)),A2))
     => ( pp(aa(real,bool,ord_less(real,zero_zero(real)),X))
       => ( pp(aa(real,bool,ord_less_eq(real,zero_zero(real)),aa(real,real,log(A2),X)))
        <=> pp(aa(real,bool,ord_less_eq(real,one_one(real)),X)) ) ) ) ).

% zero_le_log_cancel_iff
tff(fact_1513_log__le__zero__cancel__iff,axiom,
    ! [A2: real,X: real] :
      ( pp(aa(real,bool,ord_less(real,one_one(real)),A2))
     => ( pp(aa(real,bool,ord_less(real,zero_zero(real)),X))
       => ( pp(aa(real,bool,ord_less_eq(real,aa(real,real,log(A2),X)),zero_zero(real)))
        <=> pp(aa(real,bool,ord_less_eq(real,X),one_one(real))) ) ) ) ).

% log_le_zero_cancel_iff
tff(fact_1514_one__le__log__cancel__iff,axiom,
    ! [A2: real,X: real] :
      ( pp(aa(real,bool,ord_less(real,one_one(real)),A2))
     => ( pp(aa(real,bool,ord_less(real,zero_zero(real)),X))
       => ( pp(aa(real,bool,ord_less_eq(real,one_one(real)),aa(real,real,log(A2),X)))
        <=> pp(aa(real,bool,ord_less_eq(real,A2),X)) ) ) ) ).

% one_le_log_cancel_iff
tff(fact_1515_log__le__one__cancel__iff,axiom,
    ! [A2: real,X: real] :
      ( pp(aa(real,bool,ord_less(real,one_one(real)),A2))
     => ( pp(aa(real,bool,ord_less(real,zero_zero(real)),X))
       => ( pp(aa(real,bool,ord_less_eq(real,aa(real,real,log(A2),X)),one_one(real)))
        <=> pp(aa(real,bool,ord_less_eq(real,X),A2)) ) ) ) ).

% log_le_one_cancel_iff
tff(fact_1516_log__le__cancel__iff,axiom,
    ! [A2: real,X: real,Y: real] :
      ( pp(aa(real,bool,ord_less(real,one_one(real)),A2))
     => ( pp(aa(real,bool,ord_less(real,zero_zero(real)),X))
       => ( pp(aa(real,bool,ord_less(real,zero_zero(real)),Y))
         => ( pp(aa(real,bool,ord_less_eq(real,aa(real,real,log(A2),X)),aa(real,real,log(A2),Y)))
          <=> pp(aa(real,bool,ord_less_eq(real,X),Y)) ) ) ) ) ).

% log_le_cancel_iff
tff(fact_1517_log2__of__power__le,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,ord_less_eq(nat,M),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)))
     => ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),M))
       => pp(aa(real,bool,ord_less_eq(real,aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(nat,real,semiring_1_of_nat(real),M))),aa(nat,real,semiring_1_of_nat(real),N))) ) ) ).

% log2_of_power_le
tff(fact_1518_log__eq__one,axiom,
    ! [A2: real] :
      ( pp(aa(real,bool,ord_less(real,zero_zero(real)),A2))
     => ( ( A2 != one_one(real) )
       => ( aa(real,real,log(A2),A2) = one_one(real) ) ) ) ).

% log_eq_one
tff(fact_1519_log__one,axiom,
    ! [A2: real] : aa(real,real,log(A2),one_one(real)) = zero_zero(real) ).

% log_one
tff(fact_1520_zero__less__log__cancel__iff,axiom,
    ! [A2: real,X: real] :
      ( pp(aa(real,bool,ord_less(real,one_one(real)),A2))
     => ( pp(aa(real,bool,ord_less(real,zero_zero(real)),X))
       => ( pp(aa(real,bool,ord_less(real,zero_zero(real)),aa(real,real,log(A2),X)))
        <=> pp(aa(real,bool,ord_less(real,one_one(real)),X)) ) ) ) ).

% zero_less_log_cancel_iff
tff(fact_1521_log__less__zero__cancel__iff,axiom,
    ! [A2: real,X: real] :
      ( pp(aa(real,bool,ord_less(real,one_one(real)),A2))
     => ( pp(aa(real,bool,ord_less(real,zero_zero(real)),X))
       => ( pp(aa(real,bool,ord_less(real,aa(real,real,log(A2),X)),zero_zero(real)))
        <=> pp(aa(real,bool,ord_less(real,X),one_one(real))) ) ) ) ).

% log_less_zero_cancel_iff
tff(fact_1522_one__less__log__cancel__iff,axiom,
    ! [A2: real,X: real] :
      ( pp(aa(real,bool,ord_less(real,one_one(real)),A2))
     => ( pp(aa(real,bool,ord_less(real,zero_zero(real)),X))
       => ( pp(aa(real,bool,ord_less(real,one_one(real)),aa(real,real,log(A2),X)))
        <=> pp(aa(real,bool,ord_less(real,A2),X)) ) ) ) ).

% one_less_log_cancel_iff
tff(fact_1523_log__less__one__cancel__iff,axiom,
    ! [A2: real,X: real] :
      ( pp(aa(real,bool,ord_less(real,one_one(real)),A2))
     => ( pp(aa(real,bool,ord_less(real,zero_zero(real)),X))
       => ( pp(aa(real,bool,ord_less(real,aa(real,real,log(A2),X)),one_one(real)))
        <=> pp(aa(real,bool,ord_less(real,X),A2)) ) ) ) ).

% log_less_one_cancel_iff
tff(fact_1524_log__less__cancel__iff,axiom,
    ! [A2: real,X: real,Y: real] :
      ( pp(aa(real,bool,ord_less(real,one_one(real)),A2))
     => ( pp(aa(real,bool,ord_less(real,zero_zero(real)),X))
       => ( pp(aa(real,bool,ord_less(real,zero_zero(real)),Y))
         => ( pp(aa(real,bool,ord_less(real,aa(real,real,log(A2),X)),aa(real,real,log(A2),Y)))
          <=> pp(aa(real,bool,ord_less(real,X),Y)) ) ) ) ) ).

% log_less_cancel_iff
tff(fact_1525_minus__set__def,axiom,
    ! [A: $tType,A3: set(A),B4: set(A)] : aa(set(A),set(A),minus_minus(set(A),A3),B4) = collect(A,aa(fun(A,bool),fun(A,bool),minus_minus(fun(A,bool),aTP_Lamp_a(set(A),fun(A,bool),A3)),aTP_Lamp_a(set(A),fun(A,bool),B4))) ).

% minus_set_def
tff(fact_1526_set__diff__eq,axiom,
    ! [A: $tType,A3: set(A),B4: set(A)] : aa(set(A),set(A),minus_minus(set(A),A3),B4) = collect(A,aa(set(A),fun(A,bool),aTP_Lamp_af(set(A),fun(set(A),fun(A,bool)),A3),B4)) ).

% set_diff_eq
tff(fact_1527_log__base__change,axiom,
    ! [A2: real,B2: real,X: real] :
      ( pp(aa(real,bool,ord_less(real,zero_zero(real)),A2))
     => ( ( A2 != one_one(real) )
       => ( aa(real,real,log(B2),X) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,log(A2),X)),aa(real,real,log(A2),B2)) ) ) ) ).

% log_base_change
tff(fact_1528_less__log__of__power,axiom,
    ! [B2: real,N: nat,M: real] :
      ( pp(aa(real,bool,ord_less(real,aa(nat,real,aa(real,fun(nat,real),power_power(real),B2),N)),M))
     => ( pp(aa(real,bool,ord_less(real,one_one(real)),B2))
       => pp(aa(real,bool,ord_less(real,aa(nat,real,semiring_1_of_nat(real),N)),aa(real,real,log(B2),M))) ) ) ).

% less_log_of_power
tff(fact_1529_log__of__power__eq,axiom,
    ! [M: nat,B2: real,N: nat] :
      ( ( aa(nat,real,semiring_1_of_nat(real),M) = aa(nat,real,aa(real,fun(nat,real),power_power(real),B2),N) )
     => ( pp(aa(real,bool,ord_less(real,one_one(real)),B2))
       => ( aa(nat,real,semiring_1_of_nat(real),N) = aa(real,real,log(B2),aa(nat,real,semiring_1_of_nat(real),M)) ) ) ) ).

% log_of_power_eq
tff(fact_1530_log__mult,axiom,
    ! [A2: real,X: real,Y: real] :
      ( pp(aa(real,bool,ord_less(real,zero_zero(real)),A2))
     => ( ( A2 != one_one(real) )
       => ( pp(aa(real,bool,ord_less(real,zero_zero(real)),X))
         => ( pp(aa(real,bool,ord_less(real,zero_zero(real)),Y))
           => ( aa(real,real,log(A2),aa(real,real,aa(real,fun(real,real),times_times(real),X),Y)) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,log(A2),X)),aa(real,real,log(A2),Y)) ) ) ) ) ) ).

% log_mult
tff(fact_1531_le__log__of__power,axiom,
    ! [B2: real,N: nat,M: real] :
      ( pp(aa(real,bool,ord_less_eq(real,aa(nat,real,aa(real,fun(nat,real),power_power(real),B2),N)),M))
     => ( pp(aa(real,bool,ord_less(real,one_one(real)),B2))
       => pp(aa(real,bool,ord_less_eq(real,aa(nat,real,semiring_1_of_nat(real),N)),aa(real,real,log(B2),M))) ) ) ).

% le_log_of_power
tff(fact_1532_log__divide,axiom,
    ! [A2: real,X: real,Y: real] :
      ( pp(aa(real,bool,ord_less(real,zero_zero(real)),A2))
     => ( ( A2 != one_one(real) )
       => ( pp(aa(real,bool,ord_less(real,zero_zero(real)),X))
         => ( pp(aa(real,bool,ord_less(real,zero_zero(real)),Y))
           => ( aa(real,real,log(A2),aa(real,real,aa(real,fun(real,real),divide_divide(real),X),Y)) = aa(real,real,minus_minus(real,aa(real,real,log(A2),X)),aa(real,real,log(A2),Y)) ) ) ) ) ) ).

% log_divide
tff(fact_1533_log__base__pow,axiom,
    ! [A2: real,N: nat,X: real] :
      ( pp(aa(real,bool,ord_less(real,zero_zero(real)),A2))
     => ( aa(real,real,log(aa(nat,real,aa(real,fun(nat,real),power_power(real),A2),N)),X) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,log(A2),X)),aa(nat,real,semiring_1_of_nat(real),N)) ) ) ).

% log_base_pow
tff(fact_1534_log__nat__power,axiom,
    ! [X: real,B2: real,N: nat] :
      ( pp(aa(real,bool,ord_less(real,zero_zero(real)),X))
     => ( aa(real,real,log(B2),aa(nat,real,aa(real,fun(nat,real),power_power(real),X),N)) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),N)),aa(real,real,log(B2),X)) ) ) ).

% log_nat_power
tff(fact_1535_log2__of__power__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( M = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N) )
     => ( aa(nat,real,semiring_1_of_nat(real),N) = aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(nat,real,semiring_1_of_nat(real),M)) ) ) ).

% log2_of_power_eq
tff(fact_1536_log__of__power__less,axiom,
    ! [M: nat,B2: real,N: nat] :
      ( pp(aa(real,bool,ord_less(real,aa(nat,real,semiring_1_of_nat(real),M)),aa(nat,real,aa(real,fun(nat,real),power_power(real),B2),N)))
     => ( pp(aa(real,bool,ord_less(real,one_one(real)),B2))
       => ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),M))
         => pp(aa(real,bool,ord_less(real,aa(real,real,log(B2),aa(nat,real,semiring_1_of_nat(real),M))),aa(nat,real,semiring_1_of_nat(real),N))) ) ) ) ).

% log_of_power_less
tff(fact_1537_log__of__power__le,axiom,
    ! [M: nat,B2: real,N: nat] :
      ( pp(aa(real,bool,ord_less_eq(real,aa(nat,real,semiring_1_of_nat(real),M)),aa(nat,real,aa(real,fun(nat,real),power_power(real),B2),N)))
     => ( pp(aa(real,bool,ord_less(real,one_one(real)),B2))
       => ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),M))
         => pp(aa(real,bool,ord_less_eq(real,aa(real,real,log(B2),aa(nat,real,semiring_1_of_nat(real),M))),aa(nat,real,semiring_1_of_nat(real),N))) ) ) ) ).

% log_of_power_le
tff(fact_1538_less__log2__of__power,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa(nat,bool,ord_less(nat,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)),M))
     => pp(aa(real,bool,ord_less(real,aa(nat,real,semiring_1_of_nat(real),N)),aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(nat,real,semiring_1_of_nat(real),M)))) ) ).

% less_log2_of_power
tff(fact_1539_le__log2__of__power,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa(nat,bool,ord_less_eq(nat,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)),M))
     => pp(aa(real,bool,ord_less_eq(real,aa(nat,real,semiring_1_of_nat(real),N)),aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(nat,real,semiring_1_of_nat(real),M)))) ) ).

% le_log2_of_power
tff(fact_1540_log2__of__power__less,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,ord_less(nat,M),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)))
     => ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),M))
       => pp(aa(real,bool,ord_less(real,aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(nat,real,semiring_1_of_nat(real),M))),aa(nat,real,semiring_1_of_nat(real),N))) ) ) ).

% log2_of_power_less
tff(fact_1541_arcosh__1,axiom,
    ! [A: $tType] :
      ( ln(A)
     => ( aa(A,A,arcosh(A),one_one(A)) = zero_zero(A) ) ) ).

% arcosh_1
tff(fact_1542_inrange,axiom,
    ! [T: vEBT_VEBT,N: nat] :
      ( vEBT_invar_vebt(T,N)
     => pp(aa(set(nat),bool,ord_less_eq(set(nat),vEBT_VEBT_set_vebt(T)),set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)),one_one(nat))))) ) ).

% inrange
tff(fact_1543_pos__mult__pos__ge,axiom,
    ! [X: int,N: int] :
      ( pp(aa(int,bool,ord_less(int,zero_zero(int)),X))
     => ( pp(aa(int,bool,ord_less_eq(int,zero_zero(int)),N))
       => pp(aa(int,bool,ord_less_eq(int,aa(int,int,aa(int,fun(int,int),times_times(int),N),one_one(int))),aa(int,int,aa(int,fun(int,int),times_times(int),N),X))) ) ) ).

% pos_mult_pos_ge
tff(fact_1544_heigt__uplog__rel,axiom,
    ! [T: vEBT_VEBT,N: nat] :
      ( vEBT_invar_vebt(T,N)
     => ( aa(nat,int,semiring_1_of_nat(int),aa(vEBT_VEBT,nat,vEBT_VEBT_height,T)) = archimedean_ceiling(real,aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(nat,real,semiring_1_of_nat(real),N))) ) ) ).

% heigt_uplog_rel
tff(fact_1545_lemma__termdiff3,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [H: A,Z: A,K6: real,N: nat] :
          ( ( H != zero_zero(A) )
         => ( pp(aa(real,bool,ord_less_eq(real,real_V7770717601297561774m_norm(A,Z)),K6))
           => ( pp(aa(real,bool,ord_less_eq(real,real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Z),H))),K6))
             => pp(aa(real,bool,ord_less_eq(real,real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,minus_minus(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Z),H)),N)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Z),N))),H)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),N)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Z),aa(nat,nat,minus_minus(nat,N),aa(nat,nat,suc,zero_zero(nat)))))))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),N)),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,minus_minus(nat,N),aa(nat,nat,suc,zero_zero(nat)))))),aa(nat,real,aa(real,fun(nat,real),power_power(real),K6),aa(nat,nat,minus_minus(nat,N),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),real_V7770717601297561774m_norm(A,H)))) ) ) ) ) ).

% lemma_termdiff3
tff(fact_1546_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)
tff(fact_1547_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)
tff(fact_1548_valid__insert__both__member__options__add,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( vEBT_invar_vebt(T,N)
     => ( pp(aa(nat,bool,ord_less(nat,X),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)))
       => pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,vEBT_vebt_insert(T,X)),X)) ) ) ).

% valid_insert_both_member_options_add
tff(fact_1549_both__member__options__equiv__member,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( vEBT_invar_vebt(T,N)
     => ( pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,T),X))
      <=> pp(aa(nat,bool,vEBT_vebt_member(T),X)) ) ) ).

% both_member_options_equiv_member
tff(fact_1550_valid__member__both__member__options,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( vEBT_invar_vebt(T,N)
     => ( pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,T),X))
       => pp(aa(nat,bool,vEBT_vebt_member(T),X)) ) ) ).

% valid_member_both_member_options
tff(fact_1551_both__member__options__def,axiom,
    ! [T: vEBT_VEBT,X: nat] :
      ( pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,T),X))
    <=> ( vEBT_V5719532721284313246member(T,X)
        | vEBT_VEBT_membermima(T,X) ) ) ).

% both_member_options_def
tff(fact_1552_valid__insert__both__member__options__pres,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat,Y: nat] :
      ( vEBT_invar_vebt(T,N)
     => ( pp(aa(nat,bool,ord_less(nat,X),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)))
       => ( pp(aa(nat,bool,ord_less(nat,Y),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)))
         => ( pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,T),X))
           => pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,vEBT_vebt_insert(T,Y)),X)) ) ) ) ) ).

% valid_insert_both_member_options_pres
tff(fact_1553_ceiling__zero,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ( archimedean_ceiling(A,zero_zero(A)) = zero_zero(int) ) ) ).

% ceiling_zero
tff(fact_1554_ceiling__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V: num] : archimedean_ceiling(A,aa(num,A,numeral_numeral(A),V)) = aa(num,int,numeral_numeral(int),V) ) ).

% ceiling_numeral
tff(fact_1555_ceiling__one,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ( archimedean_ceiling(A,one_one(A)) = one_one(int) ) ) ).

% ceiling_one
tff(fact_1556_ceiling__of__nat,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [N: nat] : archimedean_ceiling(A,aa(nat,A,semiring_1_of_nat(A),N)) = aa(nat,int,semiring_1_of_nat(int),N) ) ).

% ceiling_of_nat
tff(fact_1557_ceiling__le__zero,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( pp(aa(int,bool,ord_less_eq(int,archimedean_ceiling(A,X)),zero_zero(int)))
        <=> pp(aa(A,bool,ord_less_eq(A,X),zero_zero(A))) ) ) ).

% ceiling_le_zero
tff(fact_1558_ceiling__le__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,V: num] :
          ( pp(aa(int,bool,ord_less_eq(int,archimedean_ceiling(A,X)),aa(num,int,numeral_numeral(int),V)))
        <=> pp(aa(A,bool,ord_less_eq(A,X),aa(num,A,numeral_numeral(A),V))) ) ) ).

% ceiling_le_numeral
tff(fact_1559_zero__less__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( pp(aa(int,bool,ord_less(int,zero_zero(int)),archimedean_ceiling(A,X)))
        <=> pp(aa(A,bool,ord_less(A,zero_zero(A)),X)) ) ) ).

% zero_less_ceiling
tff(fact_1560_numeral__less__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V: num,X: A] :
          ( pp(aa(int,bool,ord_less(int,aa(num,int,numeral_numeral(int),V)),archimedean_ceiling(A,X)))
        <=> pp(aa(A,bool,ord_less(A,aa(num,A,numeral_numeral(A),V)),X)) ) ) ).

% numeral_less_ceiling
tff(fact_1561_ceiling__less__one,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( pp(aa(int,bool,ord_less(int,archimedean_ceiling(A,X)),one_one(int)))
        <=> pp(aa(A,bool,ord_less_eq(A,X),zero_zero(A))) ) ) ).

% ceiling_less_one
tff(fact_1562_one__le__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( pp(aa(int,bool,ord_less_eq(int,one_one(int)),archimedean_ceiling(A,X)))
        <=> pp(aa(A,bool,ord_less(A,zero_zero(A)),X)) ) ) ).

% one_le_ceiling
tff(fact_1563_ceiling__add__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,V: num] : archimedean_ceiling(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),aa(num,A,numeral_numeral(A),V))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),archimedean_ceiling(A,X)),aa(num,int,numeral_numeral(int),V)) ) ).

% ceiling_add_numeral
tff(fact_1564_ceiling__le__one,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( pp(aa(int,bool,ord_less_eq(int,archimedean_ceiling(A,X)),one_one(int)))
        <=> pp(aa(A,bool,ord_less_eq(A,X),one_one(A))) ) ) ).

% ceiling_le_one
tff(fact_1565_one__less__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( pp(aa(int,bool,ord_less(int,one_one(int)),archimedean_ceiling(A,X)))
        <=> pp(aa(A,bool,ord_less(A,one_one(A)),X)) ) ) ).

% one_less_ceiling
tff(fact_1566_ceiling__add__one,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] : archimedean_ceiling(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),one_one(A))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),archimedean_ceiling(A,X)),one_one(int)) ) ).

% ceiling_add_one
tff(fact_1567_ceiling__diff__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,V: num] : archimedean_ceiling(A,aa(A,A,minus_minus(A,X),aa(num,A,numeral_numeral(A),V))) = aa(int,int,minus_minus(int,archimedean_ceiling(A,X)),aa(num,int,numeral_numeral(int),V)) ) ).

% ceiling_diff_numeral
tff(fact_1568_ceiling__numeral__power,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: num,N: nat] : archimedean_ceiling(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),X)),N)) = aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),X)),N) ) ).

% ceiling_numeral_power
tff(fact_1569_ceiling__diff__one,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] : archimedean_ceiling(A,aa(A,A,minus_minus(A,X),one_one(A))) = aa(int,int,minus_minus(int,archimedean_ceiling(A,X)),one_one(int)) ) ).

% ceiling_diff_one
tff(fact_1570_ceiling__less__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,V: num] :
          ( pp(aa(int,bool,ord_less(int,archimedean_ceiling(A,X)),aa(num,int,numeral_numeral(int),V)))
        <=> pp(aa(A,bool,ord_less_eq(A,X),aa(A,A,minus_minus(A,aa(num,A,numeral_numeral(A),V)),one_one(A)))) ) ) ).

% ceiling_less_numeral
tff(fact_1571_numeral__le__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V: num,X: A] :
          ( pp(aa(int,bool,ord_less_eq(int,aa(num,int,numeral_numeral(int),V)),archimedean_ceiling(A,X)))
        <=> pp(aa(A,bool,ord_less(A,aa(A,A,minus_minus(A,aa(num,A,numeral_numeral(A),V)),one_one(A))),X)) ) ) ).

% numeral_le_ceiling
tff(fact_1572_set__vebt__def,axiom,
    ! [T: vEBT_VEBT] : vEBT_set_vebt(T) = collect(nat,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,T)) ).

% set_vebt_def
tff(fact_1573_ceiling__mono,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Y: A,X: A] :
          ( pp(aa(A,bool,ord_less_eq(A,Y),X))
         => pp(aa(int,bool,ord_less_eq(int,archimedean_ceiling(A,Y)),archimedean_ceiling(A,X))) ) ) ).

% ceiling_mono
tff(fact_1574_all__nat__less,axiom,
    ! [N: nat,P: fun(nat,bool)] :
      ( ! [M6: nat] :
          ( pp(aa(nat,bool,ord_less_eq(nat,M6),N))
         => pp(aa(nat,bool,P,M6)) )
    <=> ! [X2: nat] :
          ( pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),X2),set_or1337092689740270186AtMost(nat,zero_zero(nat),N)))
         => pp(aa(nat,bool,P,X2)) ) ) ).

% all_nat_less
tff(fact_1575_ex__nat__less,axiom,
    ! [N: nat,P: fun(nat,bool)] :
      ( ? [M6: nat] :
          ( pp(aa(nat,bool,ord_less_eq(nat,M6),N))
          & pp(aa(nat,bool,P,M6)) )
    <=> ? [X2: nat] :
          ( pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),X2),set_or1337092689740270186AtMost(nat,zero_zero(nat),N)))
          & pp(aa(nat,bool,P,X2)) ) ) ).

% ex_nat_less
tff(fact_1576_ceiling__add__le,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,Y: A] : pp(aa(int,bool,ord_less_eq(int,archimedean_ceiling(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y))),aa(int,int,aa(int,fun(int,int),plus_plus(int),archimedean_ceiling(A,X)),archimedean_ceiling(A,Y)))) ) ).

% ceiling_add_le
tff(fact_1577_mult__ceiling__le,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),A2))
         => ( pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),B2))
           => pp(aa(int,bool,ord_less_eq(int,archimedean_ceiling(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2))),aa(int,int,aa(int,fun(int,int),times_times(int),archimedean_ceiling(A,A2)),archimedean_ceiling(A,B2)))) ) ) ) ).

% mult_ceiling_le
tff(fact_1578_lemma__NBseq__def2,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [X6: fun(A,B)] :
          ( ? [K7: real] :
              ( pp(aa(real,bool,ord_less(real,zero_zero(real)),K7))
              & ! [N5: A] : pp(aa(real,bool,ord_less_eq(real,real_V7770717601297561774m_norm(B,aa(A,B,X6,N5))),K7)) )
        <=> ? [N8: nat] :
            ! [N5: A] : pp(aa(real,bool,ord_less(real,real_V7770717601297561774m_norm(B,aa(A,B,X6,N5))),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,N8)))) ) ) ).

% lemma_NBseq_def2
tff(fact_1579_lemma__NBseq__def,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [X6: fun(A,B)] :
          ( ? [K7: real] :
              ( pp(aa(real,bool,ord_less(real,zero_zero(real)),K7))
              & ! [N5: A] : pp(aa(real,bool,ord_less_eq(real,real_V7770717601297561774m_norm(B,aa(A,B,X6,N5))),K7)) )
        <=> ? [N8: nat] :
            ! [N5: A] : pp(aa(real,bool,ord_less_eq(real,real_V7770717601297561774m_norm(B,aa(A,B,X6,N5))),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,N8)))) ) ) ).

% lemma_NBseq_def
tff(fact_1580_p2__eq__1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: nat] :
          ( ( aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),N) = one_one(word(A)) )
        <=> ( N = zero_zero(nat) ) ) ) ).

% p2_eq_1
tff(fact_1581_ceiling__log__nat__eq__if,axiom,
    ! [B2: nat,N: nat,K: nat] :
      ( pp(aa(nat,bool,ord_less(nat,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),N)),K))
     => ( pp(aa(nat,bool,ord_less_eq(nat,K),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat)))))
       => ( pp(aa(nat,bool,ord_less_eq(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),B2))
         => ( archimedean_ceiling(real,aa(real,real,log(aa(nat,real,semiring_1_of_nat(real),B2)),aa(nat,real,semiring_1_of_nat(real),K))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(nat,int,semiring_1_of_nat(int),N)),one_one(int)) ) ) ) ) ).

% ceiling_log_nat_eq_if
tff(fact_1582_ceiling__log2__div2,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,ord_less_eq(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N))
     => ( archimedean_ceiling(real,aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(nat,real,semiring_1_of_nat(real),N))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),archimedean_ceiling(real,aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,minus_minus(nat,N),one_one(nat))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),one_one(nat)))))),one_one(int)) ) ) ).

% ceiling_log2_div2
tff(fact_1583_ceiling__log__nat__eq__powr__iff,axiom,
    ! [B2: nat,K: nat,N: nat] :
      ( pp(aa(nat,bool,ord_less_eq(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),B2))
     => ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),K))
       => ( ( archimedean_ceiling(real,aa(real,real,log(aa(nat,real,semiring_1_of_nat(real),B2)),aa(nat,real,semiring_1_of_nat(real),K))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(nat,int,semiring_1_of_nat(int),N)),one_one(int)) )
        <=> ( pp(aa(nat,bool,ord_less(nat,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),N)),K))
            & pp(aa(nat,bool,ord_less_eq(nat,K),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat))))) ) ) ) ) ).

% ceiling_log_nat_eq_powr_iff
tff(fact_1584_word__less__two__pow__divD,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),N: nat,M: nat] :
          ( pp(aa(word(A),bool,ord_less(word(A),X),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),N)),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),M))))
         => ( pp(aa(nat,bool,ord_less_eq(nat,M),N))
            & pp(aa(word(A),bool,ord_less(word(A),X),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),aa(nat,nat,minus_minus(nat,N),M)))) ) ) ) ).

% word_less_two_pow_divD
tff(fact_1585_word__unat__power,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: nat] : aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),N) = aa(nat,word(A),semiring_1_of_nat(word(A)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)) ) ).

% word_unat_power
tff(fact_1586_less__1__helper,axiom,
    ! [N: int,M: int] :
      ( pp(aa(int,bool,ord_less_eq(int,N),M))
     => pp(aa(int,bool,ord_less(int,aa(int,int,minus_minus(int,N),one_one(int))),M)) ) ).

% less_1_helper
tff(fact_1587_norm__divide__numeral,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [A2: A,W: num] : real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(num,A,numeral_numeral(A),W))) = aa(real,real,aa(real,fun(real,real),divide_divide(real),real_V7770717601297561774m_norm(A,A2)),aa(num,real,numeral_numeral(real),W)) ) ).

% norm_divide_numeral
tff(fact_1588_norm__mult__numeral2,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [A2: A,W: num] : real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(num,A,numeral_numeral(A),W))) = aa(real,real,aa(real,fun(real,real),times_times(real),real_V7770717601297561774m_norm(A,A2)),aa(num,real,numeral_numeral(real),W)) ) ).

% norm_mult_numeral2
tff(fact_1589_norm__mult__numeral1,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [W: num,A2: A] : real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W)),A2)) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),W)),real_V7770717601297561774m_norm(A,A2)) ) ).

% norm_mult_numeral1
tff(fact_1590_norm__le__zero__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X: A] :
          ( pp(aa(real,bool,ord_less_eq(real,real_V7770717601297561774m_norm(A,X)),zero_zero(real)))
        <=> ( X = zero_zero(A) ) ) ) ).

% norm_le_zero_iff
tff(fact_1591_zero__less__norm__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X: A] :
          ( pp(aa(real,bool,ord_less(real,zero_zero(real)),real_V7770717601297561774m_norm(A,X)))
        <=> ( X != zero_zero(A) ) ) ) ).

% zero_less_norm_iff
tff(fact_1592_norm__of__nat,axiom,
    ! [A: $tType] :
      ( real_V2822296259951069270ebra_1(A)
     => ! [N: nat] : real_V7770717601297561774m_norm(A,aa(nat,A,semiring_1_of_nat(A),N)) = aa(nat,real,semiring_1_of_nat(real),N) ) ).

% norm_of_nat
tff(fact_1593_norm__numeral,axiom,
    ! [A: $tType] :
      ( real_V2822296259951069270ebra_1(A)
     => ! [W: num] : real_V7770717601297561774m_norm(A,aa(num,A,numeral_numeral(A),W)) = aa(num,real,numeral_numeral(real),W) ) ).

% norm_numeral
tff(fact_1594_norm__one,axiom,
    ! [A: $tType] :
      ( real_V2822296259951069270ebra_1(A)
     => ( real_V7770717601297561774m_norm(A,one_one(A)) = one_one(real) ) ) ).

% norm_one
tff(fact_1595_div__of__0__id,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),zero_zero(word(A))),N) = zero_zero(word(A)) ) ).

% div_of_0_id
tff(fact_1596_norm__eq__zero,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X: A] :
          ( ( real_V7770717601297561774m_norm(A,X) = zero_zero(real) )
        <=> ( X = zero_zero(A) ) ) ) ).

% norm_eq_zero
tff(fact_1597_norm__zero,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ( real_V7770717601297561774m_norm(A,zero_zero(A)) = zero_zero(real) ) ) ).

% norm_zero
tff(fact_1598_div__to__mult__word__lt,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A),Z: word(A)] :
          ( pp(aa(word(A),bool,ord_less_eq(word(A),X),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),Y),Z)))
         => pp(aa(word(A),bool,ord_less_eq(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),X),Z)),Y)) ) ) ).

% div_to_mult_word_lt
tff(fact_1599_More__Word_Oword__l__diffs_I2_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),X: word(A),Z: word(A)] :
          ( pp(aa(word(A),bool,ord_less(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),X)),Z))
         => ( pp(aa(word(A),bool,ord_less_eq(word(A),X),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),X)))
           => pp(aa(word(A),bool,ord_less(word(A),W),aa(word(A),word(A),minus_minus(word(A),Z),X))) ) ) ) ).

% More_Word.word_l_diffs(2)
tff(fact_1600_More__Word_Oword__l__diffs_I1_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),Z: word(A),X: word(A)] :
          ( pp(aa(word(A),bool,ord_less(word(A),W),aa(word(A),word(A),minus_minus(word(A),Z),X)))
         => ( pp(aa(word(A),bool,ord_less_eq(word(A),X),Z))
           => pp(aa(word(A),bool,ord_less(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),X)),Z)) ) ) ) ).

% More_Word.word_l_diffs(1)
tff(fact_1601_word__diff__ls_H_I2_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),W: word(A),X: word(A)] :
          ( pp(aa(word(A),bool,ord_less(word(A),Y),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),X)))
         => ( pp(aa(word(A),bool,ord_less_eq(word(A),X),Y))
           => pp(aa(word(A),bool,ord_less(word(A),aa(word(A),word(A),minus_minus(word(A),Y),X)),W)) ) ) ) ).

% word_diff_ls'(2)
tff(fact_1602_word__diff__ls_H_I1_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),X: word(A),W: word(A)] :
          ( pp(aa(word(A),bool,ord_less(word(A),aa(word(A),word(A),minus_minus(word(A),Y),X)),W))
         => ( pp(aa(word(A),bool,ord_less_eq(word(A),X),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),X)))
           => pp(aa(word(A),bool,ord_less(word(A),Y),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),X))) ) ) ) ).

% word_diff_ls'(1)
tff(fact_1603_word__l__diffs_H_I2_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),X: word(A),Z: word(A)] :
          ( pp(aa(word(A),bool,ord_less(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),X)),Z))
         => ( pp(aa(word(A),bool,ord_less_eq(word(A),X),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),X)))
           => pp(aa(word(A),bool,ord_less(word(A),aa(word(A),word(A),minus_minus(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),X)),X)),aa(word(A),word(A),minus_minus(word(A),Z),X))) ) ) ) ).

% word_l_diffs'(2)
tff(fact_1604_word__l__diffs_H_I1_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),X: word(A),Z: word(A)] :
          ( pp(aa(word(A),bool,ord_less(word(A),aa(word(A),word(A),minus_minus(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),X)),X)),aa(word(A),word(A),minus_minus(word(A),Z),X)))
         => ( pp(aa(word(A),bool,ord_less_eq(word(A),X),Z))
           => pp(aa(word(A),bool,ord_less(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),X)),Z)) ) ) ) ).

% word_l_diffs'(1)
tff(fact_1605_word__diff__ls_H_H_I2_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),W: word(A),X: word(A)] :
          ( pp(aa(word(A),bool,ord_less(word(A),Y),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),X)))
         => ( pp(aa(word(A),bool,ord_less_eq(word(A),X),Y))
           => pp(aa(word(A),bool,ord_less(word(A),aa(word(A),word(A),minus_minus(word(A),Y),X)),aa(word(A),word(A),minus_minus(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),X)),X))) ) ) ) ).

% word_diff_ls''(2)
tff(fact_1606_word__diff__ls_H_H_I1_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),X: word(A),W: word(A)] :
          ( pp(aa(word(A),bool,ord_less(word(A),aa(word(A),word(A),minus_minus(word(A),Y),X)),aa(word(A),word(A),minus_minus(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),X)),X)))
         => ( pp(aa(word(A),bool,ord_less_eq(word(A),X),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),X)))
           => pp(aa(word(A),bool,ord_less(word(A),Y),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),X))) ) ) ) ).

% word_diff_ls''(1)
tff(fact_1607_lt1__neq0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A)] :
          ( pp(aa(word(A),bool,ord_less_eq(word(A),one_one(word(A))),X))
        <=> ( X != zero_zero(word(A)) ) ) ) ).

% lt1_neq0
tff(fact_1608_less__1__simp,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: word(A),M: word(A)] :
          ( pp(aa(word(A),bool,ord_less(word(A),aa(word(A),word(A),minus_minus(word(A),N),one_one(word(A)))),M))
        <=> ( pp(aa(word(A),bool,ord_less_eq(word(A),N),M))
            & ( N != zero_zero(word(A)) ) ) ) ) ).

% less_1_simp
tff(fact_1609_le__m1__iff__lt,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A)] :
          ( pp(aa(word(A),bool,ord_less(word(A),zero_zero(word(A))),X))
        <=> ( pp(aa(word(A),bool,ord_less_eq(word(A),Y),aa(word(A),word(A),minus_minus(word(A),X),one_one(word(A)))))
          <=> pp(aa(word(A),bool,ord_less(word(A),Y),X)) ) ) ) ).

% le_m1_iff_lt
tff(fact_1610_word__le__plus,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A),C2: word(A)] :
          ( pp(aa(word(A),bool,ord_less(word(A),A2),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),A2),B2)))
         => ( pp(aa(word(A),bool,ord_less(word(A),C2),B2))
           => pp(aa(word(A),bool,ord_less_eq(word(A),A2),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),A2),C2))) ) ) ) ).

% word_le_plus
tff(fact_1611_word__overflow,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A)] :
          ( pp(aa(word(A),bool,ord_less(word(A),X),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),X),one_one(word(A)))))
          | ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),X),one_one(word(A))) = zero_zero(word(A)) ) ) ) ).

% word_overflow
tff(fact_1612_word__sub__1__le,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A)] :
          ( ( X != zero_zero(word(A)) )
         => pp(aa(word(A),bool,ord_less_eq(word(A),aa(word(A),word(A),minus_minus(word(A),X),one_one(word(A)))),X)) ) ) ).

% word_sub_1_le
tff(fact_1613_word__diff__less,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: word(A),M: word(A)] :
          ( pp(aa(word(A),bool,ord_less(word(A),zero_zero(word(A))),N))
         => ( pp(aa(word(A),bool,ord_less(word(A),zero_zero(word(A))),M))
           => ( pp(aa(word(A),bool,ord_less_eq(word(A),N),M))
             => pp(aa(word(A),bool,ord_less(word(A),aa(word(A),word(A),minus_minus(word(A),M),N)),M)) ) ) ) ) ).

% word_diff_less
tff(fact_1614_word__le__plus__1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),N: word(A),A2: word(A)] :
          ( pp(aa(word(A),bool,ord_less(word(A),Y),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Y),N)))
         => ( pp(aa(word(A),bool,ord_less(word(A),A2),N))
           => pp(aa(word(A),bool,ord_less_eq(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Y),A2)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Y),A2)),one_one(word(A))))) ) ) ) ).

% word_le_plus_1
tff(fact_1615_word__must__wrap,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),N: word(A)] :
          ( pp(aa(word(A),bool,ord_less_eq(word(A),X),aa(word(A),word(A),minus_minus(word(A),N),one_one(word(A)))))
         => ( pp(aa(word(A),bool,ord_less_eq(word(A),N),X))
           => ( N = zero_zero(word(A)) ) ) ) ) ).

% word_must_wrap
tff(fact_1616_plus__one__helper,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),N: word(A)] :
          ( pp(aa(word(A),bool,ord_less(word(A),X),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),N),one_one(word(A)))))
         => pp(aa(word(A),bool,ord_less_eq(word(A),X),N)) ) ) ).

% plus_one_helper
tff(fact_1617_plus__one__helper2,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),N: word(A)] :
          ( pp(aa(word(A),bool,ord_less_eq(word(A),X),N))
         => ( ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),N),one_one(word(A))) != zero_zero(word(A)) )
           => pp(aa(word(A),bool,ord_less(word(A),X),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),N),one_one(word(A))))) ) ) ) ).

% plus_one_helper2
tff(fact_1618_le__step__down__word,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [I: word(A),N: word(A)] :
          ( pp(aa(word(A),bool,ord_less_eq(word(A),I),N))
         => ( ( I != N )
           => pp(aa(word(A),bool,ord_less_eq(word(A),I),aa(word(A),word(A),minus_minus(word(A),N),one_one(word(A))))) ) ) ) ).

% le_step_down_word
tff(fact_1619_word__gr0__conv__Suc,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [M: word(A)] :
          ( pp(aa(word(A),bool,ord_less(word(A),zero_zero(word(A))),M))
         => ? [N2: word(A)] : M = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),N2),one_one(word(A))) ) ) ).

% word_gr0_conv_Suc
tff(fact_1620_word__less__nowrapI_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Z: word(A),K: word(A)] :
          ( pp(aa(word(A),bool,ord_less_eq(word(A),X),aa(word(A),word(A),minus_minus(word(A),Z),K)))
         => ( pp(aa(word(A),bool,ord_less_eq(word(A),K),Z))
           => ( pp(aa(word(A),bool,ord_less(word(A),zero_zero(word(A))),K))
             => pp(aa(word(A),bool,ord_less(word(A),X),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),X),K))) ) ) ) ) ).

% word_less_nowrapI'
tff(fact_1621_le__step__down__word__2,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A)] :
          ( pp(aa(word(A),bool,ord_less_eq(word(A),X),Y))
         => ( ( X != Y )
           => pp(aa(word(A),bool,ord_less_eq(word(A),X),aa(word(A),word(A),minus_minus(word(A),Y),one_one(word(A))))) ) ) ) ).

% le_step_down_word_2
tff(fact_1622_less__is__non__zero__p1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),K: word(A)] :
          ( pp(aa(word(A),bool,ord_less(word(A),A2),K))
         => ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),A2),one_one(word(A))) != zero_zero(word(A)) ) ) ) ).

% less_is_non_zero_p1
tff(fact_1623_word__le__minus__one__leq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A)] :
          ( pp(aa(word(A),bool,ord_less(word(A),X),Y))
         => pp(aa(word(A),bool,ord_less_eq(word(A),X),aa(word(A),word(A),minus_minus(word(A),Y),one_one(word(A))))) ) ) ).

% word_le_minus_one_leq
tff(fact_1624_word__leq__le__minus__one,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A)] :
          ( pp(aa(word(A),bool,ord_less_eq(word(A),X),Y))
         => ( ( X != zero_zero(word(A)) )
           => pp(aa(word(A),bool,ord_less(word(A),aa(word(A),word(A),minus_minus(word(A),X),one_one(word(A)))),Y)) ) ) ) ).

% word_leq_le_minus_one
tff(fact_1625_word__leq__minus__one__le,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),X: word(A)] :
          ( ( Y != zero_zero(word(A)) )
         => ( pp(aa(word(A),bool,ord_less_eq(word(A),X),aa(word(A),word(A),minus_minus(word(A),Y),one_one(word(A)))))
           => pp(aa(word(A),bool,ord_less(word(A),X),Y)) ) ) ) ).

% word_leq_minus_one_le
tff(fact_1626_word__minus__one__le__leq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A)] :
          ( pp(aa(word(A),bool,ord_less(word(A),aa(word(A),word(A),minus_minus(word(A),X),one_one(word(A)))),Y))
         => pp(aa(word(A),bool,ord_less_eq(word(A),X),Y)) ) ) ).

% word_minus_one_le_leq
tff(fact_1627_word__plus__one__nonzero,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A)] :
          ( pp(aa(word(A),bool,ord_less_eq(word(A),X),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),X),Y)))
         => ( ( Y != zero_zero(word(A)) )
           => ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),X),one_one(word(A))) != zero_zero(word(A)) ) ) ) ) ).

% word_plus_one_nonzero
tff(fact_1628_word__sub__plus__one__nonzero,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N3: word(A),N: word(A)] :
          ( pp(aa(word(A),bool,ord_less_eq(word(A),N3),N))
         => ( ( N3 != zero_zero(word(A)) )
           => ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),aa(word(A),word(A),minus_minus(word(A),N),N3)),one_one(word(A))) != zero_zero(word(A)) ) ) ) ) ).

% word_sub_plus_one_nonzero
tff(fact_1629_word__plus__strict__mono__right,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),Z: word(A),X: word(A)] :
          ( pp(aa(word(A),bool,ord_less(word(A),Y),Z))
         => ( pp(aa(word(A),bool,ord_less_eq(word(A),X),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),X),Z)))
           => pp(aa(word(A),bool,ord_less(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),X),Y)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),X),Z))) ) ) ) ).

% word_plus_strict_mono_right
tff(fact_1630_word__less__cases,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A)] :
          ( pp(aa(word(A),bool,ord_less(word(A),X),Y))
         => ( ( X = aa(word(A),word(A),minus_minus(word(A),Y),one_one(word(A))) )
            | pp(aa(word(A),bool,ord_less(word(A),X),aa(word(A),word(A),minus_minus(word(A),Y),one_one(word(A))))) ) ) ) ).

% word_less_cases
tff(fact_1631_word__gt__a__gt__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),N: word(A)] :
          ( pp(aa(word(A),bool,ord_less(word(A),A2),N))
         => pp(aa(word(A),bool,ord_less(word(A),zero_zero(word(A))),N)) ) ) ).

% word_gt_a_gt_0
tff(fact_1632_gt0__iff__gem1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A)] :
          ( pp(aa(word(A),bool,ord_less(word(A),zero_zero(word(A))),X))
        <=> pp(aa(word(A),bool,ord_less(word(A),aa(word(A),word(A),minus_minus(word(A),X),one_one(word(A)))),X)) ) ) ).

% gt0_iff_gem1
tff(fact_1633_div__less__dividend__word,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),N: word(A)] :
          ( ( X != zero_zero(word(A)) )
         => ( ( N != one_one(word(A)) )
           => pp(aa(word(A),bool,ord_less(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),X),N)),X)) ) ) ) ).

% div_less_dividend_word
tff(fact_1634_word__div__lt__eq__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A)] :
          ( pp(aa(word(A),bool,ord_less(word(A),X),Y))
         => ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),X),Y) = zero_zero(word(A)) ) ) ) ).

% word_div_lt_eq_0
tff(fact_1635_word__less__div,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A)] :
          ( ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),X),Y) = zero_zero(word(A)) )
         => ( ( Y = zero_zero(word(A)) )
            | pp(aa(word(A),bool,ord_less(word(A),X),Y)) ) ) ) ).

% word_less_div
tff(fact_1636_word__div__less,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),V: word(A)] :
          ( pp(aa(word(A),bool,ord_less(word(A),W),V))
         => ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),W),V) = zero_zero(word(A)) ) ) ) ).

% word_div_less
tff(fact_1637_More__Word_Oword__div__mult,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [C2: word(A),A2: word(A),B2: word(A)] :
          ( pp(aa(word(A),bool,ord_less(word(A),zero_zero(word(A))),C2))
         => ( pp(aa(word(A),bool,ord_less(word(A),A2),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),B2),C2)))
           => pp(aa(word(A),bool,ord_less(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),A2),C2)),B2)) ) ) ) ).

% More_Word.word_div_mult
tff(fact_1638_word__div__sub,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),X: word(A)] :
          ( pp(aa(word(A),bool,ord_less_eq(word(A),Y),X))
         => ( pp(aa(word(A),bool,ord_less(word(A),zero_zero(word(A))),Y))
           => ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),aa(word(A),word(A),minus_minus(word(A),X),Y)),Y) = aa(word(A),word(A),minus_minus(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),X),Y)),one_one(word(A))) ) ) ) ) ).

% word_div_sub
tff(fact_1639_word__le__add,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A)] :
          ( pp(aa(word(A),bool,ord_less_eq(word(A),X),Y))
         => ? [N2: nat] : Y = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),X),aa(nat,word(A),semiring_1_of_nat(word(A)),N2)) ) ) ).

% word_le_add
tff(fact_1640_word__1__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A),X: nat] :
          ( pp(aa(word(A),bool,ord_less_eq(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),A2),one_one(word(A)))),B2))
         => ( pp(aa(word(A),bool,ord_less(word(A),A2),aa(nat,word(A),semiring_1_of_nat(word(A)),X)))
           => pp(aa(word(A),bool,ord_less(word(A),A2),B2)) ) ) ) ).

% word_1_0
tff(fact_1641_word__div__1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),N),one_one(word(A))) = N ) ).

% word_div_1
tff(fact_1642_div__by__0__word,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),X),zero_zero(word(A))) = zero_zero(word(A)) ) ).

% div_by_0_word
tff(fact_1643_norm__not__less__zero,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X: A] : ~ pp(aa(real,bool,ord_less(real,real_V7770717601297561774m_norm(A,X)),zero_zero(real))) ) ).

% norm_not_less_zero
tff(fact_1644_norm__mult,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [X: A,Y: A] : real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),times_times(A),X),Y)) = aa(real,real,aa(real,fun(real,real),times_times(real),real_V7770717601297561774m_norm(A,X)),real_V7770717601297561774m_norm(A,Y)) ) ).

% norm_mult
tff(fact_1645_norm__ge__zero,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X: A] : pp(aa(real,bool,ord_less_eq(real,zero_zero(real)),real_V7770717601297561774m_norm(A,X))) ) ).

% norm_ge_zero
tff(fact_1646_norm__divide,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [A2: A,B2: A] : real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),real_V7770717601297561774m_norm(A,A2)),real_V7770717601297561774m_norm(A,B2)) ) ).

% norm_divide
tff(fact_1647_norm__power,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [X: A,N: nat] : real_V7770717601297561774m_norm(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),X),N)) = aa(nat,real,aa(real,fun(nat,real),power_power(real),real_V7770717601297561774m_norm(A,X)),N) ) ).

% norm_power
tff(fact_1648_power__eq__imp__eq__norm,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [W: A,N: nat,Z: A] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),W),N) = aa(nat,A,aa(A,fun(nat,A),power_power(A),Z),N) )
         => ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),N))
           => ( real_V7770717601297561774m_norm(A,W) = real_V7770717601297561774m_norm(A,Z) ) ) ) ) ).

% power_eq_imp_eq_norm
tff(fact_1649_nonzero__norm__divide,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [B2: A,A2: A] :
          ( ( B2 != zero_zero(A) )
         => ( real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),real_V7770717601297561774m_norm(A,A2)),real_V7770717601297561774m_norm(A,B2)) ) ) ) ).

% nonzero_norm_divide
tff(fact_1650_norm__mult__less,axiom,
    ! [A: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [X: A,R2: real,Y: A,S3: real] :
          ( pp(aa(real,bool,ord_less(real,real_V7770717601297561774m_norm(A,X)),R2))
         => ( pp(aa(real,bool,ord_less(real,real_V7770717601297561774m_norm(A,Y)),S3))
           => pp(aa(real,bool,ord_less(real,real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),times_times(A),X),Y))),aa(real,real,aa(real,fun(real,real),times_times(real),R2),S3))) ) ) ) ).

% norm_mult_less
tff(fact_1651_norm__mult__ineq,axiom,
    ! [A: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [X: A,Y: A] : pp(aa(real,bool,ord_less_eq(real,real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),times_times(A),X),Y))),aa(real,real,aa(real,fun(real,real),times_times(real),real_V7770717601297561774m_norm(A,X)),real_V7770717601297561774m_norm(A,Y)))) ) ).

% norm_mult_ineq
tff(fact_1652_norm__triangle__lt,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X: A,Y: A,E: real] :
          ( pp(aa(real,bool,ord_less(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),real_V7770717601297561774m_norm(A,X)),real_V7770717601297561774m_norm(A,Y))),E))
         => pp(aa(real,bool,ord_less(real,real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y))),E)) ) ) ).

% norm_triangle_lt
tff(fact_1653_norm__add__less,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X: A,R2: real,Y: A,S3: real] :
          ( pp(aa(real,bool,ord_less(real,real_V7770717601297561774m_norm(A,X)),R2))
         => ( pp(aa(real,bool,ord_less(real,real_V7770717601297561774m_norm(A,Y)),S3))
           => pp(aa(real,bool,ord_less(real,real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y))),aa(real,real,aa(real,fun(real,real),plus_plus(real),R2),S3))) ) ) ) ).

% norm_add_less
tff(fact_1654_norm__triangle__mono,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [A2: A,R2: real,B2: A,S3: real] :
          ( pp(aa(real,bool,ord_less_eq(real,real_V7770717601297561774m_norm(A,A2)),R2))
         => ( pp(aa(real,bool,ord_less_eq(real,real_V7770717601297561774m_norm(A,B2)),S3))
           => pp(aa(real,bool,ord_less_eq(real,real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2))),aa(real,real,aa(real,fun(real,real),plus_plus(real),R2),S3))) ) ) ) ).

% norm_triangle_mono
tff(fact_1655_norm__triangle__ineq,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X: A,Y: A] : pp(aa(real,bool,ord_less_eq(real,real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y))),aa(real,real,aa(real,fun(real,real),plus_plus(real),real_V7770717601297561774m_norm(A,X)),real_V7770717601297561774m_norm(A,Y)))) ) ).

% norm_triangle_ineq
tff(fact_1656_norm__triangle__le,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X: A,Y: A,E: real] :
          ( pp(aa(real,bool,ord_less_eq(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),real_V7770717601297561774m_norm(A,X)),real_V7770717601297561774m_norm(A,Y))),E))
         => pp(aa(real,bool,ord_less_eq(real,real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y))),E)) ) ) ).

% norm_triangle_le
tff(fact_1657_norm__add__leD,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [A2: A,B2: A,C2: real] :
          ( pp(aa(real,bool,ord_less_eq(real,real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2))),C2))
         => pp(aa(real,bool,ord_less_eq(real,real_V7770717601297561774m_norm(A,B2)),aa(real,real,aa(real,fun(real,real),plus_plus(real),real_V7770717601297561774m_norm(A,A2)),C2))) ) ) ).

% norm_add_leD
tff(fact_1658_norm__diff__triangle__less,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X: A,Y: A,E1: real,Z: A,E22: real] :
          ( pp(aa(real,bool,ord_less(real,real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,X),Y))),E1))
         => ( pp(aa(real,bool,ord_less(real,real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,Y),Z))),E22))
           => pp(aa(real,bool,ord_less(real,real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,X),Z))),aa(real,real,aa(real,fun(real,real),plus_plus(real),E1),E22))) ) ) ) ).

% norm_diff_triangle_less
tff(fact_1659_norm__power__ineq,axiom,
    ! [A: $tType] :
      ( real_V2822296259951069270ebra_1(A)
     => ! [X: A,N: nat] : pp(aa(real,bool,ord_less_eq(real,real_V7770717601297561774m_norm(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),X),N))),aa(nat,real,aa(real,fun(nat,real),power_power(real),real_V7770717601297561774m_norm(A,X)),N))) ) ).

% norm_power_ineq
tff(fact_1660_norm__triangle__le__diff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X: A,Y: A,E: real] :
          ( pp(aa(real,bool,ord_less_eq(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),real_V7770717601297561774m_norm(A,X)),real_V7770717601297561774m_norm(A,Y))),E))
         => pp(aa(real,bool,ord_less_eq(real,real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,X),Y))),E)) ) ) ).

% norm_triangle_le_diff
tff(fact_1661_norm__diff__triangle__le,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X: A,Y: A,E1: real,Z: A,E22: real] :
          ( pp(aa(real,bool,ord_less_eq(real,real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,X),Y))),E1))
         => ( pp(aa(real,bool,ord_less_eq(real,real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,Y),Z))),E22))
           => pp(aa(real,bool,ord_less_eq(real,real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,X),Z))),aa(real,real,aa(real,fun(real,real),plus_plus(real),E1),E22))) ) ) ) ).

% norm_diff_triangle_le
tff(fact_1662_norm__triangle__ineq4,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [A2: A,B2: A] : pp(aa(real,bool,ord_less_eq(real,real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,A2),B2))),aa(real,real,aa(real,fun(real,real),plus_plus(real),real_V7770717601297561774m_norm(A,A2)),real_V7770717601297561774m_norm(A,B2)))) ) ).

% norm_triangle_ineq4
tff(fact_1663_norm__triangle__sub,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X: A,Y: A] : pp(aa(real,bool,ord_less_eq(real,real_V7770717601297561774m_norm(A,X)),aa(real,real,aa(real,fun(real,real),plus_plus(real),real_V7770717601297561774m_norm(A,Y)),real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,X),Y))))) ) ).

% norm_triangle_sub
tff(fact_1664_norm__diff__ineq,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [A2: A,B2: A] : pp(aa(real,bool,ord_less_eq(real,aa(real,real,minus_minus(real,real_V7770717601297561774m_norm(A,A2)),real_V7770717601297561774m_norm(A,B2))),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)))) ) ).

% norm_diff_ineq
tff(fact_1665_norm__triangle__ineq2,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [A2: A,B2: A] : pp(aa(real,bool,ord_less_eq(real,aa(real,real,minus_minus(real,real_V7770717601297561774m_norm(A,A2)),real_V7770717601297561774m_norm(A,B2))),real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,A2),B2)))) ) ).

% norm_triangle_ineq2
tff(fact_1666_power__eq__1__iff,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [W: A,N: nat] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),W),N) = one_one(A) )
         => ( ( real_V7770717601297561774m_norm(A,W) = one_one(real) )
            | ( N = zero_zero(nat) ) ) ) ) ).

% power_eq_1_iff
tff(fact_1667_norm__diff__triangle__ineq,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [A2: A,B2: A,C2: A,D: A] : pp(aa(real,bool,ord_less_eq(real,real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),D)))),aa(real,real,aa(real,fun(real,real),plus_plus(real),real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,A2),C2))),real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,B2),D))))) ) ).

% norm_diff_triangle_ineq
tff(fact_1668_square__norm__one,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [X: A] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(num,nat,numeral_numeral(nat),bit0(one2))) = one_one(A) )
         => ( real_V7770717601297561774m_norm(A,X) = one_one(real) ) ) ) ).

% square_norm_one
tff(fact_1669_norm__power__diff,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_mult(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Z: A,W: A,M: nat] :
          ( pp(aa(real,bool,ord_less_eq(real,real_V7770717601297561774m_norm(A,Z)),one_one(real)))
         => ( pp(aa(real,bool,ord_less_eq(real,real_V7770717601297561774m_norm(A,W)),one_one(real)))
           => pp(aa(real,bool,ord_less_eq(real,real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),Z),M)),aa(nat,A,aa(A,fun(nat,A),power_power(A),W),M)))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),M)),real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,Z),W))))) ) ) ) ).

% norm_power_diff
tff(fact_1670_word__less__sub1__numberof,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: num] :
          ( ( aa(num,word(A),numeral_numeral(word(A)),W) != zero_zero(word(A)) )
         => ( pp(aa(word(A),bool,ord_less(word(A),one_one(word(A))),aa(num,word(A),numeral_numeral(word(A)),W)))
          <=> pp(aa(word(A),bool,ord_less(word(A),zero_zero(word(A))),aa(word(A),word(A),minus_minus(word(A),aa(num,word(A),numeral_numeral(word(A)),W)),one_one(word(A))))) ) ) ) ).

% word_less_sub1_numberof
tff(fact_1671_atLeastatMost__subset__iff,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A2: A,B2: A,C2: A,D: A] :
          ( pp(aa(set(A),bool,ord_less_eq(set(A),set_or1337092689740270186AtMost(A,A2,B2)),set_or1337092689740270186AtMost(A,C2,D)))
        <=> ( ~ pp(aa(A,bool,ord_less_eq(A,A2),B2))
            | ( pp(aa(A,bool,ord_less_eq(A,C2),A2))
              & pp(aa(A,bool,ord_less_eq(A,B2),D)) ) ) ) ) ).

% atLeastatMost_subset_iff
tff(fact_1672_word__less__1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A)] :
          ( pp(aa(word(A),bool,ord_less(word(A),X),one_one(word(A))))
        <=> ( X = zero_zero(word(A)) ) ) ) ).

% word_less_1
tff(fact_1673_word__gt__0__no,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: num] :
          ( pp(aa(word(A),bool,ord_less(word(A),zero_zero(word(A))),aa(num,word(A),numeral_numeral(word(A)),Y)))
        <=> ( zero_zero(word(A)) != aa(num,word(A),numeral_numeral(word(A)),Y) ) ) ) ).

% word_gt_0_no
tff(fact_1674_word__le__sub1__numberof,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: num] :
          ( ( aa(num,word(A),numeral_numeral(word(A)),W) != zero_zero(word(A)) )
         => ( pp(aa(word(A),bool,ord_less_eq(word(A),one_one(word(A))),aa(num,word(A),numeral_numeral(word(A)),W)))
          <=> pp(aa(word(A),bool,ord_less_eq(word(A),zero_zero(word(A))),aa(word(A),word(A),minus_minus(word(A),aa(num,word(A),numeral_numeral(word(A)),W)),one_one(word(A))))) ) ) ) ).

% word_le_sub1_numberof
tff(fact_1675_log__ceil__idem,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,ord_less_eq(real,one_one(real)),X))
     => ( archimedean_ceiling(real,aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),X)) = archimedean_ceiling(real,aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(int,real,ring_1_of_int(real),archimedean_ceiling(real,X)))) ) ) ).

% log_ceil_idem
tff(fact_1676_Icc__eq__Icc,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [L: A,H: A,L2: A,H2: A] :
          ( ( set_or1337092689740270186AtMost(A,L,H) = set_or1337092689740270186AtMost(A,L2,H2) )
        <=> ( ( ( L = L2 )
              & ( H = H2 ) )
            | ( ~ pp(aa(A,bool,ord_less_eq(A,L),H))
              & ~ pp(aa(A,bool,ord_less_eq(A,L2),H2)) ) ) ) ) ).

% Icc_eq_Icc
tff(fact_1677_atLeastAtMost__iff,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [I: A,L: A,U: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),I),set_or1337092689740270186AtMost(A,L,U)))
        <=> ( pp(aa(A,bool,ord_less_eq(A,L),I))
            & pp(aa(A,bool,ord_less_eq(A,I),U)) ) ) ) ).

% atLeastAtMost_iff
tff(fact_1678_word__coorder_Oextremum__unique,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A)] :
          ( pp(aa(word(A),bool,ord_less_eq(word(A),A2),zero_zero(word(A))))
        <=> ( A2 = zero_zero(word(A)) ) ) ) ).

% word_coorder.extremum_unique
tff(fact_1679_word__le__0__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A)] :
          ( pp(aa(word(A),bool,ord_less_eq(word(A),X),zero_zero(word(A))))
        <=> ( X = zero_zero(word(A)) ) ) ) ).

% word_le_0_iff
tff(fact_1680_of__int__ceiling__cancel,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( ( aa(int,A,ring_1_of_int(A),archimedean_ceiling(A,X)) = X )
        <=> ? [N5: int] : X = aa(int,A,ring_1_of_int(A),N5) ) ) ).

% of_int_ceiling_cancel
tff(fact_1681_of__int__0,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ( aa(int,A,ring_1_of_int(A),zero_zero(int)) = zero_zero(A) ) ) ).

% of_int_0
tff(fact_1682_of__int__0__eq__iff,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [Z: int] :
          ( ( zero_zero(A) = aa(int,A,ring_1_of_int(A),Z) )
        <=> ( Z = zero_zero(int) ) ) ) ).

% of_int_0_eq_iff
tff(fact_1683_of__int__eq__0__iff,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [Z: int] :
          ( ( aa(int,A,ring_1_of_int(A),Z) = zero_zero(A) )
        <=> ( Z = zero_zero(int) ) ) ) ).

% of_int_eq_0_iff
tff(fact_1684_of__int__eq__numeral__iff,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [Z: int,N: num] :
          ( ( aa(int,A,ring_1_of_int(A),Z) = aa(num,A,numeral_numeral(A),N) )
        <=> ( Z = aa(num,int,numeral_numeral(int),N) ) ) ) ).

% of_int_eq_numeral_iff
tff(fact_1685_of__int__numeral,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [K: num] : aa(int,A,ring_1_of_int(A),aa(num,int,numeral_numeral(int),K)) = aa(num,A,numeral_numeral(A),K) ) ).

% of_int_numeral
tff(fact_1686_of__int__le__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [W: int,Z: int] :
          ( pp(aa(A,bool,ord_less_eq(A,aa(int,A,ring_1_of_int(A),W)),aa(int,A,ring_1_of_int(A),Z)))
        <=> pp(aa(int,bool,ord_less_eq(int,W),Z)) ) ) ).

% of_int_le_iff
tff(fact_1687_of__int__1,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ( aa(int,A,ring_1_of_int(A),one_one(int)) = one_one(A) ) ) ).

% of_int_1
tff(fact_1688_of__int__eq__1__iff,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [Z: int] :
          ( ( aa(int,A,ring_1_of_int(A),Z) = one_one(A) )
        <=> ( Z = one_one(int) ) ) ) ).

% of_int_eq_1_iff
tff(fact_1689_of__int__add,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [W: int,Z: int] : aa(int,A,ring_1_of_int(A),aa(int,int,aa(int,fun(int,int),plus_plus(int),W),Z)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(int,A,ring_1_of_int(A),W)),aa(int,A,ring_1_of_int(A),Z)) ) ).

% of_int_add
tff(fact_1690_of__int__mult,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [W: int,Z: int] : aa(int,A,ring_1_of_int(A),aa(int,int,aa(int,fun(int,int),times_times(int),W),Z)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(int,A,ring_1_of_int(A),W)),aa(int,A,ring_1_of_int(A),Z)) ) ).

% of_int_mult
tff(fact_1691_of__int__of__nat__eq,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [N: nat] : aa(int,A,ring_1_of_int(A),aa(nat,int,semiring_1_of_nat(int),N)) = aa(nat,A,semiring_1_of_nat(A),N) ) ).

% of_int_of_nat_eq
tff(fact_1692_of__int__power__eq__of__int__cancel__iff,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [X: int,B2: int,W: nat] :
          ( ( aa(int,A,ring_1_of_int(A),X) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(int,A,ring_1_of_int(A),B2)),W) )
        <=> ( X = aa(nat,int,aa(int,fun(nat,int),power_power(int),B2),W) ) ) ) ).

% of_int_power_eq_of_int_cancel_iff
tff(fact_1693_of__int__eq__of__int__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [B2: int,W: nat,X: int] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(int,A,ring_1_of_int(A),B2)),W) = aa(int,A,ring_1_of_int(A),X) )
        <=> ( aa(nat,int,aa(int,fun(nat,int),power_power(int),B2),W) = X ) ) ) ).

% of_int_eq_of_int_power_cancel_iff
tff(fact_1694_of__int__power,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Z: int,N: nat] : aa(int,A,ring_1_of_int(A),aa(nat,int,aa(int,fun(nat,int),power_power(int),Z),N)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(int,A,ring_1_of_int(A),Z)),N) ) ).

% of_int_power
tff(fact_1695_ceiling__add__of__int,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,Z: int] : archimedean_ceiling(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),aa(int,A,ring_1_of_int(A),Z))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),archimedean_ceiling(A,X)),Z) ) ).

% ceiling_add_of_int
tff(fact_1696_of__int__le__0__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: int] :
          ( pp(aa(A,bool,ord_less_eq(A,aa(int,A,ring_1_of_int(A),Z)),zero_zero(A)))
        <=> pp(aa(int,bool,ord_less_eq(int,Z),zero_zero(int))) ) ) ).

% of_int_le_0_iff
tff(fact_1697_of__int__0__le__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: int] :
          ( pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),aa(int,A,ring_1_of_int(A),Z)))
        <=> pp(aa(int,bool,ord_less_eq(int,zero_zero(int)),Z)) ) ) ).

% of_int_0_le_iff
tff(fact_1698_of__int__numeral__le__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [N: num,Z: int] :
          ( pp(aa(A,bool,ord_less_eq(A,aa(num,A,numeral_numeral(A),N)),aa(int,A,ring_1_of_int(A),Z)))
        <=> pp(aa(int,bool,ord_less_eq(int,aa(num,int,numeral_numeral(int),N)),Z)) ) ) ).

% of_int_numeral_le_iff
tff(fact_1699_of__int__le__numeral__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: int,N: num] :
          ( pp(aa(A,bool,ord_less_eq(A,aa(int,A,ring_1_of_int(A),Z)),aa(num,A,numeral_numeral(A),N)))
        <=> pp(aa(int,bool,ord_less_eq(int,Z),aa(num,int,numeral_numeral(int),N))) ) ) ).

% of_int_le_numeral_iff
tff(fact_1700_of__int__less__0__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: int] :
          ( pp(aa(A,bool,ord_less(A,aa(int,A,ring_1_of_int(A),Z)),zero_zero(A)))
        <=> pp(aa(int,bool,ord_less(int,Z),zero_zero(int))) ) ) ).

% of_int_less_0_iff
tff(fact_1701_of__int__0__less__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: int] :
          ( pp(aa(A,bool,ord_less(A,zero_zero(A)),aa(int,A,ring_1_of_int(A),Z)))
        <=> pp(aa(int,bool,ord_less(int,zero_zero(int)),Z)) ) ) ).

% of_int_0_less_iff
tff(fact_1702_of__int__less__numeral__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: int,N: num] :
          ( pp(aa(A,bool,ord_less(A,aa(int,A,ring_1_of_int(A),Z)),aa(num,A,numeral_numeral(A),N)))
        <=> pp(aa(int,bool,ord_less(int,Z),aa(num,int,numeral_numeral(int),N))) ) ) ).

% of_int_less_numeral_iff
tff(fact_1703_of__int__numeral__less__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [N: num,Z: int] :
          ( pp(aa(A,bool,ord_less(A,aa(num,A,numeral_numeral(A),N)),aa(int,A,ring_1_of_int(A),Z)))
        <=> pp(aa(int,bool,ord_less(int,aa(num,int,numeral_numeral(int),N)),Z)) ) ) ).

% of_int_numeral_less_iff
tff(fact_1704_of__int__le__1__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: int] :
          ( pp(aa(A,bool,ord_less_eq(A,aa(int,A,ring_1_of_int(A),Z)),one_one(A)))
        <=> pp(aa(int,bool,ord_less_eq(int,Z),one_one(int))) ) ) ).

% of_int_le_1_iff
tff(fact_1705_of__int__1__le__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: int] :
          ( pp(aa(A,bool,ord_less_eq(A,one_one(A)),aa(int,A,ring_1_of_int(A),Z)))
        <=> pp(aa(int,bool,ord_less_eq(int,one_one(int)),Z)) ) ) ).

% of_int_1_le_iff
tff(fact_1706_of__int__less__1__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: int] :
          ( pp(aa(A,bool,ord_less(A,aa(int,A,ring_1_of_int(A),Z)),one_one(A)))
        <=> pp(aa(int,bool,ord_less(int,Z),one_one(int))) ) ) ).

% of_int_less_1_iff
tff(fact_1707_of__int__1__less__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: int] :
          ( pp(aa(A,bool,ord_less(A,one_one(A)),aa(int,A,ring_1_of_int(A),Z)))
        <=> pp(aa(int,bool,ord_less(int,one_one(int)),Z)) ) ) ).

% of_int_1_less_iff
tff(fact_1708_of__int__eq__numeral__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [Y: int,X: num,N: nat] :
          ( ( aa(int,A,ring_1_of_int(A),Y) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),X)),N) )
        <=> ( Y = aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),X)),N) ) ) ) ).

% of_int_eq_numeral_power_cancel_iff
tff(fact_1709_numeral__power__eq__of__int__cancel__iff,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [X: num,N: nat,Y: int] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),X)),N) = aa(int,A,ring_1_of_int(A),Y) )
        <=> ( aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),X)),N) = Y ) ) ) ).

% numeral_power_eq_of_int_cancel_iff
tff(fact_1710_of__int__power__le__of__int__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: int,B2: int,W: nat] :
          ( pp(aa(A,bool,ord_less_eq(A,aa(int,A,ring_1_of_int(A),X)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(int,A,ring_1_of_int(A),B2)),W)))
        <=> pp(aa(int,bool,ord_less_eq(int,X),aa(nat,int,aa(int,fun(nat,int),power_power(int),B2),W))) ) ) ).

% of_int_power_le_of_int_cancel_iff
tff(fact_1711_of__int__le__of__int__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [B2: int,W: nat,X: int] :
          ( pp(aa(A,bool,ord_less_eq(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(int,A,ring_1_of_int(A),B2)),W)),aa(int,A,ring_1_of_int(A),X)))
        <=> pp(aa(int,bool,ord_less_eq(int,aa(nat,int,aa(int,fun(nat,int),power_power(int),B2),W)),X)) ) ) ).

% of_int_le_of_int_power_cancel_iff
tff(fact_1712_of__int__less__of__int__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [B2: int,W: nat,X: int] :
          ( pp(aa(A,bool,ord_less(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(int,A,ring_1_of_int(A),B2)),W)),aa(int,A,ring_1_of_int(A),X)))
        <=> pp(aa(int,bool,ord_less(int,aa(nat,int,aa(int,fun(nat,int),power_power(int),B2),W)),X)) ) ) ).

% of_int_less_of_int_power_cancel_iff
tff(fact_1713_of__int__power__less__of__int__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: int,B2: int,W: nat] :
          ( pp(aa(A,bool,ord_less(A,aa(int,A,ring_1_of_int(A),X)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(int,A,ring_1_of_int(A),B2)),W)))
        <=> pp(aa(int,bool,ord_less(int,X),aa(nat,int,aa(int,fun(nat,int),power_power(int),B2),W))) ) ) ).

% of_int_power_less_of_int_cancel_iff
tff(fact_1714_of__int__le__numeral__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: int,X: num,N: nat] :
          ( pp(aa(A,bool,ord_less_eq(A,aa(int,A,ring_1_of_int(A),A2)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),X)),N)))
        <=> pp(aa(int,bool,ord_less_eq(int,A2),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),X)),N))) ) ) ).

% of_int_le_numeral_power_cancel_iff
tff(fact_1715_numeral__power__le__of__int__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: num,N: nat,A2: int] :
          ( pp(aa(A,bool,ord_less_eq(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),X)),N)),aa(int,A,ring_1_of_int(A),A2)))
        <=> pp(aa(int,bool,ord_less_eq(int,aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),X)),N)),A2)) ) ) ).

% numeral_power_le_of_int_cancel_iff
tff(fact_1716_of__int__less__numeral__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: int,X: num,N: nat] :
          ( pp(aa(A,bool,ord_less(A,aa(int,A,ring_1_of_int(A),A2)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),X)),N)))
        <=> pp(aa(int,bool,ord_less(int,A2),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),X)),N))) ) ) ).

% of_int_less_numeral_power_cancel_iff
tff(fact_1717_numeral__power__less__of__int__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: num,N: nat,A2: int] :
          ( pp(aa(A,bool,ord_less(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),X)),N)),aa(int,A,ring_1_of_int(A),A2)))
        <=> pp(aa(int,bool,ord_less(int,aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),X)),N)),A2)) ) ) ).

% numeral_power_less_of_int_cancel_iff
tff(fact_1718_plus__minus__no__overflow__ab,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Ab: word(A),C2: word(A)] :
          ( pp(aa(word(A),bool,ord_less_eq(word(A),X),aa(word(A),word(A),minus_minus(word(A),Ab),C2)))
         => ( pp(aa(word(A),bool,ord_less_eq(word(A),C2),Ab))
           => pp(aa(word(A),bool,ord_less_eq(word(A),X),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),X),C2))) ) ) ) ).

% plus_minus_no_overflow_ab
tff(fact_1719_plus__minus__not__NULL__ab,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Ab: word(A),C2: word(A)] :
          ( pp(aa(word(A),bool,ord_less_eq(word(A),X),aa(word(A),word(A),minus_minus(word(A),Ab),C2)))
         => ( pp(aa(word(A),bool,ord_less_eq(word(A),C2),Ab))
           => ( ( C2 != zero_zero(word(A)) )
             => ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),X),C2) != zero_zero(word(A)) ) ) ) ) ) ).

% plus_minus_not_NULL_ab
tff(fact_1720_neq__0__no__wrap,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A)] :
          ( pp(aa(word(A),bool,ord_less_eq(word(A),X),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),X),Y)))
         => ( ( X != zero_zero(word(A)) )
           => ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),X),Y) != zero_zero(word(A)) ) ) ) ) ).

% neq_0_no_wrap
tff(fact_1721_le__minus_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),C2: word(A),B2: word(A)] :
          ( pp(aa(word(A),bool,ord_less_eq(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),A2),C2)),B2))
         => ( pp(aa(word(A),bool,ord_less_eq(word(A),A2),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),A2),C2)))
           => pp(aa(word(A),bool,ord_less_eq(word(A),C2),aa(word(A),word(A),minus_minus(word(A),B2),A2))) ) ) ) ).

% le_minus'
tff(fact_1722_le__plus_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A),C2: word(A)] :
          ( pp(aa(word(A),bool,ord_less_eq(word(A),A2),B2))
         => ( pp(aa(word(A),bool,ord_less_eq(word(A),C2),aa(word(A),word(A),minus_minus(word(A),B2),A2)))
           => pp(aa(word(A),bool,ord_less_eq(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),A2),C2)),B2)) ) ) ) ).

% le_plus'
tff(fact_1723_le__plus,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [C2: word(A),B2: word(A),A2: word(A)] :
          ( pp(aa(word(A),bool,ord_less_eq(word(A),C2),aa(word(A),word(A),minus_minus(word(A),B2),A2)))
         => ( pp(aa(word(A),bool,ord_less_eq(word(A),A2),B2))
           => pp(aa(word(A),bool,ord_less_eq(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),A2),C2)),B2)) ) ) ) ).

% le_plus
tff(fact_1724_word__diff__ls_H_H_I3_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),X: word(A),W: word(A)] :
          ( pp(aa(word(A),bool,ord_less_eq(word(A),aa(word(A),word(A),minus_minus(word(A),Y),X)),aa(word(A),word(A),minus_minus(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),X)),X)))
         => ( pp(aa(word(A),bool,ord_less_eq(word(A),X),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),X)))
           => pp(aa(word(A),bool,ord_less_eq(word(A),Y),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),X))) ) ) ) ).

% word_diff_ls''(3)
tff(fact_1725_word__diff__ls_H_H_I4_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),W: word(A),X: word(A)] :
          ( pp(aa(word(A),bool,ord_less_eq(word(A),Y),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),X)))
         => ( pp(aa(word(A),bool,ord_less_eq(word(A),X),Y))
           => pp(aa(word(A),bool,ord_less_eq(word(A),aa(word(A),word(A),minus_minus(word(A),Y),X)),aa(word(A),word(A),minus_minus(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),X)),X))) ) ) ) ).

% word_diff_ls''(4)
tff(fact_1726_word__l__diffs_H_I3_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),X: word(A),Z: word(A)] :
          ( pp(aa(word(A),bool,ord_less_eq(word(A),aa(word(A),word(A),minus_minus(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),X)),X)),aa(word(A),word(A),minus_minus(word(A),Z),X)))
         => ( pp(aa(word(A),bool,ord_less_eq(word(A),X),Z))
           => pp(aa(word(A),bool,ord_less_eq(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),X)),Z)) ) ) ) ).

% word_l_diffs'(3)
tff(fact_1727_word__l__diffs_H_I4_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),X: word(A),Z: word(A)] :
          ( pp(aa(word(A),bool,ord_less_eq(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),X)),Z))
         => ( pp(aa(word(A),bool,ord_less_eq(word(A),X),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),X)))
           => pp(aa(word(A),bool,ord_less_eq(word(A),aa(word(A),word(A),minus_minus(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),X)),X)),aa(word(A),word(A),minus_minus(word(A),Z),X))) ) ) ) ).

% word_l_diffs'(4)
tff(fact_1728_word__diff__ls_H_I3_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),X: word(A),W: word(A)] :
          ( pp(aa(word(A),bool,ord_less_eq(word(A),aa(word(A),word(A),minus_minus(word(A),Y),X)),W))
         => ( pp(aa(word(A),bool,ord_less_eq(word(A),X),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),X)))
           => pp(aa(word(A),bool,ord_less_eq(word(A),Y),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),X))) ) ) ) ).

% word_diff_ls'(3)
tff(fact_1729_word__diff__ls_H_I4_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),W: word(A),X: word(A)] :
          ( pp(aa(word(A),bool,ord_less_eq(word(A),Y),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),X)))
         => ( pp(aa(word(A),bool,ord_less_eq(word(A),X),Y))
           => pp(aa(word(A),bool,ord_less_eq(word(A),aa(word(A),word(A),minus_minus(word(A),Y),X)),W)) ) ) ) ).

% word_diff_ls'(4)
tff(fact_1730_More__Word_Oword__l__diffs_I3_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),Z: word(A),X: word(A)] :
          ( pp(aa(word(A),bool,ord_less_eq(word(A),W),aa(word(A),word(A),minus_minus(word(A),Z),X)))
         => ( pp(aa(word(A),bool,ord_less_eq(word(A),X),Z))
           => pp(aa(word(A),bool,ord_less_eq(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),X)),Z)) ) ) ) ).

% More_Word.word_l_diffs(3)
tff(fact_1731_More__Word_Oword__l__diffs_I4_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),X: word(A),Z: word(A)] :
          ( pp(aa(word(A),bool,ord_less_eq(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),X)),Z))
         => ( pp(aa(word(A),bool,ord_less_eq(word(A),X),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),X)))
           => pp(aa(word(A),bool,ord_less_eq(word(A),W),aa(word(A),word(A),minus_minus(word(A),Z),X))) ) ) ) ).

% More_Word.word_l_diffs(4)
tff(fact_1732_word__plus__mcs_I3_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [V: word(A),Xb: word(A),X: word(A),W: word(A),Xa2: word(A)] :
          ( pp(aa(word(A),bool,ord_less_eq(word(A),aa(word(A),word(A),minus_minus(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),V),Xb)),X)),aa(word(A),word(A),minus_minus(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xa2)),X)))
         => ( pp(aa(word(A),bool,ord_less_eq(word(A),X),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xa2)))
           => pp(aa(word(A),bool,ord_less_eq(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),V),Xb)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xa2))) ) ) ) ).

% word_plus_mcs(3)
tff(fact_1733_word__plus__mcs_I4_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [V: word(A),Xb: word(A),W: word(A),Xa2: word(A),X: word(A)] :
          ( pp(aa(word(A),bool,ord_less_eq(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),V),Xb)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xa2)))
         => ( pp(aa(word(A),bool,ord_less_eq(word(A),X),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),V),Xb)))
           => pp(aa(word(A),bool,ord_less_eq(word(A),aa(word(A),word(A),minus_minus(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),V),Xb)),X)),aa(word(A),word(A),minus_minus(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xa2)),X))) ) ) ) ).

% word_plus_mcs(4)
tff(fact_1734_Word_Oword__l__diffs_I3_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),Xa2: word(A),X: word(A),Z: word(A)] :
          ( pp(aa(word(A),bool,ord_less_eq(word(A),aa(word(A),word(A),minus_minus(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xa2)),X)),aa(word(A),word(A),minus_minus(word(A),Z),X)))
         => ( pp(aa(word(A),bool,ord_less_eq(word(A),X),Z))
           => pp(aa(word(A),bool,ord_less_eq(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xa2)),Z)) ) ) ) ).

% Word.word_l_diffs(3)
tff(fact_1735_Word_Oword__l__diffs_I4_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),Xa2: word(A),Z: word(A),X: word(A)] :
          ( pp(aa(word(A),bool,ord_less_eq(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xa2)),Z))
         => ( pp(aa(word(A),bool,ord_less_eq(word(A),X),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xa2)))
           => pp(aa(word(A),bool,ord_less_eq(word(A),aa(word(A),word(A),minus_minus(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xa2)),X)),aa(word(A),word(A),minus_minus(word(A),Z),X))) ) ) ) ).

% Word.word_l_diffs(4)
tff(fact_1736_word__diff__ls_I3_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),X: word(A),W: word(A),Xa2: word(A)] :
          ( pp(aa(word(A),bool,ord_less_eq(word(A),aa(word(A),word(A),minus_minus(word(A),Y),X)),aa(word(A),word(A),minus_minus(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xa2)),X)))
         => ( pp(aa(word(A),bool,ord_less_eq(word(A),X),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xa2)))
           => pp(aa(word(A),bool,ord_less_eq(word(A),Y),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xa2))) ) ) ) ).

% word_diff_ls(3)
tff(fact_1737_word__diff__ls_I4_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),W: word(A),Xa2: word(A),X: word(A)] :
          ( pp(aa(word(A),bool,ord_less_eq(word(A),Y),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xa2)))
         => ( pp(aa(word(A),bool,ord_less_eq(word(A),X),Y))
           => pp(aa(word(A),bool,ord_less_eq(word(A),aa(word(A),word(A),minus_minus(word(A),Y),X)),aa(word(A),word(A),minus_minus(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xa2)),X))) ) ) ) ).

% word_diff_ls(4)
tff(fact_1738_olen__add__eqv,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),X: word(A)] :
          ( pp(aa(word(A),bool,ord_less_eq(word(A),Y),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Y),X)))
        <=> pp(aa(word(A),bool,ord_less_eq(word(A),X),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Y),X))) ) ) ).

% olen_add_eqv
tff(fact_1739_le__no__overflow,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),B2: word(A),A2: word(A)] :
          ( pp(aa(word(A),bool,ord_less_eq(word(A),X),B2))
         => ( pp(aa(word(A),bool,ord_less_eq(word(A),A2),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),A2),B2)))
           => pp(aa(word(A),bool,ord_less_eq(word(A),X),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),A2),B2))) ) ) ) ).

% le_no_overflow
tff(fact_1740_word__random,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [P2: word(A),X5: word(A),X: word(A)] :
          ( pp(aa(word(A),bool,ord_less_eq(word(A),P2),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),P2),X5)))
         => ( pp(aa(word(A),bool,ord_less_eq(word(A),X),X5))
           => pp(aa(word(A),bool,ord_less_eq(word(A),P2),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),P2),X))) ) ) ) ).

% word_random
tff(fact_1741_word__sub__mono2,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A),C2: word(A),D: word(A)] :
          ( pp(aa(word(A),bool,ord_less_eq(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),A2),B2)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),C2),D)))
         => ( pp(aa(word(A),bool,ord_less_eq(word(A),C2),A2))
           => ( pp(aa(word(A),bool,ord_less_eq(word(A),B2),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),A2),B2)))
             => ( pp(aa(word(A),bool,ord_less_eq(word(A),D),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),C2),D)))
               => pp(aa(word(A),bool,ord_less_eq(word(A),B2),D)) ) ) ) ) ) ).

% word_sub_mono2
tff(fact_1742_word__le__plus__either,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A),Z: word(A)] :
          ( ( pp(aa(word(A),bool,ord_less_eq(word(A),X),Y))
            | pp(aa(word(A),bool,ord_less_eq(word(A),X),Z)) )
         => ( pp(aa(word(A),bool,ord_less_eq(word(A),Y),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Y),Z)))
           => pp(aa(word(A),bool,ord_less_eq(word(A),X),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Y),Z))) ) ) ) ).

% word_le_plus_either
tff(fact_1743_word__plus__mcs__3,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [V: word(A),W: word(A),X: word(A)] :
          ( pp(aa(word(A),bool,ord_less_eq(word(A),V),W))
         => ( pp(aa(word(A),bool,ord_less_eq(word(A),X),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),X)))
           => pp(aa(word(A),bool,ord_less_eq(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),V),X)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),X))) ) ) ) ).

% word_plus_mcs_3
tff(fact_1744_word__plus__mcs__4,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [V: word(A),X: word(A),W: word(A)] :
          ( pp(aa(word(A),bool,ord_less_eq(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),V),X)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),X)))
         => ( pp(aa(word(A),bool,ord_less_eq(word(A),X),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),V),X)))
           => pp(aa(word(A),bool,ord_less_eq(word(A),V),W)) ) ) ) ).

% word_plus_mcs_4
tff(fact_1745_word__plus__mono__right,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),Z: word(A),X: word(A)] :
          ( pp(aa(word(A),bool,ord_less_eq(word(A),Y),Z))
         => ( pp(aa(word(A),bool,ord_less_eq(word(A),X),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),X),Z)))
           => pp(aa(word(A),bool,ord_less_eq(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),X),Y)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),X),Z))) ) ) ) ).

% word_plus_mono_right
tff(fact_1746_word__add__le__iff2,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [I: word(A),K: word(A),J: word(A)] :
          ( pp(aa(word(A),bool,ord_less_eq(word(A),I),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),I),K)))
         => ( pp(aa(word(A),bool,ord_less_eq(word(A),J),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),J),K)))
           => ( pp(aa(word(A),bool,ord_less_eq(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),I),K)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),J),K)))
            <=> pp(aa(word(A),bool,ord_less_eq(word(A),I),J)) ) ) ) ) ).

% word_add_le_iff2
tff(fact_1747_word__plus__mcs__4_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),V: word(A),W: word(A)] :
          ( pp(aa(word(A),bool,ord_less_eq(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),X),V)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),X),W)))
         => ( pp(aa(word(A),bool,ord_less_eq(word(A),X),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),X),V)))
           => pp(aa(word(A),bool,ord_less_eq(word(A),V),W)) ) ) ) ).

% word_plus_mcs_4'
tff(fact_1748_word__plus__mono__right2,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A),C2: word(A)] :
          ( pp(aa(word(A),bool,ord_less_eq(word(A),A2),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),A2),B2)))
         => ( pp(aa(word(A),bool,ord_less_eq(word(A),C2),B2))
           => pp(aa(word(A),bool,ord_less_eq(word(A),A2),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),A2),C2))) ) ) ) ).

% word_plus_mono_right2
tff(fact_1749_word__add__increasing,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [P2: word(A),W: word(A),X: word(A)] :
          ( pp(aa(word(A),bool,ord_less_eq(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),P2),W)),X))
         => ( pp(aa(word(A),bool,ord_less_eq(word(A),P2),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),P2),W)))
           => pp(aa(word(A),bool,ord_less_eq(word(A),P2),X)) ) ) ) ).

% word_add_increasing
tff(fact_1750_word__plus__mono__left,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),Z: word(A),X: word(A)] :
          ( pp(aa(word(A),bool,ord_less_eq(word(A),Y),Z))
         => ( pp(aa(word(A),bool,ord_less_eq(word(A),X),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),X),Z)))
           => pp(aa(word(A),bool,ord_less_eq(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Y),X)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Z),X))) ) ) ) ).

% word_plus_mono_left
tff(fact_1751_word__coorder_Oextremum,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A)] : pp(aa(word(A),bool,ord_less_eq(word(A),zero_zero(word(A))),A2)) ) ).

% word_coorder.extremum
tff(fact_1752_word__coorder_Oextremum__uniqueI,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A)] :
          ( pp(aa(word(A),bool,ord_less_eq(word(A),A2),zero_zero(word(A))))
         => ( A2 = zero_zero(word(A)) ) ) ) ).

% word_coorder.extremum_uniqueI
tff(fact_1753_word__zero__le,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A)] : pp(aa(word(A),bool,ord_less_eq(word(A),zero_zero(word(A))),Y)) ) ).

% word_zero_le
tff(fact_1754_ex__le__of__int,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [X: A] :
        ? [Z2: int] : pp(aa(A,bool,ord_less_eq(A,X),aa(int,A,ring_1_of_int(A),Z2))) ) ).

% ex_le_of_int
tff(fact_1755_mult__of__int__commute,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [X: int,Y: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(int,A,ring_1_of_int(A),X)),Y) = aa(A,A,aa(A,fun(A,A),times_times(A),Y),aa(int,A,ring_1_of_int(A),X)) ) ).

% mult_of_int_commute
tff(fact_1756_word__subset__less,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),R2: word(A),Y: word(A),S3: word(A)] :
          ( pp(aa(set(word(A)),bool,ord_less_eq(set(word(A)),set_or1337092689740270186AtMost(word(A),X,aa(word(A),word(A),minus_minus(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),X),R2)),one_one(word(A))))),set_or1337092689740270186AtMost(word(A),Y,aa(word(A),word(A),minus_minus(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Y),S3)),one_one(word(A))))))
         => ( pp(aa(word(A),bool,ord_less_eq(word(A),X),aa(word(A),word(A),minus_minus(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),X),R2)),one_one(word(A)))))
           => ( pp(aa(word(A),bool,ord_less_eq(word(A),Y),aa(word(A),word(A),minus_minus(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Y),S3)),one_one(word(A)))))
             => ( ( S3 != zero_zero(word(A)) )
               => pp(aa(word(A),bool,ord_less_eq(word(A),R2),S3)) ) ) ) ) ) ).

% word_subset_less
tff(fact_1757_le__of__int__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] : pp(aa(A,bool,ord_less_eq(A,X),aa(int,A,ring_1_of_int(A),archimedean_ceiling(A,X)))) ) ).

% le_of_int_ceiling
tff(fact_1758_bset_I1_J,axiom,
    ! [D4: int,B4: set(int),P: fun(int,bool),Q: fun(int,bool)] :
      ( ! [X3: int] :
          ( ! [Xa: int] :
              ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xa),set_or1337092689740270186AtMost(int,one_one(int),D4)))
             => ! [Xb2: int] :
                  ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xb2),B4))
                 => ( X3 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb2),Xa) ) ) )
         => ( pp(aa(int,bool,P,X3))
           => pp(aa(int,bool,P,aa(int,int,minus_minus(int,X3),D4))) ) )
     => ( ! [X3: int] :
            ( ! [Xa: int] :
                ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xa),set_or1337092689740270186AtMost(int,one_one(int),D4)))
               => ! [Xb2: int] :
                    ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xb2),B4))
                   => ( X3 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb2),Xa) ) ) )
           => ( pp(aa(int,bool,Q,X3))
             => pp(aa(int,bool,Q,aa(int,int,minus_minus(int,X3),D4))) ) )
       => ! [X4: int] :
            ( ! [Xa3: int] :
                ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xa3),set_or1337092689740270186AtMost(int,one_one(int),D4)))
               => ! [Xb3: int] :
                    ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xb3),B4))
                   => ( X4 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb3),Xa3) ) ) )
           => ( ( pp(aa(int,bool,P,X4))
                & pp(aa(int,bool,Q,X4)) )
             => ( pp(aa(int,bool,P,aa(int,int,minus_minus(int,X4),D4)))
                & pp(aa(int,bool,Q,aa(int,int,minus_minus(int,X4),D4))) ) ) ) ) ) ).

% bset(1)
tff(fact_1759_bset_I2_J,axiom,
    ! [D4: int,B4: set(int),P: fun(int,bool),Q: fun(int,bool)] :
      ( ! [X3: int] :
          ( ! [Xa: int] :
              ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xa),set_or1337092689740270186AtMost(int,one_one(int),D4)))
             => ! [Xb2: int] :
                  ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xb2),B4))
                 => ( X3 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb2),Xa) ) ) )
         => ( pp(aa(int,bool,P,X3))
           => pp(aa(int,bool,P,aa(int,int,minus_minus(int,X3),D4))) ) )
     => ( ! [X3: int] :
            ( ! [Xa: int] :
                ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xa),set_or1337092689740270186AtMost(int,one_one(int),D4)))
               => ! [Xb2: int] :
                    ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xb2),B4))
                   => ( X3 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb2),Xa) ) ) )
           => ( pp(aa(int,bool,Q,X3))
             => pp(aa(int,bool,Q,aa(int,int,minus_minus(int,X3),D4))) ) )
       => ! [X4: int] :
            ( ! [Xa3: int] :
                ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xa3),set_or1337092689740270186AtMost(int,one_one(int),D4)))
               => ! [Xb3: int] :
                    ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xb3),B4))
                   => ( X4 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb3),Xa3) ) ) )
           => ( ( pp(aa(int,bool,P,X4))
                | pp(aa(int,bool,Q,X4)) )
             => ( pp(aa(int,bool,P,aa(int,int,minus_minus(int,X4),D4)))
                | pp(aa(int,bool,Q,aa(int,int,minus_minus(int,X4),D4))) ) ) ) ) ) ).

% bset(2)
tff(fact_1760_aset_I1_J,axiom,
    ! [D4: int,A3: set(int),P: fun(int,bool),Q: fun(int,bool)] :
      ( ! [X3: int] :
          ( ! [Xa: int] :
              ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xa),set_or1337092689740270186AtMost(int,one_one(int),D4)))
             => ! [Xb2: int] :
                  ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xb2),A3))
                 => ( X3 != aa(int,int,minus_minus(int,Xb2),Xa) ) ) )
         => ( pp(aa(int,bool,P,X3))
           => pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),X3),D4))) ) )
     => ( ! [X3: int] :
            ( ! [Xa: int] :
                ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xa),set_or1337092689740270186AtMost(int,one_one(int),D4)))
               => ! [Xb2: int] :
                    ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xb2),A3))
                   => ( X3 != aa(int,int,minus_minus(int,Xb2),Xa) ) ) )
           => ( pp(aa(int,bool,Q,X3))
             => pp(aa(int,bool,Q,aa(int,int,aa(int,fun(int,int),plus_plus(int),X3),D4))) ) )
       => ! [X4: int] :
            ( ! [Xa3: int] :
                ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xa3),set_or1337092689740270186AtMost(int,one_one(int),D4)))
               => ! [Xb3: int] :
                    ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xb3),A3))
                   => ( X4 != aa(int,int,minus_minus(int,Xb3),Xa3) ) ) )
           => ( ( pp(aa(int,bool,P,X4))
                & pp(aa(int,bool,Q,X4)) )
             => ( pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),X4),D4)))
                & pp(aa(int,bool,Q,aa(int,int,aa(int,fun(int,int),plus_plus(int),X4),D4))) ) ) ) ) ) ).

% aset(1)
tff(fact_1761_aset_I2_J,axiom,
    ! [D4: int,A3: set(int),P: fun(int,bool),Q: fun(int,bool)] :
      ( ! [X3: int] :
          ( ! [Xa: int] :
              ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xa),set_or1337092689740270186AtMost(int,one_one(int),D4)))
             => ! [Xb2: int] :
                  ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xb2),A3))
                 => ( X3 != aa(int,int,minus_minus(int,Xb2),Xa) ) ) )
         => ( pp(aa(int,bool,P,X3))
           => pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),X3),D4))) ) )
     => ( ! [X3: int] :
            ( ! [Xa: int] :
                ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xa),set_or1337092689740270186AtMost(int,one_one(int),D4)))
               => ! [Xb2: int] :
                    ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xb2),A3))
                   => ( X3 != aa(int,int,minus_minus(int,Xb2),Xa) ) ) )
           => ( pp(aa(int,bool,Q,X3))
             => pp(aa(int,bool,Q,aa(int,int,aa(int,fun(int,int),plus_plus(int),X3),D4))) ) )
       => ! [X4: int] :
            ( ! [Xa3: int] :
                ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xa3),set_or1337092689740270186AtMost(int,one_one(int),D4)))
               => ! [Xb3: int] :
                    ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xb3),A3))
                   => ( X4 != aa(int,int,minus_minus(int,Xb3),Xa3) ) ) )
           => ( ( pp(aa(int,bool,P,X4))
                | pp(aa(int,bool,Q,X4)) )
             => ( pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),X4),D4)))
                | pp(aa(int,bool,Q,aa(int,int,aa(int,fun(int,int),plus_plus(int),X4),D4))) ) ) ) ) ) ).

% aset(2)
tff(fact_1762_ceiling__le,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,A2: int] :
          ( pp(aa(A,bool,ord_less_eq(A,X),aa(int,A,ring_1_of_int(A),A2)))
         => pp(aa(int,bool,ord_less_eq(int,archimedean_ceiling(A,X)),A2)) ) ) ).

% ceiling_le
tff(fact_1763_ceiling__le__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,Z: int] :
          ( pp(aa(int,bool,ord_less_eq(int,archimedean_ceiling(A,X)),Z))
        <=> pp(aa(A,bool,ord_less_eq(A,X),aa(int,A,ring_1_of_int(A),Z))) ) ) ).

% ceiling_le_iff
tff(fact_1764_real__of__int__div4,axiom,
    ! [N: int,X: int] : pp(aa(real,bool,ord_less_eq(real,aa(int,real,ring_1_of_int(real),aa(int,int,aa(int,fun(int,int),divide_divide(int),N),X))),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(int,real,ring_1_of_int(real),N)),aa(int,real,ring_1_of_int(real),X)))) ).

% real_of_int_div4
tff(fact_1765_real__of__int__div,axiom,
    ! [D: int,N: int] :
      ( pp(aa(int,bool,dvd_dvd(int,D),N))
     => ( aa(int,real,ring_1_of_int(real),aa(int,int,aa(int,fun(int,int),divide_divide(int),N),D)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(int,real,ring_1_of_int(real),N)),aa(int,real,ring_1_of_int(real),D)) ) ) ).

% real_of_int_div
tff(fact_1766_le__minus,axiom,
    ! [Aa: $tType,A: $tType] :
      ( ( type_len(A)
        & order(Aa) )
     => ! [Y: Aa,X: Aa,A2: word(A),C2: word(A),B2: word(A)] :
          ( pp(aa(Aa,bool,ord_less_eq(Aa,Y),X))
         => ( pp(aa(word(A),bool,ord_less_eq(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),A2),C2)),B2))
           => ( pp(aa(word(A),bool,ord_less_eq(word(A),A2),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),A2),C2)))
             => pp(aa(word(A),bool,ord_less_eq(word(A),C2),aa(word(A),word(A),minus_minus(word(A),B2),A2))) ) ) ) ) ).

% le_minus
tff(fact_1767_bounded__Max__nat,axiom,
    ! [P: fun(nat,bool),X: nat,M7: nat] :
      ( pp(aa(nat,bool,P,X))
     => ( ! [X3: nat] :
            ( pp(aa(nat,bool,P,X3))
           => pp(aa(nat,bool,ord_less_eq(nat,X3),M7)) )
       => ~ ! [M3: nat] :
              ( pp(aa(nat,bool,P,M3))
             => ~ ! [X4: nat] :
                    ( pp(aa(nat,bool,P,X4))
                   => pp(aa(nat,bool,ord_less_eq(nat,X4),M3)) ) ) ) ) ).

% bounded_Max_nat
tff(fact_1768_word__induct2,axiom,
    ! [B: $tType] :
      ( type_len(B)
     => ! [P: fun(word(B),bool),N: word(B)] :
          ( pp(aa(word(B),bool,P,zero_zero(word(B))))
         => ( ! [N2: word(B)] :
                ( ( aa(word(B),word(B),aa(word(B),fun(word(B),word(B)),plus_plus(word(B)),one_one(word(B))),N2) != zero_zero(word(B)) )
               => ( pp(aa(word(B),bool,P,N2))
                 => pp(aa(word(B),bool,P,aa(word(B),word(B),aa(word(B),fun(word(B),word(B)),plus_plus(word(B)),one_one(word(B))),N2))) ) )
           => pp(aa(word(B),bool,P,N)) ) ) ) ).

% word_induct2
tff(fact_1769_word__induct,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [P: fun(word(A),bool),M: word(A)] :
          ( pp(aa(word(A),bool,P,zero_zero(word(A))))
         => ( ! [N2: word(A)] :
                ( pp(aa(word(A),bool,P,N2))
               => pp(aa(word(A),bool,P,aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),one_one(word(A))),N2))) )
           => pp(aa(word(A),bool,P,M)) ) ) ) ).

% word_induct
tff(fact_1770_word__le__sub1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A)] :
          ( ( X != zero_zero(word(A)) )
         => ( pp(aa(word(A),bool,ord_less_eq(word(A),one_one(word(A))),X))
          <=> pp(aa(word(A),bool,ord_less_eq(word(A),zero_zero(word(A))),aa(word(A),word(A),minus_minus(word(A),X),one_one(word(A))))) ) ) ) ).

% word_le_sub1
tff(fact_1771_plus__le__left__cancel__wrap,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y5: word(A),Y: word(A)] :
          ( pp(aa(word(A),bool,ord_less(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),X),Y5)),X))
         => ( pp(aa(word(A),bool,ord_less(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),X),Y)),X))
           => ( pp(aa(word(A),bool,ord_less(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),X),Y5)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),X),Y)))
            <=> pp(aa(word(A),bool,ord_less(word(A),Y5),Y)) ) ) ) ) ).

% plus_le_left_cancel_wrap
tff(fact_1772_word__greater__zero__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A)] :
          ( pp(aa(word(A),bool,ord_less(word(A),zero_zero(word(A))),A2))
        <=> ( A2 != zero_zero(word(A)) ) ) ) ).

% word_greater_zero_iff
tff(fact_1773_word__neq__0__conv,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] :
          ( ( W != zero_zero(word(A)) )
        <=> pp(aa(word(A),bool,ord_less(word(A),zero_zero(word(A))),W)) ) ) ).

% word_neq_0_conv
tff(fact_1774_word__gt__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A)] :
          ( pp(aa(word(A),bool,ord_less(word(A),zero_zero(word(A))),Y))
        <=> ( zero_zero(word(A)) != Y ) ) ) ).

% word_gt_0
tff(fact_1775_word__coorder_Oextremum__strict,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A)] : ~ pp(aa(word(A),bool,ord_less(word(A),A2),zero_zero(word(A)))) ) ).

% word_coorder.extremum_strict
tff(fact_1776_word__not__simps_I1_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A)] : ~ pp(aa(word(A),bool,ord_less(word(A),X),zero_zero(word(A)))) ) ).

% word_not_simps(1)
tff(fact_1777_plus__le__left__cancel__nowrap,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y5: word(A),Y: word(A)] :
          ( pp(aa(word(A),bool,ord_less_eq(word(A),X),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),X),Y5)))
         => ( pp(aa(word(A),bool,ord_less_eq(word(A),X),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),X),Y)))
           => ( pp(aa(word(A),bool,ord_less(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),X),Y5)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),X),Y)))
            <=> pp(aa(word(A),bool,ord_less(word(A),Y5),Y)) ) ) ) ) ).

% plus_le_left_cancel_nowrap
tff(fact_1778_plus__minus__no__overflow,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Ab: word(A),C2: word(A)] :
          ( pp(aa(word(A),bool,ord_less(word(A),X),aa(word(A),word(A),minus_minus(word(A),Ab),C2)))
         => ( pp(aa(word(A),bool,ord_less_eq(word(A),C2),Ab))
           => pp(aa(word(A),bool,ord_less_eq(word(A),X),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),X),C2))) ) ) ) ).

% plus_minus_no_overflow
tff(fact_1779_word__less__sub__right,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A),Z: word(A)] :
          ( pp(aa(word(A),bool,ord_less(word(A),X),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Y),Z)))
         => ( pp(aa(word(A),bool,ord_less_eq(word(A),Y),X))
           => pp(aa(word(A),bool,ord_less(word(A),aa(word(A),word(A),minus_minus(word(A),X),Y)),Z)) ) ) ) ).

% word_less_sub_right
tff(fact_1780_word__less__add__right,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A),Z: word(A)] :
          ( pp(aa(word(A),bool,ord_less(word(A),X),aa(word(A),word(A),minus_minus(word(A),Y),Z)))
         => ( pp(aa(word(A),bool,ord_less_eq(word(A),Z),Y))
           => pp(aa(word(A),bool,ord_less(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),X),Z)),Y)) ) ) ) ).

% word_less_add_right
tff(fact_1781_plus__minus__not__NULL,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Ab: word(A),C2: word(A)] :
          ( pp(aa(word(A),bool,ord_less(word(A),X),aa(word(A),word(A),minus_minus(word(A),Ab),C2)))
         => ( pp(aa(word(A),bool,ord_less_eq(word(A),C2),Ab))
           => ( ( C2 != zero_zero(word(A)) )
             => ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),X),C2) != zero_zero(word(A)) ) ) ) ) ) ).

% plus_minus_not_NULL
tff(fact_1782_word__less__nowrapI,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Z: word(A),K: word(A)] :
          ( pp(aa(word(A),bool,ord_less(word(A),X),aa(word(A),word(A),minus_minus(word(A),Z),K)))
         => ( pp(aa(word(A),bool,ord_less_eq(word(A),K),Z))
           => ( pp(aa(word(A),bool,ord_less(word(A),zero_zero(word(A))),K))
             => pp(aa(word(A),bool,ord_less(word(A),X),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),X),K))) ) ) ) ) ).

% word_less_nowrapI
tff(fact_1783_sub__wrap,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Z: word(A)] :
          ( pp(aa(word(A),bool,ord_less_eq(word(A),X),aa(word(A),word(A),minus_minus(word(A),X),Z)))
        <=> ( ( Z = zero_zero(word(A)) )
            | pp(aa(word(A),bool,ord_less(word(A),X),Z)) ) ) ) ).

% sub_wrap
tff(fact_1784_word__plus__mcs_I1_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [V: word(A),Xb: word(A),X: word(A),W: word(A),Xa2: word(A)] :
          ( pp(aa(word(A),bool,ord_less(word(A),aa(word(A),word(A),minus_minus(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),V),Xb)),X)),aa(word(A),word(A),minus_minus(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xa2)),X)))
         => ( pp(aa(word(A),bool,ord_less_eq(word(A),X),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xa2)))
           => pp(aa(word(A),bool,ord_less(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),V),Xb)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xa2))) ) ) ) ).

% word_plus_mcs(1)
tff(fact_1785_word__plus__mcs_I2_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [V: word(A),Xb: word(A),W: word(A),Xa2: word(A),X: word(A)] :
          ( pp(aa(word(A),bool,ord_less(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),V),Xb)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xa2)))
         => ( pp(aa(word(A),bool,ord_less_eq(word(A),X),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),V),Xb)))
           => pp(aa(word(A),bool,ord_less(word(A),aa(word(A),word(A),minus_minus(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),V),Xb)),X)),aa(word(A),word(A),minus_minus(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xa2)),X))) ) ) ) ).

% word_plus_mcs(2)
tff(fact_1786_Word_Oword__l__diffs_I1_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),Xa2: word(A),X: word(A),Z: word(A)] :
          ( pp(aa(word(A),bool,ord_less(word(A),aa(word(A),word(A),minus_minus(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xa2)),X)),aa(word(A),word(A),minus_minus(word(A),Z),X)))
         => ( pp(aa(word(A),bool,ord_less_eq(word(A),X),Z))
           => pp(aa(word(A),bool,ord_less(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xa2)),Z)) ) ) ) ).

% Word.word_l_diffs(1)
tff(fact_1787_Word_Oword__l__diffs_I2_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),Xa2: word(A),Z: word(A),X: word(A)] :
          ( pp(aa(word(A),bool,ord_less(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xa2)),Z))
         => ( pp(aa(word(A),bool,ord_less_eq(word(A),X),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xa2)))
           => pp(aa(word(A),bool,ord_less(word(A),aa(word(A),word(A),minus_minus(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xa2)),X)),aa(word(A),word(A),minus_minus(word(A),Z),X))) ) ) ) ).

% Word.word_l_diffs(2)
tff(fact_1788_word__diff__ls_I1_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),X: word(A),W: word(A),Xa2: word(A)] :
          ( pp(aa(word(A),bool,ord_less(word(A),aa(word(A),word(A),minus_minus(word(A),Y),X)),aa(word(A),word(A),minus_minus(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xa2)),X)))
         => ( pp(aa(word(A),bool,ord_less_eq(word(A),X),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xa2)))
           => pp(aa(word(A),bool,ord_less(word(A),Y),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xa2))) ) ) ) ).

% word_diff_ls(1)
tff(fact_1789_word__diff__ls_I2_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),W: word(A),Xa2: word(A),X: word(A)] :
          ( pp(aa(word(A),bool,ord_less(word(A),Y),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xa2)))
         => ( pp(aa(word(A),bool,ord_less_eq(word(A),X),Y))
           => pp(aa(word(A),bool,ord_less(word(A),aa(word(A),word(A),minus_minus(word(A),Y),X)),aa(word(A),word(A),minus_minus(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xa2)),X))) ) ) ) ).

% word_diff_ls(2)
tff(fact_1790_of__int__nonneg,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: int] :
          ( pp(aa(int,bool,ord_less_eq(int,zero_zero(int)),Z))
         => pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),aa(int,A,ring_1_of_int(A),Z))) ) ) ).

% of_int_nonneg
tff(fact_1791_of__int__pos,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: int] :
          ( pp(aa(int,bool,ord_less(int,zero_zero(int)),Z))
         => pp(aa(A,bool,ord_less(A,zero_zero(A)),aa(int,A,ring_1_of_int(A),Z))) ) ) ).

% of_int_pos
tff(fact_1792_word__div__mult__le,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A)] : pp(aa(word(A),bool,ord_less_eq(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),A2),B2)),B2)),A2)) ) ).

% word_div_mult_le
tff(fact_1793_floor__exists,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [X: A] :
        ? [Z2: int] :
          ( pp(aa(A,bool,ord_less_eq(A,aa(int,A,ring_1_of_int(A),Z2)),X))
          & pp(aa(A,bool,ord_less(A,X),aa(int,A,ring_1_of_int(A),aa(int,int,aa(int,fun(int,int),plus_plus(int),Z2),one_one(int))))) ) ) ).

% floor_exists
tff(fact_1794_floor__exists1,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [X: A] :
        ? [X3: int] :
          ( pp(aa(A,bool,ord_less_eq(A,aa(int,A,ring_1_of_int(A),X3)),X))
          & pp(aa(A,bool,ord_less(A,X),aa(int,A,ring_1_of_int(A),aa(int,int,aa(int,fun(int,int),plus_plus(int),X3),one_one(int)))))
          & ! [Y4: int] :
              ( ( pp(aa(A,bool,ord_less_eq(A,aa(int,A,ring_1_of_int(A),Y4)),X))
                & pp(aa(A,bool,ord_less(A,X),aa(int,A,ring_1_of_int(A),aa(int,int,aa(int,fun(int,int),plus_plus(int),Y4),one_one(int))))) )
             => ( Y4 = X3 ) ) ) ) ).

% floor_exists1
tff(fact_1795_of__int__ceiling__le__add__one,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [R2: A] : pp(aa(A,bool,ord_less_eq(A,aa(int,A,ring_1_of_int(A),archimedean_ceiling(A,R2))),aa(A,A,aa(A,fun(A,A),plus_plus(A),R2),one_one(A)))) ) ).

% of_int_ceiling_le_add_one
tff(fact_1796_of__int__ceiling__diff__one__le,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [R2: A] : pp(aa(A,bool,ord_less_eq(A,aa(A,A,minus_minus(A,aa(int,A,ring_1_of_int(A),archimedean_ceiling(A,R2))),one_one(A))),R2)) ) ).

% of_int_ceiling_diff_one_le
tff(fact_1797_of__nat__less__of__int__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [N: nat,X: int] :
          ( pp(aa(A,bool,ord_less(A,aa(nat,A,semiring_1_of_nat(A),N)),aa(int,A,ring_1_of_int(A),X)))
        <=> pp(aa(int,bool,ord_less(int,aa(nat,int,semiring_1_of_nat(int),N)),X)) ) ) ).

% of_nat_less_of_int_iff
tff(fact_1798_bset_I9_J,axiom,
    ! [D: int,D4: int,B4: set(int),T: int] :
      ( pp(aa(int,bool,dvd_dvd(int,D),D4))
     => ! [X4: int] :
          ( ! [Xa3: int] :
              ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xa3),set_or1337092689740270186AtMost(int,one_one(int),D4)))
             => ! [Xb3: int] :
                  ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xb3),B4))
                 => ( X4 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb3),Xa3) ) ) )
         => ( pp(aa(int,bool,dvd_dvd(int,D),aa(int,int,aa(int,fun(int,int),plus_plus(int),X4),T)))
           => pp(aa(int,bool,dvd_dvd(int,D),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,minus_minus(int,X4),D4)),T))) ) ) ) ).

% bset(9)
tff(fact_1799_bset_I10_J,axiom,
    ! [D: int,D4: int,B4: set(int),T: int] :
      ( pp(aa(int,bool,dvd_dvd(int,D),D4))
     => ! [X4: int] :
          ( ! [Xa3: int] :
              ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xa3),set_or1337092689740270186AtMost(int,one_one(int),D4)))
             => ! [Xb3: int] :
                  ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xb3),B4))
                 => ( X4 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb3),Xa3) ) ) )
         => ( ~ pp(aa(int,bool,dvd_dvd(int,D),aa(int,int,aa(int,fun(int,int),plus_plus(int),X4),T)))
           => ~ pp(aa(int,bool,dvd_dvd(int,D),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,minus_minus(int,X4),D4)),T))) ) ) ) ).

% bset(10)
tff(fact_1800_aset_I9_J,axiom,
    ! [D: int,D4: int,A3: set(int),T: int] :
      ( pp(aa(int,bool,dvd_dvd(int,D),D4))
     => ! [X4: int] :
          ( ! [Xa3: int] :
              ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xa3),set_or1337092689740270186AtMost(int,one_one(int),D4)))
             => ! [Xb3: int] :
                  ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xb3),A3))
                 => ( X4 != aa(int,int,minus_minus(int,Xb3),Xa3) ) ) )
         => ( pp(aa(int,bool,dvd_dvd(int,D),aa(int,int,aa(int,fun(int,int),plus_plus(int),X4),T)))
           => pp(aa(int,bool,dvd_dvd(int,D),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),X4),D4)),T))) ) ) ) ).

% aset(9)
tff(fact_1801_aset_I10_J,axiom,
    ! [D: int,D4: int,A3: set(int),T: int] :
      ( pp(aa(int,bool,dvd_dvd(int,D),D4))
     => ! [X4: int] :
          ( ! [Xa3: int] :
              ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xa3),set_or1337092689740270186AtMost(int,one_one(int),D4)))
             => ! [Xb3: int] :
                  ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xb3),A3))
                 => ( X4 != aa(int,int,minus_minus(int,Xb3),Xa3) ) ) )
         => ( ~ pp(aa(int,bool,dvd_dvd(int,D),aa(int,int,aa(int,fun(int,int),plus_plus(int),X4),T)))
           => ~ pp(aa(int,bool,dvd_dvd(int,D),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),X4),D4)),T))) ) ) ) ).

% aset(10)
tff(fact_1802_int__le__real__less,axiom,
    ! [N: int,M: int] :
      ( pp(aa(int,bool,ord_less_eq(int,N),M))
    <=> pp(aa(real,bool,ord_less(real,aa(int,real,ring_1_of_int(real),N)),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(int,real,ring_1_of_int(real),M)),one_one(real)))) ) ).

% int_le_real_less
tff(fact_1803_int__less__real__le,axiom,
    ! [N: int,M: int] :
      ( pp(aa(int,bool,ord_less(int,N),M))
    <=> pp(aa(real,bool,ord_less_eq(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(int,real,ring_1_of_int(real),N)),one_one(real))),aa(int,real,ring_1_of_int(real),M))) ) ).

% int_less_real_le
tff(fact_1804_ceiling__split,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [P: fun(int,bool),T: A] :
          ( pp(aa(int,bool,P,archimedean_ceiling(A,T)))
        <=> ! [I3: int] :
              ( ( pp(aa(A,bool,ord_less(A,aa(A,A,minus_minus(A,aa(int,A,ring_1_of_int(A),I3)),one_one(A))),T))
                & pp(aa(A,bool,ord_less_eq(A,T),aa(int,A,ring_1_of_int(A),I3))) )
             => pp(aa(int,bool,P,I3)) ) ) ) ).

% ceiling_split
tff(fact_1805_ceiling__eq__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,A2: int] :
          ( ( archimedean_ceiling(A,X) = A2 )
        <=> ( pp(aa(A,bool,ord_less(A,aa(A,A,minus_minus(A,aa(int,A,ring_1_of_int(A),A2)),one_one(A))),X))
            & pp(aa(A,bool,ord_less_eq(A,X),aa(int,A,ring_1_of_int(A),A2))) ) ) ) ).

% ceiling_eq_iff
tff(fact_1806_ceiling__unique,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Z: int,X: A] :
          ( pp(aa(A,bool,ord_less(A,aa(A,A,minus_minus(A,aa(int,A,ring_1_of_int(A),Z)),one_one(A))),X))
         => ( pp(aa(A,bool,ord_less_eq(A,X),aa(int,A,ring_1_of_int(A),Z)))
           => ( archimedean_ceiling(A,X) = Z ) ) ) ) ).

% ceiling_unique
tff(fact_1807_ceiling__correct,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( pp(aa(A,bool,ord_less(A,aa(A,A,minus_minus(A,aa(int,A,ring_1_of_int(A),archimedean_ceiling(A,X))),one_one(A))),X))
          & pp(aa(A,bool,ord_less_eq(A,X),aa(int,A,ring_1_of_int(A),archimedean_ceiling(A,X)))) ) ) ).

% ceiling_correct
tff(fact_1808_ceiling__less__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,Z: int] :
          ( pp(aa(int,bool,ord_less(int,archimedean_ceiling(A,X)),Z))
        <=> pp(aa(A,bool,ord_less_eq(A,X),aa(A,A,minus_minus(A,aa(int,A,ring_1_of_int(A),Z)),one_one(A)))) ) ) ).

% ceiling_less_iff
tff(fact_1809_le__ceiling__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Z: int,X: A] :
          ( pp(aa(int,bool,ord_less_eq(int,Z),archimedean_ceiling(A,X)))
        <=> pp(aa(A,bool,ord_less(A,aa(A,A,minus_minus(A,aa(int,A,ring_1_of_int(A),Z)),one_one(A))),X)) ) ) ).

% le_ceiling_iff
tff(fact_1810_bset_I3_J,axiom,
    ! [D4: int,T: int,B4: set(int)] :
      ( pp(aa(int,bool,ord_less(int,zero_zero(int)),D4))
     => ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),aa(int,int,minus_minus(int,T),one_one(int))),B4))
       => ! [X4: int] :
            ( ! [Xa3: int] :
                ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xa3),set_or1337092689740270186AtMost(int,one_one(int),D4)))
               => ! [Xb3: int] :
                    ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xb3),B4))
                   => ( X4 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb3),Xa3) ) ) )
           => ( ( X4 = T )
             => ( aa(int,int,minus_minus(int,X4),D4) = T ) ) ) ) ) ).

% bset(3)
tff(fact_1811_bset_I4_J,axiom,
    ! [D4: int,T: int,B4: set(int)] :
      ( pp(aa(int,bool,ord_less(int,zero_zero(int)),D4))
     => ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),T),B4))
       => ! [X4: int] :
            ( ! [Xa3: int] :
                ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xa3),set_or1337092689740270186AtMost(int,one_one(int),D4)))
               => ! [Xb3: int] :
                    ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xb3),B4))
                   => ( X4 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb3),Xa3) ) ) )
           => ( ( X4 != T )
             => ( aa(int,int,minus_minus(int,X4),D4) != T ) ) ) ) ) ).

% bset(4)
tff(fact_1812_bset_I5_J,axiom,
    ! [D4: int,B4: set(int),T: int] :
      ( pp(aa(int,bool,ord_less(int,zero_zero(int)),D4))
     => ! [X4: int] :
          ( ! [Xa3: int] :
              ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xa3),set_or1337092689740270186AtMost(int,one_one(int),D4)))
             => ! [Xb3: int] :
                  ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xb3),B4))
                 => ( X4 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb3),Xa3) ) ) )
         => ( pp(aa(int,bool,ord_less(int,X4),T))
           => pp(aa(int,bool,ord_less(int,aa(int,int,minus_minus(int,X4),D4)),T)) ) ) ) ).

% bset(5)
tff(fact_1813_bset_I7_J,axiom,
    ! [D4: int,T: int,B4: set(int)] :
      ( pp(aa(int,bool,ord_less(int,zero_zero(int)),D4))
     => ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),T),B4))
       => ! [X4: int] :
            ( ! [Xa3: int] :
                ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xa3),set_or1337092689740270186AtMost(int,one_one(int),D4)))
               => ! [Xb3: int] :
                    ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xb3),B4))
                   => ( X4 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb3),Xa3) ) ) )
           => ( pp(aa(int,bool,ord_less(int,T),X4))
             => pp(aa(int,bool,ord_less(int,T),aa(int,int,minus_minus(int,X4),D4))) ) ) ) ) ).

% bset(7)
tff(fact_1814_aset_I3_J,axiom,
    ! [D4: int,T: int,A3: set(int)] :
      ( pp(aa(int,bool,ord_less(int,zero_zero(int)),D4))
     => ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),T),one_one(int))),A3))
       => ! [X4: int] :
            ( ! [Xa3: int] :
                ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xa3),set_or1337092689740270186AtMost(int,one_one(int),D4)))
               => ! [Xb3: int] :
                    ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xb3),A3))
                   => ( X4 != aa(int,int,minus_minus(int,Xb3),Xa3) ) ) )
           => ( ( X4 = T )
             => ( aa(int,int,aa(int,fun(int,int),plus_plus(int),X4),D4) = T ) ) ) ) ) ).

% aset(3)
tff(fact_1815_aset_I4_J,axiom,
    ! [D4: int,T: int,A3: set(int)] :
      ( pp(aa(int,bool,ord_less(int,zero_zero(int)),D4))
     => ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),T),A3))
       => ! [X4: int] :
            ( ! [Xa3: int] :
                ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xa3),set_or1337092689740270186AtMost(int,one_one(int),D4)))
               => ! [Xb3: int] :
                    ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xb3),A3))
                   => ( X4 != aa(int,int,minus_minus(int,Xb3),Xa3) ) ) )
           => ( ( X4 != T )
             => ( aa(int,int,aa(int,fun(int,int),plus_plus(int),X4),D4) != T ) ) ) ) ) ).

% aset(4)
tff(fact_1816_aset_I5_J,axiom,
    ! [D4: int,T: int,A3: set(int)] :
      ( pp(aa(int,bool,ord_less(int,zero_zero(int)),D4))
     => ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),T),A3))
       => ! [X4: int] :
            ( ! [Xa3: int] :
                ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xa3),set_or1337092689740270186AtMost(int,one_one(int),D4)))
               => ! [Xb3: int] :
                    ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xb3),A3))
                   => ( X4 != aa(int,int,minus_minus(int,Xb3),Xa3) ) ) )
           => ( pp(aa(int,bool,ord_less(int,X4),T))
             => pp(aa(int,bool,ord_less(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),X4),D4)),T)) ) ) ) ) ).

% aset(5)
tff(fact_1817_aset_I7_J,axiom,
    ! [D4: int,A3: set(int),T: int] :
      ( pp(aa(int,bool,ord_less(int,zero_zero(int)),D4))
     => ! [X4: int] :
          ( ! [Xa3: int] :
              ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xa3),set_or1337092689740270186AtMost(int,one_one(int),D4)))
             => ! [Xb3: int] :
                  ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xb3),A3))
                 => ( X4 != aa(int,int,minus_minus(int,Xb3),Xa3) ) ) )
         => ( pp(aa(int,bool,ord_less(int,T),X4))
           => pp(aa(int,bool,ord_less(int,T),aa(int,int,aa(int,fun(int,int),plus_plus(int),X4),D4))) ) ) ) ).

% aset(7)
tff(fact_1818_periodic__finite__ex,axiom,
    ! [D: int,P: fun(int,bool)] :
      ( pp(aa(int,bool,ord_less(int,zero_zero(int)),D))
     => ( ! [X3: int,K3: int] :
            ( pp(aa(int,bool,P,X3))
          <=> pp(aa(int,bool,P,aa(int,int,minus_minus(int,X3),aa(int,int,aa(int,fun(int,int),times_times(int),K3),D)))) )
       => ( ? [X_13: int] : pp(aa(int,bool,P,X_13))
        <=> ? [X2: int] :
              ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),X2),set_or1337092689740270186AtMost(int,one_one(int),D)))
              & pp(aa(int,bool,P,X2)) ) ) ) ) ).

% periodic_finite_ex
tff(fact_1819_real__of__int__div2,axiom,
    ! [N: int,X: int] : pp(aa(real,bool,ord_less_eq(real,zero_zero(real)),aa(real,real,minus_minus(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(int,real,ring_1_of_int(real),N)),aa(int,real,ring_1_of_int(real),X))),aa(int,real,ring_1_of_int(real),aa(int,int,aa(int,fun(int,int),divide_divide(int),N),X))))) ).

% real_of_int_div2
tff(fact_1820_real__of__int__div3,axiom,
    ! [N: int,X: int] : pp(aa(real,bool,ord_less_eq(real,aa(real,real,minus_minus(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(int,real,ring_1_of_int(real),N)),aa(int,real,ring_1_of_int(real),X))),aa(int,real,ring_1_of_int(real),aa(int,int,aa(int,fun(int,int),divide_divide(int),N),X)))),one_one(real))) ).

% real_of_int_div3
tff(fact_1821_ceiling__divide__upper,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Q2: A,P2: A] :
          ( pp(aa(A,bool,ord_less(A,zero_zero(A)),Q2))
         => pp(aa(A,bool,ord_less_eq(A,P2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(int,A,ring_1_of_int(A),archimedean_ceiling(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),P2),Q2)))),Q2))) ) ) ).

% ceiling_divide_upper
tff(fact_1822_even__of__int__iff,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [K: int] :
          ( pp(aa(A,bool,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),aa(int,A,ring_1_of_int(A),K)))
        <=> pp(aa(int,bool,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),K)) ) ) ).

% even_of_int_iff
tff(fact_1823_bset_I6_J,axiom,
    ! [D4: int,B4: set(int),T: int] :
      ( pp(aa(int,bool,ord_less(int,zero_zero(int)),D4))
     => ! [X4: int] :
          ( ! [Xa3: int] :
              ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xa3),set_or1337092689740270186AtMost(int,one_one(int),D4)))
             => ! [Xb3: int] :
                  ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xb3),B4))
                 => ( X4 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb3),Xa3) ) ) )
         => ( pp(aa(int,bool,ord_less_eq(int,X4),T))
           => pp(aa(int,bool,ord_less_eq(int,aa(int,int,minus_minus(int,X4),D4)),T)) ) ) ) ).

% bset(6)
tff(fact_1824_bset_I8_J,axiom,
    ! [D4: int,T: int,B4: set(int)] :
      ( pp(aa(int,bool,ord_less(int,zero_zero(int)),D4))
     => ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),aa(int,int,minus_minus(int,T),one_one(int))),B4))
       => ! [X4: int] :
            ( ! [Xa3: int] :
                ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xa3),set_or1337092689740270186AtMost(int,one_one(int),D4)))
               => ! [Xb3: int] :
                    ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xb3),B4))
                   => ( X4 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb3),Xa3) ) ) )
           => ( pp(aa(int,bool,ord_less_eq(int,T),X4))
             => pp(aa(int,bool,ord_less_eq(int,T),aa(int,int,minus_minus(int,X4),D4))) ) ) ) ) ).

% bset(8)
tff(fact_1825_aset_I6_J,axiom,
    ! [D4: int,T: int,A3: set(int)] :
      ( pp(aa(int,bool,ord_less(int,zero_zero(int)),D4))
     => ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),T),one_one(int))),A3))
       => ! [X4: int] :
            ( ! [Xa3: int] :
                ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xa3),set_or1337092689740270186AtMost(int,one_one(int),D4)))
               => ! [Xb3: int] :
                    ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xb3),A3))
                   => ( X4 != aa(int,int,minus_minus(int,Xb3),Xa3) ) ) )
           => ( pp(aa(int,bool,ord_less_eq(int,X4),T))
             => pp(aa(int,bool,ord_less_eq(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),X4),D4)),T)) ) ) ) ) ).

% aset(6)
tff(fact_1826_aset_I8_J,axiom,
    ! [D4: int,A3: set(int),T: int] :
      ( pp(aa(int,bool,ord_less(int,zero_zero(int)),D4))
     => ! [X4: int] :
          ( ! [Xa3: int] :
              ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xa3),set_or1337092689740270186AtMost(int,one_one(int),D4)))
             => ! [Xb3: int] :
                  ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xb3),A3))
                 => ( X4 != aa(int,int,minus_minus(int,Xb3),Xa3) ) ) )
         => ( pp(aa(int,bool,ord_less_eq(int,T),X4))
           => pp(aa(int,bool,ord_less_eq(int,T),aa(int,int,aa(int,fun(int,int),plus_plus(int),X4),D4))) ) ) ) ).

% aset(8)
tff(fact_1827_cppi,axiom,
    ! [D4: int,P: fun(int,bool),P3: fun(int,bool),A3: set(int)] :
      ( pp(aa(int,bool,ord_less(int,zero_zero(int)),D4))
     => ( ? [Z3: int] :
          ! [X3: int] :
            ( pp(aa(int,bool,ord_less(int,Z3),X3))
           => ( pp(aa(int,bool,P,X3))
            <=> pp(aa(int,bool,P3,X3)) ) )
       => ( ! [X3: int] :
              ( ! [Xa: int] :
                  ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xa),set_or1337092689740270186AtMost(int,one_one(int),D4)))
                 => ! [Xb2: int] :
                      ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xb2),A3))
                     => ( X3 != aa(int,int,minus_minus(int,Xb2),Xa) ) ) )
             => ( pp(aa(int,bool,P,X3))
               => pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),X3),D4))) ) )
         => ( ! [X3: int,K3: int] :
                ( pp(aa(int,bool,P3,X3))
              <=> pp(aa(int,bool,P3,aa(int,int,minus_minus(int,X3),aa(int,int,aa(int,fun(int,int),times_times(int),K3),D4)))) )
           => ( ? [X_13: int] : pp(aa(int,bool,P,X_13))
            <=> ( ? [X2: int] :
                    ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),X2),set_or1337092689740270186AtMost(int,one_one(int),D4)))
                    & pp(aa(int,bool,P3,X2)) )
                | ? [X2: int] :
                    ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),X2),set_or1337092689740270186AtMost(int,one_one(int),D4)))
                    & ? [Xa4: int] :
                        ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xa4),A3))
                        & pp(aa(int,bool,P,aa(int,int,minus_minus(int,Xa4),X2))) ) ) ) ) ) ) ) ) ).

% cppi
tff(fact_1828_cpmi,axiom,
    ! [D4: int,P: fun(int,bool),P3: fun(int,bool),B4: set(int)] :
      ( pp(aa(int,bool,ord_less(int,zero_zero(int)),D4))
     => ( ? [Z3: int] :
          ! [X3: int] :
            ( pp(aa(int,bool,ord_less(int,X3),Z3))
           => ( pp(aa(int,bool,P,X3))
            <=> pp(aa(int,bool,P3,X3)) ) )
       => ( ! [X3: int] :
              ( ! [Xa: int] :
                  ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xa),set_or1337092689740270186AtMost(int,one_one(int),D4)))
                 => ! [Xb2: int] :
                      ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xb2),B4))
                     => ( X3 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb2),Xa) ) ) )
             => ( pp(aa(int,bool,P,X3))
               => pp(aa(int,bool,P,aa(int,int,minus_minus(int,X3),D4))) ) )
         => ( ! [X3: int,K3: int] :
                ( pp(aa(int,bool,P3,X3))
              <=> pp(aa(int,bool,P3,aa(int,int,minus_minus(int,X3),aa(int,int,aa(int,fun(int,int),times_times(int),K3),D4)))) )
           => ( ? [X_13: int] : pp(aa(int,bool,P,X_13))
            <=> ( ? [X2: int] :
                    ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),X2),set_or1337092689740270186AtMost(int,one_one(int),D4)))
                    & pp(aa(int,bool,P3,X2)) )
                | ? [X2: int] :
                    ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),X2),set_or1337092689740270186AtMost(int,one_one(int),D4)))
                    & ? [Xa4: int] :
                        ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xa4),B4))
                        & pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),Xa4),X2))) ) ) ) ) ) ) ) ) ).

% cpmi
tff(fact_1829_ceiling__divide__lower,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Q2: A,P2: A] :
          ( pp(aa(A,bool,ord_less(A,zero_zero(A)),Q2))
         => pp(aa(A,bool,ord_less(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,minus_minus(A,aa(int,A,ring_1_of_int(A),archimedean_ceiling(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),P2),Q2)))),one_one(A))),Q2)),P2)) ) ) ).

% ceiling_divide_lower
tff(fact_1830_ceiling__eq,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [N: int,X: A] :
          ( pp(aa(A,bool,ord_less(A,aa(int,A,ring_1_of_int(A),N)),X))
         => ( pp(aa(A,bool,ord_less_eq(A,X),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(int,A,ring_1_of_int(A),N)),one_one(A))))
           => ( archimedean_ceiling(A,X) = aa(int,int,aa(int,fun(int,int),plus_plus(int),N),one_one(int)) ) ) ) ) ).

% ceiling_eq
tff(fact_1831_Abs__fnat__hom__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( zero_zero(word(A)) = aa(nat,word(A),semiring_1_of_nat(word(A)),zero_zero(nat)) ) ) ).

% Abs_fnat_hom_0
tff(fact_1832_word__induct__less,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [P: fun(word(A),bool),M: word(A)] :
          ( pp(aa(word(A),bool,P,zero_zero(word(A))))
         => ( ! [N2: word(A)] :
                ( pp(aa(word(A),bool,ord_less(word(A),N2),M))
               => ( pp(aa(word(A),bool,P,N2))
                 => pp(aa(word(A),bool,P,aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),one_one(word(A))),N2))) ) )
           => pp(aa(word(A),bool,P,M)) ) ) ) ).

% word_induct_less
tff(fact_1833_word__less__sub1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A)] :
          ( ( X != zero_zero(word(A)) )
         => ( pp(aa(word(A),bool,ord_less(word(A),one_one(word(A))),X))
          <=> pp(aa(word(A),bool,ord_less(word(A),zero_zero(word(A))),aa(word(A),word(A),minus_minus(word(A),X),one_one(word(A))))) ) ) ) ).

% word_less_sub1
tff(fact_1834_inc__le,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [I: word(A),M: word(A)] :
          ( pp(aa(word(A),bool,ord_less(word(A),I),M))
         => pp(aa(word(A),bool,ord_less_eq(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),I),one_one(word(A)))),M)) ) ) ).

% inc_le
tff(fact_1835_inc__i,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [I: word(A),M: word(A)] :
          ( pp(aa(word(A),bool,ord_less_eq(word(A),one_one(word(A))),I))
         => ( pp(aa(word(A),bool,ord_less(word(A),I),M))
           => ( pp(aa(word(A),bool,ord_less_eq(word(A),one_one(word(A))),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),I),one_one(word(A)))))
              & pp(aa(word(A),bool,ord_less_eq(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),I),one_one(word(A)))),M)) ) ) ) ) ).

% inc_i
tff(fact_1836_Abs__fnat__hom__mult,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: nat,B2: nat] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),aa(nat,word(A),semiring_1_of_nat(word(A)),A2)),aa(nat,word(A),semiring_1_of_nat(word(A)),B2)) = aa(nat,word(A),semiring_1_of_nat(word(A)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A2),B2)) ) ).

% Abs_fnat_hom_mult
tff(fact_1837_div__word__self,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] :
          ( ( W != zero_zero(word(A)) )
         => ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),W),W) = one_one(word(A)) ) ) ) ).

% div_word_self
tff(fact_1838_div__lt__mult,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [I: word(A),K: word(A),X: word(A)] :
          ( pp(aa(word(A),bool,ord_less(word(A),I),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),K),X)))
         => ( pp(aa(word(A),bool,ord_less(word(A),zero_zero(word(A))),X))
           => pp(aa(word(A),bool,ord_less(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),I),X)),K)) ) ) ) ).

% div_lt_mult
tff(fact_1839_div__le__mult,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [I: word(A),K: word(A),X: word(A)] :
          ( pp(aa(word(A),bool,ord_less_eq(word(A),I),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),K),X)))
         => ( pp(aa(word(A),bool,ord_less(word(A),zero_zero(word(A))),X))
           => pp(aa(word(A),bool,ord_less_eq(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),I),X)),K)) ) ) ) ).

% div_le_mult
tff(fact_1840_Abs__fnat__hom__add,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [A2: nat,B2: nat] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),A2)),aa(nat,A,semiring_1_of_nat(A),B2)) = aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A2),B2)) ) ).

% Abs_fnat_hom_add
tff(fact_1841_of__nat__gt__0,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [K: nat] :
          ( ( aa(nat,A,semiring_1_of_nat(A),K) != zero_zero(A) )
         => pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),K)) ) ) ).

% of_nat_gt_0
tff(fact_1842_atLeastatMost__psubset__iff,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A2: A,B2: A,C2: A,D: A] :
          ( pp(aa(set(A),bool,ord_less(set(A),set_or1337092689740270186AtMost(A,A2,B2)),set_or1337092689740270186AtMost(A,C2,D)))
        <=> ( ( ~ pp(aa(A,bool,ord_less_eq(A,A2),B2))
              | ( pp(aa(A,bool,ord_less_eq(A,C2),A2))
                & pp(aa(A,bool,ord_less_eq(A,B2),D))
                & ( pp(aa(A,bool,ord_less(A,C2),A2))
                  | pp(aa(A,bool,ord_less(A,B2),D)) ) ) )
            & pp(aa(A,bool,ord_less_eq(A,C2),D)) ) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_1843_Abs__fnat__hom__1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( one_one(word(A)) = aa(nat,word(A),semiring_1_of_nat(word(A)),aa(nat,nat,suc,zero_zero(nat))) ) ) ).

% Abs_fnat_hom_1
tff(fact_1844_vebt__insert_Osimps_I3_J,axiom,
    ! [Info: option(product_prod(nat,nat)),Ts: list(vEBT_VEBT),S3: vEBT_VEBT,X: nat] : vEBT_vebt_insert(vEBT_Node(Info,aa(nat,nat,suc,zero_zero(nat)),Ts,S3),X) = vEBT_Node(Info,aa(nat,nat,suc,zero_zero(nat)),Ts,S3) ).

% vebt_insert.simps(3)
tff(fact_1845_low__inv,axiom,
    ! [X: nat,N: nat,Y: nat] :
      ( pp(aa(nat,bool,ord_less(nat,X),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)))
     => ( vEBT_VEBT_low(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Y),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N))),X),N) = X ) ) ).

% low_inv
tff(fact_1846_Suc__if__eq,axiom,
    ! [A: $tType,F2: fun(nat,A),H: fun(nat,A),G: A,N: nat] :
      ( ! [N2: nat] : aa(nat,A,F2,aa(nat,nat,suc,N2)) = aa(nat,A,H,N2)
     => ( ( aa(nat,A,F2,zero_zero(nat)) = G )
       => ( ( ( N = zero_zero(nat) )
           => ( aa(nat,A,F2,N) = G ) )
          & ( ( N != zero_zero(nat) )
           => ( aa(nat,A,F2,N) = aa(nat,A,H,aa(nat,nat,minus_minus(nat,N),one_one(nat))) ) ) ) ) ) ).

% Suc_if_eq
tff(fact_1847_height__compose__child,axiom,
    ! [T: vEBT_VEBT,TreeList: list(vEBT_VEBT),Info: option(product_prod(nat,nat)),Deg: nat,Summary: vEBT_VEBT] :
      ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),member(vEBT_VEBT),T),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList)))
     => pp(aa(nat,bool,ord_less_eq(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),aa(vEBT_VEBT,nat,vEBT_VEBT_height,T))),aa(vEBT_VEBT,nat,vEBT_VEBT_height,vEBT_Node(Info,Deg,TreeList,Summary)))) ) ).

% height_compose_child
tff(fact_1848_round__unique,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,Y: int] :
          ( pp(aa(A,bool,ord_less(A,aa(A,A,minus_minus(A,X),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),bit0(one2))))),aa(int,A,ring_1_of_int(A),Y)))
         => ( pp(aa(A,bool,ord_less_eq(A,aa(int,A,ring_1_of_int(A),Y)),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),bit0(one2))))))
           => ( archimedean_round(A,X) = Y ) ) ) ) ).

% round_unique
tff(fact_1849_mult__le__cancel__iff2,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: A,X: A,Y: A] :
          ( pp(aa(A,bool,ord_less(A,zero_zero(A)),Z))
         => ( pp(aa(A,bool,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),times_times(A),Z),X)),aa(A,A,aa(A,fun(A,A),times_times(A),Z),Y)))
          <=> pp(aa(A,bool,ord_less_eq(A,X),Y)) ) ) ) ).

% mult_le_cancel_iff2
tff(fact_1850_mult__le__cancel__iff1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: A,X: A,Y: A] :
          ( pp(aa(A,bool,ord_less(A,zero_zero(A)),Z))
         => ( pp(aa(A,bool,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),times_times(A),X),Z)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Z)))
          <=> pp(aa(A,bool,ord_less_eq(A,X),Y)) ) ) ) ).

% mult_le_cancel_iff1
tff(fact_1851_bit__split__inv,axiom,
    ! [X: nat,D: nat] : vEBT_VEBT_bit_concat(vEBT_VEBT_high(X,D),vEBT_VEBT_low(X,D),D) = X ).

% bit_split_inv
tff(fact_1852_word__of__int__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Bin: num] : aa(int,word(A),ring_1_of_int(word(A)),aa(num,int,numeral_numeral(int),Bin)) = aa(num,word(A),numeral_numeral(word(A)),Bin) ) ).

% word_of_int_numeral
tff(fact_1853_word__of__int__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( aa(int,word(A),ring_1_of_int(word(A)),zero_zero(int)) = zero_zero(word(A)) ) ) ).

% word_of_int_0
tff(fact_1854_word__of__int__1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( aa(int,word(A),ring_1_of_int(word(A)),one_one(int)) = one_one(word(A)) ) ) ).

% word_of_int_1
tff(fact_1855_Ball__set__replicate,axiom,
    ! [A: $tType,N: nat,A2: A,P: fun(A,bool)] :
      ( ! [X2: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X2),aa(list(A),set(A),set2(A),replicate(A,N,A2))))
         => pp(aa(A,bool,P,X2)) )
    <=> ( pp(aa(A,bool,P,A2))
        | ( N = zero_zero(nat) ) ) ) ).

% Ball_set_replicate
tff(fact_1856_Bex__set__replicate,axiom,
    ! [A: $tType,N: nat,A2: A,P: fun(A,bool)] :
      ( ? [X2: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X2),aa(list(A),set(A),set2(A),replicate(A,N,A2))))
          & pp(aa(A,bool,P,X2)) )
    <=> ( pp(aa(A,bool,P,A2))
        & ( N != zero_zero(nat) ) ) ) ).

% Bex_set_replicate
tff(fact_1857_in__set__replicate,axiom,
    ! [A: $tType,X: A,N: nat,Y: A] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(list(A),set(A),set2(A),replicate(A,N,Y))))
    <=> ( ( X = Y )
        & ( N != zero_zero(nat) ) ) ) ).

% in_set_replicate
tff(fact_1858_round__0,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ( archimedean_round(A,zero_zero(A)) = zero_zero(int) ) ) ).

% round_0
tff(fact_1859_round__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [N: num] : archimedean_round(A,aa(num,A,numeral_numeral(A),N)) = aa(num,int,numeral_numeral(int),N) ) ).

% round_numeral
tff(fact_1860_round__1,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ( archimedean_round(A,one_one(A)) = one_one(int) ) ) ).

% round_1
tff(fact_1861_round__of__nat,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [N: nat] : archimedean_round(A,aa(nat,A,semiring_1_of_nat(A),N)) = aa(nat,int,semiring_1_of_nat(int),N) ) ).

% round_of_nat
tff(fact_1862_subset__code_I1_J,axiom,
    ! [A: $tType,Xs: list(A),B4: set(A)] :
      ( pp(aa(set(A),bool,ord_less_eq(set(A),aa(list(A),set(A),set2(A),Xs)),B4))
    <=> ! [X2: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X2),aa(list(A),set(A),set2(A),Xs)))
         => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X2),B4)) ) ) ).

% subset_code(1)
tff(fact_1863_word__numeral__alt,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [B2: num] : aa(num,word(A),numeral_numeral(word(A)),B2) = aa(int,word(A),ring_1_of_int(word(A)),aa(num,int,numeral_numeral(int),B2)) ) ).

% word_numeral_alt
tff(fact_1864_wi__hom__add,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: int,B2: int] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),aa(int,word(A),ring_1_of_int(word(A)),A2)),aa(int,word(A),ring_1_of_int(word(A)),B2)) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(int,fun(int,int),plus_plus(int),A2),B2)) ) ).

% wi_hom_add
tff(fact_1865_word__of__int__power__hom,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: int,N: nat] : aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(int,word(A),ring_1_of_int(word(A)),A2)),N) = aa(int,word(A),ring_1_of_int(word(A)),aa(nat,int,aa(int,fun(nat,int),power_power(int),A2),N)) ) ).

% word_of_int_power_hom
tff(fact_1866_wi__hom__mult,axiom,
    ! [C: $tType] :
      ( type_len(C)
     => ! [A2: int,B2: int] : aa(word(C),word(C),aa(word(C),fun(word(C),word(C)),times_times(word(C)),aa(int,word(C),ring_1_of_int(word(C)),A2)),aa(int,word(C),ring_1_of_int(word(C)),B2)) = aa(int,word(C),ring_1_of_int(word(C)),aa(int,int,aa(int,fun(int,int),times_times(int),A2),B2)) ) ).

% wi_hom_mult
tff(fact_1867_round__mono,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,ord_less_eq(A,X),Y))
         => pp(aa(int,bool,ord_less_eq(int,archimedean_round(A,X)),archimedean_round(A,Y))) ) ) ).

% round_mono
tff(fact_1868_word__of__int__2p,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: nat] : aa(int,word(A),ring_1_of_int(word(A)),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),N)) = aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),N) ) ).

% word_of_int_2p
tff(fact_1869_VEBT__internal_Oexp__split__high__low_I2_J,axiom,
    ! [X: nat,N: nat,M: nat] :
      ( pp(aa(nat,bool,ord_less(nat,X),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),M))))
     => ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),N))
       => ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),M))
         => pp(aa(nat,bool,ord_less(nat,vEBT_VEBT_low(X,N)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N))) ) ) ) ).

% VEBT_internal.exp_split_high_low(2)
tff(fact_1870_mult__less__iff1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: A,X: A,Y: A] :
          ( pp(aa(A,bool,ord_less(A,zero_zero(A)),Z))
         => ( pp(aa(A,bool,ord_less(A,aa(A,A,aa(A,fun(A,A),times_times(A),X),Z)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Z)))
          <=> pp(aa(A,bool,ord_less(A,X),Y)) ) ) ) ).

% mult_less_iff1
tff(fact_1871_of__int__round__le,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] : pp(aa(A,bool,ord_less_eq(A,aa(int,A,ring_1_of_int(A),archimedean_round(A,X))),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),bit0(one2)))))) ) ).

% of_int_round_le
tff(fact_1872_of__int__round__ge,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] : pp(aa(A,bool,ord_less_eq(A,aa(A,A,minus_minus(A,X),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),bit0(one2))))),aa(int,A,ring_1_of_int(A),archimedean_round(A,X)))) ) ).

% of_int_round_ge
tff(fact_1873_of__int__round__gt,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] : pp(aa(A,bool,ord_less(A,aa(A,A,minus_minus(A,X),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),bit0(one2))))),aa(int,A,ring_1_of_int(A),archimedean_round(A,X)))) ) ).

% of_int_round_gt
tff(fact_1874_vebt__insert_Osimps_I2_J,axiom,
    ! [Info: option(product_prod(nat,nat)),Ts: list(vEBT_VEBT),S3: vEBT_VEBT,X: nat] : vEBT_vebt_insert(vEBT_Node(Info,zero_zero(nat),Ts,S3),X) = vEBT_Node(Info,zero_zero(nat),Ts,S3) ).

% vebt_insert.simps(2)
tff(fact_1875_both__member__options__ding,axiom,
    ! [Info: option(product_prod(nat,nat)),Deg: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,N: nat,X: nat] :
      ( vEBT_invar_vebt(vEBT_Node(Info,Deg,TreeList,Summary),N)
     => ( pp(aa(nat,bool,ord_less(nat,X),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg)))
       => ( pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),vEBT_VEBT_low(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))))
         => pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,vEBT_Node(Info,Deg,TreeList,Summary)),X)) ) ) ) ).

% both_member_options_ding
tff(fact_1876_set__n__deg__not__0,axiom,
    ! [TreeList: list(vEBT_VEBT),N: nat,M: nat] :
      ( ! [X3: vEBT_VEBT] :
          ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),member(vEBT_VEBT),X3),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList)))
         => vEBT_invar_vebt(X3,N) )
     => ( ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M) )
       => pp(aa(nat,bool,ord_less_eq(nat,one_one(nat)),N)) ) ) ).

% set_n_deg_not_0
tff(fact_1877_artanh__0,axiom,
    ! [A: $tType] :
      ( ( real_V3459762299906320749_field(A)
        & ln(A) )
     => ( aa(A,A,artanh(A),zero_zero(A)) = zero_zero(A) ) ) ).

% artanh_0
tff(fact_1878_arsinh__0,axiom,
    ! [A: $tType] :
      ( ln(A)
     => ( aa(A,A,arsinh(A),zero_zero(A)) = zero_zero(A) ) ) ).

% arsinh_0
tff(fact_1879_unset__bit__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A] : aa(A,A,aa(nat,fun(A,A),bit_se2638667681897837118et_bit(A),zero_zero(nat)),A2) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(num,A,numeral_numeral(A),bit0(one2)))) ) ).

% unset_bit_0
tff(fact_1880_low__def,axiom,
    ! [X: nat,N: nat] : vEBT_VEBT_low(X,N) = modulo_modulo(nat,X,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)) ).

% low_def
tff(fact_1881_vebt__buildup_Oelims,axiom,
    ! [X: nat,Y: vEBT_VEBT] :
      ( ( vEBT_vebt_buildup(X) = Y )
     => ( ( ( X = zero_zero(nat) )
         => ( Y != vEBT_Leaf(fFalse,fFalse) ) )
       => ( ( ( X = aa(nat,nat,suc,zero_zero(nat)) )
           => ( Y != vEBT_Leaf(fFalse,fFalse) ) )
         => ~ ! [Va2: nat] :
                ( ( X = aa(nat,nat,suc,aa(nat,nat,suc,Va2)) )
               => ~ ( ( pp(aa(nat,bool,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,aa(nat,nat,suc,Va2))))
                     => ( Y = vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),replicate(vEBT_VEBT,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),vEBT_vebt_buildup(aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_vebt_buildup(aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) )
                    & ( ~ pp(aa(nat,bool,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,aa(nat,nat,suc,Va2))))
                     => ( Y = vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),replicate(vEBT_VEBT,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_vebt_buildup(aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_vebt_buildup(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))) ) ) ) ) ) ) ) ).

% vebt_buildup.elims
tff(fact_1882_intind,axiom,
    ! [A: $tType,I: nat,N: nat,P: fun(A,bool),X: A] :
      ( pp(aa(nat,bool,ord_less(nat,I),N))
     => ( pp(aa(A,bool,P,X))
       => pp(aa(A,bool,P,aa(nat,A,nth(A,replicate(A,N,X)),I))) ) ) ).

% intind
tff(fact_1883_Leaf__0__not,axiom,
    ! [A2: bool,B2: bool] : ~ vEBT_invar_vebt(vEBT_Leaf(A2,B2),zero_zero(nat)) ).

% Leaf_0_not
tff(fact_1884_deg__1__Leafy,axiom,
    ! [T: vEBT_VEBT,N: nat] :
      ( vEBT_invar_vebt(T,N)
     => ( ( N = one_one(nat) )
       => ? [A4: bool,B3: bool] : T = vEBT_Leaf(A4,B3) ) ) ).

% deg_1_Leafy
tff(fact_1885_deg__1__Leaf,axiom,
    ! [T: vEBT_VEBT] :
      ( vEBT_invar_vebt(T,one_one(nat))
     => ? [A4: bool,B3: bool] : T = vEBT_Leaf(A4,B3) ) ).

% deg_1_Leaf
tff(fact_1886_deg1Leaf,axiom,
    ! [T: vEBT_VEBT] :
      ( vEBT_invar_vebt(T,one_one(nat))
    <=> ? [A6: bool,B7: bool] : T = vEBT_Leaf(A6,B7) ) ).

% deg1Leaf
tff(fact_1887_inthall,axiom,
    ! [A: $tType,Xs: list(A),P: fun(A,bool),N: nat] :
      ( ! [X3: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),aa(list(A),set(A),set2(A),Xs)))
         => pp(aa(A,bool,P,X3)) )
     => ( pp(aa(nat,bool,ord_less(nat,N),aa(list(A),nat,size_size(list(A)),Xs)))
       => pp(aa(A,bool,P,aa(nat,A,nth(A,Xs),N))) ) ) ).

% inthall
tff(fact_1888_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
tff(fact_1889_VEBT_Oinject_I2_J,axiom,
    ! [X21: bool,X222: bool,Y21: bool,Y22: bool] :
      ( ( vEBT_Leaf(X21,X222) = vEBT_Leaf(Y21,Y22) )
    <=> ( ( pp(X21)
        <=> pp(Y21) )
        & ( pp(X222)
        <=> pp(Y22) ) ) ) ).

% VEBT.inject(2)
tff(fact_1890_bits__mod__0,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A] : modulo_modulo(A,zero_zero(A),A2) = zero_zero(A) ) ).

% bits_mod_0
tff(fact_1891_mod__self,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [A2: A] : modulo_modulo(A,A2,A2) = zero_zero(A) ) ).

% mod_self
tff(fact_1892_mod__by__0,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [A2: A] : modulo_modulo(A,A2,zero_zero(A)) = A2 ) ).

% mod_by_0
tff(fact_1893_mod__0,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [A2: A] : modulo_modulo(A,zero_zero(A),A2) = zero_zero(A) ) ).

% mod_0
tff(fact_1894_mod__add__self1,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [B2: A,A2: A] : modulo_modulo(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A2),B2) = modulo_modulo(A,A2,B2) ) ).

% mod_add_self1
tff(fact_1895_mod__add__self2,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A2: A,B2: A] : modulo_modulo(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2),B2) = modulo_modulo(A,A2,B2) ) ).

% mod_add_self2
tff(fact_1896_minus__mod__self2,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [A2: A,B2: A] : modulo_modulo(A,aa(A,A,minus_minus(A,A2),B2),B2) = modulo_modulo(A,A2,B2) ) ).

% minus_mod_self2
tff(fact_1897_unset__bit__nonnegative__int__iff,axiom,
    ! [N: nat,K: int] :
      ( pp(aa(int,bool,ord_less_eq(int,zero_zero(int)),aa(int,int,aa(nat,fun(int,int),bit_se2638667681897837118et_bit(int),N),K)))
    <=> pp(aa(int,bool,ord_less_eq(int,zero_zero(int)),K)) ) ).

% unset_bit_nonnegative_int_iff
tff(fact_1898_unset__bit__negative__int__iff,axiom,
    ! [N: nat,K: int] :
      ( pp(aa(int,bool,ord_less(int,aa(int,int,aa(nat,fun(int,int),bit_se2638667681897837118et_bit(int),N),K)),zero_zero(int)))
    <=> pp(aa(int,bool,ord_less(int,K),zero_zero(int))) ) ).

% unset_bit_negative_int_iff
tff(fact_1899_mod__less,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,ord_less(nat,M),N))
     => ( modulo_modulo(nat,M,N) = M ) ) ).

% mod_less
tff(fact_1900_length__replicate,axiom,
    ! [A: $tType,N: nat,X: A] : aa(list(A),nat,size_size(list(A)),replicate(A,N,X)) = N ).

% length_replicate
tff(fact_1901_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
tff(fact_1902_mod__by__1,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [A2: A] : modulo_modulo(A,A2,one_one(A)) = zero_zero(A) ) ).

% mod_by_1
tff(fact_1903_mod__mult__self1__is__0,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [B2: A,A2: A] : modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),B2),A2),B2) = zero_zero(A) ) ).

% mod_mult_self1_is_0
tff(fact_1904_mod__mult__self2__is__0,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A2: A,B2: A] : modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2),B2) = zero_zero(A) ) ).

% mod_mult_self2_is_0
tff(fact_1905_mod__div__trivial,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),modulo_modulo(A,A2,B2)),B2) = zero_zero(A) ) ).

% mod_div_trivial
tff(fact_1906_bits__mod__div__trivial,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),modulo_modulo(A,A2,B2)),B2) = zero_zero(A) ) ).

% bits_mod_div_trivial
tff(fact_1907_mod__mult__self1,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A2: A,C2: A,B2: A] : modulo_modulo(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)),B2) = modulo_modulo(A,A2,B2) ) ).

% mod_mult_self1
tff(fact_1908_mod__mult__self2,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A2: A,B2: A,C2: A] : modulo_modulo(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)),B2) = modulo_modulo(A,A2,B2) ) ).

% mod_mult_self2
tff(fact_1909_mod__mult__self3,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [C2: A,B2: A,A2: A] : modulo_modulo(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)),A2),B2) = modulo_modulo(A,A2,B2) ) ).

% mod_mult_self3
tff(fact_1910_mod__mult__self4,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [B2: A,C2: A,A2: A] : modulo_modulo(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)),A2),B2) = modulo_modulo(A,A2,B2) ) ).

% mod_mult_self4
tff(fact_1911_dvd__imp__mod__0,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,dvd_dvd(A,A2),B2))
         => ( modulo_modulo(A,B2,A2) = zero_zero(A) ) ) ) ).

% dvd_imp_mod_0
tff(fact_1912_mod__by__Suc__0,axiom,
    ! [M: nat] : modulo_modulo(nat,M,aa(nat,nat,suc,zero_zero(nat))) = zero_zero(nat) ).

% mod_by_Suc_0
tff(fact_1913_nth__replicate,axiom,
    ! [A: $tType,I: nat,N: nat,X: A] :
      ( pp(aa(nat,bool,ord_less(nat,I),N))
     => ( aa(nat,A,nth(A,replicate(A,N,X)),I) = X ) ) ).

% nth_replicate
tff(fact_1914_Suc__mod__mult__self4,axiom,
    ! [N: nat,K: nat,M: nat] : modulo_modulo(nat,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),K)),M)),N) = modulo_modulo(nat,aa(nat,nat,suc,M),N) ).

% Suc_mod_mult_self4
tff(fact_1915_Suc__mod__mult__self3,axiom,
    ! [K: nat,N: nat,M: nat] : modulo_modulo(nat,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),N)),M)),N) = modulo_modulo(nat,aa(nat,nat,suc,M),N) ).

% Suc_mod_mult_self3
tff(fact_1916_Suc__mod__mult__self2,axiom,
    ! [M: nat,N: nat,K: nat] : modulo_modulo(nat,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),K))),N) = modulo_modulo(nat,aa(nat,nat,suc,M),N) ).

% Suc_mod_mult_self2
tff(fact_1917_Suc__mod__mult__self1,axiom,
    ! [M: nat,K: nat,N: nat] : modulo_modulo(nat,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),N))),N) = modulo_modulo(nat,aa(nat,nat,suc,M),N) ).

% Suc_mod_mult_self1
tff(fact_1918_one__mod__two__eq__one,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ( modulo_modulo(A,one_one(A),aa(num,A,numeral_numeral(A),bit0(one2))) = one_one(A) ) ) ).

% one_mod_two_eq_one
tff(fact_1919_bits__one__mod__two__eq__one,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ( modulo_modulo(A,one_one(A),aa(num,A,numeral_numeral(A),bit0(one2))) = one_one(A) ) ) ).

% bits_one_mod_two_eq_one
tff(fact_1920_even__mod__2__iff,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A2: A] :
          ( pp(aa(A,bool,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),modulo_modulo(A,A2,aa(num,A,numeral_numeral(A),bit0(one2)))))
        <=> pp(aa(A,bool,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A2)) ) ) ).

% even_mod_2_iff
tff(fact_1921_mod2__Suc__Suc,axiom,
    ! [M: nat] : modulo_modulo(nat,aa(nat,nat,suc,aa(nat,nat,suc,M)),aa(num,nat,numeral_numeral(nat),bit0(one2))) = modulo_modulo(nat,M,aa(num,nat,numeral_numeral(nat),bit0(one2))) ).

% mod2_Suc_Suc
tff(fact_1922_Suc__times__numeral__mod__eq,axiom,
    ! [K: num,N: nat] :
      ( ( aa(num,nat,numeral_numeral(nat),K) != one_one(nat) )
     => ( modulo_modulo(nat,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),K)),N)),aa(num,nat,numeral_numeral(nat),K)) = one_one(nat) ) ) ).

% Suc_times_numeral_mod_eq
tff(fact_1923_not__mod__2__eq__1__eq__0,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A2: A] :
          ( ( modulo_modulo(A,A2,aa(num,A,numeral_numeral(A),bit0(one2))) != one_one(A) )
        <=> ( modulo_modulo(A,A2,aa(num,A,numeral_numeral(A),bit0(one2))) = zero_zero(A) ) ) ) ).

% not_mod_2_eq_1_eq_0
tff(fact_1924_not__mod__2__eq__0__eq__1,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A2: A] :
          ( ( modulo_modulo(A,A2,aa(num,A,numeral_numeral(A),bit0(one2))) != zero_zero(A) )
        <=> ( modulo_modulo(A,A2,aa(num,A,numeral_numeral(A),bit0(one2))) = one_one(A) ) ) ) ).

% not_mod_2_eq_0_eq_1
tff(fact_1925_not__mod2__eq__Suc__0__eq__0,axiom,
    ! [N: nat] :
      ( ( modulo_modulo(nat,N,aa(num,nat,numeral_numeral(nat),bit0(one2))) != aa(nat,nat,suc,zero_zero(nat)) )
    <=> ( modulo_modulo(nat,N,aa(num,nat,numeral_numeral(nat),bit0(one2))) = zero_zero(nat) ) ) ).

% not_mod2_eq_Suc_0_eq_0
tff(fact_1926_add__self__mod__2,axiom,
    ! [M: nat] : modulo_modulo(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),M),aa(num,nat,numeral_numeral(nat),bit0(one2))) = zero_zero(nat) ).

% add_self_mod_2
tff(fact_1927_Suc__mod__eq__add3__mod__numeral,axiom,
    ! [M: nat,V: num] : modulo_modulo(nat,aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,suc,M))),aa(num,nat,numeral_numeral(nat),V)) = modulo_modulo(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2))),M),aa(num,nat,numeral_numeral(nat),V)) ).

% Suc_mod_eq_add3_mod_numeral
tff(fact_1928_mod__Suc__eq__mod__add3,axiom,
    ! [M: nat,N: nat] : modulo_modulo(nat,M,aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,suc,N)))) = modulo_modulo(nat,M,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2))),N)) ).

% mod_Suc_eq_mod_add3
tff(fact_1929_mod2__gr__0,axiom,
    ! [M: nat] :
      ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),modulo_modulo(nat,M,aa(num,nat,numeral_numeral(nat),bit0(one2)))))
    <=> ( modulo_modulo(nat,M,aa(num,nat,numeral_numeral(nat),bit0(one2))) = one_one(nat) ) ) ).

% mod2_gr_0
tff(fact_1930_even__succ__mod__exp,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A,N: nat] :
          ( pp(aa(A,bool,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A2))
         => ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),N))
           => ( modulo_modulo(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),A2),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),modulo_modulo(A,A2,aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N))) ) ) ) ) ).

% even_succ_mod_exp
tff(fact_1931_of__nat__mod,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [M: nat,N: nat] : aa(nat,A,semiring_1_of_nat(A),modulo_modulo(nat,M,N)) = modulo_modulo(A,aa(nat,A,semiring_1_of_nat(A),M),aa(nat,A,semiring_1_of_nat(A),N)) ) ).

% of_nat_mod
tff(fact_1932_size__neq__size__imp__neq,axiom,
    ! [A: $tType] :
      ( size(A)
     => ! [X: A,Y: A] :
          ( ( aa(A,nat,size_size(A),X) != aa(A,nat,size_size(A),Y) )
         => ( X != Y ) ) ) ).

% size_neq_size_imp_neq
tff(fact_1933_mod__add__eq,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A2: A,C2: A,B2: A] : modulo_modulo(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),modulo_modulo(A,A2,C2)),modulo_modulo(A,B2,C2)),C2) = modulo_modulo(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2),C2) ) ).

% mod_add_eq
tff(fact_1934_mod__add__cong,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A2: A,C2: A,A5: A,B2: A,B6: A] :
          ( ( modulo_modulo(A,A2,C2) = modulo_modulo(A,A5,C2) )
         => ( ( modulo_modulo(A,B2,C2) = modulo_modulo(A,B6,C2) )
           => ( modulo_modulo(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2),C2) = modulo_modulo(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A5),B6),C2) ) ) ) ) ).

% mod_add_cong
tff(fact_1935_mod__add__left__eq,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A2: A,C2: A,B2: A] : modulo_modulo(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),modulo_modulo(A,A2,C2)),B2),C2) = modulo_modulo(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2),C2) ) ).

% mod_add_left_eq
tff(fact_1936_mod__add__right__eq,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A2: A,B2: A,C2: A] : modulo_modulo(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),modulo_modulo(A,B2,C2)),C2) = modulo_modulo(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2),C2) ) ).

% mod_add_right_eq
tff(fact_1937_mod__mult__right__eq,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A2: A,B2: A,C2: A] : modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),modulo_modulo(A,B2,C2)),C2) = modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2),C2) ) ).

% mod_mult_right_eq
tff(fact_1938_mod__mult__left__eq,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A2: A,C2: A,B2: A] : modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),modulo_modulo(A,A2,C2)),B2),C2) = modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2),C2) ) ).

% mod_mult_left_eq
tff(fact_1939_mult__mod__right,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [C2: A,A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),C2),modulo_modulo(A,A2,B2)) = modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)) ) ).

% mult_mod_right
tff(fact_1940_mod__mult__mult2,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A2: A,C2: A,B2: A] : modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),times_times(A),modulo_modulo(A,A2,B2)),C2) ) ).

% mod_mult_mult2
tff(fact_1941_mod__mult__cong,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A2: A,C2: A,A5: A,B2: A,B6: A] :
          ( ( modulo_modulo(A,A2,C2) = modulo_modulo(A,A5,C2) )
         => ( ( modulo_modulo(A,B2,C2) = modulo_modulo(A,B6,C2) )
           => ( modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2),C2) = modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),A5),B6),C2) ) ) ) ) ).

% mod_mult_cong
tff(fact_1942_mod__mult__eq,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A2: A,C2: A,B2: A] : modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),modulo_modulo(A,A2,C2)),modulo_modulo(A,B2,C2)),C2) = modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2),C2) ) ).

% mod_mult_eq
tff(fact_1943_mod__diff__right__eq,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [A2: A,B2: A,C2: A] : modulo_modulo(A,aa(A,A,minus_minus(A,A2),modulo_modulo(A,B2,C2)),C2) = modulo_modulo(A,aa(A,A,minus_minus(A,A2),B2),C2) ) ).

% mod_diff_right_eq
tff(fact_1944_mod__diff__left__eq,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [A2: A,C2: A,B2: A] : modulo_modulo(A,aa(A,A,minus_minus(A,modulo_modulo(A,A2,C2)),B2),C2) = modulo_modulo(A,aa(A,A,minus_minus(A,A2),B2),C2) ) ).

% mod_diff_left_eq
tff(fact_1945_mod__diff__cong,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [A2: A,C2: A,A5: A,B2: A,B6: A] :
          ( ( modulo_modulo(A,A2,C2) = modulo_modulo(A,A5,C2) )
         => ( ( modulo_modulo(A,B2,C2) = modulo_modulo(A,B6,C2) )
           => ( modulo_modulo(A,aa(A,A,minus_minus(A,A2),B2),C2) = modulo_modulo(A,aa(A,A,minus_minus(A,A5),B6),C2) ) ) ) ) ).

% mod_diff_cong
tff(fact_1946_mod__diff__eq,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [A2: A,C2: A,B2: A] : modulo_modulo(A,aa(A,A,minus_minus(A,modulo_modulo(A,A2,C2)),modulo_modulo(A,B2,C2)),C2) = modulo_modulo(A,aa(A,A,minus_minus(A,A2),B2),C2) ) ).

% mod_diff_eq
tff(fact_1947_power__mod,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A2: A,B2: A,N: nat] : modulo_modulo(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),modulo_modulo(A,A2,B2)),N),B2) = modulo_modulo(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),N),B2) ) ).

% power_mod
tff(fact_1948_mod__mod__cancel,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [C2: A,B2: A,A2: A] :
          ( pp(aa(A,bool,dvd_dvd(A,C2),B2))
         => ( modulo_modulo(A,modulo_modulo(A,A2,B2),C2) = modulo_modulo(A,A2,C2) ) ) ) ).

% mod_mod_cancel
tff(fact_1949_dvd__mod,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [K: A,M: A,N: A] :
          ( pp(aa(A,bool,dvd_dvd(A,K),M))
         => ( pp(aa(A,bool,dvd_dvd(A,K),N))
           => pp(aa(A,bool,dvd_dvd(A,K),modulo_modulo(A,M,N))) ) ) ) ).

% dvd_mod
tff(fact_1950_dvd__mod__imp__dvd,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [C2: A,A2: A,B2: A] :
          ( pp(aa(A,bool,dvd_dvd(A,C2),modulo_modulo(A,A2,B2)))
         => ( pp(aa(A,bool,dvd_dvd(A,C2),B2))
           => pp(aa(A,bool,dvd_dvd(A,C2),A2)) ) ) ) ).

% dvd_mod_imp_dvd
tff(fact_1951_dvd__mod__iff,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [C2: A,B2: A,A2: A] :
          ( pp(aa(A,bool,dvd_dvd(A,C2),B2))
         => ( pp(aa(A,bool,dvd_dvd(A,C2),modulo_modulo(A,A2,B2)))
          <=> pp(aa(A,bool,dvd_dvd(A,C2),A2)) ) ) ) ).

% dvd_mod_iff
tff(fact_1952_mod__Suc__Suc__eq,axiom,
    ! [M: nat,N: nat] : modulo_modulo(nat,aa(nat,nat,suc,aa(nat,nat,suc,modulo_modulo(nat,M,N))),N) = modulo_modulo(nat,aa(nat,nat,suc,aa(nat,nat,suc,M)),N) ).

% mod_Suc_Suc_eq
tff(fact_1953_mod__Suc__eq,axiom,
    ! [M: nat,N: nat] : modulo_modulo(nat,aa(nat,nat,suc,modulo_modulo(nat,M,N)),N) = modulo_modulo(nat,aa(nat,nat,suc,M),N) ).

% mod_Suc_eq
tff(fact_1954_VEBT_Oexhaust,axiom,
    ! [Y: vEBT_VEBT] :
      ( ! [X112: option(product_prod(nat,nat)),X122: nat,X132: list(vEBT_VEBT),X142: vEBT_VEBT] : Y != vEBT_Node(X112,X122,X132,X142)
     => ~ ! [X212: bool,X223: bool] : Y != vEBT_Leaf(X212,X223) ) ).

% VEBT.exhaust
tff(fact_1955_VEBT_Odistinct_I1_J,axiom,
    ! [X11: option(product_prod(nat,nat)),X12: nat,X13: list(vEBT_VEBT),X14: vEBT_VEBT,X21: bool,X222: bool] : vEBT_Node(X11,X12,X13,X14) != vEBT_Leaf(X21,X222) ).

% VEBT.distinct(1)
tff(fact_1956_mod__plus__right,axiom,
    ! [A2: nat,X: nat,M: nat,B2: nat] :
      ( ( modulo_modulo(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A2),X),M) = modulo_modulo(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),B2),X),M) )
    <=> ( modulo_modulo(nat,A2,M) = modulo_modulo(nat,B2,M) ) ) ).

% mod_plus_right
tff(fact_1957_mod__less__eq__dividend,axiom,
    ! [M: nat,N: nat] : pp(aa(nat,bool,ord_less_eq(nat,modulo_modulo(nat,M,N)),M)) ).

% mod_less_eq_dividend
tff(fact_1958_VEBT__internal_Omembermima_Osimps_I1_J,axiom,
    ! [Uu: bool,Uv: bool,Uw: nat] : ~ vEBT_VEBT_membermima(vEBT_Leaf(Uu,Uv),Uw) ).

% VEBT_internal.membermima.simps(1)
tff(fact_1959_unset__bit__less__eq,axiom,
    ! [N: nat,K: int] : pp(aa(int,bool,ord_less_eq(int,aa(int,int,aa(nat,fun(int,int),bit_se2638667681897837118et_bit(int),N),K)),K)) ).

% unset_bit_less_eq
tff(fact_1960_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),A2))
         => pp(aa(A,bool,ord_less_eq(A,modulo_modulo(A,A2,B2)),A2)) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
tff(fact_1961_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [B2: A,A2: A] :
          ( pp(aa(A,bool,ord_less(A,zero_zero(A)),B2))
         => pp(aa(A,bool,ord_less(A,modulo_modulo(A,A2,B2)),B2)) ) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_bound
tff(fact_1962_cong__exp__iff__simps_I9_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M: num,Q2: num,N: num] :
          ( ( modulo_modulo(A,aa(num,A,numeral_numeral(A),bit0(M)),aa(num,A,numeral_numeral(A),bit0(Q2))) = modulo_modulo(A,aa(num,A,numeral_numeral(A),bit0(N)),aa(num,A,numeral_numeral(A),bit0(Q2))) )
        <=> ( modulo_modulo(A,aa(num,A,numeral_numeral(A),M),aa(num,A,numeral_numeral(A),Q2)) = modulo_modulo(A,aa(num,A,numeral_numeral(A),N),aa(num,A,numeral_numeral(A),Q2)) ) ) ) ).

% cong_exp_iff_simps(9)
tff(fact_1963_cong__exp__iff__simps_I4_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M: num,N: num] : modulo_modulo(A,aa(num,A,numeral_numeral(A),M),aa(num,A,numeral_numeral(A),one2)) = modulo_modulo(A,aa(num,A,numeral_numeral(A),N),aa(num,A,numeral_numeral(A),one2)) ) ).

% cong_exp_iff_simps(4)
tff(fact_1964_mod__eq__self__iff__div__eq__0,axiom,
    ! [A: $tType] :
      ( euclid3725896446679973847miring(A)
     => ! [A2: A,B2: A] :
          ( ( modulo_modulo(A,A2,B2) = A2 )
        <=> ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2) = zero_zero(A) ) ) ) ).

% mod_eq_self_iff_div_eq_0
tff(fact_1965_mod__eqE,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [A2: A,C2: A,B2: A] :
          ( ( modulo_modulo(A,A2,C2) = modulo_modulo(A,B2,C2) )
         => ~ ! [D2: A] : B2 != aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),C2),D2)) ) ) ).

% mod_eqE
tff(fact_1966_div__add1__eq,axiom,
    ! [A: $tType] :
      ( euclid3128863361964157862miring(A)
     => ! [A2: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),C2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2))),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),modulo_modulo(A,A2,C2)),modulo_modulo(A,B2,C2))),C2)) ) ).

% div_add1_eq
tff(fact_1967_mod__0__imp__dvd,axiom,
    ! [A: $tType] :
      ( semiring_modulo(A)
     => ! [A2: A,B2: A] :
          ( ( modulo_modulo(A,A2,B2) = zero_zero(A) )
         => pp(aa(A,bool,dvd_dvd(A,B2),A2)) ) ) ).

% mod_0_imp_dvd
tff(fact_1968_dvd__eq__mod__eq__0,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,dvd_dvd(A,A2),B2))
        <=> ( modulo_modulo(A,B2,A2) = zero_zero(A) ) ) ) ).

% dvd_eq_mod_eq_0
tff(fact_1969_mod__eq__0__iff__dvd,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [A2: A,B2: A] :
          ( ( modulo_modulo(A,A2,B2) = zero_zero(A) )
        <=> pp(aa(A,bool,dvd_dvd(A,B2),A2)) ) ) ).

% mod_eq_0_iff_dvd
tff(fact_1970_dvd__minus__mod,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [B2: A,A2: A] : pp(aa(A,bool,dvd_dvd(A,B2),aa(A,A,minus_minus(A,A2),modulo_modulo(A,A2,B2)))) ) ).

% dvd_minus_mod
tff(fact_1971_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) )
        <=> pp(aa(A,bool,dvd_dvd(A,C2),aa(A,A,minus_minus(A,A2),B2))) ) ) ).

% mod_eq_dvd_iff
tff(fact_1972_mod__Suc,axiom,
    ! [M: nat,N: nat] :
      ( ( ( aa(nat,nat,suc,modulo_modulo(nat,M,N)) = N )
       => ( modulo_modulo(nat,aa(nat,nat,suc,M),N) = zero_zero(nat) ) )
      & ( ( aa(nat,nat,suc,modulo_modulo(nat,M,N)) != N )
       => ( modulo_modulo(nat,aa(nat,nat,suc,M),N) = aa(nat,nat,suc,modulo_modulo(nat,M,N)) ) ) ) ).

% mod_Suc
tff(fact_1973_mod__induct,axiom,
    ! [P: fun(nat,bool),N: nat,P2: nat,M: nat] :
      ( pp(aa(nat,bool,P,N))
     => ( pp(aa(nat,bool,ord_less(nat,N),P2))
       => ( pp(aa(nat,bool,ord_less(nat,M),P2))
         => ( ! [N2: nat] :
                ( pp(aa(nat,bool,ord_less(nat,N2),P2))
               => ( pp(aa(nat,bool,P,N2))
                 => pp(aa(nat,bool,P,modulo_modulo(nat,aa(nat,nat,suc,N2),P2))) ) )
           => pp(aa(nat,bool,P,M)) ) ) ) ) ).

% mod_induct
tff(fact_1974_nat__mod__lem,axiom,
    ! [N: nat,B2: nat] :
      ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),N))
     => ( pp(aa(nat,bool,ord_less(nat,B2),N))
      <=> ( modulo_modulo(nat,B2,N) = B2 ) ) ) ).

% nat_mod_lem
tff(fact_1975_gcd__nat__induct,axiom,
    ! [P: fun(nat,fun(nat,bool)),M: nat,N: nat] :
      ( ! [M3: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),P,M3),zero_zero(nat)))
     => ( ! [M3: nat,N2: nat] :
            ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),N2))
           => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),P,N2),modulo_modulo(nat,M3,N2)))
             => pp(aa(nat,bool,aa(nat,fun(nat,bool),P,M3),N2)) ) )
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),P,M),N)) ) ) ).

% gcd_nat_induct
tff(fact_1976_mod__less__divisor,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),N))
     => pp(aa(nat,bool,ord_less(nat,modulo_modulo(nat,M,N)),N)) ) ).

% mod_less_divisor
tff(fact_1977_mod__Suc__le__divisor,axiom,
    ! [M: nat,N: nat] : pp(aa(nat,bool,ord_less_eq(nat,modulo_modulo(nat,M,aa(nat,nat,suc,N))),N)) ).

% mod_Suc_le_divisor
tff(fact_1978_word__rot__lem,axiom,
    ! [L: nat,K: nat,D: nat,N: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),L),K) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),D),modulo_modulo(nat,K,L)) )
     => ( pp(aa(nat,bool,ord_less(nat,N),L))
       => ( modulo_modulo(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),D),N),L) = N ) ) ) ).

% word_rot_lem
tff(fact_1979_nat__minus__mod,axiom,
    ! [N: nat,M: nat] : modulo_modulo(nat,aa(nat,nat,minus_minus(nat,N),modulo_modulo(nat,N,M)),M) = zero_zero(nat) ).

% nat_minus_mod
tff(fact_1980_mod__geq,axiom,
    ! [M: nat,N: nat] :
      ( ~ pp(aa(nat,bool,ord_less(nat,M),N))
     => ( modulo_modulo(nat,M,N) = modulo_modulo(nat,aa(nat,nat,minus_minus(nat,M),N),N) ) ) ).

% mod_geq
tff(fact_1981_mod__if,axiom,
    ! [M: nat,N: nat] :
      ( ( pp(aa(nat,bool,ord_less(nat,M),N))
       => ( modulo_modulo(nat,M,N) = M ) )
      & ( ~ pp(aa(nat,bool,ord_less(nat,M),N))
       => ( modulo_modulo(nat,M,N) = modulo_modulo(nat,aa(nat,nat,minus_minus(nat,M),N),N) ) ) ) ).

% mod_if
tff(fact_1982_mod__eq__0D,axiom,
    ! [M: nat,D: nat] :
      ( ( modulo_modulo(nat,M,D) = zero_zero(nat) )
     => ? [Q3: nat] : M = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),D),Q3) ) ).

% mod_eq_0D
tff(fact_1983_nat__minus__mod__plus__right,axiom,
    ! [N: nat,X: nat,M: nat] : modulo_modulo(nat,aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),X)),modulo_modulo(nat,N,M)),M) = modulo_modulo(nat,X,M) ).

% nat_minus_mod_plus_right
tff(fact_1984_le__mod__geq,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa(nat,bool,ord_less_eq(nat,N),M))
     => ( modulo_modulo(nat,M,N) = modulo_modulo(nat,aa(nat,nat,minus_minus(nat,M),N),N) ) ) ).

% le_mod_geq
tff(fact_1985_msrevs_I2_J,axiom,
    ! [K: nat,N: nat,M: nat] : modulo_modulo(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),N)),M),N) = modulo_modulo(nat,M,N) ).

% msrevs(2)
tff(fact_1986_nat__mod__eq__iff,axiom,
    ! [X: nat,N: nat,Y: nat] :
      ( ( modulo_modulo(nat,X,N) = modulo_modulo(nat,Y,N) )
    <=> ? [Q1: nat,Q22: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),X),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),Q1)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Y),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),Q22)) ) ).

% nat_mod_eq_iff
tff(fact_1987_replicate__length__same,axiom,
    ! [A: $tType,Xs: list(A),X: A] :
      ( ! [X3: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),aa(list(A),set(A),set2(A),Xs)))
         => ( X3 = X ) )
     => ( replicate(A,aa(list(A),nat,size_size(list(A)),Xs),X) = Xs ) ) ).

% replicate_length_same
tff(fact_1988_replicate__eqI,axiom,
    ! [A: $tType,Xs: list(A),N: nat,X: A] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = N )
     => ( ! [Y3: A] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Y3),aa(list(A),set(A),set2(A),Xs)))
           => ( Y3 = X ) )
       => ( Xs = replicate(A,N,X) ) ) ) ).

% replicate_eqI
tff(fact_1989_vebt__buildup_Osimps_I1_J,axiom,
    vEBT_vebt_buildup(zero_zero(nat)) = vEBT_Leaf(fFalse,fFalse) ).

% vebt_buildup.simps(1)
tff(fact_1990_VEBT__internal_Ocnt_Osimps_I1_J,axiom,
    ! [A2: bool,B2: bool] : aa(vEBT_VEBT,real,vEBT_VEBT_cnt,vEBT_Leaf(A2,B2)) = one_one(real) ).

% VEBT_internal.cnt.simps(1)
tff(fact_1991_VEBT__internal_Ocnt_H_Osimps_I1_J,axiom,
    ! [A2: bool,B2: bool] : aa(vEBT_VEBT,nat,vEBT_VEBT_cnt2,vEBT_Leaf(A2,B2)) = one_one(nat) ).

% VEBT_internal.cnt'.simps(1)
tff(fact_1992_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),A2))
         => ( pp(aa(A,bool,ord_less(A,A2),B2))
           => ( modulo_modulo(A,A2,B2) = A2 ) ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_less
tff(fact_1993_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [B2: A,A2: A] :
          ( pp(aa(A,bool,ord_less(A,zero_zero(A)),B2))
         => pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),modulo_modulo(A,A2,B2))) ) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_sign
tff(fact_1994_cong__exp__iff__simps_I2_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [N: num,Q2: num] :
          ( ( modulo_modulo(A,aa(num,A,numeral_numeral(A),bit0(N)),aa(num,A,numeral_numeral(A),bit0(Q2))) = zero_zero(A) )
        <=> ( modulo_modulo(A,aa(num,A,numeral_numeral(A),N),aa(num,A,numeral_numeral(A),Q2)) = zero_zero(A) ) ) ) ).

% cong_exp_iff_simps(2)
tff(fact_1995_cong__exp__iff__simps_I1_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [N: num] : modulo_modulo(A,aa(num,A,numeral_numeral(A),N),aa(num,A,numeral_numeral(A),one2)) = zero_zero(A) ) ).

% cong_exp_iff_simps(1)
tff(fact_1996_cong__exp__iff__simps_I6_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Q2: num,N: num] : modulo_modulo(A,aa(num,A,numeral_numeral(A),one2),aa(num,A,numeral_numeral(A),bit0(Q2))) != modulo_modulo(A,aa(num,A,numeral_numeral(A),bit0(N)),aa(num,A,numeral_numeral(A),bit0(Q2))) ) ).

% cong_exp_iff_simps(6)
tff(fact_1997_cong__exp__iff__simps_I8_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M: num,Q2: num] : modulo_modulo(A,aa(num,A,numeral_numeral(A),bit0(M)),aa(num,A,numeral_numeral(A),bit0(Q2))) != modulo_modulo(A,aa(num,A,numeral_numeral(A),one2),aa(num,A,numeral_numeral(A),bit0(Q2))) ) ).

% cong_exp_iff_simps(8)
tff(fact_1998_cong__exp__iff__simps_I10_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M: num,Q2: num,N: num] : modulo_modulo(A,aa(num,A,numeral_numeral(A),bit0(M)),aa(num,A,numeral_numeral(A),bit0(Q2))) != modulo_modulo(A,aa(num,A,numeral_numeral(A),aa(num,num,bit1,N)),aa(num,A,numeral_numeral(A),bit0(Q2))) ) ).

% cong_exp_iff_simps(10)
tff(fact_1999_cong__exp__iff__simps_I12_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M: num,Q2: num,N: num] : modulo_modulo(A,aa(num,A,numeral_numeral(A),aa(num,num,bit1,M)),aa(num,A,numeral_numeral(A),bit0(Q2))) != modulo_modulo(A,aa(num,A,numeral_numeral(A),bit0(N)),aa(num,A,numeral_numeral(A),bit0(Q2))) ) ).

% cong_exp_iff_simps(12)
tff(fact_2000_cong__exp__iff__simps_I13_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M: num,Q2: num,N: num] :
          ( ( modulo_modulo(A,aa(num,A,numeral_numeral(A),aa(num,num,bit1,M)),aa(num,A,numeral_numeral(A),bit0(Q2))) = modulo_modulo(A,aa(num,A,numeral_numeral(A),aa(num,num,bit1,N)),aa(num,A,numeral_numeral(A),bit0(Q2))) )
        <=> ( modulo_modulo(A,aa(num,A,numeral_numeral(A),M),aa(num,A,numeral_numeral(A),Q2)) = modulo_modulo(A,aa(num,A,numeral_numeral(A),N),aa(num,A,numeral_numeral(A),Q2)) ) ) ) ).

% cong_exp_iff_simps(13)
tff(fact_2001_div__mult1__eq,axiom,
    ! [A: $tType] :
      ( euclid3128863361964157862miring(A)
     => ! [A2: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2))),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),modulo_modulo(A,B2,C2))),C2)) ) ).

% div_mult1_eq
tff(fact_2002_mult__div__mod__eq,axiom,
    ! [A: $tType] :
      ( semiring_modulo(A)
     => ! [B2: A,A2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2))),modulo_modulo(A,A2,B2)) = A2 ) ).

% mult_div_mod_eq
tff(fact_2003_mod__mult__div__eq,axiom,
    ! [A: $tType] :
      ( semiring_modulo(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),modulo_modulo(A,A2,B2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2))) = A2 ) ).

% mod_mult_div_eq
tff(fact_2004_mod__div__mult__eq,axiom,
    ! [A: $tType] :
      ( semiring_modulo(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),modulo_modulo(A,A2,B2)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)),B2)) = A2 ) ).

% mod_div_mult_eq
tff(fact_2005_div__mult__mod__eq,axiom,
    ! [A: $tType] :
      ( semiring_modulo(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)),B2)),modulo_modulo(A,A2,B2)) = A2 ) ).

% div_mult_mod_eq
tff(fact_2006_mod__div__decomp,axiom,
    ! [A: $tType] :
      ( semiring_modulo(A)
     => ! [A2: A,B2: A] : A2 = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)),B2)),modulo_modulo(A,A2,B2)) ) ).

% mod_div_decomp
tff(fact_2007_cancel__div__mod__rules_I1_J,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [A2: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)),B2)),modulo_modulo(A,A2,B2))),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C2) ) ).

% cancel_div_mod_rules(1)
tff(fact_2008_cancel__div__mod__rules_I2_J,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [B2: A,A2: A,C2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2))),modulo_modulo(A,A2,B2))),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C2) ) ).

% cancel_div_mod_rules(2)
tff(fact_2009_unit__imp__mod__eq__0,axiom,
    ! [A: $tType] :
      ( euclid3725896446679973847miring(A)
     => ! [B2: A,A2: A] :
          ( pp(aa(A,bool,dvd_dvd(A,B2),one_one(A)))
         => ( modulo_modulo(A,A2,B2) = zero_zero(A) ) ) ) ).

% unit_imp_mod_eq_0
tff(fact_2010_zmde,axiom,
    ! [A: $tType] :
      ( ( group_add(A)
        & semiring_modulo(A) )
     => ! [B2: A,A2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),B2),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)) = aa(A,A,minus_minus(A,A2),modulo_modulo(A,A2,B2)) ) ).

% zmde
tff(fact_2011_minus__mult__div__eq__mod,axiom,
    ! [A: $tType] :
      ( semiring_modulo(A)
     => ! [A2: A,B2: A] : aa(A,A,minus_minus(A,A2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2))) = modulo_modulo(A,A2,B2) ) ).

% minus_mult_div_eq_mod
tff(fact_2012_minus__mod__eq__mult__div,axiom,
    ! [A: $tType] :
      ( semiring_modulo(A)
     => ! [A2: A,B2: A] : aa(A,A,minus_minus(A,A2),modulo_modulo(A,A2,B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),B2),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)) ) ).

% minus_mod_eq_mult_div
tff(fact_2013_minus__mod__eq__div__mult,axiom,
    ! [A: $tType] :
      ( semiring_modulo(A)
     => ! [A2: A,B2: A] : aa(A,A,minus_minus(A,A2),modulo_modulo(A,A2,B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)),B2) ) ).

% minus_mod_eq_div_mult
tff(fact_2014_minus__div__mult__eq__mod,axiom,
    ! [A: $tType] :
      ( semiring_modulo(A)
     => ! [A2: A,B2: A] : aa(A,A,minus_minus(A,A2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)),B2)) = modulo_modulo(A,A2,B2) ) ).

% minus_div_mult_eq_mod
tff(fact_2015_mod__le__divisor,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),N))
     => pp(aa(nat,bool,ord_less_eq(nat,modulo_modulo(nat,M,N)),N)) ) ).

% mod_le_divisor
tff(fact_2016_div__less__mono,axiom,
    ! [A3: nat,B4: nat,N: nat] :
      ( pp(aa(nat,bool,ord_less(nat,A3),B4))
     => ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),N))
       => ( ( modulo_modulo(nat,A3,N) = zero_zero(nat) )
         => ( ( modulo_modulo(nat,B4,N) = zero_zero(nat) )
           => pp(aa(nat,bool,ord_less(nat,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),A3),N)),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),B4),N))) ) ) ) ) ).

% div_less_mono
tff(fact_2017_mod__nat__add,axiom,
    ! [X: nat,Z: nat,Y: nat] :
      ( pp(aa(nat,bool,ord_less(nat,X),Z))
     => ( pp(aa(nat,bool,ord_less(nat,Y),Z))
       => ( ( pp(aa(nat,bool,ord_less(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),X),Y)),Z))
           => ( modulo_modulo(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),X),Y),Z) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),X),Y) ) )
          & ( ~ pp(aa(nat,bool,ord_less(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),X),Y)),Z))
           => ( modulo_modulo(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),X),Y),Z) = aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),X),Y)),Z) ) ) ) ) ) ).

% mod_nat_add
tff(fact_2018_invar__vebt_Ointros_I1_J,axiom,
    ! [A2: bool,B2: bool] : vEBT_invar_vebt(vEBT_Leaf(A2,B2),aa(nat,nat,suc,zero_zero(nat))) ).

% invar_vebt.intros(1)
tff(fact_2019_mod__greater__zero__iff__not__dvd,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),modulo_modulo(nat,M,N)))
    <=> ~ pp(aa(nat,bool,dvd_dvd(nat,N),M)) ) ).

% mod_greater_zero_iff_not_dvd
tff(fact_2020_nat__mod__eq__lemma,axiom,
    ! [X: nat,N: nat,Y: nat] :
      ( ( modulo_modulo(nat,X,N) = modulo_modulo(nat,Y,N) )
     => ( pp(aa(nat,bool,ord_less_eq(nat,Y),X))
       => ? [Q3: nat] : X = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Y),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),Q3)) ) ) ).

% nat_mod_eq_lemma
tff(fact_2021_mod__eq__nat2E,axiom,
    ! [M: nat,Q2: nat,N: nat] :
      ( ( modulo_modulo(nat,M,Q2) = modulo_modulo(nat,N,Q2) )
     => ( pp(aa(nat,bool,ord_less_eq(nat,M),N))
       => ~ ! [S2: nat] : N != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Q2),S2)) ) ) ).

% mod_eq_nat2E
tff(fact_2022_mod__eq__nat1E,axiom,
    ! [M: nat,Q2: nat,N: nat] :
      ( ( modulo_modulo(nat,M,Q2) = modulo_modulo(nat,N,Q2) )
     => ( pp(aa(nat,bool,ord_less_eq(nat,N),M))
       => ~ ! [S2: nat] : M != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Q2),S2)) ) ) ).

% mod_eq_nat1E
tff(fact_2023_length__pos__if__in__set,axiom,
    ! [A: $tType,X: A,Xs: list(A)] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(list(A),set(A),set2(A),Xs)))
     => pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),aa(list(A),nat,size_size(list(A)),Xs))) ) ).

% length_pos_if_in_set
tff(fact_2024_div__mod__decomp,axiom,
    ! [A3: nat,N: nat] : A3 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),A3),N)),N)),modulo_modulo(nat,A3,N)) ).

% div_mod_decomp
tff(fact_2025_mod__mult2__eq,axiom,
    ! [M: nat,N: nat,Q2: nat] : modulo_modulo(nat,M,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),Q2)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),modulo_modulo(nat,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),N),Q2))),modulo_modulo(nat,M,N)) ).

% mod_mult2_eq
tff(fact_2026_modulo__nat__def,axiom,
    ! [M: nat,N: nat] : modulo_modulo(nat,M,N) = aa(nat,nat,minus_minus(nat,M),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),N)),N)) ).

% modulo_nat_def
tff(fact_2027_mod__eq__dvd__iff__nat,axiom,
    ! [N: nat,M: nat,Q2: nat] :
      ( pp(aa(nat,bool,ord_less_eq(nat,N),M))
     => ( ( modulo_modulo(nat,M,Q2) = modulo_modulo(nat,N,Q2) )
      <=> pp(aa(nat,bool,dvd_dvd(nat,Q2),aa(nat,nat,minus_minus(nat,M),N))) ) ) ).

% mod_eq_dvd_iff_nat
tff(fact_2028_vebt__buildup_Osimps_I2_J,axiom,
    vEBT_vebt_buildup(aa(nat,nat,suc,zero_zero(nat))) = vEBT_Leaf(fFalse,fFalse) ).

% vebt_buildup.simps(2)
tff(fact_2029_vebt__insert_Osimps_I1_J,axiom,
    ! [X: nat,A2: bool,B2: bool] :
      ( ( ( X = zero_zero(nat) )
       => ( vEBT_vebt_insert(vEBT_Leaf(A2,B2),X) = vEBT_Leaf(fTrue,B2) ) )
      & ( ( X != zero_zero(nat) )
       => ( ( ( X = one_one(nat) )
           => ( vEBT_vebt_insert(vEBT_Leaf(A2,B2),X) = vEBT_Leaf(A2,fTrue) ) )
          & ( ( X != one_one(nat) )
           => ( vEBT_vebt_insert(vEBT_Leaf(A2,B2),X) = vEBT_Leaf(A2,B2) ) ) ) ) ) ).

% vebt_insert.simps(1)
tff(fact_2030_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: bool,B2: bool,N: nat] : vEBT_V1232361888498592333_e_t_e(vEBT_Leaf(A2,B2),aa(nat,nat,suc,aa(nat,nat,suc,N))) = one_one(nat) ).

% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.simps(3)
tff(fact_2031_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: bool,B2: bool,N: nat] : vEBT_T_d_e_l_e_t_e(vEBT_Leaf(A2,B2),aa(nat,nat,suc,aa(nat,nat,suc,N))) = one_one(nat) ).

% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.simps(3)
tff(fact_2032_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: bool,B2: bool] : 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)
tff(fact_2033_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: bool,B2: bool] : 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)
tff(fact_2034_VEBT__internal_Onaive__member_Osimps_I1_J,axiom,
    ! [A2: bool,B2: bool,X: nat] :
      ( vEBT_V5719532721284313246member(vEBT_Leaf(A2,B2),X)
    <=> ( ( ( X = zero_zero(nat) )
         => pp(A2) )
        & ( ( X != zero_zero(nat) )
         => ( ( ( X = one_one(nat) )
             => pp(B2) )
            & ( X = one_one(nat) ) ) ) ) ) ).

% VEBT_internal.naive_member.simps(1)
tff(fact_2035_cong__exp__iff__simps_I3_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [N: num,Q2: num] : modulo_modulo(A,aa(num,A,numeral_numeral(A),aa(num,num,bit1,N)),aa(num,A,numeral_numeral(A),bit0(Q2))) != zero_zero(A) ) ).

% cong_exp_iff_simps(3)
tff(fact_2036_mod__mult2__eq_H,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [A2: A,M: nat,N: nat] : modulo_modulo(A,A2,aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),M)),aa(nat,A,semiring_1_of_nat(A),N))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),M)),modulo_modulo(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(nat,A,semiring_1_of_nat(A),M)),aa(nat,A,semiring_1_of_nat(A),N)))),modulo_modulo(A,A2,aa(nat,A,semiring_1_of_nat(A),M))) ) ).

% mod_mult2_eq'
tff(fact_2037_even__even__mod__4__iff,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N))
    <=> pp(aa(nat,bool,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),modulo_modulo(nat,N,aa(num,nat,numeral_numeral(nat),bit0(bit0(one2)))))) ) ).

% even_even_mod_4_iff
tff(fact_2038_unset__bit__Suc,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A2: A] : aa(A,A,aa(nat,fun(A,A),bit_se2638667681897837118et_bit(A),aa(nat,nat,suc,N)),A2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),modulo_modulo(A,A2,aa(num,A,numeral_numeral(A),bit0(one2)))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(nat,fun(A,A),bit_se2638667681897837118et_bit(A),N),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(num,A,numeral_numeral(A),bit0(one2)))))) ) ).

% unset_bit_Suc
tff(fact_2039_field__char__0__class_Oof__nat__div,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [M: nat,N: nat] : aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),N)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,minus_minus(A,aa(nat,A,semiring_1_of_nat(A),M)),aa(nat,A,semiring_1_of_nat(A),modulo_modulo(nat,M,N)))),aa(nat,A,semiring_1_of_nat(A),N)) ) ).

% field_char_0_class.of_nat_div
tff(fact_2040_mod__lemma,axiom,
    ! [C2: nat,R2: nat,B2: nat,Q2: nat] :
      ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),C2))
     => ( pp(aa(nat,bool,ord_less(nat,R2),B2))
       => pp(aa(nat,bool,ord_less(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),modulo_modulo(nat,Q2,C2))),R2)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),C2))) ) ) ).

% mod_lemma
tff(fact_2041_split__mod,axiom,
    ! [P: fun(nat,bool),M: nat,N: nat] :
      ( pp(aa(nat,bool,P,modulo_modulo(nat,M,N)))
    <=> ( ( ( N = zero_zero(nat) )
         => pp(aa(nat,bool,P,M)) )
        & ( ( N != zero_zero(nat) )
         => ! [I3: nat,J3: nat] :
              ( pp(aa(nat,bool,ord_less(nat,J3),N))
             => ( ( M = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),I3)),J3) )
               => pp(aa(nat,bool,P,J3)) ) ) ) ) ) ).

% split_mod
tff(fact_2042_VEBT__internal_Onaive__member_Oelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: bool] :
      ( ( vEBT_V5719532721284313246member(X,Xa2)
      <=> pp(Y) )
     => ( ! [A4: bool,B3: bool] :
            ( ( X = vEBT_Leaf(A4,B3) )
           => ( pp(Y)
            <=> ~ ( ( ( Xa2 = zero_zero(nat) )
                   => pp(A4) )
                  & ( ( Xa2 != zero_zero(nat) )
                   => ( ( ( Xa2 = one_one(nat) )
                       => pp(B3) )
                      & ( Xa2 = one_one(nat) ) ) ) ) ) )
       => ( ( ? [Uu2: option(product_prod(nat,nat)),Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT] : X = vEBT_Node(Uu2,zero_zero(nat),Uv2,Uw2)
           => pp(Y) )
         => ~ ! [Uy2: option(product_prod(nat,nat)),V3: nat,TreeList2: list(vEBT_VEBT)] :
                ( ? [S2: vEBT_VEBT] : X = vEBT_Node(Uy2,aa(nat,nat,suc,V3),TreeList2,S2)
               => ( pp(Y)
                <=> ~ ( ( pp(aa(nat,bool,ord_less(nat,vEBT_VEBT_high(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V3)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)))
                       => vEBT_V5719532721284313246member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),vEBT_VEBT_high(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V3)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_VEBT_low(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V3)),aa(num,nat,numeral_numeral(nat),bit0(one2))))) )
                      & pp(aa(nat,bool,ord_less(nat,vEBT_VEBT_high(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V3)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2))) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.elims(1)
tff(fact_2043_VEBT__internal_Onaive__member_Oelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( vEBT_V5719532721284313246member(X,Xa2)
     => ( ! [A4: bool,B3: bool] :
            ( ( X = vEBT_Leaf(A4,B3) )
           => ~ ( ( ( Xa2 = zero_zero(nat) )
                 => pp(A4) )
                & ( ( Xa2 != zero_zero(nat) )
                 => ( ( ( Xa2 = one_one(nat) )
                     => pp(B3) )
                    & ( Xa2 = one_one(nat) ) ) ) ) )
       => ~ ! [Uy2: option(product_prod(nat,nat)),V3: nat,TreeList2: list(vEBT_VEBT)] :
              ( ? [S2: vEBT_VEBT] : X = vEBT_Node(Uy2,aa(nat,nat,suc,V3),TreeList2,S2)
             => ~ ( ( pp(aa(nat,bool,ord_less(nat,vEBT_VEBT_high(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V3)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)))
                   => vEBT_V5719532721284313246member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),vEBT_VEBT_high(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V3)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_VEBT_low(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V3)),aa(num,nat,numeral_numeral(nat),bit0(one2))))) )
                  & pp(aa(nat,bool,ord_less(nat,vEBT_VEBT_high(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V3)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2))) ) ) ) ) ).

% VEBT_internal.naive_member.elims(2)
tff(fact_2044_VEBT__internal_Onaive__member_Oelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ~ vEBT_V5719532721284313246member(X,Xa2)
     => ( ! [A4: bool,B3: bool] :
            ( ( X = vEBT_Leaf(A4,B3) )
           => ( ( ( Xa2 = zero_zero(nat) )
               => pp(A4) )
              & ( ( Xa2 != zero_zero(nat) )
               => ( ( ( Xa2 = one_one(nat) )
                   => pp(B3) )
                  & ( Xa2 = one_one(nat) ) ) ) ) )
       => ( ! [Uu2: option(product_prod(nat,nat)),Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT] : X != vEBT_Node(Uu2,zero_zero(nat),Uv2,Uw2)
         => ~ ! [Uy2: option(product_prod(nat,nat)),V3: nat,TreeList2: list(vEBT_VEBT)] :
                ( ? [S2: vEBT_VEBT] : X = vEBT_Node(Uy2,aa(nat,nat,suc,V3),TreeList2,S2)
               => ( ( pp(aa(nat,bool,ord_less(nat,vEBT_VEBT_high(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V3)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)))
                   => vEBT_V5719532721284313246member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),vEBT_VEBT_high(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V3)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_VEBT_low(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V3)),aa(num,nat,numeral_numeral(nat),bit0(one2))))) )
                  & pp(aa(nat,bool,ord_less(nat,vEBT_VEBT_high(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V3)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2))) ) ) ) ) ) ).

% VEBT_internal.naive_member.elims(3)
tff(fact_2045_diff__mod__le,axiom,
    ! [A2: nat,D: nat,B2: nat] :
      ( pp(aa(nat,bool,ord_less(nat,A2),D))
     => ( pp(aa(nat,bool,dvd_dvd(nat,B2),D))
       => pp(aa(nat,bool,ord_less_eq(nat,aa(nat,nat,minus_minus(nat,A2),modulo_modulo(nat,A2,B2))),aa(nat,nat,minus_minus(nat,D),B2))) ) ) ).

% diff_mod_le
tff(fact_2046_mod__nat__eqI,axiom,
    ! [R2: nat,N: nat,M: nat] :
      ( pp(aa(nat,bool,ord_less(nat,R2),N))
     => ( pp(aa(nat,bool,ord_less_eq(nat,R2),M))
       => ( pp(aa(nat,bool,dvd_dvd(nat,N),aa(nat,nat,minus_minus(nat,M),R2)))
         => ( modulo_modulo(nat,M,N) = R2 ) ) ) ) ).

% mod_nat_eqI
tff(fact_2047_real__of__nat__div__aux,axiom,
    ! [X: nat,D: nat] : aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(nat,real,semiring_1_of_nat(real),X)),aa(nat,real,semiring_1_of_nat(real),D)) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),X),D))),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(nat,real,semiring_1_of_nat(real),modulo_modulo(nat,X,D))),aa(nat,real,semiring_1_of_nat(real),D))) ).

% real_of_nat_div_aux
tff(fact_2048_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: bool,B2: bool] : vEBT_T_d_e_l_e_t_e(vEBT_Leaf(A2,B2),aa(nat,nat,suc,zero_zero(nat))) = one_one(nat) ).

% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.simps(2)
tff(fact_2049_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: bool,B2: bool] : vEBT_V1232361888498592333_e_t_e(vEBT_Leaf(A2,B2),aa(nat,nat,suc,zero_zero(nat))) = one_one(nat) ).

% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.simps(2)
tff(fact_2050_VEBT__internal_Ospace_H_Osimps_I1_J,axiom,
    ! [A2: bool,B2: bool] : aa(vEBT_VEBT,nat,vEBT_VEBT_space2,vEBT_Leaf(A2,B2)) = aa(num,nat,numeral_numeral(nat),bit0(bit0(one2))) ).

% VEBT_internal.space'.simps(1)
tff(fact_2051_VEBT__internal_Ospace_Osimps_I1_J,axiom,
    ! [A2: bool,B2: bool] : aa(vEBT_VEBT,nat,vEBT_VEBT_space,vEBT_Leaf(A2,B2)) = aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2)) ).

% VEBT_internal.space.simps(1)
tff(fact_2052_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [C2: A,A2: A,B2: A] :
          ( pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),C2))
         => ( modulo_modulo(A,A2,aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),modulo_modulo(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2),C2))),modulo_modulo(A,A2,B2)) ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_mult2_eq
tff(fact_2053_cong__exp__iff__simps_I7_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Q2: num,N: num] :
          ( ( modulo_modulo(A,aa(num,A,numeral_numeral(A),one2),aa(num,A,numeral_numeral(A),bit0(Q2))) = modulo_modulo(A,aa(num,A,numeral_numeral(A),aa(num,num,bit1,N)),aa(num,A,numeral_numeral(A),bit0(Q2))) )
        <=> ( modulo_modulo(A,aa(num,A,numeral_numeral(A),N),aa(num,A,numeral_numeral(A),Q2)) = zero_zero(A) ) ) ) ).

% cong_exp_iff_simps(7)
tff(fact_2054_cong__exp__iff__simps_I11_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M: num,Q2: num] :
          ( ( modulo_modulo(A,aa(num,A,numeral_numeral(A),aa(num,num,bit1,M)),aa(num,A,numeral_numeral(A),bit0(Q2))) = modulo_modulo(A,aa(num,A,numeral_numeral(A),one2),aa(num,A,numeral_numeral(A),bit0(Q2))) )
        <=> ( modulo_modulo(A,aa(num,A,numeral_numeral(A),M),aa(num,A,numeral_numeral(A),Q2)) = zero_zero(A) ) ) ) ).

% cong_exp_iff_simps(11)
tff(fact_2055_even__iff__mod__2__eq__zero,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A2: A] :
          ( pp(aa(A,bool,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A2))
        <=> ( modulo_modulo(A,A2,aa(num,A,numeral_numeral(A),bit0(one2))) = zero_zero(A) ) ) ) ).

% even_iff_mod_2_eq_zero
tff(fact_2056_odd__iff__mod__2__eq__one,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A2: A] :
          ( ~ pp(aa(A,bool,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A2))
        <=> ( modulo_modulo(A,A2,aa(num,A,numeral_numeral(A),bit0(one2))) = one_one(A) ) ) ) ).

% odd_iff_mod_2_eq_one
tff(fact_2057_Suc__mod__eq__add3__mod,axiom,
    ! [M: nat,N: nat] : modulo_modulo(nat,aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,suc,M))),N) = modulo_modulo(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2))),M),N) ).

% Suc_mod_eq_add3_mod
tff(fact_2058_Suc__times__mod__eq,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,ord_less(nat,aa(nat,nat,suc,zero_zero(nat))),M))
     => ( modulo_modulo(nat,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N)),M) = one_one(nat) ) ) ).

% Suc_times_mod_eq
tff(fact_2059_VEBT__internal_Onaive__member_Osimps_I3_J,axiom,
    ! [Uy: option(product_prod(nat,nat)),V: nat,TreeList: list(vEBT_VEBT),S3: vEBT_VEBT,X: nat] :
      ( vEBT_V5719532721284313246member(vEBT_Node(Uy,aa(nat,nat,suc,V),TreeList,S3),X)
    <=> ( ( pp(aa(nat,bool,ord_less(nat,vEBT_VEBT_high(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)))
         => vEBT_V5719532721284313246member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_VEBT_low(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V)),aa(num,nat,numeral_numeral(nat),bit0(one2))))) )
        & pp(aa(nat,bool,ord_less(nat,vEBT_VEBT_high(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))) ) ) ).

% VEBT_internal.naive_member.simps(3)
tff(fact_2060_divmod__digit__0_I2_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [B2: A,A2: A] :
          ( pp(aa(A,bool,ord_less(A,zero_zero(A)),B2))
         => ( pp(aa(A,bool,ord_less(A,modulo_modulo(A,A2,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),B2))),B2))
           => ( modulo_modulo(A,A2,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),B2)) = modulo_modulo(A,A2,B2) ) ) ) ) ).

% divmod_digit_0(2)
tff(fact_2061_bits__stable__imp__add__self,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(num,A,numeral_numeral(A),bit0(one2))) = A2 )
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),modulo_modulo(A,A2,aa(num,A,numeral_numeral(A),bit0(one2)))) = zero_zero(A) ) ) ) ).

% bits_stable_imp_add_self
tff(fact_2062_parity__cases,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A2: A] :
          ( ( pp(aa(A,bool,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A2))
           => ( modulo_modulo(A,A2,aa(num,A,numeral_numeral(A),bit0(one2))) != zero_zero(A) ) )
         => ~ ( ~ pp(aa(A,bool,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A2))
             => ( modulo_modulo(A,A2,aa(num,A,numeral_numeral(A),bit0(one2))) != one_one(A) ) ) ) ) ).

% parity_cases
tff(fact_2063_mod2__eq__if,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A2: A] :
          ( ( pp(aa(A,bool,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A2))
           => ( modulo_modulo(A,A2,aa(num,A,numeral_numeral(A),bit0(one2))) = zero_zero(A) ) )
          & ( ~ pp(aa(A,bool,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A2))
           => ( modulo_modulo(A,A2,aa(num,A,numeral_numeral(A),bit0(one2))) = one_one(A) ) ) ) ) ).

% mod2_eq_if
tff(fact_2064_div__exp__mod__exp__eq,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A,N: nat,M: nat] : modulo_modulo(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),M)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),modulo_modulo(A,A2,aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),M)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N)) ) ).

% div_exp_mod_exp_eq
tff(fact_2065_VEBT__internal_Omembermima_Osimps_I5_J,axiom,
    ! [V: nat,TreeList: list(vEBT_VEBT),Vd: vEBT_VEBT,X: nat] :
      ( vEBT_VEBT_membermima(vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,V),TreeList,Vd),X)
    <=> ( ( pp(aa(nat,bool,ord_less(nat,vEBT_VEBT_high(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)))
         => vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_VEBT_low(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V)),aa(num,nat,numeral_numeral(nat),bit0(one2))))) )
        & pp(aa(nat,bool,ord_less(nat,vEBT_VEBT_high(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))) ) ) ).

% VEBT_internal.membermima.simps(5)
tff(fact_2066_power__mod__div,axiom,
    ! [X: nat,N: nat,M: nat] : aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),modulo_modulo(nat,X,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M)) = modulo_modulo(nat,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),X),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,minus_minus(nat,N),M))) ).

% power_mod_div
tff(fact_2067_verit__le__mono__div,axiom,
    ! [A3: nat,B4: nat,N: nat] :
      ( pp(aa(nat,bool,ord_less(nat,A3),B4))
     => ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),N))
       => pp(aa(nat,bool,ord_less_eq(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),A3),N)),if(nat,aa(nat,bool,fequal(nat,modulo_modulo(nat,B4,N)),zero_zero(nat)),one_one(nat),zero_zero(nat)))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),B4),N))) ) ) ).

% verit_le_mono_div
tff(fact_2068_divmod__digit__0_I1_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [B2: A,A2: A] :
          ( pp(aa(A,bool,ord_less(A,zero_zero(A)),B2))
         => ( pp(aa(A,bool,ord_less(A,modulo_modulo(A,A2,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),B2))),B2))
           => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),B2))) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2) ) ) ) ) ).

% divmod_digit_0(1)
tff(fact_2069_mult__exp__mod__exp__eq,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [M: nat,N: nat,A2: A] :
          ( pp(aa(nat,bool,ord_less_eq(nat,M),N))
         => ( modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),M)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N)) = aa(A,A,aa(A,fun(A,A),times_times(A),modulo_modulo(A,A2,aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(nat,nat,minus_minus(nat,N),M)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),M)) ) ) ) ).

% mult_exp_mod_exp_eq
tff(fact_2070_mod__double__modulus,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M: A,X: A] :
          ( pp(aa(A,bool,ord_less(A,zero_zero(A)),M))
         => ( pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),X))
           => ( ( modulo_modulo(A,X,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),M)) = modulo_modulo(A,X,M) )
              | ( modulo_modulo(A,X,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),M)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),modulo_modulo(A,X,M)),M) ) ) ) ) ) ).

% mod_double_modulus
tff(fact_2071_divmod__digit__1_I2_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),A2))
         => ( pp(aa(A,bool,ord_less(A,zero_zero(A)),B2))
           => ( pp(aa(A,bool,ord_less_eq(A,B2),modulo_modulo(A,A2,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),B2))))
             => ( aa(A,A,minus_minus(A,modulo_modulo(A,A2,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),B2))),B2) = modulo_modulo(A,A2,B2) ) ) ) ) ) ).

% divmod_digit_1(2)
tff(fact_2072_set__bit__Suc,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A2: A] : aa(A,A,aa(nat,fun(A,A),bit_se5668285175392031749et_bit(A),aa(nat,nat,suc,N)),A2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),modulo_modulo(A,A2,aa(num,A,numeral_numeral(A),bit0(one2)))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(nat,fun(A,A),bit_se5668285175392031749et_bit(A),N),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(num,A,numeral_numeral(A),bit0(one2)))))) ) ).

% set_bit_Suc
tff(fact_2073_even__mod__4__div__2,axiom,
    ! [N: nat] :
      ( ( modulo_modulo(nat,N,aa(num,nat,numeral_numeral(nat),bit0(bit0(one2)))) = aa(nat,nat,suc,zero_zero(nat)) )
     => pp(aa(nat,bool,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,minus_minus(nat,N),aa(nat,nat,suc,zero_zero(nat)))),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) ).

% even_mod_4_div_2
tff(fact_2074_even__unset__bit__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,A2: A] :
          ( pp(aa(A,bool,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(nat,fun(A,A),bit_se2638667681897837118et_bit(A),M),A2)))
        <=> ( pp(aa(A,bool,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A2))
            | ( M = zero_zero(nat) ) ) ) ) ).

% even_unset_bit_iff
tff(fact_2075_odd__mod__4__div__2,axiom,
    ! [N: nat] :
      ( ( modulo_modulo(nat,N,aa(num,nat,numeral_numeral(nat),bit0(bit0(one2)))) = aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2)) )
     => ~ pp(aa(nat,bool,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,minus_minus(nat,N),aa(nat,nat,suc,zero_zero(nat)))),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) ).

% odd_mod_4_div_2
tff(fact_2076_divmod__digit__1_I1_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),A2))
         => ( pp(aa(A,bool,ord_less(A,zero_zero(A)),B2))
           => ( pp(aa(A,bool,ord_less_eq(A,B2),modulo_modulo(A,A2,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),B2))))
             => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),B2)))),one_one(A)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2) ) ) ) ) ) ).

% divmod_digit_1(1)
tff(fact_2077_invar__vebt_Ointros_I2_J,axiom,
    ! [TreeList: list(vEBT_VEBT),N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat] :
      ( ! [X3: vEBT_VEBT] :
          ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),member(vEBT_VEBT),X3),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList)))
         => vEBT_invar_vebt(X3,N) )
     => ( vEBT_invar_vebt(Summary,M)
       => ( ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M) )
         => ( ( M = N )
           => ( ( Deg = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),M) )
             => ( ~ ? [X_1: nat] : pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,Summary),X_1))
               => ( ! [X3: vEBT_VEBT] :
                      ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),member(vEBT_VEBT),X3),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList)))
                     => ~ ? [X_1: nat] : pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,X3),X_1)) )
                 => vEBT_invar_vebt(vEBT_Node(none(product_prod(nat,nat)),Deg,TreeList,Summary),Deg) ) ) ) ) ) ) ) ).

% invar_vebt.intros(2)
tff(fact_2078_invar__vebt_Ointros_I3_J,axiom,
    ! [TreeList: list(vEBT_VEBT),N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat] :
      ( ! [X3: vEBT_VEBT] :
          ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),member(vEBT_VEBT),X3),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList)))
         => vEBT_invar_vebt(X3,N) )
     => ( vEBT_invar_vebt(Summary,M)
       => ( ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M) )
         => ( ( M = aa(nat,nat,suc,N) )
           => ( ( Deg = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),M) )
             => ( ~ ? [X_1: nat] : pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,Summary),X_1))
               => ( ! [X3: vEBT_VEBT] :
                      ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),member(vEBT_VEBT),X3),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList)))
                     => ~ ? [X_1: nat] : pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,X3),X_1)) )
                 => vEBT_invar_vebt(vEBT_Node(none(product_prod(nat,nat)),Deg,TreeList,Summary),Deg) ) ) ) ) ) ) ) ).

% invar_vebt.intros(3)
tff(fact_2079_in__children__def,axiom,
    ! [N: nat,TreeList: list(vEBT_VEBT),X: nat] :
      ( vEBT_V5917875025757280293ildren(N,TreeList,X)
    <=> pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(X,N))),vEBT_VEBT_low(X,N))) ) ).

% in_children_def
tff(fact_2080_mod__exhaust__less__4,axiom,
    ! [M: nat] :
      ( ( modulo_modulo(nat,M,aa(num,nat,numeral_numeral(nat),bit0(bit0(one2)))) = zero_zero(nat) )
      | ( modulo_modulo(nat,M,aa(num,nat,numeral_numeral(nat),bit0(bit0(one2)))) = one_one(nat) )
      | ( modulo_modulo(nat,M,aa(num,nat,numeral_numeral(nat),bit0(bit0(one2)))) = aa(num,nat,numeral_numeral(nat),bit0(one2)) )
      | ( modulo_modulo(nat,M,aa(num,nat,numeral_numeral(nat),bit0(bit0(one2)))) = aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2)) ) ) ).

% mod_exhaust_less_4
tff(fact_2081_artanh__def,axiom,
    ! [A: $tType] :
      ( ( real_V3459762299906320749_field(A)
        & ln(A) )
     => ! [X: A] : aa(A,A,artanh(A),X) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,ln_ln(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),X)),aa(A,A,minus_minus(A,one_one(A)),X)))),aa(num,A,numeral_numeral(A),bit0(one2))) ) ).

% artanh_def
tff(fact_2082_vebt__member_Osimps_I1_J,axiom,
    ! [A2: bool,B2: bool,X: nat] :
      ( pp(aa(nat,bool,vEBT_vebt_member(vEBT_Leaf(A2,B2)),X))
    <=> ( ( ( X = zero_zero(nat) )
         => pp(A2) )
        & ( ( X != zero_zero(nat) )
         => ( ( ( X = one_one(nat) )
             => pp(B2) )
            & ( X = one_one(nat) ) ) ) ) ) ).

% vebt_member.simps(1)
tff(fact_2083_flip__bit__Suc,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A2: A] : bit_se8732182000553998342ip_bit(A,aa(nat,nat,suc,N),A2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),modulo_modulo(A,A2,aa(num,A,numeral_numeral(A),bit0(one2)))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),bit_se8732182000553998342ip_bit(A,N,aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(num,A,numeral_numeral(A),bit0(one2)))))) ) ).

% flip_bit_Suc
tff(fact_2084_is__succ__in__set__def,axiom,
    ! [Xs: set(nat),X: nat,Y: nat] :
      ( vEBT_is_succ_in_set(Xs,X,Y)
    <=> ( pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),Y),Xs))
        & pp(aa(nat,bool,ord_less(nat,X),Y))
        & ! [X2: nat] :
            ( pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),X2),Xs))
           => ( pp(aa(nat,bool,ord_less(nat,X),X2))
             => pp(aa(nat,bool,ord_less_eq(nat,Y),X2)) ) ) ) ) ).

% is_succ_in_set_def
tff(fact_2085_is__pred__in__set__def,axiom,
    ! [Xs: set(nat),X: nat,Y: nat] :
      ( vEBT_is_pred_in_set(Xs,X,Y)
    <=> ( pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),Y),Xs))
        & pp(aa(nat,bool,ord_less(nat,Y),X))
        & ! [X2: nat] :
            ( pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),X2),Xs))
           => ( pp(aa(nat,bool,ord_less(nat,X2),X))
             => pp(aa(nat,bool,ord_less_eq(nat,X2),Y)) ) ) ) ) ).

% is_pred_in_set_def
tff(fact_2086_mod__word__self,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] : modulo_modulo(word(A),W,W) = zero_zero(word(A)) ) ).

% mod_word_self
tff(fact_2087_word__size__gt__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] : pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),aa(word(A),nat,size_size(word(A)),W))) ) ).

% word_size_gt_0
tff(fact_2088_ln__inj__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa(real,bool,ord_less(real,zero_zero(real)),X))
     => ( pp(aa(real,bool,ord_less(real,zero_zero(real)),Y))
       => ( ( aa(real,real,ln_ln(real),X) = aa(real,real,ln_ln(real),Y) )
        <=> ( X = Y ) ) ) ) ).

% ln_inj_iff
tff(fact_2089_ln__less__cancel__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa(real,bool,ord_less(real,zero_zero(real)),X))
     => ( pp(aa(real,bool,ord_less(real,zero_zero(real)),Y))
       => ( pp(aa(real,bool,ord_less(real,aa(real,real,ln_ln(real),X)),aa(real,real,ln_ln(real),Y)))
        <=> pp(aa(real,bool,ord_less(real,X),Y)) ) ) ) ).

% ln_less_cancel_iff
tff(fact_2090_flip__bit__nonnegative__int__iff,axiom,
    ! [N: nat,K: int] :
      ( pp(aa(int,bool,ord_less_eq(int,zero_zero(int)),bit_se8732182000553998342ip_bit(int,N,K)))
    <=> pp(aa(int,bool,ord_less_eq(int,zero_zero(int)),K)) ) ).

% flip_bit_nonnegative_int_iff
tff(fact_2091_flip__bit__negative__int__iff,axiom,
    ! [N: nat,K: int] :
      ( pp(aa(int,bool,ord_less(int,bit_se8732182000553998342ip_bit(int,N,K)),zero_zero(int)))
    <=> pp(aa(int,bool,ord_less(int,K),zero_zero(int))) ) ).

% flip_bit_negative_int_iff
tff(fact_2092_ln__le__cancel__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa(real,bool,ord_less(real,zero_zero(real)),X))
     => ( pp(aa(real,bool,ord_less(real,zero_zero(real)),Y))
       => ( pp(aa(real,bool,ord_less_eq(real,aa(real,real,ln_ln(real),X)),aa(real,real,ln_ln(real),Y)))
        <=> pp(aa(real,bool,ord_less_eq(real,X),Y)) ) ) ) ).

% ln_le_cancel_iff
tff(fact_2093_ln__less__zero__iff,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,ord_less(real,zero_zero(real)),X))
     => ( pp(aa(real,bool,ord_less(real,aa(real,real,ln_ln(real),X)),zero_zero(real)))
      <=> pp(aa(real,bool,ord_less(real,X),one_one(real))) ) ) ).

% ln_less_zero_iff
tff(fact_2094_ln__gt__zero__iff,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,ord_less(real,zero_zero(real)),X))
     => ( pp(aa(real,bool,ord_less(real,zero_zero(real)),aa(real,real,ln_ln(real),X)))
      <=> pp(aa(real,bool,ord_less(real,one_one(real)),X)) ) ) ).

% ln_gt_zero_iff
tff(fact_2095_ln__eq__zero__iff,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,ord_less(real,zero_zero(real)),X))
     => ( ( aa(real,real,ln_ln(real),X) = zero_zero(real) )
      <=> ( X = one_one(real) ) ) ) ).

% ln_eq_zero_iff
tff(fact_2096_ln__one,axiom,
    ! [A: $tType] :
      ( ln(A)
     => ( aa(A,A,ln_ln(A),one_one(A)) = zero_zero(A) ) ) ).

% ln_one
tff(fact_2097_mod__neg__neg__trivial,axiom,
    ! [K: int,L: int] :
      ( pp(aa(int,bool,ord_less_eq(int,K),zero_zero(int)))
     => ( pp(aa(int,bool,ord_less(int,L),K))
       => ( modulo_modulo(int,K,L) = K ) ) ) ).

% mod_neg_neg_trivial
tff(fact_2098_mod__pos__pos__trivial,axiom,
    ! [K: int,L: int] :
      ( pp(aa(int,bool,ord_less_eq(int,zero_zero(int)),K))
     => ( pp(aa(int,bool,ord_less(int,K),L))
       => ( modulo_modulo(int,K,L) = K ) ) ) ).

% mod_pos_pos_trivial
tff(fact_2099_zmod__numeral__Bit0,axiom,
    ! [V: num,W: num] : modulo_modulo(int,aa(num,int,numeral_numeral(int),bit0(V)),aa(num,int,numeral_numeral(int),bit0(W))) = aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),modulo_modulo(int,aa(num,int,numeral_numeral(int),V),aa(num,int,numeral_numeral(int),W))) ).

% zmod_numeral_Bit0
tff(fact_2100_ln__ge__zero__iff,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,ord_less(real,zero_zero(real)),X))
     => ( pp(aa(real,bool,ord_less_eq(real,zero_zero(real)),aa(real,real,ln_ln(real),X)))
      <=> pp(aa(real,bool,ord_less_eq(real,one_one(real)),X)) ) ) ).

% ln_ge_zero_iff
tff(fact_2101_ln__le__zero__iff,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,ord_less(real,zero_zero(real)),X))
     => ( pp(aa(real,bool,ord_less_eq(real,aa(real,real,ln_ln(real),X)),zero_zero(real)))
      <=> pp(aa(real,bool,ord_less_eq(real,X),one_one(real))) ) ) ).

% ln_le_zero_iff
tff(fact_2102_one__mod__exp__eq__one,axiom,
    ! [N: nat] : modulo_modulo(int,one_one(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),N))) = one_one(int) ).

% one_mod_exp_eq_one
tff(fact_2103_zmod__numeral__Bit1,axiom,
    ! [V: num,W: num] : modulo_modulo(int,aa(num,int,numeral_numeral(int),aa(num,num,bit1,V)),aa(num,int,numeral_numeral(int),bit0(W))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),modulo_modulo(int,aa(num,int,numeral_numeral(int),V),aa(num,int,numeral_numeral(int),W)))),one_one(int)) ).

% zmod_numeral_Bit1
tff(fact_2104_size__0__same_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),V: word(A)] :
          ( ( aa(word(A),nat,size_size(word(A)),W) = zero_zero(nat) )
         => ( W = V ) ) ) ).

% size_0_same'
tff(fact_2105_lens__not__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] : aa(word(A),nat,size_size(word(A)),W) != zero_zero(nat) ) ).

% lens_not_0
tff(fact_2106_size__0__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),V: word(A)] :
          ( ( aa(word(A),nat,size_size(word(A)),W) = zero_zero(nat) )
         => ( V = W ) ) ) ).

% size_0_eq
tff(fact_2107_word__mod__by__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [K: word(A)] : modulo_modulo(word(A),K,zero_zero(word(A))) = K ) ).

% word_mod_by_0
tff(fact_2108_zmod__helper,axiom,
    ! [N: int,M: int,K: int,A2: int] :
      ( ( modulo_modulo(int,N,M) = K )
     => ( modulo_modulo(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),N),A2),M) = modulo_modulo(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),K),A2),M) ) ) ).

% zmod_helper
tff(fact_2109_mod__plus__cong,axiom,
    ! [B2: int,B6: int,X: int,X5: int,Y: int,Y5: int,Z5: int] :
      ( ( B2 = B6 )
     => ( ( modulo_modulo(int,X,B6) = modulo_modulo(int,X5,B6) )
       => ( ( modulo_modulo(int,Y,B6) = modulo_modulo(int,Y5,B6) )
         => ( ( aa(int,int,aa(int,fun(int,int),plus_plus(int),X5),Y5) = Z5 )
           => ( modulo_modulo(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),X),Y),B2) = modulo_modulo(int,Z5,B6) ) ) ) ) ) ).

% mod_plus_cong
tff(fact_2110_ln__less__self,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,ord_less(real,zero_zero(real)),X))
     => pp(aa(real,bool,ord_less(real,aa(real,real,ln_ln(real),X)),X)) ) ).

% ln_less_self
tff(fact_2111_log__def,axiom,
    ! [A2: real,X: real] : aa(real,real,log(A2),X) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,ln_ln(real),X)),aa(real,real,ln_ln(real),A2)) ).

% log_def
tff(fact_2112_zmod__le__nonneg__dividend,axiom,
    ! [M: int,K: int] :
      ( pp(aa(int,bool,ord_less_eq(int,zero_zero(int)),M))
     => pp(aa(int,bool,ord_less_eq(int,modulo_modulo(int,M,K)),M)) ) ).

% zmod_le_nonneg_dividend
tff(fact_2113_neg__mod__bound,axiom,
    ! [L: int,K: int] :
      ( pp(aa(int,bool,ord_less(int,L),zero_zero(int)))
     => pp(aa(int,bool,ord_less(int,L),modulo_modulo(int,K,L))) ) ).

% neg_mod_bound
tff(fact_2114_Euclidean__Division_Opos__mod__bound,axiom,
    ! [L: int,K: int] :
      ( pp(aa(int,bool,ord_less(int,zero_zero(int)),L))
     => pp(aa(int,bool,ord_less(int,modulo_modulo(int,K,L)),L)) ) ).

% Euclidean_Division.pos_mod_bound
tff(fact_2115_zmod__eq__0__iff,axiom,
    ! [M: int,D: int] :
      ( ( modulo_modulo(int,M,D) = zero_zero(int) )
    <=> ? [Q4: int] : M = aa(int,int,aa(int,fun(int,int),times_times(int),D),Q4) ) ).

% zmod_eq_0_iff
tff(fact_2116_zmod__eq__0D,axiom,
    ! [M: int,D: int] :
      ( ( modulo_modulo(int,M,D) = zero_zero(int) )
     => ? [Q3: int] : M = aa(int,int,aa(int,fun(int,int),times_times(int),D),Q3) ) ).

% zmod_eq_0D
tff(fact_2117_word__mod__less__divisor,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: word(A),M: word(A)] :
          ( pp(aa(word(A),bool,ord_less(word(A),zero_zero(word(A))),N))
         => pp(aa(word(A),bool,ord_less(word(A),modulo_modulo(word(A),M,N)),N)) ) ) ).

% word_mod_less_divisor
tff(fact_2118_zmod__int,axiom,
    ! [A2: nat,B2: nat] : aa(nat,int,semiring_1_of_nat(int),modulo_modulo(nat,A2,B2)) = modulo_modulo(int,aa(nat,int,semiring_1_of_nat(int),A2),aa(nat,int,semiring_1_of_nat(int),B2)) ).

% zmod_int
tff(fact_2119_ln__bound,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,ord_less(real,zero_zero(real)),X))
     => pp(aa(real,bool,ord_less_eq(real,aa(real,real,ln_ln(real),X)),X)) ) ).

% ln_bound
tff(fact_2120_ln__gt__zero__imp__gt__one,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,ord_less(real,zero_zero(real)),aa(real,real,ln_ln(real),X)))
     => ( pp(aa(real,bool,ord_less(real,zero_zero(real)),X))
       => pp(aa(real,bool,ord_less(real,one_one(real)),X)) ) ) ).

% ln_gt_zero_imp_gt_one
tff(fact_2121_ln__less__zero,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,ord_less(real,zero_zero(real)),X))
     => ( pp(aa(real,bool,ord_less(real,X),one_one(real)))
       => pp(aa(real,bool,ord_less(real,aa(real,real,ln_ln(real),X)),zero_zero(real))) ) ) ).

% ln_less_zero
tff(fact_2122_ln__gt__zero,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,ord_less(real,one_one(real)),X))
     => pp(aa(real,bool,ord_less(real,zero_zero(real)),aa(real,real,ln_ln(real),X))) ) ).

% ln_gt_zero
tff(fact_2123_ln__ge__zero,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,ord_less_eq(real,one_one(real)),X))
     => pp(aa(real,bool,ord_less_eq(real,zero_zero(real)),aa(real,real,ln_ln(real),X))) ) ).

% ln_ge_zero
tff(fact_2124_int__mod__eq,axiom,
    ! [B2: int,N: int,A2: int] :
      ( pp(aa(int,bool,ord_less_eq(int,zero_zero(int)),B2))
     => ( pp(aa(int,bool,ord_less(int,B2),N))
       => ( ( modulo_modulo(int,A2,N) = modulo_modulo(int,B2,N) )
         => ( modulo_modulo(int,A2,N) = B2 ) ) ) ) ).

% int_mod_eq
tff(fact_2125_int__mod__lem,axiom,
    ! [N: int,B2: int] :
      ( pp(aa(int,bool,ord_less(int,zero_zero(int)),N))
     => ( ( pp(aa(int,bool,ord_less_eq(int,zero_zero(int)),B2))
          & pp(aa(int,bool,ord_less(int,B2),N)) )
      <=> ( modulo_modulo(int,B2,N) = B2 ) ) ) ).

% int_mod_lem
tff(fact_2126_neg__mod__sign,axiom,
    ! [L: int,K: int] :
      ( pp(aa(int,bool,ord_less(int,L),zero_zero(int)))
     => pp(aa(int,bool,ord_less_eq(int,modulo_modulo(int,K,L)),zero_zero(int))) ) ).

% neg_mod_sign
tff(fact_2127_Euclidean__Division_Opos__mod__sign,axiom,
    ! [L: int,K: int] :
      ( pp(aa(int,bool,ord_less(int,zero_zero(int)),L))
     => pp(aa(int,bool,ord_less_eq(int,zero_zero(int)),modulo_modulo(int,K,L))) ) ).

% Euclidean_Division.pos_mod_sign
tff(fact_2128_zmod__trivial__iff,axiom,
    ! [I: int,K: int] :
      ( ( modulo_modulo(int,I,K) = I )
    <=> ( ( K = zero_zero(int) )
        | ( pp(aa(int,bool,ord_less_eq(int,zero_zero(int)),I))
          & pp(aa(int,bool,ord_less(int,I),K)) )
        | ( pp(aa(int,bool,ord_less_eq(int,I),zero_zero(int)))
          & pp(aa(int,bool,ord_less(int,K),I)) ) ) ) ).

% zmod_trivial_iff
tff(fact_2129_pos__mod__conj,axiom,
    ! [B2: int,A2: int] :
      ( pp(aa(int,bool,ord_less(int,zero_zero(int)),B2))
     => ( pp(aa(int,bool,ord_less_eq(int,zero_zero(int)),modulo_modulo(int,A2,B2)))
        & pp(aa(int,bool,ord_less(int,modulo_modulo(int,A2,B2)),B2)) ) ) ).

% pos_mod_conj
tff(fact_2130_neg__mod__conj,axiom,
    ! [B2: int,A2: int] :
      ( pp(aa(int,bool,ord_less(int,B2),zero_zero(int)))
     => ( pp(aa(int,bool,ord_less_eq(int,modulo_modulo(int,A2,B2)),zero_zero(int)))
        & pp(aa(int,bool,ord_less(int,B2),modulo_modulo(int,A2,B2))) ) ) ).

% neg_mod_conj
tff(fact_2131_int__mod__ge,axiom,
    ! [A2: int,N: int] :
      ( pp(aa(int,bool,ord_less(int,A2),N))
     => ( pp(aa(int,bool,ord_less(int,zero_zero(int)),N))
       => pp(aa(int,bool,ord_less_eq(int,A2),modulo_modulo(int,A2,N))) ) ) ).

% int_mod_ge
tff(fact_2132_int__mod__le_H,axiom,
    ! [B2: int,N: int] :
      ( pp(aa(int,bool,ord_less_eq(int,zero_zero(int)),aa(int,int,minus_minus(int,B2),N)))
     => pp(aa(int,bool,ord_less_eq(int,modulo_modulo(int,B2,N)),aa(int,int,minus_minus(int,B2),N))) ) ).

% int_mod_le'
tff(fact_2133_nonneg__mod__div,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa(int,bool,ord_less_eq(int,zero_zero(int)),A2))
     => ( pp(aa(int,bool,ord_less_eq(int,zero_zero(int)),B2))
       => ( pp(aa(int,bool,ord_less_eq(int,zero_zero(int)),modulo_modulo(int,A2,B2)))
          & pp(aa(int,bool,ord_less_eq(int,zero_zero(int)),aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),B2))) ) ) ) ).

% nonneg_mod_div
tff(fact_2134_zdiv__mono__strict,axiom,
    ! [A3: int,B4: int,N: int] :
      ( pp(aa(int,bool,ord_less(int,A3),B4))
     => ( pp(aa(int,bool,ord_less(int,zero_zero(int)),N))
       => ( ( modulo_modulo(int,A3,N) = zero_zero(int) )
         => ( ( modulo_modulo(int,B4,N) = zero_zero(int) )
           => pp(aa(int,bool,ord_less(int,aa(int,int,aa(int,fun(int,int),divide_divide(int),A3),N)),aa(int,int,aa(int,fun(int,int),divide_divide(int),B4),N))) ) ) ) ) ).

% zdiv_mono_strict
tff(fact_2135_div__mod__decomp__int,axiom,
    ! [A3: int,N: int] : A3 = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A3),N)),N)),modulo_modulo(int,A3,N)) ).

% div_mod_decomp_int
tff(fact_2136_mod__div__equality__div__eq,axiom,
    ! [A2: int,B2: int] : aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),B2)),B2) = aa(int,int,minus_minus(int,A2),modulo_modulo(int,A2,B2)) ).

% mod_div_equality_div_eq
tff(fact_2137_word__mod__div__equality,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: word(A),B2: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),N),B2)),B2)),modulo_modulo(word(A),N,B2)) = N ) ).

% word_mod_div_equality
tff(fact_2138_ln__2__less__1,axiom,
    pp(aa(real,bool,ord_less(real,aa(real,real,ln_ln(real),aa(num,real,numeral_numeral(real),bit0(one2)))),one_one(real))) ).

% ln_2_less_1
tff(fact_2139_num_Osize_I4_J,axiom,
    aa(num,nat,size_size(num),one2) = zero_zero(nat) ).

% num.size(4)
tff(fact_2140_ln__ge__zero__imp__ge__one,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,ord_less_eq(real,zero_zero(real)),aa(real,real,ln_ln(real),X)))
     => ( pp(aa(real,bool,ord_less(real,zero_zero(real)),X))
       => pp(aa(real,bool,ord_less_eq(real,one_one(real)),X)) ) ) ).

% ln_ge_zero_imp_ge_one
tff(fact_2141_VEBT_Osize_I4_J,axiom,
    ! [X21: bool,X222: bool] : aa(vEBT_VEBT,nat,size_size(vEBT_VEBT),vEBT_Leaf(X21,X222)) = zero_zero(nat) ).

% VEBT.size(4)
tff(fact_2142_ln__add__one__self__le__self,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,ord_less_eq(real,zero_zero(real)),X))
     => pp(aa(real,bool,ord_less_eq(real,aa(real,real,ln_ln(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),X))),X)) ) ).

% ln_add_one_self_le_self
tff(fact_2143_ln__mult,axiom,
    ! [X: real,Y: real] :
      ( pp(aa(real,bool,ord_less(real,zero_zero(real)),X))
     => ( pp(aa(real,bool,ord_less(real,zero_zero(real)),Y))
       => ( aa(real,real,ln_ln(real),aa(real,real,aa(real,fun(real,real),times_times(real),X),Y)) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,ln_ln(real),X)),aa(real,real,ln_ln(real),Y)) ) ) ) ).

% ln_mult
tff(fact_2144_ln__eq__minus__one,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,ord_less(real,zero_zero(real)),X))
     => ( ( aa(real,real,ln_ln(real),X) = aa(real,real,minus_minus(real,X),one_one(real)) )
       => ( X = one_one(real) ) ) ) ).

% ln_eq_minus_one
tff(fact_2145_ln__div,axiom,
    ! [X: real,Y: real] :
      ( pp(aa(real,bool,ord_less(real,zero_zero(real)),X))
     => ( pp(aa(real,bool,ord_less(real,zero_zero(real)),Y))
       => ( aa(real,real,ln_ln(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),X),Y)) = aa(real,real,minus_minus(real,aa(real,real,ln_ln(real),X)),aa(real,real,ln_ln(real),Y)) ) ) ) ).

% ln_div
tff(fact_2146_pos__mod__bound2,axiom,
    ! [A2: int] : pp(aa(int,bool,ord_less(int,modulo_modulo(int,A2,aa(num,int,numeral_numeral(int),bit0(one2)))),aa(num,int,numeral_numeral(int),bit0(one2)))) ).

% pos_mod_bound2
tff(fact_2147_int__mod__ge_H,axiom,
    ! [B2: int,N: int] :
      ( pp(aa(int,bool,ord_less(int,B2),zero_zero(int)))
     => ( pp(aa(int,bool,ord_less(int,zero_zero(int)),N))
       => pp(aa(int,bool,ord_less_eq(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),B2),N)),modulo_modulo(int,B2,N))) ) ) ).

% int_mod_ge'
tff(fact_2148_mod__pos__neg__trivial,axiom,
    ! [K: int,L: int] :
      ( pp(aa(int,bool,ord_less(int,zero_zero(int)),K))
     => ( pp(aa(int,bool,ord_less_eq(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),K),L)),zero_zero(int)))
       => ( modulo_modulo(int,K,L) = aa(int,int,aa(int,fun(int,int),plus_plus(int),K),L) ) ) ) ).

% mod_pos_neg_trivial
tff(fact_2149_mod__pos__geq,axiom,
    ! [L: int,K: int] :
      ( pp(aa(int,bool,ord_less(int,zero_zero(int)),L))
     => ( pp(aa(int,bool,ord_less_eq(int,L),K))
       => ( modulo_modulo(int,K,L) = modulo_modulo(int,aa(int,int,minus_minus(int,K),L),L) ) ) ) ).

% mod_pos_geq
tff(fact_2150_mod__int__pos__iff,axiom,
    ! [K: int,L: int] :
      ( pp(aa(int,bool,ord_less_eq(int,zero_zero(int)),modulo_modulo(int,K,L)))
    <=> ( pp(aa(int,bool,dvd_dvd(int,L),K))
        | ( ( L = zero_zero(int) )
          & pp(aa(int,bool,ord_less_eq(int,zero_zero(int)),K)) )
        | pp(aa(int,bool,ord_less(int,zero_zero(int)),L)) ) ) ).

% mod_int_pos_iff
tff(fact_2151_real__of__int__div__aux,axiom,
    ! [X: int,D: int] : aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(int,real,ring_1_of_int(real),X)),aa(int,real,ring_1_of_int(real),D)) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(int,real,ring_1_of_int(real),aa(int,int,aa(int,fun(int,int),divide_divide(int),X),D))),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(int,real,ring_1_of_int(real),modulo_modulo(int,X,D))),aa(int,real,ring_1_of_int(real),D))) ).

% real_of_int_div_aux
tff(fact_2152_ln__le__minus__one,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,ord_less(real,zero_zero(real)),X))
     => pp(aa(real,bool,ord_less_eq(real,aa(real,real,ln_ln(real),X)),aa(real,real,minus_minus(real,X),one_one(real)))) ) ).

% ln_le_minus_one
tff(fact_2153_ln__diff__le,axiom,
    ! [X: real,Y: real] :
      ( pp(aa(real,bool,ord_less(real,zero_zero(real)),X))
     => ( pp(aa(real,bool,ord_less(real,zero_zero(real)),Y))
       => pp(aa(real,bool,ord_less_eq(real,aa(real,real,minus_minus(real,aa(real,real,ln_ln(real),X)),aa(real,real,ln_ln(real),Y))),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,minus_minus(real,X),Y)),Y))) ) ) ).

% ln_diff_le
tff(fact_2154_ln__realpow,axiom,
    ! [X: real,N: nat] :
      ( pp(aa(real,bool,ord_less(real,zero_zero(real)),X))
     => ( aa(real,real,ln_ln(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),X),N)) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),N)),aa(real,real,ln_ln(real),X)) ) ) ).

% ln_realpow
tff(fact_2155_pos__mod__sign2,axiom,
    ! [A2: int] : pp(aa(int,bool,ord_less_eq(int,zero_zero(int)),modulo_modulo(int,A2,aa(num,int,numeral_numeral(int),bit0(one2))))) ).

% pos_mod_sign2
tff(fact_2156_nmod2,axiom,
    ! [N: int] :
      ( ( modulo_modulo(int,N,aa(num,int,numeral_numeral(int),bit0(one2))) = zero_zero(int) )
      | ( modulo_modulo(int,N,aa(num,int,numeral_numeral(int),bit0(one2))) = one_one(int) ) ) ).

% nmod2
tff(fact_2157_mod__2__neq__1__eq__eq__0,axiom,
    ! [K: int] :
      ( ( modulo_modulo(int,K,aa(num,int,numeral_numeral(int),bit0(one2))) != one_one(int) )
    <=> ( modulo_modulo(int,K,aa(num,int,numeral_numeral(int),bit0(one2))) = zero_zero(int) ) ) ).

% mod_2_neq_1_eq_eq_0
tff(fact_2158_mod__exp__less__eq__exp,axiom,
    ! [A2: int,N: nat] : pp(aa(int,bool,ord_less(int,modulo_modulo(int,A2,aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),N))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),N))) ).

% mod_exp_less_eq_exp
tff(fact_2159_mod__power__lem,axiom,
    ! [A2: int,M: nat,N: nat] :
      ( pp(aa(int,bool,ord_less(int,one_one(int)),A2))
     => ( ( pp(aa(nat,bool,ord_less_eq(nat,M),N))
         => ( modulo_modulo(int,aa(nat,int,aa(int,fun(nat,int),power_power(int),A2),N),aa(nat,int,aa(int,fun(nat,int),power_power(int),A2),M)) = zero_zero(int) ) )
        & ( ~ pp(aa(nat,bool,ord_less_eq(nat,M),N))
         => ( modulo_modulo(int,aa(nat,int,aa(int,fun(nat,int),power_power(int),A2),N),aa(nat,int,aa(int,fun(nat,int),power_power(int),A2),M)) = aa(nat,int,aa(int,fun(nat,int),power_power(int),A2),N) ) ) ) ) ).

% mod_power_lem
tff(fact_2160_int__mod__pos__eq,axiom,
    ! [A2: int,B2: int,Q2: int,R2: int] :
      ( ( A2 = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B2),Q2)),R2) )
     => ( pp(aa(int,bool,ord_less_eq(int,zero_zero(int)),R2))
       => ( pp(aa(int,bool,ord_less(int,R2),B2))
         => ( modulo_modulo(int,A2,B2) = R2 ) ) ) ) ).

% int_mod_pos_eq
tff(fact_2161_int__mod__neg__eq,axiom,
    ! [A2: int,B2: int,Q2: int,R2: int] :
      ( ( A2 = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B2),Q2)),R2) )
     => ( pp(aa(int,bool,ord_less_eq(int,R2),zero_zero(int)))
       => ( pp(aa(int,bool,ord_less(int,B2),R2))
         => ( modulo_modulo(int,A2,B2) = R2 ) ) ) ) ).

% int_mod_neg_eq
tff(fact_2162_split__zmod,axiom,
    ! [P: fun(int,bool),N: int,K: int] :
      ( pp(aa(int,bool,P,modulo_modulo(int,N,K)))
    <=> ( ( ( K = zero_zero(int) )
         => pp(aa(int,bool,P,N)) )
        & ( pp(aa(int,bool,ord_less(int,zero_zero(int)),K))
         => ! [I3: int,J3: int] :
              ( ( pp(aa(int,bool,ord_less_eq(int,zero_zero(int)),J3))
                & pp(aa(int,bool,ord_less(int,J3),K))
                & ( N = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),K),I3)),J3) ) )
             => pp(aa(int,bool,P,J3)) ) )
        & ( pp(aa(int,bool,ord_less(int,K),zero_zero(int)))
         => ! [I3: int,J3: int] :
              ( ( pp(aa(int,bool,ord_less(int,K),J3))
                & pp(aa(int,bool,ord_less_eq(int,J3),zero_zero(int)))
                & ( N = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),K),I3)),J3) ) )
             => pp(aa(int,bool,P,J3)) ) ) ) ) ).

% split_zmod
tff(fact_2163_mod__add__if__z,axiom,
    ! [X: int,Z: int,Y: int] :
      ( pp(aa(int,bool,ord_less(int,X),Z))
     => ( pp(aa(int,bool,ord_less(int,Y),Z))
       => ( pp(aa(int,bool,ord_less_eq(int,zero_zero(int)),Y))
         => ( pp(aa(int,bool,ord_less_eq(int,zero_zero(int)),X))
           => ( pp(aa(int,bool,ord_less_eq(int,zero_zero(int)),Z))
             => ( ( pp(aa(int,bool,ord_less(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),X),Y)),Z))
                 => ( modulo_modulo(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),X),Y),Z) = aa(int,int,aa(int,fun(int,int),plus_plus(int),X),Y) ) )
                & ( ~ pp(aa(int,bool,ord_less(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),X),Y)),Z))
                 => ( modulo_modulo(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),X),Y),Z) = aa(int,int,minus_minus(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),X),Y)),Z) ) ) ) ) ) ) ) ) ).

% mod_add_if_z
tff(fact_2164_mod__sub__if__z,axiom,
    ! [X: int,Z: int,Y: int] :
      ( pp(aa(int,bool,ord_less(int,X),Z))
     => ( pp(aa(int,bool,ord_less(int,Y),Z))
       => ( pp(aa(int,bool,ord_less_eq(int,zero_zero(int)),Y))
         => ( pp(aa(int,bool,ord_less_eq(int,zero_zero(int)),X))
           => ( pp(aa(int,bool,ord_less_eq(int,zero_zero(int)),Z))
             => ( ( pp(aa(int,bool,ord_less_eq(int,Y),X))
                 => ( modulo_modulo(int,aa(int,int,minus_minus(int,X),Y),Z) = aa(int,int,minus_minus(int,X),Y) ) )
                & ( ~ pp(aa(int,bool,ord_less_eq(int,Y),X))
                 => ( modulo_modulo(int,aa(int,int,minus_minus(int,X),Y),Z) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,minus_minus(int,X),Y)),Z) ) ) ) ) ) ) ) ) ).

% mod_sub_if_z
tff(fact_2165_zmod__zmult2__eq,axiom,
    ! [C2: int,A2: int,B2: int] :
      ( pp(aa(int,bool,ord_less_eq(int,zero_zero(int)),C2))
     => ( modulo_modulo(int,A2,aa(int,int,aa(int,fun(int,int),times_times(int),B2),C2)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B2),modulo_modulo(int,aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),B2),C2))),modulo_modulo(int,A2,B2)) ) ) ).

% zmod_zmult2_eq
tff(fact_2166_log__eq__div__ln__mult__log,axiom,
    ! [A2: real,B2: real,X: real] :
      ( pp(aa(real,bool,ord_less(real,zero_zero(real)),A2))
     => ( ( A2 != one_one(real) )
       => ( pp(aa(real,bool,ord_less(real,zero_zero(real)),B2))
         => ( ( B2 != one_one(real) )
           => ( pp(aa(real,bool,ord_less(real,zero_zero(real)),X))
             => ( aa(real,real,log(A2),X) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,ln_ln(real),B2)),aa(real,real,ln_ln(real),A2))),aa(real,real,log(B2),X)) ) ) ) ) ) ) ).

% log_eq_div_ln_mult_log
tff(fact_2167_axxmod2,axiom,
    ! [X: int] :
      ( ( modulo_modulo(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),X)),X),aa(num,int,numeral_numeral(int),bit0(one2))) = one_one(int) )
      & ( modulo_modulo(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),zero_zero(int)),X)),X),aa(num,int,numeral_numeral(int),bit0(one2))) = zero_zero(int) ) ) ).

% axxmod2
tff(fact_2168_z1pmod2,axiom,
    ! [B2: int] : modulo_modulo(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),B2)),one_one(int)),aa(num,int,numeral_numeral(int),bit0(one2))) = one_one(int) ).

% z1pmod2
tff(fact_2169_verit__le__mono__div__int,axiom,
    ! [A3: int,B4: int,N: int] :
      ( pp(aa(int,bool,ord_less(int,A3),B4))
     => ( pp(aa(int,bool,ord_less(int,zero_zero(int)),N))
       => pp(aa(int,bool,ord_less_eq(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A3),N)),if(int,aa(int,bool,fequal(int,modulo_modulo(int,B4,N)),zero_zero(int)),one_one(int),zero_zero(int)))),aa(int,int,aa(int,fun(int,int),divide_divide(int),B4),N))) ) ) ).

% verit_le_mono_div_int
tff(fact_2170_split__pos__lemma,axiom,
    ! [K: int,P: fun(int,fun(int,bool)),N: int] :
      ( pp(aa(int,bool,ord_less(int,zero_zero(int)),K))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),P,aa(int,int,aa(int,fun(int,int),divide_divide(int),N),K)),modulo_modulo(int,N,K)))
      <=> ! [I3: int,J3: int] :
            ( ( pp(aa(int,bool,ord_less_eq(int,zero_zero(int)),J3))
              & pp(aa(int,bool,ord_less(int,J3),K))
              & ( N = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),K),I3)),J3) ) )
           => pp(aa(int,bool,aa(int,fun(int,bool),P,I3),J3)) ) ) ) ).

% split_pos_lemma
tff(fact_2171_split__neg__lemma,axiom,
    ! [K: int,P: fun(int,fun(int,bool)),N: int] :
      ( pp(aa(int,bool,ord_less(int,K),zero_zero(int)))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),P,aa(int,int,aa(int,fun(int,int),divide_divide(int),N),K)),modulo_modulo(int,N,K)))
      <=> ! [I3: int,J3: int] :
            ( ( pp(aa(int,bool,ord_less(int,K),J3))
              & pp(aa(int,bool,ord_less_eq(int,J3),zero_zero(int)))
              & ( N = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),K),I3)),J3) ) )
           => pp(aa(int,bool,aa(int,fun(int,bool),P,I3),J3)) ) ) ) ).

% split_neg_lemma
tff(fact_2172_num_Osize_I5_J,axiom,
    ! [X23: num] : aa(num,nat,size_size(num),bit0(X23)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,size_size(num),X23)),aa(nat,nat,suc,zero_zero(nat))) ).

% num.size(5)
tff(fact_2173_num_Osize_I6_J,axiom,
    ! [X33: num] : aa(num,nat,size_size(num),aa(num,num,bit1,X33)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,size_size(num),X33)),aa(nat,nat,suc,zero_zero(nat))) ).

% num.size(6)
tff(fact_2174_p1mod22k,axiom,
    ! [B2: int,N: nat] : modulo_modulo(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),B2)),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),N))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),modulo_modulo(int,B2,aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),N)))),one_one(int)) ).

% p1mod22k
tff(fact_2175_p1mod22k_H,axiom,
    ! [B2: int,N: nat] : modulo_modulo(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),B2)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),N))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),modulo_modulo(int,B2,aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),N)))) ).

% p1mod22k'
tff(fact_2176_eq__diff__eq_H,axiom,
    ! [X: real,Y: real,Z: real] :
      ( ( X = aa(real,real,minus_minus(real,Y),Z) )
    <=> ( Y = aa(real,real,aa(real,fun(real,real),plus_plus(real),X),Z) ) ) ).

% eq_diff_eq'
tff(fact_2177_pos__zmod__mult__2,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa(int,bool,ord_less_eq(int,zero_zero(int)),A2))
     => ( modulo_modulo(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),B2)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),A2)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),modulo_modulo(int,B2,A2))) ) ) ).

% pos_zmod_mult_2
tff(fact_2178_emep1,axiom,
    ! [N: int,D: int] :
      ( pp(aa(int,bool,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),N))
     => ( pp(aa(int,bool,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),D))
       => ( pp(aa(int,bool,ord_less_eq(int,zero_zero(int)),D))
         => ( modulo_modulo(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),N),one_one(int)),D) = aa(int,int,aa(int,fun(int,int),plus_plus(int),modulo_modulo(int,N,D)),one_one(int)) ) ) ) ) ).

% emep1
tff(fact_2179_eme1p,axiom,
    ! [N: int,D: int] :
      ( pp(aa(int,bool,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),N))
     => ( pp(aa(int,bool,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),D))
       => ( pp(aa(int,bool,ord_less_eq(int,zero_zero(int)),D))
         => ( modulo_modulo(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),N),D) = aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),modulo_modulo(int,N,D)) ) ) ) ) ).

% eme1p
tff(fact_2180_even__flip__bit__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,A2: A] :
          ( pp(aa(A,bool,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),bit_se8732182000553998342ip_bit(A,M,A2)))
        <=> ~ ( pp(aa(A,bool,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A2))
            <=> ( M = zero_zero(nat) ) ) ) ) ).

% even_flip_bit_iff
tff(fact_2181_ln__one__plus__pos__lower__bound,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,ord_less_eq(real,zero_zero(real)),X))
     => ( pp(aa(real,bool,ord_less_eq(real,X),one_one(real)))
       => pp(aa(real,bool,ord_less_eq(real,aa(real,real,minus_minus(real,X),aa(nat,real,aa(real,fun(nat,real),power_power(real),X),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(real,real,ln_ln(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),X)))) ) ) ).

% ln_one_plus_pos_lower_bound
tff(fact_2182_sb__inc__lem,axiom,
    ! [A2: int,K: nat] :
      ( pp(aa(int,bool,ord_less(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),A2),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),K))),zero_zero(int)))
     => pp(aa(int,bool,ord_less_eq(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),A2),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),K))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,suc,K)))),modulo_modulo(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),A2),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),K)),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,suc,K))))) ) ).

% sb_inc_lem
tff(fact_2183_neg__zmod__mult__2,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa(int,bool,ord_less_eq(int,A2),zero_zero(int)))
     => ( modulo_modulo(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),B2)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),A2)) = aa(int,int,minus_minus(int,aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),modulo_modulo(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),B2),one_one(int)),A2))),one_one(int)) ) ) ).

% neg_zmod_mult_2
tff(fact_2184_vebt__member_Osimps_I2_J,axiom,
    ! [Uu: nat,Uv: list(vEBT_VEBT),Uw: vEBT_VEBT,X: nat] : ~ pp(aa(nat,bool,vEBT_vebt_member(vEBT_Node(none(product_prod(nat,nat)),Uu,Uv,Uw)),X)) ).

% vebt_member.simps(2)
tff(fact_2185_foldr__zero,axiom,
    ! [Xs: list(nat),D: nat] :
      ( ! [I2: nat] :
          ( pp(aa(nat,bool,ord_less(nat,I2),aa(list(nat),nat,size_size(list(nat)),Xs)))
         => pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),aa(nat,nat,nth(nat,Xs),I2))) )
     => pp(aa(nat,bool,ord_less_eq(nat,aa(list(nat),nat,size_size(list(nat)),Xs)),aa(nat,nat,minus_minus(nat,foldr(nat,nat,plus_plus(nat),Xs,D)),D))) ) ).

% foldr_zero
tff(fact_2186_foldr__same,axiom,
    ! [Xs: list(real),Y: real] :
      ( ! [X3: real,Y3: real] :
          ( pp(aa(set(real),bool,aa(real,fun(set(real),bool),member(real),X3),aa(list(real),set(real),set2(real),Xs)))
         => ( pp(aa(set(real),bool,aa(real,fun(set(real),bool),member(real),Y3),aa(list(real),set(real),set2(real),Xs)))
           => ( X3 = Y3 ) ) )
     => ( ! [X3: real] :
            ( pp(aa(set(real),bool,aa(real,fun(set(real),bool),member(real),X3),aa(list(real),set(real),set2(real),Xs)))
           => ( X3 = Y ) )
       => ( foldr(real,real,plus_plus(real),Xs,zero_zero(real)) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),aa(list(real),nat,size_size(list(real)),Xs))),Y) ) ) ) ).

% foldr_same
tff(fact_2187_member__bound__size__univ,axiom,
    ! [T: vEBT_VEBT,N: nat,U: real,X: nat] :
      ( vEBT_invar_vebt(T,N)
     => ( ( U = aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(num,real,numeral_numeral(real),bit0(one2))),N) )
       => pp(aa(real,bool,ord_less_eq(real,aa(nat,real,semiring_1_of_nat(real),vEBT_T_m_e_m_b_e_r(T,X))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(num,real,numeral_numeral(real),bit0(aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,one2)))))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,one2))))),aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),U)))))) ) ) ).

% member_bound_size_univ
tff(fact_2188_even__word__def,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( even_word(A) = dvd_dvd(word(A),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))) ) ) ).

% even_word_def
tff(fact_2189_obtain__set__succ,axiom,
    ! [X: nat,Z: nat,A3: set(nat),B4: set(nat)] :
      ( pp(aa(nat,bool,ord_less(nat,X),Z))
     => ( vEBT_VEBT_max_in_set(A3,Z)
       => ( pp(aa(set(nat),bool,finite_finite2(nat),B4))
         => ( ( A3 = B4 )
           => ? [X_1: nat] : vEBT_is_succ_in_set(A3,X,X_1) ) ) ) ) ).

% obtain_set_succ
tff(fact_2190_obtain__set__pred,axiom,
    ! [Z: nat,X: nat,A3: set(nat)] :
      ( pp(aa(nat,bool,ord_less(nat,Z),X))
     => ( vEBT_VEBT_min_in_set(A3,Z)
       => ( pp(aa(set(nat),bool,finite_finite2(nat),A3))
         => ? [X_1: nat] : vEBT_is_pred_in_set(A3,X,X_1) ) ) ) ).

% obtain_set_pred
tff(fact_2191_ceiling__log__eq__powr__iff,axiom,
    ! [X: real,B2: real,K: nat] :
      ( pp(aa(real,bool,ord_less(real,zero_zero(real)),X))
     => ( pp(aa(real,bool,ord_less(real,one_one(real)),B2))
       => ( ( archimedean_ceiling(real,aa(real,real,log(B2),X)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(nat,int,semiring_1_of_nat(int),K)),one_one(int)) )
        <=> ( pp(aa(real,bool,ord_less(real,powr(real,B2,aa(nat,real,semiring_1_of_nat(real),K))),X))
            & pp(aa(real,bool,ord_less_eq(real,X),powr(real,B2,aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),one_one(nat)))))) ) ) ) ) ).

% ceiling_log_eq_powr_iff
tff(fact_2192_foldr0,axiom,
    ! [Xs: list(real),C2: real,D: real] : foldr(real,real,plus_plus(real),Xs,aa(real,real,aa(real,fun(real,real),plus_plus(real),C2),D)) = aa(real,real,aa(real,fun(real,real),plus_plus(real),foldr(real,real,plus_plus(real),Xs,D)),C2) ).

% foldr0
tff(fact_2193_foldr__one,axiom,
    ! [D: nat,Ys: list(nat)] : pp(aa(nat,bool,ord_less_eq(nat,D),foldr(nat,nat,plus_plus(nat),Ys,D))) ).

% foldr_one
tff(fact_2194_set__vebt__finite,axiom,
    ! [T: vEBT_VEBT,N: nat] :
      ( vEBT_invar_vebt(T,N)
     => pp(aa(set(nat),bool,finite_finite2(nat),vEBT_VEBT_set_vebt(T))) ) ).

% set_vebt_finite
tff(fact_2195_pred__none__empty,axiom,
    ! [Xs: set(nat),A2: nat] :
      ( ~ ? [X_1: nat] : vEBT_is_pred_in_set(Xs,A2,X_1)
     => ( pp(aa(set(nat),bool,finite_finite2(nat),Xs))
       => ~ ? [X4: nat] :
              ( pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),X4),Xs))
              & pp(aa(nat,bool,ord_less(nat,X4),A2)) ) ) ) ).

% pred_none_empty
tff(fact_2196_succ__none__empty,axiom,
    ! [Xs: set(nat),A2: nat] :
      ( ~ ? [X_1: nat] : vEBT_is_succ_in_set(Xs,A2,X_1)
     => ( pp(aa(set(nat),bool,finite_finite2(nat),Xs))
       => ~ ? [X4: nat] :
              ( pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),X4),Xs))
              & pp(aa(nat,bool,ord_less(nat,A2),X4)) ) ) ) ).

% succ_none_empty
tff(fact_2197_foldr__same__int,axiom,
    ! [Xs: list(nat),Y: nat] :
      ( ! [X3: nat,Y3: nat] :
          ( pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),X3),aa(list(nat),set(nat),set2(nat),Xs)))
         => ( pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),Y3),aa(list(nat),set(nat),set2(nat),Xs)))
           => ( X3 = Y3 ) ) )
     => ( ! [X3: nat] :
            ( pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),X3),aa(list(nat),set(nat),set2(nat),Xs)))
           => ( X3 = Y ) )
       => ( foldr(nat,nat,plus_plus(nat),Xs,zero_zero(nat)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(list(nat),nat,size_size(list(nat)),Xs)),Y) ) ) ) ).

% foldr_same_int
tff(fact_2198_foldr__mono,axiom,
    ! [Xs: list(nat),Ys: list(nat),C2: nat,D: nat] :
      ( ( aa(list(nat),nat,size_size(list(nat)),Xs) = aa(list(nat),nat,size_size(list(nat)),Ys) )
     => ( ! [I2: nat] :
            ( pp(aa(nat,bool,ord_less(nat,I2),aa(list(nat),nat,size_size(list(nat)),Xs)))
           => pp(aa(nat,bool,ord_less(nat,aa(nat,nat,nth(nat,Xs),I2)),aa(nat,nat,nth(nat,Ys),I2))) )
       => ( pp(aa(nat,bool,ord_less_eq(nat,C2),D))
         => pp(aa(nat,bool,ord_less_eq(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),foldr(nat,nat,plus_plus(nat),Xs,C2)),aa(list(nat),nat,size_size(list(nat)),Ys))),foldr(nat,nat,plus_plus(nat),Ys,D))) ) ) ) ).

% foldr_mono
tff(fact_2199_powr__0,axiom,
    ! [A: $tType] :
      ( ln(A)
     => ! [Z: A] : powr(A,zero_zero(A),Z) = zero_zero(A) ) ).

% powr_0
tff(fact_2200_powr__eq__0__iff,axiom,
    ! [A: $tType] :
      ( ln(A)
     => ! [W: A,Z: A] :
          ( ( powr(A,W,Z) = zero_zero(A) )
        <=> ( W = zero_zero(A) ) ) ) ).

% powr_eq_0_iff
tff(fact_2201_powr__one__eq__one,axiom,
    ! [A: $tType] :
      ( ln(A)
     => ! [A2: A] : powr(A,one_one(A),A2) = one_one(A) ) ).

% powr_one_eq_one
tff(fact_2202_powr__zero__eq__one,axiom,
    ! [A: $tType] :
      ( ln(A)
     => ! [X: A] :
          ( ( ( X = zero_zero(A) )
           => ( powr(A,X,zero_zero(A)) = zero_zero(A) ) )
          & ( ( X != zero_zero(A) )
           => ( powr(A,X,zero_zero(A)) = one_one(A) ) ) ) ) ).

% powr_zero_eq_one
tff(fact_2203_powr__gt__zero,axiom,
    ! [X: real,A2: real] :
      ( pp(aa(real,bool,ord_less(real,zero_zero(real)),powr(real,X,A2)))
    <=> ( X != zero_zero(real) ) ) ).

% powr_gt_zero
tff(fact_2204_powr__nonneg__iff,axiom,
    ! [A2: real,X: real] :
      ( pp(aa(real,bool,ord_less_eq(real,powr(real,A2,X)),zero_zero(real)))
    <=> ( A2 = zero_zero(real) ) ) ).

% powr_nonneg_iff
tff(fact_2205_powr__less__cancel__iff,axiom,
    ! [X: real,A2: real,B2: real] :
      ( pp(aa(real,bool,ord_less(real,one_one(real)),X))
     => ( pp(aa(real,bool,ord_less(real,powr(real,X,A2)),powr(real,X,B2)))
      <=> pp(aa(real,bool,ord_less(real,A2),B2)) ) ) ).

% powr_less_cancel_iff
tff(fact_2206_powr__eq__one__iff,axiom,
    ! [A2: real,X: real] :
      ( pp(aa(real,bool,ord_less(real,one_one(real)),A2))
     => ( ( powr(real,A2,X) = one_one(real) )
      <=> ( X = zero_zero(real) ) ) ) ).

% powr_eq_one_iff
tff(fact_2207_powr__one__gt__zero__iff,axiom,
    ! [X: real] :
      ( ( powr(real,X,one_one(real)) = X )
    <=> pp(aa(real,bool,ord_less_eq(real,zero_zero(real)),X)) ) ).

% powr_one_gt_zero_iff
tff(fact_2208_powr__one,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,ord_less_eq(real,zero_zero(real)),X))
     => ( powr(real,X,one_one(real)) = X ) ) ).

% powr_one
tff(fact_2209_powr__le__cancel__iff,axiom,
    ! [X: real,A2: real,B2: real] :
      ( pp(aa(real,bool,ord_less(real,one_one(real)),X))
     => ( pp(aa(real,bool,ord_less_eq(real,powr(real,X,A2)),powr(real,X,B2)))
      <=> pp(aa(real,bool,ord_less_eq(real,A2),B2)) ) ) ).

% powr_le_cancel_iff
tff(fact_2210_numeral__powr__numeral__real,axiom,
    ! [M: num,N: num] : powr(real,aa(num,real,numeral_numeral(real),M),aa(num,real,numeral_numeral(real),N)) = aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(num,real,numeral_numeral(real),M)),aa(num,nat,numeral_numeral(nat),N)) ).

% numeral_powr_numeral_real
tff(fact_2211_foldr__length,axiom,
    ! [A: $tType,L: list(A)] : foldr(A,nat,aTP_Lamp_ag(A,fun(nat,nat)),L,zero_zero(nat)) = aa(list(A),nat,size_size(list(A)),L) ).

% foldr_length
tff(fact_2212_log__powr__cancel,axiom,
    ! [A2: real,Y: real] :
      ( pp(aa(real,bool,ord_less(real,zero_zero(real)),A2))
     => ( ( A2 != one_one(real) )
       => ( aa(real,real,log(A2),powr(real,A2,Y)) = Y ) ) ) ).

% log_powr_cancel
tff(fact_2213_powr__log__cancel,axiom,
    ! [A2: real,X: real] :
      ( pp(aa(real,bool,ord_less(real,zero_zero(real)),A2))
     => ( ( A2 != one_one(real) )
       => ( pp(aa(real,bool,ord_less(real,zero_zero(real)),X))
         => ( powr(real,A2,aa(real,real,log(A2),X)) = X ) ) ) ) ).

% powr_log_cancel
tff(fact_2214_powr__numeral,axiom,
    ! [X: real,N: num] :
      ( pp(aa(real,bool,ord_less_eq(real,zero_zero(real)),X))
     => ( powr(real,X,aa(num,real,numeral_numeral(real),N)) = aa(nat,real,aa(real,fun(nat,real),power_power(real),X),aa(num,nat,numeral_numeral(nat),N)) ) ) ).

% powr_numeral
tff(fact_2215_finite__if__eq__beyond__finite,axiom,
    ! [A: $tType,S: set(A),S4: set(A)] :
      ( pp(aa(set(A),bool,finite_finite2(A),S))
     => pp(aa(set(set(A)),bool,finite_finite2(set(A)),collect(set(A),aa(set(A),fun(set(A),bool),aTP_Lamp_ah(set(A),fun(set(A),fun(set(A),bool)),S),S4)))) ) ).

% finite_if_eq_beyond_finite
tff(fact_2216_powr__powr,axiom,
    ! [X: real,A2: real,B2: real] : powr(real,powr(real,X,A2),B2) = powr(real,X,aa(real,real,aa(real,fun(real,real),times_times(real),A2),B2)) ).

% powr_powr
tff(fact_2217_finite__nat__set__iff__bounded__le,axiom,
    ! [N4: set(nat)] :
      ( pp(aa(set(nat),bool,finite_finite2(nat),N4))
    <=> ? [M6: nat] :
        ! [X2: nat] :
          ( pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),X2),N4))
         => pp(aa(nat,bool,ord_less_eq(nat,X2),M6)) ) ) ).

% finite_nat_set_iff_bounded_le
tff(fact_2218_finite__M__bounded__by__nat,axiom,
    ! [P: fun(nat,bool),I: nat] : pp(aa(set(nat),bool,finite_finite2(nat),collect(nat,aa(nat,fun(nat,bool),aTP_Lamp_ai(fun(nat,bool),fun(nat,fun(nat,bool)),P),I)))) ).

% finite_M_bounded_by_nat
tff(fact_2219_finite__less__ub,axiom,
    ! [F2: fun(nat,nat),U: nat] :
      ( ! [N2: nat] : pp(aa(nat,bool,ord_less_eq(nat,N2),aa(nat,nat,F2,N2)))
     => pp(aa(set(nat),bool,finite_finite2(nat),collect(nat,aa(nat,fun(nat,bool),aTP_Lamp_aj(fun(nat,nat),fun(nat,fun(nat,bool)),F2),U)))) ) ).

% finite_less_ub
tff(fact_2220_finite__lists__length__eq,axiom,
    ! [A: $tType,A3: set(A),N: nat] :
      ( pp(aa(set(A),bool,finite_finite2(A),A3))
     => pp(aa(set(list(A)),bool,finite_finite2(list(A)),collect(list(A),aa(nat,fun(list(A),bool),aTP_Lamp_ak(set(A),fun(nat,fun(list(A),bool)),A3),N)))) ) ).

% finite_lists_length_eq
tff(fact_2221_powr__less__mono2__neg,axiom,
    ! [A2: real,X: real,Y: real] :
      ( pp(aa(real,bool,ord_less(real,A2),zero_zero(real)))
     => ( pp(aa(real,bool,ord_less(real,zero_zero(real)),X))
       => ( pp(aa(real,bool,ord_less(real,X),Y))
         => pp(aa(real,bool,ord_less(real,powr(real,Y,A2)),powr(real,X,A2))) ) ) ) ).

% powr_less_mono2_neg
tff(fact_2222_powr__non__neg,axiom,
    ! [A2: real,X: real] : ~ pp(aa(real,bool,ord_less(real,powr(real,A2,X)),zero_zero(real))) ).

% powr_non_neg
tff(fact_2223_powr__mono2,axiom,
    ! [A2: real,X: real,Y: real] :
      ( pp(aa(real,bool,ord_less_eq(real,zero_zero(real)),A2))
     => ( pp(aa(real,bool,ord_less_eq(real,zero_zero(real)),X))
       => ( pp(aa(real,bool,ord_less_eq(real,X),Y))
         => pp(aa(real,bool,ord_less_eq(real,powr(real,X,A2)),powr(real,Y,A2))) ) ) ) ).

% powr_mono2
tff(fact_2224_powr__ge__pzero,axiom,
    ! [X: real,Y: real] : pp(aa(real,bool,ord_less_eq(real,zero_zero(real)),powr(real,X,Y))) ).

% powr_ge_pzero
tff(fact_2225_powr__less__cancel,axiom,
    ! [X: real,A2: real,B2: real] :
      ( pp(aa(real,bool,ord_less(real,powr(real,X,A2)),powr(real,X,B2)))
     => ( pp(aa(real,bool,ord_less(real,one_one(real)),X))
       => pp(aa(real,bool,ord_less(real,A2),B2)) ) ) ).

% powr_less_cancel
tff(fact_2226_powr__less__mono,axiom,
    ! [A2: real,B2: real,X: real] :
      ( pp(aa(real,bool,ord_less(real,A2),B2))
     => ( pp(aa(real,bool,ord_less(real,one_one(real)),X))
       => pp(aa(real,bool,ord_less(real,powr(real,X,A2)),powr(real,X,B2))) ) ) ).

% powr_less_mono
tff(fact_2227_powr__mono,axiom,
    ! [A2: real,B2: real,X: real] :
      ( pp(aa(real,bool,ord_less_eq(real,A2),B2))
     => ( pp(aa(real,bool,ord_less_eq(real,one_one(real)),X))
       => pp(aa(real,bool,ord_less_eq(real,powr(real,X,A2)),powr(real,X,B2))) ) ) ).

% powr_mono
tff(fact_2228_ln__powr,axiom,
    ! [X: real,Y: real] :
      ( ( X != zero_zero(real) )
     => ( aa(real,real,ln_ln(real),powr(real,X,Y)) = aa(real,real,aa(real,fun(real,real),times_times(real),Y),aa(real,real,ln_ln(real),X)) ) ) ).

% ln_powr
tff(fact_2229_finite__lists__length__le,axiom,
    ! [A: $tType,A3: set(A),N: nat] :
      ( pp(aa(set(A),bool,finite_finite2(A),A3))
     => pp(aa(set(list(A)),bool,finite_finite2(list(A)),collect(list(A),aa(nat,fun(list(A),bool),aTP_Lamp_al(set(A),fun(nat,fun(list(A),bool)),A3),N)))) ) ).

% finite_lists_length_le
tff(fact_2230_powr__less__mono2,axiom,
    ! [A2: real,X: real,Y: real] :
      ( pp(aa(real,bool,ord_less(real,zero_zero(real)),A2))
     => ( pp(aa(real,bool,ord_less_eq(real,zero_zero(real)),X))
       => ( pp(aa(real,bool,ord_less(real,X),Y))
         => pp(aa(real,bool,ord_less(real,powr(real,X,A2)),powr(real,Y,A2))) ) ) ) ).

% powr_less_mono2
tff(fact_2231_powr__mono2_H,axiom,
    ! [A2: real,X: real,Y: real] :
      ( pp(aa(real,bool,ord_less_eq(real,A2),zero_zero(real)))
     => ( pp(aa(real,bool,ord_less(real,zero_zero(real)),X))
       => ( pp(aa(real,bool,ord_less_eq(real,X),Y))
         => pp(aa(real,bool,ord_less_eq(real,powr(real,Y,A2)),powr(real,X,A2))) ) ) ) ).

% powr_mono2'
tff(fact_2232_powr__inj,axiom,
    ! [A2: real,X: real,Y: real] :
      ( pp(aa(real,bool,ord_less(real,zero_zero(real)),A2))
     => ( ( A2 != one_one(real) )
       => ( ( powr(real,A2,X) = powr(real,A2,Y) )
        <=> ( X = Y ) ) ) ) ).

% powr_inj
tff(fact_2233_gr__one__powr,axiom,
    ! [X: real,Y: real] :
      ( pp(aa(real,bool,ord_less(real,one_one(real)),X))
     => ( pp(aa(real,bool,ord_less(real,zero_zero(real)),Y))
       => pp(aa(real,bool,ord_less(real,one_one(real)),powr(real,X,Y))) ) ) ).

% gr_one_powr
tff(fact_2234_ge__one__powr__ge__zero,axiom,
    ! [X: real,A2: real] :
      ( pp(aa(real,bool,ord_less_eq(real,one_one(real)),X))
     => ( pp(aa(real,bool,ord_less_eq(real,zero_zero(real)),A2))
       => pp(aa(real,bool,ord_less_eq(real,one_one(real)),powr(real,X,A2))) ) ) ).

% ge_one_powr_ge_zero
tff(fact_2235_powr__mono__both,axiom,
    ! [A2: real,B2: real,X: real,Y: real] :
      ( pp(aa(real,bool,ord_less_eq(real,zero_zero(real)),A2))
     => ( pp(aa(real,bool,ord_less_eq(real,A2),B2))
       => ( pp(aa(real,bool,ord_less_eq(real,one_one(real)),X))
         => ( pp(aa(real,bool,ord_less_eq(real,X),Y))
           => pp(aa(real,bool,ord_less_eq(real,powr(real,X,A2)),powr(real,Y,B2))) ) ) ) ) ).

% powr_mono_both
tff(fact_2236_powr__le1,axiom,
    ! [A2: real,X: real] :
      ( pp(aa(real,bool,ord_less_eq(real,zero_zero(real)),A2))
     => ( pp(aa(real,bool,ord_less_eq(real,zero_zero(real)),X))
       => ( pp(aa(real,bool,ord_less_eq(real,X),one_one(real)))
         => pp(aa(real,bool,ord_less_eq(real,powr(real,X,A2)),one_one(real))) ) ) ) ).

% powr_le1
tff(fact_2237_powr__divide,axiom,
    ! [X: real,Y: real,A2: real] :
      ( pp(aa(real,bool,ord_less_eq(real,zero_zero(real)),X))
     => ( pp(aa(real,bool,ord_less_eq(real,zero_zero(real)),Y))
       => ( powr(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),X),Y),A2) = aa(real,real,aa(real,fun(real,real),divide_divide(real),powr(real,X,A2)),powr(real,Y,A2)) ) ) ) ).

% powr_divide
tff(fact_2238_powr__mult,axiom,
    ! [X: real,Y: real,A2: real] :
      ( pp(aa(real,bool,ord_less_eq(real,zero_zero(real)),X))
     => ( pp(aa(real,bool,ord_less_eq(real,zero_zero(real)),Y))
       => ( powr(real,aa(real,real,aa(real,fun(real,real),times_times(real),X),Y),A2) = aa(real,real,aa(real,fun(real,real),times_times(real),powr(real,X,A2)),powr(real,Y,A2)) ) ) ) ).

% powr_mult
tff(fact_2239_log__base__powr,axiom,
    ! [A2: real,B2: real,X: real] :
      ( ( A2 != zero_zero(real) )
     => ( aa(real,real,log(powr(real,A2,B2)),X) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,log(A2),X)),B2) ) ) ).

% log_base_powr
tff(fact_2240_log__powr,axiom,
    ! [X: real,B2: real,Y: real] :
      ( ( X != zero_zero(real) )
     => ( aa(real,real,log(B2),powr(real,X,Y)) = aa(real,real,aa(real,fun(real,real),times_times(real),Y),aa(real,real,log(B2),X)) ) ) ).

% log_powr
tff(fact_2241_powr__add,axiom,
    ! [A: $tType] :
      ( ( real_V3459762299906320749_field(A)
        & ln(A) )
     => ! [X: A,A2: A,B2: A] : powr(A,X,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),powr(A,X,A2)),powr(A,X,B2)) ) ).

% powr_add
tff(fact_2242_powr__diff,axiom,
    ! [A: $tType] :
      ( ( real_V3459762299906320749_field(A)
        & ln(A) )
     => ! [W: A,Z1: A,Z22: A] : powr(A,W,aa(A,A,minus_minus(A,Z1),Z22)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),powr(A,W,Z1)),powr(A,W,Z22)) ) ).

% powr_diff
tff(fact_2243_finite__divisors__nat,axiom,
    ! [M: nat] :
      ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),M))
     => pp(aa(set(nat),bool,finite_finite2(nat),collect(nat,aTP_Lamp_am(nat,fun(nat,bool),M)))) ) ).

% finite_divisors_nat
tff(fact_2244_subset__eq__atLeast0__atMost__finite,axiom,
    ! [N4: set(nat),N: nat] :
      ( pp(aa(set(nat),bool,ord_less_eq(set(nat),N4),set_or1337092689740270186AtMost(nat,zero_zero(nat),N)))
     => pp(aa(set(nat),bool,finite_finite2(nat),N4)) ) ).

% subset_eq_atLeast0_atMost_finite
tff(fact_2245_powr__realpow,axiom,
    ! [X: real,N: nat] :
      ( pp(aa(real,bool,ord_less(real,zero_zero(real)),X))
     => ( powr(real,X,aa(nat,real,semiring_1_of_nat(real),N)) = aa(nat,real,aa(real,fun(nat,real),power_power(real),X),N) ) ) ).

% powr_realpow
tff(fact_2246_powr__less__iff,axiom,
    ! [B2: real,X: real,Y: real] :
      ( pp(aa(real,bool,ord_less(real,one_one(real)),B2))
     => ( pp(aa(real,bool,ord_less(real,zero_zero(real)),X))
       => ( pp(aa(real,bool,ord_less(real,powr(real,B2,Y)),X))
        <=> pp(aa(real,bool,ord_less(real,Y),aa(real,real,log(B2),X))) ) ) ) ).

% powr_less_iff
tff(fact_2247_less__powr__iff,axiom,
    ! [B2: real,X: real,Y: real] :
      ( pp(aa(real,bool,ord_less(real,one_one(real)),B2))
     => ( pp(aa(real,bool,ord_less(real,zero_zero(real)),X))
       => ( pp(aa(real,bool,ord_less(real,X),powr(real,B2,Y)))
        <=> pp(aa(real,bool,ord_less(real,aa(real,real,log(B2),X)),Y)) ) ) ) ).

% less_powr_iff
tff(fact_2248_log__less__iff,axiom,
    ! [B2: real,X: real,Y: real] :
      ( pp(aa(real,bool,ord_less(real,one_one(real)),B2))
     => ( pp(aa(real,bool,ord_less(real,zero_zero(real)),X))
       => ( pp(aa(real,bool,ord_less(real,aa(real,real,log(B2),X)),Y))
        <=> pp(aa(real,bool,ord_less(real,X),powr(real,B2,Y))) ) ) ) ).

% log_less_iff
tff(fact_2249_less__log__iff,axiom,
    ! [B2: real,X: real,Y: real] :
      ( pp(aa(real,bool,ord_less(real,one_one(real)),B2))
     => ( pp(aa(real,bool,ord_less(real,zero_zero(real)),X))
       => ( pp(aa(real,bool,ord_less(real,Y),aa(real,real,log(B2),X)))
        <=> pp(aa(real,bool,ord_less(real,powr(real,B2,Y)),X)) ) ) ) ).

% less_log_iff
tff(fact_2250_foldr__length__aux,axiom,
    ! [A: $tType,L: list(A),A2: nat] : foldr(A,nat,aTP_Lamp_ag(A,fun(nat,nat)),L,A2) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A2),aa(list(A),nat,size_size(list(A)),L)) ).

% foldr_length_aux
tff(fact_2251_finite__roots__unity,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [N: nat] :
          ( pp(aa(nat,bool,ord_less_eq(nat,one_one(nat)),N))
         => pp(aa(set(A),bool,finite_finite2(A),collect(A,aTP_Lamp_an(nat,fun(A,bool),N)))) ) ) ).

% finite_roots_unity
tff(fact_2252_ln__powr__bound,axiom,
    ! [X: real,A2: real] :
      ( pp(aa(real,bool,ord_less_eq(real,one_one(real)),X))
     => ( pp(aa(real,bool,ord_less(real,zero_zero(real)),A2))
       => pp(aa(real,bool,ord_less_eq(real,aa(real,real,ln_ln(real),X)),aa(real,real,aa(real,fun(real,real),divide_divide(real),powr(real,X,A2)),A2))) ) ) ).

% ln_powr_bound
tff(fact_2253_ln__powr__bound2,axiom,
    ! [X: real,A2: real] :
      ( pp(aa(real,bool,ord_less(real,one_one(real)),X))
     => ( pp(aa(real,bool,ord_less(real,zero_zero(real)),A2))
       => pp(aa(real,bool,ord_less_eq(real,powr(real,aa(real,real,ln_ln(real),X),A2)),aa(real,real,aa(real,fun(real,real),times_times(real),powr(real,A2,A2)),X))) ) ) ).

% ln_powr_bound2
tff(fact_2254_powr__mult__base,axiom,
    ! [X: real,Y: real] :
      ( pp(aa(real,bool,ord_less_eq(real,zero_zero(real)),X))
     => ( aa(real,real,aa(real,fun(real,real),times_times(real),X),powr(real,X,Y)) = powr(real,X,aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),Y)) ) ) ).

% powr_mult_base
tff(fact_2255_powr__le__iff,axiom,
    ! [B2: real,X: real,Y: real] :
      ( pp(aa(real,bool,ord_less(real,one_one(real)),B2))
     => ( pp(aa(real,bool,ord_less(real,zero_zero(real)),X))
       => ( pp(aa(real,bool,ord_less_eq(real,powr(real,B2,Y)),X))
        <=> pp(aa(real,bool,ord_less_eq(real,Y),aa(real,real,log(B2),X))) ) ) ) ).

% powr_le_iff
tff(fact_2256_le__powr__iff,axiom,
    ! [B2: real,X: real,Y: real] :
      ( pp(aa(real,bool,ord_less(real,one_one(real)),B2))
     => ( pp(aa(real,bool,ord_less(real,zero_zero(real)),X))
       => ( pp(aa(real,bool,ord_less_eq(real,X),powr(real,B2,Y)))
        <=> pp(aa(real,bool,ord_less_eq(real,aa(real,real,log(B2),X)),Y)) ) ) ) ).

% le_powr_iff
tff(fact_2257_log__le__iff,axiom,
    ! [B2: real,X: real,Y: real] :
      ( pp(aa(real,bool,ord_less(real,one_one(real)),B2))
     => ( pp(aa(real,bool,ord_less(real,zero_zero(real)),X))
       => ( pp(aa(real,bool,ord_less_eq(real,aa(real,real,log(B2),X)),Y))
        <=> pp(aa(real,bool,ord_less_eq(real,X),powr(real,B2,Y))) ) ) ) ).

% log_le_iff
tff(fact_2258_le__log__iff,axiom,
    ! [B2: real,X: real,Y: real] :
      ( pp(aa(real,bool,ord_less(real,one_one(real)),B2))
     => ( pp(aa(real,bool,ord_less(real,zero_zero(real)),X))
       => ( pp(aa(real,bool,ord_less_eq(real,Y),aa(real,real,log(B2),X)))
        <=> pp(aa(real,bool,ord_less_eq(real,powr(real,B2,Y)),X)) ) ) ) ).

% le_log_iff
tff(fact_2259_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,X: nat] : vEBT_T_m_e_m_b_e_r(vEBT_Node(none(product_prod(nat,nat)),Uu,Uv,Uw),X) = aa(num,nat,numeral_numeral(nat),bit0(one2)) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.simps(2)
tff(fact_2260_log__add__eq__powr,axiom,
    ! [B2: real,X: real,Y: real] :
      ( pp(aa(real,bool,ord_less(real,zero_zero(real)),B2))
     => ( ( B2 != one_one(real) )
       => ( pp(aa(real,bool,ord_less(real,zero_zero(real)),X))
         => ( aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,log(B2),X)),Y) = aa(real,real,log(B2),aa(real,real,aa(real,fun(real,real),times_times(real),X),powr(real,B2,Y))) ) ) ) ) ).

% log_add_eq_powr
tff(fact_2261_add__log__eq__powr,axiom,
    ! [B2: real,X: real,Y: real] :
      ( pp(aa(real,bool,ord_less(real,zero_zero(real)),B2))
     => ( ( B2 != one_one(real) )
       => ( pp(aa(real,bool,ord_less(real,zero_zero(real)),X))
         => ( aa(real,real,aa(real,fun(real,real),plus_plus(real),Y),aa(real,real,log(B2),X)) = aa(real,real,log(B2),aa(real,real,aa(real,fun(real,real),times_times(real),powr(real,B2,Y)),X)) ) ) ) ) ).

% add_log_eq_powr
tff(fact_2262_minus__log__eq__powr,axiom,
    ! [B2: real,X: real,Y: real] :
      ( pp(aa(real,bool,ord_less(real,zero_zero(real)),B2))
     => ( ( B2 != one_one(real) )
       => ( pp(aa(real,bool,ord_less(real,zero_zero(real)),X))
         => ( aa(real,real,minus_minus(real,Y),aa(real,real,log(B2),X)) = aa(real,real,log(B2),aa(real,real,aa(real,fun(real,real),divide_divide(real),powr(real,B2,Y)),X)) ) ) ) ) ).

% minus_log_eq_powr
tff(fact_2263_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: bool,B2: bool,X: nat] : vEBT_T_m_e_m_b_e_r(vEBT_Leaf(A2,B2),X) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),if(nat,aa(nat,bool,fequal(nat,X),zero_zero(nat)),one_one(nat),aa(nat,nat,aa(nat,fun(nat,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)
tff(fact_2264_member__bound__height,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( vEBT_invar_vebt(T,N)
     => pp(aa(nat,bool,ord_less_eq(nat,vEBT_T_m_e_m_b_e_r(T,X)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),aa(vEBT_VEBT,nat,vEBT_VEBT_height,T))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,one2))))))) ) ).

% member_bound_height
tff(fact_2265_finite__Collect__le__nat,axiom,
    ! [K: nat] : pp(aa(set(nat),bool,finite_finite2(nat),collect(nat,aTP_Lamp_ao(nat,fun(nat,bool),K)))) ).

% finite_Collect_le_nat
tff(fact_2266_finite__Collect__less__nat,axiom,
    ! [K: nat] : pp(aa(set(nat),bool,finite_finite2(nat),collect(nat,aTP_Lamp_ap(nat,fun(nat,bool),K)))) ).

% finite_Collect_less_nat
tff(fact_2267_finite__Collect__subsets,axiom,
    ! [A: $tType,A3: set(A)] :
      ( pp(aa(set(A),bool,finite_finite2(A),A3))
     => pp(aa(set(set(A)),bool,finite_finite2(set(A)),collect(set(A),aTP_Lamp_aq(set(A),fun(set(A),bool),A3)))) ) ).

% finite_Collect_subsets
tff(fact_2268_succ__bound__size__univ,axiom,
    ! [T: vEBT_VEBT,N: nat,U: real,X: nat] :
      ( vEBT_invar_vebt(T,N)
     => ( ( U = aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(num,real,numeral_numeral(real),bit0(one2))),N) )
       => pp(aa(real,bool,ord_less_eq(real,aa(nat,real,semiring_1_of_nat(real),vEBT_T_s_u_c_c(T,X))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(num,real,numeral_numeral(real),bit0(aa(num,num,bit1,aa(num,num,bit1,bit0(aa(num,num,bit1,one2))))))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),aa(num,num,bit1,aa(num,num,bit1,bit0(aa(num,num,bit1,one2)))))),aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),U)))))) ) ) ).

% succ_bound_size_univ
tff(fact_2269_list__every__elemnt__bound__sum__bound__real,axiom,
    ! [A: $tType,Xs: list(A),F2: fun(A,real),Bound: real,I: real] :
      ( ! [X3: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),aa(list(A),set(A),set2(A),Xs)))
         => pp(aa(real,bool,ord_less_eq(real,aa(A,real,F2,X3)),Bound)) )
     => pp(aa(real,bool,ord_less_eq(real,foldr(real,real,plus_plus(real),map(A,real,F2,Xs),I)),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),aa(list(A),nat,size_size(list(A)),Xs))),Bound)),I))) ) ).

% list_every_elemnt_bound_sum_bound_real
tff(fact_2270_insert__bound__size__univ,axiom,
    ! [T: vEBT_VEBT,N: nat,U: real,X: nat] :
      ( vEBT_invar_vebt(T,N)
     => ( ( U = aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(num,real,numeral_numeral(real),bit0(one2))),N) )
       => pp(aa(real,bool,ord_less_eq(real,aa(nat,real,semiring_1_of_nat(real),vEBT_T_i_n_s_e_r_t(T,X))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(num,real,numeral_numeral(real),bit0(aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,bit0(one2))))))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,bit0(one2)))))),aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),U)))))) ) ) ).

% insert_bound_size_univ
tff(fact_2271_pred__bound__size__univ,axiom,
    ! [T: vEBT_VEBT,N: nat,U: real,X: nat] :
      ( vEBT_invar_vebt(T,N)
     => ( ( U = aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(num,real,numeral_numeral(real),bit0(one2))),N) )
       => pp(aa(real,bool,ord_less_eq(real,aa(nat,real,semiring_1_of_nat(real),vEBT_T_p_r_e_d(T,X))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(num,real,numeral_numeral(real),bit0(aa(num,num,bit1,bit0(aa(num,num,bit1,aa(num,num,bit1,one2))))))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),aa(num,num,bit1,bit0(aa(num,num,bit1,aa(num,num,bit1,one2)))))),aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),U)))))) ) ) ).

% pred_bound_size_univ
tff(fact_2272_finite__Collect__disjI,axiom,
    ! [A: $tType,P: fun(A,bool),Q: fun(A,bool)] :
      ( pp(aa(set(A),bool,finite_finite2(A),collect(A,aa(fun(A,bool),fun(A,bool),aTP_Lamp_ar(fun(A,bool),fun(fun(A,bool),fun(A,bool)),P),Q))))
    <=> ( pp(aa(set(A),bool,finite_finite2(A),collect(A,P)))
        & pp(aa(set(A),bool,finite_finite2(A),collect(A,Q))) ) ) ).

% finite_Collect_disjI
tff(fact_2273_finite__Collect__conjI,axiom,
    ! [A: $tType,P: fun(A,bool),Q: fun(A,bool)] :
      ( ( pp(aa(set(A),bool,finite_finite2(A),collect(A,P)))
        | pp(aa(set(A),bool,finite_finite2(A),collect(A,Q))) )
     => pp(aa(set(A),bool,finite_finite2(A),collect(A,aa(fun(A,bool),fun(A,bool),aTP_Lamp_as(fun(A,bool),fun(fun(A,bool),fun(A,bool)),P),Q)))) ) ).

% finite_Collect_conjI
tff(fact_2274_map__ident,axiom,
    ! [A: $tType,X4: list(A)] : map(A,A,aTP_Lamp_at(A,A),X4) = X4 ).

% map_ident
tff(fact_2275_finite__interval__int1,axiom,
    ! [A2: int,B2: int] : pp(aa(set(int),bool,finite_finite2(int),collect(int,aa(int,fun(int,bool),aTP_Lamp_au(int,fun(int,fun(int,bool)),A2),B2)))) ).

% finite_interval_int1
tff(fact_2276_finite__interval__int4,axiom,
    ! [A2: int,B2: int] : pp(aa(set(int),bool,finite_finite2(int),collect(int,aa(int,fun(int,bool),aTP_Lamp_av(int,fun(int,fun(int,bool)),A2),B2)))) ).

% finite_interval_int4
tff(fact_2277_listsum__bound,axiom,
    ! [A: $tType,Xs: list(A),F2: fun(A,real),Y: real] :
      ( ! [X3: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),aa(list(A),set(A),set2(A),Xs)))
         => pp(aa(real,bool,ord_less_eq(real,zero_zero(real)),aa(A,real,F2,X3))) )
     => pp(aa(real,bool,ord_less_eq(real,Y),foldr(real,real,plus_plus(real),map(A,real,F2,Xs),Y))) ) ).

% listsum_bound
tff(fact_2278_f__g__map__foldr__bound,axiom,
    ! [A: $tType,Xs: list(A),F2: fun(A,real),C2: real,G: fun(A,real),D: real] :
      ( ! [X3: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),aa(list(A),set(A),set2(A),Xs)))
         => pp(aa(real,bool,ord_less_eq(real,aa(A,real,F2,X3)),aa(real,real,aa(real,fun(real,real),times_times(real),C2),aa(A,real,G,X3)))) )
     => pp(aa(real,bool,ord_less_eq(real,foldr(real,real,plus_plus(real),map(A,real,F2,Xs),D)),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,aa(real,fun(real,real),times_times(real),C2),foldr(real,real,plus_plus(real),map(A,real,G,Xs),zero_zero(real)))),D))) ) ).

% f_g_map_foldr_bound
tff(fact_2279_map__replicate,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),N: nat,X: B] : map(B,A,F2,replicate(B,N,X)) = replicate(A,N,aa(B,A,F2,X)) ).

% map_replicate
tff(fact_2280_finite__interval__int3,axiom,
    ! [A2: int,B2: int] : pp(aa(set(int),bool,finite_finite2(int),collect(int,aa(int,fun(int,bool),aTP_Lamp_aw(int,fun(int,fun(int,bool)),A2),B2)))) ).

% finite_interval_int3
tff(fact_2281_finite__interval__int2,axiom,
    ! [A2: int,B2: int] : pp(aa(set(int),bool,finite_finite2(int),collect(int,aa(int,fun(int,bool),aTP_Lamp_ax(int,fun(int,fun(int,bool)),A2),B2)))) ).

% finite_interval_int2
tff(fact_2282_list_Omap__ident,axiom,
    ! [A: $tType,T: list(A)] : map(A,A,aTP_Lamp_at(A,A),T) = T ).

% list.map_ident
tff(fact_2283_map__replicate__const,axiom,
    ! [B: $tType,A: $tType,K: A,Lst: list(B)] : map(B,A,aTP_Lamp_ay(A,fun(B,A),K),Lst) = replicate(A,aa(list(B),nat,size_size(list(B)),Lst),K) ).

% map_replicate_const
tff(fact_2284_finite__divisors__int,axiom,
    ! [I: int] :
      ( ( I != zero_zero(int) )
     => pp(aa(set(int),bool,finite_finite2(int),collect(int,aTP_Lamp_az(int,fun(int,bool),I)))) ) ).

% finite_divisors_int
tff(fact_2285_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),S3: vEBT_VEBT,X: nat] : vEBT_T_i_n_s_e_r_t(vEBT_Node(Info,zero_zero(nat),Ts,S3),X) = one_one(nat) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.simps(2)
tff(fact_2286_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Osimps_I1_J,axiom,
    ! [Uu: bool,Uv: bool] : 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)
tff(fact_2287_pigeonhole__infinite__rel,axiom,
    ! [A: $tType,B: $tType,A3: set(A),B4: set(B),R: fun(A,fun(B,bool))] :
      ( ~ pp(aa(set(A),bool,finite_finite2(A),A3))
     => ( pp(aa(set(B),bool,finite_finite2(B),B4))
       => ( ! [X3: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),A3))
             => ? [Xa: B] :
                  ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),Xa),B4))
                  & pp(aa(B,bool,aa(A,fun(B,bool),R,X3),Xa)) ) )
         => ? [X3: B] :
              ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),B4))
              & ~ pp(aa(set(A),bool,finite_finite2(A),collect(A,aa(B,fun(A,bool),aa(fun(A,fun(B,bool)),fun(B,fun(A,bool)),aTP_Lamp_ba(set(A),fun(fun(A,fun(B,bool)),fun(B,fun(A,bool))),A3),R),X3)))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_2288_not__finite__existsD,axiom,
    ! [A: $tType,P: fun(A,bool)] :
      ( ~ pp(aa(set(A),bool,finite_finite2(A),collect(A,P)))
     => ? [X_1: A] : pp(aa(A,bool,P,X_1)) ) ).

% not_finite_existsD
tff(fact_2289_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Osimps_I2_J,axiom,
    ! [Uv: bool,Uw: bool,N: nat] : vEBT_T_s_u_c_c(vEBT_Leaf(Uv,Uw),aa(nat,nat,suc,N)) = one_one(nat) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.simps(2)
tff(fact_2290_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)
tff(fact_2291_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)
tff(fact_2292_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),S3: vEBT_VEBT,X: nat] : vEBT_T_i_n_s_e_r_t(vEBT_Node(Info,aa(nat,nat,suc,zero_zero(nat)),Ts,S3),X) = one_one(nat) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.simps(3)
tff(fact_2293_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Osimps_I3_J,axiom,
    ! [A2: bool,B2: bool,Va: nat] : vEBT_T_p_r_e_d(vEBT_Leaf(A2,B2),aa(nat,nat,suc,aa(nat,nat,suc,Va))) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),if(nat,B2,one_one(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),one_one(nat)))) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.simps(3)
tff(fact_2294_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: bool,B2: bool,X: nat] : vEBT_T_i_n_s_e_r_t(vEBT_Leaf(A2,B2),X) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),if(nat,aa(nat,bool,fequal(nat,X),zero_zero(nat)),one_one(nat),aa(nat,nat,aa(nat,fun(nat,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)
tff(fact_2295_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Osimps_I1_J,axiom,
    ! [Uu: bool,B2: bool] : vEBT_T_s_u_c_c(vEBT_Leaf(Uu,B2),zero_zero(nat)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),one_one(nat)) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.simps(1)
tff(fact_2296_VEBT__internal_Ocnt_Osimps_I2_J,axiom,
    ! [Info: option(product_prod(nat,nat)),Deg: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] : aa(vEBT_VEBT,real,vEBT_VEBT_cnt,vEBT_Node(Info,Deg,TreeList,Summary)) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),aa(vEBT_VEBT,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)
tff(fact_2297_finite__has__minimal2,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A3: set(A),A2: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A3))
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),A3))
           => ? [X3: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),A3))
                & pp(aa(A,bool,ord_less_eq(A,X3),A2))
                & ! [Xa: A] :
                    ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Xa),A3))
                   => ( pp(aa(A,bool,ord_less_eq(A,Xa),X3))
                     => ( X3 = Xa ) ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_2298_finite__has__maximal2,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A3: set(A),A2: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A3))
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),A3))
           => ? [X3: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),A3))
                & pp(aa(A,bool,ord_less_eq(A,A2),X3))
                & ! [Xa: A] :
                    ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Xa),A3))
                   => ( pp(aa(A,bool,ord_less_eq(A,X3),Xa))
                     => ( X3 = Xa ) ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_2299_rev__finite__subset,axiom,
    ! [A: $tType,B4: set(A),A3: set(A)] :
      ( pp(aa(set(A),bool,finite_finite2(A),B4))
     => ( pp(aa(set(A),bool,ord_less_eq(set(A),A3),B4))
       => pp(aa(set(A),bool,finite_finite2(A),A3)) ) ) ).

% rev_finite_subset
tff(fact_2300_infinite__super,axiom,
    ! [A: $tType,S: set(A),T3: set(A)] :
      ( pp(aa(set(A),bool,ord_less_eq(set(A),S),T3))
     => ( ~ pp(aa(set(A),bool,finite_finite2(A),S))
       => ~ pp(aa(set(A),bool,finite_finite2(A),T3)) ) ) ).

% infinite_super
tff(fact_2301_finite__subset,axiom,
    ! [A: $tType,A3: set(A),B4: set(A)] :
      ( pp(aa(set(A),bool,ord_less_eq(set(A),A3),B4))
     => ( pp(aa(set(A),bool,finite_finite2(A),B4))
       => pp(aa(set(A),bool,finite_finite2(A),A3)) ) ) ).

% finite_subset
tff(fact_2302_VEBT__internal_Ocnt_Oelims,axiom,
    ! [X: vEBT_VEBT,Y: real] :
      ( ( aa(vEBT_VEBT,real,vEBT_VEBT_cnt,X) = Y )
     => ( ( ? [A4: bool,B3: bool] : X = vEBT_Leaf(A4,B3)
         => ( Y != one_one(real) ) )
       => ~ ! [Info2: option(product_prod(nat,nat)),Deg2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
              ( ( X = vEBT_Node(Info2,Deg2,TreeList2,Summary2) )
             => ( Y != aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),aa(vEBT_VEBT,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
tff(fact_2303_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Osimps_I2_J,axiom,
    ! [A2: bool,Uw: bool] : vEBT_T_p_r_e_d(vEBT_Leaf(A2,Uw),aa(nat,nat,suc,zero_zero(nat))) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),one_one(nat)) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.simps(2)
tff(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_Osimps_I4_J,axiom,
    ! [V: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,X: nat] : vEBT_T_i_n_s_e_r_t(vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,V)),TreeList,Summary),X) = aa(num,nat,numeral_numeral(nat),bit0(one2)) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.simps(4)
tff(fact_2305_insersimp,axiom,
    ! [T: vEBT_VEBT,N: nat,Y: nat] :
      ( vEBT_invar_vebt(T,N)
     => ( ~ ? [X_1: nat] : pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,T),X_1))
       => pp(aa(nat,bool,ord_less_eq(nat,vEBT_T_i_n_s_e_r_t(T,Y)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2)))) ) ) ).

% insersimp
tff(fact_2306_pred__bound__height,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( vEBT_invar_vebt(T,N)
     => pp(aa(nat,bool,ord_less_eq(nat,vEBT_T_p_r_e_d(T,X)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),aa(vEBT_VEBT,nat,vEBT_VEBT_height,T))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,bit0(aa(num,num,bit1,aa(num,num,bit1,one2)))))))) ) ).

% pred_bound_height
tff(fact_2307_insert__bound__height,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( vEBT_invar_vebt(T,N)
     => pp(aa(nat,bool,ord_less_eq(nat,vEBT_T_i_n_s_e_r_t(T,X)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),aa(vEBT_VEBT,nat,vEBT_VEBT_height,T))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,bit0(one2)))))))) ) ).

% insert_bound_height
tff(fact_2308_succ__bound__height,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( vEBT_invar_vebt(T,N)
     => pp(aa(nat,bool,ord_less_eq(nat,vEBT_T_s_u_c_c(T,X)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),aa(vEBT_VEBT,nat,vEBT_VEBT_height,T))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,aa(num,num,bit1,bit0(aa(num,num,bit1,one2)))))))) ) ).

% succ_bound_height
tff(fact_2309_finite__nth__roots,axiom,
    ! [N: nat,C2: complex] :
      ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),N))
     => pp(aa(set(complex),bool,finite_finite2(complex),collect(complex,aa(complex,fun(complex,bool),aTP_Lamp_bb(nat,fun(complex,fun(complex,bool)),N),C2)))) ) ).

% finite_nth_roots
tff(fact_2310_real__nat__list,axiom,
    ! [A: $tType,F2: fun(A,nat),Xs: list(A),C2: nat] : aa(nat,real,semiring_1_of_nat(real),foldr(nat,nat,plus_plus(nat),map(A,nat,F2,Xs),C2)) = foldr(real,real,plus_plus(real),map(A,real,aTP_Lamp_bc(fun(A,nat),fun(A,real),F2),Xs),aa(nat,real,semiring_1_of_nat(real),C2)) ).

% real_nat_list
tff(fact_2311_diff__preserves__multiset,axiom,
    ! [A: $tType,M7: fun(A,nat),N4: fun(A,nat)] :
      ( pp(aa(set(A),bool,finite_finite2(A),collect(A,aTP_Lamp_bd(fun(A,nat),fun(A,bool),M7))))
     => pp(aa(set(A),bool,finite_finite2(A),collect(A,aa(fun(A,nat),fun(A,bool),aTP_Lamp_be(fun(A,nat),fun(fun(A,nat),fun(A,bool)),M7),N4)))) ) ).

% diff_preserves_multiset
tff(fact_2312_add__mset__in__multiset,axiom,
    ! [A: $tType,M7: fun(A,nat),A2: A] :
      ( pp(aa(set(A),bool,finite_finite2(A),collect(A,aTP_Lamp_bd(fun(A,nat),fun(A,bool),M7))))
     => pp(aa(set(A),bool,finite_finite2(A),collect(A,aa(A,fun(A,bool),aTP_Lamp_bf(fun(A,nat),fun(A,fun(A,bool)),M7),A2)))) ) ).

% add_mset_in_multiset
tff(fact_2313_list__every__elemnt__bound__sum__bound,axiom,
    ! [A: $tType,Xs: list(A),F2: fun(A,nat),Bound: nat,I: nat] :
      ( ! [X3: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),aa(list(A),set(A),set2(A),Xs)))
         => pp(aa(nat,bool,ord_less_eq(nat,aa(A,nat,F2,X3)),Bound)) )
     => pp(aa(nat,bool,ord_less_eq(nat,foldr(nat,nat,plus_plus(nat),map(A,nat,F2,Xs),I)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(list(A),nat,size_size(list(A)),Xs)),Bound)),I))) ) ).

% list_every_elemnt_bound_sum_bound
tff(fact_2314_succ__bound__size__univ_H,axiom,
    ! [T: vEBT_VEBT,N: nat,U: real,X: nat] :
      ( vEBT_invar_vebt(T,N)
     => ( ( U = aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(num,real,numeral_numeral(real),bit0(one2))),N) )
       => pp(aa(real,bool,ord_less_eq(real,aa(nat,real,semiring_1_of_nat(real),vEBT_T_s_u_c_c2(T,X))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(num,real,numeral_numeral(real),bit0(one2))),aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),U))))) ) ) ).

% succ_bound_size_univ'
tff(fact_2315_size__empty,axiom,
    ! [A: $tType] : aa(multiset(A),nat,size_size(multiset(A)),zero_zero(multiset(A))) = zero_zero(nat) ).

% size_empty
tff(fact_2316_size__eq__0__iff__empty,axiom,
    ! [A: $tType,M7: multiset(A)] :
      ( ( aa(multiset(A),nat,size_size(multiset(A)),M7) = zero_zero(nat) )
    <=> ( M7 = zero_zero(multiset(A)) ) ) ).

% size_eq_0_iff_empty
tff(fact_2317_size__union,axiom,
    ! [A: $tType,M7: multiset(A),N4: multiset(A)] : aa(multiset(A),nat,size_size(multiset(A)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),M7),N4)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(multiset(A),nat,size_size(multiset(A)),M7)),aa(multiset(A),nat,size_size(multiset(A)),N4)) ).

% size_union
tff(fact_2318_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Osimps_I2_J,axiom,
    ! [Uv: bool,Uw: bool,N: nat] : vEBT_T_s_u_c_c2(vEBT_Leaf(Uv,Uw),aa(nat,nat,suc,N)) = one_one(nat) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.simps(2)
tff(fact_2319_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Osimps_I1_J,axiom,
    ! [Uu: bool,B2: bool] : 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)
tff(fact_2320_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)
tff(fact_2321_VEBT__internal_Ocnt_H_Osimps_I2_J,axiom,
    ! [Info: option(product_prod(nat,nat)),Deg: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] : aa(vEBT_VEBT,nat,vEBT_VEBT_cnt2,vEBT_Node(Info,Deg,TreeList,Summary)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),aa(vEBT_VEBT,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)
tff(fact_2322_nonempty__has__size,axiom,
    ! [A: $tType,S: multiset(A)] :
      ( ( S != zero_zero(multiset(A)) )
    <=> pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),aa(multiset(A),nat,size_size(multiset(A)),S))) ) ).

% nonempty_has_size
tff(fact_2323_diff__size__le__size__Diff,axiom,
    ! [A: $tType,M7: multiset(A),M8: multiset(A)] : pp(aa(nat,bool,ord_less_eq(nat,aa(nat,nat,minus_minus(nat,aa(multiset(A),nat,size_size(multiset(A)),M7)),aa(multiset(A),nat,size_size(multiset(A)),M8))),aa(multiset(A),nat,size_size(multiset(A)),aa(multiset(A),multiset(A),minus_minus(multiset(A),M7),M8)))) ).

% diff_size_le_size_Diff
tff(fact_2324_VEBT__internal_Ocnt_H_Oelims,axiom,
    ! [X: vEBT_VEBT,Y: nat] :
      ( ( aa(vEBT_VEBT,nat,vEBT_VEBT_cnt2,X) = Y )
     => ( ( ? [A4: bool,B3: bool] : X = vEBT_Leaf(A4,B3)
         => ( Y != one_one(nat) ) )
       => ~ ! [Info2: option(product_prod(nat,nat)),Deg2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
              ( ( X = vEBT_Node(Info2,Deg2,TreeList2,Summary2) )
             => ( Y != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),aa(vEBT_VEBT,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
tff(fact_2325_complex__mod__triangle__ineq2,axiom,
    ! [B2: complex,A2: complex] : pp(aa(real,bool,ord_less_eq(real,aa(real,real,minus_minus(real,real_V7770717601297561774m_norm(complex,aa(complex,complex,aa(complex,fun(complex,complex),plus_plus(complex),B2),A2))),real_V7770717601297561774m_norm(complex,B2))),real_V7770717601297561774m_norm(complex,A2))) ).

% complex_mod_triangle_ineq2
tff(fact_2326_succ_H__bound__height,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( vEBT_invar_vebt(T,N)
     => pp(aa(nat,bool,ord_less_eq(nat,vEBT_T_s_u_c_c2(T,X)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),aa(vEBT_VEBT,nat,vEBT_VEBT_height,T)))) ) ).

% succ'_bound_height
tff(fact_2327_VEBT__internal_Ospace_H_Osimps_I2_J,axiom,
    ! [Info: option(product_prod(nat,nat)),Deg: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] : aa(vEBT_VEBT,nat,vEBT_VEBT_space2,vEBT_Node(Info,Deg,TreeList,Summary)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(aa(num,num,bit1,one2)))),aa(vEBT_VEBT,nat,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)
tff(fact_2328_VEBT__internal_Ospace_H_Oelims,axiom,
    ! [X: vEBT_VEBT,Y: nat] :
      ( ( aa(vEBT_VEBT,nat,vEBT_VEBT_space2,X) = Y )
     => ( ( ? [A4: bool,B3: bool] : X = vEBT_Leaf(A4,B3)
         => ( Y != aa(num,nat,numeral_numeral(nat),bit0(bit0(one2))) ) )
       => ~ ! [Info2: option(product_prod(nat,nat)),Deg2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
              ( ( X = vEBT_Node(Info2,Deg2,TreeList2,Summary2) )
             => ( Y != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(aa(num,num,bit1,one2)))),aa(vEBT_VEBT,nat,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
tff(fact_2329_VEBT__internal_Ospace_Osimps_I2_J,axiom,
    ! [Info: option(product_prod(nat,nat)),Deg: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] : aa(vEBT_VEBT,nat,vEBT_VEBT_space,vEBT_Node(Info,Deg,TreeList,Summary)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,bit0(one2)))),aa(vEBT_VEBT,nat,vEBT_VEBT_space,Summary))),aa(list(vEBT_VEBT),nat,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)
tff(fact_2330_VEBT__internal_Ospace_Oelims,axiom,
    ! [X: vEBT_VEBT,Y: nat] :
      ( ( aa(vEBT_VEBT,nat,vEBT_VEBT_space,X) = Y )
     => ( ( ? [A4: bool,B3: bool] : X = vEBT_Leaf(A4,B3)
         => ( Y != aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2)) ) )
       => ~ ! [Info2: option(product_prod(nat,nat)),Deg2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
              ( ( X = vEBT_Node(Info2,Deg2,TreeList2,Summary2) )
             => ( Y != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,bit0(one2)))),aa(vEBT_VEBT,nat,vEBT_VEBT_space,Summary2))),aa(list(vEBT_VEBT),nat,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
tff(fact_2331_filter__preserves__multiset,axiom,
    ! [A: $tType,M7: fun(A,nat),P: fun(A,bool)] :
      ( pp(aa(set(A),bool,finite_finite2(A),collect(A,aTP_Lamp_bd(fun(A,nat),fun(A,bool),M7))))
     => pp(aa(set(A),bool,finite_finite2(A),collect(A,aa(fun(A,bool),fun(A,bool),aTP_Lamp_bg(fun(A,nat),fun(fun(A,bool),fun(A,bool)),M7),P)))) ) ).

% filter_preserves_multiset
tff(fact_2332_pred__bound__size__univ_H,axiom,
    ! [T: vEBT_VEBT,N: nat,U: real,X: nat] :
      ( vEBT_invar_vebt(T,N)
     => ( ( U = aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(num,real,numeral_numeral(real),bit0(one2))),N) )
       => pp(aa(real,bool,ord_less_eq(real,aa(nat,real,semiring_1_of_nat(real),vEBT_T_p_r_e_d2(T,X))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(num,real,numeral_numeral(real),bit0(one2))),aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),U))))) ) ) ).

% pred_bound_size_univ'
tff(fact_2333_tanh__ln__real,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,ord_less(real,zero_zero(real)),X))
     => ( aa(real,real,tanh(real),aa(real,real,ln_ln(real),X)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,minus_minus(real,aa(nat,real,aa(real,fun(nat,real),power_power(real),X),aa(num,nat,numeral_numeral(nat),bit0(one2)))),one_one(real))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),X),aa(num,nat,numeral_numeral(nat),bit0(one2)))),one_one(real))) ) ) ).

% tanh_ln_real
tff(fact_2334_ln__one__minus__pos__lower__bound,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,ord_less_eq(real,zero_zero(real)),X))
     => ( pp(aa(real,bool,ord_less_eq(real,X),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),bit0(one2)))))
       => pp(aa(real,bool,ord_less_eq(real,aa(real,real,minus_minus(real,aa(real,real,uminus_uminus(real),X)),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),aa(nat,real,aa(real,fun(nat,real),power_power(real),X),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(real,real,ln_ln(real),aa(real,real,minus_minus(real,one_one(real)),X)))) ) ) ).

% ln_one_minus_pos_lower_bound
tff(fact_2335_arcosh__def,axiom,
    ! [A: $tType] :
      ( ln(A)
     => ! [X: A] : aa(A,A,arcosh(A),X) = aa(A,A,ln_ln(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),powr(A,aa(A,A,minus_minus(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(num,nat,numeral_numeral(nat),bit0(one2)))),one_one(A)),aa(real,A,real_Vector_of_real(A),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),bit0(one2))))))) ) ).

% arcosh_def
tff(fact_2336_abs__ln__one__plus__x__minus__x__bound,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,ord_less_eq(real,aa(real,real,abs_abs(real),X)),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),bit0(one2)))))
     => pp(aa(real,bool,ord_less_eq(real,aa(real,real,abs_abs(real),aa(real,real,minus_minus(real,aa(real,real,ln_ln(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),X))),X))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),aa(nat,real,aa(real,fun(nat,real),power_power(real),X),aa(num,nat,numeral_numeral(nat),bit0(one2)))))) ) ).

% abs_ln_one_plus_x_minus_x_bound
tff(fact_2337_prod_Ofinite__Collect__op,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [I5: set(B),X: fun(B,A),Y: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),collect(B,aa(fun(B,A),fun(B,bool),aTP_Lamp_bh(set(B),fun(fun(B,A),fun(B,bool)),I5),X))))
         => ( pp(aa(set(B),bool,finite_finite2(B),collect(B,aa(fun(B,A),fun(B,bool),aTP_Lamp_bh(set(B),fun(fun(B,A),fun(B,bool)),I5),Y))))
           => pp(aa(set(B),bool,finite_finite2(B),collect(B,aa(fun(B,A),fun(B,bool),aa(fun(B,A),fun(fun(B,A),fun(B,bool)),aTP_Lamp_bi(set(B),fun(fun(B,A),fun(fun(B,A),fun(B,bool))),I5),X),Y)))) ) ) ) ).

% prod.finite_Collect_op
tff(fact_2338_add_Oinverse__inverse,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A] : aa(A,A,uminus_uminus(A),aa(A,A,uminus_uminus(A),A2)) = A2 ) ).

% add.inverse_inverse
tff(fact_2339_neg__equal__iff__equal,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A,B2: A] :
          ( ( aa(A,A,uminus_uminus(A),A2) = aa(A,A,uminus_uminus(A),B2) )
        <=> ( A2 = B2 ) ) ) ).

% neg_equal_iff_equal
tff(fact_2340_abs__abs,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [A2: A] : aa(A,A,abs_abs(A),aa(A,A,abs_abs(A),A2)) = aa(A,A,abs_abs(A),A2) ) ).

% abs_abs
tff(fact_2341_abs__idempotent,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A] : aa(A,A,abs_abs(A),aa(A,A,abs_abs(A),A2)) = aa(A,A,abs_abs(A),A2) ) ).

% abs_idempotent
tff(fact_2342_add_Oinverse__neutral,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ( aa(A,A,uminus_uminus(A),zero_zero(A)) = zero_zero(A) ) ) ).

% add.inverse_neutral
tff(fact_2343_neg__0__equal__iff__equal,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A] :
          ( ( zero_zero(A) = aa(A,A,uminus_uminus(A),A2) )
        <=> ( zero_zero(A) = A2 ) ) ) ).

% neg_0_equal_iff_equal
tff(fact_2344_neg__equal__0__iff__equal,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A] :
          ( ( aa(A,A,uminus_uminus(A),A2) = zero_zero(A) )
        <=> ( A2 = zero_zero(A) ) ) ) ).

% neg_equal_0_iff_equal
tff(fact_2345_equal__neg__zero,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A2: A] :
          ( ( A2 = aa(A,A,uminus_uminus(A),A2) )
        <=> ( A2 = zero_zero(A) ) ) ) ).

% equal_neg_zero
tff(fact_2346_neg__equal__zero,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A2: A] :
          ( ( aa(A,A,uminus_uminus(A),A2) = A2 )
        <=> ( A2 = zero_zero(A) ) ) ) ).

% neg_equal_zero
tff(fact_2347_neg__le__iff__le,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [B2: A,A2: A] :
          ( pp(aa(A,bool,ord_less_eq(A,aa(A,A,uminus_uminus(A),B2)),aa(A,A,uminus_uminus(A),A2)))
        <=> pp(aa(A,bool,ord_less_eq(A,A2),B2)) ) ) ).

% neg_le_iff_le
tff(fact_2348_neg__less__iff__less,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [B2: A,A2: A] :
          ( pp(aa(A,bool,ord_less(A,aa(A,A,uminus_uminus(A),B2)),aa(A,A,uminus_uminus(A),A2)))
        <=> pp(aa(A,bool,ord_less(A,A2),B2)) ) ) ).

% neg_less_iff_less
tff(fact_2349_neg__numeral__eq__iff,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [M: num,N: num] :
          ( ( aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M)) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N)) )
        <=> ( M = N ) ) ) ).

% neg_numeral_eq_iff
tff(fact_2350_minus__add__distrib,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A2: A,B2: A] : aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),A2)),aa(A,A,uminus_uminus(A),B2)) ) ).

% minus_add_distrib
tff(fact_2351_minus__add__cancel,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),A2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)) = B2 ) ).

% minus_add_cancel
tff(fact_2352_add__minus__cancel,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),A2)),B2)) = B2 ) ).

% add_minus_cancel
tff(fact_2353_mult__minus__left,axiom,
    ! [A: $tType] :
      ( ring(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),A2)),B2) = aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)) ) ).

% mult_minus_left
tff(fact_2354_minus__mult__minus,axiom,
    ! [A: $tType] :
      ( ring(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),A2)),aa(A,A,uminus_uminus(A),B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2) ) ).

% minus_mult_minus
tff(fact_2355_mult__minus__right,axiom,
    ! [A: $tType] :
      ( ring(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(A,A,uminus_uminus(A),B2)) = aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)) ) ).

% mult_minus_right
tff(fact_2356_minus__diff__eq,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A,B2: A] : aa(A,A,uminus_uminus(A),aa(A,A,minus_minus(A,A2),B2)) = aa(A,A,minus_minus(A,B2),A2) ) ).

% minus_diff_eq
tff(fact_2357_div__minus__minus,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,uminus_uminus(A),A2)),aa(A,A,uminus_uminus(A),B2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2) ) ).

% div_minus_minus
tff(fact_2358_abs__0,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ( aa(A,A,abs_abs(A),zero_zero(A)) = zero_zero(A) ) ) ).

% abs_0
tff(fact_2359_abs__0__eq,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A] :
          ( ( zero_zero(A) = aa(A,A,abs_abs(A),A2) )
        <=> ( A2 = zero_zero(A) ) ) ) ).

% abs_0_eq
tff(fact_2360_abs__eq__0,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A] :
          ( ( aa(A,A,abs_abs(A),A2) = zero_zero(A) )
        <=> ( A2 = zero_zero(A) ) ) ) ).

% abs_eq_0
tff(fact_2361_abs__zero,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ( aa(A,A,abs_abs(A),zero_zero(A)) = zero_zero(A) ) ) ).

% abs_zero
tff(fact_2362_dvd__minus__iff,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,dvd_dvd(A,X),aa(A,A,uminus_uminus(A),Y)))
        <=> pp(aa(A,bool,dvd_dvd(A,X),Y)) ) ) ).

% dvd_minus_iff
tff(fact_2363_minus__dvd__iff,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,dvd_dvd(A,aa(A,A,uminus_uminus(A),X)),Y))
        <=> pp(aa(A,bool,dvd_dvd(A,X),Y)) ) ) ).

% minus_dvd_iff
tff(fact_2364_abs__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [N: num] : aa(A,A,abs_abs(A),aa(num,A,numeral_numeral(A),N)) = aa(num,A,numeral_numeral(A),N) ) ).

% abs_numeral
tff(fact_2365_abs__add__abs,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A,B2: A] : aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,abs_abs(A),A2)),aa(A,A,abs_abs(A),B2))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,abs_abs(A),A2)),aa(A,A,abs_abs(A),B2)) ) ).

% abs_add_abs
tff(fact_2366_abs__1,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ( aa(A,A,abs_abs(A),one_one(A)) = one_one(A) ) ) ).

% abs_1
tff(fact_2367_mod__minus__minus,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [A2: A,B2: A] : modulo_modulo(A,aa(A,A,uminus_uminus(A),A2),aa(A,A,uminus_uminus(A),B2)) = aa(A,A,uminus_uminus(A),modulo_modulo(A,A2,B2)) ) ).

% mod_minus_minus
tff(fact_2368_abs__mult__self__eq,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,abs_abs(A),A2)),aa(A,A,abs_abs(A),A2)) = aa(A,A,aa(A,fun(A,A),times_times(A),A2),A2) ) ).

% abs_mult_self_eq
tff(fact_2369_abs__divide,axiom,
    ! [A: $tType] :
      ( field_abs_sgn(A)
     => ! [A2: A,B2: A] : aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,abs_abs(A),A2)),aa(A,A,abs_abs(A),B2)) ) ).

% abs_divide
tff(fact_2370_abs__minus,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [A2: A] : aa(A,A,abs_abs(A),aa(A,A,uminus_uminus(A),A2)) = aa(A,A,abs_abs(A),A2) ) ).

% abs_minus
tff(fact_2371_abs__minus__cancel,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A] : aa(A,A,abs_abs(A),aa(A,A,uminus_uminus(A),A2)) = aa(A,A,abs_abs(A),A2) ) ).

% abs_minus_cancel
tff(fact_2372_abs__dvd__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: A,K: A] :
          ( pp(aa(A,bool,dvd_dvd(A,aa(A,A,abs_abs(A),M)),K))
        <=> pp(aa(A,bool,dvd_dvd(A,M),K)) ) ) ).

% abs_dvd_iff
tff(fact_2373_dvd__abs__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: A,K: A] :
          ( pp(aa(A,bool,dvd_dvd(A,M),aa(A,A,abs_abs(A),K)))
        <=> pp(aa(A,bool,dvd_dvd(A,M),K)) ) ) ).

% dvd_abs_iff
tff(fact_2374_abs__of__nat,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [N: nat] : aa(A,A,abs_abs(A),aa(nat,A,semiring_1_of_nat(A),N)) = aa(nat,A,semiring_1_of_nat(A),N) ) ).

% abs_of_nat
tff(fact_2375_subset__mset_Oadd__eq__0__iff__both__eq__0,axiom,
    ! [A: $tType,X: multiset(A),Y: multiset(A)] :
      ( ( aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),X),Y) = zero_zero(multiset(A)) )
    <=> ( ( X = zero_zero(multiset(A)) )
        & ( Y = zero_zero(multiset(A)) ) ) ) ).

% subset_mset.add_eq_0_iff_both_eq_0
tff(fact_2376_subset__mset_Ozero__eq__add__iff__both__eq__0,axiom,
    ! [A: $tType,X: multiset(A),Y: multiset(A)] :
      ( ( zero_zero(multiset(A)) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),X),Y) )
    <=> ( ( X = zero_zero(multiset(A)) )
        & ( Y = zero_zero(multiset(A)) ) ) ) ).

% subset_mset.zero_eq_add_iff_both_eq_0
tff(fact_2377_empty__eq__union,axiom,
    ! [A: $tType,M7: multiset(A),N4: multiset(A)] :
      ( ( zero_zero(multiset(A)) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),M7),N4) )
    <=> ( ( M7 = zero_zero(multiset(A)) )
        & ( N4 = zero_zero(multiset(A)) ) ) ) ).

% empty_eq_union
tff(fact_2378_union__eq__empty,axiom,
    ! [A: $tType,M7: multiset(A),N4: multiset(A)] :
      ( ( aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),M7),N4) = zero_zero(multiset(A)) )
    <=> ( ( M7 = zero_zero(multiset(A)) )
        & ( N4 = zero_zero(multiset(A)) ) ) ) ).

% union_eq_empty
tff(fact_2379_diff__diff__add__mset,axiom,
    ! [A: $tType,M7: multiset(A),N4: multiset(A),P: multiset(A)] : aa(multiset(A),multiset(A),minus_minus(multiset(A),aa(multiset(A),multiset(A),minus_minus(multiset(A),M7),N4)),P) = aa(multiset(A),multiset(A),minus_minus(multiset(A),M7),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),N4),P)) ).

% diff_diff_add_mset
tff(fact_2380_tanh__0,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ( aa(A,A,tanh(A),zero_zero(A)) = zero_zero(A) ) ) ).

% tanh_0
tff(fact_2381_tanh__real__zero__iff,axiom,
    ! [X: real] :
      ( ( aa(real,real,tanh(real),X) = zero_zero(real) )
    <=> ( X = zero_zero(real) ) ) ).

% tanh_real_zero_iff
tff(fact_2382_tanh__real__le__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa(real,bool,ord_less_eq(real,aa(real,real,tanh(real),X)),aa(real,real,tanh(real),Y)))
    <=> pp(aa(real,bool,ord_less_eq(real,X),Y)) ) ).

% tanh_real_le_iff
tff(fact_2383_neg__0__le__iff__le,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A] :
          ( pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),aa(A,A,uminus_uminus(A),A2)))
        <=> pp(aa(A,bool,ord_less_eq(A,A2),zero_zero(A))) ) ) ).

% neg_0_le_iff_le
tff(fact_2384_neg__le__0__iff__le,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A] :
          ( pp(aa(A,bool,ord_less_eq(A,aa(A,A,uminus_uminus(A),A2)),zero_zero(A)))
        <=> pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),A2)) ) ) ).

% neg_le_0_iff_le
tff(fact_2385_less__eq__neg__nonpos,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A2: A] :
          ( pp(aa(A,bool,ord_less_eq(A,A2),aa(A,A,uminus_uminus(A),A2)))
        <=> pp(aa(A,bool,ord_less_eq(A,A2),zero_zero(A))) ) ) ).

% less_eq_neg_nonpos
tff(fact_2386_neg__less__eq__nonneg,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A2: A] :
          ( pp(aa(A,bool,ord_less_eq(A,aa(A,A,uminus_uminus(A),A2)),A2))
        <=> pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),A2)) ) ) ).

% neg_less_eq_nonneg
tff(fact_2387_less__neg__neg,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A2: A] :
          ( pp(aa(A,bool,ord_less(A,A2),aa(A,A,uminus_uminus(A),A2)))
        <=> pp(aa(A,bool,ord_less(A,A2),zero_zero(A))) ) ) ).

% less_neg_neg
tff(fact_2388_neg__less__pos,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A2: A] :
          ( pp(aa(A,bool,ord_less(A,aa(A,A,uminus_uminus(A),A2)),A2))
        <=> pp(aa(A,bool,ord_less(A,zero_zero(A)),A2)) ) ) ).

% neg_less_pos
tff(fact_2389_neg__0__less__iff__less,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A] :
          ( pp(aa(A,bool,ord_less(A,zero_zero(A)),aa(A,A,uminus_uminus(A),A2)))
        <=> pp(aa(A,bool,ord_less(A,A2),zero_zero(A))) ) ) ).

% neg_0_less_iff_less
tff(fact_2390_neg__less__0__iff__less,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A] :
          ( pp(aa(A,bool,ord_less(A,aa(A,A,uminus_uminus(A),A2)),zero_zero(A)))
        <=> pp(aa(A,bool,ord_less(A,zero_zero(A)),A2)) ) ) ).

% neg_less_0_iff_less
tff(fact_2391_add_Oright__inverse,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa(A,A,uminus_uminus(A),A2)) = zero_zero(A) ) ).

% add.right_inverse
tff(fact_2392_ab__left__minus,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),A2)),A2) = zero_zero(A) ) ).

% ab_left_minus
tff(fact_2393_diff__0,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A] : aa(A,A,minus_minus(A,zero_zero(A)),A2) = aa(A,A,uminus_uminus(A),A2) ) ).

% diff_0
tff(fact_2394_verit__minus__simplify_I3_J,axiom,
    ! [B: $tType] :
      ( group_add(B)
     => ! [B2: B] : aa(B,B,minus_minus(B,zero_zero(B)),B2) = aa(B,B,uminus_uminus(B),B2) ) ).

% verit_minus_simplify(3)
tff(fact_2395_add__neg__numeral__simps_I3_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [M: num,N: num] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N))) = aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),M)),aa(num,A,numeral_numeral(A),N))) ) ).

% add_neg_numeral_simps(3)
tff(fact_2396_mult__minus1__right,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Z: A] : aa(A,A,aa(A,fun(A,A),times_times(A),Z),aa(A,A,uminus_uminus(A),one_one(A))) = aa(A,A,uminus_uminus(A),Z) ) ).

% mult_minus1_right
tff(fact_2397_mult__minus1,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Z: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),one_one(A))),Z) = aa(A,A,uminus_uminus(A),Z) ) ).

% mult_minus1
tff(fact_2398_diff__minus__eq__add,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A,B2: A] : aa(A,A,minus_minus(A,A2),aa(A,A,uminus_uminus(A),B2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2) ) ).

% diff_minus_eq_add
tff(fact_2399_uminus__add__conv__diff,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),A2)),B2) = aa(A,A,minus_minus(A,B2),A2) ) ).

% uminus_add_conv_diff
tff(fact_2400_divide__minus1,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),X),aa(A,A,uminus_uminus(A),one_one(A))) = aa(A,A,uminus_uminus(A),X) ) ).

% divide_minus1
tff(fact_2401_div__minus1__right,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(A,A,uminus_uminus(A),one_one(A))) = aa(A,A,uminus_uminus(A),A2) ) ).

% div_minus1_right
tff(fact_2402_abs__of__nonneg,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A] :
          ( pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),A2))
         => ( aa(A,A,abs_abs(A),A2) = A2 ) ) ) ).

% abs_of_nonneg
tff(fact_2403_abs__le__self__iff,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A] :
          ( pp(aa(A,bool,ord_less_eq(A,aa(A,A,abs_abs(A),A2)),A2))
        <=> pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),A2)) ) ) ).

% abs_le_self_iff
tff(fact_2404_abs__le__zero__iff,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A] :
          ( pp(aa(A,bool,ord_less_eq(A,aa(A,A,abs_abs(A),A2)),zero_zero(A)))
        <=> ( A2 = zero_zero(A) ) ) ) ).

% abs_le_zero_iff
tff(fact_2405_zero__less__abs__iff,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A] :
          ( pp(aa(A,bool,ord_less(A,zero_zero(A)),aa(A,A,abs_abs(A),A2)))
        <=> ( A2 != zero_zero(A) ) ) ) ).

% zero_less_abs_iff
tff(fact_2406_minus__mod__self1,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [B2: A,A2: A] : modulo_modulo(A,aa(A,A,minus_minus(A,B2),A2),B2) = modulo_modulo(A,aa(A,A,uminus_uminus(A),A2),B2) ) ).

% minus_mod_self1
tff(fact_2407_abs__neg__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [N: num] : aa(A,A,abs_abs(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N))) = aa(num,A,numeral_numeral(A),N) ) ).

% abs_neg_numeral
tff(fact_2408_abs__neg__one,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ( aa(A,A,abs_abs(A),aa(A,A,uminus_uminus(A),one_one(A))) = one_one(A) ) ) ).

% abs_neg_one
tff(fact_2409_abs__power__minus,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,N: nat] : aa(A,A,abs_abs(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),A2)),N)) = aa(A,A,abs_abs(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),N)) ) ).

% abs_power_minus
tff(fact_2410_of__real__eq__0__iff,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [X: real] :
          ( ( aa(real,A,real_Vector_of_real(A),X) = zero_zero(A) )
        <=> ( X = zero_zero(real) ) ) ) ).

% of_real_eq_0_iff
tff(fact_2411_of__real__0,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ( aa(real,A,real_Vector_of_real(A),zero_zero(real)) = zero_zero(A) ) ) ).

% of_real_0
tff(fact_2412_of__real__eq__1__iff,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [X: real] :
          ( ( aa(real,A,real_Vector_of_real(A),X) = one_one(A) )
        <=> ( X = one_one(real) ) ) ) ).

% of_real_eq_1_iff
tff(fact_2413_of__real__1,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ( aa(real,A,real_Vector_of_real(A),one_one(real)) = one_one(A) ) ) ).

% of_real_1
tff(fact_2414_of__real__numeral,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [W: num] : aa(real,A,real_Vector_of_real(A),aa(num,real,numeral_numeral(real),W)) = aa(num,A,numeral_numeral(A),W) ) ).

% of_real_numeral
tff(fact_2415_of__real__mult,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [X: real,Y: real] : aa(real,A,real_Vector_of_real(A),aa(real,real,aa(real,fun(real,real),times_times(real),X),Y)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(real,A,real_Vector_of_real(A),X)),aa(real,A,real_Vector_of_real(A),Y)) ) ).

% of_real_mult
tff(fact_2416_of__real__divide,axiom,
    ! [A: $tType] :
      ( real_V5047593784448816457lgebra(A)
     => ! [X: real,Y: real] : aa(real,A,real_Vector_of_real(A),aa(real,real,aa(real,fun(real,real),divide_divide(real),X),Y)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(real,A,real_Vector_of_real(A),X)),aa(real,A,real_Vector_of_real(A),Y)) ) ).

% of_real_divide
tff(fact_2417_of__real__add,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [X: real,Y: real] : aa(real,A,real_Vector_of_real(A),aa(real,real,aa(real,fun(real,real),plus_plus(real),X),Y)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(real,A,real_Vector_of_real(A),X)),aa(real,A,real_Vector_of_real(A),Y)) ) ).

% of_real_add
tff(fact_2418_of__real__power,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [X: real,N: nat] : aa(real,A,real_Vector_of_real(A),aa(nat,real,aa(real,fun(nat,real),power_power(real),X),N)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(real,A,real_Vector_of_real(A),X)),N) ) ).

% of_real_power
tff(fact_2419_real__add__minus__iff,axiom,
    ! [X: real,A2: real] :
      ( ( aa(real,real,aa(real,fun(real,real),plus_plus(real),X),aa(real,real,uminus_uminus(real),A2)) = zero_zero(real) )
    <=> ( X = A2 ) ) ).

% real_add_minus_iff
tff(fact_2420_of__real__of__nat__eq,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [N: nat] : aa(real,A,real_Vector_of_real(A),aa(nat,real,semiring_1_of_nat(real),N)) = aa(nat,A,semiring_1_of_nat(A),N) ) ).

% of_real_of_nat_eq
tff(fact_2421_tanh__real__pos__iff,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,ord_less(real,zero_zero(real)),aa(real,real,tanh(real),X)))
    <=> pp(aa(real,bool,ord_less(real,zero_zero(real)),X)) ) ).

% tanh_real_pos_iff
tff(fact_2422_tanh__real__neg__iff,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,ord_less(real,aa(real,real,tanh(real),X)),zero_zero(real)))
    <=> pp(aa(real,bool,ord_less(real,X),zero_zero(real))) ) ).

% tanh_real_neg_iff
tff(fact_2423_tanh__real__nonpos__iff,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,ord_less_eq(real,aa(real,real,tanh(real),X)),zero_zero(real)))
    <=> pp(aa(real,bool,ord_less_eq(real,X),zero_zero(real))) ) ).

% tanh_real_nonpos_iff
tff(fact_2424_tanh__real__nonneg__iff,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,ord_less_eq(real,zero_zero(real)),aa(real,real,tanh(real),X)))
    <=> pp(aa(real,bool,ord_less_eq(real,zero_zero(real)),X)) ) ).

% tanh_real_nonneg_iff
tff(fact_2425_add__neg__numeral__special_I8_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),one_one(A))),one_one(A)) = zero_zero(A) ) ) ).

% add_neg_numeral_special(8)
tff(fact_2426_add__neg__numeral__special_I7_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(A,A,uminus_uminus(A),one_one(A))) = zero_zero(A) ) ) ).

% add_neg_numeral_special(7)
tff(fact_2427_diff__numeral__special_I12_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ( aa(A,A,minus_minus(A,aa(A,A,uminus_uminus(A),one_one(A))),aa(A,A,uminus_uminus(A),one_one(A))) = zero_zero(A) ) ) ).

% diff_numeral_special(12)
tff(fact_2428_neg__one__eq__numeral__iff,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [N: num] :
          ( ( aa(A,A,uminus_uminus(A),one_one(A)) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N)) )
        <=> ( N = one2 ) ) ) ).

% neg_one_eq_numeral_iff
tff(fact_2429_numeral__eq__neg__one__iff,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [N: num] :
          ( ( aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N)) = aa(A,A,uminus_uminus(A),one_one(A)) )
        <=> ( N = one2 ) ) ) ).

% numeral_eq_neg_one_iff
tff(fact_2430_minus__one__mult__self,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [N: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),N)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),N)) = one_one(A) ) ).

% minus_one_mult_self
tff(fact_2431_left__minus__one__mult__self,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [N: nat,A2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),N)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),N)),A2)) = A2 ) ).

% left_minus_one_mult_self
tff(fact_2432_mod__minus1__right,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [A2: A] : modulo_modulo(A,A2,aa(A,A,uminus_uminus(A),one_one(A))) = zero_zero(A) ) ).

% mod_minus1_right
tff(fact_2433_divide__le__0__abs__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(A,A,abs_abs(A),B2))),zero_zero(A)))
        <=> ( pp(aa(A,bool,ord_less_eq(A,A2),zero_zero(A)))
            | ( B2 = zero_zero(A) ) ) ) ) ).

% divide_le_0_abs_iff
tff(fact_2434_zero__le__divide__abs__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(A,A,abs_abs(A),B2))))
        <=> ( pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),A2))
            | ( B2 = zero_zero(A) ) ) ) ) ).

% zero_le_divide_abs_iff
tff(fact_2435_abs__of__nonpos,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A] :
          ( pp(aa(A,bool,ord_less_eq(A,A2),zero_zero(A)))
         => ( aa(A,A,abs_abs(A),A2) = aa(A,A,uminus_uminus(A),A2) ) ) ) ).

% abs_of_nonpos
tff(fact_2436_norm__neg__numeral,axiom,
    ! [A: $tType] :
      ( real_V2822296259951069270ebra_1(A)
     => ! [W: num] : real_V7770717601297561774m_norm(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))) = aa(num,real,numeral_numeral(real),W) ) ).

% norm_neg_numeral
tff(fact_2437_semiring__norm_I168_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [V: num,W: num,Y: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))),Y)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),plus_plus(num),V),W)))),Y) ) ).

% semiring_norm(168)
tff(fact_2438_ceiling__neg__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V: num] : archimedean_ceiling(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))) = aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V)) ) ).

% ceiling_neg_numeral
tff(fact_2439_diff__numeral__simps_I3_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [M: num,N: num] : aa(A,A,minus_minus(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))),aa(num,A,numeral_numeral(A),N)) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),plus_plus(num),M),N))) ) ).

% diff_numeral_simps(3)
tff(fact_2440_diff__numeral__simps_I2_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [M: num,N: num] : aa(A,A,minus_minus(A,aa(num,A,numeral_numeral(A),M)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N))) = aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),plus_plus(num),M),N)) ) ).

% diff_numeral_simps(2)
tff(fact_2441_mult__neg__numeral__simps_I3_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [M: num,N: num] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),M)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N))) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),times_times(num),M),N))) ) ).

% mult_neg_numeral_simps(3)
tff(fact_2442_mult__neg__numeral__simps_I2_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [M: num,N: num] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))),aa(num,A,numeral_numeral(A),N)) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),times_times(num),M),N))) ) ).

% mult_neg_numeral_simps(2)
tff(fact_2443_mult__neg__numeral__simps_I1_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [M: num,N: num] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N))) = aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),times_times(num),M),N)) ) ).

% mult_neg_numeral_simps(1)
tff(fact_2444_semiring__norm_I172_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [V: num,W: num,Y: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))),Y)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),times_times(num),V),W))),Y) ) ).

% semiring_norm(172)
tff(fact_2445_semiring__norm_I171_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [V: num,W: num,Y: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),V)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))),Y)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),times_times(num),V),W)))),Y) ) ).

% semiring_norm(171)
tff(fact_2446_semiring__norm_I170_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [V: num,W: num,Y: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W)),Y)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),times_times(num),V),W)))),Y) ) ).

% semiring_norm(170)
tff(fact_2447_artanh__minus__real,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,ord_less(real,aa(real,real,abs_abs(real),X)),one_one(real)))
     => ( aa(real,real,artanh(real),aa(real,real,uminus_uminus(real),X)) = aa(real,real,uminus_uminus(real),aa(real,real,artanh(real),X)) ) ) ).

% artanh_minus_real
tff(fact_2448_neg__numeral__le__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num,N: num] :
          ( pp(aa(A,bool,ord_less_eq(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N))))
        <=> pp(aa(num,bool,ord_less_eq(num,N),M)) ) ) ).

% neg_numeral_le_iff
tff(fact_2449_neg__numeral__less__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num,N: num] :
          ( pp(aa(A,bool,ord_less(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N))))
        <=> pp(aa(num,bool,ord_less(num,N),M)) ) ) ).

% neg_numeral_less_iff
tff(fact_2450_round__neg__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [N: num] : archimedean_round(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N))) = aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),N)) ) ).

% round_neg_numeral
tff(fact_2451_not__neg__one__le__neg__numeral__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num] :
          ( ~ pp(aa(A,bool,ord_less_eq(A,aa(A,A,uminus_uminus(A),one_one(A))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))))
        <=> ( M != one2 ) ) ) ).

% not_neg_one_le_neg_numeral_iff
tff(fact_2452_neg__numeral__less__neg__one__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num] :
          ( pp(aa(A,bool,ord_less(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))),aa(A,A,uminus_uminus(A),one_one(A))))
        <=> ( M != one2 ) ) ) ).

% neg_numeral_less_neg_one_iff
tff(fact_2453_divide__eq__eq__numeral1_I2_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B2: A,W: num,A2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))) = A2 )
        <=> ( ( ( aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W)) != zero_zero(A) )
             => ( B2 = aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))) ) )
            & ( ( aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W)) = zero_zero(A) )
             => ( A2 = zero_zero(A) ) ) ) ) ) ).

% divide_eq_eq_numeral1(2)
tff(fact_2454_eq__divide__eq__numeral1_I2_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,B2: A,W: num] :
          ( ( A2 = aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))) )
        <=> ( ( ( aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W)) != zero_zero(A) )
             => ( aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))) = B2 ) )
            & ( ( aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W)) = zero_zero(A) )
             => ( A2 = zero_zero(A) ) ) ) ) ) ).

% eq_divide_eq_numeral1(2)
tff(fact_2455_divide__le__eq__numeral1_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,W: num,A2: A] :
          ( pp(aa(A,bool,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W)))),A2))
        <=> pp(aa(A,bool,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W)))),B2)) ) ) ).

% divide_le_eq_numeral1(2)
tff(fact_2456_le__divide__eq__numeral1_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A,W: num] :
          ( pp(aa(A,bool,ord_less_eq(A,A2),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W)))))
        <=> pp(aa(A,bool,ord_less_eq(A,B2),aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))))) ) ) ).

% le_divide_eq_numeral1(2)
tff(fact_2457_divide__less__eq__numeral1_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,W: num,A2: A] :
          ( pp(aa(A,bool,ord_less(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W)))),A2))
        <=> pp(aa(A,bool,ord_less(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W)))),B2)) ) ) ).

% divide_less_eq_numeral1(2)
tff(fact_2458_less__divide__eq__numeral1_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A,W: num] :
          ( pp(aa(A,bool,ord_less(A,A2),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W)))))
        <=> pp(aa(A,bool,ord_less(A,B2),aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))))) ) ) ).

% less_divide_eq_numeral1(2)
tff(fact_2459_power2__minus,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [A2: A] : aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),A2)),aa(num,nat,numeral_numeral(nat),bit0(one2))) = aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(num,nat,numeral_numeral(nat),bit0(one2))) ) ).

% power2_minus
tff(fact_2460_zero__less__power__abs__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,N: nat] :
          ( pp(aa(A,bool,ord_less(A,zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,abs_abs(A),A2)),N)))
        <=> ( ( A2 != zero_zero(A) )
            | ( N = zero_zero(nat) ) ) ) ) ).

% zero_less_power_abs_iff
tff(fact_2461_abs__power2,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A] : aa(A,A,abs_abs(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(num,nat,numeral_numeral(nat),bit0(one2))) ) ).

% abs_power2
tff(fact_2462_power2__abs,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A] : aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,abs_abs(A),A2)),aa(num,nat,numeral_numeral(nat),bit0(one2))) = aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(num,nat,numeral_numeral(nat),bit0(one2))) ) ).

% power2_abs
tff(fact_2463_of__real__neg__numeral,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [W: num] : aa(real,A,real_Vector_of_real(A),aa(real,real,uminus_uminus(real),aa(num,real,numeral_numeral(real),W))) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W)) ) ).

% of_real_neg_numeral
tff(fact_2464_add__neg__numeral__special_I9_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),one_one(A))),aa(A,A,uminus_uminus(A),one_one(A))) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),bit0(one2))) ) ) ).

% add_neg_numeral_special(9)
tff(fact_2465_diff__numeral__special_I11_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ( aa(A,A,minus_minus(A,one_one(A)),aa(A,A,uminus_uminus(A),one_one(A))) = aa(num,A,numeral_numeral(A),bit0(one2)) ) ) ).

% diff_numeral_special(11)
tff(fact_2466_diff__numeral__special_I10_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ( aa(A,A,minus_minus(A,aa(A,A,uminus_uminus(A),one_one(A))),one_one(A)) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),bit0(one2))) ) ) ).

% diff_numeral_special(10)
tff(fact_2467_minus__1__div__2__eq,axiom,
    ! [A: $tType] :
      ( euclid8789492081693882211th_nat(A)
     => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,uminus_uminus(A),one_one(A))),aa(num,A,numeral_numeral(A),bit0(one2))) = aa(A,A,uminus_uminus(A),one_one(A)) ) ) ).

% minus_1_div_2_eq
tff(fact_2468_bits__minus__1__mod__2__eq,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ( modulo_modulo(A,aa(A,A,uminus_uminus(A),one_one(A)),aa(num,A,numeral_numeral(A),bit0(one2))) = one_one(A) ) ) ).

% bits_minus_1_mod_2_eq
tff(fact_2469_minus__1__mod__2__eq,axiom,
    ! [A: $tType] :
      ( euclid8789492081693882211th_nat(A)
     => ( modulo_modulo(A,aa(A,A,uminus_uminus(A),one_one(A)),aa(num,A,numeral_numeral(A),bit0(one2))) = one_one(A) ) ) ).

% minus_1_mod_2_eq
tff(fact_2470_Power_Oring__1__class_Opower__minus__even,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [A2: A,N: nat] : aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),A2)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)) ) ).

% Power.ring_1_class.power_minus_even
tff(fact_2471_power__minus__odd,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [N: nat,A2: A] :
          ( ~ pp(aa(nat,bool,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N))
         => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),A2)),N) = aa(A,A,uminus_uminus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),N)) ) ) ) ).

% power_minus_odd
tff(fact_2472_Parity_Oring__1__class_Opower__minus__even,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [N: nat,A2: A] :
          ( pp(aa(nat,bool,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N))
         => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),A2)),N) = aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),N) ) ) ) ).

% Parity.ring_1_class.power_minus_even
tff(fact_2473_power__even__abs__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [W: num,A2: A] :
          ( pp(aa(nat,bool,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(num,nat,numeral_numeral(nat),W)))
         => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,abs_abs(A),A2)),aa(num,nat,numeral_numeral(nat),W)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(num,nat,numeral_numeral(nat),W)) ) ) ) ).

% power_even_abs_numeral
tff(fact_2474_diff__numeral__special_I3_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [N: num] : aa(A,A,minus_minus(A,one_one(A)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N))) = aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),plus_plus(num),one2),N)) ) ).

% diff_numeral_special(3)
tff(fact_2475_diff__numeral__special_I4_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [M: num] : aa(A,A,minus_minus(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))),one_one(A)) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),plus_plus(num),M),one2))) ) ).

% diff_numeral_special(4)
tff(fact_2476_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [Y: int,X: num,N: nat] :
          ( ( aa(int,A,ring_1_of_int(A),Y) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),X))),N) )
        <=> ( Y = aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),X))),N) ) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
tff(fact_2477_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [X: num,N: nat,Y: int] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),X))),N) = aa(int,A,ring_1_of_int(A),Y) )
        <=> ( aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),X))),N) = Y ) ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
tff(fact_2478_ceiling__le__neg__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,V: num] :
          ( pp(aa(int,bool,ord_less_eq(int,archimedean_ceiling(A,X)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V))))
        <=> pp(aa(A,bool,ord_less_eq(A,X),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V)))) ) ) ).

% ceiling_le_neg_numeral
tff(fact_2479_neg__numeral__less__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V: num,X: A] :
          ( pp(aa(int,bool,ord_less(int,aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V))),archimedean_ceiling(A,X)))
        <=> pp(aa(A,bool,ord_less(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))),X)) ) ) ).

% neg_numeral_less_ceiling
tff(fact_2480_ceiling__less__zero,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( pp(aa(int,bool,ord_less(int,archimedean_ceiling(A,X)),zero_zero(int)))
        <=> pp(aa(A,bool,ord_less_eq(A,X),aa(A,A,uminus_uminus(A),one_one(A)))) ) ) ).

% ceiling_less_zero
tff(fact_2481_zero__le__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( pp(aa(int,bool,ord_less_eq(int,zero_zero(int)),archimedean_ceiling(A,X)))
        <=> pp(aa(A,bool,ord_less(A,aa(A,A,uminus_uminus(A),one_one(A))),X)) ) ) ).

% zero_le_ceiling
tff(fact_2482_power__minus1__even,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [N: nat] : aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)) = one_one(A) ) ).

% power_minus1_even
tff(fact_2483_neg__one__odd__power,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [N: nat] :
          ( ~ pp(aa(nat,bool,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N))
         => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),N) = aa(A,A,uminus_uminus(A),one_one(A)) ) ) ) ).

% neg_one_odd_power
tff(fact_2484_neg__one__even__power,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [N: nat] :
          ( pp(aa(nat,bool,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N))
         => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),N) = one_one(A) ) ) ) ).

% neg_one_even_power
tff(fact_2485_norm__of__real__add1,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [X: real] : real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(real,A,real_Vector_of_real(A),X)),one_one(A))) = aa(real,real,abs_abs(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),X),one_one(real))) ) ).

% norm_of_real_add1
tff(fact_2486_norm__of__real__addn,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [X: real,B2: num] : real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(real,A,real_Vector_of_real(A),X)),aa(num,A,numeral_numeral(A),B2))) = aa(real,real,abs_abs(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),X),aa(num,real,numeral_numeral(real),B2))) ) ).

% norm_of_real_addn
tff(fact_2487_ceiling__less__neg__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,V: num] :
          ( pp(aa(int,bool,ord_less(int,archimedean_ceiling(A,X)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V))))
        <=> pp(aa(A,bool,ord_less_eq(A,X),aa(A,A,minus_minus(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))),one_one(A)))) ) ) ).

% ceiling_less_neg_numeral
tff(fact_2488_neg__numeral__le__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V: num,X: A] :
          ( pp(aa(int,bool,ord_less_eq(int,aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V))),archimedean_ceiling(A,X)))
        <=> pp(aa(A,bool,ord_less(A,aa(A,A,minus_minus(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))),one_one(A))),X)) ) ) ).

% neg_numeral_le_ceiling
tff(fact_2489_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: num,N: nat,A2: int] :
          ( pp(aa(A,bool,ord_less_eq(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),X))),N)),aa(int,A,ring_1_of_int(A),A2)))
        <=> pp(aa(int,bool,ord_less_eq(int,aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),X))),N)),A2)) ) ) ).

% neg_numeral_power_le_of_int_cancel_iff
tff(fact_2490_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: int,X: num,N: nat] :
          ( pp(aa(A,bool,ord_less_eq(A,aa(int,A,ring_1_of_int(A),A2)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),X))),N)))
        <=> pp(aa(int,bool,ord_less_eq(int,A2),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),X))),N))) ) ) ).

% of_int_le_neg_numeral_power_cancel_iff
tff(fact_2491_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: int,X: num,N: nat] :
          ( pp(aa(A,bool,ord_less(A,aa(int,A,ring_1_of_int(A),A2)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),X))),N)))
        <=> pp(aa(int,bool,ord_less(int,A2),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),X))),N))) ) ) ).

% of_int_less_neg_numeral_power_cancel_iff
tff(fact_2492_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: num,N: nat,A2: int] :
          ( pp(aa(A,bool,ord_less(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),X))),N)),aa(int,A,ring_1_of_int(A),A2)))
        <=> pp(aa(int,bool,ord_less(int,aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),X))),N)),A2)) ) ) ).

% neg_numeral_power_less_of_int_cancel_iff
tff(fact_2493_square__powr__half,axiom,
    ! [X: real] : powr(real,aa(nat,real,aa(real,fun(nat,real),power_power(real),X),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),bit0(one2)))) = aa(real,real,abs_abs(real),X) ).

% square_powr_half
tff(fact_2494_Multiset_Odiff__add,axiom,
    ! [A: $tType,M7: multiset(A),N4: multiset(A),Q: multiset(A)] : aa(multiset(A),multiset(A),minus_minus(multiset(A),M7),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),N4),Q)) = aa(multiset(A),multiset(A),minus_minus(multiset(A),aa(multiset(A),multiset(A),minus_minus(multiset(A),M7),N4)),Q) ).

% Multiset.diff_add
tff(fact_2495_diff__union__cancelL,axiom,
    ! [A: $tType,N4: multiset(A),M7: multiset(A)] : aa(multiset(A),multiset(A),minus_minus(multiset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),N4),M7)),N4) = M7 ).

% diff_union_cancelL
tff(fact_2496_diff__union__cancelR,axiom,
    ! [A: $tType,M7: multiset(A),N4: multiset(A)] : aa(multiset(A),multiset(A),minus_minus(multiset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),M7),N4)),N4) = M7 ).

% diff_union_cancelR
tff(fact_2497_Multiset_Odiff__cancel,axiom,
    ! [A: $tType,A3: multiset(A)] : aa(multiset(A),multiset(A),minus_minus(multiset(A),A3),A3) = zero_zero(multiset(A)) ).

% Multiset.diff_cancel
tff(fact_2498_diff__empty,axiom,
    ! [A: $tType,M7: multiset(A)] :
      ( ( aa(multiset(A),multiset(A),minus_minus(multiset(A),M7),zero_zero(multiset(A))) = M7 )
      & ( aa(multiset(A),multiset(A),minus_minus(multiset(A),zero_zero(multiset(A))),M7) = zero_zero(multiset(A)) ) ) ).

% diff_empty
tff(fact_2499_union__assoc,axiom,
    ! [A: $tType,M7: multiset(A),N4: multiset(A),K6: multiset(A)] : aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),M7),N4)),K6) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),M7),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),N4),K6)) ).

% union_assoc
tff(fact_2500_union__lcomm,axiom,
    ! [A: $tType,M7: multiset(A),N4: multiset(A),K6: multiset(A)] : aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),M7),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),N4),K6)) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),N4),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),M7),K6)) ).

% union_lcomm
tff(fact_2501_union__commute,axiom,
    ! [A: $tType,M7: multiset(A),N4: multiset(A)] : aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),M7),N4) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),N4),M7) ).

% union_commute
tff(fact_2502_union__left__cancel,axiom,
    ! [A: $tType,K6: multiset(A),M7: multiset(A),N4: multiset(A)] :
      ( ( aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),K6),M7) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),K6),N4) )
    <=> ( M7 = N4 ) ) ).

% union_left_cancel
tff(fact_2503_union__right__cancel,axiom,
    ! [A: $tType,M7: multiset(A),K6: multiset(A),N4: multiset(A)] :
      ( ( aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),M7),K6) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),N4),K6) )
    <=> ( M7 = N4 ) ) ).

% union_right_cancel
tff(fact_2504_multi__union__self__other__eq,axiom,
    ! [A: $tType,A3: multiset(A),X6: multiset(A),Y6: multiset(A)] :
      ( ( aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A3),X6) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A3),Y6) )
     => ( X6 = Y6 ) ) ).

% multi_union_self_other_eq
tff(fact_2505_empty__neutral_I2_J,axiom,
    ! [A: $tType,X: multiset(A)] : aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),X),zero_zero(multiset(A))) = X ).

% empty_neutral(2)
tff(fact_2506_empty__neutral_I1_J,axiom,
    ! [A: $tType,X: multiset(A)] : aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),zero_zero(multiset(A))),X) = X ).

% empty_neutral(1)
tff(fact_2507_union__le__mono1,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [B4: multiset(A),D4: multiset(A),C6: multiset(A)] :
          ( pp(aa(multiset(A),bool,ord_less(multiset(A),B4),D4))
         => pp(aa(multiset(A),bool,ord_less(multiset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),B4),C6)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),D4),C6))) ) ) ).

% union_le_mono1
tff(fact_2508_union__le__mono2,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [B4: multiset(A),D4: multiset(A),C6: multiset(A)] :
          ( pp(aa(multiset(A),bool,ord_less(multiset(A),B4),D4))
         => pp(aa(multiset(A),bool,ord_less(multiset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),C6),B4)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),C6),D4))) ) ) ).

% union_le_mono2
tff(fact_2509_union__less__mono,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A3: multiset(A),C6: multiset(A),B4: multiset(A),D4: multiset(A)] :
          ( pp(aa(multiset(A),bool,ord_less(multiset(A),A3),C6))
         => ( pp(aa(multiset(A),bool,ord_less(multiset(A),B4),D4))
           => pp(aa(multiset(A),bool,ord_less(multiset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A3),B4)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),C6),D4))) ) ) ) ).

% union_less_mono
tff(fact_2510_mset__distrib,axiom,
    ! [A: $tType,A3: multiset(A),B4: multiset(A),M7: multiset(A),N4: multiset(A)] :
      ( ( aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A3),B4) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),M7),N4) )
     => ~ ! [Am: multiset(A),An: multiset(A)] :
            ( ( A3 = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),Am),An) )
           => ! [Bm: multiset(A),Bn: multiset(A)] :
                ( ( B4 = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),Bm),Bn) )
               => ( ( M7 = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),Am),Bm) )
                 => ( N4 != aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),An),Bn) ) ) ) ) ) ).

% mset_distrib
tff(fact_2511_union__diff__assoc,axiom,
    ! [A: $tType,C6: multiset(A),B4: multiset(A),A3: multiset(A)] :
      ( ( aa(multiset(A),multiset(A),minus_minus(multiset(A),C6),B4) = zero_zero(multiset(A)) )
     => ( aa(multiset(A),multiset(A),minus_minus(multiset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A3),B4)),C6) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A3),aa(multiset(A),multiset(A),minus_minus(multiset(A),B4),C6)) ) ) ).

% union_diff_assoc
tff(fact_2512_abs__eq__iff,axiom,
    ! [A: $tType] :
      ( linordered_ring(A)
     => ! [X: A,Y: A] :
          ( ( aa(A,A,abs_abs(A),X) = aa(A,A,abs_abs(A),Y) )
        <=> ( ( X = Y )
            | ( X = aa(A,A,uminus_uminus(A),Y) ) ) ) ) ).

% abs_eq_iff
tff(fact_2513_equation__minus__iff,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A,B2: A] :
          ( ( A2 = aa(A,A,uminus_uminus(A),B2) )
        <=> ( B2 = aa(A,A,uminus_uminus(A),A2) ) ) ) ).

% equation_minus_iff
tff(fact_2514_minus__equation__iff,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A,B2: A] :
          ( ( aa(A,A,uminus_uminus(A),A2) = B2 )
        <=> ( aa(A,A,uminus_uminus(A),B2) = A2 ) ) ) ).

% minus_equation_iff
tff(fact_2515_abs__less__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,ord_less(A,aa(A,A,abs_abs(A),A2)),B2))
        <=> ( pp(aa(A,bool,ord_less(A,A2),B2))
            & pp(aa(A,bool,ord_less(A,aa(A,A,uminus_uminus(A),A2)),B2)) ) ) ) ).

% abs_less_iff
tff(fact_2516_abs__leI,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,ord_less_eq(A,A2),B2))
         => ( pp(aa(A,bool,ord_less_eq(A,aa(A,A,uminus_uminus(A),A2)),B2))
           => pp(aa(A,bool,ord_less_eq(A,aa(A,A,abs_abs(A),A2)),B2)) ) ) ) ).

% abs_leI
tff(fact_2517_abs__le__D2,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,ord_less_eq(A,aa(A,A,abs_abs(A),A2)),B2))
         => pp(aa(A,bool,ord_less_eq(A,aa(A,A,uminus_uminus(A),A2)),B2)) ) ) ).

% abs_le_D2
tff(fact_2518_abs__le__iff,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,ord_less_eq(A,aa(A,A,abs_abs(A),A2)),B2))
        <=> ( pp(aa(A,bool,ord_less_eq(A,A2),B2))
            & pp(aa(A,bool,ord_less_eq(A,aa(A,A,uminus_uminus(A),A2)),B2)) ) ) ) ).

% abs_le_iff
tff(fact_2519_abs__ge__minus__self,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A] : pp(aa(A,bool,ord_less_eq(A,aa(A,A,uminus_uminus(A),A2)),aa(A,A,abs_abs(A),A2))) ) ).

% abs_ge_minus_self
tff(fact_2520_abs__if__raw,axiom,
    ! [A: $tType] :
      ( abs_if(A)
     => ! [X4: A] :
          ( ( pp(aa(A,bool,ord_less(A,X4),zero_zero(A)))
           => ( aa(A,A,abs_abs(A),X4) = aa(A,A,uminus_uminus(A),X4) ) )
          & ( ~ pp(aa(A,bool,ord_less(A,X4),zero_zero(A)))
           => ( aa(A,A,abs_abs(A),X4) = X4 ) ) ) ) ).

% abs_if_raw
tff(fact_2521_abs__if,axiom,
    ! [A: $tType] :
      ( abs_if(A)
     => ! [A2: A] :
          ( ( pp(aa(A,bool,ord_less(A,A2),zero_zero(A)))
           => ( aa(A,A,abs_abs(A),A2) = aa(A,A,uminus_uminus(A),A2) ) )
          & ( ~ pp(aa(A,bool,ord_less(A,A2),zero_zero(A)))
           => ( aa(A,A,abs_abs(A),A2) = A2 ) ) ) ) ).

% abs_if
tff(fact_2522_abs__of__neg,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A] :
          ( pp(aa(A,bool,ord_less(A,A2),zero_zero(A)))
         => ( aa(A,A,abs_abs(A),A2) = aa(A,A,uminus_uminus(A),A2) ) ) ) ).

% abs_of_neg
tff(fact_2523_abs__minus__le__zero,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A] : pp(aa(A,bool,ord_less_eq(A,aa(A,A,uminus_uminus(A),aa(A,A,abs_abs(A),A2))),zero_zero(A))) ) ).

% abs_minus_le_zero
tff(fact_2524_eq__abs__iff_H,axiom,
    ! [A: $tType] :
      ( linordered_ring(A)
     => ! [A2: A,B2: A] :
          ( ( A2 = aa(A,A,abs_abs(A),B2) )
        <=> ( pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),A2))
            & ( ( B2 = A2 )
              | ( B2 = aa(A,A,uminus_uminus(A),A2) ) ) ) ) ) ).

% eq_abs_iff'
tff(fact_2525_abs__eq__iff_H,axiom,
    ! [A: $tType] :
      ( linordered_ring(A)
     => ! [A2: A,B2: A] :
          ( ( aa(A,A,abs_abs(A),A2) = B2 )
        <=> ( pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),B2))
            & ( ( A2 = B2 )
              | ( A2 = aa(A,A,uminus_uminus(A),B2) ) ) ) ) ) ).

% abs_eq_iff'
tff(fact_2526_abs__real__def,axiom,
    ! [A2: real] :
      ( ( pp(aa(real,bool,ord_less(real,A2),zero_zero(real)))
       => ( aa(real,real,abs_abs(real),A2) = aa(real,real,uminus_uminus(real),A2) ) )
      & ( ~ pp(aa(real,bool,ord_less(real,A2),zero_zero(real)))
       => ( aa(real,real,abs_abs(real),A2) = A2 ) ) ) ).

% abs_real_def
tff(fact_2527_abs__eq__0__iff,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [A2: A] :
          ( ( aa(A,A,abs_abs(A),A2) = zero_zero(A) )
        <=> ( A2 = zero_zero(A) ) ) ) ).

% abs_eq_0_iff
tff(fact_2528_abs__le__D1,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,ord_less_eq(A,aa(A,A,abs_abs(A),A2)),B2))
         => pp(aa(A,bool,ord_less_eq(A,A2),B2)) ) ) ).

% abs_le_D1
tff(fact_2529_abs__ge__self,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A] : pp(aa(A,bool,ord_less_eq(A,A2),aa(A,A,abs_abs(A),A2))) ) ).

% abs_ge_self
tff(fact_2530_abs__one,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ( aa(A,A,abs_abs(A),one_one(A)) = one_one(A) ) ) ).

% abs_one
tff(fact_2531_abs__mult,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [A2: A,B2: A] : aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,abs_abs(A),A2)),aa(A,A,abs_abs(A),B2)) ) ).

% abs_mult
tff(fact_2532_abs__minus__commute,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A,B2: A] : aa(A,A,abs_abs(A),aa(A,A,minus_minus(A,A2),B2)) = aa(A,A,abs_abs(A),aa(A,A,minus_minus(A,B2),A2)) ) ).

% abs_minus_commute
tff(fact_2533_power__abs,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,N: nat] : aa(A,A,abs_abs(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),N)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,abs_abs(A),A2)),N) ) ).

% power_abs
tff(fact_2534_tanh__real__gt__neg1,axiom,
    ! [X: real] : pp(aa(real,bool,ord_less(real,aa(real,real,uminus_uminus(real),one_one(real))),aa(real,real,tanh(real),X))) ).

% tanh_real_gt_neg1
tff(fact_2535_dvd__if__abs__eq,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [L: A,K: A] :
          ( ( aa(A,A,abs_abs(A),L) = aa(A,A,abs_abs(A),K) )
         => pp(aa(A,bool,dvd_dvd(A,L),K)) ) ) ).

% dvd_if_abs_eq
tff(fact_2536_le__minus__iff,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,ord_less_eq(A,A2),aa(A,A,uminus_uminus(A),B2)))
        <=> pp(aa(A,bool,ord_less_eq(A,B2),aa(A,A,uminus_uminus(A),A2))) ) ) ).

% le_minus_iff
tff(fact_2537_minus__le__iff,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,ord_less_eq(A,aa(A,A,uminus_uminus(A),A2)),B2))
        <=> pp(aa(A,bool,ord_less_eq(A,aa(A,A,uminus_uminus(A),B2)),A2)) ) ) ).

% minus_le_iff
tff(fact_2538_le__imp__neg__le,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,ord_less_eq(A,A2),B2))
         => pp(aa(A,bool,ord_less_eq(A,aa(A,A,uminus_uminus(A),B2)),aa(A,A,uminus_uminus(A),A2))) ) ) ).

% le_imp_neg_le
tff(fact_2539_less__minus__iff,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,ord_less(A,A2),aa(A,A,uminus_uminus(A),B2)))
        <=> pp(aa(A,bool,ord_less(A,B2),aa(A,A,uminus_uminus(A),A2))) ) ) ).

% less_minus_iff
tff(fact_2540_minus__less__iff,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,ord_less(A,aa(A,A,uminus_uminus(A),A2)),B2))
        <=> pp(aa(A,bool,ord_less(A,aa(A,A,uminus_uminus(A),B2)),A2)) ) ) ).

% minus_less_iff
tff(fact_2541_neg__numeral__neq__numeral,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [M: num,N: num] : aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M)) != aa(num,A,numeral_numeral(A),N) ) ).

% neg_numeral_neq_numeral
tff(fact_2542_numeral__neq__neg__numeral,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [M: num,N: num] : aa(num,A,numeral_numeral(A),M) != aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N)) ) ).

% numeral_neq_neg_numeral
tff(fact_2543_add_Oinverse__distrib__swap,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A,B2: A] : aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,uminus_uminus(A),A2)) ) ).

% add.inverse_distrib_swap
tff(fact_2544_group__cancel_Oneg1,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A3: A,K: A,A2: A] :
          ( ( A3 = aa(A,A,aa(A,fun(A,A),plus_plus(A),K),A2) )
         => ( aa(A,A,uminus_uminus(A),A3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),K)),aa(A,A,uminus_uminus(A),A2)) ) ) ) ).

% group_cancel.neg1
tff(fact_2545_is__num__normalize_I8_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [A2: A,B2: A] : aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,uminus_uminus(A),A2)) ) ).

% is_num_normalize(8)
tff(fact_2546_one__neq__neg__one,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ( one_one(A) != aa(A,A,uminus_uminus(A),one_one(A)) ) ) ).

% one_neq_neg_one
tff(fact_2547_square__eq__iff,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [A2: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),A2),A2) = aa(A,A,aa(A,fun(A,A),times_times(A),B2),B2) )
        <=> ( ( A2 = B2 )
            | ( A2 = aa(A,A,uminus_uminus(A),B2) ) ) ) ) ).

% square_eq_iff
tff(fact_2548_minus__mult__commute,axiom,
    ! [A: $tType] :
      ( ring(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),A2)),B2) = aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(A,A,uminus_uminus(A),B2)) ) ).

% minus_mult_commute
tff(fact_2549_minus__diff__commute,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [B2: A,A2: A] : aa(A,A,minus_minus(A,aa(A,A,uminus_uminus(A),B2)),A2) = aa(A,A,minus_minus(A,aa(A,A,uminus_uminus(A),A2)),B2) ) ).

% minus_diff_commute
tff(fact_2550_minus__diff__minus,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A2: A,B2: A] : aa(A,A,minus_minus(A,aa(A,A,uminus_uminus(A),A2)),aa(A,A,uminus_uminus(A),B2)) = aa(A,A,uminus_uminus(A),aa(A,A,minus_minus(A,A2),B2)) ) ).

% minus_diff_minus
tff(fact_2551_minus__divide__right,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A2: A,B2: A] : aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(A,A,uminus_uminus(A),B2)) ) ).

% minus_divide_right
tff(fact_2552_minus__divide__divide,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,uminus_uminus(A),A2)),aa(A,A,uminus_uminus(A),B2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2) ) ).

% minus_divide_divide
tff(fact_2553_minus__divide__left,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,B2: A] : aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,uminus_uminus(A),A2)),B2) ) ).

% minus_divide_left
tff(fact_2554_div__minus__right,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(A,A,uminus_uminus(A),B2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,uminus_uminus(A),A2)),B2) ) ).

% div_minus_right
tff(fact_2555_mod__minus__eq,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [A2: A,B2: A] : modulo_modulo(A,aa(A,A,uminus_uminus(A),modulo_modulo(A,A2,B2)),B2) = modulo_modulo(A,aa(A,A,uminus_uminus(A),A2),B2) ) ).

% mod_minus_eq
tff(fact_2556_euclidean__ring__cancel__class_Omod__minus__cong,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [A2: A,B2: A,A5: A] :
          ( ( modulo_modulo(A,A2,B2) = modulo_modulo(A,A5,B2) )
         => ( modulo_modulo(A,aa(A,A,uminus_uminus(A),A2),B2) = modulo_modulo(A,aa(A,A,uminus_uminus(A),A5),B2) ) ) ) ).

% euclidean_ring_cancel_class.mod_minus_cong
tff(fact_2557_mod__minus__right,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [A2: A,B2: A] : modulo_modulo(A,A2,aa(A,A,uminus_uminus(A),B2)) = aa(A,A,uminus_uminus(A),modulo_modulo(A,aa(A,A,uminus_uminus(A),A2),B2)) ) ).

% mod_minus_right
tff(fact_2558_complex__mod__minus__le__complex__mod,axiom,
    ! [X: complex] : pp(aa(real,bool,ord_less_eq(real,aa(real,real,uminus_uminus(real),real_V7770717601297561774m_norm(complex,X))),real_V7770717601297561774m_norm(complex,X))) ).

% complex_mod_minus_le_complex_mod
tff(fact_2559_of__int__neg__numeral,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [K: num] : aa(int,A,ring_1_of_int(A),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),K))) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),K)) ) ).

% of_int_neg_numeral
tff(fact_2560_norm__of__real__diff,axiom,
    ! [A: $tType] :
      ( real_V2822296259951069270ebra_1(A)
     => ! [B2: real,A2: real] : pp(aa(real,bool,ord_less_eq(real,real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,aa(real,A,real_Vector_of_real(A),B2)),aa(real,A,real_Vector_of_real(A),A2)))),aa(real,real,abs_abs(real),aa(real,real,minus_minus(real,B2),A2)))) ) ).

% norm_of_real_diff
tff(fact_2561_abs__ge__zero,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A] : pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),aa(A,A,abs_abs(A),A2))) ) ).

% abs_ge_zero
tff(fact_2562_abs__of__pos,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A] :
          ( pp(aa(A,bool,ord_less(A,zero_zero(A)),A2))
         => ( aa(A,A,abs_abs(A),A2) = A2 ) ) ) ).

% abs_of_pos
tff(fact_2563_abs__not__less__zero,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A] : ~ pp(aa(A,bool,ord_less(A,aa(A,A,abs_abs(A),A2)),zero_zero(A))) ) ).

% abs_not_less_zero
tff(fact_2564_abs__triangle__ineq,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A,B2: A] : pp(aa(A,bool,ord_less_eq(A,aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2))),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,abs_abs(A),A2)),aa(A,A,abs_abs(A),B2)))) ) ).

% abs_triangle_ineq
tff(fact_2565_abs__mult__less,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,C2: A,B2: A,D: A] :
          ( pp(aa(A,bool,ord_less(A,aa(A,A,abs_abs(A),A2)),C2))
         => ( pp(aa(A,bool,ord_less(A,aa(A,A,abs_abs(A),B2)),D))
           => pp(aa(A,bool,ord_less(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,abs_abs(A),A2)),aa(A,A,abs_abs(A),B2))),aa(A,A,aa(A,fun(A,A),times_times(A),C2),D))) ) ) ) ).

% abs_mult_less
tff(fact_2566_abs__triangle__ineq2,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A,B2: A] : pp(aa(A,bool,ord_less_eq(A,aa(A,A,minus_minus(A,aa(A,A,abs_abs(A),A2)),aa(A,A,abs_abs(A),B2))),aa(A,A,abs_abs(A),aa(A,A,minus_minus(A,A2),B2)))) ) ).

% abs_triangle_ineq2
tff(fact_2567_abs__triangle__ineq3,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A,B2: A] : pp(aa(A,bool,ord_less_eq(A,aa(A,A,abs_abs(A),aa(A,A,minus_minus(A,aa(A,A,abs_abs(A),A2)),aa(A,A,abs_abs(A),B2)))),aa(A,A,abs_abs(A),aa(A,A,minus_minus(A,A2),B2)))) ) ).

% abs_triangle_ineq3
tff(fact_2568_abs__triangle__ineq2__sym,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A,B2: A] : pp(aa(A,bool,ord_less_eq(A,aa(A,A,minus_minus(A,aa(A,A,abs_abs(A),A2)),aa(A,A,abs_abs(A),B2))),aa(A,A,abs_abs(A),aa(A,A,minus_minus(A,B2),A2)))) ) ).

% abs_triangle_ineq2_sym
tff(fact_2569_nonzero__abs__divide,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,A2: A] :
          ( ( B2 != zero_zero(A) )
         => ( aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,abs_abs(A),A2)),aa(A,A,abs_abs(A),B2)) ) ) ) ).

% nonzero_abs_divide
tff(fact_2570_zero__neq__neg__numeral,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [N: num] : zero_zero(A) != aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N)) ) ).

% zero_neq_neg_numeral
tff(fact_2571_neg__numeral__le__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num,N: num] : pp(aa(A,bool,ord_less_eq(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))),aa(num,A,numeral_numeral(A),N))) ) ).

% neg_numeral_le_numeral
tff(fact_2572_not__numeral__le__neg__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num,N: num] : ~ pp(aa(A,bool,ord_less_eq(A,aa(num,A,numeral_numeral(A),M)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N)))) ) ).

% not_numeral_le_neg_numeral
tff(fact_2573_not__numeral__less__neg__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num,N: num] : ~ pp(aa(A,bool,ord_less(A,aa(num,A,numeral_numeral(A),M)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N)))) ) ).

% not_numeral_less_neg_numeral
tff(fact_2574_neg__numeral__less__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num,N: num] : pp(aa(A,bool,ord_less(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))),aa(num,A,numeral_numeral(A),N))) ) ).

% neg_numeral_less_numeral
tff(fact_2575_add__eq__0__iff,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2) = zero_zero(A) )
        <=> ( B2 = aa(A,A,uminus_uminus(A),A2) ) ) ) ).

% add_eq_0_iff
tff(fact_2576_ab__group__add__class_Oab__left__minus,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),A2)),A2) = zero_zero(A) ) ).

% ab_group_add_class.ab_left_minus
tff(fact_2577_add_Oinverse__unique,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2) = zero_zero(A) )
         => ( aa(A,A,uminus_uminus(A),A2) = B2 ) ) ) ).

% add.inverse_unique
tff(fact_2578_eq__neg__iff__add__eq__0,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A,B2: A] :
          ( ( A2 = aa(A,A,uminus_uminus(A),B2) )
        <=> ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2) = zero_zero(A) ) ) ) ).

% eq_neg_iff_add_eq_0
tff(fact_2579_neg__eq__iff__add__eq__0,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A,B2: A] :
          ( ( aa(A,A,uminus_uminus(A),A2) = B2 )
        <=> ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2) = zero_zero(A) ) ) ) ).

% neg_eq_iff_add_eq_0
tff(fact_2580_zero__neq__neg__one,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ( zero_zero(A) != aa(A,A,uminus_uminus(A),one_one(A)) ) ) ).

% zero_neq_neg_one
tff(fact_2581_le__minus__one__simps_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => pp(aa(A,bool,ord_less_eq(A,aa(A,A,uminus_uminus(A),one_one(A))),one_one(A))) ) ).

% le_minus_one_simps(2)
tff(fact_2582_le__minus__one__simps_I4_J,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ~ pp(aa(A,bool,ord_less_eq(A,one_one(A)),aa(A,A,uminus_uminus(A),one_one(A)))) ) ).

% le_minus_one_simps(4)
tff(fact_2583_less__minus__one__simps_I4_J,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ~ pp(aa(A,bool,ord_less(A,one_one(A)),aa(A,A,uminus_uminus(A),one_one(A)))) ) ).

% less_minus_one_simps(4)
tff(fact_2584_less__minus__one__simps_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => pp(aa(A,bool,ord_less(A,aa(A,A,uminus_uminus(A),one_one(A))),one_one(A))) ) ).

% less_minus_one_simps(2)
tff(fact_2585_one__neq__neg__numeral,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [N: num] : one_one(A) != aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N)) ) ).

% one_neq_neg_numeral
tff(fact_2586_numeral__neq__neg__one,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [N: num] : aa(num,A,numeral_numeral(A),N) != aa(A,A,uminus_uminus(A),one_one(A)) ) ).

% numeral_neq_neg_one
tff(fact_2587_numeral__times__minus__swap,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [W: num,X: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W)),aa(A,A,uminus_uminus(A),X)) = aa(A,A,aa(A,fun(A,A),times_times(A),X),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))) ) ).

% numeral_times_minus_swap
tff(fact_2588_nonzero__minus__divide__right,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B2: A,A2: A] :
          ( ( B2 != zero_zero(A) )
         => ( aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(A,A,uminus_uminus(A),B2)) ) ) ) ).

% nonzero_minus_divide_right
tff(fact_2589_nonzero__minus__divide__divide,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B2: A,A2: A] :
          ( ( B2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,uminus_uminus(A),A2)),aa(A,A,uminus_uminus(A),B2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2) ) ) ) ).

% nonzero_minus_divide_divide
tff(fact_2590_square__eq__1__iff,axiom,
    ! [A: $tType] :
      ( ring_15535105094025558882visors(A)
     => ! [X: A] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),X),X) = one_one(A) )
        <=> ( ( X = one_one(A) )
            | ( X = aa(A,A,uminus_uminus(A),one_one(A)) ) ) ) ) ).

% square_eq_1_iff
tff(fact_2591_group__cancel_Osub2,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [B4: A,K: A,B2: A,A2: A] :
          ( ( B4 = aa(A,A,aa(A,fun(A,A),plus_plus(A),K),B2) )
         => ( aa(A,A,minus_minus(A,A2),B4) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),K)),aa(A,A,minus_minus(A,A2),B2)) ) ) ) ).

% group_cancel.sub2
tff(fact_2592_diff__conv__add__uminus,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A,B2: A] : aa(A,A,minus_minus(A,A2),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa(A,A,uminus_uminus(A),B2)) ) ).

% diff_conv_add_uminus
tff(fact_2593_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A2: A,B2: A] : aa(A,A,minus_minus(A,A2),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa(A,A,uminus_uminus(A),B2)) ) ).

% ab_group_add_class.ab_diff_conv_add_uminus
tff(fact_2594_dvd__div__neg,axiom,
    ! [A: $tType] :
      ( idom_divide(A)
     => ! [B2: A,A2: A] :
          ( pp(aa(A,bool,dvd_dvd(A,B2),A2))
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(A,A,uminus_uminus(A),B2)) = aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)) ) ) ) ).

% dvd_div_neg
tff(fact_2595_dvd__neg__div,axiom,
    ! [A: $tType] :
      ( idom_divide(A)
     => ! [B2: A,A2: A] :
          ( pp(aa(A,bool,dvd_dvd(A,B2),A2))
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,uminus_uminus(A),A2)),B2) = aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)) ) ) ) ).

% dvd_neg_div
tff(fact_2596_real__minus__mult__self__le,axiom,
    ! [U: real,X: real] : pp(aa(real,bool,ord_less_eq(real,aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),times_times(real),U),U))),aa(real,real,aa(real,fun(real,real),times_times(real),X),X))) ).

% real_minus_mult_self_le
tff(fact_2597_minus__real__def,axiom,
    ! [X: real,Y: real] : aa(real,real,minus_minus(real,X),Y) = aa(real,real,aa(real,fun(real,real),plus_plus(real),X),aa(real,real,uminus_uminus(real),Y)) ).

% minus_real_def
tff(fact_2598_tanh__real__lt__1,axiom,
    ! [X: real] : pp(aa(real,bool,ord_less(real,aa(real,real,tanh(real),X)),one_one(real))) ).

% tanh_real_lt_1
tff(fact_2599_dense__eq0__I,axiom,
    ! [A: $tType] :
      ( ( ordere166539214618696060dd_abs(A)
        & dense_linorder(A) )
     => ! [X: A] :
          ( ! [E2: A] :
              ( pp(aa(A,bool,ord_less(A,zero_zero(A)),E2))
             => pp(aa(A,bool,ord_less_eq(A,aa(A,A,abs_abs(A),X)),E2)) )
         => ( X = zero_zero(A) ) ) ) ).

% dense_eq0_I
tff(fact_2600_abs__mult__pos,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),X))
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,abs_abs(A),Y)),X) = aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),times_times(A),Y),X)) ) ) ) ).

% abs_mult_pos
tff(fact_2601_abs__eq__mult,axiom,
    ! [A: $tType] :
      ( ordered_ring_abs(A)
     => ! [A2: A,B2: A] :
          ( ( ( pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),A2))
              | pp(aa(A,bool,ord_less_eq(A,A2),zero_zero(A))) )
            & ( pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),B2))
              | pp(aa(A,bool,ord_less_eq(A,B2),zero_zero(A))) ) )
         => ( aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,abs_abs(A),A2)),aa(A,A,abs_abs(A),B2)) ) ) ) ).

% abs_eq_mult
tff(fact_2602_zero__le__power__abs,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,N: nat] : pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,abs_abs(A),A2)),N))) ) ).

% zero_le_power_abs
tff(fact_2603_abs__div__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Y: A,X: A] :
          ( pp(aa(A,bool,ord_less(A,zero_zero(A)),Y))
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,abs_abs(A),X)),Y) = aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y)) ) ) ) ).

% abs_div_pos
tff(fact_2604_abs__diff__triangle__ineq,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A,B2: A,C2: A,D: A] : pp(aa(A,bool,ord_less_eq(A,aa(A,A,abs_abs(A),aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),D)))),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,abs_abs(A),aa(A,A,minus_minus(A,A2),C2))),aa(A,A,abs_abs(A),aa(A,A,minus_minus(A,B2),D))))) ) ).

% abs_diff_triangle_ineq
tff(fact_2605_abs__triangle__ineq4,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A,B2: A] : pp(aa(A,bool,ord_less_eq(A,aa(A,A,abs_abs(A),aa(A,A,minus_minus(A,A2),B2))),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,abs_abs(A),A2)),aa(A,A,abs_abs(A),B2)))) ) ).

% abs_triangle_ineq4
tff(fact_2606_abs__diff__le__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A,A2: A,R2: A] :
          ( pp(aa(A,bool,ord_less_eq(A,aa(A,A,abs_abs(A),aa(A,A,minus_minus(A,X),A2))),R2))
        <=> ( pp(aa(A,bool,ord_less_eq(A,aa(A,A,minus_minus(A,A2),R2)),X))
            & pp(aa(A,bool,ord_less_eq(A,X),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),R2))) ) ) ) ).

% abs_diff_le_iff
tff(fact_2607_abs__diff__less__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A,A2: A,R2: A] :
          ( pp(aa(A,bool,ord_less(A,aa(A,A,abs_abs(A),aa(A,A,minus_minus(A,X),A2))),R2))
        <=> ( pp(aa(A,bool,ord_less(A,aa(A,A,minus_minus(A,A2),R2)),X))
            & pp(aa(A,bool,ord_less(A,X),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),R2))) ) ) ) ).

% abs_diff_less_iff
tff(fact_2608_nonzero__of__real__divide,axiom,
    ! [A: $tType] :
      ( real_V7773925162809079976_field(A)
     => ! [Y: real,X: real] :
          ( ( Y != zero_zero(real) )
         => ( aa(real,A,real_Vector_of_real(A),aa(real,real,aa(real,fun(real,real),divide_divide(real),X),Y)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(real,A,real_Vector_of_real(A),X)),aa(real,A,real_Vector_of_real(A),Y)) ) ) ) ).

% nonzero_of_real_divide
tff(fact_2609_neg__numeral__le__zero,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [N: num] : pp(aa(A,bool,ord_less_eq(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N))),zero_zero(A))) ) ).

% neg_numeral_le_zero
tff(fact_2610_not__zero__le__neg__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [N: num] : ~ pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N)))) ) ).

% not_zero_le_neg_numeral
tff(fact_2611_not__zero__less__neg__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [N: num] : ~ pp(aa(A,bool,ord_less(A,zero_zero(A)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N)))) ) ).

% not_zero_less_neg_numeral
tff(fact_2612_neg__numeral__less__zero,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [N: num] : pp(aa(A,bool,ord_less(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N))),zero_zero(A))) ) ).

% neg_numeral_less_zero
tff(fact_2613_le__minus__one__simps_I3_J,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ~ pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),aa(A,A,uminus_uminus(A),one_one(A)))) ) ).

% le_minus_one_simps(3)
tff(fact_2614_le__minus__one__simps_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => pp(aa(A,bool,ord_less_eq(A,aa(A,A,uminus_uminus(A),one_one(A))),zero_zero(A))) ) ).

% le_minus_one_simps(1)
tff(fact_2615_less__minus__one__simps_I3_J,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ~ pp(aa(A,bool,ord_less(A,zero_zero(A)),aa(A,A,uminus_uminus(A),one_one(A)))) ) ).

% less_minus_one_simps(3)
tff(fact_2616_less__minus__one__simps_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => pp(aa(A,bool,ord_less(A,aa(A,A,uminus_uminus(A),one_one(A))),zero_zero(A))) ) ).

% less_minus_one_simps(1)
tff(fact_2617_neg__numeral__le__one,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num] : pp(aa(A,bool,ord_less_eq(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))),one_one(A))) ) ).

% neg_numeral_le_one
tff(fact_2618_neg__one__le__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num] : pp(aa(A,bool,ord_less_eq(A,aa(A,A,uminus_uminus(A),one_one(A))),aa(num,A,numeral_numeral(A),M))) ) ).

% neg_one_le_numeral
tff(fact_2619_neg__numeral__le__neg__one,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num] : pp(aa(A,bool,ord_less_eq(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))),aa(A,A,uminus_uminus(A),one_one(A)))) ) ).

% neg_numeral_le_neg_one
tff(fact_2620_not__numeral__le__neg__one,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num] : ~ pp(aa(A,bool,ord_less_eq(A,aa(num,A,numeral_numeral(A),M)),aa(A,A,uminus_uminus(A),one_one(A)))) ) ).

% not_numeral_le_neg_one
tff(fact_2621_not__one__le__neg__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num] : ~ pp(aa(A,bool,ord_less_eq(A,one_one(A)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M)))) ) ).

% not_one_le_neg_numeral
tff(fact_2622_not__neg__one__less__neg__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num] : ~ pp(aa(A,bool,ord_less(A,aa(A,A,uminus_uminus(A),one_one(A))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M)))) ) ).

% not_neg_one_less_neg_numeral
tff(fact_2623_not__one__less__neg__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num] : ~ pp(aa(A,bool,ord_less(A,one_one(A)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M)))) ) ).

% not_one_less_neg_numeral
tff(fact_2624_not__numeral__less__neg__one,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num] : ~ pp(aa(A,bool,ord_less(A,aa(num,A,numeral_numeral(A),M)),aa(A,A,uminus_uminus(A),one_one(A)))) ) ).

% not_numeral_less_neg_one
tff(fact_2625_neg__one__less__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num] : pp(aa(A,bool,ord_less(A,aa(A,A,uminus_uminus(A),one_one(A))),aa(num,A,numeral_numeral(A),M))) ) ).

% neg_one_less_numeral
tff(fact_2626_neg__numeral__less__one,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num] : pp(aa(A,bool,ord_less(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))),one_one(A))) ) ).

% neg_numeral_less_one
tff(fact_2627_uminus__numeral__One,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ( aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),one2)) = aa(A,A,uminus_uminus(A),one_one(A)) ) ) ).

% uminus_numeral_One
tff(fact_2628_mult__1s__ring__1_I2_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [B2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),B2),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),one2))) = aa(A,A,uminus_uminus(A),B2) ) ).

% mult_1s_ring_1(2)
tff(fact_2629_mult__1s__ring__1_I1_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [B2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),one2))),B2) = aa(A,A,uminus_uminus(A),B2) ) ).

% mult_1s_ring_1(1)
tff(fact_2630_divide__eq__minus__1__iff,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A2: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2) = aa(A,A,uminus_uminus(A),one_one(A)) )
        <=> ( ( B2 != zero_zero(A) )
            & ( A2 = aa(A,A,uminus_uminus(A),B2) ) ) ) ) ).

% divide_eq_minus_1_iff
tff(fact_2631_eq__minus__divide__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,B2: A,C2: A] :
          ( ( A2 = aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)) )
        <=> ( ( ( C2 != zero_zero(A) )
             => ( aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2) = aa(A,A,uminus_uminus(A),B2) ) )
            & ( ( C2 = zero_zero(A) )
             => ( A2 = zero_zero(A) ) ) ) ) ) ).

% eq_minus_divide_eq
tff(fact_2632_minus__divide__eq__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B2: A,C2: A,A2: A] :
          ( ( aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)) = A2 )
        <=> ( ( ( C2 != zero_zero(A) )
             => ( aa(A,A,uminus_uminus(A),B2) = aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2) ) )
            & ( ( C2 = zero_zero(A) )
             => ( A2 = zero_zero(A) ) ) ) ) ) ).

% minus_divide_eq_eq
tff(fact_2633_nonzero__neg__divide__eq__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B2: A,A2: A,C2: A] :
          ( ( B2 != zero_zero(A) )
         => ( ( aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)) = C2 )
          <=> ( aa(A,A,uminus_uminus(A),A2) = aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2) ) ) ) ) ).

% nonzero_neg_divide_eq_eq
tff(fact_2634_nonzero__neg__divide__eq__eq2,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B2: A,C2: A,A2: A] :
          ( ( B2 != zero_zero(A) )
         => ( ( C2 = aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)) )
          <=> ( aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2) = aa(A,A,uminus_uminus(A),A2) ) ) ) ) ).

% nonzero_neg_divide_eq_eq2
tff(fact_2635_power__minus,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [A2: A,N: nat] : aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),A2)),N) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),N)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),N)) ) ).

% power_minus
tff(fact_2636_power__minus__Bit0,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [X: A,K: num] : aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),X)),aa(num,nat,numeral_numeral(nat),bit0(K))) = aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(num,nat,numeral_numeral(nat),bit0(K))) ) ).

% power_minus_Bit0
tff(fact_2637_power__minus__Bit1,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [X: A,K: num] : aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),X)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,K))) = aa(A,A,uminus_uminus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,K)))) ) ).

% power_minus_Bit1
tff(fact_2638_lemma__interval__lt,axiom,
    ! [A2: real,X: real,B2: real] :
      ( pp(aa(real,bool,ord_less(real,A2),X))
     => ( pp(aa(real,bool,ord_less(real,X),B2))
       => ? [D2: real] :
            ( pp(aa(real,bool,ord_less(real,zero_zero(real)),D2))
            & ! [Y4: real] :
                ( pp(aa(real,bool,ord_less(real,aa(real,real,abs_abs(real),aa(real,real,minus_minus(real,X),Y4))),D2))
               => ( pp(aa(real,bool,ord_less(real,A2),Y4))
                  & pp(aa(real,bool,ord_less(real,Y4),B2)) ) ) ) ) ) ).

% lemma_interval_lt
tff(fact_2639_norm__uminus__minus,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X: A,Y: A] : real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,aa(A,A,uminus_uminus(A),X)),Y)) = real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y)) ) ).

% norm_uminus_minus
tff(fact_2640_sin__bound__lemma,axiom,
    ! [X: real,Y: real,U: real,V: real] :
      ( ( X = Y )
     => ( pp(aa(real,bool,ord_less_eq(real,aa(real,real,abs_abs(real),U)),V))
       => pp(aa(real,bool,ord_less_eq(real,aa(real,real,abs_abs(real),aa(real,real,minus_minus(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),X),U)),Y))),V)) ) ) ).

% sin_bound_lemma
tff(fact_2641_powr__minus__divide,axiom,
    ! [A: $tType] :
      ( ( real_V3459762299906320749_field(A)
        & ln(A) )
     => ! [X: A,A2: A] : powr(A,X,aa(A,A,uminus_uminus(A),A2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),powr(A,X,A2)) ) ).

% powr_minus_divide
tff(fact_2642_real__0__less__add__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa(real,bool,ord_less(real,zero_zero(real)),aa(real,real,aa(real,fun(real,real),plus_plus(real),X),Y)))
    <=> pp(aa(real,bool,ord_less(real,aa(real,real,uminus_uminus(real),X)),Y)) ) ).

% real_0_less_add_iff
tff(fact_2643_real__add__less__0__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa(real,bool,ord_less(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),X),Y)),zero_zero(real)))
    <=> pp(aa(real,bool,ord_less(real,Y),aa(real,real,uminus_uminus(real),X))) ) ).

% real_add_less_0_iff
tff(fact_2644_real__add__le__0__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa(real,bool,ord_less_eq(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),X),Y)),zero_zero(real)))
    <=> pp(aa(real,bool,ord_less_eq(real,Y),aa(real,real,uminus_uminus(real),X))) ) ).

% real_add_le_0_iff
tff(fact_2645_real__0__le__add__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa(real,bool,ord_less_eq(real,zero_zero(real)),aa(real,real,aa(real,fun(real,real),plus_plus(real),X),Y)))
    <=> pp(aa(real,bool,ord_less_eq(real,aa(real,real,uminus_uminus(real),X)),Y)) ) ).

% real_0_le_add_iff
tff(fact_2646_divide__powr__uminus,axiom,
    ! [A2: real,B2: real,C2: real] : aa(real,real,aa(real,fun(real,real),divide_divide(real),A2),powr(real,B2,C2)) = aa(real,real,aa(real,fun(real,real),times_times(real),A2),powr(real,B2,aa(real,real,uminus_uminus(real),C2))) ).

% divide_powr_uminus
tff(fact_2647_abs__add__one__gt__zero,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A] : pp(aa(A,bool,ord_less(A,zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(A,A,abs_abs(A),X)))) ) ).

% abs_add_one_gt_zero
tff(fact_2648_norm__less__p1,axiom,
    ! [A: $tType] :
      ( real_V2822296259951069270ebra_1(A)
     => ! [X: A] : pp(aa(real,bool,ord_less(real,real_V7770717601297561774m_norm(A,X)),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(real,A,real_Vector_of_real(A),real_V7770717601297561774m_norm(A,X))),one_one(A))))) ) ).

% norm_less_p1
tff(fact_2649_of__int__leD,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [N: int,X: A] :
          ( pp(aa(A,bool,ord_less_eq(A,aa(A,A,abs_abs(A),aa(int,A,ring_1_of_int(A),N))),X))
         => ( ( N = zero_zero(int) )
            | pp(aa(A,bool,ord_less_eq(A,one_one(A)),X)) ) ) ) ).

% of_int_leD
tff(fact_2650_of__int__lessD,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [N: int,X: A] :
          ( pp(aa(A,bool,ord_less(A,aa(A,A,abs_abs(A),aa(int,A,ring_1_of_int(A),N))),X))
         => ( ( N = zero_zero(int) )
            | pp(aa(A,bool,ord_less(A,one_one(A)),X)) ) ) ) ).

% of_int_lessD
tff(fact_2651_pos__minus__divide__less__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,B2: A,A2: A] :
          ( pp(aa(A,bool,ord_less(A,zero_zero(A)),C2))
         => ( pp(aa(A,bool,ord_less(A,aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2))),A2))
          <=> pp(aa(A,bool,ord_less(A,aa(A,A,uminus_uminus(A),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2))) ) ) ) ).

% pos_minus_divide_less_eq
tff(fact_2652_pos__less__minus__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,A2: A,B2: A] :
          ( pp(aa(A,bool,ord_less(A,zero_zero(A)),C2))
         => ( pp(aa(A,bool,ord_less(A,A2),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2))))
          <=> pp(aa(A,bool,ord_less(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,uminus_uminus(A),B2))) ) ) ) ).

% pos_less_minus_divide_eq
tff(fact_2653_neg__minus__divide__less__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,B2: A,A2: A] :
          ( pp(aa(A,bool,ord_less(A,C2),zero_zero(A)))
         => ( pp(aa(A,bool,ord_less(A,aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2))),A2))
          <=> pp(aa(A,bool,ord_less(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,uminus_uminus(A),B2))) ) ) ) ).

% neg_minus_divide_less_eq
tff(fact_2654_neg__less__minus__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,A2: A,B2: A] :
          ( pp(aa(A,bool,ord_less(A,C2),zero_zero(A)))
         => ( pp(aa(A,bool,ord_less(A,A2),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2))))
          <=> pp(aa(A,bool,ord_less(A,aa(A,A,uminus_uminus(A),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2))) ) ) ) ).

% neg_less_minus_divide_eq
tff(fact_2655_minus__divide__less__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,C2: A,A2: A] :
          ( pp(aa(A,bool,ord_less(A,aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2))),A2))
        <=> ( ( pp(aa(A,bool,ord_less(A,zero_zero(A)),C2))
             => pp(aa(A,bool,ord_less(A,aa(A,A,uminus_uminus(A),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2))) )
            & ( ~ pp(aa(A,bool,ord_less(A,zero_zero(A)),C2))
             => ( ( pp(aa(A,bool,ord_less(A,C2),zero_zero(A)))
                 => pp(aa(A,bool,ord_less(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,uminus_uminus(A),B2))) )
                & ( ~ pp(aa(A,bool,ord_less(A,C2),zero_zero(A)))
                 => pp(aa(A,bool,ord_less(A,zero_zero(A)),A2)) ) ) ) ) ) ) ).

% minus_divide_less_eq
tff(fact_2656_less__minus__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,ord_less(A,A2),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2))))
        <=> ( ( pp(aa(A,bool,ord_less(A,zero_zero(A)),C2))
             => pp(aa(A,bool,ord_less(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,uminus_uminus(A),B2))) )
            & ( ~ pp(aa(A,bool,ord_less(A,zero_zero(A)),C2))
             => ( ( pp(aa(A,bool,ord_less(A,C2),zero_zero(A)))
                 => pp(aa(A,bool,ord_less(A,aa(A,A,uminus_uminus(A),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2))) )
                & ( ~ pp(aa(A,bool,ord_less(A,C2),zero_zero(A)))
                 => pp(aa(A,bool,ord_less(A,A2),zero_zero(A))) ) ) ) ) ) ) ).

% less_minus_divide_eq
tff(fact_2657_divide__eq__eq__numeral_I2_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B2: A,C2: A,W: num] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W)) )
        <=> ( ( ( C2 != zero_zero(A) )
             => ( B2 = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))),C2) ) )
            & ( ( C2 = zero_zero(A) )
             => ( aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W)) = zero_zero(A) ) ) ) ) ) ).

% divide_eq_eq_numeral(2)
tff(fact_2658_eq__divide__eq__numeral_I2_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [W: num,B2: A,C2: A] :
          ( ( aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2) )
        <=> ( ( ( C2 != zero_zero(A) )
             => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))),C2) = B2 ) )
            & ( ( C2 = zero_zero(A) )
             => ( aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W)) = zero_zero(A) ) ) ) ) ) ).

% eq_divide_eq_numeral(2)
tff(fact_2659_add__divide__eq__if__simps_I3_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Z: A,A2: A,B2: A] :
          ( ( ( Z = zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),Z))),B2) = B2 ) )
          & ( ( Z != zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),Z))),B2) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),Z))),Z) ) ) ) ) ).

% add_divide_eq_if_simps(3)
tff(fact_2660_minus__divide__add__eq__iff,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Z: A,X: A,Y: A] :
          ( ( Z != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Z))),Y) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),X)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Z))),Z) ) ) ) ).

% minus_divide_add_eq_iff
tff(fact_2661_minus__divide__diff__eq__iff,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Z: A,X: A,Y: A] :
          ( ( Z != zero_zero(A) )
         => ( aa(A,A,minus_minus(A,aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Z))),Y) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,minus_minus(A,aa(A,A,uminus_uminus(A),X)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Z))),Z) ) ) ) ).

% minus_divide_diff_eq_iff
tff(fact_2662_add__divide__eq__if__simps_I5_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Z: A,A2: A,B2: A] :
          ( ( ( Z = zero_zero(A) )
           => ( aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),Z)),B2) = aa(A,A,uminus_uminus(A),B2) ) )
          & ( ( Z != zero_zero(A) )
           => ( aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),Z)),B2) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,minus_minus(A,A2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),Z))),Z) ) ) ) ) ).

% add_divide_eq_if_simps(5)
tff(fact_2663_add__divide__eq__if__simps_I6_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Z: A,A2: A,B2: A] :
          ( ( ( Z = zero_zero(A) )
           => ( aa(A,A,minus_minus(A,aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),Z))),B2) = aa(A,A,uminus_uminus(A),B2) ) )
          & ( ( Z != zero_zero(A) )
           => ( aa(A,A,minus_minus(A,aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),Z))),B2) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,minus_minus(A,aa(A,A,uminus_uminus(A),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),Z))),Z) ) ) ) ) ).

% add_divide_eq_if_simps(6)
tff(fact_2664_even__minus,axiom,
    ! [A: $tType] :
      ( ring_parity(A)
     => ! [A2: A] :
          ( pp(aa(A,bool,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,uminus_uminus(A),A2)))
        <=> pp(aa(A,bool,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A2)) ) ) ).

% even_minus
tff(fact_2665_power2__eq__iff,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [X: A,Y: A] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(num,nat,numeral_numeral(nat),bit0(one2))) = aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),aa(num,nat,numeral_numeral(nat),bit0(one2))) )
        <=> ( ( X = Y )
            | ( X = aa(A,A,uminus_uminus(A),Y) ) ) ) ) ).

% power2_eq_iff
tff(fact_2666_lemma__interval,axiom,
    ! [A2: real,X: real,B2: real] :
      ( pp(aa(real,bool,ord_less(real,A2),X))
     => ( pp(aa(real,bool,ord_less(real,X),B2))
       => ? [D2: real] :
            ( pp(aa(real,bool,ord_less(real,zero_zero(real)),D2))
            & ! [Y4: real] :
                ( pp(aa(real,bool,ord_less(real,aa(real,real,abs_abs(real),aa(real,real,minus_minus(real,X),Y4))),D2))
               => ( pp(aa(real,bool,ord_less_eq(real,A2),Y4))
                  & pp(aa(real,bool,ord_less_eq(real,Y4),B2)) ) ) ) ) ) ).

% lemma_interval
tff(fact_2667_round__diff__minimal,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Z: A,M: int] : pp(aa(A,bool,ord_less_eq(A,aa(A,A,abs_abs(A),aa(A,A,minus_minus(A,Z),aa(int,A,ring_1_of_int(A),archimedean_round(A,Z))))),aa(A,A,abs_abs(A),aa(A,A,minus_minus(A,Z),aa(int,A,ring_1_of_int(A),M))))) ) ).

% round_diff_minimal
tff(fact_2668_norm__triangle__ineq3,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [A2: A,B2: A] : pp(aa(real,bool,ord_less_eq(real,aa(real,real,abs_abs(real),aa(real,real,minus_minus(real,real_V7770717601297561774m_norm(A,A2)),real_V7770717601297561774m_norm(A,B2)))),real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,A2),B2)))) ) ).

% norm_triangle_ineq3
tff(fact_2669_abs__le__square__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,ord_less_eq(A,aa(A,A,abs_abs(A),X)),aa(A,A,abs_abs(A),Y)))
        <=> pp(aa(A,bool,ord_less_eq(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) ) ).

% abs_le_square_iff
tff(fact_2670_abs__square__eq__1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(num,nat,numeral_numeral(nat),bit0(one2))) = one_one(A) )
        <=> ( aa(A,A,abs_abs(A),X) = one_one(A) ) ) ) ).

% abs_square_eq_1
tff(fact_2671_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Osimps_I3_J,axiom,
    ! [A2: bool,B2: bool,Va: nat] : vEBT_T_p_r_e_d2(vEBT_Leaf(A2,B2),aa(nat,nat,suc,aa(nat,nat,suc,Va))) = one_one(nat) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.simps(3)
tff(fact_2672_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Osimps_I1_J,axiom,
    ! [Uu: bool,Uv: bool] : 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)
tff(fact_2673_power__even__abs,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [N: nat,A2: A] :
          ( pp(aa(nat,bool,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N))
         => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,abs_abs(A),A2)),N) = aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),N) ) ) ) ).

% power_even_abs
tff(fact_2674_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)
tff(fact_2675_le__minus__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,ord_less_eq(A,A2),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2))))
        <=> ( ( pp(aa(A,bool,ord_less(A,zero_zero(A)),C2))
             => pp(aa(A,bool,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,uminus_uminus(A),B2))) )
            & ( ~ pp(aa(A,bool,ord_less(A,zero_zero(A)),C2))
             => ( ( pp(aa(A,bool,ord_less(A,C2),zero_zero(A)))
                 => pp(aa(A,bool,ord_less_eq(A,aa(A,A,uminus_uminus(A),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2))) )
                & ( ~ pp(aa(A,bool,ord_less(A,C2),zero_zero(A)))
                 => pp(aa(A,bool,ord_less_eq(A,A2),zero_zero(A))) ) ) ) ) ) ) ).

% le_minus_divide_eq
tff(fact_2676_minus__divide__le__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,C2: A,A2: A] :
          ( pp(aa(A,bool,ord_less_eq(A,aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2))),A2))
        <=> ( ( pp(aa(A,bool,ord_less(A,zero_zero(A)),C2))
             => pp(aa(A,bool,ord_less_eq(A,aa(A,A,uminus_uminus(A),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2))) )
            & ( ~ pp(aa(A,bool,ord_less(A,zero_zero(A)),C2))
             => ( ( pp(aa(A,bool,ord_less(A,C2),zero_zero(A)))
                 => pp(aa(A,bool,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,uminus_uminus(A),B2))) )
                & ( ~ pp(aa(A,bool,ord_less(A,C2),zero_zero(A)))
                 => pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),A2)) ) ) ) ) ) ) ).

% minus_divide_le_eq
tff(fact_2677_neg__le__minus__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,A2: A,B2: A] :
          ( pp(aa(A,bool,ord_less(A,C2),zero_zero(A)))
         => ( pp(aa(A,bool,ord_less_eq(A,A2),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2))))
          <=> pp(aa(A,bool,ord_less_eq(A,aa(A,A,uminus_uminus(A),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2))) ) ) ) ).

% neg_le_minus_divide_eq
tff(fact_2678_neg__minus__divide__le__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,B2: A,A2: A] :
          ( pp(aa(A,bool,ord_less(A,C2),zero_zero(A)))
         => ( pp(aa(A,bool,ord_less_eq(A,aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2))),A2))
          <=> pp(aa(A,bool,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,uminus_uminus(A),B2))) ) ) ) ).

% neg_minus_divide_le_eq
tff(fact_2679_pos__le__minus__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,A2: A,B2: A] :
          ( pp(aa(A,bool,ord_less(A,zero_zero(A)),C2))
         => ( pp(aa(A,bool,ord_less_eq(A,A2),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2))))
          <=> pp(aa(A,bool,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,uminus_uminus(A),B2))) ) ) ) ).

% pos_le_minus_divide_eq
tff(fact_2680_pos__minus__divide__le__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,B2: A,A2: A] :
          ( pp(aa(A,bool,ord_less(A,zero_zero(A)),C2))
         => ( pp(aa(A,bool,ord_less_eq(A,aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2))),A2))
          <=> pp(aa(A,bool,ord_less_eq(A,aa(A,A,uminus_uminus(A),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2))) ) ) ) ).

% pos_minus_divide_le_eq
tff(fact_2681_divide__less__eq__numeral_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,C2: A,W: num] :
          ( pp(aa(A,bool,ord_less(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))))
        <=> ( ( pp(aa(A,bool,ord_less(A,zero_zero(A)),C2))
             => pp(aa(A,bool,ord_less(A,B2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))),C2))) )
            & ( ~ pp(aa(A,bool,ord_less(A,zero_zero(A)),C2))
             => ( ( pp(aa(A,bool,ord_less(A,C2),zero_zero(A)))
                 => pp(aa(A,bool,ord_less(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))),C2)),B2)) )
                & ( ~ pp(aa(A,bool,ord_less(A,C2),zero_zero(A)))
                 => pp(aa(A,bool,ord_less(A,zero_zero(A)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W)))) ) ) ) ) ) ) ).

% divide_less_eq_numeral(2)
tff(fact_2682_less__divide__eq__numeral_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [W: num,B2: A,C2: A] :
          ( pp(aa(A,bool,ord_less(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)))
        <=> ( ( pp(aa(A,bool,ord_less(A,zero_zero(A)),C2))
             => pp(aa(A,bool,ord_less(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))),C2)),B2)) )
            & ( ~ pp(aa(A,bool,ord_less(A,zero_zero(A)),C2))
             => ( ( pp(aa(A,bool,ord_less(A,C2),zero_zero(A)))
                 => pp(aa(A,bool,ord_less(A,B2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))),C2))) )
                & ( ~ pp(aa(A,bool,ord_less(A,C2),zero_zero(A)))
                 => pp(aa(A,bool,ord_less(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))),zero_zero(A))) ) ) ) ) ) ) ).

% less_divide_eq_numeral(2)
tff(fact_2683_power2__eq__1__iff,axiom,
    ! [A: $tType] :
      ( ring_15535105094025558882visors(A)
     => ! [A2: A] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(num,nat,numeral_numeral(nat),bit0(one2))) = one_one(A) )
        <=> ( ( A2 = one_one(A) )
            | ( A2 = aa(A,A,uminus_uminus(A),one_one(A)) ) ) ) ) ).

% power2_eq_1_iff
tff(fact_2684_uminus__power__if,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [N: nat,A2: A] :
          ( ( pp(aa(nat,bool,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N))
           => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),A2)),N) = aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),N) ) )
          & ( ~ pp(aa(nat,bool,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N))
           => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),A2)),N) = aa(A,A,uminus_uminus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),N)) ) ) ) ) ).

% uminus_power_if
tff(fact_2685_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [K: nat,N: nat] :
          ( pp(aa(nat,bool,ord_less_eq(nat,K),N))
         => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),K)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),aa(nat,nat,minus_minus(nat,N),K)) ) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
tff(fact_2686_realpow__square__minus__le,axiom,
    ! [U: real,X: real] : pp(aa(real,bool,ord_less_eq(real,aa(real,real,uminus_uminus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),U),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(nat,real,aa(real,fun(nat,real),power_power(real),X),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ).

% realpow_square_minus_le
tff(fact_2687_powr__neg__one,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,ord_less(real,zero_zero(real)),X))
     => ( powr(real,X,aa(real,real,uminus_uminus(real),one_one(real))) = aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),X) ) ) ).

% powr_neg_one
tff(fact_2688_ln__add__one__self__le__self2,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,ord_less(real,aa(real,real,uminus_uminus(real),one_one(real))),X))
     => pp(aa(real,bool,ord_less_eq(real,aa(real,real,ln_ln(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),X))),X)) ) ).

% ln_add_one_self_le_self2
tff(fact_2689_power2__le__iff__abs__le,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Y: A,X: A] :
          ( pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),Y))
         => ( pp(aa(A,bool,ord_less_eq(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),aa(num,nat,numeral_numeral(nat),bit0(one2)))))
          <=> pp(aa(A,bool,ord_less_eq(A,aa(A,A,abs_abs(A),X)),Y)) ) ) ) ).

% power2_le_iff_abs_le
tff(fact_2690_abs__sqrt__wlog,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [P: fun(A,fun(A,bool)),X: A] :
          ( ! [X3: A] :
              ( pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),X3))
             => pp(aa(A,bool,aa(A,fun(A,bool),P,X3),aa(nat,A,aa(A,fun(nat,A),power_power(A),X3),aa(num,nat,numeral_numeral(nat),bit0(one2))))) )
         => pp(aa(A,bool,aa(A,fun(A,bool),P,aa(A,A,abs_abs(A),X)),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) ) ).

% abs_sqrt_wlog
tff(fact_2691_abs__square__le__1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A] :
          ( pp(aa(A,bool,ord_less_eq(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(num,nat,numeral_numeral(nat),bit0(one2)))),one_one(A)))
        <=> pp(aa(A,bool,ord_less_eq(A,aa(A,A,abs_abs(A),X)),one_one(A))) ) ) ).

% abs_square_le_1
tff(fact_2692_abs__square__less__1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A] :
          ( pp(aa(A,bool,ord_less(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(num,nat,numeral_numeral(nat),bit0(one2)))),one_one(A)))
        <=> pp(aa(A,bool,ord_less(A,aa(A,A,abs_abs(A),X)),one_one(A))) ) ) ).

% abs_square_less_1
tff(fact_2693_power__mono__even,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [N: nat,A2: A,B2: A] :
          ( pp(aa(nat,bool,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N))
         => ( pp(aa(A,bool,ord_less_eq(A,aa(A,A,abs_abs(A),A2)),aa(A,A,abs_abs(A),B2)))
           => pp(aa(A,bool,ord_less_eq(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),N)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),N))) ) ) ) ).

% power_mono_even
tff(fact_2694_divide__le__eq__numeral_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,C2: A,W: num] :
          ( pp(aa(A,bool,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))))
        <=> ( ( pp(aa(A,bool,ord_less(A,zero_zero(A)),C2))
             => pp(aa(A,bool,ord_less_eq(A,B2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))),C2))) )
            & ( ~ pp(aa(A,bool,ord_less(A,zero_zero(A)),C2))
             => ( ( pp(aa(A,bool,ord_less(A,C2),zero_zero(A)))
                 => pp(aa(A,bool,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))),C2)),B2)) )
                & ( ~ pp(aa(A,bool,ord_less(A,C2),zero_zero(A)))
                 => pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W)))) ) ) ) ) ) ) ).

% divide_le_eq_numeral(2)
tff(fact_2695_le__divide__eq__numeral_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [W: num,B2: A,C2: A] :
          ( pp(aa(A,bool,ord_less_eq(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)))
        <=> ( ( pp(aa(A,bool,ord_less(A,zero_zero(A)),C2))
             => pp(aa(A,bool,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))),C2)),B2)) )
            & ( ~ pp(aa(A,bool,ord_less(A,zero_zero(A)),C2))
             => ( ( pp(aa(A,bool,ord_less(A,C2),zero_zero(A)))
                 => pp(aa(A,bool,ord_less_eq(A,B2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))),C2))) )
                & ( ~ pp(aa(A,bool,ord_less(A,C2),zero_zero(A)))
                 => pp(aa(A,bool,ord_less_eq(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))),zero_zero(A))) ) ) ) ) ) ) ).

% le_divide_eq_numeral(2)
tff(fact_2696_square__le__1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A] :
          ( pp(aa(A,bool,ord_less_eq(A,aa(A,A,uminus_uminus(A),one_one(A))),X))
         => ( pp(aa(A,bool,ord_less_eq(A,X),one_one(A)))
           => pp(aa(A,bool,ord_less_eq(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(num,nat,numeral_numeral(nat),bit0(one2)))),one_one(A))) ) ) ) ).

% square_le_1
tff(fact_2697_minus__power__mult__self,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [A2: A,N: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),A2)),N)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),A2)),N)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)) ) ).

% minus_power_mult_self
tff(fact_2698_minus__one__power__iff,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [N: nat] :
          ( ( pp(aa(nat,bool,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N))
           => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),N) = one_one(A) ) )
          & ( ~ pp(aa(nat,bool,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N))
           => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),N) = aa(A,A,uminus_uminus(A),one_one(A)) ) ) ) ) ).

% minus_one_power_iff
tff(fact_2699_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Osimps_I2_J,axiom,
    ! [A2: bool,Uw: bool] : vEBT_T_p_r_e_d2(vEBT_Leaf(A2,Uw),aa(nat,nat,suc,zero_zero(nat))) = one_one(nat) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.simps(2)
tff(fact_2700_Bernoulli__inequality,axiom,
    ! [X: real,N: nat] :
      ( pp(aa(real,bool,ord_less_eq(real,aa(real,real,uminus_uminus(real),one_one(real))),X))
     => pp(aa(real,bool,ord_less_eq(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),N)),X))),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),X)),N))) ) ).

% Bernoulli_inequality
tff(fact_2701_ln__one__minus__pos__upper__bound,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,ord_less_eq(real,zero_zero(real)),X))
     => ( pp(aa(real,bool,ord_less(real,X),one_one(real)))
       => pp(aa(real,bool,ord_less_eq(real,aa(real,real,ln_ln(real),aa(real,real,minus_minus(real,one_one(real)),X))),aa(real,real,uminus_uminus(real),X))) ) ) ).

% ln_one_minus_pos_upper_bound
tff(fact_2702_abs__ln__one__plus__x__minus__x__bound__nonpos,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,ord_less_eq(real,aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),bit0(one2))))),X))
     => ( pp(aa(real,bool,ord_less_eq(real,X),zero_zero(real)))
       => pp(aa(real,bool,ord_less_eq(real,aa(real,real,abs_abs(real),aa(real,real,minus_minus(real,aa(real,real,ln_ln(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),X))),X))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),aa(nat,real,aa(real,fun(nat,real),power_power(real),X),aa(num,nat,numeral_numeral(nat),bit0(one2)))))) ) ) ).

% abs_ln_one_plus_x_minus_x_bound_nonpos
tff(fact_2703_power__minus1__odd,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [N: nat] : aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N))) = aa(A,A,uminus_uminus(A),one_one(A)) ) ).

% power_minus1_odd
tff(fact_2704_log__minus__eq__powr,axiom,
    ! [B2: real,X: real,Y: real] :
      ( pp(aa(real,bool,ord_less(real,zero_zero(real)),B2))
     => ( ( B2 != one_one(real) )
       => ( pp(aa(real,bool,ord_less(real,zero_zero(real)),X))
         => ( aa(real,real,minus_minus(real,aa(real,real,log(B2),X)),Y) = aa(real,real,log(B2),aa(real,real,aa(real,fun(real,real),times_times(real),X),powr(real,B2,aa(real,real,uminus_uminus(real),Y)))) ) ) ) ) ).

% log_minus_eq_powr
tff(fact_2705_pred__bound__height_H,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( vEBT_invar_vebt(T,N)
     => pp(aa(nat,bool,ord_less_eq(nat,vEBT_T_p_r_e_d2(T,X)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),aa(vEBT_VEBT,nat,vEBT_VEBT_height,T)))) ) ).

% pred_bound_height'
tff(fact_2706_powr__neg__numeral,axiom,
    ! [X: real,N: num] :
      ( pp(aa(real,bool,ord_less(real,zero_zero(real)),X))
     => ( powr(real,X,aa(real,real,uminus_uminus(real),aa(num,real,numeral_numeral(real),N))) = aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(nat,real,aa(real,fun(nat,real),power_power(real),X),aa(num,nat,numeral_numeral(nat),N))) ) ) ).

% powr_neg_numeral
tff(fact_2707_of__int__round__abs__le,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] : pp(aa(A,bool,ord_less_eq(A,aa(A,A,abs_abs(A),aa(A,A,minus_minus(A,aa(int,A,ring_1_of_int(A),archimedean_round(A,X))),X))),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),bit0(one2))))) ) ).

% of_int_round_abs_le
tff(fact_2708_round__unique_H,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,N: int] :
          ( pp(aa(A,bool,ord_less(A,aa(A,A,abs_abs(A),aa(A,A,minus_minus(A,X),aa(int,A,ring_1_of_int(A),N)))),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),bit0(one2)))))
         => ( archimedean_round(A,X) = N ) ) ) ).

% round_unique'
tff(fact_2709_abs__ln__one__plus__x__minus__x__bound__nonneg,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,ord_less_eq(real,zero_zero(real)),X))
     => ( pp(aa(real,bool,ord_less_eq(real,X),one_one(real)))
       => pp(aa(real,bool,ord_less_eq(real,aa(real,real,abs_abs(real),aa(real,real,minus_minus(real,aa(real,real,ln_ln(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),X))),X))),aa(nat,real,aa(real,fun(nat,real),power_power(real),X),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) ) ).

% abs_ln_one_plus_x_minus_x_bound_nonneg
tff(fact_2710_arsinh__def,axiom,
    ! [A: $tType] :
      ( ln(A)
     => ! [X: A] : aa(A,A,arsinh(A),X) = aa(A,A,ln_ln(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),powr(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(num,nat,numeral_numeral(nat),bit0(one2)))),one_one(A)),aa(real,A,real_Vector_of_real(A),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),bit0(one2))))))) ) ).

% arsinh_def
tff(fact_2711_sum_Ofinite__Collect__op,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [I5: set(B),X: fun(B,A),Y: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),collect(B,aa(fun(B,A),fun(B,bool),aTP_Lamp_bj(set(B),fun(fun(B,A),fun(B,bool)),I5),X))))
         => ( pp(aa(set(B),bool,finite_finite2(B),collect(B,aa(fun(B,A),fun(B,bool),aTP_Lamp_bj(set(B),fun(fun(B,A),fun(B,bool)),I5),Y))))
           => pp(aa(set(B),bool,finite_finite2(B),collect(B,aa(fun(B,A),fun(B,bool),aa(fun(B,A),fun(fun(B,A),fun(B,bool)),aTP_Lamp_bk(set(B),fun(fun(B,A),fun(fun(B,A),fun(B,bool))),I5),X),Y)))) ) ) ) ).

% sum.finite_Collect_op
tff(fact_2712_of__int__code__if,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [K: int] :
          ( ( ( K = zero_zero(int) )
           => ( aa(int,A,ring_1_of_int(A),K) = zero_zero(A) ) )
          & ( ( K != zero_zero(int) )
           => ( ( pp(aa(int,bool,ord_less(int,K),zero_zero(int)))
               => ( aa(int,A,ring_1_of_int(A),K) = aa(A,A,uminus_uminus(A),aa(int,A,ring_1_of_int(A),aa(int,int,uminus_uminus(int),K))) ) )
              & ( ~ pp(aa(int,bool,ord_less(int,K),zero_zero(int)))
               => ( aa(int,A,ring_1_of_int(A),K) = if(A,aa(int,bool,fequal(int,modulo_modulo(int,K,aa(num,int,numeral_numeral(int),bit0(one2)))),zero_zero(int)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(int,A,ring_1_of_int(A),aa(int,int,aa(int,fun(int,int),divide_divide(int),K),aa(num,int,numeral_numeral(int),bit0(one2))))),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(int,A,ring_1_of_int(A),aa(int,int,aa(int,fun(int,int),divide_divide(int),K),aa(num,int,numeral_numeral(int),bit0(one2)))))),one_one(A))) ) ) ) ) ) ) ).

% of_int_code_if
tff(fact_2713_compl__le__compl__iff,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,ord_less_eq(A,aa(A,A,uminus_uminus(A),X)),aa(A,A,uminus_uminus(A),Y)))
        <=> pp(aa(A,bool,ord_less_eq(A,Y),X)) ) ) ).

% compl_le_compl_iff
tff(fact_2714_monoseq__arctan__series,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,ord_less_eq(real,aa(real,real,abs_abs(real),X)),one_one(real)))
     => topological_monoseq(real,aTP_Lamp_bl(real,fun(nat,real),X)) ) ).

% monoseq_arctan_series
tff(fact_2715_ln__series,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,ord_less(real,zero_zero(real)),X))
     => ( pp(aa(real,bool,ord_less(real,X),aa(num,real,numeral_numeral(real),bit0(one2))))
       => ( aa(real,real,ln_ln(real),X) = suminf(real,aTP_Lamp_bm(real,fun(nat,real),X)) ) ) ) ).

% ln_series
tff(fact_2716_dbl__dec__simps_I4_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ( neg_numeral_dbl_dec(A,aa(A,A,uminus_uminus(A),one_one(A))) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),aa(num,num,bit1,one2))) ) ) ).

% dbl_dec_simps(4)
tff(fact_2717_signed__take__bit__rec,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [N: nat,A2: A] :
          ( ( ( N = zero_zero(nat) )
           => ( aa(A,A,bit_ri4674362597316999326ke_bit(A,N),A2) = aa(A,A,uminus_uminus(A),modulo_modulo(A,A2,aa(num,A,numeral_numeral(A),bit0(one2)))) ) )
          & ( ( N != zero_zero(nat) )
           => ( aa(A,A,bit_ri4674362597316999326ke_bit(A,N),A2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),modulo_modulo(A,A2,aa(num,A,numeral_numeral(A),bit0(one2)))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,bit_ri4674362597316999326ke_bit(A,aa(nat,nat,minus_minus(nat,N),one_one(nat))),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(num,A,numeral_numeral(A),bit0(one2)))))) ) ) ) ) ).

% signed_take_bit_rec
tff(fact_2718_Compl__subset__Compl__iff,axiom,
    ! [A: $tType,A3: set(A),B4: set(A)] :
      ( pp(aa(set(A),bool,ord_less_eq(set(A),aa(set(A),set(A),uminus_uminus(set(A)),A3)),aa(set(A),set(A),uminus_uminus(set(A)),B4)))
    <=> pp(aa(set(A),bool,ord_less_eq(set(A),B4),A3)) ) ).

% Compl_subset_Compl_iff
tff(fact_2719_Compl__anti__mono,axiom,
    ! [A: $tType,A3: set(A),B4: set(A)] :
      ( pp(aa(set(A),bool,ord_less_eq(set(A),A3),B4))
     => pp(aa(set(A),bool,ord_less_eq(set(A),aa(set(A),set(A),uminus_uminus(set(A)),B4)),aa(set(A),set(A),uminus_uminus(set(A)),A3))) ) ).

% Compl_anti_mono
tff(fact_2720_signed__take__bit__of__0,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [N: nat] : aa(A,A,bit_ri4674362597316999326ke_bit(A,N),zero_zero(A)) = zero_zero(A) ) ).

% signed_take_bit_of_0
tff(fact_2721_max__word__max,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: word(A)] : pp(aa(word(A),bool,ord_less_eq(word(A),N),aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A))))) ) ).

% max_word_max
tff(fact_2722_word__n1__ge,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A)] : pp(aa(word(A),bool,ord_less_eq(word(A),Y),aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A))))) ) ).

% word_n1_ge
tff(fact_2723_word__order_Oextremum__unique,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A)] :
          ( pp(aa(word(A),bool,ord_less_eq(word(A),aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A)))),A2))
        <=> ( A2 = aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A))) ) ) ) ).

% word_order.extremum_unique
tff(fact_2724_word__order_Oextremum,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A)] : pp(aa(word(A),bool,ord_less_eq(word(A),A2),aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A))))) ) ).

% word_order.extremum
tff(fact_2725_max__word__less__eq__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] :
          ( pp(aa(word(A),bool,ord_less_eq(word(A),aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A)))),W))
        <=> ( W = aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A))) ) ) ) ).

% max_word_less_eq_iff
tff(fact_2726_word__minus__one__le,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A)] :
          ( pp(aa(word(A),bool,ord_less_eq(word(A),aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A)))),X))
        <=> ( X = aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A))) ) ) ) ).

% word_minus_one_le
tff(fact_2727_zdvd1__eq,axiom,
    ! [X: int] :
      ( pp(aa(int,bool,dvd_dvd(int,X),one_one(int)))
    <=> ( aa(int,int,abs_abs(int),X) = one_one(int) ) ) ).

% zdvd1_eq
tff(fact_2728_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)
tff(fact_2729_signed__take__bit__Suc__1,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [N: nat] : aa(A,A,bit_ri4674362597316999326ke_bit(A,aa(nat,nat,suc,N)),one_one(A)) = one_one(A) ) ).

% signed_take_bit_Suc_1
tff(fact_2730_signed__take__bit__numeral__of__1,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [K: num] : aa(A,A,bit_ri4674362597316999326ke_bit(A,aa(num,nat,numeral_numeral(nat),K)),one_one(A)) = one_one(A) ) ).

% signed_take_bit_numeral_of_1
tff(fact_2731_signed__take__bit__of__minus__1,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [N: nat] : aa(A,A,bit_ri4674362597316999326ke_bit(A,N),aa(A,A,uminus_uminus(A),one_one(A))) = aa(A,A,uminus_uminus(A),one_one(A)) ) ).

% signed_take_bit_of_minus_1
tff(fact_2732_negative__eq__positive,axiom,
    ! [N: nat,M: nat] :
      ( ( aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),N)) = aa(nat,int,semiring_1_of_nat(int),M) )
    <=> ( ( N = zero_zero(nat) )
        & ( M = zero_zero(nat) ) ) ) ).

% negative_eq_positive
tff(fact_2733_zabs__less__one__iff,axiom,
    ! [Z: int] :
      ( pp(aa(int,bool,ord_less(int,aa(int,int,abs_abs(int),Z)),one_one(int)))
    <=> ( Z = zero_zero(int) ) ) ).

% zabs_less_one_iff
tff(fact_2734_negative__zle,axiom,
    ! [N: nat,M: nat] : pp(aa(int,bool,ord_less_eq(int,aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),N))),aa(nat,int,semiring_1_of_nat(int),M))) ).

% negative_zle
tff(fact_2735_negative__zless,axiom,
    ! [N: nat,M: nat] : pp(aa(int,bool,ord_less(int,aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,suc,N)))),aa(nat,int,semiring_1_of_nat(int),M))) ).

% negative_zless
tff(fact_2736_word__of__int__neg__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Bin: num] : aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Bin))) = aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),Bin)) ) ).

% word_of_int_neg_numeral
tff(fact_2737_word__of__int__neg__1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,uminus_uminus(int),one_one(int))) = aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A))) ) ) ).

% word_of_int_neg_1
tff(fact_2738_dbl__dec__simps_I2_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ( neg_numeral_dbl_dec(A,zero_zero(A)) = aa(A,A,uminus_uminus(A),one_one(A)) ) ) ).

% dbl_dec_simps(2)
tff(fact_2739_signed__take__bit__Suc__bit0,axiom,
    ! [N: nat,K: num] : aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,suc,N)),aa(num,int,numeral_numeral(int),bit0(K))) = aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,N),aa(num,int,numeral_numeral(int),K))),aa(num,int,numeral_numeral(int),bit0(one2))) ).

% signed_take_bit_Suc_bit0
tff(fact_2740_int__div__minus__is__minus1,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa(int,bool,ord_less(int,A2),zero_zero(int)))
     => ( ( aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),B2) = aa(int,int,uminus_uminus(int),A2) )
      <=> ( B2 = aa(int,int,uminus_uminus(int),one_one(int)) ) ) ) ).

% int_div_minus_is_minus1
tff(fact_2741_powser__zero,axiom,
    ! [A: $tType] :
      ( real_V2822296259951069270ebra_1(A)
     => ! [F2: fun(nat,A)] : suminf(A,aTP_Lamp_bn(fun(nat,A),fun(nat,A),F2)) = aa(nat,A,F2,zero_zero(nat)) ) ).

% powser_zero
tff(fact_2742_signed__take__bit__Suc__minus__bit0,axiom,
    ! [N: nat,K: num] : aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,suc,N)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit0(K)))) = aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,N),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),K)))),aa(num,int,numeral_numeral(int),bit0(one2))) ).

% signed_take_bit_Suc_minus_bit0
tff(fact_2743_ceiling__divide__eq__div__numeral,axiom,
    ! [A2: num,B2: num] : archimedean_ceiling(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(num,real,numeral_numeral(real),A2)),aa(num,real,numeral_numeral(real),B2))) = aa(int,int,uminus_uminus(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),A2))),aa(num,int,numeral_numeral(int),B2))) ).

% ceiling_divide_eq_div_numeral
tff(fact_2744_signed__take__bit__0,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A2: A] : aa(A,A,bit_ri4674362597316999326ke_bit(A,zero_zero(nat)),A2) = aa(A,A,uminus_uminus(A),modulo_modulo(A,A2,aa(num,A,numeral_numeral(A),bit0(one2)))) ) ).

% signed_take_bit_0
tff(fact_2745_ceiling__minus__divide__eq__div__numeral,axiom,
    ! [A2: num,B2: num] : archimedean_ceiling(real,aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(num,real,numeral_numeral(real),A2)),aa(num,real,numeral_numeral(real),B2)))) = aa(int,int,uminus_uminus(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(num,int,numeral_numeral(int),A2)),aa(num,int,numeral_numeral(int),B2))) ).

% ceiling_minus_divide_eq_div_numeral
tff(fact_2746_signed__take__bit__Suc__bit1,axiom,
    ! [N: nat,K: num] : aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,suc,N)),aa(num,int,numeral_numeral(int),aa(num,num,bit1,K))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,N),aa(num,int,numeral_numeral(int),K))),aa(num,int,numeral_numeral(int),bit0(one2)))),one_one(int)) ).

% signed_take_bit_Suc_bit1
tff(fact_2747_signed__take__bit__Suc__minus__bit1,axiom,
    ! [N: nat,K: num] : aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,suc,N)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,K)))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,N),aa(int,int,minus_minus(int,aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),K))),one_one(int)))),aa(num,int,numeral_numeral(int),bit0(one2)))),one_one(int)) ).

% signed_take_bit_Suc_minus_bit1
tff(fact_2748_signed__take__bit__minus,axiom,
    ! [N: nat,K: int] : aa(int,int,bit_ri4674362597316999326ke_bit(int,N),aa(int,int,uminus_uminus(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,N),K))) = aa(int,int,bit_ri4674362597316999326ke_bit(int,N),aa(int,int,uminus_uminus(int),K)) ).

% signed_take_bit_minus
tff(fact_2749_zabs__def,axiom,
    ! [I: int] :
      ( ( pp(aa(int,bool,ord_less(int,I),zero_zero(int)))
       => ( aa(int,int,abs_abs(int),I) = aa(int,int,uminus_uminus(int),I) ) )
      & ( ~ pp(aa(int,bool,ord_less(int,I),zero_zero(int)))
       => ( aa(int,int,abs_abs(int),I) = I ) ) ) ).

% zabs_def
tff(fact_2750_signed__take__bit__add,axiom,
    ! [N: nat,K: int,L: int] : aa(int,int,bit_ri4674362597316999326ke_bit(int,N),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,N),K)),aa(int,int,bit_ri4674362597316999326ke_bit(int,N),L))) = aa(int,int,bit_ri4674362597316999326ke_bit(int,N),aa(int,int,aa(int,fun(int,int),plus_plus(int),K),L)) ).

% signed_take_bit_add
tff(fact_2751_signed__take__bit__mult,axiom,
    ! [N: nat,K: int,L: int] : aa(int,int,bit_ri4674362597316999326ke_bit(int,N),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,N),K)),aa(int,int,bit_ri4674362597316999326ke_bit(int,N),L))) = aa(int,int,bit_ri4674362597316999326ke_bit(int,N),aa(int,int,aa(int,fun(int,int),times_times(int),K),L)) ).

% signed_take_bit_mult
tff(fact_2752_signed__take__bit__diff,axiom,
    ! [N: nat,K: int,L: int] : aa(int,int,bit_ri4674362597316999326ke_bit(int,N),aa(int,int,minus_minus(int,aa(int,int,bit_ri4674362597316999326ke_bit(int,N),K)),aa(int,int,bit_ri4674362597316999326ke_bit(int,N),L))) = aa(int,int,bit_ri4674362597316999326ke_bit(int,N),aa(int,int,minus_minus(int,K),L)) ).

% signed_take_bit_diff
tff(fact_2753_uminus__int__code_I1_J,axiom,
    aa(int,int,uminus_uminus(int),zero_zero(int)) = zero_zero(int) ).

% uminus_int_code(1)
tff(fact_2754_int__cases2,axiom,
    ! [Z: int] :
      ( ! [N2: nat] : Z != aa(nat,int,semiring_1_of_nat(int),N2)
     => ~ ! [N2: nat] : Z != aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),N2)) ) ).

% int_cases2
tff(fact_2755_word__neg__numeral__alt,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [B2: num] : aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),B2)) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),B2))) ) ).

% word_neg_numeral_alt
tff(fact_2756_abs__zmult__eq__1,axiom,
    ! [M: int,N: int] :
      ( ( aa(int,int,abs_abs(int),aa(int,int,aa(int,fun(int,int),times_times(int),M),N)) = one_one(int) )
     => ( aa(int,int,abs_abs(int),M) = one_one(int) ) ) ).

% abs_zmult_eq_1
tff(fact_2757_int__of__nat__induct,axiom,
    ! [P: fun(int,bool),Z: int] :
      ( ! [N2: nat] : pp(aa(int,bool,P,aa(nat,int,semiring_1_of_nat(int),N2)))
     => ( ! [N2: nat] : pp(aa(int,bool,P,aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,suc,N2)))))
       => pp(aa(int,bool,P,Z)) ) ) ).

% int_of_nat_induct
tff(fact_2758_int__cases,axiom,
    ! [Z: int] :
      ( ! [N2: nat] : Z != aa(nat,int,semiring_1_of_nat(int),N2)
     => ~ ! [N2: nat] : Z != aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,suc,N2))) ) ).

% int_cases
tff(fact_2759_abs__div,axiom,
    ! [Y: int,X: int] :
      ( pp(aa(int,bool,dvd_dvd(int,Y),X))
     => ( aa(int,int,abs_abs(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),X),Y)) = aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,abs_abs(int),X)),aa(int,int,abs_abs(int),Y)) ) ) ).

% abs_div
tff(fact_2760_zmult__eq__1__iff,axiom,
    ! [M: int,N: int] :
      ( ( aa(int,int,aa(int,fun(int,int),times_times(int),M),N) = one_one(int) )
    <=> ( ( ( M = one_one(int) )
          & ( N = one_one(int) ) )
        | ( ( M = aa(int,int,uminus_uminus(int),one_one(int)) )
          & ( N = aa(int,int,uminus_uminus(int),one_one(int)) ) ) ) ) ).

% zmult_eq_1_iff
tff(fact_2761_pos__zmult__eq__1__iff__lemma,axiom,
    ! [M: int,N: int] :
      ( ( aa(int,int,aa(int,fun(int,int),times_times(int),M),N) = one_one(int) )
     => ( ( M = one_one(int) )
        | ( M = aa(int,int,uminus_uminus(int),one_one(int)) ) ) ) ).

% pos_zmult_eq_1_iff_lemma
tff(fact_2762_minus__int__code_I2_J,axiom,
    ! [L: int] : aa(int,int,minus_minus(int,zero_zero(int)),L) = aa(int,int,uminus_uminus(int),L) ).

% minus_int_code(2)
tff(fact_2763_zmod__zminus1__not__zero,axiom,
    ! [K: int,L: int] :
      ( ( modulo_modulo(int,aa(int,int,uminus_uminus(int),K),L) != zero_zero(int) )
     => ( modulo_modulo(int,K,L) != zero_zero(int) ) ) ).

% zmod_zminus1_not_zero
tff(fact_2764_zmod__zminus2__not__zero,axiom,
    ! [K: int,L: int] :
      ( ( modulo_modulo(int,K,aa(int,int,uminus_uminus(int),L)) != zero_zero(int) )
     => ( modulo_modulo(int,K,L) != zero_zero(int) ) ) ).

% zmod_zminus2_not_zero
tff(fact_2765_not__int__zless__negative,axiom,
    ! [N: nat,M: nat] : ~ pp(aa(int,bool,ord_less(int,aa(nat,int,semiring_1_of_nat(int),N)),aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),M)))) ).

% not_int_zless_negative
tff(fact_2766_word__order_Oextremum__uniqueI,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A)] :
          ( pp(aa(word(A),bool,ord_less_eq(word(A),aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A)))),A2))
         => ( A2 = aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A))) ) ) ) ).

% word_order.extremum_uniqueI
tff(fact_2767_max__word__not__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A))) != zero_zero(word(A)) ) ) ).

% max_word_not_0
tff(fact_2768_word__order_Onot__eq__extremum,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A)] :
          ( ( A2 != aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A))) )
        <=> pp(aa(word(A),bool,ord_less(word(A),A2),aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A))))) ) ) ).

% word_order.not_eq_extremum
tff(fact_2769_word__order_Oextremum__strict,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A)] : ~ pp(aa(word(A),bool,ord_less(word(A),aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A)))),A2)) ) ).

% word_order.extremum_strict
tff(fact_2770_word__not__simps_I3_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A)] : ~ pp(aa(word(A),bool,ord_less(word(A),aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A)))),Y)) ) ).

% word_not_simps(3)
tff(fact_2771_max__word__not__less,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A)] : ~ pp(aa(word(A),bool,ord_less(word(A),aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A)))),X)) ) ).

% max_word_not_less
tff(fact_2772_signed__take__bit__int__less__eq__self__iff,axiom,
    ! [N: nat,K: int] :
      ( pp(aa(int,bool,ord_less_eq(int,aa(int,int,bit_ri4674362597316999326ke_bit(int,N),K)),K))
    <=> pp(aa(int,bool,ord_less_eq(int,aa(int,int,uminus_uminus(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),N))),K)) ) ).

% signed_take_bit_int_less_eq_self_iff
tff(fact_2773_signed__take__bit__int__greater__eq__minus__exp,axiom,
    ! [N: nat,K: int] : pp(aa(int,bool,ord_less_eq(int,aa(int,int,uminus_uminus(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),N))),aa(int,int,bit_ri4674362597316999326ke_bit(int,N),K))) ).

% signed_take_bit_int_greater_eq_minus_exp
tff(fact_2774_signed__take__bit__int__greater__self__iff,axiom,
    ! [K: int,N: nat] :
      ( pp(aa(int,bool,ord_less(int,K),aa(int,int,bit_ri4674362597316999326ke_bit(int,N),K)))
    <=> pp(aa(int,bool,ord_less(int,K),aa(int,int,uminus_uminus(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),N)))) ) ).

% signed_take_bit_int_greater_self_iff
tff(fact_2775_signed__take__bit__int__eq__self__iff,axiom,
    ! [N: nat,K: int] :
      ( ( aa(int,int,bit_ri4674362597316999326ke_bit(int,N),K) = K )
    <=> ( pp(aa(int,bool,ord_less_eq(int,aa(int,int,uminus_uminus(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),N))),K))
        & pp(aa(int,bool,ord_less(int,K),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),N))) ) ) ).

% signed_take_bit_int_eq_self_iff
tff(fact_2776_signed__take__bit__int__eq__self,axiom,
    ! [N: nat,K: int] :
      ( pp(aa(int,bool,ord_less_eq(int,aa(int,int,uminus_uminus(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),N))),K))
     => ( pp(aa(int,bool,ord_less(int,K),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),N)))
       => ( aa(int,int,bit_ri4674362597316999326ke_bit(int,N),K) = K ) ) ) ).

% signed_take_bit_int_eq_self
tff(fact_2777_abs__mod__less,axiom,
    ! [L: int,K: int] :
      ( ( L != zero_zero(int) )
     => pp(aa(int,bool,ord_less(int,aa(int,int,abs_abs(int),modulo_modulo(int,K,L))),aa(int,int,abs_abs(int),L))) ) ).

% abs_mod_less
tff(fact_2778_dvd__imp__le__int,axiom,
    ! [I: int,D: int] :
      ( ( I != zero_zero(int) )
     => ( pp(aa(int,bool,dvd_dvd(int,D),I))
       => pp(aa(int,bool,ord_less_eq(int,aa(int,int,abs_abs(int),D)),aa(int,int,abs_abs(int),I))) ) ) ).

% dvd_imp_le_int
tff(fact_2779_int__cases4,axiom,
    ! [M: int] :
      ( ! [N2: nat] : M != aa(nat,int,semiring_1_of_nat(int),N2)
     => ~ ! [N2: nat] :
            ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),N2))
           => ( M != aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),N2)) ) ) ) ).

% int_cases4
tff(fact_2780_int__zle__neg,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa(int,bool,ord_less_eq(int,aa(nat,int,semiring_1_of_nat(int),N)),aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),M))))
    <=> ( ( N = zero_zero(nat) )
        & ( M = zero_zero(nat) ) ) ) ).

% int_zle_neg
tff(fact_2781_nonpos__int__cases,axiom,
    ! [K: int] :
      ( pp(aa(int,bool,ord_less_eq(int,K),zero_zero(int)))
     => ~ ! [N2: nat] : K != aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),N2)) ) ).

% nonpos_int_cases
tff(fact_2782_negative__zle__0,axiom,
    ! [N: nat] : pp(aa(int,bool,ord_less_eq(int,aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),N))),zero_zero(int))) ).

% negative_zle_0
tff(fact_2783_overflow__plus__one__self,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [P2: word(A)] :
          ( pp(aa(word(A),bool,ord_less_eq(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),one_one(word(A))),P2)),P2))
        <=> ( P2 = aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A))) ) ) ) ).

% overflow_plus_one_self
tff(fact_2784_plus__1__less,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A)] :
          ( pp(aa(word(A),bool,ord_less_eq(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),X),one_one(word(A)))),X))
        <=> ( X = aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A))) ) ) ) ).

% plus_1_less
tff(fact_2785_max__word__wrap,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A)] :
          ( ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),X),one_one(word(A))) = zero_zero(word(A)) )
         => ( X = aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A))) ) ) ) ).

% max_word_wrap
tff(fact_2786_word__add__no__overflow,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A)] :
          ( pp(aa(word(A),bool,ord_less(word(A),X),aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A)))))
         => pp(aa(word(A),bool,ord_less(word(A),X),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),X),one_one(word(A))))) ) ) ).

% word_add_no_overflow
tff(fact_2787_less__x__plus__1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A)] :
          ( ( X != aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A))) )
         => ( pp(aa(word(A),bool,ord_less(word(A),Y),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),X),one_one(word(A)))))
          <=> ( pp(aa(word(A),bool,ord_less(word(A),Y),X))
              | ( Y = X ) ) ) ) ) ).

% less_x_plus_1
tff(fact_2788_zmod__zminus1__eq__if,axiom,
    ! [A2: int,B2: int] :
      ( ( ( modulo_modulo(int,A2,B2) = zero_zero(int) )
       => ( modulo_modulo(int,aa(int,int,uminus_uminus(int),A2),B2) = zero_zero(int) ) )
      & ( ( modulo_modulo(int,A2,B2) != zero_zero(int) )
       => ( modulo_modulo(int,aa(int,int,uminus_uminus(int),A2),B2) = aa(int,int,minus_minus(int,B2),modulo_modulo(int,A2,B2)) ) ) ) ).

% zmod_zminus1_eq_if
tff(fact_2789_zmod__zminus2__eq__if,axiom,
    ! [A2: int,B2: int] :
      ( ( ( modulo_modulo(int,A2,B2) = zero_zero(int) )
       => ( modulo_modulo(int,A2,aa(int,int,uminus_uminus(int),B2)) = zero_zero(int) ) )
      & ( ( modulo_modulo(int,A2,B2) != zero_zero(int) )
       => ( modulo_modulo(int,A2,aa(int,int,uminus_uminus(int),B2)) = aa(int,int,minus_minus(int,modulo_modulo(int,A2,B2)),B2) ) ) ) ).

% zmod_zminus2_eq_if
tff(fact_2790_no__plus__overflow__neg,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A)] :
          ( pp(aa(word(A),bool,ord_less(word(A),X),aa(word(A),word(A),uminus_uminus(word(A)),Y)))
         => pp(aa(word(A),bool,ord_less_eq(word(A),X),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),X),Y))) ) ) ).

% no_plus_overflow_neg
tff(fact_2791_signed__take__bit__int__greater__eq,axiom,
    ! [K: int,N: nat] :
      ( pp(aa(int,bool,ord_less(int,K),aa(int,int,uminus_uminus(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),N))))
     => pp(aa(int,bool,ord_less_eq(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),K),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,suc,N)))),aa(int,int,bit_ri4674362597316999326ke_bit(int,N),K))) ) ).

% signed_take_bit_int_greater_eq
tff(fact_2792_zdvd__mult__cancel1,axiom,
    ! [M: int,N: int] :
      ( ( M != zero_zero(int) )
     => ( pp(aa(int,bool,dvd_dvd(int,aa(int,int,aa(int,fun(int,int),times_times(int),M),N)),M))
      <=> ( aa(int,int,abs_abs(int),N) = one_one(int) ) ) ) ).

% zdvd_mult_cancel1
tff(fact_2793_int__cases3,axiom,
    ! [K: int] :
      ( ( K != zero_zero(int) )
     => ( ! [N2: nat] :
            ( ( K = aa(nat,int,semiring_1_of_nat(int),N2) )
           => ~ pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),N2)) )
       => ~ ! [N2: nat] :
              ( ( K = aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),N2)) )
             => ~ pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),N2)) ) ) ) ).

% int_cases3
tff(fact_2794_not__zle__0__negative,axiom,
    ! [N: nat] : ~ pp(aa(int,bool,ord_less_eq(int,zero_zero(int)),aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,suc,N))))) ).

% not_zle_0_negative
tff(fact_2795_negD,axiom,
    ! [X: int] :
      ( pp(aa(int,bool,ord_less(int,X),zero_zero(int)))
     => ? [N2: nat] : X = aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,suc,N2))) ) ).

% negD
tff(fact_2796_negative__zless__0,axiom,
    ! [N: nat] : pp(aa(int,bool,ord_less(int,aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,suc,N)))),zero_zero(int))) ).

% negative_zless_0
tff(fact_2797_verit__less__mono__div__int2,axiom,
    ! [A3: int,B4: int,N: int] :
      ( pp(aa(int,bool,ord_less_eq(int,A3),B4))
     => ( pp(aa(int,bool,ord_less(int,zero_zero(int)),aa(int,int,uminus_uminus(int),N)))
       => pp(aa(int,bool,ord_less_eq(int,aa(int,int,aa(int,fun(int,int),divide_divide(int),B4),N)),aa(int,int,aa(int,fun(int,int),divide_divide(int),A3),N))) ) ) ).

% verit_less_mono_div_int2
tff(fact_2798_div__eq__minus1,axiom,
    ! [B2: int] :
      ( pp(aa(int,bool,ord_less(int,zero_zero(int)),B2))
     => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,uminus_uminus(int),one_one(int))),B2) = aa(int,int,uminus_uminus(int),one_one(int)) ) ) ).

% div_eq_minus1
tff(fact_2799_word__le__make__less,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),X: word(A)] :
          ( ( Y != aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A))) )
         => ( pp(aa(word(A),bool,ord_less_eq(word(A),X),Y))
          <=> pp(aa(word(A),bool,ord_less(word(A),X),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Y),one_one(word(A))))) ) ) ) ).

% word_le_make_less
tff(fact_2800_word__Suc__leq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [K: word(A),X: word(A)] :
          ( ( K != aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A))) )
         => ( pp(aa(word(A),bool,ord_less(word(A),X),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),K),one_one(word(A)))))
          <=> pp(aa(word(A),bool,ord_less_eq(word(A),X),K)) ) ) ) ).

% word_Suc_leq
tff(fact_2801_word__Suc__le,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),K: word(A)] :
          ( ( X != aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A))) )
         => ( pp(aa(word(A),bool,ord_less_eq(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),X),one_one(word(A)))),K))
          <=> pp(aa(word(A),bool,ord_less(word(A),X),K)) ) ) ) ).

% word_Suc_le
tff(fact_2802_ceiling__divide__eq__div,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [A2: int,B2: int] : archimedean_ceiling(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(int,A,ring_1_of_int(A),A2)),aa(int,A,ring_1_of_int(A),B2))) = aa(int,int,uminus_uminus(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,uminus_uminus(int),A2)),B2)) ) ).

% ceiling_divide_eq_div
tff(fact_2803_signed__take__bit__int__less__exp,axiom,
    ! [N: nat,K: int] : pp(aa(int,bool,ord_less(int,aa(int,int,bit_ri4674362597316999326ke_bit(int,N),K)),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),N))) ).

% signed_take_bit_int_less_exp
tff(fact_2804_even__signed__take__bit__iff,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [M: nat,A2: A] :
          ( pp(aa(A,bool,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,bit_ri4674362597316999326ke_bit(A,M),A2)))
        <=> pp(aa(A,bool,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A2)) ) ) ).

% even_signed_take_bit_iff
tff(fact_2805_even__abs__add__iff,axiom,
    ! [K: int,L: int] :
      ( pp(aa(int,bool,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,abs_abs(int),K)),L)))
    <=> pp(aa(int,bool,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),aa(int,int,aa(int,fun(int,int),plus_plus(int),K),L))) ) ).

% even_abs_add_iff
tff(fact_2806_even__add__abs__iff,axiom,
    ! [K: int,L: int] :
      ( pp(aa(int,bool,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),aa(int,int,aa(int,fun(int,int),plus_plus(int),K),aa(int,int,abs_abs(int),L))))
    <=> pp(aa(int,bool,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),aa(int,int,aa(int,fun(int,int),plus_plus(int),K),L))) ) ).

% even_add_abs_iff
tff(fact_2807_neg__int__cases,axiom,
    ! [K: int] :
      ( pp(aa(int,bool,ord_less(int,K),zero_zero(int)))
     => ~ ! [N2: nat] :
            ( ( K = aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),N2)) )
           => ~ pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),N2)) ) ) ).

% neg_int_cases
tff(fact_2808_minus__mod__int__eq,axiom,
    ! [L: int,K: int] :
      ( pp(aa(int,bool,ord_less_eq(int,zero_zero(int)),L))
     => ( modulo_modulo(int,aa(int,int,uminus_uminus(int),K),L) = aa(int,int,minus_minus(int,aa(int,int,minus_minus(int,L),one_one(int))),modulo_modulo(int,aa(int,int,minus_minus(int,K),one_one(int)),L)) ) ) ).

% minus_mod_int_eq
tff(fact_2809_zmod__minus1,axiom,
    ! [B2: int] :
      ( pp(aa(int,bool,ord_less(int,zero_zero(int)),B2))
     => ( modulo_modulo(int,aa(int,int,uminus_uminus(int),one_one(int)),B2) = aa(int,int,minus_minus(int,B2),one_one(int)) ) ) ).

% zmod_minus1
tff(fact_2810_zdiv__zminus1__eq__if,axiom,
    ! [B2: int,A2: int] :
      ( ( B2 != zero_zero(int) )
     => ( ( ( modulo_modulo(int,A2,B2) = zero_zero(int) )
         => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,uminus_uminus(int),A2)),B2) = aa(int,int,uminus_uminus(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),B2)) ) )
        & ( ( modulo_modulo(int,A2,B2) != zero_zero(int) )
         => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,uminus_uminus(int),A2)),B2) = aa(int,int,minus_minus(int,aa(int,int,uminus_uminus(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),B2))),one_one(int)) ) ) ) ) ).

% zdiv_zminus1_eq_if
tff(fact_2811_zdiv__zminus2__eq__if,axiom,
    ! [B2: int,A2: int] :
      ( ( B2 != zero_zero(int) )
     => ( ( ( modulo_modulo(int,A2,B2) = zero_zero(int) )
         => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),aa(int,int,uminus_uminus(int),B2)) = aa(int,int,uminus_uminus(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),B2)) ) )
        & ( ( modulo_modulo(int,A2,B2) != zero_zero(int) )
         => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),aa(int,int,uminus_uminus(int),B2)) = aa(int,int,minus_minus(int,aa(int,int,uminus_uminus(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),B2))),one_one(int)) ) ) ) ) ).

% zdiv_zminus2_eq_if
tff(fact_2812_monoseq__realpow,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,ord_less_eq(real,zero_zero(real)),X))
     => ( pp(aa(real,bool,ord_less_eq(real,X),one_one(real)))
       => topological_monoseq(real,aa(real,fun(nat,real),power_power(real),X)) ) ) ).

% monoseq_realpow
tff(fact_2813_signed__take__bit__int__less__self__iff,axiom,
    ! [N: nat,K: int] :
      ( pp(aa(int,bool,ord_less(int,aa(int,int,bit_ri4674362597316999326ke_bit(int,N),K)),K))
    <=> pp(aa(int,bool,ord_less_eq(int,aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),N)),K)) ) ).

% signed_take_bit_int_less_self_iff
tff(fact_2814_signed__take__bit__int__greater__eq__self__iff,axiom,
    ! [K: int,N: nat] :
      ( pp(aa(int,bool,ord_less_eq(int,K),aa(int,int,bit_ri4674362597316999326ke_bit(int,N),K)))
    <=> pp(aa(int,bool,ord_less(int,K),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),N))) ) ).

% signed_take_bit_int_greater_eq_self_iff
tff(fact_2815_nat__intermed__int__val,axiom,
    ! [M: nat,N: nat,F2: fun(nat,int),K: int] :
      ( ! [I2: nat] :
          ( ( pp(aa(nat,bool,ord_less_eq(nat,M),I2))
            & pp(aa(nat,bool,ord_less(nat,I2),N)) )
         => pp(aa(int,bool,ord_less_eq(int,aa(int,int,abs_abs(int),aa(int,int,minus_minus(int,aa(nat,int,F2,aa(nat,nat,suc,I2))),aa(nat,int,F2,I2)))),one_one(int))) )
     => ( pp(aa(nat,bool,ord_less_eq(nat,M),N))
       => ( pp(aa(int,bool,ord_less_eq(int,aa(nat,int,F2,M)),K))
         => ( pp(aa(int,bool,ord_less_eq(int,K),aa(nat,int,F2,N)))
           => ? [I2: nat] :
                ( pp(aa(nat,bool,ord_less_eq(nat,M),I2))
                & pp(aa(nat,bool,ord_less_eq(nat,I2),N))
                & ( aa(nat,int,F2,I2) = K ) ) ) ) ) ) ).

% nat_intermed_int_val
tff(fact_2816_dbl__dec__def,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [X: A] : neg_numeral_dbl_dec(A,X) = aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),X)),one_one(A)) ) ).

% dbl_dec_def
tff(fact_2817_incr__lemma,axiom,
    ! [D: int,Z: int,X: int] :
      ( pp(aa(int,bool,ord_less(int,zero_zero(int)),D))
     => pp(aa(int,bool,ord_less(int,Z),aa(int,int,aa(int,fun(int,int),plus_plus(int),X),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,abs_abs(int),aa(int,int,minus_minus(int,X),Z))),one_one(int))),D)))) ) ).

% incr_lemma
tff(fact_2818_decr__lemma,axiom,
    ! [D: int,X: int,Z: int] :
      ( pp(aa(int,bool,ord_less(int,zero_zero(int)),D))
     => pp(aa(int,bool,ord_less(int,aa(int,int,minus_minus(int,X),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,abs_abs(int),aa(int,int,minus_minus(int,X),Z))),one_one(int))),D))),Z)) ) ).

% decr_lemma
tff(fact_2819_minus__1__div__exp__eq__int,axiom,
    ! [N: nat] : aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,uminus_uminus(int),one_one(int))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),N)) = aa(int,int,uminus_uminus(int),one_one(int)) ).

% minus_1_div_exp_eq_int
tff(fact_2820_div__pos__neg__trivial,axiom,
    ! [K: int,L: int] :
      ( pp(aa(int,bool,ord_less(int,zero_zero(int)),K))
     => ( pp(aa(int,bool,ord_less_eq(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),K),L)),zero_zero(int)))
       => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),K),L) = aa(int,int,uminus_uminus(int),one_one(int)) ) ) ) ).

% div_pos_neg_trivial
tff(fact_2821_compl__mono,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,ord_less_eq(A,X),Y))
         => pp(aa(A,bool,ord_less_eq(A,aa(A,A,uminus_uminus(A),Y)),aa(A,A,uminus_uminus(A),X))) ) ) ).

% compl_mono
tff(fact_2822_compl__le__swap1,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Y: A,X: A] :
          ( pp(aa(A,bool,ord_less_eq(A,Y),aa(A,A,uminus_uminus(A),X)))
         => pp(aa(A,bool,ord_less_eq(A,X),aa(A,A,uminus_uminus(A),Y))) ) ) ).

% compl_le_swap1
tff(fact_2823_compl__le__swap2,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Y: A,X: A] :
          ( pp(aa(A,bool,ord_less_eq(A,aa(A,A,uminus_uminus(A),Y)),X))
         => pp(aa(A,bool,ord_less_eq(A,aa(A,A,uminus_uminus(A),X)),Y)) ) ) ).

% compl_le_swap2
tff(fact_2824_signed__take__bit__int__less__eq,axiom,
    ! [N: nat,K: int] :
      ( pp(aa(int,bool,ord_less_eq(int,aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),N)),K))
     => pp(aa(int,bool,ord_less_eq(int,aa(int,int,bit_ri4674362597316999326ke_bit(int,N),K)),aa(int,int,minus_minus(int,K),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,suc,N))))) ) ).

% signed_take_bit_int_less_eq
tff(fact_2825_nat__ivt__aux,axiom,
    ! [N: nat,F2: fun(nat,int),K: int] :
      ( ! [I2: nat] :
          ( pp(aa(nat,bool,ord_less(nat,I2),N))
         => pp(aa(int,bool,ord_less_eq(int,aa(int,int,abs_abs(int),aa(int,int,minus_minus(int,aa(nat,int,F2,aa(nat,nat,suc,I2))),aa(nat,int,F2,I2)))),one_one(int))) )
     => ( pp(aa(int,bool,ord_less_eq(int,aa(nat,int,F2,zero_zero(nat))),K))
       => ( pp(aa(int,bool,ord_less_eq(int,K),aa(nat,int,F2,N)))
         => ? [I2: nat] :
              ( pp(aa(nat,bool,ord_less_eq(nat,I2),N))
              & ( aa(nat,int,F2,I2) = K ) ) ) ) ) ).

% nat_ivt_aux
tff(fact_2826_int__bit__induct,axiom,
    ! [P: fun(int,bool),K: int] :
      ( pp(aa(int,bool,P,zero_zero(int)))
     => ( pp(aa(int,bool,P,aa(int,int,uminus_uminus(int),one_one(int))))
       => ( ! [K3: int] :
              ( pp(aa(int,bool,P,K3))
             => ( ( K3 != zero_zero(int) )
               => pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),times_times(int),K3),aa(num,int,numeral_numeral(int),bit0(one2))))) ) )
         => ( ! [K3: int] :
                ( pp(aa(int,bool,P,K3))
               => ( ( K3 != aa(int,int,uminus_uminus(int),one_one(int)) )
                 => pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),K3),aa(num,int,numeral_numeral(int),bit0(one2)))))) ) )
           => pp(aa(int,bool,P,K)) ) ) ) ) ).

% int_bit_induct
tff(fact_2827_m1mod2k,axiom,
    ! [N: nat] : modulo_modulo(int,aa(int,int,uminus_uminus(int),one_one(int)),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),N)) = aa(int,int,minus_minus(int,aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),N)),one_one(int)) ).

% m1mod2k
tff(fact_2828_nat0__intermed__int__val,axiom,
    ! [N: nat,F2: fun(nat,int),K: int] :
      ( ! [I2: nat] :
          ( pp(aa(nat,bool,ord_less(nat,I2),N))
         => pp(aa(int,bool,ord_less_eq(int,aa(int,int,abs_abs(int),aa(int,int,minus_minus(int,aa(nat,int,F2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I2),one_one(nat)))),aa(nat,int,F2,I2)))),one_one(int))) )
     => ( pp(aa(int,bool,ord_less_eq(int,aa(nat,int,F2,zero_zero(nat))),K))
       => ( pp(aa(int,bool,ord_less_eq(int,K),aa(nat,int,F2,N)))
         => ? [I2: nat] :
              ( pp(aa(nat,bool,ord_less_eq(nat,I2),N))
              & ( aa(nat,int,F2,I2) = K ) ) ) ) ) ).

% nat0_intermed_int_val
tff(fact_2829_sb__dec__lem_H,axiom,
    ! [K: nat,A2: int] :
      ( pp(aa(int,bool,ord_less_eq(int,aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),K)),A2))
     => pp(aa(int,bool,ord_less_eq(int,modulo_modulo(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),A2),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),K)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),K)))),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,uminus_uminus(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),K))),A2))) ) ).

% sb_dec_lem'
tff(fact_2830_m1mod22k,axiom,
    ! [N: nat] : modulo_modulo(int,aa(int,int,uminus_uminus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),N))) = aa(int,int,minus_minus(int,aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),N))),one_one(int)) ).

% m1mod22k
tff(fact_2831_sb__inc__lem_H,axiom,
    ! [A2: int,K: nat] :
      ( pp(aa(int,bool,ord_less(int,A2),aa(int,int,uminus_uminus(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),K))))
     => pp(aa(int,bool,ord_less_eq(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),A2),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),K))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,suc,K)))),modulo_modulo(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),A2),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),K)),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,suc,K))))) ) ).

% sb_inc_lem'
tff(fact_2832_sb__dec__lem,axiom,
    ! [K: nat,A2: int] :
      ( pp(aa(int,bool,ord_less_eq(int,zero_zero(int)),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,uminus_uminus(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),K))),A2)))
     => pp(aa(int,bool,ord_less_eq(int,modulo_modulo(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),A2),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),K)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),K)))),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,uminus_uminus(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),K))),A2))) ) ).

% sb_dec_lem
tff(fact_2833_signed__take__bit__Suc,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [N: nat,A2: A] : aa(A,A,bit_ri4674362597316999326ke_bit(A,aa(nat,nat,suc,N)),A2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),modulo_modulo(A,A2,aa(num,A,numeral_numeral(A),bit0(one2)))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,bit_ri4674362597316999326ke_bit(A,N),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(num,A,numeral_numeral(A),bit0(one2)))))) ) ).

% signed_take_bit_Suc
tff(fact_2834_suminf__geometric,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [C2: A] :
          ( pp(aa(real,bool,ord_less(real,real_V7770717601297561774m_norm(A,C2)),one_one(real)))
         => ( suminf(A,aa(A,fun(nat,A),power_power(A),C2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(A,A,minus_minus(A,one_one(A)),C2)) ) ) ) ).

% suminf_geometric
tff(fact_2835_suminf__zero,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topological_t2_space(A) )
     => ( suminf(A,aTP_Lamp_bo(nat,A)) = zero_zero(A) ) ) ).

% suminf_zero
tff(fact_2836_arctan__series,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,ord_less_eq(real,aa(real,real,abs_abs(real),X)),one_one(real)))
     => ( aa(real,real,arctan,X) = suminf(real,aTP_Lamp_bp(real,fun(nat,real),X)) ) ) ).

% arctan_series
tff(fact_2837_pi__series,axiom,
    aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(bit0(one2)))) = suminf(real,aTP_Lamp_bq(nat,real)) ).

% pi_series
tff(fact_2838_dbl__simps_I4_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ( neg_numeral_dbl(A,aa(A,A,uminus_uminus(A),one_one(A))) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),bit0(one2))) ) ) ).

% dbl_simps(4)
tff(fact_2839_summable__arctan__series,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,ord_less_eq(real,aa(real,real,abs_abs(real),X)),one_one(real)))
     => summable(real,aTP_Lamp_bp(real,fun(nat,real),X)) ) ).

% summable_arctan_series
tff(fact_2840_arctan__eq__zero__iff,axiom,
    ! [X: real] :
      ( ( aa(real,real,arctan,X) = zero_zero(real) )
    <=> ( X = zero_zero(real) ) ) ).

% arctan_eq_zero_iff
tff(fact_2841_arctan__zero__zero,axiom,
    aa(real,real,arctan,zero_zero(real)) = zero_zero(real) ).

% arctan_zero_zero
tff(fact_2842_dbl__simps_I2_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ( neg_numeral_dbl(A,zero_zero(A)) = zero_zero(A) ) ) ).

% dbl_simps(2)
tff(fact_2843_summable__single,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [I: nat,F2: fun(nat,A)] : summable(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_br(nat,fun(fun(nat,A),fun(nat,A)),I),F2)) ) ).

% summable_single
tff(fact_2844_summable__zero,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => summable(A,aTP_Lamp_bs(nat,A)) ) ).

% summable_zero
tff(fact_2845_summable__iff__shift,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A),K: nat] :
          ( summable(A,aa(nat,fun(nat,A),aTP_Lamp_bt(fun(nat,A),fun(nat,fun(nat,A)),F2),K))
        <=> summable(A,F2) ) ) ).

% summable_iff_shift
tff(fact_2846_zero__less__arctan__iff,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,ord_less(real,zero_zero(real)),aa(real,real,arctan,X)))
    <=> pp(aa(real,bool,ord_less(real,zero_zero(real)),X)) ) ).

% zero_less_arctan_iff
tff(fact_2847_arctan__less__zero__iff,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,ord_less(real,aa(real,real,arctan,X)),zero_zero(real)))
    <=> pp(aa(real,bool,ord_less(real,X),zero_zero(real))) ) ).

% arctan_less_zero_iff
tff(fact_2848_zero__le__arctan__iff,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,ord_less_eq(real,zero_zero(real)),aa(real,real,arctan,X)))
    <=> pp(aa(real,bool,ord_less_eq(real,zero_zero(real)),X)) ) ).

% zero_le_arctan_iff
tff(fact_2849_arctan__le__zero__iff,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,ord_less_eq(real,aa(real,real,arctan,X)),zero_zero(real)))
    <=> pp(aa(real,bool,ord_less_eq(real,X),zero_zero(real))) ) ).

% arctan_le_zero_iff
tff(fact_2850_dbl__simps_I5_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [K: num] : neg_numeral_dbl(A,aa(num,A,numeral_numeral(A),K)) = aa(num,A,numeral_numeral(A),bit0(K)) ) ).

% dbl_simps(5)
tff(fact_2851_summable__cmult__iff,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [C2: A,F2: fun(nat,A)] :
          ( summable(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_bu(A,fun(fun(nat,A),fun(nat,A)),C2),F2))
        <=> ( ( C2 = zero_zero(A) )
            | summable(A,F2) ) ) ) ).

% summable_cmult_iff
tff(fact_2852_dbl__simps_I1_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [K: num] : neg_numeral_dbl(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),K))) = aa(A,A,uminus_uminus(A),neg_numeral_dbl(A,aa(num,A,numeral_numeral(A),K))) ) ).

% dbl_simps(1)
tff(fact_2853_summable__divide__iff,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(nat,A),C2: A] :
          ( summable(A,aa(A,fun(nat,A),aTP_Lamp_bv(fun(nat,A),fun(A,fun(nat,A)),F2),C2))
        <=> ( ( C2 = zero_zero(A) )
            | summable(A,F2) ) ) ) ).

% summable_divide_iff
tff(fact_2854_summable__If__finite__set,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [A3: set(nat),F2: fun(nat,A)] :
          ( pp(aa(set(nat),bool,finite_finite2(nat),A3))
         => summable(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_bw(set(nat),fun(fun(nat,A),fun(nat,A)),A3),F2)) ) ) ).

% summable_If_finite_set
tff(fact_2855_summable__If__finite,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [P: fun(nat,bool),F2: fun(nat,A)] :
          ( pp(aa(set(nat),bool,finite_finite2(nat),collect(nat,P)))
         => summable(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_bx(fun(nat,bool),fun(fun(nat,A),fun(nat,A)),P),F2)) ) ) ).

% summable_If_finite
tff(fact_2856_dbl__simps_I3_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ( neg_numeral_dbl(A,one_one(A)) = aa(num,A,numeral_numeral(A),bit0(one2)) ) ) ).

% dbl_simps(3)
tff(fact_2857_summable__geometric__iff,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [C2: A] :
          ( summable(A,aa(A,fun(nat,A),power_power(A),C2))
        <=> pp(aa(real,bool,ord_less(real,real_V7770717601297561774m_norm(A,C2)),one_one(real))) ) ) ).

% summable_geometric_iff
tff(fact_2858_Compl__eq,axiom,
    ! [A: $tType,A3: set(A)] : aa(set(A),set(A),uminus_uminus(set(A)),A3) = collect(A,aTP_Lamp_by(set(A),fun(A,bool),A3)) ).

% Compl_eq
tff(fact_2859_Collect__neg__eq,axiom,
    ! [A: $tType,P: fun(A,bool)] : collect(A,aTP_Lamp_bz(fun(A,bool),fun(A,bool),P)) = aa(set(A),set(A),uminus_uminus(set(A)),collect(A,P)) ).

% Collect_neg_eq
tff(fact_2860_uminus__set__def,axiom,
    ! [A: $tType,A3: set(A)] : aa(set(A),set(A),uminus_uminus(set(A)),A3) = collect(A,aa(fun(A,bool),fun(A,bool),uminus_uminus(fun(A,bool)),aTP_Lamp_a(set(A),fun(A,bool),A3))) ).

% uminus_set_def
tff(fact_2861_summable__rabs__cancel,axiom,
    ! [F2: fun(nat,real)] :
      ( summable(real,aTP_Lamp_ca(fun(nat,real),fun(nat,real),F2))
     => summable(real,F2) ) ).

% summable_rabs_cancel
tff(fact_2862_summable__minus__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A)] :
          ( summable(A,aTP_Lamp_cb(fun(nat,A),fun(nat,A),F2))
        <=> summable(A,F2) ) ) ).

% summable_minus_iff
tff(fact_2863_summable__minus,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A)] :
          ( summable(A,F2)
         => summable(A,aTP_Lamp_cb(fun(nat,A),fun(nat,A),F2)) ) ) ).

% summable_minus
tff(fact_2864_summable__comparison__test_H,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [G: fun(nat,real),N4: nat,F2: fun(nat,A)] :
          ( summable(real,G)
         => ( ! [N2: nat] :
                ( pp(aa(nat,bool,ord_less_eq(nat,N4),N2))
               => pp(aa(real,bool,ord_less_eq(real,real_V7770717601297561774m_norm(A,aa(nat,A,F2,N2))),aa(nat,real,G,N2))) )
           => summable(A,F2) ) ) ) ).

% summable_comparison_test'
tff(fact_2865_summable__comparison__test,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [F2: fun(nat,A),G: fun(nat,real)] :
          ( ? [N9: nat] :
            ! [N2: nat] :
              ( pp(aa(nat,bool,ord_less_eq(nat,N9),N2))
             => pp(aa(real,bool,ord_less_eq(real,real_V7770717601297561774m_norm(A,aa(nat,A,F2,N2))),aa(nat,real,G,N2))) )
         => ( summable(real,G)
           => summable(A,F2) ) ) ) ).

% summable_comparison_test
tff(fact_2866_summable__mult2,axiom,
    ! [A: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [F2: fun(nat,A),C2: A] :
          ( summable(A,F2)
         => summable(A,aa(A,fun(nat,A),aTP_Lamp_cc(fun(nat,A),fun(A,fun(nat,A)),F2),C2)) ) ) ).

% summable_mult2
tff(fact_2867_summable__mult,axiom,
    ! [A: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [F2: fun(nat,A),C2: A] :
          ( summable(A,F2)
         => summable(A,aa(A,fun(nat,A),aTP_Lamp_cd(fun(nat,A),fun(A,fun(nat,A)),F2),C2)) ) ) ).

% summable_mult
tff(fact_2868_summable__divide,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(nat,A),C2: A] :
          ( summable(A,F2)
         => summable(A,aa(A,fun(nat,A),aTP_Lamp_bv(fun(nat,A),fun(A,fun(nat,A)),F2),C2)) ) ) ).

% summable_divide
tff(fact_2869_summable__Suc__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A)] :
          ( summable(A,aTP_Lamp_ce(fun(nat,A),fun(nat,A),F2))
        <=> summable(A,F2) ) ) ).

% summable_Suc_iff
tff(fact_2870_summable__diff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A),G: fun(nat,A)] :
          ( summable(A,F2)
         => ( summable(A,G)
           => summable(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_cf(fun(nat,A),fun(fun(nat,A),fun(nat,A)),F2),G)) ) ) ) ).

% summable_diff
tff(fact_2871_summable__ignore__initial__segment,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A),K: nat] :
          ( summable(A,F2)
         => summable(A,aa(nat,fun(nat,A),aTP_Lamp_bt(fun(nat,A),fun(nat,fun(nat,A)),F2),K)) ) ) ).

% summable_ignore_initial_segment
tff(fact_2872_summable__of__real,axiom,
    ! [A: $tType] :
      ( ( real_V2191834092415804123ebra_1(A)
        & real_V822414075346904944vector(A) )
     => ! [X6: fun(nat,real)] :
          ( summable(real,X6)
         => summable(A,aTP_Lamp_cg(fun(nat,real),fun(nat,A),X6)) ) ) ).

% summable_of_real
tff(fact_2873_summable__norm__cancel,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [F2: fun(nat,A)] :
          ( summable(real,aTP_Lamp_ch(fun(nat,A),fun(nat,real),F2))
         => summable(A,F2) ) ) ).

% summable_norm_cancel
tff(fact_2874_summable__add,axiom,
    ! [A: $tType] :
      ( ( topolo5987344860129210374id_add(A)
        & topological_t2_space(A) )
     => ! [F2: fun(nat,A),G: fun(nat,A)] :
          ( summable(A,F2)
         => ( summable(A,G)
           => summable(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_ci(fun(nat,A),fun(fun(nat,A),fun(nat,A)),F2),G)) ) ) ) ).

% summable_add
tff(fact_2875_summable__const__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [C2: A] :
          ( summable(A,aTP_Lamp_cj(A,fun(nat,A),C2))
        <=> ( C2 = zero_zero(A) ) ) ) ).

% summable_const_iff
tff(fact_2876_pi__neq__zero,axiom,
    pi != zero_zero(real) ).

% pi_neq_zero
tff(fact_2877_arctan__le__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa(real,bool,ord_less_eq(real,aa(real,real,arctan,X)),aa(real,real,arctan,Y)))
    <=> pp(aa(real,bool,ord_less_eq(real,X),Y)) ) ).

% arctan_le_iff
tff(fact_2878_arctan__monotone_H,axiom,
    ! [X: real,Y: real] :
      ( pp(aa(real,bool,ord_less_eq(real,X),Y))
     => pp(aa(real,bool,ord_less_eq(real,aa(real,real,arctan,X)),aa(real,real,arctan,Y))) ) ).

% arctan_monotone'
tff(fact_2879_suminf__le,axiom,
    ! [A: $tType] :
      ( ( ordere6911136660526730532id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F2: fun(nat,A),G: fun(nat,A)] :
          ( ! [N2: nat] : pp(aa(A,bool,ord_less_eq(A,aa(nat,A,F2,N2)),aa(nat,A,G,N2)))
         => ( summable(A,F2)
           => ( summable(A,G)
             => pp(aa(A,bool,ord_less_eq(A,suminf(A,F2)),suminf(A,G))) ) ) ) ) ).

% suminf_le
tff(fact_2880_summable__finite,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [N4: set(nat),F2: fun(nat,A)] :
          ( pp(aa(set(nat),bool,finite_finite2(nat),N4))
         => ( ! [N2: nat] :
                ( ~ pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),N2),N4))
               => ( aa(nat,A,F2,N2) = zero_zero(A) ) )
           => summable(A,F2) ) ) ) ).

% summable_finite
tff(fact_2881_summable__mult__D,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [C2: A,F2: fun(nat,A)] :
          ( summable(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_bu(A,fun(fun(nat,A),fun(nat,A)),C2),F2))
         => ( ( C2 != zero_zero(A) )
           => summable(A,F2) ) ) ) ).

% summable_mult_D
tff(fact_2882_summable__zero__power,axiom,
    ! [A: $tType] :
      ( ( comm_ring_1(A)
        & topolo4958980785337419405_space(A) )
     => summable(A,aa(A,fun(nat,A),power_power(A),zero_zero(A))) ) ).

% summable_zero_power
tff(fact_2883_suminf__add,axiom,
    ! [A: $tType] :
      ( ( topolo5987344860129210374id_add(A)
        & topological_t2_space(A) )
     => ! [F2: fun(nat,A),G: fun(nat,A)] :
          ( summable(A,F2)
         => ( summable(A,G)
           => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),suminf(A,F2)),suminf(A,G)) = suminf(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_ci(fun(nat,A),fun(fun(nat,A),fun(nat,A)),F2),G)) ) ) ) ) ).

% suminf_add
tff(fact_2884_suminf__mult2,axiom,
    ! [A: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [F2: fun(nat,A),C2: A] :
          ( summable(A,F2)
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),suminf(A,F2)),C2) = suminf(A,aa(A,fun(nat,A),aTP_Lamp_cc(fun(nat,A),fun(A,fun(nat,A)),F2),C2)) ) ) ) ).

% suminf_mult2
tff(fact_2885_suminf__mult,axiom,
    ! [A: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [F2: fun(nat,A),C2: A] :
          ( summable(A,F2)
         => ( suminf(A,aa(A,fun(nat,A),aTP_Lamp_cd(fun(nat,A),fun(A,fun(nat,A)),F2),C2)) = aa(A,A,aa(A,fun(A,A),times_times(A),C2),suminf(A,F2)) ) ) ) ).

% suminf_mult
tff(fact_2886_suminf__diff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A),G: fun(nat,A)] :
          ( summable(A,F2)
         => ( summable(A,G)
           => ( aa(A,A,minus_minus(A,suminf(A,F2)),suminf(A,G)) = suminf(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_cf(fun(nat,A),fun(fun(nat,A),fun(nat,A)),F2),G)) ) ) ) ) ).

% suminf_diff
tff(fact_2887_suminf__divide,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(nat,A),C2: A] :
          ( summable(A,F2)
         => ( suminf(A,aa(A,fun(nat,A),aTP_Lamp_bv(fun(nat,A),fun(A,fun(nat,A)),F2),C2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),suminf(A,F2)),C2) ) ) ) ).

% suminf_divide
tff(fact_2888_suminf__minus,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A)] :
          ( summable(A,F2)
         => ( suminf(A,aTP_Lamp_cb(fun(nat,A),fun(nat,A),F2)) = aa(A,A,uminus_uminus(A),suminf(A,F2)) ) ) ) ).

% suminf_minus
tff(fact_2889_powser__insidea,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [F2: fun(nat,A),X: A,Z: A] :
          ( summable(A,aa(A,fun(nat,A),aTP_Lamp_ck(fun(nat,A),fun(A,fun(nat,A)),F2),X))
         => ( pp(aa(real,bool,ord_less(real,real_V7770717601297561774m_norm(A,Z)),real_V7770717601297561774m_norm(A,X)))
           => summable(real,aa(A,fun(nat,real),aTP_Lamp_cl(fun(nat,A),fun(A,fun(nat,real)),F2),Z)) ) ) ) ).

% powser_insidea
tff(fact_2890_suminf__nonneg,axiom,
    ! [A: $tType] :
      ( ( ordere6911136660526730532id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F2: fun(nat,A)] :
          ( summable(A,F2)
         => ( ! [N2: nat] : pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),aa(nat,A,F2,N2)))
           => pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),suminf(A,F2))) ) ) ) ).

% suminf_nonneg
tff(fact_2891_suminf__eq__zero__iff,axiom,
    ! [A: $tType] :
      ( ( ordere6911136660526730532id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F2: fun(nat,A)] :
          ( summable(A,F2)
         => ( ! [N2: nat] : pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),aa(nat,A,F2,N2)))
           => ( ( suminf(A,F2) = zero_zero(A) )
            <=> ! [N5: nat] : aa(nat,A,F2,N5) = zero_zero(A) ) ) ) ) ).

% suminf_eq_zero_iff
tff(fact_2892_suminf__pos,axiom,
    ! [A: $tType] :
      ( ( ordere6911136660526730532id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F2: fun(nat,A)] :
          ( summable(A,F2)
         => ( ! [N2: nat] : pp(aa(A,bool,ord_less(A,zero_zero(A)),aa(nat,A,F2,N2)))
           => pp(aa(A,bool,ord_less(A,zero_zero(A)),suminf(A,F2))) ) ) ) ).

% suminf_pos
tff(fact_2893_summable__0__powser,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [F2: fun(nat,A)] : summable(A,aTP_Lamp_cm(fun(nat,A),fun(nat,A),F2)) ) ).

% summable_0_powser
tff(fact_2894_summable__zero__power_H,axiom,
    ! [A: $tType] :
      ( ( ring_1(A)
        & topolo4958980785337419405_space(A) )
     => ! [F2: fun(nat,A)] : summable(A,aTP_Lamp_cn(fun(nat,A),fun(nat,A),F2)) ) ).

% summable_zero_power'
tff(fact_2895_powser__split__head_I3_J,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V8999393235501362500lgebra(A) )
     => ! [F2: fun(nat,A),Z: A] :
          ( summable(A,aa(A,fun(nat,A),aTP_Lamp_co(fun(nat,A),fun(A,fun(nat,A)),F2),Z))
         => summable(A,aa(A,fun(nat,A),aTP_Lamp_cp(fun(nat,A),fun(A,fun(nat,A)),F2),Z)) ) ) ).

% powser_split_head(3)
tff(fact_2896_summable__powser__split__head,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [F2: fun(nat,A),Z: A] :
          ( summable(A,aa(A,fun(nat,A),aTP_Lamp_cq(fun(nat,A),fun(A,fun(nat,A)),F2),Z))
        <=> summable(A,aa(A,fun(nat,A),aTP_Lamp_ck(fun(nat,A),fun(A,fun(nat,A)),F2),Z)) ) ) ).

% summable_powser_split_head
tff(fact_2897_summable__powser__ignore__initial__segment,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [F2: fun(nat,A),M: nat,Z: A] :
          ( summable(A,aa(A,fun(nat,A),aa(nat,fun(A,fun(nat,A)),aTP_Lamp_cr(fun(nat,A),fun(nat,fun(A,fun(nat,A))),F2),M),Z))
        <=> summable(A,aa(A,fun(nat,A),aTP_Lamp_ck(fun(nat,A),fun(A,fun(nat,A)),F2),Z)) ) ) ).

% summable_powser_ignore_initial_segment
tff(fact_2898_pi__gt__zero,axiom,
    pp(aa(real,bool,ord_less(real,zero_zero(real)),pi)) ).

% pi_gt_zero
tff(fact_2899_pi__not__less__zero,axiom,
    ~ pp(aa(real,bool,ord_less(real,pi),zero_zero(real))) ).

% pi_not_less_zero
tff(fact_2900_pi__ge__zero,axiom,
    pp(aa(real,bool,ord_less_eq(real,zero_zero(real)),pi)) ).

% pi_ge_zero
tff(fact_2901_dbl__def,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [X: A] : neg_numeral_dbl(A,X) = aa(A,A,aa(A,fun(A,A),plus_plus(A),X),X) ) ).

% dbl_def
tff(fact_2902_summable__norm__comparison__test,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A),G: fun(nat,real)] :
          ( ? [N9: nat] :
            ! [N2: nat] :
              ( pp(aa(nat,bool,ord_less_eq(nat,N9),N2))
             => pp(aa(real,bool,ord_less_eq(real,real_V7770717601297561774m_norm(A,aa(nat,A,F2,N2))),aa(nat,real,G,N2))) )
         => ( summable(real,G)
           => summable(real,aTP_Lamp_cs(fun(nat,A),fun(nat,real),F2)) ) ) ) ).

% summable_norm_comparison_test
tff(fact_2903_summable__rabs__comparison__test,axiom,
    ! [F2: fun(nat,real),G: fun(nat,real)] :
      ( ? [N9: nat] :
        ! [N2: nat] :
          ( pp(aa(nat,bool,ord_less_eq(nat,N9),N2))
         => pp(aa(real,bool,ord_less_eq(real,aa(real,real,abs_abs(real),aa(nat,real,F2,N2))),aa(nat,real,G,N2))) )
     => ( summable(real,G)
       => summable(real,aTP_Lamp_ca(fun(nat,real),fun(nat,real),F2)) ) ) ).

% summable_rabs_comparison_test
tff(fact_2904_summable__rabs,axiom,
    ! [F2: fun(nat,real)] :
      ( summable(real,aTP_Lamp_ca(fun(nat,real),fun(nat,real),F2))
     => pp(aa(real,bool,ord_less_eq(real,aa(real,real,abs_abs(real),suminf(real,F2))),suminf(real,aTP_Lamp_ca(fun(nat,real),fun(nat,real),F2)))) ) ).

% summable_rabs
tff(fact_2905_suminf__of__real,axiom,
    ! [A: $tType] :
      ( ( real_V2191834092415804123ebra_1(A)
        & real_V822414075346904944vector(A) )
     => ! [X6: fun(nat,real)] :
          ( summable(real,X6)
         => ( aa(real,A,real_Vector_of_real(A),suminf(real,X6)) = suminf(A,aTP_Lamp_cg(fun(nat,real),fun(nat,A),X6)) ) ) ) ).

% suminf_of_real
tff(fact_2906_arctan__ubound,axiom,
    ! [Y: real] : pp(aa(real,bool,ord_less(real,aa(real,real,arctan,Y)),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))) ).

% arctan_ubound
tff(fact_2907_arctan__one,axiom,
    aa(real,real,arctan,one_one(real)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(bit0(one2)))) ).

% arctan_one
tff(fact_2908_suminf__pos__iff,axiom,
    ! [A: $tType] :
      ( ( ordere6911136660526730532id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F2: fun(nat,A)] :
          ( summable(A,F2)
         => ( ! [N2: nat] : pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),aa(nat,A,F2,N2)))
           => ( pp(aa(A,bool,ord_less(A,zero_zero(A)),suminf(A,F2)))
            <=> ? [I3: nat] : pp(aa(A,bool,ord_less(A,zero_zero(A)),aa(nat,A,F2,I3))) ) ) ) ) ).

% suminf_pos_iff
tff(fact_2909_suminf__pos2,axiom,
    ! [A: $tType] :
      ( ( ordere6911136660526730532id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F2: fun(nat,A),I: nat] :
          ( summable(A,F2)
         => ( ! [N2: nat] : pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),aa(nat,A,F2,N2)))
           => ( pp(aa(A,bool,ord_less(A,zero_zero(A)),aa(nat,A,F2,I)))
             => pp(aa(A,bool,ord_less(A,zero_zero(A)),suminf(A,F2))) ) ) ) ) ).

% suminf_pos2
tff(fact_2910_suminf__split__head,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A)] :
          ( summable(A,F2)
         => ( suminf(A,aTP_Lamp_ce(fun(nat,A),fun(nat,A),F2)) = aa(A,A,minus_minus(A,suminf(A,F2)),aa(nat,A,F2,zero_zero(nat))) ) ) ) ).

% suminf_split_head
tff(fact_2911_summable__geometric,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [C2: A] :
          ( pp(aa(real,bool,ord_less(real,real_V7770717601297561774m_norm(A,C2)),one_one(real)))
         => summable(A,aa(A,fun(nat,A),power_power(A),C2)) ) ) ).

% summable_geometric
tff(fact_2912_complete__algebra__summable__geometric,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [X: A] :
          ( pp(aa(real,bool,ord_less(real,real_V7770717601297561774m_norm(A,X)),one_one(real)))
         => summable(A,aa(A,fun(nat,A),power_power(A),X)) ) ) ).

% complete_algebra_summable_geometric
tff(fact_2913_arctan__bounded,axiom,
    ! [Y: real] :
      ( pp(aa(real,bool,ord_less(real,aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),aa(real,real,arctan,Y)))
      & pp(aa(real,bool,ord_less(real,aa(real,real,arctan,Y)),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))) ) ).

% arctan_bounded
tff(fact_2914_arctan__lbound,axiom,
    ! [Y: real] : pp(aa(real,bool,ord_less(real,aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),aa(real,real,arctan,Y))) ).

% arctan_lbound
tff(fact_2915_summable__norm,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [F2: fun(nat,A)] :
          ( summable(real,aTP_Lamp_ch(fun(nat,A),fun(nat,real),F2))
         => pp(aa(real,bool,ord_less_eq(real,real_V7770717601297561774m_norm(A,suminf(A,F2))),suminf(real,aTP_Lamp_ch(fun(nat,A),fun(nat,real),F2)))) ) ) ).

% summable_norm
tff(fact_2916_powser__inside,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V8999393235501362500lgebra(A) )
     => ! [F2: fun(nat,A),X: A,Z: A] :
          ( summable(A,aa(A,fun(nat,A),aTP_Lamp_co(fun(nat,A),fun(A,fun(nat,A)),F2),X))
         => ( pp(aa(real,bool,ord_less(real,real_V7770717601297561774m_norm(A,Z)),real_V7770717601297561774m_norm(A,X)))
           => summable(A,aa(A,fun(nat,A),aTP_Lamp_co(fun(nat,A),fun(A,fun(nat,A)),F2),Z)) ) ) ) ).

% powser_inside
tff(fact_2917_machin__Euler,axiom,
    aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),aa(num,num,bit1,bit0(one2)))),aa(real,real,arctan,aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),aa(num,num,bit1,aa(num,num,bit1,one2))))))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),aa(real,real,arctan,aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(num,real,numeral_numeral(real),aa(num,num,bit1,one2))),aa(num,real,numeral_numeral(real),aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,bit0(bit0(one2))))))))))) = aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(bit0(one2)))) ).

% machin_Euler
tff(fact_2918_machin,axiom,
    aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(bit0(one2)))) = aa(real,real,minus_minus(real,aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(bit0(one2)))),aa(real,real,arctan,aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),aa(num,num,bit1,bit0(one2))))))),aa(real,real,arctan,aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,bit0(aa(num,num,bit1,aa(num,num,bit1,one2))))))))))) ).

% machin
tff(fact_2919_pi__less__4,axiom,
    pp(aa(real,bool,ord_less(real,pi),aa(num,real,numeral_numeral(real),bit0(bit0(one2))))) ).

% pi_less_4
tff(fact_2920_pi__ge__two,axiom,
    pp(aa(real,bool,ord_less_eq(real,aa(num,real,numeral_numeral(real),bit0(one2))),pi)) ).

% pi_ge_two
tff(fact_2921_pi__half__neq__two,axiom,
    aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))) != aa(num,real,numeral_numeral(real),bit0(one2)) ).

% pi_half_neq_two
tff(fact_2922_powser__split__head_I1_J,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V8999393235501362500lgebra(A) )
     => ! [F2: fun(nat,A),Z: A] :
          ( summable(A,aa(A,fun(nat,A),aTP_Lamp_co(fun(nat,A),fun(A,fun(nat,A)),F2),Z))
         => ( suminf(A,aa(A,fun(nat,A),aTP_Lamp_co(fun(nat,A),fun(A,fun(nat,A)),F2),Z)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,F2,zero_zero(nat))),aa(A,A,aa(A,fun(A,A),times_times(A),suminf(A,aa(A,fun(nat,A),aTP_Lamp_cp(fun(nat,A),fun(A,fun(nat,A)),F2),Z))),Z)) ) ) ) ).

% powser_split_head(1)
tff(fact_2923_powser__split__head_I2_J,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V8999393235501362500lgebra(A) )
     => ! [F2: fun(nat,A),Z: A] :
          ( summable(A,aa(A,fun(nat,A),aTP_Lamp_co(fun(nat,A),fun(A,fun(nat,A)),F2),Z))
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),suminf(A,aa(A,fun(nat,A),aTP_Lamp_cp(fun(nat,A),fun(A,fun(nat,A)),F2),Z))),Z) = aa(A,A,minus_minus(A,suminf(A,aa(A,fun(nat,A),aTP_Lamp_co(fun(nat,A),fun(A,fun(nat,A)),F2),Z))),aa(nat,A,F2,zero_zero(nat))) ) ) ) ).

% powser_split_head(2)
tff(fact_2924_suminf__exist__split,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [R2: real,F2: fun(nat,A)] :
          ( pp(aa(real,bool,ord_less(real,zero_zero(real)),R2))
         => ( summable(A,F2)
           => ? [N10: nat] :
              ! [N11: nat] :
                ( pp(aa(nat,bool,ord_less_eq(nat,N10),N11))
               => pp(aa(real,bool,ord_less(real,real_V7770717601297561774m_norm(A,suminf(A,aa(nat,fun(nat,A),aTP_Lamp_bt(fun(nat,A),fun(nat,fun(nat,A)),F2),N11)))),R2)) ) ) ) ) ).

% suminf_exist_split
tff(fact_2925_summable__power__series,axiom,
    ! [F2: fun(nat,real),Z: real] :
      ( ! [I2: nat] : pp(aa(real,bool,ord_less_eq(real,aa(nat,real,F2,I2)),one_one(real)))
     => ( ! [I2: nat] : pp(aa(real,bool,ord_less_eq(real,zero_zero(real)),aa(nat,real,F2,I2)))
       => ( pp(aa(real,bool,ord_less_eq(real,zero_zero(real)),Z))
         => ( pp(aa(real,bool,ord_less(real,Z),one_one(real)))
           => summable(real,aa(real,fun(nat,real),aTP_Lamp_ct(fun(nat,real),fun(real,fun(nat,real)),F2),Z)) ) ) ) ) ).

% summable_power_series
tff(fact_2926_Abel__lemma,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [R2: real,R0: real,A2: fun(nat,A),M7: real] :
          ( pp(aa(real,bool,ord_less_eq(real,zero_zero(real)),R2))
         => ( pp(aa(real,bool,ord_less(real,R2),R0))
           => ( ! [N2: nat] : pp(aa(real,bool,ord_less_eq(real,aa(real,real,aa(real,fun(real,real),times_times(real),real_V7770717601297561774m_norm(A,aa(nat,A,A2,N2))),aa(nat,real,aa(real,fun(nat,real),power_power(real),R0),N2))),M7))
             => summable(real,aa(fun(nat,A),fun(nat,real),aTP_Lamp_cu(real,fun(fun(nat,A),fun(nat,real)),R2),A2)) ) ) ) ) ).

% Abel_lemma
tff(fact_2927_summable__ratio__test,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [C2: real,N4: nat,F2: fun(nat,A)] :
          ( pp(aa(real,bool,ord_less(real,C2),one_one(real)))
         => ( ! [N2: nat] :
                ( pp(aa(nat,bool,ord_less_eq(nat,N4),N2))
               => pp(aa(real,bool,ord_less_eq(real,real_V7770717601297561774m_norm(A,aa(nat,A,F2,aa(nat,nat,suc,N2)))),aa(real,real,aa(real,fun(real,real),times_times(real),C2),real_V7770717601297561774m_norm(A,aa(nat,A,F2,N2))))) )
           => summable(A,F2) ) ) ) ).

% summable_ratio_test
tff(fact_2928_pi__half__neq__zero,axiom,
    aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))) != zero_zero(real) ).

% pi_half_neq_zero
tff(fact_2929_pi__half__less__two,axiom,
    pp(aa(real,bool,ord_less(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2)))),aa(num,real,numeral_numeral(real),bit0(one2)))) ).

% pi_half_less_two
tff(fact_2930_pi__half__le__two,axiom,
    pp(aa(real,bool,ord_less_eq(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2)))),aa(num,real,numeral_numeral(real),bit0(one2)))) ).

% pi_half_le_two
tff(fact_2931_pi__half__gt__zero,axiom,
    pp(aa(real,bool,ord_less(real,zero_zero(real)),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))) ).

% pi_half_gt_zero
tff(fact_2932_pi__half__ge__zero,axiom,
    pp(aa(real,bool,ord_less_eq(real,zero_zero(real)),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))) ).

% pi_half_ge_zero
tff(fact_2933_m2pi__less__pi,axiom,
    pp(aa(real,bool,ord_less(real,aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),pi))),pi)) ).

% m2pi_less_pi
tff(fact_2934_minus__pi__half__less__zero,axiom,
    pp(aa(real,bool,ord_less(real,aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),zero_zero(real))) ).

% minus_pi_half_less_zero
tff(fact_2935_arctan__add,axiom,
    ! [X: real,Y: real] :
      ( pp(aa(real,bool,ord_less_eq(real,aa(real,real,abs_abs(real),X)),one_one(real)))
     => ( pp(aa(real,bool,ord_less(real,aa(real,real,abs_abs(real),Y)),one_one(real)))
       => ( aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,arctan,X)),aa(real,real,arctan,Y)) = aa(real,real,arctan,aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),X),Y)),aa(real,real,minus_minus(real,one_one(real)),aa(real,real,aa(real,fun(real,real),times_times(real),X),Y)))) ) ) ) ).

% arctan_add
tff(fact_2936_arctan__double,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,ord_less(real,aa(real,real,abs_abs(real),X)),one_one(real)))
     => ( aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),aa(real,real,arctan,X)) = aa(real,real,arctan,aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),X)),aa(real,real,minus_minus(real,one_one(real)),aa(nat,real,aa(real,fun(nat,real),power_power(real),X),aa(num,nat,numeral_numeral(nat),bit0(one2)))))) ) ) ).

% arctan_double
tff(fact_2937_sin__cos__npi,axiom,
    ! [N: nat] : sin(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)))),pi)),aa(num,real,numeral_numeral(real),bit0(one2)))) = aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,uminus_uminus(real),one_one(real))),N) ).

% sin_cos_npi
tff(fact_2938_signed__take__bit__numeral__minus__bit1,axiom,
    ! [L: num,K: num] : aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(num,nat,numeral_numeral(nat),L)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,K)))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,pred_numeral(L)),aa(int,int,minus_minus(int,aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),K))),one_one(int)))),aa(num,int,numeral_numeral(int),bit0(one2)))),one_one(int)) ).

% signed_take_bit_numeral_minus_bit1
tff(fact_2939_cos__pi__eq__zero,axiom,
    ! [M: nat] : cos(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),times_times(real),pi),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M))))),aa(num,real,numeral_numeral(real),bit0(one2)))) = zero_zero(real) ).

% cos_pi_eq_zero
tff(fact_2940_sum__gp,axiom,
    ! [A: $tType] :
      ( ( division_ring(A)
        & comm_ring(A) )
     => ! [N: nat,M: nat,X: A] :
          ( ( pp(aa(nat,bool,ord_less(nat,N),M))
           => ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),power_power(A),X)),set_or1337092689740270186AtMost(nat,M,N)) = zero_zero(A) ) )
          & ( ~ pp(aa(nat,bool,ord_less(nat,N),M))
           => ( ( ( X = one_one(A) )
               => ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),power_power(A),X)),set_or1337092689740270186AtMost(nat,M,N)) = aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat))),M)) ) )
              & ( ( X != one_one(A) )
               => ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),power_power(A),X)),set_or1337092689740270186AtMost(nat,M,N)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,minus_minus(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),X),M)),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(nat,nat,suc,N)))),aa(A,A,minus_minus(A,one_one(A)),X)) ) ) ) ) ) ) ).

% sum_gp
tff(fact_2941_signed__take__bit__numeral__bit1,axiom,
    ! [L: num,K: num] : aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(num,nat,numeral_numeral(nat),L)),aa(num,int,numeral_numeral(int),aa(num,num,bit1,K))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,pred_numeral(L)),aa(num,int,numeral_numeral(int),K))),aa(num,int,numeral_numeral(int),bit0(one2)))),one_one(int)) ).

% signed_take_bit_numeral_bit1
tff(fact_2942_summable__complex__of__real,axiom,
    ! [F2: fun(nat,real)] :
      ( summable(complex,aTP_Lamp_cv(fun(nat,real),fun(nat,complex),F2))
    <=> summable(real,F2) ) ).

% summable_complex_of_real
tff(fact_2943_sin__zero,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ( sin(A,zero_zero(A)) = zero_zero(A) ) ) ).

% sin_zero
tff(fact_2944_sum_Oneutral__const,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ! [A3: set(B)] : aa(set(B),A,groups7311177749621191930dd_sum(B,A,aTP_Lamp_cw(B,A)),A3) = zero_zero(A) ) ).

% sum.neutral_const
tff(fact_2945_of__nat__sum,axiom,
    ! [A: $tType,B: $tType] :
      ( semiring_1(A)
     => ! [F2: fun(B,nat),A3: set(B)] : aa(nat,A,semiring_1_of_nat(A),aa(set(B),nat,groups7311177749621191930dd_sum(B,nat,F2),A3)) = aa(set(B),A,groups7311177749621191930dd_sum(B,A,aTP_Lamp_cx(fun(B,nat),fun(B,A),F2)),A3) ) ).

% of_nat_sum
tff(fact_2946_of__int__sum,axiom,
    ! [A: $tType,B: $tType] :
      ( ring_1(A)
     => ! [F2: fun(B,int),A3: set(B)] : aa(int,A,ring_1_of_int(A),aa(set(B),int,groups7311177749621191930dd_sum(B,int,F2),A3)) = aa(set(B),A,groups7311177749621191930dd_sum(B,A,aTP_Lamp_cy(fun(B,int),fun(B,A),F2)),A3) ) ).

% of_int_sum
tff(fact_2947_abs__sum__abs,axiom,
    ! [B: $tType,A: $tType] :
      ( ordere166539214618696060dd_abs(B)
     => ! [F2: fun(A,B),A3: set(A)] : aa(B,B,abs_abs(B),aa(set(A),B,groups7311177749621191930dd_sum(A,B,aTP_Lamp_cz(fun(A,B),fun(A,B),F2)),A3)) = aa(set(A),B,groups7311177749621191930dd_sum(A,B,aTP_Lamp_cz(fun(A,B),fun(A,B),F2)),A3) ) ).

% abs_sum_abs
tff(fact_2948_of__real__sum,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [F2: fun(B,real),S3: set(B)] : aa(real,A,real_Vector_of_real(A),aa(set(B),real,groups7311177749621191930dd_sum(B,real,F2),S3)) = aa(set(B),A,groups7311177749621191930dd_sum(B,A,aTP_Lamp_da(fun(B,real),fun(B,A),F2)),S3) ) ).

% of_real_sum
tff(fact_2949_sum_Oinfinite,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ! [A3: set(B),G: fun(B,A)] :
          ( ~ pp(aa(set(B),bool,finite_finite2(B),A3))
         => ( aa(set(B),A,groups7311177749621191930dd_sum(B,A,G),A3) = zero_zero(A) ) ) ) ).

% sum.infinite
tff(fact_2950_sum__eq__0__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [F3: set(B),F2: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),F3))
         => ( ( aa(set(B),A,groups7311177749621191930dd_sum(B,A,F2),F3) = zero_zero(A) )
          <=> ! [X2: B] :
                ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X2),F3))
               => ( aa(B,A,F2,X2) = zero_zero(A) ) ) ) ) ) ).

% sum_eq_0_iff
tff(fact_2951_cos__zero,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ( cos(A,zero_zero(A)) = one_one(A) ) ) ).

% cos_zero
tff(fact_2952_pred__numeral__simps_I1_J,axiom,
    pred_numeral(one2) = zero_zero(nat) ).

% pred_numeral_simps(1)
tff(fact_2953_eq__numeral__Suc,axiom,
    ! [K: num,N: nat] :
      ( ( aa(num,nat,numeral_numeral(nat),K) = aa(nat,nat,suc,N) )
    <=> ( pred_numeral(K) = N ) ) ).

% eq_numeral_Suc
tff(fact_2954_Suc__eq__numeral,axiom,
    ! [N: nat,K: num] :
      ( ( aa(nat,nat,suc,N) = aa(num,nat,numeral_numeral(nat),K) )
    <=> ( N = pred_numeral(K) ) ) ).

% Suc_eq_numeral
tff(fact_2955_sin__pi,axiom,
    sin(real,pi) = zero_zero(real) ).

% sin_pi
tff(fact_2956_sum_Odelta,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ! [S: set(B),A2: B,B2: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),S))
         => ( ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A2),S))
             => ( aa(set(B),A,groups7311177749621191930dd_sum(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_db(B,fun(fun(B,A),fun(B,A)),A2),B2)),S) = aa(B,A,B2,A2) ) )
            & ( ~ pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A2),S))
             => ( aa(set(B),A,groups7311177749621191930dd_sum(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_db(B,fun(fun(B,A),fun(B,A)),A2),B2)),S) = zero_zero(A) ) ) ) ) ) ).

% sum.delta
tff(fact_2957_sum_Odelta_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ! [S: set(B),A2: B,B2: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),S))
         => ( ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A2),S))
             => ( aa(set(B),A,groups7311177749621191930dd_sum(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_dc(B,fun(fun(B,A),fun(B,A)),A2),B2)),S) = aa(B,A,B2,A2) ) )
            & ( ~ pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A2),S))
             => ( aa(set(B),A,groups7311177749621191930dd_sum(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_dc(B,fun(fun(B,A),fun(B,A)),A2),B2)),S) = zero_zero(A) ) ) ) ) ) ).

% sum.delta'
tff(fact_2958_sum__abs,axiom,
    ! [B: $tType,A: $tType] :
      ( ordere166539214618696060dd_abs(B)
     => ! [F2: fun(A,B),A3: set(A)] : pp(aa(B,bool,ord_less_eq(B,aa(B,B,abs_abs(B),aa(set(A),B,groups7311177749621191930dd_sum(A,B,F2),A3))),aa(set(A),B,groups7311177749621191930dd_sum(A,B,aTP_Lamp_cz(fun(A,B),fun(A,B),F2)),A3))) ) ).

% sum_abs
tff(fact_2959_pred__numeral__simps_I3_J,axiom,
    ! [K: num] : pred_numeral(aa(num,num,bit1,K)) = aa(num,nat,numeral_numeral(nat),bit0(K)) ).

% pred_numeral_simps(3)
tff(fact_2960_less__numeral__Suc,axiom,
    ! [K: num,N: nat] :
      ( pp(aa(nat,bool,ord_less(nat,aa(num,nat,numeral_numeral(nat),K)),aa(nat,nat,suc,N)))
    <=> pp(aa(nat,bool,ord_less(nat,pred_numeral(K)),N)) ) ).

% less_numeral_Suc
tff(fact_2961_less__Suc__numeral,axiom,
    ! [N: nat,K: num] :
      ( pp(aa(nat,bool,ord_less(nat,aa(nat,nat,suc,N)),aa(num,nat,numeral_numeral(nat),K)))
    <=> pp(aa(nat,bool,ord_less(nat,N),pred_numeral(K))) ) ).

% less_Suc_numeral
tff(fact_2962_le__numeral__Suc,axiom,
    ! [K: num,N: nat] :
      ( pp(aa(nat,bool,ord_less_eq(nat,aa(num,nat,numeral_numeral(nat),K)),aa(nat,nat,suc,N)))
    <=> pp(aa(nat,bool,ord_less_eq(nat,pred_numeral(K)),N)) ) ).

% le_numeral_Suc
tff(fact_2963_le__Suc__numeral,axiom,
    ! [N: nat,K: num] :
      ( pp(aa(nat,bool,ord_less_eq(nat,aa(nat,nat,suc,N)),aa(num,nat,numeral_numeral(nat),K)))
    <=> pp(aa(nat,bool,ord_less_eq(nat,N),pred_numeral(K))) ) ).

% le_Suc_numeral
tff(fact_2964_diff__numeral__Suc,axiom,
    ! [K: num,N: nat] : aa(nat,nat,minus_minus(nat,aa(num,nat,numeral_numeral(nat),K)),aa(nat,nat,suc,N)) = aa(nat,nat,minus_minus(nat,pred_numeral(K)),N) ).

% diff_numeral_Suc
tff(fact_2965_diff__Suc__numeral,axiom,
    ! [N: nat,K: num] : aa(nat,nat,minus_minus(nat,aa(nat,nat,suc,N)),aa(num,nat,numeral_numeral(nat),K)) = aa(nat,nat,minus_minus(nat,N),pred_numeral(K)) ).

% diff_Suc_numeral
tff(fact_2966_sin__of__real__pi,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ( sin(A,aa(real,A,real_Vector_of_real(A),pi)) = zero_zero(A) ) ) ).

% sin_of_real_pi
tff(fact_2967_cos__pi,axiom,
    cos(real,pi) = aa(real,real,uminus_uminus(real),one_one(real)) ).

% cos_pi
tff(fact_2968_cos__periodic__pi2,axiom,
    ! [X: real] : cos(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),pi),X)) = aa(real,real,uminus_uminus(real),cos(real,X)) ).

% cos_periodic_pi2
tff(fact_2969_cos__periodic__pi,axiom,
    ! [X: real] : cos(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),X),pi)) = aa(real,real,uminus_uminus(real),cos(real,X)) ).

% cos_periodic_pi
tff(fact_2970_sin__periodic__pi2,axiom,
    ! [X: real] : sin(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),pi),X)) = aa(real,real,uminus_uminus(real),sin(real,X)) ).

% sin_periodic_pi2
tff(fact_2971_sin__periodic__pi,axiom,
    ! [X: real] : sin(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),X),pi)) = aa(real,real,uminus_uminus(real),sin(real,X)) ).

% sin_periodic_pi
tff(fact_2972_sum__abs__ge__zero,axiom,
    ! [B: $tType,A: $tType] :
      ( ordere166539214618696060dd_abs(B)
     => ! [F2: fun(A,B),A3: set(A)] : pp(aa(B,bool,ord_less_eq(B,zero_zero(B)),aa(set(A),B,groups7311177749621191930dd_sum(A,B,aTP_Lamp_cz(fun(A,B),fun(A,B),F2)),A3))) ) ).

% sum_abs_ge_zero
tff(fact_2973_sin__cos__squared__add3,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),cos(A,X)),cos(A,X))),aa(A,A,aa(A,fun(A,A),times_times(A),sin(A,X)),sin(A,X))) = one_one(A) ) ).

% sin_cos_squared_add3
tff(fact_2974_cos__of__real__pi,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ( cos(A,aa(real,A,real_Vector_of_real(A),pi)) = aa(A,A,uminus_uminus(A),one_one(A)) ) ) ).

% cos_of_real_pi
tff(fact_2975_sin__npi2,axiom,
    ! [N: nat] : sin(real,aa(real,real,aa(real,fun(real,real),times_times(real),pi),aa(nat,real,semiring_1_of_nat(real),N))) = zero_zero(real) ).

% sin_npi2
tff(fact_2976_sin__npi,axiom,
    ! [N: nat] : sin(real,aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),N)),pi)) = zero_zero(real) ).

% sin_npi
tff(fact_2977_sin__npi__int,axiom,
    ! [N: int] : sin(real,aa(real,real,aa(real,fun(real,real),times_times(real),pi),aa(int,real,ring_1_of_int(real),N))) = zero_zero(real) ).

% sin_npi_int
tff(fact_2978_sum_Ocl__ivl__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [N: nat,M: nat,G: fun(nat,A)] :
          ( ( pp(aa(nat,bool,ord_less(nat,aa(nat,nat,suc,N)),M))
           => ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or1337092689740270186AtMost(nat,M,aa(nat,nat,suc,N))) = zero_zero(A) ) )
          & ( ~ pp(aa(nat,bool,ord_less(nat,aa(nat,nat,suc,N)),M))
           => ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or1337092689740270186AtMost(nat,M,aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or1337092689740270186AtMost(nat,M,N))),aa(nat,A,G,aa(nat,nat,suc,N))) ) ) ) ) ).

% sum.cl_ivl_Suc
tff(fact_2979_sum__zero__power,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: set(nat),C2: fun(nat,A)] :
          ( ( ( pp(aa(set(nat),bool,finite_finite2(nat),A3))
              & pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),zero_zero(nat)),A3)) )
           => ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_dd(fun(nat,A),fun(nat,A),C2)),A3) = aa(nat,A,C2,zero_zero(nat)) ) )
          & ( ~ ( pp(aa(set(nat),bool,finite_finite2(nat),A3))
                & pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),zero_zero(nat)),A3)) )
           => ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_dd(fun(nat,A),fun(nat,A),C2)),A3) = zero_zero(A) ) ) ) ) ).

% sum_zero_power
tff(fact_2980_cos__pi__half,axiom,
    cos(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2)))) = zero_zero(real) ).

% cos_pi_half
tff(fact_2981_sin__two__pi,axiom,
    sin(real,aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),pi)) = zero_zero(real) ).

% sin_two_pi
tff(fact_2982_cos__two__pi,axiom,
    cos(real,aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),pi)) = one_one(real) ).

% cos_two_pi
tff(fact_2983_sin__pi__half,axiom,
    sin(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2)))) = one_one(real) ).

% sin_pi_half
tff(fact_2984_signed__take__bit__numeral__bit0,axiom,
    ! [L: num,K: num] : aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(num,nat,numeral_numeral(nat),L)),aa(num,int,numeral_numeral(int),bit0(K))) = aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,pred_numeral(L)),aa(num,int,numeral_numeral(int),K))),aa(num,int,numeral_numeral(int),bit0(one2))) ).

% signed_take_bit_numeral_bit0
tff(fact_2985_cos__periodic,axiom,
    ! [X: real] : cos(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),X),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),pi))) = cos(real,X) ).

% cos_periodic
tff(fact_2986_sin__periodic,axiom,
    ! [X: real] : sin(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),X),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),pi))) = sin(real,X) ).

% sin_periodic
tff(fact_2987_cos__2pi__minus,axiom,
    ! [X: real] : cos(real,aa(real,real,minus_minus(real,aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),pi)),X)) = cos(real,X) ).

% cos_2pi_minus
tff(fact_2988_cos__npi2,axiom,
    ! [N: nat] : cos(real,aa(real,real,aa(real,fun(real,real),times_times(real),pi),aa(nat,real,semiring_1_of_nat(real),N))) = aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,uminus_uminus(real),one_one(real))),N) ).

% cos_npi2
tff(fact_2989_cos__npi,axiom,
    ! [N: nat] : cos(real,aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),N)),pi)) = aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,uminus_uminus(real),one_one(real))),N) ).

% cos_npi
tff(fact_2990_sum__zero__power_H,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A3: set(nat),C2: fun(nat,A),D: fun(nat,A)] :
          ( ( ( pp(aa(set(nat),bool,finite_finite2(nat),A3))
              & pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),zero_zero(nat)),A3)) )
           => ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_de(fun(nat,A),fun(fun(nat,A),fun(nat,A)),C2),D)),A3) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,C2,zero_zero(nat))),aa(nat,A,D,zero_zero(nat))) ) )
          & ( ~ ( pp(aa(set(nat),bool,finite_finite2(nat),A3))
                & pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),zero_zero(nat)),A3)) )
           => ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_de(fun(nat,A),fun(fun(nat,A),fun(nat,A)),C2),D)),A3) = zero_zero(A) ) ) ) ) ).

% sum_zero_power'
tff(fact_2991_sin__cos__squared__add2,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),cos(A,X)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),sin(A,X)),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = one_one(A) ) ).

% sin_cos_squared_add2
tff(fact_2992_sin__cos__squared__add,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),sin(A,X)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),cos(A,X)),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = one_one(A) ) ).

% sin_cos_squared_add
tff(fact_2993_cos__of__real__pi__half,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V7773925162809079976_field(A)
        & real_V2822296259951069270ebra_1(A) )
     => ( cos(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(real,A,real_Vector_of_real(A),pi)),aa(num,A,numeral_numeral(A),bit0(one2)))) = zero_zero(A) ) ) ).

% cos_of_real_pi_half
tff(fact_2994_sin__of__real__pi__half,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V7773925162809079976_field(A)
        & real_V2822296259951069270ebra_1(A) )
     => ( sin(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(real,A,real_Vector_of_real(A),pi)),aa(num,A,numeral_numeral(A),bit0(one2)))) = one_one(A) ) ) ).

% sin_of_real_pi_half
tff(fact_2995_signed__take__bit__numeral__minus__bit0,axiom,
    ! [L: num,K: num] : aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(num,nat,numeral_numeral(nat),L)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit0(K)))) = aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,pred_numeral(L)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),K)))),aa(num,int,numeral_numeral(int),bit0(one2))) ).

% signed_take_bit_numeral_minus_bit0
tff(fact_2996_sin__2npi,axiom,
    ! [N: nat] : sin(real,aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),aa(nat,real,semiring_1_of_nat(real),N))),pi)) = zero_zero(real) ).

% sin_2npi
tff(fact_2997_cos__2npi,axiom,
    ! [N: nat] : cos(real,aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),aa(nat,real,semiring_1_of_nat(real),N))),pi)) = one_one(real) ).

% cos_2npi
tff(fact_2998_sin__2pi__minus,axiom,
    ! [X: real] : sin(real,aa(real,real,minus_minus(real,aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),pi)),X)) = aa(real,real,uminus_uminus(real),sin(real,X)) ).

% sin_2pi_minus
tff(fact_2999_sin__int__2pin,axiom,
    ! [N: int] : sin(real,aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),pi)),aa(int,real,ring_1_of_int(real),N))) = zero_zero(real) ).

% sin_int_2pin
tff(fact_3000_cos__int__2pin,axiom,
    ! [N: int] : cos(real,aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),pi)),aa(int,real,ring_1_of_int(real),N))) = one_one(real) ).

% cos_int_2pin
tff(fact_3001_cos__3over2__pi,axiom,
    cos(real,aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(num,real,numeral_numeral(real),aa(num,num,bit1,one2))),aa(num,real,numeral_numeral(real),bit0(one2)))),pi)) = zero_zero(real) ).

% cos_3over2_pi
tff(fact_3002_sin__3over2__pi,axiom,
    sin(real,aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(num,real,numeral_numeral(real),aa(num,num,bit1,one2))),aa(num,real,numeral_numeral(real),bit0(one2)))),pi)) = aa(real,real,uminus_uminus(real),one_one(real)) ).

% sin_3over2_pi
tff(fact_3003_cos__npi__int,axiom,
    ! [N: int] :
      ( ( pp(aa(int,bool,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),N))
       => ( cos(real,aa(real,real,aa(real,fun(real,real),times_times(real),pi),aa(int,real,ring_1_of_int(real),N))) = one_one(real) ) )
      & ( ~ pp(aa(int,bool,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),N))
       => ( cos(real,aa(real,real,aa(real,fun(real,real),times_times(real),pi),aa(int,real,ring_1_of_int(real),N))) = aa(real,real,uminus_uminus(real),one_one(real)) ) ) ) ).

% cos_npi_int
tff(fact_3004_dvd__sum,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_semiring_1(A)
     => ! [A3: set(B),D: A,F2: fun(B,A)] :
          ( ! [A4: B] :
              ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A4),A3))
             => pp(aa(A,bool,dvd_dvd(A,D),aa(B,A,F2,A4))) )
         => pp(aa(A,bool,dvd_dvd(A,D),aa(set(B),A,groups7311177749621191930dd_sum(B,A,F2),A3))) ) ) ).

% dvd_sum
tff(fact_3005_sin__add,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A,Y: A] : sin(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),sin(A,X)),cos(A,Y))),aa(A,A,aa(A,fun(A,A),times_times(A),cos(A,X)),sin(A,Y))) ) ).

% sin_add
tff(fact_3006_sin__diff,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A,Y: A] : sin(A,aa(A,A,minus_minus(A,X),Y)) = aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),sin(A,X)),cos(A,Y))),aa(A,A,aa(A,fun(A,A),times_times(A),cos(A,X)),sin(A,Y))) ) ).

% sin_diff
tff(fact_3007_sum_Oswap,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(B,fun(C,A)),B4: set(C),A3: set(B)] : aa(set(B),A,groups7311177749621191930dd_sum(B,A,aa(set(C),fun(B,A),aTP_Lamp_df(fun(B,fun(C,A)),fun(set(C),fun(B,A)),G),B4)),A3) = aa(set(C),A,groups7311177749621191930dd_sum(C,A,aa(set(B),fun(C,A),aTP_Lamp_dh(fun(B,fun(C,A)),fun(set(B),fun(C,A)),G),A3)),B4) ) ).

% sum.swap
tff(fact_3008_cos__one__sin__zero,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] :
          ( ( cos(A,X) = one_one(A) )
         => ( sin(A,X) = zero_zero(A) ) ) ) ).

% cos_one_sin_zero
tff(fact_3009_sum__norm__le,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [S: set(B),F2: fun(B,A),G: fun(B,real)] :
          ( ! [X3: B] :
              ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),S))
             => pp(aa(real,bool,ord_less_eq(real,real_V7770717601297561774m_norm(A,aa(B,A,F2,X3))),aa(B,real,G,X3))) )
         => pp(aa(real,bool,ord_less_eq(real,real_V7770717601297561774m_norm(A,aa(set(B),A,groups7311177749621191930dd_sum(B,A,F2),S))),aa(set(B),real,groups7311177749621191930dd_sum(B,real,G),S))) ) ) ).

% sum_norm_le
tff(fact_3010_polar__Ex,axiom,
    ! [X: real,Y: real] :
    ? [R3: real,A4: real] :
      ( ( X = aa(real,real,aa(real,fun(real,real),times_times(real),R3),cos(real,A4)) )
      & ( Y = aa(real,real,aa(real,fun(real,real),times_times(real),R3),sin(real,A4)) ) ) ).

% polar_Ex
tff(fact_3011_sum_Oneutral,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ! [A3: set(B),G: fun(B,A)] :
          ( ! [X3: B] :
              ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),A3))
             => ( aa(B,A,G,X3) = zero_zero(A) ) )
         => ( aa(set(B),A,groups7311177749621191930dd_sum(B,A,G),A3) = zero_zero(A) ) ) ) ).

% sum.neutral
tff(fact_3012_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(B,A),A3: set(B)] :
          ( ( aa(set(B),A,groups7311177749621191930dd_sum(B,A,G),A3) != zero_zero(A) )
         => ~ ! [A4: B] :
                ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A4),A3))
               => ( aa(B,A,G,A4) = zero_zero(A) ) ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_3013_norm__sum,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(B,A),A3: set(B)] : pp(aa(real,bool,ord_less_eq(real,real_V7770717601297561774m_norm(A,aa(set(B),A,groups7311177749621191930dd_sum(B,A,F2),A3))),aa(set(B),real,groups7311177749621191930dd_sum(B,real,aTP_Lamp_di(fun(B,A),fun(B,real),F2)),A3))) ) ).

% norm_sum
tff(fact_3014_cos__add,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A,Y: A] : cos(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y)) = aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),cos(A,X)),cos(A,Y))),aa(A,A,aa(A,fun(A,A),times_times(A),sin(A,X)),sin(A,Y))) ) ).

% cos_add
tff(fact_3015_cos__diff,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A,Y: A] : cos(A,aa(A,A,minus_minus(A,X),Y)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),cos(A,X)),cos(A,Y))),aa(A,A,aa(A,fun(A,A),times_times(A),sin(A,X)),sin(A,Y))) ) ).

% cos_diff
tff(fact_3016_sin__zero__norm__cos__one,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] :
          ( ( sin(A,X) = zero_zero(A) )
         => ( real_V7770717601297561774m_norm(A,cos(A,X)) = one_one(real) ) ) ) ).

% sin_zero_norm_cos_one
tff(fact_3017_sum__mono,axiom,
    ! [A: $tType,B: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [K6: set(B),F2: fun(B,A),G: fun(B,A)] :
          ( ! [I2: B] :
              ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I2),K6))
             => pp(aa(A,bool,ord_less_eq(A,aa(B,A,F2,I2)),aa(B,A,G,I2))) )
         => pp(aa(A,bool,ord_less_eq(A,aa(set(B),A,groups7311177749621191930dd_sum(B,A,F2),K6)),aa(set(B),A,groups7311177749621191930dd_sum(B,A,G),K6))) ) ) ).

% sum_mono
tff(fact_3018_sum_Odistrib,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(B,A),H: fun(B,A),A3: set(B)] : aa(set(B),A,groups7311177749621191930dd_sum(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_dj(fun(B,A),fun(fun(B,A),fun(B,A)),G),H)),A3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(B),A,groups7311177749621191930dd_sum(B,A,G),A3)),aa(set(B),A,groups7311177749621191930dd_sum(B,A,H),A3)) ) ).

% sum.distrib
tff(fact_3019_sum__distrib__left,axiom,
    ! [A: $tType,B: $tType] :
      ( semiring_0(A)
     => ! [R2: A,F2: fun(B,A),A3: set(B)] : aa(A,A,aa(A,fun(A,A),times_times(A),R2),aa(set(B),A,groups7311177749621191930dd_sum(B,A,F2),A3)) = aa(set(B),A,groups7311177749621191930dd_sum(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_dk(A,fun(fun(B,A),fun(B,A)),R2),F2)),A3) ) ).

% sum_distrib_left
tff(fact_3020_sum__distrib__right,axiom,
    ! [A: $tType,B: $tType] :
      ( semiring_0(A)
     => ! [F2: fun(B,A),A3: set(B),R2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(B),A,groups7311177749621191930dd_sum(B,A,F2),A3)),R2) = aa(set(B),A,groups7311177749621191930dd_sum(B,A,aa(A,fun(B,A),aTP_Lamp_dl(fun(B,A),fun(A,fun(B,A)),F2),R2)),A3) ) ).

% sum_distrib_right
tff(fact_3021_sum__product,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( semiring_0(B)
     => ! [F2: fun(A,B),A3: set(A),G: fun(C,B),B4: set(C)] : aa(B,B,aa(B,fun(B,B),times_times(B),aa(set(A),B,groups7311177749621191930dd_sum(A,B,F2),A3)),aa(set(C),B,groups7311177749621191930dd_sum(C,B,G),B4)) = aa(set(A),B,groups7311177749621191930dd_sum(A,B,aa(set(C),fun(A,B),aa(fun(C,B),fun(set(C),fun(A,B)),aTP_Lamp_dn(fun(A,B),fun(fun(C,B),fun(set(C),fun(A,B))),F2),G),B4)),A3) ) ).

% sum_product
tff(fact_3022_sum__subtractf,axiom,
    ! [A: $tType,B: $tType] :
      ( ab_group_add(A)
     => ! [F2: fun(B,A),G: fun(B,A),A3: set(B)] : aa(set(B),A,groups7311177749621191930dd_sum(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_do(fun(B,A),fun(fun(B,A),fun(B,A)),F2),G)),A3) = aa(A,A,minus_minus(A,aa(set(B),A,groups7311177749621191930dd_sum(B,A,F2),A3)),aa(set(B),A,groups7311177749621191930dd_sum(B,A,G),A3)) ) ).

% sum_subtractf
tff(fact_3023_sum__divide__distrib,axiom,
    ! [A: $tType,B: $tType] :
      ( field(A)
     => ! [F2: fun(B,A),A3: set(B),R2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(set(B),A,groups7311177749621191930dd_sum(B,A,F2),A3)),R2) = aa(set(B),A,groups7311177749621191930dd_sum(B,A,aa(A,fun(B,A),aTP_Lamp_dp(fun(B,A),fun(A,fun(B,A)),F2),R2)),A3) ) ).

% sum_divide_distrib
tff(fact_3024_sum__negf,axiom,
    ! [A: $tType,B: $tType] :
      ( ab_group_add(A)
     => ! [F2: fun(B,A),A3: set(B)] : aa(set(B),A,groups7311177749621191930dd_sum(B,A,aTP_Lamp_dq(fun(B,A),fun(B,A),F2)),A3) = aa(A,A,uminus_uminus(A),aa(set(B),A,groups7311177749621191930dd_sum(B,A,F2),A3)) ) ).

% sum_negf
tff(fact_3025_sum_Oswap__restrict,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comm_monoid_add(A)
     => ! [A3: set(B),B4: set(C),G: fun(B,fun(C,A)),R: fun(B,fun(C,bool))] :
          ( pp(aa(set(B),bool,finite_finite2(B),A3))
         => ( pp(aa(set(C),bool,finite_finite2(C),B4))
           => ( aa(set(B),A,groups7311177749621191930dd_sum(B,A,aa(fun(B,fun(C,bool)),fun(B,A),aa(fun(B,fun(C,A)),fun(fun(B,fun(C,bool)),fun(B,A)),aTP_Lamp_ds(set(C),fun(fun(B,fun(C,A)),fun(fun(B,fun(C,bool)),fun(B,A))),B4),G),R)),A3) = aa(set(C),A,groups7311177749621191930dd_sum(C,A,aa(fun(B,fun(C,bool)),fun(C,A),aa(fun(B,fun(C,A)),fun(fun(B,fun(C,bool)),fun(C,A)),aTP_Lamp_du(set(B),fun(fun(B,fun(C,A)),fun(fun(B,fun(C,bool)),fun(C,A))),A3),G),R)),B4) ) ) ) ) ).

% sum.swap_restrict
tff(fact_3026_mod__sum__eq,axiom,
    ! [B: $tType,A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [F2: fun(B,A),A2: A,A3: set(B)] : modulo_modulo(A,aa(set(B),A,groups7311177749621191930dd_sum(B,A,aa(A,fun(B,A),aTP_Lamp_dv(fun(B,A),fun(A,fun(B,A)),F2),A2)),A3),A2) = modulo_modulo(A,aa(set(B),A,groups7311177749621191930dd_sum(B,A,F2),A3),A2) ) ).

% mod_sum_eq
tff(fact_3027_sin__zero__abs__cos__one,axiom,
    ! [X: real] :
      ( ( sin(real,X) = zero_zero(real) )
     => ( aa(real,real,abs_abs(real),cos(real,X)) = one_one(real) ) ) ).

% sin_zero_abs_cos_one
tff(fact_3028_summable__sum,axiom,
    ! [A: $tType,I6: $tType] :
      ( ( topolo5987344860129210374id_add(A)
        & topological_t2_space(A) )
     => ! [I5: set(I6),F2: fun(I6,fun(nat,A))] :
          ( ! [I2: I6] :
              ( pp(aa(set(I6),bool,aa(I6,fun(set(I6),bool),member(I6),I2),I5))
             => summable(A,aa(I6,fun(nat,A),F2,I2)) )
         => summable(A,aa(fun(I6,fun(nat,A)),fun(nat,A),aTP_Lamp_dx(set(I6),fun(fun(I6,fun(nat,A)),fun(nat,A)),I5),F2)) ) ) ).

% summable_sum
tff(fact_3029_sin__double,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] : sin(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),X)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),sin(A,X))),cos(A,X)) ) ).

% sin_double
tff(fact_3030_sum__nonpos,axiom,
    ! [B: $tType,A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A3: set(B),F2: fun(B,A)] :
          ( ! [X3: B] :
              ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),A3))
             => pp(aa(A,bool,ord_less_eq(A,aa(B,A,F2,X3)),zero_zero(A))) )
         => pp(aa(A,bool,ord_less_eq(A,aa(set(B),A,groups7311177749621191930dd_sum(B,A,F2),A3)),zero_zero(A))) ) ) ).

% sum_nonpos
tff(fact_3031_sum__nonneg,axiom,
    ! [A: $tType,B: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A3: set(B),F2: fun(B,A)] :
          ( ! [X3: B] :
              ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),A3))
             => pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),aa(B,A,F2,X3))) )
         => pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),aa(set(B),A,groups7311177749621191930dd_sum(B,A,F2),A3))) ) ) ).

% sum_nonneg
tff(fact_3032_sincos__principal__value,axiom,
    ! [X: real] :
    ? [Y3: real] :
      ( pp(aa(real,bool,ord_less(real,aa(real,real,uminus_uminus(real),pi)),Y3))
      & pp(aa(real,bool,ord_less_eq(real,Y3),pi))
      & ( sin(real,Y3) = sin(real,X) )
      & ( cos(real,Y3) = cos(real,X) ) ) ).

% sincos_principal_value
tff(fact_3033_sum__mono__inv,axiom,
    ! [A: $tType,I6: $tType] :
      ( ordere8940638589300402666id_add(A)
     => ! [F2: fun(I6,A),I5: set(I6),G: fun(I6,A),I: I6] :
          ( ( aa(set(I6),A,groups7311177749621191930dd_sum(I6,A,F2),I5) = aa(set(I6),A,groups7311177749621191930dd_sum(I6,A,G),I5) )
         => ( ! [I2: I6] :
                ( pp(aa(set(I6),bool,aa(I6,fun(set(I6),bool),member(I6),I2),I5))
               => pp(aa(A,bool,ord_less_eq(A,aa(I6,A,F2,I2)),aa(I6,A,G,I2))) )
           => ( pp(aa(set(I6),bool,aa(I6,fun(set(I6),bool),member(I6),I),I5))
             => ( pp(aa(set(I6),bool,finite_finite2(I6),I5))
               => ( aa(I6,A,F2,I) = aa(I6,A,G,I) ) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_3034_sum__cong__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [A3: set(nat),F2: fun(nat,A),G: fun(nat,A)] :
          ( ~ pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),zero_zero(nat)),A3))
         => ( ! [X3: nat] :
                ( pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),aa(nat,nat,suc,X3)),A3))
               => ( aa(nat,A,F2,aa(nat,nat,suc,X3)) = aa(nat,A,G,aa(nat,nat,suc,X3)) ) )
           => ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F2),A3) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),A3) ) ) ) ) ).

% sum_cong_Suc
tff(fact_3035_sin__x__le__x,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,ord_less_eq(real,zero_zero(real)),X))
     => pp(aa(real,bool,ord_less_eq(real,sin(real,X)),X)) ) ).

% sin_x_le_x
tff(fact_3036_sin__le__one,axiom,
    ! [X: real] : pp(aa(real,bool,ord_less_eq(real,sin(real,X)),one_one(real))) ).

% sin_le_one
tff(fact_3037_cos__le__one,axiom,
    ! [X: real] : pp(aa(real,bool,ord_less_eq(real,cos(real,X)),one_one(real))) ).

% cos_le_one
tff(fact_3038_abs__sin__x__le__abs__x,axiom,
    ! [X: real] : pp(aa(real,bool,ord_less_eq(real,aa(real,real,abs_abs(real),sin(real,X))),aa(real,real,abs_abs(real),X))) ).

% abs_sin_x_le_abs_x
tff(fact_3039_cos__arctan__not__zero,axiom,
    ! [X: real] : cos(real,aa(real,real,arctan,X)) != zero_zero(real) ).

% cos_arctan_not_zero
tff(fact_3040_numeral__eq__Suc,axiom,
    ! [K: num] : aa(num,nat,numeral_numeral(nat),K) = aa(nat,nat,suc,pred_numeral(K)) ).

% numeral_eq_Suc
tff(fact_3041_sum_Ointer__filter,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [A3: set(B),G: fun(B,A),P: fun(B,bool)] :
          ( pp(aa(set(B),bool,finite_finite2(B),A3))
         => ( aa(set(B),A,groups7311177749621191930dd_sum(B,A,G),collect(B,aa(fun(B,bool),fun(B,bool),aTP_Lamp_dy(set(B),fun(fun(B,bool),fun(B,bool)),A3),P))) = aa(set(B),A,groups7311177749621191930dd_sum(B,A,aa(fun(B,bool),fun(B,A),aTP_Lamp_dz(fun(B,A),fun(fun(B,bool),fun(B,A)),G),P)),A3) ) ) ) ).

% sum.inter_filter
tff(fact_3042_cos__int__times__real,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [M: int,X: real] : cos(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(int,A,ring_1_of_int(A),M)),aa(real,A,real_Vector_of_real(A),X))) = aa(real,A,real_Vector_of_real(A),cos(real,aa(real,real,aa(real,fun(real,real),times_times(real),aa(int,real,ring_1_of_int(real),M)),X))) ) ).

% cos_int_times_real
tff(fact_3043_sin__cos__le1,axiom,
    ! [X: real,Y: real] : pp(aa(real,bool,ord_less_eq(real,aa(real,real,abs_abs(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,aa(real,fun(real,real),times_times(real),sin(real,X)),sin(real,Y))),aa(real,real,aa(real,fun(real,real),times_times(real),cos(real,X)),cos(real,Y))))),one_one(real))) ).

% sin_cos_le1
tff(fact_3044_sum_Oshift__bounds__cl__Suc__ivl,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),M: nat,N: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M),aa(nat,nat,suc,N))) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_ea(fun(nat,A),fun(nat,A),G)),set_or1337092689740270186AtMost(nat,M,N)) ) ).

% sum.shift_bounds_cl_Suc_ivl
tff(fact_3045_sin__int__times__real,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [M: int,X: real] : sin(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(int,A,ring_1_of_int(A),M)),aa(real,A,real_Vector_of_real(A),X))) = aa(real,A,real_Vector_of_real(A),sin(real,aa(real,real,aa(real,fun(real,real),times_times(real),aa(int,real,ring_1_of_int(real),M)),X))) ) ).

% sin_int_times_real
tff(fact_3046_sum_Oshift__bounds__cl__nat__ivl,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),M: nat,K: nat,N: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),K),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),K))) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aTP_Lamp_eb(fun(nat,A),fun(nat,fun(nat,A)),G),K)),set_or1337092689740270186AtMost(nat,M,N)) ) ).

% sum.shift_bounds_cl_nat_ivl
tff(fact_3047_suminf__sum,axiom,
    ! [A: $tType,I6: $tType] :
      ( ( topolo5987344860129210374id_add(A)
        & topological_t2_space(A) )
     => ! [I5: set(I6),F2: fun(I6,fun(nat,A))] :
          ( ! [I2: I6] :
              ( pp(aa(set(I6),bool,aa(I6,fun(set(I6),bool),member(I6),I2),I5))
             => summable(A,aa(I6,fun(nat,A),F2,I2)) )
         => ( suminf(A,aa(fun(I6,fun(nat,A)),fun(nat,A),aTP_Lamp_dx(set(I6),fun(fun(I6,fun(nat,A)),fun(nat,A)),I5),F2)) = aa(set(I6),A,groups7311177749621191930dd_sum(I6,A,aTP_Lamp_ec(fun(I6,fun(nat,A)),fun(I6,A),F2)),I5) ) ) ) ).

% suminf_sum
tff(fact_3048_sin__squared__eq,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] : aa(nat,A,aa(A,fun(nat,A),power_power(A),sin(A,X)),aa(num,nat,numeral_numeral(nat),bit0(one2))) = aa(A,A,minus_minus(A,one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),cos(A,X)),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ) ).

% sin_squared_eq
tff(fact_3049_cos__squared__eq,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] : aa(nat,A,aa(A,fun(nat,A),power_power(A),cos(A,X)),aa(num,nat,numeral_numeral(nat),bit0(one2))) = aa(A,A,minus_minus(A,one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),sin(A,X)),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ) ).

% cos_squared_eq
tff(fact_3050_sum__le__included,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [S3: set(B),T: set(C),G: fun(C,A),I: fun(C,B),F2: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),S3))
         => ( pp(aa(set(C),bool,finite_finite2(C),T))
           => ( ! [X3: C] :
                  ( pp(aa(set(C),bool,aa(C,fun(set(C),bool),member(C),X3),T))
                 => pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),aa(C,A,G,X3))) )
             => ( ! [X3: B] :
                    ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),S3))
                   => ? [Xa: C] :
                        ( pp(aa(set(C),bool,aa(C,fun(set(C),bool),member(C),Xa),T))
                        & ( aa(C,B,I,Xa) = X3 )
                        & pp(aa(A,bool,ord_less_eq(A,aa(B,A,F2,X3)),aa(C,A,G,Xa))) ) )
               => pp(aa(A,bool,ord_less_eq(A,aa(set(B),A,groups7311177749621191930dd_sum(B,A,F2),S3)),aa(set(C),A,groups7311177749621191930dd_sum(C,A,G),T))) ) ) ) ) ) ).

% sum_le_included
tff(fact_3051_sum__nonneg__eq__0__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A3: set(B),F2: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),A3))
         => ( ! [X3: B] :
                ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),A3))
               => pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),aa(B,A,F2,X3))) )
           => ( ( aa(set(B),A,groups7311177749621191930dd_sum(B,A,F2),A3) = zero_zero(A) )
            <=> ! [X2: B] :
                  ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X2),A3))
                 => ( aa(B,A,F2,X2) = zero_zero(A) ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_3052_sum__strict__mono__ex1,axiom,
    ! [A: $tType,I6: $tType] :
      ( ordere8940638589300402666id_add(A)
     => ! [A3: set(I6),F2: fun(I6,A),G: fun(I6,A)] :
          ( pp(aa(set(I6),bool,finite_finite2(I6),A3))
         => ( ! [X3: I6] :
                ( pp(aa(set(I6),bool,aa(I6,fun(set(I6),bool),member(I6),X3),A3))
               => pp(aa(A,bool,ord_less_eq(A,aa(I6,A,F2,X3)),aa(I6,A,G,X3))) )
           => ( ? [X4: I6] :
                  ( pp(aa(set(I6),bool,aa(I6,fun(set(I6),bool),member(I6),X4),A3))
                  & pp(aa(A,bool,ord_less(A,aa(I6,A,F2,X4)),aa(I6,A,G,X4))) )
             => pp(aa(A,bool,ord_less(A,aa(set(I6),A,groups7311177749621191930dd_sum(I6,A,F2),A3)),aa(set(I6),A,groups7311177749621191930dd_sum(I6,A,G),A3))) ) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_3053_sum_Orelated,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [R: fun(A,fun(A,bool)),S: set(B),H: fun(B,A),G: fun(B,A)] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),R,zero_zero(A)),zero_zero(A)))
         => ( ! [X1: A,Y1: A,X22: A,Y23: A] :
                ( ( pp(aa(A,bool,aa(A,fun(A,bool),R,X1),X22))
                  & pp(aa(A,bool,aa(A,fun(A,bool),R,Y1),Y23)) )
               => pp(aa(A,bool,aa(A,fun(A,bool),R,aa(A,A,aa(A,fun(A,A),plus_plus(A),X1),Y1)),aa(A,A,aa(A,fun(A,A),plus_plus(A),X22),Y23))) )
           => ( pp(aa(set(B),bool,finite_finite2(B),S))
             => ( ! [X3: B] :
                    ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),S))
                   => pp(aa(A,bool,aa(A,fun(A,bool),R,aa(B,A,H,X3)),aa(B,A,G,X3))) )
               => pp(aa(A,bool,aa(A,fun(A,bool),R,aa(set(B),A,groups7311177749621191930dd_sum(B,A,H),S)),aa(set(B),A,groups7311177749621191930dd_sum(B,A,G),S))) ) ) ) ) ) ).

% sum.related
tff(fact_3054_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comm_monoid_add(A)
     => ! [S5: set(B),T4: set(C),S: set(B),I: fun(C,B),J: fun(B,C),T3: set(C),G: fun(B,A),H: fun(C,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),S5))
         => ( pp(aa(set(C),bool,finite_finite2(C),T4))
           => ( ! [A4: B] :
                  ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A4),aa(set(B),set(B),minus_minus(set(B),S),S5)))
                 => ( aa(C,B,I,aa(B,C,J,A4)) = A4 ) )
             => ( ! [A4: B] :
                    ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A4),aa(set(B),set(B),minus_minus(set(B),S),S5)))
                   => pp(aa(set(C),bool,aa(C,fun(set(C),bool),member(C),aa(B,C,J,A4)),aa(set(C),set(C),minus_minus(set(C),T3),T4))) )
               => ( ! [B3: C] :
                      ( pp(aa(set(C),bool,aa(C,fun(set(C),bool),member(C),B3),aa(set(C),set(C),minus_minus(set(C),T3),T4)))
                     => ( aa(B,C,J,aa(C,B,I,B3)) = B3 ) )
                 => ( ! [B3: C] :
                        ( pp(aa(set(C),bool,aa(C,fun(set(C),bool),member(C),B3),aa(set(C),set(C),minus_minus(set(C),T3),T4)))
                       => pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),aa(C,B,I,B3)),aa(set(B),set(B),minus_minus(set(B),S),S5))) )
                   => ( ! [A4: B] :
                          ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A4),S5))
                         => ( aa(B,A,G,A4) = zero_zero(A) ) )
                     => ( ! [B3: C] :
                            ( pp(aa(set(C),bool,aa(C,fun(set(C),bool),member(C),B3),T4))
                           => ( aa(C,A,H,B3) = zero_zero(A) ) )
                       => ( ! [A4: B] :
                              ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A4),S))
                             => ( aa(C,A,H,aa(B,C,J,A4)) = aa(B,A,G,A4) ) )
                         => ( aa(set(B),A,groups7311177749621191930dd_sum(B,A,G),S) = aa(set(C),A,groups7311177749621191930dd_sum(C,A,H),T3) ) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_3055_sin__gt__zero,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,ord_less(real,zero_zero(real)),X))
     => ( pp(aa(real,bool,ord_less(real,X),pi))
       => pp(aa(real,bool,ord_less(real,zero_zero(real)),sin(real,X))) ) ) ).

% sin_gt_zero
tff(fact_3056_sin__x__ge__neg__x,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,ord_less_eq(real,zero_zero(real)),X))
     => pp(aa(real,bool,ord_less_eq(real,aa(real,real,uminus_uminus(real),X)),sin(real,X))) ) ).

% sin_x_ge_neg_x
tff(fact_3057_sin__ge__zero,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,ord_less_eq(real,zero_zero(real)),X))
     => ( pp(aa(real,bool,ord_less_eq(real,X),pi))
       => pp(aa(real,bool,ord_less_eq(real,zero_zero(real)),sin(real,X))) ) ) ).

% sin_ge_zero
tff(fact_3058_sin__ge__minus__one,axiom,
    ! [X: real] : pp(aa(real,bool,ord_less_eq(real,aa(real,real,uminus_uminus(real),one_one(real))),sin(real,X))) ).

% sin_ge_minus_one
tff(fact_3059_cos__monotone__0__pi__le,axiom,
    ! [Y: real,X: real] :
      ( pp(aa(real,bool,ord_less_eq(real,zero_zero(real)),Y))
     => ( pp(aa(real,bool,ord_less_eq(real,Y),X))
       => ( pp(aa(real,bool,ord_less_eq(real,X),pi))
         => pp(aa(real,bool,ord_less_eq(real,cos(real,X)),cos(real,Y))) ) ) ) ).

% cos_monotone_0_pi_le
tff(fact_3060_cos__mono__le__eq,axiom,
    ! [X: real,Y: real] :
      ( pp(aa(real,bool,ord_less_eq(real,zero_zero(real)),X))
     => ( pp(aa(real,bool,ord_less_eq(real,X),pi))
       => ( pp(aa(real,bool,ord_less_eq(real,zero_zero(real)),Y))
         => ( pp(aa(real,bool,ord_less_eq(real,Y),pi))
           => ( pp(aa(real,bool,ord_less_eq(real,cos(real,X)),cos(real,Y)))
            <=> pp(aa(real,bool,ord_less_eq(real,Y),X)) ) ) ) ) ) ).

% cos_mono_le_eq
tff(fact_3061_cos__inj__pi,axiom,
    ! [X: real,Y: real] :
      ( pp(aa(real,bool,ord_less_eq(real,zero_zero(real)),X))
     => ( pp(aa(real,bool,ord_less_eq(real,X),pi))
       => ( pp(aa(real,bool,ord_less_eq(real,zero_zero(real)),Y))
         => ( pp(aa(real,bool,ord_less_eq(real,Y),pi))
           => ( ( cos(real,X) = cos(real,Y) )
             => ( X = Y ) ) ) ) ) ) ).

% cos_inj_pi
tff(fact_3062_cos__ge__minus__one,axiom,
    ! [X: real] : pp(aa(real,bool,ord_less_eq(real,aa(real,real,uminus_uminus(real),one_one(real))),cos(real,X))) ).

% cos_ge_minus_one
tff(fact_3063_abs__sin__le__one,axiom,
    ! [X: real] : pp(aa(real,bool,ord_less_eq(real,aa(real,real,abs_abs(real),sin(real,X))),one_one(real))) ).

% abs_sin_le_one
tff(fact_3064_abs__cos__le__one,axiom,
    ! [X: real] : pp(aa(real,bool,ord_less_eq(real,aa(real,real,abs_abs(real),cos(real,X))),one_one(real))) ).

% abs_cos_le_one
tff(fact_3065_sin__times__sin,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [W: A,Z: A] : aa(A,A,aa(A,fun(A,A),times_times(A),sin(A,W)),sin(A,Z)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,minus_minus(A,cos(A,aa(A,A,minus_minus(A,W),Z))),cos(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),W),Z)))),aa(num,A,numeral_numeral(A),bit0(one2))) ) ).

% sin_times_sin
tff(fact_3066_sin__times__cos,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [W: A,Z: A] : aa(A,A,aa(A,fun(A,A),times_times(A),sin(A,W)),cos(A,Z)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),sin(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),W),Z))),sin(A,aa(A,A,minus_minus(A,W),Z)))),aa(num,A,numeral_numeral(A),bit0(one2))) ) ).

% sin_times_cos
tff(fact_3067_cos__times__sin,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [W: A,Z: A] : aa(A,A,aa(A,fun(A,A),times_times(A),cos(A,W)),sin(A,Z)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,minus_minus(A,sin(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),W),Z))),sin(A,aa(A,A,minus_minus(A,W),Z)))),aa(num,A,numeral_numeral(A),bit0(one2))) ) ).

% cos_times_sin
tff(fact_3068_sin__plus__sin,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [W: A,Z: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),sin(A,W)),sin(A,Z)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),sin(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),W),Z)),aa(num,A,numeral_numeral(A),bit0(one2)))))),cos(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,minus_minus(A,W),Z)),aa(num,A,numeral_numeral(A),bit0(one2))))) ) ).

% sin_plus_sin
tff(fact_3069_sin__diff__sin,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [W: A,Z: A] : aa(A,A,minus_minus(A,sin(A,W)),sin(A,Z)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),sin(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,minus_minus(A,W),Z)),aa(num,A,numeral_numeral(A),bit0(one2)))))),cos(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),W),Z)),aa(num,A,numeral_numeral(A),bit0(one2))))) ) ).

% sin_diff_sin
tff(fact_3070_cos__diff__cos,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [W: A,Z: A] : aa(A,A,minus_minus(A,cos(A,W)),cos(A,Z)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),sin(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),W),Z)),aa(num,A,numeral_numeral(A),bit0(one2)))))),sin(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,minus_minus(A,Z),W)),aa(num,A,numeral_numeral(A),bit0(one2))))) ) ).

% cos_diff_cos
tff(fact_3071_cos__double,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] : cos(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),X)) = aa(A,A,minus_minus(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),cos(A,X)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),sin(A,X)),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ) ).

% cos_double
tff(fact_3072_sum__nonneg__0,axiom,
    ! [B: $tType,A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [S3: set(B),F2: fun(B,A),I: B] :
          ( pp(aa(set(B),bool,finite_finite2(B),S3))
         => ( ! [I2: B] :
                ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I2),S3))
               => pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),aa(B,A,F2,I2))) )
           => ( ( aa(set(B),A,groups7311177749621191930dd_sum(B,A,F2),S3) = zero_zero(A) )
             => ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I),S3))
               => ( aa(B,A,F2,I) = zero_zero(A) ) ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_3073_sum__nonneg__leq__bound,axiom,
    ! [B: $tType,A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [S3: set(B),F2: fun(B,A),B4: A,I: B] :
          ( pp(aa(set(B),bool,finite_finite2(B),S3))
         => ( ! [I2: B] :
                ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I2),S3))
               => pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),aa(B,A,F2,I2))) )
           => ( ( aa(set(B),A,groups7311177749621191930dd_sum(B,A,F2),S3) = B4 )
             => ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I),S3))
               => pp(aa(A,bool,ord_less_eq(A,aa(B,A,F2,I)),B4)) ) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_3074_sin__cos__eq,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] : sin(A,X) = cos(A,aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(real,A,real_Vector_of_real(A),pi)),aa(num,A,numeral_numeral(A),bit0(one2)))),X)) ) ).

% sin_cos_eq
tff(fact_3075_cos__sin__eq,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] : cos(A,X) = sin(A,aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(real,A,real_Vector_of_real(A),pi)),aa(num,A,numeral_numeral(A),bit0(one2)))),X)) ) ).

% cos_sin_eq
tff(fact_3076_sum_Osetdiff__irrelevant,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [A3: set(B),G: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),A3))
         => ( aa(set(B),A,groups7311177749621191930dd_sum(B,A,G),aa(set(B),set(B),minus_minus(set(B),A3),collect(B,aTP_Lamp_ed(fun(B,A),fun(B,bool),G)))) = aa(set(B),A,groups7311177749621191930dd_sum(B,A,G),A3) ) ) ) ).

% sum.setdiff_irrelevant
tff(fact_3077_pred__numeral__def,axiom,
    ! [K: num] : pred_numeral(K) = aa(nat,nat,minus_minus(nat,aa(num,nat,numeral_numeral(nat),K)),one_one(nat)) ).

% pred_numeral_def
tff(fact_3078_sum__power__add,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [X: A,M: nat,I5: set(nat)] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aTP_Lamp_ee(A,fun(nat,fun(nat,A)),X),M)),I5) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),M)),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),power_power(A),X)),I5)) ) ).

% sum_power_add
tff(fact_3079_sum_OatLeastAtMost__rev,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),N: nat,M: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or1337092689740270186AtMost(nat,N,M)) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aa(nat,fun(nat,fun(nat,A)),aTP_Lamp_ef(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),G),N),M)),set_or1337092689740270186AtMost(nat,N,M)) ) ).

% sum.atLeastAtMost_rev
tff(fact_3080_suminf__finite,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topological_t2_space(A) )
     => ! [N4: set(nat),F2: fun(nat,A)] :
          ( pp(aa(set(nat),bool,finite_finite2(nat),N4))
         => ( ! [N2: nat] :
                ( ~ pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),N2),N4))
               => ( aa(nat,A,F2,N2) = zero_zero(A) ) )
           => ( suminf(A,F2) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F2),N4) ) ) ) ) ).

% suminf_finite
tff(fact_3081_cos__double__sin,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [W: A] : cos(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),W)) = aa(A,A,minus_minus(A,one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(nat,A,aa(A,fun(nat,A),power_power(A),sin(A,W)),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) ).

% cos_double_sin
tff(fact_3082_minus__sin__cos__eq,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] : aa(A,A,uminus_uminus(A),sin(A,X)) = cos(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(real,A,real_Vector_of_real(A),pi)),aa(num,A,numeral_numeral(A),bit0(one2))))) ) ).

% minus_sin_cos_eq
tff(fact_3083_sum__pos2,axiom,
    ! [A: $tType,B: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [I5: set(B),I: B,F2: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),I5))
         => ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I),I5))
           => ( pp(aa(A,bool,ord_less(A,zero_zero(A)),aa(B,A,F2,I)))
             => ( ! [I2: B] :
                    ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I2),I5))
                   => pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),aa(B,A,F2,I2))) )
               => pp(aa(A,bool,ord_less(A,zero_zero(A)),aa(set(B),A,groups7311177749621191930dd_sum(B,A,F2),I5))) ) ) ) ) ) ).

% sum_pos2
tff(fact_3084_sum_Omono__neutral__cong__right,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [T3: set(B),S: set(B),G: fun(B,A),H: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),T3))
         => ( pp(aa(set(B),bool,ord_less_eq(set(B),S),T3))
           => ( ! [X3: B] :
                  ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),aa(set(B),set(B),minus_minus(set(B),T3),S)))
                 => ( aa(B,A,G,X3) = zero_zero(A) ) )
             => ( ! [X3: B] :
                    ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),S))
                   => ( aa(B,A,G,X3) = aa(B,A,H,X3) ) )
               => ( aa(set(B),A,groups7311177749621191930dd_sum(B,A,G),T3) = aa(set(B),A,groups7311177749621191930dd_sum(B,A,H),S) ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_3085_sum_Omono__neutral__cong__left,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [T3: set(B),S: set(B),H: fun(B,A),G: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),T3))
         => ( pp(aa(set(B),bool,ord_less_eq(set(B),S),T3))
           => ( ! [X3: B] :
                  ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),aa(set(B),set(B),minus_minus(set(B),T3),S)))
                 => ( aa(B,A,H,X3) = zero_zero(A) ) )
             => ( ! [X3: B] :
                    ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),S))
                   => ( aa(B,A,G,X3) = aa(B,A,H,X3) ) )
               => ( aa(set(B),A,groups7311177749621191930dd_sum(B,A,G),S) = aa(set(B),A,groups7311177749621191930dd_sum(B,A,H),T3) ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_3086_sum_Omono__neutral__right,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [T3: set(B),S: set(B),G: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),T3))
         => ( pp(aa(set(B),bool,ord_less_eq(set(B),S),T3))
           => ( ! [X3: B] :
                  ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),aa(set(B),set(B),minus_minus(set(B),T3),S)))
                 => ( aa(B,A,G,X3) = zero_zero(A) ) )
             => ( aa(set(B),A,groups7311177749621191930dd_sum(B,A,G),T3) = aa(set(B),A,groups7311177749621191930dd_sum(B,A,G),S) ) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_3087_sum_Omono__neutral__left,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [T3: set(B),S: set(B),G: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),T3))
         => ( pp(aa(set(B),bool,ord_less_eq(set(B),S),T3))
           => ( ! [X3: B] :
                  ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),aa(set(B),set(B),minus_minus(set(B),T3),S)))
                 => ( aa(B,A,G,X3) = zero_zero(A) ) )
             => ( aa(set(B),A,groups7311177749621191930dd_sum(B,A,G),S) = aa(set(B),A,groups7311177749621191930dd_sum(B,A,G),T3) ) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_3088_sum_Osame__carrierI,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [C6: set(B),A3: set(B),B4: set(B),G: fun(B,A),H: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),C6))
         => ( pp(aa(set(B),bool,ord_less_eq(set(B),A3),C6))
           => ( pp(aa(set(B),bool,ord_less_eq(set(B),B4),C6))
             => ( ! [A4: B] :
                    ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A4),aa(set(B),set(B),minus_minus(set(B),C6),A3)))
                   => ( aa(B,A,G,A4) = zero_zero(A) ) )
               => ( ! [B3: B] :
                      ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),B3),aa(set(B),set(B),minus_minus(set(B),C6),B4)))
                     => ( aa(B,A,H,B3) = zero_zero(A) ) )
                 => ( ( aa(set(B),A,groups7311177749621191930dd_sum(B,A,G),C6) = aa(set(B),A,groups7311177749621191930dd_sum(B,A,H),C6) )
                   => ( aa(set(B),A,groups7311177749621191930dd_sum(B,A,G),A3) = aa(set(B),A,groups7311177749621191930dd_sum(B,A,H),B4) ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_3089_sum_Osame__carrier,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [C6: set(B),A3: set(B),B4: set(B),G: fun(B,A),H: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),C6))
         => ( pp(aa(set(B),bool,ord_less_eq(set(B),A3),C6))
           => ( pp(aa(set(B),bool,ord_less_eq(set(B),B4),C6))
             => ( ! [A4: B] :
                    ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A4),aa(set(B),set(B),minus_minus(set(B),C6),A3)))
                   => ( aa(B,A,G,A4) = zero_zero(A) ) )
               => ( ! [B3: B] :
                      ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),B3),aa(set(B),set(B),minus_minus(set(B),C6),B4)))
                     => ( aa(B,A,H,B3) = zero_zero(A) ) )
                 => ( ( aa(set(B),A,groups7311177749621191930dd_sum(B,A,G),A3) = aa(set(B),A,groups7311177749621191930dd_sum(B,A,H),B4) )
                  <=> ( aa(set(B),A,groups7311177749621191930dd_sum(B,A,G),C6) = aa(set(B),A,groups7311177749621191930dd_sum(B,A,H),C6) ) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_3090_sum_Osubset__diff,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [B4: set(B),A3: set(B),G: fun(B,A)] :
          ( pp(aa(set(B),bool,ord_less_eq(set(B),B4),A3))
         => ( pp(aa(set(B),bool,finite_finite2(B),A3))
           => ( aa(set(B),A,groups7311177749621191930dd_sum(B,A,G),A3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(B),A,groups7311177749621191930dd_sum(B,A,G),aa(set(B),set(B),minus_minus(set(B),A3),B4))),aa(set(B),A,groups7311177749621191930dd_sum(B,A,G),B4)) ) ) ) ) ).

% sum.subset_diff
tff(fact_3091_sum__diff,axiom,
    ! [A: $tType,B: $tType] :
      ( ab_group_add(A)
     => ! [A3: set(B),B4: set(B),F2: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),A3))
         => ( pp(aa(set(B),bool,ord_less_eq(set(B),B4),A3))
           => ( aa(set(B),A,groups7311177749621191930dd_sum(B,A,F2),aa(set(B),set(B),minus_minus(set(B),A3),B4)) = aa(A,A,minus_minus(A,aa(set(B),A,groups7311177749621191930dd_sum(B,A,F2),A3)),aa(set(B),A,groups7311177749621191930dd_sum(B,A,F2),B4)) ) ) ) ) ).

% sum_diff
tff(fact_3092_sum__shift__lb__Suc0__0,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [F2: fun(nat,A),K: nat] :
          ( ( aa(nat,A,F2,zero_zero(nat)) = zero_zero(A) )
         => ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F2),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,zero_zero(nat)),K)) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F2),set_or1337092689740270186AtMost(nat,zero_zero(nat),K)) ) ) ) ).

% sum_shift_lb_Suc0_0
tff(fact_3093_cos__two__neq__zero,axiom,
    cos(real,aa(num,real,numeral_numeral(real),bit0(one2))) != zero_zero(real) ).

% cos_two_neq_zero
tff(fact_3094_sum_OatLeast0__atMost__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),N: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or1337092689740270186AtMost(nat,zero_zero(nat),N))),aa(nat,A,G,aa(nat,nat,suc,N))) ) ).

% sum.atLeast0_atMost_Suc
tff(fact_3095_sum_OatLeast__Suc__atMost,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [M: nat,N: nat,G: fun(nat,A)] :
          ( pp(aa(nat,bool,ord_less_eq(nat,M),N))
         => ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or1337092689740270186AtMost(nat,M,N)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G,M)),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M),N))) ) ) ) ).

% sum.atLeast_Suc_atMost
tff(fact_3096_sum_Onat__ivl__Suc_H,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [M: nat,N: nat,G: fun(nat,A)] :
          ( pp(aa(nat,bool,ord_less_eq(nat,M),aa(nat,nat,suc,N)))
         => ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or1337092689740270186AtMost(nat,M,aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G,aa(nat,nat,suc,N))),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or1337092689740270186AtMost(nat,M,N))) ) ) ) ).

% sum.nat_ivl_Suc'
tff(fact_3097_cos__monotone__0__pi,axiom,
    ! [Y: real,X: real] :
      ( pp(aa(real,bool,ord_less_eq(real,zero_zero(real)),Y))
     => ( pp(aa(real,bool,ord_less(real,Y),X))
       => ( pp(aa(real,bool,ord_less_eq(real,X),pi))
         => pp(aa(real,bool,ord_less(real,cos(real,X)),cos(real,Y))) ) ) ) ).

% cos_monotone_0_pi
tff(fact_3098_cos__mono__less__eq,axiom,
    ! [X: real,Y: real] :
      ( pp(aa(real,bool,ord_less_eq(real,zero_zero(real)),X))
     => ( pp(aa(real,bool,ord_less_eq(real,X),pi))
       => ( pp(aa(real,bool,ord_less_eq(real,zero_zero(real)),Y))
         => ( pp(aa(real,bool,ord_less_eq(real,Y),pi))
           => ( pp(aa(real,bool,ord_less(real,cos(real,X)),cos(real,Y)))
            <=> pp(aa(real,bool,ord_less(real,Y),X)) ) ) ) ) ) ).

% cos_mono_less_eq
tff(fact_3099_sin__eq__0__pi,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,ord_less(real,aa(real,real,uminus_uminus(real),pi)),X))
     => ( pp(aa(real,bool,ord_less(real,X),pi))
       => ( ( sin(real,X) = zero_zero(real) )
         => ( X = zero_zero(real) ) ) ) ) ).

% sin_eq_0_pi
tff(fact_3100_sin__zero__pi__iff,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,ord_less(real,aa(real,real,abs_abs(real),X)),pi))
     => ( ( sin(real,X) = zero_zero(real) )
      <=> ( X = zero_zero(real) ) ) ) ).

% sin_zero_pi_iff
tff(fact_3101_cos__monotone__minus__pi__0_H,axiom,
    ! [Y: real,X: real] :
      ( pp(aa(real,bool,ord_less_eq(real,aa(real,real,uminus_uminus(real),pi)),Y))
     => ( pp(aa(real,bool,ord_less_eq(real,Y),X))
       => ( pp(aa(real,bool,ord_less_eq(real,X),zero_zero(real)))
         => pp(aa(real,bool,ord_less_eq(real,cos(real,Y)),cos(real,X))) ) ) ) ).

% cos_monotone_minus_pi_0'
tff(fact_3102_sin__zero__iff__int2,axiom,
    ! [X: real] :
      ( ( sin(real,X) = zero_zero(real) )
    <=> ? [I3: int] : X = aa(real,real,aa(real,fun(real,real),times_times(real),aa(int,real,ring_1_of_int(real),I3)),pi) ) ).

% sin_zero_iff_int2
tff(fact_3103_sincos__total__pi,axiom,
    ! [Y: real,X: real] :
      ( pp(aa(real,bool,ord_less_eq(real,zero_zero(real)),Y))
     => ( ( aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),X),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Y),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = one_one(real) )
       => ? [T5: real] :
            ( pp(aa(real,bool,ord_less_eq(real,zero_zero(real)),T5))
            & pp(aa(real,bool,ord_less_eq(real,T5),pi))
            & ( X = cos(real,T5) )
            & ( Y = sin(real,T5) ) ) ) ) ).

% sincos_total_pi
tff(fact_3104_sum_OSuc__reindex__ivl,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [M: nat,N: nat,G: fun(nat,A)] :
          ( pp(aa(nat,bool,ord_less_eq(nat,M),N))
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or1337092689740270186AtMost(nat,M,N))),aa(nat,A,G,aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G,M)),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_ea(fun(nat,A),fun(nat,A),G)),set_or1337092689740270186AtMost(nat,M,N))) ) ) ) ).

% sum.Suc_reindex_ivl
tff(fact_3105_sum__Suc__diff,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [M: nat,N: nat,F2: fun(nat,A)] :
          ( pp(aa(nat,bool,ord_less_eq(nat,M),aa(nat,nat,suc,N)))
         => ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_eg(fun(nat,A),fun(nat,A),F2)),set_or1337092689740270186AtMost(nat,M,N)) = aa(A,A,minus_minus(A,aa(nat,A,F2,aa(nat,nat,suc,N))),aa(nat,A,F2,M)) ) ) ) ).

% sum_Suc_diff
tff(fact_3106_sin__expansion__lemma,axiom,
    ! [X: real,M: nat] : sin(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),X),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,M))),pi)),aa(num,real,numeral_numeral(real),bit0(one2))))) = cos(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),X),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),M)),pi)),aa(num,real,numeral_numeral(real),bit0(one2))))) ).

% sin_expansion_lemma
tff(fact_3107_sum__mono2,axiom,
    ! [A: $tType,B: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [B4: set(B),A3: set(B),F2: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),B4))
         => ( pp(aa(set(B),bool,ord_less_eq(set(B),A3),B4))
           => ( ! [B3: B] :
                  ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),B3),aa(set(B),set(B),minus_minus(set(B),B4),A3)))
                 => pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),aa(B,A,F2,B3))) )
             => pp(aa(A,bool,ord_less_eq(A,aa(set(B),A,groups7311177749621191930dd_sum(B,A,F2),A3)),aa(set(B),A,groups7311177749621191930dd_sum(B,A,F2),B4))) ) ) ) ) ).

% sum_mono2
tff(fact_3108_cos__expansion__lemma,axiom,
    ! [X: real,M: nat] : cos(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),X),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,M))),pi)),aa(num,real,numeral_numeral(real),bit0(one2))))) = aa(real,real,uminus_uminus(real),sin(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),X),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),M)),pi)),aa(num,real,numeral_numeral(real),bit0(one2)))))) ).

% cos_expansion_lemma
tff(fact_3109_sin__gt__zero__02,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,ord_less(real,zero_zero(real)),X))
     => ( pp(aa(real,bool,ord_less(real,X),aa(num,real,numeral_numeral(real),bit0(one2))))
       => pp(aa(real,bool,ord_less(real,zero_zero(real)),sin(real,X))) ) ) ).

% sin_gt_zero_02
tff(fact_3110_cos__two__less__zero,axiom,
    pp(aa(real,bool,ord_less(real,cos(real,aa(num,real,numeral_numeral(real),bit0(one2)))),zero_zero(real))) ).

% cos_two_less_zero
tff(fact_3111_cos__is__zero,axiom,
    ? [X3: real] :
      ( pp(aa(real,bool,ord_less_eq(real,zero_zero(real)),X3))
      & pp(aa(real,bool,ord_less_eq(real,X3),aa(num,real,numeral_numeral(real),bit0(one2))))
      & ( cos(real,X3) = zero_zero(real) )
      & ! [Y4: real] :
          ( ( pp(aa(real,bool,ord_less_eq(real,zero_zero(real)),Y4))
            & pp(aa(real,bool,ord_less_eq(real,Y4),aa(num,real,numeral_numeral(real),bit0(one2))))
            & ( cos(real,Y4) = zero_zero(real) ) )
         => ( Y4 = X3 ) ) ) ).

% cos_is_zero
tff(fact_3112_cos__two__le__zero,axiom,
    pp(aa(real,bool,ord_less_eq(real,cos(real,aa(num,real,numeral_numeral(real),bit0(one2)))),zero_zero(real))) ).

% cos_two_le_zero
tff(fact_3113_sum_Oub__add__nat,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [M: nat,N: nat,G: fun(nat,A),P2: nat] :
          ( pp(aa(nat,bool,ord_less_eq(nat,M),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat))))
         => ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or1337092689740270186AtMost(nat,M,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),P2))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or1337092689740270186AtMost(nat,M,N))),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),P2)))) ) ) ) ).

% sum.ub_add_nat
tff(fact_3114_cos__monotone__minus__pi__0,axiom,
    ! [Y: real,X: real] :
      ( pp(aa(real,bool,ord_less_eq(real,aa(real,real,uminus_uminus(real),pi)),Y))
     => ( pp(aa(real,bool,ord_less(real,Y),X))
       => ( pp(aa(real,bool,ord_less_eq(real,X),zero_zero(real)))
         => pp(aa(real,bool,ord_less(real,cos(real,Y)),cos(real,X))) ) ) ) ).

% cos_monotone_minus_pi_0
tff(fact_3115_cos__total,axiom,
    ! [Y: real] :
      ( pp(aa(real,bool,ord_less_eq(real,aa(real,real,uminus_uminus(real),one_one(real))),Y))
     => ( pp(aa(real,bool,ord_less_eq(real,Y),one_one(real)))
       => ? [X3: real] :
            ( pp(aa(real,bool,ord_less_eq(real,zero_zero(real)),X3))
            & pp(aa(real,bool,ord_less_eq(real,X3),pi))
            & ( cos(real,X3) = Y )
            & ! [Y4: real] :
                ( ( pp(aa(real,bool,ord_less_eq(real,zero_zero(real)),Y4))
                  & pp(aa(real,bool,ord_less_eq(real,Y4),pi))
                  & ( cos(real,Y4) = Y ) )
               => ( Y4 = X3 ) ) ) ) ) ).

% cos_total
tff(fact_3116_sum__le__suminf,axiom,
    ! [A: $tType] :
      ( ( ordere6911136660526730532id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F2: fun(nat,A),I5: set(nat)] :
          ( summable(A,F2)
         => ( pp(aa(set(nat),bool,finite_finite2(nat),I5))
           => ( ! [N2: nat] :
                  ( pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),N2),aa(set(nat),set(nat),uminus_uminus(set(nat)),I5)))
                 => pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),aa(nat,A,F2,N2))) )
             => pp(aa(A,bool,ord_less_eq(A,aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F2),I5)),suminf(A,F2))) ) ) ) ) ).

% sum_le_suminf
tff(fact_3117_sincos__total__pi__half,axiom,
    ! [X: real,Y: real] :
      ( pp(aa(real,bool,ord_less_eq(real,zero_zero(real)),X))
     => ( pp(aa(real,bool,ord_less_eq(real,zero_zero(real)),Y))
       => ( ( aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),X),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Y),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = one_one(real) )
         => ? [T5: real] :
              ( pp(aa(real,bool,ord_less_eq(real,zero_zero(real)),T5))
              & pp(aa(real,bool,ord_less_eq(real,T5),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2)))))
              & ( X = cos(real,T5) )
              & ( Y = sin(real,T5) ) ) ) ) ) ).

% sincos_total_pi_half
tff(fact_3118_sincos__total__2pi__le,axiom,
    ! [X: real,Y: real] :
      ( ( aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),X),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Y),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = one_one(real) )
     => ? [T5: real] :
          ( pp(aa(real,bool,ord_less_eq(real,zero_zero(real)),T5))
          & pp(aa(real,bool,ord_less_eq(real,T5),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),pi)))
          & ( X = cos(real,T5) )
          & ( Y = sin(real,T5) ) ) ) ).

% sincos_total_2pi_le
tff(fact_3119_sum__strict__mono2,axiom,
    ! [B: $tType,A: $tType] :
      ( ordere8940638589300402666id_add(B)
     => ! [B4: set(A),A3: set(A),B2: A,F2: fun(A,B)] :
          ( pp(aa(set(A),bool,finite_finite2(A),B4))
         => ( pp(aa(set(A),bool,ord_less_eq(set(A),A3),B4))
           => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),B2),aa(set(A),set(A),minus_minus(set(A),B4),A3)))
             => ( pp(aa(B,bool,ord_less(B,zero_zero(B)),aa(A,B,F2,B2)))
               => ( ! [X3: A] :
                      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),B4))
                     => pp(aa(B,bool,ord_less_eq(B,zero_zero(B)),aa(A,B,F2,X3))) )
                 => pp(aa(B,bool,ord_less(B,aa(set(A),B,groups7311177749621191930dd_sum(A,B,F2),A3)),aa(set(A),B,groups7311177749621191930dd_sum(A,B,F2),B4))) ) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_3120_sincos__total__2pi,axiom,
    ! [X: real,Y: real] :
      ( ( aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),X),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Y),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = one_one(real) )
     => ~ ! [T5: real] :
            ( pp(aa(real,bool,ord_less_eq(real,zero_zero(real)),T5))
           => ( pp(aa(real,bool,ord_less(real,T5),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),pi)))
             => ( ( X = cos(real,T5) )
               => ( Y != sin(real,T5) ) ) ) ) ) ).

% sincos_total_2pi
tff(fact_3121_convex__sum__bound__le,axiom,
    ! [A: $tType,B: $tType] :
      ( linordered_idom(B)
     => ! [I5: set(A),X: fun(A,B),A2: fun(A,B),B2: B,Delta: B] :
          ( ! [I2: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),I2),I5))
             => pp(aa(B,bool,ord_less_eq(B,zero_zero(B)),aa(A,B,X,I2))) )
         => ( ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,X),I5) = one_one(B) )
           => ( ! [I2: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),I2),I5))
                 => pp(aa(B,bool,ord_less_eq(B,aa(B,B,abs_abs(B),aa(B,B,minus_minus(B,aa(A,B,A2,I2)),B2))),Delta)) )
             => pp(aa(B,bool,ord_less_eq(B,aa(B,B,abs_abs(B),aa(B,B,minus_minus(B,aa(set(A),B,groups7311177749621191930dd_sum(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_eh(fun(A,B),fun(fun(A,B),fun(A,B)),X),A2)),I5)),B2))),Delta)) ) ) ) ) ).

% convex_sum_bound_le
tff(fact_3122_sin__pi__divide__n__ge__0,axiom,
    ! [N: nat] :
      ( ( N != zero_zero(nat) )
     => pp(aa(real,bool,ord_less_eq(real,zero_zero(real)),sin(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(nat,real,semiring_1_of_nat(real),N))))) ) ).

% sin_pi_divide_n_ge_0
tff(fact_3123_sum__natinterval__diff,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [M: nat,N: nat,F2: fun(nat,A)] :
          ( ( pp(aa(nat,bool,ord_less_eq(nat,M),N))
           => ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_ei(fun(nat,A),fun(nat,A),F2)),set_or1337092689740270186AtMost(nat,M,N)) = aa(A,A,minus_minus(A,aa(nat,A,F2,M)),aa(nat,A,F2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat)))) ) )
          & ( ~ pp(aa(nat,bool,ord_less_eq(nat,M),N))
           => ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_ei(fun(nat,A),fun(nat,A),F2)),set_or1337092689740270186AtMost(nat,M,N)) = zero_zero(A) ) ) ) ) ).

% sum_natinterval_diff
tff(fact_3124_sum__telescope_H_H,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [M: nat,N: nat,F2: fun(nat,A)] :
          ( pp(aa(nat,bool,ord_less_eq(nat,M),N))
         => ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_ej(fun(nat,A),fun(nat,A),F2)),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M),N)) = aa(A,A,minus_minus(A,aa(nat,A,F2,N)),aa(nat,A,F2,M)) ) ) ) ).

% sum_telescope''
tff(fact_3125_cos__times__cos,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [W: A,Z: A] : aa(A,A,aa(A,fun(A,A),times_times(A),cos(A,W)),cos(A,Z)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),cos(A,aa(A,A,minus_minus(A,W),Z))),cos(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),W),Z)))),aa(num,A,numeral_numeral(A),bit0(one2))) ) ).

% cos_times_cos
tff(fact_3126_cos__plus__cos,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [W: A,Z: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),cos(A,W)),cos(A,Z)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),cos(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),W),Z)),aa(num,A,numeral_numeral(A),bit0(one2)))))),cos(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,minus_minus(A,W),Z)),aa(num,A,numeral_numeral(A),bit0(one2))))) ) ).

% cos_plus_cos
tff(fact_3127_summable__partial__sum__bound,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [F2: fun(nat,A),E: real] :
          ( summable(A,F2)
         => ( pp(aa(real,bool,ord_less(real,zero_zero(real)),E))
           => ~ ! [N10: nat] :
                  ~ ! [M4: nat] :
                      ( pp(aa(nat,bool,ord_less_eq(nat,N10),M4))
                     => ! [N11: nat] : pp(aa(real,bool,ord_less(real,real_V7770717601297561774m_norm(A,aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F2),set_or1337092689740270186AtMost(nat,M4,N11)))),E)) ) ) ) ) ).

% summable_partial_sum_bound
tff(fact_3128_sin__gt__zero2,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,ord_less(real,zero_zero(real)),X))
     => ( pp(aa(real,bool,ord_less(real,X),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2)))))
       => pp(aa(real,bool,ord_less(real,zero_zero(real)),sin(real,X))) ) ) ).

% sin_gt_zero2
tff(fact_3129_sin__lt__zero,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,ord_less(real,pi),X))
     => ( pp(aa(real,bool,ord_less(real,X),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),pi)))
       => pp(aa(real,bool,ord_less(real,sin(real,X)),zero_zero(real))) ) ) ).

% sin_lt_zero
tff(fact_3130_sin__30,axiom,
    sin(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(aa(num,num,bit1,one2))))) = aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),bit0(one2))) ).

% sin_30
tff(fact_3131_cos__double__less__one,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,ord_less(real,zero_zero(real)),X))
     => ( pp(aa(real,bool,ord_less(real,X),aa(num,real,numeral_numeral(real),bit0(one2))))
       => pp(aa(real,bool,ord_less(real,cos(real,aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),X))),one_one(real))) ) ) ).

% cos_double_less_one
tff(fact_3132_cos__gt__zero,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,ord_less(real,zero_zero(real)),X))
     => ( pp(aa(real,bool,ord_less(real,X),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2)))))
       => pp(aa(real,bool,ord_less(real,zero_zero(real)),cos(real,X))) ) ) ).

% cos_gt_zero
tff(fact_3133_sin__monotone__2pi__le,axiom,
    ! [Y: real,X: real] :
      ( pp(aa(real,bool,ord_less_eq(real,aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),Y))
     => ( pp(aa(real,bool,ord_less_eq(real,Y),X))
       => ( pp(aa(real,bool,ord_less_eq(real,X),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2)))))
         => pp(aa(real,bool,ord_less_eq(real,sin(real,Y)),sin(real,X))) ) ) ) ).

% sin_monotone_2pi_le
tff(fact_3134_sin__mono__le__eq,axiom,
    ! [X: real,Y: real] :
      ( pp(aa(real,bool,ord_less_eq(real,aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),X))
     => ( pp(aa(real,bool,ord_less_eq(real,X),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2)))))
       => ( pp(aa(real,bool,ord_less_eq(real,aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),Y))
         => ( pp(aa(real,bool,ord_less_eq(real,Y),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2)))))
           => ( pp(aa(real,bool,ord_less_eq(real,sin(real,X)),sin(real,Y)))
            <=> pp(aa(real,bool,ord_less_eq(real,X),Y)) ) ) ) ) ) ).

% sin_mono_le_eq
tff(fact_3135_sin__inj__pi,axiom,
    ! [X: real,Y: real] :
      ( pp(aa(real,bool,ord_less_eq(real,aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),X))
     => ( pp(aa(real,bool,ord_less_eq(real,X),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2)))))
       => ( pp(aa(real,bool,ord_less_eq(real,aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),Y))
         => ( pp(aa(real,bool,ord_less_eq(real,Y),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2)))))
           => ( ( sin(real,X) = sin(real,Y) )
             => ( X = Y ) ) ) ) ) ) ).

% sin_inj_pi
tff(fact_3136_cos__60,axiom,
    cos(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit1,one2)))) = aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),bit0(one2))) ).

% cos_60
tff(fact_3137_mask__eq__sum__exp,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [N: nat] : aa(A,A,minus_minus(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N)),one_one(A)) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2)))),collect(nat,aTP_Lamp_ap(nat,fun(nat,bool),N))) ) ).

% mask_eq_sum_exp
tff(fact_3138_cos__one__2pi__int,axiom,
    ! [X: real] :
      ( ( cos(real,X) = one_one(real) )
    <=> ? [X2: int] : X = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(int,real,ring_1_of_int(real),X2)),aa(num,real,numeral_numeral(real),bit0(one2)))),pi) ) ).

% cos_one_2pi_int
tff(fact_3139_sum__gp__multiplied,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [M: nat,N: nat,X: A] :
          ( pp(aa(nat,bool,ord_less_eq(nat,M),N))
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,minus_minus(A,one_one(A)),X)),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),power_power(A),X)),set_or1337092689740270186AtMost(nat,M,N))) = aa(A,A,minus_minus(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),X),M)),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(nat,nat,suc,N))) ) ) ) ).

% sum_gp_multiplied
tff(fact_3140_sum_Oin__pairs,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),M: nat,N: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)))) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_ek(fun(nat,A),fun(nat,A),G)),set_or1337092689740270186AtMost(nat,M,N)) ) ).

% sum.in_pairs
tff(fact_3141_cos__double__cos,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [W: A] : cos(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),W)) = aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(nat,A,aa(A,fun(nat,A),power_power(A),cos(A,W)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),one_one(A)) ) ).

% cos_double_cos
tff(fact_3142_cos__treble__cos,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] : cos(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit1,one2))),X)) = aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(bit0(one2)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),cos(A,X)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2))))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit1,one2))),cos(A,X))) ) ).

% cos_treble_cos
tff(fact_3143_sin__le__zero,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,ord_less_eq(real,pi),X))
     => ( pp(aa(real,bool,ord_less(real,X),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),pi)))
       => pp(aa(real,bool,ord_less_eq(real,sin(real,X)),zero_zero(real))) ) ) ).

% sin_le_zero
tff(fact_3144_sin__less__zero,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,ord_less(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,uminus_uminus(real),pi)),aa(num,real,numeral_numeral(real),bit0(one2)))),X))
     => ( pp(aa(real,bool,ord_less(real,X),zero_zero(real)))
       => pp(aa(real,bool,ord_less(real,sin(real,X)),zero_zero(real))) ) ) ).

% sin_less_zero
tff(fact_3145_sin__mono__less__eq,axiom,
    ! [X: real,Y: real] :
      ( pp(aa(real,bool,ord_less_eq(real,aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),X))
     => ( pp(aa(real,bool,ord_less_eq(real,X),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2)))))
       => ( pp(aa(real,bool,ord_less_eq(real,aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),Y))
         => ( pp(aa(real,bool,ord_less_eq(real,Y),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2)))))
           => ( pp(aa(real,bool,ord_less(real,sin(real,X)),sin(real,Y)))
            <=> pp(aa(real,bool,ord_less(real,X),Y)) ) ) ) ) ) ).

% sin_mono_less_eq
tff(fact_3146_sin__monotone__2pi,axiom,
    ! [Y: real,X: real] :
      ( pp(aa(real,bool,ord_less_eq(real,aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),Y))
     => ( pp(aa(real,bool,ord_less(real,Y),X))
       => ( pp(aa(real,bool,ord_less_eq(real,X),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2)))))
         => pp(aa(real,bool,ord_less(real,sin(real,Y)),sin(real,X))) ) ) ) ).

% sin_monotone_2pi
tff(fact_3147_sin__total,axiom,
    ! [Y: real] :
      ( pp(aa(real,bool,ord_less_eq(real,aa(real,real,uminus_uminus(real),one_one(real))),Y))
     => ( pp(aa(real,bool,ord_less_eq(real,Y),one_one(real)))
       => ? [X3: real] :
            ( pp(aa(real,bool,ord_less_eq(real,aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),X3))
            & pp(aa(real,bool,ord_less_eq(real,X3),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2)))))
            & ( sin(real,X3) = Y )
            & ! [Y4: real] :
                ( ( pp(aa(real,bool,ord_less_eq(real,aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),Y4))
                  & pp(aa(real,bool,ord_less_eq(real,Y4),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2)))))
                  & ( sin(real,Y4) = Y ) )
               => ( Y4 = X3 ) ) ) ) ) ).

% sin_total
tff(fact_3148_cos__gt__zero__pi,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,ord_less(real,aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),X))
     => ( pp(aa(real,bool,ord_less(real,X),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2)))))
       => pp(aa(real,bool,ord_less(real,zero_zero(real)),cos(real,X))) ) ) ).

% cos_gt_zero_pi
tff(fact_3149_cos__ge__zero,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,ord_less_eq(real,aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),X))
     => ( pp(aa(real,bool,ord_less_eq(real,X),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2)))))
       => pp(aa(real,bool,ord_less_eq(real,zero_zero(real)),cos(real,X))) ) ) ).

% cos_ge_zero
tff(fact_3150_cos__one__2pi,axiom,
    ! [X: real] :
      ( ( cos(real,X) = one_one(real) )
    <=> ( ? [X2: nat] : X = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),X2)),aa(num,real,numeral_numeral(real),bit0(one2)))),pi)
        | ? [X2: nat] : X = aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),X2)),aa(num,real,numeral_numeral(real),bit0(one2)))),pi)) ) ) ).

% cos_one_2pi
tff(fact_3151_mask__eq__sum__exp__nat,axiom,
    ! [N: nat] : aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)),aa(nat,nat,suc,zero_zero(nat))) = aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2)))),collect(nat,aTP_Lamp_ap(nat,fun(nat,bool),N))) ).

% mask_eq_sum_exp_nat
tff(fact_3152_gauss__sum__nat,axiom,
    ! [N: nat] : aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,aTP_Lamp_el(nat,nat)),set_or1337092689740270186AtMost(nat,zero_zero(nat),N)) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),aa(nat,nat,suc,N))),aa(num,nat,numeral_numeral(nat),bit0(one2))) ).

% gauss_sum_nat
tff(fact_3153_sin__pi__divide__n__gt__0,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,ord_less_eq(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N))
     => pp(aa(real,bool,ord_less(real,zero_zero(real)),sin(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(nat,real,semiring_1_of_nat(real),N))))) ) ).

% sin_pi_divide_n_gt_0
tff(fact_3154_double__arith__series,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A2: A,D: A,N: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_em(A,fun(A,fun(nat,A)),A2),D)),set_or1337092689740270186AtMost(nat,zero_zero(nat),N))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),N)),one_one(A))),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),N)),D))) ) ).

% double_arith_series
tff(fact_3155_double__gauss__sum,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [N: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,semiring_1_of_nat(A)),set_or1337092689740270186AtMost(nat,zero_zero(nat),N))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),N)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),N)),one_one(A))) ) ).

% double_gauss_sum
tff(fact_3156_arith__series__nat,axiom,
    ! [A2: nat,D: nat,N: nat] : aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,aa(nat,fun(nat,nat),aTP_Lamp_en(nat,fun(nat,fun(nat,nat)),A2),D)),set_or1337092689740270186AtMost(nat,zero_zero(nat),N)) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,N)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),A2)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),D)))),aa(num,nat,numeral_numeral(nat),bit0(one2))) ).

% arith_series_nat
tff(fact_3157_Sum__Icc__nat,axiom,
    ! [M: nat,N: nat] : aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,aTP_Lamp_el(nat,nat)),set_or1337092689740270186AtMost(nat,M,N)) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat)))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),aa(nat,nat,minus_minus(nat,M),one_one(nat))))),aa(num,nat,numeral_numeral(nat),bit0(one2))) ).

% Sum_Icc_nat
tff(fact_3158_double__gauss__sum__from__Suc__0,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [N: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,semiring_1_of_nat(A)),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,zero_zero(nat)),N))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),N)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),N)),one_one(A))) ) ).

% double_gauss_sum_from_Suc_0
tff(fact_3159_arith__series,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [A2: A,D: A,N: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_eo(A,fun(A,fun(nat,A)),A2),D)),set_or1337092689740270186AtMost(nat,zero_zero(nat),N)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),N)),one_one(A))),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),N)),D)))),aa(num,A,numeral_numeral(A),bit0(one2))) ) ).

% arith_series
tff(fact_3160_gauss__sum,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [N: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,semiring_1_of_nat(A)),set_or1337092689740270186AtMost(nat,zero_zero(nat),N)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),N)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),N)),one_one(A)))),aa(num,A,numeral_numeral(A),bit0(one2))) ) ).

% gauss_sum
tff(fact_3161_sum__gp__offset,axiom,
    ! [A: $tType] :
      ( ( division_ring(A)
        & comm_ring(A) )
     => ! [X: A,M: nat,N: nat] :
          ( ( ( X = one_one(A) )
           => ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),power_power(A),X)),set_or1337092689740270186AtMost(nat,M,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),N)),one_one(A)) ) )
          & ( ( X != one_one(A) )
           => ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),power_power(A),X)),set_or1337092689740270186AtMost(nat,M,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N))) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),M)),aa(A,A,minus_minus(A,one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(nat,nat,suc,N))))),aa(A,A,minus_minus(A,one_one(A)),X)) ) ) ) ) ).

% sum_gp_offset
tff(fact_3162_sin__zero__iff__int,axiom,
    ! [X: real] :
      ( ( sin(real,X) = zero_zero(real) )
    <=> ? [I3: int] :
          ( pp(aa(int,bool,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),I3))
          & ( X = aa(real,real,aa(real,fun(real,real),times_times(real),aa(int,real,ring_1_of_int(real),I3)),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2)))) ) ) ) ).

% sin_zero_iff_int
tff(fact_3163_cos__zero__iff__int,axiom,
    ! [X: real] :
      ( ( cos(real,X) = zero_zero(real) )
    <=> ? [I3: int] :
          ( ~ pp(aa(int,bool,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),I3))
          & ( X = aa(real,real,aa(real,fun(real,real),times_times(real),aa(int,real,ring_1_of_int(real),I3)),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2)))) ) ) ) ).

% cos_zero_iff_int
tff(fact_3164_sin__zero__lemma,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,ord_less_eq(real,zero_zero(real)),X))
     => ( ( sin(real,X) = zero_zero(real) )
       => ? [N2: nat] :
            ( pp(aa(nat,bool,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N2))
            & ( X = aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),N2)),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2)))) ) ) ) ) ).

% sin_zero_lemma
tff(fact_3165_gauss__sum__from__Suc__0,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [N: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,semiring_1_of_nat(A)),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,zero_zero(nat)),N)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),N)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),N)),one_one(A)))),aa(num,A,numeral_numeral(A),bit0(one2))) ) ).

% gauss_sum_from_Suc_0
tff(fact_3166_sin__zero__iff,axiom,
    ! [X: real] :
      ( ( sin(real,X) = zero_zero(real) )
    <=> ( ? [N5: nat] :
            ( pp(aa(nat,bool,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N5))
            & ( X = aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),N5)),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2)))) ) )
        | ? [N5: nat] :
            ( pp(aa(nat,bool,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N5))
            & ( X = aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),N5)),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))) ) ) ) ) ).

% sin_zero_iff
tff(fact_3167_cos__zero__lemma,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,ord_less_eq(real,zero_zero(real)),X))
     => ( ( cos(real,X) = zero_zero(real) )
       => ? [N2: nat] :
            ( ~ pp(aa(nat,bool,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N2))
            & ( X = aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),N2)),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2)))) ) ) ) ) ).

% cos_zero_lemma
tff(fact_3168_cos__zero__iff,axiom,
    ! [X: real] :
      ( ( cos(real,X) = zero_zero(real) )
    <=> ( ? [N5: nat] :
            ( ~ pp(aa(nat,bool,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N5))
            & ( X = aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),N5)),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2)))) ) )
        | ? [N5: nat] :
            ( ~ pp(aa(nat,bool,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N5))
            & ( X = aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),N5)),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))) ) ) ) ) ).

% cos_zero_iff
tff(fact_3169_lemma__termdiff2,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [H: A,Z: A,N: nat] :
          ( ( H != zero_zero(A) )
         => ( aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,minus_minus(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Z),H)),N)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Z),N))),H)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),N)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Z),aa(nat,nat,minus_minus(nat,N),aa(nat,nat,suc,zero_zero(nat)))))) = aa(A,A,aa(A,fun(A,A),times_times(A),H),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_eq(A,fun(A,fun(nat,fun(nat,A))),H),Z),N)),set_ord_lessThan(nat,aa(nat,nat,minus_minus(nat,N),aa(nat,nat,suc,zero_zero(nat)))))) ) ) ) ).

% lemma_termdiff2
tff(fact_3170_tan__double,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] :
          ( ( cos(A,X) != zero_zero(A) )
         => ( ( cos(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),X)) != zero_zero(A) )
           => ( aa(A,A,tan(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),X)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,tan(A),X))),aa(A,A,minus_minus(A,one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,tan(A),X)),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) ) ) ) ).

% tan_double
tff(fact_3171_complex__unimodular__polar,axiom,
    ! [Z: complex] :
      ( ( real_V7770717601297561774m_norm(complex,Z) = one_one(real) )
     => ~ ! [T5: real] :
            ( pp(aa(real,bool,ord_less_eq(real,zero_zero(real)),T5))
           => ( pp(aa(real,bool,ord_less(real,T5),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),pi)))
             => ( Z != complex2(cos(real,T5),sin(real,T5)) ) ) ) ) ).

% complex_unimodular_polar
tff(fact_3172_dbl__inc__simps_I3_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ( neg_numeral_dbl_inc(A,one_one(A)) = aa(num,A,numeral_numeral(A),aa(num,num,bit1,one2)) ) ) ).

% dbl_inc_simps(3)
tff(fact_3173_geometric__deriv__sums,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Z: A] :
          ( pp(aa(real,bool,ord_less(real,real_V7770717601297561774m_norm(A,Z)),one_one(real)))
         => pp(aa(A,bool,sums(A,aTP_Lamp_er(A,fun(nat,A),Z)),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,minus_minus(A,one_one(A)),Z)),aa(num,nat,numeral_numeral(nat),bit0(one2)))))) ) ) ).

% geometric_deriv_sums
tff(fact_3174_tan__pi,axiom,
    aa(real,real,tan(real),pi) = zero_zero(real) ).

% tan_pi
tff(fact_3175_tan__zero,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ( aa(A,A,tan(A),zero_zero(A)) = zero_zero(A) ) ) ).

% tan_zero
tff(fact_3176_tan__periodic__pi,axiom,
    ! [X: real] : aa(real,real,tan(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),X),pi)) = aa(real,real,tan(real),X) ).

% tan_periodic_pi
tff(fact_3177_sums__zero,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => pp(aa(A,bool,sums(A,aTP_Lamp_bs(nat,A)),zero_zero(A))) ) ).

% sums_zero
tff(fact_3178_lessThan__subset__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y: A] :
          ( pp(aa(set(A),bool,ord_less_eq(set(A),set_ord_lessThan(A,X)),set_ord_lessThan(A,Y)))
        <=> pp(aa(A,bool,ord_less_eq(A,X),Y)) ) ) ).

% lessThan_subset_iff
tff(fact_3179_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)
tff(fact_3180_dbl__inc__simps_I4_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ( neg_numeral_dbl_inc(A,aa(A,A,uminus_uminus(A),one_one(A))) = aa(A,A,uminus_uminus(A),one_one(A)) ) ) ).

% dbl_inc_simps(4)
tff(fact_3181_dbl__inc__simps_I5_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [K: num] : neg_numeral_dbl_inc(A,aa(num,A,numeral_numeral(A),K)) = aa(num,A,numeral_numeral(A),aa(num,num,bit1,K)) ) ).

% dbl_inc_simps(5)
tff(fact_3182_sum_OlessThan__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),N: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_ord_lessThan(nat,aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_ord_lessThan(nat,N))),aa(nat,A,G,N)) ) ).

% sum.lessThan_Suc
tff(fact_3183_tan__npi,axiom,
    ! [N: nat] : aa(real,real,tan(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),N)),pi)) = zero_zero(real) ).

% tan_npi
tff(fact_3184_tan__periodic__n,axiom,
    ! [X: real,N: num] : aa(real,real,tan(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),X),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),N)),pi))) = aa(real,real,tan(real),X) ).

% tan_periodic_n
tff(fact_3185_tan__periodic__nat,axiom,
    ! [X: real,N: nat] : aa(real,real,tan(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),X),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),N)),pi))) = aa(real,real,tan(real),X) ).

% tan_periodic_nat
tff(fact_3186_tan__periodic__int,axiom,
    ! [X: real,I: int] : aa(real,real,tan(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),X),aa(real,real,aa(real,fun(real,real),times_times(real),aa(int,real,ring_1_of_int(real),I)),pi))) = aa(real,real,tan(real),X) ).

% tan_periodic_int
tff(fact_3187_dbl__inc__simps_I1_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [K: num] : neg_numeral_dbl_inc(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),K))) = aa(A,A,uminus_uminus(A),neg_numeral_dbl_dec(A,aa(num,A,numeral_numeral(A),K))) ) ).

% dbl_inc_simps(1)
tff(fact_3188_dbl__dec__simps_I1_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [K: num] : neg_numeral_dbl_dec(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),K))) = aa(A,A,uminus_uminus(A),neg_numeral_dbl_inc(A,aa(num,A,numeral_numeral(A),K))) ) ).

% dbl_dec_simps(1)
tff(fact_3189_powser__sums__zero__iff,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [A2: fun(nat,A),X: A] :
          ( pp(aa(A,bool,sums(A,aTP_Lamp_cm(fun(nat,A),fun(nat,A),A2)),X))
        <=> ( aa(nat,A,A2,zero_zero(nat)) = X ) ) ) ).

% powser_sums_zero_iff
tff(fact_3190_norm__cos__sin,axiom,
    ! [T: real] : real_V7770717601297561774m_norm(complex,complex2(cos(real,T),sin(real,T))) = one_one(real) ).

% norm_cos_sin
tff(fact_3191_tan__periodic,axiom,
    ! [X: real] : aa(real,real,tan(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),X),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),pi))) = aa(real,real,tan(real),X) ).

% tan_periodic
tff(fact_3192_Complex__sum_H,axiom,
    ! [A: $tType,F2: fun(A,real),S3: set(A)] : aa(set(A),complex,groups7311177749621191930dd_sum(A,complex,aTP_Lamp_es(fun(A,real),fun(A,complex),F2)),S3) = complex2(aa(set(A),real,groups7311177749621191930dd_sum(A,real,F2),S3),zero_zero(real)) ).

% Complex_sum'
tff(fact_3193_int__sum,axiom,
    ! [B: $tType,F2: fun(B,nat),A3: set(B)] : aa(nat,int,semiring_1_of_nat(int),aa(set(B),nat,groups7311177749621191930dd_sum(B,nat,F2),A3)) = aa(set(B),int,groups7311177749621191930dd_sum(B,int,aTP_Lamp_et(fun(B,nat),fun(B,int),F2)),A3) ).

% int_sum
tff(fact_3194_sum__diff__distrib,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Q: fun(A,nat),P: fun(A,nat),N: A] :
          ( ! [X3: A] : pp(aa(nat,bool,ord_less_eq(nat,aa(A,nat,Q,X3)),aa(A,nat,P,X3)))
         => ( aa(nat,nat,minus_minus(nat,aa(set(A),nat,groups7311177749621191930dd_sum(A,nat,P),set_ord_lessThan(A,N))),aa(set(A),nat,groups7311177749621191930dd_sum(A,nat,Q),set_ord_lessThan(A,N))) = aa(set(A),nat,groups7311177749621191930dd_sum(A,nat,aa(fun(A,nat),fun(A,nat),aTP_Lamp_eu(fun(A,nat),fun(fun(A,nat),fun(A,nat)),Q),P)),set_ord_lessThan(A,N)) ) ) ) ).

% sum_diff_distrib
tff(fact_3195_sums__0,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [F2: fun(nat,A)] :
          ( ! [N2: nat] : aa(nat,A,F2,N2) = zero_zero(A)
         => pp(aa(A,bool,sums(A,F2),zero_zero(A))) ) ) ).

% sums_0
tff(fact_3196_sums__of__real__iff,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [F2: fun(nat,real),C2: real] :
          ( pp(aa(A,bool,sums(A,aTP_Lamp_ev(fun(nat,real),fun(nat,A),F2)),aa(real,A,real_Vector_of_real(A),C2)))
        <=> pp(aa(real,bool,sums(real,F2),C2)) ) ) ).

% sums_of_real_iff
tff(fact_3197_sums__of__real,axiom,
    ! [A: $tType] :
      ( ( real_V2191834092415804123ebra_1(A)
        & real_V822414075346904944vector(A) )
     => ! [X6: fun(nat,real),A2: real] :
          ( pp(aa(real,bool,sums(real,X6),A2))
         => pp(aa(A,bool,sums(A,aTP_Lamp_cg(fun(nat,real),fun(nat,A),X6)),aa(real,A,real_Vector_of_real(A),A2))) ) ) ).

% sums_of_real
tff(fact_3198_sums__le,axiom,
    ! [A: $tType] :
      ( ( ordere6911136660526730532id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F2: fun(nat,A),G: fun(nat,A),S3: A,T: A] :
          ( ! [N2: nat] : pp(aa(A,bool,ord_less_eq(A,aa(nat,A,F2,N2)),aa(nat,A,G,N2)))
         => ( pp(aa(A,bool,sums(A,F2),S3))
           => ( pp(aa(A,bool,sums(A,G),T))
             => pp(aa(A,bool,ord_less_eq(A,S3),T)) ) ) ) ) ).

% sums_le
tff(fact_3199_complex__of__real__def,axiom,
    ! [R2: real] : aa(real,complex,real_Vector_of_real(complex),R2) = complex2(R2,zero_zero(real)) ).

% complex_of_real_def
tff(fact_3200_complex__of__real__code,axiom,
    ! [X4: real] : aa(real,complex,real_Vector_of_real(complex),X4) = complex2(X4,zero_zero(real)) ).

% complex_of_real_code
tff(fact_3201_complex__eq__cancel__iff2,axiom,
    ! [X: real,Y: real,Xa2: real] :
      ( ( complex2(X,Y) = aa(real,complex,real_Vector_of_real(complex),Xa2) )
    <=> ( ( X = Xa2 )
        & ( Y = zero_zero(real) ) ) ) ).

% complex_eq_cancel_iff2
tff(fact_3202_Complex__eq__0,axiom,
    ! [A2: real,B2: real] :
      ( ( complex2(A2,B2) = zero_zero(complex) )
    <=> ( ( A2 = zero_zero(real) )
        & ( B2 = zero_zero(real) ) ) ) ).

% Complex_eq_0
tff(fact_3203_zero__complex_Ocode,axiom,
    zero_zero(complex) = complex2(zero_zero(real),zero_zero(real)) ).

% zero_complex.code
tff(fact_3204_Complex__mult__complex__of__real,axiom,
    ! [X: real,Y: real,R2: real] : aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),complex2(X,Y)),aa(real,complex,real_Vector_of_real(complex),R2)) = complex2(aa(real,real,aa(real,fun(real,real),times_times(real),X),R2),aa(real,real,aa(real,fun(real,real),times_times(real),Y),R2)) ).

% Complex_mult_complex_of_real
tff(fact_3205_complex__of__real__mult__Complex,axiom,
    ! [R2: real,X: real,Y: real] : aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),aa(real,complex,real_Vector_of_real(complex),R2)),complex2(X,Y)) = complex2(aa(real,real,aa(real,fun(real,real),times_times(real),R2),X),aa(real,real,aa(real,fun(real,real),times_times(real),R2),Y)) ).

% complex_of_real_mult_Complex
tff(fact_3206_sums__single,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [I: nat,F2: fun(nat,A)] : pp(aa(A,bool,sums(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_br(nat,fun(fun(nat,A),fun(nat,A)),I),F2)),aa(nat,A,F2,I))) ) ).

% sums_single
tff(fact_3207_sums__add,axiom,
    ! [A: $tType] :
      ( ( topolo5987344860129210374id_add(A)
        & topological_t2_space(A) )
     => ! [F2: fun(nat,A),A2: A,G: fun(nat,A),B2: A] :
          ( pp(aa(A,bool,sums(A,F2),A2))
         => ( pp(aa(A,bool,sums(A,G),B2))
           => pp(aa(A,bool,sums(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_ci(fun(nat,A),fun(fun(nat,A),fun(nat,A)),F2),G)),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2))) ) ) ) ).

% sums_add
tff(fact_3208_sums__mult2,axiom,
    ! [A: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [F2: fun(nat,A),A2: A,C2: A] :
          ( pp(aa(A,bool,sums(A,F2),A2))
         => pp(aa(A,bool,sums(A,aa(A,fun(nat,A),aTP_Lamp_cc(fun(nat,A),fun(A,fun(nat,A)),F2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2))) ) ) ).

% sums_mult2
tff(fact_3209_sums__mult,axiom,
    ! [A: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [F2: fun(nat,A),A2: A,C2: A] :
          ( pp(aa(A,bool,sums(A,F2),A2))
         => pp(aa(A,bool,sums(A,aa(A,fun(nat,A),aTP_Lamp_cd(fun(nat,A),fun(A,fun(nat,A)),F2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2))) ) ) ).

% sums_mult
tff(fact_3210_sums__diff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A),A2: A,G: fun(nat,A),B2: A] :
          ( pp(aa(A,bool,sums(A,F2),A2))
         => ( pp(aa(A,bool,sums(A,G),B2))
           => pp(aa(A,bool,sums(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_cf(fun(nat,A),fun(fun(nat,A),fun(nat,A)),F2),G)),aa(A,A,minus_minus(A,A2),B2))) ) ) ) ).

% sums_diff
tff(fact_3211_sums__divide,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(nat,A),A2: A,C2: A] :
          ( pp(aa(A,bool,sums(A,F2),A2))
         => pp(aa(A,bool,sums(A,aa(A,fun(nat,A),aTP_Lamp_bv(fun(nat,A),fun(A,fun(nat,A)),F2),C2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),C2))) ) ) ).

% sums_divide
tff(fact_3212_sums__minus,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A),A2: A] :
          ( pp(aa(A,bool,sums(A,F2),A2))
         => pp(aa(A,bool,sums(A,aTP_Lamp_cb(fun(nat,A),fun(nat,A),F2)),aa(A,A,uminus_uminus(A),A2))) ) ) ).

% sums_minus
tff(fact_3213_sums__sum,axiom,
    ! [A: $tType,I6: $tType] :
      ( ( topolo5987344860129210374id_add(A)
        & topological_t2_space(A) )
     => ! [I5: set(I6),F2: fun(I6,fun(nat,A)),X: fun(I6,A)] :
          ( ! [I2: I6] :
              ( pp(aa(set(I6),bool,aa(I6,fun(set(I6),bool),member(I6),I2),I5))
             => pp(aa(A,bool,sums(A,aa(I6,fun(nat,A),F2,I2)),aa(I6,A,X,I2))) )
         => pp(aa(A,bool,sums(A,aa(fun(I6,fun(nat,A)),fun(nat,A),aTP_Lamp_dx(set(I6),fun(fun(I6,fun(nat,A)),fun(nat,A)),I5),F2)),aa(set(I6),A,groups7311177749621191930dd_sum(I6,A,X),I5))) ) ) ).

% sums_sum
tff(fact_3214_lessThan__def,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [U: A] : set_ord_lessThan(A,U) = collect(A,aTP_Lamp_ew(A,fun(A,bool),U)) ) ).

% lessThan_def
tff(fact_3215_sums__iff__shift,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A),N: nat,S3: A] :
          ( pp(aa(A,bool,sums(A,aa(nat,fun(nat,A),aTP_Lamp_bt(fun(nat,A),fun(nat,fun(nat,A)),F2),N)),S3))
        <=> pp(aa(A,bool,sums(A,F2),aa(A,A,aa(A,fun(A,A),plus_plus(A),S3),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F2),set_ord_lessThan(nat,N))))) ) ) ).

% sums_iff_shift
tff(fact_3216_sums__split__initial__segment,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A),S3: A,N: nat] :
          ( pp(aa(A,bool,sums(A,F2),S3))
         => pp(aa(A,bool,sums(A,aa(nat,fun(nat,A),aTP_Lamp_bt(fun(nat,A),fun(nat,fun(nat,A)),F2),N)),aa(A,A,minus_minus(A,S3),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F2),set_ord_lessThan(nat,N))))) ) ) ).

% sums_split_initial_segment
tff(fact_3217_sums__iff__shift_H,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A),N: nat,S3: A] :
          ( pp(aa(A,bool,sums(A,aa(nat,fun(nat,A),aTP_Lamp_bt(fun(nat,A),fun(nat,fun(nat,A)),F2),N)),aa(A,A,minus_minus(A,S3),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F2),set_ord_lessThan(nat,N)))))
        <=> pp(aa(A,bool,sums(A,F2),S3)) ) ) ).

% sums_iff_shift'
tff(fact_3218_sum__eq__empty__iff,axiom,
    ! [A: $tType,B: $tType,A3: set(A),F2: fun(A,multiset(B))] :
      ( pp(aa(set(A),bool,finite_finite2(A),A3))
     => ( ( aa(set(A),multiset(B),groups7311177749621191930dd_sum(A,multiset(B),F2),A3) = zero_zero(multiset(B)) )
      <=> ! [X2: A] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X2),A3))
           => ( aa(A,multiset(B),F2,X2) = zero_zero(multiset(B)) ) ) ) ) ).

% sum_eq_empty_iff
tff(fact_3219_one__complex_Ocode,axiom,
    one_one(complex) = complex2(one_one(real),zero_zero(real)) ).

% one_complex.code
tff(fact_3220_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
tff(fact_3221_sums__mult__iff,axiom,
    ! [A: $tType] :
      ( ( field(A)
        & real_V4412858255891104859lgebra(A) )
     => ! [C2: A,F2: fun(nat,A),D: A] :
          ( ( C2 != zero_zero(A) )
         => ( pp(aa(A,bool,sums(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_ex(A,fun(fun(nat,A),fun(nat,A)),C2),F2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),D)))
          <=> pp(aa(A,bool,sums(A,F2),D)) ) ) ) ).

% sums_mult_iff
tff(fact_3222_sums__mult2__iff,axiom,
    ! [A: $tType] :
      ( ( field(A)
        & real_V4412858255891104859lgebra(A) )
     => ! [C2: A,F2: fun(nat,A),D: A] :
          ( ( C2 != zero_zero(A) )
         => ( pp(aa(A,bool,sums(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_ey(A,fun(fun(nat,A),fun(nat,A)),C2),F2)),aa(A,A,aa(A,fun(A,A),times_times(A),D),C2)))
          <=> pp(aa(A,bool,sums(A,F2),D)) ) ) ) ).

% sums_mult2_iff
tff(fact_3223_Complex__eq__numeral,axiom,
    ! [A2: real,B2: real,W: num] :
      ( ( complex2(A2,B2) = aa(num,complex,numeral_numeral(complex),W) )
    <=> ( ( A2 = aa(num,real,numeral_numeral(real),W) )
        & ( B2 = zero_zero(real) ) ) ) ).

% Complex_eq_numeral
tff(fact_3224_sum__subtractf__nat,axiom,
    ! [A: $tType,A3: set(A),G: fun(A,nat),F2: fun(A,nat)] :
      ( ! [X3: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),A3))
         => pp(aa(nat,bool,ord_less_eq(nat,aa(A,nat,G,X3)),aa(A,nat,F2,X3))) )
     => ( aa(set(A),nat,groups7311177749621191930dd_sum(A,nat,aa(fun(A,nat),fun(A,nat),aTP_Lamp_ez(fun(A,nat),fun(fun(A,nat),fun(A,nat)),G),F2)),A3) = aa(nat,nat,minus_minus(nat,aa(set(A),nat,groups7311177749621191930dd_sum(A,nat,F2),A3)),aa(set(A),nat,groups7311177749621191930dd_sum(A,nat,G),A3)) ) ) ).

% sum_subtractf_nat
tff(fact_3225_Complex__add__complex__of__real,axiom,
    ! [X: real,Y: real,R2: real] : aa(complex,complex,aa(complex,fun(complex,complex),plus_plus(complex),complex2(X,Y)),aa(real,complex,real_Vector_of_real(complex),R2)) = complex2(aa(real,real,aa(real,fun(real,real),plus_plus(real),X),R2),Y) ).

% Complex_add_complex_of_real
tff(fact_3226_complex__of__real__add__Complex,axiom,
    ! [R2: real,X: real,Y: real] : aa(complex,complex,aa(complex,fun(complex,complex),plus_plus(complex),aa(real,complex,real_Vector_of_real(complex),R2)),complex2(X,Y)) = complex2(aa(real,real,aa(real,fun(real,real),plus_plus(real),R2),X),Y) ).

% complex_of_real_add_Complex
tff(fact_3227_complex__add,axiom,
    ! [A2: real,B2: real,C2: real,D: real] : aa(complex,complex,aa(complex,fun(complex,complex),plus_plus(complex),complex2(A2,B2)),complex2(C2,D)) = complex2(aa(real,real,aa(real,fun(real,real),plus_plus(real),A2),C2),aa(real,real,aa(real,fun(real,real),plus_plus(real),B2),D)) ).

% complex_add
tff(fact_3228_sum__SucD,axiom,
    ! [A: $tType,F2: fun(A,nat),A3: set(A),N: nat] :
      ( ( aa(set(A),nat,groups7311177749621191930dd_sum(A,nat,F2),A3) = aa(nat,nat,suc,N) )
     => ? [X3: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),A3))
          & pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),aa(A,nat,F2,X3))) ) ) ).

% sum_SucD
tff(fact_3229_sum__eq__Suc0__iff,axiom,
    ! [A: $tType,A3: set(A),F2: fun(A,nat)] :
      ( pp(aa(set(A),bool,finite_finite2(A),A3))
     => ( ( aa(set(A),nat,groups7311177749621191930dd_sum(A,nat,F2),A3) = aa(nat,nat,suc,zero_zero(nat)) )
      <=> ? [X2: A] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X2),A3))
            & ( aa(A,nat,F2,X2) = aa(nat,nat,suc,zero_zero(nat)) )
            & ! [Xa4: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Xa4),A3))
               => ( ( X2 != Xa4 )
                 => ( aa(A,nat,F2,Xa4) = zero_zero(nat) ) ) ) ) ) ) ).

% sum_eq_Suc0_iff
tff(fact_3230_sum__eq__1__iff,axiom,
    ! [A: $tType,A3: set(A),F2: fun(A,nat)] :
      ( pp(aa(set(A),bool,finite_finite2(A),A3))
     => ( ( aa(set(A),nat,groups7311177749621191930dd_sum(A,nat,F2),A3) = one_one(nat) )
      <=> ? [X2: A] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X2),A3))
            & ( aa(A,nat,F2,X2) = one_one(nat) )
            & ! [Xa4: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Xa4),A3))
               => ( ( X2 != Xa4 )
                 => ( aa(A,nat,F2,Xa4) = zero_zero(nat) ) ) ) ) ) ) ).

% sum_eq_1_iff
tff(fact_3231_sums__mult__D,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [C2: A,F2: fun(nat,A),A2: A] :
          ( pp(aa(A,bool,sums(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_bu(A,fun(fun(nat,A),fun(nat,A)),C2),F2)),A2))
         => ( ( C2 != zero_zero(A) )
           => pp(aa(A,bool,sums(A,F2),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),C2))) ) ) ) ).

% sums_mult_D
tff(fact_3232_sums__Suc__imp,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A),S3: A] :
          ( ( aa(nat,A,F2,zero_zero(nat)) = zero_zero(A) )
         => ( pp(aa(A,bool,sums(A,aTP_Lamp_ce(fun(nat,A),fun(nat,A),F2)),S3))
           => pp(aa(A,bool,sums(A,F2),S3)) ) ) ) ).

% sums_Suc_imp
tff(fact_3233_sums__Suc__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A),S3: A] :
          ( pp(aa(A,bool,sums(A,aTP_Lamp_ce(fun(nat,A),fun(nat,A),F2)),S3))
        <=> pp(aa(A,bool,sums(A,F2),aa(A,A,aa(A,fun(A,A),plus_plus(A),S3),aa(nat,A,F2,zero_zero(nat))))) ) ) ).

% sums_Suc_iff
tff(fact_3234_sums__Suc,axiom,
    ! [A: $tType] :
      ( ( topolo5987344860129210374id_add(A)
        & topological_t2_space(A) )
     => ! [F2: fun(nat,A),L: A] :
          ( pp(aa(A,bool,sums(A,aTP_Lamp_fa(fun(nat,A),fun(nat,A),F2)),L))
         => pp(aa(A,bool,sums(A,F2),aa(A,A,aa(A,fun(A,A),plus_plus(A),L),aa(nat,A,F2,zero_zero(nat))))) ) ) ).

% sums_Suc
tff(fact_3235_sums__zero__iff__shift,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [N: nat,F2: fun(nat,A),S3: A] :
          ( ! [I2: nat] :
              ( pp(aa(nat,bool,ord_less(nat,I2),N))
             => ( aa(nat,A,F2,I2) = zero_zero(A) ) )
         => ( pp(aa(A,bool,sums(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_fb(nat,fun(fun(nat,A),fun(nat,A)),N),F2)),S3))
          <=> pp(aa(A,bool,sums(A,F2),S3)) ) ) ) ).

% sums_zero_iff_shift
tff(fact_3236_Complex__eq__neg__1,axiom,
    ! [A2: real,B2: real] :
      ( ( complex2(A2,B2) = aa(complex,complex,uminus_uminus(complex),one_one(complex)) )
    <=> ( ( A2 = aa(real,real,uminus_uminus(real),one_one(real)) )
        & ( B2 = zero_zero(real) ) ) ) ).

% Complex_eq_neg_1
tff(fact_3237_Complex__eq__neg__numeral,axiom,
    ! [A2: real,B2: real,W: num] :
      ( ( complex2(A2,B2) = aa(complex,complex,uminus_uminus(complex),aa(num,complex,numeral_numeral(complex),W)) )
    <=> ( ( A2 = aa(real,real,uminus_uminus(real),aa(num,real,numeral_numeral(real),W)) )
        & ( B2 = zero_zero(real) ) ) ) ).

% Complex_eq_neg_numeral
tff(fact_3238_sums__finite,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [N4: set(nat),F2: fun(nat,A)] :
          ( pp(aa(set(nat),bool,finite_finite2(nat),N4))
         => ( ! [N2: nat] :
                ( ~ pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),N2),N4))
               => ( aa(nat,A,F2,N2) = zero_zero(A) ) )
           => pp(aa(A,bool,sums(A,F2),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F2),N4))) ) ) ) ).

% sums_finite
tff(fact_3239_sums__If__finite,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [P: fun(nat,bool),F2: fun(nat,A)] :
          ( pp(aa(set(nat),bool,finite_finite2(nat),collect(nat,P)))
         => pp(aa(A,bool,sums(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_bx(fun(nat,bool),fun(fun(nat,A),fun(nat,A)),P),F2)),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F2),collect(nat,P)))) ) ) ).

% sums_If_finite
tff(fact_3240_sums__If__finite__set,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [A3: set(nat),F2: fun(nat,A)] :
          ( pp(aa(set(nat),bool,finite_finite2(nat),A3))
         => pp(aa(A,bool,sums(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_bw(set(nat),fun(fun(nat,A),fun(nat,A)),A3),F2)),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F2),A3))) ) ) ).

% sums_If_finite_set
tff(fact_3241_complex__mult,axiom,
    ! [A2: real,B2: real,C2: real,D: real] : aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),complex2(A2,B2)),complex2(C2,D)) = complex2(aa(real,real,minus_minus(real,aa(real,real,aa(real,fun(real,real),times_times(real),A2),C2)),aa(real,real,aa(real,fun(real,real),times_times(real),B2),D)),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,aa(real,fun(real,real),times_times(real),A2),D)),aa(real,real,aa(real,fun(real,real),times_times(real),B2),C2))) ).

% complex_mult
tff(fact_3242_sum_Onat__diff__reindex,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),N: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aTP_Lamp_fc(fun(nat,A),fun(nat,fun(nat,A)),G),N)),set_ord_lessThan(nat,N)) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_ord_lessThan(nat,N)) ) ).

% sum.nat_diff_reindex
tff(fact_3243_sum__nth__roots,axiom,
    ! [N: nat,C2: complex] :
      ( pp(aa(nat,bool,ord_less(nat,one_one(nat)),N))
     => ( aa(set(complex),complex,groups7311177749621191930dd_sum(complex,complex,aTP_Lamp_fd(complex,complex)),collect(complex,aa(complex,fun(complex,bool),aTP_Lamp_bb(nat,fun(complex,fun(complex,bool)),N),C2))) = zero_zero(complex) ) ) ).

% sum_nth_roots
tff(fact_3244_sum__roots__unity,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,ord_less(nat,one_one(nat)),N))
     => ( aa(set(complex),complex,groups7311177749621191930dd_sum(complex,complex,aTP_Lamp_fd(complex,complex)),collect(complex,aTP_Lamp_fe(nat,fun(complex,bool),N))) = zero_zero(complex) ) ) ).

% sum_roots_unity
tff(fact_3245_powser__sums__if,axiom,
    ! [A: $tType] :
      ( ( ring_1(A)
        & topolo4958980785337419405_space(A) )
     => ! [M: nat,Z: A] : pp(aa(A,bool,sums(A,aa(A,fun(nat,A),aTP_Lamp_ff(nat,fun(A,fun(nat,A)),M),Z)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Z),M))) ) ).

% powser_sums_if
tff(fact_3246_powser__sums__zero,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [A2: fun(nat,A)] : pp(aa(A,bool,sums(A,aTP_Lamp_cm(fun(nat,A),fun(nat,A),A2)),aa(nat,A,A2,zero_zero(nat)))) ) ).

% powser_sums_zero
tff(fact_3247_sum__diff__nat,axiom,
    ! [A: $tType,B4: set(A),A3: set(A),F2: fun(A,nat)] :
      ( pp(aa(set(A),bool,finite_finite2(A),B4))
     => ( pp(aa(set(A),bool,ord_less_eq(set(A),B4),A3))
       => ( aa(set(A),nat,groups7311177749621191930dd_sum(A,nat,F2),aa(set(A),set(A),minus_minus(set(A),A3),B4)) = aa(nat,nat,minus_minus(nat,aa(set(A),nat,groups7311177749621191930dd_sum(A,nat,F2),A3)),aa(set(A),nat,groups7311177749621191930dd_sum(A,nat,F2),B4)) ) ) ) ).

% sum_diff_nat
tff(fact_3248_tan__def,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X4: A] : aa(A,A,tan(A),X4) = aa(A,A,aa(A,fun(A,A),divide_divide(A),sin(A,X4)),cos(A,X4)) ) ).

% tan_def
tff(fact_3249_sums__If__finite__set_H,axiom,
    ! [A: $tType] :
      ( ( topolo1287966508704411220up_add(A)
        & topological_t2_space(A) )
     => ! [G: fun(nat,A),S: A,A3: set(nat),S5: A,F2: fun(nat,A)] :
          ( pp(aa(A,bool,sums(A,G),S))
         => ( pp(aa(set(nat),bool,finite_finite2(nat),A3))
           => ( ( S5 = aa(A,A,aa(A,fun(A,A),plus_plus(A),S),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_fg(fun(nat,A),fun(fun(nat,A),fun(nat,A)),G),F2)),A3)) )
             => pp(aa(A,bool,sums(A,aa(fun(nat,A),fun(nat,A),aa(set(nat),fun(fun(nat,A),fun(nat,A)),aTP_Lamp_fh(fun(nat,A),fun(set(nat),fun(fun(nat,A),fun(nat,A))),G),A3),F2)),S5)) ) ) ) ) ).

% sums_If_finite_set'
tff(fact_3250_suminf__le__const,axiom,
    ! [A: $tType] :
      ( ( ordere6911136660526730532id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F2: fun(nat,A),X: A] :
          ( summable(A,F2)
         => ( ! [N2: nat] : pp(aa(A,bool,ord_less_eq(A,aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F2),set_ord_lessThan(nat,N2))),X))
           => pp(aa(A,bool,ord_less_eq(A,suminf(A,F2)),X)) ) ) ) ).

% suminf_le_const
tff(fact_3251_dbl__inc__def,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [X: A] : neg_numeral_dbl_inc(A,X) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),X)),one_one(A)) ) ).

% dbl_inc_def
tff(fact_3252_sum_OlessThan__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),N: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_ord_lessThan(nat,aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G,zero_zero(nat))),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_ea(fun(nat,A),fun(nat,A),G)),set_ord_lessThan(nat,N))) ) ).

% sum.lessThan_Suc_shift
tff(fact_3253_sum__lessThan__telescope_H,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [F2: fun(nat,A),M: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_fi(fun(nat,A),fun(nat,A),F2)),set_ord_lessThan(nat,M)) = aa(A,A,minus_minus(A,aa(nat,A,F2,zero_zero(nat))),aa(nat,A,F2,M)) ) ).

% sum_lessThan_telescope'
tff(fact_3254_sum__lessThan__telescope,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [F2: fun(nat,A),M: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_eg(fun(nat,A),fun(nat,A),F2)),set_ord_lessThan(nat,M)) = aa(A,A,minus_minus(A,aa(nat,A,F2,M)),aa(nat,A,F2,zero_zero(nat))) ) ).

% sum_lessThan_telescope
tff(fact_3255_sumr__diff__mult__const2,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [F2: fun(nat,A),N: nat,R2: A] : aa(A,A,minus_minus(A,aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F2),set_ord_lessThan(nat,N))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),N)),R2)) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_fj(fun(nat,A),fun(A,fun(nat,A)),F2),R2)),set_ord_lessThan(nat,N)) ) ).

% sumr_diff_mult_const2
tff(fact_3256_summableI__nonneg__bounded,axiom,
    ! [A: $tType] :
      ( ( condit6923001295902523014norder(A)
        & ordere6911136660526730532id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F2: fun(nat,A),X: A] :
          ( ! [N2: nat] : pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),aa(nat,A,F2,N2)))
         => ( ! [N2: nat] : pp(aa(A,bool,ord_less_eq(A,aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F2),set_ord_lessThan(nat,N2))),X))
           => summable(A,F2) ) ) ) ).

% summableI_nonneg_bounded
tff(fact_3257_sum_OatLeast1__atMost__eq,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),N: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,zero_zero(nat)),N)) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_ea(fun(nat,A),fun(nat,A),G)),set_ord_lessThan(nat,N)) ) ).

% sum.atLeast1_atMost_eq
tff(fact_3258_power__diff__1__eq,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [X: A,N: nat] : aa(A,A,minus_minus(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),X),N)),one_one(A)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,minus_minus(A,X),one_one(A))),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),power_power(A),X)),set_ord_lessThan(nat,N))) ) ).

% power_diff_1_eq
tff(fact_3259_one__diff__power__eq,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [X: A,N: nat] : aa(A,A,minus_minus(A,one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,minus_minus(A,one_one(A)),X)),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),power_power(A),X)),set_ord_lessThan(nat,N))) ) ).

% one_diff_power_eq
tff(fact_3260_geometric__sum,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [X: A,N: nat] :
          ( ( X != one_one(A) )
         => ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),power_power(A),X)),set_ord_lessThan(nat,N)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,minus_minus(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),X),N)),one_one(A))),aa(A,A,minus_minus(A,X),one_one(A))) ) ) ) ).

% geometric_sum
tff(fact_3261_suminf__split__initial__segment,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A),K: nat] :
          ( summable(A,F2)
         => ( suminf(A,F2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),suminf(A,aa(nat,fun(nat,A),aTP_Lamp_bt(fun(nat,A),fun(nat,fun(nat,A)),F2),K))),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F2),set_ord_lessThan(nat,K))) ) ) ) ).

% suminf_split_initial_segment
tff(fact_3262_suminf__minus__initial__segment,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A),K: nat] :
          ( summable(A,F2)
         => ( suminf(A,aa(nat,fun(nat,A),aTP_Lamp_bt(fun(nat,A),fun(nat,fun(nat,A)),F2),K)) = aa(A,A,minus_minus(A,suminf(A,F2)),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F2),set_ord_lessThan(nat,K))) ) ) ) ).

% suminf_minus_initial_segment
tff(fact_3263_tan__45,axiom,
    aa(real,real,tan(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(bit0(one2))))) = one_one(real) ).

% tan_45
tff(fact_3264_sum__less__suminf,axiom,
    ! [A: $tType] :
      ( ( ordere8940638589300402666id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F2: fun(nat,A),N: nat] :
          ( summable(A,F2)
         => ( ! [M3: nat] :
                ( pp(aa(nat,bool,ord_less_eq(nat,N),M3))
               => pp(aa(A,bool,ord_less(A,zero_zero(A)),aa(nat,A,F2,M3))) )
           => pp(aa(A,bool,ord_less(A,aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F2),set_ord_lessThan(nat,N))),suminf(A,F2))) ) ) ) ).

% sum_less_suminf
tff(fact_3265_lemma__termdiff1,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [Z: A,H: A,M: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_fk(A,fun(A,fun(nat,fun(nat,A))),Z),H),M)),set_ord_lessThan(nat,M)) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_fl(A,fun(A,fun(nat,fun(nat,A))),Z),H),M)),set_ord_lessThan(nat,M)) ) ).

% lemma_termdiff1
tff(fact_3266_sum__gp__strict,axiom,
    ! [A: $tType] :
      ( ( division_ring(A)
        & comm_ring(A) )
     => ! [X: A,N: nat] :
          ( ( ( X = one_one(A) )
           => ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),power_power(A),X)),set_ord_lessThan(nat,N)) = aa(nat,A,semiring_1_of_nat(A),N) ) )
          & ( ( X != one_one(A) )
           => ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),power_power(A),X)),set_ord_lessThan(nat,N)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,minus_minus(A,one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),N))),aa(A,A,minus_minus(A,one_one(A)),X)) ) ) ) ) ).

% sum_gp_strict
tff(fact_3267_power__diff__sumr2,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [X: A,N: nat,Y: A] : aa(A,A,minus_minus(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),X),N)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,minus_minus(A,X),Y)),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aa(nat,fun(A,fun(nat,A)),aTP_Lamp_fm(A,fun(nat,fun(A,fun(nat,A))),X),N),Y)),set_ord_lessThan(nat,N))) ) ).

% power_diff_sumr2
tff(fact_3268_diff__power__eq__sum,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [X: A,N: nat,Y: A] : aa(A,A,minus_minus(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(nat,nat,suc,N))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,minus_minus(A,X),Y)),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aa(nat,fun(A,fun(nat,A)),aTP_Lamp_fn(A,fun(nat,fun(A,fun(nat,A))),X),N),Y)),set_ord_lessThan(nat,aa(nat,nat,suc,N)))) ) ).

% diff_power_eq_sum
tff(fact_3269_geometric__sums,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [C2: A] :
          ( pp(aa(real,bool,ord_less(real,real_V7770717601297561774m_norm(A,C2)),one_one(real)))
         => pp(aa(A,bool,sums(A,aa(A,fun(nat,A),power_power(A),C2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(A,A,minus_minus(A,one_one(A)),C2)))) ) ) ).

% geometric_sums
tff(fact_3270_power__half__series,axiom,
    pp(aa(real,bool,sums(real,aTP_Lamp_fo(nat,real)),one_one(real))) ).

% power_half_series
tff(fact_3271_lemma__tan__total,axiom,
    ! [Y: real] :
      ( pp(aa(real,bool,ord_less(real,zero_zero(real)),Y))
     => ? [X3: real] :
          ( pp(aa(real,bool,ord_less(real,zero_zero(real)),X3))
          & pp(aa(real,bool,ord_less(real,X3),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2)))))
          & pp(aa(real,bool,ord_less(real,Y),aa(real,real,tan(real),X3))) ) ) ).

% lemma_tan_total
tff(fact_3272_tan__gt__zero,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,ord_less(real,zero_zero(real)),X))
     => ( pp(aa(real,bool,ord_less(real,X),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2)))))
       => pp(aa(real,bool,ord_less(real,zero_zero(real)),aa(real,real,tan(real),X))) ) ) ).

% tan_gt_zero
tff(fact_3273_real__sum__nat__ivl__bounded2,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [N: nat,F2: fun(nat,A),K6: A,K: nat] :
          ( ! [P4: nat] :
              ( pp(aa(nat,bool,ord_less(nat,P4),N))
             => pp(aa(A,bool,ord_less_eq(A,aa(nat,A,F2,P4)),K6)) )
         => ( pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),K6))
           => pp(aa(A,bool,ord_less_eq(A,aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F2),set_ord_lessThan(nat,aa(nat,nat,minus_minus(nat,N),K)))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),N)),K6))) ) ) ) ).

% real_sum_nat_ivl_bounded2
tff(fact_3274_lemma__tan__total1,axiom,
    ! [Y: real] :
    ? [X3: real] :
      ( pp(aa(real,bool,ord_less(real,aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),X3))
      & pp(aa(real,bool,ord_less(real,X3),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2)))))
      & ( aa(real,real,tan(real),X3) = Y ) ) ).

% lemma_tan_total1
tff(fact_3275_tan__mono__lt__eq,axiom,
    ! [X: real,Y: real] :
      ( pp(aa(real,bool,ord_less(real,aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),X))
     => ( pp(aa(real,bool,ord_less(real,X),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2)))))
       => ( pp(aa(real,bool,ord_less(real,aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),Y))
         => ( pp(aa(real,bool,ord_less(real,Y),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2)))))
           => ( pp(aa(real,bool,ord_less(real,aa(real,real,tan(real),X)),aa(real,real,tan(real),Y)))
            <=> pp(aa(real,bool,ord_less(real,X),Y)) ) ) ) ) ) ).

% tan_mono_lt_eq
tff(fact_3276_tan__monotone_H,axiom,
    ! [Y: real,X: real] :
      ( pp(aa(real,bool,ord_less(real,aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),Y))
     => ( pp(aa(real,bool,ord_less(real,Y),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2)))))
       => ( pp(aa(real,bool,ord_less(real,aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),X))
         => ( pp(aa(real,bool,ord_less(real,X),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2)))))
           => ( pp(aa(real,bool,ord_less(real,Y),X))
            <=> pp(aa(real,bool,ord_less(real,aa(real,real,tan(real),Y)),aa(real,real,tan(real),X))) ) ) ) ) ) ).

% tan_monotone'
tff(fact_3277_tan__monotone,axiom,
    ! [Y: real,X: real] :
      ( pp(aa(real,bool,ord_less(real,aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),Y))
     => ( pp(aa(real,bool,ord_less(real,Y),X))
       => ( pp(aa(real,bool,ord_less(real,X),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2)))))
         => pp(aa(real,bool,ord_less(real,aa(real,real,tan(real),Y)),aa(real,real,tan(real),X))) ) ) ) ).

% tan_monotone
tff(fact_3278_tan__total,axiom,
    ! [Y: real] :
    ? [X3: real] :
      ( pp(aa(real,bool,ord_less(real,aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),X3))
      & pp(aa(real,bool,ord_less(real,X3),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2)))))
      & ( aa(real,real,tan(real),X3) = Y )
      & ! [Y4: real] :
          ( ( pp(aa(real,bool,ord_less(real,aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),Y4))
            & pp(aa(real,bool,ord_less(real,Y4),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2)))))
            & ( aa(real,real,tan(real),Y4) = Y ) )
         => ( Y4 = X3 ) ) ) ).

% tan_total
tff(fact_3279_tan__minus__45,axiom,
    aa(real,real,tan(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(bit0(one2)))))) = aa(real,real,uminus_uminus(real),one_one(real)) ).

% tan_minus_45
tff(fact_3280_tan__inverse,axiom,
    ! [Y: real] : aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(real,real,tan(real),Y)) = aa(real,real,tan(real),aa(real,real,minus_minus(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2)))),Y)) ).

% tan_inverse
tff(fact_3281_sum__less__suminf2,axiom,
    ! [A: $tType] :
      ( ( ordere8940638589300402666id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F2: fun(nat,A),N: nat,I: nat] :
          ( summable(A,F2)
         => ( ! [M3: nat] :
                ( pp(aa(nat,bool,ord_less_eq(nat,N),M3))
               => pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),aa(nat,A,F2,M3))) )
           => ( pp(aa(nat,bool,ord_less_eq(nat,N),I))
             => ( pp(aa(A,bool,ord_less(A,zero_zero(A)),aa(nat,A,F2,I)))
               => pp(aa(A,bool,ord_less(A,aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F2),set_ord_lessThan(nat,N))),suminf(A,F2))) ) ) ) ) ) ).

% sum_less_suminf2
tff(fact_3282_one__diff__power__eq_H,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [X: A,N: nat] : aa(A,A,minus_minus(A,one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,minus_minus(A,one_one(A)),X)),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aTP_Lamp_fp(A,fun(nat,fun(nat,A)),X),N)),set_ord_lessThan(nat,N))) ) ).

% one_diff_power_eq'
tff(fact_3283_add__tan__eq,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A,Y: A] :
          ( ( cos(A,X) != zero_zero(A) )
         => ( ( cos(A,Y) != zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,tan(A),X)),aa(A,A,tan(A),Y)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),sin(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y))),aa(A,A,aa(A,fun(A,A),times_times(A),cos(A,X)),cos(A,Y))) ) ) ) ) ).

% add_tan_eq
tff(fact_3284_sums__if_H,axiom,
    ! [G: fun(nat,real),X: real] :
      ( pp(aa(real,bool,sums(real,G),X))
     => pp(aa(real,bool,sums(real,aTP_Lamp_fq(fun(nat,real),fun(nat,real),G)),X)) ) ).

% sums_if'
tff(fact_3285_sums__if,axiom,
    ! [G: fun(nat,real),X: real,F2: fun(nat,real),Y: real] :
      ( pp(aa(real,bool,sums(real,G),X))
     => ( pp(aa(real,bool,sums(real,F2),Y))
       => pp(aa(real,bool,sums(real,aa(fun(nat,real),fun(nat,real),aTP_Lamp_fr(fun(nat,real),fun(fun(nat,real),fun(nat,real)),G),F2)),aa(real,real,aa(real,fun(real,real),plus_plus(real),X),Y))) ) ) ).

% sums_if
tff(fact_3286_tan__pos__pi2__le,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,ord_less_eq(real,zero_zero(real)),X))
     => ( pp(aa(real,bool,ord_less(real,X),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2)))))
       => pp(aa(real,bool,ord_less_eq(real,zero_zero(real)),aa(real,real,tan(real),X))) ) ) ).

% tan_pos_pi2_le
tff(fact_3287_tan__total__pos,axiom,
    ! [Y: real] :
      ( pp(aa(real,bool,ord_less_eq(real,zero_zero(real)),Y))
     => ? [X3: real] :
          ( pp(aa(real,bool,ord_less_eq(real,zero_zero(real)),X3))
          & pp(aa(real,bool,ord_less(real,X3),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2)))))
          & ( aa(real,real,tan(real),X3) = Y ) ) ) ).

% tan_total_pos
tff(fact_3288_tan__less__zero,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,ord_less(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,uminus_uminus(real),pi)),aa(num,real,numeral_numeral(real),bit0(one2)))),X))
     => ( pp(aa(real,bool,ord_less(real,X),zero_zero(real)))
       => pp(aa(real,bool,ord_less(real,aa(real,real,tan(real),X)),zero_zero(real))) ) ) ).

% tan_less_zero
tff(fact_3289_tan__mono__le__eq,axiom,
    ! [X: real,Y: real] :
      ( pp(aa(real,bool,ord_less(real,aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),X))
     => ( pp(aa(real,bool,ord_less(real,X),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2)))))
       => ( pp(aa(real,bool,ord_less(real,aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),Y))
         => ( pp(aa(real,bool,ord_less(real,Y),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2)))))
           => ( pp(aa(real,bool,ord_less_eq(real,aa(real,real,tan(real),X)),aa(real,real,tan(real),Y)))
            <=> pp(aa(real,bool,ord_less_eq(real,X),Y)) ) ) ) ) ) ).

% tan_mono_le_eq
tff(fact_3290_tan__mono__le,axiom,
    ! [X: real,Y: real] :
      ( pp(aa(real,bool,ord_less(real,aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),X))
     => ( pp(aa(real,bool,ord_less_eq(real,X),Y))
       => ( pp(aa(real,bool,ord_less(real,Y),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2)))))
         => pp(aa(real,bool,ord_less_eq(real,aa(real,real,tan(real),X)),aa(real,real,tan(real),Y))) ) ) ) ).

% tan_mono_le
tff(fact_3291_tan__bound__pi2,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,ord_less(real,aa(real,real,abs_abs(real),X)),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(bit0(one2))))))
     => pp(aa(real,bool,ord_less(real,aa(real,real,abs_abs(real),aa(real,real,tan(real),X))),one_one(real))) ) ).

% tan_bound_pi2
tff(fact_3292_arctan,axiom,
    ! [Y: real] :
      ( pp(aa(real,bool,ord_less(real,aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),aa(real,real,arctan,Y)))
      & pp(aa(real,bool,ord_less(real,aa(real,real,arctan,Y)),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2)))))
      & ( aa(real,real,tan(real),aa(real,real,arctan,Y)) = Y ) ) ).

% arctan
tff(fact_3293_arctan__tan,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,ord_less(real,aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),X))
     => ( pp(aa(real,bool,ord_less(real,X),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2)))))
       => ( aa(real,real,arctan,aa(real,real,tan(real),X)) = X ) ) ) ).

% arctan_tan
tff(fact_3294_arctan__unique,axiom,
    ! [X: real,Y: real] :
      ( pp(aa(real,bool,ord_less(real,aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),X))
     => ( pp(aa(real,bool,ord_less(real,X),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2)))))
       => ( ( aa(real,real,tan(real),X) = Y )
         => ( aa(real,real,arctan,Y) = X ) ) ) ) ).

% arctan_unique
tff(fact_3295_sum__split__even__odd,axiom,
    ! [F2: fun(nat,real),G: fun(nat,real),N: nat] : aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aa(fun(nat,real),fun(nat,real),aTP_Lamp_fs(fun(nat,real),fun(fun(nat,real),fun(nat,real)),F2),G)),set_ord_lessThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N))) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_ft(fun(nat,real),fun(nat,real),F2)),set_ord_lessThan(nat,N))),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_fu(fun(nat,real),fun(nat,real),G)),set_ord_lessThan(nat,N))) ).

% sum_split_even_odd
tff(fact_3296_lemma__tan__add1,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A,Y: A] :
          ( ( cos(A,X) != zero_zero(A) )
         => ( ( cos(A,Y) != zero_zero(A) )
           => ( aa(A,A,minus_minus(A,one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,tan(A),X)),aa(A,A,tan(A),Y))) = aa(A,A,aa(A,fun(A,A),divide_divide(A),cos(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y))),aa(A,A,aa(A,fun(A,A),times_times(A),cos(A,X)),cos(A,Y))) ) ) ) ) ).

% lemma_tan_add1
tff(fact_3297_tan__diff,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A,Y: A] :
          ( ( cos(A,X) != zero_zero(A) )
         => ( ( cos(A,Y) != zero_zero(A) )
           => ( ( cos(A,aa(A,A,minus_minus(A,X),Y)) != zero_zero(A) )
             => ( aa(A,A,tan(A),aa(A,A,minus_minus(A,X),Y)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,minus_minus(A,aa(A,A,tan(A),X)),aa(A,A,tan(A),Y))),aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,tan(A),X)),aa(A,A,tan(A),Y)))) ) ) ) ) ) ).

% tan_diff
tff(fact_3298_tan__add,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A,Y: A] :
          ( ( cos(A,X) != zero_zero(A) )
         => ( ( cos(A,Y) != zero_zero(A) )
           => ( ( cos(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y)) != zero_zero(A) )
             => ( aa(A,A,tan(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,tan(A),X)),aa(A,A,tan(A),Y))),aa(A,A,minus_minus(A,one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,tan(A),X)),aa(A,A,tan(A),Y)))) ) ) ) ) ) ).

% tan_add
tff(fact_3299_tan__total__pi4,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,ord_less(real,aa(real,real,abs_abs(real),X)),one_one(real)))
     => ? [Z2: real] :
          ( pp(aa(real,bool,ord_less(real,aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(bit0(one2)))))),Z2))
          & pp(aa(real,bool,ord_less(real,Z2),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(bit0(one2))))))
          & ( aa(real,real,tan(real),Z2) = X ) ) ) ).

% tan_total_pi4
tff(fact_3300_Sum__Icc__int,axiom,
    ! [M: int,N: int] :
      ( pp(aa(int,bool,ord_less_eq(int,M),N))
     => ( aa(set(int),int,groups7311177749621191930dd_sum(int,int,aTP_Lamp_fv(int,int)),set_or1337092689740270186AtMost(int,M,N)) = aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,minus_minus(int,aa(int,int,aa(int,fun(int,int),times_times(int),N),aa(int,int,aa(int,fun(int,int),plus_plus(int),N),one_one(int)))),aa(int,int,aa(int,fun(int,int),times_times(int),M),aa(int,int,minus_minus(int,M),one_one(int))))),aa(num,int,numeral_numeral(int),bit0(one2))) ) ) ).

% Sum_Icc_int
tff(fact_3301_sum__pos__lt__pair,axiom,
    ! [F2: fun(nat,real),K: nat] :
      ( summable(real,F2)
     => ( ! [D2: nat] : pp(aa(real,bool,ord_less(real,zero_zero(real)),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,F2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,aa(nat,nat,suc,zero_zero(nat)))),D2)))),aa(nat,real,F2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,aa(nat,nat,suc,zero_zero(nat)))),D2)),one_one(nat)))))))
       => pp(aa(real,bool,ord_less(real,aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,F2),set_ord_lessThan(nat,K))),suminf(real,F2))) ) ) ).

% sum_pos_lt_pair
tff(fact_3302_tan__half,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] : aa(A,A,tan(A),X) = aa(A,A,aa(A,fun(A,A),divide_divide(A),sin(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),X))),aa(A,A,aa(A,fun(A,A),plus_plus(A),cos(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),X))),one_one(A))) ) ).

% tan_half
tff(fact_3303_sum__bounds__lt__plus1,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [F2: fun(nat,A),Mm: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_ea(fun(nat,A),fun(nat,A),F2)),set_ord_lessThan(nat,Mm)) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F2),set_or1337092689740270186AtMost(nat,one_one(nat),Mm)) ) ).

% sum_bounds_lt_plus1
tff(fact_3304_sumr__cos__zero__one,axiom,
    ! [N: nat] : aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_fw(nat,real)),set_ord_lessThan(nat,aa(nat,nat,suc,N))) = one_one(real) ).

% sumr_cos_zero_one
tff(fact_3305_diffs__equiv,axiom,
    ! [A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & ring_1(A) )
     => ! [C2: fun(nat,A),X: A] :
          ( summable(A,aa(A,fun(nat,A),aTP_Lamp_fx(fun(nat,A),fun(A,fun(nat,A)),C2),X))
         => pp(aa(A,bool,sums(A,aa(A,fun(nat,A),aTP_Lamp_fy(fun(nat,A),fun(A,fun(nat,A)),C2),X)),suminf(A,aa(A,fun(nat,A),aTP_Lamp_fx(fun(nat,A),fun(A,fun(nat,A)),C2),X)))) ) ) ).

% diffs_equiv
tff(fact_3306_sin__paired,axiom,
    ! [X: real] : pp(aa(real,bool,sums(real,aTP_Lamp_fz(real,fun(nat,real),X)),sin(real,X))) ).

% sin_paired
tff(fact_3307_sin__tan,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,ord_less(real,aa(real,real,abs_abs(real),X)),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2)))))
     => ( sin(real,X) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,tan(real),X)),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,tan(real),X)),aa(num,nat,numeral_numeral(nat),bit0(one2)))))) ) ) ).

% sin_tan
tff(fact_3308_real__sqrt__eq__iff,axiom,
    ! [X: real,Y: real] :
      ( ( aa(real,real,sqrt,X) = aa(real,real,sqrt,Y) )
    <=> ( X = Y ) ) ).

% real_sqrt_eq_iff
tff(fact_3309_of__nat__fact,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [N: nat] : aa(nat,A,semiring_1_of_nat(A),semiring_char_0_fact(nat,N)) = semiring_char_0_fact(A,N) ) ).

% of_nat_fact
tff(fact_3310_real__sqrt__eq__zero__cancel__iff,axiom,
    ! [X: real] :
      ( ( aa(real,real,sqrt,X) = zero_zero(real) )
    <=> ( X = zero_zero(real) ) ) ).

% real_sqrt_eq_zero_cancel_iff
tff(fact_3311_real__sqrt__zero,axiom,
    aa(real,real,sqrt,zero_zero(real)) = zero_zero(real) ).

% real_sqrt_zero
tff(fact_3312_real__sqrt__less__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa(real,bool,ord_less(real,aa(real,real,sqrt,X)),aa(real,real,sqrt,Y)))
    <=> pp(aa(real,bool,ord_less(real,X),Y)) ) ).

% real_sqrt_less_iff
tff(fact_3313_real__sqrt__le__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa(real,bool,ord_less_eq(real,aa(real,real,sqrt,X)),aa(real,real,sqrt,Y)))
    <=> pp(aa(real,bool,ord_less_eq(real,X),Y)) ) ).

% real_sqrt_le_iff
tff(fact_3314_real__sqrt__eq__1__iff,axiom,
    ! [X: real] :
      ( ( aa(real,real,sqrt,X) = one_one(real) )
    <=> ( X = one_one(real) ) ) ).

% real_sqrt_eq_1_iff
tff(fact_3315_real__sqrt__one,axiom,
    aa(real,real,sqrt,one_one(real)) = one_one(real) ).

% real_sqrt_one
tff(fact_3316_fact__0,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ( semiring_char_0_fact(A,zero_zero(nat)) = one_one(A) ) ) ).

% fact_0
tff(fact_3317_fact__1,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ( semiring_char_0_fact(A,one_one(nat)) = one_one(A) ) ) ).

% fact_1
tff(fact_3318_real__sqrt__lt__0__iff,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,ord_less(real,aa(real,real,sqrt,X)),zero_zero(real)))
    <=> pp(aa(real,bool,ord_less(real,X),zero_zero(real))) ) ).

% real_sqrt_lt_0_iff
tff(fact_3319_real__sqrt__gt__0__iff,axiom,
    ! [Y: real] :
      ( pp(aa(real,bool,ord_less(real,zero_zero(real)),aa(real,real,sqrt,Y)))
    <=> pp(aa(real,bool,ord_less(real,zero_zero(real)),Y)) ) ).

% real_sqrt_gt_0_iff
tff(fact_3320_real__sqrt__ge__0__iff,axiom,
    ! [Y: real] :
      ( pp(aa(real,bool,ord_less_eq(real,zero_zero(real)),aa(real,real,sqrt,Y)))
    <=> pp(aa(real,bool,ord_less_eq(real,zero_zero(real)),Y)) ) ).

% real_sqrt_ge_0_iff
tff(fact_3321_real__sqrt__le__0__iff,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,ord_less_eq(real,aa(real,real,sqrt,X)),zero_zero(real)))
    <=> pp(aa(real,bool,ord_less_eq(real,X),zero_zero(real))) ) ).

% real_sqrt_le_0_iff
tff(fact_3322_real__sqrt__lt__1__iff,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,ord_less(real,aa(real,real,sqrt,X)),one_one(real)))
    <=> pp(aa(real,bool,ord_less(real,X),one_one(real))) ) ).

% real_sqrt_lt_1_iff
tff(fact_3323_real__sqrt__gt__1__iff,axiom,
    ! [Y: real] :
      ( pp(aa(real,bool,ord_less(real,one_one(real)),aa(real,real,sqrt,Y)))
    <=> pp(aa(real,bool,ord_less(real,one_one(real)),Y)) ) ).

% real_sqrt_gt_1_iff
tff(fact_3324_real__sqrt__ge__1__iff,axiom,
    ! [Y: real] :
      ( pp(aa(real,bool,ord_less_eq(real,one_one(real)),aa(real,real,sqrt,Y)))
    <=> pp(aa(real,bool,ord_less_eq(real,one_one(real)),Y)) ) ).

% real_sqrt_ge_1_iff
tff(fact_3325_real__sqrt__le__1__iff,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,ord_less_eq(real,aa(real,real,sqrt,X)),one_one(real)))
    <=> pp(aa(real,bool,ord_less_eq(real,X),one_one(real))) ) ).

% real_sqrt_le_1_iff
tff(fact_3326_real__sqrt__mult__self,axiom,
    ! [A2: real] : aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,sqrt,A2)),aa(real,real,sqrt,A2)) = aa(real,real,abs_abs(real),A2) ).

% real_sqrt_mult_self
tff(fact_3327_real__sqrt__abs2,axiom,
    ! [X: real] : aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),times_times(real),X),X)) = aa(real,real,abs_abs(real),X) ).

% real_sqrt_abs2
tff(fact_3328_cos__coeff__0,axiom,
    cos_coeff(zero_zero(nat)) = one_one(real) ).

% cos_coeff_0
tff(fact_3329_fact__Suc__0,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ( semiring_char_0_fact(A,aa(nat,nat,suc,zero_zero(nat))) = one_one(A) ) ) ).

% fact_Suc_0
tff(fact_3330_fact__Suc,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [N: nat] : semiring_char_0_fact(A,aa(nat,nat,suc,N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,N))),semiring_char_0_fact(A,N)) ) ).

% fact_Suc
tff(fact_3331_real__sqrt__four,axiom,
    aa(real,real,sqrt,aa(num,real,numeral_numeral(real),bit0(bit0(one2)))) = aa(num,real,numeral_numeral(real),bit0(one2)) ).

% real_sqrt_four
tff(fact_3332_fact__2,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ( semiring_char_0_fact(A,aa(num,nat,numeral_numeral(nat),bit0(one2))) = aa(num,A,numeral_numeral(A),bit0(one2)) ) ) ).

% fact_2
tff(fact_3333_real__sqrt__abs,axiom,
    ! [X: real] : aa(real,real,sqrt,aa(nat,real,aa(real,fun(nat,real),power_power(real),X),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = aa(real,real,abs_abs(real),X) ).

% real_sqrt_abs
tff(fact_3334_real__sqrt__pow2__iff,axiom,
    ! [X: real] :
      ( ( aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,sqrt,X)),aa(num,nat,numeral_numeral(nat),bit0(one2))) = X )
    <=> pp(aa(real,bool,ord_less_eq(real,zero_zero(real)),X)) ) ).

% real_sqrt_pow2_iff
tff(fact_3335_real__sqrt__pow2,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,ord_less_eq(real,zero_zero(real)),X))
     => ( aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,sqrt,X)),aa(num,nat,numeral_numeral(nat),bit0(one2))) = X ) ) ).

% real_sqrt_pow2
tff(fact_3336_real__sqrt__sum__squares__mult__squared__eq,axiom,
    ! [X: real,Y: real,Xa2: real,Ya: real] : aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),X),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Y),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xa2),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Ya),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),aa(num,nat,numeral_numeral(nat),bit0(one2))) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),X),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Y),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xa2),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Ya),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ).

% real_sqrt_sum_squares_mult_squared_eq
tff(fact_3337_diffs__of__real,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [F2: fun(nat,real),X4: nat] : aa(nat,A,diffs(A,aTP_Lamp_ga(fun(nat,real),fun(nat,A),F2)),X4) = aa(real,A,real_Vector_of_real(A),aa(nat,real,diffs(real,F2),X4)) ) ).

% diffs_of_real
tff(fact_3338_real__sqrt__minus,axiom,
    ! [X: real] : aa(real,real,sqrt,aa(real,real,uminus_uminus(real),X)) = aa(real,real,uminus_uminus(real),aa(real,real,sqrt,X)) ).

% real_sqrt_minus
tff(fact_3339_fact__nonzero,axiom,
    ! [A: $tType] :
      ( ( semiring_char_0(A)
        & semiri3467727345109120633visors(A) )
     => ! [N: nat] : semiring_char_0_fact(A,N) != zero_zero(A) ) ).

% fact_nonzero
tff(fact_3340_real__sqrt__less__mono,axiom,
    ! [X: real,Y: real] :
      ( pp(aa(real,bool,ord_less(real,X),Y))
     => pp(aa(real,bool,ord_less(real,aa(real,real,sqrt,X)),aa(real,real,sqrt,Y))) ) ).

% real_sqrt_less_mono
tff(fact_3341_real__sqrt__le__mono,axiom,
    ! [X: real,Y: real] :
      ( pp(aa(real,bool,ord_less_eq(real,X),Y))
     => pp(aa(real,bool,ord_less_eq(real,aa(real,real,sqrt,X)),aa(real,real,sqrt,Y))) ) ).

% real_sqrt_le_mono
tff(fact_3342_real__sqrt__power,axiom,
    ! [X: real,K: nat] : aa(real,real,sqrt,aa(nat,real,aa(real,fun(nat,real),power_power(real),X),K)) = aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,sqrt,X)),K) ).

% real_sqrt_power
tff(fact_3343_real__sqrt__mult,axiom,
    ! [X: real,Y: real] : aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),times_times(real),X),Y)) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,sqrt,X)),aa(real,real,sqrt,Y)) ).

% real_sqrt_mult
tff(fact_3344_real__sqrt__divide,axiom,
    ! [X: real,Y: real] : aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),divide_divide(real),X),Y)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,sqrt,X)),aa(real,real,sqrt,Y)) ).

% real_sqrt_divide
tff(fact_3345_real__sqrt__gt__zero,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,ord_less(real,zero_zero(real)),X))
     => pp(aa(real,bool,ord_less(real,zero_zero(real)),aa(real,real,sqrt,X))) ) ).

% real_sqrt_gt_zero
tff(fact_3346_real__sqrt__ge__zero,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,ord_less_eq(real,zero_zero(real)),X))
     => pp(aa(real,bool,ord_less_eq(real,zero_zero(real)),aa(real,real,sqrt,X))) ) ).

% real_sqrt_ge_zero
tff(fact_3347_real__sqrt__eq__zero__cancel,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,ord_less_eq(real,zero_zero(real)),X))
     => ( ( aa(real,real,sqrt,X) = zero_zero(real) )
       => ( X = zero_zero(real) ) ) ) ).

% real_sqrt_eq_zero_cancel
tff(fact_3348_real__sqrt__ge__one,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,ord_less_eq(real,one_one(real)),X))
     => pp(aa(real,bool,ord_less_eq(real,one_one(real)),aa(real,real,sqrt,X))) ) ).

% real_sqrt_ge_one
tff(fact_3349_fact__ge__zero,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [N: nat] : pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),semiring_char_0_fact(A,N))) ) ).

% fact_ge_zero
tff(fact_3350_fact__not__neg,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [N: nat] : ~ pp(aa(A,bool,ord_less(A,semiring_char_0_fact(A,N)),zero_zero(A))) ) ).

% fact_not_neg
tff(fact_3351_fact__gt__zero,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [N: nat] : pp(aa(A,bool,ord_less(A,zero_zero(A)),semiring_char_0_fact(A,N))) ) ).

% fact_gt_zero
tff(fact_3352_fact__ge__1,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [N: nat] : pp(aa(A,bool,ord_less_eq(A,one_one(A)),semiring_char_0_fact(A,N))) ) ).

% fact_ge_1
tff(fact_3353_fact__fact__dvd__fact,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [K: nat,N: nat] : pp(aa(A,bool,dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),semiring_char_0_fact(A,K)),semiring_char_0_fact(A,N))),semiring_char_0_fact(A,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),N)))) ) ).

% fact_fact_dvd_fact
tff(fact_3354_fact__mono,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [M: nat,N: nat] :
          ( pp(aa(nat,bool,ord_less_eq(nat,M),N))
         => pp(aa(A,bool,ord_less_eq(A,semiring_char_0_fact(A,M)),semiring_char_0_fact(A,N))) ) ) ).

% fact_mono
tff(fact_3355_diffs__minus,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [C2: fun(nat,A),X4: nat] : aa(nat,A,diffs(A,aTP_Lamp_gb(fun(nat,A),fun(nat,A),C2)),X4) = aa(A,A,uminus_uminus(A),aa(nat,A,diffs(A,C2),X4)) ) ).

% diffs_minus
tff(fact_3356_fact__dvd,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [N: nat,M: nat] :
          ( pp(aa(nat,bool,ord_less_eq(nat,N),M))
         => pp(aa(A,bool,dvd_dvd(A,semiring_char_0_fact(A,N)),semiring_char_0_fact(A,M))) ) ) ).

% fact_dvd
tff(fact_3357_choose__dvd,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [K: nat,N: nat] :
          ( pp(aa(nat,bool,ord_less_eq(nat,K),N))
         => pp(aa(A,bool,dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),semiring_char_0_fact(A,K)),semiring_char_0_fact(A,aa(nat,nat,minus_minus(nat,N),K)))),semiring_char_0_fact(A,N))) ) ) ).

% choose_dvd
tff(fact_3358_real__div__sqrt,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,ord_less_eq(real,zero_zero(real)),X))
     => ( aa(real,real,aa(real,fun(real,real),divide_divide(real),X),aa(real,real,sqrt,X)) = aa(real,real,sqrt,X) ) ) ).

% real_div_sqrt
tff(fact_3359_sqrt__add__le__add__sqrt,axiom,
    ! [X: real,Y: real] :
      ( pp(aa(real,bool,ord_less_eq(real,zero_zero(real)),X))
     => ( pp(aa(real,bool,ord_less_eq(real,zero_zero(real)),Y))
       => pp(aa(real,bool,ord_less_eq(real,aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),X),Y))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,sqrt,X)),aa(real,real,sqrt,Y)))) ) ) ).

% sqrt_add_le_add_sqrt
tff(fact_3360_le__real__sqrt__sumsq,axiom,
    ! [X: real,Y: real] : pp(aa(real,bool,ord_less_eq(real,X),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,aa(real,fun(real,real),times_times(real),X),X)),aa(real,real,aa(real,fun(real,real),times_times(real),Y),Y))))) ).

% le_real_sqrt_sumsq
tff(fact_3361_fact__less__mono,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [M: nat,N: nat] :
          ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),M))
         => ( pp(aa(nat,bool,ord_less(nat,M),N))
           => pp(aa(A,bool,ord_less(A,semiring_char_0_fact(A,M)),semiring_char_0_fact(A,N))) ) ) ) ).

% fact_less_mono
tff(fact_3362_fact__mod,axiom,
    ! [A: $tType] :
      ( ( linordered_semidom(A)
        & semidom_modulo(A) )
     => ! [M: nat,N: nat] :
          ( pp(aa(nat,bool,ord_less_eq(nat,M),N))
         => ( modulo_modulo(A,semiring_char_0_fact(A,N),semiring_char_0_fact(A,M)) = zero_zero(A) ) ) ) ).

% fact_mod
tff(fact_3363_fact__le__power,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [N: nat] : pp(aa(A,bool,ord_less_eq(A,semiring_char_0_fact(A,N)),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),N),N)))) ) ).

% fact_le_power
tff(fact_3364_sqrt2__less__2,axiom,
    pp(aa(real,bool,ord_less(real,aa(real,real,sqrt,aa(num,real,numeral_numeral(real),bit0(one2)))),aa(num,real,numeral_numeral(real),bit0(one2)))) ).

% sqrt2_less_2
tff(fact_3365_diffs__def,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [C2: fun(nat,A),X4: nat] : aa(nat,A,diffs(A,C2),X4) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,X4))),aa(nat,A,C2,aa(nat,nat,suc,X4))) ) ).

% diffs_def
tff(fact_3366_fact__numeral,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [K: num] : semiring_char_0_fact(A,aa(num,nat,numeral_numeral(nat),K)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),K)),semiring_char_0_fact(A,pred_numeral(K))) ) ).

% fact_numeral
tff(fact_3367_real__less__rsqrt,axiom,
    ! [X: real,Y: real] :
      ( pp(aa(real,bool,ord_less(real,aa(nat,real,aa(real,fun(nat,real),power_power(real),X),aa(num,nat,numeral_numeral(nat),bit0(one2)))),Y))
     => pp(aa(real,bool,ord_less(real,X),aa(real,real,sqrt,Y))) ) ).

% real_less_rsqrt
tff(fact_3368_real__le__rsqrt,axiom,
    ! [X: real,Y: real] :
      ( pp(aa(real,bool,ord_less_eq(real,aa(nat,real,aa(real,fun(nat,real),power_power(real),X),aa(num,nat,numeral_numeral(nat),bit0(one2)))),Y))
     => pp(aa(real,bool,ord_less_eq(real,X),aa(real,real,sqrt,Y))) ) ).

% real_le_rsqrt
tff(fact_3369_sqrt__le__D,axiom,
    ! [X: real,Y: real] :
      ( pp(aa(real,bool,ord_less_eq(real,aa(real,real,sqrt,X)),Y))
     => pp(aa(real,bool,ord_less_eq(real,X),aa(nat,real,aa(real,fun(nat,real),power_power(real),Y),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) ).

% sqrt_le_D
tff(fact_3370_termdiff__converges__all,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [C2: fun(nat,A),X: A] :
          ( ! [X3: A] : summable(A,aa(A,fun(nat,A),aTP_Lamp_gc(fun(nat,A),fun(A,fun(nat,A)),C2),X3))
         => summable(A,aa(A,fun(nat,A),aTP_Lamp_gd(fun(nat,A),fun(A,fun(nat,A)),C2),X)) ) ) ).

% termdiff_converges_all
tff(fact_3371_tan__60,axiom,
    aa(real,real,tan(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit1,one2)))) = aa(real,real,sqrt,aa(num,real,numeral_numeral(real),aa(num,num,bit1,one2))) ).

% tan_60
tff(fact_3372_square__fact__le__2__fact,axiom,
    ! [N: nat] : pp(aa(real,bool,ord_less_eq(real,aa(real,real,aa(real,fun(real,real),times_times(real),semiring_char_0_fact(real,N)),semiring_char_0_fact(real,N))),semiring_char_0_fact(real,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)))) ).

% square_fact_le_2_fact
tff(fact_3373_real__sqrt__unique,axiom,
    ! [Y: real,X: real] :
      ( ( aa(nat,real,aa(real,fun(nat,real),power_power(real),Y),aa(num,nat,numeral_numeral(nat),bit0(one2))) = X )
     => ( pp(aa(real,bool,ord_less_eq(real,zero_zero(real)),Y))
       => ( aa(real,real,sqrt,X) = Y ) ) ) ).

% real_sqrt_unique
tff(fact_3374_real__le__lsqrt,axiom,
    ! [X: real,Y: real] :
      ( pp(aa(real,bool,ord_less_eq(real,zero_zero(real)),X))
     => ( pp(aa(real,bool,ord_less_eq(real,zero_zero(real)),Y))
       => ( pp(aa(real,bool,ord_less_eq(real,X),aa(nat,real,aa(real,fun(nat,real),power_power(real),Y),aa(num,nat,numeral_numeral(nat),bit0(one2)))))
         => pp(aa(real,bool,ord_less_eq(real,aa(real,real,sqrt,X)),Y)) ) ) ) ).

% real_le_lsqrt
tff(fact_3375_lemma__real__divide__sqrt__less,axiom,
    ! [U: real] :
      ( pp(aa(real,bool,ord_less(real,zero_zero(real)),U))
     => pp(aa(real,bool,ord_less(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),U),aa(real,real,sqrt,aa(num,real,numeral_numeral(real),bit0(one2))))),U)) ) ).

% lemma_real_divide_sqrt_less
tff(fact_3376_real__sqrt__sum__squares__eq__cancel2,axiom,
    ! [X: real,Y: real] :
      ( ( aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),X),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Y),aa(num,nat,numeral_numeral(nat),bit0(one2))))) = Y )
     => ( X = zero_zero(real) ) ) ).

% real_sqrt_sum_squares_eq_cancel2
tff(fact_3377_real__sqrt__sum__squares__eq__cancel,axiom,
    ! [X: real,Y: real] :
      ( ( aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),X),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Y),aa(num,nat,numeral_numeral(nat),bit0(one2))))) = X )
     => ( Y = zero_zero(real) ) ) ).

% real_sqrt_sum_squares_eq_cancel
tff(fact_3378_real__sqrt__sum__squares__triangle__ineq,axiom,
    ! [A2: real,C2: real,B2: real,D: real] : pp(aa(real,bool,ord_less_eq(real,aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),A2),C2)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),B2),D)),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),A2),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),B2),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),C2),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),D),aa(num,nat,numeral_numeral(nat),bit0(one2)))))))) ).

% real_sqrt_sum_squares_triangle_ineq
tff(fact_3379_real__sqrt__sum__squares__ge2,axiom,
    ! [Y: real,X: real] : pp(aa(real,bool,ord_less_eq(real,Y),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),X),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Y),aa(num,nat,numeral_numeral(nat),bit0(one2))))))) ).

% real_sqrt_sum_squares_ge2
tff(fact_3380_real__sqrt__sum__squares__ge1,axiom,
    ! [X: real,Y: real] : pp(aa(real,bool,ord_less_eq(real,X),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),X),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Y),aa(num,nat,numeral_numeral(nat),bit0(one2))))))) ).

% real_sqrt_sum_squares_ge1
tff(fact_3381_sqrt__ge__absD,axiom,
    ! [X: real,Y: real] :
      ( pp(aa(real,bool,ord_less_eq(real,aa(real,real,abs_abs(real),X)),aa(real,real,sqrt,Y)))
     => pp(aa(real,bool,ord_less_eq(real,aa(nat,real,aa(real,fun(nat,real),power_power(real),X),aa(num,nat,numeral_numeral(nat),bit0(one2)))),Y)) ) ).

% sqrt_ge_absD
tff(fact_3382_cos__45,axiom,
    cos(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(bit0(one2))))) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,sqrt,aa(num,real,numeral_numeral(real),bit0(one2)))),aa(num,real,numeral_numeral(real),bit0(one2))) ).

% cos_45
tff(fact_3383_sin__45,axiom,
    sin(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(bit0(one2))))) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,sqrt,aa(num,real,numeral_numeral(real),bit0(one2)))),aa(num,real,numeral_numeral(real),bit0(one2))) ).

% sin_45
tff(fact_3384_cos__coeff__def,axiom,
    ! [X4: nat] :
      ( ( pp(aa(nat,bool,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),X4))
       => ( cos_coeff(X4) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,uminus_uminus(real),one_one(real))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),X4),aa(num,nat,numeral_numeral(nat),bit0(one2))))),semiring_char_0_fact(real,X4)) ) )
      & ( ~ pp(aa(nat,bool,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),X4))
       => ( cos_coeff(X4) = zero_zero(real) ) ) ) ).

% cos_coeff_def
tff(fact_3385_fact__num__eq__if,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [M: nat] :
          ( ( ( M = zero_zero(nat) )
           => ( semiring_char_0_fact(A,M) = one_one(A) ) )
          & ( ( M != zero_zero(nat) )
           => ( semiring_char_0_fact(A,M) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),M)),semiring_char_0_fact(A,aa(nat,nat,minus_minus(nat,M),one_one(nat)))) ) ) ) ) ).

% fact_num_eq_if
tff(fact_3386_fact__reduce,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [N: nat] :
          ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),N))
         => ( semiring_char_0_fact(A,N) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),N)),semiring_char_0_fact(A,aa(nat,nat,minus_minus(nat,N),one_one(nat)))) ) ) ) ).

% fact_reduce
tff(fact_3387_real__less__lsqrt,axiom,
    ! [X: real,Y: real] :
      ( pp(aa(real,bool,ord_less_eq(real,zero_zero(real)),X))
     => ( pp(aa(real,bool,ord_less_eq(real,zero_zero(real)),Y))
       => ( pp(aa(real,bool,ord_less(real,X),aa(nat,real,aa(real,fun(nat,real),power_power(real),Y),aa(num,nat,numeral_numeral(nat),bit0(one2)))))
         => pp(aa(real,bool,ord_less(real,aa(real,real,sqrt,X)),Y)) ) ) ) ).

% real_less_lsqrt
tff(fact_3388_sqrt__sum__squares__le__sum,axiom,
    ! [X: real,Y: real] :
      ( pp(aa(real,bool,ord_less_eq(real,zero_zero(real)),X))
     => ( pp(aa(real,bool,ord_less_eq(real,zero_zero(real)),Y))
       => pp(aa(real,bool,ord_less_eq(real,aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),X),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Y),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(real,real,aa(real,fun(real,real),plus_plus(real),X),Y))) ) ) ).

% sqrt_sum_squares_le_sum
tff(fact_3389_sqrt__even__pow2,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N))
     => ( aa(real,real,sqrt,aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(num,real,numeral_numeral(real),bit0(one2))),N)) = aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(num,real,numeral_numeral(real),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ) ) ).

% sqrt_even_pow2
tff(fact_3390_tan__30,axiom,
    aa(real,real,tan(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(aa(num,num,bit1,one2))))) = aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(real,real,sqrt,aa(num,real,numeral_numeral(real),aa(num,num,bit1,one2)))) ).

% tan_30
tff(fact_3391_real__sqrt__ge__abs1,axiom,
    ! [X: real,Y: real] : pp(aa(real,bool,ord_less_eq(real,aa(real,real,abs_abs(real),X)),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),X),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Y),aa(num,nat,numeral_numeral(nat),bit0(one2))))))) ).

% real_sqrt_ge_abs1
tff(fact_3392_real__sqrt__ge__abs2,axiom,
    ! [Y: real,X: real] : pp(aa(real,bool,ord_less_eq(real,aa(real,real,abs_abs(real),Y)),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),X),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Y),aa(num,nat,numeral_numeral(nat),bit0(one2))))))) ).

% real_sqrt_ge_abs2
tff(fact_3393_sqrt__sum__squares__le__sum__abs,axiom,
    ! [X: real,Y: real] : pp(aa(real,bool,ord_less_eq(real,aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),X),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Y),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,abs_abs(real),X)),aa(real,real,abs_abs(real),Y)))) ).

% sqrt_sum_squares_le_sum_abs
tff(fact_3394_ln__sqrt,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,ord_less(real,zero_zero(real)),X))
     => ( aa(real,real,ln_ln(real),aa(real,real,sqrt,X)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,ln_ln(real),X)),aa(num,real,numeral_numeral(real),bit0(one2))) ) ) ).

% ln_sqrt
tff(fact_3395_cos__30,axiom,
    cos(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(aa(num,num,bit1,one2))))) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,sqrt,aa(num,real,numeral_numeral(real),aa(num,num,bit1,one2)))),aa(num,real,numeral_numeral(real),bit0(one2))) ).

% cos_30
tff(fact_3396_sin__60,axiom,
    sin(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit1,one2)))) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,sqrt,aa(num,real,numeral_numeral(real),aa(num,num,bit1,one2)))),aa(num,real,numeral_numeral(real),bit0(one2))) ).

% sin_60
tff(fact_3397_arsinh__real__def,axiom,
    ! [X: real] : aa(real,real,arsinh(real),X) = aa(real,real,ln_ln(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),X),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),X),aa(num,nat,numeral_numeral(nat),bit0(one2)))),one_one(real))))) ).

% arsinh_real_def
tff(fact_3398_complex__norm,axiom,
    ! [X: real,Y: real] : real_V7770717601297561774m_norm(complex,complex2(X,Y)) = aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),X),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Y),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ).

% complex_norm
tff(fact_3399_Maclaurin__zero,axiom,
    ! [A: $tType] :
      ( zero(A)
     => ! [X: real,N: nat,Diff: fun(nat,fun(A,real))] :
          ( ( X = zero_zero(real) )
         => ( ( N != zero_zero(nat) )
           => ( aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aa(fun(nat,fun(A,real)),fun(nat,real),aTP_Lamp_ge(real,fun(fun(nat,fun(A,real)),fun(nat,real)),X),Diff)),set_ord_lessThan(nat,N)) = aa(A,real,aa(nat,fun(A,real),Diff,zero_zero(nat)),zero_zero(A)) ) ) ) ) ).

% Maclaurin_zero
tff(fact_3400_Maclaurin__lemma,axiom,
    ! [H: real,F2: fun(real,real),J: fun(nat,real),N: nat] :
      ( pp(aa(real,bool,ord_less(real,zero_zero(real)),H))
     => ? [B8: real] : aa(real,real,F2,H) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aa(fun(nat,real),fun(nat,real),aTP_Lamp_gf(real,fun(fun(nat,real),fun(nat,real)),H),J)),set_ord_lessThan(nat,N))),aa(real,real,aa(real,fun(real,real),times_times(real),B8),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),H),N)),semiring_char_0_fact(real,N)))) ) ).

% Maclaurin_lemma
tff(fact_3401_termdiff__converges,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A,K6: real,C2: fun(nat,A)] :
          ( pp(aa(real,bool,ord_less(real,real_V7770717601297561774m_norm(A,X)),K6))
         => ( ! [X3: A] :
                ( pp(aa(real,bool,ord_less(real,real_V7770717601297561774m_norm(A,X3)),K6))
               => summable(A,aa(A,fun(nat,A),aTP_Lamp_gc(fun(nat,A),fun(A,fun(nat,A)),C2),X3)) )
           => summable(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_gg(A,fun(fun(nat,A),fun(nat,A)),X),C2)) ) ) ) ).

% termdiff_converges
tff(fact_3402_real__sqrt__power__even,axiom,
    ! [N: nat,X: real] :
      ( pp(aa(nat,bool,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N))
     => ( pp(aa(real,bool,ord_less_eq(real,zero_zero(real)),X))
       => ( aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,sqrt,X)),N) = aa(nat,real,aa(real,fun(nat,real),power_power(real),X),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ) ) ) ).

% real_sqrt_power_even
tff(fact_3403_arsinh__real__aux,axiom,
    ! [X: real] : pp(aa(real,bool,ord_less(real,zero_zero(real)),aa(real,real,aa(real,fun(real,real),plus_plus(real),X),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),X),aa(num,nat,numeral_numeral(nat),bit0(one2)))),one_one(real)))))) ).

% arsinh_real_aux
tff(fact_3404_real__sqrt__sum__squares__mult__ge__zero,axiom,
    ! [X: real,Y: real,Xa2: real,Ya: real] : pp(aa(real,bool,ord_less_eq(real,zero_zero(real)),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),X),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Y),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xa2),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Ya),aa(num,nat,numeral_numeral(nat),bit0(one2)))))))) ).

% real_sqrt_sum_squares_mult_ge_zero
tff(fact_3405_arith__geo__mean__sqrt,axiom,
    ! [X: real,Y: real] :
      ( pp(aa(real,bool,ord_less_eq(real,zero_zero(real)),X))
     => ( pp(aa(real,bool,ord_less_eq(real,zero_zero(real)),Y))
       => pp(aa(real,bool,ord_less_eq(real,aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),times_times(real),X),Y))),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),X),Y)),aa(num,real,numeral_numeral(real),bit0(one2))))) ) ) ).

% arith_geo_mean_sqrt
tff(fact_3406_powr__half__sqrt,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,ord_less_eq(real,zero_zero(real)),X))
     => ( powr(real,X,aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),bit0(one2)))) = aa(real,real,sqrt,X) ) ) ).

% powr_half_sqrt
tff(fact_3407_cos__x__y__le__one,axiom,
    ! [X: real,Y: real] : pp(aa(real,bool,ord_less_eq(real,aa(real,real,abs_abs(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),X),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),X),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Y),aa(num,nat,numeral_numeral(nat),bit0(one2)))))))),one_one(real))) ).

% cos_x_y_le_one
tff(fact_3408_real__sqrt__sum__squares__less,axiom,
    ! [X: real,U: real,Y: real] :
      ( pp(aa(real,bool,ord_less(real,aa(real,real,abs_abs(real),X)),aa(real,real,aa(real,fun(real,real),divide_divide(real),U),aa(real,real,sqrt,aa(num,real,numeral_numeral(real),bit0(one2))))))
     => ( pp(aa(real,bool,ord_less(real,aa(real,real,abs_abs(real),Y)),aa(real,real,aa(real,fun(real,real),divide_divide(real),U),aa(real,real,sqrt,aa(num,real,numeral_numeral(real),bit0(one2))))))
       => pp(aa(real,bool,ord_less(real,aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),X),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Y),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),U)) ) ) ).

% real_sqrt_sum_squares_less
tff(fact_3409_Maclaurin__cos__expansion,axiom,
    ! [X: real,N: nat] :
    ? [T5: real] :
      ( pp(aa(real,bool,ord_less_eq(real,aa(real,real,abs_abs(real),T5)),aa(real,real,abs_abs(real),X)))
      & ( cos(real,X) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_gh(real,fun(nat,real),X)),set_ord_lessThan(nat,N))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),cos(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),T5),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),bit0(one2)))),aa(nat,real,semiring_1_of_nat(real),N))),pi)))),semiring_char_0_fact(real,N))),aa(nat,real,aa(real,fun(nat,real),power_power(real),X),N))) ) ) ).

% Maclaurin_cos_expansion
tff(fact_3410_arcosh__real__def,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,ord_less_eq(real,one_one(real)),X))
     => ( aa(real,real,arcosh(real),X) = aa(real,real,ln_ln(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),X),aa(real,real,sqrt,aa(real,real,minus_minus(real,aa(nat,real,aa(real,fun(nat,real),power_power(real),X),aa(num,nat,numeral_numeral(nat),bit0(one2)))),one_one(real))))) ) ) ).

% arcosh_real_def
tff(fact_3411_cos__arctan,axiom,
    ! [X: real] : cos(real,aa(real,real,arctan,X)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),aa(nat,real,aa(real,fun(nat,real),power_power(real),X),aa(num,nat,numeral_numeral(nat),bit0(one2)))))) ).

% cos_arctan
tff(fact_3412_sin__arctan,axiom,
    ! [X: real] : sin(real,aa(real,real,arctan,X)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),X),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),aa(nat,real,aa(real,fun(nat,real),power_power(real),X),aa(num,nat,numeral_numeral(nat),bit0(one2)))))) ).

% sin_arctan
tff(fact_3413_sqrt__sum__squares__half__less,axiom,
    ! [X: real,U: real,Y: real] :
      ( pp(aa(real,bool,ord_less(real,X),aa(real,real,aa(real,fun(real,real),divide_divide(real),U),aa(num,real,numeral_numeral(real),bit0(one2)))))
     => ( pp(aa(real,bool,ord_less(real,Y),aa(real,real,aa(real,fun(real,real),divide_divide(real),U),aa(num,real,numeral_numeral(real),bit0(one2)))))
       => ( pp(aa(real,bool,ord_less_eq(real,zero_zero(real)),X))
         => ( pp(aa(real,bool,ord_less_eq(real,zero_zero(real)),Y))
           => pp(aa(real,bool,ord_less(real,aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),X),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Y),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),U)) ) ) ) ) ).

% sqrt_sum_squares_half_less
tff(fact_3414_sin__cos__sqrt,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,ord_less_eq(real,zero_zero(real)),sin(real,X)))
     => ( sin(real,X) = aa(real,real,sqrt,aa(real,real,minus_minus(real,one_one(real)),aa(nat,real,aa(real,fun(nat,real),power_power(real),cos(real,X)),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) ) ).

% sin_cos_sqrt
tff(fact_3415_arctan__half,axiom,
    ! [X: real] : aa(real,real,arctan,X) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),aa(real,real,arctan,aa(real,real,aa(real,fun(real,real),divide_divide(real),X),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),aa(nat,real,aa(real,fun(nat,real),power_power(real),X),aa(num,nat,numeral_numeral(nat),bit0(one2))))))))) ).

% arctan_half
tff(fact_3416_Maclaurin__minus__cos__expansion,axiom,
    ! [N: nat,X: real] :
      ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),N))
     => ( pp(aa(real,bool,ord_less(real,X),zero_zero(real)))
       => ? [T5: real] :
            ( pp(aa(real,bool,ord_less(real,X),T5))
            & pp(aa(real,bool,ord_less(real,T5),zero_zero(real)))
            & ( cos(real,X) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_gh(real,fun(nat,real),X)),set_ord_lessThan(nat,N))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),cos(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),T5),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),bit0(one2)))),aa(nat,real,semiring_1_of_nat(real),N))),pi)))),semiring_char_0_fact(real,N))),aa(nat,real,aa(real,fun(nat,real),power_power(real),X),N))) ) ) ) ) ).

% Maclaurin_minus_cos_expansion
tff(fact_3417_Maclaurin__cos__expansion2,axiom,
    ! [X: real,N: nat] :
      ( pp(aa(real,bool,ord_less(real,zero_zero(real)),X))
     => ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),N))
       => ? [T5: real] :
            ( pp(aa(real,bool,ord_less(real,zero_zero(real)),T5))
            & pp(aa(real,bool,ord_less(real,T5),X))
            & ( cos(real,X) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_gh(real,fun(nat,real),X)),set_ord_lessThan(nat,N))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),cos(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),T5),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),bit0(one2)))),aa(nat,real,semiring_1_of_nat(real),N))),pi)))),semiring_char_0_fact(real,N))),aa(nat,real,aa(real,fun(nat,real),power_power(real),X),N))) ) ) ) ) ).

% Maclaurin_cos_expansion2
tff(fact_3418_cos__paired,axiom,
    ! [X: real] : pp(aa(real,bool,sums(real,aTP_Lamp_gi(real,fun(nat,real),X)),cos(real,X))) ).

% cos_paired
tff(fact_3419_cos__tan,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,ord_less(real,aa(real,real,abs_abs(real),X)),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2)))))
     => ( cos(real,X) = aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,tan(real),X)),aa(num,nat,numeral_numeral(nat),bit0(one2)))))) ) ) ).

% cos_tan
tff(fact_3420_Maclaurin__sin__expansion3,axiom,
    ! [N: nat,X: real] :
      ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),N))
     => ( pp(aa(real,bool,ord_less(real,zero_zero(real)),X))
       => ? [T5: real] :
            ( pp(aa(real,bool,ord_less(real,zero_zero(real)),T5))
            & pp(aa(real,bool,ord_less(real,T5),X))
            & ( sin(real,X) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_gj(real,fun(nat,real),X)),set_ord_lessThan(nat,N))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),sin(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),T5),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),bit0(one2)))),aa(nat,real,semiring_1_of_nat(real),N))),pi)))),semiring_char_0_fact(real,N))),aa(nat,real,aa(real,fun(nat,real),power_power(real),X),N))) ) ) ) ) ).

% Maclaurin_sin_expansion3
tff(fact_3421_Maclaurin__sin__expansion4,axiom,
    ! [X: real,N: nat] :
      ( pp(aa(real,bool,ord_less(real,zero_zero(real)),X))
     => ? [T5: real] :
          ( pp(aa(real,bool,ord_less(real,zero_zero(real)),T5))
          & pp(aa(real,bool,ord_less_eq(real,T5),X))
          & ( sin(real,X) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_gj(real,fun(nat,real),X)),set_ord_lessThan(nat,N))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),sin(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),T5),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),bit0(one2)))),aa(nat,real,semiring_1_of_nat(real),N))),pi)))),semiring_char_0_fact(real,N))),aa(nat,real,aa(real,fun(nat,real),power_power(real),X),N))) ) ) ) ).

% Maclaurin_sin_expansion4
tff(fact_3422_Maclaurin__sin__expansion2,axiom,
    ! [X: real,N: nat] :
    ? [T5: real] :
      ( pp(aa(real,bool,ord_less_eq(real,aa(real,real,abs_abs(real),T5)),aa(real,real,abs_abs(real),X)))
      & ( sin(real,X) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_gj(real,fun(nat,real),X)),set_ord_lessThan(nat,N))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),sin(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),T5),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),bit0(one2)))),aa(nat,real,semiring_1_of_nat(real),N))),pi)))),semiring_char_0_fact(real,N))),aa(nat,real,aa(real,fun(nat,real),power_power(real),X),N))) ) ) ).

% Maclaurin_sin_expansion2
tff(fact_3423_Maclaurin__sin__expansion,axiom,
    ! [X: real,N: nat] :
    ? [T5: real] : sin(real,X) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_gj(real,fun(nat,real),X)),set_ord_lessThan(nat,N))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),sin(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),T5),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),bit0(one2)))),aa(nat,real,semiring_1_of_nat(real),N))),pi)))),semiring_char_0_fact(real,N))),aa(nat,real,aa(real,fun(nat,real),power_power(real),X),N))) ).

% Maclaurin_sin_expansion
tff(fact_3424_sin__coeff__def,axiom,
    ! [X4: nat] :
      ( ( pp(aa(nat,bool,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),X4))
       => ( sin_coeff(X4) = zero_zero(real) ) )
      & ( ~ pp(aa(nat,bool,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),X4))
       => ( sin_coeff(X4) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,uminus_uminus(real),one_one(real))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,minus_minus(nat,X4),aa(nat,nat,suc,zero_zero(nat)))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),semiring_char_0_fact(real,X4)) ) ) ) ).

% sin_coeff_def
tff(fact_3425_sin__coeff__0,axiom,
    sin_coeff(zero_zero(nat)) = zero_zero(real) ).

% sin_coeff_0
tff(fact_3426_fact__ge__self,axiom,
    ! [N: nat] : pp(aa(nat,bool,ord_less_eq(nat,N),semiring_char_0_fact(nat,N))) ).

% fact_ge_self
tff(fact_3427_fact__mono__nat,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,ord_less_eq(nat,M),N))
     => pp(aa(nat,bool,ord_less_eq(nat,semiring_char_0_fact(nat,M)),semiring_char_0_fact(nat,N))) ) ).

% fact_mono_nat
tff(fact_3428_fact__less__mono__nat,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),M))
     => ( pp(aa(nat,bool,ord_less(nat,M),N))
       => pp(aa(nat,bool,ord_less(nat,semiring_char_0_fact(nat,M)),semiring_char_0_fact(nat,N))) ) ) ).

% fact_less_mono_nat
tff(fact_3429_fact__ge__Suc__0__nat,axiom,
    ! [N: nat] : pp(aa(nat,bool,ord_less_eq(nat,aa(nat,nat,suc,zero_zero(nat))),semiring_char_0_fact(nat,N))) ).

% fact_ge_Suc_0_nat
tff(fact_3430_dvd__fact,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,ord_less_eq(nat,one_one(nat)),M))
     => ( pp(aa(nat,bool,ord_less_eq(nat,M),N))
       => pp(aa(nat,bool,dvd_dvd(nat,M),semiring_char_0_fact(nat,N))) ) ) ).

% dvd_fact
tff(fact_3431_fact__diff__Suc,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa(nat,bool,ord_less(nat,N),aa(nat,nat,suc,M)))
     => ( semiring_char_0_fact(nat,aa(nat,nat,minus_minus(nat,aa(nat,nat,suc,M)),N)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,minus_minus(nat,aa(nat,nat,suc,M)),N)),semiring_char_0_fact(nat,aa(nat,nat,minus_minus(nat,M),N))) ) ) ).

% fact_diff_Suc
tff(fact_3432_fact__div__fact__le__pow,axiom,
    ! [R2: nat,N: nat] :
      ( pp(aa(nat,bool,ord_less_eq(nat,R2),N))
     => pp(aa(nat,bool,ord_less_eq(nat,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),semiring_char_0_fact(nat,N)),semiring_char_0_fact(nat,aa(nat,nat,minus_minus(nat,N),R2)))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),N),R2))) ) ).

% fact_div_fact_le_pow
tff(fact_3433_sin__coeff__Suc,axiom,
    ! [N: nat] : sin_coeff(aa(nat,nat,suc,N)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),cos_coeff(N)),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,N))) ).

% sin_coeff_Suc
tff(fact_3434_cos__coeff__Suc,axiom,
    ! [N: nat] : cos_coeff(aa(nat,nat,suc,N)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,uminus_uminus(real),sin_coeff(N))),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,N))) ).

% cos_coeff_Suc
tff(fact_3435_Maclaurin__exp__lt,axiom,
    ! [X: real,N: nat] :
      ( ( X != zero_zero(real) )
     => ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),N))
       => ? [T5: real] :
            ( pp(aa(real,bool,ord_less(real,zero_zero(real)),aa(real,real,abs_abs(real),T5)))
            & pp(aa(real,bool,ord_less(real,aa(real,real,abs_abs(real),T5)),aa(real,real,abs_abs(real),X)))
            & ( aa(real,real,exp(real),X) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_gk(real,fun(nat,real),X)),set_ord_lessThan(nat,N))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,exp(real),T5)),semiring_char_0_fact(real,N))),aa(nat,real,aa(real,fun(nat,real),power_power(real),X),N))) ) ) ) ) ).

% Maclaurin_exp_lt
tff(fact_3436_cos__arcsin,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,ord_less_eq(real,aa(real,real,uminus_uminus(real),one_one(real))),X))
     => ( pp(aa(real,bool,ord_less_eq(real,X),one_one(real)))
       => ( cos(real,aa(real,real,arcsin,X)) = aa(real,real,sqrt,aa(real,real,minus_minus(real,one_one(real)),aa(nat,real,aa(real,fun(nat,real),power_power(real),X),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) ) ) ).

% cos_arcsin
tff(fact_3437_sin__arccos__abs,axiom,
    ! [Y: real] :
      ( pp(aa(real,bool,ord_less_eq(real,aa(real,real,abs_abs(real),Y)),one_one(real)))
     => ( sin(real,aa(real,real,arccos,Y)) = aa(real,real,sqrt,aa(real,real,minus_minus(real,one_one(real)),aa(nat,real,aa(real,fun(nat,real),power_power(real),Y),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) ) ).

% sin_arccos_abs
tff(fact_3438_sin__arccos,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,ord_less_eq(real,aa(real,real,uminus_uminus(real),one_one(real))),X))
     => ( pp(aa(real,bool,ord_less_eq(real,X),one_one(real)))
       => ( sin(real,aa(real,real,arccos,X)) = aa(real,real,sqrt,aa(real,real,minus_minus(real,one_one(real)),aa(nat,real,aa(real,fun(nat,real),power_power(real),X),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) ) ) ).

% sin_arccos
tff(fact_3439_VEBT__internal_Oheight_Osimps_I1_J,axiom,
    ! [A2: bool,B2: bool] : aa(vEBT_VEBT,nat,vEBT_VEBT_height,vEBT_Leaf(A2,B2)) = zero_zero(nat) ).

% VEBT_internal.height.simps(1)
tff(fact_3440_exp__le__cancel__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa(real,bool,ord_less_eq(real,aa(real,real,exp(real),X)),aa(real,real,exp(real),Y)))
    <=> pp(aa(real,bool,ord_less_eq(real,X),Y)) ) ).

% exp_le_cancel_iff
tff(fact_3441_arcsin__0,axiom,
    aa(real,real,arcsin,zero_zero(real)) = zero_zero(real) ).

% arcsin_0
tff(fact_3442_exp__zero,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ( aa(A,A,exp(A),zero_zero(A)) = one_one(A) ) ) ).

% exp_zero
tff(fact_3443_exp__eq__one__iff,axiom,
    ! [X: real] :
      ( ( aa(real,real,exp(real),X) = one_one(real) )
    <=> ( X = zero_zero(real) ) ) ).

% exp_eq_one_iff
tff(fact_3444_arccos__1,axiom,
    aa(real,real,arccos,one_one(real)) = zero_zero(real) ).

% arccos_1
tff(fact_3445_exp__less__one__iff,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,ord_less(real,aa(real,real,exp(real),X)),one_one(real)))
    <=> pp(aa(real,bool,ord_less(real,X),zero_zero(real))) ) ).

% exp_less_one_iff
tff(fact_3446_one__less__exp__iff,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,ord_less(real,one_one(real)),aa(real,real,exp(real),X)))
    <=> pp(aa(real,bool,ord_less(real,zero_zero(real)),X)) ) ).

% one_less_exp_iff
tff(fact_3447_one__le__exp__iff,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,ord_less_eq(real,one_one(real)),aa(real,real,exp(real),X)))
    <=> pp(aa(real,bool,ord_less_eq(real,zero_zero(real)),X)) ) ).

% one_le_exp_iff
tff(fact_3448_exp__le__one__iff,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,ord_less_eq(real,aa(real,real,exp(real),X)),one_one(real)))
    <=> pp(aa(real,bool,ord_less_eq(real,X),zero_zero(real))) ) ).

% exp_le_one_iff
tff(fact_3449_exp__ln,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,ord_less(real,zero_zero(real)),X))
     => ( aa(real,real,exp(real),aa(real,real,ln_ln(real),X)) = X ) ) ).

% exp_ln
tff(fact_3450_exp__ln__iff,axiom,
    ! [X: real] :
      ( ( aa(real,real,exp(real),aa(real,real,ln_ln(real),X)) = X )
    <=> pp(aa(real,bool,ord_less(real,zero_zero(real)),X)) ) ).

% exp_ln_iff
tff(fact_3451_arccos__minus__1,axiom,
    aa(real,real,arccos,aa(real,real,uminus_uminus(real),one_one(real))) = pi ).

% arccos_minus_1
tff(fact_3452_cos__arccos,axiom,
    ! [Y: real] :
      ( pp(aa(real,bool,ord_less_eq(real,aa(real,real,uminus_uminus(real),one_one(real))),Y))
     => ( pp(aa(real,bool,ord_less_eq(real,Y),one_one(real)))
       => ( cos(real,aa(real,real,arccos,Y)) = Y ) ) ) ).

% cos_arccos
tff(fact_3453_sin__arcsin,axiom,
    ! [Y: real] :
      ( pp(aa(real,bool,ord_less_eq(real,aa(real,real,uminus_uminus(real),one_one(real))),Y))
     => ( pp(aa(real,bool,ord_less_eq(real,Y),one_one(real)))
       => ( sin(real,aa(real,real,arcsin,Y)) = Y ) ) ) ).

% sin_arcsin
tff(fact_3454_arccos__0,axiom,
    aa(real,real,arccos,zero_zero(real)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))) ).

% arccos_0
tff(fact_3455_arcsin__1,axiom,
    aa(real,real,arcsin,one_one(real)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))) ).

% arcsin_1
tff(fact_3456_arcsin__minus__1,axiom,
    aa(real,real,arcsin,aa(real,real,uminus_uminus(real),one_one(real))) = aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2)))) ).

% arcsin_minus_1
tff(fact_3457_norm__exp,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [X: A] : pp(aa(real,bool,ord_less_eq(real,real_V7770717601297561774m_norm(A,aa(A,A,exp(A),X))),aa(real,real,exp(real),real_V7770717601297561774m_norm(A,X)))) ) ).

% norm_exp
tff(fact_3458_exp__not__eq__zero,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [X: A] : aa(A,A,exp(A),X) != zero_zero(A) ) ).

% exp_not_eq_zero
tff(fact_3459_exp__times__arg__commute,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,exp(A),A3)),A3) = aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,exp(A),A3)) ) ).

% exp_times_arg_commute
tff(fact_3460_not__exp__less__zero,axiom,
    ! [X: real] : ~ pp(aa(real,bool,ord_less(real,aa(real,real,exp(real),X)),zero_zero(real))) ).

% not_exp_less_zero
tff(fact_3461_exp__gt__zero,axiom,
    ! [X: real] : pp(aa(real,bool,ord_less(real,zero_zero(real)),aa(real,real,exp(real),X))) ).

% exp_gt_zero
tff(fact_3462_exp__total,axiom,
    ! [Y: real] :
      ( pp(aa(real,bool,ord_less(real,zero_zero(real)),Y))
     => ? [X3: real] : aa(real,real,exp(real),X3) = Y ) ).

% exp_total
tff(fact_3463_exp__ge__zero,axiom,
    ! [X: real] : pp(aa(real,bool,ord_less_eq(real,zero_zero(real)),aa(real,real,exp(real),X))) ).

% exp_ge_zero
tff(fact_3464_not__exp__le__zero,axiom,
    ! [X: real] : ~ pp(aa(real,bool,ord_less_eq(real,aa(real,real,exp(real),X)),zero_zero(real))) ).

% not_exp_le_zero
tff(fact_3465_mult__exp__exp,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A,Y: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,exp(A),X)),aa(A,A,exp(A),Y)) = aa(A,A,exp(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y)) ) ).

% mult_exp_exp
tff(fact_3466_exp__add__commuting,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [X: A,Y: A] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),X),Y) = aa(A,A,aa(A,fun(A,A),times_times(A),Y),X) )
         => ( aa(A,A,exp(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,exp(A),X)),aa(A,A,exp(A),Y)) ) ) ) ).

% exp_add_commuting
tff(fact_3467_exp__diff,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A,Y: A] : aa(A,A,exp(A),aa(A,A,minus_minus(A,X),Y)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,exp(A),X)),aa(A,A,exp(A),Y)) ) ).

% exp_diff
tff(fact_3468_exp__gt__one,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,ord_less(real,zero_zero(real)),X))
     => pp(aa(real,bool,ord_less(real,one_one(real)),aa(real,real,exp(real),X))) ) ).

% exp_gt_one
tff(fact_3469_exp__ge__add__one__self,axiom,
    ! [X: real] : pp(aa(real,bool,ord_less_eq(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),X)),aa(real,real,exp(real),X))) ).

% exp_ge_add_one_self
tff(fact_3470_exp__minus__inverse,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,exp(A),X)),aa(A,A,exp(A),aa(A,A,uminus_uminus(A),X))) = one_one(A) ) ).

% exp_minus_inverse
tff(fact_3471_exp__of__nat2__mult,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A,N: nat] : aa(A,A,exp(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),aa(nat,A,semiring_1_of_nat(A),N))) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,exp(A),X)),N) ) ).

% exp_of_nat2_mult
tff(fact_3472_exp__of__nat__mult,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [N: nat,X: A] : aa(A,A,exp(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),N)),X)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,exp(A),X)),N) ) ).

% exp_of_nat_mult
tff(fact_3473_log__ln,axiom,
    ! [X: real] : aa(real,real,ln_ln(real),X) = aa(real,real,log(aa(real,real,exp(real),one_one(real))),X) ).

% log_ln
tff(fact_3474_arccos__le__arccos,axiom,
    ! [X: real,Y: real] :
      ( pp(aa(real,bool,ord_less_eq(real,aa(real,real,uminus_uminus(real),one_one(real))),X))
     => ( pp(aa(real,bool,ord_less_eq(real,X),Y))
       => ( pp(aa(real,bool,ord_less_eq(real,Y),one_one(real)))
         => pp(aa(real,bool,ord_less_eq(real,aa(real,real,arccos,Y)),aa(real,real,arccos,X))) ) ) ) ).

% arccos_le_arccos
tff(fact_3475_arccos__le__mono,axiom,
    ! [X: real,Y: real] :
      ( pp(aa(real,bool,ord_less_eq(real,aa(real,real,abs_abs(real),X)),one_one(real)))
     => ( pp(aa(real,bool,ord_less_eq(real,aa(real,real,abs_abs(real),Y)),one_one(real)))
       => ( pp(aa(real,bool,ord_less_eq(real,aa(real,real,arccos,X)),aa(real,real,arccos,Y)))
        <=> pp(aa(real,bool,ord_less_eq(real,Y),X)) ) ) ) ).

% arccos_le_mono
tff(fact_3476_arccos__eq__iff,axiom,
    ! [X: real,Y: real] :
      ( ( pp(aa(real,bool,ord_less_eq(real,aa(real,real,abs_abs(real),X)),one_one(real)))
        & pp(aa(real,bool,ord_less_eq(real,aa(real,real,abs_abs(real),Y)),one_one(real))) )
     => ( ( aa(real,real,arccos,X) = aa(real,real,arccos,Y) )
      <=> ( X = Y ) ) ) ).

% arccos_eq_iff
tff(fact_3477_arcsin__le__arcsin,axiom,
    ! [X: real,Y: real] :
      ( pp(aa(real,bool,ord_less_eq(real,aa(real,real,uminus_uminus(real),one_one(real))),X))
     => ( pp(aa(real,bool,ord_less_eq(real,X),Y))
       => ( pp(aa(real,bool,ord_less_eq(real,Y),one_one(real)))
         => pp(aa(real,bool,ord_less_eq(real,aa(real,real,arcsin,X)),aa(real,real,arcsin,Y))) ) ) ) ).

% arcsin_le_arcsin
tff(fact_3478_arcsin__minus,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,ord_less_eq(real,aa(real,real,uminus_uminus(real),one_one(real))),X))
     => ( pp(aa(real,bool,ord_less_eq(real,X),one_one(real)))
       => ( aa(real,real,arcsin,aa(real,real,uminus_uminus(real),X)) = aa(real,real,uminus_uminus(real),aa(real,real,arcsin,X)) ) ) ) ).

% arcsin_minus
tff(fact_3479_arcsin__le__mono,axiom,
    ! [X: real,Y: real] :
      ( pp(aa(real,bool,ord_less_eq(real,aa(real,real,abs_abs(real),X)),one_one(real)))
     => ( pp(aa(real,bool,ord_less_eq(real,aa(real,real,abs_abs(real),Y)),one_one(real)))
       => ( pp(aa(real,bool,ord_less_eq(real,aa(real,real,arcsin,X)),aa(real,real,arcsin,Y)))
        <=> pp(aa(real,bool,ord_less_eq(real,X),Y)) ) ) ) ).

% arcsin_le_mono
tff(fact_3480_arcsin__eq__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa(real,bool,ord_less_eq(real,aa(real,real,abs_abs(real),X)),one_one(real)))
     => ( pp(aa(real,bool,ord_less_eq(real,aa(real,real,abs_abs(real),Y)),one_one(real)))
       => ( ( aa(real,real,arcsin,X) = aa(real,real,arcsin,Y) )
        <=> ( X = Y ) ) ) ) ).

% arcsin_eq_iff
tff(fact_3481_exp__ge__add__one__self__aux,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,ord_less_eq(real,zero_zero(real)),X))
     => pp(aa(real,bool,ord_less_eq(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),X)),aa(real,real,exp(real),X))) ) ).

% exp_ge_add_one_self_aux
tff(fact_3482_lemma__exp__total,axiom,
    ! [Y: real] :
      ( pp(aa(real,bool,ord_less_eq(real,one_one(real)),Y))
     => ? [X3: real] :
          ( pp(aa(real,bool,ord_less_eq(real,zero_zero(real)),X3))
          & pp(aa(real,bool,ord_less_eq(real,X3),aa(real,real,minus_minus(real,Y),one_one(real))))
          & ( aa(real,real,exp(real),X3) = Y ) ) ) ).

% lemma_exp_total
tff(fact_3483_ln__ge__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa(real,bool,ord_less(real,zero_zero(real)),X))
     => ( pp(aa(real,bool,ord_less_eq(real,Y),aa(real,real,ln_ln(real),X)))
      <=> pp(aa(real,bool,ord_less_eq(real,aa(real,real,exp(real),Y)),X)) ) ) ).

% ln_ge_iff
tff(fact_3484_ln__x__over__x__mono,axiom,
    ! [X: real,Y: real] :
      ( pp(aa(real,bool,ord_less_eq(real,aa(real,real,exp(real),one_one(real))),X))
     => ( pp(aa(real,bool,ord_less_eq(real,X),Y))
       => pp(aa(real,bool,ord_less_eq(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,ln_ln(real),Y)),Y)),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,ln_ln(real),X)),X))) ) ) ).

% ln_x_over_x_mono
tff(fact_3485_powr__def,axiom,
    ! [A: $tType] :
      ( ln(A)
     => ! [X: A,A2: A] :
          ( ( ( X = zero_zero(A) )
           => ( powr(A,X,A2) = zero_zero(A) ) )
          & ( ( X != zero_zero(A) )
           => ( powr(A,X,A2) = aa(A,A,exp(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(A,A,ln_ln(A),X))) ) ) ) ) ).

% powr_def
tff(fact_3486_arccos__lbound,axiom,
    ! [Y: real] :
      ( pp(aa(real,bool,ord_less_eq(real,aa(real,real,uminus_uminus(real),one_one(real))),Y))
     => ( pp(aa(real,bool,ord_less_eq(real,Y),one_one(real)))
       => pp(aa(real,bool,ord_less_eq(real,zero_zero(real)),aa(real,real,arccos,Y))) ) ) ).

% arccos_lbound
tff(fact_3487_arccos__less__arccos,axiom,
    ! [X: real,Y: real] :
      ( pp(aa(real,bool,ord_less_eq(real,aa(real,real,uminus_uminus(real),one_one(real))),X))
     => ( pp(aa(real,bool,ord_less(real,X),Y))
       => ( pp(aa(real,bool,ord_less_eq(real,Y),one_one(real)))
         => pp(aa(real,bool,ord_less(real,aa(real,real,arccos,Y)),aa(real,real,arccos,X))) ) ) ) ).

% arccos_less_arccos
tff(fact_3488_arccos__less__mono,axiom,
    ! [X: real,Y: real] :
      ( pp(aa(real,bool,ord_less_eq(real,aa(real,real,abs_abs(real),X)),one_one(real)))
     => ( pp(aa(real,bool,ord_less_eq(real,aa(real,real,abs_abs(real),Y)),one_one(real)))
       => ( pp(aa(real,bool,ord_less(real,aa(real,real,arccos,X)),aa(real,real,arccos,Y)))
        <=> pp(aa(real,bool,ord_less(real,Y),X)) ) ) ) ).

% arccos_less_mono
tff(fact_3489_exp__le,axiom,
    pp(aa(real,bool,ord_less_eq(real,aa(real,real,exp(real),one_one(real))),aa(num,real,numeral_numeral(real),aa(num,num,bit1,one2)))) ).

% exp_le
tff(fact_3490_arccos__ubound,axiom,
    ! [Y: real] :
      ( pp(aa(real,bool,ord_less_eq(real,aa(real,real,uminus_uminus(real),one_one(real))),Y))
     => ( pp(aa(real,bool,ord_less_eq(real,Y),one_one(real)))
       => pp(aa(real,bool,ord_less_eq(real,aa(real,real,arccos,Y)),pi)) ) ) ).

% arccos_ubound
tff(fact_3491_arccos__cos,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,ord_less_eq(real,zero_zero(real)),X))
     => ( pp(aa(real,bool,ord_less_eq(real,X),pi))
       => ( aa(real,real,arccos,cos(real,X)) = X ) ) ) ).

% arccos_cos
tff(fact_3492_arcsin__less__arcsin,axiom,
    ! [X: real,Y: real] :
      ( pp(aa(real,bool,ord_less_eq(real,aa(real,real,uminus_uminus(real),one_one(real))),X))
     => ( pp(aa(real,bool,ord_less(real,X),Y))
       => ( pp(aa(real,bool,ord_less_eq(real,Y),one_one(real)))
         => pp(aa(real,bool,ord_less(real,aa(real,real,arcsin,X)),aa(real,real,arcsin,Y))) ) ) ) ).

% arcsin_less_arcsin
tff(fact_3493_arcsin__less__mono,axiom,
    ! [X: real,Y: real] :
      ( pp(aa(real,bool,ord_less_eq(real,aa(real,real,abs_abs(real),X)),one_one(real)))
     => ( pp(aa(real,bool,ord_less_eq(real,aa(real,real,abs_abs(real),Y)),one_one(real)))
       => ( pp(aa(real,bool,ord_less(real,aa(real,real,arcsin,X)),aa(real,real,arcsin,Y)))
        <=> pp(aa(real,bool,ord_less(real,X),Y)) ) ) ) ).

% arcsin_less_mono
tff(fact_3494_cos__arccos__abs,axiom,
    ! [Y: real] :
      ( pp(aa(real,bool,ord_less_eq(real,aa(real,real,abs_abs(real),Y)),one_one(real)))
     => ( cos(real,aa(real,real,arccos,Y)) = Y ) ) ).

% cos_arccos_abs
tff(fact_3495_arccos__cos__eq__abs,axiom,
    ! [Theta: real] :
      ( pp(aa(real,bool,ord_less_eq(real,aa(real,real,abs_abs(real),Theta)),pi))
     => ( aa(real,real,arccos,cos(real,Theta)) = aa(real,real,abs_abs(real),Theta) ) ) ).

% arccos_cos_eq_abs
tff(fact_3496_exp__divide__power__eq,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [N: nat,X: A] :
          ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),N))
         => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,exp(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),aa(nat,A,semiring_1_of_nat(A),N)))),N) = aa(A,A,exp(A),X) ) ) ) ).

% exp_divide_power_eq
tff(fact_3497_tanh__altdef,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] : aa(A,A,tanh(A),X) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,minus_minus(A,aa(A,A,exp(A),X)),aa(A,A,exp(A),aa(A,A,uminus_uminus(A),X)))),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,exp(A),X)),aa(A,A,exp(A),aa(A,A,uminus_uminus(A),X)))) ) ).

% tanh_altdef
tff(fact_3498_exp__half__le2,axiom,
    pp(aa(real,bool,ord_less_eq(real,aa(real,real,exp(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),bit0(one2))))),aa(num,real,numeral_numeral(real),bit0(one2)))) ).

% exp_half_le2
tff(fact_3499_arccos__lt__bounded,axiom,
    ! [Y: real] :
      ( pp(aa(real,bool,ord_less(real,aa(real,real,uminus_uminus(real),one_one(real))),Y))
     => ( pp(aa(real,bool,ord_less(real,Y),one_one(real)))
       => ( pp(aa(real,bool,ord_less(real,zero_zero(real)),aa(real,real,arccos,Y)))
          & pp(aa(real,bool,ord_less(real,aa(real,real,arccos,Y)),pi)) ) ) ) ).

% arccos_lt_bounded
tff(fact_3500_arccos__bounded,axiom,
    ! [Y: real] :
      ( pp(aa(real,bool,ord_less_eq(real,aa(real,real,uminus_uminus(real),one_one(real))),Y))
     => ( pp(aa(real,bool,ord_less_eq(real,Y),one_one(real)))
       => ( pp(aa(real,bool,ord_less_eq(real,zero_zero(real)),aa(real,real,arccos,Y)))
          & pp(aa(real,bool,ord_less_eq(real,aa(real,real,arccos,Y)),pi)) ) ) ) ).

% arccos_bounded
tff(fact_3501_exp__double,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Z: A] : aa(A,A,exp(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),Z)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,exp(A),Z)),aa(num,nat,numeral_numeral(nat),bit0(one2))) ) ).

% exp_double
tff(fact_3502_sin__arccos__nonzero,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,ord_less(real,aa(real,real,uminus_uminus(real),one_one(real))),X))
     => ( pp(aa(real,bool,ord_less(real,X),one_one(real)))
       => ( sin(real,aa(real,real,arccos,X)) != zero_zero(real) ) ) ) ).

% sin_arccos_nonzero
tff(fact_3503_arccos__cos2,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,ord_less_eq(real,X),zero_zero(real)))
     => ( pp(aa(real,bool,ord_less_eq(real,aa(real,real,uminus_uminus(real),pi)),X))
       => ( aa(real,real,arccos,cos(real,X)) = aa(real,real,uminus_uminus(real),X) ) ) ) ).

% arccos_cos2
tff(fact_3504_arccos__minus,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,ord_less_eq(real,aa(real,real,uminus_uminus(real),one_one(real))),X))
     => ( pp(aa(real,bool,ord_less_eq(real,X),one_one(real)))
       => ( aa(real,real,arccos,aa(real,real,uminus_uminus(real),X)) = aa(real,real,minus_minus(real,pi),aa(real,real,arccos,X)) ) ) ) ).

% arccos_minus
tff(fact_3505_cos__arcsin__nonzero,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,ord_less(real,aa(real,real,uminus_uminus(real),one_one(real))),X))
     => ( pp(aa(real,bool,ord_less(real,X),one_one(real)))
       => ( cos(real,aa(real,real,arcsin,X)) != zero_zero(real) ) ) ) ).

% cos_arcsin_nonzero
tff(fact_3506_arccos,axiom,
    ! [Y: real] :
      ( pp(aa(real,bool,ord_less_eq(real,aa(real,real,uminus_uminus(real),one_one(real))),Y))
     => ( pp(aa(real,bool,ord_less_eq(real,Y),one_one(real)))
       => ( pp(aa(real,bool,ord_less_eq(real,zero_zero(real)),aa(real,real,arccos,Y)))
          & pp(aa(real,bool,ord_less_eq(real,aa(real,real,arccos,Y)),pi))
          & ( cos(real,aa(real,real,arccos,Y)) = Y ) ) ) ) ).

% arccos
tff(fact_3507_exp__bound__half,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Z: A] :
          ( pp(aa(real,bool,ord_less_eq(real,real_V7770717601297561774m_norm(A,Z)),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),bit0(one2)))))
         => pp(aa(real,bool,ord_less_eq(real,real_V7770717601297561774m_norm(A,aa(A,A,exp(A),Z))),aa(num,real,numeral_numeral(real),bit0(one2)))) ) ) ).

% exp_bound_half
tff(fact_3508_arccos__minus__abs,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,ord_less_eq(real,aa(real,real,abs_abs(real),X)),one_one(real)))
     => ( aa(real,real,arccos,aa(real,real,uminus_uminus(real),X)) = aa(real,real,minus_minus(real,pi),aa(real,real,arccos,X)) ) ) ).

% arccos_minus_abs
tff(fact_3509_VEBT__internal_Oheight_Ocases,axiom,
    ! [X: vEBT_VEBT] :
      ( ! [A4: bool,B3: bool] : X != vEBT_Leaf(A4,B3)
     => ~ ! [Uu2: option(product_prod(nat,nat)),Deg2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] : X != vEBT_Node(Uu2,Deg2,TreeList2,Summary2) ) ).

% VEBT_internal.height.cases
tff(fact_3510_exp__bound,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,ord_less_eq(real,zero_zero(real)),X))
     => ( pp(aa(real,bool,ord_less_eq(real,X),one_one(real)))
       => pp(aa(real,bool,ord_less_eq(real,aa(real,real,exp(real),X)),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),X)),aa(nat,real,aa(real,fun(nat,real),power_power(real),X),aa(num,nat,numeral_numeral(nat),bit0(one2)))))) ) ) ).

% exp_bound
tff(fact_3511_real__exp__bound__lemma,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,ord_less_eq(real,zero_zero(real)),X))
     => ( pp(aa(real,bool,ord_less_eq(real,X),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),bit0(one2)))))
       => pp(aa(real,bool,ord_less_eq(real,aa(real,real,exp(real),X)),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),X)))) ) ) ).

% real_exp_bound_lemma
tff(fact_3512_exp__ge__one__plus__x__over__n__power__n,axiom,
    ! [N: nat,X: real] :
      ( pp(aa(real,bool,ord_less_eq(real,aa(real,real,uminus_uminus(real),aa(nat,real,semiring_1_of_nat(real),N))),X))
     => ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),N))
       => pp(aa(real,bool,ord_less_eq(real,aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),aa(real,real,aa(real,fun(real,real),divide_divide(real),X),aa(nat,real,semiring_1_of_nat(real),N)))),N)),aa(real,real,exp(real),X))) ) ) ).

% exp_ge_one_plus_x_over_n_power_n
tff(fact_3513_exp__ge__one__minus__x__over__n__power__n,axiom,
    ! [X: real,N: nat] :
      ( pp(aa(real,bool,ord_less_eq(real,X),aa(nat,real,semiring_1_of_nat(real),N)))
     => ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),N))
       => pp(aa(real,bool,ord_less_eq(real,aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,minus_minus(real,one_one(real)),aa(real,real,aa(real,fun(real,real),divide_divide(real),X),aa(nat,real,semiring_1_of_nat(real),N)))),N)),aa(real,real,exp(real),aa(real,real,uminus_uminus(real),X)))) ) ) ).

% exp_ge_one_minus_x_over_n_power_n
tff(fact_3514_arccos__le__pi2,axiom,
    ! [Y: real] :
      ( pp(aa(real,bool,ord_less_eq(real,zero_zero(real)),Y))
     => ( pp(aa(real,bool,ord_less_eq(real,Y),one_one(real)))
       => pp(aa(real,bool,ord_less_eq(real,aa(real,real,arccos,Y)),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))) ) ) ).

% arccos_le_pi2
tff(fact_3515_exp__bound__lemma,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Z: A] :
          ( pp(aa(real,bool,ord_less_eq(real,real_V7770717601297561774m_norm(A,Z)),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),bit0(one2)))))
         => pp(aa(real,bool,ord_less_eq(real,real_V7770717601297561774m_norm(A,aa(A,A,exp(A),Z))),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),real_V7770717601297561774m_norm(A,Z))))) ) ) ).

% exp_bound_lemma
tff(fact_3516_arcsin__lt__bounded,axiom,
    ! [Y: real] :
      ( pp(aa(real,bool,ord_less(real,aa(real,real,uminus_uminus(real),one_one(real))),Y))
     => ( pp(aa(real,bool,ord_less(real,Y),one_one(real)))
       => ( pp(aa(real,bool,ord_less(real,aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),aa(real,real,arcsin,Y)))
          & pp(aa(real,bool,ord_less(real,aa(real,real,arcsin,Y)),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))) ) ) ) ).

% arcsin_lt_bounded
tff(fact_3517_arcsin__lbound,axiom,
    ! [Y: real] :
      ( pp(aa(real,bool,ord_less_eq(real,aa(real,real,uminus_uminus(real),one_one(real))),Y))
     => ( pp(aa(real,bool,ord_less_eq(real,Y),one_one(real)))
       => pp(aa(real,bool,ord_less_eq(real,aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),aa(real,real,arcsin,Y))) ) ) ).

% arcsin_lbound
tff(fact_3518_arcsin__ubound,axiom,
    ! [Y: real] :
      ( pp(aa(real,bool,ord_less_eq(real,aa(real,real,uminus_uminus(real),one_one(real))),Y))
     => ( pp(aa(real,bool,ord_less_eq(real,Y),one_one(real)))
       => pp(aa(real,bool,ord_less_eq(real,aa(real,real,arcsin,Y)),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))) ) ) ).

% arcsin_ubound
tff(fact_3519_arcsin__bounded,axiom,
    ! [Y: real] :
      ( pp(aa(real,bool,ord_less_eq(real,aa(real,real,uminus_uminus(real),one_one(real))),Y))
     => ( pp(aa(real,bool,ord_less_eq(real,Y),one_one(real)))
       => ( pp(aa(real,bool,ord_less_eq(real,aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),aa(real,real,arcsin,Y)))
          & pp(aa(real,bool,ord_less_eq(real,aa(real,real,arcsin,Y)),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))) ) ) ) ).

% arcsin_bounded
tff(fact_3520_Maclaurin__exp__le,axiom,
    ! [X: real,N: nat] :
    ? [T5: real] :
      ( pp(aa(real,bool,ord_less_eq(real,aa(real,real,abs_abs(real),T5)),aa(real,real,abs_abs(real),X)))
      & ( aa(real,real,exp(real),X) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_gk(real,fun(nat,real),X)),set_ord_lessThan(nat,N))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,exp(real),T5)),semiring_char_0_fact(real,N))),aa(nat,real,aa(real,fun(nat,real),power_power(real),X),N))) ) ) ).

% Maclaurin_exp_le
tff(fact_3521_exp__lower__Taylor__quadratic,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,ord_less_eq(real,zero_zero(real)),X))
     => pp(aa(real,bool,ord_less_eq(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),X)),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),X),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(num,real,numeral_numeral(real),bit0(one2))))),aa(real,real,exp(real),X))) ) ).

% exp_lower_Taylor_quadratic
tff(fact_3522_arcsin__sin,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,ord_less_eq(real,aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),X))
     => ( pp(aa(real,bool,ord_less_eq(real,X),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2)))))
       => ( aa(real,real,arcsin,sin(real,X)) = X ) ) ) ).

% arcsin_sin
tff(fact_3523_log__base__10__eq2,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,ord_less(real,zero_zero(real)),X))
     => ( aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(aa(num,num,bit1,bit0(one2))))),X) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(aa(num,num,bit1,bit0(one2))))),aa(real,real,exp(real),one_one(real)))),aa(real,real,ln_ln(real),X)) ) ) ).

% log_base_10_eq2
tff(fact_3524_tanh__real__altdef,axiom,
    ! [X: real] : aa(real,real,tanh(real),X) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,minus_minus(real,one_one(real)),aa(real,real,exp(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,uminus_uminus(real),aa(num,real,numeral_numeral(real),bit0(one2)))),X)))),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),aa(real,real,exp(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,uminus_uminus(real),aa(num,real,numeral_numeral(real),bit0(one2)))),X)))) ).

% tanh_real_altdef
tff(fact_3525_log__base__10__eq1,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,ord_less(real,zero_zero(real)),X))
     => ( aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(aa(num,num,bit1,bit0(one2))))),X) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,ln_ln(real),aa(real,real,exp(real),one_one(real)))),aa(real,real,ln_ln(real),aa(num,real,numeral_numeral(real),bit0(aa(num,num,bit1,bit0(one2))))))),aa(real,real,ln_ln(real),X)) ) ) ).

% log_base_10_eq1
tff(fact_3526_arcsin,axiom,
    ! [Y: real] :
      ( pp(aa(real,bool,ord_less_eq(real,aa(real,real,uminus_uminus(real),one_one(real))),Y))
     => ( pp(aa(real,bool,ord_less_eq(real,Y),one_one(real)))
       => ( pp(aa(real,bool,ord_less_eq(real,aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),aa(real,real,arcsin,Y)))
          & pp(aa(real,bool,ord_less_eq(real,aa(real,real,arcsin,Y)),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2)))))
          & ( sin(real,aa(real,real,arcsin,Y)) = Y ) ) ) ) ).

% arcsin
tff(fact_3527_arcsin__pi,axiom,
    ! [Y: real] :
      ( pp(aa(real,bool,ord_less_eq(real,aa(real,real,uminus_uminus(real),one_one(real))),Y))
     => ( pp(aa(real,bool,ord_less_eq(real,Y),one_one(real)))
       => ( pp(aa(real,bool,ord_less_eq(real,aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),aa(real,real,arcsin,Y)))
          & pp(aa(real,bool,ord_less_eq(real,aa(real,real,arcsin,Y)),pi))
          & ( sin(real,aa(real,real,arcsin,Y)) = Y ) ) ) ) ).

% arcsin_pi
tff(fact_3528_arcsin__le__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa(real,bool,ord_less_eq(real,aa(real,real,uminus_uminus(real),one_one(real))),X))
     => ( pp(aa(real,bool,ord_less_eq(real,X),one_one(real)))
       => ( pp(aa(real,bool,ord_less_eq(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,uminus_uminus(real),pi)),aa(num,real,numeral_numeral(real),bit0(one2)))),Y))
         => ( pp(aa(real,bool,ord_less_eq(real,Y),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2)))))
           => ( pp(aa(real,bool,ord_less_eq(real,aa(real,real,arcsin,X)),Y))
            <=> pp(aa(real,bool,ord_less_eq(real,X),sin(real,Y))) ) ) ) ) ) ).

% arcsin_le_iff
tff(fact_3529_le__arcsin__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa(real,bool,ord_less_eq(real,aa(real,real,uminus_uminus(real),one_one(real))),X))
     => ( pp(aa(real,bool,ord_less_eq(real,X),one_one(real)))
       => ( pp(aa(real,bool,ord_less_eq(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,uminus_uminus(real),pi)),aa(num,real,numeral_numeral(real),bit0(one2)))),Y))
         => ( pp(aa(real,bool,ord_less_eq(real,Y),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2)))))
           => ( pp(aa(real,bool,ord_less_eq(real,Y),aa(real,real,arcsin,X)))
            <=> pp(aa(real,bool,ord_less_eq(real,sin(real,Y)),X)) ) ) ) ) ) ).

% le_arcsin_iff
tff(fact_3530_arccos__cos__eq__abs__2pi,axiom,
    ! [Theta: real] :
      ~ ! [K3: int] : aa(real,real,arccos,cos(real,Theta)) != aa(real,real,abs_abs(real),aa(real,real,minus_minus(real,Theta),aa(real,real,aa(real,fun(real,real),times_times(real),aa(int,real,ring_1_of_int(real),K3)),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),pi)))) ).

% arccos_cos_eq_abs_2pi
tff(fact_3531_monoseq__def,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X6: fun(nat,A)] :
          ( topological_monoseq(A,X6)
        <=> ( ! [M6: nat,N5: nat] :
                ( pp(aa(nat,bool,ord_less_eq(nat,M6),N5))
               => pp(aa(A,bool,ord_less_eq(A,aa(nat,A,X6,M6)),aa(nat,A,X6,N5))) )
            | ! [M6: nat,N5: nat] :
                ( pp(aa(nat,bool,ord_less_eq(nat,M6),N5))
               => pp(aa(A,bool,ord_less_eq(A,aa(nat,A,X6,N5)),aa(nat,A,X6,M6))) ) ) ) ) ).

% monoseq_def
tff(fact_3532_monoI2,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X6: fun(nat,A)] :
          ( ! [M3: nat,N2: nat] :
              ( pp(aa(nat,bool,ord_less_eq(nat,M3),N2))
             => pp(aa(A,bool,ord_less_eq(A,aa(nat,A,X6,N2)),aa(nat,A,X6,M3))) )
         => topological_monoseq(A,X6) ) ) ).

% monoI2
tff(fact_3533_monoI1,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X6: fun(nat,A)] :
          ( ! [M3: nat,N2: nat] :
              ( pp(aa(nat,bool,ord_less_eq(nat,M3),N2))
             => pp(aa(A,bool,ord_less_eq(A,aa(nat,A,X6,M3)),aa(nat,A,X6,N2))) )
         => topological_monoseq(A,X6) ) ) ).

% monoI1
tff(fact_3534_monoseq__Suc,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X6: fun(nat,A)] :
          ( topological_monoseq(A,X6)
        <=> ( ! [N5: nat] : pp(aa(A,bool,ord_less_eq(A,aa(nat,A,X6,N5)),aa(nat,A,X6,aa(nat,nat,suc,N5))))
            | ! [N5: nat] : pp(aa(A,bool,ord_less_eq(A,aa(nat,A,X6,aa(nat,nat,suc,N5))),aa(nat,A,X6,N5))) ) ) ) ).

% monoseq_Suc
tff(fact_3535_mono__SucI2,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X6: fun(nat,A)] :
          ( ! [N2: nat] : pp(aa(A,bool,ord_less_eq(A,aa(nat,A,X6,aa(nat,nat,suc,N2))),aa(nat,A,X6,N2)))
         => topological_monoseq(A,X6) ) ) ).

% mono_SucI2
tff(fact_3536_complex__exp__exists,axiom,
    ! [Z: complex] :
    ? [A4: complex,R3: real] : Z = aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),aa(real,complex,real_Vector_of_real(complex),R3)),aa(complex,complex,exp(complex),A4)) ).

% complex_exp_exists
tff(fact_3537_monoseq__minus,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: fun(nat,A)] :
          ( topological_monoseq(A,A2)
         => topological_monoseq(A,aTP_Lamp_gl(fun(nat,A),fun(nat,A),A2)) ) ) ).

% monoseq_minus
tff(fact_3538_mono__SucI1,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X6: fun(nat,A)] :
          ( ! [N2: nat] : pp(aa(A,bool,ord_less_eq(A,aa(nat,A,X6,N2)),aa(nat,A,X6,aa(nat,nat,suc,N2))))
         => topological_monoseq(A,X6) ) ) ).

% mono_SucI1
tff(fact_3539_pochhammer__double,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Z: A,N: nat] : comm_s3205402744901411588hammer(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),Z),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)))),comm_s3205402744901411588hammer(A,Z,N))),comm_s3205402744901411588hammer(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Z),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),bit0(one2)))),N)) ) ).

% pochhammer_double
tff(fact_3540_of__nat__code,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [N: nat] : aa(nat,A,semiring_1_of_nat(A),N) = semiri8178284476397505188at_aux(A,aTP_Lamp_gm(A,A),N,zero_zero(A)) ) ).

% of_nat_code
tff(fact_3541_floor__log__nat__eq__powr__iff,axiom,
    ! [B2: nat,K: nat,N: nat] :
      ( pp(aa(nat,bool,ord_less_eq(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),B2))
     => ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),K))
       => ( ( archim6421214686448440834_floor(real,aa(real,real,log(aa(nat,real,semiring_1_of_nat(real),B2)),aa(nat,real,semiring_1_of_nat(real),K))) = aa(nat,int,semiring_1_of_nat(int),N) )
        <=> ( pp(aa(nat,bool,ord_less_eq(nat,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),N)),K))
            & pp(aa(nat,bool,ord_less(nat,K),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat))))) ) ) ) ) ).

% floor_log_nat_eq_powr_iff
tff(fact_3542_Maclaurin__sin__bound,axiom,
    ! [X: real,N: nat] : pp(aa(real,bool,ord_less_eq(real,aa(real,real,abs_abs(real),aa(real,real,minus_minus(real,sin(real,X)),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_gj(real,fun(nat,real),X)),set_ord_lessThan(nat,N))))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,inverse_inverse(real),semiring_char_0_fact(real,N))),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,abs_abs(real),X)),N)))) ).

% Maclaurin_sin_bound
tff(fact_3543_gchoose__row__sum__weighted,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [R2: A,M: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_gn(A,fun(nat,A),R2)),set_or1337092689740270186AtMost(nat,zero_zero(nat),M)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,M))),aa(num,A,numeral_numeral(A),bit0(one2)))),aa(nat,A,gbinomial(A,R2),aa(nat,nat,suc,M))) ) ).

% gchoose_row_sum_weighted
tff(fact_3544_inverse__inverse__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A] : aa(A,A,inverse_inverse(A),aa(A,A,inverse_inverse(A),A2)) = A2 ) ).

% inverse_inverse_eq
tff(fact_3545_inverse__eq__iff__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,B2: A] :
          ( ( aa(A,A,inverse_inverse(A),A2) = aa(A,A,inverse_inverse(A),B2) )
        <=> ( A2 = B2 ) ) ) ).

% inverse_eq_iff_eq
tff(fact_3546_inverse__zero,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ( aa(A,A,inverse_inverse(A),zero_zero(A)) = zero_zero(A) ) ) ).

% inverse_zero
tff(fact_3547_inverse__nonzero__iff__nonzero,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A] :
          ( ( aa(A,A,inverse_inverse(A),A2) = zero_zero(A) )
        <=> ( A2 = zero_zero(A) ) ) ) ).

% inverse_nonzero_iff_nonzero
tff(fact_3548_inverse__eq__1__iff,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [X: A] :
          ( ( aa(A,A,inverse_inverse(A),X) = one_one(A) )
        <=> ( X = one_one(A) ) ) ) ).

% inverse_eq_1_iff
tff(fact_3549_inverse__1,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ( aa(A,A,inverse_inverse(A),one_one(A)) = one_one(A) ) ) ).

% inverse_1
tff(fact_3550_inverse__mult__distrib,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A2: A,B2: A] : aa(A,A,inverse_inverse(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,inverse_inverse(A),A2)),aa(A,A,inverse_inverse(A),B2)) ) ).

% inverse_mult_distrib
tff(fact_3551_inverse__divide,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A2: A,B2: A] : aa(A,A,inverse_inverse(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A2) ) ).

% inverse_divide
tff(fact_3552_inverse__minus__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A] : aa(A,A,inverse_inverse(A),aa(A,A,uminus_uminus(A),A2)) = aa(A,A,uminus_uminus(A),aa(A,A,inverse_inverse(A),A2)) ) ).

% inverse_minus_eq
tff(fact_3553_abs__inverse,axiom,
    ! [A: $tType] :
      ( field_abs_sgn(A)
     => ! [A2: A] : aa(A,A,abs_abs(A),aa(A,A,inverse_inverse(A),A2)) = aa(A,A,inverse_inverse(A),aa(A,A,abs_abs(A),A2)) ) ).

% abs_inverse
tff(fact_3554_of__int__floor__cancel,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( ( aa(int,A,ring_1_of_int(A),archim6421214686448440834_floor(A,X)) = X )
        <=> ? [N5: int] : X = aa(int,A,ring_1_of_int(A),N5) ) ) ).

% of_int_floor_cancel
tff(fact_3555_gbinomial__1,axiom,
    ! [A: $tType] :
      ( ( semiring_char_0(A)
        & semidom_divide(A) )
     => ! [A2: A] : aa(nat,A,gbinomial(A,A2),one_one(nat)) = A2 ) ).

% gbinomial_1
tff(fact_3556_pochhammer__1,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A2: A] : comm_s3205402744901411588hammer(A,A2,one_one(nat)) = A2 ) ).

% pochhammer_1
tff(fact_3557_inverse__nonpositive__iff__nonpositive,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( pp(aa(A,bool,ord_less_eq(A,aa(A,A,inverse_inverse(A),A2)),zero_zero(A)))
        <=> pp(aa(A,bool,ord_less_eq(A,A2),zero_zero(A))) ) ) ).

% inverse_nonpositive_iff_nonpositive
tff(fact_3558_inverse__nonnegative__iff__nonnegative,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),aa(A,A,inverse_inverse(A),A2)))
        <=> pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),A2)) ) ) ).

% inverse_nonnegative_iff_nonnegative
tff(fact_3559_inverse__positive__iff__positive,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( pp(aa(A,bool,ord_less(A,zero_zero(A)),aa(A,A,inverse_inverse(A),A2)))
        <=> pp(aa(A,bool,ord_less(A,zero_zero(A)),A2)) ) ) ).

% inverse_positive_iff_positive
tff(fact_3560_inverse__negative__iff__negative,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( pp(aa(A,bool,ord_less(A,aa(A,A,inverse_inverse(A),A2)),zero_zero(A)))
        <=> pp(aa(A,bool,ord_less(A,A2),zero_zero(A))) ) ) ).

% inverse_negative_iff_negative
tff(fact_3561_inverse__less__iff__less__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,ord_less(A,A2),zero_zero(A)))
         => ( pp(aa(A,bool,ord_less(A,B2),zero_zero(A)))
           => ( pp(aa(A,bool,ord_less(A,aa(A,A,inverse_inverse(A),A2)),aa(A,A,inverse_inverse(A),B2)))
            <=> pp(aa(A,bool,ord_less(A,B2),A2)) ) ) ) ) ).

% inverse_less_iff_less_neg
tff(fact_3562_inverse__less__iff__less,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,ord_less(A,zero_zero(A)),A2))
         => ( pp(aa(A,bool,ord_less(A,zero_zero(A)),B2))
           => ( pp(aa(A,bool,ord_less(A,aa(A,A,inverse_inverse(A),A2)),aa(A,A,inverse_inverse(A),B2)))
            <=> pp(aa(A,bool,ord_less(A,B2),A2)) ) ) ) ) ).

% inverse_less_iff_less
tff(fact_3563_floor__zero,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ( archim6421214686448440834_floor(A,zero_zero(A)) = zero_zero(int) ) ) ).

% floor_zero
tff(fact_3564_floor__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V: num] : archim6421214686448440834_floor(A,aa(num,A,numeral_numeral(A),V)) = aa(num,int,numeral_numeral(int),V) ) ).

% floor_numeral
tff(fact_3565_gbinomial__0_I2_J,axiom,
    ! [B: $tType] :
      ( ( semiring_char_0(B)
        & semidom_divide(B) )
     => ! [K: nat] : aa(nat,B,gbinomial(B,zero_zero(B)),aa(nat,nat,suc,K)) = zero_zero(B) ) ).

% gbinomial_0(2)
tff(fact_3566_floor__one,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ( archim6421214686448440834_floor(A,one_one(A)) = one_one(int) ) ) ).

% floor_one
tff(fact_3567_gbinomial__0_I1_J,axiom,
    ! [A: $tType] :
      ( ( semiring_char_0(A)
        & semidom_divide(A) )
     => ! [A2: A] : aa(nat,A,gbinomial(A,A2),zero_zero(nat)) = one_one(A) ) ).

% gbinomial_0(1)
tff(fact_3568_gbinomial__Suc0,axiom,
    ! [A: $tType] :
      ( ( semiring_char_0(A)
        & semidom_divide(A) )
     => ! [A2: A] : aa(nat,A,gbinomial(A,A2),aa(nat,nat,suc,zero_zero(nat))) = A2 ) ).

% gbinomial_Suc0
tff(fact_3569_pochhammer__0,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A2: A] : comm_s3205402744901411588hammer(A,A2,zero_zero(nat)) = one_one(A) ) ).

% pochhammer_0
tff(fact_3570_pochhammer__Suc0,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A2: A] : comm_s3205402744901411588hammer(A,A2,aa(nat,nat,suc,zero_zero(nat))) = A2 ) ).

% pochhammer_Suc0
tff(fact_3571_floor__of__nat,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [N: nat] : archim6421214686448440834_floor(A,aa(nat,A,semiring_1_of_nat(A),N)) = aa(nat,int,semiring_1_of_nat(int),N) ) ).

% floor_of_nat
tff(fact_3572_inverse__le__iff__le,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,ord_less(A,zero_zero(A)),A2))
         => ( pp(aa(A,bool,ord_less(A,zero_zero(A)),B2))
           => ( pp(aa(A,bool,ord_less_eq(A,aa(A,A,inverse_inverse(A),A2)),aa(A,A,inverse_inverse(A),B2)))
            <=> pp(aa(A,bool,ord_less_eq(A,B2),A2)) ) ) ) ) ).

% inverse_le_iff_le
tff(fact_3573_inverse__le__iff__le__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,ord_less(A,A2),zero_zero(A)))
         => ( pp(aa(A,bool,ord_less(A,B2),zero_zero(A)))
           => ( pp(aa(A,bool,ord_less_eq(A,aa(A,A,inverse_inverse(A),A2)),aa(A,A,inverse_inverse(A),B2)))
            <=> pp(aa(A,bool,ord_less_eq(A,B2),A2)) ) ) ) ) ).

% inverse_le_iff_le_neg
tff(fact_3574_left__inverse,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A] :
          ( ( A2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,inverse_inverse(A),A2)),A2) = one_one(A) ) ) ) ).

% left_inverse
tff(fact_3575_right__inverse,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A] :
          ( ( A2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(A,A,inverse_inverse(A),A2)) = one_one(A) ) ) ) ).

% right_inverse
tff(fact_3576_inverse__eq__divide__numeral,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [W: num] : aa(A,A,inverse_inverse(A),aa(num,A,numeral_numeral(A),W)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),W)) ) ).

% inverse_eq_divide_numeral
tff(fact_3577_inverse__eq__divide__neg__numeral,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [W: num] : aa(A,A,inverse_inverse(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))) = aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))) ) ).

% inverse_eq_divide_neg_numeral
tff(fact_3578_zero__le__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( pp(aa(int,bool,ord_less_eq(int,zero_zero(int)),archim6421214686448440834_floor(A,X)))
        <=> pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),X)) ) ) ).

% zero_le_floor
tff(fact_3579_numeral__le__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V: num,X: A] :
          ( pp(aa(int,bool,ord_less_eq(int,aa(num,int,numeral_numeral(int),V)),archim6421214686448440834_floor(A,X)))
        <=> pp(aa(A,bool,ord_less_eq(A,aa(num,A,numeral_numeral(A),V)),X)) ) ) ).

% numeral_le_floor
tff(fact_3580_floor__less__zero,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( pp(aa(int,bool,ord_less(int,archim6421214686448440834_floor(A,X)),zero_zero(int)))
        <=> pp(aa(A,bool,ord_less(A,X),zero_zero(A))) ) ) ).

% floor_less_zero
tff(fact_3581_floor__less__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,V: num] :
          ( pp(aa(int,bool,ord_less(int,archim6421214686448440834_floor(A,X)),aa(num,int,numeral_numeral(int),V)))
        <=> pp(aa(A,bool,ord_less(A,X),aa(num,A,numeral_numeral(A),V))) ) ) ).

% floor_less_numeral
tff(fact_3582_zero__less__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( pp(aa(int,bool,ord_less(int,zero_zero(int)),archim6421214686448440834_floor(A,X)))
        <=> pp(aa(A,bool,ord_less_eq(A,one_one(A)),X)) ) ) ).

% zero_less_floor
tff(fact_3583_floor__le__zero,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( pp(aa(int,bool,ord_less_eq(int,archim6421214686448440834_floor(A,X)),zero_zero(int)))
        <=> pp(aa(A,bool,ord_less(A,X),one_one(A))) ) ) ).

% floor_le_zero
tff(fact_3584_one__le__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( pp(aa(int,bool,ord_less_eq(int,one_one(int)),archim6421214686448440834_floor(A,X)))
        <=> pp(aa(A,bool,ord_less_eq(A,one_one(A)),X)) ) ) ).

% one_le_floor
tff(fact_3585_floor__neg__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V: num] : archim6421214686448440834_floor(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))) = aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V)) ) ).

% floor_neg_numeral
tff(fact_3586_floor__less__one,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( pp(aa(int,bool,ord_less(int,archim6421214686448440834_floor(A,X)),one_one(int)))
        <=> pp(aa(A,bool,ord_less(A,X),one_one(A))) ) ) ).

% floor_less_one
tff(fact_3587_floor__diff__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,V: num] : archim6421214686448440834_floor(A,aa(A,A,minus_minus(A,X),aa(num,A,numeral_numeral(A),V))) = aa(int,int,minus_minus(int,archim6421214686448440834_floor(A,X)),aa(num,int,numeral_numeral(int),V)) ) ).

% floor_diff_numeral
tff(fact_3588_floor__numeral__power,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: num,N: nat] : archim6421214686448440834_floor(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),X)),N)) = aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),X)),N) ) ).

% floor_numeral_power
tff(fact_3589_floor__diff__one,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] : archim6421214686448440834_floor(A,aa(A,A,minus_minus(A,X),one_one(A))) = aa(int,int,minus_minus(int,archim6421214686448440834_floor(A,X)),one_one(int)) ) ).

% floor_diff_one
tff(fact_3590_floor__divide__eq__div__numeral,axiom,
    ! [A2: num,B2: num] : archim6421214686448440834_floor(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(num,real,numeral_numeral(real),A2)),aa(num,real,numeral_numeral(real),B2))) = aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(num,int,numeral_numeral(int),A2)),aa(num,int,numeral_numeral(int),B2)) ).

% floor_divide_eq_div_numeral
tff(fact_3591_numeral__less__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V: num,X: A] :
          ( pp(aa(int,bool,ord_less(int,aa(num,int,numeral_numeral(int),V)),archim6421214686448440834_floor(A,X)))
        <=> pp(aa(A,bool,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),V)),one_one(A))),X)) ) ) ).

% numeral_less_floor
tff(fact_3592_floor__le__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,V: num] :
          ( pp(aa(int,bool,ord_less_eq(int,archim6421214686448440834_floor(A,X)),aa(num,int,numeral_numeral(int),V)))
        <=> pp(aa(A,bool,ord_less(A,X),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),V)),one_one(A)))) ) ) ).

% floor_le_numeral
tff(fact_3593_one__less__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( pp(aa(int,bool,ord_less(int,one_one(int)),archim6421214686448440834_floor(A,X)))
        <=> pp(aa(A,bool,ord_less_eq(A,aa(num,A,numeral_numeral(A),bit0(one2))),X)) ) ) ).

% one_less_floor
tff(fact_3594_floor__le__one,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( pp(aa(int,bool,ord_less_eq(int,archim6421214686448440834_floor(A,X)),one_one(int)))
        <=> pp(aa(A,bool,ord_less(A,X),aa(num,A,numeral_numeral(A),bit0(one2)))) ) ) ).

% floor_le_one
tff(fact_3595_neg__numeral__le__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V: num,X: A] :
          ( pp(aa(int,bool,ord_less_eq(int,aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V))),archim6421214686448440834_floor(A,X)))
        <=> pp(aa(A,bool,ord_less_eq(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))),X)) ) ) ).

% neg_numeral_le_floor
tff(fact_3596_floor__less__neg__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,V: num] :
          ( pp(aa(int,bool,ord_less(int,archim6421214686448440834_floor(A,X)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V))))
        <=> pp(aa(A,bool,ord_less(A,X),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V)))) ) ) ).

% floor_less_neg_numeral
tff(fact_3597_floor__one__divide__eq__div__numeral,axiom,
    ! [B2: num] : archim6421214686448440834_floor(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),B2))) = aa(int,int,aa(int,fun(int,int),divide_divide(int),one_one(int)),aa(num,int,numeral_numeral(int),B2)) ).

% floor_one_divide_eq_div_numeral
tff(fact_3598_floor__minus__divide__eq__div__numeral,axiom,
    ! [A2: num,B2: num] : archim6421214686448440834_floor(real,aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(num,real,numeral_numeral(real),A2)),aa(num,real,numeral_numeral(real),B2)))) = aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),A2))),aa(num,int,numeral_numeral(int),B2)) ).

% floor_minus_divide_eq_div_numeral
tff(fact_3599_neg__numeral__less__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V: num,X: A] :
          ( pp(aa(int,bool,ord_less(int,aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V))),archim6421214686448440834_floor(A,X)))
        <=> pp(aa(A,bool,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))),one_one(A))),X)) ) ) ).

% neg_numeral_less_floor
tff(fact_3600_floor__le__neg__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,V: num] :
          ( pp(aa(int,bool,ord_less_eq(int,archim6421214686448440834_floor(A,X)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V))))
        <=> pp(aa(A,bool,ord_less(A,X),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))),one_one(A)))) ) ) ).

% floor_le_neg_numeral
tff(fact_3601_floor__minus__one__divide__eq__div__numeral,axiom,
    ! [B2: num] : archim6421214686448440834_floor(real,aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),B2)))) = aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,uminus_uminus(int),one_one(int))),aa(num,int,numeral_numeral(int),B2)) ).

% floor_minus_one_divide_eq_div_numeral
tff(fact_3602_power__inverse,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,N: nat] : aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,inverse_inverse(A),A2)),N) = aa(A,A,inverse_inverse(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),N)) ) ).

% power_inverse
tff(fact_3603_pochhammer__of__nat,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [X: nat,N: nat] : comm_s3205402744901411588hammer(A,aa(nat,A,semiring_1_of_nat(A),X),N) = aa(nat,A,semiring_1_of_nat(A),comm_s3205402744901411588hammer(nat,X,N)) ) ).

% pochhammer_of_nat
tff(fact_3604_inverse__eq__imp__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,B2: A] :
          ( ( aa(A,A,inverse_inverse(A),A2) = aa(A,A,inverse_inverse(A),B2) )
         => ( A2 = B2 ) ) ) ).

% inverse_eq_imp_eq
tff(fact_3605_real__sqrt__inverse,axiom,
    ! [X: real] : aa(real,real,sqrt,aa(real,real,inverse_inverse(real),X)) = aa(real,real,inverse_inverse(real),aa(real,real,sqrt,X)) ).

% real_sqrt_inverse
tff(fact_3606_of__nat__gbinomial,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [N: nat,K: nat] : aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,gbinomial(nat,N),K)) = aa(nat,A,gbinomial(A,aa(nat,A,semiring_1_of_nat(A),N)),K) ) ).

% of_nat_gbinomial
tff(fact_3607_nonzero__norm__inverse,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [A2: A] :
          ( ( A2 != zero_zero(A) )
         => ( real_V7770717601297561774m_norm(A,aa(A,A,inverse_inverse(A),A2)) = aa(real,real,inverse_inverse(real),real_V7770717601297561774m_norm(A,A2)) ) ) ) ).

% nonzero_norm_inverse
tff(fact_3608_mult__commute__imp__mult__inverse__commute,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Y: A,X: A] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),Y),X) = aa(A,A,aa(A,fun(A,A),times_times(A),X),Y) )
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,inverse_inverse(A),Y)),X) = aa(A,A,aa(A,fun(A,A),times_times(A),X),aa(A,A,inverse_inverse(A),Y)) ) ) ) ).

% mult_commute_imp_mult_inverse_commute
tff(fact_3609_nonzero__imp__inverse__nonzero,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A] :
          ( ( A2 != zero_zero(A) )
         => ( aa(A,A,inverse_inverse(A),A2) != zero_zero(A) ) ) ) ).

% nonzero_imp_inverse_nonzero
tff(fact_3610_nonzero__inverse__inverse__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A] :
          ( ( A2 != zero_zero(A) )
         => ( aa(A,A,inverse_inverse(A),aa(A,A,inverse_inverse(A),A2)) = A2 ) ) ) ).

% nonzero_inverse_inverse_eq
tff(fact_3611_nonzero__inverse__eq__imp__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,B2: A] :
          ( ( aa(A,A,inverse_inverse(A),A2) = aa(A,A,inverse_inverse(A),B2) )
         => ( ( A2 != zero_zero(A) )
           => ( ( B2 != zero_zero(A) )
             => ( A2 = B2 ) ) ) ) ) ).

% nonzero_inverse_eq_imp_eq
tff(fact_3612_inverse__zero__imp__zero,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A] :
          ( ( aa(A,A,inverse_inverse(A),A2) = zero_zero(A) )
         => ( A2 = zero_zero(A) ) ) ) ).

% inverse_zero_imp_zero
tff(fact_3613_field__class_Ofield__inverse__zero,axiom,
    ! [A: $tType] :
      ( field(A)
     => ( aa(A,A,inverse_inverse(A),zero_zero(A)) = zero_zero(A) ) ) ).

% field_class.field_inverse_zero
tff(fact_3614_nonzero__of__real__inverse,axiom,
    ! [A: $tType] :
      ( real_V5047593784448816457lgebra(A)
     => ! [X: real] :
          ( ( X != zero_zero(real) )
         => ( aa(real,A,real_Vector_of_real(A),aa(real,real,inverse_inverse(real),X)) = aa(A,A,inverse_inverse(A),aa(real,A,real_Vector_of_real(A),X)) ) ) ) ).

% nonzero_of_real_inverse
tff(fact_3615_norm__inverse__le__norm,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [R2: real,X: A] :
          ( pp(aa(real,bool,ord_less_eq(real,R2),real_V7770717601297561774m_norm(A,X)))
         => ( pp(aa(real,bool,ord_less(real,zero_zero(real)),R2))
           => pp(aa(real,bool,ord_less_eq(real,real_V7770717601297561774m_norm(A,aa(A,A,inverse_inverse(A),X))),aa(real,real,inverse_inverse(real),R2))) ) ) ) ).

% norm_inverse_le_norm
tff(fact_3616_of__int__floor__le,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] : pp(aa(A,bool,ord_less_eq(A,aa(int,A,ring_1_of_int(A),archim6421214686448440834_floor(A,X))),X)) ) ).

% of_int_floor_le
tff(fact_3617_floor__mono,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,ord_less_eq(A,X),Y))
         => pp(aa(int,bool,ord_less_eq(int,archim6421214686448440834_floor(A,X)),archim6421214686448440834_floor(A,Y))) ) ) ).

% floor_mono
tff(fact_3618_positive__imp__inverse__positive,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( pp(aa(A,bool,ord_less(A,zero_zero(A)),A2))
         => pp(aa(A,bool,ord_less(A,zero_zero(A)),aa(A,A,inverse_inverse(A),A2))) ) ) ).

% positive_imp_inverse_positive
tff(fact_3619_negative__imp__inverse__negative,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( pp(aa(A,bool,ord_less(A,A2),zero_zero(A)))
         => pp(aa(A,bool,ord_less(A,aa(A,A,inverse_inverse(A),A2)),zero_zero(A))) ) ) ).

% negative_imp_inverse_negative
tff(fact_3620_inverse__positive__imp__positive,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( pp(aa(A,bool,ord_less(A,zero_zero(A)),aa(A,A,inverse_inverse(A),A2)))
         => ( ( A2 != zero_zero(A) )
           => pp(aa(A,bool,ord_less(A,zero_zero(A)),A2)) ) ) ) ).

% inverse_positive_imp_positive
tff(fact_3621_inverse__negative__imp__negative,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( pp(aa(A,bool,ord_less(A,aa(A,A,inverse_inverse(A),A2)),zero_zero(A)))
         => ( ( A2 != zero_zero(A) )
           => pp(aa(A,bool,ord_less(A,A2),zero_zero(A))) ) ) ) ).

% inverse_negative_imp_negative
tff(fact_3622_less__imp__inverse__less__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,ord_less(A,A2),B2))
         => ( pp(aa(A,bool,ord_less(A,B2),zero_zero(A)))
           => pp(aa(A,bool,ord_less(A,aa(A,A,inverse_inverse(A),B2)),aa(A,A,inverse_inverse(A),A2))) ) ) ) ).

% less_imp_inverse_less_neg
tff(fact_3623_inverse__less__imp__less__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,ord_less(A,aa(A,A,inverse_inverse(A),A2)),aa(A,A,inverse_inverse(A),B2)))
         => ( pp(aa(A,bool,ord_less(A,B2),zero_zero(A)))
           => pp(aa(A,bool,ord_less(A,B2),A2)) ) ) ) ).

% inverse_less_imp_less_neg
tff(fact_3624_less__imp__inverse__less,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,ord_less(A,A2),B2))
         => ( pp(aa(A,bool,ord_less(A,zero_zero(A)),A2))
           => pp(aa(A,bool,ord_less(A,aa(A,A,inverse_inverse(A),B2)),aa(A,A,inverse_inverse(A),A2))) ) ) ) ).

% less_imp_inverse_less
tff(fact_3625_inverse__less__imp__less,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,ord_less(A,aa(A,A,inverse_inverse(A),A2)),aa(A,A,inverse_inverse(A),B2)))
         => ( pp(aa(A,bool,ord_less(A,zero_zero(A)),A2))
           => pp(aa(A,bool,ord_less(A,B2),A2)) ) ) ) ).

% inverse_less_imp_less
tff(fact_3626_nonzero__inverse__mult__distrib,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,B2: A] :
          ( ( A2 != zero_zero(A) )
         => ( ( B2 != zero_zero(A) )
           => ( aa(A,A,inverse_inverse(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,inverse_inverse(A),B2)),aa(A,A,inverse_inverse(A),A2)) ) ) ) ) ).

% nonzero_inverse_mult_distrib
tff(fact_3627_inverse__numeral__1,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ( aa(A,A,inverse_inverse(A),aa(num,A,numeral_numeral(A),one2)) = aa(num,A,numeral_numeral(A),one2) ) ) ).

% inverse_numeral_1
tff(fact_3628_inverse__unique,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2) = one_one(A) )
         => ( aa(A,A,inverse_inverse(A),A2) = B2 ) ) ) ).

% inverse_unique
tff(fact_3629_nonzero__inverse__minus__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A] :
          ( ( A2 != zero_zero(A) )
         => ( aa(A,A,inverse_inverse(A),aa(A,A,uminus_uminus(A),A2)) = aa(A,A,uminus_uminus(A),aa(A,A,inverse_inverse(A),A2)) ) ) ) ).

% nonzero_inverse_minus_eq
tff(fact_3630_inverse__eq__divide,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A] : aa(A,A,inverse_inverse(A),A2) = aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),A2) ) ).

% inverse_eq_divide
tff(fact_3631_divide__inverse__commute,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,inverse_inverse(A),B2)),A2) ) ).

% divide_inverse_commute
tff(fact_3632_divide__inverse,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2) = aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(A,A,inverse_inverse(A),B2)) ) ).

% divide_inverse
tff(fact_3633_field__class_Ofield__divide__inverse,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2) = aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(A,A,inverse_inverse(A),B2)) ) ).

% field_class.field_divide_inverse
tff(fact_3634_power__mult__power__inverse__commute,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [X: A,M: nat,N: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),M)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,inverse_inverse(A),X)),N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,inverse_inverse(A),X)),N)),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),M)) ) ).

% power_mult_power_inverse_commute
tff(fact_3635_power__mult__inverse__distrib,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [X: A,M: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),M)),aa(A,A,inverse_inverse(A),X)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,inverse_inverse(A),X)),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),M)) ) ).

% power_mult_inverse_distrib
tff(fact_3636_mult__inverse__of__nat__commute,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Xa2: nat,X: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,inverse_inverse(A),aa(nat,A,semiring_1_of_nat(A),Xa2))),X) = aa(A,A,aa(A,fun(A,A),times_times(A),X),aa(A,A,inverse_inverse(A),aa(nat,A,semiring_1_of_nat(A),Xa2))) ) ).

% mult_inverse_of_nat_commute
tff(fact_3637_nonzero__abs__inverse,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( ( A2 != zero_zero(A) )
         => ( aa(A,A,abs_abs(A),aa(A,A,inverse_inverse(A),A2)) = aa(A,A,inverse_inverse(A),aa(A,A,abs_abs(A),A2)) ) ) ) ).

% nonzero_abs_inverse
tff(fact_3638_mult__inverse__of__int__commute,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Xa2: int,X: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,inverse_inverse(A),aa(int,A,ring_1_of_int(A),Xa2))),X) = aa(A,A,aa(A,fun(A,A),times_times(A),X),aa(A,A,inverse_inverse(A),aa(int,A,ring_1_of_int(A),Xa2))) ) ).

% mult_inverse_of_int_commute
tff(fact_3639_divide__real__def,axiom,
    ! [X: real,Y: real] : aa(real,real,aa(real,fun(real,real),divide_divide(real),X),Y) = aa(real,real,aa(real,fun(real,real),times_times(real),X),aa(real,real,inverse_inverse(real),Y)) ).

% divide_real_def
tff(fact_3640_pochhammer__pos,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [X: A,N: nat] :
          ( pp(aa(A,bool,ord_less(A,zero_zero(A)),X))
         => pp(aa(A,bool,ord_less(A,zero_zero(A)),comm_s3205402744901411588hammer(A,X,N))) ) ) ).

% pochhammer_pos
tff(fact_3641_pochhammer__neq__0__mono,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A,M: nat,N: nat] :
          ( ( comm_s3205402744901411588hammer(A,A2,M) != zero_zero(A) )
         => ( pp(aa(nat,bool,ord_less_eq(nat,N),M))
           => ( comm_s3205402744901411588hammer(A,A2,N) != zero_zero(A) ) ) ) ) ).

% pochhammer_neq_0_mono
tff(fact_3642_pochhammer__eq__0__mono,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A,N: nat,M: nat] :
          ( ( comm_s3205402744901411588hammer(A,A2,N) = zero_zero(A) )
         => ( pp(aa(nat,bool,ord_less_eq(nat,N),M))
           => ( comm_s3205402744901411588hammer(A,A2,M) = zero_zero(A) ) ) ) ) ).

% pochhammer_eq_0_mono
tff(fact_3643_pochhammer__fact,axiom,
    ! [A: $tType] :
      ( ( semiring_char_0(A)
        & comm_semiring_1(A) )
     => ! [N: nat] : semiring_char_0_fact(A,N) = comm_s3205402744901411588hammer(A,one_one(A),N) ) ).

% pochhammer_fact
tff(fact_3644_exp__fdiffs,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X4: nat] : aa(nat,A,diffs(A,aTP_Lamp_go(nat,A)),X4) = aa(A,A,inverse_inverse(A),semiring_char_0_fact(A,X4)) ) ).

% exp_fdiffs
tff(fact_3645_inverse__le__imp__le,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,ord_less_eq(A,aa(A,A,inverse_inverse(A),A2)),aa(A,A,inverse_inverse(A),B2)))
         => ( pp(aa(A,bool,ord_less(A,zero_zero(A)),A2))
           => pp(aa(A,bool,ord_less_eq(A,B2),A2)) ) ) ) ).

% inverse_le_imp_le
tff(fact_3646_le__imp__inverse__le,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,ord_less_eq(A,A2),B2))
         => ( pp(aa(A,bool,ord_less(A,zero_zero(A)),A2))
           => pp(aa(A,bool,ord_less_eq(A,aa(A,A,inverse_inverse(A),B2)),aa(A,A,inverse_inverse(A),A2))) ) ) ) ).

% le_imp_inverse_le
tff(fact_3647_inverse__le__imp__le__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,ord_less_eq(A,aa(A,A,inverse_inverse(A),A2)),aa(A,A,inverse_inverse(A),B2)))
         => ( pp(aa(A,bool,ord_less(A,B2),zero_zero(A)))
           => pp(aa(A,bool,ord_less_eq(A,B2),A2)) ) ) ) ).

% inverse_le_imp_le_neg
tff(fact_3648_le__imp__inverse__le__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,ord_less_eq(A,A2),B2))
         => ( pp(aa(A,bool,ord_less(A,B2),zero_zero(A)))
           => pp(aa(A,bool,ord_less_eq(A,aa(A,A,inverse_inverse(A),B2)),aa(A,A,inverse_inverse(A),A2))) ) ) ) ).

% le_imp_inverse_le_neg
tff(fact_3649_inverse__le__1__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A] :
          ( pp(aa(A,bool,ord_less_eq(A,aa(A,A,inverse_inverse(A),X)),one_one(A)))
        <=> ( pp(aa(A,bool,ord_less_eq(A,X),zero_zero(A)))
            | pp(aa(A,bool,ord_less_eq(A,one_one(A)),X)) ) ) ) ).

% inverse_le_1_iff
tff(fact_3650_one__less__inverse__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A] :
          ( pp(aa(A,bool,ord_less(A,one_one(A)),aa(A,A,inverse_inverse(A),X)))
        <=> ( pp(aa(A,bool,ord_less(A,zero_zero(A)),X))
            & pp(aa(A,bool,ord_less(A,X),one_one(A))) ) ) ) ).

% one_less_inverse_iff
tff(fact_3651_one__less__inverse,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( pp(aa(A,bool,ord_less(A,zero_zero(A)),A2))
         => ( pp(aa(A,bool,ord_less(A,A2),one_one(A)))
           => pp(aa(A,bool,ord_less(A,one_one(A)),aa(A,A,inverse_inverse(A),A2))) ) ) ) ).

% one_less_inverse
tff(fact_3652_le__floor__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Z: int,X: A] :
          ( pp(aa(int,bool,ord_less_eq(int,Z),archim6421214686448440834_floor(A,X)))
        <=> pp(aa(A,bool,ord_less_eq(A,aa(int,A,ring_1_of_int(A),Z)),X)) ) ) ).

% le_floor_iff
tff(fact_3653_inverse__add,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A2: A,B2: A] :
          ( ( A2 != zero_zero(A) )
         => ( ( B2 != zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,inverse_inverse(A),A2)),aa(A,A,inverse_inverse(A),B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),aa(A,A,inverse_inverse(A),A2))),aa(A,A,inverse_inverse(A),B2)) ) ) ) ) ).

% inverse_add
tff(fact_3654_division__ring__inverse__add,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,B2: A] :
          ( ( A2 != zero_zero(A) )
         => ( ( B2 != zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,inverse_inverse(A),A2)),aa(A,A,inverse_inverse(A),B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,inverse_inverse(A),A2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2))),aa(A,A,inverse_inverse(A),B2)) ) ) ) ) ).

% division_ring_inverse_add
tff(fact_3655_field__class_Ofield__inverse,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A2: A] :
          ( ( A2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,inverse_inverse(A),A2)),A2) = one_one(A) ) ) ) ).

% field_class.field_inverse
tff(fact_3656_division__ring__inverse__diff,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,B2: A] :
          ( ( A2 != zero_zero(A) )
         => ( ( B2 != zero_zero(A) )
           => ( aa(A,A,minus_minus(A,aa(A,A,inverse_inverse(A),A2)),aa(A,A,inverse_inverse(A),B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,inverse_inverse(A),A2)),aa(A,A,minus_minus(A,B2),A2))),aa(A,A,inverse_inverse(A),B2)) ) ) ) ) ).

% division_ring_inverse_diff
tff(fact_3657_nonzero__inverse__eq__divide,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A] :
          ( ( A2 != zero_zero(A) )
         => ( aa(A,A,inverse_inverse(A),A2) = aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),A2) ) ) ) ).

% nonzero_inverse_eq_divide
tff(fact_3658_floor__add__int,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,Z: int] : aa(int,int,aa(int,fun(int,int),plus_plus(int),archim6421214686448440834_floor(A,X)),Z) = archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),aa(int,A,ring_1_of_int(A),Z))) ) ).

% floor_add_int
tff(fact_3659_int__add__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Z: int,X: A] : aa(int,int,aa(int,fun(int,int),plus_plus(int),Z),archim6421214686448440834_floor(A,X)) = archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(int,A,ring_1_of_int(A),Z)),X)) ) ).

% int_add_floor
tff(fact_3660_le__floor__add,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,Y: A] : pp(aa(int,bool,ord_less_eq(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),archim6421214686448440834_floor(A,X)),archim6421214686448440834_floor(A,Y))),archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y)))) ) ).

% le_floor_add
tff(fact_3661_floor__power,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,N: nat] :
          ( ( X = aa(int,A,ring_1_of_int(A),archim6421214686448440834_floor(A,X)) )
         => ( archim6421214686448440834_floor(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),X),N)) = aa(nat,int,aa(int,fun(nat,int),power_power(int),archim6421214686448440834_floor(A,X)),N) ) ) ) ).

% floor_power
tff(fact_3662_floor__divide__of__int__eq,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [K: int,L: int] : archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(int,A,ring_1_of_int(A),K)),aa(int,A,ring_1_of_int(A),L))) = aa(int,int,aa(int,fun(int,int),divide_divide(int),K),L) ) ).

% floor_divide_of_int_eq
tff(fact_3663_gbinomial__pochhammer,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A,K: nat] : aa(nat,A,gbinomial(A,A2),K) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),K)),comm_s3205402744901411588hammer(A,aa(A,A,uminus_uminus(A),A2),K))),semiring_char_0_fact(A,K)) ) ).

% gbinomial_pochhammer
tff(fact_3664_gbinomial__pochhammer_H,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A,K: nat] : aa(nat,A,gbinomial(A,A2),K) = aa(A,A,aa(A,fun(A,A),divide_divide(A),comm_s3205402744901411588hammer(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,minus_minus(A,A2),aa(nat,A,semiring_1_of_nat(A),K))),one_one(A)),K)),semiring_char_0_fact(A,K)) ) ).

% gbinomial_pochhammer'
tff(fact_3665_gbinomial__Suc__Suc,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A,K: nat] : aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),one_one(A))),aa(nat,nat,suc,K)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,gbinomial(A,A2),K)),aa(nat,A,gbinomial(A,A2),aa(nat,nat,suc,K))) ) ).

% gbinomial_Suc_Suc
tff(fact_3666_gbinomial__of__nat__symmetric,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,N: nat] :
          ( pp(aa(nat,bool,ord_less_eq(nat,K),N))
         => ( aa(nat,A,gbinomial(A,aa(nat,A,semiring_1_of_nat(A),N)),K) = aa(nat,A,gbinomial(A,aa(nat,A,semiring_1_of_nat(A),N)),aa(nat,nat,minus_minus(nat,N),K)) ) ) ) ).

% gbinomial_of_nat_symmetric
tff(fact_3667_pochhammer__nonneg,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [X: A,N: nat] :
          ( pp(aa(A,bool,ord_less(A,zero_zero(A)),X))
         => pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),comm_s3205402744901411588hammer(A,X,N))) ) ) ).

% pochhammer_nonneg
tff(fact_3668_of__nat__aux_Osimps_I2_J,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [Inc: fun(A,A),N: nat,I: A] : semiri8178284476397505188at_aux(A,Inc,aa(nat,nat,suc,N),I) = semiri8178284476397505188at_aux(A,Inc,N,aa(A,A,Inc,I)) ) ).

% of_nat_aux.simps(2)
tff(fact_3669_inverse__powr,axiom,
    ! [Y: real,A2: real] :
      ( pp(aa(real,bool,ord_less_eq(real,zero_zero(real)),Y))
     => ( powr(real,aa(real,real,inverse_inverse(real),Y),A2) = aa(real,real,inverse_inverse(real),powr(real,Y,A2)) ) ) ).

% inverse_powr
tff(fact_3670_of__nat__aux_Osimps_I1_J,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [Inc: fun(A,A),I: A] : semiri8178284476397505188at_aux(A,Inc,zero_zero(nat),I) = I ) ).

% of_nat_aux.simps(1)
tff(fact_3671_pochhammer__0__left,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [N: nat] :
          ( ( ( N = zero_zero(nat) )
           => ( comm_s3205402744901411588hammer(A,zero_zero(A),N) = one_one(A) ) )
          & ( ( N != zero_zero(nat) )
           => ( comm_s3205402744901411588hammer(A,zero_zero(A),N) = zero_zero(A) ) ) ) ) ).

% pochhammer_0_left
tff(fact_3672_one__le__inverse,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( pp(aa(A,bool,ord_less(A,zero_zero(A)),A2))
         => ( pp(aa(A,bool,ord_less_eq(A,A2),one_one(A)))
           => pp(aa(A,bool,ord_less_eq(A,one_one(A)),aa(A,A,inverse_inverse(A),A2))) ) ) ) ).

% one_le_inverse
tff(fact_3673_inverse__less__1__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A] :
          ( pp(aa(A,bool,ord_less(A,aa(A,A,inverse_inverse(A),X)),one_one(A)))
        <=> ( pp(aa(A,bool,ord_less_eq(A,X),zero_zero(A)))
            | pp(aa(A,bool,ord_less(A,one_one(A)),X)) ) ) ) ).

% inverse_less_1_iff
tff(fact_3674_one__le__inverse__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A] :
          ( pp(aa(A,bool,ord_less_eq(A,one_one(A)),aa(A,A,inverse_inverse(A),X)))
        <=> ( pp(aa(A,bool,ord_less(A,zero_zero(A)),X))
            & pp(aa(A,bool,ord_less_eq(A,X),one_one(A))) ) ) ) ).

% one_le_inverse_iff
tff(fact_3675_inverse__less__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,ord_less(A,aa(A,A,inverse_inverse(A),A2)),aa(A,A,inverse_inverse(A),B2)))
        <=> ( ( pp(aa(A,bool,ord_less(A,zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)))
             => pp(aa(A,bool,ord_less(A,B2),A2)) )
            & ( pp(aa(A,bool,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),zero_zero(A)))
             => pp(aa(A,bool,ord_less(A,A2),B2)) ) ) ) ) ).

% inverse_less_iff
tff(fact_3676_inverse__le__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,ord_less_eq(A,aa(A,A,inverse_inverse(A),A2)),aa(A,A,inverse_inverse(A),B2)))
        <=> ( ( pp(aa(A,bool,ord_less(A,zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)))
             => pp(aa(A,bool,ord_less_eq(A,B2),A2)) )
            & ( pp(aa(A,bool,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),zero_zero(A)))
             => pp(aa(A,bool,ord_less_eq(A,A2),B2)) ) ) ) ) ).

% inverse_le_iff
tff(fact_3677_one__add__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] : aa(int,int,aa(int,fun(int,int),plus_plus(int),archim6421214686448440834_floor(A,X)),one_one(int)) = archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),one_one(A))) ) ).

% one_add_floor
tff(fact_3678_inverse__diff__inverse,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,B2: A] :
          ( ( A2 != zero_zero(A) )
         => ( ( B2 != zero_zero(A) )
           => ( aa(A,A,minus_minus(A,aa(A,A,inverse_inverse(A),A2)),aa(A,A,inverse_inverse(A),B2)) = aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,inverse_inverse(A),A2)),aa(A,A,minus_minus(A,A2),B2))),aa(A,A,inverse_inverse(A),B2))) ) ) ) ) ).

% inverse_diff_inverse
tff(fact_3679_reals__Archimedean,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [X: A] :
          ( pp(aa(A,bool,ord_less(A,zero_zero(A)),X))
         => ? [N2: nat] : pp(aa(A,bool,ord_less(A,aa(A,A,inverse_inverse(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,N2)))),X)) ) ) ).

% reals_Archimedean
tff(fact_3680_floor__divide__of__nat__eq,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [M: nat,N: nat] : archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,semiring_1_of_nat(A),M)),aa(nat,A,semiring_1_of_nat(A),N))) = aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),N)) ) ).

% floor_divide_of_nat_eq
tff(fact_3681_ceiling__altdef,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( ( ( X = aa(int,A,ring_1_of_int(A),archim6421214686448440834_floor(A,X)) )
           => ( archimedean_ceiling(A,X) = archim6421214686448440834_floor(A,X) ) )
          & ( ( X != aa(int,A,ring_1_of_int(A),archim6421214686448440834_floor(A,X)) )
           => ( archimedean_ceiling(A,X) = aa(int,int,aa(int,fun(int,int),plus_plus(int),archim6421214686448440834_floor(A,X)),one_one(int)) ) ) ) ) ).

% ceiling_altdef
tff(fact_3682_gbinomial__addition__formula,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A,K: nat] : aa(nat,A,gbinomial(A,A2),aa(nat,nat,suc,K)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,gbinomial(A,aa(A,A,minus_minus(A,A2),one_one(A))),aa(nat,nat,suc,K))),aa(nat,A,gbinomial(A,aa(A,A,minus_minus(A,A2),one_one(A))),K)) ) ).

% gbinomial_addition_formula
tff(fact_3683_ceiling__diff__floor__le__1,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] : pp(aa(int,bool,ord_less_eq(int,aa(int,int,minus_minus(int,archimedean_ceiling(A,X)),archim6421214686448440834_floor(A,X))),one_one(int))) ) ).

% ceiling_diff_floor_le_1
tff(fact_3684_gbinomial__absorb__comp,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A,K: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,minus_minus(A,A2),aa(nat,A,semiring_1_of_nat(A),K))),aa(nat,A,gbinomial(A,A2),K)) = aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(nat,A,gbinomial(A,aa(A,A,minus_minus(A,A2),one_one(A))),K)) ) ).

% gbinomial_absorb_comp
tff(fact_3685_gbinomial__mult__1_H,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A,K: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,gbinomial(A,A2),K)),A2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),K)),aa(nat,A,gbinomial(A,A2),K))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,K))),aa(nat,A,gbinomial(A,A2),aa(nat,nat,suc,K)))) ) ).

% gbinomial_mult_1'
tff(fact_3686_gbinomial__mult__1,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A,K: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(nat,A,gbinomial(A,A2),K)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),K)),aa(nat,A,gbinomial(A,A2),K))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,K))),aa(nat,A,gbinomial(A,A2),aa(nat,nat,suc,K)))) ) ).

% gbinomial_mult_1
tff(fact_3687_gbinomial__ge__n__over__k__pow__k,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [K: nat,A2: A] :
          ( pp(aa(A,bool,ord_less_eq(A,aa(nat,A,semiring_1_of_nat(A),K)),A2))
         => pp(aa(A,bool,ord_less_eq(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(nat,A,semiring_1_of_nat(A),K))),K)),aa(nat,A,gbinomial(A,A2),K))) ) ) ).

% gbinomial_ge_n_over_k_pow_k
tff(fact_3688_floor__eq,axiom,
    ! [N: int,X: real] :
      ( pp(aa(real,bool,ord_less(real,aa(int,real,ring_1_of_int(real),N)),X))
     => ( pp(aa(real,bool,ord_less(real,X),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(int,real,ring_1_of_int(real),N)),one_one(real))))
       => ( archim6421214686448440834_floor(real,X) = N ) ) ) ).

% floor_eq
tff(fact_3689_real__of__int__floor__add__one__gt,axiom,
    ! [R2: real] : pp(aa(real,bool,ord_less(real,R2),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(int,real,ring_1_of_int(real),archim6421214686448440834_floor(real,R2))),one_one(real)))) ).

% real_of_int_floor_add_one_gt
tff(fact_3690_real__of__int__floor__add__one__ge,axiom,
    ! [R2: real] : pp(aa(real,bool,ord_less_eq(real,R2),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(int,real,ring_1_of_int(real),archim6421214686448440834_floor(real,R2))),one_one(real)))) ).

% real_of_int_floor_add_one_ge
tff(fact_3691_forall__pos__mono__1,axiom,
    ! [P: fun(real,bool),E: real] :
      ( ! [D2: real,E2: real] :
          ( pp(aa(real,bool,ord_less(real,D2),E2))
         => ( pp(aa(real,bool,P,D2))
           => pp(aa(real,bool,P,E2)) ) )
     => ( ! [N2: nat] : pp(aa(real,bool,P,aa(real,real,inverse_inverse(real),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,N2)))))
       => ( pp(aa(real,bool,ord_less(real,zero_zero(real)),E))
         => pp(aa(real,bool,P,E)) ) ) ) ).

% forall_pos_mono_1
tff(fact_3692_real__arch__inverse,axiom,
    ! [E: real] :
      ( pp(aa(real,bool,ord_less(real,zero_zero(real)),E))
    <=> ? [N5: nat] :
          ( ( N5 != zero_zero(nat) )
          & pp(aa(real,bool,ord_less(real,zero_zero(real)),aa(real,real,inverse_inverse(real),aa(nat,real,semiring_1_of_nat(real),N5))))
          & pp(aa(real,bool,ord_less(real,aa(real,real,inverse_inverse(real),aa(nat,real,semiring_1_of_nat(real),N5))),E)) ) ) ).

% real_arch_inverse
tff(fact_3693_forall__pos__mono,axiom,
    ! [P: fun(real,bool),E: real] :
      ( ! [D2: real,E2: real] :
          ( pp(aa(real,bool,ord_less(real,D2),E2))
         => ( pp(aa(real,bool,P,D2))
           => pp(aa(real,bool,P,E2)) ) )
     => ( ! [N2: nat] :
            ( ( N2 != zero_zero(nat) )
           => pp(aa(real,bool,P,aa(real,real,inverse_inverse(real),aa(nat,real,semiring_1_of_nat(real),N2)))) )
       => ( pp(aa(real,bool,ord_less(real,zero_zero(real)),E))
         => pp(aa(real,bool,P,E)) ) ) ) ).

% forall_pos_mono
tff(fact_3694_real__of__int__floor__gt__diff__one,axiom,
    ! [R2: real] : pp(aa(real,bool,ord_less(real,aa(real,real,minus_minus(real,R2),one_one(real))),aa(int,real,ring_1_of_int(real),archim6421214686448440834_floor(real,R2)))) ).

% real_of_int_floor_gt_diff_one
tff(fact_3695_real__of__int__floor__ge__diff__one,axiom,
    ! [R2: real] : pp(aa(real,bool,ord_less_eq(real,aa(real,real,minus_minus(real,R2),one_one(real))),aa(int,real,ring_1_of_int(real),archim6421214686448440834_floor(real,R2)))) ).

% real_of_int_floor_ge_diff_one
tff(fact_3696_sqrt__divide__self__eq,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,ord_less_eq(real,zero_zero(real)),X))
     => ( aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,sqrt,X)),X) = aa(real,real,inverse_inverse(real),aa(real,real,sqrt,X)) ) ) ).

% sqrt_divide_self_eq
tff(fact_3697_ln__inverse,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,ord_less(real,zero_zero(real)),X))
     => ( aa(real,real,ln_ln(real),aa(real,real,inverse_inverse(real),X)) = aa(real,real,uminus_uminus(real),aa(real,real,ln_ln(real),X)) ) ) ).

% ln_inverse
tff(fact_3698_pochhammer__rec,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A2: A,N: nat] : comm_s3205402744901411588hammer(A,A2,aa(nat,nat,suc,N)) = aa(A,A,aa(A,fun(A,A),times_times(A),A2),comm_s3205402744901411588hammer(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),one_one(A)),N)) ) ).

% pochhammer_rec
tff(fact_3699_pochhammer__rec_H,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Z: A,N: nat] : comm_s3205402744901411588hammer(A,Z,aa(nat,nat,suc,N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Z),aa(nat,A,semiring_1_of_nat(A),N))),comm_s3205402744901411588hammer(A,Z,N)) ) ).

% pochhammer_rec'
tff(fact_3700_pochhammer__Suc,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A2: A,N: nat] : comm_s3205402744901411588hammer(A,A2,aa(nat,nat,suc,N)) = aa(A,A,aa(A,fun(A,A),times_times(A),comm_s3205402744901411588hammer(A,A2,N)),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa(nat,A,semiring_1_of_nat(A),N))) ) ).

% pochhammer_Suc
tff(fact_3701_pochhammer__eq__0__iff,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A,N: nat] :
          ( ( comm_s3205402744901411588hammer(A,A2,N) = zero_zero(A) )
        <=> ? [K2: nat] :
              ( pp(aa(nat,bool,ord_less(nat,K2),N))
              & ( A2 = aa(A,A,uminus_uminus(A),aa(nat,A,semiring_1_of_nat(A),K2)) ) ) ) ) ).

% pochhammer_eq_0_iff
tff(fact_3702_pochhammer__of__nat__eq__0__iff,axiom,
    ! [A: $tType] :
      ( ( ring_char_0(A)
        & idom(A) )
     => ! [N: nat,K: nat] :
          ( ( comm_s3205402744901411588hammer(A,aa(A,A,uminus_uminus(A),aa(nat,A,semiring_1_of_nat(A),N)),K) = zero_zero(A) )
        <=> pp(aa(nat,bool,ord_less(nat,N),K)) ) ) ).

% pochhammer_of_nat_eq_0_iff
tff(fact_3703_pochhammer__of__nat__eq__0__lemma,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [N: nat,K: nat] :
          ( pp(aa(nat,bool,ord_less(nat,N),K))
         => ( comm_s3205402744901411588hammer(A,aa(A,A,uminus_uminus(A),aa(nat,A,semiring_1_of_nat(A),N)),K) = zero_zero(A) ) ) ) ).

% pochhammer_of_nat_eq_0_lemma
tff(fact_3704_pochhammer__product_H,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Z: A,N: nat,M: nat] : comm_s3205402744901411588hammer(A,Z,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),M)) = aa(A,A,aa(A,fun(A,A),times_times(A),comm_s3205402744901411588hammer(A,Z,N)),comm_s3205402744901411588hammer(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Z),aa(nat,A,semiring_1_of_nat(A),N)),M)) ) ).

% pochhammer_product'
tff(fact_3705_pochhammer__of__nat__eq__0__lemma_H,axiom,
    ! [A: $tType] :
      ( ( ring_char_0(A)
        & idom(A) )
     => ! [K: nat,N: nat] :
          ( pp(aa(nat,bool,ord_less_eq(nat,K),N))
         => ( comm_s3205402744901411588hammer(A,aa(A,A,uminus_uminus(A),aa(nat,A,semiring_1_of_nat(A),N)),K) != zero_zero(A) ) ) ) ).

% pochhammer_of_nat_eq_0_lemma'
tff(fact_3706_summable__exp,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] : summable(A,aTP_Lamp_gp(A,fun(nat,A),X)) ) ).

% summable_exp
tff(fact_3707_floor__unique,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Z: int,X: A] :
          ( pp(aa(A,bool,ord_less_eq(A,aa(int,A,ring_1_of_int(A),Z)),X))
         => ( pp(aa(A,bool,ord_less(A,X),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(int,A,ring_1_of_int(A),Z)),one_one(A))))
           => ( archim6421214686448440834_floor(A,X) = Z ) ) ) ) ).

% floor_unique
tff(fact_3708_floor__eq__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,A2: int] :
          ( ( archim6421214686448440834_floor(A,X) = A2 )
        <=> ( pp(aa(A,bool,ord_less_eq(A,aa(int,A,ring_1_of_int(A),A2)),X))
            & pp(aa(A,bool,ord_less(A,X),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(int,A,ring_1_of_int(A),A2)),one_one(A)))) ) ) ) ).

% floor_eq_iff
tff(fact_3709_floor__split,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [P: fun(int,bool),T: A] :
          ( pp(aa(int,bool,P,archim6421214686448440834_floor(A,T)))
        <=> ! [I3: int] :
              ( ( pp(aa(A,bool,ord_less_eq(A,aa(int,A,ring_1_of_int(A),I3)),T))
                & pp(aa(A,bool,ord_less(A,T),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(int,A,ring_1_of_int(A),I3)),one_one(A)))) )
             => pp(aa(int,bool,P,I3)) ) ) ) ).

% floor_split
tff(fact_3710_less__floor__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Z: int,X: A] :
          ( pp(aa(int,bool,ord_less(int,Z),archim6421214686448440834_floor(A,X)))
        <=> pp(aa(A,bool,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(int,A,ring_1_of_int(A),Z)),one_one(A))),X)) ) ) ).

% less_floor_iff
tff(fact_3711_le__mult__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),A2))
         => ( pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),B2))
           => pp(aa(int,bool,ord_less_eq(int,aa(int,int,aa(int,fun(int,int),times_times(int),archim6421214686448440834_floor(A,A2)),archim6421214686448440834_floor(A,B2))),archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)))) ) ) ) ).

% le_mult_floor
tff(fact_3712_floor__le__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,Z: int] :
          ( pp(aa(int,bool,ord_less_eq(int,archim6421214686448440834_floor(A,X)),Z))
        <=> pp(aa(A,bool,ord_less(A,X),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(int,A,ring_1_of_int(A),Z)),one_one(A)))) ) ) ).

% floor_le_iff
tff(fact_3713_floor__correct,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( pp(aa(A,bool,ord_less_eq(A,aa(int,A,ring_1_of_int(A),archim6421214686448440834_floor(A,X))),X))
          & pp(aa(A,bool,ord_less(A,X),aa(int,A,ring_1_of_int(A),aa(int,int,aa(int,fun(int,int),plus_plus(int),archim6421214686448440834_floor(A,X)),one_one(int))))) ) ) ).

% floor_correct
tff(fact_3714_ex__inverse__of__nat__less,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [X: A] :
          ( pp(aa(A,bool,ord_less(A,zero_zero(A)),X))
         => ? [N2: nat] :
              ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),N2))
              & pp(aa(A,bool,ord_less(A,aa(A,A,inverse_inverse(A),aa(nat,A,semiring_1_of_nat(A),N2))),X)) ) ) ) ).

% ex_inverse_of_nat_less
tff(fact_3715_power__diff__conv__inverse,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [X: A,M: nat,N: nat] :
          ( ( X != zero_zero(A) )
         => ( pp(aa(nat,bool,ord_less_eq(nat,M),N))
           => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(nat,nat,minus_minus(nat,N),M)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),N)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,inverse_inverse(A),X)),M)) ) ) ) ) ).

% power_diff_conv_inverse
tff(fact_3716_Suc__times__gbinomial,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,A2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,K))),aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),one_one(A))),aa(nat,nat,suc,K))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),one_one(A))),aa(nat,A,gbinomial(A,A2),K)) ) ).

% Suc_times_gbinomial
tff(fact_3717_gbinomial__absorption,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,A2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,K))),aa(nat,A,gbinomial(A,A2),aa(nat,nat,suc,K))) = aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(nat,A,gbinomial(A,aa(A,A,minus_minus(A,A2),one_one(A))),K)) ) ).

% gbinomial_absorption
tff(fact_3718_floor__eq2,axiom,
    ! [N: int,X: real] :
      ( pp(aa(real,bool,ord_less_eq(real,aa(int,real,ring_1_of_int(real),N)),X))
     => ( pp(aa(real,bool,ord_less(real,X),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(int,real,ring_1_of_int(real),N)),one_one(real))))
       => ( archim6421214686448440834_floor(real,X) = N ) ) ) ).

% floor_eq2
tff(fact_3719_floor__divide__real__eq__div,axiom,
    ! [B2: int,A2: real] :
      ( pp(aa(int,bool,ord_less_eq(int,zero_zero(int)),B2))
     => ( archim6421214686448440834_floor(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),A2),aa(int,real,ring_1_of_int(real),B2))) = aa(int,int,aa(int,fun(int,int),divide_divide(int),archim6421214686448440834_floor(real,A2)),B2) ) ) ).

% floor_divide_real_eq_div
tff(fact_3720_gbinomial__trinomial__revision,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,M: nat,A2: A] :
          ( pp(aa(nat,bool,ord_less_eq(nat,K),M))
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,gbinomial(A,A2),M)),aa(nat,A,gbinomial(A,aa(nat,A,semiring_1_of_nat(A),M)),K)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,gbinomial(A,A2),K)),aa(nat,A,gbinomial(A,aa(A,A,minus_minus(A,A2),aa(nat,A,semiring_1_of_nat(A),K))),aa(nat,nat,minus_minus(nat,M),K))) ) ) ) ).

% gbinomial_trinomial_revision
tff(fact_3721_log__inverse,axiom,
    ! [A2: real,X: real] :
      ( pp(aa(real,bool,ord_less(real,zero_zero(real)),A2))
     => ( ( A2 != one_one(real) )
       => ( pp(aa(real,bool,ord_less(real,zero_zero(real)),X))
         => ( aa(real,real,log(A2),aa(real,real,inverse_inverse(real),X)) = aa(real,real,uminus_uminus(real),aa(real,real,log(A2),X)) ) ) ) ) ).

% log_inverse
tff(fact_3722_pochhammer__product,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [M: nat,N: nat,Z: A] :
          ( pp(aa(nat,bool,ord_less_eq(nat,M),N))
         => ( comm_s3205402744901411588hammer(A,Z,N) = aa(A,A,aa(A,fun(A,A),times_times(A),comm_s3205402744901411588hammer(A,Z,M)),comm_s3205402744901411588hammer(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Z),aa(nat,A,semiring_1_of_nat(A),M)),aa(nat,nat,minus_minus(nat,N),M))) ) ) ) ).

% pochhammer_product
tff(fact_3723_floor__divide__lower,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Q2: A,P2: A] :
          ( pp(aa(A,bool,ord_less(A,zero_zero(A)),Q2))
         => pp(aa(A,bool,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(int,A,ring_1_of_int(A),archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),P2),Q2)))),Q2)),P2)) ) ) ).

% floor_divide_lower
tff(fact_3724_gbinomial__factors,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A,K: nat] : aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),one_one(A))),aa(nat,nat,suc,K)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),one_one(A))),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,K)))),aa(nat,A,gbinomial(A,A2),K)) ) ).

% gbinomial_factors
tff(fact_3725_gbinomial__rec,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A,K: nat] : aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),one_one(A))),aa(nat,nat,suc,K)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,gbinomial(A,A2),K)),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),one_one(A))),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,K)))) ) ).

% gbinomial_rec
tff(fact_3726_gbinomial__negated__upper,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A,K: nat] : aa(nat,A,gbinomial(A,A2),K) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),K)),aa(nat,A,gbinomial(A,aa(A,A,minus_minus(A,aa(A,A,minus_minus(A,aa(nat,A,semiring_1_of_nat(A),K)),A2)),one_one(A))),K)) ) ).

% gbinomial_negated_upper
tff(fact_3727_gbinomial__index__swap,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,N: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),K)),aa(nat,A,gbinomial(A,aa(A,A,minus_minus(A,aa(A,A,uminus_uminus(A),aa(nat,A,semiring_1_of_nat(A),N))),one_one(A))),K)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),N)),aa(nat,A,gbinomial(A,aa(A,A,minus_minus(A,aa(A,A,uminus_uminus(A),aa(nat,A,semiring_1_of_nat(A),K))),one_one(A))),N)) ) ).

% gbinomial_index_swap
tff(fact_3728_exp__plus__inverse__exp,axiom,
    ! [X: real] : pp(aa(real,bool,ord_less_eq(real,aa(num,real,numeral_numeral(real),bit0(one2))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,exp(real),X)),aa(real,real,inverse_inverse(real),aa(real,real,exp(real),X))))) ).

% exp_plus_inverse_exp
tff(fact_3729_pochhammer__absorb__comp,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [R2: A,K: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,minus_minus(A,R2),aa(nat,A,semiring_1_of_nat(A),K))),comm_s3205402744901411588hammer(A,aa(A,A,uminus_uminus(A),R2),K)) = aa(A,A,aa(A,fun(A,A),times_times(A),R2),comm_s3205402744901411588hammer(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),R2)),one_one(A)),K)) ) ).

% pochhammer_absorb_comp
tff(fact_3730_floor__divide__upper,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Q2: A,P2: A] :
          ( pp(aa(A,bool,ord_less(A,zero_zero(A)),Q2))
         => pp(aa(A,bool,ord_less(A,P2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(int,A,ring_1_of_int(A),archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),P2),Q2)))),one_one(A))),Q2))) ) ) ).

% floor_divide_upper
tff(fact_3731_pochhammer__same,axiom,
    ! [A: $tType] :
      ( ( semiring_char_0(A)
        & comm_ring_1(A)
        & semiri3467727345109120633visors(A) )
     => ! [N: nat] : comm_s3205402744901411588hammer(A,aa(A,A,uminus_uminus(A),aa(nat,A,semiring_1_of_nat(A),N)),N) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),N)),semiring_char_0_fact(A,N)) ) ).

% pochhammer_same
tff(fact_3732_round__def,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] : archimedean_round(A,X) = archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),bit0(one2))))) ) ).

% round_def
tff(fact_3733_gbinomial__minus,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A,K: nat] : aa(nat,A,gbinomial(A,aa(A,A,uminus_uminus(A),A2)),K) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),K)),aa(nat,A,gbinomial(A,aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa(nat,A,semiring_1_of_nat(A),K))),one_one(A))),K)) ) ).

% gbinomial_minus
tff(fact_3734_gbinomial__reduce__nat,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,A2: A] :
          ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),K))
         => ( aa(nat,A,gbinomial(A,A2),K) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,gbinomial(A,aa(A,A,minus_minus(A,A2),one_one(A))),aa(nat,nat,minus_minus(nat,K),one_one(nat)))),aa(nat,A,gbinomial(A,aa(A,A,minus_minus(A,A2),one_one(A))),K)) ) ) ) ).

% gbinomial_reduce_nat
tff(fact_3735_plus__inverse__ge__2,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,ord_less(real,zero_zero(real)),X))
     => pp(aa(real,bool,ord_less_eq(real,aa(num,real,numeral_numeral(real),bit0(one2))),aa(real,real,aa(real,fun(real,real),plus_plus(real),X),aa(real,real,inverse_inverse(real),X)))) ) ).

% plus_inverse_ge_2
tff(fact_3736_real__inv__sqrt__pow2,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,ord_less(real,zero_zero(real)),X))
     => ( aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,inverse_inverse(real),aa(real,real,sqrt,X))),aa(num,nat,numeral_numeral(nat),bit0(one2))) = aa(real,real,inverse_inverse(real),X) ) ) ).

% real_inv_sqrt_pow2
tff(fact_3737_tan__cot,axiom,
    ! [X: real] : aa(real,real,tan(real),aa(real,real,minus_minus(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2)))),X)) = aa(real,real,inverse_inverse(real),aa(real,real,tan(real),X)) ).

% tan_cot
tff(fact_3738_pochhammer__minus_H,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [B2: A,K: nat] : comm_s3205402744901411588hammer(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,minus_minus(A,B2),aa(nat,A,semiring_1_of_nat(A),K))),one_one(A)),K) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),K)),comm_s3205402744901411588hammer(A,aa(A,A,uminus_uminus(A),B2),K)) ) ).

% pochhammer_minus'
tff(fact_3739_pochhammer__minus,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [B2: A,K: nat] : comm_s3205402744901411588hammer(A,aa(A,A,uminus_uminus(A),B2),K) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),K)),comm_s3205402744901411588hammer(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,minus_minus(A,B2),aa(nat,A,semiring_1_of_nat(A),K))),one_one(A)),K)) ) ).

% pochhammer_minus
tff(fact_3740_real__le__x__sinh,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,ord_less_eq(real,zero_zero(real)),X))
     => pp(aa(real,bool,ord_less_eq(real,X),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,minus_minus(real,aa(real,real,exp(real),X)),aa(real,real,inverse_inverse(real),aa(real,real,exp(real),X)))),aa(num,real,numeral_numeral(real),bit0(one2))))) ) ).

% real_le_x_sinh
tff(fact_3741_real__le__abs__sinh,axiom,
    ! [X: real] : pp(aa(real,bool,ord_less_eq(real,aa(real,real,abs_abs(real),X)),aa(real,real,abs_abs(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,minus_minus(real,aa(real,real,exp(real),X)),aa(real,real,inverse_inverse(real),aa(real,real,exp(real),X)))),aa(num,real,numeral_numeral(real),bit0(one2)))))) ).

% real_le_abs_sinh
tff(fact_3742_gbinomial__sum__up__index,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,N: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_gq(nat,fun(nat,A),K)),set_or1337092689740270186AtMost(nat,zero_zero(nat),N)) = aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),N)),one_one(A))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),one_one(nat))) ) ).

% gbinomial_sum_up_index
tff(fact_3743_tan__sec,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] :
          ( ( cos(A,X) != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,tan(A),X)),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,inverse_inverse(A),cos(A,X))),aa(num,nat,numeral_numeral(nat),bit0(one2))) ) ) ) ).

% tan_sec
tff(fact_3744_floor__log__eq__powr__iff,axiom,
    ! [X: real,B2: real,K: int] :
      ( pp(aa(real,bool,ord_less(real,zero_zero(real)),X))
     => ( pp(aa(real,bool,ord_less(real,one_one(real)),B2))
       => ( ( archim6421214686448440834_floor(real,aa(real,real,log(B2),X)) = K )
        <=> ( pp(aa(real,bool,ord_less_eq(real,powr(real,B2,aa(int,real,ring_1_of_int(real),K))),X))
            & pp(aa(real,bool,ord_less(real,X),powr(real,B2,aa(int,real,ring_1_of_int(real),aa(int,int,aa(int,fun(int,int),plus_plus(int),K),one_one(int)))))) ) ) ) ) ).

% floor_log_eq_powr_iff
tff(fact_3745_gbinomial__absorption_H,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,A2: A] :
          ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),K))
         => ( aa(nat,A,gbinomial(A,A2),K) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(nat,A,semiring_1_of_nat(A),K))),aa(nat,A,gbinomial(A,aa(A,A,minus_minus(A,A2),one_one(A))),aa(nat,nat,minus_minus(nat,K),one_one(nat)))) ) ) ) ).

% gbinomial_absorption'
tff(fact_3746_floor__log2__div2,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,ord_less_eq(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N))
     => ( archim6421214686448440834_floor(real,aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(nat,real,semiring_1_of_nat(real),N))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),archim6421214686448440834_floor(real,aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),one_one(int)) ) ) ).

% floor_log2_div2
tff(fact_3747_fact__double,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [N: nat] : semiring_char_0_fact(A,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N))),comm_s3205402744901411588hammer(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),bit0(one2))),N))),semiring_char_0_fact(A,N)) ) ).

% fact_double
tff(fact_3748_floor__log__nat__eq__if,axiom,
    ! [B2: nat,N: nat,K: nat] :
      ( pp(aa(nat,bool,ord_less_eq(nat,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),N)),K))
     => ( pp(aa(nat,bool,ord_less(nat,K),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat)))))
       => ( pp(aa(nat,bool,ord_less_eq(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),B2))
         => ( archim6421214686448440834_floor(real,aa(real,real,log(aa(nat,real,semiring_1_of_nat(real),B2)),aa(nat,real,semiring_1_of_nat(real),K))) = aa(nat,int,semiring_1_of_nat(int),N) ) ) ) ) ).

% floor_log_nat_eq_if
tff(fact_3749_gbinomial__code,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,A2: A] :
          ( ( ( K = zero_zero(nat) )
           => ( aa(nat,A,gbinomial(A,A2),K) = one_one(A) ) )
          & ( ( K != zero_zero(nat) )
           => ( aa(nat,A,gbinomial(A,A2),K) = aa(A,A,aa(A,fun(A,A),divide_divide(A),set_fo6178422350223883121st_nat(A,aTP_Lamp_gr(A,fun(nat,fun(A,A)),A2),zero_zero(nat),aa(nat,nat,minus_minus(nat,K),one_one(nat)),one_one(A))),semiring_char_0_fact(A,K)) ) ) ) ) ).

% gbinomial_code
tff(fact_3750_pochhammer__times__pochhammer__half,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Z: A,N: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),comm_s3205402744901411588hammer(A,Z,aa(nat,nat,suc,N))),comm_s3205402744901411588hammer(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Z),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),bit0(one2)))),aa(nat,nat,suc,N))) = groups7121269368397514597t_prod(nat,A,aTP_Lamp_gs(A,fun(nat,A),Z),set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)),one_one(nat)))) ) ).

% pochhammer_times_pochhammer_half
tff(fact_3751_pochhammer__code,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [N: nat,A2: A] :
          ( ( ( N = zero_zero(nat) )
           => ( comm_s3205402744901411588hammer(A,A2,N) = one_one(A) ) )
          & ( ( N != zero_zero(nat) )
           => ( comm_s3205402744901411588hammer(A,A2,N) = set_fo6178422350223883121st_nat(A,aTP_Lamp_gt(A,fun(nat,fun(A,A)),A2),zero_zero(nat),aa(nat,nat,minus_minus(nat,N),one_one(nat)),one_one(A)) ) ) ) ) ).

% pochhammer_code
tff(fact_3752_exp__two__pi__i,axiom,
    aa(complex,complex,exp(complex),aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),aa(num,complex,numeral_numeral(complex),bit0(one2))),aa(real,complex,real_Vector_of_real(complex),pi))),imaginary_unit)) = one_one(complex) ).

% exp_two_pi_i
tff(fact_3753_exp__two__pi__i_H,axiom,
    aa(complex,complex,exp(complex),aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),imaginary_unit),aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),aa(real,complex,real_Vector_of_real(complex),pi)),aa(num,complex,numeral_numeral(complex),bit0(one2))))) = one_one(complex) ).

% exp_two_pi_i'
tff(fact_3754_prod_Oneutral__const,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A3: set(B)] : groups7121269368397514597t_prod(B,A,aTP_Lamp_gu(B,A),A3) = one_one(A) ) ).

% prod.neutral_const
tff(fact_3755_of__nat__prod,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_semiring_1(A)
     => ! [F2: fun(B,nat),A3: set(B)] : aa(nat,A,semiring_1_of_nat(A),groups7121269368397514597t_prod(B,nat,F2,A3)) = groups7121269368397514597t_prod(B,A,aTP_Lamp_gv(fun(B,nat),fun(B,A),F2),A3) ) ).

% of_nat_prod
tff(fact_3756_of__int__prod,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_ring_1(A)
     => ! [F2: fun(B,int),A3: set(B)] : aa(int,A,ring_1_of_int(A),groups7121269368397514597t_prod(B,int,F2,A3)) = groups7121269368397514597t_prod(B,A,aTP_Lamp_gw(fun(B,int),fun(B,A),F2),A3) ) ).

% of_int_prod
tff(fact_3757_of__real__prod,axiom,
    ! [A: $tType,B: $tType] :
      ( ( comm_monoid_mult(A)
        & real_V2191834092415804123ebra_1(A) )
     => ! [F2: fun(B,real),S3: set(B)] : aa(real,A,real_Vector_of_real(A),groups7121269368397514597t_prod(B,real,F2,S3)) = groups7121269368397514597t_prod(B,A,aTP_Lamp_gx(fun(B,real),fun(B,A),F2),S3) ) ).

% of_real_prod
tff(fact_3758_prod__zero__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( semidom(A)
     => ! [A3: set(B),F2: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),A3))
         => ( ( groups7121269368397514597t_prod(B,A,F2,A3) = zero_zero(A) )
          <=> ? [X2: B] :
                ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X2),A3))
                & ( aa(B,A,F2,X2) = zero_zero(A) ) ) ) ) ) ).

% prod_zero_iff
tff(fact_3759_prod_Oinfinite,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A3: set(B),G: fun(B,A)] :
          ( ~ pp(aa(set(B),bool,finite_finite2(B),A3))
         => ( groups7121269368397514597t_prod(B,A,G,A3) = one_one(A) ) ) ) ).

% prod.infinite
tff(fact_3760_dvd__prod__eqI,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_semiring_1(A)
     => ! [A3: set(B),A2: B,B2: A,F2: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),A3))
         => ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A2),A3))
           => ( ( B2 = aa(B,A,F2,A2) )
             => pp(aa(A,bool,dvd_dvd(A,B2),groups7121269368397514597t_prod(B,A,F2,A3))) ) ) ) ) ).

% dvd_prod_eqI
tff(fact_3761_dvd__prodI,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_semiring_1(A)
     => ! [A3: set(B),A2: B,F2: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),A3))
         => ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A2),A3))
           => pp(aa(A,bool,dvd_dvd(A,aa(B,A,F2,A2)),groups7121269368397514597t_prod(B,A,F2,A3))) ) ) ) ).

% dvd_prodI
tff(fact_3762_norm__ii,axiom,
    real_V7770717601297561774m_norm(complex,imaginary_unit) = one_one(real) ).

% norm_ii
tff(fact_3763_prod_Odelta,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(A)
     => ! [S: set(B),A2: B,B2: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),S))
         => ( ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A2),S))
             => ( groups7121269368397514597t_prod(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_gy(B,fun(fun(B,A),fun(B,A)),A2),B2),S) = aa(B,A,B2,A2) ) )
            & ( ~ pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A2),S))
             => ( groups7121269368397514597t_prod(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_gy(B,fun(fun(B,A),fun(B,A)),A2),B2),S) = one_one(A) ) ) ) ) ) ).

% prod.delta
tff(fact_3764_prod_Odelta_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(A)
     => ! [S: set(B),A2: B,B2: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),S))
         => ( ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A2),S))
             => ( groups7121269368397514597t_prod(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_gz(B,fun(fun(B,A),fun(B,A)),A2),B2),S) = aa(B,A,B2,A2) ) )
            & ( ~ pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A2),S))
             => ( groups7121269368397514597t_prod(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_gz(B,fun(fun(B,A),fun(B,A)),A2),B2),S) = one_one(A) ) ) ) ) ) ).

% prod.delta'
tff(fact_3765_divide__i,axiom,
    ! [X: complex] : aa(complex,complex,aa(complex,fun(complex,complex),divide_divide(complex),X),imaginary_unit) = aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),aa(complex,complex,uminus_uminus(complex),imaginary_unit)),X) ).

% divide_i
tff(fact_3766_complex__i__mult__minus,axiom,
    ! [X: complex] : aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),imaginary_unit),aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),imaginary_unit),X)) = aa(complex,complex,uminus_uminus(complex),X) ).

% complex_i_mult_minus
tff(fact_3767_prod_OlessThan__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),N: nat] : groups7121269368397514597t_prod(nat,A,G,set_ord_lessThan(nat,aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),times_times(A),groups7121269368397514597t_prod(nat,A,G,set_ord_lessThan(nat,N))),aa(nat,A,G,N)) ) ).

% prod.lessThan_Suc
tff(fact_3768_i__squared,axiom,
    aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),imaginary_unit),imaginary_unit) = aa(complex,complex,uminus_uminus(complex),one_one(complex)) ).

% i_squared
tff(fact_3769_divide__numeral__i,axiom,
    ! [Z: complex,N: num] : aa(complex,complex,aa(complex,fun(complex,complex),divide_divide(complex),Z),aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),aa(num,complex,numeral_numeral(complex),N)),imaginary_unit)) = aa(complex,complex,aa(complex,fun(complex,complex),divide_divide(complex),aa(complex,complex,uminus_uminus(complex),aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),imaginary_unit),Z))),aa(num,complex,numeral_numeral(complex),N)) ).

% divide_numeral_i
tff(fact_3770_prod_Ocl__ivl__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [N: nat,M: nat,G: fun(nat,A)] :
          ( ( pp(aa(nat,bool,ord_less(nat,aa(nat,nat,suc,N)),M))
           => ( groups7121269368397514597t_prod(nat,A,G,set_or1337092689740270186AtMost(nat,M,aa(nat,nat,suc,N))) = one_one(A) ) )
          & ( ~ pp(aa(nat,bool,ord_less(nat,aa(nat,nat,suc,N)),M))
           => ( groups7121269368397514597t_prod(nat,A,G,set_or1337092689740270186AtMost(nat,M,aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),times_times(A),groups7121269368397514597t_prod(nat,A,G,set_or1337092689740270186AtMost(nat,M,N))),aa(nat,A,G,aa(nat,nat,suc,N))) ) ) ) ) ).

% prod.cl_ivl_Suc
tff(fact_3771_power2__i,axiom,
    aa(nat,complex,aa(complex,fun(nat,complex),power_power(complex),imaginary_unit),aa(num,nat,numeral_numeral(nat),bit0(one2))) = aa(complex,complex,uminus_uminus(complex),one_one(complex)) ).

% power2_i
tff(fact_3772_exp__pi__i_H,axiom,
    aa(complex,complex,exp(complex),aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),imaginary_unit),aa(real,complex,real_Vector_of_real(complex),pi))) = aa(complex,complex,uminus_uminus(complex),one_one(complex)) ).

% exp_pi_i'
tff(fact_3773_exp__pi__i,axiom,
    aa(complex,complex,exp(complex),aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),aa(real,complex,real_Vector_of_real(complex),pi)),imaginary_unit)) = aa(complex,complex,uminus_uminus(complex),one_one(complex)) ).

% exp_pi_i
tff(fact_3774_i__even__power,axiom,
    ! [N: nat] : aa(nat,complex,aa(complex,fun(nat,complex),power_power(complex),imaginary_unit),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = aa(nat,complex,aa(complex,fun(nat,complex),power_power(complex),aa(complex,complex,uminus_uminus(complex),one_one(complex))),N) ).

% i_even_power
tff(fact_3775_divide__complex__def,axiom,
    ! [X: complex,Y: complex] : aa(complex,complex,aa(complex,fun(complex,complex),divide_divide(complex),X),Y) = aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),X),aa(complex,complex,inverse_inverse(complex),Y)) ).

% divide_complex_def
tff(fact_3776_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(B,A),A3: set(B)] :
          ( ( groups7121269368397514597t_prod(B,A,G,A3) != one_one(A) )
         => ~ ! [A4: B] :
                ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A4),A3))
               => ( aa(B,A,G,A4) = one_one(A) ) ) ) ) ).

% prod.not_neutral_contains_not_neutral
tff(fact_3777_prod_Oneutral,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A3: set(B),G: fun(B,A)] :
          ( ! [X3: B] :
              ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),A3))
             => ( aa(B,A,G,X3) = one_one(A) ) )
         => ( groups7121269368397514597t_prod(B,A,G,A3) = one_one(A) ) ) ) ).

% prod.neutral
tff(fact_3778_prod__dvd__prod,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [A3: set(B),F2: fun(B,A),G: fun(B,A)] :
          ( ! [A4: B] :
              ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A4),A3))
             => pp(aa(A,bool,dvd_dvd(A,aa(B,A,F2,A4)),aa(B,A,G,A4))) )
         => pp(aa(A,bool,dvd_dvd(A,groups7121269368397514597t_prod(B,A,F2,A3)),groups7121269368397514597t_prod(B,A,G,A3))) ) ) ).

% prod_dvd_prod
tff(fact_3779_prod__norm,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V8999393235501362500lgebra(B)
        & comm_semiring_1(B) )
     => ! [F2: fun(A,B),A3: set(A)] : groups7121269368397514597t_prod(A,real,aTP_Lamp_ha(fun(A,B),fun(A,real),F2),A3) = real_V7770717601297561774m_norm(B,groups7121269368397514597t_prod(A,B,F2,A3)) ) ).

% prod_norm
tff(fact_3780_prod_Oswap,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(B,fun(C,A)),B4: set(C),A3: set(B)] : groups7121269368397514597t_prod(B,A,aa(set(C),fun(B,A),aTP_Lamp_hb(fun(B,fun(C,A)),fun(set(C),fun(B,A)),G),B4),A3) = groups7121269368397514597t_prod(C,A,aa(set(B),fun(C,A),aTP_Lamp_hd(fun(B,fun(C,A)),fun(set(B),fun(C,A)),G),A3),B4) ) ).

% prod.swap
tff(fact_3781_norm__prod__le,axiom,
    ! [A: $tType,B: $tType] :
      ( ( comm_monoid_mult(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [F2: fun(B,A),A3: set(B)] : pp(aa(real,bool,ord_less_eq(real,real_V7770717601297561774m_norm(A,groups7121269368397514597t_prod(B,A,F2,A3))),groups7121269368397514597t_prod(B,real,aTP_Lamp_he(fun(B,A),fun(B,real),F2),A3))) ) ).

% norm_prod_le
tff(fact_3782_prod_Odistrib,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(B,A),H: fun(B,A),A3: set(B)] : groups7121269368397514597t_prod(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_hf(fun(B,A),fun(fun(B,A),fun(B,A)),G),H),A3) = aa(A,A,aa(A,fun(A,A),times_times(A),groups7121269368397514597t_prod(B,A,G,A3)),groups7121269368397514597t_prod(B,A,H,A3)) ) ).

% prod.distrib
tff(fact_3783_prod__dividef,axiom,
    ! [A: $tType,B: $tType] :
      ( field(A)
     => ! [F2: fun(B,A),G: fun(B,A),A3: set(B)] : groups7121269368397514597t_prod(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_hg(fun(B,A),fun(fun(B,A),fun(B,A)),F2),G),A3) = aa(A,A,aa(A,fun(A,A),divide_divide(A),groups7121269368397514597t_prod(B,A,F2,A3)),groups7121269368397514597t_prod(B,A,G,A3)) ) ).

% prod_dividef
tff(fact_3784_prod__power__distrib,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_semiring_1(B)
     => ! [F2: fun(A,B),A3: set(A),N: nat] : aa(nat,B,aa(B,fun(nat,B),power_power(B),groups7121269368397514597t_prod(A,B,F2,A3)),N) = groups7121269368397514597t_prod(A,B,aa(nat,fun(A,B),aTP_Lamp_hh(fun(A,B),fun(nat,fun(A,B)),F2),N),A3) ) ).

% prod_power_distrib
tff(fact_3785_prod_Oswap__restrict,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comm_monoid_mult(A)
     => ! [A3: set(B),B4: set(C),G: fun(B,fun(C,A)),R: fun(B,fun(C,bool))] :
          ( pp(aa(set(B),bool,finite_finite2(B),A3))
         => ( pp(aa(set(C),bool,finite_finite2(C),B4))
           => ( groups7121269368397514597t_prod(B,A,aa(fun(B,fun(C,bool)),fun(B,A),aa(fun(B,fun(C,A)),fun(fun(B,fun(C,bool)),fun(B,A)),aTP_Lamp_hi(set(C),fun(fun(B,fun(C,A)),fun(fun(B,fun(C,bool)),fun(B,A))),B4),G),R),A3) = groups7121269368397514597t_prod(C,A,aa(fun(B,fun(C,bool)),fun(C,A),aa(fun(B,fun(C,A)),fun(fun(B,fun(C,bool)),fun(C,A)),aTP_Lamp_hj(set(B),fun(fun(B,fun(C,A)),fun(fun(B,fun(C,bool)),fun(C,A))),A3),G),R),B4) ) ) ) ) ).

% prod.swap_restrict
tff(fact_3786_mod__prod__eq,axiom,
    ! [B: $tType,A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [F2: fun(B,A),A2: A,A3: set(B)] : modulo_modulo(A,groups7121269368397514597t_prod(B,A,aa(A,fun(B,A),aTP_Lamp_dv(fun(B,A),fun(A,fun(B,A)),F2),A2),A3),A2) = modulo_modulo(A,groups7121269368397514597t_prod(B,A,F2,A3),A2) ) ).

% mod_prod_eq
tff(fact_3787_abs__prod,axiom,
    ! [A: $tType,B: $tType] :
      ( linordered_field(A)
     => ! [F2: fun(B,A),A3: set(B)] : aa(A,A,abs_abs(A),groups7121269368397514597t_prod(B,A,F2,A3)) = groups7121269368397514597t_prod(B,A,aTP_Lamp_hk(fun(B,A),fun(B,A),F2),A3) ) ).

% abs_prod
tff(fact_3788_complex__i__not__zero,axiom,
    imaginary_unit != zero_zero(complex) ).

% complex_i_not_zero
tff(fact_3789_complex__i__not__one,axiom,
    imaginary_unit != one_one(complex) ).

% complex_i_not_one
tff(fact_3790_prod__mono,axiom,
    ! [A: $tType,B: $tType] :
      ( linordered_semidom(A)
     => ! [A3: set(B),F2: fun(B,A),G: fun(B,A)] :
          ( ! [I2: B] :
              ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I2),A3))
             => ( pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),aa(B,A,F2,I2)))
                & pp(aa(A,bool,ord_less_eq(A,aa(B,A,F2,I2)),aa(B,A,G,I2))) ) )
         => pp(aa(A,bool,ord_less_eq(A,groups7121269368397514597t_prod(B,A,F2,A3)),groups7121269368397514597t_prod(B,A,G,A3))) ) ) ).

% prod_mono
tff(fact_3791_prod__nonneg,axiom,
    ! [A: $tType,B: $tType] :
      ( linordered_semidom(A)
     => ! [A3: set(B),F2: fun(B,A)] :
          ( ! [X3: B] :
              ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),A3))
             => pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),aa(B,A,F2,X3))) )
         => pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),groups7121269368397514597t_prod(B,A,F2,A3))) ) ) ).

% prod_nonneg
tff(fact_3792_prod__pos,axiom,
    ! [A: $tType,B: $tType] :
      ( linordered_semidom(A)
     => ! [A3: set(B),F2: fun(B,A)] :
          ( ! [X3: B] :
              ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),A3))
             => pp(aa(A,bool,ord_less(A,zero_zero(A)),aa(B,A,F2,X3))) )
         => pp(aa(A,bool,ord_less(A,zero_zero(A)),groups7121269368397514597t_prod(B,A,F2,A3))) ) ) ).

% prod_pos
tff(fact_3793_prod__ge__1,axiom,
    ! [A: $tType,B: $tType] :
      ( linord181362715937106298miring(A)
     => ! [A3: set(B),F2: fun(B,A)] :
          ( ! [X3: B] :
              ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),A3))
             => pp(aa(A,bool,ord_less_eq(A,one_one(A)),aa(B,A,F2,X3))) )
         => pp(aa(A,bool,ord_less_eq(A,one_one(A)),groups7121269368397514597t_prod(B,A,F2,A3))) ) ) ).

% prod_ge_1
tff(fact_3794_prod__zero,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_semiring_1(A)
     => ! [A3: set(B),F2: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),A3))
         => ( ? [X4: B] :
                ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X4),A3))
                & ( aa(B,A,F2,X4) = zero_zero(A) ) )
           => ( groups7121269368397514597t_prod(B,A,F2,A3) = zero_zero(A) ) ) ) ) ).

% prod_zero
tff(fact_3795_prod__atLeastAtMost__code,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [F2: fun(nat,A),A2: nat,B2: nat] : groups7121269368397514597t_prod(nat,A,F2,set_or1337092689740270186AtMost(nat,A2,B2)) = set_fo6178422350223883121st_nat(A,aTP_Lamp_hl(fun(nat,A),fun(nat,fun(A,A)),F2),A2,B2,one_one(A)) ) ).

% prod_atLeastAtMost_code
tff(fact_3796_prod_Ointer__filter,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [A3: set(B),G: fun(B,A),P: fun(B,bool)] :
          ( pp(aa(set(B),bool,finite_finite2(B),A3))
         => ( groups7121269368397514597t_prod(B,A,G,collect(B,aa(fun(B,bool),fun(B,bool),aTP_Lamp_dy(set(B),fun(fun(B,bool),fun(B,bool)),A3),P))) = groups7121269368397514597t_prod(B,A,aa(fun(B,bool),fun(B,A),aTP_Lamp_hm(fun(B,A),fun(fun(B,bool),fun(B,A)),G),P),A3) ) ) ) ).

% prod.inter_filter
tff(fact_3797_prod_Oshift__bounds__cl__Suc__ivl,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),M: nat,N: nat] : groups7121269368397514597t_prod(nat,A,G,set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M),aa(nat,nat,suc,N))) = groups7121269368397514597t_prod(nat,A,aTP_Lamp_hn(fun(nat,A),fun(nat,A),G),set_or1337092689740270186AtMost(nat,M,N)) ) ).

% prod.shift_bounds_cl_Suc_ivl
tff(fact_3798_power__sum,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [C2: A,F2: fun(B,nat),A3: set(B)] : aa(nat,A,aa(A,fun(nat,A),power_power(A),C2),aa(set(B),nat,groups7311177749621191930dd_sum(B,nat,F2),A3)) = groups7121269368397514597t_prod(B,A,aa(fun(B,nat),fun(B,A),aTP_Lamp_ho(A,fun(fun(B,nat),fun(B,A)),C2),F2),A3) ) ).

% power_sum
tff(fact_3799_prod_Oshift__bounds__cl__nat__ivl,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),M: nat,K: nat,N: nat] : groups7121269368397514597t_prod(nat,A,G,set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),K),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),K))) = groups7121269368397514597t_prod(nat,A,aa(nat,fun(nat,A),aTP_Lamp_hp(fun(nat,A),fun(nat,fun(nat,A)),G),K),set_or1337092689740270186AtMost(nat,M,N)) ) ).

% prod.shift_bounds_cl_nat_ivl
tff(fact_3800_prod__le__1,axiom,
    ! [B: $tType,A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [A3: set(B),F2: fun(B,A)] :
          ( ! [X3: B] :
              ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),A3))
             => ( pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),aa(B,A,F2,X3)))
                & pp(aa(A,bool,ord_less_eq(A,aa(B,A,F2,X3)),one_one(A))) ) )
         => pp(aa(A,bool,ord_less_eq(A,groups7121269368397514597t_prod(B,A,F2,A3)),one_one(A))) ) ) ).

% prod_le_1
tff(fact_3801_prod_Orelated,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [R: fun(A,fun(A,bool)),S: set(B),H: fun(B,A),G: fun(B,A)] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),R,one_one(A)),one_one(A)))
         => ( ! [X1: A,Y1: A,X22: A,Y23: A] :
                ( ( pp(aa(A,bool,aa(A,fun(A,bool),R,X1),X22))
                  & pp(aa(A,bool,aa(A,fun(A,bool),R,Y1),Y23)) )
               => pp(aa(A,bool,aa(A,fun(A,bool),R,aa(A,A,aa(A,fun(A,A),times_times(A),X1),Y1)),aa(A,A,aa(A,fun(A,A),times_times(A),X22),Y23))) )
           => ( pp(aa(set(B),bool,finite_finite2(B),S))
             => ( ! [X3: B] :
                    ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),S))
                   => pp(aa(A,bool,aa(A,fun(A,bool),R,aa(B,A,H,X3)),aa(B,A,G,X3))) )
               => pp(aa(A,bool,aa(A,fun(A,bool),R,groups7121269368397514597t_prod(B,A,H,S)),groups7121269368397514597t_prod(B,A,G,S))) ) ) ) ) ) ).

% prod.related
tff(fact_3802_prod__dvd__prod__subset2,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_semiring_1(A)
     => ! [B4: set(B),A3: set(B),F2: fun(B,A),G: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),B4))
         => ( pp(aa(set(B),bool,ord_less_eq(set(B),A3),B4))
           => ( ! [A4: B] :
                  ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A4),A3))
                 => pp(aa(A,bool,dvd_dvd(A,aa(B,A,F2,A4)),aa(B,A,G,A4))) )
             => pp(aa(A,bool,dvd_dvd(A,groups7121269368397514597t_prod(B,A,F2,A3)),groups7121269368397514597t_prod(B,A,G,B4))) ) ) ) ) ).

% prod_dvd_prod_subset2
tff(fact_3803_prod__dvd__prod__subset,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [B4: set(B),A3: set(B),F2: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),B4))
         => ( pp(aa(set(B),bool,ord_less_eq(set(B),A3),B4))
           => pp(aa(A,bool,dvd_dvd(A,groups7121269368397514597t_prod(B,A,F2,A3)),groups7121269368397514597t_prod(B,A,F2,B4))) ) ) ) ).

% prod_dvd_prod_subset
tff(fact_3804_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comm_monoid_mult(A)
     => ! [S5: set(B),T4: set(C),S: set(B),I: fun(C,B),J: fun(B,C),T3: set(C),G: fun(B,A),H: fun(C,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),S5))
         => ( pp(aa(set(C),bool,finite_finite2(C),T4))
           => ( ! [A4: B] :
                  ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A4),aa(set(B),set(B),minus_minus(set(B),S),S5)))
                 => ( aa(C,B,I,aa(B,C,J,A4)) = A4 ) )
             => ( ! [A4: B] :
                    ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A4),aa(set(B),set(B),minus_minus(set(B),S),S5)))
                   => pp(aa(set(C),bool,aa(C,fun(set(C),bool),member(C),aa(B,C,J,A4)),aa(set(C),set(C),minus_minus(set(C),T3),T4))) )
               => ( ! [B3: C] :
                      ( pp(aa(set(C),bool,aa(C,fun(set(C),bool),member(C),B3),aa(set(C),set(C),minus_minus(set(C),T3),T4)))
                     => ( aa(B,C,J,aa(C,B,I,B3)) = B3 ) )
                 => ( ! [B3: C] :
                        ( pp(aa(set(C),bool,aa(C,fun(set(C),bool),member(C),B3),aa(set(C),set(C),minus_minus(set(C),T3),T4)))
                       => pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),aa(C,B,I,B3)),aa(set(B),set(B),minus_minus(set(B),S),S5))) )
                   => ( ! [A4: B] :
                          ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A4),S5))
                         => ( aa(B,A,G,A4) = one_one(A) ) )
                     => ( ! [B3: C] :
                            ( pp(aa(set(C),bool,aa(C,fun(set(C),bool),member(C),B3),T4))
                           => ( aa(C,A,H,B3) = one_one(A) ) )
                       => ( ! [A4: B] :
                              ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A4),S))
                             => ( aa(C,A,H,aa(B,C,J,A4)) = aa(B,A,G,A4) ) )
                         => ( groups7121269368397514597t_prod(B,A,G,S) = groups7121269368397514597t_prod(C,A,H,T3) ) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
tff(fact_3805_i__times__eq__iff,axiom,
    ! [W: complex,Z: complex] :
      ( ( aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),imaginary_unit),W) = Z )
    <=> ( W = aa(complex,complex,uminus_uminus(complex),aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),imaginary_unit),Z)) ) ) ).

% i_times_eq_iff
tff(fact_3806_prod_Osetdiff__irrelevant,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [A3: set(B),G: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),A3))
         => ( groups7121269368397514597t_prod(B,A,G,aa(set(B),set(B),minus_minus(set(B),A3),collect(B,aTP_Lamp_hq(fun(B,A),fun(B,bool),G)))) = groups7121269368397514597t_prod(B,A,G,A3) ) ) ) ).

% prod.setdiff_irrelevant
tff(fact_3807_exp__sum,axiom,
    ! [B: $tType,A: $tType] :
      ( ( comm_monoid_mult(B)
        & real_Vector_banach(B)
        & real_V2822296259951069270ebra_1(B) )
     => ! [I5: set(A),F2: fun(A,B)] :
          ( pp(aa(set(A),bool,finite_finite2(A),I5))
         => ( aa(B,B,exp(B),aa(set(A),B,groups7311177749621191930dd_sum(A,B,F2),I5)) = groups7121269368397514597t_prod(A,B,aTP_Lamp_hr(fun(A,B),fun(A,B),F2),I5) ) ) ) ).

% exp_sum
tff(fact_3808_prod_Onat__diff__reindex,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),N: nat] : groups7121269368397514597t_prod(nat,A,aa(nat,fun(nat,A),aTP_Lamp_hs(fun(nat,A),fun(nat,fun(nat,A)),G),N),set_ord_lessThan(nat,N)) = groups7121269368397514597t_prod(nat,A,G,set_ord_lessThan(nat,N)) ) ).

% prod.nat_diff_reindex
tff(fact_3809_prod_OatLeastAtMost__rev,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),N: nat,M: nat] : groups7121269368397514597t_prod(nat,A,G,set_or1337092689740270186AtMost(nat,N,M)) = groups7121269368397514597t_prod(nat,A,aa(nat,fun(nat,A),aa(nat,fun(nat,fun(nat,A)),aTP_Lamp_ht(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),G),N),M),set_or1337092689740270186AtMost(nat,N,M)) ) ).

% prod.atLeastAtMost_rev
tff(fact_3810_less__1__prod2,axiom,
    ! [B: $tType,A: $tType] :
      ( linordered_idom(B)
     => ! [I5: set(A),I: A,F2: fun(A,B)] :
          ( pp(aa(set(A),bool,finite_finite2(A),I5))
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),I),I5))
           => ( pp(aa(B,bool,ord_less(B,one_one(B)),aa(A,B,F2,I)))
             => ( ! [I2: A] :
                    ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),I2),I5))
                   => pp(aa(B,bool,ord_less_eq(B,one_one(B)),aa(A,B,F2,I2))) )
               => pp(aa(B,bool,ord_less(B,one_one(B)),groups7121269368397514597t_prod(A,B,F2,I5))) ) ) ) ) ) ).

% less_1_prod2
tff(fact_3811_Complex__eq__i,axiom,
    ! [X: real,Y: real] :
      ( ( complex2(X,Y) = imaginary_unit )
    <=> ( ( X = zero_zero(real) )
        & ( Y = one_one(real) ) ) ) ).

% Complex_eq_i
tff(fact_3812_imaginary__unit_Ocode,axiom,
    imaginary_unit = complex2(zero_zero(real),one_one(real)) ).

% imaginary_unit.code
tff(fact_3813_prod_Osame__carrier,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [C6: set(B),A3: set(B),B4: set(B),G: fun(B,A),H: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),C6))
         => ( pp(aa(set(B),bool,ord_less_eq(set(B),A3),C6))
           => ( pp(aa(set(B),bool,ord_less_eq(set(B),B4),C6))
             => ( ! [A4: B] :
                    ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A4),aa(set(B),set(B),minus_minus(set(B),C6),A3)))
                   => ( aa(B,A,G,A4) = one_one(A) ) )
               => ( ! [B3: B] :
                      ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),B3),aa(set(B),set(B),minus_minus(set(B),C6),B4)))
                     => ( aa(B,A,H,B3) = one_one(A) ) )
                 => ( ( groups7121269368397514597t_prod(B,A,G,A3) = groups7121269368397514597t_prod(B,A,H,B4) )
                  <=> ( groups7121269368397514597t_prod(B,A,G,C6) = groups7121269368397514597t_prod(B,A,H,C6) ) ) ) ) ) ) ) ) ).

% prod.same_carrier
tff(fact_3814_prod_Osame__carrierI,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [C6: set(B),A3: set(B),B4: set(B),G: fun(B,A),H: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),C6))
         => ( pp(aa(set(B),bool,ord_less_eq(set(B),A3),C6))
           => ( pp(aa(set(B),bool,ord_less_eq(set(B),B4),C6))
             => ( ! [A4: B] :
                    ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A4),aa(set(B),set(B),minus_minus(set(B),C6),A3)))
                   => ( aa(B,A,G,A4) = one_one(A) ) )
               => ( ! [B3: B] :
                      ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),B3),aa(set(B),set(B),minus_minus(set(B),C6),B4)))
                     => ( aa(B,A,H,B3) = one_one(A) ) )
                 => ( ( groups7121269368397514597t_prod(B,A,G,C6) = groups7121269368397514597t_prod(B,A,H,C6) )
                   => ( groups7121269368397514597t_prod(B,A,G,A3) = groups7121269368397514597t_prod(B,A,H,B4) ) ) ) ) ) ) ) ) ).

% prod.same_carrierI
tff(fact_3815_prod_Omono__neutral__left,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [T3: set(B),S: set(B),G: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),T3))
         => ( pp(aa(set(B),bool,ord_less_eq(set(B),S),T3))
           => ( ! [X3: B] :
                  ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),aa(set(B),set(B),minus_minus(set(B),T3),S)))
                 => ( aa(B,A,G,X3) = one_one(A) ) )
             => ( groups7121269368397514597t_prod(B,A,G,S) = groups7121269368397514597t_prod(B,A,G,T3) ) ) ) ) ) ).

% prod.mono_neutral_left
tff(fact_3816_prod_Omono__neutral__right,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [T3: set(B),S: set(B),G: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),T3))
         => ( pp(aa(set(B),bool,ord_less_eq(set(B),S),T3))
           => ( ! [X3: B] :
                  ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),aa(set(B),set(B),minus_minus(set(B),T3),S)))
                 => ( aa(B,A,G,X3) = one_one(A) ) )
             => ( groups7121269368397514597t_prod(B,A,G,T3) = groups7121269368397514597t_prod(B,A,G,S) ) ) ) ) ) ).

% prod.mono_neutral_right
tff(fact_3817_prod_Omono__neutral__cong__left,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [T3: set(B),S: set(B),H: fun(B,A),G: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),T3))
         => ( pp(aa(set(B),bool,ord_less_eq(set(B),S),T3))
           => ( ! [X3: B] :
                  ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),aa(set(B),set(B),minus_minus(set(B),T3),S)))
                 => ( aa(B,A,H,X3) = one_one(A) ) )
             => ( ! [X3: B] :
                    ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),S))
                   => ( aa(B,A,G,X3) = aa(B,A,H,X3) ) )
               => ( groups7121269368397514597t_prod(B,A,G,S) = groups7121269368397514597t_prod(B,A,H,T3) ) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
tff(fact_3818_prod_Omono__neutral__cong__right,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [T3: set(B),S: set(B),G: fun(B,A),H: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),T3))
         => ( pp(aa(set(B),bool,ord_less_eq(set(B),S),T3))
           => ( ! [X3: B] :
                  ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),aa(set(B),set(B),minus_minus(set(B),T3),S)))
                 => ( aa(B,A,G,X3) = one_one(A) ) )
             => ( ! [X3: B] :
                    ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),S))
                   => ( aa(B,A,G,X3) = aa(B,A,H,X3) ) )
               => ( groups7121269368397514597t_prod(B,A,G,T3) = groups7121269368397514597t_prod(B,A,H,S) ) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
tff(fact_3819_prod_Osubset__diff,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [B4: set(B),A3: set(B),G: fun(B,A)] :
          ( pp(aa(set(B),bool,ord_less_eq(set(B),B4),A3))
         => ( pp(aa(set(B),bool,finite_finite2(B),A3))
           => ( groups7121269368397514597t_prod(B,A,G,A3) = aa(A,A,aa(A,fun(A,A),times_times(A),groups7121269368397514597t_prod(B,A,G,aa(set(B),set(B),minus_minus(set(B),A3),B4))),groups7121269368397514597t_prod(B,A,G,B4)) ) ) ) ) ).

% prod.subset_diff
tff(fact_3820_prod_OatLeast0__atMost__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),N: nat] : groups7121269368397514597t_prod(nat,A,G,set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),times_times(A),groups7121269368397514597t_prod(nat,A,G,set_or1337092689740270186AtMost(nat,zero_zero(nat),N))),aa(nat,A,G,aa(nat,nat,suc,N))) ) ).

% prod.atLeast0_atMost_Suc
tff(fact_3821_prod_Onat__ivl__Suc_H,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [M: nat,N: nat,G: fun(nat,A)] :
          ( pp(aa(nat,bool,ord_less_eq(nat,M),aa(nat,nat,suc,N)))
         => ( groups7121269368397514597t_prod(nat,A,G,set_or1337092689740270186AtMost(nat,M,aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G,aa(nat,nat,suc,N))),groups7121269368397514597t_prod(nat,A,G,set_or1337092689740270186AtMost(nat,M,N))) ) ) ) ).

% prod.nat_ivl_Suc'
tff(fact_3822_prod_OatLeast__Suc__atMost,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [M: nat,N: nat,G: fun(nat,A)] :
          ( pp(aa(nat,bool,ord_less_eq(nat,M),N))
         => ( groups7121269368397514597t_prod(nat,A,G,set_or1337092689740270186AtMost(nat,M,N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G,M)),groups7121269368397514597t_prod(nat,A,G,set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M),N))) ) ) ) ).

% prod.atLeast_Suc_atMost
tff(fact_3823_Complex__mult__i,axiom,
    ! [A2: real,B2: real] : aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),complex2(A2,B2)),imaginary_unit) = complex2(aa(real,real,uminus_uminus(real),B2),A2) ).

% Complex_mult_i
tff(fact_3824_i__mult__Complex,axiom,
    ! [A2: real,B2: real] : aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),imaginary_unit),complex2(A2,B2)) = complex2(aa(real,real,uminus_uminus(real),B2),A2) ).

% i_mult_Complex
tff(fact_3825_prod_OlessThan__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),N: nat] : groups7121269368397514597t_prod(nat,A,G,set_ord_lessThan(nat,aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G,zero_zero(nat))),groups7121269368397514597t_prod(nat,A,aTP_Lamp_hn(fun(nat,A),fun(nat,A),G),set_ord_lessThan(nat,N))) ) ).

% prod.lessThan_Suc_shift
tff(fact_3826_prod_OSuc__reindex__ivl,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [M: nat,N: nat,G: fun(nat,A)] :
          ( pp(aa(nat,bool,ord_less_eq(nat,M),N))
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),groups7121269368397514597t_prod(nat,A,G,set_or1337092689740270186AtMost(nat,M,N))),aa(nat,A,G,aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G,M)),groups7121269368397514597t_prod(nat,A,aTP_Lamp_hn(fun(nat,A),fun(nat,A),G),set_or1337092689740270186AtMost(nat,M,N))) ) ) ) ).

% prod.Suc_reindex_ivl
tff(fact_3827_prod_OatLeast1__atMost__eq,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),N: nat] : groups7121269368397514597t_prod(nat,A,G,set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,zero_zero(nat)),N)) = groups7121269368397514597t_prod(nat,A,aTP_Lamp_hn(fun(nat,A),fun(nat,A),G),set_ord_lessThan(nat,N)) ) ).

% prod.atLeast1_atMost_eq
tff(fact_3828_fact__prod,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [N: nat] : semiring_char_0_fact(A,N) = aa(nat,A,semiring_1_of_nat(A),groups7121269368397514597t_prod(nat,nat,aTP_Lamp_el(nat,nat),set_or1337092689740270186AtMost(nat,one_one(nat),N))) ) ).

% fact_prod
tff(fact_3829_even__prod__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( semiring_parity(A)
     => ! [A3: set(B),F2: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),A3))
         => ( pp(aa(A,bool,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),groups7121269368397514597t_prod(B,A,F2,A3)))
          <=> ? [X2: B] :
                ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X2),A3))
                & pp(aa(A,bool,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),aa(B,A,F2,X2))) ) ) ) ) ).

% even_prod_iff
tff(fact_3830_prod_Oub__add__nat,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [M: nat,N: nat,G: fun(nat,A),P2: nat] :
          ( pp(aa(nat,bool,ord_less_eq(nat,M),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat))))
         => ( groups7121269368397514597t_prod(nat,A,G,set_or1337092689740270186AtMost(nat,M,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),P2))) = aa(A,A,aa(A,fun(A,A),times_times(A),groups7121269368397514597t_prod(nat,A,G,set_or1337092689740270186AtMost(nat,M,N))),groups7121269368397514597t_prod(nat,A,G,set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),P2)))) ) ) ) ).

% prod.ub_add_nat
tff(fact_3831_fold__atLeastAtMost__nat_Osimps,axiom,
    ! [A: $tType,B2: nat,A2: nat,F2: fun(nat,fun(A,A)),Acc: A] :
      ( ( pp(aa(nat,bool,ord_less(nat,B2),A2))
       => ( set_fo6178422350223883121st_nat(A,F2,A2,B2,Acc) = Acc ) )
      & ( ~ pp(aa(nat,bool,ord_less(nat,B2),A2))
       => ( set_fo6178422350223883121st_nat(A,F2,A2,B2,Acc) = set_fo6178422350223883121st_nat(A,F2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A2),one_one(nat)),B2,aa(A,A,aa(nat,fun(A,A),F2,A2),Acc)) ) ) ) ).

% fold_atLeastAtMost_nat.simps
tff(fact_3832_fold__atLeastAtMost__nat_Oelims,axiom,
    ! [A: $tType,X: fun(nat,fun(A,A)),Xa2: nat,Xb: nat,Xc: A,Y: A] :
      ( ( set_fo6178422350223883121st_nat(A,X,Xa2,Xb,Xc) = Y )
     => ( ( pp(aa(nat,bool,ord_less(nat,Xb),Xa2))
         => ( Y = Xc ) )
        & ( ~ pp(aa(nat,bool,ord_less(nat,Xb),Xa2))
         => ( Y = set_fo6178422350223883121st_nat(A,X,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Xa2),one_one(nat)),Xb,aa(A,A,aa(nat,fun(A,A),X,Xa2),Xc)) ) ) ) ) ).

% fold_atLeastAtMost_nat.elims
tff(fact_3833_i__complex__of__real,axiom,
    ! [R2: real] : aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),imaginary_unit),aa(real,complex,real_Vector_of_real(complex),R2)) = complex2(zero_zero(real),R2) ).

% i_complex_of_real
tff(fact_3834_complex__of__real__i,axiom,
    ! [R2: real] : aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),aa(real,complex,real_Vector_of_real(complex),R2)),imaginary_unit) = complex2(zero_zero(real),R2) ).

% complex_of_real_i
tff(fact_3835_norm__prod__diff,axiom,
    ! [A: $tType,I6: $tType] :
      ( ( comm_monoid_mult(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [I5: set(I6),Z: fun(I6,A),W: fun(I6,A)] :
          ( ! [I2: I6] :
              ( pp(aa(set(I6),bool,aa(I6,fun(set(I6),bool),member(I6),I2),I5))
             => pp(aa(real,bool,ord_less_eq(real,real_V7770717601297561774m_norm(A,aa(I6,A,Z,I2))),one_one(real))) )
         => ( ! [I2: I6] :
                ( pp(aa(set(I6),bool,aa(I6,fun(set(I6),bool),member(I6),I2),I5))
               => pp(aa(real,bool,ord_less_eq(real,real_V7770717601297561774m_norm(A,aa(I6,A,W,I2))),one_one(real))) )
           => pp(aa(real,bool,ord_less_eq(real,real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,groups7121269368397514597t_prod(I6,A,Z,I5)),groups7121269368397514597t_prod(I6,A,W,I5)))),aa(set(I6),real,groups7311177749621191930dd_sum(I6,real,aa(fun(I6,A),fun(I6,real),aTP_Lamp_hu(fun(I6,A),fun(fun(I6,A),fun(I6,real)),Z),W)),I5))) ) ) ) ).

% norm_prod_diff
tff(fact_3836_fact__eq__fact__times,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa(nat,bool,ord_less_eq(nat,N),M))
     => ( semiring_char_0_fact(nat,M) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),semiring_char_0_fact(nat,N)),groups7121269368397514597t_prod(nat,nat,aTP_Lamp_el(nat,nat),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,N),M))) ) ) ).

% fact_eq_fact_times
tff(fact_3837_Complex__eq,axiom,
    ! [A2: real,B2: real] : complex2(A2,B2) = aa(complex,complex,aa(complex,fun(complex,complex),plus_plus(complex),aa(real,complex,real_Vector_of_real(complex),A2)),aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),imaginary_unit),aa(real,complex,real_Vector_of_real(complex),B2))) ).

% Complex_eq
tff(fact_3838_prod__mono2,axiom,
    ! [B: $tType,A: $tType] :
      ( linordered_idom(B)
     => ! [B4: set(A),A3: set(A),F2: fun(A,B)] :
          ( pp(aa(set(A),bool,finite_finite2(A),B4))
         => ( pp(aa(set(A),bool,ord_less_eq(set(A),A3),B4))
           => ( ! [B3: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),B3),aa(set(A),set(A),minus_minus(set(A),B4),A3)))
                 => pp(aa(B,bool,ord_less_eq(B,one_one(B)),aa(A,B,F2,B3))) )
             => ( ! [A4: A] :
                    ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A4),A3))
                   => pp(aa(B,bool,ord_less_eq(B,zero_zero(B)),aa(A,B,F2,A4))) )
               => pp(aa(B,bool,ord_less_eq(B,groups7121269368397514597t_prod(A,B,F2,A3)),groups7121269368397514597t_prod(A,B,F2,B4))) ) ) ) ) ) ).

% prod_mono2
tff(fact_3839_pochhammer__Suc__prod,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A2: A,N: nat] : comm_s3205402744901411588hammer(A,A2,aa(nat,nat,suc,N)) = groups7121269368397514597t_prod(nat,A,aTP_Lamp_hv(A,fun(nat,A),A2),set_or1337092689740270186AtMost(nat,zero_zero(nat),N)) ) ).

% pochhammer_Suc_prod
tff(fact_3840_pochhammer__prod__rev,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A2: A,N: nat] : comm_s3205402744901411588hammer(A,A2,N) = groups7121269368397514597t_prod(nat,A,aa(nat,fun(nat,A),aTP_Lamp_hw(A,fun(nat,fun(nat,A)),A2),N),set_or1337092689740270186AtMost(nat,one_one(nat),N)) ) ).

% pochhammer_prod_rev
tff(fact_3841_fact__div__fact,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa(nat,bool,ord_less_eq(nat,N),M))
     => ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),semiring_char_0_fact(nat,M)),semiring_char_0_fact(nat,N)) = groups7121269368397514597t_prod(nat,nat,aTP_Lamp_el(nat,nat),set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat)),M)) ) ) ).

% fact_div_fact
tff(fact_3842_complex__split__polar,axiom,
    ! [Z: complex] :
    ? [R3: real,A4: real] : Z = aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),aa(real,complex,real_Vector_of_real(complex),R3)),aa(complex,complex,aa(complex,fun(complex,complex),plus_plus(complex),aa(real,complex,real_Vector_of_real(complex),cos(real,A4))),aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),imaginary_unit),aa(real,complex,real_Vector_of_real(complex),sin(real,A4))))) ).

% complex_split_polar
tff(fact_3843_prod_Oin__pairs,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),M: nat,N: nat] : groups7121269368397514597t_prod(nat,A,G,set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)))) = groups7121269368397514597t_prod(nat,A,aTP_Lamp_hx(fun(nat,A),fun(nat,A),G),set_or1337092689740270186AtMost(nat,M,N)) ) ).

% prod.in_pairs
tff(fact_3844_sum__atLeastAtMost__code,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [F2: fun(nat,A),A2: nat,B2: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F2),set_or1337092689740270186AtMost(nat,A2,B2)) = set_fo6178422350223883121st_nat(A,aTP_Lamp_hy(fun(nat,A),fun(nat,fun(A,A)),F2),A2,B2,zero_zero(A)) ) ).

% sum_atLeastAtMost_code
tff(fact_3845_pochhammer__Suc__prod__rev,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A2: A,N: nat] : comm_s3205402744901411588hammer(A,A2,aa(nat,nat,suc,N)) = groups7121269368397514597t_prod(nat,A,aa(nat,fun(nat,A),aTP_Lamp_hw(A,fun(nat,fun(nat,A)),A2),N),set_or1337092689740270186AtMost(nat,zero_zero(nat),N)) ) ).

% pochhammer_Suc_prod_rev
tff(fact_3846_complex__inverse,axiom,
    ! [A2: real,B2: real] : aa(complex,complex,inverse_inverse(complex),complex2(A2,B2)) = complex2(aa(real,real,aa(real,fun(real,real),divide_divide(real),A2),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),A2),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),B2),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,uminus_uminus(real),B2)),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),A2),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),B2),aa(num,nat,numeral_numeral(nat),bit0(one2)))))) ).

% complex_inverse
tff(fact_3847_gbinomial__Suc,axiom,
    ! [A: $tType] :
      ( ( semiring_char_0(A)
        & semidom_divide(A) )
     => ! [A2: A,K: nat] : aa(nat,A,gbinomial(A,A2),aa(nat,nat,suc,K)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),groups7121269368397514597t_prod(nat,A,aTP_Lamp_hz(A,fun(nat,A),A2),set_or1337092689740270186AtMost(nat,zero_zero(nat),K))),semiring_char_0_fact(A,aa(nat,nat,suc,K))) ) ).

% gbinomial_Suc
tff(fact_3848_cmod__unit__one,axiom,
    ! [A2: real] : real_V7770717601297561774m_norm(complex,aa(complex,complex,aa(complex,fun(complex,complex),plus_plus(complex),aa(real,complex,real_Vector_of_real(complex),cos(real,A2))),aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),imaginary_unit),aa(real,complex,real_Vector_of_real(complex),sin(real,A2))))) = one_one(real) ).

% cmod_unit_one
tff(fact_3849_cmod__complex__polar,axiom,
    ! [R2: real,A2: real] : real_V7770717601297561774m_norm(complex,aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),aa(real,complex,real_Vector_of_real(complex),R2)),aa(complex,complex,aa(complex,fun(complex,complex),plus_plus(complex),aa(real,complex,real_Vector_of_real(complex),cos(real,A2))),aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),imaginary_unit),aa(real,complex,real_Vector_of_real(complex),sin(real,A2)))))) = aa(real,real,abs_abs(real),R2) ).

% cmod_complex_polar
tff(fact_3850_fact__code,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [N: nat] : semiring_char_0_fact(A,N) = aa(nat,A,semiring_1_of_nat(A),set_fo6178422350223883121st_nat(nat,times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2)),N,one_one(nat))) ) ).

% fact_code
tff(fact_3851_Arg__minus__ii,axiom,
    arg(aa(complex,complex,uminus_uminus(complex),imaginary_unit)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,uminus_uminus(real),pi)),aa(num,real,numeral_numeral(real),bit0(one2))) ).

% Arg_minus_ii
tff(fact_3852_csqrt__ii,axiom,
    csqrt(imaginary_unit) = aa(complex,complex,aa(complex,fun(complex,complex),divide_divide(complex),aa(complex,complex,aa(complex,fun(complex,complex),plus_plus(complex),one_one(complex)),imaginary_unit)),aa(real,complex,real_Vector_of_real(complex),aa(real,real,sqrt,aa(num,real,numeral_numeral(real),bit0(one2))))) ).

% csqrt_ii
tff(fact_3853_Arg__ii,axiom,
    arg(imaginary_unit) = aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))) ).

% Arg_ii
tff(fact_3854_gbinomial__partial__row__sum,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A,M: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_gn(A,fun(nat,A),A2)),set_ord_atMost(nat,M)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),M)),one_one(A))),aa(num,A,numeral_numeral(A),bit0(one2)))),aa(nat,A,gbinomial(A,A2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),one_one(nat)))) ) ).

% gbinomial_partial_row_sum
tff(fact_3855_atMost__iff,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [I: A,K: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),I),set_ord_atMost(A,K)))
        <=> pp(aa(A,bool,ord_less_eq(A,I),K)) ) ) ).

% atMost_iff
tff(fact_3856_csqrt__0,axiom,
    csqrt(zero_zero(complex)) = zero_zero(complex) ).

% csqrt_0
tff(fact_3857_csqrt__eq__0,axiom,
    ! [Z: complex] :
      ( ( csqrt(Z) = zero_zero(complex) )
    <=> ( Z = zero_zero(complex) ) ) ).

% csqrt_eq_0
tff(fact_3858_csqrt__1,axiom,
    csqrt(one_one(complex)) = one_one(complex) ).

% csqrt_1
tff(fact_3859_csqrt__eq__1,axiom,
    ! [Z: complex] :
      ( ( csqrt(Z) = one_one(complex) )
    <=> ( Z = one_one(complex) ) ) ).

% csqrt_eq_1
tff(fact_3860_atMost__subset__iff,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [X: A,Y: A] :
          ( pp(aa(set(A),bool,ord_less_eq(set(A),set_ord_atMost(A,X)),set_ord_atMost(A,Y)))
        <=> pp(aa(A,bool,ord_less_eq(A,X),Y)) ) ) ).

% atMost_subset_iff
tff(fact_3861_prod__eq__1__iff,axiom,
    ! [A: $tType,A3: set(A),F2: fun(A,nat)] :
      ( pp(aa(set(A),bool,finite_finite2(A),A3))
     => ( ( groups7121269368397514597t_prod(A,nat,F2,A3) = one_one(nat) )
      <=> ! [X2: A] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X2),A3))
           => ( aa(A,nat,F2,X2) = one_one(nat) ) ) ) ) ).

% prod_eq_1_iff
tff(fact_3862_Icc__subset__Iic__iff,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [L: A,H: A,H2: A] :
          ( pp(aa(set(A),bool,ord_less_eq(set(A),set_or1337092689740270186AtMost(A,L,H)),set_ord_atMost(A,H2)))
        <=> ( ~ pp(aa(A,bool,ord_less_eq(A,L),H))
            | pp(aa(A,bool,ord_less_eq(A,H),H2)) ) ) ) ).

% Icc_subset_Iic_iff
tff(fact_3863_sum_OatMost__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),N: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_ord_atMost(nat,aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_ord_atMost(nat,N))),aa(nat,A,G,aa(nat,nat,suc,N))) ) ).

% sum.atMost_Suc
tff(fact_3864_prod_OatMost__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),N: nat] : groups7121269368397514597t_prod(nat,A,G,set_ord_atMost(nat,aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),times_times(A),groups7121269368397514597t_prod(nat,A,G,set_ord_atMost(nat,N))),aa(nat,A,G,aa(nat,nat,suc,N))) ) ).

% prod.atMost_Suc
tff(fact_3865_prod__pos__nat__iff,axiom,
    ! [A: $tType,A3: set(A),F2: fun(A,nat)] :
      ( pp(aa(set(A),bool,finite_finite2(A),A3))
     => ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),groups7121269368397514597t_prod(A,nat,F2,A3)))
      <=> ! [X2: A] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X2),A3))
           => pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),aa(A,nat,F2,X2))) ) ) ) ).

% prod_pos_nat_iff
tff(fact_3866_power2__csqrt,axiom,
    ! [Z: complex] : aa(nat,complex,aa(complex,fun(nat,complex),power_power(complex),csqrt(Z)),aa(num,nat,numeral_numeral(nat),bit0(one2))) = Z ).

% power2_csqrt
tff(fact_3867_int__prod,axiom,
    ! [B: $tType,F2: fun(B,nat),A3: set(B)] : aa(nat,int,semiring_1_of_nat(int),groups7121269368397514597t_prod(B,nat,F2,A3)) = groups7121269368397514597t_prod(B,int,aTP_Lamp_et(fun(B,nat),fun(B,int),F2),A3) ).

% int_prod
tff(fact_3868_atMost__def,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [U: A] : set_ord_atMost(A,U) = collect(A,aTP_Lamp_ia(A,fun(A,bool),U)) ) ).

% atMost_def
tff(fact_3869_atMost__atLeast0,axiom,
    ! [N: nat] : set_ord_atMost(nat,N) = set_or1337092689740270186AtMost(nat,zero_zero(nat),N) ).

% atMost_atLeast0
tff(fact_3870_lessThan__Suc__atMost,axiom,
    ! [K: nat] : set_ord_lessThan(nat,aa(nat,nat,suc,K)) = set_ord_atMost(nat,K) ).

% lessThan_Suc_atMost
tff(fact_3871_not__Iic__le__Icc,axiom,
    ! [A: $tType] :
      ( no_bot(A)
     => ! [H: A,L2: A,H2: A] : ~ pp(aa(set(A),bool,ord_less_eq(set(A),set_ord_atMost(A,H)),set_or1337092689740270186AtMost(A,L2,H2))) ) ).

% not_Iic_le_Icc
tff(fact_3872_Iic__subset__Iio__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(set(A),bool,ord_less_eq(set(A),set_ord_atMost(A,A2)),set_ord_lessThan(A,B2)))
        <=> pp(aa(A,bool,ord_less(A,A2),B2)) ) ) ).

% Iic_subset_Iio_iff
tff(fact_3873_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,aTP_Lamp_fv(int,int),set_or1337092689740270186AtMost(int,aa(nat,int,semiring_1_of_nat(int),I),aa(nat,int,semiring_1_of_nat(int),J))) ).

% prod_int_eq
tff(fact_3874_Arg__zero,axiom,
    arg(zero_zero(complex)) = zero_zero(real) ).

% Arg_zero
tff(fact_3875_sum_OatMost__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),N: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_ord_atMost(nat,aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G,zero_zero(nat))),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_ea(fun(nat,A),fun(nat,A),G)),set_ord_atMost(nat,N))) ) ).

% sum.atMost_Suc_shift
tff(fact_3876_sum__telescope,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [F2: fun(nat,A),I: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_fi(fun(nat,A),fun(nat,A),F2)),set_ord_atMost(nat,I)) = aa(A,A,minus_minus(A,aa(nat,A,F2,zero_zero(nat))),aa(nat,A,F2,aa(nat,nat,suc,I))) ) ).

% sum_telescope
tff(fact_3877_polyfun__eq__coeffs,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [C2: fun(nat,A),N: nat,D: fun(nat,A)] :
          ( ! [X2: A] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_ib(fun(nat,A),fun(A,fun(nat,A)),C2),X2)),set_ord_atMost(nat,N)) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_ib(fun(nat,A),fun(A,fun(nat,A)),D),X2)),set_ord_atMost(nat,N))
        <=> ! [I3: nat] :
              ( pp(aa(nat,bool,ord_less_eq(nat,I3),N))
             => ( aa(nat,A,C2,I3) = aa(nat,A,D,I3) ) ) ) ) ).

% polyfun_eq_coeffs
tff(fact_3878_prod_OatMost__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),N: nat] : groups7121269368397514597t_prod(nat,A,G,set_ord_atMost(nat,aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G,zero_zero(nat))),groups7121269368397514597t_prod(nat,A,aTP_Lamp_hn(fun(nat,A),fun(nat,A),G),set_ord_atMost(nat,N))) ) ).

% prod.atMost_Suc_shift
tff(fact_3879_bounded__imp__summable,axiom,
    ! [A: $tType] :
      ( ( condit6923001295902523014norder(A)
        & linord2810124833399127020strict(A)
        & topolo1944317154257567458pology(A) )
     => ! [A2: fun(nat,A),B4: A] :
          ( ! [N2: nat] : pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),aa(nat,A,A2,N2)))
         => ( ! [N2: nat] : pp(aa(A,bool,ord_less_eq(A,aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,A2),set_ord_atMost(nat,N2))),B4))
           => summable(A,A2) ) ) ) ).

% bounded_imp_summable
tff(fact_3880_prod__int__plus__eq,axiom,
    ! [I: nat,J: nat] : groups7121269368397514597t_prod(nat,int,semiring_1_of_nat(int),set_or1337092689740270186AtMost(nat,I,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),J))) = groups7121269368397514597t_prod(int,int,aTP_Lamp_fv(int,int),set_or1337092689740270186AtMost(int,aa(nat,int,semiring_1_of_nat(int),I),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),J)))) ).

% prod_int_plus_eq
tff(fact_3881_sum_Onested__swap_H,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [A2: fun(nat,fun(nat,A)),N: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_ic(fun(nat,fun(nat,A)),fun(nat,A),A2)),set_ord_atMost(nat,N)) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aTP_Lamp_ie(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),A2),N)),set_ord_lessThan(nat,N)) ) ).

% sum.nested_swap'
tff(fact_3882_prod_Onested__swap_H,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A2: fun(nat,fun(nat,A)),N: nat] : groups7121269368397514597t_prod(nat,A,aTP_Lamp_if(fun(nat,fun(nat,A)),fun(nat,A),A2),set_ord_atMost(nat,N)) = groups7121269368397514597t_prod(nat,A,aa(nat,fun(nat,A),aTP_Lamp_ih(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),A2),N),set_ord_lessThan(nat,N)) ) ).

% prod.nested_swap'
tff(fact_3883_zero__polynom__imp__zero__coeffs,axiom,
    ! [A: $tType] :
      ( ( ab_semigroup_mult(A)
        & real_V8999393235501362500lgebra(A) )
     => ! [C2: fun(nat,A),N: nat,K: nat] :
          ( ! [W2: A] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_ii(fun(nat,A),fun(A,fun(nat,A)),C2),W2)),set_ord_atMost(nat,N)) = zero_zero(A)
         => ( pp(aa(nat,bool,ord_less_eq(nat,K),N))
           => ( aa(nat,A,C2,K) = zero_zero(A) ) ) ) ) ).

% zero_polynom_imp_zero_coeffs
tff(fact_3884_polyfun__eq__0,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [C2: fun(nat,A),N: nat] :
          ( ! [X2: A] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_ib(fun(nat,A),fun(A,fun(nat,A)),C2),X2)),set_ord_atMost(nat,N)) = zero_zero(A)
        <=> ! [I3: nat] :
              ( pp(aa(nat,bool,ord_less_eq(nat,I3),N))
             => ( aa(nat,A,C2,I3) = zero_zero(A) ) ) ) ) ).

% polyfun_eq_0
tff(fact_3885_ln__prod,axiom,
    ! [A: $tType,I5: set(A),F2: fun(A,real)] :
      ( pp(aa(set(A),bool,finite_finite2(A),I5))
     => ( ! [I2: A] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),I2),I5))
           => pp(aa(real,bool,ord_less(real,zero_zero(real)),aa(A,real,F2,I2))) )
       => ( aa(real,real,ln_ln(real),groups7121269368397514597t_prod(A,real,F2,I5)) = aa(set(A),real,groups7311177749621191930dd_sum(A,real,aTP_Lamp_ij(fun(A,real),fun(A,real),F2)),I5) ) ) ) ).

% ln_prod
tff(fact_3886_sum_OatMost__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),N: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_ord_atMost(nat,N)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G,zero_zero(nat))),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_ea(fun(nat,A),fun(nat,A),G)),set_ord_lessThan(nat,N))) ) ).

% sum.atMost_shift
tff(fact_3887_sum__up__index__split,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [F2: fun(nat,A),M: nat,N: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F2),set_ord_atMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F2),set_ord_atMost(nat,M))),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F2),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N)))) ) ).

% sum_up_index_split
tff(fact_3888_prod_OatMost__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),N: nat] : groups7121269368397514597t_prod(nat,A,G,set_ord_atMost(nat,N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G,zero_zero(nat))),groups7121269368397514597t_prod(nat,A,aTP_Lamp_hn(fun(nat,A),fun(nat,A),G),set_ord_lessThan(nat,N))) ) ).

% prod.atMost_shift
tff(fact_3889_gbinomial__parallel__sum,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A,N: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_ik(A,fun(nat,A),A2)),set_ord_atMost(nat,N)) = aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa(nat,A,semiring_1_of_nat(A),N))),one_one(A))),N) ) ).

% gbinomial_parallel_sum
tff(fact_3890_sum__gp__basic,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [X: A,N: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,minus_minus(A,one_one(A)),X)),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),power_power(A),X)),set_ord_atMost(nat,N))) = aa(A,A,minus_minus(A,one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(nat,nat,suc,N))) ) ).

% sum_gp_basic
tff(fact_3891_polyfun__roots__finite,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [C2: fun(nat,A),K: nat,N: nat] :
          ( ( aa(nat,A,C2,K) != zero_zero(A) )
         => ( pp(aa(nat,bool,ord_less_eq(nat,K),N))
           => pp(aa(set(A),bool,finite_finite2(A),collect(A,aa(nat,fun(A,bool),aTP_Lamp_il(fun(nat,A),fun(nat,fun(A,bool)),C2),N)))) ) ) ) ).

% polyfun_roots_finite
tff(fact_3892_polyfun__finite__roots,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [C2: fun(nat,A),N: nat] :
          ( pp(aa(set(A),bool,finite_finite2(A),collect(A,aa(nat,fun(A,bool),aTP_Lamp_il(fun(nat,A),fun(nat,fun(A,bool)),C2),N))))
        <=> ? [I3: nat] :
              ( pp(aa(nat,bool,ord_less_eq(nat,I3),N))
              & ( aa(nat,A,C2,I3) != zero_zero(A) ) ) ) ) ).

% polyfun_finite_roots
tff(fact_3893_of__real__sqrt,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,ord_less_eq(real,zero_zero(real)),X))
     => ( aa(real,complex,real_Vector_of_real(complex),aa(real,real,sqrt,X)) = csqrt(aa(real,complex,real_Vector_of_real(complex),X)) ) ) ).

% of_real_sqrt
tff(fact_3894_polyfun__linear__factor__root,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [C2: fun(nat,A),A2: A,N: nat] :
          ( ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_im(fun(nat,A),fun(A,fun(nat,A)),C2),A2)),set_ord_atMost(nat,N)) = zero_zero(A) )
         => ~ ! [B3: fun(nat,A)] :
                ~ ! [Z3: A] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_im(fun(nat,A),fun(A,fun(nat,A)),C2),Z3)),set_ord_atMost(nat,N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,minus_minus(A,Z3),A2)),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_im(fun(nat,A),fun(A,fun(nat,A)),B3),Z3)),set_ord_lessThan(nat,N))) ) ) ).

% polyfun_linear_factor_root
tff(fact_3895_polyfun__linear__factor,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [C2: fun(nat,A),N: nat,A2: A] :
        ? [B3: fun(nat,A)] :
        ! [Z3: A] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_im(fun(nat,A),fun(A,fun(nat,A)),C2),Z3)),set_ord_atMost(nat,N)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,minus_minus(A,Z3),A2)),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_im(fun(nat,A),fun(A,fun(nat,A)),B3),Z3)),set_ord_lessThan(nat,N)))),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_im(fun(nat,A),fun(A,fun(nat,A)),C2),A2)),set_ord_atMost(nat,N))) ) ).

% polyfun_linear_factor
tff(fact_3896_sum__power__shift,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [M: nat,N: nat,X: A] :
          ( pp(aa(nat,bool,ord_less_eq(nat,M),N))
         => ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),power_power(A),X)),set_or1337092689740270186AtMost(nat,M,N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),M)),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),power_power(A),X)),set_ord_atMost(nat,aa(nat,nat,minus_minus(nat,N),M)))) ) ) ) ).

% sum_power_shift
tff(fact_3897_summable__Cauchy__product,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V4412858255891104859lgebra(A) )
     => ! [A2: fun(nat,A),B2: fun(nat,A)] :
          ( summable(real,aTP_Lamp_in(fun(nat,A),fun(nat,real),A2))
         => ( summable(real,aTP_Lamp_in(fun(nat,A),fun(nat,real),B2))
           => summable(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_ip(fun(nat,A),fun(fun(nat,A),fun(nat,A)),A2),B2)) ) ) ) ).

% summable_Cauchy_product
tff(fact_3898_Cauchy__product,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V4412858255891104859lgebra(A) )
     => ! [A2: fun(nat,A),B2: fun(nat,A)] :
          ( summable(real,aTP_Lamp_in(fun(nat,A),fun(nat,real),A2))
         => ( summable(real,aTP_Lamp_in(fun(nat,A),fun(nat,real),B2))
           => ( aa(A,A,aa(A,fun(A,A),times_times(A),suminf(A,A2)),suminf(A,B2)) = suminf(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_ip(fun(nat,A),fun(fun(nat,A),fun(nat,A)),A2),B2)) ) ) ) ) ).

% Cauchy_product
tff(fact_3899_Arg__bounded,axiom,
    ! [Z: complex] :
      ( pp(aa(real,bool,ord_less(real,aa(real,real,uminus_uminus(real),pi)),arg(Z)))
      & pp(aa(real,bool,ord_less_eq(real,arg(Z)),pi)) ) ).

% Arg_bounded
tff(fact_3900_sum_Oin__pairs__0,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),N: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_ord_atMost(nat,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)))) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_ek(fun(nat,A),fun(nat,A),G)),set_ord_atMost(nat,N)) ) ).

% sum.in_pairs_0
tff(fact_3901_polynomial__product,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [M: nat,A2: fun(nat,A),N: nat,B2: fun(nat,A),X: A] :
          ( ! [I2: nat] :
              ( pp(aa(nat,bool,ord_less(nat,M),I2))
             => ( aa(nat,A,A2,I2) = zero_zero(A) ) )
         => ( ! [J2: nat] :
                ( pp(aa(nat,bool,ord_less(nat,N),J2))
               => ( aa(nat,A,B2,J2) = zero_zero(A) ) )
           => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_im(fun(nat,A),fun(A,fun(nat,A)),A2),X)),set_ord_atMost(nat,M))),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_im(fun(nat,A),fun(A,fun(nat,A)),B2),X)),set_ord_atMost(nat,N))) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aa(fun(nat,A),fun(A,fun(nat,A)),aTP_Lamp_ir(fun(nat,A),fun(fun(nat,A),fun(A,fun(nat,A))),A2),B2),X)),set_ord_atMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N))) ) ) ) ) ).

% polynomial_product
tff(fact_3902_prod_Oin__pairs__0,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),N: nat] : groups7121269368397514597t_prod(nat,A,G,set_ord_atMost(nat,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)))) = groups7121269368397514597t_prod(nat,A,aTP_Lamp_hx(fun(nat,A),fun(nat,A),G),set_ord_atMost(nat,N)) ) ).

% prod.in_pairs_0
tff(fact_3903_polyfun__eq__const,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [C2: fun(nat,A),N: nat,K: A] :
          ( ! [X2: A] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_ib(fun(nat,A),fun(A,fun(nat,A)),C2),X2)),set_ord_atMost(nat,N)) = K
        <=> ( ( aa(nat,A,C2,zero_zero(nat)) = K )
            & ! [X2: nat] :
                ( pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),X2),set_or1337092689740270186AtMost(nat,one_one(nat),N)))
               => ( aa(nat,A,C2,X2) = zero_zero(A) ) ) ) ) ) ).

% polyfun_eq_const
tff(fact_3904_gbinomial__sum__lower__neg,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A,M: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_is(A,fun(nat,A),A2)),set_ord_atMost(nat,M)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),M)),aa(nat,A,gbinomial(A,aa(A,A,minus_minus(A,A2),one_one(A))),M)) ) ).

% gbinomial_sum_lower_neg
tff(fact_3905_polynomial__product__nat,axiom,
    ! [M: nat,A2: fun(nat,nat),N: nat,B2: fun(nat,nat),X: nat] :
      ( ! [I2: nat] :
          ( pp(aa(nat,bool,ord_less(nat,M),I2))
         => ( aa(nat,nat,A2,I2) = zero_zero(nat) ) )
     => ( ! [J2: nat] :
            ( pp(aa(nat,bool,ord_less(nat,N),J2))
           => ( aa(nat,nat,B2,J2) = zero_zero(nat) ) )
       => ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,aa(nat,fun(nat,nat),aTP_Lamp_it(fun(nat,nat),fun(nat,fun(nat,nat)),A2),X)),set_ord_atMost(nat,M))),aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,aa(nat,fun(nat,nat),aTP_Lamp_it(fun(nat,nat),fun(nat,fun(nat,nat)),B2),X)),set_ord_atMost(nat,N))) = aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,aa(nat,fun(nat,nat),aa(fun(nat,nat),fun(nat,fun(nat,nat)),aTP_Lamp_iv(fun(nat,nat),fun(fun(nat,nat),fun(nat,fun(nat,nat))),A2),B2),X)),set_ord_atMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N))) ) ) ) ).

% polynomial_product_nat
tff(fact_3906_Cauchy__product__sums,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V4412858255891104859lgebra(A) )
     => ! [A2: fun(nat,A),B2: fun(nat,A)] :
          ( summable(real,aTP_Lamp_in(fun(nat,A),fun(nat,real),A2))
         => ( summable(real,aTP_Lamp_in(fun(nat,A),fun(nat,real),B2))
           => pp(aa(A,bool,sums(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_ip(fun(nat,A),fun(fun(nat,A),fun(nat,A)),A2),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),suminf(A,A2)),suminf(A,B2)))) ) ) ) ).

% Cauchy_product_sums
tff(fact_3907_sum_Ozero__middle,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [P2: nat,K: nat,G: fun(nat,A),H: fun(nat,A)] :
          ( pp(aa(nat,bool,ord_less_eq(nat,one_one(nat)),P2))
         => ( pp(aa(nat,bool,ord_less_eq(nat,K),P2))
           => ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(fun(nat,A),fun(nat,A),aa(fun(nat,A),fun(fun(nat,A),fun(nat,A)),aTP_Lamp_iw(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),K),G),H)),set_ord_atMost(nat,P2)) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(fun(nat,A),fun(nat,A),aa(fun(nat,A),fun(fun(nat,A),fun(nat,A)),aTP_Lamp_ix(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),K),G),H)),set_ord_atMost(nat,aa(nat,nat,minus_minus(nat,P2),aa(nat,nat,suc,zero_zero(nat))))) ) ) ) ) ).

% sum.zero_middle
tff(fact_3908_prod_Ozero__middle,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [P2: nat,K: nat,G: fun(nat,A),H: fun(nat,A)] :
          ( pp(aa(nat,bool,ord_less_eq(nat,one_one(nat)),P2))
         => ( pp(aa(nat,bool,ord_less_eq(nat,K),P2))
           => ( groups7121269368397514597t_prod(nat,A,aa(fun(nat,A),fun(nat,A),aa(fun(nat,A),fun(fun(nat,A),fun(nat,A)),aTP_Lamp_iy(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),K),G),H),set_ord_atMost(nat,P2)) = groups7121269368397514597t_prod(nat,A,aa(fun(nat,A),fun(nat,A),aa(fun(nat,A),fun(fun(nat,A),fun(nat,A)),aTP_Lamp_iz(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),K),G),H),set_ord_atMost(nat,aa(nat,nat,minus_minus(nat,P2),aa(nat,nat,suc,zero_zero(nat))))) ) ) ) ) ).

% prod.zero_middle
tff(fact_3909_gbinomial__partial__sum__poly,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [M: nat,A2: A,X: A,Y: A] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aa(A,fun(A,fun(nat,A)),aa(A,fun(A,fun(A,fun(nat,A))),aTP_Lamp_ja(nat,fun(A,fun(A,fun(A,fun(nat,A)))),M),A2),X),Y)),set_ord_atMost(nat,M)) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aa(A,fun(A,fun(nat,A)),aa(A,fun(A,fun(A,fun(nat,A))),aTP_Lamp_jb(nat,fun(A,fun(A,fun(A,fun(nat,A)))),M),A2),X),Y)),set_ord_atMost(nat,M)) ) ).

% gbinomial_partial_sum_poly
tff(fact_3910_root__polyfun,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [N: nat,Z: A,A2: A] :
          ( pp(aa(nat,bool,ord_less_eq(nat,one_one(nat)),N))
         => ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),Z),N) = A2 )
          <=> ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aa(A,fun(A,fun(nat,A)),aTP_Lamp_jc(nat,fun(A,fun(A,fun(nat,A))),N),Z),A2)),set_ord_atMost(nat,N)) = zero_zero(A) ) ) ) ) ).

% root_polyfun
tff(fact_3911_sum__gp0,axiom,
    ! [A: $tType] :
      ( ( division_ring(A)
        & comm_ring(A) )
     => ! [X: A,N: nat] :
          ( ( ( X = one_one(A) )
           => ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),power_power(A),X)),set_ord_atMost(nat,N)) = aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat))) ) )
          & ( ( X != one_one(A) )
           => ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),power_power(A),X)),set_ord_atMost(nat,N)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,minus_minus(A,one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(nat,nat,suc,N)))),aa(A,A,minus_minus(A,one_one(A)),X)) ) ) ) ) ).

% sum_gp0
tff(fact_3912_gbinomial__sum__nat__pow2,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [M: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_jd(nat,fun(nat,A),M)),set_ord_atMost(nat,M)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),M) ) ).

% gbinomial_sum_nat_pow2
tff(fact_3913_gbinomial__partial__sum__poly__xpos,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [M: nat,A2: A,X: A,Y: A] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aa(A,fun(A,fun(nat,A)),aa(A,fun(A,fun(A,fun(nat,A))),aTP_Lamp_ja(nat,fun(A,fun(A,fun(A,fun(nat,A)))),M),A2),X),Y)),set_ord_atMost(nat,M)) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aa(A,fun(A,fun(nat,A)),aa(A,fun(A,fun(A,fun(nat,A))),aTP_Lamp_je(nat,fun(A,fun(A,fun(A,fun(nat,A)))),M),A2),X),Y)),set_ord_atMost(nat,M)) ) ).

% gbinomial_partial_sum_poly_xpos
tff(fact_3914_polyfun__diff__alt,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [N: nat,A2: fun(nat,A),X: A,Y: A] :
          ( pp(aa(nat,bool,ord_less_eq(nat,one_one(nat)),N))
         => ( aa(A,A,minus_minus(A,aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_im(fun(nat,A),fun(A,fun(nat,A)),A2),X)),set_ord_atMost(nat,N))),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_im(fun(nat,A),fun(A,fun(nat,A)),A2),Y)),set_ord_atMost(nat,N))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,minus_minus(A,X),Y)),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aa(A,fun(A,fun(nat,A)),aa(fun(nat,A),fun(A,fun(A,fun(nat,A))),aTP_Lamp_jg(nat,fun(fun(nat,A),fun(A,fun(A,fun(nat,A)))),N),A2),X),Y)),set_ord_lessThan(nat,N))) ) ) ) ).

% polyfun_diff_alt
tff(fact_3915_polyfun__extremal__lemma,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [E: real,C2: fun(nat,A),N: nat] :
          ( pp(aa(real,bool,ord_less(real,zero_zero(real)),E))
         => ? [M9: real] :
            ! [Z3: A] :
              ( pp(aa(real,bool,ord_less_eq(real,M9),real_V7770717601297561774m_norm(A,Z3)))
             => pp(aa(real,bool,ord_less_eq(real,real_V7770717601297561774m_norm(A,aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_ck(fun(nat,A),fun(A,fun(nat,A)),C2),Z3)),set_ord_atMost(nat,N)))),aa(real,real,aa(real,fun(real,real),times_times(real),E),aa(nat,real,aa(real,fun(nat,real),power_power(real),real_V7770717601297561774m_norm(A,Z3)),aa(nat,nat,suc,N))))) ) ) ) ).

% polyfun_extremal_lemma
tff(fact_3916_polyfun__diff,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [N: nat,A2: fun(nat,A),X: A,Y: A] :
          ( pp(aa(nat,bool,ord_less_eq(nat,one_one(nat)),N))
         => ( aa(A,A,minus_minus(A,aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_im(fun(nat,A),fun(A,fun(nat,A)),A2),X)),set_ord_atMost(nat,N))),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_im(fun(nat,A),fun(A,fun(nat,A)),A2),Y)),set_ord_atMost(nat,N))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,minus_minus(A,X),Y)),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aa(A,fun(A,fun(nat,A)),aa(fun(nat,A),fun(A,fun(A,fun(nat,A))),aTP_Lamp_ji(nat,fun(fun(nat,A),fun(A,fun(A,fun(nat,A)))),N),A2),X),Y)),set_ord_lessThan(nat,N))) ) ) ) ).

% polyfun_diff
tff(fact_3917_gbinomial__r__part__sum,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [M: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(nat,A,semiring_1_of_nat(A),M))),one_one(A)))),set_ord_atMost(nat,M)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M)) ) ).

% gbinomial_r_part_sum
tff(fact_3918_cis__minus__pi__half,axiom,
    cis(aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))) = aa(complex,complex,uminus_uminus(complex),imaginary_unit) ).

% cis_minus_pi_half
tff(fact_3919_binomial__code,axiom,
    ! [N: nat,K: nat] :
      ( ( pp(aa(nat,bool,ord_less(nat,N),K))
       => ( aa(nat,nat,binomial(N),K) = zero_zero(nat) ) )
      & ( ~ pp(aa(nat,bool,ord_less(nat,N),K))
       => ( ( pp(aa(nat,bool,ord_less(nat,N),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),K)))
           => ( aa(nat,nat,binomial(N),K) = aa(nat,nat,binomial(N),aa(nat,nat,minus_minus(nat,N),K)) ) )
          & ( ~ pp(aa(nat,bool,ord_less(nat,N),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),K)))
           => ( aa(nat,nat,binomial(N),K) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),set_fo6178422350223883121st_nat(nat,times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,minus_minus(nat,N),K)),one_one(nat)),N,one_one(nat))),semiring_char_0_fact(nat,K)) ) ) ) ) ) ).

% binomial_code
tff(fact_3920_choose__even__sum,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [N: nat] :
          ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),N))
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_jj(nat,fun(nat,A),N)),set_ord_atMost(nat,N))) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N) ) ) ) ).

% choose_even_sum
tff(fact_3921_choose__odd__sum,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [N: nat] :
          ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),N))
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_jk(nat,fun(nat,A),N)),set_ord_atMost(nat,N))) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N) ) ) ) ).

% choose_odd_sum
tff(fact_3922_binomial__Suc__n,axiom,
    ! [N: nat] : aa(nat,nat,binomial(aa(nat,nat,suc,N)),N) = aa(nat,nat,suc,N) ).

% binomial_Suc_n
tff(fact_3923_binomial__n__n,axiom,
    ! [N: nat] : aa(nat,nat,binomial(N),N) = one_one(nat) ).

% binomial_n_n
tff(fact_3924_binomial__0__Suc,axiom,
    ! [K: nat] : aa(nat,nat,binomial(zero_zero(nat)),aa(nat,nat,suc,K)) = zero_zero(nat) ).

% binomial_0_Suc
tff(fact_3925_binomial__1,axiom,
    ! [N: nat] : aa(nat,nat,binomial(N),aa(nat,nat,suc,zero_zero(nat))) = N ).

% binomial_1
tff(fact_3926_binomial__eq__0__iff,axiom,
    ! [N: nat,K: nat] :
      ( ( aa(nat,nat,binomial(N),K) = zero_zero(nat) )
    <=> pp(aa(nat,bool,ord_less(nat,N),K)) ) ).

% binomial_eq_0_iff
tff(fact_3927_binomial__Suc__Suc,axiom,
    ! [N: nat,K: nat] : aa(nat,nat,binomial(aa(nat,nat,suc,N)),aa(nat,nat,suc,K)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,binomial(N),K)),aa(nat,nat,binomial(N),aa(nat,nat,suc,K))) ).

% binomial_Suc_Suc
tff(fact_3928_binomial__n__0,axiom,
    ! [N: nat] : aa(nat,nat,binomial(N),zero_zero(nat)) = one_one(nat) ).

% binomial_n_0
tff(fact_3929_norm__cis,axiom,
    ! [A2: real] : real_V7770717601297561774m_norm(complex,cis(A2)) = one_one(real) ).

% norm_cis
tff(fact_3930_cis__zero,axiom,
    cis(zero_zero(real)) = one_one(complex) ).

% cis_zero
tff(fact_3931_zero__less__binomial__iff,axiom,
    ! [N: nat,K: nat] :
      ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),aa(nat,nat,binomial(N),K)))
    <=> pp(aa(nat,bool,ord_less_eq(nat,K),N)) ) ).

% zero_less_binomial_iff
tff(fact_3932_cis__pi,axiom,
    cis(pi) = aa(complex,complex,uminus_uminus(complex),one_one(complex)) ).

% cis_pi
tff(fact_3933_cis__pi__half,axiom,
    cis(aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2)))) = imaginary_unit ).

% cis_pi_half
tff(fact_3934_cis__2pi,axiom,
    cis(aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),pi)) = one_one(complex) ).

% cis_2pi
tff(fact_3935_choose__one,axiom,
    ! [N: nat] : aa(nat,nat,binomial(N),one_one(nat)) = N ).

% choose_one
tff(fact_3936_cis__neq__zero,axiom,
    ! [A2: real] : cis(A2) != zero_zero(complex) ).

% cis_neq_zero
tff(fact_3937_binomial__eq__0,axiom,
    ! [N: nat,K: nat] :
      ( pp(aa(nat,bool,ord_less(nat,N),K))
     => ( aa(nat,nat,binomial(N),K) = zero_zero(nat) ) ) ).

% binomial_eq_0
tff(fact_3938_Suc__times__binomial,axiom,
    ! [K: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,K)),aa(nat,nat,binomial(aa(nat,nat,suc,N)),aa(nat,nat,suc,K))) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,N)),aa(nat,nat,binomial(N),K)) ).

% Suc_times_binomial
tff(fact_3939_Suc__times__binomial__eq,axiom,
    ! [N: nat,K: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,N)),aa(nat,nat,binomial(N),K)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,binomial(aa(nat,nat,suc,N)),aa(nat,nat,suc,K))),aa(nat,nat,suc,K)) ).

% Suc_times_binomial_eq
tff(fact_3940_binomial__symmetric,axiom,
    ! [K: nat,N: nat] :
      ( pp(aa(nat,bool,ord_less_eq(nat,K),N))
     => ( aa(nat,nat,binomial(N),K) = aa(nat,nat,binomial(N),aa(nat,nat,minus_minus(nat,N),K)) ) ) ).

% binomial_symmetric
tff(fact_3941_choose__mult__lemma,axiom,
    ! [M: nat,R2: nat,K: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),R2)),K)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),K))),aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),K)),K)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),R2)),K)),K)),aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),R2)),M)) ).

% choose_mult_lemma
tff(fact_3942_binomial__le__pow,axiom,
    ! [R2: nat,N: nat] :
      ( pp(aa(nat,bool,ord_less_eq(nat,R2),N))
     => pp(aa(nat,bool,ord_less_eq(nat,aa(nat,nat,binomial(N),R2)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),N),R2))) ) ).

% binomial_le_pow
tff(fact_3943_binomial__gbinomial,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [N: nat,K: nat] : aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,binomial(N),K)) = aa(nat,A,gbinomial(A,aa(nat,A,semiring_1_of_nat(A),N)),K) ) ).

% binomial_gbinomial
tff(fact_3944_cis__mult,axiom,
    ! [A2: real,B2: real] : aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),cis(A2)),cis(B2)) = cis(aa(real,real,aa(real,fun(real,real),plus_plus(real),A2),B2)) ).

% cis_mult
tff(fact_3945_zero__less__binomial,axiom,
    ! [K: nat,N: nat] :
      ( pp(aa(nat,bool,ord_less_eq(nat,K),N))
     => pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),aa(nat,nat,binomial(N),K))) ) ).

% zero_less_binomial
tff(fact_3946_Suc__times__binomial__add,axiom,
    ! [A2: nat,B2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,A2)),aa(nat,nat,binomial(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A2),B2))),aa(nat,nat,suc,A2))) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,B2)),aa(nat,nat,binomial(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A2),B2))),A2)) ).

% Suc_times_binomial_add
tff(fact_3947_binomial__Suc__Suc__eq__times,axiom,
    ! [N: nat,K: nat] : aa(nat,nat,binomial(aa(nat,nat,suc,N)),aa(nat,nat,suc,K)) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,N)),aa(nat,nat,binomial(N),K))),aa(nat,nat,suc,K)) ).

% binomial_Suc_Suc_eq_times
tff(fact_3948_choose__mult,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa(nat,bool,ord_less_eq(nat,K),M))
     => ( pp(aa(nat,bool,ord_less_eq(nat,M),N))
       => ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,binomial(N),M)),aa(nat,nat,binomial(M),K)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,binomial(N),K)),aa(nat,nat,binomial(aa(nat,nat,minus_minus(nat,N),K)),aa(nat,nat,minus_minus(nat,M),K))) ) ) ) ).

% choose_mult
tff(fact_3949_binomial__absorb__comp,axiom,
    ! [N: nat,K: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,minus_minus(nat,N),K)),aa(nat,nat,binomial(N),K)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),aa(nat,nat,binomial(aa(nat,nat,minus_minus(nat,N),one_one(nat))),K)) ).

% binomial_absorb_comp
tff(fact_3950_sum__choose__upper,axiom,
    ! [M: nat,N: nat] : aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,aTP_Lamp_jl(nat,fun(nat,nat),M)),set_ord_atMost(nat,N)) = aa(nat,nat,binomial(aa(nat,nat,suc,N)),aa(nat,nat,suc,M)) ).

% sum_choose_upper
tff(fact_3951_word__lessThan__Suc__atMost,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [K: word(A)] :
          ( ( K != aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A))) )
         => ( set_ord_lessThan(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),K),one_one(word(A)))) = set_ord_atMost(word(A),K) ) ) ) ).

% word_lessThan_Suc_atMost
tff(fact_3952_binomial__absorption,axiom,
    ! [K: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,K)),aa(nat,nat,binomial(N),aa(nat,nat,suc,K))) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),aa(nat,nat,binomial(aa(nat,nat,minus_minus(nat,N),one_one(nat))),K)) ).

% binomial_absorption
tff(fact_3953_binomial__fact__lemma,axiom,
    ! [K: nat,N: nat] :
      ( pp(aa(nat,bool,ord_less_eq(nat,K),N))
     => ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),semiring_char_0_fact(nat,K)),semiring_char_0_fact(nat,aa(nat,nat,minus_minus(nat,N),K)))),aa(nat,nat,binomial(N),K)) = semiring_char_0_fact(nat,N) ) ) ).

% binomial_fact_lemma
tff(fact_3954_DeMoivre,axiom,
    ! [A2: real,N: nat] : aa(nat,complex,aa(complex,fun(nat,complex),power_power(complex),cis(A2)),N) = cis(aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),N)),A2)) ).

% DeMoivre
tff(fact_3955_sum__choose__lower,axiom,
    ! [R2: nat,N: nat] : aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,aTP_Lamp_jm(nat,fun(nat,nat),R2)),set_ord_atMost(nat,N)) = aa(nat,nat,binomial(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),R2),N))),N) ).

% sum_choose_lower
tff(fact_3956_choose__rising__sum_I2_J,axiom,
    ! [N: nat,M: nat] : aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,aTP_Lamp_jn(nat,fun(nat,nat),N)),set_ord_atMost(nat,M)) = aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),M)),one_one(nat))),M) ).

% choose_rising_sum(2)
tff(fact_3957_choose__rising__sum_I1_J,axiom,
    ! [N: nat,M: nat] : aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,aTP_Lamp_jn(nat,fun(nat,nat),N)),set_ord_atMost(nat,M)) = aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),M)),one_one(nat))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat))) ).

% choose_rising_sum(1)
tff(fact_3958_binomial__ge__n__over__k__pow__k,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [K: nat,N: nat] :
          ( pp(aa(nat,bool,ord_less_eq(nat,K),N))
         => pp(aa(A,bool,ord_less_eq(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,semiring_1_of_nat(A),N)),aa(nat,A,semiring_1_of_nat(A),K))),K)),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,binomial(N),K)))) ) ) ).

% binomial_ge_n_over_k_pow_k
tff(fact_3959_binomial__maximum,axiom,
    ! [N: nat,K: nat] : pp(aa(nat,bool,ord_less_eq(nat,aa(nat,nat,binomial(N),K)),aa(nat,nat,binomial(N),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2)))))) ).

% binomial_maximum
tff(fact_3960_binomial__antimono,axiom,
    ! [K: nat,K4: nat,N: nat] :
      ( pp(aa(nat,bool,ord_less_eq(nat,K),K4))
     => ( pp(aa(nat,bool,ord_less_eq(nat,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2)))),K))
       => ( pp(aa(nat,bool,ord_less_eq(nat,K4),N))
         => pp(aa(nat,bool,ord_less_eq(nat,aa(nat,nat,binomial(N),K4)),aa(nat,nat,binomial(N),K))) ) ) ) ).

% binomial_antimono
tff(fact_3961_binomial__mono,axiom,
    ! [K: nat,K4: nat,N: nat] :
      ( pp(aa(nat,bool,ord_less_eq(nat,K),K4))
     => ( pp(aa(nat,bool,ord_less_eq(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),K4)),N))
       => pp(aa(nat,bool,ord_less_eq(nat,aa(nat,nat,binomial(N),K)),aa(nat,nat,binomial(N),K4))) ) ) ).

% binomial_mono
tff(fact_3962_binomial__maximum_H,axiom,
    ! [N: nat,K: nat] : pp(aa(nat,bool,ord_less_eq(nat,aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)),K)),aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)),N))) ).

% binomial_maximum'
tff(fact_3963_binomial__le__pow2,axiom,
    ! [N: nat,K: nat] : pp(aa(nat,bool,ord_less_eq(nat,aa(nat,nat,binomial(N),K)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N))) ).

% binomial_le_pow2
tff(fact_3964_choose__reduce__nat,axiom,
    ! [N: nat,K: nat] :
      ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),N))
     => ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),K))
       => ( aa(nat,nat,binomial(N),K) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,binomial(aa(nat,nat,minus_minus(nat,N),one_one(nat))),aa(nat,nat,minus_minus(nat,K),one_one(nat)))),aa(nat,nat,binomial(aa(nat,nat,minus_minus(nat,N),one_one(nat))),K)) ) ) ) ).

% choose_reduce_nat
tff(fact_3965_times__binomial__minus1__eq,axiom,
    ! [K: nat,N: nat] :
      ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),K))
     => ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),aa(nat,nat,binomial(N),K)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),aa(nat,nat,binomial(aa(nat,nat,minus_minus(nat,N),one_one(nat))),aa(nat,nat,minus_minus(nat,K),one_one(nat)))) ) ) ).

% times_binomial_minus1_eq
tff(fact_3966_binomial__altdef__nat,axiom,
    ! [K: nat,N: nat] :
      ( pp(aa(nat,bool,ord_less_eq(nat,K),N))
     => ( aa(nat,nat,binomial(N),K) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),semiring_char_0_fact(nat,N)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),semiring_char_0_fact(nat,K)),semiring_char_0_fact(nat,aa(nat,nat,minus_minus(nat,N),K)))) ) ) ).

% binomial_altdef_nat
tff(fact_3967_sum__choose__diagonal,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,ord_less_eq(nat,M),N))
     => ( aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,aa(nat,fun(nat,nat),aTP_Lamp_jo(nat,fun(nat,fun(nat,nat)),M),N)),set_ord_atMost(nat,M)) = aa(nat,nat,binomial(aa(nat,nat,suc,N)),M) ) ) ).

% sum_choose_diagonal
tff(fact_3968_vandermonde,axiom,
    ! [M: nat,N: nat,R2: nat] : aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,aa(nat,fun(nat,nat),aa(nat,fun(nat,fun(nat,nat)),aTP_Lamp_jp(nat,fun(nat,fun(nat,fun(nat,nat))),M),N),R2)),set_ord_atMost(nat,R2)) = aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N)),R2) ).

% vandermonde
tff(fact_3969_binomial__less__binomial__Suc,axiom,
    ! [K: nat,N: nat] :
      ( pp(aa(nat,bool,ord_less(nat,K),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2)))))
     => pp(aa(nat,bool,ord_less(nat,aa(nat,nat,binomial(N),K)),aa(nat,nat,binomial(N),aa(nat,nat,suc,K)))) ) ).

% binomial_less_binomial_Suc
tff(fact_3970_binomial__strict__mono,axiom,
    ! [K: nat,K4: nat,N: nat] :
      ( pp(aa(nat,bool,ord_less(nat,K),K4))
     => ( pp(aa(nat,bool,ord_less_eq(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),K4)),N))
       => pp(aa(nat,bool,ord_less(nat,aa(nat,nat,binomial(N),K)),aa(nat,nat,binomial(N),K4))) ) ) ).

% binomial_strict_mono
tff(fact_3971_binomial__strict__antimono,axiom,
    ! [K: nat,K4: nat,N: nat] :
      ( pp(aa(nat,bool,ord_less(nat,K),K4))
     => ( pp(aa(nat,bool,ord_less_eq(nat,N),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),K)))
       => ( pp(aa(nat,bool,ord_less_eq(nat,K4),N))
         => pp(aa(nat,bool,ord_less(nat,aa(nat,nat,binomial(N),K4)),aa(nat,nat,binomial(N),K))) ) ) ) ).

% binomial_strict_antimono
tff(fact_3972_central__binomial__odd,axiom,
    ! [N: nat] :
      ( ~ pp(aa(nat,bool,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N))
     => ( aa(nat,nat,binomial(N),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2))))) = aa(nat,nat,binomial(N),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ) ) ).

% central_binomial_odd
tff(fact_3973_binomial__addition__formula,axiom,
    ! [N: nat,K: nat] :
      ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),N))
     => ( aa(nat,nat,binomial(N),aa(nat,nat,suc,K)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,binomial(aa(nat,nat,minus_minus(nat,N),one_one(nat))),aa(nat,nat,suc,K))),aa(nat,nat,binomial(aa(nat,nat,minus_minus(nat,N),one_one(nat))),K)) ) ) ).

% binomial_addition_formula
tff(fact_3974_binomial__fact,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,N: nat] :
          ( pp(aa(nat,bool,ord_less_eq(nat,K),N))
         => ( aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,binomial(N),K)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),semiring_char_0_fact(A,N)),aa(A,A,aa(A,fun(A,A),times_times(A),semiring_char_0_fact(A,K)),semiring_char_0_fact(A,aa(nat,nat,minus_minus(nat,N),K)))) ) ) ) ).

% binomial_fact
tff(fact_3975_fact__binomial,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,N: nat] :
          ( pp(aa(nat,bool,ord_less_eq(nat,K),N))
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),semiring_char_0_fact(A,K)),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,binomial(N),K))) = aa(A,A,aa(A,fun(A,A),divide_divide(A),semiring_char_0_fact(A,N)),semiring_char_0_fact(A,aa(nat,nat,minus_minus(nat,N),K))) ) ) ) ).

% fact_binomial
tff(fact_3976_cis__conv__exp,axiom,
    ! [B2: real] : cis(B2) = aa(complex,complex,exp(complex),aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),imaginary_unit),aa(real,complex,real_Vector_of_real(complex),B2))) ).

% cis_conv_exp
tff(fact_3977_choose__row__sum,axiom,
    ! [N: nat] : aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,binomial(N)),set_ord_atMost(nat,N)) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N) ).

% choose_row_sum
tff(fact_3978_binomial,axiom,
    ! [A2: nat,B2: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A2),B2)),N) = aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,aa(nat,fun(nat,nat),aa(nat,fun(nat,fun(nat,nat)),aTP_Lamp_jq(nat,fun(nat,fun(nat,fun(nat,nat))),A2),B2),N)),set_ord_atMost(nat,N)) ).

% binomial
tff(fact_3979_choose__two,axiom,
    ! [N: nat] : aa(nat,nat,binomial(N),aa(num,nat,numeral_numeral(nat),bit0(one2))) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),aa(nat,nat,minus_minus(nat,N),one_one(nat)))),aa(num,nat,numeral_numeral(nat),bit0(one2))) ).

% choose_two
tff(fact_3980_binomial__ring,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A2: A,B2: A,N: nat] : aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),N) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_jr(A,fun(A,fun(nat,fun(nat,A))),A2),B2),N)),set_ord_atMost(nat,N)) ) ).

% binomial_ring
tff(fact_3981_pochhammer__binomial__sum,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [A2: A,B2: A,N: nat] : comm_s3205402744901411588hammer(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2),N) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_js(A,fun(A,fun(nat,fun(nat,A))),A2),B2),N)),set_ord_atMost(nat,N)) ) ).

% pochhammer_binomial_sum
tff(fact_3982_choose__square__sum,axiom,
    ! [N: nat] : aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,aTP_Lamp_jt(nat,fun(nat,nat),N)),set_ord_atMost(nat,N)) = aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)),N) ).

% choose_square_sum
tff(fact_3983_choose__alternating__linear__sum,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [N: nat] :
          ( ( N != one_one(nat) )
         => ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_ju(nat,fun(nat,A),N)),set_ord_atMost(nat,N)) = zero_zero(A) ) ) ) ).

% choose_alternating_linear_sum
tff(fact_3984_binomial__r__part__sum,axiom,
    ! [M: nat] : aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M)),one_one(nat)))),set_ord_atMost(nat,M)) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M)) ).

% binomial_r_part_sum
tff(fact_3985_choose__linear__sum,axiom,
    ! [N: nat] : aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,aTP_Lamp_jv(nat,fun(nat,nat),N)),set_ord_atMost(nat,N)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,minus_minus(nat,N),one_one(nat)))) ).

% choose_linear_sum
tff(fact_3986_choose__alternating__sum,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [N: nat] :
          ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),N))
         => ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_jw(nat,fun(nat,A),N)),set_ord_atMost(nat,N)) = zero_zero(A) ) ) ) ).

% choose_alternating_sum
tff(fact_3987_central__binomial__lower__bound,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),N))
     => pp(aa(real,bool,ord_less_eq(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(num,real,numeral_numeral(real),bit0(bit0(one2)))),N)),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),aa(nat,real,semiring_1_of_nat(real),N)))),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)),N)))) ) ).

% central_binomial_lower_bound
tff(fact_3988_sin__x__sin__y,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A,Y: A] : pp(aa(A,bool,sums(A,aa(A,fun(nat,A),aTP_Lamp_jy(A,fun(A,fun(nat,A)),X),Y)),aa(A,A,aa(A,fun(A,A),times_times(A),sin(A,X)),sin(A,Y)))) ) ).

% sin_x_sin_y
tff(fact_3989_sums__cos__x__plus__y,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A,Y: A] : pp(aa(A,bool,sums(A,aa(A,fun(nat,A),aTP_Lamp_ka(A,fun(A,fun(nat,A)),X),Y)),cos(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y)))) ) ).

% sums_cos_x_plus_y
tff(fact_3990_cos__x__cos__y,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A,Y: A] : pp(aa(A,bool,sums(A,aa(A,fun(nat,A),aTP_Lamp_kc(A,fun(A,fun(nat,A)),X),Y)),aa(A,A,aa(A,fun(A,A),times_times(A),cos(A,X)),cos(A,Y)))) ) ).

% cos_x_cos_y
tff(fact_3991_bij__betw__roots__unity,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),N))
     => bij_betw(nat,complex,aTP_Lamp_kd(nat,fun(nat,complex),N),set_ord_lessThan(nat,N),collect(complex,aTP_Lamp_fe(nat,fun(complex,bool),N))) ) ).

% bij_betw_roots_unity
tff(fact_3992_of__nat__id,axiom,
    ! [N: nat] : aa(nat,nat,semiring_1_of_nat(nat),N) = N ).

% of_nat_id
tff(fact_3993_scaleR__zero__right,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [A2: real] : aa(A,A,real_V8093663219630862766scaleR(A,A2),zero_zero(A)) = zero_zero(A) ) ).

% scaleR_zero_right
tff(fact_3994_scaleR__cancel__right,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [A2: real,X: A,B2: real] :
          ( ( aa(A,A,real_V8093663219630862766scaleR(A,A2),X) = aa(A,A,real_V8093663219630862766scaleR(A,B2),X) )
        <=> ( ( A2 = B2 )
            | ( X = zero_zero(A) ) ) ) ) ).

% scaleR_cancel_right
tff(fact_3995_mult__scaleR__left,axiom,
    ! [A: $tType] :
      ( real_V6157519004096292374lgebra(A)
     => ! [A2: real,X: A,Y: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,real_V8093663219630862766scaleR(A,A2),X)),Y) = aa(A,A,real_V8093663219630862766scaleR(A,A2),aa(A,A,aa(A,fun(A,A),times_times(A),X),Y)) ) ).

% mult_scaleR_left
tff(fact_3996_mult__scaleR__right,axiom,
    ! [A: $tType] :
      ( real_V6157519004096292374lgebra(A)
     => ! [X: A,A2: real,Y: A] : aa(A,A,aa(A,fun(A,A),times_times(A),X),aa(A,A,real_V8093663219630862766scaleR(A,A2),Y)) = aa(A,A,real_V8093663219630862766scaleR(A,A2),aa(A,A,aa(A,fun(A,A),times_times(A),X),Y)) ) ).

% mult_scaleR_right
tff(fact_3997_scaleR__cancel__left,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [A2: real,X: A,Y: A] :
          ( ( aa(A,A,real_V8093663219630862766scaleR(A,A2),X) = aa(A,A,real_V8093663219630862766scaleR(A,A2),Y) )
        <=> ( ( X = Y )
            | ( A2 = zero_zero(real) ) ) ) ) ).

% scaleR_cancel_left
tff(fact_3998_scaleR__one,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [X: A] : aa(A,A,real_V8093663219630862766scaleR(A,one_one(real)),X) = X ) ).

% scaleR_one
tff(fact_3999_scaleR__scaleR,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [A2: real,B2: real,X: A] : aa(A,A,real_V8093663219630862766scaleR(A,A2),aa(A,A,real_V8093663219630862766scaleR(A,B2),X)) = aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,aa(real,fun(real,real),times_times(real),A2),B2)),X) ) ).

% scaleR_scaleR
tff(fact_4000_scaleR__eq__0__iff,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [A2: real,X: A] :
          ( ( aa(A,A,real_V8093663219630862766scaleR(A,A2),X) = zero_zero(A) )
        <=> ( ( A2 = zero_zero(real) )
            | ( X = zero_zero(A) ) ) ) ) ).

% scaleR_eq_0_iff
tff(fact_4001_scaleR__zero__left,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [X: A] : aa(A,A,real_V8093663219630862766scaleR(A,zero_zero(real)),X) = zero_zero(A) ) ).

% scaleR_zero_left
tff(fact_4002_scaleR__eq__iff,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [B2: A,U: real,A2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),aa(A,A,real_V8093663219630862766scaleR(A,U),A2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa(A,A,real_V8093663219630862766scaleR(A,U),B2)) )
        <=> ( ( A2 = B2 )
            | ( U = one_one(real) ) ) ) ) ).

% scaleR_eq_iff
tff(fact_4003_scaleR__power,axiom,
    ! [A: $tType] :
      ( real_V2822296259951069270ebra_1(A)
     => ! [X: real,Y: A,N: nat] : aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,real_V8093663219630862766scaleR(A,X),Y)),N) = aa(A,A,real_V8093663219630862766scaleR(A,aa(nat,real,aa(real,fun(nat,real),power_power(real),X),N)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),N)) ) ).

% scaleR_power
tff(fact_4004_scaleR__minus1__left,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [X: A] : aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,uminus_uminus(real),one_one(real))),X) = aa(A,A,uminus_uminus(A),X) ) ).

% scaleR_minus1_left
tff(fact_4005_scaleR__collapse,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [U: real,A2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,minus_minus(real,one_one(real)),U)),A2)),aa(A,A,real_V8093663219630862766scaleR(A,U),A2)) = A2 ) ).

% scaleR_collapse
tff(fact_4006_norm__scaleR,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [A2: real,X: A] : real_V7770717601297561774m_norm(A,aa(A,A,real_V8093663219630862766scaleR(A,A2),X)) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,abs_abs(real),A2)),real_V7770717601297561774m_norm(A,X)) ) ).

% norm_scaleR
tff(fact_4007_scaleR__times,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [U: num,W: num,A2: A] : aa(A,A,real_V8093663219630862766scaleR(A,aa(num,real,numeral_numeral(real),U)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W)),A2)) = aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),U)),aa(num,real,numeral_numeral(real),W))),A2) ) ).

% scaleR_times
tff(fact_4008_inverse__scaleR__times,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [V: num,W: num,A2: A] : aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),V))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W)),A2)) = aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(num,real,numeral_numeral(real),W)),aa(num,real,numeral_numeral(real),V))),A2) ) ).

% inverse_scaleR_times
tff(fact_4009_fraction__scaleR__times,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [U: num,V: num,W: num,A2: A] : aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(num,real,numeral_numeral(real),U)),aa(num,real,numeral_numeral(real),V))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W)),A2)) = aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),U)),aa(num,real,numeral_numeral(real),W))),aa(num,real,numeral_numeral(real),V))),A2) ) ).

% fraction_scaleR_times
tff(fact_4010_scaleR__half__double,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [A2: A] : aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),bit0(one2)))),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),A2)) = A2 ) ).

% scaleR_half_double
tff(fact_4011_prod_Oreindex__bij__betw,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comm_monoid_mult(A)
     => ! [H: fun(B,C),S: set(B),T3: set(C),G: fun(C,A)] :
          ( bij_betw(B,C,H,S,T3)
         => ( groups7121269368397514597t_prod(B,A,aa(fun(C,A),fun(B,A),aTP_Lamp_ke(fun(B,C),fun(fun(C,A),fun(B,A)),H),G),S) = groups7121269368397514597t_prod(C,A,G,T3) ) ) ) ).

% prod.reindex_bij_betw
tff(fact_4012_sum_Oreindex__bij__betw,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comm_monoid_add(A)
     => ! [H: fun(B,C),S: set(B),T3: set(C),G: fun(C,A)] :
          ( bij_betw(B,C,H,S,T3)
         => ( aa(set(B),A,groups7311177749621191930dd_sum(B,A,aa(fun(C,A),fun(B,A),aTP_Lamp_kf(fun(B,C),fun(fun(C,A),fun(B,A)),H),G)),S) = aa(set(C),A,groups7311177749621191930dd_sum(C,A,G),T3) ) ) ) ).

% sum.reindex_bij_betw
tff(fact_4013_scaleR__right__distrib,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [A2: real,X: A,Y: A] : aa(A,A,real_V8093663219630862766scaleR(A,A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,real_V8093663219630862766scaleR(A,A2),X)),aa(A,A,real_V8093663219630862766scaleR(A,A2),Y)) ) ).

% scaleR_right_distrib
tff(fact_4014_scaleR__right__imp__eq,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [X: A,A2: real,B2: real] :
          ( ( X != zero_zero(A) )
         => ( ( aa(A,A,real_V8093663219630862766scaleR(A,A2),X) = aa(A,A,real_V8093663219630862766scaleR(A,B2),X) )
           => ( A2 = B2 ) ) ) ) ).

% scaleR_right_imp_eq
tff(fact_4015_real__scaleR__def,axiom,
    ! [A2: real,X: real] : aa(real,real,real_V8093663219630862766scaleR(real,A2),X) = aa(real,real,aa(real,fun(real,real),times_times(real),A2),X) ).

% real_scaleR_def
tff(fact_4016_scaleR__left__imp__eq,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [A2: real,X: A,Y: A] :
          ( ( A2 != zero_zero(real) )
         => ( ( aa(A,A,real_V8093663219630862766scaleR(A,A2),X) = aa(A,A,real_V8093663219630862766scaleR(A,A2),Y) )
           => ( X = Y ) ) ) ) ).

% scaleR_left_imp_eq
tff(fact_4017_scaleR__sum__right,axiom,
    ! [A: $tType,C: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [A2: real,F2: fun(C,A),A3: set(C)] : aa(A,A,real_V8093663219630862766scaleR(A,A2),aa(set(C),A,groups7311177749621191930dd_sum(C,A,F2),A3)) = aa(set(C),A,groups7311177749621191930dd_sum(C,A,aa(fun(C,A),fun(C,A),aTP_Lamp_kg(real,fun(fun(C,A),fun(C,A)),A2),F2)),A3) ) ).

% scaleR_sum_right
tff(fact_4018_scaleR__right_Osum,axiom,
    ! [A: $tType,C: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [A2: real,G: fun(C,A),A3: set(C)] : aa(A,A,real_V8093663219630862766scaleR(A,A2),aa(set(C),A,groups7311177749621191930dd_sum(C,A,G),A3)) = aa(set(C),A,groups7311177749621191930dd_sum(C,A,aa(fun(C,A),fun(C,A),aTP_Lamp_kg(real,fun(fun(C,A),fun(C,A)),A2),G)),A3) ) ).

% scaleR_right.sum
tff(fact_4019_summable__scaleR__right,axiom,
    ! [B: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [X6: fun(nat,B),R2: real] :
          ( summable(B,X6)
         => summable(B,aa(real,fun(nat,B),aTP_Lamp_kh(fun(nat,B),fun(real,fun(nat,B)),X6),R2)) ) ) ).

% summable_scaleR_right
tff(fact_4020_sums__scaleR__right,axiom,
    ! [B: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [X6: fun(nat,B),A2: B,R2: real] :
          ( pp(aa(B,bool,sums(B,X6),A2))
         => pp(aa(B,bool,sums(B,aa(real,fun(nat,B),aTP_Lamp_kh(fun(nat,B),fun(real,fun(nat,B)),X6),R2)),aa(B,B,real_V8093663219630862766scaleR(B,R2),A2))) ) ) ).

% sums_scaleR_right
tff(fact_4021_scaleR__left_Oadd,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [X: real,Y: real,Xa2: A] : aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,aa(real,fun(real,real),plus_plus(real),X),Y)),Xa2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,real_V8093663219630862766scaleR(A,X),Xa2)),aa(A,A,real_V8093663219630862766scaleR(A,Y),Xa2)) ) ).

% scaleR_left.add
tff(fact_4022_scaleR__left__distrib,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [A2: real,B2: real,X: A] : aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,aa(real,fun(real,real),plus_plus(real),A2),B2)),X) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,real_V8093663219630862766scaleR(A,A2),X)),aa(A,A,real_V8093663219630862766scaleR(A,B2),X)) ) ).

% scaleR_left_distrib
tff(fact_4023_of__real__def,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [R2: real] : aa(real,A,real_Vector_of_real(A),R2) = aa(A,A,real_V8093663219630862766scaleR(A,R2),one_one(A)) ) ).

% of_real_def
tff(fact_4024_scaleR__conv__of__real,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [R2: real,X: A] : aa(A,A,real_V8093663219630862766scaleR(A,R2),X) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(real,A,real_Vector_of_real(A),R2)),X) ) ).

% scaleR_conv_of_real
tff(fact_4025_complex__scaleR,axiom,
    ! [R2: real,A2: real,B2: real] : aa(complex,complex,real_V8093663219630862766scaleR(complex,R2),complex2(A2,B2)) = complex2(aa(real,real,aa(real,fun(real,real),times_times(real),R2),A2),aa(real,real,aa(real,fun(real,real),times_times(real),R2),B2)) ).

% complex_scaleR
tff(fact_4026_scaleR__sum__left,axiom,
    ! [A: $tType,C: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [F2: fun(C,real),A3: set(C),X: A] : aa(A,A,real_V8093663219630862766scaleR(A,aa(set(C),real,groups7311177749621191930dd_sum(C,real,F2),A3)),X) = aa(set(C),A,groups7311177749621191930dd_sum(C,A,aa(A,fun(C,A),aTP_Lamp_ki(fun(C,real),fun(A,fun(C,A)),F2),X)),A3) ) ).

% scaleR_sum_left
tff(fact_4027_scaleR__left_Osum,axiom,
    ! [A: $tType,C: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [G: fun(C,real),A3: set(C),X: A] : aa(A,A,real_V8093663219630862766scaleR(A,aa(set(C),real,groups7311177749621191930dd_sum(C,real,G),A3)),X) = aa(set(C),A,groups7311177749621191930dd_sum(C,A,aa(A,fun(C,A),aTP_Lamp_ki(fun(C,real),fun(A,fun(C,A)),G),X)),A3) ) ).

% scaleR_left.sum
tff(fact_4028_suminf__scaleR__right,axiom,
    ! [B: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [X6: fun(nat,B),R2: real] :
          ( summable(B,X6)
         => ( aa(B,B,real_V8093663219630862766scaleR(B,R2),suminf(B,X6)) = suminf(B,aa(real,fun(nat,B),aTP_Lamp_kh(fun(nat,B),fun(real,fun(nat,B)),X6),R2)) ) ) ) ).

% suminf_scaleR_right
tff(fact_4029_summable__scaleR__left,axiom,
    ! [B: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [X6: fun(nat,real),X: B] :
          ( summable(real,X6)
         => summable(B,aa(B,fun(nat,B),aTP_Lamp_kj(fun(nat,real),fun(B,fun(nat,B)),X6),X)) ) ) ).

% summable_scaleR_left
tff(fact_4030_sums__scaleR__left,axiom,
    ! [B: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [X6: fun(nat,real),A2: real,X: B] :
          ( pp(aa(real,bool,sums(real,X6),A2))
         => pp(aa(B,bool,sums(B,aa(B,fun(nat,B),aTP_Lamp_kj(fun(nat,real),fun(B,fun(nat,B)),X6),X)),aa(B,B,real_V8093663219630862766scaleR(B,A2),X))) ) ) ).

% sums_scaleR_left
tff(fact_4031_scaleR__right__mono,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A2: real,B2: real,X: A] :
          ( pp(aa(real,bool,ord_less_eq(real,A2),B2))
         => ( pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),X))
           => pp(aa(A,bool,ord_less_eq(A,aa(A,A,real_V8093663219630862766scaleR(A,A2),X)),aa(A,A,real_V8093663219630862766scaleR(A,B2),X))) ) ) ) ).

% scaleR_right_mono
tff(fact_4032_scaleR__right__mono__neg,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [B2: real,A2: real,C2: A] :
          ( pp(aa(real,bool,ord_less_eq(real,B2),A2))
         => ( pp(aa(A,bool,ord_less_eq(A,C2),zero_zero(A)))
           => pp(aa(A,bool,ord_less_eq(A,aa(A,A,real_V8093663219630862766scaleR(A,A2),C2)),aa(A,A,real_V8093663219630862766scaleR(A,B2),C2))) ) ) ) ).

% scaleR_right_mono_neg
tff(fact_4033_scaleR__le__cancel__left__pos,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,A2: A,B2: A] :
          ( pp(aa(real,bool,ord_less(real,zero_zero(real)),C2))
         => ( pp(aa(A,bool,ord_less_eq(A,aa(A,A,real_V8093663219630862766scaleR(A,C2),A2)),aa(A,A,real_V8093663219630862766scaleR(A,C2),B2)))
          <=> pp(aa(A,bool,ord_less_eq(A,A2),B2)) ) ) ) ).

% scaleR_le_cancel_left_pos
tff(fact_4034_scaleR__le__cancel__left__neg,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,A2: A,B2: A] :
          ( pp(aa(real,bool,ord_less(real,C2),zero_zero(real)))
         => ( pp(aa(A,bool,ord_less_eq(A,aa(A,A,real_V8093663219630862766scaleR(A,C2),A2)),aa(A,A,real_V8093663219630862766scaleR(A,C2),B2)))
          <=> pp(aa(A,bool,ord_less_eq(A,B2),A2)) ) ) ) ).

% scaleR_le_cancel_left_neg
tff(fact_4035_scaleR__le__cancel__left,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,A2: A,B2: A] :
          ( pp(aa(A,bool,ord_less_eq(A,aa(A,A,real_V8093663219630862766scaleR(A,C2),A2)),aa(A,A,real_V8093663219630862766scaleR(A,C2),B2)))
        <=> ( ( pp(aa(real,bool,ord_less(real,zero_zero(real)),C2))
             => pp(aa(A,bool,ord_less_eq(A,A2),B2)) )
            & ( pp(aa(real,bool,ord_less(real,C2),zero_zero(real)))
             => pp(aa(A,bool,ord_less_eq(A,B2),A2)) ) ) ) ) ).

% scaleR_le_cancel_left
tff(fact_4036_scaleR__left__mono,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [X: A,Y: A,A2: real] :
          ( pp(aa(A,bool,ord_less_eq(A,X),Y))
         => ( pp(aa(real,bool,ord_less_eq(real,zero_zero(real)),A2))
           => pp(aa(A,bool,ord_less_eq(A,aa(A,A,real_V8093663219630862766scaleR(A,A2),X)),aa(A,A,real_V8093663219630862766scaleR(A,A2),Y))) ) ) ) ).

% scaleR_left_mono
tff(fact_4037_scaleR__left__mono__neg,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [B2: A,A2: A,C2: real] :
          ( pp(aa(A,bool,ord_less_eq(A,B2),A2))
         => ( pp(aa(real,bool,ord_less_eq(real,C2),zero_zero(real)))
           => pp(aa(A,bool,ord_less_eq(A,aa(A,A,real_V8093663219630862766scaleR(A,C2),A2)),aa(A,A,real_V8093663219630862766scaleR(A,C2),B2))) ) ) ) ).

% scaleR_left_mono_neg
tff(fact_4038_eq__vector__fraction__iff,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [X: A,U: real,V: real,A2: A] :
          ( ( X = aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,aa(real,fun(real,real),divide_divide(real),U),V)),A2) )
        <=> ( ( ( V = zero_zero(real) )
             => ( X = zero_zero(A) ) )
            & ( ( V != zero_zero(real) )
             => ( aa(A,A,real_V8093663219630862766scaleR(A,V),X) = aa(A,A,real_V8093663219630862766scaleR(A,U),A2) ) ) ) ) ) ).

% eq_vector_fraction_iff
tff(fact_4039_vector__fraction__eq__iff,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [U: real,V: real,A2: A,X: A] :
          ( ( aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,aa(real,fun(real,real),divide_divide(real),U),V)),A2) = X )
        <=> ( ( ( V = zero_zero(real) )
             => ( X = zero_zero(A) ) )
            & ( ( V != zero_zero(real) )
             => ( aa(A,A,real_V8093663219630862766scaleR(A,U),A2) = aa(A,A,real_V8093663219630862766scaleR(A,V),X) ) ) ) ) ) ).

% vector_fraction_eq_iff
tff(fact_4040_Real__Vector__Spaces_Ole__add__iff1,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A2: real,E: A,C2: A,B2: real,D: A] :
          ( pp(aa(A,bool,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,real_V8093663219630862766scaleR(A,A2),E)),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,real_V8093663219630862766scaleR(A,B2),E)),D)))
        <=> pp(aa(A,bool,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,minus_minus(real,A2),B2)),E)),C2)),D)) ) ) ).

% Real_Vector_Spaces.le_add_iff1
tff(fact_4041_Real__Vector__Spaces_Ole__add__iff2,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A2: real,E: A,C2: A,B2: real,D: A] :
          ( pp(aa(A,bool,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,real_V8093663219630862766scaleR(A,A2),E)),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,real_V8093663219630862766scaleR(A,B2),E)),D)))
        <=> pp(aa(A,bool,ord_less_eq(A,C2),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,minus_minus(real,B2),A2)),E)),D))) ) ) ).

% Real_Vector_Spaces.le_add_iff2
tff(fact_4042_zero__le__scaleR__iff,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A2: real,B2: A] :
          ( pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),aa(A,A,real_V8093663219630862766scaleR(A,A2),B2)))
        <=> ( ( pp(aa(real,bool,ord_less(real,zero_zero(real)),A2))
              & pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),B2)) )
            | ( pp(aa(real,bool,ord_less(real,A2),zero_zero(real)))
              & pp(aa(A,bool,ord_less_eq(A,B2),zero_zero(A))) )
            | ( A2 = zero_zero(real) ) ) ) ) ).

% zero_le_scaleR_iff
tff(fact_4043_scaleR__le__0__iff,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A2: real,B2: A] :
          ( pp(aa(A,bool,ord_less_eq(A,aa(A,A,real_V8093663219630862766scaleR(A,A2),B2)),zero_zero(A)))
        <=> ( ( pp(aa(real,bool,ord_less(real,zero_zero(real)),A2))
              & pp(aa(A,bool,ord_less_eq(A,B2),zero_zero(A))) )
            | ( pp(aa(real,bool,ord_less(real,A2),zero_zero(real)))
              & pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),B2)) )
            | ( A2 = zero_zero(real) ) ) ) ) ).

% scaleR_le_0_iff
tff(fact_4044_suminf__scaleR__left,axiom,
    ! [B: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [X6: fun(nat,real),X: B] :
          ( summable(real,X6)
         => ( aa(B,B,real_V8093663219630862766scaleR(B,suminf(real,X6)),X) = suminf(B,aa(B,fun(nat,B),aTP_Lamp_kj(fun(nat,real),fun(B,fun(nat,B)),X6),X)) ) ) ) ).

% suminf_scaleR_left
tff(fact_4045_scaleR__nonpos__nonpos,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A2: real,B2: A] :
          ( pp(aa(real,bool,ord_less_eq(real,A2),zero_zero(real)))
         => ( pp(aa(A,bool,ord_less_eq(A,B2),zero_zero(A)))
           => pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),aa(A,A,real_V8093663219630862766scaleR(A,A2),B2))) ) ) ) ).

% scaleR_nonpos_nonpos
tff(fact_4046_scaleR__nonpos__nonneg,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A2: real,X: A] :
          ( pp(aa(real,bool,ord_less_eq(real,A2),zero_zero(real)))
         => ( pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),X))
           => pp(aa(A,bool,ord_less_eq(A,aa(A,A,real_V8093663219630862766scaleR(A,A2),X)),zero_zero(A))) ) ) ) ).

% scaleR_nonpos_nonneg
tff(fact_4047_scaleR__nonneg__nonpos,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A2: real,X: A] :
          ( pp(aa(real,bool,ord_less_eq(real,zero_zero(real)),A2))
         => ( pp(aa(A,bool,ord_less_eq(A,X),zero_zero(A)))
           => pp(aa(A,bool,ord_less_eq(A,aa(A,A,real_V8093663219630862766scaleR(A,A2),X)),zero_zero(A))) ) ) ) ).

% scaleR_nonneg_nonpos
tff(fact_4048_scaleR__nonneg__nonneg,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A2: real,X: A] :
          ( pp(aa(real,bool,ord_less_eq(real,zero_zero(real)),A2))
         => ( pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),X))
           => pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),aa(A,A,real_V8093663219630862766scaleR(A,A2),X))) ) ) ) ).

% scaleR_nonneg_nonneg
tff(fact_4049_split__scaleR__pos__le,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A2: real,B2: A] :
          ( ( ( pp(aa(real,bool,ord_less_eq(real,zero_zero(real)),A2))
              & pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),B2)) )
            | ( pp(aa(real,bool,ord_less_eq(real,A2),zero_zero(real)))
              & pp(aa(A,bool,ord_less_eq(A,B2),zero_zero(A))) ) )
         => pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),aa(A,A,real_V8093663219630862766scaleR(A,A2),B2))) ) ) ).

% split_scaleR_pos_le
tff(fact_4050_split__scaleR__neg__le,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A2: real,X: A] :
          ( ( ( pp(aa(real,bool,ord_less_eq(real,zero_zero(real)),A2))
              & pp(aa(A,bool,ord_less_eq(A,X),zero_zero(A))) )
            | ( pp(aa(real,bool,ord_less_eq(real,A2),zero_zero(real)))
              & pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),X)) ) )
         => pp(aa(A,bool,ord_less_eq(A,aa(A,A,real_V8093663219630862766scaleR(A,A2),X)),zero_zero(A))) ) ) ).

% split_scaleR_neg_le
tff(fact_4051_scaleR__mono_H,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A2: real,B2: real,C2: A,D: A] :
          ( pp(aa(real,bool,ord_less_eq(real,A2),B2))
         => ( pp(aa(A,bool,ord_less_eq(A,C2),D))
           => ( pp(aa(real,bool,ord_less_eq(real,zero_zero(real)),A2))
             => ( pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),C2))
               => pp(aa(A,bool,ord_less_eq(A,aa(A,A,real_V8093663219630862766scaleR(A,A2),C2)),aa(A,A,real_V8093663219630862766scaleR(A,B2),D))) ) ) ) ) ) ).

% scaleR_mono'
tff(fact_4052_scaleR__mono,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A2: real,B2: real,X: A,Y: A] :
          ( pp(aa(real,bool,ord_less_eq(real,A2),B2))
         => ( pp(aa(A,bool,ord_less_eq(A,X),Y))
           => ( pp(aa(real,bool,ord_less_eq(real,zero_zero(real)),B2))
             => ( pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),X))
               => pp(aa(A,bool,ord_less_eq(A,aa(A,A,real_V8093663219630862766scaleR(A,A2),X)),aa(A,A,real_V8093663219630862766scaleR(A,B2),Y))) ) ) ) ) ) ).

% scaleR_mono
tff(fact_4053_scaleR__2,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [X: A] : aa(A,A,real_V8093663219630862766scaleR(A,aa(num,real,numeral_numeral(real),bit0(one2))),X) = aa(A,A,aa(A,fun(A,A),plus_plus(A),X),X) ) ).

% scaleR_2
tff(fact_4054_scaleR__left__le__one__le,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [X: A,A2: real] :
          ( pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),X))
         => ( pp(aa(real,bool,ord_less_eq(real,A2),one_one(real)))
           => pp(aa(A,bool,ord_less_eq(A,aa(A,A,real_V8093663219630862766scaleR(A,A2),X)),X)) ) ) ) ).

% scaleR_left_le_one_le
tff(fact_4055_sum_Oreindex__bij__betw__not__neutral,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comm_monoid_add(A)
     => ! [S5: set(B),T4: set(C),H: fun(B,C),S: set(B),T3: set(C),G: fun(C,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),S5))
         => ( pp(aa(set(C),bool,finite_finite2(C),T4))
           => ( bij_betw(B,C,H,aa(set(B),set(B),minus_minus(set(B),S),S5),aa(set(C),set(C),minus_minus(set(C),T3),T4))
             => ( ! [A4: B] :
                    ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A4),S5))
                   => ( aa(C,A,G,aa(B,C,H,A4)) = zero_zero(A) ) )
               => ( ! [B3: C] :
                      ( pp(aa(set(C),bool,aa(C,fun(set(C),bool),member(C),B3),T4))
                     => ( aa(C,A,G,B3) = zero_zero(A) ) )
                 => ( aa(set(B),A,groups7311177749621191930dd_sum(B,A,aa(fun(C,A),fun(B,A),aTP_Lamp_kf(fun(B,C),fun(fun(C,A),fun(B,A)),H),G)),S) = aa(set(C),A,groups7311177749621191930dd_sum(C,A,G),T3) ) ) ) ) ) ) ) ).

% sum.reindex_bij_betw_not_neutral
tff(fact_4056_prod_Oreindex__bij__betw__not__neutral,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comm_monoid_mult(A)
     => ! [S5: set(B),T4: set(C),H: fun(B,C),S: set(B),T3: set(C),G: fun(C,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),S5))
         => ( pp(aa(set(C),bool,finite_finite2(C),T4))
           => ( bij_betw(B,C,H,aa(set(B),set(B),minus_minus(set(B),S),S5),aa(set(C),set(C),minus_minus(set(C),T3),T4))
             => ( ! [A4: B] :
                    ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A4),S5))
                   => ( aa(C,A,G,aa(B,C,H,A4)) = one_one(A) ) )
               => ( ! [B3: C] :
                      ( pp(aa(set(C),bool,aa(C,fun(set(C),bool),member(C),B3),T4))
                     => ( aa(C,A,G,B3) = one_one(A) ) )
                 => ( groups7121269368397514597t_prod(B,A,aa(fun(C,A),fun(B,A),aTP_Lamp_ke(fun(B,C),fun(fun(C,A),fun(B,A)),H),G),S) = groups7121269368397514597t_prod(C,A,G,T3) ) ) ) ) ) ) ) ).

% prod.reindex_bij_betw_not_neutral
tff(fact_4057_real__vector__affinity__eq,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [M: real,X: A,C2: A,Y: A] :
          ( ( M != zero_zero(real) )
         => ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,real_V8093663219630862766scaleR(A,M),X)),C2) = Y )
          <=> ( X = aa(A,A,minus_minus(A,aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),M)),Y)),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),M)),C2)) ) ) ) ) ).

% real_vector_affinity_eq
tff(fact_4058_real__vector__eq__affinity,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [M: real,Y: A,X: A,C2: A] :
          ( ( M != zero_zero(real) )
         => ( ( Y = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,real_V8093663219630862766scaleR(A,M),X)),C2) )
          <=> ( aa(A,A,minus_minus(A,aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),M)),Y)),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),M)),C2)) = X ) ) ) ) ).

% real_vector_eq_affinity
tff(fact_4059_pos__divideR__le__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,B2: A,A2: A] :
          ( pp(aa(real,bool,ord_less(real,zero_zero(real)),C2))
         => ( pp(aa(A,bool,ord_less_eq(A,aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),B2)),A2))
          <=> pp(aa(A,bool,ord_less_eq(A,B2),aa(A,A,real_V8093663219630862766scaleR(A,C2),A2))) ) ) ) ).

% pos_divideR_le_eq
tff(fact_4060_pos__le__divideR__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,A2: A,B2: A] :
          ( pp(aa(real,bool,ord_less(real,zero_zero(real)),C2))
         => ( pp(aa(A,bool,ord_less_eq(A,A2),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),B2)))
          <=> pp(aa(A,bool,ord_less_eq(A,aa(A,A,real_V8093663219630862766scaleR(A,C2),A2)),B2)) ) ) ) ).

% pos_le_divideR_eq
tff(fact_4061_neg__divideR__le__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,B2: A,A2: A] :
          ( pp(aa(real,bool,ord_less(real,C2),zero_zero(real)))
         => ( pp(aa(A,bool,ord_less_eq(A,aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),B2)),A2))
          <=> pp(aa(A,bool,ord_less_eq(A,aa(A,A,real_V8093663219630862766scaleR(A,C2),A2)),B2)) ) ) ) ).

% neg_divideR_le_eq
tff(fact_4062_neg__le__divideR__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,A2: A,B2: A] :
          ( pp(aa(real,bool,ord_less(real,C2),zero_zero(real)))
         => ( pp(aa(A,bool,ord_less_eq(A,A2),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),B2)))
          <=> pp(aa(A,bool,ord_less_eq(A,B2),aa(A,A,real_V8093663219630862766scaleR(A,C2),A2))) ) ) ) ).

% neg_le_divideR_eq
tff(fact_4063_neg__less__divideR__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,A2: A,B2: A] :
          ( pp(aa(real,bool,ord_less(real,C2),zero_zero(real)))
         => ( pp(aa(A,bool,ord_less(A,A2),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),B2)))
          <=> pp(aa(A,bool,ord_less(A,B2),aa(A,A,real_V8093663219630862766scaleR(A,C2),A2))) ) ) ) ).

% neg_less_divideR_eq
tff(fact_4064_neg__divideR__less__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,B2: A,A2: A] :
          ( pp(aa(real,bool,ord_less(real,C2),zero_zero(real)))
         => ( pp(aa(A,bool,ord_less(A,aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),B2)),A2))
          <=> pp(aa(A,bool,ord_less(A,aa(A,A,real_V8093663219630862766scaleR(A,C2),A2)),B2)) ) ) ) ).

% neg_divideR_less_eq
tff(fact_4065_pos__less__divideR__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,A2: A,B2: A] :
          ( pp(aa(real,bool,ord_less(real,zero_zero(real)),C2))
         => ( pp(aa(A,bool,ord_less(A,A2),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),B2)))
          <=> pp(aa(A,bool,ord_less(A,aa(A,A,real_V8093663219630862766scaleR(A,C2),A2)),B2)) ) ) ) ).

% pos_less_divideR_eq
tff(fact_4066_pos__divideR__less__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,B2: A,A2: A] :
          ( pp(aa(real,bool,ord_less(real,zero_zero(real)),C2))
         => ( pp(aa(A,bool,ord_less(A,aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),B2)),A2))
          <=> pp(aa(A,bool,ord_less(A,B2),aa(A,A,real_V8093663219630862766scaleR(A,C2),A2))) ) ) ) ).

% pos_divideR_less_eq
tff(fact_4067_nonzero__inverse__scaleR__distrib,axiom,
    ! [A: $tType] :
      ( real_V5047593784448816457lgebra(A)
     => ! [A2: real,X: A] :
          ( ( A2 != zero_zero(real) )
         => ( ( X != zero_zero(A) )
           => ( aa(A,A,inverse_inverse(A),aa(A,A,real_V8093663219630862766scaleR(A,A2),X)) = aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),A2)),aa(A,A,inverse_inverse(A),X)) ) ) ) ) ).

% nonzero_inverse_scaleR_distrib
tff(fact_4068_summable__exp__generic,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [X: A] : summable(A,aTP_Lamp_kk(A,fun(nat,A),X)) ) ).

% summable_exp_generic
tff(fact_4069_sin__converges,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [X: A] : pp(aa(A,bool,sums(A,aTP_Lamp_kl(A,fun(nat,A),X)),sin(A,X))) ) ).

% sin_converges
tff(fact_4070_sin__def,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [X4: A] : sin(A,X4) = suminf(A,aTP_Lamp_kl(A,fun(nat,A),X4)) ) ).

% sin_def
tff(fact_4071_cos__converges,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [X: A] : pp(aa(A,bool,sums(A,aTP_Lamp_km(A,fun(nat,A),X)),cos(A,X))) ) ).

% cos_converges
tff(fact_4072_cos__def,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [X4: A] : cos(A,X4) = suminf(A,aTP_Lamp_km(A,fun(nat,A),X4)) ) ).

% cos_def
tff(fact_4073_summable__norm__sin,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [X: A] : summable(real,aTP_Lamp_kn(A,fun(nat,real),X)) ) ).

% summable_norm_sin
tff(fact_4074_summable__norm__cos,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [X: A] : summable(real,aTP_Lamp_ko(A,fun(nat,real),X)) ) ).

% summable_norm_cos
tff(fact_4075_pos__le__minus__divideR__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,A2: A,B2: A] :
          ( pp(aa(real,bool,ord_less(real,zero_zero(real)),C2))
         => ( pp(aa(A,bool,ord_less_eq(A,A2),aa(A,A,uminus_uminus(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),B2))))
          <=> pp(aa(A,bool,ord_less_eq(A,aa(A,A,real_V8093663219630862766scaleR(A,C2),A2)),aa(A,A,uminus_uminus(A),B2))) ) ) ) ).

% pos_le_minus_divideR_eq
tff(fact_4076_pos__minus__divideR__le__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,B2: A,A2: A] :
          ( pp(aa(real,bool,ord_less(real,zero_zero(real)),C2))
         => ( pp(aa(A,bool,ord_less_eq(A,aa(A,A,uminus_uminus(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),B2))),A2))
          <=> pp(aa(A,bool,ord_less_eq(A,aa(A,A,uminus_uminus(A),B2)),aa(A,A,real_V8093663219630862766scaleR(A,C2),A2))) ) ) ) ).

% pos_minus_divideR_le_eq
tff(fact_4077_neg__le__minus__divideR__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,A2: A,B2: A] :
          ( pp(aa(real,bool,ord_less(real,C2),zero_zero(real)))
         => ( pp(aa(A,bool,ord_less_eq(A,A2),aa(A,A,uminus_uminus(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),B2))))
          <=> pp(aa(A,bool,ord_less_eq(A,aa(A,A,uminus_uminus(A),B2)),aa(A,A,real_V8093663219630862766scaleR(A,C2),A2))) ) ) ) ).

% neg_le_minus_divideR_eq
tff(fact_4078_neg__minus__divideR__le__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,B2: A,A2: A] :
          ( pp(aa(real,bool,ord_less(real,C2),zero_zero(real)))
         => ( pp(aa(A,bool,ord_less_eq(A,aa(A,A,uminus_uminus(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),B2))),A2))
          <=> pp(aa(A,bool,ord_less_eq(A,aa(A,A,real_V8093663219630862766scaleR(A,C2),A2)),aa(A,A,uminus_uminus(A),B2))) ) ) ) ).

% neg_minus_divideR_le_eq
tff(fact_4079_pos__less__minus__divideR__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,A2: A,B2: A] :
          ( pp(aa(real,bool,ord_less(real,zero_zero(real)),C2))
         => ( pp(aa(A,bool,ord_less(A,A2),aa(A,A,uminus_uminus(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),B2))))
          <=> pp(aa(A,bool,ord_less(A,aa(A,A,real_V8093663219630862766scaleR(A,C2),A2)),aa(A,A,uminus_uminus(A),B2))) ) ) ) ).

% pos_less_minus_divideR_eq
tff(fact_4080_pos__minus__divideR__less__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,B2: A,A2: A] :
          ( pp(aa(real,bool,ord_less(real,zero_zero(real)),C2))
         => ( pp(aa(A,bool,ord_less(A,aa(A,A,uminus_uminus(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),B2))),A2))
          <=> pp(aa(A,bool,ord_less(A,aa(A,A,uminus_uminus(A),B2)),aa(A,A,real_V8093663219630862766scaleR(A,C2),A2))) ) ) ) ).

% pos_minus_divideR_less_eq
tff(fact_4081_neg__less__minus__divideR__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,A2: A,B2: A] :
          ( pp(aa(real,bool,ord_less(real,C2),zero_zero(real)))
         => ( pp(aa(A,bool,ord_less(A,A2),aa(A,A,uminus_uminus(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),B2))))
          <=> pp(aa(A,bool,ord_less(A,aa(A,A,uminus_uminus(A),B2)),aa(A,A,real_V8093663219630862766scaleR(A,C2),A2))) ) ) ) ).

% neg_less_minus_divideR_eq
tff(fact_4082_neg__minus__divideR__less__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,B2: A,A2: A] :
          ( pp(aa(real,bool,ord_less(real,C2),zero_zero(real)))
         => ( pp(aa(A,bool,ord_less(A,aa(A,A,uminus_uminus(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),B2))),A2))
          <=> pp(aa(A,bool,ord_less(A,aa(A,A,real_V8093663219630862766scaleR(A,C2),A2)),aa(A,A,uminus_uminus(A),B2))) ) ) ) ).

% neg_minus_divideR_less_eq
tff(fact_4083_exp__converges,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [X: A] : pp(aa(A,bool,sums(A,aTP_Lamp_kk(A,fun(nat,A),X)),aa(A,A,exp(A),X))) ) ).

% exp_converges
tff(fact_4084_exp__def,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [X4: A] : aa(A,A,exp(A),X4) = suminf(A,aTP_Lamp_kk(A,fun(nat,A),X4)) ) ).

% exp_def
tff(fact_4085_summable__norm__exp,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [X: A] : summable(real,aTP_Lamp_kp(A,fun(nat,real),X)) ) ).

% summable_norm_exp
tff(fact_4086_sin__minus__converges,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [X: A] : pp(aa(A,bool,sums(A,aTP_Lamp_kq(A,fun(nat,A),X)),sin(A,X))) ) ).

% sin_minus_converges
tff(fact_4087_cos__minus__converges,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [X: A] : pp(aa(A,bool,sums(A,aTP_Lamp_kr(A,fun(nat,A),X)),cos(A,X))) ) ).

% cos_minus_converges
tff(fact_4088_exp__series__add__commuting,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [X: A,Y: A,N: nat] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),X),Y) = aa(A,A,aa(A,fun(A,A),times_times(A),Y),X) )
         => ( aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),semiring_char_0_fact(real,N))),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y)),N)) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_ks(A,fun(A,fun(nat,fun(nat,A))),X),Y),N)),set_ord_atMost(nat,N)) ) ) ) ).

% exp_series_add_commuting
tff(fact_4089_exp__first__term,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [X: A] : aa(A,A,exp(A),X) = aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),suminf(A,aTP_Lamp_kt(A,fun(nat,A),X))) ) ).

% exp_first_term
tff(fact_4090_exp__first__terms,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [X: A,K: nat] : aa(A,A,exp(A),X) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_kk(A,fun(nat,A),X)),set_ord_lessThan(nat,K))),suminf(A,aa(nat,fun(nat,A),aTP_Lamp_ku(A,fun(nat,fun(nat,A)),X),K))) ) ).

% exp_first_terms
tff(fact_4091_exp__first__two__terms,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [X: A] : aa(A,A,exp(A),X) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),X)),suminf(A,aTP_Lamp_kv(A,fun(nat,A),X))) ) ).

% exp_first_two_terms
tff(fact_4092_sinh__converges,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [X: A] : pp(aa(A,bool,sums(A,aTP_Lamp_kw(A,fun(nat,A),X)),sinh(A,X))) ) ).

% sinh_converges
tff(fact_4093_round__altdef,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( ( pp(aa(A,bool,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),bit0(one2)))),archimedean_frac(A,X)))
           => ( archimedean_round(A,X) = archimedean_ceiling(A,X) ) )
          & ( ~ pp(aa(A,bool,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),bit0(one2)))),archimedean_frac(A,X)))
           => ( archimedean_round(A,X) = archim6421214686448440834_floor(A,X) ) ) ) ) ).

% round_altdef
tff(fact_4094_cosh__converges,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [X: A] : pp(aa(A,bool,sums(A,aTP_Lamp_kx(A,fun(nat,A),X)),cosh(A,X))) ) ).

% cosh_converges
tff(fact_4095_sinh__real__zero__iff,axiom,
    ! [X: real] :
      ( ( sinh(real,X) = zero_zero(real) )
    <=> ( X = zero_zero(real) ) ) ).

% sinh_real_zero_iff
tff(fact_4096_sinh__real__le__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa(real,bool,ord_less_eq(real,sinh(real,X)),sinh(real,Y)))
    <=> pp(aa(real,bool,ord_less_eq(real,X),Y)) ) ).

% sinh_real_le_iff
tff(fact_4097_sinh__real__pos__iff,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,ord_less(real,zero_zero(real)),sinh(real,X)))
    <=> pp(aa(real,bool,ord_less(real,zero_zero(real)),X)) ) ).

% sinh_real_pos_iff
tff(fact_4098_sinh__real__neg__iff,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,ord_less(real,sinh(real,X)),zero_zero(real)))
    <=> pp(aa(real,bool,ord_less(real,X),zero_zero(real))) ) ).

% sinh_real_neg_iff
tff(fact_4099_sinh__real__nonpos__iff,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,ord_less_eq(real,sinh(real,X)),zero_zero(real)))
    <=> pp(aa(real,bool,ord_less_eq(real,X),zero_zero(real))) ) ).

% sinh_real_nonpos_iff
tff(fact_4100_sinh__real__nonneg__iff,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,ord_less_eq(real,zero_zero(real)),sinh(real,X)))
    <=> pp(aa(real,bool,ord_less_eq(real,zero_zero(real)),X)) ) ).

% sinh_real_nonneg_iff
tff(fact_4101_sinh__0,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ( sinh(A,zero_zero(A)) = zero_zero(A) ) ) ).

% sinh_0
tff(fact_4102_cosh__0,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ( cosh(A,zero_zero(A)) = one_one(A) ) ) ).

% cosh_0
tff(fact_4103_frac__of__int,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Z: int] : archimedean_frac(A,aa(int,A,ring_1_of_int(A),Z)) = zero_zero(A) ) ).

% frac_of_int
tff(fact_4104_tanh__def,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] : aa(A,A,tanh(A),X) = aa(A,A,aa(A,fun(A,A),divide_divide(A),sinh(A,X)),cosh(A,X)) ) ).

% tanh_def
tff(fact_4105_cosh__plus__sinh,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),cosh(A,X)),sinh(A,X)) = aa(A,A,exp(A),X) ) ).

% cosh_plus_sinh
tff(fact_4106_sinh__plus__cosh,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),sinh(A,X)),cosh(A,X)) = aa(A,A,exp(A),X) ) ).

% sinh_plus_cosh
tff(fact_4107_cosh__real__nonzero,axiom,
    ! [X: real] : cosh(real,X) != zero_zero(real) ).

% cosh_real_nonzero
tff(fact_4108_sinh__diff,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A,Y: A] : sinh(A,aa(A,A,minus_minus(A,X),Y)) = aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),sinh(A,X)),cosh(A,Y))),aa(A,A,aa(A,fun(A,A),times_times(A),cosh(A,X)),sinh(A,Y))) ) ).

% sinh_diff
tff(fact_4109_cosh__diff,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A,Y: A] : cosh(A,aa(A,A,minus_minus(A,X),Y)) = aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),cosh(A,X)),cosh(A,Y))),aa(A,A,aa(A,fun(A,A),times_times(A),sinh(A,X)),sinh(A,Y))) ) ).

% cosh_diff
tff(fact_4110_sinh__add,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A,Y: A] : sinh(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),sinh(A,X)),cosh(A,Y))),aa(A,A,aa(A,fun(A,A),times_times(A),cosh(A,X)),sinh(A,Y))) ) ).

% sinh_add
tff(fact_4111_cosh__add,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A,Y: A] : cosh(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),cosh(A,X)),cosh(A,Y))),aa(A,A,aa(A,fun(A,A),times_times(A),sinh(A,X)),sinh(A,Y))) ) ).

% cosh_add
tff(fact_4112_sinh__le__cosh__real,axiom,
    ! [X: real] : pp(aa(real,bool,ord_less_eq(real,sinh(real,X)),cosh(real,X))) ).

% sinh_le_cosh_real
tff(fact_4113_cosh__real__pos,axiom,
    ! [X: real] : pp(aa(real,bool,ord_less(real,zero_zero(real)),cosh(real,X))) ).

% cosh_real_pos
tff(fact_4114_cosh__real__nonpos__le__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa(real,bool,ord_less_eq(real,X),zero_zero(real)))
     => ( pp(aa(real,bool,ord_less_eq(real,Y),zero_zero(real)))
       => ( pp(aa(real,bool,ord_less_eq(real,cosh(real,X)),cosh(real,Y)))
        <=> pp(aa(real,bool,ord_less_eq(real,Y),X)) ) ) ) ).

% cosh_real_nonpos_le_iff
tff(fact_4115_cosh__real__nonneg__le__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa(real,bool,ord_less_eq(real,zero_zero(real)),X))
     => ( pp(aa(real,bool,ord_less_eq(real,zero_zero(real)),Y))
       => ( pp(aa(real,bool,ord_less_eq(real,cosh(real,X)),cosh(real,Y)))
        <=> pp(aa(real,bool,ord_less_eq(real,X),Y)) ) ) ) ).

% cosh_real_nonneg_le_iff
tff(fact_4116_cosh__real__nonneg,axiom,
    ! [X: real] : pp(aa(real,bool,ord_less_eq(real,zero_zero(real)),cosh(real,X))) ).

% cosh_real_nonneg
tff(fact_4117_sinh__double,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] : sinh(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),X)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),sinh(A,X))),cosh(A,X)) ) ).

% sinh_double
tff(fact_4118_cosh__real__ge__1,axiom,
    ! [X: real] : pp(aa(real,bool,ord_less_eq(real,one_one(real)),cosh(real,X))) ).

% cosh_real_ge_1
tff(fact_4119_frac__ge__0,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] : pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),archimedean_frac(A,X))) ) ).

% frac_ge_0
tff(fact_4120_frac__lt__1,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] : pp(aa(A,bool,ord_less(A,archimedean_frac(A,X)),one_one(A))) ) ).

% frac_lt_1
tff(fact_4121_frac__1__eq,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] : archimedean_frac(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),one_one(A))) = archimedean_frac(A,X) ) ).

% frac_1_eq
tff(fact_4122_cosh__real__strict__mono,axiom,
    ! [X: real,Y: real] :
      ( pp(aa(real,bool,ord_less_eq(real,zero_zero(real)),X))
     => ( pp(aa(real,bool,ord_less(real,X),Y))
       => pp(aa(real,bool,ord_less(real,cosh(real,X)),cosh(real,Y))) ) ) ).

% cosh_real_strict_mono
tff(fact_4123_cosh__real__nonneg__less__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa(real,bool,ord_less_eq(real,zero_zero(real)),X))
     => ( pp(aa(real,bool,ord_less_eq(real,zero_zero(real)),Y))
       => ( pp(aa(real,bool,ord_less(real,cosh(real,X)),cosh(real,Y)))
        <=> pp(aa(real,bool,ord_less(real,X),Y)) ) ) ) ).

% cosh_real_nonneg_less_iff
tff(fact_4124_cosh__real__nonpos__less__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa(real,bool,ord_less_eq(real,X),zero_zero(real)))
     => ( pp(aa(real,bool,ord_less_eq(real,Y),zero_zero(real)))
       => ( pp(aa(real,bool,ord_less(real,cosh(real,X)),cosh(real,Y)))
        <=> pp(aa(real,bool,ord_less(real,Y),X)) ) ) ) ).

% cosh_real_nonpos_less_iff
tff(fact_4125_cosh__square__eq,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] : aa(nat,A,aa(A,fun(nat,A),power_power(A),cosh(A,X)),aa(num,nat,numeral_numeral(nat),bit0(one2))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),sinh(A,X)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),one_one(A)) ) ).

% cosh_square_eq
tff(fact_4126_sinh__square__eq,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] : aa(nat,A,aa(A,fun(nat,A),power_power(A),sinh(A,X)),aa(num,nat,numeral_numeral(nat),bit0(one2))) = aa(A,A,minus_minus(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),cosh(A,X)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),one_one(A)) ) ).

% sinh_square_eq
tff(fact_4127_hyperbolic__pythagoras,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] : aa(A,A,minus_minus(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),cosh(A,X)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),sinh(A,X)),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = one_one(A) ) ).

% hyperbolic_pythagoras
tff(fact_4128_arcosh__cosh__real,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,ord_less_eq(real,zero_zero(real)),X))
     => ( aa(real,real,arcosh(real),cosh(real,X)) = X ) ) ).

% arcosh_cosh_real
tff(fact_4129_cosh__double,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] : cosh(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),X)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),cosh(A,X)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),sinh(A,X)),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ) ).

% cosh_double
tff(fact_4130_frac__eq,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( ( archimedean_frac(A,X) = X )
        <=> ( pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),X))
            & pp(aa(A,bool,ord_less(A,X),one_one(A))) ) ) ) ).

% frac_eq
tff(fact_4131_frac__add,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,Y: A] :
          ( ( pp(aa(A,bool,ord_less(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),archimedean_frac(A,X)),archimedean_frac(A,Y))),one_one(A)))
           => ( archimedean_frac(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),archimedean_frac(A,X)),archimedean_frac(A,Y)) ) )
          & ( ~ pp(aa(A,bool,ord_less(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),archimedean_frac(A,X)),archimedean_frac(A,Y))),one_one(A)))
           => ( archimedean_frac(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y)) = aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),archimedean_frac(A,X)),archimedean_frac(A,Y))),one_one(A)) ) ) ) ) ).

% frac_add
tff(fact_4132_tanh__add,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A,Y: A] :
          ( ( cosh(A,X) != zero_zero(A) )
         => ( ( cosh(A,Y) != zero_zero(A) )
           => ( aa(A,A,tanh(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,tanh(A),X)),aa(A,A,tanh(A),Y))),aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,tanh(A),X)),aa(A,A,tanh(A),Y)))) ) ) ) ) ).

% tanh_add
tff(fact_4133_cosh__field__def,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Z: A] : cosh(A,Z) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,exp(A),Z)),aa(A,A,exp(A),aa(A,A,uminus_uminus(A),Z)))),aa(num,A,numeral_numeral(A),bit0(one2))) ) ).

% cosh_field_def
tff(fact_4134_floor__add,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,Y: A] :
          ( ( pp(aa(A,bool,ord_less(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),archimedean_frac(A,X)),archimedean_frac(A,Y))),one_one(A)))
           => ( archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),archim6421214686448440834_floor(A,X)),archim6421214686448440834_floor(A,Y)) ) )
          & ( ~ pp(aa(A,bool,ord_less(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),archimedean_frac(A,X)),archimedean_frac(A,Y))),one_one(A)))
           => ( archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),archim6421214686448440834_floor(A,X)),archim6421214686448440834_floor(A,Y))),one_one(int)) ) ) ) ) ).

% floor_add
tff(fact_4135_sinh__field__def,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Z: A] : sinh(A,Z) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,minus_minus(A,aa(A,A,exp(A),Z)),aa(A,A,exp(A),aa(A,A,uminus_uminus(A),Z)))),aa(num,A,numeral_numeral(A),bit0(one2))) ) ).

% sinh_field_def
tff(fact_4136_cosh__zero__iff,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] :
          ( ( cosh(A,X) = zero_zero(A) )
        <=> ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,exp(A),X)),aa(num,nat,numeral_numeral(nat),bit0(one2))) = aa(A,A,uminus_uminus(A),one_one(A)) ) ) ) ).

% cosh_zero_iff
tff(fact_4137_cosh__def,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [X: A] : cosh(A,X) = aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),aa(num,real,numeral_numeral(real),bit0(one2)))),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,exp(A),X)),aa(A,A,exp(A),aa(A,A,uminus_uminus(A),X)))) ) ).

% cosh_def
tff(fact_4138_cosh__ln__real,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,ord_less(real,zero_zero(real)),X))
     => ( cosh(real,aa(real,real,ln_ln(real),X)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),X),aa(real,real,inverse_inverse(real),X))),aa(num,real,numeral_numeral(real),bit0(one2))) ) ) ).

% cosh_ln_real
tff(fact_4139_sinh__def,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [X: A] : sinh(A,X) = aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),aa(num,real,numeral_numeral(real),bit0(one2)))),aa(A,A,minus_minus(A,aa(A,A,exp(A),X)),aa(A,A,exp(A),aa(A,A,uminus_uminus(A),X)))) ) ).

% sinh_def
tff(fact_4140_sinh__ln__real,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,ord_less(real,zero_zero(real)),X))
     => ( sinh(real,aa(real,real,ln_ln(real),X)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,minus_minus(real,X),aa(real,real,inverse_inverse(real),X))),aa(num,real,numeral_numeral(real),bit0(one2))) ) ) ).

% sinh_ln_real
tff(fact_4141_bij__betw__nth__root__unity,axiom,
    ! [C2: complex,N: nat] :
      ( ( C2 != zero_zero(complex) )
     => ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),N))
       => bij_betw(complex,complex,aa(complex,fun(complex,complex),times_times(complex),aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),aa(real,complex,real_Vector_of_real(complex),aa(real,real,root(N),real_V7770717601297561774m_norm(complex,C2)))),cis(aa(real,real,aa(real,fun(real,real),divide_divide(real),arg(C2)),aa(nat,real,semiring_1_of_nat(real),N))))),collect(complex,aTP_Lamp_fe(nat,fun(complex,bool),N)),collect(complex,aa(nat,fun(complex,bool),aTP_Lamp_ky(complex,fun(nat,fun(complex,bool)),C2),N))) ) ) ).

% bij_betw_nth_root_unity
tff(fact_4142_cot__less__zero,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,ord_less(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,uminus_uminus(real),pi)),aa(num,real,numeral_numeral(real),bit0(one2)))),X))
     => ( pp(aa(real,bool,ord_less(real,X),zero_zero(real)))
       => pp(aa(real,bool,ord_less(real,aa(real,real,cot(real),X)),zero_zero(real))) ) ) ).

% cot_less_zero
tff(fact_4143_real__root__zero,axiom,
    ! [N: nat] : aa(real,real,root(N),zero_zero(real)) = zero_zero(real) ).

% real_root_zero
tff(fact_4144_cot__zero,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ( aa(A,A,cot(A),zero_zero(A)) = zero_zero(A) ) ) ).

% cot_zero
tff(fact_4145_real__root__Suc__0,axiom,
    ! [X: real] : aa(real,real,root(aa(nat,nat,suc,zero_zero(nat))),X) = X ).

% real_root_Suc_0
tff(fact_4146_real__root__eq__iff,axiom,
    ! [N: nat,X: real,Y: real] :
      ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),N))
     => ( ( aa(real,real,root(N),X) = aa(real,real,root(N),Y) )
      <=> ( X = Y ) ) ) ).

% real_root_eq_iff
tff(fact_4147_root__0,axiom,
    ! [X: real] : aa(real,real,root(zero_zero(nat)),X) = zero_zero(real) ).

% root_0
tff(fact_4148_cot__pi,axiom,
    aa(real,real,cot(real),pi) = zero_zero(real) ).

% cot_pi
tff(fact_4149_real__root__eq__0__iff,axiom,
    ! [N: nat,X: real] :
      ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),N))
     => ( ( aa(real,real,root(N),X) = zero_zero(real) )
      <=> ( X = zero_zero(real) ) ) ) ).

% real_root_eq_0_iff
tff(fact_4150_real__root__less__iff,axiom,
    ! [N: nat,X: real,Y: real] :
      ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),N))
     => ( pp(aa(real,bool,ord_less(real,aa(real,real,root(N),X)),aa(real,real,root(N),Y)))
      <=> pp(aa(real,bool,ord_less(real,X),Y)) ) ) ).

% real_root_less_iff
tff(fact_4151_real__root__le__iff,axiom,
    ! [N: nat,X: real,Y: real] :
      ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),N))
     => ( pp(aa(real,bool,ord_less_eq(real,aa(real,real,root(N),X)),aa(real,real,root(N),Y)))
      <=> pp(aa(real,bool,ord_less_eq(real,X),Y)) ) ) ).

% real_root_le_iff
tff(fact_4152_real__root__eq__1__iff,axiom,
    ! [N: nat,X: real] :
      ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),N))
     => ( ( aa(real,real,root(N),X) = one_one(real) )
      <=> ( X = one_one(real) ) ) ) ).

% real_root_eq_1_iff
tff(fact_4153_real__root__one,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),N))
     => ( aa(real,real,root(N),one_one(real)) = one_one(real) ) ) ).

% real_root_one
tff(fact_4154_real__root__gt__0__iff,axiom,
    ! [N: nat,Y: real] :
      ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),N))
     => ( pp(aa(real,bool,ord_less(real,zero_zero(real)),aa(real,real,root(N),Y)))
      <=> pp(aa(real,bool,ord_less(real,zero_zero(real)),Y)) ) ) ).

% real_root_gt_0_iff
tff(fact_4155_real__root__lt__0__iff,axiom,
    ! [N: nat,X: real] :
      ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),N))
     => ( pp(aa(real,bool,ord_less(real,aa(real,real,root(N),X)),zero_zero(real)))
      <=> pp(aa(real,bool,ord_less(real,X),zero_zero(real))) ) ) ).

% real_root_lt_0_iff
tff(fact_4156_real__root__le__0__iff,axiom,
    ! [N: nat,X: real] :
      ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),N))
     => ( pp(aa(real,bool,ord_less_eq(real,aa(real,real,root(N),X)),zero_zero(real)))
      <=> pp(aa(real,bool,ord_less_eq(real,X),zero_zero(real))) ) ) ).

% real_root_le_0_iff
tff(fact_4157_real__root__ge__0__iff,axiom,
    ! [N: nat,Y: real] :
      ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),N))
     => ( pp(aa(real,bool,ord_less_eq(real,zero_zero(real)),aa(real,real,root(N),Y)))
      <=> pp(aa(real,bool,ord_less_eq(real,zero_zero(real)),Y)) ) ) ).

% real_root_ge_0_iff
tff(fact_4158_real__root__lt__1__iff,axiom,
    ! [N: nat,X: real] :
      ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),N))
     => ( pp(aa(real,bool,ord_less(real,aa(real,real,root(N),X)),one_one(real)))
      <=> pp(aa(real,bool,ord_less(real,X),one_one(real))) ) ) ).

% real_root_lt_1_iff
tff(fact_4159_real__root__gt__1__iff,axiom,
    ! [N: nat,Y: real] :
      ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),N))
     => ( pp(aa(real,bool,ord_less(real,one_one(real)),aa(real,real,root(N),Y)))
      <=> pp(aa(real,bool,ord_less(real,one_one(real)),Y)) ) ) ).

% real_root_gt_1_iff
tff(fact_4160_real__root__le__1__iff,axiom,
    ! [N: nat,X: real] :
      ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),N))
     => ( pp(aa(real,bool,ord_less_eq(real,aa(real,real,root(N),X)),one_one(real)))
      <=> pp(aa(real,bool,ord_less_eq(real,X),one_one(real))) ) ) ).

% real_root_le_1_iff
tff(fact_4161_real__root__ge__1__iff,axiom,
    ! [N: nat,Y: real] :
      ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),N))
     => ( pp(aa(real,bool,ord_less_eq(real,one_one(real)),aa(real,real,root(N),Y)))
      <=> pp(aa(real,bool,ord_less_eq(real,one_one(real)),Y)) ) ) ).

% real_root_ge_1_iff
tff(fact_4162_cot__npi,axiom,
    ! [N: nat] : aa(real,real,cot(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),N)),pi)) = zero_zero(real) ).

% cot_npi
tff(fact_4163_real__root__pow__pos2,axiom,
    ! [N: nat,X: real] :
      ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),N))
     => ( pp(aa(real,bool,ord_less_eq(real,zero_zero(real)),X))
       => ( aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,root(N),X)),N) = X ) ) ) ).

% real_root_pow_pos2
tff(fact_4164_cot__periodic,axiom,
    ! [X: real] : aa(real,real,cot(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),X),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),pi))) = aa(real,real,cot(real),X) ).

% cot_periodic
tff(fact_4165_real__root__commute,axiom,
    ! [M: nat,N: nat,X: real] : aa(real,real,root(M),aa(real,real,root(N),X)) = aa(real,real,root(N),aa(real,real,root(M),X)) ).

% real_root_commute
tff(fact_4166_real__root__mult,axiom,
    ! [N: nat,X: real,Y: real] : aa(real,real,root(N),aa(real,real,aa(real,fun(real,real),times_times(real),X),Y)) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,root(N),X)),aa(real,real,root(N),Y)) ).

% real_root_mult
tff(fact_4167_real__root__mult__exp,axiom,
    ! [M: nat,N: nat,X: real] : aa(real,real,root(aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N)),X) = aa(real,real,root(M),aa(real,real,root(N),X)) ).

% real_root_mult_exp
tff(fact_4168_real__root__minus,axiom,
    ! [N: nat,X: real] : aa(real,real,root(N),aa(real,real,uminus_uminus(real),X)) = aa(real,real,uminus_uminus(real),aa(real,real,root(N),X)) ).

% real_root_minus
tff(fact_4169_real__root__divide,axiom,
    ! [N: nat,X: real,Y: real] : aa(real,real,root(N),aa(real,real,aa(real,fun(real,real),divide_divide(real),X),Y)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,root(N),X)),aa(real,real,root(N),Y)) ).

% real_root_divide
tff(fact_4170_real__root__inverse,axiom,
    ! [N: nat,X: real] : aa(real,real,root(N),aa(real,real,inverse_inverse(real),X)) = aa(real,real,inverse_inverse(real),aa(real,real,root(N),X)) ).

% real_root_inverse
tff(fact_4171_real__root__pos__pos__le,axiom,
    ! [X: real,N: nat] :
      ( pp(aa(real,bool,ord_less_eq(real,zero_zero(real)),X))
     => pp(aa(real,bool,ord_less_eq(real,zero_zero(real)),aa(real,real,root(N),X))) ) ).

% real_root_pos_pos_le
tff(fact_4172_real__root__less__mono,axiom,
    ! [N: nat,X: real,Y: real] :
      ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),N))
     => ( pp(aa(real,bool,ord_less(real,X),Y))
       => pp(aa(real,bool,ord_less(real,aa(real,real,root(N),X)),aa(real,real,root(N),Y))) ) ) ).

% real_root_less_mono
tff(fact_4173_real__root__le__mono,axiom,
    ! [N: nat,X: real,Y: real] :
      ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),N))
     => ( pp(aa(real,bool,ord_less_eq(real,X),Y))
       => pp(aa(real,bool,ord_less_eq(real,aa(real,real,root(N),X)),aa(real,real,root(N),Y))) ) ) ).

% real_root_le_mono
tff(fact_4174_real__root__power,axiom,
    ! [N: nat,X: real,K: nat] :
      ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),N))
     => ( aa(real,real,root(N),aa(nat,real,aa(real,fun(nat,real),power_power(real),X),K)) = aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,root(N),X)),K) ) ) ).

% real_root_power
tff(fact_4175_real__root__abs,axiom,
    ! [N: nat,X: real] :
      ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),N))
     => ( aa(real,real,root(N),aa(real,real,abs_abs(real),X)) = aa(real,real,abs_abs(real),aa(real,real,root(N),X)) ) ) ).

% real_root_abs
tff(fact_4176_real__root__gt__zero,axiom,
    ! [N: nat,X: real] :
      ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),N))
     => ( pp(aa(real,bool,ord_less(real,zero_zero(real)),X))
       => pp(aa(real,bool,ord_less(real,zero_zero(real)),aa(real,real,root(N),X))) ) ) ).

% real_root_gt_zero
tff(fact_4177_real__root__strict__decreasing,axiom,
    ! [N: nat,N4: nat,X: real] :
      ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),N))
     => ( pp(aa(nat,bool,ord_less(nat,N),N4))
       => ( pp(aa(real,bool,ord_less(real,one_one(real)),X))
         => pp(aa(real,bool,ord_less(real,aa(real,real,root(N4),X)),aa(real,real,root(N),X))) ) ) ) ).

% real_root_strict_decreasing
tff(fact_4178_sqrt__def,axiom,
    sqrt = root(aa(num,nat,numeral_numeral(nat),bit0(one2))) ).

% sqrt_def
tff(fact_4179_root__abs__power,axiom,
    ! [N: nat,Y: real] :
      ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),N))
     => ( aa(real,real,abs_abs(real),aa(real,real,root(N),aa(nat,real,aa(real,fun(nat,real),power_power(real),Y),N))) = aa(real,real,abs_abs(real),Y) ) ) ).

% root_abs_power
tff(fact_4180_real__root__pos__pos,axiom,
    ! [N: nat,X: real] :
      ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),N))
     => ( pp(aa(real,bool,ord_less(real,zero_zero(real)),X))
       => pp(aa(real,bool,ord_less_eq(real,zero_zero(real)),aa(real,real,root(N),X))) ) ) ).

% real_root_pos_pos
tff(fact_4181_real__root__strict__increasing,axiom,
    ! [N: nat,N4: nat,X: real] :
      ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),N))
     => ( pp(aa(nat,bool,ord_less(nat,N),N4))
       => ( pp(aa(real,bool,ord_less(real,zero_zero(real)),X))
         => ( pp(aa(real,bool,ord_less(real,X),one_one(real)))
           => pp(aa(real,bool,ord_less(real,aa(real,real,root(N),X)),aa(real,real,root(N4),X))) ) ) ) ) ).

% real_root_strict_increasing
tff(fact_4182_real__root__decreasing,axiom,
    ! [N: nat,N4: nat,X: real] :
      ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),N))
     => ( pp(aa(nat,bool,ord_less_eq(nat,N),N4))
       => ( pp(aa(real,bool,ord_less_eq(real,one_one(real)),X))
         => pp(aa(real,bool,ord_less_eq(real,aa(real,real,root(N4),X)),aa(real,real,root(N),X))) ) ) ) ).

% real_root_decreasing
tff(fact_4183_odd__real__root__power__cancel,axiom,
    ! [N: nat,X: real] :
      ( ~ pp(aa(nat,bool,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N))
     => ( aa(real,real,root(N),aa(nat,real,aa(real,fun(nat,real),power_power(real),X),N)) = X ) ) ).

% odd_real_root_power_cancel
tff(fact_4184_odd__real__root__unique,axiom,
    ! [N: nat,Y: real,X: real] :
      ( ~ pp(aa(nat,bool,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N))
     => ( ( aa(nat,real,aa(real,fun(nat,real),power_power(real),Y),N) = X )
       => ( aa(real,real,root(N),X) = Y ) ) ) ).

% odd_real_root_unique
tff(fact_4185_odd__real__root__pow,axiom,
    ! [N: nat,X: real] :
      ( ~ pp(aa(nat,bool,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N))
     => ( aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,root(N),X)),N) = X ) ) ).

% odd_real_root_pow
tff(fact_4186_real__root__pow__pos,axiom,
    ! [N: nat,X: real] :
      ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),N))
     => ( pp(aa(real,bool,ord_less(real,zero_zero(real)),X))
       => ( aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,root(N),X)),N) = X ) ) ) ).

% real_root_pow_pos
tff(fact_4187_real__root__pos__unique,axiom,
    ! [N: nat,Y: real,X: real] :
      ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),N))
     => ( pp(aa(real,bool,ord_less_eq(real,zero_zero(real)),Y))
       => ( ( aa(nat,real,aa(real,fun(nat,real),power_power(real),Y),N) = X )
         => ( aa(real,real,root(N),X) = Y ) ) ) ) ).

% real_root_pos_unique
tff(fact_4188_real__root__power__cancel,axiom,
    ! [N: nat,X: real] :
      ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),N))
     => ( pp(aa(real,bool,ord_less_eq(real,zero_zero(real)),X))
       => ( aa(real,real,root(N),aa(nat,real,aa(real,fun(nat,real),power_power(real),X),N)) = X ) ) ) ).

% real_root_power_cancel
tff(fact_4189_cot__def,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X4: A] : aa(A,A,cot(A),X4) = aa(A,A,aa(A,fun(A,A),divide_divide(A),cos(A,X4)),sin(A,X4)) ) ).

% cot_def
tff(fact_4190_real__root__increasing,axiom,
    ! [N: nat,N4: nat,X: real] :
      ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),N))
     => ( pp(aa(nat,bool,ord_less_eq(nat,N),N4))
       => ( pp(aa(real,bool,ord_less_eq(real,zero_zero(real)),X))
         => ( pp(aa(real,bool,ord_less_eq(real,X),one_one(real)))
           => pp(aa(real,bool,ord_less_eq(real,aa(real,real,root(N),X)),aa(real,real,root(N4),X))) ) ) ) ) ).

% real_root_increasing
tff(fact_4191_log__root,axiom,
    ! [N: nat,A2: real,B2: real] :
      ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),N))
     => ( pp(aa(real,bool,ord_less(real,zero_zero(real)),A2))
       => ( aa(real,real,log(B2),aa(real,real,root(N),A2)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,log(B2),A2)),aa(nat,real,semiring_1_of_nat(real),N)) ) ) ) ).

% log_root
tff(fact_4192_log__base__root,axiom,
    ! [N: nat,B2: real,X: real] :
      ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),N))
     => ( pp(aa(real,bool,ord_less(real,zero_zero(real)),B2))
       => ( aa(real,real,log(aa(real,real,root(N),B2)),X) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),N)),aa(real,real,log(B2),X)) ) ) ) ).

% log_base_root
tff(fact_4193_ln__root,axiom,
    ! [N: nat,B2: real] :
      ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),N))
     => ( pp(aa(real,bool,ord_less(real,zero_zero(real)),B2))
       => ( aa(real,real,ln_ln(real),aa(real,real,root(N),B2)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,ln_ln(real),B2)),aa(nat,real,semiring_1_of_nat(real),N)) ) ) ) ).

% ln_root
tff(fact_4194_root__powr__inverse,axiom,
    ! [N: nat,X: real] :
      ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),N))
     => ( pp(aa(real,bool,ord_less(real,zero_zero(real)),X))
       => ( aa(real,real,root(N),X) = powr(real,X,aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(nat,real,semiring_1_of_nat(real),N))) ) ) ) ).

% root_powr_inverse
tff(fact_4195_cot__gt__zero,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,ord_less(real,zero_zero(real)),X))
     => ( pp(aa(real,bool,ord_less(real,X),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2)))))
       => pp(aa(real,bool,ord_less(real,zero_zero(real)),aa(real,real,cot(real),X))) ) ) ).

% cot_gt_zero
tff(fact_4196_tan__cot_H,axiom,
    ! [X: real] : aa(real,real,tan(real),aa(real,real,minus_minus(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2)))),X)) = aa(real,real,cot(real),X) ).

% tan_cot'
tff(fact_4197_sint__range__size,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] :
          ( pp(aa(int,bool,ord_less_eq(int,aa(int,int,uminus_uminus(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,minus_minus(nat,aa(word(A),nat,size_size(word(A)),W)),aa(nat,nat,suc,zero_zero(nat)))))),ring_1_signed(A,int,W)))
          & pp(aa(int,bool,ord_less(int,ring_1_signed(A,int,W)),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,minus_minus(nat,aa(word(A),nat,size_size(word(A)),W)),aa(nat,nat,suc,zero_zero(nat)))))) ) ) ).

% sint_range_size
tff(fact_4198_lowi__def,axiom,
    ! [X: nat,N: nat] : vEBT_VEBT_lowi(X,N) = heap_Time_return(nat,modulo_modulo(nat,X,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N))) ).

% lowi_def
tff(fact_4199_ex__has__greatest__nat__lemma,axiom,
    ! [A: $tType,P: fun(A,bool),K: A,F2: fun(A,nat),N: nat] :
      ( pp(aa(A,bool,P,K))
     => ( ! [X3: A] :
            ( pp(aa(A,bool,P,X3))
           => ? [Y4: A] :
                ( pp(aa(A,bool,P,Y4))
                & ~ pp(aa(nat,bool,ord_less_eq(nat,aa(A,nat,F2,Y4)),aa(A,nat,F2,X3))) ) )
       => ? [Y3: A] :
            ( pp(aa(A,bool,P,Y3))
            & ~ pp(aa(nat,bool,ord_less(nat,aa(A,nat,F2,Y3)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(A,nat,F2,K)),N))) ) ) ) ).

% ex_has_greatest_nat_lemma
tff(fact_4200_option_Osize_I3_J,axiom,
    ! [A: $tType] : aa(option(A),nat,size_size(option(A)),none(A)) = aa(nat,nat,suc,zero_zero(nat)) ).

% option.size(3)
tff(fact_4201_signed__0,axiom,
    ! [B: $tType,A: $tType] :
      ( ( ring_1(A)
        & type_len(B) )
     => ( ring_1_signed(B,A,zero_zero(word(B))) = zero_zero(A) ) ) ).

% signed_0
tff(fact_4202_More__Word_Osint__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A)] :
          ( ( ring_1_signed(A,int,X) = zero_zero(int) )
        <=> ( X = zero_zero(word(A)) ) ) ) ).

% More_Word.sint_0
tff(fact_4203_highi__def,axiom,
    ! [X: nat,N: nat] : vEBT_VEBT_highi(X,N) = heap_Time_return(nat,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),X),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N))) ).

% highi_def
tff(fact_4204_signed__minus__1,axiom,
    ! [B: $tType,A: $tType] :
      ( ( ring_1(A)
        & type_len(B) )
     => ( ring_1_signed(B,A,aa(word(B),word(B),uminus_uminus(word(B)),one_one(word(B)))) = aa(A,A,uminus_uminus(A),one_one(A)) ) ) ).

% signed_minus_1
tff(fact_4205_sint__minus1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A)] :
          ( ( ring_1_signed(A,int,X) = aa(int,int,uminus_uminus(int),one_one(int)) )
        <=> ( X = aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A))) ) ) ) ).

% sint_minus1
tff(fact_4206_Word_Osint__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( ring_1_signed(A,int,zero_zero(word(A))) = zero_zero(int) ) ) ).

% Word.sint_0
tff(fact_4207_signed__eq__0__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & ring_char_0(A) )
     => ! [W: word(B)] :
          ( ( ring_1_signed(B,A,W) = zero_zero(A) )
        <=> ( W = zero_zero(word(B)) ) ) ) ).

% signed_eq_0_iff
tff(fact_4208_option_Osize__neq,axiom,
    ! [A: $tType,X: option(A)] : aa(option(A),nat,size_size(option(A)),X) != zero_zero(nat) ).

% option.size_neq
tff(fact_4209_ex__has__least__nat,axiom,
    ! [A: $tType,P: fun(A,bool),K: A,M: fun(A,nat)] :
      ( pp(aa(A,bool,P,K))
     => ? [X3: A] :
          ( pp(aa(A,bool,P,X3))
          & ! [Y4: A] :
              ( pp(aa(A,bool,P,Y4))
             => pp(aa(nat,bool,ord_less_eq(nat,aa(A,nat,M,X3)),aa(A,nat,M,Y4))) ) ) ) ).

% ex_has_least_nat
tff(fact_4210_sint__n1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( ring_1_signed(A,int,aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A)))) = aa(int,int,uminus_uminus(int),one_one(int)) ) ) ).

% sint_n1
tff(fact_4211_Lattices__Big_Oex__has__greatest__nat,axiom,
    ! [A: $tType,P: fun(A,bool),K: A,F2: fun(A,nat),B2: nat] :
      ( pp(aa(A,bool,P,K))
     => ( ! [Y3: A] :
            ( pp(aa(A,bool,P,Y3))
           => pp(aa(nat,bool,ord_less(nat,aa(A,nat,F2,Y3)),B2)) )
       => ? [X3: A] :
            ( pp(aa(A,bool,P,X3))
            & ! [Y4: A] :
                ( pp(aa(A,bool,P,Y4))
               => pp(aa(nat,bool,ord_less_eq(nat,aa(A,nat,F2,Y4)),aa(A,nat,F2,X3))) ) ) ) ) ).

% Lattices_Big.ex_has_greatest_nat
tff(fact_4212_sint__above__size,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),X: int] :
          ( pp(aa(int,bool,ord_less_eq(int,aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,minus_minus(nat,aa(word(A),nat,size_size(word(A)),W)),one_one(nat)))),X))
         => pp(aa(int,bool,ord_less(int,ring_1_signed(A,int,W)),X)) ) ) ).

% sint_above_size
tff(fact_4213_sint__below__size,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: int,W: word(A)] :
          ( pp(aa(int,bool,ord_less_eq(int,X),aa(int,int,uminus_uminus(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,minus_minus(nat,aa(word(A),nat,size_size(word(A)),W)),one_one(nat))))))
         => pp(aa(int,bool,ord_less_eq(int,X),ring_1_signed(A,int,W))) ) ) ).

% sint_below_size
tff(fact_4214_set__decode__0,axiom,
    ! [X: nat] :
      ( pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),zero_zero(nat)),nat_set_decode(X)))
    <=> ~ pp(aa(nat,bool,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),X)) ) ).

% set_decode_0
tff(fact_4215_Comparator__Generator_OAll__less__Suc,axiom,
    ! [X: nat,P: fun(nat,bool)] :
      ( ! [I3: nat] :
          ( pp(aa(nat,bool,ord_less(nat,I3),aa(nat,nat,suc,X)))
         => pp(aa(nat,bool,P,I3)) )
    <=> ( pp(aa(nat,bool,P,zero_zero(nat)))
        & ! [I3: nat] :
            ( pp(aa(nat,bool,ord_less(nat,I3),X))
           => pp(aa(nat,bool,P,aa(nat,nat,suc,I3))) ) ) ) ).

% Comparator_Generator.All_less_Suc
tff(fact_4216_forall__finite_I2_J,axiom,
    ! [P: fun(nat,bool)] :
      ( ! [I3: nat] :
          ( pp(aa(nat,bool,ord_less(nat,I3),aa(nat,nat,suc,zero_zero(nat))))
         => pp(aa(nat,bool,P,I3)) )
    <=> pp(aa(nat,bool,P,zero_zero(nat))) ) ).

% forall_finite(2)
tff(fact_4217_forall__finite_I3_J,axiom,
    ! [X: nat,P: fun(nat,bool)] :
      ( ! [I3: nat] :
          ( pp(aa(nat,bool,ord_less(nat,I3),aa(nat,nat,suc,aa(nat,nat,suc,X))))
         => pp(aa(nat,bool,P,I3)) )
    <=> ( pp(aa(nat,bool,P,zero_zero(nat)))
        & ! [I3: nat] :
            ( pp(aa(nat,bool,ord_less(nat,I3),aa(nat,nat,suc,X)))
           => pp(aa(nat,bool,P,aa(nat,nat,suc,I3))) ) ) ) ).

% forall_finite(3)
tff(fact_4218_set__decode__Suc,axiom,
    ! [N: nat,X: nat] :
      ( pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),aa(nat,nat,suc,N)),nat_set_decode(X)))
    <=> pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),N),nat_set_decode(aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),X),aa(num,nat,numeral_numeral(nat),bit0(one2)))))) ) ).

% set_decode_Suc
tff(fact_4219_scast__0,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ( ring_1_signed(B,word(A),zero_zero(word(B))) = zero_zero(word(A)) ) ) ).

% scast_0
tff(fact_4220_finite__set__decode,axiom,
    ! [N: nat] : pp(aa(set(nat),bool,finite_finite2(nat),nat_set_decode(N))) ).

% finite_set_decode
tff(fact_4221_scast__n1,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ( ring_1_signed(B,word(A),aa(word(B),word(B),uminus_uminus(word(B)),one_one(word(B)))) = aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A))) ) ) ).

% scast_n1
tff(fact_4222_subset__decode__imp__le,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(set(nat),bool,ord_less_eq(set(nat),nat_set_decode(M)),nat_set_decode(N)))
     => pp(aa(nat,bool,ord_less_eq(nat,M),N)) ) ).

% subset_decode_imp_le
tff(fact_4223_forall__finite_I1_J,axiom,
    ! [P: fun(nat,bool),I4: nat] :
      ( pp(aa(nat,bool,ord_less(nat,I4),zero_zero(nat)))
     => pp(aa(nat,bool,P,I4)) ) ).

% forall_finite(1)
tff(fact_4224_set__decode__def,axiom,
    ! [X: nat] : nat_set_decode(X) = collect(nat,aTP_Lamp_kz(nat,fun(nat,bool),X)) ).

% set_decode_def
tff(fact_4225_flip__bit__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A] : bit_se8732182000553998342ip_bit(A,zero_zero(nat),A2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(bool,A,zero_neq_one_of_bool(A),aa(A,bool,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A2))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(num,A,numeral_numeral(A),bit0(one2))))) ) ).

% flip_bit_0
tff(fact_4226_take__bit__word__Bit1__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: num,M: num] : aa(word(A),word(A),bit_se2584673776208193580ke_bit(word(A),aa(num,nat,numeral_numeral(nat),N)),aa(num,word(A),numeral_numeral(word(A)),aa(num,num,bit1,M))) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),one_one(word(A))),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),aa(word(A),word(A),bit_se2584673776208193580ke_bit(word(A),pred_numeral(N)),aa(num,word(A),numeral_numeral(word(A)),M)))) ) ).

% take_bit_word_Bit1_eq
tff(fact_4227_uint32_Osize__eq,axiom,
    ! [P2: uint32] : aa(uint32,nat,size_size(uint32),P2) = aa(num,nat,numeral_numeral(nat),bit0(bit0(bit0(bit0(bit0(one2)))))) ).

% uint32.size_eq
tff(fact_4228_add__scale__eq__noteq,axiom,
    ! [A: $tType] :
      ( semiri1453513574482234551roduct(A)
     => ! [R2: A,A2: A,B2: A,C2: A,D: A] :
          ( ( R2 != zero_zero(A) )
         => ( ( ( A2 = B2 )
              & ( C2 != D ) )
           => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),R2),C2)) != aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),R2),D)) ) ) ) ) ).

% add_scale_eq_noteq
tff(fact_4229_take__bit__of__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat] : aa(A,A,bit_se2584673776208193580ke_bit(A,N),zero_zero(A)) = zero_zero(A) ) ).

% take_bit_of_0
tff(fact_4230_of__bool__eq__0__iff,axiom,
    ! [A: $tType] :
      ( zero_neq_one(A)
     => ! [P: bool] :
          ( ( aa(bool,A,zero_neq_one_of_bool(A),P) = zero_zero(A) )
        <=> ~ pp(P) ) ) ).

% of_bool_eq_0_iff
tff(fact_4231_of__bool__eq_I1_J,axiom,
    ! [A: $tType] :
      ( zero_neq_one(A)
     => ( aa(bool,A,zero_neq_one_of_bool(A),fFalse) = zero_zero(A) ) ) ).

% of_bool_eq(1)
tff(fact_4232_of__bool__less__eq__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [P: bool,Q: bool] :
          ( pp(aa(A,bool,ord_less_eq(A,aa(bool,A,zero_neq_one_of_bool(A),P)),aa(bool,A,zero_neq_one_of_bool(A),Q)))
        <=> ( pp(P)
           => pp(Q) ) ) ) ).

% of_bool_less_eq_iff
tff(fact_4233_of__bool__less__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [P: bool,Q: bool] :
          ( pp(aa(A,bool,ord_less(A,aa(bool,A,zero_neq_one_of_bool(A),P)),aa(bool,A,zero_neq_one_of_bool(A),Q)))
        <=> ( ~ pp(P)
            & pp(Q) ) ) ) ).

% of_bool_less_iff
tff(fact_4234_of__bool__eq__1__iff,axiom,
    ! [A: $tType] :
      ( zero_neq_one(A)
     => ! [P: bool] :
          ( ( aa(bool,A,zero_neq_one_of_bool(A),P) = one_one(A) )
        <=> pp(P) ) ) ).

% of_bool_eq_1_iff
tff(fact_4235_of__bool__eq_I2_J,axiom,
    ! [A: $tType] :
      ( zero_neq_one(A)
     => ( aa(bool,A,zero_neq_one_of_bool(A),fTrue) = one_one(A) ) ) ).

% of_bool_eq(2)
tff(fact_4236_of__nat__of__bool,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [P: bool] : aa(nat,A,semiring_1_of_nat(A),aa(bool,nat,zero_neq_one_of_bool(nat),P)) = aa(bool,A,zero_neq_one_of_bool(A),P) ) ).

% of_nat_of_bool
tff(fact_4237_abs__bool__eq,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [P: bool] : aa(A,A,abs_abs(A),aa(bool,A,zero_neq_one_of_bool(A),P)) = aa(bool,A,zero_neq_one_of_bool(A),P) ) ).

% abs_bool_eq
tff(fact_4238_div__word__one,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),one_one(word(A))),W) = aa(bool,word(A),zero_neq_one_of_bool(word(A)),aa(word(A),bool,fequal(word(A),W),one_one(word(A)))) ) ).

% div_word_one
tff(fact_4239_take__bit__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A] : aa(A,A,bit_se2584673776208193580ke_bit(A,zero_zero(nat)),A2) = zero_zero(A) ) ).

% take_bit_0
tff(fact_4240_take__bit__Suc__1,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [N: nat] : aa(A,A,bit_se2584673776208193580ke_bit(A,aa(nat,nat,suc,N)),one_one(A)) = one_one(A) ) ).

% take_bit_Suc_1
tff(fact_4241_take__bit__numeral__1,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [L: num] : aa(A,A,bit_se2584673776208193580ke_bit(A,aa(num,nat,numeral_numeral(nat),L)),one_one(A)) = one_one(A) ) ).

% take_bit_numeral_1
tff(fact_4242_zero__less__of__bool__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [P: bool] :
          ( pp(aa(A,bool,ord_less(A,zero_zero(A)),aa(bool,A,zero_neq_one_of_bool(A),P)))
        <=> pp(P) ) ) ).

% zero_less_of_bool_iff
tff(fact_4243_of__bool__less__one__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [P: bool] :
          ( pp(aa(A,bool,ord_less(A,aa(bool,A,zero_neq_one_of_bool(A),P)),one_one(A)))
        <=> ~ pp(P) ) ) ).

% of_bool_less_one_iff
tff(fact_4244_Suc__0__mod__eq,axiom,
    ! [N: nat] : modulo_modulo(nat,aa(nat,nat,suc,zero_zero(nat)),N) = aa(bool,nat,zero_neq_one_of_bool(nat),aa(bool,bool,fNot,aa(nat,bool,fequal(nat,N),aa(nat,nat,suc,zero_zero(nat))))) ).

% Suc_0_mod_eq
tff(fact_4245_of__bool__not__iff,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [P: bool] : aa(bool,A,zero_neq_one_of_bool(A),aa(bool,bool,fNot,P)) = aa(A,A,minus_minus(A,one_one(A)),aa(bool,A,zero_neq_one_of_bool(A),P)) ) ).

% of_bool_not_iff
tff(fact_4246_div__word__by__minus__1__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),W),aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A)))) = aa(bool,word(A),zero_neq_one_of_bool(word(A)),aa(word(A),bool,fequal(word(A),W),aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A))))) ) ).

% div_word_by_minus_1_eq
tff(fact_4247_take__bit__of__1__eq__0__iff,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [N: nat] :
          ( ( aa(A,A,bit_se2584673776208193580ke_bit(A,N),one_one(A)) = zero_zero(A) )
        <=> ( N = zero_zero(nat) ) ) ) ).

% take_bit_of_1_eq_0_iff
tff(fact_4248_mod__word__one,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] : modulo_modulo(word(A),one_one(word(A)),W) = aa(word(A),word(A),minus_minus(word(A),one_one(word(A))),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),W),aa(bool,word(A),zero_neq_one_of_bool(word(A)),aa(word(A),bool,fequal(word(A),W),one_one(word(A)))))) ) ).

% mod_word_one
tff(fact_4249_odd__of__bool__self,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [P2: bool] :
          ( ~ pp(aa(A,bool,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),aa(bool,A,zero_neq_one_of_bool(A),P2)))
        <=> pp(P2) ) ) ).

% odd_of_bool_self
tff(fact_4250_take__bit__of__1,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat] : aa(A,A,bit_se2584673776208193580ke_bit(A,N),one_one(A)) = aa(bool,A,zero_neq_one_of_bool(A),aa(nat,bool,ord_less(nat,zero_zero(nat)),N)) ) ).

% take_bit_of_1
tff(fact_4251_mod__word__by__minus__1__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] : modulo_modulo(word(A),W,aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A)))) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),W),aa(bool,word(A),zero_neq_one_of_bool(word(A)),aa(word(A),bool,ord_less(word(A),W),aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A)))))) ) ).

% mod_word_by_minus_1_eq
tff(fact_4252_even__take__bit__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A2: A] :
          ( pp(aa(A,bool,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,bit_se2584673776208193580ke_bit(A,N),A2)))
        <=> ( ( N = zero_zero(nat) )
            | pp(aa(A,bool,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A2)) ) ) ) ).

% even_take_bit_eq
tff(fact_4253_of__bool__half__eq__0,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [B2: bool] : aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(bool,A,zero_neq_one_of_bool(A),B2)),aa(num,A,numeral_numeral(A),bit0(one2))) = zero_zero(A) ) ).

% of_bool_half_eq_0
tff(fact_4254_take__bit__Suc__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A] : aa(A,A,bit_se2584673776208193580ke_bit(A,aa(nat,nat,suc,zero_zero(nat))),A2) = modulo_modulo(A,A2,aa(num,A,numeral_numeral(A),bit0(one2))) ) ).

% take_bit_Suc_0
tff(fact_4255_take__bit__word__Bit0__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: num,M: num] : aa(word(A),word(A),bit_se2584673776208193580ke_bit(word(A),aa(num,nat,numeral_numeral(nat),N)),aa(num,word(A),numeral_numeral(word(A)),bit0(M))) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),aa(word(A),word(A),bit_se2584673776208193580ke_bit(word(A),pred_numeral(N)),aa(num,word(A),numeral_numeral(word(A)),M))) ) ).

% take_bit_word_Bit0_eq
tff(fact_4256_bits__1__div__exp,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [N: nat] : aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N)) = aa(bool,A,zero_neq_one_of_bool(A),aa(nat,bool,fequal(nat,N),zero_zero(nat))) ) ).

% bits_1_div_exp
tff(fact_4257_one__div__2__pow__eq,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [N: nat] : aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N)) = aa(bool,A,zero_neq_one_of_bool(A),aa(nat,bool,fequal(nat,N),zero_zero(nat))) ) ).

% one_div_2_pow_eq
tff(fact_4258_take__bit__of__exp,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [M: nat,N: nat] : aa(A,A,bit_se2584673776208193580ke_bit(A,M),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(bool,A,zero_neq_one_of_bool(A),aa(nat,bool,ord_less(nat,N),M))),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N)) ) ).

% take_bit_of_exp
tff(fact_4259_take__bit__of__2,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [N: nat] : aa(A,A,bit_se2584673776208193580ke_bit(A,N),aa(num,A,numeral_numeral(A),bit0(one2))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(bool,A,zero_neq_one_of_bool(A),aa(nat,bool,ord_less_eq(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N))),aa(num,A,numeral_numeral(A),bit0(one2))) ) ).

% take_bit_of_2
tff(fact_4260_take__bit__word__minus__Bit0__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: num,M: num] : aa(word(A),word(A),bit_se2584673776208193580ke_bit(word(A),aa(num,nat,numeral_numeral(nat),N)),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(M)))) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),aa(word(A),word(A),bit_se2584673776208193580ke_bit(word(A),pred_numeral(N)),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),M)))) ) ).

% take_bit_word_minus_Bit0_eq
tff(fact_4261_one__mod__2__pow__eq,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [N: nat] : modulo_modulo(A,one_one(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N)) = aa(bool,A,zero_neq_one_of_bool(A),aa(nat,bool,ord_less(nat,zero_zero(nat)),N)) ) ).

% one_mod_2_pow_eq
tff(fact_4262_take__bit__of__nat,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,M: nat] : aa(A,A,bit_se2584673776208193580ke_bit(A,N),aa(nat,A,semiring_1_of_nat(A),M)) = aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,bit_se2584673776208193580ke_bit(nat,N),M)) ) ).

% take_bit_of_nat
tff(fact_4263_of__bool__conj,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [P: bool,Q: bool] : aa(bool,A,zero_neq_one_of_bool(A),fconj(P,Q)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(bool,A,zero_neq_one_of_bool(A),P)),aa(bool,A,zero_neq_one_of_bool(A),Q)) ) ).

% of_bool_conj
tff(fact_4264_take__bit__add,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [N: nat,A2: A,B2: A] : aa(A,A,bit_se2584673776208193580ke_bit(A,N),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,bit_se2584673776208193580ke_bit(A,N),A2)),aa(A,A,bit_se2584673776208193580ke_bit(A,N),B2))) = aa(A,A,bit_se2584673776208193580ke_bit(A,N),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)) ) ).

% take_bit_add
tff(fact_4265_take__bit__of__int,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [N: nat,K: int] : aa(A,A,bit_se2584673776208193580ke_bit(A,N),aa(int,A,ring_1_of_int(A),K)) = aa(int,A,ring_1_of_int(A),aa(int,int,bit_se2584673776208193580ke_bit(int,N),K)) ) ).

% take_bit_of_int
tff(fact_4266_of__bool__eq__iff,axiom,
    ! [A: $tType] :
      ( zero_neq_one(A)
     => ! [P2: bool,Q2: bool] :
          ( ( aa(bool,A,zero_neq_one_of_bool(A),P2) = aa(bool,A,zero_neq_one_of_bool(A),Q2) )
        <=> ( pp(P2)
          <=> pp(Q2) ) ) ) ).

% of_bool_eq_iff
tff(fact_4267_take__bit__tightened,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A2: A,B2: A,M: nat] :
          ( ( aa(A,A,bit_se2584673776208193580ke_bit(A,N),A2) = aa(A,A,bit_se2584673776208193580ke_bit(A,N),B2) )
         => ( pp(aa(nat,bool,ord_less_eq(nat,M),N))
           => ( aa(A,A,bit_se2584673776208193580ke_bit(A,M),A2) = aa(A,A,bit_se2584673776208193580ke_bit(A,M),B2) ) ) ) ) ).

% take_bit_tightened
tff(fact_4268_zero__less__eq__of__bool,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [P: bool] : pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),aa(bool,A,zero_neq_one_of_bool(A),P))) ) ).

% zero_less_eq_of_bool
tff(fact_4269_of__bool__def,axiom,
    ! [A: $tType] :
      ( zero_neq_one(A)
     => ! [P2: bool] :
          ( ( pp(P2)
           => ( aa(bool,A,zero_neq_one_of_bool(A),P2) = one_one(A) ) )
          & ( ~ pp(P2)
           => ( aa(bool,A,zero_neq_one_of_bool(A),P2) = zero_zero(A) ) ) ) ) ).

% of_bool_def
tff(fact_4270_split__of__bool,axiom,
    ! [A: $tType] :
      ( zero_neq_one(A)
     => ! [P: fun(A,bool),P2: bool] :
          ( pp(aa(A,bool,P,aa(bool,A,zero_neq_one_of_bool(A),P2)))
        <=> ( ( pp(P2)
             => pp(aa(A,bool,P,one_one(A))) )
            & ( ~ pp(P2)
             => pp(aa(A,bool,P,zero_zero(A))) ) ) ) ) ).

% split_of_bool
tff(fact_4271_split__of__bool__asm,axiom,
    ! [A: $tType] :
      ( zero_neq_one(A)
     => ! [P: fun(A,bool),P2: bool] :
          ( pp(aa(A,bool,P,aa(bool,A,zero_neq_one_of_bool(A),P2)))
        <=> ~ ( ( pp(P2)
                & ~ pp(aa(A,bool,P,one_one(A))) )
              | ( ~ pp(P2)
                & ~ pp(aa(A,bool,P,zero_zero(A))) ) ) ) ) ).

% split_of_bool_asm
tff(fact_4272_of__bool__less__eq__one,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [P: bool] : pp(aa(A,bool,ord_less_eq(A,aa(bool,A,zero_neq_one_of_bool(A),P)),one_one(A))) ) ).

% of_bool_less_eq_one
tff(fact_4273_signed__take__bit__eq__iff__take__bit__eq,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [N: nat,A2: A,B2: A] :
          ( ( aa(A,A,bit_ri4674362597316999326ke_bit(A,N),A2) = aa(A,A,bit_ri4674362597316999326ke_bit(A,N),B2) )
        <=> ( aa(A,A,bit_se2584673776208193580ke_bit(A,aa(nat,nat,suc,N)),A2) = aa(A,A,bit_se2584673776208193580ke_bit(A,aa(nat,nat,suc,N)),B2) ) ) ) ).

% signed_take_bit_eq_iff_take_bit_eq
tff(fact_4274_signed__take__bit__take__bit,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [M: nat,N: nat,A2: A] : aa(A,A,bit_ri4674362597316999326ke_bit(A,M),aa(A,A,bit_se2584673776208193580ke_bit(A,N),A2)) = aa(A,A,if(fun(A,A),aa(nat,bool,ord_less_eq(nat,N),M),bit_se2584673776208193580ke_bit(A,N),bit_ri4674362597316999326ke_bit(A,M)),A2) ) ).

% signed_take_bit_take_bit
tff(fact_4275_take__bit__unset__bit__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,M: nat,A2: A] :
          ( ( pp(aa(nat,bool,ord_less_eq(nat,N),M))
           => ( aa(A,A,bit_se2584673776208193580ke_bit(A,N),aa(A,A,aa(nat,fun(A,A),bit_se2638667681897837118et_bit(A),M),A2)) = aa(A,A,bit_se2584673776208193580ke_bit(A,N),A2) ) )
          & ( ~ pp(aa(nat,bool,ord_less_eq(nat,N),M))
           => ( aa(A,A,bit_se2584673776208193580ke_bit(A,N),aa(A,A,aa(nat,fun(A,A),bit_se2638667681897837118et_bit(A),M),A2)) = aa(A,A,aa(nat,fun(A,A),bit_se2638667681897837118et_bit(A),M),aa(A,A,bit_se2584673776208193580ke_bit(A,N),A2)) ) ) ) ) ).

% take_bit_unset_bit_eq
tff(fact_4276_take__bit__set__bit__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,M: nat,A2: A] :
          ( ( pp(aa(nat,bool,ord_less_eq(nat,N),M))
           => ( aa(A,A,bit_se2584673776208193580ke_bit(A,N),aa(A,A,aa(nat,fun(A,A),bit_se5668285175392031749et_bit(A),M),A2)) = aa(A,A,bit_se2584673776208193580ke_bit(A,N),A2) ) )
          & ( ~ pp(aa(nat,bool,ord_less_eq(nat,N),M))
           => ( aa(A,A,bit_se2584673776208193580ke_bit(A,N),aa(A,A,aa(nat,fun(A,A),bit_se5668285175392031749et_bit(A),M),A2)) = aa(A,A,aa(nat,fun(A,A),bit_se5668285175392031749et_bit(A),M),aa(A,A,bit_se2584673776208193580ke_bit(A,N),A2)) ) ) ) ) ).

% take_bit_set_bit_eq
tff(fact_4277_take__bit__flip__bit__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,M: nat,A2: A] :
          ( ( pp(aa(nat,bool,ord_less_eq(nat,N),M))
           => ( aa(A,A,bit_se2584673776208193580ke_bit(A,N),bit_se8732182000553998342ip_bit(A,M,A2)) = aa(A,A,bit_se2584673776208193580ke_bit(A,N),A2) ) )
          & ( ~ pp(aa(nat,bool,ord_less_eq(nat,N),M))
           => ( aa(A,A,bit_se2584673776208193580ke_bit(A,N),bit_se8732182000553998342ip_bit(A,M,A2)) = bit_se8732182000553998342ip_bit(A,M,aa(A,A,bit_se2584673776208193580ke_bit(A,N),A2)) ) ) ) ) ).

% take_bit_flip_bit_eq
tff(fact_4278_take__bit__signed__take__bit,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [M: nat,N: nat,A2: A] :
          ( pp(aa(nat,bool,ord_less_eq(nat,M),aa(nat,nat,suc,N)))
         => ( aa(A,A,bit_se2584673776208193580ke_bit(A,M),aa(A,A,bit_ri4674362597316999326ke_bit(A,N),A2)) = aa(A,A,bit_se2584673776208193580ke_bit(A,M),A2) ) ) ) ).

% take_bit_signed_take_bit
tff(fact_4279_take__bit__Suc__bit0,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [N: nat,K: num] : aa(A,A,bit_se2584673776208193580ke_bit(A,aa(nat,nat,suc,N)),aa(num,A,numeral_numeral(A),bit0(K))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,bit_se2584673776208193580ke_bit(A,N),aa(num,A,numeral_numeral(A),K))),aa(num,A,numeral_numeral(A),bit0(one2))) ) ).

% take_bit_Suc_bit0
tff(fact_4280_take__bit__eq__mod,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A2: A] : aa(A,A,bit_se2584673776208193580ke_bit(A,N),A2) = modulo_modulo(A,A2,aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N)) ) ).

% take_bit_eq_mod
tff(fact_4281_of__bool__odd__eq__mod__2,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A2: A] : aa(bool,A,zero_neq_one_of_bool(A),aa(bool,bool,fNot,aa(A,bool,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A2))) = modulo_modulo(A,A2,aa(num,A,numeral_numeral(A),bit0(one2))) ) ).

% of_bool_odd_eq_mod_2
tff(fact_4282_take__bit__eq__0__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A2: A] :
          ( ( aa(A,A,bit_se2584673776208193580ke_bit(A,N),A2) = zero_zero(A) )
        <=> pp(aa(A,bool,dvd_dvd(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N)),A2)) ) ) ).

% take_bit_eq_0_iff
tff(fact_4283_bin__last__bintrunc,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [L: nat,N: A] :
          ( ~ pp(aa(A,bool,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,bit_se2584673776208193580ke_bit(A,L),N)))
        <=> ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),L))
            & ~ pp(aa(A,bool,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),N)) ) ) ) ).

% bin_last_bintrunc
tff(fact_4284_take__bit__numeral__bit0,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [L: num,K: num] : aa(A,A,bit_se2584673776208193580ke_bit(A,aa(num,nat,numeral_numeral(nat),L)),aa(num,A,numeral_numeral(A),bit0(K))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,bit_se2584673776208193580ke_bit(A,pred_numeral(L)),aa(num,A,numeral_numeral(A),K))),aa(num,A,numeral_numeral(A),bit0(one2))) ) ).

% take_bit_numeral_bit0
tff(fact_4285_bits__induct,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [P: fun(A,bool),A2: A] :
          ( ! [A4: A] :
              ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A4),aa(num,A,numeral_numeral(A),bit0(one2))) = A4 )
             => pp(aa(A,bool,P,A4)) )
         => ( ! [A4: A,B3: bool] :
                ( pp(aa(A,bool,P,A4))
               => ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(bool,A,zero_neq_one_of_bool(A),B3)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),A4))),aa(num,A,numeral_numeral(A),bit0(one2))) = A4 )
                 => pp(aa(A,bool,P,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(bool,A,zero_neq_one_of_bool(A),B3)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),A4)))) ) )
           => pp(aa(A,bool,P,A2)) ) ) ) ).

% bits_induct
tff(fact_4286_exp__mod__exp,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [M: nat,N: nat] : modulo_modulo(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),M),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(bool,A,zero_neq_one_of_bool(A),aa(nat,bool,ord_less(nat,M),N))),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),M)) ) ).

% exp_mod_exp
tff(fact_4287_add__0__iff,axiom,
    ! [A: $tType] :
      ( semiri1453513574482234551roduct(A)
     => ! [B2: A,A2: A] :
          ( ( B2 = aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A2) )
        <=> ( A2 = zero_zero(A) ) ) ) ).

% add_0_iff
tff(fact_4288_crossproduct__eq,axiom,
    ! [A: $tType] :
      ( semiri1453513574482234551roduct(A)
     => ! [W: A,Y: A,X: A,Z: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),W),Y)),aa(A,A,aa(A,fun(A,A),times_times(A),X),Z)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),W),Z)),aa(A,A,aa(A,fun(A,A),times_times(A),X),Y)) )
        <=> ( ( W = X )
            | ( Y = Z ) ) ) ) ).

% crossproduct_eq
tff(fact_4289_crossproduct__noteq,axiom,
    ! [A: $tType] :
      ( semiri1453513574482234551roduct(A)
     => ! [A2: A,B2: A,C2: A,D: A] :
          ( ( ( A2 != B2 )
            & ( C2 != D ) )
        <=> ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),D)) != aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),D)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) ) ) ) ).

% crossproduct_noteq
tff(fact_4290_take__bit__Suc__bit1,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [N: nat,K: num] : aa(A,A,bit_se2584673776208193580ke_bit(A,aa(nat,nat,suc,N)),aa(num,A,numeral_numeral(A),aa(num,num,bit1,K))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,bit_se2584673776208193580ke_bit(A,N),aa(num,A,numeral_numeral(A),K))),aa(num,A,numeral_numeral(A),bit0(one2)))),one_one(A)) ) ).

% take_bit_Suc_bit1
tff(fact_4291_take__bit__Suc__minus__1__eq,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [N: nat] : aa(A,A,bit_se2584673776208193580ke_bit(A,aa(nat,nat,suc,N)),aa(A,A,uminus_uminus(A),one_one(A))) = aa(A,A,minus_minus(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(nat,nat,suc,N))),one_one(A)) ) ).

% take_bit_Suc_minus_1_eq
tff(fact_4292_take__bit__numeral__minus__1__eq,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [K: num] : aa(A,A,bit_se2584673776208193580ke_bit(A,aa(num,nat,numeral_numeral(nat),K)),aa(A,A,uminus_uminus(A),one_one(A))) = aa(A,A,minus_minus(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(num,nat,numeral_numeral(nat),K))),one_one(A)) ) ).

% take_bit_numeral_minus_1_eq
tff(fact_4293_take__bit__Suc,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A2: A] : aa(A,A,bit_se2584673776208193580ke_bit(A,aa(nat,nat,suc,N)),A2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,bit_se2584673776208193580ke_bit(A,N),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(num,A,numeral_numeral(A),bit0(one2))))),aa(num,A,numeral_numeral(A),bit0(one2)))),modulo_modulo(A,A2,aa(num,A,numeral_numeral(A),bit0(one2)))) ) ).

% take_bit_Suc
tff(fact_4294_stable__imp__take__bit__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,N: nat] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(num,A,numeral_numeral(A),bit0(one2))) = A2 )
         => ( ( pp(aa(A,bool,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A2))
             => ( aa(A,A,bit_se2584673776208193580ke_bit(A,N),A2) = zero_zero(A) ) )
            & ( ~ pp(aa(A,bool,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A2))
             => ( aa(A,A,bit_se2584673776208193580ke_bit(A,N),A2) = aa(A,A,minus_minus(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N)),one_one(A)) ) ) ) ) ) ).

% stable_imp_take_bit_eq
tff(fact_4295_take__bit__numeral__bit1,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [L: num,K: num] : aa(A,A,bit_se2584673776208193580ke_bit(A,aa(num,nat,numeral_numeral(nat),L)),aa(num,A,numeral_numeral(A),aa(num,num,bit1,K))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,bit_se2584673776208193580ke_bit(A,pred_numeral(L)),aa(num,A,numeral_numeral(A),K))),aa(num,A,numeral_numeral(A),bit0(one2)))),one_one(A)) ) ).

% take_bit_numeral_bit1
tff(fact_4296_exp__div__exp__eq,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [M: nat,N: nat] : aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),M)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(bool,A,zero_neq_one_of_bool(A),fconj(aa(bool,bool,fNot,aa(A,bool,fequal(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),M)),zero_zero(A))),aa(nat,bool,ord_less_eq(nat,N),M)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(nat,nat,minus_minus(nat,M),N))) ) ).

% exp_div_exp_eq
tff(fact_4297_take__bit__rec,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A2: A] :
          ( ( ( N = zero_zero(nat) )
           => ( aa(A,A,bit_se2584673776208193580ke_bit(A,N),A2) = zero_zero(A) ) )
          & ( ( N != zero_zero(nat) )
           => ( aa(A,A,bit_se2584673776208193580ke_bit(A,N),A2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,bit_se2584673776208193580ke_bit(A,aa(nat,nat,minus_minus(nat,N),one_one(nat))),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(num,A,numeral_numeral(A),bit0(one2))))),aa(num,A,numeral_numeral(A),bit0(one2)))),modulo_modulo(A,A2,aa(num,A,numeral_numeral(A),bit0(one2)))) ) ) ) ) ).

% take_bit_rec
tff(fact_4298_take__bit__word__minus__Bit1__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: num,M: num] : aa(word(A),word(A),bit_se2584673776208193580ke_bit(word(A),aa(num,nat,numeral_numeral(nat),N)),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),aa(num,num,bit1,M)))) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),one_one(word(A))),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),aa(word(A),word(A),bit_se2584673776208193580ke_bit(word(A),pred_numeral(N)),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),inc(M)))))) ) ).

% take_bit_word_minus_Bit1_eq
tff(fact_4299_slice__nth,axiom,
    ! [A: $tType,From: nat,To: nat,Xs: list(A),I: nat] :
      ( pp(aa(nat,bool,ord_less(nat,From),To))
     => ( pp(aa(nat,bool,ord_less_eq(nat,To),aa(list(A),nat,size_size(list(A)),Xs)))
       => ( pp(aa(nat,bool,ord_less(nat,I),aa(nat,nat,minus_minus(nat,To),From)))
         => ( aa(nat,A,nth(A,slice(A,From,To,Xs)),I) = aa(nat,A,nth(A,Xs),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),From),I)) ) ) ) ) ).

% slice_nth
tff(fact_4300_word__2p__lem,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: nat,W: word(A)] :
          ( pp(aa(nat,bool,ord_less(nat,N),aa(word(A),nat,size_size(word(A)),W)))
         => ( pp(aa(word(A),bool,ord_less(word(A),W),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),N)))
          <=> pp(aa(int,bool,ord_less(int,aa(word(A),int,semiring_1_unsigned(A,int),W)),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),N))) ) ) ) ).

% word_2p_lem
tff(fact_4301_powr__int,axiom,
    ! [X: real,I: int] :
      ( pp(aa(real,bool,ord_less(real,zero_zero(real)),X))
     => ( ( pp(aa(int,bool,ord_less_eq(int,zero_zero(int)),I))
         => ( powr(real,X,aa(int,real,ring_1_of_int(real),I)) = aa(nat,real,aa(real,fun(nat,real),power_power(real),X),nat2(I)) ) )
        & ( ~ pp(aa(int,bool,ord_less_eq(int,zero_zero(int)),I))
         => ( powr(real,X,aa(int,real,ring_1_of_int(real),I)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(nat,real,aa(real,fun(nat,real),power_power(real),X),nat2(aa(int,int,uminus_uminus(int),I)))) ) ) ) ) ).

% powr_int
tff(fact_4302_Word_Oof__nat__unat,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & semiring_1(A) )
     => ! [W: word(B)] : aa(nat,A,semiring_1_of_nat(A),aa(word(B),nat,semiring_1_unsigned(B,nat),W)) = aa(word(B),A,semiring_1_unsigned(B,A),W) ) ).

% Word.of_nat_unat
tff(fact_4303_nat__int,axiom,
    ! [N: nat] : nat2(aa(nat,int,semiring_1_of_nat(int),N)) = N ).

% nat_int
tff(fact_4304_unsigned__0,axiom,
    ! [B: $tType,A: $tType] :
      ( ( semiring_1(A)
        & type_len(B) )
     => ( aa(word(B),A,semiring_1_unsigned(B,A),zero_zero(word(B))) = zero_zero(A) ) ) ).

% unsigned_0
tff(fact_4305_unsigned__1,axiom,
    ! [B: $tType,A: $tType] :
      ( ( semiring_1(A)
        & type_len(B) )
     => ( aa(word(B),A,semiring_1_unsigned(B,A),one_one(word(B))) = one_one(A) ) ) ).

% unsigned_1
tff(fact_4306_nat__numeral,axiom,
    ! [K: num] : nat2(aa(num,int,numeral_numeral(int),K)) = aa(num,nat,numeral_numeral(nat),K) ).

% nat_numeral
tff(fact_4307_uint__nonnegative,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] : pp(aa(int,bool,ord_less_eq(int,zero_zero(int)),aa(word(A),int,semiring_1_unsigned(A,int),W))) ) ).

% uint_nonnegative
tff(fact_4308_uint__le__0__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A)] :
          ( pp(aa(int,bool,ord_less_eq(int,aa(word(A),int,semiring_1_unsigned(A,int),X)),zero_zero(int)))
        <=> ( aa(word(A),int,semiring_1_unsigned(A,int),X) = zero_zero(int) ) ) ) ).

% uint_le_0_iff
tff(fact_4309_uint__ge__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A)] : pp(aa(int,bool,ord_less_eq(int,zero_zero(int)),aa(word(A),int,semiring_1_unsigned(A,int),X))) ) ).

% uint_ge_0
tff(fact_4310_uint__lt__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A)] : ~ pp(aa(int,bool,ord_less(int,aa(word(A),int,semiring_1_unsigned(A,int),X)),zero_zero(int))) ) ).

% uint_lt_0
tff(fact_4311_pred__numeral__inc,axiom,
    ! [K: num] : pred_numeral(inc(K)) = aa(num,nat,numeral_numeral(nat),K) ).

% pred_numeral_inc
tff(fact_4312_slice__complete,axiom,
    ! [A: $tType,Xs: list(A)] : slice(A,zero_zero(nat),aa(list(A),nat,size_size(list(A)),Xs),Xs) = Xs ).

% slice_complete
tff(fact_4313_nat__1,axiom,
    nat2(one_one(int)) = aa(nat,nat,suc,zero_zero(nat)) ).

% nat_1
tff(fact_4314_nat__le__0,axiom,
    ! [Z: int] :
      ( pp(aa(int,bool,ord_less_eq(int,Z),zero_zero(int)))
     => ( nat2(Z) = zero_zero(nat) ) ) ).

% nat_le_0
tff(fact_4315_nat__0__iff,axiom,
    ! [I: int] :
      ( ( nat2(I) = zero_zero(nat) )
    <=> pp(aa(int,bool,ord_less_eq(int,I),zero_zero(int))) ) ).

% nat_0_iff
tff(fact_4316_nat__neg__numeral,axiom,
    ! [K: num] : nat2(aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),K))) = zero_zero(nat) ).

% nat_neg_numeral
tff(fact_4317_zless__nat__conj,axiom,
    ! [W: int,Z: int] :
      ( pp(aa(nat,bool,ord_less(nat,nat2(W)),nat2(Z)))
    <=> ( pp(aa(int,bool,ord_less(int,zero_zero(int)),Z))
        & pp(aa(int,bool,ord_less(int,W),Z)) ) ) ).

% zless_nat_conj
tff(fact_4318_of__nat__nat__take__bit__eq,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [N: nat,K: int] : aa(nat,A,semiring_1_of_nat(A),nat2(aa(int,int,bit_se2584673776208193580ke_bit(int,N),K))) = aa(int,A,ring_1_of_int(A),aa(int,int,bit_se2584673776208193580ke_bit(int,N),K)) ) ).

% of_nat_nat_take_bit_eq
tff(fact_4319_nat__zminus__int,axiom,
    ! [N: nat] : nat2(aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),N))) = zero_zero(nat) ).

% nat_zminus_int
tff(fact_4320_int__nat__eq,axiom,
    ! [Z: int] :
      ( ( pp(aa(int,bool,ord_less_eq(int,zero_zero(int)),Z))
       => ( aa(nat,int,semiring_1_of_nat(int),nat2(Z)) = Z ) )
      & ( ~ pp(aa(int,bool,ord_less_eq(int,zero_zero(int)),Z))
       => ( aa(nat,int,semiring_1_of_nat(int),nat2(Z)) = zero_zero(int) ) ) ) ).

% int_nat_eq
tff(fact_4321_slice__len,axiom,
    ! [A: $tType,From: nat,To: nat,Xs: list(A)] :
      ( pp(aa(nat,bool,ord_less_eq(nat,From),To))
     => ( pp(aa(nat,bool,ord_less_eq(nat,To),aa(list(A),nat,size_size(list(A)),Xs)))
       => ( aa(list(A),nat,size_size(list(A)),slice(A,From,To,Xs)) = aa(nat,nat,minus_minus(nat,To),From) ) ) ) ).

% slice_len
tff(fact_4322_zero__less__nat__eq,axiom,
    ! [Z: int] :
      ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),nat2(Z)))
    <=> pp(aa(int,bool,ord_less(int,zero_zero(int)),Z)) ) ).

% zero_less_nat_eq
tff(fact_4323_add__neg__numeral__special_I6_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [M: num] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))),aa(A,A,uminus_uminus(A),one_one(A))) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),inc(M))) ) ).

% add_neg_numeral_special(6)
tff(fact_4324_add__neg__numeral__special_I5_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [N: num] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),one_one(A))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N))) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),inc(N))) ) ).

% add_neg_numeral_special(5)
tff(fact_4325_diff__numeral__special_I6_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [M: num] : aa(A,A,minus_minus(A,aa(num,A,numeral_numeral(A),M)),aa(A,A,uminus_uminus(A),one_one(A))) = aa(num,A,numeral_numeral(A),inc(M)) ) ).

% diff_numeral_special(6)
tff(fact_4326_diff__numeral__special_I5_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [N: num] : aa(A,A,minus_minus(A,aa(A,A,uminus_uminus(A),one_one(A))),aa(num,A,numeral_numeral(A),N)) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),inc(N))) ) ).

% diff_numeral_special(5)
tff(fact_4327_diff__nat__numeral,axiom,
    ! [V: num,V4: num] : aa(nat,nat,minus_minus(nat,aa(num,nat,numeral_numeral(nat),V)),aa(num,nat,numeral_numeral(nat),V4)) = nat2(aa(int,int,minus_minus(int,aa(num,int,numeral_numeral(int),V)),aa(num,int,numeral_numeral(int),V4))) ).

% diff_nat_numeral
tff(fact_4328_of__nat__nat,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Z: int] :
          ( pp(aa(int,bool,ord_less_eq(int,zero_zero(int)),Z))
         => ( aa(nat,A,semiring_1_of_nat(A),nat2(Z)) = aa(int,A,ring_1_of_int(A),Z) ) ) ) ).

% of_nat_nat
tff(fact_4329_numeral__power__eq__nat__cancel__iff,axiom,
    ! [X: num,N: nat,Y: int] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),X)),N) = nat2(Y) )
    <=> ( aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),X)),N) = Y ) ) ).

% numeral_power_eq_nat_cancel_iff
tff(fact_4330_nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y: int,X: num,N: nat] :
      ( ( nat2(Y) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),X)),N) )
    <=> ( Y = aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),X)),N) ) ) ).

% nat_eq_numeral_power_cancel_iff
tff(fact_4331_nat__abs__dvd__iff,axiom,
    ! [K: int,N: nat] :
      ( pp(aa(nat,bool,dvd_dvd(nat,nat2(aa(int,int,abs_abs(int),K))),N))
    <=> pp(aa(int,bool,dvd_dvd(int,K),aa(nat,int,semiring_1_of_nat(int),N))) ) ).

% nat_abs_dvd_iff
tff(fact_4332_dvd__nat__abs__iff,axiom,
    ! [N: nat,K: int] :
      ( pp(aa(nat,bool,dvd_dvd(nat,N),nat2(aa(int,int,abs_abs(int),K))))
    <=> pp(aa(int,bool,dvd_dvd(int,aa(nat,int,semiring_1_of_nat(int),N)),K)) ) ).

% dvd_nat_abs_iff
tff(fact_4333_word__div__no,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num,B2: num] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),aa(num,word(A),numeral_numeral(word(A)),A2)),aa(num,word(A),numeral_numeral(word(A)),B2)) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(word(A),int,semiring_1_unsigned(A,int),aa(num,word(A),numeral_numeral(word(A)),A2))),aa(word(A),int,semiring_1_unsigned(A,int),aa(num,word(A),numeral_numeral(word(A)),B2)))) ) ).

% word_div_no
tff(fact_4334_take__bit__of__Suc__0,axiom,
    ! [N: nat] : aa(nat,nat,bit_se2584673776208193580ke_bit(nat,N),aa(nat,nat,suc,zero_zero(nat))) = aa(bool,nat,zero_neq_one_of_bool(nat),aa(nat,bool,ord_less(nat,zero_zero(nat)),N)) ).

% take_bit_of_Suc_0
tff(fact_4335_nat__ceiling__le__eq,axiom,
    ! [X: real,A2: nat] :
      ( pp(aa(nat,bool,ord_less_eq(nat,nat2(archimedean_ceiling(real,X))),A2))
    <=> pp(aa(real,bool,ord_less_eq(real,X),aa(nat,real,semiring_1_of_nat(real),A2))) ) ).

% nat_ceiling_le_eq
tff(fact_4336_one__less__nat__eq,axiom,
    ! [Z: int] :
      ( pp(aa(nat,bool,ord_less(nat,aa(nat,nat,suc,zero_zero(nat))),nat2(Z)))
    <=> pp(aa(int,bool,ord_less(int,one_one(int)),Z)) ) ).

% one_less_nat_eq
tff(fact_4337_nat__numeral__diff__1,axiom,
    ! [V: num] : aa(nat,nat,minus_minus(nat,aa(num,nat,numeral_numeral(nat),V)),one_one(nat)) = nat2(aa(int,int,minus_minus(int,aa(num,int,numeral_numeral(int),V)),one_one(int))) ).

% nat_numeral_diff_1
tff(fact_4338_numeral__power__less__nat__cancel__iff,axiom,
    ! [X: num,N: nat,A2: int] :
      ( pp(aa(nat,bool,ord_less(nat,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),X)),N)),nat2(A2)))
    <=> pp(aa(int,bool,ord_less(int,aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),X)),N)),A2)) ) ).

% numeral_power_less_nat_cancel_iff
tff(fact_4339_nat__less__numeral__power__cancel__iff,axiom,
    ! [A2: int,X: num,N: nat] :
      ( pp(aa(nat,bool,ord_less(nat,nat2(A2)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),X)),N)))
    <=> pp(aa(int,bool,ord_less(int,A2),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),X)),N))) ) ).

% nat_less_numeral_power_cancel_iff
tff(fact_4340_numeral__power__le__nat__cancel__iff,axiom,
    ! [X: num,N: nat,A2: int] :
      ( pp(aa(nat,bool,ord_less_eq(nat,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),X)),N)),nat2(A2)))
    <=> pp(aa(int,bool,ord_less_eq(int,aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),X)),N)),A2)) ) ).

% numeral_power_le_nat_cancel_iff
tff(fact_4341_nat__le__numeral__power__cancel__iff,axiom,
    ! [A2: int,X: num,N: nat] :
      ( pp(aa(nat,bool,ord_less_eq(nat,nat2(A2)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),X)),N)))
    <=> pp(aa(int,bool,ord_less_eq(int,A2),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),X)),N))) ) ).

% nat_le_numeral_power_cancel_iff
tff(fact_4342_take__bit__nat__eq,axiom,
    ! [K: int,N: nat] :
      ( pp(aa(int,bool,ord_less_eq(int,zero_zero(int)),K))
     => ( aa(nat,nat,bit_se2584673776208193580ke_bit(nat,N),nat2(K)) = nat2(aa(int,int,bit_se2584673776208193580ke_bit(int,N),K)) ) ) ).

% take_bit_nat_eq
tff(fact_4343_nat__take__bit__eq,axiom,
    ! [K: int,N: nat] :
      ( pp(aa(int,bool,ord_less_eq(int,zero_zero(int)),K))
     => ( nat2(aa(int,int,bit_se2584673776208193580ke_bit(int,N),K)) = aa(nat,nat,bit_se2584673776208193580ke_bit(nat,N),nat2(K)) ) ) ).

% nat_take_bit_eq
tff(fact_4344_take__bit__minus,axiom,
    ! [N: nat,K: int] : aa(int,int,bit_se2584673776208193580ke_bit(int,N),aa(int,int,uminus_uminus(int),aa(int,int,bit_se2584673776208193580ke_bit(int,N),K))) = aa(int,int,bit_se2584673776208193580ke_bit(int,N),aa(int,int,uminus_uminus(int),K)) ).

% take_bit_minus
tff(fact_4345_take__bit__mult,axiom,
    ! [N: nat,K: int,L: int] : aa(int,int,bit_se2584673776208193580ke_bit(int,N),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,bit_se2584673776208193580ke_bit(int,N),K)),aa(int,int,bit_se2584673776208193580ke_bit(int,N),L))) = aa(int,int,bit_se2584673776208193580ke_bit(int,N),aa(int,int,aa(int,fun(int,int),times_times(int),K),L)) ).

% take_bit_mult
tff(fact_4346_take__bit__diff,axiom,
    ! [N: nat,K: int,L: int] : aa(int,int,bit_se2584673776208193580ke_bit(int,N),aa(int,int,minus_minus(int,aa(int,int,bit_se2584673776208193580ke_bit(int,N),K)),aa(int,int,bit_se2584673776208193580ke_bit(int,N),L))) = aa(int,int,bit_se2584673776208193580ke_bit(int,N),aa(int,int,minus_minus(int,K),L)) ).

% take_bit_diff
tff(fact_4347_take__bit__tightened__less__eq__nat,axiom,
    ! [M: nat,N: nat,Q2: nat] :
      ( pp(aa(nat,bool,ord_less_eq(nat,M),N))
     => pp(aa(nat,bool,ord_less_eq(nat,aa(nat,nat,bit_se2584673776208193580ke_bit(nat,M),Q2)),aa(nat,nat,bit_se2584673776208193580ke_bit(nat,N),Q2))) ) ).

% take_bit_tightened_less_eq_nat
tff(fact_4348_take__bit__nat__less__eq__self,axiom,
    ! [N: nat,M: nat] : pp(aa(nat,bool,ord_less_eq(nat,aa(nat,nat,bit_se2584673776208193580ke_bit(nat,N),M)),M)) ).

% take_bit_nat_less_eq_self
tff(fact_4349_num__induct,axiom,
    ! [P: fun(num,bool),X: num] :
      ( pp(aa(num,bool,P,one2))
     => ( ! [X3: num] :
            ( pp(aa(num,bool,P,X3))
           => pp(aa(num,bool,P,inc(X3))) )
       => pp(aa(num,bool,P,X)) ) ) ).

% num_induct
tff(fact_4350_add__inc,axiom,
    ! [X: num,Y: num] : aa(num,num,aa(num,fun(num,num),plus_plus(num),X),inc(Y)) = inc(aa(num,num,aa(num,fun(num,num),plus_plus(num),X),Y)) ).

% add_inc
tff(fact_4351_nat__numeral__as__int,axiom,
    ! [X4: num] : aa(num,nat,numeral_numeral(nat),X4) = nat2(aa(num,int,numeral_numeral(int),X4)) ).

% nat_numeral_as_int
tff(fact_4352_nat__zero__as__int,axiom,
    zero_zero(nat) = nat2(zero_zero(int)) ).

% nat_zero_as_int
tff(fact_4353_nat__mono,axiom,
    ! [X: int,Y: int] :
      ( pp(aa(int,bool,ord_less_eq(int,X),Y))
     => pp(aa(nat,bool,ord_less_eq(nat,nat2(X)),nat2(Y))) ) ).

% nat_mono
tff(fact_4354_nat__one__as__int,axiom,
    one_one(nat) = nat2(one_one(int)) ).

% nat_one_as_int
tff(fact_4355_eq__nat__nat__iff,axiom,
    ! [Z: int,Z5: int] :
      ( pp(aa(int,bool,ord_less_eq(int,zero_zero(int)),Z))
     => ( pp(aa(int,bool,ord_less_eq(int,zero_zero(int)),Z5))
       => ( ( nat2(Z) = nat2(Z5) )
        <=> ( Z = Z5 ) ) ) ) ).

% eq_nat_nat_iff
tff(fact_4356_all__nat,axiom,
    ! [P: fun(nat,bool)] :
      ( ! [X_13: nat] : pp(aa(nat,bool,P,X_13))
    <=> ! [X2: int] :
          ( pp(aa(int,bool,ord_less_eq(int,zero_zero(int)),X2))
         => pp(aa(nat,bool,P,nat2(X2))) ) ) ).

% all_nat
tff(fact_4357_ex__nat,axiom,
    ! [P: fun(nat,bool)] :
      ( ? [X_13: nat] : pp(aa(nat,bool,P,X_13))
    <=> ? [X2: int] :
          ( pp(aa(int,bool,ord_less_eq(int,zero_zero(int)),X2))
          & pp(aa(nat,bool,P,nat2(X2))) ) ) ).

% ex_nat
tff(fact_4358_uint__0__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A)] :
          ( ( aa(word(A),int,semiring_1_unsigned(A,int),X) = zero_zero(int) )
        <=> ( X = zero_zero(word(A)) ) ) ) ).

% uint_0_iff
tff(fact_4359_uint__0__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( aa(word(A),int,semiring_1_unsigned(A,int),zero_zero(word(A))) = zero_zero(int) ) ) ).

% uint_0_eq
tff(fact_4360_uint__1__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( aa(word(A),int,semiring_1_unsigned(A,int),one_one(word(A))) = one_one(int) ) ) ).

% uint_1_eq
tff(fact_4361_take__bit__tightened__less__eq__int,axiom,
    ! [M: nat,N: nat,K: int] :
      ( pp(aa(nat,bool,ord_less_eq(nat,M),N))
     => pp(aa(int,bool,ord_less_eq(int,aa(int,int,bit_se2584673776208193580ke_bit(int,M),K)),aa(int,int,bit_se2584673776208193580ke_bit(int,N),K))) ) ).

% take_bit_tightened_less_eq_int
tff(fact_4362_take__bit__int__less__eq__self__iff,axiom,
    ! [N: nat,K: int] :
      ( pp(aa(int,bool,ord_less_eq(int,aa(int,int,bit_se2584673776208193580ke_bit(int,N),K)),K))
    <=> pp(aa(int,bool,ord_less_eq(int,zero_zero(int)),K)) ) ).

% take_bit_int_less_eq_self_iff
tff(fact_4363_take__bit__nonnegative,axiom,
    ! [N: nat,K: int] : pp(aa(int,bool,ord_less_eq(int,zero_zero(int)),aa(int,int,bit_se2584673776208193580ke_bit(int,N),K))) ).

% take_bit_nonnegative
tff(fact_4364_take__bit__int__greater__self__iff,axiom,
    ! [K: int,N: nat] :
      ( pp(aa(int,bool,ord_less(int,K),aa(int,int,bit_se2584673776208193580ke_bit(int,N),K)))
    <=> pp(aa(int,bool,ord_less(int,K),zero_zero(int))) ) ).

% take_bit_int_greater_self_iff
tff(fact_4365_not__take__bit__negative,axiom,
    ! [N: nat,K: int] : ~ pp(aa(int,bool,ord_less(int,aa(int,int,bit_se2584673776208193580ke_bit(int,N),K)),zero_zero(int))) ).

% not_take_bit_negative
tff(fact_4366_unsigned__greater__eq,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & unique1627219031080169319umeral(A) )
     => ! [W: word(B)] : pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),aa(word(B),A,semiring_1_unsigned(B,A),W))) ) ).

% unsigned_greater_eq
tff(fact_4367_uint__div,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A)] : aa(word(A),int,semiring_1_unsigned(A,int),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),X),Y)) = aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(word(A),int,semiring_1_unsigned(A,int),X)),aa(word(A),int,semiring_1_unsigned(A,int),Y)) ) ).

% uint_div
tff(fact_4368_uint__div__distrib,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [V: word(A),W: word(A)] : aa(word(A),int,semiring_1_unsigned(A,int),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),V),W)) = aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(word(A),int,semiring_1_unsigned(A,int),V)),aa(word(A),int,semiring_1_unsigned(A,int),W)) ) ).

% uint_div_distrib
tff(fact_4369_word__less__eq__iff__unsigned,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & linordered_semidom(A) )
     => ! [A2: word(B),B2: word(B)] :
          ( pp(aa(word(B),bool,ord_less_eq(word(B),A2),B2))
        <=> pp(aa(A,bool,ord_less_eq(A,aa(word(B),A,semiring_1_unsigned(B,A),A2)),aa(word(B),A,semiring_1_unsigned(B,A),B2))) ) ) ).

% word_less_eq_iff_unsigned
tff(fact_4370_unsigned__eq__0__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & semiring_char_0(A) )
     => ! [W: word(B)] :
          ( ( aa(word(B),A,semiring_1_unsigned(B,A),W) = zero_zero(A) )
        <=> ( W = zero_zero(word(B)) ) ) ) ).

% unsigned_eq_0_iff
tff(fact_4371_unset__bit__nat__def,axiom,
    ! [M: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se2638667681897837118et_bit(nat),M),N) = nat2(aa(int,int,aa(nat,fun(int,int),bit_se2638667681897837118et_bit(int),M),aa(nat,int,semiring_1_of_nat(int),N))) ).

% unset_bit_nat_def
tff(fact_4372_inc_Osimps_I1_J,axiom,
    inc(one2) = bit0(one2) ).

% inc.simps(1)
tff(fact_4373_inc_Osimps_I2_J,axiom,
    ! [X: num] : inc(bit0(X)) = aa(num,num,bit1,X) ).

% inc.simps(2)
tff(fact_4374_inc_Osimps_I3_J,axiom,
    ! [X: num] : inc(aa(num,num,bit1,X)) = bit0(inc(X)) ).

% inc.simps(3)
tff(fact_4375_add__One,axiom,
    ! [X: num] : aa(num,num,aa(num,fun(num,num),plus_plus(num),X),one2) = inc(X) ).

% add_One
tff(fact_4376_mult__inc,axiom,
    ! [X: num,Y: num] : aa(num,num,aa(num,fun(num,num),times_times(num),X),inc(Y)) = aa(num,num,aa(num,fun(num,num),plus_plus(num),aa(num,num,aa(num,fun(num,num),times_times(num),X),Y)),X) ).

% mult_inc
tff(fact_4377_nat__mono__iff,axiom,
    ! [Z: int,W: int] :
      ( pp(aa(int,bool,ord_less(int,zero_zero(int)),Z))
     => ( pp(aa(nat,bool,ord_less(nat,nat2(W)),nat2(Z)))
      <=> pp(aa(int,bool,ord_less(int,W),Z)) ) ) ).

% nat_mono_iff
tff(fact_4378_zless__nat__eq__int__zless,axiom,
    ! [M: nat,Z: int] :
      ( pp(aa(nat,bool,ord_less(nat,M),nat2(Z)))
    <=> pp(aa(int,bool,ord_less(int,aa(nat,int,semiring_1_of_nat(int),M)),Z)) ) ).

% zless_nat_eq_int_zless
tff(fact_4379_nat__le__iff,axiom,
    ! [X: int,N: nat] :
      ( pp(aa(nat,bool,ord_less_eq(nat,nat2(X)),N))
    <=> pp(aa(int,bool,ord_less_eq(int,X),aa(nat,int,semiring_1_of_nat(int),N))) ) ).

% nat_le_iff
tff(fact_4380_of__nat__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [R2: A] : pp(aa(A,bool,ord_less_eq(A,R2),aa(nat,A,semiring_1_of_nat(A),nat2(archimedean_ceiling(A,R2))))) ) ).

% of_nat_ceiling
tff(fact_4381_nat__int__add,axiom,
    ! [A2: nat,B2: nat] : nat2(aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(nat,int,semiring_1_of_nat(int),A2)),aa(nat,int,semiring_1_of_nat(int),B2))) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A2),B2) ).

% nat_int_add
tff(fact_4382_int__eq__iff,axiom,
    ! [M: nat,Z: int] :
      ( ( aa(nat,int,semiring_1_of_nat(int),M) = Z )
    <=> ( ( M = nat2(Z) )
        & pp(aa(int,bool,ord_less_eq(int,zero_zero(int)),Z)) ) ) ).

% int_eq_iff
tff(fact_4383_nat__0__le,axiom,
    ! [Z: int] :
      ( pp(aa(int,bool,ord_less_eq(int,zero_zero(int)),Z))
     => ( aa(nat,int,semiring_1_of_nat(int),nat2(Z)) = Z ) ) ).

% nat_0_le
tff(fact_4384_int__minus,axiom,
    ! [N: nat,M: nat] : aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,minus_minus(nat,N),M)) = aa(nat,int,semiring_1_of_nat(int),nat2(aa(int,int,minus_minus(int,aa(nat,int,semiring_1_of_nat(int),N)),aa(nat,int,semiring_1_of_nat(int),M)))) ).

% int_minus
tff(fact_4385_uint__add__ge0,axiom,
    ! [Aa: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(Aa) )
     => ! [Xa2: word(Aa),X: word(A)] : pp(aa(int,bool,ord_less_eq(int,zero_zero(int)),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(word(Aa),int,semiring_1_unsigned(Aa,int),Xa2)),aa(word(A),int,semiring_1_unsigned(A,int),X)))) ) ).

% uint_add_ge0
tff(fact_4386_uint__mult__ge0,axiom,
    ! [Aa: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(Aa) )
     => ! [Xa2: word(Aa),X: word(A)] : pp(aa(int,bool,ord_less_eq(int,zero_zero(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(word(Aa),int,semiring_1_unsigned(Aa,int),Xa2)),aa(word(A),int,semiring_1_unsigned(A,int),X)))) ) ).

% uint_mult_ge0
tff(fact_4387_nat__abs__mult__distrib,axiom,
    ! [W: int,Z: int] : nat2(aa(int,int,abs_abs(int),aa(int,int,aa(int,fun(int,int),times_times(int),W),Z))) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),nat2(aa(int,int,abs_abs(int),W))),nat2(aa(int,int,abs_abs(int),Z))) ).

% nat_abs_mult_distrib
tff(fact_4388_uint__add__le,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A)] : pp(aa(int,bool,ord_less_eq(int,aa(word(A),int,semiring_1_unsigned(A,int),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),X),Y))),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(word(A),int,semiring_1_unsigned(A,int),X)),aa(word(A),int,semiring_1_unsigned(A,int),Y)))) ) ).

% uint_add_le
tff(fact_4389_uint__plus__simple__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A)] :
          ( pp(aa(word(A),bool,ord_less_eq(word(A),X),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),X),Y)))
        <=> ( aa(word(A),int,semiring_1_unsigned(A,int),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),X),Y)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(word(A),int,semiring_1_unsigned(A,int),X)),aa(word(A),int,semiring_1_unsigned(A,int),Y)) ) ) ) ).

% uint_plus_simple_iff
tff(fact_4390_uint__plus__simple,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A)] :
          ( pp(aa(word(A),bool,ord_less_eq(word(A),X),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),X),Y)))
         => ( aa(word(A),int,semiring_1_unsigned(A,int),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),X),Y)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(word(A),int,semiring_1_unsigned(A,int),X)),aa(word(A),int,semiring_1_unsigned(A,int),Y)) ) ) ) ).

% uint_plus_simple
tff(fact_4391_word__add__def,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),A2),B2) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(word(A),int,semiring_1_unsigned(A,int),A2)),aa(word(A),int,semiring_1_unsigned(A,int),B2))) ) ).

% word_add_def
tff(fact_4392_take__bit__decr__eq,axiom,
    ! [N: nat,K: int] :
      ( ( aa(int,int,bit_se2584673776208193580ke_bit(int,N),K) != zero_zero(int) )
     => ( aa(int,int,bit_se2584673776208193580ke_bit(int,N),aa(int,int,minus_minus(int,K),one_one(int))) = aa(int,int,minus_minus(int,aa(int,int,bit_se2584673776208193580ke_bit(int,N),K)),one_one(int)) ) ) ).

% take_bit_decr_eq
tff(fact_4393_word__arith__power__alt,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),N: nat] : aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),A2),N) = aa(int,word(A),ring_1_of_int(word(A)),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(word(A),int,semiring_1_unsigned(A,int),A2)),N)) ) ).

% word_arith_power_alt
tff(fact_4394_word__mult__def,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),A2),B2) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(word(A),int,semiring_1_unsigned(A,int),A2)),aa(word(A),int,semiring_1_unsigned(A,int),B2))) ) ).

% word_mult_def
tff(fact_4395_word__div__def,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),A2),B2) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(word(A),int,semiring_1_unsigned(A,int),A2)),aa(word(A),int,semiring_1_unsigned(A,int),B2))) ) ).

% word_div_def
tff(fact_4396_real__nat__ceiling__ge,axiom,
    ! [X: real] : pp(aa(real,bool,ord_less_eq(real,X),aa(nat,real,semiring_1_of_nat(real),nat2(archimedean_ceiling(real,X))))) ).

% real_nat_ceiling_ge
tff(fact_4397_nat__plus__as__int,axiom,
    ! [X4: nat,Xa: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),X4),Xa) = nat2(aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(nat,int,semiring_1_of_nat(int),X4)),aa(nat,int,semiring_1_of_nat(int),Xa))) ).

% nat_plus_as_int
tff(fact_4398_nat__times__as__int,axiom,
    ! [X4: nat,Xa: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),X4),Xa) = nat2(aa(int,int,aa(int,fun(int,int),times_times(int),aa(nat,int,semiring_1_of_nat(int),X4)),aa(nat,int,semiring_1_of_nat(int),Xa))) ).

% nat_times_as_int
tff(fact_4399_nat__minus__as__int,axiom,
    ! [X4: nat,Xa: nat] : aa(nat,nat,minus_minus(nat,X4),Xa) = nat2(aa(int,int,minus_minus(int,aa(nat,int,semiring_1_of_nat(int),X4)),aa(nat,int,semiring_1_of_nat(int),Xa))) ).

% nat_minus_as_int
tff(fact_4400_nat__div__as__int,axiom,
    ! [X4: nat,Xa: nat] : aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),X4),Xa) = nat2(aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(nat,int,semiring_1_of_nat(int),X4)),aa(nat,int,semiring_1_of_nat(int),Xa))) ).

% nat_div_as_int
tff(fact_4401_nat__mod__as__int,axiom,
    ! [X4: nat,Xa: nat] : modulo_modulo(nat,X4,Xa) = nat2(modulo_modulo(int,aa(nat,int,semiring_1_of_nat(int),X4),aa(nat,int,semiring_1_of_nat(int),Xa))) ).

% nat_mod_as_int
tff(fact_4402_numeral__inc,axiom,
    ! [A: $tType] :
      ( numeral(A)
     => ! [X: num] : aa(num,A,numeral_numeral(A),inc(X)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),X)),one_one(A)) ) ).

% numeral_inc
tff(fact_4403_of__nat__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [R2: A] :
          ( pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),R2))
         => pp(aa(A,bool,ord_less_eq(A,aa(nat,A,semiring_1_of_nat(A),nat2(archim6421214686448440834_floor(A,R2)))),R2)) ) ) ).

% of_nat_floor
tff(fact_4404_nat__less__eq__zless,axiom,
    ! [W: int,Z: int] :
      ( pp(aa(int,bool,ord_less_eq(int,zero_zero(int)),W))
     => ( pp(aa(nat,bool,ord_less(nat,nat2(W)),nat2(Z)))
      <=> pp(aa(int,bool,ord_less(int,W),Z)) ) ) ).

% nat_less_eq_zless
tff(fact_4405_nat__eq__iff2,axiom,
    ! [M: nat,W: int] :
      ( ( M = nat2(W) )
    <=> ( ( pp(aa(int,bool,ord_less_eq(int,zero_zero(int)),W))
         => ( W = aa(nat,int,semiring_1_of_nat(int),M) ) )
        & ( ~ pp(aa(int,bool,ord_less_eq(int,zero_zero(int)),W))
         => ( M = zero_zero(nat) ) ) ) ) ).

% nat_eq_iff2
tff(fact_4406_nat__eq__iff,axiom,
    ! [W: int,M: nat] :
      ( ( nat2(W) = M )
    <=> ( ( pp(aa(int,bool,ord_less_eq(int,zero_zero(int)),W))
         => ( W = aa(nat,int,semiring_1_of_nat(int),M) ) )
        & ( ~ pp(aa(int,bool,ord_less_eq(int,zero_zero(int)),W))
         => ( M = zero_zero(nat) ) ) ) ) ).

% nat_eq_iff
tff(fact_4407_split__nat,axiom,
    ! [P: fun(nat,bool),I: int] :
      ( pp(aa(nat,bool,P,nat2(I)))
    <=> ( ! [N5: nat] :
            ( ( I = aa(nat,int,semiring_1_of_nat(int),N5) )
           => pp(aa(nat,bool,P,N5)) )
        & ( pp(aa(int,bool,ord_less(int,I),zero_zero(int)))
         => pp(aa(nat,bool,P,zero_zero(nat))) ) ) ) ).

% split_nat
tff(fact_4408_nat__le__eq__zle,axiom,
    ! [W: int,Z: int] :
      ( ( pp(aa(int,bool,ord_less(int,zero_zero(int)),W))
        | pp(aa(int,bool,ord_less_eq(int,zero_zero(int)),Z)) )
     => ( pp(aa(nat,bool,ord_less_eq(nat,nat2(W)),nat2(Z)))
      <=> pp(aa(int,bool,ord_less_eq(int,W),Z)) ) ) ).

% nat_le_eq_zle
tff(fact_4409_nat__add__distrib,axiom,
    ! [Z: int,Z5: int] :
      ( pp(aa(int,bool,ord_less_eq(int,zero_zero(int)),Z))
     => ( pp(aa(int,bool,ord_less_eq(int,zero_zero(int)),Z5))
       => ( nat2(aa(int,int,aa(int,fun(int,int),plus_plus(int),Z),Z5)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),nat2(Z)),nat2(Z5)) ) ) ) ).

% nat_add_distrib
tff(fact_4410_le__nat__iff,axiom,
    ! [K: int,N: nat] :
      ( pp(aa(int,bool,ord_less_eq(int,zero_zero(int)),K))
     => ( pp(aa(nat,bool,ord_less_eq(nat,N),nat2(K)))
      <=> pp(aa(int,bool,ord_less_eq(int,aa(nat,int,semiring_1_of_nat(int),N)),K)) ) ) ).

% le_nat_iff
tff(fact_4411_Suc__as__int,axiom,
    ! [X4: nat] : aa(nat,nat,suc,X4) = nat2(aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(nat,int,semiring_1_of_nat(int),X4)),one_one(int))) ).

% Suc_as_int
tff(fact_4412_le__mult__nat__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [A2: A,B2: A] : pp(aa(nat,bool,ord_less_eq(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),nat2(archim6421214686448440834_floor(A,A2))),nat2(archim6421214686448440834_floor(A,B2)))),nat2(archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2))))) ) ).

% le_mult_nat_floor
tff(fact_4413_nat__mult__distrib,axiom,
    ! [Z: int,Z5: int] :
      ( pp(aa(int,bool,ord_less_eq(int,zero_zero(int)),Z))
     => ( nat2(aa(int,int,aa(int,fun(int,int),times_times(int),Z),Z5)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),nat2(Z)),nat2(Z5)) ) ) ).

% nat_mult_distrib
tff(fact_4414_nat__diff__distrib_H,axiom,
    ! [X: int,Y: int] :
      ( pp(aa(int,bool,ord_less_eq(int,zero_zero(int)),X))
     => ( pp(aa(int,bool,ord_less_eq(int,zero_zero(int)),Y))
       => ( nat2(aa(int,int,minus_minus(int,X),Y)) = aa(nat,nat,minus_minus(nat,nat2(X)),nat2(Y)) ) ) ) ).

% nat_diff_distrib'
tff(fact_4415_nat__diff__distrib,axiom,
    ! [Z5: int,Z: int] :
      ( pp(aa(int,bool,ord_less_eq(int,zero_zero(int)),Z5))
     => ( pp(aa(int,bool,ord_less_eq(int,Z5),Z))
       => ( nat2(aa(int,int,minus_minus(int,Z),Z5)) = aa(nat,nat,minus_minus(nat,nat2(Z)),nat2(Z5)) ) ) ) ).

% nat_diff_distrib
tff(fact_4416_nat__abs__triangle__ineq,axiom,
    ! [K: int,L: int] : pp(aa(nat,bool,ord_less_eq(nat,nat2(aa(int,int,abs_abs(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),K),L)))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),nat2(aa(int,int,abs_abs(int),K))),nat2(aa(int,int,abs_abs(int),L))))) ).

% nat_abs_triangle_ineq
tff(fact_4417_nat__div__distrib,axiom,
    ! [X: int,Y: int] :
      ( pp(aa(int,bool,ord_less_eq(int,zero_zero(int)),X))
     => ( nat2(aa(int,int,aa(int,fun(int,int),divide_divide(int),X),Y)) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),nat2(X)),nat2(Y)) ) ) ).

% nat_div_distrib
tff(fact_4418_nat__div__distrib_H,axiom,
    ! [Y: int,X: int] :
      ( pp(aa(int,bool,ord_less_eq(int,zero_zero(int)),Y))
     => ( nat2(aa(int,int,aa(int,fun(int,int),divide_divide(int),X),Y)) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),nat2(X)),nat2(Y)) ) ) ).

% nat_div_distrib'
tff(fact_4419_nat__power__eq,axiom,
    ! [Z: int,N: nat] :
      ( pp(aa(int,bool,ord_less_eq(int,zero_zero(int)),Z))
     => ( nat2(aa(nat,int,aa(int,fun(nat,int),power_power(int),Z),N)) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),nat2(Z)),N) ) ) ).

% nat_power_eq
tff(fact_4420_nat__mod__distrib,axiom,
    ! [X: int,Y: int] :
      ( pp(aa(int,bool,ord_less_eq(int,zero_zero(int)),X))
     => ( pp(aa(int,bool,ord_less_eq(int,zero_zero(int)),Y))
       => ( nat2(modulo_modulo(int,X,Y)) = modulo_modulo(nat,nat2(X),nat2(Y)) ) ) ) ).

% nat_mod_distrib
tff(fact_4421_udvd__incr__lem0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Up: int,Uq: int,N: int,K6: word(A),N3: int] :
          ( pp(aa(int,bool,ord_less(int,Up),Uq))
         => ( ( Up = aa(int,int,aa(int,fun(int,int),times_times(int),N),aa(word(A),int,semiring_1_unsigned(A,int),K6)) )
           => ( ( Uq = aa(int,int,aa(int,fun(int,int),times_times(int),N3),aa(word(A),int,semiring_1_unsigned(A,int),K6)) )
             => pp(aa(int,bool,ord_less_eq(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),Up),aa(word(A),int,semiring_1_unsigned(A,int),K6))),Uq)) ) ) ) ) ).

% udvd_incr_lem0
tff(fact_4422_udvd__incr__lem,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Up: int,Uq: int,Ua: int,N: int,K6: word(A),N3: int] :
          ( pp(aa(int,bool,ord_less(int,Up),Uq))
         => ( ( Up = aa(int,int,aa(int,fun(int,int),plus_plus(int),Ua),aa(int,int,aa(int,fun(int,int),times_times(int),N),aa(word(A),int,semiring_1_unsigned(A,int),K6))) )
           => ( ( Uq = aa(int,int,aa(int,fun(int,int),plus_plus(int),Ua),aa(int,int,aa(int,fun(int,int),times_times(int),N3),aa(word(A),int,semiring_1_unsigned(A,int),K6))) )
             => pp(aa(int,bool,ord_less_eq(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),Up),aa(word(A),int,semiring_1_unsigned(A,int),K6))),Uq)) ) ) ) ) ).

% udvd_incr_lem
tff(fact_4423_word__of__int__nat,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: int] :
          ( pp(aa(int,bool,ord_less_eq(int,zero_zero(int)),X))
         => ( aa(int,word(A),ring_1_of_int(word(A)),X) = aa(nat,word(A),semiring_1_of_nat(word(A)),nat2(X)) ) ) ) ).

% word_of_int_nat
tff(fact_4424_div__abs__eq__div__nat,axiom,
    ! [K: int,L: int] : aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,abs_abs(int),K)),aa(int,int,abs_abs(int),L)) = aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),nat2(aa(int,int,abs_abs(int),K))),nat2(aa(int,int,abs_abs(int),L)))) ).

% div_abs_eq_div_nat
tff(fact_4425_nat__floor__neg,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,ord_less_eq(real,X),zero_zero(real)))
     => ( nat2(archim6421214686448440834_floor(real,X)) = zero_zero(nat) ) ) ).

% nat_floor_neg
tff(fact_4426_mod__abs__eq__div__nat,axiom,
    ! [K: int,L: int] : modulo_modulo(int,aa(int,int,abs_abs(int),K),aa(int,int,abs_abs(int),L)) = aa(nat,int,semiring_1_of_nat(int),modulo_modulo(nat,nat2(aa(int,int,abs_abs(int),K)),nat2(aa(int,int,abs_abs(int),L)))) ).

% mod_abs_eq_div_nat
tff(fact_4427_udvd__incr0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [P2: word(A),Q2: word(A),N: int,K6: word(A),N3: int] :
          ( pp(aa(word(A),bool,ord_less(word(A),P2),Q2))
         => ( ( aa(word(A),int,semiring_1_unsigned(A,int),P2) = aa(int,int,aa(int,fun(int,int),times_times(int),N),aa(word(A),int,semiring_1_unsigned(A,int),K6)) )
           => ( ( aa(word(A),int,semiring_1_unsigned(A,int),Q2) = aa(int,int,aa(int,fun(int,int),times_times(int),N3),aa(word(A),int,semiring_1_unsigned(A,int),K6)) )
             => pp(aa(word(A),bool,ord_less_eq(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),P2),K6)),Q2)) ) ) ) ) ).

% udvd_incr0
tff(fact_4428_udvd__decr0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [P2: word(A),Q2: word(A),N: int,K6: word(A),N3: int] :
          ( pp(aa(word(A),bool,ord_less(word(A),P2),Q2))
         => ( ( aa(word(A),int,semiring_1_unsigned(A,int),P2) = aa(int,int,aa(int,fun(int,int),times_times(int),N),aa(word(A),int,semiring_1_unsigned(A,int),K6)) )
           => ( ( aa(word(A),int,semiring_1_unsigned(A,int),Q2) = aa(int,int,aa(int,fun(int,int),times_times(int),N3),aa(word(A),int,semiring_1_unsigned(A,int),K6)) )
             => ( ( aa(word(A),int,semiring_1_unsigned(A,int),Q2) = aa(int,int,aa(int,fun(int,int),times_times(int),N3),aa(word(A),int,semiring_1_unsigned(A,int),K6)) )
               => pp(aa(word(A),bool,ord_less_eq(word(A),P2),aa(word(A),word(A),minus_minus(word(A),Q2),K6))) ) ) ) ) ) ).

% udvd_decr0
tff(fact_4429_floor__eq3,axiom,
    ! [N: nat,X: real] :
      ( pp(aa(real,bool,ord_less(real,aa(nat,real,semiring_1_of_nat(real),N)),X))
     => ( pp(aa(real,bool,ord_less(real,X),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,N))))
       => ( nat2(archim6421214686448440834_floor(real,X)) = N ) ) ) ).

% floor_eq3
tff(fact_4430_le__nat__floor,axiom,
    ! [X: nat,A2: real] :
      ( pp(aa(real,bool,ord_less_eq(real,aa(nat,real,semiring_1_of_nat(real),X)),A2))
     => pp(aa(nat,bool,ord_less_eq(nat,X),nat2(archim6421214686448440834_floor(real,A2)))) ) ).

% le_nat_floor
tff(fact_4431_take__bit__nat__eq__self__iff,axiom,
    ! [N: nat,M: nat] :
      ( ( aa(nat,nat,bit_se2584673776208193580ke_bit(nat,N),M) = M )
    <=> pp(aa(nat,bool,ord_less(nat,M),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N))) ) ).

% take_bit_nat_eq_self_iff
tff(fact_4432_take__bit__nat__less__exp,axiom,
    ! [N: nat,M: nat] : pp(aa(nat,bool,ord_less(nat,aa(nat,nat,bit_se2584673776208193580ke_bit(nat,N),M)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N))) ).

% take_bit_nat_less_exp
tff(fact_4433_take__bit__nat__eq__self,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,ord_less(nat,M),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)))
     => ( aa(nat,nat,bit_se2584673776208193580ke_bit(nat,N),M) = M ) ) ).

% take_bit_nat_eq_self
tff(fact_4434_take__bit__nat__def,axiom,
    ! [N: nat,M: nat] : aa(nat,nat,bit_se2584673776208193580ke_bit(nat,N),M) = modulo_modulo(nat,M,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)) ).

% take_bit_nat_def
tff(fact_4435_nat__2,axiom,
    nat2(aa(num,int,numeral_numeral(int),bit0(one2))) = aa(nat,nat,suc,aa(nat,nat,suc,zero_zero(nat))) ).

% nat_2
tff(fact_4436_take__bit__Suc__minus__bit1,axiom,
    ! [N: nat,K: num] : aa(int,int,bit_se2584673776208193580ke_bit(int,aa(nat,nat,suc,N)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,K)))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,bit_se2584673776208193580ke_bit(int,N),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),inc(K))))),aa(num,int,numeral_numeral(int),bit0(one2)))),one_one(int)) ).

% take_bit_Suc_minus_bit1
tff(fact_4437_Suc__nat__eq__nat__zadd1,axiom,
    ! [Z: int] :
      ( pp(aa(int,bool,ord_less_eq(int,zero_zero(int)),Z))
     => ( aa(nat,nat,suc,nat2(Z)) = nat2(aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),Z)) ) ) ).

% Suc_nat_eq_nat_zadd1
tff(fact_4438_nat__less__iff,axiom,
    ! [W: int,M: nat] :
      ( pp(aa(int,bool,ord_less_eq(int,zero_zero(int)),W))
     => ( pp(aa(nat,bool,ord_less(nat,nat2(W)),M))
      <=> pp(aa(int,bool,ord_less(int,W),aa(nat,int,semiring_1_of_nat(int),M))) ) ) ).

% nat_less_iff
tff(fact_4439_nat__mult__distrib__neg,axiom,
    ! [Z: int,Z5: int] :
      ( pp(aa(int,bool,ord_less_eq(int,Z),zero_zero(int)))
     => ( nat2(aa(int,int,aa(int,fun(int,int),times_times(int),Z),Z5)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),nat2(aa(int,int,uminus_uminus(int),Z))),nat2(aa(int,int,uminus_uminus(int),Z5))) ) ) ).

% nat_mult_distrib_neg
tff(fact_4440_nat__abs__int__diff,axiom,
    ! [A2: nat,B2: nat] :
      ( ( pp(aa(nat,bool,ord_less_eq(nat,A2),B2))
       => ( nat2(aa(int,int,abs_abs(int),aa(int,int,minus_minus(int,aa(nat,int,semiring_1_of_nat(int),A2)),aa(nat,int,semiring_1_of_nat(int),B2)))) = aa(nat,nat,minus_minus(nat,B2),A2) ) )
      & ( ~ pp(aa(nat,bool,ord_less_eq(nat,A2),B2))
       => ( nat2(aa(int,int,abs_abs(int),aa(int,int,minus_minus(int,aa(nat,int,semiring_1_of_nat(int),A2)),aa(nat,int,semiring_1_of_nat(int),B2)))) = aa(nat,nat,minus_minus(nat,A2),B2) ) ) ) ).

% nat_abs_int_diff
tff(fact_4441_take__bit__int__less__exp,axiom,
    ! [N: nat,K: int] : pp(aa(int,bool,ord_less(int,aa(int,int,bit_se2584673776208193580ke_bit(int,N),K)),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),N))) ).

% take_bit_int_less_exp
tff(fact_4442_take__bit__int__def,axiom,
    ! [N: nat,K: int] : aa(int,int,bit_se2584673776208193580ke_bit(int,N),K) = modulo_modulo(int,K,aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),N)) ).

% take_bit_int_def
tff(fact_4443_floor__eq4,axiom,
    ! [N: nat,X: real] :
      ( pp(aa(real,bool,ord_less_eq(real,aa(nat,real,semiring_1_of_nat(real),N)),X))
     => ( pp(aa(real,bool,ord_less(real,X),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,N))))
       => ( nat2(archim6421214686448440834_floor(real,X)) = N ) ) ) ).

% floor_eq4
tff(fact_4444_udvd__incr_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [P2: word(A),Q2: word(A),Ua: int,N: int,K6: word(A),N3: int] :
          ( pp(aa(word(A),bool,ord_less(word(A),P2),Q2))
         => ( ( aa(word(A),int,semiring_1_unsigned(A,int),P2) = aa(int,int,aa(int,fun(int,int),plus_plus(int),Ua),aa(int,int,aa(int,fun(int,int),times_times(int),N),aa(word(A),int,semiring_1_unsigned(A,int),K6))) )
           => ( ( aa(word(A),int,semiring_1_unsigned(A,int),Q2) = aa(int,int,aa(int,fun(int,int),plus_plus(int),Ua),aa(int,int,aa(int,fun(int,int),times_times(int),N3),aa(word(A),int,semiring_1_unsigned(A,int),K6))) )
             => pp(aa(word(A),bool,ord_less_eq(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),P2),K6)),Q2)) ) ) ) ) ).

% udvd_incr'
tff(fact_4445_udvd__decr_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [P2: word(A),Q2: word(A),Ua: int,N: int,K6: word(A),N3: int] :
          ( pp(aa(word(A),bool,ord_less(word(A),P2),Q2))
         => ( ( aa(word(A),int,semiring_1_unsigned(A,int),P2) = aa(int,int,aa(int,fun(int,int),plus_plus(int),Ua),aa(int,int,aa(int,fun(int,int),times_times(int),N),aa(word(A),int,semiring_1_unsigned(A,int),K6))) )
           => ( ( aa(word(A),int,semiring_1_unsigned(A,int),Q2) = aa(int,int,aa(int,fun(int,int),plus_plus(int),Ua),aa(int,int,aa(int,fun(int,int),times_times(int),N3),aa(word(A),int,semiring_1_unsigned(A,int),K6))) )
             => ( ( aa(word(A),int,semiring_1_unsigned(A,int),Q2) = aa(int,int,aa(int,fun(int,int),plus_plus(int),Ua),aa(int,int,aa(int,fun(int,int),times_times(int),N3),aa(word(A),int,semiring_1_unsigned(A,int),K6))) )
               => pp(aa(word(A),bool,ord_less_eq(word(A),P2),aa(word(A),word(A),minus_minus(word(A),Q2),K6))) ) ) ) ) ) ).

% udvd_decr'
tff(fact_4446_diff__nat__eq__if,axiom,
    ! [Z5: int,Z: int] :
      ( ( pp(aa(int,bool,ord_less(int,Z5),zero_zero(int)))
       => ( aa(nat,nat,minus_minus(nat,nat2(Z)),nat2(Z5)) = nat2(Z) ) )
      & ( ~ pp(aa(int,bool,ord_less(int,Z5),zero_zero(int)))
       => ( aa(nat,nat,minus_minus(nat,nat2(Z)),nat2(Z5)) = if(nat,aa(int,bool,ord_less(int,aa(int,int,minus_minus(int,Z),Z5)),zero_zero(int)),zero_zero(nat),nat2(aa(int,int,minus_minus(int,Z),Z5))) ) ) ) ).

% diff_nat_eq_if
tff(fact_4447_take__bit__numeral__minus__bit1,axiom,
    ! [L: num,K: num] : aa(int,int,bit_se2584673776208193580ke_bit(int,aa(num,nat,numeral_numeral(nat),L)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,K)))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,bit_se2584673776208193580ke_bit(int,pred_numeral(L)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),inc(K))))),aa(num,int,numeral_numeral(int),bit0(one2)))),one_one(int)) ).

% take_bit_numeral_minus_bit1
tff(fact_4448_take__bit__nat__less__self__iff,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa(nat,bool,ord_less(nat,aa(nat,nat,bit_se2584673776208193580ke_bit(nat,N),M)),M))
    <=> pp(aa(nat,bool,ord_less_eq(nat,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)),M)) ) ).

% take_bit_nat_less_self_iff
tff(fact_4449_of__int__of__nat,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [K: int] :
          ( ( pp(aa(int,bool,ord_less(int,K),zero_zero(int)))
           => ( aa(int,A,ring_1_of_int(A),K) = aa(A,A,uminus_uminus(A),aa(nat,A,semiring_1_of_nat(A),nat2(aa(int,int,uminus_uminus(int),K)))) ) )
          & ( ~ pp(aa(int,bool,ord_less(int,K),zero_zero(int)))
           => ( aa(int,A,ring_1_of_int(A),K) = aa(nat,A,semiring_1_of_nat(A),nat2(K)) ) ) ) ) ).

% of_int_of_nat
tff(fact_4450_nat__dvd__iff,axiom,
    ! [Z: int,M: nat] :
      ( pp(aa(nat,bool,dvd_dvd(nat,nat2(Z)),M))
    <=> ( ( pp(aa(int,bool,ord_less_eq(int,zero_zero(int)),Z))
         => pp(aa(int,bool,dvd_dvd(int,Z),aa(nat,int,semiring_1_of_nat(int),M))) )
        & ( ~ pp(aa(int,bool,ord_less_eq(int,zero_zero(int)),Z))
         => ( M = zero_zero(nat) ) ) ) ) ).

% nat_dvd_iff
tff(fact_4451_take__bit__Suc__minus__bit0,axiom,
    ! [N: nat,K: num] : aa(int,int,bit_se2584673776208193580ke_bit(int,aa(nat,nat,suc,N)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit0(K)))) = aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,bit_se2584673776208193580ke_bit(int,N),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),K)))),aa(num,int,numeral_numeral(int),bit0(one2))) ).

% take_bit_Suc_minus_bit0
tff(fact_4452_take__bit__int__less__self__iff,axiom,
    ! [N: nat,K: int] :
      ( pp(aa(int,bool,ord_less(int,aa(int,int,bit_se2584673776208193580ke_bit(int,N),K)),K))
    <=> pp(aa(int,bool,ord_less_eq(int,aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),N)),K)) ) ).

% take_bit_int_less_self_iff
tff(fact_4453_take__bit__int__greater__eq__self__iff,axiom,
    ! [K: int,N: nat] :
      ( pp(aa(int,bool,ord_less_eq(int,K),aa(int,int,bit_se2584673776208193580ke_bit(int,N),K)))
    <=> pp(aa(int,bool,ord_less(int,K),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),N))) ) ).

% take_bit_int_greater_eq_self_iff
tff(fact_4454_take__bit__int__eq__self__iff,axiom,
    ! [N: nat,K: int] :
      ( ( aa(int,int,bit_se2584673776208193580ke_bit(int,N),K) = K )
    <=> ( pp(aa(int,bool,ord_less_eq(int,zero_zero(int)),K))
        & pp(aa(int,bool,ord_less(int,K),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),N))) ) ) ).

% take_bit_int_eq_self_iff
tff(fact_4455_take__bit__int__eq__self,axiom,
    ! [K: int,N: nat] :
      ( pp(aa(int,bool,ord_less_eq(int,zero_zero(int)),K))
     => ( pp(aa(int,bool,ord_less(int,K),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),N)))
       => ( aa(int,int,bit_se2584673776208193580ke_bit(int,N),K) = K ) ) ) ).

% take_bit_int_eq_self
tff(fact_4456_take__bit__incr__eq,axiom,
    ! [N: nat,K: int] :
      ( ( aa(int,int,bit_se2584673776208193580ke_bit(int,N),K) != aa(int,int,minus_minus(int,aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),N)),one_one(int)) )
     => ( aa(int,int,bit_se2584673776208193580ke_bit(int,N),aa(int,int,aa(int,fun(int,int),plus_plus(int),K),one_one(int))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,bit_se2584673776208193580ke_bit(int,N),K)) ) ) ).

% take_bit_incr_eq
tff(fact_4457_take__bit__numeral__minus__bit0,axiom,
    ! [L: num,K: num] : aa(int,int,bit_se2584673776208193580ke_bit(int,aa(num,nat,numeral_numeral(nat),L)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit0(K)))) = aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,bit_se2584673776208193580ke_bit(int,pred_numeral(L)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),K)))),aa(num,int,numeral_numeral(int),bit0(one2))) ).

% take_bit_numeral_minus_bit0
tff(fact_4458_even__nat__iff,axiom,
    ! [K: int] :
      ( pp(aa(int,bool,ord_less_eq(int,zero_zero(int)),K))
     => ( pp(aa(nat,bool,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),nat2(K)))
      <=> pp(aa(int,bool,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),K)) ) ) ).

% even_nat_iff
tff(fact_4459_uint__range__size,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] :
          ( pp(aa(int,bool,ord_less_eq(int,zero_zero(int)),aa(word(A),int,semiring_1_unsigned(A,int),W)))
          & pp(aa(int,bool,ord_less(int,aa(word(A),int,semiring_1_unsigned(A,int),W)),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(word(A),nat,size_size(word(A)),W)))) ) ) ).

% uint_range_size
tff(fact_4460_uint__2p,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: nat] :
          ( pp(aa(word(A),bool,ord_less(word(A),zero_zero(word(A))),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),N)))
         => ( aa(word(A),int,semiring_1_unsigned(A,int),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),N)) = aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),N) ) ) ) ).

% uint_2p
tff(fact_4461_take__bit__int__less__eq,axiom,
    ! [N: nat,K: int] :
      ( pp(aa(int,bool,ord_less_eq(int,aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),N)),K))
     => ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),N))
       => pp(aa(int,bool,ord_less_eq(int,aa(int,int,bit_se2584673776208193580ke_bit(int,N),K)),aa(int,int,minus_minus(int,K),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),N)))) ) ) ).

% take_bit_int_less_eq
tff(fact_4462_take__bit__int__greater__eq,axiom,
    ! [K: int,N: nat] :
      ( pp(aa(int,bool,ord_less(int,K),zero_zero(int)))
     => pp(aa(int,bool,ord_less_eq(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),K),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),N))),aa(int,int,bit_se2584673776208193580ke_bit(int,N),K))) ) ).

% take_bit_int_greater_eq
tff(fact_4463_signed__take__bit__eq__take__bit__shift,axiom,
    ! [N: nat,K: int] : aa(int,int,bit_ri4674362597316999326ke_bit(int,N),K) = aa(int,int,minus_minus(int,aa(int,int,bit_se2584673776208193580ke_bit(int,aa(nat,nat,suc,N)),aa(int,int,aa(int,fun(int,int),plus_plus(int),K),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),N)))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),N)) ).

% signed_take_bit_eq_take_bit_shift
tff(fact_4464_no__plus__overflow__uint__size,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A)] :
          ( pp(aa(word(A),bool,ord_less_eq(word(A),X),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),X),Y)))
        <=> pp(aa(int,bool,ord_less(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(word(A),int,semiring_1_unsigned(A,int),X)),aa(word(A),int,semiring_1_unsigned(A,int),Y))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(word(A),nat,size_size(word(A)),X)))) ) ) ).

% no_plus_overflow_uint_size
tff(fact_4465_uint__plus__if__size,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A)] :
          ( ( pp(aa(int,bool,ord_less(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(word(A),int,semiring_1_unsigned(A,int),X)),aa(word(A),int,semiring_1_unsigned(A,int),Y))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(word(A),nat,size_size(word(A)),X))))
           => ( aa(word(A),int,semiring_1_unsigned(A,int),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),X),Y)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(word(A),int,semiring_1_unsigned(A,int),X)),aa(word(A),int,semiring_1_unsigned(A,int),Y)) ) )
          & ( ~ pp(aa(int,bool,ord_less(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(word(A),int,semiring_1_unsigned(A,int),X)),aa(word(A),int,semiring_1_unsigned(A,int),Y))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(word(A),nat,size_size(word(A)),X))))
           => ( aa(word(A),int,semiring_1_unsigned(A,int),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),X),Y)) = aa(int,int,minus_minus(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(word(A),int,semiring_1_unsigned(A,int),X)),aa(word(A),int,semiring_1_unsigned(A,int),Y))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(word(A),nat,size_size(word(A)),X))) ) ) ) ) ).

% uint_plus_if_size
tff(fact_4466_uint__sub__if__size,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),X: word(A)] :
          ( ( pp(aa(int,bool,ord_less_eq(int,aa(word(A),int,semiring_1_unsigned(A,int),Y)),aa(word(A),int,semiring_1_unsigned(A,int),X)))
           => ( aa(word(A),int,semiring_1_unsigned(A,int),aa(word(A),word(A),minus_minus(word(A),X),Y)) = aa(int,int,minus_minus(int,aa(word(A),int,semiring_1_unsigned(A,int),X)),aa(word(A),int,semiring_1_unsigned(A,int),Y)) ) )
          & ( ~ pp(aa(int,bool,ord_less_eq(int,aa(word(A),int,semiring_1_unsigned(A,int),Y)),aa(word(A),int,semiring_1_unsigned(A,int),X)))
           => ( aa(word(A),int,semiring_1_unsigned(A,int),aa(word(A),word(A),minus_minus(word(A),X),Y)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,minus_minus(int,aa(word(A),int,semiring_1_unsigned(A,int),X)),aa(word(A),int,semiring_1_unsigned(A,int),Y))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(word(A),nat,size_size(word(A)),X))) ) ) ) ) ).

% uint_sub_if_size
tff(fact_4467_powr__real__of__int,axiom,
    ! [X: real,N: int] :
      ( pp(aa(real,bool,ord_less(real,zero_zero(real)),X))
     => ( ( pp(aa(int,bool,ord_less_eq(int,zero_zero(int)),N))
         => ( powr(real,X,aa(int,real,ring_1_of_int(real),N)) = aa(nat,real,aa(real,fun(nat,real),power_power(real),X),nat2(N)) ) )
        & ( ~ pp(aa(int,bool,ord_less_eq(int,zero_zero(int)),N))
         => ( powr(real,X,aa(int,real,ring_1_of_int(real),N)) = aa(real,real,inverse_inverse(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),X),nat2(aa(int,int,uminus_uminus(int),N)))) ) ) ) ) ).

% powr_real_of_int
tff(fact_4468_take__bit__minus__small__eq,axiom,
    ! [K: int,N: nat] :
      ( pp(aa(int,bool,ord_less(int,zero_zero(int)),K))
     => ( pp(aa(int,bool,ord_less_eq(int,K),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),N)))
       => ( aa(int,int,bit_se2584673776208193580ke_bit(int,N),aa(int,int,uminus_uminus(int),K)) = aa(int,int,minus_minus(int,aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),N)),K) ) ) ) ).

% take_bit_minus_small_eq
tff(fact_4469_arctan__def,axiom,
    ! [Y: real] : aa(real,real,arctan,Y) = the(real,aTP_Lamp_la(real,fun(real,bool),Y)) ).

% arctan_def
tff(fact_4470_arcsin__def,axiom,
    ! [Y: real] : aa(real,real,arcsin,Y) = the(real,aTP_Lamp_lb(real,fun(real,bool),Y)) ).

% arcsin_def
tff(fact_4471_divide__int__def,axiom,
    ! [L: int,K: int] :
      ( ( ( L = zero_zero(int) )
       => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),K),L) = zero_zero(int) ) )
      & ( ( L != zero_zero(int) )
       => ( ( ( sgn_sgn(int,K) = sgn_sgn(int,L) )
           => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),K),L) = aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),nat2(aa(int,int,abs_abs(int),K))),nat2(aa(int,int,abs_abs(int),L)))) ) )
          & ( ( sgn_sgn(int,K) != sgn_sgn(int,L) )
           => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),K),L) = aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),nat2(aa(int,int,abs_abs(int),K))),nat2(aa(int,int,abs_abs(int),L)))),aa(bool,nat,zero_neq_one_of_bool(nat),aa(bool,bool,fNot,aa(int,bool,dvd_dvd(int,L),K)))))) ) ) ) ) ) ).

% divide_int_def
tff(fact_4472_even__set__encode__iff,axiom,
    ! [A3: set(nat)] :
      ( pp(aa(set(nat),bool,finite_finite2(nat),A3))
     => ( pp(aa(nat,bool,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(set(nat),nat,nat_set_encode,A3)))
      <=> ~ pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),zero_zero(nat)),A3)) ) ) ).

% even_set_encode_iff
tff(fact_4473_sgn__sgn,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [A2: A] : sgn_sgn(A,sgn_sgn(A,A2)) = sgn_sgn(A,A2) ) ).

% sgn_sgn
tff(fact_4474_sgn__0,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ( sgn_sgn(A,zero_zero(A)) = zero_zero(A) ) ) ).

% sgn_0
tff(fact_4475_sgn__zero,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ( sgn_sgn(A,zero_zero(A)) = zero_zero(A) ) ) ).

% sgn_zero
tff(fact_4476_sgn__1,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ( sgn_sgn(A,one_one(A)) = one_one(A) ) ) ).

% sgn_1
tff(fact_4477_sgn__one,axiom,
    ! [A: $tType] :
      ( real_V2822296259951069270ebra_1(A)
     => ( sgn_sgn(A,one_one(A)) = one_one(A) ) ) ).

% sgn_one
tff(fact_4478_sgn__divide,axiom,
    ! [A: $tType] :
      ( field_abs_sgn(A)
     => ! [A2: A,B2: A] : sgn_sgn(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),sgn_sgn(A,A2)),sgn_sgn(A,B2)) ) ).

% sgn_divide
tff(fact_4479_idom__abs__sgn__class_Osgn__minus,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [A2: A] : sgn_sgn(A,aa(A,A,uminus_uminus(A),A2)) = aa(A,A,uminus_uminus(A),sgn_sgn(A,A2)) ) ).

% idom_abs_sgn_class.sgn_minus
tff(fact_4480_power__sgn,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,N: nat] : sgn_sgn(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),N)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),sgn_sgn(A,A2)),N) ) ).

% power_sgn
tff(fact_4481_inverse__sgn,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] : aa(A,A,inverse_inverse(A),sgn_sgn(A,A2)) = sgn_sgn(A,A2) ) ).

% inverse_sgn
tff(fact_4482_sgn__inverse,axiom,
    ! [A: $tType] :
      ( field_abs_sgn(A)
     => ! [A2: A] : sgn_sgn(A,aa(A,A,inverse_inverse(A),A2)) = aa(A,A,inverse_inverse(A),sgn_sgn(A,A2)) ) ).

% sgn_inverse
tff(fact_4483_set__decode__inverse,axiom,
    ! [N: nat] : aa(set(nat),nat,nat_set_encode,nat_set_decode(N)) = N ).

% set_decode_inverse
tff(fact_4484_sgn__less,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A] :
          ( pp(aa(A,bool,ord_less(A,sgn_sgn(A,A2)),zero_zero(A)))
        <=> pp(aa(A,bool,ord_less(A,A2),zero_zero(A))) ) ) ).

% sgn_less
tff(fact_4485_sgn__greater,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A] :
          ( pp(aa(A,bool,ord_less(A,zero_zero(A)),sgn_sgn(A,A2)))
        <=> pp(aa(A,bool,ord_less(A,zero_zero(A)),A2)) ) ) ).

% sgn_greater
tff(fact_4486_divide__sgn,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),sgn_sgn(A,B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),A2),sgn_sgn(A,B2)) ) ).

% divide_sgn
tff(fact_4487_sgn__uint__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] : sgn_sgn(int,aa(word(A),int,semiring_1_unsigned(A,int),W)) = aa(bool,int,zero_neq_one_of_bool(int),aa(bool,bool,fNot,aa(word(A),bool,fequal(word(A),W),zero_zero(word(A))))) ) ).

% sgn_uint_eq
tff(fact_4488_set__encode__inverse,axiom,
    ! [A3: set(nat)] :
      ( pp(aa(set(nat),bool,finite_finite2(nat),A3))
     => ( nat_set_decode(aa(set(nat),nat,nat_set_encode,A3)) = A3 ) ) ).

% set_encode_inverse
tff(fact_4489_sgn__pos,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A] :
          ( pp(aa(A,bool,ord_less(A,zero_zero(A)),A2))
         => ( sgn_sgn(A,A2) = one_one(A) ) ) ) ).

% sgn_pos
tff(fact_4490_abs__sgn__eq__1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A] :
          ( ( A2 != zero_zero(A) )
         => ( aa(A,A,abs_abs(A),sgn_sgn(A,A2)) = one_one(A) ) ) ) ).

% abs_sgn_eq_1
tff(fact_4491_sgn__mult__self__eq,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),sgn_sgn(A,A2)),sgn_sgn(A,A2)) = aa(bool,A,zero_neq_one_of_bool(A),aa(bool,bool,fNot,aa(A,bool,fequal(A,A2),zero_zero(A)))) ) ).

% sgn_mult_self_eq
tff(fact_4492_sgn__abs,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [A2: A] : aa(A,A,abs_abs(A),sgn_sgn(A,A2)) = aa(bool,A,zero_neq_one_of_bool(A),aa(bool,bool,fNot,aa(A,bool,fequal(A,A2),zero_zero(A)))) ) ).

% sgn_abs
tff(fact_4493_idom__abs__sgn__class_Oabs__sgn,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [A2: A] : sgn_sgn(A,aa(A,A,abs_abs(A),A2)) = aa(bool,A,zero_neq_one_of_bool(A),aa(bool,bool,fNot,aa(A,bool,fequal(A,A2),zero_zero(A)))) ) ).

% idom_abs_sgn_class.abs_sgn
tff(fact_4494_sgn__mult__dvd__iff,axiom,
    ! [R2: int,L: int,K: int] :
      ( pp(aa(int,bool,dvd_dvd(int,aa(int,int,aa(int,fun(int,int),times_times(int),sgn_sgn(int,R2)),L)),K))
    <=> ( pp(aa(int,bool,dvd_dvd(int,L),K))
        & ( ( R2 = zero_zero(int) )
         => ( K = zero_zero(int) ) ) ) ) ).

% sgn_mult_dvd_iff
tff(fact_4495_mult__sgn__dvd__iff,axiom,
    ! [L: int,R2: int,K: int] :
      ( pp(aa(int,bool,dvd_dvd(int,aa(int,int,aa(int,fun(int,int),times_times(int),L),sgn_sgn(int,R2))),K))
    <=> ( pp(aa(int,bool,dvd_dvd(int,L),K))
        & ( ( R2 = zero_zero(int) )
         => ( K = zero_zero(int) ) ) ) ) ).

% mult_sgn_dvd_iff
tff(fact_4496_dvd__sgn__mult__iff,axiom,
    ! [L: int,R2: int,K: int] :
      ( pp(aa(int,bool,dvd_dvd(int,L),aa(int,int,aa(int,fun(int,int),times_times(int),sgn_sgn(int,R2)),K)))
    <=> ( pp(aa(int,bool,dvd_dvd(int,L),K))
        | ( R2 = zero_zero(int) ) ) ) ).

% dvd_sgn_mult_iff
tff(fact_4497_dvd__mult__sgn__iff,axiom,
    ! [L: int,K: int,R2: int] :
      ( pp(aa(int,bool,dvd_dvd(int,L),aa(int,int,aa(int,fun(int,int),times_times(int),K),sgn_sgn(int,R2))))
    <=> ( pp(aa(int,bool,dvd_dvd(int,L),K))
        | ( R2 = zero_zero(int) ) ) ) ).

% dvd_mult_sgn_iff
tff(fact_4498_sgn__neg,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A] :
          ( pp(aa(A,bool,ord_less(A,A2),zero_zero(A)))
         => ( sgn_sgn(A,A2) = aa(A,A,uminus_uminus(A),one_one(A)) ) ) ) ).

% sgn_neg
tff(fact_4499_sgn__of__nat,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [N: nat] : sgn_sgn(A,aa(nat,A,semiring_1_of_nat(A),N)) = aa(bool,A,zero_neq_one_of_bool(A),aa(nat,bool,ord_less(nat,zero_zero(nat)),N)) ) ).

% sgn_of_nat
tff(fact_4500_Suc__unat__minus__one,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A)] :
          ( ( X != zero_zero(word(A)) )
         => ( aa(nat,nat,suc,aa(word(A),nat,semiring_1_unsigned(A,nat),aa(word(A),word(A),minus_minus(word(A),X),one_one(word(A))))) = aa(word(A),nat,semiring_1_unsigned(A,nat),X) ) ) ) ).

% Suc_unat_minus_one
tff(fact_4501_word__unat__Rep__inject1,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [X: word(A)] :
          ( ( aa(word(A),nat,semiring_1_unsigned(A,nat),X) = aa(word(B),nat,semiring_1_unsigned(B,nat),one_one(word(B))) )
        <=> ( X = one_one(word(A)) ) ) ) ).

% word_unat_Rep_inject1
tff(fact_4502_ucast__0,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ( aa(word(B),word(A),semiring_1_unsigned(B,word(A)),zero_zero(word(B))) = zero_zero(word(A)) ) ) ).

% ucast_0
tff(fact_4503_ucast__0__I,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(A) )
     => ! [X: word(A)] :
          ( ( X = zero_zero(word(A)) )
         => ( aa(word(A),word(B),semiring_1_unsigned(A,word(B)),X) = zero_zero(word(B)) ) ) ) ).

% ucast_0_I
tff(fact_4504_ucast__1,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(A) )
     => ( aa(word(A),word(B),semiring_1_unsigned(A,word(B)),one_one(word(A))) = one_one(word(B)) ) ) ).

% ucast_1
tff(fact_4505_ucast__nat__def,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [X: word(A)] : aa(nat,word(B),semiring_1_of_nat(word(B)),aa(word(A),nat,semiring_1_unsigned(A,nat),X)) = aa(word(A),word(B),semiring_1_unsigned(A,word(B)),X) ) ).

% ucast_nat_def
tff(fact_4506_sgn__zero__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X: A] :
          ( ( sgn_sgn(A,X) = zero_zero(A) )
        <=> ( X = zero_zero(A) ) ) ) ).

% sgn_zero_iff
tff(fact_4507_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
tff(fact_4508_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
tff(fact_4509_Real__Vector__Spaces_Osgn__mult,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [X: A,Y: A] : sgn_sgn(A,aa(A,A,aa(A,fun(A,A),times_times(A),X),Y)) = aa(A,A,aa(A,fun(A,A),times_times(A),sgn_sgn(A,X)),sgn_sgn(A,Y)) ) ).

% Real_Vector_Spaces.sgn_mult
tff(fact_4510_sgn__mult,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [A2: A,B2: A] : sgn_sgn(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),sgn_sgn(A,A2)),sgn_sgn(A,B2)) ) ).

% sgn_mult
tff(fact_4511_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,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)) = sgn_sgn(A,A2) ) ) ) ).

% same_sgn_sgn_add
tff(fact_4512_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) = aa(A,A,uminus_uminus(A),sgn_sgn(A,B2)) ) ) ) ) ) ).

% sgn_not_eq_imp
tff(fact_4513_sgn__minus__1,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ( sgn_sgn(A,aa(A,A,uminus_uminus(A),one_one(A))) = aa(A,A,uminus_uminus(A),one_one(A)) ) ) ).

% sgn_minus_1
tff(fact_4514_int__sgnE,axiom,
    ! [K: int] :
      ~ ! [N2: nat,L3: int] : K != aa(int,int,aa(int,fun(int,int),times_times(int),sgn_sgn(int,L3)),aa(nat,int,semiring_1_of_nat(int),N2)) ).

% int_sgnE
tff(fact_4515_same__sgn__abs__add,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [B2: A,A2: A] :
          ( ( sgn_sgn(A,B2) = sgn_sgn(A,A2) )
         => ( aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,abs_abs(A),A2)),aa(A,A,abs_abs(A),B2)) ) ) ) ).

% same_sgn_abs_add
tff(fact_4516_linordered__idom__class_Oabs__sgn,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [K: A] : aa(A,A,abs_abs(A),K) = aa(A,A,aa(A,fun(A,A),times_times(A),K),sgn_sgn(A,K)) ) ).

% linordered_idom_class.abs_sgn
tff(fact_4517_abs__mult__sgn,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,abs_abs(A),A2)),sgn_sgn(A,A2)) = A2 ) ).

% abs_mult_sgn
tff(fact_4518_sgn__mult__abs,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),sgn_sgn(A,A2)),aa(A,A,abs_abs(A),A2)) = A2 ) ).

% sgn_mult_abs
tff(fact_4519_mult__sgn__abs,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),times_times(A),sgn_sgn(A,X)),aa(A,A,abs_abs(A),X)) = X ) ).

% mult_sgn_abs
tff(fact_4520_unat__eq__zero,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A)] :
          ( ( aa(word(A),nat,semiring_1_unsigned(A,nat),X) = zero_zero(nat) )
        <=> ( X = zero_zero(word(A)) ) ) ) ).

% unat_eq_zero
tff(fact_4521_unat__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( aa(word(A),nat,semiring_1_unsigned(A,nat),zero_zero(word(A))) = zero_zero(nat) ) ) ).

% unat_0
tff(fact_4522_word__le__nat__alt,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A)] :
          ( pp(aa(word(A),bool,ord_less_eq(word(A),A2),B2))
        <=> pp(aa(nat,bool,ord_less_eq(nat,aa(word(A),nat,semiring_1_unsigned(A,nat),A2)),aa(word(A),nat,semiring_1_unsigned(A,nat),B2))) ) ) ).

% word_le_nat_alt
tff(fact_4523_unat__1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( aa(word(A),nat,semiring_1_unsigned(A,nat),one_one(word(A))) = one_one(nat) ) ) ).

% unat_1
tff(fact_4524_div__eq__sgn__abs,axiom,
    ! [K: int,L: int] :
      ( ( sgn_sgn(int,K) = sgn_sgn(int,L) )
     => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),K),L) = aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,abs_abs(int),K)),aa(int,int,abs_abs(int),L)) ) ) ).

% div_eq_sgn_abs
tff(fact_4525_uint__nat,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] : aa(word(A),int,semiring_1_unsigned(A,int),W) = aa(nat,int,semiring_1_of_nat(int),aa(word(A),nat,semiring_1_unsigned(A,nat),W)) ) ).

% uint_nat
tff(fact_4526_le__unat__uoi,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: nat,Z: word(A)] :
          ( pp(aa(nat,bool,ord_less_eq(nat,Y),aa(word(A),nat,semiring_1_unsigned(A,nat),Z)))
         => ( aa(word(A),nat,semiring_1_unsigned(A,nat),aa(nat,word(A),semiring_1_of_nat(word(A)),Y)) = Y ) ) ) ).

% le_unat_uoi
tff(fact_4527_set__encode__eq,axiom,
    ! [A3: set(nat),B4: set(nat)] :
      ( pp(aa(set(nat),bool,finite_finite2(nat),A3))
     => ( pp(aa(set(nat),bool,finite_finite2(nat),B4))
       => ( ( aa(set(nat),nat,nat_set_encode,A3) = aa(set(nat),nat,nat_set_encode,B4) )
        <=> ( A3 = B4 ) ) ) ) ).

% set_encode_eq
tff(fact_4528_uno__simps_I2_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Z: word(A),N: nat] : aa(word(A),nat,semiring_1_unsigned(A,nat),aa(nat,word(A),semiring_1_of_nat(word(A)),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),Z)),N))) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),Z)),N) ) ).

% uno_simps(2)
tff(fact_4529_unat__div,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A)] : aa(word(A),nat,semiring_1_unsigned(A,nat),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),X),Y)) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),X)),aa(word(A),nat,semiring_1_unsigned(A,nat),Y)) ) ).

% unat_div
tff(fact_4530_unat__div__distrib,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [V: word(A),W: word(A)] : aa(word(A),nat,semiring_1_unsigned(A,nat),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),V),W)) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),V)),aa(word(A),nat,semiring_1_unsigned(A,nat),W)) ) ).

% unat_div_distrib
tff(fact_4531_unat__of__nat__less,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A),C2: nat] :
          ( pp(aa(word(A),bool,ord_less(word(A),A2),B2))
         => ( ( aa(word(A),nat,semiring_1_unsigned(A,nat),B2) = C2 )
           => pp(aa(word(A),bool,ord_less(word(A),A2),aa(nat,word(A),semiring_1_of_nat(word(A)),C2))) ) ) ) ).

% unat_of_nat_less
tff(fact_4532_ln__real__def,axiom,
    ! [X: real] : aa(real,real,ln_ln(real),X) = the(real,aTP_Lamp_lc(real,fun(real,bool),X)) ).

% ln_real_def
tff(fact_4533_suminf__def,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [F2: fun(nat,A)] : suminf(A,F2) = the(A,sums(A,F2)) ) ).

% suminf_def
tff(fact_4534_sgn__1__pos,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A] :
          ( ( sgn_sgn(A,A2) = one_one(A) )
        <=> pp(aa(A,bool,ord_less(A,zero_zero(A)),A2)) ) ) ).

% sgn_1_pos
tff(fact_4535_abs__sgn__eq,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A] :
          ( ( ( A2 = zero_zero(A) )
           => ( aa(A,A,abs_abs(A),sgn_sgn(A,A2)) = zero_zero(A) ) )
          & ( ( A2 != zero_zero(A) )
           => ( aa(A,A,abs_abs(A),sgn_sgn(A,A2)) = one_one(A) ) ) ) ) ).

% abs_sgn_eq
tff(fact_4536_unat__eq__1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A)] :
          ( ( aa(word(A),nat,semiring_1_unsigned(A,nat),X) = aa(nat,nat,suc,zero_zero(nat)) )
        <=> ( X = one_one(word(A)) ) ) ) ).

% unat_eq_1
tff(fact_4537_sgn__mod,axiom,
    ! [L: int,K: int] :
      ( ( L != zero_zero(int) )
     => ( ~ pp(aa(int,bool,dvd_dvd(int,L),K))
       => ( sgn_sgn(int,modulo_modulo(int,K,L)) = sgn_sgn(int,L) ) ) ) ).

% sgn_mod
tff(fact_4538_unat__gt__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A)] :
          ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),aa(word(A),nat,semiring_1_unsigned(A,nat),X)))
        <=> ( X != zero_zero(word(A)) ) ) ) ).

% unat_gt_0
tff(fact_4539_un__ui__le,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(A) )
     => ! [A2: word(A),B2: word(B)] :
          ( pp(aa(nat,bool,ord_less_eq(nat,aa(word(A),nat,semiring_1_unsigned(A,nat),A2)),aa(word(B),nat,semiring_1_unsigned(B,nat),B2)))
        <=> pp(aa(int,bool,ord_less_eq(int,aa(word(A),int,semiring_1_unsigned(A,int),A2)),aa(word(B),int,semiring_1_unsigned(B,int),B2))) ) ) ).

% un_ui_le
tff(fact_4540_unat__plus__simple,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A)] :
          ( pp(aa(word(A),bool,ord_less_eq(word(A),X),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),X),Y)))
        <=> ( aa(word(A),nat,semiring_1_unsigned(A,nat),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),X),Y)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),X)),aa(word(A),nat,semiring_1_unsigned(A,nat),Y)) ) ) ) ).

% unat_plus_simple
tff(fact_4541_word__of__nat__less,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: nat,X: word(A)] :
          ( pp(aa(nat,bool,ord_less(nat,N),aa(word(A),nat,semiring_1_unsigned(A,nat),X)))
         => pp(aa(word(A),bool,ord_less(word(A),aa(nat,word(A),semiring_1_of_nat(word(A)),N)),X)) ) ) ).

% word_of_nat_less
tff(fact_4542_unat__less__helper,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),N: nat] :
          ( pp(aa(word(A),bool,ord_less(word(A),X),aa(nat,word(A),semiring_1_of_nat(word(A)),N)))
         => pp(aa(nat,bool,ord_less(nat,aa(word(A),nat,semiring_1_unsigned(A,nat),X)),N)) ) ) ).

% unat_less_helper
tff(fact_4543_word__unat__less__le,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: nat] :
          ( pp(aa(word(A),bool,ord_less_eq(word(A),A2),aa(nat,word(A),semiring_1_of_nat(word(A)),B2)))
         => pp(aa(nat,bool,ord_less_eq(nat,aa(word(A),nat,semiring_1_unsigned(A,nat),A2)),B2)) ) ) ).

% word_unat_less_le
tff(fact_4544_word__of__nat__le,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: nat,X: word(A)] :
          ( pp(aa(nat,bool,ord_less_eq(nat,N),aa(word(A),nat,semiring_1_unsigned(A,nat),X)))
         => pp(aa(word(A),bool,ord_less_eq(word(A),aa(nat,word(A),semiring_1_of_nat(word(A)),N)),X)) ) ) ).

% word_of_nat_le
tff(fact_4545_word__arith__nat__add,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),A2),B2) = aa(nat,word(A),semiring_1_of_nat(word(A)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),A2)),aa(word(A),nat,semiring_1_unsigned(A,nat),B2))) ) ).

% word_arith_nat_add
tff(fact_4546_word__arith__nat__mult,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),A2),B2) = aa(nat,word(A),semiring_1_of_nat(word(A)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),A2)),aa(word(A),nat,semiring_1_unsigned(A,nat),B2))) ) ).

% word_arith_nat_mult
tff(fact_4547_word__arith__nat__div,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),A2),B2) = aa(nat,word(A),semiring_1_of_nat(word(A)),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),A2)),aa(word(A),nat,semiring_1_unsigned(A,nat),B2))) ) ).

% word_arith_nat_div
tff(fact_4548_word__arith__nat__mod,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A)] : modulo_modulo(word(A),A2,B2) = aa(nat,word(A),semiring_1_of_nat(word(A)),modulo_modulo(nat,aa(word(A),nat,semiring_1_unsigned(A,nat),A2),aa(word(A),nat,semiring_1_unsigned(A,nat),B2))) ) ).

% word_arith_nat_mod
tff(fact_4549_ln__neg__is__const,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,ord_less_eq(real,X),zero_zero(real)))
     => ( aa(real,real,ln_ln(real),X) = the(real,aTP_Lamp_ld(real,bool)) ) ) ).

% ln_neg_is_const
tff(fact_4550_set__encode__inf,axiom,
    ! [A3: set(nat)] :
      ( ~ pp(aa(set(nat),bool,finite_finite2(nat),A3))
     => ( aa(set(nat),nat,nat_set_encode,A3) = zero_zero(nat) ) ) ).

% set_encode_inf
tff(fact_4551_sgn__if,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A] :
          ( ( ( X = zero_zero(A) )
           => ( sgn_sgn(A,X) = zero_zero(A) ) )
          & ( ( X != zero_zero(A) )
           => ( ( pp(aa(A,bool,ord_less(A,zero_zero(A)),X))
               => ( sgn_sgn(A,X) = one_one(A) ) )
              & ( ~ pp(aa(A,bool,ord_less(A,zero_zero(A)),X))
               => ( sgn_sgn(A,X) = aa(A,A,uminus_uminus(A),one_one(A)) ) ) ) ) ) ) ).

% sgn_if
tff(fact_4552_sgn__1__neg,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A] :
          ( ( sgn_sgn(A,A2) = aa(A,A,uminus_uminus(A),one_one(A)) )
        <=> pp(aa(A,bool,ord_less(A,A2),zero_zero(A))) ) ) ).

% sgn_1_neg
tff(fact_4553_zsgn__def,axiom,
    ! [I: int] :
      ( ( ( I = zero_zero(int) )
       => ( sgn_sgn(int,I) = zero_zero(int) ) )
      & ( ( I != zero_zero(int) )
       => ( ( pp(aa(int,bool,ord_less(int,zero_zero(int)),I))
           => ( sgn_sgn(int,I) = one_one(int) ) )
          & ( ~ pp(aa(int,bool,ord_less(int,zero_zero(int)),I))
           => ( sgn_sgn(int,I) = aa(int,int,uminus_uminus(int),one_one(int)) ) ) ) ) ) ).

% zsgn_def
tff(fact_4554_norm__sgn,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X: A] :
          ( ( ( X = zero_zero(A) )
           => ( real_V7770717601297561774m_norm(A,sgn_sgn(A,X)) = zero_zero(real) ) )
          & ( ( X != zero_zero(A) )
           => ( real_V7770717601297561774m_norm(A,sgn_sgn(A,X)) = one_one(real) ) ) ) ) ).

% norm_sgn
tff(fact_4555_div__sgn__abs__cancel,axiom,
    ! [V: int,K: int,L: int] :
      ( ( V != zero_zero(int) )
     => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,aa(int,fun(int,int),times_times(int),sgn_sgn(int,V)),aa(int,int,abs_abs(int),K))),aa(int,int,aa(int,fun(int,int),times_times(int),sgn_sgn(int,V)),aa(int,int,abs_abs(int),L))) = aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,abs_abs(int),K)),aa(int,int,abs_abs(int),L)) ) ) ).

% div_sgn_abs_cancel
tff(fact_4556_unat__1__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A)] :
          ( pp(aa(word(A),bool,ord_less_eq(word(A),one_one(word(A))),X))
        <=> pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),aa(word(A),nat,semiring_1_unsigned(A,nat),X))) ) ) ).

% unat_1_0
tff(fact_4557_unat__max__word__pos,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),aa(word(A),nat,semiring_1_unsigned(A,nat),aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A)))))) ) ).

% unat_max_word_pos
tff(fact_4558_div__dvd__sgn__abs,axiom,
    ! [L: int,K: int] :
      ( pp(aa(int,bool,dvd_dvd(int,L),K))
     => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),K),L) = aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,aa(int,fun(int,int),times_times(int),sgn_sgn(int,K)),sgn_sgn(int,L))),aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,abs_abs(int),K)),aa(int,int,abs_abs(int),L))) ) ) ).

% div_dvd_sgn_abs
tff(fact_4559_unatSuc2,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: word(A)] :
          ( ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),N),one_one(word(A))) != zero_zero(word(A)) )
         => ( aa(word(A),nat,semiring_1_unsigned(A,nat),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),N),one_one(word(A)))) = aa(nat,nat,suc,aa(word(A),nat,semiring_1_unsigned(A,nat),N)) ) ) ) ).

% unatSuc2
tff(fact_4560_unatSuc,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: word(A)] :
          ( ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),one_one(word(A))),N) != zero_zero(word(A)) )
         => ( aa(word(A),nat,semiring_1_unsigned(A,nat),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),one_one(word(A))),N)) = aa(nat,nat,suc,aa(word(A),nat,semiring_1_unsigned(A,nat),N)) ) ) ) ).

% unatSuc
tff(fact_4561_Suc__unat__diff__1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A)] :
          ( pp(aa(word(A),bool,ord_less_eq(word(A),one_one(word(A))),X))
         => ( aa(nat,nat,suc,aa(word(A),nat,semiring_1_unsigned(A,nat),aa(word(A),word(A),minus_minus(word(A),X),one_one(word(A))))) = aa(word(A),nat,semiring_1_unsigned(A,nat),X) ) ) ) ).

% Suc_unat_diff_1
tff(fact_4562_unat__Suc2,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: word(A)] :
          ( ( N != aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A))) )
         => ( aa(word(A),nat,semiring_1_unsigned(A,nat),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),N),one_one(word(A)))) = aa(nat,nat,suc,aa(word(A),nat,semiring_1_unsigned(A,nat),N)) ) ) ) ).

% unat_Suc2
tff(fact_4563_uno__simps_I1_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Z: word(A),M: nat] :
          ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),aa(word(A),nat,semiring_1_unsigned(A,nat),Z)))
         => ( aa(word(A),nat,semiring_1_unsigned(A,nat),aa(nat,word(A),semiring_1_of_nat(word(A)),modulo_modulo(nat,M,aa(word(A),nat,semiring_1_unsigned(A,nat),Z)))) = modulo_modulo(nat,M,aa(word(A),nat,semiring_1_unsigned(A,nat),Z)) ) ) ) ).

% uno_simps(1)
tff(fact_4564_measure__unat,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [P2: word(A)] :
          ( ( P2 != zero_zero(word(A)) )
         => pp(aa(nat,bool,ord_less(nat,aa(word(A),nat,semiring_1_unsigned(A,nat),aa(word(A),word(A),minus_minus(word(A),P2),one_one(word(A))))),aa(word(A),nat,semiring_1_unsigned(A,nat),P2))) ) ) ).

% measure_unat
tff(fact_4565_word__overflow__unat,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A)] :
          ( ( aa(word(A),nat,semiring_1_unsigned(A,nat),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),X),one_one(word(A)))) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),X)),one_one(nat)) )
          | ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),X),one_one(word(A))) = zero_zero(word(A)) ) ) ) ).

% word_overflow_unat
tff(fact_4566_unat__minus__one,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] :
          ( ( W != zero_zero(word(A)) )
         => ( aa(word(A),nat,semiring_1_unsigned(A,nat),aa(word(A),word(A),minus_minus(word(A),W),one_one(word(A)))) = aa(nat,nat,minus_minus(nat,aa(word(A),nat,semiring_1_unsigned(A,nat),W)),one_one(nat)) ) ) ) ).

% unat_minus_one
tff(fact_4567_arccos__def,axiom,
    ! [Y: real] : aa(real,real,arccos,Y) = the(real,aTP_Lamp_le(real,fun(real,bool),Y)) ).

% arccos_def
tff(fact_4568_div__noneq__sgn__abs,axiom,
    ! [L: int,K: int] :
      ( ( L != zero_zero(int) )
     => ( ( sgn_sgn(int,K) != sgn_sgn(int,L) )
       => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),K),L) = aa(int,int,minus_minus(int,aa(int,int,uminus_uminus(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,abs_abs(int),K)),aa(int,int,abs_abs(int),L)))),aa(bool,int,zero_neq_one_of_bool(int),aa(bool,bool,fNot,aa(int,bool,dvd_dvd(int,L),K)))) ) ) ) ).

% div_noneq_sgn_abs
tff(fact_4569_lt__plus__1__le__word,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: nat,MaxBound: word(A),X: word(A)] :
          ( pp(aa(nat,bool,ord_less(nat,N),aa(word(A),nat,semiring_1_unsigned(A,nat),MaxBound)))
         => ( pp(aa(word(A),bool,ord_less(word(A),X),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),one_one(word(A))),aa(nat,word(A),semiring_1_of_nat(word(A)),N))))
          <=> pp(aa(word(A),bool,ord_less_eq(word(A),X),aa(nat,word(A),semiring_1_of_nat(word(A)),N))) ) ) ) ).

% lt_plus_1_le_word
tff(fact_4570_even__word__imp__odd__next,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A)] :
          ( pp(aa(nat,bool,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(word(A),nat,semiring_1_unsigned(A,nat),X)))
         => ( ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),X),one_one(word(A))) = zero_zero(word(A)) )
            | ~ pp(aa(nat,bool,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(word(A),nat,semiring_1_unsigned(A,nat),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),X),one_one(word(A)))))) ) ) ) ).

% even_word_imp_odd_next
tff(fact_4571_odd__word__imp__even__next,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A)] :
          ( ~ pp(aa(nat,bool,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(word(A),nat,semiring_1_unsigned(A,nat),X)))
         => ( ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),X),one_one(word(A))) = zero_zero(word(A)) )
            | pp(aa(nat,bool,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(word(A),nat,semiring_1_unsigned(A,nat),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),X),one_one(word(A)))))) ) ) ) ).

% odd_word_imp_even_next
tff(fact_4572_word__div__eq__1__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: word(A),M: word(A)] :
          ( ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),N),M) = one_one(word(A)) )
        <=> ( pp(aa(word(A),bool,ord_less_eq(word(A),M),N))
            & pp(aa(nat,bool,ord_less(nat,aa(word(A),nat,semiring_1_unsigned(A,nat),N)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(word(A),nat,semiring_1_unsigned(A,nat),M)))) ) ) ) ).

% word_div_eq_1_iff
tff(fact_4573_of__nat__eq__size,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: nat,W: word(A)] :
          ( ( aa(nat,word(A),semiring_1_of_nat(word(A)),N) = W )
        <=> ? [Q4: nat] : N = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),W)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Q4),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(word(A),nat,size_size(word(A)),W)))) ) ) ).

% of_nat_eq_size
tff(fact_4574_set__encode__def,axiom,
    nat_set_encode = groups7311177749621191930dd_sum(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ).

% set_encode_def
tff(fact_4575_pi__half,axiom,
    aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))) = the(real,aTP_Lamp_lf(real,bool)) ).

% pi_half
tff(fact_4576_pi__def,axiom,
    pi = aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),the(real,aTP_Lamp_lf(real,bool))) ).

% pi_def
tff(fact_4577_unat__plus__if__size,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A)] :
          ( ( pp(aa(nat,bool,ord_less(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),X)),aa(word(A),nat,semiring_1_unsigned(A,nat),Y))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(word(A),nat,size_size(word(A)),X))))
           => ( aa(word(A),nat,semiring_1_unsigned(A,nat),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),X),Y)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),X)),aa(word(A),nat,semiring_1_unsigned(A,nat),Y)) ) )
          & ( ~ pp(aa(nat,bool,ord_less(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),X)),aa(word(A),nat,semiring_1_unsigned(A,nat),Y))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(word(A),nat,size_size(word(A)),X))))
           => ( aa(word(A),nat,semiring_1_unsigned(A,nat),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),X),Y)) = aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),X)),aa(word(A),nat,semiring_1_unsigned(A,nat),Y))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(word(A),nat,size_size(word(A)),X))) ) ) ) ) ).

% unat_plus_if_size
tff(fact_4578_modulo__int__def,axiom,
    ! [L: int,K: int] :
      ( ( ( L = zero_zero(int) )
       => ( modulo_modulo(int,K,L) = K ) )
      & ( ( L != zero_zero(int) )
       => ( ( ( sgn_sgn(int,K) = sgn_sgn(int,L) )
           => ( modulo_modulo(int,K,L) = aa(int,int,aa(int,fun(int,int),times_times(int),sgn_sgn(int,L)),aa(nat,int,semiring_1_of_nat(int),modulo_modulo(nat,nat2(aa(int,int,abs_abs(int),K)),nat2(aa(int,int,abs_abs(int),L))))) ) )
          & ( ( sgn_sgn(int,K) != sgn_sgn(int,L) )
           => ( modulo_modulo(int,K,L) = aa(int,int,aa(int,fun(int,int),times_times(int),sgn_sgn(int,L)),aa(int,int,minus_minus(int,aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,abs_abs(int),L)),aa(bool,int,zero_neq_one_of_bool(int),aa(bool,bool,fNot,aa(int,bool,dvd_dvd(int,L),K))))),aa(nat,int,semiring_1_of_nat(int),modulo_modulo(nat,nat2(aa(int,int,abs_abs(int),K)),nat2(aa(int,int,abs_abs(int),L)))))) ) ) ) ) ) ).

% modulo_int_def
tff(fact_4579_unat__sub__if__size,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),X: word(A)] :
          ( ( pp(aa(nat,bool,ord_less_eq(nat,aa(word(A),nat,semiring_1_unsigned(A,nat),Y)),aa(word(A),nat,semiring_1_unsigned(A,nat),X)))
           => ( aa(word(A),nat,semiring_1_unsigned(A,nat),aa(word(A),word(A),minus_minus(word(A),X),Y)) = aa(nat,nat,minus_minus(nat,aa(word(A),nat,semiring_1_unsigned(A,nat),X)),aa(word(A),nat,semiring_1_unsigned(A,nat),Y)) ) )
          & ( ~ pp(aa(nat,bool,ord_less_eq(nat,aa(word(A),nat,semiring_1_unsigned(A,nat),Y)),aa(word(A),nat,semiring_1_unsigned(A,nat),X)))
           => ( aa(word(A),nat,semiring_1_unsigned(A,nat),aa(word(A),word(A),minus_minus(word(A),X),Y)) = aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),X)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(word(A),nat,size_size(word(A)),X)))),aa(word(A),nat,semiring_1_unsigned(A,nat),Y)) ) ) ) ) ).

% unat_sub_if_size
tff(fact_4580_no__plus__overflow__unat__size,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A)] :
          ( pp(aa(word(A),bool,ord_less_eq(word(A),X),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),X),Y)))
        <=> pp(aa(nat,bool,ord_less(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),X)),aa(word(A),nat,semiring_1_unsigned(A,nat),Y))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(word(A),nat,size_size(word(A)),X)))) ) ) ).

% no_plus_overflow_unat_size
tff(fact_4581_divide__int__unfold,axiom,
    ! [L: int,K: int,N: nat,M: nat] :
      ( ( ( ( sgn_sgn(int,L) = zero_zero(int) )
          | ( sgn_sgn(int,K) = zero_zero(int) )
          | ( N = zero_zero(nat) ) )
       => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,aa(int,fun(int,int),times_times(int),sgn_sgn(int,K)),aa(nat,int,semiring_1_of_nat(int),M))),aa(int,int,aa(int,fun(int,int),times_times(int),sgn_sgn(int,L)),aa(nat,int,semiring_1_of_nat(int),N))) = zero_zero(int) ) )
      & ( ~ ( ( sgn_sgn(int,L) = zero_zero(int) )
            | ( sgn_sgn(int,K) = zero_zero(int) )
            | ( N = zero_zero(nat) ) )
       => ( ( ( sgn_sgn(int,K) = sgn_sgn(int,L) )
           => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,aa(int,fun(int,int),times_times(int),sgn_sgn(int,K)),aa(nat,int,semiring_1_of_nat(int),M))),aa(int,int,aa(int,fun(int,int),times_times(int),sgn_sgn(int,L)),aa(nat,int,semiring_1_of_nat(int),N))) = aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),N)) ) )
          & ( ( sgn_sgn(int,K) != sgn_sgn(int,L) )
           => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,aa(int,fun(int,int),times_times(int),sgn_sgn(int,K)),aa(nat,int,semiring_1_of_nat(int),M))),aa(int,int,aa(int,fun(int,int),times_times(int),sgn_sgn(int,L)),aa(nat,int,semiring_1_of_nat(int),N))) = aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),N)),aa(bool,nat,zero_neq_one_of_bool(nat),aa(bool,bool,fNot,aa(nat,bool,dvd_dvd(nat,N),M)))))) ) ) ) ) ) ).

% divide_int_unfold
tff(fact_4582_modulo__int__unfold,axiom,
    ! [L: int,K: int,N: nat,M: nat] :
      ( ( ( ( sgn_sgn(int,L) = zero_zero(int) )
          | ( sgn_sgn(int,K) = zero_zero(int) )
          | ( N = zero_zero(nat) ) )
       => ( modulo_modulo(int,aa(int,int,aa(int,fun(int,int),times_times(int),sgn_sgn(int,K)),aa(nat,int,semiring_1_of_nat(int),M)),aa(int,int,aa(int,fun(int,int),times_times(int),sgn_sgn(int,L)),aa(nat,int,semiring_1_of_nat(int),N))) = aa(int,int,aa(int,fun(int,int),times_times(int),sgn_sgn(int,K)),aa(nat,int,semiring_1_of_nat(int),M)) ) )
      & ( ~ ( ( sgn_sgn(int,L) = zero_zero(int) )
            | ( sgn_sgn(int,K) = zero_zero(int) )
            | ( N = zero_zero(nat) ) )
       => ( ( ( sgn_sgn(int,K) = sgn_sgn(int,L) )
           => ( modulo_modulo(int,aa(int,int,aa(int,fun(int,int),times_times(int),sgn_sgn(int,K)),aa(nat,int,semiring_1_of_nat(int),M)),aa(int,int,aa(int,fun(int,int),times_times(int),sgn_sgn(int,L)),aa(nat,int,semiring_1_of_nat(int),N))) = aa(int,int,aa(int,fun(int,int),times_times(int),sgn_sgn(int,L)),aa(nat,int,semiring_1_of_nat(int),modulo_modulo(nat,M,N))) ) )
          & ( ( sgn_sgn(int,K) != sgn_sgn(int,L) )
           => ( modulo_modulo(int,aa(int,int,aa(int,fun(int,int),times_times(int),sgn_sgn(int,K)),aa(nat,int,semiring_1_of_nat(int),M)),aa(int,int,aa(int,fun(int,int),times_times(int),sgn_sgn(int,L)),aa(nat,int,semiring_1_of_nat(int),N))) = aa(int,int,aa(int,fun(int,int),times_times(int),sgn_sgn(int,L)),aa(int,int,minus_minus(int,aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),aa(bool,nat,zero_neq_one_of_bool(nat),aa(bool,bool,fNot,aa(nat,bool,dvd_dvd(nat,N),M)))))),aa(nat,int,semiring_1_of_nat(int),modulo_modulo(nat,M,N)))) ) ) ) ) ) ).

% modulo_int_unfold
tff(fact_4583_sgn__div__eq__sgn__mult,axiom,
    ! [A2: int,B2: int] :
      ( ( aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),B2) != zero_zero(int) )
     => ( sgn_sgn(int,aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),B2)) = sgn_sgn(int,aa(int,int,aa(int,fun(int,int),times_times(int),A2),B2)) ) ) ).

% sgn_div_eq_sgn_mult
tff(fact_4584_the__sym__eq__trivial,axiom,
    ! [A: $tType,X: A] : the(A,fequal(A,X)) = X ).

% the_sym_eq_trivial
tff(fact_4585_the__eq__trivial,axiom,
    ! [A: $tType,A2: A] : the(A,aTP_Lamp_lg(A,fun(A,bool),A2)) = A2 ).

% the_eq_trivial
tff(fact_4586_the__equality,axiom,
    ! [A: $tType,P: fun(A,bool),A2: A] :
      ( pp(aa(A,bool,P,A2))
     => ( ! [X3: A] :
            ( pp(aa(A,bool,P,X3))
           => ( X3 = A2 ) )
       => ( the(A,P) = A2 ) ) ) ).

% the_equality
tff(fact_4587_sgn__le__0__iff,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,ord_less_eq(real,sgn_sgn(real,X)),zero_zero(real)))
    <=> pp(aa(real,bool,ord_less_eq(real,X),zero_zero(real))) ) ).

% sgn_le_0_iff
tff(fact_4588_zero__le__sgn__iff,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,ord_less_eq(real,zero_zero(real)),sgn_sgn(real,X)))
    <=> pp(aa(real,bool,ord_less_eq(real,zero_zero(real)),X)) ) ).

% zero_le_sgn_iff
tff(fact_4589_real__sgn__eq,axiom,
    ! [X: real] : sgn_sgn(real,X) = aa(real,real,aa(real,fun(real,real),divide_divide(real),X),aa(real,real,abs_abs(real),X)) ).

% real_sgn_eq
tff(fact_4590_sgn__root,axiom,
    ! [N: nat,X: real] :
      ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),N))
     => ( sgn_sgn(real,aa(real,real,root(N),X)) = sgn_sgn(real,X) ) ) ).

% sgn_root
tff(fact_4591_cis__Arg,axiom,
    ! [Z: complex] :
      ( ( Z != zero_zero(complex) )
     => ( cis(arg(Z)) = sgn_sgn(complex,Z) ) ) ).

% cis_Arg
tff(fact_4592_sgn__real__def,axiom,
    ! [A2: real] :
      ( ( ( A2 = zero_zero(real) )
       => ( sgn_sgn(real,A2) = zero_zero(real) ) )
      & ( ( A2 != zero_zero(real) )
       => ( ( pp(aa(real,bool,ord_less(real,zero_zero(real)),A2))
           => ( sgn_sgn(real,A2) = one_one(real) ) )
          & ( ~ pp(aa(real,bool,ord_less(real,zero_zero(real)),A2))
           => ( sgn_sgn(real,A2) = aa(real,real,uminus_uminus(real),one_one(real)) ) ) ) ) ) ).

% sgn_real_def
tff(fact_4593_sgn__power__injE,axiom,
    ! [A2: real,N: nat,X: real,B2: real] :
      ( ( aa(real,real,aa(real,fun(real,real),times_times(real),sgn_sgn(real,A2)),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,abs_abs(real),A2)),N)) = X )
     => ( ( X = aa(real,real,aa(real,fun(real,real),times_times(real),sgn_sgn(real,B2)),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,abs_abs(real),B2)),N)) )
       => ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),N))
         => ( A2 = B2 ) ) ) ) ).

% sgn_power_injE
tff(fact_4594_theI,axiom,
    ! [A: $tType,P: fun(A,bool),A2: A] :
      ( pp(aa(A,bool,P,A2))
     => ( ! [X3: A] :
            ( pp(aa(A,bool,P,X3))
           => ( X3 = A2 ) )
       => pp(aa(A,bool,P,the(A,P))) ) ) ).

% theI
tff(fact_4595_theI_H,axiom,
    ! [A: $tType,P: fun(A,bool)] :
      ( ? [X4: A] :
          ( pp(aa(A,bool,P,X4))
          & ! [Y3: A] :
              ( pp(aa(A,bool,P,Y3))
             => ( Y3 = X4 ) ) )
     => pp(aa(A,bool,P,the(A,P))) ) ).

% theI'
tff(fact_4596_theI2,axiom,
    ! [A: $tType,P: fun(A,bool),A2: A,Q: fun(A,bool)] :
      ( pp(aa(A,bool,P,A2))
     => ( ! [X3: A] :
            ( pp(aa(A,bool,P,X3))
           => ( X3 = A2 ) )
       => ( ! [X3: A] :
              ( pp(aa(A,bool,P,X3))
             => pp(aa(A,bool,Q,X3)) )
         => pp(aa(A,bool,Q,the(A,P))) ) ) ) ).

% theI2
tff(fact_4597_If__def,axiom,
    ! [A: $tType,P: bool,X: A,Y: A] :
      ( ( pp(P)
       => ( X = the(A,aa(A,fun(A,bool),aa(A,fun(A,fun(A,bool)),aTP_Lamp_lh(bool,fun(A,fun(A,fun(A,bool))),P),X),Y)) ) )
      & ( ~ pp(P)
       => ( Y = the(A,aa(A,fun(A,bool),aa(A,fun(A,fun(A,bool)),aTP_Lamp_lh(bool,fun(A,fun(A,fun(A,bool))),P),X),Y)) ) ) ) ).

% If_def
tff(fact_4598_the1I2,axiom,
    ! [A: $tType,P: fun(A,bool),Q: fun(A,bool)] :
      ( ? [X4: A] :
          ( pp(aa(A,bool,P,X4))
          & ! [Y3: A] :
              ( pp(aa(A,bool,P,Y3))
             => ( Y3 = X4 ) ) )
     => ( ! [X3: A] :
            ( pp(aa(A,bool,P,X3))
           => pp(aa(A,bool,Q,X3)) )
       => pp(aa(A,bool,Q,the(A,P))) ) ) ).

% the1I2
tff(fact_4599_the1__equality,axiom,
    ! [A: $tType,P: fun(A,bool),A2: A] :
      ( ? [X4: A] :
          ( pp(aa(A,bool,P,X4))
          & ! [Y3: A] :
              ( pp(aa(A,bool,P,Y3))
             => ( Y3 = X4 ) ) )
     => ( pp(aa(A,bool,P,A2))
       => ( the(A,P) = A2 ) ) ) ).

% the1_equality
tff(fact_4600_sgn__power__root,axiom,
    ! [N: nat,X: real] :
      ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),N))
     => ( aa(real,real,aa(real,fun(real,real),times_times(real),sgn_sgn(real,aa(real,real,root(N),X))),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,abs_abs(real),aa(real,real,root(N),X))),N)) = X ) ) ).

% sgn_power_root
tff(fact_4601_root__sgn__power,axiom,
    ! [N: nat,Y: real] :
      ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),N))
     => ( aa(real,real,root(N),aa(real,real,aa(real,fun(real,real),times_times(real),sgn_sgn(real,Y)),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,abs_abs(real),Y)),N))) = Y ) ) ).

% root_sgn_power
tff(fact_4602_cis__Arg__unique,axiom,
    ! [Z: complex,X: real] :
      ( ( sgn_sgn(complex,Z) = cis(X) )
     => ( pp(aa(real,bool,ord_less(real,aa(real,real,uminus_uminus(real),pi)),X))
       => ( pp(aa(real,bool,ord_less_eq(real,X),pi))
         => ( arg(Z) = X ) ) ) ) ).

% cis_Arg_unique
tff(fact_4603_split__root,axiom,
    ! [P: fun(real,bool),N: nat,X: real] :
      ( pp(aa(real,bool,P,aa(real,real,root(N),X)))
    <=> ( ( ( N = zero_zero(nat) )
         => pp(aa(real,bool,P,zero_zero(real))) )
        & ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),N))
         => ! [Y7: real] :
              ( ( aa(real,real,aa(real,fun(real,real),times_times(real),sgn_sgn(real,Y7)),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,abs_abs(real),Y7)),N)) = X )
             => pp(aa(real,bool,P,Y7)) ) ) ) ) ).

% split_root
tff(fact_4604_floor__real__def,axiom,
    ! [X: real] : archim6421214686448440834_floor(real,X) = the(int,aTP_Lamp_li(real,fun(int,bool),X)) ).

% floor_real_def
tff(fact_4605_Arg__correct,axiom,
    ! [Z: complex] :
      ( ( Z != zero_zero(complex) )
     => ( ( sgn_sgn(complex,Z) = cis(arg(Z)) )
        & pp(aa(real,bool,ord_less(real,aa(real,real,uminus_uminus(real),pi)),arg(Z)))
        & pp(aa(real,bool,ord_less_eq(real,arg(Z)),pi)) ) ) ).

% Arg_correct
tff(fact_4606_arctan__inverse,axiom,
    ! [X: real] :
      ( ( X != zero_zero(real) )
     => ( aa(real,real,arctan,aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),X)) = aa(real,real,minus_minus(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),times_times(real),sgn_sgn(real,X)),pi)),aa(num,real,numeral_numeral(real),bit0(one2)))),aa(real,real,arctan,X)) ) ) ).

% arctan_inverse
tff(fact_4607_signed__take__bit__eq__take__bit__minus,axiom,
    ! [N: nat,K: int] : aa(int,int,bit_ri4674362597316999326ke_bit(int,N),K) = aa(int,int,minus_minus(int,aa(int,int,bit_se2584673776208193580ke_bit(int,aa(nat,nat,suc,N)),K)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,suc,N))),aa(bool,int,zero_neq_one_of_bool(int),aa(nat,bool,bit_se5641148757651400278ts_bit(int,K),N)))) ).

% signed_take_bit_eq_take_bit_minus
tff(fact_4608_num_Osize__gen_I3_J,axiom,
    ! [X33: num] : size_num(aa(num,num,bit1,X33)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),size_num(X33)),aa(nat,nat,suc,zero_zero(nat))) ).

% num.size_gen(3)
tff(fact_4609_mi__eq__ma__no__ch,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] :
      ( vEBT_invar_vebt(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary),Deg)
     => ( ( Mi = Ma )
       => ( ! [X4: vEBT_VEBT] :
              ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),member(vEBT_VEBT),X4),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList)))
             => ~ ? [X_12: nat] : pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,X4),X_12)) )
          & ~ ? [X_12: nat] : pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,Summary),X_12)) ) ) ) ).

% mi_eq_ma_no_ch
tff(fact_4610_insert__simp__mima,axiom,
    ! [X: nat,Mi: nat,Ma: nat,Deg: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] :
      ( ( ( X = Mi )
        | ( X = Ma ) )
     => ( pp(aa(nat,bool,ord_less_eq(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg))
       => ( vEBT_vebt_insert(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary),X) = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary) ) ) ) ).

% insert_simp_mima
tff(fact_4611_not__None__eq,axiom,
    ! [A: $tType,X: option(A)] :
      ( ( X != none(A) )
    <=> ? [Y7: A] : X = aa(A,option(A),some(A),Y7) ) ).

% not_None_eq
tff(fact_4612_not__Some__eq,axiom,
    ! [A: $tType,X: option(A)] :
      ( ! [Y7: A] : X != aa(A,option(A),some(A),Y7)
    <=> ( X = none(A) ) ) ).

% not_Some_eq
tff(fact_4613_tdeletemimi,axiom,
    ! [Deg: nat,Mi: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,X: nat] :
      ( pp(aa(nat,bool,ord_less_eq(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg))
     => pp(aa(nat,bool,ord_less_eq(nat,vEBT_T_d_e_l_e_t_e(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Mi)),Deg,TreeList,Summary),X)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,bit0(bit0(one2)))))) ) ).

% tdeletemimi
tff(fact_4614_tdeletemimi_H,axiom,
    ! [Deg: nat,Mi: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,X: nat] :
      ( pp(aa(nat,bool,ord_less_eq(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg))
     => pp(aa(nat,bool,ord_less_eq(nat,vEBT_V1232361888498592333_e_t_e(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Mi)),Deg,TreeList,Summary),X)),one_one(nat))) ) ).

% tdeletemimi'
tff(fact_4615_mi__ma__2__deg,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,N: nat] :
      ( vEBT_invar_vebt(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary),N)
     => ( pp(aa(nat,bool,ord_less_eq(nat,Mi),Ma))
        & pp(aa(nat,bool,ord_less(nat,Ma),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg))) ) ) ).

% mi_ma_2_deg
tff(fact_4616_not__Some__eq2,axiom,
    ! [B: $tType,A: $tType,V: option(product_prod(A,B))] :
      ( ! [X2: A,Y7: B] : V != aa(product_prod(A,B),option(product_prod(A,B)),some(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X2),Y7))
    <=> ( V = none(product_prod(A,B)) ) ) ).

% not_Some_eq2
tff(fact_4617_both__member__options__from__complete__tree__to__child,axiom,
    ! [Deg: nat,Mi: nat,Ma: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,X: nat] :
      ( pp(aa(nat,bool,ord_less_eq(nat,one_one(nat)),Deg))
     => ( pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary)),X))
       => ( pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),vEBT_VEBT_low(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))))
          | ( X = Mi )
          | ( X = Ma ) ) ) ) ).

% both_member_options_from_complete_tree_to_child
tff(fact_4618_member__inv,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,X: nat] :
      ( pp(aa(nat,bool,vEBT_vebt_member(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary)),X))
     => ( pp(aa(nat,bool,ord_less_eq(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg))
        & ( ( X = Mi )
          | ( X = Ma )
          | ( pp(aa(nat,bool,ord_less(nat,X),Ma))
            & pp(aa(nat,bool,ord_less(nat,Mi),X))
            & pp(aa(nat,bool,ord_less(nat,vEBT_VEBT_high(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)))
            & pp(aa(nat,bool,vEBT_vebt_member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),vEBT_VEBT_low(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))) ) ) ) ) ).

% member_inv
tff(fact_4619_bit__numeral__Bit0__Suc__iff,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [M: num,N: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,aa(num,A,numeral_numeral(A),bit0(M))),aa(nat,nat,suc,N)))
        <=> pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,aa(num,A,numeral_numeral(A),M)),N)) ) ) ).

% bit_numeral_Bit0_Suc_iff
tff(fact_4620_both__member__options__from__chilf__to__complete__tree,axiom,
    ! [X: nat,Deg: nat,TreeList: list(vEBT_VEBT),Mi: nat,Ma: nat,Summary: vEBT_VEBT] :
      ( pp(aa(nat,bool,ord_less(nat,vEBT_VEBT_high(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)))
     => ( pp(aa(nat,bool,ord_less_eq(nat,one_one(nat)),Deg))
       => ( pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),vEBT_VEBT_low(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))))
         => pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary)),X)) ) ) ) ).

% both_member_options_from_chilf_to_complete_tree
tff(fact_4621_bit__numeral__Bit1__Suc__iff,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [M: num,N: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,aa(num,A,numeral_numeral(A),aa(num,num,bit1,M))),aa(nat,nat,suc,N)))
        <=> pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,aa(num,A,numeral_numeral(A),M)),N)) ) ) ).

% bit_numeral_Bit1_Suc_iff
tff(fact_4622_signed__take__bit__nonnegative__iff,axiom,
    ! [N: nat,K: int] :
      ( pp(aa(int,bool,ord_less_eq(int,zero_zero(int)),aa(int,int,bit_ri4674362597316999326ke_bit(int,N),K)))
    <=> ~ pp(aa(nat,bool,bit_se5641148757651400278ts_bit(int,K),N)) ) ).

% signed_take_bit_nonnegative_iff
tff(fact_4623_signed__take__bit__negative__iff,axiom,
    ! [N: nat,K: int] :
      ( pp(aa(int,bool,ord_less(int,aa(int,int,bit_ri4674362597316999326ke_bit(int,N),K)),zero_zero(int)))
    <=> pp(aa(nat,bool,bit_se5641148757651400278ts_bit(int,K),N)) ) ).

% signed_take_bit_negative_iff
tff(fact_4624_bit__minus__numeral__Bit0__Suc__iff,axiom,
    ! [W: num,N: nat] :
      ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(int,aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit0(W)))),aa(nat,nat,suc,N)))
    <=> pp(aa(nat,bool,bit_se5641148757651400278ts_bit(int,aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),W))),N)) ) ).

% bit_minus_numeral_Bit0_Suc_iff
tff(fact_4625_bit__numeral__simps_I2_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [W: num,N: num] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,aa(num,A,numeral_numeral(A),bit0(W))),aa(num,nat,numeral_numeral(nat),N)))
        <=> pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,aa(num,A,numeral_numeral(A),W)),pred_numeral(N))) ) ) ).

% bit_numeral_simps(2)
tff(fact_4626_bit__minus__numeral__Bit1__Suc__iff,axiom,
    ! [W: num,N: nat] :
      ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(int,aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,W)))),aa(nat,nat,suc,N)))
    <=> ~ pp(aa(nat,bool,bit_se5641148757651400278ts_bit(int,aa(num,int,numeral_numeral(int),W)),N)) ) ).

% bit_minus_numeral_Bit1_Suc_iff
tff(fact_4627_bit__numeral__simps_I3_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [W: num,N: num] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,aa(num,A,numeral_numeral(A),aa(num,num,bit1,W))),aa(num,nat,numeral_numeral(nat),N)))
        <=> pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,aa(num,A,numeral_numeral(A),W)),pred_numeral(N))) ) ) ).

% bit_numeral_simps(3)
tff(fact_4628_bit__0,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,A2),zero_zero(nat)))
        <=> ~ pp(aa(A,bool,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A2)) ) ) ).

% bit_0
tff(fact_4629_bit__minus__numeral__int_I1_J,axiom,
    ! [W: num,N: num] :
      ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(int,aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit0(W)))),aa(num,nat,numeral_numeral(nat),N)))
    <=> pp(aa(nat,bool,bit_se5641148757651400278ts_bit(int,aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),W))),pred_numeral(N))) ) ).

% bit_minus_numeral_int(1)
tff(fact_4630_bit__minus__numeral__int_I2_J,axiom,
    ! [W: num,N: num] :
      ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(int,aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,W)))),aa(num,nat,numeral_numeral(nat),N)))
    <=> ~ pp(aa(nat,bool,bit_se5641148757651400278ts_bit(int,aa(num,int,numeral_numeral(int),W)),pred_numeral(N))) ) ).

% bit_minus_numeral_int(2)
tff(fact_4631_bit__mod__2__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A,N: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,modulo_modulo(A,A2,aa(num,A,numeral_numeral(A),bit0(one2)))),N))
        <=> ( ( N = zero_zero(nat) )
            & ~ pp(aa(A,bool,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A2)) ) ) ) ).

% bit_mod_2_iff
tff(fact_4632_bin__nth__minus__Bit0,axiom,
    ! [N: nat,W: num] :
      ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),N))
     => ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(int,aa(num,int,numeral_numeral(int),bit0(W))),N))
      <=> pp(aa(nat,bool,bit_se5641148757651400278ts_bit(int,aa(num,int,numeral_numeral(int),W)),aa(nat,nat,minus_minus(nat,N),one_one(nat)))) ) ) ).

% bin_nth_minus_Bit0
tff(fact_4633_bin__nth__minus__Bit1,axiom,
    ! [N: nat,W: num] :
      ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),N))
     => ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(int,aa(num,int,numeral_numeral(int),aa(num,num,bit1,W))),N))
      <=> pp(aa(nat,bool,bit_se5641148757651400278ts_bit(int,aa(num,int,numeral_numeral(int),W)),aa(nat,nat,minus_minus(nat,N),one_one(nat)))) ) ) ).

% bin_nth_minus_Bit1
tff(fact_4634_bit__of__nat__iff__bit,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [M: nat,N: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,aa(nat,A,semiring_1_of_nat(A),M)),N))
        <=> pp(aa(nat,bool,bit_se5641148757651400278ts_bit(nat,M),N)) ) ) ).

% bit_of_nat_iff_bit
tff(fact_4635_bit__numeral__iff,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [M: num,N: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,aa(num,A,numeral_numeral(A),M)),N))
        <=> pp(aa(nat,bool,bit_se5641148757651400278ts_bit(nat,aa(num,nat,numeral_numeral(nat),M)),N)) ) ) ).

% bit_numeral_iff
tff(fact_4636_bit__disjunctive__add__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A,B2: A,N: nat] :
          ( ! [N2: nat] :
              ( ~ pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,A2),N2))
              | ~ pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,B2),N2)) )
         => ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),N))
          <=> ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,A2),N))
              | pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,B2),N)) ) ) ) ) ).

% bit_disjunctive_add_iff
tff(fact_4637_prod__decode__aux_Ocases,axiom,
    ! [X: product_prod(nat,nat)] :
      ~ ! [K3: nat,M3: nat] : X != aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),K3),M3) ).

% prod_decode_aux.cases
tff(fact_4638_le__some__optE,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [M: A,X: option(A)] :
          ( pp(aa(option(A),bool,ord_less_eq(option(A),aa(A,option(A),some(A),M)),X))
         => ~ ! [M10: A] :
                ( ( X = aa(A,option(A),some(A),M10) )
               => ~ pp(aa(A,bool,ord_less_eq(A,M),M10)) ) ) ) ).

% le_some_optE
tff(fact_4639_option_Odistinct_I1_J,axiom,
    ! [A: $tType,X23: A] : none(A) != aa(A,option(A),some(A),X23) ).

% option.distinct(1)
tff(fact_4640_option_OdiscI,axiom,
    ! [A: $tType,Option: option(A),X23: A] :
      ( ( Option = aa(A,option(A),some(A),X23) )
     => ( Option != none(A) ) ) ).

% option.discI
tff(fact_4641_option_Oexhaust,axiom,
    ! [A: $tType,Y: option(A)] :
      ( ( Y != none(A) )
     => ~ ! [X22: A] : Y != aa(A,option(A),some(A),X22) ) ).

% option.exhaust
tff(fact_4642_split__option__ex,axiom,
    ! [A: $tType,P: fun(option(A),bool)] :
      ( ? [X_13: option(A)] : pp(aa(option(A),bool,P,X_13))
    <=> ( pp(aa(option(A),bool,P,none(A)))
        | ? [X2: A] : pp(aa(option(A),bool,P,aa(A,option(A),some(A),X2))) ) ) ).

% split_option_ex
tff(fact_4643_split__option__all,axiom,
    ! [A: $tType,P: fun(option(A),bool)] :
      ( ! [X_13: option(A)] : pp(aa(option(A),bool,P,X_13))
    <=> ( pp(aa(option(A),bool,P,none(A)))
        & ! [X2: A] : pp(aa(option(A),bool,P,aa(A,option(A),some(A),X2))) ) ) ).

% split_option_all
tff(fact_4644_combine__options__cases,axiom,
    ! [A: $tType,B: $tType,X: option(A),P: fun(option(A),fun(option(B),bool)),Y: option(B)] :
      ( ( ( X = none(A) )
       => pp(aa(option(B),bool,aa(option(A),fun(option(B),bool),P,X),Y)) )
     => ( ( ( Y = none(B) )
         => pp(aa(option(B),bool,aa(option(A),fun(option(B),bool),P,X),Y)) )
       => ( ! [A4: A,B3: B] :
              ( ( X = aa(A,option(A),some(A),A4) )
             => ( ( Y = aa(B,option(B),some(B),B3) )
               => pp(aa(option(B),bool,aa(option(A),fun(option(B),bool),P,X),Y)) ) )
         => pp(aa(option(B),bool,aa(option(A),fun(option(B),bool),P,X),Y)) ) ) ) ).

% combine_options_cases
tff(fact_4645_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t_Ocases,axiom,
    ! [X: vEBT_VEBT] :
      ( ! [A4: bool,B3: bool] : X != vEBT_Leaf(A4,B3)
     => ( ! [Uu2: nat,Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT] : X != vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw2)
       => ~ ! [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list(vEBT_VEBT),Uz2: vEBT_VEBT] : X != vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),Ux2,Uy2,Uz2) ) ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>t.cases
tff(fact_4646_VEBT__internal_Omembermima_Osimps_I3_J,axiom,
    ! [Mi: nat,Ma: nat,Va: list(vEBT_VEBT),Vb: vEBT_VEBT,X: nat] :
      ( vEBT_VEBT_membermima(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),zero_zero(nat),Va,Vb),X)
    <=> ( ( X = Mi )
        | ( X = Ma ) ) ) ).

% VEBT_internal.membermima.simps(3)
tff(fact_4647_bit__unset__bit__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,A2: A,N: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,aa(A,A,aa(nat,fun(A,A),bit_se2638667681897837118et_bit(A),M),A2)),N))
        <=> ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,A2),N))
            & ( M != N ) ) ) ) ).

% bit_unset_bit_iff
tff(fact_4648_not__bit__1__Suc,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat] : ~ pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,one_one(A)),aa(nat,nat,suc,N))) ) ).

% not_bit_1_Suc
tff(fact_4649_bit__1__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [N: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,one_one(A)),N))
        <=> ( N = zero_zero(nat) ) ) ) ).

% bit_1_iff
tff(fact_4650_bit__numeral__simps_I1_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [N: num] : ~ pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,one_one(A)),aa(num,nat,numeral_numeral(nat),N))) ) ).

% bit_numeral_simps(1)
tff(fact_4651_bit__take__bit__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,A2: A,N: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,aa(A,A,bit_se2584673776208193580ke_bit(A,M),A2)),N))
        <=> ( pp(aa(nat,bool,ord_less(nat,N),M))
            & pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,A2),N)) ) ) ) ).

% bit_take_bit_iff
tff(fact_4652_bit__of__bool__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [B2: bool,N: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,aa(bool,A,zero_neq_one_of_bool(A),B2)),N))
        <=> ( pp(B2)
            & ( N = zero_zero(nat) ) ) ) ) ).

% bit_of_bool_iff
tff(fact_4653_signed__take__bit__eq__if__positive,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A2: A,N: nat] :
          ( ~ pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,A2),N))
         => ( aa(A,A,bit_ri4674362597316999326ke_bit(A,N),A2) = aa(A,A,bit_se2584673776208193580ke_bit(A,N),A2) ) ) ) ).

% signed_take_bit_eq_if_positive
tff(fact_4654_T_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,X: nat] : vEBT_T_d_e_l_e_t_e(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),zero_zero(nat),TreeList,Summary),X) = one_one(nat) ).

% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.simps(5)
tff(fact_4655_VEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,X: nat] : vEBT_V1232361888498592333_e_t_e(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),zero_zero(nat),TreeList,Summary),X) = one_one(nat) ).

% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.simps(5)
tff(fact_4656_vebt__insert_Osimps_I4_J,axiom,
    ! [V: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,X: nat] : vEBT_vebt_insert(vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,V)),TreeList,Summary),X) = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),X),X)),aa(nat,nat,suc,aa(nat,nat,suc,V)),TreeList,Summary) ).

% vebt_insert.simps(4)
tff(fact_4657_VEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H_Osimps_I6_J,axiom,
    ! [Mi: nat,Ma: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,X: nat] : vEBT_V1232361888498592333_e_t_e(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,zero_zero(nat)),TreeList,Summary),X) = one_one(nat) ).

% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.simps(6)
tff(fact_4658_T_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_Osimps_I6_J,axiom,
    ! [Mi: nat,Ma: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,X: nat] : vEBT_T_d_e_l_e_t_e(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,zero_zero(nat)),TreeList,Summary),X) = one_one(nat) ).

% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.simps(6)
tff(fact_4659_height__node,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,N: nat] :
      ( vEBT_invar_vebt(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary),N)
     => pp(aa(nat,bool,ord_less_eq(nat,one_one(nat)),aa(vEBT_VEBT,nat,vEBT_VEBT_height,vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary)))) ) ).

% height_node
tff(fact_4660_bit__not__int__iff_H,axiom,
    ! [K: int,N: nat] :
      ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(int,aa(int,int,minus_minus(int,aa(int,int,uminus_uminus(int),K)),one_one(int))),N))
    <=> ~ pp(aa(nat,bool,bit_se5641148757651400278ts_bit(int,K),N)) ) ).

% bit_not_int_iff'
tff(fact_4661_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062N_092_060_094sub_062u_092_060_094sub_062l_092_060_094sub_062l_Ocases,axiom,
    ! [X: vEBT_VEBT] :
      ( ( X != vEBT_Leaf(fFalse,fFalse) )
     => ( ! [Uv2: bool] : X != vEBT_Leaf(fTrue,Uv2)
       => ( ! [Uu2: bool] : X != vEBT_Leaf(Uu2,fTrue)
         => ( ! [Uw2: nat,Ux2: list(vEBT_VEBT),Uy2: vEBT_VEBT] : X != vEBT_Node(none(product_prod(nat,nat)),Uw2,Ux2,Uy2)
           => ~ ! [Uz2: product_prod(nat,nat),Va3: nat,Vb2: list(vEBT_VEBT),Vc: vEBT_VEBT] : X != vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),Uz2),Va3,Vb2,Vc) ) ) ) ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.cases
tff(fact_4662_flip__bit__eq__if,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A2: A] : bit_se8732182000553998342ip_bit(A,N,A2) = aa(A,A,aa(nat,fun(A,A),if(fun(nat,fun(A,A)),aa(nat,bool,bit_se5641148757651400278ts_bit(A,A2),N),bit_se2638667681897837118et_bit(A),bit_se5668285175392031749et_bit(A)),N),A2) ) ).

% flip_bit_eq_if
tff(fact_4663_vebt__member_Osimps_I3_J,axiom,
    ! [V: product_prod(nat,nat),Uy: list(vEBT_VEBT),Uz: vEBT_VEBT,X: nat] : ~ pp(aa(nat,bool,vEBT_vebt_member(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V),zero_zero(nat),Uy,Uz)),X)) ).

% vebt_member.simps(3)
tff(fact_4664_option_Osize_I4_J,axiom,
    ! [A: $tType,X23: A] : aa(option(A),nat,size_size(option(A)),aa(A,option(A),some(A),X23)) = aa(nat,nat,suc,zero_zero(nat)) ).

% option.size(4)
tff(fact_4665_bit__imp__take__bit__positive,axiom,
    ! [N: nat,M: nat,K: int] :
      ( pp(aa(nat,bool,ord_less(nat,N),M))
     => ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(int,K),N))
       => pp(aa(int,bool,ord_less(int,zero_zero(int)),aa(int,int,bit_se2584673776208193580ke_bit(int,M),K))) ) ) ).

% bit_imp_take_bit_positive
tff(fact_4666_vebt__member_Osimps_I4_J,axiom,
    ! [V: product_prod(nat,nat),Vb: list(vEBT_VEBT),Vc2: vEBT_VEBT,X: nat] : ~ pp(aa(nat,bool,vEBT_vebt_member(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V),aa(nat,nat,suc,zero_zero(nat)),Vb,Vc2)),X)) ).

% vebt_member.simps(4)
tff(fact_4667_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(aa(product_prod(nat,nat),option(product_prod(nat,nat)),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)
tff(fact_4668_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(aa(product_prod(nat,nat),option(product_prod(nat,nat)),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)
tff(fact_4669_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(aa(product_prod(nat,nat),option(product_prod(nat,nat)),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)
tff(fact_4670_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(aa(product_prod(nat,nat),option(product_prod(nat,nat)),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)
tff(fact_4671_exp__eq__0__imp__not__bit,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [N: nat,A2: A] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N) = zero_zero(A) )
         => ~ pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,A2),N)) ) ) ).

% exp_eq_0_imp_not_bit
tff(fact_4672_bit__Suc,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A,N: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,A2),aa(nat,nat,suc,N)))
        <=> pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(num,A,numeral_numeral(A),bit0(one2)))),N)) ) ) ).

% bit_Suc
tff(fact_4673_bit__iff__idd__imp__stable,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A] :
          ( ! [N2: nat] :
              ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,A2),N2))
            <=> ~ pp(aa(A,bool,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A2)) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(num,A,numeral_numeral(A),bit0(one2))) = A2 ) ) ) ).

% bit_iff_idd_imp_stable
tff(fact_4674_stable__imp__bit__iff__odd,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A,N: nat] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(num,A,numeral_numeral(A),bit0(one2))) = A2 )
         => ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,A2),N))
          <=> ~ pp(aa(A,bool,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A2)) ) ) ) ).

% stable_imp_bit_iff_odd
tff(fact_4675_int__bit__bound,axiom,
    ! [K: int] :
      ~ ! [N2: nat] :
          ( ! [M4: nat] :
              ( pp(aa(nat,bool,ord_less_eq(nat,N2),M4))
             => ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(int,K),M4))
              <=> pp(aa(nat,bool,bit_se5641148757651400278ts_bit(int,K),N2)) ) )
         => ~ ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),N2))
             => ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(int,K),aa(nat,nat,minus_minus(nat,N2),one_one(nat))))
              <=> ~ pp(aa(nat,bool,bit_se5641148757651400278ts_bit(int,K),N2)) ) ) ) ).

% int_bit_bound
tff(fact_4676_num_Osize__gen_I1_J,axiom,
    size_num(one2) = zero_zero(nat) ).

% num.size_gen(1)
tff(fact_4677_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(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V),aa(nat,nat,suc,zero_zero(nat)),Vh,Vi),Vj) = one_one(nat) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.simps(6)
tff(fact_4678_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(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V),aa(nat,nat,suc,zero_zero(nat)),Vh,Vi),Vj) = one_one(nat) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.simps(6)
tff(fact_4679_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(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V),aa(nat,nat,suc,zero_zero(nat)),Vg,Vh),Vi) = one_one(nat) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.simps(5)
tff(fact_4680_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(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V),aa(nat,nat,suc,zero_zero(nat)),Vg,Vh),Vi) = one_one(nat) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.simps(5)
tff(fact_4681_bit__iff__odd,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A,N: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,A2),N))
        <=> ~ pp(aa(A,bool,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N)))) ) ) ).

% bit_iff_odd
tff(fact_4682_bit__int__def,axiom,
    ! [K: int,N: nat] :
      ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(int,K),N))
    <=> ~ pp(aa(int,bool,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),aa(int,int,aa(int,fun(int,int),divide_divide(int),K),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),N)))) ) ).

% bit_int_def
tff(fact_4683_even__bit__succ__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A,N: nat] :
          ( pp(aa(A,bool,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A2))
         => ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),A2)),N))
          <=> ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,A2),N))
              | ( N = zero_zero(nat) ) ) ) ) ) ).

% even_bit_succ_iff
tff(fact_4684_odd__bit__iff__bit__pred,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A,N: nat] :
          ( ~ pp(aa(A,bool,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A2))
         => ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,A2),N))
          <=> ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,aa(A,A,minus_minus(A,A2),one_one(A))),N))
              | ( N = zero_zero(nat) ) ) ) ) ) ).

% odd_bit_iff_bit_pred
tff(fact_4685_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,X: nat] : vEBT_T_m_e_m_b_e_r(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V),zero_zero(nat),Uy,Uz),X) = aa(num,nat,numeral_numeral(nat),bit0(one2)) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.simps(3)
tff(fact_4686_vebt__member_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,Va: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,X: nat] :
      ( pp(aa(nat,bool,vEBT_vebt_member(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList,Summary)),X))
    <=> ( ( X != Mi )
       => ( ( X != Ma )
         => ( ~ pp(aa(nat,bool,ord_less(nat,X),Mi))
            & ( ~ pp(aa(nat,bool,ord_less(nat,X),Mi))
             => ( ~ pp(aa(nat,bool,ord_less(nat,Ma),X))
                & ( ~ pp(aa(nat,bool,ord_less(nat,Ma),X))
                 => ( ( pp(aa(nat,bool,ord_less(nat,vEBT_VEBT_high(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)))
                     => pp(aa(nat,bool,vEBT_vebt_member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),vEBT_VEBT_low(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))) )
                    & pp(aa(nat,bool,ord_less(nat,vEBT_VEBT_high(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))) ) ) ) ) ) ) ) ) ).

% vebt_member.simps(5)
tff(fact_4687_VEBT__internal_Omembermima_Osimps_I4_J,axiom,
    ! [Mi: nat,Ma: nat,V: nat,TreeList: list(vEBT_VEBT),Vc2: vEBT_VEBT,X: nat] :
      ( vEBT_VEBT_membermima(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,V),TreeList,Vc2),X)
    <=> ( ( X = Mi )
        | ( X = Ma )
        | ( ( pp(aa(nat,bool,ord_less(nat,vEBT_VEBT_high(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)))
           => vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_VEBT_low(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V)),aa(num,nat,numeral_numeral(nat),bit0(one2))))) )
          & pp(aa(nat,bool,ord_less(nat,vEBT_VEBT_high(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))) ) ) ) ).

% VEBT_internal.membermima.simps(4)
tff(fact_4688_bit__sum__mult__2__cases,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,B2: A,N: nat] :
          ( ! [J2: nat] : ~ pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,A2),aa(nat,nat,suc,J2)))
         => ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),B2))),N))
          <=> ( ( ( N = zero_zero(nat) )
               => ~ pp(aa(A,bool,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A2)) )
              & ( ( N != zero_zero(nat) )
               => pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),B2)),N)) ) ) ) ) ) ).

% bit_sum_mult_2_cases
tff(fact_4689_bit__rec,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A,N: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,A2),N))
        <=> ( ( ( N = zero_zero(nat) )
             => ~ pp(aa(A,bool,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A2)) )
            & ( ( N != zero_zero(nat) )
             => pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(num,A,numeral_numeral(A),bit0(one2)))),aa(nat,nat,minus_minus(nat,N),one_one(nat)))) ) ) ) ) ).

% bit_rec
tff(fact_4690_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,X: nat] : vEBT_T_m_e_m_b_e_r(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V),aa(nat,nat,suc,zero_zero(nat)),Vb,Vc2),X) = aa(num,nat,numeral_numeral(nat),bit0(one2)) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.simps(4)
tff(fact_4691_VEBT__internal_Omembermima_Oelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( vEBT_VEBT_membermima(X,Xa2)
     => ( ! [Mi2: nat,Ma2: nat] :
            ( ? [Va3: list(vEBT_VEBT),Vb2: vEBT_VEBT] : X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),zero_zero(nat),Va3,Vb2)
           => ~ ( ( Xa2 = Mi2 )
                | ( Xa2 = Ma2 ) ) )
       => ( ! [Mi2: nat,Ma2: nat,V3: nat,TreeList2: list(vEBT_VEBT)] :
              ( ? [Vc: vEBT_VEBT] : X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,V3),TreeList2,Vc)
             => ~ ( ( Xa2 = Mi2 )
                  | ( Xa2 = Ma2 )
                  | ( ( pp(aa(nat,bool,ord_less(nat,vEBT_VEBT_high(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V3)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)))
                     => vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),vEBT_VEBT_high(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V3)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_VEBT_low(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V3)),aa(num,nat,numeral_numeral(nat),bit0(one2))))) )
                    & pp(aa(nat,bool,ord_less(nat,vEBT_VEBT_high(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V3)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2))) ) ) )
         => ~ ! [V3: nat,TreeList2: list(vEBT_VEBT)] :
                ( ? [Vd2: vEBT_VEBT] : X = vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,V3),TreeList2,Vd2)
               => ~ ( ( pp(aa(nat,bool,ord_less(nat,vEBT_VEBT_high(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V3)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)))
                     => vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),vEBT_VEBT_high(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V3)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_VEBT_low(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V3)),aa(num,nat,numeral_numeral(nat),bit0(one2))))) )
                    & pp(aa(nat,bool,ord_less(nat,vEBT_VEBT_high(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V3)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2))) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(2)
tff(fact_4692_invar__vebt_Ointros_I4_J,axiom,
    ! [TreeList: list(vEBT_VEBT),N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat,Mi: nat,Ma: nat] :
      ( ! [X3: vEBT_VEBT] :
          ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),member(vEBT_VEBT),X3),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList)))
         => vEBT_invar_vebt(X3,N) )
     => ( vEBT_invar_vebt(Summary,M)
       => ( ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M) )
         => ( ( M = N )
           => ( ( Deg = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),M) )
             => ( ! [I2: nat] :
                    ( pp(aa(nat,bool,ord_less(nat,I2),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M)))
                   => ( ? [X_13: nat] : pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),I2)),X_13))
                    <=> pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,Summary),I2)) ) )
               => ( ( ( Mi = Ma )
                   => ! [X3: vEBT_VEBT] :
                        ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),member(vEBT_VEBT),X3),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList)))
                       => ~ ? [X_1: nat] : pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,X3),X_1)) ) )
                 => ( pp(aa(nat,bool,ord_less_eq(nat,Mi),Ma))
                   => ( pp(aa(nat,bool,ord_less(nat,Ma),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg)))
                     => ( ( ( Mi != Ma )
                         => ! [I2: nat] :
                              ( pp(aa(nat,bool,ord_less(nat,I2),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M)))
                             => ( ( ( vEBT_VEBT_high(Ma,N) = I2 )
                                 => pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),I2)),vEBT_VEBT_low(Ma,N))) )
                                & ! [X3: nat] :
                                    ( ( ( vEBT_VEBT_high(X3,N) = I2 )
                                      & pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),I2)),vEBT_VEBT_low(X3,N))) )
                                   => ( pp(aa(nat,bool,ord_less(nat,Mi),X3))
                                      & pp(aa(nat,bool,ord_less_eq(nat,X3),Ma)) ) ) ) ) )
                       => vEBT_invar_vebt(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary),Deg) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.intros(4)
tff(fact_4693_vebt__member_Oelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( pp(aa(nat,bool,vEBT_vebt_member(X),Xa2))
     => ( ! [A4: bool,B3: bool] :
            ( ( X = vEBT_Leaf(A4,B3) )
           => ~ ( ( ( Xa2 = zero_zero(nat) )
                 => pp(A4) )
                & ( ( Xa2 != zero_zero(nat) )
                 => ( ( ( Xa2 = one_one(nat) )
                     => pp(B3) )
                    & ( Xa2 = one_one(nat) ) ) ) ) )
       => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list(vEBT_VEBT)] :
              ( ? [Summary2: vEBT_VEBT] : X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary2)
             => ~ ( ( Xa2 != Mi2 )
                 => ( ( Xa2 != Ma2 )
                   => ( ~ pp(aa(nat,bool,ord_less(nat,Xa2),Mi2))
                      & ( ~ pp(aa(nat,bool,ord_less(nat,Xa2),Mi2))
                       => ( ~ pp(aa(nat,bool,ord_less(nat,Ma2),Xa2))
                          & ( ~ pp(aa(nat,bool,ord_less(nat,Ma2),Xa2))
                           => ( ( pp(aa(nat,bool,ord_less(nat,vEBT_VEBT_high(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)))
                               => pp(aa(nat,bool,vEBT_vebt_member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),vEBT_VEBT_high(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),vEBT_VEBT_low(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))) )
                              & pp(aa(nat,bool,ord_less(nat,vEBT_VEBT_high(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2))) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(2)
tff(fact_4694_VEBT__internal_Omembermima_Oelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ~ vEBT_VEBT_membermima(X,Xa2)
     => ( ! [Uu2: bool,Uv2: bool] : X != vEBT_Leaf(Uu2,Uv2)
       => ( ! [Ux2: list(vEBT_VEBT),Uy2: vEBT_VEBT] : X != vEBT_Node(none(product_prod(nat,nat)),zero_zero(nat),Ux2,Uy2)
         => ( ! [Mi2: nat,Ma2: nat] :
                ( ? [Va3: list(vEBT_VEBT),Vb2: vEBT_VEBT] : X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),zero_zero(nat),Va3,Vb2)
               => ( ( Xa2 = Mi2 )
                  | ( Xa2 = Ma2 ) ) )
           => ( ! [Mi2: nat,Ma2: nat,V3: nat,TreeList2: list(vEBT_VEBT)] :
                  ( ? [Vc: vEBT_VEBT] : X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,V3),TreeList2,Vc)
                 => ( ( Xa2 = Mi2 )
                    | ( Xa2 = Ma2 )
                    | ( ( pp(aa(nat,bool,ord_less(nat,vEBT_VEBT_high(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V3)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)))
                       => vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),vEBT_VEBT_high(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V3)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_VEBT_low(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V3)),aa(num,nat,numeral_numeral(nat),bit0(one2))))) )
                      & pp(aa(nat,bool,ord_less(nat,vEBT_VEBT_high(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V3)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2))) ) ) )
             => ~ ! [V3: nat,TreeList2: list(vEBT_VEBT)] :
                    ( ? [Vd2: vEBT_VEBT] : X = vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,V3),TreeList2,Vd2)
                   => ( ( pp(aa(nat,bool,ord_less(nat,vEBT_VEBT_high(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V3)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)))
                       => vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),vEBT_VEBT_high(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V3)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_VEBT_low(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V3)),aa(num,nat,numeral_numeral(nat),bit0(one2))))) )
                      & pp(aa(nat,bool,ord_less(nat,vEBT_VEBT_high(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V3)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2))) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(3)
tff(fact_4695_VEBT__internal_Omembermima_Oelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: bool] :
      ( ( vEBT_VEBT_membermima(X,Xa2)
      <=> pp(Y) )
     => ( ( ? [Uu2: bool,Uv2: bool] : X = vEBT_Leaf(Uu2,Uv2)
         => pp(Y) )
       => ( ( ? [Ux2: list(vEBT_VEBT),Uy2: vEBT_VEBT] : X = vEBT_Node(none(product_prod(nat,nat)),zero_zero(nat),Ux2,Uy2)
           => pp(Y) )
         => ( ! [Mi2: nat,Ma2: nat] :
                ( ? [Va3: list(vEBT_VEBT),Vb2: vEBT_VEBT] : X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),zero_zero(nat),Va3,Vb2)
               => ( pp(Y)
                <=> ~ ( ( Xa2 = Mi2 )
                      | ( Xa2 = Ma2 ) ) ) )
           => ( ! [Mi2: nat,Ma2: nat,V3: nat,TreeList2: list(vEBT_VEBT)] :
                  ( ? [Vc: vEBT_VEBT] : X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,V3),TreeList2,Vc)
                 => ( pp(Y)
                  <=> ~ ( ( Xa2 = Mi2 )
                        | ( Xa2 = Ma2 )
                        | ( ( pp(aa(nat,bool,ord_less(nat,vEBT_VEBT_high(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V3)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)))
                           => vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),vEBT_VEBT_high(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V3)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_VEBT_low(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V3)),aa(num,nat,numeral_numeral(nat),bit0(one2))))) )
                          & pp(aa(nat,bool,ord_less(nat,vEBT_VEBT_high(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V3)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2))) ) ) ) )
             => ~ ! [V3: nat,TreeList2: list(vEBT_VEBT)] :
                    ( ? [Vd2: vEBT_VEBT] : X = vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,V3),TreeList2,Vd2)
                   => ( pp(Y)
                    <=> ~ ( ( pp(aa(nat,bool,ord_less(nat,vEBT_VEBT_high(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V3)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)))
                           => vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),vEBT_VEBT_high(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V3)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_VEBT_low(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V3)),aa(num,nat,numeral_numeral(nat),bit0(one2))))) )
                          & pp(aa(nat,bool,ord_less(nat,vEBT_VEBT_high(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V3)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2))) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(1)
tff(fact_4696_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,X: nat] : vEBT_T_m_e_m_b_e_r(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList,Summary),X) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),if(nat,aa(nat,bool,fequal(nat,X),Mi),one_one(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),if(nat,aa(nat,bool,fequal(nat,X),Ma),one_one(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),if(nat,aa(nat,bool,ord_less(nat,X),Mi),one_one(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),if(nat,aa(nat,bool,ord_less(nat,Ma),X),one_one(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,bit0(bit0(one2))))),if(nat,aa(nat,bool,ord_less(nat,vEBT_VEBT_high(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),vEBT_T_m_e_m_b_e_r(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_VEBT_low(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),one_one(nat))))))))))) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.simps(5)
tff(fact_4697_invar__vebt_Ointros_I5_J,axiom,
    ! [TreeList: list(vEBT_VEBT),N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat,Mi: nat,Ma: nat] :
      ( ! [X3: vEBT_VEBT] :
          ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),member(vEBT_VEBT),X3),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList)))
         => vEBT_invar_vebt(X3,N) )
     => ( vEBT_invar_vebt(Summary,M)
       => ( ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M) )
         => ( ( M = aa(nat,nat,suc,N) )
           => ( ( Deg = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),M) )
             => ( ! [I2: nat] :
                    ( pp(aa(nat,bool,ord_less(nat,I2),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M)))
                   => ( ? [X_13: nat] : pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),I2)),X_13))
                    <=> pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,Summary),I2)) ) )
               => ( ( ( Mi = Ma )
                   => ! [X3: vEBT_VEBT] :
                        ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),member(vEBT_VEBT),X3),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList)))
                       => ~ ? [X_1: nat] : pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,X3),X_1)) ) )
                 => ( pp(aa(nat,bool,ord_less_eq(nat,Mi),Ma))
                   => ( pp(aa(nat,bool,ord_less(nat,Ma),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg)))
                     => ( ( ( Mi != Ma )
                         => ! [I2: nat] :
                              ( pp(aa(nat,bool,ord_less(nat,I2),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M)))
                             => ( ( ( vEBT_VEBT_high(Ma,N) = I2 )
                                 => pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),I2)),vEBT_VEBT_low(Ma,N))) )
                                & ! [X3: nat] :
                                    ( ( ( vEBT_VEBT_high(X3,N) = I2 )
                                      & pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),I2)),vEBT_VEBT_low(X3,N))) )
                                   => ( pp(aa(nat,bool,ord_less(nat,Mi),X3))
                                      & pp(aa(nat,bool,ord_less_eq(nat,X3),Ma)) ) ) ) ) )
                       => vEBT_invar_vebt(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary),Deg) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.intros(5)
tff(fact_4698_Bit__Operations_Oset__bit__eq,axiom,
    ! [N: nat,K: int] : aa(int,int,aa(nat,fun(int,int),bit_se5668285175392031749et_bit(int),N),K) = aa(int,int,aa(int,fun(int,int),plus_plus(int),K),aa(int,int,aa(int,fun(int,int),times_times(int),aa(bool,int,zero_neq_one_of_bool(int),aa(bool,bool,fNot,aa(nat,bool,bit_se5641148757651400278ts_bit(int,K),N)))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),N))) ).

% Bit_Operations.set_bit_eq
tff(fact_4699_unset__bit__eq,axiom,
    ! [N: nat,K: int] : aa(int,int,aa(nat,fun(int,int),bit_se2638667681897837118et_bit(int),N),K) = aa(int,int,minus_minus(int,K),aa(int,int,aa(int,fun(int,int),times_times(int),aa(bool,int,zero_neq_one_of_bool(int),aa(nat,bool,bit_se5641148757651400278ts_bit(int,K),N))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),N))) ).

% unset_bit_eq
tff(fact_4700_vebt__member_Oelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ~ pp(aa(nat,bool,vEBT_vebt_member(X),Xa2))
     => ( ! [A4: bool,B3: bool] :
            ( ( X = vEBT_Leaf(A4,B3) )
           => ( ( ( Xa2 = zero_zero(nat) )
               => pp(A4) )
              & ( ( Xa2 != zero_zero(nat) )
               => ( ( ( Xa2 = one_one(nat) )
                   => pp(B3) )
                  & ( Xa2 = one_one(nat) ) ) ) ) )
       => ( ! [Uu2: nat,Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT] : X != vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw2)
         => ( ! [V3: product_prod(nat,nat),Uy2: list(vEBT_VEBT),Uz2: vEBT_VEBT] : X != vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),zero_zero(nat),Uy2,Uz2)
           => ( ! [V3: product_prod(nat,nat),Vb2: list(vEBT_VEBT),Vc: vEBT_VEBT] : X != vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),aa(nat,nat,suc,zero_zero(nat)),Vb2,Vc)
             => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list(vEBT_VEBT)] :
                    ( ? [Summary2: vEBT_VEBT] : X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary2)
                   => ( ( Xa2 != Mi2 )
                     => ( ( Xa2 != Ma2 )
                       => ( ~ pp(aa(nat,bool,ord_less(nat,Xa2),Mi2))
                          & ( ~ pp(aa(nat,bool,ord_less(nat,Xa2),Mi2))
                           => ( ~ pp(aa(nat,bool,ord_less(nat,Ma2),Xa2))
                              & ( ~ pp(aa(nat,bool,ord_less(nat,Ma2),Xa2))
                               => ( ( pp(aa(nat,bool,ord_less(nat,vEBT_VEBT_high(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)))
                                   => pp(aa(nat,bool,vEBT_vebt_member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),vEBT_VEBT_high(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),vEBT_VEBT_low(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))) )
                                  & pp(aa(nat,bool,ord_less(nat,vEBT_VEBT_high(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2))) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(3)
tff(fact_4701_vebt__member_Oelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: bool] :
      ( ( pp(aa(nat,bool,vEBT_vebt_member(X),Xa2))
      <=> pp(Y) )
     => ( ! [A4: bool,B3: bool] :
            ( ( X = vEBT_Leaf(A4,B3) )
           => ( pp(Y)
            <=> ~ ( ( ( Xa2 = zero_zero(nat) )
                   => pp(A4) )
                  & ( ( Xa2 != zero_zero(nat) )
                   => ( ( ( Xa2 = one_one(nat) )
                       => pp(B3) )
                      & ( Xa2 = one_one(nat) ) ) ) ) ) )
       => ( ( ? [Uu2: nat,Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT] : X = vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw2)
           => pp(Y) )
         => ( ( ? [V3: product_prod(nat,nat),Uy2: list(vEBT_VEBT),Uz2: vEBT_VEBT] : X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),zero_zero(nat),Uy2,Uz2)
             => pp(Y) )
           => ( ( ? [V3: product_prod(nat,nat),Vb2: list(vEBT_VEBT),Vc: vEBT_VEBT] : X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),aa(nat,nat,suc,zero_zero(nat)),Vb2,Vc)
               => pp(Y) )
             => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list(vEBT_VEBT)] :
                    ( ? [Summary2: vEBT_VEBT] : X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary2)
                   => ( pp(Y)
                    <=> ~ ( ( Xa2 != Mi2 )
                         => ( ( Xa2 != Ma2 )
                           => ( ~ pp(aa(nat,bool,ord_less(nat,Xa2),Mi2))
                              & ( ~ pp(aa(nat,bool,ord_less(nat,Xa2),Mi2))
                               => ( ~ pp(aa(nat,bool,ord_less(nat,Ma2),Xa2))
                                  & ( ~ pp(aa(nat,bool,ord_less(nat,Ma2),Xa2))
                                   => ( ( pp(aa(nat,bool,ord_less(nat,vEBT_VEBT_high(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)))
                                       => pp(aa(nat,bool,vEBT_vebt_member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),vEBT_VEBT_high(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),vEBT_VEBT_low(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))) )
                                      & pp(aa(nat,bool,ord_less(nat,vEBT_VEBT_high(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2))) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(1)
tff(fact_4702_take__bit__Suc__from__most,axiom,
    ! [N: nat,K: int] : aa(int,int,bit_se2584673776208193580ke_bit(int,aa(nat,nat,suc,N)),K) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),N)),aa(bool,int,zero_neq_one_of_bool(int),aa(nat,bool,bit_se5641148757651400278ts_bit(int,K),N)))),aa(int,int,bit_se2584673776208193580ke_bit(int,N),K)) ).

% take_bit_Suc_from_most
tff(fact_4703_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_Oelims,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: nat] :
      ( ( vEBT_T_m_e_m_b_e_r(X,Xa2) = Y )
     => ( ( ? [A4: bool,B3: bool] : X = vEBT_Leaf(A4,B3)
         => ( Y != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),if(nat,aa(nat,bool,fequal(nat,Xa2),zero_zero(nat)),one_one(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),one_one(nat)))) ) )
       => ( ( ? [Uu2: nat,Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT] : X = vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw2)
           => ( Y != aa(num,nat,numeral_numeral(nat),bit0(one2)) ) )
         => ( ( ? [V3: product_prod(nat,nat),Uy2: list(vEBT_VEBT),Uz2: vEBT_VEBT] : X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),zero_zero(nat),Uy2,Uz2)
             => ( Y != aa(num,nat,numeral_numeral(nat),bit0(one2)) ) )
           => ( ( ? [V3: product_prod(nat,nat),Vb2: list(vEBT_VEBT),Vc: vEBT_VEBT] : X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),aa(nat,nat,suc,zero_zero(nat)),Vb2,Vc)
               => ( Y != aa(num,nat,numeral_numeral(nat),bit0(one2)) ) )
             => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list(vEBT_VEBT)] :
                    ( ? [Summary2: vEBT_VEBT] : X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary2)
                   => ( Y != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),if(nat,aa(nat,bool,fequal(nat,Xa2),Mi2),one_one(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),if(nat,aa(nat,bool,fequal(nat,Xa2),Ma2),one_one(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),if(nat,aa(nat,bool,ord_less(nat,Xa2),Mi2),one_one(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),if(nat,aa(nat,bool,ord_less(nat,Ma2),Xa2),one_one(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,bit0(bit0(one2))))),if(nat,aa(nat,bool,ord_less(nat,vEBT_VEBT_high(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),vEBT_T_m_e_m_b_e_r(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),vEBT_VEBT_high(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_VEBT_low(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),one_one(nat))))))))))) ) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.elims
tff(fact_4704_num_Osize__gen_I2_J,axiom,
    ! [X23: num] : size_num(bit0(X23)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),size_num(X23)),aa(nat,nat,suc,zero_zero(nat))) ).

% num.size_gen(2)
tff(fact_4705_invar__vebt_Ocases,axiom,
    ! [A1: vEBT_VEBT,A22: nat] :
      ( vEBT_invar_vebt(A1,A22)
     => ( ( ? [A4: bool,B3: bool] : A1 = vEBT_Leaf(A4,B3)
         => ( A22 != aa(nat,nat,suc,zero_zero(nat)) ) )
       => ( ! [TreeList2: list(vEBT_VEBT),N2: nat,Summary2: vEBT_VEBT,M3: nat,Deg2: nat] :
              ( ( A1 = vEBT_Node(none(product_prod(nat,nat)),Deg2,TreeList2,Summary2) )
             => ( ( A22 = Deg2 )
               => ( ! [X4: vEBT_VEBT] :
                      ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),member(vEBT_VEBT),X4),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2)))
                     => vEBT_invar_vebt(X4,N2) )
                 => ( vEBT_invar_vebt(Summary2,M3)
                   => ( ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M3) )
                     => ( ( M3 = N2 )
                       => ( ( Deg2 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N2),M3) )
                         => ( ~ ? [X_12: nat] : pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,Summary2),X_12))
                           => ~ ! [X4: vEBT_VEBT] :
                                  ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),member(vEBT_VEBT),X4),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2)))
                                 => ~ ? [X_12: nat] : pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,X4),X_12)) ) ) ) ) ) ) ) ) )
         => ( ! [TreeList2: list(vEBT_VEBT),N2: nat,Summary2: vEBT_VEBT,M3: nat,Deg2: nat] :
                ( ( A1 = vEBT_Node(none(product_prod(nat,nat)),Deg2,TreeList2,Summary2) )
               => ( ( A22 = Deg2 )
                 => ( ! [X4: vEBT_VEBT] :
                        ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),member(vEBT_VEBT),X4),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2)))
                       => vEBT_invar_vebt(X4,N2) )
                   => ( vEBT_invar_vebt(Summary2,M3)
                     => ( ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M3) )
                       => ( ( M3 = aa(nat,nat,suc,N2) )
                         => ( ( Deg2 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N2),M3) )
                           => ( ~ ? [X_12: nat] : pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,Summary2),X_12))
                             => ~ ! [X4: vEBT_VEBT] :
                                    ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),member(vEBT_VEBT),X4),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2)))
                                   => ~ ? [X_12: nat] : pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,X4),X_12)) ) ) ) ) ) ) ) ) )
           => ( ! [TreeList2: list(vEBT_VEBT),N2: nat,Summary2: vEBT_VEBT,M3: nat,Deg2: nat,Mi2: nat,Ma2: nat] :
                  ( ( A1 = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),Deg2,TreeList2,Summary2) )
                 => ( ( A22 = Deg2 )
                   => ( ! [X4: vEBT_VEBT] :
                          ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),member(vEBT_VEBT),X4),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2)))
                         => vEBT_invar_vebt(X4,N2) )
                     => ( vEBT_invar_vebt(Summary2,M3)
                       => ( ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M3) )
                         => ( ( M3 = N2 )
                           => ( ( Deg2 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N2),M3) )
                             => ( ! [I4: nat] :
                                    ( pp(aa(nat,bool,ord_less(nat,I4),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M3)))
                                   => ( ? [X_13: nat] : pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),I4)),X_13))
                                    <=> pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,Summary2),I4)) ) )
                               => ( ( ( Mi2 = Ma2 )
                                   => ! [X4: vEBT_VEBT] :
                                        ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),member(vEBT_VEBT),X4),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2)))
                                       => ~ ? [X_12: nat] : pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,X4),X_12)) ) )
                                 => ( pp(aa(nat,bool,ord_less_eq(nat,Mi2),Ma2))
                                   => ( pp(aa(nat,bool,ord_less(nat,Ma2),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg2)))
                                     => ~ ( ( Mi2 != Ma2 )
                                         => ! [I4: nat] :
                                              ( pp(aa(nat,bool,ord_less(nat,I4),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M3)))
                                             => ( ( ( vEBT_VEBT_high(Ma2,N2) = I4 )
                                                 => pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),I4)),vEBT_VEBT_low(Ma2,N2))) )
                                                & ! [X4: nat] :
                                                    ( ( ( vEBT_VEBT_high(X4,N2) = I4 )
                                                      & pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),I4)),vEBT_VEBT_low(X4,N2))) )
                                                   => ( pp(aa(nat,bool,ord_less(nat,Mi2),X4))
                                                      & pp(aa(nat,bool,ord_less_eq(nat,X4),Ma2)) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
             => ~ ! [TreeList2: list(vEBT_VEBT),N2: nat,Summary2: vEBT_VEBT,M3: nat,Deg2: nat,Mi2: nat,Ma2: nat] :
                    ( ( A1 = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),Deg2,TreeList2,Summary2) )
                   => ( ( A22 = Deg2 )
                     => ( ! [X4: vEBT_VEBT] :
                            ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),member(vEBT_VEBT),X4),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2)))
                           => vEBT_invar_vebt(X4,N2) )
                       => ( vEBT_invar_vebt(Summary2,M3)
                         => ( ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M3) )
                           => ( ( M3 = aa(nat,nat,suc,N2) )
                             => ( ( Deg2 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N2),M3) )
                               => ( ! [I4: nat] :
                                      ( pp(aa(nat,bool,ord_less(nat,I4),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M3)))
                                     => ( ? [X_13: nat] : pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),I4)),X_13))
                                      <=> pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,Summary2),I4)) ) )
                                 => ( ( ( Mi2 = Ma2 )
                                     => ! [X4: vEBT_VEBT] :
                                          ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),member(vEBT_VEBT),X4),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2)))
                                         => ~ ? [X_12: nat] : pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,X4),X_12)) ) )
                                   => ( pp(aa(nat,bool,ord_less_eq(nat,Mi2),Ma2))
                                     => ( pp(aa(nat,bool,ord_less(nat,Ma2),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg2)))
                                       => ~ ( ( Mi2 != Ma2 )
                                           => ! [I4: nat] :
                                                ( pp(aa(nat,bool,ord_less(nat,I4),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M3)))
                                               => ( ( ( vEBT_VEBT_high(Ma2,N2) = I4 )
                                                   => pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),I4)),vEBT_VEBT_low(Ma2,N2))) )
                                                  & ! [X4: nat] :
                                                      ( ( ( vEBT_VEBT_high(X4,N2) = I4 )
                                                        & pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),I4)),vEBT_VEBT_low(X4,N2))) )
                                                     => ( pp(aa(nat,bool,ord_less(nat,Mi2),X4))
                                                        & pp(aa(nat,bool,ord_less_eq(nat,X4),Ma2)) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.cases
tff(fact_4706_invar__vebt_Osimps,axiom,
    ! [A1: vEBT_VEBT,A22: nat] :
      ( vEBT_invar_vebt(A1,A22)
    <=> ( ( ? [A6: bool,B7: bool] : A1 = vEBT_Leaf(A6,B7)
          & ( A22 = aa(nat,nat,suc,zero_zero(nat)) ) )
        | ? [TreeList3: list(vEBT_VEBT),N5: nat,Summary3: vEBT_VEBT] :
            ( ( A1 = vEBT_Node(none(product_prod(nat,nat)),A22,TreeList3,Summary3) )
            & ! [X2: vEBT_VEBT] :
                ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),member(vEBT_VEBT),X2),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList3)))
               => vEBT_invar_vebt(X2,N5) )
            & vEBT_invar_vebt(Summary3,N5)
            & ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList3) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N5) )
            & ( A22 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N5),N5) )
            & ~ ? [X_13: nat] : pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,Summary3),X_13))
            & ! [X2: vEBT_VEBT] :
                ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),member(vEBT_VEBT),X2),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList3)))
               => ~ ? [X_13: nat] : pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,X2),X_13)) ) )
        | ? [TreeList3: list(vEBT_VEBT),N5: nat,Summary3: vEBT_VEBT] :
            ( ( A1 = vEBT_Node(none(product_prod(nat,nat)),A22,TreeList3,Summary3) )
            & ! [X2: vEBT_VEBT] :
                ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),member(vEBT_VEBT),X2),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList3)))
               => vEBT_invar_vebt(X2,N5) )
            & vEBT_invar_vebt(Summary3,aa(nat,nat,suc,N5))
            & ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList3) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,N5)) )
            & ( A22 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N5),aa(nat,nat,suc,N5)) )
            & ~ ? [X_13: nat] : pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,Summary3),X_13))
            & ! [X2: vEBT_VEBT] :
                ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),member(vEBT_VEBT),X2),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList3)))
               => ~ ? [X_13: nat] : pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,X2),X_13)) ) )
        | ? [TreeList3: list(vEBT_VEBT),N5: nat,Summary3: vEBT_VEBT,Mi3: nat,Ma3: nat] :
            ( ( A1 = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi3),Ma3)),A22,TreeList3,Summary3) )
            & ! [X2: vEBT_VEBT] :
                ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),member(vEBT_VEBT),X2),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList3)))
               => vEBT_invar_vebt(X2,N5) )
            & vEBT_invar_vebt(Summary3,N5)
            & ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList3) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N5) )
            & ( A22 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N5),N5) )
            & ! [I3: nat] :
                ( pp(aa(nat,bool,ord_less(nat,I3),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N5)))
               => ( ? [X_13: nat] : pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList3),I3)),X_13))
                <=> pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,Summary3),I3)) ) )
            & ( ( Mi3 = Ma3 )
             => ! [X2: vEBT_VEBT] :
                  ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),member(vEBT_VEBT),X2),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList3)))
                 => ~ ? [X_13: nat] : pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,X2),X_13)) ) )
            & pp(aa(nat,bool,ord_less_eq(nat,Mi3),Ma3))
            & pp(aa(nat,bool,ord_less(nat,Ma3),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),A22)))
            & ( ( Mi3 != Ma3 )
             => ! [I3: nat] :
                  ( pp(aa(nat,bool,ord_less(nat,I3),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N5)))
                 => ( ( ( vEBT_VEBT_high(Ma3,N5) = I3 )
                     => pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList3),I3)),vEBT_VEBT_low(Ma3,N5))) )
                    & ! [X2: nat] :
                        ( ( ( vEBT_VEBT_high(X2,N5) = I3 )
                          & pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList3),I3)),vEBT_VEBT_low(X2,N5))) )
                       => ( pp(aa(nat,bool,ord_less(nat,Mi3),X2))
                          & pp(aa(nat,bool,ord_less_eq(nat,X2),Ma3)) ) ) ) ) ) )
        | ? [TreeList3: list(vEBT_VEBT),N5: nat,Summary3: vEBT_VEBT,Mi3: nat,Ma3: nat] :
            ( ( A1 = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi3),Ma3)),A22,TreeList3,Summary3) )
            & ! [X2: vEBT_VEBT] :
                ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),member(vEBT_VEBT),X2),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList3)))
               => vEBT_invar_vebt(X2,N5) )
            & vEBT_invar_vebt(Summary3,aa(nat,nat,suc,N5))
            & ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList3) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,N5)) )
            & ( A22 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N5),aa(nat,nat,suc,N5)) )
            & ! [I3: nat] :
                ( pp(aa(nat,bool,ord_less(nat,I3),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,N5))))
               => ( ? [X_13: nat] : pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList3),I3)),X_13))
                <=> pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,Summary3),I3)) ) )
            & ( ( Mi3 = Ma3 )
             => ! [X2: vEBT_VEBT] :
                  ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),member(vEBT_VEBT),X2),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList3)))
                 => ~ ? [X_13: nat] : pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,X2),X_13)) ) )
            & pp(aa(nat,bool,ord_less_eq(nat,Mi3),Ma3))
            & pp(aa(nat,bool,ord_less(nat,Ma3),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),A22)))
            & ( ( Mi3 != Ma3 )
             => ! [I3: nat] :
                  ( pp(aa(nat,bool,ord_less(nat,I3),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,N5))))
                 => ( ( ( vEBT_VEBT_high(Ma3,N5) = I3 )
                     => pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList3),I3)),vEBT_VEBT_low(Ma3,N5))) )
                    & ! [X2: nat] :
                        ( ( ( vEBT_VEBT_high(X2,N5) = I3 )
                          & pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList3),I3)),vEBT_VEBT_low(X2,N5))) )
                       => ( pp(aa(nat,bool,ord_less(nat,Mi3),X2))
                          & pp(aa(nat,bool,ord_less_eq(nat,X2),Ma3)) ) ) ) ) ) ) ) ) ).

% invar_vebt.simps
tff(fact_4707_div__half__nat,axiom,
    ! [Y: nat,X: nat] :
      ( ( Y != zero_zero(nat) )
     => ( aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),X),Y)),modulo_modulo(nat,X,Y)) = if(product_prod(nat,nat),aa(nat,bool,ord_less_eq(nat,Y),aa(nat,nat,minus_minus(nat,X),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),X),aa(num,nat,numeral_numeral(nat),bit0(one2)))),Y))),Y))),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),X),aa(num,nat,numeral_numeral(nat),bit0(one2)))),Y))),one_one(nat))),aa(nat,nat,minus_minus(nat,aa(nat,nat,minus_minus(nat,X),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),X),aa(num,nat,numeral_numeral(nat),bit0(one2)))),Y))),Y))),Y)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),X),aa(num,nat,numeral_numeral(nat),bit0(one2)))),Y))),aa(nat,nat,minus_minus(nat,X),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),X),aa(num,nat,numeral_numeral(nat),bit0(one2)))),Y))),Y)))) ) ) ).

% div_half_nat
tff(fact_4708_pred__list__to__short,axiom,
    ! [Deg: nat,X: nat,Ma: nat,TreeList: list(vEBT_VEBT),Mi: nat,Summary: vEBT_VEBT] :
      ( pp(aa(nat,bool,ord_less_eq(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg))
     => ( pp(aa(nat,bool,ord_less_eq(nat,X),Ma))
       => ( pp(aa(nat,bool,ord_less_eq(nat,aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),vEBT_VEBT_high(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))))
         => ( vEBT_vebt_pred(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary),X) = none(nat) ) ) ) ) ).

% pred_list_to_short
tff(fact_4709_succ__list__to__short,axiom,
    ! [Deg: nat,Mi: nat,X: nat,TreeList: list(vEBT_VEBT),Ma: nat,Summary: vEBT_VEBT] :
      ( pp(aa(nat,bool,ord_less_eq(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg))
     => ( pp(aa(nat,bool,ord_less_eq(nat,Mi),X))
       => ( pp(aa(nat,bool,ord_less_eq(nat,aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),vEBT_VEBT_high(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))))
         => ( vEBT_vebt_succ(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary),X) = none(nat) ) ) ) ) ).

% succ_list_to_short
tff(fact_4710_power__shift,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),X),Y) = Z )
    <=> ( aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_power,aa(nat,option(nat),some(nat),X)),aa(nat,option(nat),some(nat),Y)) = aa(nat,option(nat),some(nat),Z) ) ) ).

% power_shift
tff(fact_4711_succ__corr,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat,Sx: nat] :
      ( vEBT_invar_vebt(T,N)
     => ( ( vEBT_vebt_succ(T,X) = aa(nat,option(nat),some(nat),Sx) )
      <=> vEBT_is_succ_in_set(vEBT_VEBT_set_vebt(T),X,Sx) ) ) ).

% succ_corr
tff(fact_4712_pred__corr,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat,Px: nat] :
      ( vEBT_invar_vebt(T,N)
     => ( ( vEBT_vebt_pred(T,X) = aa(nat,option(nat),some(nat),Px) )
      <=> vEBT_is_pred_in_set(vEBT_VEBT_set_vebt(T),X,Px) ) ) ).

% pred_corr
tff(fact_4713_succ__correct,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat,Sx: nat] :
      ( vEBT_invar_vebt(T,N)
     => ( ( vEBT_vebt_succ(T,X) = aa(nat,option(nat),some(nat),Sx) )
      <=> vEBT_is_succ_in_set(vEBT_set_vebt(T),X,Sx) ) ) ).

% succ_correct
tff(fact_4714_pred__correct,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat,Sx: nat] :
      ( vEBT_invar_vebt(T,N)
     => ( ( vEBT_vebt_pred(T,X) = aa(nat,option(nat),some(nat),Sx) )
      <=> vEBT_is_pred_in_set(vEBT_set_vebt(T),X,Sx) ) ) ).

% pred_correct
tff(fact_4715_helpypredd,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat,Y: nat] :
      ( vEBT_invar_vebt(T,N)
     => ( ( vEBT_vebt_pred(T,X) = aa(nat,option(nat),some(nat),Y) )
       => pp(aa(nat,bool,ord_less(nat,Y),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N))) ) ) ).

% helpypredd
tff(fact_4716_helpyd,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat,Y: nat] :
      ( vEBT_invar_vebt(T,N)
     => ( ( vEBT_vebt_succ(T,X) = aa(nat,option(nat),some(nat),Y) )
       => pp(aa(nat,bool,ord_less(nat,Y),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N))) ) ) ).

% helpyd
tff(fact_4717_geqmaxNone,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,N: nat,X: nat] :
      ( vEBT_invar_vebt(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary),N)
     => ( pp(aa(nat,bool,ord_less_eq(nat,Ma),X))
       => ( vEBT_vebt_succ(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary),X) = none(nat) ) ) ) ).

% geqmaxNone
tff(fact_4718_test__bit__1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(word(A),one_one(word(A))),N))
        <=> ( N = zero_zero(nat) ) ) ) ).

% test_bit_1
tff(fact_4719_succ__min,axiom,
    ! [Deg: nat,X: nat,Mi: nat,Ma: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] :
      ( pp(aa(nat,bool,ord_less_eq(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg))
     => ( pp(aa(nat,bool,ord_less(nat,X),Mi))
       => ( vEBT_vebt_succ(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary),X) = aa(nat,option(nat),some(nat),Mi) ) ) ) ).

% succ_min
tff(fact_4720_pred__max,axiom,
    ! [Deg: nat,Ma: nat,X: nat,Mi: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] :
      ( pp(aa(nat,bool,ord_less_eq(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg))
     => ( pp(aa(nat,bool,ord_less(nat,Ma),X))
       => ( vEBT_vebt_pred(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary),X) = aa(nat,option(nat),some(nat),Ma) ) ) ) ).

% pred_max
tff(fact_4721_map__nth__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xs: list(nat)] : map(nat,bool,bit_se5641148757651400278ts_bit(word(A),zero_zero(word(A))),Xs) = replicate(bool,aa(list(nat),nat,size_size(list(nat)),Xs),fFalse) ) ).

% map_nth_0
tff(fact_4722_lesseq__shift,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa(nat,bool,ord_less_eq(nat,X),Y))
    <=> vEBT_VEBT_lesseq(aa(nat,option(nat),some(nat),X),aa(nat,option(nat),some(nat),Y)) ) ).

% lesseq_shift
tff(fact_4723_vebt__pred_Osimps_I2_J,axiom,
    ! [A2: bool,Uw: bool] :
      ( ( pp(A2)
       => ( vEBT_vebt_pred(vEBT_Leaf(A2,Uw),aa(nat,nat,suc,zero_zero(nat))) = aa(nat,option(nat),some(nat),zero_zero(nat)) ) )
      & ( ~ pp(A2)
       => ( vEBT_vebt_pred(vEBT_Leaf(A2,Uw),aa(nat,nat,suc,zero_zero(nat))) = none(nat) ) ) ) ).

% vebt_pred.simps(2)
tff(fact_4724_word__exists__nth,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] :
          ( ( W != zero_zero(word(A)) )
         => ? [X_1: nat] : pp(aa(nat,bool,bit_se5641148757651400278ts_bit(word(A),W),X_1)) ) ) ).

% word_exists_nth
tff(fact_4725_nth__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: nat] : ~ pp(aa(nat,bool,bit_se5641148757651400278ts_bit(word(A),zero_zero(word(A))),N)) ) ).

% nth_0
tff(fact_4726_finite__bit__word,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] : pp(aa(set(nat),bool,finite_finite2(nat),collect(nat,bit_se5641148757651400278ts_bit(word(A),W)))) ) ).

% finite_bit_word
tff(fact_4727_vebt__succ_Osimps_I1_J,axiom,
    ! [B2: bool,Uu: bool] :
      ( ( pp(B2)
       => ( vEBT_vebt_succ(vEBT_Leaf(Uu,B2),zero_zero(nat)) = aa(nat,option(nat),some(nat),one_one(nat)) ) )
      & ( ~ pp(B2)
       => ( vEBT_vebt_succ(vEBT_Leaf(Uu,B2),zero_zero(nat)) = none(nat) ) ) ) ).

% vebt_succ.simps(1)
tff(fact_4728_bit__Suc__0__iff,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(nat,aa(nat,nat,suc,zero_zero(nat))),N))
    <=> ( N = zero_zero(nat) ) ) ).

% bit_Suc_0_iff
tff(fact_4729_not__bit__Suc__0__Suc,axiom,
    ! [N: nat] : ~ pp(aa(nat,bool,bit_se5641148757651400278ts_bit(nat,aa(nat,nat,suc,zero_zero(nat))),aa(nat,nat,suc,N))) ).

% not_bit_Suc_0_Suc
tff(fact_4730_vebt__pred_Osimps_I3_J,axiom,
    ! [B2: bool,A2: bool,Va: nat] :
      ( ( pp(B2)
       => ( vEBT_vebt_pred(vEBT_Leaf(A2,B2),aa(nat,nat,suc,aa(nat,nat,suc,Va))) = aa(nat,option(nat),some(nat),one_one(nat)) ) )
      & ( ~ pp(B2)
       => ( ( pp(A2)
           => ( vEBT_vebt_pred(vEBT_Leaf(A2,B2),aa(nat,nat,suc,aa(nat,nat,suc,Va))) = aa(nat,option(nat),some(nat),zero_zero(nat)) ) )
          & ( ~ pp(A2)
           => ( vEBT_vebt_pred(vEBT_Leaf(A2,B2),aa(nat,nat,suc,aa(nat,nat,suc,Va))) = none(nat) ) ) ) ) ) ).

% vebt_pred.simps(3)
tff(fact_4731_VEBT__internal_Oreplicatei_Ocases,axiom,
    ! [A: $tType,X: product_prod(nat,heap_Time_Heap(A))] :
      ( ! [X3: heap_Time_Heap(A)] : X != aa(heap_Time_Heap(A),product_prod(nat,heap_Time_Heap(A)),aa(nat,fun(heap_Time_Heap(A),product_prod(nat,heap_Time_Heap(A))),product_Pair(nat,heap_Time_Heap(A)),zero_zero(nat)),X3)
     => ~ ! [N2: nat,X3: heap_Time_Heap(A)] : X != aa(heap_Time_Heap(A),product_prod(nat,heap_Time_Heap(A)),aa(nat,fun(heap_Time_Heap(A),product_prod(nat,heap_Time_Heap(A))),product_Pair(nat,heap_Time_Heap(A)),aa(nat,nat,suc,N2)),X3) ) ).

% VEBT_internal.replicatei.cases
tff(fact_4732_test__bit__over,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),N: nat] :
          ( pp(aa(nat,bool,ord_less_eq(nat,aa(word(A),nat,size_size(word(A)),X)),N))
         => ~ pp(aa(nat,bool,bit_se5641148757651400278ts_bit(word(A),X),N)) ) ) ).

% test_bit_over
tff(fact_4733_VEBT__internal_Ovalid_H_Ocases,axiom,
    ! [X: product_prod(vEBT_VEBT,nat)] :
      ( ! [Uu2: bool,Uv2: bool,D2: nat] : X != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf(Uu2,Uv2)),D2)
     => ~ ! [Mima: option(product_prod(nat,nat)),Deg2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT,Deg3: nat] : X != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(Mima,Deg2,TreeList2,Summary2)),Deg3) ) ).

% VEBT_internal.valid'.cases
tff(fact_4734_vebt__succ_Osimps_I2_J,axiom,
    ! [Uv: bool,Uw: bool,N: nat] : vEBT_vebt_succ(vEBT_Leaf(Uv,Uw),aa(nat,nat,suc,N)) = none(nat) ).

% vebt_succ.simps(2)
tff(fact_4735_vebt__pred_Osimps_I1_J,axiom,
    ! [Uu: bool,Uv: bool] : vEBT_vebt_pred(vEBT_Leaf(Uu,Uv),zero_zero(nat)) = none(nat) ).

% vebt_pred.simps(1)
tff(fact_4736_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)
tff(fact_4737_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)
tff(fact_4738_not__bit__Suc__0__numeral,axiom,
    ! [N: num] : ~ pp(aa(nat,bool,bit_se5641148757651400278ts_bit(nat,aa(nat,nat,suc,zero_zero(nat))),aa(num,nat,numeral_numeral(nat),N))) ).

% not_bit_Suc_0_numeral
tff(fact_4739_xor__num_Ocases,axiom,
    ! [X: product_prod(num,num)] :
      ( ( X != aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),one2),one2) )
     => ( ! [N2: num] : X != aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),one2),bit0(N2))
       => ( ! [N2: num] : X != aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),one2),aa(num,num,bit1,N2))
         => ( ! [M3: num] : X != aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),bit0(M3)),one2)
           => ( ! [M3: num,N2: num] : X != aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),bit0(M3)),bit0(N2))
             => ( ! [M3: num,N2: num] : X != aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),bit0(M3)),aa(num,num,bit1,N2))
               => ( ! [M3: num] : X != aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit1,M3)),one2)
                 => ( ! [M3: num,N2: num] : X != aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit1,M3)),bit0(N2))
                   => ~ ! [M3: num,N2: num] : X != aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit1,M3)),aa(num,num,bit1,N2)) ) ) ) ) ) ) ) ) ).

% xor_num.cases
tff(fact_4740_lsb__this__or__next,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A)] :
          ( ~ pp(aa(nat,bool,bit_se5641148757651400278ts_bit(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),X),one_one(word(A)))),zero_zero(nat)))
         => pp(aa(nat,bool,bit_se5641148757651400278ts_bit(word(A),X),zero_zero(nat))) ) ) ).

% lsb_this_or_next
tff(fact_4741_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(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V),zero_zero(nat),Vc2,Vd),Ve) = none(nat) ).

% vebt_succ.simps(4)
tff(fact_4742_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(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V),zero_zero(nat),Vd,Ve),Vf) = none(nat) ).

% vebt_pred.simps(5)
tff(fact_4743_bit__nat__iff,axiom,
    ! [K: int,N: nat] :
      ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(nat,nat2(K)),N))
    <=> ( pp(aa(int,bool,ord_less_eq(int,zero_zero(int)),K))
        & pp(aa(nat,bool,bit_se5641148757651400278ts_bit(int,K),N)) ) ) ).

% bit_nat_iff
tff(fact_4744_overflow__imp__lsb,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A)] :
          ( ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),X),one_one(word(A))) = zero_zero(word(A)) )
         => pp(aa(nat,bool,bit_se5641148757651400278ts_bit(word(A),X),zero_zero(nat))) ) ) ).

% overflow_imp_lsb
tff(fact_4745_VEBT__internal_Onaive__member_Ocases,axiom,
    ! [X: product_prod(vEBT_VEBT,nat)] :
      ( ! [A4: bool,B3: bool,X3: nat] : X != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),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] : X != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),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),S2: vEBT_VEBT,X3: nat] : X != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(Uy2,aa(nat,nat,suc,V3),TreeList2,S2)),X3) ) ) ).

% VEBT_internal.naive_member.cases
tff(fact_4746_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(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V),aa(nat,nat,suc,zero_zero(nat)),Vh,Vi),Vj) = none(nat) ).

% vebt_pred.simps(6)
tff(fact_4747_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(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V),aa(nat,nat,suc,zero_zero(nat)),Vg,Vh),Vi) = none(nat) ).

% vebt_succ.simps(5)
tff(fact_4748_bang__is__le,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),M: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(word(A),X),M))
         => pp(aa(word(A),bool,ord_less_eq(word(A),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),M)),X)) ) ) ).

% bang_is_le
tff(fact_4749_bit__nat__def,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(nat,M),N))
    <=> ~ pp(aa(nat,bool,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)))) ) ).

% bit_nat_def
tff(fact_4750_odd__iff__lsb,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A)] :
          ( ~ pp(aa(nat,bool,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(word(A),nat,semiring_1_unsigned(A,nat),X)))
        <=> pp(aa(nat,bool,bit_se5641148757651400278ts_bit(word(A),X),zero_zero(nat))) ) ) ).

% odd_iff_lsb
tff(fact_4751_VEBT__internal_Omembermima_Ocases,axiom,
    ! [X: product_prod(vEBT_VEBT,nat)] :
      ( ! [Uu2: bool,Uv2: bool,Uw2: nat] : X != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf(Uu2,Uv2)),Uw2)
     => ( ! [Ux2: list(vEBT_VEBT),Uy2: vEBT_VEBT,Uz2: nat] : X != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),zero_zero(nat),Ux2,Uy2)),Uz2)
       => ( ! [Mi2: nat,Ma2: nat,Va3: list(vEBT_VEBT),Vb2: vEBT_VEBT,X3: nat] : X != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),zero_zero(nat),Va3,Vb2)),X3)
         => ( ! [Mi2: nat,Ma2: nat,V3: nat,TreeList2: list(vEBT_VEBT),Vc: vEBT_VEBT,X3: nat] : X != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,V3),TreeList2,Vc)),X3)
           => ~ ! [V3: nat,TreeList2: list(vEBT_VEBT),Vd2: vEBT_VEBT,X3: nat] : X != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,V3),TreeList2,Vd2)),X3) ) ) ) ) ).

% VEBT_internal.membermima.cases
tff(fact_4752_VEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H_Ocases,axiom,
    ! [X: product_prod(vEBT_VEBT,nat)] :
      ( ! [A4: bool,B3: bool] : X != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf(A4,B3)),zero_zero(nat))
     => ( ! [A4: bool,B3: bool] : X != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf(A4,B3)),aa(nat,nat,suc,zero_zero(nat)))
       => ( ! [A4: bool,B3: bool,N2: nat] : X != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf(A4,B3)),aa(nat,nat,suc,aa(nat,nat,suc,N2)))
         => ( ! [Deg2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT,Uu2: nat] : X != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),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] : X != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,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] : X != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,zero_zero(nat)),TreeList2,Summary2)),X3)
               => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT,X3: nat] : X != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary2)),X3) ) ) ) ) ) ) ).

% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.cases
tff(fact_4753_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Ocases,axiom,
    ! [X: product_prod(vEBT_VEBT,nat)] :
      ( ! [Uu2: bool,Uv2: bool] : X != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf(Uu2,Uv2)),zero_zero(nat))
     => ( ! [A4: bool,Uw2: bool] : X != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf(A4,Uw2)),aa(nat,nat,suc,zero_zero(nat)))
       => ( ! [A4: bool,B3: bool,Va2: nat] : X != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf(A4,B3)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)))
         => ( ! [Uy2: nat,Uz2: list(vEBT_VEBT),Va3: vEBT_VEBT,Vb2: nat] : X != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),Uy2,Uz2,Va3)),Vb2)
           => ( ! [V3: product_prod(nat,nat),Vd2: list(vEBT_VEBT),Ve2: vEBT_VEBT,Vf2: nat] : X != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),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] : X != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),aa(nat,nat,suc,zero_zero(nat)),Vh2,Vi2)),Vj2)
               => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT,X3: nat] : X != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary2)),X3) ) ) ) ) ) ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.cases
tff(fact_4754_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Ocases,axiom,
    ! [X: product_prod(vEBT_VEBT,nat)] :
      ( ! [Uu2: bool,B3: bool] : X != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf(Uu2,B3)),zero_zero(nat))
     => ( ! [Uv2: bool,Uw2: bool,N2: nat] : X != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf(Uv2,Uw2)),aa(nat,nat,suc,N2))
       => ( ! [Ux2: nat,Uy2: list(vEBT_VEBT),Uz2: vEBT_VEBT,Va3: nat] : X != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),Ux2,Uy2,Uz2)),Va3)
         => ( ! [V3: product_prod(nat,nat),Vc: list(vEBT_VEBT),Vd2: vEBT_VEBT,Ve2: nat] : X != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),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] : X != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),aa(nat,nat,suc,zero_zero(nat)),Vg2,Vh2)),Vi2)
             => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT,X3: nat] : X != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary2)),X3) ) ) ) ) ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.cases
tff(fact_4755_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H_Ocases,axiom,
    ! [X: product_prod(vEBT_VEBT,nat)] :
      ( ! [A4: bool,B3: bool,X3: nat] : X != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf(A4,B3)),X3)
     => ( ! [Info2: option(product_prod(nat,nat)),Ts2: list(vEBT_VEBT),S2: vEBT_VEBT,X3: nat] : X != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(Info2,zero_zero(nat),Ts2,S2)),X3)
       => ( ! [Info2: option(product_prod(nat,nat)),Ts2: list(vEBT_VEBT),S2: vEBT_VEBT,X3: nat] : X != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(Info2,aa(nat,nat,suc,zero_zero(nat)),Ts2,S2)),X3)
         => ( ! [V3: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT,X3: nat] : X != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,V3)),TreeList2,Summary2)),X3)
           => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT,X3: nat] : X != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary2)),X3) ) ) ) ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.cases
tff(fact_4756_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H_Ocases,axiom,
    ! [X: product_prod(vEBT_VEBT,nat)] :
      ( ! [A4: bool,B3: bool,X3: nat] : X != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf(A4,B3)),X3)
     => ( ! [Uu2: nat,Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT,X3: nat] : X != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),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] : X != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),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] : X != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),aa(nat,nat,suc,zero_zero(nat)),Vb2,Vc)),X3)
           => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT,X3: nat] : X != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary2)),X3) ) ) ) ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.cases
tff(fact_4757_divmod__step__eq,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [L: num,R2: A,Q2: A] :
          ( ( pp(aa(A,bool,ord_less_eq(A,aa(num,A,numeral_numeral(A),L)),R2))
           => ( unique1321980374590559556d_step(A,L,aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Q2),R2)) = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),Q2)),one_one(A))),aa(A,A,minus_minus(A,R2),aa(num,A,numeral_numeral(A),L))) ) )
          & ( ~ pp(aa(A,bool,ord_less_eq(A,aa(num,A,numeral_numeral(A),L)),R2))
           => ( unique1321980374590559556d_step(A,L,aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Q2),R2)) = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),Q2)),R2) ) ) ) ) ).

% divmod_step_eq
tff(fact_4758_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H_Oelims,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: nat] :
      ( ( vEBT_T_m_e_m_b_e_r2(X,Xa2) = Y )
     => ( ( ? [A4: bool,B3: bool] : X = vEBT_Leaf(A4,B3)
         => ( Y != one_one(nat) ) )
       => ( ( ? [Uu2: nat,Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT] : X = vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw2)
           => ( Y != one_one(nat) ) )
         => ( ( ? [V3: product_prod(nat,nat),Uy2: list(vEBT_VEBT),Uz2: vEBT_VEBT] : X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),zero_zero(nat),Uy2,Uz2)
             => ( Y != one_one(nat) ) )
           => ( ( ? [V3: product_prod(nat,nat),Vb2: list(vEBT_VEBT),Vc: vEBT_VEBT] : X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),aa(nat,nat,suc,zero_zero(nat)),Vb2,Vc)
               => ( Y != one_one(nat) ) )
             => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list(vEBT_VEBT)] :
                    ( ? [Summary2: vEBT_VEBT] : X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary2)
                   => ( Y != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),if(nat,aa(nat,bool,fequal(nat,Xa2),Mi2),zero_zero(nat),if(nat,aa(nat,bool,fequal(nat,Xa2),Ma2),zero_zero(nat),if(nat,aa(nat,bool,ord_less(nat,Xa2),Mi2),zero_zero(nat),if(nat,aa(nat,bool,ord_less(nat,Ma2),Xa2),zero_zero(nat),if(nat,fconj(aa(nat,bool,ord_less(nat,Mi2),Xa2),aa(nat,bool,ord_less(nat,Xa2),Ma2)),if(nat,aa(nat,bool,ord_less(nat,vEBT_VEBT_high(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),vEBT_T_m_e_m_b_e_r2(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),vEBT_VEBT_high(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_VEBT_low(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,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
tff(fact_4759_less__eq__option__Some__None,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [X: A] : ~ pp(aa(option(A),bool,ord_less_eq(option(A),aa(A,option(A),some(A),X)),none(A))) ) ).

% less_eq_option_Some_None
tff(fact_4760_less__option__None,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [X: option(A)] : ~ pp(aa(option(A),bool,ord_less(option(A),X),none(A))) ) ).

% less_option_None
tff(fact_4761_less__option__None__Some__code,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [X: A] : pp(aa(option(A),bool,ord_less(option(A),none(A)),aa(A,option(A),some(A),X))) ) ).

% less_option_None_Some_code
tff(fact_4762_less__eq__option__None__code,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [X: option(A)] : pp(aa(option(A),bool,ord_less_eq(option(A),none(A)),X)) ) ).

% less_eq_option_None_code
tff(fact_4763_less__eq__option__Some,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [X: A,Y: A] :
          ( pp(aa(option(A),bool,ord_less_eq(option(A),aa(A,option(A),some(A),X)),aa(A,option(A),some(A),Y)))
        <=> pp(aa(A,bool,ord_less_eq(A,X),Y)) ) ) ).

% less_eq_option_Some
tff(fact_4764_local_Opower__def,axiom,
    vEBT_VEBT_power = vEBT_V2048590022279873568_shift(nat,power_power(nat)) ).

% local.power_def
tff(fact_4765_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: bool,B2: bool,X: nat] : vEBT_T_m_e_m_b_e_r2(vEBT_Leaf(A2,B2),X) = one_one(nat) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.simps(1)
tff(fact_4766_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,X: nat] : vEBT_T_m_e_m_b_e_r2(vEBT_Node(none(product_prod(nat,nat)),Uu,Uv,Uw),X) = one_one(nat) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.simps(2)
tff(fact_4767_less__option__None__Some,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [X: A] : pp(aa(option(A),bool,ord_less(option(A),none(A)),aa(A,option(A),some(A),X))) ) ).

% less_option_None_Some
tff(fact_4768_less__option__None__is__Some,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [X: option(A)] :
          ( pp(aa(option(A),bool,ord_less(option(A),none(A)),X))
         => ? [Z2: A] : X = aa(A,option(A),some(A),Z2) ) ) ).

% less_option_None_is_Some
tff(fact_4769_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,X: nat] : vEBT_T_m_e_m_b_e_r2(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V),zero_zero(nat),Uy,Uz),X) = one_one(nat) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.simps(3)
tff(fact_4770_less__eq__option__None,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [X: option(A)] : pp(aa(option(A),bool,ord_less_eq(option(A),none(A)),X)) ) ).

% less_eq_option_None
tff(fact_4771_less__eq__option__None__is__None,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [X: option(A)] :
          ( pp(aa(option(A),bool,ord_less_eq(option(A),X),none(A)))
         => ( X = none(A) ) ) ) ).

% less_eq_option_None_is_None
tff(fact_4772_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,X: nat] : vEBT_T_m_e_m_b_e_r2(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V),aa(nat,nat,suc,zero_zero(nat)),Vb,Vc2),X) = one_one(nat) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.simps(4)
tff(fact_4773_member__bound__height_H,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( vEBT_invar_vebt(T,N)
     => pp(aa(nat,bool,ord_less_eq(nat,vEBT_T_m_e_m_b_e_r2(T,X)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),aa(vEBT_VEBT,nat,vEBT_VEBT_height,T)))) ) ).

% member_bound_height'
tff(fact_4774_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,X: nat] : vEBT_T_m_e_m_b_e_r2(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList,Summary),X) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),if(nat,aa(nat,bool,fequal(nat,X),Mi),zero_zero(nat),if(nat,aa(nat,bool,fequal(nat,X),Ma),zero_zero(nat),if(nat,aa(nat,bool,ord_less(nat,X),Mi),zero_zero(nat),if(nat,aa(nat,bool,ord_less(nat,Ma),X),zero_zero(nat),if(nat,fconj(aa(nat,bool,ord_less(nat,Mi),X),aa(nat,bool,ord_less(nat,X),Ma)),if(nat,aa(nat,bool,ord_less(nat,vEBT_VEBT_high(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),vEBT_T_m_e_m_b_e_r2(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_VEBT_low(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,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)
tff(fact_4775_mintlistlength,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,N: nat] :
      ( vEBT_invar_vebt(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary),N)
     => ( ( Mi != Ma )
       => ( pp(aa(nat,bool,ord_less(nat,Mi),Ma))
          & ? [M3: nat] :
              ( ( aa(nat,option(nat),some(nat),M3) = vEBT_vebt_mint(Summary) )
              & pp(aa(nat,bool,ord_less(nat,M3),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,minus_minus(nat,N),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2))))))) ) ) ) ) ).

% mintlistlength
tff(fact_4776_del__single__cont,axiom,
    ! [X: nat,Mi: nat,Ma: nat,Deg: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] :
      ( ( ( X = Mi )
        & ( X = Ma ) )
     => ( pp(aa(nat,bool,ord_less_eq(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg))
       => ( vEBT_vebt_delete(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary),X) = vEBT_Node(none(product_prod(nat,nat)),Deg,TreeList,Summary) ) ) ) ).

% del_single_cont
tff(fact_4777_delt__out__of__range,axiom,
    ! [X: nat,Mi: nat,Ma: nat,Deg: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] :
      ( ( pp(aa(nat,bool,ord_less(nat,X),Mi))
        | pp(aa(nat,bool,ord_less(nat,Ma),X)) )
     => ( pp(aa(nat,bool,ord_less_eq(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg))
       => ( vEBT_vebt_delete(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary),X) = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary) ) ) ) ).

% delt_out_of_range
tff(fact_4778_delete__pres__valid,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( vEBT_invar_vebt(T,N)
     => vEBT_invar_vebt(vEBT_vebt_delete(T,X),N) ) ).

% delete_pres_valid
tff(fact_4779_dele__bmo__cont__corr,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat,Y: nat] :
      ( vEBT_invar_vebt(T,N)
     => ( pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,vEBT_vebt_delete(T,X)),Y))
      <=> ( ( X != Y )
          & pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,T),Y)) ) ) ) ).

% dele_bmo_cont_corr
tff(fact_4780_dele__member__cont__corr,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat,Y: nat] :
      ( vEBT_invar_vebt(T,N)
     => ( pp(aa(nat,bool,vEBT_vebt_member(vEBT_vebt_delete(T,X)),Y))
      <=> ( ( X != Y )
          & pp(aa(nat,bool,vEBT_vebt_member(T),Y)) ) ) ) ).

% dele_member_cont_corr
tff(fact_4781_mint__member,axiom,
    ! [T: vEBT_VEBT,N: nat,Maxi: nat] :
      ( vEBT_invar_vebt(T,N)
     => ( ( vEBT_vebt_mint(T) = aa(nat,option(nat),some(nat),Maxi) )
       => pp(aa(nat,bool,vEBT_vebt_member(T),Maxi)) ) ) ).

% mint_member
tff(fact_4782_mint__corr__help,axiom,
    ! [T: vEBT_VEBT,N: nat,Mini: nat,X: nat] :
      ( vEBT_invar_vebt(T,N)
     => ( ( vEBT_vebt_mint(T) = aa(nat,option(nat),some(nat),Mini) )
       => ( pp(aa(nat,bool,vEBT_vebt_member(T),X))
         => pp(aa(nat,bool,ord_less_eq(nat,Mini),X)) ) ) ) ).

% mint_corr_help
tff(fact_4783_mint__corr,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( vEBT_invar_vebt(T,N)
     => ( ( vEBT_vebt_mint(T) = aa(nat,option(nat),some(nat),X) )
       => vEBT_VEBT_min_in_set(vEBT_VEBT_set_vebt(T),X) ) ) ).

% mint_corr
tff(fact_4784_mint__sound,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( vEBT_invar_vebt(T,N)
     => ( vEBT_VEBT_min_in_set(vEBT_VEBT_set_vebt(T),X)
       => ( vEBT_vebt_mint(T) = aa(nat,option(nat),some(nat),X) ) ) ) ).

% mint_sound
tff(fact_4785_misiz,axiom,
    ! [T: vEBT_VEBT,N: nat,M: nat] :
      ( vEBT_invar_vebt(T,N)
     => ( ( aa(nat,option(nat),some(nat),M) = vEBT_vebt_mint(T) )
       => pp(aa(nat,bool,ord_less(nat,M),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N))) ) ) ).

% misiz
tff(fact_4786_nested__mint,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,N: nat,Va: nat] :
      ( vEBT_invar_vebt(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary),N)
     => ( ( N = aa(nat,nat,suc,aa(nat,nat,suc,Va)) )
       => ( ~ pp(aa(nat,bool,ord_less(nat,Ma),Mi))
         => ( ( Ma != Mi )
           => pp(aa(nat,bool,ord_less(nat,vEBT_VEBT_high(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),the2(nat,vEBT_vebt_mint(Summary))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Va),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),the2(nat,vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),the2(nat,vEBT_vebt_mint(Summary)))))),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Va),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))) ) ) ) ) ).

% nested_mint
tff(fact_4787_vebt__mint_Oelims,axiom,
    ! [X: vEBT_VEBT,Y: option(nat)] :
      ( ( vEBT_vebt_mint(X) = Y )
     => ( ! [A4: bool,B3: bool] :
            ( ( X = vEBT_Leaf(A4,B3) )
           => ~ ( ( pp(A4)
                 => ( Y = aa(nat,option(nat),some(nat),zero_zero(nat)) ) )
                & ( ~ pp(A4)
                 => ( ( pp(B3)
                     => ( Y = aa(nat,option(nat),some(nat),one_one(nat)) ) )
                    & ( ~ pp(B3)
                     => ( Y = none(nat) ) ) ) ) ) )
       => ( ( ? [Uu2: nat,Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT] : X = vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw2)
           => ( Y != none(nat) ) )
         => ~ ! [Mi2: nat] :
                ( ? [Ma2: nat,Ux2: nat,Uy2: list(vEBT_VEBT),Uz2: vEBT_VEBT] : X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),Ux2,Uy2,Uz2)
               => ( Y != aa(nat,option(nat),some(nat),Mi2) ) ) ) ) ) ).

% vebt_mint.elims
tff(fact_4788_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Oelims,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: nat] :
      ( ( vEBT_T_p_r_e_d2(X,Xa2) = Y )
     => ( ( ? [Uu2: bool,Uv2: bool] : X = vEBT_Leaf(Uu2,Uv2)
         => ( ( Xa2 = zero_zero(nat) )
           => ( Y != one_one(nat) ) ) )
       => ( ( ? [A4: bool,Uw2: bool] : X = vEBT_Leaf(A4,Uw2)
           => ( ( Xa2 = aa(nat,nat,suc,zero_zero(nat)) )
             => ( Y != one_one(nat) ) ) )
         => ( ( ? [A4: bool,B3: bool] : X = vEBT_Leaf(A4,B3)
             => ( ? [Va2: nat] : Xa2 = aa(nat,nat,suc,aa(nat,nat,suc,Va2))
               => ( Y != one_one(nat) ) ) )
           => ( ( ? [Uy2: nat,Uz2: list(vEBT_VEBT),Va3: vEBT_VEBT] : X = vEBT_Node(none(product_prod(nat,nat)),Uy2,Uz2,Va3)
               => ( Y != one_one(nat) ) )
             => ( ( ? [V3: product_prod(nat,nat),Vd2: list(vEBT_VEBT),Ve2: vEBT_VEBT] : X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),zero_zero(nat),Vd2,Ve2)
                 => ( Y != one_one(nat) ) )
               => ( ( ? [V3: product_prod(nat,nat),Vh2: list(vEBT_VEBT),Vi2: vEBT_VEBT] : X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),aa(nat,nat,suc,zero_zero(nat)),Vh2,Vi2)
                   => ( Y != one_one(nat) ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                        ( ( X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary2) )
                       => ~ ( ( pp(aa(nat,bool,ord_less(nat,Ma2),Xa2))
                             => ( Y = one_one(nat) ) )
                            & ( ~ pp(aa(nat,bool,ord_less(nat,Ma2),Xa2))
                             => ( Y = if(nat,aa(nat,bool,ord_less(nat,vEBT_VEBT_high(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),if(nat,fconj(aa(bool,bool,fNot,aa(option(nat),bool,fequal(option(nat),vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),vEBT_VEBT_high(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),none(nat))),vEBT_VEBT_greater(aa(nat,option(nat),some(nat),vEBT_VEBT_low(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),vEBT_VEBT_high(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),vEBT_T_p_r_e_d2(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),vEBT_VEBT_high(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_VEBT_low(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_T_p_r_e_d2(Summary2,vEBT_VEBT_high(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),one_one(nat))),one_one(nat)) ) ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.elims
tff(fact_4789_greater__shift,axiom,
    ! [Y: nat,X: nat] :
      ( pp(aa(nat,bool,ord_less(nat,Y),X))
    <=> pp(vEBT_VEBT_greater(aa(nat,option(nat),some(nat),X),aa(nat,option(nat),some(nat),Y))) ) ).

% greater_shift
tff(fact_4790_option_Ocollapse,axiom,
    ! [A: $tType,Option: option(A)] :
      ( ( Option != none(A) )
     => ( aa(A,option(A),some(A),the2(A,Option)) = Option ) ) ).

% option.collapse
tff(fact_4791_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
tff(fact_4792_option_Oexhaust__sel,axiom,
    ! [A: $tType,Option: option(A)] :
      ( ( Option != none(A) )
     => ( Option = aa(A,option(A),some(A),the2(A,Option)) ) ) ).

% option.exhaust_sel
tff(fact_4793_VEBT__internal_Ooption__comp__shift_Ocases,axiom,
    ! [A: $tType,X: product_prod(fun(A,fun(A,bool)),product_prod(option(A),option(A)))] :
      ( ! [Uu2: fun(A,fun(A,bool)),Uv2: option(A)] : X != aa(product_prod(option(A),option(A)),product_prod(fun(A,fun(A,bool)),product_prod(option(A),option(A))),aa(fun(A,fun(A,bool)),fun(product_prod(option(A),option(A)),product_prod(fun(A,fun(A,bool)),product_prod(option(A),option(A)))),product_Pair(fun(A,fun(A,bool)),product_prod(option(A),option(A))),Uu2),aa(option(A),product_prod(option(A),option(A)),aa(option(A),fun(option(A),product_prod(option(A),option(A))),product_Pair(option(A),option(A)),none(A)),Uv2))
     => ( ! [Uw2: fun(A,fun(A,bool)),V3: A] : X != aa(product_prod(option(A),option(A)),product_prod(fun(A,fun(A,bool)),product_prod(option(A),option(A))),aa(fun(A,fun(A,bool)),fun(product_prod(option(A),option(A)),product_prod(fun(A,fun(A,bool)),product_prod(option(A),option(A)))),product_Pair(fun(A,fun(A,bool)),product_prod(option(A),option(A))),Uw2),aa(option(A),product_prod(option(A),option(A)),aa(option(A),fun(option(A),product_prod(option(A),option(A))),product_Pair(option(A),option(A)),aa(A,option(A),some(A),V3)),none(A)))
       => ~ ! [F4: fun(A,fun(A,bool)),X3: A,Y3: A] : X != aa(product_prod(option(A),option(A)),product_prod(fun(A,fun(A,bool)),product_prod(option(A),option(A))),aa(fun(A,fun(A,bool)),fun(product_prod(option(A),option(A)),product_prod(fun(A,fun(A,bool)),product_prod(option(A),option(A)))),product_Pair(fun(A,fun(A,bool)),product_prod(option(A),option(A))),F4),aa(option(A),product_prod(option(A),option(A)),aa(option(A),fun(option(A),product_prod(option(A),option(A))),product_Pair(option(A),option(A)),aa(A,option(A),some(A),X3)),aa(A,option(A),some(A),Y3))) ) ) ).

% VEBT_internal.option_comp_shift.cases
tff(fact_4794_VEBT__internal_Ooption__shift_Ocases,axiom,
    ! [A: $tType,X: product_prod(fun(A,fun(A,A)),product_prod(option(A),option(A)))] :
      ( ! [Uu2: fun(A,fun(A,A)),Uv2: option(A)] : X != aa(product_prod(option(A),option(A)),product_prod(fun(A,fun(A,A)),product_prod(option(A),option(A))),aa(fun(A,fun(A,A)),fun(product_prod(option(A),option(A)),product_prod(fun(A,fun(A,A)),product_prod(option(A),option(A)))),product_Pair(fun(A,fun(A,A)),product_prod(option(A),option(A))),Uu2),aa(option(A),product_prod(option(A),option(A)),aa(option(A),fun(option(A),product_prod(option(A),option(A))),product_Pair(option(A),option(A)),none(A)),Uv2))
     => ( ! [Uw2: fun(A,fun(A,A)),V3: A] : X != aa(product_prod(option(A),option(A)),product_prod(fun(A,fun(A,A)),product_prod(option(A),option(A))),aa(fun(A,fun(A,A)),fun(product_prod(option(A),option(A)),product_prod(fun(A,fun(A,A)),product_prod(option(A),option(A)))),product_Pair(fun(A,fun(A,A)),product_prod(option(A),option(A))),Uw2),aa(option(A),product_prod(option(A),option(A)),aa(option(A),fun(option(A),product_prod(option(A),option(A))),product_Pair(option(A),option(A)),aa(A,option(A),some(A),V3)),none(A)))
       => ~ ! [F4: fun(A,fun(A,A)),A4: A,B3: A] : X != aa(product_prod(option(A),option(A)),product_prod(fun(A,fun(A,A)),product_prod(option(A),option(A))),aa(fun(A,fun(A,A)),fun(product_prod(option(A),option(A)),product_prod(fun(A,fun(A,A)),product_prod(option(A),option(A)))),product_Pair(fun(A,fun(A,A)),product_prod(option(A),option(A))),F4),aa(option(A),product_prod(option(A),option(A)),aa(option(A),fun(option(A),product_prod(option(A),option(A))),product_Pair(option(A),option(A)),aa(A,option(A),some(A),A4)),aa(A,option(A),some(A),B3))) ) ) ).

% VEBT_internal.option_shift.cases
tff(fact_4795_VEBT__internal_Ooption__shift_Osimps_I1_J,axiom,
    ! [A: $tType,Uu: fun(A,fun(A,A)),Uv: option(A)] : aa(option(A),option(A),aa(option(A),fun(option(A),option(A)),vEBT_V2048590022279873568_shift(A,Uu),none(A)),Uv) = none(A) ).

% VEBT_internal.option_shift.simps(1)
tff(fact_4796_VEBT__internal_Ooption__shift_Osimps_I2_J,axiom,
    ! [A: $tType,Uw: fun(A,fun(A,A)),V: A] : aa(option(A),option(A),aa(option(A),fun(option(A),option(A)),vEBT_V2048590022279873568_shift(A,Uw),aa(A,option(A),some(A),V)),none(A)) = none(A) ).

% VEBT_internal.option_shift.simps(2)
tff(fact_4797_VEBT__internal_Ooption__shift_Oelims,axiom,
    ! [A: $tType,X: fun(A,fun(A,A)),Xa2: option(A),Xb: option(A),Y: option(A)] :
      ( ( aa(option(A),option(A),aa(option(A),fun(option(A),option(A)),vEBT_V2048590022279873568_shift(A,X),Xa2),Xb) = Y )
     => ( ( ( Xa2 = none(A) )
         => ( Y != none(A) ) )
       => ( ( ? [V3: A] : Xa2 = aa(A,option(A),some(A),V3)
           => ( ( Xb = none(A) )
             => ( Y != none(A) ) ) )
         => ~ ! [A4: A] :
                ( ( Xa2 = aa(A,option(A),some(A),A4) )
               => ! [B3: A] :
                    ( ( Xb = aa(A,option(A),some(A),B3) )
                   => ( Y != aa(A,option(A),some(A),aa(A,A,aa(A,fun(A,A),X,A4),B3)) ) ) ) ) ) ) ).

% VEBT_internal.option_shift.elims
tff(fact_4798_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)
tff(fact_4799_vebt__mint_Osimps_I3_J,axiom,
    ! [Mi: nat,Ma: nat,Ux: nat,Uy: list(vEBT_VEBT),Uz: vEBT_VEBT] : vEBT_vebt_mint(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Ux,Uy,Uz)) = aa(nat,option(nat),some(nat),Mi) ).

% vebt_mint.simps(3)
tff(fact_4800_vebt__mint_Osimps_I1_J,axiom,
    ! [A2: bool,B2: bool] :
      ( ( pp(A2)
       => ( vEBT_vebt_mint(vEBT_Leaf(A2,B2)) = aa(nat,option(nat),some(nat),zero_zero(nat)) ) )
      & ( ~ pp(A2)
       => ( ( pp(B2)
           => ( vEBT_vebt_mint(vEBT_Leaf(A2,B2)) = aa(nat,option(nat),some(nat),one_one(nat)) ) )
          & ( ~ pp(B2)
           => ( vEBT_vebt_mint(vEBT_Leaf(A2,B2)) = none(nat) ) ) ) ) ) ).

% vebt_mint.simps(1)
tff(fact_4801_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Osimps_I7_J,axiom,
    ! [Ma: nat,X: nat,Mi: nat,Va: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] :
      ( ( pp(aa(nat,bool,ord_less(nat,Ma),X))
       => ( vEBT_T_p_r_e_d2(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList,Summary),X) = one_one(nat) ) )
      & ( ~ pp(aa(nat,bool,ord_less(nat,Ma),X))
       => ( vEBT_T_p_r_e_d2(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList,Summary),X) = if(nat,aa(nat,bool,ord_less(nat,vEBT_VEBT_high(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),if(nat,fconj(aa(bool,bool,fNot,aa(option(nat),bool,fequal(option(nat),vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),none(nat))),vEBT_VEBT_greater(aa(nat,option(nat),some(nat),vEBT_VEBT_low(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),vEBT_T_p_r_e_d2(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_VEBT_low(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_T_p_r_e_d2(Summary,vEBT_VEBT_high(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),one_one(nat))),one_one(nat)) ) ) ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.simps(7)
tff(fact_4802_summaxma,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] :
      ( vEBT_invar_vebt(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,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,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ) ) ) ).

% summaxma
tff(fact_4803_minNull__delete__time__bound,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( vEBT_invar_vebt(T,N)
     => ( pp(vEBT_VEBT_minNull(vEBT_vebt_delete(T,X)))
       => pp(aa(nat,bool,ord_less_eq(nat,vEBT_T_d_e_l_e_t_e(T,X)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,bit0(bit0(one2)))))) ) ) ).

% minNull_delete_time_bound
tff(fact_4804_mul__def,axiom,
    vEBT_VEBT_mul = vEBT_V2048590022279873568_shift(nat,times_times(nat)) ).

% mul_def
tff(fact_4805_not__min__Null__member,axiom,
    ! [T: vEBT_VEBT] :
      ( ~ pp(vEBT_VEBT_minNull(T))
     => ? [X_1: nat] : pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,T),X_1)) ) ).

% not_min_Null_member
tff(fact_4806_min__Null__member,axiom,
    ! [T: vEBT_VEBT,X: nat] :
      ( pp(vEBT_VEBT_minNull(T))
     => ~ pp(aa(nat,bool,vEBT_vebt_member(T),X)) ) ).

% min_Null_member
tff(fact_4807_maxbmo,axiom,
    ! [T: vEBT_VEBT,X: nat] :
      ( ( vEBT_vebt_maxt(T) = aa(nat,option(nat),some(nat),X) )
     => pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,T),X)) ) ).

% maxbmo
tff(fact_4808_minNullmin,axiom,
    ! [T: vEBT_VEBT] :
      ( pp(vEBT_VEBT_minNull(T))
     => ( vEBT_vebt_mint(T) = none(nat) ) ) ).

% minNullmin
tff(fact_4809_minminNull,axiom,
    ! [T: vEBT_VEBT] :
      ( ( vEBT_vebt_mint(T) = none(nat) )
     => pp(vEBT_VEBT_minNull(T)) ) ).

% minminNull
tff(fact_4810_mul__shift,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),X),Y) = Z )
    <=> ( aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_mul,aa(nat,option(nat),some(nat),X)),aa(nat,option(nat),some(nat),Y)) = aa(nat,option(nat),some(nat),Z) ) ) ).

% mul_shift
tff(fact_4811_maxt__member,axiom,
    ! [T: vEBT_VEBT,N: nat,Maxi: nat] :
      ( vEBT_invar_vebt(T,N)
     => ( ( vEBT_vebt_maxt(T) = aa(nat,option(nat),some(nat),Maxi) )
       => pp(aa(nat,bool,vEBT_vebt_member(T),Maxi)) ) ) ).

% maxt_member
tff(fact_4812_maxt__corr__help,axiom,
    ! [T: vEBT_VEBT,N: nat,Maxi: nat,X: nat] :
      ( vEBT_invar_vebt(T,N)
     => ( ( vEBT_vebt_maxt(T) = aa(nat,option(nat),some(nat),Maxi) )
       => ( pp(aa(nat,bool,vEBT_vebt_member(T),X))
         => pp(aa(nat,bool,ord_less_eq(nat,X),Maxi)) ) ) ) ).

% maxt_corr_help
tff(fact_4813_maxt__corr,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( vEBT_invar_vebt(T,N)
     => ( ( vEBT_vebt_maxt(T) = aa(nat,option(nat),some(nat),X) )
       => vEBT_VEBT_max_in_set(vEBT_VEBT_set_vebt(T),X) ) ) ).

% maxt_corr
tff(fact_4814_maxt__sound,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( vEBT_invar_vebt(T,N)
     => ( vEBT_VEBT_max_in_set(vEBT_VEBT_set_vebt(T),X)
       => ( vEBT_vebt_maxt(T) = aa(nat,option(nat),some(nat),X) ) ) ) ).

% maxt_sound
tff(fact_4815_minNull__delete__time__bound_H,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( vEBT_invar_vebt(T,N)
     => ( pp(vEBT_VEBT_minNull(vEBT_vebt_delete(T,X)))
       => pp(aa(nat,bool,ord_less_eq(nat,vEBT_V1232361888498592333_e_t_e(T,X)),one_one(nat))) ) ) ).

% minNull_delete_time_bound'
tff(fact_4816_VEBT__internal_OminNull_Osimps_I5_J,axiom,
    ! [Uz: product_prod(nat,nat),Va: nat,Vb: list(vEBT_VEBT),Vc2: vEBT_VEBT] : ~ pp(vEBT_VEBT_minNull(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),Uz),Va,Vb,Vc2))) ).

% VEBT_internal.minNull.simps(5)
tff(fact_4817_VEBT__internal_OminNull_Osimps_I4_J,axiom,
    ! [Uw: nat,Ux: list(vEBT_VEBT),Uy: vEBT_VEBT] : pp(vEBT_VEBT_minNull(vEBT_Node(none(product_prod(nat,nat)),Uw,Ux,Uy))) ).

% VEBT_internal.minNull.simps(4)
tff(fact_4818_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)
tff(fact_4819_VEBT__internal_OminNull_Oelims_I3_J,axiom,
    ! [X: vEBT_VEBT] :
      ( ~ pp(vEBT_VEBT_minNull(X))
     => ( ! [Uv2: bool] : X != vEBT_Leaf(fTrue,Uv2)
       => ( ! [Uu2: bool] : X != vEBT_Leaf(Uu2,fTrue)
         => ~ ! [Uz2: product_prod(nat,nat),Va3: nat,Vb2: list(vEBT_VEBT),Vc: vEBT_VEBT] : X != vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),Uz2),Va3,Vb2,Vc) ) ) ) ).

% VEBT_internal.minNull.elims(3)
tff(fact_4820_VEBT__internal_OminNull_Oelims_I2_J,axiom,
    ! [X: vEBT_VEBT] :
      ( pp(vEBT_VEBT_minNull(X))
     => ( ( X != vEBT_Leaf(fFalse,fFalse) )
       => ~ ! [Uw2: nat,Ux2: list(vEBT_VEBT),Uy2: vEBT_VEBT] : X != vEBT_Node(none(product_prod(nat,nat)),Uw2,Ux2,Uy2) ) ) ).

% VEBT_internal.minNull.elims(2)
tff(fact_4821_vebt__maxt_Osimps_I3_J,axiom,
    ! [Mi: nat,Ma: nat,Ux: nat,Uy: list(vEBT_VEBT),Uz: vEBT_VEBT] : vEBT_vebt_maxt(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Ux,Uy,Uz)) = aa(nat,option(nat),some(nat),Ma) ).

% vebt_maxt.simps(3)
tff(fact_4822_VEBT__internal_OminNull_Oelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Y: bool] :
      ( ( pp(vEBT_VEBT_minNull(X))
      <=> pp(Y) )
     => ( ( ( X = vEBT_Leaf(fFalse,fFalse) )
         => ~ pp(Y) )
       => ( ( ? [Uv2: bool] : X = vEBT_Leaf(fTrue,Uv2)
           => pp(Y) )
         => ( ( ? [Uu2: bool] : X = vEBT_Leaf(Uu2,fTrue)
             => pp(Y) )
           => ( ( ? [Uw2: nat,Ux2: list(vEBT_VEBT),Uy2: vEBT_VEBT] : X = vEBT_Node(none(product_prod(nat,nat)),Uw2,Ux2,Uy2)
               => ~ pp(Y) )
             => ~ ( ? [Uz2: product_prod(nat,nat),Va3: nat,Vb2: list(vEBT_VEBT),Vc: vEBT_VEBT] : X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),Uz2),Va3,Vb2,Vc)
                 => pp(Y) ) ) ) ) ) ) ).

% VEBT_internal.minNull.elims(1)
tff(fact_4823_vebt__maxt_Osimps_I1_J,axiom,
    ! [B2: bool,A2: bool] :
      ( ( pp(B2)
       => ( vEBT_vebt_maxt(vEBT_Leaf(A2,B2)) = aa(nat,option(nat),some(nat),one_one(nat)) ) )
      & ( ~ pp(B2)
       => ( ( pp(A2)
           => ( vEBT_vebt_maxt(vEBT_Leaf(A2,B2)) = aa(nat,option(nat),some(nat),zero_zero(nat)) ) )
          & ( ~ pp(A2)
           => ( vEBT_vebt_maxt(vEBT_Leaf(A2,B2)) = none(nat) ) ) ) ) ) ).

% vebt_maxt.simps(1)
tff(fact_4824_insertsimp,axiom,
    ! [T: vEBT_VEBT,N: nat,L: nat] :
      ( vEBT_invar_vebt(T,N)
     => ( pp(vEBT_VEBT_minNull(T))
       => pp(aa(nat,bool,ord_less_eq(nat,vEBT_T_i_n_s_e_r_t(T,L)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2)))) ) ) ).

% insertsimp
tff(fact_4825_vebt__maxt_Oelims,axiom,
    ! [X: vEBT_VEBT,Y: option(nat)] :
      ( ( vEBT_vebt_maxt(X) = Y )
     => ( ! [A4: bool,B3: bool] :
            ( ( X = vEBT_Leaf(A4,B3) )
           => ~ ( ( pp(B3)
                 => ( Y = aa(nat,option(nat),some(nat),one_one(nat)) ) )
                & ( ~ pp(B3)
                 => ( ( pp(A4)
                     => ( Y = aa(nat,option(nat),some(nat),zero_zero(nat)) ) )
                    & ( ~ pp(A4)
                     => ( Y = none(nat) ) ) ) ) ) )
       => ( ( ? [Uu2: nat,Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT] : X = vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw2)
           => ( Y != none(nat) ) )
         => ~ ! [Mi2: nat,Ma2: nat] :
                ( ? [Ux2: nat,Uy2: list(vEBT_VEBT),Uz2: vEBT_VEBT] : X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),Ux2,Uy2,Uz2)
               => ( Y != aa(nat,option(nat),some(nat),Ma2) ) ) ) ) ) ).

% vebt_maxt.elims
tff(fact_4826_pred__less__length__list,axiom,
    ! [Deg: nat,X: nat,Ma: nat,TreeList: list(vEBT_VEBT),Mi: nat,Summary: vEBT_VEBT] :
      ( pp(aa(nat,bool,ord_less_eq(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg))
     => ( pp(aa(nat,bool,ord_less_eq(nat,X),Ma))
       => ( pp(aa(nat,bool,ord_less(nat,vEBT_VEBT_high(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)))
         => ( vEBT_vebt_pred(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary),X) = if(option(nat),fconj(aa(bool,bool,fNot,aa(option(nat),bool,fequal(option(nat),vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),none(nat))),vEBT_VEBT_greater(aa(nat,option(nat),some(nat),vEBT_VEBT_low(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))))),aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_add,aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_mul,aa(nat,option(nat),some(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,option(nat),some(nat),vEBT_VEBT_high(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),vEBT_vebt_pred(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_VEBT_low(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),if(option(nat),aa(option(nat),bool,fequal(option(nat),vEBT_vebt_pred(Summary,vEBT_VEBT_high(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),none(nat)),if(option(nat),aa(nat,bool,ord_less(nat,Mi),X),aa(nat,option(nat),some(nat),Mi),none(nat)),aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_add,aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_mul,aa(nat,option(nat),some(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),vEBT_vebt_pred(Summary,vEBT_VEBT_high(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),the2(nat,vEBT_vebt_pred(Summary,vEBT_VEBT_high(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))))))))) ) ) ) ) ).

% pred_less_length_list
tff(fact_4827_pred__lesseq__max,axiom,
    ! [Deg: nat,X: nat,Ma: nat,Mi: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] :
      ( pp(aa(nat,bool,ord_less_eq(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg))
     => ( pp(aa(nat,bool,ord_less_eq(nat,X),Ma))
       => ( vEBT_vebt_pred(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary),X) = if(option(nat),aa(nat,bool,ord_less(nat,vEBT_VEBT_high(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),if(option(nat),fconj(aa(bool,bool,fNot,aa(option(nat),bool,fequal(option(nat),vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),none(nat))),vEBT_VEBT_greater(aa(nat,option(nat),some(nat),vEBT_VEBT_low(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))))),aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_add,aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_mul,aa(nat,option(nat),some(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,option(nat),some(nat),vEBT_VEBT_high(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),vEBT_vebt_pred(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_VEBT_low(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),if(option(nat),aa(option(nat),bool,fequal(option(nat),vEBT_vebt_pred(Summary,vEBT_VEBT_high(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),none(nat)),if(option(nat),aa(nat,bool,ord_less(nat,Mi),X),aa(nat,option(nat),some(nat),Mi),none(nat)),aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_add,aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_mul,aa(nat,option(nat),some(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),vEBT_vebt_pred(Summary,vEBT_VEBT_high(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),the2(nat,vEBT_vebt_pred(Summary,vEBT_VEBT_high(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))))))))),none(nat)) ) ) ) ).

% pred_lesseq_max
tff(fact_4828_del__x__mia,axiom,
    ! [X: nat,Mi: nat,Ma: nat,Deg: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] :
      ( ( ( X = Mi )
        & pp(aa(nat,bool,ord_less(nat,X),Ma)) )
     => ( ( Mi != Ma )
       => ( pp(aa(nat,bool,ord_less_eq(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg))
         => ( vEBT_vebt_delete(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary),X) = if(vEBT_VEBT,aa(nat,bool,ord_less(nat,vEBT_VEBT_high(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),the2(nat,vEBT_vebt_mint(Summary))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),the2(nat,vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),the2(nat,vEBT_vebt_mint(Summary)))))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),if(vEBT_VEBT,vEBT_VEBT_minNull(vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),the2(nat,vEBT_vebt_mint(Summary))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),the2(nat,vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),the2(nat,vEBT_vebt_mint(Summary)))))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_VEBT_low(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),the2(nat,vEBT_vebt_mint(Summary))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),the2(nat,vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),the2(nat,vEBT_vebt_mint(Summary)))))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),the2(nat,vEBT_vebt_mint(Summary))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),the2(nat,vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),the2(nat,vEBT_vebt_mint(Summary))))))),if(nat,aa(nat,bool,fequal(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),the2(nat,vEBT_vebt_mint(Summary))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),the2(nat,vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),the2(nat,vEBT_vebt_mint(Summary))))))),Ma),if(nat,aa(option(nat),bool,fequal(option(nat),vEBT_vebt_maxt(vEBT_vebt_delete(Summary,vEBT_VEBT_high(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),the2(nat,vEBT_vebt_mint(Summary))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),the2(nat,vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),the2(nat,vEBT_vebt_mint(Summary)))))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),none(nat)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),the2(nat,vEBT_vebt_mint(Summary))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),the2(nat,vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),the2(nat,vEBT_vebt_mint(Summary)))))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))),the2(nat,vEBT_vebt_maxt(vEBT_vebt_delete(Summary,vEBT_VEBT_high(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),the2(nat,vEBT_vebt_mint(Summary))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),the2(nat,vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),the2(nat,vEBT_vebt_mint(Summary)))))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))))))),the2(nat,vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,list_update(vEBT_VEBT,TreeList,vEBT_VEBT_high(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),the2(nat,vEBT_vebt_mint(Summary))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),the2(nat,vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),the2(nat,vEBT_vebt_mint(Summary)))))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))),vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),the2(nat,vEBT_vebt_mint(Summary))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),the2(nat,vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),the2(nat,vEBT_vebt_mint(Summary)))))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_VEBT_low(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),the2(nat,vEBT_vebt_mint(Summary))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),the2(nat,vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),the2(nat,vEBT_vebt_mint(Summary)))))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),the2(nat,vEBT_vebt_maxt(vEBT_vebt_delete(Summary,vEBT_VEBT_high(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),the2(nat,vEBT_vebt_mint(Summary))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),the2(nat,vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),the2(nat,vEBT_vebt_mint(Summary)))))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))))))))),Ma))),Deg,list_update(vEBT_VEBT,TreeList,vEBT_VEBT_high(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),the2(nat,vEBT_vebt_mint(Summary))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),the2(nat,vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),the2(nat,vEBT_vebt_mint(Summary)))))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))),vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),the2(nat,vEBT_vebt_mint(Summary))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),the2(nat,vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),the2(nat,vEBT_vebt_mint(Summary)))))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_VEBT_low(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),the2(nat,vEBT_vebt_mint(Summary))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),the2(nat,vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),the2(nat,vEBT_vebt_mint(Summary)))))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),vEBT_vebt_delete(Summary,vEBT_VEBT_high(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),the2(nat,vEBT_vebt_mint(Summary))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),the2(nat,vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),the2(nat,vEBT_vebt_mint(Summary)))))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),the2(nat,vEBT_vebt_mint(Summary))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),the2(nat,vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),the2(nat,vEBT_vebt_mint(Summary))))))),if(nat,aa(nat,bool,fequal(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),the2(nat,vEBT_vebt_mint(Summary))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),the2(nat,vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),the2(nat,vEBT_vebt_mint(Summary))))))),Ma),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),vEBT_VEBT_high(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),the2(nat,vEBT_vebt_mint(Summary))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),the2(nat,vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),the2(nat,vEBT_vebt_mint(Summary)))))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),the2(nat,vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,list_update(vEBT_VEBT,TreeList,vEBT_VEBT_high(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),the2(nat,vEBT_vebt_mint(Summary))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),the2(nat,vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),the2(nat,vEBT_vebt_mint(Summary)))))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))),vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),the2(nat,vEBT_vebt_mint(Summary))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),the2(nat,vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),the2(nat,vEBT_vebt_mint(Summary)))))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_VEBT_low(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),the2(nat,vEBT_vebt_mint(Summary))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),the2(nat,vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),the2(nat,vEBT_vebt_mint(Summary)))))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),vEBT_VEBT_high(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),the2(nat,vEBT_vebt_mint(Summary))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),the2(nat,vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),the2(nat,vEBT_vebt_mint(Summary)))))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))))),Ma))),Deg,list_update(vEBT_VEBT,TreeList,vEBT_VEBT_high(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),the2(nat,vEBT_vebt_mint(Summary))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),the2(nat,vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),the2(nat,vEBT_vebt_mint(Summary)))))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))),vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),the2(nat,vEBT_vebt_mint(Summary))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),the2(nat,vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),the2(nat,vEBT_vebt_mint(Summary)))))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_VEBT_low(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),the2(nat,vEBT_vebt_mint(Summary))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),the2(nat,vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),the2(nat,vEBT_vebt_mint(Summary)))))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),Summary)),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary)) ) ) ) ) ).

% del_x_mia
tff(fact_4829_add__shift,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),X),Y) = Z )
    <=> ( aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_add,aa(nat,option(nat),some(nat),X)),aa(nat,option(nat),some(nat),Y)) = aa(nat,option(nat),some(nat),Z) ) ) ).

% add_shift
tff(fact_4830_add__def,axiom,
    vEBT_VEBT_add = vEBT_V2048590022279873568_shift(nat,plus_plus(nat)) ).

% add_def
tff(fact_4831_list__update__beyond,axiom,
    ! [A: $tType,Xs: list(A),I: nat,X: A] :
      ( pp(aa(nat,bool,ord_less_eq(nat,aa(list(A),nat,size_size(list(A)),Xs)),I))
     => ( list_update(A,Xs,I,X) = Xs ) ) ).

% list_update_beyond
tff(fact_4832_del__x__not__mi__newnode__not__nil,axiom,
    ! [Mi: nat,X: nat,Ma: nat,Deg: nat,H: nat,L: nat,Newnode: vEBT_VEBT,TreeList: list(vEBT_VEBT),Newlist: list(vEBT_VEBT),Summary: vEBT_VEBT] :
      ( ( pp(aa(nat,bool,ord_less(nat,Mi),X))
        & pp(aa(nat,bool,ord_less_eq(nat,X),Ma)) )
     => ( ( Mi != Ma )
       => ( pp(aa(nat,bool,ord_less_eq(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg))
         => ( ( vEBT_VEBT_high(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = H )
           => ( ( vEBT_VEBT_low(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = L )
             => ( ( Newnode = vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),H),L) )
               => ( ~ pp(vEBT_VEBT_minNull(Newnode))
                 => ( ( Newlist = list_update(vEBT_VEBT,TreeList,H,Newnode) )
                   => ( pp(aa(nat,bool,ord_less(nat,vEBT_VEBT_high(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)))
                     => ( vEBT_vebt_delete(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary),X) = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),if(nat,aa(nat,bool,fequal(nat,X),Ma),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),H),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),the2(nat,vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,Newlist),H)))),Ma))),Deg,Newlist,Summary) ) ) ) ) ) ) ) ) ) ) ).

% del_x_not_mi_newnode_not_nil
tff(fact_4833_del__x__mi__lets__in__not__minNull,axiom,
    ! [X: nat,Mi: nat,Ma: nat,Deg: nat,Xn: nat,H: nat,Summary: vEBT_VEBT,TreeList: list(vEBT_VEBT),L: nat,Newnode: vEBT_VEBT,Newlist: list(vEBT_VEBT)] :
      ( ( ( X = Mi )
        & pp(aa(nat,bool,ord_less(nat,X),Ma)) )
     => ( ( Mi != Ma )
       => ( pp(aa(nat,bool,ord_less_eq(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg))
         => ( ( vEBT_VEBT_high(Xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = H )
           => ( ( Xn = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),the2(nat,vEBT_vebt_mint(Summary))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),the2(nat,vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),the2(nat,vEBT_vebt_mint(Summary)))))) )
             => ( ( vEBT_VEBT_low(Xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = L )
               => ( pp(aa(nat,bool,ord_less(nat,vEBT_VEBT_high(Xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)))
                 => ( ( Newnode = vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),H),L) )
                   => ( ( Newlist = list_update(vEBT_VEBT,TreeList,H,Newnode) )
                     => ( ~ pp(vEBT_VEBT_minNull(Newnode))
                       => ( vEBT_vebt_delete(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary),X) = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Xn),if(nat,aa(nat,bool,fequal(nat,Xn),Ma),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),H),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),the2(nat,vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,Newlist),H)))),Ma))),Deg,Newlist,Summary) ) ) ) ) ) ) ) ) ) ) ) ).

% del_x_mi_lets_in_not_minNull
tff(fact_4834_del__x__not__mi,axiom,
    ! [Mi: nat,X: nat,Ma: nat,Deg: nat,H: nat,L: nat,Newnode: vEBT_VEBT,TreeList: list(vEBT_VEBT),Newlist: list(vEBT_VEBT),Summary: vEBT_VEBT] :
      ( ( pp(aa(nat,bool,ord_less(nat,Mi),X))
        & pp(aa(nat,bool,ord_less_eq(nat,X),Ma)) )
     => ( ( Mi != Ma )
       => ( pp(aa(nat,bool,ord_less_eq(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg))
         => ( ( vEBT_VEBT_high(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = H )
           => ( ( vEBT_VEBT_low(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = L )
             => ( ( Newnode = vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),H),L) )
               => ( ( Newlist = list_update(vEBT_VEBT,TreeList,H,Newnode) )
                 => ( pp(aa(nat,bool,ord_less(nat,vEBT_VEBT_high(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)))
                   => ( ( pp(vEBT_VEBT_minNull(Newnode))
                       => ( vEBT_vebt_delete(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary),X) = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),if(nat,aa(nat,bool,fequal(nat,X),Ma),if(nat,aa(option(nat),bool,fequal(option(nat),vEBT_vebt_maxt(vEBT_vebt_delete(Summary,H))),none(nat)),Mi,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))),the2(nat,vEBT_vebt_maxt(vEBT_vebt_delete(Summary,H))))),the2(nat,vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,Newlist),the2(nat,vEBT_vebt_maxt(vEBT_vebt_delete(Summary,H)))))))),Ma))),Deg,Newlist,vEBT_vebt_delete(Summary,H)) ) )
                      & ( ~ pp(vEBT_VEBT_minNull(Newnode))
                       => ( vEBT_vebt_delete(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary),X) = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),if(nat,aa(nat,bool,fequal(nat,X),Ma),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),H),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),the2(nat,vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,Newlist),H)))),Ma))),Deg,Newlist,Summary) ) ) ) ) ) ) ) ) ) ) ) ).

% del_x_not_mi
tff(fact_4835_del__x__not__mi__new__node__nil,axiom,
    ! [Mi: nat,X: nat,Ma: nat,Deg: nat,H: nat,L: nat,Newnode: vEBT_VEBT,TreeList: list(vEBT_VEBT),Sn: vEBT_VEBT,Summary: vEBT_VEBT,Newlist: list(vEBT_VEBT)] :
      ( ( pp(aa(nat,bool,ord_less(nat,Mi),X))
        & pp(aa(nat,bool,ord_less_eq(nat,X),Ma)) )
     => ( ( Mi != Ma )
       => ( pp(aa(nat,bool,ord_less_eq(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg))
         => ( ( vEBT_VEBT_high(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = H )
           => ( ( vEBT_VEBT_low(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = L )
             => ( ( Newnode = vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),H),L) )
               => ( pp(vEBT_VEBT_minNull(Newnode))
                 => ( ( Sn = vEBT_vebt_delete(Summary,H) )
                   => ( ( Newlist = list_update(vEBT_VEBT,TreeList,H,Newnode) )
                     => ( pp(aa(nat,bool,ord_less(nat,vEBT_VEBT_high(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)))
                       => ( vEBT_vebt_delete(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary),X) = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),if(nat,aa(nat,bool,fequal(nat,X),Ma),if(nat,aa(option(nat),bool,fequal(option(nat),vEBT_vebt_maxt(Sn)),none(nat)),Mi,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))),the2(nat,vEBT_vebt_maxt(Sn)))),the2(nat,vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,Newlist),the2(nat,vEBT_vebt_maxt(Sn))))))),Ma))),Deg,Newlist,Sn) ) ) ) ) ) ) ) ) ) ) ) ).

% del_x_not_mi_new_node_nil
tff(fact_4836_del__x__not__mia,axiom,
    ! [Mi: nat,X: nat,Ma: nat,Deg: nat,H: nat,L: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] :
      ( ( pp(aa(nat,bool,ord_less(nat,Mi),X))
        & pp(aa(nat,bool,ord_less_eq(nat,X),Ma)) )
     => ( ( Mi != Ma )
       => ( pp(aa(nat,bool,ord_less_eq(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg))
         => ( ( vEBT_VEBT_high(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = H )
           => ( ( vEBT_VEBT_low(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = L )
             => ( pp(aa(nat,bool,ord_less(nat,vEBT_VEBT_high(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)))
               => ( vEBT_vebt_delete(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary),X) = if(vEBT_VEBT,vEBT_VEBT_minNull(vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),H),L)),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),if(nat,aa(nat,bool,fequal(nat,X),Ma),if(nat,aa(option(nat),bool,fequal(option(nat),vEBT_vebt_maxt(vEBT_vebt_delete(Summary,H))),none(nat)),Mi,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))),the2(nat,vEBT_vebt_maxt(vEBT_vebt_delete(Summary,H))))),the2(nat,vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,list_update(vEBT_VEBT,TreeList,H,vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),H),L))),the2(nat,vEBT_vebt_maxt(vEBT_vebt_delete(Summary,H)))))))),Ma))),Deg,list_update(vEBT_VEBT,TreeList,H,vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),H),L)),vEBT_vebt_delete(Summary,H)),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),if(nat,aa(nat,bool,fequal(nat,X),Ma),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),H),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),the2(nat,vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,list_update(vEBT_VEBT,TreeList,H,vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),H),L))),H)))),Ma))),Deg,list_update(vEBT_VEBT,TreeList,H,vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),H),L)),Summary)) ) ) ) ) ) ) ) ).

% del_x_not_mia
tff(fact_4837_del__in__range,axiom,
    ! [Mi: nat,X: nat,Ma: nat,Deg: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] :
      ( ( pp(aa(nat,bool,ord_less_eq(nat,Mi),X))
        & pp(aa(nat,bool,ord_less_eq(nat,X),Ma)) )
     => ( ( Mi != Ma )
       => ( pp(aa(nat,bool,ord_less_eq(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg))
         => ( vEBT_vebt_delete(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary),X) = 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ebt_mint(Summary))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),the2(nat,vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),the2(nat,vEBT_vebt_mint(Summary)))))),X),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))))))),the2(nat,vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,list_update(vEBT_VEBT,TreeList,vEBT_VEBT_high(if(nat,aa(nat,bool,fequal(nat,X),Mi),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),the2(nat,vEBT_vebt_mint(Summary))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),the2(nat,vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),the2(nat,vEBT_vebt_mint(Summary)))))),X),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))),vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(if(nat,aa(nat,bool,fequal(nat,X),Mi),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),the2(nat,vEBT_vebt_mint(Summary))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),the2(nat,vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),the2(nat,vEBT_vebt_mint(Summary)))))),X),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_VEBT_low(if(nat,aa(nat,bool,fequal(nat,X),Mi),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),the2(nat,vEBT_vebt_mint(Summary))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),the2(nat,vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),the2(nat,vEBT_vebt_mint(Summary)))))),X),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),the2(nat,vEBT_vebt_maxt(vEBT_vebt_delete(Summary,vEBT_VEBT_high(if(nat,aa(nat,bool,fequal(nat,X),Mi),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),the2(nat,vEBT_vebt_mint(Summary))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),the2(nat,vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),the2(nat,vEBT_vebt_mint(Summary)))))),X),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))))))))),Ma))),Deg,list_update(vEBT_VEBT,TreeList,vEBT_VEBT_high(if(nat,aa(nat,bool,fequal(nat,X),Mi),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),the2(nat,vEBT_vebt_mint(Summary))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),the2(nat,vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),the2(nat,vEBT_vebt_mint(Summary)))))),X),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))),vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(if(nat,aa(nat,bool,fequal(nat,X),Mi),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),the2(nat,vEBT_vebt_mint(Summary))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),the2(nat,vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),the2(nat,vEBT_vebt_mint(Summary)))))),X),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_VEBT_low(if(nat,aa(nat,bool,fequal(nat,X),Mi),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),the2(nat,vEBT_vebt_mint(Summary))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),the2(nat,vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),the2(nat,vEBT_vebt_mint(Summary)))))),X),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),vEBT_vebt_delete(Summary,vEBT_VEBT_high(if(nat,aa(nat,bool,fequal(nat,X),Mi),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),the2(nat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BT,TreeList),the2(nat,vEBT_vebt_mint(Summary)))))),X),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))))),Ma))),Deg,list_update(vEBT_VEBT,TreeList,vEBT_VEBT_high(if(nat,aa(nat,bool,fequal(nat,X),Mi),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),the2(nat,vEBT_vebt_mint(Summary))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),the2(nat,vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),the2(nat,vEBT_vebt_mint(Summary)))))),X),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))),vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(if(nat,aa(nat,bool,fequal(nat,X),Mi),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),the2(nat,vEBT_vebt_mint(Summary))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),the2(nat,vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),the2(nat,vEBT_vebt_mint(Summary)))))),X),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_VEBT_low(if(nat,aa(nat,bool,fequal(nat,X),Mi),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),the2(nat,vEBT_vebt_mint(Summary))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),the2(nat,vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),the2(nat,vEBT_vebt_mint(Summary)))))),X),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),Summary)),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary)) ) ) ) ) ).

% del_in_range
tff(fact_4838_del__x__mi,axiom,
    ! [X: nat,Mi: nat,Ma: nat,Deg: nat,Xn: nat,H: nat,Summary: vEBT_VEBT,TreeList: list(vEBT_VEBT),L: nat] :
      ( ( ( X = Mi )
        & pp(aa(nat,bool,ord_less(nat,X),Ma)) )
     => ( ( Mi != Ma )
       => ( pp(aa(nat,bool,ord_less_eq(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg))
         => ( ( vEBT_VEBT_high(Xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = H )
           => ( ( Xn = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),the2(nat,vEBT_vebt_mint(Summary))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),the2(nat,vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),the2(nat,vEBT_vebt_mint(Summary)))))) )
             => ( ( vEBT_VEBT_low(Xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = L )
               => ( pp(aa(nat,bool,ord_less(nat,vEBT_VEBT_high(Xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)))
                 => ( vEBT_vebt_delete(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary),X) = if(vEBT_VEBT,vEBT_VEBT_minNull(vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),H),L)),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Xn),if(nat,aa(nat,bool,fequal(nat,Xn),Ma),if(nat,aa(option(nat),bool,fequal(option(nat),vEBT_vebt_maxt(vEBT_vebt_delete(Summary,H))),none(nat)),Xn,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))),the2(nat,vEBT_vebt_maxt(vEBT_vebt_delete(Summary,H))))),the2(nat,vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,list_update(vEBT_VEBT,TreeList,H,vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),H),L))),the2(nat,vEBT_vebt_maxt(vEBT_vebt_delete(Summary,H)))))))),Ma))),Deg,list_update(vEBT_VEBT,TreeList,H,vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),H),L)),vEBT_vebt_delete(Summary,H)),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Xn),if(nat,aa(nat,bool,fequal(nat,Xn),Ma),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),H),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),the2(nat,vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,list_update(vEBT_VEBT,TreeList,H,vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),H),L))),H)))),Ma))),Deg,list_update(vEBT_VEBT,TreeList,H,vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),H),L)),Summary)) ) ) ) ) ) ) ) ) ).

% del_x_mi
tff(fact_4839_del__x__mi__lets__in,axiom,
    ! [X: nat,Mi: nat,Ma: nat,Deg: nat,Xn: nat,H: nat,Summary: vEBT_VEBT,TreeList: list(vEBT_VEBT),L: nat,Newnode: vEBT_VEBT,Newlist: list(vEBT_VEBT)] :
      ( ( ( X = Mi )
        & pp(aa(nat,bool,ord_less(nat,X),Ma)) )
     => ( ( Mi != Ma )
       => ( pp(aa(nat,bool,ord_less_eq(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg))
         => ( ( vEBT_VEBT_high(Xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = H )
           => ( ( Xn = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),the2(nat,vEBT_vebt_mint(Summary))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),the2(nat,vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),the2(nat,vEBT_vebt_mint(Summary)))))) )
             => ( ( vEBT_VEBT_low(Xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = L )
               => ( pp(aa(nat,bool,ord_less(nat,vEBT_VEBT_high(Xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)))
                 => ( ( Newnode = vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),H),L) )
                   => ( ( Newlist = list_update(vEBT_VEBT,TreeList,H,Newnode) )
                     => ( ( pp(vEBT_VEBT_minNull(Newnode))
                         => ( vEBT_vebt_delete(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary),X) = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Xn),if(nat,aa(nat,bool,fequal(nat,Xn),Ma),if(nat,aa(option(nat),bool,fequal(option(nat),vEBT_vebt_maxt(vEBT_vebt_delete(Summary,H))),none(nat)),Xn,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))),the2(nat,vEBT_vebt_maxt(vEBT_vebt_delete(Summary,H))))),the2(nat,vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,Newlist),the2(nat,vEBT_vebt_maxt(vEBT_vebt_delete(Summary,H)))))))),Ma))),Deg,Newlist,vEBT_vebt_delete(Summary,H)) ) )
                        & ( ~ pp(vEBT_VEBT_minNull(Newnode))
                         => ( vEBT_vebt_delete(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary),X) = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Xn),if(nat,aa(nat,bool,fequal(nat,Xn),Ma),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),H),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),the2(nat,vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,Newlist),H)))),Ma))),Deg,Newlist,Summary) ) ) ) ) ) ) ) ) ) ) ) ) ).

% del_x_mi_lets_in
tff(fact_4840_del__x__mi__lets__in__minNull,axiom,
    ! [X: nat,Mi: nat,Ma: nat,Deg: nat,Xn: nat,H: nat,Summary: vEBT_VEBT,TreeList: list(vEBT_VEBT),L: nat,Newnode: vEBT_VEBT,Newlist: list(vEBT_VEBT),Sn: vEBT_VEBT] :
      ( ( ( X = Mi )
        & pp(aa(nat,bool,ord_less(nat,X),Ma)) )
     => ( ( Mi != Ma )
       => ( pp(aa(nat,bool,ord_less_eq(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg))
         => ( ( vEBT_VEBT_high(Xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = H )
           => ( ( Xn = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),the2(nat,vEBT_vebt_mint(Summary))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),the2(nat,vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),the2(nat,vEBT_vebt_mint(Summary)))))) )
             => ( ( vEBT_VEBT_low(Xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = L )
               => ( pp(aa(nat,bool,ord_less(nat,vEBT_VEBT_high(Xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)))
                 => ( ( Newnode = vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),H),L) )
                   => ( ( Newlist = list_update(vEBT_VEBT,TreeList,H,Newnode) )
                     => ( pp(vEBT_VEBT_minNull(Newnode))
                       => ( ( Sn = vEBT_vebt_delete(Summary,H) )
                         => ( vEBT_vebt_delete(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary),X) = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Xn),if(nat,aa(nat,bool,fequal(nat,Xn),Ma),if(nat,aa(option(nat),bool,fequal(option(nat),vEBT_vebt_maxt(Sn)),none(nat)),Xn,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))),the2(nat,vEBT_vebt_maxt(Sn)))),the2(nat,vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,Newlist),the2(nat,vEBT_vebt_maxt(Sn))))))),Ma))),Deg,Newlist,Sn) ) ) ) ) ) ) ) ) ) ) ) ) ).

% del_x_mi_lets_in_minNull
tff(fact_4841_set__update__subsetI,axiom,
    ! [A: $tType,Xs: list(A),A3: set(A),X: A,I: nat] :
      ( pp(aa(set(A),bool,ord_less_eq(set(A),aa(list(A),set(A),set2(A),Xs)),A3))
     => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A3))
       => pp(aa(set(A),bool,ord_less_eq(set(A),aa(list(A),set(A),set2(A),list_update(A,Xs,I,X))),A3)) ) ) ).

% set_update_subsetI
tff(fact_4842_VEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H_Osimps_I7_J,axiom,
    ! [X: nat,Mi: nat,Ma: nat,Va: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] :
      ( ( ( pp(aa(nat,bool,ord_less(nat,X),Mi))
          | pp(aa(nat,bool,ord_less(nat,Ma),X)) )
       => ( vEBT_V1232361888498592333_e_t_e(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList,Summary),X) = one_one(nat) ) )
      & ( ~ ( pp(aa(nat,bool,ord_less(nat,X),Mi))
            | pp(aa(nat,bool,ord_less(nat,Ma),X)) )
       => ( ( ( ( X = Mi )
              & ( X = Ma ) )
           => ( vEBT_V1232361888498592333_e_t_e(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList,Summary),X) = one_one(nat) ) )
          & ( ~ ( ( X = Mi )
                & ( X = Ma ) )
           => ( vEBT_V1232361888498592333_e_t_e(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList,Summary),X) = if(nat,aa(nat,bool,ord_less(nat,vEBT_VEBT_high(if(nat,aa(nat,bool,fequal(nat,X),Mi),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),the2(nat,vEBT_vebt_mint(Summary))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),the2(nat,vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),the2(nat,vEBT_vebt_mint(Summary)))))),X),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_V1232361888498592333_e_t_e(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(if(nat,aa(nat,bool,fequal(nat,X),Mi),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),the2(nat,vEBT_vebt_mint(Summary))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),the2(nat,vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),the2(nat,vEBT_vebt_mint(Summary)))))),X),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_VEBT_low(if(nat,aa(nat,bool,fequal(nat,X),Mi),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),the2(nat,vEBT_vebt_mint(Summary))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),the2(nat,vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),the2(nat,vEBT_vebt_mint(Summary)))))),X),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),if(nat,vEBT_VEBT_minNull(vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(if(nat,aa(nat,bool,fequal(nat,X),Mi),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),the2(nat,vEBT_vebt_mint(Summary))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),the2(nat,vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),the2(nat,vEBT_vebt_mint(Summary)))))),X),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_VEBT_low(if(nat,aa(nat,bool,fequal(nat,X),Mi),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),the2(nat,vEBT_vebt_mint(Summary))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),the2(nat,vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),the2(nat,vEBT_vebt_mint(Summary)))))),X),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),vEBT_V1232361888498592333_e_t_e(Summary,vEBT_VEBT_high(if(nat,aa(nat,bool,fequal(nat,X),Mi),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),the2(nat,vEBT_vebt_mint(Summary))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),the2(nat,vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),the2(nat,vEBT_vebt_mint(Summary)))))),X),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,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)
tff(fact_4843_VEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H_Oelims,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: nat] :
      ( ( vEBT_V1232361888498592333_e_t_e(X,Xa2) = Y )
     => ( ( ? [A4: bool,B3: bool] : X = vEBT_Leaf(A4,B3)
         => ( ( Xa2 = zero_zero(nat) )
           => ( Y != one_one(nat) ) ) )
       => ( ( ? [A4: bool,B3: bool] : X = vEBT_Leaf(A4,B3)
           => ( ( Xa2 = aa(nat,nat,suc,zero_zero(nat)) )
             => ( Y != one_one(nat) ) ) )
         => ( ( ? [A4: bool,B3: bool] : X = vEBT_Leaf(A4,B3)
             => ( ? [N2: nat] : Xa2 = aa(nat,nat,suc,aa(nat,nat,suc,N2))
               => ( Y != one_one(nat) ) ) )
           => ( ( ? [Deg2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] : X = vEBT_Node(none(product_prod(nat,nat)),Deg2,TreeList2,Summary2)
               => ( Y != one_one(nat) ) )
             => ( ( ? [Mi2: nat,Ma2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] : X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),zero_zero(nat),TreeList2,Summary2)
                 => ( Y != one_one(nat) ) )
               => ( ( ? [Mi2: nat,Ma2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] : X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,zero_zero(nat)),TreeList2,Summary2)
                   => ( Y != one_one(nat) ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                        ( ( X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary2) )
                       => ~ ( ( ( pp(aa(nat,bool,ord_less(nat,Xa2),Mi2))
                                | pp(aa(nat,bool,ord_less(nat,Ma2),Xa2)) )
                             => ( Y = one_one(nat) ) )
                            & ( ~ ( pp(aa(nat,bool,ord_less(nat,Xa2),Mi2))
                                  | pp(aa(nat,bool,ord_less(nat,Ma2),Xa2)) )
                             => ( ( ( ( Xa2 = Mi2 )
                                    & ( Xa2 = Ma2 ) )
                                 => ( Y = one_one(nat) ) )
                                & ( ~ ( ( Xa2 = Mi2 )
                                      & ( Xa2 = Ma2 ) )
                                 => ( Y = if(nat,aa(nat,bool,ord_less(nat,vEBT_VEBT_high(if(nat,aa(nat,bool,fequal(nat,Xa2),Mi2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),the2(nat,vEBT_vebt_mint(Summary2))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),the2(nat,vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),the2(nat,vEBT_vebt_mint(Summary2)))))),Xa2),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_V1232361888498592333_e_t_e(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),vEBT_VEBT_high(if(nat,aa(nat,bool,fequal(nat,Xa2),Mi2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),the2(nat,vEBT_vebt_mint(Summary2))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),the2(nat,vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),the2(nat,vEBT_vebt_mint(Summary2)))))),Xa2),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_VEBT_low(if(nat,aa(nat,bool,fequal(nat,Xa2),Mi2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),the2(nat,vEBT_vebt_mint(Summary2))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),the2(nat,vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),the2(nat,vEBT_vebt_mint(Summary2)))))),Xa2),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),if(nat,vEBT_VEBT_minNull(vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),vEBT_VEBT_high(if(nat,aa(nat,bool,fequal(nat,Xa2),Mi2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),the2(nat,vEBT_vebt_mint(Summary2))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),the2(nat,vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),the2(nat,vEBT_vebt_mint(Summary2)))))),Xa2),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_VEBT_low(if(nat,aa(nat,bool,fequal(nat,Xa2),Mi2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),the2(nat,vEBT_vebt_mint(Summary2))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),the2(nat,vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),the2(nat,vEBT_vebt_mint(Summary2)))))),Xa2),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),vEBT_V1232361888498592333_e_t_e(Summary2,vEBT_VEBT_high(if(nat,aa(nat,bool,fequal(nat,Xa2),Mi2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),the2(nat,vEBT_vebt_mint(Summary2))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),the2(nat,vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),the2(nat,vEBT_vebt_mint(Summary2)))))),Xa2),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,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
tff(fact_4844_vebt__pred_Osimps_I7_J,axiom,
    ! [Ma: nat,X: nat,Mi: nat,Va: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] :
      ( ( pp(aa(nat,bool,ord_less(nat,Ma),X))
       => ( vEBT_vebt_pred(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList,Summary),X) = aa(nat,option(nat),some(nat),Ma) ) )
      & ( ~ pp(aa(nat,bool,ord_less(nat,Ma),X))
       => ( vEBT_vebt_pred(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList,Summary),X) = if(option(nat),aa(nat,bool,ord_less(nat,vEBT_VEBT_high(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),if(option(nat),fconj(aa(bool,bool,fNot,aa(option(nat),bool,fequal(option(nat),vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),none(nat))),vEBT_VEBT_greater(aa(nat,option(nat),some(nat),vEBT_VEBT_low(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))))),aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_add,aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_mul,aa(nat,option(nat),some(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,option(nat),some(nat),vEBT_VEBT_high(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),vEBT_vebt_pred(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_VEBT_low(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),if(option(nat),aa(option(nat),bool,fequal(option(nat),vEBT_vebt_pred(Summary,vEBT_VEBT_high(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),none(nat)),if(option(nat),aa(nat,bool,ord_less(nat,Mi),X),aa(nat,option(nat),some(nat),Mi),none(nat)),aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_add,aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_mul,aa(nat,option(nat),some(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),vEBT_vebt_pred(Summary,vEBT_VEBT_high(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),the2(nat,vEBT_vebt_pred(Summary,vEBT_VEBT_high(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2))))))))))),none(nat)) ) ) ) ).

% vebt_pred.simps(7)
tff(fact_4845_vebt__pred_Oelims,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: option(nat)] :
      ( ( vEBT_vebt_pred(X,Xa2) = Y )
     => ( ( ? [Uu2: bool,Uv2: bool] : X = vEBT_Leaf(Uu2,Uv2)
         => ( ( Xa2 = zero_zero(nat) )
           => ( Y != none(nat) ) ) )
       => ( ! [A4: bool] :
              ( ? [Uw2: bool] : X = vEBT_Leaf(A4,Uw2)
             => ( ( Xa2 = aa(nat,nat,suc,zero_zero(nat)) )
               => ~ ( ( pp(A4)
                     => ( Y = aa(nat,option(nat),some(nat),zero_zero(nat)) ) )
                    & ( ~ pp(A4)
                     => ( Y = none(nat) ) ) ) ) )
         => ( ! [A4: bool,B3: bool] :
                ( ( X = vEBT_Leaf(A4,B3) )
               => ( ? [Va2: nat] : Xa2 = aa(nat,nat,suc,aa(nat,nat,suc,Va2))
                 => ~ ( ( pp(B3)
                       => ( Y = aa(nat,option(nat),some(nat),one_one(nat)) ) )
                      & ( ~ pp(B3)
                       => ( ( pp(A4)
                           => ( Y = aa(nat,option(nat),some(nat),zero_zero(nat)) ) )
                          & ( ~ pp(A4)
                           => ( Y = none(nat) ) ) ) ) ) ) )
           => ( ( ? [Uy2: nat,Uz2: list(vEBT_VEBT),Va3: vEBT_VEBT] : X = vEBT_Node(none(product_prod(nat,nat)),Uy2,Uz2,Va3)
               => ( Y != none(nat) ) )
             => ( ( ? [V3: product_prod(nat,nat),Vd2: list(vEBT_VEBT),Ve2: vEBT_VEBT] : X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),zero_zero(nat),Vd2,Ve2)
                 => ( Y != none(nat) ) )
               => ( ( ? [V3: product_prod(nat,nat),Vh2: list(vEBT_VEBT),Vi2: vEBT_VEBT] : X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),aa(nat,nat,suc,zero_zero(nat)),Vh2,Vi2)
                   => ( Y != none(nat) ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                        ( ( X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary2) )
                       => ~ ( ( pp(aa(nat,bool,ord_less(nat,Ma2),Xa2))
                             => ( Y = aa(nat,option(nat),some(nat),Ma2) ) )
                            & ( ~ pp(aa(nat,bool,ord_less(nat,Ma2),Xa2))
                             => ( Y = if(option(nat),aa(nat,bool,ord_less(nat,vEBT_VEBT_high(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),if(option(nat),fconj(aa(bool,bool,fNot,aa(option(nat),bool,fequal(option(nat),vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),vEBT_VEBT_high(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),none(nat))),vEBT_VEBT_greater(aa(nat,option(nat),some(nat),vEBT_VEBT_low(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),vEBT_VEBT_high(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))))),aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_add,aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_mul,aa(nat,option(nat),some(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,option(nat),some(nat),vEBT_VEBT_high(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),vEBT_vebt_pred(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),vEBT_VEBT_high(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_VEBT_low(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),if(option(nat),aa(option(nat),bool,fequal(option(nat),vEBT_vebt_pred(Summary2,vEBT_VEBT_high(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),none(nat)),if(option(nat),aa(nat,bool,ord_less(nat,Mi2),Xa2),aa(nat,option(nat),some(nat),Mi2),none(nat)),aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_add,aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_mul,aa(nat,option(nat),some(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),vEBT_vebt_pred(Summary2,vEBT_VEBT_high(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),the2(nat,vEBT_vebt_pred(Summary2,vEBT_VEBT_high(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))))))))))),none(nat)) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_pred.elims
tff(fact_4846_succ__less__length__list,axiom,
    ! [Deg: nat,Mi: nat,X: nat,TreeList: list(vEBT_VEBT),Ma: nat,Summary: vEBT_VEBT] :
      ( pp(aa(nat,bool,ord_less_eq(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg))
     => ( pp(aa(nat,bool,ord_less_eq(nat,Mi),X))
       => ( pp(aa(nat,bool,ord_less(nat,vEBT_VEBT_high(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)))
         => ( vEBT_vebt_succ(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary),X) = if(option(nat),fconj(aa(bool,bool,fNot,aa(option(nat),bool,fequal(option(nat),vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),none(nat))),vEBT_VEBT_less(aa(nat,option(nat),some(nat),vEBT_VEBT_low(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))))),aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_add,aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_mul,aa(nat,option(nat),some(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,option(nat),some(nat),vEBT_VEBT_high(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),vEBT_vebt_succ(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_VEBT_low(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),if(option(nat),aa(option(nat),bool,fequal(option(nat),vEBT_vebt_succ(Summary,vEBT_VEBT_high(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),none(nat)),none(nat),aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_add,aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_mul,aa(nat,option(nat),some(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),vEBT_vebt_succ(Summary,vEBT_VEBT_high(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),the2(nat,vEBT_vebt_succ(Summary,vEBT_VEBT_high(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))))))))) ) ) ) ) ).

% succ_less_length_list
tff(fact_4847_succ__greatereq__min,axiom,
    ! [Deg: nat,Mi: nat,X: nat,Ma: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] :
      ( pp(aa(nat,bool,ord_less_eq(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg))
     => ( pp(aa(nat,bool,ord_less_eq(nat,Mi),X))
       => ( vEBT_vebt_succ(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary),X) = if(option(nat),aa(nat,bool,ord_less(nat,vEBT_VEBT_high(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),if(option(nat),fconj(aa(bool,bool,fNot,aa(option(nat),bool,fequal(option(nat),vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),none(nat))),vEBT_VEBT_less(aa(nat,option(nat),some(nat),vEBT_VEBT_low(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))))),aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_add,aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_mul,aa(nat,option(nat),some(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,option(nat),some(nat),vEBT_VEBT_high(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),vEBT_vebt_succ(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_VEBT_low(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),if(option(nat),aa(option(nat),bool,fequal(option(nat),vEBT_vebt_succ(Summary,vEBT_VEBT_high(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),none(nat)),none(nat),aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_add,aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_mul,aa(nat,option(nat),some(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),vEBT_vebt_succ(Summary,vEBT_VEBT_high(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),the2(nat,vEBT_vebt_succ(Summary,vEBT_VEBT_high(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))))))))),none(nat)) ) ) ) ).

% succ_greatereq_min
tff(fact_4848_vebt__delete_Oelims,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: vEBT_VEBT] :
      ( ( vEBT_vebt_delete(X,Xa2) = Y )
     => ( ! [A4: bool,B3: bool] :
            ( ( X = vEBT_Leaf(A4,B3) )
           => ( ( Xa2 = zero_zero(nat) )
             => ( Y != vEBT_Leaf(fFalse,B3) ) ) )
       => ( ! [A4: bool] :
              ( ? [B3: bool] : X = vEBT_Leaf(A4,B3)
             => ( ( Xa2 = aa(nat,nat,suc,zero_zero(nat)) )
               => ( Y != vEBT_Leaf(A4,fFalse) ) ) )
         => ( ! [A4: bool,B3: bool] :
                ( ( X = vEBT_Leaf(A4,B3) )
               => ( ? [N2: nat] : Xa2 = aa(nat,nat,suc,aa(nat,nat,suc,N2))
                 => ( Y != vEBT_Leaf(A4,B3) ) ) )
           => ( ! [Deg2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                  ( ( X = vEBT_Node(none(product_prod(nat,nat)),Deg2,TreeList2,Summary2) )
                 => ( Y != vEBT_Node(none(product_prod(nat,nat)),Deg2,TreeList2,Summary2) ) )
             => ( ! [Mi2: nat,Ma2: nat,TrLst: list(vEBT_VEBT),Smry: vEBT_VEBT] :
                    ( ( X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),zero_zero(nat),TrLst,Smry) )
                   => ( Y != vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,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] :
                      ( ( X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,zero_zero(nat)),Tr,Sm) )
                     => ( Y != vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,zero_zero(nat)),Tr,Sm) ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                        ( ( X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary2) )
                       => ~ ( ( ( pp(aa(nat,bool,ord_less(nat,Xa2),Mi2))
                                | pp(aa(nat,bool,ord_less(nat,Ma2),Xa2)) )
                             => ( Y = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary2) ) )
                            & ( ~ ( pp(aa(nat,bool,ord_less(nat,Xa2),Mi2))
                                  | pp(aa(nat,bool,ord_less(nat,Ma2),Xa2)) )
                             => ( ( ( ( Xa2 = Mi2 )
                                    & ( Xa2 = Ma2 ) )
                                 => ( Y = vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary2) ) )
                                & ( ~ ( ( Xa2 = Mi2 )
                                      & ( Xa2 = Ma2 ) )
                                 => ( Y = if(vEBT_VEBT,aa(nat,bool,ord_less(nat,vEBT_VEBT_high(if(nat,aa(nat,bool,fequal(nat,Xa2),Mi2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),the2(nat,vEBT_vebt_mint(Summary2))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),the2(nat,vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),the2(nat,vEBT_vebt_mint(Summary2)))))),Xa2),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),if(vEBT_VEBT,vEBT_VEBT_minNull(vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),vEBT_VEBT_high(if(nat,aa(nat,bool,fequal(nat,Xa2),Mi2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),the2(nat,vEBT_vebt_mint(Summary2))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),the2(nat,vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),the2(nat,vEBT_vebt_mint(Summary2)))))),Xa2),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_VEBT_low(if(nat,aa(nat,bool,fequal(nat,Xa2),Mi2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),the2(nat,vEBT_vebt_mint(Summary2))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),the2(nat,vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),the2(nat,vEBT_vebt_mint(Summary2)))))),Xa2),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),if(nat,aa(nat,bool,fequal(nat,Xa2),Mi2),if(nat,aa(nat,bool,fequal(nat,Xa2),Mi2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),the2(nat,vEBT_vebt_mint(Summary2))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),the2(nat,vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),the2(nat,vEBT_vebt_mint(Summary2)))))),Xa2),Mi2)),if(nat,fconj(aa(bool,bool,aa(bool,fun(bool,bool),fimplies,aa(nat,bool,fequal(nat,Xa2),Mi2)),aa(nat,bool,fequal(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),the2(nat,vEBT_vebt_mint(Summary2))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),the2(nat,vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),the2(nat,vEBT_vebt_mint(Summary2))))))),Ma2)),aa(bool,bool,aa(bool,fun(bool,bool),fimplies,aa(bool,bool,fNot,aa(nat,bool,fequal(nat,Xa2),Mi2))),aa(nat,bool,fequal(nat,Xa2),Ma2))),if(nat,aa(option(nat),bool,fequal(option(nat),vEBT_vebt_maxt(vEBT_vebt_delete(Summary2,vEBT_VEBT_high(if(nat,aa(nat,bool,fequal(nat,Xa2),Mi2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),the2(nat,vEBT_vebt_mint(Summary2))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),the2(nat,vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),the2(nat,vEBT_vebt_mint(Summary2)))))),Xa2),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),none(nat)),if(nat,aa(nat,bool,fequal(nat,Xa2),Mi2),if(nat,aa(nat,bool,fequal(nat,Xa2),Mi2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),the2(nat,vEBT_vebt_mint(Summary2))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),the2(nat,vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),the2(nat,vEBT_vebt_mint(Summary2)))))),Xa2),Mi2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat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) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_delete.elims
tff(fact_4849_less__shift,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa(nat,bool,ord_less(nat,X),Y))
    <=> pp(vEBT_VEBT_less(aa(nat,option(nat),some(nat),X),aa(nat,option(nat),some(nat),Y))) ) ).

% less_shift
tff(fact_4850_vebt__delete_Osimps_I3_J,axiom,
    ! [A2: bool,B2: bool,N: nat] : vEBT_vebt_delete(vEBT_Leaf(A2,B2),aa(nat,nat,suc,aa(nat,nat,suc,N))) = vEBT_Leaf(A2,B2) ).

% vebt_delete.simps(3)
tff(fact_4851_vebt__delete_Osimps_I1_J,axiom,
    ! [A2: bool,B2: bool] : vEBT_vebt_delete(vEBT_Leaf(A2,B2),zero_zero(nat)) = vEBT_Leaf(fFalse,B2) ).

% vebt_delete.simps(1)
tff(fact_4852_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)
tff(fact_4853_vebt__delete_Osimps_I2_J,axiom,
    ! [A2: bool,B2: bool] : vEBT_vebt_delete(vEBT_Leaf(A2,B2),aa(nat,nat,suc,zero_zero(nat))) = vEBT_Leaf(A2,fFalse) ).

% vebt_delete.simps(2)
tff(fact_4854_vebt__delete_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,TrLst2: list(vEBT_VEBT),Smry2: vEBT_VEBT,X: nat] : vEBT_vebt_delete(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),zero_zero(nat),TrLst2,Smry2),X) = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),zero_zero(nat),TrLst2,Smry2) ).

% vebt_delete.simps(5)
tff(fact_4855_vebt__delete_Osimps_I6_J,axiom,
    ! [Mi: nat,Ma: nat,Tr2: list(vEBT_VEBT),Sm2: vEBT_VEBT,X: nat] : vEBT_vebt_delete(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,zero_zero(nat)),Tr2,Sm2),X) = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,zero_zero(nat)),Tr2,Sm2) ).

% vebt_delete.simps(6)
tff(fact_4856_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Osimps_I6_J,axiom,
    ! [X: nat,Mi: nat,Ma: nat,Va: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] :
      ( ( pp(aa(nat,bool,ord_less(nat,X),Mi))
       => ( vEBT_T_s_u_c_c2(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList,Summary),X) = one_one(nat) ) )
      & ( ~ pp(aa(nat,bool,ord_less(nat,X),Mi))
       => ( vEBT_T_s_u_c_c2(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList,Summary),X) = if(nat,aa(nat,bool,ord_less(nat,vEBT_VEBT_high(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),if(nat,fconj(aa(bool,bool,fNot,aa(option(nat),bool,fequal(option(nat),vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),none(nat))),vEBT_VEBT_less(aa(nat,option(nat),some(nat),vEBT_VEBT_low(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),vEBT_T_s_u_c_c2(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_VEBT_low(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_T_s_u_c_c2(Summary,vEBT_VEBT_high(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),one_one(nat))),one_one(nat)) ) ) ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.simps(6)
tff(fact_4857_vebt__succ_Osimps_I6_J,axiom,
    ! [X: nat,Mi: nat,Ma: nat,Va: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] :
      ( ( pp(aa(nat,bool,ord_less(nat,X),Mi))
       => ( vEBT_vebt_succ(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList,Summary),X) = aa(nat,option(nat),some(nat),Mi) ) )
      & ( ~ pp(aa(nat,bool,ord_less(nat,X),Mi))
       => ( vEBT_vebt_succ(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList,Summary),X) = if(option(nat),aa(nat,bool,ord_less(nat,vEBT_VEBT_high(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),if(option(nat),fconj(aa(bool,bool,fNot,aa(option(nat),bool,fequal(option(nat),vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),none(nat))),vEBT_VEBT_less(aa(nat,option(nat),some(nat),vEBT_VEBT_low(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))))),aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_add,aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_mul,aa(nat,option(nat),some(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,option(nat),some(nat),vEBT_VEBT_high(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),vEBT_vebt_succ(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_VEBT_low(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),if(option(nat),aa(option(nat),bool,fequal(option(nat),vEBT_vebt_succ(Summary,vEBT_VEBT_high(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),none(nat)),none(nat),aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_add,aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_mul,aa(nat,option(nat),some(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),vEBT_vebt_succ(Summary,vEBT_VEBT_high(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),the2(nat,vEBT_vebt_succ(Summary,vEBT_VEBT_high(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2))))))))))),none(nat)) ) ) ) ).

% vebt_succ.simps(6)
tff(fact_4858_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Oelims,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: nat] :
      ( ( vEBT_T_s_u_c_c2(X,Xa2) = Y )
     => ( ( ? [Uu2: bool,B3: bool] : X = vEBT_Leaf(Uu2,B3)
         => ( ( Xa2 = zero_zero(nat) )
           => ( Y != one_one(nat) ) ) )
       => ( ( ? [Uv2: bool,Uw2: bool] : X = vEBT_Leaf(Uv2,Uw2)
           => ( ? [N2: nat] : Xa2 = aa(nat,nat,suc,N2)
             => ( Y != one_one(nat) ) ) )
         => ( ( ? [Ux2: nat,Uy2: list(vEBT_VEBT),Uz2: vEBT_VEBT] : X = vEBT_Node(none(product_prod(nat,nat)),Ux2,Uy2,Uz2)
             => ( Y != one_one(nat) ) )
           => ( ( ? [V3: product_prod(nat,nat),Vc: list(vEBT_VEBT),Vd2: vEBT_VEBT] : X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),zero_zero(nat),Vc,Vd2)
               => ( Y != one_one(nat) ) )
             => ( ( ? [V3: product_prod(nat,nat),Vg2: list(vEBT_VEBT),Vh2: vEBT_VEBT] : X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),aa(nat,nat,suc,zero_zero(nat)),Vg2,Vh2)
                 => ( Y != one_one(nat) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                      ( ( X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary2) )
                     => ~ ( ( pp(aa(nat,bool,ord_less(nat,Xa2),Mi2))
                           => ( Y = one_one(nat) ) )
                          & ( ~ pp(aa(nat,bool,ord_less(nat,Xa2),Mi2))
                           => ( Y = if(nat,aa(nat,bool,ord_less(nat,vEBT_VEBT_high(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),if(nat,fconj(aa(bool,bool,fNot,aa(option(nat),bool,fequal(option(nat),vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),vEBT_VEBT_high(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),none(nat))),vEBT_VEBT_less(aa(nat,option(nat),some(nat),vEBT_VEBT_low(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),vEBT_VEBT_high(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),vEBT_T_s_u_c_c2(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),vEBT_VEBT_high(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_VEBT_low(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_T_s_u_c_c2(Summary2,vEBT_VEBT_high(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),one_one(nat))),one_one(nat)) ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.elims
tff(fact_4859_vebt__succ_Oelims,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: option(nat)] :
      ( ( vEBT_vebt_succ(X,Xa2) = Y )
     => ( ! [Uu2: bool,B3: bool] :
            ( ( X = vEBT_Leaf(Uu2,B3) )
           => ( ( Xa2 = zero_zero(nat) )
             => ~ ( ( pp(B3)
                   => ( Y = aa(nat,option(nat),some(nat),one_one(nat)) ) )
                  & ( ~ pp(B3)
                   => ( Y = none(nat) ) ) ) ) )
       => ( ( ? [Uv2: bool,Uw2: bool] : X = vEBT_Leaf(Uv2,Uw2)
           => ( ? [N2: nat] : Xa2 = aa(nat,nat,suc,N2)
             => ( Y != none(nat) ) ) )
         => ( ( ? [Ux2: nat,Uy2: list(vEBT_VEBT),Uz2: vEBT_VEBT] : X = vEBT_Node(none(product_prod(nat,nat)),Ux2,Uy2,Uz2)
             => ( Y != none(nat) ) )
           => ( ( ? [V3: product_prod(nat,nat),Vc: list(vEBT_VEBT),Vd2: vEBT_VEBT] : X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),zero_zero(nat),Vc,Vd2)
               => ( Y != none(nat) ) )
             => ( ( ? [V3: product_prod(nat,nat),Vg2: list(vEBT_VEBT),Vh2: vEBT_VEBT] : X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),aa(nat,nat,suc,zero_zero(nat)),Vg2,Vh2)
                 => ( Y != none(nat) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                      ( ( X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary2) )
                     => ~ ( ( pp(aa(nat,bool,ord_less(nat,Xa2),Mi2))
                           => ( Y = aa(nat,option(nat),some(nat),Mi2) ) )
                          & ( ~ pp(aa(nat,bool,ord_less(nat,Xa2),Mi2))
                           => ( Y = if(option(nat),aa(nat,bool,ord_less(nat,vEBT_VEBT_high(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),if(option(nat),fconj(aa(bool,bool,fNot,aa(option(nat),bool,fequal(option(nat),vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),vEBT_VEBT_high(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),none(nat))),vEBT_VEBT_less(aa(nat,option(nat),some(nat),vEBT_VEBT_low(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),vEBT_VEBT_high(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))))),aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_add,aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_mul,aa(nat,option(nat),some(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,option(nat),some(nat),vEBT_VEBT_high(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),vEBT_vebt_succ(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),vEBT_VEBT_high(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_VEBT_low(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),if(option(nat),aa(option(nat),bool,fequal(option(nat),vEBT_vebt_succ(Summary2,vEBT_VEBT_high(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),none(nat)),none(nat),aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_add,aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_mul,aa(nat,option(nat),some(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),vEBT_vebt_succ(Summary2,vEBT_VEBT_high(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),the2(nat,vEBT_vebt_succ(Summary2,vEBT_VEBT_high(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))))))))))),none(nat)) ) ) ) ) ) ) ) ) ) ) ).

% vebt_succ.elims
tff(fact_4860_vebt__delete_Osimps_I7_J,axiom,
    ! [X: nat,Mi: nat,Ma: nat,Va: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] :
      ( ( ( pp(aa(nat,bool,ord_less(nat,X),Mi))
          | pp(aa(nat,bool,ord_less(nat,Ma),X)) )
       => ( vEBT_vebt_delete(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList,Summary),X) = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList,Summary) ) )
      & ( ~ ( pp(aa(nat,bool,ord_less(nat,X),Mi))
            | pp(aa(nat,bool,ord_less(nat,Ma),X)) )
       => ( ( ( ( X = Mi )
              & ( X = Ma ) )
           => ( vEBT_vebt_delete(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList,Summary),X) = vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList,Summary) ) )
          & ( ~ ( ( X = Mi )
                & ( X = Ma ) )
           => ( vEBT_vebt_delete(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList,Summary),X) = 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al_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),the2(nat,vEBT_vebt_maxt(vEBT_vebt_delete(Summary,vEBT_VEBT_high(if(nat,aa(nat,bool,fequal(nat,X),Mi),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),the2(nat,vEBT_vebt_mint(Summary))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),the2(nat,vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),the2(nat,vEBT_vebt_mint(Summary)))))),X),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2))))))))),the2(nat,vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,list_update(vEBT_VEBT,TreeList,vEBT_VEBT_high(if(nat,aa(nat,bool,fequal(nat,X),Mi),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),the2(nat,vEBT_vebt_mint(Summary))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),the2(nat,vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),the2(nat,vEBT_vebt_mint(Summary)))))),X),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(if(nat,aa(nat,bool,fequal(nat,X),Mi),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),the2(nat,vEBT_vebt_mint(Summary))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),the2(nat,vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),the2(nat,vEBT_vebt_mint(Summary)))))),X),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_VEBT_low(if(nat,aa(nat,bool,fequal(nat,X),Mi),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),the2(nat,vEBT_vebt_mint(Summary))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),the2(nat,vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),the2(nat,vEBT_vebt_mint(Summary)))))),X),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),the2(nat,vEBT_vebt_maxt(vEBT_vebt_delete(Summary,vEBT_VEBT_high(if(nat,aa(nat,bool,fequal(nat,X),Mi),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),the2(nat,vEBT_vebt_mint(Summary))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),the2(nat,vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),the2(nat,vEBT_vebt_mint(Summary)))))),X),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))))))))),Ma))),aa(nat,nat,suc,aa(nat,nat,suc,Va)),list_update(vEBT_VEBT,TreeList,vEBT_VEBT_high(if(nat,aa(nat,bool,fequal(nat,X),Mi),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),the2(nat,vEBT_vebt_mint(Summary))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),the2(nat,vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),the2(nat,vEBT_vebt_mint(Summary)))))),X),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(if(nat,aa(nat,bool,fequal(nat,X),Mi),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),the2(nat,vEBT_vebt_mint(Summary))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),the2(nat,vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),the2(nat,vEBT_vebt_mint(Summary)))))),X),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat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vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),the2(nat,vEBT_vebt_mint(Summary)))))),X),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_VEBT_low(if(nat,aa(nat,bool,fequal(nat,X),Mi),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),the2(nat,vEBT_vebt_mint(Summary))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),the2(nat,vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),the2(nat,vEBT_vebt_mint(Summary)))))),X),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),vEBT_VEBT_high(if(nat,aa(nat,bool,fequal(nat,X),Mi),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),the2(nat,vEBT_vebt_mint(Summary))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),the2(nat,vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),the2(nat,vEBT_vebt_mint(Summary)))))),X),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))))),Ma))),aa(nat,nat,suc,aa(nat,nat,suc,Va)),list_update(vEBT_VEBT,TreeList,vEBT_VEBT_high(if(nat,aa(nat,bool,fequal(nat,X),Mi),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),the2(nat,vEBT_vebt_mint(Summary))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),the2(nat,vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),the2(nat,vEBT_vebt_mint(Summary)))))),X),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(if(nat,aa(nat,bool,fequal(nat,X),Mi),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),the2(nat,vEBT_vebt_mint(Summary))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),the2(nat,vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),the2(nat,vEBT_vebt_mint(Summary)))))),X),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_VEBT_low(if(nat,aa(nat,bool,fequal(nat,X),Mi),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),the2(nat,vEBT_vebt_mint(Summary))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),the2(nat,vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),the2(nat,vEBT_vebt_mint(Summary)))))),X),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),Summary)),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList,Summary)) ) ) ) ) ) ).

% vebt_delete.simps(7)
tff(fact_4861_insert__simp__norm,axiom,
    ! [X: nat,Deg: nat,TreeList: list(vEBT_VEBT),Mi: nat,Ma: nat,Summary: vEBT_VEBT] :
      ( pp(aa(nat,bool,ord_less(nat,vEBT_VEBT_high(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)))
     => ( pp(aa(nat,bool,ord_less(nat,Mi),X))
       => ( pp(aa(nat,bool,ord_less_eq(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg))
         => ( ( X != Ma )
           => ( vEBT_vebt_insert(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary),X) = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),aa(nat,nat,ord_max(nat,X),Ma))),Deg,list_update(vEBT_VEBT,TreeList,vEBT_VEBT_high(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))),vEBT_vebt_insert(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_VEBT_low(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),if(vEBT_VEBT,vEBT_VEBT_minNull(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),vEBT_vebt_insert(Summary,vEBT_VEBT_high(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))),Summary)) ) ) ) ) ) ).

% insert_simp_norm
tff(fact_4862_insert__simp__excp,axiom,
    ! [Mi: nat,Deg: nat,TreeList: list(vEBT_VEBT),X: nat,Ma: nat,Summary: vEBT_VEBT] :
      ( pp(aa(nat,bool,ord_less(nat,vEBT_VEBT_high(Mi,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)))
     => ( pp(aa(nat,bool,ord_less(nat,X),Mi))
       => ( pp(aa(nat,bool,ord_less_eq(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg))
         => ( ( X != Ma )
           => ( vEBT_vebt_insert(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary),X) = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),X),aa(nat,nat,ord_max(nat,Mi),Ma))),Deg,list_update(vEBT_VEBT,TreeList,vEBT_VEBT_high(Mi,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))),vEBT_vebt_insert(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(Mi,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_VEBT_low(Mi,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),if(vEBT_VEBT,vEBT_VEBT_minNull(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(Mi,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),vEBT_vebt_insert(Summary,vEBT_VEBT_high(Mi,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))),Summary)) ) ) ) ) ) ).

% insert_simp_excp
tff(fact_4863_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_Oelims,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: nat] :
      ( ( vEBT_T_i_n_s_e_r_t(X,Xa2) = Y )
     => ( ( ? [A4: bool,B3: bool] : X = vEBT_Leaf(A4,B3)
         => ( Y != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),if(nat,aa(nat,bool,fequal(nat,Xa2),zero_zero(nat)),one_one(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),one_one(nat)))) ) )
       => ( ( ? [Info2: option(product_prod(nat,nat)),Ts2: list(vEBT_VEBT),S2: vEBT_VEBT] : X = vEBT_Node(Info2,zero_zero(nat),Ts2,S2)
           => ( Y != one_one(nat) ) )
         => ( ( ? [Info2: option(product_prod(nat,nat)),Ts2: list(vEBT_VEBT),S2: vEBT_VEBT] : X = vEBT_Node(Info2,aa(nat,nat,suc,zero_zero(nat)),Ts2,S2)
             => ( Y != one_one(nat) ) )
           => ( ( ? [V3: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] : X = vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,V3)),TreeList2,Summary2)
               => ( Y != aa(num,nat,numeral_numeral(nat),bit0(one2)) ) )
             => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                    ( ( X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary2) )
                   => ( Y != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,aa(num,num,bit1,bit0(bit0(one2)))))),if(nat,fconj(aa(nat,bool,ord_less(nat,vEBT_VEBT_high(if(nat,aa(nat,bool,ord_less(nat,Xa2),Mi2),Mi2,Xa2),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),aa(bool,bool,fNot,fdisj(aa(nat,bool,fequal(nat,Xa2),Mi2),aa(nat,bool,fequal(nat,Xa2),Ma2)))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_T_i_n_s_e_r_t(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),vEBT_VEBT_high(if(nat,aa(nat,bool,ord_less(nat,Xa2),Mi2),Mi2,Xa2),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_VEBT_low(if(nat,aa(nat,bool,ord_less(nat,Xa2),Mi2),Mi2,Xa2),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),vEBT_T_m_i_n_N_u_l_l(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),vEBT_VEBT_high(if(nat,aa(nat,bool,ord_less(nat,Xa2),Mi2),Mi2,Xa2),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))))),if(nat,vEBT_VEBT_minNull(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),vEBT_VEBT_high(if(nat,aa(nat,bool,ord_less(nat,Xa2),Mi2),Mi2,Xa2),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),vEBT_T_i_n_s_e_r_t(Summary2,vEBT_VEBT_high(if(nat,aa(nat,bool,ord_less(nat,Xa2),Mi2),Mi2,Xa2),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,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
tff(fact_4864_max__Suc__Suc,axiom,
    ! [M: nat,N: nat] : aa(nat,nat,ord_max(nat,aa(nat,nat,suc,M)),aa(nat,nat,suc,N)) = aa(nat,nat,suc,aa(nat,nat,ord_max(nat,M),N)) ).

% max_Suc_Suc
tff(fact_4865_max__nat_Oeq__neutr__iff,axiom,
    ! [A2: nat,B2: nat] :
      ( ( aa(nat,nat,ord_max(nat,A2),B2) = zero_zero(nat) )
    <=> ( ( A2 = zero_zero(nat) )
        & ( B2 = zero_zero(nat) ) ) ) ).

% max_nat.eq_neutr_iff
tff(fact_4866_max__nat_Oleft__neutral,axiom,
    ! [A2: nat] : aa(nat,nat,ord_max(nat,zero_zero(nat)),A2) = A2 ).

% max_nat.left_neutral
tff(fact_4867_max__nat_Oneutr__eq__iff,axiom,
    ! [A2: nat,B2: nat] :
      ( ( zero_zero(nat) = aa(nat,nat,ord_max(nat,A2),B2) )
    <=> ( ( A2 = zero_zero(nat) )
        & ( B2 = zero_zero(nat) ) ) ) ).

% max_nat.neutr_eq_iff
tff(fact_4868_max__nat_Oright__neutral,axiom,
    ! [A2: nat] : aa(nat,nat,ord_max(nat,A2),zero_zero(nat)) = A2 ).

% max_nat.right_neutral
tff(fact_4869_max__0L,axiom,
    ! [N: nat] : aa(nat,nat,ord_max(nat,zero_zero(nat)),N) = N ).

% max_0L
tff(fact_4870_max__0R,axiom,
    ! [N: nat] : aa(nat,nat,ord_max(nat,N),zero_zero(nat)) = N ).

% max_0R
tff(fact_4871_of__bool__or__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [P: bool,Q: bool] : aa(bool,A,zero_neq_one_of_bool(A),fdisj(P,Q)) = aa(A,A,ord_max(A,aa(bool,A,zero_neq_one_of_bool(A),P)),aa(bool,A,zero_neq_one_of_bool(A),Q)) ) ).

% of_bool_or_iff
tff(fact_4872_max__0__1_I3_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [X: num] : aa(A,A,ord_max(A,zero_zero(A)),aa(num,A,numeral_numeral(A),X)) = aa(num,A,numeral_numeral(A),X) ) ).

% max_0_1(3)
tff(fact_4873_max__0__1_I4_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [X: num] : aa(A,A,ord_max(A,aa(num,A,numeral_numeral(A),X)),zero_zero(A)) = aa(num,A,numeral_numeral(A),X) ) ).

% max_0_1(4)
tff(fact_4874_max__number__of_I1_J,axiom,
    ! [A: $tType] :
      ( ( numeral(A)
        & ord(A) )
     => ! [U: num,V: num] :
          ( ( pp(aa(A,bool,ord_less_eq(A,aa(num,A,numeral_numeral(A),U)),aa(num,A,numeral_numeral(A),V)))
           => ( aa(A,A,ord_max(A,aa(num,A,numeral_numeral(A),U)),aa(num,A,numeral_numeral(A),V)) = aa(num,A,numeral_numeral(A),V) ) )
          & ( ~ pp(aa(A,bool,ord_less_eq(A,aa(num,A,numeral_numeral(A),U)),aa(num,A,numeral_numeral(A),V)))
           => ( aa(A,A,ord_max(A,aa(num,A,numeral_numeral(A),U)),aa(num,A,numeral_numeral(A),V)) = aa(num,A,numeral_numeral(A),U) ) ) ) ) ).

% max_number_of(1)
tff(fact_4875_max__0__1_I2_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ( aa(A,A,ord_max(A,one_one(A)),zero_zero(A)) = one_one(A) ) ) ).

% max_0_1(2)
tff(fact_4876_max__0__1_I1_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ( aa(A,A,ord_max(A,zero_zero(A)),one_one(A)) = one_one(A) ) ) ).

% max_0_1(1)
tff(fact_4877_max__0__1_I5_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [X: num] : aa(A,A,ord_max(A,one_one(A)),aa(num,A,numeral_numeral(A),X)) = aa(num,A,numeral_numeral(A),X) ) ).

% max_0_1(5)
tff(fact_4878_max__0__1_I6_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [X: num] : aa(A,A,ord_max(A,aa(num,A,numeral_numeral(A),X)),one_one(A)) = aa(num,A,numeral_numeral(A),X) ) ).

% max_0_1(6)
tff(fact_4879_max__number__of_I4_J,axiom,
    ! [A: $tType] :
      ( ( uminus(A)
        & numeral(A)
        & ord(A) )
     => ! [U: num,V: num] :
          ( ( pp(aa(A,bool,ord_less_eq(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))))
           => ( aa(A,A,ord_max(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V)) ) )
          & ( ~ pp(aa(A,bool,ord_less_eq(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))))
           => ( aa(A,A,ord_max(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U)) ) ) ) ) ).

% max_number_of(4)
tff(fact_4880_max__number__of_I3_J,axiom,
    ! [A: $tType] :
      ( ( uminus(A)
        & numeral(A)
        & ord(A) )
     => ! [U: num,V: num] :
          ( ( pp(aa(A,bool,ord_less_eq(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U))),aa(num,A,numeral_numeral(A),V)))
           => ( aa(A,A,ord_max(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U))),aa(num,A,numeral_numeral(A),V)) = aa(num,A,numeral_numeral(A),V) ) )
          & ( ~ pp(aa(A,bool,ord_less_eq(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U))),aa(num,A,numeral_numeral(A),V)))
           => ( aa(A,A,ord_max(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U))),aa(num,A,numeral_numeral(A),V)) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U)) ) ) ) ) ).

% max_number_of(3)
tff(fact_4881_max__number__of_I2_J,axiom,
    ! [A: $tType] :
      ( ( uminus(A)
        & numeral(A)
        & ord(A) )
     => ! [U: num,V: num] :
          ( ( pp(aa(A,bool,ord_less_eq(A,aa(num,A,numeral_numeral(A),U)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))))
           => ( aa(A,A,ord_max(A,aa(num,A,numeral_numeral(A),U)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V)) ) )
          & ( ~ pp(aa(A,bool,ord_less_eq(A,aa(num,A,numeral_numeral(A),U)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))))
           => ( aa(A,A,ord_max(A,aa(num,A,numeral_numeral(A),U)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))) = aa(num,A,numeral_numeral(A),U) ) ) ) ) ).

% max_number_of(2)
tff(fact_4882_max__numeral__Suc,axiom,
    ! [K: num,N: nat] : aa(nat,nat,ord_max(nat,aa(num,nat,numeral_numeral(nat),K)),aa(nat,nat,suc,N)) = aa(nat,nat,suc,aa(nat,nat,ord_max(nat,pred_numeral(K)),N)) ).

% max_numeral_Suc
tff(fact_4883_max__Suc__numeral,axiom,
    ! [N: nat,K: num] : aa(nat,nat,ord_max(nat,aa(nat,nat,suc,N)),aa(num,nat,numeral_numeral(nat),K)) = aa(nat,nat,suc,aa(nat,nat,ord_max(nat,N),pred_numeral(K))) ).

% max_Suc_numeral
tff(fact_4884_max__add__distrib__right,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [X: A,Y: A,Z: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),X),aa(A,A,ord_max(A,Y),Z)) = aa(A,A,ord_max(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y)),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Z)) ) ).

% max_add_distrib_right
tff(fact_4885_max__add__distrib__left,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [X: A,Y: A,Z: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,ord_max(A,X),Y)),Z) = aa(A,A,ord_max(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Z)),aa(A,A,aa(A,fun(A,A),plus_plus(A),Y),Z)) ) ).

% max_add_distrib_left
tff(fact_4886_nat__add__max__right,axiom,
    ! [M: nat,N: nat,Q2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),aa(nat,nat,ord_max(nat,N),Q2)) = aa(nat,nat,ord_max(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Q2)) ).

% nat_add_max_right
tff(fact_4887_nat__add__max__left,axiom,
    ! [M: nat,N: nat,Q2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,ord_max(nat,M),N)),Q2) = aa(nat,nat,ord_max(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Q2)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),Q2)) ).

% nat_add_max_left
tff(fact_4888_max__def__raw,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [X4: A,Xa: A] :
          ( ( pp(aa(A,bool,ord_less_eq(A,X4),Xa))
           => ( aa(A,A,ord_max(A,X4),Xa) = Xa ) )
          & ( ~ pp(aa(A,bool,ord_less_eq(A,X4),Xa))
           => ( aa(A,A,ord_max(A,X4),Xa) = X4 ) ) ) ) ).

% max_def_raw
tff(fact_4889_of__nat__max,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [X: nat,Y: nat] : aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,ord_max(nat,X),Y)) = aa(A,A,ord_max(A,aa(nat,A,semiring_1_of_nat(A),X)),aa(nat,A,semiring_1_of_nat(A),Y)) ) ).

% of_nat_max
tff(fact_4890_nat__mult__max__left,axiom,
    ! [M: nat,N: nat,Q2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,ord_max(nat,M),N)),Q2) = aa(nat,nat,ord_max(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),Q2)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),Q2)) ).

% nat_mult_max_left
tff(fact_4891_nat__mult__max__right,axiom,
    ! [M: nat,N: nat,Q2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),aa(nat,nat,ord_max(nat,N),Q2)) = aa(nat,nat,ord_max(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),Q2)) ).

% nat_mult_max_right
tff(fact_4892_max__diff__distrib__left,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [X: A,Y: A,Z: A] : aa(A,A,minus_minus(A,aa(A,A,ord_max(A,X),Y)),Z) = aa(A,A,ord_max(A,aa(A,A,minus_minus(A,X),Z)),aa(A,A,minus_minus(A,Y),Z)) ) ).

% max_diff_distrib_left
tff(fact_4893_nat__minus__add__max,axiom,
    ! [N: nat,M: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,minus_minus(nat,N),M)),M) = aa(nat,nat,ord_max(nat,N),M) ).

% nat_minus_add_max
tff(fact_4894_minNull__bound,axiom,
    ! [T: vEBT_VEBT] : pp(aa(nat,bool,ord_less_eq(nat,vEBT_T_m_i_n_N_u_l_l(T)),one_one(nat))) ).

% minNull_bound
tff(fact_4895_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: bool] : vEBT_T_m_i_n_N_u_l_l(vEBT_Leaf(Uu,fTrue)) = one_one(nat) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.simps(3)
tff(fact_4896_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: bool] : vEBT_T_m_i_n_N_u_l_l(vEBT_Leaf(fTrue,Uv)) = one_one(nat) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.simps(2)
tff(fact_4897_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(fFalse,fFalse)) = one_one(nat) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.simps(1)
tff(fact_4898_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(aa(product_prod(nat,nat),option(product_prod(nat,nat)),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)
tff(fact_4899_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)
tff(fact_4900_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062N_092_060_094sub_062u_092_060_094sub_062l_092_060_094sub_062l_Oelims,axiom,
    ! [X: vEBT_VEBT,Y: nat] :
      ( ( vEBT_T_m_i_n_N_u_l_l(X) = Y )
     => ( ( ( X = vEBT_Leaf(fFalse,fFalse) )
         => ( Y != one_one(nat) ) )
       => ( ( ? [Uv2: bool] : X = vEBT_Leaf(fTrue,Uv2)
           => ( Y != one_one(nat) ) )
         => ( ( ? [Uu2: bool] : X = vEBT_Leaf(Uu2,fTrue)
             => ( Y != one_one(nat) ) )
           => ( ( ? [Uw2: nat,Ux2: list(vEBT_VEBT),Uy2: vEBT_VEBT] : X = vEBT_Node(none(product_prod(nat,nat)),Uw2,Ux2,Uy2)
               => ( Y != one_one(nat) ) )
             => ~ ( ? [Uz2: product_prod(nat,nat),Va3: nat,Vb2: list(vEBT_VEBT),Vc: vEBT_VEBT] : X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),Uz2),Va3,Vb2,Vc)
                 => ( Y != one_one(nat) ) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.elims
tff(fact_4901_vebt__insert_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,Va: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,X: nat] : vEBT_vebt_insert(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList,Summary),X) = if(vEBT_VEBT,fconj(aa(nat,bool,ord_less(nat,vEBT_VEBT_high(if(nat,aa(nat,bool,ord_less(nat,X),Mi),Mi,X),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),aa(bool,bool,fNot,fdisj(aa(nat,bool,fequal(nat,X),Mi),aa(nat,bool,fequal(nat,X),Ma)))),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),if(nat,aa(nat,bool,ord_less(nat,X),Mi),X,Mi)),aa(nat,nat,ord_max(nat,if(nat,aa(nat,bool,ord_less(nat,X),Mi),Mi,X)),Ma))),aa(nat,nat,suc,aa(nat,nat,suc,Va)),list_update(vEBT_VEBT,TreeList,vEBT_VEBT_high(if(nat,aa(nat,bool,ord_less(nat,X),Mi),Mi,X),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),vEBT_vebt_insert(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(if(nat,aa(nat,bool,ord_less(nat,X),Mi),Mi,X),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_VEBT_low(if(nat,aa(nat,bool,ord_less(nat,X),Mi),Mi,X),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),if(vEBT_VEBT,vEBT_VEBT_minNull(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(if(nat,aa(nat,bool,ord_less(nat,X),Mi),Mi,X),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),vEBT_vebt_insert(Summary,vEBT_VEBT_high(if(nat,aa(nat,bool,ord_less(nat,X),Mi),Mi,X),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),Summary)),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList,Summary)) ).

% vebt_insert.simps(5)
tff(fact_4902_vebt__insert_Oelims,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: vEBT_VEBT] :
      ( ( vEBT_vebt_insert(X,Xa2) = Y )
     => ( ! [A4: bool,B3: bool] :
            ( ( X = vEBT_Leaf(A4,B3) )
           => ~ ( ( ( Xa2 = zero_zero(nat) )
                 => ( Y = vEBT_Leaf(fTrue,B3) ) )
                & ( ( Xa2 != zero_zero(nat) )
                 => ( ( ( Xa2 = one_one(nat) )
                     => ( Y = vEBT_Leaf(A4,fTrue) ) )
                    & ( ( Xa2 != one_one(nat) )
                     => ( Y = vEBT_Leaf(A4,B3) ) ) ) ) ) )
       => ( ! [Info2: option(product_prod(nat,nat)),Ts2: list(vEBT_VEBT),S2: vEBT_VEBT] :
              ( ( X = vEBT_Node(Info2,zero_zero(nat),Ts2,S2) )
             => ( Y != vEBT_Node(Info2,zero_zero(nat),Ts2,S2) ) )
         => ( ! [Info2: option(product_prod(nat,nat)),Ts2: list(vEBT_VEBT),S2: vEBT_VEBT] :
                ( ( X = vEBT_Node(Info2,aa(nat,nat,suc,zero_zero(nat)),Ts2,S2) )
               => ( Y != vEBT_Node(Info2,aa(nat,nat,suc,zero_zero(nat)),Ts2,S2) ) )
           => ( ! [V3: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                  ( ( X = vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,V3)),TreeList2,Summary2) )
                 => ( Y != vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Xa2),Xa2)),aa(nat,nat,suc,aa(nat,nat,suc,V3)),TreeList2,Summary2) ) )
             => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                    ( ( X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary2) )
                   => ( Y != if(vEBT_VEBT,fconj(aa(nat,bool,ord_less(nat,vEBT_VEBT_high(if(nat,aa(nat,bool,ord_less(nat,Xa2),Mi2),Mi2,Xa2),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),aa(bool,bool,fNot,fdisj(aa(nat,bool,fequal(nat,Xa2),Mi2),aa(nat,bool,fequal(nat,Xa2),Ma2)))),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),if(nat,aa(nat,bool,ord_less(nat,Xa2),Mi2),Xa2,Mi2)),aa(nat,nat,ord_max(nat,if(nat,aa(nat,bool,ord_less(nat,Xa2),Mi2),Mi2,Xa2)),Ma2))),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),list_update(vEBT_VEBT,TreeList2,vEBT_VEBT_high(if(nat,aa(nat,bool,ord_less(nat,Xa2),Mi2),Mi2,Xa2),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),vEBT_vebt_insert(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),vEBT_VEBT_high(if(nat,aa(nat,bool,ord_less(nat,Xa2),Mi2),Mi2,Xa2),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_VEBT_low(if(nat,aa(nat,bool,ord_less(nat,Xa2),Mi2),Mi2,Xa2),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),if(vEBT_VEBT,vEBT_VEBT_minNull(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),vEBT_VEBT_high(if(nat,aa(nat,bool,ord_less(nat,Xa2),Mi2),Mi2,Xa2),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),vEBT_vebt_insert(Summary2,vEBT_VEBT_high(if(nat,aa(nat,bool,ord_less(nat,Xa2),Mi2),Mi2,Xa2),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),Summary2)),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary2)) ) ) ) ) ) ) ) ).

% vebt_insert.elims
tff(fact_4903_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,X: nat] : vEBT_T_i_n_s_e_r_t(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList,Summary),X) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,aa(num,num,bit1,bit0(bit0(one2)))))),if(nat,fconj(aa(nat,bool,ord_less(nat,vEBT_VEBT_high(if(nat,aa(nat,bool,ord_less(nat,X),Mi),Mi,X),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),aa(bool,bool,fNot,fdisj(aa(nat,bool,fequal(nat,X),Mi),aa(nat,bool,fequal(nat,X),Ma)))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_T_i_n_s_e_r_t(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(if(nat,aa(nat,bool,ord_less(nat,X),Mi),Mi,X),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_VEBT_low(if(nat,aa(nat,bool,ord_less(nat,X),Mi),Mi,X),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),vEBT_T_m_i_n_N_u_l_l(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(if(nat,aa(nat,bool,ord_less(nat,X),Mi),Mi,X),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))))),if(nat,vEBT_VEBT_minNull(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(if(nat,aa(nat,bool,ord_less(nat,X),Mi),Mi,X),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),vEBT_T_i_n_s_e_r_t(Summary,vEBT_VEBT_high(if(nat,aa(nat,bool,ord_less(nat,X),Mi),Mi,X),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,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)
tff(fact_4904_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H_Oelims,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: nat] :
      ( ( vEBT_T_i_n_s_e_r_t2(X,Xa2) = Y )
     => ( ( ? [A4: bool,B3: bool] : X = vEBT_Leaf(A4,B3)
         => ( Y != one_one(nat) ) )
       => ( ( ? [Info2: option(product_prod(nat,nat)),Ts2: list(vEBT_VEBT),S2: vEBT_VEBT] : X = vEBT_Node(Info2,zero_zero(nat),Ts2,S2)
           => ( Y != one_one(nat) ) )
         => ( ( ? [Info2: option(product_prod(nat,nat)),Ts2: list(vEBT_VEBT),S2: vEBT_VEBT] : X = vEBT_Node(Info2,aa(nat,nat,suc,zero_zero(nat)),Ts2,S2)
             => ( Y != one_one(nat) ) )
           => ( ( ? [V3: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] : X = vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,V3)),TreeList2,Summary2)
               => ( Y != one_one(nat) ) )
             => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                    ( ( X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary2) )
                   => ( Y != if(nat,fconj(aa(nat,bool,ord_less(nat,vEBT_VEBT_high(if(nat,aa(nat,bool,ord_less(nat,Xa2),Mi2),Mi2,Xa2),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),aa(bool,bool,fNot,fdisj(aa(nat,bool,fequal(nat,Xa2),Mi2),aa(nat,bool,fequal(nat,Xa2),Ma2)))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_T_i_n_s_e_r_t2(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),vEBT_VEBT_high(if(nat,aa(nat,bool,ord_less(nat,Xa2),Mi2),Mi2,Xa2),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_VEBT_low(if(nat,aa(nat,bool,ord_less(nat,Xa2),Mi2),Mi2,Xa2),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),if(nat,vEBT_VEBT_minNull(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),vEBT_VEBT_high(if(nat,aa(nat,bool,ord_less(nat,Xa2),Mi2),Mi2,Xa2),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),vEBT_T_i_n_s_e_r_t2(Summary2,vEBT_VEBT_high(if(nat,aa(nat,bool,ord_less(nat,Xa2),Mi2),Mi2,Xa2),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,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
tff(fact_4905_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,X: nat] : vEBT_T_i_n_s_e_r_t2(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList,Summary),X) = if(nat,fconj(aa(nat,bool,ord_less(nat,vEBT_VEBT_high(if(nat,aa(nat,bool,ord_less(nat,X),Mi),Mi,X),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),aa(bool,bool,fNot,fdisj(aa(nat,bool,fequal(nat,X),Mi),aa(nat,bool,fequal(nat,X),Ma)))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_T_i_n_s_e_r_t2(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(if(nat,aa(nat,bool,ord_less(nat,X),Mi),Mi,X),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_VEBT_low(if(nat,aa(nat,bool,ord_less(nat,X),Mi),Mi,X),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),if(nat,vEBT_VEBT_minNull(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(if(nat,aa(nat,bool,ord_less(nat,X),Mi),Mi,X),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),vEBT_T_i_n_s_e_r_t2(Summary,vEBT_VEBT_high(if(nat,aa(nat,bool,ord_less(nat,X),Mi),Mi,X),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,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)
tff(fact_4906_T_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_Oelims,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: nat] :
      ( ( vEBT_T_d_e_l_e_t_e(X,Xa2) = Y )
     => ( ( ? [A4: bool,B3: bool] : X = vEBT_Leaf(A4,B3)
         => ( ( Xa2 = zero_zero(nat) )
           => ( Y != one_one(nat) ) ) )
       => ( ( ? [A4: bool,B3: bool] : X = vEBT_Leaf(A4,B3)
           => ( ( Xa2 = aa(nat,nat,suc,zero_zero(nat)) )
             => ( Y != one_one(nat) ) ) )
         => ( ( ? [A4: bool,B3: bool] : X = vEBT_Leaf(A4,B3)
             => ( ? [N2: nat] : Xa2 = aa(nat,nat,suc,aa(nat,nat,suc,N2))
               => ( Y != one_one(nat) ) ) )
           => ( ( ? [Deg2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] : X = vEBT_Node(none(product_prod(nat,nat)),Deg2,TreeList2,Summary2)
               => ( Y != one_one(nat) ) )
             => ( ( ? [Mi2: nat,Ma2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] : X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),zero_zero(nat),TreeList2,Summary2)
                 => ( Y != one_one(nat) ) )
               => ( ( ? [Mi2: nat,Ma2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] : X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,zero_zero(nat)),TreeList2,Summary2)
                   => ( Y != one_one(nat) ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                        ( ( X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary2) )
                       => ( Y != 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) ) ) ) ) ) ) ) ) ).

% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.elims
tff(fact_4907_max__enat__simps_I3_J,axiom,
    ! [Q2: extended_enat] : aa(extended_enat,extended_enat,ord_max(extended_enat,zero_zero(extended_enat)),Q2) = Q2 ).

% max_enat_simps(3)
tff(fact_4908_max__enat__simps_I2_J,axiom,
    ! [Q2: extended_enat] : aa(extended_enat,extended_enat,ord_max(extended_enat,Q2),zero_zero(extended_enat)) = Q2 ).

% max_enat_simps(2)
tff(fact_4909_max__lt,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A),C2: word(A)] : aa(word(A),nat,semiring_1_unsigned(A,nat),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),aa(word(A),word(A),ord_max(word(A),A2),B2)),C2)) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),aa(word(A),word(A),ord_max(word(A),A2),B2))),aa(word(A),nat,semiring_1_unsigned(A,nat),C2)) ) ).

% max_lt
tff(fact_4910_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: bool,B2: bool,X: nat] : vEBT_T_i_n_s_e_r_t2(vEBT_Leaf(A2,B2),X) = one_one(nat) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.simps(1)
tff(fact_4911_T_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062x_092_060_094sub_062t_Osimps_I1_J,axiom,
    ! [A2: bool,B2: bool] : vEBT_T_m_a_x_t(vEBT_Leaf(A2,B2)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),if(nat,B2,one_one(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),one_one(nat)))) ).

% T\<^sub>m\<^sub>a\<^sub>x\<^sub>t.simps(1)
tff(fact_4912_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)
tff(fact_4913_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)
tff(fact_4914_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),S3: vEBT_VEBT,X: nat] : vEBT_T_i_n_s_e_r_t2(vEBT_Node(Info,zero_zero(nat),Ts,S3),X) = one_one(nat) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.simps(2)
tff(fact_4915_maxt__bound,axiom,
    ! [T: vEBT_VEBT] : pp(aa(nat,bool,ord_less_eq(nat,vEBT_T_m_a_x_t(T)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2)))) ).

% maxt_bound
tff(fact_4916_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(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,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)
tff(fact_4917_mint__bound,axiom,
    ! [T: vEBT_VEBT] : pp(aa(nat,bool,ord_less_eq(nat,vEBT_T_m_i_n_t(T)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2)))) ).

% mint_bound
tff(fact_4918_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t_Osimps_I1_J,axiom,
    ! [A2: bool,B2: bool] : vEBT_T_m_i_n_t(vEBT_Leaf(A2,B2)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),if(nat,A2,zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),one_one(nat)))) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>t.simps(1)
tff(fact_4919_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(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,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)
tff(fact_4920_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),S3: vEBT_VEBT,X: nat] : vEBT_T_i_n_s_e_r_t2(vEBT_Node(Info,aa(nat,nat,suc,zero_zero(nat)),Ts,S3),X) = one_one(nat) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.simps(3)
tff(fact_4921_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H_Osimps_I4_J,axiom,
    ! [V: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,X: nat] : vEBT_T_i_n_s_e_r_t2(vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,V)),TreeList,Summary),X) = one_one(nat) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.simps(4)
tff(fact_4922_insersimp_H,axiom,
    ! [T: vEBT_VEBT,N: nat,Y: nat] :
      ( vEBT_invar_vebt(T,N)
     => ( ~ ? [X_1: nat] : pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,T),X_1))
       => pp(aa(nat,bool,ord_less_eq(nat,vEBT_T_i_n_s_e_r_t2(T,Y)),one_one(nat))) ) ) ).

% insersimp'
tff(fact_4923_insertsimp_H,axiom,
    ! [T: vEBT_VEBT,N: nat,L: nat] :
      ( vEBT_invar_vebt(T,N)
     => ( pp(vEBT_VEBT_minNull(T))
       => pp(aa(nat,bool,ord_less_eq(nat,vEBT_T_i_n_s_e_r_t2(T,L)),one_one(nat))) ) ) ).

% insertsimp'
tff(fact_4924_insert_H__bound__height,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( vEBT_invar_vebt(T,N)
     => pp(aa(nat,bool,ord_less_eq(nat,vEBT_T_i_n_s_e_r_t2(T,X)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),aa(vEBT_VEBT,nat,vEBT_VEBT_height,T)))) ) ).

% insert'_bound_height
tff(fact_4925_T_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062x_092_060_094sub_062t_Oelims,axiom,
    ! [X: vEBT_VEBT,Y: nat] :
      ( ( vEBT_T_m_a_x_t(X) = Y )
     => ( ! [A4: bool,B3: bool] :
            ( ( X = vEBT_Leaf(A4,B3) )
           => ( Y != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),if(nat,B3,one_one(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),one_one(nat)))) ) )
       => ( ( ? [Uu2: nat,Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT] : X = vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw2)
           => ( Y != one_one(nat) ) )
         => ~ ( ? [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list(vEBT_VEBT),Uz2: vEBT_VEBT] : X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),Ux2,Uy2,Uz2)
             => ( Y != one_one(nat) ) ) ) ) ) ).

% T\<^sub>m\<^sub>a\<^sub>x\<^sub>t.elims
tff(fact_4926_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t_Oelims,axiom,
    ! [X: vEBT_VEBT,Y: nat] :
      ( ( vEBT_T_m_i_n_t(X) = Y )
     => ( ! [A4: bool] :
            ( ? [B3: bool] : X = vEBT_Leaf(A4,B3)
           => ( Y != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),if(nat,A4,zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),one_one(nat)))) ) )
       => ( ( ? [Uu2: nat,Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT] : X = vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw2)
           => ( Y != one_one(nat) ) )
         => ~ ( ? [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list(vEBT_VEBT),Uz2: vEBT_VEBT] : X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),Ux2,Uy2,Uz2)
             => ( Y != one_one(nat) ) ) ) ) ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>t.elims
tff(fact_4927_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,X: nat] : vEBT_T_p_r_e_d(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList,Summary),X) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),if(nat,aa(nat,bool,ord_less(nat,Ma),X),one_one(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(aa(num,num,bit1,bit0(one2))))),one_one(nat))),if(nat,aa(nat,bool,ord_less(nat,vEBT_VEBT_high(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),vEBT_T_m_i_n_t(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2)))),if(nat,fconj(aa(bool,bool,fNot,aa(option(nat),bool,fequal(option(nat),vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),none(nat))),vEBT_VEBT_greater(aa(nat,option(nat),some(nat),vEBT_VEBT_low(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(one2)))),vEBT_T_p_r_e_d(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_VEBT_low(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),vEBT_T_p_r_e_d(Summary,vEBT_VEBT_high(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),one_one(nat))),if(nat,aa(option(nat),bool,fequal(option(nat),vEBT_vebt_pred(Summary,vEBT_VEBT_high(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),none(nat)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),one_one(nat)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(one2)))),vEBT_T_m_a_x_t(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),the2(nat,vEBT_vebt_pred(Summary,vEBT_VEBT_high(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2))))))))))))),one_one(nat))))) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.simps(7)
tff(fact_4928_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,X: nat] : vEBT_T_s_u_c_c(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList,Summary),X) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),if(nat,aa(nat,bool,ord_less(nat,X),Mi),one_one(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(aa(num,num,bit1,bit0(one2))))),if(nat,aa(nat,bool,ord_less(nat,vEBT_VEBT_high(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),vEBT_T_m_a_x_t(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2))),if(nat,fconj(aa(bool,bool,fNot,aa(option(nat),bool,fequal(option(nat),vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),none(nat))),vEBT_VEBT_less(aa(nat,option(nat),some(nat),vEBT_VEBT_low(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(one2)))),vEBT_T_s_u_c_c(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_VEBT_low(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),vEBT_T_s_u_c_c(Summary,vEBT_VEBT_high(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),one_one(nat))),if(nat,aa(option(nat),bool,fequal(option(nat),vEBT_vebt_succ(Summary,vEBT_VEBT_high(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),none(nat)),one_one(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(one2)))),vEBT_T_m_i_n_t(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),the2(nat,vEBT_vebt_succ(Summary,vEBT_VEBT_high(X,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))))))))))),one_one(nat))))) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.simps(6)
tff(fact_4929_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Oelims,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: nat] :
      ( ( vEBT_T_p_r_e_d(X,Xa2) = Y )
     => ( ( ? [Uu2: bool,Uv2: bool] : X = vEBT_Leaf(Uu2,Uv2)
         => ( ( Xa2 = zero_zero(nat) )
           => ( Y != one_one(nat) ) ) )
       => ( ( ? [A4: bool,Uw2: bool] : X = vEBT_Leaf(A4,Uw2)
           => ( ( Xa2 = aa(nat,nat,suc,zero_zero(nat)) )
             => ( Y != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),one_one(nat)) ) ) )
         => ( ! [A4: bool,B3: bool] :
                ( ( X = vEBT_Leaf(A4,B3) )
               => ( ? [Va2: nat] : Xa2 = aa(nat,nat,suc,aa(nat,nat,suc,Va2))
                 => ( Y != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),if(nat,B3,one_one(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),one_one(nat)))) ) ) )
           => ( ( ? [Uy2: nat,Uz2: list(vEBT_VEBT),Va3: vEBT_VEBT] : X = vEBT_Node(none(product_prod(nat,nat)),Uy2,Uz2,Va3)
               => ( Y != one_one(nat) ) )
             => ( ( ? [V3: product_prod(nat,nat),Vd2: list(vEBT_VEBT),Ve2: vEBT_VEBT] : X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),zero_zero(nat),Vd2,Ve2)
                 => ( Y != one_one(nat) ) )
               => ( ( ? [V3: product_prod(nat,nat),Vh2: list(vEBT_VEBT),Vi2: vEBT_VEBT] : X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),aa(nat,nat,suc,zero_zero(nat)),Vh2,Vi2)
                   => ( Y != one_one(nat) ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                        ( ( X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary2) )
                       => ( Y != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),if(nat,aa(nat,bool,ord_less(nat,Ma2),Xa2),one_one(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(aa(num,num,bit1,bit0(one2))))),one_one(nat))),if(nat,aa(nat,bool,ord_less(nat,vEBT_VEBT_high(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),vEBT_T_m_i_n_t(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),vEBT_VEBT_high(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2)))),if(nat,fconj(aa(bool,bool,fNot,aa(option(nat),bool,fequal(option(nat),vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),vEBT_VEBT_high(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),none(nat))),vEBT_VEBT_greater(aa(nat,option(nat),some(nat),vEBT_VEBT_low(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),vEBT_VEBT_high(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(one2)))),vEBT_T_p_r_e_d(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),vEBT_VEBT_high(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_VEBT_low(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),vEBT_T_p_r_e_d(Summary2,vEBT_VEBT_high(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),one_one(nat))),if(nat,aa(option(nat),bool,fequal(option(nat),vEBT_vebt_pred(Summary2,vEBT_VEBT_high(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),none(nat)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),one_one(nat)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(one2)))),vEBT_T_m_a_x_t(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),the2(nat,vEBT_vebt_pred(Summary2,vEBT_VEBT_high(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))))))))))))),one_one(nat))))) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.elims
tff(fact_4930_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Oelims,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: nat] :
      ( ( vEBT_T_s_u_c_c(X,Xa2) = Y )
     => ( ( ? [Uu2: bool,B3: bool] : X = vEBT_Leaf(Uu2,B3)
         => ( ( Xa2 = zero_zero(nat) )
           => ( Y != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),one_one(nat)) ) ) )
       => ( ( ? [Uv2: bool,Uw2: bool] : X = vEBT_Leaf(Uv2,Uw2)
           => ( ? [N2: nat] : Xa2 = aa(nat,nat,suc,N2)
             => ( Y != one_one(nat) ) ) )
         => ( ( ? [Ux2: nat,Uy2: list(vEBT_VEBT),Uz2: vEBT_VEBT] : X = vEBT_Node(none(product_prod(nat,nat)),Ux2,Uy2,Uz2)
             => ( Y != one_one(nat) ) )
           => ( ( ? [V3: product_prod(nat,nat),Vc: list(vEBT_VEBT),Vd2: vEBT_VEBT] : X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),zero_zero(nat),Vc,Vd2)
               => ( Y != one_one(nat) ) )
             => ( ( ? [V3: product_prod(nat,nat),Vg2: list(vEBT_VEBT),Vh2: vEBT_VEBT] : X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),aa(nat,nat,suc,zero_zero(nat)),Vg2,Vh2)
                 => ( Y != one_one(nat) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                      ( ( X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary2) )
                     => ( Y != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),if(nat,aa(nat,bool,ord_less(nat,Xa2),Mi2),one_one(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(aa(num,num,bit1,bit0(one2))))),if(nat,aa(nat,bool,ord_less(nat,vEBT_VEBT_high(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),vEBT_T_m_a_x_t(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),vEBT_VEBT_high(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2))),if(nat,fconj(aa(bool,bool,fNot,aa(option(nat),bool,fequal(option(nat),vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),vEBT_VEBT_high(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),none(nat))),vEBT_VEBT_less(aa(nat,option(nat),some(nat),vEBT_VEBT_low(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),vEBT_VEBT_high(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(one2)))),vEBT_T_s_u_c_c(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),vEBT_VEBT_high(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_VEBT_low(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),vEBT_T_s_u_c_c(Summary2,vEBT_VEBT_high(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),one_one(nat))),if(nat,aa(option(nat),bool,fequal(option(nat),vEBT_vebt_succ(Summary2,vEBT_VEBT_high(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),none(nat)),one_one(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(one2)))),vEBT_T_m_i_n_t(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),the2(nat,vEBT_vebt_succ(Summary2,vEBT_VEBT_high(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))))))))))),one_one(nat))))) ) ) ) ) ) ) ) ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.elims
tff(fact_4931_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,X: nat] : vEBT_T_d_e_l_e_t_e(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList,Summary),X) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2))),if(nat,fdisj(aa(nat,bool,ord_less(nat,X),Mi),aa(nat,bool,ord_less(nat,Ma),X)),one_one(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2))),if(nat,fconj(aa(nat,bool,fequal(nat,X),Mi),aa(nat,bool,fequal(nat,X),Ma)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,bit0(aa(num,num,bit1,one2))))),if(nat,aa(nat,bool,fequal(nat,X),Mi),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_T_m_i_n_t(Summary)),vEBT_T_m_i_n_t(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),the2(nat,vEBT_vebt_mint(Summary)))))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,aa(num,num,bit1,one2)))),one_one(nat)))),one_one(nat))),if(nat,aa(nat,bool,ord_less(nat,vEBT_VEBT_high(if(nat,aa(nat,bool,fequal(nat,X),Mi),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),the2(nat,vEBT_vebt_mint(Summary))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),the2(nat,vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),the2(nat,vEBT_vebt_mint(Summary)))))),X),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(one2)))),vEBT_T_d_e_l_e_t_e(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(if(nat,aa(nat,bool,fequal(nat,X),Mi),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),the2(nat,vEBT_vebt_mint(Summary))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),the2(nat,vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),the2(nat,vEBT_vebt_mint(Summary)))))),X),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_VEBT_low(if(nat,aa(nat,bool,fequal(nat,X),Mi),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),the2(nat,vEBT_vebt_mint(Summary))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),the2(nat,vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),the2(nat,vEBT_vebt_mint(Summary)))))),X),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),vEBT_T_m_i_n_N_u_l_l(vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(if(nat,aa(nat,bool,fequal(nat,X),Mi),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),the2(nat,vEBT_vebt_mint(Summary))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),the2(nat,vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),the2(nat,vEBT_vebt_mint(Summary)))))),X),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_VEBT_low(if(nat,aa(nat,bool,fequal(nat,X),Mi),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),the2(nat,vEBT_vebt_mint(Summary))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),the2(nat,vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),the2(nat,vEBT_vebt_mint(Summary)))))),X),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))))),if(nat,vEBT_VEBT_minNull(vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(if(nat,aa(nat,bool,fequal(nat,X),Mi),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),the2(nat,vEBT_vebt_mint(Summary))),aa(nat,nat,aa(nat,fun(na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).

% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.simps(7)
tff(fact_4932_T_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_Opelims,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: nat] :
      ( ( vEBT_T_d_e_l_e_t_e(X,Xa2) = Y )
     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_T8441311223069195367_e_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),X),Xa2))
       => ( ! [A4: bool,B3: bool] :
              ( ( X = vEBT_Leaf(A4,B3) )
             => ( ( Xa2 = zero_zero(nat) )
               => ( ( Y = one_one(nat) )
                 => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T8441311223069195367_e_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf(A4,B3)),zero_zero(nat))) ) ) )
         => ( ! [A4: bool,B3: bool] :
                ( ( X = vEBT_Leaf(A4,B3) )
               => ( ( Xa2 = aa(nat,nat,suc,zero_zero(nat)) )
                 => ( ( Y = one_one(nat) )
                   => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T8441311223069195367_e_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf(A4,B3)),aa(nat,nat,suc,zero_zero(nat)))) ) ) )
           => ( ! [A4: bool,B3: bool] :
                  ( ( X = vEBT_Leaf(A4,B3) )
                 => ! [N2: nat] :
                      ( ( Xa2 = aa(nat,nat,suc,aa(nat,nat,suc,N2)) )
                     => ( ( Y = one_one(nat) )
                       => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T8441311223069195367_e_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf(A4,B3)),aa(nat,nat,suc,aa(nat,nat,suc,N2)))) ) ) )
             => ( ! [Deg2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                    ( ( X = vEBT_Node(none(product_prod(nat,nat)),Deg2,TreeList2,Summary2) )
                   => ( ( Y = one_one(nat) )
                     => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T8441311223069195367_e_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),Deg2,TreeList2,Summary2)),Xa2)) ) )
               => ( ! [Mi2: nat,Ma2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                      ( ( X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),zero_zero(nat),TreeList2,Summary2) )
                     => ( ( Y = one_one(nat) )
                       => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T8441311223069195367_e_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),zero_zero(nat),TreeList2,Summary2)),Xa2)) ) )
                 => ( ! [Mi2: nat,Ma2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                        ( ( X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,zero_zero(nat)),TreeList2,Summary2) )
                       => ( ( Y = one_one(nat) )
                         => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T8441311223069195367_e_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,zero_zero(nat)),TreeList2,Summary2)),Xa2)) ) )
                   => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                          ( ( X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary2) )
                         => ( ( Y = 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e2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),the2(nat,vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),the2(nat,vEBT_vebt_mint(Summary2)))))),Xa2),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),vEBT_VEBT_high(if(nat,aa(nat,bool,fequal(nat,Xa2),Mi2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),the2(nat,vEBT_vebt_mint(Summary2))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),the2(nat,vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),the2(nat,vEBT_vebt_mint(Summary2)))))),Xa2),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),one_one(nat)))))),one_one(nat))))))) )
                           => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T8441311223069195367_e_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary2)),Xa2)) ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.pelims
tff(fact_4933_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Opelims,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: nat] :
      ( ( vEBT_T_s_u_c_c(X,Xa2) = Y )
     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_T_s_u_c_c_rel2,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),X),Xa2))
       => ( ! [Uu2: bool,B3: bool] :
              ( ( X = vEBT_Leaf(Uu2,B3) )
             => ( ( Xa2 = zero_zero(nat) )
               => ( ( Y = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),one_one(nat)) )
                 => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T_s_u_c_c_rel2,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf(Uu2,B3)),zero_zero(nat))) ) ) )
         => ( ! [Uv2: bool,Uw2: bool] :
                ( ( X = vEBT_Leaf(Uv2,Uw2) )
               => ! [N2: nat] :
                    ( ( Xa2 = aa(nat,nat,suc,N2) )
                   => ( ( Y = one_one(nat) )
                     => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T_s_u_c_c_rel2,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf(Uv2,Uw2)),aa(nat,nat,suc,N2))) ) ) )
           => ( ! [Ux2: nat,Uy2: list(vEBT_VEBT),Uz2: vEBT_VEBT] :
                  ( ( X = vEBT_Node(none(product_prod(nat,nat)),Ux2,Uy2,Uz2) )
                 => ( ( Y = one_one(nat) )
                   => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T_s_u_c_c_rel2,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),Ux2,Uy2,Uz2)),Xa2)) ) )
             => ( ! [V3: product_prod(nat,nat),Vc: list(vEBT_VEBT),Vd2: vEBT_VEBT] :
                    ( ( X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),zero_zero(nat),Vc,Vd2) )
                   => ( ( Y = one_one(nat) )
                     => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T_s_u_c_c_rel2,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),zero_zero(nat),Vc,Vd2)),Xa2)) ) )
               => ( ! [V3: product_prod(nat,nat),Vg2: list(vEBT_VEBT),Vh2: vEBT_VEBT] :
                      ( ( X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),aa(nat,nat,suc,zero_zero(nat)),Vg2,Vh2) )
                     => ( ( Y = one_one(nat) )
                       => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T_s_u_c_c_rel2,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),aa(nat,nat,suc,zero_zero(nat)),Vg2,Vh2)),Xa2)) ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                        ( ( X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary2) )
                       => ( ( Y = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),if(nat,aa(nat,bool,ord_less(nat,Xa2),Mi2),one_one(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(aa(num,num,bit1,bit0(one2))))),if(nat,aa(nat,bool,ord_less(nat,vEBT_VEBT_high(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),vEBT_T_m_a_x_t(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),vEBT_VEBT_high(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2))),if(nat,fconj(aa(bool,bool,fNot,aa(option(nat),bool,fequal(option(nat),vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),vEBT_VEBT_high(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),none(nat))),vEBT_VEBT_less(aa(nat,option(nat),some(nat),vEBT_VEBT_low(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),vEBT_VEBT_high(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(one2)))),vEBT_T_s_u_c_c(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),vEBT_VEBT_high(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_VEBT_low(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),vEBT_T_s_u_c_c(Summary2,vEBT_VEBT_high(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),one_one(nat))),if(nat,aa(option(nat),bool,fequal(option(nat),vEBT_vebt_succ(Summary2,vEBT_VEBT_high(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),none(nat)),one_one(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(one2)))),vEBT_T_m_i_n_t(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),the2(nat,vEBT_vebt_succ(Summary2,vEBT_VEBT_high(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))))))))))),one_one(nat))))) )
                         => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T_s_u_c_c_rel2,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary2)),Xa2)) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.pelims
tff(fact_4934_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Opelims,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: nat] :
      ( ( vEBT_T_p_r_e_d(X,Xa2) = Y )
     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_T_p_r_e_d_rel2,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),X),Xa2))
       => ( ! [Uu2: bool,Uv2: bool] :
              ( ( X = vEBT_Leaf(Uu2,Uv2) )
             => ( ( Xa2 = zero_zero(nat) )
               => ( ( Y = one_one(nat) )
                 => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T_p_r_e_d_rel2,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf(Uu2,Uv2)),zero_zero(nat))) ) ) )
         => ( ! [A4: bool,Uw2: bool] :
                ( ( X = vEBT_Leaf(A4,Uw2) )
               => ( ( Xa2 = aa(nat,nat,suc,zero_zero(nat)) )
                 => ( ( Y = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),one_one(nat)) )
                   => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T_p_r_e_d_rel2,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf(A4,Uw2)),aa(nat,nat,suc,zero_zero(nat)))) ) ) )
           => ( ! [A4: bool,B3: bool] :
                  ( ( X = vEBT_Leaf(A4,B3) )
                 => ! [Va2: nat] :
                      ( ( Xa2 = aa(nat,nat,suc,aa(nat,nat,suc,Va2)) )
                     => ( ( Y = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),if(nat,B3,one_one(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),one_one(nat)))) )
                       => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T_p_r_e_d_rel2,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf(A4,B3)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)))) ) ) )
             => ( ! [Uy2: nat,Uz2: list(vEBT_VEBT),Va3: vEBT_VEBT] :
                    ( ( X = vEBT_Node(none(product_prod(nat,nat)),Uy2,Uz2,Va3) )
                   => ( ( Y = one_one(nat) )
                     => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T_p_r_e_d_rel2,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),Uy2,Uz2,Va3)),Xa2)) ) )
               => ( ! [V3: product_prod(nat,nat),Vd2: list(vEBT_VEBT),Ve2: vEBT_VEBT] :
                      ( ( X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),zero_zero(nat),Vd2,Ve2) )
                     => ( ( Y = one_one(nat) )
                       => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T_p_r_e_d_rel2,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),zero_zero(nat),Vd2,Ve2)),Xa2)) ) )
                 => ( ! [V3: product_prod(nat,nat),Vh2: list(vEBT_VEBT),Vi2: vEBT_VEBT] :
                        ( ( X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),aa(nat,nat,suc,zero_zero(nat)),Vh2,Vi2) )
                       => ( ( Y = one_one(nat) )
                         => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T_p_r_e_d_rel2,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),aa(nat,nat,suc,zero_zero(nat)),Vh2,Vi2)),Xa2)) ) )
                   => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                          ( ( X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary2) )
                         => ( ( Y = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),if(nat,aa(nat,bool,ord_less(nat,Ma2),Xa2),one_one(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(aa(num,num,bit1,bit0(one2))))),one_one(nat))),if(nat,aa(nat,bool,ord_less(nat,vEBT_VEBT_high(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),vEBT_T_m_i_n_t(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),vEBT_VEBT_high(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2)))),if(nat,fconj(aa(bool,bool,fNot,aa(option(nat),bool,fequal(option(nat),vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),vEBT_VEBT_high(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),none(nat))),vEBT_VEBT_greater(aa(nat,option(nat),some(nat),vEBT_VEBT_low(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),vEBT_VEBT_high(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(one2)))),vEBT_T_p_r_e_d(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),vEBT_VEBT_high(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_VEBT_low(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),vEBT_T_p_r_e_d(Summary2,vEBT_VEBT_high(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),one_one(nat))),if(nat,aa(option(nat),bool,fequal(option(nat),vEBT_vebt_pred(Summary2,vEBT_VEBT_high(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),none(nat)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),one_one(nat)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(one2)))),vEBT_T_m_a_x_t(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),the2(nat,vEBT_vebt_pred(Summary2,vEBT_VEBT_high(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))))))))))))),one_one(nat))))) )
                           => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T_p_r_e_d_rel2,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary2)),Xa2)) ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.pelims
tff(fact_4935_vebt__succ_Opelims,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: option(nat)] :
      ( ( vEBT_vebt_succ(X,Xa2) = Y )
     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_succ_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),X),Xa2))
       => ( ! [Uu2: bool,B3: bool] :
              ( ( X = vEBT_Leaf(Uu2,B3) )
             => ( ( Xa2 = zero_zero(nat) )
               => ( ( ( pp(B3)
                     => ( Y = aa(nat,option(nat),some(nat),one_one(nat)) ) )
                    & ( ~ pp(B3)
                     => ( Y = none(nat) ) ) )
                 => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_succ_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf(Uu2,B3)),zero_zero(nat))) ) ) )
         => ( ! [Uv2: bool,Uw2: bool] :
                ( ( X = vEBT_Leaf(Uv2,Uw2) )
               => ! [N2: nat] :
                    ( ( Xa2 = aa(nat,nat,suc,N2) )
                   => ( ( Y = none(nat) )
                     => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_succ_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf(Uv2,Uw2)),aa(nat,nat,suc,N2))) ) ) )
           => ( ! [Ux2: nat,Uy2: list(vEBT_VEBT),Uz2: vEBT_VEBT] :
                  ( ( X = vEBT_Node(none(product_prod(nat,nat)),Ux2,Uy2,Uz2) )
                 => ( ( Y = none(nat) )
                   => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_succ_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),Ux2,Uy2,Uz2)),Xa2)) ) )
             => ( ! [V3: product_prod(nat,nat),Vc: list(vEBT_VEBT),Vd2: vEBT_VEBT] :
                    ( ( X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),zero_zero(nat),Vc,Vd2) )
                   => ( ( Y = none(nat) )
                     => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_succ_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),zero_zero(nat),Vc,Vd2)),Xa2)) ) )
               => ( ! [V3: product_prod(nat,nat),Vg2: list(vEBT_VEBT),Vh2: vEBT_VEBT] :
                      ( ( X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),aa(nat,nat,suc,zero_zero(nat)),Vg2,Vh2) )
                     => ( ( Y = none(nat) )
                       => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_succ_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),aa(nat,nat,suc,zero_zero(nat)),Vg2,Vh2)),Xa2)) ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                        ( ( X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary2) )
                       => ( ( ( pp(aa(nat,bool,ord_less(nat,Xa2),Mi2))
                             => ( Y = aa(nat,option(nat),some(nat),Mi2) ) )
                            & ( ~ pp(aa(nat,bool,ord_less(nat,Xa2),Mi2))
                             => ( Y = if(option(nat),aa(nat,bool,ord_less(nat,vEBT_VEBT_high(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),if(option(nat),fconj(aa(bool,bool,fNot,aa(option(nat),bool,fequal(option(nat),vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),vEBT_VEBT_high(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),none(nat))),vEBT_VEBT_less(aa(nat,option(nat),some(nat),vEBT_VEBT_low(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),vEBT_VEBT_high(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))))),aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_add,aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_mul,aa(nat,option(nat),some(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,option(nat),some(nat),vEBT_VEBT_high(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),vEBT_vebt_succ(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),vEBT_VEBT_high(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_VEBT_low(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),if(option(nat),aa(option(nat),bool,fequal(option(nat),vEBT_vebt_succ(Summary2,vEBT_VEBT_high(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),none(nat)),none(nat),aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_add,aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_mul,aa(nat,option(nat),some(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),vEBT_vebt_succ(Summary2,vEBT_VEBT_high(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),the2(nat,vEBT_vebt_succ(Summary2,vEBT_VEBT_high(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))))))))))),none(nat)) ) ) )
                         => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_succ_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary2)),Xa2)) ) ) ) ) ) ) ) ) ) ).

% vebt_succ.pelims
tff(fact_4936_vebt__pred_Opelims,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: option(nat)] :
      ( ( vEBT_vebt_pred(X,Xa2) = Y )
     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_pred_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),X),Xa2))
       => ( ! [Uu2: bool,Uv2: bool] :
              ( ( X = vEBT_Leaf(Uu2,Uv2) )
             => ( ( Xa2 = zero_zero(nat) )
               => ( ( Y = none(nat) )
                 => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_pred_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf(Uu2,Uv2)),zero_zero(nat))) ) ) )
         => ( ! [A4: bool,Uw2: bool] :
                ( ( X = vEBT_Leaf(A4,Uw2) )
               => ( ( Xa2 = aa(nat,nat,suc,zero_zero(nat)) )
                 => ( ( ( pp(A4)
                       => ( Y = aa(nat,option(nat),some(nat),zero_zero(nat)) ) )
                      & ( ~ pp(A4)
                       => ( Y = none(nat) ) ) )
                   => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_pred_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf(A4,Uw2)),aa(nat,nat,suc,zero_zero(nat)))) ) ) )
           => ( ! [A4: bool,B3: bool] :
                  ( ( X = vEBT_Leaf(A4,B3) )
                 => ! [Va2: nat] :
                      ( ( Xa2 = aa(nat,nat,suc,aa(nat,nat,suc,Va2)) )
                     => ( ( ( pp(B3)
                           => ( Y = aa(nat,option(nat),some(nat),one_one(nat)) ) )
                          & ( ~ pp(B3)
                           => ( ( pp(A4)
                               => ( Y = aa(nat,option(nat),some(nat),zero_zero(nat)) ) )
                              & ( ~ pp(A4)
                               => ( Y = none(nat) ) ) ) ) )
                       => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_pred_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf(A4,B3)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)))) ) ) )
             => ( ! [Uy2: nat,Uz2: list(vEBT_VEBT),Va3: vEBT_VEBT] :
                    ( ( X = vEBT_Node(none(product_prod(nat,nat)),Uy2,Uz2,Va3) )
                   => ( ( Y = none(nat) )
                     => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_pred_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),Uy2,Uz2,Va3)),Xa2)) ) )
               => ( ! [V3: product_prod(nat,nat),Vd2: list(vEBT_VEBT),Ve2: vEBT_VEBT] :
                      ( ( X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),zero_zero(nat),Vd2,Ve2) )
                     => ( ( Y = none(nat) )
                       => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_pred_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),zero_zero(nat),Vd2,Ve2)),Xa2)) ) )
                 => ( ! [V3: product_prod(nat,nat),Vh2: list(vEBT_VEBT),Vi2: vEBT_VEBT] :
                        ( ( X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),aa(nat,nat,suc,zero_zero(nat)),Vh2,Vi2) )
                       => ( ( Y = none(nat) )
                         => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_pred_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),aa(nat,nat,suc,zero_zero(nat)),Vh2,Vi2)),Xa2)) ) )
                   => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                          ( ( X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary2) )
                         => ( ( ( pp(aa(nat,bool,ord_less(nat,Ma2),Xa2))
                               => ( Y = aa(nat,option(nat),some(nat),Ma2) ) )
                              & ( ~ pp(aa(nat,bool,ord_less(nat,Ma2),Xa2))
                               => ( Y = if(option(nat),aa(nat,bool,ord_less(nat,vEBT_VEBT_high(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),if(option(nat),fconj(aa(bool,bool,fNot,aa(option(nat),bool,fequal(option(nat),vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),vEBT_VEBT_high(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),none(nat))),vEBT_VEBT_greater(aa(nat,option(nat),some(nat),vEBT_VEBT_low(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),vEBT_VEBT_high(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))))),aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_add,aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_mul,aa(nat,option(nat),some(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,option(nat),some(nat),vEBT_VEBT_high(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),vEBT_vebt_pred(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),vEBT_VEBT_high(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_VEBT_low(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),if(option(nat),aa(option(nat),bool,fequal(option(nat),vEBT_vebt_pred(Summary2,vEBT_VEBT_high(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),none(nat)),if(option(nat),aa(nat,bool,ord_less(nat,Mi2),Xa2),aa(nat,option(nat),some(nat),Mi2),none(nat)),aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_add,aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_mul,aa(nat,option(nat),some(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),vEBT_vebt_pred(Summary2,vEBT_VEBT_high(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),the2(nat,vEBT_vebt_pred(Summary2,vEBT_VEBT_high(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))))))))))),none(nat)) ) ) )
                           => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_pred_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary2)),Xa2)) ) ) ) ) ) ) ) ) ) ) ).

% vebt_pred.pelims
tff(fact_4937_VEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H_Opelims,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: nat] :
      ( ( vEBT_V1232361888498592333_e_t_e(X,Xa2) = Y )
     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_V6368547301243506412_e_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),X),Xa2))
       => ( ! [A4: bool,B3: bool] :
              ( ( X = vEBT_Leaf(A4,B3) )
             => ( ( Xa2 = zero_zero(nat) )
               => ( ( Y = one_one(nat) )
                 => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_V6368547301243506412_e_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf(A4,B3)),zero_zero(nat))) ) ) )
         => ( ! [A4: bool,B3: bool] :
                ( ( X = vEBT_Leaf(A4,B3) )
               => ( ( Xa2 = aa(nat,nat,suc,zero_zero(nat)) )
                 => ( ( Y = one_one(nat) )
                   => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_V6368547301243506412_e_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf(A4,B3)),aa(nat,nat,suc,zero_zero(nat)))) ) ) )
           => ( ! [A4: bool,B3: bool] :
                  ( ( X = vEBT_Leaf(A4,B3) )
                 => ! [N2: nat] :
                      ( ( Xa2 = aa(nat,nat,suc,aa(nat,nat,suc,N2)) )
                     => ( ( Y = one_one(nat) )
                       => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_V6368547301243506412_e_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf(A4,B3)),aa(nat,nat,suc,aa(nat,nat,suc,N2)))) ) ) )
             => ( ! [Deg2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                    ( ( X = vEBT_Node(none(product_prod(nat,nat)),Deg2,TreeList2,Summary2) )
                   => ( ( Y = one_one(nat) )
                     => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_V6368547301243506412_e_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),Deg2,TreeList2,Summary2)),Xa2)) ) )
               => ( ! [Mi2: nat,Ma2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                      ( ( X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),zero_zero(nat),TreeList2,Summary2) )
                     => ( ( Y = one_one(nat) )
                       => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_V6368547301243506412_e_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),zero_zero(nat),TreeList2,Summary2)),Xa2)) ) )
                 => ( ! [Mi2: nat,Ma2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                        ( ( X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,zero_zero(nat)),TreeList2,Summary2) )
                       => ( ( Y = one_one(nat) )
                         => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_V6368547301243506412_e_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,zero_zero(nat)),TreeList2,Summary2)),Xa2)) ) )
                   => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                          ( ( X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary2) )
                         => ( ( ( ( pp(aa(nat,bool,ord_less(nat,Xa2),Mi2))
                                  | pp(aa(nat,bool,ord_less(nat,Ma2),Xa2)) )
                               => ( Y = one_one(nat) ) )
                              & ( ~ ( pp(aa(nat,bool,ord_less(nat,Xa2),Mi2))
                                    | pp(aa(nat,bool,ord_less(nat,Ma2),Xa2)) )
                               => ( ( ( ( Xa2 = Mi2 )
                                      & ( Xa2 = Ma2 ) )
                                   => ( Y = one_one(nat) ) )
                                  & ( ~ ( ( Xa2 = Mi2 )
                                        & ( Xa2 = Ma2 ) )
                                   => ( Y = if(nat,aa(nat,bool,ord_less(nat,vEBT_VEBT_high(if(nat,aa(nat,bool,fequal(nat,Xa2),Mi2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),the2(nat,vEBT_vebt_mint(Summary2))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),the2(nat,vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),the2(nat,vEBT_vebt_mint(Summary2)))))),Xa2),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_V1232361888498592333_e_t_e(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),vEBT_VEBT_high(if(nat,aa(nat,bool,fequal(nat,Xa2),Mi2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),the2(nat,vEBT_vebt_mint(Summary2))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),the2(nat,vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),the2(nat,vEBT_vebt_mint(Summary2)))))),Xa2),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_VEBT_low(if(nat,aa(nat,bool,fequal(nat,Xa2),Mi2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),the2(nat,vEBT_vebt_mint(Summary2))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),the2(nat,vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),the2(nat,vEBT_vebt_mint(Summary2)))))),Xa2),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),if(nat,vEBT_VEBT_minNull(vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),vEBT_VEBT_high(if(nat,aa(nat,bool,fequal(nat,Xa2),Mi2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),the2(nat,vEBT_vebt_mint(Summary2))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),the2(nat,vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),the2(nat,vEBT_vebt_mint(Summary2)))))),Xa2),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_VEBT_low(if(nat,aa(nat,bool,fequal(nat,Xa2),Mi2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),the2(nat,vEBT_vebt_mint(Summary2))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),the2(nat,vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),the2(nat,vEBT_vebt_mint(Summary2)))))),Xa2),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),vEBT_V1232361888498592333_e_t_e(Summary2,vEBT_VEBT_high(if(nat,aa(nat,bool,fequal(nat,Xa2),Mi2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),the2(nat,vEBT_vebt_mint(Summary2))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),the2(nat,vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),the2(nat,vEBT_vebt_mint(Summary2)))))),Xa2),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),one_one(nat))),one_one(nat)) ) ) ) ) )
                           => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_V6368547301243506412_e_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary2)),Xa2)) ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.pelims
tff(fact_4938_vebt__delete_Opelims,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: vEBT_VEBT] :
      ( ( vEBT_vebt_delete(X,Xa2) = Y )
     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_delete_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),X),Xa2))
       => ( ! [A4: bool,B3: bool] :
              ( ( X = vEBT_Leaf(A4,B3) )
             => ( ( Xa2 = zero_zero(nat) )
               => ( ( Y = vEBT_Leaf(fFalse,B3) )
                 => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_delete_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf(A4,B3)),zero_zero(nat))) ) ) )
         => ( ! [A4: bool,B3: bool] :
                ( ( X = vEBT_Leaf(A4,B3) )
               => ( ( Xa2 = aa(nat,nat,suc,zero_zero(nat)) )
                 => ( ( Y = vEBT_Leaf(A4,fFalse) )
                   => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_delete_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf(A4,B3)),aa(nat,nat,suc,zero_zero(nat)))) ) ) )
           => ( ! [A4: bool,B3: bool] :
                  ( ( X = vEBT_Leaf(A4,B3) )
                 => ! [N2: nat] :
                      ( ( Xa2 = aa(nat,nat,suc,aa(nat,nat,suc,N2)) )
                     => ( ( Y = vEBT_Leaf(A4,B3) )
                       => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_delete_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf(A4,B3)),aa(nat,nat,suc,aa(nat,nat,suc,N2)))) ) ) )
             => ( ! [Deg2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                    ( ( X = vEBT_Node(none(product_prod(nat,nat)),Deg2,TreeList2,Summary2) )
                   => ( ( Y = vEBT_Node(none(product_prod(nat,nat)),Deg2,TreeList2,Summary2) )
                     => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_delete_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),Deg2,TreeList2,Summary2)),Xa2)) ) )
               => ( ! [Mi2: nat,Ma2: nat,TrLst: list(vEBT_VEBT),Smry: vEBT_VEBT] :
                      ( ( X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),zero_zero(nat),TrLst,Smry) )
                     => ( ( Y = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,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,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),zero_zero(nat),TrLst,Smry)),Xa2)) ) )
                 => ( ! [Mi2: nat,Ma2: nat,Tr: list(vEBT_VEBT),Sm: vEBT_VEBT] :
                        ( ( X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,zero_zero(nat)),Tr,Sm) )
                       => ( ( Y = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,zero_zero(nat)),Tr,Sm) )
                         => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_delete_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,zero_zero(nat)),Tr,Sm)),Xa2)) ) )
                   => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                          ( ( X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary2) )
                         => ( ( ( ( pp(aa(nat,bool,ord_less(nat,Xa2),Mi2))
                                  | pp(aa(nat,bool,ord_less(nat,Ma2),Xa2)) )
                               => ( Y = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary2) ) )
                              & ( ~ ( pp(aa(nat,bool,ord_less(nat,Xa2),Mi2))
                                    | pp(aa(nat,bool,ord_less(nat,Ma2),Xa2)) )
                               => ( ( ( ( Xa2 = Mi2 )
                                      & ( Xa2 = Ma2 ) )
                                   => ( Y = vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary2) ) )
                                  & ( ~ ( ( Xa2 = Mi2 )
                                        & ( Xa2 = Ma2 ) )
                                   => ( Y = 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at,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),the2(nat,vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),the2(nat,vEBT_vebt_mint(Summary2)))))),Xa2),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_VEBT_low(if(nat,aa(nat,bool,fequal(nat,Xa2),Mi2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),the2(nat,vEBT_vebt_mint(Summary2))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),the2(nat,vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),the2(nat,vEBT_vebt_mint(Summary2)))))),Xa2),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),vEBT_VEBT_high(if(nat,aa(nat,bool,fequal(nat,Xa2),Mi2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),the2(nat,vEBT_vebt_mint(Summary2))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),the2(nat,vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),the2(nat,vEBT_vebt_mint(Summary2)))))),Xa2),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))))),Ma2))),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),list_update(vEBT_VEBT,TreeList2,vEBT_VEBT_high(if(nat,aa(nat,bool,fequal(nat,Xa2),Mi2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),the2(nat,vEBT_vebt_mint(Summary2))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),the2(nat,vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),the2(nat,vEBT_vebt_mint(Summary2)))))),Xa2),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),vEBT_VEBT_high(if(nat,aa(nat,bool,fequal(nat,Xa2),Mi2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),the2(nat,vEBT_vebt_mint(Summary2))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),the2(nat,vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),the2(nat,vEBT_vebt_mint(Summary2)))))),Xa2),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_VEBT_low(if(nat,aa(nat,bool,fequal(nat,Xa2),Mi2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),the2(nat,vEBT_vebt_mint(Summary2))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),the2(nat,vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),the2(nat,vEBT_vebt_mint(Summary2)))))),Xa2),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),Summary2)),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary2)) ) ) ) ) )
                           => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_delete_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary2)),Xa2)) ) ) ) ) ) ) ) ) ) ) ).

% vebt_delete.pelims
tff(fact_4939_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Opelims,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: nat] :
      ( ( vEBT_T_s_u_c_c2(X,Xa2) = Y )
     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_T_s_u_c_c_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),X),Xa2))
       => ( ! [Uu2: bool,B3: bool] :
              ( ( X = vEBT_Leaf(Uu2,B3) )
             => ( ( Xa2 = zero_zero(nat) )
               => ( ( Y = one_one(nat) )
                 => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T_s_u_c_c_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf(Uu2,B3)),zero_zero(nat))) ) ) )
         => ( ! [Uv2: bool,Uw2: bool] :
                ( ( X = vEBT_Leaf(Uv2,Uw2) )
               => ! [N2: nat] :
                    ( ( Xa2 = aa(nat,nat,suc,N2) )
                   => ( ( Y = one_one(nat) )
                     => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T_s_u_c_c_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf(Uv2,Uw2)),aa(nat,nat,suc,N2))) ) ) )
           => ( ! [Ux2: nat,Uy2: list(vEBT_VEBT),Uz2: vEBT_VEBT] :
                  ( ( X = vEBT_Node(none(product_prod(nat,nat)),Ux2,Uy2,Uz2) )
                 => ( ( Y = one_one(nat) )
                   => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T_s_u_c_c_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),Ux2,Uy2,Uz2)),Xa2)) ) )
             => ( ! [V3: product_prod(nat,nat),Vc: list(vEBT_VEBT),Vd2: vEBT_VEBT] :
                    ( ( X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),zero_zero(nat),Vc,Vd2) )
                   => ( ( Y = one_one(nat) )
                     => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T_s_u_c_c_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),zero_zero(nat),Vc,Vd2)),Xa2)) ) )
               => ( ! [V3: product_prod(nat,nat),Vg2: list(vEBT_VEBT),Vh2: vEBT_VEBT] :
                      ( ( X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),aa(nat,nat,suc,zero_zero(nat)),Vg2,Vh2) )
                     => ( ( Y = one_one(nat) )
                       => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T_s_u_c_c_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),aa(nat,nat,suc,zero_zero(nat)),Vg2,Vh2)),Xa2)) ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                        ( ( X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary2) )
                       => ( ( ( pp(aa(nat,bool,ord_less(nat,Xa2),Mi2))
                             => ( Y = one_one(nat) ) )
                            & ( ~ pp(aa(nat,bool,ord_less(nat,Xa2),Mi2))
                             => ( Y = if(nat,aa(nat,bool,ord_less(nat,vEBT_VEBT_high(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),if(nat,fconj(aa(bool,bool,fNot,aa(option(nat),bool,fequal(option(nat),vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),vEBT_VEBT_high(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),none(nat))),vEBT_VEBT_less(aa(nat,option(nat),some(nat),vEBT_VEBT_low(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),vEBT_VEBT_high(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),vEBT_T_s_u_c_c2(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),vEBT_VEBT_high(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_VEBT_low(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_T_s_u_c_c2(Summary2,vEBT_VEBT_high(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),one_one(nat))),one_one(nat)) ) ) )
                         => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T_s_u_c_c_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary2)),Xa2)) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.pelims
tff(fact_4940_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_Opelims,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: nat] :
      ( ( vEBT_T_i_n_s_e_r_t(X,Xa2) = Y )
     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_T9217963907923527482_t_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),X),Xa2))
       => ( ! [A4: bool,B3: bool] :
              ( ( X = vEBT_Leaf(A4,B3) )
             => ( ( Y = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),if(nat,aa(nat,bool,fequal(nat,Xa2),zero_zero(nat)),one_one(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),one_one(nat)))) )
               => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T9217963907923527482_t_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf(A4,B3)),Xa2)) ) )
         => ( ! [Info2: option(product_prod(nat,nat)),Ts2: list(vEBT_VEBT),S2: vEBT_VEBT] :
                ( ( X = vEBT_Node(Info2,zero_zero(nat),Ts2,S2) )
               => ( ( Y = one_one(nat) )
                 => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T9217963907923527482_t_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(Info2,zero_zero(nat),Ts2,S2)),Xa2)) ) )
           => ( ! [Info2: option(product_prod(nat,nat)),Ts2: list(vEBT_VEBT),S2: vEBT_VEBT] :
                  ( ( X = vEBT_Node(Info2,aa(nat,nat,suc,zero_zero(nat)),Ts2,S2) )
                 => ( ( Y = one_one(nat) )
                   => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T9217963907923527482_t_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(Info2,aa(nat,nat,suc,zero_zero(nat)),Ts2,S2)),Xa2)) ) )
             => ( ! [V3: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                    ( ( X = vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,V3)),TreeList2,Summary2) )
                   => ( ( Y = aa(num,nat,numeral_numeral(nat),bit0(one2)) )
                     => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T9217963907923527482_t_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,V3)),TreeList2,Summary2)),Xa2)) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                      ( ( X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary2) )
                     => ( ( Y = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,aa(num,num,bit1,bit0(bit0(one2)))))),if(nat,fconj(aa(nat,bool,ord_less(nat,vEBT_VEBT_high(if(nat,aa(nat,bool,ord_less(nat,Xa2),Mi2),Mi2,Xa2),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),aa(bool,bool,fNot,fdisj(aa(nat,bool,fequal(nat,Xa2),Mi2),aa(nat,bool,fequal(nat,Xa2),Ma2)))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_T_i_n_s_e_r_t(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),vEBT_VEBT_high(if(nat,aa(nat,bool,ord_less(nat,Xa2),Mi2),Mi2,Xa2),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_VEBT_low(if(nat,aa(nat,bool,ord_less(nat,Xa2),Mi2),Mi2,Xa2),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),vEBT_T_m_i_n_N_u_l_l(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),vEBT_VEBT_high(if(nat,aa(nat,bool,ord_less(nat,Xa2),Mi2),Mi2,Xa2),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))))),if(nat,vEBT_VEBT_minNull(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),vEBT_VEBT_high(if(nat,aa(nat,bool,ord_less(nat,Xa2),Mi2),Mi2,Xa2),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),vEBT_T_i_n_s_e_r_t(Summary2,vEBT_VEBT_high(if(nat,aa(nat,bool,ord_less(nat,Xa2),Mi2),Mi2,Xa2),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),one_one(nat))),one_one(nat))) )
                       => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T9217963907923527482_t_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary2)),Xa2)) ) ) ) ) ) ) ) ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.pelims
tff(fact_4941_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Opelims,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: nat] :
      ( ( vEBT_T_p_r_e_d2(X,Xa2) = Y )
     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_T_p_r_e_d_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),X),Xa2))
       => ( ! [Uu2: bool,Uv2: bool] :
              ( ( X = vEBT_Leaf(Uu2,Uv2) )
             => ( ( Xa2 = zero_zero(nat) )
               => ( ( Y = one_one(nat) )
                 => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T_p_r_e_d_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf(Uu2,Uv2)),zero_zero(nat))) ) ) )
         => ( ! [A4: bool,Uw2: bool] :
                ( ( X = vEBT_Leaf(A4,Uw2) )
               => ( ( Xa2 = aa(nat,nat,suc,zero_zero(nat)) )
                 => ( ( Y = one_one(nat) )
                   => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T_p_r_e_d_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf(A4,Uw2)),aa(nat,nat,suc,zero_zero(nat)))) ) ) )
           => ( ! [A4: bool,B3: bool] :
                  ( ( X = vEBT_Leaf(A4,B3) )
                 => ! [Va2: nat] :
                      ( ( Xa2 = aa(nat,nat,suc,aa(nat,nat,suc,Va2)) )
                     => ( ( Y = one_one(nat) )
                       => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T_p_r_e_d_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf(A4,B3)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)))) ) ) )
             => ( ! [Uy2: nat,Uz2: list(vEBT_VEBT),Va3: vEBT_VEBT] :
                    ( ( X = vEBT_Node(none(product_prod(nat,nat)),Uy2,Uz2,Va3) )
                   => ( ( Y = one_one(nat) )
                     => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T_p_r_e_d_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),Uy2,Uz2,Va3)),Xa2)) ) )
               => ( ! [V3: product_prod(nat,nat),Vd2: list(vEBT_VEBT),Ve2: vEBT_VEBT] :
                      ( ( X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),zero_zero(nat),Vd2,Ve2) )
                     => ( ( Y = one_one(nat) )
                       => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T_p_r_e_d_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),zero_zero(nat),Vd2,Ve2)),Xa2)) ) )
                 => ( ! [V3: product_prod(nat,nat),Vh2: list(vEBT_VEBT),Vi2: vEBT_VEBT] :
                        ( ( X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),aa(nat,nat,suc,zero_zero(nat)),Vh2,Vi2) )
                       => ( ( Y = one_one(nat) )
                         => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T_p_r_e_d_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),aa(nat,nat,suc,zero_zero(nat)),Vh2,Vi2)),Xa2)) ) )
                   => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                          ( ( X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary2) )
                         => ( ( ( pp(aa(nat,bool,ord_less(nat,Ma2),Xa2))
                               => ( Y = one_one(nat) ) )
                              & ( ~ pp(aa(nat,bool,ord_less(nat,Ma2),Xa2))
                               => ( Y = if(nat,aa(nat,bool,ord_less(nat,vEBT_VEBT_high(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),if(nat,fconj(aa(bool,bool,fNot,aa(option(nat),bool,fequal(option(nat),vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),vEBT_VEBT_high(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),none(nat))),vEBT_VEBT_greater(aa(nat,option(nat),some(nat),vEBT_VEBT_low(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),vEBT_VEBT_high(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),vEBT_T_p_r_e_d2(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),vEBT_VEBT_high(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_VEBT_low(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_T_p_r_e_d2(Summary2,vEBT_VEBT_high(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),one_one(nat))),one_one(nat)) ) ) )
                           => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T_p_r_e_d_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary2)),Xa2)) ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.pelims
tff(fact_4942_vebt__insert_Opelims,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: vEBT_VEBT] :
      ( ( vEBT_vebt_insert(X,Xa2) = Y )
     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_insert_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),X),Xa2))
       => ( ! [A4: bool,B3: bool] :
              ( ( X = vEBT_Leaf(A4,B3) )
             => ( ( ( ( Xa2 = zero_zero(nat) )
                   => ( Y = vEBT_Leaf(fTrue,B3) ) )
                  & ( ( Xa2 != zero_zero(nat) )
                   => ( ( ( Xa2 = one_one(nat) )
                       => ( Y = vEBT_Leaf(A4,fTrue) ) )
                      & ( ( Xa2 != one_one(nat) )
                       => ( Y = vEBT_Leaf(A4,B3) ) ) ) ) )
               => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_insert_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf(A4,B3)),Xa2)) ) )
         => ( ! [Info2: option(product_prod(nat,nat)),Ts2: list(vEBT_VEBT),S2: vEBT_VEBT] :
                ( ( X = vEBT_Node(Info2,zero_zero(nat),Ts2,S2) )
               => ( ( Y = vEBT_Node(Info2,zero_zero(nat),Ts2,S2) )
                 => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_insert_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(Info2,zero_zero(nat),Ts2,S2)),Xa2)) ) )
           => ( ! [Info2: option(product_prod(nat,nat)),Ts2: list(vEBT_VEBT),S2: vEBT_VEBT] :
                  ( ( X = vEBT_Node(Info2,aa(nat,nat,suc,zero_zero(nat)),Ts2,S2) )
                 => ( ( Y = vEBT_Node(Info2,aa(nat,nat,suc,zero_zero(nat)),Ts2,S2) )
                   => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_insert_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(Info2,aa(nat,nat,suc,zero_zero(nat)),Ts2,S2)),Xa2)) ) )
             => ( ! [V3: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                    ( ( X = vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,V3)),TreeList2,Summary2) )
                   => ( ( Y = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Xa2),Xa2)),aa(nat,nat,suc,aa(nat,nat,suc,V3)),TreeList2,Summary2) )
                     => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_insert_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,V3)),TreeList2,Summary2)),Xa2)) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                      ( ( X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary2) )
                     => ( ( Y = if(vEBT_VEBT,fconj(aa(nat,bool,ord_less(nat,vEBT_VEBT_high(if(nat,aa(nat,bool,ord_less(nat,Xa2),Mi2),Mi2,Xa2),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),aa(bool,bool,fNot,fdisj(aa(nat,bool,fequal(nat,Xa2),Mi2),aa(nat,bool,fequal(nat,Xa2),Ma2)))),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),if(nat,aa(nat,bool,ord_less(nat,Xa2),Mi2),Xa2,Mi2)),aa(nat,nat,ord_max(nat,if(nat,aa(nat,bool,ord_less(nat,Xa2),Mi2),Mi2,Xa2)),Ma2))),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),list_update(vEBT_VEBT,TreeList2,vEBT_VEBT_high(if(nat,aa(nat,bool,ord_less(nat,Xa2),Mi2),Mi2,Xa2),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),vEBT_vebt_insert(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),vEBT_VEBT_high(if(nat,aa(nat,bool,ord_less(nat,Xa2),Mi2),Mi2,Xa2),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_VEBT_low(if(nat,aa(nat,bool,ord_less(nat,Xa2),Mi2),Mi2,Xa2),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),if(vEBT_VEBT,vEBT_VEBT_minNull(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),vEBT_VEBT_high(if(nat,aa(nat,bool,ord_less(nat,Xa2),Mi2),Mi2,Xa2),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),vEBT_vebt_insert(Summary2,vEBT_VEBT_high(if(nat,aa(nat,bool,ord_less(nat,Xa2),Mi2),Mi2,Xa2),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),Summary2)),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary2)) )
                       => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_insert_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary2)),Xa2)) ) ) ) ) ) ) ) ) ).

% vebt_insert.pelims
tff(fact_4943_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H_Opelims,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: nat] :
      ( ( vEBT_T_i_n_s_e_r_t2(X,Xa2) = Y )
     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_T5076183648494686801_t_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),X),Xa2))
       => ( ! [A4: bool,B3: bool] :
              ( ( X = vEBT_Leaf(A4,B3) )
             => ( ( Y = one_one(nat) )
               => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T5076183648494686801_t_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf(A4,B3)),Xa2)) ) )
         => ( ! [Info2: option(product_prod(nat,nat)),Ts2: list(vEBT_VEBT),S2: vEBT_VEBT] :
                ( ( X = vEBT_Node(Info2,zero_zero(nat),Ts2,S2) )
               => ( ( Y = one_one(nat) )
                 => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T5076183648494686801_t_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(Info2,zero_zero(nat),Ts2,S2)),Xa2)) ) )
           => ( ! [Info2: option(product_prod(nat,nat)),Ts2: list(vEBT_VEBT),S2: vEBT_VEBT] :
                  ( ( X = vEBT_Node(Info2,aa(nat,nat,suc,zero_zero(nat)),Ts2,S2) )
                 => ( ( Y = one_one(nat) )
                   => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T5076183648494686801_t_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(Info2,aa(nat,nat,suc,zero_zero(nat)),Ts2,S2)),Xa2)) ) )
             => ( ! [V3: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                    ( ( X = vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,V3)),TreeList2,Summary2) )
                   => ( ( Y = one_one(nat) )
                     => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T5076183648494686801_t_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,V3)),TreeList2,Summary2)),Xa2)) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                      ( ( X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary2) )
                     => ( ( Y = if(nat,fconj(aa(nat,bool,ord_less(nat,vEBT_VEBT_high(if(nat,aa(nat,bool,ord_less(nat,Xa2),Mi2),Mi2,Xa2),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),aa(bool,bool,fNot,fdisj(aa(nat,bool,fequal(nat,Xa2),Mi2),aa(nat,bool,fequal(nat,Xa2),Ma2)))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_T_i_n_s_e_r_t2(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),vEBT_VEBT_high(if(nat,aa(nat,bool,ord_less(nat,Xa2),Mi2),Mi2,Xa2),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_VEBT_low(if(nat,aa(nat,bool,ord_less(nat,Xa2),Mi2),Mi2,Xa2),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),if(nat,vEBT_VEBT_minNull(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),vEBT_VEBT_high(if(nat,aa(nat,bool,ord_less(nat,Xa2),Mi2),Mi2,Xa2),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),vEBT_T_i_n_s_e_r_t2(Summary2,vEBT_VEBT_high(if(nat,aa(nat,bool,ord_less(nat,Xa2),Mi2),Mi2,Xa2),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),one_one(nat))),one_one(nat)) )
                       => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T5076183648494686801_t_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary2)),Xa2)) ) ) ) ) ) ) ) ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.pelims
tff(fact_4944_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_Opelims,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: nat] :
      ( ( vEBT_T_m_e_m_b_e_r(X,Xa2) = Y )
     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_T5837161174952499735_r_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),X),Xa2))
       => ( ! [A4: bool,B3: bool] :
              ( ( X = vEBT_Leaf(A4,B3) )
             => ( ( Y = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),if(nat,aa(nat,bool,fequal(nat,Xa2),zero_zero(nat)),one_one(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),one_one(nat)))) )
               => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T5837161174952499735_r_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf(A4,B3)),Xa2)) ) )
         => ( ! [Uu2: nat,Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT] :
                ( ( X = vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw2) )
               => ( ( Y = aa(num,nat,numeral_numeral(nat),bit0(one2)) )
                 => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T5837161174952499735_r_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw2)),Xa2)) ) )
           => ( ! [V3: product_prod(nat,nat),Uy2: list(vEBT_VEBT),Uz2: vEBT_VEBT] :
                  ( ( X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),zero_zero(nat),Uy2,Uz2) )
                 => ( ( Y = aa(num,nat,numeral_numeral(nat),bit0(one2)) )
                   => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T5837161174952499735_r_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),zero_zero(nat),Uy2,Uz2)),Xa2)) ) )
             => ( ! [V3: product_prod(nat,nat),Vb2: list(vEBT_VEBT),Vc: vEBT_VEBT] :
                    ( ( X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),aa(nat,nat,suc,zero_zero(nat)),Vb2,Vc) )
                   => ( ( Y = aa(num,nat,numeral_numeral(nat),bit0(one2)) )
                     => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T5837161174952499735_r_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),aa(nat,nat,suc,zero_zero(nat)),Vb2,Vc)),Xa2)) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                      ( ( X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary2) )
                     => ( ( Y = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),if(nat,aa(nat,bool,fequal(nat,Xa2),Mi2),one_one(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),if(nat,aa(nat,bool,fequal(nat,Xa2),Ma2),one_one(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),if(nat,aa(nat,bool,ord_less(nat,Xa2),Mi2),one_one(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),if(nat,aa(nat,bool,ord_less(nat,Ma2),Xa2),one_one(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,bit0(bit0(one2))))),if(nat,aa(nat,bool,ord_less(nat,vEBT_VEBT_high(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),vEBT_T_m_e_m_b_e_r(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),vEBT_VEBT_high(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_VEBT_low(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),one_one(nat))))))))))) )
                       => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T5837161174952499735_r_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary2)),Xa2)) ) ) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.pelims
tff(fact_4945_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H_Opelims,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: nat] :
      ( ( vEBT_T_m_e_m_b_e_r2(X,Xa2) = Y )
     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_T8099345112685741742_r_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),X),Xa2))
       => ( ! [A4: bool,B3: bool] :
              ( ( X = vEBT_Leaf(A4,B3) )
             => ( ( Y = one_one(nat) )
               => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T8099345112685741742_r_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf(A4,B3)),Xa2)) ) )
         => ( ! [Uu2: nat,Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT] :
                ( ( X = vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw2) )
               => ( ( Y = one_one(nat) )
                 => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T8099345112685741742_r_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw2)),Xa2)) ) )
           => ( ! [V3: product_prod(nat,nat),Uy2: list(vEBT_VEBT),Uz2: vEBT_VEBT] :
                  ( ( X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),zero_zero(nat),Uy2,Uz2) )
                 => ( ( Y = one_one(nat) )
                   => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T8099345112685741742_r_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),zero_zero(nat),Uy2,Uz2)),Xa2)) ) )
             => ( ! [V3: product_prod(nat,nat),Vb2: list(vEBT_VEBT),Vc: vEBT_VEBT] :
                    ( ( X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),aa(nat,nat,suc,zero_zero(nat)),Vb2,Vc) )
                   => ( ( Y = one_one(nat) )
                     => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T8099345112685741742_r_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),aa(nat,nat,suc,zero_zero(nat)),Vb2,Vc)),Xa2)) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                      ( ( X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary2) )
                     => ( ( Y = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),if(nat,aa(nat,bool,fequal(nat,Xa2),Mi2),zero_zero(nat),if(nat,aa(nat,bool,fequal(nat,Xa2),Ma2),zero_zero(nat),if(nat,aa(nat,bool,ord_less(nat,Xa2),Mi2),zero_zero(nat),if(nat,aa(nat,bool,ord_less(nat,Ma2),Xa2),zero_zero(nat),if(nat,fconj(aa(nat,bool,ord_less(nat,Mi2),Xa2),aa(nat,bool,ord_less(nat,Xa2),Ma2)),if(nat,aa(nat,bool,ord_less(nat,vEBT_VEBT_high(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),vEBT_T_m_e_m_b_e_r2(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),vEBT_VEBT_high(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_VEBT_low(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),zero_zero(nat)),zero_zero(nat))))))) )
                       => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T8099345112685741742_r_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary2)),Xa2)) ) ) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.pelims
tff(fact_4946_vebt__member_Opelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ~ pp(aa(nat,bool,vEBT_vebt_member(X),Xa2))
     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_member_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),X),Xa2))
       => ( ! [A4: bool,B3: bool] :
              ( ( X = vEBT_Leaf(A4,B3) )
             => ( accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_member_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf(A4,B3)),Xa2))
               => ( ( ( Xa2 = zero_zero(nat) )
                   => pp(A4) )
                  & ( ( Xa2 != zero_zero(nat) )
                   => ( ( ( Xa2 = one_one(nat) )
                       => pp(B3) )
                      & ( Xa2 = one_one(nat) ) ) ) ) ) )
         => ( ! [Uu2: nat,Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT] :
                ( ( X = vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw2) )
               => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_member_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw2)),Xa2)) )
           => ( ! [V3: product_prod(nat,nat),Uy2: list(vEBT_VEBT),Uz2: vEBT_VEBT] :
                  ( ( X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),zero_zero(nat),Uy2,Uz2) )
                 => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_member_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),zero_zero(nat),Uy2,Uz2)),Xa2)) )
             => ( ! [V3: product_prod(nat,nat),Vb2: list(vEBT_VEBT),Vc: vEBT_VEBT] :
                    ( ( X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),aa(nat,nat,suc,zero_zero(nat)),Vb2,Vc) )
                   => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_member_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),aa(nat,nat,suc,zero_zero(nat)),Vb2,Vc)),Xa2)) )
               => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                      ( ( X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary2) )
                     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_member_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary2)),Xa2))
                       => ( ( Xa2 != Mi2 )
                         => ( ( Xa2 != Ma2 )
                           => ( ~ pp(aa(nat,bool,ord_less(nat,Xa2),Mi2))
                              & ( ~ pp(aa(nat,bool,ord_less(nat,Xa2),Mi2))
                               => ( ~ pp(aa(nat,bool,ord_less(nat,Ma2),Xa2))
                                  & ( ~ pp(aa(nat,bool,ord_less(nat,Ma2),Xa2))
                                   => ( ( pp(aa(nat,bool,ord_less(nat,vEBT_VEBT_high(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)))
                                       => pp(aa(nat,bool,vEBT_vebt_member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),vEBT_VEBT_high(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),vEBT_VEBT_low(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))) )
                                      & pp(aa(nat,bool,ord_less(nat,vEBT_VEBT_high(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2))) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(3)
tff(fact_4947_vebt__member_Opelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( pp(aa(nat,bool,vEBT_vebt_member(X),Xa2))
     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_member_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),X),Xa2))
       => ( ! [A4: bool,B3: bool] :
              ( ( X = vEBT_Leaf(A4,B3) )
             => ( accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_member_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf(A4,B3)),Xa2))
               => ~ ( ( ( Xa2 = zero_zero(nat) )
                     => pp(A4) )
                    & ( ( Xa2 != zero_zero(nat) )
                     => ( ( ( Xa2 = one_one(nat) )
                         => pp(B3) )
                        & ( Xa2 = one_one(nat) ) ) ) ) ) )
         => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                ( ( X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary2) )
               => ( accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_member_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary2)),Xa2))
                 => ~ ( ( Xa2 != Mi2 )
                     => ( ( Xa2 != Ma2 )
                       => ( ~ pp(aa(nat,bool,ord_less(nat,Xa2),Mi2))
                          & ( ~ pp(aa(nat,bool,ord_less(nat,Xa2),Mi2))
                           => ( ~ pp(aa(nat,bool,ord_less(nat,Ma2),Xa2))
                              & ( ~ pp(aa(nat,bool,ord_less(nat,Ma2),Xa2))
                               => ( ( pp(aa(nat,bool,ord_less(nat,vEBT_VEBT_high(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)))
                                   => pp(aa(nat,bool,vEBT_vebt_member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),vEBT_VEBT_high(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),vEBT_VEBT_low(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))) )
                                  & pp(aa(nat,bool,ord_less(nat,vEBT_VEBT_high(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2))) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(2)
tff(fact_4948_vebt__member_Opelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: bool] :
      ( ( pp(aa(nat,bool,vEBT_vebt_member(X),Xa2))
      <=> pp(Y) )
     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_member_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),X),Xa2))
       => ( ! [A4: bool,B3: bool] :
              ( ( X = vEBT_Leaf(A4,B3) )
             => ( ( pp(Y)
                <=> ( ( ( Xa2 = zero_zero(nat) )
                     => pp(A4) )
                    & ( ( Xa2 != zero_zero(nat) )
                     => ( ( ( Xa2 = one_one(nat) )
                         => pp(B3) )
                        & ( Xa2 = one_one(nat) ) ) ) ) )
               => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_member_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf(A4,B3)),Xa2)) ) )
         => ( ! [Uu2: nat,Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT] :
                ( ( X = vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw2) )
               => ( ~ pp(Y)
                 => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_member_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw2)),Xa2)) ) )
           => ( ! [V3: product_prod(nat,nat),Uy2: list(vEBT_VEBT),Uz2: vEBT_VEBT] :
                  ( ( X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),zero_zero(nat),Uy2,Uz2) )
                 => ( ~ pp(Y)
                   => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_member_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),zero_zero(nat),Uy2,Uz2)),Xa2)) ) )
             => ( ! [V3: product_prod(nat,nat),Vb2: list(vEBT_VEBT),Vc: vEBT_VEBT] :
                    ( ( X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),aa(nat,nat,suc,zero_zero(nat)),Vb2,Vc) )
                   => ( ~ pp(Y)
                     => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_member_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),aa(nat,nat,suc,zero_zero(nat)),Vb2,Vc)),Xa2)) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                      ( ( X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary2) )
                     => ( ( pp(Y)
                        <=> ( ( Xa2 != Mi2 )
                           => ( ( Xa2 != Ma2 )
                             => ( ~ pp(aa(nat,bool,ord_less(nat,Xa2),Mi2))
                                & ( ~ pp(aa(nat,bool,ord_less(nat,Xa2),Mi2))
                                 => ( ~ pp(aa(nat,bool,ord_less(nat,Ma2),Xa2))
                                    & ( ~ pp(aa(nat,bool,ord_less(nat,Ma2),Xa2))
                                     => ( ( pp(aa(nat,bool,ord_less(nat,vEBT_VEBT_high(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)))
                                         => pp(aa(nat,bool,vEBT_vebt_member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),vEBT_VEBT_high(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),vEBT_VEBT_low(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))) )
                                        & pp(aa(nat,bool,ord_less(nat,vEBT_VEBT_high(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2))) ) ) ) ) ) ) ) )
                       => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_member_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary2)),Xa2)) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(1)
tff(fact_4949_VEBT__internal_Onaive__member_Opelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: bool] :
      ( ( vEBT_V5719532721284313246member(X,Xa2)
      <=> pp(Y) )
     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_V5765760719290551771er_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),X),Xa2))
       => ( ! [A4: bool,B3: bool] :
              ( ( X = vEBT_Leaf(A4,B3) )
             => ( ( pp(Y)
                <=> ( ( ( Xa2 = zero_zero(nat) )
                     => pp(A4) )
                    & ( ( Xa2 != zero_zero(nat) )
                     => ( ( ( Xa2 = one_one(nat) )
                         => pp(B3) )
                        & ( Xa2 = one_one(nat) ) ) ) ) )
               => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_V5765760719290551771er_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf(A4,B3)),Xa2)) ) )
         => ( ! [Uu2: option(product_prod(nat,nat)),Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT] :
                ( ( X = vEBT_Node(Uu2,zero_zero(nat),Uv2,Uw2) )
               => ( ~ pp(Y)
                 => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_V5765760719290551771er_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(Uu2,zero_zero(nat),Uv2,Uw2)),Xa2)) ) )
           => ~ ! [Uy2: option(product_prod(nat,nat)),V3: nat,TreeList2: list(vEBT_VEBT),S2: vEBT_VEBT] :
                  ( ( X = vEBT_Node(Uy2,aa(nat,nat,suc,V3),TreeList2,S2) )
                 => ( ( pp(Y)
                    <=> ( ( pp(aa(nat,bool,ord_less(nat,vEBT_VEBT_high(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V3)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)))
                         => vEBT_V5719532721284313246member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),vEBT_VEBT_high(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V3)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_VEBT_low(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V3)),aa(num,nat,numeral_numeral(nat),bit0(one2))))) )
                        & pp(aa(nat,bool,ord_less(nat,vEBT_VEBT_high(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V3)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2))) ) )
                   => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_V5765760719290551771er_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(Uy2,aa(nat,nat,suc,V3),TreeList2,S2)),Xa2)) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(1)
tff(fact_4950_VEBT__internal_Onaive__member_Opelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( vEBT_V5719532721284313246member(X,Xa2)
     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_V5765760719290551771er_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),X),Xa2))
       => ( ! [A4: bool,B3: bool] :
              ( ( X = vEBT_Leaf(A4,B3) )
             => ( accp(product_prod(vEBT_VEBT,nat),vEBT_V5765760719290551771er_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf(A4,B3)),Xa2))
               => ~ ( ( ( Xa2 = zero_zero(nat) )
                     => pp(A4) )
                    & ( ( Xa2 != zero_zero(nat) )
                     => ( ( ( Xa2 = one_one(nat) )
                         => pp(B3) )
                        & ( Xa2 = one_one(nat) ) ) ) ) ) )
         => ~ ! [Uy2: option(product_prod(nat,nat)),V3: nat,TreeList2: list(vEBT_VEBT),S2: vEBT_VEBT] :
                ( ( X = vEBT_Node(Uy2,aa(nat,nat,suc,V3),TreeList2,S2) )
               => ( accp(product_prod(vEBT_VEBT,nat),vEBT_V5765760719290551771er_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(Uy2,aa(nat,nat,suc,V3),TreeList2,S2)),Xa2))
                 => ~ ( ( pp(aa(nat,bool,ord_less(nat,vEBT_VEBT_high(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V3)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)))
                       => vEBT_V5719532721284313246member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),vEBT_VEBT_high(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V3)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_VEBT_low(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V3)),aa(num,nat,numeral_numeral(nat),bit0(one2))))) )
                      & pp(aa(nat,bool,ord_less(nat,vEBT_VEBT_high(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V3)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2))) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(2)
tff(fact_4951_VEBT__internal_Onaive__member_Opelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ~ vEBT_V5719532721284313246member(X,Xa2)
     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_V5765760719290551771er_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),X),Xa2))
       => ( ! [A4: bool,B3: bool] :
              ( ( X = vEBT_Leaf(A4,B3) )
             => ( accp(product_prod(vEBT_VEBT,nat),vEBT_V5765760719290551771er_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf(A4,B3)),Xa2))
               => ( ( ( Xa2 = zero_zero(nat) )
                   => pp(A4) )
                  & ( ( Xa2 != zero_zero(nat) )
                   => ( ( ( Xa2 = one_one(nat) )
                       => pp(B3) )
                      & ( Xa2 = one_one(nat) ) ) ) ) ) )
         => ( ! [Uu2: option(product_prod(nat,nat)),Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT] :
                ( ( X = vEBT_Node(Uu2,zero_zero(nat),Uv2,Uw2) )
               => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_V5765760719290551771er_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(Uu2,zero_zero(nat),Uv2,Uw2)),Xa2)) )
           => ~ ! [Uy2: option(product_prod(nat,nat)),V3: nat,TreeList2: list(vEBT_VEBT),S2: vEBT_VEBT] :
                  ( ( X = vEBT_Node(Uy2,aa(nat,nat,suc,V3),TreeList2,S2) )
                 => ( accp(product_prod(vEBT_VEBT,nat),vEBT_V5765760719290551771er_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(Uy2,aa(nat,nat,suc,V3),TreeList2,S2)),Xa2))
                   => ( ( pp(aa(nat,bool,ord_less(nat,vEBT_VEBT_high(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V3)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)))
                       => vEBT_V5719532721284313246member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),vEBT_VEBT_high(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V3)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_VEBT_low(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V3)),aa(num,nat,numeral_numeral(nat),bit0(one2))))) )
                      & pp(aa(nat,bool,ord_less(nat,vEBT_VEBT_high(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V3)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2))) ) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(3)
tff(fact_4952_VEBT__internal_Omembermima_Opelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ~ vEBT_VEBT_membermima(X,Xa2)
     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),X),Xa2))
       => ( ! [Uu2: bool,Uv2: bool] :
              ( ( X = vEBT_Leaf(Uu2,Uv2) )
             => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf(Uu2,Uv2)),Xa2)) )
         => ( ! [Ux2: list(vEBT_VEBT),Uy2: vEBT_VEBT] :
                ( ( X = vEBT_Node(none(product_prod(nat,nat)),zero_zero(nat),Ux2,Uy2) )
               => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),zero_zero(nat),Ux2,Uy2)),Xa2)) )
           => ( ! [Mi2: nat,Ma2: nat,Va3: list(vEBT_VEBT),Vb2: vEBT_VEBT] :
                  ( ( X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),zero_zero(nat),Va3,Vb2) )
                 => ( accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),zero_zero(nat),Va3,Vb2)),Xa2))
                   => ( ( Xa2 = Mi2 )
                      | ( Xa2 = Ma2 ) ) ) )
             => ( ! [Mi2: nat,Ma2: nat,V3: nat,TreeList2: list(vEBT_VEBT),Vc: vEBT_VEBT] :
                    ( ( X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,V3),TreeList2,Vc) )
                   => ( accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,V3),TreeList2,Vc)),Xa2))
                     => ( ( Xa2 = Mi2 )
                        | ( Xa2 = Ma2 )
                        | ( ( pp(aa(nat,bool,ord_less(nat,vEBT_VEBT_high(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V3)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)))
                           => vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),vEBT_VEBT_high(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V3)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_VEBT_low(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V3)),aa(num,nat,numeral_numeral(nat),bit0(one2))))) )
                          & pp(aa(nat,bool,ord_less(nat,vEBT_VEBT_high(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V3)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2))) ) ) ) )
               => ~ ! [V3: nat,TreeList2: list(vEBT_VEBT),Vd2: vEBT_VEBT] :
                      ( ( X = vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,V3),TreeList2,Vd2) )
                     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,V3),TreeList2,Vd2)),Xa2))
                       => ( ( pp(aa(nat,bool,ord_less(nat,vEBT_VEBT_high(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V3)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)))
                           => vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),vEBT_VEBT_high(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V3)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_VEBT_low(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V3)),aa(num,nat,numeral_numeral(nat),bit0(one2))))) )
                          & pp(aa(nat,bool,ord_less(nat,vEBT_VEBT_high(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V3)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2))) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(3)
tff(fact_4953_VEBT__internal_Omembermima_Opelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: bool] :
      ( ( vEBT_VEBT_membermima(X,Xa2)
      <=> pp(Y) )
     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),X),Xa2))
       => ( ! [Uu2: bool,Uv2: bool] :
              ( ( X = vEBT_Leaf(Uu2,Uv2) )
             => ( ~ pp(Y)
               => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf(Uu2,Uv2)),Xa2)) ) )
         => ( ! [Ux2: list(vEBT_VEBT),Uy2: vEBT_VEBT] :
                ( ( X = vEBT_Node(none(product_prod(nat,nat)),zero_zero(nat),Ux2,Uy2) )
               => ( ~ pp(Y)
                 => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),zero_zero(nat),Ux2,Uy2)),Xa2)) ) )
           => ( ! [Mi2: nat,Ma2: nat,Va3: list(vEBT_VEBT),Vb2: vEBT_VEBT] :
                  ( ( X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),zero_zero(nat),Va3,Vb2) )
                 => ( ( pp(Y)
                    <=> ( ( Xa2 = Mi2 )
                        | ( Xa2 = Ma2 ) ) )
                   => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),zero_zero(nat),Va3,Vb2)),Xa2)) ) )
             => ( ! [Mi2: nat,Ma2: nat,V3: nat,TreeList2: list(vEBT_VEBT),Vc: vEBT_VEBT] :
                    ( ( X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,V3),TreeList2,Vc) )
                   => ( ( pp(Y)
                      <=> ( ( Xa2 = Mi2 )
                          | ( Xa2 = Ma2 )
                          | ( ( pp(aa(nat,bool,ord_less(nat,vEBT_VEBT_high(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V3)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)))
                             => vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),vEBT_VEBT_high(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V3)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_VEBT_low(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V3)),aa(num,nat,numeral_numeral(nat),bit0(one2))))) )
                            & pp(aa(nat,bool,ord_less(nat,vEBT_VEBT_high(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V3)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2))) ) ) )
                     => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,V3),TreeList2,Vc)),Xa2)) ) )
               => ~ ! [V3: nat,TreeList2: list(vEBT_VEBT),Vd2: vEBT_VEBT] :
                      ( ( X = vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,V3),TreeList2,Vd2) )
                     => ( ( pp(Y)
                        <=> ( ( pp(aa(nat,bool,ord_less(nat,vEBT_VEBT_high(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V3)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)))
                             => vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),vEBT_VEBT_high(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V3)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_VEBT_low(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V3)),aa(num,nat,numeral_numeral(nat),bit0(one2))))) )
                            & pp(aa(nat,bool,ord_less(nat,vEBT_VEBT_high(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V3)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2))) ) )
                       => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,V3),TreeList2,Vd2)),Xa2)) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(1)
tff(fact_4954_VEBT__internal_Omembermima_Opelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( vEBT_VEBT_membermima(X,Xa2)
     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),X),Xa2))
       => ( ! [Mi2: nat,Ma2: nat,Va3: list(vEBT_VEBT),Vb2: vEBT_VEBT] :
              ( ( X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),zero_zero(nat),Va3,Vb2) )
             => ( accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),zero_zero(nat),Va3,Vb2)),Xa2))
               => ~ ( ( Xa2 = Mi2 )
                    | ( Xa2 = Ma2 ) ) ) )
         => ( ! [Mi2: nat,Ma2: nat,V3: nat,TreeList2: list(vEBT_VEBT),Vc: vEBT_VEBT] :
                ( ( X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,V3),TreeList2,Vc) )
               => ( accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,V3),TreeList2,Vc)),Xa2))
                 => ~ ( ( Xa2 = Mi2 )
                      | ( Xa2 = Ma2 )
                      | ( ( pp(aa(nat,bool,ord_less(nat,vEBT_VEBT_high(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V3)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)))
                         => vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),vEBT_VEBT_high(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V3)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_VEBT_low(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V3)),aa(num,nat,numeral_numeral(nat),bit0(one2))))) )
                        & pp(aa(nat,bool,ord_less(nat,vEBT_VEBT_high(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V3)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2))) ) ) ) )
           => ~ ! [V3: nat,TreeList2: list(vEBT_VEBT),Vd2: vEBT_VEBT] :
                  ( ( X = vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,V3),TreeList2,Vd2) )
                 => ( accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,V3),TreeList2,Vd2)),Xa2))
                   => ~ ( ( pp(aa(nat,bool,ord_less(nat,vEBT_VEBT_high(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V3)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)))
                         => vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),vEBT_VEBT_high(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V3)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_VEBT_low(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V3)),aa(num,nat,numeral_numeral(nat),bit0(one2))))) )
                        & pp(aa(nat,bool,ord_less(nat,vEBT_VEBT_high(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V3)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2))) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(2)
tff(fact_4955_succ__empty,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( vEBT_invar_vebt(T,N)
     => ( ( vEBT_vebt_succ(T,X) = none(nat) )
      <=> ( collect(nat,aa(nat,fun(nat,bool),aTP_Lamp_lj(vEBT_VEBT,fun(nat,fun(nat,bool)),T),X)) = bot_bot(set(nat)) ) ) ) ).

% succ_empty
tff(fact_4956_pred__empty,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( vEBT_invar_vebt(T,N)
     => ( ( vEBT_vebt_pred(T,X) = none(nat) )
      <=> ( collect(nat,aa(nat,fun(nat,bool),aTP_Lamp_lk(vEBT_VEBT,fun(nat,fun(nat,bool)),T),X)) = bot_bot(set(nat)) ) ) ) ).

% pred_empty
tff(fact_4957_max_Oabsorb1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,A2: A] :
          ( pp(aa(A,bool,ord_less_eq(A,B2),A2))
         => ( aa(A,A,ord_max(A,A2),B2) = A2 ) ) ) ).

% max.absorb1
tff(fact_4958_buildup__gives__empty,axiom,
    ! [N: nat] : vEBT_VEBT_set_vebt(vEBT_vebt_buildup(N)) = bot_bot(set(nat)) ).

% buildup_gives_empty
tff(fact_4959_mint__corr__help__empty,axiom,
    ! [T: vEBT_VEBT,N: nat] :
      ( vEBT_invar_vebt(T,N)
     => ( ( vEBT_vebt_mint(T) = none(nat) )
       => ( vEBT_VEBT_set_vebt(T) = bot_bot(set(nat)) ) ) ) ).

% mint_corr_help_empty
tff(fact_4960_maxt__corr__help__empty,axiom,
    ! [T: vEBT_VEBT,N: nat] :
      ( vEBT_invar_vebt(T,N)
     => ( ( vEBT_vebt_maxt(T) = none(nat) )
       => ( vEBT_VEBT_set_vebt(T) = bot_bot(set(nat)) ) ) ) ).

% maxt_corr_help_empty
tff(fact_4961_empty__subsetI,axiom,
    ! [A: $tType,A3: set(A)] : pp(aa(set(A),bool,ord_less_eq(set(A),bot_bot(set(A))),A3)) ).

% empty_subsetI
tff(fact_4962_subset__empty,axiom,
    ! [A: $tType,A3: set(A)] :
      ( pp(aa(set(A),bool,ord_less_eq(set(A),A3),bot_bot(set(A))))
    <=> ( A3 = bot_bot(set(A)) ) ) ).

% subset_empty
tff(fact_4963_max_Obounded__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,C2: A,A2: A] :
          ( pp(aa(A,bool,ord_less_eq(A,aa(A,A,ord_max(A,B2),C2)),A2))
        <=> ( pp(aa(A,bool,ord_less_eq(A,B2),A2))
            & pp(aa(A,bool,ord_less_eq(A,C2),A2)) ) ) ) ).

% max.bounded_iff
tff(fact_4964_max_Oabsorb2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,ord_less_eq(A,A2),B2))
         => ( aa(A,A,ord_max(A,A2),B2) = B2 ) ) ) ).

% max.absorb2
tff(fact_4965_sum_Oempty,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(B,A)] : aa(set(B),A,groups7311177749621191930dd_sum(B,A,G),bot_bot(set(B))) = zero_zero(A) ) ).

% sum.empty
tff(fact_4966_atLeastatMost__empty__iff2,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A2: A,B2: A] :
          ( ( bot_bot(set(A)) = set_or1337092689740270186AtMost(A,A2,B2) )
        <=> ~ pp(aa(A,bool,ord_less_eq(A,A2),B2)) ) ) ).

% atLeastatMost_empty_iff2
tff(fact_4967_atLeastatMost__empty__iff,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A2: A,B2: A] :
          ( ( set_or1337092689740270186AtMost(A,A2,B2) = bot_bot(set(A)) )
        <=> ~ pp(aa(A,bool,ord_less_eq(A,A2),B2)) ) ) ).

% atLeastatMost_empty_iff
tff(fact_4968_prod_Oempty,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(B,A)] : groups7121269368397514597t_prod(B,A,G,bot_bot(set(B))) = one_one(A) ) ).

% prod.empty
tff(fact_4969_Diff__eq__empty__iff,axiom,
    ! [A: $tType,A3: set(A),B4: set(A)] :
      ( ( aa(set(A),set(A),minus_minus(set(A),A3),B4) = bot_bot(set(A)) )
    <=> pp(aa(set(A),bool,ord_less_eq(set(A),A3),B4)) ) ).

% Diff_eq_empty_iff
tff(fact_4970_lessThan__0,axiom,
    set_ord_lessThan(nat,zero_zero(nat)) = bot_bot(set(nat)) ).

% lessThan_0
tff(fact_4971_set__decode__zero,axiom,
    nat_set_decode(zero_zero(nat)) = bot_bot(set(nat)) ).

% set_decode_zero
tff(fact_4972_set__encode__empty,axiom,
    aa(set(nat),nat,nat_set_encode,bot_bot(set(nat))) = zero_zero(nat) ).

% set_encode_empty
tff(fact_4973_Set_Oempty__def,axiom,
    ! [A: $tType] : bot_bot(set(A)) = collect(A,aTP_Lamp_ll(A,bool)) ).

% Set.empty_def
tff(fact_4974_diff__shunt__var,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [X: A,Y: A] :
          ( ( aa(A,A,minus_minus(A,X),Y) = bot_bot(A) )
        <=> pp(aa(A,bool,ord_less_eq(A,X),Y)) ) ) ).

% diff_shunt_var
tff(fact_4975_finite__has__maximal,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A3: set(A)] :
          ( pp(aa(set(A),bool,finite_finite2(A),A3))
         => ( ( A3 != bot_bot(set(A)) )
           => ? [X3: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),A3))
                & ! [Xa: A] :
                    ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Xa),A3))
                   => ( pp(aa(A,bool,ord_less_eq(A,X3),Xa))
                     => ( X3 = Xa ) ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_4976_finite__has__minimal,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A3: set(A)] :
          ( pp(aa(set(A),bool,finite_finite2(A),A3))
         => ( ( A3 != bot_bot(set(A)) )
           => ? [X3: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),A3))
                & ! [Xa: A] :
                    ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Xa),A3))
                   => ( pp(aa(A,bool,ord_less_eq(A,Xa),X3))
                     => ( X3 = Xa ) ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_4977_subset__minus__empty,axiom,
    ! [A: $tType,A3: set(A),B4: set(A)] :
      ( pp(aa(set(A),bool,ord_less_eq(set(A),A3),B4))
     => ( aa(set(A),set(A),minus_minus(set(A),A3),B4) = bot_bot(set(A)) ) ) ).

% subset_minus_empty
tff(fact_4978_lessThan__empty__iff,axiom,
    ! [N: nat] :
      ( ( set_ord_lessThan(nat,N) = bot_bot(set(nat)) )
    <=> ( N = zero_zero(nat) ) ) ).

% lessThan_empty_iff
tff(fact_4979_subset__Compl__self__eq,axiom,
    ! [A: $tType,A3: set(A)] :
      ( pp(aa(set(A),bool,ord_less_eq(set(A),A3),aa(set(A),set(A),uminus_uminus(set(A)),A3)))
    <=> ( A3 = bot_bot(set(A)) ) ) ).

% subset_Compl_self_eq
tff(fact_4980_sum__pos,axiom,
    ! [A: $tType,B: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [I5: set(B),F2: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),I5))
         => ( ( I5 != bot_bot(set(B)) )
           => ( ! [I2: B] :
                  ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I2),I5))
                 => pp(aa(A,bool,ord_less(A,zero_zero(A)),aa(B,A,F2,I2))) )
             => pp(aa(A,bool,ord_less(A,zero_zero(A)),aa(set(B),A,groups7311177749621191930dd_sum(B,A,F2),I5))) ) ) ) ) ).

% sum_pos
tff(fact_4981_less__1__prod,axiom,
    ! [B: $tType,A: $tType] :
      ( linordered_idom(B)
     => ! [I5: set(A),F2: fun(A,B)] :
          ( pp(aa(set(A),bool,finite_finite2(A),I5))
         => ( ( I5 != bot_bot(set(A)) )
           => ( ! [I2: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),I2),I5))
                 => pp(aa(B,bool,ord_less(B,one_one(B)),aa(A,B,F2,I2))) )
             => pp(aa(B,bool,ord_less(B,one_one(B)),groups7121269368397514597t_prod(A,B,F2,I5))) ) ) ) ) ).

% less_1_prod
tff(fact_4982_prod__mono__strict,axiom,
    ! [A: $tType,B: $tType] :
      ( linordered_semidom(A)
     => ! [A3: set(B),F2: fun(B,A),G: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),A3))
         => ( ! [I2: B] :
                ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I2),A3))
               => ( pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),aa(B,A,F2,I2)))
                  & pp(aa(A,bool,ord_less(A,aa(B,A,F2,I2)),aa(B,A,G,I2))) ) )
           => ( ( A3 != bot_bot(set(B)) )
             => pp(aa(A,bool,ord_less(A,groups7121269368397514597t_prod(B,A,F2,A3)),groups7121269368397514597t_prod(B,A,G,A3))) ) ) ) ) ).

% prod_mono_strict
tff(fact_4983_max_Omono,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [C2: A,A2: A,D: A,B2: A] :
          ( pp(aa(A,bool,ord_less_eq(A,C2),A2))
         => ( pp(aa(A,bool,ord_less_eq(A,D),B2))
           => pp(aa(A,bool,ord_less_eq(A,aa(A,A,ord_max(A,C2),D)),aa(A,A,ord_max(A,A2),B2))) ) ) ) ).

% max.mono
tff(fact_4984_max_OorderE,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,A2: A] :
          ( pp(aa(A,bool,ord_less_eq(A,B2),A2))
         => ( A2 = aa(A,A,ord_max(A,A2),B2) ) ) ) ).

% max.orderE
tff(fact_4985_max_OorderI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A] :
          ( ( A2 = aa(A,A,ord_max(A,A2),B2) )
         => pp(aa(A,bool,ord_less_eq(A,B2),A2)) ) ) ).

% max.orderI
tff(fact_4986_max_OboundedE,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,C2: A,A2: A] :
          ( pp(aa(A,bool,ord_less_eq(A,aa(A,A,ord_max(A,B2),C2)),A2))
         => ~ ( pp(aa(A,bool,ord_less_eq(A,B2),A2))
             => ~ pp(aa(A,bool,ord_less_eq(A,C2),A2)) ) ) ) ).

% max.boundedE
tff(fact_4987_max_OboundedI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,A2: A,C2: A] :
          ( pp(aa(A,bool,ord_less_eq(A,B2),A2))
         => ( pp(aa(A,bool,ord_less_eq(A,C2),A2))
           => pp(aa(A,bool,ord_less_eq(A,aa(A,A,ord_max(A,B2),C2)),A2)) ) ) ) ).

% max.boundedI
tff(fact_4988_max_Oorder__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,A2: A] :
          ( pp(aa(A,bool,ord_less_eq(A,B2),A2))
        <=> ( A2 = aa(A,A,ord_max(A,A2),B2) ) ) ) ).

% max.order_iff
tff(fact_4989_max_Ocobounded1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A] : pp(aa(A,bool,ord_less_eq(A,A2),aa(A,A,ord_max(A,A2),B2))) ) ).

% max.cobounded1
tff(fact_4990_max_Ocobounded2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,A2: A] : pp(aa(A,bool,ord_less_eq(A,B2),aa(A,A,ord_max(A,A2),B2))) ) ).

% max.cobounded2
tff(fact_4991_le__max__iff__disj,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Z: A,X: A,Y: A] :
          ( pp(aa(A,bool,ord_less_eq(A,Z),aa(A,A,ord_max(A,X),Y)))
        <=> ( pp(aa(A,bool,ord_less_eq(A,Z),X))
            | pp(aa(A,bool,ord_less_eq(A,Z),Y)) ) ) ) ).

% le_max_iff_disj
tff(fact_4992_max_Oabsorb__iff1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,A2: A] :
          ( pp(aa(A,bool,ord_less_eq(A,B2),A2))
        <=> ( aa(A,A,ord_max(A,A2),B2) = A2 ) ) ) ).

% max.absorb_iff1
tff(fact_4993_max_Oabsorb__iff2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,ord_less_eq(A,A2),B2))
        <=> ( aa(A,A,ord_max(A,A2),B2) = B2 ) ) ) ).

% max.absorb_iff2
tff(fact_4994_max_OcoboundedI1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [C2: A,A2: A,B2: A] :
          ( pp(aa(A,bool,ord_less_eq(A,C2),A2))
         => pp(aa(A,bool,ord_less_eq(A,C2),aa(A,A,ord_max(A,A2),B2))) ) ) ).

% max.coboundedI1
tff(fact_4995_max_OcoboundedI2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [C2: A,B2: A,A2: A] :
          ( pp(aa(A,bool,ord_less_eq(A,C2),B2))
         => pp(aa(A,bool,ord_less_eq(A,C2),aa(A,A,ord_max(A,A2),B2))) ) ) ).

% max.coboundedI2
tff(fact_4996_delete__correct,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( vEBT_invar_vebt(T,N)
     => ( vEBT_VEBT_set_vebt(vEBT_vebt_delete(T,X)) = aa(set(nat),set(nat),minus_minus(set(nat),vEBT_set_vebt(T)),insert(nat,X,bot_bot(set(nat)))) ) ) ).

% delete_correct
tff(fact_4997_floor__rat__def,axiom,
    ! [X: rat] : archim6421214686448440834_floor(rat,X) = the(int,aTP_Lamp_lm(rat,fun(int,bool),X)) ).

% floor_rat_def
tff(fact_4998_divmod__algorithm__code_I8_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M: num,N: num] :
          ( ( pp(aa(num,bool,ord_less(num,M),N))
           => ( unique8689654367752047608divmod(A,aa(num,num,bit1,M),aa(num,num,bit1,N)) = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),zero_zero(A)),aa(num,A,numeral_numeral(A),aa(num,num,bit1,M))) ) )
          & ( ~ pp(aa(num,bool,ord_less(num,M),N))
           => ( unique8689654367752047608divmod(A,aa(num,num,bit1,M),aa(num,num,bit1,N)) = unique1321980374590559556d_step(A,aa(num,num,bit1,N),unique8689654367752047608divmod(A,aa(num,num,bit1,M),bit0(aa(num,num,bit1,N)))) ) ) ) ) ).

% divmod_algorithm_code(8)
tff(fact_4999_delete__correct_H,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( vEBT_invar_vebt(T,N)
     => ( vEBT_VEBT_set_vebt(vEBT_vebt_delete(T,X)) = aa(set(nat),set(nat),minus_minus(set(nat),vEBT_VEBT_set_vebt(T)),insert(nat,X,bot_bot(set(nat)))) ) ) ).

% delete_correct'
tff(fact_5000_bit__0__eq,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ( bit_se5641148757651400278ts_bit(A,zero_zero(A)) = bot_bot(fun(nat,bool)) ) ) ).

% bit_0_eq
tff(fact_5001_insert__subset,axiom,
    ! [A: $tType,X: A,A3: set(A),B4: set(A)] :
      ( pp(aa(set(A),bool,ord_less_eq(set(A),insert(A,X,A3)),B4))
    <=> ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),B4))
        & pp(aa(set(A),bool,ord_less_eq(set(A),A3),B4)) ) ) ).

% insert_subset
tff(fact_5002_singleton__conv,axiom,
    ! [A: $tType,A2: A] : collect(A,aTP_Lamp_lg(A,fun(A,bool),A2)) = insert(A,A2,bot_bot(set(A))) ).

% singleton_conv
tff(fact_5003_singleton__conv2,axiom,
    ! [A: $tType,A2: A] : collect(A,fequal(A,A2)) = insert(A,A2,bot_bot(set(A))) ).

% singleton_conv2
tff(fact_5004_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 )
        & pp(aa(set(A),bool,ord_less_eq(set(A),A3),insert(A,B2,bot_bot(set(A))))) ) ) ).

% singleton_insert_inj_eq'
tff(fact_5005_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 )
        & pp(aa(set(A),bool,ord_less_eq(set(A),A3),insert(A,B2,bot_bot(set(A))))) ) ) ).

% singleton_insert_inj_eq
tff(fact_5006_sum_Oinsert,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [A3: set(B),X: B,G: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),A3))
         => ( ~ pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X),A3))
           => ( aa(set(B),A,groups7311177749621191930dd_sum(B,A,G),insert(B,X,A3)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(B,A,G,X)),aa(set(B),A,groups7311177749621191930dd_sum(B,A,G),A3)) ) ) ) ) ).

% sum.insert
tff(fact_5007_prod_Oinsert,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [A3: set(B),X: B,G: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),A3))
         => ( ~ pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X),A3))
           => ( groups7121269368397514597t_prod(B,A,G,insert(B,X,A3)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(B,A,G,X)),groups7121269368397514597t_prod(B,A,G,A3)) ) ) ) ) ).

% prod.insert
tff(fact_5008_subset__Compl__singleton,axiom,
    ! [A: $tType,A3: set(A),B2: A] :
      ( pp(aa(set(A),bool,ord_less_eq(set(A),A3),aa(set(A),set(A),uminus_uminus(set(A)),insert(A,B2,bot_bot(set(A))))))
    <=> ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),B2),A3)) ) ).

% subset_Compl_singleton
tff(fact_5009_atMost__0,axiom,
    set_ord_atMost(nat,zero_zero(nat)) = insert(nat,zero_zero(nat),bot_bot(set(nat))) ).

% atMost_0
tff(fact_5010_divmod__algorithm__code_I2_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M: num] : unique8689654367752047608divmod(A,M,one2) = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(num,A,numeral_numeral(A),M)),zero_zero(A)) ) ).

% divmod_algorithm_code(2)
tff(fact_5011_set__replicate,axiom,
    ! [A: $tType,N: nat,X: A] :
      ( ( N != zero_zero(nat) )
     => ( aa(list(A),set(A),set2(A),replicate(A,N,X)) = insert(A,X,bot_bot(set(A))) ) ) ).

% set_replicate
tff(fact_5012_divmod__algorithm__code_I3_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [N: num] : unique8689654367752047608divmod(A,one2,bit0(N)) = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),zero_zero(A)),aa(num,A,numeral_numeral(A),one2)) ) ).

% divmod_algorithm_code(3)
tff(fact_5013_divmod__algorithm__code_I4_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [N: num] : unique8689654367752047608divmod(A,one2,aa(num,num,bit1,N)) = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),zero_zero(A)),aa(num,A,numeral_numeral(A),one2)) ) ).

% divmod_algorithm_code(4)
tff(fact_5014_set__encode__insert,axiom,
    ! [A3: set(nat),N: nat] :
      ( pp(aa(set(nat),bool,finite_finite2(nat),A3))
     => ( ~ pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),N),A3))
       => ( aa(set(nat),nat,nat_set_encode,insert(nat,N,A3)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)),aa(set(nat),nat,nat_set_encode,A3)) ) ) ) ).

% set_encode_insert
tff(fact_5015_divmod__algorithm__code_I7_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M: num,N: num] :
          ( ( pp(aa(num,bool,ord_less_eq(num,M),N))
           => ( unique8689654367752047608divmod(A,bit0(M),aa(num,num,bit1,N)) = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),zero_zero(A)),aa(num,A,numeral_numeral(A),bit0(M))) ) )
          & ( ~ pp(aa(num,bool,ord_less_eq(num,M),N))
           => ( unique8689654367752047608divmod(A,bit0(M),aa(num,num,bit1,N)) = unique1321980374590559556d_step(A,aa(num,num,bit1,N),unique8689654367752047608divmod(A,bit0(M),bit0(aa(num,num,bit1,N)))) ) ) ) ) ).

% divmod_algorithm_code(7)
tff(fact_5016_bot__option__def,axiom,
    ! [A: $tType] :
      ( order(A)
     => ( bot_bot(option(A)) = none(A) ) ) ).

% bot_option_def
tff(fact_5017_bot__nat__def,axiom,
    bot_bot(nat) = zero_zero(nat) ).

% bot_nat_def
tff(fact_5018_bot__enat__def,axiom,
    bot_bot(extended_enat) = zero_zero(extended_enat) ).

% bot_enat_def
tff(fact_5019_Collect__conv__if2,axiom,
    ! [A: $tType,P: fun(A,bool),A2: A] :
      ( ( pp(aa(A,bool,P,A2))
       => ( collect(A,aa(A,fun(A,bool),aTP_Lamp_ln(fun(A,bool),fun(A,fun(A,bool)),P),A2)) = insert(A,A2,bot_bot(set(A))) ) )
      & ( ~ pp(aa(A,bool,P,A2))
       => ( collect(A,aa(A,fun(A,bool),aTP_Lamp_ln(fun(A,bool),fun(A,fun(A,bool)),P),A2)) = bot_bot(set(A)) ) ) ) ).

% Collect_conv_if2
tff(fact_5020_Collect__conv__if,axiom,
    ! [A: $tType,P: fun(A,bool),A2: A] :
      ( ( pp(aa(A,bool,P,A2))
       => ( collect(A,aa(A,fun(A,bool),aTP_Lamp_lo(fun(A,bool),fun(A,fun(A,bool)),P),A2)) = insert(A,A2,bot_bot(set(A))) ) )
      & ( ~ pp(aa(A,bool,P,A2))
       => ( collect(A,aa(A,fun(A,bool),aTP_Lamp_lo(fun(A,bool),fun(A,fun(A,bool)),P),A2)) = bot_bot(set(A)) ) ) ) ).

% Collect_conv_if
tff(fact_5021_abs__rat__def,axiom,
    ! [A2: rat] :
      ( ( pp(aa(rat,bool,ord_less(rat,A2),zero_zero(rat)))
       => ( aa(rat,rat,abs_abs(rat),A2) = aa(rat,rat,uminus_uminus(rat),A2) ) )
      & ( ~ pp(aa(rat,bool,ord_less(rat,A2),zero_zero(rat)))
       => ( aa(rat,rat,abs_abs(rat),A2) = A2 ) ) ) ).

% abs_rat_def
tff(fact_5022_obtain__pos__sum,axiom,
    ! [R2: rat] :
      ( pp(aa(rat,bool,ord_less(rat,zero_zero(rat)),R2))
     => ~ ! [S2: rat] :
            ( pp(aa(rat,bool,ord_less(rat,zero_zero(rat)),S2))
           => ! [T5: rat] :
                ( pp(aa(rat,bool,ord_less(rat,zero_zero(rat)),T5))
               => ( R2 != aa(rat,rat,aa(rat,fun(rat,rat),plus_plus(rat),S2),T5) ) ) ) ) ).

% obtain_pos_sum
tff(fact_5023_sgn__rat__def,axiom,
    ! [A2: rat] :
      ( ( ( A2 = zero_zero(rat) )
       => ( sgn_sgn(rat,A2) = zero_zero(rat) ) )
      & ( ( A2 != zero_zero(rat) )
       => ( ( pp(aa(rat,bool,ord_less(rat,zero_zero(rat)),A2))
           => ( sgn_sgn(rat,A2) = one_one(rat) ) )
          & ( ~ pp(aa(rat,bool,ord_less(rat,zero_zero(rat)),A2))
           => ( sgn_sgn(rat,A2) = aa(rat,rat,uminus_uminus(rat),one_one(rat)) ) ) ) ) ) ).

% sgn_rat_def
tff(fact_5024_subset__insertI2,axiom,
    ! [A: $tType,A3: set(A),B4: set(A),B2: A] :
      ( pp(aa(set(A),bool,ord_less_eq(set(A),A3),B4))
     => pp(aa(set(A),bool,ord_less_eq(set(A),A3),insert(A,B2,B4))) ) ).

% subset_insertI2
tff(fact_5025_subset__insertI,axiom,
    ! [A: $tType,B4: set(A),A2: A] : pp(aa(set(A),bool,ord_less_eq(set(A),B4),insert(A,A2,B4))) ).

% subset_insertI
tff(fact_5026_subset__insert,axiom,
    ! [A: $tType,X: A,A3: set(A),B4: set(A)] :
      ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A3))
     => ( pp(aa(set(A),bool,ord_less_eq(set(A),A3),insert(A,X,B4)))
      <=> pp(aa(set(A),bool,ord_less_eq(set(A),A3),B4)) ) ) ).

% subset_insert
tff(fact_5027_insert__mono,axiom,
    ! [A: $tType,C6: set(A),D4: set(A),A2: A] :
      ( pp(aa(set(A),bool,ord_less_eq(set(A),C6),D4))
     => pp(aa(set(A),bool,ord_less_eq(set(A),insert(A,A2,C6)),insert(A,A2,D4))) ) ).

% insert_mono
tff(fact_5028_insert__compr,axiom,
    ! [A: $tType,A2: A,B4: set(A)] : insert(A,A2,B4) = collect(A,aa(set(A),fun(A,bool),aTP_Lamp_lp(A,fun(set(A),fun(A,bool)),A2),B4)) ).

% insert_compr
tff(fact_5029_insert__Collect,axiom,
    ! [A: $tType,A2: A,P: fun(A,bool)] : insert(A,A2,collect(A,P)) = collect(A,aa(fun(A,bool),fun(A,bool),aTP_Lamp_lq(A,fun(fun(A,bool),fun(A,bool)),A2),P)) ).

% insert_Collect
tff(fact_5030_subset__singleton__iff,axiom,
    ! [A: $tType,X6: set(A),A2: A] :
      ( pp(aa(set(A),bool,ord_less_eq(set(A),X6),insert(A,A2,bot_bot(set(A)))))
    <=> ( ( X6 = bot_bot(set(A)) )
        | ( X6 = insert(A,A2,bot_bot(set(A))) ) ) ) ).

% subset_singleton_iff
tff(fact_5031_subset__singletonD,axiom,
    ! [A: $tType,A3: set(A),X: A] :
      ( pp(aa(set(A),bool,ord_less_eq(set(A),A3),insert(A,X,bot_bot(set(A)))))
     => ( ( A3 = bot_bot(set(A)) )
        | ( A3 = insert(A,X,bot_bot(set(A))) ) ) ) ).

% subset_singletonD
tff(fact_5032_subset__Diff__insert,axiom,
    ! [A: $tType,A3: set(A),B4: set(A),X: A,C6: set(A)] :
      ( pp(aa(set(A),bool,ord_less_eq(set(A),A3),aa(set(A),set(A),minus_minus(set(A),B4),insert(A,X,C6))))
    <=> ( pp(aa(set(A),bool,ord_less_eq(set(A),A3),aa(set(A),set(A),minus_minus(set(A),B4),C6)))
        & ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A3)) ) ) ).

% subset_Diff_insert
tff(fact_5033_lessThan__Suc,axiom,
    ! [K: nat] : set_ord_lessThan(nat,aa(nat,nat,suc,K)) = insert(nat,K,set_ord_lessThan(nat,K)) ).

% lessThan_Suc
tff(fact_5034_atMost__Suc,axiom,
    ! [K: nat] : set_ord_atMost(nat,aa(nat,nat,suc,K)) = insert(nat,aa(nat,nat,suc,K),set_ord_atMost(nat,K)) ).

% atMost_Suc
tff(fact_5035_finite__ranking__induct,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [S: set(B),P: fun(set(B),bool),F2: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),S))
         => ( pp(aa(set(B),bool,P,bot_bot(set(B))))
           => ( ! [X3: B,S6: set(B)] :
                  ( pp(aa(set(B),bool,finite_finite2(B),S6))
                 => ( ! [Y4: B] :
                        ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),Y4),S6))
                       => pp(aa(A,bool,ord_less_eq(A,aa(B,A,F2,Y4)),aa(B,A,F2,X3))) )
                   => ( pp(aa(set(B),bool,P,S6))
                     => pp(aa(set(B),bool,P,insert(B,X3,S6))) ) ) )
             => pp(aa(set(B),bool,P,S)) ) ) ) ) ).

% finite_ranking_induct
tff(fact_5036_sum_Oinsert__if,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [A3: set(B),X: B,G: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),A3))
         => ( ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X),A3))
             => ( aa(set(B),A,groups7311177749621191930dd_sum(B,A,G),insert(B,X,A3)) = aa(set(B),A,groups7311177749621191930dd_sum(B,A,G),A3) ) )
            & ( ~ pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X),A3))
             => ( aa(set(B),A,groups7311177749621191930dd_sum(B,A,G),insert(B,X,A3)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(B,A,G,X)),aa(set(B),A,groups7311177749621191930dd_sum(B,A,G),A3)) ) ) ) ) ) ).

% sum.insert_if
tff(fact_5037_finite__subset__induct,axiom,
    ! [A: $tType,F3: set(A),A3: set(A),P: fun(set(A),bool)] :
      ( pp(aa(set(A),bool,finite_finite2(A),F3))
     => ( pp(aa(set(A),bool,ord_less_eq(set(A),F3),A3))
       => ( pp(aa(set(A),bool,P,bot_bot(set(A))))
         => ( ! [A4: A,F5: set(A)] :
                ( pp(aa(set(A),bool,finite_finite2(A),F5))
               => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A4),A3))
                 => ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A4),F5))
                   => ( pp(aa(set(A),bool,P,F5))
                     => pp(aa(set(A),bool,P,insert(A,A4,F5))) ) ) ) )
           => pp(aa(set(A),bool,P,F3)) ) ) ) ) ).

% finite_subset_induct
tff(fact_5038_finite__subset__induct_H,axiom,
    ! [A: $tType,F3: set(A),A3: set(A),P: fun(set(A),bool)] :
      ( pp(aa(set(A),bool,finite_finite2(A),F3))
     => ( pp(aa(set(A),bool,ord_less_eq(set(A),F3),A3))
       => ( pp(aa(set(A),bool,P,bot_bot(set(A))))
         => ( ! [A4: A,F5: set(A)] :
                ( pp(aa(set(A),bool,finite_finite2(A),F5))
               => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A4),A3))
                 => ( pp(aa(set(A),bool,ord_less_eq(set(A),F5),A3))
                   => ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A4),F5))
                     => ( pp(aa(set(A),bool,P,F5))
                       => pp(aa(set(A),bool,P,insert(A,A4,F5))) ) ) ) ) )
           => pp(aa(set(A),bool,P,F3)) ) ) ) ) ).

% finite_subset_induct'
tff(fact_5039_prod_Oinsert__if,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [A3: set(B),X: B,G: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),A3))
         => ( ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X),A3))
             => ( groups7121269368397514597t_prod(B,A,G,insert(B,X,A3)) = groups7121269368397514597t_prod(B,A,G,A3) ) )
            & ( ~ pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X),A3))
             => ( groups7121269368397514597t_prod(B,A,G,insert(B,X,A3)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(B,A,G,X)),groups7121269368397514597t_prod(B,A,G,A3)) ) ) ) ) ) ).

% prod.insert_if
tff(fact_5040_Diff__single__insert,axiom,
    ! [A: $tType,A3: set(A),X: A,B4: set(A)] :
      ( pp(aa(set(A),bool,ord_less_eq(set(A),aa(set(A),set(A),minus_minus(set(A),A3),insert(A,X,bot_bot(set(A))))),B4))
     => pp(aa(set(A),bool,ord_less_eq(set(A),A3),insert(A,X,B4))) ) ).

% Diff_single_insert
tff(fact_5041_subset__insert__iff,axiom,
    ! [A: $tType,A3: set(A),X: A,B4: set(A)] :
      ( pp(aa(set(A),bool,ord_less_eq(set(A),A3),insert(A,X,B4)))
    <=> ( ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A3))
         => pp(aa(set(A),bool,ord_less_eq(set(A),aa(set(A),set(A),minus_minus(set(A),A3),insert(A,X,bot_bot(set(A))))),B4)) )
        & ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A3))
         => pp(aa(set(A),bool,ord_less_eq(set(A),A3),B4)) ) ) ) ).

% subset_insert_iff
tff(fact_5042_atLeast0__atMost__Suc,axiom,
    ! [N: nat] : set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,suc,N)) = insert(nat,aa(nat,nat,suc,N),set_or1337092689740270186AtMost(nat,zero_zero(nat),N)) ).

% atLeast0_atMost_Suc
tff(fact_5043_Icc__eq__insert__lb__nat,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,ord_less_eq(nat,M),N))
     => ( set_or1337092689740270186AtMost(nat,M,N) = insert(nat,M,set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M),N)) ) ) ).

% Icc_eq_insert_lb_nat
tff(fact_5044_atLeastAtMostSuc__conv,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,ord_less_eq(nat,M),aa(nat,nat,suc,N)))
     => ( set_or1337092689740270186AtMost(nat,M,aa(nat,nat,suc,N)) = insert(nat,aa(nat,nat,suc,N),set_or1337092689740270186AtMost(nat,M,N)) ) ) ).

% atLeastAtMostSuc_conv
tff(fact_5045_atLeastAtMost__insertL,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,ord_less_eq(nat,M),N))
     => ( insert(nat,M,set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M),N)) = set_or1337092689740270186AtMost(nat,M,N) ) ) ).

% atLeastAtMost_insertL
tff(fact_5046_set__update__subset__insert,axiom,
    ! [A: $tType,Xs: list(A),I: nat,X: A] : pp(aa(set(A),bool,ord_less_eq(set(A),aa(list(A),set(A),set2(A),list_update(A,Xs,I,X))),insert(A,X,aa(list(A),set(A),set2(A),Xs)))) ).

% set_update_subset_insert
tff(fact_5047_lessThan__nat__numeral,axiom,
    ! [K: num] : set_ord_lessThan(nat,aa(num,nat,numeral_numeral(nat),K)) = insert(nat,pred_numeral(K),set_ord_lessThan(nat,pred_numeral(K))) ).

% lessThan_nat_numeral
tff(fact_5048_atMost__nat__numeral,axiom,
    ! [K: num] : set_ord_atMost(nat,aa(num,nat,numeral_numeral(nat),K)) = insert(nat,aa(num,nat,numeral_numeral(nat),K),set_ord_atMost(nat,pred_numeral(K))) ).

% atMost_nat_numeral
tff(fact_5049_divmod__int__def,axiom,
    ! [M: num,N: num] : unique8689654367752047608divmod(int,M,N) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(num,int,numeral_numeral(int),M)),aa(num,int,numeral_numeral(int),N))),modulo_modulo(int,aa(num,int,numeral_numeral(int),M),aa(num,int,numeral_numeral(int),N))) ).

% divmod_int_def
tff(fact_5050_remove__induct,axiom,
    ! [A: $tType,P: fun(set(A),bool),B4: set(A)] :
      ( pp(aa(set(A),bool,P,bot_bot(set(A))))
     => ( ( ~ pp(aa(set(A),bool,finite_finite2(A),B4))
         => pp(aa(set(A),bool,P,B4)) )
       => ( ! [A7: set(A)] :
              ( pp(aa(set(A),bool,finite_finite2(A),A7))
             => ( ( A7 != bot_bot(set(A)) )
               => ( pp(aa(set(A),bool,ord_less_eq(set(A),A7),B4))
                 => ( ! [X4: A] :
                        ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),A7))
                       => pp(aa(set(A),bool,P,aa(set(A),set(A),minus_minus(set(A),A7),insert(A,X4,bot_bot(set(A)))))) )
                   => pp(aa(set(A),bool,P,A7)) ) ) ) )
         => pp(aa(set(A),bool,P,B4)) ) ) ) ).

% remove_induct
tff(fact_5051_finite__remove__induct,axiom,
    ! [A: $tType,B4: set(A),P: fun(set(A),bool)] :
      ( pp(aa(set(A),bool,finite_finite2(A),B4))
     => ( pp(aa(set(A),bool,P,bot_bot(set(A))))
       => ( ! [A7: set(A)] :
              ( pp(aa(set(A),bool,finite_finite2(A),A7))
             => ( ( A7 != bot_bot(set(A)) )
               => ( pp(aa(set(A),bool,ord_less_eq(set(A),A7),B4))
                 => ( ! [X4: A] :
                        ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),A7))
                       => pp(aa(set(A),bool,P,aa(set(A),set(A),minus_minus(set(A),A7),insert(A,X4,bot_bot(set(A)))))) )
                   => pp(aa(set(A),bool,P,A7)) ) ) ) )
         => pp(aa(set(A),bool,P,B4)) ) ) ) ).

% finite_remove_induct
tff(fact_5052_set__replicate__Suc,axiom,
    ! [A: $tType,N: nat,X: A] : aa(list(A),set(A),set2(A),replicate(A,aa(nat,nat,suc,N),X)) = insert(A,X,bot_bot(set(A))) ).

% set_replicate_Suc
tff(fact_5053_psubset__insert__iff,axiom,
    ! [A: $tType,A3: set(A),X: A,B4: set(A)] :
      ( pp(aa(set(A),bool,ord_less(set(A),A3),insert(A,X,B4)))
    <=> ( ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),B4))
         => pp(aa(set(A),bool,ord_less(set(A),A3),B4)) )
        & ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),B4))
         => ( ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A3))
             => pp(aa(set(A),bool,ord_less(set(A),aa(set(A),set(A),minus_minus(set(A),A3),insert(A,X,bot_bot(set(A))))),B4)) )
            & ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A3))
             => pp(aa(set(A),bool,ord_less_eq(set(A),A3),B4)) ) ) ) ) ) ).

% psubset_insert_iff
tff(fact_5054_set__replicate__conv__if,axiom,
    ! [A: $tType,N: nat,X: A] :
      ( ( ( N = zero_zero(nat) )
       => ( aa(list(A),set(A),set2(A),replicate(A,N,X)) = bot_bot(set(A)) ) )
      & ( ( N != zero_zero(nat) )
       => ( aa(list(A),set(A),set2(A),replicate(A,N,X)) = insert(A,X,bot_bot(set(A))) ) ) ) ).

% set_replicate_conv_if
tff(fact_5055_sum_Oinsert__remove,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [A3: set(B),G: fun(B,A),X: B] :
          ( pp(aa(set(B),bool,finite_finite2(B),A3))
         => ( aa(set(B),A,groups7311177749621191930dd_sum(B,A,G),insert(B,X,A3)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(B,A,G,X)),aa(set(B),A,groups7311177749621191930dd_sum(B,A,G),aa(set(B),set(B),minus_minus(set(B),A3),insert(B,X,bot_bot(set(B)))))) ) ) ) ).

% sum.insert_remove
tff(fact_5056_sum_Oremove,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [A3: set(B),X: B,G: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),A3))
         => ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X),A3))
           => ( aa(set(B),A,groups7311177749621191930dd_sum(B,A,G),A3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(B,A,G,X)),aa(set(B),A,groups7311177749621191930dd_sum(B,A,G),aa(set(B),set(B),minus_minus(set(B),A3),insert(B,X,bot_bot(set(B)))))) ) ) ) ) ).

% sum.remove
tff(fact_5057_prod_Oremove,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [A3: set(B),X: B,G: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),A3))
         => ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X),A3))
           => ( groups7121269368397514597t_prod(B,A,G,A3) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(B,A,G,X)),groups7121269368397514597t_prod(B,A,G,aa(set(B),set(B),minus_minus(set(B),A3),insert(B,X,bot_bot(set(B)))))) ) ) ) ) ).

% prod.remove
tff(fact_5058_prod_Oinsert__remove,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [A3: set(B),G: fun(B,A),X: B] :
          ( pp(aa(set(B),bool,finite_finite2(B),A3))
         => ( groups7121269368397514597t_prod(B,A,G,insert(B,X,A3)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(B,A,G,X)),groups7121269368397514597t_prod(B,A,G,aa(set(B),set(B),minus_minus(set(B),A3),insert(B,X,bot_bot(set(B)))))) ) ) ) ).

% prod.insert_remove
tff(fact_5059_atLeast1__atMost__eq__remove0,axiom,
    ! [N: nat] : set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,zero_zero(nat)),N) = aa(set(nat),set(nat),minus_minus(set(nat),set_ord_atMost(nat,N)),insert(nat,zero_zero(nat),bot_bot(set(nat)))) ).

% atLeast1_atMost_eq_remove0
tff(fact_5060_sum_Odelta__remove,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [S: set(B),A2: B,B2: fun(B,A),C2: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),S))
         => ( ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A2),S))
             => ( aa(set(B),A,groups7311177749621191930dd_sum(B,A,aa(fun(B,A),fun(B,A),aa(fun(B,A),fun(fun(B,A),fun(B,A)),aTP_Lamp_lr(B,fun(fun(B,A),fun(fun(B,A),fun(B,A))),A2),B2),C2)),S) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(B,A,B2,A2)),aa(set(B),A,groups7311177749621191930dd_sum(B,A,C2),aa(set(B),set(B),minus_minus(set(B),S),insert(B,A2,bot_bot(set(B)))))) ) )
            & ( ~ pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A2),S))
             => ( aa(set(B),A,groups7311177749621191930dd_sum(B,A,aa(fun(B,A),fun(B,A),aa(fun(B,A),fun(fun(B,A),fun(B,A)),aTP_Lamp_lr(B,fun(fun(B,A),fun(fun(B,A),fun(B,A))),A2),B2),C2)),S) = aa(set(B),A,groups7311177749621191930dd_sum(B,A,C2),aa(set(B),set(B),minus_minus(set(B),S),insert(B,A2,bot_bot(set(B))))) ) ) ) ) ) ).

% sum.delta_remove
tff(fact_5061_prod_Odelta__remove,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [S: set(B),A2: B,B2: fun(B,A),C2: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),S))
         => ( ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A2),S))
             => ( groups7121269368397514597t_prod(B,A,aa(fun(B,A),fun(B,A),aa(fun(B,A),fun(fun(B,A),fun(B,A)),aTP_Lamp_ls(B,fun(fun(B,A),fun(fun(B,A),fun(B,A))),A2),B2),C2),S) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(B,A,B2,A2)),groups7121269368397514597t_prod(B,A,C2,aa(set(B),set(B),minus_minus(set(B),S),insert(B,A2,bot_bot(set(B)))))) ) )
            & ( ~ pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A2),S))
             => ( groups7121269368397514597t_prod(B,A,aa(fun(B,A),fun(B,A),aa(fun(B,A),fun(fun(B,A),fun(B,A)),aTP_Lamp_ls(B,fun(fun(B,A),fun(fun(B,A),fun(B,A))),A2),B2),C2),S) = groups7121269368397514597t_prod(B,A,C2,aa(set(B),set(B),minus_minus(set(B),S),insert(B,A2,bot_bot(set(B))))) ) ) ) ) ) ).

% prod.delta_remove
tff(fact_5062_divmod__def,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M: num,N: num] : unique8689654367752047608divmod(A,M,N) = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(num,A,numeral_numeral(A),M)),aa(num,A,numeral_numeral(A),N))),modulo_modulo(A,aa(num,A,numeral_numeral(A),M),aa(num,A,numeral_numeral(A),N))) ) ).

% divmod_def
tff(fact_5063_divmod_H__nat__def,axiom,
    ! [M: num,N: num] : unique8689654367752047608divmod(nat,M,N) = aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(num,nat,numeral_numeral(nat),M)),aa(num,nat,numeral_numeral(nat),N))),modulo_modulo(nat,aa(num,nat,numeral_numeral(nat),M),aa(num,nat,numeral_numeral(nat),N))) ).

% divmod'_nat_def
tff(fact_5064_member__le__sum,axiom,
    ! [B: $tType,C: $tType] :
      ( ( ordere6911136660526730532id_add(B)
        & semiring_1(B) )
     => ! [I: C,A3: set(C),F2: fun(C,B)] :
          ( pp(aa(set(C),bool,aa(C,fun(set(C),bool),member(C),I),A3))
         => ( ! [X3: C] :
                ( pp(aa(set(C),bool,aa(C,fun(set(C),bool),member(C),X3),aa(set(C),set(C),minus_minus(set(C),A3),insert(C,I,bot_bot(set(C))))))
               => pp(aa(B,bool,ord_less_eq(B,zero_zero(B)),aa(C,B,F2,X3))) )
           => ( pp(aa(set(C),bool,finite_finite2(C),A3))
             => pp(aa(B,bool,ord_less_eq(B,aa(C,B,F2,I)),aa(set(C),B,groups7311177749621191930dd_sum(C,B,F2),A3))) ) ) ) ) ).

% member_le_sum
tff(fact_5065_prod__diff1,axiom,
    ! [A: $tType,B: $tType] :
      ( semidom_divide(A)
     => ! [A3: set(B),F2: fun(B,A),A2: B] :
          ( pp(aa(set(B),bool,finite_finite2(B),A3))
         => ( ( aa(B,A,F2,A2) != zero_zero(A) )
           => ( ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A2),A3))
               => ( groups7121269368397514597t_prod(B,A,F2,aa(set(B),set(B),minus_minus(set(B),A3),insert(B,A2,bot_bot(set(B))))) = aa(A,A,aa(A,fun(A,A),divide_divide(A),groups7121269368397514597t_prod(B,A,F2,A3)),aa(B,A,F2,A2)) ) )
              & ( ~ pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A2),A3))
               => ( groups7121269368397514597t_prod(B,A,F2,aa(set(B),set(B),minus_minus(set(B),A3),insert(B,A2,bot_bot(set(B))))) = groups7121269368397514597t_prod(B,A,F2,A3) ) ) ) ) ) ) ).

% prod_diff1
tff(fact_5066_sinh__zero__iff,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] :
          ( ( sinh(A,X) = zero_zero(A) )
        <=> pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(A,A,exp(A),X)),insert(A,one_one(A),insert(A,aa(A,A,uminus_uminus(A),one_one(A)),bot_bot(set(A)))))) ) ) ).

% sinh_zero_iff
tff(fact_5067_set__decode__plus__power__2,axiom,
    ! [N: nat,Z: nat] :
      ( ~ pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),N),nat_set_decode(Z)))
     => ( nat_set_decode(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)),Z)) = insert(nat,N,nat_set_decode(Z)) ) ) ).

% set_decode_plus_power_2
tff(fact_5068_divmod__divmod__step,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M: num,N: num] :
          ( ( pp(aa(num,bool,ord_less(num,M),N))
           => ( unique8689654367752047608divmod(A,M,N) = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),zero_zero(A)),aa(num,A,numeral_numeral(A),M)) ) )
          & ( ~ pp(aa(num,bool,ord_less(num,M),N))
           => ( unique8689654367752047608divmod(A,M,N) = unique1321980374590559556d_step(A,N,unique8689654367752047608divmod(A,M,bit0(N))) ) ) ) ) ).

% divmod_divmod_step
tff(fact_5069_dvd__numeral__simp,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M: num,N: num] :
          ( pp(aa(A,bool,dvd_dvd(A,aa(num,A,numeral_numeral(A),M)),aa(num,A,numeral_numeral(A),N)))
        <=> unique5940410009612947441es_aux(A,unique8689654367752047608divmod(A,N,M)) ) ) ).

% dvd_numeral_simp
tff(fact_5070_minus__one__div__numeral,axiom,
    ! [N: num] : aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,uminus_uminus(int),one_one(int))),aa(num,int,numeral_numeral(int),N)) = aa(int,int,uminus_uminus(int),adjust_div(unique8689654367752047608divmod(int,one2,N))) ).

% minus_one_div_numeral
tff(fact_5071_one__div__minus__numeral,axiom,
    ! [N: num] : aa(int,int,aa(int,fun(int,int),divide_divide(int),one_one(int)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),N))) = aa(int,int,uminus_uminus(int),adjust_div(unique8689654367752047608divmod(int,one2,N))) ).

% one_div_minus_numeral
tff(fact_5072_divides__aux__eq,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Q2: A,R2: A] :
          ( unique5940410009612947441es_aux(A,aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Q2),R2))
        <=> ( R2 = zero_zero(A) ) ) ) ).

% divides_aux_eq
tff(fact_5073_Divides_Oadjust__div__eq,axiom,
    ! [Q2: int,R2: int] : adjust_div(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q2),R2)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),Q2),aa(bool,int,zero_neq_one_of_bool(int),aa(bool,bool,fNot,aa(int,bool,fequal(int,R2),zero_zero(int))))) ).

% Divides.adjust_div_eq
tff(fact_5074_numeral__div__minus__numeral,axiom,
    ! [M: num,N: num] : aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(num,int,numeral_numeral(int),M)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),N))) = aa(int,int,uminus_uminus(int),adjust_div(unique8689654367752047608divmod(int,M,N))) ).

% numeral_div_minus_numeral
tff(fact_5075_minus__numeral__div__numeral,axiom,
    ! [M: num,N: num] : aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),M))),aa(num,int,numeral_numeral(int),N)) = aa(int,int,uminus_uminus(int),adjust_div(unique8689654367752047608divmod(int,M,N))) ).

% minus_numeral_div_numeral
tff(fact_5076_diff__rat__def,axiom,
    ! [Q2: rat,R2: rat] : aa(rat,rat,minus_minus(rat,Q2),R2) = aa(rat,rat,aa(rat,fun(rat,rat),plus_plus(rat),Q2),aa(rat,rat,uminus_uminus(rat),R2)) ).

% diff_rat_def
tff(fact_5077_atLeastAtMostPlus1__int__conv,axiom,
    ! [M: int,N: int] :
      ( pp(aa(int,bool,ord_less_eq(int,M),aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),N)))
     => ( set_or1337092689740270186AtMost(int,M,aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),N)) = insert(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),N),set_or1337092689740270186AtMost(int,M,N)) ) ) ).

% atLeastAtMostPlus1_int_conv
tff(fact_5078_simp__from__to,axiom,
    ! [J: int,I: int] :
      ( ( pp(aa(int,bool,ord_less(int,J),I))
       => ( set_or1337092689740270186AtMost(int,I,J) = bot_bot(set(int)) ) )
      & ( ~ pp(aa(int,bool,ord_less(int,J),I))
       => ( set_or1337092689740270186AtMost(int,I,J) = insert(int,I,set_or1337092689740270186AtMost(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),I),one_one(int)),J)) ) ) ) ).

% simp_from_to
tff(fact_5079_rel__of__empty,axiom,
    ! [B: $tType,A: $tType,P: fun(product_prod(A,B),bool)] : rel_of(A,B,aTP_Lamp_lt(A,option(B)),P) = bot_bot(set(product_prod(A,B))) ).

% rel_of_empty
tff(fact_5080_neg__eucl__rel__int__mult__2,axiom,
    ! [B2: int,A2: int,Q2: int,R2: int] :
      ( pp(aa(int,bool,ord_less_eq(int,B2),zero_zero(int)))
     => ( eucl_rel_int(aa(int,int,aa(int,fun(int,int),plus_plus(int),A2),one_one(int)),B2,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q2),R2))
       => eucl_rel_int(aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),A2)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),B2),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q2),aa(int,int,minus_minus(int,aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),R2)),one_one(int)))) ) ) ).

% neg_eucl_rel_int_mult_2
tff(fact_5081_pos__eucl__rel__int__mult__2,axiom,
    ! [B2: int,A2: int,Q2: int,R2: int] :
      ( pp(aa(int,bool,ord_less_eq(int,zero_zero(int)),B2))
     => ( eucl_rel_int(A2,B2,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q2),R2))
       => eucl_rel_int(aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),A2)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),B2),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q2),aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),R2)))) ) ) ).

% pos_eucl_rel_int_mult_2
tff(fact_5082_unique__remainder,axiom,
    ! [A2: int,B2: int,Q2: int,R2: int,Q5: int,R4: int] :
      ( eucl_rel_int(A2,B2,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q2),R2))
     => ( eucl_rel_int(A2,B2,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q5),R4))
       => ( R2 = R4 ) ) ) ).

% unique_remainder
tff(fact_5083_unique__quotient,axiom,
    ! [A2: int,B2: int,Q2: int,R2: int,Q5: int,R4: int] :
      ( eucl_rel_int(A2,B2,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q2),R2))
     => ( eucl_rel_int(A2,B2,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q5),R4))
       => ( Q2 = Q5 ) ) ) ).

% unique_quotient
tff(fact_5084_eucl__rel__int__by0,axiom,
    ! [K: int] : eucl_rel_int(K,zero_zero(int),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),K)) ).

% eucl_rel_int_by0
tff(fact_5085_mod__int__unique,axiom,
    ! [K: int,L: int,Q2: int,R2: int] :
      ( eucl_rel_int(K,L,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q2),R2))
     => ( modulo_modulo(int,K,L) = R2 ) ) ).

% mod_int_unique
tff(fact_5086_div__int__unique,axiom,
    ! [K: int,L: int,Q2: int,R2: int] :
      ( eucl_rel_int(K,L,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q2),R2))
     => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),K),L) = Q2 ) ) ).

% div_int_unique
tff(fact_5087_eucl__rel__int__dividesI,axiom,
    ! [L: int,K: int,Q2: int] :
      ( ( L != zero_zero(int) )
     => ( ( K = aa(int,int,aa(int,fun(int,int),times_times(int),Q2),L) )
       => eucl_rel_int(K,L,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q2),zero_zero(int))) ) ) ).

% eucl_rel_int_dividesI
tff(fact_5088_eucl__rel__int,axiom,
    ! [K: int,L: int] : eucl_rel_int(K,L,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,aa(int,fun(int,int),divide_divide(int),K),L)),modulo_modulo(int,K,L))) ).

% eucl_rel_int
tff(fact_5089_zminus1__lemma,axiom,
    ! [A2: int,B2: int,Q2: int,R2: int] :
      ( eucl_rel_int(A2,B2,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q2),R2))
     => ( ( B2 != zero_zero(int) )
       => eucl_rel_int(aa(int,int,uminus_uminus(int),A2),B2,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),if(int,aa(int,bool,fequal(int,R2),zero_zero(int)),aa(int,int,uminus_uminus(int),Q2),aa(int,int,minus_minus(int,aa(int,int,uminus_uminus(int),Q2)),one_one(int)))),if(int,aa(int,bool,fequal(int,R2),zero_zero(int)),zero_zero(int),aa(int,int,minus_minus(int,B2),R2)))) ) ) ).

% zminus1_lemma
tff(fact_5090_eucl__rel__int__iff,axiom,
    ! [K: int,L: int,Q2: int,R2: int] :
      ( eucl_rel_int(K,L,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q2),R2))
    <=> ( ( K = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),L),Q2)),R2) )
        & ( pp(aa(int,bool,ord_less(int,zero_zero(int)),L))
         => ( pp(aa(int,bool,ord_less_eq(int,zero_zero(int)),R2))
            & pp(aa(int,bool,ord_less(int,R2),L)) ) )
        & ( ~ pp(aa(int,bool,ord_less(int,zero_zero(int)),L))
         => ( ( pp(aa(int,bool,ord_less(int,L),zero_zero(int)))
             => ( pp(aa(int,bool,ord_less(int,L),R2))
                & pp(aa(int,bool,ord_less_eq(int,R2),zero_zero(int))) ) )
            & ( ~ pp(aa(int,bool,ord_less(int,L),zero_zero(int)))
             => ( Q2 = zero_zero(int) ) ) ) ) ) ) ).

% eucl_rel_int_iff
tff(fact_5091_eucl__rel__int__remainderI,axiom,
    ! [R2: int,L: int,K: int,Q2: int] :
      ( ( sgn_sgn(int,R2) = sgn_sgn(int,L) )
     => ( pp(aa(int,bool,ord_less(int,aa(int,int,abs_abs(int),R2)),aa(int,int,abs_abs(int),L)))
       => ( ( K = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),Q2),L)),R2) )
         => eucl_rel_int(K,L,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q2),R2)) ) ) ) ).

% eucl_rel_int_remainderI
tff(fact_5092_eucl__rel__int_Osimps,axiom,
    ! [A1: int,A22: int,A32: product_prod(int,int)] :
      ( eucl_rel_int(A1,A22,A32)
    <=> ( ? [K2: int] :
            ( ( A1 = K2 )
            & ( A22 = zero_zero(int) )
            & ( A32 = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),K2) ) )
        | ? [L4: int,K2: int,Q4: int] :
            ( ( A1 = K2 )
            & ( A22 = L4 )
            & ( A32 = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q4),zero_zero(int)) )
            & ( L4 != zero_zero(int) )
            & ( K2 = aa(int,int,aa(int,fun(int,int),times_times(int),Q4),L4) ) )
        | ? [R5: int,L4: int,K2: int,Q4: int] :
            ( ( A1 = K2 )
            & ( A22 = L4 )
            & ( A32 = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q4),R5) )
            & ( sgn_sgn(int,R5) = sgn_sgn(int,L4) )
            & pp(aa(int,bool,ord_less(int,aa(int,int,abs_abs(int),R5)),aa(int,int,abs_abs(int),L4)))
            & ( K2 = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),Q4),L4)),R5) ) ) ) ) ).

% eucl_rel_int.simps
tff(fact_5093_eucl__rel__int_Ocases,axiom,
    ! [A1: int,A22: int,A32: product_prod(int,int)] :
      ( eucl_rel_int(A1,A22,A32)
     => ( ( ( A22 = zero_zero(int) )
         => ( A32 != aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),A1) ) )
       => ( ! [Q3: int] :
              ( ( A32 = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q3),zero_zero(int)) )
             => ( ( A22 != zero_zero(int) )
               => ( A1 != aa(int,int,aa(int,fun(int,int),times_times(int),Q3),A22) ) ) )
         => ~ ! [R3: int,Q3: int] :
                ( ( A32 = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q3),R3) )
               => ( ( sgn_sgn(int,R3) = sgn_sgn(int,A22) )
                 => ( pp(aa(int,bool,ord_less(int,aa(int,int,abs_abs(int),R3)),aa(int,int,abs_abs(int),A22)))
                   => ( A1 != aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),Q3),A22)),R3) ) ) ) ) ) ) ) ).

% eucl_rel_int.cases
tff(fact_5094_and__int_Oelims,axiom,
    ! [X: int,Xa2: int,Y: int] :
      ( ( aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),X),Xa2) = Y )
     => ( ( ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),X),insert(int,zero_zero(int),insert(int,aa(int,int,uminus_uminus(int),one_one(int)),bot_bot(set(int))))))
            & pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xa2),insert(int,zero_zero(int),insert(int,aa(int,int,uminus_uminus(int),one_one(int)),bot_bot(set(int)))))) )
         => ( Y = aa(int,int,uminus_uminus(int),aa(bool,int,zero_neq_one_of_bool(int),fconj(aa(bool,bool,fNot,aa(int,bool,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),X)),aa(bool,bool,fNot,aa(int,bool,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),Xa2))))) ) )
        & ( ~ ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),X),insert(int,zero_zero(int),insert(int,aa(int,int,uminus_uminus(int),one_one(int)),bot_bot(set(int))))))
              & pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xa2),insert(int,zero_zero(int),insert(int,aa(int,int,uminus_uminus(int),one_one(int)),bot_bot(set(int)))))) )
         => ( Y = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(bool,int,zero_neq_one_of_bool(int),fconj(aa(bool,bool,fNot,aa(int,bool,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),X)),aa(bool,bool,fNot,aa(int,bool,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),Xa2))))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),X),aa(num,int,numeral_numeral(int),bit0(one2)))),aa(int,int,aa(int,fun(int,int),divide_divide(int),Xa2),aa(num,int,numeral_numeral(int),bit0(one2)))))) ) ) ) ) ).

% and_int.elims
tff(fact_5095_and__int_Osimps,axiom,
    ! [K: int,L: int] :
      ( ( ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),K),insert(int,zero_zero(int),insert(int,aa(int,int,uminus_uminus(int),one_one(int)),bot_bot(set(int))))))
          & pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),L),insert(int,zero_zero(int),insert(int,aa(int,int,uminus_uminus(int),one_one(int)),bot_bot(set(int)))))) )
       => ( aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),K),L) = aa(int,int,uminus_uminus(int),aa(bool,int,zero_neq_one_of_bool(int),fconj(aa(bool,bool,fNot,aa(int,bool,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),K)),aa(bool,bool,fNot,aa(int,bool,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),L))))) ) )
      & ( ~ ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),K),insert(int,zero_zero(int),insert(int,aa(int,int,uminus_uminus(int),one_one(int)),bot_bot(set(int))))))
            & pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),L),insert(int,zero_zero(int),insert(int,aa(int,int,uminus_uminus(int),one_one(int)),bot_bot(set(int)))))) )
       => ( aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),K),L) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(bool,int,zero_neq_one_of_bool(int),fconj(aa(bool,bool,fNot,aa(int,bool,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),K)),aa(bool,bool,fNot,aa(int,bool,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),L))))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),K),aa(num,int,numeral_numeral(int),bit0(one2)))),aa(int,int,aa(int,fun(int,int),divide_divide(int),L),aa(num,int,numeral_numeral(int),bit0(one2)))))) ) ) ) ).

% and_int.simps
tff(fact_5096_product__nth,axiom,
    ! [A: $tType,B: $tType,N: nat,Xs: list(A),Ys: list(B)] :
      ( pp(aa(nat,bool,ord_less(nat,N),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(list(A),nat,size_size(list(A)),Xs)),aa(list(B),nat,size_size(list(B)),Ys))))
     => ( aa(nat,product_prod(A,B),nth(product_prod(A,B),product(A,B,Xs,Ys)),N) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(nat,A,nth(A,Xs),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(list(B),nat,size_size(list(B)),Ys)))),aa(nat,B,nth(B,Ys),modulo_modulo(nat,N,aa(list(B),nat,size_size(list(B)),Ys)))) ) ) ).

% product_nth
tff(fact_5097_and_Oright__idem,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A2),B2)),B2) = aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A2),B2) ) ).

% and.right_idem
tff(fact_5098_and_Oleft__idem,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A2),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A2),B2) ) ).

% and.left_idem
tff(fact_5099_and_Oidem,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A2),A2) = A2 ) ).

% and.idem
tff(fact_5100_and__zero__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A2),zero_zero(A)) = zero_zero(A) ) ).

% and_zero_eq
tff(fact_5101_zero__and__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),zero_zero(A)),A2) = zero_zero(A) ) ).

% zero_and_eq
tff(fact_5102_bit_Oconj__zero__left,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),zero_zero(A)),X) = zero_zero(A) ) ).

% bit.conj_zero_left
tff(fact_5103_bit_Oconj__zero__right,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),X),zero_zero(A)) = zero_zero(A) ) ).

% bit.conj_zero_right
tff(fact_5104_take__bit__and,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A2: A,B2: A] : aa(A,A,bit_se2584673776208193580ke_bit(A,N),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(A,A,bit_se2584673776208193580ke_bit(A,N),A2)),aa(A,A,bit_se2584673776208193580ke_bit(A,N),B2)) ) ).

% take_bit_and
tff(fact_5105_bit_Oconj__one__right,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),X),aa(A,A,uminus_uminus(A),one_one(A))) = X ) ).

% bit.conj_one_right
tff(fact_5106_and_Oright__neutral,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A2),aa(A,A,uminus_uminus(A),one_one(A))) = A2 ) ).

% and.right_neutral
tff(fact_5107_and_Oleft__neutral,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(A,A,uminus_uminus(A),one_one(A))),A2) = A2 ) ).

% and.left_neutral
tff(fact_5108_and__nonnegative__int__iff,axiom,
    ! [K: int,L: int] :
      ( pp(aa(int,bool,ord_less_eq(int,zero_zero(int)),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),K),L)))
    <=> ( pp(aa(int,bool,ord_less_eq(int,zero_zero(int)),K))
        | pp(aa(int,bool,ord_less_eq(int,zero_zero(int)),L)) ) ) ).

% and_nonnegative_int_iff
tff(fact_5109_and__negative__int__iff,axiom,
    ! [K: int,L: int] :
      ( pp(aa(int,bool,ord_less(int,aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),K),L)),zero_zero(int)))
    <=> ( pp(aa(int,bool,ord_less(int,K),zero_zero(int)))
        & pp(aa(int,bool,ord_less(int,L),zero_zero(int))) ) ) ).

% and_negative_int_iff
tff(fact_5110_length__product,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys: list(B)] : aa(list(product_prod(A,B)),nat,size_size(list(product_prod(A,B))),product(A,B,Xs,Ys)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(list(A),nat,size_size(list(A)),Xs)),aa(list(B),nat,size_size(list(B)),Ys)) ).

% length_product
tff(fact_5111_and__numerals_I2_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Y: num] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),one_one(A)),aa(num,A,numeral_numeral(A),aa(num,num,bit1,Y))) = one_one(A) ) ).

% and_numerals(2)
tff(fact_5112_and__numerals_I8_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [X: num] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(num,A,numeral_numeral(A),aa(num,num,bit1,X))),one_one(A)) = one_one(A) ) ).

% and_numerals(8)
tff(fact_5113_and__numerals_I1_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Y: num] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),one_one(A)),aa(num,A,numeral_numeral(A),bit0(Y))) = zero_zero(A) ) ).

% and_numerals(1)
tff(fact_5114_and__numerals_I5_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [X: num] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(num,A,numeral_numeral(A),bit0(X))),one_one(A)) = zero_zero(A) ) ).

% and_numerals(5)
tff(fact_5115_and__numerals_I3_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [X: num,Y: num] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(num,A,numeral_numeral(A),bit0(X))),aa(num,A,numeral_numeral(A),bit0(Y))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(num,A,numeral_numeral(A),X)),aa(num,A,numeral_numeral(A),Y))) ) ).

% and_numerals(3)
tff(fact_5116_and__minus__numerals_I2_J,axiom,
    ! [N: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),one_one(int)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,N)))) = one_one(int) ).

% and_minus_numerals(2)
tff(fact_5117_and__minus__numerals_I6_J,axiom,
    ! [N: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,N)))),one_one(int)) = one_one(int) ).

% and_minus_numerals(6)
tff(fact_5118_and__numerals_I4_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [X: num,Y: num] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(num,A,numeral_numeral(A),bit0(X))),aa(num,A,numeral_numeral(A),aa(num,num,bit1,Y))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(num,A,numeral_numeral(A),X)),aa(num,A,numeral_numeral(A),Y))) ) ).

% and_numerals(4)
tff(fact_5119_and__numerals_I6_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [X: num,Y: num] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(num,A,numeral_numeral(A),aa(num,num,bit1,X))),aa(num,A,numeral_numeral(A),bit0(Y))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(num,A,numeral_numeral(A),X)),aa(num,A,numeral_numeral(A),Y))) ) ).

% and_numerals(6)
tff(fact_5120_and__minus__numerals_I5_J,axiom,
    ! [N: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit0(N)))),one_one(int)) = zero_zero(int) ).

% and_minus_numerals(5)
tff(fact_5121_and__minus__numerals_I1_J,axiom,
    ! [N: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),one_one(int)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit0(N)))) = zero_zero(int) ).

% and_minus_numerals(1)
tff(fact_5122_and__numerals_I7_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [X: num,Y: num] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(num,A,numeral_numeral(A),aa(num,num,bit1,X))),aa(num,A,numeral_numeral(A),aa(num,num,bit1,Y))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(num,A,numeral_numeral(A),X)),aa(num,A,numeral_numeral(A),Y)))) ) ).

% and_numerals(7)
tff(fact_5123_of__int__and__eq,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [K: int,L: int] : aa(int,A,ring_1_of_int(A),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),K),L)) = aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(int,A,ring_1_of_int(A),K)),aa(int,A,ring_1_of_int(A),L)) ) ).

% of_int_and_eq
tff(fact_5124_bit__and__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,B2: A,N: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A2),B2)),N))
        <=> ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,A2),N))
            & pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,B2),N)) ) ) ) ).

% bit_and_iff
tff(fact_5125_bit__and__int__iff,axiom,
    ! [K: int,L: int,N: nat] :
      ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(int,aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),K),L)),N))
    <=> ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(int,K),N))
        & pp(aa(nat,bool,bit_se5641148757651400278ts_bit(int,L),N)) ) ) ).

% bit_and_int_iff
tff(fact_5126_and_Oleft__commute,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [B2: A,A2: A,C2: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),B2),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A2),C2)) = aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A2),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),B2),C2)) ) ).

% and.left_commute
tff(fact_5127_and_Ocommute,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A2),B2) = aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),B2),A2) ) ).

% and.commute
tff(fact_5128_and_Oassoc,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A2),B2)),C2) = aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A2),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),B2),C2)) ) ).

% and.assoc
tff(fact_5129_of__nat__and__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,N: nat] : aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344872417868541ns_and(nat),M),N)) = aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(nat,A,semiring_1_of_nat(A),M)),aa(nat,A,semiring_1_of_nat(A),N)) ) ).

% of_nat_and_eq
tff(fact_5130_and__eq__minus__1__iff,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A2: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A2),B2) = aa(A,A,uminus_uminus(A),one_one(A)) )
        <=> ( ( A2 = aa(A,A,uminus_uminus(A),one_one(A)) )
            & ( B2 = aa(A,A,uminus_uminus(A),one_one(A)) ) ) ) ) ).

% and_eq_minus_1_iff
tff(fact_5131_AND__upper2_H,axiom,
    ! [Y: int,Z: int,X: int] :
      ( pp(aa(int,bool,ord_less_eq(int,zero_zero(int)),Y))
     => ( pp(aa(int,bool,ord_less_eq(int,Y),Z))
       => pp(aa(int,bool,ord_less_eq(int,aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),X),Y)),Z)) ) ) ).

% AND_upper2'
tff(fact_5132_AND__upper1_H,axiom,
    ! [Y: int,Z: int,Ya: int] :
      ( pp(aa(int,bool,ord_less_eq(int,zero_zero(int)),Y))
     => ( pp(aa(int,bool,ord_less_eq(int,Y),Z))
       => pp(aa(int,bool,ord_less_eq(int,aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),Y),Ya)),Z)) ) ) ).

% AND_upper1'
tff(fact_5133_AND__upper2,axiom,
    ! [Y: int,X: int] :
      ( pp(aa(int,bool,ord_less_eq(int,zero_zero(int)),Y))
     => pp(aa(int,bool,ord_less_eq(int,aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),X),Y)),Y)) ) ).

% AND_upper2
tff(fact_5134_AND__upper1,axiom,
    ! [X: int,Y: int] :
      ( pp(aa(int,bool,ord_less_eq(int,zero_zero(int)),X))
     => pp(aa(int,bool,ord_less_eq(int,aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),X),Y)),X)) ) ).

% AND_upper1
tff(fact_5135_AND__lower,axiom,
    ! [X: int,Y: int] :
      ( pp(aa(int,bool,ord_less_eq(int,zero_zero(int)),X))
     => pp(aa(int,bool,ord_less_eq(int,zero_zero(int)),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),X),Y))) ) ).

% AND_lower
tff(fact_5136_AND__upper2_H_H,axiom,
    ! [Y: int,Z: int,X: int] :
      ( pp(aa(int,bool,ord_less_eq(int,zero_zero(int)),Y))
     => ( pp(aa(int,bool,ord_less(int,Y),Z))
       => pp(aa(int,bool,ord_less(int,aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),X),Y)),Z)) ) ) ).

% AND_upper2''
tff(fact_5137_AND__upper1_H_H,axiom,
    ! [Y: int,Z: int,Ya: int] :
      ( pp(aa(int,bool,ord_less_eq(int,zero_zero(int)),Y))
     => ( pp(aa(int,bool,ord_less(int,Y),Z))
       => pp(aa(int,bool,ord_less(int,aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),Y),Ya)),Z)) ) ) ).

% AND_upper1''
tff(fact_5138_and__less__eq,axiom,
    ! [L: int,K: int] :
      ( pp(aa(int,bool,ord_less(int,L),zero_zero(int)))
     => pp(aa(int,bool,ord_less_eq(int,aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),K),L)),K)) ) ).

% and_less_eq
tff(fact_5139_even__and__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A2),B2)))
        <=> ( pp(aa(A,bool,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A2))
            | pp(aa(A,bool,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),B2)) ) ) ) ).

% even_and_iff
tff(fact_5140_even__and__iff__int,axiom,
    ! [K: int,L: int] :
      ( pp(aa(int,bool,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),K),L)))
    <=> ( pp(aa(int,bool,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),K))
        | pp(aa(int,bool,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),L)) ) ) ).

% even_and_iff_int
tff(fact_5141_one__and__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),one_one(A)),A2) = modulo_modulo(A,A2,aa(num,A,numeral_numeral(A),bit0(one2))) ) ).

% one_and_eq
tff(fact_5142_and__one__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A2),one_one(A)) = modulo_modulo(A,A2,aa(num,A,numeral_numeral(A),bit0(one2))) ) ).

% and_one_eq
tff(fact_5143_and__exp__eq__0__iff__not__bit,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,N: nat] :
          ( ( aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A2),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N)) = zero_zero(A) )
        <=> ~ pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,A2),N)) ) ) ).

% and_exp_eq_0_iff_not_bit
tff(fact_5144_and__int__rec,axiom,
    ! [K: int,L: int] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),K),L) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(bool,int,zero_neq_one_of_bool(int),fconj(aa(bool,bool,fNot,aa(int,bool,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),K)),aa(bool,bool,fNot,aa(int,bool,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),L))))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),K),aa(num,int,numeral_numeral(int),bit0(one2)))),aa(int,int,aa(int,fun(int,int),divide_divide(int),L),aa(num,int,numeral_numeral(int),bit0(one2)))))) ).

% and_int_rec
tff(fact_5145_and__int__unfold,axiom,
    ! [K: int,L: int] :
      ( ( ( ( K = zero_zero(int) )
          | ( L = zero_zero(int) ) )
       => ( aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),K),L) = zero_zero(int) ) )
      & ( ~ ( ( K = zero_zero(int) )
            | ( L = zero_zero(int) ) )
       => ( ( ( K = aa(int,int,uminus_uminus(int),one_one(int)) )
           => ( aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),K),L) = L ) )
          & ( ( K != aa(int,int,uminus_uminus(int),one_one(int)) )
           => ( ( ( L = aa(int,int,uminus_uminus(int),one_one(int)) )
               => ( aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),K),L) = K ) )
              & ( ( L != aa(int,int,uminus_uminus(int),one_one(int)) )
               => ( aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),K),L) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),modulo_modulo(int,K,aa(num,int,numeral_numeral(int),bit0(one2)))),modulo_modulo(int,L,aa(num,int,numeral_numeral(int),bit0(one2))))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),K),aa(num,int,numeral_numeral(int),bit0(one2)))),aa(int,int,aa(int,fun(int,int),divide_divide(int),L),aa(num,int,numeral_numeral(int),bit0(one2)))))) ) ) ) ) ) ) ) ).

% and_int_unfold
tff(fact_5146_sgn__integer__code,axiom,
    ! [K: code_integer] :
      ( ( ( K = zero_zero(code_integer) )
       => ( sgn_sgn(code_integer,K) = zero_zero(code_integer) ) )
      & ( ( K != zero_zero(code_integer) )
       => ( ( pp(aa(code_integer,bool,ord_less(code_integer,K),zero_zero(code_integer)))
           => ( sgn_sgn(code_integer,K) = aa(code_integer,code_integer,uminus_uminus(code_integer),one_one(code_integer)) ) )
          & ( ~ pp(aa(code_integer,bool,ord_less(code_integer,K),zero_zero(code_integer)))
           => ( sgn_sgn(code_integer,K) = one_one(code_integer) ) ) ) ) ) ).

% sgn_integer_code
tff(fact_5147_and__int_Opsimps,axiom,
    ! [K: int,L: int] :
      ( accp(product_prod(int,int),bit_and_int_rel,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),K),L))
     => ( ( ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),K),insert(int,zero_zero(int),insert(int,aa(int,int,uminus_uminus(int),one_one(int)),bot_bot(set(int))))))
            & pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),L),insert(int,zero_zero(int),insert(int,aa(int,int,uminus_uminus(int),one_one(int)),bot_bot(set(int)))))) )
         => ( aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),K),L) = aa(int,int,uminus_uminus(int),aa(bool,int,zero_neq_one_of_bool(int),fconj(aa(bool,bool,fNot,aa(int,bool,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),K)),aa(bool,bool,fNot,aa(int,bool,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),L))))) ) )
        & ( ~ ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),K),insert(int,zero_zero(int),insert(int,aa(int,int,uminus_uminus(int),one_one(int)),bot_bot(set(int))))))
              & pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),L),insert(int,zero_zero(int),insert(int,aa(int,int,uminus_uminus(int),one_one(int)),bot_bot(set(int)))))) )
         => ( aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),K),L) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(bool,int,zero_neq_one_of_bool(int),fconj(aa(bool,bool,fNot,aa(int,bool,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),K)),aa(bool,bool,fNot,aa(int,bool,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),L))))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),K),aa(num,int,numeral_numeral(int),bit0(one2)))),aa(int,int,aa(int,fun(int,int),divide_divide(int),L),aa(num,int,numeral_numeral(int),bit0(one2)))))) ) ) ) ) ).

% and_int.psimps
tff(fact_5148_and__int_Opelims,axiom,
    ! [X: int,Xa2: int,Y: int] :
      ( ( aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),X),Xa2) = Y )
     => ( accp(product_prod(int,int),bit_and_int_rel,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),X),Xa2))
       => ~ ( ( ( ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),X),insert(int,zero_zero(int),insert(int,aa(int,int,uminus_uminus(int),one_one(int)),bot_bot(set(int))))))
                  & pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xa2),insert(int,zero_zero(int),insert(int,aa(int,int,uminus_uminus(int),one_one(int)),bot_bot(set(int)))))) )
               => ( Y = aa(int,int,uminus_uminus(int),aa(bool,int,zero_neq_one_of_bool(int),fconj(aa(bool,bool,fNot,aa(int,bool,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),X)),aa(bool,bool,fNot,aa(int,bool,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),Xa2))))) ) )
              & ( ~ ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),X),insert(int,zero_zero(int),insert(int,aa(int,int,uminus_uminus(int),one_one(int)),bot_bot(set(int))))))
                    & pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xa2),insert(int,zero_zero(int),insert(int,aa(int,int,uminus_uminus(int),one_one(int)),bot_bot(set(int)))))) )
               => ( Y = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(bool,int,zero_neq_one_of_bool(int),fconj(aa(bool,bool,fNot,aa(int,bool,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),X)),aa(bool,bool,fNot,aa(int,bool,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),Xa2))))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),X),aa(num,int,numeral_numeral(int),bit0(one2)))),aa(int,int,aa(int,fun(int,int),divide_divide(int),Xa2),aa(num,int,numeral_numeral(int),bit0(one2)))))) ) ) )
           => ~ accp(product_prod(int,int),bit_and_int_rel,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),X),Xa2)) ) ) ) ).

% and_int.pelims
tff(fact_5149_word__bitwise__m1__simps_I2_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A)))),X) = X ) ).

% word_bitwise_m1_simps(2)
tff(fact_5150_word__and__max,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),X),aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A)))) = X ) ).

% word_and_max
tff(fact_5151_word__no__log__defs_I3_J,axiom,
    ! [C: $tType] :
      ( type_len(C)
     => ! [A2: num,B2: num] : aa(word(C),word(C),aa(word(C),fun(word(C),word(C)),bit_se5824344872417868541ns_and(word(C)),aa(num,word(C),numeral_numeral(word(C)),A2)),aa(num,word(C),numeral_numeral(word(C)),B2)) = aa(int,word(C),ring_1_of_int(word(C)),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(num,int,numeral_numeral(int),A2)),aa(num,int,numeral_numeral(int),B2))) ) ).

% word_no_log_defs(3)
tff(fact_5152_and__nat__numerals_I3_J,axiom,
    ! [X: num] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344872417868541ns_and(nat),aa(num,nat,numeral_numeral(nat),bit0(X))),aa(nat,nat,suc,zero_zero(nat))) = zero_zero(nat) ).

% and_nat_numerals(3)
tff(fact_5153_and__nat__numerals_I1_J,axiom,
    ! [Y: num] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344872417868541ns_and(nat),aa(nat,nat,suc,zero_zero(nat))),aa(num,nat,numeral_numeral(nat),bit0(Y))) = zero_zero(nat) ).

% and_nat_numerals(1)
tff(fact_5154_and__nat__numerals_I4_J,axiom,
    ! [X: num] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344872417868541ns_and(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,X))),aa(nat,nat,suc,zero_zero(nat))) = one_one(nat) ).

% and_nat_numerals(4)
tff(fact_5155_and__nat__numerals_I2_J,axiom,
    ! [Y: num] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344872417868541ns_and(nat),aa(nat,nat,suc,zero_zero(nat))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,Y))) = one_one(nat) ).

% and_nat_numerals(2)
tff(fact_5156_word__bitwise__1__simps_I2_J,axiom,
    ! [B: $tType] :
      ( type_len(B)
     => ! [B2: num] : aa(word(B),word(B),aa(word(B),fun(word(B),word(B)),bit_se5824344872417868541ns_and(word(B)),one_one(word(B))),aa(num,word(B),numeral_numeral(word(B)),B2)) = aa(int,word(B),ring_1_of_int(word(B)),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),one_one(int)),aa(num,int,numeral_numeral(int),B2))) ) ).

% word_bitwise_1_simps(2)
tff(fact_5157_word__bitwise__1__simps_I4_J,axiom,
    ! [D6: $tType] :
      ( type_len(D6)
     => ! [A2: num] : aa(word(D6),word(D6),aa(word(D6),fun(word(D6),word(D6)),bit_se5824344872417868541ns_and(word(D6)),aa(num,word(D6),numeral_numeral(word(D6)),A2)),one_one(word(D6))) = aa(int,word(D6),ring_1_of_int(word(D6)),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(num,int,numeral_numeral(int),A2)),one_one(int))) ) ).

% word_bitwise_1_simps(4)
tff(fact_5158_word__no__log__defs_I6_J,axiom,
    ! [F: $tType] :
      ( type_len(F)
     => ! [A2: num,B2: num] : aa(word(F),word(F),aa(word(F),fun(word(F),word(F)),bit_se5824344872417868541ns_and(word(F)),aa(word(F),word(F),uminus_uminus(word(F)),aa(num,word(F),numeral_numeral(word(F)),A2))),aa(word(F),word(F),uminus_uminus(word(F)),aa(num,word(F),numeral_numeral(word(F)),B2))) = aa(int,word(F),ring_1_of_int(word(F)),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),A2))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),B2)))) ) ).

% word_no_log_defs(6)
tff(fact_5159_word__no__log__defs_I5_J,axiom,
    ! [E3: $tType] :
      ( type_len(E3)
     => ! [A2: num,B2: num] : aa(word(E3),word(E3),aa(word(E3),fun(word(E3),word(E3)),bit_se5824344872417868541ns_and(word(E3)),aa(word(E3),word(E3),uminus_uminus(word(E3)),aa(num,word(E3),numeral_numeral(word(E3)),A2))),aa(num,word(E3),numeral_numeral(word(E3)),B2)) = aa(int,word(E3),ring_1_of_int(word(E3)),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),A2))),aa(num,int,numeral_numeral(int),B2))) ) ).

% word_no_log_defs(5)
tff(fact_5160_word__no__log__defs_I4_J,axiom,
    ! [D6: $tType] :
      ( type_len(D6)
     => ! [A2: num,B2: num] : aa(word(D6),word(D6),aa(word(D6),fun(word(D6),word(D6)),bit_se5824344872417868541ns_and(word(D6)),aa(num,word(D6),numeral_numeral(word(D6)),A2)),aa(word(D6),word(D6),uminus_uminus(word(D6)),aa(num,word(D6),numeral_numeral(word(D6)),B2))) = aa(int,word(D6),ring_1_of_int(word(D6)),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(num,int,numeral_numeral(int),A2)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),B2)))) ) ).

% word_no_log_defs(4)
tff(fact_5161_Suc__0__and__eq,axiom,
    ! [N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344872417868541ns_and(nat),aa(nat,nat,suc,zero_zero(nat))),N) = modulo_modulo(nat,N,aa(num,nat,numeral_numeral(nat),bit0(one2))) ).

% Suc_0_and_eq
tff(fact_5162_and__Suc__0__eq,axiom,
    ! [N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344872417868541ns_and(nat),N),aa(nat,nat,suc,zero_zero(nat))) = modulo_modulo(nat,N,aa(num,nat,numeral_numeral(nat),bit0(one2))) ).

% and_Suc_0_eq
tff(fact_5163_word__bitwise__1__simps_I3_J,axiom,
    ! [C: $tType] :
      ( type_len(C)
     => ! [B2: num] : aa(word(C),word(C),aa(word(C),fun(word(C),word(C)),bit_se5824344872417868541ns_and(word(C)),one_one(word(C))),aa(word(C),word(C),uminus_uminus(word(C)),aa(num,word(C),numeral_numeral(word(C)),B2))) = aa(int,word(C),ring_1_of_int(word(C)),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),one_one(int)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),B2)))) ) ).

% word_bitwise_1_simps(3)
tff(fact_5164_word__bitwise__1__simps_I5_J,axiom,
    ! [E3: $tType] :
      ( type_len(E3)
     => ! [A2: num] : aa(word(E3),word(E3),aa(word(E3),fun(word(E3),word(E3)),bit_se5824344872417868541ns_and(word(E3)),aa(word(E3),word(E3),uminus_uminus(word(E3)),aa(num,word(E3),numeral_numeral(word(E3)),A2))),one_one(word(E3))) = aa(int,word(E3),ring_1_of_int(word(E3)),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),A2))),one_one(int))) ) ).

% word_bitwise_1_simps(5)
tff(fact_5165_word__log__esimps_I1_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),X),zero_zero(word(A))) = zero_zero(word(A)) ) ).

% word_log_esimps(1)
tff(fact_5166_word__log__esimps_I7_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),zero_zero(word(A))),X) = zero_zero(word(A)) ) ).

% word_log_esimps(7)
tff(fact_5167_word__and__max__word,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),A2: word(A)] :
          ( ( X = aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A))) )
         => ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),A2),X) = A2 ) ) ) ).

% word_and_max_word
tff(fact_5168_and__nat__def,axiom,
    ! [M: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344872417868541ns_and(nat),M),N) = nat2(aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(nat,int,semiring_1_of_nat(int),M)),aa(nat,int,semiring_1_of_nat(int),N))) ).

% and_nat_def
tff(fact_5169_zero__natural_Orsp,axiom,
    zero_zero(nat) = zero_zero(nat) ).

% zero_natural.rsp
tff(fact_5170_one__natural_Orsp,axiom,
    one_one(nat) = one_one(nat) ).

% one_natural.rsp
tff(fact_5171_zero__integer_Orsp,axiom,
    zero_zero(int) = zero_zero(int) ).

% zero_integer.rsp
tff(fact_5172_one__integer_Orsp,axiom,
    one_one(int) = one_one(int) ).

% one_integer.rsp
tff(fact_5173_less__eq__integer__code_I1_J,axiom,
    pp(aa(code_integer,bool,ord_less_eq(code_integer,zero_zero(code_integer)),zero_zero(code_integer))) ).

% less_eq_integer_code(1)
tff(fact_5174_uminus__integer__code_I1_J,axiom,
    aa(code_integer,code_integer,uminus_uminus(code_integer),zero_zero(code_integer)) = zero_zero(code_integer) ).

% uminus_integer_code(1)
tff(fact_5175_times__integer__code_I1_J,axiom,
    ! [K: code_integer] : aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),times_times(code_integer),K),zero_zero(code_integer)) = zero_zero(code_integer) ).

% times_integer_code(1)
tff(fact_5176_times__integer__code_I2_J,axiom,
    ! [L: code_integer] : aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),times_times(code_integer),zero_zero(code_integer)),L) = zero_zero(code_integer) ).

% times_integer_code(2)
tff(fact_5177_word__and__1,axiom,
    ! [B: $tType] :
      ( type_len(B)
     => ! [N: word(B)] :
          ( ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(word(B),N),zero_zero(nat)))
           => ( aa(word(B),word(B),aa(word(B),fun(word(B),word(B)),bit_se5824344872417868541ns_and(word(B)),N),one_one(word(B))) = one_one(word(B)) ) )
          & ( ~ pp(aa(nat,bool,bit_se5641148757651400278ts_bit(word(B),N),zero_zero(nat)))
           => ( aa(word(B),word(B),aa(word(B),fun(word(B),word(B)),bit_se5824344872417868541ns_and(word(B)),N),one_one(word(B))) = zero_zero(word(B)) ) ) ) ) ).

% word_and_1
tff(fact_5178_even__word__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A)] :
          ( pp(aa(word(A),bool,even_word(A),A2))
        <=> ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),A2),one_one(word(A))) = zero_zero(word(A)) ) ) ) ).

% even_word_iff
tff(fact_5179_nth__is__and__neq__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),N: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(word(A),X),N))
        <=> ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),X),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),N)) != zero_zero(word(A)) ) ) ) ).

% nth_is_and_neq_0
tff(fact_5180_and__neq__0__is__nth,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),N: nat,X: word(A)] :
          ( ( Y = aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),N) )
         => ( ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),X),Y) != zero_zero(word(A)) )
          <=> pp(aa(nat,bool,bit_se5641148757651400278ts_bit(word(A),X),N)) ) ) ) ).

% and_neq_0_is_nth
tff(fact_5181_and__nat__unfold,axiom,
    ! [M: nat,N: nat] :
      ( ( ( ( M = zero_zero(nat) )
          | ( N = zero_zero(nat) ) )
       => ( aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344872417868541ns_and(nat),M),N) = zero_zero(nat) ) )
      & ( ~ ( ( M = zero_zero(nat) )
            | ( N = zero_zero(nat) ) )
       => ( aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344872417868541ns_and(nat),M),N) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),modulo_modulo(nat,M,aa(num,nat,numeral_numeral(nat),bit0(one2)))),modulo_modulo(nat,N,aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344872417868541ns_and(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2)))))) ) ) ) ).

% and_nat_unfold
tff(fact_5182_and__nat__rec,axiom,
    ! [M: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344872417868541ns_and(nat),M),N) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(bool,nat,zero_neq_one_of_bool(nat),fconj(aa(bool,bool,fNot,aa(nat,bool,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),M)),aa(bool,bool,fNot,aa(nat,bool,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N))))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344872417868541ns_and(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2)))))) ).

% and_nat_rec
tff(fact_5183_abs__integer__code,axiom,
    ! [K: code_integer] :
      ( ( pp(aa(code_integer,bool,ord_less(code_integer,K),zero_zero(code_integer)))
       => ( aa(code_integer,code_integer,abs_abs(code_integer),K) = aa(code_integer,code_integer,uminus_uminus(code_integer),K) ) )
      & ( ~ pp(aa(code_integer,bool,ord_less(code_integer,K),zero_zero(code_integer)))
       => ( aa(code_integer,code_integer,abs_abs(code_integer),K) = K ) ) ) ).

% abs_integer_code
tff(fact_5184_less__integer__code_I1_J,axiom,
    ~ pp(aa(code_integer,bool,ord_less(code_integer,zero_zero(code_integer)),zero_zero(code_integer))) ).

% less_integer_code(1)
tff(fact_5185_and__int_Opinduct,axiom,
    ! [A0: int,A1: int,P: fun(int,fun(int,bool))] :
      ( accp(product_prod(int,int),bit_and_int_rel,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),A0),A1))
     => ( ! [K3: int,L3: int] :
            ( accp(product_prod(int,int),bit_and_int_rel,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),K3),L3))
           => ( ( ~ ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),K3),insert(int,zero_zero(int),insert(int,aa(int,int,uminus_uminus(int),one_one(int)),bot_bot(set(int))))))
                    & pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),L3),insert(int,zero_zero(int),insert(int,aa(int,int,uminus_uminus(int),one_one(int)),bot_bot(set(int)))))) )
               => pp(aa(int,bool,aa(int,fun(int,bool),P,aa(int,int,aa(int,fun(int,int),divide_divide(int),K3),aa(num,int,numeral_numeral(int),bit0(one2)))),aa(int,int,aa(int,fun(int,int),divide_divide(int),L3),aa(num,int,numeral_numeral(int),bit0(one2))))) )
             => pp(aa(int,bool,aa(int,fun(int,bool),P,K3),L3)) ) )
       => pp(aa(int,bool,aa(int,fun(int,bool),P,A0),A1)) ) ) ).

% and_int.pinduct
tff(fact_5186_plus__integer__code_I1_J,axiom,
    ! [K: code_integer] : aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),K),zero_zero(code_integer)) = K ).

% plus_integer_code(1)
tff(fact_5187_plus__integer__code_I2_J,axiom,
    ! [L: code_integer] : aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),zero_zero(code_integer)),L) = L ).

% plus_integer_code(2)
tff(fact_5188_minus__integer__code_I2_J,axiom,
    ! [L: code_integer] : aa(code_integer,code_integer,minus_minus(code_integer,zero_zero(code_integer)),L) = aa(code_integer,code_integer,uminus_uminus(code_integer),L) ).

% minus_integer_code(2)
tff(fact_5189_minus__integer__code_I1_J,axiom,
    ! [K: code_integer] : aa(code_integer,code_integer,minus_minus(code_integer,K),zero_zero(code_integer)) = K ).

% minus_integer_code(1)
tff(fact_5190_upto_Opinduct,axiom,
    ! [A0: int,A1: int,P: fun(int,fun(int,bool))] :
      ( accp(product_prod(int,int),upto_rel,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),A0),A1))
     => ( ! [I2: int,J2: int] :
            ( accp(product_prod(int,int),upto_rel,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),I2),J2))
           => ( ( pp(aa(int,bool,ord_less_eq(int,I2),J2))
               => pp(aa(int,bool,aa(int,fun(int,bool),P,aa(int,int,aa(int,fun(int,int),plus_plus(int),I2),one_one(int))),J2)) )
             => pp(aa(int,bool,aa(int,fun(int,bool),P,I2),J2)) ) )
       => pp(aa(int,bool,aa(int,fun(int,bool),P,A0),A1)) ) ) ).

% upto.pinduct
tff(fact_5191_integer__of__int__code,axiom,
    ! [K: int] :
      ( ( pp(aa(int,bool,ord_less(int,K),zero_zero(int)))
       => ( code_integer_of_int(K) = aa(code_integer,code_integer,uminus_uminus(code_integer),code_integer_of_int(aa(int,int,uminus_uminus(int),K))) ) )
      & ( ~ pp(aa(int,bool,ord_less(int,K),zero_zero(int)))
       => ( ( ( K = zero_zero(int) )
           => ( code_integer_of_int(K) = zero_zero(code_integer) ) )
          & ( ( K != zero_zero(int) )
           => ( code_integer_of_int(K) = if(code_integer,aa(int,bool,fequal(int,modulo_modulo(int,K,aa(num,int,numeral_numeral(int),bit0(one2)))),zero_zero(int)),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),times_times(code_integer),aa(num,code_integer,numeral_numeral(code_integer),bit0(one2))),code_integer_of_int(aa(int,int,aa(int,fun(int,int),divide_divide(int),K),aa(num,int,numeral_numeral(int),bit0(one2))))),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),times_times(code_integer),aa(num,code_integer,numeral_numeral(code_integer),bit0(one2))),code_integer_of_int(aa(int,int,aa(int,fun(int,int),divide_divide(int),K),aa(num,int,numeral_numeral(int),bit0(one2)))))),one_one(code_integer))) ) ) ) ) ) ).

% integer_of_int_code
tff(fact_5192_Arg__def,axiom,
    ! [Z: complex] :
      ( ( ( Z = zero_zero(complex) )
       => ( arg(Z) = zero_zero(real) ) )
      & ( ( Z != zero_zero(complex) )
       => ( arg(Z) = fChoice(real,aTP_Lamp_lu(complex,fun(real,bool),Z)) ) ) ) ).

% Arg_def
tff(fact_5193_some__insert__self,axiom,
    ! [A: $tType,S: set(A)] :
      ( ( S != bot_bot(set(A)) )
     => ( insert(A,fChoice(A,aTP_Lamp_a(set(A),fun(A,bool),S)),S) = S ) ) ).

% some_insert_self
tff(fact_5194_divide__integer_Oabs__eq,axiom,
    ! [Xa2: int,X: int] : aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),divide_divide(code_integer),code_integer_of_int(Xa2)),code_integer_of_int(X)) = code_integer_of_int(aa(int,int,aa(int,fun(int,int),divide_divide(int),Xa2),X)) ).

% divide_integer.abs_eq
tff(fact_5195_times__integer_Oabs__eq,axiom,
    ! [Xa2: int,X: int] : aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),times_times(code_integer),code_integer_of_int(Xa2)),code_integer_of_int(X)) = code_integer_of_int(aa(int,int,aa(int,fun(int,int),times_times(int),Xa2),X)) ).

% times_integer.abs_eq
tff(fact_5196_some__elem,axiom,
    ! [A: $tType,S: set(A)] :
      ( ( S != bot_bot(set(A)) )
     => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),fChoice(A,aTP_Lamp_a(set(A),fun(A,bool),S))),S)) ) ).

% some_elem
tff(fact_5197_verit__sko__forall__indirect2,axiom,
    ! [A: $tType,X: A,P: fun(A,bool),P3: fun(A,bool)] :
      ( ( X = fChoice(A,aTP_Lamp_bz(fun(A,bool),fun(A,bool),P)) )
     => ( ! [X3: A] :
            ( pp(aa(A,bool,P,X3))
          <=> pp(aa(A,bool,P3,X3)) )
       => ( ! [X_13: A] : pp(aa(A,bool,P3,X_13))
        <=> pp(aa(A,bool,P,X)) ) ) ) ).

% verit_sko_forall_indirect2
tff(fact_5198_verit__sko__forall__indirect,axiom,
    ! [A: $tType,X: A,P: fun(A,bool)] :
      ( ( X = fChoice(A,aTP_Lamp_bz(fun(A,bool),fun(A,bool),P)) )
     => ( ! [X_13: A] : pp(aa(A,bool,P,X_13))
      <=> pp(aa(A,bool,P,X)) ) ) ).

% verit_sko_forall_indirect
tff(fact_5199_verit__sko__ex__indirect2,axiom,
    ! [A: $tType,X: A,P: fun(A,bool),P3: fun(A,bool)] :
      ( ( X = fChoice(A,P) )
     => ( ! [X3: A] :
            ( pp(aa(A,bool,P,X3))
          <=> pp(aa(A,bool,P3,X3)) )
       => ( ? [X_13: A] : pp(aa(A,bool,P3,X_13))
        <=> pp(aa(A,bool,P,X)) ) ) ) ).

% verit_sko_ex_indirect2
tff(fact_5200_verit__sko__ex__indirect,axiom,
    ! [A: $tType,X: A,P: fun(A,bool)] :
      ( ( X = fChoice(A,P) )
     => ( ? [X_13: A] : pp(aa(A,bool,P,X_13))
      <=> pp(aa(A,bool,P,X)) ) ) ).

% verit_sko_ex_indirect
tff(fact_5201_verit__sko__forall_H_H,axiom,
    ! [A: $tType,B4: A,A3: A,P: fun(A,bool)] :
      ( ( B4 = A3 )
     => ( ( fChoice(A,P) = A3 )
      <=> ( fChoice(A,P) = B4 ) ) ) ).

% verit_sko_forall''
tff(fact_5202_verit__sko__forall_H,axiom,
    ! [A: $tType,P: fun(A,bool),A3: bool] :
      ( ( pp(aa(A,bool,P,fChoice(A,aTP_Lamp_bz(fun(A,bool),fun(A,bool),P))))
      <=> pp(A3) )
     => ( ! [X_13: A] : pp(aa(A,bool,P,X_13))
      <=> pp(A3) ) ) ).

% verit_sko_forall'
tff(fact_5203_verit__sko__forall,axiom,
    ! [A: $tType,P: fun(A,bool)] :
      ( ! [X_13: A] : pp(aa(A,bool,P,X_13))
    <=> pp(aa(A,bool,P,fChoice(A,aTP_Lamp_bz(fun(A,bool),fun(A,bool),P)))) ) ).

% verit_sko_forall
tff(fact_5204_verit__sko__ex_H,axiom,
    ! [A: $tType,P: fun(A,bool),A3: bool] :
      ( ( pp(aa(A,bool,P,fChoice(A,P)))
      <=> pp(A3) )
     => ( ? [X_13: A] : pp(aa(A,bool,P,X_13))
      <=> pp(A3) ) ) ).

% verit_sko_ex'
tff(fact_5205_zero__integer__def,axiom,
    zero_zero(code_integer) = code_integer_of_int(zero_zero(int)) ).

% zero_integer_def
tff(fact_5206_one__integer__def,axiom,
    one_one(code_integer) = code_integer_of_int(one_one(int)) ).

% one_integer_def
tff(fact_5207_plus__integer_Oabs__eq,axiom,
    ! [Xa2: int,X: int] : aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),code_integer_of_int(Xa2)),code_integer_of_int(X)) = code_integer_of_int(aa(int,int,aa(int,fun(int,int),plus_plus(int),Xa2),X)) ).

% plus_integer.abs_eq
tff(fact_5208_some__equality,axiom,
    ! [A: $tType,P: fun(A,bool),A2: A] :
      ( pp(aa(A,bool,P,A2))
     => ( ! [X3: A] :
            ( pp(aa(A,bool,P,X3))
           => ( X3 = A2 ) )
       => ( fChoice(A,P) = A2 ) ) ) ).

% some_equality
tff(fact_5209_some__eq__trivial,axiom,
    ! [A: $tType,X: A] : fChoice(A,aTP_Lamp_lg(A,fun(A,bool),X)) = X ).

% some_eq_trivial
tff(fact_5210_some__sym__eq__trivial,axiom,
    ! [A: $tType,X: A] : fChoice(A,fequal(A,X)) = X ).

% some_sym_eq_trivial
tff(fact_5211_some1__equality,axiom,
    ! [A: $tType,P: fun(A,bool),A2: A] :
      ( ? [X4: A] :
          ( pp(aa(A,bool,P,X4))
          & ! [Y3: A] :
              ( pp(aa(A,bool,P,Y3))
             => ( Y3 = X4 ) ) )
     => ( pp(aa(A,bool,P,A2))
       => ( fChoice(A,P) = A2 ) ) ) ).

% some1_equality
tff(fact_5212_some__eq__ex,axiom,
    ! [A: $tType,P: fun(A,bool)] :
      ( pp(aa(A,bool,P,fChoice(A,P)))
    <=> ? [X_13: A] : pp(aa(A,bool,P,X_13)) ) ).

% some_eq_ex
tff(fact_5213_someI2__bex,axiom,
    ! [A: $tType,A3: set(A),P: fun(A,bool),Q: fun(A,bool)] :
      ( ? [X4: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),A3))
          & pp(aa(A,bool,P,X4)) )
     => ( ! [X3: A] :
            ( ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),A3))
              & pp(aa(A,bool,P,X3)) )
           => pp(aa(A,bool,Q,X3)) )
       => pp(aa(A,bool,Q,fChoice(A,aa(fun(A,bool),fun(A,bool),aTP_Lamp_ae(set(A),fun(fun(A,bool),fun(A,bool)),A3),P)))) ) ) ).

% someI2_bex
tff(fact_5214_someI2__ex,axiom,
    ! [A: $tType,P: fun(A,bool),Q: fun(A,bool)] :
      ( ? [X_12: A] : pp(aa(A,bool,P,X_12))
     => ( ! [X3: A] :
            ( pp(aa(A,bool,P,X3))
           => pp(aa(A,bool,Q,X3)) )
       => pp(aa(A,bool,Q,fChoice(A,P))) ) ) ).

% someI2_ex
tff(fact_5215_someI__ex,axiom,
    ! [A: $tType,P: fun(A,bool)] :
      ( ? [X_12: A] : pp(aa(A,bool,P,X_12))
     => pp(aa(A,bool,P,fChoice(A,P))) ) ).

% someI_ex
tff(fact_5216_someI2,axiom,
    ! [A: $tType,P: fun(A,bool),A2: A,Q: fun(A,bool)] :
      ( pp(aa(A,bool,P,A2))
     => ( ! [X3: A] :
            ( pp(aa(A,bool,P,X3))
           => pp(aa(A,bool,Q,X3)) )
       => pp(aa(A,bool,Q,fChoice(A,P))) ) ) ).

% someI2
tff(fact_5217_dependent__nat__choice,axiom,
    ! [A: $tType,P: fun(nat,fun(A,bool)),Q: fun(nat,fun(A,fun(A,bool)))] :
      ( ? [X_12: A] : pp(aa(A,bool,aa(nat,fun(A,bool),P,zero_zero(nat)),X_12))
     => ( ! [X3: A,N2: nat] :
            ( pp(aa(A,bool,aa(nat,fun(A,bool),P,N2),X3))
           => ? [Y4: A] :
                ( pp(aa(A,bool,aa(nat,fun(A,bool),P,aa(nat,nat,suc,N2)),Y4))
                & pp(aa(A,bool,aa(A,fun(A,bool),aa(nat,fun(A,fun(A,bool)),Q,N2),X3),Y4)) ) )
       => ? [F4: fun(nat,A)] :
          ! [N11: nat] :
            ( pp(aa(A,bool,aa(nat,fun(A,bool),P,N11),aa(nat,A,F4,N11)))
            & pp(aa(A,bool,aa(A,fun(A,bool),aa(nat,fun(A,fun(A,bool)),Q,N11),aa(nat,A,F4,N11)),aa(nat,A,F4,aa(nat,nat,suc,N11)))) ) ) ) ).

% dependent_nat_choice
tff(fact_5218_some__in__eq,axiom,
    ! [A: $tType,A3: set(A)] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),fChoice(A,aTP_Lamp_a(set(A),fun(A,bool),A3))),A3))
    <=> ( A3 != bot_bot(set(A)) ) ) ).

% some_in_eq
tff(fact_5219_and__mask__arith_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: nat,W: word(A)] :
          ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),N))
         => ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),W),bit_se2239418461657761734s_mask(word(A),N)) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),W),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),aa(nat,nat,minus_minus(nat,aa(word(A),nat,size_size(word(A)),W)),N)))),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),aa(nat,nat,minus_minus(nat,aa(word(A),nat,size_size(word(A)),W)),N))) ) ) ) ).

% and_mask_arith'
tff(fact_5220_fun__of__rel__def,axiom,
    ! [A: $tType,B: $tType,R: set(product_prod(B,A)),X: B] : fun_of_rel(B,A,R,X) = fChoice(A,aa(B,fun(A,bool),aTP_Lamp_lv(set(product_prod(B,A)),fun(B,fun(A,bool)),R),X)) ).

% fun_of_rel_def
tff(fact_5221_divmod__BitM__2__eq,axiom,
    ! [M: num] : unique8689654367752047608divmod(int,bitM(M),bit0(one2)) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,minus_minus(int,aa(num,int,numeral_numeral(int),M)),one_one(int))),one_one(int)) ).

% divmod_BitM_2_eq
tff(fact_5222_mask__eq__0__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat] :
          ( ( bit_se2239418461657761734s_mask(A,N) = zero_zero(A) )
        <=> ( N = zero_zero(nat) ) ) ) ).

% mask_eq_0_iff
tff(fact_5223_mask__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ( bit_se2239418461657761734s_mask(A,zero_zero(nat)) = zero_zero(A) ) ) ).

% mask_0
tff(fact_5224_dbl__dec__simps_I5_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [K: num] : neg_numeral_dbl_dec(A,aa(num,A,numeral_numeral(A),K)) = aa(num,A,numeral_numeral(A),bitM(K)) ) ).

% dbl_dec_simps(5)
tff(fact_5225_Word_Omask__Suc__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ( bit_se2239418461657761734s_mask(A,aa(nat,nat,suc,zero_zero(nat))) = one_one(A) ) ) ).

% Word.mask_Suc_0
tff(fact_5226_take__bit__minus__one__eq__mask,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [N: nat] : aa(A,A,bit_se2584673776208193580ke_bit(A,N),aa(A,A,uminus_uminus(A),one_one(A))) = bit_se2239418461657761734s_mask(A,N) ) ).

% take_bit_minus_one_eq_mask
tff(fact_5227_pred__numeral__simps_I2_J,axiom,
    ! [K: num] : pred_numeral(bit0(K)) = aa(num,nat,numeral_numeral(nat),bitM(K)) ).

% pred_numeral_simps(2)
tff(fact_5228_of__int__mask__eq,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [N: nat] : aa(int,A,ring_1_of_int(A),bit_se2239418461657761734s_mask(int,N)) = bit_se2239418461657761734s_mask(A,N) ) ).

% of_int_mask_eq
tff(fact_5229_semiring__norm_I26_J,axiom,
    bitM(one2) = one2 ).

% semiring_norm(26)
tff(fact_5230_of__nat__mask__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat] : aa(nat,A,semiring_1_of_nat(A),bit_se2239418461657761734s_mask(nat,N)) = bit_se2239418461657761734s_mask(A,N) ) ).

% of_nat_mask_eq
tff(fact_5231_ex__mask__1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ? [X3: nat] : bit_se2239418461657761734s_mask(word(A),X3) = one_one(word(A)) ) ).

% ex_mask_1
tff(fact_5232_mask__twice2,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: nat,M: nat,X: word(A)] :
          ( pp(aa(nat,bool,ord_less_eq(nat,N),M))
         => ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),X),bit_se2239418461657761734s_mask(word(A),M))),bit_se2239418461657761734s_mask(word(A),N)) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),X),bit_se2239418461657761734s_mask(word(A),N)) ) ) ) ).

% mask_twice2
tff(fact_5233_mask__eqs_I1_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),N: nat,B2: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),A2),bit_se2239418461657761734s_mask(word(A),N))),B2)),bit_se2239418461657761734s_mask(word(A),N)) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),A2),B2)),bit_se2239418461657761734s_mask(word(A),N)) ) ).

% mask_eqs(1)
tff(fact_5234_mask__eqs_I2_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A),N: nat] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),A2),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),B2),bit_se2239418461657761734s_mask(word(A),N)))),bit_se2239418461657761734s_mask(word(A),N)) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),A2),B2)),bit_se2239418461657761734s_mask(word(A),N)) ) ).

% mask_eqs(2)
tff(fact_5235_mask__eqs_I7_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),N: nat,B2: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),A2),bit_se2239418461657761734s_mask(word(A),N))),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),B2),bit_se2239418461657761734s_mask(word(A),N)))),bit_se2239418461657761734s_mask(word(A),N)) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),A2),B2)),bit_se2239418461657761734s_mask(word(A),N)) ) ).

% mask_eqs(7)
tff(fact_5236_mask__power__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),N: nat,K: nat] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),X),bit_se2239418461657761734s_mask(word(A),N))),K)),bit_se2239418461657761734s_mask(word(A),N)) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),X),K)),bit_se2239418461657761734s_mask(word(A),N)) ) ).

% mask_power_eq
tff(fact_5237_mask__eqs_I5_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A),N: nat] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),A2),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),B2),bit_se2239418461657761734s_mask(word(A),N)))),bit_se2239418461657761734s_mask(word(A),N)) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),A2),B2)),bit_se2239418461657761734s_mask(word(A),N)) ) ).

% mask_eqs(5)
tff(fact_5238_mask__eqs_I6_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [B2: word(A),N: nat,A2: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),B2),bit_se2239418461657761734s_mask(word(A),N))),A2)),bit_se2239418461657761734s_mask(word(A),N)) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),B2),A2)),bit_se2239418461657761734s_mask(word(A),N)) ) ).

% mask_eqs(6)
tff(fact_5239_mask__eqs_I9_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),N: nat,B2: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),A2),bit_se2239418461657761734s_mask(word(A),N))),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),B2),bit_se2239418461657761734s_mask(word(A),N)))),bit_se2239418461657761734s_mask(word(A),N)) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),A2),B2)),bit_se2239418461657761734s_mask(word(A),N)) ) ).

% mask_eqs(9)
tff(fact_5240_semiring__norm_I27_J,axiom,
    ! [N: num] : bitM(bit0(N)) = aa(num,num,bit1,bitM(N)) ).

% semiring_norm(27)
tff(fact_5241_semiring__norm_I28_J,axiom,
    ! [N: num] : bitM(aa(num,num,bit1,N)) = aa(num,num,bit1,bit0(N)) ).

% semiring_norm(28)
tff(fact_5242_mask__1,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ( bit_se2239418461657761734s_mask(A,one_one(nat)) = one_one(A) ) ) ).

% mask_1
tff(fact_5243_take__bit__eq__mask,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A2: A] : aa(A,A,bit_se2584673776208193580ke_bit(A,N),A2) = aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A2),bit_se2239418461657761734s_mask(A,N)) ) ).

% take_bit_eq_mask
tff(fact_5244_inc__BitM__eq,axiom,
    ! [N: num] : inc(bitM(N)) = bit0(N) ).

% inc_BitM_eq
tff(fact_5245_BitM__inc__eq,axiom,
    ! [N: num] : bitM(inc(N)) = aa(num,num,bit1,N) ).

% BitM_inc_eq
tff(fact_5246_More__Word_Omask__Suc__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( bit_se2239418461657761734s_mask(word(A),aa(nat,nat,suc,zero_zero(nat))) = one_one(word(A)) ) ) ).

% More_Word.mask_Suc_0
tff(fact_5247_eval__nat__numeral_I2_J,axiom,
    ! [N: num] : aa(num,nat,numeral_numeral(nat),bit0(N)) = aa(nat,nat,suc,aa(num,nat,numeral_numeral(nat),bitM(N))) ).

% eval_nat_numeral(2)
tff(fact_5248_one__plus__BitM,axiom,
    ! [N: num] : aa(num,num,aa(num,fun(num,num),plus_plus(num),one2),bitM(N)) = bit0(N) ).

% one_plus_BitM
tff(fact_5249_BitM__plus__one,axiom,
    ! [N: num] : aa(num,num,aa(num,fun(num,num),plus_plus(num),bitM(N)),one2) = bit0(N) ).

% BitM_plus_one
tff(fact_5250_mask__bin,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: nat] : bit_se2239418461657761734s_mask(word(A),N) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,bit_se2584673776208193580ke_bit(int,N),aa(int,int,uminus_uminus(int),one_one(int)))) ) ).

% mask_bin
tff(fact_5251_numeral__BitM,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [N: num] : aa(num,A,numeral_numeral(A),bitM(N)) = aa(A,A,minus_minus(A,aa(num,A,numeral_numeral(A),bit0(N))),one_one(A)) ) ).

% numeral_BitM
tff(fact_5252_odd__numeral__BitM,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [W: num] : ~ pp(aa(A,bool,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),aa(num,A,numeral_numeral(A),bitM(W)))) ) ).

% odd_numeral_BitM
tff(fact_5253_word__1FF__is__mask,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( aa(num,word(A),numeral_numeral(word(A)),aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,one2))))))))) = bit_se2239418461657761734s_mask(word(A),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,bit0(bit0(one2))))) ) ) ).

% word_1FF_is_mask
tff(fact_5254_word__FF__is__mask,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( aa(num,word(A),numeral_numeral(word(A)),aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,one2)))))))) = bit_se2239418461657761734s_mask(word(A),aa(num,nat,numeral_numeral(nat),bit0(bit0(bit0(one2))))) ) ) ).

% word_FF_is_mask
tff(fact_5255_and__mask__no,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [I: num,N: nat] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),aa(num,word(A),numeral_numeral(word(A)),I)),bit_se2239418461657761734s_mask(word(A),N)) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,bit_se2584673776208193580ke_bit(int,N),aa(num,int,numeral_numeral(int),I))) ) ).

% and_mask_no
tff(fact_5256_semiring__bit__operations__class_Oeven__mask__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat] :
          ( pp(aa(A,bool,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),bit_se2239418461657761734s_mask(A,N)))
        <=> ( N = zero_zero(nat) ) ) ) ).

% semiring_bit_operations_class.even_mask_iff
tff(fact_5257_mask__plus__1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: nat] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),bit_se2239418461657761734s_mask(word(A),N)),one_one(word(A))) = aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),N) ) ).

% mask_plus_1
tff(fact_5258_less__mask__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),N: nat] :
          ( pp(aa(word(A),bool,ord_less(word(A),X),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),N)))
         => ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),X),bit_se2239418461657761734s_mask(word(A),N)) = X ) ) ) ).

% less_mask_eq
tff(fact_5259_mask__eq__decr__exp,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: nat] : bit_se2239418461657761734s_mask(word(A),N) = aa(word(A),word(A),minus_minus(word(A),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),N)),one_one(word(A))) ) ).

% mask_eq_decr_exp
tff(fact_5260_mask__eq__exp__minus__1,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat] : bit_se2239418461657761734s_mask(A,N) = aa(A,A,minus_minus(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N)),one_one(A)) ) ).

% mask_eq_exp_minus_1
tff(fact_5261_mask__eq__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),N: nat] :
          ( ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),W),bit_se2239418461657761734s_mask(word(A),N)) = W )
        <=> pp(aa(int,bool,ord_less(int,aa(word(A),int,semiring_1_unsigned(A,int),W)),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),N))) ) ) ).

% mask_eq_iff
tff(fact_5262_and__mask__lt__2p,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),N: nat] : pp(aa(int,bool,ord_less(int,aa(word(A),int,semiring_1_unsigned(A,int),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),W),bit_se2239418461657761734s_mask(word(A),N)))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),N))) ) ).

% and_mask_lt_2p
tff(fact_5263_mask__Suc__rec,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: nat] : bit_se2239418461657761734s_mask(word(A),aa(nat,nat,suc,N)) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),bit_se2239418461657761734s_mask(word(A),N))),one_one(word(A))) ) ).

% mask_Suc_rec
tff(fact_5264_is__aligned__AND__less__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [U: word(A),N: nat,V: word(A)] :
          ( ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),U),bit_se2239418461657761734s_mask(word(A),N)) = zero_zero(word(A)) )
         => ( pp(aa(word(A),bool,ord_less(word(A),V),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),N)))
           => ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),U),V) = zero_zero(word(A)) ) ) ) ) ).

% is_aligned_AND_less_0
tff(fact_5265_add__mask__fold,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),N: nat] : aa(word(A),word(A),minus_minus(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),X),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),N))),one_one(word(A))) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),X),bit_se2239418461657761734s_mask(word(A),N)) ) ).

% add_mask_fold
tff(fact_5266_and__mask__dvd__nat,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: nat,W: word(A)] :
          ( pp(aa(nat,bool,dvd_dvd(nat,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)),aa(word(A),nat,semiring_1_unsigned(A,nat),W)))
        <=> ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),W),bit_se2239418461657761734s_mask(word(A),N)) = zero_zero(word(A)) ) ) ) ).

% and_mask_dvd_nat
tff(fact_5267_and__mask__mod__2p,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),N: nat] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),W),bit_se2239418461657761734s_mask(word(A),N)) = aa(int,word(A),ring_1_of_int(word(A)),modulo_modulo(int,aa(word(A),int,semiring_1_unsigned(A,int),W),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),N))) ) ).

% and_mask_mod_2p
tff(fact_5268_and__mask__dvd,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: nat,W: word(A)] :
          ( pp(aa(int,bool,dvd_dvd(int,aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),N)),aa(word(A),int,semiring_1_unsigned(A,int),W)))
        <=> ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),W),bit_se2239418461657761734s_mask(word(A),N)) = zero_zero(word(A)) ) ) ) ).

% and_mask_dvd
tff(fact_5269_and__mask__less__size,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: nat,X: word(A)] :
          ( pp(aa(nat,bool,ord_less(nat,N),aa(word(A),nat,size_size(word(A)),X)))
         => pp(aa(word(A),bool,ord_less(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),X),bit_se2239418461657761734s_mask(word(A),N))),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),N))) ) ) ).

% and_mask_less_size
tff(fact_5270_mask__eq__iff__w2p,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: nat,W: word(A)] :
          ( pp(aa(nat,bool,ord_less(nat,N),aa(word(A),nat,size_size(word(A)),W)))
         => ( ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),W),bit_se2239418461657761734s_mask(word(A),N)) = W )
          <=> pp(aa(word(A),bool,ord_less(word(A),W),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),N))) ) ) ) ).

% mask_eq_iff_w2p
tff(fact_5271_word__and__mask__le__2pm1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),N: nat] : pp(aa(word(A),bool,ord_less_eq(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),W),bit_se2239418461657761734s_mask(word(A),N))),aa(word(A),word(A),minus_minus(word(A),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),N)),one_one(word(A))))) ) ).

% word_and_mask_le_2pm1
tff(fact_5272_word__mod__2p__is__mask,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: nat,X: word(A)] :
          ( pp(aa(word(A),bool,ord_less(word(A),zero_zero(word(A))),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),N)))
         => ( modulo_modulo(word(A),X,aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),N)) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),X),bit_se2239418461657761734s_mask(word(A),N)) ) ) ) ).

% word_mod_2p_is_mask
tff(fact_5273_word__unat__mask__lt,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [M: nat,W: word(A)] :
          ( pp(aa(nat,bool,ord_less_eq(nat,M),aa(word(A),nat,size_size(word(A)),W)))
         => pp(aa(nat,bool,ord_less(nat,aa(word(A),nat,semiring_1_unsigned(A,nat),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),W),bit_se2239418461657761734s_mask(word(A),M)))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M))) ) ) ).

% word_unat_mask_lt
tff(fact_5274_and__mask__arith,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),N: nat] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),W),bit_se2239418461657761734s_mask(word(A),N)) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),W),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),aa(nat,nat,minus_minus(nat,aa(word(A),nat,size_size(word(A)),W)),N)))),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),aa(nat,nat,minus_minus(nat,aa(word(A),nat,size_size(word(A)),W)),N))) ) ).

% and_mask_arith
tff(fact_5275_mask__numeral,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: num] : bit_se2239418461657761734s_mask(A,aa(num,nat,numeral_numeral(nat),N)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),bit_se2239418461657761734s_mask(A,pred_numeral(N)))) ) ).

% mask_numeral
tff(fact_5276_option_Osize__gen_I2_J,axiom,
    ! [A: $tType,X: fun(A,nat),X23: A] : size_option(A,X,aa(A,option(A),some(A),X23)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(A,nat,X,X23)),aa(nat,nat,suc,zero_zero(nat))) ).

% option.size_gen(2)
tff(fact_5277_divmod__algorithm__code_I6_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M: num,N: num] : unique8689654367752047608divmod(A,aa(num,num,bit1,M),bit0(N)) = aa(product_prod(A,A),product_prod(A,A),product_case_prod(A,A,product_prod(A,A),aTP_Lamp_lw(A,fun(A,product_prod(A,A)))),unique8689654367752047608divmod(A,M,N)) ) ).

% divmod_algorithm_code(6)
tff(fact_5278_neg__mask__is__div_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: nat,W: word(A)] :
          ( pp(aa(nat,bool,ord_less(nat,N),aa(word(A),nat,size_size(word(A)),W)))
         => ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),W),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),bit_se2239418461657761734s_mask(word(A),N))) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),W),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),N))),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),N)) ) ) ) ).

% neg_mask_is_div'
tff(fact_5279_bit_Odouble__compl,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [X: A] : aa(A,A,bit_ri4277139882892585799ns_not(A),aa(A,A,bit_ri4277139882892585799ns_not(A),X)) = X ) ).

% bit.double_compl
tff(fact_5280_bit_Ocompl__eq__compl__iff,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [X: A,Y: A] :
          ( ( aa(A,A,bit_ri4277139882892585799ns_not(A),X) = aa(A,A,bit_ri4277139882892585799ns_not(A),Y) )
        <=> ( X = Y ) ) ) ).

% bit.compl_eq_compl_iff
tff(fact_5281_bit_Oconj__cancel__right,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),X),aa(A,A,bit_ri4277139882892585799ns_not(A),X)) = zero_zero(A) ) ).

% bit.conj_cancel_right
tff(fact_5282_bit_Oconj__cancel__left,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(A,A,bit_ri4277139882892585799ns_not(A),X)),X) = zero_zero(A) ) ).

% bit.conj_cancel_left
tff(fact_5283_word__and__not,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),X),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),X)) = zero_zero(word(A)) ) ).

% word_and_not
tff(fact_5284_and__and__not,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),A2),B2)),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),B2)) = zero_zero(word(A)) ) ).

% and_and_not
tff(fact_5285_mask__nat__positive__iff,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),bit_se2239418461657761734s_mask(nat,N)))
    <=> pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),N)) ) ).

% mask_nat_positive_iff
tff(fact_5286_bit_Ocompl__one,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ( aa(A,A,bit_ri4277139882892585799ns_not(A),aa(A,A,uminus_uminus(A),one_one(A))) = zero_zero(A) ) ) ).

% bit.compl_one
tff(fact_5287_bit_Ocompl__zero,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ( aa(A,A,bit_ri4277139882892585799ns_not(A),zero_zero(A)) = aa(A,A,uminus_uminus(A),one_one(A)) ) ) ).

% bit.compl_zero
tff(fact_5288_NOT__mask__AND__mask,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [W: A,N: nat] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),W),bit_se2239418461657761734s_mask(A,N))),aa(A,A,bit_ri4277139882892585799ns_not(A),bit_se2239418461657761734s_mask(A,N))) = zero_zero(A) ) ).

% NOT_mask_AND_mask
tff(fact_5289_minus__not__numeral__eq,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [N: num] : aa(A,A,uminus_uminus(A),aa(A,A,bit_ri4277139882892585799ns_not(A),aa(num,A,numeral_numeral(A),N))) = aa(num,A,numeral_numeral(A),inc(N)) ) ).

% minus_not_numeral_eq
tff(fact_5290_word__bitwise__m1__simps_I1_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A)))) = zero_zero(word(A)) ) ) ).

% word_bitwise_m1_simps(1)
tff(fact_5291_word__add__not,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),X),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),X)) = aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A))) ) ).

% word_add_not
tff(fact_5292_even__not__iff,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A2: A] :
          ( pp(aa(A,bool,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,bit_ri4277139882892585799ns_not(A),A2)))
        <=> ~ pp(aa(A,bool,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A2)) ) ) ).

% even_not_iff
tff(fact_5293_not__one__eq,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ( aa(A,A,bit_ri4277139882892585799ns_not(A),one_one(A)) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),bit0(one2))) ) ) ).

% not_one_eq
tff(fact_5294_compl__of__1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),one_one(word(A))) = aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))) ) ) ).

% compl_of_1
tff(fact_5295_divmod__algorithm__code_I5_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M: num,N: num] : unique8689654367752047608divmod(A,bit0(M),bit0(N)) = aa(product_prod(A,A),product_prod(A,A),product_case_prod(A,A,product_prod(A,A),aTP_Lamp_lx(A,fun(A,product_prod(A,A)))),unique8689654367752047608divmod(A,M,N)) ) ).

% divmod_algorithm_code(5)
tff(fact_5296_nat__mask__eq,axiom,
    ! [N: nat] : nat2(bit_se2239418461657761734s_mask(int,N)) = bit_se2239418461657761734s_mask(nat,N) ).

% nat_mask_eq
tff(fact_5297_less__eq__mask,axiom,
    ! [N: nat] : pp(aa(nat,bool,ord_less_eq(nat,N),bit_se2239418461657761734s_mask(nat,N))) ).

% less_eq_mask
tff(fact_5298_of__int__not__numeral,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [K: num] : aa(int,A,ring_1_of_int(A),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),K))) = aa(A,A,bit_ri4277139882892585799ns_not(A),aa(num,A,numeral_numeral(A),K)) ) ).

% of_int_not_numeral
tff(fact_5299_take__bit__not__iff,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [N: nat,A2: A,B2: A] :
          ( ( aa(A,A,bit_se2584673776208193580ke_bit(A,N),aa(A,A,bit_ri4277139882892585799ns_not(A),A2)) = aa(A,A,bit_se2584673776208193580ke_bit(A,N),aa(A,A,bit_ri4277139882892585799ns_not(A),B2)) )
        <=> ( aa(A,A,bit_se2584673776208193580ke_bit(A,N),A2) = aa(A,A,bit_se2584673776208193580ke_bit(A,N),B2) ) ) ) ).

% take_bit_not_iff
tff(fact_5300_take__bit__not__take__bit,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [N: nat,A2: A] : aa(A,A,bit_se2584673776208193580ke_bit(A,N),aa(A,A,bit_ri4277139882892585799ns_not(A),aa(A,A,bit_se2584673776208193580ke_bit(A,N),A2))) = aa(A,A,bit_se2584673776208193580ke_bit(A,N),aa(A,A,bit_ri4277139882892585799ns_not(A),A2)) ) ).

% take_bit_not_take_bit
tff(fact_5301_nested__case__prod__simp,axiom,
    ! [D6: $tType,A: $tType,C: $tType,B: $tType,F2: fun(B,fun(C,fun(D6,A))),X: product_prod(B,C),Y: D6] : aa(D6,A,aa(product_prod(B,C),fun(D6,A),product_case_prod(B,C,fun(D6,A),F2),X),Y) = aa(product_prod(B,C),A,product_case_prod(B,C,A,aa(D6,fun(B,fun(C,A)),aTP_Lamp_ly(fun(B,fun(C,fun(D6,A))),fun(D6,fun(B,fun(C,A))),F2),Y)),X) ).

% nested_case_prod_simp
tff(fact_5302_of__int__not__eq,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [K: int] : aa(int,A,ring_1_of_int(A),aa(int,int,bit_ri4277139882892585799ns_not(int),K)) = aa(A,A,bit_ri4277139882892585799ns_not(A),aa(int,A,ring_1_of_int(A),K)) ) ).

% of_int_not_eq
tff(fact_5303_not__add__distrib,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A2: A,B2: A] : aa(A,A,bit_ri4277139882892585799ns_not(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)) = aa(A,A,minus_minus(A,aa(A,A,bit_ri4277139882892585799ns_not(A),A2)),B2) ) ).

% not_add_distrib
tff(fact_5304_not__diff__distrib,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A2: A,B2: A] : aa(A,A,bit_ri4277139882892585799ns_not(A),aa(A,A,minus_minus(A,A2),B2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,bit_ri4277139882892585799ns_not(A),A2)),B2) ) ).

% not_diff_distrib
tff(fact_5305_mask__eq__0__eq__x,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),W: word(A)] :
          ( ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),X),W) = zero_zero(word(A)) )
        <=> ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),X),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),W)) = X ) ) ) ).

% mask_eq_0_eq_x
tff(fact_5306_mask__eq__x__eq__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),W: word(A)] :
          ( ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),X),W) = X )
        <=> ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),X),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),W)) = zero_zero(word(A)) ) ) ) ).

% mask_eq_x_eq_0
tff(fact_5307_word__plus__and__or__coroll2,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),W: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),X),W)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),X),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),W))) = X ) ).

% word_plus_and_or_coroll2
tff(fact_5308_mask__nonnegative__int,axiom,
    ! [N: nat] : pp(aa(int,bool,ord_less_eq(int,zero_zero(int)),bit_se2239418461657761734s_mask(int,N))) ).

% mask_nonnegative_int
tff(fact_5309_not__mask__negative__int,axiom,
    ! [N: nat] : ~ pp(aa(int,bool,ord_less(int,bit_se2239418461657761734s_mask(int,N)),zero_zero(int))) ).

% not_mask_negative_int
tff(fact_5310_minus__eq__not__plus__1,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A2: A] : aa(A,A,uminus_uminus(A),A2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,bit_ri4277139882892585799ns_not(A),A2)),one_one(A)) ) ).

% minus_eq_not_plus_1
tff(fact_5311_not__eq__complement,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A2: A] : aa(A,A,bit_ri4277139882892585799ns_not(A),A2) = aa(A,A,minus_minus(A,aa(A,A,uminus_uminus(A),A2)),one_one(A)) ) ).

% not_eq_complement
tff(fact_5312_minus__eq__not__minus__1,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A2: A] : aa(A,A,uminus_uminus(A),A2) = aa(A,A,bit_ri4277139882892585799ns_not(A),aa(A,A,minus_minus(A,A2),one_one(A))) ) ).

% minus_eq_not_minus_1
tff(fact_5313_disjunctive__diff,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [B2: A,A2: A] :
          ( ! [N2: nat] :
              ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,B2),N2))
             => pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,A2),N2)) )
         => ( aa(A,A,minus_minus(A,A2),B2) = aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A2),aa(A,A,bit_ri4277139882892585799ns_not(A),B2)) ) ) ) ).

% disjunctive_diff
tff(fact_5314_take__bit__not__eq__mask__diff,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [N: nat,A2: A] : aa(A,A,bit_se2584673776208193580ke_bit(A,N),aa(A,A,bit_ri4277139882892585799ns_not(A),A2)) = aa(A,A,minus_minus(A,bit_se2239418461657761734s_mask(A,N)),aa(A,A,bit_se2584673776208193580ke_bit(A,N),A2)) ) ).

% take_bit_not_eq_mask_diff
tff(fact_5315_mask__lower__twice,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: nat,M: nat,X: word(A)] :
          ( pp(aa(nat,bool,ord_less_eq(nat,N),M))
         => ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),X),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),bit_se2239418461657761734s_mask(word(A),N)))),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),bit_se2239418461657761734s_mask(word(A),M))) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),X),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),bit_se2239418461657761734s_mask(word(A),M))) ) ) ) ).

% mask_lower_twice
tff(fact_5316_mask__out__first__mask__some,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),N: nat,Y: word(A),M: nat] :
          ( ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),X),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),bit_se2239418461657761734s_mask(word(A),N))) = Y )
         => ( pp(aa(nat,bool,ord_less_eq(nat,N),M))
           => ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),X),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),bit_se2239418461657761734s_mask(word(A),M))) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Y),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),bit_se2239418461657761734s_mask(word(A),M))) ) ) ) ) ).

% mask_out_first_mask_some
tff(fact_5317_NOT__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A)] : aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),X) = aa(word(A),word(A),minus_minus(word(A),aa(word(A),word(A),uminus_uminus(word(A)),X)),one_one(word(A))) ) ).

% NOT_eq
tff(fact_5318_mask__AND__NOT__mask,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),N: nat] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),W),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),bit_se2239418461657761734s_mask(word(A),N)))),bit_se2239418461657761734s_mask(word(A),N)) = zero_zero(word(A)) ) ).

% mask_AND_NOT_mask
tff(fact_5319_AND__NOT__mask__plus__AND__mask__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),N: nat] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),W),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),bit_se2239418461657761734s_mask(word(A),N)))),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),W),bit_se2239418461657761734s_mask(word(A),N))) = W ) ).

% AND_NOT_mask_plus_AND_mask_eq
tff(fact_5320_minus__numeral__inc__eq,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [N: num] : aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),inc(N))) = aa(A,A,bit_ri4277139882892585799ns_not(A),aa(num,A,numeral_numeral(A),N)) ) ).

% minus_numeral_inc_eq
tff(fact_5321_less__mask,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,ord_less(nat,aa(nat,nat,suc,zero_zero(nat))),N))
     => pp(aa(nat,bool,ord_less(nat,N),bit_se2239418461657761734s_mask(nat,N))) ) ).

% less_mask
tff(fact_5322_not__numeral__Bit0__eq,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [N: num] : aa(A,A,bit_ri4277139882892585799ns_not(A),aa(num,A,numeral_numeral(A),bit0(N))) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),aa(num,num,bit1,N))) ) ).

% not_numeral_Bit0_eq
tff(fact_5323_take__bit__not__mask__eq__0,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [M: nat,N: nat] :
          ( pp(aa(nat,bool,ord_less_eq(nat,M),N))
         => ( aa(A,A,bit_se2584673776208193580ke_bit(A,M),aa(A,A,bit_ri4277139882892585799ns_not(A),bit_se2239418461657761734s_mask(A,N))) = zero_zero(A) ) ) ) ).

% take_bit_not_mask_eq_0
tff(fact_5324_not__numeral__BitM__eq,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [N: num] : aa(A,A,bit_ri4277139882892585799ns_not(A),aa(num,A,numeral_numeral(A),bitM(N))) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),bit0(N))) ) ).

% not_numeral_BitM_eq
tff(fact_5325_multiple__mask__trivia,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [M: nat,N: nat,X: word(A)] :
          ( pp(aa(nat,bool,ord_less_eq(nat,M),N))
         => ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),X),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),bit_se2239418461657761734s_mask(word(A),N)))),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),X),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),bit_se2239418461657761734s_mask(word(A),N)),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),bit_se2239418461657761734s_mask(word(A),M))))) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),X),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),bit_se2239418461657761734s_mask(word(A),M))) ) ) ) ).

% multiple_mask_trivia
tff(fact_5326_and__mask__0__iff__le__mask,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),N: nat] :
          ( ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),W),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),bit_se2239418461657761734s_mask(word(A),N))) = zero_zero(word(A)) )
        <=> pp(aa(word(A),bool,ord_less_eq(word(A),W),bit_se2239418461657761734s_mask(word(A),N))) ) ) ).

% and_mask_0_iff_le_mask
tff(fact_5327_take__bit__eq__mask__iff,axiom,
    ! [N: nat,K: int] :
      ( ( aa(int,int,bit_se2584673776208193580ke_bit(int,N),K) = bit_se2239418461657761734s_mask(int,N) )
    <=> ( aa(int,int,bit_se2584673776208193580ke_bit(int,N),aa(int,int,aa(int,fun(int,int),plus_plus(int),K),one_one(int))) = zero_zero(int) ) ) ).

% take_bit_eq_mask_iff
tff(fact_5328_Suc__mask__eq__exp,axiom,
    ! [N: nat] : aa(nat,nat,suc,bit_se2239418461657761734s_mask(nat,N)) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N) ).

% Suc_mask_eq_exp
tff(fact_5329_mask__nat__less__exp,axiom,
    ! [N: nat] : pp(aa(nat,bool,ord_less(nat,bit_se2239418461657761734s_mask(nat,N)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N))) ).

% mask_nat_less_exp
tff(fact_5330_bit__not__iff__eq,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A2: A,N: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,aa(A,A,bit_ri4277139882892585799ns_not(A),A2)),N))
        <=> ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N) != zero_zero(A) )
            & ~ pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,A2),N)) ) ) ) ).

% bit_not_iff_eq
tff(fact_5331_minus__exp__eq__not__mask,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [N: nat] : aa(A,A,uminus_uminus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N)) = aa(A,A,bit_ri4277139882892585799ns_not(A),bit_se2239418461657761734s_mask(A,N)) ) ).

% minus_exp_eq_not_mask
tff(fact_5332_NOT__mask,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: nat] : aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),bit_se2239418461657761734s_mask(word(A),N)) = aa(word(A),word(A),uminus_uminus(word(A)),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),N)) ) ).

% NOT_mask
tff(fact_5333_mask__half__int,axiom,
    ! [N: nat] : aa(int,int,aa(int,fun(int,int),divide_divide(int),bit_se2239418461657761734s_mask(int,N)),aa(num,int,numeral_numeral(int),bit0(one2))) = bit_se2239418461657761734s_mask(int,aa(nat,nat,minus_minus(nat,N),one_one(nat))) ).

% mask_half_int
tff(fact_5334_mask__int__def,axiom,
    ! [N: nat] : bit_se2239418461657761734s_mask(int,N) = aa(int,int,minus_minus(int,aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),N)),one_one(int)) ).

% mask_int_def
tff(fact_5335_mask__nat__def,axiom,
    ! [N: nat] : bit_se2239418461657761734s_mask(nat,N) = aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)),one_one(nat)) ).

% mask_nat_def
tff(fact_5336_neg__mask__is__div,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),N: nat] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),W),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),bit_se2239418461657761734s_mask(word(A),N))) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),W),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),N))),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),N)) ) ).

% neg_mask_is_div
tff(fact_5337_option_Osize__gen_I1_J,axiom,
    ! [A: $tType,X: fun(A,nat)] : size_option(A,X,none(A)) = aa(nat,nat,suc,zero_zero(nat)) ).

% option.size_gen(1)
tff(fact_5338_divmod__step__nat__def,axiom,
    ! [L: num,Qr: product_prod(nat,nat)] : unique1321980374590559556d_step(nat,L,Qr) = aa(product_prod(nat,nat),product_prod(nat,nat),product_case_prod(nat,nat,product_prod(nat,nat),aTP_Lamp_lz(num,fun(nat,fun(nat,product_prod(nat,nat))),L)),Qr) ).

% divmod_step_nat_def
tff(fact_5339_take__bit__eq__mask__iff__exp__dvd,axiom,
    ! [N: nat,K: int] :
      ( ( aa(int,int,bit_se2584673776208193580ke_bit(int,N),K) = bit_se2239418461657761734s_mask(int,N) )
    <=> pp(aa(int,bool,dvd_dvd(int,aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),N)),aa(int,int,aa(int,fun(int,int),plus_plus(int),K),one_one(int)))) ) ).

% take_bit_eq_mask_iff_exp_dvd
tff(fact_5340_divmod__step__int__def,axiom,
    ! [L: num,Qr: product_prod(int,int)] : unique1321980374590559556d_step(int,L,Qr) = aa(product_prod(int,int),product_prod(int,int),product_case_prod(int,int,product_prod(int,int),aTP_Lamp_ma(num,fun(int,fun(int,product_prod(int,int))),L)),Qr) ).

% divmod_step_int_def
tff(fact_5341_divmod__step__integer__def,axiom,
    ! [L: num,Qr: product_prod(code_integer,code_integer)] : unique1321980374590559556d_step(code_integer,L,Qr) = aa(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer),product_case_prod(code_integer,code_integer,product_prod(code_integer,code_integer),aTP_Lamp_mb(num,fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),L)),Qr) ).

% divmod_step_integer_def
tff(fact_5342_divmod__step__def,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [L: num,Qr: product_prod(A,A)] : unique1321980374590559556d_step(A,L,Qr) = aa(product_prod(A,A),product_prod(A,A),product_case_prod(A,A,product_prod(A,A),aTP_Lamp_mc(num,fun(A,fun(A,product_prod(A,A))),L)),Qr) ) ).

% divmod_step_def
tff(fact_5343_case__prod__conv,axiom,
    ! [B: $tType,A: $tType,C: $tType,F2: fun(B,fun(C,A)),A2: B,B2: C] : aa(product_prod(B,C),A,product_case_prod(B,C,A,F2),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),A2),B2)) = aa(C,A,aa(B,fun(C,A),F2,A2),B2) ).

% case_prod_conv
tff(fact_5344_divmod__nat__if,axiom,
    ! [N: nat,M: nat] :
      ( ( ( ( N = zero_zero(nat) )
          | pp(aa(nat,bool,ord_less(nat,M),N)) )
       => ( divmod_nat(M,N) = aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),zero_zero(nat)),M) ) )
      & ( ~ ( ( N = zero_zero(nat) )
            | pp(aa(nat,bool,ord_less(nat,M),N)) )
       => ( divmod_nat(M,N) = aa(product_prod(nat,nat),product_prod(nat,nat),product_case_prod(nat,nat,product_prod(nat,nat),aTP_Lamp_md(nat,fun(nat,product_prod(nat,nat)))),divmod_nat(aa(nat,nat,minus_minus(nat,M),N),N)) ) ) ) ).

% divmod_nat_if
tff(fact_5345_neg__mask__add__mask,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),N: nat] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),X),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),bit_se2239418461657761734s_mask(word(A),N)))),aa(word(A),word(A),minus_minus(word(A),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),N)),one_one(word(A)))) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se1065995026697491101ons_or(word(A)),X),bit_se2239418461657761734s_mask(word(A),N)) ) ).

% neg_mask_add_mask
tff(fact_5346_or_Oright__idem,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A2),B2)),B2) = aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A2),B2) ) ).

% or.right_idem
tff(fact_5347_or_Oleft__idem,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A2),aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A2),B2) ) ).

% or.left_idem
tff(fact_5348_or_Oidem,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A2),A2) = A2 ) ).

% or.idem
tff(fact_5349_case__prodI,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,fun(B,bool)),A2: A,B2: B] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),F2,A2),B2))
     => pp(aa(product_prod(A,B),bool,product_case_prod(A,B,bool,F2),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A2),B2))) ) ).

% case_prodI
tff(fact_5350_case__prodI2,axiom,
    ! [B: $tType,A: $tType,P2: product_prod(A,B),C2: fun(A,fun(B,bool))] :
      ( ! [A4: A,B3: B] :
          ( ( P2 = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B3) )
         => pp(aa(B,bool,aa(A,fun(B,bool),C2,A4),B3)) )
     => pp(aa(product_prod(A,B),bool,product_case_prod(A,B,bool,C2),P2)) ) ).

% case_prodI2
tff(fact_5351_case__prodI2_H,axiom,
    ! [A: $tType,B: $tType,C: $tType,P2: product_prod(A,B),C2: fun(A,fun(B,fun(C,bool))),X: C] :
      ( ! [A4: A,B3: B] :
          ( ( aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B3) = P2 )
         => pp(aa(C,bool,aa(B,fun(C,bool),aa(A,fun(B,fun(C,bool)),C2,A4),B3),X)) )
     => pp(aa(C,bool,aa(product_prod(A,B),fun(C,bool),product_case_prod(A,B,fun(C,bool),C2),P2),X)) ) ).

% case_prodI2'
tff(fact_5352_mem__case__prodI,axiom,
    ! [A: $tType,B: $tType,C: $tType,Z: A,C2: fun(B,fun(C,set(A))),A2: B,B2: C] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Z),aa(C,set(A),aa(B,fun(C,set(A)),C2,A2),B2)))
     => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Z),aa(product_prod(B,C),set(A),product_case_prod(B,C,set(A),C2),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),A2),B2)))) ) ).

% mem_case_prodI
tff(fact_5353_mem__case__prodI2,axiom,
    ! [C: $tType,B: $tType,A: $tType,P2: product_prod(A,B),Z: C,C2: fun(A,fun(B,set(C)))] :
      ( ! [A4: A,B3: B] :
          ( ( P2 = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B3) )
         => pp(aa(set(C),bool,aa(C,fun(set(C),bool),member(C),Z),aa(B,set(C),aa(A,fun(B,set(C)),C2,A4),B3))) )
     => pp(aa(set(C),bool,aa(C,fun(set(C),bool),member(C),Z),aa(product_prod(A,B),set(C),product_case_prod(A,B,set(C),C2),P2))) ) ).

% mem_case_prodI2
tff(fact_5354_The__split__eq,axiom,
    ! [A: $tType,B: $tType,X: A,Y: B] : the(product_prod(A,B),product_case_prod(A,B,bool,aa(B,fun(A,fun(B,bool)),aTP_Lamp_me(A,fun(B,fun(A,fun(B,bool))),X),Y))) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y) ).

% The_split_eq
tff(fact_5355_Eps__case__prod__eq,axiom,
    ! [A: $tType,B: $tType,X: A,Y: B] : fChoice(product_prod(A,B),product_case_prod(A,B,bool,aa(B,fun(A,fun(B,bool)),aTP_Lamp_me(A,fun(B,fun(A,fun(B,bool))),X),Y))) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y) ).

% Eps_case_prod_eq
tff(fact_5356_or_Oleft__neutral,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),zero_zero(A)),A2) = A2 ) ).

% or.left_neutral
tff(fact_5357_or_Oright__neutral,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A2),zero_zero(A)) = A2 ) ).

% or.right_neutral
tff(fact_5358_take__bit__or,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A2: A,B2: A] : aa(A,A,bit_se2584673776208193580ke_bit(A,N),aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(A,A,bit_se2584673776208193580ke_bit(A,N),A2)),aa(A,A,bit_se2584673776208193580ke_bit(A,N),B2)) ) ).

% take_bit_or
tff(fact_5359_bit_Odisj__one__right,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),X),aa(A,A,uminus_uminus(A),one_one(A))) = aa(A,A,uminus_uminus(A),one_one(A)) ) ).

% bit.disj_one_right
tff(fact_5360_bit_Odisj__one__left,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(A,A,uminus_uminus(A),one_one(A))),X) = aa(A,A,uminus_uminus(A),one_one(A)) ) ).

% bit.disj_one_left
tff(fact_5361_bit_Ode__Morgan__conj,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [X: A,Y: A] : aa(A,A,bit_ri4277139882892585799ns_not(A),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),X),Y)) = aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(A,A,bit_ri4277139882892585799ns_not(A),X)),aa(A,A,bit_ri4277139882892585799ns_not(A),Y)) ) ).

% bit.de_Morgan_conj
tff(fact_5362_bit_Ode__Morgan__disj,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [X: A,Y: A] : aa(A,A,bit_ri4277139882892585799ns_not(A),aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),X),Y)) = aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(A,A,bit_ri4277139882892585799ns_not(A),X)),aa(A,A,bit_ri4277139882892585799ns_not(A),Y)) ) ).

% bit.de_Morgan_disj
tff(fact_5363_word__plus__and__or,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),X),Y)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se1065995026697491101ons_or(word(A)),X),Y)) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),X),Y) ) ).

% word_plus_and_or
tff(fact_5364_word__or__max,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se1065995026697491101ons_or(word(A)),X),aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A)))) = aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A))) ) ).

% word_or_max
tff(fact_5365_word__bitwise__m1__simps_I4_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se1065995026697491101ons_or(word(A)),aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A)))),X) = aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A))) ) ).

% word_bitwise_m1_simps(4)
tff(fact_5366_or__numerals_I2_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Y: num] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),one_one(A)),aa(num,A,numeral_numeral(A),aa(num,num,bit1,Y))) = aa(num,A,numeral_numeral(A),aa(num,num,bit1,Y)) ) ).

% or_numerals(2)
tff(fact_5367_or__numerals_I8_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [X: num] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(num,A,numeral_numeral(A),aa(num,num,bit1,X))),one_one(A)) = aa(num,A,numeral_numeral(A),aa(num,num,bit1,X)) ) ).

% or_numerals(8)
tff(fact_5368_bit_Odisj__cancel__left,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(A,A,bit_ri4277139882892585799ns_not(A),X)),X) = aa(A,A,uminus_uminus(A),one_one(A)) ) ).

% bit.disj_cancel_left
tff(fact_5369_bit_Odisj__cancel__right,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),X),aa(A,A,bit_ri4277139882892585799ns_not(A),X)) = aa(A,A,uminus_uminus(A),one_one(A)) ) ).

% bit.disj_cancel_right
tff(fact_5370_not__nonnegative__int__iff,axiom,
    ! [K: int] :
      ( pp(aa(int,bool,ord_less_eq(int,zero_zero(int)),aa(int,int,bit_ri4277139882892585799ns_not(int),K)))
    <=> pp(aa(int,bool,ord_less(int,K),zero_zero(int))) ) ).

% not_nonnegative_int_iff
tff(fact_5371_not__negative__int__iff,axiom,
    ! [K: int] :
      ( pp(aa(int,bool,ord_less(int,aa(int,int,bit_ri4277139882892585799ns_not(int),K)),zero_zero(int)))
    <=> pp(aa(int,bool,ord_less_eq(int,zero_zero(int)),K)) ) ).

% not_negative_int_iff
tff(fact_5372_word__or__not,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se1065995026697491101ons_or(word(A)),X),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),X)) = aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A))) ) ).

% word_or_not
tff(fact_5373_or__numerals_I3_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [X: num,Y: num] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(num,A,numeral_numeral(A),bit0(X))),aa(num,A,numeral_numeral(A),bit0(Y))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(num,A,numeral_numeral(A),X)),aa(num,A,numeral_numeral(A),Y))) ) ).

% or_numerals(3)
tff(fact_5374_or__numerals_I1_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Y: num] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),one_one(A)),aa(num,A,numeral_numeral(A),bit0(Y))) = aa(num,A,numeral_numeral(A),aa(num,num,bit1,Y)) ) ).

% or_numerals(1)
tff(fact_5375_or__numerals_I5_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [X: num] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(num,A,numeral_numeral(A),bit0(X))),one_one(A)) = aa(num,A,numeral_numeral(A),aa(num,num,bit1,X)) ) ).

% or_numerals(5)
tff(fact_5376_word__no__log__defs_I1_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num] : aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),aa(num,word(A),numeral_numeral(word(A)),A2)) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),A2))) ) ).

% word_no_log_defs(1)
tff(fact_5377_word__of__int__not__numeral__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Bin: num] : aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),Bin))) = aa(word(A),word(A),minus_minus(word(A),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),Bin))),one_one(word(A))) ) ).

% word_of_int_not_numeral_eq
tff(fact_5378_word__no__log__defs_I2_J,axiom,
    ! [B: $tType] :
      ( type_len(B)
     => ! [A2: num] : aa(word(B),word(B),bit_ri4277139882892585799ns_not(word(B)),aa(word(B),word(B),uminus_uminus(word(B)),aa(num,word(B),numeral_numeral(word(B)),A2))) = aa(int,word(B),ring_1_of_int(word(B)),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),A2)))) ) ).

% word_no_log_defs(2)
tff(fact_5379_or__numerals_I4_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [X: num,Y: num] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(num,A,numeral_numeral(A),bit0(X))),aa(num,A,numeral_numeral(A),aa(num,num,bit1,Y))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(num,A,numeral_numeral(A),X)),aa(num,A,numeral_numeral(A),Y)))) ) ).

% or_numerals(4)
tff(fact_5380_or__numerals_I6_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [X: num,Y: num] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(num,A,numeral_numeral(A),aa(num,num,bit1,X))),aa(num,A,numeral_numeral(A),bit0(Y))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(num,A,numeral_numeral(A),X)),aa(num,A,numeral_numeral(A),Y)))) ) ).

% or_numerals(6)
tff(fact_5381_or__numerals_I7_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [X: num,Y: num] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(num,A,numeral_numeral(A),aa(num,num,bit1,X))),aa(num,A,numeral_numeral(A),aa(num,num,bit1,Y))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(num,A,numeral_numeral(A),X)),aa(num,A,numeral_numeral(A),Y)))) ) ).

% or_numerals(7)
tff(fact_5382_case__prodD,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,fun(B,bool)),A2: A,B2: B] :
      ( pp(aa(product_prod(A,B),bool,product_case_prod(A,B,bool,F2),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A2),B2)))
     => pp(aa(B,bool,aa(A,fun(B,bool),F2,A2),B2)) ) ).

% case_prodD
tff(fact_5383_case__prodE,axiom,
    ! [A: $tType,B: $tType,C2: fun(A,fun(B,bool)),P2: product_prod(A,B)] :
      ( pp(aa(product_prod(A,B),bool,product_case_prod(A,B,bool,C2),P2))
     => ~ ! [X3: A,Y3: B] :
            ( ( P2 = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X3),Y3) )
           => ~ pp(aa(B,bool,aa(A,fun(B,bool),C2,X3),Y3)) ) ) ).

% case_prodE
tff(fact_5384_case__prodD_H,axiom,
    ! [B: $tType,A: $tType,C: $tType,R: fun(A,fun(B,fun(C,bool))),A2: A,B2: B,C2: C] :
      ( pp(aa(C,bool,aa(product_prod(A,B),fun(C,bool),product_case_prod(A,B,fun(C,bool),R),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A2),B2)),C2))
     => pp(aa(C,bool,aa(B,fun(C,bool),aa(A,fun(B,fun(C,bool)),R,A2),B2),C2)) ) ).

% case_prodD'
tff(fact_5385_case__prodE_H,axiom,
    ! [B: $tType,A: $tType,C: $tType,C2: fun(A,fun(B,fun(C,bool))),P2: product_prod(A,B),Z: C] :
      ( pp(aa(C,bool,aa(product_prod(A,B),fun(C,bool),product_case_prod(A,B,fun(C,bool),C2),P2),Z))
     => ~ ! [X3: A,Y3: B] :
            ( ( P2 = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X3),Y3) )
           => ~ pp(aa(C,bool,aa(B,fun(C,bool),aa(A,fun(B,fun(C,bool)),C2,X3),Y3),Z)) ) ) ).

% case_prodE'
tff(fact_5386_bit__not__int__iff,axiom,
    ! [K: int,N: nat] :
      ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(int,aa(int,int,bit_ri4277139882892585799ns_not(int),K)),N))
    <=> ~ pp(aa(nat,bool,bit_se5641148757651400278ts_bit(int,K),N)) ) ).

% bit_not_int_iff
tff(fact_5387_split__paired__Eps,axiom,
    ! [B: $tType,A: $tType,P: fun(product_prod(A,B),bool)] : fChoice(product_prod(A,B),P) = fChoice(product_prod(A,B),product_case_prod(A,B,bool,aTP_Lamp_mf(fun(product_prod(A,B),bool),fun(A,fun(B,bool)),P))) ).

% split_paired_Eps
tff(fact_5388_disjunctive__add,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,B2: A] :
          ( ! [N2: nat] :
              ( ~ pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,A2),N2))
              | ~ pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,B2),N2)) )
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2) = aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A2),B2) ) ) ) ).

% disjunctive_add
tff(fact_5389_of__int__or__eq,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [K: int,L: int] : aa(int,A,ring_1_of_int(A),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),K),L)) = aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(int,A,ring_1_of_int(A),K)),aa(int,A,ring_1_of_int(A),L)) ) ).

% of_int_or_eq
tff(fact_5390_bit__or__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,B2: A,N: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A2),B2)),N))
        <=> ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,A2),N))
            | pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,B2),N)) ) ) ) ).

% bit_or_iff
tff(fact_5391_or_Oleft__commute,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [B2: A,A2: A,C2: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),B2),aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A2),C2)) = aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A2),aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),B2),C2)) ) ).

% or.left_commute
tff(fact_5392_or_Ocommute,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A2),B2) = aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),B2),A2) ) ).

% or.commute
tff(fact_5393_or_Oassoc,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A2),B2)),C2) = aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A2),aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),B2),C2)) ) ).

% or.assoc
tff(fact_5394_of__nat__or__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,N: nat] : aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),M),N)) = aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(nat,A,semiring_1_of_nat(A),M)),aa(nat,A,semiring_1_of_nat(A),N)) ) ).

% of_nat_or_eq
tff(fact_5395_bit_Odisj__zero__right,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),X),zero_zero(A)) = X ) ).

% bit.disj_zero_right
tff(fact_5396_or__eq__0__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A2),B2) = zero_zero(A) )
        <=> ( ( A2 = zero_zero(A) )
            & ( B2 = zero_zero(A) ) ) ) ) ).

% or_eq_0_iff
tff(fact_5397_word__log__esimps_I3_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se1065995026697491101ons_or(word(A)),X),zero_zero(word(A))) = X ) ).

% word_log_esimps(3)
tff(fact_5398_word__log__esimps_I9_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se1065995026697491101ons_or(word(A)),zero_zero(word(A))),X) = X ) ).

% word_log_esimps(9)
tff(fact_5399_word__or__zero,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A)] :
          ( ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se1065995026697491101ons_or(word(A)),A2),B2) = zero_zero(word(A)) )
        <=> ( ( A2 = zero_zero(word(A)) )
            & ( B2 = zero_zero(word(A)) ) ) ) ) ).

% word_or_zero
tff(fact_5400_bit_Oconj__disj__distrib,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [X: A,Y: A,Z: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),X),aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),Y),Z)) = aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),X),Y)),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),X),Z)) ) ).

% bit.conj_disj_distrib
tff(fact_5401_bit_Odisj__conj__distrib,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [X: A,Y: A,Z: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),X),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),Y),Z)) = aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),X),Y)),aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),X),Z)) ) ).

% bit.disj_conj_distrib
tff(fact_5402_bit_Oconj__disj__distrib2,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Y: A,Z: A,X: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),Y),Z)),X) = aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),Y),X)),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),Z),X)) ) ).

% bit.conj_disj_distrib2
tff(fact_5403_bit_Odisj__conj__distrib2,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Y: A,Z: A,X: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),Y),Z)),X) = aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),Y),X)),aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),Z),X)) ) ).

% bit.disj_conj_distrib2
tff(fact_5404_and__eq__not__not__or,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A2),B2) = aa(A,A,bit_ri4277139882892585799ns_not(A),aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(A,A,bit_ri4277139882892585799ns_not(A),A2)),aa(A,A,bit_ri4277139882892585799ns_not(A),B2))) ) ).

% and_eq_not_not_or
tff(fact_5405_or__eq__not__not__and,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A2),B2) = aa(A,A,bit_ri4277139882892585799ns_not(A),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(A,A,bit_ri4277139882892585799ns_not(A),A2)),aa(A,A,bit_ri4277139882892585799ns_not(A),B2))) ) ).

% or_eq_not_not_and
tff(fact_5406_sum_Otriangle__reindex__eq,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,fun(nat,A)),N: nat] : aa(set(product_prod(nat,nat)),A,groups7311177749621191930dd_sum(product_prod(nat,nat),A,product_case_prod(nat,nat,A,G)),collect(product_prod(nat,nat),product_case_prod(nat,nat,bool,aTP_Lamp_mg(nat,fun(nat,fun(nat,bool)),N)))) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_mi(fun(nat,fun(nat,A)),fun(nat,A),G)),set_ord_atMost(nat,N)) ) ).

% sum.triangle_reindex_eq
tff(fact_5407_prod_Otriangle__reindex__eq,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,fun(nat,A)),N: 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,bool,aTP_Lamp_mg(nat,fun(nat,fun(nat,bool)),N)))) = groups7121269368397514597t_prod(nat,A,aTP_Lamp_mk(fun(nat,fun(nat,A)),fun(nat,A),G),set_ord_atMost(nat,N)) ) ).

% prod.triangle_reindex_eq
tff(fact_5408_sum_Otriangle__reindex,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,fun(nat,A)),N: nat] : aa(set(product_prod(nat,nat)),A,groups7311177749621191930dd_sum(product_prod(nat,nat),A,product_case_prod(nat,nat,A,G)),collect(product_prod(nat,nat),product_case_prod(nat,nat,bool,aTP_Lamp_ml(nat,fun(nat,fun(nat,bool)),N)))) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_mi(fun(nat,fun(nat,A)),fun(nat,A),G)),set_ord_lessThan(nat,N)) ) ).

% sum.triangle_reindex
tff(fact_5409_prod_Otriangle__reindex,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,fun(nat,A)),N: 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,bool,aTP_Lamp_ml(nat,fun(nat,fun(nat,bool)),N)))) = groups7121269368397514597t_prod(nat,A,aTP_Lamp_mk(fun(nat,fun(nat,A)),fun(nat,A),G),set_ord_lessThan(nat,N)) ) ).

% prod.triangle_reindex
tff(fact_5410_not__int__def,axiom,
    ! [K: int] : aa(int,int,bit_ri4277139882892585799ns_not(int),K) = aa(int,int,minus_minus(int,aa(int,int,uminus_uminus(int),K)),one_one(int)) ).

% not_int_def
tff(fact_5411_and__not__numerals_I1_J,axiom,
    aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),one_one(int)),aa(int,int,bit_ri4277139882892585799ns_not(int),one_one(int))) = zero_zero(int) ).

% and_not_numerals(1)
tff(fact_5412_word__plus__and__or__coroll,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A)] :
          ( ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),X),Y) = zero_zero(word(A)) )
         => ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),X),Y) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se1065995026697491101ons_or(word(A)),X),Y) ) ) ) ).

% word_plus_and_or_coroll
tff(fact_5413_even__or__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A2),B2)))
        <=> ( pp(aa(A,bool,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A2))
            & pp(aa(A,bool,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),B2)) ) ) ) ).

% even_or_iff
tff(fact_5414_bit_Ocomplement__unique,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A2: A,X: A,Y: A] :
          ( ( aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A2),X) = zero_zero(A) )
         => ( ( aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A2),X) = aa(A,A,uminus_uminus(A),one_one(A)) )
           => ( ( aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A2),Y) = zero_zero(A) )
             => ( ( aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A2),Y) = aa(A,A,uminus_uminus(A),one_one(A)) )
               => ( X = Y ) ) ) ) ) ) ).

% bit.complement_unique
tff(fact_5415_not__int__div__2,axiom,
    ! [K: int] : aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,bit_ri4277139882892585799ns_not(int),K)),aa(num,int,numeral_numeral(int),bit0(one2))) = aa(int,int,bit_ri4277139882892585799ns_not(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),K),aa(num,int,numeral_numeral(int),bit0(one2)))) ).

% not_int_div_2
tff(fact_5416_even__not__iff__int,axiom,
    ! [K: int] :
      ( pp(aa(int,bool,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),aa(int,int,bit_ri4277139882892585799ns_not(int),K)))
    <=> ~ pp(aa(int,bool,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),K)) ) ).

% even_not_iff_int
tff(fact_5417_and__not__numerals_I4_J,axiom,
    ! [M: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(num,int,numeral_numeral(int),bit0(M))),aa(int,int,bit_ri4277139882892585799ns_not(int),one_one(int))) = aa(num,int,numeral_numeral(int),bit0(M)) ).

% and_not_numerals(4)
tff(fact_5418_and__not__numerals_I2_J,axiom,
    ! [N: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),one_one(int)),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),bit0(N)))) = one_one(int) ).

% and_not_numerals(2)
tff(fact_5419_bit__minus__int__iff,axiom,
    ! [K: int,N: nat] :
      ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(int,aa(int,int,uminus_uminus(int),K)),N))
    <=> pp(aa(nat,bool,bit_se5641148757651400278ts_bit(int,aa(int,int,bit_ri4277139882892585799ns_not(int),aa(int,int,minus_minus(int,K),one_one(int)))),N)) ) ).

% bit_minus_int_iff
tff(fact_5420_mask__subsume,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: nat,M: nat,X: word(A),Y: word(A)] :
          ( pp(aa(nat,bool,ord_less_eq(nat,N),M))
         => ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se1065995026697491101ons_or(word(A)),X),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Y),bit_se2239418461657761734s_mask(word(A),N)))),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),bit_se2239418461657761734s_mask(word(A),M))) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),X),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),bit_se2239418461657761734s_mask(word(A),M))) ) ) ) ).

% mask_subsume
tff(fact_5421_rel__of__def,axiom,
    ! [B: $tType,A: $tType,M: fun(A,option(B)),P: fun(product_prod(A,B),bool)] : rel_of(A,B,M,P) = collect(product_prod(A,B),product_case_prod(A,B,bool,aa(fun(product_prod(A,B),bool),fun(A,fun(B,bool)),aTP_Lamp_mm(fun(A,option(B)),fun(fun(product_prod(A,B),bool),fun(A,fun(B,bool))),M),P))) ).

% rel_of_def
tff(fact_5422_bit_Ocompl__unique,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [X: A,Y: A] :
          ( ( aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),X),Y) = zero_zero(A) )
         => ( ( aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),X),Y) = aa(A,A,uminus_uminus(A),one_one(A)) )
           => ( aa(A,A,bit_ri4277139882892585799ns_not(A),X) = Y ) ) ) ) ).

% bit.compl_unique
tff(fact_5423_and__not__numerals_I5_J,axiom,
    ! [M: num,N: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(num,int,numeral_numeral(int),bit0(M))),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),bit0(N)))) = aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(num,int,numeral_numeral(int),M)),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),N)))) ).

% and_not_numerals(5)
tff(fact_5424_prod_Ocase__distrib,axiom,
    ! [C: $tType,D6: $tType,B: $tType,A: $tType,H: fun(C,D6),F2: fun(A,fun(B,C)),Prod: product_prod(A,B)] : aa(C,D6,H,aa(product_prod(A,B),C,product_case_prod(A,B,C,F2),Prod)) = aa(product_prod(A,B),D6,product_case_prod(A,B,D6,aa(fun(A,fun(B,C)),fun(A,fun(B,D6)),aTP_Lamp_mn(fun(C,D6),fun(fun(A,fun(B,C)),fun(A,fun(B,D6))),H),F2)),Prod) ).

% prod.case_distrib
tff(fact_5425_and__not__numerals_I7_J,axiom,
    ! [M: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,M))),aa(int,int,bit_ri4277139882892585799ns_not(int),one_one(int))) = aa(num,int,numeral_numeral(int),bit0(M)) ).

% and_not_numerals(7)
tff(fact_5426_and__not__numerals_I3_J,axiom,
    ! [N: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),one_one(int)),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,N)))) = zero_zero(int) ).

% and_not_numerals(3)
tff(fact_5427_signed__take__bit__eq__if__negative,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A2: A,N: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,A2),N))
         => ( aa(A,A,bit_ri4674362597316999326ke_bit(A,N),A2) = aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(A,A,bit_se2584673776208193580ke_bit(A,N),A2)),aa(A,A,bit_ri4277139882892585799ns_not(A),bit_se2239418461657761734s_mask(A,N))) ) ) ) ).

% signed_take_bit_eq_if_negative
tff(fact_5428_Divides_Oadjust__div__def,axiom,
    ! [Qr: product_prod(int,int)] : adjust_div(Qr) = aa(product_prod(int,int),int,product_case_prod(int,int,int,aTP_Lamp_mo(int,fun(int,int))),Qr) ).

% Divides.adjust_div_def
tff(fact_5429_mask__Suc__exp,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat] : bit_se2239418461657761734s_mask(A,aa(nat,nat,suc,N)) = aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N)),bit_se2239418461657761734s_mask(A,N)) ) ).

% mask_Suc_exp
tff(fact_5430_and__not__numerals_I6_J,axiom,
    ! [M: num,N: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(num,int,numeral_numeral(int),bit0(M))),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,N)))) = aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(num,int,numeral_numeral(int),M)),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),N)))) ).

% and_not_numerals(6)
tff(fact_5431_and__not__numerals_I9_J,axiom,
    ! [M: num,N: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,M))),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,N)))) = aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(num,int,numeral_numeral(int),M)),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),N)))) ).

% and_not_numerals(9)
tff(fact_5432_mask__Suc__double,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat] : bit_se2239418461657761734s_mask(A,aa(nat,nat,suc,N)) = aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),bit_se2239418461657761734s_mask(A,N))) ) ).

% mask_Suc_double
tff(fact_5433_or__one__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A2),one_one(A)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa(bool,A,zero_neq_one_of_bool(A),aa(A,bool,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A2))) ) ).

% or_one_eq
tff(fact_5434_one__or__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),one_one(A)),A2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa(bool,A,zero_neq_one_of_bool(A),aa(A,bool,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A2))) ) ).

% one_or_eq
tff(fact_5435_divmod__nat__def,axiom,
    ! [M: nat,N: nat] : divmod_nat(M,N) = aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),N)),modulo_modulo(nat,M,N)) ).

% divmod_nat_def
tff(fact_5436_signed__take__bit__def,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [N: nat,A2: A] : aa(A,A,bit_ri4674362597316999326ke_bit(A,N),A2) = aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(A,A,bit_se2584673776208193580ke_bit(A,N),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(bool,A,zero_neq_one_of_bool(A),aa(nat,bool,bit_se5641148757651400278ts_bit(A,A2),N))),aa(A,A,bit_ri4277139882892585799ns_not(A),bit_se2239418461657761734s_mask(A,N)))) ) ).

% signed_take_bit_def
tff(fact_5437_and__not__numerals_I8_J,axiom,
    ! [M: num,N: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,M))),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),bit0(N)))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(num,int,numeral_numeral(int),M)),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),N))))) ).

% and_not_numerals(8)
tff(fact_5438_not__int__rec,axiom,
    ! [K: int] : aa(int,int,bit_ri4277139882892585799ns_not(int),K) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(bool,int,zero_neq_one_of_bool(int),aa(int,bool,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),K))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),K),aa(num,int,numeral_numeral(int),bit0(one2)))))) ).

% not_int_rec
tff(fact_5439_cond__case__prod__eta,axiom,
    ! [C: $tType,B: $tType,A: $tType,F2: fun(A,fun(B,C)),G: fun(product_prod(A,B),C)] :
      ( ! [X3: A,Y3: B] : aa(B,C,aa(A,fun(B,C),F2,X3),Y3) = aa(product_prod(A,B),C,G,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X3),Y3))
     => ( product_case_prod(A,B,C,F2) = G ) ) ).

% cond_case_prod_eta
tff(fact_5440_case__prod__eta,axiom,
    ! [C: $tType,B: $tType,A: $tType,F2: fun(product_prod(A,B),C)] : product_case_prod(A,B,C,aTP_Lamp_mp(fun(product_prod(A,B),C),fun(A,fun(B,C)),F2)) = F2 ).

% case_prod_eta
tff(fact_5441_case__prodE2,axiom,
    ! [B: $tType,A: $tType,C: $tType,Q: fun(A,bool),P: fun(B,fun(C,A)),Z: product_prod(B,C)] :
      ( pp(aa(A,bool,Q,aa(product_prod(B,C),A,product_case_prod(B,C,A,P),Z)))
     => ~ ! [X3: B,Y3: C] :
            ( ( Z = aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),X3),Y3) )
           => ~ pp(aa(A,bool,Q,aa(C,A,aa(B,fun(C,A),P,X3),Y3))) ) ) ).

% case_prodE2
tff(fact_5442_of__nat__code__if,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [N: nat] :
          ( ( ( N = zero_zero(nat) )
           => ( aa(nat,A,semiring_1_of_nat(A),N) = zero_zero(A) ) )
          & ( ( N != zero_zero(nat) )
           => ( aa(nat,A,semiring_1_of_nat(A),N) = aa(product_prod(nat,nat),A,product_case_prod(nat,nat,A,aTP_Lamp_mq(nat,fun(nat,A))),divmod_nat(N,aa(num,nat,numeral_numeral(nat),bit0(one2)))) ) ) ) ) ).

% of_nat_code_if
tff(fact_5443_int__not__code_I1_J,axiom,
    aa(int,int,bit_ri4277139882892585799ns_not(int),zero_zero(int)) = aa(int,int,uminus_uminus(int),one_one(int)) ).

% int_not_code(1)
tff(fact_5444_bitNOT__integer__code,axiom,
    ! [I: code_integer] : aa(code_integer,code_integer,bit_ri4277139882892585799ns_not(code_integer),I) = aa(code_integer,code_integer,minus_minus(code_integer,aa(code_integer,code_integer,uminus_uminus(code_integer),I)),one_one(code_integer)) ).

% bitNOT_integer_code
tff(fact_5445_split__part,axiom,
    ! [B: $tType,A: $tType,P: bool,Q: fun(A,fun(B,bool)),X4: product_prod(A,B)] :
      ( pp(aa(product_prod(A,B),bool,product_case_prod(A,B,bool,aa(fun(A,fun(B,bool)),fun(A,fun(B,bool)),aTP_Lamp_mr(bool,fun(fun(A,fun(B,bool)),fun(A,fun(B,bool))),P),Q)),X4))
    <=> ( pp(P)
        & pp(aa(product_prod(A,B),bool,product_case_prod(A,B,bool,Q),X4)) ) ) ).

% split_part
tff(fact_5446_or__nonnegative__int__iff,axiom,
    ! [K: int,L: int] :
      ( pp(aa(int,bool,ord_less_eq(int,zero_zero(int)),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),K),L)))
    <=> ( pp(aa(int,bool,ord_less_eq(int,zero_zero(int)),K))
        & pp(aa(int,bool,ord_less_eq(int,zero_zero(int)),L)) ) ) ).

% or_nonnegative_int_iff
tff(fact_5447_or__negative__int__iff,axiom,
    ! [K: int,L: int] :
      ( pp(aa(int,bool,ord_less(int,aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),K),L)),zero_zero(int)))
    <=> ( pp(aa(int,bool,ord_less(int,K),zero_zero(int)))
        | pp(aa(int,bool,ord_less(int,L),zero_zero(int))) ) ) ).

% or_negative_int_iff
tff(fact_5448_or__minus__numerals_I6_J,axiom,
    ! [N: num] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,N)))),one_one(int)) = aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,N))) ).

% or_minus_numerals(6)
tff(fact_5449_or__minus__numerals_I2_J,axiom,
    ! [N: num] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),one_one(int)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,N)))) = aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,N))) ).

% or_minus_numerals(2)
tff(fact_5450_or__nat__numerals_I4_J,axiom,
    ! [X: num] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,X))),aa(nat,nat,suc,zero_zero(nat))) = aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,X)) ).

% or_nat_numerals(4)
tff(fact_5451_or__nat__numerals_I2_J,axiom,
    ! [Y: num] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),aa(nat,nat,suc,zero_zero(nat))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,Y))) = aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,Y)) ).

% or_nat_numerals(2)
tff(fact_5452_word__no__log__defs_I7_J,axiom,
    ! [G2: $tType] :
      ( type_len(G2)
     => ! [A2: num,B2: num] : aa(word(G2),word(G2),aa(word(G2),fun(word(G2),word(G2)),bit_se1065995026697491101ons_or(word(G2)),aa(num,word(G2),numeral_numeral(word(G2)),A2)),aa(num,word(G2),numeral_numeral(word(G2)),B2)) = aa(int,word(G2),ring_1_of_int(word(G2)),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(num,int,numeral_numeral(int),A2)),aa(num,int,numeral_numeral(int),B2))) ) ).

% word_no_log_defs(7)
tff(fact_5453_or__nat__numerals_I3_J,axiom,
    ! [X: num] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),aa(num,nat,numeral_numeral(nat),bit0(X))),aa(nat,nat,suc,zero_zero(nat))) = aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,X)) ).

% or_nat_numerals(3)
tff(fact_5454_or__nat__numerals_I1_J,axiom,
    ! [Y: num] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),aa(nat,nat,suc,zero_zero(nat))),aa(num,nat,numeral_numeral(nat),bit0(Y))) = aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,Y)) ).

% or_nat_numerals(1)
tff(fact_5455_word__bitwise__1__simps_I6_J,axiom,
    ! [F: $tType] :
      ( type_len(F)
     => ! [B2: num] : aa(word(F),word(F),aa(word(F),fun(word(F),word(F)),bit_se1065995026697491101ons_or(word(F)),one_one(word(F))),aa(num,word(F),numeral_numeral(word(F)),B2)) = aa(int,word(F),ring_1_of_int(word(F)),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),one_one(int)),aa(num,int,numeral_numeral(int),B2))) ) ).

% word_bitwise_1_simps(6)
tff(fact_5456_word__bitwise__1__simps_I8_J,axiom,
    ! [H3: $tType] :
      ( type_len(H3)
     => ! [A2: num] : aa(word(H3),word(H3),aa(word(H3),fun(word(H3),word(H3)),bit_se1065995026697491101ons_or(word(H3)),aa(num,word(H3),numeral_numeral(word(H3)),A2)),one_one(word(H3))) = aa(int,word(H3),ring_1_of_int(word(H3)),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(num,int,numeral_numeral(int),A2)),one_one(int))) ) ).

% word_bitwise_1_simps(8)
tff(fact_5457_word__no__log__defs_I10_J,axiom,
    ! [J4: $tType] :
      ( type_len(J4)
     => ! [A2: num,B2: num] : aa(word(J4),word(J4),aa(word(J4),fun(word(J4),word(J4)),bit_se1065995026697491101ons_or(word(J4)),aa(word(J4),word(J4),uminus_uminus(word(J4)),aa(num,word(J4),numeral_numeral(word(J4)),A2))),aa(word(J4),word(J4),uminus_uminus(word(J4)),aa(num,word(J4),numeral_numeral(word(J4)),B2))) = aa(int,word(J4),ring_1_of_int(word(J4)),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),A2))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),B2)))) ) ).

% word_no_log_defs(10)
tff(fact_5458_word__no__log__defs_I9_J,axiom,
    ! [I6: $tType] :
      ( type_len(I6)
     => ! [A2: num,B2: num] : aa(word(I6),word(I6),aa(word(I6),fun(word(I6),word(I6)),bit_se1065995026697491101ons_or(word(I6)),aa(word(I6),word(I6),uminus_uminus(word(I6)),aa(num,word(I6),numeral_numeral(word(I6)),A2))),aa(num,word(I6),numeral_numeral(word(I6)),B2)) = aa(int,word(I6),ring_1_of_int(word(I6)),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),A2))),aa(num,int,numeral_numeral(int),B2))) ) ).

% word_no_log_defs(9)
tff(fact_5459_word__no__log__defs_I8_J,axiom,
    ! [H3: $tType] :
      ( type_len(H3)
     => ! [A2: num,B2: num] : aa(word(H3),word(H3),aa(word(H3),fun(word(H3),word(H3)),bit_se1065995026697491101ons_or(word(H3)),aa(num,word(H3),numeral_numeral(word(H3)),A2)),aa(word(H3),word(H3),uminus_uminus(word(H3)),aa(num,word(H3),numeral_numeral(word(H3)),B2))) = aa(int,word(H3),ring_1_of_int(word(H3)),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(num,int,numeral_numeral(int),A2)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),B2)))) ) ).

% word_no_log_defs(8)
tff(fact_5460_and__minus__minus__numerals,axiom,
    ! [M: num,N: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),M))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),N))) = aa(int,int,bit_ri4277139882892585799ns_not(int),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(int,int,minus_minus(int,aa(num,int,numeral_numeral(int),M)),one_one(int))),aa(int,int,minus_minus(int,aa(num,int,numeral_numeral(int),N)),one_one(int)))) ).

% and_minus_minus_numerals
tff(fact_5461_or__minus__minus__numerals,axiom,
    ! [M: num,N: num] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),M))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),N))) = aa(int,int,bit_ri4277139882892585799ns_not(int),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(int,int,minus_minus(int,aa(num,int,numeral_numeral(int),M)),one_one(int))),aa(int,int,minus_minus(int,aa(num,int,numeral_numeral(int),N)),one_one(int)))) ).

% or_minus_minus_numerals
tff(fact_5462_word__bitwise__1__simps_I7_J,axiom,
    ! [G2: $tType] :
      ( type_len(G2)
     => ! [B2: num] : aa(word(G2),word(G2),aa(word(G2),fun(word(G2),word(G2)),bit_se1065995026697491101ons_or(word(G2)),one_one(word(G2))),aa(word(G2),word(G2),uminus_uminus(word(G2)),aa(num,word(G2),numeral_numeral(word(G2)),B2))) = aa(int,word(G2),ring_1_of_int(word(G2)),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),one_one(int)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),B2)))) ) ).

% word_bitwise_1_simps(7)
tff(fact_5463_word__bitwise__1__simps_I9_J,axiom,
    ! [I6: $tType] :
      ( type_len(I6)
     => ! [A2: num] : aa(word(I6),word(I6),aa(word(I6),fun(word(I6),word(I6)),bit_se1065995026697491101ons_or(word(I6)),aa(word(I6),word(I6),uminus_uminus(word(I6)),aa(num,word(I6),numeral_numeral(word(I6)),A2))),one_one(word(I6))) = aa(int,word(I6),ring_1_of_int(word(I6)),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),A2))),one_one(int))) ) ).

% word_bitwise_1_simps(9)
tff(fact_5464_prod_Odisc__eq__case,axiom,
    ! [B: $tType,A: $tType,Prod: product_prod(A,B)] : pp(aa(product_prod(A,B),bool,product_case_prod(A,B,bool,aTP_Lamp_ms(A,fun(B,bool))),Prod)) ).

% prod.disc_eq_case
tff(fact_5465_bit__or__int__iff,axiom,
    ! [K: int,L: int,N: nat] :
      ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(int,aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),K),L)),N))
    <=> ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(int,K),N))
        | pp(aa(nat,bool,bit_se5641148757651400278ts_bit(int,L),N)) ) ) ).

% bit_or_int_iff
tff(fact_5466_or__nat__def,axiom,
    ! [M: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),M),N) = nat2(aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(nat,int,semiring_1_of_nat(int),M)),aa(nat,int,semiring_1_of_nat(int),N))) ).

% or_nat_def
tff(fact_5467_int__or__code_I2_J,axiom,
    ! [I: int] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),I),zero_zero(int)) = I ).

% int_or_code(2)
tff(fact_5468_int__or__code_I1_J,axiom,
    ! [J: int] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),zero_zero(int)),J) = J ).

% int_or_code(1)
tff(fact_5469_OR__lower,axiom,
    ! [X: int,Y: int] :
      ( pp(aa(int,bool,ord_less_eq(int,zero_zero(int)),X))
     => ( pp(aa(int,bool,ord_less_eq(int,zero_zero(int)),Y))
       => pp(aa(int,bool,ord_less_eq(int,zero_zero(int)),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),X),Y))) ) ) ).

% OR_lower
tff(fact_5470_or__greater__eq,axiom,
    ! [L: int,K: int] :
      ( pp(aa(int,bool,ord_less_eq(int,zero_zero(int)),L))
     => pp(aa(int,bool,ord_less_eq(int,K),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),K),L))) ) ).

% or_greater_eq
tff(fact_5471_plus__and__or,axiom,
    ! [X: int,Y: int] : aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),X),Y)),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),X),Y)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),X),Y) ).

% plus_and_or
tff(fact_5472_or__int__def,axiom,
    ! [K: int,L: int] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),K),L) = aa(int,int,bit_ri4277139882892585799ns_not(int),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(int,int,bit_ri4277139882892585799ns_not(int),K)),aa(int,int,bit_ri4277139882892585799ns_not(int),L))) ).

% or_int_def
tff(fact_5473_or__not__numerals_I1_J,axiom,
    aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),one_one(int)),aa(int,int,bit_ri4277139882892585799ns_not(int),one_one(int))) = aa(int,int,bit_ri4277139882892585799ns_not(int),zero_zero(int)) ).

% or_not_numerals(1)
tff(fact_5474_or__not__numerals_I2_J,axiom,
    ! [N: num] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),one_one(int)),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),bit0(N)))) = aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),bit0(N))) ).

% or_not_numerals(2)
tff(fact_5475_or__not__numerals_I4_J,axiom,
    ! [M: num] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(num,int,numeral_numeral(int),bit0(M))),aa(int,int,bit_ri4277139882892585799ns_not(int),one_one(int))) = aa(int,int,bit_ri4277139882892585799ns_not(int),one_one(int)) ).

% or_not_numerals(4)
tff(fact_5476_or__not__numerals_I3_J,axiom,
    ! [N: num] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),one_one(int)),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,N)))) = aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),bit0(N))) ).

% or_not_numerals(3)
tff(fact_5477_or__not__numerals_I7_J,axiom,
    ! [M: num] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,M))),aa(int,int,bit_ri4277139882892585799ns_not(int),one_one(int))) = aa(int,int,bit_ri4277139882892585799ns_not(int),zero_zero(int)) ).

% or_not_numerals(7)
tff(fact_5478_or__not__numerals_I6_J,axiom,
    ! [M: num,N: num] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(num,int,numeral_numeral(int),bit0(M))),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,N)))) = aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(num,int,numeral_numeral(int),M)),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),N)))) ).

% or_not_numerals(6)
tff(fact_5479_OR__upper,axiom,
    ! [X: int,N: nat,Y: int] :
      ( pp(aa(int,bool,ord_less_eq(int,zero_zero(int)),X))
     => ( pp(aa(int,bool,ord_less(int,X),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),N)))
       => ( pp(aa(int,bool,ord_less(int,Y),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),N)))
         => pp(aa(int,bool,ord_less(int,aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),X),Y)),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),N))) ) ) ) ).

% OR_upper
tff(fact_5480_or__not__numerals_I5_J,axiom,
    ! [M: num,N: num] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(num,int,numeral_numeral(int),bit0(M))),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),bit0(N)))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(num,int,numeral_numeral(int),M)),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),N))))) ).

% or_not_numerals(5)
tff(fact_5481_test__bit__int__code_I1_J,axiom,
    ! [N: nat] : ~ pp(aa(nat,bool,bit_se5641148757651400278ts_bit(int,zero_zero(int)),N)) ).

% test_bit_int_code(1)
tff(fact_5482_int__and__code_I1_J,axiom,
    ! [J: int] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),zero_zero(int)),J) = zero_zero(int) ).

% int_and_code(1)
tff(fact_5483_int__and__code_I2_J,axiom,
    ! [I: int] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),I),zero_zero(int)) = zero_zero(int) ).

% int_and_code(2)
tff(fact_5484_Suc__0__or__eq,axiom,
    ! [N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),aa(nat,nat,suc,zero_zero(nat))),N) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),aa(bool,nat,zero_neq_one_of_bool(nat),aa(nat,bool,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N))) ).

% Suc_0_or_eq
tff(fact_5485_or__Suc__0__eq,axiom,
    ! [N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),N),aa(nat,nat,suc,zero_zero(nat))) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),aa(bool,nat,zero_neq_one_of_bool(nat),aa(nat,bool,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N))) ).

% or_Suc_0_eq
tff(fact_5486_or__nat__rec,axiom,
    ! [M: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),M),N) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(bool,nat,zero_neq_one_of_bool(nat),fdisj(aa(bool,bool,fNot,aa(nat,bool,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),M)),aa(bool,bool,fNot,aa(nat,bool,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N))))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2)))))) ).

% or_nat_rec
tff(fact_5487_or__not__numerals_I8_J,axiom,
    ! [M: num,N: num] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,M))),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),bit0(N)))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(num,int,numeral_numeral(int),M)),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),N))))) ).

% or_not_numerals(8)
tff(fact_5488_or__not__numerals_I9_J,axiom,
    ! [M: num,N: num] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,M))),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,N)))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(num,int,numeral_numeral(int),M)),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),N))))) ).

% or_not_numerals(9)
tff(fact_5489_or__int__rec,axiom,
    ! [K: int,L: int] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),K),L) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(bool,int,zero_neq_one_of_bool(int),fdisj(aa(bool,bool,fNot,aa(int,bool,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),K)),aa(bool,bool,fNot,aa(int,bool,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),L))))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),K),aa(num,int,numeral_numeral(int),bit0(one2)))),aa(int,int,aa(int,fun(int,int),divide_divide(int),L),aa(num,int,numeral_numeral(int),bit0(one2)))))) ).

% or_int_rec
tff(fact_5490_or__nat__unfold,axiom,
    ! [M: nat,N: nat] :
      ( ( ( M = zero_zero(nat) )
       => ( aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),M),N) = N ) )
      & ( ( M != zero_zero(nat) )
       => ( ( ( N = zero_zero(nat) )
           => ( aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),M),N) = M ) )
          & ( ( N != zero_zero(nat) )
           => ( aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),M),N) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,ord_max(nat,modulo_modulo(nat,M,aa(num,nat,numeral_numeral(nat),bit0(one2)))),modulo_modulo(nat,N,aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2)))))) ) ) ) ) ) ).

% or_nat_unfold
tff(fact_5491_or__int__unfold,axiom,
    ! [K: int,L: int] :
      ( ( ( ( K = aa(int,int,uminus_uminus(int),one_one(int)) )
          | ( L = aa(int,int,uminus_uminus(int),one_one(int)) ) )
       => ( aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),K),L) = aa(int,int,uminus_uminus(int),one_one(int)) ) )
      & ( ~ ( ( K = aa(int,int,uminus_uminus(int),one_one(int)) )
            | ( L = aa(int,int,uminus_uminus(int),one_one(int)) ) )
       => ( ( ( K = zero_zero(int) )
           => ( aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),K),L) = L ) )
          & ( ( K != zero_zero(int) )
           => ( ( ( L = zero_zero(int) )
               => ( aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),K),L) = K ) )
              & ( ( L != zero_zero(int) )
               => ( aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),K),L) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,ord_max(int,modulo_modulo(int,K,aa(num,int,numeral_numeral(int),bit0(one2)))),modulo_modulo(int,L,aa(num,int,numeral_numeral(int),bit0(one2))))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),K),aa(num,int,numeral_numeral(int),bit0(one2)))),aa(int,int,aa(int,fun(int,int),divide_divide(int),L),aa(num,int,numeral_numeral(int),bit0(one2)))))) ) ) ) ) ) ) ) ).

% or_int_unfold
tff(fact_5492_Bit__integer_Oabs__eq,axiom,
    ! [Xa2: int,X: bool] : bits_Bit_integer(code_integer_of_int(Xa2),X) = code_integer_of_int(aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(bool,int,zero_neq_one_of_bool(int),X)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),Xa2))) ).

% Bit_integer.abs_eq
tff(fact_5493_xor__int__unfold,axiom,
    ! [K: int,L: int] :
      ( ( ( K = aa(int,int,uminus_uminus(int),one_one(int)) )
       => ( aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),K),L) = aa(int,int,bit_ri4277139882892585799ns_not(int),L) ) )
      & ( ( K != aa(int,int,uminus_uminus(int),one_one(int)) )
       => ( ( ( L = aa(int,int,uminus_uminus(int),one_one(int)) )
           => ( aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),K),L) = aa(int,int,bit_ri4277139882892585799ns_not(int),K) ) )
          & ( ( L != aa(int,int,uminus_uminus(int),one_one(int)) )
           => ( ( ( K = zero_zero(int) )
               => ( aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),K),L) = L ) )
              & ( ( K != zero_zero(int) )
               => ( ( ( L = zero_zero(int) )
                   => ( aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),K),L) = K ) )
                  & ( ( L != zero_zero(int) )
                   => ( aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),K),L) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,abs_abs(int),aa(int,int,minus_minus(int,modulo_modulo(int,K,aa(num,int,numeral_numeral(int),bit0(one2)))),modulo_modulo(int,L,aa(num,int,numeral_numeral(int),bit0(one2)))))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),K),aa(num,int,numeral_numeral(int),bit0(one2)))),aa(int,int,aa(int,fun(int,int),divide_divide(int),L),aa(num,int,numeral_numeral(int),bit0(one2)))))) ) ) ) ) ) ) ) ) ) ).

% xor_int_unfold
tff(fact_5494_or__minus__numerals_I5_J,axiom,
    ! [N: num] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit0(N)))),one_one(int)) = aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit_or_not_num_neg(one2,bitM(N)))) ).

% or_minus_numerals(5)
tff(fact_5495_bit_Oxor__left__self,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [X: A,Y: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),X),aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),X),Y)) = Y ) ).

% bit.xor_left_self
tff(fact_5496_bit_Oxor__self,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),X),X) = zero_zero(A) ) ).

% bit.xor_self
tff(fact_5497_xor__self__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),A2),A2) = zero_zero(A) ) ).

% xor_self_eq
tff(fact_5498_xor_Oleft__neutral,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),zero_zero(A)),A2) = A2 ) ).

% xor.left_neutral
tff(fact_5499_xor_Oright__neutral,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),A2),zero_zero(A)) = A2 ) ).

% xor.right_neutral
tff(fact_5500_take__bit__xor,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A2: A,B2: A] : aa(A,A,bit_se2584673776208193580ke_bit(A,N),aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(A,A,bit_se2584673776208193580ke_bit(A,N),A2)),aa(A,A,bit_se2584673776208193580ke_bit(A,N),B2)) ) ).

% take_bit_xor
tff(fact_5501_bit_Oxor__compl__left,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [X: A,Y: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(A,A,bit_ri4277139882892585799ns_not(A),X)),Y) = aa(A,A,bit_ri4277139882892585799ns_not(A),aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),X),Y)) ) ).

% bit.xor_compl_left
tff(fact_5502_bit_Oxor__compl__right,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [X: A,Y: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),X),aa(A,A,bit_ri4277139882892585799ns_not(A),Y)) = aa(A,A,bit_ri4277139882892585799ns_not(A),aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),X),Y)) ) ).

% bit.xor_compl_right
tff(fact_5503_xor__nonnegative__int__iff,axiom,
    ! [K: int,L: int] :
      ( pp(aa(int,bool,ord_less_eq(int,zero_zero(int)),aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),K),L)))
    <=> ( pp(aa(int,bool,ord_less_eq(int,zero_zero(int)),K))
      <=> pp(aa(int,bool,ord_less_eq(int,zero_zero(int)),L)) ) ) ).

% xor_nonnegative_int_iff
tff(fact_5504_xor__negative__int__iff,axiom,
    ! [K: int,L: int] :
      ( pp(aa(int,bool,ord_less(int,aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),K),L)),zero_zero(int)))
    <=> ~ ( pp(aa(int,bool,ord_less(int,K),zero_zero(int)))
        <=> pp(aa(int,bool,ord_less(int,L),zero_zero(int))) ) ) ).

% xor_negative_int_iff
tff(fact_5505_bit_Oxor__one__left,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(A,A,uminus_uminus(A),one_one(A))),X) = aa(A,A,bit_ri4277139882892585799ns_not(A),X) ) ).

% bit.xor_one_left
tff(fact_5506_bit_Oxor__one__right,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),X),aa(A,A,uminus_uminus(A),one_one(A))) = aa(A,A,bit_ri4277139882892585799ns_not(A),X) ) ).

% bit.xor_one_right
tff(fact_5507_bit_Oxor__cancel__left,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(A,A,bit_ri4277139882892585799ns_not(A),X)),X) = aa(A,A,uminus_uminus(A),one_one(A)) ) ).

% bit.xor_cancel_left
tff(fact_5508_bit_Oxor__cancel__right,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),X),aa(A,A,bit_ri4277139882892585799ns_not(A),X)) = aa(A,A,uminus_uminus(A),one_one(A)) ) ).

% bit.xor_cancel_right
tff(fact_5509_xor__numerals_I3_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [X: num,Y: num] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(num,A,numeral_numeral(A),bit0(X))),aa(num,A,numeral_numeral(A),bit0(Y))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(num,A,numeral_numeral(A),X)),aa(num,A,numeral_numeral(A),Y))) ) ).

% xor_numerals(3)
tff(fact_5510_xor__numerals_I1_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Y: num] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),one_one(A)),aa(num,A,numeral_numeral(A),bit0(Y))) = aa(num,A,numeral_numeral(A),aa(num,num,bit1,Y)) ) ).

% xor_numerals(1)
tff(fact_5511_xor__numerals_I2_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Y: num] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),one_one(A)),aa(num,A,numeral_numeral(A),aa(num,num,bit1,Y))) = aa(num,A,numeral_numeral(A),bit0(Y)) ) ).

% xor_numerals(2)
tff(fact_5512_xor__numerals_I5_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [X: num] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(num,A,numeral_numeral(A),bit0(X))),one_one(A)) = aa(num,A,numeral_numeral(A),aa(num,num,bit1,X)) ) ).

% xor_numerals(5)
tff(fact_5513_xor__numerals_I8_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [X: num] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(num,A,numeral_numeral(A),aa(num,num,bit1,X))),one_one(A)) = aa(num,A,numeral_numeral(A),bit0(X)) ) ).

% xor_numerals(8)
tff(fact_5514_xor__numerals_I7_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [X: num,Y: num] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(num,A,numeral_numeral(A),aa(num,num,bit1,X))),aa(num,A,numeral_numeral(A),aa(num,num,bit1,Y))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(num,A,numeral_numeral(A),X)),aa(num,A,numeral_numeral(A),Y))) ) ).

% xor_numerals(7)
tff(fact_5515_or__minus__numerals_I8_J,axiom,
    ! [N: num,M: num] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,N)))),aa(num,int,numeral_numeral(int),M)) = aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit_or_not_num_neg(M,bit0(N)))) ).

% or_minus_numerals(8)
tff(fact_5516_or__minus__numerals_I4_J,axiom,
    ! [M: num,N: num] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(num,int,numeral_numeral(int),M)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,N)))) = aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit_or_not_num_neg(M,bit0(N)))) ).

% or_minus_numerals(4)
tff(fact_5517_or__minus__numerals_I3_J,axiom,
    ! [M: num,N: num] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(num,int,numeral_numeral(int),M)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit0(N)))) = aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit_or_not_num_neg(M,bitM(N)))) ).

% or_minus_numerals(3)
tff(fact_5518_or__minus__numerals_I7_J,axiom,
    ! [N: num,M: num] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit0(N)))),aa(num,int,numeral_numeral(int),M)) = aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit_or_not_num_neg(M,bitM(N)))) ).

% or_minus_numerals(7)
tff(fact_5519_xor__numerals_I4_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [X: num,Y: num] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(num,A,numeral_numeral(A),bit0(X))),aa(num,A,numeral_numeral(A),aa(num,num,bit1,Y))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(num,A,numeral_numeral(A),X)),aa(num,A,numeral_numeral(A),Y)))) ) ).

% xor_numerals(4)
tff(fact_5520_xor__numerals_I6_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [X: num,Y: num] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(num,A,numeral_numeral(A),aa(num,num,bit1,X))),aa(num,A,numeral_numeral(A),bit0(Y))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(num,A,numeral_numeral(A),X)),aa(num,A,numeral_numeral(A),Y)))) ) ).

% xor_numerals(6)
tff(fact_5521_or__minus__numerals_I1_J,axiom,
    ! [N: num] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),one_one(int)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit0(N)))) = aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit_or_not_num_neg(one2,bitM(N)))) ).

% or_minus_numerals(1)
tff(fact_5522_int__xor__code_I1_J,axiom,
    ! [J: int] : aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),zero_zero(int)),J) = J ).

% int_xor_code(1)
tff(fact_5523_int__xor__code_I2_J,axiom,
    ! [I: int] : aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),I),zero_zero(int)) = I ).

% int_xor_code(2)
tff(fact_5524_XOR__lower,axiom,
    ! [X: int,Y: int] :
      ( pp(aa(int,bool,ord_less_eq(int,zero_zero(int)),X))
     => ( pp(aa(int,bool,ord_less_eq(int,zero_zero(int)),Y))
       => pp(aa(int,bool,ord_less_eq(int,zero_zero(int)),aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),X),Y))) ) ) ).

% XOR_lower
tff(fact_5525_of__int__xor__eq,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [K: int,L: int] : aa(int,A,ring_1_of_int(A),aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),K),L)) = aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(int,A,ring_1_of_int(A),K)),aa(int,A,ring_1_of_int(A),L)) ) ).

% of_int_xor_eq
tff(fact_5526_bit__xor__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,B2: A,N: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),A2),B2)),N))
        <=> ~ ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,A2),N))
            <=> pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,B2),N)) ) ) ) ).

% bit_xor_iff
tff(fact_5527_bit__xor__int__iff,axiom,
    ! [K: int,L: int,N: nat] :
      ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(int,aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),K),L)),N))
    <=> ~ ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(int,K),N))
        <=> pp(aa(nat,bool,bit_se5641148757651400278ts_bit(int,L),N)) ) ) ).

% bit_xor_int_iff
tff(fact_5528_or__not__num__neg_Osimps_I1_J,axiom,
    bit_or_not_num_neg(one2,one2) = one2 ).

% or_not_num_neg.simps(1)
tff(fact_5529_xor_Oleft__commute,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [B2: A,A2: A,C2: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),B2),aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),A2),C2)) = aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),A2),aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),B2),C2)) ) ).

% xor.left_commute
tff(fact_5530_xor_Ocommute,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),A2),B2) = aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),B2),A2) ) ).

% xor.commute
tff(fact_5531_xor_Oassoc,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),A2),B2)),C2) = aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),A2),aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),B2),C2)) ) ).

% xor.assoc
tff(fact_5532_of__nat__xor__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,N: nat] : aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),M),N)) = aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(nat,A,semiring_1_of_nat(A),M)),aa(nat,A,semiring_1_of_nat(A),N)) ) ).

% of_nat_xor_eq
tff(fact_5533_bit_Oconj__xor__distrib,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [X: A,Y: A,Z: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),X),aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),Y),Z)) = aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),X),Y)),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),X),Z)) ) ).

% bit.conj_xor_distrib
tff(fact_5534_bit_Oconj__xor__distrib2,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Y: A,Z: A,X: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),Y),Z)),X) = aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),Y),X)),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),Z),X)) ) ).

% bit.conj_xor_distrib2
tff(fact_5535_or__not__num__neg_Osimps_I4_J,axiom,
    ! [N: num] : bit_or_not_num_neg(bit0(N),one2) = bit0(one2) ).

% or_not_num_neg.simps(4)
tff(fact_5536_or__not__num__neg_Osimps_I6_J,axiom,
    ! [N: num,M: num] : bit_or_not_num_neg(bit0(N),aa(num,num,bit1,M)) = bit0(bit_or_not_num_neg(N,M)) ).

% or_not_num_neg.simps(6)
tff(fact_5537_or__not__num__neg_Osimps_I3_J,axiom,
    ! [M: num] : bit_or_not_num_neg(one2,aa(num,num,bit1,M)) = aa(num,num,bit1,M) ).

% or_not_num_neg.simps(3)
tff(fact_5538_or__not__num__neg_Osimps_I7_J,axiom,
    ! [N: num] : bit_or_not_num_neg(aa(num,num,bit1,N),one2) = one2 ).

% or_not_num_neg.simps(7)
tff(fact_5539_or__not__num__neg_Osimps_I5_J,axiom,
    ! [N: num,M: num] : bit_or_not_num_neg(bit0(N),bit0(M)) = bitM(bit_or_not_num_neg(N,M)) ).

% or_not_num_neg.simps(5)
tff(fact_5540_or__not__num__neg_Osimps_I9_J,axiom,
    ! [N: num,M: num] : bit_or_not_num_neg(aa(num,num,bit1,N),aa(num,num,bit1,M)) = bitM(bit_or_not_num_neg(N,M)) ).

% or_not_num_neg.simps(9)
tff(fact_5541_xor__int__def,axiom,
    ! [K: int,L: int] : aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),K),L) = aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),K),aa(int,int,bit_ri4277139882892585799ns_not(int),L))),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(int,int,bit_ri4277139882892585799ns_not(int),K)),L)) ).

% xor_int_def
tff(fact_5542_bit_Oxor__def2,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [X: A,Y: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),X),Y) = aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),X),Y)),aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(A,A,bit_ri4277139882892585799ns_not(A),X)),aa(A,A,bit_ri4277139882892585799ns_not(A),Y))) ) ).

% bit.xor_def2
tff(fact_5543_bit_Oxor__def,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [X: A,Y: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),X),Y) = aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),X),aa(A,A,bit_ri4277139882892585799ns_not(A),Y))),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(A,A,bit_ri4277139882892585799ns_not(A),X)),Y)) ) ).

% bit.xor_def
tff(fact_5544_or__not__num__neg_Osimps_I2_J,axiom,
    ! [M: num] : bit_or_not_num_neg(one2,bit0(M)) = aa(num,num,bit1,M) ).

% or_not_num_neg.simps(2)
tff(fact_5545_or__not__num__neg_Osimps_I8_J,axiom,
    ! [N: num,M: num] : bit_or_not_num_neg(aa(num,num,bit1,N),bit0(M)) = bitM(bit_or_not_num_neg(N,M)) ).

% or_not_num_neg.simps(8)
tff(fact_5546_even__xor__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),A2),B2)))
        <=> ( pp(aa(A,bool,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A2))
          <=> pp(aa(A,bool,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),B2)) ) ) ) ).

% even_xor_iff
tff(fact_5547_or__not__num__neg_Oelims,axiom,
    ! [X: num,Xa2: num,Y: num] :
      ( ( bit_or_not_num_neg(X,Xa2) = Y )
     => ( ( ( X = one2 )
         => ( ( Xa2 = one2 )
           => ( Y != one2 ) ) )
       => ( ( ( X = one2 )
           => ! [M3: num] :
                ( ( Xa2 = bit0(M3) )
               => ( Y != aa(num,num,bit1,M3) ) ) )
         => ( ( ( X = one2 )
             => ! [M3: num] :
                  ( ( Xa2 = aa(num,num,bit1,M3) )
                 => ( Y != aa(num,num,bit1,M3) ) ) )
           => ( ( ? [N2: num] : X = bit0(N2)
               => ( ( Xa2 = one2 )
                 => ( Y != bit0(one2) ) ) )
             => ( ! [N2: num] :
                    ( ( X = bit0(N2) )
                   => ! [M3: num] :
                        ( ( Xa2 = bit0(M3) )
                       => ( Y != bitM(bit_or_not_num_neg(N2,M3)) ) ) )
               => ( ! [N2: num] :
                      ( ( X = bit0(N2) )
                     => ! [M3: num] :
                          ( ( Xa2 = aa(num,num,bit1,M3) )
                         => ( Y != bit0(bit_or_not_num_neg(N2,M3)) ) ) )
                 => ( ( ? [N2: num] : X = aa(num,num,bit1,N2)
                     => ( ( Xa2 = one2 )
                       => ( Y != one2 ) ) )
                   => ( ! [N2: num] :
                          ( ( X = aa(num,num,bit1,N2) )
                         => ! [M3: num] :
                              ( ( Xa2 = bit0(M3) )
                             => ( Y != bitM(bit_or_not_num_neg(N2,M3)) ) ) )
                     => ~ ! [N2: num] :
                            ( ( X = aa(num,num,bit1,N2) )
                           => ! [M3: num] :
                                ( ( Xa2 = aa(num,num,bit1,M3) )
                               => ( Y != bitM(bit_or_not_num_neg(N2,M3)) ) ) ) ) ) ) ) ) ) ) ) ) ).

% or_not_num_neg.elims
tff(fact_5548_numeral__or__not__num__eq,axiom,
    ! [M: num,N: num] : aa(num,int,numeral_numeral(int),bit_or_not_num_neg(M,N)) = aa(int,int,uminus_uminus(int),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(num,int,numeral_numeral(int),M)),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),N)))) ).

% numeral_or_not_num_eq
tff(fact_5549_int__numeral__not__or__num__neg,axiom,
    ! [M: num,N: num] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),M))),aa(num,int,numeral_numeral(int),N)) = aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit_or_not_num_neg(N,M))) ).

% int_numeral_not_or_num_neg
tff(fact_5550_int__numeral__or__not__num__neg,axiom,
    ! [M: num,N: num] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(num,int,numeral_numeral(int),M)),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),N))) = aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit_or_not_num_neg(M,N))) ).

% int_numeral_or_not_num_neg
tff(fact_5551_XOR__upper,axiom,
    ! [X: int,N: nat,Y: int] :
      ( pp(aa(int,bool,ord_less_eq(int,zero_zero(int)),X))
     => ( pp(aa(int,bool,ord_less(int,X),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),N)))
       => ( pp(aa(int,bool,ord_less(int,Y),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),N)))
         => pp(aa(int,bool,ord_less(int,aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),X),Y)),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),N))) ) ) ) ).

% XOR_upper
tff(fact_5552_xor__int__rec,axiom,
    ! [K: int,L: int] : aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),K),L) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(bool,int,zero_neq_one_of_bool(int),aa(bool,bool,fNot,aa(bool,bool,fequal(bool,aa(bool,bool,fNot,aa(int,bool,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),K))),aa(bool,bool,fNot,aa(int,bool,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),L)))))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),K),aa(num,int,numeral_numeral(int),bit0(one2)))),aa(int,int,aa(int,fun(int,int),divide_divide(int),L),aa(num,int,numeral_numeral(int),bit0(one2)))))) ).

% xor_int_rec
tff(fact_5553_xor__one__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),A2),one_one(A)) = aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa(bool,A,zero_neq_one_of_bool(A),aa(A,bool,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A2)))),aa(bool,A,zero_neq_one_of_bool(A),aa(bool,bool,fNot,aa(A,bool,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A2)))) ) ).

% xor_one_eq
tff(fact_5554_one__xor__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),one_one(A)),A2) = aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa(bool,A,zero_neq_one_of_bool(A),aa(A,bool,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A2)))),aa(bool,A,zero_neq_one_of_bool(A),aa(bool,bool,fNot,aa(A,bool,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A2)))) ) ).

% one_xor_eq
tff(fact_5555_int__ge__less__than__def,axiom,
    ! [D: int] : int_ge_less_than(D) = collect(product_prod(int,int),product_case_prod(int,int,bool,aTP_Lamp_mt(int,fun(int,fun(int,bool)),D))) ).

% int_ge_less_than_def
tff(fact_5556_int__ge__less__than2__def,axiom,
    ! [D: int] : int_ge_less_than2(D) = collect(product_prod(int,int),product_case_prod(int,int,bool,aTP_Lamp_mu(int,fun(int,fun(int,bool)),D))) ).

% int_ge_less_than2_def
tff(fact_5557_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),aa(A,A,uminus_uminus(A),one_one(A)),bit_se5824344971392196577ns_xor(A)) ) ).

% bit.abstract_boolean_algebra_sym_diff_axioms
tff(fact_5558_word__bitwise__m1__simps_I7_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344971392196577ns_xor(word(A)),X),aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A)))) = aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),X) ) ).

% word_bitwise_m1_simps(7)
tff(fact_5559_word__bitwise__m1__simps_I6_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344971392196577ns_xor(word(A)),aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A)))),X) = aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),X) ) ).

% word_bitwise_m1_simps(6)
tff(fact_5560_word__no__log__defs_I11_J,axiom,
    ! [K8: $tType] :
      ( type_len(K8)
     => ! [A2: num,B2: num] : aa(word(K8),word(K8),aa(word(K8),fun(word(K8),word(K8)),bit_se5824344971392196577ns_xor(word(K8)),aa(num,word(K8),numeral_numeral(word(K8)),A2)),aa(num,word(K8),numeral_numeral(word(K8)),B2)) = aa(int,word(K8),ring_1_of_int(word(K8)),aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),aa(num,int,numeral_numeral(int),A2)),aa(num,int,numeral_numeral(int),B2))) ) ).

% word_no_log_defs(11)
tff(fact_5561_xor__nat__numerals_I4_J,axiom,
    ! [X: num] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,X))),aa(nat,nat,suc,zero_zero(nat))) = aa(num,nat,numeral_numeral(nat),bit0(X)) ).

% xor_nat_numerals(4)
tff(fact_5562_xor__nat__numerals_I3_J,axiom,
    ! [X: num] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),aa(num,nat,numeral_numeral(nat),bit0(X))),aa(nat,nat,suc,zero_zero(nat))) = aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,X)) ).

% xor_nat_numerals(3)
tff(fact_5563_xor__nat__numerals_I2_J,axiom,
    ! [Y: num] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),aa(nat,nat,suc,zero_zero(nat))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,Y))) = aa(num,nat,numeral_numeral(nat),bit0(Y)) ).

% xor_nat_numerals(2)
tff(fact_5564_xor__nat__numerals_I1_J,axiom,
    ! [Y: num] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),aa(nat,nat,suc,zero_zero(nat))),aa(num,nat,numeral_numeral(nat),bit0(Y))) = aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,Y)) ).

% xor_nat_numerals(1)
tff(fact_5565_word__bitwise__1__simps_I10_J,axiom,
    ! [J4: $tType] :
      ( type_len(J4)
     => ! [B2: num] : aa(word(J4),word(J4),aa(word(J4),fun(word(J4),word(J4)),bit_se5824344971392196577ns_xor(word(J4)),one_one(word(J4))),aa(num,word(J4),numeral_numeral(word(J4)),B2)) = aa(int,word(J4),ring_1_of_int(word(J4)),aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),one_one(int)),aa(num,int,numeral_numeral(int),B2))) ) ).

% word_bitwise_1_simps(10)
tff(fact_5566_word__bitwise__1__simps_I12_J,axiom,
    ! [L5: $tType] :
      ( type_len(L5)
     => ! [A2: num] : aa(word(L5),word(L5),aa(word(L5),fun(word(L5),word(L5)),bit_se5824344971392196577ns_xor(word(L5)),aa(num,word(L5),numeral_numeral(word(L5)),A2)),one_one(word(L5))) = aa(int,word(L5),ring_1_of_int(word(L5)),aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),aa(num,int,numeral_numeral(int),A2)),one_one(int))) ) ).

% word_bitwise_1_simps(12)
tff(fact_5567_word__no__log__defs_I14_J,axiom,
    ! [N12: $tType] :
      ( type_len(N12)
     => ! [A2: num,B2: num] : aa(word(N12),word(N12),aa(word(N12),fun(word(N12),word(N12)),bit_se5824344971392196577ns_xor(word(N12)),aa(word(N12),word(N12),uminus_uminus(word(N12)),aa(num,word(N12),numeral_numeral(word(N12)),A2))),aa(word(N12),word(N12),uminus_uminus(word(N12)),aa(num,word(N12),numeral_numeral(word(N12)),B2))) = aa(int,word(N12),ring_1_of_int(word(N12)),aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),A2))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),B2)))) ) ).

% word_no_log_defs(14)
tff(fact_5568_word__no__log__defs_I13_J,axiom,
    ! [M11: $tType] :
      ( type_len(M11)
     => ! [A2: num,B2: num] : aa(word(M11),word(M11),aa(word(M11),fun(word(M11),word(M11)),bit_se5824344971392196577ns_xor(word(M11)),aa(word(M11),word(M11),uminus_uminus(word(M11)),aa(num,word(M11),numeral_numeral(word(M11)),A2))),aa(num,word(M11),numeral_numeral(word(M11)),B2)) = aa(int,word(M11),ring_1_of_int(word(M11)),aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),A2))),aa(num,int,numeral_numeral(int),B2))) ) ).

% word_no_log_defs(13)
tff(fact_5569_word__no__log__defs_I12_J,axiom,
    ! [L5: $tType] :
      ( type_len(L5)
     => ! [A2: num,B2: num] : aa(word(L5),word(L5),aa(word(L5),fun(word(L5),word(L5)),bit_se5824344971392196577ns_xor(word(L5)),aa(num,word(L5),numeral_numeral(word(L5)),A2)),aa(word(L5),word(L5),uminus_uminus(word(L5)),aa(num,word(L5),numeral_numeral(word(L5)),B2))) = aa(int,word(L5),ring_1_of_int(word(L5)),aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),aa(num,int,numeral_numeral(int),A2)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),B2)))) ) ).

% word_no_log_defs(12)
tff(fact_5570_word__bitwise__1__simps_I11_J,axiom,
    ! [K8: $tType] :
      ( type_len(K8)
     => ! [B2: num] : aa(word(K8),word(K8),aa(word(K8),fun(word(K8),word(K8)),bit_se5824344971392196577ns_xor(word(K8)),one_one(word(K8))),aa(word(K8),word(K8),uminus_uminus(word(K8)),aa(num,word(K8),numeral_numeral(word(K8)),B2))) = aa(int,word(K8),ring_1_of_int(word(K8)),aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),one_one(int)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),B2)))) ) ).

% word_bitwise_1_simps(11)
tff(fact_5571_word__bitwise__1__simps_I13_J,axiom,
    ! [M11: $tType] :
      ( type_len(M11)
     => ! [A2: num] : aa(word(M11),word(M11),aa(word(M11),fun(word(M11),word(M11)),bit_se5824344971392196577ns_xor(word(M11)),aa(word(M11),word(M11),uminus_uminus(word(M11)),aa(num,word(M11),numeral_numeral(word(M11)),A2))),one_one(word(M11))) = aa(int,word(M11),ring_1_of_int(word(M11)),aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),A2))),one_one(int))) ) ).

% word_bitwise_1_simps(13)
tff(fact_5572_word__log__esimps_I5_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344971392196577ns_xor(word(A)),X),zero_zero(word(A))) = X ) ).

% word_log_esimps(5)
tff(fact_5573_word__log__esimps_I11_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344971392196577ns_xor(word(A)),zero_zero(word(A))),X) = X ) ).

% word_log_esimps(11)
tff(fact_5574_word__bw__same_I3_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344971392196577ns_xor(word(A)),X),X) = zero_zero(word(A)) ) ).

% word_bw_same(3)
tff(fact_5575_xor__nat__def,axiom,
    ! [M: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),M),N) = nat2(aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),aa(nat,int,semiring_1_of_nat(int),M)),aa(nat,int,semiring_1_of_nat(int),N))) ).

% xor_nat_def
tff(fact_5576_bit__twiddle__max,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344971392196577ns_xor(word(A)),X),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344971392196577ns_xor(word(A)),X),Y)),if(word(A),aa(word(A),bool,ord_less(word(A),X),Y),aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A))),zero_zero(word(A))))) = aa(word(A),word(A),ord_max(word(A),X),Y) ) ).

% bit_twiddle_max
tff(fact_5577_xor__nat__unfold,axiom,
    ! [M: nat,N: nat] :
      ( ( ( M = zero_zero(nat) )
       => ( aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),M),N) = N ) )
      & ( ( M != zero_zero(nat) )
       => ( ( ( N = zero_zero(nat) )
           => ( aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),M),N) = M ) )
          & ( ( N != zero_zero(nat) )
           => ( aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),M),N) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),modulo_modulo(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),modulo_modulo(nat,M,aa(num,nat,numeral_numeral(nat),bit0(one2)))),modulo_modulo(nat,N,aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2)))))) ) ) ) ) ) ).

% xor_nat_unfold
tff(fact_5578_xor__nat__rec,axiom,
    ! [M: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),M),N) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(bool,nat,zero_neq_one_of_bool(nat),aa(bool,bool,fNot,aa(bool,bool,fequal(bool,aa(bool,bool,fNot,aa(nat,bool,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),M))),aa(bool,bool,fNot,aa(nat,bool,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N)))))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2)))))) ).

% xor_nat_rec
tff(fact_5579_Suc__0__xor__eq,axiom,
    ! [N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),aa(nat,nat,suc,zero_zero(nat))),N) = aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),aa(bool,nat,zero_neq_one_of_bool(nat),aa(nat,bool,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N)))),aa(bool,nat,zero_neq_one_of_bool(nat),aa(bool,bool,fNot,aa(nat,bool,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N)))) ).

% Suc_0_xor_eq
tff(fact_5580_xor__Suc__0__eq,axiom,
    ! [N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),N),aa(nat,nat,suc,zero_zero(nat))) = aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),aa(bool,nat,zero_neq_one_of_bool(nat),aa(nat,bool,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N)))),aa(bool,nat,zero_neq_one_of_bool(nat),aa(bool,bool,fNot,aa(nat,bool,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N)))) ).

% xor_Suc_0_eq
tff(fact_5581_word__ops__lsb,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A)] :
          ( ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se1065995026697491101ons_or(word(A)),X),Y)),zero_zero(nat)))
          <=> ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(word(A),X),zero_zero(nat)))
              | pp(aa(nat,bool,bit_se5641148757651400278ts_bit(word(A),Y),zero_zero(nat))) ) )
          & ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),X),Y)),zero_zero(nat)))
          <=> ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(word(A),X),zero_zero(nat)))
              & pp(aa(nat,bool,bit_se5641148757651400278ts_bit(word(A),Y),zero_zero(nat))) ) )
          & ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344971392196577ns_xor(word(A)),X),Y)),zero_zero(nat)))
          <=> ~ ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(word(A),X),zero_zero(nat)))
              <=> pp(aa(nat,bool,bit_se5641148757651400278ts_bit(word(A),Y),zero_zero(nat))) ) )
          & ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(word(A),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),X)),zero_zero(nat)))
          <=> ~ pp(aa(nat,bool,bit_se5641148757651400278ts_bit(word(A),X),zero_zero(nat))) ) ) ) ).

% word_ops_lsb
tff(fact_5582_hash__code__prod__simps,axiom,
    ! [A: $tType,B: $tType,H_a: fun(A,uint32),H_b: fun(B,uint32),X: A,Xa2: B] : hash_hash_code_prod(A,B,H_a,H_b,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Xa2)) = aa(uint32,uint32,aa(uint32,fun(uint32,uint32),plus_plus(uint32),aa(uint32,uint32,aa(uint32,fun(uint32,uint32),times_times(uint32),aa(A,uint32,H_a,X)),aa(num,uint32,numeral_numeral(uint32),bit0(aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,bit0(aa(num,num,bit1,aa(num,num,bit1,bit0(aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,bit0(bit0(aa(num,num,bit1,aa(num,num,bit1,bit0(bit0(bit0(aa(num,num,bit1,bit0(aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,bit0(bit0(aa(num,num,bit1,aa(num,num,bit1,one2))))))))))))))))))))))))))))))))),aa(uint32,uint32,aa(uint32,fun(uint32,uint32),plus_plus(uint32),aa(uint32,uint32,aa(uint32,fun(uint32,uint32),times_times(uint32),aa(B,uint32,H_b,Xa2)),aa(num,uint32,numeral_numeral(uint32),aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,bit0(aa(num,num,bit1,bit0(bit0(bit0(bit0(aa(num,num,bit1,aa(num,num,bit1,bit0(bit0(aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,bit0(aa(num,num,bit1,bit0(bit0(bit0(bit0(aa(num,num,bit1,aa(num,num,bit1,bit0(aa(num,num,bit1,aa(num,num,bit1,bit0(aa(num,num,bit1,one2))))))))))))))))))))))))))))))))),aa(num,uint32,numeral_numeral(uint32),bit0(bit0(bit0(bit0(bit0(aa(num,num,bit1,bit0(aa(num,num,bit1,bit0(aa(num,num,bit1,bit0(aa(num,num,bit1,aa(num,num,bit1,bit0(aa(num,num,bit1,bit0(bit0(bit0(bit0(aa(num,num,bit1,bit0(aa(num,num,bit1,aa(num,num,bit1,bit0(aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,bit0(bit0(aa(num,num,bit1,one2))))))))))))))))))))))))))))))))) ).

% hash_code_prod_simps
tff(fact_5583_sum__count__set,axiom,
    ! [A: $tType,Xs: list(A),X6: set(A)] :
      ( pp(aa(set(A),bool,ord_less_eq(set(A),aa(list(A),set(A),set2(A),Xs)),X6))
     => ( pp(aa(set(A),bool,finite_finite2(A),X6))
       => ( aa(set(A),nat,groups7311177749621191930dd_sum(A,nat,count_list(A,Xs)),X6) = aa(list(A),nat,size_size(list(A)),Xs) ) ) ) ).

% sum_count_set
tff(fact_5584_count__notin,axiom,
    ! [A: $tType,X: A,Xs: list(A)] :
      ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(list(A),set(A),set2(A),Xs)))
     => ( aa(A,nat,count_list(A,Xs),X) = zero_zero(nat) ) ) ).

% count_notin
tff(fact_5585_count__le__length,axiom,
    ! [A: $tType,Xs: list(A),X: A] : pp(aa(nat,bool,ord_less_eq(nat,aa(A,nat,count_list(A,Xs),X)),aa(list(A),nat,size_size(list(A)),Xs))) ).

% count_le_length
tff(fact_5586_hash__code__option__simps_I2_J,axiom,
    ! [A: $tType,H_a: fun(A,uint32),X: A] : hash_h1887023736457453652option(A,H_a,aa(A,option(A),some(A),X)) = aa(uint32,uint32,aa(uint32,fun(uint32,uint32),plus_plus(uint32),aa(uint32,uint32,aa(uint32,fun(uint32,uint32),times_times(uint32),aa(A,uint32,H_a,X)),aa(num,uint32,numeral_numeral(uint32),aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,bit0(aa(num,num,bit1,bit0(bit0(aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,bit0(bit0(aa(num,num,bit1,aa(num,num,bit1,bit0(bit0(bit0(aa(num,num,bit1,bit0(aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,bit0(bit0(bit0(aa(num,num,bit1,bit0(one2)))))))))))))))))))))))))))))))),aa(num,uint32,numeral_numeral(uint32),bit0(bit0(bit0(aa(num,num,bit1,bit0(bit0(bit0(bit0(bit0(aa(num,num,bit1,aa(num,num,bit1,bit0(aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,bit0(aa(num,num,bit1,aa(num,num,bit1,bit0(aa(num,num,bit1,bit0(aa(num,num,bit1,aa(num,num,bit1,bit0(aa(num,num,bit1,bit0(bit0(bit0(one2))))))))))))))))))))))))))))))) ).

% hash_code_option_simps(2)
tff(fact_5587_hash__code__option__simps_I1_J,axiom,
    ! [A: $tType,H_a: fun(A,uint32)] : hash_h1887023736457453652option(A,H_a,none(A)) = aa(num,uint32,numeral_numeral(uint32),aa(num,num,bit1,aa(num,num,bit1,bit0(aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,bit0(aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,bit0(bit0(bit0(aa(num,num,bit1,bit0(bit0(bit0(bit0(bit0(bit0(aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,bit0(aa(num,num,bit1,bit0(aa(num,num,bit1,one2))))))))))))))))))))))))))))))) ).

% hash_code_option_simps(1)
tff(fact_5588_word__lsb__neg__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Bin: num] :
          ( pp(aa(word(A),bool,least_8051144512741203767sb_lsb(word(A)),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),Bin))))
        <=> ~ pp(aa(int,bool,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Bin)))) ) ) ).

% word_lsb_neg_numeral
tff(fact_5589_word__lsb__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Bin: num] :
          ( pp(aa(word(A),bool,least_8051144512741203767sb_lsb(word(A)),aa(num,word(A),numeral_numeral(word(A)),Bin)))
        <=> ~ pp(aa(int,bool,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),aa(num,int,numeral_numeral(int),Bin))) ) ) ).

% word_lsb_numeral
tff(fact_5590_word__lsb__alt,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] :
          ( pp(aa(word(A),bool,least_8051144512741203767sb_lsb(word(A)),W))
        <=> pp(aa(nat,bool,bit_se5641148757651400278ts_bit(word(A),W),zero_zero(nat))) ) ) ).

% word_lsb_alt
tff(fact_5591_word__lsb__1__0,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(A) )
     => ( pp(aa(word(A),bool,least_8051144512741203767sb_lsb(word(A)),one_one(word(A))))
        & ~ pp(aa(word(B),bool,least_8051144512741203767sb_lsb(word(B)),zero_zero(word(B)))) ) ) ).

% word_lsb_1_0
tff(fact_5592_lsb__word__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X4: word(A)] :
          ( pp(aa(word(A),bool,least_8051144512741203767sb_lsb(word(A)),X4))
        <=> ~ pp(aa(word(A),bool,dvd_dvd(word(A),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),X4)) ) ) ).

% lsb_word_eq
tff(fact_5593_word__lsb__def,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A)] :
          ( pp(aa(word(A),bool,least_8051144512741203767sb_lsb(word(A)),A2))
        <=> ~ pp(aa(int,bool,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),aa(word(A),int,semiring_1_unsigned(A,int),A2))) ) ) ).

% word_lsb_def
tff(fact_5594_word__lsb__nat,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] :
          ( pp(aa(word(A),bool,least_8051144512741203767sb_lsb(word(A)),W))
        <=> ( modulo_modulo(nat,aa(word(A),nat,semiring_1_unsigned(A,nat),W),aa(num,nat,numeral_numeral(nat),bit0(one2))) = one_one(nat) ) ) ) ).

% word_lsb_nat
tff(fact_5595_word__lsb__int,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] :
          ( pp(aa(word(A),bool,least_8051144512741203767sb_lsb(word(A)),W))
        <=> ( modulo_modulo(int,aa(word(A),int,semiring_1_unsigned(A,int),W),aa(num,int,numeral_numeral(int),bit0(one2))) = one_one(int) ) ) ) ).

% word_lsb_int
tff(fact_5596_lsb__odd,axiom,
    ! [A: $tType] :
      ( least_6119777620449941438nt_lsb(A)
     => ! [X4: A] :
          ( pp(aa(A,bool,least_8051144512741203767sb_lsb(A),X4))
        <=> ~ pp(aa(A,bool,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),X4)) ) ) ).

% lsb_odd
tff(fact_5597_dup__1,axiom,
    code_dup(one_one(code_integer)) = aa(num,code_integer,numeral_numeral(code_integer),bit0(one2)) ).

% dup_1
tff(fact_5598_power__numeral,axiom,
    ! [A: $tType] :
      ( semiring_numeral(A)
     => ! [K: num,L: num] : aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),K)),aa(num,nat,numeral_numeral(nat),L)) = aa(num,A,numeral_numeral(A),pow(K,L)) ) ).

% power_numeral
tff(fact_5599_VEBT__internal_OT__vebt__buildupi_H_Opelims,axiom,
    ! [X: nat,Y: int] :
      ( ( vEBT_V9176841429113362141ildupi(X) = Y )
     => ( accp(nat,vEBT_V3352910403632780892pi_rel,X)
       => ( ( ( X = zero_zero(nat) )
           => ( ( Y = one_one(int) )
             => ~ accp(nat,vEBT_V3352910403632780892pi_rel,zero_zero(nat)) ) )
         => ( ( ( X = aa(nat,nat,suc,zero_zero(nat)) )
             => ( ( Y = one_one(int) )
               => ~ accp(nat,vEBT_V3352910403632780892pi_rel,aa(nat,nat,suc,zero_zero(nat))) ) )
           => ~ ! [N2: nat] :
                  ( ( X = aa(nat,nat,suc,aa(nat,nat,suc,N2)) )
                 => ( ( ( pp(aa(nat,bool,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N2))
                       => ( Y = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,one2))),aa(int,int,aa(int,fun(int,int),plus_plus(int),vEBT_V9176841429113362141ildupi(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N2),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(bit0(one2)))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N2),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(int,int,aa(int,fun(int,int),times_times(int),vEBT_V9176841429113362141ildupi(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N2),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N2),aa(num,nat,numeral_numeral(nat),bit0(one2))))))))) ) )
                      & ( ~ pp(aa(nat,bool,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N2))
                       => ( Y = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,one2))),aa(int,int,aa(int,fun(int,int),plus_plus(int),vEBT_V9176841429113362141ildupi(aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N2),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(bit0(bit0(one2))))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N2),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(bit0(one2)))),aa(int,int,aa(int,fun(int,int),times_times(int),vEBT_V9176841429113362141ildupi(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N2),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N2),aa(num,nat,numeral_numeral(nat),bit0(one2))))))))) ) ) )
                   => ~ accp(nat,vEBT_V3352910403632780892pi_rel,aa(nat,nat,suc,aa(nat,nat,suc,N2))) ) ) ) ) ) ) ).

% VEBT_internal.T_vebt_buildupi'.pelims
tff(fact_5600_int__lsb__numeral_I1_J,axiom,
    ~ pp(aa(int,bool,least_8051144512741203767sb_lsb(int),zero_zero(int))) ).

% int_lsb_numeral(1)
tff(fact_5601_int__lsb__numeral_I2_J,axiom,
    pp(aa(int,bool,least_8051144512741203767sb_lsb(int),one_one(int))) ).

% int_lsb_numeral(2)
tff(fact_5602_int__lsb__numeral_I6_J,axiom,
    ! [W: num] : ~ pp(aa(int,bool,least_8051144512741203767sb_lsb(int),aa(num,int,numeral_numeral(int),bit0(W)))) ).

% int_lsb_numeral(6)
tff(fact_5603_int__lsb__numeral_I3_J,axiom,
    pp(aa(int,bool,least_8051144512741203767sb_lsb(int),aa(num,int,numeral_numeral(int),one2))) ).

% int_lsb_numeral(3)
tff(fact_5604_int__lsb__numeral_I7_J,axiom,
    ! [W: num] : pp(aa(int,bool,least_8051144512741203767sb_lsb(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,W)))) ).

% int_lsb_numeral(7)
tff(fact_5605_int__lsb__numeral_I4_J,axiom,
    pp(aa(int,bool,least_8051144512741203767sb_lsb(int),aa(int,int,uminus_uminus(int),one_one(int)))) ).

% int_lsb_numeral(4)
tff(fact_5606_int__lsb__numeral_I8_J,axiom,
    ! [W: num] : ~ pp(aa(int,bool,least_8051144512741203767sb_lsb(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit0(W))))) ).

% int_lsb_numeral(8)
tff(fact_5607_int__lsb__numeral_I5_J,axiom,
    pp(aa(int,bool,least_8051144512741203767sb_lsb(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),one2)))) ).

% int_lsb_numeral(5)
tff(fact_5608_int__lsb__numeral_I9_J,axiom,
    ! [W: num] : pp(aa(int,bool,least_8051144512741203767sb_lsb(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,W))))) ).

% int_lsb_numeral(9)
tff(fact_5609_Code__Numeral_Odup__code_I1_J,axiom,
    code_dup(zero_zero(code_integer)) = zero_zero(code_integer) ).

% Code_Numeral.dup_code(1)
tff(fact_5610_lsb__integer__code,axiom,
    ! [X: code_integer] :
      ( pp(aa(code_integer,bool,least_8051144512741203767sb_lsb(code_integer),X))
    <=> pp(aa(nat,bool,bit_se5641148757651400278ts_bit(code_integer,X),zero_zero(nat))) ) ).

% lsb_integer_code
tff(fact_5611_pow_Osimps_I1_J,axiom,
    ! [X: num] : pow(X,one2) = X ).

% pow.simps(1)
tff(fact_5612_lsb__int__def,axiom,
    ! [I: int] :
      ( pp(aa(int,bool,least_8051144512741203767sb_lsb(int),I))
    <=> pp(aa(nat,bool,bit_se5641148757651400278ts_bit(int,I),zero_zero(nat))) ) ).

% lsb_int_def
tff(fact_5613_dup_Oabs__eq,axiom,
    ! [X: int] : code_dup(code_integer_of_int(X)) = code_integer_of_int(aa(int,int,aa(int,fun(int,int),plus_plus(int),X),X)) ).

% dup.abs_eq
tff(fact_5614_bin__last__conv__lsb,axiom,
    aTP_Lamp_mv(int,bool) = least_8051144512741203767sb_lsb(int) ).

% bin_last_conv_lsb
tff(fact_5615_vebt__buildup_Opelims,axiom,
    ! [X: nat,Y: vEBT_VEBT] :
      ( ( vEBT_vebt_buildup(X) = Y )
     => ( accp(nat,vEBT_v4011308405150292612up_rel,X)
       => ( ( ( X = zero_zero(nat) )
           => ( ( Y = vEBT_Leaf(fFalse,fFalse) )
             => ~ accp(nat,vEBT_v4011308405150292612up_rel,zero_zero(nat)) ) )
         => ( ( ( X = aa(nat,nat,suc,zero_zero(nat)) )
             => ( ( Y = vEBT_Leaf(fFalse,fFalse) )
               => ~ accp(nat,vEBT_v4011308405150292612up_rel,aa(nat,nat,suc,zero_zero(nat))) ) )
           => ~ ! [Va2: nat] :
                  ( ( X = aa(nat,nat,suc,aa(nat,nat,suc,Va2)) )
                 => ( ( ( pp(aa(nat,bool,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,aa(nat,nat,suc,Va2))))
                       => ( Y = vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),replicate(vEBT_VEBT,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),vEBT_vebt_buildup(aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_vebt_buildup(aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) )
                      & ( ~ pp(aa(nat,bool,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,aa(nat,nat,suc,Va2))))
                       => ( Y = vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),replicate(vEBT_VEBT,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_vebt_buildup(aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_vebt_buildup(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))) ) ) )
                   => ~ accp(nat,vEBT_v4011308405150292612up_rel,aa(nat,nat,suc,aa(nat,nat,suc,Va2))) ) ) ) ) ) ) ).

% vebt_buildup.pelims
tff(fact_5616_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_Opelims,axiom,
    ! [X: nat,Y: nat] :
      ( ( vEBT_V8646137997579335489_i_l_d(X) = Y )
     => ( accp(nat,vEBT_V5144397997797733112_d_rel,X)
       => ( ( ( X = zero_zero(nat) )
           => ( ( Y = aa(num,nat,numeral_numeral(nat),bit0(bit0(one2))) )
             => ~ accp(nat,vEBT_V5144397997797733112_d_rel,zero_zero(nat)) ) )
         => ( ( ( X = aa(nat,nat,suc,zero_zero(nat)) )
             => ( ( Y = aa(num,nat,numeral_numeral(nat),bit0(bit0(one2))) )
               => ~ accp(nat,vEBT_V5144397997797733112_d_rel,aa(nat,nat,suc,zero_zero(nat))) ) )
           => ~ ! [Va2: nat] :
                  ( ( X = aa(nat,nat,suc,aa(nat,nat,suc,Va2)) )
                 => ( ( ( pp(aa(nat,bool,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,aa(nat,nat,suc,Va2))))
                       => ( Y = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(aa(num,num,bit1,bit0(one2))))),vEBT_V8646137997579335489_i_l_d(aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_V8646137997579335489_i_l_d(aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))))))) ) )
                      & ( ~ pp(aa(nat,bool,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,aa(nat,nat,suc,Va2))))
                       => ( Y = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(aa(num,num,bit1,one2))))),vEBT_V8646137997579335489_i_l_d(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),vEBT_V8646137997579335489_i_l_d(aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))) ) ) )
                   => ~ accp(nat,vEBT_V5144397997797733112_d_rel,aa(nat,nat,suc,aa(nat,nat,suc,Va2))) ) ) ) ) ) ) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d.pelims
tff(fact_5617_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_092_060_094sub_062u_092_060_094sub_062p_Opelims,axiom,
    ! [X: nat,Y: nat] :
      ( ( vEBT_V8346862874174094_d_u_p(X) = Y )
     => ( accp(nat,vEBT_V1247956027447740395_p_rel,X)
       => ( ( ( X = zero_zero(nat) )
           => ( ( Y = aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2)) )
             => ~ accp(nat,vEBT_V1247956027447740395_p_rel,zero_zero(nat)) ) )
         => ( ( ( X = aa(nat,nat,suc,zero_zero(nat)) )
             => ( ( Y = aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2)) )
               => ~ accp(nat,vEBT_V1247956027447740395_p_rel,aa(nat,nat,suc,zero_zero(nat))) ) )
           => ~ ! [Va2: nat] :
                  ( ( X = aa(nat,nat,suc,aa(nat,nat,suc,Va2)) )
                 => ( ( ( pp(aa(nat,bool,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,aa(nat,nat,suc,Va2))))
                       => ( Y = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,bit0(bit0(one2))))),vEBT_V8346862874174094_d_u_p(aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_V8346862874174094_d_u_p(aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),one_one(nat))))) ) )
                      & ( ~ pp(aa(nat,bool,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,aa(nat,nat,suc,Va2))))
                       => ( Y = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,aa(num,num,bit1,bit0(one2))))),vEBT_V8346862874174094_d_u_p(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_V8346862874174094_d_u_p(aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),one_one(nat)))) ) ) )
                   => ~ accp(nat,vEBT_V1247956027447740395_p_rel,aa(nat,nat,suc,aa(nat,nat,suc,Va2))) ) ) ) ) ) ) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d\<^sub>u\<^sub>p.pelims
tff(fact_5618_VEBT__internal_OTb_Opelims,axiom,
    ! [X: nat,Y: int] :
      ( ( vEBT_VEBT_Tb(X) = Y )
     => ( accp(nat,vEBT_VEBT_Tb_rel2,X)
       => ( ( ( X = zero_zero(nat) )
           => ( ( Y = aa(num,int,numeral_numeral(int),aa(num,num,bit1,one2)) )
             => ~ accp(nat,vEBT_VEBT_Tb_rel2,zero_zero(nat)) ) )
         => ( ( ( X = aa(nat,nat,suc,zero_zero(nat)) )
             => ( ( Y = aa(num,int,numeral_numeral(int),aa(num,num,bit1,one2)) )
               => ~ accp(nat,vEBT_VEBT_Tb_rel2,aa(nat,nat,suc,zero_zero(nat))) ) )
           => ~ ! [N2: nat] :
                  ( ( X = aa(nat,nat,suc,aa(nat,nat,suc,N2)) )
                 => ( ( ( pp(aa(nat,bool,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N2))
                       => ( Y = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,bit0(one2)))),vEBT_VEBT_Tb(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N2),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),aa(int,int,aa(int,fun(int,int),times_times(int),vEBT_VEBT_Tb(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N2),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N2),aa(num,nat,numeral_numeral(nat),bit0(one2))))))) ) )
                      & ( ~ pp(aa(nat,bool,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N2))
                       => ( Y = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,bit0(one2)))),vEBT_VEBT_Tb(aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N2),aa(num,nat,numeral_numeral(nat),bit0(one2)))))))),aa(int,int,aa(int,fun(int,int),times_times(int),vEBT_VEBT_Tb(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N2),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N2),aa(num,nat,numeral_numeral(nat),bit0(one2)))))))) ) ) )
                   => ~ accp(nat,vEBT_VEBT_Tb_rel2,aa(nat,nat,suc,aa(nat,nat,suc,N2))) ) ) ) ) ) ) ).

% VEBT_internal.Tb.pelims
tff(fact_5619_VEBT__internal_OTb_H_Opelims,axiom,
    ! [X: nat,Y: nat] :
      ( ( vEBT_VEBT_Tb2(X) = Y )
     => ( accp(nat,vEBT_VEBT_Tb_rel,X)
       => ( ( ( X = zero_zero(nat) )
           => ( ( Y = aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2)) )
             => ~ accp(nat,vEBT_VEBT_Tb_rel,zero_zero(nat)) ) )
         => ( ( ( X = aa(nat,nat,suc,zero_zero(nat)) )
             => ( ( Y = aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2)) )
               => ~ accp(nat,vEBT_VEBT_Tb_rel,aa(nat,nat,suc,zero_zero(nat))) ) )
           => ~ ! [N2: nat] :
                  ( ( X = aa(nat,nat,suc,aa(nat,nat,suc,N2)) )
                 => ( ( ( pp(aa(nat,bool,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N2))
                       => ( Y = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,bit0(one2)))),vEBT_VEBT_Tb2(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N2),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),vEBT_VEBT_Tb2(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N2),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N2),aa(num,nat,numeral_numeral(nat),bit0(one2))))))) ) )
                      & ( ~ pp(aa(nat,bool,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N2))
                       => ( Y = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,bit0(one2)))),vEBT_VEBT_Tb2(aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N2),aa(num,nat,numeral_numeral(nat),bit0(one2)))))))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),vEBT_VEBT_Tb2(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N2),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N2),aa(num,nat,numeral_numeral(nat),bit0(one2)))))))) ) ) )
                   => ~ accp(nat,vEBT_VEBT_Tb_rel,aa(nat,nat,suc,aa(nat,nat,suc,N2))) ) ) ) ) ) ) ).

% VEBT_internal.Tb'.pelims
tff(fact_5620_VEBT__internal_OT__vebt__buildupi_Opelims,axiom,
    ! [X: nat,Y: nat] :
      ( ( vEBT_V441764108873111860ildupi(X) = Y )
     => ( accp(nat,vEBT_V2957053500504383685pi_rel,X)
       => ( ( ( X = zero_zero(nat) )
           => ( ( Y = aa(nat,nat,suc,zero_zero(nat)) )
             => ~ accp(nat,vEBT_V2957053500504383685pi_rel,zero_zero(nat)) ) )
         => ( ( ( X = aa(nat,nat,suc,zero_zero(nat)) )
             => ( ( Y = aa(nat,nat,suc,zero_zero(nat)) )
               => ~ accp(nat,vEBT_V2957053500504383685pi_rel,aa(nat,nat,suc,zero_zero(nat))) ) )
           => ~ ! [N2: nat] :
                  ( ( X = aa(nat,nat,suc,aa(nat,nat,suc,N2)) )
                 => ( ( ( pp(aa(nat,bool,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N2))
                       => ( Y = aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_V441764108873111860ildupi(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N2),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(one2)))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N2),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),vEBT_V441764108873111860ildupi(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N2),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N2),aa(num,nat,numeral_numeral(nat),bit0(one2))))))))))) ) )
                      & ( ~ pp(aa(nat,bool,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N2))
                       => ( Y = aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_V441764108873111860ildupi(aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N2),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(bit0(one2))))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N2),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(one2)))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),vEBT_V441764108873111860ildupi(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N2),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N2),aa(num,nat,numeral_numeral(nat),bit0(one2))))))))))) ) ) )
                   => ~ accp(nat,vEBT_V2957053500504383685pi_rel,aa(nat,nat,suc,aa(nat,nat,suc,N2))) ) ) ) ) ) ) ).

% VEBT_internal.T_vebt_buildupi.pelims
tff(fact_5621_cis__multiple__2pi,axiom,
    ! [N: real] :
      ( pp(aa(set(real),bool,aa(real,fun(set(real),bool),member(real),N),ring_1_Ints(real)))
     => ( cis(aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),pi)),N)) = one_one(complex) ) ) ).

% cis_multiple_2pi
tff(fact_5622_ran__nth__set__encoding__conv,axiom,
    ! [A: $tType,L: list(A)] : ran(nat,A,aTP_Lamp_mw(list(A),fun(nat,option(A)),L)) = aa(list(A),set(A),set2(A),L) ).

% ran_nth_set_encoding_conv
tff(fact_5623_rat__inverse__code,axiom,
    ! [P2: rat] : quotient_of(aa(rat,rat,inverse_inverse(rat),P2)) = aa(product_prod(int,int),product_prod(int,int),product_case_prod(int,int,product_prod(int,int),aTP_Lamp_mx(int,fun(int,product_prod(int,int)))),quotient_of(P2)) ).

% rat_inverse_code
tff(fact_5624_frac__eq__0__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( ( archimedean_frac(A,X) = zero_zero(A) )
        <=> pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),ring_1_Ints(A))) ) ) ).

% frac_eq_0_iff
tff(fact_5625_floor__add2,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,Y: A] :
          ( ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),ring_1_Ints(A)))
            | pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Y),ring_1_Ints(A))) )
         => ( archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),archim6421214686448440834_floor(A,X)),archim6421214686448440834_floor(A,Y)) ) ) ) ).

% floor_add2
tff(fact_5626_frac__gt__0__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( pp(aa(A,bool,ord_less(A,zero_zero(A)),archimedean_frac(A,X)))
        <=> ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),ring_1_Ints(A))) ) ) ).

% frac_gt_0_iff
tff(fact_5627_quotient__of__number_I3_J,axiom,
    ! [K: num] : quotient_of(aa(num,rat,numeral_numeral(rat),K)) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(num,int,numeral_numeral(int),K)),one_one(int)) ).

% quotient_of_number(3)
tff(fact_5628_rat__one__code,axiom,
    quotient_of(one_one(rat)) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),one_one(int)),one_one(int)) ).

% rat_one_code
tff(fact_5629_map__update__eta__repair_I2_J,axiom,
    ! [B: $tType,A: $tType,M: fun(A,option(B)),K: A,V: B] :
      ( ( aa(A,option(B),M,K) = none(B) )
     => ( ran(A,B,aa(B,fun(A,option(B)),aa(A,fun(B,fun(A,option(B))),aTP_Lamp_my(fun(A,option(B)),fun(A,fun(B,fun(A,option(B)))),M),K),V)) = insert(B,V,ran(A,B,M)) ) ) ).

% map_update_eta_repair(2)
tff(fact_5630_rat__zero__code,axiom,
    quotient_of(zero_zero(rat)) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),one_one(int)) ).

% rat_zero_code
tff(fact_5631_quotient__of__number_I5_J,axiom,
    ! [K: num] : quotient_of(aa(rat,rat,uminus_uminus(rat),aa(num,rat,numeral_numeral(rat),K))) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),K))),one_one(int)) ).

% quotient_of_number(5)
tff(fact_5632_quotient__of__number_I4_J,axiom,
    quotient_of(aa(rat,rat,uminus_uminus(rat),one_one(rat))) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,uminus_uminus(int),one_one(int))),one_one(int)) ).

% quotient_of_number(4)
tff(fact_5633_divide__rat__def,axiom,
    ! [Q2: rat,R2: rat] : aa(rat,rat,aa(rat,fun(rat,rat),divide_divide(rat),Q2),R2) = aa(rat,rat,aa(rat,fun(rat,rat),times_times(rat),Q2),aa(rat,rat,inverse_inverse(rat),R2)) ).

% divide_rat_def
tff(fact_5634_Ints__add,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),ring_1_Ints(A)))
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),B2),ring_1_Ints(A)))
           => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),ring_1_Ints(A))) ) ) ) ).

% Ints_add
tff(fact_5635_Ints__numeral,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [N: num] : pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(num,A,numeral_numeral(A),N)),ring_1_Ints(A))) ) ).

% Ints_numeral
tff(fact_5636_Ints__mult,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),ring_1_Ints(A)))
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),B2),ring_1_Ints(A)))
           => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),ring_1_Ints(A))) ) ) ) ).

% Ints_mult
tff(fact_5637_Ints__0,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),zero_zero(A)),ring_1_Ints(A))) ) ).

% Ints_0
tff(fact_5638_Ints__1,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),one_one(A)),ring_1_Ints(A))) ) ).

% Ints_1
tff(fact_5639_Ints__of__nat,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [N: nat] : pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(nat,A,semiring_1_of_nat(A),N)),ring_1_Ints(A))) ) ).

% Ints_of_nat
tff(fact_5640_Ints__power,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [A2: A,N: nat] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),ring_1_Ints(A)))
         => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),N)),ring_1_Ints(A))) ) ) ).

% Ints_power
tff(fact_5641_Ints__double__eq__0__iff,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [A2: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),ring_1_Ints(A)))
         => ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),A2) = zero_zero(A) )
          <=> ( A2 = zero_zero(A) ) ) ) ) ).

% Ints_double_eq_0_iff
tff(fact_5642_finite__int__segment,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [A2: A,B2: A] : pp(aa(set(A),bool,finite_finite2(A),collect(A,aa(A,fun(A,bool),aTP_Lamp_mz(A,fun(A,fun(A,bool)),A2),B2)))) ) ).

% finite_int_segment
tff(fact_5643_Ints__odd__nonzero,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [A2: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),ring_1_Ints(A)))
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),A2)),A2) != zero_zero(A) ) ) ) ).

% Ints_odd_nonzero
tff(fact_5644_of__int__divide__in__Ints,axiom,
    ! [A: $tType] :
      ( idom_divide(A)
     => ! [B2: int,A2: int] :
          ( pp(aa(int,bool,dvd_dvd(int,B2),A2))
         => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(int,A,ring_1_of_int(A),A2)),aa(int,A,ring_1_of_int(A),B2))),ring_1_Ints(A))) ) ) ).

% of_int_divide_in_Ints
tff(fact_5645_quotient__of__denom__pos,axiom,
    ! [R2: rat,P2: int,Q2: int] :
      ( ( quotient_of(R2) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),P2),Q2) )
     => pp(aa(int,bool,ord_less(int,zero_zero(int)),Q2)) ) ).

% quotient_of_denom_pos
tff(fact_5646_finite__abs__int__segment,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [A2: A] : pp(aa(set(A),bool,finite_finite2(A),collect(A,aTP_Lamp_na(A,fun(A,bool),A2)))) ) ).

% finite_abs_int_segment
tff(fact_5647_Ints__odd__less__0,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),ring_1_Ints(A)))
         => ( pp(aa(A,bool,ord_less(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),A2)),A2)),zero_zero(A)))
          <=> pp(aa(A,bool,ord_less(A,A2),zero_zero(A))) ) ) ) ).

% Ints_odd_less_0
tff(fact_5648_Ints__nonzero__abs__ge1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),ring_1_Ints(A)))
         => ( ( X != zero_zero(A) )
           => pp(aa(A,bool,ord_less_eq(A,one_one(A)),aa(A,A,abs_abs(A),X))) ) ) ) ).

% Ints_nonzero_abs_ge1
tff(fact_5649_Ints__nonzero__abs__less1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),ring_1_Ints(A)))
         => ( pp(aa(A,bool,ord_less(A,aa(A,A,abs_abs(A),X)),one_one(A)))
           => ( X = zero_zero(A) ) ) ) ) ).

% Ints_nonzero_abs_less1
tff(fact_5650_Ints__eq__abs__less1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A,Y: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),ring_1_Ints(A)))
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Y),ring_1_Ints(A)))
           => ( ( X = Y )
            <=> pp(aa(A,bool,ord_less(A,aa(A,A,abs_abs(A),aa(A,A,minus_minus(A,X),Y))),one_one(A))) ) ) ) ) ).

% Ints_eq_abs_less1
tff(fact_5651_sin__times__pi__eq__0,axiom,
    ! [X: real] :
      ( ( sin(real,aa(real,real,aa(real,fun(real,real),times_times(real),X),pi)) = zero_zero(real) )
    <=> pp(aa(set(real),bool,aa(real,fun(set(real),bool),member(real),X),ring_1_Ints(real))) ) ).

% sin_times_pi_eq_0
tff(fact_5652_frac__neg,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),ring_1_Ints(A)))
           => ( archimedean_frac(A,aa(A,A,uminus_uminus(A),X)) = zero_zero(A) ) )
          & ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),ring_1_Ints(A)))
           => ( archimedean_frac(A,aa(A,A,uminus_uminus(A),X)) = aa(A,A,minus_minus(A,one_one(A)),archimedean_frac(A,X)) ) ) ) ) ).

% frac_neg
tff(fact_5653_rat__uminus__code,axiom,
    ! [P2: rat] : quotient_of(aa(rat,rat,uminus_uminus(rat),P2)) = aa(product_prod(int,int),product_prod(int,int),product_case_prod(int,int,product_prod(int,int),aTP_Lamp_nb(int,fun(int,product_prod(int,int)))),quotient_of(P2)) ).

% rat_uminus_code
tff(fact_5654_rat__less__code,axiom,
    ! [P2: rat,Q2: rat] :
      ( pp(aa(rat,bool,ord_less(rat,P2),Q2))
    <=> pp(aa(product_prod(int,int),bool,product_case_prod(int,int,bool,aTP_Lamp_nd(rat,fun(int,fun(int,bool)),Q2)),quotient_of(P2))) ) ).

% rat_less_code
tff(fact_5655_rat__floor__code,axiom,
    ! [P2: rat] : archim6421214686448440834_floor(rat,P2) = aa(product_prod(int,int),int,product_case_prod(int,int,int,divide_divide(int)),quotient_of(P2)) ).

% rat_floor_code
tff(fact_5656_rat__abs__code,axiom,
    ! [P2: rat] : quotient_of(aa(rat,rat,abs_abs(rat),P2)) = aa(product_prod(int,int),product_prod(int,int),product_case_prod(int,int,product_prod(int,int),aTP_Lamp_ne(int,fun(int,product_prod(int,int)))),quotient_of(P2)) ).

% rat_abs_code
tff(fact_5657_rat__less__eq__code,axiom,
    ! [P2: rat,Q2: rat] :
      ( pp(aa(rat,bool,ord_less_eq(rat,P2),Q2))
    <=> pp(aa(product_prod(int,int),bool,product_case_prod(int,int,bool,aTP_Lamp_ng(rat,fun(int,fun(int,bool)),Q2)),quotient_of(P2))) ) ).

% rat_less_eq_code
tff(fact_5658_le__mult__floor__Ints,axiom,
    ! [A: $tType,B: $tType] :
      ( ( archim2362893244070406136eiling(B)
        & linordered_idom(A) )
     => ! [A2: B,B2: B] :
          ( pp(aa(B,bool,ord_less_eq(B,zero_zero(B)),A2))
         => ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A2),ring_1_Ints(B)))
           => pp(aa(A,bool,ord_less_eq(A,aa(int,A,ring_1_of_int(A),aa(int,int,aa(int,fun(int,int),times_times(int),archim6421214686448440834_floor(B,A2)),archim6421214686448440834_floor(B,B2)))),aa(int,A,ring_1_of_int(A),archim6421214686448440834_floor(B,aa(B,B,aa(B,fun(B,B),times_times(B),A2),B2))))) ) ) ) ).

% le_mult_floor_Ints
tff(fact_5659_frac__unique__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,A2: A] :
          ( ( archimedean_frac(A,X) = A2 )
        <=> ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(A,A,minus_minus(A,X),A2)),ring_1_Ints(A)))
            & pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),A2))
            & pp(aa(A,bool,ord_less(A,A2),one_one(A))) ) ) ) ).

% frac_unique_iff
tff(fact_5660_mult__ceiling__le__Ints,axiom,
    ! [A: $tType,B: $tType] :
      ( ( archim2362893244070406136eiling(B)
        & linordered_idom(A) )
     => ! [A2: B,B2: B] :
          ( pp(aa(B,bool,ord_less_eq(B,zero_zero(B)),A2))
         => ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A2),ring_1_Ints(B)))
           => pp(aa(A,bool,ord_less_eq(A,aa(int,A,ring_1_of_int(A),archimedean_ceiling(B,aa(B,B,aa(B,fun(B,B),times_times(B),A2),B2)))),aa(int,A,ring_1_of_int(A),aa(int,int,aa(int,fun(int,int),times_times(int),archimedean_ceiling(B,A2)),archimedean_ceiling(B,B2))))) ) ) ) ).

% mult_ceiling_le_Ints
tff(fact_5661_sin__integer__2pi,axiom,
    ! [N: real] :
      ( pp(aa(set(real),bool,aa(real,fun(set(real),bool),member(real),N),ring_1_Ints(real)))
     => ( sin(real,aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),pi)),N)) = zero_zero(real) ) ) ).

% sin_integer_2pi
tff(fact_5662_cos__integer__2pi,axiom,
    ! [N: real] :
      ( pp(aa(set(real),bool,aa(real,fun(set(real),bool),member(real),N),ring_1_Ints(real)))
     => ( cos(real,aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),pi)),N)) = one_one(real) ) ) ).

% cos_integer_2pi
tff(fact_5663_ran__empty,axiom,
    ! [B: $tType,A: $tType] : ran(B,A,aTP_Lamp_nh(B,option(A))) = bot_bot(set(A)) ).

% ran_empty
tff(fact_5664_quotient__of__int,axiom,
    ! [A2: int] : quotient_of(of_int(A2)) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),A2),one_one(int)) ).

% quotient_of_int
tff(fact_5665_rat__minus__code,axiom,
    ! [P2: rat,Q2: rat] : quotient_of(aa(rat,rat,minus_minus(rat,P2),Q2)) = aa(product_prod(int,int),product_prod(int,int),product_case_prod(int,int,product_prod(int,int),aTP_Lamp_nj(rat,fun(int,fun(int,product_prod(int,int))),Q2)),quotient_of(P2)) ).

% rat_minus_code
tff(fact_5666_normalize__denom__zero,axiom,
    ! [P2: int] : normalize(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),P2),zero_zero(int))) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),one_one(int)) ).

% normalize_denom_zero
tff(fact_5667_normalize__negative,axiom,
    ! [Q2: int,P2: int] :
      ( pp(aa(int,bool,ord_less(int,Q2),zero_zero(int)))
     => ( normalize(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),P2),Q2)) = normalize(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,uminus_uminus(int),P2)),aa(int,int,uminus_uminus(int),Q2))) ) ) ).

% normalize_negative
tff(fact_5668_normalize__denom__pos,axiom,
    ! [R2: product_prod(int,int),P2: int,Q2: int] :
      ( ( normalize(R2) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),P2),Q2) )
     => pp(aa(int,bool,ord_less(int,zero_zero(int)),Q2)) ) ).

% normalize_denom_pos
tff(fact_5669_normalize__crossproduct,axiom,
    ! [Q2: int,S3: int,P2: int,R2: int] :
      ( ( Q2 != zero_zero(int) )
     => ( ( S3 != zero_zero(int) )
       => ( ( normalize(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),P2),Q2)) = normalize(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),R2),S3)) )
         => ( aa(int,int,aa(int,fun(int,int),times_times(int),P2),S3) = aa(int,int,aa(int,fun(int,int),times_times(int),R2),Q2) ) ) ) ) ).

% normalize_crossproduct
tff(fact_5670_rat__times__code,axiom,
    ! [P2: rat,Q2: rat] : quotient_of(aa(rat,rat,aa(rat,fun(rat,rat),times_times(rat),P2),Q2)) = aa(product_prod(int,int),product_prod(int,int),product_case_prod(int,int,product_prod(int,int),aTP_Lamp_nl(rat,fun(int,fun(int,product_prod(int,int))),Q2)),quotient_of(P2)) ).

% rat_times_code
tff(fact_5671_rat__divide__code,axiom,
    ! [P2: rat,Q2: rat] : quotient_of(aa(rat,rat,aa(rat,fun(rat,rat),divide_divide(rat),P2),Q2)) = aa(product_prod(int,int),product_prod(int,int),product_case_prod(int,int,product_prod(int,int),aTP_Lamp_nn(rat,fun(int,fun(int,product_prod(int,int))),Q2)),quotient_of(P2)) ).

% rat_divide_code
tff(fact_5672_rat__plus__code,axiom,
    ! [P2: rat,Q2: rat] : quotient_of(aa(rat,rat,aa(rat,fun(rat,rat),plus_plus(rat),P2),Q2)) = aa(product_prod(int,int),product_prod(int,int),product_case_prod(int,int,product_prod(int,int),aTP_Lamp_np(rat,fun(int,fun(int,product_prod(int,int))),Q2)),quotient_of(P2)) ).

% rat_plus_code
tff(fact_5673_Frct__code__post_I5_J,axiom,
    ! [K: num] : frct(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),one_one(int)),aa(num,int,numeral_numeral(int),K))) = aa(rat,rat,aa(rat,fun(rat,rat),divide_divide(rat),one_one(rat)),aa(num,rat,numeral_numeral(rat),K)) ).

% Frct_code_post(5)
tff(fact_5674_setceilmax,axiom,
    ! [S3: vEBT_VEBT,M: nat,Listy: list(vEBT_VEBT),N: nat] :
      ( vEBT_invar_vebt(S3,M)
     => ( ! [X3: vEBT_VEBT] :
            ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),member(vEBT_VEBT),X3),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),Listy)))
           => vEBT_invar_vebt(X3,N) )
       => ( ( M = aa(nat,nat,suc,N) )
         => ( ! [X3: vEBT_VEBT] :
                ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),member(vEBT_VEBT),X3),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),Listy)))
               => ( aa(nat,int,semiring_1_of_nat(int),aa(vEBT_VEBT,nat,vEBT_VEBT_height,X3)) = archimedean_ceiling(real,aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(nat,real,semiring_1_of_nat(real),N))) ) )
           => ( ( aa(nat,int,semiring_1_of_nat(int),aa(vEBT_VEBT,nat,vEBT_VEBT_height,S3)) = archimedean_ceiling(real,aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(nat,real,semiring_1_of_nat(real),M))) )
             => ( aa(nat,int,semiring_1_of_nat(int),lattic643756798349783984er_Max(nat,aa(set(vEBT_VEBT),set(nat),image(vEBT_VEBT,nat,vEBT_VEBT_height),insert(vEBT_VEBT,S3,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),Listy))))) = archimedean_ceiling(real,aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(nat,real,semiring_1_of_nat(real),M))) ) ) ) ) ) ) ).

% setceilmax
tff(fact_5675_VEBT__internal_Ospace_Opelims,axiom,
    ! [X: vEBT_VEBT,Y: nat] :
      ( ( aa(vEBT_VEBT,nat,vEBT_VEBT_space,X) = Y )
     => ( accp(vEBT_VEBT,vEBT_VEBT_space_rel2,X)
       => ( ! [A4: bool,B3: bool] :
              ( ( X = vEBT_Leaf(A4,B3) )
             => ( ( Y = aa(num,nat,numeral_numeral(nat),aa(num,num,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] :
                ( ( X = vEBT_Node(Info2,Deg2,TreeList2,Summary2) )
               => ( ( Y = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,bit0(one2)))),aa(vEBT_VEBT,nat,vEBT_VEBT_space,Summary2))),aa(list(vEBT_VEBT),nat,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
tff(fact_5676_image__ident,axiom,
    ! [A: $tType,Y6: set(A)] : aa(set(A),set(A),image(A,A,aTP_Lamp_at(A,A)),Y6) = Y6 ).

% image_ident
tff(fact_5677_height__compose__list,axiom,
    ! [T: vEBT_VEBT,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] :
      ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),member(vEBT_VEBT),T),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList)))
     => pp(aa(nat,bool,ord_less_eq(nat,aa(vEBT_VEBT,nat,vEBT_VEBT_height,T)),lattic643756798349783984er_Max(nat,aa(set(vEBT_VEBT),set(nat),image(vEBT_VEBT,nat,vEBT_VEBT_height),insert(vEBT_VEBT,Summary,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList)))))) ) ).

% height_compose_list
tff(fact_5678_max__ins__scaled,axiom,
    ! [N: nat,X14: vEBT_VEBT,M: nat,X13: list(vEBT_VEBT)] : pp(aa(nat,bool,ord_less_eq(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),aa(vEBT_VEBT,nat,vEBT_VEBT_height,X14))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),lattic643756798349783984er_Max(nat,insert(nat,aa(vEBT_VEBT,nat,vEBT_VEBT_height,X14),aa(set(vEBT_VEBT),set(nat),image(vEBT_VEBT,nat,vEBT_VEBT_height),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),X13)))))))) ).

% max_ins_scaled
tff(fact_5679_height__i__max,axiom,
    ! [I: nat,X13: list(vEBT_VEBT),Foo: nat] :
      ( pp(aa(nat,bool,ord_less(nat,I),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),X13)))
     => pp(aa(nat,bool,ord_less_eq(nat,aa(vEBT_VEBT,nat,vEBT_VEBT_height,aa(nat,vEBT_VEBT,nth(vEBT_VEBT,X13),I))),aa(nat,nat,ord_max(nat,Foo),lattic643756798349783984er_Max(nat,aa(set(vEBT_VEBT),set(nat),image(vEBT_VEBT,nat,vEBT_VEBT_height),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),X13)))))) ) ).

% height_i_max
tff(fact_5680_image__add__0,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [S: set(A)] : aa(set(A),set(A),image(A,A,aa(A,fun(A,A),plus_plus(A),zero_zero(A))),S) = S ) ).

% image_add_0
tff(fact_5681_image__add__atLeastAtMost,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [K: A,I: A,J: A] : aa(set(A),set(A),image(A,A,aa(A,fun(A,A),plus_plus(A),K)),set_or1337092689740270186AtMost(A,I,J)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),K)) ) ).

% image_add_atLeastAtMost
tff(fact_5682_image__add__atMost,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [C2: A,A2: A] : aa(set(A),set(A),image(A,A,aa(A,fun(A,A),plus_plus(A),C2)),set_ord_atMost(A,A2)) = set_ord_atMost(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),A2)) ) ).

% image_add_atMost
tff(fact_5683_max__idx__list,axiom,
    ! [I: nat,X13: list(vEBT_VEBT),N: nat,X14: vEBT_VEBT] :
      ( pp(aa(nat,bool,ord_less(nat,I),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),X13)))
     => pp(aa(nat,bool,ord_less_eq(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),aa(vEBT_VEBT,nat,vEBT_VEBT_height,aa(nat,vEBT_VEBT,nth(vEBT_VEBT,X13),I)))),aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),aa(nat,nat,ord_max(nat,aa(vEBT_VEBT,nat,vEBT_VEBT_height,X14)),lattic643756798349783984er_Max(nat,aa(set(vEBT_VEBT),set(nat),image(vEBT_VEBT,nat,vEBT_VEBT_height),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),X13))))))))) ) ).

% max_idx_list
tff(fact_5684_image__add__atLeastAtMost_H,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [K: A,I: A,J: A] : aa(set(A),set(A),image(A,A,aTP_Lamp_nq(A,fun(A,A),K)),set_or1337092689740270186AtMost(A,I,J)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),K)) ) ).

% image_add_atLeastAtMost'
tff(fact_5685_image__minus__const__atLeastAtMost_H,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [D: A,A2: A,B2: A] : aa(set(A),set(A),image(A,A,aTP_Lamp_nr(A,fun(A,A),D)),set_or1337092689740270186AtMost(A,A2,B2)) = set_or1337092689740270186AtMost(A,aa(A,A,minus_minus(A,A2),D),aa(A,A,minus_minus(A,B2),D)) ) ).

% image_minus_const_atLeastAtMost'
tff(fact_5686_Max__divisors__self__nat,axiom,
    ! [N: nat] :
      ( ( N != zero_zero(nat) )
     => ( lattic643756798349783984er_Max(nat,collect(nat,aTP_Lamp_am(nat,fun(nat,bool),N))) = N ) ) ).

% Max_divisors_self_nat
tff(fact_5687_Max_Obounded__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A3))
         => ( ( A3 != bot_bot(set(A)) )
           => ( pp(aa(A,bool,ord_less_eq(A,lattic643756798349783984er_Max(A,A3)),X))
            <=> ! [X2: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X2),A3))
                 => pp(aa(A,bool,ord_less_eq(A,X2),X)) ) ) ) ) ) ).

% Max.bounded_iff
tff(fact_5688_Max__const,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(A)
     => ! [A3: set(B),C2: A] :
          ( pp(aa(set(B),bool,finite_finite2(B),A3))
         => ( ( A3 != bot_bot(set(B)) )
           => ( lattic643756798349783984er_Max(A,aa(set(B),set(A),image(B,A,aTP_Lamp_ns(A,fun(B,A),C2)),A3)) = C2 ) ) ) ) ).

% Max_const
tff(fact_5689_image__mult__atLeastAtMost,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [D: A,A2: A,B2: A] :
          ( pp(aa(A,bool,ord_less(A,zero_zero(A)),D))
         => ( aa(set(A),set(A),image(A,A,aa(A,fun(A,A),times_times(A),D)),set_or1337092689740270186AtMost(A,A2,B2)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),times_times(A),D),A2),aa(A,A,aa(A,fun(A,A),times_times(A),D),B2)) ) ) ) ).

% image_mult_atLeastAtMost
tff(fact_5690_image__divide__atLeastAtMost,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [D: A,A2: A,B2: A] :
          ( pp(aa(A,bool,ord_less(A,zero_zero(A)),D))
         => ( aa(set(A),set(A),image(A,A,aTP_Lamp_nt(A,fun(A,A),D)),set_or1337092689740270186AtMost(A,A2,B2)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),D),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),D)) ) ) ) ).

% image_divide_atLeastAtMost
tff(fact_5691_Max_OcoboundedI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),A2: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A3))
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),A3))
           => pp(aa(A,bool,ord_less_eq(A,A2),lattic643756798349783984er_Max(A,A3))) ) ) ) ).

% Max.coboundedI
tff(fact_5692_Max__eq__if,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),B4: set(A)] :
          ( pp(aa(set(A),bool,finite_finite2(A),A3))
         => ( pp(aa(set(A),bool,finite_finite2(A),B4))
           => ( ! [X3: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),A3))
                 => ? [Xa: A] :
                      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Xa),B4))
                      & pp(aa(A,bool,ord_less_eq(A,X3),Xa)) ) )
             => ( ! [X3: A] :
                    ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),B4))
                   => ? [Xa: A] :
                        ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Xa),A3))
                        & pp(aa(A,bool,ord_less_eq(A,X3),Xa)) ) )
               => ( lattic643756798349783984er_Max(A,A3) = lattic643756798349783984er_Max(A,B4) ) ) ) ) ) ) ).

% Max_eq_if
tff(fact_5693_Max__eqI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A3))
         => ( ! [Y3: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Y3),A3))
               => pp(aa(A,bool,ord_less_eq(A,Y3),X)) )
           => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A3))
             => ( lattic643756798349783984er_Max(A,A3) = X ) ) ) ) ) ).

% Max_eqI
tff(fact_5694_Max__ge,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A3))
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A3))
           => pp(aa(A,bool,ord_less_eq(A,X),lattic643756798349783984er_Max(A,A3))) ) ) ) ).

% Max_ge
tff(fact_5695_Max__add__commute,axiom,
    ! [B: $tType,A: $tType] :
      ( linord4140545234300271783up_add(A)
     => ! [S: set(B),F2: fun(B,A),K: A] :
          ( pp(aa(set(B),bool,finite_finite2(B),S))
         => ( ( S != bot_bot(set(B)) )
           => ( lattic643756798349783984er_Max(A,aa(set(B),set(A),image(B,A,aa(A,fun(B,A),aTP_Lamp_nu(fun(B,A),fun(A,fun(B,A)),F2),K)),S)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),lattic643756798349783984er_Max(A,aa(set(B),set(A),image(B,A,F2),S))),K) ) ) ) ) ).

% Max_add_commute
tff(fact_5696_image__diff__subset,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),A3: set(B),B4: set(B)] : pp(aa(set(A),bool,ord_less_eq(set(A),aa(set(A),set(A),minus_minus(set(A),aa(set(B),set(A),image(B,A,F2),A3)),aa(set(B),set(A),image(B,A,F2),B4))),aa(set(B),set(A),image(B,A,F2),aa(set(B),set(B),minus_minus(set(B),A3),B4)))) ).

% image_diff_subset
tff(fact_5697_all__subset__image,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),A3: set(B),P: fun(set(A),bool)] :
      ( ! [B9: set(A)] :
          ( pp(aa(set(A),bool,ord_less_eq(set(A),B9),aa(set(B),set(A),image(B,A,F2),A3)))
         => pp(aa(set(A),bool,P,B9)) )
    <=> ! [B9: set(B)] :
          ( pp(aa(set(B),bool,ord_less_eq(set(B),B9),A3))
         => pp(aa(set(A),bool,P,aa(set(B),set(A),image(B,A,F2),B9))) ) ) ).

% all_subset_image
tff(fact_5698_subset__image__iff,axiom,
    ! [A: $tType,B: $tType,B4: set(A),F2: fun(B,A),A3: set(B)] :
      ( pp(aa(set(A),bool,ord_less_eq(set(A),B4),aa(set(B),set(A),image(B,A,F2),A3)))
    <=> ? [AA: set(B)] :
          ( pp(aa(set(B),bool,ord_less_eq(set(B),AA),A3))
          & ( B4 = aa(set(B),set(A),image(B,A,F2),AA) ) ) ) ).

% subset_image_iff
tff(fact_5699_image__subset__iff,axiom,
    ! [B: $tType,A: $tType,F2: fun(B,A),A3: set(B),B4: set(A)] :
      ( pp(aa(set(A),bool,ord_less_eq(set(A),aa(set(B),set(A),image(B,A,F2),A3)),B4))
    <=> ! [X2: B] :
          ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X2),A3))
         => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(B,A,F2,X2)),B4)) ) ) ).

% image_subset_iff
tff(fact_5700_subset__imageE,axiom,
    ! [A: $tType,B: $tType,B4: set(A),F2: fun(B,A),A3: set(B)] :
      ( pp(aa(set(A),bool,ord_less_eq(set(A),B4),aa(set(B),set(A),image(B,A,F2),A3)))
     => ~ ! [C7: set(B)] :
            ( pp(aa(set(B),bool,ord_less_eq(set(B),C7),A3))
           => ( B4 != aa(set(B),set(A),image(B,A,F2),C7) ) ) ) ).

% subset_imageE
tff(fact_5701_image__subsetI,axiom,
    ! [A: $tType,B: $tType,A3: set(A),F2: fun(A,B),B4: set(B)] :
      ( ! [X3: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),A3))
         => pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),aa(A,B,F2,X3)),B4)) )
     => pp(aa(set(B),bool,ord_less_eq(set(B),aa(set(A),set(B),image(A,B,F2),A3)),B4)) ) ).

% image_subsetI
tff(fact_5702_image__mono,axiom,
    ! [B: $tType,A: $tType,A3: set(A),B4: set(A),F2: fun(A,B)] :
      ( pp(aa(set(A),bool,ord_less_eq(set(A),A3),B4))
     => pp(aa(set(B),bool,ord_less_eq(set(B),aa(set(A),set(B),image(A,B,F2),A3)),aa(set(A),set(B),image(A,B,F2),B4))) ) ).

% image_mono
tff(fact_5703_imageE,axiom,
    ! [A: $tType,B: $tType,B2: A,F2: fun(B,A),A3: set(B)] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),B2),aa(set(B),set(A),image(B,A,F2),A3)))
     => ~ ! [X3: B] :
            ( ( B2 = aa(B,A,F2,X3) )
           => ~ pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),A3)) ) ) ).

% imageE
tff(fact_5704_image__image,axiom,
    ! [B: $tType,A: $tType,C: $tType,F2: fun(B,A),G: fun(C,B),A3: set(C)] : aa(set(B),set(A),image(B,A,F2),aa(set(C),set(B),image(C,B,G),A3)) = aa(set(C),set(A),image(C,A,aa(fun(C,B),fun(C,A),aTP_Lamp_nv(fun(B,A),fun(fun(C,B),fun(C,A)),F2),G)),A3) ).

% image_image
tff(fact_5705_Compr__image__eq,axiom,
    ! [B: $tType,A: $tType,F2: fun(B,A),A3: set(B),P: fun(A,bool)] : collect(A,aa(fun(A,bool),fun(A,bool),aa(set(B),fun(fun(A,bool),fun(A,bool)),aTP_Lamp_nw(fun(B,A),fun(set(B),fun(fun(A,bool),fun(A,bool))),F2),A3),P)) = aa(set(B),set(A),image(B,A,F2),collect(B,aa(fun(A,bool),fun(B,bool),aa(set(B),fun(fun(A,bool),fun(B,bool)),aTP_Lamp_nx(fun(B,A),fun(set(B),fun(fun(A,bool),fun(B,bool))),F2),A3),P))) ).

% Compr_image_eq
tff(fact_5706_pigeonhole__infinite,axiom,
    ! [B: $tType,A: $tType,A3: set(A),F2: fun(A,B)] :
      ( ~ pp(aa(set(A),bool,finite_finite2(A),A3))
     => ( pp(aa(set(B),bool,finite_finite2(B),aa(set(A),set(B),image(A,B,F2),A3)))
       => ? [X3: A] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),A3))
            & ~ pp(aa(set(A),bool,finite_finite2(A),collect(A,aa(A,fun(A,bool),aa(fun(A,B),fun(A,fun(A,bool)),aTP_Lamp_ny(set(A),fun(fun(A,B),fun(A,fun(A,bool))),A3),F2),X3)))) ) ) ) ).

% pigeonhole_infinite
tff(fact_5707_finite__surj,axiom,
    ! [A: $tType,B: $tType,A3: set(A),B4: set(B),F2: fun(A,B)] :
      ( pp(aa(set(A),bool,finite_finite2(A),A3))
     => ( pp(aa(set(B),bool,ord_less_eq(set(B),B4),aa(set(A),set(B),image(A,B,F2),A3)))
       => pp(aa(set(B),bool,finite_finite2(B),B4)) ) ) ).

% finite_surj
tff(fact_5708_finite__subset__image,axiom,
    ! [A: $tType,B: $tType,B4: set(A),F2: fun(B,A),A3: set(B)] :
      ( pp(aa(set(A),bool,finite_finite2(A),B4))
     => ( pp(aa(set(A),bool,ord_less_eq(set(A),B4),aa(set(B),set(A),image(B,A,F2),A3)))
       => ? [C7: set(B)] :
            ( pp(aa(set(B),bool,ord_less_eq(set(B),C7),A3))
            & pp(aa(set(B),bool,finite_finite2(B),C7))
            & ( B4 = aa(set(B),set(A),image(B,A,F2),C7) ) ) ) ) ).

% finite_subset_image
tff(fact_5709_ex__finite__subset__image,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),A3: set(B),P: fun(set(A),bool)] :
      ( ? [B9: set(A)] :
          ( pp(aa(set(A),bool,finite_finite2(A),B9))
          & pp(aa(set(A),bool,ord_less_eq(set(A),B9),aa(set(B),set(A),image(B,A,F2),A3)))
          & pp(aa(set(A),bool,P,B9)) )
    <=> ? [B9: set(B)] :
          ( pp(aa(set(B),bool,finite_finite2(B),B9))
          & pp(aa(set(B),bool,ord_less_eq(set(B),B9),A3))
          & pp(aa(set(A),bool,P,aa(set(B),set(A),image(B,A,F2),B9))) ) ) ).

% ex_finite_subset_image
tff(fact_5710_all__finite__subset__image,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),A3: set(B),P: fun(set(A),bool)] :
      ( ! [B9: set(A)] :
          ( ( pp(aa(set(A),bool,finite_finite2(A),B9))
            & pp(aa(set(A),bool,ord_less_eq(set(A),B9),aa(set(B),set(A),image(B,A,F2),A3))) )
         => pp(aa(set(A),bool,P,B9)) )
    <=> ! [B9: set(B)] :
          ( ( pp(aa(set(B),bool,finite_finite2(B),B9))
            & pp(aa(set(B),bool,ord_less_eq(set(B),B9),A3)) )
         => pp(aa(set(A),bool,P,aa(set(B),set(A),image(B,A,F2),B9))) ) ) ).

% all_finite_subset_image
tff(fact_5711_image__constant__conv,axiom,
    ! [B: $tType,A: $tType,A3: set(B),C2: A] :
      ( ( ( A3 = bot_bot(set(B)) )
       => ( aa(set(B),set(A),image(B,A,aTP_Lamp_ay(A,fun(B,A),C2)),A3) = bot_bot(set(A)) ) )
      & ( ( A3 != bot_bot(set(B)) )
       => ( aa(set(B),set(A),image(B,A,aTP_Lamp_ay(A,fun(B,A),C2)),A3) = insert(A,C2,bot_bot(set(A))) ) ) ) ).

% image_constant_conv
tff(fact_5712_image__constant,axiom,
    ! [A: $tType,B: $tType,X: A,A3: set(A),C2: B] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A3))
     => ( aa(set(A),set(B),image(A,B,aTP_Lamp_nz(B,fun(A,B),C2)),A3) = insert(B,C2,bot_bot(set(B))) ) ) ).

% image_constant
tff(fact_5713_sum_Oimage__gen,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [S: set(B),H: fun(B,A),G: fun(B,C)] :
          ( pp(aa(set(B),bool,finite_finite2(B),S))
         => ( aa(set(B),A,groups7311177749621191930dd_sum(B,A,H),S) = aa(set(C),A,groups7311177749621191930dd_sum(C,A,aa(fun(B,C),fun(C,A),aa(fun(B,A),fun(fun(B,C),fun(C,A)),aTP_Lamp_ob(set(B),fun(fun(B,A),fun(fun(B,C),fun(C,A))),S),H),G)),aa(set(B),set(C),image(B,C,G),S)) ) ) ) ).

% sum.image_gen
tff(fact_5714_prod_Oimage__gen,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [S: set(B),H: fun(B,A),G: fun(B,C)] :
          ( pp(aa(set(B),bool,finite_finite2(B),S))
         => ( groups7121269368397514597t_prod(B,A,H,S) = groups7121269368397514597t_prod(C,A,aa(fun(B,C),fun(C,A),aa(fun(B,A),fun(fun(B,C),fun(C,A)),aTP_Lamp_oc(set(B),fun(fun(B,A),fun(fun(B,C),fun(C,A))),S),H),G),aa(set(B),set(C),image(B,C,G),S)) ) ) ) ).

% prod.image_gen
tff(fact_5715_Max__eq__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),M: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A3))
         => ( ( A3 != bot_bot(set(A)) )
           => ( ( lattic643756798349783984er_Max(A,A3) = M )
            <=> ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),M),A3))
                & ! [X2: A] :
                    ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X2),A3))
                   => pp(aa(A,bool,ord_less_eq(A,X2),M)) ) ) ) ) ) ) ).

% Max_eq_iff
tff(fact_5716_Max__ge__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A3))
         => ( ( A3 != bot_bot(set(A)) )
           => ( pp(aa(A,bool,ord_less_eq(A,X),lattic643756798349783984er_Max(A,A3)))
            <=> ? [X2: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X2),A3))
                  & pp(aa(A,bool,ord_less_eq(A,X),X2)) ) ) ) ) ) ).

% Max_ge_iff
tff(fact_5717_eq__Max__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),M: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A3))
         => ( ( A3 != bot_bot(set(A)) )
           => ( ( M = lattic643756798349783984er_Max(A,A3) )
            <=> ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),M),A3))
                & ! [X2: A] :
                    ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X2),A3))
                   => pp(aa(A,bool,ord_less_eq(A,X2),M)) ) ) ) ) ) ) ).

% eq_Max_iff
tff(fact_5718_Max_OboundedE,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A3))
         => ( ( A3 != bot_bot(set(A)) )
           => ( pp(aa(A,bool,ord_less_eq(A,lattic643756798349783984er_Max(A,A3)),X))
             => ! [A8: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A8),A3))
                 => pp(aa(A,bool,ord_less_eq(A,A8),X)) ) ) ) ) ) ).

% Max.boundedE
tff(fact_5719_Max_OboundedI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A3))
         => ( ( A3 != bot_bot(set(A)) )
           => ( ! [A4: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A4),A3))
                 => pp(aa(A,bool,ord_less_eq(A,A4),X)) )
             => pp(aa(A,bool,ord_less_eq(A,lattic643756798349783984er_Max(A,A3)),X)) ) ) ) ) ).

% Max.boundedI
tff(fact_5720_Max__insert2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),A2: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A3))
         => ( ! [B3: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),B3),A3))
               => pp(aa(A,bool,ord_less_eq(A,B3),A2)) )
           => ( lattic643756798349783984er_Max(A,insert(A,A2,A3)) = A2 ) ) ) ) ).

% Max_insert2
tff(fact_5721_VEBT__internal_Oheight_Osimps_I2_J,axiom,
    ! [Uu: option(product_prod(nat,nat)),Deg: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] : aa(vEBT_VEBT,nat,vEBT_VEBT_height,vEBT_Node(Uu,Deg,TreeList,Summary)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),lattic643756798349783984er_Max(nat,aa(set(vEBT_VEBT),set(nat),image(vEBT_VEBT,nat,vEBT_VEBT_height),insert(vEBT_VEBT,Summary,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))))) ).

% VEBT_internal.height.simps(2)
tff(fact_5722_Max_Oinfinite,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A)] :
          ( ~ pp(aa(set(A),bool,finite_finite2(A),A3))
         => ( lattic643756798349783984er_Max(A,A3) = the2(A,none(A)) ) ) ) ).

% Max.infinite
tff(fact_5723_sum_Ogroup,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [S: set(B),T3: set(C),G: fun(B,C),H: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),S))
         => ( pp(aa(set(C),bool,finite_finite2(C),T3))
           => ( pp(aa(set(C),bool,ord_less_eq(set(C),aa(set(B),set(C),image(B,C,G),S)),T3))
             => ( aa(set(C),A,groups7311177749621191930dd_sum(C,A,aa(fun(B,A),fun(C,A),aa(fun(B,C),fun(fun(B,A),fun(C,A)),aTP_Lamp_od(set(B),fun(fun(B,C),fun(fun(B,A),fun(C,A))),S),G),H)),T3) = aa(set(B),A,groups7311177749621191930dd_sum(B,A,H),S) ) ) ) ) ) ).

% sum.group
tff(fact_5724_prod_Ogroup,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [S: set(B),T3: set(C),G: fun(B,C),H: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),S))
         => ( pp(aa(set(C),bool,finite_finite2(C),T3))
           => ( pp(aa(set(C),bool,ord_less_eq(set(C),aa(set(B),set(C),image(B,C,G),S)),T3))
             => ( groups7121269368397514597t_prod(C,A,aa(fun(B,A),fun(C,A),aa(fun(B,C),fun(fun(B,A),fun(C,A)),aTP_Lamp_oe(set(B),fun(fun(B,C),fun(fun(B,A),fun(C,A))),S),G),H),T3) = groups7121269368397514597t_prod(B,A,H,S) ) ) ) ) ) ).

% prod.group
tff(fact_5725_scaleR__image__atLeastAtMost,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,X: A,Y: A] :
          ( pp(aa(real,bool,ord_less(real,zero_zero(real)),C2))
         => ( aa(set(A),set(A),image(A,A,real_V8093663219630862766scaleR(A,C2)),set_or1337092689740270186AtMost(A,X,Y)) = set_or1337092689740270186AtMost(A,aa(A,A,real_V8093663219630862766scaleR(A,C2),X),aa(A,A,real_V8093663219630862766scaleR(A,C2),Y)) ) ) ) ).

% scaleR_image_atLeastAtMost
tff(fact_5726_Max_Osubset__imp,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),B4: set(A)] :
          ( pp(aa(set(A),bool,ord_less_eq(set(A),A3),B4))
         => ( ( A3 != bot_bot(set(A)) )
           => ( pp(aa(set(A),bool,finite_finite2(A),B4))
             => pp(aa(A,bool,ord_less_eq(A,lattic643756798349783984er_Max(A,A3)),lattic643756798349783984er_Max(A,B4))) ) ) ) ) ).

% Max.subset_imp
tff(fact_5727_Max__mono,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [M7: set(A),N4: set(A)] :
          ( pp(aa(set(A),bool,ord_less_eq(set(A),M7),N4))
         => ( ( M7 != bot_bot(set(A)) )
           => ( pp(aa(set(A),bool,finite_finite2(A),N4))
             => pp(aa(A,bool,ord_less_eq(A,lattic643756798349783984er_Max(A,M7)),lattic643756798349783984er_Max(A,N4))) ) ) ) ) ).

% Max_mono
tff(fact_5728_VEBT__internal_Oheight_Oelims,axiom,
    ! [X: vEBT_VEBT,Y: nat] :
      ( ( aa(vEBT_VEBT,nat,vEBT_VEBT_height,X) = Y )
     => ( ( ? [A4: bool,B3: bool] : X = vEBT_Leaf(A4,B3)
         => ( Y != zero_zero(nat) ) )
       => ~ ! [Uu2: option(product_prod(nat,nat)),Deg2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
              ( ( X = vEBT_Node(Uu2,Deg2,TreeList2,Summary2) )
             => ( Y != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),lattic643756798349783984er_Max(nat,aa(set(vEBT_VEBT),set(nat),image(vEBT_VEBT,nat,vEBT_VEBT_height),insert(vEBT_VEBT,Summary2,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2))))) ) ) ) ) ).

% VEBT_internal.height.elims
tff(fact_5729_Max_Osubset,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),B4: set(A)] :
          ( pp(aa(set(A),bool,finite_finite2(A),A3))
         => ( ( B4 != bot_bot(set(A)) )
           => ( pp(aa(set(A),bool,ord_less_eq(set(A),B4),A3))
             => ( aa(A,A,ord_max(A,lattic643756798349783984er_Max(A,B4)),lattic643756798349783984er_Max(A,A3)) = lattic643756798349783984er_Max(A,A3) ) ) ) ) ) ).

% Max.subset
tff(fact_5730_divide__nat__def,axiom,
    ! [N: nat,M: nat] :
      ( ( ( N = zero_zero(nat) )
       => ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),N) = zero_zero(nat) ) )
      & ( ( N != zero_zero(nat) )
       => ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),N) = lattic643756798349783984er_Max(nat,collect(nat,aa(nat,fun(nat,bool),aTP_Lamp_of(nat,fun(nat,fun(nat,bool)),N),M))) ) ) ) ).

% divide_nat_def
tff(fact_5731_Frct__code__post_I1_J,axiom,
    ! [A2: int] : frct(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),A2)) = zero_zero(rat) ).

% Frct_code_post(1)
tff(fact_5732_Frct__code__post_I2_J,axiom,
    ! [A2: int] : frct(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),A2),zero_zero(int))) = zero_zero(rat) ).

% Frct_code_post(2)
tff(fact_5733_Frct__code__post_I3_J,axiom,
    frct(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),one_one(int)),one_one(int))) = one_one(rat) ).

% Frct_code_post(3)
tff(fact_5734_image__mult__atLeastAtMost__if,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,X: A,Y: A] :
          ( ( pp(aa(A,bool,ord_less(A,zero_zero(A)),C2))
           => ( aa(set(A),set(A),image(A,A,aa(A,fun(A,A),times_times(A),C2)),set_or1337092689740270186AtMost(A,X,Y)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),times_times(A),C2),X),aa(A,A,aa(A,fun(A,A),times_times(A),C2),Y)) ) )
          & ( ~ pp(aa(A,bool,ord_less(A,zero_zero(A)),C2))
           => ( ( pp(aa(A,bool,ord_less_eq(A,X),Y))
               => ( aa(set(A),set(A),image(A,A,aa(A,fun(A,A),times_times(A),C2)),set_or1337092689740270186AtMost(A,X,Y)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),times_times(A),C2),Y),aa(A,A,aa(A,fun(A,A),times_times(A),C2),X)) ) )
              & ( ~ pp(aa(A,bool,ord_less_eq(A,X),Y))
               => ( aa(set(A),set(A),image(A,A,aa(A,fun(A,A),times_times(A),C2)),set_or1337092689740270186AtMost(A,X,Y)) = bot_bot(set(A)) ) ) ) ) ) ) ).

% image_mult_atLeastAtMost_if
tff(fact_5735_image__mult__atLeastAtMost__if_H,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y: A,C2: A] :
          ( ( pp(aa(A,bool,ord_less_eq(A,X),Y))
           => ( ( pp(aa(A,bool,ord_less(A,zero_zero(A)),C2))
               => ( aa(set(A),set(A),image(A,A,aTP_Lamp_og(A,fun(A,A),C2)),set_or1337092689740270186AtMost(A,X,Y)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),times_times(A),X),C2),aa(A,A,aa(A,fun(A,A),times_times(A),Y),C2)) ) )
              & ( ~ pp(aa(A,bool,ord_less(A,zero_zero(A)),C2))
               => ( aa(set(A),set(A),image(A,A,aTP_Lamp_og(A,fun(A,A),C2)),set_or1337092689740270186AtMost(A,X,Y)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),times_times(A),Y),C2),aa(A,A,aa(A,fun(A,A),times_times(A),X),C2)) ) ) ) )
          & ( ~ pp(aa(A,bool,ord_less_eq(A,X),Y))
           => ( aa(set(A),set(A),image(A,A,aTP_Lamp_og(A,fun(A,A),C2)),set_or1337092689740270186AtMost(A,X,Y)) = bot_bot(set(A)) ) ) ) ) ).

% image_mult_atLeastAtMost_if'
tff(fact_5736_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)) )
           => ( aa(set(A),set(A),image(A,A,aa(A,fun(A,A),aTP_Lamp_oh(A,fun(A,fun(A,A)),M),C2)),set_or1337092689740270186AtMost(A,A2,B2)) = bot_bot(set(A)) ) )
          & ( ( set_or1337092689740270186AtMost(A,A2,B2) != bot_bot(set(A)) )
           => ( ( pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),M))
               => ( aa(set(A),set(A),image(A,A,aa(A,fun(A,A),aTP_Lamp_oh(A,fun(A,fun(A,A)),M),C2)),set_or1337092689740270186AtMost(A,A2,B2)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),M),A2)),C2),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),M),B2)),C2)) ) )
              & ( ~ pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),M))
               => ( aa(set(A),set(A),image(A,A,aa(A,fun(A,A),aTP_Lamp_oh(A,fun(A,fun(A,A)),M),C2)),set_or1337092689740270186AtMost(A,A2,B2)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),M),B2)),C2),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),M),A2)),C2)) ) ) ) ) ) ) ).

% image_affinity_atLeastAtMost
tff(fact_5737_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)) )
           => ( aa(set(A),set(A),image(A,A,aa(A,fun(A,A),aTP_Lamp_oi(A,fun(A,fun(A,A)),M),C2)),set_or1337092689740270186AtMost(A,A2,B2)) = bot_bot(set(A)) ) )
          & ( ( set_or1337092689740270186AtMost(A,A2,B2) != bot_bot(set(A)) )
           => ( ( pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),M))
               => ( aa(set(A),set(A),image(A,A,aa(A,fun(A,A),aTP_Lamp_oi(A,fun(A,fun(A,A)),M),C2)),set_or1337092689740270186AtMost(A,A2,B2)) = set_or1337092689740270186AtMost(A,aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),M),A2)),C2),aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),M),B2)),C2)) ) )
              & ( ~ pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),M))
               => ( aa(set(A),set(A),image(A,A,aa(A,fun(A,A),aTP_Lamp_oi(A,fun(A,fun(A,A)),M),C2)),set_or1337092689740270186AtMost(A,A2,B2)) = set_or1337092689740270186AtMost(A,aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),M),B2)),C2),aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),M),A2)),C2)) ) ) ) ) ) ) ).

% image_affinity_atLeastAtMost_diff
tff(fact_5738_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)) )
           => ( aa(set(A),set(A),image(A,A,aa(A,fun(A,A),aTP_Lamp_oj(A,fun(A,fun(A,A)),M),C2)),set_or1337092689740270186AtMost(A,A2,B2)) = bot_bot(set(A)) ) )
          & ( ( set_or1337092689740270186AtMost(A,A2,B2) != bot_bot(set(A)) )
           => ( ( pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),M))
               => ( aa(set(A),set(A),image(A,A,aa(A,fun(A,A),aTP_Lamp_oj(A,fun(A,fun(A,A)),M),C2)),set_or1337092689740270186AtMost(A,A2,B2)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),M)),C2),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),M)),C2)) ) )
              & ( ~ pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),M))
               => ( aa(set(A),set(A),image(A,A,aa(A,fun(A,A),aTP_Lamp_oj(A,fun(A,fun(A,A)),M),C2)),set_or1337092689740270186AtMost(A,A2,B2)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),M)),C2),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),M)),C2)) ) ) ) ) ) ) ).

% image_affinity_atLeastAtMost_div
tff(fact_5739_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)) )
           => ( aa(set(A),set(A),image(A,A,aa(A,fun(A,A),aTP_Lamp_ok(A,fun(A,fun(A,A)),M),C2)),set_or1337092689740270186AtMost(A,A2,B2)) = bot_bot(set(A)) ) )
          & ( ( set_or1337092689740270186AtMost(A,A2,B2) != bot_bot(set(A)) )
           => ( ( pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),M))
               => ( aa(set(A),set(A),image(A,A,aa(A,fun(A,A),aTP_Lamp_ok(A,fun(A,fun(A,A)),M),C2)),set_or1337092689740270186AtMost(A,A2,B2)) = set_or1337092689740270186AtMost(A,aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),M)),C2),aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),M)),C2)) ) )
              & ( ~ pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),M))
               => ( aa(set(A),set(A),image(A,A,aa(A,fun(A,A),aTP_Lamp_ok(A,fun(A,fun(A,A)),M),C2)),set_or1337092689740270186AtMost(A,A2,B2)) = set_or1337092689740270186AtMost(A,aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),M)),C2),aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),M)),C2)) ) ) ) ) ) ) ).

% image_affinity_atLeastAtMost_div_diff
tff(fact_5740_Frct__code__post_I4_J,axiom,
    ! [K: num] : frct(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(num,int,numeral_numeral(int),K)),one_one(int))) = aa(num,rat,numeral_numeral(rat),K) ).

% Frct_code_post(4)
tff(fact_5741_Frct__code__post_I6_J,axiom,
    ! [K: num,L: num] : frct(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(num,int,numeral_numeral(int),K)),aa(num,int,numeral_numeral(int),L))) = aa(rat,rat,aa(rat,fun(rat,rat),divide_divide(rat),aa(num,rat,numeral_numeral(rat),K)),aa(num,rat,numeral_numeral(rat),L)) ).

% Frct_code_post(6)
tff(fact_5742_VEBT__internal_Ospace_H_Opelims,axiom,
    ! [X: vEBT_VEBT,Y: nat] :
      ( ( aa(vEBT_VEBT,nat,vEBT_VEBT_space2,X) = Y )
     => ( accp(vEBT_VEBT,vEBT_VEBT_space_rel,X)
       => ( ! [A4: bool,B3: bool] :
              ( ( X = vEBT_Leaf(A4,B3) )
             => ( ( Y = aa(num,nat,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] :
                ( ( X = vEBT_Node(Info2,Deg2,TreeList2,Summary2) )
               => ( ( Y = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(aa(num,num,bit1,one2)))),aa(vEBT_VEBT,nat,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
tff(fact_5743_VEBT__internal_Oheight_Opelims,axiom,
    ! [X: vEBT_VEBT,Y: nat] :
      ( ( aa(vEBT_VEBT,nat,vEBT_VEBT_height,X) = Y )
     => ( accp(vEBT_VEBT,vEBT_VEBT_height_rel,X)
       => ( ! [A4: bool,B3: bool] :
              ( ( X = vEBT_Leaf(A4,B3) )
             => ( ( Y = 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] :
                ( ( X = vEBT_Node(Uu2,Deg2,TreeList2,Summary2) )
               => ( ( Y = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),lattic643756798349783984er_Max(nat,aa(set(vEBT_VEBT),set(nat),image(vEBT_VEBT,nat,vEBT_VEBT_height),insert(vEBT_VEBT,Summary2,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2))))) )
                 => ~ accp(vEBT_VEBT,vEBT_VEBT_height_rel,vEBT_Node(Uu2,Deg2,TreeList2,Summary2)) ) ) ) ) ) ).

% VEBT_internal.height.pelims
tff(fact_5744_VEBT__internal_Ocnt_Opelims,axiom,
    ! [X: vEBT_VEBT,Y: real] :
      ( ( aa(vEBT_VEBT,real,vEBT_VEBT_cnt,X) = Y )
     => ( accp(vEBT_VEBT,vEBT_VEBT_cnt_rel2,X)
       => ( ! [A4: bool,B3: bool] :
              ( ( X = vEBT_Leaf(A4,B3) )
             => ( ( Y = 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] :
                ( ( X = vEBT_Node(Info2,Deg2,TreeList2,Summary2) )
               => ( ( Y = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),aa(vEBT_VEBT,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
tff(fact_5745_bij__betw__Suc,axiom,
    ! [M7: set(nat),N4: set(nat)] :
      ( bij_betw(nat,nat,suc,M7,N4)
    <=> ( aa(set(nat),set(nat),image(nat,nat,suc),M7) = N4 ) ) ).

% bij_betw_Suc
tff(fact_5746_image__Suc__atLeastAtMost,axiom,
    ! [I: nat,J: nat] : aa(set(nat),set(nat),image(nat,nat,suc),set_or1337092689740270186AtMost(nat,I,J)) = set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,I),aa(nat,nat,suc,J)) ).

% image_Suc_atLeastAtMost
tff(fact_5747_bij__betw__of__nat,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [N4: set(nat),A3: set(A)] :
          ( bij_betw(nat,A,semiring_1_of_nat(A),N4,A3)
        <=> ( aa(set(nat),set(A),image(nat,A,semiring_1_of_nat(A)),N4) = A3 ) ) ) ).

% bij_betw_of_nat
tff(fact_5748_pair__imageI,axiom,
    ! [C: $tType,B: $tType,A: $tType,A2: A,B2: B,A3: set(product_prod(A,B)),F2: fun(A,fun(B,C))] :
      ( pp(aa(set(product_prod(A,B)),bool,aa(product_prod(A,B),fun(set(product_prod(A,B)),bool),member(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A2),B2)),A3))
     => pp(aa(set(C),bool,aa(C,fun(set(C),bool),member(C),aa(B,C,aa(A,fun(B,C),F2,A2),B2)),aa(set(product_prod(A,B)),set(C),image(product_prod(A,B),C,product_case_prod(A,B,C,F2)),A3))) ) ).

% pair_imageI
tff(fact_5749_Max__divisors__self__int,axiom,
    ! [N: int] :
      ( ( N != zero_zero(int) )
     => ( lattic643756798349783984er_Max(int,collect(int,aTP_Lamp_az(int,fun(int,bool),N))) = aa(int,int,abs_abs(int),N) ) ) ).

% Max_divisors_self_int
tff(fact_5750_None__notin__image__Some,axiom,
    ! [A: $tType,A3: set(A)] : ~ pp(aa(set(option(A)),bool,aa(option(A),fun(set(option(A)),bool),member(option(A)),none(A)),aa(set(A),set(option(A)),image(A,option(A),some(A)),A3))) ).

% None_notin_image_Some
tff(fact_5751_zero__notin__Suc__image,axiom,
    ! [A3: set(nat)] : ~ pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),zero_zero(nat)),aa(set(nat),set(nat),image(nat,nat,suc),A3))) ).

% zero_notin_Suc_image
tff(fact_5752_finite__conv__nat__seg__image,axiom,
    ! [A: $tType,A3: set(A)] :
      ( pp(aa(set(A),bool,finite_finite2(A),A3))
    <=> ? [N5: nat,F6: fun(nat,A)] : A3 = aa(set(nat),set(A),image(nat,A,F6),collect(nat,aTP_Lamp_ap(nat,fun(nat,bool),N5))) ) ).

% finite_conv_nat_seg_image
tff(fact_5753_nat__seg__image__imp__finite,axiom,
    ! [A: $tType,A3: set(A),F2: fun(nat,A),N: nat] :
      ( ( A3 = aa(set(nat),set(A),image(nat,A,F2),collect(nat,aTP_Lamp_ap(nat,fun(nat,bool),N))) )
     => pp(aa(set(A),bool,finite_finite2(A),A3)) ) ).

% nat_seg_image_imp_finite
tff(fact_5754_image__int__atLeastAtMost,axiom,
    ! [A2: nat,B2: nat] : aa(set(nat),set(int),image(nat,int,semiring_1_of_nat(int)),set_or1337092689740270186AtMost(nat,A2,B2)) = set_or1337092689740270186AtMost(int,aa(nat,int,semiring_1_of_nat(int),A2),aa(nat,int,semiring_1_of_nat(int),B2)) ).

% image_int_atLeastAtMost
tff(fact_5755_image__Suc__lessThan,axiom,
    ! [N: nat] : aa(set(nat),set(nat),image(nat,nat,suc),set_ord_lessThan(nat,N)) = set_or1337092689740270186AtMost(nat,one_one(nat),N) ).

% image_Suc_lessThan
tff(fact_5756_image__Suc__atMost,axiom,
    ! [N: nat] : aa(set(nat),set(nat),image(nat,nat,suc),set_ord_atMost(nat,N)) = set_or1337092689740270186AtMost(nat,one_one(nat),aa(nat,nat,suc,N)) ).

% image_Suc_atMost
tff(fact_5757_atLeast0__atMost__Suc__eq__insert__0,axiom,
    ! [N: nat] : set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,suc,N)) = insert(nat,zero_zero(nat),aa(set(nat),set(nat),image(nat,nat,suc),set_or1337092689740270186AtMost(nat,zero_zero(nat),N))) ).

% atLeast0_atMost_Suc_eq_insert_0
tff(fact_5758_lessThan__Suc__eq__insert__0,axiom,
    ! [N: nat] : set_ord_lessThan(nat,aa(nat,nat,suc,N)) = insert(nat,zero_zero(nat),aa(set(nat),set(nat),image(nat,nat,suc),set_ord_lessThan(nat,N))) ).

% lessThan_Suc_eq_insert_0
tff(fact_5759_atMost__Suc__eq__insert__0,axiom,
    ! [N: nat] : set_ord_atMost(nat,aa(nat,nat,suc,N)) = insert(nat,zero_zero(nat),aa(set(nat),set(nat),image(nat,nat,suc),set_ord_atMost(nat,N))) ).

% atMost_Suc_eq_insert_0
tff(fact_5760_VEBT__internal_Ocnt_H_Opelims,axiom,
    ! [X: vEBT_VEBT,Y: nat] :
      ( ( aa(vEBT_VEBT,nat,vEBT_VEBT_cnt2,X) = Y )
     => ( accp(vEBT_VEBT,vEBT_VEBT_cnt_rel,X)
       => ( ! [A4: bool,B3: bool] :
              ( ( X = vEBT_Leaf(A4,B3) )
             => ( ( Y = 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] :
                ( ( X = vEBT_Node(Info2,Deg2,TreeList2,Summary2) )
               => ( ( Y = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),aa(vEBT_VEBT,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
tff(fact_5761_vebt__maxt_Opelims,axiom,
    ! [X: vEBT_VEBT,Y: option(nat)] :
      ( ( vEBT_vebt_maxt(X) = Y )
     => ( accp(vEBT_VEBT,vEBT_vebt_maxt_rel,X)
       => ( ! [A4: bool,B3: bool] :
              ( ( X = vEBT_Leaf(A4,B3) )
             => ( ( ( pp(B3)
                   => ( Y = aa(nat,option(nat),some(nat),one_one(nat)) ) )
                  & ( ~ pp(B3)
                   => ( ( pp(A4)
                       => ( Y = aa(nat,option(nat),some(nat),zero_zero(nat)) ) )
                      & ( ~ pp(A4)
                       => ( Y = none(nat) ) ) ) ) )
               => ~ accp(vEBT_VEBT,vEBT_vebt_maxt_rel,vEBT_Leaf(A4,B3)) ) )
         => ( ! [Uu2: nat,Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT] :
                ( ( X = vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw2) )
               => ( ( Y = 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] :
                  ( ( X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),Ux2,Uy2,Uz2) )
                 => ( ( Y = aa(nat,option(nat),some(nat),Ma2) )
                   => ~ accp(vEBT_VEBT,vEBT_vebt_maxt_rel,vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),Ux2,Uy2,Uz2)) ) ) ) ) ) ) ).

% vebt_maxt.pelims
tff(fact_5762_vebt__mint_Opelims,axiom,
    ! [X: vEBT_VEBT,Y: option(nat)] :
      ( ( vEBT_vebt_mint(X) = Y )
     => ( accp(vEBT_VEBT,vEBT_vebt_mint_rel,X)
       => ( ! [A4: bool,B3: bool] :
              ( ( X = vEBT_Leaf(A4,B3) )
             => ( ( ( pp(A4)
                   => ( Y = aa(nat,option(nat),some(nat),zero_zero(nat)) ) )
                  & ( ~ pp(A4)
                   => ( ( pp(B3)
                       => ( Y = aa(nat,option(nat),some(nat),one_one(nat)) ) )
                      & ( ~ pp(B3)
                       => ( Y = none(nat) ) ) ) ) )
               => ~ accp(vEBT_VEBT,vEBT_vebt_mint_rel,vEBT_Leaf(A4,B3)) ) )
         => ( ! [Uu2: nat,Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT] :
                ( ( X = vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw2) )
               => ( ( Y = 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] :
                  ( ( X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),Ux2,Uy2,Uz2) )
                 => ( ( Y = aa(nat,option(nat),some(nat),Mi2) )
                   => ~ accp(vEBT_VEBT,vEBT_vebt_mint_rel,vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),Ux2,Uy2,Uz2)) ) ) ) ) ) ) ).

% vebt_mint.pelims
tff(fact_5763_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t_Opelims,axiom,
    ! [X: vEBT_VEBT,Y: nat] :
      ( ( vEBT_T_m_i_n_t(X) = Y )
     => ( accp(vEBT_VEBT,vEBT_T_m_i_n_t_rel,X)
       => ( ! [A4: bool,B3: bool] :
              ( ( X = vEBT_Leaf(A4,B3) )
             => ( ( Y = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),if(nat,A4,zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,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] :
                ( ( X = vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw2) )
               => ( ( Y = 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] :
                  ( ( X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),Ux2,Uy2,Uz2) )
                 => ( ( Y = one_one(nat) )
                   => ~ accp(vEBT_VEBT,vEBT_T_m_i_n_t_rel,vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),Ux2,Uy2,Uz2)) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>t.pelims
tff(fact_5764_bij__betw__add,axiom,
    ! [A: $tType] :
      ( cancel_semigroup_add(A)
     => ! [A2: A,A3: set(A),B4: set(A)] :
          ( bij_betw(A,A,aa(A,fun(A,A),plus_plus(A),A2),A3,B4)
        <=> ( aa(set(A),set(A),image(A,A,aa(A,fun(A,A),plus_plus(A),A2)),A3) = B4 ) ) ) ).

% bij_betw_add
tff(fact_5765_T_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062x_092_060_094sub_062t_Opelims,axiom,
    ! [X: vEBT_VEBT,Y: nat] :
      ( ( vEBT_T_m_a_x_t(X) = Y )
     => ( accp(vEBT_VEBT,vEBT_T_m_a_x_t_rel,X)
       => ( ! [A4: bool,B3: bool] :
              ( ( X = vEBT_Leaf(A4,B3) )
             => ( ( Y = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),if(nat,B3,one_one(nat),aa(nat,nat,aa(nat,fun(nat,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] :
                ( ( X = vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw2) )
               => ( ( Y = 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] :
                  ( ( X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),Ux2,Uy2,Uz2) )
                 => ( ( Y = one_one(nat) )
                   => ~ accp(vEBT_VEBT,vEBT_T_m_a_x_t_rel,vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),Ux2,Uy2,Uz2)) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>a\<^sub>x\<^sub>t.pelims
tff(fact_5766_translation__diff,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A2: A,S3: set(A),T: set(A)] : aa(set(A),set(A),image(A,A,aa(A,fun(A,A),plus_plus(A),A2)),aa(set(A),set(A),minus_minus(set(A),S3),T)) = aa(set(A),set(A),minus_minus(set(A),aa(set(A),set(A),image(A,A,aa(A,fun(A,A),plus_plus(A),A2)),S3)),aa(set(A),set(A),image(A,A,aa(A,fun(A,A),plus_plus(A),A2)),T)) ) ).

% translation_diff
tff(fact_5767_bij__betw__subset,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,B),A3: set(A),A9: set(B),B4: set(A),B10: set(B)] :
      ( bij_betw(A,B,F2,A3,A9)
     => ( pp(aa(set(A),bool,ord_less_eq(set(A),B4),A3))
       => ( ( aa(set(A),set(B),image(A,B,F2),B4) = B10 )
         => bij_betw(A,B,F2,B4,B10) ) ) ) ).

% bij_betw_subset
tff(fact_5768_bij__betw__byWitness,axiom,
    ! [A: $tType,B: $tType,A3: set(A),F7: fun(B,A),F2: fun(A,B),A9: set(B)] :
      ( ! [X3: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),A3))
         => ( aa(B,A,F7,aa(A,B,F2,X3)) = X3 ) )
     => ( ! [X3: B] :
            ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),A9))
           => ( aa(A,B,F2,aa(B,A,F7,X3)) = X3 ) )
       => ( pp(aa(set(B),bool,ord_less_eq(set(B),aa(set(A),set(B),image(A,B,F2),A3)),A9))
         => ( pp(aa(set(A),bool,ord_less_eq(set(A),aa(set(B),set(A),image(B,A,F7),A9)),A3))
           => bij_betw(A,B,F2,A3,A9) ) ) ) ) ).

% bij_betw_byWitness
tff(fact_5769_translation__Compl,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A2: A,T: set(A)] : aa(set(A),set(A),image(A,A,aa(A,fun(A,A),plus_plus(A),A2)),aa(set(A),set(A),uminus_uminus(set(A)),T)) = aa(set(A),set(A),uminus_uminus(set(A)),aa(set(A),set(A),image(A,A,aa(A,fun(A,A),plus_plus(A),A2)),T)) ) ).

% translation_Compl
tff(fact_5770_translation__subtract__diff,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A2: A,S3: set(A),T: set(A)] : aa(set(A),set(A),image(A,A,aTP_Lamp_ol(A,fun(A,A),A2)),aa(set(A),set(A),minus_minus(set(A),S3),T)) = aa(set(A),set(A),minus_minus(set(A),aa(set(A),set(A),image(A,A,aTP_Lamp_ol(A,fun(A,A),A2)),S3)),aa(set(A),set(A),image(A,A,aTP_Lamp_ol(A,fun(A,A),A2)),T)) ) ).

% translation_subtract_diff
tff(fact_5771_translation__subtract__Compl,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A2: A,T: set(A)] : aa(set(A),set(A),image(A,A,aTP_Lamp_ol(A,fun(A,A),A2)),aa(set(A),set(A),uminus_uminus(set(A)),T)) = aa(set(A),set(A),uminus_uminus(set(A)),aa(set(A),set(A),image(A,A,aTP_Lamp_ol(A,fun(A,A),A2)),T)) ) ).

% translation_subtract_Compl
tff(fact_5772_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062N_092_060_094sub_062u_092_060_094sub_062l_092_060_094sub_062l_Opelims,axiom,
    ! [X: vEBT_VEBT,Y: nat] :
      ( ( vEBT_T_m_i_n_N_u_l_l(X) = Y )
     => ( accp(vEBT_VEBT,vEBT_T5462971552011256508_l_rel,X)
       => ( ( ( X = vEBT_Leaf(fFalse,fFalse) )
           => ( ( Y = one_one(nat) )
             => ~ accp(vEBT_VEBT,vEBT_T5462971552011256508_l_rel,vEBT_Leaf(fFalse,fFalse)) ) )
         => ( ! [Uv2: bool] :
                ( ( X = vEBT_Leaf(fTrue,Uv2) )
               => ( ( Y = one_one(nat) )
                 => ~ accp(vEBT_VEBT,vEBT_T5462971552011256508_l_rel,vEBT_Leaf(fTrue,Uv2)) ) )
           => ( ! [Uu2: bool] :
                  ( ( X = vEBT_Leaf(Uu2,fTrue) )
                 => ( ( Y = one_one(nat) )
                   => ~ accp(vEBT_VEBT,vEBT_T5462971552011256508_l_rel,vEBT_Leaf(Uu2,fTrue)) ) )
             => ( ! [Uw2: nat,Ux2: list(vEBT_VEBT),Uy2: vEBT_VEBT] :
                    ( ( X = vEBT_Node(none(product_prod(nat,nat)),Uw2,Ux2,Uy2) )
                   => ( ( Y = 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),Va3: nat,Vb2: list(vEBT_VEBT),Vc: vEBT_VEBT] :
                      ( ( X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),Uz2),Va3,Vb2,Vc) )
                     => ( ( Y = one_one(nat) )
                       => ~ accp(vEBT_VEBT,vEBT_T5462971552011256508_l_rel,vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),Uz2),Va3,Vb2,Vc)) ) ) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.pelims
tff(fact_5773_VEBT__internal_OminNull_Opelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Y: bool] :
      ( ( pp(vEBT_VEBT_minNull(X))
      <=> pp(Y) )
     => ( accp(vEBT_VEBT,vEBT_V6963167321098673237ll_rel,X)
       => ( ( ( X = vEBT_Leaf(fFalse,fFalse) )
           => ( pp(Y)
             => ~ accp(vEBT_VEBT,vEBT_V6963167321098673237ll_rel,vEBT_Leaf(fFalse,fFalse)) ) )
         => ( ! [Uv2: bool] :
                ( ( X = vEBT_Leaf(fTrue,Uv2) )
               => ( ~ pp(Y)
                 => ~ accp(vEBT_VEBT,vEBT_V6963167321098673237ll_rel,vEBT_Leaf(fTrue,Uv2)) ) )
           => ( ! [Uu2: bool] :
                  ( ( X = vEBT_Leaf(Uu2,fTrue) )
                 => ( ~ pp(Y)
                   => ~ accp(vEBT_VEBT,vEBT_V6963167321098673237ll_rel,vEBT_Leaf(Uu2,fTrue)) ) )
             => ( ! [Uw2: nat,Ux2: list(vEBT_VEBT),Uy2: vEBT_VEBT] :
                    ( ( X = vEBT_Node(none(product_prod(nat,nat)),Uw2,Ux2,Uy2) )
                   => ( pp(Y)
                     => ~ accp(vEBT_VEBT,vEBT_V6963167321098673237ll_rel,vEBT_Node(none(product_prod(nat,nat)),Uw2,Ux2,Uy2)) ) )
               => ~ ! [Uz2: product_prod(nat,nat),Va3: nat,Vb2: list(vEBT_VEBT),Vc: vEBT_VEBT] :
                      ( ( X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),Uz2),Va3,Vb2,Vc) )
                     => ( ~ pp(Y)
                       => ~ accp(vEBT_VEBT,vEBT_V6963167321098673237ll_rel,vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),Uz2),Va3,Vb2,Vc)) ) ) ) ) ) ) ) ) ).

% VEBT_internal.minNull.pelims(1)
tff(fact_5774_sofl__test,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A)] :
          ( ( aa(int,int,aa(int,fun(int,int),plus_plus(int),ring_1_signed(A,int,X)),ring_1_signed(A,int,Y)) = ring_1_signed(A,int,aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),X),Y)) )
        <=> ( bit_se4197421643247451524op_bit(word(A),aa(nat,nat,minus_minus(nat,aa(word(A),nat,size_size(word(A)),X)),one_one(nat)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344971392196577ns_xor(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),X),Y)),X)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344971392196577ns_xor(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),X),Y)),Y))) = zero_zero(word(A)) ) ) ) ).

% sofl_test
tff(fact_5775_drop__bit__of__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat] : bit_se4197421643247451524op_bit(A,N,zero_zero(A)) = zero_zero(A) ) ).

% drop_bit_of_0
tff(fact_5776_drop__bit__drop__bit,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,N: nat,A2: A] : bit_se4197421643247451524op_bit(A,M,bit_se4197421643247451524op_bit(A,N,A2)) = bit_se4197421643247451524op_bit(A,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N),A2) ) ).

% drop_bit_drop_bit
tff(fact_5777_drop__bit__and,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A2: A,B2: A] : bit_se4197421643247451524op_bit(A,N,aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),bit_se4197421643247451524op_bit(A,N,A2)),bit_se4197421643247451524op_bit(A,N,B2)) ) ).

% drop_bit_and
tff(fact_5778_drop__bit__or,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A2: A,B2: A] : bit_se4197421643247451524op_bit(A,N,aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),bit_se4197421643247451524op_bit(A,N,A2)),bit_se4197421643247451524op_bit(A,N,B2)) ) ).

% drop_bit_or
tff(fact_5779_drop__bit__xor,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A2: A,B2: A] : bit_se4197421643247451524op_bit(A,N,aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),bit_se4197421643247451524op_bit(A,N,A2)),bit_se4197421643247451524op_bit(A,N,B2)) ) ).

% drop_bit_xor
tff(fact_5780_drop__bit__of__bool,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,B2: bool] : bit_se4197421643247451524op_bit(A,N,aa(bool,A,zero_neq_one_of_bool(A),B2)) = aa(bool,A,zero_neq_one_of_bool(A),fconj(aa(nat,bool,fequal(nat,N),zero_zero(nat)),B2)) ) ).

% drop_bit_of_bool
tff(fact_5781_drop__bit__Suc__bit0,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [N: nat,K: num] : bit_se4197421643247451524op_bit(A,aa(nat,nat,suc,N),aa(num,A,numeral_numeral(A),bit0(K))) = bit_se4197421643247451524op_bit(A,N,aa(num,A,numeral_numeral(A),K)) ) ).

% drop_bit_Suc_bit0
tff(fact_5782_drop__bit__Suc__bit1,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [N: nat,K: num] : bit_se4197421643247451524op_bit(A,aa(nat,nat,suc,N),aa(num,A,numeral_numeral(A),aa(num,num,bit1,K))) = bit_se4197421643247451524op_bit(A,N,aa(num,A,numeral_numeral(A),K)) ) ).

% drop_bit_Suc_bit1
tff(fact_5783_drop__bit__of__1,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat] : bit_se4197421643247451524op_bit(A,N,one_one(A)) = aa(bool,A,zero_neq_one_of_bool(A),aa(nat,bool,fequal(nat,N),zero_zero(nat))) ) ).

% drop_bit_of_1
tff(fact_5784_drop__bit__numeral__bit0,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [L: num,K: num] : bit_se4197421643247451524op_bit(A,aa(num,nat,numeral_numeral(nat),L),aa(num,A,numeral_numeral(A),bit0(K))) = bit_se4197421643247451524op_bit(A,pred_numeral(L),aa(num,A,numeral_numeral(A),K)) ) ).

% drop_bit_numeral_bit0
tff(fact_5785_drop__bit__numeral__bit1,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [L: num,K: num] : bit_se4197421643247451524op_bit(A,aa(num,nat,numeral_numeral(nat),L),aa(num,A,numeral_numeral(A),aa(num,num,bit1,K))) = bit_se4197421643247451524op_bit(A,pred_numeral(L),aa(num,A,numeral_numeral(A),K)) ) ).

% drop_bit_numeral_bit1
tff(fact_5786_take__bit__eq__self__iff__drop__bit__eq__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A2: A] :
          ( ( aa(A,A,bit_se2584673776208193580ke_bit(A,N),A2) = A2 )
        <=> ( bit_se4197421643247451524op_bit(A,N,A2) = zero_zero(A) ) ) ) ).

% take_bit_eq_self_iff_drop_bit_eq_0
tff(fact_5787_of__nat__drop__bit,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [M: nat,N: nat] : aa(nat,A,semiring_1_of_nat(A),bit_se4197421643247451524op_bit(nat,M,N)) = bit_se4197421643247451524op_bit(A,M,aa(nat,A,semiring_1_of_nat(A),N)) ) ).

% of_nat_drop_bit
tff(fact_5788_drop__bit__of__nat,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [N: nat,M: nat] : bit_se4197421643247451524op_bit(A,N,aa(nat,A,semiring_1_of_nat(A),M)) = aa(nat,A,semiring_1_of_nat(A),bit_se4197421643247451524op_bit(nat,N,M)) ) ).

% drop_bit_of_nat
tff(fact_5789_take__bit__drop__bit,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,N: nat,A2: A] : aa(A,A,bit_se2584673776208193580ke_bit(A,M),bit_se4197421643247451524op_bit(A,N,A2)) = bit_se4197421643247451524op_bit(A,N,aa(A,A,bit_se2584673776208193580ke_bit(A,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N)),A2)) ) ).

% take_bit_drop_bit
tff(fact_5790_drop__bit__take__bit,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,N: nat,A2: A] : bit_se4197421643247451524op_bit(A,M,aa(A,A,bit_se2584673776208193580ke_bit(A,N),A2)) = aa(A,A,bit_se2584673776208193580ke_bit(A,aa(nat,nat,minus_minus(nat,N),M)),bit_se4197421643247451524op_bit(A,M,A2)) ) ).

% drop_bit_take_bit
tff(fact_5791_bit__word__iff__drop__bit__and,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),N: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(word(A),A2),N))
        <=> ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),bit_se4197421643247451524op_bit(word(A),N,A2)),one_one(word(A))) = one_one(word(A)) ) ) ) ).

% bit_word_iff_drop_bit_and
tff(fact_5792_bit__iff__and__drop__bit__eq__1,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,N: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,A2),N))
        <=> ( aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),bit_se4197421643247451524op_bit(A,N,A2)),one_one(A)) = one_one(A) ) ) ) ).

% bit_iff_and_drop_bit_eq_1
tff(fact_5793_drop__bit__half,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A2: A] : bit_se4197421643247451524op_bit(A,N,aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(num,A,numeral_numeral(A),bit0(one2)))) = aa(A,A,aa(A,fun(A,A),divide_divide(A),bit_se4197421643247451524op_bit(A,N,A2)),aa(num,A,numeral_numeral(A),bit0(one2))) ) ).

% drop_bit_half
tff(fact_5794_stable__imp__drop__bit__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,N: nat] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(num,A,numeral_numeral(A),bit0(one2))) = A2 )
         => ( bit_se4197421643247451524op_bit(A,N,A2) = A2 ) ) ) ).

% stable_imp_drop_bit_eq
tff(fact_5795_drop__bit__Suc,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A2: A] : bit_se4197421643247451524op_bit(A,aa(nat,nat,suc,N),A2) = bit_se4197421643247451524op_bit(A,N,aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(num,A,numeral_numeral(A),bit0(one2)))) ) ).

% drop_bit_Suc
tff(fact_5796_drop__bit__eq__div,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A2: A] : bit_se4197421643247451524op_bit(A,N,A2) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N)) ) ).

% drop_bit_eq_div
tff(fact_5797_bit__iff__odd__drop__bit,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,N: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,A2),N))
        <=> ~ pp(aa(A,bool,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),bit_se4197421643247451524op_bit(A,N,A2))) ) ) ).

% bit_iff_odd_drop_bit
tff(fact_5798_even__drop__bit__iff__not__bit,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A2: A] :
          ( pp(aa(A,bool,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),bit_se4197421643247451524op_bit(A,N,A2)))
        <=> ~ pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,A2),N)) ) ) ).

% even_drop_bit_iff_not_bit
tff(fact_5799_drop__bit__rec,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A2: A] :
          ( ( ( N = zero_zero(nat) )
           => ( bit_se4197421643247451524op_bit(A,N,A2) = A2 ) )
          & ( ( N != zero_zero(nat) )
           => ( bit_se4197421643247451524op_bit(A,N,A2) = bit_se4197421643247451524op_bit(A,aa(nat,nat,minus_minus(nat,N),one_one(nat)),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(num,A,numeral_numeral(A),bit0(one2)))) ) ) ) ) ).

% drop_bit_rec
tff(fact_5800_VEBT__internal_OminNull_Opelims_I3_J,axiom,
    ! [X: vEBT_VEBT] :
      ( ~ pp(vEBT_VEBT_minNull(X))
     => ( accp(vEBT_VEBT,vEBT_V6963167321098673237ll_rel,X)
       => ( ! [Uv2: bool] :
              ( ( X = vEBT_Leaf(fTrue,Uv2) )
             => ~ accp(vEBT_VEBT,vEBT_V6963167321098673237ll_rel,vEBT_Leaf(fTrue,Uv2)) )
         => ( ! [Uu2: bool] :
                ( ( X = vEBT_Leaf(Uu2,fTrue) )
               => ~ accp(vEBT_VEBT,vEBT_V6963167321098673237ll_rel,vEBT_Leaf(Uu2,fTrue)) )
           => ~ ! [Uz2: product_prod(nat,nat),Va3: nat,Vb2: list(vEBT_VEBT),Vc: vEBT_VEBT] :
                  ( ( X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),Uz2),Va3,Vb2,Vc) )
                 => ~ accp(vEBT_VEBT,vEBT_V6963167321098673237ll_rel,vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),Uz2),Va3,Vb2,Vc)) ) ) ) ) ) ).

% VEBT_internal.minNull.pelims(3)
tff(fact_5801_VEBT__internal_OminNull_Opelims_I2_J,axiom,
    ! [X: vEBT_VEBT] :
      ( pp(vEBT_VEBT_minNull(X))
     => ( accp(vEBT_VEBT,vEBT_V6963167321098673237ll_rel,X)
       => ( ( ( X = vEBT_Leaf(fFalse,fFalse) )
           => ~ accp(vEBT_VEBT,vEBT_V6963167321098673237ll_rel,vEBT_Leaf(fFalse,fFalse)) )
         => ~ ! [Uw2: nat,Ux2: list(vEBT_VEBT),Uy2: vEBT_VEBT] :
                ( ( X = 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)
tff(fact_5802_div__half__word,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),X: word(A)] :
          ( ( Y != zero_zero(word(A)) )
         => ( aa(word(A),product_prod(word(A),word(A)),aa(word(A),fun(word(A),product_prod(word(A),word(A))),product_Pair(word(A),word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),X),Y)),modulo_modulo(word(A),X,Y)) = if(product_prod(word(A),word(A)),aa(word(A),bool,ord_less_eq(word(A),Y),aa(word(A),word(A),minus_minus(word(A),X),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),bit_se4730199178511100633sh_bit(word(A),one_one(nat),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),bit_se4197421643247451524op_bit(word(A),one_one(nat),X)),Y))),Y))),aa(word(A),product_prod(word(A),word(A)),aa(word(A),fun(word(A),product_prod(word(A),word(A))),product_Pair(word(A),word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),bit_se4730199178511100633sh_bit(word(A),one_one(nat),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),bit_se4197421643247451524op_bit(word(A),one_one(nat),X)),Y))),one_one(word(A)))),aa(word(A),word(A),minus_minus(word(A),aa(word(A),word(A),minus_minus(word(A),X),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),bit_se4730199178511100633sh_bit(word(A),one_one(nat),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),bit_se4197421643247451524op_bit(word(A),one_one(nat),X)),Y))),Y))),Y)),aa(word(A),product_prod(word(A),word(A)),aa(word(A),fun(word(A),product_prod(word(A),word(A))),product_Pair(word(A),word(A)),bit_se4730199178511100633sh_bit(word(A),one_one(nat),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),bit_se4197421643247451524op_bit(word(A),one_one(nat),X)),Y))),aa(word(A),word(A),minus_minus(word(A),X),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),bit_se4730199178511100633sh_bit(word(A),one_one(nat),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),bit_se4197421643247451524op_bit(word(A),one_one(nat),X)),Y))),Y)))) ) ) ) ).

% div_half_word
tff(fact_5803_pred__subset__eq,axiom,
    ! [A: $tType,R: set(A),S: set(A)] :
      ( pp(aa(fun(A,bool),bool,ord_less_eq(fun(A,bool),aTP_Lamp_a(set(A),fun(A,bool),R)),aTP_Lamp_a(set(A),fun(A,bool),S)))
    <=> pp(aa(set(A),bool,ord_less_eq(set(A),R),S)) ) ).

% pred_subset_eq
tff(fact_5804_case__prod__app,axiom,
    ! [D6: $tType,A: $tType,C: $tType,B: $tType,F2: fun(B,fun(C,fun(D6,A))),X: product_prod(B,C),Y: D6] : aa(D6,A,aa(product_prod(B,C),fun(D6,A),product_case_prod(B,C,fun(D6,A),F2),X),Y) = aa(product_prod(B,C),A,product_case_prod(B,C,A,aa(D6,fun(B,fun(C,A)),aTP_Lamp_ly(fun(B,fun(C,fun(D6,A))),fun(D6,fun(B,fun(C,A))),F2),Y)),X) ).

% case_prod_app
tff(fact_5805_push__bit__of__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat] : bit_se4730199178511100633sh_bit(A,N,zero_zero(A)) = zero_zero(A) ) ).

% push_bit_of_0
tff(fact_5806_push__bit__eq__0__iff,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [N: nat,A2: A] :
          ( ( bit_se4730199178511100633sh_bit(A,N,A2) = zero_zero(A) )
        <=> ( A2 = zero_zero(A) ) ) ) ).

% push_bit_eq_0_iff
tff(fact_5807_drop__bit__nonnegative__int__iff,axiom,
    ! [N: nat,K: int] :
      ( pp(aa(int,bool,ord_less_eq(int,zero_zero(int)),bit_se4197421643247451524op_bit(int,N,K)))
    <=> pp(aa(int,bool,ord_less_eq(int,zero_zero(int)),K)) ) ).

% drop_bit_nonnegative_int_iff
tff(fact_5808_drop__bit__negative__int__iff,axiom,
    ! [N: nat,K: int] :
      ( pp(aa(int,bool,ord_less(int,bit_se4197421643247451524op_bit(int,N,K)),zero_zero(int)))
    <=> pp(aa(int,bool,ord_less(int,K),zero_zero(int))) ) ).

% drop_bit_negative_int_iff
tff(fact_5809_drop__bit__minus__one,axiom,
    ! [N: nat] : bit_se4197421643247451524op_bit(int,N,aa(int,int,uminus_uminus(int),one_one(int))) = aa(int,int,uminus_uminus(int),one_one(int)) ).

% drop_bit_minus_one
tff(fact_5810_push__bit__push__bit,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,N: nat,A2: A] : bit_se4730199178511100633sh_bit(A,M,bit_se4730199178511100633sh_bit(A,N,A2)) = bit_se4730199178511100633sh_bit(A,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N),A2) ) ).

% push_bit_push_bit
tff(fact_5811_push__bit__and,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A2: A,B2: A] : bit_se4730199178511100633sh_bit(A,N,aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),bit_se4730199178511100633sh_bit(A,N,A2)),bit_se4730199178511100633sh_bit(A,N,B2)) ) ).

% push_bit_and
tff(fact_5812_push__bit__or,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A2: A,B2: A] : bit_se4730199178511100633sh_bit(A,N,aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),bit_se4730199178511100633sh_bit(A,N,A2)),bit_se4730199178511100633sh_bit(A,N,B2)) ) ).

% push_bit_or
tff(fact_5813_push__bit__xor,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A2: A,B2: A] : bit_se4730199178511100633sh_bit(A,N,aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),bit_se4730199178511100633sh_bit(A,N,A2)),bit_se4730199178511100633sh_bit(A,N,B2)) ) ).

% push_bit_xor
tff(fact_5814_drop__bit__Suc__minus__bit0,axiom,
    ! [N: nat,K: num] : bit_se4197421643247451524op_bit(int,aa(nat,nat,suc,N),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit0(K)))) = bit_se4197421643247451524op_bit(int,N,aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),K))) ).

% drop_bit_Suc_minus_bit0
tff(fact_5815_push__bit__Suc__numeral,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,K: num] : bit_se4730199178511100633sh_bit(A,aa(nat,nat,suc,N),aa(num,A,numeral_numeral(A),K)) = bit_se4730199178511100633sh_bit(A,N,aa(num,A,numeral_numeral(A),bit0(K))) ) ).

% push_bit_Suc_numeral
tff(fact_5816_drop__bit__of__Suc__0,axiom,
    ! [N: nat] : bit_se4197421643247451524op_bit(nat,N,aa(nat,nat,suc,zero_zero(nat))) = aa(bool,nat,zero_neq_one_of_bool(nat),aa(nat,bool,fequal(nat,N),zero_zero(nat))) ).

% drop_bit_of_Suc_0
tff(fact_5817_push__bit__Suc__minus__numeral,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [N: nat,K: num] : bit_se4730199178511100633sh_bit(A,aa(nat,nat,suc,N),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),K))) = bit_se4730199178511100633sh_bit(A,N,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),bit0(K)))) ) ).

% push_bit_Suc_minus_numeral
tff(fact_5818_push__bit__minus__one__eq__not__mask,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [N: nat] : bit_se4730199178511100633sh_bit(A,N,aa(A,A,uminus_uminus(A),one_one(A))) = aa(A,A,bit_ri4277139882892585799ns_not(A),bit_se2239418461657761734s_mask(A,N)) ) ).

% push_bit_minus_one_eq_not_mask
tff(fact_5819_drop__bit__numeral__minus__bit0,axiom,
    ! [L: num,K: num] : bit_se4197421643247451524op_bit(int,aa(num,nat,numeral_numeral(nat),L),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit0(K)))) = bit_se4197421643247451524op_bit(int,pred_numeral(L),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),K))) ).

% drop_bit_numeral_minus_bit0
tff(fact_5820_push__bit__numeral,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [L: num,K: num] : bit_se4730199178511100633sh_bit(A,aa(num,nat,numeral_numeral(nat),L),aa(num,A,numeral_numeral(A),K)) = bit_se4730199178511100633sh_bit(A,pred_numeral(L),aa(num,A,numeral_numeral(A),bit0(K))) ) ).

% push_bit_numeral
tff(fact_5821_drop__bit__Suc__minus__bit1,axiom,
    ! [N: nat,K: num] : bit_se4197421643247451524op_bit(int,aa(nat,nat,suc,N),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,K)))) = bit_se4197421643247451524op_bit(int,N,aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),inc(K)))) ).

% drop_bit_Suc_minus_bit1
tff(fact_5822_push__bit__Suc,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A2: A] : bit_se4730199178511100633sh_bit(A,aa(nat,nat,suc,N),A2) = bit_se4730199178511100633sh_bit(A,N,aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(num,A,numeral_numeral(A),bit0(one2)))) ) ).

% push_bit_Suc
tff(fact_5823_push__bit__of__1,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat] : bit_se4730199178511100633sh_bit(A,N,one_one(A)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N) ) ).

% push_bit_of_1
tff(fact_5824_even__push__bit__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A2: A] :
          ( pp(aa(A,bool,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),bit_se4730199178511100633sh_bit(A,N,A2)))
        <=> ( ( N != zero_zero(nat) )
            | pp(aa(A,bool,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A2)) ) ) ) ).

% even_push_bit_iff
tff(fact_5825_push__bit__minus__numeral,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [L: num,K: num] : bit_se4730199178511100633sh_bit(A,aa(num,nat,numeral_numeral(nat),L),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),K))) = bit_se4730199178511100633sh_bit(A,pred_numeral(L),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),bit0(K)))) ) ).

% push_bit_minus_numeral
tff(fact_5826_drop__bit__numeral__minus__bit1,axiom,
    ! [L: num,K: num] : bit_se4197421643247451524op_bit(int,aa(num,nat,numeral_numeral(nat),L),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,K)))) = bit_se4197421643247451524op_bit(int,pred_numeral(L),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),inc(K)))) ).

% drop_bit_numeral_minus_bit1
tff(fact_5827_push__bit__numeral__minus__1,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [N: num] : bit_se4730199178511100633sh_bit(A,aa(num,nat,numeral_numeral(nat),N),aa(A,A,uminus_uminus(A),one_one(A))) = aa(A,A,uminus_uminus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(num,nat,numeral_numeral(nat),N))) ) ).

% push_bit_numeral_minus_1
tff(fact_5828_drop__bit__nat__eq,axiom,
    ! [N: nat,K: int] : bit_se4197421643247451524op_bit(nat,N,nat2(K)) = nat2(bit_se4197421643247451524op_bit(int,N,K)) ).

% drop_bit_nat_eq
tff(fact_5829_drop__bit__int__code_I2_J,axiom,
    ! [N: nat] : bit_se4197421643247451524op_bit(int,aa(nat,nat,suc,N),zero_zero(int)) = zero_zero(int) ).

% drop_bit_int_code(2)
tff(fact_5830_drop__bit__int__code_I1_J,axiom,
    ! [I: int] : bit_se4197421643247451524op_bit(int,zero_zero(nat),I) = I ).

% drop_bit_int_code(1)
tff(fact_5831_take__bit__push__bit,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,N: nat,A2: A] : aa(A,A,bit_se2584673776208193580ke_bit(A,M),bit_se4730199178511100633sh_bit(A,N,A2)) = bit_se4730199178511100633sh_bit(A,N,aa(A,A,bit_se2584673776208193580ke_bit(A,aa(nat,nat,minus_minus(nat,M),N)),A2)) ) ).

% take_bit_push_bit
tff(fact_5832_push__bit__take__bit,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,N: nat,A2: A] : bit_se4730199178511100633sh_bit(A,M,aa(A,A,bit_se2584673776208193580ke_bit(A,N),A2)) = aa(A,A,bit_se2584673776208193580ke_bit(A,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N)),bit_se4730199178511100633sh_bit(A,M,A2)) ) ).

% push_bit_take_bit
tff(fact_5833_push__bit__minus,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [N: nat,A2: A] : bit_se4730199178511100633sh_bit(A,N,aa(A,A,uminus_uminus(A),A2)) = aa(A,A,uminus_uminus(A),bit_se4730199178511100633sh_bit(A,N,A2)) ) ).

% push_bit_minus
tff(fact_5834_push__bit__of__int,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [N: nat,K: int] : bit_se4730199178511100633sh_bit(A,N,aa(int,A,ring_1_of_int(A),K)) = aa(int,A,ring_1_of_int(A),bit_se4730199178511100633sh_bit(int,N,K)) ) ).

% push_bit_of_int
tff(fact_5835_of__nat__push__bit,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,N: nat] : aa(nat,A,semiring_1_of_nat(A),bit_se4730199178511100633sh_bit(nat,M,N)) = bit_se4730199178511100633sh_bit(A,M,aa(nat,A,semiring_1_of_nat(A),N)) ) ).

% of_nat_push_bit
tff(fact_5836_push__bit__of__nat,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,M: nat] : bit_se4730199178511100633sh_bit(A,N,aa(nat,A,semiring_1_of_nat(A),M)) = aa(nat,A,semiring_1_of_nat(A),bit_se4730199178511100633sh_bit(nat,N,M)) ) ).

% push_bit_of_nat
tff(fact_5837_push__bit__add,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A2: A,B2: A] : bit_se4730199178511100633sh_bit(A,N,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),bit_se4730199178511100633sh_bit(A,N,A2)),bit_se4730199178511100633sh_bit(A,N,B2)) ) ).

% push_bit_add
tff(fact_5838_div__push__bit__of__1__eq__drop__bit,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,N: nat] : aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),bit_se4730199178511100633sh_bit(A,N,one_one(A))) = bit_se4197421643247451524op_bit(A,N,A2) ) ).

% div_push_bit_of_1_eq_drop_bit
tff(fact_5839_bits__ident,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),bit_se4730199178511100633sh_bit(A,N,bit_se4197421643247451524op_bit(A,N,A2))),aa(A,A,bit_se2584673776208193580ke_bit(A,N),A2)) = A2 ) ).

% bits_ident
tff(fact_5840_set__bit__eq__or,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A2: A] : aa(A,A,aa(nat,fun(A,A),bit_se5668285175392031749et_bit(A),N),A2) = aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A2),bit_se4730199178511100633sh_bit(A,N,one_one(A))) ) ).

% set_bit_eq_or
tff(fact_5841_flip__bit__eq__xor,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A2: A] : bit_se8732182000553998342ip_bit(A,N,A2) = aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),A2),bit_se4730199178511100633sh_bit(A,N,one_one(A))) ) ).

% flip_bit_eq_xor
tff(fact_5842_push__bit__double,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A2: A] : bit_se4730199178511100633sh_bit(A,N,aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(num,A,numeral_numeral(A),bit0(one2)))) = aa(A,A,aa(A,fun(A,A),times_times(A),bit_se4730199178511100633sh_bit(A,N,A2)),aa(num,A,numeral_numeral(A),bit0(one2))) ) ).

% push_bit_double
tff(fact_5843_bit__iff__and__push__bit__not__eq__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,N: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,A2),N))
        <=> ( aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A2),bit_se4730199178511100633sh_bit(A,N,one_one(A))) != zero_zero(A) ) ) ) ).

% bit_iff_and_push_bit_not_eq_0
tff(fact_5844_push__bit__mask__eq,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [M: nat,N: nat] : bit_se4730199178511100633sh_bit(A,M,bit_se2239418461657761734s_mask(A,N)) = aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),bit_se2239418461657761734s_mask(A,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),M))),aa(A,A,bit_ri4277139882892585799ns_not(A),bit_se2239418461657761734s_mask(A,M))) ) ).

% push_bit_mask_eq
tff(fact_5845_unset__bit__eq__and__not,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [N: nat,A2: A] : aa(A,A,aa(nat,fun(A,A),bit_se2638667681897837118et_bit(A),N),A2) = aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A2),aa(A,A,bit_ri4277139882892585799ns_not(A),bit_se4730199178511100633sh_bit(A,N,one_one(A)))) ) ).

% unset_bit_eq_and_not
tff(fact_5846_shiftr__integer__conv__div__pow2,axiom,
    ! [N: nat,X: code_integer] : bit_se4197421643247451524op_bit(code_integer,N,X) = aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),divide_divide(code_integer),X),aa(nat,code_integer,aa(code_integer,fun(nat,code_integer),power_power(code_integer),aa(num,code_integer,numeral_numeral(code_integer),bit0(one2))),N)) ).

% shiftr_integer_conv_div_pow2
tff(fact_5847_fun__cong__unused__0,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( zero(B)
     => ! [F2: fun(fun(A,B),C),G: C] :
          ( ! [X3: fun(A,B)] : aa(fun(A,B),C,F2,X3) = G
         => ( aa(fun(A,B),C,F2,aTP_Lamp_om(A,B)) = G ) ) ) ).

% fun_cong_unused_0
tff(fact_5848_pred__subset__eq2,axiom,
    ! [B: $tType,A: $tType,R: set(product_prod(A,B)),S: set(product_prod(A,B))] :
      ( pp(aa(fun(A,fun(B,bool)),bool,ord_less_eq(fun(A,fun(B,bool)),aTP_Lamp_on(set(product_prod(A,B)),fun(A,fun(B,bool)),R)),aTP_Lamp_on(set(product_prod(A,B)),fun(A,fun(B,bool)),S)))
    <=> pp(aa(set(product_prod(A,B)),bool,ord_less_eq(set(product_prod(A,B)),R),S)) ) ).

% pred_subset_eq2
tff(fact_5849_pred__equals__eq2,axiom,
    ! [B: $tType,A: $tType,R: set(product_prod(A,B)),S: set(product_prod(A,B))] :
      ( ! [X2: A,Xa4: B] :
          ( pp(aa(set(product_prod(A,B)),bool,aa(product_prod(A,B),fun(set(product_prod(A,B)),bool),member(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X2),Xa4)),R))
        <=> pp(aa(set(product_prod(A,B)),bool,aa(product_prod(A,B),fun(set(product_prod(A,B)),bool),member(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X2),Xa4)),S)) )
    <=> ( R = S ) ) ).

% pred_equals_eq2
tff(fact_5850_bin__rest__code,axiom,
    ! [I: int] : aa(int,int,aa(int,fun(int,int),divide_divide(int),I),aa(num,int,numeral_numeral(int),bit0(one2))) = bit_se4197421643247451524op_bit(int,one_one(nat),I) ).

% bin_rest_code
tff(fact_5851_drop__bit__int__def,axiom,
    ! [N: nat,K: int] : bit_se4197421643247451524op_bit(int,N,K) = aa(int,int,aa(int,fun(int,int),divide_divide(int),K),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),N)) ).

% drop_bit_int_def
tff(fact_5852_push__bit__eq__mult,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A2: A] : bit_se4730199178511100633sh_bit(A,N,A2) = aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N)) ) ).

% push_bit_eq_mult
tff(fact_5853_exp__dvdE,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A2: A] :
          ( pp(aa(A,bool,dvd_dvd(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N)),A2))
         => ~ ! [B3: A] : A2 != bit_se4730199178511100633sh_bit(A,N,B3) ) ) ).

% exp_dvdE
tff(fact_5854_drop__bit__nat__def,axiom,
    ! [N: nat,M: nat] : bit_se4197421643247451524op_bit(nat,N,M) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)) ).

% drop_bit_nat_def
tff(fact_5855_slice__eq__mask,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [N: nat,M: nat,A2: A] : bit_se4730199178511100633sh_bit(A,N,aa(A,A,bit_se2584673776208193580ke_bit(A,M),bit_se4197421643247451524op_bit(A,N,A2))) = aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A2),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),bit_se2239418461657761734s_mask(A,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N))),aa(A,A,bit_ri4277139882892585799ns_not(A),bit_se2239418461657761734s_mask(A,N)))) ) ).

% slice_eq_mask
tff(fact_5856_word__and__mask__or__conv__and__mask,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: word(A),Index: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(word(A),N),Index))
         => ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se1065995026697491101ons_or(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),N),bit_se2239418461657761734s_mask(word(A),Index))),bit_se4730199178511100633sh_bit(word(A),Index,one_one(word(A)))) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),N),bit_se2239418461657761734s_mask(word(A),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Index),one_one(nat)))) ) ) ) ).

% word_and_mask_or_conv_and_mask
tff(fact_5857_signed__take__bit__code,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [N: nat,A2: A] : aa(A,A,bit_ri4674362597316999326ke_bit(A,N),A2) = if(A,aa(nat,bool,bit_se5641148757651400278ts_bit(A,aa(A,A,bit_se2584673776208193580ke_bit(A,aa(nat,nat,suc,N)),A2)),N),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,bit_se2584673776208193580ke_bit(A,aa(nat,nat,suc,N)),A2)),bit_se4730199178511100633sh_bit(A,aa(nat,nat,suc,N),aa(A,A,uminus_uminus(A),one_one(A)))),aa(A,A,bit_se2584673776208193580ke_bit(A,aa(nat,nat,suc,N)),A2)) ) ).

% signed_take_bit_code
tff(fact_5858_bot__empty__eq2,axiom,
    ! [B: $tType,A: $tType,X4: A,Xa: B] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),bot_bot(fun(A,fun(B,bool))),X4),Xa))
    <=> pp(aa(set(product_prod(A,B)),bool,aa(product_prod(A,B),fun(set(product_prod(A,B)),bool),member(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X4),Xa)),bot_bot(set(product_prod(A,B))))) ) ).

% bot_empty_eq2
tff(fact_5859_case__prod__Pair__iden,axiom,
    ! [B: $tType,A: $tType,P2: product_prod(A,B)] : aa(product_prod(A,B),product_prod(A,B),product_case_prod(A,B,product_prod(A,B),product_Pair(A,B)),P2) = P2 ).

% case_prod_Pair_iden
tff(fact_5860_set__bits__aux__Suc,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [F2: fun(nat,bool),N: nat,W: word(A)] : code_T2661198915054445665ts_aux(A,F2,aa(nat,nat,suc,N),W) = code_T2661198915054445665ts_aux(A,F2,N,aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se1065995026697491101ons_or(word(A)),bit_se4730199178511100633sh_bit(word(A),one_one(nat),W)),if(word(A),aa(nat,bool,F2,N),one_one(word(A)),zero_zero(word(A))))) ) ).

% set_bits_aux_Suc
tff(fact_5861_set__bits__aux__rec,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: nat,F2: fun(nat,bool),W: word(A)] :
          ( ( ( N = zero_zero(nat) )
           => ( code_T2661198915054445665ts_aux(A,F2,N,W) = W ) )
          & ( ( N != zero_zero(nat) )
           => ( code_T2661198915054445665ts_aux(A,F2,N,W) = code_T2661198915054445665ts_aux(A,F2,aa(nat,nat,minus_minus(nat,N),one_one(nat)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se1065995026697491101ons_or(word(A)),bit_se4730199178511100633sh_bit(word(A),one_one(nat),W)),if(word(A),aa(nat,bool,F2,aa(nat,nat,minus_minus(nat,N),one_one(nat))),one_one(word(A)),zero_zero(word(A))))) ) ) ) ) ).

% set_bits_aux_rec
tff(fact_5862_test__bit__split__eq,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) = aa(word(B),product_prod(word(A),word(B)),aa(word(A),fun(word(B),product_prod(word(A),word(B))),product_Pair(word(A),word(B)),A2),B2) )
        <=> ( ! [N5: nat] :
                ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(word(B),B2),N5))
              <=> ( pp(aa(nat,bool,ord_less(nat,N5),aa(word(B),nat,size_size(word(B)),B2)))
                  & pp(aa(nat,bool,bit_se5641148757651400278ts_bit(word(C),C2),N5)) ) )
            & ! [M6: nat] :
                ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(word(A),A2),M6))
              <=> ( pp(aa(nat,bool,ord_less(nat,M6),aa(word(A),nat,size_size(word(A)),A2)))
                  & pp(aa(nat,bool,bit_se5641148757651400278ts_bit(word(C),C2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M6),aa(word(B),nat,size_size(word(B)),B2)))) ) ) ) ) ) ).

% test_bit_split_eq
tff(fact_5863_set__bits__aux__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [F2: fun(nat,bool),W: word(A)] : code_T2661198915054445665ts_aux(A,F2,zero_zero(nat),W) = W ) ).

% set_bits_aux_0
tff(fact_5864_push__bit__nonnegative__int__iff,axiom,
    ! [N: nat,K: int] :
      ( pp(aa(int,bool,ord_less_eq(int,zero_zero(int)),bit_se4730199178511100633sh_bit(int,N,K)))
    <=> pp(aa(int,bool,ord_less_eq(int,zero_zero(int)),K)) ) ).

% push_bit_nonnegative_int_iff
tff(fact_5865_push__bit__negative__int__iff,axiom,
    ! [N: nat,K: int] :
      ( pp(aa(int,bool,ord_less(int,bit_se4730199178511100633sh_bit(int,N,K)),zero_zero(int)))
    <=> pp(aa(int,bool,ord_less(int,K),zero_zero(int))) ) ).

% push_bit_negative_int_iff
tff(fact_5866_push__bit__of__Suc__0,axiom,
    ! [N: nat] : bit_se4730199178511100633sh_bit(nat,N,aa(nat,nat,suc,zero_zero(nat))) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N) ).

% push_bit_of_Suc_0
tff(fact_5867_drop__bit__push__bit__int,axiom,
    ! [M: nat,N: nat,K: int] : bit_se4197421643247451524op_bit(int,M,bit_se4730199178511100633sh_bit(int,N,K)) = bit_se4197421643247451524op_bit(int,aa(nat,nat,minus_minus(nat,M),N),bit_se4730199178511100633sh_bit(int,aa(nat,nat,minus_minus(nat,N),M),K)) ).

% drop_bit_push_bit_int
tff(fact_5868_push__bit__int__code_I1_J,axiom,
    ! [I: int] : bit_se4730199178511100633sh_bit(int,zero_zero(nat),I) = I ).

% push_bit_int_code(1)
tff(fact_5869_push__bit__nat__eq,axiom,
    ! [N: nat,K: int] : bit_se4730199178511100633sh_bit(nat,N,nat2(K)) = nat2(bit_se4730199178511100633sh_bit(int,N,K)) ).

% push_bit_nat_eq
tff(fact_5870_flip__bit__nat__def,axiom,
    ! [M: nat,N: nat] : bit_se8732182000553998342ip_bit(nat,M,N) = aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),N),bit_se4730199178511100633sh_bit(nat,M,one_one(nat))) ).

% flip_bit_nat_def
tff(fact_5871_set__bit__nat__def,axiom,
    ! [M: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5668285175392031749et_bit(nat),M),N) = aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),N),bit_se4730199178511100633sh_bit(nat,M,one_one(nat))) ).

% set_bit_nat_def
tff(fact_5872_Bit__integer__code_I1_J,axiom,
    ! [I: code_integer] : bits_Bit_integer(I,fFalse) = bit_se4730199178511100633sh_bit(code_integer,one_one(nat),I) ).

% Bit_integer_code(1)
tff(fact_5873_bit__push__bit__iff__int,axiom,
    ! [M: nat,K: int,N: nat] :
      ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(int,bit_se4730199178511100633sh_bit(int,M,K)),N))
    <=> ( pp(aa(nat,bool,ord_less_eq(nat,M),N))
        & pp(aa(nat,bool,bit_se5641148757651400278ts_bit(int,K),aa(nat,nat,minus_minus(nat,N),M))) ) ) ).

% bit_push_bit_iff_int
tff(fact_5874_Bit__Operations_Oset__bit__int__def,axiom,
    ! [N: nat,K: int] : aa(int,int,aa(nat,fun(int,int),bit_se5668285175392031749et_bit(int),N),K) = aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),K),bit_se4730199178511100633sh_bit(int,N,one_one(int))) ).

% Bit_Operations.set_bit_int_def
tff(fact_5875_bit__push__bit__iff__nat,axiom,
    ! [M: nat,Q2: nat,N: nat] :
      ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(nat,bit_se4730199178511100633sh_bit(nat,M,Q2)),N))
    <=> ( pp(aa(nat,bool,ord_less_eq(nat,M),N))
        & pp(aa(nat,bool,bit_se5641148757651400278ts_bit(nat,Q2),aa(nat,nat,minus_minus(nat,N),M))) ) ) ).

% bit_push_bit_iff_nat
tff(fact_5876_flip__bit__int__def,axiom,
    ! [N: nat,K: int] : bit_se8732182000553998342ip_bit(int,N,K) = aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),K),bit_se4730199178511100633sh_bit(int,N,one_one(int))) ).

% flip_bit_int_def
tff(fact_5877_shiftl__integer__conv__mult__pow2,axiom,
    ! [N: nat,X: code_integer] : bit_se4730199178511100633sh_bit(code_integer,N,X) = aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),times_times(code_integer),X),aa(nat,code_integer,aa(code_integer,fun(nat,code_integer),power_power(code_integer),aa(num,code_integer,numeral_numeral(code_integer),bit0(one2))),N)) ).

% shiftl_integer_conv_mult_pow2
tff(fact_5878_unset__bit__int__def,axiom,
    ! [N: nat,K: int] : aa(int,int,aa(nat,fun(int,int),bit_se2638667681897837118et_bit(int),N),K) = aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),K),aa(int,int,bit_ri4277139882892585799ns_not(int),bit_se4730199178511100633sh_bit(int,N,one_one(int)))) ).

% unset_bit_int_def
tff(fact_5879_push__bit__int__def,axiom,
    ! [N: nat,K: int] : bit_se4730199178511100633sh_bit(int,N,K) = aa(int,int,aa(int,fun(int,int),times_times(int),K),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),N)) ).

% push_bit_int_def
tff(fact_5880_Bit__integer__code_I2_J,axiom,
    ! [I: code_integer] : bits_Bit_integer(I,fTrue) = aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),bit_se4730199178511100633sh_bit(code_integer,one_one(nat),I)),one_one(code_integer)) ).

% Bit_integer_code(2)
tff(fact_5881_push__bit__nat__def,axiom,
    ! [N: nat,M: nat] : bit_se4730199178511100633sh_bit(nat,N,M) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)) ).

% push_bit_nat_def
tff(fact_5882_push__bit__minus__one,axiom,
    ! [N: nat] : bit_se4730199178511100633sh_bit(int,N,aa(int,int,uminus_uminus(int),one_one(int))) = aa(int,int,uminus_uminus(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),N)) ).

% push_bit_minus_one
tff(fact_5883_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) = aa(word(B),product_prod(word(A),word(B)),aa(word(A),fun(word(B),product_prod(word(A),word(B))),product_Pair(word(A),word(B)),A2),B2) )
         => ( ! [N11: nat] :
                ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(word(B),B2),N11))
              <=> ( pp(aa(nat,bool,ord_less(nat,N11),aa(word(B),nat,size_size(word(B)),B2)))
                  & pp(aa(nat,bool,bit_se5641148757651400278ts_bit(word(C),C2),N11)) ) )
            & ! [M4: nat] :
                ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(word(A),A2),M4))
              <=> ( pp(aa(nat,bool,ord_less(nat,M4),aa(word(A),nat,size_size(word(A)),A2)))
                  & pp(aa(nat,bool,bit_se5641148757651400278ts_bit(word(C),C2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M4),aa(word(B),nat,size_size(word(B)),B2)))) ) ) ) ) ) ).

% test_bit_split
tff(fact_5884_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) = aa(word(B),product_prod(word(A),word(B)),aa(word(A),fun(word(B),product_prod(word(A),word(B))),product_Pair(word(A),word(B)),A2),B2) )
         => ! [N11: nat,M4: nat] :
              ( ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(word(B),B2),N11))
              <=> ( pp(aa(nat,bool,ord_less(nat,N11),aa(word(B),nat,size_size(word(B)),B2)))
                  & pp(aa(nat,bool,bit_se5641148757651400278ts_bit(word(C),C2),N11)) ) )
              & ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(word(A),A2),M4))
              <=> ( pp(aa(nat,bool,ord_less(nat,M4),aa(word(A),nat,size_size(word(A)),A2)))
                  & pp(aa(nat,bool,bit_se5641148757651400278ts_bit(word(C),C2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M4),aa(word(B),nat,size_size(word(B)),B2)))) ) ) ) ) ) ).

% test_bit_split'
tff(fact_5885_bin__rest__integer_Oabs__eq,axiom,
    ! [X: int] : bits_b2549910563261871055nteger(code_integer_of_int(X)) = code_integer_of_int(aa(int,int,aa(int,fun(int,int),divide_divide(int),X),aa(num,int,numeral_numeral(int),bit0(one2)))) ).

% bin_rest_integer.abs_eq
tff(fact_5886_the__elem__def,axiom,
    ! [A: $tType,X6: set(A)] : the_elem(A,X6) = the(A,aTP_Lamp_oo(set(A),fun(A,bool),X6)) ).

% the_elem_def
tff(fact_5887_concat__bit__Suc,axiom,
    ! [N: nat,K: int,L: int] : aa(int,int,bit_concat_bit(aa(nat,nat,suc,N),K),L) = aa(int,int,aa(int,fun(int,int),plus_plus(int),modulo_modulo(int,K,aa(num,int,numeral_numeral(int),bit0(one2)))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(int,int,bit_concat_bit(N,aa(int,int,aa(int,fun(int,int),divide_divide(int),K),aa(num,int,numeral_numeral(int),bit0(one2)))),L))) ).

% concat_bit_Suc
tff(fact_5888_concat__bit__0,axiom,
    ! [K: int,L: int] : aa(int,int,bit_concat_bit(zero_zero(nat),K),L) = L ).

% concat_bit_0
tff(fact_5889_concat__bit__of__zero__2,axiom,
    ! [N: nat,K: int] : aa(int,int,bit_concat_bit(N,K),zero_zero(int)) = aa(int,int,bit_se2584673776208193580ke_bit(int,N),K) ).

% concat_bit_of_zero_2
tff(fact_5890_concat__bit__nonnegative__iff,axiom,
    ! [N: nat,K: int,L: int] :
      ( pp(aa(int,bool,ord_less_eq(int,zero_zero(int)),aa(int,int,bit_concat_bit(N,K),L)))
    <=> pp(aa(int,bool,ord_less_eq(int,zero_zero(int)),L)) ) ).

% concat_bit_nonnegative_iff
tff(fact_5891_concat__bit__negative__iff,axiom,
    ! [N: nat,K: int,L: int] :
      ( pp(aa(int,bool,ord_less(int,aa(int,int,bit_concat_bit(N,K),L)),zero_zero(int)))
    <=> pp(aa(int,bool,ord_less(int,L),zero_zero(int))) ) ).

% concat_bit_negative_iff
tff(fact_5892_concat__bit__of__zero__1,axiom,
    ! [N: nat,L: int] : aa(int,int,bit_concat_bit(N,zero_zero(int)),L) = bit_se4730199178511100633sh_bit(int,N,L) ).

% concat_bit_of_zero_1
tff(fact_5893_concat__bit__take__bit__eq,axiom,
    ! [N: nat,B2: int] : bit_concat_bit(N,aa(int,int,bit_se2584673776208193580ke_bit(int,N),B2)) = bit_concat_bit(N,B2) ).

% concat_bit_take_bit_eq
tff(fact_5894_concat__bit__eq__iff,axiom,
    ! [N: nat,K: int,L: int,R2: int,S3: int] :
      ( ( aa(int,int,bit_concat_bit(N,K),L) = aa(int,int,bit_concat_bit(N,R2),S3) )
    <=> ( ( aa(int,int,bit_se2584673776208193580ke_bit(int,N),K) = aa(int,int,bit_se2584673776208193580ke_bit(int,N),R2) )
        & ( L = S3 ) ) ) ).

% concat_bit_eq_iff
tff(fact_5895_concat__bit__assoc,axiom,
    ! [N: nat,K: int,M: nat,L: int,R2: int] : aa(int,int,bit_concat_bit(N,K),aa(int,int,bit_concat_bit(M,L),R2)) = aa(int,int,bit_concat_bit(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N),aa(int,int,bit_concat_bit(N,K),L)),R2) ).

% concat_bit_assoc
tff(fact_5896_concat__bit__eq,axiom,
    ! [N: nat,K: int,L: int] : aa(int,int,bit_concat_bit(N,K),L) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,bit_se2584673776208193580ke_bit(int,N),K)),bit_se4730199178511100633sh_bit(int,N,L)) ).

% concat_bit_eq
tff(fact_5897_concat__bit__def,axiom,
    ! [N: nat,K: int,L: int] : aa(int,int,bit_concat_bit(N,K),L) = aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(int,int,bit_se2584673776208193580ke_bit(int,N),K)),bit_se4730199178511100633sh_bit(int,N,L)) ).

% concat_bit_def
tff(fact_5898_bit__concat__bit__iff,axiom,
    ! [M: nat,K: int,L: int,N: nat] :
      ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(int,aa(int,int,bit_concat_bit(M,K),L)),N))
    <=> ( ( pp(aa(nat,bool,ord_less(nat,N),M))
          & pp(aa(nat,bool,bit_se5641148757651400278ts_bit(int,K),N)) )
        | ( pp(aa(nat,bool,ord_less_eq(nat,M),N))
          & pp(aa(nat,bool,bit_se5641148757651400278ts_bit(int,L),aa(nat,nat,minus_minus(nat,N),M))) ) ) ) ).

% bit_concat_bit_iff
tff(fact_5899_signed__take__bit__eq__concat__bit,axiom,
    ! [N: nat,K: int] : aa(int,int,bit_ri4674362597316999326ke_bit(int,N),K) = aa(int,int,bit_concat_bit(N,K),aa(int,int,uminus_uminus(int),aa(bool,int,zero_neq_one_of_bool(int),aa(nat,bool,bit_se5641148757651400278ts_bit(int,K),N)))) ).

% signed_take_bit_eq_concat_bit
tff(fact_5900_bin__rest__integer__code,axiom,
    ! [I: code_integer] : bits_b2549910563261871055nteger(I) = aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),divide_divide(code_integer),I),aa(num,code_integer,numeral_numeral(code_integer),bit0(one2))) ).

% bin_rest_integer_code
tff(fact_5901_bitXOR__integer__unfold,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( ( X = zero_zero(code_integer) )
       => ( aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),bit_se5824344971392196577ns_xor(code_integer),X),Y) = Y ) )
      & ( ( X != zero_zero(code_integer) )
       => ( ( ( X = aa(code_integer,code_integer,uminus_uminus(code_integer),one_one(code_integer)) )
           => ( aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),bit_se5824344971392196577ns_xor(code_integer),X),Y) = aa(code_integer,code_integer,bit_ri4277139882892585799ns_not(code_integer),Y) ) )
          & ( ( X != aa(code_integer,code_integer,uminus_uminus(code_integer),one_one(code_integer)) )
           => ( aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),bit_se5824344971392196577ns_xor(code_integer),X),Y) = bits_Bit_integer(aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),bit_se5824344971392196577ns_xor(code_integer),bits_b2549910563261871055nteger(X)),bits_b2549910563261871055nteger(Y)),aa(bool,bool,fequal(bool,aa(bool,bool,fNot,bits_b8758750999018896077nteger(X))),bits_b8758750999018896077nteger(Y))) ) ) ) ) ) ).

% bitXOR_integer_unfold
tff(fact_5902_bitOR__integer__unfold,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( ( X = zero_zero(code_integer) )
       => ( aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),bit_se1065995026697491101ons_or(code_integer),X),Y) = Y ) )
      & ( ( X != zero_zero(code_integer) )
       => ( ( ( X = aa(code_integer,code_integer,uminus_uminus(code_integer),one_one(code_integer)) )
           => ( aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),bit_se1065995026697491101ons_or(code_integer),X),Y) = aa(code_integer,code_integer,uminus_uminus(code_integer),one_one(code_integer)) ) )
          & ( ( X != aa(code_integer,code_integer,uminus_uminus(code_integer),one_one(code_integer)) )
           => ( aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),bit_se1065995026697491101ons_or(code_integer),X),Y) = bits_Bit_integer(aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),bit_se1065995026697491101ons_or(code_integer),bits_b2549910563261871055nteger(X)),bits_b2549910563261871055nteger(Y)),fdisj(bits_b8758750999018896077nteger(X),bits_b8758750999018896077nteger(Y))) ) ) ) ) ) ).

% bitOR_integer_unfold
tff(fact_5903_bitAND__integer__unfold,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( ( X = zero_zero(code_integer) )
       => ( aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),bit_se5824344872417868541ns_and(code_integer),X),Y) = zero_zero(code_integer) ) )
      & ( ( X != zero_zero(code_integer) )
       => ( ( ( X = aa(code_integer,code_integer,uminus_uminus(code_integer),one_one(code_integer)) )
           => ( aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),bit_se5824344872417868541ns_and(code_integer),X),Y) = Y ) )
          & ( ( X != aa(code_integer,code_integer,uminus_uminus(code_integer),one_one(code_integer)) )
           => ( aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),bit_se5824344872417868541ns_and(code_integer),X),Y) = bits_Bit_integer(aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),bit_se5824344872417868541ns_and(code_integer),bits_b2549910563261871055nteger(X)),bits_b2549910563261871055nteger(Y)),fconj(bits_b8758750999018896077nteger(X),bits_b8758750999018896077nteger(Y))) ) ) ) ) ) ).

% bitAND_integer_unfold
tff(fact_5904_bitval__bin__last__integer,axiom,
    ! [I: code_integer] : aa(bool,code_integer,zero_neq_one_of_bool(code_integer),bits_b8758750999018896077nteger(I)) = aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),bit_se5824344872417868541ns_and(code_integer),I),one_one(code_integer)) ).

% bitval_bin_last_integer
tff(fact_5905_bin__last__integer__code,axiom,
    ! [I: code_integer] :
      ( pp(bits_b8758750999018896077nteger(I))
    <=> ( aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),bit_se5824344872417868541ns_and(code_integer),I),one_one(code_integer)) != zero_zero(code_integer) ) ) ).

% bin_last_integer_code
tff(fact_5906_bin__last__integer__nbe,axiom,
    ! [I: code_integer] :
      ( pp(bits_b8758750999018896077nteger(I))
    <=> ( modulo_modulo(code_integer,I,aa(num,code_integer,numeral_numeral(code_integer),bit0(one2))) != zero_zero(code_integer) ) ) ).

% bin_last_integer_nbe
tff(fact_5907_bin__last__integer_Oabs__eq,axiom,
    ! [X: int] :
      ( pp(bits_b8758750999018896077nteger(code_integer_of_int(X)))
    <=> ~ pp(aa(int,bool,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),X)) ) ).

% bin_last_integer.abs_eq
tff(fact_5908_set__bit__integer__conv__masks,axiom,
    ! [B2: bool,X: code_integer,I: nat] :
      ( ( pp(B2)
       => ( generi7602027413899671122et_bit(code_integer,X,I,B2) = aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),bit_se1065995026697491101ons_or(code_integer),X),bit_se4730199178511100633sh_bit(code_integer,I,one_one(code_integer))) ) )
      & ( ~ pp(B2)
       => ( generi7602027413899671122et_bit(code_integer,X,I,B2) = aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),bit_se5824344872417868541ns_and(code_integer),X),aa(code_integer,code_integer,bit_ri4277139882892585799ns_not(code_integer),bit_se4730199178511100633sh_bit(code_integer,I,one_one(code_integer)))) ) ) ) ).

% set_bit_integer_conv_masks
tff(fact_5909_Uint32__code,axiom,
    ! [I: code_integer] : aa(code_integer,uint32,uint322,I) = if(uint32,aa(nat,bool,bit_se5641148757651400278ts_bit(code_integer,aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),bit_se5824344872417868541ns_and(code_integer),I),aa(num,code_integer,numeral_numeral(code_integer),aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,one2)))))))))))))))))))))))))))))))))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,one2)))))),uint32_signed(aa(code_integer,code_integer,minus_minus(code_integer,aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),bit_se5824344872417868541ns_and(code_integer),I),aa(num,code_integer,numeral_numeral(code_integer),aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,one2)))))))))))))))))))))))))))))))))),aa(num,code_integer,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(bit0(one2))))))))))))))))))))))))))))))))))),uint32_signed(aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),bit_se5824344872417868541ns_and(code_integer),I),aa(num,code_integer,numeral_numeral(code_integer),aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,one2))))))))))))))))))))))))))))))))))) ).

% Uint32_code
tff(fact_5910_word__cat__split__size,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ( type_len(C)
        & type_len(A)
        & type_len(B) )
     => ! [T: word(A),U: word(B),V: word(C)] :
          ( pp(aa(nat,bool,ord_less_eq(nat,aa(word(A),nat,size_size(word(A)),T)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(word(B),nat,size_size(word(B)),U)),aa(word(C),nat,size_size(word(C)),V))))
         => ( ( aa(word(C),product_prod(word(B),word(C)),aa(word(B),fun(word(C),product_prod(word(B),word(C))),product_Pair(word(B),word(C)),U),V) = word_split(A,B,C,T) )
           => ( T = word_cat(B,C,A,U,V) ) ) ) ) ).

% word_cat_split_size
tff(fact_5911_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)] :
          ( pp(aa(nat,bool,ord_less_eq(nat,aa(word(A),nat,size_size(word(A)),W)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(word(B),nat,size_size(word(B)),U)),aa(word(C),nat,size_size(word(C)),V))))
         => ( ( word_split(A,B,C,W) = aa(word(C),product_prod(word(B),word(C)),aa(word(B),fun(word(C),product_prod(word(B),word(C))),product_Pair(word(B),word(C)),U),V) )
           => ( word_cat(B,C,A,U,V) = W ) ) ) ) ).

% word_cat_split_alt
tff(fact_5912_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) )
         => ( pp(aa(nat,bool,ord_less_eq(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(word(B),nat,size_size(word(B)),U)),aa(word(C),nat,size_size(word(C)),V))),aa(word(A),nat,size_size(word(A)),W)))
           => ( word_split(A,B,C,W) = aa(word(C),product_prod(word(B),word(C)),aa(word(B),fun(word(C),product_prod(word(B),word(C))),product_Pair(word(B),word(C)),U),V) ) ) ) ) ).

% word_split_cat_alt
tff(fact_5913_bit__set__bit__iff__2n,axiom,
    ! [A: $tType] :
      ( generic_set_set_bit(A)
     => ! [A2: A,M: nat,B2: bool,N: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,generi7602027413899671122et_bit(A,A2,M,B2)),N))
        <=> ( ( ( M = N )
             => pp(B2) )
            & ( ( M != N )
             => pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,A2),N)) )
            & ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N) != zero_zero(A) ) ) ) ) ).

% bit_set_bit_iff_2n
tff(fact_5914_Uint32__signed__def,axiom,
    ! [I: code_integer] :
      ( ( ( pp(aa(code_integer,bool,ord_less(code_integer,I),aa(code_integer,code_integer,uminus_uminus(code_integer),aa(num,code_integer,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)))))))))))))))))))))))))))))))))))
          | pp(aa(code_integer,bool,ord_less_eq(code_integer,aa(num,code_integer,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))))))))))))))))))))))))))))))))),I)) )
       => ( uint32_signed(I) = aa(code_integer,uint32,aa(fun(code_integer,uint32),fun(code_integer,uint32),undefined(fun(fun(code_integer,uint32),fun(code_integer,uint32))),uint322),I) ) )
      & ( ~ ( pp(aa(code_integer,bool,ord_less(code_integer,I),aa(code_integer,code_integer,uminus_uminus(code_integer),aa(num,code_integer,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)))))))))))))))))))))))))))))))))))
            | pp(aa(code_integer,bool,ord_less_eq(code_integer,aa(num,code_integer,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))))))))))))))))))))))))))))))))),I)) )
       => ( uint32_signed(I) = aa(code_integer,uint32,uint322,I) ) ) ) ).

% Uint32_signed_def
tff(fact_5915_cat__slices,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(A)
        & type_len(C) )
     => ! [A2: word(A),N: nat,C2: word(B),B2: word(C)] :
          ( ( A2 = slice2(B,A,N,C2) )
         => ( ( B2 = slice2(B,C,zero_zero(nat),C2) )
           => ( ( N = aa(word(C),nat,size_size(word(C)),B2) )
             => ( pp(aa(nat,bool,ord_less_eq(nat,aa(word(B),nat,size_size(word(B)),C2)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(word(A),nat,size_size(word(A)),A2)),aa(word(C),nat,size_size(word(C)),B2))))
               => ( word_cat(A,C,B,A2,B2) = C2 ) ) ) ) ) ) ).

% cat_slices
tff(fact_5916_slice__0,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [N: nat] : slice2(B,A,N,zero_zero(word(B))) = zero_zero(word(A)) ) ).

% slice_0
tff(fact_5917_set__bit__beyond,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),N: nat,B2: bool] :
          ( pp(aa(nat,bool,ord_less_eq(nat,aa(word(A),nat,size_size(word(A)),X)),N))
         => ( generi7602027413899671122et_bit(word(A),X,N,B2) = X ) ) ) ).

% set_bit_beyond
tff(fact_5918_slice__id,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [T: word(A)] : slice2(A,A,zero_zero(nat),T) = T ) ).

% slice_id
tff(fact_5919_word__set__nth__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),N: nat,B2: bool] :
          ( ( generi7602027413899671122et_bit(word(A),W,N,B2) = W )
        <=> ( ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(word(A),W),N))
            <=> pp(B2) )
            | pp(aa(nat,bool,ord_less_eq(nat,aa(word(A),nat,size_size(word(A)),W)),N)) ) ) ) ).

% word_set_nth_iff
tff(fact_5920_ucast__slice,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(A) )
     => ! [W: word(B)] : aa(word(B),word(A),semiring_1_unsigned(B,word(A)),W) = slice2(B,A,zero_zero(nat),W) ) ).

% ucast_slice
tff(fact_5921_slice__cat2,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [A2: word(B),T: word(A)] : slice2(A,A,zero_zero(nat),word_cat(B,A,A,A2,T)) = T ) ).

% slice_cat2
tff(fact_5922_one__bit__pow,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: nat] : generi7602027413899671122et_bit(word(A),zero_zero(word(A)),N,fTrue) = aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),N) ) ).

% one_bit_pow
tff(fact_5923_int__set__bit__True__conv__OR,axiom,
    ! [I: int,N: nat] : generi7602027413899671122et_bit(int,I,N,fTrue) = aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),I),bit_se4730199178511100633sh_bit(int,N,one_one(int))) ).

% int_set_bit_True_conv_OR
tff(fact_5924_int__set__bit__False__conv__NAND,axiom,
    ! [I: int,N: nat] : generi7602027413899671122et_bit(int,I,N,fFalse) = aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),I),aa(int,int,bit_ri4277139882892585799ns_not(int),bit_se4730199178511100633sh_bit(int,N,one_one(int)))) ).

% int_set_bit_False_conv_NAND
tff(fact_5925_slice__cat1,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(A)
        & type_len(C) )
     => ! [A2: word(B),B2: word(C)] :
          ( pp(aa(nat,bool,ord_less_eq(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(word(B),nat,size_size(word(B)),A2)),aa(word(C),nat,size_size(word(C)),B2))),aa(word(A),nat,size_size(word(A)),word_cat(B,C,A,A2,B2))))
         => ( slice2(A,B,aa(word(C),nat,size_size(word(C)),B2),word_cat(B,C,A,A2,B2)) = A2 ) ) ) ).

% slice_cat1
tff(fact_5926_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) = aa(word(B),product_prod(word(A),word(B)),aa(word(A),fun(word(B),product_prod(word(A),word(B))),product_Pair(word(A),word(B)),U),V) )
         => ( ( U = slice2(C,A,aa(word(B),nat,size_size(word(B)),V),W) )
            & ( V = slice2(C,B,zero_zero(nat),W) ) ) ) ) ).

% split_slices
tff(fact_5927_int__set__bit__conv__ops,axiom,
    ! [B2: bool,I: int,N: nat] :
      ( ( pp(B2)
       => ( generi7602027413899671122et_bit(int,I,N,B2) = aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),I),bit_se4730199178511100633sh_bit(int,N,one_one(int))) ) )
      & ( ~ pp(B2)
       => ( generi7602027413899671122et_bit(int,I,N,B2) = aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),I),aa(int,int,bit_ri4277139882892585799ns_not(int),bit_se4730199178511100633sh_bit(int,N,one_one(int)))) ) ) ) ).

% int_set_bit_conv_ops
tff(fact_5928_sum__diff1_H__aux,axiom,
    ! [B: $tType,A: $tType] :
      ( ab_group_add(B)
     => ! [F3: set(A),I5: set(A),F2: fun(A,B),I: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),F3))
         => ( pp(aa(set(A),bool,ord_less_eq(set(A),collect(A,aa(fun(A,B),fun(A,bool),aTP_Lamp_op(set(A),fun(fun(A,B),fun(A,bool)),I5),F2))),F3))
           => ( ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),I),I5))
               => ( groups1027152243600224163dd_sum(A,B,F2,aa(set(A),set(A),minus_minus(set(A),I5),insert(A,I,bot_bot(set(A))))) = aa(B,B,minus_minus(B,groups1027152243600224163dd_sum(A,B,F2,I5)),aa(A,B,F2,I)) ) )
              & ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),I),I5))
               => ( groups1027152243600224163dd_sum(A,B,F2,aa(set(A),set(A),minus_minus(set(A),I5),insert(A,I,bot_bot(set(A))))) = groups1027152243600224163dd_sum(A,B,F2,I5) ) ) ) ) ) ) ).

% sum_diff1'_aux
tff(fact_5929_bit__horner__sum__bit__iff,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Bs: list(bool),N: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,groups4207007520872428315er_sum(bool,A,zero_neq_one_of_bool(A),aa(num,A,numeral_numeral(A),bit0(one2)),Bs)),N))
        <=> ( pp(aa(nat,bool,ord_less(nat,N),aa(list(bool),nat,size_size(list(bool)),Bs)))
            & pp(aa(nat,bool,nth(bool,Bs),N)) ) ) ) ).

% bit_horner_sum_bit_iff
tff(fact_5930_sdiv__word__min,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A)] : pp(aa(int,bool,ord_less_eq(int,aa(int,int,uminus_uminus(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,minus_minus(nat,aa(word(A),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
tff(fact_5931_int__sdiv__simps_I2_J,axiom,
    ! [A2: int] : signed7115095781618012415divide(int,A2,zero_zero(int)) = zero_zero(int) ).

% int_sdiv_simps(2)
tff(fact_5932_sdiv__int__0__div,axiom,
    ! [X: int] : signed7115095781618012415divide(int,zero_zero(int),X) = zero_zero(int) ).

% sdiv_int_0_div
tff(fact_5933_sdiv__int__div__0,axiom,
    ! [X: int] : signed7115095781618012415divide(int,X,zero_zero(int)) = zero_zero(int) ).

% sdiv_int_div_0
tff(fact_5934_int__sdiv__simps_I1_J,axiom,
    ! [A2: int] : signed7115095781618012415divide(int,A2,one_one(int)) = A2 ).

% int_sdiv_simps(1)
tff(fact_5935_sum_Oempty_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ! [P2: fun(B,A)] : groups1027152243600224163dd_sum(B,A,P2,bot_bot(set(B))) = zero_zero(A) ) ).

% sum.empty'
tff(fact_5936_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
tff(fact_5937_int__sdiv__simps_I3_J,axiom,
    ! [A2: int] : signed7115095781618012415divide(int,A2,aa(int,int,uminus_uminus(int),one_one(int))) = aa(int,int,uminus_uminus(int),A2) ).

% int_sdiv_simps(3)
tff(fact_5938_sdiv__int__numeral__numeral,axiom,
    ! [M: num,N: num] : signed7115095781618012415divide(int,aa(num,int,numeral_numeral(int),M),aa(num,int,numeral_numeral(int),N)) = aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(num,int,numeral_numeral(int),M)),aa(num,int,numeral_numeral(int),N)) ).

% sdiv_int_numeral_numeral
tff(fact_5939_int__sdiv__negated__is__minus1,axiom,
    ! [A2: int,B2: int] :
      ( ( A2 != zero_zero(int) )
     => ( ( signed7115095781618012415divide(int,A2,B2) = aa(int,int,uminus_uminus(int),A2) )
      <=> ( B2 = aa(int,int,uminus_uminus(int),one_one(int)) ) ) ) ).

% int_sdiv_negated_is_minus1
tff(fact_5940_sum_Oinsert_H,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [I5: set(B),P2: fun(B,A),I: B] :
          ( pp(aa(set(B),bool,finite_finite2(B),collect(B,aa(fun(B,A),fun(B,bool),aTP_Lamp_bj(set(B),fun(fun(B,A),fun(B,bool)),I5),P2))))
         => ( ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I),I5))
             => ( groups1027152243600224163dd_sum(B,A,P2,insert(B,I,I5)) = groups1027152243600224163dd_sum(B,A,P2,I5) ) )
            & ( ~ pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I),I5))
             => ( groups1027152243600224163dd_sum(B,A,P2,insert(B,I,I5)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(B,A,P2,I)),groups1027152243600224163dd_sum(B,A,P2,I5)) ) ) ) ) ) ).

% sum.insert'
tff(fact_5941_sum_Onon__neutral_H,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(B,A),I5: set(B)] : groups1027152243600224163dd_sum(B,A,G,collect(B,aa(set(B),fun(B,bool),aTP_Lamp_oq(fun(B,A),fun(set(B),fun(B,bool)),G),I5))) = groups1027152243600224163dd_sum(B,A,G,I5) ) ).

% sum.non_neutral'
tff(fact_5942_sum_Odistrib__triv_H,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [I5: set(B),G: fun(B,A),H: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),I5))
         => ( groups1027152243600224163dd_sum(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_dj(fun(B,A),fun(fun(B,A),fun(B,A)),G),H),I5) = aa(A,A,aa(A,fun(A,A),plus_plus(A),groups1027152243600224163dd_sum(B,A,G,I5)),groups1027152243600224163dd_sum(B,A,H,I5)) ) ) ) ).

% sum.distrib_triv'
tff(fact_5943_sum_Omono__neutral__cong__right_H,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [S: set(B),T3: set(B),G: fun(B,A),H: fun(B,A)] :
          ( pp(aa(set(B),bool,ord_less_eq(set(B),S),T3))
         => ( ! [X3: B] :
                ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),aa(set(B),set(B),minus_minus(set(B),T3),S)))
               => ( aa(B,A,G,X3) = zero_zero(A) ) )
           => ( ! [X3: B] :
                  ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),S))
                 => ( aa(B,A,G,X3) = aa(B,A,H,X3) ) )
             => ( groups1027152243600224163dd_sum(B,A,G,T3) = groups1027152243600224163dd_sum(B,A,H,S) ) ) ) ) ) ).

% sum.mono_neutral_cong_right'
tff(fact_5944_sum_Omono__neutral__cong__left_H,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [S: set(B),T3: set(B),H: fun(B,A),G: fun(B,A)] :
          ( pp(aa(set(B),bool,ord_less_eq(set(B),S),T3))
         => ( ! [I2: B] :
                ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I2),aa(set(B),set(B),minus_minus(set(B),T3),S)))
               => ( aa(B,A,H,I2) = zero_zero(A) ) )
           => ( ! [X3: B] :
                  ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),S))
                 => ( aa(B,A,G,X3) = aa(B,A,H,X3) ) )
             => ( groups1027152243600224163dd_sum(B,A,G,S) = groups1027152243600224163dd_sum(B,A,H,T3) ) ) ) ) ) ).

% sum.mono_neutral_cong_left'
tff(fact_5945_sum_Omono__neutral__right_H,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [S: set(B),T3: set(B),G: fun(B,A)] :
          ( pp(aa(set(B),bool,ord_less_eq(set(B),S),T3))
         => ( ! [X3: B] :
                ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),aa(set(B),set(B),minus_minus(set(B),T3),S)))
               => ( aa(B,A,G,X3) = zero_zero(A) ) )
           => ( groups1027152243600224163dd_sum(B,A,G,T3) = groups1027152243600224163dd_sum(B,A,G,S) ) ) ) ) ).

% sum.mono_neutral_right'
tff(fact_5946_sum_Omono__neutral__left_H,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [S: set(B),T3: set(B),G: fun(B,A)] :
          ( pp(aa(set(B),bool,ord_less_eq(set(B),S),T3))
         => ( ! [X3: B] :
                ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),aa(set(B),set(B),minus_minus(set(B),T3),S)))
               => ( aa(B,A,G,X3) = zero_zero(A) ) )
           => ( groups1027152243600224163dd_sum(B,A,G,S) = groups1027152243600224163dd_sum(B,A,G,T3) ) ) ) ) ).

% sum.mono_neutral_left'
tff(fact_5947_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,aa(int,int,aa(int,fun(int,int),times_times(int),A2),B2)) ) ) ).

% sgn_sdiv_eq_sgn_mult
tff(fact_5948_sum_Odistrib_H,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [I5: set(B),G: fun(B,A),H: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),collect(B,aa(fun(B,A),fun(B,bool),aTP_Lamp_bj(set(B),fun(fun(B,A),fun(B,bool)),I5),G))))
         => ( pp(aa(set(B),bool,finite_finite2(B),collect(B,aa(fun(B,A),fun(B,bool),aTP_Lamp_bj(set(B),fun(fun(B,A),fun(B,bool)),I5),H))))
           => ( groups1027152243600224163dd_sum(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_dj(fun(B,A),fun(fun(B,A),fun(B,A)),G),H),I5) = aa(A,A,aa(A,fun(A,A),plus_plus(A),groups1027152243600224163dd_sum(B,A,G,I5)),groups1027152243600224163dd_sum(B,A,H,I5)) ) ) ) ) ).

% sum.distrib'
tff(fact_5949_sum_OG__def,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ! [I5: set(B),P2: fun(B,A)] :
          ( ( pp(aa(set(B),bool,finite_finite2(B),collect(B,aa(fun(B,A),fun(B,bool),aTP_Lamp_bj(set(B),fun(fun(B,A),fun(B,bool)),I5),P2))))
           => ( groups1027152243600224163dd_sum(B,A,P2,I5) = aa(set(B),A,groups7311177749621191930dd_sum(B,A,P2),collect(B,aa(fun(B,A),fun(B,bool),aTP_Lamp_bj(set(B),fun(fun(B,A),fun(B,bool)),I5),P2))) ) )
          & ( ~ pp(aa(set(B),bool,finite_finite2(B),collect(B,aa(fun(B,A),fun(B,bool),aTP_Lamp_bj(set(B),fun(fun(B,A),fun(B,bool)),I5),P2))))
           => ( groups1027152243600224163dd_sum(B,A,P2,I5) = zero_zero(A) ) ) ) ) ).

% sum.G_def
tff(fact_5950_signed__divide__int__def,axiom,
    ! [K: int,L: int] : signed7115095781618012415divide(int,K,L) = aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,aa(int,fun(int,int),times_times(int),sgn_sgn(int,K)),sgn_sgn(int,L))),aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,abs_abs(int),K)),aa(int,int,abs_abs(int),L))) ).

% signed_divide_int_def
tff(fact_5951_sum__diff1_H,axiom,
    ! [B: $tType,A: $tType] :
      ( ab_group_add(B)
     => ! [I5: set(A),F2: fun(A,B),I: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),collect(A,aa(fun(A,B),fun(A,bool),aTP_Lamp_op(set(A),fun(fun(A,B),fun(A,bool)),I5),F2))))
         => ( ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),I),I5))
             => ( groups1027152243600224163dd_sum(A,B,F2,aa(set(A),set(A),minus_minus(set(A),I5),insert(A,I,bot_bot(set(A))))) = aa(B,B,minus_minus(B,groups1027152243600224163dd_sum(A,B,F2,I5)),aa(A,B,F2,I)) ) )
            & ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),I),I5))
             => ( groups1027152243600224163dd_sum(A,B,F2,aa(set(A),set(A),minus_minus(set(A),I5),insert(A,I,bot_bot(set(A))))) = groups1027152243600224163dd_sum(A,B,F2,I5) ) ) ) ) ) ).

% sum_diff1'
tff(fact_5952_horner__sum__of__bool__2__less,axiom,
    ! [Bs: list(bool)] : pp(aa(int,bool,ord_less(int,groups4207007520872428315er_sum(bool,int,zero_neq_one_of_bool(int),aa(num,int,numeral_numeral(int),bit0(one2)),Bs)),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(list(bool),nat,size_size(list(bool)),Bs)))) ).

% horner_sum_of_bool_2_less
tff(fact_5953_sdiv__word__max,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A)] : pp(aa(int,bool,ord_less_eq(int,signed7115095781618012415divide(int,ring_1_signed(A,int,A2),ring_1_signed(A,int,B2))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,minus_minus(nat,aa(word(A),nat,size_size(word(A)),A2)),aa(nat,nat,suc,zero_zero(nat)))))) ) ).

% sdiv_word_max
tff(fact_5954_horner__sum__foldr,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_semiring_0(A)
     => ! [F2: fun(B,A),A2: A,Xs: list(B)] : groups4207007520872428315er_sum(B,A,F2,A2,Xs) = foldr(B,A,aa(A,fun(B,fun(A,A)),aTP_Lamp_or(fun(B,A),fun(A,fun(B,fun(A,A))),F2),A2),Xs,zero_zero(A)) ) ).

% horner_sum_foldr
tff(fact_5955_horner__sum__bit__eq__take__bit,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,N: nat] : groups4207007520872428315er_sum(bool,A,zero_neq_one_of_bool(A),aa(num,A,numeral_numeral(A),bit0(one2)),map(nat,bool,bit_se5641148757651400278ts_bit(A,A2),upt(zero_zero(nat),N))) = aa(A,A,bit_se2584673776208193580ke_bit(A,N),A2) ) ).

% horner_sum_bit_eq_take_bit
tff(fact_5956_Cauchy__iff2,axiom,
    ! [X6: fun(nat,real)] :
      ( topolo3814608138187158403Cauchy(real,X6)
    <=> ! [J3: nat] :
        ? [M12: nat] :
        ! [M6: nat] :
          ( pp(aa(nat,bool,ord_less_eq(nat,M12),M6))
         => ! [N5: nat] :
              ( pp(aa(nat,bool,ord_less_eq(nat,M12),N5))
             => pp(aa(real,bool,ord_less(real,aa(real,real,abs_abs(real),aa(real,real,minus_minus(real,aa(nat,real,X6,M6)),aa(nat,real,X6,N5)))),aa(real,real,inverse_inverse(real),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,J3))))) ) ) ) ).

% Cauchy_iff2
tff(fact_5957_nth__upt,axiom,
    ! [I: nat,K: nat,J: nat] :
      ( pp(aa(nat,bool,ord_less(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),K)),J))
     => ( aa(nat,nat,nth(nat,upt(I,J)),K) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),K) ) ) ).

% nth_upt
tff(fact_5958_map__add__upt,axiom,
    ! [N: nat,M: nat] : map(nat,nat,aTP_Lamp_os(nat,fun(nat,nat),N),upt(zero_zero(nat),M)) = upt(N,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N)) ).

% map_add_upt
tff(fact_5959_map__add__upt_H,axiom,
    ! [Ofs: nat,A2: nat,B2: nat] : map(nat,nat,aTP_Lamp_os(nat,fun(nat,nat),Ofs),upt(A2,B2)) = upt(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A2),Ofs),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),B2),Ofs)) ).

% map_add_upt'
tff(fact_5960_map__Suc__upt,axiom,
    ! [M: nat,N: nat] : map(nat,nat,suc,upt(M,N)) = upt(aa(nat,nat,suc,M),aa(nat,nat,suc,N)) ).

% map_Suc_upt
tff(fact_5961_map__decr__upt,axiom,
    ! [M: nat,N: nat] : map(nat,nat,aTP_Lamp_ot(nat,nat),upt(aa(nat,nat,suc,M),aa(nat,nat,suc,N))) = upt(M,N) ).

% map_decr_upt
tff(fact_5962_atLeastAtMost__upt,axiom,
    ! [N: nat,M: nat] : set_or1337092689740270186AtMost(nat,N,M) = aa(list(nat),set(nat),set2(nat),upt(N,aa(nat,nat,suc,M))) ).

% atLeastAtMost_upt
tff(fact_5963_atLeast__upt,axiom,
    ! [N: nat] : set_ord_lessThan(nat,N) = aa(list(nat),set(nat),set2(nat),upt(zero_zero(nat),N)) ).

% atLeast_upt
tff(fact_5964_map__replicate__trivial,axiom,
    ! [A: $tType,X: A,I: nat] : map(nat,A,aTP_Lamp_ou(A,fun(nat,A),X),upt(zero_zero(nat),I)) = replicate(A,I,X) ).

% map_replicate_trivial
tff(fact_5965_atMost__upto,axiom,
    ! [N: nat] : set_ord_atMost(nat,N) = aa(list(nat),set(nat),set2(nat),upt(zero_zero(nat),aa(nat,nat,suc,N))) ).

% atMost_upto
tff(fact_5966_map__bit__range__eq__if__take__bit__eq,axiom,
    ! [N: nat,K: int,L: int] :
      ( ( aa(int,int,bit_se2584673776208193580ke_bit(int,N),K) = aa(int,int,bit_se2584673776208193580ke_bit(int,N),L) )
     => ( map(nat,bool,bit_se5641148757651400278ts_bit(int,K),upt(zero_zero(nat),N)) = map(nat,bool,bit_se5641148757651400278ts_bit(int,L),upt(zero_zero(nat),N)) ) ) ).

% map_bit_range_eq_if_take_bit_eq
tff(fact_5967_map__nth,axiom,
    ! [A: $tType,Xs: list(A)] : map(nat,A,nth(A,Xs),upt(zero_zero(nat),aa(list(A),nat,size_size(list(A)),Xs))) = Xs ).

% map_nth
tff(fact_5968_nth__map__upt,axiom,
    ! [A: $tType,I: nat,N: nat,M: nat,F2: fun(nat,A)] :
      ( pp(aa(nat,bool,ord_less(nat,I),aa(nat,nat,minus_minus(nat,N),M)))
     => ( aa(nat,A,nth(A,map(nat,A,F2,upt(M,N))),I) = aa(nat,A,F2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),I)) ) ) ).

% nth_map_upt
tff(fact_5969_map__upt__eqI,axiom,
    ! [A: $tType,Xs: list(A),N: nat,M: nat,F2: fun(nat,A)] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(nat,nat,minus_minus(nat,N),M) )
     => ( ! [I2: nat] :
            ( pp(aa(nat,bool,ord_less(nat,I2),aa(list(A),nat,size_size(list(A)),Xs)))
           => ( aa(nat,A,nth(A,Xs),I2) = aa(nat,A,F2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),I2)) ) )
       => ( map(nat,A,F2,upt(M,N)) = Xs ) ) ) ).

% map_upt_eqI
tff(fact_5970_CauchyD,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X6: fun(nat,A),E: real] :
          ( topolo3814608138187158403Cauchy(A,X6)
         => ( pp(aa(real,bool,ord_less(real,zero_zero(real)),E))
           => ? [M9: nat] :
              ! [M4: nat] :
                ( pp(aa(nat,bool,ord_less_eq(nat,M9),M4))
               => ! [N11: nat] :
                    ( pp(aa(nat,bool,ord_less_eq(nat,M9),N11))
                   => pp(aa(real,bool,ord_less(real,real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,aa(nat,A,X6,M4)),aa(nat,A,X6,N11)))),E)) ) ) ) ) ) ).

% CauchyD
tff(fact_5971_CauchyI,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X6: fun(nat,A)] :
          ( ! [E2: real] :
              ( pp(aa(real,bool,ord_less(real,zero_zero(real)),E2))
             => ? [M13: nat] :
                ! [M3: nat] :
                  ( pp(aa(nat,bool,ord_less_eq(nat,M13),M3))
                 => ! [N2: nat] :
                      ( pp(aa(nat,bool,ord_less_eq(nat,M13),N2))
                     => pp(aa(real,bool,ord_less(real,real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,aa(nat,A,X6,M3)),aa(nat,A,X6,N2)))),E2)) ) ) )
         => topolo3814608138187158403Cauchy(A,X6) ) ) ).

% CauchyI
tff(fact_5972_Cauchy__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X6: fun(nat,A)] :
          ( topolo3814608138187158403Cauchy(A,X6)
        <=> ! [E4: real] :
              ( pp(aa(real,bool,ord_less(real,zero_zero(real)),E4))
             => ? [M12: nat] :
                ! [M6: nat] :
                  ( pp(aa(nat,bool,ord_less_eq(nat,M12),M6))
                 => ! [N5: nat] :
                      ( pp(aa(nat,bool,ord_less_eq(nat,M12),N5))
                     => pp(aa(real,bool,ord_less(real,real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,aa(nat,A,X6,M6)),aa(nat,A,X6,N5)))),E4)) ) ) ) ) ) ).

% Cauchy_iff
tff(fact_5973_order__refl,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [X: A] : pp(aa(A,bool,ord_less_eq(A,X),X)) ) ).

% order_refl
tff(fact_5974_dual__order_Orefl,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A2: A] : pp(aa(A,bool,ord_less_eq(A,A2),A2)) ) ).

% dual_order.refl
tff(fact_5975_order__antisym__conv,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Y: A,X: A] :
          ( pp(aa(A,bool,ord_less_eq(A,Y),X))
         => ( pp(aa(A,bool,ord_less_eq(A,X),Y))
          <=> ( X = Y ) ) ) ) ).

% order_antisym_conv
tff(fact_5976_linorder__le__cases,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y: A] :
          ( ~ pp(aa(A,bool,ord_less_eq(A,X),Y))
         => pp(aa(A,bool,ord_less_eq(A,Y),X)) ) ) ).

% linorder_le_cases
tff(fact_5977_ord__le__eq__subst,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ord(B)
        & ord(A) )
     => ! [A2: A,B2: A,F2: fun(A,B),C2: B] :
          ( pp(aa(A,bool,ord_less_eq(A,A2),B2))
         => ( ( aa(A,B,F2,B2) = C2 )
           => ( ! [X3: A,Y3: A] :
                  ( pp(aa(A,bool,ord_less_eq(A,X3),Y3))
                 => pp(aa(B,bool,ord_less_eq(B,aa(A,B,F2,X3)),aa(A,B,F2,Y3))) )
             => pp(aa(B,bool,ord_less_eq(B,aa(A,B,F2,A2)),C2)) ) ) ) ) ).

% ord_le_eq_subst
tff(fact_5978_ord__eq__le__subst,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ord(B)
        & ord(A) )
     => ! [A2: A,F2: fun(B,A),B2: B,C2: B] :
          ( ( A2 = aa(B,A,F2,B2) )
         => ( pp(aa(B,bool,ord_less_eq(B,B2),C2))
           => ( ! [X3: B,Y3: B] :
                  ( pp(aa(B,bool,ord_less_eq(B,X3),Y3))
                 => pp(aa(A,bool,ord_less_eq(A,aa(B,A,F2,X3)),aa(B,A,F2,Y3))) )
             => pp(aa(A,bool,ord_less_eq(A,A2),aa(B,A,F2,C2))) ) ) ) ) ).

% ord_eq_le_subst
tff(fact_5979_linorder__linear,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,ord_less_eq(A,X),Y))
          | pp(aa(A,bool,ord_less_eq(A,Y),X)) ) ) ).

% linorder_linear
tff(fact_5980_order__eq__refl,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [X: A,Y: A] :
          ( ( X = Y )
         => pp(aa(A,bool,ord_less_eq(A,X),Y)) ) ) ).

% order_eq_refl
tff(fact_5981_order__subst2,axiom,
    ! [A: $tType,C: $tType] :
      ( ( order(C)
        & order(A) )
     => ! [A2: A,B2: A,F2: fun(A,C),C2: C] :
          ( pp(aa(A,bool,ord_less_eq(A,A2),B2))
         => ( pp(aa(C,bool,ord_less_eq(C,aa(A,C,F2,B2)),C2))
           => ( ! [X3: A,Y3: A] :
                  ( pp(aa(A,bool,ord_less_eq(A,X3),Y3))
                 => pp(aa(C,bool,ord_less_eq(C,aa(A,C,F2,X3)),aa(A,C,F2,Y3))) )
             => pp(aa(C,bool,ord_less_eq(C,aa(A,C,F2,A2)),C2)) ) ) ) ) ).

% order_subst2
tff(fact_5982_order__subst1,axiom,
    ! [A: $tType,B: $tType] :
      ( ( order(B)
        & order(A) )
     => ! [A2: A,F2: fun(B,A),B2: B,C2: B] :
          ( pp(aa(A,bool,ord_less_eq(A,A2),aa(B,A,F2,B2)))
         => ( pp(aa(B,bool,ord_less_eq(B,B2),C2))
           => ( ! [X3: B,Y3: B] :
                  ( pp(aa(B,bool,ord_less_eq(B,X3),Y3))
                 => pp(aa(A,bool,ord_less_eq(A,aa(B,A,F2,X3)),aa(B,A,F2,Y3))) )
             => pp(aa(A,bool,ord_less_eq(A,A2),aa(B,A,F2,C2))) ) ) ) ) ).

% order_subst1
tff(fact_5983_Orderings_Oorder__eq__iff,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A2: A,B2: A] :
          ( ( A2 = B2 )
        <=> ( pp(aa(A,bool,ord_less_eq(A,A2),B2))
            & pp(aa(A,bool,ord_less_eq(A,B2),A2)) ) ) ) ).

% Orderings.order_eq_iff
tff(fact_5984_antisym,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,ord_less_eq(A,A2),B2))
         => ( pp(aa(A,bool,ord_less_eq(A,B2),A2))
           => ( A2 = B2 ) ) ) ) ).

% antisym
tff(fact_5985_dual__order_Otrans,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [B2: A,A2: A,C2: A] :
          ( pp(aa(A,bool,ord_less_eq(A,B2),A2))
         => ( pp(aa(A,bool,ord_less_eq(A,C2),B2))
           => pp(aa(A,bool,ord_less_eq(A,C2),A2)) ) ) ) ).

% dual_order.trans
tff(fact_5986_dual__order_Oantisym,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [B2: A,A2: A] :
          ( pp(aa(A,bool,ord_less_eq(A,B2),A2))
         => ( pp(aa(A,bool,ord_less_eq(A,A2),B2))
           => ( A2 = B2 ) ) ) ) ).

% dual_order.antisym
tff(fact_5987_dual__order_Oeq__iff,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A2: A,B2: A] :
          ( ( A2 = B2 )
        <=> ( pp(aa(A,bool,ord_less_eq(A,B2),A2))
            & pp(aa(A,bool,ord_less_eq(A,A2),B2)) ) ) ) ).

% dual_order.eq_iff
tff(fact_5988_linorder__wlog,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [P: fun(A,fun(A,bool)),A2: A,B2: A] :
          ( ! [A4: A,B3: A] :
              ( pp(aa(A,bool,ord_less_eq(A,A4),B3))
             => pp(aa(A,bool,aa(A,fun(A,bool),P,A4),B3)) )
         => ( ! [A4: A,B3: A] :
                ( pp(aa(A,bool,aa(A,fun(A,bool),P,B3),A4))
               => pp(aa(A,bool,aa(A,fun(A,bool),P,A4),B3)) )
           => pp(aa(A,bool,aa(A,fun(A,bool),P,A2),B2)) ) ) ) ).

% linorder_wlog
tff(fact_5989_order__trans,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [X: A,Y: A,Z: A] :
          ( pp(aa(A,bool,ord_less_eq(A,X),Y))
         => ( pp(aa(A,bool,ord_less_eq(A,Y),Z))
           => pp(aa(A,bool,ord_less_eq(A,X),Z)) ) ) ) ).

% order_trans
tff(fact_5990_order_Otrans,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,ord_less_eq(A,A2),B2))
         => ( pp(aa(A,bool,ord_less_eq(A,B2),C2))
           => pp(aa(A,bool,ord_less_eq(A,A2),C2)) ) ) ) ).

% order.trans
tff(fact_5991_order__antisym,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,ord_less_eq(A,X),Y))
         => ( pp(aa(A,bool,ord_less_eq(A,Y),X))
           => ( X = Y ) ) ) ) ).

% order_antisym
tff(fact_5992_ord__le__eq__trans,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,ord_less_eq(A,A2),B2))
         => ( ( B2 = C2 )
           => pp(aa(A,bool,ord_less_eq(A,A2),C2)) ) ) ) ).

% ord_le_eq_trans
tff(fact_5993_ord__eq__le__trans,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [A2: A,B2: A,C2: A] :
          ( ( A2 = B2 )
         => ( pp(aa(A,bool,ord_less_eq(A,B2),C2))
           => pp(aa(A,bool,ord_less_eq(A,A2),C2)) ) ) ) ).

% ord_eq_le_trans
tff(fact_5994_order__class_Oorder__eq__iff,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X: A,Y: A] :
          ( ( X = Y )
        <=> ( pp(aa(A,bool,ord_less_eq(A,X),Y))
            & pp(aa(A,bool,ord_less_eq(A,Y),X)) ) ) ) ).

% order_class.order_eq_iff
tff(fact_5995_le__cases3,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y: A,Z: A] :
          ( ( pp(aa(A,bool,ord_less_eq(A,X),Y))
           => ~ pp(aa(A,bool,ord_less_eq(A,Y),Z)) )
         => ( ( pp(aa(A,bool,ord_less_eq(A,Y),X))
             => ~ pp(aa(A,bool,ord_less_eq(A,X),Z)) )
           => ( ( pp(aa(A,bool,ord_less_eq(A,X),Z))
               => ~ pp(aa(A,bool,ord_less_eq(A,Z),Y)) )
             => ( ( pp(aa(A,bool,ord_less_eq(A,Z),Y))
                 => ~ pp(aa(A,bool,ord_less_eq(A,Y),X)) )
               => ( ( pp(aa(A,bool,ord_less_eq(A,Y),Z))
                   => ~ pp(aa(A,bool,ord_less_eq(A,Z),X)) )
                 => ~ ( pp(aa(A,bool,ord_less_eq(A,Z),X))
                     => ~ pp(aa(A,bool,ord_less_eq(A,X),Y)) ) ) ) ) ) ) ) ).

% le_cases3
tff(fact_5996_nle__le,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A] :
          ( ~ pp(aa(A,bool,ord_less_eq(A,A2),B2))
        <=> ( pp(aa(A,bool,ord_less_eq(A,B2),A2))
            & ( B2 != A2 ) ) ) ) ).

% nle_le
tff(fact_5997_order__le__imp__less__or__eq,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,ord_less_eq(A,X),Y))
         => ( pp(aa(A,bool,ord_less(A,X),Y))
            | ( X = Y ) ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_5998_linorder__le__less__linear,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,ord_less_eq(A,X),Y))
          | pp(aa(A,bool,ord_less(A,Y),X)) ) ) ).

% linorder_le_less_linear
tff(fact_5999_order__less__le__subst2,axiom,
    ! [A: $tType,C: $tType] :
      ( ( order(C)
        & order(A) )
     => ! [A2: A,B2: A,F2: fun(A,C),C2: C] :
          ( pp(aa(A,bool,ord_less(A,A2),B2))
         => ( pp(aa(C,bool,ord_less_eq(C,aa(A,C,F2,B2)),C2))
           => ( ! [X3: A,Y3: A] :
                  ( pp(aa(A,bool,ord_less(A,X3),Y3))
                 => pp(aa(C,bool,ord_less(C,aa(A,C,F2,X3)),aa(A,C,F2,Y3))) )
             => pp(aa(C,bool,ord_less(C,aa(A,C,F2,A2)),C2)) ) ) ) ) ).

% order_less_le_subst2
tff(fact_6000_order__less__le__subst1,axiom,
    ! [A: $tType,B: $tType] :
      ( ( order(B)
        & order(A) )
     => ! [A2: A,F2: fun(B,A),B2: B,C2: B] :
          ( pp(aa(A,bool,ord_less(A,A2),aa(B,A,F2,B2)))
         => ( pp(aa(B,bool,ord_less_eq(B,B2),C2))
           => ( ! [X3: B,Y3: B] :
                  ( pp(aa(B,bool,ord_less_eq(B,X3),Y3))
                 => pp(aa(A,bool,ord_less_eq(A,aa(B,A,F2,X3)),aa(B,A,F2,Y3))) )
             => pp(aa(A,bool,ord_less(A,A2),aa(B,A,F2,C2))) ) ) ) ) ).

% order_less_le_subst1
tff(fact_6001_order__le__less__subst2,axiom,
    ! [A: $tType,C: $tType] :
      ( ( order(C)
        & order(A) )
     => ! [A2: A,B2: A,F2: fun(A,C),C2: C] :
          ( pp(aa(A,bool,ord_less_eq(A,A2),B2))
         => ( pp(aa(C,bool,ord_less(C,aa(A,C,F2,B2)),C2))
           => ( ! [X3: A,Y3: A] :
                  ( pp(aa(A,bool,ord_less_eq(A,X3),Y3))
                 => pp(aa(C,bool,ord_less_eq(C,aa(A,C,F2,X3)),aa(A,C,F2,Y3))) )
             => pp(aa(C,bool,ord_less(C,aa(A,C,F2,A2)),C2)) ) ) ) ) ).

% order_le_less_subst2
tff(fact_6002_order__le__less__subst1,axiom,
    ! [A: $tType,B: $tType] :
      ( ( order(B)
        & order(A) )
     => ! [A2: A,F2: fun(B,A),B2: B,C2: B] :
          ( pp(aa(A,bool,ord_less_eq(A,A2),aa(B,A,F2,B2)))
         => ( pp(aa(B,bool,ord_less(B,B2),C2))
           => ( ! [X3: B,Y3: B] :
                  ( pp(aa(B,bool,ord_less(B,X3),Y3))
                 => pp(aa(A,bool,ord_less(A,aa(B,A,F2,X3)),aa(B,A,F2,Y3))) )
             => pp(aa(A,bool,ord_less(A,A2),aa(B,A,F2,C2))) ) ) ) ) ).

% order_le_less_subst1
tff(fact_6003_order__less__le__trans,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [X: A,Y: A,Z: A] :
          ( pp(aa(A,bool,ord_less(A,X),Y))
         => ( pp(aa(A,bool,ord_less_eq(A,Y),Z))
           => pp(aa(A,bool,ord_less(A,X),Z)) ) ) ) ).

% order_less_le_trans
tff(fact_6004_order__le__less__trans,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [X: A,Y: A,Z: A] :
          ( pp(aa(A,bool,ord_less_eq(A,X),Y))
         => ( pp(aa(A,bool,ord_less(A,Y),Z))
           => pp(aa(A,bool,ord_less(A,X),Z)) ) ) ) ).

% order_le_less_trans
tff(fact_6005_order__neq__le__trans,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A2: A,B2: A] :
          ( ( A2 != B2 )
         => ( pp(aa(A,bool,ord_less_eq(A,A2),B2))
           => pp(aa(A,bool,ord_less(A,A2),B2)) ) ) ) ).

% order_neq_le_trans
tff(fact_6006_order__le__neq__trans,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,ord_less_eq(A,A2),B2))
         => ( ( A2 != B2 )
           => pp(aa(A,bool,ord_less(A,A2),B2)) ) ) ) ).

% order_le_neq_trans
tff(fact_6007_order__less__imp__le,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,ord_less(A,X),Y))
         => pp(aa(A,bool,ord_less_eq(A,X),Y)) ) ) ).

% order_less_imp_le
tff(fact_6008_linorder__not__less,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y: A] :
          ( ~ pp(aa(A,bool,ord_less(A,X),Y))
        <=> pp(aa(A,bool,ord_less_eq(A,Y),X)) ) ) ).

% linorder_not_less
tff(fact_6009_linorder__not__le,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y: A] :
          ( ~ pp(aa(A,bool,ord_less_eq(A,X),Y))
        <=> pp(aa(A,bool,ord_less(A,Y),X)) ) ) ).

% linorder_not_le
tff(fact_6010_order__less__le,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,ord_less(A,X),Y))
        <=> ( pp(aa(A,bool,ord_less_eq(A,X),Y))
            & ( X != Y ) ) ) ) ).

% order_less_le
tff(fact_6011_order__le__less,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,ord_less_eq(A,X),Y))
        <=> ( pp(aa(A,bool,ord_less(A,X),Y))
            | ( X = Y ) ) ) ) ).

% order_le_less
tff(fact_6012_dual__order_Ostrict__implies__order,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [B2: A,A2: A] :
          ( pp(aa(A,bool,ord_less(A,B2),A2))
         => pp(aa(A,bool,ord_less_eq(A,B2),A2)) ) ) ).

% dual_order.strict_implies_order
tff(fact_6013_order_Ostrict__implies__order,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,ord_less(A,A2),B2))
         => pp(aa(A,bool,ord_less_eq(A,A2),B2)) ) ) ).

% order.strict_implies_order
tff(fact_6014_dual__order_Ostrict__iff__not,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [B2: A,A2: A] :
          ( pp(aa(A,bool,ord_less(A,B2),A2))
        <=> ( pp(aa(A,bool,ord_less_eq(A,B2),A2))
            & ~ pp(aa(A,bool,ord_less_eq(A,A2),B2)) ) ) ) ).

% dual_order.strict_iff_not
tff(fact_6015_dual__order_Ostrict__trans2,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [B2: A,A2: A,C2: A] :
          ( pp(aa(A,bool,ord_less(A,B2),A2))
         => ( pp(aa(A,bool,ord_less_eq(A,C2),B2))
           => pp(aa(A,bool,ord_less(A,C2),A2)) ) ) ) ).

% dual_order.strict_trans2
tff(fact_6016_dual__order_Ostrict__trans1,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [B2: A,A2: A,C2: A] :
          ( pp(aa(A,bool,ord_less_eq(A,B2),A2))
         => ( pp(aa(A,bool,ord_less(A,C2),B2))
           => pp(aa(A,bool,ord_less(A,C2),A2)) ) ) ) ).

% dual_order.strict_trans1
tff(fact_6017_dual__order_Ostrict__iff__order,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [B2: A,A2: A] :
          ( pp(aa(A,bool,ord_less(A,B2),A2))
        <=> ( pp(aa(A,bool,ord_less_eq(A,B2),A2))
            & ( A2 != B2 ) ) ) ) ).

% dual_order.strict_iff_order
tff(fact_6018_dual__order_Oorder__iff__strict,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [B2: A,A2: A] :
          ( pp(aa(A,bool,ord_less_eq(A,B2),A2))
        <=> ( pp(aa(A,bool,ord_less(A,B2),A2))
            | ( A2 = B2 ) ) ) ) ).

% dual_order.order_iff_strict
tff(fact_6019_dense__le__bounded,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [X: A,Y: A,Z: A] :
          ( pp(aa(A,bool,ord_less(A,X),Y))
         => ( ! [W2: A] :
                ( pp(aa(A,bool,ord_less(A,X),W2))
               => ( pp(aa(A,bool,ord_less(A,W2),Y))
                 => pp(aa(A,bool,ord_less_eq(A,W2),Z)) ) )
           => pp(aa(A,bool,ord_less_eq(A,Y),Z)) ) ) ) ).

% dense_le_bounded
tff(fact_6020_dense__ge__bounded,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [Z: A,X: A,Y: A] :
          ( pp(aa(A,bool,ord_less(A,Z),X))
         => ( ! [W2: A] :
                ( pp(aa(A,bool,ord_less(A,Z),W2))
               => ( pp(aa(A,bool,ord_less(A,W2),X))
                 => pp(aa(A,bool,ord_less_eq(A,Y),W2)) ) )
           => pp(aa(A,bool,ord_less_eq(A,Y),Z)) ) ) ) ).

% dense_ge_bounded
tff(fact_6021_order_Ostrict__iff__not,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,ord_less(A,A2),B2))
        <=> ( pp(aa(A,bool,ord_less_eq(A,A2),B2))
            & ~ pp(aa(A,bool,ord_less_eq(A,B2),A2)) ) ) ) ).

% order.strict_iff_not
tff(fact_6022_order_Ostrict__trans2,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,ord_less(A,A2),B2))
         => ( pp(aa(A,bool,ord_less_eq(A,B2),C2))
           => pp(aa(A,bool,ord_less(A,A2),C2)) ) ) ) ).

% order.strict_trans2
tff(fact_6023_order_Ostrict__trans1,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,ord_less_eq(A,A2),B2))
         => ( pp(aa(A,bool,ord_less(A,B2),C2))
           => pp(aa(A,bool,ord_less(A,A2),C2)) ) ) ) ).

% order.strict_trans1
tff(fact_6024_order_Ostrict__iff__order,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,ord_less(A,A2),B2))
        <=> ( pp(aa(A,bool,ord_less_eq(A,A2),B2))
            & ( A2 != B2 ) ) ) ) ).

% order.strict_iff_order
tff(fact_6025_order_Oorder__iff__strict,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,ord_less_eq(A,A2),B2))
        <=> ( pp(aa(A,bool,ord_less(A,A2),B2))
            | ( A2 = B2 ) ) ) ) ).

% order.order_iff_strict
tff(fact_6026_not__le__imp__less,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Y: A,X: A] :
          ( ~ pp(aa(A,bool,ord_less_eq(A,Y),X))
         => pp(aa(A,bool,ord_less(A,X),Y)) ) ) ).

% not_le_imp_less
tff(fact_6027_less__le__not__le,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,ord_less(A,X),Y))
        <=> ( pp(aa(A,bool,ord_less_eq(A,X),Y))
            & ~ pp(aa(A,bool,ord_less_eq(A,Y),X)) ) ) ) ).

% less_le_not_le
tff(fact_6028_dense__le,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [Y: A,Z: A] :
          ( ! [X3: A] :
              ( pp(aa(A,bool,ord_less(A,X3),Y))
             => pp(aa(A,bool,ord_less_eq(A,X3),Z)) )
         => pp(aa(A,bool,ord_less_eq(A,Y),Z)) ) ) ).

% dense_le
tff(fact_6029_dense__ge,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [Z: A,Y: A] :
          ( ! [X3: A] :
              ( pp(aa(A,bool,ord_less(A,Z),X3))
             => pp(aa(A,bool,ord_less_eq(A,Y),X3)) )
         => pp(aa(A,bool,ord_less_eq(A,Y),Z)) ) ) ).

% dense_ge
tff(fact_6030_antisym__conv2,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,ord_less_eq(A,X),Y))
         => ( ~ pp(aa(A,bool,ord_less(A,X),Y))
          <=> ( X = Y ) ) ) ) ).

% antisym_conv2
tff(fact_6031_antisym__conv1,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X: A,Y: A] :
          ( ~ pp(aa(A,bool,ord_less(A,X),Y))
         => ( pp(aa(A,bool,ord_less_eq(A,X),Y))
          <=> ( X = Y ) ) ) ) ).

% antisym_conv1
tff(fact_6032_nless__le,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A2: A,B2: A] :
          ( ~ pp(aa(A,bool,ord_less(A,A2),B2))
        <=> ( ~ pp(aa(A,bool,ord_less_eq(A,A2),B2))
            | ( A2 = B2 ) ) ) ) ).

% nless_le
tff(fact_6033_leI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y: A] :
          ( ~ pp(aa(A,bool,ord_less(A,X),Y))
         => pp(aa(A,bool,ord_less_eq(A,Y),X)) ) ) ).

% leI
tff(fact_6034_leD,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Y: A,X: A] :
          ( pp(aa(A,bool,ord_less_eq(A,Y),X))
         => ~ pp(aa(A,bool,ord_less(A,X),Y)) ) ) ).

% leD
tff(fact_6035_bot_Oextremum__uniqueI,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ! [A2: A] :
          ( pp(aa(A,bool,ord_less_eq(A,A2),bot_bot(A)))
         => ( A2 = bot_bot(A) ) ) ) ).

% bot.extremum_uniqueI
tff(fact_6036_bot_Oextremum__unique,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ! [A2: A] :
          ( pp(aa(A,bool,ord_less_eq(A,A2),bot_bot(A)))
        <=> ( A2 = bot_bot(A) ) ) ) ).

% bot.extremum_unique
tff(fact_6037_bot_Oextremum,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ! [A2: A] : pp(aa(A,bool,ord_less_eq(A,bot_bot(A)),A2)) ) ).

% bot.extremum
tff(fact_6038_max__def,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [A2: A,B2: A] :
          ( ( pp(aa(A,bool,ord_less_eq(A,A2),B2))
           => ( aa(A,A,ord_max(A,A2),B2) = B2 ) )
          & ( ~ pp(aa(A,bool,ord_less_eq(A,A2),B2))
           => ( aa(A,A,ord_max(A,A2),B2) = A2 ) ) ) ) ).

% max_def
tff(fact_6039_max__absorb1,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Y: A,X: A] :
          ( pp(aa(A,bool,ord_less_eq(A,Y),X))
         => ( aa(A,A,ord_max(A,X),Y) = X ) ) ) ).

% max_absorb1
tff(fact_6040_max__absorb2,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,ord_less_eq(A,X),Y))
         => ( aa(A,A,ord_max(A,X),Y) = Y ) ) ) ).

% max_absorb2
tff(fact_6041_le__fun__def,axiom,
    ! [B: $tType,A: $tType] :
      ( ord(B)
     => ! [F2: fun(A,B),G: fun(A,B)] :
          ( pp(aa(fun(A,B),bool,ord_less_eq(fun(A,B),F2),G))
        <=> ! [X2: A] : pp(aa(B,bool,ord_less_eq(B,aa(A,B,F2,X2)),aa(A,B,G,X2))) ) ) ).

% le_fun_def
tff(fact_6042_le__funI,axiom,
    ! [B: $tType,A: $tType] :
      ( ord(B)
     => ! [F2: fun(A,B),G: fun(A,B)] :
          ( ! [X3: A] : pp(aa(B,bool,ord_less_eq(B,aa(A,B,F2,X3)),aa(A,B,G,X3)))
         => pp(aa(fun(A,B),bool,ord_less_eq(fun(A,B),F2),G)) ) ) ).

% le_funI
tff(fact_6043_le__funE,axiom,
    ! [B: $tType,A: $tType] :
      ( ord(B)
     => ! [F2: fun(A,B),G: fun(A,B),X: A] :
          ( pp(aa(fun(A,B),bool,ord_less_eq(fun(A,B),F2),G))
         => pp(aa(B,bool,ord_less_eq(B,aa(A,B,F2,X)),aa(A,B,G,X))) ) ) ).

% le_funE
tff(fact_6044_le__funD,axiom,
    ! [B: $tType,A: $tType] :
      ( ord(B)
     => ! [F2: fun(A,B),G: fun(A,B),X: A] :
          ( pp(aa(fun(A,B),bool,ord_less_eq(fun(A,B),F2),G))
         => pp(aa(B,bool,ord_less_eq(B,aa(A,B,F2,X)),aa(A,B,G,X))) ) ) ).

% le_funD
tff(fact_6045_Sum__Ico__nat,axiom,
    ! [M: nat,N: nat] : aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,aTP_Lamp_el(nat,nat)),set_or7035219750837199246ssThan(nat,M,N)) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),aa(nat,nat,minus_minus(nat,N),one_one(nat)))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),aa(nat,nat,minus_minus(nat,M),one_one(nat))))),aa(num,nat,numeral_numeral(nat),bit0(one2))) ).

% Sum_Ico_nat
tff(fact_6046_VEBT_Osize_I3_J,axiom,
    ! [X11: option(product_prod(nat,nat)),X12: nat,X13: list(vEBT_VEBT),X14: vEBT_VEBT] : aa(vEBT_VEBT,nat,size_size(vEBT_VEBT),vEBT_Node(X11,X12,X13,X14)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),size_list(vEBT_VEBT,size_size(vEBT_VEBT),X13)),aa(vEBT_VEBT,nat,size_size(vEBT_VEBT),X14))),aa(nat,nat,suc,zero_zero(nat))) ).

% VEBT.size(3)
tff(fact_6047_atLeastLessThan__iff,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [I: A,L: A,U: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),I),set_or7035219750837199246ssThan(A,L,U)))
        <=> ( pp(aa(A,bool,ord_less_eq(A,L),I))
            & pp(aa(A,bool,ord_less(A,I),U)) ) ) ) ).

% atLeastLessThan_iff
tff(fact_6048_atLeastLessThan__empty,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [B2: A,A2: A] :
          ( pp(aa(A,bool,ord_less_eq(A,B2),A2))
         => ( set_or7035219750837199246ssThan(A,A2,B2) = bot_bot(set(A)) ) ) ) ).

% atLeastLessThan_empty
tff(fact_6049_ivl__subset,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [I: A,J: A,M: A,N: A] :
          ( pp(aa(set(A),bool,ord_less_eq(set(A),set_or7035219750837199246ssThan(A,I,J)),set_or7035219750837199246ssThan(A,M,N)))
        <=> ( pp(aa(A,bool,ord_less_eq(A,J),I))
            | ( pp(aa(A,bool,ord_less_eq(A,M),I))
              & pp(aa(A,bool,ord_less_eq(A,J),N)) ) ) ) ) ).

% ivl_subset
tff(fact_6050_image__add__atLeastLessThan,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [K: A,I: A,J: A] : aa(set(A),set(A),image(A,A,aa(A,fun(A,A),plus_plus(A),K)),set_or7035219750837199246ssThan(A,I,J)) = set_or7035219750837199246ssThan(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),K)) ) ).

% image_add_atLeastLessThan
tff(fact_6051_ivl__diff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [I: A,N: A,M: A] :
          ( pp(aa(A,bool,ord_less_eq(A,I),N))
         => ( aa(set(A),set(A),minus_minus(set(A),set_or7035219750837199246ssThan(A,I,M)),set_or7035219750837199246ssThan(A,I,N)) = set_or7035219750837199246ssThan(A,N,M) ) ) ) ).

% ivl_diff
tff(fact_6052_image__Suc__atLeastLessThan,axiom,
    ! [I: nat,J: nat] : aa(set(nat),set(nat),image(nat,nat,suc),set_or7035219750837199246ssThan(nat,I,J)) = set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,I),aa(nat,nat,suc,J)) ).

% image_Suc_atLeastLessThan
tff(fact_6053_image__add__atLeastLessThan_H,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [K: A,I: A,J: A] : aa(set(A),set(A),image(A,A,aTP_Lamp_nq(A,fun(A,A),K)),set_or7035219750837199246ssThan(A,I,J)) = set_or7035219750837199246ssThan(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),K)) ) ).

% image_add_atLeastLessThan'
tff(fact_6054_atLeastLessThan__singleton,axiom,
    ! [M: nat] : set_or7035219750837199246ssThan(nat,M,aa(nat,nat,suc,M)) = insert(nat,M,bot_bot(set(nat))) ).

% atLeastLessThan_singleton
tff(fact_6055_sum_Oop__ivl__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [N: nat,M: nat,G: fun(nat,A)] :
          ( ( pp(aa(nat,bool,ord_less(nat,N),M))
           => ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or7035219750837199246ssThan(nat,M,aa(nat,nat,suc,N))) = zero_zero(A) ) )
          & ( ~ pp(aa(nat,bool,ord_less(nat,N),M))
           => ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or7035219750837199246ssThan(nat,M,aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or7035219750837199246ssThan(nat,M,N))),aa(nat,A,G,N)) ) ) ) ) ).

% sum.op_ivl_Suc
tff(fact_6056_prod_Oop__ivl__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [N: nat,M: nat,G: fun(nat,A)] :
          ( ( pp(aa(nat,bool,ord_less(nat,N),M))
           => ( groups7121269368397514597t_prod(nat,A,G,set_or7035219750837199246ssThan(nat,M,aa(nat,nat,suc,N))) = one_one(A) ) )
          & ( ~ pp(aa(nat,bool,ord_less(nat,N),M))
           => ( groups7121269368397514597t_prod(nat,A,G,set_or7035219750837199246ssThan(nat,M,aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),times_times(A),groups7121269368397514597t_prod(nat,A,G,set_or7035219750837199246ssThan(nat,M,N))),aa(nat,A,G,N)) ) ) ) ) ).

% prod.op_ivl_Suc
tff(fact_6057_nth__image__indices,axiom,
    ! [A: $tType,L: list(A)] : aa(set(nat),set(A),image(nat,A,nth(A,L)),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(list(A),nat,size_size(list(A)),L))) = aa(list(A),set(A),set2(A),L) ).

% nth_image_indices
tff(fact_6058_all__nat__less__eq,axiom,
    ! [N: nat,P: fun(nat,bool)] :
      ( ! [M6: nat] :
          ( pp(aa(nat,bool,ord_less(nat,M6),N))
         => pp(aa(nat,bool,P,M6)) )
    <=> ! [X2: nat] :
          ( pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),X2),set_or7035219750837199246ssThan(nat,zero_zero(nat),N)))
         => pp(aa(nat,bool,P,X2)) ) ) ).

% all_nat_less_eq
tff(fact_6059_ex__nat__less__eq,axiom,
    ! [N: nat,P: fun(nat,bool)] :
      ( ? [M6: nat] :
          ( pp(aa(nat,bool,ord_less(nat,M6),N))
          & pp(aa(nat,bool,P,M6)) )
    <=> ? [X2: nat] :
          ( pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),X2),set_or7035219750837199246ssThan(nat,zero_zero(nat),N)))
          & pp(aa(nat,bool,P,X2)) ) ) ).

% ex_nat_less_eq
tff(fact_6060_lessThan__atLeast0,axiom,
    ! [N: nat] : set_ord_lessThan(nat,N) = set_or7035219750837199246ssThan(nat,zero_zero(nat),N) ).

% lessThan_atLeast0
tff(fact_6061_atLeastLessThanSuc__atLeastAtMost,axiom,
    ! [L: nat,U: nat] : set_or7035219750837199246ssThan(nat,L,aa(nat,nat,suc,U)) = set_or1337092689740270186AtMost(nat,L,U) ).

% atLeastLessThanSuc_atLeastAtMost
tff(fact_6062_atLeastLessThan0,axiom,
    ! [M: nat] : set_or7035219750837199246ssThan(nat,M,zero_zero(nat)) = bot_bot(set(nat)) ).

% atLeastLessThan0
tff(fact_6063_atLeastLessThan__subset__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A,C2: A,D: A] :
          ( pp(aa(set(A),bool,ord_less_eq(set(A),set_or7035219750837199246ssThan(A,A2,B2)),set_or7035219750837199246ssThan(A,C2,D)))
         => ( pp(aa(A,bool,ord_less_eq(A,B2),A2))
            | ( pp(aa(A,bool,ord_less_eq(A,C2),A2))
              & pp(aa(A,bool,ord_less_eq(A,B2),D)) ) ) ) ) ).

% atLeastLessThan_subset_iff
tff(fact_6064_sum_Oshift__bounds__Suc__ivl,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),M: nat,N: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,M),aa(nat,nat,suc,N))) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_ea(fun(nat,A),fun(nat,A),G)),set_or7035219750837199246ssThan(nat,M,N)) ) ).

% sum.shift_bounds_Suc_ivl
tff(fact_6065_sum_Oshift__bounds__nat__ivl,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),M: nat,K: nat,N: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or7035219750837199246ssThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),K),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),K))) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aTP_Lamp_eb(fun(nat,A),fun(nat,fun(nat,A)),G),K)),set_or7035219750837199246ssThan(nat,M,N)) ) ).

% sum.shift_bounds_nat_ivl
tff(fact_6066_prod_Oshift__bounds__Suc__ivl,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),M: nat,N: nat] : groups7121269368397514597t_prod(nat,A,G,set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,M),aa(nat,nat,suc,N))) = groups7121269368397514597t_prod(nat,A,aTP_Lamp_hn(fun(nat,A),fun(nat,A),G),set_or7035219750837199246ssThan(nat,M,N)) ) ).

% prod.shift_bounds_Suc_ivl
tff(fact_6067_prod_Oshift__bounds__nat__ivl,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),M: nat,K: nat,N: nat] : groups7121269368397514597t_prod(nat,A,G,set_or7035219750837199246ssThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),K),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),K))) = groups7121269368397514597t_prod(nat,A,aa(nat,fun(nat,A),aTP_Lamp_hp(fun(nat,A),fun(nat,fun(nat,A)),G),K),set_or7035219750837199246ssThan(nat,M,N)) ) ).

% prod.shift_bounds_nat_ivl
tff(fact_6068_sum_Oivl__cong,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ord(B)
        & comm_monoid_add(A) )
     => ! [A2: B,C2: B,B2: B,D: B,G: fun(B,A),H: fun(B,A)] :
          ( ( A2 = C2 )
         => ( ( B2 = D )
           => ( ! [X3: B] :
                  ( pp(aa(B,bool,ord_less_eq(B,C2),X3))
                 => ( pp(aa(B,bool,ord_less(B,X3),D))
                   => ( aa(B,A,G,X3) = aa(B,A,H,X3) ) ) )
             => ( aa(set(B),A,groups7311177749621191930dd_sum(B,A,G),set_or7035219750837199246ssThan(B,A2,B2)) = aa(set(B),A,groups7311177749621191930dd_sum(B,A,H),set_or7035219750837199246ssThan(B,C2,D)) ) ) ) ) ) ).

% sum.ivl_cong
tff(fact_6069_prod_Oivl__cong,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ord(B)
        & comm_monoid_mult(A) )
     => ! [A2: B,C2: B,B2: B,D: B,G: fun(B,A),H: fun(B,A)] :
          ( ( A2 = C2 )
         => ( ( B2 = D )
           => ( ! [X3: B] :
                  ( pp(aa(B,bool,ord_less_eq(B,C2),X3))
                 => ( pp(aa(B,bool,ord_less(B,X3),D))
                   => ( aa(B,A,G,X3) = aa(B,A,H,X3) ) ) )
             => ( groups7121269368397514597t_prod(B,A,G,set_or7035219750837199246ssThan(B,A2,B2)) = groups7121269368397514597t_prod(B,A,H,set_or7035219750837199246ssThan(B,C2,D)) ) ) ) ) ) ).

% prod.ivl_cong
tff(fact_6070_sum_OatLeastLessThan__concat,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [M: nat,N: nat,P2: nat,G: fun(nat,A)] :
          ( pp(aa(nat,bool,ord_less_eq(nat,M),N))
         => ( pp(aa(nat,bool,ord_less_eq(nat,N),P2))
           => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or7035219750837199246ssThan(nat,M,N))),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or7035219750837199246ssThan(nat,N,P2))) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or7035219750837199246ssThan(nat,M,P2)) ) ) ) ) ).

% sum.atLeastLessThan_concat
tff(fact_6071_sum__diff__nat__ivl,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [M: nat,N: nat,P2: nat,F2: fun(nat,A)] :
          ( pp(aa(nat,bool,ord_less_eq(nat,M),N))
         => ( pp(aa(nat,bool,ord_less_eq(nat,N),P2))
           => ( aa(A,A,minus_minus(A,aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F2),set_or7035219750837199246ssThan(nat,M,P2))),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F2),set_or7035219750837199246ssThan(nat,M,N))) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F2),set_or7035219750837199246ssThan(nat,N,P2)) ) ) ) ) ).

% sum_diff_nat_ivl
tff(fact_6072_size__list__pointwise,axiom,
    ! [A: $tType,Xs: list(A),F2: fun(A,nat),G: fun(A,nat)] :
      ( ! [X3: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),aa(list(A),set(A),set2(A),Xs)))
         => pp(aa(nat,bool,ord_less_eq(nat,aa(A,nat,F2,X3)),aa(A,nat,G,X3))) )
     => pp(aa(nat,bool,ord_less_eq(nat,size_list(A,F2,Xs)),size_list(A,G,Xs))) ) ).

% size_list_pointwise
tff(fact_6073_size__list__estimation_H,axiom,
    ! [A: $tType,X: A,Xs: list(A),Y: nat,F2: fun(A,nat)] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(list(A),set(A),set2(A),Xs)))
     => ( pp(aa(nat,bool,ord_less_eq(nat,Y),aa(A,nat,F2,X)))
       => pp(aa(nat,bool,ord_less_eq(nat,Y),size_list(A,F2,Xs))) ) ) ).

% size_list_estimation'
tff(fact_6074_prod_OatLeastLessThan__concat,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [M: nat,N: nat,P2: nat,G: fun(nat,A)] :
          ( pp(aa(nat,bool,ord_less_eq(nat,M),N))
         => ( pp(aa(nat,bool,ord_less_eq(nat,N),P2))
           => ( aa(A,A,aa(A,fun(A,A),times_times(A),groups7121269368397514597t_prod(nat,A,G,set_or7035219750837199246ssThan(nat,M,N))),groups7121269368397514597t_prod(nat,A,G,set_or7035219750837199246ssThan(nat,N,P2))) = groups7121269368397514597t_prod(nat,A,G,set_or7035219750837199246ssThan(nat,M,P2)) ) ) ) ) ).

% prod.atLeastLessThan_concat
tff(fact_6075_atLeast0__lessThan__Suc,axiom,
    ! [N: nat] : set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,suc,N)) = insert(nat,N,set_or7035219750837199246ssThan(nat,zero_zero(nat),N)) ).

% atLeast0_lessThan_Suc
tff(fact_6076_subset__eq__atLeast0__lessThan__finite,axiom,
    ! [N4: set(nat),N: nat] :
      ( pp(aa(set(nat),bool,ord_less_eq(set(nat),N4),set_or7035219750837199246ssThan(nat,zero_zero(nat),N)))
     => pp(aa(set(nat),bool,finite_finite2(nat),N4)) ) ).

% subset_eq_atLeast0_lessThan_finite
tff(fact_6077_atLeastAtMost__subseteq__atLeastLessThan__iff,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A2: A,B2: A,C2: A,D: A] :
          ( pp(aa(set(A),bool,ord_less_eq(set(A),set_or1337092689740270186AtMost(A,A2,B2)),set_or7035219750837199246ssThan(A,C2,D)))
        <=> ( pp(aa(A,bool,ord_less_eq(A,A2),B2))
           => ( pp(aa(A,bool,ord_less_eq(A,C2),A2))
              & pp(aa(A,bool,ord_less(A,B2),D)) ) ) ) ) ).

% atLeastAtMost_subseteq_atLeastLessThan_iff
tff(fact_6078_atLeastLessThan__subseteq__atLeastAtMost__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A2: A,B2: A,C2: A,D: A] :
          ( pp(aa(set(A),bool,ord_less_eq(set(A),set_or7035219750837199246ssThan(A,A2,B2)),set_or1337092689740270186AtMost(A,C2,D)))
        <=> ( pp(aa(A,bool,ord_less(A,A2),B2))
           => ( pp(aa(A,bool,ord_less_eq(A,C2),A2))
              & pp(aa(A,bool,ord_less_eq(A,B2),D)) ) ) ) ) ).

% atLeastLessThan_subseteq_atLeastAtMost_iff
tff(fact_6079_sum__shift__lb__Suc0__0__upt,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [F2: fun(nat,A),K: nat] :
          ( ( aa(nat,A,F2,zero_zero(nat)) = zero_zero(A) )
         => ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F2),set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,zero_zero(nat)),K)) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F2),set_or7035219750837199246ssThan(nat,zero_zero(nat),K)) ) ) ) ).

% sum_shift_lb_Suc0_0_upt
tff(fact_6080_sum_OatLeast0__lessThan__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),N: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or7035219750837199246ssThan(nat,zero_zero(nat),N))),aa(nat,A,G,N)) ) ).

% sum.atLeast0_lessThan_Suc
tff(fact_6081_sum_OatLeast__Suc__lessThan,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [M: nat,N: nat,G: fun(nat,A)] :
          ( pp(aa(nat,bool,ord_less(nat,M),N))
         => ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or7035219750837199246ssThan(nat,M,N)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G,M)),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,M),N))) ) ) ) ).

% sum.atLeast_Suc_lessThan
tff(fact_6082_sum_OatLeastLessThan__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [A2: nat,B2: nat,G: fun(nat,A)] :
          ( pp(aa(nat,bool,ord_less_eq(nat,A2),B2))
         => ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or7035219750837199246ssThan(nat,A2,aa(nat,nat,suc,B2))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or7035219750837199246ssThan(nat,A2,B2))),aa(nat,A,G,B2)) ) ) ) ).

% sum.atLeastLessThan_Suc
tff(fact_6083_prod_OatLeast0__lessThan__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),N: nat] : groups7121269368397514597t_prod(nat,A,G,set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),times_times(A),groups7121269368397514597t_prod(nat,A,G,set_or7035219750837199246ssThan(nat,zero_zero(nat),N))),aa(nat,A,G,N)) ) ).

% prod.atLeast0_lessThan_Suc
tff(fact_6084_prod_OatLeast__Suc__lessThan,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [M: nat,N: nat,G: fun(nat,A)] :
          ( pp(aa(nat,bool,ord_less(nat,M),N))
         => ( groups7121269368397514597t_prod(nat,A,G,set_or7035219750837199246ssThan(nat,M,N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G,M)),groups7121269368397514597t_prod(nat,A,G,set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,M),N))) ) ) ) ).

% prod.atLeast_Suc_lessThan
tff(fact_6085_prod_OatLeastLessThan__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A2: nat,B2: nat,G: fun(nat,A)] :
          ( pp(aa(nat,bool,ord_less_eq(nat,A2),B2))
         => ( groups7121269368397514597t_prod(nat,A,G,set_or7035219750837199246ssThan(nat,A2,aa(nat,nat,suc,B2))) = aa(A,A,aa(A,fun(A,A),times_times(A),groups7121269368397514597t_prod(nat,A,G,set_or7035219750837199246ssThan(nat,A2,B2))),aa(nat,A,G,B2)) ) ) ) ).

% prod.atLeastLessThan_Suc
tff(fact_6086_sum_Olast__plus,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [M: nat,N: nat,G: fun(nat,A)] :
          ( pp(aa(nat,bool,ord_less_eq(nat,M),N))
         => ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or1337092689740270186AtMost(nat,M,N)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G,N)),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or7035219750837199246ssThan(nat,M,N))) ) ) ) ).

% sum.last_plus
tff(fact_6087_prod_Olast__plus,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [M: nat,N: nat,G: fun(nat,A)] :
          ( pp(aa(nat,bool,ord_less_eq(nat,M),N))
         => ( groups7121269368397514597t_prod(nat,A,G,set_or1337092689740270186AtMost(nat,M,N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G,N)),groups7121269368397514597t_prod(nat,A,G,set_or7035219750837199246ssThan(nat,M,N))) ) ) ) ).

% prod.last_plus
tff(fact_6088_atLeastLessThanSuc,axiom,
    ! [M: nat,N: nat] :
      ( ( pp(aa(nat,bool,ord_less_eq(nat,M),N))
       => ( set_or7035219750837199246ssThan(nat,M,aa(nat,nat,suc,N)) = insert(nat,N,set_or7035219750837199246ssThan(nat,M,N)) ) )
      & ( ~ pp(aa(nat,bool,ord_less_eq(nat,M),N))
       => ( set_or7035219750837199246ssThan(nat,M,aa(nat,nat,suc,N)) = bot_bot(set(nat)) ) ) ) ).

% atLeastLessThanSuc
tff(fact_6089_sum__Suc__diff_H,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [M: nat,N: nat,F2: fun(nat,A)] :
          ( pp(aa(nat,bool,ord_less_eq(nat,M),N))
         => ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_eg(fun(nat,A),fun(nat,A),F2)),set_or7035219750837199246ssThan(nat,M,N)) = aa(A,A,minus_minus(A,aa(nat,A,F2,N)),aa(nat,A,F2,M)) ) ) ) ).

% sum_Suc_diff'
tff(fact_6090_atLeast0__lessThan__Suc__eq__insert__0,axiom,
    ! [N: nat] : set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,suc,N)) = insert(nat,zero_zero(nat),aa(set(nat),set(nat),image(nat,nat,suc),set_or7035219750837199246ssThan(nat,zero_zero(nat),N))) ).

% atLeast0_lessThan_Suc_eq_insert_0
tff(fact_6091_sum_OatLeastLessThan__rev,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),N: nat,M: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or7035219750837199246ssThan(nat,N,M)) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aa(nat,fun(nat,fun(nat,A)),aTP_Lamp_ov(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),G),N),M)),set_or7035219750837199246ssThan(nat,N,M)) ) ).

% sum.atLeastLessThan_rev
tff(fact_6092_sum_Onested__swap,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [A2: fun(nat,fun(nat,A)),N: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_ow(fun(nat,fun(nat,A)),fun(nat,A),A2)),set_or1337092689740270186AtMost(nat,zero_zero(nat),N)) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aTP_Lamp_ie(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),A2),N)),set_or7035219750837199246ssThan(nat,zero_zero(nat),N)) ) ).

% sum.nested_swap
tff(fact_6093_prod_OatLeastLessThan__rev,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),N: nat,M: nat] : groups7121269368397514597t_prod(nat,A,G,set_or7035219750837199246ssThan(nat,N,M)) = groups7121269368397514597t_prod(nat,A,aa(nat,fun(nat,A),aa(nat,fun(nat,fun(nat,A)),aTP_Lamp_ox(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),G),N),M),set_or7035219750837199246ssThan(nat,N,M)) ) ).

% prod.atLeastLessThan_rev
tff(fact_6094_prod_Onested__swap,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A2: fun(nat,fun(nat,A)),N: nat] : groups7121269368397514597t_prod(nat,A,aTP_Lamp_oy(fun(nat,fun(nat,A)),fun(nat,A),A2),set_or1337092689740270186AtMost(nat,zero_zero(nat),N)) = groups7121269368397514597t_prod(nat,A,aa(nat,fun(nat,A),aTP_Lamp_ih(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),A2),N),set_or7035219750837199246ssThan(nat,zero_zero(nat),N)) ) ).

% prod.nested_swap
tff(fact_6095_sum_Onat__group,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),K: nat,N: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aTP_Lamp_oz(fun(nat,A),fun(nat,fun(nat,A)),G),K)),set_ord_lessThan(nat,N)) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_ord_lessThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),K))) ) ).

% sum.nat_group
tff(fact_6096_prod_Onat__group,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),K: nat,N: nat] : groups7121269368397514597t_prod(nat,A,aa(nat,fun(nat,A),aTP_Lamp_pa(fun(nat,A),fun(nat,fun(nat,A)),G),K),set_ord_lessThan(nat,N)) = groups7121269368397514597t_prod(nat,A,G,set_ord_lessThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),K))) ) ).

% prod.nat_group
tff(fact_6097_prod__Suc__fact,axiom,
    ! [N: nat] : groups7121269368397514597t_prod(nat,nat,suc,set_or7035219750837199246ssThan(nat,zero_zero(nat),N)) = semiring_char_0_fact(nat,N) ).

% prod_Suc_fact
tff(fact_6098_prod__Suc__Suc__fact,axiom,
    ! [N: nat] : groups7121269368397514597t_prod(nat,nat,suc,set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,zero_zero(nat)),N)) = semiring_char_0_fact(nat,N) ).

% prod_Suc_Suc_fact
tff(fact_6099_sum_Ohead__if,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [N: nat,M: nat,G: fun(nat,A)] :
          ( ( pp(aa(nat,bool,ord_less(nat,N),M))
           => ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or1337092689740270186AtMost(nat,M,N)) = zero_zero(A) ) )
          & ( ~ pp(aa(nat,bool,ord_less(nat,N),M))
           => ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or1337092689740270186AtMost(nat,M,N)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or7035219750837199246ssThan(nat,M,N))),aa(nat,A,G,N)) ) ) ) ) ).

% sum.head_if
tff(fact_6100_prod_Ohead__if,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [N: nat,M: nat,G: fun(nat,A)] :
          ( ( pp(aa(nat,bool,ord_less(nat,N),M))
           => ( groups7121269368397514597t_prod(nat,A,G,set_or1337092689740270186AtMost(nat,M,N)) = one_one(A) ) )
          & ( ~ pp(aa(nat,bool,ord_less(nat,N),M))
           => ( groups7121269368397514597t_prod(nat,A,G,set_or1337092689740270186AtMost(nat,M,N)) = aa(A,A,aa(A,fun(A,A),times_times(A),groups7121269368397514597t_prod(nat,A,G,set_or7035219750837199246ssThan(nat,M,N))),aa(nat,A,G,N)) ) ) ) ) ).

% prod.head_if
tff(fact_6101_atLeastLessThan__nat__numeral,axiom,
    ! [M: nat,K: num] :
      ( ( pp(aa(nat,bool,ord_less_eq(nat,M),pred_numeral(K)))
       => ( set_or7035219750837199246ssThan(nat,M,aa(num,nat,numeral_numeral(nat),K)) = insert(nat,pred_numeral(K),set_or7035219750837199246ssThan(nat,M,pred_numeral(K))) ) )
      & ( ~ pp(aa(nat,bool,ord_less_eq(nat,M),pred_numeral(K)))
       => ( set_or7035219750837199246ssThan(nat,M,aa(num,nat,numeral_numeral(nat),K)) = bot_bot(set(nat)) ) ) ) ).

% atLeastLessThan_nat_numeral
tff(fact_6102_fact__prod__Suc,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [N: nat] : semiring_char_0_fact(A,N) = aa(nat,A,semiring_1_of_nat(A),groups7121269368397514597t_prod(nat,nat,suc,set_or7035219750837199246ssThan(nat,zero_zero(nat),N))) ) ).

% fact_prod_Suc
tff(fact_6103_sum_OatLeastLessThan__rev__at__least__Suc__atMost,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),N: nat,M: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or7035219750837199246ssThan(nat,N,M)) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aa(nat,fun(nat,fun(nat,A)),aTP_Lamp_ef(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),G),N),M)),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,N),M)) ) ).

% sum.atLeastLessThan_rev_at_least_Suc_atMost
tff(fact_6104_prod_OatLeastLessThan__rev__at__least__Suc__atMost,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),N: nat,M: nat] : groups7121269368397514597t_prod(nat,A,G,set_or7035219750837199246ssThan(nat,N,M)) = groups7121269368397514597t_prod(nat,A,aa(nat,fun(nat,A),aa(nat,fun(nat,fun(nat,A)),aTP_Lamp_ht(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),G),N),M),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,N),M)) ) ).

% prod.atLeastLessThan_rev_at_least_Suc_atMost
tff(fact_6105_pochhammer__prod,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A2: A,N: nat] : comm_s3205402744901411588hammer(A,A2,N) = groups7121269368397514597t_prod(nat,A,aTP_Lamp_hv(A,fun(nat,A),A2),set_or7035219750837199246ssThan(nat,zero_zero(nat),N)) ) ).

% pochhammer_prod
tff(fact_6106_fact__prod__rev,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [N: nat] : semiring_char_0_fact(A,N) = aa(nat,A,semiring_1_of_nat(A),groups7121269368397514597t_prod(nat,nat,minus_minus(nat,N),set_or7035219750837199246ssThan(nat,zero_zero(nat),N))) ) ).

% fact_prod_rev
tff(fact_6107_summable__Cauchy,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [F2: fun(nat,A)] :
          ( summable(A,F2)
        <=> ! [E4: real] :
              ( pp(aa(real,bool,ord_less(real,zero_zero(real)),E4))
             => ? [N8: nat] :
                ! [M6: nat] :
                  ( pp(aa(nat,bool,ord_less_eq(nat,N8),M6))
                 => ! [N5: nat] : pp(aa(real,bool,ord_less(real,real_V7770717601297561774m_norm(A,aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F2),set_or7035219750837199246ssThan(nat,M6,N5)))),E4)) ) ) ) ) ).

% summable_Cauchy
tff(fact_6108_sums__group,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [F2: fun(nat,A),S3: A,K: nat] :
          ( pp(aa(A,bool,sums(A,F2),S3))
         => ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),K))
           => pp(aa(A,bool,sums(A,aa(nat,fun(nat,A),aTP_Lamp_pb(fun(nat,A),fun(nat,fun(nat,A)),F2),K)),S3)) ) ) ) ).

% sums_group
tff(fact_6109_atLeast1__lessThan__eq__remove0,axiom,
    ! [N: nat] : set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,zero_zero(nat)),N) = aa(set(nat),set(nat),minus_minus(set(nat),set_ord_lessThan(nat,N)),insert(nat,zero_zero(nat),bot_bot(set(nat)))) ).

% atLeast1_lessThan_eq_remove0
tff(fact_6110_take__bit__sum,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [N: nat,A2: A] : aa(A,A,bit_se2584673776208193580ke_bit(A,N),A2) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_pc(A,fun(nat,A),A2)),set_or7035219750837199246ssThan(nat,zero_zero(nat),N)) ) ).

% take_bit_sum
tff(fact_6111_image__minus__const__atLeastLessThan__nat,axiom,
    ! [C2: nat,Y: nat,X: nat] :
      ( ( pp(aa(nat,bool,ord_less(nat,C2),Y))
       => ( aa(set(nat),set(nat),image(nat,nat,aTP_Lamp_pd(nat,fun(nat,nat),C2)),set_or7035219750837199246ssThan(nat,X,Y)) = set_or7035219750837199246ssThan(nat,aa(nat,nat,minus_minus(nat,X),C2),aa(nat,nat,minus_minus(nat,Y),C2)) ) )
      & ( ~ pp(aa(nat,bool,ord_less(nat,C2),Y))
       => ( ( pp(aa(nat,bool,ord_less(nat,X),Y))
           => ( aa(set(nat),set(nat),image(nat,nat,aTP_Lamp_pd(nat,fun(nat,nat),C2)),set_or7035219750837199246ssThan(nat,X,Y)) = insert(nat,zero_zero(nat),bot_bot(set(nat))) ) )
          & ( ~ pp(aa(nat,bool,ord_less(nat,X),Y))
           => ( aa(set(nat),set(nat),image(nat,nat,aTP_Lamp_pd(nat,fun(nat,nat),C2)),set_or7035219750837199246ssThan(nat,X,Y)) = bot_bot(set(nat)) ) ) ) ) ) ).

% image_minus_const_atLeastLessThan_nat
tff(fact_6112_fact__split,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [K: nat,N: nat] :
          ( pp(aa(nat,bool,ord_less_eq(nat,K),N))
         => ( semiring_char_0_fact(A,N) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),groups7121269368397514597t_prod(nat,nat,suc,set_or7035219750837199246ssThan(nat,aa(nat,nat,minus_minus(nat,N),K),N)))),semiring_char_0_fact(A,aa(nat,nat,minus_minus(nat,N),K))) ) ) ) ).

% fact_split
tff(fact_6113_binomial__altdef__of__nat,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,N: nat] :
          ( pp(aa(nat,bool,ord_less_eq(nat,K),N))
         => ( aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,binomial(N),K)) = groups7121269368397514597t_prod(nat,A,aa(nat,fun(nat,A),aTP_Lamp_pe(nat,fun(nat,fun(nat,A)),K),N),set_or7035219750837199246ssThan(nat,zero_zero(nat),K)) ) ) ) ).

% binomial_altdef_of_nat
tff(fact_6114_gbinomial__altdef__of__nat,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A,K: nat] : aa(nat,A,gbinomial(A,A2),K) = groups7121269368397514597t_prod(nat,A,aa(nat,fun(nat,A),aTP_Lamp_pf(A,fun(nat,fun(nat,A)),A2),K),set_or7035219750837199246ssThan(nat,zero_zero(nat),K)) ) ).

% gbinomial_altdef_of_nat
tff(fact_6115_horner__sum__eq__sum,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_semiring_1(A)
     => ! [F2: fun(B,A),A2: A,Xs: list(B)] : groups4207007520872428315er_sum(B,A,F2,A2,Xs) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(list(B),fun(nat,A),aa(A,fun(list(B),fun(nat,A)),aTP_Lamp_pg(fun(B,A),fun(A,fun(list(B),fun(nat,A))),F2),A2),Xs)),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(list(B),nat,size_size(list(B)),Xs))) ) ).

% horner_sum_eq_sum
tff(fact_6116_gbinomial__mult__fact,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,A2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),semiring_char_0_fact(A,K)),aa(nat,A,gbinomial(A,A2),K)) = groups7121269368397514597t_prod(nat,A,aTP_Lamp_ph(A,fun(nat,A),A2),set_or7035219750837199246ssThan(nat,zero_zero(nat),K)) ) ).

% gbinomial_mult_fact
tff(fact_6117_gbinomial__mult__fact_H,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A,K: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,gbinomial(A,A2),K)),semiring_char_0_fact(A,K)) = groups7121269368397514597t_prod(nat,A,aTP_Lamp_ph(A,fun(nat,A),A2),set_or7035219750837199246ssThan(nat,zero_zero(nat),K)) ) ).

% gbinomial_mult_fact'
tff(fact_6118_gbinomial__prod__rev,axiom,
    ! [A: $tType] :
      ( ( semiring_char_0(A)
        & semidom_divide(A) )
     => ! [A2: A,K: nat] : aa(nat,A,gbinomial(A,A2),K) = aa(A,A,aa(A,fun(A,A),divide_divide(A),groups7121269368397514597t_prod(nat,A,aTP_Lamp_hz(A,fun(nat,A),A2),set_or7035219750837199246ssThan(nat,zero_zero(nat),K))),semiring_char_0_fact(A,K)) ) ).

% gbinomial_prod_rev
tff(fact_6119_sum__power2,axiom,
    ! [K: nat] : aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2)))),set_or7035219750837199246ssThan(nat,zero_zero(nat),K)) = aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),K)),one_one(nat)) ).

% sum_power2
tff(fact_6120_Chebyshev__sum__upper,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [N: nat,A2: fun(nat,A),B2: fun(nat,A)] :
          ( ! [I2: nat,J2: nat] :
              ( pp(aa(nat,bool,ord_less_eq(nat,I2),J2))
             => ( pp(aa(nat,bool,ord_less(nat,J2),N))
               => pp(aa(A,bool,ord_less_eq(A,aa(nat,A,A2,I2)),aa(nat,A,A2,J2))) ) )
         => ( ! [I2: nat,J2: nat] :
                ( pp(aa(nat,bool,ord_less_eq(nat,I2),J2))
               => ( pp(aa(nat,bool,ord_less(nat,J2),N))
                 => pp(aa(A,bool,ord_less_eq(A,aa(nat,A,B2,J2)),aa(nat,A,B2,I2))) ) )
           => pp(aa(A,bool,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),N)),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_pi(fun(nat,A),fun(fun(nat,A),fun(nat,A)),A2),B2)),set_or7035219750837199246ssThan(nat,zero_zero(nat),N)))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,A2),set_or7035219750837199246ssThan(nat,zero_zero(nat),N))),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,B2),set_or7035219750837199246ssThan(nat,zero_zero(nat),N))))) ) ) ) ).

% Chebyshev_sum_upper
tff(fact_6121_Chebyshev__sum__upper__nat,axiom,
    ! [N: nat,A2: fun(nat,nat),B2: fun(nat,nat)] :
      ( ! [I2: nat,J2: nat] :
          ( pp(aa(nat,bool,ord_less_eq(nat,I2),J2))
         => ( pp(aa(nat,bool,ord_less(nat,J2),N))
           => pp(aa(nat,bool,ord_less_eq(nat,aa(nat,nat,A2,I2)),aa(nat,nat,A2,J2))) ) )
     => ( ! [I2: nat,J2: nat] :
            ( pp(aa(nat,bool,ord_less_eq(nat,I2),J2))
           => ( pp(aa(nat,bool,ord_less(nat,J2),N))
             => pp(aa(nat,bool,ord_less_eq(nat,aa(nat,nat,B2,J2)),aa(nat,nat,B2,I2))) ) )
       => pp(aa(nat,bool,ord_less_eq(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,aa(fun(nat,nat),fun(nat,nat),aTP_Lamp_pj(fun(nat,nat),fun(fun(nat,nat),fun(nat,nat)),A2),B2)),set_or7035219750837199246ssThan(nat,zero_zero(nat),N)))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,A2),set_or7035219750837199246ssThan(nat,zero_zero(nat),N))),aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,B2),set_or7035219750837199246ssThan(nat,zero_zero(nat),N))))) ) ) ).

% Chebyshev_sum_upper_nat
tff(fact_6122_finite__atLeastZeroLessThan__integer,axiom,
    ! [U: code_integer] : pp(aa(set(code_integer),bool,finite_finite2(code_integer),set_or7035219750837199246ssThan(code_integer,zero_zero(code_integer),U))) ).

% finite_atLeastZeroLessThan_integer
tff(fact_6123_finite__atLeastZeroLessThan__int,axiom,
    ! [U: int] : pp(aa(set(int),bool,finite_finite2(int),set_or7035219750837199246ssThan(int,zero_zero(int),U))) ).

% finite_atLeastZeroLessThan_int
tff(fact_6124_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
tff(fact_6125_atLeastLessThanPlusOne__atLeastAtMost__integer,axiom,
    ! [L: code_integer,U: code_integer] : set_or7035219750837199246ssThan(code_integer,L,aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),U),one_one(code_integer))) = set_or1337092689740270186AtMost(code_integer,L,U) ).

% atLeastLessThanPlusOne_atLeastAtMost_integer
tff(fact_6126_atLeastLessThanPlusOne__atLeastAtMost__int,axiom,
    ! [L: int,U: int] : set_or7035219750837199246ssThan(int,L,aa(int,int,aa(int,fun(int,int),plus_plus(int),U),one_one(int))) = set_or1337092689740270186AtMost(int,L,U) ).

% atLeastLessThanPlusOne_atLeastAtMost_int
tff(fact_6127_image__add__int__atLeastLessThan,axiom,
    ! [L: int,U: int] : aa(set(int),set(int),image(int,int,aTP_Lamp_pk(int,fun(int,int),L)),set_or7035219750837199246ssThan(int,zero_zero(int),aa(int,int,minus_minus(int,U),L))) = set_or7035219750837199246ssThan(int,L,U) ).

% image_add_int_atLeastLessThan
tff(fact_6128_image__int__atLeastLessThan,axiom,
    ! [A2: nat,B2: nat] : aa(set(nat),set(int),image(nat,int,semiring_1_of_nat(int)),set_or7035219750837199246ssThan(nat,A2,B2)) = set_or7035219750837199246ssThan(int,aa(nat,int,semiring_1_of_nat(int),A2),aa(nat,int,semiring_1_of_nat(int),B2)) ).

% image_int_atLeastLessThan
tff(fact_6129_image__add__integer__atLeastLessThan,axiom,
    ! [L: code_integer,U: code_integer] : aa(set(code_integer),set(code_integer),image(code_integer,code_integer,aTP_Lamp_pl(code_integer,fun(code_integer,code_integer),L)),set_or7035219750837199246ssThan(code_integer,zero_zero(code_integer),aa(code_integer,code_integer,minus_minus(code_integer,U),L))) = set_or7035219750837199246ssThan(code_integer,L,U) ).

% image_add_integer_atLeastLessThan
tff(fact_6130_word__atLeastLessThan__Suc__atLeastAtMost,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [U: word(A),L: word(A)] :
          ( ( U != aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A))) )
         => ( set_or7035219750837199246ssThan(word(A),L,aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),U),one_one(word(A)))) = set_or1337092689740270186AtMost(word(A),L,U) ) ) ) ).

% word_atLeastLessThan_Suc_atLeastAtMost
tff(fact_6131_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,aa(word(A),word(A),minus_minus(word(A),B2),one_one(word(A)))) = set_or7035219750837199246ssThan(word(A),A2,B2) ) ) ) ).

% word_range_minus_1
tff(fact_6132_image__atLeastZeroLessThan__int,axiom,
    ! [U: int] :
      ( pp(aa(int,bool,ord_less_eq(int,zero_zero(int)),U))
     => ( set_or7035219750837199246ssThan(int,zero_zero(int),U) = aa(set(nat),set(int),image(nat,int,semiring_1_of_nat(int)),set_ord_lessThan(nat,nat2(U))) ) ) ).

% image_atLeastZeroLessThan_int
tff(fact_6133_VEBT_Osize__gen_I1_J,axiom,
    ! [X11: option(product_prod(nat,nat)),X12: nat,X13: list(vEBT_VEBT),X14: vEBT_VEBT] : aa(vEBT_VEBT,nat,vEBT_size_VEBT,vEBT_Node(X11,X12,X13,X14)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),size_list(vEBT_VEBT,vEBT_size_VEBT,X13)),aa(vEBT_VEBT,nat,vEBT_size_VEBT,X14))),aa(nat,nat,suc,zero_zero(nat))) ).

% VEBT.size_gen(1)
tff(fact_6134_int__of__nat__def,axiom,
    code_T6385005292777649522of_nat = semiring_1_of_nat(int) ).

% int_of_nat_def
tff(fact_6135_VEBT_Osize__gen_I2_J,axiom,
    ! [X21: bool,X222: bool] : aa(vEBT_VEBT,nat,vEBT_size_VEBT,vEBT_Leaf(X21,X222)) = zero_zero(nat) ).

% VEBT.size_gen(2)
tff(fact_6136_image__Collect__subsetI,axiom,
    ! [A: $tType,B: $tType,P: fun(A,bool),F2: fun(A,B),B4: set(B)] :
      ( ! [X3: A] :
          ( pp(aa(A,bool,P,X3))
         => pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),aa(A,B,F2,X3)),B4)) )
     => pp(aa(set(B),bool,ord_less_eq(set(B),aa(set(A),set(B),image(A,B,F2),collect(A,P))),B4)) ) ).

% image_Collect_subsetI
tff(fact_6137_length__product__lists,axiom,
    ! [B: $tType,Xss: list(list(B))] : aa(list(list(B)),nat,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
tff(fact_6138_Collect__restrict,axiom,
    ! [A: $tType,X6: set(A),P: fun(A,bool)] : pp(aa(set(A),bool,ord_less_eq(set(A),collect(A,aa(fun(A,bool),fun(A,bool),aTP_Lamp_ae(set(A),fun(fun(A,bool),fun(A,bool)),X6),P))),X6)) ).

% Collect_restrict
tff(fact_6139_prop__restrict,axiom,
    ! [A: $tType,X: A,Z6: set(A),X6: set(A),P: fun(A,bool)] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),Z6))
     => ( pp(aa(set(A),bool,ord_less_eq(set(A),Z6),collect(A,aa(fun(A,bool),fun(A,bool),aTP_Lamp_ae(set(A),fun(fun(A,bool),fun(A,bool)),X6),P))))
       => pp(aa(A,bool,P,X)) ) ) ).

% prop_restrict
tff(fact_6140_subset__emptyI,axiom,
    ! [A: $tType,A3: set(A)] :
      ( ! [X3: A] : ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),A3))
     => pp(aa(set(A),bool,ord_less_eq(set(A),A3),bot_bot(set(A)))) ) ).

% subset_emptyI
tff(fact_6141_insert__subsetI,axiom,
    ! [A: $tType,X: A,A3: set(A),X6: set(A)] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A3))
     => ( pp(aa(set(A),bool,ord_less_eq(set(A),X6),A3))
       => pp(aa(set(A),bool,ord_less_eq(set(A),insert(A,X,X6)),A3)) ) ) ).

% insert_subsetI
tff(fact_6142_length__subseqs,axiom,
    ! [A: $tType,Xs: list(A)] : aa(list(list(A)),nat,size_size(list(list(A))),subseqs(A,Xs)) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(list(A),nat,size_size(list(A)),Xs)) ).

% length_subseqs
tff(fact_6143_length__mul__elem,axiom,
    ! [A: $tType,Xs: list(list(A)),N: nat] :
      ( ! [X3: list(A)] :
          ( pp(aa(set(list(A)),bool,aa(list(A),fun(set(list(A)),bool),member(list(A)),X3),aa(list(list(A)),set(list(A)),set2(list(A)),Xs)))
         => ( aa(list(A),nat,size_size(list(A)),X3) = N ) )
     => ( aa(list(A),nat,size_size(list(A)),concat(A,Xs)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(list(list(A)),nat,size_size(list(list(A))),Xs)),N) ) ) ).

% length_mul_elem
tff(fact_6144_product__concat__map,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),Ys: list(B)] : product(A,B,Xs,Ys) = concat(product_prod(A,B),map(A,list(product_prod(A,B)),aTP_Lamp_pm(list(B),fun(A,list(product_prod(A,B))),Ys),Xs)) ).

% product_concat_map
tff(fact_6145_subset__subseqs,axiom,
    ! [A: $tType,X6: set(A),Xs: list(A)] :
      ( pp(aa(set(A),bool,ord_less_eq(set(A),X6),aa(list(A),set(A),set2(A),Xs)))
     => pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),X6),aa(set(list(A)),set(set(A)),image(list(A),set(A),set2(A)),aa(list(list(A)),set(list(A)),set2(list(A)),subseqs(A,Xs))))) ) ).

% subset_subseqs
tff(fact_6146_set__n__lists,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] : aa(list(list(A)),set(list(A)),set2(list(A)),n_lists(A,N,Xs)) = collect(list(A),aa(list(A),fun(list(A),bool),aTP_Lamp_pn(nat,fun(list(A),fun(list(A),bool)),N),Xs)) ).

% set_n_lists
tff(fact_6147_smod__int__range,axiom,
    ! [B2: int,A2: int] :
      ( ( B2 != zero_zero(int) )
     => pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),signed6721504322012087516modulo(int,A2,B2)),set_or1337092689740270186AtMost(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,uminus_uminus(int),aa(int,int,abs_abs(int),B2))),one_one(int)),aa(int,int,minus_minus(int,aa(int,int,abs_abs(int),B2)),one_one(int))))) ) ).

% smod_int_range
tff(fact_6148_smod__int__0__mod,axiom,
    ! [X: int] : signed6721504322012087516modulo(int,zero_zero(int),X) = zero_zero(int) ).

% smod_int_0_mod
tff(fact_6149_smod__int__mod__0,axiom,
    ! [X: int] : signed6721504322012087516modulo(int,X,zero_zero(int)) = X ).

% smod_int_mod_0
tff(fact_6150_smod__int__numeral__numeral,axiom,
    ! [M: num,N: num] : signed6721504322012087516modulo(int,aa(num,int,numeral_numeral(int),M),aa(num,int,numeral_numeral(int),N)) = modulo_modulo(int,aa(num,int,numeral_numeral(int),M),aa(num,int,numeral_numeral(int),N)) ).

% smod_int_numeral_numeral
tff(fact_6151_smod__int__compares_I1_J,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa(int,bool,ord_less_eq(int,zero_zero(int)),A2))
     => ( pp(aa(int,bool,ord_less(int,zero_zero(int)),B2))
       => pp(aa(int,bool,ord_less(int,signed6721504322012087516modulo(int,A2,B2)),B2)) ) ) ).

% smod_int_compares(1)
tff(fact_6152_smod__int__compares_I2_J,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa(int,bool,ord_less_eq(int,zero_zero(int)),A2))
     => ( pp(aa(int,bool,ord_less(int,zero_zero(int)),B2))
       => pp(aa(int,bool,ord_less_eq(int,zero_zero(int)),signed6721504322012087516modulo(int,A2,B2))) ) ) ).

% smod_int_compares(2)
tff(fact_6153_smod__int__compares_I4_J,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa(int,bool,ord_less_eq(int,A2),zero_zero(int)))
     => ( pp(aa(int,bool,ord_less(int,zero_zero(int)),B2))
       => pp(aa(int,bool,ord_less_eq(int,signed6721504322012087516modulo(int,A2,B2)),zero_zero(int))) ) ) ).

% smod_int_compares(4)
tff(fact_6154_smod__int__compares_I6_J,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa(int,bool,ord_less_eq(int,zero_zero(int)),A2))
     => ( pp(aa(int,bool,ord_less(int,B2),zero_zero(int)))
       => pp(aa(int,bool,ord_less_eq(int,zero_zero(int)),signed6721504322012087516modulo(int,A2,B2))) ) ) ).

% smod_int_compares(6)
tff(fact_6155_smod__int__compares_I7_J,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa(int,bool,ord_less_eq(int,A2),zero_zero(int)))
     => ( pp(aa(int,bool,ord_less(int,B2),zero_zero(int)))
       => pp(aa(int,bool,ord_less_eq(int,signed6721504322012087516modulo(int,A2,B2)),zero_zero(int))) ) ) ).

% smod_int_compares(7)
tff(fact_6156_smod__int__compares_I8_J,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa(int,bool,ord_less_eq(int,A2),zero_zero(int)))
     => ( pp(aa(int,bool,ord_less(int,B2),zero_zero(int)))
       => pp(aa(int,bool,ord_less_eq(int,B2),signed6721504322012087516modulo(int,A2,B2))) ) ) ).

% smod_int_compares(8)
tff(fact_6157_smod__mod__positive,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa(int,bool,ord_less_eq(int,zero_zero(int)),A2))
     => ( pp(aa(int,bool,ord_less_eq(int,zero_zero(int)),B2))
       => ( signed6721504322012087516modulo(int,A2,B2) = modulo_modulo(int,A2,B2) ) ) ) ).

% smod_mod_positive
tff(fact_6158_signed__modulo__int__def,axiom,
    ! [K: int,L: int] : signed6721504322012087516modulo(int,K,L) = aa(int,int,minus_minus(int,K),aa(int,int,aa(int,fun(int,int),times_times(int),signed7115095781618012415divide(int,K,L)),L)) ).

% signed_modulo_int_def
tff(fact_6159_smod__int__compares_I3_J,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa(int,bool,ord_less_eq(int,A2),zero_zero(int)))
     => ( pp(aa(int,bool,ord_less(int,zero_zero(int)),B2))
       => pp(aa(int,bool,ord_less(int,aa(int,int,uminus_uminus(int),B2)),signed6721504322012087516modulo(int,A2,B2))) ) ) ).

% smod_int_compares(3)
tff(fact_6160_smod__int__compares_I5_J,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa(int,bool,ord_less_eq(int,zero_zero(int)),A2))
     => ( pp(aa(int,bool,ord_less(int,B2),zero_zero(int)))
       => pp(aa(int,bool,ord_less(int,signed6721504322012087516modulo(int,A2,B2)),aa(int,int,uminus_uminus(int),B2))) ) ) ).

% smod_int_compares(5)
tff(fact_6161_smod__int__alt__def,axiom,
    ! [A2: int,B2: int] : signed6721504322012087516modulo(int,A2,B2) = aa(int,int,aa(int,fun(int,int),times_times(int),sgn_sgn(int,A2)),modulo_modulo(int,aa(int,int,abs_abs(int),A2),aa(int,int,abs_abs(int),B2))) ).

% smod_int_alt_def
tff(fact_6162_length__n__lists,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] : aa(list(list(A)),nat,size_size(list(list(A))),n_lists(A,N,Xs)) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(list(A),nat,size_size(list(A)),Xs)),N) ).

% length_n_lists
tff(fact_6163_product__code,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),Ys: list(B)] : product_product(A,B,aa(list(A),set(A),set2(A),Xs),aa(list(B),set(B),set2(B),Ys)) = aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),concat(product_prod(A,B),map(A,list(product_prod(A,B)),aTP_Lamp_pm(list(B),fun(A,list(product_prod(A,B))),Ys),Xs))) ).

% product_code
tff(fact_6164_valid__eq,axiom,
    ! [T: vEBT_VEBT,D: nat] :
      ( vEBT_VEBT_valid(T,D)
    <=> vEBT_invar_vebt(T,D) ) ).

% valid_eq
tff(fact_6165_valid__eq2,axiom,
    ! [T: vEBT_VEBT,D: nat] :
      ( vEBT_VEBT_valid(T,D)
     => vEBT_invar_vebt(T,D) ) ).

% valid_eq2
tff(fact_6166_valid__eq1,axiom,
    ! [T: vEBT_VEBT,D: nat] :
      ( vEBT_invar_vebt(T,D)
     => vEBT_VEBT_valid(T,D) ) ).

% valid_eq1
tff(fact_6167_VEBT__internal_Ovalid_H_Osimps_I1_J,axiom,
    ! [Uu: bool,Uv: bool,D: nat] :
      ( vEBT_VEBT_valid(vEBT_Leaf(Uu,Uv),D)
    <=> ( D = one_one(nat) ) ) ).

% VEBT_internal.valid'.simps(1)
tff(fact_6168_Multiset_Ois__empty__def,axiom,
    ! [A: $tType,A3: multiset(A)] :
      ( is_empty(A,A3)
    <=> ( A3 = zero_zero(multiset(A)) ) ) ).

% Multiset.is_empty_def
tff(fact_6169_mod__word__minus__1__minus__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [B2: num] : modulo_modulo(word(A),aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A))),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),B2))) = aa(int,word(A),ring_1_of_int(word(A)),modulo_modulo(int,aa(int,int,minus_minus(int,aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A)))),one_one(int)),aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),B2))))) ) ).

% mod_word_minus_1_minus_numeral
tff(fact_6170_drop__bit__word__beyond,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: nat,W: word(A)] :
          ( pp(aa(nat,bool,ord_less_eq(nat,type_len0_len_of(A,type2(A))),N))
         => ( bit_se4197421643247451524op_bit(word(A),N,W) = zero_zero(word(A)) ) ) ) ).

% drop_bit_word_beyond
tff(fact_6171_push__bit__word__beyond,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: nat,W: word(A)] :
          ( pp(aa(nat,bool,ord_less_eq(nat,type_len0_len_of(A,type2(A))),N))
         => ( bit_se4730199178511100633sh_bit(word(A),N,W) = zero_zero(word(A)) ) ) ) ).

% push_bit_word_beyond
tff(fact_6172_uint__bintrunc,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Bin: num] : aa(word(A),int,semiring_1_unsigned(A,int),aa(num,word(A),numeral_numeral(word(A)),Bin)) = aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),aa(num,int,numeral_numeral(int),Bin)) ) ).

% uint_bintrunc
tff(fact_6173_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))) = aa(A,A,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
tff(fact_6174_word__exp__length__eq__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),type_len0_len_of(A,type2(A))) = zero_zero(word(A)) ) ) ).

% word_exp_length_eq_0
tff(fact_6175_less__eq__word__numeral__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num,B2: num] :
          ( pp(aa(word(A),bool,ord_less_eq(word(A),aa(num,word(A),numeral_numeral(word(A)),A2)),aa(num,word(A),numeral_numeral(word(A)),B2)))
        <=> pp(aa(int,bool,ord_less_eq(int,aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),aa(num,int,numeral_numeral(int),A2))),aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),aa(num,int,numeral_numeral(int),B2)))) ) ) ).

% less_eq_word_numeral_numeral
tff(fact_6176_less__word__numeral__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num,B2: num] :
          ( pp(aa(word(A),bool,ord_less(word(A),aa(num,word(A),numeral_numeral(word(A)),A2)),aa(num,word(A),numeral_numeral(word(A)),B2)))
        <=> pp(aa(int,bool,ord_less(int,aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),aa(num,int,numeral_numeral(int),A2))),aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),aa(num,int,numeral_numeral(int),B2)))) ) ) ).

% less_word_numeral_numeral
tff(fact_6177_unat__bintrunc,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Bin: num] : aa(word(A),nat,semiring_1_unsigned(A,nat),aa(num,word(A),numeral_numeral(word(A)),Bin)) = nat2(aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),aa(num,int,numeral_numeral(int),Bin))) ) ).

% unat_bintrunc
tff(fact_6178_bit__numeral__word__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: num,N: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(word(A),aa(num,word(A),numeral_numeral(word(A)),W)),N))
        <=> ( pp(aa(nat,bool,ord_less(nat,N),type_len0_len_of(A,type2(A))))
            & pp(aa(nat,bool,bit_se5641148757651400278ts_bit(int,aa(num,int,numeral_numeral(int),W)),N)) ) ) ) ).

% bit_numeral_word_iff
tff(fact_6179_ucast__bintr,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [W: num] : aa(word(A),word(B),semiring_1_unsigned(A,word(B)),aa(num,word(A),numeral_numeral(word(A)),W)) = aa(int,word(B),ring_1_of_int(word(B)),aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),aa(num,int,numeral_numeral(int),W))) ) ).

% ucast_bintr
tff(fact_6180_unsigned__numeral,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & semiring_1(A) )
     => ! [N: num] : aa(word(B),A,semiring_1_unsigned(B,A),aa(num,word(B),numeral_numeral(word(B)),N)) = aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,bit_se2584673776208193580ke_bit(nat,type_len0_len_of(B,type2(B))),aa(num,nat,numeral_numeral(nat),N))) ) ).

% unsigned_numeral
tff(fact_6181_unat__lt2p,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A)] : pp(aa(nat,bool,ord_less(nat,aa(word(A),nat,semiring_1_unsigned(A,nat),X)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))) ) ).

% unat_lt2p
tff(fact_6182_uint__bounded,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] : pp(aa(int,bool,ord_less(int,aa(word(A),int,semiring_1_unsigned(A,int),W)),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A))))) ) ).

% uint_bounded
tff(fact_6183_uint__lt2p,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A)] : pp(aa(int,bool,ord_less(int,aa(word(A),int,semiring_1_unsigned(A,int),X)),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A))))) ) ).

% uint_lt2p
tff(fact_6184_exp__eq__zero__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: nat] :
          ( ( aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),N) = zero_zero(word(A)) )
        <=> pp(aa(nat,bool,ord_less_eq(nat,type_len0_len_of(A,type2(A))),N)) ) ) ).

% exp_eq_zero_iff
tff(fact_6185_of__nat__2p,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( aa(nat,word(A),semiring_1_of_nat(word(A)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A)))) = zero_zero(word(A)) ) ) ).

% of_nat_2p
tff(fact_6186_signed__take__bit__word__Suc__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: nat,K: num] : aa(word(A),word(A),bit_ri4674362597316999326ke_bit(word(A),aa(nat,nat,suc,N)),aa(num,word(A),numeral_numeral(word(A)),K)) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,suc,N)),aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),aa(num,int,numeral_numeral(int),K)))) ) ).

% signed_take_bit_word_Suc_numeral
tff(fact_6187_signed__take__bit__word__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: num,K: num] : aa(word(A),word(A),bit_ri4674362597316999326ke_bit(word(A),aa(num,nat,numeral_numeral(nat),N)),aa(num,word(A),numeral_numeral(word(A)),K)) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(num,nat,numeral_numeral(nat),N)),aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),aa(num,int,numeral_numeral(int),K)))) ) ).

% signed_take_bit_word_numeral
tff(fact_6188_sint__sbintrunc,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Bin: num] : ring_1_signed(A,int,aa(num,word(A),numeral_numeral(word(A)),Bin)) = aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat))),aa(num,int,numeral_numeral(int),Bin)) ) ).

% sint_sbintrunc
tff(fact_6189_uint__bintrunc__neg,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Bin: num] : aa(word(A),int,semiring_1_unsigned(A,int),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),Bin))) = aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Bin))) ) ).

% uint_bintrunc_neg
tff(fact_6190_div__word__numeral__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num,B2: num] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),aa(num,word(A),numeral_numeral(word(A)),A2)),aa(num,word(A),numeral_numeral(word(A)),B2)) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),aa(num,int,numeral_numeral(int),A2))),aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),aa(num,int,numeral_numeral(int),B2)))) ) ).

% div_word_numeral_numeral
tff(fact_6191_mod__word__numeral__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num,B2: num] : modulo_modulo(word(A),aa(num,word(A),numeral_numeral(word(A)),A2),aa(num,word(A),numeral_numeral(word(A)),B2)) = aa(int,word(A),ring_1_of_int(word(A)),modulo_modulo(int,aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),aa(num,int,numeral_numeral(int),A2)),aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),aa(num,int,numeral_numeral(int),B2)))) ) ).

% mod_word_numeral_numeral
tff(fact_6192_signed__numeral,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & ring_1(A) )
     => ! [N: num] : ring_1_signed(B,A,aa(num,word(B),numeral_numeral(word(B)),N)) = aa(int,A,ring_1_of_int(A),aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,minus_minus(nat,type_len0_len_of(B,type2(B))),one_one(nat))),aa(num,int,numeral_numeral(int),N))) ) ).

% signed_numeral
tff(fact_6193_less__eq__word__minus__numeral__minus__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num,B2: num] :
          ( pp(aa(word(A),bool,ord_less_eq(word(A),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),A2))),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),B2))))
        <=> pp(aa(int,bool,ord_less_eq(int,aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),A2)))),aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),B2))))) ) ) ).

% less_eq_word_minus_numeral_minus_numeral
tff(fact_6194_less__eq__word__numeral__minus__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num,B2: num] :
          ( pp(aa(word(A),bool,ord_less_eq(word(A),aa(num,word(A),numeral_numeral(word(A)),A2)),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),B2))))
        <=> pp(aa(int,bool,ord_less_eq(int,aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),aa(num,int,numeral_numeral(int),A2))),aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),B2))))) ) ) ).

% less_eq_word_numeral_minus_numeral
tff(fact_6195_less__eq__word__minus__numeral__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num,B2: num] :
          ( pp(aa(word(A),bool,ord_less_eq(word(A),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),A2))),aa(num,word(A),numeral_numeral(word(A)),B2)))
        <=> pp(aa(int,bool,ord_less_eq(int,aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),A2)))),aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),aa(num,int,numeral_numeral(int),B2)))) ) ) ).

% less_eq_word_minus_numeral_numeral
tff(fact_6196_less__word__minus__numeral__minus__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num,B2: num] :
          ( pp(aa(word(A),bool,ord_less(word(A),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),A2))),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),B2))))
        <=> pp(aa(int,bool,ord_less(int,aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),A2)))),aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),B2))))) ) ) ).

% less_word_minus_numeral_minus_numeral
tff(fact_6197_less__word__numeral__minus__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num,B2: num] :
          ( pp(aa(word(A),bool,ord_less(word(A),aa(num,word(A),numeral_numeral(word(A)),A2)),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),B2))))
        <=> pp(aa(int,bool,ord_less(int,aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),aa(num,int,numeral_numeral(int),A2))),aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),B2))))) ) ) ).

% less_word_numeral_minus_numeral
tff(fact_6198_less__word__minus__numeral__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num,B2: num] :
          ( pp(aa(word(A),bool,ord_less(word(A),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),A2))),aa(num,word(A),numeral_numeral(word(A)),B2)))
        <=> pp(aa(int,bool,ord_less(int,aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),A2)))),aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),aa(num,int,numeral_numeral(int),B2)))) ) ) ).

% less_word_minus_numeral_numeral
tff(fact_6199_unat__bintrunc__neg,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Bin: num] : aa(word(A),nat,semiring_1_unsigned(A,nat),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),Bin))) = nat2(aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Bin)))) ) ).

% unat_bintrunc_neg
tff(fact_6200_bit__neg__numeral__word__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: num,N: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(word(A),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),W))),N))
        <=> ( pp(aa(nat,bool,ord_less(nat,N),type_len0_len_of(A,type2(A))))
            & pp(aa(nat,bool,bit_se5641148757651400278ts_bit(int,aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),W))),N)) ) ) ) ).

% bit_neg_numeral_word_iff
tff(fact_6201_drop__bit__word__Suc__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: nat,K: num] : bit_se4197421643247451524op_bit(word(A),aa(nat,nat,suc,N),aa(num,word(A),numeral_numeral(word(A)),K)) = aa(int,word(A),ring_1_of_int(word(A)),bit_se4197421643247451524op_bit(int,aa(nat,nat,suc,N),aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),aa(num,int,numeral_numeral(int),K)))) ) ).

% drop_bit_word_Suc_numeral
tff(fact_6202_drop__bit__word__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: num,K: num] : bit_se4197421643247451524op_bit(word(A),aa(num,nat,numeral_numeral(nat),N),aa(num,word(A),numeral_numeral(word(A)),K)) = aa(int,word(A),ring_1_of_int(word(A)),bit_se4197421643247451524op_bit(int,aa(num,nat,numeral_numeral(nat),N),aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),aa(num,int,numeral_numeral(int),K)))) ) ).

% drop_bit_word_numeral
tff(fact_6203_unat__power__lower,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: nat] :
          ( pp(aa(nat,bool,ord_less(nat,N),type_len0_len_of(A,type2(A))))
         => ( aa(word(A),nat,semiring_1_unsigned(A,nat),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),N)) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N) ) ) ) ).

% unat_power_lower
tff(fact_6204_unsigned__neg__numeral,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & semiring_1(A) )
     => ! [N: num] : aa(word(B),A,semiring_1_unsigned(B,A),aa(word(B),word(B),uminus_uminus(word(B)),aa(num,word(B),numeral_numeral(word(B)),N))) = aa(nat,A,semiring_1_of_nat(A),nat2(aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(B,type2(B))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),N))))) ) ).

% unsigned_neg_numeral
tff(fact_6205_signed__take__bit__word__Suc__minus__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: nat,K: num] : aa(word(A),word(A),bit_ri4674362597316999326ke_bit(word(A),aa(nat,nat,suc,N)),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),K))) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,suc,N)),aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),K))))) ) ).

% signed_take_bit_word_Suc_minus_numeral
tff(fact_6206_signed__take__bit__word__minus__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: num,K: num] : aa(word(A),word(A),bit_ri4674362597316999326ke_bit(word(A),aa(num,nat,numeral_numeral(nat),N)),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),K))) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(num,nat,numeral_numeral(nat),N)),aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),K))))) ) ).

% signed_take_bit_word_minus_numeral
tff(fact_6207_sint__sbintrunc__neg,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Bin: num] : ring_1_signed(A,int,aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),Bin))) = aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Bin))) ) ).

% sint_sbintrunc_neg
tff(fact_6208_scast__sbintr,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [W: num] : ring_1_signed(A,word(B),aa(num,word(A),numeral_numeral(word(A)),W)) = aa(int,word(B),ring_1_of_int(word(B)),aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat)))),aa(num,int,numeral_numeral(int),W))) ) ).

% scast_sbintr
tff(fact_6209_drop__bit__numeral__bit0__1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [K: num] : bit_se4197421643247451524op_bit(word(A),aa(nat,nat,suc,zero_zero(nat)),aa(num,word(A),numeral_numeral(word(A)),K)) = aa(int,word(A),ring_1_of_int(word(A)),bit_se4197421643247451524op_bit(int,aa(nat,nat,suc,zero_zero(nat)),aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),aa(num,int,numeral_numeral(int),K)))) ) ).

% drop_bit_numeral_bit0_1
tff(fact_6210_div__word__minus__numeral__minus__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num,B2: num] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),A2))),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),B2))) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),A2)))),aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),B2))))) ) ).

% div_word_minus_numeral_minus_numeral
tff(fact_6211_div__word__numeral__minus__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num,B2: num] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),aa(num,word(A),numeral_numeral(word(A)),A2)),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),B2))) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),aa(num,int,numeral_numeral(int),A2))),aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),B2))))) ) ).

% div_word_numeral_minus_numeral
tff(fact_6212_div__word__minus__numeral__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num,B2: num] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),A2))),aa(num,word(A),numeral_numeral(word(A)),B2)) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),A2)))),aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),aa(num,int,numeral_numeral(int),B2)))) ) ).

% div_word_minus_numeral_numeral
tff(fact_6213_mod__word__minus__numeral__minus__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num,B2: num] : modulo_modulo(word(A),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),A2)),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),B2))) = aa(int,word(A),ring_1_of_int(word(A)),modulo_modulo(int,aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),A2))),aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),B2))))) ) ).

% mod_word_minus_numeral_minus_numeral
tff(fact_6214_mod__word__numeral__minus__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num,B2: num] : modulo_modulo(word(A),aa(num,word(A),numeral_numeral(word(A)),A2),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),B2))) = aa(int,word(A),ring_1_of_int(word(A)),modulo_modulo(int,aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),aa(num,int,numeral_numeral(int),A2)),aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),B2))))) ) ).

% mod_word_numeral_minus_numeral
tff(fact_6215_mod__word__minus__numeral__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num,B2: num] : modulo_modulo(word(A),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),A2)),aa(num,word(A),numeral_numeral(word(A)),B2)) = aa(int,word(A),ring_1_of_int(word(A)),modulo_modulo(int,aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),A2))),aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),aa(num,int,numeral_numeral(int),B2)))) ) ).

% mod_word_minus_numeral_numeral
tff(fact_6216_word__less__sub__le,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: nat,X: word(A)] :
          ( pp(aa(nat,bool,ord_less(nat,N),type_len0_len_of(A,type2(A))))
         => ( pp(aa(word(A),bool,ord_less_eq(word(A),X),aa(word(A),word(A),minus_minus(word(A),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),N)),one_one(word(A)))))
          <=> pp(aa(word(A),bool,ord_less(word(A),X),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),N))) ) ) ) ).

% word_less_sub_le
tff(fact_6217_signed__neg__numeral,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & ring_1(A) )
     => ! [N: num] : ring_1_signed(B,A,aa(word(B),word(B),uminus_uminus(word(B)),aa(num,word(B),numeral_numeral(word(B)),N))) = aa(int,A,ring_1_of_int(A),aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,minus_minus(nat,type_len0_len_of(B,type2(B))),one_one(nat))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),N)))) ) ).

% signed_neg_numeral
tff(fact_6218_drop__bit__word__Suc__minus__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: nat,K: num] : bit_se4197421643247451524op_bit(word(A),aa(nat,nat,suc,N),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),K))) = aa(int,word(A),ring_1_of_int(word(A)),bit_se4197421643247451524op_bit(int,aa(nat,nat,suc,N),aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),K))))) ) ).

% drop_bit_word_Suc_minus_numeral
tff(fact_6219_drop__bit__word__minus__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: num,K: num] : bit_se4197421643247451524op_bit(word(A),aa(num,nat,numeral_numeral(nat),N),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),K))) = aa(int,word(A),ring_1_of_int(word(A)),bit_se4197421643247451524op_bit(int,aa(num,nat,numeral_numeral(nat),N),aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),K))))) ) ).

% drop_bit_word_minus_numeral
tff(fact_6220_less__word__numeral__minus__1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num] :
          ( pp(aa(word(A),bool,ord_less(word(A),aa(num,word(A),numeral_numeral(word(A)),A2)),aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A)))))
        <=> pp(aa(int,bool,ord_less(int,aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),aa(num,int,numeral_numeral(int),A2))),aa(int,int,minus_minus(int,aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A)))),one_one(int)))) ) ) ).

% less_word_numeral_minus_1
tff(fact_6221_less__word__minus__numeral__minus__1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num] :
          ( pp(aa(word(A),bool,ord_less(word(A),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),A2))),aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A)))))
        <=> pp(aa(int,bool,ord_less(int,aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),A2)))),aa(int,int,minus_minus(int,aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A)))),one_one(int)))) ) ) ).

% less_word_minus_numeral_minus_1
tff(fact_6222_div__word__minus__1__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [B2: num] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A)))),aa(num,word(A),numeral_numeral(word(A)),B2)) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,minus_minus(int,aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A)))),one_one(int))),aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),aa(num,int,numeral_numeral(int),B2)))) ) ).

% div_word_minus_1_numeral
tff(fact_6223_mod__word__minus__1__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [B2: num] : modulo_modulo(word(A),aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A))),aa(num,word(A),numeral_numeral(word(A)),B2)) = aa(int,word(A),ring_1_of_int(word(A)),modulo_modulo(int,aa(int,int,minus_minus(int,aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A)))),one_one(int)),aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),aa(num,int,numeral_numeral(int),B2)))) ) ).

% mod_word_minus_1_numeral
tff(fact_6224_div__word__minus__1__minus__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [B2: num] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A)))),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),B2))) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,minus_minus(int,aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A)))),one_one(int))),aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),B2))))) ) ).

% div_word_minus_1_minus_numeral
tff(fact_6225_word__cat__inj,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(C)
        & type_len(A) )
     => ! [A2: word(A),B2: word(B),C2: word(A),D: word(B)] :
          ( pp(aa(nat,bool,ord_less_eq(nat,aa(nat,nat,aa(nat,fun(nat,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,D) )
          <=> ( ( A2 = C2 )
              & ( B2 = D ) ) ) ) ) ).

% word_cat_inj
tff(fact_6226_mask__over__length,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: nat] :
          ( pp(aa(nat,bool,ord_less_eq(nat,type_len0_len_of(A,type2(A))),N))
         => ( bit_se2239418461657761734s_mask(word(A),N) = aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A))) ) ) ) ).

% mask_over_length
tff(fact_6227_bintr__uint,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: nat,W: word(A)] :
          ( pp(aa(nat,bool,ord_less_eq(nat,type_len0_len_of(A,type2(A))),N))
         => ( aa(int,int,bit_se2584673776208193580ke_bit(int,N),aa(word(A),int,semiring_1_unsigned(A,int),W)) = aa(word(A),int,semiring_1_unsigned(A,int),W) ) ) ) ).

% bintr_uint
tff(fact_6228_up__ucast__inj,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [X: word(A),Y: word(A)] :
          ( ( aa(word(A),word(B),semiring_1_unsigned(A,word(B)),X) = aa(word(A),word(B),semiring_1_unsigned(A,word(B)),Y) )
         => ( pp(aa(nat,bool,ord_less_eq(nat,type_len0_len_of(A,type2(A))),type_len0_len_of(B,type2(B))))
           => ( X = Y ) ) ) ) ).

% up_ucast_inj
tff(fact_6229_ucast__le__ucast,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [X: word(A),Y: word(A)] :
          ( pp(aa(nat,bool,ord_less_eq(nat,type_len0_len_of(A,type2(A))),type_len0_len_of(B,type2(B))))
         => ( pp(aa(word(B),bool,ord_less_eq(word(B),aa(word(A),word(B),semiring_1_unsigned(A,word(B)),X)),aa(word(A),word(B),semiring_1_unsigned(A,word(B)),Y)))
          <=> pp(aa(word(A),bool,ord_less_eq(word(A),X),Y)) ) ) ) ).

% ucast_le_ucast
tff(fact_6230_ucast__ucast__eq,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(A)
        & type_len(C) )
     => ! [X: word(A),Y: word(B)] :
          ( ( aa(word(A),word(C),semiring_1_unsigned(A,word(C)),X) = aa(word(A),word(C),semiring_1_unsigned(A,word(C)),aa(word(B),word(A),semiring_1_unsigned(B,word(A)),Y)) )
         => ( pp(aa(nat,bool,ord_less_eq(nat,type_len0_len_of(A,type2(A))),type_len0_len_of(B,type2(B))))
           => ( pp(aa(nat,bool,ord_less_eq(nat,type_len0_len_of(B,type2(B))),type_len0_len_of(C,type2(C))))
             => ( X = aa(word(B),word(A),semiring_1_unsigned(B,word(A)),Y) ) ) ) ) ) ).

% ucast_ucast_eq
tff(fact_6231_up__ucast__inj__eq,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [X: word(A),Y: word(A)] :
          ( pp(aa(nat,bool,ord_less_eq(nat,type_len0_len_of(A,type2(A))),type_len0_len_of(B,type2(B))))
         => ( ( aa(word(A),word(B),semiring_1_unsigned(A,word(B)),X) = aa(word(A),word(B),semiring_1_unsigned(A,word(B)),Y) )
          <=> ( X = Y ) ) ) ) ).

% up_ucast_inj_eq
tff(fact_6232_ucast__up__mono__le,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [X: word(A),Y: word(A)] :
          ( pp(aa(nat,bool,ord_less_eq(nat,type_len0_len_of(A,type2(A))),type_len0_len_of(B,type2(B))))
         => ( pp(aa(word(A),bool,ord_less_eq(word(A),X),Y))
           => pp(aa(word(B),bool,ord_less_eq(word(B),aa(word(A),word(B),semiring_1_unsigned(A,word(B)),X)),aa(word(A),word(B),semiring_1_unsigned(A,word(B)),Y))) ) ) ) ).

% ucast_up_mono_le
tff(fact_6233_eq__ucast__ucast__eq,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(A) )
     => ! [X: word(A),Y: word(B)] :
          ( pp(aa(nat,bool,ord_less_eq(nat,type_len0_len_of(B,type2(B))),type_len0_len_of(A,type2(A))))
         => ( ( X = aa(word(B),word(A),semiring_1_unsigned(B,word(A)),Y) )
           => ( aa(word(A),word(B),semiring_1_unsigned(A,word(B)),X) = Y ) ) ) ) ).

% eq_ucast_ucast_eq
tff(fact_6234_unat__ucast__up__simp,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [X: word(A)] :
          ( pp(aa(nat,bool,ord_less_eq(nat,type_len0_len_of(A,type2(A))),type_len0_len_of(B,type2(B))))
         => ( aa(word(B),nat,semiring_1_unsigned(B,nat),aa(word(A),word(B),semiring_1_unsigned(A,word(B)),X)) = aa(word(A),nat,semiring_1_unsigned(A,nat),X) ) ) ) ).

% unat_ucast_up_simp
tff(fact_6235_wi__bintr,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: nat,W: int] :
          ( pp(aa(nat,bool,ord_less_eq(nat,type_len0_len_of(A,type2(A))),N))
         => ( aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,bit_se2584673776208193580ke_bit(int,N),W)) = aa(int,word(A),ring_1_of_int(word(A)),W) ) ) ) ).

% wi_bintr
tff(fact_6236_take__bit__word__beyond__length__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: nat,W: word(A)] :
          ( pp(aa(nat,bool,ord_less_eq(nat,type_len0_len_of(A,type2(A))),N))
         => ( aa(word(A),word(A),bit_se2584673776208193580ke_bit(word(A),N),W) = W ) ) ) ).

% take_bit_word_beyond_length_eq
tff(fact_6237_up__scast__inj__eq,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [X: word(A),Y: word(A)] :
          ( pp(aa(nat,bool,ord_less_eq(nat,type_len0_len_of(A,type2(A))),type_len0_len_of(B,type2(B))))
         => ( ( ring_1_signed(A,word(B),X) = ring_1_signed(A,word(B),Y) )
          <=> ( X = Y ) ) ) ) ).

% up_scast_inj_eq
tff(fact_6238_ucast__up__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [X: word(A),Y: word(A)] :
          ( pp(aa(nat,bool,ord_less_eq(nat,type_len0_len_of(A,type2(A))),type_len0_len_of(B,type2(B))))
         => ( pp(aa(word(A),bool,ord_less(word(A),X),Y))
           => pp(aa(word(B),bool,ord_less(word(B),aa(word(A),word(B),semiring_1_unsigned(A,word(B)),X)),aa(word(A),word(B),semiring_1_unsigned(A,word(B)),Y))) ) ) ) ).

% ucast_up_mono
tff(fact_6239_ucast__less__ucast,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [X: word(A),Y: word(A)] :
          ( pp(aa(nat,bool,ord_less_eq(nat,type_len0_len_of(A,type2(A))),type_len0_len_of(B,type2(B))))
         => ( pp(aa(word(B),bool,ord_less(word(B),aa(word(A),word(B),semiring_1_unsigned(A,word(B)),X)),aa(word(A),word(B),semiring_1_unsigned(A,word(B)),Y)))
          <=> pp(aa(word(A),bool,ord_less(word(A),X),Y)) ) ) ) ).

% ucast_less_ucast
tff(fact_6240_less__ucast__ucast__less,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(A) )
     => ! [X: word(A),Y: word(B)] :
          ( pp(aa(nat,bool,ord_less_eq(nat,type_len0_len_of(B,type2(B))),type_len0_len_of(A,type2(A))))
         => ( pp(aa(word(A),bool,ord_less(word(A),X),aa(word(B),word(A),semiring_1_unsigned(B,word(A)),Y)))
           => pp(aa(word(B),bool,ord_less(word(B),aa(word(A),word(B),semiring_1_unsigned(A,word(B)),X)),Y)) ) ) ) ).

% less_ucast_ucast_less
tff(fact_6241_ucast__ucast__add,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(A) )
     => ! [X: word(A),Y: word(B)] :
          ( pp(aa(nat,bool,ord_less_eq(nat,type_len0_len_of(A,type2(A))),type_len0_len_of(B,type2(B))))
         => ( aa(word(B),word(A),semiring_1_unsigned(B,word(A)),aa(word(B),word(B),aa(word(B),fun(word(B),word(B)),plus_plus(word(B)),aa(word(A),word(B),semiring_1_unsigned(A,word(B)),X)),Y)) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),X),aa(word(B),word(A),semiring_1_unsigned(B,word(A)),Y)) ) ) ) ).

% ucast_ucast_add
tff(fact_6242_ucast__sub__ucast,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [Y: word(A),X: word(A)] :
          ( pp(aa(word(A),bool,ord_less_eq(word(A),Y),X))
         => ( pp(aa(nat,bool,ord_less_eq(nat,type_len0_len_of(A,type2(A))),type_len0_len_of(B,type2(B))))
           => ( aa(word(A),word(B),semiring_1_unsigned(A,word(B)),aa(word(A),word(A),minus_minus(word(A),X),Y)) = aa(word(B),word(B),minus_minus(word(B),aa(word(A),word(B),semiring_1_unsigned(A,word(B)),X)),aa(word(A),word(B),semiring_1_unsigned(A,word(B)),Y)) ) ) ) ) ).

% ucast_sub_ucast
tff(fact_6243_word__of__nat__less__eq__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [M: nat,N: nat] :
          ( pp(aa(word(A),bool,ord_less_eq(word(A),aa(nat,word(A),semiring_1_of_nat(word(A)),M)),aa(nat,word(A),semiring_1_of_nat(word(A)),N)))
        <=> pp(aa(nat,bool,ord_less_eq(nat,aa(nat,nat,bit_se2584673776208193580ke_bit(nat,type_len0_len_of(A,type2(A))),M)),aa(nat,nat,bit_se2584673776208193580ke_bit(nat,type_len0_len_of(A,type2(A))),N))) ) ) ).

% word_of_nat_less_eq_iff
tff(fact_6244_ucast__drop__bit__eq,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [N: nat,W: word(A)] :
          ( pp(aa(nat,bool,ord_less_eq(nat,type_len0_len_of(A,type2(A))),type_len0_len_of(B,type2(B))))
         => ( aa(word(A),word(B),semiring_1_unsigned(A,word(B)),bit_se4197421643247451524op_bit(word(A),N,W)) = bit_se4197421643247451524op_bit(word(B),N,aa(word(A),word(B),semiring_1_unsigned(A,word(B)),W)) ) ) ) ).

% ucast_drop_bit_eq
tff(fact_6245_ucast__mask__drop,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(A) )
     => ! [N: nat,X: word(B)] :
          ( pp(aa(nat,bool,ord_less_eq(nat,type_len0_len_of(A,type2(A))),N))
         => ( aa(word(B),word(A),semiring_1_unsigned(B,word(A)),aa(word(B),word(B),aa(word(B),fun(word(B),word(B)),bit_se5824344872417868541ns_and(word(B)),X),bit_se2239418461657761734s_mask(word(B),N))) = aa(word(B),word(A),semiring_1_unsigned(B,word(A)),X) ) ) ) ).

% ucast_mask_drop
tff(fact_6246_nth__slice,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [N: nat,W: word(B),M: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(word(A),slice2(B,A,N,W)),M))
        <=> ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(word(B),W),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N)))
            & pp(aa(nat,bool,ord_less(nat,M),type_len0_len_of(A,type2(A)))) ) ) ) ).

% nth_slice
tff(fact_6247_unsigned__of__nat,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & semiring_1(A) )
     => ! [N: nat] : aa(word(B),A,semiring_1_unsigned(B,A),aa(nat,word(B),semiring_1_of_nat(word(B)),N)) = aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,bit_se2584673776208193580ke_bit(nat,type_len0_len_of(B,type2(B))),N)) ) ).

% unsigned_of_nat
tff(fact_6248_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))) = aa(word(B),word(B),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
tff(fact_6249_max__test__bit,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(word(A),aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A)))),N))
        <=> pp(aa(nat,bool,ord_less(nat,N),type_len0_len_of(A,type2(A)))) ) ) ).

% max_test_bit
tff(fact_6250_test__bit__1_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(word(A),one_one(word(A))),N))
        <=> ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),type_len0_len_of(A,type2(A))))
            & ( N = zero_zero(nat) ) ) ) ) ).

% test_bit_1'
tff(fact_6251_uint__word__arith__bintrs_I7_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( aa(word(A),int,semiring_1_unsigned(A,int),zero_zero(word(A))) = aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),zero_zero(int)) ) ) ).

% uint_word_arith_bintrs(7)
tff(fact_6252_uint__word__arith__bintrs_I8_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( aa(word(A),int,semiring_1_unsigned(A,int),one_one(word(A))) = aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),one_one(int)) ) ) ).

% uint_word_arith_bintrs(8)
tff(fact_6253_max__word__mask,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A))) = bit_se2239418461657761734s_mask(word(A),type_len0_len_of(A,type2(A))) ) ) ).

% max_word_mask
tff(fact_6254_word__of__nat__eq__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [M: nat,N: nat] :
          ( ( aa(nat,word(A),semiring_1_of_nat(word(A)),M) = aa(nat,word(A),semiring_1_of_nat(word(A)),N) )
        <=> ( aa(nat,nat,bit_se2584673776208193580ke_bit(nat,type_len0_len_of(A,type2(A))),M) = aa(nat,nat,bit_se2584673776208193580ke_bit(nat,type_len0_len_of(A,type2(A))),N) ) ) ) ).

% word_of_nat_eq_iff
tff(fact_6255_degenerate__word,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A)] :
          ( ( type_len0_len_of(A,type2(A)) = one_one(nat) )
         => ( ( X = zero_zero(word(A)) )
            | ( X = one_one(word(A)) ) ) ) ) ).

% degenerate_word
tff(fact_6256_size__0__same,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),V: word(A)] :
          ( ( type_len0_len_of(A,type2(A)) = zero_zero(nat) )
         => ( W = V ) ) ) ).

% size_0_same
tff(fact_6257_one__word_Orsp,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),one_one(int)) = aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),one_one(int)) ) ) ).

% one_word.rsp
tff(fact_6258_zero__word_Orsp,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),zero_zero(int)) = aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),zero_zero(int)) ) ) ).

% zero_word.rsp
tff(fact_6259_uint__word__arith__bintrs_I1_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A)] : aa(word(A),int,semiring_1_unsigned(A,int),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),A2),B2)) = aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(word(A),int,semiring_1_unsigned(A,int),A2)),aa(word(A),int,semiring_1_unsigned(A,int),B2))) ) ).

% uint_word_arith_bintrs(1)
tff(fact_6260_unsigned__minus__1__eq__mask,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & bit_se359711467146920520ations(A) )
     => ( aa(word(B),A,semiring_1_unsigned(B,A),aa(word(B),word(B),uminus_uminus(word(B)),one_one(word(B)))) = bit_se2239418461657761734s_mask(A,type_len0_len_of(B,type2(B))) ) ) ).

% unsigned_minus_1_eq_mask
tff(fact_6261_uint__word__arith__bintrs_I3_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A)] : aa(word(A),int,semiring_1_unsigned(A,int),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),A2),B2)) = aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(word(A),int,semiring_1_unsigned(A,int),A2)),aa(word(A),int,semiring_1_unsigned(A,int),B2))) ) ).

% uint_word_arith_bintrs(3)
tff(fact_6262_num__abs__bintr,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: num] : aa(num,word(A),numeral_numeral(word(A)),X) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),aa(num,int,numeral_numeral(int),X))) ) ).

% num_abs_bintr
tff(fact_6263_num__of__bintr_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num,B2: num] :
          ( ( aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),aa(num,int,numeral_numeral(int),A2)) = aa(num,int,numeral_numeral(int),B2) )
         => ( aa(num,word(A),numeral_numeral(word(A)),A2) = aa(num,word(A),numeral_numeral(word(A)),B2) ) ) ) ).

% num_of_bintr'
tff(fact_6264_uint32_Osize__eq__length,axiom,
    aa(num,nat,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
tff(fact_6265_word__of__nat__less__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [M: nat,N: nat] :
          ( pp(aa(word(A),bool,ord_less(word(A),aa(nat,word(A),semiring_1_of_nat(word(A)),M)),aa(nat,word(A),semiring_1_of_nat(word(A)),N)))
        <=> pp(aa(nat,bool,ord_less(nat,aa(nat,nat,bit_se2584673776208193580ke_bit(nat,type_len0_len_of(A,type2(A))),M)),aa(nat,nat,bit_se2584673776208193580ke_bit(nat,type_len0_len_of(A,type2(A))),N))) ) ) ).

% word_of_nat_less_iff
tff(fact_6266_word__cat__bin_H,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(C)
        & type_len(A) )
     => ! [V: word(A),W: word(B)] : word_cat(A,B,C,V,W) = aa(word(C),word(C),aa(word(C),fun(word(C),word(C)),plus_plus(word(C)),bit_se4730199178511100633sh_bit(word(C),type_len0_len_of(B,type2(B)),aa(word(A),word(C),semiring_1_unsigned(A,word(C)),V))),aa(word(B),word(C),semiring_1_unsigned(B,word(C)),W)) ) ).

% word_cat_bin'
tff(fact_6267_up__scast__inj,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [X: word(A),Y: word(A)] :
          ( ( ring_1_signed(A,word(B),X) = ring_1_signed(A,word(B),Y) )
         => ( pp(aa(nat,bool,ord_less_eq(nat,aa(word(A),nat,size_size(word(A)),X)),type_len0_len_of(B,type2(B))))
           => ( X = Y ) ) ) ) ).

% up_scast_inj
tff(fact_6268_two__power__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: nat,M: nat] :
          ( pp(aa(nat,bool,ord_less(nat,N),type_len0_len_of(A,type2(A))))
         => ( pp(aa(nat,bool,ord_less(nat,M),type_len0_len_of(A,type2(A))))
           => ( ( aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),N) = aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),M) )
            <=> ( N = M ) ) ) ) ) ).

% two_power_eq
tff(fact_6269_signed__take__bit__decr__length__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( ( bit_ri3973907225187159222ations(B)
        & type_len(A) )
     => ! [K: B,L: B] :
          ( ( aa(B,B,bit_ri4674362597316999326ke_bit(B,aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat)))),K) = aa(B,B,bit_ri4674362597316999326ke_bit(B,aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat)))),L) )
        <=> ( aa(B,B,bit_se2584673776208193580ke_bit(B,type_len0_len_of(A,type2(A))),K) = aa(B,B,bit_se2584673776208193580ke_bit(B,type_len0_len_of(A,type2(A))),L) ) ) ) ).

% signed_take_bit_decr_length_iff
tff(fact_6270_num__of__sbintr_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num,B2: num] :
          ( ( aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat))),aa(num,int,numeral_numeral(int),A2)) = aa(num,int,numeral_numeral(int),B2) )
         => ( aa(num,word(A),numeral_numeral(word(A)),A2) = aa(num,word(A),numeral_numeral(word(A)),B2) ) ) ) ).

% num_of_sbintr'
tff(fact_6271_unsigned__of__int,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & semiring_1(A) )
     => ! [K: int] : aa(word(B),A,semiring_1_unsigned(B,A),aa(int,word(B),ring_1_of_int(word(B)),K)) = aa(nat,A,semiring_1_of_nat(A),nat2(aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(B,type2(B))),K))) ) ).

% unsigned_of_int
tff(fact_6272_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))) = aa(int,int,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
tff(fact_6273_bin__nth__sint,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: nat,W: word(A)] :
          ( pp(aa(nat,bool,ord_less_eq(nat,type_len0_len_of(A,type2(A))),N))
         => ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(int,ring_1_signed(A,int,W)),N))
          <=> pp(aa(nat,bool,bit_se5641148757651400278ts_bit(int,ring_1_signed(A,int,W)),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat)))) ) ) ) ).

% bin_nth_sint
tff(fact_6274_sint__sbintrunc_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Bin: int] : ring_1_signed(A,int,aa(int,word(A),ring_1_of_int(word(A)),Bin)) = aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat))),Bin) ) ).

% sint_sbintrunc'
tff(fact_6275_neg__mask__test__bit,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: nat,M: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(word(A),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),bit_se2239418461657761734s_mask(word(A),N))),M))
        <=> ( pp(aa(nat,bool,ord_less_eq(nat,N),M))
            & pp(aa(nat,bool,ord_less(nat,M),type_len0_len_of(A,type2(A)))) ) ) ) ).

% neg_mask_test_bit
tff(fact_6276_word__of__int__bin__cat__eq__iff,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(C)
        & type_len(B) )
     => ! [B2: word(B),A2: word(A),D: word(B),C2: word(A)] :
          ( pp(aa(nat,bool,ord_less_eq(nat,aa(nat,nat,aa(nat,fun(nat,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))))
         => ( ( aa(int,word(C),ring_1_of_int(word(C)),aa(int,int,bit_concat_bit(type_len0_len_of(B,type2(B)),aa(word(B),int,semiring_1_unsigned(B,int),B2)),aa(word(A),int,semiring_1_unsigned(A,int),A2))) = aa(int,word(C),ring_1_of_int(word(C)),aa(int,int,bit_concat_bit(type_len0_len_of(B,type2(B)),aa(word(B),int,semiring_1_unsigned(B,int),D)),aa(word(A),int,semiring_1_unsigned(A,int),C2))) )
          <=> ( ( B2 = D )
              & ( A2 = C2 ) ) ) ) ) ).

% word_of_int_bin_cat_eq_iff
tff(fact_6277_mask__exceed,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: nat,X: word(A)] :
          ( pp(aa(nat,bool,ord_less_eq(nat,type_len0_len_of(A,type2(A))),N))
         => ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),X),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),bit_se2239418461657761734s_mask(word(A),N))) = zero_zero(word(A)) ) ) ) ).

% mask_exceed
tff(fact_6278_unsigned__less,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & unique1627219031080169319umeral(A) )
     => ! [W: word(B)] : pp(aa(A,bool,ord_less(A,aa(word(B),A,semiring_1_unsigned(B,A),W)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),type_len0_len_of(B,type2(B))))) ) ).

% unsigned_less
tff(fact_6279_not__degenerate__imp__2__neq__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( pp(aa(nat,bool,ord_less(nat,one_one(nat)),type_len0_len_of(A,type2(A))))
       => ( aa(num,word(A),numeral_numeral(word(A)),bit0(one2)) != zero_zero(word(A)) ) ) ) ).

% not_degenerate_imp_2_neq_0
tff(fact_6280_word__nat__cases,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A)] :
          ~ ! [N2: nat] :
              ( ( X = aa(nat,word(A),semiring_1_of_nat(word(A)),N2) )
             => ~ pp(aa(nat,bool,ord_less(nat,N2),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))) ) ) ).

% word_nat_cases
tff(fact_6281_word__nchotomy,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W3: word(A)] :
        ? [N2: nat] :
          ( ( W3 = aa(nat,word(A),semiring_1_of_nat(word(A)),N2) )
          & pp(aa(nat,bool,ord_less(nat,N2),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))) ) ) ).

% word_nchotomy
tff(fact_6282_word__of__nat__inj,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: nat,Y: nat] :
          ( pp(aa(nat,bool,ord_less(nat,X),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A)))))
         => ( pp(aa(nat,bool,ord_less(nat,Y),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A)))))
           => ( ( aa(nat,word(A),semiring_1_of_nat(word(A)),X) = aa(nat,word(A),semiring_1_of_nat(word(A)),Y) )
             => ( X = Y ) ) ) ) ) ).

% word_of_nat_inj
tff(fact_6283_of__nat__inj,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: nat,Y: nat] :
          ( pp(aa(nat,bool,ord_less(nat,X),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A)))))
         => ( pp(aa(nat,bool,ord_less(nat,Y),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A)))))
           => ( ( aa(nat,word(A),semiring_1_of_nat(word(A)),X) = aa(nat,word(A),semiring_1_of_nat(word(A)),Y) )
            <=> ( X = Y ) ) ) ) ) ).

% of_nat_inj
tff(fact_6284_More__Word_Opower__not__zero,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: nat] :
          ( pp(aa(nat,bool,ord_less(nat,N),type_len0_len_of(A,type2(A))))
         => ( aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),N) != zero_zero(word(A)) ) ) ) ).

% More_Word.power_not_zero
tff(fact_6285_word__power__increasing,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: nat,Y: nat] :
          ( pp(aa(word(A),bool,ord_less(word(A),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),X)),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Y)))
         => ( pp(aa(nat,bool,ord_less(nat,X),type_len0_len_of(A,type2(A))))
           => ( pp(aa(nat,bool,ord_less(nat,Y),type_len0_len_of(A,type2(A))))
             => pp(aa(nat,bool,ord_less(nat,X),Y)) ) ) ) ) ).

% word_power_increasing
tff(fact_6286_power__overflow,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: nat] :
          ( pp(aa(nat,bool,ord_less_eq(nat,type_len0_len_of(A,type2(A))),N))
         => ( aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),N) = zero_zero(word(A)) ) ) ) ).

% power_overflow
tff(fact_6287_nth__w2p__same,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(word(A),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),N)),N))
        <=> pp(aa(nat,bool,ord_less(nat,N),type_len0_len_of(A,type2(A)))) ) ) ).

% nth_w2p_same
tff(fact_6288_nth__w2p,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: nat,M: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(word(A),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),N)),M))
        <=> ( ( M = N )
            & pp(aa(nat,bool,ord_less(nat,M),type_len0_len_of(A,type2(A)))) ) ) ) ).

% nth_w2p
tff(fact_6289_uint__idem,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] : modulo_modulo(int,aa(word(A),int,semiring_1_unsigned(A,int),W),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A)))) = aa(word(A),int,semiring_1_unsigned(A,int),W) ) ).

% uint_idem
tff(fact_6290_word__of__int__2p__len,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( aa(int,word(A),ring_1_of_int(word(A)),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A)))) = zero_zero(word(A)) ) ) ).

% word_of_int_2p_len
tff(fact_6291_of__nat__neq__iff__word,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: nat,Y: nat] :
          ( ( modulo_modulo(nat,X,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A)))) != modulo_modulo(nat,Y,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A)))) )
         => ( ( aa(nat,word(A),semiring_1_of_nat(word(A)),X) != aa(nat,word(A),semiring_1_of_nat(word(A)),Y) )
          <=> ( X != Y ) ) ) ) ).

% of_nat_neq_iff_word
tff(fact_6292_sint__uint,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] : ring_1_signed(A,int,W) = aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat)))),aa(word(A),int,semiring_1_unsigned(A,int),W)) ) ).

% sint_uint
tff(fact_6293_num__abs__sbintr,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: num] : aa(num,word(A),numeral_numeral(word(A)),X) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat))),aa(num,int,numeral_numeral(int),X))) ) ).

% num_abs_sbintr
tff(fact_6294_ucast__mono,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(A) )
     => ! [X: word(B),Y: word(B)] :
          ( pp(aa(word(B),bool,ord_less(word(B),X),Y))
         => ( pp(aa(word(B),bool,ord_less(word(B),Y),aa(nat,word(B),aa(word(B),fun(nat,word(B)),power_power(word(B)),aa(num,word(B),numeral_numeral(word(B)),bit0(one2))),type_len0_len_of(A,type2(A)))))
           => pp(aa(word(A),bool,ord_less(word(A),aa(word(B),word(A),semiring_1_unsigned(B,word(A)),X)),aa(word(B),word(A),semiring_1_unsigned(B,word(A)),Y))) ) ) ) ).

% ucast_mono
tff(fact_6295_ucast__ucast__len,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [X: word(A)] :
          ( pp(aa(word(A),bool,ord_less(word(A),X),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),type_len0_len_of(B,type2(B)))))
         => ( aa(word(B),word(A),semiring_1_unsigned(B,word(A)),aa(word(A),word(B),semiring_1_unsigned(A,word(B)),X)) = X ) ) ) ).

% ucast_ucast_len
tff(fact_6296_sint__word__ariths_I7_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( ring_1_signed(A,int,zero_zero(word(A))) = aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat))),zero_zero(int)) ) ) ).

% sint_word_ariths(7)
tff(fact_6297_sint__word__ariths_I8_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( ring_1_signed(A,int,one_one(word(A))) = aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat))),one_one(int)) ) ) ).

% sint_word_ariths(8)
tff(fact_6298_sint__word__ariths_I1_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A)] : ring_1_signed(A,int,aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),A2),B2)) = aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat))),aa(int,int,aa(int,fun(int,int),plus_plus(int),ring_1_signed(A,int,A2)),ring_1_signed(A,int,B2))) ) ).

% sint_word_ariths(1)
tff(fact_6299_nth__sint,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),N: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(int,ring_1_signed(A,int,W)),N))
        <=> ( ( pp(aa(nat,bool,ord_less(nat,N),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat))))
             => pp(aa(nat,bool,bit_se5641148757651400278ts_bit(word(A),W),N)) )
            & ( ~ pp(aa(nat,bool,ord_less(nat,N),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat))))
             => pp(aa(nat,bool,bit_se5641148757651400278ts_bit(word(A),W),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat)))) ) ) ) ) ).

% nth_sint
tff(fact_6300_bit__sint__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),N: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(int,ring_1_signed(A,int,W)),N))
        <=> ( ( pp(aa(nat,bool,ord_less_eq(nat,type_len0_len_of(A,type2(A))),N))
              & pp(aa(nat,bool,bit_se5641148757651400278ts_bit(word(A),W),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat)))) )
            | pp(aa(nat,bool,bit_se5641148757651400278ts_bit(word(A),W),N)) ) ) ) ).

% bit_sint_iff
tff(fact_6301_sint__word__ariths_I4_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A)] : ring_1_signed(A,int,aa(word(A),word(A),uminus_uminus(word(A)),A2)) = aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat))),aa(int,int,uminus_uminus(int),ring_1_signed(A,int,A2))) ) ).

% sint_word_ariths(4)
tff(fact_6302_drop__bit__eq__zero__iff__not__bit__last,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] :
          ( ( bit_se4197421643247451524op_bit(word(A),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat))),W) = zero_zero(word(A)) )
        <=> ~ pp(aa(nat,bool,bit_se5641148757651400278ts_bit(word(A),W),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat))))) ) ) ).

% drop_bit_eq_zero_iff_not_bit_last
tff(fact_6303_sint__word__ariths_I2_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A)] : ring_1_signed(A,int,aa(word(A),word(A),minus_minus(word(A),A2),B2)) = aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat))),aa(int,int,minus_minus(int,ring_1_signed(A,int,A2)),ring_1_signed(A,int,B2))) ) ).

% sint_word_ariths(2)
tff(fact_6304_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)) = aa(A,A,bit_ri4674362597316999326ke_bit(A,aa(nat,nat,minus_minus(nat,type_len0_len_of(C,type2(C))),aa(nat,nat,suc,zero_zero(nat)))),ring_1_signed(B,A,W)) ) ).

% signed_scast_eq
tff(fact_6305_sint__word__ariths_I3_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A)] : ring_1_signed(A,int,aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),A2),B2)) = aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat))),aa(int,int,aa(int,fun(int,int),times_times(int),ring_1_signed(A,int,A2)),ring_1_signed(A,int,B2))) ) ).

% sint_word_ariths(3)
tff(fact_6306_less__Suc__unat__less__bound,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: nat,X: word(A)] :
          ( pp(aa(nat,bool,ord_less(nat,N),aa(nat,nat,suc,aa(word(A),nat,semiring_1_unsigned(A,nat),X))))
         => pp(aa(nat,bool,ord_less(nat,N),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))) ) ) ).

% less_Suc_unat_less_bound
tff(fact_6307_uint__2__id,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( pp(aa(nat,bool,ord_less_eq(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
       => ( aa(word(A),int,semiring_1_unsigned(A,int),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))) = aa(num,int,numeral_numeral(int),bit0(one2)) ) ) ) ).

% uint_2_id
tff(fact_6308_lt2p__lem,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: nat,W: word(A)] :
          ( pp(aa(nat,bool,ord_less_eq(nat,type_len0_len_of(A,type2(A))),N))
         => pp(aa(int,bool,ord_less(int,aa(word(A),int,semiring_1_unsigned(A,int),W)),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),N))) ) ) ).

% lt2p_lem
tff(fact_6309_power__le__mono,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: nat,M: nat] :
          ( pp(aa(word(A),bool,ord_less_eq(word(A),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),N)),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),M)))
         => ( pp(aa(nat,bool,ord_less(nat,N),type_len0_len_of(A,type2(A))))
           => ( pp(aa(nat,bool,ord_less(nat,M),type_len0_len_of(A,type2(A))))
             => pp(aa(nat,bool,ord_less_eq(nat,N),M)) ) ) ) ) ).

% power_le_mono
tff(fact_6310_two__power__increasing,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: nat,M: nat] :
          ( pp(aa(nat,bool,ord_less_eq(nat,N),M))
         => ( pp(aa(nat,bool,ord_less(nat,M),type_len0_len_of(A,type2(A))))
           => pp(aa(word(A),bool,ord_less_eq(word(A),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),N)),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),M))) ) ) ) ).

% two_power_increasing
tff(fact_6311_unat__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [B2: num] : aa(word(A),nat,semiring_1_unsigned(A,nat),aa(num,word(A),numeral_numeral(word(A)),B2)) = modulo_modulo(nat,aa(num,nat,numeral_numeral(nat),B2),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A)))) ) ).

% unat_numeral
tff(fact_6312_of__nat__mono__maybe_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: nat,Y: nat] :
          ( pp(aa(nat,bool,ord_less(nat,X),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A)))))
         => ( pp(aa(nat,bool,ord_less(nat,Y),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A)))))
           => ( pp(aa(nat,bool,ord_less(nat,Y),X))
            <=> pp(aa(word(A),bool,ord_less(word(A),aa(nat,word(A),semiring_1_of_nat(word(A)),Y)),aa(nat,word(A),semiring_1_of_nat(word(A)),X))) ) ) ) ) ).

% of_nat_mono_maybe'
tff(fact_6313_of__nat__mono__maybe,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: nat,Y: nat] :
          ( pp(aa(nat,bool,ord_less(nat,X),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A)))))
         => ( pp(aa(nat,bool,ord_less(nat,Y),X))
           => pp(aa(word(A),bool,ord_less(word(A),aa(nat,word(A),semiring_1_of_nat(word(A)),Y)),aa(nat,word(A),semiring_1_of_nat(word(A)),X))) ) ) ) ).

% of_nat_mono_maybe
tff(fact_6314_unat__split__asm,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [P: fun(nat,bool),X: word(A)] :
          ( pp(aa(nat,bool,P,aa(word(A),nat,semiring_1_unsigned(A,nat),X)))
        <=> ~ ? [N5: nat] :
                ( ( aa(nat,word(A),semiring_1_of_nat(word(A)),N5) = X )
                & pp(aa(nat,bool,ord_less(nat,N5),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A)))))
                & ~ pp(aa(nat,bool,P,N5)) ) ) ) ).

% unat_split_asm
tff(fact_6315_of__nat__inverse,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [R2: nat,A2: word(A)] :
          ( ( aa(nat,word(A),semiring_1_of_nat(word(A)),R2) = A2 )
         => ( pp(aa(nat,bool,ord_less(nat,R2),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A)))))
           => ( aa(word(A),nat,semiring_1_unsigned(A,nat),A2) = R2 ) ) ) ) ).

% of_nat_inverse
tff(fact_6316_unat__split,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [P: fun(nat,bool),X: word(A)] :
          ( pp(aa(nat,bool,P,aa(word(A),nat,semiring_1_unsigned(A,nat),X)))
        <=> ! [N5: nat] :
              ( ( ( aa(nat,word(A),semiring_1_of_nat(word(A)),N5) = X )
                & pp(aa(nat,bool,ord_less(nat,N5),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))) )
             => pp(aa(nat,bool,P,N5)) ) ) ) ).

% unat_split
tff(fact_6317_unat__of__nat__len,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: nat] :
          ( pp(aa(nat,bool,ord_less(nat,X),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A)))))
         => ( aa(word(A),nat,semiring_1_unsigned(A,nat),aa(nat,word(A),semiring_1_of_nat(word(A)),X)) = X ) ) ) ).

% unat_of_nat_len
tff(fact_6318_unat__eq__of__nat,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: nat,X: word(A)] :
          ( pp(aa(nat,bool,ord_less(nat,N),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A)))))
         => ( ( aa(word(A),nat,semiring_1_unsigned(A,nat),X) = N )
          <=> ( X = aa(nat,word(A),semiring_1_of_nat(word(A)),N) ) ) ) ) ).

% unat_eq_of_nat
tff(fact_6319_x__less__2__0__1_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A)] :
          ( ( type_len0_len_of(A,type2(A)) != one_one(nat) )
         => ( pp(aa(word(A),bool,ord_less(word(A),X),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))))
           => ( ( X = zero_zero(word(A)) )
              | ( X = one_one(word(A)) ) ) ) ) ) ).

% x_less_2_0_1'
tff(fact_6320_test__bit__2p,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: nat,M: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(word(A),aa(int,word(A),ring_1_of_int(word(A)),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),N))),M))
        <=> ( ( M = N )
            & pp(aa(nat,bool,ord_less(nat,M),type_len0_len_of(A,type2(A)))) ) ) ) ).

% test_bit_2p
tff(fact_6321_word__1__le__power,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: nat] :
          ( pp(aa(nat,bool,ord_less(nat,N),type_len0_len_of(A,type2(A))))
         => pp(aa(word(A),bool,ord_less_eq(word(A),one_one(word(A))),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),N))) ) ) ).

% word_1_le_power
tff(fact_6322_Word_Oof__nat__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [M: nat] :
          ( ( aa(nat,word(A),semiring_1_of_nat(word(A)),M) = zero_zero(word(A)) )
        <=> ? [Q4: nat] : M = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Q4),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A)))) ) ) ).

% Word.of_nat_0
tff(fact_6323_uint__sub__lt2p,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [X: word(A),Y: word(B)] : pp(aa(int,bool,ord_less(int,aa(int,int,minus_minus(int,aa(word(A),int,semiring_1_unsigned(A,int),X)),aa(word(B),int,semiring_1_unsigned(B,int),Y))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A))))) ) ).

% uint_sub_lt2p
tff(fact_6324_uint__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [B2: num] : aa(word(A),int,semiring_1_unsigned(A,int),aa(num,word(A),numeral_numeral(word(A)),B2)) = modulo_modulo(int,aa(num,int,numeral_numeral(int),B2),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A)))) ) ).

% uint_numeral
tff(fact_6325_p2__gt__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: nat] :
          ( pp(aa(word(A),bool,ord_less(word(A),zero_zero(word(A))),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),N)))
        <=> pp(aa(nat,bool,ord_less(nat,N),type_len0_len_of(A,type2(A)))) ) ) ).

% p2_gt_0
tff(fact_6326_ucast__of__nat__small,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(A) )
     => ! [X: nat] :
          ( pp(aa(nat,bool,ord_less(nat,X),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A)))))
         => ( aa(word(A),word(B),semiring_1_unsigned(A,word(B)),aa(nat,word(A),semiring_1_of_nat(word(A)),X)) = aa(nat,word(B),semiring_1_of_nat(word(B)),X) ) ) ) ).

% ucast_of_nat_small
tff(fact_6327_bit__last__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(word(A),W),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat)))))
        <=> pp(aa(int,bool,ord_less(int,ring_1_signed(A,int,W)),zero_zero(int))) ) ) ).

% bit_last_iff
tff(fact_6328_word__of__int__minus,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [I: int] : aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,minus_minus(int,aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A)))),I)) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,uminus_uminus(int),I)) ) ).

% word_of_int_minus
tff(fact_6329_word__of__nat__eq__0__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: nat] :
          ( ( aa(nat,word(A),semiring_1_of_nat(word(A)),N) = zero_zero(word(A)) )
        <=> pp(aa(nat,bool,dvd_dvd(nat,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A)))),N)) ) ) ).

% word_of_nat_eq_0_iff
tff(fact_6330_unat__ucast,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [X: word(B)] : aa(word(A),nat,semiring_1_unsigned(A,nat),aa(word(B),word(A),semiring_1_unsigned(B,word(A)),X)) = modulo_modulo(nat,aa(word(B),nat,semiring_1_unsigned(B,nat),X),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A)))) ) ).

% unat_ucast
tff(fact_6331_uint__word__of__int,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [K: int] : aa(word(A),int,semiring_1_unsigned(A,int),aa(int,word(A),ring_1_of_int(word(A)),K)) = modulo_modulo(int,K,aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A)))) ) ).

% uint_word_of_int
tff(fact_6332_mask__lt__2pn,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: nat] :
          ( pp(aa(nat,bool,ord_less(nat,N),type_len0_len_of(A,type2(A))))
         => pp(aa(word(A),bool,ord_less(word(A),bit_se2239418461657761734s_mask(word(A),N)),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),N))) ) ) ).

% mask_lt_2pn
tff(fact_6333_unat__of__nat,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: nat] : aa(word(A),nat,semiring_1_unsigned(A,nat),aa(nat,word(A),semiring_1_of_nat(word(A)),X)) = modulo_modulo(nat,X,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A)))) ) ).

% unat_of_nat
tff(fact_6334_signed__of__int,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & ring_1(A) )
     => ! [N: int] : ring_1_signed(B,A,aa(int,word(B),ring_1_of_int(word(B)),N)) = aa(int,A,ring_1_of_int(A),aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,minus_minus(nat,type_len0_len_of(B,type2(B))),aa(nat,nat,suc,zero_zero(nat)))),N)) ) ).

% signed_of_int
tff(fact_6335_ucast__less,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(A) )
     => ! [X: word(B)] :
          ( pp(aa(nat,bool,ord_less(nat,type_len0_len_of(B,type2(B))),type_len0_len_of(A,type2(A))))
         => pp(aa(word(A),bool,ord_less(word(A),aa(word(B),word(A),semiring_1_unsigned(B,word(A)),X)),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),type_len0_len_of(B,type2(B))))) ) ) ).

% ucast_less
tff(fact_6336_word__of__int__eq__0__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [K: int] :
          ( ( aa(int,word(A),ring_1_of_int(word(A)),K) = zero_zero(word(A)) )
        <=> pp(aa(int,bool,dvd_dvd(int,aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A)))),K)) ) ) ).

% word_of_int_eq_0_iff
tff(fact_6337_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,aa(word(B),word(C),semiring_1_unsigned(B,word(C)),W)) = aa(A,A,bit_ri4674362597316999326ke_bit(A,aa(nat,nat,minus_minus(nat,type_len0_len_of(C,type2(C))),aa(nat,nat,suc,zero_zero(nat)))),aa(word(B),A,semiring_1_unsigned(B,A),W)) ) ).

% signed_ucast_eq
tff(fact_6338_of__nat__n__less__equal__power__2,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: nat] :
          ( pp(aa(nat,bool,ord_less(nat,N),type_len0_len_of(A,type2(A))))
         => pp(aa(word(A),bool,ord_less(word(A),aa(nat,word(A),semiring_1_of_nat(word(A)),N)),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),N))) ) ) ).

% of_nat_n_less_equal_power_2
tff(fact_6339_upper__trivial,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A)] :
          ( ( X != aa(word(A),word(A),minus_minus(word(A),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),type_len0_len_of(A,type2(A)))),one_one(word(A))) )
         => pp(aa(word(A),bool,ord_less(word(A),X),aa(word(A),word(A),minus_minus(word(A),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),type_len0_len_of(A,type2(A)))),one_one(word(A))))) ) ) ).

% upper_trivial
tff(fact_6340_complement__nth__w2p,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N3: nat,N: nat] :
          ( pp(aa(nat,bool,ord_less(nat,N3),type_len0_len_of(A,type2(A))))
         => ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(word(A),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),N))),N3))
          <=> ( N3 != N ) ) ) ) ).

% complement_nth_w2p
tff(fact_6341_bit__word__scast__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [W: word(A),N: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(word(B),ring_1_signed(A,word(B),W)),N))
        <=> ( pp(aa(nat,bool,ord_less(nat,N),type_len0_len_of(B,type2(B))))
            & ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(word(A),W),N))
              | ( pp(aa(nat,bool,ord_less_eq(nat,type_len0_len_of(A,type2(A))),N))
                & pp(aa(nat,bool,bit_se5641148757651400278ts_bit(word(A),W),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat))))) ) ) ) ) ) ).

% bit_word_scast_iff
tff(fact_6342_minus__one__word,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A))) = aa(word(A),word(A),minus_minus(word(A),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),type_len0_len_of(A,type2(A)))),one_one(word(A))) ) ) ).

% minus_one_word
tff(fact_6343_word__power__less__diff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: nat,Q2: word(A),M: nat] :
          ( pp(aa(word(A),bool,ord_less(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),N)),Q2)),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),M)))
         => ( pp(aa(word(A),bool,ord_less(word(A),Q2),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),N))))
           => pp(aa(word(A),bool,ord_less(word(A),Q2),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),aa(nat,nat,minus_minus(nat,M),N)))) ) ) ) ).

% word_power_less_diff
tff(fact_6344_ucast__mono__le,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [X: word(A),Y: word(A)] :
          ( pp(aa(word(A),bool,ord_less_eq(word(A),X),Y))
         => ( pp(aa(word(A),bool,ord_less(word(A),Y),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),type_len0_len_of(B,type2(B)))))
           => pp(aa(word(B),bool,ord_less_eq(word(B),aa(word(A),word(B),semiring_1_unsigned(A,word(B)),X)),aa(word(A),word(B),semiring_1_unsigned(A,word(B)),Y))) ) ) ) ).

% ucast_mono_le
tff(fact_6345_unat__word__ariths_I7_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A)] : aa(word(A),nat,semiring_1_unsigned(A,nat),modulo_modulo(word(A),A2,B2)) = modulo_modulo(nat,modulo_modulo(nat,aa(word(A),nat,semiring_1_unsigned(A,nat),A2),aa(word(A),nat,semiring_1_unsigned(A,nat),B2)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A)))) ) ).

% unat_word_ariths(7)
tff(fact_6346_take__bit__word__eq__self__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: nat,W: word(A)] :
          ( ( aa(word(A),word(A),bit_se2584673776208193580ke_bit(word(A),N),W) = W )
        <=> ( pp(aa(nat,bool,ord_less_eq(nat,type_len0_len_of(A,type2(A))),N))
            | pp(aa(word(A),bool,ord_less(word(A),W),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),N))) ) ) ) ).

% take_bit_word_eq_self_iff
tff(fact_6347_signed__push__bit__eq,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & bit_ri3973907225187159222ations(A) )
     => ! [N: nat,W: word(B)] : ring_1_signed(B,A,bit_se4730199178511100633sh_bit(word(B),N,W)) = aa(A,A,bit_ri4674362597316999326ke_bit(A,aa(nat,nat,minus_minus(nat,type_len0_len_of(B,type2(B))),aa(nat,nat,suc,zero_zero(nat)))),bit_se4730199178511100633sh_bit(A,N,ring_1_signed(B,A,W))) ) ).

% signed_push_bit_eq
tff(fact_6348_msb0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A),I: nat] :
          ( ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se1065995026697491101ons_or(word(A)),X),Y)),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,suc,I))))
          <=> ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(word(A),X),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,suc,I))))
              | pp(aa(nat,bool,bit_se5641148757651400278ts_bit(word(A),Y),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,suc,I)))) ) )
          & ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),X),Y)),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,suc,I))))
          <=> ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(word(A),X),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,suc,I))))
              & pp(aa(nat,bool,bit_se5641148757651400278ts_bit(word(A),Y),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,suc,I)))) ) )
          & ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344971392196577ns_xor(word(A)),X),Y)),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,suc,I))))
          <=> ~ ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(word(A),X),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,suc,I))))
              <=> pp(aa(nat,bool,bit_se5641148757651400278ts_bit(word(A),Y),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,suc,I)))) ) )
          & ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(word(A),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),X)),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,suc,I))))
          <=> ~ pp(aa(nat,bool,bit_se5641148757651400278ts_bit(word(A),X),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,suc,I)))) ) ) ) ).

% msb0
tff(fact_6349_unat__add__lem,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A)] :
          ( pp(aa(nat,bool,ord_less(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),X)),aa(word(A),nat,semiring_1_unsigned(A,nat),Y))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A)))))
        <=> ( aa(word(A),nat,semiring_1_unsigned(A,nat),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),X),Y)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),X)),aa(word(A),nat,semiring_1_unsigned(A,nat),Y)) ) ) ) ).

% unat_add_lem
tff(fact_6350_unat__add__lem_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A)] :
          ( pp(aa(nat,bool,ord_less(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),X)),aa(word(A),nat,semiring_1_unsigned(A,nat),Y))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A)))))
         => ( aa(word(A),nat,semiring_1_unsigned(A,nat),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),X),Y)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),X)),aa(word(A),nat,semiring_1_unsigned(A,nat),Y)) ) ) ) ).

% unat_add_lem'
tff(fact_6351_Word_Oof__nat__neq__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [K: nat] :
          ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),K))
         => ( pp(aa(nat,bool,ord_less(nat,K),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A)))))
           => ( aa(nat,word(A),semiring_1_of_nat(word(A)),K) != zero_zero(word(A)) ) ) ) ) ).

% Word.of_nat_neq_0
tff(fact_6352_More__Word_Oof__nat__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: nat] :
          ( ( aa(nat,word(A),semiring_1_of_nat(word(A)),N) = zero_zero(word(A)) )
         => ( pp(aa(nat,bool,ord_less(nat,N),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A)))))
           => ( N = zero_zero(nat) ) ) ) ) ).

% More_Word.of_nat_0
tff(fact_6353_of__nat__mono__maybe__le,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: nat,Y: nat] :
          ( pp(aa(nat,bool,ord_less(nat,X),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A)))))
         => ( pp(aa(nat,bool,ord_less(nat,Y),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A)))))
           => ( pp(aa(nat,bool,ord_less_eq(nat,Y),X))
            <=> pp(aa(word(A),bool,ord_less_eq(word(A),aa(nat,word(A),semiring_1_of_nat(word(A)),Y)),aa(nat,word(A),semiring_1_of_nat(word(A)),X))) ) ) ) ) ).

% of_nat_mono_maybe_le
tff(fact_6354_unat__word__ariths_I4_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( aa(word(A),nat,semiring_1_unsigned(A,nat),zero_zero(word(A))) = modulo_modulo(nat,zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A)))) ) ) ).

% unat_word_ariths(4)
tff(fact_6355_uint__range_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A)] :
          ( pp(aa(int,bool,ord_less_eq(int,zero_zero(int)),aa(word(A),int,semiring_1_unsigned(A,int),X)))
          & pp(aa(int,bool,ord_less(int,aa(word(A),int,semiring_1_unsigned(A,int),X)),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A))))) ) ) ).

% uint_range'
tff(fact_6356_bool__mask_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A)] :
          ( pp(aa(nat,bool,ord_less(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
         => ( pp(aa(word(A),bool,ord_less(word(A),zero_zero(word(A))),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),X),one_one(word(A)))))
          <=> ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),X),one_one(word(A))) = one_one(word(A)) ) ) ) ) ).

% bool_mask'
tff(fact_6357_unat__word__ariths_I1_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A)] : aa(word(A),nat,semiring_1_unsigned(A,nat),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),A2),B2)) = modulo_modulo(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),A2)),aa(word(A),nat,semiring_1_unsigned(A,nat),B2)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A)))) ) ).

% unat_word_ariths(1)
tff(fact_6358_sint__lt,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A)] : pp(aa(int,bool,ord_less(int,ring_1_signed(A,int,X)),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat))))) ) ).

% sint_lt
tff(fact_6359_word__int__cases,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A)] :
          ~ ! [N2: int] :
              ( ( X = aa(int,word(A),ring_1_of_int(word(A)),N2) )
             => ( pp(aa(int,bool,ord_less_eq(int,zero_zero(int)),N2))
               => ~ pp(aa(int,bool,ord_less(int,N2),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A))))) ) ) ) ).

% word_int_cases
tff(fact_6360_word__of__int__inj,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: int,Y: int] :
          ( ( pp(aa(int,bool,ord_less_eq(int,zero_zero(int)),X))
            & pp(aa(int,bool,ord_less(int,X),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A))))) )
         => ( ( pp(aa(int,bool,ord_less_eq(int,zero_zero(int)),Y))
              & pp(aa(int,bool,ord_less(int,Y),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A))))) )
           => ( ( aa(int,word(A),ring_1_of_int(word(A)),X) = aa(int,word(A),ring_1_of_int(word(A)),Y) )
            <=> ( X = Y ) ) ) ) ) ).

% word_of_int_inj
tff(fact_6361_ucast__mono__le_H,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [Y: word(A),X: word(A)] :
          ( pp(aa(nat,bool,ord_less(nat,aa(word(A),nat,semiring_1_unsigned(A,nat),Y)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(B,type2(B)))))
         => ( pp(aa(nat,bool,ord_less(nat,type_len0_len_of(B,type2(B))),type_len0_len_of(A,type2(A))))
           => ( pp(aa(word(A),bool,ord_less_eq(word(A),X),Y))
             => pp(aa(word(B),bool,ord_less_eq(word(B),aa(word(A),word(B),semiring_1_unsigned(A,word(B)),X)),aa(word(A),word(B),semiring_1_unsigned(A,word(B)),Y))) ) ) ) ) ).

% ucast_mono_le'
tff(fact_6362_of__nat__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: nat,W: word(A)] :
          ( ( aa(nat,word(A),semiring_1_of_nat(word(A)),N) = W )
        <=> ? [Q4: nat] : N = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),W)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Q4),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))) ) ) ).

% of_nat_eq
tff(fact_6363_unat__mult__lem,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A)] :
          ( pp(aa(nat,bool,ord_less(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),X)),aa(word(A),nat,semiring_1_unsigned(A,nat),Y))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A)))))
        <=> ( aa(word(A),nat,semiring_1_unsigned(A,nat),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),X),Y)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),X)),aa(word(A),nat,semiring_1_unsigned(A,nat),Y)) ) ) ) ).

% unat_mult_lem
tff(fact_6364_unat__ucast__no__overflow__le,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(A) )
     => ! [B2: word(B),F2: word(A)] :
          ( pp(aa(nat,bool,ord_less(nat,aa(word(B),nat,semiring_1_unsigned(B,nat),B2)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A)))))
         => ( pp(aa(nat,bool,ord_less(nat,type_len0_len_of(A,type2(A))),type_len0_len_of(B,type2(B))))
           => ( pp(aa(word(B),bool,ord_less(word(B),aa(word(A),word(B),semiring_1_unsigned(A,word(B)),F2)),B2))
            <=> pp(aa(nat,bool,ord_less(nat,aa(word(A),nat,semiring_1_unsigned(A,nat),F2)),aa(word(B),nat,semiring_1_unsigned(B,nat),B2))) ) ) ) ) ).

% unat_ucast_no_overflow_le
tff(fact_6365_uint__m2p__not__non__neg,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A)] : ~ pp(aa(int,bool,ord_less_eq(int,zero_zero(int)),aa(int,int,minus_minus(int,aa(word(A),int,semiring_1_unsigned(A,int),X)),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A)))))) ) ).

% uint_m2p_not_non_neg
tff(fact_6366_uint__m2p__neg,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A)] : pp(aa(int,bool,ord_less(int,aa(int,int,minus_minus(int,aa(word(A),int,semiring_1_unsigned(A,int),X)),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A))))),zero_zero(int))) ) ).

% uint_m2p_neg
tff(fact_6367_uint__power__lower,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: nat] :
          ( pp(aa(nat,bool,ord_less(nat,N),type_len0_len_of(A,type2(A))))
         => ( aa(word(A),int,semiring_1_unsigned(A,int),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),N)) = aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),N) ) ) ) ).

% uint_power_lower
tff(fact_6368_unat__ucast__less__no__overflow__simp,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: nat,F2: word(A)] :
          ( pp(aa(nat,bool,ord_less(nat,N),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A)))))
         => ( pp(aa(nat,bool,ord_less(nat,aa(word(A),nat,semiring_1_unsigned(A,nat),F2)),N))
          <=> pp(aa(word(A),bool,ord_less(word(A),F2),aa(nat,word(A),semiring_1_of_nat(word(A)),N))) ) ) ) ).

% unat_ucast_less_no_overflow_simp
tff(fact_6369_unat__ucast__less__no__overflow,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: nat,F2: word(A)] :
          ( pp(aa(nat,bool,ord_less(nat,N),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A)))))
         => ( pp(aa(nat,bool,ord_less(nat,aa(word(A),nat,semiring_1_unsigned(A,nat),F2)),N))
           => pp(aa(word(A),bool,ord_less(word(A),F2),aa(nat,word(A),semiring_1_of_nat(word(A)),N))) ) ) ) ).

% unat_ucast_less_no_overflow
tff(fact_6370_less__2p__is__upper__bits__unset,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [P2: word(A),N: nat] :
          ( pp(aa(word(A),bool,ord_less(word(A),P2),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),N)))
        <=> ( pp(aa(nat,bool,ord_less(nat,N),type_len0_len_of(A,type2(A))))
            & ! [N13: nat] :
                ( pp(aa(nat,bool,ord_less_eq(nat,N),N13))
               => ( pp(aa(nat,bool,ord_less(nat,N13),type_len0_len_of(A,type2(A))))
                 => ~ pp(aa(nat,bool,bit_se5641148757651400278ts_bit(word(A),P2),N13)) ) ) ) ) ) ).

% less_2p_is_upper_bits_unset
tff(fact_6371_upper__bits__unset__is__l2p,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: nat,P2: word(A)] :
          ( pp(aa(nat,bool,ord_less(nat,N),type_len0_len_of(A,type2(A))))
         => ( ! [N13: nat] :
                ( pp(aa(nat,bool,ord_less_eq(nat,N),N13))
               => ( pp(aa(nat,bool,ord_less(nat,N13),type_len0_len_of(A,type2(A))))
                 => ~ pp(aa(nat,bool,bit_se5641148757651400278ts_bit(word(A),P2),N13)) ) )
          <=> pp(aa(word(A),bool,ord_less(word(A),P2),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),N))) ) ) ) ).

% upper_bits_unset_is_l2p
tff(fact_6372_nth__bounded,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),N: nat,M: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(word(A),X),N))
         => ( pp(aa(word(A),bool,ord_less(word(A),X),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),M)))
           => ( pp(aa(nat,bool,ord_less_eq(nat,M),type_len0_len_of(A,type2(A))))
             => pp(aa(nat,bool,ord_less(nat,N),M)) ) ) ) ) ).

% nth_bounded
tff(fact_6373_uint__add__lem,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A)] :
          ( pp(aa(int,bool,ord_less(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(word(A),int,semiring_1_unsigned(A,int),X)),aa(word(A),int,semiring_1_unsigned(A,int),Y))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A)))))
        <=> ( aa(word(A),int,semiring_1_unsigned(A,int),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),X),Y)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(word(A),int,semiring_1_unsigned(A,int),X)),aa(word(A),int,semiring_1_unsigned(A,int),Y)) ) ) ) ).

% uint_add_lem
tff(fact_6374_uint__word__ariths_I7_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( aa(word(A),int,semiring_1_unsigned(A,int),zero_zero(word(A))) = modulo_modulo(int,zero_zero(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A)))) ) ) ).

% uint_word_ariths(7)
tff(fact_6375_uint__word__ariths_I8_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( aa(word(A),int,semiring_1_unsigned(A,int),one_one(word(A))) = modulo_modulo(int,one_one(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A)))) ) ) ).

% uint_word_ariths(8)
tff(fact_6376_wi__le,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: int,M: int] :
          ( pp(aa(word(A),bool,ord_less_eq(word(A),aa(int,word(A),ring_1_of_int(word(A)),N)),aa(int,word(A),ring_1_of_int(word(A)),M)))
        <=> pp(aa(int,bool,ord_less_eq(int,modulo_modulo(int,N,aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A))))),modulo_modulo(int,M,aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A)))))) ) ) ).

% wi_le
tff(fact_6377_uint__word__ariths_I1_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A)] : aa(word(A),int,semiring_1_unsigned(A,int),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),A2),B2)) = modulo_modulo(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(word(A),int,semiring_1_unsigned(A,int),A2)),aa(word(A),int,semiring_1_unsigned(A,int),B2)),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A)))) ) ).

% uint_word_ariths(1)
tff(fact_6378_word__2p__mult__inc,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: nat,M: nat] :
          ( pp(aa(word(A),bool,ord_less(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),N))),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),M))))
         => ( pp(aa(nat,bool,ord_less(nat,aa(nat,nat,suc,N)),type_len0_len_of(A,type2(A))))
           => pp(aa(word(A),bool,ord_less(word(A),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),N)),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),M))) ) ) ) ).

% word_2p_mult_inc
tff(fact_6379_wi__less,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: int,M: int] :
          ( pp(aa(word(A),bool,ord_less(word(A),aa(int,word(A),ring_1_of_int(word(A)),N)),aa(int,word(A),ring_1_of_int(word(A)),M)))
        <=> pp(aa(int,bool,ord_less(int,modulo_modulo(int,N,aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A))))),modulo_modulo(int,M,aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A)))))) ) ) ).

% wi_less
tff(fact_6380_power__2__ge__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: nat] :
          ( pp(aa(word(A),bool,ord_less(word(A),aa(word(A),word(A),minus_minus(word(A),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),N)),one_one(word(A)))),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),N)))
        <=> pp(aa(nat,bool,ord_less(nat,N),type_len0_len_of(A,type2(A)))) ) ) ).

% power_2_ge_iff
tff(fact_6381_word__power__less__1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Sz: nat] :
          ( pp(aa(nat,bool,ord_less(nat,Sz),type_len0_len_of(A,type2(A))))
         => pp(aa(word(A),bool,ord_less(word(A),aa(word(A),word(A),minus_minus(word(A),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Sz)),one_one(word(A)))),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Sz))) ) ) ).

% word_power_less_1
tff(fact_6382_uint__word__ariths_I4_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A)] : aa(word(A),int,semiring_1_unsigned(A,int),aa(word(A),word(A),uminus_uminus(word(A)),A2)) = modulo_modulo(int,aa(int,int,uminus_uminus(int),aa(word(A),int,semiring_1_unsigned(A,int),A2)),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A)))) ) ).

% uint_word_ariths(4)
tff(fact_6383_le__mask__iff__lt__2n,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: nat,W: word(A)] :
          ( pp(aa(nat,bool,ord_less(nat,N),type_len0_len_of(A,type2(A))))
        <=> ( pp(aa(word(A),bool,ord_less_eq(word(A),W),bit_se2239418461657761734s_mask(word(A),N)))
          <=> pp(aa(word(A),bool,ord_less(word(A),W),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),N))) ) ) ) ).

% le_mask_iff_lt_2n
tff(fact_6384_unat__word__ariths_I2_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A)] : aa(word(A),nat,semiring_1_unsigned(A,nat),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),A2),B2)) = modulo_modulo(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),A2)),aa(word(A),nat,semiring_1_unsigned(A,nat),B2)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A)))) ) ).

% unat_word_ariths(2)
tff(fact_6385_unat__word__ariths_I6_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A)] : aa(word(A),nat,semiring_1_unsigned(A,nat),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),A2),B2)) = modulo_modulo(nat,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),A2)),aa(word(A),nat,semiring_1_unsigned(A,nat),B2)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A)))) ) ).

% unat_word_ariths(6)
tff(fact_6386_eq__mask__less,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),N: nat] :
          ( ( W = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),W),bit_se2239418461657761734s_mask(word(A),N)) )
         => ( pp(aa(nat,bool,ord_less(nat,N),type_len0_len_of(A,type2(A))))
           => pp(aa(word(A),bool,ord_less(word(A),W),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),N))) ) ) ) ).

% eq_mask_less
tff(fact_6387_and__mask__less_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: nat,W: word(A)] :
          ( pp(aa(nat,bool,ord_less(nat,N),type_len0_len_of(A,type2(A))))
         => pp(aa(word(A),bool,ord_less(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),W),bit_se2239418461657761734s_mask(word(A),N))),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),N))) ) ) ).

% and_mask_less'
tff(fact_6388_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))) != aa(int,int,uminus_uminus(int),one_one(int)) ) ) )
           => ~ ( pp(aa(nat,bool,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
tff(fact_6389_uint__word__ariths_I2_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A)] : aa(word(A),int,semiring_1_unsigned(A,int),aa(word(A),word(A),minus_minus(word(A),A2),B2)) = modulo_modulo(int,aa(int,int,minus_minus(int,aa(word(A),int,semiring_1_unsigned(A,int),A2)),aa(word(A),int,semiring_1_unsigned(A,int),B2)),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A)))) ) ).

% uint_word_ariths(2)
tff(fact_6390_uint__mult__lem,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A)] :
          ( pp(aa(int,bool,ord_less(int,aa(int,int,aa(int,fun(int,int),times_times(int),aa(word(A),int,semiring_1_unsigned(A,int),X)),aa(word(A),int,semiring_1_unsigned(A,int),Y))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A)))))
        <=> ( aa(word(A),int,semiring_1_unsigned(A,int),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),X),Y)) = aa(int,int,aa(int,fun(int,int),times_times(int),aa(word(A),int,semiring_1_unsigned(A,int),X)),aa(word(A),int,semiring_1_unsigned(A,int),Y)) ) ) ) ).

% uint_mult_lem
tff(fact_6391_uint__word__ariths_I3_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A)] : aa(word(A),int,semiring_1_unsigned(A,int),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),A2),B2)) = modulo_modulo(int,aa(int,int,aa(int,fun(int,int),times_times(int),aa(word(A),int,semiring_1_unsigned(A,int),A2)),aa(word(A),int,semiring_1_unsigned(A,int),B2)),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A)))) ) ).

% uint_word_ariths(3)
tff(fact_6392_signed__of__nat,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & ring_1(A) )
     => ! [N: nat] : ring_1_signed(B,A,aa(nat,word(B),semiring_1_of_nat(word(B)),N)) = aa(int,A,ring_1_of_int(A),aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,minus_minus(nat,type_len0_len_of(B,type2(B))),aa(nat,nat,suc,zero_zero(nat)))),aa(nat,int,semiring_1_of_nat(int),N))) ) ).

% signed_of_nat
tff(fact_6393_word__power__mod__div,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: nat,M: nat,X: word(A)] :
          ( pp(aa(nat,bool,ord_less(nat,N),type_len0_len_of(A,type2(A))))
         => ( pp(aa(nat,bool,ord_less(nat,M),type_len0_len_of(A,type2(A))))
           => ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),modulo_modulo(word(A),X,aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),N))),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),M)) = modulo_modulo(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),X),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),M)),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),aa(nat,nat,minus_minus(nat,N),M))) ) ) ) ) ).

% word_power_mod_div
tff(fact_6394_scast__1_H,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ( ring_1_signed(A,word(B),one_one(word(A))) = aa(int,word(B),ring_1_of_int(word(B)),aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat)))),one_one(int))) ) ) ).

% scast_1'
tff(fact_6395_msb1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A)] :
          ( ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se1065995026697491101ons_or(word(A)),X),Y)),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat)))))
          <=> ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(word(A),X),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat)))))
              | pp(aa(nat,bool,bit_se5641148757651400278ts_bit(word(A),Y),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat))))) ) )
          & ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),X),Y)),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat)))))
          <=> ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(word(A),X),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat)))))
              & pp(aa(nat,bool,bit_se5641148757651400278ts_bit(word(A),Y),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat))))) ) )
          & ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344971392196577ns_xor(word(A)),X),Y)),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat)))))
          <=> ~ ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(word(A),X),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat)))))
              <=> pp(aa(nat,bool,bit_se5641148757651400278ts_bit(word(A),Y),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat))))) ) )
          & ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(word(A),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),X)),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat)))))
          <=> ~ pp(aa(nat,bool,bit_se5641148757651400278ts_bit(word(A),X),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat))))) ) ) ) ).

% msb1
tff(fact_6396_unat__plus__if_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A)] :
          ( ( pp(aa(nat,bool,ord_less(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),A2)),aa(word(A),nat,semiring_1_unsigned(A,nat),B2))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A)))))
           => ( aa(word(A),nat,semiring_1_unsigned(A,nat),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),A2),B2)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),A2)),aa(word(A),nat,semiring_1_unsigned(A,nat),B2)) ) )
          & ( ~ pp(aa(nat,bool,ord_less(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),A2)),aa(word(A),nat,semiring_1_unsigned(A,nat),B2))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A)))))
           => ( aa(word(A),nat,semiring_1_unsigned(A,nat),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),A2),B2)) = aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),A2)),aa(word(A),nat,semiring_1_unsigned(A,nat),B2))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A)))) ) ) ) ) ).

% unat_plus_if'
tff(fact_6397_unat__word__ariths_I5_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( aa(word(A),nat,semiring_1_unsigned(A,nat),one_one(word(A))) = modulo_modulo(nat,aa(nat,nat,suc,zero_zero(nat)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A)))) ) ) ).

% unat_word_ariths(5)
tff(fact_6398_sint__less,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] : pp(aa(int,bool,ord_less(int,ring_1_signed(A,int,W)),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat)))))) ) ).

% sint_less
tff(fact_6399_unat__sub__if_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),X: word(A)] :
          ( ( pp(aa(nat,bool,ord_less_eq(nat,aa(word(A),nat,semiring_1_unsigned(A,nat),Y)),aa(word(A),nat,semiring_1_unsigned(A,nat),X)))
           => ( aa(word(A),nat,semiring_1_unsigned(A,nat),aa(word(A),word(A),minus_minus(word(A),X),Y)) = aa(nat,nat,minus_minus(nat,aa(word(A),nat,semiring_1_unsigned(A,nat),X)),aa(word(A),nat,semiring_1_unsigned(A,nat),Y)) ) )
          & ( ~ pp(aa(nat,bool,ord_less_eq(nat,aa(word(A),nat,semiring_1_unsigned(A,nat),Y)),aa(word(A),nat,semiring_1_unsigned(A,nat),X)))
           => ( aa(word(A),nat,semiring_1_unsigned(A,nat),aa(word(A),word(A),minus_minus(word(A),X),Y)) = aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),X)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))),aa(word(A),nat,semiring_1_unsigned(A,nat),Y)) ) ) ) ) ).

% unat_sub_if'
tff(fact_6400_no__olen__add__nat,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A)] :
          ( pp(aa(word(A),bool,ord_less_eq(word(A),X),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),X),Y)))
        <=> pp(aa(nat,bool,ord_less(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),X)),aa(word(A),nat,semiring_1_unsigned(A,nat),Y))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))) ) ) ).

% no_olen_add_nat
tff(fact_6401_word__add__le__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [I: word(A),K: word(A),J: word(A)] :
          ( pp(aa(nat,bool,ord_less(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),I)),aa(word(A),nat,semiring_1_unsigned(A,nat),K))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A)))))
         => ( pp(aa(nat,bool,ord_less(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),J)),aa(word(A),nat,semiring_1_unsigned(A,nat),K))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A)))))
           => ( pp(aa(word(A),bool,ord_less_eq(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),I),K)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),J),K)))
            <=> pp(aa(word(A),bool,ord_less_eq(word(A),I),J)) ) ) ) ) ).

% word_add_le_iff
tff(fact_6402_word__add__le__dest,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [I: word(A),K: word(A),J: word(A)] :
          ( pp(aa(word(A),bool,ord_less_eq(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),I),K)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),J),K)))
         => ( pp(aa(nat,bool,ord_less(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),I)),aa(word(A),nat,semiring_1_unsigned(A,nat),K))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A)))))
           => ( pp(aa(nat,bool,ord_less(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),J)),aa(word(A),nat,semiring_1_unsigned(A,nat),K))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A)))))
             => pp(aa(word(A),bool,ord_less_eq(word(A),I),J)) ) ) ) ) ).

% word_add_le_dest
tff(fact_6403_word__add__le__mono1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [I: word(A),J: word(A),K: word(A)] :
          ( pp(aa(word(A),bool,ord_less_eq(word(A),I),J))
         => ( pp(aa(nat,bool,ord_less(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),J)),aa(word(A),nat,semiring_1_unsigned(A,nat),K))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A)))))
           => pp(aa(word(A),bool,ord_less_eq(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),I),K)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),J),K))) ) ) ) ).

% word_add_le_mono1
tff(fact_6404_word__add__le__mono2,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [I: word(A),J: word(A),K: word(A)] :
          ( pp(aa(word(A),bool,ord_less_eq(word(A),I),J))
         => ( pp(aa(nat,bool,ord_less(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),J)),aa(word(A),nat,semiring_1_unsigned(A,nat),K))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A)))))
           => pp(aa(word(A),bool,ord_less_eq(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),K),I)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),K),J))) ) ) ) ).

% word_add_le_mono2
tff(fact_6405_word__add__less__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [I: word(A),K: word(A),J: word(A)] :
          ( pp(aa(nat,bool,ord_less(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),I)),aa(word(A),nat,semiring_1_unsigned(A,nat),K))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A)))))
         => ( pp(aa(nat,bool,ord_less(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),J)),aa(word(A),nat,semiring_1_unsigned(A,nat),K))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A)))))
           => ( pp(aa(word(A),bool,ord_less(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),I),K)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),J),K)))
            <=> pp(aa(word(A),bool,ord_less(word(A),I),J)) ) ) ) ) ).

% word_add_less_iff
tff(fact_6406_word__add__less__dest,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [I: word(A),K: word(A),J: word(A)] :
          ( pp(aa(word(A),bool,ord_less(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),I),K)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),J),K)))
         => ( pp(aa(nat,bool,ord_less(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),I)),aa(word(A),nat,semiring_1_unsigned(A,nat),K))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A)))))
           => ( pp(aa(nat,bool,ord_less(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),J)),aa(word(A),nat,semiring_1_unsigned(A,nat),K))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A)))))
             => pp(aa(word(A),bool,ord_less(word(A),I),J)) ) ) ) ) ).

% word_add_less_dest
tff(fact_6407_word__add__less__mono1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [I: word(A),J: word(A),K: word(A)] :
          ( pp(aa(word(A),bool,ord_less(word(A),I),J))
         => ( pp(aa(nat,bool,ord_less(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),J)),aa(word(A),nat,semiring_1_unsigned(A,nat),K))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A)))))
           => pp(aa(word(A),bool,ord_less(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),I),K)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),J),K))) ) ) ) ).

% word_add_less_mono1
tff(fact_6408_unat__minus__one__word,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( aa(word(A),nat,semiring_1_unsigned(A,nat),aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A)))) = aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A)))),one_one(nat)) ) ) ).

% unat_minus_one_word
tff(fact_6409_sint__ge,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A)] : pp(aa(int,bool,ord_less_eq(int,aa(int,int,uminus_uminus(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat))))),ring_1_signed(A,int,X))) ) ).

% sint_ge
tff(fact_6410_unat__less__power,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Sz: nat,K: word(A)] :
          ( pp(aa(nat,bool,ord_less(nat,Sz),type_len0_len_of(A,type2(A))))
         => ( pp(aa(word(A),bool,ord_less(word(A),K),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Sz)))
           => pp(aa(nat,bool,ord_less(nat,aa(word(A),nat,semiring_1_unsigned(A,nat),K)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Sz))) ) ) ) ).

% unat_less_power
tff(fact_6411_uint__split,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [P: fun(int,bool),X: word(A)] :
          ( pp(aa(int,bool,P,aa(word(A),int,semiring_1_unsigned(A,int),X)))
        <=> ! [I3: int] :
              ( ( ( aa(int,word(A),ring_1_of_int(word(A)),I3) = X )
                & pp(aa(int,bool,ord_less_eq(int,zero_zero(int)),I3))
                & pp(aa(int,bool,ord_less(int,I3),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A))))) )
             => pp(aa(int,bool,P,I3)) ) ) ) ).

% uint_split
tff(fact_6412_uint__split__asm,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [P: fun(int,bool),X: word(A)] :
          ( pp(aa(int,bool,P,aa(word(A),int,semiring_1_unsigned(A,int),X)))
        <=> ~ ? [I3: int] :
                ( ( aa(int,word(A),ring_1_of_int(word(A)),I3) = X )
                & pp(aa(int,bool,ord_less_eq(int,zero_zero(int)),I3))
                & pp(aa(int,bool,ord_less(int,I3),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A)))))
                & ~ pp(aa(int,bool,P,I3)) ) ) ) ).

% uint_split_asm
tff(fact_6413_word__of__int__inverse,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [R2: int,A2: word(A)] :
          ( ( aa(int,word(A),ring_1_of_int(word(A)),R2) = A2 )
         => ( pp(aa(int,bool,ord_less_eq(int,zero_zero(int)),R2))
           => ( pp(aa(int,bool,ord_less(int,R2),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A)))))
             => ( aa(word(A),int,semiring_1_unsigned(A,int),A2) = R2 ) ) ) ) ) ).

% word_of_int_inverse
tff(fact_6414_word__mult__less__dest,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [I: word(A),K: word(A),J: word(A)] :
          ( pp(aa(word(A),bool,ord_less(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),I),K)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),J),K)))
         => ( pp(aa(nat,bool,ord_less(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),I)),aa(word(A),nat,semiring_1_unsigned(A,nat),K))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A)))))
           => ( pp(aa(nat,bool,ord_less(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),J)),aa(word(A),nat,semiring_1_unsigned(A,nat),K))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A)))))
             => pp(aa(word(A),bool,ord_less(word(A),I),J)) ) ) ) ) ).

% word_mult_less_dest
tff(fact_6415_div__lt_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [I: word(A),K: word(A),X: word(A)] :
          ( pp(aa(word(A),bool,ord_less_eq(word(A),I),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),K),X)))
         => pp(aa(nat,bool,ord_less(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),I)),aa(word(A),nat,semiring_1_unsigned(A,nat),X))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))) ) ) ).

% div_lt'
tff(fact_6416_double__eq__zero__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A)] :
          ( ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),A2) = zero_zero(word(A)) )
        <=> ( ( A2 = zero_zero(word(A)) )
            | ( A2 = aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat)))) ) ) ) ) ).

% double_eq_zero_iff
tff(fact_6417_div__lt_H_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [I: word(A),K: word(A),X: word(A)] :
          ( pp(aa(word(A),bool,ord_less(word(A),I),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),K),X)))
         => pp(aa(nat,bool,ord_less(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),I)),aa(word(A),nat,semiring_1_unsigned(A,nat),X))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))) ) ) ).

% div_lt''
tff(fact_6418_word__le__exists_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A)] :
          ( pp(aa(word(A),bool,ord_less_eq(word(A),X),Y))
         => ? [Z2: word(A)] :
              ( ( Y = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),X),Z2) )
              & pp(aa(int,bool,ord_less(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(word(A),int,semiring_1_unsigned(A,int),X)),aa(word(A),int,semiring_1_unsigned(A,int),Z2))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A))))) ) ) ) ).

% word_le_exists'
tff(fact_6419_no__olen__add_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A)] :
          ( pp(aa(word(A),bool,ord_less_eq(word(A),X),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Y),X)))
        <=> pp(aa(int,bool,ord_less(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(word(A),int,semiring_1_unsigned(A,int),Y)),aa(word(A),int,semiring_1_unsigned(A,int),X))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A))))) ) ) ).

% no_olen_add'
tff(fact_6420_no__olen__add,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A)] :
          ( pp(aa(word(A),bool,ord_less_eq(word(A),X),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),X),Y)))
        <=> pp(aa(int,bool,ord_less(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(word(A),int,semiring_1_unsigned(A,int),X)),aa(word(A),int,semiring_1_unsigned(A,int),Y))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A))))) ) ) ).

% no_olen_add
tff(fact_6421_More__Word_Oof__nat__power,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [P2: nat,X: nat] :
          ( pp(aa(nat,bool,ord_less(nat,P2),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),X)))
         => ( pp(aa(nat,bool,ord_less(nat,X),type_len0_len_of(A,type2(A))))
           => pp(aa(word(A),bool,ord_less(word(A),aa(nat,word(A),semiring_1_of_nat(word(A)),P2)),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),X))) ) ) ) ).

% More_Word.of_nat_power
tff(fact_6422_uint__plus__if_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A)] :
          ( ( pp(aa(int,bool,ord_less(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(word(A),int,semiring_1_unsigned(A,int),A2)),aa(word(A),int,semiring_1_unsigned(A,int),B2))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A)))))
           => ( aa(word(A),int,semiring_1_unsigned(A,int),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),A2),B2)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(word(A),int,semiring_1_unsigned(A,int),A2)),aa(word(A),int,semiring_1_unsigned(A,int),B2)) ) )
          & ( ~ pp(aa(int,bool,ord_less(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(word(A),int,semiring_1_unsigned(A,int),A2)),aa(word(A),int,semiring_1_unsigned(A,int),B2))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A)))))
           => ( aa(word(A),int,semiring_1_unsigned(A,int),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),A2),B2)) = aa(int,int,minus_minus(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(word(A),int,semiring_1_unsigned(A,int),A2)),aa(word(A),int,semiring_1_unsigned(A,int),B2))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A)))) ) ) ) ) ).

% uint_plus_if'
tff(fact_6423_word__less__power__trans2,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: word(A),M: nat,K: nat] :
          ( pp(aa(word(A),bool,ord_less(word(A),N),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),aa(nat,nat,minus_minus(nat,M),K))))
         => ( pp(aa(nat,bool,ord_less_eq(nat,K),M))
           => ( pp(aa(nat,bool,ord_less(nat,M),type_len0_len_of(A,type2(A))))
             => pp(aa(word(A),bool,ord_less(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),N),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),K))),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),M))) ) ) ) ) ).

% word_less_power_trans2
tff(fact_6424_word__less__power__trans,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: word(A),M: nat,K: nat] :
          ( pp(aa(word(A),bool,ord_less(word(A),N),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),aa(nat,nat,minus_minus(nat,M),K))))
         => ( pp(aa(nat,bool,ord_less_eq(nat,K),M))
           => ( pp(aa(nat,bool,ord_less(nat,M),type_len0_len_of(A,type2(A))))
             => pp(aa(word(A),bool,ord_less(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),K)),N)),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),M))) ) ) ) ) ).

% word_less_power_trans
tff(fact_6425_uint__sub__if_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [B2: word(A),A2: word(A)] :
          ( ( pp(aa(int,bool,ord_less_eq(int,aa(word(A),int,semiring_1_unsigned(A,int),B2)),aa(word(A),int,semiring_1_unsigned(A,int),A2)))
           => ( aa(word(A),int,semiring_1_unsigned(A,int),aa(word(A),word(A),minus_minus(word(A),A2),B2)) = aa(int,int,minus_minus(int,aa(word(A),int,semiring_1_unsigned(A,int),A2)),aa(word(A),int,semiring_1_unsigned(A,int),B2)) ) )
          & ( ~ pp(aa(int,bool,ord_less_eq(int,aa(word(A),int,semiring_1_unsigned(A,int),B2)),aa(word(A),int,semiring_1_unsigned(A,int),A2)))
           => ( aa(word(A),int,semiring_1_unsigned(A,int),aa(word(A),word(A),minus_minus(word(A),A2),B2)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,minus_minus(int,aa(word(A),int,semiring_1_unsigned(A,int),A2)),aa(word(A),int,semiring_1_unsigned(A,int),B2))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A)))) ) ) ) ) ).

% uint_sub_if'
tff(fact_6426_uint__neg__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [B2: num] : aa(word(A),int,semiring_1_unsigned(A,int),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),B2))) = modulo_modulo(int,aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),B2)),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A)))) ) ).

% uint_neg_numeral
tff(fact_6427_word__less__two__pow__divI,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),N: nat,M: nat] :
          ( pp(aa(word(A),bool,ord_less(word(A),X),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),aa(nat,nat,minus_minus(nat,N),M))))
         => ( pp(aa(nat,bool,ord_less_eq(nat,M),N))
           => ( pp(aa(nat,bool,ord_less(nat,N),type_len0_len_of(A,type2(A))))
             => pp(aa(word(A),bool,ord_less(word(A),X),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),N)),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),M)))) ) ) ) ) ).

% word_less_two_pow_divI
tff(fact_6428_word__power__nonzero,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),N: nat] :
          ( pp(aa(word(A),bool,ord_less(word(A),X),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),N))))
         => ( pp(aa(nat,bool,ord_less(nat,N),type_len0_len_of(A,type2(A))))
           => ( ( X != zero_zero(word(A)) )
             => ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),X),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),N)) != zero_zero(word(A)) ) ) ) ) ) ).

% word_power_nonzero
tff(fact_6429_div__lt__uint_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [I: word(A),K: word(A),X: word(A)] :
          ( pp(aa(word(A),bool,ord_less_eq(word(A),I),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),K),X)))
         => pp(aa(int,bool,ord_less(int,aa(int,int,aa(int,fun(int,int),times_times(int),aa(word(A),int,semiring_1_unsigned(A,int),I)),aa(word(A),int,semiring_1_unsigned(A,int),X))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A))))) ) ) ).

% div_lt_uint'
tff(fact_6430_mult__pow2__inj,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [M: nat,N: nat,X: word(A),Y: word(A)] :
          ( pp(aa(nat,bool,ord_less_eq(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N)),type_len0_len_of(A,type2(A))))
         => ( pp(aa(word(A),bool,ord_less_eq(word(A),X),bit_se2239418461657761734s_mask(word(A),M)))
           => ( pp(aa(word(A),bool,ord_less_eq(word(A),Y),bit_se2239418461657761734s_mask(word(A),M)))
             => ( ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),X),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),N)) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),Y),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),N)) )
               => ( X = Y ) ) ) ) ) ) ).

% mult_pow2_inj
tff(fact_6431_div__lt__uint_H_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [I: word(A),K: word(A),X: word(A)] :
          ( pp(aa(word(A),bool,ord_less(word(A),I),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),K),X)))
         => pp(aa(int,bool,ord_less(int,aa(int,int,aa(int,fun(int,int),times_times(int),aa(word(A),int,semiring_1_unsigned(A,int),I)),aa(word(A),int,semiring_1_unsigned(A,int),X))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A))))) ) ) ).

% div_lt_uint''
tff(fact_6432_push__bit__word__eq__nonzero,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),M: nat,N: nat] :
          ( pp(aa(word(A),bool,ord_less(word(A),W),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),M)))
         => ( pp(aa(nat,bool,ord_less(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N)),type_len0_len_of(A,type2(A))))
           => ( ( W != zero_zero(word(A)) )
             => ( bit_se4730199178511100633sh_bit(word(A),N,W) != zero_zero(word(A)) ) ) ) ) ) ).

% push_bit_word_eq_nonzero
tff(fact_6433_uint__and__mask__or__full,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: word(A),Mask1: word(A),Mask2: int] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(word(A),N),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat))))
         => ( ( Mask1 = bit_se2239418461657761734s_mask(word(A),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat))) )
           => ( ( Mask2 = bit_se4730199178511100633sh_bit(int,aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat)),one_one(int)) )
             => ( aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(word(A),int,semiring_1_unsigned(A,int),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),N),Mask1))),Mask2) = aa(word(A),int,semiring_1_unsigned(A,int),N) ) ) ) ) ) ).

% uint_and_mask_or_full
tff(fact_6434_sint__greater__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] : pp(aa(int,bool,ord_less_eq(int,aa(int,int,uminus_uminus(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat)))))),ring_1_signed(A,int,W))) ) ).

% sint_greater_eq
tff(fact_6435_int__eq__sint,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: nat] :
          ( pp(aa(nat,bool,ord_less(nat,X),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat)))))
         => ( ring_1_signed(A,int,aa(nat,word(A),semiring_1_of_nat(word(A)),X)) = aa(nat,int,semiring_1_of_nat(int),X) ) ) ) ).

% int_eq_sint
tff(fact_6436_word__mult__less__cancel,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [K: word(A),I: word(A),J: word(A)] :
          ( pp(aa(word(A),bool,ord_less(word(A),zero_zero(word(A))),K))
         => ( pp(aa(nat,bool,ord_less(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),I)),aa(word(A),nat,semiring_1_unsigned(A,nat),K))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A)))))
           => ( pp(aa(nat,bool,ord_less(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),J)),aa(word(A),nat,semiring_1_unsigned(A,nat),K))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A)))))
             => ( pp(aa(word(A),bool,ord_less(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),I),K)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),J),K)))
              <=> pp(aa(word(A),bool,ord_less(word(A),I),J)) ) ) ) ) ) ).

% word_mult_less_cancel
tff(fact_6437_word__mult__less__mono1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [I: word(A),J: word(A),K: word(A)] :
          ( pp(aa(word(A),bool,ord_less(word(A),I),J))
         => ( pp(aa(word(A),bool,ord_less(word(A),zero_zero(word(A))),K))
           => ( pp(aa(nat,bool,ord_less(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),J)),aa(word(A),nat,semiring_1_unsigned(A,nat),K))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A)))))
             => pp(aa(word(A),bool,ord_less(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),I),K)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),J),K))) ) ) ) ) ).

% word_mult_less_mono1
tff(fact_6438_smod__word__max,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A)] : pp(aa(int,bool,ord_less(int,signed6721504322012087516modulo(int,ring_1_signed(A,int,A2),ring_1_signed(A,int,B2))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat)))))) ) ).

% smod_word_max
tff(fact_6439_le__2p__upper__bits,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [P2: word(A),N: nat] :
          ( pp(aa(word(A),bool,ord_less_eq(word(A),P2),aa(word(A),word(A),minus_minus(word(A),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),N)),one_one(word(A)))))
         => ( pp(aa(nat,bool,ord_less(nat,N),type_len0_len_of(A,type2(A))))
           => ! [N6: nat] :
                ( pp(aa(nat,bool,ord_less_eq(nat,N),N6))
               => ( pp(aa(nat,bool,ord_less(nat,N6),type_len0_len_of(A,type2(A))))
                 => ~ pp(aa(nat,bool,bit_se5641148757651400278ts_bit(word(A),P2),N6)) ) ) ) ) ) ).

% le_2p_upper_bits
tff(fact_6440_le2p__bits__unset,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [P2: word(A),N: nat] :
          ( pp(aa(word(A),bool,ord_less_eq(word(A),P2),aa(word(A),word(A),minus_minus(word(A),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),N)),one_one(word(A)))))
         => ! [N6: nat] :
              ( pp(aa(nat,bool,ord_less_eq(nat,N),N6))
             => ( pp(aa(nat,bool,ord_less(nat,N6),type_len0_len_of(A,type2(A))))
               => ~ pp(aa(nat,bool,bit_se5641148757651400278ts_bit(word(A),P2),N6)) ) ) ) ) ).

% le2p_bits_unset
tff(fact_6441_word__add__offset__less,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),N: nat,X: word(A),M: nat,Sz: nat] :
          ( pp(aa(word(A),bool,ord_less(word(A),Y),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),N)))
         => ( pp(aa(word(A),bool,ord_less(word(A),X),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),M)))
           => ( pp(aa(nat,bool,ord_less(nat,Sz),type_len0_len_of(A,type2(A))))
             => ( pp(aa(word(A),bool,ord_less(word(A),X),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),N))))
               => ( ( Sz = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N) )
                 => pp(aa(word(A),bool,ord_less(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),X),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),N))),Y)),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Sz))) ) ) ) ) ) ) ).

% word_add_offset_less
tff(fact_6442_bit__horner__sum__uint__exp__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Ws: list(word(A)),N: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(int,groups4207007520872428315er_sum(word(A),int,semiring_1_unsigned(A,int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A))),Ws)),N))
        <=> ( pp(aa(nat,bool,ord_less(nat,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),type_len0_len_of(A,type2(A)))),aa(list(word(A)),nat,size_size(list(word(A))),Ws)))
            & pp(aa(nat,bool,bit_se5641148757651400278ts_bit(word(A),aa(nat,word(A),nth(word(A),Ws),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),type_len0_len_of(A,type2(A))))),modulo_modulo(nat,N,type_len0_len_of(A,type2(A))))) ) ) ) ).

% bit_horner_sum_uint_exp_iff
tff(fact_6443_div__power__helper,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: nat,Y: nat] :
          ( pp(aa(nat,bool,ord_less_eq(nat,X),Y))
         => ( pp(aa(nat,bool,ord_less(nat,Y),type_len0_len_of(A,type2(A))))
           => ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),aa(word(A),word(A),minus_minus(word(A),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Y)),one_one(word(A)))),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),X)) = aa(word(A),word(A),minus_minus(word(A),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),aa(nat,nat,minus_minus(nat,Y),X))),one_one(word(A))) ) ) ) ) ).

% div_power_helper
tff(fact_6444_even__mult__exp__div__word__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),M: nat,N: nat] :
          ( pp(aa(word(A),bool,dvd_dvd(word(A),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),A2),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),M))),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),N))))
        <=> ~ ( pp(aa(nat,bool,ord_less_eq(nat,M),N))
              & pp(aa(nat,bool,ord_less(nat,N),type_len0_len_of(A,type2(A))))
              & ~ pp(aa(word(A),bool,dvd_dvd(word(A),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),A2),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),aa(nat,nat,minus_minus(nat,N),M))))) ) ) ) ).

% even_mult_exp_div_word_iff
tff(fact_6445_Suc__2p__unat__mask,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: nat,K: nat] :
          ( pp(aa(nat,bool,ord_less(nat,N),type_len0_len_of(A,type2(A))))
         => ( aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)),K)),aa(word(A),nat,semiring_1_unsigned(A,nat),bit_se2239418461657761734s_mask(word(A),N)))) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),one_one(nat))) ) ) ) ).

% Suc_2p_unat_mask
tff(fact_6446_sint__of__nat__le,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [B2: nat,A2: nat] :
          ( pp(aa(nat,bool,ord_less(nat,B2),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat)))))
         => ( pp(aa(nat,bool,ord_less_eq(nat,A2),B2))
           => pp(aa(int,bool,ord_less_eq(int,ring_1_signed(A,int,aa(nat,word(A),semiring_1_of_nat(word(A)),A2))),ring_1_signed(A,int,aa(nat,word(A),semiring_1_of_nat(word(A)),B2)))) ) ) ) ).

% sint_of_nat_le
tff(fact_6447_sint__of__nat__ge__zero,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: nat] :
          ( pp(aa(nat,bool,ord_less(nat,X),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat)))))
         => pp(aa(int,bool,ord_less_eq(int,zero_zero(int)),ring_1_signed(A,int,aa(nat,word(A),semiring_1_of_nat(word(A)),X)))) ) ) ).

% sint_of_nat_ge_zero
tff(fact_6448_sint__int__max__plus__1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( ring_1_signed(A,int,aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat))))) = aa(int,int,uminus_uminus(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat))))) ) ) ).

% sint_int_max_plus_1
tff(fact_6449_sint__of__int__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: int] :
          ( pp(aa(int,bool,ord_less_eq(int,aa(int,int,uminus_uminus(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat))))),X))
         => ( pp(aa(int,bool,ord_less(int,X),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat)))))
           => ( ring_1_signed(A,int,aa(int,word(A),ring_1_of_int(word(A)),X)) = X ) ) ) ) ).

% sint_of_int_eq
tff(fact_6450_sint__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [B2: num] : ring_1_signed(A,int,aa(num,word(A),numeral_numeral(word(A)),B2)) = aa(int,int,minus_minus(int,modulo_modulo(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(num,int,numeral_numeral(int),B2)),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat)))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A))))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat)))) ) ).

% sint_numeral
tff(fact_6451_word__mult__le__mono1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [I: word(A),J: word(A),K: word(A)] :
          ( pp(aa(word(A),bool,ord_less_eq(word(A),I),J))
         => ( pp(aa(word(A),bool,ord_less(word(A),zero_zero(word(A))),K))
           => ( pp(aa(nat,bool,ord_less(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),J)),aa(word(A),nat,semiring_1_unsigned(A,nat),K))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A)))))
             => pp(aa(word(A),bool,ord_less_eq(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),I),K)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),J),K))) ) ) ) ) ).

% word_mult_le_mono1
tff(fact_6452_word__mult__le__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [K: word(A),I: word(A),J: word(A)] :
          ( pp(aa(word(A),bool,ord_less(word(A),zero_zero(word(A))),K))
         => ( pp(aa(nat,bool,ord_less(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),I)),aa(word(A),nat,semiring_1_unsigned(A,nat),K))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A)))))
           => ( pp(aa(nat,bool,ord_less(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),J)),aa(word(A),nat,semiring_1_unsigned(A,nat),K))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A)))))
             => ( pp(aa(word(A),bool,ord_less_eq(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),I),K)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),J),K)))
              <=> pp(aa(word(A),bool,ord_less_eq(word(A),I),J)) ) ) ) ) ) ).

% word_mult_le_iff
tff(fact_6453_smod__word__min,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A)] : pp(aa(int,bool,ord_less_eq(int,aa(int,int,uminus_uminus(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat)))))),signed6721504322012087516modulo(int,ring_1_signed(A,int,A2),ring_1_signed(A,int,B2)))) ) ).

% smod_word_min
tff(fact_6454_int__word__sint,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: int] : ring_1_signed(A,int,aa(int,word(A),ring_1_of_int(word(A)),X)) = aa(int,int,minus_minus(int,modulo_modulo(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),X),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat)))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A))))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat)))) ) ).

% int_word_sint
tff(fact_6455_Word_Oword__div__mult,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),X: word(A)] :
          ( pp(aa(word(A),bool,ord_less(word(A),zero_zero(word(A))),Y))
         => ( pp(aa(nat,bool,ord_less(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),X)),aa(word(A),nat,semiring_1_unsigned(A,nat),Y))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A)))))
           => ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),X),Y)),Y) = X ) ) ) ) ).

% Word.word_div_mult
tff(fact_6456_of__nat__less__two__pow__div__set,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: nat,M: nat] :
          ( pp(aa(nat,bool,ord_less(nat,N),type_len0_len_of(A,type2(A))))
         => ( collect(word(A),aa(nat,fun(word(A),bool),aTP_Lamp_po(nat,fun(nat,fun(word(A),bool)),N),M)) = aa(set(nat),set(word(A)),image(nat,word(A),semiring_1_of_nat(word(A))),collect(nat,aa(nat,fun(nat,bool),aTP_Lamp_pp(nat,fun(nat,fun(nat,bool)),N),M))) ) ) ) ).

% of_nat_less_two_pow_div_set
tff(fact_6457_sint__int__min,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( ring_1_signed(A,int,aa(word(A),word(A),uminus_uminus(word(A)),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat)))))) = aa(int,int,uminus_uminus(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat))))) ) ) ).

% sint_int_min
tff(fact_6458_word__bit__induct,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [P: fun(word(A),bool),A2: word(A)] :
          ( pp(aa(word(A),bool,P,zero_zero(word(A))))
         => ( ! [A4: word(A)] :
                ( pp(aa(word(A),bool,P,A4))
               => ( pp(aa(word(A),bool,ord_less(word(A),zero_zero(word(A))),A4))
                 => ( pp(aa(word(A),bool,ord_less(word(A),A4),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat))))))
                   => pp(aa(word(A),bool,P,aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),A4))) ) ) )
           => ( ! [A4: word(A)] :
                  ( pp(aa(word(A),bool,P,A4))
                 => ( pp(aa(word(A),bool,ord_less(word(A),A4),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat))))))
                   => pp(aa(word(A),bool,P,aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),one_one(word(A))),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),A4)))) ) )
             => pp(aa(word(A),bool,P,A2)) ) ) ) ) ).

% word_bit_induct
tff(fact_6459_word__less__power__trans__ofnat,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: nat,M: nat,K: nat] :
          ( pp(aa(nat,bool,ord_less(nat,N),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,minus_minus(nat,M),K))))
         => ( pp(aa(nat,bool,ord_less_eq(nat,K),M))
           => ( pp(aa(nat,bool,ord_less(nat,M),type_len0_len_of(A,type2(A))))
             => pp(aa(word(A),bool,ord_less(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),aa(nat,word(A),semiring_1_of_nat(word(A)),N)),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),K))),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),M))) ) ) ) ) ).

% word_less_power_trans_ofnat
tff(fact_6460_unat__mult__power__lem,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [K: nat,Sz: nat] :
          ( pp(aa(nat,bool,ord_less(nat,K),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),Sz))))
         => ( aa(word(A),nat,semiring_1_unsigned(A,nat),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Sz)),aa(nat,word(A),semiring_1_of_nat(word(A)),K))) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Sz)),K) ) ) ) ).

% unat_mult_power_lem
tff(fact_6461_bit__word__half__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: bool] :
          ( pp(aa(word(A),bool,ord_less(word(A),A2),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat))))))
         => ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),aa(bool,word(A),zero_neq_one_of_bool(word(A)),B2)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),A2),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))))),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))) = A2 ) ) ) ).

% bit_word_half_eq
tff(fact_6462_word__of__int__via__signed,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Mask: int,Shift: int,Index: nat,Overflow: int,Least: int,I: int,Arbitrary1: fun(int,word(A)),Arbitrary2: fun(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 = aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat)) )
             => ( ( Overflow = bit_se4730199178511100633sh_bit(int,aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat)),one_one(int)) )
               => ( ( Least = aa(int,int,uminus_uminus(int),Overflow) )
                 => ( aa(int,word(A),ring_1_of_int(word(A)),I) = if(word(A),aa(nat,bool,bit_se5641148757651400278ts_bit(int,aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),I),Mask)),Index),if(word(A),fdisj(aa(int,bool,ord_less(int,aa(int,int,minus_minus(int,aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),I),Mask)),Shift)),Least),aa(int,bool,ord_less_eq(int,Overflow),aa(int,int,minus_minus(int,aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),I),Mask)),Shift))),aa(int,word(A),Arbitrary1,aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),I),Mask)),aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,minus_minus(int,aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),I),Mask)),Shift))),if(word(A),fdisj(aa(int,bool,ord_less(int,aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),I),Mask)),Least),aa(int,bool,ord_less_eq(int,Overflow),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),I),Mask))),aa(int,word(A),Arbitrary2,aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),I),Mask)),aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),I),Mask)))) ) ) ) ) ) ) ) ).

% word_of_int_via_signed
tff(fact_6463_Suc__div__unat__helper,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Sz: nat,Us: nat] :
          ( pp(aa(nat,bool,ord_less(nat,Sz),type_len0_len_of(A,type2(A))))
         => ( pp(aa(nat,bool,ord_less_eq(nat,Us),Sz))
           => ( aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,minus_minus(nat,Sz),Us)) = aa(nat,nat,suc,aa(word(A),nat,semiring_1_unsigned(A,nat),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),aa(word(A),word(A),minus_minus(word(A),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Sz)),one_one(word(A)))),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Us)))) ) ) ) ) ).

% Suc_div_unat_helper
tff(fact_6464_alignUp__div__helper,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [K: nat,N: nat,X: word(A),A2: word(A)] :
          ( pp(aa(nat,bool,ord_less(nat,K),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),N))))
         => ( ( X = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),N)),aa(nat,word(A),semiring_1_of_nat(word(A)),K)) )
           => ( pp(aa(word(A),bool,ord_less_eq(word(A),A2),X))
             => ( pp(aa(nat,bool,ord_less(nat,N),type_len0_len_of(A,type2(A))))
               => ( ( modulo_modulo(word(A),A2,aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),N)) != zero_zero(word(A)) )
                 => pp(aa(word(A),bool,ord_less(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),A2),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),N))),aa(nat,word(A),semiring_1_of_nat(word(A)),K))) ) ) ) ) ) ) ).

% alignUp_div_helper
tff(fact_6465_decr__length__less__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: nat] :
          ( pp(aa(nat,bool,ord_less(nat,aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat)))),N))
        <=> pp(aa(nat,bool,ord_less_eq(nat,type_len0_len_of(A,type2(A))),N)) ) ) ).

% decr_length_less_iff
tff(fact_6466_less__eq__decr__length__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: nat] :
          ( pp(aa(nat,bool,ord_less_eq(nat,N),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat)))))
        <=> pp(aa(nat,bool,ord_less(nat,N),type_len0_len_of(A,type2(A)))) ) ) ).

% less_eq_decr_length_iff
tff(fact_6467_len__gt__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),type_len0_len_of(A,type2(A)))) ) ).

% len_gt_0
tff(fact_6468_length__not__greater__eq__2__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( ~ pp(aa(nat,bool,ord_less_eq(nat,aa(num,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
tff(fact_6469_len__num0,axiom,
    ! [Uu: itself(numeral_num0)] : type_len0_len_of(numeral_num0,Uu) = zero_zero(nat) ).

% len_num0
tff(fact_6470_len__num1,axiom,
    ! [Uu: itself(numeral_num1)] : type_len0_len_of(numeral_num1,Uu) = one_one(nat) ).

% len_num1
tff(fact_6471_len__of__finite__1__def,axiom,
    ! [X: itself(finite_1)] : type_len0_len_of(finite_1,X) = one_one(nat) ).

% len_of_finite_1_def
tff(fact_6472_len__of__finite__3__def,axiom,
    ! [X: itself(finite_3)] : type_len0_len_of(finite_3,X) = aa(num,nat,numeral_numeral(nat),bit0(bit0(one2))) ).

% len_of_finite_3_def
tff(fact_6473_len__of__finite__2__def,axiom,
    ! [X: itself(finite_2)] : type_len0_len_of(finite_2,X) = aa(num,nat,numeral_numeral(nat),bit0(one2)) ).

% len_of_finite_2_def
tff(fact_6474_len__not__eq__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( type_len0_len_of(A,type2(A)) != zero_zero(nat) ) ) ).

% len_not_eq_0
tff(fact_6475_len__bit0,axiom,
    ! [A: $tType] :
      ( type_len0(A)
     => ! [Uu: itself(numeral_bit0(A))] : type_len0_len_of(numeral_bit0(A),Uu) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))) ) ).

% len_bit0
tff(fact_6476_two__less__eq__exp__length,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & linordered_idom(A) )
     => pp(aa(A,bool,ord_less_eq(A,aa(num,A,numeral_numeral(A),bit0(one2))),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),type_len0_len_of(B,type2(B))))) ) ).

% two_less_eq_exp_length
tff(fact_6477_len__bit1,axiom,
    ! [A: $tType] :
      ( type_len0(A)
     => ! [Uu: itself(numeral_bit1(A))] : type_len0_len_of(numeral_bit1(A),Uu) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A)))),one_one(nat)) ) ).

% len_bit1
tff(fact_6478_divmod__via__sdivmod,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),X: word(A)] :
          ( ( Y != zero_zero(word(A)) )
         => ( ( pp(aa(word(A),bool,ord_less_eq(word(A),bit_se4730199178511100633sh_bit(word(A),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat)),one_one(word(A)))),Y))
             => ( ( pp(aa(word(A),bool,ord_less(word(A),X),Y))
                 => ( aa(word(A),product_prod(word(A),word(A)),aa(word(A),fun(word(A),product_prod(word(A),word(A))),product_Pair(word(A),word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),X),Y)),modulo_modulo(word(A),X,Y)) = aa(word(A),product_prod(word(A),word(A)),aa(word(A),fun(word(A),product_prod(word(A),word(A))),product_Pair(word(A),word(A)),zero_zero(word(A))),X) ) )
                & ( ~ pp(aa(word(A),bool,ord_less(word(A),X),Y))
                 => ( aa(word(A),product_prod(word(A),word(A)),aa(word(A),fun(word(A),product_prod(word(A),word(A))),product_Pair(word(A),word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),X),Y)),modulo_modulo(word(A),X,Y)) = aa(word(A),product_prod(word(A),word(A)),aa(word(A),fun(word(A),product_prod(word(A),word(A))),product_Pair(word(A),word(A)),one_one(word(A))),aa(word(A),word(A),minus_minus(word(A),X),Y)) ) ) ) )
            & ( ~ pp(aa(word(A),bool,ord_less_eq(word(A),bit_se4730199178511100633sh_bit(word(A),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat)),one_one(word(A)))),Y))
             => ( aa(word(A),product_prod(word(A),word(A)),aa(word(A),fun(word(A),product_prod(word(A),word(A))),product_Pair(word(A),word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),X),Y)),modulo_modulo(word(A),X,Y)) = if(product_prod(word(A),word(A)),aa(word(A),bool,ord_less_eq(word(A),Y),aa(word(A),word(A),minus_minus(word(A),X),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),bit_se4730199178511100633sh_bit(word(A),one_one(nat),signed7115095781618012415divide(word(A),bit_se4197421643247451524op_bit(word(A),one_one(nat),X),Y))),Y))),aa(word(A),product_prod(word(A),word(A)),aa(word(A),fun(word(A),product_prod(word(A),word(A))),product_Pair(word(A),word(A)),aa(word(A),word(A),aa(word(A),fun(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),X),Y))),one_one(word(A)))),aa(word(A),word(A),minus_minus(word(A),aa(word(A),word(A),minus_minus(word(A),X),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),bit_se4730199178511100633sh_bit(word(A),one_one(nat),signed7115095781618012415divide(word(A),bit_se4197421643247451524op_bit(word(A),one_one(nat),X),Y))),Y))),Y)),aa(word(A),product_prod(word(A),word(A)),aa(word(A),fun(word(A),product_prod(word(A),word(A))),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),X),Y))),aa(word(A),word(A),minus_minus(word(A),X),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),bit_se4730199178511100633sh_bit(word(A),one_one(nat),signed7115095781618012415divide(word(A),bit_se4197421643247451524op_bit(word(A),one_one(nat),X),Y))),Y)))) ) ) ) ) ) ).

% divmod_via_sdivmod
tff(fact_6479_signed__drop__bit__word__minus__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: num,K: num] : signed_drop_bit(A,aa(num,nat,numeral_numeral(nat),N),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),K))) = aa(int,word(A),ring_1_of_int(word(A)),bit_se4197421643247451524op_bit(int,aa(num,nat,numeral_numeral(nat),N),aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),K))))) ) ).

% signed_drop_bit_word_minus_numeral
tff(fact_6480_signed__drop__bit__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] : signed_drop_bit(A,zero_zero(nat),W) = W ) ).

% signed_drop_bit_0
tff(fact_6481_signed__drop__bit__signed__drop__bit,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [M: nat,N: nat,W: word(A)] : signed_drop_bit(A,M,signed_drop_bit(A,N,W)) = signed_drop_bit(A,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N),W) ) ).

% signed_drop_bit_signed_drop_bit
tff(fact_6482_signed__drop__bit__of__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: nat] : signed_drop_bit(A,N,zero_zero(word(A))) = zero_zero(word(A)) ) ).

% signed_drop_bit_of_0
tff(fact_6483_word__sdiv__div1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A)] : signed7115095781618012415divide(word(A),A2,one_one(word(A))) = A2 ) ).

% word_sdiv_div1
tff(fact_6484_word__sdiv__div0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A)] : signed7115095781618012415divide(word(A),A2,zero_zero(word(A))) = zero_zero(word(A)) ) ).

% word_sdiv_div0
tff(fact_6485_signed__drop__bit__of__minus__1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: nat] : signed_drop_bit(A,N,aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A)))) = aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A))) ) ).

% signed_drop_bit_of_minus_1
tff(fact_6486_word__sdiv__div__minus1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A)] : signed7115095781618012415divide(word(A),A2,aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A)))) = aa(word(A),word(A),uminus_uminus(word(A)),A2) ) ).

% word_sdiv_div_minus1
tff(fact_6487_minus__one__sdiv__word__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] : signed7115095781618012415divide(word(A),aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A))),W) = aa(word(A),word(A),uminus_uminus(word(A)),signed7115095781618012415divide(word(A),one_one(word(A)),W)) ) ).

% minus_one_sdiv_word_eq
tff(fact_6488_one__sdiv__word__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] : signed7115095781618012415divide(word(A),one_one(word(A)),W) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),aa(bool,word(A),zero_neq_one_of_bool(word(A)),fdisj(aa(word(A),bool,fequal(word(A),W),one_one(word(A))),aa(word(A),bool,fequal(word(A),W),aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A))))))),W) ) ).

% one_sdiv_word_eq
tff(fact_6489_signed__drop__bit__of__1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: nat] : signed_drop_bit(A,N,one_one(word(A))) = aa(bool,word(A),zero_neq_one_of_bool(word(A)),fdisj(aa(nat,bool,fequal(nat,type_len0_len_of(A,type2(A))),one_one(nat)),aa(nat,bool,fequal(nat,N),zero_zero(nat)))) ) ).

% signed_drop_bit_of_1
tff(fact_6490_sdiv__word__numeral__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num,B2: num] : signed7115095781618012415divide(word(A),aa(num,word(A),numeral_numeral(word(A)),A2),aa(num,word(A),numeral_numeral(word(A)),B2)) = aa(int,word(A),ring_1_of_int(word(A)),signed7115095781618012415divide(int,aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat))),aa(num,int,numeral_numeral(int),A2)),aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat))),aa(num,int,numeral_numeral(int),B2)))) ) ).

% sdiv_word_numeral_numeral
tff(fact_6491_signed__drop__bit__word__Suc__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: nat,K: num] : signed_drop_bit(A,aa(nat,nat,suc,N),aa(num,word(A),numeral_numeral(word(A)),K)) = aa(int,word(A),ring_1_of_int(word(A)),bit_se4197421643247451524op_bit(int,aa(nat,nat,suc,N),aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat))),aa(num,int,numeral_numeral(int),K)))) ) ).

% signed_drop_bit_word_Suc_numeral
tff(fact_6492_signed__drop__bit__word__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: num,K: num] : signed_drop_bit(A,aa(num,nat,numeral_numeral(nat),N),aa(num,word(A),numeral_numeral(word(A)),K)) = aa(int,word(A),ring_1_of_int(word(A)),bit_se4197421643247451524op_bit(int,aa(num,nat,numeral_numeral(nat),N),aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat))),aa(num,int,numeral_numeral(int),K)))) ) ).

% signed_drop_bit_word_numeral
tff(fact_6493_sdiv__word__minus__numeral__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num,B2: num] : signed7115095781618012415divide(word(A),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),A2)),aa(num,word(A),numeral_numeral(word(A)),B2)) = aa(int,word(A),ring_1_of_int(word(A)),signed7115095781618012415divide(int,aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),A2))),aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat))),aa(num,int,numeral_numeral(int),B2)))) ) ).

% sdiv_word_minus_numeral_numeral
tff(fact_6494_sdiv__word__numeral__minus__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num,B2: num] : signed7115095781618012415divide(word(A),aa(num,word(A),numeral_numeral(word(A)),A2),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),B2))) = aa(int,word(A),ring_1_of_int(word(A)),signed7115095781618012415divide(int,aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat))),aa(num,int,numeral_numeral(int),A2)),aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),B2))))) ) ).

% sdiv_word_numeral_minus_numeral
tff(fact_6495_sdiv__word__minus__numeral__minus__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num,B2: num] : signed7115095781618012415divide(word(A),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),A2)),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),B2))) = aa(int,word(A),ring_1_of_int(word(A)),signed7115095781618012415divide(int,aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),A2))),aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),B2))))) ) ).

% sdiv_word_minus_numeral_minus_numeral
tff(fact_6496_signed__drop__bit__word__Suc__minus__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: nat,K: num] : signed_drop_bit(A,aa(nat,nat,suc,N),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),K))) = aa(int,word(A),ring_1_of_int(word(A)),bit_se4197421643247451524op_bit(int,aa(nat,nat,suc,N),aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),K))))) ) ).

% signed_drop_bit_word_Suc_minus_numeral
tff(fact_6497_word__sdiv__numerals__lhs_I3_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] : signed7115095781618012415divide(word(A),one_one(word(A)),W) = aa(int,word(A),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)
tff(fact_6498_word__sdiv__numerals_I9_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( signed7115095781618012415divide(word(A),one_one(word(A)),one_one(word(A))) = aa(int,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)
tff(fact_6499_word__sdiv__numerals_I5_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( signed7115095781618012415divide(word(A),zero_zero(word(A)),zero_zero(word(A))) = aa(int,word(A),ring_1_of_int(word(A)),signed7115095781618012415divide(int,ring_1_signed(A,int,zero_zero(word(A))),ring_1_signed(A,int,zero_zero(word(A))))) ) ) ).

% word_sdiv_numerals(5)
tff(fact_6500_word__sdiv__numerals__lhs_I2_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] : signed7115095781618012415divide(word(A),zero_zero(word(A)),W) = aa(int,word(A),ring_1_of_int(word(A)),signed7115095781618012415divide(int,ring_1_signed(A,int,zero_zero(word(A))),ring_1_signed(A,int,W))) ) ).

% word_sdiv_numerals_lhs(2)
tff(fact_6501_word__sdiv__numerals_I3_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: num] : signed7115095781618012415divide(word(A),one_one(word(A)),aa(num,word(A),numeral_numeral(word(A)),Y)) = aa(int,word(A),ring_1_of_int(word(A)),signed7115095781618012415divide(int,ring_1_signed(A,int,one_one(word(A))),ring_1_signed(A,int,aa(num,word(A),numeral_numeral(word(A)),Y)))) ) ).

% word_sdiv_numerals(3)
tff(fact_6502_word__sdiv__numerals_I7_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: num] : signed7115095781618012415divide(word(A),aa(num,word(A),numeral_numeral(word(A)),X),one_one(word(A))) = aa(int,word(A),ring_1_of_int(word(A)),signed7115095781618012415divide(int,ring_1_signed(A,int,aa(num,word(A),numeral_numeral(word(A)),X)),ring_1_signed(A,int,one_one(word(A))))) ) ).

% word_sdiv_numerals(7)
tff(fact_6503_word__sdiv__numerals_I2_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: num] : signed7115095781618012415divide(word(A),zero_zero(word(A)),aa(num,word(A),numeral_numeral(word(A)),Y)) = aa(int,word(A),ring_1_of_int(word(A)),signed7115095781618012415divide(int,ring_1_signed(A,int,zero_zero(word(A))),ring_1_signed(A,int,aa(num,word(A),numeral_numeral(word(A)),Y)))) ) ).

% word_sdiv_numerals(2)
tff(fact_6504_word__sdiv__numerals_I4_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: num] : signed7115095781618012415divide(word(A),aa(num,word(A),numeral_numeral(word(A)),X),zero_zero(word(A))) = aa(int,word(A),ring_1_of_int(word(A)),signed7115095781618012415divide(int,ring_1_signed(A,int,aa(num,word(A),numeral_numeral(word(A)),X)),ring_1_signed(A,int,zero_zero(word(A))))) ) ).

% word_sdiv_numerals(4)
tff(fact_6505_word__sdiv__numerals_I6_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( signed7115095781618012415divide(word(A),one_one(word(A)),zero_zero(word(A))) = aa(int,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)
tff(fact_6506_word__sdiv__numerals_I8_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( signed7115095781618012415divide(word(A),zero_zero(word(A)),one_one(word(A))) = aa(int,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)
tff(fact_6507_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)) = aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,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
tff(fact_6508_bit__signed__drop__bit__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [M: nat,W: word(A),N: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(word(A),signed_drop_bit(A,M,W)),N))
        <=> pp(aa(nat,bool,bit_se5641148757651400278ts_bit(word(A),W),if(nat,fconj(aa(nat,bool,ord_less_eq(nat,aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),M)),N),aa(nat,bool,ord_less(nat,N),type_len0_len_of(A,type2(A)))),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N)))) ) ) ).

% bit_signed_drop_bit_iff
tff(fact_6509_signed__drop__bit__beyond,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: nat,W: word(A)] :
          ( pp(aa(nat,bool,ord_less_eq(nat,type_len0_len_of(A,type2(A))),N))
         => ( ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(word(A),W),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat)))))
             => ( signed_drop_bit(A,N,W) = aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A))) ) )
            & ( ~ pp(aa(nat,bool,bit_se5641148757651400278ts_bit(word(A),W),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat)))))
             => ( signed_drop_bit(A,N,W) = zero_zero(word(A)) ) ) ) ) ) ).

% signed_drop_bit_beyond
tff(fact_6510_word__int__split,axiom,
    ! [B: $tType,A: $tType] :
      ( type_len(B)
     => ! [P: fun(A,bool),F2: fun(int,A),X: word(B)] :
          ( pp(aa(A,bool,P,word_int_case(A,B,F2,X)))
        <=> ! [I3: int] :
              ( ( ( X = aa(int,word(B),ring_1_of_int(word(B)),I3) )
                & pp(aa(int,bool,ord_less_eq(int,zero_zero(int)),I3))
                & pp(aa(int,bool,ord_less(int,I3),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(B,type2(B))))) )
             => pp(aa(A,bool,P,aa(int,A,F2,I3))) ) ) ) ).

% word_int_split
tff(fact_6511_word__int__split__asm,axiom,
    ! [B: $tType,A: $tType] :
      ( type_len(B)
     => ! [P: fun(A,bool),F2: fun(int,A),X: word(B)] :
          ( pp(aa(A,bool,P,word_int_case(A,B,F2,X)))
        <=> ~ ? [N5: int] :
                ( ( X = aa(int,word(B),ring_1_of_int(word(B)),N5) )
                & pp(aa(int,bool,ord_less_eq(int,zero_zero(int)),N5))
                & pp(aa(int,bool,ord_less(int,N5),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(B,type2(B)))))
                & ~ pp(aa(A,bool,P,aa(int,A,F2,N5))) ) ) ) ).

% word_int_split_asm
tff(fact_6512_word__int__case__wi,axiom,
    ! [A: $tType,B: $tType] :
      ( type_len(B)
     => ! [F2: fun(int,A),I: int] : word_int_case(A,B,F2,aa(int,word(B),ring_1_of_int(word(B)),I)) = aa(int,A,F2,modulo_modulo(int,I,aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(B,type2(B))))) ) ).

% word_int_case_wi
tff(fact_6513_smod__word__minus__numeral__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num,B2: num] : signed6721504322012087516modulo(word(A),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),A2)),aa(num,word(A),numeral_numeral(word(A)),B2)) = aa(int,word(A),ring_1_of_int(word(A)),signed6721504322012087516modulo(int,aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),A2))),aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat))),aa(num,int,numeral_numeral(int),B2)))) ) ).

% smod_word_minus_numeral_numeral
tff(fact_6514_smod__word__numeral__minus__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num,B2: num] : signed6721504322012087516modulo(word(A),aa(num,word(A),numeral_numeral(word(A)),A2),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),B2))) = aa(int,word(A),ring_1_of_int(word(A)),signed6721504322012087516modulo(int,aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat))),aa(num,int,numeral_numeral(int),A2)),aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),B2))))) ) ).

% smod_word_numeral_minus_numeral
tff(fact_6515_smod__word__zero,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] : signed6721504322012087516modulo(word(A),W,zero_zero(word(A))) = W ) ).

% smod_word_zero
tff(fact_6516_smod__word__0__mod,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A)] : signed6721504322012087516modulo(word(A),zero_zero(word(A)),X) = zero_zero(word(A)) ) ).

% smod_word_0_mod
tff(fact_6517_smod__word__mod__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A)] : signed6721504322012087516modulo(word(A),X,zero_zero(word(A))) = X ) ).

% smod_word_mod_0
tff(fact_6518_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
tff(fact_6519_minus__one__smod__word__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] : signed6721504322012087516modulo(word(A),aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A))),W) = aa(word(A),word(A),uminus_uminus(word(A)),signed6721504322012087516modulo(word(A),one_one(word(A)),W)) ) ).

% minus_one_smod_word_eq
tff(fact_6520_smod__word__minus__one,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] : signed6721504322012087516modulo(word(A),W,aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A)))) = zero_zero(word(A)) ) ).

% smod_word_minus_one
tff(fact_6521_one__smod__word__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] : signed6721504322012087516modulo(word(A),one_one(word(A)),W) = aa(word(A),word(A),minus_minus(word(A),one_one(word(A))),aa(bool,word(A),zero_neq_one_of_bool(word(A)),fdisj(aa(word(A),bool,fequal(word(A),W),one_one(word(A))),aa(word(A),bool,fequal(word(A),W),aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A))))))) ) ).

% one_smod_word_eq
tff(fact_6522_smod__word__numeral__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num,B2: num] : signed6721504322012087516modulo(word(A),aa(num,word(A),numeral_numeral(word(A)),A2),aa(num,word(A),numeral_numeral(word(A)),B2)) = aa(int,word(A),ring_1_of_int(word(A)),signed6721504322012087516modulo(int,aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat))),aa(num,int,numeral_numeral(int),A2)),aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat))),aa(num,int,numeral_numeral(int),B2)))) ) ).

% smod_word_numeral_numeral
tff(fact_6523_smod__word__minus__numeral__minus__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num,B2: num] : signed6721504322012087516modulo(word(A),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),A2)),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),B2))) = aa(int,word(A),ring_1_of_int(word(A)),signed6721504322012087516modulo(int,aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),A2))),aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),B2))))) ) ).

% smod_word_minus_numeral_minus_numeral
tff(fact_6524_sdiv__smod__id,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),signed7115095781618012415divide(word(A),A2,B2)),B2)),signed6721504322012087516modulo(word(A),A2,B2)) = A2 ) ).

% sdiv_smod_id
tff(fact_6525_smod__word__alt__def,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A)] : signed6721504322012087516modulo(word(A),A2,B2) = aa(word(A),word(A),minus_minus(word(A),A2),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),signed7115095781618012415divide(word(A),A2,B2)),B2)) ) ).

% smod_word_alt_def
tff(fact_6526_word__smod__numerals_I9_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( signed6721504322012087516modulo(word(A),one_one(word(A)),one_one(word(A))) = aa(int,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)
tff(fact_6527_word__smod__numerals__lhs_I3_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] : signed6721504322012087516modulo(word(A),one_one(word(A)),W) = aa(int,word(A),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)
tff(fact_6528_word__smod__numerals__lhs_I2_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] : signed6721504322012087516modulo(word(A),zero_zero(word(A)),W) = aa(int,word(A),ring_1_of_int(word(A)),signed6721504322012087516modulo(int,ring_1_signed(A,int,zero_zero(word(A))),ring_1_signed(A,int,W))) ) ).

% word_smod_numerals_lhs(2)
tff(fact_6529_word__smod__numerals_I5_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( signed6721504322012087516modulo(word(A),zero_zero(word(A)),zero_zero(word(A))) = aa(int,word(A),ring_1_of_int(word(A)),signed6721504322012087516modulo(int,ring_1_signed(A,int,zero_zero(word(A))),ring_1_signed(A,int,zero_zero(word(A))))) ) ) ).

% word_smod_numerals(5)
tff(fact_6530_word__smod__numerals_I3_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: num] : signed6721504322012087516modulo(word(A),one_one(word(A)),aa(num,word(A),numeral_numeral(word(A)),Y)) = aa(int,word(A),ring_1_of_int(word(A)),signed6721504322012087516modulo(int,ring_1_signed(A,int,one_one(word(A))),ring_1_signed(A,int,aa(num,word(A),numeral_numeral(word(A)),Y)))) ) ).

% word_smod_numerals(3)
tff(fact_6531_word__smod__numerals_I7_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: num] : signed6721504322012087516modulo(word(A),aa(num,word(A),numeral_numeral(word(A)),X),one_one(word(A))) = aa(int,word(A),ring_1_of_int(word(A)),signed6721504322012087516modulo(int,ring_1_signed(A,int,aa(num,word(A),numeral_numeral(word(A)),X)),ring_1_signed(A,int,one_one(word(A))))) ) ).

% word_smod_numerals(7)
tff(fact_6532_word__smod__numerals_I2_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: num] : signed6721504322012087516modulo(word(A),zero_zero(word(A)),aa(num,word(A),numeral_numeral(word(A)),Y)) = aa(int,word(A),ring_1_of_int(word(A)),signed6721504322012087516modulo(int,ring_1_signed(A,int,zero_zero(word(A))),ring_1_signed(A,int,aa(num,word(A),numeral_numeral(word(A)),Y)))) ) ).

% word_smod_numerals(2)
tff(fact_6533_word__smod__numerals_I4_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: num] : signed6721504322012087516modulo(word(A),aa(num,word(A),numeral_numeral(word(A)),X),zero_zero(word(A))) = aa(int,word(A),ring_1_of_int(word(A)),signed6721504322012087516modulo(int,ring_1_signed(A,int,aa(num,word(A),numeral_numeral(word(A)),X)),ring_1_signed(A,int,zero_zero(word(A))))) ) ).

% word_smod_numerals(4)
tff(fact_6534_word__smod__numerals_I8_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( signed6721504322012087516modulo(word(A),zero_zero(word(A)),one_one(word(A))) = aa(int,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)
tff(fact_6535_word__smod__numerals_I6_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( signed6721504322012087516modulo(word(A),one_one(word(A)),zero_zero(word(A))) = aa(int,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)
tff(fact_6536_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)) = aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,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
tff(fact_6537_uint__word__ariths_I6_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A)] : aa(word(A),int,semiring_1_unsigned(A,int),word_pred(A,A2)) = modulo_modulo(int,aa(int,int,minus_minus(int,aa(word(A),int,semiring_1_unsigned(A,int),A2)),one_one(int)),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A)))) ) ).

% uint_word_ariths(6)
tff(fact_6538_uint__word__ariths_I5_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A)] : aa(word(A),int,semiring_1_unsigned(A,int),word_succ(A,A2)) = modulo_modulo(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(word(A),int,semiring_1_unsigned(A,int),A2)),one_one(int)),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A)))) ) ).

% uint_word_ariths(5)
tff(fact_6539_succ__pred__no_I1_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: num] : word_succ(A,aa(num,word(A),numeral_numeral(word(A)),W)) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),aa(num,word(A),numeral_numeral(word(A)),W)),one_one(word(A))) ) ).

% succ_pred_no(1)
tff(fact_6540_succ__pred__no_I2_J,axiom,
    ! [B: $tType] :
      ( type_len(B)
     => ! [W: num] : word_pred(B,aa(num,word(B),numeral_numeral(word(B)),W)) = aa(word(B),word(B),minus_minus(word(B),aa(num,word(B),numeral_numeral(word(B)),W)),one_one(word(B))) ) ).

% succ_pred_no(2)
tff(fact_6541_succ__pred__no_I3_J,axiom,
    ! [C: $tType] :
      ( type_len(C)
     => ! [W: num] : word_succ(C,aa(word(C),word(C),uminus_uminus(word(C)),aa(num,word(C),numeral_numeral(word(C)),W))) = aa(word(C),word(C),aa(word(C),fun(word(C),word(C)),plus_plus(word(C)),aa(word(C),word(C),uminus_uminus(word(C)),aa(num,word(C),numeral_numeral(word(C)),W))),one_one(word(C))) ) ).

% succ_pred_no(3)
tff(fact_6542_succ__pred__no_I4_J,axiom,
    ! [D6: $tType] :
      ( type_len(D6)
     => ! [W: num] : word_pred(D6,aa(word(D6),word(D6),uminus_uminus(word(D6)),aa(num,word(D6),numeral_numeral(word(D6)),W))) = aa(word(D6),word(D6),minus_minus(word(D6),aa(word(D6),word(D6),uminus_uminus(word(D6)),aa(num,word(D6),numeral_numeral(word(D6)),W))),one_one(word(D6))) ) ).

% succ_pred_no(4)
tff(fact_6543_word__sp__01,axiom,
    ! [C: $tType,A: $tType,B: $tType,D6: $tType] :
      ( ( type_len(D6)
        & type_len(B)
        & type_len(A)
        & type_len(C) )
     => ( ( word_succ(A,aa(word(A),word(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))) = aa(word(C),word(C),uminus_uminus(word(C)),one_one(word(C))) )
        & ( word_pred(D6,one_one(word(D6))) = zero_zero(word(D6)) ) ) ) ).

% word_sp_01
tff(fact_6544_Abs__fnat__hom__Suc,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: nat] : word_succ(A,aa(nat,word(A),semiring_1_of_nat(word(A)),A2)) = aa(nat,word(A),semiring_1_of_nat(word(A)),aa(nat,nat,suc,A2)) ) ).

% Abs_fnat_hom_Suc
tff(fact_6545_word__succ__p1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A)] : word_succ(A,A2) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),A2),one_one(word(A))) ) ).

% word_succ_p1
tff(fact_6546_word__m1__ge,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A)] : pp(aa(word(A),bool,ord_less_eq(word(A),Y),word_pred(A,zero_zero(word(A))))) ) ).

% word_m1_ge
tff(fact_6547_word__not__simps_I2_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A)] : ~ pp(aa(word(A),bool,ord_less(word(A),word_pred(A,zero_zero(word(A)))),Y)) ) ).

% word_not_simps(2)
tff(fact_6548_word__pred__m1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A)] : word_pred(A,A2) = aa(word(A),word(A),minus_minus(word(A),A2),one_one(word(A))) ) ).

% word_pred_m1
tff(fact_6549_word__mult__succ,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),word_succ(A,A2)),B2) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),B2),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),A2),B2)) ) ).

% word_mult_succ
tff(fact_6550_wi__hom__succ,axiom,
    ! [E3: $tType] :
      ( type_len(E3)
     => ! [A2: int] : word_succ(E3,aa(int,word(E3),ring_1_of_int(word(E3)),A2)) = aa(int,word(E3),ring_1_of_int(word(E3)),aa(int,int,aa(int,fun(int,int),plus_plus(int),A2),one_one(int))) ) ).

% wi_hom_succ
tff(fact_6551_word__arith__nat__Suc,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A)] : word_succ(A,A2) = aa(nat,word(A),semiring_1_of_nat(word(A)),aa(nat,nat,suc,aa(word(A),nat,semiring_1_unsigned(A,nat),A2))) ) ).

% word_arith_nat_Suc
tff(fact_6552_wi__hom__pred,axiom,
    ! [F: $tType] :
      ( type_len(F)
     => ! [A2: int] : word_pred(F,aa(int,word(F),ring_1_of_int(word(F)),A2)) = aa(int,word(F),ring_1_of_int(word(F)),aa(int,int,minus_minus(int,A2),one_one(int))) ) ).

% wi_hom_pred
tff(fact_6553_word__succ__alt,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A)] : word_succ(A,A2) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(word(A),int,semiring_1_unsigned(A,int),A2)),one_one(int))) ) ).

% word_succ_alt
tff(fact_6554_word__pred__0__n1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( word_pred(A,zero_zero(word(A))) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,uminus_uminus(int),one_one(int))) ) ) ).

% word_pred_0_n1
tff(fact_6555_word__pred__alt,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A)] : word_pred(A,A2) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,minus_minus(int,aa(word(A),int,semiring_1_unsigned(A,int),A2)),one_one(int))) ) ).

% word_pred_alt
tff(fact_6556_uint__word__arith__bintrs_I5_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A)] : aa(word(A),int,semiring_1_unsigned(A,int),word_succ(A,A2)) = aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(word(A),int,semiring_1_unsigned(A,int),A2)),one_one(int))) ) ).

% uint_word_arith_bintrs(5)
tff(fact_6557_uint__word__arith__bintrs_I6_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A)] : aa(word(A),int,semiring_1_unsigned(A,int),word_pred(A,A2)) = aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),aa(int,int,minus_minus(int,aa(word(A),int,semiring_1_unsigned(A,int),A2)),one_one(int))) ) ).

% uint_word_arith_bintrs(6)
tff(fact_6558_sint__word__ariths_I5_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A)] : ring_1_signed(A,int,word_succ(A,A2)) = aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat))),aa(int,int,aa(int,fun(int,int),plus_plus(int),ring_1_signed(A,int,A2)),one_one(int))) ) ).

% sint_word_ariths(5)
tff(fact_6559_sint__word__ariths_I6_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A)] : ring_1_signed(A,int,word_pred(A,A2)) = aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat))),aa(int,int,minus_minus(int,ring_1_signed(A,int,A2)),one_one(int))) ) ).

% sint_word_ariths(6)
tff(fact_6560_unat__word__ariths_I3_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A)] : aa(word(A),nat,semiring_1_unsigned(A,nat),word_succ(A,A2)) = modulo_modulo(nat,aa(nat,nat,suc,aa(word(A),nat,semiring_1_unsigned(A,nat),A2)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A)))) ) ).

% unat_word_ariths(3)
tff(fact_6561_bit__word__roti__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [K: int,W: word(A),N: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(word(A),word_roti(A,K,W)),N))
        <=> ( pp(aa(nat,bool,ord_less(nat,N),type_len0_len_of(A,type2(A))))
            & pp(aa(nat,bool,bit_se5641148757651400278ts_bit(word(A),W),nat2(modulo_modulo(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(nat,int,semiring_1_of_nat(int),N)),K),aa(nat,int,semiring_1_of_nat(int),type_len0_len_of(A,type2(A))))))) ) ) ) ).

% bit_word_roti_iff
tff(fact_6562_sless__eq__word__minus__numeral__minus__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num,B2: num] :
          ( word_sle(A,aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),A2)),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),B2)))
        <=> pp(aa(int,bool,ord_less_eq(int,aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),A2)))),aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),B2))))) ) ) ).

% sless_eq_word_minus_numeral_minus_numeral
tff(fact_6563_word__roti__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] : word_roti(A,zero_zero(int),W) = W ) ).

% word_roti_0
tff(fact_6564_word__roti__0_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: int] : word_roti(A,N,zero_zero(word(A))) = zero_zero(word(A)) ) ).

% word_roti_0'
tff(fact_6565_extra__sle__sless__unfolds_I6_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: num] :
          ( word_sle(A,aa(num,word(A),numeral_numeral(word(A)),N),one_one(word(A)))
        <=> pp(aa(int,bool,ord_less_eq(int,ring_1_signed(A,int,aa(num,word(A),numeral_numeral(word(A)),N))),ring_1_signed(A,int,one_one(word(A))))) ) ) ).

% extra_sle_sless_unfolds(6)
tff(fact_6566_extra__sle__sless__unfolds_I4_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: num] :
          ( word_sle(A,one_one(word(A)),aa(num,word(A),numeral_numeral(word(A)),N))
        <=> pp(aa(int,bool,ord_less_eq(int,ring_1_signed(A,int,one_one(word(A)))),ring_1_signed(A,int,aa(num,word(A),numeral_numeral(word(A)),N)))) ) ) ).

% extra_sle_sless_unfolds(4)
tff(fact_6567_extra__sle__sless__unfolds_I2_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: num] :
          ( word_sle(A,zero_zero(word(A)),aa(num,word(A),numeral_numeral(word(A)),N))
        <=> pp(aa(int,bool,ord_less_eq(int,ring_1_signed(A,int,zero_zero(word(A)))),ring_1_signed(A,int,aa(num,word(A),numeral_numeral(word(A)),N)))) ) ) ).

% extra_sle_sless_unfolds(2)
tff(fact_6568_extra__sle__sless__unfolds_I5_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: num] :
          ( word_sle(A,aa(num,word(A),numeral_numeral(word(A)),N),zero_zero(word(A)))
        <=> pp(aa(int,bool,ord_less_eq(int,ring_1_signed(A,int,aa(num,word(A),numeral_numeral(word(A)),N))),ring_1_signed(A,int,zero_zero(word(A))))) ) ) ).

% extra_sle_sless_unfolds(5)
tff(fact_6569_extra__sle__sless__unfolds_I1_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( word_sle(A,zero_zero(word(A)),one_one(word(A)))
      <=> pp(aa(int,bool,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)
tff(fact_6570_extra__sle__sless__unfolds_I3_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( word_sle(A,one_one(word(A)),zero_zero(word(A)))
      <=> pp(aa(int,bool,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)
tff(fact_6571_sless__eq__word__numeral__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num,B2: num] :
          ( word_sle(A,aa(num,word(A),numeral_numeral(word(A)),A2),aa(num,word(A),numeral_numeral(word(A)),B2))
        <=> pp(aa(int,bool,ord_less_eq(int,aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat))),aa(num,int,numeral_numeral(int),A2))),aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat))),aa(num,int,numeral_numeral(int),B2)))) ) ) ).

% sless_eq_word_numeral_numeral
tff(fact_6572_sless__eq__word__minus__numeral__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num,B2: num] :
          ( word_sle(A,aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),A2)),aa(num,word(A),numeral_numeral(word(A)),B2))
        <=> pp(aa(int,bool,ord_less_eq(int,aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),A2)))),aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat))),aa(num,int,numeral_numeral(int),B2)))) ) ) ).

% sless_eq_word_minus_numeral_numeral
tff(fact_6573_sless__eq__word__numeral__minus__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num,B2: num] :
          ( word_sle(A,aa(num,word(A),numeral_numeral(word(A)),A2),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),B2)))
        <=> pp(aa(int,bool,ord_less_eq(int,aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat))),aa(num,int,numeral_numeral(int),A2))),aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),B2))))) ) ) ).

% sless_eq_word_numeral_minus_numeral
tff(fact_6574_signed_Olift__Suc__mono__le,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [F2: fun(nat,word(A)),N: nat,N3: nat] :
          ( ! [N2: nat] : word_sle(A,aa(nat,word(A),F2,N2),aa(nat,word(A),F2,aa(nat,nat,suc,N2)))
         => ( pp(aa(nat,bool,ord_less_eq(nat,N),N3))
           => word_sle(A,aa(nat,word(A),F2,N),aa(nat,word(A),F2,N3)) ) ) ) ).

% signed.lift_Suc_mono_le
tff(fact_6575_signed_Olift__Suc__antimono__le,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [F2: fun(nat,word(A)),N: nat,N3: nat] :
          ( ! [N2: nat] : word_sle(A,aa(nat,word(A),F2,aa(nat,nat,suc,N2)),aa(nat,word(A),F2,N2))
         => ( pp(aa(nat,bool,ord_less_eq(nat,N),N3))
           => word_sle(A,aa(nat,word(A),F2,N3),aa(nat,word(A),F2,N)) ) ) ) ).

% signed.lift_Suc_antimono_le
tff(fact_6576_word__roti__add,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [M: int,N: int,W: word(A)] : word_roti(A,aa(int,int,aa(int,fun(int,int),plus_plus(int),M),N),W) = word_roti(A,M,word_roti(A,N,W)) ) ).

% word_roti_add
tff(fact_6577_word__roti__conv__mod_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: int,W: word(A)] : word_roti(A,N,W) = word_roti(A,modulo_modulo(int,N,aa(nat,int,semiring_1_of_nat(int),aa(word(A),nat,size_size(word(A)),W))),W) ) ).

% word_roti_conv_mod'
tff(fact_6578_word__roti__conv__mod,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: int,W: word(A)] : word_roti(A,N,W) = word_roti(A,modulo_modulo(int,N,aa(nat,int,semiring_1_of_nat(int),type_len0_len_of(A,type2(A)))),W) ) ).

% word_roti_conv_mod
tff(fact_6579_word__0__sle__from__less,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A)] :
          ( pp(aa(word(A),bool,ord_less(word(A),X),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat)))))
         => word_sle(A,zero_zero(word(A)),X) ) ) ).

% word_0_sle_from_less
tff(fact_6580_sless__word__minus__numeral__minus__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num,B2: num] :
          ( word_sless(A,aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),A2)),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),B2)))
        <=> pp(aa(int,bool,ord_less(int,aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),A2)))),aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),B2))))) ) ) ).

% sless_word_minus_numeral_minus_numeral
tff(fact_6581_sless__word__numeral__minus__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num,B2: num] :
          ( word_sless(A,aa(num,word(A),numeral_numeral(word(A)),A2),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),B2)))
        <=> pp(aa(int,bool,ord_less(int,aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat))),aa(num,int,numeral_numeral(int),A2))),aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),B2))))) ) ) ).

% sless_word_numeral_minus_numeral
tff(fact_6582_extra__sle__sless__unfolds_I10_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: num] :
          ( word_sless(A,one_one(word(A)),aa(num,word(A),numeral_numeral(word(A)),N))
        <=> pp(aa(int,bool,ord_less(int,ring_1_signed(A,int,one_one(word(A)))),ring_1_signed(A,int,aa(num,word(A),numeral_numeral(word(A)),N)))) ) ) ).

% extra_sle_sless_unfolds(10)
tff(fact_6583_extra__sle__sless__unfolds_I12_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: num] :
          ( word_sless(A,aa(num,word(A),numeral_numeral(word(A)),N),one_one(word(A)))
        <=> pp(aa(int,bool,ord_less(int,ring_1_signed(A,int,aa(num,word(A),numeral_numeral(word(A)),N))),ring_1_signed(A,int,one_one(word(A))))) ) ) ).

% extra_sle_sless_unfolds(12)
tff(fact_6584_extra__sle__sless__unfolds_I8_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: num] :
          ( word_sless(A,zero_zero(word(A)),aa(num,word(A),numeral_numeral(word(A)),N))
        <=> pp(aa(int,bool,ord_less(int,ring_1_signed(A,int,zero_zero(word(A)))),ring_1_signed(A,int,aa(num,word(A),numeral_numeral(word(A)),N)))) ) ) ).

% extra_sle_sless_unfolds(8)
tff(fact_6585_extra__sle__sless__unfolds_I11_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: num] :
          ( word_sless(A,aa(num,word(A),numeral_numeral(word(A)),N),zero_zero(word(A)))
        <=> pp(aa(int,bool,ord_less(int,ring_1_signed(A,int,aa(num,word(A),numeral_numeral(word(A)),N))),ring_1_signed(A,int,zero_zero(word(A))))) ) ) ).

% extra_sle_sless_unfolds(11)
tff(fact_6586_extra__sle__sless__unfolds_I7_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( word_sless(A,zero_zero(word(A)),one_one(word(A)))
      <=> pp(aa(int,bool,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)
tff(fact_6587_extra__sle__sless__unfolds_I9_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( word_sless(A,one_one(word(A)),zero_zero(word(A)))
      <=> pp(aa(int,bool,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)
tff(fact_6588_sless__word__numeral__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num,B2: num] :
          ( word_sless(A,aa(num,word(A),numeral_numeral(word(A)),A2),aa(num,word(A),numeral_numeral(word(A)),B2))
        <=> pp(aa(int,bool,ord_less(int,aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat))),aa(num,int,numeral_numeral(int),A2))),aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat))),aa(num,int,numeral_numeral(int),B2)))) ) ) ).

% sless_word_numeral_numeral
tff(fact_6589_sless__word__minus__numeral__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num,B2: num] :
          ( word_sless(A,aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),A2)),aa(num,word(A),numeral_numeral(word(A)),B2))
        <=> pp(aa(int,bool,ord_less(int,aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),A2)))),aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat))),aa(num,int,numeral_numeral(int),B2)))) ) ) ).

% sless_word_minus_numeral_numeral
tff(fact_6590_signed_Olift__Suc__mono__less,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [F2: fun(nat,word(A)),N: nat,N3: nat] :
          ( ! [N2: nat] : word_sless(A,aa(nat,word(A),F2,N2),aa(nat,word(A),F2,aa(nat,nat,suc,N2)))
         => ( pp(aa(nat,bool,ord_less(nat,N),N3))
           => word_sless(A,aa(nat,word(A),F2,N),aa(nat,word(A),F2,N3)) ) ) ) ).

% signed.lift_Suc_mono_less
tff(fact_6591_signed_Olift__Suc__mono__less__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [F2: fun(nat,word(A)),N: nat,M: nat] :
          ( ! [N2: nat] : word_sless(A,aa(nat,word(A),F2,N2),aa(nat,word(A),F2,aa(nat,nat,suc,N2)))
         => ( word_sless(A,aa(nat,word(A),F2,N),aa(nat,word(A),F2,M))
          <=> pp(aa(nat,bool,ord_less(nat,N),M)) ) ) ) ).

% signed.lift_Suc_mono_less_iff
tff(fact_6592_word__sless__sint__le,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A)] :
          ( word_sless(A,X,Y)
         => pp(aa(int,bool,ord_less_eq(int,ring_1_signed(A,int,X)),aa(int,int,minus_minus(int,ring_1_signed(A,int,Y)),one_one(int)))) ) ) ).

% word_sless_sint_le
tff(fact_6593_take__bit__word__minus__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: num,K: num] : aa(word(A),word(A),bit_se2584673776208193580ke_bit(word(A),aa(num,nat,numeral_numeral(nat),N)),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),K))) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,bit_se2584673776208193580ke_bit(int,ord_min(nat,type_len0_len_of(A,type2(A)),aa(num,nat,numeral_numeral(nat),N))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),K)))) ) ).

% take_bit_word_minus_numeral
tff(fact_6594_take__bit__word__Suc__minus__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: nat,K: num] : aa(word(A),word(A),bit_se2584673776208193580ke_bit(word(A),aa(nat,nat,suc,N)),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),K))) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,bit_se2584673776208193580ke_bit(int,ord_min(nat,type_len0_len_of(A,type2(A)),aa(nat,nat,suc,N))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),K)))) ) ).

% take_bit_word_Suc_minus_numeral
tff(fact_6595_min__eq__arg_I1_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [M: A,N: A] :
          ( ( ord_min(A,M,N) = M )
        <=> pp(aa(A,bool,ord_less_eq(A,M),N)) ) ) ).

% min_eq_arg(1)
tff(fact_6596_min__eq__arg_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [M: A,N: A] :
          ( ( ord_min(A,M,N) = N )
        <=> pp(aa(A,bool,ord_less_eq(A,N),M)) ) ) ).

% min_eq_arg(2)
tff(fact_6597_min__arg__le_I1_J,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [N: A,M: A] :
          ( pp(aa(A,bool,ord_less_eq(A,N),ord_min(A,M,N)))
        <=> ( ord_min(A,M,N) = N ) ) ) ).

% min_arg_le(1)
tff(fact_6598_min__arg__le_I2_J,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [M: A,N: A] :
          ( pp(aa(A,bool,ord_less_eq(A,M),ord_min(A,M,N)))
        <=> ( ord_min(A,M,N) = M ) ) ) ).

% min_arg_le(2)
tff(fact_6599_min_Oabsorb1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,ord_less_eq(A,A2),B2))
         => ( ord_min(A,A2,B2) = A2 ) ) ) ).

% min.absorb1
tff(fact_6600_min_Oabsorb2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,A2: A] :
          ( pp(aa(A,bool,ord_less_eq(A,B2),A2))
         => ( ord_min(A,A2,B2) = B2 ) ) ) ).

% min.absorb2
tff(fact_6601_min_Obounded__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,ord_less_eq(A,A2),ord_min(A,B2,C2)))
        <=> ( pp(aa(A,bool,ord_less_eq(A,A2),B2))
            & pp(aa(A,bool,ord_less_eq(A,A2),C2)) ) ) ) ).

% min.bounded_iff
tff(fact_6602_min__Suc__Suc,axiom,
    ! [M: nat,N: nat] : ord_min(nat,aa(nat,nat,suc,M),aa(nat,nat,suc,N)) = aa(nat,nat,suc,ord_min(nat,M,N)) ).

% min_Suc_Suc
tff(fact_6603_min__0L,axiom,
    ! [N: nat] : ord_min(nat,zero_zero(nat),N) = zero_zero(nat) ).

% min_0L
tff(fact_6604_min__0R,axiom,
    ! [N: nat] : ord_min(nat,N,zero_zero(nat)) = zero_zero(nat) ).

% min_0R
tff(fact_6605_take__bit__take__bit,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,N: nat,A2: A] : aa(A,A,bit_se2584673776208193580ke_bit(A,M),aa(A,A,bit_se2584673776208193580ke_bit(A,N),A2)) = aa(A,A,bit_se2584673776208193580ke_bit(A,ord_min(nat,M,N)),A2) ) ).

% take_bit_take_bit
tff(fact_6606_signed__take__bit__signed__take__bit,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [M: nat,N: nat,A2: A] : aa(A,A,bit_ri4674362597316999326ke_bit(A,M),aa(A,A,bit_ri4674362597316999326ke_bit(A,N),A2)) = aa(A,A,bit_ri4674362597316999326ke_bit(A,ord_min(nat,M,N)),A2) ) ).

% signed_take_bit_signed_take_bit
tff(fact_6607_min__0__1_I3_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [X: num] : ord_min(A,zero_zero(A),aa(num,A,numeral_numeral(A),X)) = zero_zero(A) ) ).

% min_0_1(3)
tff(fact_6608_min__0__1_I4_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [X: num] : ord_min(A,aa(num,A,numeral_numeral(A),X),zero_zero(A)) = zero_zero(A) ) ).

% min_0_1(4)
tff(fact_6609_min__number__of_I1_J,axiom,
    ! [A: $tType] :
      ( ( numeral(A)
        & ord(A) )
     => ! [U: num,V: num] :
          ( ( pp(aa(A,bool,ord_less_eq(A,aa(num,A,numeral_numeral(A),U)),aa(num,A,numeral_numeral(A),V)))
           => ( ord_min(A,aa(num,A,numeral_numeral(A),U),aa(num,A,numeral_numeral(A),V)) = aa(num,A,numeral_numeral(A),U) ) )
          & ( ~ pp(aa(A,bool,ord_less_eq(A,aa(num,A,numeral_numeral(A),U)),aa(num,A,numeral_numeral(A),V)))
           => ( ord_min(A,aa(num,A,numeral_numeral(A),U),aa(num,A,numeral_numeral(A),V)) = aa(num,A,numeral_numeral(A),V) ) ) ) ) ).

% min_number_of(1)
tff(fact_6610_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)
tff(fact_6611_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)
tff(fact_6612_min__0__1_I5_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [X: num] : ord_min(A,one_one(A),aa(num,A,numeral_numeral(A),X)) = one_one(A) ) ).

% min_0_1(5)
tff(fact_6613_min__0__1_I6_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [X: num] : ord_min(A,aa(num,A,numeral_numeral(A),X),one_one(A)) = one_one(A) ) ).

% min_0_1(6)
tff(fact_6614_min__Suc__gt_I1_J,axiom,
    ! [A2: nat,B2: nat] :
      ( pp(aa(nat,bool,ord_less(nat,A2),B2))
     => ( ord_min(nat,aa(nat,nat,suc,A2),B2) = aa(nat,nat,suc,A2) ) ) ).

% min_Suc_gt(1)
tff(fact_6615_min__Suc__gt_I2_J,axiom,
    ! [A2: nat,B2: nat] :
      ( pp(aa(nat,bool,ord_less(nat,A2),B2))
     => ( ord_min(nat,B2,aa(nat,nat,suc,A2)) = aa(nat,nat,suc,A2) ) ) ).

% min_Suc_gt(2)
tff(fact_6616_rev__min__pm1,axiom,
    ! [A2: nat,B2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,minus_minus(nat,A2),B2)),ord_min(nat,B2,A2)) = A2 ).

% rev_min_pm1
tff(fact_6617_rev__min__pm,axiom,
    ! [B2: nat,A2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),ord_min(nat,B2,A2)),aa(nat,nat,minus_minus(nat,A2),B2)) = A2 ).

% rev_min_pm
tff(fact_6618_min__pm1,axiom,
    ! [A2: nat,B2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,minus_minus(nat,A2),B2)),ord_min(nat,A2,B2)) = A2 ).

% min_pm1
tff(fact_6619_min__pm,axiom,
    ! [A2: nat,B2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),ord_min(nat,A2,B2)),aa(nat,nat,minus_minus(nat,A2),B2)) = A2 ).

% min_pm
tff(fact_6620_take__bit__of__mask,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,N: nat] : aa(A,A,bit_se2584673776208193580ke_bit(A,M),bit_se2239418461657761734s_mask(A,N)) = bit_se2239418461657761734s_mask(A,ord_min(nat,M,N)) ) ).

% take_bit_of_mask
tff(fact_6621_min__number__of_I2_J,axiom,
    ! [A: $tType] :
      ( ( uminus(A)
        & numeral(A)
        & ord(A) )
     => ! [U: num,V: num] :
          ( ( pp(aa(A,bool,ord_less_eq(A,aa(num,A,numeral_numeral(A),U)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))))
           => ( ord_min(A,aa(num,A,numeral_numeral(A),U),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))) = aa(num,A,numeral_numeral(A),U) ) )
          & ( ~ pp(aa(A,bool,ord_less_eq(A,aa(num,A,numeral_numeral(A),U)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))))
           => ( ord_min(A,aa(num,A,numeral_numeral(A),U),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V)) ) ) ) ) ).

% min_number_of(2)
tff(fact_6622_min__number__of_I3_J,axiom,
    ! [A: $tType] :
      ( ( uminus(A)
        & numeral(A)
        & ord(A) )
     => ! [U: num,V: num] :
          ( ( pp(aa(A,bool,ord_less_eq(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U))),aa(num,A,numeral_numeral(A),V)))
           => ( ord_min(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U)),aa(num,A,numeral_numeral(A),V)) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U)) ) )
          & ( ~ pp(aa(A,bool,ord_less_eq(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U))),aa(num,A,numeral_numeral(A),V)))
           => ( ord_min(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U)),aa(num,A,numeral_numeral(A),V)) = aa(num,A,numeral_numeral(A),V) ) ) ) ) ).

% min_number_of(3)
tff(fact_6623_min__number__of_I4_J,axiom,
    ! [A: $tType] :
      ( ( uminus(A)
        & numeral(A)
        & ord(A) )
     => ! [U: num,V: num] :
          ( ( pp(aa(A,bool,ord_less_eq(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))))
           => ( ord_min(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U)) ) )
          & ( ~ pp(aa(A,bool,ord_less_eq(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))))
           => ( ord_min(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V)) ) ) ) ) ).

% min_number_of(4)
tff(fact_6624_min__Suc__numeral,axiom,
    ! [N: nat,K: num] : ord_min(nat,aa(nat,nat,suc,N),aa(num,nat,numeral_numeral(nat),K)) = aa(nat,nat,suc,ord_min(nat,N,pred_numeral(K))) ).

% min_Suc_numeral
tff(fact_6625_min__numeral__Suc,axiom,
    ! [K: num,N: nat] : ord_min(nat,aa(num,nat,numeral_numeral(nat),K),aa(nat,nat,suc,N)) = aa(nat,nat,suc,ord_min(nat,pred_numeral(K),N)) ).

% min_numeral_Suc
tff(fact_6626_take__bit__word__Suc__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: nat,K: num] : aa(word(A),word(A),bit_se2584673776208193580ke_bit(word(A),aa(nat,nat,suc,N)),aa(num,word(A),numeral_numeral(word(A)),K)) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,bit_se2584673776208193580ke_bit(int,ord_min(nat,type_len0_len_of(A,type2(A)),aa(nat,nat,suc,N))),aa(num,int,numeral_numeral(int),K))) ) ).

% take_bit_word_Suc_numeral
tff(fact_6627_take__bit__word__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: num,K: num] : aa(word(A),word(A),bit_se2584673776208193580ke_bit(word(A),aa(num,nat,numeral_numeral(nat),N)),aa(num,word(A),numeral_numeral(word(A)),K)) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,bit_se2584673776208193580ke_bit(int,ord_min(nat,type_len0_len_of(A,type2(A)),aa(num,nat,numeral_numeral(nat),N))),aa(num,int,numeral_numeral(int),K))) ) ).

% take_bit_word_numeral
tff(fact_6628_min__absorb2,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Y: A,X: A] :
          ( pp(aa(A,bool,ord_less_eq(A,Y),X))
         => ( ord_min(A,X,Y) = Y ) ) ) ).

% min_absorb2
tff(fact_6629_min__absorb1,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,ord_less_eq(A,X),Y))
         => ( ord_min(A,X,Y) = X ) ) ) ).

% min_absorb1
tff(fact_6630_min__def,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [A2: A,B2: A] :
          ( ( pp(aa(A,bool,ord_less_eq(A,A2),B2))
           => ( ord_min(A,A2,B2) = A2 ) )
          & ( ~ pp(aa(A,bool,ord_less_eq(A,A2),B2))
           => ( ord_min(A,A2,B2) = B2 ) ) ) ) ).

% min_def
tff(fact_6631_min__def__raw,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [X4: A,Xa: A] :
          ( ( pp(aa(A,bool,ord_less_eq(A,X4),Xa))
           => ( ord_min(A,X4,Xa) = X4 ) )
          & ( ~ pp(aa(A,bool,ord_less_eq(A,X4),Xa))
           => ( ord_min(A,X4,Xa) = Xa ) ) ) ) ).

% min_def_raw
tff(fact_6632_min__le__iff__disj,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y: A,Z: A] :
          ( pp(aa(A,bool,ord_less_eq(A,ord_min(A,X,Y)),Z))
        <=> ( pp(aa(A,bool,ord_less_eq(A,X),Z))
            | pp(aa(A,bool,ord_less_eq(A,Y),Z)) ) ) ) ).

% min_le_iff_disj
tff(fact_6633_min_OcoboundedI2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,C2: A,A2: A] :
          ( pp(aa(A,bool,ord_less_eq(A,B2),C2))
         => pp(aa(A,bool,ord_less_eq(A,ord_min(A,A2,B2)),C2)) ) ) ).

% min.coboundedI2
tff(fact_6634_min_OcoboundedI1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,C2: A,B2: A] :
          ( pp(aa(A,bool,ord_less_eq(A,A2),C2))
         => pp(aa(A,bool,ord_less_eq(A,ord_min(A,A2,B2)),C2)) ) ) ).

% min.coboundedI1
tff(fact_6635_min_Oabsorb__iff2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,A2: A] :
          ( pp(aa(A,bool,ord_less_eq(A,B2),A2))
        <=> ( ord_min(A,A2,B2) = B2 ) ) ) ).

% min.absorb_iff2
tff(fact_6636_min_Oabsorb__iff1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,ord_less_eq(A,A2),B2))
        <=> ( ord_min(A,A2,B2) = A2 ) ) ) ).

% min.absorb_iff1
tff(fact_6637_min_Ocobounded2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A] : pp(aa(A,bool,ord_less_eq(A,ord_min(A,A2,B2)),B2)) ) ).

% min.cobounded2
tff(fact_6638_min_Ocobounded1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A] : pp(aa(A,bool,ord_less_eq(A,ord_min(A,A2,B2)),A2)) ) ).

% min.cobounded1
tff(fact_6639_min_Oorder__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,ord_less_eq(A,A2),B2))
        <=> ( A2 = ord_min(A,A2,B2) ) ) ) ).

% min.order_iff
tff(fact_6640_min_OboundedI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,ord_less_eq(A,A2),B2))
         => ( pp(aa(A,bool,ord_less_eq(A,A2),C2))
           => pp(aa(A,bool,ord_less_eq(A,A2),ord_min(A,B2,C2))) ) ) ) ).

% min.boundedI
tff(fact_6641_min_OboundedE,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,ord_less_eq(A,A2),ord_min(A,B2,C2)))
         => ~ ( pp(aa(A,bool,ord_less_eq(A,A2),B2))
             => ~ pp(aa(A,bool,ord_less_eq(A,A2),C2)) ) ) ) ).

% min.boundedE
tff(fact_6642_min_OorderI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A] :
          ( ( A2 = ord_min(A,A2,B2) )
         => pp(aa(A,bool,ord_less_eq(A,A2),B2)) ) ) ).

% min.orderI
tff(fact_6643_min_OorderE,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,ord_less_eq(A,A2),B2))
         => ( A2 = ord_min(A,A2,B2) ) ) ) ).

% min.orderE
tff(fact_6644_min_Omono,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,C2: A,B2: A,D: A] :
          ( pp(aa(A,bool,ord_less_eq(A,A2),C2))
         => ( pp(aa(A,bool,ord_less_eq(A,B2),D))
           => pp(aa(A,bool,ord_less_eq(A,ord_min(A,A2,B2)),ord_min(A,C2,D))) ) ) ) ).

% min.mono
tff(fact_6645_min__diff,axiom,
    ! [M: nat,I: nat,N: nat] : ord_min(nat,aa(nat,nat,minus_minus(nat,M),I),aa(nat,nat,minus_minus(nat,N),I)) = aa(nat,nat,minus_minus(nat,ord_min(nat,M,N)),I) ).

% min_diff
tff(fact_6646_min__diff__distrib__left,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [X: A,Y: A,Z: A] : aa(A,A,minus_minus(A,ord_min(A,X,Y)),Z) = ord_min(A,aa(A,A,minus_minus(A,X),Z),aa(A,A,minus_minus(A,Y),Z)) ) ).

% min_diff_distrib_left
tff(fact_6647_of__nat__min,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [X: nat,Y: nat] : aa(nat,A,semiring_1_of_nat(A),ord_min(nat,X,Y)) = ord_min(A,aa(nat,A,semiring_1_of_nat(A),X),aa(nat,A,semiring_1_of_nat(A),Y)) ) ).

% of_nat_min
tff(fact_6648_minus__min__eq__max,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [X: A,Y: A] : aa(A,A,uminus_uminus(A),ord_min(A,X,Y)) = aa(A,A,ord_max(A,aa(A,A,uminus_uminus(A),X)),aa(A,A,uminus_uminus(A),Y)) ) ).

% minus_min_eq_max
tff(fact_6649_minus__max__eq__min,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [X: A,Y: A] : aa(A,A,uminus_uminus(A),aa(A,A,ord_max(A,X),Y)) = ord_min(A,aa(A,A,uminus_uminus(A),X),aa(A,A,uminus_uminus(A),Y)) ) ).

% minus_max_eq_min
tff(fact_6650_min__add__distrib__left,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [X: A,Y: A,Z: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),ord_min(A,X,Y)),Z) = ord_min(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Z),aa(A,A,aa(A,fun(A,A),plus_plus(A),Y),Z)) ) ).

% min_add_distrib_left
tff(fact_6651_min__add__distrib__right,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [X: A,Y: A,Z: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),X),ord_min(A,Y,Z)) = ord_min(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Z)) ) ).

% min_add_distrib_right
tff(fact_6652_nat__mult__min__left,axiom,
    ! [M: nat,N: nat,Q2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),ord_min(nat,M,N)),Q2) = ord_min(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),Q2),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),Q2)) ).

% nat_mult_min_left
tff(fact_6653_nat__mult__min__right,axiom,
    ! [M: nat,N: nat,Q2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),ord_min(nat,N,Q2)) = ord_min(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),Q2)) ).

% nat_mult_min_right
tff(fact_6654_concat__bit__assoc__sym,axiom,
    ! [M: nat,N: nat,K: int,L: int,R2: int] : aa(int,int,bit_concat_bit(M,aa(int,int,bit_concat_bit(N,K),L)),R2) = aa(int,int,bit_concat_bit(ord_min(nat,M,N),K),aa(int,int,bit_concat_bit(aa(nat,nat,minus_minus(nat,M),N),L),R2)) ).

% concat_bit_assoc_sym
tff(fact_6655_take__bit__concat__bit__eq,axiom,
    ! [M: nat,N: nat,K: int,L: int] : aa(int,int,bit_se2584673776208193580ke_bit(int,M),aa(int,int,bit_concat_bit(N,K),L)) = aa(int,int,bit_concat_bit(ord_min(nat,M,N),K),aa(int,int,bit_se2584673776208193580ke_bit(int,aa(nat,nat,minus_minus(nat,M),N)),L)) ).

% take_bit_concat_bit_eq
tff(fact_6656_max__mult__distrib__left,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [P2: A,X: A,Y: A] :
          ( ( pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),P2))
           => ( aa(A,A,aa(A,fun(A,A),times_times(A),P2),aa(A,A,ord_max(A,X),Y)) = aa(A,A,ord_max(A,aa(A,A,aa(A,fun(A,A),times_times(A),P2),X)),aa(A,A,aa(A,fun(A,A),times_times(A),P2),Y)) ) )
          & ( ~ pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),P2))
           => ( aa(A,A,aa(A,fun(A,A),times_times(A),P2),aa(A,A,ord_max(A,X),Y)) = ord_min(A,aa(A,A,aa(A,fun(A,A),times_times(A),P2),X),aa(A,A,aa(A,fun(A,A),times_times(A),P2),Y)) ) ) ) ) ).

% max_mult_distrib_left
tff(fact_6657_min__mult__distrib__left,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [P2: A,X: A,Y: A] :
          ( ( pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),P2))
           => ( aa(A,A,aa(A,fun(A,A),times_times(A),P2),ord_min(A,X,Y)) = ord_min(A,aa(A,A,aa(A,fun(A,A),times_times(A),P2),X),aa(A,A,aa(A,fun(A,A),times_times(A),P2),Y)) ) )
          & ( ~ pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),P2))
           => ( aa(A,A,aa(A,fun(A,A),times_times(A),P2),ord_min(A,X,Y)) = aa(A,A,ord_max(A,aa(A,A,aa(A,fun(A,A),times_times(A),P2),X)),aa(A,A,aa(A,fun(A,A),times_times(A),P2),Y)) ) ) ) ) ).

% min_mult_distrib_left
tff(fact_6658_max__mult__distrib__right,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [P2: A,X: A,Y: A] :
          ( ( pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),P2))
           => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,ord_max(A,X),Y)),P2) = aa(A,A,ord_max(A,aa(A,A,aa(A,fun(A,A),times_times(A),X),P2)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),P2)) ) )
          & ( ~ pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),P2))
           => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,ord_max(A,X),Y)),P2) = ord_min(A,aa(A,A,aa(A,fun(A,A),times_times(A),X),P2),aa(A,A,aa(A,fun(A,A),times_times(A),Y),P2)) ) ) ) ) ).

% max_mult_distrib_right
tff(fact_6659_min__mult__distrib__right,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [P2: A,X: A,Y: A] :
          ( ( pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),P2))
           => ( aa(A,A,aa(A,fun(A,A),times_times(A),ord_min(A,X,Y)),P2) = ord_min(A,aa(A,A,aa(A,fun(A,A),times_times(A),X),P2),aa(A,A,aa(A,fun(A,A),times_times(A),Y),P2)) ) )
          & ( ~ pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),P2))
           => ( aa(A,A,aa(A,fun(A,A),times_times(A),ord_min(A,X,Y)),P2) = aa(A,A,ord_max(A,aa(A,A,aa(A,fun(A,A),times_times(A),X),P2)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),P2)) ) ) ) ) ).

% min_mult_distrib_right
tff(fact_6660_min__divide__distrib__right,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [P2: A,X: A,Y: A] :
          ( ( pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),P2))
           => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),ord_min(A,X,Y)),P2) = ord_min(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),X),P2),aa(A,A,aa(A,fun(A,A),divide_divide(A),Y),P2)) ) )
          & ( ~ pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),P2))
           => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),ord_min(A,X,Y)),P2) = aa(A,A,ord_max(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),X),P2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),Y),P2)) ) ) ) ) ).

% min_divide_distrib_right
tff(fact_6661_max__divide__distrib__right,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [P2: A,X: A,Y: A] :
          ( ( pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),P2))
           => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,ord_max(A,X),Y)),P2) = aa(A,A,ord_max(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),X),P2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),Y),P2)) ) )
          & ( ~ pp(aa(A,bool,ord_less_eq(A,zero_zero(A)),P2))
           => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,ord_max(A,X),Y)),P2) = ord_min(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),X),P2),aa(A,A,aa(A,fun(A,A),divide_divide(A),Y),P2)) ) ) ) ) ).

% max_divide_distrib_right
tff(fact_6662_mod__exp__eq,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A,M: nat,N: nat] : modulo_modulo(A,modulo_modulo(A,A2,aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),M)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N)) = modulo_modulo(A,A2,aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),ord_min(nat,M,N))) ) ).

% mod_exp_eq
tff(fact_6663_mod__mod__power,axiom,
    ! [K: nat,M: nat,N: nat] : modulo_modulo(nat,modulo_modulo(nat,K,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)) = modulo_modulo(nat,K,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),ord_min(nat,M,N))) ).

% mod_mod_power
tff(fact_6664_mask__mod__exp,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [N: nat,M: nat] : modulo_modulo(A,aa(A,A,minus_minus(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N)),one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),M)) = aa(A,A,minus_minus(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),ord_min(nat,M,N))),one_one(A)) ) ).

% mask_mod_exp
tff(fact_6665_bit__slice__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [M: nat,W: word(A),N: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(word(B),slice2(A,B,M,W)),N))
        <=> ( pp(aa(nat,bool,ord_less(nat,N),ord_min(nat,type_len0_len_of(B,type2(B)),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),M))))
            & pp(aa(nat,bool,bit_se5641148757651400278ts_bit(word(A),W),aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),type_len0_len_of(A,type2(A)))),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),M)))) ) ) ) ).

% bit_slice_iff
tff(fact_6666_unat__mask,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: nat] : aa(word(A),nat,semiring_1_unsigned(A,nat),bit_se2239418461657761734s_mask(word(A),N)) = aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),ord_min(nat,N,type_len0_len_of(A,type2(A))))),one_one(nat)) ) ).

% unat_mask
tff(fact_6667_bit__horner__sum__bit__word__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Bs: list(bool),N: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(word(A),groups4207007520872428315er_sum(bool,word(A),zero_neq_one_of_bool(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2)),Bs)),N))
        <=> ( pp(aa(nat,bool,ord_less(nat,N),ord_min(nat,type_len0_len_of(A,type2(A)),aa(list(bool),nat,size_size(list(bool)),Bs))))
            & pp(aa(nat,bool,nth(bool,Bs),N)) ) ) ) ).

% bit_horner_sum_bit_word_iff
tff(fact_6668_bit__slice1__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [M: nat,W: word(A),N: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(word(B),slice1(A,B,M,W)),N))
        <=> ( pp(aa(nat,bool,ord_less_eq(nat,aa(nat,nat,minus_minus(nat,M),type_len0_len_of(A,type2(A)))),N))
            & pp(aa(nat,bool,ord_less(nat,N),ord_min(nat,type_len0_len_of(B,type2(B)),M)))
            & pp(aa(nat,bool,bit_se5641148757651400278ts_bit(word(A),W),aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),M))),aa(nat,nat,minus_minus(nat,M),type_len0_len_of(A,type2(A)))))) ) ) ) ).

% bit_slice1_iff
tff(fact_6669_drop__bit__exp__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,N: nat] : bit_se4197421643247451524op_bit(A,M,aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(bool,A,zero_neq_one_of_bool(A),fconj(aa(nat,bool,ord_less_eq(nat,M),N),bit_se6407376104438227557le_bit(A,type2(A),N)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(nat,nat,minus_minus(nat,N),M))) ) ).

% drop_bit_exp_eq
tff(fact_6670_min__enat__simps_I3_J,axiom,
    ! [Q2: extended_enat] : ord_min(extended_enat,zero_zero(extended_enat),Q2) = zero_zero(extended_enat) ).

% min_enat_simps(3)
tff(fact_6671_min__enat__simps_I2_J,axiom,
    ! [Q2: extended_enat] : ord_min(extended_enat,Q2,zero_zero(extended_enat)) = zero_zero(extended_enat) ).

% min_enat_simps(2)
tff(fact_6672_possible__bit__min,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [Tyrep: itself(A),I: nat,J: nat] :
          ( pp(bit_se6407376104438227557le_bit(A,Tyrep,ord_min(nat,I,J)))
        <=> ( pp(bit_se6407376104438227557le_bit(A,Tyrep,I))
            | pp(bit_se6407376104438227557le_bit(A,Tyrep,J)) ) ) ) ).

% possible_bit_min
tff(fact_6673_slice1__0,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [N: nat] : slice1(B,A,N,zero_zero(word(B))) = zero_zero(word(A)) ) ).

% slice1_0
tff(fact_6674_bit__minus__1__iff,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [N: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,aa(A,A,uminus_uminus(A),one_one(A))),N))
        <=> pp(bit_se6407376104438227557le_bit(A,type2(A),N)) ) ) ).

% bit_minus_1_iff
tff(fact_6675_bit__minus__2__iff,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [N: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),bit0(one2)))),N))
        <=> ( pp(bit_se6407376104438227557le_bit(A,type2(A),N))
            & pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),N)) ) ) ) ).

% bit_minus_2_iff
tff(fact_6676_possible__bit__0,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [Ty: itself(A)] : pp(bit_se6407376104438227557le_bit(A,Ty,zero_zero(nat))) ) ).

% possible_bit_0
tff(fact_6677_possible__bit__less__imp,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [Tyrep: itself(A),I: nat,J: nat] :
          ( pp(bit_se6407376104438227557le_bit(A,Tyrep,I))
         => ( pp(aa(nat,bool,ord_less_eq(nat,J),I))
           => pp(bit_se6407376104438227557le_bit(A,Tyrep,J)) ) ) ) ).

% possible_bit_less_imp
tff(fact_6678_bit__imp__possible__bit,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A,N: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,A2),N))
         => pp(bit_se6407376104438227557le_bit(A,type2(A),N)) ) ) ).

% bit_imp_possible_bit
tff(fact_6679_impossible__bit,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [N: nat,A2: A] :
          ( ~ pp(bit_se6407376104438227557le_bit(A,type2(A),N))
         => ~ pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,A2),N)) ) ) ).

% impossible_bit
tff(fact_6680_bit__eq__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A,B2: A] :
          ( ( A2 = B2 )
        <=> ! [N5: nat] :
              ( pp(bit_se6407376104438227557le_bit(A,type2(A),N5))
             => ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,A2),N5))
              <=> pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,B2),N5)) ) ) ) ) ).

% bit_eq_iff
tff(fact_6681_bit__eqI,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A,B2: A] :
          ( ! [N2: nat] :
              ( pp(bit_se6407376104438227557le_bit(A,type2(A),N2))
             => ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,A2),N2))
              <=> pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,B2),N2)) ) )
         => ( A2 = B2 ) ) ) ).

% bit_eqI
tff(fact_6682_bit__not__iff,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A2: A,N: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,aa(A,A,bit_ri4277139882892585799ns_not(A),A2)),N))
        <=> ( pp(bit_se6407376104438227557le_bit(A,type2(A),N))
            & ~ pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,A2),N)) ) ) ) ).

% bit_not_iff
tff(fact_6683_semiring__bit__operations__class_Obit__set__bit__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,A2: A,N: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,aa(A,A,aa(nat,fun(A,A),bit_se5668285175392031749et_bit(A),M),A2)),N))
        <=> ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,A2),N))
            | ( ( M = N )
              & pp(bit_se6407376104438227557le_bit(A,type2(A),N)) ) ) ) ) ).

% semiring_bit_operations_class.bit_set_bit_iff
tff(fact_6684_bit__flip__bit__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,A2: A,N: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,bit_se8732182000553998342ip_bit(A,M,A2)),N))
        <=> ( ( ( M = N )
            <=> ~ pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,A2),N)) )
            & pp(bit_se6407376104438227557le_bit(A,type2(A),N)) ) ) ) ).

% bit_flip_bit_iff
tff(fact_6685_bit__mask__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,N: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,bit_se2239418461657761734s_mask(A,M)),N))
        <=> ( pp(bit_se6407376104438227557le_bit(A,type2(A),N))
            & pp(aa(nat,bool,ord_less(nat,N),M)) ) ) ) ).

% bit_mask_iff
tff(fact_6686_bit__of__int__iff,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [K: int,N: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,aa(int,A,ring_1_of_int(A),K)),N))
        <=> ( pp(bit_se6407376104438227557le_bit(A,type2(A),N))
            & pp(aa(nat,bool,bit_se5641148757651400278ts_bit(int,K),N)) ) ) ) ).

% bit_of_int_iff
tff(fact_6687_bit__signed__take__bit__iff,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [M: nat,A2: A,N: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,aa(A,A,bit_ri4674362597316999326ke_bit(A,M),A2)),N))
        <=> ( pp(bit_se6407376104438227557le_bit(A,type2(A),N))
            & pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,A2),ord_min(nat,M,N))) ) ) ) ).

% bit_signed_take_bit_iff
tff(fact_6688_bit__of__nat__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [M: nat,N: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,aa(nat,A,semiring_1_of_nat(A),M)),N))
        <=> ( pp(bit_se6407376104438227557le_bit(A,type2(A),N))
            & pp(aa(nat,bool,bit_se5641148757651400278ts_bit(nat,M),N)) ) ) ) ).

% bit_of_nat_iff
tff(fact_6689_possible__bit__def,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [Tyrep: itself(A),N: nat] :
          ( pp(bit_se6407376104438227557le_bit(A,Tyrep,N))
        <=> ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N) != zero_zero(A) ) ) ) ).

% possible_bit_def
tff(fact_6690_bit__minus__iff,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A2: A,N: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,aa(A,A,uminus_uminus(A),A2)),N))
        <=> ( pp(bit_se6407376104438227557le_bit(A,type2(A),N))
            & ~ pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,aa(A,A,minus_minus(A,A2),one_one(A))),N)) ) ) ) ).

% bit_minus_iff
tff(fact_6691_bit__twiddle__min,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),X: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344971392196577ns_xor(word(A)),Y),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344971392196577ns_xor(word(A)),X),Y)),if(word(A),aa(word(A),bool,ord_less(word(A),X),Y),aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A))),zero_zero(word(A))))) = ord_min(word(A),X,Y) ) ).

% bit_twiddle_min
tff(fact_6692_bit__push__bit__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,A2: A,N: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,bit_se4730199178511100633sh_bit(A,M,A2)),N))
        <=> ( pp(aa(nat,bool,ord_less_eq(nat,M),N))
            & pp(bit_se6407376104438227557le_bit(A,type2(A),N))
            & pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,A2),aa(nat,nat,minus_minus(nat,N),M))) ) ) ) ).

% bit_push_bit_iff
tff(fact_6693_fold__possible__bit,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [N: nat] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N) = zero_zero(A) )
        <=> ~ pp(bit_se6407376104438227557le_bit(A,type2(A),N)) ) ) ).

% fold_possible_bit
tff(fact_6694_bit__exp__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [M: nat,N: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),M)),N))
        <=> ( pp(bit_se6407376104438227557le_bit(A,type2(A),N))
            & ( M = N ) ) ) ) ).

% bit_exp_iff
tff(fact_6695_bit__2__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [N: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,aa(num,A,numeral_numeral(A),bit0(one2))),N))
        <=> ( pp(bit_se6407376104438227557le_bit(A,type2(A),one_one(nat)))
            & ( N = one_one(nat) ) ) ) ) ).

% bit_2_iff
tff(fact_6696_bit__not__exp__iff,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [M: nat,N: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,aa(A,A,bit_ri4277139882892585799ns_not(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),M))),N))
        <=> ( pp(bit_se6407376104438227557le_bit(A,type2(A),N))
            & ( N != M ) ) ) ) ).

% bit_not_exp_iff
tff(fact_6697_bit__minus__exp__iff,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [M: nat,N: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,aa(A,A,uminus_uminus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),M))),N))
        <=> ( pp(bit_se6407376104438227557le_bit(A,type2(A),N))
            & pp(aa(nat,bool,ord_less_eq(nat,M),N)) ) ) ) ).

% bit_minus_exp_iff
tff(fact_6698_bit__mask__sub__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [M: nat,N: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,aa(A,A,minus_minus(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),M)),one_one(A))),N))
        <=> ( pp(bit_se6407376104438227557le_bit(A,type2(A),N))
            & pp(aa(nat,bool,ord_less(nat,N),M)) ) ) ) ).

% bit_mask_sub_iff
tff(fact_6699_bit__double__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A,N: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),A2)),N))
        <=> ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,A2),aa(nat,nat,minus_minus(nat,N),one_one(nat))))
            & ( N != zero_zero(nat) )
            & pp(bit_se6407376104438227557le_bit(A,type2(A),N)) ) ) ) ).

% bit_double_iff
tff(fact_6700_bit__signed__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & bit_ri3973907225187159222ations(A) )
     => ! [W: word(B),N: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,ring_1_signed(B,A,W)),N))
        <=> ( pp(bit_se6407376104438227557le_bit(A,type2(A),N))
            & pp(aa(nat,bool,bit_se5641148757651400278ts_bit(word(B),W),ord_min(nat,aa(nat,nat,minus_minus(nat,type_len0_len_of(B,type2(B))),aa(nat,nat,suc,zero_zero(nat))),N))) ) ) ) ).

% bit_signed_iff
tff(fact_6701_word__set__bits__def,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [P: fun(nat,bool)] : bit_bi4170147762399595738t_bits(word(A),P) = groups4207007520872428315er_sum(bool,word(A),zero_neq_one_of_bool(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2)),map(nat,bool,P,upt(zero_zero(nat),type_len0_len_of(A,type2(A))))) ) ).

% word_set_bits_def
tff(fact_6702_UNIV__word__eq__word__of__nat,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( top_top(set(word(A))) = aa(set(nat),set(word(A)),image(nat,word(A),semiring_1_of_nat(word(A))),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))) ) ) ).

% UNIV_word_eq_word_of_nat
tff(fact_6703_Collect__const,axiom,
    ! [A: $tType,P: bool] :
      ( ( pp(P)
       => ( collect(A,aTP_Lamp_pq(bool,fun(A,bool),P)) = top_top(set(A)) ) )
      & ( ~ pp(P)
       => ( collect(A,aTP_Lamp_pq(bool,fun(A,bool),P)) = bot_bot(set(A)) ) ) ) ).

% Collect_const
tff(fact_6704_finite__Collect__not,axiom,
    ! [A: $tType,P: fun(A,bool)] :
      ( pp(aa(set(A),bool,finite_finite2(A),collect(A,P)))
     => ( pp(aa(set(A),bool,finite_finite2(A),collect(A,aTP_Lamp_bz(fun(A,bool),fun(A,bool),P))))
      <=> pp(aa(set(A),bool,finite_finite2(A),top_top(set(A)))) ) ) ).

% finite_Collect_not
tff(fact_6705_set__bits__False__eq,axiom,
    ! [A: $tType] :
      ( bit_bi6583157726757044596ension(A)
     => ( bit_bi4170147762399595738t_bits(A,aTP_Lamp_pr(nat,bool)) = zero_zero(A) ) ) ).

% set_bits_False_eq
tff(fact_6706_set__bits__K__False,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( bit_bi4170147762399595738t_bits(word(A),aTP_Lamp_pr(nat,bool)) = zero_zero(word(A)) ) ) ).

% set_bits_K_False
tff(fact_6707_surj__plus,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A2: A] : aa(set(A),set(A),image(A,A,aa(A,fun(A,A),plus_plus(A),A2)),top_top(set(A))) = top_top(set(A)) ) ).

% surj_plus
tff(fact_6708_range__add,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A] : aa(set(A),set(A),image(A,A,aa(A,fun(A,A),plus_plus(A),A2)),top_top(set(A))) = top_top(set(A)) ) ).

% range_add
tff(fact_6709_surj__diff__right,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A2: A] : aa(set(A),set(A),image(A,A,aTP_Lamp_ol(A,fun(A,A),A2)),top_top(set(A))) = top_top(set(A)) ) ).

% surj_diff_right
tff(fact_6710_range__constant,axiom,
    ! [B: $tType,A: $tType,X: A] : aa(set(B),set(A),image(B,A,aTP_Lamp_ay(A,fun(B,A),X)),top_top(set(B))) = insert(A,X,bot_bot(set(A))) ).

% range_constant
tff(fact_6711_finite__range__imageI,axiom,
    ! [A: $tType,C: $tType,B: $tType,G: fun(B,A),F2: fun(A,C)] :
      ( pp(aa(set(A),bool,finite_finite2(A),aa(set(B),set(A),image(B,A,G),top_top(set(B)))))
     => pp(aa(set(C),bool,finite_finite2(C),aa(set(B),set(C),image(B,C,aa(fun(A,C),fun(B,C),aTP_Lamp_ps(fun(B,A),fun(fun(A,C),fun(B,C)),G),F2)),top_top(set(B))))) ) ).

% finite_range_imageI
tff(fact_6712_UNIV__option__conv,axiom,
    ! [A: $tType] : top_top(set(option(A))) = insert(option(A),none(A),aa(set(A),set(option(A)),image(A,option(A),some(A)),top_top(set(A)))) ).

% UNIV_option_conv
tff(fact_6713_range__composition,axiom,
    ! [A: $tType,C: $tType,B: $tType,F2: fun(C,A),G: fun(B,C)] : aa(set(B),set(A),image(B,A,aa(fun(B,C),fun(B,A),aTP_Lamp_pt(fun(C,A),fun(fun(B,C),fun(B,A)),F2),G)),top_top(set(B))) = aa(set(C),set(A),image(C,A,F2),aa(set(B),set(C),image(B,C,G),top_top(set(B)))) ).

% range_composition
tff(fact_6714_rangeE,axiom,
    ! [A: $tType,B: $tType,B2: A,F2: fun(B,A)] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),B2),aa(set(B),set(A),image(B,A,F2),top_top(set(B)))))
     => ~ ! [X3: B] : B2 != aa(B,A,F2,X3) ) ).

% rangeE
tff(fact_6715_range__subsetD,axiom,
    ! [B: $tType,A: $tType,F2: fun(B,A),B4: set(A),I: B] :
      ( pp(aa(set(A),bool,ord_less_eq(set(A),aa(set(B),set(A),image(B,A,F2),top_top(set(B)))),B4))
     => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(B,A,F2,I)),B4)) ) ).

% range_subsetD
tff(fact_6716_not__UNIV__le__Icc,axiom,
    ! [A: $tType] :
      ( no_top(A)
     => ! [L: A,H: A] : ~ pp(aa(set(A),bool,ord_less_eq(set(A),top_top(set(A))),set_or1337092689740270186AtMost(A,L,H))) ) ).

% not_UNIV_le_Icc
tff(fact_6717_top__greatest,axiom,
    ! [A: $tType] :
      ( order_top(A)
     => ! [A2: A] : pp(aa(A,bool,ord_less_eq(A,A2),top_top(A))) ) ).

% top_greatest
tff(fact_6718_top_Oextremum__unique,axiom,
    ! [A: $tType] :
      ( order_top(A)
     => ! [A2: A] :
          ( pp(aa(A,bool,ord_less_eq(A,top_top(A)),A2))
        <=> ( A2 = top_top(A) ) ) ) ).

% top.extremum_unique
tff(fact_6719_top_Oextremum__uniqueI,axiom,
    ! [A: $tType] :
      ( order_top(A)
     => ! [A2: A] :
          ( pp(aa(A,bool,ord_less_eq(A,top_top(A)),A2))
         => ( A2 = top_top(A) ) ) ) ).

% top.extremum_uniqueI
tff(fact_6720_subset__UNIV,axiom,
    ! [A: $tType,A3: set(A)] : pp(aa(set(A),bool,ord_less_eq(set(A),A3),top_top(set(A)))) ).

% subset_UNIV
tff(fact_6721_UNIV__def,axiom,
    ! [A: $tType] : top_top(set(A)) = collect(A,aTP_Lamp_pu(A,bool)) ).

% UNIV_def
tff(fact_6722_not__UNIV__le__Iic,axiom,
    ! [A: $tType] :
      ( no_top(A)
     => ! [H: A] : ~ pp(aa(set(A),bool,ord_less_eq(set(A),top_top(set(A))),set_ord_atMost(A,H))) ) ).

% not_UNIV_le_Iic
tff(fact_6723_surj__Compl__image__subset,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),A3: set(B)] :
      ( ( aa(set(B),set(A),image(B,A,F2),top_top(set(B))) = top_top(set(A)) )
     => pp(aa(set(A),bool,ord_less_eq(set(A),aa(set(A),set(A),uminus_uminus(set(A)),aa(set(B),set(A),image(B,A,F2),A3))),aa(set(B),set(A),image(B,A,F2),aa(set(B),set(B),uminus_uminus(set(B)),A3)))) ) ).

% surj_Compl_image_subset
tff(fact_6724_notin__range__Some,axiom,
    ! [A: $tType,X: option(A)] :
      ( ~ pp(aa(set(option(A)),bool,aa(option(A),fun(set(option(A)),bool),member(option(A)),X),aa(set(A),set(option(A)),image(A,option(A),some(A)),top_top(set(A)))))
    <=> ( X = none(A) ) ) ).

% notin_range_Some
tff(fact_6725_word__of__int__conv__set__bits,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [I: int] : aa(int,word(A),ring_1_of_int(word(A)),I) = bit_bi4170147762399595738t_bits(word(A),bit_se5641148757651400278ts_bit(int,I)) ) ).

% word_of_int_conv_set_bits
tff(fact_6726_set__bits__conv__set__bits__aux,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [F2: fun(nat,bool)] : bit_bi4170147762399595738t_bits(word(A),F2) = code_T2661198915054445665ts_aux(A,F2,type_len0_len_of(A,type2(A)),zero_zero(word(A))) ) ).

% set_bits_conv_set_bits_aux
tff(fact_6727_word__test__bit__set__bits,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [F2: fun(nat,bool),N: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(word(A),bit_bi4170147762399595738t_bits(word(A),F2)),N))
        <=> ( pp(aa(nat,bool,ord_less(nat,N),type_len0_len_of(A,type2(A))))
            & pp(aa(nat,bool,F2,N)) ) ) ) ).

% word_test_bit_set_bits
tff(fact_6728_range__uint,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( aa(set(word(A)),set(int),image(word(A),int,semiring_1_unsigned(A,int)),top_top(set(word(A)))) = set_or7035219750837199246ssThan(int,zero_zero(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A)))) ) ) ).

% range_uint
tff(fact_6729_UNIV__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( top_top(set(word(A))) = aa(set(int),set(word(A)),image(int,word(A),ring_1_of_int(word(A))),set_or7035219750837199246ssThan(int,zero_zero(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A))))) ) ) ).

% UNIV_eq
tff(fact_6730_ucast__range__less,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(A) )
     => ( pp(aa(nat,bool,ord_less(nat,type_len0_len_of(A,type2(A))),type_len0_len_of(B,type2(B))))
       => ( aa(set(word(A)),set(word(B)),image(word(A),word(B),semiring_1_unsigned(A,word(B))),top_top(set(word(A)))) = collect(word(B),aTP_Lamp_pv(word(B),bool)) ) ) ) ).

% ucast_range_less
tff(fact_6731_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),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A))),aa(list(word(A)),list(word(A)),cons(word(A),W),Ws)) = aa(int,int,bit_concat_bit(type_len0_len_of(A,type2(A)),aa(word(A),int,semiring_1_unsigned(A,int),W)),groups4207007520872428315er_sum(word(A),int,semiring_1_unsigned(A,int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A))),Ws)) ) ).

% horner_sum_uint_exp_Cons_eq
tff(fact_6732_word__roti_Oabs__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa2: int,X: int] : word_roti(A,Xa2,word2(A,X)) = word2(A,aa(int,int,bit_concat_bit(aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),nat2(modulo_modulo(int,Xa2,aa(nat,int,semiring_1_of_nat(int),type_len0_len_of(A,type2(A)))))),bit_se4197421643247451524op_bit(int,nat2(modulo_modulo(int,Xa2,aa(nat,int,semiring_1_of_nat(int),type_len0_len_of(A,type2(A))))),aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),X))),aa(int,int,bit_se2584673776208193580ke_bit(int,nat2(modulo_modulo(int,Xa2,aa(nat,int,semiring_1_of_nat(int),type_len0_len_of(A,type2(A)))))),X))) ) ).

% word_roti.abs_eq
tff(fact_6733_int__set__bits__K__False,axiom,
    bit_bi4170147762399595738t_bits(int,aTP_Lamp_pr(nat,bool)) = zero_zero(int) ).

% int_set_bits_K_False
tff(fact_6734_range__mult,axiom,
    ! [A2: real] :
      ( ( ( A2 = zero_zero(real) )
       => ( aa(set(real),set(real),image(real,real,aa(real,fun(real,real),times_times(real),A2)),top_top(set(real))) = insert(real,zero_zero(real),bot_bot(set(real))) ) )
      & ( ( A2 != zero_zero(real) )
       => ( aa(set(real),set(real),image(real,real,aa(real,fun(real,real),times_times(real),A2)),top_top(set(real))) = top_top(set(real)) ) ) ) ).

% range_mult
tff(fact_6735_int__set__bits__K__True,axiom,
    bit_bi4170147762399595738t_bits(int,aTP_Lamp_pw(nat,bool)) = aa(int,int,uminus_uminus(int),one_one(int)) ).

% int_set_bits_K_True
tff(fact_6736_Collect__const__case__prod,axiom,
    ! [B: $tType,A: $tType,P: bool] :
      ( ( pp(P)
       => ( collect(product_prod(A,B),product_case_prod(A,B,bool,aTP_Lamp_px(bool,fun(A,fun(B,bool)),P))) = top_top(set(product_prod(A,B))) ) )
      & ( ~ pp(P)
       => ( collect(product_prod(A,B),product_case_prod(A,B,bool,aTP_Lamp_px(bool,fun(A,fun(B,bool)),P))) = bot_bot(set(product_prod(A,B))) ) ) ) ).

% Collect_const_case_prod
tff(fact_6737_length__nth__simps_I4_J,axiom,
    ! [B: $tType,X: B,Xs: list(B),N: nat] : aa(nat,B,nth(B,aa(list(B),list(B),cons(B,X),Xs)),aa(nat,nat,suc,N)) = aa(nat,B,nth(B,Xs),N) ).

% length_nth_simps(4)
tff(fact_6738_nth__Cons__Suc,axiom,
    ! [A: $tType,X: A,Xs: list(A),N: nat] : aa(nat,A,nth(A,aa(list(A),list(A),cons(A,X),Xs)),aa(nat,nat,suc,N)) = aa(nat,A,nth(A,Xs),N) ).

% nth_Cons_Suc
tff(fact_6739_length__nth__simps_I3_J,axiom,
    ! [B: $tType,X: B,Xs: list(B)] : aa(nat,B,nth(B,aa(list(B),list(B),cons(B,X),Xs)),zero_zero(nat)) = X ).

% length_nth_simps(3)
tff(fact_6740_nth__Cons__0,axiom,
    ! [A: $tType,X: A,Xs: list(A)] : aa(nat,A,nth(A,aa(list(A),list(A),cons(A,X),Xs)),zero_zero(nat)) = X ).

% nth_Cons_0
tff(fact_6741_horner__sum__simps_I2_J,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_semiring_0(A)
     => ! [F2: fun(B,A),A2: A,X: B,Xs: list(B)] : groups4207007520872428315er_sum(B,A,F2,A2,aa(list(B),list(B),cons(B,X),Xs)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(B,A,F2,X)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),groups4207007520872428315er_sum(B,A,F2,A2,Xs))) ) ).

% horner_sum_simps(2)
tff(fact_6742_nth__Cons__numeral,axiom,
    ! [A: $tType,X: A,Xs: list(A),V: num] : aa(nat,A,nth(A,aa(list(A),list(A),cons(A,X),Xs)),aa(num,nat,numeral_numeral(nat),V)) = aa(nat,A,nth(A,Xs),aa(nat,nat,minus_minus(nat,aa(num,nat,numeral_numeral(nat),V)),one_one(nat))) ).

% nth_Cons_numeral
tff(fact_6743_nth__Cons__pos,axiom,
    ! [A: $tType,N: nat,X: A,Xs: list(A)] :
      ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),N))
     => ( aa(nat,A,nth(A,aa(list(A),list(A),cons(A,X),Xs)),N) = aa(nat,A,nth(A,Xs),aa(nat,nat,minus_minus(nat,N),one_one(nat))) ) ) ).

% nth_Cons_pos
tff(fact_6744_top__empty__eq2,axiom,
    ! [B: $tType,A: $tType,X4: A,Xa: B] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),top_top(fun(A,fun(B,bool))),X4),Xa))
    <=> pp(aa(set(product_prod(A,B)),bool,aa(product_prod(A,B),fun(set(product_prod(A,B)),bool),member(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X4),Xa)),top_top(set(product_prod(A,B))))) ) ).

% top_empty_eq2
tff(fact_6745_product__lists_Osimps_I2_J,axiom,
    ! [A: $tType,Xs: list(A),Xss: list(list(A))] : product_lists(A,aa(list(list(A)),list(list(A)),cons(list(A),Xs),Xss)) = concat(list(A),map(A,list(list(A)),aTP_Lamp_py(list(list(A)),fun(A,list(list(A))),Xss),Xs)) ).

% product_lists.simps(2)
tff(fact_6746_times__word_Oabs__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa2: int,X: int] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),word2(A,Xa2)),word2(A,X)) = word2(A,aa(int,int,aa(int,fun(int,int),times_times(int),Xa2),X)) ) ).

% times_word.abs_eq
tff(fact_6747_plus__word_Oabs__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa2: int,X: int] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),word2(A,Xa2)),word2(A,X)) = word2(A,aa(int,int,aa(int,fun(int,int),plus_plus(int),Xa2),X)) ) ).

% plus_word.abs_eq
tff(fact_6748_zero__word__def,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( zero_zero(word(A)) = word2(A,zero_zero(int)) ) ) ).

% zero_word_def
tff(fact_6749_one__word__def,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( one_one(word(A)) = word2(A,one_one(int)) ) ) ).

% one_word_def
tff(fact_6750_replicate__Suc,axiom,
    ! [A: $tType,N: nat,X: A] : replicate(A,aa(nat,nat,suc,N),X) = aa(list(A),list(A),cons(A,X),replicate(A,N,X)) ).

% replicate_Suc
tff(fact_6751_list__update__code_I2_J,axiom,
    ! [A: $tType,X: A,Xs: list(A),Y: A] : list_update(A,aa(list(A),list(A),cons(A,X),Xs),zero_zero(nat),Y) = aa(list(A),list(A),cons(A,Y),Xs) ).

% list_update_code(2)
tff(fact_6752_list__update__code_I3_J,axiom,
    ! [A: $tType,X: A,Xs: list(A),I: nat,Y: A] : list_update(A,aa(list(A),list(A),cons(A,X),Xs),aa(nat,nat,suc,I),Y) = aa(list(A),list(A),cons(A,X),list_update(A,Xs,I,Y)) ).

% list_update_code(3)
tff(fact_6753_set__subset__Cons,axiom,
    ! [A: $tType,Xs: list(A),X: A] : pp(aa(set(A),bool,ord_less_eq(set(A),aa(list(A),set(A),set2(A),Xs)),aa(list(A),set(A),set2(A),aa(list(A),list(A),cons(A,X),Xs)))) ).

% set_subset_Cons
tff(fact_6754_Suc__length__conv,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] :
      ( ( aa(nat,nat,suc,N) = aa(list(A),nat,size_size(list(A)),Xs) )
    <=> ? [Y7: A,Ys2: list(A)] :
          ( ( Xs = aa(list(A),list(A),cons(A,Y7),Ys2) )
          & ( aa(list(A),nat,size_size(list(A)),Ys2) = N ) ) ) ).

% Suc_length_conv
tff(fact_6755_length__Suc__conv,axiom,
    ! [A: $tType,Xs: list(A),N: nat] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(nat,nat,suc,N) )
    <=> ? [Y7: A,Ys2: list(A)] :
          ( ( Xs = aa(list(A),list(A),cons(A,Y7),Ys2) )
          & ( aa(list(A),nat,size_size(list(A)),Ys2) = N ) ) ) ).

% length_Suc_conv
tff(fact_6756_length__nth__simps_I2_J,axiom,
    ! [B: $tType,X: B,Xs: list(B)] : aa(list(B),nat,size_size(list(B)),aa(list(B),list(B),cons(B,X),Xs)) = aa(nat,nat,suc,aa(list(B),nat,size_size(list(B)),Xs)) ).

% length_nth_simps(2)
tff(fact_6757_impossible__Cons,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),X: A] :
      ( pp(aa(nat,bool,ord_less_eq(nat,aa(list(A),nat,size_size(list(A)),Xs)),aa(list(A),nat,size_size(list(A)),Ys)))
     => ( Xs != aa(list(A),list(A),cons(A,X),Ys) ) ) ).

% impossible_Cons
tff(fact_6758_Suc__le__length__iff,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] :
      ( pp(aa(nat,bool,ord_less_eq(nat,aa(nat,nat,suc,N)),aa(list(A),nat,size_size(list(A)),Xs)))
    <=> ? [X2: A,Ys2: list(A)] :
          ( ( Xs = aa(list(A),list(A),cons(A,X2),Ys2) )
          & pp(aa(nat,bool,ord_less_eq(nat,N),aa(list(A),nat,size_size(list(A)),Ys2))) ) ) ).

% Suc_le_length_iff
tff(fact_6759_word__succ_Oabs__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: int] : word_succ(A,word2(A,X)) = word2(A,aa(int,int,aa(int,fun(int,int),plus_plus(int),X),one_one(int))) ) ).

% word_succ.abs_eq
tff(fact_6760_word__pred_Oabs__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: int] : word_pred(A,word2(A,X)) = word2(A,aa(int,int,minus_minus(int,X),one_one(int))) ) ).

% word_pred.abs_eq
tff(fact_6761_count__list_Osimps_I2_J,axiom,
    ! [A: $tType,X: A,Y: A,Xs: list(A)] :
      ( ( ( X = Y )
       => ( aa(A,nat,count_list(A,aa(list(A),list(A),cons(A,X),Xs)),Y) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(A,nat,count_list(A,Xs),Y)),one_one(nat)) ) )
      & ( ( X != Y )
       => ( aa(A,nat,count_list(A,aa(list(A),list(A),cons(A,X),Xs)),Y) = aa(A,nat,count_list(A,Xs),Y) ) ) ) ).

% count_list.simps(2)
tff(fact_6762_int__in__range__abs,axiom,
    ! [N: nat] : pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),aa(nat,int,semiring_1_of_nat(int),N)),aa(set(int),set(int),image(int,int,abs_abs(int)),top_top(set(int))))) ).

% int_in_range_abs
tff(fact_6763_list_Osize_I4_J,axiom,
    ! [A: $tType,X21: A,X222: list(A)] : aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),cons(A,X21),X222)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(list(A),nat,size_size(list(A)),X222)),aa(nat,nat,suc,zero_zero(nat))) ).

% list.size(4)
tff(fact_6764_nth__Cons_H,axiom,
    ! [A: $tType,N: nat,X: A,Xs: list(A)] :
      ( ( ( N = zero_zero(nat) )
       => ( aa(nat,A,nth(A,aa(list(A),list(A),cons(A,X),Xs)),N) = X ) )
      & ( ( N != zero_zero(nat) )
       => ( aa(nat,A,nth(A,aa(list(A),list(A),cons(A,X),Xs)),N) = aa(nat,A,nth(A,Xs),aa(nat,nat,minus_minus(nat,N),one_one(nat))) ) ) ) ).

% nth_Cons'
tff(fact_6765_n__lists_Osimps_I2_J,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] : n_lists(A,aa(nat,nat,suc,N),Xs) = concat(list(A),map(list(A),list(list(A)),aTP_Lamp_qa(list(A),fun(list(A),list(list(A))),Xs),n_lists(A,N,Xs))) ).

% n_lists.simps(2)
tff(fact_6766_list_Osize__gen_I2_J,axiom,
    ! [A: $tType,X: fun(A,nat),X21: A,X222: list(A)] : size_list(A,X,aa(list(A),list(A),cons(A,X21),X222)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(A,nat,X,X21)),size_list(A,X,X222))),aa(nat,nat,suc,zero_zero(nat))) ).

% list.size_gen(2)
tff(fact_6767_map__upt__Suc,axiom,
    ! [A: $tType,F2: fun(nat,A),N: nat] : map(nat,A,F2,upt(zero_zero(nat),aa(nat,nat,suc,N))) = aa(list(A),list(A),cons(A,aa(nat,A,F2,zero_zero(nat))),map(nat,A,aTP_Lamp_qb(fun(nat,A),fun(nat,A),F2),upt(zero_zero(nat),N))) ).

% map_upt_Suc
tff(fact_6768_divide__word_Oabs__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa2: int,X: int] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),word2(A,Xa2)),word2(A,X)) = word2(A,aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),Xa2)),aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),X))) ) ).

% divide_word.abs_eq
tff(fact_6769_nth__non__equal__first__eq,axiom,
    ! [A: $tType,X: A,Y: A,Xs: list(A),N: nat] :
      ( ( X != Y )
     => ( ( aa(nat,A,nth(A,aa(list(A),list(A),cons(A,X),Xs)),N) = Y )
      <=> ( ( aa(nat,A,nth(A,Xs),aa(nat,nat,minus_minus(nat,N),one_one(nat))) = Y )
          & pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),N)) ) ) ) ).

% nth_non_equal_first_eq
tff(fact_6770_nth__equal__first__eq,axiom,
    ! [A: $tType,X: A,Xs: list(A),N: nat] :
      ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(list(A),set(A),set2(A),Xs)))
     => ( pp(aa(nat,bool,ord_less_eq(nat,N),aa(list(A),nat,size_size(list(A)),Xs)))
       => ( ( aa(nat,A,nth(A,aa(list(A),list(A),cons(A,X),Xs)),N) = X )
        <=> ( N = zero_zero(nat) ) ) ) ) ).

% nth_equal_first_eq
tff(fact_6771_Cons__replicate__eq,axiom,
    ! [A: $tType,X: A,Xs: list(A),N: nat,Y: A] :
      ( ( aa(list(A),list(A),cons(A,X),Xs) = replicate(A,N,Y) )
    <=> ( ( X = Y )
        & pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),N))
        & ( Xs = replicate(A,aa(nat,nat,minus_minus(nat,N),one_one(nat)),X) ) ) ) ).

% Cons_replicate_eq
tff(fact_6772_slice__Cons,axiom,
    ! [A: $tType,Begin: nat,End: nat,X: A,Xs: list(A)] :
      ( ( ( ( Begin = zero_zero(nat) )
          & pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),End)) )
       => ( slice(A,Begin,End,aa(list(A),list(A),cons(A,X),Xs)) = aa(list(A),list(A),cons(A,X),slice(A,Begin,aa(nat,nat,minus_minus(nat,End),one_one(nat)),Xs)) ) )
      & ( ~ ( ( Begin = zero_zero(nat) )
            & pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),End)) )
       => ( slice(A,Begin,End,aa(list(A),list(A),cons(A,X),Xs)) = slice(A,aa(nat,nat,minus_minus(nat,Begin),one_one(nat)),aa(nat,nat,minus_minus(nat,End),one_one(nat)),Xs) ) ) ) ).

% slice_Cons
tff(fact_6773_unsigned_Oabs__eq,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & semiring_1(A) )
     => ! [X: int] : aa(word(B),A,semiring_1_unsigned(B,A),word2(B,X)) = aa(nat,A,semiring_1_of_nat(A),nat2(aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(B,type2(B))),X))) ) ).

% unsigned.abs_eq
tff(fact_6774_range__mod,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),N))
     => ( aa(set(nat),set(nat),image(nat,nat,aTP_Lamp_qc(nat,fun(nat,nat),N)),top_top(set(nat))) = set_or7035219750837199246ssThan(nat,zero_zero(nat),N) ) ) ).

% range_mod
tff(fact_6775_word__sle_Oabs__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa2: int,X: int] :
          ( word_sle(A,word2(A,Xa2),word2(A,X))
        <=> pp(aa(int,bool,ord_less_eq(int,aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat)))),Xa2)),aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat)))),X))) ) ) ).

% word_sle.abs_eq
tff(fact_6776_word__sless_Oabs__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa2: int,X: int] :
          ( word_sless(A,word2(A,Xa2),word2(A,X))
        <=> pp(aa(int,bool,ord_less(int,aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat)))),Xa2)),aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat)))),X))) ) ) ).

% word_sless.abs_eq
tff(fact_6777_signed_Oabs__eq,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & ring_1(A) )
     => ! [X: int] : ring_1_signed(B,A,word2(B,X)) = aa(int,A,ring_1_of_int(A),aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,minus_minus(nat,type_len0_len_of(B,type2(B))),aa(nat,nat,suc,zero_zero(nat)))),X)) ) ).

% signed.abs_eq
tff(fact_6778_signed__drop__bit_Oabs__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa2: nat,X: int] : signed_drop_bit(A,Xa2,word2(A,X)) = word2(A,bit_se4197421643247451524op_bit(int,Xa2,aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat)))),X))) ) ).

% signed_drop_bit.abs_eq
tff(fact_6779_signed__modulo__word_Oabs__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa2: int,X: int] : signed6721504322012087516modulo(word(A),word2(A,Xa2),word2(A,X)) = word2(A,signed6721504322012087516modulo(int,aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat)))),Xa2),aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat)))),X))) ) ).

% signed_modulo_word.abs_eq
tff(fact_6780_signed__divide__word_Oabs__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa2: int,X: int] : signed7115095781618012415divide(word(A),word2(A,Xa2),word2(A,X)) = word2(A,signed7115095781618012415divide(int,aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat)))),Xa2),aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat)))),X))) ) ).

% signed_divide_word.abs_eq
tff(fact_6781_hash__code__list__simps_I2_J,axiom,
    ! [A: $tType,H_a: fun(A,uint32),X: A,Xa2: list(A)] : hash_hash_code_list(A,H_a,aa(list(A),list(A),cons(A,X),Xa2)) = aa(uint32,uint32,aa(uint32,fun(uint32,uint32),plus_plus(uint32),aa(uint32,uint32,aa(uint32,fun(uint32,uint32),times_times(uint32),aa(A,uint32,H_a,X)),aa(num,uint32,numeral_numeral(uint32),bit0(bit0(aa(num,num,bit1,aa(num,num,bit1,bit0(aa(num,num,bit1,bit0(bit0(bit0(bit0(bit0(aa(num,num,bit1,bit0(aa(num,num,bit1,bit0(aa(num,num,bit1,bit0(bit0(aa(num,num,bit1,bit0(bit0(aa(num,num,bit1,bit0(aa(num,num,bit1,bit0(aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,bit0(bit0(one2))))))))))))))))))))))))))))))))),aa(uint32,uint32,aa(uint32,fun(uint32,uint32),plus_plus(uint32),aa(uint32,uint32,aa(uint32,fun(uint32,uint32),times_times(uint32),hash_hash_code_list(A,H_a,Xa2)),aa(num,uint32,numeral_numeral(uint32),aa(num,num,bit1,bit0(aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,bit0(bit0(aa(num,num,bit1,bit0(aa(num,num,bit1,aa(num,num,bit1,bit0(aa(num,num,bit1,aa(num,num,bit1,bit0(bit0(aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,bit0(bit0(bit0(bit0(aa(num,num,bit1,bit0(bit0(bit0(aa(num,num,bit1,bit0(bit0(one2))))))))))))))))))))))))))))))))),aa(num,uint32,numeral_numeral(uint32),bit0(aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,bit0(bit0(bit0(bit0(aa(num,num,bit1,bit0(aa(num,num,bit1,bit0(bit0(bit0(aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,bit0(bit0(aa(num,num,bit1,bit0(aa(num,num,bit1,aa(num,num,bit1,bit0(bit0(aa(num,num,bit1,bit0(bit0(bit0(bit0(one2))))))))))))))))))))))))))))))))) ).

% hash_code_list_simps(2)
tff(fact_6782_signed__cast_Oabs__eq,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [X: int] : signed_cast(A,B,word2(A,X)) = word2(B,aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat)))),X)) ) ).

% signed_cast.abs_eq
tff(fact_6783_upt__conv__Cons__Cons,axiom,
    ! [M: nat,N: nat,Ns: list(nat),Q2: nat] :
      ( ( aa(list(nat),list(nat),cons(nat,M),aa(list(nat),list(nat),cons(nat,N),Ns)) = upt(M,Q2) )
    <=> ( aa(list(nat),list(nat),cons(nat,N),Ns) = upt(aa(nat,nat,suc,M),Q2) ) ) ).

% upt_conv_Cons_Cons
tff(fact_6784_upt__conv__Cons,axiom,
    ! [I: nat,J: nat] :
      ( pp(aa(nat,bool,ord_less(nat,I),J))
     => ( upt(I,J) = aa(list(nat),list(nat),cons(nat,I),upt(aa(nat,nat,suc,I),J)) ) ) ).

% upt_conv_Cons
tff(fact_6785_upt__eq__Cons__conv,axiom,
    ! [I: nat,J: nat,X: nat,Xs: list(nat)] :
      ( ( upt(I,J) = aa(list(nat),list(nat),cons(nat,X),Xs) )
    <=> ( pp(aa(nat,bool,ord_less(nat,I),J))
        & ( I = X )
        & ( upt(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),one_one(nat)),J) = Xs ) ) ) ).

% upt_eq_Cons_conv
tff(fact_6786_UNIV__nat__eq,axiom,
    top_top(set(nat)) = insert(nat,zero_zero(nat),aa(set(nat),set(nat),image(nat,nat,suc),top_top(set(nat)))) ).

% UNIV_nat_eq
tff(fact_6787_the__signed__int_Oabs__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: int] : the_signed_int(A,word2(A,X)) = aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat)))),X) ) ).

% the_signed_int.abs_eq
tff(fact_6788_of__nat_Oabs__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: nat] : of_nat(A,X) = word2(A,aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,bit_se2584673776208193580ke_bit(nat,type_len0_len_of(A,type2(A))),X))) ) ).

% of_nat.abs_eq
tff(fact_6789_bin__last__set__bits,axiom,
    ! [F2: fun(nat,bool)] :
      ( bit_wf_set_bits_int(F2)
     => ( ~ pp(aa(int,bool,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),bit_bi4170147762399595738t_bits(int,F2)))
      <=> pp(aa(nat,bool,F2,zero_zero(nat))) ) ) ).

% bin_last_set_bits
tff(fact_6790_wf__set__bits__int__Suc,axiom,
    ! [F2: fun(nat,bool)] :
      ( bit_wf_set_bits_int(aTP_Lamp_qd(fun(nat,bool),fun(nat,bool),F2))
    <=> bit_wf_set_bits_int(F2) ) ).

% wf_set_bits_int_Suc
tff(fact_6791_wf__set__bits__int__const,axiom,
    ! [B2: bool] : bit_wf_set_bits_int(aTP_Lamp_qe(bool,fun(nat,bool),B2)) ).

% wf_set_bits_int_const
tff(fact_6792_ones,axiom,
    ! [N: nat,F2: fun(nat,bool)] :
      ( ! [N7: nat] :
          ( pp(aa(nat,bool,ord_less_eq(nat,N),N7))
         => pp(aa(nat,bool,F2,N7)) )
     => bit_wf_set_bits_int(F2) ) ).

% ones
tff(fact_6793_wf__set__bits__int_Ocases,axiom,
    ! [F2: fun(nat,bool)] :
      ( bit_wf_set_bits_int(F2)
     => ( ! [N2: nat] :
            ~ ! [N6: nat] :
                ( pp(aa(nat,bool,ord_less_eq(nat,N2),N6))
               => ~ pp(aa(nat,bool,F2,N6)) )
       => ~ ! [N2: nat] :
              ~ ! [N6: nat] :
                  ( pp(aa(nat,bool,ord_less_eq(nat,N2),N6))
                 => pp(aa(nat,bool,F2,N6)) ) ) ) ).

% wf_set_bits_int.cases
tff(fact_6794_wf__set__bits__int_Osimps,axiom,
    ! [F2: fun(nat,bool)] :
      ( bit_wf_set_bits_int(F2)
    <=> ( ? [N5: nat] :
          ! [N13: nat] :
            ( pp(aa(nat,bool,ord_less_eq(nat,N5),N13))
           => ~ pp(aa(nat,bool,F2,N13)) )
        | ? [N5: nat] :
          ! [N13: nat] :
            ( pp(aa(nat,bool,ord_less_eq(nat,N5),N13))
           => pp(aa(nat,bool,F2,N13)) ) ) ) ).

% wf_set_bits_int.simps
tff(fact_6795_zeros,axiom,
    ! [N: nat,F2: fun(nat,bool)] :
      ( ! [N7: nat] :
          ( pp(aa(nat,bool,ord_less_eq(nat,N),N7))
         => ~ pp(aa(nat,bool,F2,N7)) )
     => bit_wf_set_bits_int(F2) ) ).

% zeros
tff(fact_6796_wf__set__bits__int__simps,axiom,
    ! [F2: fun(nat,bool)] :
      ( bit_wf_set_bits_int(F2)
    <=> ? [N5: nat] :
          ( ! [N13: nat] :
              ( pp(aa(nat,bool,ord_less_eq(nat,N5),N13))
             => ~ pp(aa(nat,bool,F2,N13)) )
          | ! [N13: nat] :
              ( pp(aa(nat,bool,ord_less_eq(nat,N5),N13))
             => pp(aa(nat,bool,F2,N13)) ) ) ) ).

% wf_set_bits_int_simps
tff(fact_6797_root__def,axiom,
    ! [N: nat,X: real] :
      ( ( ( N = zero_zero(nat) )
       => ( aa(real,real,root(N),X) = zero_zero(real) ) )
      & ( ( N != zero_zero(nat) )
       => ( aa(real,real,root(N),X) = the_inv_into(real,real,top_top(set(real)),aTP_Lamp_qf(nat,fun(real,real),N),X) ) ) ) ).

% root_def
tff(fact_6798_length__Cons,axiom,
    ! [A: $tType,X: A,Xs: list(A)] : aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),cons(A,X),Xs)) = aa(nat,nat,suc,aa(list(A),nat,size_size(list(A)),Xs)) ).

% length_Cons
tff(fact_6799_the__inv__into__def,axiom,
    ! [A: $tType,B: $tType,A3: set(A),F2: fun(A,B),X4: B] : the_inv_into(A,B,A3,F2,X4) = the(A,aa(B,fun(A,bool),aa(fun(A,B),fun(B,fun(A,bool)),aTP_Lamp_qg(set(A),fun(fun(A,B),fun(B,fun(A,bool))),A3),F2),X4)) ).

% the_inv_into_def
tff(fact_6800_subset__CollectI,axiom,
    ! [A: $tType,B4: set(A),A3: set(A),Q: fun(A,bool),P: fun(A,bool)] :
      ( pp(aa(set(A),bool,ord_less_eq(set(A),B4),A3))
     => ( ! [X3: A] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),B4))
           => ( pp(aa(A,bool,Q,X3))
             => pp(aa(A,bool,P,X3)) ) )
       => pp(aa(set(A),bool,ord_less_eq(set(A),collect(A,aa(fun(A,bool),fun(A,bool),aTP_Lamp_ae(set(A),fun(fun(A,bool),fun(A,bool)),B4),Q))),collect(A,aa(fun(A,bool),fun(A,bool),aTP_Lamp_ae(set(A),fun(fun(A,bool),fun(A,bool)),A3),P)))) ) ) ).

% subset_CollectI
tff(fact_6801_subset__Collect__iff,axiom,
    ! [A: $tType,B4: set(A),A3: set(A),P: fun(A,bool)] :
      ( pp(aa(set(A),bool,ord_less_eq(set(A),B4),A3))
     => ( pp(aa(set(A),bool,ord_less_eq(set(A),B4),collect(A,aa(fun(A,bool),fun(A,bool),aTP_Lamp_ae(set(A),fun(fun(A,bool),fun(A,bool)),A3),P))))
      <=> ! [X2: A] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X2),B4))
           => pp(aa(A,bool,P,X2)) ) ) ) ).

% subset_Collect_iff
tff(fact_6802_int__set__bits__unfold__BIT,axiom,
    ! [F2: fun(nat,bool)] :
      ( bit_wf_set_bits_int(F2)
     => ( bit_bi4170147762399595738t_bits(int,F2) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(bool,int,zero_neq_one_of_bool(int),aa(nat,bool,F2,zero_zero(nat)))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),bit_bi4170147762399595738t_bits(int,comp(nat,bool,nat,F2,suc)))) ) ) ).

% int_set_bits_unfold_BIT
tff(fact_6803_bit__word__rotl__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [M: nat,W: word(A),N: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(word(A),word_rotl(A,M,W)),N))
        <=> ( pp(aa(nat,bool,ord_less(nat,N),type_len0_len_of(A,type2(A))))
            & pp(aa(nat,bool,bit_se5641148757651400278ts_bit(word(A),W),modulo_modulo(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),aa(nat,nat,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
tff(fact_6804_bin__rest__set__bits,axiom,
    ! [F2: fun(nat,bool)] :
      ( bit_wf_set_bits_int(F2)
     => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),bit_bi4170147762399595738t_bits(int,F2)),aa(num,int,numeral_numeral(int),bit0(one2))) = bit_bi4170147762399595738t_bits(int,comp(nat,bool,nat,F2,suc)) ) ) ).

% bin_rest_set_bits
tff(fact_6805_sum__comp__morphism,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( ( comm_monoid_add(B)
        & comm_monoid_add(A) )
     => ! [H: fun(B,A),G: fun(C,B),A3: set(C)] :
          ( ( aa(B,A,H,zero_zero(B)) = zero_zero(A) )
         => ( ! [X3: B,Y3: B] : aa(B,A,H,aa(B,B,aa(B,fun(B,B),plus_plus(B),X3),Y3)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(B,A,H,X3)),aa(B,A,H,Y3))
           => ( aa(set(C),A,groups7311177749621191930dd_sum(C,A,comp(B,A,C,H,G)),A3) = aa(B,A,H,aa(set(C),B,groups7311177749621191930dd_sum(C,B,G),A3)) ) ) ) ) ).

% sum_comp_morphism
tff(fact_6806_bij__betw__comp__iff2,axiom,
    ! [C: $tType,A: $tType,B: $tType,F7: fun(A,B),A9: set(A),A10: set(B),F2: fun(C,A),A3: set(C)] :
      ( bij_betw(A,B,F7,A9,A10)
     => ( pp(aa(set(A),bool,ord_less_eq(set(A),aa(set(C),set(A),image(C,A,F2),A3)),A9))
       => ( bij_betw(C,A,F2,A3,A9)
        <=> bij_betw(C,B,comp(A,B,C,F7,F2),A3,A10) ) ) ) ).

% bij_betw_comp_iff2
tff(fact_6807_sum_OatLeast__Suc__lessThan__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),M: nat,N: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,M),aa(nat,nat,suc,N))) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,comp(nat,A,nat,G,suc)),set_or7035219750837199246ssThan(nat,M,N)) ) ).

% sum.atLeast_Suc_lessThan_Suc_shift
tff(fact_6808_sum_OatLeast__Suc__atMost__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),M: nat,N: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M),aa(nat,nat,suc,N))) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,comp(nat,A,nat,G,suc)),set_or1337092689740270186AtMost(nat,M,N)) ) ).

% sum.atLeast_Suc_atMost_Suc_shift
tff(fact_6809_bit__drop__bit__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A2: A] : bit_se5641148757651400278ts_bit(A,bit_se4197421643247451524op_bit(A,N,A2)) = comp(nat,bool,nat,bit_se5641148757651400278ts_bit(A,A2),aa(nat,fun(nat,nat),plus_plus(nat),N)) ) ).

% bit_drop_bit_eq
tff(fact_6810_sum_OatLeastLessThan__shift__bounds,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),M: nat,K: nat,N: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or7035219750837199246ssThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),K),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),K))) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,comp(nat,A,nat,G,aa(nat,fun(nat,nat),plus_plus(nat),K))),set_or7035219750837199246ssThan(nat,M,N)) ) ).

% sum.atLeastLessThan_shift_bounds
tff(fact_6811_sum_OatLeastAtMost__shift__bounds,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),M: nat,K: nat,N: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),K),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),K))) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,comp(nat,A,nat,G,aa(nat,fun(nat,nat),plus_plus(nat),K))),set_or1337092689740270186AtMost(nat,M,N)) ) ).

% sum.atLeastAtMost_shift_bounds
tff(fact_6812_prod_OatLeast__Suc__lessThan__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),M: nat,N: nat] : groups7121269368397514597t_prod(nat,A,G,set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,M),aa(nat,nat,suc,N))) = groups7121269368397514597t_prod(nat,A,comp(nat,A,nat,G,suc),set_or7035219750837199246ssThan(nat,M,N)) ) ).

% prod.atLeast_Suc_lessThan_Suc_shift
tff(fact_6813_prod_OatLeast__Suc__atMost__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),M: nat,N: nat] : groups7121269368397514597t_prod(nat,A,G,set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M),aa(nat,nat,suc,N))) = groups7121269368397514597t_prod(nat,A,comp(nat,A,nat,G,suc),set_or1337092689740270186AtMost(nat,M,N)) ) ).

% prod.atLeast_Suc_atMost_Suc_shift
tff(fact_6814_prod_OatLeastLessThan__shift__bounds,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),M: nat,K: nat,N: nat] : groups7121269368397514597t_prod(nat,A,G,set_or7035219750837199246ssThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),K),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),K))) = groups7121269368397514597t_prod(nat,A,comp(nat,A,nat,G,aa(nat,fun(nat,nat),plus_plus(nat),K)),set_or7035219750837199246ssThan(nat,M,N)) ) ).

% prod.atLeastLessThan_shift_bounds
tff(fact_6815_prod_OatLeastAtMost__shift__bounds,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),M: nat,K: nat,N: nat] : groups7121269368397514597t_prod(nat,A,G,set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),K),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),K))) = groups7121269368397514597t_prod(nat,A,comp(nat,A,nat,G,aa(nat,fun(nat,nat),plus_plus(nat),K)),set_or1337092689740270186AtMost(nat,M,N)) ) ).

% prod.atLeastAtMost_shift_bounds
tff(fact_6816_summable__inverse__divide,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(nat,A),C2: A] :
          ( summable(A,comp(A,A,nat,inverse_inverse(A),F2))
         => summable(A,aa(A,fun(nat,A),aTP_Lamp_qh(fun(nat,A),fun(A,fun(nat,A)),F2),C2)) ) ) ).

% summable_inverse_divide
tff(fact_6817_sum_Oreindex__nontrivial,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [A3: set(B),H: fun(B,C),G: fun(C,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),A3))
         => ( ! [X3: B,Y3: B] :
                ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),A3))
               => ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),Y3),A3))
                 => ( ( X3 != Y3 )
                   => ( ( aa(B,C,H,X3) = aa(B,C,H,Y3) )
                     => ( aa(C,A,G,aa(B,C,H,X3)) = zero_zero(A) ) ) ) ) )
           => ( aa(set(C),A,groups7311177749621191930dd_sum(C,A,G),aa(set(B),set(C),image(B,C,H),A3)) = aa(set(B),A,groups7311177749621191930dd_sum(B,A,comp(C,A,B,G,H)),A3) ) ) ) ) ).

% sum.reindex_nontrivial
tff(fact_6818_prod_Oreindex__nontrivial,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [A3: set(B),H: fun(B,C),G: fun(C,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),A3))
         => ( ! [X3: B,Y3: B] :
                ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),A3))
               => ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),Y3),A3))
                 => ( ( X3 != Y3 )
                   => ( ( aa(B,C,H,X3) = aa(B,C,H,Y3) )
                     => ( aa(C,A,G,aa(B,C,H,X3)) = one_one(A) ) ) ) ) )
           => ( groups7121269368397514597t_prod(C,A,G,aa(set(B),set(C),image(B,C,H),A3)) = groups7121269368397514597t_prod(B,A,comp(C,A,B,G,H),A3) ) ) ) ) ).

% prod.reindex_nontrivial
tff(fact_6819_sum__image__le,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( ordere6911136660526730532id_add(B)
     => ! [I5: set(C),G: fun(A,B),F2: fun(C,A)] :
          ( pp(aa(set(C),bool,finite_finite2(C),I5))
         => ( ! [I2: C] :
                ( pp(aa(set(C),bool,aa(C,fun(set(C),bool),member(C),I2),I5))
               => pp(aa(B,bool,ord_less_eq(B,zero_zero(B)),aa(A,B,G,aa(C,A,F2,I2)))) )
           => pp(aa(B,bool,ord_less_eq(B,aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),aa(set(C),set(A),image(C,A,F2),I5))),aa(set(C),B,groups7311177749621191930dd_sum(C,B,comp(A,B,C,G,F2)),I5))) ) ) ) ).

% sum_image_le
tff(fact_6820_sum_OatLeast0__lessThan__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),N: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G,zero_zero(nat))),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,comp(nat,A,nat,G,suc)),set_or7035219750837199246ssThan(nat,zero_zero(nat),N))) ) ).

% sum.atLeast0_lessThan_Suc_shift
tff(fact_6821_sum_OatLeast0__atMost__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),N: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G,zero_zero(nat))),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,comp(nat,A,nat,G,suc)),set_or1337092689740270186AtMost(nat,zero_zero(nat),N))) ) ).

% sum.atLeast0_atMost_Suc_shift
tff(fact_6822_prod_OatLeast0__lessThan__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),N: nat] : groups7121269368397514597t_prod(nat,A,G,set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G,zero_zero(nat))),groups7121269368397514597t_prod(nat,A,comp(nat,A,nat,G,suc),set_or7035219750837199246ssThan(nat,zero_zero(nat),N))) ) ).

% prod.atLeast0_lessThan_Suc_shift
tff(fact_6823_prod_OatLeast0__atMost__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),N: nat] : groups7121269368397514597t_prod(nat,A,G,set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G,zero_zero(nat))),groups7121269368397514597t_prod(nat,A,comp(nat,A,nat,G,suc),set_or1337092689740270186AtMost(nat,zero_zero(nat),N))) ) ).

% prod.atLeast0_atMost_Suc_shift
tff(fact_6824_sum_OatLeastLessThan__shift__0,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),M: nat,N: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or7035219750837199246ssThan(nat,M,N)) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,comp(nat,A,nat,G,aa(nat,fun(nat,nat),plus_plus(nat),M))),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,minus_minus(nat,N),M))) ) ).

% sum.atLeastLessThan_shift_0
tff(fact_6825_prod_OatLeastLessThan__shift__0,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),M: nat,N: nat] : groups7121269368397514597t_prod(nat,A,G,set_or7035219750837199246ssThan(nat,M,N)) = groups7121269368397514597t_prod(nat,A,comp(nat,A,nat,G,aa(nat,fun(nat,nat),plus_plus(nat),M)),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,minus_minus(nat,N),M))) ) ).

% prod.atLeastLessThan_shift_0
tff(fact_6826_sum_OatLeast__lessThan__pred__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),M: nat,N: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,comp(nat,A,nat,G,aTP_Lamp_ot(nat,nat))),set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,M),aa(nat,nat,suc,N))) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or7035219750837199246ssThan(nat,M,N)) ) ).

% sum.atLeast_lessThan_pred_shift
tff(fact_6827_sum_OatLeast__atMost__pred__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),M: nat,N: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,comp(nat,A,nat,G,aTP_Lamp_ot(nat,nat))),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M),aa(nat,nat,suc,N))) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or1337092689740270186AtMost(nat,M,N)) ) ).

% sum.atLeast_atMost_pred_shift
tff(fact_6828_prod_OatLeast__lessThan__pred__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),M: nat,N: nat] : groups7121269368397514597t_prod(nat,A,comp(nat,A,nat,G,aTP_Lamp_ot(nat,nat)),set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,M),aa(nat,nat,suc,N))) = groups7121269368397514597t_prod(nat,A,G,set_or7035219750837199246ssThan(nat,M,N)) ) ).

% prod.atLeast_lessThan_pred_shift
tff(fact_6829_prod_OatLeast__atMost__pred__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),M: nat,N: nat] : groups7121269368397514597t_prod(nat,A,comp(nat,A,nat,G,aTP_Lamp_ot(nat,nat)),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M),aa(nat,nat,suc,N))) = groups7121269368397514597t_prod(nat,A,G,set_or1337092689740270186AtMost(nat,M,N)) ) ).

% prod.atLeast_atMost_pred_shift
tff(fact_6830_sum_OatLeastAtMost__shift__0,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [M: nat,N: nat,G: fun(nat,A)] :
          ( pp(aa(nat,bool,ord_less_eq(nat,M),N))
         => ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or1337092689740270186AtMost(nat,M,N)) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,comp(nat,A,nat,G,aa(nat,fun(nat,nat),plus_plus(nat),M))),set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,minus_minus(nat,N),M))) ) ) ) ).

% sum.atLeastAtMost_shift_0
tff(fact_6831_prod_OatLeastAtMost__shift__0,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [M: nat,N: nat,G: fun(nat,A)] :
          ( pp(aa(nat,bool,ord_less_eq(nat,M),N))
         => ( groups7121269368397514597t_prod(nat,A,G,set_or1337092689740270186AtMost(nat,M,N)) = groups7121269368397514597t_prod(nat,A,comp(nat,A,nat,G,aa(nat,fun(nat,nat),plus_plus(nat),M)),set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,minus_minus(nat,N),M))) ) ) ) ).

% prod.atLeastAtMost_shift_0
tff(fact_6832_word__roti__eq__word__rotr__word__rotl,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [I: int,W: word(A)] :
          ( ( pp(aa(int,bool,ord_less_eq(int,zero_zero(int)),I))
           => ( word_roti(A,I,W) = word_rotr(A,nat2(I),W) ) )
          & ( ~ pp(aa(int,bool,ord_less_eq(int,zero_zero(int)),I))
           => ( word_roti(A,I,W) = word_rotl(A,nat2(aa(int,int,uminus_uminus(int),I)),W) ) ) ) ) ).

% word_roti_eq_word_rotr_word_rotl
tff(fact_6833_CHAR__pos__iff,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),semiri4206861660011772517g_char(A,type2(A))))
      <=> ? [N5: nat] :
            ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),N5))
            & ( aa(nat,A,semiring_1_of_nat(A),N5) = zero_zero(A) ) ) ) ) ).

% CHAR_pos_iff
tff(fact_6834_CHAR__eq__0,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ( semiri4206861660011772517g_char(A,type2(A)) = zero_zero(nat) ) ) ).

% CHAR_eq_0
tff(fact_6835_of__nat__CHAR,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ( aa(nat,A,semiring_1_of_nat(A),semiri4206861660011772517g_char(A,type2(A))) = zero_zero(A) ) ) ).

% of_nat_CHAR
tff(fact_6836_card_Ocomp__fun__commute__on,axiom,
    comp(nat,nat,nat,suc,suc) = comp(nat,nat,nat,suc,suc) ).

% card.comp_fun_commute_on
tff(fact_6837_word__rotr__word__rotr__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [M: nat,N: nat,W: word(A)] : word_rotr(A,M,word_rotr(A,N,W)) = word_rotr(A,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N),W) ) ).

% word_rotr_word_rotr_eq
tff(fact_6838_word__rotx__0,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(A) )
     => ! [I: nat] :
          ( ( word_rotr(A,I,zero_zero(word(A))) = zero_zero(word(A)) )
          & ( word_rotl(B,I,zero_zero(word(B))) = zero_zero(word(B)) ) ) ) ).

% word_rotx_0
tff(fact_6839_sum_OatLeast__int__atMost__int__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(int,A),M: nat,N: nat] : aa(set(int),A,groups7311177749621191930dd_sum(int,A,G),set_or1337092689740270186AtMost(int,aa(nat,int,semiring_1_of_nat(int),M),aa(nat,int,semiring_1_of_nat(int),N))) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,comp(int,A,nat,G,semiring_1_of_nat(int))),set_or1337092689740270186AtMost(nat,M,N)) ) ).

% sum.atLeast_int_atMost_int_shift
tff(fact_6840_sum_OatLeast__int__lessThan__int__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(int,A),M: nat,N: nat] : aa(set(int),A,groups7311177749621191930dd_sum(int,A,G),set_or7035219750837199246ssThan(int,aa(nat,int,semiring_1_of_nat(int),M),aa(nat,int,semiring_1_of_nat(int),N))) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,comp(int,A,nat,G,semiring_1_of_nat(int))),set_or7035219750837199246ssThan(nat,M,N)) ) ).

% sum.atLeast_int_lessThan_int_shift
tff(fact_6841_prod_OatLeast__int__atMost__int__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(int,A),M: nat,N: nat] : groups7121269368397514597t_prod(int,A,G,set_or1337092689740270186AtMost(int,aa(nat,int,semiring_1_of_nat(int),M),aa(nat,int,semiring_1_of_nat(int),N))) = groups7121269368397514597t_prod(nat,A,comp(int,A,nat,G,semiring_1_of_nat(int)),set_or1337092689740270186AtMost(nat,M,N)) ) ).

% prod.atLeast_int_atMost_int_shift
tff(fact_6842_prod_OatLeast__int__lessThan__int__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(int,A),M: nat,N: nat] : groups7121269368397514597t_prod(int,A,G,set_or7035219750837199246ssThan(int,aa(nat,int,semiring_1_of_nat(int),M),aa(nat,int,semiring_1_of_nat(int),N))) = groups7121269368397514597t_prod(nat,A,comp(int,A,nat,G,semiring_1_of_nat(int)),set_or7035219750837199246ssThan(nat,M,N)) ) ).

% prod.atLeast_int_lessThan_int_shift
tff(fact_6843_of__nat__eq__0__iff__char__dvd,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [N: nat] :
          ( ( aa(nat,A,semiring_1_of_nat(A),N) = zero_zero(A) )
        <=> pp(aa(nat,bool,dvd_dvd(nat,semiri4206861660011772517g_char(A,type2(A))),N)) ) ) ).

% of_nat_eq_0_iff_char_dvd
tff(fact_6844_CHAR__eqI,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [C2: nat] :
          ( ( aa(nat,A,semiring_1_of_nat(A),C2) = zero_zero(A) )
         => ( ! [X3: nat] :
                ( ( aa(nat,A,semiring_1_of_nat(A),X3) = zero_zero(A) )
               => pp(aa(nat,bool,dvd_dvd(nat,C2),X3)) )
           => ( semiri4206861660011772517g_char(A,type2(A)) = C2 ) ) ) ) ).

% CHAR_eqI
tff(fact_6845_bit__word__rotr__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [M: nat,W: word(A),N: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(word(A),word_rotr(A,M,W)),N))
        <=> ( pp(aa(nat,bool,ord_less(nat,N),type_len0_len_of(A,type2(A))))
            & pp(aa(nat,bool,bit_se5641148757651400278ts_bit(word(A),W),modulo_modulo(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),M),type_len0_len_of(A,type2(A))))) ) ) ) ).

% bit_word_rotr_iff
tff(fact_6846_CHAR__eq0__iff,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ( ( semiri4206861660011772517g_char(A,type2(A)) = zero_zero(nat) )
      <=> ! [N5: nat] :
            ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),N5))
           => ( aa(nat,A,semiring_1_of_nat(A),N5) != zero_zero(A) ) ) ) ) ).

% CHAR_eq0_iff
tff(fact_6847_CHAR__eq__posI,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [C2: nat] :
          ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),C2))
         => ( ( aa(nat,A,semiring_1_of_nat(A),C2) = zero_zero(A) )
           => ( ! [X3: nat] :
                  ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),X3))
                 => ( pp(aa(nat,bool,ord_less(nat,X3),C2))
                   => ( aa(nat,A,semiring_1_of_nat(A),X3) != zero_zero(A) ) ) )
             => ( semiri4206861660011772517g_char(A,type2(A)) = C2 ) ) ) ) ) ).

% CHAR_eq_posI
tff(fact_6848_Code__Target__Int_Onegative__def,axiom,
    code_Target_negative = comp(int,int,num,uminus_uminus(int),numeral_numeral(int)) ).

% Code_Target_Int.negative_def
tff(fact_6849_bit__sshiftr__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),M: nat,N: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(word(A),bit_Sh8784991116023147202shiftr(A,W,M)),N))
        <=> pp(aa(nat,bool,bit_se5641148757651400278ts_bit(word(A),W),if(nat,fconj(aa(nat,bool,ord_less_eq(nat,aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),M)),N),aa(nat,bool,ord_less(nat,N),type_len0_len_of(A,type2(A)))),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N)))) ) ) ).

% bit_sshiftr_iff
tff(fact_6850_sshiftr__of__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] : bit_Sh8784991116023147202shiftr(A,W,zero_zero(nat)) = W ) ).

% sshiftr_of_0
tff(fact_6851_sshiftr__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: nat] : bit_Sh8784991116023147202shiftr(A,zero_zero(word(A)),N) = zero_zero(word(A)) ) ).

% sshiftr_0
tff(fact_6852_sshiftr__numeral__Suc,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [M: num,N: nat] : bit_Sh8784991116023147202shiftr(A,aa(num,word(A),numeral_numeral(word(A)),M),aa(nat,nat,suc,N)) = signed_drop_bit(A,aa(nat,nat,suc,N),aa(num,word(A),numeral_numeral(word(A)),M)) ) ).

% sshiftr_numeral_Suc
tff(fact_6853_sshiftr__numeral__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [M: num,N: num] : bit_Sh8784991116023147202shiftr(A,aa(num,word(A),numeral_numeral(word(A)),M),aa(num,nat,numeral_numeral(nat),N)) = signed_drop_bit(A,aa(num,nat,numeral_numeral(nat),N),aa(num,word(A),numeral_numeral(word(A)),M)) ) ).

% sshiftr_numeral_numeral
tff(fact_6854_sshiftr__minus__numeral__Suc,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [M: num,N: nat] : bit_Sh8784991116023147202shiftr(A,aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),M)),aa(nat,nat,suc,N)) = signed_drop_bit(A,aa(nat,nat,suc,N),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),M))) ) ).

% sshiftr_minus_numeral_Suc
tff(fact_6855_sshiftr__minus__numeral__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [M: num,N: num] : bit_Sh8784991116023147202shiftr(A,aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),M)),aa(num,nat,numeral_numeral(nat),N)) = signed_drop_bit(A,aa(num,nat,numeral_numeral(nat),N),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),M))) ) ).

% sshiftr_minus_numeral_numeral
tff(fact_6856_sshiftr__1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: nat] : bit_Sh8784991116023147202shiftr(A,one_one(word(A)),N) = aa(bool,word(A),zero_neq_one_of_bool(word(A)),fdisj(aa(nat,bool,fequal(nat,type_len0_len_of(A,type2(A))),one_one(nat)),aa(nat,bool,fequal(nat,N),zero_zero(nat)))) ) ).

% sshiftr_1
tff(fact_6857_shiftl__Suc__0,axiom,
    ! [N: nat] : bit_Sh4282982442137083160shiftl(nat,aa(nat,nat,suc,zero_zero(nat)),N) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N) ).

% shiftl_Suc_0
tff(fact_6858_shiftl__minus__1__numeral,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [N: num] : bit_Sh4282982442137083160shiftl(A,aa(A,A,uminus_uminus(A),one_one(A)),aa(num,nat,numeral_numeral(nat),N)) = aa(A,A,bit_ri4277139882892585799ns_not(A),bit_se2239418461657761734s_mask(A,aa(num,nat,numeral_numeral(nat),N))) ) ).

% shiftl_minus_1_numeral
tff(fact_6859_shiftl__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat] : bit_Sh4282982442137083160shiftl(A,zero_zero(A),N) = zero_zero(A) ) ).

% shiftl_0
tff(fact_6860_shiftl__of__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A] : bit_Sh4282982442137083160shiftl(A,A2,zero_zero(nat)) = A2 ) ).

% shiftl_of_0
tff(fact_6861_shiftl__numeral__Suc,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: num,N: nat] : bit_Sh4282982442137083160shiftl(A,aa(num,A,numeral_numeral(A),M),aa(nat,nat,suc,N)) = bit_se4730199178511100633sh_bit(A,aa(nat,nat,suc,N),aa(num,A,numeral_numeral(A),M)) ) ).

% shiftl_numeral_Suc
tff(fact_6862_shiftl__numeral__numeral,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: num,N: num] : bit_Sh4282982442137083160shiftl(A,aa(num,A,numeral_numeral(A),M),aa(num,nat,numeral_numeral(nat),N)) = bit_se4730199178511100633sh_bit(A,aa(num,nat,numeral_numeral(nat),N),aa(num,A,numeral_numeral(A),M)) ) ).

% shiftl_numeral_numeral
tff(fact_6863_shiftl__minus__numeral__Suc,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [M: num,N: nat] : bit_Sh4282982442137083160shiftl(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M)),aa(nat,nat,suc,N)) = bit_se4730199178511100633sh_bit(A,aa(nat,nat,suc,N),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))) ) ).

% shiftl_minus_numeral_Suc
tff(fact_6864_shiftl__minus__numeral__numeral,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [M: num,N: num] : bit_Sh4282982442137083160shiftl(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M)),aa(num,nat,numeral_numeral(nat),N)) = bit_se4730199178511100633sh_bit(A,aa(num,nat,numeral_numeral(nat),N),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))) ) ).

% shiftl_minus_numeral_numeral
tff(fact_6865_shiftl__of__Suc,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,N: nat] : bit_Sh4282982442137083160shiftl(A,A2,aa(nat,nat,suc,N)) = bit_Sh4282982442137083160shiftl(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(num,A,numeral_numeral(A),bit0(one2))),N) ) ).

% shiftl_of_Suc
tff(fact_6866_shiftl__1,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat] : bit_Sh4282982442137083160shiftl(A,one_one(A),N) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N) ) ).

% shiftl_1
tff(fact_6867_one__bit__shiftl,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: nat] : generi7602027413899671122et_bit(word(A),zero_zero(word(A)),N,fTrue) = bit_Sh4282982442137083160shiftl(word(A),one_one(word(A)),N) ) ).

% one_bit_shiftl
tff(fact_6868_shiftl__eq__mult,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [X: A,N: nat] : bit_Sh4282982442137083160shiftl(A,X,N) = aa(A,A,aa(A,fun(A,A),times_times(A),X),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N)) ) ).

% shiftl_eq_mult
tff(fact_6869_bit__shiftl__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,M: nat,N: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,bit_Sh4282982442137083160shiftl(A,A2,M)),N))
        <=> ( pp(aa(nat,bool,ord_less_eq(nat,M),N))
            & pp(bit_se6407376104438227557le_bit(A,type2(A),N))
            & pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,A2),aa(nat,nat,minus_minus(nat,N),M))) ) ) ) ).

% bit_shiftl_iff
tff(fact_6870_shiftr__minus__numeral__numeral,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [M: num,N: num] : bit_Sh4282982442137083166shiftr(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M)),aa(num,nat,numeral_numeral(nat),N)) = bit_se4197421643247451524op_bit(A,aa(num,nat,numeral_numeral(nat),N),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))) ) ).

% shiftr_minus_numeral_numeral
tff(fact_6871_shiftr__minus__numeral__Suc,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [M: num,N: nat] : bit_Sh4282982442137083166shiftr(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M)),aa(nat,nat,suc,N)) = bit_se4197421643247451524op_bit(A,aa(nat,nat,suc,N),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))) ) ).

% shiftr_minus_numeral_Suc
tff(fact_6872_shiftr__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat] : bit_Sh4282982442137083166shiftr(A,zero_zero(A),N) = zero_zero(A) ) ).

% shiftr_0
tff(fact_6873_shiftr__of__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A] : bit_Sh4282982442137083166shiftr(A,A2,zero_zero(nat)) = A2 ) ).

% shiftr_of_0
tff(fact_6874_shiftr__Suc__0,axiom,
    ! [N: nat] : bit_Sh4282982442137083166shiftr(nat,aa(nat,nat,suc,zero_zero(nat)),N) = aa(bool,nat,zero_neq_one_of_bool(nat),aa(nat,bool,fequal(nat,N),zero_zero(nat))) ).

% shiftr_Suc_0
tff(fact_6875_shiftr__numeral__Suc,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: num,N: nat] : bit_Sh4282982442137083166shiftr(A,aa(num,A,numeral_numeral(A),M),aa(nat,nat,suc,N)) = bit_se4197421643247451524op_bit(A,aa(nat,nat,suc,N),aa(num,A,numeral_numeral(A),M)) ) ).

% shiftr_numeral_Suc
tff(fact_6876_shiftr__numeral__numeral,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: num,N: num] : bit_Sh4282982442137083166shiftr(A,aa(num,A,numeral_numeral(A),M),aa(num,nat,numeral_numeral(nat),N)) = bit_se4197421643247451524op_bit(A,aa(num,nat,numeral_numeral(nat),N),aa(num,A,numeral_numeral(A),M)) ) ).

% shiftr_numeral_numeral
tff(fact_6877_shiftr__1,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat] : bit_Sh4282982442137083166shiftr(A,one_one(A),N) = aa(bool,A,zero_neq_one_of_bool(A),aa(nat,bool,fequal(nat,N),zero_zero(nat))) ) ).

% shiftr_1
tff(fact_6878_bit__shiftr__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,N: nat] : bit_se5641148757651400278ts_bit(A,bit_Sh4282982442137083166shiftr(A,A2,N)) = comp(nat,bool,nat,bit_se5641148757651400278ts_bit(A,A2),aa(nat,fun(nat,nat),plus_plus(nat),N)) ) ).

% bit_shiftr_eq
tff(fact_6879_shiftr__eq__div,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [X: A,N: nat] : bit_Sh4282982442137083166shiftr(A,X,N) = aa(A,A,aa(A,fun(A,A),divide_divide(A),X),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N)) ) ).

% shiftr_eq_div
tff(fact_6880_bit__revcast__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [W: word(A),N: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(word(B),revcast(A,B,W)),N))
        <=> ( pp(aa(nat,bool,ord_less_eq(nat,aa(nat,nat,minus_minus(nat,type_len0_len_of(B,type2(B))),type_len0_len_of(A,type2(A)))),N))
            & pp(aa(nat,bool,ord_less(nat,N),type_len0_len_of(B,type2(B))))
            & pp(aa(nat,bool,bit_se5641148757651400278ts_bit(word(A),W),aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),type_len0_len_of(B,type2(B))))),aa(nat,nat,minus_minus(nat,type_len0_len_of(B,type2(B))),type_len0_len_of(A,type2(A)))))) ) ) ) ).

% bit_revcast_iff
tff(fact_6881_upto__aux__rec,axiom,
    ! [J: int,I: int,Js: list(int)] :
      ( ( pp(aa(int,bool,ord_less(int,J),I))
       => ( upto_aux(I,J,Js) = Js ) )
      & ( ~ pp(aa(int,bool,ord_less(int,J),I))
       => ( upto_aux(I,J,Js) = upto_aux(I,aa(int,int,minus_minus(int,J),one_one(int)),aa(list(int),list(int),cons(int,J),Js)) ) ) ) ).

% upto_aux_rec
tff(fact_6882_word__msb__neg__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: num] :
          ( pp(most_s684356279273892711sb_msb(word(A),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),W))))
        <=> pp(aa(nat,bool,bit_se5641148757651400278ts_bit(int,aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),W))),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat)))) ) ) ).

% word_msb_neg_numeral
tff(fact_6883_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),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A))))) ) ).

% inj_on_word_of_int
tff(fact_6884_inj__mult__left,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [A2: A] :
          ( inj_on(A,A,aa(A,fun(A,A),times_times(A),A2),top_top(set(A)))
        <=> ( A2 != zero_zero(A) ) ) ) ).

% inj_mult_left
tff(fact_6885_inj__divide__right,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A2: A] :
          ( inj_on(A,A,aTP_Lamp_qi(A,fun(A,A),A2),top_top(set(A)))
        <=> ( A2 != zero_zero(A) ) ) ) ).

% inj_divide_right
tff(fact_6886_word__msb__1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( pp(most_s684356279273892711sb_msb(word(A),one_one(word(A))))
      <=> ( type_len0_len_of(A,type2(A)) = one_one(nat) ) ) ) ).

% word_msb_1
tff(fact_6887_word__msb__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: num] :
          ( pp(most_s684356279273892711sb_msb(word(A),aa(num,word(A),numeral_numeral(word(A)),W)))
        <=> pp(aa(nat,bool,bit_se5641148757651400278ts_bit(int,aa(num,int,numeral_numeral(int),W)),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat)))) ) ) ).

% word_msb_numeral
tff(fact_6888_msb__word__iff__sless__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] :
          ( pp(most_s684356279273892711sb_msb(word(A),W))
        <=> word_sless(A,W,zero_zero(word(A))) ) ) ).

% msb_word_iff_sless_0
tff(fact_6889_finite__inverse__image__gen,axiom,
    ! [A: $tType,B: $tType,A3: set(A),F2: fun(B,A),D4: set(B)] :
      ( pp(aa(set(A),bool,finite_finite2(A),A3))
     => ( inj_on(B,A,F2,D4)
       => pp(aa(set(B),bool,finite_finite2(B),collect(B,aa(set(B),fun(B,bool),aa(fun(B,A),fun(set(B),fun(B,bool)),aTP_Lamp_qj(set(A),fun(fun(B,A),fun(set(B),fun(B,bool))),A3),F2),D4)))) ) ) ).

% finite_inverse_image_gen
tff(fact_6890_linorder__inj__onI,axiom,
    ! [B: $tType,A: $tType] :
      ( order(A)
     => ! [A3: set(A),F2: fun(A,B)] :
          ( ! [X3: A,Y3: A] :
              ( pp(aa(A,bool,ord_less(A,X3),Y3))
             => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),A3))
               => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Y3),A3))
                 => ( aa(A,B,F2,X3) != aa(A,B,F2,Y3) ) ) ) )
         => ( ! [X3: A,Y3: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),A3))
               => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Y3),A3))
                 => ( pp(aa(A,bool,ord_less_eq(A,X3),Y3))
                    | pp(aa(A,bool,ord_less_eq(A,Y3),X3)) ) ) )
           => inj_on(A,B,F2,A3) ) ) ) ).

% linorder_inj_onI
tff(fact_6891_inj__on__subset,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),A3: set(A),B4: set(A)] :
      ( inj_on(A,B,F2,A3)
     => ( pp(aa(set(A),bool,ord_less_eq(set(A),B4),A3))
       => inj_on(A,B,F2,B4) ) ) ).

% inj_on_subset
tff(fact_6892_subset__inj__on,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),B4: set(A),A3: set(A)] :
      ( inj_on(A,B,F2,B4)
     => ( pp(aa(set(A),bool,ord_less_eq(set(A),A3),B4))
       => inj_on(A,B,F2,A3) ) ) ).

% subset_inj_on
tff(fact_6893_word__msb__n1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => pp(most_s684356279273892711sb_msb(word(A),aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A))))) ) ).

% word_msb_n1
tff(fact_6894_word__msb__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ~ pp(most_s684356279273892711sb_msb(word(A),zero_zero(word(A)))) ) ).

% word_msb_0
tff(fact_6895_inj__on__mult,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [A2: A,A3: set(A)] :
          ( ( A2 != zero_zero(A) )
         => inj_on(A,A,aa(A,fun(A,A),times_times(A),A2),A3) ) ) ).

% inj_on_mult
tff(fact_6896_inj__on__add,axiom,
    ! [A: $tType] :
      ( cancel_semigroup_add(A)
     => ! [A2: A,A3: set(A)] : inj_on(A,A,aa(A,fun(A,A),plus_plus(A),A2),A3) ) ).

% inj_on_add
tff(fact_6897_inj__on__add_H,axiom,
    ! [A: $tType] :
      ( cancel_semigroup_add(A)
     => ! [A2: A,A3: set(A)] : inj_on(A,A,aTP_Lamp_qk(A,fun(A,A),A2),A3) ) ).

% inj_on_add'
tff(fact_6898_inj__on__id2,axiom,
    ! [A: $tType,A3: set(A)] : inj_on(A,A,aTP_Lamp_at(A,A),A3) ).

% inj_on_id2
tff(fact_6899_word__msb__sint,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] :
          ( pp(most_s684356279273892711sb_msb(word(A),W))
        <=> pp(aa(int,bool,ord_less(int,ring_1_signed(A,int,W)),zero_zero(int))) ) ) ).

% word_msb_sint
tff(fact_6900_inj__on__image__eq__iff,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),C6: set(A),A3: set(A),B4: set(A)] :
      ( inj_on(A,B,F2,C6)
     => ( pp(aa(set(A),bool,ord_less_eq(set(A),A3),C6))
       => ( pp(aa(set(A),bool,ord_less_eq(set(A),B4),C6))
         => ( ( aa(set(A),set(B),image(A,B,F2),A3) = aa(set(A),set(B),image(A,B,F2),B4) )
          <=> ( A3 = B4 ) ) ) ) ) ).

% inj_on_image_eq_iff
tff(fact_6901_inj__on__image__mem__iff,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),B4: set(A),A2: A,A3: set(A)] :
      ( inj_on(A,B,F2,B4)
     => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),B4))
       => ( pp(aa(set(A),bool,ord_less_eq(set(A),A3),B4))
         => ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),aa(A,B,F2,A2)),aa(set(A),set(B),image(A,B,F2),A3)))
          <=> pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),A3)) ) ) ) ) ).

% inj_on_image_mem_iff
tff(fact_6902_subset__image__inj,axiom,
    ! [A: $tType,B: $tType,S: set(A),F2: fun(B,A),T3: set(B)] :
      ( pp(aa(set(A),bool,ord_less_eq(set(A),S),aa(set(B),set(A),image(B,A,F2),T3)))
    <=> ? [U2: set(B)] :
          ( pp(aa(set(B),bool,ord_less_eq(set(B),U2),T3))
          & inj_on(B,A,F2,U2)
          & ( S = aa(set(B),set(A),image(B,A,F2),U2) ) ) ) ).

% subset_image_inj
tff(fact_6903_inj__diff__right,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A2: A] : inj_on(A,A,aTP_Lamp_ol(A,fun(A,A),A2),top_top(set(A))) ) ).

% inj_diff_right
tff(fact_6904_inj__fun,axiom,
    ! [B: $tType,C: $tType,A: $tType,F2: fun(A,B)] :
      ( inj_on(A,B,F2,top_top(set(A)))
     => inj_on(A,fun(C,B),aTP_Lamp_ql(fun(A,B),fun(A,fun(C,B)),F2),top_top(set(A))) ) ).

% inj_fun
tff(fact_6905_inj__add__left,axiom,
    ! [A: $tType] :
      ( cancel_semigroup_add(A)
     => ! [A2: A] : inj_on(A,A,aa(A,fun(A,A),plus_plus(A),A2),top_top(set(A))) ) ).

% inj_add_left
tff(fact_6906_sorted__list__of__set_Oinj__on,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => inj_on(A,A,aTP_Lamp_qm(A,A),top_top(set(A))) ) ).

% sorted_list_of_set.inj_on
tff(fact_6907_finite__Collect,axiom,
    ! [A: $tType,B: $tType,S: set(A),F2: fun(B,A)] :
      ( pp(aa(set(A),bool,finite_finite2(A),S))
     => ( inj_on(B,A,F2,top_top(set(B)))
       => pp(aa(set(B),bool,finite_finite2(B),collect(B,aa(fun(B,A),fun(B,bool),aTP_Lamp_qn(set(A),fun(fun(B,A),fun(B,bool)),S),F2)))) ) ) ).

% finite_Collect
tff(fact_6908_finite__inverse__image,axiom,
    ! [A: $tType,B: $tType,A3: set(A),F2: fun(B,A)] :
      ( pp(aa(set(A),bool,finite_finite2(A),A3))
     => ( inj_on(B,A,F2,top_top(set(B)))
       => pp(aa(set(B),bool,finite_finite2(B),collect(B,aa(fun(B,A),fun(B,bool),aTP_Lamp_qn(set(A),fun(fun(B,A),fun(B,bool)),A3),F2)))) ) ) ).

% finite_inverse_image
tff(fact_6909_prod_Oimage__eq,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(B,A),A3: set(B)] :
          ( inj_on(B,A,G,A3)
         => ( groups7121269368397514597t_prod(A,A,aTP_Lamp_qo(A,A),aa(set(B),set(A),image(B,A,G),A3)) = groups7121269368397514597t_prod(B,A,G,A3) ) ) ) ).

% prod.image_eq
tff(fact_6910_sum_Oimage__eq,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(B,A),A3: set(B)] :
          ( inj_on(B,A,G,A3)
         => ( aa(set(A),A,groups7311177749621191930dd_sum(A,A,aTP_Lamp_qp(A,A)),aa(set(B),set(A),image(B,A,G),A3)) = aa(set(B),A,groups7311177749621191930dd_sum(B,A,G),A3) ) ) ) ).

% sum.image_eq
tff(fact_6911_inj__on__iff__surj,axiom,
    ! [B: $tType,A: $tType,A3: set(A),A9: set(B)] :
      ( ( A3 != bot_bot(set(A)) )
     => ( ? [F6: fun(A,B)] :
            ( inj_on(A,B,F6,A3)
            & pp(aa(set(B),bool,ord_less_eq(set(B),aa(set(A),set(B),image(A,B,F6),A3)),A9)) )
      <=> ? [G3: fun(B,A)] : aa(set(B),set(A),image(B,A,G3),A9) = A3 ) ) ).

% inj_on_iff_surj
tff(fact_6912_inj__image__subset__iff,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),A3: set(A),B4: set(A)] :
      ( inj_on(A,B,F2,top_top(set(A)))
     => ( pp(aa(set(B),bool,ord_less_eq(set(B),aa(set(A),set(B),image(A,B,F2),A3)),aa(set(A),set(B),image(A,B,F2),B4)))
      <=> pp(aa(set(A),bool,ord_less_eq(set(A),A3),B4)) ) ) ).

% inj_image_subset_iff
tff(fact_6913_endo__inj__surj,axiom,
    ! [A: $tType,A3: set(A),F2: fun(A,A)] :
      ( pp(aa(set(A),bool,finite_finite2(A),A3))
     => ( pp(aa(set(A),bool,ord_less_eq(set(A),aa(set(A),set(A),image(A,A,F2),A3)),A3))
       => ( inj_on(A,A,F2,A3)
         => ( aa(set(A),set(A),image(A,A,F2),A3) = A3 ) ) ) ) ).

% endo_inj_surj
tff(fact_6914_inj__on__finite,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),A3: set(A),B4: set(B)] :
      ( inj_on(A,B,F2,A3)
     => ( pp(aa(set(B),bool,ord_less_eq(set(B),aa(set(A),set(B),image(A,B,F2),A3)),B4))
       => ( pp(aa(set(B),bool,finite_finite2(B),B4))
         => pp(aa(set(A),bool,finite_finite2(A),A3)) ) ) ) ).

% inj_on_finite
tff(fact_6915_finite__surj__inj,axiom,
    ! [A: $tType,A3: set(A),F2: fun(A,A)] :
      ( pp(aa(set(A),bool,finite_finite2(A),A3))
     => ( pp(aa(set(A),bool,ord_less_eq(set(A),A3),aa(set(A),set(A),image(A,A,F2),A3)))
       => inj_on(A,A,F2,A3) ) ) ).

% finite_surj_inj
tff(fact_6916_inj__on__image__set__diff,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),C6: set(A),A3: set(A),B4: set(A)] :
      ( inj_on(A,B,F2,C6)
     => ( pp(aa(set(A),bool,ord_less_eq(set(A),aa(set(A),set(A),minus_minus(set(A),A3),B4)),C6))
       => ( pp(aa(set(A),bool,ord_less_eq(set(A),B4),C6))
         => ( aa(set(A),set(B),image(A,B,F2),aa(set(A),set(A),minus_minus(set(A),A3),B4)) = aa(set(B),set(B),minus_minus(set(B),aa(set(A),set(B),image(A,B,F2),A3)),aa(set(A),set(B),image(A,B,F2),B4)) ) ) ) ) ).

% inj_on_image_set_diff
tff(fact_6917_the__inv__into__into,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),A3: set(A),X: B,B4: set(A)] :
      ( inj_on(A,B,F2,A3)
     => ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X),aa(set(A),set(B),image(A,B,F2),A3)))
       => ( pp(aa(set(A),bool,ord_less_eq(set(A),A3),B4))
         => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),the_inv_into(A,B,A3,F2,X)),B4)) ) ) ) ).

% the_inv_into_into
tff(fact_6918_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
tff(fact_6919_inj__image__Compl__subset,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),A3: set(A)] :
      ( inj_on(A,B,F2,top_top(set(A)))
     => pp(aa(set(B),bool,ord_less_eq(set(B),aa(set(A),set(B),image(A,B,F2),aa(set(A),set(A),uminus_uminus(set(A)),A3))),aa(set(B),set(B),uminus_uminus(set(B)),aa(set(A),set(B),image(A,B,F2),A3)))) ) ).

% inj_image_Compl_subset
tff(fact_6920_msb__shift,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] :
          ( pp(most_s684356279273892711sb_msb(word(A),W))
        <=> ( bit_Sh4282982442137083166shiftr(word(A),W,aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat))) != zero_zero(word(A)) ) ) ) ).

% msb_shift
tff(fact_6921_msb__word__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] :
          ( pp(most_s684356279273892711sb_msb(word(A),W))
        <=> pp(aa(nat,bool,bit_se5641148757651400278ts_bit(word(A),W),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat)))) ) ) ).

% msb_word_eq
tff(fact_6922_msb__word__iff__bit,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] :
          ( pp(most_s684356279273892711sb_msb(word(A),W))
        <=> pp(aa(nat,bool,bit_se5641148757651400278ts_bit(word(A),W),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat))))) ) ) ).

% msb_word_iff_bit
tff(fact_6923_word__msb__nth,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] :
          ( pp(most_s684356279273892711sb_msb(word(A),W))
        <=> pp(aa(nat,bool,bit_se5641148757651400278ts_bit(int,aa(word(A),int,semiring_1_unsigned(A,int),W)),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat)))) ) ) ).

% word_msb_nth
tff(fact_6924_msb__word__of__int,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: int] :
          ( pp(most_s684356279273892711sb_msb(word(A),aa(int,word(A),ring_1_of_int(word(A)),X)))
        <=> pp(aa(nat,bool,bit_se5641148757651400278ts_bit(int,X),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat)))) ) ) ).

% msb_word_of_int
tff(fact_6925_not__msb__from__less,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [V: word(A)] :
          ( pp(aa(word(A),bool,ord_less(word(A),V),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat)))))
         => ~ pp(most_s684356279273892711sb_msb(word(A),V)) ) ) ).

% not_msb_from_less
tff(fact_6926_word__sint__msb__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A)] : ring_1_signed(A,int,X) = aa(int,int,minus_minus(int,aa(word(A),int,semiring_1_unsigned(A,int),X)),if(int,most_s684356279273892711sb_msb(word(A),X),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(word(A),nat,size_size(word(A)),X)),zero_zero(int))) ) ).

% word_sint_msb_eq
tff(fact_6927_msb__big,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A)] :
          ( pp(most_s684356279273892711sb_msb(word(A),A2))
        <=> pp(aa(word(A),bool,ord_less_eq(word(A),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat))))),A2)) ) ) ).

% msb_big
tff(fact_6928_inj__on__map__inv__f,axiom,
    ! [B: $tType,A: $tType,L: list(A),A3: set(A),F2: fun(A,B)] :
      ( pp(aa(set(A),bool,ord_less_eq(set(A),aa(list(A),set(A),set2(A),L)),A3))
     => ( inj_on(A,B,F2,A3)
       => ( map(B,A,inv_on(A,B,F2,A3),map(A,B,F2,L)) = L ) ) ) ).

% inj_on_map_inv_f
tff(fact_6929_Schroeder__Bernstein,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,B),A3: set(A),B4: set(B),G: fun(B,A)] :
      ( inj_on(A,B,F2,A3)
     => ( pp(aa(set(B),bool,ord_less_eq(set(B),aa(set(A),set(B),image(A,B,F2),A3)),B4))
       => ( inj_on(B,A,G,B4)
         => ( pp(aa(set(A),bool,ord_less_eq(set(A),aa(set(B),set(A),image(B,A,G),B4)),A3))
           => ? [H4: fun(A,B)] : bij_betw(A,B,H4,A3,B4) ) ) ) ) ).

% Schroeder_Bernstein
tff(fact_6930_inj__split__Cons,axiom,
    ! [A: $tType,X6: set(product_prod(list(A),A))] : inj_on(product_prod(list(A),A),list(A),product_case_prod(list(A),A,list(A),aTP_Lamp_pz(list(A),fun(A,list(A)))),X6) ).

% inj_split_Cons
tff(fact_6931_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),aTP_Lamp_qq(A,fun(B,product_prod(B,A)))),A3) ).

% swap_inj_on
tff(fact_6932_inj__on__diff__nat,axiom,
    ! [N4: set(nat),K: nat] :
      ( ! [N2: nat] :
          ( pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),N2),N4))
         => pp(aa(nat,bool,ord_less_eq(nat,K),N2)) )
     => inj_on(nat,nat,aTP_Lamp_pd(nat,fun(nat,nat),K),N4) ) ).

% inj_on_diff_nat
tff(fact_6933_inj__Suc,axiom,
    ! [N4: set(nat)] : inj_on(nat,nat,suc,N4) ).

% inj_Suc
tff(fact_6934_inj__on__of__nat,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [N4: set(nat)] : inj_on(nat,A,semiring_1_of_nat(A),N4) ) ).

% inj_on_of_nat
tff(fact_6935_inj__Pair_I2_J,axiom,
    ! [B: $tType,A: $tType,C2: fun(A,B),S: set(A)] : inj_on(A,product_prod(B,A),aTP_Lamp_qr(fun(A,B),fun(A,product_prod(B,A)),C2),S) ).

% inj_Pair(2)
tff(fact_6936_inj__Pair_I1_J,axiom,
    ! [B: $tType,A: $tType,C2: fun(A,B),S: set(A)] : inj_on(A,product_prod(A,B),aTP_Lamp_qs(fun(A,B),fun(A,product_prod(A,B)),C2),S) ).

% inj_Pair(1)
tff(fact_6937_inj__on__convol__ident,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),X6: set(A)] : inj_on(A,product_prod(A,B),aTP_Lamp_qs(fun(A,B),fun(A,product_prod(A,B)),F2),X6) ).

% inj_on_convol_ident
tff(fact_6938_inj__on__set__encode,axiom,
    inj_on(set(nat),nat,nat_set_encode,collect(set(nat),finite_finite2(nat))) ).

% inj_on_set_encode
tff(fact_6939_inj__singleton,axiom,
    ! [A: $tType,A3: set(A)] : inj_on(A,set(A),aTP_Lamp_qt(A,set(A)),A3) ).

% inj_singleton
tff(fact_6940_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
tff(fact_6941_signed_Osorted__list__of__set_Oinj__on,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => inj_on(word(A),word(A),aTP_Lamp_qu(word(A),word(A)),top_top(set(word(A)))) ) ).

% signed.sorted_list_of_set.inj_on
tff(fact_6942_inj__graph,axiom,
    ! [B: $tType,A: $tType] : inj_on(fun(A,B),set(product_prod(A,B)),aTP_Lamp_qw(fun(A,B),set(product_prod(A,B))),top_top(set(fun(A,B)))) ).

% inj_graph
tff(fact_6943_le__rel__bool__arg__iff,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [X6: fun(bool,A),Y6: fun(bool,A)] :
          ( pp(aa(fun(bool,A),bool,ord_less_eq(fun(bool,A),X6),Y6))
        <=> ( pp(aa(A,bool,ord_less_eq(A,aa(bool,A,X6,fFalse)),aa(bool,A,Y6,fFalse)))
            & pp(aa(A,bool,ord_less_eq(A,aa(bool,A,X6,fTrue)),aa(bool,A,Y6,fTrue))) ) ) ) ).

% le_rel_bool_arg_iff
tff(fact_6944_finite__imp__nat__seg__image__inj__on,axiom,
    ! [A: $tType,A3: set(A)] :
      ( pp(aa(set(A),bool,finite_finite2(A),A3))
     => ? [N2: nat,F4: fun(nat,A)] :
          ( ( A3 = aa(set(nat),set(A),image(nat,A,F4),collect(nat,aTP_Lamp_ap(nat,fun(nat,bool),N2))) )
          & inj_on(nat,A,F4,collect(nat,aTP_Lamp_ap(nat,fun(nat,bool),N2))) ) ) ).

% finite_imp_nat_seg_image_inj_on
tff(fact_6945_finite__imp__inj__to__nat__seg,axiom,
    ! [A: $tType,A3: set(A)] :
      ( pp(aa(set(A),bool,finite_finite2(A),A3))
     => ? [F4: fun(A,nat),N2: nat] :
          ( ( aa(set(A),set(nat),image(A,nat,F4),A3) = collect(nat,aTP_Lamp_ap(nat,fun(nat,bool),N2)) )
          & inj_on(A,nat,F4,A3) ) ) ).

% finite_imp_inj_to_nat_seg
tff(fact_6946_finite__imp__inj__to__nat__seg_H,axiom,
    ! [A: $tType,A3: set(A)] :
      ( pp(aa(set(A),bool,finite_finite2(A),A3))
     => ~ ! [F4: fun(A,nat)] :
            ( ? [N2: nat] : aa(set(A),set(nat),image(A,nat,F4),A3) = collect(nat,aTP_Lamp_ap(nat,fun(nat,bool),N2))
           => ~ inj_on(A,nat,F4,A3) ) ) ).

% finite_imp_inj_to_nat_seg'
tff(fact_6947_infinite__iff__countable__subset,axiom,
    ! [A: $tType,S: set(A)] :
      ( ~ pp(aa(set(A),bool,finite_finite2(A),S))
    <=> ? [F6: fun(nat,A)] :
          ( inj_on(nat,A,F6,top_top(set(nat)))
          & pp(aa(set(A),bool,ord_less_eq(set(A),aa(set(nat),set(A),image(nat,A,F6),top_top(set(nat)))),S)) ) ) ).

% infinite_iff_countable_subset
tff(fact_6948_infinite__countable__subset,axiom,
    ! [A: $tType,S: set(A)] :
      ( ~ pp(aa(set(A),bool,finite_finite2(A),S))
     => ? [F4: fun(nat,A)] :
          ( inj_on(nat,A,F4,top_top(set(nat)))
          & pp(aa(set(A),bool,ord_less_eq(set(A),aa(set(nat),set(A),image(nat,A,F4),top_top(set(nat)))),S)) ) ) ).

% infinite_countable_subset
tff(fact_6949_inv__on__def,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,B),A3: set(A),X: B] : aa(B,A,inv_on(A,B,F2,A3),X) = fChoice(A,aa(B,fun(A,bool),aa(set(A),fun(B,fun(A,bool)),aTP_Lamp_qx(fun(A,B),fun(set(A),fun(B,fun(A,bool))),F2),A3),X)) ).

% inv_on_def
tff(fact_6950_summable__reindex,axiom,
    ! [F2: fun(nat,real),G: fun(nat,nat)] :
      ( summable(real,F2)
     => ( inj_on(nat,nat,G,top_top(set(nat)))
       => ( ! [X3: nat] : pp(aa(real,bool,ord_less_eq(real,zero_zero(real)),aa(nat,real,F2,X3)))
         => summable(real,comp(nat,real,nat,F2,G)) ) ) ) ).

% summable_reindex
tff(fact_6951_suminf__reindex__mono,axiom,
    ! [F2: fun(nat,real),G: fun(nat,nat)] :
      ( summable(real,F2)
     => ( inj_on(nat,nat,G,top_top(set(nat)))
       => ( ! [X3: nat] : pp(aa(real,bool,ord_less_eq(real,zero_zero(real)),aa(nat,real,F2,X3)))
         => pp(aa(real,bool,ord_less_eq(real,suminf(real,comp(nat,real,nat,F2,G))),suminf(real,F2))) ) ) ) ).

% suminf_reindex_mono
tff(fact_6952_suminf__reindex,axiom,
    ! [F2: fun(nat,real),G: fun(nat,nat)] :
      ( summable(real,F2)
     => ( inj_on(nat,nat,G,top_top(set(nat)))
       => ( ! [X3: nat] : pp(aa(real,bool,ord_less_eq(real,zero_zero(real)),aa(nat,real,F2,X3)))
         => ( ! [X3: nat] :
                ( ~ pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),X3),aa(set(nat),set(nat),image(nat,nat,G),top_top(set(nat)))))
               => ( aa(nat,real,F2,X3) = zero_zero(real) ) )
           => ( suminf(real,comp(nat,real,nat,F2,G)) = suminf(real,F2) ) ) ) ) ) ).

% suminf_reindex
tff(fact_6953_inj__sgn__power,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),N))
     => inj_on(real,real,aTP_Lamp_qf(nat,fun(real,real),N),top_top(set(real))) ) ).

% inj_sgn_power
tff(fact_6954_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),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))) ) ).

% inj_on_word_of_nat
tff(fact_6955_ex__subset__image__inj,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),S: set(B),P: fun(set(A),bool)] :
      ( ? [T6: set(A)] :
          ( pp(aa(set(A),bool,ord_less_eq(set(A),T6),aa(set(B),set(A),image(B,A,F2),S)))
          & pp(aa(set(A),bool,P,T6)) )
    <=> ? [T6: set(B)] :
          ( pp(aa(set(B),bool,ord_less_eq(set(B),T6),S))
          & inj_on(B,A,F2,T6)
          & pp(aa(set(A),bool,P,aa(set(B),set(A),image(B,A,F2),T6))) ) ) ).

% ex_subset_image_inj
tff(fact_6956_all__subset__image__inj,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),S: set(B),P: fun(set(A),bool)] :
      ( ! [T6: set(A)] :
          ( pp(aa(set(A),bool,ord_less_eq(set(A),T6),aa(set(B),set(A),image(B,A,F2),S)))
         => pp(aa(set(A),bool,P,T6)) )
    <=> ! [T6: set(B)] :
          ( ( pp(aa(set(B),bool,ord_less_eq(set(B),T6),S))
            & inj_on(B,A,F2,T6) )
         => pp(aa(set(A),bool,P,aa(set(B),set(A),image(B,A,F2),T6))) ) ) ).

% all_subset_image_inj
tff(fact_6957_empty__natural,axiom,
    ! [C: $tType,B: $tType,D6: $tType,A: $tType,F2: fun(A,C),G: fun(D6,B)] : comp(C,set(B),A,aTP_Lamp_qy(C,set(B)),F2) = comp(set(D6),set(B),A,image(D6,B,G),aTP_Lamp_qz(A,set(D6))) ).

% empty_natural
tff(fact_6958_DERIV__even__real__root,axiom,
    ! [N: nat,X: real] :
      ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),N))
     => ( pp(aa(nat,bool,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N))
       => ( pp(aa(real,bool,ord_less(real,X),zero_zero(real)))
         => has_field_derivative(real,root(N),aa(real,real,inverse_inverse(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,uminus_uminus(real),aa(nat,real,semiring_1_of_nat(real),N))),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,root(N),X)),aa(nat,nat,minus_minus(nat,N),aa(nat,nat,suc,zero_zero(nat)))))),topolo174197925503356063within(real,X,top_top(set(real)))) ) ) ) ).

% DERIV_even_real_root
tff(fact_6959_DERIV__subset,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),F7: A,X: A,S3: set(A),T: set(A)] :
          ( has_field_derivative(A,F2,F7,topolo174197925503356063within(A,X,S3))
         => ( pp(aa(set(A),bool,ord_less_eq(set(A),T),S3))
           => has_field_derivative(A,F2,F7,topolo174197925503356063within(A,X,T)) ) ) ) ).

% DERIV_subset
tff(fact_6960_has__field__derivative__subset,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),Y: A,X: A,S3: set(A),T: set(A)] :
          ( has_field_derivative(A,F2,Y,topolo174197925503356063within(A,X,S3))
         => ( pp(aa(set(A),bool,ord_less_eq(set(A),T),S3))
           => has_field_derivative(A,F2,Y,topolo174197925503356063within(A,X,T)) ) ) ) ).

% has_field_derivative_subset
tff(fact_6961_at__le,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [S3: set(A),T: set(A),X: A] :
          ( pp(aa(set(A),bool,ord_less_eq(set(A),S3),T))
         => pp(aa(filter(A),bool,ord_less_eq(filter(A),topolo174197925503356063within(A,X,S3)),topolo174197925503356063within(A,X,T))) ) ) ).

% at_le
tff(fact_6962_field__differentiable__minus,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),F7: A,F3: filter(A)] :
          ( has_field_derivative(A,F2,F7,F3)
         => has_field_derivative(A,aTP_Lamp_ra(fun(A,A),fun(A,A),F2),aa(A,A,uminus_uminus(A),F7),F3) ) ) ).

% field_differentiable_minus
tff(fact_6963_DERIV__minus,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D4: A,X: A,S3: set(A)] :
          ( has_field_derivative(A,F2,D4,topolo174197925503356063within(A,X,S3))
         => has_field_derivative(A,aTP_Lamp_ra(fun(A,A),fun(A,A),F2),aa(A,A,uminus_uminus(A),D4),topolo174197925503356063within(A,X,S3)) ) ) ).

% DERIV_minus
tff(fact_6964_DERIV__diff,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D4: A,X: A,S3: set(A),G: fun(A,A),E5: A] :
          ( has_field_derivative(A,F2,D4,topolo174197925503356063within(A,X,S3))
         => ( has_field_derivative(A,G,E5,topolo174197925503356063within(A,X,S3))
           => has_field_derivative(A,aa(fun(A,A),fun(A,A),aTP_Lamp_rb(fun(A,A),fun(fun(A,A),fun(A,A)),F2),G),aa(A,A,minus_minus(A,D4),E5),topolo174197925503356063within(A,X,S3)) ) ) ) ).

% DERIV_diff
tff(fact_6965_field__differentiable__diff,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),F7: A,F3: filter(A),G: fun(A,A),G4: A] :
          ( has_field_derivative(A,F2,F7,F3)
         => ( has_field_derivative(A,G,G4,F3)
           => has_field_derivative(A,aa(fun(A,A),fun(A,A),aTP_Lamp_rb(fun(A,A),fun(fun(A,A),fun(A,A)),F2),G),aa(A,A,minus_minus(A,F7),G4),F3) ) ) ) ).

% field_differentiable_diff
tff(fact_6966_has__field__derivative__scaleR__right,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D4: A,F3: filter(A),C2: real] :
          ( has_field_derivative(A,F2,D4,F3)
         => has_field_derivative(A,aa(real,fun(A,A),aTP_Lamp_rc(fun(A,A),fun(real,fun(A,A)),F2),C2),aa(A,A,real_V8093663219630862766scaleR(A,C2),D4),F3) ) ) ).

% has_field_derivative_scaleR_right
tff(fact_6967_DERIV__ident,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F3: filter(A)] : has_field_derivative(A,aTP_Lamp_rd(A,A),one_one(A),F3) ) ).

% DERIV_ident
tff(fact_6968_DERIV__const,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [K: A,F3: filter(A)] : has_field_derivative(A,aTP_Lamp_re(A,fun(A,A),K),zero_zero(A),F3) ) ).

% DERIV_const
tff(fact_6969_DERIV__add,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D4: A,X: A,S3: set(A),G: fun(A,A),E5: A] :
          ( has_field_derivative(A,F2,D4,topolo174197925503356063within(A,X,S3))
         => ( has_field_derivative(A,G,E5,topolo174197925503356063within(A,X,S3))
           => has_field_derivative(A,aa(fun(A,A),fun(A,A),aTP_Lamp_rf(fun(A,A),fun(fun(A,A),fun(A,A)),F2),G),aa(A,A,aa(A,fun(A,A),plus_plus(A),D4),E5),topolo174197925503356063within(A,X,S3)) ) ) ) ).

% DERIV_add
tff(fact_6970_field__differentiable__add,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),F7: A,F3: filter(A),G: fun(A,A),G4: A] :
          ( has_field_derivative(A,F2,F7,F3)
         => ( has_field_derivative(A,G,G4,F3)
           => has_field_derivative(A,aa(fun(A,A),fun(A,A),aTP_Lamp_rf(fun(A,A),fun(fun(A,A),fun(A,A)),F2),G),aa(A,A,aa(A,fun(A,A),plus_plus(A),F7),G4),F3) ) ) ) ).

% field_differentiable_add
tff(fact_6971_DERIV__divide,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D4: A,X: A,S3: set(A),G: fun(A,A),E5: A] :
          ( has_field_derivative(A,F2,D4,topolo174197925503356063within(A,X,S3))
         => ( has_field_derivative(A,G,E5,topolo174197925503356063within(A,X,S3))
           => ( ( aa(A,A,G,X) != zero_zero(A) )
             => has_field_derivative(A,aa(fun(A,A),fun(A,A),aTP_Lamp_rg(fun(A,A),fun(fun(A,A),fun(A,A)),F2),G),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),D4),aa(A,A,G,X))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,F2,X)),E5))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,G,X)),aa(A,A,G,X))),topolo174197925503356063within(A,X,S3)) ) ) ) ) ).

% DERIV_divide
tff(fact_6972_DERIV__mult_H,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D4: A,X: A,S3: set(A),G: fun(A,A),E5: A] :
          ( has_field_derivative(A,F2,D4,topolo174197925503356063within(A,X,S3))
         => ( has_field_derivative(A,G,E5,topolo174197925503356063within(A,X,S3))
           => has_field_derivative(A,aa(fun(A,A),fun(A,A),aTP_Lamp_rh(fun(A,A),fun(fun(A,A),fun(A,A)),F2),G),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,F2,X)),E5)),aa(A,A,aa(A,fun(A,A),times_times(A),D4),aa(A,A,G,X))),topolo174197925503356063within(A,X,S3)) ) ) ) ).

% DERIV_mult'
tff(fact_6973_DERIV__mult,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),Da: A,X: A,S3: set(A),G: fun(A,A),Db: A] :
          ( has_field_derivative(A,F2,Da,topolo174197925503356063within(A,X,S3))
         => ( has_field_derivative(A,G,Db,topolo174197925503356063within(A,X,S3))
           => has_field_derivative(A,aa(fun(A,A),fun(A,A),aTP_Lamp_rh(fun(A,A),fun(fun(A,A),fun(A,A)),F2),G),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Da),aa(A,A,G,X))),aa(A,A,aa(A,fun(A,A),times_times(A),Db),aa(A,A,F2,X))),topolo174197925503356063within(A,X,S3)) ) ) ) ).

% DERIV_mult
tff(fact_6974_has__field__derivative__cosh,axiom,
    ! [A11: $tType] :
      ( ( real_Vector_banach(A11)
        & real_V3459762299906320749_field(A11) )
     => ! [G: fun(A11,A11),Db: A11,X: A11,S3: set(A11)] :
          ( has_field_derivative(A11,G,Db,topolo174197925503356063within(A11,X,S3))
         => has_field_derivative(A11,aTP_Lamp_ri(fun(A11,A11),fun(A11,A11),G),aa(A11,A11,aa(A11,fun(A11,A11),times_times(A11),sinh(A11,aa(A11,A11,G,X))),Db),topolo174197925503356063within(A11,X,S3)) ) ) ).

% has_field_derivative_cosh
tff(fact_6975_has__field__derivative__sinh,axiom,
    ! [A11: $tType] :
      ( ( real_Vector_banach(A11)
        & real_V3459762299906320749_field(A11) )
     => ! [G: fun(A11,A11),Db: A11,X: A11,S3: set(A11)] :
          ( has_field_derivative(A11,G,Db,topolo174197925503356063within(A11,X,S3))
         => has_field_derivative(A11,aTP_Lamp_rj(fun(A11,A11),fun(A11,A11),G),aa(A11,A11,aa(A11,fun(A11,A11),times_times(A11),cosh(A11,aa(A11,A11,G,X))),Db),topolo174197925503356063within(A11,X,S3)) ) ) ).

% has_field_derivative_sinh
tff(fact_6976_DERIV__cmult,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D4: A,X: A,S3: set(A),C2: A] :
          ( has_field_derivative(A,F2,D4,topolo174197925503356063within(A,X,S3))
         => has_field_derivative(A,aa(A,fun(A,A),aTP_Lamp_rk(fun(A,A),fun(A,fun(A,A)),F2),C2),aa(A,A,aa(A,fun(A,A),times_times(A),C2),D4),topolo174197925503356063within(A,X,S3)) ) ) ).

% DERIV_cmult
tff(fact_6977_DERIV__cmult__right,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D4: A,X: A,S3: set(A),C2: A] :
          ( has_field_derivative(A,F2,D4,topolo174197925503356063within(A,X,S3))
         => has_field_derivative(A,aa(A,fun(A,A),aTP_Lamp_rl(fun(A,A),fun(A,fun(A,A)),F2),C2),aa(A,A,aa(A,fun(A,A),times_times(A),D4),C2),topolo174197925503356063within(A,X,S3)) ) ) ).

% DERIV_cmult_right
tff(fact_6978_DERIV__cdivide,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D4: A,X: A,S3: set(A),C2: A] :
          ( has_field_derivative(A,F2,D4,topolo174197925503356063within(A,X,S3))
         => has_field_derivative(A,aa(A,fun(A,A),aTP_Lamp_rm(fun(A,A),fun(A,fun(A,A)),F2),C2),aa(A,A,aa(A,fun(A,A),divide_divide(A),D4),C2),topolo174197925503356063within(A,X,S3)) ) ) ).

% DERIV_cdivide
tff(fact_6979_DERIV__inverse_H,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D4: A,X: A,S3: set(A)] :
          ( has_field_derivative(A,F2,D4,topolo174197925503356063within(A,X,S3))
         => ( ( aa(A,A,F2,X) != zero_zero(A) )
           => has_field_derivative(A,aTP_Lamp_rn(fun(A,A),fun(A,A),F2),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,inverse_inverse(A),aa(A,A,F2,X))),D4)),aa(A,A,inverse_inverse(A),aa(A,A,F2,X)))),topolo174197925503356063within(A,X,S3)) ) ) ) ).

% DERIV_inverse'
tff(fact_6980_DERIV__at__within__shift,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),Y: A,Z: A,X: A,S: set(A)] :
          ( has_field_derivative(A,F2,Y,topolo174197925503356063within(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Z),X),aa(set(A),set(A),image(A,A,aa(A,fun(A,A),plus_plus(A),Z)),S)))
        <=> has_field_derivative(A,aa(A,fun(A,A),aTP_Lamp_ro(fun(A,A),fun(A,fun(A,A)),F2),Z),Y,topolo174197925503356063within(A,X,S)) ) ) ).

% DERIV_at_within_shift
tff(fact_6981_DERIV__const__ratio__const2,axiom,
    ! [A2: real,B2: real,F2: fun(real,real),K: real] :
      ( ( A2 != B2 )
     => ( ! [X3: real] : has_field_derivative(real,F2,K,topolo174197925503356063within(real,X3,top_top(set(real))))
       => ( aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,minus_minus(real,aa(real,real,F2,B2)),aa(real,real,F2,A2))),aa(real,real,minus_minus(real,B2),A2)) = K ) ) ) ).

% DERIV_const_ratio_const2
tff(fact_6982_DERIV__fun__cos,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [G: fun(A,A),M: A,X: A] :
          ( has_field_derivative(A,G,M,topolo174197925503356063within(A,X,top_top(set(A))))
         => has_field_derivative(A,aTP_Lamp_rp(fun(A,A),fun(A,A),G),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),sin(A,aa(A,A,G,X)))),M),topolo174197925503356063within(A,X,top_top(set(A)))) ) ) ).

% DERIV_fun_cos
tff(fact_6983_DERIV__const__ratio__const,axiom,
    ! [A2: real,B2: real,F2: fun(real,real),K: real] :
      ( ( A2 != B2 )
     => ( ! [X3: real] : has_field_derivative(real,F2,K,topolo174197925503356063within(real,X3,top_top(set(real))))
       => ( aa(real,real,minus_minus(real,aa(real,real,F2,B2)),aa(real,real,F2,A2)) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,minus_minus(real,B2),A2)),K) ) ) ) ).

% DERIV_const_ratio_const
tff(fact_6984_DERIV__chain__s,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [S3: set(A),G: fun(A,A),G4: fun(A,A),F2: fun(A,A),F7: A,X: A] :
          ( ! [X3: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),S3))
             => has_field_derivative(A,G,aa(A,A,G4,X3),topolo174197925503356063within(A,X3,top_top(set(A)))) )
         => ( has_field_derivative(A,F2,F7,topolo174197925503356063within(A,X,top_top(set(A))))
           => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(A,A,F2,X)),S3))
             => has_field_derivative(A,aa(fun(A,A),fun(A,A),aTP_Lamp_rq(fun(A,A),fun(fun(A,A),fun(A,A)),G),F2),aa(A,A,aa(A,fun(A,A),times_times(A),F7),aa(A,A,G4,aa(A,A,F2,X))),topolo174197925503356063within(A,X,top_top(set(A)))) ) ) ) ) ).

% DERIV_chain_s
tff(fact_6985_DERIV__chain3,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [G: fun(A,A),G4: fun(A,A),F2: fun(A,A),F7: A,X: A] :
          ( ! [X3: A] : has_field_derivative(A,G,aa(A,A,G4,X3),topolo174197925503356063within(A,X3,top_top(set(A))))
         => ( has_field_derivative(A,F2,F7,topolo174197925503356063within(A,X,top_top(set(A))))
           => has_field_derivative(A,aa(fun(A,A),fun(A,A),aTP_Lamp_rq(fun(A,A),fun(fun(A,A),fun(A,A)),G),F2),aa(A,A,aa(A,fun(A,A),times_times(A),F7),aa(A,A,G4,aa(A,A,F2,X))),topolo174197925503356063within(A,X,top_top(set(A)))) ) ) ) ).

% DERIV_chain3
tff(fact_6986_DERIV__chain2,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),Da: A,G: fun(A,A),X: A,Db: A,S3: set(A)] :
          ( has_field_derivative(A,F2,Da,topolo174197925503356063within(A,aa(A,A,G,X),top_top(set(A))))
         => ( has_field_derivative(A,G,Db,topolo174197925503356063within(A,X,S3))
           => has_field_derivative(A,aa(fun(A,A),fun(A,A),aTP_Lamp_rq(fun(A,A),fun(fun(A,A),fun(A,A)),F2),G),aa(A,A,aa(A,fun(A,A),times_times(A),Da),Db),topolo174197925503356063within(A,X,S3)) ) ) ) ).

% DERIV_chain2
tff(fact_6987_DERIV__chain_H,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D4: A,X: A,S3: set(A),G: fun(A,A),E5: A] :
          ( has_field_derivative(A,F2,D4,topolo174197925503356063within(A,X,S3))
         => ( has_field_derivative(A,G,E5,topolo174197925503356063within(A,aa(A,A,F2,X),top_top(set(A))))
           => has_field_derivative(A,aa(fun(A,A),fun(A,A),aTP_Lamp_rr(fun(A,A),fun(fun(A,A),fun(A,A)),F2),G),aa(A,A,aa(A,fun(A,A),times_times(A),E5),D4),topolo174197925503356063within(A,X,S3)) ) ) ) ).

% DERIV_chain'
tff(fact_6988_DERIV__fun__sin,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [G: fun(A,A),M: A,X: A] :
          ( has_field_derivative(A,G,M,topolo174197925503356063within(A,X,top_top(set(A))))
         => has_field_derivative(A,aTP_Lamp_rs(fun(A,A),fun(A,A),G),aa(A,A,aa(A,fun(A,A),times_times(A),cos(A,aa(A,A,G,X))),M),topolo174197925503356063within(A,X,top_top(set(A)))) ) ) ).

% DERIV_fun_sin
tff(fact_6989_DERIV__fun__exp,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [G: fun(A,A),M: A,X: A] :
          ( has_field_derivative(A,G,M,topolo174197925503356063within(A,X,top_top(set(A))))
         => has_field_derivative(A,aTP_Lamp_rt(fun(A,A),fun(A,A),G),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,exp(A),aa(A,A,G,X))),M),topolo174197925503356063within(A,X,top_top(set(A)))) ) ) ).

% DERIV_fun_exp
tff(fact_6990_DERIV__cos__add,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [K: A,Xa2: A] : has_field_derivative(A,aTP_Lamp_ru(A,fun(A,A),K),aa(A,A,uminus_uminus(A),sin(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa2),K))),topolo174197925503356063within(A,Xa2,top_top(set(A)))) ) ).

% DERIV_cos_add
tff(fact_6991_DERIV__shift,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),Y: A,X: A,Z: A] :
          ( has_field_derivative(A,F2,Y,topolo174197925503356063within(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Z),top_top(set(A))))
        <=> has_field_derivative(A,aa(A,fun(A,A),aTP_Lamp_rv(fun(A,A),fun(A,fun(A,A)),F2),Z),Y,topolo174197925503356063within(A,X,top_top(set(A)))) ) ) ).

% DERIV_shift
tff(fact_6992_DERIV__neg__dec__right,axiom,
    ! [F2: fun(real,real),L: real,X: real] :
      ( has_field_derivative(real,F2,L,topolo174197925503356063within(real,X,top_top(set(real))))
     => ( pp(aa(real,bool,ord_less(real,L),zero_zero(real)))
       => ? [D2: real] :
            ( pp(aa(real,bool,ord_less(real,zero_zero(real)),D2))
            & ! [H5: real] :
                ( pp(aa(real,bool,ord_less(real,zero_zero(real)),H5))
               => ( pp(aa(real,bool,ord_less(real,H5),D2))
                 => pp(aa(real,bool,ord_less(real,aa(real,real,F2,aa(real,real,aa(real,fun(real,real),plus_plus(real),X),H5))),aa(real,real,F2,X))) ) ) ) ) ) ).

% DERIV_neg_dec_right
tff(fact_6993_DERIV__pos__inc__right,axiom,
    ! [F2: fun(real,real),L: real,X: real] :
      ( has_field_derivative(real,F2,L,topolo174197925503356063within(real,X,top_top(set(real))))
     => ( pp(aa(real,bool,ord_less(real,zero_zero(real)),L))
       => ? [D2: real] :
            ( pp(aa(real,bool,ord_less(real,zero_zero(real)),D2))
            & ! [H5: real] :
                ( pp(aa(real,bool,ord_less(real,zero_zero(real)),H5))
               => ( pp(aa(real,bool,ord_less(real,H5),D2))
                 => pp(aa(real,bool,ord_less(real,aa(real,real,F2,X)),aa(real,real,F2,aa(real,real,aa(real,fun(real,real),plus_plus(real),X),H5)))) ) ) ) ) ) ).

% DERIV_pos_inc_right
tff(fact_6994_DERIV__local__const,axiom,
    ! [F2: fun(real,real),L: real,X: real,D: real] :
      ( has_field_derivative(real,F2,L,topolo174197925503356063within(real,X,top_top(set(real))))
     => ( pp(aa(real,bool,ord_less(real,zero_zero(real)),D))
       => ( ! [Y3: real] :
              ( pp(aa(real,bool,ord_less(real,aa(real,real,abs_abs(real),aa(real,real,minus_minus(real,X),Y3))),D))
             => ( aa(real,real,F2,X) = aa(real,real,F2,Y3) ) )
         => ( L = zero_zero(real) ) ) ) ) ).

% DERIV_local_const
tff(fact_6995_DERIV__isconst__all,axiom,
    ! [F2: fun(real,real),X: real,Y: real] :
      ( ! [X3: real] : has_field_derivative(real,F2,zero_zero(real),topolo174197925503356063within(real,X3,top_top(set(real))))
     => ( aa(real,real,F2,X) = aa(real,real,F2,Y) ) ) ).

% DERIV_isconst_all
tff(fact_6996_DERIV__pos__inc__left,axiom,
    ! [F2: fun(real,real),L: real,X: real] :
      ( has_field_derivative(real,F2,L,topolo174197925503356063within(real,X,top_top(set(real))))
     => ( pp(aa(real,bool,ord_less(real,zero_zero(real)),L))
       => ? [D2: real] :
            ( pp(aa(real,bool,ord_less(real,zero_zero(real)),D2))
            & ! [H5: real] :
                ( pp(aa(real,bool,ord_less(real,zero_zero(real)),H5))
               => ( pp(aa(real,bool,ord_less(real,H5),D2))
                 => pp(aa(real,bool,ord_less(real,aa(real,real,F2,aa(real,real,minus_minus(real,X),H5))),aa(real,real,F2,X))) ) ) ) ) ) ).

% DERIV_pos_inc_left
tff(fact_6997_DERIV__neg__dec__left,axiom,
    ! [F2: fun(real,real),L: real,X: real] :
      ( has_field_derivative(real,F2,L,topolo174197925503356063within(real,X,top_top(set(real))))
     => ( pp(aa(real,bool,ord_less(real,L),zero_zero(real)))
       => ? [D2: real] :
            ( pp(aa(real,bool,ord_less(real,zero_zero(real)),D2))
            & ! [H5: real] :
                ( pp(aa(real,bool,ord_less(real,zero_zero(real)),H5))
               => ( pp(aa(real,bool,ord_less(real,H5),D2))
                 => pp(aa(real,bool,ord_less(real,aa(real,real,F2,X)),aa(real,real,F2,aa(real,real,minus_minus(real,X),H5)))) ) ) ) ) ) ).

% DERIV_neg_dec_left
tff(fact_6998_DERIV__mirror,axiom,
    ! [F2: fun(real,real),Y: real,X: real] :
      ( has_field_derivative(real,F2,Y,topolo174197925503356063within(real,aa(real,real,uminus_uminus(real),X),top_top(set(real))))
    <=> has_field_derivative(real,aTP_Lamp_rw(fun(real,real),fun(real,real),F2),aa(real,real,uminus_uminus(real),Y),topolo174197925503356063within(real,X,top_top(set(real)))) ) ).

% DERIV_mirror
tff(fact_6999_DERIV__ln,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,ord_less(real,zero_zero(real)),X))
     => has_field_derivative(real,ln_ln(real),aa(real,real,inverse_inverse(real),X),topolo174197925503356063within(real,X,top_top(set(real)))) ) ).

% DERIV_ln
tff(fact_7000_MVT2,axiom,
    ! [A2: real,B2: real,F2: fun(real,real),F7: fun(real,real)] :
      ( pp(aa(real,bool,ord_less(real,A2),B2))
     => ( ! [X3: real] :
            ( pp(aa(real,bool,ord_less_eq(real,A2),X3))
           => ( pp(aa(real,bool,ord_less_eq(real,X3),B2))
             => has_field_derivative(real,F2,aa(real,real,F7,X3),topolo174197925503356063within(real,X3,top_top(set(real)))) ) )
       => ? [Z2: real] :
            ( pp(aa(real,bool,ord_less(real,A2),Z2))
            & pp(aa(real,bool,ord_less(real,Z2),B2))
            & ( aa(real,real,minus_minus(real,aa(real,real,F2,B2)),aa(real,real,F2,A2)) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,minus_minus(real,B2),A2)),aa(real,real,F7,Z2)) ) ) ) ) ).

% MVT2
tff(fact_7001_DERIV__nonpos__imp__nonincreasing,axiom,
    ! [A2: real,B2: real,F2: fun(real,real)] :
      ( pp(aa(real,bool,ord_less_eq(real,A2),B2))
     => ( ! [X3: real] :
            ( pp(aa(real,bool,ord_less_eq(real,A2),X3))
           => ( pp(aa(real,bool,ord_less_eq(real,X3),B2))
             => ? [Y4: real] :
                  ( has_field_derivative(real,F2,Y4,topolo174197925503356063within(real,X3,top_top(set(real))))
                  & pp(aa(real,bool,ord_less_eq(real,Y4),zero_zero(real))) ) ) )
       => pp(aa(real,bool,ord_less_eq(real,aa(real,real,F2,B2)),aa(real,real,F2,A2))) ) ) ).

% DERIV_nonpos_imp_nonincreasing
tff(fact_7002_DERIV__nonneg__imp__nondecreasing,axiom,
    ! [A2: real,B2: real,F2: fun(real,real)] :
      ( pp(aa(real,bool,ord_less_eq(real,A2),B2))
     => ( ! [X3: real] :
            ( pp(aa(real,bool,ord_less_eq(real,A2),X3))
           => ( pp(aa(real,bool,ord_less_eq(real,X3),B2))
             => ? [Y4: real] :
                  ( has_field_derivative(real,F2,Y4,topolo174197925503356063within(real,X3,top_top(set(real))))
                  & pp(aa(real,bool,ord_less_eq(real,zero_zero(real)),Y4)) ) ) )
       => pp(aa(real,bool,ord_less_eq(real,aa(real,real,F2,A2)),aa(real,real,F2,B2))) ) ) ).

% DERIV_nonneg_imp_nondecreasing
tff(fact_7003_DERIV__pos__imp__increasing,axiom,
    ! [A2: real,B2: real,F2: fun(real,real)] :
      ( pp(aa(real,bool,ord_less(real,A2),B2))
     => ( ! [X3: real] :
            ( pp(aa(real,bool,ord_less_eq(real,A2),X3))
           => ( pp(aa(real,bool,ord_less_eq(real,X3),B2))
             => ? [Y4: real] :
                  ( has_field_derivative(real,F2,Y4,topolo174197925503356063within(real,X3,top_top(set(real))))
                  & pp(aa(real,bool,ord_less(real,zero_zero(real)),Y4)) ) ) )
       => pp(aa(real,bool,ord_less(real,aa(real,real,F2,A2)),aa(real,real,F2,B2))) ) ) ).

% DERIV_pos_imp_increasing
tff(fact_7004_DERIV__neg__imp__decreasing,axiom,
    ! [A2: real,B2: real,F2: fun(real,real)] :
      ( pp(aa(real,bool,ord_less(real,A2),B2))
     => ( ! [X3: real] :
            ( pp(aa(real,bool,ord_less_eq(real,A2),X3))
           => ( pp(aa(real,bool,ord_less_eq(real,X3),B2))
             => ? [Y4: real] :
                  ( has_field_derivative(real,F2,Y4,topolo174197925503356063within(real,X3,top_top(set(real))))
                  & pp(aa(real,bool,ord_less(real,Y4),zero_zero(real))) ) ) )
       => pp(aa(real,bool,ord_less(real,aa(real,real,F2,B2)),aa(real,real,F2,A2))) ) ) ).

% DERIV_neg_imp_decreasing
tff(fact_7005_deriv__nonneg__imp__mono,axiom,
    ! [A2: real,B2: real,G: fun(real,real),G4: fun(real,real)] :
      ( ! [X3: real] :
          ( pp(aa(set(real),bool,aa(real,fun(set(real),bool),member(real),X3),set_or1337092689740270186AtMost(real,A2,B2)))
         => has_field_derivative(real,G,aa(real,real,G4,X3),topolo174197925503356063within(real,X3,top_top(set(real)))) )
     => ( ! [X3: real] :
            ( pp(aa(set(real),bool,aa(real,fun(set(real),bool),member(real),X3),set_or1337092689740270186AtMost(real,A2,B2)))
           => pp(aa(real,bool,ord_less_eq(real,zero_zero(real)),aa(real,real,G4,X3))) )
       => ( pp(aa(real,bool,ord_less_eq(real,A2),B2))
         => pp(aa(real,bool,ord_less_eq(real,aa(real,real,G,A2)),aa(real,real,G,B2))) ) ) ) ).

% deriv_nonneg_imp_mono
tff(fact_7006_DERIV__sum,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( real_V3459762299906320749_field(B)
     => ! [S: set(A),F2: fun(B,fun(A,B)),F7: fun(C,fun(A,B)),X: C,F3: filter(B)] :
          ( ! [N2: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),N2),S))
             => has_field_derivative(B,aa(A,fun(B,B),aTP_Lamp_rx(fun(B,fun(A,B)),fun(A,fun(B,B)),F2),N2),aa(A,B,aa(C,fun(A,B),F7,X),N2),F3) )
         => has_field_derivative(B,aa(fun(B,fun(A,B)),fun(B,B),aTP_Lamp_ry(set(A),fun(fun(B,fun(A,B)),fun(B,B)),S),F2),aa(set(A),B,groups7311177749621191930dd_sum(A,B,aa(C,fun(A,B),F7,X)),S),F3) ) ) ).

% DERIV_sum
tff(fact_7007_DERIV__chain,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),Da: A,G: fun(A,A),X: A,Db: A,S3: set(A)] :
          ( has_field_derivative(A,F2,Da,topolo174197925503356063within(A,aa(A,A,G,X),top_top(set(A))))
         => ( has_field_derivative(A,G,Db,topolo174197925503356063within(A,X,S3))
           => has_field_derivative(A,comp(A,A,A,F2,G),aa(A,A,aa(A,fun(A,A),times_times(A),Da),Db),topolo174197925503356063within(A,X,S3)) ) ) ) ).

% DERIV_chain
tff(fact_7008_DERIV__image__chain,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),Da: A,G: fun(A,A),X: A,S3: set(A),Db: A] :
          ( has_field_derivative(A,F2,Da,topolo174197925503356063within(A,aa(A,A,G,X),aa(set(A),set(A),image(A,A,G),S3)))
         => ( has_field_derivative(A,G,Db,topolo174197925503356063within(A,X,S3))
           => has_field_derivative(A,comp(A,A,A,F2,G),aa(A,A,aa(A,fun(A,A),times_times(A),Da),Db),topolo174197925503356063within(A,X,S3)) ) ) ) ).

% DERIV_image_chain
tff(fact_7009_DERIV__at__within__shift__lemma,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),Y: A,Z: A,X: A,S: set(A)] :
          ( has_field_derivative(A,F2,Y,topolo174197925503356063within(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Z),X),aa(set(A),set(A),image(A,A,aa(A,fun(A,A),plus_plus(A),Z)),S)))
         => has_field_derivative(A,comp(A,A,A,F2,aa(A,fun(A,A),plus_plus(A),Z)),Y,topolo174197925503356063within(A,X,S)) ) ) ).

% DERIV_at_within_shift_lemma
tff(fact_7010_DERIV__power__Suc,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D4: A,X: A,S3: set(A),N: nat] :
          ( has_field_derivative(A,F2,D4,topolo174197925503356063within(A,X,S3))
         => has_field_derivative(A,aa(nat,fun(A,A),aTP_Lamp_rz(fun(A,A),fun(nat,fun(A,A)),F2),N),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(nat,A,semiring_1_of_nat(A),N))),aa(A,A,aa(A,fun(A,A),times_times(A),D4),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,F2,X)),N))),topolo174197925503356063within(A,X,S3)) ) ) ).

% DERIV_power_Suc
tff(fact_7011_DERIV__cmult__Id,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [C2: A,X: A,S3: set(A)] : has_field_derivative(A,aa(A,fun(A,A),times_times(A),C2),C2,topolo174197925503356063within(A,X,S3)) ) ).

% DERIV_cmult_Id
tff(fact_7012_has__real__derivative__neg__dec__right,axiom,
    ! [F2: fun(real,real),L: real,X: real,S: set(real)] :
      ( has_field_derivative(real,F2,L,topolo174197925503356063within(real,X,S))
     => ( pp(aa(real,bool,ord_less(real,L),zero_zero(real)))
       => ? [D2: real] :
            ( pp(aa(real,bool,ord_less(real,zero_zero(real)),D2))
            & ! [H5: real] :
                ( pp(aa(real,bool,ord_less(real,zero_zero(real)),H5))
               => ( pp(aa(set(real),bool,aa(real,fun(set(real),bool),member(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),X),H5)),S))
                 => ( pp(aa(real,bool,ord_less(real,H5),D2))
                   => pp(aa(real,bool,ord_less(real,aa(real,real,F2,aa(real,real,aa(real,fun(real,real),plus_plus(real),X),H5))),aa(real,real,F2,X))) ) ) ) ) ) ) ).

% has_real_derivative_neg_dec_right
tff(fact_7013_has__real__derivative__pos__inc__right,axiom,
    ! [F2: fun(real,real),L: real,X: real,S: set(real)] :
      ( has_field_derivative(real,F2,L,topolo174197925503356063within(real,X,S))
     => ( pp(aa(real,bool,ord_less(real,zero_zero(real)),L))
       => ? [D2: real] :
            ( pp(aa(real,bool,ord_less(real,zero_zero(real)),D2))
            & ! [H5: real] :
                ( pp(aa(real,bool,ord_less(real,zero_zero(real)),H5))
               => ( pp(aa(set(real),bool,aa(real,fun(set(real),bool),member(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),X),H5)),S))
                 => ( pp(aa(real,bool,ord_less(real,H5),D2))
                   => pp(aa(real,bool,ord_less(real,aa(real,real,F2,X)),aa(real,real,F2,aa(real,real,aa(real,fun(real,real),plus_plus(real),X),H5)))) ) ) ) ) ) ) ).

% has_real_derivative_pos_inc_right
tff(fact_7014_has__real__derivative__pos__inc__left,axiom,
    ! [F2: fun(real,real),L: real,X: real,S: set(real)] :
      ( has_field_derivative(real,F2,L,topolo174197925503356063within(real,X,S))
     => ( pp(aa(real,bool,ord_less(real,zero_zero(real)),L))
       => ? [D2: real] :
            ( pp(aa(real,bool,ord_less(real,zero_zero(real)),D2))
            & ! [H5: real] :
                ( pp(aa(real,bool,ord_less(real,zero_zero(real)),H5))
               => ( pp(aa(set(real),bool,aa(real,fun(set(real),bool),member(real),aa(real,real,minus_minus(real,X),H5)),S))
                 => ( pp(aa(real,bool,ord_less(real,H5),D2))
                   => pp(aa(real,bool,ord_less(real,aa(real,real,F2,aa(real,real,minus_minus(real,X),H5))),aa(real,real,F2,X))) ) ) ) ) ) ) ).

% has_real_derivative_pos_inc_left
tff(fact_7015_has__real__derivative__neg__dec__left,axiom,
    ! [F2: fun(real,real),L: real,X: real,S: set(real)] :
      ( has_field_derivative(real,F2,L,topolo174197925503356063within(real,X,S))
     => ( pp(aa(real,bool,ord_less(real,L),zero_zero(real)))
       => ? [D2: real] :
            ( pp(aa(real,bool,ord_less(real,zero_zero(real)),D2))
            & ! [H5: real] :
                ( pp(aa(real,bool,ord_less(real,zero_zero(real)),H5))
               => ( pp(aa(set(real),bool,aa(real,fun(set(real),bool),member(real),aa(real,real,minus_minus(real,X),H5)),S))
                 => ( pp(aa(real,bool,ord_less(real,H5),D2))
                   => pp(aa(real,bool,ord_less(real,aa(real,real,F2,X)),aa(real,real,F2,aa(real,real,minus_minus(real,X),H5)))) ) ) ) ) ) ) ).

% has_real_derivative_neg_dec_left
tff(fact_7016_DERIV__const__average,axiom,
    ! [A2: real,B2: real,V: fun(real,real),K: real] :
      ( ( A2 != B2 )
     => ( ! [X3: real] : has_field_derivative(real,V,K,topolo174197925503356063within(real,X3,top_top(set(real))))
       => ( aa(real,real,V,aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),A2),B2)),aa(num,real,numeral_numeral(real),bit0(one2)))) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,V,A2)),aa(real,real,V,B2))),aa(num,real,numeral_numeral(real),bit0(one2))) ) ) ) ).

% DERIV_const_average
tff(fact_7017_DERIV__power,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D4: A,X: A,S3: set(A),N: nat] :
          ( has_field_derivative(A,F2,D4,topolo174197925503356063within(A,X,S3))
         => has_field_derivative(A,aa(nat,fun(A,A),aTP_Lamp_sa(fun(A,A),fun(nat,fun(A,A)),F2),N),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),N)),aa(A,A,aa(A,fun(A,A),times_times(A),D4),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,F2,X)),aa(nat,nat,minus_minus(nat,N),aa(nat,nat,suc,zero_zero(nat)))))),topolo174197925503356063within(A,X,S3)) ) ) ).

% DERIV_power
tff(fact_7018_DERIV__inverse,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [X: A,S3: set(A)] :
          ( ( X != zero_zero(A) )
         => has_field_derivative(A,inverse_inverse(A),aa(A,A,uminus_uminus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,inverse_inverse(A),X)),aa(nat,nat,suc,aa(nat,nat,suc,zero_zero(nat))))),topolo174197925503356063within(A,X,S3)) ) ) ).

% DERIV_inverse
tff(fact_7019_DERIV__local__max,axiom,
    ! [F2: fun(real,real),L: real,X: real,D: real] :
      ( has_field_derivative(real,F2,L,topolo174197925503356063within(real,X,top_top(set(real))))
     => ( pp(aa(real,bool,ord_less(real,zero_zero(real)),D))
       => ( ! [Y3: real] :
              ( pp(aa(real,bool,ord_less(real,aa(real,real,abs_abs(real),aa(real,real,minus_minus(real,X),Y3))),D))
             => pp(aa(real,bool,ord_less_eq(real,aa(real,real,F2,Y3)),aa(real,real,F2,X))) )
         => ( L = zero_zero(real) ) ) ) ) ).

% DERIV_local_max
tff(fact_7020_DERIV__local__min,axiom,
    ! [F2: fun(real,real),L: real,X: real,D: real] :
      ( has_field_derivative(real,F2,L,topolo174197925503356063within(real,X,top_top(set(real))))
     => ( pp(aa(real,bool,ord_less(real,zero_zero(real)),D))
       => ( ! [Y3: real] :
              ( pp(aa(real,bool,ord_less(real,aa(real,real,abs_abs(real),aa(real,real,minus_minus(real,X),Y3))),D))
             => pp(aa(real,bool,ord_less_eq(real,aa(real,real,F2,X)),aa(real,real,F2,Y3))) )
         => ( L = zero_zero(real) ) ) ) ) ).

% DERIV_local_min
tff(fact_7021_DERIV__ln__divide,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,ord_less(real,zero_zero(real)),X))
     => has_field_derivative(real,ln_ln(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),X),topolo174197925503356063within(real,X,top_top(set(real)))) ) ).

% DERIV_ln_divide
tff(fact_7022_DERIV__pow,axiom,
    ! [N: nat,X: real,S3: set(real)] : has_field_derivative(real,aTP_Lamp_sb(nat,fun(real,real),N),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),N)),aa(nat,real,aa(real,fun(nat,real),power_power(real),X),aa(nat,nat,minus_minus(nat,N),aa(nat,nat,suc,zero_zero(nat))))),topolo174197925503356063within(real,X,S3)) ).

% DERIV_pow
tff(fact_7023_termdiffs__strong__converges__everywhere,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [C2: fun(nat,A),X: A] :
          ( ! [Y3: A] : summable(A,aa(A,fun(nat,A),aTP_Lamp_gc(fun(nat,A),fun(A,fun(nat,A)),C2),Y3))
         => has_field_derivative(A,aTP_Lamp_sc(fun(nat,A),fun(A,A),C2),suminf(A,aa(A,fun(nat,A),aTP_Lamp_gd(fun(nat,A),fun(A,fun(nat,A)),C2),X)),topolo174197925503356063within(A,X,top_top(set(A)))) ) ) ).

% termdiffs_strong_converges_everywhere
tff(fact_7024_DERIV__fun__pow,axiom,
    ! [G: fun(real,real),M: real,X: real,N: nat] :
      ( has_field_derivative(real,G,M,topolo174197925503356063within(real,X,top_top(set(real))))
     => has_field_derivative(real,aa(nat,fun(real,real),aTP_Lamp_sd(fun(real,real),fun(nat,fun(real,real)),G),N),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),N)),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,G,X)),aa(nat,nat,minus_minus(nat,N),one_one(nat))))),M),topolo174197925503356063within(real,X,top_top(set(real)))) ) ).

% DERIV_fun_pow
tff(fact_7025_DERIV__quotient,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D: A,X: A,S3: set(A),G: fun(A,A),E: A] :
          ( has_field_derivative(A,F2,D,topolo174197925503356063within(A,X,S3))
         => ( has_field_derivative(A,G,E,topolo174197925503356063within(A,X,S3))
           => ( ( aa(A,A,G,X) != zero_zero(A) )
             => has_field_derivative(A,aa(fun(A,A),fun(A,A),aTP_Lamp_rg(fun(A,A),fun(fun(A,A),fun(A,A)),F2),G),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),D),aa(A,A,G,X))),aa(A,A,aa(A,fun(A,A),times_times(A),E),aa(A,A,F2,X)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,G,X)),aa(nat,nat,suc,aa(nat,nat,suc,zero_zero(nat))))),topolo174197925503356063within(A,X,S3)) ) ) ) ) ).

% DERIV_quotient
tff(fact_7026_DERIV__inverse__fun,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D: A,X: A,S3: set(A)] :
          ( has_field_derivative(A,F2,D,topolo174197925503356063within(A,X,S3))
         => ( ( aa(A,A,F2,X) != zero_zero(A) )
           => has_field_derivative(A,aTP_Lamp_rn(fun(A,A),fun(A,A),F2),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),times_times(A),D),aa(A,A,inverse_inverse(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,F2,X)),aa(nat,nat,suc,aa(nat,nat,suc,zero_zero(nat))))))),topolo174197925503356063within(A,X,S3)) ) ) ) ).

% DERIV_inverse_fun
tff(fact_7027_termdiffs__sums__strong,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [K6: real,C2: fun(nat,A),F2: fun(A,A),F7: A,Z: A] :
          ( ! [Z2: A] :
              ( pp(aa(real,bool,ord_less(real,real_V7770717601297561774m_norm(A,Z2)),K6))
             => pp(aa(A,bool,sums(A,aa(A,fun(nat,A),aTP_Lamp_gc(fun(nat,A),fun(A,fun(nat,A)),C2),Z2)),aa(A,A,F2,Z2))) )
         => ( has_field_derivative(A,F2,F7,topolo174197925503356063within(A,Z,top_top(set(A))))
           => ( pp(aa(real,bool,ord_less(real,real_V7770717601297561774m_norm(A,Z)),K6))
             => pp(aa(A,bool,sums(A,aa(A,fun(nat,A),aTP_Lamp_gd(fun(nat,A),fun(A,fun(nat,A)),C2),Z)),F7)) ) ) ) ) ).

% termdiffs_sums_strong
tff(fact_7028_has__real__derivative__powr,axiom,
    ! [Z: real,R2: real] :
      ( pp(aa(real,bool,ord_less(real,zero_zero(real)),Z))
     => has_field_derivative(real,aTP_Lamp_se(real,fun(real,real),R2),aa(real,real,aa(real,fun(real,real),times_times(real),R2),powr(real,Z,aa(real,real,minus_minus(real,R2),one_one(real)))),topolo174197925503356063within(real,Z,top_top(set(real)))) ) ).

% has_real_derivative_powr
tff(fact_7029_termdiffs,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [C2: fun(nat,A),K6: A,X: A] :
          ( summable(A,aa(A,fun(nat,A),aTP_Lamp_gc(fun(nat,A),fun(A,fun(nat,A)),C2),K6))
         => ( summable(A,aa(A,fun(nat,A),aTP_Lamp_gd(fun(nat,A),fun(A,fun(nat,A)),C2),K6))
           => ( summable(A,aa(A,fun(nat,A),aTP_Lamp_sf(fun(nat,A),fun(A,fun(nat,A)),C2),K6))
             => ( pp(aa(real,bool,ord_less(real,real_V7770717601297561774m_norm(A,X)),real_V7770717601297561774m_norm(A,K6)))
               => has_field_derivative(A,aTP_Lamp_sc(fun(nat,A),fun(A,A),C2),suminf(A,aa(A,fun(nat,A),aTP_Lamp_gd(fun(nat,A),fun(A,fun(nat,A)),C2),X)),topolo174197925503356063within(A,X,top_top(set(A)))) ) ) ) ) ) ).

% termdiffs
tff(fact_7030_termdiffs__strong,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [C2: fun(nat,A),K6: A,X: A] :
          ( summable(A,aa(A,fun(nat,A),aTP_Lamp_gc(fun(nat,A),fun(A,fun(nat,A)),C2),K6))
         => ( pp(aa(real,bool,ord_less(real,real_V7770717601297561774m_norm(A,X)),real_V7770717601297561774m_norm(A,K6)))
           => has_field_derivative(A,aTP_Lamp_sc(fun(nat,A),fun(A,A),C2),suminf(A,aa(A,fun(nat,A),aTP_Lamp_gd(fun(nat,A),fun(A,fun(nat,A)),C2),X)),topolo174197925503356063within(A,X,top_top(set(A)))) ) ) ) ).

% termdiffs_strong
tff(fact_7031_termdiffs__strong_H,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [K6: real,C2: fun(nat,A),Z: A] :
          ( ! [Z2: A] :
              ( pp(aa(real,bool,ord_less(real,real_V7770717601297561774m_norm(A,Z2)),K6))
             => summable(A,aa(A,fun(nat,A),aTP_Lamp_gc(fun(nat,A),fun(A,fun(nat,A)),C2),Z2)) )
         => ( pp(aa(real,bool,ord_less(real,real_V7770717601297561774m_norm(A,Z)),K6))
           => has_field_derivative(A,aTP_Lamp_sc(fun(nat,A),fun(A,A),C2),suminf(A,aa(A,fun(nat,A),aTP_Lamp_gd(fun(nat,A),fun(A,fun(nat,A)),C2),Z)),topolo174197925503356063within(A,Z,top_top(set(A)))) ) ) ) ).

% termdiffs_strong'
tff(fact_7032_DERIV__fun__powr,axiom,
    ! [G: fun(real,real),M: real,X: real,R2: real] :
      ( has_field_derivative(real,G,M,topolo174197925503356063within(real,X,top_top(set(real))))
     => ( pp(aa(real,bool,ord_less(real,zero_zero(real)),aa(real,real,G,X)))
       => has_field_derivative(real,aa(real,fun(real,real),aTP_Lamp_sg(fun(real,real),fun(real,fun(real,real)),G),R2),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),R2),powr(real,aa(real,real,G,X),aa(real,real,minus_minus(real,R2),aa(nat,real,semiring_1_of_nat(real),one_one(nat)))))),M),topolo174197925503356063within(real,X,top_top(set(real)))) ) ) ).

% DERIV_fun_powr
tff(fact_7033_DERIV__log,axiom,
    ! [X: real,B2: real] :
      ( pp(aa(real,bool,ord_less(real,zero_zero(real)),X))
     => has_field_derivative(real,log(B2),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,ln_ln(real),B2)),X)),topolo174197925503356063within(real,X,top_top(set(real)))) ) ).

% DERIV_log
tff(fact_7034_DERIV__powr,axiom,
    ! [G: fun(real,real),M: real,X: real,F2: fun(real,real),R2: real] :
      ( has_field_derivative(real,G,M,topolo174197925503356063within(real,X,top_top(set(real))))
     => ( pp(aa(real,bool,ord_less(real,zero_zero(real)),aa(real,real,G,X)))
       => ( has_field_derivative(real,F2,R2,topolo174197925503356063within(real,X,top_top(set(real))))
         => has_field_derivative(real,aa(fun(real,real),fun(real,real),aTP_Lamp_sh(fun(real,real),fun(fun(real,real),fun(real,real)),G),F2),aa(real,real,aa(real,fun(real,real),times_times(real),powr(real,aa(real,real,G,X),aa(real,real,F2,X))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,aa(real,fun(real,real),times_times(real),R2),aa(real,real,ln_ln(real),aa(real,real,G,X)))),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),times_times(real),M),aa(real,real,F2,X))),aa(real,real,G,X)))),topolo174197925503356063within(real,X,top_top(set(real)))) ) ) ) ).

% DERIV_powr
tff(fact_7035_DERIV__tan,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] :
          ( ( cos(A,X) != zero_zero(A) )
         => has_field_derivative(A,tan(A),aa(A,A,inverse_inverse(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),cos(A,X)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),topolo174197925503356063within(A,X,top_top(set(A)))) ) ) ).

% DERIV_tan
tff(fact_7036_DERIV__real__sqrt,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,ord_less(real,zero_zero(real)),X))
     => has_field_derivative(real,sqrt,aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,inverse_inverse(real),aa(real,real,sqrt,X))),aa(num,real,numeral_numeral(real),bit0(one2))),topolo174197925503356063within(real,X,top_top(set(real)))) ) ).

% DERIV_real_sqrt
tff(fact_7037_DERIV__arctan,axiom,
    ! [X: real] : has_field_derivative(real,arctan,aa(real,real,inverse_inverse(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),aa(nat,real,aa(real,fun(nat,real),power_power(real),X),aa(num,nat,numeral_numeral(nat),bit0(one2))))),topolo174197925503356063within(real,X,top_top(set(real)))) ).

% DERIV_arctan
tff(fact_7038_arsinh__real__has__field__derivative,axiom,
    ! [X: real,A3: set(real)] : has_field_derivative(real,arsinh(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),X),aa(num,nat,numeral_numeral(nat),bit0(one2)))),one_one(real)))),topolo174197925503356063within(real,X,A3)) ).

% arsinh_real_has_field_derivative
tff(fact_7039_DERIV__cot,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] :
          ( ( sin(A,X) != zero_zero(A) )
         => has_field_derivative(A,cot(A),aa(A,A,uminus_uminus(A),aa(A,A,inverse_inverse(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),sin(A,X)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),topolo174197925503356063within(A,X,top_top(set(A)))) ) ) ).

% DERIV_cot
tff(fact_7040_conj__subset__def,axiom,
    ! [A: $tType,A3: set(A),P: fun(A,bool),Q: fun(A,bool)] :
      ( pp(aa(set(A),bool,ord_less_eq(set(A),A3),collect(A,aa(fun(A,bool),fun(A,bool),aTP_Lamp_as(fun(A,bool),fun(fun(A,bool),fun(A,bool)),P),Q))))
    <=> ( pp(aa(set(A),bool,ord_less_eq(set(A),A3),collect(A,P)))
        & pp(aa(set(A),bool,ord_less_eq(set(A),A3),collect(A,Q))) ) ) ).

% conj_subset_def
tff(fact_7041_has__field__derivative__tanh,axiom,
    ! [A11: $tType] :
      ( ( real_Vector_banach(A11)
        & real_V3459762299906320749_field(A11) )
     => ! [G: fun(A11,A11),X: A11,Db: A11,S3: set(A11)] :
          ( ( cosh(A11,aa(A11,A11,G,X)) != zero_zero(A11) )
         => ( has_field_derivative(A11,G,Db,topolo174197925503356063within(A11,X,S3))
           => has_field_derivative(A11,aTP_Lamp_si(fun(A11,A11),fun(A11,A11),G),aa(A11,A11,aa(A11,fun(A11,A11),times_times(A11),aa(A11,A11,minus_minus(A11,one_one(A11)),aa(nat,A11,aa(A11,fun(nat,A11),power_power(A11),aa(A11,A11,tanh(A11),aa(A11,A11,G,X))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),Db),topolo174197925503356063within(A11,X,S3)) ) ) ) ).

% has_field_derivative_tanh
tff(fact_7042_DERIV__real__sqrt__generic,axiom,
    ! [X: real,D4: real] :
      ( ( X != zero_zero(real) )
     => ( ( pp(aa(real,bool,ord_less(real,zero_zero(real)),X))
         => ( D4 = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,inverse_inverse(real),aa(real,real,sqrt,X))),aa(num,real,numeral_numeral(real),bit0(one2))) ) )
       => ( ( pp(aa(real,bool,ord_less(real,X),zero_zero(real)))
           => ( D4 = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,uminus_uminus(real),aa(real,real,inverse_inverse(real),aa(real,real,sqrt,X)))),aa(num,real,numeral_numeral(real),bit0(one2))) ) )
         => has_field_derivative(real,sqrt,D4,topolo174197925503356063within(real,X,top_top(set(real)))) ) ) ) ).

% DERIV_real_sqrt_generic
tff(fact_7043_arcosh__real__has__field__derivative,axiom,
    ! [X: real,A3: set(real)] :
      ( pp(aa(real,bool,ord_less(real,one_one(real)),X))
     => has_field_derivative(real,arcosh(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(real,real,sqrt,aa(real,real,minus_minus(real,aa(nat,real,aa(real,fun(nat,real),power_power(real),X),aa(num,nat,numeral_numeral(nat),bit0(one2)))),one_one(real)))),topolo174197925503356063within(real,X,A3)) ) ).

% arcosh_real_has_field_derivative
tff(fact_7044_artanh__real__has__field__derivative,axiom,
    ! [X: real,A3: set(real)] :
      ( pp(aa(real,bool,ord_less(real,aa(real,real,abs_abs(real),X)),one_one(real)))
     => has_field_derivative(real,artanh(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(real,real,minus_minus(real,one_one(real)),aa(nat,real,aa(real,fun(nat,real),power_power(real),X),aa(num,nat,numeral_numeral(nat),bit0(one2))))),topolo174197925503356063within(real,X,A3)) ) ).

% artanh_real_has_field_derivative
tff(fact_7045_conj__comp__iff,axiom,
    ! [B: $tType,A: $tType,P: fun(B,bool),Q: fun(B,bool),G: fun(A,B),X4: A] :
      ( pp(aa(A,bool,comp(B,bool,A,aa(fun(B,bool),fun(B,bool),aTP_Lamp_sj(fun(B,bool),fun(fun(B,bool),fun(B,bool)),P),Q),G),X4))
    <=> ( pp(aa(A,bool,comp(B,bool,A,P,G),X4))
        & pp(aa(A,bool,comp(B,bool,A,Q,G),X4)) ) ) ).

% conj_comp_iff
tff(fact_7046_DERIV__real__root,axiom,
    ! [N: nat,X: real] :
      ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),N))
     => ( pp(aa(real,bool,ord_less(real,zero_zero(real)),X))
       => has_field_derivative(real,root(N),aa(real,real,inverse_inverse(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),N)),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,root(N),X)),aa(nat,nat,minus_minus(nat,N),aa(nat,nat,suc,zero_zero(nat)))))),topolo174197925503356063within(real,X,top_top(set(real)))) ) ) ).

% DERIV_real_root
tff(fact_7047_DERIV__arccos,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,ord_less(real,aa(real,real,uminus_uminus(real),one_one(real))),X))
     => ( pp(aa(real,bool,ord_less(real,X),one_one(real)))
       => has_field_derivative(real,arccos,aa(real,real,inverse_inverse(real),aa(real,real,uminus_uminus(real),aa(real,real,sqrt,aa(real,real,minus_minus(real,one_one(real)),aa(nat,real,aa(real,fun(nat,real),power_power(real),X),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),topolo174197925503356063within(real,X,top_top(set(real)))) ) ) ).

% DERIV_arccos
tff(fact_7048_DERIV__arcsin,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,ord_less(real,aa(real,real,uminus_uminus(real),one_one(real))),X))
     => ( pp(aa(real,bool,ord_less(real,X),one_one(real)))
       => has_field_derivative(real,arcsin,aa(real,real,inverse_inverse(real),aa(real,real,sqrt,aa(real,real,minus_minus(real,one_one(real)),aa(nat,real,aa(real,fun(nat,real),power_power(real),X),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),topolo174197925503356063within(real,X,top_top(set(real)))) ) ) ).

% DERIV_arcsin
tff(fact_7049_Maclaurin__all__le,axiom,
    ! [Diff: fun(nat,fun(real,real)),F2: fun(real,real),X: real,N: nat] :
      ( ( aa(nat,fun(real,real),Diff,zero_zero(nat)) = F2 )
     => ( ! [M3: nat,X3: real] : has_field_derivative(real,aa(nat,fun(real,real),Diff,M3),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M3)),X3),topolo174197925503356063within(real,X3,top_top(set(real))))
       => ? [T5: real] :
            ( pp(aa(real,bool,ord_less_eq(real,aa(real,real,abs_abs(real),T5)),aa(real,real,abs_abs(real),X)))
            & ( aa(real,real,F2,X) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aa(real,fun(nat,real),aTP_Lamp_sk(fun(nat,fun(real,real)),fun(real,fun(nat,real)),Diff),X)),set_ord_lessThan(nat,N))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(nat,fun(real,real),Diff,N),T5)),semiring_char_0_fact(real,N))),aa(nat,real,aa(real,fun(nat,real),power_power(real),X),N))) ) ) ) ) ).

% Maclaurin_all_le
tff(fact_7050_Maclaurin__all__le__objl,axiom,
    ! [Diff: fun(nat,fun(real,real)),F2: fun(real,real),X: real,N: nat] :
      ( ( ( aa(nat,fun(real,real),Diff,zero_zero(nat)) = F2 )
        & ! [M3: nat,X3: real] : has_field_derivative(real,aa(nat,fun(real,real),Diff,M3),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M3)),X3),topolo174197925503356063within(real,X3,top_top(set(real)))) )
     => ? [T5: real] :
          ( pp(aa(real,bool,ord_less_eq(real,aa(real,real,abs_abs(real),T5)),aa(real,real,abs_abs(real),X)))
          & ( aa(real,real,F2,X) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aa(real,fun(nat,real),aTP_Lamp_sk(fun(nat,fun(real,real)),fun(real,fun(nat,real)),Diff),X)),set_ord_lessThan(nat,N))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(nat,fun(real,real),Diff,N),T5)),semiring_char_0_fact(real,N))),aa(nat,real,aa(real,fun(nat,real),power_power(real),X),N))) ) ) ) ).

% Maclaurin_all_le_objl
tff(fact_7051_DERIV__odd__real__root,axiom,
    ! [N: nat,X: real] :
      ( ~ pp(aa(nat,bool,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N))
     => ( ( X != zero_zero(real) )
       => has_field_derivative(real,root(N),aa(real,real,inverse_inverse(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),N)),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,root(N),X)),aa(nat,nat,minus_minus(nat,N),aa(nat,nat,suc,zero_zero(nat)))))),topolo174197925503356063within(real,X,top_top(set(real)))) ) ) ).

% DERIV_odd_real_root
tff(fact_7052_Maclaurin,axiom,
    ! [H: real,N: nat,Diff: fun(nat,fun(real,real)),F2: fun(real,real)] :
      ( pp(aa(real,bool,ord_less(real,zero_zero(real)),H))
     => ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),N))
       => ( ( aa(nat,fun(real,real),Diff,zero_zero(nat)) = F2 )
         => ( ! [M3: nat,T5: real] :
                ( ( pp(aa(nat,bool,ord_less(nat,M3),N))
                  & pp(aa(real,bool,ord_less_eq(real,zero_zero(real)),T5))
                  & pp(aa(real,bool,ord_less_eq(real,T5),H)) )
               => has_field_derivative(real,aa(nat,fun(real,real),Diff,M3),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M3)),T5),topolo174197925503356063within(real,T5,top_top(set(real)))) )
           => ? [T5: real] :
                ( pp(aa(real,bool,ord_less(real,zero_zero(real)),T5))
                & pp(aa(real,bool,ord_less(real,T5),H))
                & ( aa(real,real,F2,H) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aa(fun(nat,fun(real,real)),fun(nat,real),aTP_Lamp_sl(real,fun(fun(nat,fun(real,real)),fun(nat,real)),H),Diff)),set_ord_lessThan(nat,N))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(nat,fun(real,real),Diff,N),T5)),semiring_char_0_fact(real,N))),aa(nat,real,aa(real,fun(nat,real),power_power(real),H),N))) ) ) ) ) ) ) ).

% Maclaurin
tff(fact_7053_Maclaurin2,axiom,
    ! [H: real,Diff: fun(nat,fun(real,real)),F2: fun(real,real),N: nat] :
      ( pp(aa(real,bool,ord_less(real,zero_zero(real)),H))
     => ( ( aa(nat,fun(real,real),Diff,zero_zero(nat)) = F2 )
       => ( ! [M3: nat,T5: real] :
              ( ( pp(aa(nat,bool,ord_less(nat,M3),N))
                & pp(aa(real,bool,ord_less_eq(real,zero_zero(real)),T5))
                & pp(aa(real,bool,ord_less_eq(real,T5),H)) )
             => has_field_derivative(real,aa(nat,fun(real,real),Diff,M3),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M3)),T5),topolo174197925503356063within(real,T5,top_top(set(real)))) )
         => ? [T5: real] :
              ( pp(aa(real,bool,ord_less(real,zero_zero(real)),T5))
              & pp(aa(real,bool,ord_less_eq(real,T5),H))
              & ( aa(real,real,F2,H) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aa(fun(nat,fun(real,real)),fun(nat,real),aTP_Lamp_sl(real,fun(fun(nat,fun(real,real)),fun(nat,real)),H),Diff)),set_ord_lessThan(nat,N))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(nat,fun(real,real),Diff,N),T5)),semiring_char_0_fact(real,N))),aa(nat,real,aa(real,fun(nat,real),power_power(real),H),N))) ) ) ) ) ) ).

% Maclaurin2
tff(fact_7054_Maclaurin__minus,axiom,
    ! [H: real,N: nat,Diff: fun(nat,fun(real,real)),F2: fun(real,real)] :
      ( pp(aa(real,bool,ord_less(real,H),zero_zero(real)))
     => ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),N))
       => ( ( aa(nat,fun(real,real),Diff,zero_zero(nat)) = F2 )
         => ( ! [M3: nat,T5: real] :
                ( ( pp(aa(nat,bool,ord_less(nat,M3),N))
                  & pp(aa(real,bool,ord_less_eq(real,H),T5))
                  & pp(aa(real,bool,ord_less_eq(real,T5),zero_zero(real))) )
               => has_field_derivative(real,aa(nat,fun(real,real),Diff,M3),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M3)),T5),topolo174197925503356063within(real,T5,top_top(set(real)))) )
           => ? [T5: real] :
                ( pp(aa(real,bool,ord_less(real,H),T5))
                & pp(aa(real,bool,ord_less(real,T5),zero_zero(real)))
                & ( aa(real,real,F2,H) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aa(fun(nat,fun(real,real)),fun(nat,real),aTP_Lamp_sl(real,fun(fun(nat,fun(real,real)),fun(nat,real)),H),Diff)),set_ord_lessThan(nat,N))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(nat,fun(real,real),Diff,N),T5)),semiring_char_0_fact(real,N))),aa(nat,real,aa(real,fun(nat,real),power_power(real),H),N))) ) ) ) ) ) ) ).

% Maclaurin_minus
tff(fact_7055_Maclaurin__all__lt,axiom,
    ! [Diff: fun(nat,fun(real,real)),F2: fun(real,real),N: nat,X: real] :
      ( ( aa(nat,fun(real,real),Diff,zero_zero(nat)) = F2 )
     => ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),N))
       => ( ( X != zero_zero(real) )
         => ( ! [M3: nat,X3: real] : has_field_derivative(real,aa(nat,fun(real,real),Diff,M3),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M3)),X3),topolo174197925503356063within(real,X3,top_top(set(real))))
           => ? [T5: real] :
                ( pp(aa(real,bool,ord_less(real,zero_zero(real)),aa(real,real,abs_abs(real),T5)))
                & pp(aa(real,bool,ord_less(real,aa(real,real,abs_abs(real),T5)),aa(real,real,abs_abs(real),X)))
                & ( aa(real,real,F2,X) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aa(real,fun(nat,real),aTP_Lamp_sk(fun(nat,fun(real,real)),fun(real,fun(nat,real)),Diff),X)),set_ord_lessThan(nat,N))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(nat,fun(real,real),Diff,N),T5)),semiring_char_0_fact(real,N))),aa(nat,real,aa(real,fun(nat,real),power_power(real),X),N))) ) ) ) ) ) ) ).

% Maclaurin_all_lt
tff(fact_7056_Maclaurin__bi__le,axiom,
    ! [Diff: fun(nat,fun(real,real)),F2: fun(real,real),N: nat,X: real] :
      ( ( aa(nat,fun(real,real),Diff,zero_zero(nat)) = F2 )
     => ( ! [M3: nat,T5: real] :
            ( ( pp(aa(nat,bool,ord_less(nat,M3),N))
              & pp(aa(real,bool,ord_less_eq(real,aa(real,real,abs_abs(real),T5)),aa(real,real,abs_abs(real),X))) )
           => has_field_derivative(real,aa(nat,fun(real,real),Diff,M3),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M3)),T5),topolo174197925503356063within(real,T5,top_top(set(real)))) )
       => ? [T5: real] :
            ( pp(aa(real,bool,ord_less_eq(real,aa(real,real,abs_abs(real),T5)),aa(real,real,abs_abs(real),X)))
            & ( aa(real,real,F2,X) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aa(real,fun(nat,real),aTP_Lamp_sk(fun(nat,fun(real,real)),fun(real,fun(nat,real)),Diff),X)),set_ord_lessThan(nat,N))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(nat,fun(real,real),Diff,N),T5)),semiring_char_0_fact(real,N))),aa(nat,real,aa(real,fun(nat,real),power_power(real),X),N))) ) ) ) ) ).

% Maclaurin_bi_le
tff(fact_7057_Taylor,axiom,
    ! [N: nat,Diff: fun(nat,fun(real,real)),F2: fun(real,real),A2: real,B2: real,C2: real,X: real] :
      ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),N))
     => ( ( aa(nat,fun(real,real),Diff,zero_zero(nat)) = F2 )
       => ( ! [M3: nat,T5: real] :
              ( ( pp(aa(nat,bool,ord_less(nat,M3),N))
                & pp(aa(real,bool,ord_less_eq(real,A2),T5))
                & pp(aa(real,bool,ord_less_eq(real,T5),B2)) )
             => has_field_derivative(real,aa(nat,fun(real,real),Diff,M3),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M3)),T5),topolo174197925503356063within(real,T5,top_top(set(real)))) )
         => ( pp(aa(real,bool,ord_less_eq(real,A2),C2))
           => ( pp(aa(real,bool,ord_less_eq(real,C2),B2))
             => ( pp(aa(real,bool,ord_less_eq(real,A2),X))
               => ( pp(aa(real,bool,ord_less_eq(real,X),B2))
                 => ( ( X != C2 )
                   => ? [T5: real] :
                        ( ( pp(aa(real,bool,ord_less(real,X),C2))
                         => ( pp(aa(real,bool,ord_less(real,X),T5))
                            & pp(aa(real,bool,ord_less(real,T5),C2)) ) )
                        & ( ~ pp(aa(real,bool,ord_less(real,X),C2))
                         => ( pp(aa(real,bool,ord_less(real,C2),T5))
                            & pp(aa(real,bool,ord_less(real,T5),X)) ) )
                        & ( aa(real,real,F2,X) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aa(real,fun(nat,real),aa(real,fun(real,fun(nat,real)),aTP_Lamp_sm(fun(nat,fun(real,real)),fun(real,fun(real,fun(nat,real))),Diff),C2),X)),set_ord_lessThan(nat,N))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(nat,fun(real,real),Diff,N),T5)),semiring_char_0_fact(real,N))),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,minus_minus(real,X),C2)),N))) ) ) ) ) ) ) ) ) ) ) ).

% Taylor
tff(fact_7058_Taylor__up,axiom,
    ! [N: nat,Diff: fun(nat,fun(real,real)),F2: fun(real,real),A2: real,B2: real,C2: real] :
      ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),N))
     => ( ( aa(nat,fun(real,real),Diff,zero_zero(nat)) = F2 )
       => ( ! [M3: nat,T5: real] :
              ( ( pp(aa(nat,bool,ord_less(nat,M3),N))
                & pp(aa(real,bool,ord_less_eq(real,A2),T5))
                & pp(aa(real,bool,ord_less_eq(real,T5),B2)) )
             => has_field_derivative(real,aa(nat,fun(real,real),Diff,M3),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M3)),T5),topolo174197925503356063within(real,T5,top_top(set(real)))) )
         => ( pp(aa(real,bool,ord_less_eq(real,A2),C2))
           => ( pp(aa(real,bool,ord_less(real,C2),B2))
             => ? [T5: real] :
                  ( pp(aa(real,bool,ord_less(real,C2),T5))
                  & pp(aa(real,bool,ord_less(real,T5),B2))
                  & ( aa(real,real,F2,B2) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aa(real,fun(nat,real),aa(real,fun(real,fun(nat,real)),aTP_Lamp_sn(fun(nat,fun(real,real)),fun(real,fun(real,fun(nat,real))),Diff),B2),C2)),set_ord_lessThan(nat,N))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(nat,fun(real,real),Diff,N),T5)),semiring_char_0_fact(real,N))),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,minus_minus(real,B2),C2)),N))) ) ) ) ) ) ) ) ).

% Taylor_up
tff(fact_7059_Taylor__down,axiom,
    ! [N: nat,Diff: fun(nat,fun(real,real)),F2: fun(real,real),A2: real,B2: real,C2: real] :
      ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),N))
     => ( ( aa(nat,fun(real,real),Diff,zero_zero(nat)) = F2 )
       => ( ! [M3: nat,T5: real] :
              ( ( pp(aa(nat,bool,ord_less(nat,M3),N))
                & pp(aa(real,bool,ord_less_eq(real,A2),T5))
                & pp(aa(real,bool,ord_less_eq(real,T5),B2)) )
             => has_field_derivative(real,aa(nat,fun(real,real),Diff,M3),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M3)),T5),topolo174197925503356063within(real,T5,top_top(set(real)))) )
         => ( pp(aa(real,bool,ord_less(real,A2),C2))
           => ( pp(aa(real,bool,ord_less_eq(real,C2),B2))
             => ? [T5: real] :
                  ( pp(aa(real,bool,ord_less(real,A2),T5))
                  & pp(aa(real,bool,ord_less(real,T5),C2))
                  & ( aa(real,real,F2,A2) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aa(real,fun(nat,real),aa(real,fun(real,fun(nat,real)),aTP_Lamp_sn(fun(nat,fun(real,real)),fun(real,fun(real,fun(nat,real))),Diff),A2),C2)),set_ord_lessThan(nat,N))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(nat,fun(real,real),Diff,N),T5)),semiring_char_0_fact(real,N))),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,minus_minus(real,A2),C2)),N))) ) ) ) ) ) ) ) ).

% Taylor_down
tff(fact_7060_Maclaurin__lemma2,axiom,
    ! [N: nat,H: real,Diff: fun(nat,fun(real,real)),K: nat,B4: real] :
      ( ! [M3: nat,T5: real] :
          ( ( pp(aa(nat,bool,ord_less(nat,M3),N))
            & pp(aa(real,bool,ord_less_eq(real,zero_zero(real)),T5))
            & pp(aa(real,bool,ord_less_eq(real,T5),H)) )
         => has_field_derivative(real,aa(nat,fun(real,real),Diff,M3),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M3)),T5),topolo174197925503356063within(real,T5,top_top(set(real)))) )
     => ( ( N = aa(nat,nat,suc,K) )
       => ! [M4: nat,T7: real] :
            ( ( pp(aa(nat,bool,ord_less(nat,M4),N))
              & pp(aa(real,bool,ord_less_eq(real,zero_zero(real)),T7))
              & pp(aa(real,bool,ord_less_eq(real,T7),H)) )
           => has_field_derivative(real,aa(nat,fun(real,real),aa(real,fun(nat,fun(real,real)),aa(fun(nat,fun(real,real)),fun(real,fun(nat,fun(real,real))),aTP_Lamp_sp(nat,fun(fun(nat,fun(real,real)),fun(real,fun(nat,fun(real,real)))),N),Diff),B4),M4),aa(real,real,minus_minus(real,aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M4)),T7)),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aa(real,fun(nat,real),aa(nat,fun(real,fun(nat,real)),aTP_Lamp_sq(fun(nat,fun(real,real)),fun(nat,fun(real,fun(nat,real))),Diff),M4),T7)),set_ord_lessThan(nat,aa(nat,nat,minus_minus(nat,N),aa(nat,nat,suc,M4))))),aa(real,real,aa(real,fun(real,real),times_times(real),B4),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),T7),aa(nat,nat,minus_minus(nat,N),aa(nat,nat,suc,M4)))),semiring_char_0_fact(real,aa(nat,nat,minus_minus(nat,N),aa(nat,nat,suc,M4))))))),topolo174197925503356063within(real,T7,top_top(set(real)))) ) ) ) ).

% Maclaurin_lemma2
tff(fact_7061_DERIV__arctan__series,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,ord_less(real,aa(real,real,abs_abs(real),X)),one_one(real)))
     => has_field_derivative(real,aTP_Lamp_sr(real,real),suminf(real,aTP_Lamp_ss(real,fun(nat,real),X)),topolo174197925503356063within(real,X,top_top(set(real)))) ) ).

% DERIV_arctan_series
tff(fact_7062_DERIV__real__root__generic,axiom,
    ! [N: nat,X: real,D4: real] :
      ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),N))
     => ( ( X != zero_zero(real) )
       => ( ( pp(aa(nat,bool,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N))
           => ( pp(aa(real,bool,ord_less(real,zero_zero(real)),X))
             => ( D4 = aa(real,real,inverse_inverse(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),N)),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,root(N),X)),aa(nat,nat,minus_minus(nat,N),aa(nat,nat,suc,zero_zero(nat)))))) ) ) )
         => ( ( pp(aa(nat,bool,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N))
             => ( pp(aa(real,bool,ord_less(real,X),zero_zero(real)))
               => ( D4 = aa(real,real,uminus_uminus(real),aa(real,real,inverse_inverse(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),N)),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,root(N),X)),aa(nat,nat,minus_minus(nat,N),aa(nat,nat,suc,zero_zero(nat))))))) ) ) )
           => ( ( ~ pp(aa(nat,bool,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N))
               => ( D4 = aa(real,real,inverse_inverse(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),N)),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,root(N),X)),aa(nat,nat,minus_minus(nat,N),aa(nat,nat,suc,zero_zero(nat)))))) ) )
             => has_field_derivative(real,root(N),D4,topolo174197925503356063within(real,X,top_top(set(real)))) ) ) ) ) ) ).

% DERIV_real_root_generic
tff(fact_7063_DERIV__power__series_H,axiom,
    ! [R: real,F2: fun(nat,real),X0: real] :
      ( ! [X3: real] :
          ( pp(aa(set(real),bool,aa(real,fun(set(real),bool),member(real),X3),set_or5935395276787703475ssThan(real,aa(real,real,uminus_uminus(real),R),R)))
         => summable(real,aa(real,fun(nat,real),aTP_Lamp_st(fun(nat,real),fun(real,fun(nat,real)),F2),X3)) )
     => ( pp(aa(set(real),bool,aa(real,fun(set(real),bool),member(real),X0),set_or5935395276787703475ssThan(real,aa(real,real,uminus_uminus(real),R),R)))
       => ( pp(aa(real,bool,ord_less(real,zero_zero(real)),R))
         => has_field_derivative(real,aTP_Lamp_sv(fun(nat,real),fun(real,real),F2),suminf(real,aa(real,fun(nat,real),aTP_Lamp_st(fun(nat,real),fun(real,fun(nat,real)),F2),X0)),topolo174197925503356063within(real,X0,top_top(set(real)))) ) ) ) ).

% DERIV_power_series'
tff(fact_7064_has__derivative__arcsin,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [G: fun(A,real),X: A,G4: fun(A,real),S3: set(A)] :
          ( pp(aa(real,bool,ord_less(real,aa(real,real,uminus_uminus(real),one_one(real))),aa(A,real,G,X)))
         => ( pp(aa(real,bool,ord_less(real,aa(A,real,G,X)),one_one(real)))
           => ( has_derivative(A,real,G,G4,topolo174197925503356063within(A,X,S3))
             => has_derivative(A,real,aTP_Lamp_sw(fun(A,real),fun(A,real),G),aa(fun(A,real),fun(A,real),aa(A,fun(fun(A,real),fun(A,real)),aTP_Lamp_sx(fun(A,real),fun(A,fun(fun(A,real),fun(A,real))),G),X),G4),topolo174197925503356063within(A,X,S3)) ) ) ) ) ).

% has_derivative_arcsin
tff(fact_7065_greaterThanLessThan__empty__iff2,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A2: A,B2: A] :
          ( ( bot_bot(set(A)) = set_or5935395276787703475ssThan(A,A2,B2) )
        <=> pp(aa(A,bool,ord_less_eq(A,B2),A2)) ) ) ).

% greaterThanLessThan_empty_iff2
tff(fact_7066_greaterThanLessThan__empty__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A2: A,B2: A] :
          ( ( set_or5935395276787703475ssThan(A,A2,B2) = bot_bot(set(A)) )
        <=> pp(aa(A,bool,ord_less_eq(A,B2),A2)) ) ) ).

% greaterThanLessThan_empty_iff
tff(fact_7067_greaterThanLessThan__empty,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [L: A,K: A] :
          ( pp(aa(A,bool,ord_less_eq(A,L),K))
         => ( set_or5935395276787703475ssThan(A,K,L) = bot_bot(set(A)) ) ) ) ).

% greaterThanLessThan_empty
tff(fact_7068_has__derivative__subset,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),F7: fun(A,B),X: A,S3: set(A),T: set(A)] :
          ( has_derivative(A,B,F2,F7,topolo174197925503356063within(A,X,S3))
         => ( pp(aa(set(A),bool,ord_less_eq(set(A),T),S3))
           => has_derivative(A,B,F2,F7,topolo174197925503356063within(A,X,T)) ) ) ) ).

% has_derivative_subset
tff(fact_7069_has__derivative__scaleR,axiom,
    ! [C: $tType,D6: $tType] :
      ( ( real_V822414075346904944vector(D6)
        & real_V822414075346904944vector(C) )
     => ! [F2: fun(D6,real),F7: fun(D6,real),X: D6,S3: set(D6),G: fun(D6,C),G4: fun(D6,C)] :
          ( has_derivative(D6,real,F2,F7,topolo174197925503356063within(D6,X,S3))
         => ( has_derivative(D6,C,G,G4,topolo174197925503356063within(D6,X,S3))
           => has_derivative(D6,C,aa(fun(D6,C),fun(D6,C),aTP_Lamp_sy(fun(D6,real),fun(fun(D6,C),fun(D6,C)),F2),G),aa(fun(D6,C),fun(D6,C),aa(fun(D6,C),fun(fun(D6,C),fun(D6,C)),aa(D6,fun(fun(D6,C),fun(fun(D6,C),fun(D6,C))),aa(fun(D6,real),fun(D6,fun(fun(D6,C),fun(fun(D6,C),fun(D6,C)))),aTP_Lamp_sz(fun(D6,real),fun(fun(D6,real),fun(D6,fun(fun(D6,C),fun(fun(D6,C),fun(D6,C))))),F2),F7),X),G),G4),topolo174197925503356063within(D6,X,S3)) ) ) ) ).

% has_derivative_scaleR
tff(fact_7070_has__derivative__in__compose,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(C)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),F7: fun(A,B),X: A,S3: set(A),G: fun(B,C),G4: fun(B,C)] :
          ( has_derivative(A,B,F2,F7,topolo174197925503356063within(A,X,S3))
         => ( has_derivative(B,C,G,G4,topolo174197925503356063within(B,aa(A,B,F2,X),aa(set(A),set(B),image(A,B,F2),S3)))
           => has_derivative(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_ta(fun(A,B),fun(fun(B,C),fun(A,C)),F2),G),aa(fun(B,C),fun(A,C),aTP_Lamp_ta(fun(A,B),fun(fun(B,C),fun(A,C)),F7),G4),topolo174197925503356063within(A,X,S3)) ) ) ) ).

% has_derivative_in_compose
tff(fact_7071_has__derivative__compose,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(C)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),F7: fun(A,B),X: A,S3: set(A),G: fun(B,C),G4: fun(B,C)] :
          ( has_derivative(A,B,F2,F7,topolo174197925503356063within(A,X,S3))
         => ( has_derivative(B,C,G,G4,topolo174197925503356063within(B,aa(A,B,F2,X),top_top(set(B))))
           => has_derivative(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_ta(fun(A,B),fun(fun(B,C),fun(A,C)),F2),G),aa(fun(B,C),fun(A,C),aTP_Lamp_ta(fun(A,B),fun(fun(B,C),fun(A,C)),F7),G4),topolo174197925503356063within(A,X,S3)) ) ) ) ).

% has_derivative_compose
tff(fact_7072_has__field__derivative__imp__has__derivative,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D4: A,F3: filter(A)] :
          ( has_field_derivative(A,F2,D4,F3)
         => has_derivative(A,A,F2,aa(A,fun(A,A),times_times(A),D4),F3) ) ) ).

% has_field_derivative_imp_has_derivative
tff(fact_7073_has__derivative__imp__has__field__derivative,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D4: fun(A,A),F3: filter(A),D7: A] :
          ( has_derivative(A,A,F2,D4,F3)
         => ( ! [X3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),X3),D7) = aa(A,A,D4,X3)
           => has_field_derivative(A,F2,D7,F3) ) ) ) ).

% has_derivative_imp_has_field_derivative
tff(fact_7074_has__field__derivative__def,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D4: A,F3: filter(A)] :
          ( has_field_derivative(A,F2,D4,F3)
        <=> has_derivative(A,A,F2,aa(A,fun(A,A),times_times(A),D4),F3) ) ) ).

% has_field_derivative_def
tff(fact_7075_has__derivative__sum,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V822414075346904944vector(C) )
     => ! [I5: set(A),F2: fun(A,fun(B,C)),F7: fun(A,fun(B,C)),F3: filter(B)] :
          ( ! [I2: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),I2),I5))
             => has_derivative(B,C,aa(A,fun(B,C),F2,I2),aa(A,fun(B,C),F7,I2),F3) )
         => has_derivative(B,C,aa(fun(A,fun(B,C)),fun(B,C),aTP_Lamp_tc(set(A),fun(fun(A,fun(B,C)),fun(B,C)),I5),F2),aa(fun(A,fun(B,C)),fun(B,C),aTP_Lamp_tc(set(A),fun(fun(A,fun(B,C)),fun(B,C)),I5),F7),F3) ) ) ).

% has_derivative_sum
tff(fact_7076_has__derivative__mult__left,axiom,
    ! [A: $tType,C: $tType] :
      ( ( real_V822414075346904944vector(C)
        & real_V4412858255891104859lgebra(A) )
     => ! [G: fun(C,A),G4: fun(C,A),F3: filter(C),Y: A] :
          ( has_derivative(C,A,G,G4,F3)
         => has_derivative(C,A,aa(A,fun(C,A),aTP_Lamp_td(fun(C,A),fun(A,fun(C,A)),G),Y),aa(A,fun(C,A),aTP_Lamp_td(fun(C,A),fun(A,fun(C,A)),G4),Y),F3) ) ) ).

% has_derivative_mult_left
tff(fact_7077_has__derivative__mult__right,axiom,
    ! [A: $tType,C: $tType] :
      ( ( real_V822414075346904944vector(C)
        & real_V4412858255891104859lgebra(A) )
     => ! [G: fun(C,A),G4: fun(C,A),F3: filter(C),X: A] :
          ( has_derivative(C,A,G,G4,F3)
         => has_derivative(C,A,aa(A,fun(C,A),aTP_Lamp_te(fun(C,A),fun(A,fun(C,A)),G),X),aa(A,fun(C,A),aTP_Lamp_te(fun(C,A),fun(A,fun(C,A)),G4),X),F3) ) ) ).

% has_derivative_mult_right
tff(fact_7078_has__derivative__add,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),F7: fun(A,B),F3: filter(A),G: fun(A,B),G4: fun(A,B)] :
          ( has_derivative(A,B,F2,F7,F3)
         => ( has_derivative(A,B,G,G4,F3)
           => has_derivative(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_tf(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G),aa(fun(A,B),fun(A,B),aTP_Lamp_tf(fun(A,B),fun(fun(A,B),fun(A,B)),F7),G4),F3) ) ) ) ).

% has_derivative_add
tff(fact_7079_has__derivative__const,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [C2: B,F3: filter(A)] : has_derivative(A,B,aTP_Lamp_tg(B,fun(A,B),C2),aTP_Lamp_th(A,B),F3) ) ).

% has_derivative_const
tff(fact_7080_has__derivative__ident,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F3: filter(A)] : has_derivative(A,A,aTP_Lamp_ti(A,A),aTP_Lamp_ti(A,A),F3) ) ).

% has_derivative_ident
tff(fact_7081_has__derivative__of__real,axiom,
    ! [A: $tType,C: $tType] :
      ( ( real_V822414075346904944vector(C)
        & real_V2191834092415804123ebra_1(A)
        & real_V822414075346904944vector(A) )
     => ! [G: fun(C,real),G4: fun(C,real),F3: filter(C)] :
          ( has_derivative(C,real,G,G4,F3)
         => has_derivative(C,A,aTP_Lamp_tj(fun(C,real),fun(C,A),G),aTP_Lamp_tj(fun(C,real),fun(C,A),G4),F3) ) ) ).

% has_derivative_of_real
tff(fact_7082_has__derivative__scaleR__left,axiom,
    ! [B: $tType,C: $tType] :
      ( ( real_V822414075346904944vector(C)
        & real_V822414075346904944vector(B) )
     => ! [G: fun(C,real),G4: fun(C,real),F3: filter(C),X: B] :
          ( has_derivative(C,real,G,G4,F3)
         => has_derivative(C,B,aa(B,fun(C,B),aTP_Lamp_tk(fun(C,real),fun(B,fun(C,B)),G),X),aa(B,fun(C,B),aTP_Lamp_tk(fun(C,real),fun(B,fun(C,B)),G4),X),F3) ) ) ).

% has_derivative_scaleR_left
tff(fact_7083_has__derivative__scaleR__right,axiom,
    ! [B: $tType,C: $tType] :
      ( ( real_V822414075346904944vector(C)
        & real_V822414075346904944vector(B) )
     => ! [G: fun(C,B),G4: fun(C,B),F3: filter(C),R2: real] :
          ( has_derivative(C,B,G,G4,F3)
         => has_derivative(C,B,aa(real,fun(C,B),aTP_Lamp_tl(fun(C,B),fun(real,fun(C,B)),G),R2),aa(real,fun(C,B),aTP_Lamp_tl(fun(C,B),fun(real,fun(C,B)),G4),R2),F3) ) ) ).

% has_derivative_scaleR_right
tff(fact_7084_has__derivative__diff,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),F7: fun(A,B),F3: filter(A),G: fun(A,B),G4: fun(A,B)] :
          ( has_derivative(A,B,F2,F7,F3)
         => ( has_derivative(A,B,G,G4,F3)
           => has_derivative(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_tm(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G),aa(fun(A,B),fun(A,B),aTP_Lamp_tm(fun(A,B),fun(fun(A,B),fun(A,B)),F7),G4),F3) ) ) ) ).

% has_derivative_diff
tff(fact_7085_has__derivative__minus,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),F7: fun(A,B),F3: filter(A)] :
          ( has_derivative(A,B,F2,F7,F3)
         => has_derivative(A,B,aTP_Lamp_tn(fun(A,B),fun(A,B),F2),aTP_Lamp_tn(fun(A,B),fun(A,B),F7),F3) ) ) ).

% has_derivative_minus
tff(fact_7086_has__derivative__mult,axiom,
    ! [A: $tType,D6: $tType] :
      ( ( real_V822414075346904944vector(D6)
        & real_V4412858255891104859lgebra(A) )
     => ! [F2: fun(D6,A),F7: fun(D6,A),X: D6,S3: set(D6),G: fun(D6,A),G4: fun(D6,A)] :
          ( has_derivative(D6,A,F2,F7,topolo174197925503356063within(D6,X,S3))
         => ( has_derivative(D6,A,G,G4,topolo174197925503356063within(D6,X,S3))
           => has_derivative(D6,A,aa(fun(D6,A),fun(D6,A),aTP_Lamp_to(fun(D6,A),fun(fun(D6,A),fun(D6,A)),F2),G),aa(fun(D6,A),fun(D6,A),aa(fun(D6,A),fun(fun(D6,A),fun(D6,A)),aa(D6,fun(fun(D6,A),fun(fun(D6,A),fun(D6,A))),aa(fun(D6,A),fun(D6,fun(fun(D6,A),fun(fun(D6,A),fun(D6,A)))),aTP_Lamp_tp(fun(D6,A),fun(fun(D6,A),fun(D6,fun(fun(D6,A),fun(fun(D6,A),fun(D6,A))))),F2),F7),X),G),G4),topolo174197925503356063within(D6,X,S3)) ) ) ) ).

% has_derivative_mult
tff(fact_7087_has__derivative__zero__unique,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V822414075346904944vector(A) )
     => ! [F3: fun(A,B),X: A] :
          ( has_derivative(A,B,aTP_Lamp_th(A,B),F3,topolo174197925503356063within(A,X,top_top(set(A))))
         => ! [X4: A] : aa(A,B,F3,X4) = zero_zero(B) ) ) ).

% has_derivative_zero_unique
tff(fact_7088_has__derivative__in__compose2,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( ( real_V822414075346904944vector(C)
        & real_V822414075346904944vector(B)
        & real_V822414075346904944vector(A) )
     => ! [T: set(A),G: fun(A,B),G4: fun(A,fun(A,B)),F2: fun(C,A),S3: set(C),X: C,F7: fun(C,A)] :
          ( ! [X3: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),T))
             => has_derivative(A,B,G,aa(A,fun(A,B),G4,X3),topolo174197925503356063within(A,X3,T)) )
         => ( pp(aa(set(A),bool,ord_less_eq(set(A),aa(set(C),set(A),image(C,A,F2),S3)),T))
           => ( pp(aa(set(C),bool,aa(C,fun(set(C),bool),member(C),X),S3))
             => ( has_derivative(C,A,F2,F7,topolo174197925503356063within(C,X,S3))
               => has_derivative(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_tq(fun(A,B),fun(fun(C,A),fun(C,B)),G),F2),aa(fun(C,A),fun(C,B),aa(C,fun(fun(C,A),fun(C,B)),aa(fun(C,A),fun(C,fun(fun(C,A),fun(C,B))),aTP_Lamp_tr(fun(A,fun(A,B)),fun(fun(C,A),fun(C,fun(fun(C,A),fun(C,B)))),G4),F2),X),F7),topolo174197925503356063within(C,X,S3)) ) ) ) ) ) ).

% has_derivative_in_compose2
tff(fact_7089_has__derivative__exp,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [G: fun(A,real),G4: fun(A,real),X: A,S3: set(A)] :
          ( has_derivative(A,real,G,G4,topolo174197925503356063within(A,X,S3))
         => has_derivative(A,real,aTP_Lamp_ts(fun(A,real),fun(A,real),G),aa(A,fun(A,real),aa(fun(A,real),fun(A,fun(A,real)),aTP_Lamp_tt(fun(A,real),fun(fun(A,real),fun(A,fun(A,real))),G),G4),X),topolo174197925503356063within(A,X,S3)) ) ) ).

% has_derivative_exp
tff(fact_7090_greaterThanLessThan__subseteq__greaterThanLessThan,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A2: A,B2: A,C2: A,D: A] :
          ( pp(aa(set(A),bool,ord_less_eq(set(A),set_or5935395276787703475ssThan(A,A2,B2)),set_or5935395276787703475ssThan(A,C2,D)))
        <=> ( pp(aa(A,bool,ord_less(A,A2),B2))
           => ( pp(aa(A,bool,ord_less_eq(A,C2),A2))
              & pp(aa(A,bool,ord_less_eq(A,B2),D)) ) ) ) ) ).

% greaterThanLessThan_subseteq_greaterThanLessThan
tff(fact_7091_tanh__real__bounds,axiom,
    ! [X: real] : pp(aa(set(real),bool,aa(real,fun(set(real),bool),member(real),aa(real,real,tanh(real),X)),set_or5935395276787703475ssThan(real,aa(real,real,uminus_uminus(real),one_one(real)),one_one(real)))) ).

% tanh_real_bounds
tff(fact_7092_has__derivative__sinh,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [G: fun(A,A),Db: A,X: A,S3: set(A)] :
          ( has_derivative(A,A,G,aa(A,fun(A,A),times_times(A),Db),topolo174197925503356063within(A,X,S3))
         => has_derivative(A,A,aTP_Lamp_tu(fun(A,A),fun(A,A),G),aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),cosh(A,aa(A,A,G,X))),Db)),topolo174197925503356063within(A,X,S3)) ) ) ).

% has_derivative_sinh
tff(fact_7093_has__derivative__cosh,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [G: fun(A,A),Db: A,X: A,S3: set(A)] :
          ( has_derivative(A,A,G,aa(A,fun(A,A),times_times(A),Db),topolo174197925503356063within(A,X,S3))
         => has_derivative(A,A,aTP_Lamp_tv(fun(A,A),fun(A,A),G),aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),sinh(A,aa(A,A,G,X))),Db)),topolo174197925503356063within(A,X,S3)) ) ) ).

% has_derivative_cosh
tff(fact_7094_has__derivative__sin,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [G: fun(A,real),G4: fun(A,real),X: A,S3: set(A)] :
          ( has_derivative(A,real,G,G4,topolo174197925503356063within(A,X,S3))
         => has_derivative(A,real,aTP_Lamp_tw(fun(A,real),fun(A,real),G),aa(A,fun(A,real),aa(fun(A,real),fun(A,fun(A,real)),aTP_Lamp_tx(fun(A,real),fun(fun(A,real),fun(A,fun(A,real))),G),G4),X),topolo174197925503356063within(A,X,S3)) ) ) ).

% has_derivative_sin
tff(fact_7095_greaterThanLessThan__subseteq__atLeastAtMost__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A2: A,B2: A,C2: A,D: A] :
          ( pp(aa(set(A),bool,ord_less_eq(set(A),set_or5935395276787703475ssThan(A,A2,B2)),set_or1337092689740270186AtMost(A,C2,D)))
        <=> ( pp(aa(A,bool,ord_less(A,A2),B2))
           => ( pp(aa(A,bool,ord_less_eq(A,C2),A2))
              & pp(aa(A,bool,ord_less_eq(A,B2),D)) ) ) ) ) ).

% greaterThanLessThan_subseteq_atLeastAtMost_iff
tff(fact_7096_greaterThanLessThan__subseteq__atLeastLessThan__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A2: A,B2: A,C2: A,D: A] :
          ( pp(aa(set(A),bool,ord_less_eq(set(A),set_or5935395276787703475ssThan(A,A2,B2)),set_or7035219750837199246ssThan(A,C2,D)))
        <=> ( pp(aa(A,bool,ord_less(A,A2),B2))
           => ( pp(aa(A,bool,ord_less_eq(A,C2),A2))
              & pp(aa(A,bool,ord_less_eq(A,B2),D)) ) ) ) ) ).

% greaterThanLessThan_subseteq_atLeastLessThan_iff
tff(fact_7097_has__derivative__divide_H,axiom,
    ! [A: $tType,C: $tType] :
      ( ( real_V822414075346904944vector(C)
        & real_V3459762299906320749_field(A) )
     => ! [F2: fun(C,A),F7: fun(C,A),X: C,S: set(C),G: fun(C,A),G4: fun(C,A)] :
          ( has_derivative(C,A,F2,F7,topolo174197925503356063within(C,X,S))
         => ( has_derivative(C,A,G,G4,topolo174197925503356063within(C,X,S))
           => ( ( aa(C,A,G,X) != zero_zero(A) )
             => has_derivative(C,A,aa(fun(C,A),fun(C,A),aTP_Lamp_ty(fun(C,A),fun(fun(C,A),fun(C,A)),F2),G),aa(fun(C,A),fun(C,A),aa(fun(C,A),fun(fun(C,A),fun(C,A)),aa(C,fun(fun(C,A),fun(fun(C,A),fun(C,A))),aa(fun(C,A),fun(C,fun(fun(C,A),fun(fun(C,A),fun(C,A)))),aTP_Lamp_tz(fun(C,A),fun(fun(C,A),fun(C,fun(fun(C,A),fun(fun(C,A),fun(C,A))))),F2),F7),X),G),G4),topolo174197925503356063within(C,X,S)) ) ) ) ) ).

% has_derivative_divide'
tff(fact_7098_has__derivative__inverse_H,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [X: A,S: set(A)] :
          ( ( X != zero_zero(A) )
         => has_derivative(A,A,inverse_inverse(A),aTP_Lamp_ua(A,fun(A,A),X),topolo174197925503356063within(A,X,S)) ) ) ).

% has_derivative_inverse'
tff(fact_7099_has__derivative__inverse,axiom,
    ! [A: $tType,C: $tType] :
      ( ( real_V822414075346904944vector(C)
        & real_V8999393235501362500lgebra(A) )
     => ! [F2: fun(C,A),X: C,F7: fun(C,A),S: set(C)] :
          ( ( aa(C,A,F2,X) != zero_zero(A) )
         => ( has_derivative(C,A,F2,F7,topolo174197925503356063within(C,X,S))
           => has_derivative(C,A,aTP_Lamp_ub(fun(C,A),fun(C,A),F2),aa(fun(C,A),fun(C,A),aa(C,fun(fun(C,A),fun(C,A)),aTP_Lamp_uc(fun(C,A),fun(C,fun(fun(C,A),fun(C,A))),F2),X),F7),topolo174197925503356063within(C,X,S)) ) ) ) ).

% has_derivative_inverse
tff(fact_7100_DERIV__compose__FDERIV,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(real,real),F7: real,G: fun(A,real),X: A,G4: fun(A,real),S3: set(A)] :
          ( has_field_derivative(real,F2,F7,topolo174197925503356063within(real,aa(A,real,G,X),top_top(set(real))))
         => ( has_derivative(A,real,G,G4,topolo174197925503356063within(A,X,S3))
           => has_derivative(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_ud(fun(real,real),fun(fun(A,real),fun(A,real)),F2),G),aa(fun(A,real),fun(A,real),aTP_Lamp_ue(real,fun(fun(A,real),fun(A,real)),F7),G4),topolo174197925503356063within(A,X,S3)) ) ) ) ).

% DERIV_compose_FDERIV
tff(fact_7101_has__derivative__cos,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [G: fun(A,real),G4: fun(A,real),X: A,S3: set(A)] :
          ( has_derivative(A,real,G,G4,topolo174197925503356063within(A,X,S3))
         => has_derivative(A,real,aTP_Lamp_uf(fun(A,real),fun(A,real),G),aa(A,fun(A,real),aa(fun(A,real),fun(A,fun(A,real)),aTP_Lamp_ug(fun(A,real),fun(fun(A,real),fun(A,fun(A,real))),G),G4),X),topolo174197925503356063within(A,X,S3)) ) ) ).

% has_derivative_cos
tff(fact_7102_DERIV__isconst3,axiom,
    ! [A2: real,B2: real,X: real,Y: real,F2: fun(real,real)] :
      ( pp(aa(real,bool,ord_less(real,A2),B2))
     => ( pp(aa(set(real),bool,aa(real,fun(set(real),bool),member(real),X),set_or5935395276787703475ssThan(real,A2,B2)))
       => ( pp(aa(set(real),bool,aa(real,fun(set(real),bool),member(real),Y),set_or5935395276787703475ssThan(real,A2,B2)))
         => ( ! [X3: real] :
                ( pp(aa(set(real),bool,aa(real,fun(set(real),bool),member(real),X3),set_or5935395276787703475ssThan(real,A2,B2)))
               => has_field_derivative(real,F2,zero_zero(real),topolo174197925503356063within(real,X3,top_top(set(real)))) )
           => ( aa(real,real,F2,X) = aa(real,real,F2,Y) ) ) ) ) ) ).

% DERIV_isconst3
tff(fact_7103_has__derivative__power,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V3459762299906320749_field(B) )
     => ! [F2: fun(A,B),F7: fun(A,B),X: A,S: set(A),N: nat] :
          ( has_derivative(A,B,F2,F7,topolo174197925503356063within(A,X,S))
         => has_derivative(A,B,aa(nat,fun(A,B),aTP_Lamp_uh(fun(A,B),fun(nat,fun(A,B)),F2),N),aa(nat,fun(A,B),aa(A,fun(nat,fun(A,B)),aa(fun(A,B),fun(A,fun(nat,fun(A,B))),aTP_Lamp_ui(fun(A,B),fun(fun(A,B),fun(A,fun(nat,fun(A,B)))),F2),F7),X),N),topolo174197925503356063within(A,X,S)) ) ) ).

% has_derivative_power
tff(fact_7104_has__derivative__ln,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [G: fun(A,real),X: A,G4: fun(A,real),S3: set(A)] :
          ( pp(aa(real,bool,ord_less(real,zero_zero(real)),aa(A,real,G,X)))
         => ( has_derivative(A,real,G,G4,topolo174197925503356063within(A,X,S3))
           => has_derivative(A,real,aTP_Lamp_uj(fun(A,real),fun(A,real),G),aa(fun(A,real),fun(A,real),aa(A,fun(fun(A,real),fun(A,real)),aTP_Lamp_uk(fun(A,real),fun(A,fun(fun(A,real),fun(A,real))),G),X),G4),topolo174197925503356063within(A,X,S3)) ) ) ) ).

% has_derivative_ln
tff(fact_7105_has__derivative__divide,axiom,
    ! [A: $tType,C: $tType] :
      ( ( real_V822414075346904944vector(C)
        & real_V8999393235501362500lgebra(A) )
     => ! [F2: fun(C,A),F7: fun(C,A),X: C,S: set(C),G: fun(C,A),G4: fun(C,A)] :
          ( has_derivative(C,A,F2,F7,topolo174197925503356063within(C,X,S))
         => ( has_derivative(C,A,G,G4,topolo174197925503356063within(C,X,S))
           => ( ( aa(C,A,G,X) != zero_zero(A) )
             => has_derivative(C,A,aa(fun(C,A),fun(C,A),aTP_Lamp_ul(fun(C,A),fun(fun(C,A),fun(C,A)),F2),G),aa(fun(C,A),fun(C,A),aa(fun(C,A),fun(fun(C,A),fun(C,A)),aa(C,fun(fun(C,A),fun(fun(C,A),fun(C,A))),aa(fun(C,A),fun(C,fun(fun(C,A),fun(fun(C,A),fun(C,A)))),aTP_Lamp_um(fun(C,A),fun(fun(C,A),fun(C,fun(fun(C,A),fun(fun(C,A),fun(C,A))))),F2),F7),X),G),G4),topolo174197925503356063within(C,X,S)) ) ) ) ) ).

% has_derivative_divide
tff(fact_7106_has__derivative__prod,axiom,
    ! [B: $tType,I6: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V3459762299906320749_field(B) )
     => ! [I5: set(I6),F2: fun(I6,fun(A,B)),F7: fun(I6,fun(A,B)),X: A,S: set(A)] :
          ( ! [I2: I6] :
              ( pp(aa(set(I6),bool,aa(I6,fun(set(I6),bool),member(I6),I2),I5))
             => has_derivative(A,B,aa(I6,fun(A,B),F2,I2),aa(I6,fun(A,B),F7,I2),topolo174197925503356063within(A,X,S)) )
         => has_derivative(A,B,aa(fun(I6,fun(A,B)),fun(A,B),aTP_Lamp_uo(set(I6),fun(fun(I6,fun(A,B)),fun(A,B)),I5),F2),aa(A,fun(A,B),aa(fun(I6,fun(A,B)),fun(A,fun(A,B)),aa(fun(I6,fun(A,B)),fun(fun(I6,fun(A,B)),fun(A,fun(A,B))),aTP_Lamp_uq(set(I6),fun(fun(I6,fun(A,B)),fun(fun(I6,fun(A,B)),fun(A,fun(A,B)))),I5),F2),F7),X),topolo174197925503356063within(A,X,S)) ) ) ).

% has_derivative_prod
tff(fact_7107_has__derivative__powr,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [G: fun(A,real),G4: fun(A,real),X: A,X6: set(A),F2: fun(A,real),F7: fun(A,real)] :
          ( has_derivative(A,real,G,G4,topolo174197925503356063within(A,X,X6))
         => ( has_derivative(A,real,F2,F7,topolo174197925503356063within(A,X,X6))
           => ( pp(aa(real,bool,ord_less(real,zero_zero(real)),aa(A,real,G,X)))
             => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),X6))
               => has_derivative(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_ur(fun(A,real),fun(fun(A,real),fun(A,real)),G),F2),aa(fun(A,real),fun(A,real),aa(fun(A,real),fun(fun(A,real),fun(A,real)),aa(A,fun(fun(A,real),fun(fun(A,real),fun(A,real))),aa(fun(A,real),fun(A,fun(fun(A,real),fun(fun(A,real),fun(A,real)))),aTP_Lamp_us(fun(A,real),fun(fun(A,real),fun(A,fun(fun(A,real),fun(fun(A,real),fun(A,real))))),G),G4),X),F2),F7),topolo174197925503356063within(A,X,X6)) ) ) ) ) ) ).

% has_derivative_powr
tff(fact_7108_has__derivative__real__sqrt,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [G: fun(A,real),X: A,G4: fun(A,real),S3: set(A)] :
          ( pp(aa(real,bool,ord_less(real,zero_zero(real)),aa(A,real,G,X)))
         => ( has_derivative(A,real,G,G4,topolo174197925503356063within(A,X,S3))
           => has_derivative(A,real,aTP_Lamp_ut(fun(A,real),fun(A,real),G),aa(fun(A,real),fun(A,real),aa(A,fun(fun(A,real),fun(A,real)),aTP_Lamp_uu(fun(A,real),fun(A,fun(fun(A,real),fun(A,real))),G),X),G4),topolo174197925503356063within(A,X,S3)) ) ) ) ).

% has_derivative_real_sqrt
tff(fact_7109_has__derivative__arctan,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [G: fun(A,real),G4: fun(A,real),X: A,S3: set(A)] :
          ( has_derivative(A,real,G,G4,topolo174197925503356063within(A,X,S3))
         => has_derivative(A,real,aTP_Lamp_uv(fun(A,real),fun(A,real),G),aa(A,fun(A,real),aa(fun(A,real),fun(A,fun(A,real)),aTP_Lamp_uw(fun(A,real),fun(fun(A,real),fun(A,fun(A,real))),G),G4),X),topolo174197925503356063within(A,X,S3)) ) ) ).

% has_derivative_arctan
tff(fact_7110_has__derivative__tan,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [G: fun(A,real),X: A,G4: fun(A,real),S3: set(A)] :
          ( ( cos(real,aa(A,real,G,X)) != zero_zero(real) )
         => ( has_derivative(A,real,G,G4,topolo174197925503356063within(A,X,S3))
           => has_derivative(A,real,aTP_Lamp_ux(fun(A,real),fun(A,real),G),aa(fun(A,real),fun(A,real),aa(A,fun(fun(A,real),fun(A,real)),aTP_Lamp_uy(fun(A,real),fun(A,fun(fun(A,real),fun(A,real))),G),X),G4),topolo174197925503356063within(A,X,S3)) ) ) ) ).

% has_derivative_tan
tff(fact_7111_DERIV__series_H,axiom,
    ! [F2: fun(real,fun(nat,real)),F7: fun(real,fun(nat,real)),X0: real,A2: real,B2: real,L6: fun(nat,real)] :
      ( ! [N2: nat] : has_field_derivative(real,aa(nat,fun(real,real),aTP_Lamp_uz(fun(real,fun(nat,real)),fun(nat,fun(real,real)),F2),N2),aa(nat,real,aa(real,fun(nat,real),F7,X0),N2),topolo174197925503356063within(real,X0,top_top(set(real))))
     => ( ! [X3: real] :
            ( pp(aa(set(real),bool,aa(real,fun(set(real),bool),member(real),X3),set_or5935395276787703475ssThan(real,A2,B2)))
           => summable(real,aa(real,fun(nat,real),F2,X3)) )
       => ( pp(aa(set(real),bool,aa(real,fun(set(real),bool),member(real),X0),set_or5935395276787703475ssThan(real,A2,B2)))
         => ( summable(real,aa(real,fun(nat,real),F7,X0))
           => ( summable(real,L6)
             => ( ! [N2: nat,X3: real,Y3: real] :
                    ( pp(aa(set(real),bool,aa(real,fun(set(real),bool),member(real),X3),set_or5935395276787703475ssThan(real,A2,B2)))
                   => ( pp(aa(set(real),bool,aa(real,fun(set(real),bool),member(real),Y3),set_or5935395276787703475ssThan(real,A2,B2)))
                     => pp(aa(real,bool,ord_less_eq(real,aa(real,real,abs_abs(real),aa(real,real,minus_minus(real,aa(nat,real,aa(real,fun(nat,real),F2,X3),N2)),aa(nat,real,aa(real,fun(nat,real),F2,Y3),N2)))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,L6,N2)),aa(real,real,abs_abs(real),aa(real,real,minus_minus(real,X3),Y3))))) ) )
               => has_field_derivative(real,aTP_Lamp_va(fun(real,fun(nat,real)),fun(real,real),F2),suminf(real,aa(real,fun(nat,real),F7,X0)),topolo174197925503356063within(real,X0,top_top(set(real)))) ) ) ) ) ) ) ).

% DERIV_series'
tff(fact_7112_has__derivative__arccos,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [G: fun(A,real),X: A,G4: fun(A,real),S3: set(A)] :
          ( pp(aa(real,bool,ord_less(real,aa(real,real,uminus_uminus(real),one_one(real))),aa(A,real,G,X)))
         => ( pp(aa(real,bool,ord_less(real,aa(A,real,G,X)),one_one(real)))
           => ( has_derivative(A,real,G,G4,topolo174197925503356063within(A,X,S3))
             => has_derivative(A,real,aTP_Lamp_vb(fun(A,real),fun(A,real),G),aa(fun(A,real),fun(A,real),aa(A,fun(fun(A,real),fun(A,real)),aTP_Lamp_vc(fun(A,real),fun(A,fun(fun(A,real),fun(A,real))),G),X),G4),topolo174197925503356063within(A,X,S3)) ) ) ) ) ).

% has_derivative_arccos
tff(fact_7113_has__derivative__floor,axiom,
    ! [Aa: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & archim2362893244070406136eiling(Aa)
        & topolo2564578578187576103pology(Aa) )
     => ! [G: fun(A,real),X: A,F2: fun(real,Aa),G4: fun(A,real),S3: set(A)] :
          ( topolo3448309680560233919inuous(real,Aa,topolo174197925503356063within(real,aa(A,real,G,X),top_top(set(real))),F2)
         => ( ~ pp(aa(set(Aa),bool,aa(Aa,fun(set(Aa),bool),member(Aa),aa(real,Aa,F2,aa(A,real,G,X))),ring_1_Ints(Aa)))
           => ( has_derivative(A,real,G,G4,topolo174197925503356063within(A,X,S3))
             => has_derivative(A,real,aa(fun(real,Aa),fun(A,real),aTP_Lamp_vd(fun(A,real),fun(fun(real,Aa),fun(A,real)),G),F2),aTP_Lamp_ve(fun(A,real),fun(A,real),G4),topolo174197925503356063within(A,X,S3)) ) ) ) ) ).

% has_derivative_floor
tff(fact_7114_termdiffs__aux,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [C2: fun(nat,A),K6: A,X: A] :
          ( summable(A,aa(A,fun(nat,A),aTP_Lamp_sf(fun(nat,A),fun(A,fun(nat,A)),C2),K6))
         => ( pp(aa(real,bool,ord_less(real,real_V7770717601297561774m_norm(A,X)),real_V7770717601297561774m_norm(A,K6)))
           => filterlim(A,A,aa(A,fun(A,A),aTP_Lamp_vg(fun(nat,A),fun(A,fun(A,A)),C2),X),topolo7230453075368039082e_nhds(A,zero_zero(A)),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ) ).

% termdiffs_aux
tff(fact_7115_tendsto__const,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [K: A,F3: filter(B)] : filterlim(B,A,aTP_Lamp_vh(A,fun(B,A),K),topolo7230453075368039082e_nhds(A,K),F3) ) ).

% tendsto_const
tff(fact_7116_tendsto__ident__at,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [A2: A,S3: set(A)] : filterlim(A,A,aTP_Lamp_vi(A,A),topolo7230453075368039082e_nhds(A,A2),topolo174197925503356063within(A,A2,S3)) ) ).

% tendsto_ident_at
tff(fact_7117_tendsto__mult__left__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( ( field(A)
        & topolo4211221413907600880p_mult(A) )
     => ! [C2: A,F2: fun(B,A),L: A,F3: filter(B)] :
          ( ( C2 != zero_zero(A) )
         => ( filterlim(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_vj(A,fun(fun(B,A),fun(B,A)),C2),F2),topolo7230453075368039082e_nhds(A,aa(A,A,aa(A,fun(A,A),times_times(A),C2),L)),F3)
          <=> filterlim(B,A,F2,topolo7230453075368039082e_nhds(A,L),F3) ) ) ) ).

% tendsto_mult_left_iff
tff(fact_7118_tendsto__mult__right__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( ( field(A)
        & topolo4211221413907600880p_mult(A) )
     => ! [C2: A,F2: fun(B,A),L: A,F3: filter(B)] :
          ( ( C2 != zero_zero(A) )
         => ( filterlim(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_vk(A,fun(fun(B,A),fun(B,A)),C2),F2),topolo7230453075368039082e_nhds(A,aa(A,A,aa(A,fun(A,A),times_times(A),L),C2)),F3)
          <=> filterlim(B,A,F2,topolo7230453075368039082e_nhds(A,L),F3) ) ) ) ).

% tendsto_mult_right_iff
tff(fact_7119_power__tendsto__0__iff,axiom,
    ! [A: $tType,N: nat,F2: fun(A,real),F3: filter(A)] :
      ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),N))
     => ( filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_vl(nat,fun(fun(A,real),fun(A,real)),N),F2),topolo7230453075368039082e_nhds(real,zero_zero(real)),F3)
      <=> filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,zero_zero(real)),F3) ) ) ).

% power_tendsto_0_iff
tff(fact_7120_tendsto__within__subset,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [F2: fun(A,B),L: filter(B),X: A,S: set(A),T3: set(A)] :
          ( filterlim(A,B,F2,L,topolo174197925503356063within(A,X,S))
         => ( pp(aa(set(A),bool,ord_less_eq(set(A),T3),S))
           => filterlim(A,B,F2,L,topolo174197925503356063within(A,X,T3)) ) ) ) ).

% tendsto_within_subset
tff(fact_7121_continuous__ident,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [X: A,S: set(A)] : topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,X,S),aTP_Lamp_vm(A,A)) ) ).

% continuous_ident
tff(fact_7122_has__field__derivativeD,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D4: A,X: A,S: set(A)] :
          ( has_field_derivative(A,F2,D4,topolo174197925503356063within(A,X,S))
         => filterlim(A,A,aa(A,fun(A,A),aTP_Lamp_vn(fun(A,A),fun(A,fun(A,A)),F2),X),topolo7230453075368039082e_nhds(A,D4),topolo174197925503356063within(A,X,S)) ) ) ).

% has_field_derivativeD
tff(fact_7123_has__field__derivative__iff,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D4: A,X: A,S: set(A)] :
          ( has_field_derivative(A,F2,D4,topolo174197925503356063within(A,X,S))
        <=> filterlim(A,A,aa(A,fun(A,A),aTP_Lamp_vn(fun(A,A),fun(A,fun(A,A)),F2),X),topolo7230453075368039082e_nhds(A,D4),topolo174197925503356063within(A,X,S)) ) ) ).

% has_field_derivative_iff
tff(fact_7124_continuous__within__compose2,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( ( topological_t2_space(B)
        & topolo4958980785337419405_space(C)
        & topological_t2_space(A) )
     => ! [X: A,S3: set(A),F2: fun(A,B),G: fun(B,C)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,X,S3),F2)
         => ( topolo3448309680560233919inuous(B,C,topolo174197925503356063within(B,aa(A,B,F2,X),aa(set(A),set(B),image(A,B,F2),S3)),G)
           => topolo3448309680560233919inuous(A,C,topolo174197925503356063within(A,X,S3),aa(fun(B,C),fun(A,C),aTP_Lamp_vo(fun(A,B),fun(fun(B,C),fun(A,C)),F2),G)) ) ) ) ).

% continuous_within_compose2
tff(fact_7125_isCont__Pair,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo4958980785337419405_space(C)
        & topolo4958980785337419405_space(B) )
     => ! [A2: A,F2: fun(A,B),G: fun(A,C)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,top_top(set(A))),F2)
         => ( topolo3448309680560233919inuous(A,C,topolo174197925503356063within(A,A2,top_top(set(A))),G)
           => topolo3448309680560233919inuous(A,product_prod(B,C),topolo174197925503356063within(A,A2,top_top(set(A))),aa(fun(A,C),fun(A,product_prod(B,C)),aTP_Lamp_vp(fun(A,B),fun(fun(A,C),fun(A,product_prod(B,C))),F2),G)) ) ) ) ).

% isCont_Pair
tff(fact_7126_LIM__offset,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & topolo4958980785337419405_space(B) )
     => ! [F2: fun(A,B),L6: B,A2: A,K: A] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L6),topolo174197925503356063within(A,A2,top_top(set(A))))
         => filterlim(A,B,aa(A,fun(A,B),aTP_Lamp_vq(fun(A,B),fun(A,fun(A,B)),F2),K),topolo7230453075368039082e_nhds(B,L6),topolo174197925503356063within(A,aa(A,A,minus_minus(A,A2),K),top_top(set(A)))) ) ) ).

% LIM_offset
tff(fact_7127_LIM__offset__zero__cancel,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & topolo4958980785337419405_space(B) )
     => ! [F2: fun(A,B),A2: A,L6: B] :
          ( filterlim(A,B,aa(A,fun(A,B),aTP_Lamp_vr(fun(A,B),fun(A,fun(A,B)),F2),A2),topolo7230453075368039082e_nhds(B,L6),topolo174197925503356063within(A,zero_zero(A),top_top(set(A))))
         => filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L6),topolo174197925503356063within(A,A2,top_top(set(A)))) ) ) ).

% LIM_offset_zero_cancel
tff(fact_7128_LIM__offset__zero,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & topolo4958980785337419405_space(B) )
     => ! [F2: fun(A,B),L6: B,A2: A] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L6),topolo174197925503356063within(A,A2,top_top(set(A))))
         => filterlim(A,B,aa(A,fun(A,B),aTP_Lamp_vr(fun(A,B),fun(A,fun(A,B)),F2),A2),topolo7230453075368039082e_nhds(B,L6),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ).

% LIM_offset_zero
tff(fact_7129_LIM__isCont__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & topolo4958980785337419405_space(B) )
     => ! [F2: fun(A,B),A2: A] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,aa(A,B,F2,A2)),topolo174197925503356063within(A,A2,top_top(set(A))))
        <=> filterlim(A,B,aa(A,fun(A,B),aTP_Lamp_vr(fun(A,B),fun(A,fun(A,B)),F2),A2),topolo7230453075368039082e_nhds(B,aa(A,B,F2,A2)),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ).

% LIM_isCont_iff
tff(fact_7130_isCont__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & topolo4958980785337419405_space(B) )
     => ! [X: A,F2: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,X,top_top(set(A))),F2)
        <=> filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_vs(A,fun(fun(A,B),fun(A,B)),X),F2),topolo7230453075368039082e_nhds(B,aa(A,B,F2,X)),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ).

% isCont_iff
tff(fact_7131_LIM__not__zero,axiom,
    ! [Aa: $tType,A: $tType] :
      ( ( topolo8386298272705272623_space(A)
        & zero(Aa)
        & topological_t2_space(Aa) )
     => ! [K: Aa,A2: A] :
          ( ( K != zero_zero(Aa) )
         => ~ filterlim(A,Aa,aTP_Lamp_vt(Aa,fun(A,Aa),K),topolo7230453075368039082e_nhds(Aa,zero_zero(Aa)),topolo174197925503356063within(A,A2,top_top(set(A)))) ) ) ).

% LIM_not_zero
tff(fact_7132_isCont__norm,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V822414075346904944vector(B) )
     => ! [A2: A,F2: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,top_top(set(A))),F2)
         => topolo3448309680560233919inuous(A,real,topolo174197925503356063within(A,A2,top_top(set(A))),aTP_Lamp_vu(fun(A,B),fun(A,real),F2)) ) ) ).

% isCont_norm
tff(fact_7133_isCont__of__real,axiom,
    ! [A: $tType,C: $tType] :
      ( ( topological_t2_space(C)
        & real_V2191834092415804123ebra_1(A)
        & real_V822414075346904944vector(A) )
     => ! [A2: C,G: fun(C,real)] :
          ( topolo3448309680560233919inuous(C,real,topolo174197925503356063within(C,A2,top_top(set(C))),G)
         => topolo3448309680560233919inuous(C,A,topolo174197925503356063within(C,A2,top_top(set(C))),aTP_Lamp_vv(fun(C,real),fun(C,A),G)) ) ) ).

% isCont_of_real
tff(fact_7134_filterlim__at__If,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [F2: fun(A,B),G5: filter(B),X: A,P: fun(A,bool),G: fun(A,B)] :
          ( filterlim(A,B,F2,G5,topolo174197925503356063within(A,X,collect(A,P)))
         => ( filterlim(A,B,G,G5,topolo174197925503356063within(A,X,collect(A,aTP_Lamp_vw(fun(A,bool),fun(A,bool),P))))
           => filterlim(A,B,aa(fun(A,B),fun(A,B),aa(fun(A,bool),fun(fun(A,B),fun(A,B)),aTP_Lamp_vx(fun(A,B),fun(fun(A,bool),fun(fun(A,B),fun(A,B))),F2),P),G),G5,topolo174197925503356063within(A,X,top_top(set(A)))) ) ) ) ).

% filterlim_at_If
tff(fact_7135_isCont__o2,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( ( topological_t2_space(B)
        & topolo4958980785337419405_space(C)
        & topological_t2_space(A) )
     => ! [A2: A,F2: fun(A,B),G: fun(B,C)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,top_top(set(A))),F2)
         => ( topolo3448309680560233919inuous(B,C,topolo174197925503356063within(B,aa(A,B,F2,A2),top_top(set(B))),G)
           => topolo3448309680560233919inuous(A,C,topolo174197925503356063within(A,A2,top_top(set(A))),aa(fun(B,C),fun(A,C),aTP_Lamp_vo(fun(A,B),fun(fun(B,C),fun(A,C)),F2),G)) ) ) ) ).

% isCont_o2
tff(fact_7136_LIM__const__eq,axiom,
    ! [A: $tType,B: $tType] :
      ( ( topological_t2_space(B)
        & topolo8386298272705272623_space(A) )
     => ! [K: B,L6: B,A2: A] :
          ( filterlim(A,B,aTP_Lamp_vy(B,fun(A,B),K),topolo7230453075368039082e_nhds(B,L6),topolo174197925503356063within(A,A2,top_top(set(A))))
         => ( K = L6 ) ) ) ).

% LIM_const_eq
tff(fact_7137_tendsto__compose,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( ( topolo4958980785337419405_space(B)
        & topolo4958980785337419405_space(A) )
     => ! [G: fun(A,B),L: A,F2: fun(C,A),F3: filter(C)] :
          ( filterlim(A,B,G,topolo7230453075368039082e_nhds(B,aa(A,B,G,L)),topolo174197925503356063within(A,L,top_top(set(A))))
         => ( filterlim(C,A,F2,topolo7230453075368039082e_nhds(A,L),F3)
           => filterlim(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_vz(fun(A,B),fun(fun(C,A),fun(C,B)),G),F2),topolo7230453075368039082e_nhds(B,aa(A,B,G,L)),F3) ) ) ) ).

% tendsto_compose
tff(fact_7138_LIM__const__not__eq,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo8386298272705272623_space(A)
        & topological_t2_space(B) )
     => ! [K: B,L6: B,A2: A] :
          ( ( K != L6 )
         => ~ filterlim(A,B,aTP_Lamp_vy(B,fun(A,B),K),topolo7230453075368039082e_nhds(B,L6),topolo174197925503356063within(A,A2,top_top(set(A)))) ) ) ).

% LIM_const_not_eq
tff(fact_7139_isCont__tendsto__compose,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( ( topolo4958980785337419405_space(B)
        & topological_t2_space(A) )
     => ! [L: A,G: fun(A,B),F2: fun(C,A),F3: filter(C)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,L,top_top(set(A))),G)
         => ( filterlim(C,A,F2,topolo7230453075368039082e_nhds(A,L),F3)
           => filterlim(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_wa(fun(A,B),fun(fun(C,A),fun(C,B)),G),F2),topolo7230453075368039082e_nhds(B,aa(A,B,G,L)),F3) ) ) ) ).

% isCont_tendsto_compose
tff(fact_7140_continuous__within__compose3,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo4958980785337419405_space(B)
        & topological_t2_space(C) )
     => ! [F2: fun(C,A),X: C,G: fun(A,B),S3: set(C)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,aa(C,A,F2,X),top_top(set(A))),G)
         => ( topolo3448309680560233919inuous(C,A,topolo174197925503356063within(C,X,S3),F2)
           => topolo3448309680560233919inuous(C,B,topolo174197925503356063within(C,X,S3),aa(fun(A,B),fun(C,B),aTP_Lamp_wb(fun(C,A),fun(fun(A,B),fun(C,B)),F2),G)) ) ) ) ).

% continuous_within_compose3
tff(fact_7141_isCont__scaleR,axiom,
    ! [C: $tType,D6: $tType] :
      ( ( topological_t2_space(D6)
        & real_V822414075346904944vector(C) )
     => ! [A2: D6,F2: fun(D6,real),G: fun(D6,C)] :
          ( topolo3448309680560233919inuous(D6,real,topolo174197925503356063within(D6,A2,top_top(set(D6))),F2)
         => ( topolo3448309680560233919inuous(D6,C,topolo174197925503356063within(D6,A2,top_top(set(D6))),G)
           => topolo3448309680560233919inuous(D6,C,topolo174197925503356063within(D6,A2,top_top(set(D6))),aa(fun(D6,C),fun(D6,C),aTP_Lamp_wc(fun(D6,real),fun(fun(D6,C),fun(D6,C)),F2),G)) ) ) ) ).

% isCont_scaleR
tff(fact_7142_LIM__equal2,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & topolo4958980785337419405_space(B) )
     => ! [R: real,A2: A,F2: fun(A,B),G: fun(A,B),L: B] :
          ( pp(aa(real,bool,ord_less(real,zero_zero(real)),R))
         => ( ! [X3: A] :
                ( ( X3 != A2 )
               => ( pp(aa(real,bool,ord_less(real,real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,X3),A2))),R))
                 => ( aa(A,B,F2,X3) = aa(A,B,G,X3) ) ) )
           => ( filterlim(A,B,G,topolo7230453075368039082e_nhds(B,L),topolo174197925503356063within(A,A2,top_top(set(A))))
             => filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),topolo174197925503356063within(A,A2,top_top(set(A)))) ) ) ) ) ).

% LIM_equal2
tff(fact_7143_LIM__eq,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),L6: B,A2: A] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L6),topolo174197925503356063within(A,A2,top_top(set(A))))
        <=> ! [R5: real] :
              ( pp(aa(real,bool,ord_less(real,zero_zero(real)),R5))
             => ? [S7: real] :
                  ( pp(aa(real,bool,ord_less(real,zero_zero(real)),S7))
                  & ! [X2: A] :
                      ( ( ( X2 != A2 )
                        & pp(aa(real,bool,ord_less(real,real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,X2),A2))),S7)) )
                     => pp(aa(real,bool,ord_less(real,real_V7770717601297561774m_norm(B,aa(B,B,minus_minus(B,aa(A,B,F2,X2)),L6))),R5)) ) ) ) ) ) ).

% LIM_eq
tff(fact_7144_LIM__I,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [A2: A,F2: fun(A,B),L6: B] :
          ( ! [R3: real] :
              ( pp(aa(real,bool,ord_less(real,zero_zero(real)),R3))
             => ? [S8: real] :
                  ( pp(aa(real,bool,ord_less(real,zero_zero(real)),S8))
                  & ! [X3: A] :
                      ( ( ( X3 != A2 )
                        & pp(aa(real,bool,ord_less(real,real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,X3),A2))),S8)) )
                     => pp(aa(real,bool,ord_less(real,real_V7770717601297561774m_norm(B,aa(B,B,minus_minus(B,aa(A,B,F2,X3)),L6))),R3)) ) ) )
         => filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L6),topolo174197925503356063within(A,A2,top_top(set(A)))) ) ) ).

% LIM_I
tff(fact_7145_LIM__D,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),L6: B,A2: A,R2: real] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L6),topolo174197925503356063within(A,A2,top_top(set(A))))
         => ( pp(aa(real,bool,ord_less(real,zero_zero(real)),R2))
           => ? [S2: real] :
                ( pp(aa(real,bool,ord_less(real,zero_zero(real)),S2))
                & ! [X4: A] :
                    ( ( ( X4 != A2 )
                      & pp(aa(real,bool,ord_less(real,real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,X4),A2))),S2)) )
                   => pp(aa(real,bool,ord_less(real,real_V7770717601297561774m_norm(B,aa(B,B,minus_minus(B,aa(A,B,F2,X4)),L6))),R2)) ) ) ) ) ) ).

% LIM_D
tff(fact_7146_isCont__LIM__compose2,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & topolo4958980785337419405_space(B)
        & topolo4958980785337419405_space(C) )
     => ! [A2: A,F2: fun(A,B),G: fun(B,C),L: C] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,top_top(set(A))),F2)
         => ( filterlim(B,C,G,topolo7230453075368039082e_nhds(C,L),topolo174197925503356063within(B,aa(A,B,F2,A2),top_top(set(B))))
           => ( ? [D5: real] :
                  ( pp(aa(real,bool,ord_less(real,zero_zero(real)),D5))
                  & ! [X3: A] :
                      ( ( ( X3 != A2 )
                        & pp(aa(real,bool,ord_less(real,real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,X3),A2))),D5)) )
                     => ( aa(A,B,F2,X3) != aa(A,B,F2,A2) ) ) )
             => filterlim(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_wd(fun(A,B),fun(fun(B,C),fun(A,C)),F2),G),topolo7230453075368039082e_nhds(C,L),topolo174197925503356063within(A,A2,top_top(set(A)))) ) ) ) ) ).

% isCont_LIM_compose2
tff(fact_7147_LIM__imp__LIM,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & real_V822414075346904944vector(C)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),L: B,A2: A,G: fun(A,C),M: C] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),topolo174197925503356063within(A,A2,top_top(set(A))))
         => ( ! [X3: A] :
                ( ( X3 != A2 )
               => pp(aa(real,bool,ord_less_eq(real,real_V7770717601297561774m_norm(C,aa(C,C,minus_minus(C,aa(A,C,G,X3)),M))),real_V7770717601297561774m_norm(B,aa(B,B,minus_minus(B,aa(A,B,F2,X3)),L)))) )
           => filterlim(A,C,G,topolo7230453075368039082e_nhds(C,M),topolo174197925503356063within(A,A2,top_top(set(A)))) ) ) ) ).

% LIM_imp_LIM
tff(fact_7148_IVT2,axiom,
    ! [A: $tType,B: $tType] :
      ( ( topolo1944317154257567458pology(B)
        & topolo8458572112393995274pology(A) )
     => ! [F2: fun(A,B),B2: A,Y: B,A2: A] :
          ( pp(aa(B,bool,ord_less_eq(B,aa(A,B,F2,B2)),Y))
         => ( pp(aa(B,bool,ord_less_eq(B,Y),aa(A,B,F2,A2)))
           => ( pp(aa(A,bool,ord_less_eq(A,A2),B2))
             => ( ! [X3: A] :
                    ( ( pp(aa(A,bool,ord_less_eq(A,A2),X3))
                      & pp(aa(A,bool,ord_less_eq(A,X3),B2)) )
                   => topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,X3,top_top(set(A))),F2) )
               => ? [X3: A] :
                    ( pp(aa(A,bool,ord_less_eq(A,A2),X3))
                    & pp(aa(A,bool,ord_less_eq(A,X3),B2))
                    & ( aa(A,B,F2,X3) = Y ) ) ) ) ) ) ) ).

% IVT2
tff(fact_7149_IVT,axiom,
    ! [A: $tType,B: $tType] :
      ( ( topolo1944317154257567458pology(B)
        & topolo8458572112393995274pology(A) )
     => ! [F2: fun(A,B),A2: A,Y: B,B2: A] :
          ( pp(aa(B,bool,ord_less_eq(B,aa(A,B,F2,A2)),Y))
         => ( pp(aa(B,bool,ord_less_eq(B,Y),aa(A,B,F2,B2)))
           => ( pp(aa(A,bool,ord_less_eq(A,A2),B2))
             => ( ! [X3: A] :
                    ( ( pp(aa(A,bool,ord_less_eq(A,A2),X3))
                      & pp(aa(A,bool,ord_less_eq(A,X3),B2)) )
                   => topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,X3,top_top(set(A))),F2) )
               => ? [X3: A] :
                    ( pp(aa(A,bool,ord_less_eq(A,A2),X3))
                    & pp(aa(A,bool,ord_less_eq(A,X3),B2))
                    & ( aa(A,B,F2,X3) = Y ) ) ) ) ) ) ) ).

% IVT
tff(fact_7150_real__LIM__sandwich__zero,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [F2: fun(A,real),A2: A,G: fun(A,real)] :
          ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,zero_zero(real)),topolo174197925503356063within(A,A2,top_top(set(A))))
         => ( ! [X3: A] :
                ( ( X3 != A2 )
               => pp(aa(real,bool,ord_less_eq(real,zero_zero(real)),aa(A,real,G,X3))) )
           => ( ! [X3: A] :
                  ( ( X3 != A2 )
                 => pp(aa(real,bool,ord_less_eq(real,aa(A,real,G,X3)),aa(A,real,F2,X3))) )
             => filterlim(A,real,G,topolo7230453075368039082e_nhds(real,zero_zero(real)),topolo174197925503356063within(A,A2,top_top(set(A)))) ) ) ) ) ).

% real_LIM_sandwich_zero
tff(fact_7151_atLeastSucLessThan__greaterThanLessThan,axiom,
    ! [L: nat,U: nat] : set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,L),U) = set_or5935395276787703475ssThan(nat,L,U) ).

% atLeastSucLessThan_greaterThanLessThan
tff(fact_7152_tendsto__null__power,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V2822296259951069270ebra_1(B)
     => ! [F2: fun(A,B),F3: filter(A),N: nat] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,zero_zero(B)),F3)
         => ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),N))
           => filterlim(A,B,aa(nat,fun(A,B),aTP_Lamp_we(fun(A,B),fun(nat,fun(A,B)),F2),N),topolo7230453075368039082e_nhds(B,zero_zero(B)),F3) ) ) ) ).

% tendsto_null_power
tff(fact_7153_tendsto__power,axiom,
    ! [B: $tType,A: $tType] :
      ( ( power(B)
        & real_V4412858255891104859lgebra(B) )
     => ! [F2: fun(A,B),A2: B,F3: filter(A),N: nat] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,A2),F3)
         => filterlim(A,B,aa(nat,fun(A,B),aTP_Lamp_wf(fun(A,B),fun(nat,fun(A,B)),F2),N),topolo7230453075368039082e_nhds(B,aa(nat,B,aa(B,fun(nat,B),power_power(B),A2),N)),F3) ) ) ).

% tendsto_power
tff(fact_7154_tendsto__power__strong,axiom,
    ! [B: $tType,C: $tType] :
      ( topolo1898628316856586783d_mult(B)
     => ! [F2: fun(C,B),A2: B,F3: filter(C),G: fun(C,nat),B2: nat] :
          ( filterlim(C,B,F2,topolo7230453075368039082e_nhds(B,A2),F3)
         => ( filterlim(C,nat,G,topolo7230453075368039082e_nhds(nat,B2),F3)
           => filterlim(C,B,aa(fun(C,nat),fun(C,B),aTP_Lamp_wg(fun(C,B),fun(fun(C,nat),fun(C,B)),F2),G),topolo7230453075368039082e_nhds(B,aa(nat,B,aa(B,fun(nat,B),power_power(B),A2),B2)),F3) ) ) ) ).

% tendsto_power_strong
tff(fact_7155_continuous__power_H,axiom,
    ! [B: $tType,C: $tType] :
      ( ( topological_t2_space(C)
        & topolo1898628316856586783d_mult(B) )
     => ! [F3: filter(C),F2: fun(C,B),G: fun(C,nat)] :
          ( topolo3448309680560233919inuous(C,B,F3,F2)
         => ( topolo3448309680560233919inuous(C,nat,F3,G)
           => topolo3448309680560233919inuous(C,B,F3,aa(fun(C,nat),fun(C,B),aTP_Lamp_wh(fun(C,B),fun(fun(C,nat),fun(C,B)),F2),G)) ) ) ) ).

% continuous_power'
tff(fact_7156_continuous__power,axiom,
    ! [A: $tType,B: $tType] :
      ( ( power(B)
        & real_V4412858255891104859lgebra(B)
        & topological_t2_space(A) )
     => ! [F3: filter(A),F2: fun(A,B),N: nat] :
          ( topolo3448309680560233919inuous(A,B,F3,F2)
         => topolo3448309680560233919inuous(A,B,F3,aa(nat,fun(A,B),aTP_Lamp_wi(fun(A,B),fun(nat,fun(A,B)),F2),N)) ) ) ).

% continuous_power
tff(fact_7157_tendsto__inverse,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [F2: fun(B,A),A2: A,F3: filter(B)] :
          ( filterlim(B,A,F2,topolo7230453075368039082e_nhds(A,A2),F3)
         => ( ( A2 != zero_zero(A) )
           => filterlim(B,A,aTP_Lamp_wj(fun(B,A),fun(B,A),F2),topolo7230453075368039082e_nhds(A,aa(A,A,inverse_inverse(A),A2)),F3) ) ) ) ).

% tendsto_inverse
tff(fact_7158_tendsto__sgn,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(B,A),L: A,F3: filter(B)] :
          ( filterlim(B,A,F2,topolo7230453075368039082e_nhds(A,L),F3)
         => ( ( L != zero_zero(A) )
           => filterlim(B,A,aTP_Lamp_wk(fun(B,A),fun(B,A),F2),topolo7230453075368039082e_nhds(A,sgn_sgn(A,L)),F3) ) ) ) ).

% tendsto_sgn
tff(fact_7159_continuous__minus,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo1633459387980952147up_add(B) )
     => ! [F3: filter(A),F2: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,F3,F2)
         => topolo3448309680560233919inuous(A,B,F3,aTP_Lamp_wl(fun(A,B),fun(A,B),F2)) ) ) ).

% continuous_minus
tff(fact_7160_tendsto__minus__cancel__left,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo1633459387980952147up_add(B)
     => ! [F2: fun(A,B),Y: B,F3: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,aa(B,B,uminus_uminus(B),Y)),F3)
        <=> filterlim(A,B,aTP_Lamp_wm(fun(A,B),fun(A,B),F2),topolo7230453075368039082e_nhds(B,Y),F3) ) ) ).

% tendsto_minus_cancel_left
tff(fact_7161_tendsto__minus__cancel,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo1633459387980952147up_add(A)
     => ! [F2: fun(B,A),A2: A,F3: filter(B)] :
          ( filterlim(B,A,aTP_Lamp_wn(fun(B,A),fun(B,A),F2),topolo7230453075368039082e_nhds(A,aa(A,A,uminus_uminus(A),A2)),F3)
         => filterlim(B,A,F2,topolo7230453075368039082e_nhds(A,A2),F3) ) ) ).

% tendsto_minus_cancel
tff(fact_7162_tendsto__minus,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo1633459387980952147up_add(A)
     => ! [F2: fun(B,A),A2: A,F3: filter(B)] :
          ( filterlim(B,A,F2,topolo7230453075368039082e_nhds(A,A2),F3)
         => filterlim(B,A,aTP_Lamp_wn(fun(B,A),fun(B,A),F2),topolo7230453075368039082e_nhds(A,aa(A,A,uminus_uminus(A),A2)),F3) ) ) ).

% tendsto_minus
tff(fact_7163_Lim__transform__eq,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(B,A),G: fun(B,A),F3: filter(B),A2: A] :
          ( filterlim(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_wo(fun(B,A),fun(fun(B,A),fun(B,A)),F2),G),topolo7230453075368039082e_nhds(A,zero_zero(A)),F3)
         => ( filterlim(B,A,F2,topolo7230453075368039082e_nhds(A,A2),F3)
          <=> filterlim(B,A,G,topolo7230453075368039082e_nhds(A,A2),F3) ) ) ) ).

% Lim_transform_eq
tff(fact_7164_LIM__zero__cancel,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F2: fun(A,B),L: B,F3: filter(A)] :
          ( filterlim(A,B,aa(B,fun(A,B),aTP_Lamp_wp(fun(A,B),fun(B,fun(A,B)),F2),L),topolo7230453075368039082e_nhds(B,zero_zero(B)),F3)
         => filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),F3) ) ) ).

% LIM_zero_cancel
tff(fact_7165_Lim__transform2,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(B,A),A2: A,F3: filter(B),G: fun(B,A)] :
          ( filterlim(B,A,F2,topolo7230453075368039082e_nhds(A,A2),F3)
         => ( filterlim(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_wo(fun(B,A),fun(fun(B,A),fun(B,A)),F2),G),topolo7230453075368039082e_nhds(A,zero_zero(A)),F3)
           => filterlim(B,A,G,topolo7230453075368039082e_nhds(A,A2),F3) ) ) ) ).

% Lim_transform2
tff(fact_7166_Lim__transform,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [G: fun(B,A),A2: A,F3: filter(B),F2: fun(B,A)] :
          ( filterlim(B,A,G,topolo7230453075368039082e_nhds(A,A2),F3)
         => ( filterlim(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_wq(fun(B,A),fun(fun(B,A),fun(B,A)),G),F2),topolo7230453075368039082e_nhds(A,zero_zero(A)),F3)
           => filterlim(B,A,F2,topolo7230453075368039082e_nhds(A,A2),F3) ) ) ) ).

% Lim_transform
tff(fact_7167_LIM__zero__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F2: fun(A,B),L: B,F3: filter(A)] :
          ( filterlim(A,B,aa(B,fun(A,B),aTP_Lamp_wp(fun(A,B),fun(B,fun(A,B)),F2),L),topolo7230453075368039082e_nhds(B,zero_zero(B)),F3)
        <=> filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),F3) ) ) ).

% LIM_zero_iff
tff(fact_7168_LIM__zero,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F2: fun(A,B),L: B,F3: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),F3)
         => filterlim(A,B,aa(B,fun(A,B),aTP_Lamp_wp(fun(A,B),fun(B,fun(A,B)),F2),L),topolo7230453075368039082e_nhds(B,zero_zero(B)),F3) ) ) ).

% LIM_zero
tff(fact_7169_tendsto__diff,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo1633459387980952147up_add(A)
     => ! [F2: fun(B,A),A2: A,F3: filter(B),G: fun(B,A),B2: A] :
          ( filterlim(B,A,F2,topolo7230453075368039082e_nhds(A,A2),F3)
         => ( filterlim(B,A,G,topolo7230453075368039082e_nhds(A,B2),F3)
           => filterlim(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_wr(fun(B,A),fun(fun(B,A),fun(B,A)),F2),G),topolo7230453075368039082e_nhds(A,aa(A,A,minus_minus(A,A2),B2)),F3) ) ) ) ).

% tendsto_diff
tff(fact_7170_continuous__diff,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo1633459387980952147up_add(B) )
     => ! [F3: filter(A),F2: fun(A,B),G: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,F3,F2)
         => ( topolo3448309680560233919inuous(A,B,F3,G)
           => topolo3448309680560233919inuous(A,B,F3,aa(fun(A,B),fun(A,B),aTP_Lamp_ws(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G)) ) ) ) ).

% continuous_diff
tff(fact_7171_continuous__scaleR,axiom,
    ! [C: $tType,D6: $tType] :
      ( ( topological_t2_space(D6)
        & real_V822414075346904944vector(C) )
     => ! [F3: filter(D6),F2: fun(D6,real),G: fun(D6,C)] :
          ( topolo3448309680560233919inuous(D6,real,F3,F2)
         => ( topolo3448309680560233919inuous(D6,C,F3,G)
           => topolo3448309680560233919inuous(D6,C,F3,aa(fun(D6,C),fun(D6,C),aTP_Lamp_wc(fun(D6,real),fun(fun(D6,C),fun(D6,C)),F2),G)) ) ) ) ).

% continuous_scaleR
tff(fact_7172_tendsto__scaleR,axiom,
    ! [C: $tType,D6: $tType] :
      ( real_V822414075346904944vector(C)
     => ! [F2: fun(D6,real),A2: real,F3: filter(D6),G: fun(D6,C),B2: C] :
          ( filterlim(D6,real,F2,topolo7230453075368039082e_nhds(real,A2),F3)
         => ( filterlim(D6,C,G,topolo7230453075368039082e_nhds(C,B2),F3)
           => filterlim(D6,C,aa(fun(D6,C),fun(D6,C),aTP_Lamp_wt(fun(D6,real),fun(fun(D6,C),fun(D6,C)),F2),G),topolo7230453075368039082e_nhds(C,aa(C,C,real_V8093663219630862766scaleR(C,A2),B2)),F3) ) ) ) ).

% tendsto_scaleR
tff(fact_7173_continuous__cos,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [F3: filter(A),F2: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,F3,F2)
         => topolo3448309680560233919inuous(A,B,F3,aTP_Lamp_wu(fun(A,B),fun(A,B),F2)) ) ) ).

% continuous_cos
tff(fact_7174_continuous__const,axiom,
    ! [A: $tType,B: $tType] :
      ( ( topolo4958980785337419405_space(B)
        & topological_t2_space(A) )
     => ! [F3: filter(A),C2: B] : topolo3448309680560233919inuous(A,B,F3,aTP_Lamp_wv(B,fun(A,B),C2)) ) ).

% continuous_const
tff(fact_7175_tendsto__arsinh,axiom,
    ! [B: $tType,F2: fun(B,real),A2: real,F3: filter(B)] :
      ( filterlim(B,real,F2,topolo7230453075368039082e_nhds(real,A2),F3)
     => filterlim(B,real,aTP_Lamp_ww(fun(B,real),fun(B,real),F2),topolo7230453075368039082e_nhds(real,aa(real,real,arsinh(real),A2)),F3) ) ).

% tendsto_arsinh
tff(fact_7176_tendsto__sin,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [F2: fun(A,B),A2: B,F3: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,A2),F3)
         => filterlim(A,B,aTP_Lamp_wx(fun(A,B),fun(A,B),F2),topolo7230453075368039082e_nhds(B,sin(B,A2)),F3) ) ) ).

% tendsto_sin
tff(fact_7177_tendsto__cos,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [F2: fun(A,B),A2: B,F3: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,A2),F3)
         => filterlim(A,B,aTP_Lamp_wu(fun(A,B),fun(A,B),F2),topolo7230453075368039082e_nhds(B,cos(B,A2)),F3) ) ) ).

% tendsto_cos
tff(fact_7178_tendsto__Complex,axiom,
    ! [A: $tType,F2: fun(A,real),A2: real,F3: filter(A),G: fun(A,real),B2: real] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,A2),F3)
     => ( filterlim(A,real,G,topolo7230453075368039082e_nhds(real,B2),F3)
       => filterlim(A,complex,aa(fun(A,real),fun(A,complex),aTP_Lamp_wy(fun(A,real),fun(fun(A,real),fun(A,complex)),F2),G),topolo7230453075368039082e_nhds(complex,complex2(A2,B2)),F3) ) ) ).

% tendsto_Complex
tff(fact_7179_tendsto__of__real__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [F2: fun(B,real),C2: real,F3: filter(B)] :
          ( filterlim(B,A,aTP_Lamp_wz(fun(B,real),fun(B,A),F2),topolo7230453075368039082e_nhds(A,aa(real,A,real_Vector_of_real(A),C2)),F3)
        <=> filterlim(B,real,F2,topolo7230453075368039082e_nhds(real,C2),F3) ) ) ).

% tendsto_of_real_iff
tff(fact_7180_continuous__of__real,axiom,
    ! [A: $tType,C: $tType] :
      ( ( topological_t2_space(C)
        & real_V2191834092415804123ebra_1(A)
        & real_V822414075346904944vector(A) )
     => ! [F3: filter(C),G: fun(C,real)] :
          ( topolo3448309680560233919inuous(C,real,F3,G)
         => topolo3448309680560233919inuous(C,A,F3,aTP_Lamp_vv(fun(C,real),fun(C,A),G)) ) ) ).

% continuous_of_real
tff(fact_7181_tendsto__of__real,axiom,
    ! [A: $tType,C: $tType] :
      ( ( real_V2191834092415804123ebra_1(A)
        & real_V822414075346904944vector(A) )
     => ! [G: fun(C,real),A2: real,F3: filter(C)] :
          ( filterlim(C,real,G,topolo7230453075368039082e_nhds(real,A2),F3)
         => filterlim(C,A,aTP_Lamp_xa(fun(C,real),fun(C,A),G),topolo7230453075368039082e_nhds(A,aa(real,A,real_Vector_of_real(A),A2)),F3) ) ) ).

% tendsto_of_real
tff(fact_7182_tendsto__arctan,axiom,
    ! [A: $tType,F2: fun(A,real),X: real,F3: filter(A)] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,X),F3)
     => filterlim(A,real,aTP_Lamp_xb(fun(A,real),fun(A,real),F2),topolo7230453075368039082e_nhds(real,aa(real,real,arctan,X)),F3) ) ).

% tendsto_arctan
tff(fact_7183_continuous__norm,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V822414075346904944vector(B) )
     => ! [F3: filter(A),F2: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,F3,F2)
         => topolo3448309680560233919inuous(A,real,F3,aTP_Lamp_vu(fun(A,B),fun(A,real),F2)) ) ) ).

% continuous_norm
tff(fact_7184_tendsto__norm,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F2: fun(A,B),A2: B,F3: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,A2),F3)
         => filterlim(A,real,aTP_Lamp_xc(fun(A,B),fun(A,real),F2),topolo7230453075368039082e_nhds(real,real_V7770717601297561774m_norm(B,A2)),F3) ) ) ).

% tendsto_norm
tff(fact_7185_tendsto__real__sqrt,axiom,
    ! [A: $tType,F2: fun(A,real),X: real,F3: filter(A)] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,X),F3)
     => filterlim(A,real,aTP_Lamp_xd(fun(A,real),fun(A,real),F2),topolo7230453075368039082e_nhds(real,aa(real,real,sqrt,X)),F3) ) ).

% tendsto_real_sqrt
tff(fact_7186_tendsto__exp,axiom,
    ! [A: $tType,C: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [F2: fun(C,A),A2: A,F3: filter(C)] :
          ( filterlim(C,A,F2,topolo7230453075368039082e_nhds(A,A2),F3)
         => filterlim(C,A,aTP_Lamp_xe(fun(C,A),fun(C,A),F2),topolo7230453075368039082e_nhds(A,aa(A,A,exp(A),A2)),F3) ) ) ).

% tendsto_exp
tff(fact_7187_continuous__exp,axiom,
    ! [A: $tType,C: $tType] :
      ( ( topological_t2_space(C)
        & real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [F3: filter(C),F2: fun(C,A)] :
          ( topolo3448309680560233919inuous(C,A,F3,F2)
         => topolo3448309680560233919inuous(C,A,F3,aTP_Lamp_xf(fun(C,A),fun(C,A),F2)) ) ) ).

% continuous_exp
tff(fact_7188_continuous__sin,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [F3: filter(A),F2: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,F3,F2)
         => topolo3448309680560233919inuous(A,B,F3,aTP_Lamp_wx(fun(A,B),fun(A,B),F2)) ) ) ).

% continuous_sin
tff(fact_7189_continuous__sinh,axiom,
    ! [A: $tType,C: $tType] :
      ( ( topological_t2_space(C)
        & real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [F3: filter(C),F2: fun(C,A)] :
          ( topolo3448309680560233919inuous(C,A,F3,F2)
         => topolo3448309680560233919inuous(C,A,F3,aTP_Lamp_xg(fun(C,A),fun(C,A),F2)) ) ) ).

% continuous_sinh
tff(fact_7190_continuous__cosh,axiom,
    ! [A: $tType,C: $tType] :
      ( ( topological_t2_space(C)
        & real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [F3: filter(C),F2: fun(C,A)] :
          ( topolo3448309680560233919inuous(C,A,F3,F2)
         => topolo3448309680560233919inuous(C,A,F3,aTP_Lamp_xh(fun(C,A),fun(C,A),F2)) ) ) ).

% continuous_cosh
tff(fact_7191_tendsto__real__root,axiom,
    ! [A: $tType,F2: fun(A,real),X: real,F3: filter(A),N: nat] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,X),F3)
     => filterlim(A,real,aa(nat,fun(A,real),aTP_Lamp_xi(fun(A,real),fun(nat,fun(A,real)),F2),N),topolo7230453075368039082e_nhds(real,aa(real,real,root(N),X)),F3) ) ).

% tendsto_real_root
tff(fact_7192_tendsto__sinh,axiom,
    ! [A: $tType,C: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [F2: fun(C,A),A2: A,F3: filter(C)] :
          ( filterlim(C,A,F2,topolo7230453075368039082e_nhds(A,A2),F3)
         => filterlim(C,A,aTP_Lamp_xj(fun(C,A),fun(C,A),F2),topolo7230453075368039082e_nhds(A,sinh(A,A2)),F3) ) ) ).

% tendsto_sinh
tff(fact_7193_tendsto__cosh,axiom,
    ! [A: $tType,C: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [F2: fun(C,A),A2: A,F3: filter(C)] :
          ( filterlim(C,A,F2,topolo7230453075368039082e_nhds(A,A2),F3)
         => filterlim(C,A,aTP_Lamp_xk(fun(C,A),fun(C,A),F2),topolo7230453075368039082e_nhds(A,cosh(A,A2)),F3) ) ) ).

% tendsto_cosh
tff(fact_7194_tendsto__tan,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [F2: fun(A,A),A2: A,F3: filter(A)] :
          ( filterlim(A,A,F2,topolo7230453075368039082e_nhds(A,A2),F3)
         => ( ( cos(A,A2) != zero_zero(A) )
           => filterlim(A,A,aTP_Lamp_xl(fun(A,A),fun(A,A),F2),topolo7230453075368039082e_nhds(A,aa(A,A,tan(A),A2)),F3) ) ) ) ).

% tendsto_tan
tff(fact_7195_tendsto__norm__zero,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F2: fun(A,B),F3: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,zero_zero(B)),F3)
         => filterlim(A,real,aTP_Lamp_xc(fun(A,B),fun(A,real),F2),topolo7230453075368039082e_nhds(real,zero_zero(real)),F3) ) ) ).

% tendsto_norm_zero
tff(fact_7196_tendsto__norm__zero__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F2: fun(A,B),F3: filter(A)] :
          ( filterlim(A,real,aTP_Lamp_xc(fun(A,B),fun(A,real),F2),topolo7230453075368039082e_nhds(real,zero_zero(real)),F3)
        <=> filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,zero_zero(B)),F3) ) ) ).

% tendsto_norm_zero_iff
tff(fact_7197_tendsto__norm__zero__cancel,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F2: fun(A,B),F3: filter(A)] :
          ( filterlim(A,real,aTP_Lamp_xc(fun(A,B),fun(A,real),F2),topolo7230453075368039082e_nhds(real,zero_zero(real)),F3)
         => filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,zero_zero(B)),F3) ) ) ).

% tendsto_norm_zero_cancel
tff(fact_7198_tendsto__ln,axiom,
    ! [A: $tType,F2: fun(A,real),A2: real,F3: filter(A)] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,A2),F3)
     => ( ( A2 != zero_zero(real) )
       => filterlim(A,real,aTP_Lamp_ij(fun(A,real),fun(A,real),F2),topolo7230453075368039082e_nhds(real,aa(real,real,ln_ln(real),A2)),F3) ) ) ).

% tendsto_ln
tff(fact_7199_tendsto__powr,axiom,
    ! [A: $tType,F2: fun(A,real),A2: real,F3: filter(A),G: fun(A,real),B2: real] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,A2),F3)
     => ( filterlim(A,real,G,topolo7230453075368039082e_nhds(real,B2),F3)
       => ( ( A2 != zero_zero(real) )
         => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_xm(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G),topolo7230453075368039082e_nhds(real,powr(real,A2,B2)),F3) ) ) ) ).

% tendsto_powr
tff(fact_7200_tendsto__tanh,axiom,
    ! [A: $tType,C: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [F2: fun(C,A),A2: A,F3: filter(C)] :
          ( filterlim(C,A,F2,topolo7230453075368039082e_nhds(A,A2),F3)
         => ( ( cosh(A,A2) != zero_zero(A) )
           => filterlim(C,A,aTP_Lamp_xn(fun(C,A),fun(C,A),F2),topolo7230453075368039082e_nhds(A,aa(A,A,tanh(A),A2)),F3) ) ) ) ).

% tendsto_tanh
tff(fact_7201_tendsto__cot,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [F2: fun(A,A),A2: A,F3: filter(A)] :
          ( filterlim(A,A,F2,topolo7230453075368039082e_nhds(A,A2),F3)
         => ( ( sin(A,A2) != zero_zero(A) )
           => filterlim(A,A,aTP_Lamp_xo(fun(A,A),fun(A,A),F2),topolo7230453075368039082e_nhds(A,aa(A,A,cot(A),A2)),F3) ) ) ) ).

% tendsto_cot
tff(fact_7202_tendsto__arcosh,axiom,
    ! [B: $tType,F2: fun(B,real),A2: real,F3: filter(B)] :
      ( filterlim(B,real,F2,topolo7230453075368039082e_nhds(real,A2),F3)
     => ( pp(aa(real,bool,ord_less(real,one_one(real)),A2))
       => filterlim(B,real,aTP_Lamp_xp(fun(B,real),fun(B,real),F2),topolo7230453075368039082e_nhds(real,aa(real,real,arcosh(real),A2)),F3) ) ) ).

% tendsto_arcosh
tff(fact_7203_tendsto__rabs__zero,axiom,
    ! [A: $tType,F2: fun(A,real),F3: filter(A)] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,zero_zero(real)),F3)
     => filterlim(A,real,aTP_Lamp_xq(fun(A,real),fun(A,real),F2),topolo7230453075368039082e_nhds(real,zero_zero(real)),F3) ) ).

% tendsto_rabs_zero
tff(fact_7204_tendsto__rabs__zero__iff,axiom,
    ! [A: $tType,F2: fun(A,real),F3: filter(A)] :
      ( filterlim(A,real,aTP_Lamp_xq(fun(A,real),fun(A,real),F2),topolo7230453075368039082e_nhds(real,zero_zero(real)),F3)
    <=> filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,zero_zero(real)),F3) ) ).

% tendsto_rabs_zero_iff
tff(fact_7205_tendsto__rabs__zero__cancel,axiom,
    ! [A: $tType,F2: fun(A,real),F3: filter(A)] :
      ( filterlim(A,real,aTP_Lamp_xq(fun(A,real),fun(A,real),F2),topolo7230453075368039082e_nhds(real,zero_zero(real)),F3)
     => filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,zero_zero(real)),F3) ) ).

% tendsto_rabs_zero_cancel
tff(fact_7206_tendsto__rabs,axiom,
    ! [A: $tType,F2: fun(A,real),L: real,F3: filter(A)] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,L),F3)
     => filterlim(A,real,aTP_Lamp_xq(fun(A,real),fun(A,real),F2),topolo7230453075368039082e_nhds(real,aa(real,real,abs_abs(real),L)),F3) ) ).

% tendsto_rabs
tff(fact_7207_continuous__max,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo1944317154257567458pology(B) )
     => ! [F3: filter(A),F2: fun(A,B),G: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,F3,F2)
         => ( topolo3448309680560233919inuous(A,B,F3,G)
           => topolo3448309680560233919inuous(A,B,F3,aa(fun(A,B),fun(A,B),aTP_Lamp_xr(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G)) ) ) ) ).

% continuous_max
tff(fact_7208_tendsto__max,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [X6: fun(B,A),X: A,Net: filter(B),Y6: fun(B,A),Y: A] :
          ( filterlim(B,A,X6,topolo7230453075368039082e_nhds(A,X),Net)
         => ( filterlim(B,A,Y6,topolo7230453075368039082e_nhds(A,Y),Net)
           => filterlim(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_xs(fun(B,A),fun(fun(B,A),fun(B,A)),X6),Y6),topolo7230453075368039082e_nhds(A,aa(A,A,ord_max(A,X),Y)),Net) ) ) ) ).

% tendsto_max
tff(fact_7209_tendsto__log,axiom,
    ! [A: $tType,F2: fun(A,real),A2: real,F3: filter(A),G: fun(A,real),B2: real] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,A2),F3)
     => ( filterlim(A,real,G,topolo7230453075368039082e_nhds(real,B2),F3)
       => ( pp(aa(real,bool,ord_less(real,zero_zero(real)),A2))
         => ( ( A2 != one_one(real) )
           => ( pp(aa(real,bool,ord_less(real,zero_zero(real)),B2))
             => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_xt(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G),topolo7230453075368039082e_nhds(real,aa(real,real,log(A2),B2)),F3) ) ) ) ) ) ).

% tendsto_log
tff(fact_7210_tendsto__add,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo6943815403480290642id_add(A)
     => ! [F2: fun(B,A),A2: A,F3: filter(B),G: fun(B,A),B2: A] :
          ( filterlim(B,A,F2,topolo7230453075368039082e_nhds(A,A2),F3)
         => ( filterlim(B,A,G,topolo7230453075368039082e_nhds(A,B2),F3)
           => filterlim(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_xu(fun(B,A),fun(fun(B,A),fun(B,A)),F2),G),topolo7230453075368039082e_nhds(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),F3) ) ) ) ).

% tendsto_add
tff(fact_7211_tendsto__add__const__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo1633459387980952147up_add(A)
     => ! [C2: A,F2: fun(B,A),D: A,F3: filter(B)] :
          ( filterlim(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_xv(A,fun(fun(B,A),fun(B,A)),C2),F2),topolo7230453075368039082e_nhds(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),D)),F3)
        <=> filterlim(B,A,F2,topolo7230453075368039082e_nhds(A,D),F3) ) ) ).

% tendsto_add_const_iff
tff(fact_7212_continuous__add,axiom,
    ! [B: $tType,D6: $tType] :
      ( ( topological_t2_space(D6)
        & topolo6943815403480290642id_add(B) )
     => ! [F3: filter(D6),F2: fun(D6,B),G: fun(D6,B)] :
          ( topolo3448309680560233919inuous(D6,B,F3,F2)
         => ( topolo3448309680560233919inuous(D6,B,F3,G)
           => topolo3448309680560233919inuous(D6,B,F3,aa(fun(D6,B),fun(D6,B),aTP_Lamp_xw(fun(D6,B),fun(fun(D6,B),fun(D6,B)),F2),G)) ) ) ) ).

% continuous_add
tff(fact_7213_tendsto__add__zero,axiom,
    ! [B: $tType,D6: $tType] :
      ( topolo6943815403480290642id_add(B)
     => ! [F2: fun(D6,B),F3: filter(D6),G: fun(D6,B)] :
          ( filterlim(D6,B,F2,topolo7230453075368039082e_nhds(B,zero_zero(B)),F3)
         => ( filterlim(D6,B,G,topolo7230453075368039082e_nhds(B,zero_zero(B)),F3)
           => filterlim(D6,B,aa(fun(D6,B),fun(D6,B),aTP_Lamp_xx(fun(D6,B),fun(fun(D6,B),fun(D6,B)),F2),G),topolo7230453075368039082e_nhds(B,zero_zero(B)),F3) ) ) ) ).

% tendsto_add_zero
tff(fact_7214_tendsto__mult__zero,axiom,
    ! [A: $tType,D6: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [F2: fun(D6,A),F3: filter(D6),G: fun(D6,A)] :
          ( filterlim(D6,A,F2,topolo7230453075368039082e_nhds(A,zero_zero(A)),F3)
         => ( filterlim(D6,A,G,topolo7230453075368039082e_nhds(A,zero_zero(A)),F3)
           => filterlim(D6,A,aa(fun(D6,A),fun(D6,A),aTP_Lamp_xy(fun(D6,A),fun(fun(D6,A),fun(D6,A)),F2),G),topolo7230453075368039082e_nhds(A,zero_zero(A)),F3) ) ) ) ).

% tendsto_mult_zero
tff(fact_7215_tendsto__mult__left__zero,axiom,
    ! [A: $tType,D6: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [F2: fun(D6,A),F3: filter(D6),C2: A] :
          ( filterlim(D6,A,F2,topolo7230453075368039082e_nhds(A,zero_zero(A)),F3)
         => filterlim(D6,A,aa(A,fun(D6,A),aTP_Lamp_xz(fun(D6,A),fun(A,fun(D6,A)),F2),C2),topolo7230453075368039082e_nhds(A,zero_zero(A)),F3) ) ) ).

% tendsto_mult_left_zero
tff(fact_7216_tendsto__mult__right__zero,axiom,
    ! [A: $tType,D6: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [F2: fun(D6,A),F3: filter(D6),C2: A] :
          ( filterlim(D6,A,F2,topolo7230453075368039082e_nhds(A,zero_zero(A)),F3)
         => filterlim(D6,A,aa(A,fun(D6,A),aTP_Lamp_ya(fun(D6,A),fun(A,fun(D6,A)),F2),C2),topolo7230453075368039082e_nhds(A,zero_zero(A)),F3) ) ) ).

% tendsto_mult_right_zero
tff(fact_7217_tendsto__mult__one,axiom,
    ! [B: $tType,D6: $tType] :
      ( topolo1898628316856586783d_mult(B)
     => ! [F2: fun(D6,B),F3: filter(D6),G: fun(D6,B)] :
          ( filterlim(D6,B,F2,topolo7230453075368039082e_nhds(B,one_one(B)),F3)
         => ( filterlim(D6,B,G,topolo7230453075368039082e_nhds(B,one_one(B)),F3)
           => filterlim(D6,B,aa(fun(D6,B),fun(D6,B),aTP_Lamp_yb(fun(D6,B),fun(fun(D6,B),fun(D6,B)),F2),G),topolo7230453075368039082e_nhds(B,one_one(B)),F3) ) ) ) ).

% tendsto_mult_one
tff(fact_7218_tendsto__mult,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo4211221413907600880p_mult(A)
     => ! [F2: fun(B,A),A2: A,F3: filter(B),G: fun(B,A),B2: A] :
          ( filterlim(B,A,F2,topolo7230453075368039082e_nhds(A,A2),F3)
         => ( filterlim(B,A,G,topolo7230453075368039082e_nhds(A,B2),F3)
           => filterlim(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_yc(fun(B,A),fun(fun(B,A),fun(B,A)),F2),G),topolo7230453075368039082e_nhds(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),F3) ) ) ) ).

% tendsto_mult
tff(fact_7219_tendsto__mult__left,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo4211221413907600880p_mult(A)
     => ! [F2: fun(B,A),L: A,F3: filter(B),C2: A] :
          ( filterlim(B,A,F2,topolo7230453075368039082e_nhds(A,L),F3)
         => filterlim(B,A,aa(A,fun(B,A),aTP_Lamp_yd(fun(B,A),fun(A,fun(B,A)),F2),C2),topolo7230453075368039082e_nhds(A,aa(A,A,aa(A,fun(A,A),times_times(A),C2),L)),F3) ) ) ).

% tendsto_mult_left
tff(fact_7220_tendsto__mult__right,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo4211221413907600880p_mult(A)
     => ! [F2: fun(B,A),L: A,F3: filter(B),C2: A] :
          ( filterlim(B,A,F2,topolo7230453075368039082e_nhds(A,L),F3)
         => filterlim(B,A,aa(A,fun(B,A),aTP_Lamp_ye(fun(B,A),fun(A,fun(B,A)),F2),C2),topolo7230453075368039082e_nhds(A,aa(A,A,aa(A,fun(A,A),times_times(A),L),C2)),F3) ) ) ).

% tendsto_mult_right
tff(fact_7221_continuous__mult__right,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V4412858255891104859lgebra(A)
        & topological_t2_space(B) )
     => ! [F3: filter(B),F2: fun(B,A),C2: A] :
          ( topolo3448309680560233919inuous(B,A,F3,F2)
         => topolo3448309680560233919inuous(B,A,F3,aa(A,fun(B,A),aTP_Lamp_yf(fun(B,A),fun(A,fun(B,A)),F2),C2)) ) ) ).

% continuous_mult_right
tff(fact_7222_continuous__mult__left,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V4412858255891104859lgebra(A)
        & topological_t2_space(B) )
     => ! [F3: filter(B),F2: fun(B,A),C2: A] :
          ( topolo3448309680560233919inuous(B,A,F3,F2)
         => topolo3448309680560233919inuous(B,A,F3,aa(A,fun(B,A),aTP_Lamp_yg(fun(B,A),fun(A,fun(B,A)),F2),C2)) ) ) ).

% continuous_mult_left
tff(fact_7223_continuous__mult_H,axiom,
    ! [B: $tType,D6: $tType] :
      ( ( topological_t2_space(D6)
        & topolo4211221413907600880p_mult(B) )
     => ! [F3: filter(D6),F2: fun(D6,B),G: fun(D6,B)] :
          ( topolo3448309680560233919inuous(D6,B,F3,F2)
         => ( topolo3448309680560233919inuous(D6,B,F3,G)
           => topolo3448309680560233919inuous(D6,B,F3,aa(fun(D6,B),fun(D6,B),aTP_Lamp_yh(fun(D6,B),fun(fun(D6,B),fun(D6,B)),F2),G)) ) ) ) ).

% continuous_mult'
tff(fact_7224_continuous__mult,axiom,
    ! [A: $tType,D6: $tType] :
      ( ( topological_t2_space(D6)
        & real_V4412858255891104859lgebra(A) )
     => ! [F3: filter(D6),F2: fun(D6,A),G: fun(D6,A)] :
          ( topolo3448309680560233919inuous(D6,A,F3,F2)
         => ( topolo3448309680560233919inuous(D6,A,F3,G)
           => topolo3448309680560233919inuous(D6,A,F3,aa(fun(D6,A),fun(D6,A),aTP_Lamp_yi(fun(D6,A),fun(fun(D6,A),fun(D6,A)),F2),G)) ) ) ) ).

% continuous_mult
tff(fact_7225_tendsto__divide__zero,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(B,A),F3: filter(B),C2: A] :
          ( filterlim(B,A,F2,topolo7230453075368039082e_nhds(A,zero_zero(A)),F3)
         => filterlim(B,A,aa(A,fun(B,A),aTP_Lamp_yj(fun(B,A),fun(A,fun(B,A)),F2),C2),topolo7230453075368039082e_nhds(A,zero_zero(A)),F3) ) ) ).

% tendsto_divide_zero
tff(fact_7226_tendsto__divide,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(B,A),A2: A,F3: filter(B),G: fun(B,A),B2: A] :
          ( filterlim(B,A,F2,topolo7230453075368039082e_nhds(A,A2),F3)
         => ( filterlim(B,A,G,topolo7230453075368039082e_nhds(A,B2),F3)
           => ( ( B2 != zero_zero(A) )
             => filterlim(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_yk(fun(B,A),fun(fun(B,A),fun(B,A)),F2),G),topolo7230453075368039082e_nhds(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)),F3) ) ) ) ) ).

% tendsto_divide
tff(fact_7227_tendsto__of__int__ceiling,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( ring_1(C)
        & topolo4958980785337419405_space(C)
        & archim2362893244070406136eiling(B)
        & topolo2564578578187576103pology(B) )
     => ! [F2: fun(A,B),L: B,F3: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),F3)
         => ( ~ pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),L),ring_1_Ints(B)))
           => filterlim(A,C,aTP_Lamp_yl(fun(A,B),fun(A,C),F2),topolo7230453075368039082e_nhds(C,aa(int,C,ring_1_of_int(C),archimedean_ceiling(B,L))),F3) ) ) ) ).

% tendsto_of_int_ceiling
tff(fact_7228_tendsto__of__int__floor,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( ring_1(C)
        & topolo4958980785337419405_space(C)
        & archim2362893244070406136eiling(B)
        & topolo2564578578187576103pology(B) )
     => ! [F2: fun(A,B),L: B,F3: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),F3)
         => ( ~ pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),L),ring_1_Ints(B)))
           => filterlim(A,C,aTP_Lamp_ym(fun(A,B),fun(A,C),F2),topolo7230453075368039082e_nhds(C,aa(int,C,ring_1_of_int(C),archim6421214686448440834_floor(B,L))),F3) ) ) ) ).

% tendsto_of_int_floor
tff(fact_7229_tendsto__Pair,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(B)
        & topolo4958980785337419405_space(C) )
     => ! [F2: fun(A,B),A2: B,F3: filter(A),G: fun(A,C),B2: C] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,A2),F3)
         => ( filterlim(A,C,G,topolo7230453075368039082e_nhds(C,B2),F3)
           => filterlim(A,product_prod(B,C),aa(fun(A,C),fun(A,product_prod(B,C)),aTP_Lamp_yn(fun(A,B),fun(fun(A,C),fun(A,product_prod(B,C))),F2),G),topolo7230453075368039082e_nhds(product_prod(B,C),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),A2),B2)),F3) ) ) ) ).

% tendsto_Pair
tff(fact_7230_continuous__Pair,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo4958980785337419405_space(C)
        & topolo4958980785337419405_space(B) )
     => ! [F3: filter(A),F2: fun(A,B),G: fun(A,C)] :
          ( topolo3448309680560233919inuous(A,B,F3,F2)
         => ( topolo3448309680560233919inuous(A,C,F3,G)
           => topolo3448309680560233919inuous(A,product_prod(B,C),F3,aa(fun(A,C),fun(A,product_prod(B,C)),aTP_Lamp_vp(fun(A,B),fun(fun(A,C),fun(A,product_prod(B,C))),F2),G)) ) ) ) ).

% continuous_Pair
tff(fact_7231_tendsto__artanh,axiom,
    ! [A: $tType,F2: fun(A,real),A2: real,F3: filter(A)] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,A2),F3)
     => ( pp(aa(real,bool,ord_less(real,aa(real,real,uminus_uminus(real),one_one(real))),A2))
       => ( pp(aa(real,bool,ord_less(real,A2),one_one(real)))
         => filterlim(A,real,aTP_Lamp_yo(fun(A,real),fun(A,real),F2),topolo7230453075368039082e_nhds(real,aa(real,real,artanh(real),A2)),F3) ) ) ) ).

% tendsto_artanh
tff(fact_7232_tendsto__min,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [X6: fun(B,A),X: A,Net: filter(B),Y6: fun(B,A),Y: A] :
          ( filterlim(B,A,X6,topolo7230453075368039082e_nhds(A,X),Net)
         => ( filterlim(B,A,Y6,topolo7230453075368039082e_nhds(A,Y),Net)
           => filterlim(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_yp(fun(B,A),fun(fun(B,A),fun(B,A)),X6),Y6),topolo7230453075368039082e_nhds(A,ord_min(A,X,Y)),Net) ) ) ) ).

% tendsto_min
tff(fact_7233_continuous__min,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo1944317154257567458pology(B) )
     => ! [F3: filter(A),F2: fun(A,B),G: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,F3,F2)
         => ( topolo3448309680560233919inuous(A,B,F3,G)
           => topolo3448309680560233919inuous(A,B,F3,aa(fun(A,B),fun(A,B),aTP_Lamp_yq(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G)) ) ) ) ).

% continuous_min
tff(fact_7234_tendsto__const__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( topological_t2_space(A)
     => ! [F3: filter(B),A2: A,B2: A] :
          ( ( F3 != bot_bot(filter(B)) )
         => ( filterlim(B,A,aTP_Lamp_yr(A,fun(B,A),A2),topolo7230453075368039082e_nhds(A,B2),F3)
          <=> ( A2 = B2 ) ) ) ) ).

% tendsto_const_iff
tff(fact_7235_tendsto__prod,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( ( real_V4412858255891104859lgebra(C)
        & comm_ring_1(C) )
     => ! [S: set(A),F2: fun(A,fun(B,C)),L6: fun(A,C),F3: filter(B)] :
          ( ! [I2: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),I2),S))
             => filterlim(B,C,aa(A,fun(B,C),F2,I2),topolo7230453075368039082e_nhds(C,aa(A,C,L6,I2)),F3) )
         => filterlim(B,C,aa(fun(A,fun(B,C)),fun(B,C),aTP_Lamp_yt(set(A),fun(fun(A,fun(B,C)),fun(B,C)),S),F2),topolo7230453075368039082e_nhds(C,groups7121269368397514597t_prod(A,C,L6,S)),F3) ) ) ).

% tendsto_prod
tff(fact_7236_tendsto__prod_H,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( topolo4987421752381908075d_mult(C)
     => ! [I5: set(A),F2: fun(A,fun(B,C)),A2: fun(A,C),F3: filter(B)] :
          ( ! [I2: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),I2),I5))
             => filterlim(B,C,aa(A,fun(B,C),F2,I2),topolo7230453075368039082e_nhds(C,aa(A,C,A2,I2)),F3) )
         => filterlim(B,C,aa(fun(A,fun(B,C)),fun(B,C),aTP_Lamp_yv(set(A),fun(fun(A,fun(B,C)),fun(B,C)),I5),F2),topolo7230453075368039082e_nhds(C,groups7121269368397514597t_prod(A,C,A2,I5)),F3) ) ) ).

% tendsto_prod'
tff(fact_7237_continuous__prod_H,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( topological_t2_space(B)
        & topolo4987421752381908075d_mult(C) )
     => ! [I5: set(A),F3: filter(B),F2: fun(A,fun(B,C))] :
          ( ! [I2: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),I2),I5))
             => topolo3448309680560233919inuous(B,C,F3,aa(A,fun(B,C),F2,I2)) )
         => topolo3448309680560233919inuous(B,C,F3,aa(fun(A,fun(B,C)),fun(B,C),aTP_Lamp_yx(set(A),fun(fun(A,fun(B,C)),fun(B,C)),I5),F2)) ) ) ).

% continuous_prod'
tff(fact_7238_continuous__prod,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( topological_t2_space(B)
        & real_V4412858255891104859lgebra(C)
        & comm_ring_1(C) )
     => ! [S: set(A),F3: filter(B),F2: fun(A,fun(B,C))] :
          ( ! [I2: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),I2),S))
             => topolo3448309680560233919inuous(B,C,F3,aa(A,fun(B,C),F2,I2)) )
         => topolo3448309680560233919inuous(B,C,F3,aa(fun(A,fun(B,C)),fun(B,C),aTP_Lamp_yz(set(A),fun(fun(A,fun(B,C)),fun(B,C)),S),F2)) ) ) ).

% continuous_prod
tff(fact_7239_tendsto__sum,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( topolo5987344860129210374id_add(C)
     => ! [I5: set(A),F2: fun(A,fun(B,C)),A2: fun(A,C),F3: filter(B)] :
          ( ! [I2: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),I2),I5))
             => filterlim(B,C,aa(A,fun(B,C),F2,I2),topolo7230453075368039082e_nhds(C,aa(A,C,A2,I2)),F3) )
         => filterlim(B,C,aa(fun(A,fun(B,C)),fun(B,C),aTP_Lamp_zb(set(A),fun(fun(A,fun(B,C)),fun(B,C)),I5),F2),topolo7230453075368039082e_nhds(C,aa(set(A),C,groups7311177749621191930dd_sum(A,C,A2),I5)),F3) ) ) ).

% tendsto_sum
tff(fact_7240_continuous__sum,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( topological_t2_space(B)
        & topolo5987344860129210374id_add(C) )
     => ! [I5: set(A),F3: filter(B),F2: fun(A,fun(B,C))] :
          ( ! [I2: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),I2),I5))
             => topolo3448309680560233919inuous(B,C,F3,aa(A,fun(B,C),F2,I2)) )
         => topolo3448309680560233919inuous(B,C,F3,aa(fun(A,fun(B,C)),fun(B,C),aTP_Lamp_zd(set(A),fun(fun(A,fun(B,C)),fun(B,C)),I5),F2)) ) ) ).

% continuous_sum
tff(fact_7241_tendsto__one__prod_H,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( topolo4987421752381908075d_mult(C)
     => ! [I5: set(B),F2: fun(A,fun(B,C)),F3: filter(A)] :
          ( ! [I2: B] :
              ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I2),I5))
             => filterlim(A,C,aa(B,fun(A,C),aTP_Lamp_yu(fun(A,fun(B,C)),fun(B,fun(A,C)),F2),I2),topolo7230453075368039082e_nhds(C,one_one(C)),F3) )
         => filterlim(A,C,aa(fun(A,fun(B,C)),fun(A,C),aTP_Lamp_ze(set(B),fun(fun(A,fun(B,C)),fun(A,C)),I5),F2),topolo7230453075368039082e_nhds(C,one_one(C)),F3) ) ) ).

% tendsto_one_prod'
tff(fact_7242_tendsto__null__sum,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( topolo5987344860129210374id_add(C)
     => ! [I5: set(B),F2: fun(A,fun(B,C)),F3: filter(A)] :
          ( ! [I2: B] :
              ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I2),I5))
             => filterlim(A,C,aa(B,fun(A,C),aTP_Lamp_za(fun(A,fun(B,C)),fun(B,fun(A,C)),F2),I2),topolo7230453075368039082e_nhds(C,zero_zero(C)),F3) )
         => filterlim(A,C,aa(fun(A,fun(B,C)),fun(A,C),aTP_Lamp_zf(set(B),fun(fun(A,fun(B,C)),fun(A,C)),I5),F2),topolo7230453075368039082e_nhds(C,zero_zero(C)),F3) ) ) ).

% tendsto_null_sum
tff(fact_7243_DERIV__LIM__iff,axiom,
    ! [A: $tType] :
      ( ( inverse(A)
        & real_V822414075346904944vector(A) )
     => ! [F2: fun(A,A),A2: A,D4: A] :
          ( filterlim(A,A,aa(A,fun(A,A),aTP_Lamp_zg(fun(A,A),fun(A,fun(A,A)),F2),A2),topolo7230453075368039082e_nhds(A,D4),topolo174197925503356063within(A,zero_zero(A),top_top(set(A))))
        <=> filterlim(A,A,aa(A,fun(A,A),aTP_Lamp_zh(fun(A,A),fun(A,fun(A,A)),F2),A2),topolo7230453075368039082e_nhds(A,D4),topolo174197925503356063within(A,A2,top_top(set(A)))) ) ) ).

% DERIV_LIM_iff
tff(fact_7244_isCont__Lb__Ub,axiom,
    ! [A2: real,B2: real,F2: fun(real,real)] :
      ( pp(aa(real,bool,ord_less_eq(real,A2),B2))
     => ( ! [X3: real] :
            ( ( pp(aa(real,bool,ord_less_eq(real,A2),X3))
              & pp(aa(real,bool,ord_less_eq(real,X3),B2)) )
           => topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,X3,top_top(set(real))),F2) )
       => ? [L7: real,M9: real] :
            ( ! [X4: real] :
                ( ( pp(aa(real,bool,ord_less_eq(real,A2),X4))
                  & pp(aa(real,bool,ord_less_eq(real,X4),B2)) )
               => ( pp(aa(real,bool,ord_less_eq(real,L7),aa(real,real,F2,X4)))
                  & pp(aa(real,bool,ord_less_eq(real,aa(real,real,F2,X4)),M9)) ) )
            & ! [Y4: real] :
                ( ( pp(aa(real,bool,ord_less_eq(real,L7),Y4))
                  & pp(aa(real,bool,ord_less_eq(real,Y4),M9)) )
               => ? [X3: real] :
                    ( pp(aa(real,bool,ord_less_eq(real,A2),X3))
                    & pp(aa(real,bool,ord_less_eq(real,X3),B2))
                    & ( aa(real,real,F2,X3) = Y4 ) ) ) ) ) ) ).

% isCont_Lb_Ub
tff(fact_7245_LIM__fun__less__zero,axiom,
    ! [F2: fun(real,real),L: real,C2: real] :
      ( filterlim(real,real,F2,topolo7230453075368039082e_nhds(real,L),topolo174197925503356063within(real,C2,top_top(set(real))))
     => ( pp(aa(real,bool,ord_less(real,L),zero_zero(real)))
       => ? [R3: real] :
            ( pp(aa(real,bool,ord_less(real,zero_zero(real)),R3))
            & ! [X4: real] :
                ( ( ( X4 != C2 )
                  & pp(aa(real,bool,ord_less(real,aa(real,real,abs_abs(real),aa(real,real,minus_minus(real,C2),X4))),R3)) )
               => pp(aa(real,bool,ord_less(real,aa(real,real,F2,X4)),zero_zero(real))) ) ) ) ) ).

% LIM_fun_less_zero
tff(fact_7246_LIM__fun__not__zero,axiom,
    ! [F2: fun(real,real),L: real,C2: real] :
      ( filterlim(real,real,F2,topolo7230453075368039082e_nhds(real,L),topolo174197925503356063within(real,C2,top_top(set(real))))
     => ( ( L != zero_zero(real) )
       => ? [R3: real] :
            ( pp(aa(real,bool,ord_less(real,zero_zero(real)),R3))
            & ! [X4: real] :
                ( ( ( X4 != C2 )
                  & pp(aa(real,bool,ord_less(real,aa(real,real,abs_abs(real),aa(real,real,minus_minus(real,C2),X4))),R3)) )
               => ( aa(real,real,F2,X4) != zero_zero(real) ) ) ) ) ) ).

% LIM_fun_not_zero
tff(fact_7247_LIM__fun__gt__zero,axiom,
    ! [F2: fun(real,real),L: real,C2: real] :
      ( filterlim(real,real,F2,topolo7230453075368039082e_nhds(real,L),topolo174197925503356063within(real,C2,top_top(set(real))))
     => ( pp(aa(real,bool,ord_less(real,zero_zero(real)),L))
       => ? [R3: real] :
            ( pp(aa(real,bool,ord_less(real,zero_zero(real)),R3))
            & ! [X4: real] :
                ( ( ( X4 != C2 )
                  & pp(aa(real,bool,ord_less(real,aa(real,real,abs_abs(real),aa(real,real,minus_minus(real,C2),X4))),R3)) )
               => pp(aa(real,bool,ord_less(real,zero_zero(real)),aa(real,real,F2,X4))) ) ) ) ) ).

% LIM_fun_gt_zero
tff(fact_7248_LIM__compose2,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & topolo4958980785337419405_space(B)
        & topolo4958980785337419405_space(C) )
     => ! [F2: fun(A,B),B2: B,A2: A,G: fun(B,C),C2: C] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,B2),topolo174197925503356063within(A,A2,top_top(set(A))))
         => ( filterlim(B,C,G,topolo7230453075368039082e_nhds(C,C2),topolo174197925503356063within(B,B2,top_top(set(B))))
           => ( ? [D5: real] :
                  ( pp(aa(real,bool,ord_less(real,zero_zero(real)),D5))
                  & ! [X3: A] :
                      ( ( ( X3 != A2 )
                        & pp(aa(real,bool,ord_less(real,real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,X3),A2))),D5)) )
                     => ( aa(A,B,F2,X3) != B2 ) ) )
             => filterlim(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_wd(fun(A,B),fun(fun(B,C),fun(A,C)),F2),G),topolo7230453075368039082e_nhds(C,C2),topolo174197925503356063within(A,A2,top_top(set(A)))) ) ) ) ) ).

% LIM_compose2
tff(fact_7249_isCont__real__sqrt,axiom,
    ! [X: real] : topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,X,top_top(set(real))),sqrt) ).

% isCont_real_sqrt
tff(fact_7250_isCont__real__root,axiom,
    ! [X: real,N: nat] : topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,X,top_top(set(real))),root(N)) ).

% isCont_real_root
tff(fact_7251_continuous__at__within__divide,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V3459762299906320749_field(B) )
     => ! [A2: A,S3: set(A),F2: fun(A,B),G: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,S3),F2)
         => ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,S3),G)
           => ( ( aa(A,B,G,A2) != zero_zero(B) )
             => topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,S3),aa(fun(A,B),fun(A,B),aTP_Lamp_zi(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G)) ) ) ) ) ).

% continuous_at_within_divide
tff(fact_7252_isCont__add,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo6943815403480290642id_add(B) )
     => ! [A2: A,F2: fun(A,B),G: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,top_top(set(A))),F2)
         => ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,top_top(set(A))),G)
           => topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,top_top(set(A))),aa(fun(A,B),fun(A,B),aTP_Lamp_zj(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G)) ) ) ) ).

% isCont_add
tff(fact_7253_isCont__mult,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V4412858255891104859lgebra(B) )
     => ! [A2: A,F2: fun(A,B),G: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,top_top(set(A))),F2)
         => ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,top_top(set(A))),G)
           => topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,top_top(set(A))),aa(fun(A,B),fun(A,B),aTP_Lamp_zk(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G)) ) ) ) ).

% isCont_mult
tff(fact_7254_isCont__diff,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V822414075346904944vector(B) )
     => ! [A2: A,F2: fun(A,B),G: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,top_top(set(A))),F2)
         => ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,top_top(set(A))),G)
           => topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,top_top(set(A))),aa(fun(A,B),fun(A,B),aTP_Lamp_zl(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G)) ) ) ) ).

% isCont_diff
tff(fact_7255_isCont__minus,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V822414075346904944vector(B) )
     => ! [A2: A,F2: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,top_top(set(A))),F2)
         => topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,top_top(set(A))),aTP_Lamp_zm(fun(A,B),fun(A,B),F2)) ) ) ).

% isCont_minus
tff(fact_7256_isCont__power,axiom,
    ! [A: $tType,B: $tType] :
      ( ( power(B)
        & real_V4412858255891104859lgebra(B)
        & topological_t2_space(A) )
     => ! [A2: A,F2: fun(A,B),N: nat] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,top_top(set(A))),F2)
         => topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,top_top(set(A))),aa(nat,fun(A,B),aTP_Lamp_wi(fun(A,B),fun(nat,fun(A,B)),F2),N)) ) ) ).

% isCont_power
tff(fact_7257_continuous__at__within__inverse,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V8999393235501362500lgebra(B) )
     => ! [A2: A,S3: set(A),F2: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,S3),F2)
         => ( ( aa(A,B,F2,A2) != zero_zero(B) )
           => topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,S3),aTP_Lamp_zn(fun(A,B),fun(A,B),F2)) ) ) ) ).

% continuous_at_within_inverse
tff(fact_7258_isCont__sum,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( topological_t2_space(B)
        & topolo5987344860129210374id_add(C) )
     => ! [A3: set(A),A2: B,F2: fun(A,fun(B,C))] :
          ( ! [X3: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),A3))
             => topolo3448309680560233919inuous(B,C,topolo174197925503356063within(B,A2,top_top(set(B))),aa(A,fun(B,C),F2,X3)) )
         => topolo3448309680560233919inuous(B,C,topolo174197925503356063within(B,A2,top_top(set(B))),aa(fun(A,fun(B,C)),fun(B,C),aTP_Lamp_zd(set(A),fun(fun(A,fun(B,C)),fun(B,C)),A3),F2)) ) ) ).

% isCont_sum
tff(fact_7259_continuous__at__within__sgn,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V822414075346904944vector(B) )
     => ! [A2: A,S3: set(A),F2: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,S3),F2)
         => ( ( aa(A,B,F2,A2) != zero_zero(B) )
           => topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,S3),aTP_Lamp_zo(fun(A,B),fun(A,B),F2)) ) ) ) ).

% continuous_at_within_sgn
tff(fact_7260_isCont__cos_H,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [A2: A,F2: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,top_top(set(A))),F2)
         => topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,top_top(set(A))),aTP_Lamp_wu(fun(A,B),fun(A,B),F2)) ) ) ).

% isCont_cos'
tff(fact_7261_isCont__sin_H,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [A2: A,F2: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,top_top(set(A))),F2)
         => topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,top_top(set(A))),aTP_Lamp_wx(fun(A,B),fun(A,B),F2)) ) ) ).

% isCont_sin'
tff(fact_7262_isCont__exp_H,axiom,
    ! [A: $tType,C: $tType] :
      ( ( topological_t2_space(C)
        & real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [A2: C,F2: fun(C,A)] :
          ( topolo3448309680560233919inuous(C,A,topolo174197925503356063within(C,A2,top_top(set(C))),F2)
         => topolo3448309680560233919inuous(C,A,topolo174197925503356063within(C,A2,top_top(set(C))),aTP_Lamp_xf(fun(C,A),fun(C,A),F2)) ) ) ).

% isCont_exp'
tff(fact_7263_isCont__pochhammer,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Z: A,N: nat] : topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,Z,top_top(set(A))),aTP_Lamp_zp(nat,fun(A,A),N)) ) ).

% isCont_pochhammer
tff(fact_7264_DERIV__def,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D4: A,X: A] :
          ( has_field_derivative(A,F2,D4,topolo174197925503356063within(A,X,top_top(set(A))))
        <=> filterlim(A,A,aa(A,fun(A,A),aTP_Lamp_zq(fun(A,A),fun(A,fun(A,A)),F2),X),topolo7230453075368039082e_nhds(A,D4),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ).

% DERIV_def
tff(fact_7265_DERIV__D,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D4: A,X: A] :
          ( has_field_derivative(A,F2,D4,topolo174197925503356063within(A,X,top_top(set(A))))
         => filterlim(A,A,aa(A,fun(A,A),aTP_Lamp_zq(fun(A,A),fun(A,fun(A,A)),F2),X),topolo7230453075368039082e_nhds(A,D4),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ).

% DERIV_D
tff(fact_7266_lim__exp__minus__1,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => filterlim(A,A,aTP_Lamp_zr(A,A),topolo7230453075368039082e_nhds(A,one_one(A)),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ).

% lim_exp_minus_1
tff(fact_7267_atLeastPlusOneLessThan__greaterThanLessThan__int,axiom,
    ! [L: int,U: int] : set_or7035219750837199246ssThan(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),L),one_one(int)),U) = set_or5935395276787703475ssThan(int,L,U) ).

% atLeastPlusOneLessThan_greaterThanLessThan_int
tff(fact_7268_greaterThanLessThan__upt,axiom,
    ! [N: nat,M: nat] : set_or5935395276787703475ssThan(nat,N,M) = aa(list(nat),set(nat),set2(nat),upt(aa(nat,nat,suc,N),M)) ).

% greaterThanLessThan_upt
tff(fact_7269_lemma__termdiff4,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [K: real,F2: fun(A,B),K6: real] :
          ( pp(aa(real,bool,ord_less(real,zero_zero(real)),K))
         => ( ! [H4: A] :
                ( ( H4 != zero_zero(A) )
               => ( pp(aa(real,bool,ord_less(real,real_V7770717601297561774m_norm(A,H4)),K))
                 => pp(aa(real,bool,ord_less_eq(real,real_V7770717601297561774m_norm(B,aa(A,B,F2,H4))),aa(real,real,aa(real,fun(real,real),times_times(real),K6),real_V7770717601297561774m_norm(A,H4)))) ) )
           => filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,zero_zero(B)),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ) ).

% lemma_termdiff4
tff(fact_7270_isCont__bounded,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [A2: real,B2: real,F2: fun(real,A)] :
          ( pp(aa(real,bool,ord_less_eq(real,A2),B2))
         => ( ! [X3: real] :
                ( ( pp(aa(real,bool,ord_less_eq(real,A2),X3))
                  & pp(aa(real,bool,ord_less_eq(real,X3),B2)) )
               => topolo3448309680560233919inuous(real,A,topolo174197925503356063within(real,X3,top_top(set(real))),F2) )
           => ? [M9: A] :
              ! [X4: real] :
                ( ( pp(aa(real,bool,ord_less_eq(real,A2),X4))
                  & pp(aa(real,bool,ord_less_eq(real,X4),B2)) )
               => pp(aa(A,bool,ord_less_eq(A,aa(real,A,F2,X4)),M9)) ) ) ) ) ).

% isCont_bounded
tff(fact_7271_isCont__eq__Ub,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [A2: real,B2: real,F2: fun(real,A)] :
          ( pp(aa(real,bool,ord_less_eq(real,A2),B2))
         => ( ! [X3: real] :
                ( ( pp(aa(real,bool,ord_less_eq(real,A2),X3))
                  & pp(aa(real,bool,ord_less_eq(real,X3),B2)) )
               => topolo3448309680560233919inuous(real,A,topolo174197925503356063within(real,X3,top_top(set(real))),F2) )
           => ? [M9: A] :
                ( ! [X4: real] :
                    ( ( pp(aa(real,bool,ord_less_eq(real,A2),X4))
                      & pp(aa(real,bool,ord_less_eq(real,X4),B2)) )
                   => pp(aa(A,bool,ord_less_eq(A,aa(real,A,F2,X4)),M9)) )
                & ? [X3: real] :
                    ( pp(aa(real,bool,ord_less_eq(real,A2),X3))
                    & pp(aa(real,bool,ord_less_eq(real,X3),B2))
                    & ( aa(real,A,F2,X3) = M9 ) ) ) ) ) ) ).

% isCont_eq_Ub
tff(fact_7272_isCont__eq__Lb,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [A2: real,B2: real,F2: fun(real,A)] :
          ( pp(aa(real,bool,ord_less_eq(real,A2),B2))
         => ( ! [X3: real] :
                ( ( pp(aa(real,bool,ord_less_eq(real,A2),X3))
                  & pp(aa(real,bool,ord_less_eq(real,X3),B2)) )
               => topolo3448309680560233919inuous(real,A,topolo174197925503356063within(real,X3,top_top(set(real))),F2) )
           => ? [M9: A] :
                ( ! [X4: real] :
                    ( ( pp(aa(real,bool,ord_less_eq(real,A2),X4))
                      & pp(aa(real,bool,ord_less_eq(real,X4),B2)) )
                   => pp(aa(A,bool,ord_less_eq(A,M9),aa(real,A,F2,X4))) )
                & ? [X3: real] :
                    ( pp(aa(real,bool,ord_less_eq(real,A2),X3))
                    & pp(aa(real,bool,ord_less_eq(real,X3),B2))
                    & ( aa(real,A,F2,X3) = M9 ) ) ) ) ) ) ).

% isCont_eq_Lb
tff(fact_7273_isCont__inverse__function2,axiom,
    ! [A2: real,X: real,B2: real,G: fun(real,real),F2: fun(real,real)] :
      ( pp(aa(real,bool,ord_less(real,A2),X))
     => ( pp(aa(real,bool,ord_less(real,X),B2))
       => ( ! [Z2: real] :
              ( pp(aa(real,bool,ord_less_eq(real,A2),Z2))
             => ( pp(aa(real,bool,ord_less_eq(real,Z2),B2))
               => ( aa(real,real,G,aa(real,real,F2,Z2)) = Z2 ) ) )
         => ( ! [Z2: real] :
                ( pp(aa(real,bool,ord_less_eq(real,A2),Z2))
               => ( pp(aa(real,bool,ord_less_eq(real,Z2),B2))
                 => topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,Z2,top_top(set(real))),F2) ) )
           => topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,aa(real,real,F2,X),top_top(set(real))),G) ) ) ) ) ).

% isCont_inverse_function2
tff(fact_7274_field__has__derivative__at,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D4: A,X: A] :
          ( has_derivative(A,A,F2,aa(A,fun(A,A),times_times(A),D4),topolo174197925503356063within(A,X,top_top(set(A))))
        <=> filterlim(A,A,aa(A,fun(A,A),aTP_Lamp_zq(fun(A,A),fun(A,fun(A,A)),F2),X),topolo7230453075368039082e_nhds(A,D4),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ).

% field_has_derivative_at
tff(fact_7275_isCont__divide,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V3459762299906320749_field(B) )
     => ! [A2: A,F2: fun(A,B),G: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,top_top(set(A))),F2)
         => ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,top_top(set(A))),G)
           => ( ( aa(A,B,G,A2) != zero_zero(B) )
             => topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,top_top(set(A))),aa(fun(A,B),fun(A,B),aTP_Lamp_zi(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G)) ) ) ) ) ).

% isCont_divide
tff(fact_7276_isCont__ln,axiom,
    ! [X: real] :
      ( ( X != zero_zero(real) )
     => topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,X,top_top(set(real))),ln_ln(real)) ) ).

% isCont_ln
tff(fact_7277_isCont__sgn,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V822414075346904944vector(B) )
     => ! [A2: A,F2: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,top_top(set(A))),F2)
         => ( ( aa(A,B,F2,A2) != zero_zero(B) )
           => topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,top_top(set(A))),aTP_Lamp_zo(fun(A,B),fun(A,B),F2)) ) ) ) ).

% isCont_sgn
tff(fact_7278_filterlim__at__to__0,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(A,B),F3: filter(B),A2: A] :
          ( filterlim(A,B,F2,F3,topolo174197925503356063within(A,A2,top_top(set(A))))
        <=> filterlim(A,B,aa(A,fun(A,B),aTP_Lamp_zs(fun(A,B),fun(A,fun(A,B)),F2),A2),F3,topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ).

% filterlim_at_to_0
tff(fact_7279_continuous__within__tan,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A,S3: set(A),F2: fun(A,A)] :
          ( topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,X,S3),F2)
         => ( ( cos(A,aa(A,A,F2,X)) != zero_zero(A) )
           => topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,X,S3),aTP_Lamp_xl(fun(A,A),fun(A,A),F2)) ) ) ) ).

% continuous_within_tan
tff(fact_7280_atLeastPlusOneLessThan__greaterThanLessThan__integer,axiom,
    ! [L: code_integer,U: code_integer] : set_or7035219750837199246ssThan(code_integer,aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),L),one_one(code_integer)),U) = set_or5935395276787703475ssThan(code_integer,L,U) ).

% atLeastPlusOneLessThan_greaterThanLessThan_integer
tff(fact_7281_continuous__within__cot,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A,S3: set(A),F2: fun(A,A)] :
          ( topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,X,S3),F2)
         => ( ( sin(A,aa(A,A,F2,X)) != zero_zero(A) )
           => topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,X,S3),aTP_Lamp_xo(fun(A,A),fun(A,A),F2)) ) ) ) ).

% continuous_within_cot
tff(fact_7282_continuous__at__within__tanh,axiom,
    ! [A: $tType,C: $tType] :
      ( ( topological_t2_space(C)
        & real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: C,A3: set(C),F2: fun(C,A)] :
          ( topolo3448309680560233919inuous(C,A,topolo174197925503356063within(C,X,A3),F2)
         => ( ( cosh(A,aa(C,A,F2,X)) != zero_zero(A) )
           => topolo3448309680560233919inuous(C,A,topolo174197925503356063within(C,X,A3),aTP_Lamp_zt(fun(C,A),fun(C,A),F2)) ) ) ) ).

% continuous_at_within_tanh
tff(fact_7283_CARAT__DERIV,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),L: A,X: A] :
          ( has_field_derivative(A,F2,L,topolo174197925503356063within(A,X,top_top(set(A))))
        <=> ? [G3: fun(A,A)] :
              ( ! [Z4: A] : aa(A,A,minus_minus(A,aa(A,A,F2,Z4)),aa(A,A,F2,X)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,G3,Z4)),aa(A,A,minus_minus(A,Z4),X))
              & topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,X,top_top(set(A))),G3)
              & ( aa(A,A,G3,X) = L ) ) ) ) ).

% CARAT_DERIV
tff(fact_7284_isCont__has__Ub,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [A2: real,B2: real,F2: fun(real,A)] :
          ( pp(aa(real,bool,ord_less_eq(real,A2),B2))
         => ( ! [X3: real] :
                ( ( pp(aa(real,bool,ord_less_eq(real,A2),X3))
                  & pp(aa(real,bool,ord_less_eq(real,X3),B2)) )
               => topolo3448309680560233919inuous(real,A,topolo174197925503356063within(real,X3,top_top(set(real))),F2) )
           => ? [M9: A] :
                ( ! [X4: real] :
                    ( ( pp(aa(real,bool,ord_less_eq(real,A2),X4))
                      & pp(aa(real,bool,ord_less_eq(real,X4),B2)) )
                   => pp(aa(A,bool,ord_less_eq(A,aa(real,A,F2,X4)),M9)) )
                & ! [N9: A] :
                    ( pp(aa(A,bool,ord_less(A,N9),M9))
                   => ? [X3: real] :
                        ( pp(aa(real,bool,ord_less_eq(real,A2),X3))
                        & pp(aa(real,bool,ord_less_eq(real,X3),B2))
                        & pp(aa(A,bool,ord_less(A,N9),aa(real,A,F2,X3))) ) ) ) ) ) ) ).

% isCont_has_Ub
tff(fact_7285_filterlim__shift__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(A,B),D: A,F3: filter(B),A2: A] :
          ( filterlim(A,B,comp(A,B,A,F2,aa(A,fun(A,A),plus_plus(A),D)),F3,topolo174197925503356063within(A,aa(A,A,minus_minus(A,A2),D),top_top(set(A))))
        <=> filterlim(A,B,F2,F3,topolo174197925503356063within(A,A2,top_top(set(A)))) ) ) ).

% filterlim_shift_iff
tff(fact_7286_filterlim__shift,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(A,B),F3: filter(B),A2: A,D: A] :
          ( filterlim(A,B,F2,F3,topolo174197925503356063within(A,A2,top_top(set(A))))
         => filterlim(A,B,comp(A,B,A,F2,aa(A,fun(A,A),plus_plus(A),D)),F3,topolo174197925503356063within(A,aa(A,A,minus_minus(A,A2),D),top_top(set(A)))) ) ) ).

% filterlim_shift
tff(fact_7287_isCont__tan,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] :
          ( ( cos(A,X) != zero_zero(A) )
         => topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,X,top_top(set(A))),tan(A)) ) ) ).

% isCont_tan
tff(fact_7288_isCont__cot,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] :
          ( ( sin(A,X) != zero_zero(A) )
         => topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,X,top_top(set(A))),cot(A)) ) ) ).

% isCont_cot
tff(fact_7289_isCont__tanh,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] :
          ( ( cosh(A,X) != zero_zero(A) )
         => topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,X,top_top(set(A))),tanh(A)) ) ) ).

% isCont_tanh
tff(fact_7290_powser__limit__0__strong,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [S3: real,A2: fun(nat,A),F2: fun(A,A)] :
          ( pp(aa(real,bool,ord_less(real,zero_zero(real)),S3))
         => ( ! [X3: A] :
                ( ( X3 != zero_zero(A) )
               => ( pp(aa(real,bool,ord_less(real,real_V7770717601297561774m_norm(A,X3)),S3))
                 => pp(aa(A,bool,sums(A,aa(A,fun(nat,A),aTP_Lamp_gc(fun(nat,A),fun(A,fun(nat,A)),A2),X3)),aa(A,A,F2,X3))) ) )
           => filterlim(A,A,F2,topolo7230453075368039082e_nhds(A,aa(nat,A,A2,zero_zero(nat))),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ) ).

% powser_limit_0_strong
tff(fact_7291_powser__limit__0,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [S3: real,A2: fun(nat,A),F2: fun(A,A)] :
          ( pp(aa(real,bool,ord_less(real,zero_zero(real)),S3))
         => ( ! [X3: A] :
                ( pp(aa(real,bool,ord_less(real,real_V7770717601297561774m_norm(A,X3)),S3))
               => pp(aa(A,bool,sums(A,aa(A,fun(nat,A),aTP_Lamp_gc(fun(nat,A),fun(A,fun(nat,A)),A2),X3)),aa(A,A,F2,X3))) )
           => filterlim(A,A,F2,topolo7230453075368039082e_nhds(A,aa(nat,A,A2,zero_zero(nat))),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ) ).

% powser_limit_0
tff(fact_7292_lemma__termdiff5,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_Vector_banach(B) )
     => ! [K: real,F2: fun(nat,real),G: fun(A,fun(nat,B))] :
          ( pp(aa(real,bool,ord_less(real,zero_zero(real)),K))
         => ( summable(real,F2)
           => ( ! [H4: A,N2: nat] :
                  ( ( H4 != zero_zero(A) )
                 => ( pp(aa(real,bool,ord_less(real,real_V7770717601297561774m_norm(A,H4)),K))
                   => pp(aa(real,bool,ord_less_eq(real,real_V7770717601297561774m_norm(B,aa(nat,B,aa(A,fun(nat,B),G,H4),N2))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,F2,N2)),real_V7770717601297561774m_norm(A,H4)))) ) )
             => filterlim(A,B,aTP_Lamp_zu(fun(A,fun(nat,B)),fun(A,B),G),topolo7230453075368039082e_nhds(B,zero_zero(B)),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ) ) ).

% lemma_termdiff5
tff(fact_7293_isCont__tan_H,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [A2: A,F2: fun(A,A)] :
          ( topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,A2,top_top(set(A))),F2)
         => ( ( cos(A,aa(A,A,F2,A2)) != zero_zero(A) )
           => topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,A2,top_top(set(A))),aTP_Lamp_xl(fun(A,A),fun(A,A),F2)) ) ) ) ).

% isCont_tan'
tff(fact_7294_isCont__arcosh,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,ord_less(real,one_one(real)),X))
     => topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,X,top_top(set(real))),arcosh(real)) ) ).

% isCont_arcosh
tff(fact_7295_LIM__cos__div__sin,axiom,
    filterlim(real,real,aTP_Lamp_zv(real,real),topolo7230453075368039082e_nhds(real,zero_zero(real)),topolo174197925503356063within(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))),top_top(set(real)))) ).

% LIM_cos_div_sin
tff(fact_7296_isCont__cot_H,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [A2: A,F2: fun(A,A)] :
          ( topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,A2,top_top(set(A))),F2)
         => ( ( sin(A,aa(A,A,F2,A2)) != zero_zero(A) )
           => topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,A2,top_top(set(A))),aTP_Lamp_xo(fun(A,A),fun(A,A),F2)) ) ) ) ).

% isCont_cot'
tff(fact_7297_DERIV__inverse__function,axiom,
    ! [F2: fun(real,real),D4: real,G: fun(real,real),X: real,A2: real,B2: real] :
      ( has_field_derivative(real,F2,D4,topolo174197925503356063within(real,aa(real,real,G,X),top_top(set(real))))
     => ( ( D4 != zero_zero(real) )
       => ( pp(aa(real,bool,ord_less(real,A2),X))
         => ( pp(aa(real,bool,ord_less(real,X),B2))
           => ( ! [Y3: real] :
                  ( pp(aa(real,bool,ord_less(real,A2),Y3))
                 => ( pp(aa(real,bool,ord_less(real,Y3),B2))
                   => ( aa(real,real,F2,aa(real,real,G,Y3)) = Y3 ) ) )
             => ( topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,X,top_top(set(real))),G)
               => has_field_derivative(real,G,aa(real,real,inverse_inverse(real),D4),topolo174197925503356063within(real,X,top_top(set(real)))) ) ) ) ) ) ) ).

% DERIV_inverse_function
tff(fact_7298_isCont__polynom,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [A2: A,C2: fun(nat,A),N: nat] : topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,A2,top_top(set(A))),aa(nat,fun(A,A),aTP_Lamp_zw(fun(nat,A),fun(nat,fun(A,A)),C2),N)) ) ).

% isCont_polynom
tff(fact_7299_isCont__powser__converges__everywhere,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [C2: fun(nat,A),X: A] :
          ( ! [Y3: A] : summable(A,aa(A,fun(nat,A),aTP_Lamp_gc(fun(nat,A),fun(A,fun(nat,A)),C2),Y3))
         => topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,X,top_top(set(A))),aTP_Lamp_sc(fun(nat,A),fun(A,A),C2)) ) ) ).

% isCont_powser_converges_everywhere
tff(fact_7300_isCont__arccos,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,ord_less(real,aa(real,real,uminus_uminus(real),one_one(real))),X))
     => ( pp(aa(real,bool,ord_less(real,X),one_one(real)))
       => topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,X,top_top(set(real))),arccos) ) ) ).

% isCont_arccos
tff(fact_7301_isCont__arcsin,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,ord_less(real,aa(real,real,uminus_uminus(real),one_one(real))),X))
     => ( pp(aa(real,bool,ord_less(real,X),one_one(real)))
       => topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,X,top_top(set(real))),arcsin) ) ) ).

% isCont_arcsin
tff(fact_7302_LIM__less__bound,axiom,
    ! [B2: real,X: real,F2: fun(real,real)] :
      ( pp(aa(real,bool,ord_less(real,B2),X))
     => ( ! [X3: real] :
            ( pp(aa(set(real),bool,aa(real,fun(set(real),bool),member(real),X3),set_or5935395276787703475ssThan(real,B2,X)))
           => pp(aa(real,bool,ord_less_eq(real,zero_zero(real)),aa(real,real,F2,X3))) )
       => ( topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,X,top_top(set(real))),F2)
         => pp(aa(real,bool,ord_less_eq(real,zero_zero(real)),aa(real,real,F2,X))) ) ) ) ).

% LIM_less_bound
tff(fact_7303_isCont__artanh,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,ord_less(real,aa(real,real,uminus_uminus(real),one_one(real))),X))
     => ( pp(aa(real,bool,ord_less(real,X),one_one(real)))
       => topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,X,top_top(set(real))),artanh(real)) ) ) ).

% isCont_artanh
tff(fact_7304_isCont__inverse__function,axiom,
    ! [D: real,X: real,G: fun(real,real),F2: fun(real,real)] :
      ( pp(aa(real,bool,ord_less(real,zero_zero(real)),D))
     => ( ! [Z2: real] :
            ( pp(aa(real,bool,ord_less_eq(real,aa(real,real,abs_abs(real),aa(real,real,minus_minus(real,Z2),X))),D))
           => ( aa(real,real,G,aa(real,real,F2,Z2)) = Z2 ) )
       => ( ! [Z2: real] :
              ( pp(aa(real,bool,ord_less_eq(real,aa(real,real,abs_abs(real),aa(real,real,minus_minus(real,Z2),X))),D))
             => topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,Z2,top_top(set(real))),F2) )
         => topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,aa(real,real,F2,X),top_top(set(real))),G) ) ) ) ).

% isCont_inverse_function
tff(fact_7305_GMVT_H,axiom,
    ! [A2: real,B2: real,F2: fun(real,real),G: fun(real,real),G4: fun(real,real),F7: fun(real,real)] :
      ( pp(aa(real,bool,ord_less(real,A2),B2))
     => ( ! [Z2: real] :
            ( pp(aa(real,bool,ord_less_eq(real,A2),Z2))
           => ( pp(aa(real,bool,ord_less_eq(real,Z2),B2))
             => topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,Z2,top_top(set(real))),F2) ) )
       => ( ! [Z2: real] :
              ( pp(aa(real,bool,ord_less_eq(real,A2),Z2))
             => ( pp(aa(real,bool,ord_less_eq(real,Z2),B2))
               => topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,Z2,top_top(set(real))),G) ) )
         => ( ! [Z2: real] :
                ( pp(aa(real,bool,ord_less(real,A2),Z2))
               => ( pp(aa(real,bool,ord_less(real,Z2),B2))
                 => has_field_derivative(real,G,aa(real,real,G4,Z2),topolo174197925503356063within(real,Z2,top_top(set(real)))) ) )
           => ( ! [Z2: real] :
                  ( pp(aa(real,bool,ord_less(real,A2),Z2))
                 => ( pp(aa(real,bool,ord_less(real,Z2),B2))
                   => has_field_derivative(real,F2,aa(real,real,F7,Z2),topolo174197925503356063within(real,Z2,top_top(set(real)))) ) )
             => ? [C3: real] :
                  ( pp(aa(real,bool,ord_less(real,A2),C3))
                  & pp(aa(real,bool,ord_less(real,C3),B2))
                  & ( aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,minus_minus(real,aa(real,real,F2,B2)),aa(real,real,F2,A2))),aa(real,real,G4,C3)) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,minus_minus(real,aa(real,real,G,B2)),aa(real,real,G,A2))),aa(real,real,F7,C3)) ) ) ) ) ) ) ) ).

% GMVT'
tff(fact_7306_floor__has__real__derivative,axiom,
    ! [A: $tType] :
      ( ( archim2362893244070406136eiling(A)
        & topolo2564578578187576103pology(A) )
     => ! [X: real,F2: fun(real,A)] :
          ( topolo3448309680560233919inuous(real,A,topolo174197925503356063within(real,X,top_top(set(real))),F2)
         => ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(real,A,F2,X)),ring_1_Ints(A)))
           => has_field_derivative(real,aTP_Lamp_zx(fun(real,A),fun(real,real),F2),zero_zero(real),topolo174197925503356063within(real,X,top_top(set(real)))) ) ) ) ).

% floor_has_real_derivative
tff(fact_7307_isCont__powser,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [C2: fun(nat,A),K6: A,X: A] :
          ( summable(A,aa(A,fun(nat,A),aTP_Lamp_gc(fun(nat,A),fun(A,fun(nat,A)),C2),K6))
         => ( pp(aa(real,bool,ord_less(real,real_V7770717601297561774m_norm(A,X)),real_V7770717601297561774m_norm(A,K6)))
           => topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,X,top_top(set(A))),aTP_Lamp_sc(fun(nat,A),fun(A,A),C2)) ) ) ) ).

% isCont_powser
tff(fact_7308_isCont__powser_H,axiom,
    ! [A: $tType,Aa: $tType] :
      ( ( real_Vector_banach(Aa)
        & real_V3459762299906320749_field(Aa)
        & topological_t2_space(A) )
     => ! [A2: A,F2: fun(A,Aa),C2: fun(nat,Aa),K6: Aa] :
          ( topolo3448309680560233919inuous(A,Aa,topolo174197925503356063within(A,A2,top_top(set(A))),F2)
         => ( summable(Aa,aa(Aa,fun(nat,Aa),aTP_Lamp_zy(fun(nat,Aa),fun(Aa,fun(nat,Aa)),C2),K6))
           => ( pp(aa(real,bool,ord_less(real,real_V7770717601297561774m_norm(Aa,aa(A,Aa,F2,A2))),real_V7770717601297561774m_norm(Aa,K6)))
             => topolo3448309680560233919inuous(A,Aa,topolo174197925503356063within(A,A2,top_top(set(A))),aa(fun(nat,Aa),fun(A,Aa),aTP_Lamp_aaa(fun(A,Aa),fun(fun(nat,Aa),fun(A,Aa)),F2),C2)) ) ) ) ) ).

% isCont_powser'
tff(fact_7309_summable__Leibniz_I2_J,axiom,
    ! [A2: fun(nat,real)] :
      ( filterlim(nat,real,A2,topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat))
     => ( topological_monoseq(real,A2)
       => ( pp(aa(real,bool,ord_less(real,zero_zero(real)),aa(nat,real,A2,zero_zero(nat))))
         => ! [N11: nat] : pp(aa(set(real),bool,aa(real,fun(set(real),bool),member(real),suminf(real,aTP_Lamp_aab(fun(nat,real),fun(nat,real),A2))),set_or1337092689740270186AtMost(real,aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_aab(fun(nat,real),fun(nat,real),A2)),set_ord_lessThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N11))),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_aab(fun(nat,real),fun(nat,real),A2)),set_ord_lessThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N11)),one_one(nat))))))) ) ) ) ).

% summable_Leibniz(2)
tff(fact_7310_summable__Leibniz_I3_J,axiom,
    ! [A2: fun(nat,real)] :
      ( filterlim(nat,real,A2,topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat))
     => ( topological_monoseq(real,A2)
       => ( pp(aa(real,bool,ord_less(real,aa(nat,real,A2,zero_zero(nat))),zero_zero(real)))
         => ! [N11: nat] : pp(aa(set(real),bool,aa(real,fun(set(real),bool),member(real),suminf(real,aTP_Lamp_aab(fun(nat,real),fun(nat,real),A2))),set_or1337092689740270186AtMost(real,aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_aab(fun(nat,real),fun(nat,real),A2)),set_ord_lessThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N11)),one_one(nat)))),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_aab(fun(nat,real),fun(nat,real),A2)),set_ord_lessThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N11)))))) ) ) ) ).

% summable_Leibniz(3)
tff(fact_7311_tendsto__zero__mult__left__iff,axiom,
    ! [A: $tType] :
      ( ( field(A)
        & topolo4211221413907600880p_mult(A) )
     => ! [C2: A,A2: fun(nat,A)] :
          ( ( C2 != zero_zero(A) )
         => ( filterlim(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_aac(A,fun(fun(nat,A),fun(nat,A)),C2),A2),topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat))
          <=> filterlim(nat,A,A2,topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat)) ) ) ) ).

% tendsto_zero_mult_left_iff
tff(fact_7312_tendsto__zero__mult__right__iff,axiom,
    ! [A: $tType] :
      ( ( field(A)
        & topolo4211221413907600880p_mult(A) )
     => ! [C2: A,A2: fun(nat,A)] :
          ( ( C2 != zero_zero(A) )
         => ( filterlim(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_aad(A,fun(fun(nat,A),fun(nat,A)),C2),A2),topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat))
          <=> filterlim(nat,A,A2,topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat)) ) ) ) ).

% tendsto_zero_mult_right_iff
tff(fact_7313_tendsto__zero__divide__iff,axiom,
    ! [A: $tType] :
      ( ( field(A)
        & topolo4211221413907600880p_mult(A) )
     => ! [C2: A,A2: fun(nat,A)] :
          ( ( C2 != zero_zero(A) )
         => ( filterlim(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_aae(A,fun(fun(nat,A),fun(nat,A)),C2),A2),topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat))
          <=> filterlim(nat,A,A2,topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat)) ) ) ) ).

% tendsto_zero_divide_iff
tff(fact_7314_seq__offset__neg,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [F2: fun(nat,A),L: A,K: nat] :
          ( filterlim(nat,A,F2,topolo7230453075368039082e_nhds(A,L),at_top(nat))
         => filterlim(nat,A,aa(nat,fun(nat,A),aTP_Lamp_aaf(fun(nat,A),fun(nat,fun(nat,A)),F2),K),topolo7230453075368039082e_nhds(A,L),at_top(nat)) ) ) ).

% seq_offset_neg
tff(fact_7315_continuous__arctan,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [F3: filter(A),F2: fun(A,real)] :
          ( topolo3448309680560233919inuous(A,real,F3,F2)
         => topolo3448309680560233919inuous(A,real,F3,aTP_Lamp_aag(fun(A,real),fun(A,real),F2)) ) ) ).

% continuous_arctan
tff(fact_7316_continuous__real__sqrt,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [F3: filter(A),F2: fun(A,real)] :
          ( topolo3448309680560233919inuous(A,real,F3,F2)
         => topolo3448309680560233919inuous(A,real,F3,aTP_Lamp_aah(fun(A,real),fun(A,real),F2)) ) ) ).

% continuous_real_sqrt
tff(fact_7317_LIMSEQ__const__iff,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [K: A,L: A] :
          ( filterlim(nat,A,aTP_Lamp_aai(A,fun(nat,A),K),topolo7230453075368039082e_nhds(A,L),at_top(nat))
        <=> ( K = L ) ) ) ).

% LIMSEQ_const_iff
tff(fact_7318_LIMSEQ__imp__Suc,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [F2: fun(nat,A),L: A] :
          ( filterlim(nat,A,aTP_Lamp_aaj(fun(nat,A),fun(nat,A),F2),topolo7230453075368039082e_nhds(A,L),at_top(nat))
         => filterlim(nat,A,F2,topolo7230453075368039082e_nhds(A,L),at_top(nat)) ) ) ).

% LIMSEQ_imp_Suc
tff(fact_7319_LIMSEQ__Suc,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [F2: fun(nat,A),L: A] :
          ( filterlim(nat,A,F2,topolo7230453075368039082e_nhds(A,L),at_top(nat))
         => filterlim(nat,A,aTP_Lamp_aaj(fun(nat,A),fun(nat,A),F2),topolo7230453075368039082e_nhds(A,L),at_top(nat)) ) ) ).

% LIMSEQ_Suc
tff(fact_7320_continuous__real__root,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [F3: filter(A),F2: fun(A,real),N: nat] :
          ( topolo3448309680560233919inuous(A,real,F3,F2)
         => topolo3448309680560233919inuous(A,real,F3,aa(nat,fun(A,real),aTP_Lamp_aak(fun(A,real),fun(nat,fun(A,real)),F2),N)) ) ) ).

% continuous_real_root
tff(fact_7321_continuous__arsinh,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [F3: filter(A),F2: fun(A,real)] :
          ( topolo3448309680560233919inuous(A,real,F3,F2)
         => topolo3448309680560233919inuous(A,real,F3,aTP_Lamp_aal(fun(A,real),fun(A,real),F2)) ) ) ).

% continuous_arsinh
tff(fact_7322_continuous__rabs,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [F3: filter(A),F2: fun(A,real)] :
          ( topolo3448309680560233919inuous(A,real,F3,F2)
         => topolo3448309680560233919inuous(A,real,F3,aTP_Lamp_aam(fun(A,real),fun(A,real),F2)) ) ) ).

% continuous_rabs
tff(fact_7323_LIMSEQ__ignore__initial__segment,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [F2: fun(nat,A),A2: A,K: nat] :
          ( filterlim(nat,A,F2,topolo7230453075368039082e_nhds(A,A2),at_top(nat))
         => filterlim(nat,A,aa(nat,fun(nat,A),aTP_Lamp_aan(fun(nat,A),fun(nat,fun(nat,A)),F2),K),topolo7230453075368039082e_nhds(A,A2),at_top(nat)) ) ) ).

% LIMSEQ_ignore_initial_segment
tff(fact_7324_LIMSEQ__offset,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [F2: fun(nat,A),K: nat,A2: A] :
          ( filterlim(nat,A,aa(nat,fun(nat,A),aTP_Lamp_aan(fun(nat,A),fun(nat,fun(nat,A)),F2),K),topolo7230453075368039082e_nhds(A,A2),at_top(nat))
         => filterlim(nat,A,F2,topolo7230453075368039082e_nhds(A,A2),at_top(nat)) ) ) ).

% LIMSEQ_offset
tff(fact_7325_lim__mono,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [N4: nat,X6: fun(nat,A),Y6: fun(nat,A),X: A,Y: A] :
          ( ! [N2: nat] :
              ( pp(aa(nat,bool,ord_less_eq(nat,N4),N2))
             => pp(aa(A,bool,ord_less_eq(A,aa(nat,A,X6,N2)),aa(nat,A,Y6,N2))) )
         => ( filterlim(nat,A,X6,topolo7230453075368039082e_nhds(A,X),at_top(nat))
           => ( filterlim(nat,A,Y6,topolo7230453075368039082e_nhds(A,Y),at_top(nat))
             => pp(aa(A,bool,ord_less_eq(A,X),Y)) ) ) ) ) ).

% lim_mono
tff(fact_7326_LIMSEQ__le,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [X6: fun(nat,A),X: A,Y6: fun(nat,A),Y: A] :
          ( filterlim(nat,A,X6,topolo7230453075368039082e_nhds(A,X),at_top(nat))
         => ( filterlim(nat,A,Y6,topolo7230453075368039082e_nhds(A,Y),at_top(nat))
           => ( ? [N9: nat] :
                ! [N2: nat] :
                  ( pp(aa(nat,bool,ord_less_eq(nat,N9),N2))
                 => pp(aa(A,bool,ord_less_eq(A,aa(nat,A,X6,N2)),aa(nat,A,Y6,N2))) )
             => pp(aa(A,bool,ord_less_eq(A,X),Y)) ) ) ) ) ).

% LIMSEQ_le
tff(fact_7327_Lim__bounded,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [F2: fun(nat,A),L: A,M7: nat,C6: A] :
          ( filterlim(nat,A,F2,topolo7230453075368039082e_nhds(A,L),at_top(nat))
         => ( ! [N2: nat] :
                ( pp(aa(nat,bool,ord_less_eq(nat,M7),N2))
               => pp(aa(A,bool,ord_less_eq(A,aa(nat,A,F2,N2)),C6)) )
           => pp(aa(A,bool,ord_less_eq(A,L),C6)) ) ) ) ).

% Lim_bounded
tff(fact_7328_Lim__bounded2,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [F2: fun(nat,A),L: A,N4: nat,C6: A] :
          ( filterlim(nat,A,F2,topolo7230453075368039082e_nhds(A,L),at_top(nat))
         => ( ! [N2: nat] :
                ( pp(aa(nat,bool,ord_less_eq(nat,N4),N2))
               => pp(aa(A,bool,ord_less_eq(A,C6),aa(nat,A,F2,N2))) )
           => pp(aa(A,bool,ord_less_eq(A,C6),L)) ) ) ) ).

% Lim_bounded2
tff(fact_7329_LIMSEQ__le__const,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [X6: fun(nat,A),X: A,A2: A] :
          ( filterlim(nat,A,X6,topolo7230453075368039082e_nhds(A,X),at_top(nat))
         => ( ? [N9: nat] :
              ! [N2: nat] :
                ( pp(aa(nat,bool,ord_less_eq(nat,N9),N2))
               => pp(aa(A,bool,ord_less_eq(A,A2),aa(nat,A,X6,N2))) )
           => pp(aa(A,bool,ord_less_eq(A,A2),X)) ) ) ) ).

% LIMSEQ_le_const
tff(fact_7330_LIMSEQ__le__const2,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [X6: fun(nat,A),X: A,A2: A] :
          ( filterlim(nat,A,X6,topolo7230453075368039082e_nhds(A,X),at_top(nat))
         => ( ? [N9: nat] :
              ! [N2: nat] :
                ( pp(aa(nat,bool,ord_less_eq(nat,N9),N2))
               => pp(aa(A,bool,ord_less_eq(A,aa(nat,A,X6,N2)),A2)) )
           => pp(aa(A,bool,ord_less_eq(A,X),A2)) ) ) ) ).

% LIMSEQ_le_const2
tff(fact_7331_summable__LIMSEQ__zero,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A)] :
          ( summable(A,F2)
         => filterlim(nat,A,F2,topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat)) ) ) ).

% summable_LIMSEQ_zero
tff(fact_7332_isCont__rabs,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [A2: A,F2: fun(A,real)] :
          ( topolo3448309680560233919inuous(A,real,topolo174197925503356063within(A,A2,top_top(set(A))),F2)
         => topolo3448309680560233919inuous(A,real,topolo174197925503356063within(A,A2,top_top(set(A))),aTP_Lamp_aam(fun(A,real),fun(A,real),F2)) ) ) ).

% isCont_rabs
tff(fact_7333_continuous__at__within__powr,axiom,
    ! [C: $tType] :
      ( topological_t2_space(C)
     => ! [A2: C,S3: set(C),F2: fun(C,real),G: fun(C,real)] :
          ( topolo3448309680560233919inuous(C,real,topolo174197925503356063within(C,A2,S3),F2)
         => ( topolo3448309680560233919inuous(C,real,topolo174197925503356063within(C,A2,S3),G)
           => ( ( aa(C,real,F2,A2) != zero_zero(real) )
             => topolo3448309680560233919inuous(C,real,topolo174197925503356063within(C,A2,S3),aa(fun(C,real),fun(C,real),aTP_Lamp_aao(fun(C,real),fun(fun(C,real),fun(C,real)),F2),G)) ) ) ) ) ).

% continuous_at_within_powr
tff(fact_7334_continuous__within__ln,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [X: A,S3: set(A),F2: fun(A,real)] :
          ( topolo3448309680560233919inuous(A,real,topolo174197925503356063within(A,X,S3),F2)
         => ( ( aa(A,real,F2,X) != zero_zero(real) )
           => topolo3448309680560233919inuous(A,real,topolo174197925503356063within(A,X,S3),aTP_Lamp_aap(fun(A,real),fun(A,real),F2)) ) ) ) ).

% continuous_within_ln
tff(fact_7335_mult__nat__left__at__top,axiom,
    ! [C2: nat] :
      ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),C2))
     => filterlim(nat,nat,aa(nat,fun(nat,nat),times_times(nat),C2),at_top(nat),at_top(nat)) ) ).

% mult_nat_left_at_top
tff(fact_7336_mult__nat__right__at__top,axiom,
    ! [C2: nat] :
      ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),C2))
     => filterlim(nat,nat,aTP_Lamp_aaq(nat,fun(nat,nat),C2),at_top(nat),at_top(nat)) ) ).

% mult_nat_right_at_top
tff(fact_7337_monoseq__convergent,axiom,
    ! [X6: fun(nat,real),B4: real] :
      ( topological_monoseq(real,X6)
     => ( ! [I2: nat] : pp(aa(real,bool,ord_less_eq(real,aa(real,real,abs_abs(real),aa(nat,real,X6,I2))),B4))
       => ~ ! [L7: real] : ~ filterlim(nat,real,X6,topolo7230453075368039082e_nhds(real,L7),at_top(nat)) ) ) ).

% monoseq_convergent
tff(fact_7338_LIMSEQ__lessThan__iff__atMost,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [F2: fun(set(nat),A),X: A] :
          ( filterlim(nat,A,aTP_Lamp_aar(fun(set(nat),A),fun(nat,A),F2),topolo7230453075368039082e_nhds(A,X),at_top(nat))
        <=> filterlim(nat,A,aTP_Lamp_aas(fun(set(nat),A),fun(nat,A),F2),topolo7230453075368039082e_nhds(A,X),at_top(nat)) ) ) ).

% LIMSEQ_lessThan_iff_atMost
tff(fact_7339_LIMSEQ__root,axiom,
    filterlim(nat,real,aTP_Lamp_aat(nat,real),topolo7230453075368039082e_nhds(real,one_one(real)),at_top(nat)) ).

% LIMSEQ_root
tff(fact_7340_isCont__powr,axiom,
    ! [C: $tType] :
      ( topological_t2_space(C)
     => ! [A2: C,F2: fun(C,real),G: fun(C,real)] :
          ( topolo3448309680560233919inuous(C,real,topolo174197925503356063within(C,A2,top_top(set(C))),F2)
         => ( topolo3448309680560233919inuous(C,real,topolo174197925503356063within(C,A2,top_top(set(C))),G)
           => ( ( aa(C,real,F2,A2) != zero_zero(real) )
             => topolo3448309680560233919inuous(C,real,topolo174197925503356063within(C,A2,top_top(set(C))),aa(fun(C,real),fun(C,real),aTP_Lamp_aao(fun(C,real),fun(fun(C,real),fun(C,real)),F2),G)) ) ) ) ) ).

% isCont_powr
tff(fact_7341_isCont__ln_H,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [X: A,F2: fun(A,real)] :
          ( topolo3448309680560233919inuous(A,real,topolo174197925503356063within(A,X,top_top(set(A))),F2)
         => ( ( aa(A,real,F2,X) != zero_zero(real) )
           => topolo3448309680560233919inuous(A,real,topolo174197925503356063within(A,X,top_top(set(A))),aTP_Lamp_aap(fun(A,real),fun(A,real),F2)) ) ) ) ).

% isCont_ln'
tff(fact_7342_monoseq__le,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [A2: fun(nat,A),X: A] :
          ( topological_monoseq(A,A2)
         => ( filterlim(nat,A,A2,topolo7230453075368039082e_nhds(A,X),at_top(nat))
           => ( ( ! [N11: nat] : pp(aa(A,bool,ord_less_eq(A,aa(nat,A,A2,N11)),X))
                & ! [M4: nat,N11: nat] :
                    ( pp(aa(nat,bool,ord_less_eq(nat,M4),N11))
                   => pp(aa(A,bool,ord_less_eq(A,aa(nat,A,A2,M4)),aa(nat,A,A2,N11))) ) )
              | ( ! [N11: nat] : pp(aa(A,bool,ord_less_eq(A,X),aa(nat,A,A2,N11)))
                & ! [M4: nat,N11: nat] :
                    ( pp(aa(nat,bool,ord_less_eq(nat,M4),N11))
                   => pp(aa(A,bool,ord_less_eq(A,aa(nat,A,A2,N11)),aa(nat,A,A2,M4))) ) ) ) ) ) ) ).

% monoseq_le
tff(fact_7343_lim__const__over__n,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [A2: A] : filterlim(nat,A,aTP_Lamp_aau(A,fun(nat,A),A2),topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat)) ) ).

% lim_const_over_n
tff(fact_7344_LIMSEQ__linear,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [X6: fun(nat,A),X: A,L: nat] :
          ( filterlim(nat,A,X6,topolo7230453075368039082e_nhds(A,X),at_top(nat))
         => ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),L))
           => filterlim(nat,A,aa(nat,fun(nat,A),aTP_Lamp_aav(fun(nat,A),fun(nat,fun(nat,A)),X6),L),topolo7230453075368039082e_nhds(A,X),at_top(nat)) ) ) ) ).

% LIMSEQ_linear
tff(fact_7345_lim__inverse__n,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => filterlim(nat,A,aTP_Lamp_aaw(nat,A),topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat)) ) ).

% lim_inverse_n
tff(fact_7346_LIMSEQ__SEQ__conv2,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo3112930676232923870pology(A)
        & topolo4958980785337419405_space(B) )
     => ! [A2: A,F2: fun(A,B),L: B] :
          ( ! [S6: fun(nat,A)] :
              ( ( ! [N11: nat] : aa(nat,A,S6,N11) != A2
                & filterlim(nat,A,S6,topolo7230453075368039082e_nhds(A,A2),at_top(nat)) )
             => filterlim(nat,B,aa(fun(nat,A),fun(nat,B),aTP_Lamp_aax(fun(A,B),fun(fun(nat,A),fun(nat,B)),F2),S6),topolo7230453075368039082e_nhds(B,L),at_top(nat)) )
         => filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),topolo174197925503356063within(A,A2,top_top(set(A)))) ) ) ).

% LIMSEQ_SEQ_conv2
tff(fact_7347_LIMSEQ__SEQ__conv1,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topolo4958980785337419405_space(B) )
     => ! [F2: fun(A,B),L: B,A2: A] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),topolo174197925503356063within(A,A2,top_top(set(A))))
         => ! [S9: fun(nat,A)] :
              ( ( ! [N2: nat] : aa(nat,A,S9,N2) != A2
                & filterlim(nat,A,S9,topolo7230453075368039082e_nhds(A,A2),at_top(nat)) )
             => filterlim(nat,B,aa(fun(nat,A),fun(nat,B),aTP_Lamp_aay(fun(A,B),fun(fun(nat,A),fun(nat,B)),F2),S9),topolo7230453075368039082e_nhds(B,L),at_top(nat)) ) ) ) ).

% LIMSEQ_SEQ_conv1
tff(fact_7348_LIMSEQ__SEQ__conv,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo3112930676232923870pology(A)
        & topolo4958980785337419405_space(B) )
     => ! [A2: A,X6: fun(A,B),L6: B] :
          ( ! [S10: fun(nat,A)] :
              ( ( ! [N5: nat] : aa(nat,A,S10,N5) != A2
                & filterlim(nat,A,S10,topolo7230453075368039082e_nhds(A,A2),at_top(nat)) )
             => filterlim(nat,B,aa(fun(nat,A),fun(nat,B),aTP_Lamp_aax(fun(A,B),fun(fun(nat,A),fun(nat,B)),X6),S10),topolo7230453075368039082e_nhds(B,L6),at_top(nat)) )
        <=> filterlim(A,B,X6,topolo7230453075368039082e_nhds(B,L6),topolo174197925503356063within(A,A2,top_top(set(A)))) ) ) ).

% LIMSEQ_SEQ_conv
tff(fact_7349_telescope__summable,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A),C2: A] :
          ( filterlim(nat,A,F2,topolo7230453075368039082e_nhds(A,C2),at_top(nat))
         => summable(A,aTP_Lamp_aaz(fun(nat,A),fun(nat,A),F2)) ) ) ).

% telescope_summable
tff(fact_7350_telescope__summable_H,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A),C2: A] :
          ( filterlim(nat,A,F2,topolo7230453075368039082e_nhds(A,C2),at_top(nat))
         => summable(A,aTP_Lamp_aba(fun(nat,A),fun(nat,A),F2)) ) ) ).

% telescope_summable'
tff(fact_7351_nested__sequence__unique,axiom,
    ! [F2: fun(nat,real),G: fun(nat,real)] :
      ( ! [N2: nat] : pp(aa(real,bool,ord_less_eq(real,aa(nat,real,F2,N2)),aa(nat,real,F2,aa(nat,nat,suc,N2))))
     => ( ! [N2: nat] : pp(aa(real,bool,ord_less_eq(real,aa(nat,real,G,aa(nat,nat,suc,N2))),aa(nat,real,G,N2)))
       => ( ! [N2: nat] : pp(aa(real,bool,ord_less_eq(real,aa(nat,real,F2,N2)),aa(nat,real,G,N2)))
         => ( filterlim(nat,real,aa(fun(nat,real),fun(nat,real),aTP_Lamp_abb(fun(nat,real),fun(fun(nat,real),fun(nat,real)),F2),G),topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat))
           => ? [L3: real] :
                ( ! [N11: nat] : pp(aa(real,bool,ord_less_eq(real,aa(nat,real,F2,N11)),L3))
                & filterlim(nat,real,F2,topolo7230453075368039082e_nhds(real,L3),at_top(nat))
                & ! [N11: nat] : pp(aa(real,bool,ord_less_eq(real,L3),aa(nat,real,G,N11)))
                & filterlim(nat,real,G,topolo7230453075368039082e_nhds(real,L3),at_top(nat)) ) ) ) ) ) ).

% nested_sequence_unique
tff(fact_7352_LIMSEQ__inverse__zero,axiom,
    ! [X6: fun(nat,real)] :
      ( ! [R3: real] :
        ? [N9: nat] :
        ! [N2: nat] :
          ( pp(aa(nat,bool,ord_less_eq(nat,N9),N2))
         => pp(aa(real,bool,ord_less(real,R3),aa(nat,real,X6,N2))) )
     => filterlim(nat,real,aTP_Lamp_abc(fun(nat,real),fun(nat,real),X6),topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat)) ) ).

% LIMSEQ_inverse_zero
tff(fact_7353_lim__inverse__n_H,axiom,
    filterlim(nat,real,aTP_Lamp_abd(nat,real),topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat)) ).

% lim_inverse_n'
tff(fact_7354_LIMSEQ__inverse__real__of__nat,axiom,
    filterlim(nat,real,aTP_Lamp_abe(nat,real),topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat)) ).

% LIMSEQ_inverse_real_of_nat
tff(fact_7355_LIMSEQ__root__const,axiom,
    ! [C2: real] :
      ( pp(aa(real,bool,ord_less(real,zero_zero(real)),C2))
     => filterlim(nat,real,aTP_Lamp_abf(real,fun(nat,real),C2),topolo7230453075368039082e_nhds(real,one_one(real)),at_top(nat)) ) ).

% LIMSEQ_root_const
tff(fact_7356_LIMSEQ__inverse__real__of__nat__add,axiom,
    ! [R2: real] : filterlim(nat,real,aTP_Lamp_abg(real,fun(nat,real),R2),topolo7230453075368039082e_nhds(real,R2),at_top(nat)) ).

% LIMSEQ_inverse_real_of_nat_add
tff(fact_7357_sums__def,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [F2: fun(nat,A),S3: A] :
          ( pp(aa(A,bool,sums(A,F2),S3))
        <=> filterlim(nat,A,aTP_Lamp_abh(fun(nat,A),fun(nat,A),F2),topolo7230453075368039082e_nhds(A,S3),at_top(nat)) ) ) ).

% sums_def
tff(fact_7358_sums__def__le,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [F2: fun(nat,A),S3: A] :
          ( pp(aa(A,bool,sums(A,F2),S3))
        <=> filterlim(nat,A,aTP_Lamp_abi(fun(nat,A),fun(nat,A),F2),topolo7230453075368039082e_nhds(A,S3),at_top(nat)) ) ) ).

% sums_def_le
tff(fact_7359_continuous__at__within__log,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [A2: A,S3: set(A),F2: fun(A,real),G: fun(A,real)] :
          ( topolo3448309680560233919inuous(A,real,topolo174197925503356063within(A,A2,S3),F2)
         => ( topolo3448309680560233919inuous(A,real,topolo174197925503356063within(A,A2,S3),G)
           => ( pp(aa(real,bool,ord_less(real,zero_zero(real)),aa(A,real,F2,A2)))
             => ( ( aa(A,real,F2,A2) != one_one(real) )
               => ( pp(aa(real,bool,ord_less(real,zero_zero(real)),aa(A,real,G,A2)))
                 => topolo3448309680560233919inuous(A,real,topolo174197925503356063within(A,A2,S3),aa(fun(A,real),fun(A,real),aTP_Lamp_abj(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G)) ) ) ) ) ) ) ).

% continuous_at_within_log
tff(fact_7360_increasing__LIMSEQ,axiom,
    ! [F2: fun(nat,real),L: real] :
      ( ! [N2: nat] : pp(aa(real,bool,ord_less_eq(real,aa(nat,real,F2,N2)),aa(nat,real,F2,aa(nat,nat,suc,N2))))
     => ( ! [N2: nat] : pp(aa(real,bool,ord_less_eq(real,aa(nat,real,F2,N2)),L))
       => ( ! [E2: real] :
              ( pp(aa(real,bool,ord_less(real,zero_zero(real)),E2))
             => ? [N11: nat] : pp(aa(real,bool,ord_less_eq(real,L),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,F2,N11)),E2))) )
         => filterlim(nat,real,F2,topolo7230453075368039082e_nhds(real,L),at_top(nat)) ) ) ) ).

% increasing_LIMSEQ
tff(fact_7361_lim__1__over__n,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => filterlim(nat,A,aTP_Lamp_abk(nat,A),topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat)) ) ).

% lim_1_over_n
tff(fact_7362_LIMSEQ__n__over__Suc__n,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => filterlim(nat,A,aTP_Lamp_abl(nat,A),topolo7230453075368039082e_nhds(A,one_one(A)),at_top(nat)) ) ).

% LIMSEQ_n_over_Suc_n
tff(fact_7363_LIMSEQ__Suc__n__over__n,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => filterlim(nat,A,aTP_Lamp_abm(nat,A),topolo7230453075368039082e_nhds(A,one_one(A)),at_top(nat)) ) ).

% LIMSEQ_Suc_n_over_n
tff(fact_7364_telescope__sums,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A),C2: A] :
          ( filterlim(nat,A,F2,topolo7230453075368039082e_nhds(A,C2),at_top(nat))
         => pp(aa(A,bool,sums(A,aTP_Lamp_aaz(fun(nat,A),fun(nat,A),F2)),aa(A,A,minus_minus(A,C2),aa(nat,A,F2,zero_zero(nat))))) ) ) ).

% telescope_sums
tff(fact_7365_telescope__sums_H,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A),C2: A] :
          ( filterlim(nat,A,F2,topolo7230453075368039082e_nhds(A,C2),at_top(nat))
         => pp(aa(A,bool,sums(A,aTP_Lamp_aba(fun(nat,A),fun(nat,A),F2)),aa(A,A,minus_minus(A,aa(nat,A,F2,zero_zero(nat))),C2))) ) ) ).

% telescope_sums'
tff(fact_7366_LIMSEQ__realpow__zero,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,ord_less_eq(real,zero_zero(real)),X))
     => ( pp(aa(real,bool,ord_less(real,X),one_one(real)))
       => filterlim(nat,real,aa(real,fun(nat,real),power_power(real),X),topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat)) ) ) ).

% LIMSEQ_realpow_zero
tff(fact_7367_LIMSEQ__divide__realpow__zero,axiom,
    ! [X: real,A2: real] :
      ( pp(aa(real,bool,ord_less(real,one_one(real)),X))
     => filterlim(nat,real,aa(real,fun(nat,real),aTP_Lamp_abn(real,fun(real,fun(nat,real)),X),A2),topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat)) ) ).

% LIMSEQ_divide_realpow_zero
tff(fact_7368_LIMSEQ__abs__realpow__zero2,axiom,
    ! [C2: real] :
      ( pp(aa(real,bool,ord_less(real,aa(real,real,abs_abs(real),C2)),one_one(real)))
     => filterlim(nat,real,aa(real,fun(nat,real),power_power(real),C2),topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat)) ) ).

% LIMSEQ_abs_realpow_zero2
tff(fact_7369_LIMSEQ__abs__realpow__zero,axiom,
    ! [C2: real] :
      ( pp(aa(real,bool,ord_less(real,aa(real,real,abs_abs(real),C2)),one_one(real)))
     => filterlim(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,abs_abs(real),C2)),topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat)) ) ).

% LIMSEQ_abs_realpow_zero
tff(fact_7370_LIMSEQ__inverse__realpow__zero,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,ord_less(real,one_one(real)),X))
     => filterlim(nat,real,aTP_Lamp_abo(real,fun(nat,real),X),topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat)) ) ).

% LIMSEQ_inverse_realpow_zero
tff(fact_7371_sums__def_H,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [F2: fun(nat,A),S3: A] :
          ( pp(aa(A,bool,sums(A,F2),S3))
        <=> filterlim(nat,A,aTP_Lamp_abp(fun(nat,A),fun(nat,A),F2),topolo7230453075368039082e_nhds(A,S3),at_top(nat)) ) ) ).

% sums_def'
tff(fact_7372_LIMSEQ__inverse__real__of__nat__add__minus,axiom,
    ! [R2: real] : filterlim(nat,real,aTP_Lamp_abq(real,fun(nat,real),R2),topolo7230453075368039082e_nhds(real,R2),at_top(nat)) ).

% LIMSEQ_inverse_real_of_nat_add_minus
tff(fact_7373_root__test__convergence,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [F2: fun(nat,A),X: real] :
          ( filterlim(nat,real,aTP_Lamp_abr(fun(nat,A),fun(nat,real),F2),topolo7230453075368039082e_nhds(real,X),at_top(nat))
         => ( pp(aa(real,bool,ord_less(real,X),one_one(real)))
           => summable(A,F2) ) ) ) ).

% root_test_convergence
tff(fact_7374_summable__LIMSEQ,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topological_t2_space(A) )
     => ! [F2: fun(nat,A)] :
          ( summable(A,F2)
         => filterlim(nat,A,aTP_Lamp_abs(fun(nat,A),fun(nat,A),F2),topolo7230453075368039082e_nhds(A,suminf(A,F2)),at_top(nat)) ) ) ).

% summable_LIMSEQ
tff(fact_7375_summable__LIMSEQ_H,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topological_t2_space(A) )
     => ! [F2: fun(nat,A)] :
          ( summable(A,F2)
         => filterlim(nat,A,aTP_Lamp_abt(fun(nat,A),fun(nat,A),F2),topolo7230453075368039082e_nhds(A,suminf(A,F2)),at_top(nat)) ) ) ).

% summable_LIMSEQ'
tff(fact_7376_isCont__log,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [A2: A,F2: fun(A,real),G: fun(A,real)] :
          ( topolo3448309680560233919inuous(A,real,topolo174197925503356063within(A,A2,top_top(set(A))),F2)
         => ( topolo3448309680560233919inuous(A,real,topolo174197925503356063within(A,A2,top_top(set(A))),G)
           => ( pp(aa(real,bool,ord_less(real,zero_zero(real)),aa(A,real,F2,A2)))
             => ( ( aa(A,real,F2,A2) != one_one(real) )
               => ( pp(aa(real,bool,ord_less(real,zero_zero(real)),aa(A,real,G,A2)))
                 => topolo3448309680560233919inuous(A,real,topolo174197925503356063within(A,A2,top_top(set(A))),aa(fun(A,real),fun(A,real),aTP_Lamp_abj(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G)) ) ) ) ) ) ) ).

% isCont_log
tff(fact_7377_LIMSEQ__D,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X6: fun(nat,A),L6: A,R2: real] :
          ( filterlim(nat,A,X6,topolo7230453075368039082e_nhds(A,L6),at_top(nat))
         => ( pp(aa(real,bool,ord_less(real,zero_zero(real)),R2))
           => ? [No: nat] :
              ! [N11: nat] :
                ( pp(aa(nat,bool,ord_less_eq(nat,No),N11))
               => pp(aa(real,bool,ord_less(real,real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,aa(nat,A,X6,N11)),L6))),R2)) ) ) ) ) ).

% LIMSEQ_D
tff(fact_7378_LIMSEQ__I,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X6: fun(nat,A),L6: A] :
          ( ! [R3: real] :
              ( pp(aa(real,bool,ord_less(real,zero_zero(real)),R3))
             => ? [No2: nat] :
                ! [N2: nat] :
                  ( pp(aa(nat,bool,ord_less_eq(nat,No2),N2))
                 => pp(aa(real,bool,ord_less(real,real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,aa(nat,A,X6,N2)),L6))),R3)) ) )
         => filterlim(nat,A,X6,topolo7230453075368039082e_nhds(A,L6),at_top(nat)) ) ) ).

% LIMSEQ_I
tff(fact_7379_LIMSEQ__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X6: fun(nat,A),L6: A] :
          ( filterlim(nat,A,X6,topolo7230453075368039082e_nhds(A,L6),at_top(nat))
        <=> ! [R5: real] :
              ( pp(aa(real,bool,ord_less(real,zero_zero(real)),R5))
             => ? [No3: nat] :
                ! [N5: nat] :
                  ( pp(aa(nat,bool,ord_less_eq(nat,No3),N5))
                 => pp(aa(real,bool,ord_less(real,real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,aa(nat,A,X6,N5)),L6))),R5)) ) ) ) ) ).

% LIMSEQ_iff
tff(fact_7380_LIMSEQ__power__zero,axiom,
    ! [A: $tType] :
      ( real_V2822296259951069270ebra_1(A)
     => ! [X: A] :
          ( pp(aa(real,bool,ord_less(real,real_V7770717601297561774m_norm(A,X)),one_one(real)))
         => filterlim(nat,A,aa(A,fun(nat,A),power_power(A),X),topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat)) ) ) ).

% LIMSEQ_power_zero
tff(fact_7381_tendsto__exp__limit__sequentially,axiom,
    ! [X: real] : filterlim(nat,real,aTP_Lamp_abu(real,fun(nat,real),X),topolo7230453075368039082e_nhds(real,aa(real,real,exp(real),X)),at_top(nat)) ).

% tendsto_exp_limit_sequentially
tff(fact_7382_tendsto__power__zero,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V2822296259951069270ebra_1(A)
     => ! [F2: fun(B,nat),F3: filter(B),X: A] :
          ( filterlim(B,nat,F2,at_top(nat),F3)
         => ( pp(aa(real,bool,ord_less(real,real_V7770717601297561774m_norm(A,X)),one_one(real)))
           => filterlim(B,A,aa(A,fun(B,A),aTP_Lamp_abv(fun(B,nat),fun(A,fun(B,A)),F2),X),topolo7230453075368039082e_nhds(A,zero_zero(A)),F3) ) ) ) ).

% tendsto_power_zero
tff(fact_7383_LIMSEQ__inverse__real__of__nat__add__minus__mult,axiom,
    ! [R2: real] : filterlim(nat,real,aTP_Lamp_abw(real,fun(nat,real),R2),topolo7230453075368039082e_nhds(real,R2),at_top(nat)) ).

% LIMSEQ_inverse_real_of_nat_add_minus_mult
tff(fact_7384_LIMSEQ__norm__0,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A)] :
          ( ! [N2: nat] : pp(aa(real,bool,ord_less(real,real_V7770717601297561774m_norm(A,aa(nat,A,F2,N2))),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,N2)))))
         => filterlim(nat,A,F2,topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat)) ) ) ).

% LIMSEQ_norm_0
tff(fact_7385_summable__Leibniz_I1_J,axiom,
    ! [A2: fun(nat,real)] :
      ( filterlim(nat,real,A2,topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat))
     => ( topological_monoseq(real,A2)
       => summable(real,aTP_Lamp_aab(fun(nat,real),fun(nat,real),A2)) ) ) ).

% summable_Leibniz(1)
tff(fact_7386_field__derivative__lim__unique,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),Df: A,Z: A,S3: fun(nat,A),A2: A] :
          ( has_field_derivative(A,F2,Df,topolo174197925503356063within(A,Z,top_top(set(A))))
         => ( filterlim(nat,A,S3,topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat))
           => ( ! [N2: nat] : aa(nat,A,S3,N2) != zero_zero(A)
             => ( filterlim(nat,A,aa(fun(nat,A),fun(nat,A),aa(A,fun(fun(nat,A),fun(nat,A)),aTP_Lamp_abx(fun(A,A),fun(A,fun(fun(nat,A),fun(nat,A))),F2),Z),S3),topolo7230453075368039082e_nhds(A,A2),at_top(nat))
               => ( Df = A2 ) ) ) ) ) ) ).

% field_derivative_lim_unique
tff(fact_7387_powser__times__n__limit__0,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V8999393235501362500lgebra(A) )
     => ! [X: A] :
          ( pp(aa(real,bool,ord_less(real,real_V7770717601297561774m_norm(A,X)),one_one(real)))
         => filterlim(nat,A,aTP_Lamp_aby(A,fun(nat,A),X),topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat)) ) ) ).

% powser_times_n_limit_0
tff(fact_7388_lim__n__over__pown,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] :
          ( pp(aa(real,bool,ord_less(real,one_one(real)),real_V7770717601297561774m_norm(A,X)))
         => filterlim(nat,A,aTP_Lamp_abz(A,fun(nat,A),X),topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat)) ) ) ).

% lim_n_over_pown
tff(fact_7389_summable,axiom,
    ! [A2: fun(nat,real)] :
      ( filterlim(nat,real,A2,topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat))
     => ( ! [N2: nat] : pp(aa(real,bool,ord_less_eq(real,zero_zero(real)),aa(nat,real,A2,N2)))
       => ( ! [N2: nat] : pp(aa(real,bool,ord_less_eq(real,aa(nat,real,A2,aa(nat,nat,suc,N2))),aa(nat,real,A2,N2)))
         => summable(real,aTP_Lamp_aab(fun(nat,real),fun(nat,real),A2)) ) ) ) ).

% summable
tff(fact_7390_cos__diff__limit__1,axiom,
    ! [Theta: fun(nat,real),Theta2: real] :
      ( filterlim(nat,real,aa(real,fun(nat,real),aTP_Lamp_aca(fun(nat,real),fun(real,fun(nat,real)),Theta),Theta2),topolo7230453075368039082e_nhds(real,one_one(real)),at_top(nat))
     => ~ ! [K3: fun(nat,int)] : ~ filterlim(nat,real,aa(fun(nat,int),fun(nat,real),aTP_Lamp_acb(fun(nat,real),fun(fun(nat,int),fun(nat,real)),Theta),K3),topolo7230453075368039082e_nhds(real,Theta2),at_top(nat)) ) ).

% cos_diff_limit_1
tff(fact_7391_cos__limit__1,axiom,
    ! [Theta: fun(nat,real)] :
      ( filterlim(nat,real,aTP_Lamp_acc(fun(nat,real),fun(nat,real),Theta),topolo7230453075368039082e_nhds(real,one_one(real)),at_top(nat))
     => ? [K3: fun(nat,int)] : filterlim(nat,real,aa(fun(nat,int),fun(nat,real),aTP_Lamp_acb(fun(nat,real),fun(fun(nat,int),fun(nat,real)),Theta),K3),topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat)) ) ).

% cos_limit_1
tff(fact_7392_summable__Leibniz_I4_J,axiom,
    ! [A2: fun(nat,real)] :
      ( filterlim(nat,real,A2,topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat))
     => ( topological_monoseq(real,A2)
       => filterlim(nat,real,aTP_Lamp_acd(fun(nat,real),fun(nat,real),A2),topolo7230453075368039082e_nhds(real,suminf(real,aTP_Lamp_aab(fun(nat,real),fun(nat,real),A2))),at_top(nat)) ) ) ).

% summable_Leibniz(4)
tff(fact_7393_zeroseq__arctan__series,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,ord_less_eq(real,aa(real,real,abs_abs(real),X)),one_one(real)))
     => filterlim(nat,real,aTP_Lamp_bl(real,fun(nat,real),X),topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat)) ) ).

% zeroseq_arctan_series
tff(fact_7394_summable__Leibniz_H_I3_J,axiom,
    ! [A2: fun(nat,real)] :
      ( filterlim(nat,real,A2,topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat))
     => ( ! [N2: nat] : pp(aa(real,bool,ord_less_eq(real,zero_zero(real)),aa(nat,real,A2,N2)))
       => ( ! [N2: nat] : pp(aa(real,bool,ord_less_eq(real,aa(nat,real,A2,aa(nat,nat,suc,N2))),aa(nat,real,A2,N2)))
         => filterlim(nat,real,aTP_Lamp_acd(fun(nat,real),fun(nat,real),A2),topolo7230453075368039082e_nhds(real,suminf(real,aTP_Lamp_aab(fun(nat,real),fun(nat,real),A2))),at_top(nat)) ) ) ) ).

% summable_Leibniz'(3)
tff(fact_7395_summable__Leibniz_H_I2_J,axiom,
    ! [A2: fun(nat,real),N: nat] :
      ( filterlim(nat,real,A2,topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat))
     => ( ! [N2: nat] : pp(aa(real,bool,ord_less_eq(real,zero_zero(real)),aa(nat,real,A2,N2)))
       => ( ! [N2: nat] : pp(aa(real,bool,ord_less_eq(real,aa(nat,real,A2,aa(nat,nat,suc,N2))),aa(nat,real,A2,N2)))
         => pp(aa(real,bool,ord_less_eq(real,aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_aab(fun(nat,real),fun(nat,real),A2)),set_ord_lessThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)))),suminf(real,aTP_Lamp_aab(fun(nat,real),fun(nat,real),A2)))) ) ) ) ).

% summable_Leibniz'(2)
tff(fact_7396_sums__alternating__upper__lower,axiom,
    ! [A2: fun(nat,real)] :
      ( ! [N2: nat] : pp(aa(real,bool,ord_less_eq(real,aa(nat,real,A2,aa(nat,nat,suc,N2))),aa(nat,real,A2,N2)))
     => ( ! [N2: nat] : pp(aa(real,bool,ord_less_eq(real,zero_zero(real)),aa(nat,real,A2,N2)))
       => ( filterlim(nat,real,A2,topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat))
         => ? [L3: real] :
              ( ! [N11: nat] : pp(aa(real,bool,ord_less_eq(real,aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_aab(fun(nat,real),fun(nat,real),A2)),set_ord_lessThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N11)))),L3))
              & filterlim(nat,real,aTP_Lamp_acd(fun(nat,real),fun(nat,real),A2),topolo7230453075368039082e_nhds(real,L3),at_top(nat))
              & ! [N11: nat] : pp(aa(real,bool,ord_less_eq(real,L3),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_aab(fun(nat,real),fun(nat,real),A2)),set_ord_lessThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N11)),one_one(nat))))))
              & filterlim(nat,real,aTP_Lamp_ace(fun(nat,real),fun(nat,real),A2),topolo7230453075368039082e_nhds(real,L3),at_top(nat)) ) ) ) ) ).

% sums_alternating_upper_lower
tff(fact_7397_summable__Leibniz_I5_J,axiom,
    ! [A2: fun(nat,real)] :
      ( filterlim(nat,real,A2,topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat))
     => ( topological_monoseq(real,A2)
       => filterlim(nat,real,aTP_Lamp_ace(fun(nat,real),fun(nat,real),A2),topolo7230453075368039082e_nhds(real,suminf(real,aTP_Lamp_aab(fun(nat,real),fun(nat,real),A2))),at_top(nat)) ) ) ).

% summable_Leibniz(5)
tff(fact_7398_summable__Leibniz_H_I4_J,axiom,
    ! [A2: fun(nat,real),N: nat] :
      ( filterlim(nat,real,A2,topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat))
     => ( ! [N2: nat] : pp(aa(real,bool,ord_less_eq(real,zero_zero(real)),aa(nat,real,A2,N2)))
       => ( ! [N2: nat] : pp(aa(real,bool,ord_less_eq(real,aa(nat,real,A2,aa(nat,nat,suc,N2))),aa(nat,real,A2,N2)))
         => pp(aa(real,bool,ord_less_eq(real,suminf(real,aTP_Lamp_aab(fun(nat,real),fun(nat,real),A2))),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_aab(fun(nat,real),fun(nat,real),A2)),set_ord_lessThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)),one_one(nat)))))) ) ) ) ).

% summable_Leibniz'(4)
tff(fact_7399_summable__Leibniz_H_I5_J,axiom,
    ! [A2: fun(nat,real)] :
      ( filterlim(nat,real,A2,topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat))
     => ( ! [N2: nat] : pp(aa(real,bool,ord_less_eq(real,zero_zero(real)),aa(nat,real,A2,N2)))
       => ( ! [N2: nat] : pp(aa(real,bool,ord_less_eq(real,aa(nat,real,A2,aa(nat,nat,suc,N2))),aa(nat,real,A2,N2)))
         => filterlim(nat,real,aTP_Lamp_ace(fun(nat,real),fun(nat,real),A2),topolo7230453075368039082e_nhds(real,suminf(real,aTP_Lamp_aab(fun(nat,real),fun(nat,real),A2))),at_top(nat)) ) ) ) ).

% summable_Leibniz'(5)
tff(fact_7400_filterlim__sequentially__Suc,axiom,
    ! [A: $tType,F2: fun(nat,A),F3: filter(A)] :
      ( filterlim(nat,A,aTP_Lamp_qb(fun(nat,A),fun(nat,A),F2),F3,at_top(nat))
    <=> filterlim(nat,A,F2,F3,at_top(nat)) ) ).

% filterlim_sequentially_Suc
tff(fact_7401_has__derivative__at2,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),F7: fun(A,B),X: A] :
          ( has_derivative(A,B,F2,F7,topolo174197925503356063within(A,X,top_top(set(A))))
        <=> ( real_V3181309239436604168linear(A,B,F7)
            & filterlim(A,B,aa(A,fun(A,B),aa(fun(A,B),fun(A,fun(A,B)),aTP_Lamp_acf(fun(A,B),fun(fun(A,B),fun(A,fun(A,B))),F2),F7),X),topolo7230453075368039082e_nhds(B,zero_zero(B)),topolo174197925503356063within(A,X,top_top(set(A)))) ) ) ) ).

% has_derivative_at2
tff(fact_7402_bounded__linear_Ocontinuous,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ( topological_t2_space(C)
        & real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),F3: filter(C),G: fun(C,A)] :
          ( real_V3181309239436604168linear(A,B,F2)
         => ( topolo3448309680560233919inuous(C,A,F3,G)
           => topolo3448309680560233919inuous(C,B,F3,aa(fun(C,A),fun(C,B),aTP_Lamp_acg(fun(A,B),fun(fun(C,A),fun(C,B)),F2),G)) ) ) ) ).

% bounded_linear.continuous
tff(fact_7403_bounded__linear_Otendsto,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V822414075346904944vector(A) )
     => ! [F2: fun(A,B),G: fun(C,A),A2: A,F3: filter(C)] :
          ( real_V3181309239436604168linear(A,B,F2)
         => ( filterlim(C,A,G,topolo7230453075368039082e_nhds(A,A2),F3)
           => filterlim(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_ach(fun(A,B),fun(fun(C,A),fun(C,B)),F2),G),topolo7230453075368039082e_nhds(B,aa(A,B,F2,A2)),F3) ) ) ) ).

% bounded_linear.tendsto
tff(fact_7404_bounded__linear_Obounded,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B)] :
          ( real_V3181309239436604168linear(A,B,F2)
         => ? [K9: real] :
            ! [X4: A] : pp(aa(real,bool,ord_less_eq(real,real_V7770717601297561774m_norm(B,aa(A,B,F2,X4))),aa(real,real,aa(real,fun(real,real),times_times(real),real_V7770717601297561774m_norm(A,X4)),K9))) ) ) ).

% bounded_linear.bounded
tff(fact_7405_bounded__linear_Osuminf,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),X6: fun(nat,A)] :
          ( real_V3181309239436604168linear(A,B,F2)
         => ( summable(A,X6)
           => ( aa(A,B,F2,suminf(A,X6)) = suminf(B,aa(fun(nat,A),fun(nat,B),aTP_Lamp_aci(fun(A,B),fun(fun(nat,A),fun(nat,B)),F2),X6)) ) ) ) ) ).

% bounded_linear.suminf
tff(fact_7406_bounded__linear__minus,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B)] :
          ( real_V3181309239436604168linear(A,B,F2)
         => real_V3181309239436604168linear(A,B,aTP_Lamp_tn(fun(A,B),fun(A,B),F2)) ) ) ).

% bounded_linear_minus
tff(fact_7407_bounded__linear__sub,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),G: fun(A,B)] :
          ( real_V3181309239436604168linear(A,B,F2)
         => ( real_V3181309239436604168linear(A,B,G)
           => real_V3181309239436604168linear(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_tm(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G)) ) ) ) ).

% bounded_linear_sub
tff(fact_7408_bounded__linear__scaleR__const,axiom,
    ! [C: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V822414075346904944vector(C) )
     => ! [G: fun(C,real),X: B] :
          ( real_V3181309239436604168linear(C,real,G)
         => real_V3181309239436604168linear(C,B,aa(B,fun(C,B),aTP_Lamp_tk(fun(C,real),fun(B,fun(C,B)),G),X)) ) ) ).

% bounded_linear_scaleR_const
tff(fact_7409_bounded__linear__scaleR__left,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X: A] : real_V3181309239436604168linear(real,A,aTP_Lamp_acj(A,fun(real,A),X)) ) ).

% bounded_linear_scaleR_left
tff(fact_7410_bounded__linear__const__scaleR,axiom,
    ! [C: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V822414075346904944vector(C) )
     => ! [G: fun(C,B),R2: real] :
          ( real_V3181309239436604168linear(C,B,G)
         => real_V3181309239436604168linear(C,B,aa(real,fun(C,B),aTP_Lamp_tl(fun(C,B),fun(real,fun(C,B)),G),R2)) ) ) ).

% bounded_linear_const_scaleR
tff(fact_7411_bounded__linear__scaleR__right,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [R2: real] : real_V3181309239436604168linear(A,A,real_V8093663219630862766scaleR(A,R2)) ) ).

% bounded_linear_scaleR_right
tff(fact_7412_bounded__linear__compose,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ( real_V822414075346904944vector(C)
        & real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),G: fun(C,A)] :
          ( real_V3181309239436604168linear(A,B,F2)
         => ( real_V3181309239436604168linear(C,A,G)
           => real_V3181309239436604168linear(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_tq(fun(A,B),fun(fun(C,A),fun(C,B)),F2),G)) ) ) ) ).

% bounded_linear_compose
tff(fact_7413_bounded__linear__ident,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => real_V3181309239436604168linear(A,A,aTP_Lamp_ti(A,A)) ) ).

% bounded_linear_ident
tff(fact_7414_bounded__linear_Osummable,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),X6: fun(nat,A)] :
          ( real_V3181309239436604168linear(A,B,F2)
         => ( summable(A,X6)
           => summable(B,aa(fun(nat,A),fun(nat,B),aTP_Lamp_aci(fun(A,B),fun(fun(nat,A),fun(nat,B)),F2),X6)) ) ) ) ).

% bounded_linear.summable
tff(fact_7415_bounded__linear__of__real,axiom,
    ! [A: $tType] :
      ( ( real_V2191834092415804123ebra_1(A)
        & real_V822414075346904944vector(A) )
     => real_V3181309239436604168linear(real,A,real_Vector_of_real(A)) ) ).

% bounded_linear_of_real
tff(fact_7416_bounded__linear_Osums,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),X6: fun(nat,A),A2: A] :
          ( real_V3181309239436604168linear(A,B,F2)
         => ( pp(aa(A,bool,sums(A,X6),A2))
           => pp(aa(B,bool,sums(B,aa(fun(nat,A),fun(nat,B),aTP_Lamp_aci(fun(A,B),fun(fun(nat,A),fun(nat,B)),F2),X6)),aa(A,B,F2,A2))) ) ) ) ).

% bounded_linear.sums
tff(fact_7417_bounded__linear__zero,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => real_V3181309239436604168linear(A,B,aTP_Lamp_th(A,B)) ) ).

% bounded_linear_zero
tff(fact_7418_bounded__linear__add,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),G: fun(A,B)] :
          ( real_V3181309239436604168linear(A,B,F2)
         => ( real_V3181309239436604168linear(A,B,G)
           => real_V3181309239436604168linear(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_tf(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G)) ) ) ) ).

% bounded_linear_add
tff(fact_7419_bounded__linear__mult__right,axiom,
    ! [A: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [X: A] : real_V3181309239436604168linear(A,A,aa(A,fun(A,A),times_times(A),X)) ) ).

% bounded_linear_mult_right
tff(fact_7420_bounded__linear__mult__const,axiom,
    ! [C: $tType,A: $tType] :
      ( ( real_V4412858255891104859lgebra(A)
        & real_V822414075346904944vector(C) )
     => ! [G: fun(C,A),Y: A] :
          ( real_V3181309239436604168linear(C,A,G)
         => real_V3181309239436604168linear(C,A,aa(A,fun(C,A),aTP_Lamp_td(fun(C,A),fun(A,fun(C,A)),G),Y)) ) ) ).

% bounded_linear_mult_const
tff(fact_7421_bounded__linear__const__mult,axiom,
    ! [C: $tType,A: $tType] :
      ( ( real_V4412858255891104859lgebra(A)
        & real_V822414075346904944vector(C) )
     => ! [G: fun(C,A),X: A] :
          ( real_V3181309239436604168linear(C,A,G)
         => real_V3181309239436604168linear(C,A,aa(A,fun(C,A),aTP_Lamp_te(fun(C,A),fun(A,fun(C,A)),G),X)) ) ) ).

% bounded_linear_const_mult
tff(fact_7422_bounded__linear__mult__left,axiom,
    ! [A: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [Y: A] : real_V3181309239436604168linear(A,A,aTP_Lamp_ack(A,fun(A,A),Y)) ) ).

% bounded_linear_mult_left
tff(fact_7423_bounded__linear__divide,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Y: A] : real_V3181309239436604168linear(A,A,aTP_Lamp_acl(A,fun(A,A),Y)) ) ).

% bounded_linear_divide
tff(fact_7424_bounded__linear_OCauchy,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),X6: fun(nat,A)] :
          ( real_V3181309239436604168linear(A,B,F2)
         => ( topolo3814608138187158403Cauchy(A,X6)
           => topolo3814608138187158403Cauchy(B,aa(fun(nat,A),fun(nat,B),aTP_Lamp_aci(fun(A,B),fun(fun(nat,A),fun(nat,B)),F2),X6)) ) ) ) ).

% bounded_linear.Cauchy
tff(fact_7425_bounded__linear__sum,axiom,
    ! [B: $tType,A: $tType,I6: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [I5: set(I6),F2: fun(I6,fun(A,B))] :
          ( ! [I2: I6] :
              ( pp(aa(set(I6),bool,aa(I6,fun(set(I6),bool),member(I6),I2),I5))
             => real_V3181309239436604168linear(A,B,aa(I6,fun(A,B),F2,I2)) )
         => real_V3181309239436604168linear(A,B,aa(fun(I6,fun(A,B)),fun(A,B),aTP_Lamp_acn(set(I6),fun(fun(I6,fun(A,B)),fun(A,B)),I5),F2)) ) ) ).

% bounded_linear_sum
tff(fact_7426_real__bounded__linear,axiom,
    ! [F2: fun(real,real)] :
      ( real_V3181309239436604168linear(real,real,F2)
    <=> ? [C4: real] :
        ! [X2: real] : aa(real,real,F2,X2) = aa(real,real,aa(real,fun(real,real),times_times(real),X2),C4) ) ).

% real_bounded_linear
tff(fact_7427_bounded__linear_Ohas__derivative,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( ( real_V822414075346904944vector(C)
        & real_V822414075346904944vector(B)
        & real_V822414075346904944vector(A) )
     => ! [F2: fun(A,B),G: fun(C,A),G4: fun(C,A),F3: filter(C)] :
          ( real_V3181309239436604168linear(A,B,F2)
         => ( has_derivative(C,A,G,G4,F3)
           => has_derivative(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_tq(fun(A,B),fun(fun(C,A),fun(C,B)),F2),G),aa(fun(C,A),fun(C,B),aTP_Lamp_tq(fun(A,B),fun(fun(C,A),fun(C,B)),F2),G4),F3) ) ) ) ).

% bounded_linear.has_derivative
tff(fact_7428_bounded__linear_Otendsto__zero,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),G: fun(C,A),F3: filter(C)] :
          ( real_V3181309239436604168linear(A,B,F2)
         => ( filterlim(C,A,G,topolo7230453075368039082e_nhds(A,zero_zero(A)),F3)
           => filterlim(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_ach(fun(A,B),fun(fun(C,A),fun(C,B)),F2),G),topolo7230453075368039082e_nhds(B,zero_zero(B)),F3) ) ) ) ).

% bounded_linear.tendsto_zero
tff(fact_7429_bounded__linear_OisCont,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ( topological_t2_space(C)
        & real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),A2: C,G: fun(C,A)] :
          ( real_V3181309239436604168linear(A,B,F2)
         => ( topolo3448309680560233919inuous(C,A,topolo174197925503356063within(C,A2,top_top(set(C))),G)
           => topolo3448309680560233919inuous(C,B,topolo174197925503356063within(C,A2,top_top(set(C))),aa(fun(C,A),fun(C,B),aTP_Lamp_acg(fun(A,B),fun(fun(C,A),fun(C,B)),F2),G)) ) ) ) ).

% bounded_linear.isCont
tff(fact_7430_bounded__linear_Ononneg__bounded,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B)] :
          ( real_V3181309239436604168linear(A,B,F2)
         => ? [K9: real] :
              ( pp(aa(real,bool,ord_less_eq(real,zero_zero(real)),K9))
              & ! [X4: A] : pp(aa(real,bool,ord_less_eq(real,real_V7770717601297561774m_norm(B,aa(A,B,F2,X4))),aa(real,real,aa(real,fun(real,real),times_times(real),real_V7770717601297561774m_norm(A,X4)),K9))) ) ) ) ).

% bounded_linear.nonneg_bounded
tff(fact_7431_filterlim__ident,axiom,
    ! [A: $tType,F3: filter(A)] : filterlim(A,A,aTP_Lamp_at(A,A),F3,F3) ).

% filterlim_ident
tff(fact_7432_filterlim__compose,axiom,
    ! [B: $tType,A: $tType,C: $tType,G: fun(A,B),F32: filter(B),F22: filter(A),F2: fun(C,A),F1: filter(C)] :
      ( filterlim(A,B,G,F32,F22)
     => ( filterlim(C,A,F2,F22,F1)
       => filterlim(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_aco(fun(A,B),fun(fun(C,A),fun(C,B)),G),F2),F32,F1) ) ) ).

% filterlim_compose
tff(fact_7433_bounded__linear_Opos__bounded,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B)] :
          ( real_V3181309239436604168linear(A,B,F2)
         => ? [K9: real] :
              ( pp(aa(real,bool,ord_less(real,zero_zero(real)),K9))
              & ! [X4: A] : pp(aa(real,bool,ord_less_eq(real,real_V7770717601297561774m_norm(B,aa(A,B,F2,X4))),aa(real,real,aa(real,fun(real,real),times_times(real),real_V7770717601297561774m_norm(A,X4)),K9))) ) ) ) ).

% bounded_linear.pos_bounded
tff(fact_7434_bounded__linear__intro,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),K6: real] :
          ( ! [X3: A,Y3: A] : aa(A,B,F2,aa(A,A,aa(A,fun(A,A),plus_plus(A),X3),Y3)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,F2,X3)),aa(A,B,F2,Y3))
         => ( ! [R3: real,X3: A] : aa(A,B,F2,aa(A,A,real_V8093663219630862766scaleR(A,R3),X3)) = aa(B,B,real_V8093663219630862766scaleR(B,R3),aa(A,B,F2,X3))
           => ( ! [X3: A] : pp(aa(real,bool,ord_less_eq(real,real_V7770717601297561774m_norm(B,aa(A,B,F2,X3))),aa(real,real,aa(real,fun(real,real),times_times(real),real_V7770717601297561774m_norm(A,X3)),K6)))
             => real_V3181309239436604168linear(A,B,F2) ) ) ) ) ).

% bounded_linear_intro
tff(fact_7435_filterlim__Suc,axiom,
    filterlim(nat,nat,suc,at_top(nat),at_top(nat)) ).

% filterlim_Suc
tff(fact_7436_has__derivative__iff__norm,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),F7: fun(A,B),X: A,S3: set(A)] :
          ( has_derivative(A,B,F2,F7,topolo174197925503356063within(A,X,S3))
        <=> ( real_V3181309239436604168linear(A,B,F7)
            & filterlim(A,real,aa(A,fun(A,real),aa(fun(A,B),fun(A,fun(A,real)),aTP_Lamp_acp(fun(A,B),fun(fun(A,B),fun(A,fun(A,real))),F2),F7),X),topolo7230453075368039082e_nhds(real,zero_zero(real)),topolo174197925503356063within(A,X,S3)) ) ) ) ).

% has_derivative_iff_norm
tff(fact_7437_has__derivative__at__within,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),F7: fun(A,B),X: A,S3: set(A)] :
          ( has_derivative(A,B,F2,F7,topolo174197925503356063within(A,X,S3))
        <=> ( real_V3181309239436604168linear(A,B,F7)
            & filterlim(A,B,aa(A,fun(A,B),aa(fun(A,B),fun(A,fun(A,B)),aTP_Lamp_acq(fun(A,B),fun(fun(A,B),fun(A,fun(A,B))),F2),F7),X),topolo7230453075368039082e_nhds(B,zero_zero(B)),topolo174197925503356063within(A,X,S3)) ) ) ) ).

% has_derivative_at_within
tff(fact_7438_has__derivativeI,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F7: fun(A,B),X: A,F2: fun(A,B),S3: set(A)] :
          ( real_V3181309239436604168linear(A,B,F7)
         => ( filterlim(A,B,aa(fun(A,B),fun(A,B),aa(A,fun(fun(A,B),fun(A,B)),aTP_Lamp_acr(fun(A,B),fun(A,fun(fun(A,B),fun(A,B))),F7),X),F2),topolo7230453075368039082e_nhds(B,zero_zero(B)),topolo174197925503356063within(A,X,S3))
           => has_derivative(A,B,F2,F7,topolo174197925503356063within(A,X,S3)) ) ) ) ).

% has_derivativeI
tff(fact_7439_has__derivative__iff__Ex,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),F7: fun(A,B),X: A] :
          ( has_derivative(A,B,F2,F7,topolo174197925503356063within(A,X,top_top(set(A))))
        <=> ( real_V3181309239436604168linear(A,B,F7)
            & ? [E4: fun(A,B)] :
                ( ! [H6: A] : aa(A,B,F2,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),H6)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,F2,X)),aa(A,B,F7,H6))),aa(A,B,E4,H6))
                & filterlim(A,real,aTP_Lamp_acs(fun(A,B),fun(A,real),E4),topolo7230453075368039082e_nhds(real,zero_zero(real)),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ) ) ).

% has_derivative_iff_Ex
tff(fact_7440_has__derivative__within,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),F7: fun(A,B),X: A,S3: set(A)] :
          ( has_derivative(A,B,F2,F7,topolo174197925503356063within(A,X,S3))
        <=> ( real_V3181309239436604168linear(A,B,F7)
            & filterlim(A,B,aa(A,fun(A,B),aa(fun(A,B),fun(A,fun(A,B)),aTP_Lamp_acf(fun(A,B),fun(fun(A,B),fun(A,fun(A,B))),F2),F7),X),topolo7230453075368039082e_nhds(B,zero_zero(B)),topolo174197925503356063within(A,X,S3)) ) ) ) ).

% has_derivative_within
tff(fact_7441_has__derivative__at,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),D4: fun(A,B),X: A] :
          ( has_derivative(A,B,F2,D4,topolo174197925503356063within(A,X,top_top(set(A))))
        <=> ( real_V3181309239436604168linear(A,B,D4)
            & filterlim(A,real,aa(A,fun(A,real),aa(fun(A,B),fun(A,fun(A,real)),aTP_Lamp_act(fun(A,B),fun(fun(A,B),fun(A,fun(A,real))),F2),D4),X),topolo7230453075368039082e_nhds(real,zero_zero(real)),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ) ).

% has_derivative_at
tff(fact_7442_has__derivative__def,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),F7: fun(A,B),F3: filter(A)] :
          ( has_derivative(A,B,F2,F7,F3)
        <=> ( real_V3181309239436604168linear(A,B,F7)
            & filterlim(A,B,aa(filter(A),fun(A,B),aa(fun(A,B),fun(filter(A),fun(A,B)),aTP_Lamp_acu(fun(A,B),fun(fun(A,B),fun(filter(A),fun(A,B))),F2),F7),F3),topolo7230453075368039082e_nhds(B,zero_zero(B)),F3) ) ) ) ).

% has_derivative_def
tff(fact_7443_has__derivative__at__within__iff__Ex,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [X: A,S: set(A),F2: fun(A,B),F7: fun(A,B)] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),S))
         => ( topolo1002775350975398744n_open(A,S)
           => ( has_derivative(A,B,F2,F7,topolo174197925503356063within(A,X,S))
            <=> ( real_V3181309239436604168linear(A,B,F7)
                & ? [E4: fun(A,B)] :
                    ( ! [H6: A] :
                        ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),H6)),S))
                       => ( aa(A,B,F2,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),H6)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,F2,X)),aa(A,B,F7,H6))),aa(A,B,E4,H6)) ) )
                    & filterlim(A,real,aTP_Lamp_acs(fun(A,B),fun(A,real),E4),topolo7230453075368039082e_nhds(real,zero_zero(real)),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ) ) ) ) ).

% has_derivative_at_within_iff_Ex
tff(fact_7444_lim__const,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [A2: A] : topolo3827282254853284352ce_Lim(nat,A,at_top(nat),aTP_Lamp_aai(A,fun(nat,A),A2)) = A2 ) ).

% lim_const
tff(fact_7445_open__Collect__conj,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [P: fun(A,bool),Q: fun(A,bool)] :
          ( topolo1002775350975398744n_open(A,collect(A,P))
         => ( topolo1002775350975398744n_open(A,collect(A,Q))
           => topolo1002775350975398744n_open(A,collect(A,aa(fun(A,bool),fun(A,bool),aTP_Lamp_acv(fun(A,bool),fun(fun(A,bool),fun(A,bool)),P),Q))) ) ) ) ).

% open_Collect_conj
tff(fact_7446_open__Collect__disj,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [P: fun(A,bool),Q: fun(A,bool)] :
          ( topolo1002775350975398744n_open(A,collect(A,P))
         => ( topolo1002775350975398744n_open(A,collect(A,Q))
           => topolo1002775350975398744n_open(A,collect(A,aa(fun(A,bool),fun(A,bool),aTP_Lamp_acw(fun(A,bool),fun(fun(A,bool),fun(A,bool)),P),Q))) ) ) ) ).

% open_Collect_disj
tff(fact_7447_open__Collect__const,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [P: bool] : topolo1002775350975398744n_open(A,collect(A,aTP_Lamp_acx(bool,fun(A,bool),P))) ) ).

% open_Collect_const
tff(fact_7448_open__right,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [S: set(A),X: A,Y: A] :
          ( topolo1002775350975398744n_open(A,S)
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),S))
           => ( pp(aa(A,bool,ord_less(A,X),Y))
             => ? [B3: A] :
                  ( pp(aa(A,bool,ord_less(A,X),B3))
                  & pp(aa(set(A),bool,ord_less_eq(set(A),set_or7035219750837199246ssThan(A,X,B3)),S)) ) ) ) ) ) ).

% open_right
tff(fact_7449_at__within__open__subset,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [A2: A,S: set(A),T3: set(A)] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),S))
         => ( topolo1002775350975398744n_open(A,S)
           => ( pp(aa(set(A),bool,ord_less_eq(set(A),S),T3))
             => ( topolo174197925503356063within(A,A2,T3) = topolo174197925503356063within(A,A2,top_top(set(A))) ) ) ) ) ) ).

% at_within_open_subset
tff(fact_7450_openI,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [S: set(A)] :
          ( ! [X3: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),S))
             => ? [T8: set(A)] :
                  ( topolo1002775350975398744n_open(A,T8)
                  & pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),T8))
                  & pp(aa(set(A),bool,ord_less_eq(set(A),T8),S)) ) )
         => topolo1002775350975398744n_open(A,S) ) ) ).

% openI
tff(fact_7451_open__subopen,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [S: set(A)] :
          ( topolo1002775350975398744n_open(A,S)
        <=> ! [X2: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X2),S))
             => ? [T6: set(A)] :
                  ( topolo1002775350975398744n_open(A,T6)
                  & pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X2),T6))
                  & pp(aa(set(A),bool,ord_less_eq(set(A),T6),S)) ) ) ) ) ).

% open_subopen
tff(fact_7452_first__countable__basis,axiom,
    ! [A: $tType] :
      ( topolo3112930676232923870pology(A)
     => ! [X: A] :
        ? [A7: fun(nat,set(A))] :
          ( ! [I4: nat] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(nat,set(A),A7,I4)))
              & topolo1002775350975398744n_open(A,aa(nat,set(A),A7,I4)) )
          & ! [S9: set(A)] :
              ( ( topolo1002775350975398744n_open(A,S9)
                & pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),S9)) )
             => ? [I2: nat] : pp(aa(set(A),bool,ord_less_eq(set(A),aa(nat,set(A),A7,I2)),S9)) ) ) ) ).

% first_countable_basis
tff(fact_7453_Lim__ident__at,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [X: A,S3: set(A)] :
          ( ( topolo174197925503356063within(A,X,S3) != bot_bot(filter(A)) )
         => ( topolo3827282254853284352ce_Lim(A,A,topolo174197925503356063within(A,X,S3),aTP_Lamp_vm(A,A)) = X ) ) ) ).

% Lim_ident_at
tff(fact_7454_lim__explicit,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [F2: fun(nat,A),F0: A] :
          ( filterlim(nat,A,F2,topolo7230453075368039082e_nhds(A,F0),at_top(nat))
        <=> ! [S10: set(A)] :
              ( topolo1002775350975398744n_open(A,S10)
             => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),F0),S10))
               => ? [N8: nat] :
                  ! [N5: nat] :
                    ( pp(aa(nat,bool,ord_less_eq(nat,N8),N5))
                   => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(nat,A,F2,N5)),S10)) ) ) ) ) ) ).

% lim_explicit
tff(fact_7455_continuous__divide,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V3459762299906320749_field(B) )
     => ! [F3: filter(A),F2: fun(A,B),G: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,F3,F2)
         => ( topolo3448309680560233919inuous(A,B,F3,G)
           => ( ( aa(A,B,G,topolo3827282254853284352ce_Lim(A,A,F3,aTP_Lamp_vm(A,A))) != zero_zero(B) )
             => topolo3448309680560233919inuous(A,B,F3,aa(fun(A,B),fun(A,B),aTP_Lamp_zi(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G)) ) ) ) ) ).

% continuous_divide
tff(fact_7456_continuous__inverse,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V8999393235501362500lgebra(B) )
     => ! [F3: filter(A),F2: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,F3,F2)
         => ( ( aa(A,B,F2,topolo3827282254853284352ce_Lim(A,A,F3,aTP_Lamp_vm(A,A))) != zero_zero(B) )
           => topolo3448309680560233919inuous(A,B,F3,aTP_Lamp_zn(fun(A,B),fun(A,B),F2)) ) ) ) ).

% continuous_inverse
tff(fact_7457_continuous__def,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo4958980785337419405_space(B) )
     => ! [F3: filter(A),F2: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,F3,F2)
        <=> filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,aa(A,B,F2,topolo3827282254853284352ce_Lim(A,A,F3,aTP_Lamp_vm(A,A)))),F3) ) ) ).

% continuous_def
tff(fact_7458_continuous__sgn,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V822414075346904944vector(B) )
     => ! [F3: filter(A),F2: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,F3,F2)
         => ( ( aa(A,B,F2,topolo3827282254853284352ce_Lim(A,A,F3,aTP_Lamp_vm(A,A))) != zero_zero(B) )
           => topolo3448309680560233919inuous(A,B,F3,aTP_Lamp_zo(fun(A,B),fun(A,B),F2)) ) ) ) ).

% continuous_sgn
tff(fact_7459_t2__space__class_OLim__def,axiom,
    ! [A: $tType,F: $tType] :
      ( topological_t2_space(A)
     => ! [A3: filter(F),F2: fun(F,A)] : topolo3827282254853284352ce_Lim(F,A,A3,F2) = the(A,aa(fun(F,A),fun(A,bool),aTP_Lamp_acy(filter(F),fun(fun(F,A),fun(A,bool)),A3),F2)) ) ).

% t2_space_class.Lim_def
tff(fact_7460_continuous__powr,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [F3: filter(A),F2: fun(A,real),G: fun(A,real)] :
          ( topolo3448309680560233919inuous(A,real,F3,F2)
         => ( topolo3448309680560233919inuous(A,real,F3,G)
           => ( ( aa(A,real,F2,topolo3827282254853284352ce_Lim(A,A,F3,aTP_Lamp_vm(A,A))) != zero_zero(real) )
             => topolo3448309680560233919inuous(A,real,F3,aa(fun(A,real),fun(A,real),aTP_Lamp_acz(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G)) ) ) ) ) ).

% continuous_powr
tff(fact_7461_continuous__ln,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [F3: filter(A),F2: fun(A,real)] :
          ( topolo3448309680560233919inuous(A,real,F3,F2)
         => ( ( aa(A,real,F2,topolo3827282254853284352ce_Lim(A,A,F3,aTP_Lamp_vm(A,A))) != zero_zero(real) )
           => topolo3448309680560233919inuous(A,real,F3,aTP_Lamp_aap(fun(A,real),fun(A,real),F2)) ) ) ) ).

% continuous_ln
tff(fact_7462_suminf__eq__lim,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topological_t2_space(A) )
     => ! [F2: fun(nat,A)] : suminf(A,F2) = topolo3827282254853284352ce_Lim(nat,A,at_top(nat),aTP_Lamp_abs(fun(nat,A),fun(nat,A),F2)) ) ).

% suminf_eq_lim
tff(fact_7463_Topological__Spaces_Olim__def,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [X6: fun(nat,A)] : topolo3827282254853284352ce_Lim(nat,A,at_top(nat),X6) = the(A,aTP_Lamp_ada(fun(nat,A),fun(A,bool),X6)) ) ).

% Topological_Spaces.lim_def
tff(fact_7464_continuous__tan,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [F3: filter(A),F2: fun(A,A)] :
          ( topolo3448309680560233919inuous(A,A,F3,F2)
         => ( ( cos(A,aa(A,A,F2,topolo3827282254853284352ce_Lim(A,A,F3,aTP_Lamp_adb(A,A)))) != zero_zero(A) )
           => topolo3448309680560233919inuous(A,A,F3,aTP_Lamp_xl(fun(A,A),fun(A,A),F2)) ) ) ) ).

% continuous_tan
tff(fact_7465_continuous__cot,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [F3: filter(A),F2: fun(A,A)] :
          ( topolo3448309680560233919inuous(A,A,F3,F2)
         => ( ( sin(A,aa(A,A,F2,topolo3827282254853284352ce_Lim(A,A,F3,aTP_Lamp_adb(A,A)))) != zero_zero(A) )
           => topolo3448309680560233919inuous(A,A,F3,aTP_Lamp_xo(fun(A,A),fun(A,A),F2)) ) ) ) ).

% continuous_cot
tff(fact_7466_continuous__tanh,axiom,
    ! [A: $tType,C: $tType] :
      ( ( topological_t2_space(C)
        & real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [F3: filter(C),F2: fun(C,A)] :
          ( topolo3448309680560233919inuous(C,A,F3,F2)
         => ( ( cosh(A,aa(C,A,F2,topolo3827282254853284352ce_Lim(C,C,F3,aTP_Lamp_adc(C,C)))) != zero_zero(A) )
           => topolo3448309680560233919inuous(C,A,F3,aTP_Lamp_zt(fun(C,A),fun(C,A),F2)) ) ) ) ).

% continuous_tanh
tff(fact_7467_continuous__arcosh,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [F3: filter(A),F2: fun(A,real)] :
          ( topolo3448309680560233919inuous(A,real,F3,F2)
         => ( pp(aa(real,bool,ord_less(real,one_one(real)),aa(A,real,F2,topolo3827282254853284352ce_Lim(A,A,F3,aTP_Lamp_vm(A,A)))))
           => topolo3448309680560233919inuous(A,real,F3,aTP_Lamp_add(fun(A,real),fun(A,real),F2)) ) ) ) ).

% continuous_arcosh
tff(fact_7468_continuous__log,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [F3: filter(A),F2: fun(A,real),G: fun(A,real)] :
          ( topolo3448309680560233919inuous(A,real,F3,F2)
         => ( topolo3448309680560233919inuous(A,real,F3,G)
           => ( pp(aa(real,bool,ord_less(real,zero_zero(real)),aa(A,real,F2,topolo3827282254853284352ce_Lim(A,A,F3,aTP_Lamp_vm(A,A)))))
             => ( ( aa(A,real,F2,topolo3827282254853284352ce_Lim(A,A,F3,aTP_Lamp_vm(A,A))) != one_one(real) )
               => ( pp(aa(real,bool,ord_less(real,zero_zero(real)),aa(A,real,G,topolo3827282254853284352ce_Lim(A,A,F3,aTP_Lamp_vm(A,A)))))
                 => topolo3448309680560233919inuous(A,real,F3,aa(fun(A,real),fun(A,real),aTP_Lamp_abj(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G)) ) ) ) ) ) ) ).

% continuous_log
tff(fact_7469_continuous__artanh,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [F3: filter(A),F2: fun(A,real)] :
          ( topolo3448309680560233919inuous(A,real,F3,F2)
         => ( pp(aa(set(real),bool,aa(real,fun(set(real),bool),member(real),aa(A,real,F2,topolo3827282254853284352ce_Lim(A,A,F3,aTP_Lamp_vm(A,A)))),set_or5935395276787703475ssThan(real,aa(real,real,uminus_uminus(real),one_one(real)),one_one(real))))
           => topolo3448309680560233919inuous(A,real,F3,aTP_Lamp_ade(fun(A,real),fun(A,real),F2)) ) ) ) ).

% continuous_artanh
tff(fact_7470_tendsto__offset__zero__iff,axiom,
    ! [C: $tType,D6: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & topolo4958980785337419405_space(D6)
        & zero(C) )
     => ! [A2: A,S: set(A),F2: fun(A,D6),L6: D6] :
          ( nO_MATCH(C,A,zero_zero(C),A2)
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),S))
           => ( topolo1002775350975398744n_open(A,S)
             => ( filterlim(A,D6,F2,topolo7230453075368039082e_nhds(D6,L6),topolo174197925503356063within(A,A2,S))
              <=> filterlim(A,D6,aa(fun(A,D6),fun(A,D6),aTP_Lamp_adf(A,fun(fun(A,D6),fun(A,D6)),A2),F2),topolo7230453075368039082e_nhds(D6,L6),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ) ) ) ).

% tendsto_offset_zero_iff
tff(fact_7471_has__derivativeI__sandwich,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [E: real,F7: fun(A,B),S3: set(A),X: A,F2: fun(A,B),H7: fun(A,real)] :
          ( pp(aa(real,bool,ord_less(real,zero_zero(real)),E))
         => ( real_V3181309239436604168linear(A,B,F7)
           => ( ! [Y3: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Y3),S3))
                 => ( ( Y3 != X )
                   => ( pp(aa(real,bool,ord_less(real,real_V557655796197034286t_dist(A,Y3,X)),E))
                     => pp(aa(real,bool,ord_less_eq(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),real_V7770717601297561774m_norm(B,aa(B,B,minus_minus(B,aa(B,B,minus_minus(B,aa(A,B,F2,Y3)),aa(A,B,F2,X))),aa(A,B,F7,aa(A,A,minus_minus(A,Y3),X))))),real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,Y3),X)))),aa(A,real,H7,Y3))) ) ) )
             => ( filterlim(A,real,H7,topolo7230453075368039082e_nhds(real,zero_zero(real)),topolo174197925503356063within(A,X,S3))
               => has_derivative(A,B,F2,F7,topolo174197925503356063within(A,X,S3)) ) ) ) ) ) ).

% has_derivativeI_sandwich
tff(fact_7472_dist__add__cancel2,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [B2: A,A2: A,C2: A] : real_V557655796197034286t_dist(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),A2)) = real_V557655796197034286t_dist(A,B2,C2) ) ).

% dist_add_cancel2
tff(fact_7473_dist__add__cancel,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [A2: A,B2: A,C2: A] : real_V557655796197034286t_dist(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C2)) = real_V557655796197034286t_dist(A,B2,C2) ) ).

% dist_add_cancel
tff(fact_7474_dist__eq__0__iff,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X: A,Y: A] :
          ( ( real_V557655796197034286t_dist(A,X,Y) = zero_zero(real) )
        <=> ( X = Y ) ) ) ).

% dist_eq_0_iff
tff(fact_7475_dist__self,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X: A] : real_V557655796197034286t_dist(A,X,X) = zero_zero(real) ) ).

% dist_self
tff(fact_7476_dist__0__norm,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X: A] : real_V557655796197034286t_dist(A,zero_zero(A),X) = real_V7770717601297561774m_norm(A,X) ) ).

% dist_0_norm
tff(fact_7477_zero__less__dist__iff,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X: A,Y: A] :
          ( pp(aa(real,bool,ord_less(real,zero_zero(real)),real_V557655796197034286t_dist(A,X,Y)))
        <=> ( X != Y ) ) ) ).

% zero_less_dist_iff
tff(fact_7478_dist__le__zero__iff,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X: A,Y: A] :
          ( pp(aa(real,bool,ord_less_eq(real,real_V557655796197034286t_dist(A,X,Y)),zero_zero(real)))
        <=> ( X = Y ) ) ) ).

% dist_le_zero_iff
tff(fact_7479_div__add__self2__no__field,axiom,
    ! [B: $tType,A: $tType] :
      ( ( euclid4440199948858584721cancel(A)
        & field(B) )
     => ! [X: B,B2: A,A2: A] :
          ( nO_MATCH(B,A,X,B2)
         => ( ( B2 != zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)),one_one(A)) ) ) ) ) ).

% div_add_self2_no_field
tff(fact_7480_div__add__self1__no__field,axiom,
    ! [B: $tType,A: $tType] :
      ( ( euclid4440199948858584721cancel(A)
        & field(B) )
     => ! [X: B,B2: A,A2: A] :
          ( nO_MATCH(B,A,X,B2)
         => ( ( B2 != zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A2)),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)),one_one(A)) ) ) ) ) ).

% div_add_self1_no_field
tff(fact_7481_dist__scaleR,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X: real,A2: A,Y: real] : real_V557655796197034286t_dist(A,aa(A,A,real_V8093663219630862766scaleR(A,X),A2),aa(A,A,real_V8093663219630862766scaleR(A,Y),A2)) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,abs_abs(real),aa(real,real,minus_minus(real,X),Y))),real_V7770717601297561774m_norm(A,A2)) ) ).

% dist_scaleR
tff(fact_7482_open__dist,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [S: set(A)] :
          ( topolo1002775350975398744n_open(A,S)
        <=> ! [X2: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X2),S))
             => ? [E4: real] :
                  ( pp(aa(real,bool,ord_less(real,zero_zero(real)),E4))
                  & ! [Y7: A] :
                      ( pp(aa(real,bool,ord_less(real,real_V557655796197034286t_dist(A,Y7,X2)),E4))
                     => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Y7),S)) ) ) ) ) ) ).

% open_dist
tff(fact_7483_open__ball,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X: A,D: real] : topolo1002775350975398744n_open(A,collect(A,aa(real,fun(A,bool),aTP_Lamp_adg(A,fun(real,fun(A,bool)),X),D))) ) ).

% open_ball
tff(fact_7484_continuous__dist,axiom,
    ! [A: $tType,D6: $tType] :
      ( ( topological_t2_space(D6)
        & real_V7819770556892013058_space(A) )
     => ! [F3: filter(D6),F2: fun(D6,A),G: fun(D6,A)] :
          ( topolo3448309680560233919inuous(D6,A,F3,F2)
         => ( topolo3448309680560233919inuous(D6,A,F3,G)
           => topolo3448309680560233919inuous(D6,real,F3,aa(fun(D6,A),fun(D6,real),aTP_Lamp_adh(fun(D6,A),fun(fun(D6,A),fun(D6,real)),F2),G)) ) ) ) ).

% continuous_dist
tff(fact_7485_abs__dist__diff__le,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [A2: A,B2: A,C2: A] : pp(aa(real,bool,ord_less_eq(real,aa(real,real,abs_abs(real),aa(real,real,minus_minus(real,real_V557655796197034286t_dist(A,A2,B2)),real_V557655796197034286t_dist(A,B2,C2)))),real_V557655796197034286t_dist(A,A2,C2))) ) ).

% abs_dist_diff_le
tff(fact_7486_dist__triangle__le,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X: A,Z: A,Y: A,E: real] :
          ( pp(aa(real,bool,ord_less_eq(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),real_V557655796197034286t_dist(A,X,Z)),real_V557655796197034286t_dist(A,Y,Z))),E))
         => pp(aa(real,bool,ord_less_eq(real,real_V557655796197034286t_dist(A,X,Y)),E)) ) ) ).

% dist_triangle_le
tff(fact_7487_dist__triangle3,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X: A,Y: A,A2: A] : pp(aa(real,bool,ord_less_eq(real,real_V557655796197034286t_dist(A,X,Y)),aa(real,real,aa(real,fun(real,real),plus_plus(real),real_V557655796197034286t_dist(A,A2,X)),real_V557655796197034286t_dist(A,A2,Y)))) ) ).

% dist_triangle3
tff(fact_7488_dist__triangle2,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X: A,Y: A,Z: A] : pp(aa(real,bool,ord_less_eq(real,real_V557655796197034286t_dist(A,X,Y)),aa(real,real,aa(real,fun(real,real),plus_plus(real),real_V557655796197034286t_dist(A,X,Z)),real_V557655796197034286t_dist(A,Y,Z)))) ) ).

% dist_triangle2
tff(fact_7489_dist__triangle,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X: A,Z: A,Y: A] : pp(aa(real,bool,ord_less_eq(real,real_V557655796197034286t_dist(A,X,Z)),aa(real,real,aa(real,fun(real,real),plus_plus(real),real_V557655796197034286t_dist(A,X,Y)),real_V557655796197034286t_dist(A,Y,Z)))) ) ).

% dist_triangle
tff(fact_7490_zero__le__dist,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X: A,Y: A] : pp(aa(real,bool,ord_less_eq(real,zero_zero(real)),real_V557655796197034286t_dist(A,X,Y))) ) ).

% zero_le_dist
tff(fact_7491_dist__triangle__less__add,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X15: A,Y: A,E1: real,X23: A,E22: real] :
          ( pp(aa(real,bool,ord_less(real,real_V557655796197034286t_dist(A,X15,Y)),E1))
         => ( pp(aa(real,bool,ord_less(real,real_V557655796197034286t_dist(A,X23,Y)),E22))
           => pp(aa(real,bool,ord_less(real,real_V557655796197034286t_dist(A,X15,X23)),aa(real,real,aa(real,fun(real,real),plus_plus(real),E1),E22))) ) ) ) ).

% dist_triangle_less_add
tff(fact_7492_dist__triangle__lt,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X: A,Z: A,Y: A,E: real] :
          ( pp(aa(real,bool,ord_less(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),real_V557655796197034286t_dist(A,X,Z)),real_V557655796197034286t_dist(A,Y,Z))),E))
         => pp(aa(real,bool,ord_less(real,real_V557655796197034286t_dist(A,X,Y)),E)) ) ) ).

% dist_triangle_lt
tff(fact_7493_norm__conv__dist,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X: A] : real_V7770717601297561774m_norm(A,X) = real_V557655796197034286t_dist(A,X,zero_zero(A)) ) ).

% norm_conv_dist
tff(fact_7494_dist__not__less__zero,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X: A,Y: A] : ~ pp(aa(real,bool,ord_less(real,real_V557655796197034286t_dist(A,X,Y)),zero_zero(real))) ) ).

% dist_not_less_zero
tff(fact_7495_dist__pos__lt,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X: A,Y: A] :
          ( ( X != Y )
         => pp(aa(real,bool,ord_less(real,zero_zero(real)),real_V557655796197034286t_dist(A,X,Y))) ) ) ).

% dist_pos_lt
tff(fact_7496_has__field__derivative__transform__within,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),F7: A,A2: A,S: set(A),D: real,G: fun(A,A)] :
          ( has_field_derivative(A,F2,F7,topolo174197925503356063within(A,A2,S))
         => ( pp(aa(real,bool,ord_less(real,zero_zero(real)),D))
           => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),S))
             => ( ! [X3: A] :
                    ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),S))
                   => ( pp(aa(real,bool,ord_less(real,real_V557655796197034286t_dist(A,X3,A2)),D))
                     => ( aa(A,A,F2,X3) = aa(A,A,G,X3) ) ) )
               => has_field_derivative(A,G,F7,topolo174197925503356063within(A,A2,S)) ) ) ) ) ) ).

% has_field_derivative_transform_within
tff(fact_7497_has__derivative__transform__within,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),F7: fun(A,B),X: A,S3: set(A),D: real,G: fun(A,B)] :
          ( has_derivative(A,B,F2,F7,topolo174197925503356063within(A,X,S3))
         => ( pp(aa(real,bool,ord_less(real,zero_zero(real)),D))
           => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),S3))
             => ( ! [X7: A] :
                    ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X7),S3))
                   => ( pp(aa(real,bool,ord_less(real,real_V557655796197034286t_dist(A,X7,X)),D))
                     => ( aa(A,B,F2,X7) = aa(A,B,G,X7) ) ) )
               => has_derivative(A,B,G,F7,topolo174197925503356063within(A,X,S3)) ) ) ) ) ) ).

% has_derivative_transform_within
tff(fact_7498_Cauchy__def,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X6: fun(nat,A)] :
          ( topolo3814608138187158403Cauchy(A,X6)
        <=> ! [E4: real] :
              ( pp(aa(real,bool,ord_less(real,zero_zero(real)),E4))
             => ? [M12: nat] :
                ! [M6: nat] :
                  ( pp(aa(nat,bool,ord_less_eq(nat,M12),M6))
                 => ! [N5: nat] :
                      ( pp(aa(nat,bool,ord_less_eq(nat,M12),N5))
                     => pp(aa(real,bool,ord_less(real,real_V557655796197034286t_dist(A,aa(nat,A,X6,M6),aa(nat,A,X6,N5))),E4)) ) ) ) ) ) ).

% Cauchy_def
tff(fact_7499_Cauchy__altdef2,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [S3: fun(nat,A)] :
          ( topolo3814608138187158403Cauchy(A,S3)
        <=> ! [E4: real] :
              ( pp(aa(real,bool,ord_less(real,zero_zero(real)),E4))
             => ? [N8: nat] :
                ! [N5: nat] :
                  ( pp(aa(nat,bool,ord_less_eq(nat,N8),N5))
                 => pp(aa(real,bool,ord_less(real,real_V557655796197034286t_dist(A,aa(nat,A,S3,N5),aa(nat,A,S3,N8))),E4)) ) ) ) ) ).

% Cauchy_altdef2
tff(fact_7500_metric__CauchyD,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X6: fun(nat,A),E: real] :
          ( topolo3814608138187158403Cauchy(A,X6)
         => ( pp(aa(real,bool,ord_less(real,zero_zero(real)),E))
           => ? [M9: nat] :
              ! [M4: nat] :
                ( pp(aa(nat,bool,ord_less_eq(nat,M9),M4))
               => ! [N11: nat] :
                    ( pp(aa(nat,bool,ord_less_eq(nat,M9),N11))
                   => pp(aa(real,bool,ord_less(real,real_V557655796197034286t_dist(A,aa(nat,A,X6,M4),aa(nat,A,X6,N11))),E)) ) ) ) ) ) ).

% metric_CauchyD
tff(fact_7501_metric__CauchyI,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X6: fun(nat,A)] :
          ( ! [E2: real] :
              ( pp(aa(real,bool,ord_less(real,zero_zero(real)),E2))
             => ? [M13: nat] :
                ! [M3: nat] :
                  ( pp(aa(nat,bool,ord_less_eq(nat,M13),M3))
                 => ! [N2: nat] :
                      ( pp(aa(nat,bool,ord_less_eq(nat,M13),N2))
                     => pp(aa(real,bool,ord_less(real,real_V557655796197034286t_dist(A,aa(nat,A,X6,M3),aa(nat,A,X6,N2))),E2)) ) ) )
         => topolo3814608138187158403Cauchy(A,X6) ) ) ).

% metric_CauchyI
tff(fact_7502_tendsto__dist,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [F2: fun(B,A),L: A,F3: filter(B),G: fun(B,A),M: A] :
          ( filterlim(B,A,F2,topolo7230453075368039082e_nhds(A,L),F3)
         => ( filterlim(B,A,G,topolo7230453075368039082e_nhds(A,M),F3)
           => filterlim(B,real,aa(fun(B,A),fun(B,real),aTP_Lamp_adi(fun(B,A),fun(fun(B,A),fun(B,real)),F2),G),topolo7230453075368039082e_nhds(real,real_V557655796197034286t_dist(A,L,M)),F3) ) ) ) ).

% tendsto_dist
tff(fact_7503_scale__right__distrib__NO__MATCH,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [X: A,Y: A,A2: real] :
          ( nO_MATCH(A,real,aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y),A2)
         => ( aa(A,A,real_V8093663219630862766scaleR(A,A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,real_V8093663219630862766scaleR(A,A2),X)),aa(A,A,real_V8093663219630862766scaleR(A,A2),Y)) ) ) ) ).

% scale_right_distrib_NO_MATCH
tff(fact_7504_scale__right__diff__distrib__NO__MATCH,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [X: A,Y: A,A2: real] :
          ( nO_MATCH(A,real,aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y),A2)
         => ( aa(A,A,real_V8093663219630862766scaleR(A,A2),aa(A,A,minus_minus(A,X),Y)) = aa(A,A,minus_minus(A,aa(A,A,real_V8093663219630862766scaleR(A,A2),X)),aa(A,A,real_V8093663219630862766scaleR(A,A2),Y)) ) ) ) ).

% scale_right_diff_distrib_NO_MATCH
tff(fact_7505_distrib__left__NO__MATCH,axiom,
    ! [B: $tType,A: $tType] :
      ( semiring(A)
     => ! [X: B,Y: B,A2: A,B2: A,C2: A] :
          ( nO_MATCH(B,A,aa(B,B,aa(B,fun(B,B),divide_divide(B),X),Y),A2)
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)) ) ) ) ).

% distrib_left_NO_MATCH
tff(fact_7506_distrib__right__NO__MATCH,axiom,
    ! [B: $tType,A: $tType] :
      ( semiring(A)
     => ! [X: B,Y: B,C2: A,A2: A,B2: A] :
          ( nO_MATCH(B,A,aa(B,B,aa(B,fun(B,B),divide_divide(B),X),Y),C2)
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) ) ) ) ).

% distrib_right_NO_MATCH
tff(fact_7507_left__diff__distrib__NO__MATCH,axiom,
    ! [B: $tType,A: $tType] :
      ( ring(A)
     => ! [X: B,Y: B,C2: A,A2: A,B2: A] :
          ( nO_MATCH(B,A,aa(B,B,aa(B,fun(B,B),divide_divide(B),X),Y),C2)
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,minus_minus(A,A2),B2)),C2) = aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) ) ) ) ).

% left_diff_distrib_NO_MATCH
tff(fact_7508_right__diff__distrib__NO__MATCH,axiom,
    ! [B: $tType,A: $tType] :
      ( ring(A)
     => ! [X: B,Y: B,A2: A,B2: A,C2: A] :
          ( nO_MATCH(B,A,aa(B,B,aa(B,fun(B,B),divide_divide(B),X),Y),A2)
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(A,A,minus_minus(A,B2),C2)) = aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)) ) ) ) ).

% right_diff_distrib_NO_MATCH
tff(fact_7509_dist__triangle__half__r,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [Y: A,X15: A,E: real,X23: A] :
          ( pp(aa(real,bool,ord_less(real,real_V557655796197034286t_dist(A,Y,X15)),aa(real,real,aa(real,fun(real,real),divide_divide(real),E),aa(num,real,numeral_numeral(real),bit0(one2)))))
         => ( pp(aa(real,bool,ord_less(real,real_V557655796197034286t_dist(A,Y,X23)),aa(real,real,aa(real,fun(real,real),divide_divide(real),E),aa(num,real,numeral_numeral(real),bit0(one2)))))
           => pp(aa(real,bool,ord_less(real,real_V557655796197034286t_dist(A,X15,X23)),E)) ) ) ) ).

% dist_triangle_half_r
tff(fact_7510_dist__triangle__half__l,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X15: A,Y: A,E: real,X23: A] :
          ( pp(aa(real,bool,ord_less(real,real_V557655796197034286t_dist(A,X15,Y)),aa(real,real,aa(real,fun(real,real),divide_divide(real),E),aa(num,real,numeral_numeral(real),bit0(one2)))))
         => ( pp(aa(real,bool,ord_less(real,real_V557655796197034286t_dist(A,X23,Y)),aa(real,real,aa(real,fun(real,real),divide_divide(real),E),aa(num,real,numeral_numeral(real),bit0(one2)))))
           => pp(aa(real,bool,ord_less(real,real_V557655796197034286t_dist(A,X15,X23)),E)) ) ) ) ).

% dist_triangle_half_l
tff(fact_7511_metric__LIM__imp__LIM,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( ( topolo4958980785337419405_space(C)
        & real_V7819770556892013058_space(B)
        & real_V7819770556892013058_space(A) )
     => ! [F2: fun(C,A),L: A,A2: C,G: fun(C,B),M: B] :
          ( filterlim(C,A,F2,topolo7230453075368039082e_nhds(A,L),topolo174197925503356063within(C,A2,top_top(set(C))))
         => ( ! [X3: C] :
                ( ( X3 != A2 )
               => pp(aa(real,bool,ord_less_eq(real,real_V557655796197034286t_dist(B,aa(C,B,G,X3),M)),real_V557655796197034286t_dist(A,aa(C,A,F2,X3),L))) )
           => filterlim(C,B,G,topolo7230453075368039082e_nhds(B,M),topolo174197925503356063within(C,A2,top_top(set(C)))) ) ) ) ).

% metric_LIM_imp_LIM
tff(fact_7512_dist__triangle__third,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X15: A,X23: A,E: real,X33: A,X42: A] :
          ( pp(aa(real,bool,ord_less(real,real_V557655796197034286t_dist(A,X15,X23)),aa(real,real,aa(real,fun(real,real),divide_divide(real),E),aa(num,real,numeral_numeral(real),aa(num,num,bit1,one2)))))
         => ( pp(aa(real,bool,ord_less(real,real_V557655796197034286t_dist(A,X23,X33)),aa(real,real,aa(real,fun(real,real),divide_divide(real),E),aa(num,real,numeral_numeral(real),aa(num,num,bit1,one2)))))
           => ( pp(aa(real,bool,ord_less(real,real_V557655796197034286t_dist(A,X33,X42)),aa(real,real,aa(real,fun(real,real),divide_divide(real),E),aa(num,real,numeral_numeral(real),aa(num,num,bit1,one2)))))
             => pp(aa(real,bool,ord_less(real,real_V557655796197034286t_dist(A,X15,X42)),E)) ) ) ) ) ).

% dist_triangle_third
tff(fact_7513_Lim__transform__within,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V7819770556892013058_space(A)
        & topolo4958980785337419405_space(B) )
     => ! [F2: fun(A,B),L: B,X: A,S: set(A),D: real,G: fun(A,B)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),topolo174197925503356063within(A,X,S))
         => ( pp(aa(real,bool,ord_less(real,zero_zero(real)),D))
           => ( ! [X7: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X7),S))
                 => ( pp(aa(real,bool,ord_less(real,zero_zero(real)),real_V557655796197034286t_dist(A,X7,X)))
                   => ( pp(aa(real,bool,ord_less(real,real_V557655796197034286t_dist(A,X7,X)),D))
                     => ( aa(A,B,F2,X7) = aa(A,B,G,X7) ) ) ) )
             => filterlim(A,B,G,topolo7230453075368039082e_nhds(B,L),topolo174197925503356063within(A,X,S)) ) ) ) ) ).

% Lim_transform_within
tff(fact_7514_filterlim__transform__within,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [G: fun(A,B),G5: filter(B),X: A,S: set(A),F3: filter(B),D: real,F2: fun(A,B)] :
          ( filterlim(A,B,G,G5,topolo174197925503356063within(A,X,S))
         => ( pp(aa(filter(B),bool,ord_less_eq(filter(B),G5),F3))
           => ( pp(aa(real,bool,ord_less(real,zero_zero(real)),D))
             => ( ! [X7: A] :
                    ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X7),S))
                   => ( pp(aa(real,bool,ord_less(real,zero_zero(real)),real_V557655796197034286t_dist(A,X7,X)))
                     => ( pp(aa(real,bool,ord_less(real,real_V557655796197034286t_dist(A,X7,X)),D))
                       => ( aa(A,B,F2,X7) = aa(A,B,G,X7) ) ) ) )
               => filterlim(A,B,F2,F3,topolo174197925503356063within(A,X,S)) ) ) ) ) ) ).

% filterlim_transform_within
tff(fact_7515_CauchyI_H,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X6: fun(nat,A)] :
          ( ! [E2: real] :
              ( pp(aa(real,bool,ord_less(real,zero_zero(real)),E2))
             => ? [M13: nat] :
                ! [M3: nat] :
                  ( pp(aa(nat,bool,ord_less_eq(nat,M13),M3))
                 => ! [N2: nat] :
                      ( pp(aa(nat,bool,ord_less(nat,M3),N2))
                     => pp(aa(real,bool,ord_less(real,real_V557655796197034286t_dist(A,aa(nat,A,X6,M3),aa(nat,A,X6,N2))),E2)) ) ) )
         => topolo3814608138187158403Cauchy(A,X6) ) ) ).

% CauchyI'
tff(fact_7516_Cauchy__altdef,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [F2: fun(nat,A)] :
          ( topolo3814608138187158403Cauchy(A,F2)
        <=> ! [E4: real] :
              ( pp(aa(real,bool,ord_less(real,zero_zero(real)),E4))
             => ? [M12: nat] :
                ! [M6: nat] :
                  ( pp(aa(nat,bool,ord_less_eq(nat,M12),M6))
                 => ! [N5: nat] :
                      ( pp(aa(nat,bool,ord_less(nat,M6),N5))
                     => pp(aa(real,bool,ord_less(real,real_V557655796197034286t_dist(A,aa(nat,A,F2,M6),aa(nat,A,F2,N5))),E4)) ) ) ) ) ) ).

% Cauchy_altdef
tff(fact_7517_dist__of__nat,axiom,
    ! [A: $tType] :
      ( real_V2822296259951069270ebra_1(A)
     => ! [M: nat,N: nat] : real_V557655796197034286t_dist(A,aa(nat,A,semiring_1_of_nat(A),M),aa(nat,A,semiring_1_of_nat(A),N)) = aa(int,real,ring_1_of_int(real),aa(int,int,abs_abs(int),aa(int,int,minus_minus(int,aa(nat,int,semiring_1_of_nat(int),M)),aa(nat,int,semiring_1_of_nat(int),N)))) ) ).

% dist_of_nat
tff(fact_7518_tendsto__dist__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V7819770556892013058_space(B)
     => ! [F2: fun(A,B),L: B,F3: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),F3)
        <=> filterlim(A,real,aa(B,fun(A,real),aTP_Lamp_adj(fun(A,B),fun(B,fun(A,real)),F2),L),topolo7230453075368039082e_nhds(real,zero_zero(real)),F3) ) ) ).

% tendsto_dist_iff
tff(fact_7519_metric__LIM__equal2,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V7819770556892013058_space(A)
        & topolo4958980785337419405_space(B) )
     => ! [G: fun(A,B),L: B,A2: A,R: real,F2: fun(A,B)] :
          ( filterlim(A,B,G,topolo7230453075368039082e_nhds(B,L),topolo174197925503356063within(A,A2,top_top(set(A))))
         => ( pp(aa(real,bool,ord_less(real,zero_zero(real)),R))
           => ( ! [X3: A] :
                  ( ( X3 != A2 )
                 => ( pp(aa(real,bool,ord_less(real,real_V557655796197034286t_dist(A,X3,A2)),R))
                   => ( aa(A,B,F2,X3) = aa(A,B,G,X3) ) ) )
             => filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),topolo174197925503356063within(A,A2,top_top(set(A)))) ) ) ) ) ).

% metric_LIM_equal2
tff(fact_7520_metric__LIM__I,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V7819770556892013058_space(A)
        & real_V7819770556892013058_space(B) )
     => ! [A2: A,F2: fun(A,B),L6: B] :
          ( ! [R3: real] :
              ( pp(aa(real,bool,ord_less(real,zero_zero(real)),R3))
             => ? [S8: real] :
                  ( pp(aa(real,bool,ord_less(real,zero_zero(real)),S8))
                  & ! [X3: A] :
                      ( ( ( X3 != A2 )
                        & pp(aa(real,bool,ord_less(real,real_V557655796197034286t_dist(A,X3,A2)),S8)) )
                     => pp(aa(real,bool,ord_less(real,real_V557655796197034286t_dist(B,aa(A,B,F2,X3),L6)),R3)) ) ) )
         => filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L6),topolo174197925503356063within(A,A2,top_top(set(A)))) ) ) ).

% metric_LIM_I
tff(fact_7521_metric__LIM__D,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V7819770556892013058_space(A)
        & real_V7819770556892013058_space(B) )
     => ! [F2: fun(A,B),L6: B,A2: A,R2: real] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L6),topolo174197925503356063within(A,A2,top_top(set(A))))
         => ( pp(aa(real,bool,ord_less(real,zero_zero(real)),R2))
           => ? [S2: real] :
                ( pp(aa(real,bool,ord_less(real,zero_zero(real)),S2))
                & ! [X4: A] :
                    ( ( ( X4 != A2 )
                      & pp(aa(real,bool,ord_less(real,real_V557655796197034286t_dist(A,X4,A2)),S2)) )
                   => pp(aa(real,bool,ord_less(real,real_V557655796197034286t_dist(B,aa(A,B,F2,X4),L6)),R2)) ) ) ) ) ) ).

% metric_LIM_D
tff(fact_7522_LIM__def,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V7819770556892013058_space(A)
        & real_V7819770556892013058_space(B) )
     => ! [F2: fun(A,B),L6: B,A2: A] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L6),topolo174197925503356063within(A,A2,top_top(set(A))))
        <=> ! [R5: real] :
              ( pp(aa(real,bool,ord_less(real,zero_zero(real)),R5))
             => ? [S7: real] :
                  ( pp(aa(real,bool,ord_less(real,zero_zero(real)),S7))
                  & ! [X2: A] :
                      ( ( ( X2 != A2 )
                        & pp(aa(real,bool,ord_less(real,real_V557655796197034286t_dist(A,X2,A2)),S7)) )
                     => pp(aa(real,bool,ord_less(real,real_V557655796197034286t_dist(B,aa(A,B,F2,X2),L6)),R5)) ) ) ) ) ) ).

% LIM_def
tff(fact_7523_lim__sequentially,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X6: fun(nat,A),L6: A] :
          ( filterlim(nat,A,X6,topolo7230453075368039082e_nhds(A,L6),at_top(nat))
        <=> ! [R5: real] :
              ( pp(aa(real,bool,ord_less(real,zero_zero(real)),R5))
             => ? [No3: nat] :
                ! [N5: nat] :
                  ( pp(aa(nat,bool,ord_less_eq(nat,No3),N5))
                 => pp(aa(real,bool,ord_less(real,real_V557655796197034286t_dist(A,aa(nat,A,X6,N5),L6)),R5)) ) ) ) ) ).

% lim_sequentially
tff(fact_7524_metric__LIMSEQ__I,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X6: fun(nat,A),L6: A] :
          ( ! [R3: real] :
              ( pp(aa(real,bool,ord_less(real,zero_zero(real)),R3))
             => ? [No2: nat] :
                ! [N2: nat] :
                  ( pp(aa(nat,bool,ord_less_eq(nat,No2),N2))
                 => pp(aa(real,bool,ord_less(real,real_V557655796197034286t_dist(A,aa(nat,A,X6,N2),L6)),R3)) ) )
         => filterlim(nat,A,X6,topolo7230453075368039082e_nhds(A,L6),at_top(nat)) ) ) ).

% metric_LIMSEQ_I
tff(fact_7525_metric__LIMSEQ__D,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X6: fun(nat,A),L6: A,R2: real] :
          ( filterlim(nat,A,X6,topolo7230453075368039082e_nhds(A,L6),at_top(nat))
         => ( pp(aa(real,bool,ord_less(real,zero_zero(real)),R2))
           => ? [No: nat] :
              ! [N11: nat] :
                ( pp(aa(nat,bool,ord_less_eq(nat,No),N11))
               => pp(aa(real,bool,ord_less(real,real_V557655796197034286t_dist(A,aa(nat,A,X6,N11),L6)),R2)) ) ) ) ) ).

% metric_LIMSEQ_D
tff(fact_7526_power__minus_H,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [X: A,N: nat] :
          ( nO_MATCH(A,A,one_one(A),X)
         => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),X)),N) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),N)),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),N)) ) ) ) ).

% power_minus'
tff(fact_7527_metric__Cauchy__iff2,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X6: fun(nat,A)] :
          ( topolo3814608138187158403Cauchy(A,X6)
        <=> ! [J3: nat] :
            ? [M12: nat] :
            ! [M6: nat] :
              ( pp(aa(nat,bool,ord_less_eq(nat,M12),M6))
             => ! [N5: nat] :
                  ( pp(aa(nat,bool,ord_less_eq(nat,M12),N5))
                 => pp(aa(real,bool,ord_less(real,real_V557655796197034286t_dist(A,aa(nat,A,X6,M6),aa(nat,A,X6,N5))),aa(real,real,inverse_inverse(real),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,J3))))) ) ) ) ) ).

% metric_Cauchy_iff2
tff(fact_7528_scale__left__distrib__NO__MATCH,axiom,
    ! [C: $tType,A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [X: A,Y: A,C2: C,A2: real,B2: real] :
          ( nO_MATCH(A,C,aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y),C2)
         => ( aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,aa(real,fun(real,real),plus_plus(real),A2),B2)),X) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,real_V8093663219630862766scaleR(A,A2),X)),aa(A,A,real_V8093663219630862766scaleR(A,B2),X)) ) ) ) ).

% scale_left_distrib_NO_MATCH
tff(fact_7529_metric__LIM__compose2,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( real_V7819770556892013058_space(A)
        & topolo4958980785337419405_space(B)
        & topolo4958980785337419405_space(C) )
     => ! [F2: fun(A,B),B2: B,A2: A,G: fun(B,C),C2: C] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,B2),topolo174197925503356063within(A,A2,top_top(set(A))))
         => ( filterlim(B,C,G,topolo7230453075368039082e_nhds(C,C2),topolo174197925503356063within(B,B2,top_top(set(B))))
           => ( ? [D5: real] :
                  ( pp(aa(real,bool,ord_less(real,zero_zero(real)),D5))
                  & ! [X3: A] :
                      ( ( ( X3 != A2 )
                        & pp(aa(real,bool,ord_less(real,real_V557655796197034286t_dist(A,X3,A2)),D5)) )
                     => ( aa(A,B,F2,X3) != B2 ) ) )
             => filterlim(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_adk(fun(A,B),fun(fun(B,C),fun(A,C)),F2),G),topolo7230453075368039082e_nhds(C,C2),topolo174197925503356063within(A,A2,top_top(set(A)))) ) ) ) ) ).

% metric_LIM_compose2
tff(fact_7530_scale__left__diff__distrib__NO__MATCH,axiom,
    ! [C: $tType,A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [X: A,Y: A,C2: C,A2: real,B2: real] :
          ( nO_MATCH(A,C,aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y),C2)
         => ( aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,minus_minus(real,A2),B2)),X) = aa(A,A,minus_minus(A,aa(A,A,real_V8093663219630862766scaleR(A,A2),X)),aa(A,A,real_V8093663219630862766scaleR(A,B2),X)) ) ) ) ).

% scale_left_diff_distrib_NO_MATCH
tff(fact_7531_metric__isCont__LIM__compose2,axiom,
    ! [D6: $tType,C: $tType,A: $tType] :
      ( ( real_V7819770556892013058_space(A)
        & topolo4958980785337419405_space(C)
        & topolo4958980785337419405_space(D6) )
     => ! [A2: A,F2: fun(A,C),G: fun(C,D6),L: D6] :
          ( topolo3448309680560233919inuous(A,C,topolo174197925503356063within(A,A2,top_top(set(A))),F2)
         => ( filterlim(C,D6,G,topolo7230453075368039082e_nhds(D6,L),topolo174197925503356063within(C,aa(A,C,F2,A2),top_top(set(C))))
           => ( ? [D5: real] :
                  ( pp(aa(real,bool,ord_less(real,zero_zero(real)),D5))
                  & ! [X3: A] :
                      ( ( ( X3 != A2 )
                        & pp(aa(real,bool,ord_less(real,real_V557655796197034286t_dist(A,X3,A2)),D5)) )
                     => ( aa(A,C,F2,X3) != aa(A,C,F2,A2) ) ) )
             => filterlim(A,D6,aa(fun(C,D6),fun(A,D6),aTP_Lamp_adl(fun(A,C),fun(fun(C,D6),fun(A,D6)),F2),G),topolo7230453075368039082e_nhds(D6,L),topolo174197925503356063within(A,A2,top_top(set(A)))) ) ) ) ) ).

% metric_isCont_LIM_compose2
tff(fact_7532_LIMSEQ__iff__nz,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X6: fun(nat,A),L6: A] :
          ( filterlim(nat,A,X6,topolo7230453075368039082e_nhds(A,L6),at_top(nat))
        <=> ! [R5: real] :
              ( pp(aa(real,bool,ord_less(real,zero_zero(real)),R5))
             => ? [No3: nat] :
                  ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),No3))
                  & ! [N5: nat] :
                      ( pp(aa(nat,bool,ord_less_eq(nat,No3),N5))
                     => pp(aa(real,bool,ord_less(real,real_V557655796197034286t_dist(A,aa(nat,A,X6,N5),L6)),R5)) ) ) ) ) ) ).

% LIMSEQ_iff_nz
tff(fact_7533_LIM__offset__zero__iff,axiom,
    ! [C: $tType,D6: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & topolo4958980785337419405_space(D6)
        & zero(C) )
     => ! [A2: A,F2: fun(A,D6),L6: D6] :
          ( nO_MATCH(C,A,zero_zero(C),A2)
         => ( filterlim(A,D6,F2,topolo7230453075368039082e_nhds(D6,L6),topolo174197925503356063within(A,A2,top_top(set(A))))
          <=> filterlim(A,D6,aa(fun(A,D6),fun(A,D6),aTP_Lamp_adf(A,fun(fun(A,D6),fun(A,D6)),A2),F2),topolo7230453075368039082e_nhds(D6,L6),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ) ).

% LIM_offset_zero_iff
tff(fact_7534_tendsto__exp__limit__at__right,axiom,
    ! [X: real] : filterlim(real,real,aTP_Lamp_adm(real,fun(real,real),X),topolo7230453075368039082e_nhds(real,aa(real,real,exp(real),X)),topolo174197925503356063within(real,zero_zero(real),set_ord_greaterThan(real,zero_zero(real)))) ).

% tendsto_exp_limit_at_right
tff(fact_7535_tendsto__arctan__at__bot,axiom,
    filterlim(real,real,arctan,topolo7230453075368039082e_nhds(real,aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),at_bot(real)) ).

% tendsto_arctan_at_bot
tff(fact_7536_greaterThan__subset__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y: A] :
          ( pp(aa(set(A),bool,ord_less_eq(set(A),set_ord_greaterThan(A,X)),set_ord_greaterThan(A,Y)))
        <=> pp(aa(A,bool,ord_less_eq(A,Y),X)) ) ) ).

% greaterThan_subset_iff
tff(fact_7537_ln__at__0,axiom,
    filterlim(real,real,ln_ln(real),at_bot(real),topolo174197925503356063within(real,zero_zero(real),set_ord_greaterThan(real,zero_zero(real)))) ).

% ln_at_0
tff(fact_7538_artanh__real__at__right__1,axiom,
    filterlim(real,real,artanh(real),at_bot(real),topolo174197925503356063within(real,aa(real,real,uminus_uminus(real),one_one(real)),set_ord_greaterThan(real,aa(real,real,uminus_uminus(real),one_one(real))))) ).

% artanh_real_at_right_1
tff(fact_7539_greaterThan__def,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [L: A] : set_ord_greaterThan(A,L) = collect(A,ord_less(A,L)) ) ).

% greaterThan_def
tff(fact_7540_filterlim__at__left__to__right,axiom,
    ! [A: $tType,F2: fun(real,A),F3: filter(A),A2: real] :
      ( filterlim(real,A,F2,F3,topolo174197925503356063within(real,A2,set_ord_lessThan(real,A2)))
    <=> filterlim(real,A,aTP_Lamp_adn(fun(real,A),fun(real,A),F2),F3,topolo174197925503356063within(real,aa(real,real,uminus_uminus(real),A2),set_ord_greaterThan(real,aa(real,real,uminus_uminus(real),A2)))) ) ).

% filterlim_at_left_to_right
tff(fact_7541_filterlim__at__right__to__0,axiom,
    ! [A: $tType,F2: fun(real,A),F3: filter(A),A2: real] :
      ( filterlim(real,A,F2,F3,topolo174197925503356063within(real,A2,set_ord_greaterThan(real,A2)))
    <=> filterlim(real,A,aa(real,fun(real,A),aTP_Lamp_ado(fun(real,A),fun(real,fun(real,A)),F2),A2),F3,topolo174197925503356063within(real,zero_zero(real),set_ord_greaterThan(real,zero_zero(real)))) ) ).

% filterlim_at_right_to_0
tff(fact_7542_filterlim__tan__at__right,axiom,
    filterlim(real,real,tan(real),at_bot(real),topolo174197925503356063within(real,aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2)))),set_ord_greaterThan(real,aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))))) ).

% filterlim_tan_at_right
tff(fact_7543_exp__at__bot,axiom,
    filterlim(real,real,exp(real),topolo7230453075368039082e_nhds(real,zero_zero(real)),at_bot(real)) ).

% exp_at_bot
tff(fact_7544_filterlim__inverse__at__bot__neg,axiom,
    filterlim(real,real,inverse_inverse(real),at_bot(real),topolo174197925503356063within(real,zero_zero(real),set_ord_lessThan(real,zero_zero(real)))) ).

% filterlim_inverse_at_bot_neg
tff(fact_7545_filterlim__times__pos,axiom,
    ! [A: $tType,B: $tType] :
      ( ( linordered_field(A)
        & topolo1944317154257567458pology(A) )
     => ! [F2: fun(B,A),P2: A,F1: filter(B),C2: A,L: A] :
          ( filterlim(B,A,F2,topolo174197925503356063within(A,P2,set_ord_greaterThan(A,P2)),F1)
         => ( pp(aa(A,bool,ord_less(A,zero_zero(A)),C2))
           => ( ( L = aa(A,A,aa(A,fun(A,A),times_times(A),C2),P2) )
             => filterlim(B,A,aa(A,fun(B,A),aTP_Lamp_adp(fun(B,A),fun(A,fun(B,A)),F2),C2),topolo174197925503356063within(A,L,set_ord_greaterThan(A,L)),F1) ) ) ) ) ).

% filterlim_times_pos
tff(fact_7546_log__inj,axiom,
    ! [B2: real] :
      ( pp(aa(real,bool,ord_less(real,one_one(real)),B2))
     => inj_on(real,real,log(B2),set_ord_greaterThan(real,zero_zero(real))) ) ).

% log_inj
tff(fact_7547_tanh__real__at__bot,axiom,
    filterlim(real,real,tanh(real),topolo7230453075368039082e_nhds(real,aa(real,real,uminus_uminus(real),one_one(real))),at_bot(real)) ).

% tanh_real_at_bot
tff(fact_7548_filterlim__tendsto__pos__mult__at__bot,axiom,
    ! [A: $tType,F2: fun(A,real),C2: real,F3: filter(A),G: fun(A,real)] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,C2),F3)
     => ( pp(aa(real,bool,ord_less(real,zero_zero(real)),C2))
       => ( filterlim(A,real,G,at_bot(real),F3)
         => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_adq(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G),at_bot(real),F3) ) ) ) ).

% filterlim_tendsto_pos_mult_at_bot
tff(fact_7549_tendsto__arcosh__at__left__1,axiom,
    filterlim(real,real,arcosh(real),topolo7230453075368039082e_nhds(real,zero_zero(real)),topolo174197925503356063within(real,one_one(real),set_ord_greaterThan(real,one_one(real)))) ).

% tendsto_arcosh_at_left_1
tff(fact_7550_tendsto__at__botI__sequentially,axiom,
    ! [B: $tType] :
      ( topolo3112930676232923870pology(B)
     => ! [F2: fun(real,B),Y: B] :
          ( ! [X8: fun(nat,real)] :
              ( filterlim(nat,real,X8,at_bot(real),at_top(nat))
             => filterlim(nat,B,aa(fun(nat,real),fun(nat,B),aTP_Lamp_adr(fun(real,B),fun(fun(nat,real),fun(nat,B)),F2),X8),topolo7230453075368039082e_nhds(B,Y),at_top(nat)) )
         => filterlim(real,B,F2,topolo7230453075368039082e_nhds(B,Y),at_bot(real)) ) ) ).

% tendsto_at_botI_sequentially
tff(fact_7551_isCont__If__ge,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo1944317154257567458pology(A)
        & topolo4958980785337419405_space(B) )
     => ! [A2: A,G: fun(A,B),F2: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,set_ord_lessThan(A,A2)),G)
         => ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,aa(A,B,G,A2)),topolo174197925503356063within(A,A2,set_ord_greaterThan(A,A2)))
           => topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,top_top(set(A))),aa(fun(A,B),fun(A,B),aa(fun(A,B),fun(fun(A,B),fun(A,B)),aTP_Lamp_ads(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))),A2),G),F2)) ) ) ) ).

% isCont_If_ge
tff(fact_7552_DERIV__pos__imp__increasing__at__bot,axiom,
    ! [B2: real,F2: fun(real,real),Flim: real] :
      ( ! [X3: real] :
          ( pp(aa(real,bool,ord_less_eq(real,X3),B2))
         => ? [Y4: real] :
              ( has_field_derivative(real,F2,Y4,topolo174197925503356063within(real,X3,top_top(set(real))))
              & pp(aa(real,bool,ord_less(real,zero_zero(real)),Y4)) ) )
     => ( filterlim(real,real,F2,topolo7230453075368039082e_nhds(real,Flim),at_bot(real))
       => pp(aa(real,bool,ord_less(real,Flim),aa(real,real,F2,B2))) ) ) ).

% DERIV_pos_imp_increasing_at_bot
tff(fact_7553_filterlim__pow__at__bot__odd,axiom,
    ! [N: nat,F2: fun(real,real),F3: filter(real)] :
      ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),N))
     => ( filterlim(real,real,F2,at_bot(real),F3)
       => ( ~ pp(aa(nat,bool,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N))
         => filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_adt(nat,fun(fun(real,real),fun(real,real)),N),F2),at_bot(real),F3) ) ) ) ).

% filterlim_pow_at_bot_odd
tff(fact_7554_filterlim__pow__at__bot__even,axiom,
    ! [N: nat,F2: fun(real,real),F3: filter(real)] :
      ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),N))
     => ( filterlim(real,real,F2,at_bot(real),F3)
       => ( pp(aa(nat,bool,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N))
         => filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_adt(nat,fun(fun(real,real),fun(real,real)),N),F2),at_top(real),F3) ) ) ) ).

% filterlim_pow_at_bot_even
tff(fact_7555_lim__zero__infinity,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),L: A] :
          ( filterlim(A,A,aTP_Lamp_adu(fun(A,A),fun(A,A),F2),topolo7230453075368039082e_nhds(A,L),topolo174197925503356063within(A,zero_zero(A),top_top(set(A))))
         => filterlim(A,A,F2,topolo7230453075368039082e_nhds(A,L),at_infinity(A)) ) ) ).

% lim_zero_infinity
tff(fact_7556_filterlim__at__top__mult__at__top,axiom,
    ! [A: $tType,F2: fun(A,real),F3: filter(A),G: fun(A,real)] :
      ( filterlim(A,real,F2,at_top(real),F3)
     => ( filterlim(A,real,G,at_top(real),F3)
       => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_adq(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G),at_top(real),F3) ) ) ).

% filterlim_at_top_mult_at_top
tff(fact_7557_filterlim__at__top__add__at__top,axiom,
    ! [A: $tType,F2: fun(A,real),F3: filter(A),G: fun(A,real)] :
      ( filterlim(A,real,F2,at_top(real),F3)
     => ( filterlim(A,real,G,at_top(real),F3)
       => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_adv(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G),at_top(real),F3) ) ) ).

% filterlim_at_top_add_at_top
tff(fact_7558_exp__at__top,axiom,
    filterlim(real,real,exp(real),at_top(real),at_top(real)) ).

% exp_at_top
tff(fact_7559_sqrt__at__top,axiom,
    filterlim(real,real,sqrt,at_top(real),at_top(real)) ).

% sqrt_at_top
tff(fact_7560_ln__at__top,axiom,
    filterlim(real,real,ln_ln(real),at_top(real),at_top(real)) ).

% ln_at_top
tff(fact_7561_filterlim__at__infinity__imp__norm__at__top,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F2: fun(A,B),F3: filter(A)] :
          ( filterlim(A,B,F2,at_infinity(B),F3)
         => filterlim(A,real,aTP_Lamp_xc(fun(A,B),fun(A,real),F2),at_top(real),F3) ) ) ).

% filterlim_at_infinity_imp_norm_at_top
tff(fact_7562_filterlim__norm__at__top__imp__at__infinity,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F2: fun(A,B),F3: filter(A)] :
          ( filterlim(A,real,aTP_Lamp_xc(fun(A,B),fun(A,real),F2),at_top(real),F3)
         => filterlim(A,B,F2,at_infinity(B),F3) ) ) ).

% filterlim_norm_at_top_imp_at_infinity
tff(fact_7563_filterlim__at__infinity__conv__norm__at__top,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F2: fun(A,B),G5: filter(A)] :
          ( filterlim(A,B,F2,at_infinity(B),G5)
        <=> filterlim(A,real,aTP_Lamp_xc(fun(A,B),fun(A,real),F2),at_top(real),G5) ) ) ).

% filterlim_at_infinity_conv_norm_at_top
tff(fact_7564_filterlim__tendsto__add__at__top,axiom,
    ! [A: $tType,F2: fun(A,real),C2: real,F3: filter(A),G: fun(A,real)] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,C2),F3)
     => ( filterlim(A,real,G,at_top(real),F3)
       => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_adv(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G),at_top(real),F3) ) ) ).

% filterlim_tendsto_add_at_top
tff(fact_7565_filterlim__real__sequentially,axiom,
    filterlim(nat,real,semiring_1_of_nat(real),at_top(real),at_top(nat)) ).

% filterlim_real_sequentially
tff(fact_7566_filterlim__uminus__at__bot,axiom,
    ! [A: $tType,F2: fun(A,real),F3: filter(A)] :
      ( filterlim(A,real,F2,at_bot(real),F3)
    <=> filterlim(A,real,aTP_Lamp_adw(fun(A,real),fun(A,real),F2),at_top(real),F3) ) ).

% filterlim_uminus_at_bot
tff(fact_7567_filterlim__uminus__at__top,axiom,
    ! [A: $tType,F2: fun(A,real),F3: filter(A)] :
      ( filterlim(A,real,F2,at_top(real),F3)
    <=> filterlim(A,real,aTP_Lamp_adw(fun(A,real),fun(A,real),F2),at_bot(real),F3) ) ).

% filterlim_uminus_at_top
tff(fact_7568_filterlim__at__bot__mirror,axiom,
    ! [A: $tType,F2: fun(real,A),F3: filter(A)] :
      ( filterlim(real,A,F2,F3,at_bot(real))
    <=> filterlim(real,A,aTP_Lamp_adn(fun(real,A),fun(real,A),F2),F3,at_top(real)) ) ).

% filterlim_at_bot_mirror
tff(fact_7569_filterlim__at__top__mirror,axiom,
    ! [A: $tType,F2: fun(real,A),F3: filter(A)] :
      ( filterlim(real,A,F2,F3,at_top(real))
    <=> filterlim(real,A,aTP_Lamp_adn(fun(real,A),fun(real,A),F2),F3,at_bot(real)) ) ).

% filterlim_at_top_mirror
tff(fact_7570_filterlim__uminus__at__bot__at__top,axiom,
    filterlim(real,real,uminus_uminus(real),at_bot(real),at_top(real)) ).

% filterlim_uminus_at_bot_at_top
tff(fact_7571_filterlim__uminus__at__top__at__bot,axiom,
    filterlim(real,real,uminus_uminus(real),at_top(real),at_bot(real)) ).

% filterlim_uminus_at_top_at_bot
tff(fact_7572_filterlim__real__at__infinity__sequentially,axiom,
    filterlim(nat,real,semiring_1_of_nat(real),at_infinity(real),at_top(nat)) ).

% filterlim_real_at_infinity_sequentially
tff(fact_7573_tendsto__of__nat,axiom,
    ! [A: $tType] :
      ( real_V2822296259951069270ebra_1(A)
     => filterlim(nat,A,semiring_1_of_nat(A),at_infinity(A),at_top(nat)) ) ).

% tendsto_of_nat
tff(fact_7574_greaterThan__0,axiom,
    set_ord_greaterThan(nat,zero_zero(nat)) = aa(set(nat),set(nat),image(nat,nat,suc),top_top(set(nat))) ).

% greaterThan_0
tff(fact_7575_filterlim__pow__at__top,axiom,
    ! [A: $tType,N: nat,F2: fun(A,real),F3: filter(A)] :
      ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),N))
     => ( filterlim(A,real,F2,at_top(real),F3)
       => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_vl(nat,fun(fun(A,real),fun(A,real)),N),F2),at_top(real),F3) ) ) ).

% filterlim_pow_at_top
tff(fact_7576_greaterThan__Suc,axiom,
    ! [K: nat] : set_ord_greaterThan(nat,aa(nat,nat,suc,K)) = aa(set(nat),set(nat),minus_minus(set(nat),set_ord_greaterThan(nat,K)),insert(nat,aa(nat,nat,suc,K),bot_bot(set(nat)))) ).

% greaterThan_Suc
tff(fact_7577_tendsto__add__filterlim__at__infinity,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F2: fun(A,B),C2: B,F3: filter(A),G: fun(A,B)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,C2),F3)
         => ( filterlim(A,B,G,at_infinity(B),F3)
           => filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_adx(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G),at_infinity(B),F3) ) ) ) ).

% tendsto_add_filterlim_at_infinity
tff(fact_7578_tendsto__add__filterlim__at__infinity_H,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F2: fun(A,B),F3: filter(A),G: fun(A,B),C2: B] :
          ( filterlim(A,B,F2,at_infinity(B),F3)
         => ( filterlim(A,B,G,topolo7230453075368039082e_nhds(B,C2),F3)
           => filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_adx(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G),at_infinity(B),F3) ) ) ) ).

% tendsto_add_filterlim_at_infinity'
tff(fact_7579_tanh__real__at__top,axiom,
    filterlim(real,real,tanh(real),topolo7230453075368039082e_nhds(real,one_one(real)),at_top(real)) ).

% tanh_real_at_top
tff(fact_7580_real__tendsto__divide__at__top,axiom,
    ! [A: $tType,F2: fun(A,real),C2: real,F3: filter(A),G: fun(A,real)] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,C2),F3)
     => ( filterlim(A,real,G,at_top(real),F3)
       => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_ady(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G),topolo7230453075368039082e_nhds(real,zero_zero(real)),F3) ) ) ).

% real_tendsto_divide_at_top
tff(fact_7581_artanh__real__at__left__1,axiom,
    filterlim(real,real,artanh(real),at_top(real),topolo174197925503356063within(real,one_one(real),set_ord_lessThan(real,one_one(real)))) ).

% artanh_real_at_left_1
tff(fact_7582_tendsto__inverse__0__at__top,axiom,
    ! [A: $tType,F2: fun(A,real),F3: filter(A)] :
      ( filterlim(A,real,F2,at_top(real),F3)
     => filterlim(A,real,aTP_Lamp_adz(fun(A,real),fun(A,real),F2),topolo7230453075368039082e_nhds(real,zero_zero(real)),F3) ) ).

% tendsto_inverse_0_at_top
tff(fact_7583_filterlim__sequentially__iff__filterlim__real,axiom,
    ! [A: $tType,F2: fun(A,nat),F3: filter(A)] :
      ( filterlim(A,nat,F2,at_top(nat),F3)
    <=> filterlim(A,real,aTP_Lamp_bc(fun(A,nat),fun(A,real),F2),at_top(real),F3) ) ).

% filterlim_sequentially_iff_filterlim_real
tff(fact_7584_tendsto__inverse__0,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => filterlim(A,A,inverse_inverse(A),topolo7230453075368039082e_nhds(A,zero_zero(A)),at_infinity(A)) ) ).

% tendsto_inverse_0
tff(fact_7585_filterlim__tendsto__pos__mult__at__top,axiom,
    ! [A: $tType,F2: fun(A,real),C2: real,F3: filter(A),G: fun(A,real)] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,C2),F3)
     => ( pp(aa(real,bool,ord_less(real,zero_zero(real)),C2))
       => ( filterlim(A,real,G,at_top(real),F3)
         => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_adq(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G),at_top(real),F3) ) ) ) ).

% filterlim_tendsto_pos_mult_at_top
tff(fact_7586_filterlim__at__top__mult__tendsto__pos,axiom,
    ! [A: $tType,F2: fun(A,real),C2: real,F3: filter(A),G: fun(A,real)] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,C2),F3)
     => ( pp(aa(real,bool,ord_less(real,zero_zero(real)),C2))
       => ( filterlim(A,real,G,at_top(real),F3)
         => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_aea(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G),at_top(real),F3) ) ) ) ).

% filterlim_at_top_mult_tendsto_pos
tff(fact_7587_tendsto__neg__powr,axiom,
    ! [A: $tType,S3: real,F2: fun(A,real),F3: filter(A)] :
      ( pp(aa(real,bool,ord_less(real,S3),zero_zero(real)))
     => ( filterlim(A,real,F2,at_top(real),F3)
       => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_aeb(real,fun(fun(A,real),fun(A,real)),S3),F2),topolo7230453075368039082e_nhds(real,zero_zero(real)),F3) ) ) ).

% tendsto_neg_powr
tff(fact_7588_tendsto__mult__filterlim__at__infinity,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(B,A),C2: A,F3: filter(B),G: fun(B,A)] :
          ( filterlim(B,A,F2,topolo7230453075368039082e_nhds(A,C2),F3)
         => ( ( C2 != zero_zero(A) )
           => ( filterlim(B,A,G,at_infinity(A),F3)
             => filterlim(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_aec(fun(B,A),fun(fun(B,A),fun(B,A)),F2),G),at_infinity(A),F3) ) ) ) ) ).

% tendsto_mult_filterlim_at_infinity
tff(fact_7589_tendsto__divide__0,axiom,
    ! [A: $tType,C: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [F2: fun(C,A),C2: A,F3: filter(C),G: fun(C,A)] :
          ( filterlim(C,A,F2,topolo7230453075368039082e_nhds(A,C2),F3)
         => ( filterlim(C,A,G,at_infinity(A),F3)
           => filterlim(C,A,aa(fun(C,A),fun(C,A),aTP_Lamp_aed(fun(C,A),fun(fun(C,A),fun(C,A)),F2),G),topolo7230453075368039082e_nhds(A,zero_zero(A)),F3) ) ) ) ).

% tendsto_divide_0
tff(fact_7590_filterlim__power__at__infinity,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V8999393235501362500lgebra(B)
     => ! [F2: fun(A,B),F3: filter(A),N: nat] :
          ( filterlim(A,B,F2,at_infinity(B),F3)
         => ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),N))
           => filterlim(A,B,aa(nat,fun(A,B),aTP_Lamp_aee(fun(A,B),fun(nat,fun(A,B)),F2),N),at_infinity(B),F3) ) ) ) ).

% filterlim_power_at_infinity
tff(fact_7591_ln__x__over__x__tendsto__0,axiom,
    filterlim(real,real,aTP_Lamp_aef(real,real),topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(real)) ).

% ln_x_over_x_tendsto_0
tff(fact_7592_filterlim__at__right__to__top,axiom,
    ! [A: $tType,F2: fun(real,A),F3: filter(A)] :
      ( filterlim(real,A,F2,F3,topolo174197925503356063within(real,zero_zero(real),set_ord_greaterThan(real,zero_zero(real))))
    <=> filterlim(real,A,aTP_Lamp_aeg(fun(real,A),fun(real,A),F2),F3,at_top(real)) ) ).

% filterlim_at_right_to_top
tff(fact_7593_filterlim__at__top__to__right,axiom,
    ! [A: $tType,F2: fun(real,A),F3: filter(A)] :
      ( filterlim(real,A,F2,F3,at_top(real))
    <=> filterlim(real,A,aTP_Lamp_aeg(fun(real,A),fun(real,A),F2),F3,topolo174197925503356063within(real,zero_zero(real),set_ord_greaterThan(real,zero_zero(real)))) ) ).

% filterlim_at_top_to_right
tff(fact_7594_filterlim__inverse__at__right__top,axiom,
    filterlim(real,real,inverse_inverse(real),topolo174197925503356063within(real,zero_zero(real),set_ord_greaterThan(real,zero_zero(real))),at_top(real)) ).

% filterlim_inverse_at_right_top
tff(fact_7595_filterlim__inverse__at__top__right,axiom,
    filterlim(real,real,inverse_inverse(real),at_top(real),topolo174197925503356063within(real,zero_zero(real),set_ord_greaterThan(real,zero_zero(real)))) ).

% filterlim_inverse_at_top_right
tff(fact_7596_tendsto__at__topI__sequentially,axiom,
    ! [B: $tType] :
      ( topolo3112930676232923870pology(B)
     => ! [F2: fun(real,B),Y: B] :
          ( ! [X8: fun(nat,real)] :
              ( filterlim(nat,real,X8,at_top(real),at_top(nat))
             => filterlim(nat,B,aa(fun(nat,real),fun(nat,B),aTP_Lamp_adr(fun(real,B),fun(fun(nat,real),fun(nat,B)),F2),X8),topolo7230453075368039082e_nhds(B,Y),at_top(nat)) )
         => filterlim(real,B,F2,topolo7230453075368039082e_nhds(B,Y),at_top(real)) ) ) ).

% tendsto_at_topI_sequentially
tff(fact_7597_filterlim__inverse__at__infinity,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => filterlim(A,A,inverse_inverse(A),at_infinity(A),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ).

% filterlim_inverse_at_infinity
tff(fact_7598_filterlim__tendsto__neg__mult__at__bot,axiom,
    ! [A: $tType,F2: fun(A,real),C2: real,F3: filter(A),G: fun(A,real)] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,C2),F3)
     => ( pp(aa(real,bool,ord_less(real,C2),zero_zero(real)))
       => ( filterlim(A,real,G,at_top(real),F3)
         => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_adq(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G),at_bot(real),F3) ) ) ) ).

% filterlim_tendsto_neg_mult_at_bot
tff(fact_7599_tendsto__power__div__exp__0,axiom,
    ! [K: nat] : filterlim(real,real,aTP_Lamp_aeh(nat,fun(real,real),K),topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(real)) ).

% tendsto_power_div_exp_0
tff(fact_7600_lim__infinity__imp__sequentially,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [F2: fun(real,A),L: A] :
          ( filterlim(real,A,F2,topolo7230453075368039082e_nhds(A,L),at_infinity(real))
         => filterlim(nat,A,aTP_Lamp_aei(fun(real,A),fun(nat,A),F2),topolo7230453075368039082e_nhds(A,L),at_top(nat)) ) ) ).

% lim_infinity_imp_sequentially
tff(fact_7601_filterlim__inverse__at__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V8999393235501362500lgebra(B)
     => ! [G: fun(A,B),F3: filter(A)] :
          ( filterlim(A,B,aTP_Lamp_aej(fun(A,B),fun(A,B),G),topolo174197925503356063within(B,zero_zero(B),top_top(set(B))),F3)
        <=> filterlim(A,B,G,at_infinity(B),F3) ) ) ).

% filterlim_inverse_at_iff
tff(fact_7602_tendsto__exp__limit__at__top,axiom,
    ! [X: real] : filterlim(real,real,aTP_Lamp_aek(real,fun(real,real),X),topolo7230453075368039082e_nhds(real,aa(real,real,exp(real),X)),at_top(real)) ).

% tendsto_exp_limit_at_top
tff(fact_7603_filterlim__divide__at__infinity,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),C2: A,F3: filter(A),G: fun(A,A)] :
          ( filterlim(A,A,F2,topolo7230453075368039082e_nhds(A,C2),F3)
         => ( filterlim(A,A,G,topolo174197925503356063within(A,zero_zero(A),top_top(set(A))),F3)
           => ( ( C2 != zero_zero(A) )
             => filterlim(A,A,aa(fun(A,A),fun(A,A),aTP_Lamp_rg(fun(A,A),fun(fun(A,A),fun(A,A)),F2),G),at_infinity(A),F3) ) ) ) ) ).

% filterlim_divide_at_infinity
tff(fact_7604_filterlim__tan__at__left,axiom,
    filterlim(real,real,tan(real),at_top(real),topolo174197925503356063within(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))),set_ord_lessThan(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2)))))) ).

% filterlim_tan_at_left
tff(fact_7605_tendsto__arctan__at__top,axiom,
    filterlim(real,real,arctan,topolo7230453075368039082e_nhds(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2)))),at_top(real)) ).

% tendsto_arctan_at_top
tff(fact_7606_DERIV__neg__imp__decreasing__at__top,axiom,
    ! [B2: real,F2: fun(real,real),Flim: real] :
      ( ! [X3: real] :
          ( pp(aa(real,bool,ord_less_eq(real,B2),X3))
         => ? [Y4: real] :
              ( has_field_derivative(real,F2,Y4,topolo174197925503356063within(real,X3,top_top(set(real))))
              & pp(aa(real,bool,ord_less(real,Y4),zero_zero(real))) ) )
     => ( filterlim(real,real,F2,topolo7230453075368039082e_nhds(real,Flim),at_top(real))
       => pp(aa(real,bool,ord_less(real,Flim),aa(real,real,F2,B2))) ) ) ).

% DERIV_neg_imp_decreasing_at_top
tff(fact_7607_filterlim__realpow__sequentially__gt1,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [X: A] :
          ( pp(aa(real,bool,ord_less(real,one_one(real)),real_V7770717601297561774m_norm(A,X)))
         => filterlim(nat,A,aa(A,fun(nat,A),power_power(A),X),at_infinity(A),at_top(nat)) ) ) ).

% filterlim_realpow_sequentially_gt1
tff(fact_7608_lim__at__infinity__0,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),L: A] :
          ( filterlim(A,A,F2,topolo7230453075368039082e_nhds(A,L),at_infinity(A))
        <=> filterlim(A,A,comp(A,A,A,F2,inverse_inverse(A)),topolo7230453075368039082e_nhds(A,L),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ).

% lim_at_infinity_0
tff(fact_7609_polyfun__extremal,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [C2: fun(nat,A),K: nat,N: nat,B4: real] :
          ( ( aa(nat,A,C2,K) != zero_zero(A) )
         => ( pp(aa(nat,bool,ord_less_eq(nat,one_one(nat)),K))
           => ( pp(aa(nat,bool,ord_less_eq(nat,K),N))
             => eventually(A,aa(real,fun(A,bool),aa(nat,fun(real,fun(A,bool)),aTP_Lamp_ael(fun(nat,A),fun(nat,fun(real,fun(A,bool))),C2),N),B4),at_infinity(A)) ) ) ) ) ).

% polyfun_extremal
tff(fact_7610_lhopital__left__at__top,axiom,
    ! [G: fun(real,real),X: real,G4: fun(real,real),F2: fun(real,real),F7: fun(real,real),Y: real] :
      ( filterlim(real,real,G,at_top(real),topolo174197925503356063within(real,X,set_ord_lessThan(real,X)))
     => ( eventually(real,aTP_Lamp_aem(fun(real,real),fun(real,bool),G4),topolo174197925503356063within(real,X,set_ord_lessThan(real,X)))
       => ( eventually(real,aa(fun(real,real),fun(real,bool),aTP_Lamp_aen(fun(real,real),fun(fun(real,real),fun(real,bool)),F2),F7),topolo174197925503356063within(real,X,set_ord_lessThan(real,X)))
         => ( eventually(real,aa(fun(real,real),fun(real,bool),aTP_Lamp_aen(fun(real,real),fun(fun(real,real),fun(real,bool)),G),G4),topolo174197925503356063within(real,X,set_ord_lessThan(real,X)))
           => ( filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_aeo(fun(real,real),fun(fun(real,real),fun(real,real)),G4),F7),topolo7230453075368039082e_nhds(real,Y),topolo174197925503356063within(real,X,set_ord_lessThan(real,X)))
             => filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_aeo(fun(real,real),fun(fun(real,real),fun(real,real)),G),F2),topolo7230453075368039082e_nhds(real,Y),topolo174197925503356063within(real,X,set_ord_lessThan(real,X))) ) ) ) ) ) ).

% lhopital_left_at_top
tff(fact_7611_eventually__sequentially__Suc,axiom,
    ! [P: fun(nat,bool)] :
      ( eventually(nat,aTP_Lamp_qd(fun(nat,bool),fun(nat,bool),P),at_top(nat))
    <=> eventually(nat,P,at_top(nat)) ) ).

% eventually_sequentially_Suc
tff(fact_7612_eventually__sequentially__seg,axiom,
    ! [P: fun(nat,bool),K: nat] :
      ( eventually(nat,aa(nat,fun(nat,bool),aTP_Lamp_aep(fun(nat,bool),fun(nat,fun(nat,bool)),P),K),at_top(nat))
    <=> eventually(nat,P,at_top(nat)) ) ).

% eventually_sequentially_seg
tff(fact_7613_eventually__const,axiom,
    ! [A: $tType,F3: filter(A),P: bool] :
      ( ( F3 != bot_bot(filter(A)) )
     => ( eventually(A,aTP_Lamp_pq(bool,fun(A,bool),P),F3)
      <=> pp(P) ) ) ).

% eventually_const
tff(fact_7614_tendsto__sandwich,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo2564578578187576103pology(A)
     => ! [F2: fun(B,A),G: fun(B,A),Net: filter(B),H: fun(B,A),C2: A] :
          ( eventually(B,aa(fun(B,A),fun(B,bool),aTP_Lamp_aeq(fun(B,A),fun(fun(B,A),fun(B,bool)),F2),G),Net)
         => ( eventually(B,aa(fun(B,A),fun(B,bool),aTP_Lamp_aeq(fun(B,A),fun(fun(B,A),fun(B,bool)),G),H),Net)
           => ( filterlim(B,A,F2,topolo7230453075368039082e_nhds(A,C2),Net)
             => ( filterlim(B,A,H,topolo7230453075368039082e_nhds(A,C2),Net)
               => filterlim(B,A,G,topolo7230453075368039082e_nhds(A,C2),Net) ) ) ) ) ) ).

% tendsto_sandwich
tff(fact_7615_order__tendsto__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo2564578578187576103pology(A)
     => ! [F2: fun(B,A),X: A,F3: filter(B)] :
          ( filterlim(B,A,F2,topolo7230453075368039082e_nhds(A,X),F3)
        <=> ( ! [L4: A] :
                ( pp(aa(A,bool,ord_less(A,L4),X))
               => eventually(B,aa(A,fun(B,bool),aTP_Lamp_aer(fun(B,A),fun(A,fun(B,bool)),F2),L4),F3) )
            & ! [U3: A] :
                ( pp(aa(A,bool,ord_less(A,X),U3))
               => eventually(B,aa(A,fun(B,bool),aTP_Lamp_aes(fun(B,A),fun(A,fun(B,bool)),F2),U3),F3) ) ) ) ) ).

% order_tendsto_iff
tff(fact_7616_order__tendstoI,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo2564578578187576103pology(A)
     => ! [Y: A,F2: fun(B,A),F3: filter(B)] :
          ( ! [A4: A] :
              ( pp(aa(A,bool,ord_less(A,A4),Y))
             => eventually(B,aa(A,fun(B,bool),aTP_Lamp_aer(fun(B,A),fun(A,fun(B,bool)),F2),A4),F3) )
         => ( ! [A4: A] :
                ( pp(aa(A,bool,ord_less(A,Y),A4))
               => eventually(B,aa(A,fun(B,bool),aTP_Lamp_aes(fun(B,A),fun(A,fun(B,bool)),F2),A4),F3) )
           => filterlim(B,A,F2,topolo7230453075368039082e_nhds(A,Y),F3) ) ) ) ).

% order_tendstoI
tff(fact_7617_order__tendstoD_I1_J,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo2564578578187576103pology(A)
     => ! [F2: fun(B,A),Y: A,F3: filter(B),A2: A] :
          ( filterlim(B,A,F2,topolo7230453075368039082e_nhds(A,Y),F3)
         => ( pp(aa(A,bool,ord_less(A,A2),Y))
           => eventually(B,aa(A,fun(B,bool),aTP_Lamp_aer(fun(B,A),fun(A,fun(B,bool)),F2),A2),F3) ) ) ) ).

% order_tendstoD(1)
tff(fact_7618_order__tendstoD_I2_J,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo2564578578187576103pology(A)
     => ! [F2: fun(B,A),Y: A,F3: filter(B),A2: A] :
          ( filterlim(B,A,F2,topolo7230453075368039082e_nhds(A,Y),F3)
         => ( pp(aa(A,bool,ord_less(A,Y),A2))
           => eventually(B,aa(A,fun(B,bool),aTP_Lamp_aes(fun(B,A),fun(A,fun(B,bool)),F2),A2),F3) ) ) ) ).

% order_tendstoD(2)
tff(fact_7619_sequentially__offset,axiom,
    ! [P: fun(nat,bool),K: nat] :
      ( eventually(nat,P,at_top(nat))
     => eventually(nat,aa(nat,fun(nat,bool),aTP_Lamp_aep(fun(nat,bool),fun(nat,fun(nat,bool)),P),K),at_top(nat)) ) ).

% sequentially_offset
tff(fact_7620_summable__cong,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A),G: fun(nat,A)] :
          ( eventually(nat,aa(fun(nat,A),fun(nat,bool),aTP_Lamp_aet(fun(nat,A),fun(fun(nat,A),fun(nat,bool)),F2),G),at_top(nat))
         => ( summable(A,F2)
          <=> summable(A,G) ) ) ) ).

% summable_cong
tff(fact_7621_eventually__at__infinity,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [P: fun(A,bool)] :
          ( eventually(A,P,at_infinity(A))
        <=> ? [B7: real] :
            ! [X2: A] :
              ( pp(aa(real,bool,ord_less_eq(real,B7),real_V7770717601297561774m_norm(A,X2)))
             => pp(aa(A,bool,P,X2)) ) ) ) ).

% eventually_at_infinity
tff(fact_7622_eventually__not__equal__at__infinity,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [A2: A] : eventually(A,aTP_Lamp_aeu(A,fun(A,bool),A2),at_infinity(A)) ) ).

% eventually_not_equal_at_infinity
tff(fact_7623_eventually__ge__at__top,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [C2: A] : eventually(A,ord_less_eq(A,C2),at_top(A)) ) ).

% eventually_ge_at_top
tff(fact_7624_eventually__sequentially,axiom,
    ! [P: fun(nat,bool)] :
      ( eventually(nat,P,at_top(nat))
    <=> ? [N8: nat] :
        ! [N5: nat] :
          ( pp(aa(nat,bool,ord_less_eq(nat,N8),N5))
         => pp(aa(nat,bool,P,N5)) ) ) ).

% eventually_sequentially
tff(fact_7625_eventually__sequentiallyI,axiom,
    ! [C2: nat,P: fun(nat,bool)] :
      ( ! [X3: nat] :
          ( pp(aa(nat,bool,ord_less_eq(nat,C2),X3))
         => pp(aa(nat,bool,P,X3)) )
     => eventually(nat,P,at_top(nat)) ) ).

% eventually_sequentiallyI
tff(fact_7626_eventually__at__top__linorder,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [P: fun(A,bool)] :
          ( eventually(A,P,at_top(A))
        <=> ? [N8: A] :
            ! [N5: A] :
              ( pp(aa(A,bool,ord_less_eq(A,N8),N5))
             => pp(aa(A,bool,P,N5)) ) ) ) ).

% eventually_at_top_linorder
tff(fact_7627_eventually__at__top__linorderI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [C2: A,P: fun(A,bool)] :
          ( ! [X3: A] :
              ( pp(aa(A,bool,ord_less_eq(A,C2),X3))
             => pp(aa(A,bool,P,X3)) )
         => eventually(A,P,at_top(A)) ) ) ).

% eventually_at_top_linorderI
tff(fact_7628_le__sequentially,axiom,
    ! [F3: filter(nat)] :
      ( pp(aa(filter(nat),bool,ord_less_eq(filter(nat),F3),at_top(nat)))
    <=> ! [N8: nat] : eventually(nat,ord_less_eq(nat,N8),F3) ) ).

% le_sequentially
tff(fact_7629_eventually__gt__at__top,axiom,
    ! [A: $tType] :
      ( ( linorder(A)
        & no_top(A) )
     => ! [C2: A] : eventually(A,ord_less(A,C2),at_top(A)) ) ).

% eventually_gt_at_top
tff(fact_7630_eventually__at__top__not__equal,axiom,
    ! [A: $tType] :
      ( ( linorder(A)
        & no_top(A) )
     => ! [C2: A] : eventually(A,aTP_Lamp_aev(A,fun(A,bool),C2),at_top(A)) ) ).

% eventually_at_top_not_equal
tff(fact_7631_eventually__False__sequentially,axiom,
    ~ eventually(nat,aTP_Lamp_pr(nat,bool),at_top(nat)) ).

% eventually_False_sequentially
tff(fact_7632_eventually__compose__filterlim,axiom,
    ! [A: $tType,B: $tType,P: fun(A,bool),F3: filter(A),F2: fun(B,A),G5: filter(B)] :
      ( eventually(A,P,F3)
     => ( filterlim(B,A,F2,F3,G5)
       => eventually(B,aa(fun(B,A),fun(B,bool),aTP_Lamp_aew(fun(A,bool),fun(fun(B,A),fun(B,bool)),P),F2),G5) ) ) ).

% eventually_compose_filterlim
tff(fact_7633_filterlim__cong,axiom,
    ! [A: $tType,B: $tType,F1: filter(A),F12: filter(A),F22: filter(B),F23: filter(B),F2: fun(B,A),G: fun(B,A)] :
      ( ( F1 = F12 )
     => ( ( F22 = F23 )
       => ( eventually(B,aa(fun(B,A),fun(B,bool),aTP_Lamp_aex(fun(B,A),fun(fun(B,A),fun(B,bool)),F2),G),F22)
         => ( filterlim(B,A,F2,F1,F22)
          <=> filterlim(B,A,G,F12,F23) ) ) ) ) ).

% filterlim_cong
tff(fact_7634_filterlim__iff,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),F22: filter(B),F1: filter(A)] :
      ( filterlim(A,B,F2,F22,F1)
    <=> ! [P5: fun(B,bool)] :
          ( eventually(B,P5,F22)
         => eventually(A,aa(fun(B,bool),fun(A,bool),aTP_Lamp_aey(fun(A,B),fun(fun(B,bool),fun(A,bool)),F2),P5),F1) ) ) ).

% filterlim_iff
tff(fact_7635_Lim__transform__eventually,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo4958980785337419405_space(B)
     => ! [F2: fun(A,B),L: B,F3: filter(A),G: fun(A,B)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),F3)
         => ( eventually(A,aa(fun(A,B),fun(A,bool),aTP_Lamp_aez(fun(A,B),fun(fun(A,B),fun(A,bool)),F2),G),F3)
           => filterlim(A,B,G,topolo7230453075368039082e_nhds(B,L),F3) ) ) ) ).

% Lim_transform_eventually
tff(fact_7636_filterlim__at__infinity__imp__eventually__ne,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F2: fun(A,B),F3: filter(A),C2: B] :
          ( filterlim(A,B,F2,at_infinity(B),F3)
         => eventually(A,aa(B,fun(A,bool),aTP_Lamp_afa(fun(A,B),fun(B,fun(A,bool)),F2),C2),F3) ) ) ).

% filterlim_at_infinity_imp_eventually_ne
tff(fact_7637_tendsto__cong,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [F2: fun(B,A),G: fun(B,A),F3: filter(B),C2: A] :
          ( eventually(B,aa(fun(B,A),fun(B,bool),aTP_Lamp_afb(fun(B,A),fun(fun(B,A),fun(B,bool)),F2),G),F3)
         => ( filterlim(B,A,F2,topolo7230453075368039082e_nhds(A,C2),F3)
          <=> filterlim(B,A,G,topolo7230453075368039082e_nhds(A,C2),F3) ) ) ) ).

% tendsto_cong
tff(fact_7638_tendsto__discrete,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo8865339358273720382pology(A)
     => ! [F2: fun(B,A),Y: A,F3: filter(B)] :
          ( filterlim(B,A,F2,topolo7230453075368039082e_nhds(A,Y),F3)
        <=> eventually(B,aa(A,fun(B,bool),aTP_Lamp_afc(fun(B,A),fun(A,fun(B,bool)),F2),Y),F3) ) ) ).

% tendsto_discrete
tff(fact_7639_tendsto__imp__eventually__ne,axiom,
    ! [A: $tType,B: $tType] :
      ( topological_t1_space(A)
     => ! [F2: fun(B,A),C2: A,F3: filter(B),C8: A] :
          ( filterlim(B,A,F2,topolo7230453075368039082e_nhds(A,C2),F3)
         => ( ( C2 != C8 )
           => eventually(B,aa(A,fun(B,bool),aTP_Lamp_afd(fun(B,A),fun(A,fun(B,bool)),F2),C8),F3) ) ) ) ).

% tendsto_imp_eventually_ne
tff(fact_7640_tendsto__eventually,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [F2: fun(B,A),L: A,Net: filter(B)] :
          ( eventually(B,aa(A,fun(B,bool),aTP_Lamp_afe(fun(B,A),fun(A,fun(B,bool)),F2),L),Net)
         => filterlim(B,A,F2,topolo7230453075368039082e_nhds(A,L),Net) ) ) ).

% tendsto_eventually
tff(fact_7641_eventually__nhds__iff__sequentially,axiom,
    ! [A: $tType] :
      ( topolo3112930676232923870pology(A)
     => ! [P: fun(A,bool),A2: A] :
          ( eventually(A,P,topolo7230453075368039082e_nhds(A,A2))
        <=> ! [F6: fun(nat,A)] :
              ( filterlim(nat,A,F6,topolo7230453075368039082e_nhds(A,A2),at_top(nat))
             => eventually(nat,aa(fun(nat,A),fun(nat,bool),aTP_Lamp_aff(fun(A,bool),fun(fun(nat,A),fun(nat,bool)),P),F6),at_top(nat)) ) ) ) ).

% eventually_nhds_iff_sequentially
tff(fact_7642_filterlim__mono__eventually,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),F3: filter(B),G5: filter(A),F8: filter(B),G6: filter(A),F7: fun(A,B)] :
      ( filterlim(A,B,F2,F3,G5)
     => ( pp(aa(filter(B),bool,ord_less_eq(filter(B),F3),F8))
       => ( pp(aa(filter(A),bool,ord_less_eq(filter(A),G6),G5))
         => ( eventually(A,aa(fun(A,B),fun(A,bool),aTP_Lamp_afg(fun(A,B),fun(fun(A,B),fun(A,bool)),F2),F7),G6)
           => filterlim(A,B,F7,F8,G6) ) ) ) ) ).

% filterlim_mono_eventually
tff(fact_7643_filterlim__at__top__at__top,axiom,
    ! [B: $tType,A: $tType] :
      ( ( linorder(A)
        & linorder(B) )
     => ! [Q: fun(A,bool),F2: fun(A,B),P: fun(B,bool),G: fun(B,A)] :
          ( ! [X3: A,Y3: A] :
              ( pp(aa(A,bool,Q,X3))
             => ( pp(aa(A,bool,Q,Y3))
               => ( pp(aa(A,bool,ord_less_eq(A,X3),Y3))
                 => pp(aa(B,bool,ord_less_eq(B,aa(A,B,F2,X3)),aa(A,B,F2,Y3))) ) ) )
         => ( ! [X3: B] :
                ( pp(aa(B,bool,P,X3))
               => ( aa(A,B,F2,aa(B,A,G,X3)) = X3 ) )
           => ( ! [X3: B] :
                  ( pp(aa(B,bool,P,X3))
                 => pp(aa(A,bool,Q,aa(B,A,G,X3))) )
             => ( eventually(A,Q,at_top(A))
               => ( eventually(B,P,at_top(B))
                 => filterlim(A,B,F2,at_top(B),at_top(A)) ) ) ) ) ) ) ).

% filterlim_at_top_at_top
tff(fact_7644_sequentially__imp__eventually__at,axiom,
    ! [A: $tType] :
      ( topolo3112930676232923870pology(A)
     => ! [A2: A,P: fun(A,bool)] :
          ( ! [F4: fun(nat,A)] :
              ( ( ! [N11: nat] : aa(nat,A,F4,N11) != A2
                & filterlim(nat,A,F4,topolo7230453075368039082e_nhds(A,A2),at_top(nat)) )
             => eventually(nat,aa(fun(nat,A),fun(nat,bool),aTP_Lamp_aff(fun(A,bool),fun(fun(nat,A),fun(nat,bool)),P),F4),at_top(nat)) )
         => eventually(A,P,topolo174197925503356063within(A,A2,top_top(set(A)))) ) ) ).

% sequentially_imp_eventually_at
tff(fact_7645_sequentially__imp__eventually__within,axiom,
    ! [A: $tType] :
      ( topolo3112930676232923870pology(A)
     => ! [S3: set(A),A2: A,P: fun(A,bool)] :
          ( ! [F4: fun(nat,A)] :
              ( ( ! [N11: nat] :
                    ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(nat,A,F4,N11)),S3))
                    & ( aa(nat,A,F4,N11) != A2 ) )
                & filterlim(nat,A,F4,topolo7230453075368039082e_nhds(A,A2),at_top(nat)) )
             => eventually(nat,aa(fun(nat,A),fun(nat,bool),aTP_Lamp_aff(fun(A,bool),fun(fun(nat,A),fun(nat,bool)),P),F4),at_top(nat)) )
         => eventually(A,P,topolo174197925503356063within(A,A2,S3)) ) ) ).

% sequentially_imp_eventually_within
tff(fact_7646_filterlim__at__within__not__equal,axiom,
    ! [B: $tType,A: $tType] :
      ( topological_t2_space(B)
     => ! [F2: fun(A,B),A2: B,S3: set(B),F3: filter(A),B2: B] :
          ( filterlim(A,B,F2,topolo174197925503356063within(B,A2,S3),F3)
         => eventually(A,aa(B,fun(A,bool),aa(set(B),fun(B,fun(A,bool)),aTP_Lamp_afh(fun(A,B),fun(set(B),fun(B,fun(A,bool))),F2),S3),B2),F3) ) ) ).

% filterlim_at_within_not_equal
tff(fact_7647_filterlim__at__top,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F2: fun(A,B),F3: filter(A)] :
          ( filterlim(A,B,F2,at_top(B),F3)
        <=> ! [Z7: B] : eventually(A,aa(B,fun(A,bool),aTP_Lamp_afi(fun(A,B),fun(B,fun(A,bool)),F2),Z7),F3) ) ) ).

% filterlim_at_top
tff(fact_7648_filterlim__at__top__ge,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F2: fun(A,B),F3: filter(A),C2: B] :
          ( filterlim(A,B,F2,at_top(B),F3)
        <=> ! [Z7: B] :
              ( pp(aa(B,bool,ord_less_eq(B,C2),Z7))
             => eventually(A,aa(B,fun(A,bool),aTP_Lamp_afi(fun(A,B),fun(B,fun(A,bool)),F2),Z7),F3) ) ) ) ).

% filterlim_at_top_ge
tff(fact_7649_filterlim__at__top__mono,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F2: fun(B,A),F3: filter(B),G: fun(B,A)] :
          ( filterlim(B,A,F2,at_top(A),F3)
         => ( eventually(B,aa(fun(B,A),fun(B,bool),aTP_Lamp_afj(fun(B,A),fun(fun(B,A),fun(B,bool)),F2),G),F3)
           => filterlim(B,A,G,at_top(A),F3) ) ) ) ).

% filterlim_at_top_mono
tff(fact_7650_filterlim__at__top__dense,axiom,
    ! [B: $tType,A: $tType] :
      ( unboun7993243217541854897norder(B)
     => ! [F2: fun(A,B),F3: filter(A)] :
          ( filterlim(A,B,F2,at_top(B),F3)
        <=> ! [Z7: B] : eventually(A,aa(B,fun(A,bool),aTP_Lamp_afk(fun(A,B),fun(B,fun(A,bool)),F2),Z7),F3) ) ) ).

% filterlim_at_top_dense
tff(fact_7651_eventually__Lim__ident__at,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [P: fun(A,fun(A,bool)),X: A,X6: set(A)] :
          ( eventually(A,aa(A,fun(A,bool),P,topolo3827282254853284352ce_Lim(A,A,topolo174197925503356063within(A,X,X6),aTP_Lamp_vm(A,A))),topolo174197925503356063within(A,X,X6))
        <=> eventually(A,aa(A,fun(A,bool),P,X),topolo174197925503356063within(A,X,X6)) ) ) ).

% eventually_Lim_ident_at
tff(fact_7652_trivial__limit__def,axiom,
    ! [A: $tType,F3: filter(A)] :
      ( ( F3 = bot_bot(filter(A)) )
    <=> eventually(A,aTP_Lamp_ll(A,bool),F3) ) ).

% trivial_limit_def
tff(fact_7653_eventually__const__iff,axiom,
    ! [A: $tType,P: bool,F3: filter(A)] :
      ( eventually(A,aTP_Lamp_pq(bool,fun(A,bool),P),F3)
    <=> ( pp(P)
        | ( F3 = bot_bot(filter(A)) ) ) ) ).

% eventually_const_iff
tff(fact_7654_False__imp__not__eventually,axiom,
    ! [A: $tType,P: fun(A,bool),Net: filter(A)] :
      ( ! [X3: A] : ~ pp(aa(A,bool,P,X3))
     => ( ( Net != bot_bot(filter(A)) )
       => ~ eventually(A,P,Net) ) ) ).

% False_imp_not_eventually
tff(fact_7655_eventually__frequently__const__simps_I6_J,axiom,
    ! [A: $tType,C6: bool,P: fun(A,bool),F3: filter(A)] :
      ( eventually(A,aa(fun(A,bool),fun(A,bool),aTP_Lamp_afl(bool,fun(fun(A,bool),fun(A,bool)),C6),P),F3)
    <=> ( pp(C6)
       => eventually(A,P,F3) ) ) ).

% eventually_frequently_const_simps(6)
tff(fact_7656_eventually__frequently__const__simps_I4_J,axiom,
    ! [A: $tType,C6: bool,P: fun(A,bool),F3: filter(A)] :
      ( eventually(A,aa(fun(A,bool),fun(A,bool),aTP_Lamp_afm(bool,fun(fun(A,bool),fun(A,bool)),C6),P),F3)
    <=> ( pp(C6)
        | eventually(A,P,F3) ) ) ).

% eventually_frequently_const_simps(4)
tff(fact_7657_eventually__frequently__const__simps_I3_J,axiom,
    ! [A: $tType,P: fun(A,bool),C6: bool,F3: filter(A)] :
      ( eventually(A,aa(bool,fun(A,bool),aTP_Lamp_afn(fun(A,bool),fun(bool,fun(A,bool)),P),C6),F3)
    <=> ( eventually(A,P,F3)
        | pp(C6) ) ) ).

% eventually_frequently_const_simps(3)
tff(fact_7658_eventually__mp,axiom,
    ! [A: $tType,P: fun(A,bool),Q: fun(A,bool),F3: filter(A)] :
      ( eventually(A,aa(fun(A,bool),fun(A,bool),aTP_Lamp_afo(fun(A,bool),fun(fun(A,bool),fun(A,bool)),P),Q),F3)
     => ( eventually(A,P,F3)
       => eventually(A,Q,F3) ) ) ).

% eventually_mp
tff(fact_7659_eventually__True,axiom,
    ! [A: $tType,F3: filter(A)] : eventually(A,aTP_Lamp_pu(A,bool),F3) ).

% eventually_True
tff(fact_7660_eventually__conj,axiom,
    ! [A: $tType,P: fun(A,bool),F3: filter(A),Q: fun(A,bool)] :
      ( eventually(A,P,F3)
     => ( eventually(A,Q,F3)
       => eventually(A,aa(fun(A,bool),fun(A,bool),aTP_Lamp_as(fun(A,bool),fun(fun(A,bool),fun(A,bool)),P),Q),F3) ) ) ).

% eventually_conj
tff(fact_7661_eventually__elim2,axiom,
    ! [A: $tType,P: fun(A,bool),F3: filter(A),Q: fun(A,bool),R: fun(A,bool)] :
      ( eventually(A,P,F3)
     => ( eventually(A,Q,F3)
       => ( ! [I2: A] :
              ( pp(aa(A,bool,P,I2))
             => ( pp(aa(A,bool,Q,I2))
               => pp(aa(A,bool,R,I2)) ) )
         => eventually(A,R,F3) ) ) ) ).

% eventually_elim2
tff(fact_7662_eventually__subst,axiom,
    ! [A: $tType,P: fun(A,bool),Q: fun(A,bool),F3: filter(A)] :
      ( eventually(A,aa(fun(A,bool),fun(A,bool),aTP_Lamp_afp(fun(A,bool),fun(fun(A,bool),fun(A,bool)),P),Q),F3)
     => ( eventually(A,P,F3)
      <=> eventually(A,Q,F3) ) ) ).

% eventually_subst
tff(fact_7663_eventually__rev__mp,axiom,
    ! [A: $tType,P: fun(A,bool),F3: filter(A),Q: fun(A,bool)] :
      ( eventually(A,P,F3)
     => ( eventually(A,aa(fun(A,bool),fun(A,bool),aTP_Lamp_afo(fun(A,bool),fun(fun(A,bool),fun(A,bool)),P),Q),F3)
       => eventually(A,Q,F3) ) ) ).

% eventually_rev_mp
tff(fact_7664_eventually__conj__iff,axiom,
    ! [A: $tType,P: fun(A,bool),Q: fun(A,bool),F3: filter(A)] :
      ( eventually(A,aa(fun(A,bool),fun(A,bool),aTP_Lamp_as(fun(A,bool),fun(fun(A,bool),fun(A,bool)),P),Q),F3)
    <=> ( eventually(A,P,F3)
        & eventually(A,Q,F3) ) ) ).

% eventually_conj_iff
tff(fact_7665_not__eventually__impI,axiom,
    ! [A: $tType,P: fun(A,bool),F3: filter(A),Q: fun(A,bool)] :
      ( eventually(A,P,F3)
     => ( ~ eventually(A,Q,F3)
       => ~ eventually(A,aa(fun(A,bool),fun(A,bool),aTP_Lamp_afo(fun(A,bool),fun(fun(A,bool),fun(A,bool)),P),Q),F3) ) ) ).

% not_eventually_impI
tff(fact_7666_has__derivative__transform__eventually,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),F7: fun(A,B),X: A,S3: set(A),G: fun(A,B)] :
          ( has_derivative(A,B,F2,F7,topolo174197925503356063within(A,X,S3))
         => ( eventually(A,aa(fun(A,B),fun(A,bool),aTP_Lamp_afq(fun(A,B),fun(fun(A,B),fun(A,bool)),F2),G),topolo174197925503356063within(A,X,S3))
           => ( ( aa(A,B,F2,X) = aa(A,B,G,X) )
             => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),S3))
               => has_derivative(A,B,G,F7,topolo174197925503356063within(A,X,S3)) ) ) ) ) ) ).

% has_derivative_transform_eventually
tff(fact_7667_has__field__derivative__cong__eventually,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),G: fun(A,A),X: A,S: set(A),U: A] :
          ( eventually(A,aa(fun(A,A),fun(A,bool),aTP_Lamp_afr(fun(A,A),fun(fun(A,A),fun(A,bool)),F2),G),topolo174197925503356063within(A,X,S))
         => ( ( aa(A,A,F2,X) = aa(A,A,G,X) )
           => ( has_field_derivative(A,F2,U,topolo174197925503356063within(A,X,S))
            <=> has_field_derivative(A,G,U,topolo174197925503356063within(A,X,S)) ) ) ) ) ).

% has_field_derivative_cong_eventually
tff(fact_7668_t1__space__nhds,axiom,
    ! [A: $tType] :
      ( topological_t1_space(A)
     => ! [X: A,Y: A] :
          ( ( X != Y )
         => eventually(A,aTP_Lamp_afs(A,fun(A,bool),Y),topolo7230453075368039082e_nhds(A,X)) ) ) ).

% t1_space_nhds
tff(fact_7669_eventually__eventually,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [P: fun(A,bool),X: A] :
          ( eventually(A,aTP_Lamp_aft(fun(A,bool),fun(A,bool),P),topolo7230453075368039082e_nhds(A,X))
        <=> eventually(A,P,topolo7230453075368039082e_nhds(A,X)) ) ) ).

% eventually_eventually
tff(fact_7670_eventually__at__filter,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [P: fun(A,bool),A2: A,S3: set(A)] :
          ( eventually(A,P,topolo174197925503356063within(A,A2,S3))
        <=> eventually(A,aa(set(A),fun(A,bool),aa(A,fun(set(A),fun(A,bool)),aTP_Lamp_afu(fun(A,bool),fun(A,fun(set(A),fun(A,bool))),P),A2),S3),topolo7230453075368039082e_nhds(A,A2)) ) ) ).

% eventually_at_filter
tff(fact_7671_eventually__nhds__in__open,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [S3: set(A),X: A] :
          ( topolo1002775350975398744n_open(A,S3)
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),S3))
           => eventually(A,aTP_Lamp_afv(set(A),fun(A,bool),S3),topolo7230453075368039082e_nhds(A,X)) ) ) ) ).

% eventually_nhds_in_open
tff(fact_7672_eventually__at__bot__not__equal,axiom,
    ! [A: $tType] :
      ( ( linorder(A)
        & no_bot(A) )
     => ! [C2: A] : eventually(A,aTP_Lamp_afw(A,fun(A,bool),C2),at_bot(A)) ) ).

% eventually_at_bot_not_equal
tff(fact_7673_eventually__gt__at__bot,axiom,
    ! [A: $tType] :
      ( unboun7993243217541854897norder(A)
     => ! [C2: A] : eventually(A,aTP_Lamp_afx(A,fun(A,bool),C2),at_bot(A)) ) ).

% eventually_gt_at_bot
tff(fact_7674_eventually__le__at__bot,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [C2: A] : eventually(A,aTP_Lamp_afy(A,fun(A,bool),C2),at_bot(A)) ) ).

% eventually_le_at_bot
tff(fact_7675_eventually__at__bot__linorder,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [P: fun(A,bool)] :
          ( eventually(A,P,at_bot(A))
        <=> ? [N8: A] :
            ! [N5: A] :
              ( pp(aa(A,bool,ord_less_eq(A,N5),N8))
             => pp(aa(A,bool,P,N5)) ) ) ) ).

% eventually_at_bot_linorder
tff(fact_7676_filterlim__at,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [F2: fun(B,A),B2: A,S3: set(A),F3: filter(B)] :
          ( filterlim(B,A,F2,topolo174197925503356063within(A,B2,S3),F3)
        <=> ( eventually(B,aa(set(A),fun(B,bool),aa(A,fun(set(A),fun(B,bool)),aTP_Lamp_afz(fun(B,A),fun(A,fun(set(A),fun(B,bool))),F2),B2),S3),F3)
            & filterlim(B,A,F2,topolo7230453075368039082e_nhds(A,B2),F3) ) ) ) ).

% filterlim_at
tff(fact_7677_eventually__at__right__less,axiom,
    ! [A: $tType] :
      ( ( no_top(A)
        & topolo1944317154257567458pology(A) )
     => ! [X: A] : eventually(A,ord_less(A,X),topolo174197925503356063within(A,X,set_ord_greaterThan(A,X))) ) ).

% eventually_at_right_less
tff(fact_7678_has__field__derivative__cong__ev,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [X: A,Y: A,S: set(A),F2: fun(A,A),G: fun(A,A),U: A,V: A,T: set(A)] :
          ( ( X = Y )
         => ( eventually(A,aa(fun(A,A),fun(A,bool),aa(fun(A,A),fun(fun(A,A),fun(A,bool)),aTP_Lamp_aga(set(A),fun(fun(A,A),fun(fun(A,A),fun(A,bool))),S),F2),G),topolo7230453075368039082e_nhds(A,X))
           => ( ( U = V )
             => ( ( S = T )
               => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),S))
                 => ( has_field_derivative(A,F2,U,topolo174197925503356063within(A,X,S))
                  <=> has_field_derivative(A,G,V,topolo174197925503356063within(A,Y,T)) ) ) ) ) ) ) ) ).

% has_field_derivative_cong_ev
tff(fact_7679_topological__tendstoI,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [L: A,F2: fun(B,A),F3: filter(B)] :
          ( ! [S6: set(A)] :
              ( topolo1002775350975398744n_open(A,S6)
             => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),L),S6))
               => eventually(B,aa(set(A),fun(B,bool),aTP_Lamp_agb(fun(B,A),fun(set(A),fun(B,bool)),F2),S6),F3) ) )
         => filterlim(B,A,F2,topolo7230453075368039082e_nhds(A,L),F3) ) ) ).

% topological_tendstoI
tff(fact_7680_topological__tendstoD,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [F2: fun(B,A),L: A,F3: filter(B),S: set(A)] :
          ( filterlim(B,A,F2,topolo7230453075368039082e_nhds(A,L),F3)
         => ( topolo1002775350975398744n_open(A,S)
           => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),L),S))
             => eventually(B,aa(set(A),fun(B,bool),aTP_Lamp_agb(fun(B,A),fun(set(A),fun(B,bool)),F2),S),F3) ) ) ) ) ).

% topological_tendstoD
tff(fact_7681_tendsto__def,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [F2: fun(B,A),L: A,F3: filter(B)] :
          ( filterlim(B,A,F2,topolo7230453075368039082e_nhds(A,L),F3)
        <=> ! [S10: set(A)] :
              ( topolo1002775350975398744n_open(A,S10)
             => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),L),S10))
               => eventually(B,aa(set(A),fun(B,bool),aTP_Lamp_agb(fun(B,A),fun(set(A),fun(B,bool)),F2),S10),F3) ) ) ) ) ).

% tendsto_def
tff(fact_7682_filterlim__at__bot,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F2: fun(A,B),F3: filter(A)] :
          ( filterlim(A,B,F2,at_bot(B),F3)
        <=> ! [Z7: B] : eventually(A,aa(B,fun(A,bool),aTP_Lamp_agc(fun(A,B),fun(B,fun(A,bool)),F2),Z7),F3) ) ) ).

% filterlim_at_bot
tff(fact_7683_filterlim__at__bot__le,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F2: fun(A,B),F3: filter(A),C2: B] :
          ( filterlim(A,B,F2,at_bot(B),F3)
        <=> ! [Z7: B] :
              ( pp(aa(B,bool,ord_less_eq(B,Z7),C2))
             => eventually(A,aa(B,fun(A,bool),aTP_Lamp_agc(fun(A,B),fun(B,fun(A,bool)),F2),Z7),F3) ) ) ) ).

% filterlim_at_bot_le
tff(fact_7684_filterlim__at__bot__dense,axiom,
    ! [B: $tType,A: $tType] :
      ( ( dense_linorder(B)
        & no_bot(B) )
     => ! [F2: fun(A,B),F3: filter(A)] :
          ( filterlim(A,B,F2,at_bot(B),F3)
        <=> ! [Z7: B] : eventually(A,aa(B,fun(A,bool),aTP_Lamp_agd(fun(A,B),fun(B,fun(A,bool)),F2),Z7),F3) ) ) ).

% filterlim_at_bot_dense
tff(fact_7685_real__tendsto__sandwich,axiom,
    ! [B: $tType,F2: fun(B,real),G: fun(B,real),Net: filter(B),H: fun(B,real),C2: real] :
      ( eventually(B,aa(fun(B,real),fun(B,bool),aTP_Lamp_age(fun(B,real),fun(fun(B,real),fun(B,bool)),F2),G),Net)
     => ( eventually(B,aa(fun(B,real),fun(B,bool),aTP_Lamp_age(fun(B,real),fun(fun(B,real),fun(B,bool)),G),H),Net)
       => ( filterlim(B,real,F2,topolo7230453075368039082e_nhds(real,C2),Net)
         => ( filterlim(B,real,H,topolo7230453075368039082e_nhds(real,C2),Net)
           => filterlim(B,real,G,topolo7230453075368039082e_nhds(real,C2),Net) ) ) ) ) ).

% real_tendsto_sandwich
tff(fact_7686_countable__basis__at__decseq,axiom,
    ! [A: $tType] :
      ( topolo3112930676232923870pology(A)
     => ! [X: A] :
          ~ ! [A7: fun(nat,set(A))] :
              ( ! [I4: nat] : topolo1002775350975398744n_open(A,aa(nat,set(A),A7,I4))
             => ( ! [I4: nat] : pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(nat,set(A),A7,I4)))
               => ~ ! [S9: set(A)] :
                      ( topolo1002775350975398744n_open(A,S9)
                     => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),S9))
                       => eventually(nat,aa(set(A),fun(nat,bool),aTP_Lamp_agf(fun(nat,set(A)),fun(set(A),fun(nat,bool)),A7),S9),at_top(nat)) ) ) ) ) ) ).

% countable_basis_at_decseq
tff(fact_7687_eventually__at,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [P: fun(A,bool),A2: A,S: set(A)] :
          ( eventually(A,P,topolo174197925503356063within(A,A2,S))
        <=> ? [D3: real] :
              ( pp(aa(real,bool,ord_less(real,zero_zero(real)),D3))
              & ! [X2: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X2),S))
                 => ( ( ( X2 != A2 )
                      & pp(aa(real,bool,ord_less(real,real_V557655796197034286t_dist(A,X2,A2)),D3)) )
                   => pp(aa(A,bool,P,X2)) ) ) ) ) ) ).

% eventually_at
tff(fact_7688_eventually__nhds__metric,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [P: fun(A,bool),A2: A] :
          ( eventually(A,P,topolo7230453075368039082e_nhds(A,A2))
        <=> ? [D3: real] :
              ( pp(aa(real,bool,ord_less(real,zero_zero(real)),D3))
              & ! [X2: A] :
                  ( pp(aa(real,bool,ord_less(real,real_V557655796197034286t_dist(A,X2,A2)),D3))
                 => pp(aa(A,bool,P,X2)) ) ) ) ) ).

% eventually_nhds_metric
tff(fact_7689_eventually__at__to__0,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [P: fun(A,bool),A2: A] :
          ( eventually(A,P,topolo174197925503356063within(A,A2,top_top(set(A))))
        <=> eventually(A,aa(A,fun(A,bool),aTP_Lamp_agg(fun(A,bool),fun(A,fun(A,bool)),P),A2),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ).

% eventually_at_to_0
tff(fact_7690_decreasing__tendsto,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo2564578578187576103pology(A)
     => ! [L: A,F2: fun(B,A),F3: filter(B)] :
          ( eventually(B,aa(fun(B,A),fun(B,bool),aTP_Lamp_agh(A,fun(fun(B,A),fun(B,bool)),L),F2),F3)
         => ( ! [X3: A] :
                ( pp(aa(A,bool,ord_less(A,L),X3))
               => eventually(B,aa(A,fun(B,bool),aTP_Lamp_aes(fun(B,A),fun(A,fun(B,bool)),F2),X3),F3) )
           => filterlim(B,A,F2,topolo7230453075368039082e_nhds(A,L),F3) ) ) ) ).

% decreasing_tendsto
tff(fact_7691_increasing__tendsto,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo2564578578187576103pology(A)
     => ! [F2: fun(B,A),L: A,F3: filter(B)] :
          ( eventually(B,aa(A,fun(B,bool),aTP_Lamp_agi(fun(B,A),fun(A,fun(B,bool)),F2),L),F3)
         => ( ! [X3: A] :
                ( pp(aa(A,bool,ord_less(A,X3),L))
               => eventually(B,aa(A,fun(B,bool),aTP_Lamp_aer(fun(B,A),fun(A,fun(B,bool)),F2),X3),F3) )
           => filterlim(B,A,F2,topolo7230453075368039082e_nhds(A,L),F3) ) ) ) ).

% increasing_tendsto
tff(fact_7692_filterlim__at__top__gt,axiom,
    ! [B: $tType,A: $tType] :
      ( unboun7993243217541854897norder(B)
     => ! [F2: fun(A,B),F3: filter(A),C2: B] :
          ( filterlim(A,B,F2,at_top(B),F3)
        <=> ! [Z7: B] :
              ( pp(aa(B,bool,ord_less(B,C2),Z7))
             => eventually(A,aa(B,fun(A,bool),aTP_Lamp_agj(fun(A,B),fun(B,fun(A,bool)),F2),Z7),F3) ) ) ) ).

% filterlim_at_top_gt
tff(fact_7693_tendsto__compose__eventually,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topolo4958980785337419405_space(B) )
     => ! [G: fun(A,B),M: B,L: A,F2: fun(C,A),F3: filter(C)] :
          ( filterlim(A,B,G,topolo7230453075368039082e_nhds(B,M),topolo174197925503356063within(A,L,top_top(set(A))))
         => ( filterlim(C,A,F2,topolo7230453075368039082e_nhds(A,L),F3)
           => ( eventually(C,aa(fun(C,A),fun(C,bool),aTP_Lamp_agk(A,fun(fun(C,A),fun(C,bool)),L),F2),F3)
             => filterlim(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_vz(fun(A,B),fun(fun(C,A),fun(C,B)),G),F2),topolo7230453075368039082e_nhds(B,M),F3) ) ) ) ) ).

% tendsto_compose_eventually
tff(fact_7694_LIM__compose__eventually,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topolo4958980785337419405_space(B)
        & topolo4958980785337419405_space(C) )
     => ! [F2: fun(A,B),B2: B,A2: A,G: fun(B,C),C2: C] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,B2),topolo174197925503356063within(A,A2,top_top(set(A))))
         => ( filterlim(B,C,G,topolo7230453075368039082e_nhds(C,C2),topolo174197925503356063within(B,B2,top_top(set(B))))
           => ( eventually(A,aa(B,fun(A,bool),aTP_Lamp_agl(fun(A,B),fun(B,fun(A,bool)),F2),B2),topolo174197925503356063within(A,A2,top_top(set(A))))
             => filterlim(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_agm(fun(A,B),fun(fun(B,C),fun(A,C)),F2),G),topolo7230453075368039082e_nhds(C,C2),topolo174197925503356063within(A,A2,top_top(set(A)))) ) ) ) ) ).

% LIM_compose_eventually
tff(fact_7695_filterlim__atI,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [F2: fun(B,A),C2: A,F3: filter(B)] :
          ( filterlim(B,A,F2,topolo7230453075368039082e_nhds(A,C2),F3)
         => ( eventually(B,aa(A,fun(B,bool),aTP_Lamp_agn(fun(B,A),fun(A,fun(B,bool)),F2),C2),F3)
           => filterlim(B,A,F2,topolo174197925503356063within(A,C2,top_top(set(A))),F3) ) ) ) ).

% filterlim_atI
tff(fact_7696_isCont__cong,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo4958980785337419405_space(B) )
     => ! [F2: fun(A,B),G: fun(A,B),X: A] :
          ( eventually(A,aa(fun(A,B),fun(A,bool),aTP_Lamp_ago(fun(A,B),fun(fun(A,B),fun(A,bool)),F2),G),topolo7230453075368039082e_nhds(A,X))
         => ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,X,top_top(set(A))),F2)
          <=> topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,X,top_top(set(A))),G) ) ) ) ).

% isCont_cong
tff(fact_7697_DERIV__cong__ev,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [X: A,Y: A,F2: fun(A,A),G: fun(A,A),U: A,V: A] :
          ( ( X = Y )
         => ( eventually(A,aa(fun(A,A),fun(A,bool),aTP_Lamp_afr(fun(A,A),fun(fun(A,A),fun(A,bool)),F2),G),topolo7230453075368039082e_nhds(A,X))
           => ( ( U = V )
             => ( has_field_derivative(A,F2,U,topolo174197925503356063within(A,X,top_top(set(A))))
              <=> has_field_derivative(A,G,V,topolo174197925503356063within(A,Y,top_top(set(A)))) ) ) ) ) ) ).

% DERIV_cong_ev
tff(fact_7698_filterlim__at__bot__lt,axiom,
    ! [B: $tType,A: $tType] :
      ( unboun7993243217541854897norder(B)
     => ! [F2: fun(A,B),F3: filter(A),C2: B] :
          ( filterlim(A,B,F2,at_bot(B),F3)
        <=> ! [Z7: B] :
              ( pp(aa(B,bool,ord_less(B,Z7),C2))
             => eventually(A,aa(B,fun(A,bool),aTP_Lamp_agp(fun(A,B),fun(B,fun(A,bool)),F2),Z7),F3) ) ) ) ).

% filterlim_at_bot_lt
tff(fact_7699_tendsto__upperbound,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [F2: fun(B,A),X: A,F3: filter(B),A2: A] :
          ( filterlim(B,A,F2,topolo7230453075368039082e_nhds(A,X),F3)
         => ( eventually(B,aa(A,fun(B,bool),aTP_Lamp_agq(fun(B,A),fun(A,fun(B,bool)),F2),A2),F3)
           => ( ( F3 != bot_bot(filter(B)) )
             => pp(aa(A,bool,ord_less_eq(A,X),A2)) ) ) ) ) ).

% tendsto_upperbound
tff(fact_7700_tendsto__lowerbound,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [F2: fun(B,A),X: A,F3: filter(B),A2: A] :
          ( filterlim(B,A,F2,topolo7230453075368039082e_nhds(A,X),F3)
         => ( eventually(B,aa(A,fun(B,bool),aTP_Lamp_agr(fun(B,A),fun(A,fun(B,bool)),F2),A2),F3)
           => ( ( F3 != bot_bot(filter(B)) )
             => pp(aa(A,bool,ord_less_eq(A,A2),X)) ) ) ) ) ).

% tendsto_lowerbound
tff(fact_7701_tendsto__le,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [F3: filter(B),F2: fun(B,A),X: A,G: fun(B,A),Y: A] :
          ( ( F3 != bot_bot(filter(B)) )
         => ( filterlim(B,A,F2,topolo7230453075368039082e_nhds(A,X),F3)
           => ( filterlim(B,A,G,topolo7230453075368039082e_nhds(A,Y),F3)
             => ( eventually(B,aa(fun(B,A),fun(B,bool),aTP_Lamp_ags(fun(B,A),fun(fun(B,A),fun(B,bool)),F2),G),F3)
               => pp(aa(A,bool,ord_less_eq(A,Y),X)) ) ) ) ) ) ).

% tendsto_le
tff(fact_7702_metric__tendsto__imp__tendsto,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( ( real_V7819770556892013058_space(B)
        & real_V7819770556892013058_space(A) )
     => ! [F2: fun(C,A),A2: A,F3: filter(C),G: fun(C,B),B2: B] :
          ( filterlim(C,A,F2,topolo7230453075368039082e_nhds(A,A2),F3)
         => ( eventually(C,aa(B,fun(C,bool),aa(fun(C,B),fun(B,fun(C,bool)),aa(A,fun(fun(C,B),fun(B,fun(C,bool))),aTP_Lamp_agt(fun(C,A),fun(A,fun(fun(C,B),fun(B,fun(C,bool)))),F2),A2),G),B2),F3)
           => filterlim(C,B,G,topolo7230453075368039082e_nhds(B,B2),F3) ) ) ) ).

% metric_tendsto_imp_tendsto
tff(fact_7703_filterlim__at__infinity__imp__filterlim__at__top,axiom,
    ! [A: $tType,F2: fun(A,real),F3: filter(A)] :
      ( filterlim(A,real,F2,at_infinity(real),F3)
     => ( eventually(A,aTP_Lamp_agu(fun(A,real),fun(A,bool),F2),F3)
       => filterlim(A,real,F2,at_top(real),F3) ) ) ).

% filterlim_at_infinity_imp_filterlim_at_top
tff(fact_7704_filterlim__at__infinity__imp__filterlim__at__bot,axiom,
    ! [A: $tType,F2: fun(A,real),F3: filter(A)] :
      ( filterlim(A,real,F2,at_infinity(real),F3)
     => ( eventually(A,aTP_Lamp_agv(fun(A,real),fun(A,bool),F2),F3)
       => filterlim(A,real,F2,at_bot(real),F3) ) ) ).

% filterlim_at_infinity_imp_filterlim_at_bot
tff(fact_7705_eventually__floor__eq,axiom,
    ! [B: $tType,A: $tType] :
      ( ( archim2362893244070406136eiling(B)
        & topolo2564578578187576103pology(B) )
     => ! [F2: fun(A,B),L: B,F3: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),F3)
         => ( ~ pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),L),ring_1_Ints(B)))
           => eventually(A,aa(B,fun(A,bool),aTP_Lamp_agw(fun(A,B),fun(B,fun(A,bool)),F2),L),F3) ) ) ) ).

% eventually_floor_eq
tff(fact_7706_eventually__ceiling__eq,axiom,
    ! [B: $tType,A: $tType] :
      ( ( archim2362893244070406136eiling(B)
        & topolo2564578578187576103pology(B) )
     => ! [F2: fun(A,B),L: B,F3: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),F3)
         => ( ~ pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),L),ring_1_Ints(B)))
           => eventually(A,aa(B,fun(A,bool),aTP_Lamp_agx(fun(A,B),fun(B,fun(A,bool)),F2),L),F3) ) ) ) ).

% eventually_ceiling_eq
tff(fact_7707_eventually__at__right__to__0,axiom,
    ! [P: fun(real,bool),A2: real] :
      ( eventually(real,P,topolo174197925503356063within(real,A2,set_ord_greaterThan(real,A2)))
    <=> eventually(real,aa(real,fun(real,bool),aTP_Lamp_agy(fun(real,bool),fun(real,fun(real,bool)),P),A2),topolo174197925503356063within(real,zero_zero(real),set_ord_greaterThan(real,zero_zero(real)))) ) ).

% eventually_at_right_to_0
tff(fact_7708_eventually__at__left__to__right,axiom,
    ! [P: fun(real,bool),A2: real] :
      ( eventually(real,P,topolo174197925503356063within(real,A2,set_ord_lessThan(real,A2)))
    <=> eventually(real,aTP_Lamp_agz(fun(real,bool),fun(real,bool),P),topolo174197925503356063within(real,aa(real,real,uminus_uminus(real),A2),set_ord_greaterThan(real,aa(real,real,uminus_uminus(real),A2)))) ) ).

% eventually_at_left_to_right
tff(fact_7709_continuous__arcosh__strong,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [F3: filter(A),F2: fun(A,real)] :
          ( topolo3448309680560233919inuous(A,real,F3,F2)
         => ( eventually(A,aTP_Lamp_aha(fun(A,real),fun(A,bool),F2),F3)
           => topolo3448309680560233919inuous(A,real,F3,aTP_Lamp_add(fun(A,real),fun(A,real),F2)) ) ) ) ).

% continuous_arcosh_strong
tff(fact_7710_eventually__at__right__real,axiom,
    ! [A2: real,B2: real] :
      ( pp(aa(real,bool,ord_less(real,A2),B2))
     => eventually(real,aa(real,fun(real,bool),aTP_Lamp_ahb(real,fun(real,fun(real,bool)),A2),B2),topolo174197925503356063within(real,A2,set_ord_greaterThan(real,A2))) ) ).

% eventually_at_right_real
tff(fact_7711_eventually__at__left__real,axiom,
    ! [B2: real,A2: real] :
      ( pp(aa(real,bool,ord_less(real,B2),A2))
     => eventually(real,aa(real,fun(real,bool),aTP_Lamp_ahb(real,fun(real,fun(real,bool)),B2),A2),topolo174197925503356063within(real,A2,set_ord_lessThan(real,A2))) ) ).

% eventually_at_left_real
tff(fact_7712_eventually__at__le,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [P: fun(A,bool),A2: A,S: set(A)] :
          ( eventually(A,P,topolo174197925503356063within(A,A2,S))
        <=> ? [D3: real] :
              ( pp(aa(real,bool,ord_less(real,zero_zero(real)),D3))
              & ! [X2: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X2),S))
                 => ( ( ( X2 != A2 )
                      & pp(aa(real,bool,ord_less_eq(real,real_V557655796197034286t_dist(A,X2,A2)),D3)) )
                   => pp(aa(A,bool,P,X2)) ) ) ) ) ) ).

% eventually_at_le
tff(fact_7713_eventually__at__infinity__pos,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [P2: fun(A,bool)] :
          ( eventually(A,P2,at_infinity(A))
        <=> ? [B7: real] :
              ( pp(aa(real,bool,ord_less(real,zero_zero(real)),B7))
              & ! [X2: A] :
                  ( pp(aa(real,bool,ord_less_eq(real,B7),real_V7770717601297561774m_norm(A,X2)))
                 => pp(aa(A,bool,P2,X2)) ) ) ) ) ).

% eventually_at_infinity_pos
tff(fact_7714_tendsto__compose__at,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(B)
        & topolo4958980785337419405_space(C) )
     => ! [F2: fun(A,B),Y: B,F3: filter(A),G: fun(B,C),Z: C] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,Y),F3)
         => ( filterlim(B,C,G,topolo7230453075368039082e_nhds(C,Z),topolo174197925503356063within(B,Y,top_top(set(B))))
           => ( eventually(A,aa(C,fun(A,bool),aa(fun(B,C),fun(C,fun(A,bool)),aa(B,fun(fun(B,C),fun(C,fun(A,bool))),aTP_Lamp_ahc(fun(A,B),fun(B,fun(fun(B,C),fun(C,fun(A,bool)))),F2),Y),G),Z),F3)
             => filterlim(A,C,comp(B,C,A,G,F2),topolo7230453075368039082e_nhds(C,Z),F3) ) ) ) ) ).

% tendsto_compose_at
tff(fact_7715_tendsto__imp__filterlim__at__left,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo2564578578187576103pology(B)
     => ! [F2: fun(A,B),L6: B,F3: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L6),F3)
         => ( eventually(A,aa(B,fun(A,bool),aTP_Lamp_ahd(fun(A,B),fun(B,fun(A,bool)),F2),L6),F3)
           => filterlim(A,B,F2,topolo174197925503356063within(B,L6,set_ord_lessThan(B,L6)),F3) ) ) ) ).

% tendsto_imp_filterlim_at_left
tff(fact_7716_tendsto__imp__filterlim__at__right,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo2564578578187576103pology(B)
     => ! [F2: fun(A,B),L6: B,F3: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L6),F3)
         => ( eventually(A,aa(B,fun(A,bool),aTP_Lamp_ahe(fun(A,B),fun(B,fun(A,bool)),F2),L6),F3)
           => filterlim(A,B,F2,topolo174197925503356063within(B,L6,set_ord_greaterThan(B,L6)),F3) ) ) ) ).

% tendsto_imp_filterlim_at_right
tff(fact_7717_tendstoD,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [F2: fun(B,A),L: A,F3: filter(B),E: real] :
          ( filterlim(B,A,F2,topolo7230453075368039082e_nhds(A,L),F3)
         => ( pp(aa(real,bool,ord_less(real,zero_zero(real)),E))
           => eventually(B,aa(real,fun(B,bool),aa(A,fun(real,fun(B,bool)),aTP_Lamp_ahf(fun(B,A),fun(A,fun(real,fun(B,bool))),F2),L),E),F3) ) ) ) ).

% tendstoD
tff(fact_7718_tendstoI,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [F2: fun(B,A),L: A,F3: filter(B)] :
          ( ! [E2: real] :
              ( pp(aa(real,bool,ord_less(real,zero_zero(real)),E2))
             => eventually(B,aa(real,fun(B,bool),aa(A,fun(real,fun(B,bool)),aTP_Lamp_ahf(fun(B,A),fun(A,fun(real,fun(B,bool))),F2),L),E2),F3) )
         => filterlim(B,A,F2,topolo7230453075368039082e_nhds(A,L),F3) ) ) ).

% tendstoI
tff(fact_7719_tendsto__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [F2: fun(B,A),L: A,F3: filter(B)] :
          ( filterlim(B,A,F2,topolo7230453075368039082e_nhds(A,L),F3)
        <=> ! [E4: real] :
              ( pp(aa(real,bool,ord_less(real,zero_zero(real)),E4))
             => eventually(B,aa(real,fun(B,bool),aa(A,fun(real,fun(B,bool)),aTP_Lamp_ahf(fun(B,A),fun(A,fun(real,fun(B,bool))),F2),L),E4),F3) ) ) ) ).

% tendsto_iff
tff(fact_7720_summable__comparison__test__ev,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [F2: fun(nat,A),G: fun(nat,real)] :
          ( eventually(nat,aa(fun(nat,real),fun(nat,bool),aTP_Lamp_ahg(fun(nat,A),fun(fun(nat,real),fun(nat,bool)),F2),G),at_top(nat))
         => ( summable(real,G)
           => summable(A,F2) ) ) ) ).

% summable_comparison_test_ev
tff(fact_7721_eventually__at__right__to__top,axiom,
    ! [P: fun(real,bool)] :
      ( eventually(real,P,topolo174197925503356063within(real,zero_zero(real),set_ord_greaterThan(real,zero_zero(real))))
    <=> eventually(real,aTP_Lamp_ahh(fun(real,bool),fun(real,bool),P),at_top(real)) ) ).

% eventually_at_right_to_top
tff(fact_7722_eventually__at__top__to__right,axiom,
    ! [P: fun(real,bool)] :
      ( eventually(real,P,at_top(real))
    <=> eventually(real,aTP_Lamp_ahh(fun(real,bool),fun(real,bool),P),topolo174197925503356063within(real,zero_zero(real),set_ord_greaterThan(real,zero_zero(real)))) ) ).

% eventually_at_top_to_right
tff(fact_7723_tendsto__arcosh__strong,axiom,
    ! [B: $tType,F2: fun(B,real),A2: real,F3: filter(B)] :
      ( filterlim(B,real,F2,topolo7230453075368039082e_nhds(real,A2),F3)
     => ( pp(aa(real,bool,ord_less_eq(real,one_one(real)),A2))
       => ( eventually(B,aTP_Lamp_ahi(fun(B,real),fun(B,bool),F2),F3)
         => filterlim(B,real,aTP_Lamp_xp(fun(B,real),fun(B,real),F2),topolo7230453075368039082e_nhds(real,aa(real,real,arcosh(real),A2)),F3) ) ) ) ).

% tendsto_arcosh_strong
tff(fact_7724_filterlim__at__top__at__left,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo1944317154257567458pology(A)
        & linorder(B) )
     => ! [Q: fun(A,bool),F2: fun(A,B),P: fun(B,bool),G: fun(B,A),A2: A] :
          ( ! [X3: A,Y3: A] :
              ( pp(aa(A,bool,Q,X3))
             => ( pp(aa(A,bool,Q,Y3))
               => ( pp(aa(A,bool,ord_less_eq(A,X3),Y3))
                 => pp(aa(B,bool,ord_less_eq(B,aa(A,B,F2,X3)),aa(A,B,F2,Y3))) ) ) )
         => ( ! [X3: B] :
                ( pp(aa(B,bool,P,X3))
               => ( aa(A,B,F2,aa(B,A,G,X3)) = X3 ) )
           => ( ! [X3: B] :
                  ( pp(aa(B,bool,P,X3))
                 => pp(aa(A,bool,Q,aa(B,A,G,X3))) )
             => ( eventually(A,Q,topolo174197925503356063within(A,A2,set_ord_lessThan(A,A2)))
               => ( ! [B3: A] :
                      ( pp(aa(A,bool,Q,B3))
                     => pp(aa(A,bool,ord_less(A,B3),A2)) )
                 => ( eventually(B,P,at_top(B))
                   => filterlim(A,B,F2,at_top(B),topolo174197925503356063within(A,A2,set_ord_lessThan(A,A2))) ) ) ) ) ) ) ) ).

% filterlim_at_top_at_left
tff(fact_7725_filterlim__at__bot__at__right,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo1944317154257567458pology(A)
        & linorder(B) )
     => ! [Q: fun(A,bool),F2: fun(A,B),P: fun(B,bool),G: fun(B,A),A2: A] :
          ( ! [X3: A,Y3: A] :
              ( pp(aa(A,bool,Q,X3))
             => ( pp(aa(A,bool,Q,Y3))
               => ( pp(aa(A,bool,ord_less_eq(A,X3),Y3))
                 => pp(aa(B,bool,ord_less_eq(B,aa(A,B,F2,X3)),aa(A,B,F2,Y3))) ) ) )
         => ( ! [X3: B] :
                ( pp(aa(B,bool,P,X3))
               => ( aa(A,B,F2,aa(B,A,G,X3)) = X3 ) )
           => ( ! [X3: B] :
                  ( pp(aa(B,bool,P,X3))
                 => pp(aa(A,bool,Q,aa(B,A,G,X3))) )
             => ( eventually(A,Q,topolo174197925503356063within(A,A2,set_ord_greaterThan(A,A2)))
               => ( ! [B3: A] :
                      ( pp(aa(A,bool,Q,B3))
                     => pp(aa(A,bool,ord_less(A,A2),B3)) )
                 => ( eventually(B,P,at_bot(B))
                   => filterlim(A,B,F2,at_bot(B),topolo174197925503356063within(A,A2,set_ord_greaterThan(A,A2))) ) ) ) ) ) ) ) ).

% filterlim_at_bot_at_right
tff(fact_7726_filterlim__at__withinI,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [F2: fun(B,A),C2: A,F3: filter(B),A3: set(A)] :
          ( filterlim(B,A,F2,topolo7230453075368039082e_nhds(A,C2),F3)
         => ( eventually(B,aa(set(A),fun(B,bool),aa(A,fun(set(A),fun(B,bool)),aTP_Lamp_ahj(fun(B,A),fun(A,fun(set(A),fun(B,bool))),F2),C2),A3),F3)
           => filterlim(B,A,F2,topolo174197925503356063within(A,C2,A3),F3) ) ) ) ).

% filterlim_at_withinI
tff(fact_7727_tendsto__0__le,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(C)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),F3: filter(A),G: fun(A,C),K6: real] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,zero_zero(B)),F3)
         => ( eventually(A,aa(real,fun(A,bool),aa(fun(A,C),fun(real,fun(A,bool)),aTP_Lamp_ahk(fun(A,B),fun(fun(A,C),fun(real,fun(A,bool))),F2),G),K6),F3)
           => filterlim(A,C,G,topolo7230453075368039082e_nhds(C,zero_zero(C)),F3) ) ) ) ).

% tendsto_0_le
tff(fact_7728_eventually__floor__less,axiom,
    ! [B: $tType,A: $tType] :
      ( ( archim2362893244070406136eiling(B)
        & topolo2564578578187576103pology(B) )
     => ! [F2: fun(A,B),L: B,F3: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),F3)
         => ( ~ pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),L),ring_1_Ints(B)))
           => eventually(A,aa(B,fun(A,bool),aTP_Lamp_ahl(fun(A,B),fun(B,fun(A,bool)),F2),L),F3) ) ) ) ).

% eventually_floor_less
tff(fact_7729_filterlim__at__infinity,axiom,
    ! [A: $tType,C: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [C2: real,F2: fun(C,A),F3: filter(C)] :
          ( pp(aa(real,bool,ord_less_eq(real,zero_zero(real)),C2))
         => ( filterlim(C,A,F2,at_infinity(A),F3)
          <=> ! [R5: real] :
                ( pp(aa(real,bool,ord_less(real,C2),R5))
               => eventually(C,aa(real,fun(C,bool),aTP_Lamp_ahm(fun(C,A),fun(real,fun(C,bool)),F2),R5),F3) ) ) ) ) ).

% filterlim_at_infinity
tff(fact_7730_eventually__less__ceiling,axiom,
    ! [B: $tType,A: $tType] :
      ( ( archim2362893244070406136eiling(B)
        & topolo2564578578187576103pology(B) )
     => ! [F2: fun(A,B),L: B,F3: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),F3)
         => ( ~ pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),L),ring_1_Ints(B)))
           => eventually(A,aa(B,fun(A,bool),aTP_Lamp_ahn(fun(A,B),fun(B,fun(A,bool)),F2),L),F3) ) ) ) ).

% eventually_less_ceiling
tff(fact_7731_tendsto__zero__powrI,axiom,
    ! [A: $tType,F2: fun(A,real),F3: filter(A),G: fun(A,real),B2: real] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,zero_zero(real)),F3)
     => ( filterlim(A,real,G,topolo7230453075368039082e_nhds(real,B2),F3)
       => ( eventually(A,aTP_Lamp_aho(fun(A,real),fun(A,bool),F2),F3)
         => ( pp(aa(real,bool,ord_less(real,zero_zero(real)),B2))
           => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_xm(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G),topolo7230453075368039082e_nhds(real,zero_zero(real)),F3) ) ) ) ) ).

% tendsto_zero_powrI
tff(fact_7732_tendsto__powr2,axiom,
    ! [A: $tType,F2: fun(A,real),A2: real,F3: filter(A),G: fun(A,real),B2: real] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,A2),F3)
     => ( filterlim(A,real,G,topolo7230453075368039082e_nhds(real,B2),F3)
       => ( eventually(A,aTP_Lamp_aho(fun(A,real),fun(A,bool),F2),F3)
         => ( pp(aa(real,bool,ord_less(real,zero_zero(real)),B2))
           => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_xm(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G),topolo7230453075368039082e_nhds(real,powr(real,A2,B2)),F3) ) ) ) ) ).

% tendsto_powr2
tff(fact_7733_tendsto__powr_H,axiom,
    ! [A: $tType,F2: fun(A,real),A2: real,F3: filter(A),G: fun(A,real),B2: real] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,A2),F3)
     => ( filterlim(A,real,G,topolo7230453075368039082e_nhds(real,B2),F3)
       => ( ( ( A2 != zero_zero(real) )
            | ( pp(aa(real,bool,ord_less(real,zero_zero(real)),B2))
              & eventually(A,aTP_Lamp_aho(fun(A,real),fun(A,bool),F2),F3) ) )
         => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_xm(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G),topolo7230453075368039082e_nhds(real,powr(real,A2,B2)),F3) ) ) ) ).

% tendsto_powr'
tff(fact_7734_LIM__at__top__divide,axiom,
    ! [A: $tType,F2: fun(A,real),A2: real,F3: filter(A),G: fun(A,real)] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,A2),F3)
     => ( pp(aa(real,bool,ord_less(real,zero_zero(real)),A2))
       => ( filterlim(A,real,G,topolo7230453075368039082e_nhds(real,zero_zero(real)),F3)
         => ( eventually(A,aTP_Lamp_agu(fun(A,real),fun(A,bool),G),F3)
           => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_ady(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G),at_top(real),F3) ) ) ) ) ).

% LIM_at_top_divide
tff(fact_7735_filterlim__at__top__iff__inverse__0,axiom,
    ! [A: $tType,F2: fun(A,real),F3: filter(A)] :
      ( eventually(A,aTP_Lamp_agu(fun(A,real),fun(A,bool),F2),F3)
     => ( filterlim(A,real,F2,at_top(real),F3)
      <=> filterlim(A,real,comp(real,real,A,inverse_inverse(real),F2),topolo7230453075368039082e_nhds(real,zero_zero(real)),F3) ) ) ).

% filterlim_at_top_iff_inverse_0
tff(fact_7736_filterlim__inverse__at__top,axiom,
    ! [A: $tType,F2: fun(A,real),F3: filter(A)] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,zero_zero(real)),F3)
     => ( eventually(A,aTP_Lamp_agu(fun(A,real),fun(A,bool),F2),F3)
       => filterlim(A,real,aTP_Lamp_adz(fun(A,real),fun(A,real),F2),at_top(real),F3) ) ) ).

% filterlim_inverse_at_top
tff(fact_7737_filterlim__inverse__at__top__iff,axiom,
    ! [A: $tType,F2: fun(A,real),F3: filter(A)] :
      ( eventually(A,aTP_Lamp_agu(fun(A,real),fun(A,bool),F2),F3)
     => ( filterlim(A,real,aTP_Lamp_adz(fun(A,real),fun(A,real),F2),at_top(real),F3)
      <=> filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,zero_zero(real)),F3) ) ) ).

% filterlim_inverse_at_top_iff
tff(fact_7738_filterlim__inverse__at__bot,axiom,
    ! [A: $tType,F2: fun(A,real),F3: filter(A)] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,zero_zero(real)),F3)
     => ( eventually(A,aTP_Lamp_agv(fun(A,real),fun(A,bool),F2),F3)
       => filterlim(A,real,aTP_Lamp_adz(fun(A,real),fun(A,real),F2),at_bot(real),F3) ) ) ).

% filterlim_inverse_at_bot
tff(fact_7739_lhopital__at__top__at__top,axiom,
    ! [F2: fun(real,real),A2: real,G: fun(real,real),F7: fun(real,real),G4: fun(real,real)] :
      ( filterlim(real,real,F2,at_top(real),topolo174197925503356063within(real,A2,top_top(set(real))))
     => ( filterlim(real,real,G,at_top(real),topolo174197925503356063within(real,A2,top_top(set(real))))
       => ( eventually(real,aa(fun(real,real),fun(real,bool),aTP_Lamp_aen(fun(real,real),fun(fun(real,real),fun(real,bool)),F2),F7),topolo174197925503356063within(real,A2,top_top(set(real))))
         => ( eventually(real,aa(fun(real,real),fun(real,bool),aTP_Lamp_aen(fun(real,real),fun(fun(real,real),fun(real,bool)),G),G4),topolo174197925503356063within(real,A2,top_top(set(real))))
           => ( filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_ahp(fun(real,real),fun(fun(real,real),fun(real,real)),F7),G4),at_top(real),topolo174197925503356063within(real,A2,top_top(set(real))))
             => filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_ahp(fun(real,real),fun(fun(real,real),fun(real,real)),F2),G),at_top(real),topolo174197925503356063within(real,A2,top_top(set(real)))) ) ) ) ) ) ).

% lhopital_at_top_at_top
tff(fact_7740_lhopital,axiom,
    ! [F2: fun(real,real),X: real,G: fun(real,real),G4: fun(real,real),F7: fun(real,real),F3: filter(real)] :
      ( filterlim(real,real,F2,topolo7230453075368039082e_nhds(real,zero_zero(real)),topolo174197925503356063within(real,X,top_top(set(real))))
     => ( filterlim(real,real,G,topolo7230453075368039082e_nhds(real,zero_zero(real)),topolo174197925503356063within(real,X,top_top(set(real))))
       => ( eventually(real,aTP_Lamp_aem(fun(real,real),fun(real,bool),G),topolo174197925503356063within(real,X,top_top(set(real))))
         => ( eventually(real,aTP_Lamp_aem(fun(real,real),fun(real,bool),G4),topolo174197925503356063within(real,X,top_top(set(real))))
           => ( eventually(real,aa(fun(real,real),fun(real,bool),aTP_Lamp_aen(fun(real,real),fun(fun(real,real),fun(real,bool)),F2),F7),topolo174197925503356063within(real,X,top_top(set(real))))
             => ( eventually(real,aa(fun(real,real),fun(real,bool),aTP_Lamp_aen(fun(real,real),fun(fun(real,real),fun(real,bool)),G),G4),topolo174197925503356063within(real,X,top_top(set(real))))
               => ( filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_aeo(fun(real,real),fun(fun(real,real),fun(real,real)),G4),F7),F3,topolo174197925503356063within(real,X,top_top(set(real))))
                 => filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_ahp(fun(real,real),fun(fun(real,real),fun(real,real)),F2),G),F3,topolo174197925503356063within(real,X,top_top(set(real)))) ) ) ) ) ) ) ) ).

% lhopital
tff(fact_7741_lhopital__right__at__top__at__top,axiom,
    ! [F2: fun(real,real),A2: real,G: fun(real,real),F7: fun(real,real),G4: fun(real,real)] :
      ( filterlim(real,real,F2,at_top(real),topolo174197925503356063within(real,A2,set_ord_greaterThan(real,A2)))
     => ( filterlim(real,real,G,at_top(real),topolo174197925503356063within(real,A2,set_ord_greaterThan(real,A2)))
       => ( eventually(real,aa(fun(real,real),fun(real,bool),aTP_Lamp_aen(fun(real,real),fun(fun(real,real),fun(real,bool)),F2),F7),topolo174197925503356063within(real,A2,set_ord_greaterThan(real,A2)))
         => ( eventually(real,aa(fun(real,real),fun(real,bool),aTP_Lamp_aen(fun(real,real),fun(fun(real,real),fun(real,bool)),G),G4),topolo174197925503356063within(real,A2,set_ord_greaterThan(real,A2)))
           => ( filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_ahp(fun(real,real),fun(fun(real,real),fun(real,real)),F7),G4),at_top(real),topolo174197925503356063within(real,A2,set_ord_greaterThan(real,A2)))
             => filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_ahp(fun(real,real),fun(fun(real,real),fun(real,real)),F2),G),at_top(real),topolo174197925503356063within(real,A2,set_ord_greaterThan(real,A2))) ) ) ) ) ) ).

% lhopital_right_at_top_at_top
tff(fact_7742_lhopital__at__top__at__bot,axiom,
    ! [F2: fun(real,real),A2: real,G: fun(real,real),F7: fun(real,real),G4: fun(real,real)] :
      ( filterlim(real,real,F2,at_top(real),topolo174197925503356063within(real,A2,top_top(set(real))))
     => ( filterlim(real,real,G,at_bot(real),topolo174197925503356063within(real,A2,top_top(set(real))))
       => ( eventually(real,aa(fun(real,real),fun(real,bool),aTP_Lamp_aen(fun(real,real),fun(fun(real,real),fun(real,bool)),F2),F7),topolo174197925503356063within(real,A2,top_top(set(real))))
         => ( eventually(real,aa(fun(real,real),fun(real,bool),aTP_Lamp_aen(fun(real,real),fun(fun(real,real),fun(real,bool)),G),G4),topolo174197925503356063within(real,A2,top_top(set(real))))
           => ( filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_ahp(fun(real,real),fun(fun(real,real),fun(real,real)),F7),G4),at_bot(real),topolo174197925503356063within(real,A2,top_top(set(real))))
             => filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_ahp(fun(real,real),fun(fun(real,real),fun(real,real)),F2),G),at_bot(real),topolo174197925503356063within(real,A2,top_top(set(real)))) ) ) ) ) ) ).

% lhopital_at_top_at_bot
tff(fact_7743_lhopital__left__at__top__at__top,axiom,
    ! [F2: fun(real,real),A2: real,G: fun(real,real),F7: fun(real,real),G4: fun(real,real)] :
      ( filterlim(real,real,F2,at_top(real),topolo174197925503356063within(real,A2,set_ord_lessThan(real,A2)))
     => ( filterlim(real,real,G,at_top(real),topolo174197925503356063within(real,A2,set_ord_lessThan(real,A2)))
       => ( eventually(real,aa(fun(real,real),fun(real,bool),aTP_Lamp_aen(fun(real,real),fun(fun(real,real),fun(real,bool)),F2),F7),topolo174197925503356063within(real,A2,set_ord_lessThan(real,A2)))
         => ( eventually(real,aa(fun(real,real),fun(real,bool),aTP_Lamp_aen(fun(real,real),fun(fun(real,real),fun(real,bool)),G),G4),topolo174197925503356063within(real,A2,set_ord_lessThan(real,A2)))
           => ( filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_ahp(fun(real,real),fun(fun(real,real),fun(real,real)),F7),G4),at_top(real),topolo174197925503356063within(real,A2,set_ord_lessThan(real,A2)))
             => filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_ahp(fun(real,real),fun(fun(real,real),fun(real,real)),F2),G),at_top(real),topolo174197925503356063within(real,A2,set_ord_lessThan(real,A2))) ) ) ) ) ) ).

% lhopital_left_at_top_at_top
tff(fact_7744_lhopital__at__top,axiom,
    ! [G: fun(real,real),X: real,G4: fun(real,real),F2: fun(real,real),F7: fun(real,real),Y: real] :
      ( filterlim(real,real,G,at_top(real),topolo174197925503356063within(real,X,top_top(set(real))))
     => ( eventually(real,aTP_Lamp_aem(fun(real,real),fun(real,bool),G4),topolo174197925503356063within(real,X,top_top(set(real))))
       => ( eventually(real,aa(fun(real,real),fun(real,bool),aTP_Lamp_aen(fun(real,real),fun(fun(real,real),fun(real,bool)),F2),F7),topolo174197925503356063within(real,X,top_top(set(real))))
         => ( eventually(real,aa(fun(real,real),fun(real,bool),aTP_Lamp_aen(fun(real,real),fun(fun(real,real),fun(real,bool)),G),G4),topolo174197925503356063within(real,X,top_top(set(real))))
           => ( filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_aeo(fun(real,real),fun(fun(real,real),fun(real,real)),G4),F7),topolo7230453075368039082e_nhds(real,Y),topolo174197925503356063within(real,X,top_top(set(real))))
             => filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_aeo(fun(real,real),fun(fun(real,real),fun(real,real)),G),F2),topolo7230453075368039082e_nhds(real,Y),topolo174197925503356063within(real,X,top_top(set(real)))) ) ) ) ) ) ).

% lhopital_at_top
tff(fact_7745_lhospital__at__top__at__top,axiom,
    ! [G: fun(real,real),G4: fun(real,real),F2: fun(real,real),F7: fun(real,real),X: real] :
      ( filterlim(real,real,G,at_top(real),at_top(real))
     => ( eventually(real,aTP_Lamp_aem(fun(real,real),fun(real,bool),G4),at_top(real))
       => ( eventually(real,aa(fun(real,real),fun(real,bool),aTP_Lamp_aen(fun(real,real),fun(fun(real,real),fun(real,bool)),F2),F7),at_top(real))
         => ( eventually(real,aa(fun(real,real),fun(real,bool),aTP_Lamp_aen(fun(real,real),fun(fun(real,real),fun(real,bool)),G),G4),at_top(real))
           => ( filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_aeo(fun(real,real),fun(fun(real,real),fun(real,real)),G4),F7),topolo7230453075368039082e_nhds(real,X),at_top(real))
             => filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_aeo(fun(real,real),fun(fun(real,real),fun(real,real)),G),F2),topolo7230453075368039082e_nhds(real,X),at_top(real)) ) ) ) ) ) ).

% lhospital_at_top_at_top
tff(fact_7746_lhopital__right,axiom,
    ! [F2: fun(real,real),X: real,G: fun(real,real),G4: fun(real,real),F7: fun(real,real),F3: filter(real)] :
      ( filterlim(real,real,F2,topolo7230453075368039082e_nhds(real,zero_zero(real)),topolo174197925503356063within(real,X,set_ord_greaterThan(real,X)))
     => ( filterlim(real,real,G,topolo7230453075368039082e_nhds(real,zero_zero(real)),topolo174197925503356063within(real,X,set_ord_greaterThan(real,X)))
       => ( eventually(real,aTP_Lamp_aem(fun(real,real),fun(real,bool),G),topolo174197925503356063within(real,X,set_ord_greaterThan(real,X)))
         => ( eventually(real,aTP_Lamp_aem(fun(real,real),fun(real,bool),G4),topolo174197925503356063within(real,X,set_ord_greaterThan(real,X)))
           => ( eventually(real,aa(fun(real,real),fun(real,bool),aTP_Lamp_aen(fun(real,real),fun(fun(real,real),fun(real,bool)),F2),F7),topolo174197925503356063within(real,X,set_ord_greaterThan(real,X)))
             => ( eventually(real,aa(fun(real,real),fun(real,bool),aTP_Lamp_aen(fun(real,real),fun(fun(real,real),fun(real,bool)),G),G4),topolo174197925503356063within(real,X,set_ord_greaterThan(real,X)))
               => ( filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_aeo(fun(real,real),fun(fun(real,real),fun(real,real)),G4),F7),F3,topolo174197925503356063within(real,X,set_ord_greaterThan(real,X)))
                 => filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_ahp(fun(real,real),fun(fun(real,real),fun(real,real)),F2),G),F3,topolo174197925503356063within(real,X,set_ord_greaterThan(real,X))) ) ) ) ) ) ) ) ).

% lhopital_right
tff(fact_7747_lhopital__right__0,axiom,
    ! [F0: fun(real,real),G0: fun(real,real),G4: fun(real,real),F7: fun(real,real),F3: filter(real)] :
      ( filterlim(real,real,F0,topolo7230453075368039082e_nhds(real,zero_zero(real)),topolo174197925503356063within(real,zero_zero(real),set_ord_greaterThan(real,zero_zero(real))))
     => ( filterlim(real,real,G0,topolo7230453075368039082e_nhds(real,zero_zero(real)),topolo174197925503356063within(real,zero_zero(real),set_ord_greaterThan(real,zero_zero(real))))
       => ( eventually(real,aTP_Lamp_aem(fun(real,real),fun(real,bool),G0),topolo174197925503356063within(real,zero_zero(real),set_ord_greaterThan(real,zero_zero(real))))
         => ( eventually(real,aTP_Lamp_aem(fun(real,real),fun(real,bool),G4),topolo174197925503356063within(real,zero_zero(real),set_ord_greaterThan(real,zero_zero(real))))
           => ( eventually(real,aa(fun(real,real),fun(real,bool),aTP_Lamp_aen(fun(real,real),fun(fun(real,real),fun(real,bool)),F0),F7),topolo174197925503356063within(real,zero_zero(real),set_ord_greaterThan(real,zero_zero(real))))
             => ( eventually(real,aa(fun(real,real),fun(real,bool),aTP_Lamp_aen(fun(real,real),fun(fun(real,real),fun(real,bool)),G0),G4),topolo174197925503356063within(real,zero_zero(real),set_ord_greaterThan(real,zero_zero(real))))
               => ( filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_aeo(fun(real,real),fun(fun(real,real),fun(real,real)),G4),F7),F3,topolo174197925503356063within(real,zero_zero(real),set_ord_greaterThan(real,zero_zero(real))))
                 => filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_ahp(fun(real,real),fun(fun(real,real),fun(real,real)),F0),G0),F3,topolo174197925503356063within(real,zero_zero(real),set_ord_greaterThan(real,zero_zero(real)))) ) ) ) ) ) ) ) ).

% lhopital_right_0
tff(fact_7748_lhopital__left,axiom,
    ! [F2: fun(real,real),X: real,G: fun(real,real),G4: fun(real,real),F7: fun(real,real),F3: filter(real)] :
      ( filterlim(real,real,F2,topolo7230453075368039082e_nhds(real,zero_zero(real)),topolo174197925503356063within(real,X,set_ord_lessThan(real,X)))
     => ( filterlim(real,real,G,topolo7230453075368039082e_nhds(real,zero_zero(real)),topolo174197925503356063within(real,X,set_ord_lessThan(real,X)))
       => ( eventually(real,aTP_Lamp_aem(fun(real,real),fun(real,bool),G),topolo174197925503356063within(real,X,set_ord_lessThan(real,X)))
         => ( eventually(real,aTP_Lamp_aem(fun(real,real),fun(real,bool),G4),topolo174197925503356063within(real,X,set_ord_lessThan(real,X)))
           => ( eventually(real,aa(fun(real,real),fun(real,bool),aTP_Lamp_aen(fun(real,real),fun(fun(real,real),fun(real,bool)),F2),F7),topolo174197925503356063within(real,X,set_ord_lessThan(real,X)))
             => ( eventually(real,aa(fun(real,real),fun(real,bool),aTP_Lamp_aen(fun(real,real),fun(fun(real,real),fun(real,bool)),G),G4),topolo174197925503356063within(real,X,set_ord_lessThan(real,X)))
               => ( filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_aeo(fun(real,real),fun(fun(real,real),fun(real,real)),G4),F7),F3,topolo174197925503356063within(real,X,set_ord_lessThan(real,X)))
                 => filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_ahp(fun(real,real),fun(fun(real,real),fun(real,real)),F2),G),F3,topolo174197925503356063within(real,X,set_ord_lessThan(real,X))) ) ) ) ) ) ) ) ).

% lhopital_left
tff(fact_7749_lhopital__right__at__top__at__bot,axiom,
    ! [F2: fun(real,real),A2: real,G: fun(real,real),F7: fun(real,real),G4: fun(real,real)] :
      ( filterlim(real,real,F2,at_top(real),topolo174197925503356063within(real,A2,set_ord_greaterThan(real,A2)))
     => ( filterlim(real,real,G,at_bot(real),topolo174197925503356063within(real,A2,set_ord_greaterThan(real,A2)))
       => ( eventually(real,aa(fun(real,real),fun(real,bool),aTP_Lamp_aen(fun(real,real),fun(fun(real,real),fun(real,bool)),F2),F7),topolo174197925503356063within(real,A2,set_ord_greaterThan(real,A2)))
         => ( eventually(real,aa(fun(real,real),fun(real,bool),aTP_Lamp_aen(fun(real,real),fun(fun(real,real),fun(real,bool)),G),G4),topolo174197925503356063within(real,A2,set_ord_greaterThan(real,A2)))
           => ( filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_ahp(fun(real,real),fun(fun(real,real),fun(real,real)),F7),G4),at_bot(real),topolo174197925503356063within(real,A2,set_ord_greaterThan(real,A2)))
             => filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_ahp(fun(real,real),fun(fun(real,real),fun(real,real)),F2),G),at_bot(real),topolo174197925503356063within(real,A2,set_ord_greaterThan(real,A2))) ) ) ) ) ) ).

% lhopital_right_at_top_at_bot
tff(fact_7750_lhopital__left__at__top__at__bot,axiom,
    ! [F2: fun(real,real),A2: real,G: fun(real,real),F7: fun(real,real),G4: fun(real,real)] :
      ( filterlim(real,real,F2,at_top(real),topolo174197925503356063within(real,A2,set_ord_lessThan(real,A2)))
     => ( filterlim(real,real,G,at_bot(real),topolo174197925503356063within(real,A2,set_ord_lessThan(real,A2)))
       => ( eventually(real,aa(fun(real,real),fun(real,bool),aTP_Lamp_aen(fun(real,real),fun(fun(real,real),fun(real,bool)),F2),F7),topolo174197925503356063within(real,A2,set_ord_lessThan(real,A2)))
         => ( eventually(real,aa(fun(real,real),fun(real,bool),aTP_Lamp_aen(fun(real,real),fun(fun(real,real),fun(real,bool)),G),G4),topolo174197925503356063within(real,A2,set_ord_lessThan(real,A2)))
           => ( filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_ahp(fun(real,real),fun(fun(real,real),fun(real,real)),F7),G4),at_bot(real),topolo174197925503356063within(real,A2,set_ord_lessThan(real,A2)))
             => filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_ahp(fun(real,real),fun(fun(real,real),fun(real,real)),F2),G),at_bot(real),topolo174197925503356063within(real,A2,set_ord_lessThan(real,A2))) ) ) ) ) ) ).

% lhopital_left_at_top_at_bot
tff(fact_7751_lhopital__right__at__top,axiom,
    ! [G: fun(real,real),X: real,G4: fun(real,real),F2: fun(real,real),F7: fun(real,real),Y: real] :
      ( filterlim(real,real,G,at_top(real),topolo174197925503356063within(real,X,set_ord_greaterThan(real,X)))
     => ( eventually(real,aTP_Lamp_aem(fun(real,real),fun(real,bool),G4),topolo174197925503356063within(real,X,set_ord_greaterThan(real,X)))
       => ( eventually(real,aa(fun(real,real),fun(real,bool),aTP_Lamp_aen(fun(real,real),fun(fun(real,real),fun(real,bool)),F2),F7),topolo174197925503356063within(real,X,set_ord_greaterThan(real,X)))
         => ( eventually(real,aa(fun(real,real),fun(real,bool),aTP_Lamp_aen(fun(real,real),fun(fun(real,real),fun(real,bool)),G),G4),topolo174197925503356063within(real,X,set_ord_greaterThan(real,X)))
           => ( filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_aeo(fun(real,real),fun(fun(real,real),fun(real,real)),G4),F7),topolo7230453075368039082e_nhds(real,Y),topolo174197925503356063within(real,X,set_ord_greaterThan(real,X)))
             => filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_aeo(fun(real,real),fun(fun(real,real),fun(real,real)),G),F2),topolo7230453075368039082e_nhds(real,Y),topolo174197925503356063within(real,X,set_ord_greaterThan(real,X))) ) ) ) ) ) ).

% lhopital_right_at_top
tff(fact_7752_lhopital__right__0__at__top,axiom,
    ! [G: fun(real,real),G4: fun(real,real),F2: fun(real,real),F7: fun(real,real),X: real] :
      ( filterlim(real,real,G,at_top(real),topolo174197925503356063within(real,zero_zero(real),set_ord_greaterThan(real,zero_zero(real))))
     => ( eventually(real,aTP_Lamp_aem(fun(real,real),fun(real,bool),G4),topolo174197925503356063within(real,zero_zero(real),set_ord_greaterThan(real,zero_zero(real))))
       => ( eventually(real,aa(fun(real,real),fun(real,bool),aTP_Lamp_aen(fun(real,real),fun(fun(real,real),fun(real,bool)),F2),F7),topolo174197925503356063within(real,zero_zero(real),set_ord_greaterThan(real,zero_zero(real))))
         => ( eventually(real,aa(fun(real,real),fun(real,bool),aTP_Lamp_aen(fun(real,real),fun(fun(real,real),fun(real,bool)),G),G4),topolo174197925503356063within(real,zero_zero(real),set_ord_greaterThan(real,zero_zero(real))))
           => ( filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_aeo(fun(real,real),fun(fun(real,real),fun(real,real)),G4),F7),topolo7230453075368039082e_nhds(real,X),topolo174197925503356063within(real,zero_zero(real),set_ord_greaterThan(real,zero_zero(real))))
             => filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_aeo(fun(real,real),fun(fun(real,real),fun(real,real)),G),F2),topolo7230453075368039082e_nhds(real,X),topolo174197925503356063within(real,zero_zero(real),set_ord_greaterThan(real,zero_zero(real)))) ) ) ) ) ) ).

% lhopital_right_0_at_top
tff(fact_7753_summable__Cauchy_H,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [F2: fun(nat,A),G: fun(nat,real)] :
          ( eventually(nat,aa(fun(nat,real),fun(nat,bool),aTP_Lamp_ahq(fun(nat,A),fun(fun(nat,real),fun(nat,bool)),F2),G),at_top(nat))
         => ( filterlim(nat,real,G,topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat))
           => summable(A,F2) ) ) ) ).

% summable_Cauchy'
tff(fact_7754_Bfun__metric__def,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V7819770556892013058_space(B)
     => ! [F2: fun(A,B),F3: filter(A)] :
          ( bfun(A,B,F2,F3)
        <=> ? [Y7: B,K7: real] :
              ( pp(aa(real,bool,ord_less(real,zero_zero(real)),K7))
              & eventually(A,aa(real,fun(A,bool),aa(B,fun(real,fun(A,bool)),aTP_Lamp_ahr(fun(A,B),fun(B,fun(real,fun(A,bool))),F2),Y7),K7),F3) ) ) ) ).

% Bfun_metric_def
tff(fact_7755_eventually__all__finite,axiom,
    ! [B: $tType,A: $tType] :
      ( finite_finite(B)
     => ! [P: fun(A,fun(B,bool)),Net: filter(A)] :
          ( ! [Y3: B] : eventually(A,aa(B,fun(A,bool),aTP_Lamp_ahs(fun(A,fun(B,bool)),fun(B,fun(A,bool)),P),Y3),Net)
         => eventually(A,aTP_Lamp_aht(fun(A,fun(B,bool)),fun(A,bool),P),Net) ) ) ).

% eventually_all_finite
tff(fact_7756_finite__set__of__finite__funs,axiom,
    ! [A: $tType,B: $tType,A3: set(A),B4: set(B),D: B] :
      ( pp(aa(set(A),bool,finite_finite2(A),A3))
     => ( pp(aa(set(B),bool,finite_finite2(B),B4))
       => pp(aa(set(fun(A,B)),bool,finite_finite2(fun(A,B)),collect(fun(A,B),aa(B,fun(fun(A,B),bool),aa(set(B),fun(B,fun(fun(A,B),bool)),aTP_Lamp_ahu(set(A),fun(set(B),fun(B,fun(fun(A,B),bool))),A3),B4),D)))) ) ) ).

% finite_set_of_finite_funs
tff(fact_7757_Bfun__const,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V7819770556892013058_space(B)
     => ! [C2: B,F3: filter(A)] : bfun(A,B,aTP_Lamp_ahv(B,fun(A,B),C2),F3) ) ).

% Bfun_const
tff(fact_7758_filterlim__int__sequentially,axiom,
    filterlim(nat,int,semiring_1_of_nat(int),at_top(int),at_top(nat)) ).

% filterlim_int_sequentially
tff(fact_7759_Bseq__minus__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X6: fun(nat,A)] :
          ( bfun(nat,A,aTP_Lamp_cb(fun(nat,A),fun(nat,A),X6),at_top(nat))
        <=> bfun(nat,A,X6,at_top(nat)) ) ) ).

% Bseq_minus_iff
tff(fact_7760_Bseq__Suc__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A)] :
          ( bfun(nat,A,aTP_Lamp_ce(fun(nat,A),fun(nat,A),F2),at_top(nat))
        <=> bfun(nat,A,F2,at_top(nat)) ) ) ).

% Bseq_Suc_iff
tff(fact_7761_Bseq__subseq,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A),G: fun(nat,nat)] :
          ( bfun(nat,A,F2,at_top(nat))
         => bfun(nat,A,aa(fun(nat,nat),fun(nat,A),aTP_Lamp_ahw(fun(nat,A),fun(fun(nat,nat),fun(nat,A)),F2),G),at_top(nat)) ) ) ).

% Bseq_subseq
tff(fact_7762_Bseq__add,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A),C2: A] :
          ( bfun(nat,A,F2,at_top(nat))
         => bfun(nat,A,aa(A,fun(nat,A),aTP_Lamp_ahx(fun(nat,A),fun(A,fun(nat,A)),F2),C2),at_top(nat)) ) ) ).

% Bseq_add
tff(fact_7763_Bseq__add__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A),C2: A] :
          ( bfun(nat,A,aa(A,fun(nat,A),aTP_Lamp_ahx(fun(nat,A),fun(A,fun(nat,A)),F2),C2),at_top(nat))
        <=> bfun(nat,A,F2,at_top(nat)) ) ) ).

% Bseq_add_iff
tff(fact_7764_Bseq__offset,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X6: fun(nat,A),K: nat] :
          ( bfun(nat,A,aa(nat,fun(nat,A),aTP_Lamp_ahy(fun(nat,A),fun(nat,fun(nat,A)),X6),K),at_top(nat))
         => bfun(nat,A,X6,at_top(nat)) ) ) ).

% Bseq_offset
tff(fact_7765_Bseq__ignore__initial__segment,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X6: fun(nat,A),K: nat] :
          ( bfun(nat,A,X6,at_top(nat))
         => bfun(nat,A,aa(nat,fun(nat,A),aTP_Lamp_ahy(fun(nat,A),fun(nat,fun(nat,A)),X6),K),at_top(nat)) ) ) ).

% Bseq_ignore_initial_segment
tff(fact_7766_Bseq__mult,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(nat,A),G: fun(nat,A)] :
          ( bfun(nat,A,F2,at_top(nat))
         => ( bfun(nat,A,G,at_top(nat))
           => bfun(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_ahz(fun(nat,A),fun(fun(nat,A),fun(nat,A)),F2),G),at_top(nat)) ) ) ) ).

% Bseq_mult
tff(fact_7767_BseqI_H,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X6: fun(nat,A),K6: real] :
          ( ! [N2: nat] : pp(aa(real,bool,ord_less_eq(real,real_V7770717601297561774m_norm(A,aa(nat,A,X6,N2))),K6))
         => bfun(nat,A,X6,at_top(nat)) ) ) ).

% BseqI'
tff(fact_7768_eventually__all__ge__at__top,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [P: fun(A,bool)] :
          ( eventually(A,P,at_top(A))
         => eventually(A,aTP_Lamp_aia(fun(A,bool),fun(A,bool),P),at_top(A)) ) ) ).

% eventually_all_ge_at_top
tff(fact_7769_Bseq__cmult__iff,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [C2: A,F2: fun(nat,A)] :
          ( ( C2 != zero_zero(A) )
         => ( bfun(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_bu(A,fun(fun(nat,A),fun(nat,A)),C2),F2),at_top(nat))
          <=> bfun(nat,A,F2,at_top(nat)) ) ) ) ).

% Bseq_cmult_iff
tff(fact_7770_Bseq__eventually__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(nat,A),G: fun(nat,B)] :
          ( eventually(nat,aa(fun(nat,B),fun(nat,bool),aTP_Lamp_aib(fun(nat,A),fun(fun(nat,B),fun(nat,bool)),F2),G),at_top(nat))
         => ( bfun(nat,B,G,at_top(nat))
           => bfun(nat,A,F2,at_top(nat)) ) ) ) ).

% Bseq_eventually_mono
tff(fact_7771_filterlim__int__of__nat__at__topD,axiom,
    ! [A: $tType,F2: fun(int,A),F3: filter(A)] :
      ( filterlim(nat,A,aTP_Lamp_aic(fun(int,A),fun(nat,A),F2),F3,at_top(nat))
     => filterlim(int,A,F2,F3,at_top(int)) ) ).

% filterlim_int_of_nat_at_topD
tff(fact_7772_BseqD,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X6: fun(nat,A)] :
          ( bfun(nat,A,X6,at_top(nat))
         => ? [K9: real] :
              ( pp(aa(real,bool,ord_less(real,zero_zero(real)),K9))
              & ! [N11: nat] : pp(aa(real,bool,ord_less_eq(real,real_V7770717601297561774m_norm(A,aa(nat,A,X6,N11))),K9)) ) ) ) ).

% BseqD
tff(fact_7773_BseqE,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X6: fun(nat,A)] :
          ( bfun(nat,A,X6,at_top(nat))
         => ~ ! [K9: real] :
                ( pp(aa(real,bool,ord_less(real,zero_zero(real)),K9))
               => ~ ! [N11: nat] : pp(aa(real,bool,ord_less_eq(real,real_V7770717601297561774m_norm(A,aa(nat,A,X6,N11))),K9)) ) ) ) ).

% BseqE
tff(fact_7774_BseqI,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [K6: real,X6: fun(nat,A)] :
          ( pp(aa(real,bool,ord_less(real,zero_zero(real)),K6))
         => ( ! [N2: nat] : pp(aa(real,bool,ord_less_eq(real,real_V7770717601297561774m_norm(A,aa(nat,A,X6,N2))),K6))
           => bfun(nat,A,X6,at_top(nat)) ) ) ) ).

% BseqI
tff(fact_7775_Bseq__def,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X6: fun(nat,A)] :
          ( bfun(nat,A,X6,at_top(nat))
        <=> ? [K7: real] :
              ( pp(aa(real,bool,ord_less(real,zero_zero(real)),K7))
              & ! [N5: nat] : pp(aa(real,bool,ord_less_eq(real,real_V7770717601297561774m_norm(A,aa(nat,A,X6,N5))),K7)) ) ) ) ).

% Bseq_def
tff(fact_7776_Bseq__iff1a,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X6: fun(nat,A)] :
          ( bfun(nat,A,X6,at_top(nat))
        <=> ? [N8: nat] :
            ! [N5: nat] : pp(aa(real,bool,ord_less(real,real_V7770717601297561774m_norm(A,aa(nat,A,X6,N5))),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,N8)))) ) ) ).

% Bseq_iff1a
tff(fact_7777_Bseq__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X6: fun(nat,A)] :
          ( bfun(nat,A,X6,at_top(nat))
        <=> ? [N8: nat] :
            ! [N5: nat] : pp(aa(real,bool,ord_less_eq(real,real_V7770717601297561774m_norm(A,aa(nat,A,X6,N5))),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,N8)))) ) ) ).

% Bseq_iff
tff(fact_7778_Bseq__realpow,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,ord_less_eq(real,zero_zero(real)),X))
     => ( pp(aa(real,bool,ord_less_eq(real,X),one_one(real)))
       => bfun(nat,real,aa(real,fun(nat,real),power_power(real),X),at_top(nat)) ) ) ).

% Bseq_realpow
tff(fact_7779_BfunI,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F2: fun(A,B),K6: real,F3: filter(A)] :
          ( eventually(A,aa(real,fun(A,bool),aTP_Lamp_aid(fun(A,B),fun(real,fun(A,bool)),F2),K6),F3)
         => bfun(A,B,F2,F3) ) ) ).

% BfunI
tff(fact_7780_Bseq__iff3,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X6: fun(nat,A)] :
          ( bfun(nat,A,X6,at_top(nat))
        <=> ? [K2: real] :
              ( pp(aa(real,bool,ord_less(real,zero_zero(real)),K2))
              & ? [N8: nat] :
                ! [N5: nat] : pp(aa(real,bool,ord_less_eq(real,real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,X6,N5)),aa(A,A,uminus_uminus(A),aa(nat,A,X6,N8))))),K2)) ) ) ) ).

% Bseq_iff3
tff(fact_7781_Bseq__iff2,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X6: fun(nat,A)] :
          ( bfun(nat,A,X6,at_top(nat))
        <=> ? [K2: real] :
              ( pp(aa(real,bool,ord_less(real,zero_zero(real)),K2))
              & ? [X2: A] :
                ! [N5: nat] : pp(aa(real,bool,ord_less_eq(real,real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,X6,N5)),aa(A,A,uminus_uminus(A),X2)))),K2)) ) ) ) ).

% Bseq_iff2
tff(fact_7782_Bfun__inverse,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [F2: fun(B,A),A2: A,F3: filter(B)] :
          ( filterlim(B,A,F2,topolo7230453075368039082e_nhds(A,A2),F3)
         => ( ( A2 != zero_zero(A) )
           => bfun(B,A,aTP_Lamp_wj(fun(B,A),fun(B,A),F2),F3) ) ) ) ).

% Bfun_inverse
tff(fact_7783_BfunE,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F2: fun(A,B),F3: filter(A)] :
          ( bfun(A,B,F2,F3)
         => ~ ! [B8: real] :
                ( pp(aa(real,bool,ord_less(real,zero_zero(real)),B8))
               => ~ eventually(A,aa(real,fun(A,bool),aTP_Lamp_aid(fun(A,B),fun(real,fun(A,bool)),F2),B8),F3) ) ) ) ).

% BfunE
tff(fact_7784_Bfun__def,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F2: fun(A,B),F3: filter(A)] :
          ( bfun(A,B,F2,F3)
        <=> ? [K7: real] :
              ( pp(aa(real,bool,ord_less(real,zero_zero(real)),K7))
              & eventually(A,aa(real,fun(A,bool),aTP_Lamp_aid(fun(A,B),fun(real,fun(A,bool)),F2),K7),F3) ) ) ) ).

% Bfun_def
tff(fact_7785_summable__bounded__partials,axiom,
    ! [A: $tType] :
      ( ( real_V8037385150606011577_space(A)
        & real_V822414075346904944vector(A) )
     => ! [F2: fun(nat,A),G: fun(nat,real)] :
          ( eventually(nat,aa(fun(nat,real),fun(nat,bool),aTP_Lamp_aie(fun(nat,A),fun(fun(nat,real),fun(nat,bool)),F2),G),at_top(nat))
         => ( filterlim(nat,real,G,topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat))
           => summable(A,F2) ) ) ) ).

% summable_bounded_partials
tff(fact_7786_Greatest__def,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [P: fun(A,bool)] : order_Greatest(A,P) = the(A,aTP_Lamp_aif(fun(A,bool),fun(A,bool),P)) ) ).

% Greatest_def
tff(fact_7787_greaterThanAtMost__iff,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [I: A,L: A,U: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),I),set_or3652927894154168847AtMost(A,L,U)))
        <=> ( pp(aa(A,bool,ord_less(A,L),I))
            & pp(aa(A,bool,ord_less_eq(A,I),U)) ) ) ) ).

% greaterThanAtMost_iff
tff(fact_7788_greaterThanAtMost__empty,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [L: A,K: A] :
          ( pp(aa(A,bool,ord_less_eq(A,L),K))
         => ( set_or3652927894154168847AtMost(A,K,L) = bot_bot(set(A)) ) ) ) ).

% greaterThanAtMost_empty
tff(fact_7789_image__add__greaterThanAtMost,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [C2: A,A2: A,B2: A] : aa(set(A),set(A),image(A,A,aa(A,fun(A,A),plus_plus(A),C2)),set_or3652927894154168847AtMost(A,A2,B2)) = set_or3652927894154168847AtMost(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),B2)) ) ).

% image_add_greaterThanAtMost
tff(fact_7790_Ioc__subset__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A,C2: A,D: A] :
          ( pp(aa(set(A),bool,ord_less_eq(set(A),set_or3652927894154168847AtMost(A,A2,B2)),set_or3652927894154168847AtMost(A,C2,D)))
        <=> ( pp(aa(A,bool,ord_less_eq(A,B2),A2))
            | ( pp(aa(A,bool,ord_less_eq(A,C2),A2))
              & pp(aa(A,bool,ord_less_eq(A,B2),D)) ) ) ) ) ).

% Ioc_subset_iff
tff(fact_7791_Ioc__inj,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A,C2: A,D: A] :
          ( ( set_or3652927894154168847AtMost(A,A2,B2) = set_or3652927894154168847AtMost(A,C2,D) )
        <=> ( ( pp(aa(A,bool,ord_less_eq(A,B2),A2))
              & pp(aa(A,bool,ord_less_eq(A,D),C2)) )
            | ( ( A2 = C2 )
              & ( B2 = D ) ) ) ) ) ).

% Ioc_inj
tff(fact_7792_GreatestI__ex__nat,axiom,
    ! [P: fun(nat,bool),B2: nat] :
      ( ? [X_12: nat] : pp(aa(nat,bool,P,X_12))
     => ( ! [Y3: nat] :
            ( pp(aa(nat,bool,P,Y3))
           => pp(aa(nat,bool,ord_less_eq(nat,Y3),B2)) )
       => pp(aa(nat,bool,P,order_Greatest(nat,P))) ) ) ).

% GreatestI_ex_nat
tff(fact_7793_Greatest__le__nat,axiom,
    ! [P: fun(nat,bool),K: nat,B2: nat] :
      ( pp(aa(nat,bool,P,K))
     => ( ! [Y3: nat] :
            ( pp(aa(nat,bool,P,Y3))
           => pp(aa(nat,bool,ord_less_eq(nat,Y3),B2)) )
       => pp(aa(nat,bool,ord_less_eq(nat,K),order_Greatest(nat,P))) ) ) ).

% Greatest_le_nat
tff(fact_7794_GreatestI__nat,axiom,
    ! [P: fun(nat,bool),K: nat,B2: nat] :
      ( pp(aa(nat,bool,P,K))
     => ( ! [Y3: nat] :
            ( pp(aa(nat,bool,P,Y3))
           => pp(aa(nat,bool,ord_less_eq(nat,Y3),B2)) )
       => pp(aa(nat,bool,P,order_Greatest(nat,P))) ) ) ).

% GreatestI_nat
tff(fact_7795_open__left,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [S: set(A),X: A,Y: A] :
          ( topolo1002775350975398744n_open(A,S)
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),S))
           => ( pp(aa(A,bool,ord_less(A,Y),X))
             => ? [B3: A] :
                  ( pp(aa(A,bool,ord_less(A,B3),X))
                  & pp(aa(set(A),bool,ord_less_eq(set(A),set_or3652927894154168847AtMost(A,B3,X)),S)) ) ) ) ) ) ).

% open_left
tff(fact_7796_atLeastSucAtMost__greaterThanAtMost,axiom,
    ! [L: nat,U: nat] : set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,L),U) = set_or3652927894154168847AtMost(nat,L,U) ).

% atLeastSucAtMost_greaterThanAtMost
tff(fact_7797_greaterThanAtMost__upt,axiom,
    ! [N: nat,M: nat] : set_or3652927894154168847AtMost(nat,N,M) = aa(list(nat),set(nat),set2(nat),upt(aa(nat,nat,suc,N),aa(nat,nat,suc,M))) ).

% greaterThanAtMost_upt
tff(fact_7798_GreatestI2__order,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [P: fun(A,bool),X: A,Q: fun(A,bool)] :
          ( pp(aa(A,bool,P,X))
         => ( ! [Y3: A] :
                ( pp(aa(A,bool,P,Y3))
               => pp(aa(A,bool,ord_less_eq(A,Y3),X)) )
           => ( ! [X3: A] :
                  ( pp(aa(A,bool,P,X3))
                 => ( ! [Y4: A] :
                        ( pp(aa(A,bool,P,Y4))
                       => pp(aa(A,bool,ord_less_eq(A,Y4),X3)) )
                   => pp(aa(A,bool,Q,X3)) ) )
             => pp(aa(A,bool,Q,order_Greatest(A,P))) ) ) ) ) ).

% GreatestI2_order
tff(fact_7799_Greatest__equality,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [P: fun(A,bool),X: A] :
          ( pp(aa(A,bool,P,X))
         => ( ! [Y3: A] :
                ( pp(aa(A,bool,P,Y3))
               => pp(aa(A,bool,ord_less_eq(A,Y3),X)) )
           => ( order_Greatest(A,P) = X ) ) ) ) ).

% Greatest_equality
tff(fact_7800_sum_Ohead,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [M: nat,N: nat,G: fun(nat,A)] :
          ( pp(aa(nat,bool,ord_less_eq(nat,M),N))
         => ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or1337092689740270186AtMost(nat,M,N)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G,M)),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or3652927894154168847AtMost(nat,M,N))) ) ) ) ).

% sum.head
tff(fact_7801_prod_Ohead,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [M: nat,N: nat,G: fun(nat,A)] :
          ( pp(aa(nat,bool,ord_less_eq(nat,M),N))
         => ( groups7121269368397514597t_prod(nat,A,G,set_or1337092689740270186AtMost(nat,M,N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G,M)),groups7121269368397514597t_prod(nat,A,G,set_or3652927894154168847AtMost(nat,M,N))) ) ) ) ).

% prod.head
tff(fact_7802_greaterThanAtMost__subseteq__atLeastAtMost__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A2: A,B2: A,C2: A,D: A] :
          ( pp(aa(set(A),bool,ord_less_eq(set(A),set_or3652927894154168847AtMost(A,A2,B2)),set_or1337092689740270186AtMost(A,C2,D)))
        <=> ( pp(aa(A,bool,ord_less(A,A2),B2))
           => ( pp(aa(A,bool,ord_less_eq(A,C2),A2))
              & pp(aa(A,bool,ord_less_eq(A,B2),D)) ) ) ) ) ).

% greaterThanAtMost_subseteq_atLeastAtMost_iff
tff(fact_7803_greaterThanAtMost__subseteq__atLeastLessThan__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A2: A,B2: A,C2: A,D: A] :
          ( pp(aa(set(A),bool,ord_less_eq(set(A),set_or3652927894154168847AtMost(A,A2,B2)),set_or7035219750837199246ssThan(A,C2,D)))
        <=> ( pp(aa(A,bool,ord_less(A,A2),B2))
           => ( pp(aa(A,bool,ord_less_eq(A,C2),A2))
              & pp(aa(A,bool,ord_less(A,B2),D)) ) ) ) ) ).

% greaterThanAtMost_subseteq_atLeastLessThan_iff
tff(fact_7804_greaterThanLessThan__subseteq__greaterThanAtMost__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A2: A,B2: A,C2: A,D: A] :
          ( pp(aa(set(A),bool,ord_less_eq(set(A),set_or5935395276787703475ssThan(A,A2,B2)),set_or3652927894154168847AtMost(A,C2,D)))
        <=> ( pp(aa(A,bool,ord_less(A,A2),B2))
           => ( pp(aa(A,bool,ord_less_eq(A,C2),A2))
              & pp(aa(A,bool,ord_less_eq(A,B2),D)) ) ) ) ) ).

% greaterThanLessThan_subseteq_greaterThanAtMost_iff
tff(fact_7805_sequentially__imp__eventually__at__right,axiom,
    ! [A: $tType] :
      ( ( topolo3112930676232923870pology(A)
        & topolo1944317154257567458pology(A) )
     => ! [A2: A,B2: A,P: fun(A,bool)] :
          ( pp(aa(A,bool,ord_less(A,A2),B2))
         => ( ! [F4: fun(nat,A)] :
                ( ! [N11: nat] : pp(aa(A,bool,ord_less(A,A2),aa(nat,A,F4,N11)))
               => ( ! [N11: nat] : pp(aa(A,bool,ord_less(A,aa(nat,A,F4,N11)),B2))
                 => ( order_antimono(nat,A,F4)
                   => ( filterlim(nat,A,F4,topolo7230453075368039082e_nhds(A,A2),at_top(nat))
                     => eventually(nat,aa(fun(nat,A),fun(nat,bool),aTP_Lamp_aig(fun(A,bool),fun(fun(nat,A),fun(nat,bool)),P),F4),at_top(nat)) ) ) ) )
           => eventually(A,P,topolo174197925503356063within(A,A2,set_ord_greaterThan(A,A2))) ) ) ) ).

% sequentially_imp_eventually_at_right
tff(fact_7806_map__filter__on__comp,axiom,
    ! [A: $tType,C: $tType,B: $tType,G: fun(B,A),Y6: set(B),X6: set(A),F3: filter(B),F2: fun(A,C)] :
      ( pp(aa(set(A),bool,ord_less_eq(set(A),aa(set(B),set(A),image(B,A,G),Y6)),X6))
     => ( eventually(B,aTP_Lamp_aih(set(B),fun(B,bool),Y6),F3)
       => ( map_filter_on(A,C,X6,F2,map_filter_on(B,A,Y6,G,F3)) = map_filter_on(B,C,Y6,comp(A,C,B,F2,G),F3) ) ) ) ).

% map_filter_on_comp
tff(fact_7807_decseq__const,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [K: A] : order_antimono(nat,A,aTP_Lamp_aii(A,fun(nat,A),K)) ) ).

% decseq_const
tff(fact_7808_decseq__SucD,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A3: fun(nat,A),I: nat] :
          ( order_antimono(nat,A,A3)
         => pp(aa(A,bool,ord_less_eq(A,aa(nat,A,A3,aa(nat,nat,suc,I))),aa(nat,A,A3,I))) ) ) ).

% decseq_SucD
tff(fact_7809_decseq__SucI,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X6: fun(nat,A)] :
          ( ! [N2: nat] : pp(aa(A,bool,ord_less_eq(A,aa(nat,A,X6,aa(nat,nat,suc,N2))),aa(nat,A,X6,N2)))
         => order_antimono(nat,A,X6) ) ) ).

% decseq_SucI
tff(fact_7810_decseq__Suc__iff,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [F2: fun(nat,A)] :
          ( order_antimono(nat,A,F2)
        <=> ! [N5: nat] : pp(aa(A,bool,ord_less_eq(A,aa(nat,A,F2,aa(nat,nat,suc,N5))),aa(nat,A,F2,N5))) ) ) ).

% decseq_Suc_iff
tff(fact_7811_decseqD,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [F2: fun(nat,A),I: nat,J: nat] :
          ( order_antimono(nat,A,F2)
         => ( pp(aa(nat,bool,ord_less_eq(nat,I),J))
           => pp(aa(A,bool,ord_less_eq(A,aa(nat,A,F2,J)),aa(nat,A,F2,I))) ) ) ) ).

% decseqD
tff(fact_7812_decseq__def,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X6: fun(nat,A)] :
          ( order_antimono(nat,A,X6)
        <=> ! [M6: nat,N5: nat] :
              ( pp(aa(nat,bool,ord_less_eq(nat,M6),N5))
             => pp(aa(A,bool,ord_less_eq(A,aa(nat,A,X6,N5)),aa(nat,A,X6,M6))) ) ) ) ).

% decseq_def
tff(fact_7813_antimonoD,axiom,
    ! [B: $tType,A: $tType] :
      ( ( order(A)
        & order(B) )
     => ! [F2: fun(A,B),X: A,Y: A] :
          ( order_antimono(A,B,F2)
         => ( pp(aa(A,bool,ord_less_eq(A,X),Y))
           => pp(aa(B,bool,ord_less_eq(B,aa(A,B,F2,Y)),aa(A,B,F2,X))) ) ) ) ).

% antimonoD
tff(fact_7814_antimonoE,axiom,
    ! [B: $tType,A: $tType] :
      ( ( order(A)
        & order(B) )
     => ! [F2: fun(A,B),X: A,Y: A] :
          ( order_antimono(A,B,F2)
         => ( pp(aa(A,bool,ord_less_eq(A,X),Y))
           => pp(aa(B,bool,ord_less_eq(B,aa(A,B,F2,Y)),aa(A,B,F2,X))) ) ) ) ).

% antimonoE
tff(fact_7815_antimonoI,axiom,
    ! [B: $tType,A: $tType] :
      ( ( order(A)
        & order(B) )
     => ! [F2: fun(A,B)] :
          ( ! [X3: A,Y3: A] :
              ( pp(aa(A,bool,ord_less_eq(A,X3),Y3))
             => pp(aa(B,bool,ord_less_eq(B,aa(A,B,F2,Y3)),aa(A,B,F2,X3))) )
         => order_antimono(A,B,F2) ) ) ).

% antimonoI
tff(fact_7816_antimono__def,axiom,
    ! [B: $tType,A: $tType] :
      ( ( order(A)
        & order(B) )
     => ! [F2: fun(A,B)] :
          ( order_antimono(A,B,F2)
        <=> ! [X2: A,Y7: A] :
              ( pp(aa(A,bool,ord_less_eq(A,X2),Y7))
             => pp(aa(B,bool,ord_less_eq(B,aa(A,B,F2,Y7)),aa(A,B,F2,X2))) ) ) ) ).

% antimono_def
tff(fact_7817_atLeastPlusOneAtMost__greaterThanAtMost__int,axiom,
    ! [L: int,U: int] : set_or1337092689740270186AtMost(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),L),one_one(int)),U) = set_or3652927894154168847AtMost(int,L,U) ).

% atLeastPlusOneAtMost_greaterThanAtMost_int
tff(fact_7818_eventually__map__filter__on,axiom,
    ! [B: $tType,A: $tType,X6: set(A),F3: filter(A),P: fun(B,bool),F2: fun(A,B)] :
      ( eventually(A,aTP_Lamp_a(set(A),fun(A,bool),X6),F3)
     => ( eventually(B,P,map_filter_on(A,B,X6,F2,F3))
      <=> eventually(A,aa(fun(A,B),fun(A,bool),aa(fun(B,bool),fun(fun(A,B),fun(A,bool)),aTP_Lamp_aij(set(A),fun(fun(B,bool),fun(fun(A,B),fun(A,bool))),X6),P),F2),F3) ) ) ).

% eventually_map_filter_on
tff(fact_7819_decseq__bounded,axiom,
    ! [X6: fun(nat,real),B4: real] :
      ( order_antimono(nat,real,X6)
     => ( ! [I2: nat] : pp(aa(real,bool,ord_less_eq(real,B4),aa(nat,real,X6,I2)))
       => bfun(nat,real,X6,at_top(nat)) ) ) ).

% decseq_bounded
tff(fact_7820_atLeastPlusOneAtMost__greaterThanAtMost__integer,axiom,
    ! [L: code_integer,U: code_integer] : set_or1337092689740270186AtMost(code_integer,aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),L),one_one(code_integer)),U) = set_or3652927894154168847AtMost(code_integer,L,U) ).

% atLeastPlusOneAtMost_greaterThanAtMost_integer
tff(fact_7821_word__atLeastAtMost__Suc__greaterThanAtMost,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [M: word(A),U: word(A)] :
          ( ( M != aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A))) )
         => ( set_or3652927894154168847AtMost(word(A),M,U) = set_or1337092689740270186AtMost(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),M),one_one(word(A))),U) ) ) ) ).

% word_atLeastAtMost_Suc_greaterThanAtMost
tff(fact_7822_word__range__minus__1_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A)] :
          ( ( A2 != zero_zero(word(A)) )
         => ( set_or3652927894154168847AtMost(word(A),aa(word(A),word(A),minus_minus(word(A),A2),one_one(word(A))),B2) = set_or1337092689740270186AtMost(word(A),A2,B2) ) ) ) ).

% word_range_minus_1'
tff(fact_7823_decseq__ge,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [X6: fun(nat,A),L6: A,N: nat] :
          ( order_antimono(nat,A,X6)
         => ( filterlim(nat,A,X6,topolo7230453075368039082e_nhds(A,L6),at_top(nat))
           => pp(aa(A,bool,ord_less_eq(A,L6),aa(nat,A,X6,N))) ) ) ) ).

% decseq_ge
tff(fact_7824_decseq__convergent,axiom,
    ! [X6: fun(nat,real),B4: real] :
      ( order_antimono(nat,real,X6)
     => ( ! [I2: nat] : pp(aa(real,bool,ord_less_eq(real,B4),aa(nat,real,X6,I2)))
       => ~ ! [L7: real] :
              ( filterlim(nat,real,X6,topolo7230453075368039082e_nhds(real,L7),at_top(nat))
             => ~ ! [I4: nat] : pp(aa(real,bool,ord_less_eq(real,L7),aa(nat,real,X6,I4))) ) ) ) ).

% decseq_convergent
tff(fact_7825_tendsto__at__right__sequentially,axiom,
    ! [C: $tType,B: $tType] :
      ( ( topolo3112930676232923870pology(B)
        & topolo1944317154257567458pology(B)
        & topolo4958980785337419405_space(C) )
     => ! [A2: B,B2: B,X6: fun(B,C),L6: C] :
          ( pp(aa(B,bool,ord_less(B,A2),B2))
         => ( ! [S6: fun(nat,B)] :
                ( ! [N11: nat] : pp(aa(B,bool,ord_less(B,A2),aa(nat,B,S6,N11)))
               => ( ! [N11: nat] : pp(aa(B,bool,ord_less(B,aa(nat,B,S6,N11)),B2))
                 => ( order_antimono(nat,B,S6)
                   => ( filterlim(nat,B,S6,topolo7230453075368039082e_nhds(B,A2),at_top(nat))
                     => filterlim(nat,C,aa(fun(nat,B),fun(nat,C),aTP_Lamp_aik(fun(B,C),fun(fun(nat,B),fun(nat,C)),X6),S6),topolo7230453075368039082e_nhds(C,L6),at_top(nat)) ) ) ) )
           => filterlim(B,C,X6,topolo7230453075368039082e_nhds(C,L6),topolo174197925503356063within(B,A2,set_ord_greaterThan(B,A2))) ) ) ) ).

% tendsto_at_right_sequentially
tff(fact_7826_cauchy__filter__metric,axiom,
    ! [A: $tType] :
      ( ( real_V768167426530841204y_dist(A)
        & topolo7287701948861334536_space(A) )
     => ! [F3: filter(A)] :
          ( topolo6773858410816713723filter(A,F3)
        <=> ! [E4: real] :
              ( pp(aa(real,bool,ord_less(real,zero_zero(real)),E4))
             => ? [P5: fun(A,bool)] :
                  ( eventually(A,P5,F3)
                  & ! [X2: A,Y7: A] :
                      ( ( pp(aa(A,bool,P5,X2))
                        & pp(aa(A,bool,P5,Y7)) )
                     => pp(aa(real,bool,ord_less(real,real_V557655796197034286t_dist(A,X2,Y7)),E4)) ) ) ) ) ) ).

% cauchy_filter_metric
tff(fact_7827_GMVT,axiom,
    ! [A2: real,B2: real,F2: fun(real,real),G: fun(real,real)] :
      ( pp(aa(real,bool,ord_less(real,A2),B2))
     => ( ! [X3: real] :
            ( ( pp(aa(real,bool,ord_less_eq(real,A2),X3))
              & pp(aa(real,bool,ord_less_eq(real,X3),B2)) )
           => topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,X3,top_top(set(real))),F2) )
       => ( ! [X3: real] :
              ( ( pp(aa(real,bool,ord_less(real,A2),X3))
                & pp(aa(real,bool,ord_less(real,X3),B2)) )
             => differentiable(real,real,F2,topolo174197925503356063within(real,X3,top_top(set(real)))) )
         => ( ! [X3: real] :
                ( ( pp(aa(real,bool,ord_less_eq(real,A2),X3))
                  & pp(aa(real,bool,ord_less_eq(real,X3),B2)) )
               => topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,X3,top_top(set(real))),G) )
           => ( ! [X3: real] :
                  ( ( pp(aa(real,bool,ord_less(real,A2),X3))
                    & pp(aa(real,bool,ord_less(real,X3),B2)) )
                 => differentiable(real,real,G,topolo174197925503356063within(real,X3,top_top(set(real)))) )
             => ? [G_c: real,F_c: real,C3: real] :
                  ( has_field_derivative(real,G,G_c,topolo174197925503356063within(real,C3,top_top(set(real))))
                  & has_field_derivative(real,F2,F_c,topolo174197925503356063within(real,C3,top_top(set(real))))
                  & pp(aa(real,bool,ord_less(real,A2),C3))
                  & pp(aa(real,bool,ord_less(real,C3),B2))
                  & ( aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,minus_minus(real,aa(real,real,F2,B2)),aa(real,real,F2,A2))),G_c) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,minus_minus(real,aa(real,real,G,B2)),aa(real,real,G,A2))),F_c) ) ) ) ) ) ) ) ).

% GMVT
tff(fact_7828_differentiable__cmult__left__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V3459762299906320749_field(A) )
     => ! [C2: A,Q2: fun(B,A),T: B] :
          ( differentiable(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_ail(A,fun(fun(B,A),fun(B,A)),C2),Q2),topolo174197925503356063within(B,T,top_top(set(B))))
        <=> ( ( C2 = zero_zero(A) )
            | differentiable(B,A,Q2,topolo174197925503356063within(B,T,top_top(set(B)))) ) ) ) ).

% differentiable_cmult_left_iff
tff(fact_7829_differentiable__cmult__right__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V3459762299906320749_field(A) )
     => ! [Q2: fun(B,A),C2: A,T: B] :
          ( differentiable(B,A,aa(A,fun(B,A),aTP_Lamp_aim(fun(B,A),fun(A,fun(B,A)),Q2),C2),topolo174197925503356063within(B,T,top_top(set(B))))
        <=> ( ( C2 = zero_zero(A) )
            | differentiable(B,A,Q2,topolo174197925503356063within(B,T,top_top(set(B)))) ) ) ) ).

% differentiable_cmult_right_iff
tff(fact_7830_differentiable__within__subset,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),X: A,S3: set(A),T: set(A)] :
          ( differentiable(A,B,F2,topolo174197925503356063within(A,X,S3))
         => ( pp(aa(set(A),bool,ord_less_eq(set(A),T),S3))
           => differentiable(A,B,F2,topolo174197925503356063within(A,X,T)) ) ) ) ).

% differentiable_within_subset
tff(fact_7831_differentiable__add,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),F3: filter(A),G: fun(A,B)] :
          ( differentiable(A,B,F2,F3)
         => ( differentiable(A,B,G,F3)
           => differentiable(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_tf(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G),F3) ) ) ) ).

% differentiable_add
tff(fact_7832_differentiable__const,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [A2: B,F3: filter(A)] : differentiable(A,B,aTP_Lamp_tg(B,fun(A,B),A2),F3) ) ).

% differentiable_const
tff(fact_7833_differentiable__ident,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F3: filter(A)] : differentiable(A,A,aTP_Lamp_ti(A,A),F3) ) ).

% differentiable_ident
tff(fact_7834_differentiable__diff,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),F3: filter(A),G: fun(A,B)] :
          ( differentiable(A,B,F2,F3)
         => ( differentiable(A,B,G,F3)
           => differentiable(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_tm(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G),F3) ) ) ) ).

% differentiable_diff
tff(fact_7835_differentiable__minus,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),F3: filter(A)] :
          ( differentiable(A,B,F2,F3)
         => differentiable(A,B,aTP_Lamp_tn(fun(A,B),fun(A,B),F2),F3) ) ) ).

% differentiable_minus
tff(fact_7836_differentiable__compose,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( ( real_V822414075346904944vector(C)
        & real_V822414075346904944vector(B)
        & real_V822414075346904944vector(A) )
     => ! [F2: fun(A,B),G: fun(C,A),X: C,S3: set(C)] :
          ( differentiable(A,B,F2,topolo174197925503356063within(A,aa(C,A,G,X),top_top(set(A))))
         => ( differentiable(C,A,G,topolo174197925503356063within(C,X,S3))
           => differentiable(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_tq(fun(A,B),fun(fun(C,A),fun(C,B)),F2),G),topolo174197925503356063within(C,X,S3)) ) ) ) ).

% differentiable_compose
tff(fact_7837_differentiable__in__compose,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( ( real_V822414075346904944vector(C)
        & real_V822414075346904944vector(B)
        & real_V822414075346904944vector(A) )
     => ! [F2: fun(A,B),G: fun(C,A),X: C,S3: set(C)] :
          ( differentiable(A,B,F2,topolo174197925503356063within(A,aa(C,A,G,X),aa(set(C),set(A),image(C,A,G),S3)))
         => ( differentiable(C,A,G,topolo174197925503356063within(C,X,S3))
           => differentiable(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_tq(fun(A,B),fun(fun(C,A),fun(C,B)),F2),G),topolo174197925503356063within(C,X,S3)) ) ) ) ).

% differentiable_in_compose
tff(fact_7838_differentiable__mult,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V4412858255891104859lgebra(B) )
     => ! [F2: fun(A,B),X: A,S3: set(A),G: fun(A,B)] :
          ( differentiable(A,B,F2,topolo174197925503356063within(A,X,S3))
         => ( differentiable(A,B,G,topolo174197925503356063within(A,X,S3))
           => differentiable(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_ain(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G),topolo174197925503356063within(A,X,S3)) ) ) ) ).

% differentiable_mult
tff(fact_7839_differentiable__scaleR,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,real),X: A,S3: set(A),G: fun(A,B)] :
          ( differentiable(A,real,F2,topolo174197925503356063within(A,X,S3))
         => ( differentiable(A,B,G,topolo174197925503356063within(A,X,S3))
           => differentiable(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_aio(fun(A,real),fun(fun(A,B),fun(A,B)),F2),G),topolo174197925503356063within(A,X,S3)) ) ) ) ).

% differentiable_scaleR
tff(fact_7840_differentiable__power,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V3459762299906320749_field(B) )
     => ! [F2: fun(A,B),X: A,S3: set(A),N: nat] :
          ( differentiable(A,B,F2,topolo174197925503356063within(A,X,S3))
         => differentiable(A,B,aa(nat,fun(A,B),aTP_Lamp_uh(fun(A,B),fun(nat,fun(A,B)),F2),N),topolo174197925503356063within(A,X,S3)) ) ) ).

% differentiable_power
tff(fact_7841_differentiable__sum,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V822414075346904944vector(C) )
     => ! [S3: set(A),F2: fun(A,fun(B,C)),Net: filter(B)] :
          ( pp(aa(set(A),bool,finite_finite2(A),S3))
         => ( ! [X3: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),S3))
               => differentiable(B,C,aa(A,fun(B,C),F2,X3),Net) )
           => differentiable(B,C,aa(fun(A,fun(B,C)),fun(B,C),aTP_Lamp_tc(set(A),fun(fun(A,fun(B,C)),fun(B,C)),S3),F2),Net) ) ) ) ).

% differentiable_sum
tff(fact_7842_differentiable__divide,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V3459762299906320749_field(B) )
     => ! [F2: fun(A,B),X: A,S3: set(A),G: fun(A,B)] :
          ( differentiable(A,B,F2,topolo174197925503356063within(A,X,S3))
         => ( differentiable(A,B,G,topolo174197925503356063within(A,X,S3))
           => ( ( aa(A,B,G,X) != zero_zero(B) )
             => differentiable(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_aip(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G),topolo174197925503356063within(A,X,S3)) ) ) ) ) ).

% differentiable_divide
tff(fact_7843_differentiable__inverse,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V3459762299906320749_field(B) )
     => ! [F2: fun(A,B),X: A,S3: set(A)] :
          ( differentiable(A,B,F2,topolo174197925503356063within(A,X,S3))
         => ( ( aa(A,B,F2,X) != zero_zero(B) )
           => differentiable(A,B,aTP_Lamp_aiq(fun(A,B),fun(A,B),F2),topolo174197925503356063within(A,X,S3)) ) ) ) ).

% differentiable_inverse
tff(fact_7844_MVT,axiom,
    ! [A2: real,B2: real,F2: fun(real,real)] :
      ( pp(aa(real,bool,ord_less(real,A2),B2))
     => ( topolo81223032696312382ous_on(real,real,set_or1337092689740270186AtMost(real,A2,B2),F2)
       => ( ! [X3: real] :
              ( pp(aa(real,bool,ord_less(real,A2),X3))
             => ( pp(aa(real,bool,ord_less(real,X3),B2))
               => differentiable(real,real,F2,topolo174197925503356063within(real,X3,top_top(set(real)))) ) )
         => ? [L3: real,Z2: real] :
              ( pp(aa(real,bool,ord_less(real,A2),Z2))
              & pp(aa(real,bool,ord_less(real,Z2),B2))
              & has_field_derivative(real,F2,L3,topolo174197925503356063within(real,Z2,top_top(set(real))))
              & ( aa(real,real,minus_minus(real,aa(real,real,F2,B2)),aa(real,real,F2,A2)) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,minus_minus(real,B2),A2)),L3) ) ) ) ) ) ).

% MVT
tff(fact_7845_interval__cases,axiom,
    ! [A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [S: set(A)] :
          ( ! [A4: A,B3: A,X3: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A4),S))
             => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),B3),S))
               => ( pp(aa(A,bool,ord_less_eq(A,A4),X3))
                 => ( pp(aa(A,bool,ord_less_eq(A,X3),B3))
                   => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),S)) ) ) ) )
         => ? [A4: A,B3: A] :
              ( ( S = bot_bot(set(A)) )
              | ( S = top_top(set(A)) )
              | ( S = set_ord_lessThan(A,B3) )
              | ( S = set_ord_atMost(A,B3) )
              | ( S = set_ord_greaterThan(A,A4) )
              | ( S = set_ord_atLeast(A,A4) )
              | ( S = set_or5935395276787703475ssThan(A,A4,B3) )
              | ( S = set_or3652927894154168847AtMost(A,A4,B3) )
              | ( S = set_or7035219750837199246ssThan(A,A4,B3) )
              | ( S = set_or1337092689740270186AtMost(A,A4,B3) ) ) ) ) ).

% interval_cases
tff(fact_7846_atLeast__0,axiom,
    set_ord_atLeast(nat,zero_zero(nat)) = top_top(set(nat)) ).

% atLeast_0
tff(fact_7847_atLeast__iff,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [I: A,K: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),I),set_ord_atLeast(A,K)))
        <=> pp(aa(A,bool,ord_less_eq(A,K),I)) ) ) ).

% atLeast_iff
tff(fact_7848_atLeast__subset__iff,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [X: A,Y: A] :
          ( pp(aa(set(A),bool,ord_less_eq(set(A),set_ord_atLeast(A,X)),set_ord_atLeast(A,Y)))
        <=> pp(aa(A,bool,ord_less_eq(A,Y),X)) ) ) ).

% atLeast_subset_iff
tff(fact_7849_image__add__atLeast,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [K: A,I: A] : aa(set(A),set(A),image(A,A,aa(A,fun(A,A),plus_plus(A),K)),set_ord_atLeast(A,I)) = set_ord_atLeast(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),K),I)) ) ).

% image_add_atLeast
tff(fact_7850_Icc__subset__Ici__iff,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [L: A,H: A,L2: A] :
          ( pp(aa(set(A),bool,ord_less_eq(set(A),set_or1337092689740270186AtMost(A,L,H)),set_ord_atLeast(A,L2)))
        <=> ( ~ pp(aa(A,bool,ord_less_eq(A,L),H))
            | pp(aa(A,bool,ord_less_eq(A,L2),L)) ) ) ) ).

% Icc_subset_Ici_iff
tff(fact_7851_continuous__on__tendsto__compose,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( ( topolo4958980785337419405_space(B)
        & topolo4958980785337419405_space(A) )
     => ! [S3: set(A),F2: fun(A,B),G: fun(C,A),L: A,F3: filter(C)] :
          ( topolo81223032696312382ous_on(A,B,S3,F2)
         => ( filterlim(C,A,G,topolo7230453075368039082e_nhds(A,L),F3)
           => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),L),S3))
             => ( eventually(C,aa(fun(C,A),fun(C,bool),aTP_Lamp_air(set(A),fun(fun(C,A),fun(C,bool)),S3),G),F3)
               => filterlim(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_vz(fun(A,B),fun(fun(C,A),fun(C,B)),F2),G),topolo7230453075368039082e_nhds(B,aa(A,B,F2,L)),F3) ) ) ) ) ) ).

% continuous_on_tendsto_compose
tff(fact_7852_continuous__on__tanh,axiom,
    ! [A: $tType,C: $tType] :
      ( ( topolo4958980785337419405_space(C)
        & real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [A3: set(C),F2: fun(C,A)] :
          ( topolo81223032696312382ous_on(C,A,A3,F2)
         => ( ! [X3: C] :
                ( pp(aa(set(C),bool,aa(C,fun(set(C),bool),member(C),X3),A3))
               => ( cosh(A,aa(C,A,F2,X3)) != zero_zero(A) ) )
           => topolo81223032696312382ous_on(C,A,A3,aTP_Lamp_ais(fun(C,A),fun(C,A),F2)) ) ) ) ).

% continuous_on_tanh
tff(fact_7853_atLeast__Suc__greaterThan,axiom,
    ! [K: nat] : set_ord_atLeast(nat,aa(nat,nat,suc,K)) = set_ord_greaterThan(nat,K) ).

% atLeast_Suc_greaterThan
tff(fact_7854_continuous__on__tan,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [S3: set(A),F2: fun(A,A)] :
          ( topolo81223032696312382ous_on(A,A,S3,F2)
         => ( ! [X3: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),S3))
               => ( cos(A,aa(A,A,F2,X3)) != zero_zero(A) ) )
           => topolo81223032696312382ous_on(A,A,S3,aTP_Lamp_xl(fun(A,A),fun(A,A),F2)) ) ) ) ).

% continuous_on_tan
tff(fact_7855_bounded__linear_Ocontinuous__on,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ( topolo4958980785337419405_space(C)
        & real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),S3: set(C),G: fun(C,A)] :
          ( real_V3181309239436604168linear(A,B,F2)
         => ( topolo81223032696312382ous_on(C,A,S3,G)
           => topolo81223032696312382ous_on(C,B,S3,aa(fun(C,A),fun(C,B),aTP_Lamp_ait(fun(A,B),fun(fun(C,A),fun(C,B)),F2),G)) ) ) ) ).

% bounded_linear.continuous_on
tff(fact_7856_open__Collect__less,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topolo1944317154257567458pology(B) )
     => ! [F2: fun(A,B),G: fun(A,B)] :
          ( topolo81223032696312382ous_on(A,B,top_top(set(A)),F2)
         => ( topolo81223032696312382ous_on(A,B,top_top(set(A)),G)
           => topolo1002775350975398744n_open(A,collect(A,aa(fun(A,B),fun(A,bool),aTP_Lamp_aiu(fun(A,B),fun(fun(A,B),fun(A,bool)),F2),G))) ) ) ) ).

% open_Collect_less
tff(fact_7857_open__Collect__neq,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topological_t2_space(B) )
     => ! [F2: fun(A,B),G: fun(A,B)] :
          ( topolo81223032696312382ous_on(A,B,top_top(set(A)),F2)
         => ( topolo81223032696312382ous_on(A,B,top_top(set(A)),G)
           => topolo1002775350975398744n_open(A,collect(A,aa(fun(A,B),fun(A,bool),aTP_Lamp_aiv(fun(A,B),fun(fun(A,B),fun(A,bool)),F2),G))) ) ) ) ).

% open_Collect_neq
tff(fact_7858_continuous__on__powr,axiom,
    ! [C: $tType] :
      ( topolo4958980785337419405_space(C)
     => ! [S3: set(C),F2: fun(C,real),G: fun(C,real)] :
          ( topolo81223032696312382ous_on(C,real,S3,F2)
         => ( topolo81223032696312382ous_on(C,real,S3,G)
           => ( ! [X3: C] :
                  ( pp(aa(set(C),bool,aa(C,fun(set(C),bool),member(C),X3),S3))
                 => ( aa(C,real,F2,X3) != zero_zero(real) ) )
             => topolo81223032696312382ous_on(C,real,S3,aa(fun(C,real),fun(C,real),aTP_Lamp_aiw(fun(C,real),fun(fun(C,real),fun(C,real)),F2),G)) ) ) ) ) ).

% continuous_on_powr
tff(fact_7859_continuous__on__ln,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [S3: set(A),F2: fun(A,real)] :
          ( topolo81223032696312382ous_on(A,real,S3,F2)
         => ( ! [X3: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),S3))
               => ( aa(A,real,F2,X3) != zero_zero(real) ) )
           => topolo81223032696312382ous_on(A,real,S3,aTP_Lamp_aix(fun(A,real),fun(A,real),F2)) ) ) ) ).

% continuous_on_ln
tff(fact_7860_continuous__on__power_H,axiom,
    ! [B: $tType,C: $tType] :
      ( ( topolo4958980785337419405_space(C)
        & topolo1898628316856586783d_mult(B) )
     => ! [A3: set(C),F2: fun(C,B),G: fun(C,nat)] :
          ( topolo81223032696312382ous_on(C,B,A3,F2)
         => ( topolo81223032696312382ous_on(C,nat,A3,G)
           => topolo81223032696312382ous_on(C,B,A3,aa(fun(C,nat),fun(C,B),aTP_Lamp_aiy(fun(C,B),fun(fun(C,nat),fun(C,B)),F2),G)) ) ) ) ).

% continuous_on_power'
tff(fact_7861_continuous__on__power,axiom,
    ! [C: $tType,B: $tType] :
      ( ( power(B)
        & real_V4412858255891104859lgebra(B)
        & topolo4958980785337419405_space(C) )
     => ! [S3: set(C),F2: fun(C,B),N: nat] :
          ( topolo81223032696312382ous_on(C,B,S3,F2)
         => topolo81223032696312382ous_on(C,B,S3,aa(nat,fun(C,B),aTP_Lamp_aiz(fun(C,B),fun(nat,fun(C,B)),F2),N)) ) ) ).

% continuous_on_power
tff(fact_7862_continuous__on__sgn,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & real_V822414075346904944vector(B) )
     => ! [S3: set(A),F2: fun(A,B)] :
          ( topolo81223032696312382ous_on(A,B,S3,F2)
         => ( ! [X3: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),S3))
               => ( aa(A,B,F2,X3) != zero_zero(B) ) )
           => topolo81223032696312382ous_on(A,B,S3,aTP_Lamp_aja(fun(A,B),fun(A,B),F2)) ) ) ) ).

% continuous_on_sgn
tff(fact_7863_continuous__on__inverse,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & real_V8999393235501362500lgebra(B) )
     => ! [S3: set(A),F2: fun(A,B)] :
          ( topolo81223032696312382ous_on(A,B,S3,F2)
         => ( ! [X3: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),S3))
               => ( aa(A,B,F2,X3) != zero_zero(B) ) )
           => topolo81223032696312382ous_on(A,B,S3,aTP_Lamp_ajb(fun(A,B),fun(A,B),F2)) ) ) ) ).

% continuous_on_inverse
tff(fact_7864_continuous__on__minus,axiom,
    ! [B: $tType,C: $tType] :
      ( ( topolo4958980785337419405_space(C)
        & topolo1633459387980952147up_add(B) )
     => ! [S3: set(C),F2: fun(C,B)] :
          ( topolo81223032696312382ous_on(C,B,S3,F2)
         => topolo81223032696312382ous_on(C,B,S3,aTP_Lamp_ajc(fun(C,B),fun(C,B),F2)) ) ) ).

% continuous_on_minus
tff(fact_7865_continuous__on__diff,axiom,
    ! [B: $tType,D6: $tType] :
      ( ( topolo4958980785337419405_space(D6)
        & topolo1633459387980952147up_add(B) )
     => ! [S3: set(D6),F2: fun(D6,B),G: fun(D6,B)] :
          ( topolo81223032696312382ous_on(D6,B,S3,F2)
         => ( topolo81223032696312382ous_on(D6,B,S3,G)
           => topolo81223032696312382ous_on(D6,B,S3,aa(fun(D6,B),fun(D6,B),aTP_Lamp_ajd(fun(D6,B),fun(fun(D6,B),fun(D6,B)),F2),G)) ) ) ) ).

% continuous_on_diff
tff(fact_7866_continuous__on__scaleR,axiom,
    ! [C: $tType,D6: $tType] :
      ( ( topolo4958980785337419405_space(D6)
        & real_V822414075346904944vector(C) )
     => ! [S3: set(D6),F2: fun(D6,real),G: fun(D6,C)] :
          ( topolo81223032696312382ous_on(D6,real,S3,F2)
         => ( topolo81223032696312382ous_on(D6,C,S3,G)
           => topolo81223032696312382ous_on(D6,C,S3,aa(fun(D6,C),fun(D6,C),aTP_Lamp_aje(fun(D6,real),fun(fun(D6,C),fun(D6,C)),F2),G)) ) ) ) ).

% continuous_on_scaleR
tff(fact_7867_continuous__on__pochhammer,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [A3: set(A),N: nat] : topolo81223032696312382ous_on(A,A,A3,aTP_Lamp_zp(nat,fun(A,A),N)) ) ).

% continuous_on_pochhammer
tff(fact_7868_continuous__on__pochhammer_H,axiom,
    ! [C: $tType,A: $tType] :
      ( ( real_V3459762299906320749_field(A)
        & topolo4958980785337419405_space(C) )
     => ! [S3: set(C),F2: fun(C,A),N: nat] :
          ( topolo81223032696312382ous_on(C,A,S3,F2)
         => topolo81223032696312382ous_on(C,A,S3,aa(nat,fun(C,A),aTP_Lamp_ajf(fun(C,A),fun(nat,fun(C,A)),F2),N)) ) ) ).

% continuous_on_pochhammer'
tff(fact_7869_continuous__on__sin,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [S3: set(A),F2: fun(A,B)] :
          ( topolo81223032696312382ous_on(A,B,S3,F2)
         => topolo81223032696312382ous_on(A,B,S3,aTP_Lamp_wx(fun(A,B),fun(A,B),F2)) ) ) ).

% continuous_on_sin
tff(fact_7870_continuous__on__exp,axiom,
    ! [A: $tType,C: $tType] :
      ( ( topolo4958980785337419405_space(C)
        & real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [S3: set(C),F2: fun(C,A)] :
          ( topolo81223032696312382ous_on(C,A,S3,F2)
         => topolo81223032696312382ous_on(C,A,S3,aTP_Lamp_ajg(fun(C,A),fun(C,A),F2)) ) ) ).

% continuous_on_exp
tff(fact_7871_continuous__on__const,axiom,
    ! [A: $tType,B: $tType] :
      ( ( topolo4958980785337419405_space(B)
        & topolo4958980785337419405_space(A) )
     => ! [S3: set(A),C2: B] : topolo81223032696312382ous_on(A,B,S3,aTP_Lamp_ajh(B,fun(A,B),C2)) ) ).

% continuous_on_const
tff(fact_7872_continuous__on__id,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [S3: set(A)] : topolo81223032696312382ous_on(A,A,S3,aTP_Lamp_vi(A,A)) ) ).

% continuous_on_id
tff(fact_7873_continuous__on__of__real,axiom,
    ! [A: $tType,C: $tType] :
      ( ( topolo4958980785337419405_space(C)
        & real_V2191834092415804123ebra_1(A)
        & real_V822414075346904944vector(A) )
     => ! [S3: set(C),G: fun(C,real)] :
          ( topolo81223032696312382ous_on(C,real,S3,G)
         => topolo81223032696312382ous_on(C,A,S3,aTP_Lamp_aji(fun(C,real),fun(C,A),G)) ) ) ).

% continuous_on_of_real
tff(fact_7874_continuous__on__norm,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & real_V822414075346904944vector(B) )
     => ! [S3: set(A),F2: fun(A,B)] :
          ( topolo81223032696312382ous_on(A,B,S3,F2)
         => topolo81223032696312382ous_on(A,real,S3,aTP_Lamp_ajj(fun(A,B),fun(A,real),F2)) ) ) ).

% continuous_on_norm
tff(fact_7875_continuous__on__cos,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [S3: set(A),F2: fun(A,B)] :
          ( topolo81223032696312382ous_on(A,B,S3,F2)
         => topolo81223032696312382ous_on(A,B,S3,aTP_Lamp_wu(fun(A,B),fun(A,B),F2)) ) ) ).

% continuous_on_cos
tff(fact_7876_continuous__on__real__sqrt,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [S3: set(A),F2: fun(A,real)] :
          ( topolo81223032696312382ous_on(A,real,S3,F2)
         => topolo81223032696312382ous_on(A,real,S3,aTP_Lamp_ajk(fun(A,real),fun(A,real),F2)) ) ) ).

% continuous_on_real_sqrt
tff(fact_7877_continuous__on__arctan,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [S3: set(A),F2: fun(A,real)] :
          ( topolo81223032696312382ous_on(A,real,S3,F2)
         => topolo81223032696312382ous_on(A,real,S3,aTP_Lamp_ajl(fun(A,real),fun(A,real),F2)) ) ) ).

% continuous_on_arctan
tff(fact_7878_continuous__on__real__root,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [S3: set(A),F2: fun(A,real),N: nat] :
          ( topolo81223032696312382ous_on(A,real,S3,F2)
         => topolo81223032696312382ous_on(A,real,S3,aa(nat,fun(A,real),aTP_Lamp_ajm(fun(A,real),fun(nat,fun(A,real)),F2),N)) ) ) ).

% continuous_on_real_root
tff(fact_7879_continuous__on__sinh,axiom,
    ! [A: $tType,C: $tType] :
      ( ( topolo4958980785337419405_space(C)
        & real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [A3: set(C),F2: fun(C,A)] :
          ( topolo81223032696312382ous_on(C,A,A3,F2)
         => topolo81223032696312382ous_on(C,A,A3,aTP_Lamp_ajn(fun(C,A),fun(C,A),F2)) ) ) ).

% continuous_on_sinh
tff(fact_7880_continuous__on__cosh,axiom,
    ! [A: $tType,C: $tType] :
      ( ( topolo4958980785337419405_space(C)
        & real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [A3: set(C),F2: fun(C,A)] :
          ( topolo81223032696312382ous_on(C,A,A3,F2)
         => topolo81223032696312382ous_on(C,A,A3,aTP_Lamp_ajo(fun(C,A),fun(C,A),F2)) ) ) ).

% continuous_on_cosh
tff(fact_7881_continuous__on__arsinh_H,axiom,
    ! [A3: set(real),F2: fun(real,real)] :
      ( topolo81223032696312382ous_on(real,real,A3,F2)
     => topolo81223032696312382ous_on(real,real,A3,aTP_Lamp_ajp(fun(real,real),fun(real,real),F2)) ) ).

% continuous_on_arsinh'
tff(fact_7882_continuous__on__rabs,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [S3: set(A),F2: fun(A,real)] :
          ( topolo81223032696312382ous_on(A,real,S3,F2)
         => topolo81223032696312382ous_on(A,real,S3,aTP_Lamp_ajq(fun(A,real),fun(A,real),F2)) ) ) ).

% continuous_on_rabs
tff(fact_7883_continuous__on__max,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topolo1944317154257567458pology(B) )
     => ! [A3: set(A),F2: fun(A,B),G: fun(A,B)] :
          ( topolo81223032696312382ous_on(A,B,A3,F2)
         => ( topolo81223032696312382ous_on(A,B,A3,G)
           => topolo81223032696312382ous_on(A,B,A3,aa(fun(A,B),fun(A,B),aTP_Lamp_ajr(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G)) ) ) ) ).

% continuous_on_max
tff(fact_7884_continuous__on__add,axiom,
    ! [B: $tType,D6: $tType] :
      ( ( topolo4958980785337419405_space(D6)
        & topolo6943815403480290642id_add(B) )
     => ! [S3: set(D6),F2: fun(D6,B),G: fun(D6,B)] :
          ( topolo81223032696312382ous_on(D6,B,S3,F2)
         => ( topolo81223032696312382ous_on(D6,B,S3,G)
           => topolo81223032696312382ous_on(D6,B,S3,aa(fun(D6,B),fun(D6,B),aTP_Lamp_ajs(fun(D6,B),fun(fun(D6,B),fun(D6,B)),F2),G)) ) ) ) ).

% continuous_on_add
tff(fact_7885_continuous__on__divide,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & real_V3459762299906320749_field(B) )
     => ! [S3: set(A),F2: fun(A,B),G: fun(A,B)] :
          ( topolo81223032696312382ous_on(A,B,S3,F2)
         => ( topolo81223032696312382ous_on(A,B,S3,G)
           => ( ! [X3: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),S3))
                 => ( aa(A,B,G,X3) != zero_zero(B) ) )
             => topolo81223032696312382ous_on(A,B,S3,aa(fun(A,B),fun(A,B),aTP_Lamp_ajt(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G)) ) ) ) ) ).

% continuous_on_divide
tff(fact_7886_continuous__on__mult__right,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V4412858255891104859lgebra(A)
        & topolo4958980785337419405_space(B) )
     => ! [S3: set(B),F2: fun(B,A),C2: A] :
          ( topolo81223032696312382ous_on(B,A,S3,F2)
         => topolo81223032696312382ous_on(B,A,S3,aa(A,fun(B,A),aTP_Lamp_aju(fun(B,A),fun(A,fun(B,A)),F2),C2)) ) ) ).

% continuous_on_mult_right
tff(fact_7887_continuous__on__mult__left,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V4412858255891104859lgebra(A)
        & topolo4958980785337419405_space(B) )
     => ! [S3: set(B),F2: fun(B,A),C2: A] :
          ( topolo81223032696312382ous_on(B,A,S3,F2)
         => topolo81223032696312382ous_on(B,A,S3,aa(A,fun(B,A),aTP_Lamp_ajv(fun(B,A),fun(A,fun(B,A)),F2),C2)) ) ) ).

% continuous_on_mult_left
tff(fact_7888_continuous__on__mult_H,axiom,
    ! [B: $tType,D6: $tType] :
      ( ( topolo4958980785337419405_space(D6)
        & topolo4211221413907600880p_mult(B) )
     => ! [A3: set(D6),F2: fun(D6,B),G: fun(D6,B)] :
          ( topolo81223032696312382ous_on(D6,B,A3,F2)
         => ( topolo81223032696312382ous_on(D6,B,A3,G)
           => topolo81223032696312382ous_on(D6,B,A3,aa(fun(D6,B),fun(D6,B),aTP_Lamp_ajw(fun(D6,B),fun(fun(D6,B),fun(D6,B)),F2),G)) ) ) ) ).

% continuous_on_mult'
tff(fact_7889_continuous__on__mult,axiom,
    ! [A: $tType,D6: $tType] :
      ( ( topolo4958980785337419405_space(D6)
        & real_V4412858255891104859lgebra(A) )
     => ! [S3: set(D6),F2: fun(D6,A),G: fun(D6,A)] :
          ( topolo81223032696312382ous_on(D6,A,S3,F2)
         => ( topolo81223032696312382ous_on(D6,A,S3,G)
           => topolo81223032696312382ous_on(D6,A,S3,aa(fun(D6,A),fun(D6,A),aTP_Lamp_ajx(fun(D6,A),fun(fun(D6,A),fun(D6,A)),F2),G)) ) ) ) ).

% continuous_on_mult
tff(fact_7890_continuous__on__Pair,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topolo4958980785337419405_space(C)
        & topolo4958980785337419405_space(B) )
     => ! [S3: set(A),F2: fun(A,B),G: fun(A,C)] :
          ( topolo81223032696312382ous_on(A,B,S3,F2)
         => ( topolo81223032696312382ous_on(A,C,S3,G)
           => topolo81223032696312382ous_on(A,product_prod(B,C),S3,aa(fun(A,C),fun(A,product_prod(B,C)),aTP_Lamp_ajy(fun(A,B),fun(fun(A,C),fun(A,product_prod(B,C))),F2),G)) ) ) ) ).

% continuous_on_Pair
tff(fact_7891_continuous__on__min,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topolo1944317154257567458pology(B) )
     => ! [A3: set(A),F2: fun(A,B),G: fun(A,B)] :
          ( topolo81223032696312382ous_on(A,B,A3,F2)
         => ( topolo81223032696312382ous_on(A,B,A3,G)
           => topolo81223032696312382ous_on(A,B,A3,aa(fun(A,B),fun(A,B),aTP_Lamp_ajz(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G)) ) ) ) ).

% continuous_on_min
tff(fact_7892_continuous__on__prod_H,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(B)
        & topolo4987421752381908075d_mult(C) )
     => ! [I5: set(A),S: set(B),F2: fun(A,fun(B,C))] :
          ( ! [I2: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),I2),I5))
             => topolo81223032696312382ous_on(B,C,S,aa(A,fun(B,C),F2,I2)) )
         => topolo81223032696312382ous_on(B,C,S,aa(fun(A,fun(B,C)),fun(B,C),aTP_Lamp_akb(set(A),fun(fun(A,fun(B,C)),fun(B,C)),I5),F2)) ) ) ).

% continuous_on_prod'
tff(fact_7893_continuous__on__prod,axiom,
    ! [C: $tType,D6: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(D6)
        & real_V4412858255891104859lgebra(C)
        & comm_ring_1(C) )
     => ! [S: set(A),S3: set(D6),F2: fun(A,fun(D6,C))] :
          ( ! [I2: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),I2),S))
             => topolo81223032696312382ous_on(D6,C,S3,aa(A,fun(D6,C),F2,I2)) )
         => topolo81223032696312382ous_on(D6,C,S3,aa(fun(A,fun(D6,C)),fun(D6,C),aTP_Lamp_akd(set(A),fun(fun(A,fun(D6,C)),fun(D6,C)),S),F2)) ) ) ).

% continuous_on_prod
tff(fact_7894_continuous__on__sum,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(B)
        & topolo5987344860129210374id_add(C) )
     => ! [I5: set(A),S: set(B),F2: fun(A,fun(B,C))] :
          ( ! [I2: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),I2),I5))
             => topolo81223032696312382ous_on(B,C,S,aa(A,fun(B,C),F2,I2)) )
         => topolo81223032696312382ous_on(B,C,S,aa(fun(A,fun(B,C)),fun(B,C),aTP_Lamp_akf(set(A),fun(fun(A,fun(B,C)),fun(B,C)),I5),F2)) ) ) ).

% continuous_on_sum
tff(fact_7895_continuous__on__mult__const,axiom,
    ! [A: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [S3: set(A),C2: A] : topolo81223032696312382ous_on(A,A,S3,aa(A,fun(A,A),times_times(A),C2)) ) ).

% continuous_on_mult_const
tff(fact_7896_continuous__on__dist,axiom,
    ! [A: $tType,D6: $tType] :
      ( ( topolo4958980785337419405_space(D6)
        & real_V7819770556892013058_space(A) )
     => ! [S3: set(D6),F2: fun(D6,A),G: fun(D6,A)] :
          ( topolo81223032696312382ous_on(D6,A,S3,F2)
         => ( topolo81223032696312382ous_on(D6,A,S3,G)
           => topolo81223032696312382ous_on(D6,real,S3,aa(fun(D6,A),fun(D6,real),aTP_Lamp_akg(fun(D6,A),fun(fun(D6,A),fun(D6,real)),F2),G)) ) ) ) ).

% continuous_on_dist
tff(fact_7897_continuous__on__cot,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [S3: set(A),F2: fun(A,A)] :
          ( topolo81223032696312382ous_on(A,A,S3,F2)
         => ( ! [X3: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),S3))
               => ( sin(A,aa(A,A,F2,X3)) != zero_zero(A) ) )
           => topolo81223032696312382ous_on(A,A,S3,aTP_Lamp_xo(fun(A,A),fun(A,A),F2)) ) ) ) ).

% continuous_on_cot
tff(fact_7898_not__Ici__le__Icc,axiom,
    ! [A: $tType] :
      ( no_top(A)
     => ! [L: A,L2: A,H2: A] : ~ pp(aa(set(A),bool,ord_less_eq(set(A),set_ord_atLeast(A,L)),set_or1337092689740270186AtMost(A,L2,H2))) ) ).

% not_Ici_le_Icc
tff(fact_7899_IVT2_H,axiom,
    ! [A: $tType,B: $tType] :
      ( ( topolo1944317154257567458pology(B)
        & topolo8458572112393995274pology(A) )
     => ! [F2: fun(A,B),B2: A,Y: B,A2: A] :
          ( pp(aa(B,bool,ord_less_eq(B,aa(A,B,F2,B2)),Y))
         => ( pp(aa(B,bool,ord_less_eq(B,Y),aa(A,B,F2,A2)))
           => ( pp(aa(A,bool,ord_less_eq(A,A2),B2))
             => ( topolo81223032696312382ous_on(A,B,set_or1337092689740270186AtMost(A,A2,B2),F2)
               => ? [X3: A] :
                    ( pp(aa(A,bool,ord_less_eq(A,A2),X3))
                    & pp(aa(A,bool,ord_less_eq(A,X3),B2))
                    & ( aa(A,B,F2,X3) = Y ) ) ) ) ) ) ) ).

% IVT2'
tff(fact_7900_IVT_H,axiom,
    ! [A: $tType,B: $tType] :
      ( ( topolo1944317154257567458pology(B)
        & topolo8458572112393995274pology(A) )
     => ! [F2: fun(A,B),A2: A,Y: B,B2: A] :
          ( pp(aa(B,bool,ord_less_eq(B,aa(A,B,F2,A2)),Y))
         => ( pp(aa(B,bool,ord_less_eq(B,Y),aa(A,B,F2,B2)))
           => ( pp(aa(A,bool,ord_less_eq(A,A2),B2))
             => ( topolo81223032696312382ous_on(A,B,set_or1337092689740270186AtMost(A,A2,B2),F2)
               => ? [X3: A] :
                    ( pp(aa(A,bool,ord_less_eq(A,A2),X3))
                    & pp(aa(A,bool,ord_less_eq(A,X3),B2))
                    & ( aa(A,B,F2,X3) = Y ) ) ) ) ) ) ) ).

% IVT'
tff(fact_7901_continuous__on__arcosh,axiom,
    ! [A3: set(real)] :
      ( pp(aa(set(real),bool,ord_less_eq(set(real),A3),set_ord_atLeast(real,one_one(real))))
     => topolo81223032696312382ous_on(real,real,A3,arcosh(real)) ) ).

% continuous_on_arcosh
tff(fact_7902_continuous__on__subset,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topolo4958980785337419405_space(B) )
     => ! [S3: set(A),F2: fun(A,B),T: set(A)] :
          ( topolo81223032696312382ous_on(A,B,S3,F2)
         => ( pp(aa(set(A),bool,ord_less_eq(set(A),T),S3))
           => topolo81223032696312382ous_on(A,B,T,F2) ) ) ) ).

% continuous_on_subset
tff(fact_7903_atLeast__def,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [L: A] : set_ord_atLeast(A,L) = collect(A,ord_less_eq(A,L)) ) ).

% atLeast_def
tff(fact_7904_not__Ici__le__Iic,axiom,
    ! [A: $tType] :
      ( no_top(A)
     => ! [L: A,H2: A] : ~ pp(aa(set(A),bool,ord_less_eq(set(A),set_ord_atLeast(A,L)),set_ord_atMost(A,H2))) ) ).

% not_Ici_le_Iic
tff(fact_7905_not__Iic__le__Ici,axiom,
    ! [A: $tType] :
      ( no_bot(A)
     => ! [H: A,L2: A] : ~ pp(aa(set(A),bool,ord_less_eq(set(A),set_ord_atMost(A,H)),set_ord_atLeast(A,L2))) ) ).

% not_Iic_le_Ici
tff(fact_7906_continuous__on__compose2,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ( topolo4958980785337419405_space(C)
        & topolo4958980785337419405_space(A)
        & topolo4958980785337419405_space(B) )
     => ! [T: set(A),G: fun(A,B),S3: set(C),F2: fun(C,A)] :
          ( topolo81223032696312382ous_on(A,B,T,G)
         => ( topolo81223032696312382ous_on(C,A,S3,F2)
           => ( pp(aa(set(A),bool,ord_less_eq(set(A),aa(set(C),set(A),image(C,A,F2),S3)),T))
             => topolo81223032696312382ous_on(C,B,S3,aa(fun(C,A),fun(C,B),aTP_Lamp_akh(fun(A,B),fun(fun(C,A),fun(C,B)),G),F2)) ) ) ) ) ).

% continuous_on_compose2
tff(fact_7907_not__UNIV__le__Ici,axiom,
    ! [A: $tType] :
      ( no_bot(A)
     => ! [L: A] : ~ pp(aa(set(A),bool,ord_less_eq(set(A),top_top(set(A))),set_ord_atLeast(A,L))) ) ).

% not_UNIV_le_Ici
tff(fact_7908_continuous__onI__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo1944317154257567458pology(A)
        & dense_order(B)
        & topolo1944317154257567458pology(B) )
     => ! [F2: fun(A,B),A3: set(A)] :
          ( topolo1002775350975398744n_open(B,aa(set(A),set(B),image(A,B,F2),A3))
         => ( ! [X3: A,Y3: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),A3))
               => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Y3),A3))
                 => ( pp(aa(A,bool,ord_less_eq(A,X3),Y3))
                   => pp(aa(B,bool,ord_less_eq(B,aa(A,B,F2,X3)),aa(A,B,F2,Y3))) ) ) )
           => topolo81223032696312382ous_on(A,B,A3,F2) ) ) ) ).

% continuous_onI_mono
tff(fact_7909_Ioi__le__Ico,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A2: A] : pp(aa(set(A),bool,ord_less_eq(set(A),set_ord_greaterThan(A,A2)),set_ord_atLeast(A,A2))) ) ).

% Ioi_le_Ico
tff(fact_7910_Ici__subset__Ioi__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(set(A),bool,ord_less_eq(set(A),set_ord_atLeast(A,A2)),set_ord_greaterThan(A,B2)))
        <=> pp(aa(A,bool,ord_less(A,B2),A2)) ) ) ).

% Ici_subset_Ioi_iff
tff(fact_7911_continuous__on__arcosh_H,axiom,
    ! [A3: set(real),F2: fun(real,real)] :
      ( topolo81223032696312382ous_on(real,real,A3,F2)
     => ( ! [X3: real] :
            ( pp(aa(set(real),bool,aa(real,fun(set(real),bool),member(real),X3),A3))
           => pp(aa(real,bool,ord_less_eq(real,one_one(real)),aa(real,real,F2,X3))) )
       => topolo81223032696312382ous_on(real,real,A3,aTP_Lamp_aki(fun(real,real),fun(real,real),F2)) ) ) ).

% continuous_on_arcosh'
tff(fact_7912_continuous__image__closed__interval,axiom,
    ! [A2: real,B2: real,F2: fun(real,real)] :
      ( pp(aa(real,bool,ord_less_eq(real,A2),B2))
     => ( topolo81223032696312382ous_on(real,real,set_or1337092689740270186AtMost(real,A2,B2),F2)
       => ? [C3: real,D2: real] :
            ( ( aa(set(real),set(real),image(real,real,F2),set_or1337092689740270186AtMost(real,A2,B2)) = set_or1337092689740270186AtMost(real,C3,D2) )
            & pp(aa(real,bool,ord_less_eq(real,C3),D2)) ) ) ) ).

% continuous_image_closed_interval
tff(fact_7913_continuous__on__powr_H,axiom,
    ! [C: $tType] :
      ( topolo4958980785337419405_space(C)
     => ! [S3: set(C),F2: fun(C,real),G: fun(C,real)] :
          ( topolo81223032696312382ous_on(C,real,S3,F2)
         => ( topolo81223032696312382ous_on(C,real,S3,G)
           => ( ! [X3: C] :
                  ( pp(aa(set(C),bool,aa(C,fun(set(C),bool),member(C),X3),S3))
                 => ( pp(aa(real,bool,ord_less_eq(real,zero_zero(real)),aa(C,real,F2,X3)))
                    & ( ( aa(C,real,F2,X3) = zero_zero(real) )
                     => pp(aa(real,bool,ord_less(real,zero_zero(real)),aa(C,real,G,X3))) ) ) )
             => topolo81223032696312382ous_on(C,real,S3,aa(fun(C,real),fun(C,real),aTP_Lamp_aiw(fun(C,real),fun(fun(C,real),fun(C,real)),F2),G)) ) ) ) ) ).

% continuous_on_powr'
tff(fact_7914_continuous__on__log,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [S3: set(A),F2: fun(A,real),G: fun(A,real)] :
          ( topolo81223032696312382ous_on(A,real,S3,F2)
         => ( topolo81223032696312382ous_on(A,real,S3,G)
           => ( ! [X3: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),S3))
                 => pp(aa(real,bool,ord_less(real,zero_zero(real)),aa(A,real,F2,X3))) )
             => ( ! [X3: A] :
                    ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),S3))
                   => ( aa(A,real,F2,X3) != one_one(real) ) )
               => ( ! [X3: A] :
                      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),S3))
                     => pp(aa(real,bool,ord_less(real,zero_zero(real)),aa(A,real,G,X3))) )
                 => topolo81223032696312382ous_on(A,real,S3,aa(fun(A,real),fun(A,real),aTP_Lamp_akj(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G)) ) ) ) ) ) ) ).

% continuous_on_log
tff(fact_7915_continuous__on__arccos_H,axiom,
    topolo81223032696312382ous_on(real,real,set_or1337092689740270186AtMost(real,aa(real,real,uminus_uminus(real),one_one(real)),one_one(real)),arccos) ).

% continuous_on_arccos'
tff(fact_7916_continuous__on__arcsin_H,axiom,
    topolo81223032696312382ous_on(real,real,set_or1337092689740270186AtMost(real,aa(real,real,uminus_uminus(real),one_one(real)),one_one(real)),arcsin) ).

% continuous_on_arcsin'
tff(fact_7917_continuous__on__arccos,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [S3: set(A),F2: fun(A,real)] :
          ( topolo81223032696312382ous_on(A,real,S3,F2)
         => ( ! [X3: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),S3))
               => ( pp(aa(real,bool,ord_less_eq(real,aa(real,real,uminus_uminus(real),one_one(real))),aa(A,real,F2,X3)))
                  & pp(aa(real,bool,ord_less_eq(real,aa(A,real,F2,X3)),one_one(real))) ) )
           => topolo81223032696312382ous_on(A,real,S3,aTP_Lamp_akk(fun(A,real),fun(A,real),F2)) ) ) ) ).

% continuous_on_arccos
tff(fact_7918_continuous__on__arcsin,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [S3: set(A),F2: fun(A,real)] :
          ( topolo81223032696312382ous_on(A,real,S3,F2)
         => ( ! [X3: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),S3))
               => ( pp(aa(real,bool,ord_less_eq(real,aa(real,real,uminus_uminus(real),one_one(real))),aa(A,real,F2,X3)))
                  & pp(aa(real,bool,ord_less_eq(real,aa(A,real,F2,X3)),one_one(real))) ) )
           => topolo81223032696312382ous_on(A,real,S3,aTP_Lamp_akl(fun(A,real),fun(A,real),F2)) ) ) ) ).

% continuous_on_arcsin
tff(fact_7919_continuous__on__artanh,axiom,
    ! [A3: set(real)] :
      ( pp(aa(set(real),bool,ord_less_eq(set(real),A3),set_or5935395276787703475ssThan(real,aa(real,real,uminus_uminus(real),one_one(real)),one_one(real))))
     => topolo81223032696312382ous_on(real,real,A3,artanh(real)) ) ).

% continuous_on_artanh
tff(fact_7920_DERIV__atLeastAtMost__imp__continuous__on,axiom,
    ! [A: $tType] :
      ( ( ord(A)
        & real_V3459762299906320749_field(A) )
     => ! [A2: A,B2: A,F2: fun(A,A)] :
          ( ! [X3: A] :
              ( pp(aa(A,bool,ord_less_eq(A,A2),X3))
             => ( pp(aa(A,bool,ord_less_eq(A,X3),B2))
               => ? [Y4: A] : has_field_derivative(A,F2,Y4,topolo174197925503356063within(A,X3,top_top(set(A)))) ) )
         => topolo81223032696312382ous_on(A,A,set_or1337092689740270186AtMost(A,A2,B2),F2) ) ) ).

% DERIV_atLeastAtMost_imp_continuous_on
tff(fact_7921_continuous__on__artanh_H,axiom,
    ! [A3: set(real),F2: fun(real,real)] :
      ( topolo81223032696312382ous_on(real,real,A3,F2)
     => ( ! [X3: real] :
            ( pp(aa(set(real),bool,aa(real,fun(set(real),bool),member(real),X3),A3))
           => pp(aa(set(real),bool,aa(real,fun(set(real),bool),member(real),aa(real,real,F2,X3)),set_or5935395276787703475ssThan(real,aa(real,real,uminus_uminus(real),one_one(real)),one_one(real)))) )
       => topolo81223032696312382ous_on(real,real,A3,aTP_Lamp_akm(fun(real,real),fun(real,real),F2)) ) ) ).

% continuous_on_artanh'
tff(fact_7922_atLeast__Suc,axiom,
    ! [K: nat] : set_ord_atLeast(nat,aa(nat,nat,suc,K)) = aa(set(nat),set(nat),minus_minus(set(nat),set_ord_atLeast(nat,K)),insert(nat,K,bot_bot(set(nat)))) ).

% atLeast_Suc
tff(fact_7923_Rolle__deriv,axiom,
    ! [A2: real,B2: real,F2: fun(real,real),F7: fun(real,fun(real,real))] :
      ( pp(aa(real,bool,ord_less(real,A2),B2))
     => ( ( aa(real,real,F2,A2) = aa(real,real,F2,B2) )
       => ( topolo81223032696312382ous_on(real,real,set_or1337092689740270186AtMost(real,A2,B2),F2)
         => ( ! [X3: real] :
                ( pp(aa(real,bool,ord_less(real,A2),X3))
               => ( pp(aa(real,bool,ord_less(real,X3),B2))
                 => has_derivative(real,real,F2,aa(real,fun(real,real),F7,X3),topolo174197925503356063within(real,X3,top_top(set(real)))) ) )
           => ? [Z2: real] :
                ( pp(aa(real,bool,ord_less(real,A2),Z2))
                & pp(aa(real,bool,ord_less(real,Z2),B2))
                & ! [X4: real] : aa(real,real,aa(real,fun(real,real),F7,Z2),X4) = zero_zero(real) ) ) ) ) ) ).

% Rolle_deriv
tff(fact_7924_continuous__on__of__int__floor,axiom,
    ! [B: $tType,A: $tType] :
      ( ( archim2362893244070406136eiling(A)
        & topolo2564578578187576103pology(A)
        & ring_1(B)
        & topolo4958980785337419405_space(B) )
     => topolo81223032696312382ous_on(A,B,aa(set(A),set(A),minus_minus(set(A),top_top(set(A))),ring_1_Ints(A)),aTP_Lamp_akn(A,B)) ) ).

% continuous_on_of_int_floor
tff(fact_7925_continuous__on__of__int__ceiling,axiom,
    ! [B: $tType,A: $tType] :
      ( ( archim2362893244070406136eiling(A)
        & topolo2564578578187576103pology(A)
        & ring_1(B)
        & topolo4958980785337419405_space(B) )
     => topolo81223032696312382ous_on(A,B,aa(set(A),set(A),minus_minus(set(A),top_top(set(A))),ring_1_Ints(A)),aTP_Lamp_ako(A,B)) ) ).

% continuous_on_of_int_ceiling
tff(fact_7926_DERIV__isconst__end,axiom,
    ! [A2: real,B2: real,F2: fun(real,real)] :
      ( pp(aa(real,bool,ord_less(real,A2),B2))
     => ( topolo81223032696312382ous_on(real,real,set_or1337092689740270186AtMost(real,A2,B2),F2)
       => ( ! [X3: real] :
              ( pp(aa(real,bool,ord_less(real,A2),X3))
             => ( pp(aa(real,bool,ord_less(real,X3),B2))
               => has_field_derivative(real,F2,zero_zero(real),topolo174197925503356063within(real,X3,top_top(set(real)))) ) )
         => ( aa(real,real,F2,B2) = aa(real,real,F2,A2) ) ) ) ) ).

% DERIV_isconst_end
tff(fact_7927_DERIV__neg__imp__decreasing__open,axiom,
    ! [A2: real,B2: real,F2: fun(real,real)] :
      ( pp(aa(real,bool,ord_less(real,A2),B2))
     => ( ! [X3: real] :
            ( pp(aa(real,bool,ord_less(real,A2),X3))
           => ( pp(aa(real,bool,ord_less(real,X3),B2))
             => ? [Y4: real] :
                  ( has_field_derivative(real,F2,Y4,topolo174197925503356063within(real,X3,top_top(set(real))))
                  & pp(aa(real,bool,ord_less(real,Y4),zero_zero(real))) ) ) )
       => ( topolo81223032696312382ous_on(real,real,set_or1337092689740270186AtMost(real,A2,B2),F2)
         => pp(aa(real,bool,ord_less(real,aa(real,real,F2,B2)),aa(real,real,F2,A2))) ) ) ) ).

% DERIV_neg_imp_decreasing_open
tff(fact_7928_DERIV__pos__imp__increasing__open,axiom,
    ! [A2: real,B2: real,F2: fun(real,real)] :
      ( pp(aa(real,bool,ord_less(real,A2),B2))
     => ( ! [X3: real] :
            ( pp(aa(real,bool,ord_less(real,A2),X3))
           => ( pp(aa(real,bool,ord_less(real,X3),B2))
             => ? [Y4: real] :
                  ( has_field_derivative(real,F2,Y4,topolo174197925503356063within(real,X3,top_top(set(real))))
                  & pp(aa(real,bool,ord_less(real,zero_zero(real)),Y4)) ) ) )
       => ( topolo81223032696312382ous_on(real,real,set_or1337092689740270186AtMost(real,A2,B2),F2)
         => pp(aa(real,bool,ord_less(real,aa(real,real,F2,A2)),aa(real,real,F2,B2))) ) ) ) ).

% DERIV_pos_imp_increasing_open
tff(fact_7929_complete__interval,axiom,
    ! [A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [A2: A,B2: A,P: fun(A,bool)] :
          ( pp(aa(A,bool,ord_less(A,A2),B2))
         => ( pp(aa(A,bool,P,A2))
           => ( ~ pp(aa(A,bool,P,B2))
             => ? [C3: A] :
                  ( pp(aa(A,bool,ord_less_eq(A,A2),C3))
                  & pp(aa(A,bool,ord_less_eq(A,C3),B2))
                  & ! [X4: A] :
                      ( ( pp(aa(A,bool,ord_less_eq(A,A2),X4))
                        & pp(aa(A,bool,ord_less(A,X4),C3)) )
                     => pp(aa(A,bool,P,X4)) )
                  & ! [D5: A] :
                      ( ! [X3: A] :
                          ( ( pp(aa(A,bool,ord_less_eq(A,A2),X3))
                            & pp(aa(A,bool,ord_less(A,X3),D5)) )
                         => pp(aa(A,bool,P,X3)) )
                     => pp(aa(A,bool,ord_less_eq(A,D5),C3)) ) ) ) ) ) ) ).

% complete_interval
tff(fact_7930_DERIV__isconst2,axiom,
    ! [A2: real,B2: real,F2: fun(real,real),X: real] :
      ( pp(aa(real,bool,ord_less(real,A2),B2))
     => ( topolo81223032696312382ous_on(real,real,set_or1337092689740270186AtMost(real,A2,B2),F2)
       => ( ! [X3: real] :
              ( pp(aa(real,bool,ord_less(real,A2),X3))
             => ( pp(aa(real,bool,ord_less(real,X3),B2))
               => has_field_derivative(real,F2,zero_zero(real),topolo174197925503356063within(real,X3,top_top(set(real)))) ) )
         => ( pp(aa(real,bool,ord_less_eq(real,A2),X))
           => ( pp(aa(real,bool,ord_less_eq(real,X),B2))
             => ( aa(real,real,F2,X) = aa(real,real,F2,A2) ) ) ) ) ) ) ).

% DERIV_isconst2
tff(fact_7931_Rolle,axiom,
    ! [A2: real,B2: real,F2: fun(real,real)] :
      ( pp(aa(real,bool,ord_less(real,A2),B2))
     => ( ( aa(real,real,F2,A2) = aa(real,real,F2,B2) )
       => ( topolo81223032696312382ous_on(real,real,set_or1337092689740270186AtMost(real,A2,B2),F2)
         => ( ! [X3: real] :
                ( pp(aa(real,bool,ord_less(real,A2),X3))
               => ( pp(aa(real,bool,ord_less(real,X3),B2))
                 => differentiable(real,real,F2,topolo174197925503356063within(real,X3,top_top(set(real)))) ) )
           => ? [Z2: real] :
                ( pp(aa(real,bool,ord_less(real,A2),Z2))
                & pp(aa(real,bool,ord_less(real,Z2),B2))
                & has_field_derivative(real,F2,zero_zero(real),topolo174197925503356063within(real,Z2,top_top(set(real)))) ) ) ) ) ) ).

% Rolle
tff(fact_7932_set__bits__int__def,axiom,
    ! [F2: fun(nat,bool)] :
      ( ( ? [N11: nat] :
          ! [M3: nat] :
            ( pp(aa(nat,bool,ord_less_eq(nat,N11),M3))
           => ( pp(aa(nat,bool,F2,M3))
            <=> pp(aa(nat,bool,F2,N11)) ) )
       => ( bit_bi4170147762399595738t_bits(int,F2) = aa(int,int,bit_ri4674362597316999326ke_bit(int,ord_Least(nat,aTP_Lamp_akp(fun(nat,bool),fun(nat,bool),F2))),groups4207007520872428315er_sum(bool,int,zero_neq_one_of_bool(int),aa(num,int,numeral_numeral(int),bit0(one2)),map(nat,bool,F2,upt(zero_zero(nat),aa(nat,nat,suc,ord_Least(nat,aTP_Lamp_akp(fun(nat,bool),fun(nat,bool),F2))))))) ) )
      & ( ~ ? [N2: nat] :
            ! [M4: nat] :
              ( pp(aa(nat,bool,ord_less_eq(nat,N2),M4))
             => ( pp(aa(nat,bool,F2,M4))
              <=> pp(aa(nat,bool,F2,N2)) ) )
       => ( bit_bi4170147762399595738t_bits(int,F2) = zero_zero(int) ) ) ) ).

% set_bits_int_def
tff(fact_7933_nth__sorted__list__of__set__greaterThanLessThan,axiom,
    ! [N: nat,J: nat,I: nat] :
      ( pp(aa(nat,bool,ord_less(nat,N),aa(nat,nat,minus_minus(nat,J),aa(nat,nat,suc,I))))
     => ( aa(nat,nat,nth(nat,linord4507533701916653071of_set(nat,set_or5935395276787703475ssThan(nat,I,J))),N) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),N)) ) ) ).

% nth_sorted_list_of_set_greaterThanLessThan
tff(fact_7934_finite__Collect__bounded__ex,axiom,
    ! [B: $tType,A: $tType,P: fun(A,bool),Q: fun(B,fun(A,bool))] :
      ( pp(aa(set(A),bool,finite_finite2(A),collect(A,P)))
     => ( pp(aa(set(B),bool,finite_finite2(B),collect(B,aa(fun(B,fun(A,bool)),fun(B,bool),aTP_Lamp_akq(fun(A,bool),fun(fun(B,fun(A,bool)),fun(B,bool)),P),Q))))
      <=> ! [Y7: A] :
            ( pp(aa(A,bool,P,Y7))
           => pp(aa(set(B),bool,finite_finite2(B),collect(B,aa(A,fun(B,bool),aTP_Lamp_akr(fun(B,fun(A,bool)),fun(A,fun(B,bool)),Q),Y7)))) ) ) ) ).

% finite_Collect_bounded_ex
tff(fact_7935_eq__or__mem__image__simp,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),A2: B,B4: set(B)] : collect(A,aa(set(B),fun(A,bool),aa(B,fun(set(B),fun(A,bool)),aTP_Lamp_aks(fun(B,A),fun(B,fun(set(B),fun(A,bool))),F2),A2),B4)) = insert(A,aa(B,A,F2,A2),collect(A,aa(set(B),fun(A,bool),aTP_Lamp_akt(fun(B,A),fun(set(B),fun(A,bool)),F2),B4))) ).

% eq_or_mem_image_simp
tff(fact_7936_Least__eq__0,axiom,
    ! [P: fun(nat,bool)] :
      ( pp(aa(nat,bool,P,zero_zero(nat)))
     => ( ord_Least(nat,P) = zero_zero(nat) ) ) ).

% Least_eq_0
tff(fact_7937_continuous__on__of__real__o__iff,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [S: set(A),G: fun(A,real)] :
          ( topolo81223032696312382ous_on(A,complex,S,aTP_Lamp_aku(fun(A,real),fun(A,complex),G))
        <=> topolo81223032696312382ous_on(A,real,S,G) ) ) ).

% continuous_on_of_real_o_iff
tff(fact_7938_continuous__on__cis,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [A3: set(A),F2: fun(A,real)] :
          ( topolo81223032696312382ous_on(A,real,A3,F2)
         => topolo81223032696312382ous_on(A,complex,A3,aTP_Lamp_akv(fun(A,real),fun(A,complex),F2)) ) ) ).

% continuous_on_cis
tff(fact_7939_eventually__ex,axiom,
    ! [B: $tType,A: $tType,P: fun(A,fun(B,bool)),F3: filter(A)] :
      ( eventually(A,aTP_Lamp_akw(fun(A,fun(B,bool)),fun(A,bool),P),F3)
    <=> ? [Y8: fun(A,B)] : eventually(A,aa(fun(A,B),fun(A,bool),aTP_Lamp_akx(fun(A,fun(B,bool)),fun(fun(A,B),fun(A,bool)),P),Y8),F3) ) ).

% eventually_ex
tff(fact_7940_LeastI2__wellorder__ex,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [P: fun(A,bool),Q: fun(A,bool)] :
          ( ? [X_12: A] : pp(aa(A,bool,P,X_12))
         => ( ! [A4: A] :
                ( pp(aa(A,bool,P,A4))
               => ( ! [B11: A] :
                      ( pp(aa(A,bool,P,B11))
                     => pp(aa(A,bool,ord_less_eq(A,A4),B11)) )
                 => pp(aa(A,bool,Q,A4)) ) )
           => pp(aa(A,bool,Q,ord_Least(A,P))) ) ) ) ).

% LeastI2_wellorder_ex
tff(fact_7941_LeastI2__wellorder,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [P: fun(A,bool),A2: A,Q: fun(A,bool)] :
          ( pp(aa(A,bool,P,A2))
         => ( ! [A4: A] :
                ( pp(aa(A,bool,P,A4))
               => ( ! [B11: A] :
                      ( pp(aa(A,bool,P,B11))
                     => pp(aa(A,bool,ord_less_eq(A,A4),B11)) )
                 => pp(aa(A,bool,Q,A4)) ) )
           => pp(aa(A,bool,Q,ord_Least(A,P))) ) ) ) ).

% LeastI2_wellorder
tff(fact_7942_Least__equality,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [P: fun(A,bool),X: A] :
          ( pp(aa(A,bool,P,X))
         => ( ! [Y3: A] :
                ( pp(aa(A,bool,P,Y3))
               => pp(aa(A,bool,ord_less_eq(A,X),Y3)) )
           => ( ord_Least(A,P) = X ) ) ) ) ).

% Least_equality
tff(fact_7943_LeastI2__order,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [P: fun(A,bool),X: A,Q: fun(A,bool)] :
          ( pp(aa(A,bool,P,X))
         => ( ! [Y3: A] :
                ( pp(aa(A,bool,P,Y3))
               => pp(aa(A,bool,ord_less_eq(A,X),Y3)) )
           => ( ! [X3: A] :
                  ( pp(aa(A,bool,P,X3))
                 => ( ! [Y4: A] :
                        ( pp(aa(A,bool,P,Y4))
                       => pp(aa(A,bool,ord_less_eq(A,X3),Y4)) )
                   => pp(aa(A,bool,Q,X3)) ) )
             => pp(aa(A,bool,Q,ord_Least(A,P))) ) ) ) ) ).

% LeastI2_order
tff(fact_7944_Least1__le,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [P: fun(A,bool),Z: A] :
          ( ? [X4: A] :
              ( pp(aa(A,bool,P,X4))
              & ! [Y3: A] :
                  ( pp(aa(A,bool,P,Y3))
                 => pp(aa(A,bool,ord_less_eq(A,X4),Y3)) )
              & ! [Y3: A] :
                  ( ( pp(aa(A,bool,P,Y3))
                    & ! [Ya2: A] :
                        ( pp(aa(A,bool,P,Ya2))
                       => pp(aa(A,bool,ord_less_eq(A,Y3),Ya2)) ) )
                 => ( Y3 = X4 ) ) )
         => ( pp(aa(A,bool,P,Z))
           => pp(aa(A,bool,ord_less_eq(A,ord_Least(A,P)),Z)) ) ) ) ).

% Least1_le
tff(fact_7945_Least1I,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [P: fun(A,bool)] :
          ( ? [X4: A] :
              ( pp(aa(A,bool,P,X4))
              & ! [Y3: A] :
                  ( pp(aa(A,bool,P,Y3))
                 => pp(aa(A,bool,ord_less_eq(A,X4),Y3)) )
              & ! [Y3: A] :
                  ( ( pp(aa(A,bool,P,Y3))
                    & ! [Ya2: A] :
                        ( pp(aa(A,bool,P,Ya2))
                       => pp(aa(A,bool,ord_less_eq(A,Y3),Ya2)) ) )
                 => ( Y3 = X4 ) ) )
         => pp(aa(A,bool,P,ord_Least(A,P))) ) ) ).

% Least1I
tff(fact_7946_Least__le,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [P: fun(A,bool),K: A] :
          ( pp(aa(A,bool,P,K))
         => pp(aa(A,bool,ord_less_eq(A,ord_Least(A,P)),K)) ) ) ).

% Least_le
tff(fact_7947_Least__Suc2,axiom,
    ! [P: fun(nat,bool),N: nat,Q: fun(nat,bool),M: nat] :
      ( pp(aa(nat,bool,P,N))
     => ( pp(aa(nat,bool,Q,M))
       => ( ~ pp(aa(nat,bool,P,zero_zero(nat)))
         => ( ! [K3: nat] :
                ( pp(aa(nat,bool,P,aa(nat,nat,suc,K3)))
              <=> pp(aa(nat,bool,Q,K3)) )
           => ( ord_Least(nat,P) = aa(nat,nat,suc,ord_Least(nat,Q)) ) ) ) ) ) ).

% Least_Suc2
tff(fact_7948_full__SetCompr__eq,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A)] : collect(A,aTP_Lamp_aky(fun(B,A),fun(A,bool),F2)) = aa(set(B),set(A),image(B,A,F2),top_top(set(B))) ).

% full_SetCompr_eq
tff(fact_7949_Setcompr__eq__image,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),A3: set(B)] : collect(A,aa(set(B),fun(A,bool),aTP_Lamp_akt(fun(B,A),fun(set(B),fun(A,bool)),F2),A3)) = aa(set(B),set(A),image(B,A,F2),A3) ).

% Setcompr_eq_image
tff(fact_7950_setcompr__eq__image,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),P: fun(B,bool)] : collect(A,aa(fun(B,bool),fun(A,bool),aTP_Lamp_akz(fun(B,A),fun(fun(B,bool),fun(A,bool)),F2),P)) = aa(set(B),set(A),image(B,A,F2),collect(B,P)) ).

% setcompr_eq_image
tff(fact_7951_some__theI,axiom,
    ! [B: $tType,A: $tType,P: fun(A,fun(B,bool))] :
      ( ? [A8: A,X_12: B] : pp(aa(B,bool,aa(A,fun(B,bool),P,A8),X_12))
     => ( ! [B1: B,B22: B] :
            ( ? [A4: A] : pp(aa(B,bool,aa(A,fun(B,bool),P,A4),B1))
           => ( ? [A4: A] : pp(aa(B,bool,aa(A,fun(B,bool),P,A4),B22))
             => ( B1 = B22 ) ) )
       => pp(aa(B,bool,aa(A,fun(B,bool),P,fChoice(A,aTP_Lamp_akw(fun(A,fun(B,bool)),fun(A,bool),P))),the(B,aTP_Lamp_ala(fun(A,fun(B,bool)),fun(B,bool),P)))) ) ) ).

% some_theI
tff(fact_7952_not__less__Least,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [K: A,P: fun(A,bool)] :
          ( pp(aa(A,bool,ord_less(A,K),ord_Least(A,P)))
         => ~ pp(aa(A,bool,P,K)) ) ) ).

% not_less_Least
tff(fact_7953_LeastI,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [P: fun(A,bool),K: A] :
          ( pp(aa(A,bool,P,K))
         => pp(aa(A,bool,P,ord_Least(A,P))) ) ) ).

% LeastI
tff(fact_7954_ran__def,axiom,
    ! [B: $tType,A: $tType,M: fun(A,option(B))] : ran(A,B,M) = collect(B,aTP_Lamp_alb(fun(A,option(B)),fun(B,bool),M)) ).

% ran_def
tff(fact_7955_Least__Suc,axiom,
    ! [P: fun(nat,bool),N: nat] :
      ( pp(aa(nat,bool,P,N))
     => ( ~ pp(aa(nat,bool,P,zero_zero(nat)))
       => ( ord_Least(nat,P) = aa(nat,nat,suc,ord_Least(nat,aTP_Lamp_qd(fun(nat,bool),fun(nat,bool),P))) ) ) ) ).

% Least_Suc
tff(fact_7956_open__Collect__ex,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [P: fun(B,fun(A,bool))] :
          ( ! [I2: B] : topolo1002775350975398744n_open(A,collect(A,aa(B,fun(A,bool),P,I2)))
         => topolo1002775350975398744n_open(A,collect(A,aTP_Lamp_alc(fun(B,fun(A,bool)),fun(A,bool),P))) ) ) ).

% open_Collect_ex
tff(fact_7957_open__diagonal__complement,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => topolo1002775350975398744n_open(product_prod(A,A),collect(product_prod(A,A),aTP_Lamp_ald(product_prod(A,A),bool))) ) ).

% open_diagonal_complement
tff(fact_7958_open__subdiagonal,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => topolo1002775350975398744n_open(product_prod(A,A),collect(product_prod(A,A),aTP_Lamp_ale(product_prod(A,A),bool))) ) ).

% open_subdiagonal
tff(fact_7959_open__superdiagonal,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => topolo1002775350975398744n_open(product_prod(A,A),collect(product_prod(A,A),aTP_Lamp_alf(product_prod(A,A),bool))) ) ).

% open_superdiagonal
tff(fact_7960_finite__image__set2,axiom,
    ! [C: $tType,B: $tType,A: $tType,P: fun(A,bool),Q: fun(B,bool),F2: fun(A,fun(B,C))] :
      ( pp(aa(set(A),bool,finite_finite2(A),collect(A,P)))
     => ( pp(aa(set(B),bool,finite_finite2(B),collect(B,Q)))
       => pp(aa(set(C),bool,finite_finite2(C),collect(C,aa(fun(A,fun(B,C)),fun(C,bool),aa(fun(B,bool),fun(fun(A,fun(B,C)),fun(C,bool)),aTP_Lamp_alg(fun(A,bool),fun(fun(B,bool),fun(fun(A,fun(B,C)),fun(C,bool))),P),Q),F2)))) ) ) ).

% finite_image_set2
tff(fact_7961_finite__image__set,axiom,
    ! [B: $tType,A: $tType,P: fun(A,bool),F2: fun(A,B)] :
      ( pp(aa(set(A),bool,finite_finite2(A),collect(A,P)))
     => pp(aa(set(B),bool,finite_finite2(B),collect(B,aa(fun(A,B),fun(B,bool),aTP_Lamp_alh(fun(A,bool),fun(fun(A,B),fun(B,bool)),P),F2)))) ) ).

% finite_image_set
tff(fact_7962_set__conv__nth,axiom,
    ! [A: $tType,Xs: list(A)] : aa(list(A),set(A),set2(A),Xs) = collect(A,aTP_Lamp_ali(list(A),fun(A,bool),Xs)) ).

% set_conv_nth
tff(fact_7963_Least__def,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [P: fun(A,bool)] : ord_Least(A,P) = the(A,aTP_Lamp_alj(fun(A,bool),fun(A,bool),P)) ) ).

% Least_def
tff(fact_7964_sorted__list__of__set__greaterThanAtMost,axiom,
    ! [I: nat,J: nat] :
      ( pp(aa(nat,bool,ord_less_eq(nat,aa(nat,nat,suc,I)),J))
     => ( linord4507533701916653071of_set(nat,set_or3652927894154168847AtMost(nat,I,J)) = aa(list(nat),list(nat),cons(nat,aa(nat,nat,suc,I)),linord4507533701916653071of_set(nat,set_or3652927894154168847AtMost(nat,aa(nat,nat,suc,I),J))) ) ) ).

% sorted_list_of_set_greaterThanAtMost
tff(fact_7965_sorted__list__of__set__greaterThanLessThan,axiom,
    ! [I: nat,J: nat] :
      ( pp(aa(nat,bool,ord_less(nat,aa(nat,nat,suc,I)),J))
     => ( linord4507533701916653071of_set(nat,set_or5935395276787703475ssThan(nat,I,J)) = aa(list(nat),list(nat),cons(nat,aa(nat,nat,suc,I)),linord4507533701916653071of_set(nat,set_or5935395276787703475ssThan(nat,aa(nat,nat,suc,I),J))) ) ) ).

% sorted_list_of_set_greaterThanLessThan
tff(fact_7966_nth__sorted__list__of__set__greaterThanAtMost,axiom,
    ! [N: nat,J: nat,I: nat] :
      ( pp(aa(nat,bool,ord_less(nat,N),aa(nat,nat,minus_minus(nat,J),I)))
     => ( aa(nat,nat,nth(nat,linord4507533701916653071of_set(nat,set_or3652927894154168847AtMost(nat,I,J))),N) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),N)) ) ) ).

% nth_sorted_list_of_set_greaterThanAtMost
tff(fact_7967_pairself__image__eq,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),P: fun(B,fun(B,bool))] : aa(set(product_prod(B,B)),set(product_prod(A,A)),image(product_prod(B,B),product_prod(A,A),pairself(B,A,F2)),collect(product_prod(B,B),product_case_prod(B,B,bool,P))) = collect(product_prod(A,A),aa(fun(B,fun(B,bool)),fun(product_prod(A,A),bool),aTP_Lamp_alk(fun(B,A),fun(fun(B,fun(B,bool)),fun(product_prod(A,A),bool)),F2),P)) ).

% pairself_image_eq
tff(fact_7968_VEBT__internal_Ovalid_H_Oelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ~ vEBT_VEBT_valid(X,Xa2)
     => ( ( ? [Uu2: bool,Uv2: bool] : X = vEBT_Leaf(Uu2,Uv2)
         => ( Xa2 = one_one(nat) ) )
       => ~ ! [Mima: option(product_prod(nat,nat)),Deg2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
              ( ( X = vEBT_Node(Mima,Deg2,TreeList2,Summary2) )
             => ( ( Deg2 = Xa2 )
                & ! [X3: vEBT_VEBT] :
                    ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),member(vEBT_VEBT),X3),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2)))
                   => vEBT_VEBT_valid(X3,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg2),aa(num,nat,numeral_numeral(nat),bit0(one2)))) )
                & vEBT_VEBT_valid(Summary2,aa(nat,nat,minus_minus(nat,Deg2),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg2),aa(num,nat,numeral_numeral(nat),bit0(one2)))))
                & ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,minus_minus(nat,Deg2),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg2),aa(num,nat,numeral_numeral(nat),bit0(one2))))) )
                & pp(case_option(bool,product_prod(nat,nat),fconj(aa(bool,bool,fNot,aa(fun(nat,bool),bool,fEx(nat),aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,Summary2))),fAll(vEBT_VEBT,combs(vEBT_VEBT,bool,bool,combb(bool,fun(bool,bool),vEBT_VEBT,fimplies,combc(vEBT_VEBT,set(vEBT_VEBT),bool,member(vEBT_VEBT),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2))),combb(bool,bool,vEBT_VEBT,fNot,combb(fun(nat,bool),bool,vEBT_VEBT,fEx(nat),vEBT_V8194947554948674370ptions))))),product_case_prod(nat,nat,bool,aa(vEBT_VEBT,fun(nat,fun(nat,bool)),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,bool))),aTP_Lamp_all(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,bool)))),Deg2),TreeList2),Summary2)),Mima)) ) ) ) ) ).

% VEBT_internal.valid'.elims(3)
tff(fact_7969_less__eq__option__def,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [X: option(A),Y: option(A)] :
          ( pp(aa(option(A),bool,ord_less_eq(option(A),X),Y))
        <=> pp(case_option(bool,A,fTrue,aTP_Lamp_alm(option(A),fun(A,bool),Y),X)) ) ) ).

% less_eq_option_def
tff(fact_7970_eventually__ball__finite,axiom,
    ! [A: $tType,B: $tType,A3: set(A),P: fun(B,fun(A,bool)),Net: filter(B)] :
      ( pp(aa(set(A),bool,finite_finite2(A),A3))
     => ( ! [X3: A] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),A3))
           => eventually(B,aa(A,fun(B,bool),aTP_Lamp_akr(fun(B,fun(A,bool)),fun(A,fun(B,bool)),P),X3),Net) )
       => eventually(B,aa(fun(B,fun(A,bool)),fun(B,bool),aTP_Lamp_aln(set(A),fun(fun(B,fun(A,bool)),fun(B,bool)),A3),P),Net) ) ) ).

% eventually_ball_finite
tff(fact_7971_eventually__ball__finite__distrib,axiom,
    ! [A: $tType,B: $tType,A3: set(A),P: fun(B,fun(A,bool)),Net: filter(B)] :
      ( pp(aa(set(A),bool,finite_finite2(A),A3))
     => ( eventually(B,aa(fun(B,fun(A,bool)),fun(B,bool),aTP_Lamp_aln(set(A),fun(fun(B,fun(A,bool)),fun(B,bool)),A3),P),Net)
      <=> ! [X2: A] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X2),A3))
           => eventually(B,aa(A,fun(B,bool),aTP_Lamp_akr(fun(B,fun(A,bool)),fun(A,fun(B,bool)),P),X2),Net) ) ) ) ).

% eventually_ball_finite_distrib
tff(fact_7972_case__optionE,axiom,
    ! [A: $tType,P: bool,Q: fun(A,bool),X: option(A)] :
      ( pp(case_option(bool,A,P,Q,X))
     => ( ( ( X = none(A) )
         => ~ pp(P) )
       => ~ ! [Y3: A] :
              ( ( X = aa(A,option(A),some(A),Y3) )
             => ~ pp(aa(A,bool,Q,Y3)) ) ) ) ).

% case_optionE
tff(fact_7973_option_Ocase__distrib,axiom,
    ! [B: $tType,C: $tType,A: $tType,H: fun(B,C),F13: B,F24: fun(A,B),Option: option(A)] : aa(B,C,H,case_option(B,A,F13,F24,Option)) = case_option(C,A,aa(B,C,H,F13),aa(fun(A,B),fun(A,C),aTP_Lamp_alo(fun(B,C),fun(fun(A,B),fun(A,C)),H),F24),Option) ).

% option.case_distrib
tff(fact_7974_option_Odisc__eq__case_I1_J,axiom,
    ! [A: $tType,Option: option(A)] :
      ( ( Option = none(A) )
    <=> pp(case_option(bool,A,fTrue,aTP_Lamp_ll(A,bool),Option)) ) ).

% option.disc_eq_case(1)
tff(fact_7975_option_Odisc__eq__case_I2_J,axiom,
    ! [A: $tType,Option: option(A)] :
      ( ( Option != none(A) )
    <=> pp(case_option(bool,A,fFalse,aTP_Lamp_pu(A,bool),Option)) ) ).

% option.disc_eq_case(2)
tff(fact_7976_option_Osimps_I4_J,axiom,
    ! [A: $tType,B: $tType,F13: B,F24: fun(A,B)] : case_option(B,A,F13,F24,none(A)) = F13 ).

% option.simps(4)
tff(fact_7977_option_Ocase__eq__if,axiom,
    ! [B: $tType,A: $tType,Option: option(A),F13: B,F24: fun(A,B)] :
      ( ( ( Option = none(A) )
       => ( case_option(B,A,F13,F24,Option) = F13 ) )
      & ( ( Option != none(A) )
       => ( case_option(B,A,F13,F24,Option) = aa(A,B,F24,the2(A,Option)) ) ) ) ).

% option.case_eq_if
tff(fact_7978_Ball__comp__iff,axiom,
    ! [A: $tType,B: $tType,C: $tType,A3: fun(B,set(C)),F2: fun(C,bool),G: fun(A,B),X4: A] :
      ( pp(aa(A,bool,comp(B,bool,A,aa(fun(C,bool),fun(B,bool),aTP_Lamp_alp(fun(B,set(C)),fun(fun(C,bool),fun(B,bool)),A3),F2),G),X4))
    <=> ! [Xa4: C] :
          ( pp(aa(set(C),bool,aa(C,fun(set(C),bool),member(C),Xa4),aa(A,set(C),comp(B,set(C),A,A3,G),X4)))
         => pp(aa(C,bool,F2,Xa4)) ) ) ).

% Ball_comp_iff
tff(fact_7979_less__option__def,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [X: option(A),Y: option(A)] :
          ( pp(aa(option(A),bool,ord_less(option(A),X),Y))
        <=> pp(case_option(bool,A,fFalse,aTP_Lamp_alr(option(A),fun(A,bool),X),Y)) ) ) ).

% less_option_def
tff(fact_7980_option_Othe__def,axiom,
    ! [A: $tType,Option: option(A)] : the2(A,Option) = case_option(A,A,undefined(A),aTP_Lamp_at(A,A),Option) ).

% option.the_def
tff(fact_7981_option_Osplit__sel__asm,axiom,
    ! [B: $tType,A: $tType,P: fun(B,bool),F13: B,F24: fun(A,B),Option: option(A)] :
      ( pp(aa(B,bool,P,case_option(B,A,F13,F24,Option)))
    <=> ~ ( ( ( Option = none(A) )
            & ~ pp(aa(B,bool,P,F13)) )
          | ( ( Option = aa(A,option(A),some(A),the2(A,Option)) )
            & ~ pp(aa(B,bool,P,aa(A,B,F24,the2(A,Option)))) ) ) ) ).

% option.split_sel_asm
tff(fact_7982_option_Osplit__sel,axiom,
    ! [B: $tType,A: $tType,P: fun(B,bool),F13: B,F24: fun(A,B),Option: option(A)] :
      ( pp(aa(B,bool,P,case_option(B,A,F13,F24,Option)))
    <=> ( ( ( Option = none(A) )
         => pp(aa(B,bool,P,F13)) )
        & ( ( Option = aa(A,option(A),some(A),the2(A,Option)) )
         => pp(aa(B,bool,P,aa(A,B,F24,the2(A,Option)))) ) ) ) ).

% option.split_sel
tff(fact_7983_VEBT__internal_Ovalid_H_Osimps_I2_J,axiom,
    ! [Mima2: option(product_prod(nat,nat)),Deg: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,Deg4: nat] :
      ( vEBT_VEBT_valid(vEBT_Node(Mima2,Deg,TreeList,Summary),Deg4)
    <=> ( ( Deg = Deg4 )
        & ! [X2: vEBT_VEBT] :
            ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),member(vEBT_VEBT),X2),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList)))
           => vEBT_VEBT_valid(X2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))) )
        & vEBT_VEBT_valid(Summary,aa(nat,nat,minus_minus(nat,Deg),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))
        & ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,minus_minus(nat,Deg),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))) )
        & pp(case_option(bool,product_prod(nat,nat),fconj(aa(bool,bool,fNot,aa(fun(nat,bool),bool,fEx(nat),aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,Summary))),fAll(vEBT_VEBT,combs(vEBT_VEBT,bool,bool,combb(bool,fun(bool,bool),vEBT_VEBT,fimplies,combc(vEBT_VEBT,set(vEBT_VEBT),bool,member(vEBT_VEBT),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))),combb(bool,bool,vEBT_VEBT,fNot,combb(fun(nat,bool),bool,vEBT_VEBT,fEx(nat),vEBT_V8194947554948674370ptions))))),product_case_prod(nat,nat,bool,aa(vEBT_VEBT,fun(nat,fun(nat,bool)),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,bool))),aTP_Lamp_all(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,bool)))),Deg),TreeList),Summary)),Mima2)) ) ) ).

% VEBT_internal.valid'.simps(2)
tff(fact_7984_VEBT__internal_Ovalid_H_Oelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: bool] :
      ( ( vEBT_VEBT_valid(X,Xa2)
      <=> pp(Y) )
     => ( ( ? [Uu2: bool,Uv2: bool] : X = vEBT_Leaf(Uu2,Uv2)
         => ( pp(Y)
          <=> ( Xa2 != one_one(nat) ) ) )
       => ~ ! [Mima: option(product_prod(nat,nat)),Deg2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
              ( ( X = vEBT_Node(Mima,Deg2,TreeList2,Summary2) )
             => ( pp(Y)
              <=> ~ ( ( Deg2 = Xa2 )
                    & ! [X2: vEBT_VEBT] :
                        ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),member(vEBT_VEBT),X2),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2)))
                       => vEBT_VEBT_valid(X2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg2),aa(num,nat,numeral_numeral(nat),bit0(one2)))) )
                    & vEBT_VEBT_valid(Summary2,aa(nat,nat,minus_minus(nat,Deg2),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg2),aa(num,nat,numeral_numeral(nat),bit0(one2)))))
                    & ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,minus_minus(nat,Deg2),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg2),aa(num,nat,numeral_numeral(nat),bit0(one2))))) )
                    & pp(case_option(bool,product_prod(nat,nat),fconj(aa(bool,bool,fNot,aa(fun(nat,bool),bool,fEx(nat),aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,Summary2))),fAll(vEBT_VEBT,combs(vEBT_VEBT,bool,bool,combb(bool,fun(bool,bool),vEBT_VEBT,fimplies,combc(vEBT_VEBT,set(vEBT_VEBT),bool,member(vEBT_VEBT),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2))),combb(bool,bool,vEBT_VEBT,fNot,combb(fun(nat,bool),bool,vEBT_VEBT,fEx(nat),vEBT_V8194947554948674370ptions))))),product_case_prod(nat,nat,bool,aa(vEBT_VEBT,fun(nat,fun(nat,bool)),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,bool))),aTP_Lamp_all(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,bool)))),Deg2),TreeList2),Summary2)),Mima)) ) ) ) ) ) ).

% VEBT_internal.valid'.elims(1)
tff(fact_7985_VEBT__internal_Ovalid_H_Oelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( vEBT_VEBT_valid(X,Xa2)
     => ( ( ? [Uu2: bool,Uv2: bool] : X = vEBT_Leaf(Uu2,Uv2)
         => ( Xa2 != one_one(nat) ) )
       => ~ ! [Mima: option(product_prod(nat,nat)),Deg2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
              ( ( X = vEBT_Node(Mima,Deg2,TreeList2,Summary2) )
             => ~ ( ( Deg2 = Xa2 )
                  & ! [X4: vEBT_VEBT] :
                      ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),member(vEBT_VEBT),X4),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2)))
                     => vEBT_VEBT_valid(X4,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg2),aa(num,nat,numeral_numeral(nat),bit0(one2)))) )
                  & vEBT_VEBT_valid(Summary2,aa(nat,nat,minus_minus(nat,Deg2),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg2),aa(num,nat,numeral_numeral(nat),bit0(one2)))))
                  & ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,minus_minus(nat,Deg2),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg2),aa(num,nat,numeral_numeral(nat),bit0(one2))))) )
                  & pp(case_option(bool,product_prod(nat,nat),fconj(aa(bool,bool,fNot,aa(fun(nat,bool),bool,fEx(nat),aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,Summary2))),fAll(vEBT_VEBT,combs(vEBT_VEBT,bool,bool,combb(bool,fun(bool,bool),vEBT_VEBT,fimplies,combc(vEBT_VEBT,set(vEBT_VEBT),bool,member(vEBT_VEBT),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2))),combb(bool,bool,vEBT_VEBT,fNot,combb(fun(nat,bool),bool,vEBT_VEBT,fEx(nat),vEBT_V8194947554948674370ptions))))),product_case_prod(nat,nat,bool,aa(vEBT_VEBT,fun(nat,fun(nat,bool)),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,bool))),aTP_Lamp_all(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,bool)))),Deg2),TreeList2),Summary2)),Mima)) ) ) ) ) ).

% VEBT_internal.valid'.elims(2)
tff(fact_7986_VEBT__internal_Ovalid_H_Opelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ~ vEBT_VEBT_valid(X,Xa2)
     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_VEBT_valid_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),X),Xa2))
       => ( ! [Uu2: bool,Uv2: bool] :
              ( ( X = vEBT_Leaf(Uu2,Uv2) )
             => ( accp(product_prod(vEBT_VEBT,nat),vEBT_VEBT_valid_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf(Uu2,Uv2)),Xa2))
               => ( Xa2 = one_one(nat) ) ) )
         => ~ ! [Mima: option(product_prod(nat,nat)),Deg2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                ( ( X = vEBT_Node(Mima,Deg2,TreeList2,Summary2) )
               => ( accp(product_prod(vEBT_VEBT,nat),vEBT_VEBT_valid_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(Mima,Deg2,TreeList2,Summary2)),Xa2))
                 => ( ( Deg2 = Xa2 )
                    & ! [X3: vEBT_VEBT] :
                        ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),member(vEBT_VEBT),X3),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2)))
                       => vEBT_VEBT_valid(X3,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg2),aa(num,nat,numeral_numeral(nat),bit0(one2)))) )
                    & vEBT_VEBT_valid(Summary2,aa(nat,nat,minus_minus(nat,Deg2),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg2),aa(num,nat,numeral_numeral(nat),bit0(one2)))))
                    & ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,minus_minus(nat,Deg2),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg2),aa(num,nat,numeral_numeral(nat),bit0(one2))))) )
                    & pp(case_option(bool,product_prod(nat,nat),fconj(aa(bool,bool,fNot,aa(fun(nat,bool),bool,fEx(nat),aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,Summary2))),fAll(vEBT_VEBT,combs(vEBT_VEBT,bool,bool,combb(bool,fun(bool,bool),vEBT_VEBT,fimplies,combc(vEBT_VEBT,set(vEBT_VEBT),bool,member(vEBT_VEBT),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2))),combb(bool,bool,vEBT_VEBT,fNot,combb(fun(nat,bool),bool,vEBT_VEBT,fEx(nat),vEBT_V8194947554948674370ptions))))),product_case_prod(nat,nat,bool,aa(vEBT_VEBT,fun(nat,fun(nat,bool)),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,bool))),aTP_Lamp_all(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,bool)))),Deg2),TreeList2),Summary2)),Mima)) ) ) ) ) ) ) ).

% VEBT_internal.valid'.pelims(3)
tff(fact_7987_VEBT__internal_Ovalid_H_Opelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( vEBT_VEBT_valid(X,Xa2)
     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_VEBT_valid_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),X),Xa2))
       => ( ! [Uu2: bool,Uv2: bool] :
              ( ( X = vEBT_Leaf(Uu2,Uv2) )
             => ( accp(product_prod(vEBT_VEBT,nat),vEBT_VEBT_valid_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf(Uu2,Uv2)),Xa2))
               => ( Xa2 != one_one(nat) ) ) )
         => ~ ! [Mima: option(product_prod(nat,nat)),Deg2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                ( ( X = vEBT_Node(Mima,Deg2,TreeList2,Summary2) )
               => ( accp(product_prod(vEBT_VEBT,nat),vEBT_VEBT_valid_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(Mima,Deg2,TreeList2,Summary2)),Xa2))
                 => ~ ( ( Deg2 = Xa2 )
                      & ! [X4: vEBT_VEBT] :
                          ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),member(vEBT_VEBT),X4),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2)))
                         => vEBT_VEBT_valid(X4,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg2),aa(num,nat,numeral_numeral(nat),bit0(one2)))) )
                      & vEBT_VEBT_valid(Summary2,aa(nat,nat,minus_minus(nat,Deg2),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg2),aa(num,nat,numeral_numeral(nat),bit0(one2)))))
                      & ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,minus_minus(nat,Deg2),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg2),aa(num,nat,numeral_numeral(nat),bit0(one2))))) )
                      & pp(case_option(bool,product_prod(nat,nat),fconj(aa(bool,bool,fNot,aa(fun(nat,bool),bool,fEx(nat),aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,Summary2))),fAll(vEBT_VEBT,combs(vEBT_VEBT,bool,bool,combb(bool,fun(bool,bool),vEBT_VEBT,fimplies,combc(vEBT_VEBT,set(vEBT_VEBT),bool,member(vEBT_VEBT),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2))),combb(bool,bool,vEBT_VEBT,fNot,combb(fun(nat,bool),bool,vEBT_VEBT,fEx(nat),vEBT_V8194947554948674370ptions))))),product_case_prod(nat,nat,bool,aa(vEBT_VEBT,fun(nat,fun(nat,bool)),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,bool))),aTP_Lamp_all(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,bool)))),Deg2),TreeList2),Summary2)),Mima)) ) ) ) ) ) ) ).

% VEBT_internal.valid'.pelims(2)
tff(fact_7988_VEBT__internal_Ovalid_H_Opelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: bool] :
      ( ( vEBT_VEBT_valid(X,Xa2)
      <=> pp(Y) )
     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_VEBT_valid_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),X),Xa2))
       => ( ! [Uu2: bool,Uv2: bool] :
              ( ( X = vEBT_Leaf(Uu2,Uv2) )
             => ( ( pp(Y)
                <=> ( Xa2 = one_one(nat) ) )
               => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_VEBT_valid_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf(Uu2,Uv2)),Xa2)) ) )
         => ~ ! [Mima: option(product_prod(nat,nat)),Deg2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                ( ( X = vEBT_Node(Mima,Deg2,TreeList2,Summary2) )
               => ( ( pp(Y)
                  <=> ( ( Deg2 = Xa2 )
                      & ! [X2: vEBT_VEBT] :
                          ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),member(vEBT_VEBT),X2),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2)))
                         => vEBT_VEBT_valid(X2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg2),aa(num,nat,numeral_numeral(nat),bit0(one2)))) )
                      & vEBT_VEBT_valid(Summary2,aa(nat,nat,minus_minus(nat,Deg2),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg2),aa(num,nat,numeral_numeral(nat),bit0(one2)))))
                      & ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,minus_minus(nat,Deg2),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg2),aa(num,nat,numeral_numeral(nat),bit0(one2))))) )
                      & pp(case_option(bool,product_prod(nat,nat),fconj(aa(bool,bool,fNot,aa(fun(nat,bool),bool,fEx(nat),aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,Summary2))),fAll(vEBT_VEBT,combs(vEBT_VEBT,bool,bool,combb(bool,fun(bool,bool),vEBT_VEBT,fimplies,combc(vEBT_VEBT,set(vEBT_VEBT),bool,member(vEBT_VEBT),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2))),combb(bool,bool,vEBT_VEBT,fNot,combb(fun(nat,bool),bool,vEBT_VEBT,fEx(nat),vEBT_V8194947554948674370ptions))))),product_case_prod(nat,nat,bool,aa(vEBT_VEBT,fun(nat,fun(nat,bool)),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,bool))),aTP_Lamp_all(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,bool)))),Deg2),TreeList2),Summary2)),Mima)) ) )
                 => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_VEBT_valid_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(Mima,Deg2,TreeList2,Summary2)),Xa2)) ) ) ) ) ) ).

% VEBT_internal.valid'.pelims(1)
tff(fact_7989_take__bit__numeral__minus__numeral__word,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [M: num,N: num] : aa(word(A),word(A),bit_se2584673776208193580ke_bit(word(A),aa(num,nat,numeral_numeral(nat),M)),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),N))) = case_option(word(A),num,zero_zero(word(A)),aTP_Lamp_als(num,fun(num,word(A)),M),bit_take_bit_num(aa(num,nat,numeral_numeral(nat),M),N)) ) ).

% take_bit_numeral_minus_numeral_word
tff(fact_7990_take__bit__numeral__minus__numeral__int,axiom,
    ! [M: num,N: num] : aa(int,int,bit_se2584673776208193580ke_bit(int,aa(num,nat,numeral_numeral(nat),M)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),N))) = case_option(int,num,zero_zero(int),aTP_Lamp_alt(num,fun(num,int),M),bit_take_bit_num(aa(num,nat,numeral_numeral(nat),M),N)) ).

% take_bit_numeral_minus_numeral_int
tff(fact_7991_take__bit__num__simps_I1_J,axiom,
    ! [M: num] : bit_take_bit_num(zero_zero(nat),M) = none(num) ).

% take_bit_num_simps(1)
tff(fact_7992_take__bit__num__simps_I2_J,axiom,
    ! [N: nat] : bit_take_bit_num(aa(nat,nat,suc,N),one2) = aa(num,option(num),some(num),one2) ).

% take_bit_num_simps(2)
tff(fact_7993_take__bit__num__simps_I5_J,axiom,
    ! [R2: num] : bit_take_bit_num(aa(num,nat,numeral_numeral(nat),R2),one2) = aa(num,option(num),some(num),one2) ).

% take_bit_num_simps(5)
tff(fact_7994_take__bit__num__simps_I3_J,axiom,
    ! [N: nat,M: num] : bit_take_bit_num(aa(nat,nat,suc,N),bit0(M)) = case_option(option(num),num,none(num),aTP_Lamp_alu(num,option(num)),bit_take_bit_num(N,M)) ).

% take_bit_num_simps(3)
tff(fact_7995_take__bit__num__simps_I4_J,axiom,
    ! [N: nat,M: num] : bit_take_bit_num(aa(nat,nat,suc,N),aa(num,num,bit1,M)) = aa(num,option(num),some(num),case_option(num,num,one2,bit1,bit_take_bit_num(N,M))) ).

% take_bit_num_simps(4)
tff(fact_7996_take__bit__num__simps_I6_J,axiom,
    ! [R2: num,M: num] : bit_take_bit_num(aa(num,nat,numeral_numeral(nat),R2),bit0(M)) = case_option(option(num),num,none(num),aTP_Lamp_alu(num,option(num)),bit_take_bit_num(pred_numeral(R2),M)) ).

% take_bit_num_simps(6)
tff(fact_7997_take__bit__num__simps_I7_J,axiom,
    ! [R2: num,M: num] : bit_take_bit_num(aa(num,nat,numeral_numeral(nat),R2),aa(num,num,bit1,M)) = aa(num,option(num),some(num),case_option(num,num,one2,bit1,bit_take_bit_num(pred_numeral(R2),M))) ).

% take_bit_num_simps(7)
tff(fact_7998_take__bit__numeral__numeral,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: num,N: num] : aa(A,A,bit_se2584673776208193580ke_bit(A,aa(num,nat,numeral_numeral(nat),M)),aa(num,A,numeral_numeral(A),N)) = case_option(A,num,zero_zero(A),numeral_numeral(A),bit_take_bit_num(aa(num,nat,numeral_numeral(nat),M),N)) ) ).

% take_bit_numeral_numeral
tff(fact_7999_take__bit__num__eq__Some__imp,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,N: num,Q2: num] :
          ( ( bit_take_bit_num(M,N) = aa(num,option(num),some(num),Q2) )
         => ( aa(A,A,bit_se2584673776208193580ke_bit(A,M),aa(num,A,numeral_numeral(A),N)) = aa(num,A,numeral_numeral(A),Q2) ) ) ) ).

% take_bit_num_eq_Some_imp
tff(fact_8000_take__bit__num__eq__None__imp,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,N: num] :
          ( ( bit_take_bit_num(M,N) = none(num) )
         => ( aa(A,A,bit_se2584673776208193580ke_bit(A,M),aa(num,A,numeral_numeral(A),N)) = zero_zero(A) ) ) ) ).

% take_bit_num_eq_None_imp
tff(fact_8001_and__minus__numerals_I3_J,axiom,
    ! [M: num,N: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(num,int,numeral_numeral(int),M)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit0(N)))) = case_option(int,num,zero_zero(int),numeral_numeral(int),bit_and_not_num(M,bitM(N))) ).

% and_minus_numerals(3)
tff(fact_8002_and__minus__numerals_I7_J,axiom,
    ! [N: num,M: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit0(N)))),aa(num,int,numeral_numeral(int),M)) = case_option(int,num,zero_zero(int),numeral_numeral(int),bit_and_not_num(M,bitM(N))) ).

% and_minus_numerals(7)
tff(fact_8003_and__minus__numerals_I4_J,axiom,
    ! [M: num,N: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(num,int,numeral_numeral(int),M)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,N)))) = case_option(int,num,zero_zero(int),numeral_numeral(int),bit_and_not_num(M,bit0(N))) ).

% and_minus_numerals(4)
tff(fact_8004_and__minus__numerals_I8_J,axiom,
    ! [N: num,M: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,N)))),aa(num,int,numeral_numeral(int),M)) = case_option(int,num,zero_zero(int),numeral_numeral(int),bit_and_not_num(M,bit0(N))) ).

% and_minus_numerals(8)
tff(fact_8005_and__not__num_Osimps_I4_J,axiom,
    ! [M: num] : bit_and_not_num(bit0(M),one2) = aa(num,option(num),some(num),bit0(M)) ).

% and_not_num.simps(4)
tff(fact_8006_and__not__num_Osimps_I2_J,axiom,
    ! [N: num] : bit_and_not_num(one2,bit0(N)) = aa(num,option(num),some(num),one2) ).

% and_not_num.simps(2)
tff(fact_8007_and__not__num_Osimps_I3_J,axiom,
    ! [N: num] : bit_and_not_num(one2,aa(num,num,bit1,N)) = none(num) ).

% and_not_num.simps(3)
tff(fact_8008_and__not__num_Osimps_I1_J,axiom,
    bit_and_not_num(one2,one2) = none(num) ).

% and_not_num.simps(1)
tff(fact_8009_and__not__num_Osimps_I7_J,axiom,
    ! [M: num] : bit_and_not_num(aa(num,num,bit1,M),one2) = aa(num,option(num),some(num),bit0(M)) ).

% and_not_num.simps(7)
tff(fact_8010_and__not__num__eq__Some__iff,axiom,
    ! [M: num,N: num,Q2: num] :
      ( ( bit_and_not_num(M,N) = aa(num,option(num),some(num),Q2) )
    <=> ( aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(num,int,numeral_numeral(int),M)),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),N))) = aa(num,int,numeral_numeral(int),Q2) ) ) ).

% and_not_num_eq_Some_iff
tff(fact_8011_and__not__num_Osimps_I8_J,axiom,
    ! [M: num,N: num] : bit_and_not_num(aa(num,num,bit1,M),bit0(N)) = case_option(option(num),num,aa(num,option(num),some(num),one2),aTP_Lamp_alv(num,option(num)),bit_and_not_num(M,N)) ).

% and_not_num.simps(8)
tff(fact_8012_and__not__num__eq__None__iff,axiom,
    ! [M: num,N: num] :
      ( ( bit_and_not_num(M,N) = none(num) )
    <=> ( aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(num,int,numeral_numeral(int),M)),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),N))) = zero_zero(int) ) ) ).

% and_not_num_eq_None_iff
tff(fact_8013_int__numeral__and__not__num,axiom,
    ! [M: num,N: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(num,int,numeral_numeral(int),M)),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),N))) = case_option(int,num,zero_zero(int),numeral_numeral(int),bit_and_not_num(M,N)) ).

% int_numeral_and_not_num
tff(fact_8014_int__numeral__not__and__num,axiom,
    ! [M: num,N: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),M))),aa(num,int,numeral_numeral(int),N)) = case_option(int,num,zero_zero(int),numeral_numeral(int),bit_and_not_num(N,M)) ).

% int_numeral_not_and_num
tff(fact_8015_Code__Abstract__Nat_Otake__bit__num__code_I3_J,axiom,
    ! [N: nat,M: num] : bit_take_bit_num(N,aa(num,num,bit1,M)) = case_nat(option(num),none(num),aTP_Lamp_alw(num,fun(nat,option(num)),M),N) ).

% Code_Abstract_Nat.take_bit_num_code(3)
tff(fact_8016_disjE__realizer2,axiom,
    ! [B: $tType,A: $tType,P: bool,Q: fun(A,bool),X: option(A),R: fun(B,bool),F2: B,G: fun(A,B)] :
      ( pp(case_option(bool,A,P,Q,X))
     => ( ( pp(P)
         => pp(aa(B,bool,R,F2)) )
       => ( ! [Q3: A] :
              ( pp(aa(A,bool,Q,Q3))
             => pp(aa(B,bool,R,aa(A,B,G,Q3))) )
         => pp(aa(B,bool,R,case_option(B,A,F2,G,X))) ) ) ) ).

% disjE_realizer2
tff(fact_8017_case__nat__numeral,axiom,
    ! [A: $tType,A2: A,F2: fun(nat,A),V: num] : case_nat(A,A2,F2,aa(num,nat,numeral_numeral(nat),V)) = aa(nat,A,F2,pred_numeral(V)) ).

% case_nat_numeral
tff(fact_8018_case__nat__add__eq__if,axiom,
    ! [A: $tType,A2: A,F2: fun(nat,A),V: num,N: nat] : case_nat(A,A2,F2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),V)),N)) = aa(nat,A,F2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),pred_numeral(V)),N)) ).

% case_nat_add_eq_if
tff(fact_8019_nat_Ocase__distrib,axiom,
    ! [A: $tType,B: $tType,H: fun(A,B),F13: A,F24: fun(nat,A),Nat: nat] : aa(A,B,H,case_nat(A,F13,F24,Nat)) = case_nat(B,aa(A,B,H,F13),aa(fun(nat,A),fun(nat,B),aTP_Lamp_alx(fun(A,B),fun(fun(nat,A),fun(nat,B)),H),F24),Nat) ).

% nat.case_distrib
tff(fact_8020_old_Onat_Osimps_I5_J,axiom,
    ! [A: $tType,F13: A,F24: fun(nat,A),X23: nat] : case_nat(A,F13,F24,aa(nat,nat,suc,X23)) = aa(nat,A,F24,X23) ).

% old.nat.simps(5)
tff(fact_8021_old_Onat_Osimps_I4_J,axiom,
    ! [A: $tType,F13: A,F24: fun(nat,A)] : case_nat(A,F13,F24,zero_zero(nat)) = F13 ).

% old.nat.simps(4)
tff(fact_8022_nth__Cons,axiom,
    ! [A: $tType,X: A,Xs: list(A),N: nat] : aa(nat,A,nth(A,aa(list(A),list(A),cons(A,X),Xs)),N) = case_nat(A,X,nth(A,Xs),N) ).

% nth_Cons
tff(fact_8023_Nitpick_Ocase__nat__unfold,axiom,
    ! [A: $tType,N: nat,X: A,F2: fun(nat,A)] :
      ( ( ( N = zero_zero(nat) )
       => ( case_nat(A,X,F2,N) = X ) )
      & ( ( N != zero_zero(nat) )
       => ( case_nat(A,X,F2,N) = aa(nat,A,F2,aa(nat,nat,minus_minus(nat,N),one_one(nat))) ) ) ) ).

% Nitpick.case_nat_unfold
tff(fact_8024_Code__Abstract__Nat_Otake__bit__num__code_I1_J,axiom,
    ! [N: nat] : bit_take_bit_num(N,one2) = case_nat(option(num),none(num),aTP_Lamp_aly(nat,option(num)),N) ).

% Code_Abstract_Nat.take_bit_num_code(1)
tff(fact_8025_Code__Abstract__Nat_Otake__bit__num__code_I2_J,axiom,
    ! [N: nat,M: num] : bit_take_bit_num(N,bit0(M)) = case_nat(option(num),none(num),aTP_Lamp_alz(num,fun(nat,option(num)),M),N) ).

% Code_Abstract_Nat.take_bit_num_code(2)
tff(fact_8026_Bit__Operations_Otake__bit__num__code,axiom,
    ! [N: nat,M: num] : bit_take_bit_num(N,M) = aa(product_prod(nat,num),option(num),product_case_prod(nat,num,option(num),aTP_Lamp_amd(nat,fun(num,option(num)))),aa(num,product_prod(nat,num),aa(nat,fun(num,product_prod(nat,num)),product_Pair(nat,num),N),M)) ).

% Bit_Operations.take_bit_num_code
tff(fact_8027_take__bit__num__def,axiom,
    ! [N: nat,M: num] :
      ( ( ( aa(nat,nat,bit_se2584673776208193580ke_bit(nat,N),aa(num,nat,numeral_numeral(nat),M)) = zero_zero(nat) )
       => ( bit_take_bit_num(N,M) = none(num) ) )
      & ( ( aa(nat,nat,bit_se2584673776208193580ke_bit(nat,N),aa(num,nat,numeral_numeral(nat),M)) != zero_zero(nat) )
       => ( bit_take_bit_num(N,M) = aa(num,option(num),some(num),num_of_nat(aa(nat,nat,bit_se2584673776208193580ke_bit(nat,N),aa(num,nat,numeral_numeral(nat),M)))) ) ) ) ).

% take_bit_num_def
tff(fact_8028_num__of__nat__numeral__eq,axiom,
    ! [Q2: num] : num_of_nat(aa(num,nat,numeral_numeral(nat),Q2)) = Q2 ).

% num_of_nat_numeral_eq
tff(fact_8029_nat_Odisc__eq__case_I2_J,axiom,
    ! [Nat: nat] :
      ( ( Nat != zero_zero(nat) )
    <=> pp(case_nat(bool,fFalse,aTP_Lamp_pw(nat,bool),Nat)) ) ).

% nat.disc_eq_case(2)
tff(fact_8030_nat_Odisc__eq__case_I1_J,axiom,
    ! [Nat: nat] :
      ( ( Nat = zero_zero(nat) )
    <=> pp(case_nat(bool,fTrue,aTP_Lamp_pr(nat,bool),Nat)) ) ).

% nat.disc_eq_case(1)
tff(fact_8031_less__eq__nat_Osimps_I2_J,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,ord_less_eq(nat,aa(nat,nat,suc,M)),N))
    <=> pp(case_nat(bool,fFalse,ord_less_eq(nat,M),N)) ) ).

% less_eq_nat.simps(2)
tff(fact_8032_num__of__nat_Osimps_I1_J,axiom,
    num_of_nat(zero_zero(nat)) = one2 ).

% num_of_nat.simps(1)
tff(fact_8033_verit__eq__simplify_I17_J,axiom,
    ! [A: $tType,F13: A,F24: fun(num,A),F33: fun(num,A),X23: num] : case_num(A,F13,F24,F33,bit0(X23)) = aa(num,A,F24,X23) ).

% verit_eq_simplify(17)
tff(fact_8034_verit__eq__simplify_I16_J,axiom,
    ! [A: $tType,F13: A,F24: fun(num,A),F33: fun(num,A)] : case_num(A,F13,F24,F33,one2) = F13 ).

% verit_eq_simplify(16)
tff(fact_8035_verit__eq__simplify_I18_J,axiom,
    ! [A: $tType,F13: A,F24: fun(num,A),F33: fun(num,A),X33: num] : case_num(A,F13,F24,F33,aa(num,num,bit1,X33)) = aa(num,A,F33,X33) ).

% verit_eq_simplify(18)
tff(fact_8036_num_Ocase__distrib,axiom,
    ! [A: $tType,B: $tType,H: fun(A,B),F13: A,F24: fun(num,A),F33: fun(num,A),Num: num] : aa(A,B,H,case_num(A,F13,F24,F33,Num)) = case_num(B,aa(A,B,H,F13),aa(fun(num,A),fun(num,B),aTP_Lamp_ame(fun(A,B),fun(fun(num,A),fun(num,B)),H),F24),aa(fun(num,A),fun(num,B),aTP_Lamp_ame(fun(A,B),fun(fun(num,A),fun(num,B)),H),F33),Num) ).

% num.case_distrib
tff(fact_8037_max__Suc1,axiom,
    ! [N: nat,M: nat] : aa(nat,nat,ord_max(nat,aa(nat,nat,suc,N)),M) = case_nat(nat,aa(nat,nat,suc,N),aTP_Lamp_amf(nat,fun(nat,nat),N),M) ).

% max_Suc1
tff(fact_8038_max__Suc2,axiom,
    ! [M: nat,N: nat] : aa(nat,nat,ord_max(nat,M),aa(nat,nat,suc,N)) = case_nat(nat,aa(nat,nat,suc,N),aTP_Lamp_amg(nat,fun(nat,nat),N),M) ).

% max_Suc2
tff(fact_8039_list__update_Osimps_I2_J,axiom,
    ! [A: $tType,X: A,Xs: list(A),I: nat,V: A] : list_update(A,aa(list(A),list(A),cons(A,X),Xs),I,V) = case_nat(list(A),aa(list(A),list(A),cons(A,V),Xs),aa(A,fun(nat,list(A)),aa(list(A),fun(A,fun(nat,list(A))),aTP_Lamp_amh(A,fun(list(A),fun(A,fun(nat,list(A)))),X),Xs),V),I) ).

% list_update.simps(2)
tff(fact_8040_numeral__num__of__nat,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),N))
     => ( aa(num,nat,numeral_numeral(nat),num_of_nat(N)) = N ) ) ).

% numeral_num_of_nat
tff(fact_8041_num__of__nat__One,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,ord_less_eq(nat,N),one_one(nat)))
     => ( num_of_nat(N) = one2 ) ) ).

% num_of_nat_One
tff(fact_8042_diff__Suc,axiom,
    ! [M: nat,N: nat] : aa(nat,nat,minus_minus(nat,M),aa(nat,nat,suc,N)) = case_nat(nat,zero_zero(nat),aTP_Lamp_el(nat,nat),aa(nat,nat,minus_minus(nat,M),N)) ).

% diff_Suc
tff(fact_8043_bit__numeral__rec_I1_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [W: num,N: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,aa(num,A,numeral_numeral(A),bit0(W))),N))
        <=> pp(case_nat(bool,fFalse,bit_se5641148757651400278ts_bit(A,aa(num,A,numeral_numeral(A),W)),N)) ) ) ).

% bit_numeral_rec(1)
tff(fact_8044_bit__numeral__rec_I2_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [W: num,N: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,aa(num,A,numeral_numeral(A),aa(num,num,bit1,W))),N))
        <=> pp(case_nat(bool,fTrue,bit_se5641148757651400278ts_bit(A,aa(num,A,numeral_numeral(A),W)),N)) ) ) ).

% bit_numeral_rec(2)
tff(fact_8045_min__Suc2,axiom,
    ! [M: nat,N: nat] : ord_min(nat,M,aa(nat,nat,suc,N)) = case_nat(nat,zero_zero(nat),aTP_Lamp_ami(nat,fun(nat,nat),N),M) ).

% min_Suc2
tff(fact_8046_min__Suc1,axiom,
    ! [N: nat,M: nat] : ord_min(nat,aa(nat,nat,suc,N),M) = case_nat(nat,zero_zero(nat),aTP_Lamp_amj(nat,fun(nat,nat),N),M) ).

% min_Suc1
tff(fact_8047_numeral__num__of__nat__unfold,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [N: nat] :
          ( ( ( N = zero_zero(nat) )
           => ( aa(num,A,numeral_numeral(A),num_of_nat(N)) = one_one(A) ) )
          & ( ( N != zero_zero(nat) )
           => ( aa(num,A,numeral_numeral(A),num_of_nat(N)) = aa(nat,A,semiring_1_of_nat(A),N) ) ) ) ) ).

% numeral_num_of_nat_unfold
tff(fact_8048_num__of__nat__double,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),N))
     => ( num_of_nat(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),N)) = bit0(num_of_nat(N)) ) ) ).

% num_of_nat_double
tff(fact_8049_num__of__nat__plus__distrib,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),M))
     => ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),N))
       => ( num_of_nat(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N)) = aa(num,num,aa(num,fun(num,num),plus_plus(num),num_of_nat(M)),num_of_nat(N)) ) ) ) ).

% num_of_nat_plus_distrib
tff(fact_8050_num__of__nat_Osimps_I2_J,axiom,
    ! [N: nat] :
      ( ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),N))
       => ( num_of_nat(aa(nat,nat,suc,N)) = inc(num_of_nat(N)) ) )
      & ( ~ pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),N))
       => ( num_of_nat(aa(nat,nat,suc,N)) = one2 ) ) ) ).

% num_of_nat.simps(2)
tff(fact_8051_set__bits__int__unfold_H,axiom,
    ! [F2: fun(nat,bool)] :
      ( ( ? [N11: nat] :
          ! [N7: nat] :
            ( pp(aa(nat,bool,ord_less_eq(nat,N11),N7))
           => ~ pp(aa(nat,bool,F2,N7)) )
       => ( bit_bi4170147762399595738t_bits(int,F2) = groups4207007520872428315er_sum(bool,int,zero_neq_one_of_bool(int),aa(num,int,numeral_numeral(int),bit0(one2)),map(nat,bool,F2,upt(zero_zero(nat),ord_Least(nat,aTP_Lamp_amk(fun(nat,bool),fun(nat,bool),F2))))) ) )
      & ( ~ ? [N2: nat] :
            ! [N6: nat] :
              ( pp(aa(nat,bool,ord_less_eq(nat,N2),N6))
             => ~ pp(aa(nat,bool,F2,N6)) )
       => ( ( ? [N11: nat] :
              ! [N7: nat] :
                ( pp(aa(nat,bool,ord_less_eq(nat,N11),N7))
               => pp(aa(nat,bool,F2,N7)) )
           => ( bit_bi4170147762399595738t_bits(int,F2) = aa(int,int,bit_ri4674362597316999326ke_bit(int,ord_Least(nat,aTP_Lamp_aml(fun(nat,bool),fun(nat,bool),F2))),groups4207007520872428315er_sum(bool,int,zero_neq_one_of_bool(int),aa(num,int,numeral_numeral(int),bit0(one2)),append(bool,map(nat,bool,F2,upt(zero_zero(nat),ord_Least(nat,aTP_Lamp_aml(fun(nat,bool),fun(nat,bool),F2)))),aa(list(bool),list(bool),cons(bool,fTrue),nil(bool))))) ) )
          & ( ~ ? [N2: nat] :
                ! [N6: nat] :
                  ( pp(aa(nat,bool,ord_less_eq(nat,N2),N6))
                 => pp(aa(nat,bool,F2,N6)) )
           => ( bit_bi4170147762399595738t_bits(int,F2) = zero_zero(int) ) ) ) ) ) ).

% set_bits_int_unfold'
tff(fact_8052_Eps__Opt__def,axiom,
    ! [A: $tType,P: fun(A,bool)] :
      ( ( ? [X_12: A] : pp(aa(A,bool,P,X_12))
       => ( eps_Opt(A,P) = aa(A,option(A),some(A),fChoice(A,P)) ) )
      & ( ~ ? [X_1: A] : pp(aa(A,bool,P,X_1))
       => ( eps_Opt(A,P) = none(A) ) ) ) ).

% Eps_Opt_def
tff(fact_8053_concat__inth,axiom,
    ! [A: $tType,Xs: list(A),X: A,Ys: list(A)] : aa(nat,A,nth(A,append(A,Xs,append(A,aa(list(A),list(A),cons(A,X),nil(A)),Ys))),aa(list(A),nat,size_size(list(A)),Xs)) = X ).

% concat_inth
tff(fact_8054_some__opt__eq__trivial,axiom,
    ! [A: $tType,X: A] : eps_Opt(A,aTP_Lamp_lg(A,fun(A,bool),X)) = aa(A,option(A),some(A),X) ).

% some_opt_eq_trivial
tff(fact_8055_some__opt__false__trivial,axiom,
    ! [A: $tType] : eps_Opt(A,aTP_Lamp_ll(A,bool)) = none(A) ).

% some_opt_false_trivial
tff(fact_8056_length__0__conv,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = zero_zero(nat) )
    <=> ( Xs = nil(A) ) ) ).

% length_0_conv
tff(fact_8057_length__append,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] : aa(list(A),nat,size_size(list(A)),append(A,Xs,Ys)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(list(A),nat,size_size(list(A)),Xs)),aa(list(A),nat,size_size(list(A)),Ys)) ).

% length_append
tff(fact_8058_empty__replicate,axiom,
    ! [A: $tType,N: nat,X: A] :
      ( ( nil(A) = replicate(A,N,X) )
    <=> ( N = zero_zero(nat) ) ) ).

% empty_replicate
tff(fact_8059_replicate__empty,axiom,
    ! [A: $tType,N: nat,X: A] :
      ( ( replicate(A,N,X) = nil(A) )
    <=> ( N = zero_zero(nat) ) ) ).

% replicate_empty
tff(fact_8060_horner__sum__simps_I1_J,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_semiring_0(A)
     => ! [F2: fun(B,A),A2: A] : groups4207007520872428315er_sum(B,A,F2,A2,nil(B)) = zero_zero(A) ) ).

% horner_sum_simps(1)
tff(fact_8061_size__list__append,axiom,
    ! [A: $tType,F2: fun(A,nat),Xs: list(A),Ys: list(A)] : size_list(A,F2,append(A,Xs,Ys)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),size_list(A,F2,Xs)),size_list(A,F2,Ys)) ).

% size_list_append
tff(fact_8062_Eps__Opt__eq__None,axiom,
    ! [A: $tType,P: fun(A,bool)] :
      ( ( eps_Opt(A,P) = none(A) )
    <=> ~ ? [X_13: A] : pp(aa(A,bool,P,X_13)) ) ).

% Eps_Opt_eq_None
tff(fact_8063_length__greater__0__conv,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),aa(list(A),nat,size_size(list(A)),Xs)))
    <=> ( Xs != nil(A) ) ) ).

% length_greater_0_conv
tff(fact_8064_nth__append__length__plus,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),N: nat] : aa(nat,A,nth(A,append(A,Xs,Ys)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(list(A),nat,size_size(list(A)),Xs)),N)) = aa(nat,A,nth(A,Ys),N) ).

% nth_append_length_plus
tff(fact_8065_n__lists__Nil,axiom,
    ! [A: $tType,N: nat] :
      ( ( ( N = zero_zero(nat) )
       => ( n_lists(A,N,nil(A)) = aa(list(list(A)),list(list(A)),cons(list(A),nil(A)),nil(list(A))) ) )
      & ( ( N != zero_zero(nat) )
       => ( n_lists(A,N,nil(A)) = nil(list(A)) ) ) ) ).

% n_lists_Nil
tff(fact_8066_length__ge__1__conv,axiom,
    ! [A: $tType,L: list(A)] :
      ( pp(aa(nat,bool,ord_less_eq(nat,aa(nat,nat,suc,zero_zero(nat))),aa(list(A),nat,size_size(list(A)),L)))
    <=> ( L != nil(A) ) ) ).

% length_ge_1_conv
tff(fact_8067_concat__map__singleton,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),Xs: list(B)] : concat(A,map(B,list(A),aTP_Lamp_amm(fun(B,A),fun(B,list(A)),F2),Xs)) = map(B,A,F2,Xs) ).

% concat_map_singleton
tff(fact_8068_hash__code__list__simps_I1_J,axiom,
    ! [A: $tType,H_a: fun(A,uint32)] : hash_hash_code_list(A,H_a,nil(A)) = aa(num,uint32,numeral_numeral(uint32),bit0(bit0(aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,bit0(bit0(bit0(aa(num,num,bit1,bit0(aa(num,num,bit1,aa(num,num,bit1,bit0(bit0(aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,bit0(bit0(bit0(aa(num,num,bit1,bit0(aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,bit0(bit0(one2))))))))))))))))))))))))))))))) ).

% hash_code_list_simps(1)
tff(fact_8069_list__rest__coinc,axiom,
    ! [A: $tType,S22: list(A),S1: list(A),R1: list(A),R22: list(A)] :
      ( pp(aa(nat,bool,ord_less_eq(nat,aa(list(A),nat,size_size(list(A)),S22)),aa(list(A),nat,size_size(list(A)),S1)))
     => ( ( append(A,S1,R1) = append(A,S22,R22) )
       => ? [R1p: list(A)] : R22 = append(A,R1p,R1) ) ) ).

% list_rest_coinc
tff(fact_8070_length__compl__rev__induct,axiom,
    ! [A: $tType,P: fun(list(A),bool),L: list(A)] :
      ( pp(aa(list(A),bool,P,nil(A)))
     => ( ! [L3: list(A),E2: A] :
            ( ! [Ll: list(A)] :
                ( pp(aa(nat,bool,ord_less_eq(nat,aa(list(A),nat,size_size(list(A)),Ll)),aa(list(A),nat,size_size(list(A)),L3)))
               => pp(aa(list(A),bool,P,Ll)) )
           => pp(aa(list(A),bool,P,append(A,L3,aa(list(A),list(A),cons(A,E2),nil(A))))) )
       => pp(aa(list(A),bool,P,L)) ) ) ).

% length_compl_rev_induct
tff(fact_8071_length__compl__induct,axiom,
    ! [A: $tType,P: fun(list(A),bool),L: list(A)] :
      ( pp(aa(list(A),bool,P,nil(A)))
     => ( ! [E2: A,L3: list(A)] :
            ( ! [Ll: list(A)] :
                ( pp(aa(nat,bool,ord_less_eq(nat,aa(list(A),nat,size_size(list(A)),Ll)),aa(list(A),nat,size_size(list(A)),L3)))
               => pp(aa(list(A),bool,P,Ll)) )
           => pp(aa(list(A),bool,P,aa(list(A),list(A),cons(A,E2),L3))) )
       => pp(aa(list(A),bool,P,L)) ) ) ).

% length_compl_induct
tff(fact_8072_comm__append__is__replicate,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] :
      ( ( Xs != nil(A) )
     => ( ( Ys != nil(A) )
       => ( ( append(A,Xs,Ys) = append(A,Ys,Xs) )
         => ? [N2: nat,Zs: list(A)] :
              ( pp(aa(nat,bool,ord_less(nat,one_one(nat)),N2))
              & ( concat(A,replicate(list(A),N2,Zs)) = append(A,Xs,Ys) ) ) ) ) ) ).

% comm_append_is_replicate
tff(fact_8073_count__list_Osimps_I1_J,axiom,
    ! [A: $tType,Y: A] : aa(A,nat,count_list(A,nil(A)),Y) = zero_zero(nat) ).

% count_list.simps(1)
tff(fact_8074_replicate__0,axiom,
    ! [A: $tType,X: A] : replicate(A,zero_zero(nat),X) = nil(A) ).

% replicate_0
tff(fact_8075_replicate__add,axiom,
    ! [A: $tType,N: nat,M: nat,X: A] : replicate(A,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),M),X) = append(A,replicate(A,N,X),replicate(A,M,X)) ).

% replicate_add
tff(fact_8076_append__replicate__commute,axiom,
    ! [A: $tType,N: nat,X: A,K: nat] : append(A,replicate(A,N,X),replicate(A,K,X)) = append(A,replicate(A,K,X),replicate(A,N,X)) ).

% append_replicate_commute
tff(fact_8077_list_Osize__gen_I1_J,axiom,
    ! [A: $tType,X: fun(A,nat)] : size_list(A,X,nil(A)) = zero_zero(nat) ).

% list.size_gen(1)
tff(fact_8078_length__nth__simps_I1_J,axiom,
    ! [A: $tType] : aa(list(A),nat,size_size(list(A)),nil(A)) = zero_zero(nat) ).

% length_nth_simps(1)
tff(fact_8079_length__append__singleton,axiom,
    ! [A: $tType,Xs: list(A),X: A] : aa(list(A),nat,size_size(list(A)),append(A,Xs,aa(list(A),list(A),cons(A,X),nil(A)))) = aa(nat,nat,suc,aa(list(A),nat,size_size(list(A)),Xs)) ).

% length_append_singleton
tff(fact_8080_n__lists_Osimps_I1_J,axiom,
    ! [A: $tType,Xs: list(A)] : n_lists(A,zero_zero(nat),Xs) = aa(list(list(A)),list(list(A)),cons(list(A),nil(A)),nil(list(A))) ).

% n_lists.simps(1)
tff(fact_8081_replicate__append__same,axiom,
    ! [A: $tType,I: nat,X: A] : append(A,replicate(A,I,X),aa(list(A),list(A),cons(A,X),nil(A))) = aa(list(A),list(A),cons(A,X),replicate(A,I,X)) ).

% replicate_append_same
tff(fact_8082_replicate__app__Cons__same,axiom,
    ! [A: $tType,N: nat,X: A,Xs: list(A)] : append(A,replicate(A,N,X),aa(list(A),list(A),cons(A,X),Xs)) = aa(list(A),list(A),cons(A,X),append(A,replicate(A,N,X),Xs)) ).

% replicate_app_Cons_same
tff(fact_8083_replicate__Suc__conv__snoc,axiom,
    ! [A: $tType,N: nat,X: A] : replicate(A,aa(nat,nat,suc,N),X) = append(A,replicate(A,N,X),aa(list(A),list(A),cons(A,X),nil(A))) ).

% replicate_Suc_conv_snoc
tff(fact_8084_length__Suc__rev__conv,axiom,
    ! [A: $tType,Xs: list(A),N: nat] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(nat,nat,suc,N) )
    <=> ? [Ys2: list(A),Y7: A] :
          ( ( Xs = append(A,Ys2,aa(list(A),list(A),cons(A,Y7),nil(A))) )
          & ( aa(list(A),nat,size_size(list(A)),Ys2) = N ) ) ) ).

% length_Suc_rev_conv
tff(fact_8085_length__Suc__conv__rev,axiom,
    ! [A: $tType,Xs: list(A),N: nat] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(nat,nat,suc,N) )
    <=> ? [Y7: A,Ys2: list(A)] :
          ( ( Xs = append(A,Ys2,aa(list(A),list(A),cons(A,Y7),nil(A))) )
          & ( aa(list(A),nat,size_size(list(A)),Ys2) = N ) ) ) ).

% length_Suc_conv_rev
tff(fact_8086_list__decomp__1,axiom,
    ! [A: $tType,L: list(A)] :
      ( ( aa(list(A),nat,size_size(list(A)),L) = one_one(nat) )
     => ? [A4: A] : L = aa(list(A),list(A),cons(A,A4),nil(A)) ) ).

% list_decomp_1
tff(fact_8087_slice__prepend,axiom,
    ! [A: $tType,I: nat,K: nat,Xs: list(A),Ys: list(A)] :
      ( pp(aa(nat,bool,ord_less_eq(nat,I),K))
     => ( pp(aa(nat,bool,ord_less_eq(nat,K),aa(list(A),nat,size_size(list(A)),Xs)))
       => ( slice(A,I,K,Xs) = slice(A,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),aa(list(A),nat,size_size(list(A)),Ys)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),aa(list(A),nat,size_size(list(A)),Ys)),append(A,Ys,Xs)) ) ) ) ).

% slice_prepend
tff(fact_8088_horner__sum__append,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_semiring_1(A)
     => ! [F2: fun(B,A),A2: A,Xs: list(B),Ys: list(B)] : groups4207007520872428315er_sum(B,A,F2,A2,append(B,Xs,Ys)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),groups4207007520872428315er_sum(B,A,F2,A2,Xs)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(list(B),nat,size_size(list(B)),Xs))),groups4207007520872428315er_sum(B,A,F2,A2,Ys))) ) ).

% horner_sum_append
tff(fact_8089_list__decomp__2,axiom,
    ! [A: $tType,L: list(A)] :
      ( ( aa(list(A),nat,size_size(list(A)),L) = aa(num,nat,numeral_numeral(nat),bit0(one2)) )
     => ? [A4: A,B3: A] : L = aa(list(A),list(A),cons(A,A4),aa(list(A),list(A),cons(A,B3),nil(A))) ) ).

% list_decomp_2
tff(fact_8090_quickcheck__narrowing__samples_Onarrowing__samples_Oelims,axiom,
    ! [A: $tType] :
      ( ( code_term_of(A)
        & quickc6926020345158392990erm_of(A) )
     => ! [A_of_integer: fun(code_integer,product_prod(A,A)),Zero: A,X: code_integer,Y: list(A)] :
          ( ( code_T4080844693773952564amples(A,A_of_integer,Zero,X) = Y )
         => ( ( pp(aa(code_integer,bool,ord_less(code_integer,zero_zero(code_integer)),X))
             => ( Y = aa(product_prod(A,A),list(A),product_case_prod(A,A,list(A),aa(code_integer,fun(A,fun(A,list(A))),aa(A,fun(code_integer,fun(A,fun(A,list(A)))),aTP_Lamp_amn(fun(code_integer,product_prod(A,A)),fun(A,fun(code_integer,fun(A,fun(A,list(A))))),A_of_integer),Zero),X)),aa(code_integer,product_prod(A,A),A_of_integer,X)) ) )
            & ( ~ pp(aa(code_integer,bool,ord_less(code_integer,zero_zero(code_integer)),X))
             => ( Y = aa(list(A),list(A),cons(A,Zero),nil(A)) ) ) ) ) ) ).

% quickcheck_narrowing_samples.narrowing_samples.elims
tff(fact_8091_quickcheck__narrowing__samples_Onarrowing__samples_Osimps,axiom,
    ! [A: $tType] :
      ( ( code_term_of(A)
        & quickc6926020345158392990erm_of(A) )
     => ! [I: code_integer,A_of_integer: fun(code_integer,product_prod(A,A)),Zero: A] :
          ( ( pp(aa(code_integer,bool,ord_less(code_integer,zero_zero(code_integer)),I))
           => ( code_T4080844693773952564amples(A,A_of_integer,Zero,I) = aa(product_prod(A,A),list(A),product_case_prod(A,A,list(A),aa(A,fun(A,fun(A,list(A))),aa(fun(code_integer,product_prod(A,A)),fun(A,fun(A,fun(A,list(A)))),aTP_Lamp_amo(code_integer,fun(fun(code_integer,product_prod(A,A)),fun(A,fun(A,fun(A,list(A))))),I),A_of_integer),Zero)),aa(code_integer,product_prod(A,A),A_of_integer,I)) ) )
          & ( ~ pp(aa(code_integer,bool,ord_less(code_integer,zero_zero(code_integer)),I))
           => ( code_T4080844693773952564amples(A,A_of_integer,Zero,I) = aa(list(A),list(A),cons(A,Zero),nil(A)) ) ) ) ) ).

% quickcheck_narrowing_samples.narrowing_samples.simps
tff(fact_8092_upt__0__eq__Nil__conv,axiom,
    ! [J: nat] :
      ( ( upt(zero_zero(nat),J) = nil(nat) )
    <=> ( J = zero_zero(nat) ) ) ).

% upt_0_eq_Nil_conv
tff(fact_8093_upt__conv__Nil,axiom,
    ! [J: nat,I: nat] :
      ( pp(aa(nat,bool,ord_less_eq(nat,J),I))
     => ( upt(I,J) = nil(nat) ) ) ).

% upt_conv_Nil
tff(fact_8094_upt__merge,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( pp(aa(nat,bool,ord_less_eq(nat,I),J))
        & pp(aa(nat,bool,ord_less_eq(nat,J),K)) )
     => ( append(nat,upt(I,J),upt(J,K)) = upt(I,K) ) ) ).

% upt_merge
tff(fact_8095_upt__eq__Nil__conv,axiom,
    ! [I: nat,J: nat] :
      ( ( upt(I,J) = nil(nat) )
    <=> ( ( J = zero_zero(nat) )
        | pp(aa(nat,bool,ord_less_eq(nat,J),I)) ) ) ).

% upt_eq_Nil_conv
tff(fact_8096_sorted__list__of__set__lessThan__Suc,axiom,
    ! [K: nat] : linord4507533701916653071of_set(nat,set_ord_lessThan(nat,aa(nat,nat,suc,K))) = append(nat,linord4507533701916653071of_set(nat,set_ord_lessThan(nat,K)),aa(list(nat),list(nat),cons(nat,K),nil(nat))) ).

% sorted_list_of_set_lessThan_Suc
tff(fact_8097_sorted__list__of__set__atMost__Suc,axiom,
    ! [K: nat] : linord4507533701916653071of_set(nat,set_ord_atMost(nat,aa(nat,nat,suc,K))) = append(nat,linord4507533701916653071of_set(nat,set_ord_atMost(nat,K)),aa(list(nat),list(nat),cons(nat,aa(nat,nat,suc,K)),nil(nat))) ).

% sorted_list_of_set_atMost_Suc
tff(fact_8098_upt__rec__numeral,axiom,
    ! [M: num,N: num] :
      ( ( pp(aa(nat,bool,ord_less(nat,aa(num,nat,numeral_numeral(nat),M)),aa(num,nat,numeral_numeral(nat),N)))
       => ( upt(aa(num,nat,numeral_numeral(nat),M),aa(num,nat,numeral_numeral(nat),N)) = aa(list(nat),list(nat),cons(nat,aa(num,nat,numeral_numeral(nat),M)),upt(aa(nat,nat,suc,aa(num,nat,numeral_numeral(nat),M)),aa(num,nat,numeral_numeral(nat),N))) ) )
      & ( ~ pp(aa(nat,bool,ord_less(nat,aa(num,nat,numeral_numeral(nat),M)),aa(num,nat,numeral_numeral(nat),N)))
       => ( upt(aa(num,nat,numeral_numeral(nat),M),aa(num,nat,numeral_numeral(nat),N)) = nil(nat) ) ) ) ).

% upt_rec_numeral
tff(fact_8099_upt__eq__append__conv,axiom,
    ! [I: nat,J: nat,Xs: list(nat),Ys: list(nat)] :
      ( pp(aa(nat,bool,ord_less_eq(nat,I),J))
     => ( ( upt(I,J) = append(nat,Xs,Ys) )
      <=> ? [K2: nat] :
            ( pp(aa(nat,bool,ord_less_eq(nat,I),K2))
            & pp(aa(nat,bool,ord_less_eq(nat,K2),J))
            & ( upt(I,K2) = Xs )
            & ( upt(K2,J) = Ys ) ) ) ) ).

% upt_eq_append_conv
tff(fact_8100_upt__Suc,axiom,
    ! [I: nat,J: nat] :
      ( ( pp(aa(nat,bool,ord_less_eq(nat,I),J))
       => ( upt(I,aa(nat,nat,suc,J)) = append(nat,upt(I,J),aa(list(nat),list(nat),cons(nat,J),nil(nat))) ) )
      & ( ~ pp(aa(nat,bool,ord_less_eq(nat,I),J))
       => ( upt(I,aa(nat,nat,suc,J)) = nil(nat) ) ) ) ).

% upt_Suc
tff(fact_8101_upt__Suc__append,axiom,
    ! [I: nat,J: nat] :
      ( pp(aa(nat,bool,ord_less_eq(nat,I),J))
     => ( upt(I,aa(nat,nat,suc,J)) = append(nat,upt(I,J),aa(list(nat),list(nat),cons(nat,J),nil(nat))) ) ) ).

% upt_Suc_append
tff(fact_8102_upt__add__eq__append,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( pp(aa(nat,bool,ord_less_eq(nat,I),J))
     => ( upt(I,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),K)) = append(nat,upt(I,J),upt(J,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),K))) ) ) ).

% upt_add_eq_append
tff(fact_8103_upt__append,axiom,
    ! [I: nat,J: nat] :
      ( pp(aa(nat,bool,ord_less(nat,I),J))
     => ( append(nat,upt(zero_zero(nat),I),upt(I,J)) = upt(zero_zero(nat),J) ) ) ).

% upt_append
tff(fact_8104_upt__0,axiom,
    ! [I: nat] : upt(I,zero_zero(nat)) = nil(nat) ).

% upt_0
tff(fact_8105_list__encode_Ocases,axiom,
    ! [X: list(nat)] :
      ( ( X != nil(nat) )
     => ~ ! [X3: nat,Xs2: list(nat)] : X != aa(list(nat),list(nat),cons(nat,X3),Xs2) ) ).

% list_encode.cases
tff(fact_8106_subseqs_Osimps_I2_J,axiom,
    ! [A: $tType,X: A,Xs: list(A)] : subseqs(A,aa(list(A),list(A),cons(A,X),Xs)) = append(list(A),map(list(A),list(A),cons(A,X),subseqs(A,Xs)),subseqs(A,Xs)) ).

% subseqs.simps(2)
tff(fact_8107_upt__rec,axiom,
    ! [I: nat,J: nat] :
      ( ( pp(aa(nat,bool,ord_less(nat,I),J))
       => ( upt(I,J) = aa(list(nat),list(nat),cons(nat,I),upt(aa(nat,nat,suc,I),J)) ) )
      & ( ~ pp(aa(nat,bool,ord_less(nat,I),J))
       => ( upt(I,J) = nil(nat) ) ) ) ).

% upt_rec
tff(fact_8108_upt__eq__lel__conv,axiom,
    ! [L: nat,H: nat,Is1: list(nat),I: nat,Is2: list(nat)] :
      ( ( upt(L,H) = append(nat,Is1,aa(list(nat),list(nat),cons(nat,I),Is2)) )
    <=> ( ( Is1 = upt(L,I) )
        & ( Is2 = upt(aa(nat,nat,suc,I),H) )
        & pp(aa(nat,bool,ord_less_eq(nat,L),I))
        & pp(aa(nat,bool,ord_less(nat,I),H)) ) ) ).

% upt_eq_lel_conv
tff(fact_8109_transpose__rectangle,axiom,
    ! [A: $tType,Xs: list(list(A)),N: nat] :
      ( ( ( Xs = nil(list(A)) )
       => ( N = zero_zero(nat) ) )
     => ( ! [I2: nat] :
            ( pp(aa(nat,bool,ord_less(nat,I2),aa(list(list(A)),nat,size_size(list(list(A))),Xs)))
           => ( aa(list(A),nat,size_size(list(A)),aa(nat,list(A),nth(list(A),Xs),I2)) = N ) )
       => ( transpose(A,Xs) = map(nat,list(A),aTP_Lamp_amq(list(list(A)),fun(nat,list(A)),Xs),upt(zero_zero(nat),N)) ) ) ) ).

% transpose_rectangle
tff(fact_8110_lexn__conv,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),N: nat] : lexn(A,R2,N) = collect(product_prod(list(A),list(A)),product_case_prod(list(A),list(A),bool,aa(nat,fun(list(A),fun(list(A),bool)),aTP_Lamp_amr(set(product_prod(A,A)),fun(nat,fun(list(A),fun(list(A),bool))),R2),N))) ).

% lexn_conv
tff(fact_8111_lexn_Osimps_I1_J,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] : lexn(A,R2,zero_zero(nat)) = bot_bot(set(product_prod(list(A),list(A)))) ).

% lexn.simps(1)
tff(fact_8112_length__transpose,axiom,
    ! [A: $tType,Xs: list(list(A))] : aa(list(list(A)),nat,size_size(list(list(A))),transpose(A,Xs)) = foldr(list(A),nat,aTP_Lamp_ams(list(A),fun(nat,nat)),Xs,zero_zero(nat)) ).

% length_transpose
tff(fact_8113_lex__conv,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] : lex(A,R2) = collect(product_prod(list(A),list(A)),product_case_prod(list(A),list(A),bool,aTP_Lamp_amt(set(product_prod(A,A)),fun(list(A),fun(list(A),bool)),R2))) ).

% lex_conv
tff(fact_8114_less__eq__char__simp,axiom,
    ! [B0: bool,B12: bool,B23: bool,B32: bool,B42: bool,B52: bool,B62: bool,B72: bool,C0: bool,C1: bool,C22: bool,C32: bool,C42: bool,C52: bool,C62: bool,C72: bool] :
      ( pp(aa(char,bool,ord_less_eq(char,char2(B0,B12,B23,B32,B42,B52,B62,B72)),char2(C0,C1,C22,C32,C42,C52,C62,C72)))
    <=> pp(aa(nat,bool,ord_less_eq(nat,foldr(bool,nat,aTP_Lamp_amu(bool,fun(nat,nat)),aa(list(bool),list(bool),cons(bool,B0),aa(list(bool),list(bool),cons(bool,B12),aa(list(bool),list(bool),cons(bool,B23),aa(list(bool),list(bool),cons(bool,B32),aa(list(bool),list(bool),cons(bool,B42),aa(list(bool),list(bool),cons(bool,B52),aa(list(bool),list(bool),cons(bool,B62),aa(list(bool),list(bool),cons(bool,B72),nil(bool))))))))),zero_zero(nat))),foldr(bool,nat,aTP_Lamp_amu(bool,fun(nat,nat)),aa(list(bool),list(bool),cons(bool,C0),aa(list(bool),list(bool),cons(bool,C1),aa(list(bool),list(bool),cons(bool,C22),aa(list(bool),list(bool),cons(bool,C32),aa(list(bool),list(bool),cons(bool,C42),aa(list(bool),list(bool),cons(bool,C52),aa(list(bool),list(bool),cons(bool,C62),aa(list(bool),list(bool),cons(bool,C72),nil(bool))))))))),zero_zero(nat)))) ) ).

% less_eq_char_simp
tff(fact_8115_less__char__simp,axiom,
    ! [B0: bool,B12: bool,B23: bool,B32: bool,B42: bool,B52: bool,B62: bool,B72: bool,C0: bool,C1: bool,C22: bool,C32: bool,C42: bool,C52: bool,C62: bool,C72: bool] :
      ( pp(aa(char,bool,ord_less(char,char2(B0,B12,B23,B32,B42,B52,B62,B72)),char2(C0,C1,C22,C32,C42,C52,C62,C72)))
    <=> pp(aa(nat,bool,ord_less(nat,foldr(bool,nat,aTP_Lamp_amu(bool,fun(nat,nat)),aa(list(bool),list(bool),cons(bool,B0),aa(list(bool),list(bool),cons(bool,B12),aa(list(bool),list(bool),cons(bool,B23),aa(list(bool),list(bool),cons(bool,B32),aa(list(bool),list(bool),cons(bool,B42),aa(list(bool),list(bool),cons(bool,B52),aa(list(bool),list(bool),cons(bool,B62),aa(list(bool),list(bool),cons(bool,B72),nil(bool))))))))),zero_zero(nat))),foldr(bool,nat,aTP_Lamp_amu(bool,fun(nat,nat)),aa(list(bool),list(bool),cons(bool,C0),aa(list(bool),list(bool),cons(bool,C1),aa(list(bool),list(bool),cons(bool,C22),aa(list(bool),list(bool),cons(bool,C32),aa(list(bool),list(bool),cons(bool,C42),aa(list(bool),list(bool),cons(bool,C52),aa(list(bool),list(bool),cons(bool,C62),aa(list(bool),list(bool),cons(bool,C72),nil(bool))))))))),zero_zero(nat)))) ) ).

% less_char_simp
tff(fact_8116_integer__of__char__code,axiom,
    ! [B0: bool,B12: bool,B23: bool,B32: bool,B42: bool,B52: bool,B62: bool,B72: bool] : integer_of_char(char2(B0,B12,B23,B32,B42,B52,B62,B72)) = aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),times_times(code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),times_times(code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),times_times(code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),times_times(code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),times_times(code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),times_times(code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),times_times(code_integer),aa(bool,code_integer,zero_neq_one_of_bool(code_integer),B72)),aa(num,code_integer,numeral_numeral(code_integer),bit0(one2)))),aa(bool,code_integer,zero_neq_one_of_bool(code_integer),B62))),aa(num,code_integer,numeral_numeral(code_integer),bit0(one2)))),aa(bool,code_integer,zero_neq_one_of_bool(code_integer),B52))),aa(num,code_integer,numeral_numeral(code_integer),bit0(one2)))),aa(bool,code_integer,zero_neq_one_of_bool(code_integer),B42))),aa(num,code_integer,numeral_numeral(code_integer),bit0(one2)))),aa(bool,code_integer,zero_neq_one_of_bool(code_integer),B32))),aa(num,code_integer,numeral_numeral(code_integer),bit0(one2)))),aa(bool,code_integer,zero_neq_one_of_bool(code_integer),B23))),aa(num,code_integer,numeral_numeral(code_integer),bit0(one2)))),aa(bool,code_integer,zero_neq_one_of_bool(code_integer),B12))),aa(num,code_integer,numeral_numeral(code_integer),bit0(one2)))),aa(bool,code_integer,zero_neq_one_of_bool(code_integer),B0)) ).

% integer_of_char_code
tff(fact_8117_of__char__Char,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [B0: bool,B12: bool,B23: bool,B32: bool,B42: bool,B52: bool,B62: bool,B72: bool] : aa(char,A,comm_s6883823935334413003f_char(A),char2(B0,B12,B23,B32,B42,B52,B62,B72)) = groups4207007520872428315er_sum(bool,A,zero_neq_one_of_bool(A),aa(num,A,numeral_numeral(A),bit0(one2)),aa(list(bool),list(bool),cons(bool,B0),aa(list(bool),list(bool),cons(bool,B12),aa(list(bool),list(bool),cons(bool,B23),aa(list(bool),list(bool),cons(bool,B32),aa(list(bool),list(bool),cons(bool,B42),aa(list(bool),list(bool),cons(bool,B52),aa(list(bool),list(bool),cons(bool,B62),aa(list(bool),list(bool),cons(bool,B72),nil(bool)))))))))) ) ).

% of_char_Char
tff(fact_8118_of__char__mod__256,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [C2: char] : modulo_modulo(A,aa(char,A,comm_s6883823935334413003f_char(A),C2),aa(num,A,numeral_numeral(A),bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(one2)))))))))) = aa(char,A,comm_s6883823935334413003f_char(A),C2) ) ).

% of_char_mod_256
tff(fact_8119_nat__of__char__less__256,axiom,
    ! [C2: char] : pp(aa(nat,bool,ord_less(nat,aa(char,nat,comm_s6883823935334413003f_char(nat),C2)),aa(num,nat,numeral_numeral(nat),bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(one2))))))))))) ).

% nat_of_char_less_256
tff(fact_8120_less__eq__char__def,axiom,
    ! [C1: char,C22: char] :
      ( pp(aa(char,bool,ord_less_eq(char,C1),C22))
    <=> pp(aa(nat,bool,ord_less_eq(nat,aa(char,nat,comm_s6883823935334413003f_char(nat),C1)),aa(char,nat,comm_s6883823935334413003f_char(nat),C22))) ) ).

% less_eq_char_def
tff(fact_8121_range__nat__of__char,axiom,
    aa(set(char),set(nat),image(char,nat,comm_s6883823935334413003f_char(nat)),top_top(set(char))) = set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(one2)))))))))) ).

% range_nat_of_char
tff(fact_8122_char_Osize_I2_J,axiom,
    ! [X15: bool,X23: bool,X33: bool,X42: bool,X52: bool,X62: bool,X72: bool,X82: bool] : aa(char,nat,size_size(char),char2(X15,X23,X33,X42,X52,X62,X72,X82)) = zero_zero(nat) ).

% char.size(2)
tff(fact_8123_char__of__def,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [N: A] : aa(A,char,unique5772411509450598832har_of(A),N) = char2(aa(bool,bool,fNot,aa(A,bool,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),N)),aa(nat,bool,bit_se5641148757651400278ts_bit(A,N),one_one(nat)),aa(nat,bool,bit_se5641148757651400278ts_bit(A,N),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,bool,bit_se5641148757651400278ts_bit(A,N),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2))),aa(nat,bool,bit_se5641148757651400278ts_bit(A,N),aa(num,nat,numeral_numeral(nat),bit0(bit0(one2)))),aa(nat,bool,bit_se5641148757651400278ts_bit(A,N),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,bit0(one2)))),aa(nat,bool,bit_se5641148757651400278ts_bit(A,N),aa(num,nat,numeral_numeral(nat),bit0(aa(num,num,bit1,one2)))),aa(nat,bool,bit_se5641148757651400278ts_bit(A,N),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,aa(num,num,bit1,one2))))) ) ).

% char_of_def
tff(fact_8124_char_Osize__gen,axiom,
    ! [X15: bool,X23: bool,X33: bool,X42: bool,X52: bool,X62: bool,X72: bool,X82: bool] : size_char(char2(X15,X23,X33,X42,X52,X62,X72,X82)) = zero_zero(nat) ).

% char.size_gen
tff(fact_8125_char__of__nat,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [N: nat] : aa(A,char,unique5772411509450598832har_of(A),aa(nat,A,semiring_1_of_nat(A),N)) = aa(nat,char,unique5772411509450598832har_of(nat),N) ) ).

% char_of_nat
tff(fact_8126_char__of__quasi__inj,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [M: A,N: A] :
          ( ( aa(A,char,unique5772411509450598832har_of(A),M) = aa(A,char,unique5772411509450598832har_of(A),N) )
        <=> ( modulo_modulo(A,M,aa(num,A,numeral_numeral(A),bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(one2)))))))))) = modulo_modulo(A,N,aa(num,A,numeral_numeral(A),bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(one2)))))))))) ) ) ) ).

% char_of_quasi_inj
tff(fact_8127_char__of__mod__256,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [N: A] : aa(A,char,unique5772411509450598832har_of(A),modulo_modulo(A,N,aa(num,A,numeral_numeral(A),bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(one2))))))))))) = aa(A,char,unique5772411509450598832har_of(A),N) ) ).

% char_of_mod_256
tff(fact_8128_of__char__of,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [A2: A] : aa(char,A,comm_s6883823935334413003f_char(A),aa(A,char,unique5772411509450598832har_of(A),A2)) = modulo_modulo(A,A2,aa(num,A,numeral_numeral(A),bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(one2)))))))))) ) ).

% of_char_of
tff(fact_8129_UNIV__char__of__nat,axiom,
    top_top(set(char)) = aa(set(nat),set(char),image(nat,char,unique5772411509450598832har_of(nat)),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(one2))))))))))) ).

% UNIV_char_of_nat
tff(fact_8130_inj__on__char__of__nat,axiom,
    inj_on(nat,char,unique5772411509450598832har_of(nat),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(one2))))))))))) ).

% inj_on_char_of_nat
tff(fact_8131_char__of__take__bit__eq,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [N: nat,M: A] :
          ( pp(aa(nat,bool,ord_less_eq(nat,aa(num,nat,numeral_numeral(nat),bit0(bit0(bit0(one2))))),N))
         => ( aa(A,char,unique5772411509450598832har_of(A),aa(A,A,bit_se2584673776208193580ke_bit(A,N),M)) = aa(A,char,unique5772411509450598832har_of(A),M) ) ) ) ).

% char_of_take_bit_eq
tff(fact_8132_char__of__eq__iff,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [N: A,C2: char] :
          ( ( aa(A,char,unique5772411509450598832har_of(A),N) = C2 )
        <=> ( aa(A,A,bit_se2584673776208193580ke_bit(A,aa(num,nat,numeral_numeral(nat),bit0(bit0(bit0(one2))))),N) = aa(char,A,comm_s6883823935334413003f_char(A),C2) ) ) ) ).

% char_of_eq_iff
tff(fact_8133_String_Ochar__of__ascii__of,axiom,
    ! [C2: char] : aa(char,nat,comm_s6883823935334413003f_char(nat),ascii_of(C2)) = aa(nat,nat,bit_se2584673776208193580ke_bit(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,aa(num,num,bit1,one2)))),aa(char,nat,comm_s6883823935334413003f_char(nat),C2)) ).

% String.char_of_ascii_of
tff(fact_8134_upto_Opsimps,axiom,
    ! [I: int,J: int] :
      ( accp(product_prod(int,int),upto_rel,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),I),J))
     => ( ( pp(aa(int,bool,ord_less_eq(int,I),J))
         => ( upto(I,J) = aa(list(int),list(int),cons(int,I),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),I),one_one(int)),J)) ) )
        & ( ~ pp(aa(int,bool,ord_less_eq(int,I),J))
         => ( upto(I,J) = nil(int) ) ) ) ) ).

% upto.psimps
tff(fact_8135_nth__upto,axiom,
    ! [I: int,K: nat,J: int] :
      ( pp(aa(int,bool,ord_less_eq(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),I),aa(nat,int,semiring_1_of_nat(int),K))),J))
     => ( aa(nat,int,nth(int,upto(I,J)),K) = aa(int,int,aa(int,fun(int,int),plus_plus(int),I),aa(nat,int,semiring_1_of_nat(int),K)) ) ) ).

% nth_upto
tff(fact_8136_length__upto,axiom,
    ! [I: int,J: int] : aa(list(int),nat,size_size(list(int)),upto(I,J)) = nat2(aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,minus_minus(int,J),I)),one_one(int))) ).

% length_upto
tff(fact_8137_upto__rec__numeral_I1_J,axiom,
    ! [M: num,N: num] :
      ( ( pp(aa(int,bool,ord_less_eq(int,aa(num,int,numeral_numeral(int),M)),aa(num,int,numeral_numeral(int),N)))
       => ( upto(aa(num,int,numeral_numeral(int),M),aa(num,int,numeral_numeral(int),N)) = aa(list(int),list(int),cons(int,aa(num,int,numeral_numeral(int),M)),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(num,int,numeral_numeral(int),M)),one_one(int)),aa(num,int,numeral_numeral(int),N))) ) )
      & ( ~ pp(aa(int,bool,ord_less_eq(int,aa(num,int,numeral_numeral(int),M)),aa(num,int,numeral_numeral(int),N)))
       => ( upto(aa(num,int,numeral_numeral(int),M),aa(num,int,numeral_numeral(int),N)) = nil(int) ) ) ) ).

% upto_rec_numeral(1)
tff(fact_8138_upto__rec__numeral_I2_J,axiom,
    ! [M: num,N: num] :
      ( ( pp(aa(int,bool,ord_less_eq(int,aa(num,int,numeral_numeral(int),M)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),N))))
       => ( upto(aa(num,int,numeral_numeral(int),M),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),N))) = aa(list(int),list(int),cons(int,aa(num,int,numeral_numeral(int),M)),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(num,int,numeral_numeral(int),M)),one_one(int)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),N)))) ) )
      & ( ~ pp(aa(int,bool,ord_less_eq(int,aa(num,int,numeral_numeral(int),M)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),N))))
       => ( upto(aa(num,int,numeral_numeral(int),M),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),N))) = nil(int) ) ) ) ).

% upto_rec_numeral(2)
tff(fact_8139_upto__rec__numeral_I3_J,axiom,
    ! [M: num,N: num] :
      ( ( pp(aa(int,bool,ord_less_eq(int,aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),M))),aa(num,int,numeral_numeral(int),N)))
       => ( upto(aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),M)),aa(num,int,numeral_numeral(int),N)) = aa(list(int),list(int),cons(int,aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),M))),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),M))),one_one(int)),aa(num,int,numeral_numeral(int),N))) ) )
      & ( ~ pp(aa(int,bool,ord_less_eq(int,aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),M))),aa(num,int,numeral_numeral(int),N)))
       => ( upto(aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),M)),aa(num,int,numeral_numeral(int),N)) = nil(int) ) ) ) ).

% upto_rec_numeral(3)
tff(fact_8140_upto__rec__numeral_I4_J,axiom,
    ! [M: num,N: num] :
      ( ( pp(aa(int,bool,ord_less_eq(int,aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),M))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),N))))
       => ( upto(aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),M)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),N))) = aa(list(int),list(int),cons(int,aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),M))),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),M))),one_one(int)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),N)))) ) )
      & ( ~ pp(aa(int,bool,ord_less_eq(int,aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),M))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),N))))
       => ( upto(aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),M)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),N))) = nil(int) ) ) ) ).

% upto_rec_numeral(4)
tff(fact_8141_upto__split2,axiom,
    ! [I: int,J: int,K: int] :
      ( pp(aa(int,bool,ord_less_eq(int,I),J))
     => ( pp(aa(int,bool,ord_less_eq(int,J),K))
       => ( upto(I,K) = append(int,upto(I,J),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),J),one_one(int)),K)) ) ) ) ).

% upto_split2
tff(fact_8142_upto__split1,axiom,
    ! [I: int,J: int,K: int] :
      ( pp(aa(int,bool,ord_less_eq(int,I),J))
     => ( pp(aa(int,bool,ord_less_eq(int,J),K))
       => ( upto(I,K) = append(int,upto(I,aa(int,int,minus_minus(int,J),one_one(int))),upto(J,K)) ) ) ) ).

% upto_split1
tff(fact_8143_atLeastLessThan__upto,axiom,
    ! [I: int,J: int] : set_or7035219750837199246ssThan(int,I,J) = aa(list(int),set(int),set2(int),upto(I,aa(int,int,minus_minus(int,J),one_one(int)))) ).

% atLeastLessThan_upto
tff(fact_8144_greaterThanAtMost__upto,axiom,
    ! [I: int,J: int] : set_or3652927894154168847AtMost(int,I,J) = aa(list(int),set(int),set2(int),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),I),one_one(int)),J)) ).

% greaterThanAtMost_upto
tff(fact_8145_upto__rec1,axiom,
    ! [I: int,J: int] :
      ( pp(aa(int,bool,ord_less_eq(int,I),J))
     => ( upto(I,J) = aa(list(int),list(int),cons(int,I),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),I),one_one(int)),J)) ) ) ).

% upto_rec1
tff(fact_8146_upto_Oelims,axiom,
    ! [X: int,Xa2: int,Y: list(int)] :
      ( ( upto(X,Xa2) = Y )
     => ( ( pp(aa(int,bool,ord_less_eq(int,X),Xa2))
         => ( Y = aa(list(int),list(int),cons(int,X),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),X),one_one(int)),Xa2)) ) )
        & ( ~ pp(aa(int,bool,ord_less_eq(int,X),Xa2))
         => ( Y = nil(int) ) ) ) ) ).

% upto.elims
tff(fact_8147_upto_Osimps,axiom,
    ! [I: int,J: int] :
      ( ( pp(aa(int,bool,ord_less_eq(int,I),J))
       => ( upto(I,J) = aa(list(int),list(int),cons(int,I),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),I),one_one(int)),J)) ) )
      & ( ~ pp(aa(int,bool,ord_less_eq(int,I),J))
       => ( upto(I,J) = nil(int) ) ) ) ).

% upto.simps
tff(fact_8148_upto__rec2,axiom,
    ! [I: int,J: int] :
      ( pp(aa(int,bool,ord_less_eq(int,I),J))
     => ( upto(I,J) = append(int,upto(I,aa(int,int,minus_minus(int,J),one_one(int))),aa(list(int),list(int),cons(int,J),nil(int))) ) ) ).

% upto_rec2
tff(fact_8149_greaterThanLessThan__upto,axiom,
    ! [I: int,J: int] : set_or5935395276787703475ssThan(int,I,J) = aa(list(int),set(int),set2(int),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),I),one_one(int)),aa(int,int,minus_minus(int,J),one_one(int)))) ).

% greaterThanLessThan_upto
tff(fact_8150_upto__split3,axiom,
    ! [I: int,J: int,K: int] :
      ( pp(aa(int,bool,ord_less_eq(int,I),J))
     => ( pp(aa(int,bool,ord_less_eq(int,J),K))
       => ( upto(I,K) = append(int,upto(I,aa(int,int,minus_minus(int,J),one_one(int))),aa(list(int),list(int),cons(int,J),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),J),one_one(int)),K))) ) ) ) ).

% upto_split3
tff(fact_8151_upto_Opelims,axiom,
    ! [X: int,Xa2: int,Y: list(int)] :
      ( ( upto(X,Xa2) = Y )
     => ( accp(product_prod(int,int),upto_rel,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),X),Xa2))
       => ~ ( ( ( pp(aa(int,bool,ord_less_eq(int,X),Xa2))
               => ( Y = aa(list(int),list(int),cons(int,X),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),X),one_one(int)),Xa2)) ) )
              & ( ~ pp(aa(int,bool,ord_less_eq(int,X),Xa2))
               => ( Y = nil(int) ) ) )
           => ~ accp(product_prod(int,int),upto_rel,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),X),Xa2)) ) ) ) ).

% upto.pelims
tff(fact_8152_set__Cons__sing__Nil,axiom,
    ! [A: $tType,A3: set(A)] : set_Cons(A,A3,insert(list(A),nil(A),bot_bot(set(list(A))))) = aa(set(A),set(list(A)),image(A,list(A),aTP_Lamp_amv(A,list(A))),A3) ).

% set_Cons_sing_Nil
tff(fact_8153_list_Ocase__distrib,axiom,
    ! [B: $tType,C: $tType,A: $tType,H: fun(B,C),F13: B,F24: fun(A,fun(list(A),B)),List: list(A)] : aa(B,C,H,aa(list(A),B,case_list(B,A,F13,F24),List)) = aa(list(A),C,case_list(C,A,aa(B,C,H,F13),aa(fun(A,fun(list(A),B)),fun(A,fun(list(A),C)),aTP_Lamp_amw(fun(B,C),fun(fun(A,fun(list(A),B)),fun(A,fun(list(A),C))),H),F24)),List) ).

% list.case_distrib
tff(fact_8154_transpose_Oelims,axiom,
    ! [A: $tType,X: list(list(A)),Y: list(list(A))] :
      ( ( transpose(A,X) = Y )
     => ( ( ( X = nil(list(A)) )
         => ( Y != nil(list(A)) ) )
       => ( ! [Xss2: list(list(A))] :
              ( ( X = aa(list(list(A)),list(list(A)),cons(list(A),nil(A)),Xss2) )
             => ( Y != transpose(A,Xss2) ) )
         => ~ ! [X3: A,Xs2: list(A),Xss2: list(list(A))] :
                ( ( X = aa(list(list(A)),list(list(A)),cons(list(A),aa(list(A),list(A),cons(A,X3),Xs2)),Xss2) )
               => ( Y != aa(list(list(A)),list(list(A)),cons(list(A),aa(list(A),list(A),cons(A,X3),concat(A,map(list(A),list(A),case_list(list(A),A,nil(A),aTP_Lamp_amx(A,fun(list(A),list(A)))),Xss2)))),transpose(A,aa(list(list(A)),list(list(A)),cons(list(A),Xs2),concat(list(A),map(list(A),list(list(A)),case_list(list(list(A)),A,nil(list(A)),aTP_Lamp_amy(A,fun(list(A),list(list(A))))),Xss2))))) ) ) ) ) ) ).

% transpose.elims
tff(fact_8155_transpose_Osimps_I3_J,axiom,
    ! [A: $tType,X: A,Xs: list(A),Xss: list(list(A))] : transpose(A,aa(list(list(A)),list(list(A)),cons(list(A),aa(list(A),list(A),cons(A,X),Xs)),Xss)) = aa(list(list(A)),list(list(A)),cons(list(A),aa(list(A),list(A),cons(A,X),concat(A,map(list(A),list(A),case_list(list(A),A,nil(A),aTP_Lamp_amx(A,fun(list(A),list(A)))),Xss)))),transpose(A,aa(list(list(A)),list(list(A)),cons(list(A),Xs),concat(list(A),map(list(A),list(list(A)),case_list(list(list(A)),A,nil(list(A)),aTP_Lamp_amy(A,fun(list(A),list(list(A))))),Xss))))) ).

% transpose.simps(3)
tff(fact_8156_set__Cons__def,axiom,
    ! [A: $tType,A3: set(A),XS: set(list(A))] : set_Cons(A,A3,XS) = collect(list(A),aa(set(list(A)),fun(list(A),bool),aTP_Lamp_amz(set(A),fun(set(list(A)),fun(list(A),bool)),A3),XS)) ).

% set_Cons_def
tff(fact_8157_min__list_Oelims,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [X: list(A),Y: A] :
          ( ( min_list(A,X) = Y )
         => ( ! [X3: A,Xs2: list(A)] :
                ( ( X = aa(list(A),list(A),cons(A,X3),Xs2) )
               => ( Y != aa(list(A),A,case_list(A,A,X3,aa(list(A),fun(A,fun(list(A),A)),aTP_Lamp_ana(A,fun(list(A),fun(A,fun(list(A),A))),X3),Xs2)),Xs2) ) )
           => ~ ( ( X = nil(A) )
               => ( Y != undefined(A) ) ) ) ) ) ).

% min_list.elims
tff(fact_8158_transpose_Opsimps_I3_J,axiom,
    ! [A: $tType,X: A,Xs: list(A),Xss: list(list(A))] :
      ( accp(list(list(A)),transpose_rel(A),aa(list(list(A)),list(list(A)),cons(list(A),aa(list(A),list(A),cons(A,X),Xs)),Xss))
     => ( transpose(A,aa(list(list(A)),list(list(A)),cons(list(A),aa(list(A),list(A),cons(A,X),Xs)),Xss)) = aa(list(list(A)),list(list(A)),cons(list(A),aa(list(A),list(A),cons(A,X),concat(A,map(list(A),list(A),case_list(list(A),A,nil(A),aTP_Lamp_amx(A,fun(list(A),list(A)))),Xss)))),transpose(A,aa(list(list(A)),list(list(A)),cons(list(A),Xs),concat(list(A),map(list(A),list(list(A)),case_list(list(list(A)),A,nil(list(A)),aTP_Lamp_amy(A,fun(list(A),list(list(A))))),Xss))))) ) ) ).

% transpose.psimps(3)
tff(fact_8159_list_Odisc__eq__case_I2_J,axiom,
    ! [A: $tType,List: list(A)] :
      ( ( List != nil(A) )
    <=> pp(aa(list(A),bool,case_list(bool,A,fFalse,aTP_Lamp_anb(A,fun(list(A),bool))),List)) ) ).

% list.disc_eq_case(2)
tff(fact_8160_list_Odisc__eq__case_I1_J,axiom,
    ! [A: $tType,List: list(A)] :
      ( ( List = nil(A) )
    <=> pp(aa(list(A),bool,case_list(bool,A,fTrue,aTP_Lamp_anc(A,fun(list(A),bool))),List)) ) ).

% list.disc_eq_case(1)
tff(fact_8161_transpose_Opinduct,axiom,
    ! [A: $tType,A0: list(list(A)),P: fun(list(list(A)),bool)] :
      ( accp(list(list(A)),transpose_rel(A),A0)
     => ( ( accp(list(list(A)),transpose_rel(A),nil(list(A)))
         => pp(aa(list(list(A)),bool,P,nil(list(A)))) )
       => ( ! [Xss2: list(list(A))] :
              ( accp(list(list(A)),transpose_rel(A),aa(list(list(A)),list(list(A)),cons(list(A),nil(A)),Xss2))
             => ( pp(aa(list(list(A)),bool,P,Xss2))
               => pp(aa(list(list(A)),bool,P,aa(list(list(A)),list(list(A)),cons(list(A),nil(A)),Xss2))) ) )
         => ( ! [X3: A,Xs2: list(A),Xss2: list(list(A))] :
                ( accp(list(list(A)),transpose_rel(A),aa(list(list(A)),list(list(A)),cons(list(A),aa(list(A),list(A),cons(A,X3),Xs2)),Xss2))
               => ( pp(aa(list(list(A)),bool,P,aa(list(list(A)),list(list(A)),cons(list(A),Xs2),concat(list(A),map(list(A),list(list(A)),case_list(list(list(A)),A,nil(list(A)),aTP_Lamp_amy(A,fun(list(A),list(list(A))))),Xss2)))))
                 => pp(aa(list(list(A)),bool,P,aa(list(list(A)),list(list(A)),cons(list(A),aa(list(A),list(A),cons(A,X3),Xs2)),Xss2))) ) )
           => pp(aa(list(list(A)),bool,P,A0)) ) ) ) ) ).

% transpose.pinduct
tff(fact_8162_min__list_Osimps,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [X: A,Xs: list(A)] : min_list(A,aa(list(A),list(A),cons(A,X),Xs)) = aa(list(A),A,case_list(A,A,X,aa(list(A),fun(A,fun(list(A),A)),aTP_Lamp_ana(A,fun(list(A),fun(A,fun(list(A),A))),X),Xs)),Xs) ) ).

% min_list.simps
tff(fact_8163_transpose_Opelims,axiom,
    ! [A: $tType,X: list(list(A)),Y: list(list(A))] :
      ( ( transpose(A,X) = Y )
     => ( accp(list(list(A)),transpose_rel(A),X)
       => ( ( ( X = nil(list(A)) )
           => ( ( Y = nil(list(A)) )
             => ~ accp(list(list(A)),transpose_rel(A),nil(list(A))) ) )
         => ( ! [Xss2: list(list(A))] :
                ( ( X = aa(list(list(A)),list(list(A)),cons(list(A),nil(A)),Xss2) )
               => ( ( Y = transpose(A,Xss2) )
                 => ~ accp(list(list(A)),transpose_rel(A),aa(list(list(A)),list(list(A)),cons(list(A),nil(A)),Xss2)) ) )
           => ~ ! [X3: A,Xs2: list(A),Xss2: list(list(A))] :
                  ( ( X = aa(list(list(A)),list(list(A)),cons(list(A),aa(list(A),list(A),cons(A,X3),Xs2)),Xss2) )
                 => ( ( Y = aa(list(list(A)),list(list(A)),cons(list(A),aa(list(A),list(A),cons(A,X3),concat(A,map(list(A),list(A),case_list(list(A),A,nil(A),aTP_Lamp_amx(A,fun(list(A),list(A)))),Xss2)))),transpose(A,aa(list(list(A)),list(list(A)),cons(list(A),Xs2),concat(list(A),map(list(A),list(list(A)),case_list(list(list(A)),A,nil(list(A)),aTP_Lamp_amy(A,fun(list(A),list(list(A))))),Xss2))))) )
                   => ~ accp(list(list(A)),transpose_rel(A),aa(list(list(A)),list(list(A)),cons(list(A),aa(list(A),list(A),cons(A,X3),Xs2)),Xss2)) ) ) ) ) ) ) ).

% transpose.pelims
tff(fact_8164_min__list_Opelims,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [X: list(A),Y: A] :
          ( ( min_list(A,X) = Y )
         => ( accp(list(A),min_list_rel(A),X)
           => ( ! [X3: A,Xs2: list(A)] :
                  ( ( X = aa(list(A),list(A),cons(A,X3),Xs2) )
                 => ( ( Y = aa(list(A),A,case_list(A,A,X3,aa(list(A),fun(A,fun(list(A),A)),aTP_Lamp_ana(A,fun(list(A),fun(A,fun(list(A),A))),X3),Xs2)),Xs2) )
                   => ~ accp(list(A),min_list_rel(A),aa(list(A),list(A),cons(A,X3),Xs2)) ) )
             => ~ ( ( X = nil(A) )
                 => ( ( Y = undefined(A) )
                   => ~ accp(list(A),min_list_rel(A),nil(A)) ) ) ) ) ) ) ).

% min_list.pelims
tff(fact_8165_filter__upt__last,axiom,
    ! [A: $tType,P: fun(A,bool),L: list(A),Js: list(nat),J: nat,I: nat] :
      ( ( filter2(nat,aa(list(A),fun(nat,bool),aTP_Lamp_and(fun(A,bool),fun(list(A),fun(nat,bool)),P),L),upt(zero_zero(nat),aa(list(A),nat,size_size(list(A)),L))) = append(nat,Js,aa(list(nat),list(nat),cons(nat,J),nil(nat))) )
     => ( pp(aa(nat,bool,ord_less(nat,J),I))
       => ( pp(aa(nat,bool,ord_less(nat,I),aa(list(A),nat,size_size(list(A)),L)))
         => ~ pp(aa(A,bool,P,aa(nat,A,nth(A,L),I))) ) ) ) ).

% filter_upt_last
tff(fact_8166_filter__filter,axiom,
    ! [A: $tType,P: fun(A,bool),Q: fun(A,bool),Xs: list(A)] : filter2(A,P,filter2(A,Q,Xs)) = filter2(A,aa(fun(A,bool),fun(A,bool),aTP_Lamp_ane(fun(A,bool),fun(fun(A,bool),fun(A,bool)),P),Q),Xs) ).

% filter_filter
tff(fact_8167_set__filter,axiom,
    ! [A: $tType,P: fun(A,bool),Xs: list(A)] : aa(list(A),set(A),set2(A),filter2(A,P,Xs)) = collect(A,aa(list(A),fun(A,bool),aTP_Lamp_anf(fun(A,bool),fun(list(A),fun(A,bool)),P),Xs)) ).

% set_filter
tff(fact_8168_filter__conv__foldr,axiom,
    ! [A: $tType,P: fun(A,bool),Xs: list(A)] : filter2(A,P,Xs) = foldr(A,list(A),aTP_Lamp_ang(fun(A,bool),fun(A,fun(list(A),list(A))),P),Xs,nil(A)) ).

% filter_conv_foldr
tff(fact_8169_filter__replicate,axiom,
    ! [A: $tType,P: fun(A,bool),X: A,N: nat] :
      ( ( pp(aa(A,bool,P,X))
       => ( filter2(A,P,replicate(A,N,X)) = replicate(A,N,X) ) )
      & ( ~ pp(aa(A,bool,P,X))
       => ( filter2(A,P,replicate(A,N,X)) = nil(A) ) ) ) ).

% filter_replicate
tff(fact_8170_sum__length__filter__compl,axiom,
    ! [A: $tType,P: fun(A,bool),Xs: list(A)] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(list(A),nat,size_size(list(A)),filter2(A,P,Xs))),aa(list(A),nat,size_size(list(A)),filter2(A,aTP_Lamp_bz(fun(A,bool),fun(A,bool),P),Xs))) = aa(list(A),nat,size_size(list(A)),Xs) ).

% sum_length_filter_compl
tff(fact_8171_replicate__length__filter,axiom,
    ! [A: $tType,X: A,Xs: list(A)] : replicate(A,aa(list(A),nat,size_size(list(A)),filter2(A,fequal(A,X),Xs)),X) = filter2(A,fequal(A,X),Xs) ).

% replicate_length_filter
tff(fact_8172_length__filter__le,axiom,
    ! [A: $tType,P: fun(A,bool),Xs: list(A)] : pp(aa(nat,bool,ord_less_eq(nat,aa(list(A),nat,size_size(list(A)),filter2(A,P,Xs))),aa(list(A),nat,size_size(list(A)),Xs))) ).

% length_filter_le
tff(fact_8173_filter__is__subset,axiom,
    ! [A: $tType,P: fun(A,bool),Xs: list(A)] : pp(aa(set(A),bool,ord_less_eq(set(A),aa(list(A),set(A),set2(A),filter2(A,P,Xs))),aa(list(A),set(A),set2(A),Xs))) ).

% filter_is_subset
tff(fact_8174_inj__on__filter__key__eq,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),Y: A,Xs: list(A)] :
      ( inj_on(A,B,F2,insert(A,Y,aa(list(A),set(A),set2(A),Xs)))
     => ( filter2(A,aa(A,fun(A,bool),aTP_Lamp_anh(fun(A,B),fun(A,fun(A,bool)),F2),Y),Xs) = filter2(A,fequal(A,Y),Xs) ) ) ).

% inj_on_filter_key_eq
tff(fact_8175_set__minus__filter__out,axiom,
    ! [A: $tType,Xs: list(A),Y: A] : aa(set(A),set(A),minus_minus(set(A),aa(list(A),set(A),set2(A),Xs)),insert(A,Y,bot_bot(set(A)))) = aa(list(A),set(A),set2(A),filter2(A,aTP_Lamp_ani(A,fun(A,bool),Y),Xs)) ).

% set_minus_filter_out
tff(fact_8176_upt__filter__extend,axiom,
    ! [U: nat,U4: nat,P: fun(nat,bool)] :
      ( pp(aa(nat,bool,ord_less_eq(nat,U),U4))
     => ( ! [I2: nat] :
            ( ( pp(aa(nat,bool,ord_less_eq(nat,U),I2))
              & pp(aa(nat,bool,ord_less(nat,I2),U4)) )
           => ~ pp(aa(nat,bool,P,I2)) )
       => ( filter2(nat,P,upt(zero_zero(nat),U)) = filter2(nat,P,upt(zero_zero(nat),U4)) ) ) ) ).

% upt_filter_extend
tff(fact_8177_sort__key__by__quicksort,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F2: fun(B,A),Xs: list(B)] : linorder_sort_key(B,A,F2,Xs) = append(B,linorder_sort_key(B,A,F2,filter2(B,aa(list(B),fun(B,bool),aTP_Lamp_anj(fun(B,A),fun(list(B),fun(B,bool)),F2),Xs),Xs)),append(B,filter2(B,aa(list(B),fun(B,bool),aTP_Lamp_ank(fun(B,A),fun(list(B),fun(B,bool)),F2),Xs),Xs),linorder_sort_key(B,A,F2,filter2(B,aa(list(B),fun(B,bool),aTP_Lamp_anl(fun(B,A),fun(list(B),fun(B,bool)),F2),Xs),Xs)))) ) ).

% sort_key_by_quicksort
tff(fact_8178_sort__upt,axiom,
    ! [M: nat,N: nat] : linorder_sort_key(nat,nat,aTP_Lamp_el(nat,nat),upt(M,N)) = upt(M,N) ).

% sort_upt
tff(fact_8179_ATP_Olambda__1,axiom,
    ! [Uu: nat] : aa(nat,real,aTP_Lamp_bq(nat,real),Uu) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,uminus_uminus(real),one_one(real))),Uu)),one_one(real))),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uu),aa(num,nat,numeral_numeral(nat),bit0(one2)))),one_one(nat)))) ).

% ATP.lambda_1
tff(fact_8180_ATP_Olambda__2,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: A] : aa(A,A,aTP_Lamp_zr(A,A),Uu) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,minus_minus(A,aa(A,A,exp(A),Uu)),one_one(A))),Uu) ) ).

% ATP.lambda_2
tff(fact_8181_ATP_Olambda__3,axiom,
    ! [Uu: nat] : aa(nat,real,aTP_Lamp_fo(nat,real),Uu) = aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),bit0(one2)))),aa(nat,nat,suc,Uu)) ).

% ATP.lambda_3
tff(fact_8182_ATP_Olambda__4,axiom,
    ! [Uu: real] :
      ( pp(aa(real,bool,aTP_Lamp_lf(real,bool),Uu))
    <=> ( pp(aa(real,bool,ord_less_eq(real,zero_zero(real)),Uu))
        & pp(aa(real,bool,ord_less_eq(real,Uu),aa(num,real,numeral_numeral(real),bit0(one2))))
        & ( cos(real,Uu) = zero_zero(real) ) ) ) ).

% ATP.lambda_4
tff(fact_8183_ATP_Olambda__5,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: nat] : aa(nat,A,aTP_Lamp_abm(nat,A),Uu) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,Uu))),aa(nat,A,semiring_1_of_nat(A),Uu)) ) ).

% ATP.lambda_5
tff(fact_8184_ATP_Olambda__6,axiom,
    ! [Uu: nat] : aa(nat,real,aTP_Lamp_fw(nat,real),Uu) = aa(real,real,aa(real,fun(real,real),times_times(real),cos_coeff(Uu)),aa(nat,real,aa(real,fun(nat,real),power_power(real),zero_zero(real)),Uu)) ).

% ATP.lambda_6
tff(fact_8185_ATP_Olambda__7,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: nat] : aa(nat,A,aTP_Lamp_abl(nat,A),Uu) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,semiring_1_of_nat(A),Uu)),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,Uu))) ) ).

% ATP.lambda_7
tff(fact_8186_ATP_Olambda__8,axiom,
    ! [Uu: real] : aa(real,real,aTP_Lamp_zv(real,real),Uu) = aa(real,real,aa(real,fun(real,real),divide_divide(real),cos(real,Uu)),sin(real,Uu)) ).

% ATP.lambda_8
tff(fact_8187_ATP_Olambda__9,axiom,
    ! [Uu: real] : aa(real,real,aTP_Lamp_aef(real,real),Uu) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,ln_ln(real),Uu)),Uu) ).

% ATP.lambda_9
tff(fact_8188_ATP_Olambda__10,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [Uu: word(B)] :
          ( pp(aa(word(B),bool,aTP_Lamp_pv(word(B),bool),Uu))
        <=> pp(aa(word(B),bool,ord_less(word(B),Uu),aa(nat,word(B),aa(word(B),fun(nat,word(B)),power_power(word(B)),aa(num,word(B),numeral_numeral(word(B)),bit0(one2))),type_len0_len_of(A,type2(A))))) ) ) ).

% ATP.lambda_10
tff(fact_8189_ATP_Olambda__11,axiom,
    ! [Uu: nat] : aa(nat,real,aTP_Lamp_aat(nat,real),Uu) = aa(real,real,root(Uu),aa(nat,real,semiring_1_of_nat(real),Uu)) ).

% ATP.lambda_11
tff(fact_8190_ATP_Olambda__12,axiom,
    ! [Uu: nat] : aa(nat,nat,aTP_Lamp_ot(nat,nat),Uu) = aa(nat,nat,minus_minus(nat,Uu),aa(nat,nat,suc,zero_zero(nat))) ).

% ATP.lambda_12
tff(fact_8191_ATP_Olambda__13,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [Uu: A] : aa(A,A,aTP_Lamp_gm(A,A),Uu) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Uu),one_one(A)) ) ).

% ATP.lambda_13
tff(fact_8192_ATP_Olambda__14,axiom,
    ! [A: $tType,Uu: A] : aa(A,list(A),aTP_Lamp_amv(A,list(A)),Uu) = aa(list(A),list(A),cons(A,Uu),nil(A)) ).

% ATP.lambda_14
tff(fact_8193_ATP_Olambda__15,axiom,
    ! [A: $tType,Uu: A] : aa(A,set(A),aTP_Lamp_qt(A,set(A)),Uu) = insert(A,Uu,bot_bot(set(A))) ).

% ATP.lambda_15
tff(fact_8194_ATP_Olambda__16,axiom,
    ! [Uu: nat] : aa(nat,real,aTP_Lamp_abd(nat,real),Uu) = aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(nat,real,semiring_1_of_nat(real),Uu)) ).

% ATP.lambda_16
tff(fact_8195_ATP_Olambda__17,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: nat] : aa(nat,A,aTP_Lamp_abk(nat,A),Uu) = aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(nat,A,semiring_1_of_nat(A),Uu)) ) ).

% ATP.lambda_17
tff(fact_8196_ATP_Olambda__18,axiom,
    ! [Uu: int] :
      ( pp(aa(int,bool,aTP_Lamp_mv(int,bool),Uu))
    <=> ~ pp(aa(int,bool,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),Uu)) ) ).

% ATP.lambda_18
tff(fact_8197_ATP_Olambda__19,axiom,
    ! [Uu: real] : aa(real,real,aTP_Lamp_sr(real,real),Uu) = suminf(real,aTP_Lamp_bp(real,fun(nat,real),Uu)) ).

% ATP.lambda_19
tff(fact_8198_ATP_Olambda__20,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,B)] : aa(fun(A,B),set(product_prod(A,B)),aTP_Lamp_qw(fun(A,B),set(product_prod(A,B))),Uu) = collect(product_prod(A,B),product_case_prod(A,B,bool,aTP_Lamp_qv(fun(A,B),fun(A,fun(B,bool)),Uu))) ).

% ATP.lambda_20
tff(fact_8199_ATP_Olambda__21,axiom,
    ! [Uu: nat] : aa(nat,real,aTP_Lamp_abe(nat,real),Uu) = aa(real,real,inverse_inverse(real),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,Uu))) ).

% ATP.lambda_21
tff(fact_8200_ATP_Olambda__22,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: nat] : aa(nat,A,aTP_Lamp_go(nat,A),Uu) = aa(A,A,inverse_inverse(A),semiring_char_0_fact(A,Uu)) ) ).

% ATP.lambda_22
tff(fact_8201_ATP_Olambda__23,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: nat] : aa(nat,A,aTP_Lamp_aaw(nat,A),Uu) = aa(A,A,inverse_inverse(A),aa(nat,A,semiring_1_of_nat(A),Uu)) ) ).

% ATP.lambda_23
tff(fact_8202_ATP_Olambda__24,axiom,
    ! [A: $tType,Uu: list(A)] : aa(list(A),fun(nat,nat),aTP_Lamp_ams(list(A),fun(nat,nat)),Uu) = ord_max(nat,aa(list(A),nat,size_size(list(A)),Uu)) ).

% ATP.lambda_24
tff(fact_8203_ATP_Olambda__25,axiom,
    ! [B: $tType,A: $tType] :
      ( ( archim2362893244070406136eiling(A)
        & topolo2564578578187576103pology(A)
        & ring_1(B)
        & topolo4958980785337419405_space(B) )
     => ! [Uu: A] : aa(A,B,aTP_Lamp_akn(A,B),Uu) = aa(int,B,ring_1_of_int(B),archim6421214686448440834_floor(A,Uu)) ) ).

% ATP.lambda_25
tff(fact_8204_ATP_Olambda__26,axiom,
    ! [B: $tType,A: $tType] :
      ( ( archim2362893244070406136eiling(A)
        & topolo2564578578187576103pology(A)
        & ring_1(B)
        & topolo4958980785337419405_space(B) )
     => ! [Uu: A] : aa(A,B,aTP_Lamp_ako(A,B),Uu) = aa(int,B,ring_1_of_int(B),archimedean_ceiling(A,Uu)) ) ).

% ATP.lambda_26
tff(fact_8205_ATP_Olambda__27,axiom,
    ! [Uu: num] : aa(num,option(num),aTP_Lamp_alv(num,option(num)),Uu) = aa(num,option(num),some(num),aa(num,num,bit1,Uu)) ).

% ATP.lambda_27
tff(fact_8206_ATP_Olambda__28,axiom,
    ! [Uu: num] : aa(num,option(num),aTP_Lamp_alu(num,option(num)),Uu) = aa(num,option(num),some(num),bit0(Uu)) ).

% ATP.lambda_28
tff(fact_8207_ATP_Olambda__29,axiom,
    ! [Uu: int] : aa(int,fun(int,product_prod(int,int)),aTP_Lamp_nb(int,fun(int,product_prod(int,int))),Uu) = aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,uminus_uminus(int),Uu)) ).

% ATP.lambda_29
tff(fact_8208_ATP_Olambda__30,axiom,
    ! [Uu: int] : aa(int,fun(int,product_prod(int,int)),aTP_Lamp_ne(int,fun(int,product_prod(int,int))),Uu) = aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,abs_abs(int),Uu)) ).

% ATP.lambda_30
tff(fact_8209_ATP_Olambda__31,axiom,
    ! [Uu: nat] : aa(nat,fun(nat,product_prod(nat,nat)),aTP_Lamp_md(nat,fun(nat,product_prod(nat,nat))),Uu) = aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),aa(nat,nat,suc,Uu)) ).

% ATP.lambda_31
tff(fact_8210_ATP_Olambda__32,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [Uu: product_prod(A,A)] :
          ( pp(aa(product_prod(A,A),bool,aTP_Lamp_ale(product_prod(A,A),bool),Uu))
        <=> ? [X2: A,Y7: A] :
              ( ( Uu = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Y7) )
              & pp(aa(A,bool,ord_less(A,X2),Y7)) ) ) ) ).

% ATP.lambda_32
tff(fact_8211_ATP_Olambda__33,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [Uu: product_prod(A,A)] :
          ( pp(aa(product_prod(A,A),bool,aTP_Lamp_alf(product_prod(A,A),bool),Uu))
        <=> ? [X2: A,Y7: A] :
              ( ( Uu = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Y7) )
              & pp(aa(A,bool,ord_less(A,Y7),X2)) ) ) ) ).

% ATP.lambda_33
tff(fact_8212_ATP_Olambda__34,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Uu: product_prod(A,A)] :
          ( pp(aa(product_prod(A,A),bool,aTP_Lamp_ald(product_prod(A,A),bool),Uu))
        <=> ? [X2: A,Y7: A] :
              ( ( Uu = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Y7) )
              & ( X2 != Y7 ) ) ) ) ).

% ATP.lambda_34
tff(fact_8213_ATP_Olambda__35,axiom,
    ! [Uu: nat] : aa(nat,option(num),aTP_Lamp_aly(nat,option(num)),Uu) = aa(num,option(num),some(num),one2) ).

% ATP.lambda_35
tff(fact_8214_ATP_Olambda__36,axiom,
    ! [Uu: num,Uua: nat] : aa(nat,option(num),aTP_Lamp_amc(num,fun(nat,option(num)),Uu),Uua) = case_num(option(num),aa(num,option(num),some(num),one2),aTP_Lamp_ama(nat,fun(num,option(num)),Uua),aTP_Lamp_amb(nat,fun(num,option(num)),Uua),Uu) ).

% ATP.lambda_36
tff(fact_8215_ATP_Olambda__37,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_kx(A,fun(nat,A),Uu),Uua) = if(A,aa(nat,bool,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Uua),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),semiring_char_0_fact(real,Uua))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu),Uua)),zero_zero(A)) ) ).

% ATP.lambda_37
tff(fact_8216_ATP_Olambda__38,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Uu: nat,Uua: nat] : aa(nat,A,aTP_Lamp_jj(nat,fun(nat,A),Uu),Uua) = if(A,aa(nat,bool,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Uua),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,binomial(Uu),Uua)),zero_zero(A)) ) ).

% ATP.lambda_38
tff(fact_8217_ATP_Olambda__39,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_kw(A,fun(nat,A),Uu),Uua) = if(A,aa(nat,bool,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Uua),zero_zero(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),semiring_char_0_fact(real,Uua))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu),Uua))) ) ).

% ATP.lambda_39
tff(fact_8218_ATP_Olambda__40,axiom,
    ! [Uu: fun(nat,real),Uua: nat] : aa(nat,real,aTP_Lamp_fq(fun(nat,real),fun(nat,real),Uu),Uua) = if(real,aa(nat,bool,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Uua),zero_zero(real),aa(nat,real,Uu,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,minus_minus(nat,Uua),one_one(nat))),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ).

% ATP.lambda_40
tff(fact_8219_ATP_Olambda__41,axiom,
    ! [A: $tType,Uu: list(A),Uua: nat] : aa(nat,option(A),aTP_Lamp_mw(list(A),fun(nat,option(A)),Uu),Uua) = if(option(A),aa(nat,bool,ord_less(nat,Uua),aa(list(A),nat,size_size(list(A)),Uu)),aa(A,option(A),some(A),aa(nat,A,nth(A,Uu),Uua)),none(A)) ).

% ATP.lambda_41
tff(fact_8220_ATP_Olambda__42,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [Uu: nat,Uua: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_mq(nat,fun(nat,A)),Uu),Uua) = if(A,aa(nat,bool,fequal(nat,Uua),zero_zero(nat)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(nat,A,semiring_1_of_nat(A),Uu)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(nat,A,semiring_1_of_nat(A),Uu))),one_one(A))) ) ).

% ATP.lambda_42
tff(fact_8221_ATP_Olambda__43,axiom,
    ! [Uu: int,Uua: int] : aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aTP_Lamp_mx(int,fun(int,product_prod(int,int))),Uu),Uua) = if(product_prod(int,int),aa(int,bool,fequal(int,Uu),zero_zero(int)),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),one_one(int)),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,aa(int,fun(int,int),times_times(int),sgn_sgn(int,Uu)),Uua)),aa(int,int,abs_abs(int),Uu))) ).

% ATP.lambda_43
tff(fact_8222_ATP_Olambda__44,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Uu: nat,Uua: nat] : aa(nat,A,aTP_Lamp_jk(nat,fun(nat,A),Uu),Uua) = if(A,aa(bool,bool,fNot,aa(nat,bool,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Uua)),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,binomial(Uu),Uua)),zero_zero(A)) ) ).

% ATP.lambda_44
tff(fact_8223_ATP_Olambda__45,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Uu: fun(nat,A),Uua: A] :
          ( pp(aa(A,bool,aTP_Lamp_ada(fun(nat,A),fun(A,bool),Uu),Uua))
        <=> filterlim(nat,A,Uu,topolo7230453075368039082e_nhds(A,Uua),at_top(nat)) ) ) ).

% ATP.lambda_45
tff(fact_8224_ATP_Olambda__46,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Uu: option(A),Uua: A] :
          ( pp(aa(A,bool,aTP_Lamp_alr(option(A),fun(A,bool),Uu),Uua))
        <=> pp(case_option(bool,A,fTrue,aTP_Lamp_alq(A,fun(A,bool),Uua),Uu)) ) ) ).

% ATP.lambda_46
tff(fact_8225_ATP_Olambda__47,axiom,
    ! [Uu: nat,Uua: num] : aa(num,option(num),aa(nat,fun(num,option(num)),aTP_Lamp_amd(nat,fun(num,option(num))),Uu),Uua) = case_nat(option(num),none(num),aTP_Lamp_amc(num,fun(nat,option(num)),Uua),Uu) ).

% ATP.lambda_47
tff(fact_8226_ATP_Olambda__48,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Uu: option(A),Uua: A] :
          ( pp(aa(A,bool,aTP_Lamp_alm(option(A),fun(A,bool),Uu),Uua))
        <=> pp(case_option(bool,A,fFalse,ord_less_eq(A,Uua),Uu)) ) ) ).

% ATP.lambda_48
tff(fact_8227_ATP_Olambda__49,axiom,
    ! [Uu: nat,Uua: num] : aa(num,option(num),aTP_Lamp_ama(nat,fun(num,option(num)),Uu),Uua) = case_option(option(num),num,none(num),aTP_Lamp_alu(num,option(num)),bit_take_bit_num(Uu,Uua)) ).

% ATP.lambda_49
tff(fact_8228_ATP_Olambda__50,axiom,
    ! [Uu: num,Uua: nat] : aa(nat,option(num),aTP_Lamp_alz(num,fun(nat,option(num)),Uu),Uua) = case_option(option(num),num,none(num),aTP_Lamp_alu(num,option(num)),bit_take_bit_num(Uua,Uu)) ).

% ATP.lambda_50
tff(fact_8229_ATP_Olambda__51,axiom,
    ! [A: $tType,Uu: list(list(A)),Uua: nat] : aa(nat,list(A),aTP_Lamp_amq(list(list(A)),fun(nat,list(A)),Uu),Uua) = map(nat,A,aa(nat,fun(nat,A),aTP_Lamp_amp(list(list(A)),fun(nat,fun(nat,A)),Uu),Uua),upt(zero_zero(nat),aa(list(list(A)),nat,size_size(list(list(A))),Uu))) ).

% ATP.lambda_51
tff(fact_8230_ATP_Olambda__52,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(nat,fun(nat,A)),Uua: nat] : aa(nat,A,aTP_Lamp_mk(fun(nat,fun(nat,A)),fun(nat,A),Uu),Uua) = groups7121269368397514597t_prod(nat,A,aa(nat,fun(nat,A),aTP_Lamp_mj(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu),Uua),set_ord_atMost(nat,Uua)) ) ).

% ATP.lambda_52
tff(fact_8231_ATP_Olambda__53,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(nat,fun(nat,A)),Uua: nat] : aa(nat,A,aTP_Lamp_mi(fun(nat,fun(nat,A)),fun(nat,A),Uu),Uua) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aTP_Lamp_mh(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu),Uua)),set_ord_atMost(nat,Uua)) ) ).

% ATP.lambda_53
tff(fact_8232_ATP_Olambda__54,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_bm(real,fun(nat,real),Uu),Uua) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,uminus_uminus(real),one_one(real))),Uua)),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uua),one_one(nat)))))),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,minus_minus(real,Uu),one_one(real))),aa(nat,nat,suc,Uua))) ).

% ATP.lambda_54
tff(fact_8233_ATP_Olambda__55,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_fz(real,fun(nat,real),Uu),Uua) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,uminus_uminus(real),one_one(real))),Uua)),semiring_char_0_fact(real,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uua)),one_one(nat))))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Uu),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uua)),one_one(nat)))) ).

% ATP.lambda_55
tff(fact_8234_ATP_Olambda__56,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_gi(real,fun(nat,real),Uu),Uua) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,uminus_uminus(real),one_one(real))),Uua)),semiring_char_0_fact(real,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uua)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Uu),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uua))) ).

% ATP.lambda_56
tff(fact_8235_ATP_Olambda__57,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Uu: nat,Uua: nat] : aa(nat,A,aTP_Lamp_ju(nat,fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),Uua)),aa(nat,A,semiring_1_of_nat(A),Uua))),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,binomial(Uu),Uua))) ) ).

% ATP.lambda_57
tff(fact_8236_ATP_Olambda__58,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: nat,Uua: nat] : aa(nat,A,aTP_Lamp_jd(nat,fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,gbinomial(A,aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uu),Uua))),Uua)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),Uua)) ) ).

% ATP.lambda_58
tff(fact_8237_ATP_Olambda__59,axiom,
    ! [Uu: real,Uua: real] :
      ( pp(aa(real,bool,aTP_Lamp_lb(real,fun(real,bool),Uu),Uua))
    <=> ( pp(aa(real,bool,ord_less_eq(real,aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),Uua))
        & pp(aa(real,bool,ord_less_eq(real,Uua),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2)))))
        & ( sin(real,Uua) = Uu ) ) ) ).

% ATP.lambda_59
tff(fact_8238_ATP_Olambda__60,axiom,
    ! [Uu: real,Uua: real] :
      ( pp(aa(real,bool,aTP_Lamp_la(real,fun(real,bool),Uu),Uua))
    <=> ( pp(aa(real,bool,ord_less(real,aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),Uua))
        & pp(aa(real,bool,ord_less(real,Uua),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2)))))
        & ( aa(real,real,tan(real),Uua) = Uu ) ) ) ).

% ATP.lambda_60
tff(fact_8239_ATP_Olambda__61,axiom,
    ! [Uu: complex,Uua: real] :
      ( pp(aa(real,bool,aTP_Lamp_lu(complex,fun(real,bool),Uu),Uua))
    <=> ( ( sgn_sgn(complex,Uu) = cis(Uua) )
        & pp(aa(real,bool,ord_less(real,aa(real,real,uminus_uminus(real),pi)),Uua))
        & pp(aa(real,bool,ord_less_eq(real,Uua),pi)) ) ) ).

% ATP.lambda_61
tff(fact_8240_ATP_Olambda__62,axiom,
    ! [Uu: real,Uua: int] :
      ( pp(aa(int,bool,aTP_Lamp_li(real,fun(int,bool),Uu),Uua))
    <=> ( pp(aa(real,bool,ord_less_eq(real,aa(int,real,ring_1_of_int(real),Uua)),Uu))
        & pp(aa(real,bool,ord_less(real,Uu),aa(int,real,ring_1_of_int(real),aa(int,int,aa(int,fun(int,int),plus_plus(int),Uua),one_one(int))))) ) ) ).

% ATP.lambda_62
tff(fact_8241_ATP_Olambda__63,axiom,
    ! [Uu: rat,Uua: int] :
      ( pp(aa(int,bool,aTP_Lamp_lm(rat,fun(int,bool),Uu),Uua))
    <=> ( pp(aa(rat,bool,ord_less_eq(rat,aa(int,rat,ring_1_of_int(rat),Uua)),Uu))
        & pp(aa(rat,bool,ord_less(rat,Uu),aa(int,rat,ring_1_of_int(rat),aa(int,int,aa(int,fun(int,int),plus_plus(int),Uua),one_one(int))))) ) ) ).

% ATP.lambda_63
tff(fact_8242_ATP_Olambda__64,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_bp(real,fun(nat,real),Uu),Uua) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,uminus_uminus(real),one_one(real))),Uua)),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uua),aa(num,nat,numeral_numeral(nat),bit0(one2)))),one_one(nat))))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Uu),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uua),aa(num,nat,numeral_numeral(nat),bit0(one2)))),one_one(nat))))) ).

% ATP.lambda_64
tff(fact_8243_ATP_Olambda__65,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_ss(real,fun(nat,real),Uu),Uua) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,uminus_uminus(real),one_one(real))),Uua)),aa(nat,real,aa(real,fun(nat,real),power_power(real),Uu),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uua),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ).

% ATP.lambda_65
tff(fact_8244_ATP_Olambda__66,axiom,
    ! [Uu: fun(nat,real),Uua: nat] : aa(nat,real,aTP_Lamp_aab(fun(nat,real),fun(nat,real),Uu),Uua) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,uminus_uminus(real),one_one(real))),Uua)),aa(nat,real,Uu,Uua)) ).

% ATP.lambda_66
tff(fact_8245_ATP_Olambda__67,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Uu: nat,Uua: nat] : aa(nat,A,aTP_Lamp_jw(nat,fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),Uua)),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,binomial(Uu),Uua))) ) ).

% ATP.lambda_67
tff(fact_8246_ATP_Olambda__68,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_ik(A,fun(nat,A),Uu),Uua) = aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Uu),aa(nat,A,semiring_1_of_nat(A),Uua))),Uua) ) ).

% ATP.lambda_68
tff(fact_8247_ATP_Olambda__69,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_gn(A,fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,gbinomial(A,Uu),Uua)),aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),Uu),aa(num,A,numeral_numeral(A),bit0(one2)))),aa(nat,A,semiring_1_of_nat(A),Uua))) ) ).

% ATP.lambda_69
tff(fact_8248_ATP_Olambda__70,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_is(A,fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,gbinomial(A,Uu),Uua)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),Uua)) ) ).

% ATP.lambda_70
tff(fact_8249_ATP_Olambda__71,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_jt(nat,fun(nat,nat),Uu),Uua) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(nat,nat,binomial(Uu),Uua)),aa(num,nat,numeral_numeral(nat),bit0(one2))) ).

% ATP.lambda_71
tff(fact_8250_ATP_Olambda__72,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_gk(real,fun(nat,real),Uu),Uua) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),Uu),Uua)),semiring_char_0_fact(real,Uua)) ).

% ATP.lambda_72
tff(fact_8251_ATP_Olambda__73,axiom,
    ! [Uu: nat,Uua: real] : aa(real,real,aTP_Lamp_aeh(nat,fun(real,real),Uu),Uua) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),Uua),Uu)),aa(real,real,exp(real),Uua)) ).

% ATP.lambda_73
tff(fact_8252_ATP_Olambda__74,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_jn(nat,fun(nat,nat),Uu),Uua) = aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uu),Uua)),Uu) ).

% ATP.lambda_74
tff(fact_8253_ATP_Olambda__75,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_jm(nat,fun(nat,nat),Uu),Uua) = aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uu),Uua)),Uua) ).

% ATP.lambda_75
tff(fact_8254_ATP_Olambda__76,axiom,
    ! [Uu: nat,Uua: complex] :
      ( pp(aa(complex,bool,aTP_Lamp_fe(nat,fun(complex,bool),Uu),Uua))
    <=> ( aa(nat,complex,aa(complex,fun(nat,complex),power_power(complex),Uua),Uu) = one_one(complex) ) ) ).

% ATP.lambda_76
tff(fact_8255_ATP_Olambda__77,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [Uu: nat,Uua: A] :
          ( pp(aa(A,bool,aTP_Lamp_an(nat,fun(A,bool),Uu),Uua))
        <=> ( aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),Uu) = one_one(A) ) ) ) ).

% ATP.lambda_77
tff(fact_8256_ATP_Olambda__78,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Uu: A,Uua: A] :
          ( pp(aa(A,bool,aTP_Lamp_na(A,fun(A,bool),Uu),Uua))
        <=> ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Uua),ring_1_Ints(A)))
            & pp(aa(A,bool,ord_less_eq(A,aa(A,A,abs_abs(A),Uua)),Uu)) ) ) ) ).

% ATP.lambda_78
tff(fact_8257_ATP_Olambda__79,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_abu(real,fun(nat,real),Uu),Uua) = aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),aa(real,real,aa(real,fun(real,real),divide_divide(real),Uu),aa(nat,real,semiring_1_of_nat(real),Uua)))),Uua) ).

% ATP.lambda_79
tff(fact_8258_ATP_Olambda__80,axiom,
    ! [Uu: real,Uua: real] : aa(real,real,aTP_Lamp_adm(real,fun(real,real),Uu),Uua) = powr(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),aa(real,real,aa(real,fun(real,real),times_times(real),Uu),Uua)),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),Uua)) ).

% ATP.lambda_80
tff(fact_8259_ATP_Olambda__81,axiom,
    ! [Uu: real,Uua: real] : aa(real,real,aTP_Lamp_aek(real,fun(real,real),Uu),Uua) = powr(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),aa(real,real,aa(real,fun(real,real),divide_divide(real),Uu),Uua)),Uua) ).

% ATP.lambda_81
tff(fact_8260_ATP_Olambda__82,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_bl(real,fun(nat,real),Uu),Uua) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uua),aa(num,nat,numeral_numeral(nat),bit0(one2)))),one_one(nat))))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Uu),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uua),aa(num,nat,numeral_numeral(nat),bit0(one2)))),one_one(nat)))) ).

% ATP.lambda_82
tff(fact_8261_ATP_Olambda__83,axiom,
    ! [Uu: real,Uua: real] :
      ( pp(aa(real,bool,aTP_Lamp_le(real,fun(real,bool),Uu),Uua))
    <=> ( pp(aa(real,bool,ord_less_eq(real,zero_zero(real)),Uua))
        & pp(aa(real,bool,ord_less_eq(real,Uua),pi))
        & ( cos(real,Uua) = Uu ) ) ) ).

% ATP.lambda_83
tff(fact_8262_ATP_Olambda__84,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_hx(fun(nat,A),fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uua))),aa(nat,A,Uu,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uua)))) ) ).

% ATP.lambda_84
tff(fact_8263_ATP_Olambda__85,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_ek(fun(nat,A),fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,Uu,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uua))),aa(nat,A,Uu,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uua)))) ) ).

% ATP.lambda_85
tff(fact_8264_ATP_Olambda__86,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_aaz(fun(nat,A),fun(nat,A),Uu),Uua) = aa(A,A,minus_minus(A,aa(nat,A,Uu,aa(nat,nat,suc,Uua))),aa(nat,A,Uu,Uua)) ) ).

% ATP.lambda_86
tff(fact_8265_ATP_Olambda__87,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_eg(fun(nat,A),fun(nat,A),Uu),Uua) = aa(A,A,minus_minus(A,aa(nat,A,Uu,aa(nat,nat,suc,Uua))),aa(nat,A,Uu,Uua)) ) ).

% ATP.lambda_87
tff(fact_8266_ATP_Olambda__88,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(nat,fun(nat,A)),Uua: nat] : aa(nat,A,aTP_Lamp_oy(fun(nat,fun(nat,A)),fun(nat,A),Uu),Uua) = groups7121269368397514597t_prod(nat,A,aa(nat,fun(nat,A),Uu,Uua),set_or7035219750837199246ssThan(nat,zero_zero(nat),Uua)) ) ).

% ATP.lambda_88
tff(fact_8267_ATP_Olambda__89,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(nat,fun(nat,A)),Uua: nat] : aa(nat,A,aTP_Lamp_ow(fun(nat,fun(nat,A)),fun(nat,A),Uu),Uua) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),Uu,Uua)),set_or7035219750837199246ssThan(nat,zero_zero(nat),Uua)) ) ).

% ATP.lambda_89
tff(fact_8268_ATP_Olambda__90,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_cm(fun(nat,A),fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,Uua)),aa(nat,A,aa(A,fun(nat,A),power_power(A),zero_zero(A)),Uua)) ) ).

% ATP.lambda_90
tff(fact_8269_ATP_Olambda__91,axiom,
    ! [A: $tType] :
      ( real_V2822296259951069270ebra_1(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_bn(fun(nat,A),fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,Uua)),aa(nat,A,aa(A,fun(nat,A),power_power(A),zero_zero(A)),Uua)) ) ).

% ATP.lambda_91
tff(fact_8270_ATP_Olambda__92,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_dd(fun(nat,A),fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,Uua)),aa(nat,A,aa(A,fun(nat,A),power_power(A),zero_zero(A)),Uua)) ) ).

% ATP.lambda_92
tff(fact_8271_ATP_Olambda__93,axiom,
    ! [A: $tType] :
      ( ( ring_1(A)
        & topolo4958980785337419405_space(A) )
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_cn(fun(nat,A),fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,Uua)),aa(nat,A,aa(A,fun(nat,A),power_power(A),zero_zero(A)),Uua)) ) ).

% ATP.lambda_93
tff(fact_8272_ATP_Olambda__94,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_ej(fun(nat,A),fun(nat,A),Uu),Uua) = aa(A,A,minus_minus(A,aa(nat,A,Uu,Uua)),aa(nat,A,Uu,aa(nat,nat,minus_minus(nat,Uua),one_one(nat)))) ) ).

% ATP.lambda_94
tff(fact_8273_ATP_Olambda__95,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_ei(fun(nat,A),fun(nat,A),Uu),Uua) = aa(A,A,minus_minus(A,aa(nat,A,Uu,Uua)),aa(nat,A,Uu,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uua),one_one(nat)))) ) ).

% ATP.lambda_95
tff(fact_8274_ATP_Olambda__96,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_aba(fun(nat,A),fun(nat,A),Uu),Uua) = aa(A,A,minus_minus(A,aa(nat,A,Uu,Uua)),aa(nat,A,Uu,aa(nat,nat,suc,Uua))) ) ).

% ATP.lambda_96
tff(fact_8275_ATP_Olambda__97,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_fi(fun(nat,A),fun(nat,A),Uu),Uua) = aa(A,A,minus_minus(A,aa(nat,A,Uu,Uua)),aa(nat,A,Uu,aa(nat,nat,suc,Uua))) ) ).

% ATP.lambda_97
tff(fact_8276_ATP_Olambda__98,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Uu: fun(A,bool),Uua: A] :
          ( pp(aa(A,bool,aTP_Lamp_alj(fun(A,bool),fun(A,bool),Uu),Uua))
        <=> ( pp(aa(A,bool,Uu,Uua))
            & ! [Y7: A] :
                ( pp(aa(A,bool,Uu,Y7))
               => pp(aa(A,bool,ord_less_eq(A,Uua),Y7)) ) ) ) ) ).

% ATP.lambda_98
tff(fact_8277_ATP_Olambda__99,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Uu: fun(A,bool),Uua: A] :
          ( pp(aa(A,bool,aTP_Lamp_aif(fun(A,bool),fun(A,bool),Uu),Uua))
        <=> ( pp(aa(A,bool,Uu,Uua))
            & ! [Y7: A] :
                ( pp(aa(A,bool,Uu,Y7))
               => pp(aa(A,bool,ord_less_eq(A,Y7),Uua)) ) ) ) ) ).

% ATP.lambda_99
tff(fact_8278_ATP_Olambda__100,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(nat,fun(nat,A)),Uua: nat] : aa(nat,A,aTP_Lamp_if(fun(nat,fun(nat,A)),fun(nat,A),Uu),Uua) = groups7121269368397514597t_prod(nat,A,aa(nat,fun(nat,A),Uu,Uua),set_ord_lessThan(nat,Uua)) ) ).

% ATP.lambda_100
tff(fact_8279_ATP_Olambda__101,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(nat,fun(nat,A)),Uua: nat] : aa(nat,A,aTP_Lamp_ic(fun(nat,fun(nat,A)),fun(nat,A),Uu),Uua) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),Uu,Uua)),set_ord_lessThan(nat,Uua)) ) ).

% ATP.lambda_101
tff(fact_8280_ATP_Olambda__102,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,B),Uua: A] : aa(A,product_prod(B,A),aTP_Lamp_qr(fun(A,B),fun(A,product_prod(B,A)),Uu),Uua) = aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),aa(A,B,Uu,Uua)),Uua) ).

% ATP.lambda_102
tff(fact_8281_ATP_Olambda__103,axiom,
    ! [A: $tType,Uu: fun(A,real),Uua: A] :
      ( pp(aa(A,bool,aTP_Lamp_agv(fun(A,real),fun(A,bool),Uu),Uua))
    <=> pp(aa(real,bool,ord_less(real,aa(A,real,Uu,Uua)),zero_zero(real))) ) ).

% ATP.lambda_103
tff(fact_8282_ATP_Olambda__104,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_ve(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(A,real,Uu,Uua)),zero_zero(real)) ) ).

% ATP.lambda_104
tff(fact_8283_ATP_Olambda__105,axiom,
    ! [A: $tType,Uu: fun(A,real),Uua: A] : aa(A,complex,aTP_Lamp_es(fun(A,real),fun(A,complex),Uu),Uua) = complex2(aa(A,real,Uu,Uua),zero_zero(real)) ).

% ATP.lambda_105
tff(fact_8284_ATP_Olambda__106,axiom,
    ! [B: $tType,A: $tType,Uu: fun(B,A),Uua: B] : aa(B,list(A),aTP_Lamp_amm(fun(B,A),fun(B,list(A)),Uu),Uua) = aa(list(A),list(A),cons(A,aa(B,A,Uu,Uua)),nil(A)) ).

% ATP.lambda_106
tff(fact_8285_ATP_Olambda__107,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(B,A),Uua: B] :
          ( pp(aa(B,bool,aTP_Lamp_ed(fun(B,A),fun(B,bool),Uu),Uua))
        <=> ( aa(B,A,Uu,Uua) = zero_zero(A) ) ) ) ).

% ATP.lambda_107
tff(fact_8286_ATP_Olambda__108,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(B,A),Uua: B] :
          ( pp(aa(B,bool,aTP_Lamp_hq(fun(B,A),fun(B,bool),Uu),Uua))
        <=> ( aa(B,A,Uu,Uua) = one_one(A) ) ) ) ).

% ATP.lambda_108
tff(fact_8287_ATP_Olambda__109,axiom,
    ! [Uu: fun(nat,real),Uua: nat] : aa(nat,real,aTP_Lamp_ace(fun(nat,real),fun(nat,real),Uu),Uua) = aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_aab(fun(nat,real),fun(nat,real),Uu)),set_ord_lessThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uua)),one_one(nat)))) ).

% ATP.lambda_109
tff(fact_8288_ATP_Olambda__110,axiom,
    ! [Uu: fun(nat,real),Uua: nat] : aa(nat,real,aTP_Lamp_acd(fun(nat,real),fun(nat,real),Uu),Uua) = aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_aab(fun(nat,real),fun(nat,real),Uu)),set_ord_lessThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uua))) ).

% ATP.lambda_110
tff(fact_8289_ATP_Olambda__111,axiom,
    ! [A: $tType,Uu: list(A),Uua: list(A)] : aa(list(A),list(list(A)),aTP_Lamp_qa(list(A),fun(list(A),list(list(A))),Uu),Uua) = map(A,list(A),aa(list(A),fun(A,list(A)),aTP_Lamp_pz(list(A),fun(A,list(A))),Uua),Uu) ).

% ATP.lambda_111
tff(fact_8290_ATP_Olambda__112,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu: fun(A,B),Uua: A] : aa(A,real,aTP_Lamp_acs(fun(A,B),fun(A,real),Uu),Uua) = aa(real,real,aa(real,fun(real,real),divide_divide(real),real_V7770717601297561774m_norm(B,aa(A,B,Uu,Uua))),real_V7770717601297561774m_norm(A,Uua)) ) ).

% ATP.lambda_112
tff(fact_8291_ATP_Olambda__113,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_kv(A,fun(nat,A),Uu),Uua) = aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),semiring_char_0_fact(real,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uua),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uua),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) ).

% ATP.lambda_113
tff(fact_8292_ATP_Olambda__114,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_kt(A,fun(nat,A),Uu),Uua) = aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),semiring_char_0_fact(real,aa(nat,nat,suc,Uua)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu),aa(nat,nat,suc,Uua))) ) ).

% ATP.lambda_114
tff(fact_8293_ATP_Olambda__115,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_kk(A,fun(nat,A),Uu),Uua) = aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),semiring_char_0_fact(real,Uua))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu),Uua)) ) ).

% ATP.lambda_115
tff(fact_8294_ATP_Olambda__116,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_gp(A,fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,inverse_inverse(A),semiring_char_0_fact(A,Uua))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu),Uua)) ) ).

% ATP.lambda_116
tff(fact_8295_ATP_Olambda__117,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_er(A,fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,Uua))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu),Uua)) ) ).

% ATP.lambda_117
tff(fact_8296_ATP_Olambda__118,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Uu: num,Uua: num] : aa(num,word(A),aTP_Lamp_als(num,fun(num,word(A)),Uu),Uua) = aa(word(A),word(A),bit_se2584673776208193580ke_bit(word(A),aa(num,nat,numeral_numeral(nat),Uu)),aa(word(A),word(A),minus_minus(word(A),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),aa(num,nat,numeral_numeral(nat),Uu))),aa(num,word(A),numeral_numeral(word(A)),Uua))) ) ).

% ATP.lambda_118
tff(fact_8297_ATP_Olambda__119,axiom,
    ! [Uu: num,Uua: num] : aa(num,int,aTP_Lamp_alt(num,fun(num,int),Uu),Uua) = aa(int,int,bit_se2584673776208193580ke_bit(int,aa(num,nat,numeral_numeral(nat),Uu)),aa(int,int,minus_minus(int,aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(num,nat,numeral_numeral(nat),Uu))),aa(num,int,numeral_numeral(int),Uua))) ).

% ATP.lambda_119
tff(fact_8298_ATP_Olambda__120,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_kr(A,fun(nat,A),Uu),Uua) = aa(A,A,real_V8093663219630862766scaleR(A,cos_coeff(Uua)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),Uu)),Uua)) ) ).

% ATP.lambda_120
tff(fact_8299_ATP_Olambda__121,axiom,
    ! [Uu: nat,Uua: real] : aa(real,real,aTP_Lamp_qf(nat,fun(real,real),Uu),Uua) = aa(real,real,aa(real,fun(real,real),times_times(real),sgn_sgn(real,Uua)),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,abs_abs(real),Uua)),Uu)) ).

% ATP.lambda_121
tff(fact_8300_ATP_Olambda__122,axiom,
    ! [Uu: bool,Uua: nat] : aa(nat,nat,aa(bool,fun(nat,nat),aTP_Lamp_amu(bool,fun(nat,nat)),Uu),Uua) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(bool,nat,zero_neq_one_of_bool(nat),Uu)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uua),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ).

% ATP.lambda_122
tff(fact_8301_ATP_Olambda__123,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_kl(A,fun(nat,A),Uu),Uua) = aa(A,A,real_V8093663219630862766scaleR(A,sin_coeff(Uua)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu),Uua)) ) ).

% ATP.lambda_123
tff(fact_8302_ATP_Olambda__124,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_km(A,fun(nat,A),Uu),Uua) = aa(A,A,real_V8093663219630862766scaleR(A,cos_coeff(Uua)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu),Uua)) ) ).

% ATP.lambda_124
tff(fact_8303_ATP_Olambda__125,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_abz(A,fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,semiring_1_of_nat(A),Uua)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu),Uua)) ) ).

% ATP.lambda_125
tff(fact_8304_ATP_Olambda__126,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V8999393235501362500lgebra(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_aby(A,fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),Uua)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu),Uua)) ) ).

% ATP.lambda_126
tff(fact_8305_ATP_Olambda__127,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_gj(real,fun(nat,real),Uu),Uua) = aa(real,real,aa(real,fun(real,real),times_times(real),sin_coeff(Uua)),aa(nat,real,aa(real,fun(nat,real),power_power(real),Uu),Uua)) ).

% ATP.lambda_127
tff(fact_8306_ATP_Olambda__128,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_gh(real,fun(nat,real),Uu),Uua) = aa(real,real,aa(real,fun(real,real),times_times(real),cos_coeff(Uua)),aa(nat,real,aa(real,fun(nat,real),power_power(real),Uu),Uua)) ).

% ATP.lambda_128
tff(fact_8307_ATP_Olambda__129,axiom,
    ! [A: $tType,Uu: list(list(A)),Uua: A] : aa(A,list(list(A)),aTP_Lamp_py(list(list(A)),fun(A,list(list(A))),Uu),Uua) = map(list(A),list(A),cons(A,Uua),product_lists(A,Uu)) ).

% ATP.lambda_129
tff(fact_8308_ATP_Olambda__130,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: nat,Uua: nat] : aa(nat,A,aTP_Lamp_gq(nat,fun(nat,A),Uu),Uua) = aa(nat,A,gbinomial(A,aa(nat,A,semiring_1_of_nat(A),Uua)),Uu) ) ).

% ATP.lambda_130
tff(fact_8309_ATP_Olambda__131,axiom,
    ! [A: $tType,B: $tType,Uu: list(B),Uua: A] : aa(A,list(product_prod(A,B)),aTP_Lamp_pm(list(B),fun(A,list(product_prod(A,B))),Uu),Uua) = map(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uua),Uu) ).

% ATP.lambda_131
tff(fact_8310_ATP_Olambda__132,axiom,
    ! [Uu: real,Uua: real] :
      ( pp(aa(real,bool,aTP_Lamp_lc(real,fun(real,bool),Uu),Uua))
    <=> ( aa(real,real,exp(real),Uua) = Uu ) ) ).

% ATP.lambda_132
tff(fact_8311_ATP_Olambda__133,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Uu: A,Uua: A] : aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),aTP_Lamp_lw(A,fun(A,product_prod(A,A))),Uu),Uua) = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uu),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),Uua)),one_one(A))) ) ).

% ATP.lambda_133
tff(fact_8312_ATP_Olambda__134,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Uu: A,Uua: A] : aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),aTP_Lamp_lx(A,fun(A,product_prod(A,A))),Uu),Uua) = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uu),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),Uua)) ) ).

% ATP.lambda_134
tff(fact_8313_ATP_Olambda__135,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_gs(A,fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Uu),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,semiring_1_of_nat(A),Uua)),aa(num,A,numeral_numeral(A),bit0(one2)))) ) ).

% ATP.lambda_135
tff(fact_8314_ATP_Olambda__136,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_jv(nat,fun(nat,nat),Uu),Uua) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uua),aa(nat,nat,binomial(Uu),Uua)) ).

% ATP.lambda_136
tff(fact_8315_ATP_Olambda__137,axiom,
    ! [A: $tType,Uu: set(A),Uua: A] :
      ( pp(aa(A,bool,aTP_Lamp_oo(set(A),fun(A,bool),Uu),Uua))
    <=> ( Uu = insert(A,Uua,bot_bot(set(A))) ) ) ).

% ATP.lambda_137
tff(fact_8316_ATP_Olambda__138,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_abw(real,fun(nat,real),Uu),Uua) = aa(real,real,aa(real,fun(real,real),times_times(real),Uu),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),aa(real,real,uminus_uminus(real),aa(real,real,inverse_inverse(real),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,Uua)))))) ).

% ATP.lambda_138
tff(fact_8317_ATP_Olambda__139,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_abp(fun(nat,A),fun(nat,A),Uu),Uua) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,Uu),set_or1337092689740270186AtMost(nat,zero_zero(nat),Uua)) ) ).

% ATP.lambda_139
tff(fact_8318_ATP_Olambda__140,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,B),Uua: A] : aa(A,product_prod(A,B),aTP_Lamp_qs(fun(A,B),fun(A,product_prod(A,B)),Uu),Uua) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uua),aa(A,B,Uu,Uua)) ).

% ATP.lambda_140
tff(fact_8319_ATP_Olambda__141,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_pc(A,fun(nat,A),Uu),Uua) = bit_se4730199178511100633sh_bit(A,Uua,aa(bool,A,zero_neq_one_of_bool(A),aa(nat,bool,bit_se5641148757651400278ts_bit(A,Uu),Uua))) ) ).

% ATP.lambda_141
tff(fact_8320_ATP_Olambda__142,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,real,aTP_Lamp_abr(fun(nat,A),fun(nat,real),Uu),Uua) = aa(real,real,root(Uua),real_V7770717601297561774m_norm(A,aa(nat,A,Uu,Uua))) ) ).

% ATP.lambda_142
tff(fact_8321_ATP_Olambda__143,axiom,
    ! [Uu: int,Uua: int] : aa(int,int,aa(int,fun(int,int),aTP_Lamp_mo(int,fun(int,int)),Uu),Uua) = aa(int,int,aa(int,fun(int,int),plus_plus(int),Uu),aa(bool,int,zero_neq_one_of_bool(int),aa(bool,bool,fNot,aa(int,bool,fequal(int,Uua),zero_zero(int))))) ).

% ATP.lambda_143
tff(fact_8322_ATP_Olambda__144,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_abq(real,fun(nat,real),Uu),Uua) = aa(real,real,aa(real,fun(real,real),plus_plus(real),Uu),aa(real,real,uminus_uminus(real),aa(real,real,inverse_inverse(real),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,Uua))))) ).

% ATP.lambda_144
tff(fact_8323_ATP_Olambda__145,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_abg(real,fun(nat,real),Uu),Uua) = aa(real,real,aa(real,fun(real,real),plus_plus(real),Uu),aa(real,real,inverse_inverse(real),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,Uua)))) ).

% ATP.lambda_145
tff(fact_8324_ATP_Olambda__146,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_abh(fun(nat,A),fun(nat,A),Uu),Uua) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,Uu),set_ord_lessThan(nat,Uua)) ) ).

% ATP.lambda_146
tff(fact_8325_ATP_Olambda__147,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topological_t2_space(A) )
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_abs(fun(nat,A),fun(nat,A),Uu),Uua) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,Uu),set_ord_lessThan(nat,Uua)) ) ).

% ATP.lambda_147
tff(fact_8326_ATP_Olambda__148,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_abi(fun(nat,A),fun(nat,A),Uu),Uua) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,Uu),set_ord_atMost(nat,Uua)) ) ).

% ATP.lambda_148
tff(fact_8327_ATP_Olambda__149,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topological_t2_space(A) )
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_abt(fun(nat,A),fun(nat,A),Uu),Uua) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,Uu),set_ord_atMost(nat,Uua)) ) ).

% ATP.lambda_149
tff(fact_8328_ATP_Olambda__150,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_aau(A,fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),divide_divide(A),Uu),aa(nat,A,semiring_1_of_nat(A),Uua)) ) ).

% ATP.lambda_150
tff(fact_8329_ATP_Olambda__151,axiom,
    ! [A: $tType] :
      ( ( semiring_char_0(A)
        & semidom_divide(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_hz(A,fun(nat,A),Uu),Uua) = aa(A,A,minus_minus(A,Uu),aa(nat,A,semiring_1_of_nat(A),Uua)) ) ).

% ATP.lambda_151
tff(fact_8330_ATP_Olambda__152,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_ph(A,fun(nat,A),Uu),Uua) = aa(A,A,minus_minus(A,Uu),aa(nat,A,semiring_1_of_nat(A),Uua)) ) ).

% ATP.lambda_152
tff(fact_8331_ATP_Olambda__153,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_hv(A,fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Uu),aa(nat,A,semiring_1_of_nat(A),Uua)) ) ).

% ATP.lambda_153
tff(fact_8332_ATP_Olambda__154,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: fun(A,bool),Uua: A] :
          ( pp(aa(A,bool,aTP_Lamp_aft(fun(A,bool),fun(A,bool),Uu),Uua))
        <=> eventually(A,Uu,topolo7230453075368039082e_nhds(A,Uua)) ) ) ).

% ATP.lambda_154
tff(fact_8333_ATP_Olambda__155,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: nat,Uua: A] : aa(A,A,aTP_Lamp_zp(nat,fun(A,A),Uu),Uua) = comm_s3205402744901411588hammer(A,Uua,Uu) ) ).

% ATP.lambda_155
tff(fact_8334_ATP_Olambda__156,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: A,Uua: real] : aa(real,A,aTP_Lamp_acj(A,fun(real,A),Uu),Uua) = aa(A,A,real_V8093663219630862766scaleR(A,Uua),Uu) ) ).

% ATP.lambda_156
tff(fact_8335_ATP_Olambda__157,axiom,
    ! [A: $tType,Uu: set(A),Uua: set(A)] :
      ( pp(aa(set(A),bool,aTP_Lamp_aq(set(A),fun(set(A),bool),Uu),Uua))
    <=> pp(aa(set(A),bool,ord_less_eq(set(A),Uua),Uu)) ) ).

% ATP.lambda_157
tff(fact_8336_ATP_Olambda__158,axiom,
    ! [Uu: nat,Uua: nat] :
      ( pp(aa(nat,bool,aTP_Lamp_ao(nat,fun(nat,bool),Uu),Uua))
    <=> pp(aa(nat,bool,ord_less_eq(nat,Uua),Uu)) ) ).

% ATP.lambda_158
tff(fact_8337_ATP_Olambda__159,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu: A,Uua: A] :
          ( pp(aa(A,bool,aTP_Lamp_afy(A,fun(A,bool),Uu),Uua))
        <=> pp(aa(A,bool,ord_less_eq(A,Uua),Uu)) ) ) ).

% ATP.lambda_159
tff(fact_8338_ATP_Olambda__160,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Uu: A,Uua: A] :
          ( pp(aa(A,bool,aTP_Lamp_ia(A,fun(A,bool),Uu),Uua))
        <=> pp(aa(A,bool,ord_less_eq(A,Uua),Uu)) ) ) ).

% ATP.lambda_160
tff(fact_8339_ATP_Olambda__161,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_qc(nat,fun(nat,nat),Uu),Uua) = modulo_modulo(nat,Uua,Uu) ).

% ATP.lambda_161
tff(fact_8340_ATP_Olambda__162,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_acl(A,fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),divide_divide(A),Uua),Uu) ) ).

% ATP.lambda_162
tff(fact_8341_ATP_Olambda__163,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_nt(A,fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),divide_divide(A),Uua),Uu) ) ).

% ATP.lambda_163
tff(fact_8342_ATP_Olambda__164,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_qi(A,fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),divide_divide(A),Uua),Uu) ) ).

% ATP.lambda_164
tff(fact_8343_ATP_Olambda__165,axiom,
    ! [Uu: nat,Uua: nat] :
      ( pp(aa(nat,bool,aTP_Lamp_ap(nat,fun(nat,bool),Uu),Uua))
    <=> pp(aa(nat,bool,ord_less(nat,Uua),Uu)) ) ).

% ATP.lambda_165
tff(fact_8344_ATP_Olambda__166,axiom,
    ! [A: $tType] :
      ( unboun7993243217541854897norder(A)
     => ! [Uu: A,Uua: A] :
          ( pp(aa(A,bool,aTP_Lamp_afx(A,fun(A,bool),Uu),Uua))
        <=> pp(aa(A,bool,ord_less(A,Uua),Uu)) ) ) ).

% ATP.lambda_166
tff(fact_8345_ATP_Olambda__167,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Uu: A,Uua: A] :
          ( pp(aa(A,bool,aTP_Lamp_alq(A,fun(A,bool),Uu),Uua))
        <=> pp(aa(A,bool,ord_less(A,Uua),Uu)) ) ) ).

% ATP.lambda_167
tff(fact_8346_ATP_Olambda__168,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Uu: A,Uua: A] :
          ( pp(aa(A,bool,aTP_Lamp_ew(A,fun(A,bool),Uu),Uua))
        <=> pp(aa(A,bool,ord_less(A,Uua),Uu)) ) ) ).

% ATP.lambda_168
tff(fact_8347_ATP_Olambda__169,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_aaq(nat,fun(nat,nat),Uu),Uua) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uua),Uu) ).

% ATP.lambda_169
tff(fact_8348_ATP_Olambda__170,axiom,
    ! [A: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_ack(A,fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),Uua),Uu) ) ).

% ATP.lambda_170
tff(fact_8349_ATP_Olambda__171,axiom,
    ! [A: $tType] :
      ( ab_semigroup_mult(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_ad(A,fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),Uua),Uu) ) ).

% ATP.lambda_171
tff(fact_8350_ATP_Olambda__172,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_og(A,fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),Uua),Uu) ) ).

% ATP.lambda_172
tff(fact_8351_ATP_Olambda__173,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_pd(nat,fun(nat,nat),Uu),Uua) = aa(nat,nat,minus_minus(nat,Uua),Uu) ).

% ATP.lambda_173
tff(fact_8352_ATP_Olambda__174,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_nr(A,fun(A,A),Uu),Uua) = aa(A,A,minus_minus(A,Uua),Uu) ) ).

% ATP.lambda_174
tff(fact_8353_ATP_Olambda__175,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_ol(A,fun(A,A),Uu),Uua) = aa(A,A,minus_minus(A,Uua),Uu) ) ).

% ATP.lambda_175
tff(fact_8354_ATP_Olambda__176,axiom,
    ! [Uu: nat,Uua: real] : aa(real,real,aTP_Lamp_sb(nat,fun(real,real),Uu),Uua) = aa(nat,real,aa(real,fun(nat,real),power_power(real),Uua),Uu) ).

% ATP.lambda_176
tff(fact_8355_ATP_Olambda__177,axiom,
    ! [Uu: code_integer,Uua: code_integer] : aa(code_integer,code_integer,aTP_Lamp_pl(code_integer,fun(code_integer,code_integer),Uu),Uua) = aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),Uua),Uu) ).

% ATP.lambda_177
tff(fact_8356_ATP_Olambda__178,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_os(nat,fun(nat,nat),Uu),Uua) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uua),Uu) ).

% ATP.lambda_178
tff(fact_8357_ATP_Olambda__179,axiom,
    ! [Uu: int,Uua: int] : aa(int,int,aTP_Lamp_pk(int,fun(int,int),Uu),Uua) = aa(int,int,aa(int,fun(int,int),plus_plus(int),Uua),Uu) ).

% ATP.lambda_179
tff(fact_8358_ATP_Olambda__180,axiom,
    ! [A: $tType] :
      ( cancel_semigroup_add(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_qk(A,fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Uua),Uu) ) ).

% ATP.lambda_180
tff(fact_8359_ATP_Olambda__181,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_nq(A,fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Uua),Uu) ) ).

% ATP.lambda_181
tff(fact_8360_ATP_Olambda__182,axiom,
    ! [Uu: real,Uua: real] : aa(real,real,aTP_Lamp_se(real,fun(real,real),Uu),Uua) = powr(real,Uua,Uu) ).

% ATP.lambda_182
tff(fact_8361_ATP_Olambda__183,axiom,
    ! [Uu: nat,Uua: nat] :
      ( pp(aa(nat,bool,aTP_Lamp_am(nat,fun(nat,bool),Uu),Uua))
    <=> pp(aa(nat,bool,dvd_dvd(nat,Uua),Uu)) ) ).

% ATP.lambda_183
tff(fact_8362_ATP_Olambda__184,axiom,
    ! [Uu: int,Uua: int] :
      ( pp(aa(int,bool,aTP_Lamp_az(int,fun(int,bool),Uu),Uua))
    <=> pp(aa(int,bool,dvd_dvd(int,Uua),Uu)) ) ).

% ATP.lambda_184
tff(fact_8363_ATP_Olambda__185,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: A,Uua: A] :
          ( pp(aa(A,bool,aTP_Lamp_aa(A,fun(A,bool),Uu),Uua))
        <=> pp(aa(A,bool,dvd_dvd(A,Uua),Uu)) ) ) ).

% ATP.lambda_185
tff(fact_8364_ATP_Olambda__186,axiom,
    ! [B: $tType,A: $tType,Uu: A,Uua: B] : aa(B,product_prod(B,A),aa(A,fun(B,product_prod(B,A)),aTP_Lamp_qq(A,fun(B,product_prod(B,A))),Uu),Uua) = aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),Uua),Uu) ).

% ATP.lambda_186
tff(fact_8365_ATP_Olambda__187,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_jl(nat,fun(nat,nat),Uu),Uua) = aa(nat,nat,binomial(Uua),Uu) ).

% ATP.lambda_187
tff(fact_8366_ATP_Olambda__188,axiom,
    ! [A: $tType,Uu: list(A),Uua: A] : aa(A,list(A),aa(list(A),fun(A,list(A)),aTP_Lamp_pz(list(A),fun(A,list(A))),Uu),Uua) = aa(list(A),list(A),cons(A,Uua),Uu) ).

% ATP.lambda_188
tff(fact_8367_ATP_Olambda__189,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_abf(real,fun(nat,real),Uu),Uua) = aa(real,real,root(Uua),Uu) ).

% ATP.lambda_189
tff(fact_8368_ATP_Olambda__190,axiom,
    ! [B: $tType,Uu: set(B),Uua: B] :
      ( pp(aa(B,bool,aTP_Lamp_aih(set(B),fun(B,bool),Uu),Uua))
    <=> pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),Uua),Uu)) ) ).

% ATP.lambda_190
tff(fact_8369_ATP_Olambda__191,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: set(A),Uua: A] :
          ( pp(aa(A,bool,aTP_Lamp_afv(set(A),fun(A,bool),Uu),Uua))
        <=> pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Uua),Uu)) ) ) ).

% ATP.lambda_191
tff(fact_8370_ATP_Olambda__192,axiom,
    ! [A: $tType,Uu: set(A),Uua: A] :
      ( pp(aa(A,bool,aTP_Lamp_a(set(A),fun(A,bool),Uu),Uua))
    <=> pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Uua),Uu)) ) ).

% ATP.lambda_192
tff(fact_8371_ATP_Olambda__193,axiom,
    ! [A: $tType,Uu: A,Uua: A] :
      ( pp(aa(A,bool,aTP_Lamp_lg(A,fun(A,bool),Uu),Uua))
    <=> ( Uua = Uu ) ) ).

% ATP.lambda_193
tff(fact_8372_ATP_Olambda__194,axiom,
    ! [A: $tType,Uu: A,Uua: list(A)] : aa(list(A),list(A),aa(A,fun(list(A),list(A)),aTP_Lamp_amx(A,fun(list(A),list(A))),Uu),Uua) = aa(list(A),list(A),cons(A,Uu),nil(A)) ).

% ATP.lambda_194
tff(fact_8373_ATP_Olambda__195,axiom,
    ! [A: $tType,Uu: A,Uua: list(A)] : aa(list(A),list(list(A)),aa(A,fun(list(A),list(list(A))),aTP_Lamp_amy(A,fun(list(A),list(list(A)))),Uu),Uua) = aa(list(list(A)),list(list(A)),cons(list(A),Uua),nil(list(A))) ).

% ATP.lambda_195
tff(fact_8374_ATP_Olambda__196,axiom,
    ! [A: $tType,Uu: fun(A,real),Uua: A] :
      ( pp(aa(A,bool,aTP_Lamp_aho(fun(A,real),fun(A,bool),Uu),Uua))
    <=> pp(aa(real,bool,ord_less_eq(real,zero_zero(real)),aa(A,real,Uu,Uua))) ) ).

% ATP.lambda_196
tff(fact_8375_ATP_Olambda__197,axiom,
    ! [B: $tType,Uu: fun(B,real),Uua: B] :
      ( pp(aa(B,bool,aTP_Lamp_ahi(fun(B,real),fun(B,bool),Uu),Uua))
    <=> pp(aa(real,bool,ord_less_eq(real,one_one(real)),aa(B,real,Uu,Uua))) ) ).

% ATP.lambda_197
tff(fact_8376_ATP_Olambda__198,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Uu: fun(A,real),Uua: A] :
          ( pp(aa(A,bool,aTP_Lamp_aha(fun(A,real),fun(A,bool),Uu),Uua))
        <=> pp(aa(real,bool,ord_less_eq(real,one_one(real)),aa(A,real,Uu,Uua))) ) ) ).

% ATP.lambda_198
tff(fact_8377_ATP_Olambda__199,axiom,
    ! [A: $tType,Uu: fun(A,real),Uua: A] :
      ( pp(aa(A,bool,aTP_Lamp_agu(fun(A,real),fun(A,bool),Uu),Uua))
    <=> pp(aa(real,bool,ord_less(real,zero_zero(real)),aa(A,real,Uu,Uua))) ) ).

% ATP.lambda_199
tff(fact_8378_ATP_Olambda__200,axiom,
    ! [A: $tType,Uu: fun(A,nat),Uua: A] :
      ( pp(aa(A,bool,aTP_Lamp_bd(fun(A,nat),fun(A,bool),Uu),Uua))
    <=> pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),aa(A,nat,Uu,Uua))) ) ).

% ATP.lambda_200
tff(fact_8379_ATP_Olambda__201,axiom,
    ! [Uu: fun(nat,real),Uua: nat] : aa(nat,real,aTP_Lamp_fu(fun(nat,real),fun(nat,real),Uu),Uua) = aa(nat,real,Uu,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uua)),one_one(nat))) ).

% ATP.lambda_201
tff(fact_8380_ATP_Olambda__202,axiom,
    ! [Uu: fun(nat,real),Uua: nat] : aa(nat,real,aTP_Lamp_ft(fun(nat,real),fun(nat,real),Uu),Uua) = aa(nat,real,Uu,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uua)) ).

% ATP.lambda_202
tff(fact_8381_ATP_Olambda__203,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: fun(A,A),Uua: A] : aa(A,A,aTP_Lamp_adu(fun(A,A),fun(A,A),Uu),Uua) = aa(A,A,Uu,aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),Uua)) ) ).

% ATP.lambda_203
tff(fact_8382_ATP_Olambda__204,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: fun(set(nat),A),Uua: nat] : aa(nat,A,aTP_Lamp_aar(fun(set(nat),A),fun(nat,A),Uu),Uua) = aa(set(nat),A,Uu,set_ord_lessThan(nat,Uua)) ) ).

% ATP.lambda_204
tff(fact_8383_ATP_Olambda__205,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: fun(set(nat),A),Uua: nat] : aa(nat,A,aTP_Lamp_aas(fun(set(nat),A),fun(nat,A),Uu),Uua) = aa(set(nat),A,Uu,set_ord_atMost(nat,Uua)) ) ).

% ATP.lambda_205
tff(fact_8384_ATP_Olambda__206,axiom,
    ! [Uu: fun(real,bool),Uua: real] :
      ( pp(aa(real,bool,aTP_Lamp_ahh(fun(real,bool),fun(real,bool),Uu),Uua))
    <=> pp(aa(real,bool,Uu,aa(real,real,inverse_inverse(real),Uua))) ) ).

% ATP.lambda_206
tff(fact_8385_ATP_Olambda__207,axiom,
    ! [A: $tType,Uu: fun(real,A),Uua: real] : aa(real,A,aTP_Lamp_aeg(fun(real,A),fun(real,A),Uu),Uua) = aa(real,A,Uu,aa(real,real,inverse_inverse(real),Uua)) ).

% ATP.lambda_207
tff(fact_8386_ATP_Olambda__208,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: fun(real,A),Uua: nat] : aa(nat,A,aTP_Lamp_aei(fun(real,A),fun(nat,A),Uu),Uua) = aa(real,A,Uu,aa(nat,real,semiring_1_of_nat(real),Uua)) ) ).

% ATP.lambda_208
tff(fact_8387_ATP_Olambda__209,axiom,
    ! [A: $tType,Uu: fun(int,A),Uua: nat] : aa(nat,A,aTP_Lamp_aic(fun(int,A),fun(nat,A),Uu),Uua) = aa(int,A,Uu,aa(nat,int,semiring_1_of_nat(int),Uua)) ).

% ATP.lambda_209
tff(fact_8388_ATP_Olambda__210,axiom,
    ! [Uu: fun(real,real),Uua: real] : aa(real,real,aTP_Lamp_rw(fun(real,real),fun(real,real),Uu),Uua) = aa(real,real,Uu,aa(real,real,uminus_uminus(real),Uua)) ).

% ATP.lambda_210
tff(fact_8389_ATP_Olambda__211,axiom,
    ! [Uu: fun(real,bool),Uua: real] :
      ( pp(aa(real,bool,aTP_Lamp_agz(fun(real,bool),fun(real,bool),Uu),Uua))
    <=> pp(aa(real,bool,Uu,aa(real,real,uminus_uminus(real),Uua))) ) ).

% ATP.lambda_211
tff(fact_8390_ATP_Olambda__212,axiom,
    ! [A: $tType,Uu: fun(real,A),Uua: real] : aa(real,A,aTP_Lamp_adn(fun(real,A),fun(real,A),Uu),Uua) = aa(real,A,Uu,aa(real,real,uminus_uminus(real),Uua)) ).

% ATP.lambda_212
tff(fact_8391_ATP_Olambda__213,axiom,
    ! [Uu: fun(nat,bool),Uua: nat] :
      ( pp(aa(nat,bool,aTP_Lamp_qd(fun(nat,bool),fun(nat,bool),Uu),Uua))
    <=> pp(aa(nat,bool,Uu,aa(nat,nat,suc,Uua))) ) ).

% ATP.lambda_213
tff(fact_8392_ATP_Olambda__214,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_ce(fun(nat,A),fun(nat,A),Uu),Uua) = aa(nat,A,Uu,aa(nat,nat,suc,Uua)) ) ).

% ATP.lambda_214
tff(fact_8393_ATP_Olambda__215,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_aaj(fun(nat,A),fun(nat,A),Uu),Uua) = aa(nat,A,Uu,aa(nat,nat,suc,Uua)) ) ).

% ATP.lambda_215
tff(fact_8394_ATP_Olambda__216,axiom,
    ! [A: $tType] :
      ( ( topolo5987344860129210374id_add(A)
        & topological_t2_space(A) )
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_fa(fun(nat,A),fun(nat,A),Uu),Uua) = aa(nat,A,Uu,aa(nat,nat,suc,Uua)) ) ).

% ATP.lambda_216
tff(fact_8395_ATP_Olambda__217,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_hn(fun(nat,A),fun(nat,A),Uu),Uua) = aa(nat,A,Uu,aa(nat,nat,suc,Uua)) ) ).

% ATP.lambda_217
tff(fact_8396_ATP_Olambda__218,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_ea(fun(nat,A),fun(nat,A),Uu),Uua) = aa(nat,A,Uu,aa(nat,nat,suc,Uua)) ) ).

% ATP.lambda_218
tff(fact_8397_ATP_Olambda__219,axiom,
    ! [A: $tType,Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_qb(fun(nat,A),fun(nat,A),Uu),Uua) = aa(nat,A,Uu,aa(nat,nat,suc,Uua)) ).

% ATP.lambda_219
tff(fact_8398_ATP_Olambda__220,axiom,
    ! [Uu: nat,Uua: num] : aa(num,option(num),aTP_Lamp_amb(nat,fun(num,option(num)),Uu),Uua) = aa(num,option(num),some(num),case_option(num,num,one2,bit1,bit_take_bit_num(Uu,Uua))) ).

% ATP.lambda_220
tff(fact_8399_ATP_Olambda__221,axiom,
    ! [Uu: num,Uua: nat] : aa(nat,option(num),aTP_Lamp_alw(num,fun(nat,option(num)),Uu),Uua) = aa(num,option(num),some(num),case_option(num,num,one2,bit1,bit_take_bit_num(Uua,Uu))) ).

% ATP.lambda_221
tff(fact_8400_ATP_Olambda__222,axiom,
    ! [Uu: fun(nat,real),Uua: real] : aa(real,real,aTP_Lamp_sv(fun(nat,real),fun(real,real),Uu),Uua) = suminf(real,aa(real,fun(nat,real),aTP_Lamp_su(fun(nat,real),fun(real,fun(nat,real)),Uu),Uua)) ).

% ATP.lambda_222
tff(fact_8401_ATP_Olambda__223,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: fun(nat,A),Uua: A] : aa(A,A,aTP_Lamp_sc(fun(nat,A),fun(A,A),Uu),Uua) = suminf(A,aa(A,fun(nat,A),aTP_Lamp_gc(fun(nat,A),fun(A,fun(nat,A)),Uu),Uua)) ) ).

% ATP.lambda_223
tff(fact_8402_ATP_Olambda__224,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,complex,aTP_Lamp_kd(nat,fun(nat,complex),Uu),Uua) = cis(aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),pi)),aa(nat,real,semiring_1_of_nat(real),Uua))),aa(nat,real,semiring_1_of_nat(real),Uu))) ).

% ATP.lambda_224
tff(fact_8403_ATP_Olambda__225,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_ua(A,fun(A,A),Uu),Uua) = aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,inverse_inverse(A),Uu)),Uua)),aa(A,A,inverse_inverse(A),Uu))) ) ).

% ATP.lambda_225
tff(fact_8404_ATP_Olambda__226,axiom,
    ! [Uu: fun(real,real),Uua: real] :
      ( pp(aa(real,bool,aTP_Lamp_aem(fun(real,real),fun(real,bool),Uu),Uua))
    <=> ( aa(real,real,Uu,Uua) != zero_zero(real) ) ) ).

% ATP.lambda_226
tff(fact_8405_ATP_Olambda__227,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,real,aTP_Lamp_kp(A,fun(nat,real),Uu),Uua) = real_V7770717601297561774m_norm(A,aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),semiring_char_0_fact(real,Uua))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu),Uua))) ) ).

% ATP.lambda_227
tff(fact_8406_ATP_Olambda__228,axiom,
    ! [Uu: nat,Uua: nat] :
      ( pp(aa(nat,bool,aTP_Lamp_kz(nat,fun(nat,bool),Uu),Uua))
    <=> ~ pp(aa(nat,bool,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Uu),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uua)))) ) ).

% ATP.lambda_228
tff(fact_8407_ATP_Olambda__229,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_kq(A,fun(nat,A),Uu),Uua) = aa(A,A,uminus_uminus(A),aa(A,A,real_V8093663219630862766scaleR(A,sin_coeff(Uua)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),Uu)),Uua))) ) ).

% ATP.lambda_229
tff(fact_8408_ATP_Olambda__230,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,real,aTP_Lamp_kn(A,fun(nat,real),Uu),Uua) = real_V7770717601297561774m_norm(A,aa(A,A,real_V8093663219630862766scaleR(A,sin_coeff(Uua)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu),Uua))) ) ).

% ATP.lambda_230
tff(fact_8409_ATP_Olambda__231,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,real,aTP_Lamp_ko(A,fun(nat,real),Uu),Uua) = real_V7770717601297561774m_norm(A,aa(A,A,real_V8093663219630862766scaleR(A,cos_coeff(Uua)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu),Uua))) ) ).

% ATP.lambda_231
tff(fact_8410_ATP_Olambda__232,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: A,Uua: nat] : aa(nat,fun(A,A),aTP_Lamp_gr(A,fun(nat,fun(A,A)),Uu),Uua) = aa(A,fun(A,A),times_times(A),aa(A,A,minus_minus(A,Uu),aa(nat,A,semiring_1_of_nat(A),Uua))) ) ).

% ATP.lambda_232
tff(fact_8411_ATP_Olambda__233,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Uu: A,Uua: nat] : aa(nat,fun(A,A),aTP_Lamp_gt(A,fun(nat,fun(A,A)),Uu),Uua) = aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Uu),aa(nat,A,semiring_1_of_nat(A),Uua))) ) ).

% ATP.lambda_233
tff(fact_8412_ATP_Olambda__234,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_abo(real,fun(nat,real),Uu),Uua) = aa(real,real,inverse_inverse(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),Uu),Uua)) ).

% ATP.lambda_234
tff(fact_8413_ATP_Olambda__235,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_ru(A,fun(A,A),Uu),Uua) = cos(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Uua),Uu)) ) ).

% ATP.lambda_235
tff(fact_8414_ATP_Olambda__236,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_amj(nat,fun(nat,nat),Uu),Uua) = aa(nat,nat,suc,ord_min(nat,Uu,Uua)) ).

% ATP.lambda_236
tff(fact_8415_ATP_Olambda__237,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_ami(nat,fun(nat,nat),Uu),Uua) = aa(nat,nat,suc,ord_min(nat,Uua,Uu)) ).

% ATP.lambda_237
tff(fact_8416_ATP_Olambda__238,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_amf(nat,fun(nat,nat),Uu),Uua) = aa(nat,nat,suc,aa(nat,nat,ord_max(nat,Uu),Uua)) ).

% ATP.lambda_238
tff(fact_8417_ATP_Olambda__239,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_amg(nat,fun(nat,nat),Uu),Uua) = aa(nat,nat,suc,aa(nat,nat,ord_max(nat,Uua),Uu)) ).

% ATP.lambda_239
tff(fact_8418_ATP_Olambda__240,axiom,
    ! [A: $tType,Uu: set(A),Uua: A] :
      ( pp(aa(A,bool,aTP_Lamp_by(set(A),fun(A,bool),Uu),Uua))
    <=> ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Uua),Uu)) ) ).

% ATP.lambda_240
tff(fact_8419_ATP_Olambda__241,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: A,Uua: A] :
          ( pp(aa(A,bool,aTP_Lamp_aeu(A,fun(A,bool),Uu),Uua))
        <=> ( Uua != Uu ) ) ) ).

% ATP.lambda_241
tff(fact_8420_ATP_Olambda__242,axiom,
    ! [A: $tType] :
      ( topological_t1_space(A)
     => ! [Uu: A,Uua: A] :
          ( pp(aa(A,bool,aTP_Lamp_afs(A,fun(A,bool),Uu),Uua))
        <=> ( Uua != Uu ) ) ) ).

% ATP.lambda_242
tff(fact_8421_ATP_Olambda__243,axiom,
    ! [A: $tType] :
      ( ( linorder(A)
        & no_top(A) )
     => ! [Uu: A,Uua: A] :
          ( pp(aa(A,bool,aTP_Lamp_aev(A,fun(A,bool),Uu),Uua))
        <=> ( Uua != Uu ) ) ) ).

% ATP.lambda_243
tff(fact_8422_ATP_Olambda__244,axiom,
    ! [A: $tType] :
      ( ( linorder(A)
        & no_bot(A) )
     => ! [Uu: A,Uua: A] :
          ( pp(aa(A,bool,aTP_Lamp_afw(A,fun(A,bool),Uu),Uua))
        <=> ( Uua != Uu ) ) ) ).

% ATP.lambda_244
tff(fact_8423_ATP_Olambda__245,axiom,
    ! [A: $tType,Uu: A,Uua: A] :
      ( pp(aa(A,bool,aTP_Lamp_ani(A,fun(A,bool),Uu),Uua))
    <=> ( Uua != Uu ) ) ).

% ATP.lambda_245
tff(fact_8424_ATP_Olambda__246,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,real,aTP_Lamp_cs(fun(nat,A),fun(nat,real),Uu),Uua) = real_V7770717601297561774m_norm(A,aa(nat,A,Uu,Uua)) ) ).

% ATP.lambda_246
tff(fact_8425_ATP_Olambda__247,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V4412858255891104859lgebra(A) )
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,real,aTP_Lamp_in(fun(nat,A),fun(nat,real),Uu),Uua) = real_V7770717601297561774m_norm(A,aa(nat,A,Uu,Uua)) ) ).

% ATP.lambda_247
tff(fact_8426_ATP_Olambda__248,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,real,aTP_Lamp_ch(fun(nat,A),fun(nat,real),Uu),Uua) = real_V7770717601297561774m_norm(A,aa(nat,A,Uu,Uua)) ) ).

% ATP.lambda_248
tff(fact_8427_ATP_Olambda__249,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(B,A),Uua: B] : aa(B,real,aTP_Lamp_di(fun(B,A),fun(B,real),Uu),Uua) = real_V7770717601297561774m_norm(A,aa(B,A,Uu,Uua)) ) ).

% ATP.lambda_249
tff(fact_8428_ATP_Olambda__250,axiom,
    ! [A: $tType,B: $tType] :
      ( ( comm_monoid_mult(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu: fun(B,A),Uua: B] : aa(B,real,aTP_Lamp_he(fun(B,A),fun(B,real),Uu),Uua) = real_V7770717601297561774m_norm(A,aa(B,A,Uu,Uua)) ) ).

% ATP.lambda_250
tff(fact_8429_ATP_Olambda__251,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu: fun(A,B),Uua: A] : aa(A,real,aTP_Lamp_ajj(fun(A,B),fun(A,real),Uu),Uua) = real_V7770717601297561774m_norm(B,aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_251
tff(fact_8430_ATP_Olambda__252,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu: fun(A,B),Uua: A] : aa(A,real,aTP_Lamp_vu(fun(A,B),fun(A,real),Uu),Uua) = real_V7770717601297561774m_norm(B,aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_252
tff(fact_8431_ATP_Olambda__253,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V8999393235501362500lgebra(B)
        & comm_semiring_1(B) )
     => ! [Uu: fun(A,B),Uua: A] : aa(A,real,aTP_Lamp_ha(fun(A,B),fun(A,real),Uu),Uua) = real_V7770717601297561774m_norm(B,aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_253
tff(fact_8432_ATP_Olambda__254,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [Uu: fun(A,B),Uua: A] : aa(A,real,aTP_Lamp_xc(fun(A,B),fun(A,real),Uu),Uua) = real_V7770717601297561774m_norm(B,aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_254
tff(fact_8433_ATP_Olambda__255,axiom,
    ! [Uu: fun(nat,real),Uua: nat] : aa(nat,real,aTP_Lamp_abc(fun(nat,real),fun(nat,real),Uu),Uua) = aa(real,real,inverse_inverse(real),aa(nat,real,Uu,Uua)) ).

% ATP.lambda_255
tff(fact_8434_ATP_Olambda__256,axiom,
    ! [A: $tType,C: $tType] :
      ( ( real_V822414075346904944vector(C)
        & real_V8999393235501362500lgebra(A) )
     => ! [Uu: fun(C,A),Uua: C] : aa(C,A,aTP_Lamp_ub(fun(C,A),fun(C,A),Uu),Uua) = aa(A,A,inverse_inverse(A),aa(C,A,Uu,Uua)) ) ).

% ATP.lambda_256
tff(fact_8435_ATP_Olambda__257,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Uu: fun(B,A),Uua: B] : aa(B,A,aTP_Lamp_wj(fun(B,A),fun(B,A),Uu),Uua) = aa(A,A,inverse_inverse(A),aa(B,A,Uu,Uua)) ) ).

% ATP.lambda_257
tff(fact_8436_ATP_Olambda__258,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V3459762299906320749_field(B) )
     => ! [Uu: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_aiq(fun(A,B),fun(A,B),Uu),Uua) = aa(B,B,inverse_inverse(B),aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_258
tff(fact_8437_ATP_Olambda__259,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & real_V8999393235501362500lgebra(B) )
     => ! [Uu: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_ajb(fun(A,B),fun(A,B),Uu),Uua) = aa(B,B,inverse_inverse(B),aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_259
tff(fact_8438_ATP_Olambda__260,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: fun(A,A),Uua: A] : aa(A,A,aTP_Lamp_rn(fun(A,A),fun(A,A),Uu),Uua) = aa(A,A,inverse_inverse(A),aa(A,A,Uu,Uua)) ) ).

% ATP.lambda_260
tff(fact_8439_ATP_Olambda__261,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V8999393235501362500lgebra(B) )
     => ! [Uu: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_zn(fun(A,B),fun(A,B),Uu),Uua) = aa(B,B,inverse_inverse(B),aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_261
tff(fact_8440_ATP_Olambda__262,axiom,
    ! [A: $tType,Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_adz(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,inverse_inverse(real),aa(A,real,Uu,Uua)) ).

% ATP.lambda_262
tff(fact_8441_ATP_Olambda__263,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V8999393235501362500lgebra(B)
     => ! [Uu: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_aej(fun(A,B),fun(A,B),Uu),Uua) = aa(B,B,inverse_inverse(B),aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_263
tff(fact_8442_ATP_Olambda__264,axiom,
    ! [B: $tType,Uu: fun(B,nat),Uua: B] : aa(B,int,aTP_Lamp_et(fun(B,nat),fun(B,int),Uu),Uua) = aa(nat,int,semiring_1_of_nat(int),aa(B,nat,Uu,Uua)) ).

% ATP.lambda_264
tff(fact_8443_ATP_Olambda__265,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_semiring_1(A)
     => ! [Uu: fun(B,nat),Uua: B] : aa(B,A,aTP_Lamp_gv(fun(B,nat),fun(B,A),Uu),Uua) = aa(nat,A,semiring_1_of_nat(A),aa(B,nat,Uu,Uua)) ) ).

% ATP.lambda_265
tff(fact_8444_ATP_Olambda__266,axiom,
    ! [A: $tType,B: $tType] :
      ( semiring_1(A)
     => ! [Uu: fun(B,nat),Uua: B] : aa(B,A,aTP_Lamp_cx(fun(B,nat),fun(B,A),Uu),Uua) = aa(nat,A,semiring_1_of_nat(A),aa(B,nat,Uu,Uua)) ) ).

% ATP.lambda_266
tff(fact_8445_ATP_Olambda__267,axiom,
    ! [A: $tType,Uu: fun(A,nat),Uua: A] : aa(A,real,aTP_Lamp_bc(fun(A,nat),fun(A,real),Uu),Uua) = aa(nat,real,semiring_1_of_nat(real),aa(A,nat,Uu,Uua)) ).

% ATP.lambda_267
tff(fact_8446_ATP_Olambda__268,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_uj(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,ln_ln(real),aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_268
tff(fact_8447_ATP_Olambda__269,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_aix(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,ln_ln(real),aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_269
tff(fact_8448_ATP_Olambda__270,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_aap(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,ln_ln(real),aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_270
tff(fact_8449_ATP_Olambda__271,axiom,
    ! [A: $tType,Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_ij(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,ln_ln(real),aa(A,real,Uu,Uua)) ).

% ATP.lambda_271
tff(fact_8450_ATP_Olambda__272,axiom,
    ! [Uu: fun(nat,real),Uua: nat] : aa(nat,complex,aTP_Lamp_cv(fun(nat,real),fun(nat,complex),Uu),Uua) = aa(real,complex,real_Vector_of_real(complex),aa(nat,real,Uu,Uua)) ).

% ATP.lambda_272
tff(fact_8451_ATP_Olambda__273,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Uu: fun(nat,real),Uua: nat] : aa(nat,A,aTP_Lamp_ev(fun(nat,real),fun(nat,A),Uu),Uua) = aa(real,A,real_Vector_of_real(A),aa(nat,real,Uu,Uua)) ) ).

% ATP.lambda_273
tff(fact_8452_ATP_Olambda__274,axiom,
    ! [A: $tType] :
      ( ( real_V2191834092415804123ebra_1(A)
        & real_V822414075346904944vector(A) )
     => ! [Uu: fun(nat,real),Uua: nat] : aa(nat,A,aTP_Lamp_cg(fun(nat,real),fun(nat,A),Uu),Uua) = aa(real,A,real_Vector_of_real(A),aa(nat,real,Uu,Uua)) ) ).

% ATP.lambda_274
tff(fact_8453_ATP_Olambda__275,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [Uu: fun(nat,real),Uua: nat] : aa(nat,A,aTP_Lamp_ga(fun(nat,real),fun(nat,A),Uu),Uua) = aa(real,A,real_Vector_of_real(A),aa(nat,real,Uu,Uua)) ) ).

% ATP.lambda_275
tff(fact_8454_ATP_Olambda__276,axiom,
    ! [A: $tType,C: $tType] :
      ( ( real_V822414075346904944vector(C)
        & real_V2191834092415804123ebra_1(A)
        & real_V822414075346904944vector(A) )
     => ! [Uu: fun(C,real),Uua: C] : aa(C,A,aTP_Lamp_tj(fun(C,real),fun(C,A),Uu),Uua) = aa(real,A,real_Vector_of_real(A),aa(C,real,Uu,Uua)) ) ).

% ATP.lambda_276
tff(fact_8455_ATP_Olambda__277,axiom,
    ! [A: $tType,C: $tType] :
      ( ( topolo4958980785337419405_space(C)
        & real_V2191834092415804123ebra_1(A)
        & real_V822414075346904944vector(A) )
     => ! [Uu: fun(C,real),Uua: C] : aa(C,A,aTP_Lamp_aji(fun(C,real),fun(C,A),Uu),Uua) = aa(real,A,real_Vector_of_real(A),aa(C,real,Uu,Uua)) ) ).

% ATP.lambda_277
tff(fact_8456_ATP_Olambda__278,axiom,
    ! [A: $tType,C: $tType] :
      ( ( topological_t2_space(C)
        & real_V2191834092415804123ebra_1(A)
        & real_V822414075346904944vector(A) )
     => ! [Uu: fun(C,real),Uua: C] : aa(C,A,aTP_Lamp_vv(fun(C,real),fun(C,A),Uu),Uua) = aa(real,A,real_Vector_of_real(A),aa(C,real,Uu,Uua)) ) ).

% ATP.lambda_278
tff(fact_8457_ATP_Olambda__279,axiom,
    ! [A: $tType,C: $tType] :
      ( ( real_V2191834092415804123ebra_1(A)
        & real_V822414075346904944vector(A) )
     => ! [Uu: fun(C,real),Uua: C] : aa(C,A,aTP_Lamp_xa(fun(C,real),fun(C,A),Uu),Uua) = aa(real,A,real_Vector_of_real(A),aa(C,real,Uu,Uua)) ) ).

% ATP.lambda_279
tff(fact_8458_ATP_Olambda__280,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Uu: fun(B,real),Uua: B] : aa(B,A,aTP_Lamp_wz(fun(B,real),fun(B,A),Uu),Uua) = aa(real,A,real_Vector_of_real(A),aa(B,real,Uu,Uua)) ) ).

% ATP.lambda_280
tff(fact_8459_ATP_Olambda__281,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [Uu: fun(B,real),Uua: B] : aa(B,A,aTP_Lamp_da(fun(B,real),fun(B,A),Uu),Uua) = aa(real,A,real_Vector_of_real(A),aa(B,real,Uu,Uua)) ) ).

% ATP.lambda_281
tff(fact_8460_ATP_Olambda__282,axiom,
    ! [A: $tType,B: $tType] :
      ( ( comm_monoid_mult(A)
        & real_V2191834092415804123ebra_1(A) )
     => ! [Uu: fun(B,real),Uua: B] : aa(B,A,aTP_Lamp_gx(fun(B,real),fun(B,A),Uu),Uua) = aa(real,A,real_Vector_of_real(A),aa(B,real,Uu,Uua)) ) ).

% ATP.lambda_282
tff(fact_8461_ATP_Olambda__283,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,complex,aTP_Lamp_aku(fun(A,real),fun(A,complex),Uu),Uua) = aa(real,complex,real_Vector_of_real(complex),aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_283
tff(fact_8462_ATP_Olambda__284,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_cb(fun(nat,A),fun(nat,A),Uu),Uua) = aa(A,A,uminus_uminus(A),aa(nat,A,Uu,Uua)) ) ).

% ATP.lambda_284
tff(fact_8463_ATP_Olambda__285,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_gl(fun(nat,A),fun(nat,A),Uu),Uua) = aa(A,A,uminus_uminus(A),aa(nat,A,Uu,Uua)) ) ).

% ATP.lambda_285
tff(fact_8464_ATP_Olambda__286,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_gb(fun(nat,A),fun(nat,A),Uu),Uua) = aa(A,A,uminus_uminus(A),aa(nat,A,Uu,Uua)) ) ).

% ATP.lambda_286
tff(fact_8465_ATP_Olambda__287,axiom,
    ! [B: $tType,C: $tType] :
      ( ( topolo4958980785337419405_space(C)
        & topolo1633459387980952147up_add(B) )
     => ! [Uu: fun(C,B),Uua: C] : aa(C,B,aTP_Lamp_ajc(fun(C,B),fun(C,B),Uu),Uua) = aa(B,B,uminus_uminus(B),aa(C,B,Uu,Uua)) ) ).

% ATP.lambda_287
tff(fact_8466_ATP_Olambda__288,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo1633459387980952147up_add(A)
     => ! [Uu: fun(B,A),Uua: B] : aa(B,A,aTP_Lamp_wn(fun(B,A),fun(B,A),Uu),Uua) = aa(A,A,uminus_uminus(A),aa(B,A,Uu,Uua)) ) ).

% ATP.lambda_288
tff(fact_8467_ATP_Olambda__289,axiom,
    ! [A: $tType,B: $tType] :
      ( ab_group_add(A)
     => ! [Uu: fun(B,A),Uua: B] : aa(B,A,aTP_Lamp_dq(fun(B,A),fun(B,A),Uu),Uua) = aa(A,A,uminus_uminus(A),aa(B,A,Uu,Uua)) ) ).

% ATP.lambda_289
tff(fact_8468_ATP_Olambda__290,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_tn(fun(A,B),fun(A,B),Uu),Uua) = aa(B,B,uminus_uminus(B),aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_290
tff(fact_8469_ATP_Olambda__291,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: fun(A,A),Uua: A] : aa(A,A,aTP_Lamp_ra(fun(A,A),fun(A,A),Uu),Uua) = aa(A,A,uminus_uminus(A),aa(A,A,Uu,Uua)) ) ).

% ATP.lambda_291
tff(fact_8470_ATP_Olambda__292,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_zm(fun(A,B),fun(A,B),Uu),Uua) = aa(B,B,uminus_uminus(B),aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_292
tff(fact_8471_ATP_Olambda__293,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo1633459387980952147up_add(B) )
     => ! [Uu: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_wl(fun(A,B),fun(A,B),Uu),Uua) = aa(B,B,uminus_uminus(B),aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_293
tff(fact_8472_ATP_Olambda__294,axiom,
    ! [A: $tType,Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_adw(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,uminus_uminus(real),aa(A,real,Uu,Uua)) ).

% ATP.lambda_294
tff(fact_8473_ATP_Olambda__295,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo1633459387980952147up_add(B)
     => ! [Uu: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_wm(fun(A,B),fun(A,B),Uu),Uua) = aa(B,B,uminus_uminus(B),aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_295
tff(fact_8474_ATP_Olambda__296,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,fun(A,A),aTP_Lamp_hl(fun(nat,A),fun(nat,fun(A,A)),Uu),Uua) = aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,Uua)) ) ).

% ATP.lambda_296
tff(fact_8475_ATP_Olambda__297,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_ring_1(A)
     => ! [Uu: fun(B,int),Uua: B] : aa(B,A,aTP_Lamp_gw(fun(B,int),fun(B,A),Uu),Uua) = aa(int,A,ring_1_of_int(A),aa(B,int,Uu,Uua)) ) ).

% ATP.lambda_297
tff(fact_8476_ATP_Olambda__298,axiom,
    ! [A: $tType,B: $tType] :
      ( ring_1(A)
     => ! [Uu: fun(B,int),Uua: B] : aa(B,A,aTP_Lamp_cy(fun(B,int),fun(B,A),Uu),Uua) = aa(int,A,ring_1_of_int(A),aa(B,int,Uu,Uua)) ) ).

% ATP.lambda_298
tff(fact_8477_ATP_Olambda__299,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,fun(A,A),aTP_Lamp_hy(fun(nat,A),fun(nat,fun(A,A)),Uu),Uua) = aa(A,fun(A,A),plus_plus(A),aa(nat,A,Uu,Uua)) ) ).

% ATP.lambda_299
tff(fact_8478_ATP_Olambda__300,axiom,
    ! [Uu: fun(real,real),Uua: real] : aa(real,real,aTP_Lamp_akm(fun(real,real),fun(real,real),Uu),Uua) = aa(real,real,artanh(real),aa(real,real,Uu,Uua)) ).

% ATP.lambda_300
tff(fact_8479_ATP_Olambda__301,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_ade(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,artanh(real),aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_301
tff(fact_8480_ATP_Olambda__302,axiom,
    ! [A: $tType,Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_yo(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,artanh(real),aa(A,real,Uu,Uua)) ).

% ATP.lambda_302
tff(fact_8481_ATP_Olambda__303,axiom,
    ! [Uu: fun(real,real),Uua: real] : aa(real,real,aTP_Lamp_ajp(fun(real,real),fun(real,real),Uu),Uua) = aa(real,real,arsinh(real),aa(real,real,Uu,Uua)) ).

% ATP.lambda_303
tff(fact_8482_ATP_Olambda__304,axiom,
    ! [B: $tType,Uu: fun(B,real),Uua: B] : aa(B,real,aTP_Lamp_ww(fun(B,real),fun(B,real),Uu),Uua) = aa(real,real,arsinh(real),aa(B,real,Uu,Uua)) ).

% ATP.lambda_304
tff(fact_8483_ATP_Olambda__305,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_aal(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,arsinh(real),aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_305
tff(fact_8484_ATP_Olambda__306,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_uv(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,arctan,aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_306
tff(fact_8485_ATP_Olambda__307,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_ajl(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,arctan,aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_307
tff(fact_8486_ATP_Olambda__308,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_aag(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,arctan,aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_308
tff(fact_8487_ATP_Olambda__309,axiom,
    ! [A: $tType,Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_xb(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,arctan,aa(A,real,Uu,Uua)) ).

% ATP.lambda_309
tff(fact_8488_ATP_Olambda__310,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_sw(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,arcsin,aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_310
tff(fact_8489_ATP_Olambda__311,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_akl(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,arcsin,aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_311
tff(fact_8490_ATP_Olambda__312,axiom,
    ! [Uu: fun(real,real),Uua: real] : aa(real,real,aTP_Lamp_aki(fun(real,real),fun(real,real),Uu),Uua) = aa(real,real,arcosh(real),aa(real,real,Uu,Uua)) ).

% ATP.lambda_312
tff(fact_8491_ATP_Olambda__313,axiom,
    ! [B: $tType,Uu: fun(B,real),Uua: B] : aa(B,real,aTP_Lamp_xp(fun(B,real),fun(B,real),Uu),Uua) = aa(real,real,arcosh(real),aa(B,real,Uu,Uua)) ).

% ATP.lambda_313
tff(fact_8492_ATP_Olambda__314,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_add(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,arcosh(real),aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_314
tff(fact_8493_ATP_Olambda__315,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_vb(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,arccos,aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_315
tff(fact_8494_ATP_Olambda__316,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_akk(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,arccos,aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_316
tff(fact_8495_ATP_Olambda__317,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(B,A),Uua: B] : aa(B,A,aTP_Lamp_wk(fun(B,A),fun(B,A),Uu),Uua) = sgn_sgn(A,aa(B,A,Uu,Uua)) ) ).

% ATP.lambda_317
tff(fact_8496_ATP_Olambda__318,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_aja(fun(A,B),fun(A,B),Uu),Uua) = sgn_sgn(B,aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_318
tff(fact_8497_ATP_Olambda__319,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_zo(fun(A,B),fun(A,B),Uu),Uua) = sgn_sgn(B,aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_319
tff(fact_8498_ATP_Olambda__320,axiom,
    ! [Uu: fun(nat,real),Uua: nat] : aa(nat,real,aTP_Lamp_ca(fun(nat,real),fun(nat,real),Uu),Uua) = aa(real,real,abs_abs(real),aa(nat,real,Uu,Uua)) ).

% ATP.lambda_320
tff(fact_8499_ATP_Olambda__321,axiom,
    ! [A: $tType,B: $tType] :
      ( linordered_field(A)
     => ! [Uu: fun(B,A),Uua: B] : aa(B,A,aTP_Lamp_hk(fun(B,A),fun(B,A),Uu),Uua) = aa(A,A,abs_abs(A),aa(B,A,Uu,Uua)) ) ).

% ATP.lambda_321
tff(fact_8500_ATP_Olambda__322,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_ajq(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,abs_abs(real),aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_322
tff(fact_8501_ATP_Olambda__323,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_aam(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,abs_abs(real),aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_323
tff(fact_8502_ATP_Olambda__324,axiom,
    ! [A: $tType,Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_xq(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,abs_abs(real),aa(A,real,Uu,Uua)) ).

% ATP.lambda_324
tff(fact_8503_ATP_Olambda__325,axiom,
    ! [B: $tType,A: $tType] :
      ( ordere166539214618696060dd_abs(B)
     => ! [Uu: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_cz(fun(A,B),fun(A,B),Uu),Uua) = aa(B,B,abs_abs(B),aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_325
tff(fact_8504_ATP_Olambda__326,axiom,
    ! [A11: $tType] :
      ( ( real_Vector_banach(A11)
        & real_V3459762299906320749_field(A11) )
     => ! [Uu: fun(A11,A11),Uua: A11] : aa(A11,A11,aTP_Lamp_si(fun(A11,A11),fun(A11,A11),Uu),Uua) = aa(A11,A11,tanh(A11),aa(A11,A11,Uu,Uua)) ) ).

% ATP.lambda_326
tff(fact_8505_ATP_Olambda__327,axiom,
    ! [A: $tType,C: $tType] :
      ( ( topolo4958980785337419405_space(C)
        & real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: fun(C,A),Uua: C] : aa(C,A,aTP_Lamp_ais(fun(C,A),fun(C,A),Uu),Uua) = aa(A,A,tanh(A),aa(C,A,Uu,Uua)) ) ).

% ATP.lambda_327
tff(fact_8506_ATP_Olambda__328,axiom,
    ! [A: $tType,C: $tType] :
      ( ( topological_t2_space(C)
        & real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: fun(C,A),Uua: C] : aa(C,A,aTP_Lamp_zt(fun(C,A),fun(C,A),Uu),Uua) = aa(A,A,tanh(A),aa(C,A,Uu,Uua)) ) ).

% ATP.lambda_328
tff(fact_8507_ATP_Olambda__329,axiom,
    ! [A: $tType,C: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: fun(C,A),Uua: C] : aa(C,A,aTP_Lamp_xn(fun(C,A),fun(C,A),Uu),Uua) = aa(A,A,tanh(A),aa(C,A,Uu,Uua)) ) ).

% ATP.lambda_329
tff(fact_8508_ATP_Olambda__330,axiom,
    ! [A11: $tType] :
      ( ( real_Vector_banach(A11)
        & real_V3459762299906320749_field(A11) )
     => ! [Uu: fun(A11,A11),Uua: A11] : aa(A11,A11,aTP_Lamp_rj(fun(A11,A11),fun(A11,A11),Uu),Uua) = sinh(A11,aa(A11,A11,Uu,Uua)) ) ).

% ATP.lambda_330
tff(fact_8509_ATP_Olambda__331,axiom,
    ! [A: $tType,C: $tType] :
      ( ( topolo4958980785337419405_space(C)
        & real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: fun(C,A),Uua: C] : aa(C,A,aTP_Lamp_ajn(fun(C,A),fun(C,A),Uu),Uua) = sinh(A,aa(C,A,Uu,Uua)) ) ).

% ATP.lambda_331
tff(fact_8510_ATP_Olambda__332,axiom,
    ! [A: $tType,C: $tType] :
      ( ( topological_t2_space(C)
        & real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: fun(C,A),Uua: C] : aa(C,A,aTP_Lamp_xg(fun(C,A),fun(C,A),Uu),Uua) = sinh(A,aa(C,A,Uu,Uua)) ) ).

% ATP.lambda_332
tff(fact_8511_ATP_Olambda__333,axiom,
    ! [A: $tType,C: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: fun(C,A),Uua: C] : aa(C,A,aTP_Lamp_xj(fun(C,A),fun(C,A),Uu),Uua) = sinh(A,aa(C,A,Uu,Uua)) ) ).

% ATP.lambda_333
tff(fact_8512_ATP_Olambda__334,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: fun(A,A),Uua: A] : aa(A,A,aTP_Lamp_tu(fun(A,A),fun(A,A),Uu),Uua) = sinh(A,aa(A,A,Uu,Uua)) ) ).

% ATP.lambda_334
tff(fact_8513_ATP_Olambda__335,axiom,
    ! [A11: $tType] :
      ( ( real_Vector_banach(A11)
        & real_V3459762299906320749_field(A11) )
     => ! [Uu: fun(A11,A11),Uua: A11] : aa(A11,A11,aTP_Lamp_ri(fun(A11,A11),fun(A11,A11),Uu),Uua) = cosh(A11,aa(A11,A11,Uu,Uua)) ) ).

% ATP.lambda_335
tff(fact_8514_ATP_Olambda__336,axiom,
    ! [A: $tType,C: $tType] :
      ( ( topolo4958980785337419405_space(C)
        & real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: fun(C,A),Uua: C] : aa(C,A,aTP_Lamp_ajo(fun(C,A),fun(C,A),Uu),Uua) = cosh(A,aa(C,A,Uu,Uua)) ) ).

% ATP.lambda_336
tff(fact_8515_ATP_Olambda__337,axiom,
    ! [A: $tType,C: $tType] :
      ( ( topological_t2_space(C)
        & real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: fun(C,A),Uua: C] : aa(C,A,aTP_Lamp_xh(fun(C,A),fun(C,A),Uu),Uua) = cosh(A,aa(C,A,Uu,Uua)) ) ).

% ATP.lambda_337
tff(fact_8516_ATP_Olambda__338,axiom,
    ! [A: $tType,C: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: fun(C,A),Uua: C] : aa(C,A,aTP_Lamp_xk(fun(C,A),fun(C,A),Uu),Uua) = cosh(A,aa(C,A,Uu,Uua)) ) ).

% ATP.lambda_338
tff(fact_8517_ATP_Olambda__339,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: fun(A,A),Uua: A] : aa(A,A,aTP_Lamp_tv(fun(A,A),fun(A,A),Uu),Uua) = cosh(A,aa(A,A,Uu,Uua)) ) ).

% ATP.lambda_339
tff(fact_8518_ATP_Olambda__340,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_ux(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,tan(real),aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_340
tff(fact_8519_ATP_Olambda__341,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: fun(A,A),Uua: A] : aa(A,A,aTP_Lamp_xl(fun(A,A),fun(A,A),Uu),Uua) = aa(A,A,tan(A),aa(A,A,Uu,Uua)) ) ).

% ATP.lambda_341
tff(fact_8520_ATP_Olambda__342,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_tw(fun(A,real),fun(A,real),Uu),Uua) = sin(real,aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_342
tff(fact_8521_ATP_Olambda__343,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [Uu: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_wx(fun(A,B),fun(A,B),Uu),Uua) = sin(B,aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_343
tff(fact_8522_ATP_Olambda__344,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: fun(A,A),Uua: A] : aa(A,A,aTP_Lamp_rs(fun(A,A),fun(A,A),Uu),Uua) = sin(A,aa(A,A,Uu,Uua)) ) ).

% ATP.lambda_344
tff(fact_8523_ATP_Olambda__345,axiom,
    ! [A: $tType,C: $tType] :
      ( ( topolo4958980785337419405_space(C)
        & real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: fun(C,A),Uua: C] : aa(C,A,aTP_Lamp_ajg(fun(C,A),fun(C,A),Uu),Uua) = aa(A,A,exp(A),aa(C,A,Uu,Uua)) ) ).

% ATP.lambda_345
tff(fact_8524_ATP_Olambda__346,axiom,
    ! [A: $tType,C: $tType] :
      ( ( topological_t2_space(C)
        & real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: fun(C,A),Uua: C] : aa(C,A,aTP_Lamp_xf(fun(C,A),fun(C,A),Uu),Uua) = aa(A,A,exp(A),aa(C,A,Uu,Uua)) ) ).

% ATP.lambda_346
tff(fact_8525_ATP_Olambda__347,axiom,
    ! [A: $tType,C: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: fun(C,A),Uua: C] : aa(C,A,aTP_Lamp_xe(fun(C,A),fun(C,A),Uu),Uua) = aa(A,A,exp(A),aa(C,A,Uu,Uua)) ) ).

% ATP.lambda_347
tff(fact_8526_ATP_Olambda__348,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_ts(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,exp(real),aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_348
tff(fact_8527_ATP_Olambda__349,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: fun(A,A),Uua: A] : aa(A,A,aTP_Lamp_rt(fun(A,A),fun(A,A),Uu),Uua) = aa(A,A,exp(A),aa(A,A,Uu,Uua)) ) ).

% ATP.lambda_349
tff(fact_8528_ATP_Olambda__350,axiom,
    ! [B: $tType,A: $tType] :
      ( ( comm_monoid_mult(B)
        & real_Vector_banach(B)
        & real_V2822296259951069270ebra_1(B) )
     => ! [Uu: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_hr(fun(A,B),fun(A,B),Uu),Uua) = aa(B,B,exp(B),aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_350
tff(fact_8529_ATP_Olambda__351,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: fun(A,A),Uua: A] : aa(A,A,aTP_Lamp_xo(fun(A,A),fun(A,A),Uu),Uua) = aa(A,A,cot(A),aa(A,A,Uu,Uua)) ) ).

% ATP.lambda_351
tff(fact_8530_ATP_Olambda__352,axiom,
    ! [Uu: fun(nat,real),Uua: nat] : aa(nat,real,aTP_Lamp_acc(fun(nat,real),fun(nat,real),Uu),Uua) = cos(real,aa(nat,real,Uu,Uua)) ).

% ATP.lambda_352
tff(fact_8531_ATP_Olambda__353,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_uf(fun(A,real),fun(A,real),Uu),Uua) = cos(real,aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_353
tff(fact_8532_ATP_Olambda__354,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [Uu: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_wu(fun(A,B),fun(A,B),Uu),Uua) = cos(B,aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_354
tff(fact_8533_ATP_Olambda__355,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: fun(A,A),Uua: A] : aa(A,A,aTP_Lamp_rp(fun(A,A),fun(A,A),Uu),Uua) = cos(A,aa(A,A,Uu,Uua)) ) ).

% ATP.lambda_355
tff(fact_8534_ATP_Olambda__356,axiom,
    ! [Uu: fun(real,fun(nat,real)),Uua: real] : aa(real,real,aTP_Lamp_va(fun(real,fun(nat,real)),fun(real,real),Uu),Uua) = suminf(real,aa(real,fun(nat,real),Uu,Uua)) ).

% ATP.lambda_356
tff(fact_8535_ATP_Olambda__357,axiom,
    ! [A: $tType,I6: $tType] :
      ( ( topolo5987344860129210374id_add(A)
        & topological_t2_space(A) )
     => ! [Uu: fun(I6,fun(nat,A)),Uua: I6] : aa(I6,A,aTP_Lamp_ec(fun(I6,fun(nat,A)),fun(I6,A),Uu),Uua) = suminf(A,aa(I6,fun(nat,A),Uu,Uua)) ) ).

% ATP.lambda_357
tff(fact_8536_ATP_Olambda__358,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_Vector_banach(B) )
     => ! [Uu: fun(A,fun(nat,B)),Uua: A] : aa(A,B,aTP_Lamp_zu(fun(A,fun(nat,B)),fun(A,B),Uu),Uua) = suminf(B,aa(A,fun(nat,B),Uu,Uua)) ) ).

% ATP.lambda_358
tff(fact_8537_ATP_Olambda__359,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_ut(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,sqrt,aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_359
tff(fact_8538_ATP_Olambda__360,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_ajk(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,sqrt,aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_360
tff(fact_8539_ATP_Olambda__361,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_aah(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,sqrt,aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_361
tff(fact_8540_ATP_Olambda__362,axiom,
    ! [A: $tType,Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_xd(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,sqrt,aa(A,real,Uu,Uua)) ).

% ATP.lambda_362
tff(fact_8541_ATP_Olambda__363,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,complex,aTP_Lamp_akv(fun(A,real),fun(A,complex),Uu),Uua) = cis(aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_363
tff(fact_8542_ATP_Olambda__364,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: fun(A,bool),Uua: A] :
          ( pp(aa(A,bool,aTP_Lamp_vw(fun(A,bool),fun(A,bool),Uu),Uua))
        <=> ~ pp(aa(A,bool,Uu,Uua)) ) ) ).

% ATP.lambda_364
tff(fact_8543_ATP_Olambda__365,axiom,
    ! [A: $tType,Uu: fun(A,bool),Uua: A] :
      ( pp(aa(A,bool,aTP_Lamp_bz(fun(A,bool),fun(A,bool),Uu),Uua))
    <=> ~ pp(aa(A,bool,Uu,Uua)) ) ).

% ATP.lambda_365
tff(fact_8544_ATP_Olambda__366,axiom,
    ! [A: $tType,B: $tType] :
      ( finite_finite(B)
     => ! [Uu: fun(A,fun(B,bool)),Uua: A] :
          ( pp(aa(A,bool,aTP_Lamp_aht(fun(A,fun(B,bool)),fun(A,bool),Uu),Uua))
        <=> ! [X_13: B] : pp(aa(B,bool,aa(A,fun(B,bool),Uu,Uua),X_13)) ) ) ).

% ATP.lambda_366
tff(fact_8545_ATP_Olambda__367,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,fun(B,bool)),Uua: A] :
      ( pp(aa(A,bool,aTP_Lamp_akw(fun(A,fun(B,bool)),fun(A,bool),Uu),Uua))
    <=> ? [X_13: B] : pp(aa(B,bool,aa(A,fun(B,bool),Uu,Uua),X_13)) ) ).

% ATP.lambda_367
tff(fact_8546_ATP_Olambda__368,axiom,
    ! [A: $tType] :
      ( ( archim2362893244070406136eiling(A)
        & topolo2564578578187576103pology(A) )
     => ! [Uu: fun(real,A),Uua: real] : aa(real,real,aTP_Lamp_zx(fun(real,A),fun(real,real),Uu),Uua) = aa(int,real,ring_1_of_int(real),archim6421214686448440834_floor(A,aa(real,A,Uu,Uua))) ) ).

% ATP.lambda_368
tff(fact_8547_ATP_Olambda__369,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( archim2362893244070406136eiling(B)
        & topolo2564578578187576103pology(B)
        & ring_1(C)
        & topolo4958980785337419405_space(C) )
     => ! [Uu: fun(A,B),Uua: A] : aa(A,C,aTP_Lamp_ym(fun(A,B),fun(A,C),Uu),Uua) = aa(int,C,ring_1_of_int(C),archim6421214686448440834_floor(B,aa(A,B,Uu,Uua))) ) ).

% ATP.lambda_369
tff(fact_8548_ATP_Olambda__370,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( archim2362893244070406136eiling(B)
        & topolo2564578578187576103pology(B)
        & ring_1(C)
        & topolo4958980785337419405_space(C) )
     => ! [Uu: fun(A,B),Uua: A] : aa(A,C,aTP_Lamp_yl(fun(A,B),fun(A,C),Uu),Uua) = aa(int,C,ring_1_of_int(C),archimedean_ceiling(B,aa(A,B,Uu,Uua))) ) ).

% ATP.lambda_370
tff(fact_8549_ATP_Olambda__371,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: fun(B,fun(A,bool)),Uua: A] :
          ( pp(aa(A,bool,aTP_Lamp_alc(fun(B,fun(A,bool)),fun(A,bool),Uu),Uua))
        <=> ? [I3: B] : pp(aa(A,bool,aa(B,fun(A,bool),Uu,I3),Uua)) ) ) ).

% ATP.lambda_371
tff(fact_8550_ATP_Olambda__372,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,fun(B,bool)),Uua: B] :
      ( pp(aa(B,bool,aTP_Lamp_ala(fun(A,fun(B,bool)),fun(B,bool),Uu),Uua))
    <=> ? [A6: A] : pp(aa(B,bool,aa(A,fun(B,bool),Uu,A6),Uua)) ) ).

% ATP.lambda_372
tff(fact_8551_ATP_Olambda__373,axiom,
    ! [A: $tType,Uu: list(A),Uua: A] :
      ( pp(aa(A,bool,aTP_Lamp_ali(list(A),fun(A,bool),Uu),Uua))
    <=> ? [I3: nat] :
          ( ( Uua = aa(nat,A,nth(A,Uu),I3) )
          & pp(aa(nat,bool,ord_less(nat,I3),aa(list(A),nat,size_size(list(A)),Uu))) ) ) ).

% ATP.lambda_373
tff(fact_8552_ATP_Olambda__374,axiom,
    ! [Uu: fun(nat,bool),Uua: nat] :
      ( pp(aa(nat,bool,aTP_Lamp_akp(fun(nat,bool),fun(nat,bool),Uu),Uua))
    <=> ! [M6: nat] :
          ( pp(aa(nat,bool,ord_less_eq(nat,Uua),M6))
         => ( pp(aa(nat,bool,Uu,M6))
          <=> pp(aa(nat,bool,Uu,Uua)) ) ) ) ).

% ATP.lambda_374
tff(fact_8553_ATP_Olambda__375,axiom,
    ! [Uu: fun(nat,bool),Uua: nat] :
      ( pp(aa(nat,bool,aTP_Lamp_aml(fun(nat,bool),fun(nat,bool),Uu),Uua))
    <=> ! [N13: nat] :
          ( pp(aa(nat,bool,ord_less_eq(nat,Uua),N13))
         => pp(aa(nat,bool,Uu,N13)) ) ) ).

% ATP.lambda_375
tff(fact_8554_ATP_Olambda__376,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu: fun(A,bool),Uua: A] :
          ( pp(aa(A,bool,aTP_Lamp_aia(fun(A,bool),fun(A,bool),Uu),Uua))
        <=> ! [Y7: A] :
              ( pp(aa(A,bool,ord_less_eq(A,Uua),Y7))
             => pp(aa(A,bool,Uu,Y7)) ) ) ) ).

% ATP.lambda_376
tff(fact_8555_ATP_Olambda__377,axiom,
    ! [Uu: fun(nat,bool),Uua: nat] :
      ( pp(aa(nat,bool,aTP_Lamp_amk(fun(nat,bool),fun(nat,bool),Uu),Uua))
    <=> ! [N13: nat] :
          ( pp(aa(nat,bool,ord_less_eq(nat,Uua),N13))
         => ~ pp(aa(nat,bool,Uu,N13)) ) ) ).

% ATP.lambda_377
tff(fact_8556_ATP_Olambda__378,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,option(B)),Uua: B] :
      ( pp(aa(B,bool,aTP_Lamp_alb(fun(A,option(B)),fun(B,bool),Uu),Uua))
    <=> ? [A6: A] : aa(A,option(B),Uu,A6) = aa(B,option(B),some(B),Uua) ) ).

% ATP.lambda_378
tff(fact_8557_ATP_Olambda__379,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,A),Uua: A] :
      ( pp(aa(A,bool,aTP_Lamp_aky(fun(B,A),fun(A,bool),Uu),Uua))
    <=> ? [X2: B] : Uua = aa(B,A,Uu,X2) ) ).

% ATP.lambda_379
tff(fact_8558_ATP_Olambda__380,axiom,
    ! [Uu: fun(nat,real),Uua: fun(nat,real),Uub: nat] : aa(nat,real,aa(fun(nat,real),fun(nat,real),aTP_Lamp_fr(fun(nat,real),fun(fun(nat,real),fun(nat,real)),Uu),Uua),Uub) = if(real,aa(nat,bool,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Uub),aa(nat,real,Uua,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Uub),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,Uu,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,minus_minus(nat,Uub),one_one(nat))),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ).

% ATP.lambda_380
tff(fact_8559_ATP_Olambda__381,axiom,
    ! [Uu: fun(nat,real),Uua: fun(nat,real),Uub: nat] : aa(nat,real,aa(fun(nat,real),fun(nat,real),aTP_Lamp_fs(fun(nat,real),fun(fun(nat,real),fun(nat,real)),Uu),Uua),Uub) = if(real,aa(nat,bool,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Uub),aa(nat,real,Uu,Uub),aa(nat,real,Uua,Uub)) ).

% ATP.lambda_381
tff(fact_8560_ATP_Olambda__382,axiom,
    ! [Uu: num,Uua: code_integer,Uub: code_integer] : aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),aTP_Lamp_mb(num,fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),Uu),Uua),Uub) = if(product_prod(code_integer,code_integer),aa(code_integer,bool,ord_less_eq(code_integer,aa(num,code_integer,numeral_numeral(code_integer),Uu)),Uub),aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),times_times(code_integer),aa(num,code_integer,numeral_numeral(code_integer),bit0(one2))),Uua)),one_one(code_integer))),aa(code_integer,code_integer,minus_minus(code_integer,Uub),aa(num,code_integer,numeral_numeral(code_integer),Uu))),aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),times_times(code_integer),aa(num,code_integer,numeral_numeral(code_integer),bit0(one2))),Uua)),Uub)) ).

% ATP.lambda_382
tff(fact_8561_ATP_Olambda__383,axiom,
    ! [Uu: num,Uua: nat,Uub: nat] : aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),aTP_Lamp_lz(num,fun(nat,fun(nat,product_prod(nat,nat))),Uu),Uua),Uub) = if(product_prod(nat,nat),aa(nat,bool,ord_less_eq(nat,aa(num,nat,numeral_numeral(nat),Uu)),Uub),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uua)),one_one(nat))),aa(nat,nat,minus_minus(nat,Uub),aa(num,nat,numeral_numeral(nat),Uu))),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uua)),Uub)) ).

% ATP.lambda_383
tff(fact_8562_ATP_Olambda__384,axiom,
    ! [Uu: num,Uua: int,Uub: int] : aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aTP_Lamp_ma(num,fun(int,fun(int,product_prod(int,int))),Uu),Uua),Uub) = if(product_prod(int,int),aa(int,bool,ord_less_eq(int,aa(num,int,numeral_numeral(int),Uu)),Uub),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),Uua)),one_one(int))),aa(int,int,minus_minus(int,Uub),aa(num,int,numeral_numeral(int),Uu))),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),Uua)),Uub)) ).

% ATP.lambda_384
tff(fact_8563_ATP_Olambda__385,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Uu: num,Uua: A,Uub: A] : aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),aTP_Lamp_mc(num,fun(A,fun(A,product_prod(A,A))),Uu),Uua),Uub) = if(product_prod(A,A),aa(A,bool,ord_less_eq(A,aa(num,A,numeral_numeral(A),Uu)),Uub),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),Uua)),one_one(A))),aa(A,A,minus_minus(A,Uub),aa(num,A,numeral_numeral(A),Uu))),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),Uua)),Uub)) ) ).

% ATP.lambda_385
tff(fact_8564_ATP_Olambda__386,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [Uu: set(nat),Uua: fun(nat,A),Uub: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_bw(set(nat),fun(fun(nat,A),fun(nat,A)),Uu),Uua),Uub) = if(A,aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),Uub),Uu),aa(nat,A,Uua,Uub),zero_zero(A)) ) ).

% ATP.lambda_386
tff(fact_8565_ATP_Olambda__387,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: B,Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_dc(B,fun(fun(B,A),fun(B,A)),Uu),Uua),Uub) = if(A,aa(B,bool,fequal(B,Uu),Uub),aa(B,A,Uua,Uub),zero_zero(A)) ) ).

% ATP.lambda_387
tff(fact_8566_ATP_Olambda__388,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: B,Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_gz(B,fun(fun(B,A),fun(B,A)),Uu),Uua),Uub) = if(A,aa(B,bool,fequal(B,Uu),Uub),aa(B,A,Uua,Uub),one_one(A)) ) ).

% ATP.lambda_388
tff(fact_8567_ATP_Olambda__389,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [Uu: nat,Uua: fun(nat,A),Uub: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_br(nat,fun(fun(nat,A),fun(nat,A)),Uu),Uua),Uub) = if(A,aa(nat,bool,fequal(nat,Uub),Uu),aa(nat,A,Uua,Uub),zero_zero(A)) ) ).

% ATP.lambda_389
tff(fact_8568_ATP_Olambda__390,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: B,Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_db(B,fun(fun(B,A),fun(B,A)),Uu),Uua),Uub) = if(A,aa(B,bool,fequal(B,Uub),Uu),aa(B,A,Uua,Uub),zero_zero(A)) ) ).

% ATP.lambda_390
tff(fact_8569_ATP_Olambda__391,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: B,Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_gy(B,fun(fun(B,A),fun(B,A)),Uu),Uua),Uub) = if(A,aa(B,bool,fequal(B,Uub),Uu),aa(B,A,Uua,Uub),one_one(A)) ) ).

% ATP.lambda_391
tff(fact_8570_ATP_Olambda__392,axiom,
    ! [A: $tType,Uu: fun(A,bool),Uua: A,Uub: list(A)] : aa(list(A),list(A),aa(A,fun(list(A),list(A)),aTP_Lamp_ang(fun(A,bool),fun(A,fun(list(A),list(A))),Uu),Uua),Uub) = if(list(A),aa(A,bool,Uu,Uua),aa(list(A),list(A),cons(A,Uua),Uub),Uub) ).

% ATP.lambda_392
tff(fact_8571_ATP_Olambda__393,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [Uu: fun(nat,bool),Uua: fun(nat,A),Uub: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_bx(fun(nat,bool),fun(fun(nat,A),fun(nat,A)),Uu),Uua),Uub) = if(A,aa(nat,bool,Uu,Uub),aa(nat,A,Uua,Uub),zero_zero(A)) ) ).

% ATP.lambda_393
tff(fact_8572_ATP_Olambda__394,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(B,A),Uua: fun(B,bool),Uub: B] : aa(B,A,aa(fun(B,bool),fun(B,A),aTP_Lamp_dz(fun(B,A),fun(fun(B,bool),fun(B,A)),Uu),Uua),Uub) = if(A,aa(B,bool,Uua,Uub),aa(B,A,Uu,Uub),zero_zero(A)) ) ).

% ATP.lambda_394
tff(fact_8573_ATP_Olambda__395,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(B,A),Uua: fun(B,bool),Uub: B] : aa(B,A,aa(fun(B,bool),fun(B,A),aTP_Lamp_hm(fun(B,A),fun(fun(B,bool),fun(B,A)),Uu),Uua),Uub) = if(A,aa(B,bool,Uua,Uub),aa(B,A,Uu,Uub),one_one(A)) ) ).

% ATP.lambda_395
tff(fact_8574_ATP_Olambda__396,axiom,
    ! [Uu: fun(real,real),Uua: fun(real,real),Uub: real] :
      ( pp(aa(real,bool,aa(fun(real,real),fun(real,bool),aTP_Lamp_aen(fun(real,real),fun(fun(real,real),fun(real,bool)),Uu),Uua),Uub))
    <=> has_field_derivative(real,Uu,aa(real,real,Uua,Uub),topolo174197925503356063within(real,Uub,top_top(set(real)))) ) ).

% ATP.lambda_396
tff(fact_8575_ATP_Olambda__397,axiom,
    ! [A: $tType,F: $tType] :
      ( topological_t2_space(A)
     => ! [Uu: filter(F),Uua: fun(F,A),Uub: A] :
          ( pp(aa(A,bool,aa(fun(F,A),fun(A,bool),aTP_Lamp_acy(filter(F),fun(fun(F,A),fun(A,bool)),Uu),Uua),Uub))
        <=> filterlim(F,A,Uua,topolo7230453075368039082e_nhds(A,Uub),Uu) ) ) ).

% ATP.lambda_397
tff(fact_8576_ATP_Olambda__398,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(nat,fun(nat,A)),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_mj(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(nat,A,aa(nat,fun(nat,A),Uu,Uub),aa(nat,nat,minus_minus(nat,Uua),Uub)) ) ).

% ATP.lambda_398
tff(fact_8577_ATP_Olambda__399,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(nat,fun(nat,A)),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_mh(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(nat,A,aa(nat,fun(nat,A),Uu,Uub),aa(nat,nat,minus_minus(nat,Uua),Uub)) ) ).

% ATP.lambda_399
tff(fact_8578_ATP_Olambda__400,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,fun(B,bool)),Uua: fun(A,B),Uub: A] :
      ( pp(aa(A,bool,aa(fun(A,B),fun(A,bool),aTP_Lamp_akx(fun(A,fun(B,bool)),fun(fun(A,B),fun(A,bool)),Uu),Uua),Uub))
    <=> pp(aa(B,bool,aa(A,fun(B,bool),Uu,Uub),aa(A,B,Uua,Uub))) ) ).

% ATP.lambda_400
tff(fact_8579_ATP_Olambda__401,axiom,
    ! [Uu: fun(real,fun(nat,real)),Uua: nat,Uub: real] : aa(real,real,aa(nat,fun(real,real),aTP_Lamp_uz(fun(real,fun(nat,real)),fun(nat,fun(real,real)),Uu),Uua),Uub) = aa(nat,real,aa(real,fun(nat,real),Uu,Uub),Uua) ).

% ATP.lambda_401
tff(fact_8580_ATP_Olambda__402,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(nat,fun(nat,A)),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_ig(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(nat,A,aa(nat,fun(nat,A),Uu,Uub),Uua) ) ).

% ATP.lambda_402
tff(fact_8581_ATP_Olambda__403,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(nat,fun(nat,A)),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_id(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(nat,A,aa(nat,fun(nat,A),Uu,Uub),Uua) ) ).

% ATP.lambda_403
tff(fact_8582_ATP_Olambda__404,axiom,
    ! [I6: $tType,A: $tType] :
      ( ( topolo5987344860129210374id_add(A)
        & topological_t2_space(A) )
     => ! [Uu: fun(I6,fun(nat,A)),Uua: nat,Uub: I6] : aa(I6,A,aa(nat,fun(I6,A),aTP_Lamp_dw(fun(I6,fun(nat,A)),fun(nat,fun(I6,A)),Uu),Uua),Uub) = aa(nat,A,aa(I6,fun(nat,A),Uu,Uub),Uua) ) ).

% ATP.lambda_404
tff(fact_8583_ATP_Olambda__405,axiom,
    ! [I6: $tType,B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu: fun(I6,fun(A,B)),Uua: A,Uub: I6] : aa(I6,B,aa(A,fun(I6,B),aTP_Lamp_acm(fun(I6,fun(A,B)),fun(A,fun(I6,B)),Uu),Uua),Uub) = aa(A,B,aa(I6,fun(A,B),Uu,Uub),Uua) ) ).

% ATP.lambda_405
tff(fact_8584_ATP_Olambda__406,axiom,
    ! [I6: $tType,B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V3459762299906320749_field(B) )
     => ! [Uu: fun(I6,fun(A,B)),Uua: A,Uub: I6] : aa(I6,B,aa(A,fun(I6,B),aTP_Lamp_un(fun(I6,fun(A,B)),fun(A,fun(I6,B)),Uu),Uua),Uub) = aa(A,B,aa(I6,fun(A,B),Uu,Uub),Uua) ) ).

% ATP.lambda_406
tff(fact_8585_ATP_Olambda__407,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V3459762299906320749_field(B)
     => ! [Uu: fun(B,fun(A,B)),Uua: A,Uub: B] : aa(B,B,aa(A,fun(B,B),aTP_Lamp_rx(fun(B,fun(A,B)),fun(A,fun(B,B)),Uu),Uua),Uub) = aa(A,B,aa(B,fun(A,B),Uu,Uub),Uua) ) ).

% ATP.lambda_407
tff(fact_8586_ATP_Olambda__408,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(B,fun(C,A)),Uua: C,Uub: B] : aa(B,A,aa(C,fun(B,A),aTP_Lamp_hc(fun(B,fun(C,A)),fun(C,fun(B,A)),Uu),Uua),Uub) = aa(C,A,aa(B,fun(C,A),Uu,Uub),Uua) ) ).

% ATP.lambda_408
tff(fact_8587_ATP_Olambda__409,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(B,fun(C,A)),Uua: C,Uub: B] : aa(B,A,aa(C,fun(B,A),aTP_Lamp_dg(fun(B,fun(C,A)),fun(C,fun(B,A)),Uu),Uua),Uub) = aa(C,A,aa(B,fun(C,A),Uu,Uub),Uua) ) ).

% ATP.lambda_409
tff(fact_8588_ATP_Olambda__410,axiom,
    ! [B: $tType,A: $tType,Uu: fun(B,fun(A,bool)),Uua: A,Uub: B] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),aTP_Lamp_akr(fun(B,fun(A,bool)),fun(A,fun(B,bool)),Uu),Uua),Uub))
    <=> pp(aa(A,bool,aa(B,fun(A,bool),Uu,Uub),Uua)) ) ).

% ATP.lambda_410
tff(fact_8589_ATP_Olambda__411,axiom,
    ! [A: $tType,C: $tType,D6: $tType] :
      ( ( topolo4958980785337419405_space(D6)
        & real_V4412858255891104859lgebra(C)
        & comm_ring_1(C) )
     => ! [Uu: fun(A,fun(D6,C)),Uua: D6,Uub: A] : aa(A,C,aa(D6,fun(A,C),aTP_Lamp_akc(fun(A,fun(D6,C)),fun(D6,fun(A,C)),Uu),Uua),Uub) = aa(D6,C,aa(A,fun(D6,C),Uu,Uub),Uua) ) ).

% ATP.lambda_411
tff(fact_8590_ATP_Olambda__412,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V822414075346904944vector(C) )
     => ! [Uu: fun(A,fun(B,C)),Uua: B,Uub: A] : aa(A,C,aa(B,fun(A,C),aTP_Lamp_tb(fun(A,fun(B,C)),fun(B,fun(A,C)),Uu),Uua),Uub) = aa(B,C,aa(A,fun(B,C),Uu,Uub),Uua) ) ).

% ATP.lambda_412
tff(fact_8591_ATP_Olambda__413,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( ( topolo4958980785337419405_space(B)
        & topolo4987421752381908075d_mult(C) )
     => ! [Uu: fun(A,fun(B,C)),Uua: B,Uub: A] : aa(A,C,aa(B,fun(A,C),aTP_Lamp_aka(fun(A,fun(B,C)),fun(B,fun(A,C)),Uu),Uua),Uub) = aa(B,C,aa(A,fun(B,C),Uu,Uub),Uua) ) ).

% ATP.lambda_413
tff(fact_8592_ATP_Olambda__414,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( ( topolo4958980785337419405_space(B)
        & topolo5987344860129210374id_add(C) )
     => ! [Uu: fun(A,fun(B,C)),Uua: B,Uub: A] : aa(A,C,aa(B,fun(A,C),aTP_Lamp_ake(fun(A,fun(B,C)),fun(B,fun(A,C)),Uu),Uua),Uub) = aa(B,C,aa(A,fun(B,C),Uu,Uub),Uua) ) ).

% ATP.lambda_414
tff(fact_8593_ATP_Olambda__415,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( ( topological_t2_space(B)
        & real_V4412858255891104859lgebra(C)
        & comm_ring_1(C) )
     => ! [Uu: fun(A,fun(B,C)),Uua: B,Uub: A] : aa(A,C,aa(B,fun(A,C),aTP_Lamp_yy(fun(A,fun(B,C)),fun(B,fun(A,C)),Uu),Uua),Uub) = aa(B,C,aa(A,fun(B,C),Uu,Uub),Uua) ) ).

% ATP.lambda_415
tff(fact_8594_ATP_Olambda__416,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( ( topological_t2_space(B)
        & topolo4987421752381908075d_mult(C) )
     => ! [Uu: fun(A,fun(B,C)),Uua: B,Uub: A] : aa(A,C,aa(B,fun(A,C),aTP_Lamp_yw(fun(A,fun(B,C)),fun(B,fun(A,C)),Uu),Uua),Uub) = aa(B,C,aa(A,fun(B,C),Uu,Uub),Uua) ) ).

% ATP.lambda_416
tff(fact_8595_ATP_Olambda__417,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( ( topological_t2_space(B)
        & topolo5987344860129210374id_add(C) )
     => ! [Uu: fun(A,fun(B,C)),Uua: B,Uub: A] : aa(A,C,aa(B,fun(A,C),aTP_Lamp_zc(fun(A,fun(B,C)),fun(B,fun(A,C)),Uu),Uua),Uub) = aa(B,C,aa(A,fun(B,C),Uu,Uub),Uua) ) ).

% ATP.lambda_417
tff(fact_8596_ATP_Olambda__418,axiom,
    ! [A: $tType,B: $tType] :
      ( finite_finite(B)
     => ! [Uu: fun(A,fun(B,bool)),Uua: B,Uub: A] :
          ( pp(aa(A,bool,aa(B,fun(A,bool),aTP_Lamp_ahs(fun(A,fun(B,bool)),fun(B,fun(A,bool)),Uu),Uua),Uub))
        <=> pp(aa(B,bool,aa(A,fun(B,bool),Uu,Uub),Uua)) ) ) ).

% ATP.lambda_418
tff(fact_8597_ATP_Olambda__419,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( ( real_V4412858255891104859lgebra(C)
        & comm_ring_1(C) )
     => ! [Uu: fun(A,fun(B,C)),Uua: B,Uub: A] : aa(A,C,aa(B,fun(A,C),aTP_Lamp_ys(fun(A,fun(B,C)),fun(B,fun(A,C)),Uu),Uua),Uub) = aa(B,C,aa(A,fun(B,C),Uu,Uub),Uua) ) ).

% ATP.lambda_419
tff(fact_8598_ATP_Olambda__420,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( topolo4987421752381908075d_mult(C)
     => ! [Uu: fun(A,fun(B,C)),Uua: B,Uub: A] : aa(A,C,aa(B,fun(A,C),aTP_Lamp_yu(fun(A,fun(B,C)),fun(B,fun(A,C)),Uu),Uua),Uub) = aa(B,C,aa(A,fun(B,C),Uu,Uub),Uua) ) ).

% ATP.lambda_420
tff(fact_8599_ATP_Olambda__421,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( topolo5987344860129210374id_add(C)
     => ! [Uu: fun(A,fun(B,C)),Uua: B,Uub: A] : aa(A,C,aa(B,fun(A,C),aTP_Lamp_za(fun(A,fun(B,C)),fun(B,fun(A,C)),Uu),Uua),Uub) = aa(B,C,aa(A,fun(B,C),Uu,Uub),Uua) ) ).

% ATP.lambda_421
tff(fact_8600_ATP_Olambda__422,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: A,Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_kc(A,fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_kb(A,fun(A,fun(nat,fun(nat,A))),Uu),Uua),Uub)),set_ord_atMost(nat,Uub)) ) ).

% ATP.lambda_422
tff(fact_8601_ATP_Olambda__423,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: A,Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_ka(A,fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_jz(A,fun(A,fun(nat,fun(nat,A))),Uu),Uua),Uub)),set_ord_atMost(nat,Uub)) ) ).

% ATP.lambda_423
tff(fact_8602_ATP_Olambda__424,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: A,Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_jy(A,fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_jx(A,fun(A,fun(nat,fun(nat,A))),Uu),Uua),Uub)),set_ord_atMost(nat,Uub)) ) ).

% ATP.lambda_424
tff(fact_8603_ATP_Olambda__425,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V4412858255891104859lgebra(A) )
     => ! [Uu: fun(nat,A),Uua: fun(nat,A),Uub: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_ip(fun(nat,A),fun(fun(nat,A),fun(nat,A)),Uu),Uua),Uub) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aa(fun(nat,A),fun(nat,fun(nat,A)),aTP_Lamp_io(fun(nat,A),fun(fun(nat,A),fun(nat,fun(nat,A))),Uu),Uua),Uub)),set_ord_atMost(nat,Uub)) ) ).

% ATP.lambda_425
tff(fact_8604_ATP_Olambda__426,axiom,
    ! [A: $tType] :
      ( ( ring_1(A)
        & topolo4958980785337419405_space(A) )
     => ! [Uu: nat,Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_ff(nat,fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),if(A,aa(nat,bool,fequal(nat,Uub),Uu),one_one(A),zero_zero(A))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),Uub)) ) ).

% ATP.lambda_426
tff(fact_8605_ATP_Olambda__427,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(nat,fun(nat,A)),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_ih(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu),Uua),Uub) = groups7121269368397514597t_prod(nat,A,aa(nat,fun(nat,A),aTP_Lamp_ig(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu),Uub),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,Uub),Uua)) ) ).

% ATP.lambda_427
tff(fact_8606_ATP_Olambda__428,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(nat,fun(nat,A)),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_ie(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aTP_Lamp_id(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu),Uub)),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,Uub),Uua)) ) ).

% ATP.lambda_428
tff(fact_8607_ATP_Olambda__429,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Uu: fun(nat,A),Uua: nat,Uub: A] : aa(A,A,aa(nat,fun(A,A),aTP_Lamp_zw(fun(nat,A),fun(nat,fun(A,A)),Uu),Uua),Uub) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_ck(fun(nat,A),fun(A,fun(nat,A)),Uu),Uub)),set_ord_atMost(nat,Uua)) ) ).

% ATP.lambda_429
tff(fact_8608_ATP_Olambda__430,axiom,
    ! [Uu: rat,Uua: int,Uub: int] : aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aTP_Lamp_np(rat,fun(int,fun(int,product_prod(int,int))),Uu),Uua),Uub) = aa(product_prod(int,int),product_prod(int,int),product_case_prod(int,int,product_prod(int,int),aa(int,fun(int,fun(int,product_prod(int,int))),aTP_Lamp_no(int,fun(int,fun(int,fun(int,product_prod(int,int)))),Uua),Uub)),quotient_of(Uu)) ).

% ATP.lambda_430
tff(fact_8609_ATP_Olambda__431,axiom,
    ! [Uu: rat,Uua: int,Uub: int] : aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aTP_Lamp_nn(rat,fun(int,fun(int,product_prod(int,int))),Uu),Uua),Uub) = aa(product_prod(int,int),product_prod(int,int),product_case_prod(int,int,product_prod(int,int),aa(int,fun(int,fun(int,product_prod(int,int))),aTP_Lamp_nm(int,fun(int,fun(int,fun(int,product_prod(int,int)))),Uua),Uub)),quotient_of(Uu)) ).

% ATP.lambda_431
tff(fact_8610_ATP_Olambda__432,axiom,
    ! [Uu: rat,Uua: int,Uub: int] : aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aTP_Lamp_nl(rat,fun(int,fun(int,product_prod(int,int))),Uu),Uua),Uub) = aa(product_prod(int,int),product_prod(int,int),product_case_prod(int,int,product_prod(int,int),aa(int,fun(int,fun(int,product_prod(int,int))),aTP_Lamp_nk(int,fun(int,fun(int,fun(int,product_prod(int,int)))),Uua),Uub)),quotient_of(Uu)) ).

% ATP.lambda_432
tff(fact_8611_ATP_Olambda__433,axiom,
    ! [Uu: rat,Uua: int,Uub: int] : aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aTP_Lamp_nj(rat,fun(int,fun(int,product_prod(int,int))),Uu),Uua),Uub) = aa(product_prod(int,int),product_prod(int,int),product_case_prod(int,int,product_prod(int,int),aa(int,fun(int,fun(int,product_prod(int,int))),aTP_Lamp_ni(int,fun(int,fun(int,fun(int,product_prod(int,int)))),Uua),Uub)),quotient_of(Uu)) ).

% ATP.lambda_433
tff(fact_8612_ATP_Olambda__434,axiom,
    ! [Uu: rat,Uua: int,Uub: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),aTP_Lamp_ng(rat,fun(int,fun(int,bool)),Uu),Uua),Uub))
    <=> pp(aa(product_prod(int,int),bool,product_case_prod(int,int,bool,aa(int,fun(int,fun(int,bool)),aTP_Lamp_nf(int,fun(int,fun(int,fun(int,bool))),Uua),Uub)),quotient_of(Uu))) ) ).

% ATP.lambda_434
tff(fact_8613_ATP_Olambda__435,axiom,
    ! [Uu: rat,Uua: int,Uub: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),aTP_Lamp_nd(rat,fun(int,fun(int,bool)),Uu),Uua),Uub))
    <=> pp(aa(product_prod(int,int),bool,product_case_prod(int,int,bool,aa(int,fun(int,fun(int,bool)),aTP_Lamp_nc(int,fun(int,fun(int,fun(int,bool))),Uua),Uub)),quotient_of(Uu))) ) ).

% ATP.lambda_435
tff(fact_8614_ATP_Olambda__436,axiom,
    ! [D6: $tType,C: $tType,A: $tType] :
      ( ( real_V4412858255891104859lgebra(C)
        & comm_ring_1(C)
        & topolo4958980785337419405_space(D6) )
     => ! [Uu: set(A),Uua: fun(A,fun(D6,C)),Uub: D6] : aa(D6,C,aa(fun(A,fun(D6,C)),fun(D6,C),aTP_Lamp_akd(set(A),fun(fun(A,fun(D6,C)),fun(D6,C)),Uu),Uua),Uub) = groups7121269368397514597t_prod(A,C,aa(D6,fun(A,C),aTP_Lamp_akc(fun(A,fun(D6,C)),fun(D6,fun(A,C)),Uua),Uub),Uu) ) ).

% ATP.lambda_436
tff(fact_8615_ATP_Olambda__437,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( topolo4987421752381908075d_mult(C)
        & topolo4958980785337419405_space(B) )
     => ! [Uu: set(A),Uua: fun(A,fun(B,C)),Uub: B] : aa(B,C,aa(fun(A,fun(B,C)),fun(B,C),aTP_Lamp_akb(set(A),fun(fun(A,fun(B,C)),fun(B,C)),Uu),Uua),Uub) = groups7121269368397514597t_prod(A,C,aa(B,fun(A,C),aTP_Lamp_aka(fun(A,fun(B,C)),fun(B,fun(A,C)),Uua),Uub),Uu) ) ).

% ATP.lambda_437
tff(fact_8616_ATP_Olambda__438,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( real_V4412858255891104859lgebra(C)
        & comm_ring_1(C)
        & topological_t2_space(B) )
     => ! [Uu: set(A),Uua: fun(A,fun(B,C)),Uub: B] : aa(B,C,aa(fun(A,fun(B,C)),fun(B,C),aTP_Lamp_yz(set(A),fun(fun(A,fun(B,C)),fun(B,C)),Uu),Uua),Uub) = groups7121269368397514597t_prod(A,C,aa(B,fun(A,C),aTP_Lamp_yy(fun(A,fun(B,C)),fun(B,fun(A,C)),Uua),Uub),Uu) ) ).

% ATP.lambda_438
tff(fact_8617_ATP_Olambda__439,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( topolo4987421752381908075d_mult(C)
        & topological_t2_space(B) )
     => ! [Uu: set(A),Uua: fun(A,fun(B,C)),Uub: B] : aa(B,C,aa(fun(A,fun(B,C)),fun(B,C),aTP_Lamp_yx(set(A),fun(fun(A,fun(B,C)),fun(B,C)),Uu),Uua),Uub) = groups7121269368397514597t_prod(A,C,aa(B,fun(A,C),aTP_Lamp_yw(fun(A,fun(B,C)),fun(B,fun(A,C)),Uua),Uub),Uu) ) ).

% ATP.lambda_439
tff(fact_8618_ATP_Olambda__440,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( topolo4987421752381908075d_mult(C)
     => ! [Uu: set(A),Uua: fun(A,fun(B,C)),Uub: B] : aa(B,C,aa(fun(A,fun(B,C)),fun(B,C),aTP_Lamp_yv(set(A),fun(fun(A,fun(B,C)),fun(B,C)),Uu),Uua),Uub) = groups7121269368397514597t_prod(A,C,aa(B,fun(A,C),aTP_Lamp_yu(fun(A,fun(B,C)),fun(B,fun(A,C)),Uua),Uub),Uu) ) ).

% ATP.lambda_440
tff(fact_8619_ATP_Olambda__441,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( real_V4412858255891104859lgebra(C)
        & comm_ring_1(C) )
     => ! [Uu: set(A),Uua: fun(A,fun(B,C)),Uub: B] : aa(B,C,aa(fun(A,fun(B,C)),fun(B,C),aTP_Lamp_yt(set(A),fun(fun(A,fun(B,C)),fun(B,C)),Uu),Uua),Uub) = groups7121269368397514597t_prod(A,C,aa(B,fun(A,C),aTP_Lamp_ys(fun(A,fun(B,C)),fun(B,fun(A,C)),Uua),Uub),Uu) ) ).

% ATP.lambda_441
tff(fact_8620_ATP_Olambda__442,axiom,
    ! [A: $tType,B: $tType,I6: $tType] :
      ( ( real_V3459762299906320749_field(B)
        & real_V822414075346904944vector(A) )
     => ! [Uu: set(I6),Uua: fun(I6,fun(A,B)),Uub: A] : aa(A,B,aa(fun(I6,fun(A,B)),fun(A,B),aTP_Lamp_uo(set(I6),fun(fun(I6,fun(A,B)),fun(A,B)),Uu),Uua),Uub) = groups7121269368397514597t_prod(I6,B,aa(A,fun(I6,B),aTP_Lamp_un(fun(I6,fun(A,B)),fun(A,fun(I6,B)),Uua),Uub),Uu) ) ).

% ATP.lambda_442
tff(fact_8621_ATP_Olambda__443,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(B,fun(C,A)),Uua: set(B),Uub: C] : aa(C,A,aa(set(B),fun(C,A),aTP_Lamp_hd(fun(B,fun(C,A)),fun(set(B),fun(C,A)),Uu),Uua),Uub) = groups7121269368397514597t_prod(B,A,aa(C,fun(B,A),aTP_Lamp_hc(fun(B,fun(C,A)),fun(C,fun(B,A)),Uu),Uub),Uua) ) ).

% ATP.lambda_443
tff(fact_8622_ATP_Olambda__444,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( topolo5987344860129210374id_add(C)
        & topolo4958980785337419405_space(B) )
     => ! [Uu: set(A),Uua: fun(A,fun(B,C)),Uub: B] : aa(B,C,aa(fun(A,fun(B,C)),fun(B,C),aTP_Lamp_akf(set(A),fun(fun(A,fun(B,C)),fun(B,C)),Uu),Uua),Uub) = aa(set(A),C,groups7311177749621191930dd_sum(A,C,aa(B,fun(A,C),aTP_Lamp_ake(fun(A,fun(B,C)),fun(B,fun(A,C)),Uua),Uub)),Uu) ) ).

% ATP.lambda_444
tff(fact_8623_ATP_Olambda__445,axiom,
    ! [A: $tType,B: $tType,I6: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V822414075346904944vector(A) )
     => ! [Uu: set(I6),Uua: fun(I6,fun(A,B)),Uub: A] : aa(A,B,aa(fun(I6,fun(A,B)),fun(A,B),aTP_Lamp_acn(set(I6),fun(fun(I6,fun(A,B)),fun(A,B)),Uu),Uua),Uub) = aa(set(I6),B,groups7311177749621191930dd_sum(I6,B,aa(A,fun(I6,B),aTP_Lamp_acm(fun(I6,fun(A,B)),fun(A,fun(I6,B)),Uua),Uub)),Uu) ) ).

% ATP.lambda_445
tff(fact_8624_ATP_Olambda__446,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( topolo5987344860129210374id_add(C)
        & topological_t2_space(B) )
     => ! [Uu: set(A),Uua: fun(A,fun(B,C)),Uub: B] : aa(B,C,aa(fun(A,fun(B,C)),fun(B,C),aTP_Lamp_zd(set(A),fun(fun(A,fun(B,C)),fun(B,C)),Uu),Uua),Uub) = aa(set(A),C,groups7311177749621191930dd_sum(A,C,aa(B,fun(A,C),aTP_Lamp_zc(fun(A,fun(B,C)),fun(B,fun(A,C)),Uua),Uub)),Uu) ) ).

% ATP.lambda_446
tff(fact_8625_ATP_Olambda__447,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( topolo5987344860129210374id_add(C)
     => ! [Uu: set(A),Uua: fun(A,fun(B,C)),Uub: B] : aa(B,C,aa(fun(A,fun(B,C)),fun(B,C),aTP_Lamp_zb(set(A),fun(fun(A,fun(B,C)),fun(B,C)),Uu),Uua),Uub) = aa(set(A),C,groups7311177749621191930dd_sum(A,C,aa(B,fun(A,C),aTP_Lamp_za(fun(A,fun(B,C)),fun(B,fun(A,C)),Uua),Uub)),Uu) ) ).

% ATP.lambda_447
tff(fact_8626_ATP_Olambda__448,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(C)
        & real_V822414075346904944vector(B) )
     => ! [Uu: set(A),Uua: fun(A,fun(B,C)),Uub: B] : aa(B,C,aa(fun(A,fun(B,C)),fun(B,C),aTP_Lamp_tc(set(A),fun(fun(A,fun(B,C)),fun(B,C)),Uu),Uua),Uub) = aa(set(A),C,groups7311177749621191930dd_sum(A,C,aa(B,fun(A,C),aTP_Lamp_tb(fun(A,fun(B,C)),fun(B,fun(A,C)),Uua),Uub)),Uu) ) ).

% ATP.lambda_448
tff(fact_8627_ATP_Olambda__449,axiom,
    ! [A: $tType,I6: $tType] :
      ( ( topolo5987344860129210374id_add(A)
        & topological_t2_space(A) )
     => ! [Uu: set(I6),Uua: fun(I6,fun(nat,A)),Uub: nat] : aa(nat,A,aa(fun(I6,fun(nat,A)),fun(nat,A),aTP_Lamp_dx(set(I6),fun(fun(I6,fun(nat,A)),fun(nat,A)),Uu),Uua),Uub) = aa(set(I6),A,groups7311177749621191930dd_sum(I6,A,aa(nat,fun(I6,A),aTP_Lamp_dw(fun(I6,fun(nat,A)),fun(nat,fun(I6,A)),Uua),Uub)),Uu) ) ).

% ATP.lambda_449
tff(fact_8628_ATP_Olambda__450,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(B,fun(C,A)),Uua: set(B),Uub: C] : aa(C,A,aa(set(B),fun(C,A),aTP_Lamp_dh(fun(B,fun(C,A)),fun(set(B),fun(C,A)),Uu),Uua),Uub) = aa(set(B),A,groups7311177749621191930dd_sum(B,A,aa(C,fun(B,A),aTP_Lamp_dg(fun(B,fun(C,A)),fun(C,fun(B,A)),Uu),Uub)),Uua) ) ).

% ATP.lambda_450
tff(fact_8629_ATP_Olambda__451,axiom,
    ! [Uu: fun(nat,fun(real,real)),Uua: real,Uub: nat] : aa(nat,real,aa(real,fun(nat,real),aTP_Lamp_sk(fun(nat,fun(real,real)),fun(real,fun(nat,real)),Uu),Uua),Uub) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(nat,fun(real,real),Uu,Uub),zero_zero(real))),semiring_char_0_fact(real,Uub))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Uua),Uub)) ).

% ATP.lambda_451
tff(fact_8630_ATP_Olambda__452,axiom,
    ! [Uu: real,Uua: fun(nat,fun(real,real)),Uub: nat] : aa(nat,real,aa(fun(nat,fun(real,real)),fun(nat,real),aTP_Lamp_sl(real,fun(fun(nat,fun(real,real)),fun(nat,real)),Uu),Uua),Uub) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(nat,fun(real,real),Uua,Uub),zero_zero(real))),semiring_char_0_fact(real,Uub))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Uu),Uub)) ).

% ATP.lambda_452
tff(fact_8631_ATP_Olambda__453,axiom,
    ! [A: $tType] :
      ( zero(A)
     => ! [Uu: real,Uua: fun(nat,fun(A,real)),Uub: nat] : aa(nat,real,aa(fun(nat,fun(A,real)),fun(nat,real),aTP_Lamp_ge(real,fun(fun(nat,fun(A,real)),fun(nat,real)),Uu),Uua),Uub) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(A,real,aa(nat,fun(A,real),Uua,Uub),zero_zero(A))),semiring_char_0_fact(real,Uub))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Uu),Uub)) ) ).

% ATP.lambda_453
tff(fact_8632_ATP_Olambda__454,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [Uu: fun(nat,A),Uua: nat,Uub: A] :
          ( pp(aa(A,bool,aa(nat,fun(A,bool),aTP_Lamp_il(fun(nat,A),fun(nat,fun(A,bool)),Uu),Uua),Uub))
        <=> ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_ib(fun(nat,A),fun(A,fun(nat,A)),Uu),Uub)),set_ord_atMost(nat,Uua)) = zero_zero(A) ) ) ) ).

% ATP.lambda_454
tff(fact_8633_ATP_Olambda__455,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: fun(A,A),Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_zq(fun(A,A),fun(A,fun(A,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,minus_minus(A,aa(A,A,Uu,aa(A,A,aa(A,fun(A,A),plus_plus(A),Uua),Uub))),aa(A,A,Uu,Uua))),Uub) ) ).

% ATP.lambda_455
tff(fact_8634_ATP_Olambda__456,axiom,
    ! [A: $tType] :
      ( ( inverse(A)
        & real_V822414075346904944vector(A) )
     => ! [Uu: fun(A,A),Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_zg(fun(A,A),fun(A,fun(A,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,minus_minus(A,aa(A,A,Uu,aa(A,A,aa(A,fun(A,A),plus_plus(A),Uua),Uub))),aa(A,A,Uu,Uua))),Uub) ) ).

% ATP.lambda_456
tff(fact_8635_ATP_Olambda__457,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Uu: fun(nat,A),Uua: fun(nat,A),Uub: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_de(fun(nat,A),fun(fun(nat,A),fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,Uub)),aa(nat,A,aa(A,fun(nat,A),power_power(A),zero_zero(A)),Uub))),aa(nat,A,Uua,Uub)) ) ).

% ATP.lambda_457
tff(fact_8636_ATP_Olambda__458,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: fun(A,A),Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_vn(fun(A,A),fun(A,fun(A,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,minus_minus(A,aa(A,A,Uu,Uub)),aa(A,A,Uu,Uua))),aa(A,A,minus_minus(A,Uub),Uua)) ) ).

% ATP.lambda_458
tff(fact_8637_ATP_Olambda__459,axiom,
    ! [A: $tType] :
      ( ( inverse(A)
        & real_V822414075346904944vector(A) )
     => ! [Uu: fun(A,A),Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_zh(fun(A,A),fun(A,fun(A,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,minus_minus(A,aa(A,A,Uu,Uub)),aa(A,A,Uu,Uua))),aa(A,A,minus_minus(A,Uub),Uua)) ) ).

% ATP.lambda_459
tff(fact_8638_ATP_Olambda__460,axiom,
    ! [Uu: fun(nat,real),Uua: real,Uub: nat] : aa(nat,real,aa(real,fun(nat,real),aTP_Lamp_st(fun(nat,real),fun(real,fun(nat,real)),Uu),Uua),Uub) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,Uu,Uub)),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,Uub)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Uua),Uub)) ).

% ATP.lambda_460
tff(fact_8639_ATP_Olambda__461,axiom,
    ! [Uu: real,Uua: fun(nat,real),Uub: nat] : aa(nat,real,aa(fun(nat,real),fun(nat,real),aTP_Lamp_gf(real,fun(fun(nat,real),fun(nat,real)),Uu),Uua),Uub) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(nat,real,Uua,Uub)),semiring_char_0_fact(real,Uub))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Uu),Uub)) ).

% ATP.lambda_461
tff(fact_8640_ATP_Olambda__462,axiom,
    ! [A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & ring_1(A) )
     => ! [Uu: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_fy(fun(nat,A),fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),Uub)),aa(nat,A,Uu,Uub))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),aa(nat,nat,minus_minus(nat,Uub),aa(nat,nat,suc,zero_zero(nat))))) ) ).

% ATP.lambda_462
tff(fact_8641_ATP_Olambda__463,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: list(A),Uub: list(A)] :
      ( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),aTP_Lamp_amt(set(product_prod(A,A)),fun(list(A),fun(list(A),bool)),Uu),Uua),Uub))
    <=> ( ( aa(list(A),nat,size_size(list(A)),Uua) = aa(list(A),nat,size_size(list(A)),Uub) )
        & ? [Xys: list(A),X2: A,Y7: A,Xs3: list(A),Ys3: list(A)] :
            ( ( Uua = append(A,Xys,aa(list(A),list(A),cons(A,X2),Xs3)) )
            & ( Uub = append(A,Xys,aa(list(A),list(A),cons(A,Y7),Ys3)) )
            & pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Y7)),Uu)) ) ) ) ).

% ATP.lambda_463
tff(fact_8642_ATP_Olambda__464,axiom,
    ! [A: $tType,Uu: nat,Uua: list(A),Uub: list(A)] :
      ( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),aTP_Lamp_pn(nat,fun(list(A),fun(list(A),bool)),Uu),Uua),Uub))
    <=> ( ( aa(list(A),nat,size_size(list(A)),Uub) = Uu )
        & pp(aa(set(A),bool,ord_less_eq(set(A),aa(list(A),set(A),set2(A),Uub)),aa(list(A),set(A),set2(A),Uua))) ) ) ).

% ATP.lambda_464
tff(fact_8643_ATP_Olambda__465,axiom,
    ! [A: $tType,Uu: set(A),Uua: nat,Uub: list(A)] :
      ( pp(aa(list(A),bool,aa(nat,fun(list(A),bool),aTP_Lamp_al(set(A),fun(nat,fun(list(A),bool)),Uu),Uua),Uub))
    <=> ( pp(aa(set(A),bool,ord_less_eq(set(A),aa(list(A),set(A),set2(A),Uub)),Uu))
        & pp(aa(nat,bool,ord_less_eq(nat,aa(list(A),nat,size_size(list(A)),Uub)),Uua)) ) ) ).

% ATP.lambda_465
tff(fact_8644_ATP_Olambda__466,axiom,
    ! [A: $tType,Uu: set(A),Uua: nat,Uub: list(A)] :
      ( pp(aa(list(A),bool,aa(nat,fun(list(A),bool),aTP_Lamp_ak(set(A),fun(nat,fun(list(A),bool)),Uu),Uua),Uub))
    <=> ( pp(aa(set(A),bool,ord_less_eq(set(A),aa(list(A),set(A),set2(A),Uub)),Uu))
        & ( aa(list(A),nat,size_size(list(A)),Uub) = Uua ) ) ) ).

% ATP.lambda_466
tff(fact_8645_ATP_Olambda__467,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_sf(fun(nat,A),fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,diffs(A,diffs(A,Uu)),Uub)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),Uub)) ) ).

% ATP.lambda_467
tff(fact_8646_ATP_Olambda__468,axiom,
    ! [A: $tType,Uu: fun(A,bool),Uua: list(A),Uub: A] :
      ( pp(aa(A,bool,aa(list(A),fun(A,bool),aTP_Lamp_anf(fun(A,bool),fun(list(A),fun(A,bool)),Uu),Uua),Uub))
    <=> ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Uub),aa(list(A),set(A),set2(A),Uua)))
        & pp(aa(A,bool,Uu,Uub)) ) ) ).

% ATP.lambda_468
tff(fact_8647_ATP_Olambda__469,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: A,Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_pf(A,fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,minus_minus(A,Uu),aa(nat,A,semiring_1_of_nat(A),Uub))),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,minus_minus(nat,Uua),Uub))) ) ).

% ATP.lambda_469
tff(fact_8648_ATP_Olambda__470,axiom,
    ! [A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & ring_1(A) )
     => ! [Uu: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_fx(fun(nat,A),fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,diffs(A,Uu),Uub)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),Uub)) ) ).

% ATP.lambda_470
tff(fact_8649_ATP_Olambda__471,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_gd(fun(nat,A),fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,diffs(A,Uu),Uub)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),Uub)) ) ).

% ATP.lambda_471
tff(fact_8650_ATP_Olambda__472,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: A,Uua: fun(nat,A),Uub: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_gg(A,fun(fun(nat,A),fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,diffs(A,Uua),Uub)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu),Uub)) ) ).

% ATP.lambda_472
tff(fact_8651_ATP_Olambda__473,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] : aa(nat,nat,aa(nat,fun(nat,nat),aTP_Lamp_jo(nat,fun(nat,fun(nat,nat)),Uu),Uua),Uub) = aa(nat,nat,binomial(aa(nat,nat,minus_minus(nat,Uua),Uub)),aa(nat,nat,minus_minus(nat,Uu),Uub)) ).

% ATP.lambda_473
tff(fact_8652_ATP_Olambda__474,axiom,
    ! [A: $tType,Uu: A,Uua: set(A),Uub: A] :
      ( pp(aa(A,bool,aa(set(A),fun(A,bool),aTP_Lamp_lp(A,fun(set(A),fun(A,bool)),Uu),Uua),Uub))
    <=> ( ( Uub = Uu )
        | pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Uub),Uua)) ) ) ).

% ATP.lambda_474
tff(fact_8653_ATP_Olambda__475,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),aTP_Lamp_mt(int,fun(int,fun(int,bool)),Uu),Uua),Uub))
    <=> ( pp(aa(int,bool,ord_less_eq(int,Uu),Uua))
        & pp(aa(int,bool,ord_less(int,Uua),Uub)) ) ) ).

% ATP.lambda_475
tff(fact_8654_ATP_Olambda__476,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),aTP_Lamp_au(int,fun(int,fun(int,bool)),Uu),Uua),Uub))
    <=> ( pp(aa(int,bool,ord_less_eq(int,Uu),Uub))
        & pp(aa(int,bool,ord_less_eq(int,Uub),Uua)) ) ) ).

% ATP.lambda_476
tff(fact_8655_ATP_Olambda__477,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),aTP_Lamp_mu(int,fun(int,fun(int,bool)),Uu),Uua),Uub))
    <=> ( pp(aa(int,bool,ord_less_eq(int,Uu),Uub))
        & pp(aa(int,bool,ord_less(int,Uua),Uub)) ) ) ).

% ATP.lambda_477
tff(fact_8656_ATP_Olambda__478,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),aTP_Lamp_ax(int,fun(int,fun(int,bool)),Uu),Uua),Uub))
    <=> ( pp(aa(int,bool,ord_less_eq(int,Uu),Uub))
        & pp(aa(int,bool,ord_less(int,Uub),Uua)) ) ) ).

% ATP.lambda_478
tff(fact_8657_ATP_Olambda__479,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),aTP_Lamp_aw(int,fun(int,fun(int,bool)),Uu),Uua),Uub))
    <=> ( pp(aa(int,bool,ord_less(int,Uu),Uub))
        & pp(aa(int,bool,ord_less_eq(int,Uub),Uua)) ) ) ).

% ATP.lambda_479
tff(fact_8658_ATP_Olambda__480,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),aTP_Lamp_av(int,fun(int,fun(int,bool)),Uu),Uua),Uub))
    <=> ( pp(aa(int,bool,ord_less(int,Uu),Uub))
        & pp(aa(int,bool,ord_less(int,Uub),Uua)) ) ) ).

% ATP.lambda_480
tff(fact_8659_ATP_Olambda__481,axiom,
    ! [Uu: vEBT_VEBT,Uua: nat,Uub: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),aTP_Lamp_lj(vEBT_VEBT,fun(nat,fun(nat,bool)),Uu),Uua),Uub))
    <=> ( pp(aa(nat,bool,vEBT_vebt_member(Uu),Uub))
        & pp(aa(nat,bool,ord_less(nat,Uua),Uub)) ) ) ).

% ATP.lambda_481
tff(fact_8660_ATP_Olambda__482,axiom,
    ! [Uu: vEBT_VEBT,Uua: nat,Uub: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),aTP_Lamp_lk(vEBT_VEBT,fun(nat,fun(nat,bool)),Uu),Uua),Uub))
    <=> ( pp(aa(nat,bool,vEBT_vebt_member(Uu),Uub))
        & pp(aa(nat,bool,ord_less(nat,Uub),Uua)) ) ) ).

% ATP.lambda_482
tff(fact_8661_ATP_Olambda__483,axiom,
    ! [A: $tType,Uu: set(A),Uua: set(A),Uub: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),aTP_Lamp_ah(set(A),fun(set(A),fun(set(A),bool)),Uu),Uua),Uub))
    <=> ( aa(set(A),set(A),minus_minus(set(A),Uub),Uu) = aa(set(A),set(A),minus_minus(set(A),Uua),Uu) ) ) ).

% ATP.lambda_483
tff(fact_8662_ATP_Olambda__484,axiom,
    ! [B: $tType,Uu: set(B),Uua: fun(B,bool),Uub: B] :
      ( pp(aa(B,bool,aa(fun(B,bool),fun(B,bool),aTP_Lamp_dy(set(B),fun(fun(B,bool),fun(B,bool)),Uu),Uua),Uub))
    <=> ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),Uub),Uu))
        & pp(aa(B,bool,Uua,Uub)) ) ) ).

% ATP.lambda_484
tff(fact_8663_ATP_Olambda__485,axiom,
    ! [A: $tType,Uu: set(A),Uua: fun(A,bool),Uub: A] :
      ( pp(aa(A,bool,aa(fun(A,bool),fun(A,bool),aTP_Lamp_ae(set(A),fun(fun(A,bool),fun(A,bool)),Uu),Uua),Uub))
    <=> ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Uub),Uu))
        & pp(aa(A,bool,Uua,Uub)) ) ) ).

% ATP.lambda_485
tff(fact_8664_ATP_Olambda__486,axiom,
    ! [A: $tType,Uu: fun(A,bool),Uua: A,Uub: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_ln(fun(A,bool),fun(A,fun(A,bool)),Uu),Uua),Uub))
    <=> ( ( Uua = Uub )
        & pp(aa(A,bool,Uu,Uub)) ) ) ).

% ATP.lambda_486
tff(fact_8665_ATP_Olambda__487,axiom,
    ! [A: $tType,Uu: fun(A,bool),Uua: A,Uub: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_lo(fun(A,bool),fun(A,fun(A,bool)),Uu),Uua),Uub))
    <=> ( ( Uub = Uua )
        & pp(aa(A,bool,Uu,Uub)) ) ) ).

% ATP.lambda_487
tff(fact_8666_ATP_Olambda__488,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: set(B),Uua: fun(B,A),Uub: B] :
          ( pp(aa(B,bool,aa(fun(B,A),fun(B,bool),aTP_Lamp_bj(set(B),fun(fun(B,A),fun(B,bool)),Uu),Uua),Uub))
        <=> ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),Uub),Uu))
            & ( aa(B,A,Uua,Uub) != zero_zero(A) ) ) ) ) ).

% ATP.lambda_488
tff(fact_8667_ATP_Olambda__489,axiom,
    ! [A: $tType,B: $tType] :
      ( ab_group_add(B)
     => ! [Uu: set(A),Uua: fun(A,B),Uub: A] :
          ( pp(aa(A,bool,aa(fun(A,B),fun(A,bool),aTP_Lamp_op(set(A),fun(fun(A,B),fun(A,bool)),Uu),Uua),Uub))
        <=> ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Uub),Uu))
            & ( aa(A,B,Uua,Uub) != zero_zero(B) ) ) ) ) ).

% ATP.lambda_489
tff(fact_8668_ATP_Olambda__490,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: set(B),Uua: fun(B,A),Uub: B] :
          ( pp(aa(B,bool,aa(fun(B,A),fun(B,bool),aTP_Lamp_bh(set(B),fun(fun(B,A),fun(B,bool)),Uu),Uua),Uub))
        <=> ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),Uub),Uu))
            & ( aa(B,A,Uua,Uub) != one_one(A) ) ) ) ) ).

% ATP.lambda_490
tff(fact_8669_ATP_Olambda__491,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(B,A),Uua: set(B),Uub: B] :
          ( pp(aa(B,bool,aa(set(B),fun(B,bool),aTP_Lamp_oq(fun(B,A),fun(set(B),fun(B,bool)),Uu),Uua),Uub))
        <=> ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),Uub),Uua))
            & ( aa(B,A,Uu,Uub) != zero_zero(A) ) ) ) ) ).

% ATP.lambda_491
tff(fact_8670_ATP_Olambda__492,axiom,
    ! [A: $tType,Uu: set(A),Uua: set(A),Uub: A] :
      ( pp(aa(A,bool,aa(set(A),fun(A,bool),aTP_Lamp_af(set(A),fun(set(A),fun(A,bool)),Uu),Uua),Uub))
    <=> ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Uub),Uu))
        & ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Uub),Uua)) ) ) ).

% ATP.lambda_492
tff(fact_8671_ATP_Olambda__493,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),aTP_Lamp_of(nat,fun(nat,fun(nat,bool)),Uu),Uua),Uub))
    <=> pp(aa(nat,bool,ord_less_eq(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uub),Uu)),Uua)) ) ).

% ATP.lambda_493
tff(fact_8672_ATP_Olambda__494,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),aTP_Lamp_mg(nat,fun(nat,fun(nat,bool)),Uu),Uua),Uub))
    <=> pp(aa(nat,bool,ord_less_eq(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uua),Uub)),Uu)) ) ).

% ATP.lambda_494
tff(fact_8673_ATP_Olambda__495,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [Uu: A,Uua: real,Uub: A] :
          ( pp(aa(A,bool,aa(real,fun(A,bool),aTP_Lamp_adg(A,fun(real,fun(A,bool)),Uu),Uua),Uub))
        <=> pp(aa(real,bool,ord_less(real,real_V557655796197034286t_dist(A,Uu,Uub)),Uua)) ) ) ).

% ATP.lambda_495
tff(fact_8674_ATP_Olambda__496,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),aTP_Lamp_ml(nat,fun(nat,fun(nat,bool)),Uu),Uua),Uub))
    <=> pp(aa(nat,bool,ord_less(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uua),Uub)),Uu)) ) ).

% ATP.lambda_496
tff(fact_8675_ATP_Olambda__497,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Uu: A,Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_ok(A,fun(A,fun(A,A)),Uu),Uua),Uub) = aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),Uub),Uu)),Uua) ) ).

% ATP.lambda_497
tff(fact_8676_ATP_Olambda__498,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Uu: A,Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_oi(A,fun(A,fun(A,A)),Uu),Uua),Uub) = aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),Uu),Uub)),Uua) ) ).

% ATP.lambda_498
tff(fact_8677_ATP_Olambda__499,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Uu: A,Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_oj(A,fun(A,fun(A,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Uub),Uu)),Uua) ) ).

% ATP.lambda_499
tff(fact_8678_ATP_Olambda__500,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Uu: A,Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_oh(A,fun(A,fun(A,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Uu),Uub)),Uua) ) ).

% ATP.lambda_500
tff(fact_8679_ATP_Olambda__501,axiom,
    ! [A: $tType,B: $tType,Uu: set(product_prod(B,A)),Uua: B,Uub: A] :
      ( pp(aa(A,bool,aa(B,fun(A,bool),aTP_Lamp_lv(set(product_prod(B,A)),fun(B,fun(A,bool)),Uu),Uua),Uub))
    <=> pp(aa(set(product_prod(B,A)),bool,aa(product_prod(B,A),fun(set(product_prod(B,A)),bool),member(product_prod(B,A)),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),Uua),Uub)),Uu)) ) ).

% ATP.lambda_501
tff(fact_8680_ATP_Olambda__502,axiom,
    ! [B: $tType,A: $tType,Uu: set(product_prod(A,B)),Uua: A,Uub: B] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),aTP_Lamp_on(set(product_prod(A,B)),fun(A,fun(B,bool)),Uu),Uua),Uub))
    <=> pp(aa(set(product_prod(A,B)),bool,aa(product_prod(A,B),fun(set(product_prod(A,B)),bool),member(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uua),Uub)),Uu)) ) ).

% ATP.lambda_502
tff(fact_8681_ATP_Olambda__503,axiom,
    ! [A: $tType,Uu: list(list(A)),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_amp(list(list(A)),fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(nat,A,nth(A,aa(nat,list(A),nth(list(A),Uu),Uub)),Uua) ).

% ATP.lambda_503
tff(fact_8682_ATP_Olambda__504,axiom,
    ! [Uu: nat,Uua: complex,Uub: complex] :
      ( pp(aa(complex,bool,aa(complex,fun(complex,bool),aTP_Lamp_bb(nat,fun(complex,fun(complex,bool)),Uu),Uua),Uub))
    <=> ( aa(nat,complex,aa(complex,fun(nat,complex),power_power(complex),Uub),Uu) = Uua ) ) ).

% ATP.lambda_504
tff(fact_8683_ATP_Olambda__505,axiom,
    ! [Uu: complex,Uua: nat,Uub: complex] :
      ( pp(aa(complex,bool,aa(nat,fun(complex,bool),aTP_Lamp_ky(complex,fun(nat,fun(complex,bool)),Uu),Uua),Uub))
    <=> ( aa(nat,complex,aa(complex,fun(nat,complex),power_power(complex),Uub),Uua) = Uu ) ) ).

% ATP.lambda_505
tff(fact_8684_ATP_Olambda__506,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Uu: A,Uua: A,Uub: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_mz(A,fun(A,fun(A,bool)),Uu),Uua),Uub))
        <=> ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Uub),ring_1_Ints(A)))
            & pp(aa(A,bool,ord_less_eq(A,Uu),Uub))
            & pp(aa(A,bool,ord_less_eq(A,Uub),Uua)) ) ) ) ).

% ATP.lambda_506
tff(fact_8685_ATP_Olambda__507,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [Uu: fun(B,A),Uua: list(B),Uub: B] :
          ( pp(aa(B,bool,aa(list(B),fun(B,bool),aTP_Lamp_anl(fun(B,A),fun(list(B),fun(B,bool)),Uu),Uua),Uub))
        <=> pp(aa(A,bool,ord_less(A,aa(B,A,Uu,aa(nat,B,nth(B,Uua),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(list(B),nat,size_size(list(B)),Uua)),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(B,A,Uu,Uub))) ) ) ).

% ATP.lambda_507
tff(fact_8686_ATP_Olambda__508,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Uu: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_cq(fun(nat,A),fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,aa(nat,nat,suc,Uub))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),Uub)) ) ).

% ATP.lambda_508
tff(fact_8687_ATP_Olambda__509,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V8999393235501362500lgebra(A) )
     => ! [Uu: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_cp(fun(nat,A),fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,aa(nat,nat,suc,Uub))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),Uub)) ) ).

% ATP.lambda_509
tff(fact_8688_ATP_Olambda__510,axiom,
    ! [Uu: fun(nat,real),Uua: fun(nat,int),Uub: nat] : aa(nat,real,aa(fun(nat,int),fun(nat,real),aTP_Lamp_acb(fun(nat,real),fun(fun(nat,int),fun(nat,real)),Uu),Uua),Uub) = aa(real,real,minus_minus(real,aa(nat,real,Uu,Uub)),aa(real,real,aa(real,fun(real,real),times_times(real),aa(int,real,ring_1_of_int(real),aa(nat,int,Uua,Uub))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),pi))) ).

% ATP.lambda_510
tff(fact_8689_ATP_Olambda__511,axiom,
    ! [Uu: fun(nat,real),Uua: real,Uub: nat] : aa(nat,real,aa(real,fun(nat,real),aTP_Lamp_su(fun(nat,real),fun(real,fun(nat,real)),Uu),Uua),Uub) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,Uu,Uub)),aa(nat,real,aa(real,fun(nat,real),power_power(real),Uua),aa(nat,nat,suc,Uub))) ).

% ATP.lambda_511
tff(fact_8690_ATP_Olambda__512,axiom,
    ! [Uu: fun(nat,real),Uua: real,Uub: nat] : aa(nat,real,aa(real,fun(nat,real),aTP_Lamp_ct(fun(nat,real),fun(real,fun(nat,real)),Uu),Uua),Uub) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,Uu,Uub)),aa(nat,real,aa(real,fun(nat,real),power_power(real),Uua),Uub)) ).

% ATP.lambda_512
tff(fact_8691_ATP_Olambda__513,axiom,
    ! [Uu: fun(nat,nat),Uua: nat,Uub: nat] : aa(nat,nat,aa(nat,fun(nat,nat),aTP_Lamp_it(fun(nat,nat),fun(nat,fun(nat,nat)),Uu),Uua),Uub) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,Uu,Uub)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),Uua),Uub)) ).

% ATP.lambda_513
tff(fact_8692_ATP_Olambda__514,axiom,
    ! [Aa: $tType] :
      ( ( real_Vector_banach(Aa)
        & real_V3459762299906320749_field(Aa) )
     => ! [Uu: fun(nat,Aa),Uua: Aa,Uub: nat] : aa(nat,Aa,aa(Aa,fun(nat,Aa),aTP_Lamp_zy(fun(nat,Aa),fun(Aa,fun(nat,Aa)),Uu),Uua),Uub) = aa(Aa,Aa,aa(Aa,fun(Aa,Aa),times_times(Aa),aa(nat,Aa,Uu,Uub)),aa(nat,Aa,aa(Aa,fun(nat,Aa),power_power(Aa),Uua),Uub)) ) ).

% ATP.lambda_514
tff(fact_8693_ATP_Olambda__515,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [Uu: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_ib(fun(nat,A),fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,Uub)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),Uub)) ) ).

% ATP.lambda_515
tff(fact_8694_ATP_Olambda__516,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Uu: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_ck(fun(nat,A),fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,Uub)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),Uub)) ) ).

% ATP.lambda_516
tff(fact_8695_ATP_Olambda__517,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V8999393235501362500lgebra(A) )
     => ! [Uu: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_co(fun(nat,A),fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,Uub)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),Uub)) ) ).

% ATP.lambda_517
tff(fact_8696_ATP_Olambda__518,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_gc(fun(nat,A),fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,Uub)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),Uub)) ) ).

% ATP.lambda_518
tff(fact_8697_ATP_Olambda__519,axiom,
    ! [A: $tType] :
      ( ( ab_semigroup_mult(A)
        & real_V8999393235501362500lgebra(A) )
     => ! [Uu: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_ii(fun(nat,A),fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,Uub)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),Uub)) ) ).

% ATP.lambda_519
tff(fact_8698_ATP_Olambda__520,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [Uu: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_im(fun(nat,A),fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,Uub)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),Uub)) ) ).

% ATP.lambda_520
tff(fact_8699_ATP_Olambda__521,axiom,
    ! [Uu: fun(nat,bool),Uua: nat,Uub: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),aTP_Lamp_ai(fun(nat,bool),fun(nat,fun(nat,bool)),Uu),Uua),Uub))
    <=> ( pp(aa(nat,bool,Uu,Uub))
        & pp(aa(nat,bool,ord_less(nat,Uub),Uua)) ) ) ).

% ATP.lambda_521
tff(fact_8700_ATP_Olambda__522,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [Uu: fun(B,A),Uua: list(B),Uub: B] :
          ( pp(aa(B,bool,aa(list(B),fun(B,bool),aTP_Lamp_anj(fun(B,A),fun(list(B),fun(B,bool)),Uu),Uua),Uub))
        <=> pp(aa(A,bool,ord_less(A,aa(B,A,Uu,Uub)),aa(B,A,Uu,aa(nat,B,nth(B,Uua),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(list(B),nat,size_size(list(B)),Uua)),aa(num,nat,numeral_numeral(nat),bit0(one2))))))) ) ) ).

% ATP.lambda_522
tff(fact_8701_ATP_Olambda__523,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [Uu: fun(B,A),Uua: list(B),Uub: B] :
          ( pp(aa(B,bool,aa(list(B),fun(B,bool),aTP_Lamp_ank(fun(B,A),fun(list(B),fun(B,bool)),Uu),Uua),Uub))
        <=> ( aa(B,A,Uu,Uub) = aa(B,A,Uu,aa(nat,B,nth(B,Uua),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(list(B),nat,size_size(list(B)),Uua)),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) ) ) ).

% ATP.lambda_523
tff(fact_8702_ATP_Olambda__524,axiom,
    ! [C: $tType,D6: $tType] :
      ( ( real_V822414075346904944vector(D6)
        & real_V822414075346904944vector(C) )
     => ! [Uu: fun(D6,real),Uua: fun(D6,C),Uub: D6] : aa(D6,C,aa(fun(D6,C),fun(D6,C),aTP_Lamp_sy(fun(D6,real),fun(fun(D6,C),fun(D6,C)),Uu),Uua),Uub) = aa(C,C,real_V8093663219630862766scaleR(C,aa(D6,real,Uu,Uub)),aa(D6,C,Uua,Uub)) ) ).

% ATP.lambda_524
tff(fact_8703_ATP_Olambda__525,axiom,
    ! [C: $tType,D6: $tType] :
      ( ( topolo4958980785337419405_space(D6)
        & real_V822414075346904944vector(C) )
     => ! [Uu: fun(D6,real),Uua: fun(D6,C),Uub: D6] : aa(D6,C,aa(fun(D6,C),fun(D6,C),aTP_Lamp_aje(fun(D6,real),fun(fun(D6,C),fun(D6,C)),Uu),Uua),Uub) = aa(C,C,real_V8093663219630862766scaleR(C,aa(D6,real,Uu,Uub)),aa(D6,C,Uua,Uub)) ) ).

% ATP.lambda_525
tff(fact_8704_ATP_Olambda__526,axiom,
    ! [C: $tType,D6: $tType] :
      ( ( topological_t2_space(D6)
        & real_V822414075346904944vector(C) )
     => ! [Uu: fun(D6,real),Uua: fun(D6,C),Uub: D6] : aa(D6,C,aa(fun(D6,C),fun(D6,C),aTP_Lamp_wc(fun(D6,real),fun(fun(D6,C),fun(D6,C)),Uu),Uua),Uub) = aa(C,C,real_V8093663219630862766scaleR(C,aa(D6,real,Uu,Uub)),aa(D6,C,Uua,Uub)) ) ).

% ATP.lambda_526
tff(fact_8705_ATP_Olambda__527,axiom,
    ! [C: $tType,D6: $tType] :
      ( real_V822414075346904944vector(C)
     => ! [Uu: fun(D6,real),Uua: fun(D6,C),Uub: D6] : aa(D6,C,aa(fun(D6,C),fun(D6,C),aTP_Lamp_wt(fun(D6,real),fun(fun(D6,C),fun(D6,C)),Uu),Uua),Uub) = aa(C,C,real_V8093663219630862766scaleR(C,aa(D6,real,Uu,Uub)),aa(D6,C,Uua,Uub)) ) ).

% ATP.lambda_527
tff(fact_8706_ATP_Olambda__528,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu: fun(A,real),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_aio(fun(A,real),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,real_V8093663219630862766scaleR(B,aa(A,real,Uu,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_528
tff(fact_8707_ATP_Olambda__529,axiom,
    ! [A: $tType,D6: $tType] :
      ( ( topolo4958980785337419405_space(D6)
        & real_V7819770556892013058_space(A) )
     => ! [Uu: fun(D6,A),Uua: fun(D6,A),Uub: D6] : aa(D6,real,aa(fun(D6,A),fun(D6,real),aTP_Lamp_akg(fun(D6,A),fun(fun(D6,A),fun(D6,real)),Uu),Uua),Uub) = real_V557655796197034286t_dist(A,aa(D6,A,Uu,Uub),aa(D6,A,Uua,Uub)) ) ).

% ATP.lambda_529
tff(fact_8708_ATP_Olambda__530,axiom,
    ! [A: $tType,D6: $tType] :
      ( ( topological_t2_space(D6)
        & real_V7819770556892013058_space(A) )
     => ! [Uu: fun(D6,A),Uua: fun(D6,A),Uub: D6] : aa(D6,real,aa(fun(D6,A),fun(D6,real),aTP_Lamp_adh(fun(D6,A),fun(fun(D6,A),fun(D6,real)),Uu),Uua),Uub) = real_V557655796197034286t_dist(A,aa(D6,A,Uu,Uub),aa(D6,A,Uua,Uub)) ) ).

% ATP.lambda_530
tff(fact_8709_ATP_Olambda__531,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [Uu: fun(B,A),Uua: fun(B,A),Uub: B] : aa(B,real,aa(fun(B,A),fun(B,real),aTP_Lamp_adi(fun(B,A),fun(fun(B,A),fun(B,real)),Uu),Uua),Uub) = real_V557655796197034286t_dist(A,aa(B,A,Uu,Uub),aa(B,A,Uua,Uub)) ) ).

% ATP.lambda_531
tff(fact_8710_ATP_Olambda__532,axiom,
    ! [B: $tType,Uu: fun(B,real),Uua: fun(B,real),Uub: B] :
      ( pp(aa(B,bool,aa(fun(B,real),fun(B,bool),aTP_Lamp_age(fun(B,real),fun(fun(B,real),fun(B,bool)),Uu),Uua),Uub))
    <=> pp(aa(real,bool,ord_less_eq(real,aa(B,real,Uu,Uub)),aa(B,real,Uua,Uub))) ) ).

% ATP.lambda_532
tff(fact_8711_ATP_Olambda__533,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo2564578578187576103pology(A)
     => ! [Uu: fun(B,A),Uua: fun(B,A),Uub: B] :
          ( pp(aa(B,bool,aa(fun(B,A),fun(B,bool),aTP_Lamp_aeq(fun(B,A),fun(fun(B,A),fun(B,bool)),Uu),Uua),Uub))
        <=> pp(aa(A,bool,ord_less_eq(A,aa(B,A,Uu,Uub)),aa(B,A,Uua,Uub))) ) ) ).

% ATP.lambda_533
tff(fact_8712_ATP_Olambda__534,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [Uu: fun(B,A),Uua: fun(B,A),Uub: B] :
          ( pp(aa(B,bool,aa(fun(B,A),fun(B,bool),aTP_Lamp_afj(fun(B,A),fun(fun(B,A),fun(B,bool)),Uu),Uua),Uub))
        <=> pp(aa(A,bool,ord_less_eq(A,aa(B,A,Uu,Uub)),aa(B,A,Uua,Uub))) ) ) ).

% ATP.lambda_534
tff(fact_8713_ATP_Olambda__535,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [Uu: fun(B,A),Uua: fun(B,A),Uub: B] :
          ( pp(aa(B,bool,aa(fun(B,A),fun(B,bool),aTP_Lamp_ags(fun(B,A),fun(fun(B,A),fun(B,bool)),Uu),Uua),Uub))
        <=> pp(aa(A,bool,ord_less_eq(A,aa(B,A,Uua,Uub)),aa(B,A,Uu,Uub))) ) ) ).

% ATP.lambda_535
tff(fact_8714_ATP_Olambda__536,axiom,
    ! [Uu: fun(real,real),Uua: fun(real,real),Uub: real] : aa(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_ahp(fun(real,real),fun(fun(real,real),fun(real,real)),Uu),Uua),Uub) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,Uu,Uub)),aa(real,real,Uua,Uub)) ).

% ATP.lambda_536
tff(fact_8715_ATP_Olambda__537,axiom,
    ! [A: $tType,C: $tType] :
      ( ( real_V822414075346904944vector(C)
        & real_V8999393235501362500lgebra(A) )
     => ! [Uu: fun(C,A),Uua: fun(C,A),Uub: C] : aa(C,A,aa(fun(C,A),fun(C,A),aTP_Lamp_ul(fun(C,A),fun(fun(C,A),fun(C,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(C,A,Uu,Uub)),aa(C,A,Uua,Uub)) ) ).

% ATP.lambda_537
tff(fact_8716_ATP_Olambda__538,axiom,
    ! [A: $tType,C: $tType] :
      ( ( real_V822414075346904944vector(C)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: fun(C,A),Uua: fun(C,A),Uub: C] : aa(C,A,aa(fun(C,A),fun(C,A),aTP_Lamp_ty(fun(C,A),fun(fun(C,A),fun(C,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(C,A,Uu,Uub)),aa(C,A,Uua,Uub)) ) ).

% ATP.lambda_538
tff(fact_8717_ATP_Olambda__539,axiom,
    ! [A: $tType,C: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Uu: fun(C,A),Uua: fun(C,A),Uub: C] : aa(C,A,aa(fun(C,A),fun(C,A),aTP_Lamp_aed(fun(C,A),fun(fun(C,A),fun(C,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(C,A,Uu,Uub)),aa(C,A,Uua,Uub)) ) ).

% ATP.lambda_539
tff(fact_8718_ATP_Olambda__540,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: fun(B,A),Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_yk(fun(B,A),fun(fun(B,A),fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(B,A,Uu,Uub)),aa(B,A,Uua,Uub)) ) ).

% ATP.lambda_540
tff(fact_8719_ATP_Olambda__541,axiom,
    ! [A: $tType,B: $tType] :
      ( field(A)
     => ! [Uu: fun(B,A),Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_hg(fun(B,A),fun(fun(B,A),fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(B,A,Uu,Uub)),aa(B,A,Uua,Uub)) ) ).

% ATP.lambda_541
tff(fact_8720_ATP_Olambda__542,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V3459762299906320749_field(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_aip(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_542
tff(fact_8721_ATP_Olambda__543,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & real_V3459762299906320749_field(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_ajt(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_543
tff(fact_8722_ATP_Olambda__544,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: fun(A,A),Uua: fun(A,A),Uub: A] : aa(A,A,aa(fun(A,A),fun(A,A),aTP_Lamp_rg(fun(A,A),fun(fun(A,A),fun(A,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,Uu,Uub)),aa(A,A,Uua,Uub)) ) ).

% ATP.lambda_544
tff(fact_8723_ATP_Olambda__545,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V3459762299906320749_field(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_zi(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_545
tff(fact_8724_ATP_Olambda__546,axiom,
    ! [A: $tType,Uu: fun(A,real),Uua: fun(A,real),Uub: A] : aa(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_ady(fun(A,real),fun(fun(A,real),fun(A,real)),Uu),Uua),Uub) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(A,real,Uu,Uub)),aa(A,real,Uua,Uub)) ).

% ATP.lambda_546
tff(fact_8725_ATP_Olambda__547,axiom,
    ! [Uu: fun(real,real),Uua: fun(real,real),Uub: real] : aa(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_aeo(fun(real,real),fun(fun(real,real),fun(real,real)),Uu),Uua),Uub) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,Uua,Uub)),aa(real,real,Uu,Uub)) ).

% ATP.lambda_547
tff(fact_8726_ATP_Olambda__548,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topolo1944317154257567458pology(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] :
          ( pp(aa(A,bool,aa(fun(A,B),fun(A,bool),aTP_Lamp_aiu(fun(A,B),fun(fun(A,B),fun(A,bool)),Uu),Uua),Uub))
        <=> pp(aa(B,bool,ord_less(B,aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub))) ) ) ).

% ATP.lambda_548
tff(fact_8727_ATP_Olambda__549,axiom,
    ! [Uu: fun(nat,nat),Uua: fun(nat,nat),Uub: nat] : aa(nat,nat,aa(fun(nat,nat),fun(nat,nat),aTP_Lamp_pj(fun(nat,nat),fun(fun(nat,nat),fun(nat,nat)),Uu),Uua),Uub) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,Uu,Uub)),aa(nat,nat,Uua,Uub)) ).

% ATP.lambda_549
tff(fact_8728_ATP_Olambda__550,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: fun(nat,A),Uua: fun(nat,A),Uub: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_ahz(fun(nat,A),fun(fun(nat,A),fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,Uub)),aa(nat,A,Uua,Uub)) ) ).

% ATP.lambda_550
tff(fact_8729_ATP_Olambda__551,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Uu: fun(nat,A),Uua: fun(nat,A),Uub: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_pi(fun(nat,A),fun(fun(nat,A),fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,Uub)),aa(nat,A,Uua,Uub)) ) ).

% ATP.lambda_551
tff(fact_8730_ATP_Olambda__552,axiom,
    ! [A: $tType,D6: $tType] :
      ( ( real_V822414075346904944vector(D6)
        & real_V4412858255891104859lgebra(A) )
     => ! [Uu: fun(D6,A),Uua: fun(D6,A),Uub: D6] : aa(D6,A,aa(fun(D6,A),fun(D6,A),aTP_Lamp_to(fun(D6,A),fun(fun(D6,A),fun(D6,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(D6,A,Uu,Uub)),aa(D6,A,Uua,Uub)) ) ).

% ATP.lambda_552
tff(fact_8731_ATP_Olambda__553,axiom,
    ! [B: $tType,D6: $tType] :
      ( ( topolo4958980785337419405_space(D6)
        & topolo4211221413907600880p_mult(B) )
     => ! [Uu: fun(D6,B),Uua: fun(D6,B),Uub: D6] : aa(D6,B,aa(fun(D6,B),fun(D6,B),aTP_Lamp_ajw(fun(D6,B),fun(fun(D6,B),fun(D6,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(D6,B,Uu,Uub)),aa(D6,B,Uua,Uub)) ) ).

% ATP.lambda_553
tff(fact_8732_ATP_Olambda__554,axiom,
    ! [A: $tType,D6: $tType] :
      ( ( topolo4958980785337419405_space(D6)
        & real_V4412858255891104859lgebra(A) )
     => ! [Uu: fun(D6,A),Uua: fun(D6,A),Uub: D6] : aa(D6,A,aa(fun(D6,A),fun(D6,A),aTP_Lamp_ajx(fun(D6,A),fun(fun(D6,A),fun(D6,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(D6,A,Uu,Uub)),aa(D6,A,Uua,Uub)) ) ).

% ATP.lambda_554
tff(fact_8733_ATP_Olambda__555,axiom,
    ! [B: $tType,D6: $tType] :
      ( ( topological_t2_space(D6)
        & topolo4211221413907600880p_mult(B) )
     => ! [Uu: fun(D6,B),Uua: fun(D6,B),Uub: D6] : aa(D6,B,aa(fun(D6,B),fun(D6,B),aTP_Lamp_yh(fun(D6,B),fun(fun(D6,B),fun(D6,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(D6,B,Uu,Uub)),aa(D6,B,Uua,Uub)) ) ).

% ATP.lambda_555
tff(fact_8734_ATP_Olambda__556,axiom,
    ! [A: $tType,D6: $tType] :
      ( ( topological_t2_space(D6)
        & real_V4412858255891104859lgebra(A) )
     => ! [Uu: fun(D6,A),Uua: fun(D6,A),Uub: D6] : aa(D6,A,aa(fun(D6,A),fun(D6,A),aTP_Lamp_yi(fun(D6,A),fun(fun(D6,A),fun(D6,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(D6,A,Uu,Uub)),aa(D6,A,Uua,Uub)) ) ).

% ATP.lambda_556
tff(fact_8735_ATP_Olambda__557,axiom,
    ! [B: $tType,D6: $tType] :
      ( topolo1898628316856586783d_mult(B)
     => ! [Uu: fun(D6,B),Uua: fun(D6,B),Uub: D6] : aa(D6,B,aa(fun(D6,B),fun(D6,B),aTP_Lamp_yb(fun(D6,B),fun(fun(D6,B),fun(D6,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(D6,B,Uu,Uub)),aa(D6,B,Uua,Uub)) ) ).

% ATP.lambda_557
tff(fact_8736_ATP_Olambda__558,axiom,
    ! [A: $tType,D6: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [Uu: fun(D6,A),Uua: fun(D6,A),Uub: D6] : aa(D6,A,aa(fun(D6,A),fun(D6,A),aTP_Lamp_xy(fun(D6,A),fun(fun(D6,A),fun(D6,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(D6,A,Uu,Uub)),aa(D6,A,Uua,Uub)) ) ).

% ATP.lambda_558
tff(fact_8737_ATP_Olambda__559,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: fun(B,A),Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_aec(fun(B,A),fun(fun(B,A),fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(B,A,Uu,Uub)),aa(B,A,Uua,Uub)) ) ).

% ATP.lambda_559
tff(fact_8738_ATP_Olambda__560,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo4211221413907600880p_mult(A)
     => ! [Uu: fun(B,A),Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_yc(fun(B,A),fun(fun(B,A),fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(B,A,Uu,Uub)),aa(B,A,Uua,Uub)) ) ).

% ATP.lambda_560
tff(fact_8739_ATP_Olambda__561,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(B,A),Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_hf(fun(B,A),fun(fun(B,A),fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(B,A,Uu,Uub)),aa(B,A,Uua,Uub)) ) ).

% ATP.lambda_561
tff(fact_8740_ATP_Olambda__562,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V4412858255891104859lgebra(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_ain(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_562
tff(fact_8741_ATP_Olambda__563,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: fun(A,A),Uua: fun(A,A),Uub: A] : aa(A,A,aa(fun(A,A),fun(A,A),aTP_Lamp_rh(fun(A,A),fun(fun(A,A),fun(A,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,Uu,Uub)),aa(A,A,Uua,Uub)) ) ).

% ATP.lambda_563
tff(fact_8742_ATP_Olambda__564,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V4412858255891104859lgebra(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_zk(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_564
tff(fact_8743_ATP_Olambda__565,axiom,
    ! [A: $tType,Uu: fun(A,real),Uua: fun(A,real),Uub: A] : aa(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_adq(fun(A,real),fun(fun(A,real),fun(A,real)),Uu),Uua),Uub) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(A,real,Uu,Uub)),aa(A,real,Uua,Uub)) ).

% ATP.lambda_565
tff(fact_8744_ATP_Olambda__566,axiom,
    ! [A: $tType,Uu: fun(A,real),Uua: fun(A,real),Uub: A] : aa(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_aea(fun(A,real),fun(fun(A,real),fun(A,real)),Uu),Uua),Uub) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(A,real,Uua,Uub)),aa(A,real,Uu,Uub)) ).

% ATP.lambda_566
tff(fact_8745_ATP_Olambda__567,axiom,
    ! [B: $tType,A: $tType] :
      ( linordered_idom(B)
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_eh(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uua,Uub)),aa(A,B,Uu,Uub)) ) ).

% ATP.lambda_567
tff(fact_8746_ATP_Olambda__568,axiom,
    ! [Uu: fun(nat,real),Uua: fun(nat,real),Uub: nat] : aa(nat,real,aa(fun(nat,real),fun(nat,real),aTP_Lamp_abb(fun(nat,real),fun(fun(nat,real),fun(nat,real)),Uu),Uua),Uub) = aa(real,real,minus_minus(real,aa(nat,real,Uu,Uub)),aa(nat,real,Uua,Uub)) ).

% ATP.lambda_568
tff(fact_8747_ATP_Olambda__569,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(nat,A),Uua: fun(nat,A),Uub: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_cf(fun(nat,A),fun(fun(nat,A),fun(nat,A)),Uu),Uua),Uub) = aa(A,A,minus_minus(A,aa(nat,A,Uu,Uub)),aa(nat,A,Uua,Uub)) ) ).

% ATP.lambda_569
tff(fact_8748_ATP_Olambda__570,axiom,
    ! [B: $tType,D6: $tType] :
      ( ( topolo4958980785337419405_space(D6)
        & topolo1633459387980952147up_add(B) )
     => ! [Uu: fun(D6,B),Uua: fun(D6,B),Uub: D6] : aa(D6,B,aa(fun(D6,B),fun(D6,B),aTP_Lamp_ajd(fun(D6,B),fun(fun(D6,B),fun(D6,B)),Uu),Uua),Uub) = aa(B,B,minus_minus(B,aa(D6,B,Uu,Uub)),aa(D6,B,Uua,Uub)) ) ).

% ATP.lambda_570
tff(fact_8749_ATP_Olambda__571,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(B,A),Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_wo(fun(B,A),fun(fun(B,A),fun(B,A)),Uu),Uua),Uub) = aa(A,A,minus_minus(A,aa(B,A,Uu,Uub)),aa(B,A,Uua,Uub)) ) ).

% ATP.lambda_571
tff(fact_8750_ATP_Olambda__572,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo1633459387980952147up_add(A)
     => ! [Uu: fun(B,A),Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_wr(fun(B,A),fun(fun(B,A),fun(B,A)),Uu),Uua),Uub) = aa(A,A,minus_minus(A,aa(B,A,Uu,Uub)),aa(B,A,Uua,Uub)) ) ).

% ATP.lambda_572
tff(fact_8751_ATP_Olambda__573,axiom,
    ! [A: $tType,B: $tType] :
      ( ab_group_add(A)
     => ! [Uu: fun(B,A),Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_do(fun(B,A),fun(fun(B,A),fun(B,A)),Uu),Uua),Uub) = aa(A,A,minus_minus(A,aa(B,A,Uu,Uub)),aa(B,A,Uua,Uub)) ) ).

% ATP.lambda_573
tff(fact_8752_ATP_Olambda__574,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_tm(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,minus_minus(B,aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_574
tff(fact_8753_ATP_Olambda__575,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: fun(A,A),Uua: fun(A,A),Uub: A] : aa(A,A,aa(fun(A,A),fun(A,A),aTP_Lamp_rb(fun(A,A),fun(fun(A,A),fun(A,A)),Uu),Uua),Uub) = aa(A,A,minus_minus(A,aa(A,A,Uu,Uub)),aa(A,A,Uua,Uub)) ) ).

% ATP.lambda_575
tff(fact_8754_ATP_Olambda__576,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_zl(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,minus_minus(B,aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_576
tff(fact_8755_ATP_Olambda__577,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo1633459387980952147up_add(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_ws(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,minus_minus(B,aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_577
tff(fact_8756_ATP_Olambda__578,axiom,
    ! [A: $tType] :
      ( ( topolo1287966508704411220up_add(A)
        & topological_t2_space(A) )
     => ! [Uu: fun(nat,A),Uua: fun(nat,A),Uub: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_fg(fun(nat,A),fun(fun(nat,A),fun(nat,A)),Uu),Uua),Uub) = aa(A,A,minus_minus(A,aa(nat,A,Uua,Uub)),aa(nat,A,Uu,Uub)) ) ).

% ATP.lambda_578
tff(fact_8757_ATP_Olambda__579,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(B,A),Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_wq(fun(B,A),fun(fun(B,A),fun(B,A)),Uu),Uua),Uub) = aa(A,A,minus_minus(A,aa(B,A,Uua,Uub)),aa(B,A,Uu,Uub)) ) ).

% ATP.lambda_579
tff(fact_8758_ATP_Olambda__580,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Uu: fun(A,nat),Uua: fun(A,nat),Uub: A] : aa(A,nat,aa(fun(A,nat),fun(A,nat),aTP_Lamp_eu(fun(A,nat),fun(fun(A,nat),fun(A,nat)),Uu),Uua),Uub) = aa(nat,nat,minus_minus(nat,aa(A,nat,Uua,Uub)),aa(A,nat,Uu,Uub)) ) ).

% ATP.lambda_580
tff(fact_8759_ATP_Olambda__581,axiom,
    ! [A: $tType,Uu: fun(A,nat),Uua: fun(A,nat),Uub: A] : aa(A,nat,aa(fun(A,nat),fun(A,nat),aTP_Lamp_ez(fun(A,nat),fun(fun(A,nat),fun(A,nat)),Uu),Uua),Uub) = aa(nat,nat,minus_minus(nat,aa(A,nat,Uua,Uub)),aa(A,nat,Uu,Uub)) ).

% ATP.lambda_581
tff(fact_8760_ATP_Olambda__582,axiom,
    ! [B: $tType,C: $tType] :
      ( ( topolo4958980785337419405_space(C)
        & topolo1898628316856586783d_mult(B) )
     => ! [Uu: fun(C,B),Uua: fun(C,nat),Uub: C] : aa(C,B,aa(fun(C,nat),fun(C,B),aTP_Lamp_aiy(fun(C,B),fun(fun(C,nat),fun(C,B)),Uu),Uua),Uub) = aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(C,B,Uu,Uub)),aa(C,nat,Uua,Uub)) ) ).

% ATP.lambda_582
tff(fact_8761_ATP_Olambda__583,axiom,
    ! [B: $tType,C: $tType] :
      ( ( topological_t2_space(C)
        & topolo1898628316856586783d_mult(B) )
     => ! [Uu: fun(C,B),Uua: fun(C,nat),Uub: C] : aa(C,B,aa(fun(C,nat),fun(C,B),aTP_Lamp_wh(fun(C,B),fun(fun(C,nat),fun(C,B)),Uu),Uua),Uub) = aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(C,B,Uu,Uub)),aa(C,nat,Uua,Uub)) ) ).

% ATP.lambda_583
tff(fact_8762_ATP_Olambda__584,axiom,
    ! [B: $tType,C: $tType] :
      ( topolo1898628316856586783d_mult(B)
     => ! [Uu: fun(C,B),Uua: fun(C,nat),Uub: C] : aa(C,B,aa(fun(C,nat),fun(C,B),aTP_Lamp_wg(fun(C,B),fun(fun(C,nat),fun(C,B)),Uu),Uua),Uub) = aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(C,B,Uu,Uub)),aa(C,nat,Uua,Uub)) ) ).

% ATP.lambda_584
tff(fact_8763_ATP_Olambda__585,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [Uu: fun(B,A),Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_yp(fun(B,A),fun(fun(B,A),fun(B,A)),Uu),Uua),Uub) = ord_min(A,aa(B,A,Uu,Uub),aa(B,A,Uua,Uub)) ) ).

% ATP.lambda_585
tff(fact_8764_ATP_Olambda__586,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topolo1944317154257567458pology(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_ajz(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = ord_min(B,aa(A,B,Uu,Uub),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_586
tff(fact_8765_ATP_Olambda__587,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo1944317154257567458pology(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_yq(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = ord_min(B,aa(A,B,Uu,Uub),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_587
tff(fact_8766_ATP_Olambda__588,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [Uu: fun(B,A),Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_xs(fun(B,A),fun(fun(B,A),fun(B,A)),Uu),Uua),Uub) = aa(A,A,ord_max(A,aa(B,A,Uu,Uub)),aa(B,A,Uua,Uub)) ) ).

% ATP.lambda_588
tff(fact_8767_ATP_Olambda__589,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topolo1944317154257567458pology(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_ajr(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,ord_max(B,aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_589
tff(fact_8768_ATP_Olambda__590,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo1944317154257567458pology(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_xr(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,ord_max(B,aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_590
tff(fact_8769_ATP_Olambda__591,axiom,
    ! [A: $tType,Uu: fun(A,real),Uua: fun(A,real),Uub: A] : aa(A,complex,aa(fun(A,real),fun(A,complex),aTP_Lamp_wy(fun(A,real),fun(fun(A,real),fun(A,complex)),Uu),Uua),Uub) = complex2(aa(A,real,Uu,Uub),aa(A,real,Uua,Uub)) ).

% ATP.lambda_591
tff(fact_8770_ATP_Olambda__592,axiom,
    ! [A: $tType] :
      ( ( topolo5987344860129210374id_add(A)
        & topological_t2_space(A) )
     => ! [Uu: fun(nat,A),Uua: fun(nat,A),Uub: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_ci(fun(nat,A),fun(fun(nat,A),fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,Uu,Uub)),aa(nat,A,Uua,Uub)) ) ).

% ATP.lambda_592
tff(fact_8771_ATP_Olambda__593,axiom,
    ! [B: $tType,D6: $tType] :
      ( ( topolo4958980785337419405_space(D6)
        & topolo6943815403480290642id_add(B) )
     => ! [Uu: fun(D6,B),Uua: fun(D6,B),Uub: D6] : aa(D6,B,aa(fun(D6,B),fun(D6,B),aTP_Lamp_ajs(fun(D6,B),fun(fun(D6,B),fun(D6,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(D6,B,Uu,Uub)),aa(D6,B,Uua,Uub)) ) ).

% ATP.lambda_593
tff(fact_8772_ATP_Olambda__594,axiom,
    ! [B: $tType,D6: $tType] :
      ( ( topological_t2_space(D6)
        & topolo6943815403480290642id_add(B) )
     => ! [Uu: fun(D6,B),Uua: fun(D6,B),Uub: D6] : aa(D6,B,aa(fun(D6,B),fun(D6,B),aTP_Lamp_xw(fun(D6,B),fun(fun(D6,B),fun(D6,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(D6,B,Uu,Uub)),aa(D6,B,Uua,Uub)) ) ).

% ATP.lambda_594
tff(fact_8773_ATP_Olambda__595,axiom,
    ! [B: $tType,D6: $tType] :
      ( topolo6943815403480290642id_add(B)
     => ! [Uu: fun(D6,B),Uua: fun(D6,B),Uub: D6] : aa(D6,B,aa(fun(D6,B),fun(D6,B),aTP_Lamp_xx(fun(D6,B),fun(fun(D6,B),fun(D6,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(D6,B,Uu,Uub)),aa(D6,B,Uua,Uub)) ) ).

% ATP.lambda_595
tff(fact_8774_ATP_Olambda__596,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo6943815403480290642id_add(A)
     => ! [Uu: fun(B,A),Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_xu(fun(B,A),fun(fun(B,A),fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(B,A,Uu,Uub)),aa(B,A,Uua,Uub)) ) ).

% ATP.lambda_596
tff(fact_8775_ATP_Olambda__597,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(B,A),Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_dj(fun(B,A),fun(fun(B,A),fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(B,A,Uu,Uub)),aa(B,A,Uua,Uub)) ) ).

% ATP.lambda_597
tff(fact_8776_ATP_Olambda__598,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_tf(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_598
tff(fact_8777_ATP_Olambda__599,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: fun(A,A),Uua: fun(A,A),Uub: A] : aa(A,A,aa(fun(A,A),fun(A,A),aTP_Lamp_rf(fun(A,A),fun(fun(A,A),fun(A,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,Uu,Uub)),aa(A,A,Uua,Uub)) ) ).

% ATP.lambda_599
tff(fact_8778_ATP_Olambda__600,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo6943815403480290642id_add(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_zj(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_600
tff(fact_8779_ATP_Olambda__601,axiom,
    ! [A: $tType,Uu: fun(A,real),Uua: fun(A,real),Uub: A] : aa(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_adv(fun(A,real),fun(fun(A,real),fun(A,real)),Uu),Uua),Uub) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(A,real,Uu,Uub)),aa(A,real,Uua,Uub)) ).

% ATP.lambda_601
tff(fact_8780_ATP_Olambda__602,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_adx(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_602
tff(fact_8781_ATP_Olambda__603,axiom,
    ! [Uu: fun(real,real),Uua: fun(real,real),Uub: real] : aa(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_sh(fun(real,real),fun(fun(real,real),fun(real,real)),Uu),Uua),Uub) = powr(real,aa(real,real,Uu,Uub),aa(real,real,Uua,Uub)) ).

% ATP.lambda_603
tff(fact_8782_ATP_Olambda__604,axiom,
    ! [C: $tType] :
      ( topolo4958980785337419405_space(C)
     => ! [Uu: fun(C,real),Uua: fun(C,real),Uub: C] : aa(C,real,aa(fun(C,real),fun(C,real),aTP_Lamp_aiw(fun(C,real),fun(fun(C,real),fun(C,real)),Uu),Uua),Uub) = powr(real,aa(C,real,Uu,Uub),aa(C,real,Uua,Uub)) ) ).

% ATP.lambda_604
tff(fact_8783_ATP_Olambda__605,axiom,
    ! [C: $tType] :
      ( topological_t2_space(C)
     => ! [Uu: fun(C,real),Uua: fun(C,real),Uub: C] : aa(C,real,aa(fun(C,real),fun(C,real),aTP_Lamp_aao(fun(C,real),fun(fun(C,real),fun(C,real)),Uu),Uua),Uub) = powr(real,aa(C,real,Uu,Uub),aa(C,real,Uua,Uub)) ) ).

% ATP.lambda_605
tff(fact_8784_ATP_Olambda__606,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,real),Uua: fun(A,real),Uub: A] : aa(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_ur(fun(A,real),fun(fun(A,real),fun(A,real)),Uu),Uua),Uub) = powr(real,aa(A,real,Uu,Uub),aa(A,real,Uua,Uub)) ) ).

% ATP.lambda_606
tff(fact_8785_ATP_Olambda__607,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Uu: fun(A,real),Uua: fun(A,real),Uub: A] : aa(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_acz(fun(A,real),fun(fun(A,real),fun(A,real)),Uu),Uua),Uub) = powr(real,aa(A,real,Uu,Uub),aa(A,real,Uua,Uub)) ) ).

% ATP.lambda_607
tff(fact_8786_ATP_Olambda__608,axiom,
    ! [A: $tType,Uu: fun(A,real),Uua: fun(A,real),Uub: A] : aa(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_xm(fun(A,real),fun(fun(A,real),fun(A,real)),Uu),Uua),Uub) = powr(real,aa(A,real,Uu,Uub),aa(A,real,Uua,Uub)) ).

% ATP.lambda_608
tff(fact_8787_ATP_Olambda__609,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: fun(A,real),Uua: fun(A,real),Uub: A] : aa(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_akj(fun(A,real),fun(fun(A,real),fun(A,real)),Uu),Uua),Uub) = aa(real,real,log(aa(A,real,Uu,Uub)),aa(A,real,Uua,Uub)) ) ).

% ATP.lambda_609
tff(fact_8788_ATP_Olambda__610,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Uu: fun(A,real),Uua: fun(A,real),Uub: A] : aa(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_abj(fun(A,real),fun(fun(A,real),fun(A,real)),Uu),Uua),Uub) = aa(real,real,log(aa(A,real,Uu,Uub)),aa(A,real,Uua,Uub)) ) ).

% ATP.lambda_610
tff(fact_8789_ATP_Olambda__611,axiom,
    ! [A: $tType,Uu: fun(A,real),Uua: fun(A,real),Uub: A] : aa(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_xt(fun(A,real),fun(fun(A,real),fun(A,real)),Uu),Uua),Uub) = aa(real,real,log(aa(A,real,Uu,Uub)),aa(A,real,Uua,Uub)) ).

% ATP.lambda_611
tff(fact_8790_ATP_Olambda__612,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topolo4958980785337419405_space(C)
        & topolo4958980785337419405_space(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,C),Uub: A] : aa(A,product_prod(B,C),aa(fun(A,C),fun(A,product_prod(B,C)),aTP_Lamp_ajy(fun(A,B),fun(fun(A,C),fun(A,product_prod(B,C))),Uu),Uua),Uub) = aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),aa(A,B,Uu,Uub)),aa(A,C,Uua,Uub)) ) ).

% ATP.lambda_612
tff(fact_8791_ATP_Olambda__613,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo4958980785337419405_space(C)
        & topolo4958980785337419405_space(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,C),Uub: A] : aa(A,product_prod(B,C),aa(fun(A,C),fun(A,product_prod(B,C)),aTP_Lamp_vp(fun(A,B),fun(fun(A,C),fun(A,product_prod(B,C))),Uu),Uua),Uub) = aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),aa(A,B,Uu,Uub)),aa(A,C,Uua,Uub)) ) ).

% ATP.lambda_613
tff(fact_8792_ATP_Olambda__614,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(C)
        & topolo4958980785337419405_space(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,C),Uub: A] : aa(A,product_prod(B,C),aa(fun(A,C),fun(A,product_prod(B,C)),aTP_Lamp_yn(fun(A,B),fun(fun(A,C),fun(A,product_prod(B,C))),Uu),Uua),Uub) = aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),aa(A,B,Uu,Uub)),aa(A,C,Uua,Uub)) ) ).

% ATP.lambda_614
tff(fact_8793_ATP_Olambda__615,axiom,
    ! [A: $tType,Uu: fun(A,bool),Uua: fun(A,bool),Uub: A] :
      ( pp(aa(A,bool,aa(fun(A,bool),fun(A,bool),aTP_Lamp_afo(fun(A,bool),fun(fun(A,bool),fun(A,bool)),Uu),Uua),Uub))
    <=> ( pp(aa(A,bool,Uu,Uub))
       => pp(aa(A,bool,Uua,Uub)) ) ) ).

% ATP.lambda_615
tff(fact_8794_ATP_Olambda__616,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: fun(A,bool),Uua: fun(A,bool),Uub: A] :
          ( pp(aa(A,bool,aa(fun(A,bool),fun(A,bool),aTP_Lamp_acw(fun(A,bool),fun(fun(A,bool),fun(A,bool)),Uu),Uua),Uub))
        <=> ( pp(aa(A,bool,Uu,Uub))
            | pp(aa(A,bool,Uua,Uub)) ) ) ) ).

% ATP.lambda_616
tff(fact_8795_ATP_Olambda__617,axiom,
    ! [A: $tType,Uu: fun(A,bool),Uua: fun(A,bool),Uub: A] :
      ( pp(aa(A,bool,aa(fun(A,bool),fun(A,bool),aTP_Lamp_ar(fun(A,bool),fun(fun(A,bool),fun(A,bool)),Uu),Uua),Uub))
    <=> ( pp(aa(A,bool,Uu,Uub))
        | pp(aa(A,bool,Uua,Uub)) ) ) ).

% ATP.lambda_617
tff(fact_8796_ATP_Olambda__618,axiom,
    ! [B: $tType,Uu: fun(B,bool),Uua: fun(B,bool),Uub: B] :
      ( pp(aa(B,bool,aa(fun(B,bool),fun(B,bool),aTP_Lamp_sj(fun(B,bool),fun(fun(B,bool),fun(B,bool)),Uu),Uua),Uub))
    <=> ( pp(aa(B,bool,Uu,Uub))
        & pp(aa(B,bool,Uua,Uub)) ) ) ).

% ATP.lambda_618
tff(fact_8797_ATP_Olambda__619,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: fun(A,bool),Uua: fun(A,bool),Uub: A] :
          ( pp(aa(A,bool,aa(fun(A,bool),fun(A,bool),aTP_Lamp_acv(fun(A,bool),fun(fun(A,bool),fun(A,bool)),Uu),Uua),Uub))
        <=> ( pp(aa(A,bool,Uu,Uub))
            & pp(aa(A,bool,Uua,Uub)) ) ) ) ).

% ATP.lambda_619
tff(fact_8798_ATP_Olambda__620,axiom,
    ! [A: $tType,Uu: fun(A,bool),Uua: fun(A,bool),Uub: A] :
      ( pp(aa(A,bool,aa(fun(A,bool),fun(A,bool),aTP_Lamp_as(fun(A,bool),fun(fun(A,bool),fun(A,bool)),Uu),Uua),Uub))
    <=> ( pp(aa(A,bool,Uu,Uub))
        & pp(aa(A,bool,Uua,Uub)) ) ) ).

% ATP.lambda_620
tff(fact_8799_ATP_Olambda__621,axiom,
    ! [A: $tType,Uu: fun(A,bool),Uua: fun(A,bool),Uub: A] :
      ( pp(aa(A,bool,aa(fun(A,bool),fun(A,bool),aTP_Lamp_ane(fun(A,bool),fun(fun(A,bool),fun(A,bool)),Uu),Uua),Uub))
    <=> ( pp(aa(A,bool,Uua,Uub))
        & pp(aa(A,bool,Uu,Uub)) ) ) ).

% ATP.lambda_621
tff(fact_8800_ATP_Olambda__622,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,B),Uua: A,Uub: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_anh(fun(A,B),fun(A,fun(A,bool)),Uu),Uua),Uub))
    <=> ( aa(A,B,Uu,Uua) = aa(A,B,Uu,Uub) ) ) ).

% ATP.lambda_622
tff(fact_8801_ATP_Olambda__623,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(nat,A),Uua: fun(nat,A),Uub: nat] :
          ( pp(aa(nat,bool,aa(fun(nat,A),fun(nat,bool),aTP_Lamp_aet(fun(nat,A),fun(fun(nat,A),fun(nat,bool)),Uu),Uua),Uub))
        <=> ( aa(nat,A,Uu,Uub) = aa(nat,A,Uua,Uub) ) ) ) ).

% ATP.lambda_623
tff(fact_8802_ATP_Olambda__624,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: fun(B,A),Uua: fun(B,A),Uub: B] :
          ( pp(aa(B,bool,aa(fun(B,A),fun(B,bool),aTP_Lamp_afb(fun(B,A),fun(fun(B,A),fun(B,bool)),Uu),Uua),Uub))
        <=> ( aa(B,A,Uu,Uub) = aa(B,A,Uua,Uub) ) ) ) ).

% ATP.lambda_624
tff(fact_8803_ATP_Olambda__625,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,A),Uua: fun(B,A),Uub: B] :
      ( pp(aa(B,bool,aa(fun(B,A),fun(B,bool),aTP_Lamp_aex(fun(B,A),fun(fun(B,A),fun(B,bool)),Uu),Uua),Uub))
    <=> ( aa(B,A,Uu,Uub) = aa(B,A,Uua,Uub) ) ) ).

% ATP.lambda_625
tff(fact_8804_ATP_Olambda__626,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] :
          ( pp(aa(A,bool,aa(fun(A,B),fun(A,bool),aTP_Lamp_afq(fun(A,B),fun(fun(A,B),fun(A,bool)),Uu),Uua),Uub))
        <=> ( aa(A,B,Uu,Uub) = aa(A,B,Uua,Uub) ) ) ) ).

% ATP.lambda_626
tff(fact_8805_ATP_Olambda__627,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: fun(A,A),Uua: fun(A,A),Uub: A] :
          ( pp(aa(A,bool,aa(fun(A,A),fun(A,bool),aTP_Lamp_afr(fun(A,A),fun(fun(A,A),fun(A,bool)),Uu),Uua),Uub))
        <=> ( aa(A,A,Uu,Uub) = aa(A,A,Uua,Uub) ) ) ) ).

% ATP.lambda_627
tff(fact_8806_ATP_Olambda__628,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo4958980785337419405_space(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] :
          ( pp(aa(A,bool,aa(fun(A,B),fun(A,bool),aTP_Lamp_ago(fun(A,B),fun(fun(A,B),fun(A,bool)),Uu),Uua),Uub))
        <=> ( aa(A,B,Uu,Uub) = aa(A,B,Uua,Uub) ) ) ) ).

% ATP.lambda_628
tff(fact_8807_ATP_Olambda__629,axiom,
    ! [A: $tType,Uu: fun(A,bool),Uua: fun(A,bool),Uub: A] :
      ( pp(aa(A,bool,aa(fun(A,bool),fun(A,bool),aTP_Lamp_afp(fun(A,bool),fun(fun(A,bool),fun(A,bool)),Uu),Uua),Uub))
    <=> ( pp(aa(A,bool,Uu,Uub))
      <=> pp(aa(A,bool,Uua,Uub)) ) ) ).

% ATP.lambda_629
tff(fact_8808_ATP_Olambda__630,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo4958980785337419405_space(B)
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] :
          ( pp(aa(A,bool,aa(fun(A,B),fun(A,bool),aTP_Lamp_aez(fun(A,B),fun(fun(A,B),fun(A,bool)),Uu),Uua),Uub))
        <=> ( aa(A,B,Uu,Uub) = aa(A,B,Uua,Uub) ) ) ) ).

% ATP.lambda_630
tff(fact_8809_ATP_Olambda__631,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,B),Uua: fun(A,B),Uub: A] :
      ( pp(aa(A,bool,aa(fun(A,B),fun(A,bool),aTP_Lamp_afg(fun(A,B),fun(fun(A,B),fun(A,bool)),Uu),Uua),Uub))
    <=> ( aa(A,B,Uu,Uub) = aa(A,B,Uua,Uub) ) ) ).

% ATP.lambda_631
tff(fact_8810_ATP_Olambda__632,axiom,
    ! [A: $tType,B: $tType] :
      ( ( archim2362893244070406136eiling(B)
        & topolo2564578578187576103pology(B) )
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( pp(aa(A,bool,aa(B,fun(A,bool),aTP_Lamp_ahn(fun(A,B),fun(B,fun(A,bool)),Uu),Uua),Uub))
        <=> pp(aa(B,bool,ord_less(B,aa(A,B,Uu,Uub)),aa(int,B,ring_1_of_int(B),archimedean_ceiling(B,Uua)))) ) ) ).

% ATP.lambda_632
tff(fact_8811_ATP_Olambda__633,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: fun(A,A),Uua: nat,Uub: A] : aa(A,A,aa(nat,fun(A,A),aTP_Lamp_rz(fun(A,A),fun(nat,fun(A,A)),Uu),Uua),Uub) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,Uu,Uub)),aa(nat,nat,suc,Uua)) ) ).

% ATP.lambda_633
tff(fact_8812_ATP_Olambda__634,axiom,
    ! [C: $tType,A: $tType] :
      ( ( real_V3459762299906320749_field(A)
        & topolo4958980785337419405_space(C) )
     => ! [Uu: fun(C,A),Uua: nat,Uub: C] : aa(C,A,aa(nat,fun(C,A),aTP_Lamp_ajf(fun(C,A),fun(nat,fun(C,A)),Uu),Uua),Uub) = comm_s3205402744901411588hammer(A,aa(C,A,Uu,Uub),Uua) ) ).

% ATP.lambda_634
tff(fact_8813_ATP_Olambda__635,axiom,
    ! [B: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [Uu: fun(nat,real),Uua: B,Uub: nat] : aa(nat,B,aa(B,fun(nat,B),aTP_Lamp_kj(fun(nat,real),fun(B,fun(nat,B)),Uu),Uua),Uub) = aa(B,B,real_V8093663219630862766scaleR(B,aa(nat,real,Uu,Uub)),Uua) ) ).

% ATP.lambda_635
tff(fact_8814_ATP_Olambda__636,axiom,
    ! [C: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V822414075346904944vector(C) )
     => ! [Uu: fun(C,real),Uua: B,Uub: C] : aa(C,B,aa(B,fun(C,B),aTP_Lamp_tk(fun(C,real),fun(B,fun(C,B)),Uu),Uua),Uub) = aa(B,B,real_V8093663219630862766scaleR(B,aa(C,real,Uu,Uub)),Uua) ) ).

% ATP.lambda_636
tff(fact_8815_ATP_Olambda__637,axiom,
    ! [C: $tType,A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Uu: fun(C,real),Uua: A,Uub: C] : aa(C,A,aa(A,fun(C,A),aTP_Lamp_ki(fun(C,real),fun(A,fun(C,A)),Uu),Uua),Uub) = aa(A,A,real_V8093663219630862766scaleR(A,aa(C,real,Uu,Uub)),Uua) ) ).

% ATP.lambda_637
tff(fact_8816_ATP_Olambda__638,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(B,fun(C,A)),Uua: set(C),Uub: B] : aa(B,A,aa(set(C),fun(B,A),aTP_Lamp_hb(fun(B,fun(C,A)),fun(set(C),fun(B,A)),Uu),Uua),Uub) = groups7121269368397514597t_prod(C,A,aa(B,fun(C,A),Uu,Uub),Uua) ) ).

% ATP.lambda_638
tff(fact_8817_ATP_Olambda__639,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( topolo4987421752381908075d_mult(C)
     => ! [Uu: set(B),Uua: fun(A,fun(B,C)),Uub: A] : aa(A,C,aa(fun(A,fun(B,C)),fun(A,C),aTP_Lamp_ze(set(B),fun(fun(A,fun(B,C)),fun(A,C)),Uu),Uua),Uub) = groups7121269368397514597t_prod(B,C,aa(A,fun(B,C),Uua,Uub),Uu) ) ).

% ATP.lambda_639
tff(fact_8818_ATP_Olambda__640,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(B,fun(C,A)),Uua: set(C),Uub: B] : aa(B,A,aa(set(C),fun(B,A),aTP_Lamp_df(fun(B,fun(C,A)),fun(set(C),fun(B,A)),Uu),Uua),Uub) = aa(set(C),A,groups7311177749621191930dd_sum(C,A,aa(B,fun(C,A),Uu,Uub)),Uua) ) ).

% ATP.lambda_640
tff(fact_8819_ATP_Olambda__641,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( topolo5987344860129210374id_add(C)
     => ! [Uu: set(B),Uua: fun(A,fun(B,C)),Uub: A] : aa(A,C,aa(fun(A,fun(B,C)),fun(A,C),aTP_Lamp_zf(set(B),fun(fun(A,fun(B,C)),fun(A,C)),Uu),Uua),Uub) = aa(set(B),C,groups7311177749621191930dd_sum(B,C,aa(A,fun(B,C),Uua,Uub)),Uu) ) ).

% ATP.lambda_641
tff(fact_8820_ATP_Olambda__642,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V3459762299906320749_field(B)
     => ! [Uu: set(A),Uua: fun(B,fun(A,B)),Uub: B] : aa(B,B,aa(fun(B,fun(A,B)),fun(B,B),aTP_Lamp_ry(set(A),fun(fun(B,fun(A,B)),fun(B,B)),Uu),Uua),Uub) = aa(set(A),B,groups7311177749621191930dd_sum(A,B,aa(B,fun(A,B),Uua,Uub)),Uu) ) ).

% ATP.lambda_642
tff(fact_8821_ATP_Olambda__643,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V7819770556892013058_space(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] : aa(A,real,aa(B,fun(A,real),aTP_Lamp_adj(fun(A,B),fun(B,fun(A,real)),Uu),Uua),Uub) = real_V557655796197034286t_dist(B,aa(A,B,Uu,Uub),Uua) ) ).

% ATP.lambda_643
tff(fact_8822_ATP_Olambda__644,axiom,
    ! [A: $tType] :
      ( topolo3112930676232923870pology(A)
     => ! [Uu: fun(nat,set(A)),Uua: set(A),Uub: nat] :
          ( pp(aa(nat,bool,aa(set(A),fun(nat,bool),aTP_Lamp_agf(fun(nat,set(A)),fun(set(A),fun(nat,bool)),Uu),Uua),Uub))
        <=> pp(aa(set(A),bool,ord_less_eq(set(A),aa(nat,set(A),Uu,Uub)),Uua)) ) ) ).

% ATP.lambda_644
tff(fact_8823_ATP_Olambda__645,axiom,
    ! [Uu: fun(nat,nat),Uua: nat,Uub: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),aTP_Lamp_aj(fun(nat,nat),fun(nat,fun(nat,bool)),Uu),Uua),Uub))
    <=> pp(aa(nat,bool,ord_less_eq(nat,aa(nat,nat,Uu,Uub)),Uua)) ) ).

% ATP.lambda_645
tff(fact_8824_ATP_Olambda__646,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [Uu: fun(B,A),Uua: A,Uub: B] :
          ( pp(aa(B,bool,aa(A,fun(B,bool),aTP_Lamp_agq(fun(B,A),fun(A,fun(B,bool)),Uu),Uua),Uub))
        <=> pp(aa(A,bool,ord_less_eq(A,aa(B,A,Uu,Uub)),Uua)) ) ) ).

% ATP.lambda_646
tff(fact_8825_ATP_Olambda__647,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo2564578578187576103pology(A)
     => ! [Uu: fun(B,A),Uua: A,Uub: B] :
          ( pp(aa(B,bool,aa(A,fun(B,bool),aTP_Lamp_agi(fun(B,A),fun(A,fun(B,bool)),Uu),Uua),Uub))
        <=> pp(aa(A,bool,ord_less_eq(A,aa(B,A,Uu,Uub)),Uua)) ) ) ).

% ATP.lambda_647
tff(fact_8826_ATP_Olambda__648,axiom,
    ! [A: $tType,B: $tType] :
      ( unboun7993243217541854897norder(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( pp(aa(A,bool,aa(B,fun(A,bool),aTP_Lamp_agp(fun(A,B),fun(B,fun(A,bool)),Uu),Uua),Uub))
        <=> pp(aa(B,bool,ord_less_eq(B,aa(A,B,Uu,Uub)),Uua)) ) ) ).

% ATP.lambda_648
tff(fact_8827_ATP_Olambda__649,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( pp(aa(A,bool,aa(B,fun(A,bool),aTP_Lamp_agc(fun(A,B),fun(B,fun(A,bool)),Uu),Uua),Uub))
        <=> pp(aa(B,bool,ord_less_eq(B,aa(A,B,Uu,Uub)),Uua)) ) ) ).

% ATP.lambda_649
tff(fact_8828_ATP_Olambda__650,axiom,
    ! [B: $tType,A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [Uu: fun(B,A),Uua: A,Uub: B] : aa(B,A,aa(A,fun(B,A),aTP_Lamp_dv(fun(B,A),fun(A,fun(B,A)),Uu),Uua),Uub) = modulo_modulo(A,aa(B,A,Uu,Uub),Uua) ) ).

% ATP.lambda_650
tff(fact_8829_ATP_Olambda__651,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_bv(fun(nat,A),fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,Uu,Uub)),Uua) ) ).

% ATP.lambda_651
tff(fact_8830_ATP_Olambda__652,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: fun(B,A),Uua: A,Uub: B] : aa(B,A,aa(A,fun(B,A),aTP_Lamp_yj(fun(B,A),fun(A,fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(B,A,Uu,Uub)),Uua) ) ).

% ATP.lambda_652
tff(fact_8831_ATP_Olambda__653,axiom,
    ! [B: $tType,A: $tType] :
      ( field(A)
     => ! [Uu: fun(B,A),Uua: A,Uub: B] : aa(B,A,aa(A,fun(B,A),aTP_Lamp_dp(fun(B,A),fun(A,fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(B,A,Uu,Uub)),Uua) ) ).

% ATP.lambda_653
tff(fact_8832_ATP_Olambda__654,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: fun(A,A),Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_rm(fun(A,A),fun(A,fun(A,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,Uu,Uub)),Uua) ) ).

% ATP.lambda_654
tff(fact_8833_ATP_Olambda__655,axiom,
    ! [A: $tType] :
      ( ( field(A)
        & topolo4211221413907600880p_mult(A) )
     => ! [Uu: A,Uua: fun(nat,A),Uub: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_aae(A,fun(fun(nat,A),fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,Uua,Uub)),Uu) ) ).

% ATP.lambda_655
tff(fact_8834_ATP_Olambda__656,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo2564578578187576103pology(A)
     => ! [Uu: fun(B,A),Uua: A,Uub: B] :
          ( pp(aa(B,bool,aa(A,fun(B,bool),aTP_Lamp_aes(fun(B,A),fun(A,fun(B,bool)),Uu),Uua),Uub))
        <=> pp(aa(A,bool,ord_less(A,aa(B,A,Uu,Uub)),Uua)) ) ) ).

% ATP.lambda_656
tff(fact_8835_ATP_Olambda__657,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo2564578578187576103pology(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( pp(aa(A,bool,aa(B,fun(A,bool),aTP_Lamp_ahd(fun(A,B),fun(B,fun(A,bool)),Uu),Uua),Uub))
        <=> pp(aa(B,bool,ord_less(B,aa(A,B,Uu,Uub)),Uua)) ) ) ).

% ATP.lambda_657
tff(fact_8836_ATP_Olambda__658,axiom,
    ! [A: $tType,B: $tType] :
      ( ( dense_linorder(B)
        & no_bot(B) )
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( pp(aa(A,bool,aa(B,fun(A,bool),aTP_Lamp_agd(fun(A,B),fun(B,fun(A,bool)),Uu),Uua),Uub))
        <=> pp(aa(B,bool,ord_less(B,aa(A,B,Uu,Uub)),Uua)) ) ) ).

% ATP.lambda_658
tff(fact_8837_ATP_Olambda__659,axiom,
    ! [A: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [Uu: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_cc(fun(nat,A),fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,Uub)),Uua) ) ).

% ATP.lambda_659
tff(fact_8838_ATP_Olambda__660,axiom,
    ! [D6: $tType,A: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [Uu: fun(D6,A),Uua: A,Uub: D6] : aa(D6,A,aa(A,fun(D6,A),aTP_Lamp_xz(fun(D6,A),fun(A,fun(D6,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(D6,A,Uu,Uub)),Uua) ) ).

% ATP.lambda_660
tff(fact_8839_ATP_Olambda__661,axiom,
    ! [C: $tType,A: $tType] :
      ( ( real_V4412858255891104859lgebra(A)
        & real_V822414075346904944vector(C) )
     => ! [Uu: fun(C,A),Uua: A,Uub: C] : aa(C,A,aa(A,fun(C,A),aTP_Lamp_td(fun(C,A),fun(A,fun(C,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(C,A,Uu,Uub)),Uua) ) ).

% ATP.lambda_661
tff(fact_8840_ATP_Olambda__662,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V3459762299906320749_field(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu: fun(B,A),Uua: A,Uub: B] : aa(B,A,aa(A,fun(B,A),aTP_Lamp_aim(fun(B,A),fun(A,fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(B,A,Uu,Uub)),Uua) ) ).

% ATP.lambda_662
tff(fact_8841_ATP_Olambda__663,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V4412858255891104859lgebra(A)
        & topolo4958980785337419405_space(B) )
     => ! [Uu: fun(B,A),Uua: A,Uub: B] : aa(B,A,aa(A,fun(B,A),aTP_Lamp_aju(fun(B,A),fun(A,fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(B,A,Uu,Uub)),Uua) ) ).

% ATP.lambda_663
tff(fact_8842_ATP_Olambda__664,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V4412858255891104859lgebra(A)
        & topological_t2_space(B) )
     => ! [Uu: fun(B,A),Uua: A,Uub: B] : aa(B,A,aa(A,fun(B,A),aTP_Lamp_yf(fun(B,A),fun(A,fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(B,A,Uu,Uub)),Uua) ) ).

% ATP.lambda_664
tff(fact_8843_ATP_Olambda__665,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo4211221413907600880p_mult(A)
     => ! [Uu: fun(B,A),Uua: A,Uub: B] : aa(B,A,aa(A,fun(B,A),aTP_Lamp_ye(fun(B,A),fun(A,fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(B,A,Uu,Uub)),Uua) ) ).

% ATP.lambda_665
tff(fact_8844_ATP_Olambda__666,axiom,
    ! [B: $tType,A: $tType] :
      ( semiring_0(A)
     => ! [Uu: fun(B,A),Uua: A,Uub: B] : aa(B,A,aa(A,fun(B,A),aTP_Lamp_dl(fun(B,A),fun(A,fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(B,A,Uu,Uub)),Uua) ) ).

% ATP.lambda_666
tff(fact_8845_ATP_Olambda__667,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: fun(A,A),Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_rl(fun(A,A),fun(A,fun(A,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,Uu,Uub)),Uua) ) ).

% ATP.lambda_667
tff(fact_8846_ATP_Olambda__668,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: real,Uua: fun(A,real),Uub: A] : aa(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_ue(real,fun(fun(A,real),fun(A,real)),Uu),Uua),Uub) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(A,real,Uua,Uub)),Uu) ) ).

% ATP.lambda_668
tff(fact_8847_ATP_Olambda__669,axiom,
    ! [A: $tType] :
      ( ( field(A)
        & real_V4412858255891104859lgebra(A) )
     => ! [Uu: A,Uua: fun(nat,A),Uub: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_ey(A,fun(fun(nat,A),fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uua,Uub)),Uu) ) ).

% ATP.lambda_669
tff(fact_8848_ATP_Olambda__670,axiom,
    ! [A: $tType] :
      ( ( field(A)
        & topolo4211221413907600880p_mult(A) )
     => ! [Uu: A,Uua: fun(nat,A),Uub: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_aad(A,fun(fun(nat,A),fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uua,Uub)),Uu) ) ).

% ATP.lambda_670
tff(fact_8849_ATP_Olambda__671,axiom,
    ! [B: $tType,A: $tType] :
      ( ( field(A)
        & topolo4211221413907600880p_mult(A) )
     => ! [Uu: A,Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_vk(A,fun(fun(B,A),fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(B,A,Uua,Uub)),Uu) ) ).

% ATP.lambda_671
tff(fact_8850_ATP_Olambda__672,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Uu: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_fj(fun(nat,A),fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,minus_minus(A,aa(nat,A,Uu,Uub)),Uua) ) ).

% ATP.lambda_672
tff(fact_8851_ATP_Olambda__673,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] : aa(A,B,aa(B,fun(A,B),aTP_Lamp_wp(fun(A,B),fun(B,fun(A,B)),Uu),Uua),Uub) = aa(B,B,minus_minus(B,aa(A,B,Uu,Uub)),Uua) ) ).

% ATP.lambda_673
tff(fact_8852_ATP_Olambda__674,axiom,
    ! [Uu: fun(real,real),Uua: nat,Uub: real] : aa(real,real,aa(nat,fun(real,real),aTP_Lamp_sd(fun(real,real),fun(nat,fun(real,real)),Uu),Uua),Uub) = aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,Uu,Uub)),Uua) ).

% ATP.lambda_674
tff(fact_8853_ATP_Olambda__675,axiom,
    ! [C: $tType,B: $tType] :
      ( ( power(B)
        & real_V4412858255891104859lgebra(B)
        & topolo4958980785337419405_space(C) )
     => ! [Uu: fun(C,B),Uua: nat,Uub: C] : aa(C,B,aa(nat,fun(C,B),aTP_Lamp_aiz(fun(C,B),fun(nat,fun(C,B)),Uu),Uua),Uub) = aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(C,B,Uu,Uub)),Uua) ) ).

% ATP.lambda_675
tff(fact_8854_ATP_Olambda__676,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V3459762299906320749_field(B)
        & real_V822414075346904944vector(A) )
     => ! [Uu: fun(A,B),Uua: nat,Uub: A] : aa(A,B,aa(nat,fun(A,B),aTP_Lamp_uh(fun(A,B),fun(nat,fun(A,B)),Uu),Uua),Uub) = aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(A,B,Uu,Uub)),Uua) ) ).

% ATP.lambda_676
tff(fact_8855_ATP_Olambda__677,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: fun(A,A),Uua: nat,Uub: A] : aa(A,A,aa(nat,fun(A,A),aTP_Lamp_sa(fun(A,A),fun(nat,fun(A,A)),Uu),Uua),Uub) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,Uu,Uub)),Uua) ) ).

% ATP.lambda_677
tff(fact_8856_ATP_Olambda__678,axiom,
    ! [A: $tType,B: $tType] :
      ( ( power(B)
        & real_V4412858255891104859lgebra(B)
        & topological_t2_space(A) )
     => ! [Uu: fun(A,B),Uua: nat,Uub: A] : aa(A,B,aa(nat,fun(A,B),aTP_Lamp_wi(fun(A,B),fun(nat,fun(A,B)),Uu),Uua),Uub) = aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(A,B,Uu,Uub)),Uua) ) ).

% ATP.lambda_678
tff(fact_8857_ATP_Olambda__679,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V8999393235501362500lgebra(B)
     => ! [Uu: fun(A,B),Uua: nat,Uub: A] : aa(A,B,aa(nat,fun(A,B),aTP_Lamp_aee(fun(A,B),fun(nat,fun(A,B)),Uu),Uua),Uub) = aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(A,B,Uu,Uub)),Uua) ) ).

% ATP.lambda_679
tff(fact_8858_ATP_Olambda__680,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V2822296259951069270ebra_1(B)
     => ! [Uu: fun(A,B),Uua: nat,Uub: A] : aa(A,B,aa(nat,fun(A,B),aTP_Lamp_we(fun(A,B),fun(nat,fun(A,B)),Uu),Uua),Uub) = aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(A,B,Uu,Uub)),Uua) ) ).

% ATP.lambda_680
tff(fact_8859_ATP_Olambda__681,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_semiring_1(B)
     => ! [Uu: fun(A,B),Uua: nat,Uub: A] : aa(A,B,aa(nat,fun(A,B),aTP_Lamp_hh(fun(A,B),fun(nat,fun(A,B)),Uu),Uua),Uub) = aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(A,B,Uu,Uub)),Uua) ) ).

% ATP.lambda_681
tff(fact_8860_ATP_Olambda__682,axiom,
    ! [A: $tType,B: $tType] :
      ( ( power(B)
        & real_V4412858255891104859lgebra(B) )
     => ! [Uu: fun(A,B),Uua: nat,Uub: A] : aa(A,B,aa(nat,fun(A,B),aTP_Lamp_wf(fun(A,B),fun(nat,fun(A,B)),Uu),Uua),Uub) = aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(A,B,Uu,Uub)),Uua) ) ).

% ATP.lambda_682
tff(fact_8861_ATP_Olambda__683,axiom,
    ! [Uu: nat,Uua: fun(real,real),Uub: real] : aa(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_adt(nat,fun(fun(real,real),fun(real,real)),Uu),Uua),Uub) = aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,Uua,Uub)),Uu) ).

% ATP.lambda_683
tff(fact_8862_ATP_Olambda__684,axiom,
    ! [A: $tType,Uu: nat,Uua: fun(A,real),Uub: A] : aa(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_vl(nat,fun(fun(A,real),fun(A,real)),Uu),Uua),Uub) = aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(A,real,Uua,Uub)),Uu) ).

% ATP.lambda_684
tff(fact_8863_ATP_Olambda__685,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_ahx(fun(nat,A),fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,Uu,Uub)),Uua) ) ).

% ATP.lambda_685
tff(fact_8864_ATP_Olambda__686,axiom,
    ! [B: $tType,A: $tType] :
      ( linord4140545234300271783up_add(A)
     => ! [Uu: fun(B,A),Uua: A,Uub: B] : aa(B,A,aa(A,fun(B,A),aTP_Lamp_nu(fun(B,A),fun(A,fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(B,A,Uu,Uub)),Uua) ) ).

% ATP.lambda_686
tff(fact_8865_ATP_Olambda__687,axiom,
    ! [Uu: fun(real,real),Uua: real,Uub: real] : aa(real,real,aa(real,fun(real,real),aTP_Lamp_sg(fun(real,real),fun(real,fun(real,real)),Uu),Uua),Uub) = powr(real,aa(real,real,Uu,Uub),Uua) ).

% ATP.lambda_687
tff(fact_8866_ATP_Olambda__688,axiom,
    ! [A: $tType,Uu: real,Uua: fun(A,real),Uub: A] : aa(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_aeb(real,fun(fun(A,real),fun(A,real)),Uu),Uua),Uub) = powr(real,aa(A,real,Uua,Uub),Uu) ).

% ATP.lambda_688
tff(fact_8867_ATP_Olambda__689,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: fun(B,A),Uua: set(A),Uub: B] :
          ( pp(aa(B,bool,aa(set(A),fun(B,bool),aTP_Lamp_agb(fun(B,A),fun(set(A),fun(B,bool)),Uu),Uua),Uub))
        <=> pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(B,A,Uu,Uub)),Uua)) ) ) ).

% ATP.lambda_689
tff(fact_8868_ATP_Olambda__690,axiom,
    ! [C: $tType,A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: set(A),Uua: fun(C,A),Uub: C] :
          ( pp(aa(C,bool,aa(fun(C,A),fun(C,bool),aTP_Lamp_air(set(A),fun(fun(C,A),fun(C,bool)),Uu),Uua),Uub))
        <=> pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(C,A,Uua,Uub)),Uu)) ) ) ).

% ATP.lambda_690
tff(fact_8869_ATP_Olambda__691,axiom,
    ! [B: $tType,A: $tType,Uu: set(A),Uua: fun(B,A),Uub: B] :
      ( pp(aa(B,bool,aa(fun(B,A),fun(B,bool),aTP_Lamp_qn(set(A),fun(fun(B,A),fun(B,bool)),Uu),Uua),Uub))
    <=> pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(B,A,Uua,Uub)),Uu)) ) ).

% ATP.lambda_691
tff(fact_8870_ATP_Olambda__692,axiom,
    ! [A: $tType,Uu: fun(A,bool),Uua: bool,Uub: A] :
      ( pp(aa(A,bool,aa(bool,fun(A,bool),aTP_Lamp_afn(fun(A,bool),fun(bool,fun(A,bool)),Uu),Uua),Uub))
    <=> ( pp(aa(A,bool,Uu,Uub))
        | pp(Uua) ) ) ).

% ATP.lambda_692
tff(fact_8871_ATP_Olambda__693,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: fun(B,A),Uua: A,Uub: B] :
          ( pp(aa(B,bool,aa(A,fun(B,bool),aTP_Lamp_afe(fun(B,A),fun(A,fun(B,bool)),Uu),Uua),Uub))
        <=> ( aa(B,A,Uu,Uub) = Uua ) ) ) ).

% ATP.lambda_693
tff(fact_8872_ATP_Olambda__694,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo8865339358273720382pology(A)
     => ! [Uu: fun(B,A),Uua: A,Uub: B] :
          ( pp(aa(B,bool,aa(A,fun(B,bool),aTP_Lamp_afc(fun(B,A),fun(A,fun(B,bool)),Uu),Uua),Uub))
        <=> ( aa(B,A,Uu,Uub) = Uua ) ) ) ).

% ATP.lambda_694
tff(fact_8873_ATP_Olambda__695,axiom,
    ! [A: $tType,Uu: A,Uua: fun(A,bool),Uub: A] :
      ( pp(aa(A,bool,aa(fun(A,bool),fun(A,bool),aTP_Lamp_lq(A,fun(fun(A,bool),fun(A,bool)),Uu),Uua),Uub))
    <=> ( ( Uub != Uu )
       => pp(aa(A,bool,Uua,Uub)) ) ) ).

% ATP.lambda_695
tff(fact_8874_ATP_Olambda__696,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: nat,Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_pe(nat,fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,minus_minus(nat,Uua),Uub))),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,minus_minus(nat,Uu),Uub))) ) ).

% ATP.lambda_696
tff(fact_8875_ATP_Olambda__697,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: real,Uua: fun(nat,A),Uub: nat] : aa(nat,real,aa(fun(nat,A),fun(nat,real),aTP_Lamp_cu(real,fun(fun(nat,A),fun(nat,real)),Uu),Uua),Uub) = aa(real,real,aa(real,fun(real,real),times_times(real),real_V7770717601297561774m_norm(A,aa(nat,A,Uua,Uub))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Uu),Uub)) ) ).

% ATP.lambda_697
tff(fact_8876_ATP_Olambda__698,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [Uu: fun(nat,A),Uua: fun(nat,real),Uub: nat] :
          ( pp(aa(nat,bool,aa(fun(nat,real),fun(nat,bool),aTP_Lamp_ahg(fun(nat,A),fun(fun(nat,real),fun(nat,bool)),Uu),Uua),Uub))
        <=> pp(aa(real,bool,ord_less_eq(real,real_V7770717601297561774m_norm(A,aa(nat,A,Uu,Uub))),aa(nat,real,Uua,Uub))) ) ) ).

% ATP.lambda_698
tff(fact_8877_ATP_Olambda__699,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V822414075346904944vector(A) )
     => ! [Uu: fun(nat,A),Uua: fun(nat,B),Uub: nat] :
          ( pp(aa(nat,bool,aa(fun(nat,B),fun(nat,bool),aTP_Lamp_aib(fun(nat,A),fun(fun(nat,B),fun(nat,bool)),Uu),Uua),Uub))
        <=> pp(aa(real,bool,ord_less_eq(real,real_V7770717601297561774m_norm(A,aa(nat,A,Uu,Uub))),real_V7770717601297561774m_norm(B,aa(nat,B,Uua,Uub)))) ) ) ).

% ATP.lambda_699
tff(fact_8878_ATP_Olambda__700,axiom,
    ! [A: $tType,B: $tType] :
      ( ( archim2362893244070406136eiling(B)
        & topolo2564578578187576103pology(B) )
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( pp(aa(A,bool,aa(B,fun(A,bool),aTP_Lamp_agw(fun(A,B),fun(B,fun(A,bool)),Uu),Uua),Uub))
        <=> ( archim6421214686448440834_floor(B,aa(A,B,Uu,Uub)) = archim6421214686448440834_floor(B,Uua) ) ) ) ).

% ATP.lambda_700
tff(fact_8879_ATP_Olambda__701,axiom,
    ! [A: $tType,B: $tType] :
      ( ( archim2362893244070406136eiling(B)
        & topolo2564578578187576103pology(B) )
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( pp(aa(A,bool,aa(B,fun(A,bool),aTP_Lamp_agx(fun(A,B),fun(B,fun(A,bool)),Uu),Uua),Uub))
        <=> ( archimedean_ceiling(B,aa(A,B,Uu,Uub)) = archimedean_ceiling(B,Uua) ) ) ) ).

% ATP.lambda_701
tff(fact_8880_ATP_Olambda__702,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [Uu: fun(A,B),Uua: real,Uub: A] :
          ( pp(aa(A,bool,aa(real,fun(A,bool),aTP_Lamp_aid(fun(A,B),fun(real,fun(A,bool)),Uu),Uua),Uub))
        <=> pp(aa(real,bool,ord_less_eq(real,real_V7770717601297561774m_norm(B,aa(A,B,Uu,Uub))),Uua)) ) ) ).

% ATP.lambda_702
tff(fact_8881_ATP_Olambda__703,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu: A,Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_ku(A,fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),semiring_char_0_fact(real,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),Uua)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),Uua))) ) ).

% ATP.lambda_703
tff(fact_8882_ATP_Olambda__704,axiom,
    ! [B: $tType,A: $tType] :
      ( ( archim2362893244070406136eiling(B)
        & topolo2564578578187576103pology(B) )
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( pp(aa(A,bool,aa(B,fun(A,bool),aTP_Lamp_ahl(fun(A,B),fun(B,fun(A,bool)),Uu),Uua),Uub))
        <=> pp(aa(B,bool,ord_less(B,aa(int,B,ring_1_of_int(B),archim6421214686448440834_floor(B,Uua))),aa(A,B,Uu,Uub))) ) ) ).

% ATP.lambda_704
tff(fact_8883_ATP_Olambda__705,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Uu: nat,Uua: nat,Uub: word(A)] :
          ( pp(aa(word(A),bool,aa(nat,fun(word(A),bool),aTP_Lamp_po(nat,fun(nat,fun(word(A),bool)),Uu),Uua),Uub))
        <=> pp(aa(word(A),bool,ord_less(word(A),Uub),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Uu)),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Uua)))) ) ) ).

% ATP.lambda_705
tff(fact_8884_ATP_Olambda__706,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),aTP_Lamp_pp(nat,fun(nat,fun(nat,bool)),Uu),Uua),Uub))
    <=> pp(aa(nat,bool,ord_less(nat,Uub),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uu)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uua)))) ) ).

% ATP.lambda_706
tff(fact_8885_ATP_Olambda__707,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(nat,A),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_pa(fun(nat,A),fun(nat,fun(nat,A)),Uu),Uua),Uub) = groups7121269368397514597t_prod(nat,A,Uu,set_or7035219750837199246ssThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uub),Uua),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uub),Uua)),Uua))) ) ).

% ATP.lambda_707
tff(fact_8886_ATP_Olambda__708,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [Uu: fun(nat,A),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_pb(fun(nat,A),fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,Uu),set_or7035219750837199246ssThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uub),Uua),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uub),Uua)),Uua))) ) ).

% ATP.lambda_708
tff(fact_8887_ATP_Olambda__709,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(nat,A),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_oz(fun(nat,A),fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,Uu),set_or7035219750837199246ssThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uub),Uua),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uub),Uua)),Uua))) ) ).

% ATP.lambda_709
tff(fact_8888_ATP_Olambda__710,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [Uu: A,Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_eo(A,fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Uu),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),Uub)),Uua)) ) ).

% ATP.lambda_710
tff(fact_8889_ATP_Olambda__711,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Uu: A,Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_em(A,fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Uu),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),Uub)),Uua)) ) ).

% ATP.lambda_711
tff(fact_8890_ATP_Olambda__712,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [Uu: A,Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_fp(A,fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu),aa(nat,nat,minus_minus(nat,Uua),aa(nat,nat,suc,Uub))) ) ).

% ATP.lambda_712
tff(fact_8891_ATP_Olambda__713,axiom,
    ! [Uu: real,Uua: real,Uub: nat] : aa(nat,real,aa(real,fun(nat,real),aTP_Lamp_abn(real,fun(real,fun(nat,real)),Uu),Uua),Uub) = aa(real,real,aa(real,fun(real,real),divide_divide(real),Uua),aa(nat,real,aa(real,fun(nat,real),power_power(real),Uu),Uub)) ).

% ATP.lambda_713
tff(fact_8892_ATP_Olambda__714,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [Uu: A,Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_ee(A,fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uua),Uub)) ) ).

% ATP.lambda_714
tff(fact_8893_ATP_Olambda__715,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] : aa(nat,nat,aa(nat,fun(nat,nat),aTP_Lamp_en(nat,fun(nat,fun(nat,nat)),Uu),Uua),Uub) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uu),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uub),Uua)) ).

% ATP.lambda_715
tff(fact_8894_ATP_Olambda__716,axiom,
    ! [Uu: real,Uua: real,Uub: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),aTP_Lamp_ahb(real,fun(real,fun(real,bool)),Uu),Uua),Uub))
    <=> pp(aa(set(real),bool,aa(real,fun(set(real),bool),member(real),Uub),set_or5935395276787703475ssThan(real,Uu,Uua))) ) ).

% ATP.lambda_716
tff(fact_8895_ATP_Olambda__717,axiom,
    ! [A: $tType,C: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Uu: real,Uua: fun(C,A),Uub: C] : aa(C,A,aa(fun(C,A),fun(C,A),aTP_Lamp_kg(real,fun(fun(C,A),fun(C,A)),Uu),Uua),Uub) = aa(A,A,real_V8093663219630862766scaleR(A,Uu),aa(C,A,Uua,Uub)) ) ).

% ATP.lambda_717
tff(fact_8896_ATP_Olambda__718,axiom,
    ! [B: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [Uu: fun(nat,B),Uua: real,Uub: nat] : aa(nat,B,aa(real,fun(nat,B),aTP_Lamp_kh(fun(nat,B),fun(real,fun(nat,B)),Uu),Uua),Uub) = aa(B,B,real_V8093663219630862766scaleR(B,Uua),aa(nat,B,Uu,Uub)) ) ).

% ATP.lambda_718
tff(fact_8897_ATP_Olambda__719,axiom,
    ! [B: $tType,C: $tType] :
      ( ( real_V822414075346904944vector(C)
        & real_V822414075346904944vector(B) )
     => ! [Uu: fun(C,B),Uua: real,Uub: C] : aa(C,B,aa(real,fun(C,B),aTP_Lamp_tl(fun(C,B),fun(real,fun(C,B)),Uu),Uua),Uub) = aa(B,B,real_V8093663219630862766scaleR(B,Uua),aa(C,B,Uu,Uub)) ) ).

% ATP.lambda_719
tff(fact_8898_ATP_Olambda__720,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: fun(A,A),Uua: real,Uub: A] : aa(A,A,aa(real,fun(A,A),aTP_Lamp_rc(fun(A,A),fun(real,fun(A,A)),Uu),Uua),Uub) = aa(A,A,real_V8093663219630862766scaleR(A,Uua),aa(A,A,Uu,Uub)) ) ).

% ATP.lambda_720
tff(fact_8899_ATP_Olambda__721,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo2564578578187576103pology(A)
     => ! [Uu: A,Uua: fun(B,A),Uub: B] :
          ( pp(aa(B,bool,aa(fun(B,A),fun(B,bool),aTP_Lamp_agh(A,fun(fun(B,A),fun(B,bool)),Uu),Uua),Uub))
        <=> pp(aa(A,bool,ord_less_eq(A,Uu),aa(B,A,Uua,Uub))) ) ) ).

% ATP.lambda_721
tff(fact_8900_ATP_Olambda__722,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [Uu: fun(B,A),Uua: A,Uub: B] :
          ( pp(aa(B,bool,aa(A,fun(B,bool),aTP_Lamp_agr(fun(B,A),fun(A,fun(B,bool)),Uu),Uua),Uub))
        <=> pp(aa(A,bool,ord_less_eq(A,Uua),aa(B,A,Uu,Uub))) ) ) ).

% ATP.lambda_722
tff(fact_8901_ATP_Olambda__723,axiom,
    ! [B: $tType,A: $tType] :
      ( unboun7993243217541854897norder(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( pp(aa(A,bool,aa(B,fun(A,bool),aTP_Lamp_agj(fun(A,B),fun(B,fun(A,bool)),Uu),Uua),Uub))
        <=> pp(aa(B,bool,ord_less_eq(B,Uua),aa(A,B,Uu,Uub))) ) ) ).

% ATP.lambda_723
tff(fact_8902_ATP_Olambda__724,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( pp(aa(A,bool,aa(B,fun(A,bool),aTP_Lamp_afi(fun(A,B),fun(B,fun(A,bool)),Uu),Uua),Uub))
        <=> pp(aa(B,bool,ord_less_eq(B,Uua),aa(A,B,Uu,Uub))) ) ) ).

% ATP.lambda_724
tff(fact_8903_ATP_Olambda__725,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_qh(fun(nat,A),fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),divide_divide(A),Uua),aa(nat,A,Uu,Uub)) ) ).

% ATP.lambda_725
tff(fact_8904_ATP_Olambda__726,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo2564578578187576103pology(A)
     => ! [Uu: fun(B,A),Uua: A,Uub: B] :
          ( pp(aa(B,bool,aa(A,fun(B,bool),aTP_Lamp_aer(fun(B,A),fun(A,fun(B,bool)),Uu),Uua),Uub))
        <=> pp(aa(A,bool,ord_less(A,Uua),aa(B,A,Uu,Uub))) ) ) ).

% ATP.lambda_726
tff(fact_8905_ATP_Olambda__727,axiom,
    ! [B: $tType,A: $tType] :
      ( unboun7993243217541854897norder(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( pp(aa(A,bool,aa(B,fun(A,bool),aTP_Lamp_afk(fun(A,B),fun(B,fun(A,bool)),Uu),Uua),Uub))
        <=> pp(aa(B,bool,ord_less(B,Uua),aa(A,B,Uu,Uub))) ) ) ).

% ATP.lambda_727
tff(fact_8906_ATP_Olambda__728,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo2564578578187576103pology(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( pp(aa(A,bool,aa(B,fun(A,bool),aTP_Lamp_ahe(fun(A,B),fun(B,fun(A,bool)),Uu),Uua),Uub))
        <=> pp(aa(B,bool,ord_less(B,Uua),aa(A,B,Uu,Uub))) ) ) ).

% ATP.lambda_728
tff(fact_8907_ATP_Olambda__729,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: A,Uua: fun(nat,A),Uub: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_bu(A,fun(fun(nat,A),fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),Uu),aa(nat,A,Uua,Uub)) ) ).

% ATP.lambda_729
tff(fact_8908_ATP_Olambda__730,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: A,Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_ail(A,fun(fun(B,A),fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),Uu),aa(B,A,Uua,Uub)) ) ).

% ATP.lambda_730
tff(fact_8909_ATP_Olambda__731,axiom,
    ! [A: $tType,B: $tType] :
      ( semiring_0(A)
     => ! [Uu: A,Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_dk(A,fun(fun(B,A),fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),Uu),aa(B,A,Uua,Uub)) ) ).

% ATP.lambda_731
tff(fact_8910_ATP_Olambda__732,axiom,
    ! [A: $tType] :
      ( ( field(A)
        & real_V4412858255891104859lgebra(A) )
     => ! [Uu: A,Uua: fun(nat,A),Uub: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_ex(A,fun(fun(nat,A),fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),Uu),aa(nat,A,Uua,Uub)) ) ).

% ATP.lambda_732
tff(fact_8911_ATP_Olambda__733,axiom,
    ! [A: $tType] :
      ( ( field(A)
        & topolo4211221413907600880p_mult(A) )
     => ! [Uu: A,Uua: fun(nat,A),Uub: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_aac(A,fun(fun(nat,A),fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),Uu),aa(nat,A,Uua,Uub)) ) ).

% ATP.lambda_733
tff(fact_8912_ATP_Olambda__734,axiom,
    ! [A: $tType,B: $tType] :
      ( ( field(A)
        & topolo4211221413907600880p_mult(A) )
     => ! [Uu: A,Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_vj(A,fun(fun(B,A),fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),Uu),aa(B,A,Uua,Uub)) ) ).

% ATP.lambda_734
tff(fact_8913_ATP_Olambda__735,axiom,
    ! [A: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [Uu: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_cd(fun(nat,A),fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),Uua),aa(nat,A,Uu,Uub)) ) ).

% ATP.lambda_735
tff(fact_8914_ATP_Olambda__736,axiom,
    ! [A: $tType,D6: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [Uu: fun(D6,A),Uua: A,Uub: D6] : aa(D6,A,aa(A,fun(D6,A),aTP_Lamp_ya(fun(D6,A),fun(A,fun(D6,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),Uua),aa(D6,A,Uu,Uub)) ) ).

% ATP.lambda_736
tff(fact_8915_ATP_Olambda__737,axiom,
    ! [A: $tType,C: $tType] :
      ( ( real_V822414075346904944vector(C)
        & real_V4412858255891104859lgebra(A) )
     => ! [Uu: fun(C,A),Uua: A,Uub: C] : aa(C,A,aa(A,fun(C,A),aTP_Lamp_te(fun(C,A),fun(A,fun(C,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),Uua),aa(C,A,Uu,Uub)) ) ).

% ATP.lambda_737
tff(fact_8916_ATP_Olambda__738,axiom,
    ! [A: $tType,B: $tType] :
      ( ( topolo4958980785337419405_space(B)
        & real_V4412858255891104859lgebra(A) )
     => ! [Uu: fun(B,A),Uua: A,Uub: B] : aa(B,A,aa(A,fun(B,A),aTP_Lamp_ajv(fun(B,A),fun(A,fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),Uua),aa(B,A,Uu,Uub)) ) ).

% ATP.lambda_738
tff(fact_8917_ATP_Olambda__739,axiom,
    ! [A: $tType,B: $tType] :
      ( ( topological_t2_space(B)
        & real_V4412858255891104859lgebra(A) )
     => ! [Uu: fun(B,A),Uua: A,Uub: B] : aa(B,A,aa(A,fun(B,A),aTP_Lamp_yg(fun(B,A),fun(A,fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),Uua),aa(B,A,Uu,Uub)) ) ).

% ATP.lambda_739
tff(fact_8918_ATP_Olambda__740,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo4211221413907600880p_mult(A)
     => ! [Uu: fun(B,A),Uua: A,Uub: B] : aa(B,A,aa(A,fun(B,A),aTP_Lamp_yd(fun(B,A),fun(A,fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),Uua),aa(B,A,Uu,Uub)) ) ).

% ATP.lambda_740
tff(fact_8919_ATP_Olambda__741,axiom,
    ! [A: $tType,B: $tType] :
      ( ( linordered_field(A)
        & topolo1944317154257567458pology(A) )
     => ! [Uu: fun(B,A),Uua: A,Uub: B] : aa(B,A,aa(A,fun(B,A),aTP_Lamp_adp(fun(B,A),fun(A,fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),Uua),aa(B,A,Uu,Uub)) ) ).

% ATP.lambda_741
tff(fact_8920_ATP_Olambda__742,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: fun(A,A),Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_rk(fun(A,A),fun(A,fun(A,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),Uua),aa(A,A,Uu,Uub)) ) ).

% ATP.lambda_742
tff(fact_8921_ATP_Olambda__743,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: A,Uua: fun(B,nat),Uub: B] : aa(B,A,aa(fun(B,nat),fun(B,A),aTP_Lamp_ho(A,fun(fun(B,nat),fun(B,A)),Uu),Uua),Uub) = aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu),aa(B,nat,Uua,Uub)) ) ).

% ATP.lambda_743
tff(fact_8922_ATP_Olambda__744,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V2822296259951069270ebra_1(A)
     => ! [Uu: fun(B,nat),Uua: A,Uub: B] : aa(B,A,aa(A,fun(B,A),aTP_Lamp_abv(fun(B,nat),fun(A,fun(B,A)),Uu),Uua),Uub) = aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),aa(B,nat,Uu,Uub)) ) ).

% ATP.lambda_744
tff(fact_8923_ATP_Olambda__745,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo1633459387980952147up_add(A)
     => ! [Uu: A,Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_xv(A,fun(fun(B,A),fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Uu),aa(B,A,Uua,Uub)) ) ).

% ATP.lambda_745
tff(fact_8924_ATP_Olambda__746,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: fun(A,real),Uua: nat,Uub: A] : aa(A,real,aa(nat,fun(A,real),aTP_Lamp_ajm(fun(A,real),fun(nat,fun(A,real)),Uu),Uua),Uub) = aa(real,real,root(Uua),aa(A,real,Uu,Uub)) ) ).

% ATP.lambda_746
tff(fact_8925_ATP_Olambda__747,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Uu: fun(A,real),Uua: nat,Uub: A] : aa(A,real,aa(nat,fun(A,real),aTP_Lamp_aak(fun(A,real),fun(nat,fun(A,real)),Uu),Uua),Uub) = aa(real,real,root(Uua),aa(A,real,Uu,Uub)) ) ).

% ATP.lambda_747
tff(fact_8926_ATP_Olambda__748,axiom,
    ! [A: $tType,Uu: fun(A,real),Uua: nat,Uub: A] : aa(A,real,aa(nat,fun(A,real),aTP_Lamp_xi(fun(A,real),fun(nat,fun(A,real)),Uu),Uua),Uub) = aa(real,real,root(Uua),aa(A,real,Uu,Uub)) ).

% ATP.lambda_748
tff(fact_8927_ATP_Olambda__749,axiom,
    ! [A: $tType,Uu: bool,Uua: fun(A,bool),Uub: A] :
      ( pp(aa(A,bool,aa(fun(A,bool),fun(A,bool),aTP_Lamp_afl(bool,fun(fun(A,bool),fun(A,bool)),Uu),Uua),Uub))
    <=> ( pp(Uu)
       => pp(aa(A,bool,Uua,Uub)) ) ) ).

% ATP.lambda_749
tff(fact_8928_ATP_Olambda__750,axiom,
    ! [A: $tType,Uu: bool,Uua: fun(A,bool),Uub: A] :
      ( pp(aa(A,bool,aa(fun(A,bool),fun(A,bool),aTP_Lamp_afm(bool,fun(fun(A,bool),fun(A,bool)),Uu),Uua),Uub))
    <=> ( pp(Uu)
        | pp(aa(A,bool,Uua,Uub)) ) ) ).

% ATP.lambda_750
tff(fact_8929_ATP_Olambda__751,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,B),Uua: A,Uub: B] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),aTP_Lamp_qv(fun(A,B),fun(A,fun(B,bool)),Uu),Uua),Uub))
    <=> ( Uub = aa(A,B,Uu,Uua) ) ) ).

% ATP.lambda_751
tff(fact_8930_ATP_Olambda__752,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Uu: A,Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_hw(A,fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Uu),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,minus_minus(nat,Uua),Uub))) ) ).

% ATP.lambda_752
tff(fact_8931_ATP_Olambda__753,axiom,
    ! [A: $tType,C: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(C,A),Uua: real,Uub: C] :
          ( pp(aa(C,bool,aa(real,fun(C,bool),aTP_Lamp_ahm(fun(C,A),fun(real,fun(C,bool)),Uu),Uua),Uub))
        <=> pp(aa(real,bool,ord_less_eq(real,Uua),real_V7770717601297561774m_norm(A,aa(C,A,Uu,Uub)))) ) ) ).

% ATP.lambda_753
tff(fact_8932_ATP_Olambda__754,axiom,
    ! [A: $tType,Uu: fun(A,nat),Uua: A,Uub: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_bf(fun(A,nat),fun(A,fun(A,bool)),Uu),Uua),Uub))
    <=> pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),if(nat,aa(A,bool,fequal(A,Uub),Uua),aa(nat,nat,suc,aa(A,nat,Uu,Uub)),aa(A,nat,Uu,Uub)))) ) ).

% ATP.lambda_754
tff(fact_8933_ATP_Olambda__755,axiom,
    ! [A: $tType,Uu: fun(A,nat),Uua: fun(A,bool),Uub: A] :
      ( pp(aa(A,bool,aa(fun(A,bool),fun(A,bool),aTP_Lamp_bg(fun(A,nat),fun(fun(A,bool),fun(A,bool)),Uu),Uua),Uub))
    <=> pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),if(nat,aa(A,bool,Uua,Uub),aa(A,nat,Uu,Uub),zero_zero(nat)))) ) ).

% ATP.lambda_755
tff(fact_8934_ATP_Olambda__756,axiom,
    ! [A: $tType,Uu: fun(A,nat),Uua: fun(A,nat),Uub: A] :
      ( pp(aa(A,bool,aa(fun(A,nat),fun(A,bool),aTP_Lamp_be(fun(A,nat),fun(fun(A,nat),fun(A,bool)),Uu),Uua),Uub))
    <=> pp(aa(nat,bool,ord_less(nat,zero_zero(nat)),aa(nat,nat,minus_minus(nat,aa(A,nat,Uu,Uub)),aa(A,nat,Uua,Uub)))) ) ).

% ATP.lambda_756
tff(fact_8935_ATP_Olambda__757,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(nat,A),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_hs(fun(nat,A),fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(nat,A,Uu,aa(nat,nat,minus_minus(nat,Uua),aa(nat,nat,suc,Uub))) ) ).

% ATP.lambda_757
tff(fact_8936_ATP_Olambda__758,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(nat,A),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_fc(fun(nat,A),fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(nat,A,Uu,aa(nat,nat,minus_minus(nat,Uua),aa(nat,nat,suc,Uub))) ) ).

% ATP.lambda_758
tff(fact_8937_ATP_Olambda__759,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: fun(nat,A),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_aav(fun(nat,A),fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(nat,A,Uu,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uub),Uua)) ) ).

% ATP.lambda_759
tff(fact_8938_ATP_Olambda__760,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: fun(nat,A),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_aaf(fun(nat,A),fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(nat,A,Uu,aa(nat,nat,minus_minus(nat,Uub),Uua)) ) ).

% ATP.lambda_760
tff(fact_8939_ATP_Olambda__761,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & topolo4958980785337419405_space(B) )
     => ! [Uu: fun(A,B),Uua: A,Uub: A] : aa(A,B,aa(A,fun(A,B),aTP_Lamp_vr(fun(A,B),fun(A,fun(A,B)),Uu),Uua),Uub) = aa(A,B,Uu,aa(A,A,aa(A,fun(A,A),plus_plus(A),Uua),Uub)) ) ).

% ATP.lambda_761
tff(fact_8940_ATP_Olambda__762,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: fun(A,A),Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_ro(fun(A,A),fun(A,fun(A,A)),Uu),Uua),Uub) = aa(A,A,Uu,aa(A,A,aa(A,fun(A,A),plus_plus(A),Uua),Uub)) ) ).

% ATP.lambda_762
tff(fact_8941_ATP_Olambda__763,axiom,
    ! [Uu: fun(real,bool),Uua: real,Uub: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),aTP_Lamp_agy(fun(real,bool),fun(real,fun(real,bool)),Uu),Uua),Uub))
    <=> pp(aa(real,bool,Uu,aa(real,real,aa(real,fun(real,real),plus_plus(real),Uub),Uua))) ) ).

% ATP.lambda_763
tff(fact_8942_ATP_Olambda__764,axiom,
    ! [A: $tType,Uu: fun(real,A),Uua: real,Uub: real] : aa(real,A,aa(real,fun(real,A),aTP_Lamp_ado(fun(real,A),fun(real,fun(real,A)),Uu),Uua),Uub) = aa(real,A,Uu,aa(real,real,aa(real,fun(real,real),plus_plus(real),Uub),Uua)) ).

% ATP.lambda_764
tff(fact_8943_ATP_Olambda__765,axiom,
    ! [Uu: fun(nat,bool),Uua: nat,Uub: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),aTP_Lamp_aep(fun(nat,bool),fun(nat,fun(nat,bool)),Uu),Uua),Uub))
    <=> pp(aa(nat,bool,Uu,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),Uua))) ) ).

% ATP.lambda_765
tff(fact_8944_ATP_Olambda__766,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(nat,A),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_bt(fun(nat,A),fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(nat,A,Uu,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),Uua)) ) ).

% ATP.lambda_766
tff(fact_8945_ATP_Olambda__767,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: fun(nat,A),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_aan(fun(nat,A),fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(nat,A,Uu,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),Uua)) ) ).

% ATP.lambda_767
tff(fact_8946_ATP_Olambda__768,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [Uu: fun(nat,A),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_ahy(fun(nat,A),fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(nat,A,Uu,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),Uua)) ) ).

% ATP.lambda_768
tff(fact_8947_ATP_Olambda__769,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(nat,A),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_hp(fun(nat,A),fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(nat,A,Uu,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),Uua)) ) ).

% ATP.lambda_769
tff(fact_8948_ATP_Olambda__770,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(nat,A),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_eb(fun(nat,A),fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(nat,A,Uu,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),Uua)) ) ).

% ATP.lambda_770
tff(fact_8949_ATP_Olambda__771,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,bool),Uua: A,Uub: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_agg(fun(A,bool),fun(A,fun(A,bool)),Uu),Uua),Uub))
        <=> pp(aa(A,bool,Uu,aa(A,A,aa(A,fun(A,A),plus_plus(A),Uub),Uua))) ) ) ).

% ATP.lambda_771
tff(fact_8950_ATP_Olambda__772,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & topolo4958980785337419405_space(B) )
     => ! [Uu: fun(A,B),Uua: A,Uub: A] : aa(A,B,aa(A,fun(A,B),aTP_Lamp_vq(fun(A,B),fun(A,fun(A,B)),Uu),Uua),Uub) = aa(A,B,Uu,aa(A,A,aa(A,fun(A,A),plus_plus(A),Uub),Uua)) ) ).

% ATP.lambda_772
tff(fact_8951_ATP_Olambda__773,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,B),Uua: A,Uub: A] : aa(A,B,aa(A,fun(A,B),aTP_Lamp_zs(fun(A,B),fun(A,fun(A,B)),Uu),Uua),Uub) = aa(A,B,Uu,aa(A,A,aa(A,fun(A,A),plus_plus(A),Uub),Uua)) ) ).

% ATP.lambda_773
tff(fact_8952_ATP_Olambda__774,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: fun(A,A),Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_rv(fun(A,A),fun(A,fun(A,A)),Uu),Uua),Uub) = aa(A,A,Uu,aa(A,A,aa(A,fun(A,A),plus_plus(A),Uub),Uua)) ) ).

% ATP.lambda_774
tff(fact_8953_ATP_Olambda__775,axiom,
    ! [A: $tType,B: $tType,Uu: fun(product_prod(A,B),bool),Uua: A,Uub: B] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),aTP_Lamp_mf(fun(product_prod(A,B),bool),fun(A,fun(B,bool)),Uu),Uua),Uub))
    <=> pp(aa(product_prod(A,B),bool,Uu,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uua),Uub))) ) ).

% ATP.lambda_775
tff(fact_8954_ATP_Olambda__776,axiom,
    ! [C: $tType,A: $tType,B: $tType,Uu: fun(product_prod(A,B),C),Uua: A,Uub: B] : aa(B,C,aa(A,fun(B,C),aTP_Lamp_mp(fun(product_prod(A,B),C),fun(A,fun(B,C)),Uu),Uua),Uub) = aa(product_prod(A,B),C,Uu,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uua),Uub)) ).

% ATP.lambda_776
tff(fact_8955_ATP_Olambda__777,axiom,
    ! [A: $tType,Uu: fun(A,bool),Uua: list(A),Uub: nat] :
      ( pp(aa(nat,bool,aa(list(A),fun(nat,bool),aTP_Lamp_and(fun(A,bool),fun(list(A),fun(nat,bool)),Uu),Uua),Uub))
    <=> pp(aa(A,bool,Uu,aa(nat,A,nth(A,Uua),Uub))) ) ).

% ATP.lambda_777
tff(fact_8956_ATP_Olambda__778,axiom,
    ! [D6: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & topolo4958980785337419405_space(D6) )
     => ! [Uu: A,Uua: fun(A,D6),Uub: A] : aa(A,D6,aa(fun(A,D6),fun(A,D6),aTP_Lamp_adf(A,fun(fun(A,D6),fun(A,D6)),Uu),Uua),Uub) = aa(A,D6,Uua,aa(A,A,aa(A,fun(A,A),plus_plus(A),Uu),Uub)) ) ).

% ATP.lambda_778
tff(fact_8957_ATP_Olambda__779,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & topolo4958980785337419405_space(B) )
     => ! [Uu: A,Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_vs(A,fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(A,B,Uua,aa(A,A,aa(A,fun(A,A),plus_plus(A),Uu),Uub)) ) ).

% ATP.lambda_779
tff(fact_8958_ATP_Olambda__780,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: nat,Uua: fun(nat,A),Uub: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_fb(nat,fun(fun(nat,A),fun(nat,A)),Uu),Uua),Uub) = aa(nat,A,Uua,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),Uu)) ) ).

% ATP.lambda_780
tff(fact_8959_ATP_Olambda__781,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(real,real),Uua: fun(A,real),Uub: A] : aa(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_ud(fun(real,real),fun(fun(A,real),fun(A,real)),Uu),Uua),Uub) = aa(real,real,Uu,aa(A,real,Uua,Uub)) ) ).

% ATP.lambda_781
tff(fact_8960_ATP_Olambda__782,axiom,
    ! [B: $tType] :
      ( topolo3112930676232923870pology(B)
     => ! [Uu: fun(real,B),Uua: fun(nat,real),Uub: nat] : aa(nat,B,aa(fun(nat,real),fun(nat,B),aTP_Lamp_adr(fun(real,B),fun(fun(nat,real),fun(nat,B)),Uu),Uua),Uub) = aa(real,B,Uu,aa(nat,real,Uua,Uub)) ) ).

% ATP.lambda_782
tff(fact_8961_ATP_Olambda__783,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(nat,A),Uua: fun(nat,nat),Uub: nat] : aa(nat,A,aa(fun(nat,nat),fun(nat,A),aTP_Lamp_ahw(fun(nat,A),fun(fun(nat,nat),fun(nat,A)),Uu),Uua),Uub) = aa(nat,A,Uu,aa(nat,nat,Uua,Uub)) ) ).

% ATP.lambda_783
tff(fact_8962_ATP_Olambda__784,axiom,
    ! [A: $tType,C: $tType,B: $tType,Uu: fun(C,A),Uua: fun(B,C),Uub: B] : aa(B,A,aa(fun(B,C),fun(B,A),aTP_Lamp_pt(fun(C,A),fun(fun(B,C),fun(B,A)),Uu),Uua),Uub) = aa(C,A,Uu,aa(B,C,Uua,Uub)) ).

% ATP.lambda_784
tff(fact_8963_ATP_Olambda__785,axiom,
    ! [C: $tType,B: $tType] :
      ( ( topolo3112930676232923870pology(B)
        & topolo1944317154257567458pology(B)
        & topolo4958980785337419405_space(C) )
     => ! [Uu: fun(B,C),Uua: fun(nat,B),Uub: nat] : aa(nat,C,aa(fun(nat,B),fun(nat,C),aTP_Lamp_aik(fun(B,C),fun(fun(nat,B),fun(nat,C)),Uu),Uua),Uub) = aa(B,C,Uu,aa(nat,B,Uua,Uub)) ) ).

% ATP.lambda_785
tff(fact_8964_ATP_Olambda__786,axiom,
    ! [C: $tType,B: $tType,A: $tType,Uu: fun(B,C),Uua: fun(A,B),Uub: A] : aa(A,C,aa(fun(A,B),fun(A,C),aTP_Lamp_alo(fun(B,C),fun(fun(A,B),fun(A,C)),Uu),Uua),Uub) = aa(B,C,Uu,aa(A,B,Uua,Uub)) ).

% ATP.lambda_786
tff(fact_8965_ATP_Olambda__787,axiom,
    ! [A: $tType,B: $tType,C: $tType,Uu: fun(B,A),Uua: fun(C,B),Uub: C] : aa(C,A,aa(fun(C,B),fun(C,A),aTP_Lamp_nv(fun(B,A),fun(fun(C,B),fun(C,A)),Uu),Uua),Uub) = aa(B,A,Uu,aa(C,B,Uua,Uub)) ).

% ATP.lambda_787
tff(fact_8966_ATP_Olambda__788,axiom,
    ! [A: $tType] :
      ( ( topolo3112930676232923870pology(A)
        & topolo1944317154257567458pology(A) )
     => ! [Uu: fun(A,bool),Uua: fun(nat,A),Uub: nat] :
          ( pp(aa(nat,bool,aa(fun(nat,A),fun(nat,bool),aTP_Lamp_aig(fun(A,bool),fun(fun(nat,A),fun(nat,bool)),Uu),Uua),Uub))
        <=> pp(aa(A,bool,Uu,aa(nat,A,Uua,Uub))) ) ) ).

% ATP.lambda_788
tff(fact_8967_ATP_Olambda__789,axiom,
    ! [A: $tType] :
      ( topolo3112930676232923870pology(A)
     => ! [Uu: fun(A,bool),Uua: fun(nat,A),Uub: nat] :
          ( pp(aa(nat,bool,aa(fun(nat,A),fun(nat,bool),aTP_Lamp_aff(fun(A,bool),fun(fun(nat,A),fun(nat,bool)),Uu),Uua),Uub))
        <=> pp(aa(A,bool,Uu,aa(nat,A,Uua,Uub))) ) ) ).

% ATP.lambda_789
tff(fact_8968_ATP_Olambda__790,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo3112930676232923870pology(A)
        & topolo4958980785337419405_space(B) )
     => ! [Uu: fun(A,B),Uua: fun(nat,A),Uub: nat] : aa(nat,B,aa(fun(nat,A),fun(nat,B),aTP_Lamp_aax(fun(A,B),fun(fun(nat,A),fun(nat,B)),Uu),Uua),Uub) = aa(A,B,Uu,aa(nat,A,Uua,Uub)) ) ).

% ATP.lambda_790
tff(fact_8969_ATP_Olambda__791,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu: fun(A,B),Uua: fun(nat,A),Uub: nat] : aa(nat,B,aa(fun(nat,A),fun(nat,B),aTP_Lamp_aci(fun(A,B),fun(fun(nat,A),fun(nat,B)),Uu),Uua),Uub) = aa(A,B,Uu,aa(nat,A,Uua,Uub)) ) ).

% ATP.lambda_791
tff(fact_8970_ATP_Olambda__792,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ( real_V822414075346904944vector(C)
        & real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu: fun(A,B),Uua: fun(C,A),Uub: C] : aa(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_tq(fun(A,B),fun(fun(C,A),fun(C,B)),Uu),Uua),Uub) = aa(A,B,Uu,aa(C,A,Uua,Uub)) ) ).

% ATP.lambda_792
tff(fact_8971_ATP_Olambda__793,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ( topolo4958980785337419405_space(C)
        & real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu: fun(A,B),Uua: fun(C,A),Uub: C] : aa(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_ait(fun(A,B),fun(fun(C,A),fun(C,B)),Uu),Uua),Uub) = aa(A,B,Uu,aa(C,A,Uua,Uub)) ) ).

% ATP.lambda_793
tff(fact_8972_ATP_Olambda__794,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ( topological_t2_space(C)
        & real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu: fun(A,B),Uua: fun(C,A),Uub: C] : aa(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_acg(fun(A,B),fun(fun(C,A),fun(C,B)),Uu),Uua),Uub) = aa(A,B,Uu,aa(C,A,Uua,Uub)) ) ).

% ATP.lambda_794
tff(fact_8973_ATP_Olambda__795,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu: fun(A,B),Uua: fun(C,A),Uub: C] : aa(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_ach(fun(A,B),fun(fun(C,A),fun(C,B)),Uu),Uua),Uub) = aa(A,B,Uu,aa(C,A,Uua,Uub)) ) ).

% ATP.lambda_795
tff(fact_8974_ATP_Olambda__796,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topolo4958980785337419405_space(B) )
     => ! [Uu: fun(A,B),Uua: fun(nat,A),Uub: nat] : aa(nat,B,aa(fun(nat,A),fun(nat,B),aTP_Lamp_aay(fun(A,B),fun(fun(nat,A),fun(nat,B)),Uu),Uua),Uub) = aa(A,B,Uu,aa(nat,A,Uua,Uub)) ) ).

% ATP.lambda_796
tff(fact_8975_ATP_Olambda__797,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ( topolo4958980785337419405_space(C)
        & topolo4958980785337419405_space(A)
        & topolo4958980785337419405_space(B) )
     => ! [Uu: fun(A,B),Uua: fun(C,A),Uub: C] : aa(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_akh(fun(A,B),fun(fun(C,A),fun(C,B)),Uu),Uua),Uub) = aa(A,B,Uu,aa(C,A,Uua,Uub)) ) ).

% ATP.lambda_797
tff(fact_8976_ATP_Olambda__798,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topolo4958980785337419405_space(B) )
     => ! [Uu: fun(A,B),Uua: fun(C,A),Uub: C] : aa(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_vz(fun(A,B),fun(fun(C,A),fun(C,B)),Uu),Uua),Uub) = aa(A,B,Uu,aa(C,A,Uua,Uub)) ) ).

% ATP.lambda_798
tff(fact_8977_ATP_Olambda__799,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: fun(A,A),Uua: fun(A,A),Uub: A] : aa(A,A,aa(fun(A,A),fun(A,A),aTP_Lamp_rq(fun(A,A),fun(fun(A,A),fun(A,A)),Uu),Uua),Uub) = aa(A,A,Uu,aa(A,A,Uua,Uub)) ) ).

% ATP.lambda_799
tff(fact_8978_ATP_Olambda__800,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ( topological_t2_space(A)
        & topolo4958980785337419405_space(B) )
     => ! [Uu: fun(A,B),Uua: fun(C,A),Uub: C] : aa(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_wa(fun(A,B),fun(fun(C,A),fun(C,B)),Uu),Uua),Uub) = aa(A,B,Uu,aa(C,A,Uua,Uub)) ) ).

% ATP.lambda_800
tff(fact_8979_ATP_Olambda__801,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,bool),Uua: fun(B,A),Uub: B] :
      ( pp(aa(B,bool,aa(fun(B,A),fun(B,bool),aTP_Lamp_aew(fun(A,bool),fun(fun(B,A),fun(B,bool)),Uu),Uua),Uub))
    <=> pp(aa(A,bool,Uu,aa(B,A,Uua,Uub))) ) ).

% ATP.lambda_801
tff(fact_8980_ATP_Olambda__802,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,B),Uua: fun(num,A),Uub: num] : aa(num,B,aa(fun(num,A),fun(num,B),aTP_Lamp_ame(fun(A,B),fun(fun(num,A),fun(num,B)),Uu),Uua),Uub) = aa(A,B,Uu,aa(num,A,Uua,Uub)) ).

% ATP.lambda_802
tff(fact_8981_ATP_Olambda__803,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,B),Uua: fun(nat,A),Uub: nat] : aa(nat,B,aa(fun(nat,A),fun(nat,B),aTP_Lamp_alx(fun(A,B),fun(fun(nat,A),fun(nat,B)),Uu),Uua),Uub) = aa(A,B,Uu,aa(nat,A,Uua,Uub)) ).

% ATP.lambda_803
tff(fact_8982_ATP_Olambda__804,axiom,
    ! [B: $tType,A: $tType,C: $tType,Uu: fun(A,B),Uua: fun(C,A),Uub: C] : aa(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_aco(fun(A,B),fun(fun(C,A),fun(C,B)),Uu),Uua),Uub) = aa(A,B,Uu,aa(C,A,Uua,Uub)) ).

% ATP.lambda_804
tff(fact_8983_ATP_Olambda__805,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ( topological_t2_space(C)
        & topological_t2_space(A)
        & topolo4958980785337419405_space(B) )
     => ! [Uu: fun(C,A),Uua: fun(A,B),Uub: C] : aa(C,B,aa(fun(A,B),fun(C,B),aTP_Lamp_wb(fun(C,A),fun(fun(A,B),fun(C,B)),Uu),Uua),Uub) = aa(A,B,Uua,aa(C,A,Uu,Uub)) ) ).

% ATP.lambda_805
tff(fact_8984_ATP_Olambda__806,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(B,C),Uua: fun(C,A),Uub: B] : aa(B,A,aa(fun(C,A),fun(B,A),aTP_Lamp_ke(fun(B,C),fun(fun(C,A),fun(B,A)),Uu),Uua),Uub) = aa(C,A,Uua,aa(B,C,Uu,Uub)) ) ).

% ATP.lambda_806
tff(fact_8985_ATP_Olambda__807,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(B,C),Uua: fun(C,A),Uub: B] : aa(B,A,aa(fun(C,A),fun(B,A),aTP_Lamp_kf(fun(B,C),fun(fun(C,A),fun(B,A)),Uu),Uua),Uub) = aa(C,A,Uua,aa(B,C,Uu,Uub)) ) ).

% ATP.lambda_807
tff(fact_8986_ATP_Olambda__808,axiom,
    ! [C: $tType,A: $tType,B: $tType,Uu: fun(B,A),Uua: fun(A,C),Uub: B] : aa(B,C,aa(fun(A,C),fun(B,C),aTP_Lamp_ps(fun(B,A),fun(fun(A,C),fun(B,C)),Uu),Uua),Uub) = aa(A,C,Uua,aa(B,A,Uu,Uub)) ).

% ATP.lambda_808
tff(fact_8987_ATP_Olambda__809,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B)
        & real_V822414075346904944vector(C) )
     => ! [Uu: fun(A,B),Uua: fun(B,C),Uub: A] : aa(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_ta(fun(A,B),fun(fun(B,C),fun(A,C)),Uu),Uua),Uub) = aa(B,C,Uua,aa(A,B,Uu,Uub)) ) ).

% ATP.lambda_809
tff(fact_8988_ATP_Olambda__810,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & topolo4958980785337419405_space(B)
        & topolo4958980785337419405_space(C) )
     => ! [Uu: fun(A,B),Uua: fun(B,C),Uub: A] : aa(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_wd(fun(A,B),fun(fun(B,C),fun(A,C)),Uu),Uua),Uub) = aa(B,C,Uua,aa(A,B,Uu,Uub)) ) ).

% ATP.lambda_810
tff(fact_8989_ATP_Olambda__811,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topolo4958980785337419405_space(B)
        & topolo4958980785337419405_space(C) )
     => ! [Uu: fun(A,B),Uua: fun(B,C),Uub: A] : aa(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_agm(fun(A,B),fun(fun(B,C),fun(A,C)),Uu),Uua),Uub) = aa(B,C,Uua,aa(A,B,Uu,Uub)) ) ).

% ATP.lambda_811
tff(fact_8990_ATP_Olambda__812,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: fun(A,A),Uua: fun(A,A),Uub: A] : aa(A,A,aa(fun(A,A),fun(A,A),aTP_Lamp_rr(fun(A,A),fun(fun(A,A),fun(A,A)),Uu),Uua),Uub) = aa(A,A,Uua,aa(A,A,Uu,Uub)) ) ).

% ATP.lambda_812
tff(fact_8991_ATP_Olambda__813,axiom,
    ! [D6: $tType,C: $tType,A: $tType] :
      ( ( real_V7819770556892013058_space(A)
        & topolo4958980785337419405_space(C)
        & topolo4958980785337419405_space(D6) )
     => ! [Uu: fun(A,C),Uua: fun(C,D6),Uub: A] : aa(A,D6,aa(fun(C,D6),fun(A,D6),aTP_Lamp_adl(fun(A,C),fun(fun(C,D6),fun(A,D6)),Uu),Uua),Uub) = aa(C,D6,Uua,aa(A,C,Uu,Uub)) ) ).

% ATP.lambda_813
tff(fact_8992_ATP_Olambda__814,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( real_V7819770556892013058_space(A)
        & topolo4958980785337419405_space(B)
        & topolo4958980785337419405_space(C) )
     => ! [Uu: fun(A,B),Uua: fun(B,C),Uub: A] : aa(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_adk(fun(A,B),fun(fun(B,C),fun(A,C)),Uu),Uua),Uub) = aa(B,C,Uua,aa(A,B,Uu,Uub)) ) ).

% ATP.lambda_814
tff(fact_8993_ATP_Olambda__815,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topological_t2_space(B)
        & topolo4958980785337419405_space(C) )
     => ! [Uu: fun(A,B),Uua: fun(B,C),Uub: A] : aa(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_vo(fun(A,B),fun(fun(B,C),fun(A,C)),Uu),Uua),Uub) = aa(B,C,Uua,aa(A,B,Uu,Uub)) ) ).

% ATP.lambda_815
tff(fact_8994_ATP_Olambda__816,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,B),Uua: fun(B,bool),Uub: A] :
      ( pp(aa(A,bool,aa(fun(B,bool),fun(A,bool),aTP_Lamp_aey(fun(A,B),fun(fun(B,bool),fun(A,bool)),Uu),Uua),Uub))
    <=> pp(aa(B,bool,Uua,aa(A,B,Uu,Uub))) ) ).

% ATP.lambda_816
tff(fact_8995_ATP_Olambda__817,axiom,
    ! [C: $tType,B: $tType,A: $tType,Uu: fun(A,B),Uua: A,Uub: C] : aa(C,B,aa(A,fun(C,B),aTP_Lamp_ql(fun(A,B),fun(A,fun(C,B)),Uu),Uua),Uub) = aa(A,B,Uu,Uua) ).

% ATP.lambda_817
tff(fact_8996_ATP_Olambda__818,axiom,
    ! [Aa: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_Vector_banach(Aa)
        & real_V3459762299906320749_field(Aa) )
     => ! [Uu: fun(A,Aa),Uua: fun(nat,Aa),Uub: A] : aa(A,Aa,aa(fun(nat,Aa),fun(A,Aa),aTP_Lamp_aaa(fun(A,Aa),fun(fun(nat,Aa),fun(A,Aa)),Uu),Uua),Uub) = suminf(Aa,aa(A,fun(nat,Aa),aa(fun(nat,Aa),fun(A,fun(nat,Aa)),aTP_Lamp_zz(fun(A,Aa),fun(fun(nat,Aa),fun(A,fun(nat,Aa))),Uu),Uua),Uub)) ) ).

% ATP.lambda_818
tff(fact_8997_ATP_Olambda__819,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: fun(nat,A),Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_vg(fun(nat,A),fun(A,fun(A,A)),Uu),Uua),Uub) = suminf(A,aa(A,fun(nat,A),aa(A,fun(A,fun(nat,A)),aTP_Lamp_vf(fun(nat,A),fun(A,fun(A,fun(nat,A))),Uu),Uua),Uub)) ) ).

% ATP.lambda_819
tff(fact_8998_ATP_Olambda__820,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Uu: fun(nat,A),Uua: A,Uub: nat] : aa(nat,real,aa(A,fun(nat,real),aTP_Lamp_cl(fun(nat,A),fun(A,fun(nat,real)),Uu),Uua),Uub) = real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,Uub)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),Uub))) ) ).

% ATP.lambda_820
tff(fact_8999_ATP_Olambda__821,axiom,
    ! [A: $tType,I6: $tType] :
      ( ( comm_monoid_mult(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu: fun(I6,A),Uua: fun(I6,A),Uub: I6] : aa(I6,real,aa(fun(I6,A),fun(I6,real),aTP_Lamp_hu(fun(I6,A),fun(fun(I6,A),fun(I6,real)),Uu),Uua),Uub) = real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,aa(I6,A,Uu,Uub)),aa(I6,A,Uua,Uub))) ) ).

% ATP.lambda_821
tff(fact_9000_ATP_Olambda__822,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topological_t2_space(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] :
          ( pp(aa(A,bool,aa(fun(A,B),fun(A,bool),aTP_Lamp_aiv(fun(A,B),fun(fun(A,B),fun(A,bool)),Uu),Uua),Uub))
        <=> ( aa(A,B,Uu,Uub) != aa(A,B,Uua,Uub) ) ) ) ).

% ATP.lambda_822
tff(fact_9001_ATP_Olambda__823,axiom,
    ! [Uu: fun(nat,real),Uua: real,Uub: nat] : aa(nat,real,aa(real,fun(nat,real),aTP_Lamp_aca(fun(nat,real),fun(real,fun(nat,real)),Uu),Uua),Uub) = cos(real,aa(real,real,minus_minus(real,aa(nat,real,Uu,Uub)),Uua)) ).

% ATP.lambda_823
tff(fact_9002_ATP_Olambda__824,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: fun(B,A),Uua: A,Uub: B] :
          ( pp(aa(B,bool,aa(A,fun(B,bool),aTP_Lamp_agn(fun(B,A),fun(A,fun(B,bool)),Uu),Uua),Uub))
        <=> ( aa(B,A,Uu,Uub) != Uua ) ) ) ).

% ATP.lambda_824
tff(fact_9003_ATP_Olambda__825,axiom,
    ! [B: $tType,A: $tType] :
      ( topological_t1_space(A)
     => ! [Uu: fun(B,A),Uua: A,Uub: B] :
          ( pp(aa(B,bool,aa(A,fun(B,bool),aTP_Lamp_afd(fun(B,A),fun(A,fun(B,bool)),Uu),Uua),Uub))
        <=> ( aa(B,A,Uu,Uub) != Uua ) ) ) ).

% ATP.lambda_825
tff(fact_9004_ATP_Olambda__826,axiom,
    ! [A: $tType,B: $tType] :
      ( ( topolo4958980785337419405_space(B)
        & topolo4958980785337419405_space(A) )
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( pp(aa(A,bool,aa(B,fun(A,bool),aTP_Lamp_agl(fun(A,B),fun(B,fun(A,bool)),Uu),Uua),Uub))
        <=> ( aa(A,B,Uu,Uub) != Uua ) ) ) ).

% ATP.lambda_826
tff(fact_9005_ATP_Olambda__827,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( pp(aa(A,bool,aa(B,fun(A,bool),aTP_Lamp_afa(fun(A,B),fun(B,fun(A,bool)),Uu),Uua),Uub))
        <=> ( aa(A,B,Uu,Uub) != Uua ) ) ) ).

% ATP.lambda_827
tff(fact_9006_ATP_Olambda__828,axiom,
    ! [C: $tType,A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: A,Uua: fun(C,A),Uub: C] :
          ( pp(aa(C,bool,aa(fun(C,A),fun(C,bool),aTP_Lamp_agk(A,fun(fun(C,A),fun(C,bool)),Uu),Uua),Uub))
        <=> ( aa(C,A,Uua,Uub) != Uu ) ) ) ).

% ATP.lambda_828
tff(fact_9007_ATP_Olambda__829,axiom,
    ! [Aa: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & archim2362893244070406136eiling(Aa)
        & topolo2564578578187576103pology(Aa) )
     => ! [Uu: fun(A,real),Uua: fun(real,Aa),Uub: A] : aa(A,real,aa(fun(real,Aa),fun(A,real),aTP_Lamp_vd(fun(A,real),fun(fun(real,Aa),fun(A,real)),Uu),Uua),Uub) = aa(int,real,ring_1_of_int(real),archim6421214686448440834_floor(Aa,aa(real,Aa,Uua,aa(A,real,Uu,Uub)))) ) ).

% ATP.lambda_829
tff(fact_9008_ATP_Olambda__830,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,A),Uua: set(B),Uub: A] :
      ( pp(aa(A,bool,aa(set(B),fun(A,bool),aTP_Lamp_akt(fun(B,A),fun(set(B),fun(A,bool)),Uu),Uua),Uub))
    <=> ? [L4: B] :
          ( ( Uub = aa(B,A,Uu,L4) )
          & pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),L4),Uua)) ) ) ).

% ATP.lambda_830
tff(fact_9009_ATP_Olambda__831,axiom,
    ! [B: $tType,C: $tType,Uu: fun(B,set(C)),Uua: fun(C,bool),Uub: B] :
      ( pp(aa(B,bool,aa(fun(C,bool),fun(B,bool),aTP_Lamp_alp(fun(B,set(C)),fun(fun(C,bool),fun(B,bool)),Uu),Uua),Uub))
    <=> ! [X2: C] :
          ( pp(aa(set(C),bool,aa(C,fun(set(C),bool),member(C),X2),aa(B,set(C),Uu,Uub)))
         => pp(aa(C,bool,Uua,X2)) ) ) ).

% ATP.lambda_831
tff(fact_9010_ATP_Olambda__832,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,A),Uua: fun(B,bool),Uub: A] :
      ( pp(aa(A,bool,aa(fun(B,bool),fun(A,bool),aTP_Lamp_akz(fun(B,A),fun(fun(B,bool),fun(A,bool)),Uu),Uua),Uub))
    <=> ? [X2: B] :
          ( ( Uub = aa(B,A,Uu,X2) )
          & pp(aa(B,bool,Uua,X2)) ) ) ).

% ATP.lambda_832
tff(fact_9011_ATP_Olambda__833,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,bool),Uua: fun(A,B),Uub: B] :
      ( pp(aa(B,bool,aa(fun(A,B),fun(B,bool),aTP_Lamp_alh(fun(A,bool),fun(fun(A,B),fun(B,bool)),Uu),Uua),Uub))
    <=> ? [X2: A] :
          ( ( Uub = aa(A,B,Uua,X2) )
          & pp(aa(A,bool,Uu,X2)) ) ) ).

% ATP.lambda_833
tff(fact_9012_ATP_Olambda__834,axiom,
    ! [B: $tType,A: $tType,Uu: set(A),Uua: fun(B,fun(A,bool)),Uub: B] :
      ( pp(aa(B,bool,aa(fun(B,fun(A,bool)),fun(B,bool),aTP_Lamp_aln(set(A),fun(fun(B,fun(A,bool)),fun(B,bool)),Uu),Uua),Uub))
    <=> ! [X2: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X2),Uu))
         => pp(aa(A,bool,aa(B,fun(A,bool),Uua,Uub),X2)) ) ) ).

% ATP.lambda_834
tff(fact_9013_ATP_Olambda__835,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [Uu: fun(nat,A),Uua: fun(nat,real),Uub: nat] :
          ( pp(aa(nat,bool,aa(fun(nat,real),fun(nat,bool),aTP_Lamp_ahq(fun(nat,A),fun(fun(nat,real),fun(nat,bool)),Uu),Uua),Uub))
        <=> ! [N5: nat] :
              ( pp(aa(nat,bool,ord_less_eq(nat,Uub),N5))
             => pp(aa(real,bool,ord_less_eq(real,real_V7770717601297561774m_norm(A,aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,Uu),set_or7035219750837199246ssThan(nat,Uub,N5)))),aa(nat,real,Uua,Uub))) ) ) ) ).

% ATP.lambda_835
tff(fact_9014_ATP_Olambda__836,axiom,
    ! [A: $tType] :
      ( ( real_V8037385150606011577_space(A)
        & real_V822414075346904944vector(A) )
     => ! [Uu: fun(nat,A),Uua: fun(nat,real),Uub: nat] :
          ( pp(aa(nat,bool,aa(fun(nat,real),fun(nat,bool),aTP_Lamp_aie(fun(nat,A),fun(fun(nat,real),fun(nat,bool)),Uu),Uua),Uub))
        <=> ! [A6: nat] :
              ( pp(aa(nat,bool,ord_less_eq(nat,Uub),A6))
             => ! [B7: nat] :
                  ( pp(aa(nat,bool,ord_less(nat,A6),B7))
                 => pp(aa(real,bool,ord_less_eq(real,real_V7770717601297561774m_norm(A,aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,Uu),set_or3652927894154168847AtMost(nat,A6,B7)))),aa(nat,real,Uua,A6))) ) ) ) ) ).

% ATP.lambda_836
tff(fact_9015_ATP_Olambda__837,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,bool),Uua: fun(B,fun(A,bool)),Uub: B] :
      ( pp(aa(B,bool,aa(fun(B,fun(A,bool)),fun(B,bool),aTP_Lamp_akq(fun(A,bool),fun(fun(B,fun(A,bool)),fun(B,bool)),Uu),Uua),Uub))
    <=> ? [Y7: A] :
          ( pp(aa(A,bool,Uu,Y7))
          & pp(aa(A,bool,aa(B,fun(A,bool),Uua,Uub),Y7)) ) ) ).

% ATP.lambda_837
tff(fact_9016_ATP_Olambda__838,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,A),Uua: fun(B,fun(B,bool)),Uub: product_prod(A,A)] :
      ( pp(aa(product_prod(A,A),bool,aa(fun(B,fun(B,bool)),fun(product_prod(A,A),bool),aTP_Lamp_alk(fun(B,A),fun(fun(B,fun(B,bool)),fun(product_prod(A,A),bool)),Uu),Uua),Uub))
    <=> ? [A6: B,B7: B] :
          ( ( Uub = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(B,A,Uu,A6)),aa(B,A,Uu,B7)) )
          & pp(aa(B,bool,aa(B,fun(B,bool),Uua,A6),B7)) ) ) ).

% ATP.lambda_838
tff(fact_9017_ATP_Olambda__839,axiom,
    ! [A: $tType,Uu: set(A),Uua: set(list(A)),Uub: list(A)] :
      ( pp(aa(list(A),bool,aa(set(list(A)),fun(list(A),bool),aTP_Lamp_amz(set(A),fun(set(list(A)),fun(list(A),bool)),Uu),Uua),Uub))
    <=> ? [X2: A,Xs4: list(A)] :
          ( ( Uub = aa(list(A),list(A),cons(A,X2),Xs4) )
          & pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X2),Uu))
          & pp(aa(set(list(A)),bool,aa(list(A),fun(set(list(A)),bool),member(list(A)),Xs4),Uua)) ) ) ).

% ATP.lambda_839
tff(fact_9018_ATP_Olambda__840,axiom,
    ! [C: $tType,A: $tType,B: $tType,D6: $tType,Uu: fun(B,fun(C,fun(D6,A))),Uua: D6,Uub: B,Uuc: C] : aa(C,A,aa(B,fun(C,A),aa(D6,fun(B,fun(C,A)),aTP_Lamp_ly(fun(B,fun(C,fun(D6,A))),fun(D6,fun(B,fun(C,A))),Uu),Uua),Uub),Uuc) = aa(D6,A,aa(C,fun(D6,A),aa(B,fun(C,fun(D6,A)),Uu,Uub),Uuc),Uua) ).

% ATP.lambda_840
tff(fact_9019_ATP_Olambda__841,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: A,Uua: A,Uub: nat,Uuc: nat] : aa(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_kb(A,fun(A,fun(nat,fun(nat,A))),Uu),Uua),Uub),Uuc) = if(A,fconj(aa(nat,bool,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Uub),aa(nat,bool,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Uuc)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(int,real,ring_1_of_int(real),aa(int,int,aa(int,fun(int,int),times_times(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(int,int,uminus_uminus(int),one_one(int))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Uub),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,binomial(Uub),Uuc))))),semiring_char_0_fact(real,Uub))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu),Uuc))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),aa(nat,nat,minus_minus(nat,Uub),Uuc))),zero_zero(A)) ) ).

% ATP.lambda_841
tff(fact_9020_ATP_Olambda__842,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: A,Uua: A,Uub: nat,Uuc: nat] : aa(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_jx(A,fun(A,fun(nat,fun(nat,A))),Uu),Uua),Uub),Uuc) = if(A,fconj(aa(nat,bool,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Uub),aa(bool,bool,fNot,aa(nat,bool,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Uuc))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(int,real,ring_1_of_int(real),aa(int,int,aa(int,fun(int,int),times_times(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(int,int,uminus_uminus(int),one_one(int))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Uub),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,binomial(Uub),Uuc))))),semiring_char_0_fact(real,Uub)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu),Uuc))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),aa(nat,nat,minus_minus(nat,Uub),Uuc))),zero_zero(A)) ) ).

% ATP.lambda_842
tff(fact_9021_ATP_Olambda__843,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: A,Uua: A,Uub: nat,Uuc: nat] : aa(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_jz(A,fun(A,fun(nat,fun(nat,A))),Uu),Uua),Uub),Uuc) = if(A,aa(nat,bool,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Uub),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(int,real,ring_1_of_int(real),aa(int,int,aa(int,fun(int,int),times_times(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(int,int,uminus_uminus(int),one_one(int))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Uub),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,binomial(Uub),Uuc))))),semiring_char_0_fact(real,Uub))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu),Uuc))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),aa(nat,nat,minus_minus(nat,Uub),Uuc))),zero_zero(A)) ) ).

% ATP.lambda_843
tff(fact_9022_ATP_Olambda__844,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: nat,Uua: fun(nat,A),Uub: fun(nat,A),Uuc: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aa(fun(nat,A),fun(fun(nat,A),fun(nat,A)),aTP_Lamp_iw(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),Uu),Uua),Uub),Uuc) = if(A,aa(nat,bool,ord_less(nat,Uuc),Uu),aa(nat,A,Uua,Uuc),if(A,aa(nat,bool,fequal(nat,Uuc),Uu),zero_zero(A),aa(nat,A,Uub,aa(nat,nat,minus_minus(nat,Uuc),aa(nat,nat,suc,zero_zero(nat)))))) ) ).

% ATP.lambda_844
tff(fact_9023_ATP_Olambda__845,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: nat,Uua: fun(nat,A),Uub: fun(nat,A),Uuc: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aa(fun(nat,A),fun(fun(nat,A),fun(nat,A)),aTP_Lamp_iy(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),Uu),Uua),Uub),Uuc) = if(A,aa(nat,bool,ord_less(nat,Uuc),Uu),aa(nat,A,Uua,Uuc),if(A,aa(nat,bool,fequal(nat,Uuc),Uu),one_one(A),aa(nat,A,Uub,aa(nat,nat,minus_minus(nat,Uuc),aa(nat,nat,suc,zero_zero(nat)))))) ) ).

% ATP.lambda_845
tff(fact_9024_ATP_Olambda__846,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo1944317154257567458pology(A)
        & topolo4958980785337419405_space(B) )
     => ! [Uu: A,Uua: fun(A,B),Uub: fun(A,B),Uuc: A] : aa(A,B,aa(fun(A,B),fun(A,B),aa(fun(A,B),fun(fun(A,B),fun(A,B)),aTP_Lamp_ads(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))),Uu),Uua),Uub),Uuc) = if(B,aa(A,bool,ord_less_eq(A,Uuc),Uu),aa(A,B,Uua,Uuc),aa(A,B,Uub,Uuc)) ) ).

% ATP.lambda_846
tff(fact_9025_ATP_Olambda__847,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: nat,Uua: fun(nat,A),Uub: fun(nat,A),Uuc: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aa(fun(nat,A),fun(fun(nat,A),fun(nat,A)),aTP_Lamp_iz(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),Uu),Uua),Uub),Uuc) = if(A,aa(nat,bool,ord_less(nat,Uuc),Uu),aa(nat,A,Uua,Uuc),aa(nat,A,Uub,Uuc)) ) ).

% ATP.lambda_847
tff(fact_9026_ATP_Olambda__848,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: nat,Uua: fun(nat,A),Uub: fun(nat,A),Uuc: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aa(fun(nat,A),fun(fun(nat,A),fun(nat,A)),aTP_Lamp_ix(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),Uu),Uua),Uub),Uuc) = if(A,aa(nat,bool,ord_less(nat,Uuc),Uu),aa(nat,A,Uua,Uuc),aa(nat,A,Uub,Uuc)) ) ).

% ATP.lambda_848
tff(fact_9027_ATP_Olambda__849,axiom,
    ! [A: $tType] :
      ( ( topolo1287966508704411220up_add(A)
        & topological_t2_space(A) )
     => ! [Uu: fun(nat,A),Uua: set(nat),Uub: fun(nat,A),Uuc: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aa(set(nat),fun(fun(nat,A),fun(nat,A)),aTP_Lamp_fh(fun(nat,A),fun(set(nat),fun(fun(nat,A),fun(nat,A))),Uu),Uua),Uub),Uuc) = if(A,aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),Uuc),Uua),aa(nat,A,Uub,Uuc),aa(nat,A,Uu,Uuc)) ) ).

% ATP.lambda_849
tff(fact_9028_ATP_Olambda__850,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: B,Uua: fun(B,A),Uub: fun(B,A),Uuc: B] : aa(B,A,aa(fun(B,A),fun(B,A),aa(fun(B,A),fun(fun(B,A),fun(B,A)),aTP_Lamp_ls(B,fun(fun(B,A),fun(fun(B,A),fun(B,A))),Uu),Uua),Uub),Uuc) = if(A,aa(B,bool,fequal(B,Uuc),Uu),aa(B,A,Uua,Uuc),aa(B,A,Uub,Uuc)) ) ).

% ATP.lambda_850
tff(fact_9029_ATP_Olambda__851,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: B,Uua: fun(B,A),Uub: fun(B,A),Uuc: B] : aa(B,A,aa(fun(B,A),fun(B,A),aa(fun(B,A),fun(fun(B,A),fun(B,A)),aTP_Lamp_lr(B,fun(fun(B,A),fun(fun(B,A),fun(B,A))),Uu),Uua),Uub),Uuc) = if(A,aa(B,bool,fequal(B,Uuc),Uu),aa(B,A,Uua,Uuc),aa(B,A,Uub,Uuc)) ) ).

% ATP.lambda_851
tff(fact_9030_ATP_Olambda__852,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,option(B)),Uua: A,Uub: B,Uuc: A] : aa(A,option(B),aa(B,fun(A,option(B)),aa(A,fun(B,fun(A,option(B))),aTP_Lamp_my(fun(A,option(B)),fun(A,fun(B,fun(A,option(B)))),Uu),Uua),Uub),Uuc) = if(option(B),aa(A,bool,fequal(A,Uuc),Uua),aa(B,option(B),some(B),Uub),aa(A,option(B),Uu,Uuc)) ).

% ATP.lambda_852
tff(fact_9031_ATP_Olambda__853,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: fun(A,B),Uua: fun(A,bool),Uub: fun(A,B),Uuc: A] : aa(A,B,aa(fun(A,B),fun(A,B),aa(fun(A,bool),fun(fun(A,B),fun(A,B)),aTP_Lamp_vx(fun(A,B),fun(fun(A,bool),fun(fun(A,B),fun(A,B))),Uu),Uua),Uub),Uuc) = if(B,aa(A,bool,Uua,Uuc),aa(A,B,Uu,Uuc),aa(A,B,Uub,Uuc)) ) ).

% ATP.lambda_853
tff(fact_9032_ATP_Olambda__854,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Uu: A,Uua: A,Uub: nat,Uuc: nat] : aa(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_eq(A,fun(A,fun(nat,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_ep(A,fun(A,fun(nat,fun(nat,A))),Uu),Uua),Uub)),set_ord_lessThan(nat,aa(nat,nat,minus_minus(nat,aa(nat,nat,minus_minus(nat,Uub),aa(nat,nat,suc,zero_zero(nat)))),Uuc))) ) ).

% ATP.lambda_854
tff(fact_9033_ATP_Olambda__855,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( semiring_0(B)
     => ! [Uu: fun(A,B),Uua: fun(C,B),Uub: set(C),Uuc: A] : aa(A,B,aa(set(C),fun(A,B),aa(fun(C,B),fun(set(C),fun(A,B)),aTP_Lamp_dn(fun(A,B),fun(fun(C,B),fun(set(C),fun(A,B))),Uu),Uua),Uub),Uuc) = aa(set(C),B,groups7311177749621191930dd_sum(C,B,aa(A,fun(C,B),aa(fun(C,B),fun(A,fun(C,B)),aTP_Lamp_dm(fun(A,B),fun(fun(C,B),fun(A,fun(C,B))),Uu),Uua),Uuc)),Uub) ) ).

% ATP.lambda_855
tff(fact_9034_ATP_Olambda__856,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [Uu: nat,Uua: A,Uub: A,Uuc: nat] : aa(nat,A,aa(A,fun(nat,A),aa(A,fun(A,fun(nat,A)),aTP_Lamp_jc(nat,fun(A,fun(A,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),times_times(A),if(A,aa(nat,bool,fequal(nat,Uuc),zero_zero(nat)),aa(A,A,uminus_uminus(A),Uub),if(A,aa(nat,bool,fequal(nat,Uuc),Uu),one_one(A),zero_zero(A)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),Uuc)) ) ).

% ATP.lambda_856
tff(fact_9035_ATP_Olambda__857,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: set(B),Uua: fun(B,fun(C,A)),Uub: fun(B,fun(C,bool)),Uuc: C] : aa(C,A,aa(fun(B,fun(C,bool)),fun(C,A),aa(fun(B,fun(C,A)),fun(fun(B,fun(C,bool)),fun(C,A)),aTP_Lamp_hj(set(B),fun(fun(B,fun(C,A)),fun(fun(B,fun(C,bool)),fun(C,A))),Uu),Uua),Uub),Uuc) = groups7121269368397514597t_prod(B,A,aa(C,fun(B,A),aTP_Lamp_hc(fun(B,fun(C,A)),fun(C,fun(B,A)),Uua),Uuc),collect(B,aa(C,fun(B,bool),aa(fun(B,fun(C,bool)),fun(C,fun(B,bool)),aTP_Lamp_dt(set(B),fun(fun(B,fun(C,bool)),fun(C,fun(B,bool))),Uu),Uub),Uuc))) ) ).

% ATP.lambda_857
tff(fact_9036_ATP_Olambda__858,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: set(B),Uua: fun(B,fun(C,A)),Uub: fun(B,fun(C,bool)),Uuc: C] : aa(C,A,aa(fun(B,fun(C,bool)),fun(C,A),aa(fun(B,fun(C,A)),fun(fun(B,fun(C,bool)),fun(C,A)),aTP_Lamp_du(set(B),fun(fun(B,fun(C,A)),fun(fun(B,fun(C,bool)),fun(C,A))),Uu),Uua),Uub),Uuc) = aa(set(B),A,groups7311177749621191930dd_sum(B,A,aa(C,fun(B,A),aTP_Lamp_dg(fun(B,fun(C,A)),fun(C,fun(B,A)),Uua),Uuc)),collect(B,aa(C,fun(B,bool),aa(fun(B,fun(C,bool)),fun(C,fun(B,bool)),aTP_Lamp_dt(set(B),fun(fun(B,fun(C,bool)),fun(C,fun(B,bool))),Uu),Uub),Uuc))) ) ).

% ATP.lambda_858
tff(fact_9037_ATP_Olambda__859,axiom,
    ! [Uu: fun(nat,nat),Uua: fun(nat,nat),Uub: nat,Uuc: nat] : aa(nat,nat,aa(nat,fun(nat,nat),aa(fun(nat,nat),fun(nat,fun(nat,nat)),aTP_Lamp_iv(fun(nat,nat),fun(fun(nat,nat),fun(nat,fun(nat,nat))),Uu),Uua),Uub),Uuc) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,aa(nat,fun(nat,nat),aa(fun(nat,nat),fun(nat,fun(nat,nat)),aTP_Lamp_iu(fun(nat,nat),fun(fun(nat,nat),fun(nat,fun(nat,nat))),Uu),Uua),Uuc)),set_ord_atMost(nat,Uuc))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),Uub),Uuc)) ).

% ATP.lambda_859
tff(fact_9038_ATP_Olambda__860,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [Uu: fun(nat,A),Uua: fun(nat,A),Uub: A,Uuc: nat] : aa(nat,A,aa(A,fun(nat,A),aa(fun(nat,A),fun(A,fun(nat,A)),aTP_Lamp_ir(fun(nat,A),fun(fun(nat,A),fun(A,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aa(fun(nat,A),fun(nat,fun(nat,A)),aTP_Lamp_iq(fun(nat,A),fun(fun(nat,A),fun(nat,fun(nat,A))),Uu),Uua),Uuc)),set_ord_atMost(nat,Uuc))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uub),Uuc)) ) ).

% ATP.lambda_860
tff(fact_9039_ATP_Olambda__861,axiom,
    ! [Uu: fun(nat,fun(real,real)),Uua: nat,Uub: real,Uuc: nat] : aa(nat,real,aa(real,fun(nat,real),aa(nat,fun(real,fun(nat,real)),aTP_Lamp_sq(fun(nat,fun(real,real)),fun(nat,fun(real,fun(nat,real))),Uu),Uua),Uub),Uuc) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(nat,fun(real,real),Uu,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,suc,Uua)),Uuc)),zero_zero(real))),semiring_char_0_fact(real,Uuc))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Uub),Uuc)) ).

% ATP.lambda_861
tff(fact_9040_ATP_Olambda__862,axiom,
    ! [Uu: fun(nat,fun(real,real)),Uua: nat,Uub: real,Uuc: nat] : aa(nat,real,aa(real,fun(nat,real),aa(nat,fun(real,fun(nat,real)),aTP_Lamp_so(fun(nat,fun(real,real)),fun(nat,fun(real,fun(nat,real))),Uu),Uua),Uub),Uuc) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(nat,fun(real,real),Uu,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uua),Uuc)),zero_zero(real))),semiring_char_0_fact(real,Uuc))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Uub),Uuc)) ).

% ATP.lambda_862
tff(fact_9041_ATP_Olambda__863,axiom,
    ! [Uu: fun(nat,fun(real,real)),Uua: real,Uub: real,Uuc: nat] : aa(nat,real,aa(real,fun(nat,real),aa(real,fun(real,fun(nat,real)),aTP_Lamp_sm(fun(nat,fun(real,real)),fun(real,fun(real,fun(nat,real))),Uu),Uua),Uub),Uuc) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(nat,fun(real,real),Uu,Uuc),Uua)),semiring_char_0_fact(real,Uuc))),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,minus_minus(real,Uub),Uua)),Uuc)) ).

% ATP.lambda_863
tff(fact_9042_ATP_Olambda__864,axiom,
    ! [Uu: fun(nat,fun(real,real)),Uua: real,Uub: real,Uuc: nat] : aa(nat,real,aa(real,fun(nat,real),aa(real,fun(real,fun(nat,real)),aTP_Lamp_sn(fun(nat,fun(real,real)),fun(real,fun(real,fun(nat,real))),Uu),Uua),Uub),Uuc) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(nat,fun(real,real),Uu,Uuc),Uub)),semiring_char_0_fact(real,Uuc))),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,minus_minus(real,Uua),Uub)),Uuc)) ).

% ATP.lambda_864
tff(fact_9043_ATP_Olambda__865,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [Uu: A,Uua: A,Uub: nat,Uuc: nat] : aa(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_fk(A,fun(A,fun(nat,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Uu),Uua)),aa(nat,nat,minus_minus(nat,Uub),Uuc))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu),Uuc))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu),Uub)) ) ).

% ATP.lambda_865
tff(fact_9044_ATP_Olambda__866,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Uu: A,Uua: A,Uub: nat,Uuc: nat] : aa(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_ep(A,fun(A,fun(nat,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Uua),Uu)),Uuc)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),aa(nat,nat,minus_minus(nat,aa(nat,nat,minus_minus(nat,Uub),aa(num,nat,numeral_numeral(nat),bit0(one2)))),Uuc))) ) ).

% ATP.lambda_866
tff(fact_9045_ATP_Olambda__867,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: fun(A,A),Uua: A,Uub: fun(nat,A),Uuc: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aa(A,fun(fun(nat,A),fun(nat,A)),aTP_Lamp_abx(fun(A,A),fun(A,fun(fun(nat,A),fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,minus_minus(A,aa(A,A,Uu,aa(A,A,aa(A,fun(A,A),plus_plus(A),Uua),aa(nat,A,Uub,Uuc)))),aa(A,A,Uu,Uua))),aa(nat,A,Uub,Uuc)) ) ).

% ATP.lambda_867
tff(fact_9046_ATP_Olambda__868,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,option(B)),Uua: fun(product_prod(A,B),bool),Uub: A,Uuc: B] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),aa(fun(product_prod(A,B),bool),fun(A,fun(B,bool)),aTP_Lamp_mm(fun(A,option(B)),fun(fun(product_prod(A,B),bool),fun(A,fun(B,bool))),Uu),Uua),Uub),Uuc))
    <=> ( ( aa(A,option(B),Uu,Uub) = aa(B,option(B),some(B),Uuc) )
        & pp(aa(product_prod(A,B),bool,Uua,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uub),Uuc))) ) ) ).

% ATP.lambda_868
tff(fact_9047_ATP_Olambda__869,axiom,
    ! [A: $tType,B: $tType] :
      ( topological_t2_space(B)
     => ! [Uu: fun(A,B),Uua: set(B),Uub: B,Uuc: A] :
          ( pp(aa(A,bool,aa(B,fun(A,bool),aa(set(B),fun(B,fun(A,bool)),aTP_Lamp_afh(fun(A,B),fun(set(B),fun(B,fun(A,bool))),Uu),Uua),Uub),Uuc))
        <=> ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),aa(A,B,Uu,Uuc)),Uua))
            & ( aa(A,B,Uu,Uuc) != Uub ) ) ) ) ).

% ATP.lambda_869
tff(fact_9048_ATP_Olambda__870,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: fun(B,A),Uua: A,Uub: set(A),Uuc: B] :
          ( pp(aa(B,bool,aa(set(A),fun(B,bool),aa(A,fun(set(A),fun(B,bool)),aTP_Lamp_afz(fun(B,A),fun(A,fun(set(A),fun(B,bool))),Uu),Uua),Uub),Uuc))
        <=> ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(B,A,Uu,Uuc)),Uub))
            & ( aa(B,A,Uu,Uuc) != Uua ) ) ) ) ).

% ATP.lambda_870
tff(fact_9049_ATP_Olambda__871,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V7819770556892013058_space(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: real,Uuc: A] :
          ( pp(aa(A,bool,aa(real,fun(A,bool),aa(B,fun(real,fun(A,bool)),aTP_Lamp_ahr(fun(A,B),fun(B,fun(real,fun(A,bool))),Uu),Uua),Uub),Uuc))
        <=> pp(aa(real,bool,ord_less_eq(real,real_V557655796197034286t_dist(B,aa(A,B,Uu,Uuc),Uua)),Uub)) ) ) ).

% ATP.lambda_871
tff(fact_9050_ATP_Olambda__872,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [Uu: fun(B,A),Uua: A,Uub: real,Uuc: B] :
          ( pp(aa(B,bool,aa(real,fun(B,bool),aa(A,fun(real,fun(B,bool)),aTP_Lamp_ahf(fun(B,A),fun(A,fun(real,fun(B,bool))),Uu),Uua),Uub),Uuc))
        <=> pp(aa(real,bool,ord_less(real,real_V557655796197034286t_dist(A,aa(B,A,Uu,Uuc),Uua)),Uub)) ) ) ).

% ATP.lambda_872
tff(fact_9051_ATP_Olambda__873,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Uu: A,Uua: A,Uub: nat,Uuc: nat] : aa(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_js(A,fun(A,fun(nat,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,binomial(Uub),Uuc))),comm_s3205402744901411588hammer(A,Uu,Uuc))),comm_s3205402744901411588hammer(A,Uua,aa(nat,nat,minus_minus(nat,Uub),Uuc))) ) ).

% ATP.lambda_873
tff(fact_9052_ATP_Olambda__874,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] : aa(nat,nat,aa(nat,fun(nat,nat),aa(nat,fun(nat,fun(nat,nat)),aTP_Lamp_jq(nat,fun(nat,fun(nat,fun(nat,nat))),Uu),Uua),Uub),Uuc) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,semiring_1_of_nat(nat),aa(nat,nat,binomial(Uub),Uuc))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),Uu),Uuc))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),Uua),aa(nat,nat,minus_minus(nat,Uub),Uuc))) ).

% ATP.lambda_874
tff(fact_9053_ATP_Olambda__875,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Uu: A,Uua: A,Uub: nat,Uuc: nat] : aa(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_jr(A,fun(A,fun(nat,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,binomial(Uub),Uuc))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu),Uuc))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),aa(nat,nat,minus_minus(nat,Uub),Uuc))) ) ).

% ATP.lambda_875
tff(fact_9054_ATP_Olambda__876,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu: A,Uua: A,Uub: nat,Uuc: nat] : aa(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_ks(A,fun(A,fun(nat,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),semiring_char_0_fact(real,Uuc))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu),Uuc))),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),semiring_char_0_fact(real,aa(nat,nat,minus_minus(nat,Uub),Uuc)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),aa(nat,nat,minus_minus(nat,Uub),Uuc)))) ) ).

% ATP.lambda_876
tff(fact_9055_ATP_Olambda__877,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: nat,Uub: list(A),Uuc: list(A)] :
      ( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),aa(nat,fun(list(A),fun(list(A),bool)),aTP_Lamp_amr(set(product_prod(A,A)),fun(nat,fun(list(A),fun(list(A),bool))),Uu),Uua),Uub),Uuc))
    <=> ( ( aa(list(A),nat,size_size(list(A)),Uub) = Uua )
        & ( aa(list(A),nat,size_size(list(A)),Uuc) = Uua )
        & ? [Xys: list(A),X2: A,Y7: A,Xs3: list(A),Ys3: list(A)] :
            ( ( Uub = append(A,Xys,aa(list(A),list(A),cons(A,X2),Xs3)) )
            & ( Uuc = append(A,Xys,aa(list(A),list(A),cons(A,Y7),Ys3)) )
            & pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Y7)),Uu)) ) ) ) ).

% ATP.lambda_877
tff(fact_9056_ATP_Olambda__878,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [Uu: A,Uua: nat,Uub: A,Uuc: nat] : aa(nat,A,aa(A,fun(nat,A),aa(nat,fun(A,fun(nat,A)),aTP_Lamp_fm(A,fun(nat,fun(A,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uub),aa(nat,nat,minus_minus(nat,Uua),aa(nat,nat,suc,Uuc)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu),Uuc)) ) ).

% ATP.lambda_878
tff(fact_9057_ATP_Olambda__879,axiom,
    ! [A: $tType,Uu: bool,Uua: A,Uub: A,Uuc: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),aa(A,fun(A,fun(A,bool)),aTP_Lamp_lh(bool,fun(A,fun(A,fun(A,bool))),Uu),Uua),Uub),Uuc))
    <=> ( ( pp(Uu)
         => ( Uuc = Uua ) )
        & ( ~ pp(Uu)
         => ( Uuc = Uub ) ) ) ) ).

% ATP.lambda_879
tff(fact_9058_ATP_Olambda__880,axiom,
    ! [B: $tType,A: $tType,Uu: fun(B,A),Uua: set(B),Uub: fun(A,bool),Uuc: A] :
      ( pp(aa(A,bool,aa(fun(A,bool),fun(A,bool),aa(set(B),fun(fun(A,bool),fun(A,bool)),aTP_Lamp_nw(fun(B,A),fun(set(B),fun(fun(A,bool),fun(A,bool))),Uu),Uua),Uub),Uuc))
    <=> ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Uuc),aa(set(B),set(A),image(B,A,Uu),Uua)))
        & pp(aa(A,bool,Uub,Uuc)) ) ) ).

% ATP.lambda_880
tff(fact_9059_ATP_Olambda__881,axiom,
    ! [B: $tType,C: $tType,Uu: set(C),Uua: fun(B,fun(C,bool)),Uub: B,Uuc: C] :
      ( pp(aa(C,bool,aa(B,fun(C,bool),aa(fun(B,fun(C,bool)),fun(B,fun(C,bool)),aTP_Lamp_dr(set(C),fun(fun(B,fun(C,bool)),fun(B,fun(C,bool))),Uu),Uua),Uub),Uuc))
    <=> ( pp(aa(set(C),bool,aa(C,fun(set(C),bool),member(C),Uuc),Uu))
        & pp(aa(C,bool,aa(B,fun(C,bool),Uua,Uub),Uuc)) ) ) ).

% ATP.lambda_881
tff(fact_9060_ATP_Olambda__882,axiom,
    ! [B: $tType,C: $tType,Uu: set(B),Uua: fun(B,fun(C,bool)),Uub: C,Uuc: B] :
      ( pp(aa(B,bool,aa(C,fun(B,bool),aa(fun(B,fun(C,bool)),fun(C,fun(B,bool)),aTP_Lamp_dt(set(B),fun(fun(B,fun(C,bool)),fun(C,fun(B,bool))),Uu),Uua),Uub),Uuc))
    <=> ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),Uuc),Uu))
        & pp(aa(C,bool,aa(B,fun(C,bool),Uua,Uuc),Uub)) ) ) ).

% ATP.lambda_882
tff(fact_9061_ATP_Olambda__883,axiom,
    ! [A: $tType,B: $tType,Uu: set(A),Uua: fun(A,fun(B,bool)),Uub: B,Uuc: A] :
      ( pp(aa(A,bool,aa(B,fun(A,bool),aa(fun(A,fun(B,bool)),fun(B,fun(A,bool)),aTP_Lamp_ba(set(A),fun(fun(A,fun(B,bool)),fun(B,fun(A,bool))),Uu),Uua),Uub),Uuc))
    <=> ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Uuc),Uu))
        & pp(aa(B,bool,aa(A,fun(B,bool),Uua,Uuc),Uub)) ) ) ).

% ATP.lambda_883
tff(fact_9062_ATP_Olambda__884,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [Uu: A,Uua: A,Uub: nat,Uuc: nat] : aa(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_fl(A,fun(A,fun(nat,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu),Uuc)),aa(A,A,minus_minus(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Uu),Uua)),aa(nat,nat,minus_minus(nat,Uub),Uuc))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu),aa(nat,nat,minus_minus(nat,Uub),Uuc)))) ) ).

% ATP.lambda_884
tff(fact_9063_ATP_Olambda__885,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: set(A),Uua: fun(A,A),Uub: fun(A,A),Uuc: A] :
          ( pp(aa(A,bool,aa(fun(A,A),fun(A,bool),aa(fun(A,A),fun(fun(A,A),fun(A,bool)),aTP_Lamp_aga(set(A),fun(fun(A,A),fun(fun(A,A),fun(A,bool))),Uu),Uua),Uub),Uuc))
        <=> ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Uuc),Uu))
           => ( aa(A,A,Uua,Uuc) = aa(A,A,Uub,Uuc) ) ) ) ) ).

% ATP.lambda_885
tff(fact_9064_ATP_Olambda__886,axiom,
    ! [B: $tType,A: $tType,Uu: set(A),Uua: fun(A,B),Uub: A,Uuc: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),aa(fun(A,B),fun(A,fun(A,bool)),aTP_Lamp_ny(set(A),fun(fun(A,B),fun(A,fun(A,bool))),Uu),Uua),Uub),Uuc))
    <=> ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Uuc),Uu))
        & ( aa(A,B,Uua,Uuc) = aa(A,B,Uua,Uub) ) ) ) ).

% ATP.lambda_886
tff(fact_9065_ATP_Olambda__887,axiom,
    ! [B: $tType,C: $tType,Uu: set(B),Uua: fun(B,C),Uub: C,Uuc: B] :
      ( pp(aa(B,bool,aa(C,fun(B,bool),aa(fun(B,C),fun(C,fun(B,bool)),aTP_Lamp_oa(set(B),fun(fun(B,C),fun(C,fun(B,bool))),Uu),Uua),Uub),Uuc))
    <=> ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),Uuc),Uu))
        & ( aa(B,C,Uua,Uuc) = Uub ) ) ) ).

% ATP.lambda_887
tff(fact_9066_ATP_Olambda__888,axiom,
    ! [A: $tType,B: $tType,Uu: set(A),Uua: fun(A,B),Uub: B,Uuc: A] :
      ( pp(aa(A,bool,aa(B,fun(A,bool),aa(fun(A,B),fun(B,fun(A,bool)),aTP_Lamp_qg(set(A),fun(fun(A,B),fun(B,fun(A,bool))),Uu),Uua),Uub),Uuc))
    <=> ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Uuc),Uu))
        & ( aa(A,B,Uua,Uuc) = Uub ) ) ) ).

% ATP.lambda_888
tff(fact_9067_ATP_Olambda__889,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,B),Uua: set(A),Uub: B,Uuc: A] :
      ( pp(aa(A,bool,aa(B,fun(A,bool),aa(set(A),fun(B,fun(A,bool)),aTP_Lamp_qx(fun(A,B),fun(set(A),fun(B,fun(A,bool))),Uu),Uua),Uub),Uuc))
    <=> ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Uuc),Uua))
        & ( aa(A,B,Uu,Uuc) = Uub ) ) ) ).

% ATP.lambda_889
tff(fact_9068_ATP_Olambda__890,axiom,
    ! [B: $tType,A: $tType,Uu: set(A),Uua: fun(B,A),Uub: set(B),Uuc: B] :
      ( pp(aa(B,bool,aa(set(B),fun(B,bool),aa(fun(B,A),fun(set(B),fun(B,bool)),aTP_Lamp_qj(set(A),fun(fun(B,A),fun(set(B),fun(B,bool))),Uu),Uua),Uub),Uuc))
    <=> ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),Uuc),Uub))
        & pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(B,A,Uua,Uuc)),Uu)) ) ) ).

% ATP.lambda_890
tff(fact_9069_ATP_Olambda__891,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [Uu: A,Uua: nat,Uub: A,Uuc: nat] : aa(nat,A,aa(A,fun(nat,A),aa(nat,fun(A,fun(nat,A)),aTP_Lamp_fn(A,fun(nat,fun(A,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu),Uuc)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uub),aa(nat,nat,minus_minus(nat,Uua),Uuc))) ) ).

% ATP.lambda_891
tff(fact_9070_ATP_Olambda__892,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] : aa(nat,nat,aa(nat,fun(nat,nat),aa(nat,fun(nat,fun(nat,nat)),aTP_Lamp_jp(nat,fun(nat,fun(nat,fun(nat,nat))),Uu),Uua),Uub),Uuc) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,binomial(Uu),Uuc)),aa(nat,nat,binomial(Uua),aa(nat,nat,minus_minus(nat,Uub),Uuc))) ).

% ATP.lambda_892
tff(fact_9071_ATP_Olambda__893,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),aa(int,fun(int,fun(int,bool)),aTP_Lamp_nf(int,fun(int,fun(int,fun(int,bool))),Uu),Uua),Uub),Uuc))
    <=> pp(aa(int,bool,ord_less_eq(int,aa(int,int,aa(int,fun(int,int),times_times(int),Uu),Uuc)),aa(int,int,aa(int,fun(int,int),times_times(int),Uua),Uub))) ) ).

% ATP.lambda_893
tff(fact_9072_ATP_Olambda__894,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),aa(int,fun(int,fun(int,bool)),aTP_Lamp_nc(int,fun(int,fun(int,fun(int,bool))),Uu),Uua),Uub),Uuc))
    <=> pp(aa(int,bool,ord_less(int,aa(int,int,aa(int,fun(int,int),times_times(int),Uu),Uuc)),aa(int,int,aa(int,fun(int,int),times_times(int),Uua),Uub))) ) ).

% ATP.lambda_894
tff(fact_9073_ATP_Olambda__895,axiom,
    ! [A: $tType,B: $tType,Uu: A,Uua: B,Uub: A,Uuc: B] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),aa(B,fun(A,fun(B,bool)),aTP_Lamp_me(A,fun(B,fun(A,fun(B,bool))),Uu),Uua),Uub),Uuc))
    <=> ( ( Uu = Uub )
        & ( Uua = Uuc ) ) ) ).

% ATP.lambda_895
tff(fact_9074_ATP_Olambda__896,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,A),Uua: set(B),Uub: fun(A,bool),Uuc: B] :
      ( pp(aa(B,bool,aa(fun(A,bool),fun(B,bool),aa(set(B),fun(fun(A,bool),fun(B,bool)),aTP_Lamp_nx(fun(B,A),fun(set(B),fun(fun(A,bool),fun(B,bool))),Uu),Uua),Uub),Uuc))
    <=> ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),Uuc),Uua))
        & pp(aa(A,bool,Uub,aa(B,A,Uu,Uuc))) ) ) ).

% ATP.lambda_896
tff(fact_9075_ATP_Olambda__897,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: set(B),Uua: fun(B,A),Uub: fun(B,A),Uuc: B] :
          ( pp(aa(B,bool,aa(fun(B,A),fun(B,bool),aa(fun(B,A),fun(fun(B,A),fun(B,bool)),aTP_Lamp_bi(set(B),fun(fun(B,A),fun(fun(B,A),fun(B,bool))),Uu),Uua),Uub),Uuc))
        <=> ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),Uuc),Uu))
            & ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(B,A,Uua,Uuc)),aa(B,A,Uub,Uuc)) != one_one(A) ) ) ) ) ).

% ATP.lambda_897
tff(fact_9076_ATP_Olambda__898,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: set(B),Uua: fun(B,A),Uub: fun(B,A),Uuc: B] :
          ( pp(aa(B,bool,aa(fun(B,A),fun(B,bool),aa(fun(B,A),fun(fun(B,A),fun(B,bool)),aTP_Lamp_bk(set(B),fun(fun(B,A),fun(fun(B,A),fun(B,bool))),Uu),Uua),Uub),Uuc))
        <=> ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),Uuc),Uu))
            & ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(B,A,Uua,Uuc)),aa(B,A,Uub,Uuc)) != zero_zero(A) ) ) ) ) ).

% ATP.lambda_898
tff(fact_9077_ATP_Olambda__899,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A,Uuc: A] : aa(A,B,aa(A,fun(A,B),aa(fun(A,B),fun(A,fun(A,B)),aTP_Lamp_acf(fun(A,B),fun(fun(A,B),fun(A,fun(A,B))),Uu),Uua),Uub),Uuc) = aa(B,B,real_V8093663219630862766scaleR(B,aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,Uuc),Uub)))),aa(B,B,minus_minus(B,aa(A,B,Uu,Uuc)),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,Uu,Uub)),aa(A,B,Uua,aa(A,A,minus_minus(A,Uuc),Uub))))) ) ).

% ATP.lambda_899
tff(fact_9078_ATP_Olambda__900,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Uu: fun(nat,A),Uua: nat,Uub: A,Uuc: nat] : aa(nat,A,aa(A,fun(nat,A),aa(nat,fun(A,fun(nat,A)),aTP_Lamp_cr(fun(nat,A),fun(nat,fun(A,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uuc),Uua))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uub),Uuc)) ) ).

% ATP.lambda_900
tff(fact_9079_ATP_Olambda__901,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_semiring_1(A)
     => ! [Uu: fun(B,A),Uua: A,Uub: list(B),Uuc: nat] : aa(nat,A,aa(list(B),fun(nat,A),aa(A,fun(list(B),fun(nat,A)),aTP_Lamp_pg(fun(B,A),fun(A,fun(list(B),fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(B,A,Uu,aa(nat,B,nth(B,Uub),Uuc))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),Uuc)) ) ).

% ATP.lambda_901
tff(fact_9080_ATP_Olambda__902,axiom,
    ! [B: $tType,A: $tType,Uu: set(A),Uua: fun(B,bool),Uub: fun(A,B),Uuc: A] :
      ( pp(aa(A,bool,aa(fun(A,B),fun(A,bool),aa(fun(B,bool),fun(fun(A,B),fun(A,bool)),aTP_Lamp_aij(set(A),fun(fun(B,bool),fun(fun(A,B),fun(A,bool))),Uu),Uua),Uub),Uuc))
    <=> ( pp(aa(B,bool,Uua,aa(A,B,Uub,Uuc)))
        & pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Uuc),Uu)) ) ) ).

% ATP.lambda_902
tff(fact_9081_ATP_Olambda__903,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: fun(nat,A),Uua: A,Uub: A,Uuc: nat] : aa(nat,A,aa(A,fun(nat,A),aa(A,fun(A,fun(nat,A)),aTP_Lamp_vf(fun(nat,A),fun(A,fun(A,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,Uuc)),aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,minus_minus(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Uua),Uub)),Uuc)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),Uuc))),Uub)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),Uuc)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),aa(nat,nat,minus_minus(nat,Uuc),aa(nat,nat,suc,zero_zero(nat))))))) ) ).

% ATP.lambda_903
tff(fact_9082_ATP_Olambda__904,axiom,
    ! [A: $tType,Aa: $tType] :
      ( ( real_Vector_banach(Aa)
        & real_V3459762299906320749_field(Aa)
        & topological_t2_space(A) )
     => ! [Uu: fun(A,Aa),Uua: fun(nat,Aa),Uub: A,Uuc: nat] : aa(nat,Aa,aa(A,fun(nat,Aa),aa(fun(nat,Aa),fun(A,fun(nat,Aa)),aTP_Lamp_zz(fun(A,Aa),fun(fun(nat,Aa),fun(A,fun(nat,Aa))),Uu),Uua),Uub),Uuc) = aa(Aa,Aa,aa(Aa,fun(Aa,Aa),times_times(Aa),aa(nat,Aa,Uua,Uuc)),aa(nat,Aa,aa(Aa,fun(nat,Aa),power_power(Aa),aa(A,Aa,Uu,Uub)),Uuc)) ) ).

% ATP.lambda_904
tff(fact_9083_ATP_Olambda__905,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,real),Uua: A,Uub: fun(A,real),Uuc: A] : aa(A,real,aa(fun(A,real),fun(A,real),aa(A,fun(fun(A,real),fun(A,real)),aTP_Lamp_uu(fun(A,real),fun(A,fun(fun(A,real),fun(A,real))),Uu),Uua),Uub),Uuc) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(A,real,Uub,Uuc)),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,inverse_inverse(real),aa(real,real,sqrt,aa(A,real,Uu,Uua)))),aa(num,real,numeral_numeral(real),bit0(one2)))) ) ).

% ATP.lambda_905
tff(fact_9084_ATP_Olambda__906,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [Uu: fun(nat,A),Uua: A,Uub: nat,Uuc: nat] : aa(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_jh(fun(nat,A),fun(A,fun(nat,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,Uuc)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),aa(nat,nat,minus_minus(nat,aa(nat,nat,minus_minus(nat,Uuc),Uub)),one_one(nat)))) ) ).

% ATP.lambda_906
tff(fact_9085_ATP_Olambda__907,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: fun(B,A),Uua: A,Uub: set(A),Uuc: B] :
          ( pp(aa(B,bool,aa(set(A),fun(B,bool),aa(A,fun(set(A),fun(B,bool)),aTP_Lamp_ahj(fun(B,A),fun(A,fun(set(A),fun(B,bool))),Uu),Uua),Uub),Uuc))
        <=> pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(B,A,Uu,Uuc)),aa(set(A),set(A),minus_minus(set(A),Uub),insert(A,Uua,bot_bot(set(A)))))) ) ) ).

% ATP.lambda_907
tff(fact_9086_ATP_Olambda__908,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_semiring_0(A)
     => ! [Uu: fun(B,A),Uua: A,Uub: B,Uuc: A] : aa(A,A,aa(B,fun(A,A),aa(A,fun(B,fun(A,A)),aTP_Lamp_or(fun(B,A),fun(A,fun(B,fun(A,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(B,A,Uu,Uub)),aa(A,A,aa(A,fun(A,A),times_times(A),Uua),Uuc)) ) ).

% ATP.lambda_908
tff(fact_9087_ATP_Olambda__909,axiom,
    ! [Uu: fun(nat,nat),Uua: fun(nat,nat),Uub: nat,Uuc: nat] : aa(nat,nat,aa(nat,fun(nat,nat),aa(fun(nat,nat),fun(nat,fun(nat,nat)),aTP_Lamp_iu(fun(nat,nat),fun(fun(nat,nat),fun(nat,fun(nat,nat))),Uu),Uua),Uub),Uuc) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,Uu,Uuc)),aa(nat,nat,Uua,aa(nat,nat,minus_minus(nat,Uub),Uuc))) ).

% ATP.lambda_909
tff(fact_9088_ATP_Olambda__910,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V4412858255891104859lgebra(A) )
     => ! [Uu: fun(nat,A),Uua: fun(nat,A),Uub: nat,Uuc: nat] : aa(nat,A,aa(nat,fun(nat,A),aa(fun(nat,A),fun(nat,fun(nat,A)),aTP_Lamp_io(fun(nat,A),fun(fun(nat,A),fun(nat,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,Uuc)),aa(nat,A,Uua,aa(nat,nat,minus_minus(nat,Uub),Uuc))) ) ).

% ATP.lambda_910
tff(fact_9089_ATP_Olambda__911,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [Uu: fun(nat,A),Uua: fun(nat,A),Uub: nat,Uuc: nat] : aa(nat,A,aa(nat,fun(nat,A),aa(fun(nat,A),fun(nat,fun(nat,A)),aTP_Lamp_iq(fun(nat,A),fun(fun(nat,A),fun(nat,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,Uuc)),aa(nat,A,Uua,aa(nat,nat,minus_minus(nat,Uub),Uuc))) ) ).

% ATP.lambda_911
tff(fact_9090_ATP_Olambda__912,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( semiring_0(B)
     => ! [Uu: fun(A,B),Uua: fun(C,B),Uub: A,Uuc: C] : aa(C,B,aa(A,fun(C,B),aa(fun(C,B),fun(A,fun(C,B)),aTP_Lamp_dm(fun(A,B),fun(fun(C,B),fun(A,fun(C,B))),Uu),Uua),Uub),Uuc) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uu,Uub)),aa(C,B,Uua,Uuc)) ) ).

% ATP.lambda_912
tff(fact_9091_ATP_Olambda__913,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: set(C),Uua: fun(B,fun(C,A)),Uub: fun(B,fun(C,bool)),Uuc: B] : aa(B,A,aa(fun(B,fun(C,bool)),fun(B,A),aa(fun(B,fun(C,A)),fun(fun(B,fun(C,bool)),fun(B,A)),aTP_Lamp_hi(set(C),fun(fun(B,fun(C,A)),fun(fun(B,fun(C,bool)),fun(B,A))),Uu),Uua),Uub),Uuc) = groups7121269368397514597t_prod(C,A,aa(B,fun(C,A),Uua,Uuc),collect(C,aa(B,fun(C,bool),aa(fun(B,fun(C,bool)),fun(B,fun(C,bool)),aTP_Lamp_dr(set(C),fun(fun(B,fun(C,bool)),fun(B,fun(C,bool))),Uu),Uub),Uuc))) ) ).

% ATP.lambda_913
tff(fact_9092_ATP_Olambda__914,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: set(C),Uua: fun(B,fun(C,A)),Uub: fun(B,fun(C,bool)),Uuc: B] : aa(B,A,aa(fun(B,fun(C,bool)),fun(B,A),aa(fun(B,fun(C,A)),fun(fun(B,fun(C,bool)),fun(B,A)),aTP_Lamp_ds(set(C),fun(fun(B,fun(C,A)),fun(fun(B,fun(C,bool)),fun(B,A))),Uu),Uua),Uub),Uuc) = aa(set(C),A,groups7311177749621191930dd_sum(C,A,aa(B,fun(C,A),Uua,Uuc)),collect(C,aa(B,fun(C,bool),aa(fun(B,fun(C,bool)),fun(B,fun(C,bool)),aTP_Lamp_dr(set(C),fun(fun(B,fun(C,bool)),fun(B,fun(C,bool))),Uu),Uub),Uuc))) ) ).

% ATP.lambda_914
tff(fact_9093_ATP_Olambda__915,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,real),Uua: A,Uub: fun(A,real),Uuc: A] : aa(A,real,aa(fun(A,real),fun(A,real),aa(A,fun(fun(A,real),fun(A,real)),aTP_Lamp_uy(fun(A,real),fun(A,fun(fun(A,real),fun(A,real))),Uu),Uua),Uub),Uuc) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(A,real,Uub,Uuc)),aa(real,real,inverse_inverse(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),cos(real,aa(A,real,Uu,Uua))),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) ).

% ATP.lambda_915
tff(fact_9094_ATP_Olambda__916,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,real),Uua: fun(A,real),Uub: A,Uuc: A] : aa(A,real,aa(A,fun(A,real),aa(fun(A,real),fun(A,fun(A,real)),aTP_Lamp_uw(fun(A,real),fun(fun(A,real),fun(A,fun(A,real))),Uu),Uua),Uub),Uuc) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(A,real,Uua,Uuc)),aa(real,real,inverse_inverse(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(A,real,Uu,Uub)),aa(num,nat,numeral_numeral(nat),bit0(one2)))))) ) ).

% ATP.lambda_916
tff(fact_9095_ATP_Olambda__917,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,real),Uua: fun(A,real),Uub: A,Uuc: A] : aa(A,real,aa(A,fun(A,real),aa(fun(A,real),fun(A,fun(A,real)),aTP_Lamp_tt(fun(A,real),fun(fun(A,real),fun(A,fun(A,real))),Uu),Uua),Uub),Uuc) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(A,real,Uua,Uuc)),aa(real,real,exp(real),aa(A,real,Uu,Uub))) ) ).

% ATP.lambda_917
tff(fact_9096_ATP_Olambda__918,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,real),Uua: fun(A,real),Uub: A,Uuc: A] : aa(A,real,aa(A,fun(A,real),aa(fun(A,real),fun(A,fun(A,real)),aTP_Lamp_tx(fun(A,real),fun(fun(A,real),fun(A,fun(A,real))),Uu),Uua),Uub),Uuc) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(A,real,Uua,Uuc)),cos(real,aa(A,real,Uu,Uub))) ) ).

% ATP.lambda_918
tff(fact_9097_ATP_Olambda__919,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,real),Uua: A,Uub: fun(A,real),Uuc: A] : aa(A,real,aa(fun(A,real),fun(A,real),aa(A,fun(fun(A,real),fun(A,real)),aTP_Lamp_uk(fun(A,real),fun(A,fun(fun(A,real),fun(A,real))),Uu),Uua),Uub),Uuc) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(A,real,Uub,Uuc)),aa(real,real,inverse_inverse(real),aa(A,real,Uu,Uua))) ) ).

% ATP.lambda_919
tff(fact_9098_ATP_Olambda__920,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,real),Uua: A,Uub: fun(A,real),Uuc: A] : aa(A,real,aa(fun(A,real),fun(A,real),aa(A,fun(fun(A,real),fun(A,real)),aTP_Lamp_sx(fun(A,real),fun(A,fun(fun(A,real),fun(A,real))),Uu),Uua),Uub),Uuc) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(A,real,Uub,Uuc)),aa(real,real,inverse_inverse(real),aa(real,real,sqrt,aa(real,real,minus_minus(real,one_one(real)),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(A,real,Uu,Uua)),aa(num,nat,numeral_numeral(nat),bit0(one2))))))) ) ).

% ATP.lambda_920
tff(fact_9099_ATP_Olambda__921,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,real),Uua: fun(A,real),Uub: A,Uuc: A] : aa(A,real,aa(A,fun(A,real),aa(fun(A,real),fun(A,fun(A,real)),aTP_Lamp_ug(fun(A,real),fun(fun(A,real),fun(A,fun(A,real))),Uu),Uua),Uub),Uuc) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(A,real,Uua,Uuc)),aa(real,real,uminus_uminus(real),sin(real,aa(A,real,Uu,Uub)))) ) ).

% ATP.lambda_921
tff(fact_9100_ATP_Olambda__922,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,real),Uua: A,Uub: fun(A,real),Uuc: A] : aa(A,real,aa(fun(A,real),fun(A,real),aa(A,fun(fun(A,real),fun(A,real)),aTP_Lamp_vc(fun(A,real),fun(A,fun(fun(A,real),fun(A,real))),Uu),Uua),Uub),Uuc) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(A,real,Uub,Uuc)),aa(real,real,inverse_inverse(real),aa(real,real,uminus_uminus(real),aa(real,real,sqrt,aa(real,real,minus_minus(real,one_one(real)),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(A,real,Uu,Uua)),aa(num,nat,numeral_numeral(nat),bit0(one2)))))))) ) ).

% ATP.lambda_922
tff(fact_9101_ATP_Olambda__923,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A,Uuc: A] : aa(A,real,aa(A,fun(A,real),aa(fun(A,B),fun(A,fun(A,real)),aTP_Lamp_act(fun(A,B),fun(fun(A,B),fun(A,fun(A,real))),Uu),Uua),Uub),Uuc) = aa(real,real,aa(real,fun(real,real),divide_divide(real),real_V7770717601297561774m_norm(B,aa(B,B,minus_minus(B,aa(B,B,minus_minus(B,aa(A,B,Uu,aa(A,A,aa(A,fun(A,A),plus_plus(A),Uub),Uuc))),aa(A,B,Uu,Uub))),aa(A,B,Uua,Uuc)))),real_V7770717601297561774m_norm(A,Uuc)) ) ).

% ATP.lambda_923
tff(fact_9102_ATP_Olambda__924,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A,Uuc: A] : aa(A,real,aa(A,fun(A,real),aa(fun(A,B),fun(A,fun(A,real)),aTP_Lamp_acp(fun(A,B),fun(fun(A,B),fun(A,fun(A,real))),Uu),Uua),Uub),Uuc) = aa(real,real,aa(real,fun(real,real),divide_divide(real),real_V7770717601297561774m_norm(B,aa(B,B,minus_minus(B,aa(B,B,minus_minus(B,aa(A,B,Uu,Uuc)),aa(A,B,Uu,Uub))),aa(A,B,Uua,aa(A,A,minus_minus(A,Uuc),Uub))))),real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,Uuc),Uub))) ) ).

% ATP.lambda_924
tff(fact_9103_ATP_Olambda__925,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: fun(A,bool),Uua: A,Uub: set(A),Uuc: A] :
          ( pp(aa(A,bool,aa(set(A),fun(A,bool),aa(A,fun(set(A),fun(A,bool)),aTP_Lamp_afu(fun(A,bool),fun(A,fun(set(A),fun(A,bool))),Uu),Uua),Uub),Uuc))
        <=> ( ( Uuc != Uua )
           => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Uuc),Uub))
             => pp(aa(A,bool,Uu,Uuc)) ) ) ) ) ).

% ATP.lambda_925
tff(fact_9104_ATP_Olambda__926,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V822414075346904944vector(C) )
     => ! [Uu: fun(A,B),Uua: fun(A,C),Uub: real,Uuc: A] :
          ( pp(aa(A,bool,aa(real,fun(A,bool),aa(fun(A,C),fun(real,fun(A,bool)),aTP_Lamp_ahk(fun(A,B),fun(fun(A,C),fun(real,fun(A,bool))),Uu),Uua),Uub),Uuc))
        <=> pp(aa(real,bool,ord_less_eq(real,real_V7770717601297561774m_norm(C,aa(A,C,Uua,Uuc))),aa(real,real,aa(real,fun(real,real),times_times(real),real_V7770717601297561774m_norm(B,aa(A,B,Uu,Uuc))),Uub))) ) ) ).

% ATP.lambda_926
tff(fact_9105_ATP_Olambda__927,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: filter(A),Uuc: A] : aa(A,B,aa(filter(A),fun(A,B),aa(fun(A,B),fun(filter(A),fun(A,B)),aTP_Lamp_acu(fun(A,B),fun(fun(A,B),fun(filter(A),fun(A,B))),Uu),Uua),Uub),Uuc) = aa(B,B,real_V8093663219630862766scaleR(B,aa(real,real,inverse_inverse(real),real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,Uuc),topolo3827282254853284352ce_Lim(A,A,Uub,aTP_Lamp_ti(A,A)))))),aa(B,B,minus_minus(B,aa(B,B,minus_minus(B,aa(A,B,Uu,Uuc)),aa(A,B,Uu,topolo3827282254853284352ce_Lim(A,A,Uub,aTP_Lamp_ti(A,A))))),aa(A,B,Uua,aa(A,A,minus_minus(A,Uuc),topolo3827282254853284352ce_Lim(A,A,Uub,aTP_Lamp_ti(A,A)))))) ) ).

% ATP.lambda_927
tff(fact_9106_ATP_Olambda__928,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu: fun(A,B),Uua: A,Uub: fun(A,B),Uuc: A] : aa(A,B,aa(fun(A,B),fun(A,B),aa(A,fun(fun(A,B),fun(A,B)),aTP_Lamp_acr(fun(A,B),fun(A,fun(fun(A,B),fun(A,B))),Uu),Uua),Uub),Uuc) = aa(B,B,real_V8093663219630862766scaleR(B,aa(real,real,inverse_inverse(real),real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,Uuc),Uua)))),aa(B,B,minus_minus(B,aa(B,B,minus_minus(B,aa(A,B,Uub,Uuc)),aa(A,B,Uub,Uua))),aa(A,B,Uu,aa(A,A,minus_minus(A,Uuc),Uua)))) ) ).

% ATP.lambda_928
tff(fact_9107_ATP_Olambda__929,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A,Uuc: A] : aa(A,B,aa(A,fun(A,B),aa(fun(A,B),fun(A,fun(A,B)),aTP_Lamp_acq(fun(A,B),fun(fun(A,B),fun(A,fun(A,B))),Uu),Uua),Uub),Uuc) = aa(B,B,real_V8093663219630862766scaleR(B,aa(real,real,inverse_inverse(real),real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,Uuc),Uub)))),aa(B,B,minus_minus(B,aa(B,B,minus_minus(B,aa(A,B,Uu,Uuc)),aa(A,B,Uu,Uub))),aa(A,B,Uua,aa(A,A,minus_minus(A,Uuc),Uub)))) ) ).

% ATP.lambda_929
tff(fact_9108_ATP_Olambda__930,axiom,
    ! [A: $tType,Uu: A,Uua: list(A),Uub: A,Uuc: nat] : aa(nat,list(A),aa(A,fun(nat,list(A)),aa(list(A),fun(A,fun(nat,list(A))),aTP_Lamp_amh(A,fun(list(A),fun(A,fun(nat,list(A)))),Uu),Uua),Uub),Uuc) = aa(list(A),list(A),cons(A,Uu),list_update(A,Uua,Uuc,Uub)) ).

% ATP.lambda_930
tff(fact_9109_ATP_Olambda__931,axiom,
    ! [A: $tType,B: $tType,Uu: bool,Uua: fun(A,fun(B,bool)),Uub: A,Uuc: B] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),aa(fun(A,fun(B,bool)),fun(A,fun(B,bool)),aTP_Lamp_mr(bool,fun(fun(A,fun(B,bool)),fun(A,fun(B,bool))),Uu),Uua),Uub),Uuc))
    <=> ( pp(Uu)
        & pp(aa(B,bool,aa(A,fun(B,bool),Uua,Uub),Uuc)) ) ) ).

% ATP.lambda_931
tff(fact_9110_ATP_Olambda__932,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: set(B),Uua: fun(B,A),Uub: fun(B,C),Uuc: C] : aa(C,A,aa(fun(B,C),fun(C,A),aa(fun(B,A),fun(fun(B,C),fun(C,A)),aTP_Lamp_oc(set(B),fun(fun(B,A),fun(fun(B,C),fun(C,A))),Uu),Uua),Uub),Uuc) = groups7121269368397514597t_prod(B,A,Uua,collect(B,aa(C,fun(B,bool),aa(fun(B,C),fun(C,fun(B,bool)),aTP_Lamp_oa(set(B),fun(fun(B,C),fun(C,fun(B,bool))),Uu),Uub),Uuc))) ) ).

% ATP.lambda_932
tff(fact_9111_ATP_Olambda__933,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: set(B),Uua: fun(B,C),Uub: fun(B,A),Uuc: C] : aa(C,A,aa(fun(B,A),fun(C,A),aa(fun(B,C),fun(fun(B,A),fun(C,A)),aTP_Lamp_oe(set(B),fun(fun(B,C),fun(fun(B,A),fun(C,A))),Uu),Uua),Uub),Uuc) = groups7121269368397514597t_prod(B,A,Uub,collect(B,aa(C,fun(B,bool),aa(fun(B,C),fun(C,fun(B,bool)),aTP_Lamp_oa(set(B),fun(fun(B,C),fun(C,fun(B,bool))),Uu),Uua),Uuc))) ) ).

% ATP.lambda_933
tff(fact_9112_ATP_Olambda__934,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: set(B),Uua: fun(B,A),Uub: fun(B,C),Uuc: C] : aa(C,A,aa(fun(B,C),fun(C,A),aa(fun(B,A),fun(fun(B,C),fun(C,A)),aTP_Lamp_ob(set(B),fun(fun(B,A),fun(fun(B,C),fun(C,A))),Uu),Uua),Uub),Uuc) = aa(set(B),A,groups7311177749621191930dd_sum(B,A,Uua),collect(B,aa(C,fun(B,bool),aa(fun(B,C),fun(C,fun(B,bool)),aTP_Lamp_oa(set(B),fun(fun(B,C),fun(C,fun(B,bool))),Uu),Uub),Uuc))) ) ).

% ATP.lambda_934
tff(fact_9113_ATP_Olambda__935,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: set(B),Uua: fun(B,C),Uub: fun(B,A),Uuc: C] : aa(C,A,aa(fun(B,A),fun(C,A),aa(fun(B,C),fun(fun(B,A),fun(C,A)),aTP_Lamp_od(set(B),fun(fun(B,C),fun(fun(B,A),fun(C,A))),Uu),Uua),Uub),Uuc) = aa(set(B),A,groups7311177749621191930dd_sum(B,A,Uub),collect(B,aa(C,fun(B,bool),aa(fun(B,C),fun(C,fun(B,bool)),aTP_Lamp_oa(set(B),fun(fun(B,C),fun(C,fun(B,bool))),Uu),Uua),Uuc))) ) ).

% ATP.lambda_935
tff(fact_9114_ATP_Olambda__936,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Uu: fun(nat,A),Uua: nat,Uub: real,Uuc: A] :
          ( pp(aa(A,bool,aa(real,fun(A,bool),aa(nat,fun(real,fun(A,bool)),aTP_Lamp_ael(fun(nat,A),fun(nat,fun(real,fun(A,bool))),Uu),Uua),Uub),Uuc))
        <=> pp(aa(real,bool,ord_less_eq(real,Uub),real_V7770717601297561774m_norm(A,aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_ck(fun(nat,A),fun(A,fun(nat,A)),Uu),Uuc)),set_ord_atMost(nat,Uua))))) ) ) ).

% ATP.lambda_936
tff(fact_9115_ATP_Olambda__937,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Uu: A,Uua: list(A),Uub: A,Uuc: list(A)] : aa(list(A),A,aa(A,fun(list(A),A),aa(list(A),fun(A,fun(list(A),A)),aTP_Lamp_ana(A,fun(list(A),fun(A,fun(list(A),A))),Uu),Uua),Uub),Uuc) = ord_min(A,Uu,min_list(A,Uua)) ) ).

% ATP.lambda_937
tff(fact_9116_ATP_Olambda__938,axiom,
    ! [D6: $tType,A: $tType,C: $tType,B: $tType,Uu: fun(C,D6),Uua: fun(A,fun(B,C)),Uub: A,Uuc: B] : aa(B,D6,aa(A,fun(B,D6),aa(fun(A,fun(B,C)),fun(A,fun(B,D6)),aTP_Lamp_mn(fun(C,D6),fun(fun(A,fun(B,C)),fun(A,fun(B,D6))),Uu),Uua),Uub),Uuc) = aa(C,D6,Uu,aa(B,C,aa(A,fun(B,C),Uua,Uub),Uuc)) ).

% ATP.lambda_938
tff(fact_9117_ATP_Olambda__939,axiom,
    ! [C: $tType,B: $tType,A: $tType,Uu: fun(B,C),Uua: fun(A,fun(list(A),B)),Uub: A,Uuc: list(A)] : aa(list(A),C,aa(A,fun(list(A),C),aa(fun(A,fun(list(A),B)),fun(A,fun(list(A),C)),aTP_Lamp_amw(fun(B,C),fun(fun(A,fun(list(A),B)),fun(A,fun(list(A),C))),Uu),Uua),Uub),Uuc) = aa(B,C,Uu,aa(list(A),B,aa(A,fun(list(A),B),Uua,Uub),Uuc)) ).

% ATP.lambda_939
tff(fact_9118_ATP_Olambda__940,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(nat,A),Uua: nat,Uub: nat,Uuc: nat] : aa(nat,A,aa(nat,fun(nat,A),aa(nat,fun(nat,fun(nat,A)),aTP_Lamp_ox(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(nat,A,Uu,aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),Uua)),aa(nat,nat,suc,Uuc))) ) ).

% ATP.lambda_940
tff(fact_9119_ATP_Olambda__941,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(nat,A),Uua: nat,Uub: nat,Uuc: nat] : aa(nat,A,aa(nat,fun(nat,A),aa(nat,fun(nat,fun(nat,A)),aTP_Lamp_ov(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(nat,A,Uu,aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),Uua)),aa(nat,nat,suc,Uuc))) ) ).

% ATP.lambda_941
tff(fact_9120_ATP_Olambda__942,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(nat,A),Uua: nat,Uub: nat,Uuc: nat] : aa(nat,A,aa(nat,fun(nat,A),aa(nat,fun(nat,fun(nat,A)),aTP_Lamp_ht(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(nat,A,Uu,aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),Uua)),Uuc)) ) ).

% ATP.lambda_942
tff(fact_9121_ATP_Olambda__943,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(nat,A),Uua: nat,Uub: nat,Uuc: nat] : aa(nat,A,aa(nat,fun(nat,A),aa(nat,fun(nat,fun(nat,A)),aTP_Lamp_ef(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(nat,A,Uu,aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),Uua)),Uuc)) ) ).

% ATP.lambda_943
tff(fact_9122_ATP_Olambda__944,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] : aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aa(int,fun(int,fun(int,product_prod(int,int))),aTP_Lamp_ni(int,fun(int,fun(int,fun(int,product_prod(int,int)))),Uu),Uua),Uub),Uuc) = normalize(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,minus_minus(int,aa(int,int,aa(int,fun(int,int),times_times(int),Uu),Uuc)),aa(int,int,aa(int,fun(int,int),times_times(int),Uub),Uua))),aa(int,int,aa(int,fun(int,int),times_times(int),Uua),Uuc))) ).

% ATP.lambda_944
tff(fact_9123_ATP_Olambda__945,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] : aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aa(int,fun(int,fun(int,product_prod(int,int))),aTP_Lamp_no(int,fun(int,fun(int,fun(int,product_prod(int,int)))),Uu),Uua),Uub),Uuc) = normalize(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),Uu),Uuc)),aa(int,int,aa(int,fun(int,int),times_times(int),Uub),Uua))),aa(int,int,aa(int,fun(int,int),times_times(int),Uua),Uuc))) ).

% ATP.lambda_945
tff(fact_9124_ATP_Olambda__946,axiom,
    ! [A: $tType,C: $tType] :
      ( ( real_V822414075346904944vector(C)
        & real_V8999393235501362500lgebra(A) )
     => ! [Uu: fun(C,A),Uua: C,Uub: fun(C,A),Uuc: C] : aa(C,A,aa(fun(C,A),fun(C,A),aa(C,fun(fun(C,A),fun(C,A)),aTP_Lamp_uc(fun(C,A),fun(C,fun(fun(C,A),fun(C,A))),Uu),Uua),Uub),Uuc) = aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,inverse_inverse(A),aa(C,A,Uu,Uua))),aa(C,A,Uub,Uuc))),aa(A,A,inverse_inverse(A),aa(C,A,Uu,Uua)))) ) ).

% ATP.lambda_946
tff(fact_9125_ATP_Olambda__947,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] : aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aa(int,fun(int,fun(int,product_prod(int,int))),aTP_Lamp_nk(int,fun(int,fun(int,fun(int,product_prod(int,int)))),Uu),Uua),Uub),Uuc) = normalize(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,aa(int,fun(int,int),times_times(int),Uu),Uub)),aa(int,int,aa(int,fun(int,int),times_times(int),Uua),Uuc))) ).

% ATP.lambda_947
tff(fact_9126_ATP_Olambda__948,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] : aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aa(int,fun(int,fun(int,product_prod(int,int))),aTP_Lamp_nm(int,fun(int,fun(int,fun(int,product_prod(int,int)))),Uu),Uua),Uub),Uuc) = normalize(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,aa(int,fun(int,int),times_times(int),Uu),Uuc)),aa(int,int,aa(int,fun(int,int),times_times(int),Uua),Uub))) ).

% ATP.lambda_948
tff(fact_9127_ATP_Olambda__949,axiom,
    ! [A: $tType,B: $tType,Uu: set(A),Uua: set(B),Uub: B,Uuc: fun(A,B)] :
      ( pp(aa(fun(A,B),bool,aa(B,fun(fun(A,B),bool),aa(set(B),fun(B,fun(fun(A,B),bool)),aTP_Lamp_ahu(set(A),fun(set(B),fun(B,fun(fun(A,B),bool))),Uu),Uua),Uub),Uuc))
    <=> ! [X2: A] :
          ( ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X2),Uu))
           => pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),aa(A,B,Uuc,X2)),Uua)) )
          & ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X2),Uu))
           => ( aa(A,B,Uuc,X2) = Uub ) ) ) ) ).

% ATP.lambda_949
tff(fact_9128_ATP_Olambda__950,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,A),Uua: B,Uub: set(B),Uuc: A] :
      ( pp(aa(A,bool,aa(set(B),fun(A,bool),aa(B,fun(set(B),fun(A,bool)),aTP_Lamp_aks(fun(B,A),fun(B,fun(set(B),fun(A,bool))),Uu),Uua),Uub),Uuc))
    <=> ? [L4: B] :
          ( ( Uuc = aa(B,A,Uu,L4) )
          & ( ( L4 = Uua )
            | pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),L4),Uub)) ) ) ) ).

% ATP.lambda_950
tff(fact_9129_ATP_Olambda__951,axiom,
    ! [A: $tType,C: $tType,B: $tType,Uu: fun(A,bool),Uua: fun(B,bool),Uub: fun(A,fun(B,C)),Uuc: C] :
      ( pp(aa(C,bool,aa(fun(A,fun(B,C)),fun(C,bool),aa(fun(B,bool),fun(fun(A,fun(B,C)),fun(C,bool)),aTP_Lamp_alg(fun(A,bool),fun(fun(B,bool),fun(fun(A,fun(B,C)),fun(C,bool))),Uu),Uua),Uub),Uuc))
    <=> ? [X2: A,Y7: B] :
          ( ( Uuc = aa(B,C,aa(A,fun(B,C),Uub,X2),Y7) )
          & pp(aa(A,bool,Uu,X2))
          & pp(aa(B,bool,Uua,Y7)) ) ) ).

% ATP.lambda_951
tff(fact_9130_ATP_Olambda__952,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ( real_V822414075346904944vector(C)
        & real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu: fun(A,fun(A,B)),Uua: fun(C,A),Uub: C,Uuc: fun(C,A),Uud: C] : aa(C,B,aa(fun(C,A),fun(C,B),aa(C,fun(fun(C,A),fun(C,B)),aa(fun(C,A),fun(C,fun(fun(C,A),fun(C,B))),aTP_Lamp_tr(fun(A,fun(A,B)),fun(fun(C,A),fun(C,fun(fun(C,A),fun(C,B)))),Uu),Uua),Uub),Uuc),Uud) = aa(A,B,aa(A,fun(A,B),Uu,aa(C,A,Uua,Uub)),aa(C,A,Uuc,Uud)) ) ).

% ATP.lambda_952
tff(fact_9131_ATP_Olambda__953,axiom,
    ! [A: $tType,B: $tType,I6: $tType] :
      ( ( real_V3459762299906320749_field(B)
        & real_V822414075346904944vector(A) )
     => ! [Uu: set(I6),Uua: fun(I6,fun(A,B)),Uub: fun(I6,fun(A,B)),Uuc: A,Uud: A] : aa(A,B,aa(A,fun(A,B),aa(fun(I6,fun(A,B)),fun(A,fun(A,B)),aa(fun(I6,fun(A,B)),fun(fun(I6,fun(A,B)),fun(A,fun(A,B))),aTP_Lamp_uq(set(I6),fun(fun(I6,fun(A,B)),fun(fun(I6,fun(A,B)),fun(A,fun(A,B)))),Uu),Uua),Uub),Uuc),Uud) = aa(set(I6),B,groups7311177749621191930dd_sum(I6,B,aa(A,fun(I6,B),aa(A,fun(A,fun(I6,B)),aa(fun(I6,fun(A,B)),fun(A,fun(A,fun(I6,B))),aa(fun(I6,fun(A,B)),fun(fun(I6,fun(A,B)),fun(A,fun(A,fun(I6,B)))),aTP_Lamp_up(set(I6),fun(fun(I6,fun(A,B)),fun(fun(I6,fun(A,B)),fun(A,fun(A,fun(I6,B))))),Uu),Uua),Uub),Uuc),Uud)),Uu) ) ).

% ATP.lambda_953
tff(fact_9132_ATP_Olambda__954,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [Uu: nat,Uua: fun(nat,A),Uub: A,Uuc: A,Uud: nat] : aa(nat,A,aa(A,fun(nat,A),aa(A,fun(A,fun(nat,A)),aa(fun(nat,A),fun(A,fun(A,fun(nat,A))),aTP_Lamp_jg(nat,fun(fun(nat,A),fun(A,fun(A,fun(nat,A)))),Uu),Uua),Uub),Uuc),Uud) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aa(A,fun(A,fun(nat,fun(nat,A))),aTP_Lamp_jf(fun(nat,A),fun(A,fun(A,fun(nat,fun(nat,A)))),Uua),Uub),Uuc),Uud)),set_ord_lessThan(nat,aa(nat,nat,minus_minus(nat,Uu),Uud))) ) ).

% ATP.lambda_954
tff(fact_9133_ATP_Olambda__955,axiom,
    ! [A: $tType] :
      ( ( code_term_of(A)
        & quickc6926020345158392990erm_of(A) )
     => ! [Uu: fun(code_integer,product_prod(A,A)),Uua: A,Uub: code_integer,Uuc: A,Uud: A] : aa(A,list(A),aa(A,fun(A,list(A)),aa(code_integer,fun(A,fun(A,list(A))),aa(A,fun(code_integer,fun(A,fun(A,list(A)))),aTP_Lamp_amn(fun(code_integer,product_prod(A,A)),fun(A,fun(code_integer,fun(A,fun(A,list(A))))),Uu),Uua),Uub),Uuc),Uud) = append(A,code_T4080844693773952564amples(A,Uu,Uua,aa(code_integer,code_integer,minus_minus(code_integer,Uub),one_one(code_integer))),aa(list(A),list(A),cons(A,Uuc),aa(list(A),list(A),cons(A,Uud),nil(A)))) ) ).

% ATP.lambda_955
tff(fact_9134_ATP_Olambda__956,axiom,
    ! [A: $tType] :
      ( ( code_term_of(A)
        & quickc6926020345158392990erm_of(A) )
     => ! [Uu: code_integer,Uua: fun(code_integer,product_prod(A,A)),Uub: A,Uuc: A,Uud: A] : aa(A,list(A),aa(A,fun(A,list(A)),aa(A,fun(A,fun(A,list(A))),aa(fun(code_integer,product_prod(A,A)),fun(A,fun(A,fun(A,list(A)))),aTP_Lamp_amo(code_integer,fun(fun(code_integer,product_prod(A,A)),fun(A,fun(A,fun(A,list(A))))),Uu),Uua),Uub),Uuc),Uud) = append(A,code_T4080844693773952564amples(A,Uua,Uub,aa(code_integer,code_integer,minus_minus(code_integer,Uu),one_one(code_integer))),aa(list(A),list(A),cons(A,Uuc),aa(list(A),list(A),cons(A,Uud),nil(A)))) ) ).

% ATP.lambda_956
tff(fact_9135_ATP_Olambda__957,axiom,
    ! [Uu: nat,Uua: fun(nat,fun(real,real)),Uub: real,Uuc: nat,Uud: real] : aa(real,real,aa(nat,fun(real,real),aa(real,fun(nat,fun(real,real)),aa(fun(nat,fun(real,real)),fun(real,fun(nat,fun(real,real))),aTP_Lamp_sp(nat,fun(fun(nat,fun(real,real)),fun(real,fun(nat,fun(real,real)))),Uu),Uua),Uub),Uuc),Uud) = aa(real,real,minus_minus(real,aa(real,real,aa(nat,fun(real,real),Uua,Uuc),Uud)),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aa(real,fun(nat,real),aa(nat,fun(real,fun(nat,real)),aTP_Lamp_so(fun(nat,fun(real,real)),fun(nat,fun(real,fun(nat,real))),Uua),Uuc),Uud)),set_ord_lessThan(nat,aa(nat,nat,minus_minus(nat,Uu),Uuc)))),aa(real,real,aa(real,fun(real,real),times_times(real),Uub),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),Uud),aa(nat,nat,minus_minus(nat,Uu),Uuc))),semiring_char_0_fact(real,aa(nat,nat,minus_minus(nat,Uu),Uuc)))))) ).

% ATP.lambda_957
tff(fact_9136_ATP_Olambda__958,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [Uu: nat,Uua: fun(nat,A),Uub: A,Uuc: A,Uud: nat] : aa(nat,A,aa(A,fun(nat,A),aa(A,fun(A,fun(nat,A)),aa(fun(nat,A),fun(A,fun(A,fun(nat,A))),aTP_Lamp_ji(nat,fun(fun(nat,A),fun(A,fun(A,fun(nat,A)))),Uu),Uua),Uub),Uuc),Uud) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_jh(fun(nat,A),fun(A,fun(nat,fun(nat,A))),Uua),Uuc),Uud)),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,Uud),Uu))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uub),Uud)) ) ).

% ATP.lambda_958
tff(fact_9137_ATP_Olambda__959,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: nat,Uua: A,Uub: A,Uuc: A,Uud: nat] : aa(nat,A,aa(A,fun(nat,A),aa(A,fun(A,fun(nat,A)),aa(A,fun(A,fun(A,fun(nat,A))),aTP_Lamp_je(nat,fun(A,fun(A,fun(A,fun(nat,A)))),Uu),Uua),Uub),Uuc),Uud) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,gbinomial(A,aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),Uud)),Uua)),one_one(A))),Uud)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uub),Uud))),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Uub),Uuc)),aa(nat,nat,minus_minus(nat,Uu),Uud))) ) ).

% ATP.lambda_959
tff(fact_9138_ATP_Olambda__960,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: nat,Uua: A,Uub: A,Uuc: A,Uud: nat] : aa(nat,A,aa(A,fun(nat,A),aa(A,fun(A,fun(nat,A)),aa(A,fun(A,fun(A,fun(nat,A))),aTP_Lamp_ja(nat,fun(A,fun(A,fun(A,fun(nat,A)))),Uu),Uua),Uub),Uuc),Uud) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),Uu)),Uua)),Uud)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uub),Uud))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uuc),aa(nat,nat,minus_minus(nat,Uu),Uud))) ) ).

% ATP.lambda_960
tff(fact_9139_ATP_Olambda__961,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: nat,Uua: A,Uub: A,Uuc: A,Uud: nat] : aa(nat,A,aa(A,fun(nat,A),aa(A,fun(A,fun(nat,A)),aa(A,fun(A,fun(A,fun(nat,A))),aTP_Lamp_jb(nat,fun(A,fun(A,fun(A,fun(nat,A)))),Uu),Uua),Uub),Uuc),Uud) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,gbinomial(A,aa(A,A,uminus_uminus(A),Uua)),Uud)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),Uub)),Uud))),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Uub),Uuc)),aa(nat,nat,minus_minus(nat,Uu),Uud))) ) ).

% ATP.lambda_961
tff(fact_9140_ATP_Olambda__962,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [Uu: fun(nat,A),Uua: A,Uub: A,Uuc: nat,Uud: nat] : aa(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aa(A,fun(A,fun(nat,fun(nat,A))),aTP_Lamp_jf(fun(nat,A),fun(A,fun(A,fun(nat,fun(nat,A)))),Uu),Uua),Uub),Uuc),Uud) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uuc),Uud)),one_one(nat)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uub),Uud))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),Uuc)) ) ).

% ATP.lambda_962
tff(fact_9141_ATP_Olambda__963,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( real_V7819770556892013058_space(A)
        & real_V7819770556892013058_space(B) )
     => ! [Uu: fun(C,A),Uua: A,Uub: fun(C,B),Uuc: B,Uud: C] :
          ( pp(aa(C,bool,aa(B,fun(C,bool),aa(fun(C,B),fun(B,fun(C,bool)),aa(A,fun(fun(C,B),fun(B,fun(C,bool))),aTP_Lamp_agt(fun(C,A),fun(A,fun(fun(C,B),fun(B,fun(C,bool)))),Uu),Uua),Uub),Uuc),Uud))
        <=> pp(aa(real,bool,ord_less_eq(real,real_V557655796197034286t_dist(B,aa(C,B,Uub,Uud),Uuc)),real_V557655796197034286t_dist(A,aa(C,A,Uu,Uud),Uua))) ) ) ).

% ATP.lambda_963
tff(fact_9142_ATP_Olambda__964,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( ( topolo4958980785337419405_space(C)
        & topolo4958980785337419405_space(B) )
     => ! [Uu: fun(A,B),Uua: B,Uub: fun(B,C),Uuc: C,Uud: A] :
          ( pp(aa(A,bool,aa(C,fun(A,bool),aa(fun(B,C),fun(C,fun(A,bool)),aa(B,fun(fun(B,C),fun(C,fun(A,bool))),aTP_Lamp_ahc(fun(A,B),fun(B,fun(fun(B,C),fun(C,fun(A,bool)))),Uu),Uua),Uub),Uuc),Uud))
        <=> ( ( aa(A,B,Uu,Uud) = Uua )
           => ( aa(B,C,Uub,Uua) = Uuc ) ) ) ) ).

% ATP.lambda_964
tff(fact_9143_ATP_Olambda__965,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V3459762299906320749_field(B)
        & real_V822414075346904944vector(A) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A,Uuc: nat,Uud: A] : aa(A,B,aa(nat,fun(A,B),aa(A,fun(nat,fun(A,B)),aa(fun(A,B),fun(A,fun(nat,fun(A,B))),aTP_Lamp_ui(fun(A,B),fun(fun(A,B),fun(A,fun(nat,fun(A,B)))),Uu),Uua),Uub),Uuc),Uud) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(B,B,aa(B,fun(B,B),times_times(B),aa(nat,B,semiring_1_of_nat(B),Uuc)),aa(A,B,Uua,Uud))),aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(A,B,Uu,Uub)),aa(nat,nat,minus_minus(nat,Uuc),one_one(nat)))) ) ).

% ATP.lambda_965
tff(fact_9144_ATP_Olambda__966,axiom,
    ! [Uu: nat,Uua: list(vEBT_VEBT),Uub: vEBT_VEBT,Uuc: nat,Uud: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),aa(vEBT_VEBT,fun(nat,fun(nat,bool)),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,bool))),aTP_Lamp_all(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,bool)))),Uu),Uua),Uub),Uuc),Uud))
    <=> ( pp(aa(nat,bool,ord_less_eq(nat,Uuc),Uud))
        & pp(aa(nat,bool,ord_less(nat,Uud),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uu)))
        & ! [I3: nat] :
            ( pp(aa(nat,bool,ord_less(nat,I3),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,minus_minus(nat,Uu),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Uu),aa(num,nat,numeral_numeral(nat),bit0(one2)))))))
           => ( ? [X_13: nat] : pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,aa(nat,vEBT_VEBT,nth(vEBT_VEBT,Uua),I3)),X_13))
            <=> pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,Uub),I3)) ) )
        & ( ( Uuc = Uud )
         => ! [X2: vEBT_VEBT] :
              ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),member(vEBT_VEBT),X2),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),Uua)))
             => ~ ? [X_13: nat] : pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,X2),X_13)) ) )
        & ( ( Uuc != Uud )
         => ( vEBT_V5917875025757280293ildren(aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Uu),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uua,Uud)
            & ! [X2: nat] :
                ( pp(aa(nat,bool,ord_less(nat,X2),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uu)))
               => ( vEBT_V5917875025757280293ildren(aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Uu),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uua,X2)
                 => ( pp(aa(nat,bool,ord_less(nat,Uuc),X2))
                    & pp(aa(nat,bool,ord_less_eq(nat,X2),Uud)) ) ) ) ) ) ) ) ).

% ATP.lambda_966
tff(fact_9145_ATP_Olambda__967,axiom,
    ! [A: $tType,B: $tType,I6: $tType] :
      ( ( real_V3459762299906320749_field(B)
        & real_V822414075346904944vector(A) )
     => ! [Uu: set(I6),Uua: fun(I6,fun(A,B)),Uub: fun(I6,fun(A,B)),Uuc: A,Uud: A,Uue: I6] : aa(I6,B,aa(A,fun(I6,B),aa(A,fun(A,fun(I6,B)),aa(fun(I6,fun(A,B)),fun(A,fun(A,fun(I6,B))),aa(fun(I6,fun(A,B)),fun(fun(I6,fun(A,B)),fun(A,fun(A,fun(I6,B)))),aTP_Lamp_up(set(I6),fun(fun(I6,fun(A,B)),fun(fun(I6,fun(A,B)),fun(A,fun(A,fun(I6,B))))),Uu),Uua),Uub),Uuc),Uud),Uue) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,aa(I6,fun(A,B),Uub,Uue),Uud)),groups7121269368397514597t_prod(I6,B,aa(A,fun(I6,B),aTP_Lamp_un(fun(I6,fun(A,B)),fun(A,fun(I6,B)),Uua),Uuc),aa(set(I6),set(I6),minus_minus(set(I6),Uu),insert(I6,Uue,bot_bot(set(I6)))))) ) ).

% ATP.lambda_967
tff(fact_9146_ATP_Olambda__968,axiom,
    ! [A: $tType,C: $tType] :
      ( ( real_V822414075346904944vector(C)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: fun(C,A),Uua: fun(C,A),Uub: C,Uuc: fun(C,A),Uud: fun(C,A),Uue: C] : aa(C,A,aa(fun(C,A),fun(C,A),aa(fun(C,A),fun(fun(C,A),fun(C,A)),aa(C,fun(fun(C,A),fun(fun(C,A),fun(C,A))),aa(fun(C,A),fun(C,fun(fun(C,A),fun(fun(C,A),fun(C,A)))),aTP_Lamp_tz(fun(C,A),fun(fun(C,A),fun(C,fun(fun(C,A),fun(fun(C,A),fun(C,A))))),Uu),Uua),Uub),Uuc),Uud),Uue) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(C,A,Uua,Uue)),aa(C,A,Uuc,Uub))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(C,A,Uu,Uub)),aa(C,A,Uud,Uue)))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(C,A,Uuc,Uub)),aa(C,A,Uuc,Uub))) ) ).

% ATP.lambda_968
tff(fact_9147_ATP_Olambda__969,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,real),Uua: fun(A,real),Uub: A,Uuc: fun(A,real),Uud: fun(A,real),Uue: A] : aa(A,real,aa(fun(A,real),fun(A,real),aa(fun(A,real),fun(fun(A,real),fun(A,real)),aa(A,fun(fun(A,real),fun(fun(A,real),fun(A,real))),aa(fun(A,real),fun(A,fun(fun(A,real),fun(fun(A,real),fun(A,real)))),aTP_Lamp_us(fun(A,real),fun(fun(A,real),fun(A,fun(fun(A,real),fun(fun(A,real),fun(A,real))))),Uu),Uua),Uub),Uuc),Uud),Uue) = aa(real,real,aa(real,fun(real,real),times_times(real),powr(real,aa(A,real,Uu,Uub),aa(A,real,Uuc,Uub))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(A,real,Uud,Uue)),aa(real,real,ln_ln(real),aa(A,real,Uu,Uub)))),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(A,real,Uua,Uue)),aa(A,real,Uuc,Uub))),aa(A,real,Uu,Uub)))) ) ).

% ATP.lambda_969
tff(fact_9148_ATP_Olambda__970,axiom,
    ! [C: $tType,D6: $tType] :
      ( ( real_V822414075346904944vector(D6)
        & real_V822414075346904944vector(C) )
     => ! [Uu: fun(D6,real),Uua: fun(D6,real),Uub: D6,Uuc: fun(D6,C),Uud: fun(D6,C),Uue: D6] : aa(D6,C,aa(fun(D6,C),fun(D6,C),aa(fun(D6,C),fun(fun(D6,C),fun(D6,C)),aa(D6,fun(fun(D6,C),fun(fun(D6,C),fun(D6,C))),aa(fun(D6,real),fun(D6,fun(fun(D6,C),fun(fun(D6,C),fun(D6,C)))),aTP_Lamp_sz(fun(D6,real),fun(fun(D6,real),fun(D6,fun(fun(D6,C),fun(fun(D6,C),fun(D6,C))))),Uu),Uua),Uub),Uuc),Uud),Uue) = aa(C,C,aa(C,fun(C,C),plus_plus(C),aa(C,C,real_V8093663219630862766scaleR(C,aa(D6,real,Uu,Uub)),aa(D6,C,Uud,Uue))),aa(C,C,real_V8093663219630862766scaleR(C,aa(D6,real,Uua,Uue)),aa(D6,C,Uuc,Uub))) ) ).

% ATP.lambda_970
tff(fact_9149_ATP_Olambda__971,axiom,
    ! [A: $tType,D6: $tType] :
      ( ( real_V822414075346904944vector(D6)
        & real_V4412858255891104859lgebra(A) )
     => ! [Uu: fun(D6,A),Uua: fun(D6,A),Uub: D6,Uuc: fun(D6,A),Uud: fun(D6,A),Uue: D6] : aa(D6,A,aa(fun(D6,A),fun(D6,A),aa(fun(D6,A),fun(fun(D6,A),fun(D6,A)),aa(D6,fun(fun(D6,A),fun(fun(D6,A),fun(D6,A))),aa(fun(D6,A),fun(D6,fun(fun(D6,A),fun(fun(D6,A),fun(D6,A)))),aTP_Lamp_tp(fun(D6,A),fun(fun(D6,A),fun(D6,fun(fun(D6,A),fun(fun(D6,A),fun(D6,A))))),Uu),Uua),Uub),Uuc),Uud),Uue) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(D6,A,Uu,Uub)),aa(D6,A,Uud,Uue))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(D6,A,Uua,Uue)),aa(D6,A,Uuc,Uub))) ) ).

% ATP.lambda_971
tff(fact_9150_ATP_Olambda__972,axiom,
    ! [A: $tType,C: $tType] :
      ( ( real_V822414075346904944vector(C)
        & real_V8999393235501362500lgebra(A) )
     => ! [Uu: fun(C,A),Uua: fun(C,A),Uub: C,Uuc: fun(C,A),Uud: fun(C,A),Uue: C] : aa(C,A,aa(fun(C,A),fun(C,A),aa(fun(C,A),fun(fun(C,A),fun(C,A)),aa(C,fun(fun(C,A),fun(fun(C,A),fun(C,A))),aa(fun(C,A),fun(C,fun(fun(C,A),fun(fun(C,A),fun(C,A)))),aTP_Lamp_um(fun(C,A),fun(fun(C,A),fun(C,fun(fun(C,A),fun(fun(C,A),fun(C,A))))),Uu),Uua),Uub),Uuc),Uud),Uue) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(C,A,Uu,Uub))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,inverse_inverse(A),aa(C,A,Uuc,Uub))),aa(C,A,Uud,Uue))),aa(A,A,inverse_inverse(A),aa(C,A,Uuc,Uub))))),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(C,A,Uua,Uue)),aa(C,A,Uuc,Uub))) ) ).

% ATP.lambda_972
tff(fact_9151_ATP_Olambda__973,axiom,
    ! [B: $tType,A: $tType,Uu: bool,Uua: A,Uub: B] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),aTP_Lamp_px(bool,fun(A,fun(B,bool)),Uu),Uua),Uub))
    <=> pp(Uu) ) ).

% ATP.lambda_973
tff(fact_9152_ATP_Olambda__974,axiom,
    ! [Uu: bool,Uua: nat] :
      ( pp(aa(nat,bool,aTP_Lamp_qe(bool,fun(nat,bool),Uu),Uua))
    <=> pp(Uu) ) ).

% ATP.lambda_974
tff(fact_9153_ATP_Olambda__975,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: bool,Uua: A] :
          ( pp(aa(A,bool,aTP_Lamp_acx(bool,fun(A,bool),Uu),Uua))
        <=> pp(Uu) ) ) ).

% ATP.lambda_975
tff(fact_9154_ATP_Olambda__976,axiom,
    ! [A: $tType,Uu: bool,Uua: A] :
      ( pp(aa(A,bool,aTP_Lamp_pq(bool,fun(A,bool),Uu),Uua))
    <=> pp(Uu) ) ).

% ATP.lambda_976
tff(fact_9155_ATP_Olambda__977,axiom,
    ! [A: $tType,Aa: $tType] :
      ( ( zero(Aa)
        & topological_t2_space(Aa)
        & topolo8386298272705272623_space(A) )
     => ! [Uu: Aa,Uua: A] : aa(A,Aa,aTP_Lamp_vt(Aa,fun(A,Aa),Uu),Uua) = Uu ) ).

% ATP.lambda_977
tff(fact_9156_ATP_Olambda__978,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V822414075346904944vector(A) )
     => ! [Uu: B,Uua: A] : aa(A,B,aTP_Lamp_tg(B,fun(A,B),Uu),Uua) = Uu ) ).

% ATP.lambda_978
tff(fact_9157_ATP_Olambda__979,axiom,
    ! [A: $tType,B: $tType] :
      ( ( topolo4958980785337419405_space(B)
        & topolo4958980785337419405_space(A) )
     => ! [Uu: B,Uua: A] : aa(A,B,aTP_Lamp_ajh(B,fun(A,B),Uu),Uua) = Uu ) ).

% ATP.lambda_979
tff(fact_9158_ATP_Olambda__980,axiom,
    ! [A: $tType,B: $tType] :
      ( ( topolo4958980785337419405_space(B)
        & topological_t2_space(A) )
     => ! [Uu: B,Uua: A] : aa(A,B,aTP_Lamp_wv(B,fun(A,B),Uu),Uua) = Uu ) ).

% ATP.lambda_980
tff(fact_9159_ATP_Olambda__981,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V7819770556892013058_space(B)
     => ! [Uu: B,Uua: A] : aa(A,B,aTP_Lamp_ahv(B,fun(A,B),Uu),Uua) = Uu ) ).

% ATP.lambda_981
tff(fact_9160_ATP_Olambda__982,axiom,
    ! [A: $tType,B: $tType] :
      ( ( topological_t2_space(B)
        & topolo8386298272705272623_space(A) )
     => ! [Uu: B,Uua: A] : aa(A,B,aTP_Lamp_vy(B,fun(A,B),Uu),Uua) = Uu ) ).

% ATP.lambda_982
tff(fact_9161_ATP_Olambda__983,axiom,
    ! [A: $tType,B: $tType,Uu: B,Uua: A] : aa(A,B,aTP_Lamp_nz(B,fun(A,B),Uu),Uua) = Uu ).

% ATP.lambda_983
tff(fact_9162_ATP_Olambda__984,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_cj(A,fun(nat,A),Uu),Uua) = Uu ) ).

% ATP.lambda_984
tff(fact_9163_ATP_Olambda__985,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: A,Uua: B] : aa(B,A,aTP_Lamp_vh(A,fun(B,A),Uu),Uua) = Uu ) ).

% ATP.lambda_985
tff(fact_9164_ATP_Olambda__986,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_re(A,fun(A,A),Uu),Uua) = Uu ) ).

% ATP.lambda_986
tff(fact_9165_ATP_Olambda__987,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_aai(A,fun(nat,A),Uu),Uua) = Uu ) ).

% ATP.lambda_987
tff(fact_9166_ATP_Olambda__988,axiom,
    ! [B: $tType,A: $tType] :
      ( topological_t2_space(A)
     => ! [Uu: A,Uua: B] : aa(B,A,aTP_Lamp_yr(A,fun(B,A),Uu),Uua) = Uu ) ).

% ATP.lambda_988
tff(fact_9167_ATP_Olambda__989,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(A)
     => ! [Uu: A,Uua: B] : aa(B,A,aTP_Lamp_ns(A,fun(B,A),Uu),Uua) = Uu ) ).

% ATP.lambda_989
tff(fact_9168_ATP_Olambda__990,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_aii(A,fun(nat,A),Uu),Uua) = Uu ) ).

% ATP.lambda_990
tff(fact_9169_ATP_Olambda__991,axiom,
    ! [A: $tType,Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_ou(A,fun(nat,A),Uu),Uua) = Uu ).

% ATP.lambda_991
tff(fact_9170_ATP_Olambda__992,axiom,
    ! [B: $tType,A: $tType,Uu: A,Uua: B] : aa(B,A,aTP_Lamp_ay(A,fun(B,A),Uu),Uua) = Uu ).

% ATP.lambda_992
tff(fact_9171_ATP_Olambda__993,axiom,
    ! [A: $tType,Uu: A,Uua: list(A)] :
      ( pp(aa(list(A),bool,aa(A,fun(list(A),bool),aTP_Lamp_anc(A,fun(list(A),bool)),Uu),Uua))
    <=> $false ) ).

% ATP.lambda_993
tff(fact_9172_ATP_Olambda__994,axiom,
    ! [A: $tType,Uu: A,Uua: list(A)] :
      ( pp(aa(list(A),bool,aa(A,fun(list(A),bool),aTP_Lamp_anb(A,fun(list(A),bool)),Uu),Uua))
    <=> $true ) ).

% ATP.lambda_994
tff(fact_9173_ATP_Olambda__995,axiom,
    ! [B: $tType,A: $tType,Uu: A,Uua: B] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),aTP_Lamp_ms(A,fun(B,bool)),Uu),Uua))
    <=> $true ) ).

% ATP.lambda_995
tff(fact_9174_ATP_Olambda__996,axiom,
    ! [Uu: complex] : aa(complex,complex,aTP_Lamp_fd(complex,complex),Uu) = Uu ).

% ATP.lambda_996
tff(fact_9175_ATP_Olambda__997,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Uu: word(A)] : aa(word(A),word(A),aTP_Lamp_qu(word(A),word(A)),Uu) = Uu ) ).

% ATP.lambda_997
tff(fact_9176_ATP_Olambda__998,axiom,
    ! [Uu: nat] : aa(nat,nat,aTP_Lamp_el(nat,nat),Uu) = Uu ).

% ATP.lambda_998
tff(fact_9177_ATP_Olambda__999,axiom,
    ! [Uu: int] : aa(int,int,aTP_Lamp_fv(int,int),Uu) = Uu ).

% ATP.lambda_999
tff(fact_9178_ATP_Olambda__1000,axiom,
    ! [C: $tType] :
      ( topological_t2_space(C)
     => ! [Uu: C] : aa(C,C,aTP_Lamp_adc(C,C),Uu) = Uu ) ).

% ATP.lambda_1000
tff(fact_9179_ATP_Olambda__1001,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: A] : aa(A,A,aTP_Lamp_ti(A,A),Uu) = Uu ) ).

% ATP.lambda_1001
tff(fact_9180_ATP_Olambda__1002,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: A] : aa(A,A,aTP_Lamp_vi(A,A),Uu) = Uu ) ).

% ATP.lambda_1002
tff(fact_9181_ATP_Olambda__1003,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: A] : aa(A,A,aTP_Lamp_rd(A,A),Uu) = Uu ) ).

% ATP.lambda_1003
tff(fact_9182_ATP_Olambda__1004,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Uu: A] : aa(A,A,aTP_Lamp_vm(A,A),Uu) = Uu ) ).

% ATP.lambda_1004
tff(fact_9183_ATP_Olambda__1005,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: A] : aa(A,A,aTP_Lamp_adb(A,A),Uu) = Uu ) ).

% ATP.lambda_1005
tff(fact_9184_ATP_Olambda__1006,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: A] : aa(A,A,aTP_Lamp_qo(A,A),Uu) = Uu ) ).

% ATP.lambda_1006
tff(fact_9185_ATP_Olambda__1007,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: A] : aa(A,A,aTP_Lamp_qp(A,A),Uu) = Uu ) ).

% ATP.lambda_1007
tff(fact_9186_ATP_Olambda__1008,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu: A] : aa(A,A,aTP_Lamp_qm(A,A),Uu) = Uu ) ).

% ATP.lambda_1008
tff(fact_9187_ATP_Olambda__1009,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [Uu: A] : aa(A,A,aTP_Lamp_ab(A,A),Uu) = Uu ) ).

% ATP.lambda_1009
tff(fact_9188_ATP_Olambda__1010,axiom,
    ! [A: $tType,Uu: A] : aa(A,A,aTP_Lamp_at(A,A),Uu) = Uu ).

% ATP.lambda_1010
tff(fact_9189_ATP_Olambda__1011,axiom,
    ! [C: $tType,B: $tType,Uu: C] : aa(C,set(B),aTP_Lamp_qy(C,set(B)),Uu) = bot_bot(set(B)) ).

% ATP.lambda_1011
tff(fact_9190_ATP_Olambda__1012,axiom,
    ! [A: $tType,D6: $tType,Uu: A] : aa(A,set(D6),aTP_Lamp_qz(A,set(D6)),Uu) = bot_bot(set(D6)) ).

% ATP.lambda_1012
tff(fact_9191_ATP_Olambda__1013,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [Uu: nat] : aa(nat,A,aTP_Lamp_bs(nat,A),Uu) = zero_zero(A) ) ).

% ATP.lambda_1013
tff(fact_9192_ATP_Olambda__1014,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topological_t2_space(A) )
     => ! [Uu: nat] : aa(nat,A,aTP_Lamp_bo(nat,A),Uu) = zero_zero(A) ) ).

% ATP.lambda_1014
tff(fact_9193_ATP_Olambda__1015,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: B] : aa(B,A,aTP_Lamp_cw(B,A),Uu) = zero_zero(A) ) ).

% ATP.lambda_1015
tff(fact_9194_ATP_Olambda__1016,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V822414075346904944vector(A) )
     => ! [Uu: A] : aa(A,B,aTP_Lamp_th(A,B),Uu) = zero_zero(B) ) ).

% ATP.lambda_1016
tff(fact_9195_ATP_Olambda__1017,axiom,
    ! [A: $tType] :
      ( mult_zero(A)
     => ! [Uu: A] : aa(A,A,aTP_Lamp_ac(A,A),Uu) = zero_zero(A) ) ).

% ATP.lambda_1017
tff(fact_9196_ATP_Olambda__1018,axiom,
    ! [A: $tType,B: $tType] :
      ( zero(B)
     => ! [Uu: A] : aa(A,B,aTP_Lamp_om(A,B),Uu) = zero_zero(B) ) ).

% ATP.lambda_1018
tff(fact_9197_ATP_Olambda__1019,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: B] : aa(B,A,aTP_Lamp_gu(B,A),Uu) = one_one(A) ) ).

% ATP.lambda_1019
tff(fact_9198_ATP_Olambda__1020,axiom,
    ! [B: $tType,A: $tType,Uu: B] : aa(B,option(A),aTP_Lamp_nh(B,option(A)),Uu) = none(A) ).

% ATP.lambda_1020
tff(fact_9199_ATP_Olambda__1021,axiom,
    ! [A: $tType,B: $tType,Uu: A] : aa(A,option(B),aTP_Lamp_lt(A,option(B)),Uu) = none(B) ).

% ATP.lambda_1021
tff(fact_9200_ATP_Olambda__1022,axiom,
    ! [Uu: real] :
      ( pp(aa(real,bool,aTP_Lamp_ld(real,bool),Uu))
    <=> $false ) ).

% ATP.lambda_1022
tff(fact_9201_ATP_Olambda__1023,axiom,
    ! [Uu: nat] :
      ( pp(aa(nat,bool,aTP_Lamp_pr(nat,bool),Uu))
    <=> $false ) ).

% ATP.lambda_1023
tff(fact_9202_ATP_Olambda__1024,axiom,
    ! [A: $tType,Uu: A] :
      ( pp(aa(A,bool,aTP_Lamp_ll(A,bool),Uu))
    <=> $false ) ).

% ATP.lambda_1024
tff(fact_9203_ATP_Olambda__1025,axiom,
    ! [Uu: nat] :
      ( pp(aa(nat,bool,aTP_Lamp_pw(nat,bool),Uu))
    <=> $true ) ).

% ATP.lambda_1025
tff(fact_9204_ATP_Olambda__1026,axiom,
    ! [A: $tType,Uu: A] :
      ( pp(aa(A,bool,aTP_Lamp_pu(A,bool),Uu))
    <=> $true ) ).

% ATP.lambda_1026
tff(fact_9205_ATP_Olambda__1027,axiom,
    ! [A: $tType,Uu: A] : aa(A,fun(nat,nat),aTP_Lamp_ag(A,fun(nat,nat)),Uu) = suc ).

% ATP.lambda_1027

% Type constructors (1243)
tff(tcon_VEBT__Definitions_OVEBT___Typerep_Otyperep,axiom,
    typerep(vEBT_VEBT) ).

tff(tcon_Heap__Time__Monad_OHeap___Typerep_Otyperep_1,axiom,
    ! [A11: $tType] :
      ( typerep(A11)
     => typerep(heap_Time_Heap(A11)) ) ).

tff(tcon_Code__Numeral_Ointeger___Typerep_Otyperep_2,axiom,
    typerep(code_integer) ).

tff(tcon_Product__Type_Oprod___Typerep_Otyperep_3,axiom,
    ! [A11: $tType,A12: $tType] :
      ( ( typerep(A11)
        & typerep(A12) )
     => typerep(product_prod(A11,A12)) ) ).

tff(tcon_Numeral__Type_Onum1___Typerep_Otyperep_4,axiom,
    typerep(numeral_num1) ).

tff(tcon_Numeral__Type_Onum0___Typerep_Otyperep_5,axiom,
    typerep(numeral_num0) ).

tff(tcon_Numeral__Type_Obit1___Typerep_Otyperep_6,axiom,
    ! [A11: $tType] :
      ( typerep(A11)
     => typerep(numeral_bit1(A11)) ) ).

tff(tcon_Numeral__Type_Obit0___Typerep_Otyperep_7,axiom,
    ! [A11: $tType] :
      ( typerep(A11)
     => typerep(numeral_bit0(A11)) ) ).

tff(tcon_Multiset_Omultiset___Typerep_Otyperep_8,axiom,
    ! [A11: $tType] :
      ( typerep(A11)
     => typerep(multiset(A11)) ) ).

tff(tcon_Extended__Nat_Oenat___Typerep_Otyperep_9,axiom,
    typerep(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Complete__Lattices_Ocomplete__linorder,axiom,
    comple5582772986160207858norder(extended_enat) ).

tff(tcon_Complex_Ocomplex___Typerep_Otyperep_10,axiom,
    typerep(complex) ).

tff(tcon_Uint32_Ouint32___Typerep_Otyperep_11,axiom,
    typerep(uint32) ).

tff(tcon_Option_Ooption___Typerep_Otyperep_12,axiom,
    ! [A11: $tType] :
      ( typerep(A11)
     => typerep(option(A11)) ) ).

tff(tcon_Option_Ooption___Complete__Lattices_Ocomplete__linorder_13,axiom,
    ! [A11: $tType] :
      ( comple5582772986160207858norder(A11)
     => comple5582772986160207858norder(option(A11)) ) ).

tff(tcon_Filter_Ofilter___Typerep_Otyperep_14,axiom,
    ! [A11: $tType] :
      ( typerep(A11)
     => typerep(filter(A11)) ) ).

tff(tcon_Enum_Ofinite__3___Typerep_Otyperep_15,axiom,
    typerep(finite_3) ).

tff(tcon_Enum_Ofinite__3___Complete__Lattices_Ocomplete__linorder_16,axiom,
    comple5582772986160207858norder(finite_3) ).

tff(tcon_Enum_Ofinite__2___Typerep_Otyperep_17,axiom,
    typerep(finite_2) ).

tff(tcon_Enum_Ofinite__2___Complete__Lattices_Ocomplete__linorder_18,axiom,
    comple5582772986160207858norder(finite_2) ).

tff(tcon_Enum_Ofinite__1___Typerep_Otyperep_19,axiom,
    typerep(finite_1) ).

tff(tcon_Enum_Ofinite__1___Complete__Lattices_Ocomplete__linorder_20,axiom,
    comple5582772986160207858norder(finite_1) ).

tff(tcon_String_Ochar___Typerep_Otyperep_21,axiom,
    typerep(char) ).

tff(tcon_Word_Oword___Typerep_Otyperep_22,axiom,
    ! [A11: $tType] :
      ( typerep(A11)
     => typerep(word(A11)) ) ).

tff(tcon_Real_Oreal___Typerep_Otyperep_23,axiom,
    typerep(real) ).

tff(tcon_List_Olist___Typerep_Otyperep_24,axiom,
    ! [A11: $tType] :
      ( typerep(A11)
     => typerep(list(A11)) ) ).

tff(tcon_HOL_Obool___Typerep_Otyperep_25,axiom,
    typerep(bool) ).

tff(tcon_Set_Oset___Typerep_Otyperep_26,axiom,
    ! [A11: $tType] :
      ( typerep(A11)
     => typerep(set(A11)) ) ).

tff(tcon_Rat_Orat___Typerep_Otyperep_27,axiom,
    typerep(rat) ).

tff(tcon_Num_Onum___Typerep_Otyperep_28,axiom,
    typerep(num) ).

tff(tcon_Nat_Onat___Typerep_Otyperep_29,axiom,
    typerep(nat) ).

tff(tcon_Int_Oint___Typerep_Otyperep_30,axiom,
    typerep(int) ).

tff(tcon_fun___Typerep_Otyperep_31,axiom,
    ! [A11: $tType,A12: $tType] :
      ( ( typerep(A11)
        & typerep(A12) )
     => typerep(fun(A11,A12)) ) ).

tff(tcon_fun___Boolean__Algebras_Oboolean__algebra,axiom,
    ! [A11: $tType,A12: $tType] :
      ( boolea8198339166811842893lgebra(A12)
     => boolea8198339166811842893lgebra(fun(A11,A12)) ) ).

tff(tcon_fun___Code__Evaluation_Oterm__of,axiom,
    ! [A11: $tType,A12: $tType] :
      ( ( typerep(A11)
        & typerep(A12) )
     => code_term_of(fun(A11,A12)) ) ).

tff(tcon_fun___Orderings_Oorder__top,axiom,
    ! [A11: $tType,A12: $tType] :
      ( order_top(A12)
     => order_top(fun(A11,A12)) ) ).

tff(tcon_fun___Orderings_Oorder__bot,axiom,
    ! [A11: $tType,A12: $tType] :
      ( order_bot(A12)
     => order_bot(fun(A11,A12)) ) ).

tff(tcon_fun___Orderings_Opreorder,axiom,
    ! [A11: $tType,A12: $tType] :
      ( preorder(A12)
     => preorder(fun(A11,A12)) ) ).

tff(tcon_fun___Finite__Set_Ofinite,axiom,
    ! [A11: $tType,A12: $tType] :
      ( ( finite_finite(A11)
        & finite_finite(A12) )
     => finite_finite(fun(A11,A12)) ) ).

tff(tcon_fun___Orderings_Oorder,axiom,
    ! [A11: $tType,A12: $tType] :
      ( order(A12)
     => order(fun(A11,A12)) ) ).

tff(tcon_fun___Orderings_Oord,axiom,
    ! [A11: $tType,A12: $tType] :
      ( ord(A12)
     => ord(fun(A11,A12)) ) ).

tff(tcon_fun___Groups_Ouminus,axiom,
    ! [A11: $tType,A12: $tType] :
      ( uminus(A12)
     => uminus(fun(A11,A12)) ) ).

tff(tcon_Int_Oint___Conditionally__Complete__Lattices_Oconditionally__complete__linorder,axiom,
    condit6923001295902523014norder(int) ).

tff(tcon_Int_Oint___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations,axiom,
    bit_un5681908812861735899ations(int) ).

tff(tcon_Int_Oint___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct,axiom,
    semiri1453513574482234551roduct(int) ).

tff(tcon_Int_Oint___Euclidean__Division_Ounique__euclidean__semiring__with__nat,axiom,
    euclid5411537665997757685th_nat(int) ).

tff(tcon_Int_Oint___Euclidean__Division_Ounique__euclidean__ring__with__nat,axiom,
    euclid8789492081693882211th_nat(int) ).

tff(tcon_Int_Oint___Groups_Oordered__ab__semigroup__monoid__add__imp__le,axiom,
    ordere1937475149494474687imp_le(int) ).

tff(tcon_Int_Oint___Euclidean__Division_Ounique__euclidean__semiring,axiom,
    euclid3128863361964157862miring(int) ).

tff(tcon_Int_Oint___Euclidean__Division_Oeuclidean__semiring__cancel,axiom,
    euclid4440199948858584721cancel(int) ).

tff(tcon_Int_Oint___Divides_Ounique__euclidean__semiring__numeral,axiom,
    unique1627219031080169319umeral(int) ).

tff(tcon_Int_Oint___Euclidean__Division_Oeuclidean__ring__cancel,axiom,
    euclid8851590272496341667cancel(int) ).

tff(tcon_Int_Oint___Rings_Osemiring__no__zero__divisors__cancel,axiom,
    semiri6575147826004484403cancel(int) ).

tff(tcon_Int_Oint___Groups_Ostrict__ordered__ab__semigroup__add,axiom,
    strict9044650504122735259up_add(int) ).

tff(tcon_Int_Oint___Groups_Oordered__cancel__ab__semigroup__add,axiom,
    ordere580206878836729694up_add(int) ).

tff(tcon_Int_Oint___Groups_Oordered__ab__semigroup__add__imp__le,axiom,
    ordere2412721322843649153imp_le(int) ).

tff(tcon_Int_Oint___Bit__Operations_Osemiring__bit__operations,axiom,
    bit_se359711467146920520ations(int) ).

tff(tcon_Int_Oint___Rings_Olinordered__comm__semiring__strict,axiom,
    linord2810124833399127020strict(int) ).

tff(tcon_Int_Oint___Groups_Ostrict__ordered__comm__monoid__add,axiom,
    strict7427464778891057005id_add(int) ).

tff(tcon_Int_Oint___Groups_Oordered__cancel__comm__monoid__add,axiom,
    ordere8940638589300402666id_add(int) ).

tff(tcon_Int_Oint___Euclidean__Division_Oeuclidean__semiring,axiom,
    euclid3725896446679973847miring(int) ).

tff(tcon_Int_Oint___Topological__Spaces_Otopological__space,axiom,
    topolo4958980785337419405_space(int) ).

tff(tcon_Int_Oint___Topological__Spaces_Olinorder__topology,axiom,
    topolo1944317154257567458pology(int) ).

tff(tcon_Int_Oint___Topological__Spaces_Odiscrete__topology,axiom,
    topolo8865339358273720382pology(int) ).

tff(tcon_Int_Oint___Quickcheck__Narrowing_Opartial__term__of,axiom,
    quickc6926020345158392990erm_of(int) ).

tff(tcon_Int_Oint___Limits_Otopological__comm__monoid__mult,axiom,
    topolo4987421752381908075d_mult(int) ).

tff(tcon_Int_Oint___Bit__Comprehension_Obit__comprehension,axiom,
    bit_bi6583157726757044596ension(int) ).

tff(tcon_Int_Oint___Rings_Olinordered__semiring__1__strict,axiom,
    linord715952674999750819strict(int) ).

tff(tcon_Int_Oint___Limits_Otopological__comm__monoid__add,axiom,
    topolo5987344860129210374id_add(int) ).

tff(tcon_Int_Oint___Groups_Olinordered__ab__semigroup__add,axiom,
    linord4140545234300271783up_add(int) ).

tff(tcon_Int_Oint___Bit__Operations_Oring__bit__operations,axiom,
    bit_ri3973907225187159222ations(int) ).

tff(tcon_Int_Oint___Topological__Spaces_Oorder__topology,axiom,
    topolo2564578578187576103pology(int) ).

tff(tcon_Int_Oint___Rings_Osemiring__1__no__zero__divisors,axiom,
    semiri2026040879449505780visors(int) ).

tff(tcon_Int_Oint___Rings_Olinordered__nonzero__semiring,axiom,
    linord181362715937106298miring(int) ).

tff(tcon_Int_Oint___Limits_Otopological__semigroup__mult,axiom,
    topolo4211221413907600880p_mult(int) ).

tff(tcon_Int_Oint___Euclidean__Division_Oeuclidean__ring,axiom,
    euclid5891614535332579305n_ring(int) ).

tff(tcon_Int_Oint___Rings_Olinordered__semiring__strict,axiom,
    linord8928482502909563296strict(int) ).

tff(tcon_Int_Oint___Rings_Osemiring__no__zero__divisors,axiom,
    semiri3467727345109120633visors(int) ).

tff(tcon_Int_Oint___Groups_Oordered__ab__semigroup__add,axiom,
    ordere6658533253407199908up_add(int) ).

tff(tcon_Int_Oint___Groups_Oordered__ab__group__add__abs,axiom,
    ordere166539214618696060dd_abs(int) ).

tff(tcon_Int_Oint___Limits_Otopological__monoid__mult,axiom,
    topolo1898628316856586783d_mult(int) ).

tff(tcon_Int_Oint___Groups_Oordered__comm__monoid__add,axiom,
    ordere6911136660526730532id_add(int) ).

tff(tcon_Int_Oint___Groups_Olinordered__ab__group__add,axiom,
    linord5086331880401160121up_add(int) ).

tff(tcon_Int_Oint___Groups_Ocancel__ab__semigroup__add,axiom,
    cancel2418104881723323429up_add(int) ).

tff(tcon_Int_Oint___Rings_Oring__1__no__zero__divisors,axiom,
    ring_15535105094025558882visors(int) ).

tff(tcon_Int_Oint___Limits_Otopological__monoid__add,axiom,
    topolo6943815403480290642id_add(int) ).

tff(tcon_Int_Oint___Groups_Ocancel__comm__monoid__add,axiom,
    cancel1802427076303600483id_add(int) ).

tff(tcon_Int_Oint___Rings_Olinordered__ring__strict,axiom,
    linord4710134922213307826strict(int) ).

tff(tcon_Int_Oint___Rings_Ocomm__semiring__1__cancel,axiom,
    comm_s4317794764714335236cancel(int) ).

tff(tcon_Int_Oint___Bit__Operations_Osemiring__bits,axiom,
    bit_semiring_bits(int) ).

tff(tcon_Int_Oint___Topological__Spaces_Ot2__space,axiom,
    topological_t2_space(int) ).

tff(tcon_Int_Oint___Topological__Spaces_Ot1__space,axiom,
    topological_t1_space(int) ).

tff(tcon_Int_Oint___Rings_Oordered__comm__semiring,axiom,
    ordere2520102378445227354miring(int) ).

tff(tcon_Int_Oint___Rings_Olinordered__semiring__1,axiom,
    linord6961819062388156250ring_1(int) ).

tff(tcon_Int_Oint___Groups_Oordered__ab__group__add,axiom,
    ordered_ab_group_add(int) ).

tff(tcon_Int_Oint___Groups_Ocancel__semigroup__add,axiom,
    cancel_semigroup_add(int) ).

tff(tcon_Int_Oint___Rings_Olinordered__semiring,axiom,
    linordered_semiring(int) ).

tff(tcon_Int_Oint___Least__significant__bit_Olsb,axiom,
    least_6119777620449941438nt_lsb(int) ).

tff(tcon_Int_Oint___Rings_Oordered__semiring__0,axiom,
    ordered_semiring_0(int) ).

tff(tcon_Int_Oint___Rings_Olinordered__semidom,axiom,
    linordered_semidom(int) ).

tff(tcon_Int_Oint___Groups_Oab__semigroup__mult,axiom,
    ab_semigroup_mult(int) ).

tff(tcon_Int_Oint___Rings_Osemiring__1__cancel,axiom,
    semiring_1_cancel(int) ).

tff(tcon_Int_Oint___Rings_Oalgebraic__semidom,axiom,
    algebraic_semidom(int) ).

tff(tcon_Int_Oint___Groups_Ocomm__monoid__mult,axiom,
    comm_monoid_mult(int) ).

tff(tcon_Int_Oint___Groups_Oab__semigroup__add,axiom,
    ab_semigroup_add(int) ).

tff(tcon_Int_Oint___Generic__set__bit_Oset__bit,axiom,
    generic_set_set_bit(int) ).

tff(tcon_Int_Oint___Code__Evaluation_Oterm__of_32,axiom,
    code_term_of(int) ).

tff(tcon_Int_Oint___Rings_Oordered__semiring,axiom,
    ordered_semiring(int) ).

tff(tcon_Int_Oint___Rings_Oordered__ring__abs,axiom,
    ordered_ring_abs(int) ).

tff(tcon_Int_Oint___Parity_Osemiring__parity,axiom,
    semiring_parity(int) ).

tff(tcon_Int_Oint___Groups_Ocomm__monoid__add,axiom,
    comm_monoid_add(int) ).

tff(tcon_Int_Oint___Rings_Osemiring__modulo,axiom,
    semiring_modulo(int) ).

tff(tcon_Int_Oint___Rings_Olinordered__ring,axiom,
    linordered_ring(int) ).

tff(tcon_Int_Oint___Rings_Olinordered__idom,axiom,
    linordered_idom(int) ).

tff(tcon_Int_Oint___Rings_Ocomm__semiring__1,axiom,
    comm_semiring_1(int) ).

tff(tcon_Int_Oint___Rings_Ocomm__semiring__0,axiom,
    comm_semiring_0(int) ).

tff(tcon_Int_Oint___Groups_Osemigroup__mult,axiom,
    semigroup_mult(int) ).

tff(tcon_Int_Oint___Rings_Osemidom__modulo,axiom,
    semidom_modulo(int) ).

tff(tcon_Int_Oint___Rings_Osemidom__divide,axiom,
    semidom_divide(int) ).

tff(tcon_Int_Oint___Num_Osemiring__numeral,axiom,
    semiring_numeral(int) ).

tff(tcon_Int_Oint___Groups_Osemigroup__add,axiom,
    semigroup_add(int) ).

tff(tcon_Int_Oint___Rings_Ozero__less__one,axiom,
    zero_less_one(int) ).

tff(tcon_Int_Oint___Rings_Ocomm__semiring,axiom,
    comm_semiring(int) ).

tff(tcon_Int_Oint___Nat_Osemiring__char__0,axiom,
    semiring_char_0(int) ).

tff(tcon_Int_Oint___Groups_Oab__group__add,axiom,
    ab_group_add(int) ).

tff(tcon_Int_Oint___Rings_Ozero__neq__one,axiom,
    zero_neq_one(int) ).

tff(tcon_Int_Oint___Rings_Oordered__ring,axiom,
    ordered_ring(int) ).

tff(tcon_Int_Oint___Rings_Oidom__abs__sgn,axiom,
    idom_abs_sgn(int) ).

tff(tcon_Int_Oint___Parity_Oring__parity,axiom,
    ring_parity(int) ).

tff(tcon_Int_Oint___Orderings_Opreorder_33,axiom,
    preorder(int) ).

tff(tcon_Int_Oint___Orderings_Olinorder,axiom,
    linorder(int) ).

tff(tcon_Int_Oint___Groups_Omonoid__mult,axiom,
    monoid_mult(int) ).

tff(tcon_Int_Oint___Rings_Oidom__modulo,axiom,
    idom_modulo(int) ).

tff(tcon_Int_Oint___Rings_Oidom__divide,axiom,
    idom_divide(int) ).

tff(tcon_Int_Oint___Rings_Ocomm__ring__1,axiom,
    comm_ring_1(int) ).

tff(tcon_Int_Oint___Groups_Omonoid__add,axiom,
    monoid_add(int) ).

tff(tcon_Int_Oint___Rings_Osemiring__1,axiom,
    semiring_1(int) ).

tff(tcon_Int_Oint___Rings_Osemiring__0,axiom,
    semiring_0(int) ).

tff(tcon_Int_Oint___Orderings_Ono__top,axiom,
    no_top(int) ).

tff(tcon_Int_Oint___Orderings_Ono__bot,axiom,
    no_bot(int) ).

tff(tcon_Int_Oint___Groups_Ogroup__add,axiom,
    group_add(int) ).

tff(tcon_Int_Oint___GCD_Osemiring__gcd,axiom,
    semiring_gcd(int) ).

tff(tcon_Int_Oint___Rings_Omult__zero,axiom,
    mult_zero(int) ).

tff(tcon_Int_Oint___Rings_Ocomm__ring,axiom,
    comm_ring(int) ).

tff(tcon_Int_Oint___Orderings_Oorder_34,axiom,
    order(int) ).

tff(tcon_Int_Oint___Num_Oneg__numeral,axiom,
    neg_numeral(int) ).

tff(tcon_Int_Oint___Nat_Oring__char__0,axiom,
    ring_char_0(int) ).

tff(tcon_Int_Oint___Rings_Osemiring,axiom,
    semiring(int) ).

tff(tcon_Int_Oint___Rings_Osemidom,axiom,
    semidom(int) ).

tff(tcon_Int_Oint___Orderings_Oord_35,axiom,
    ord(int) ).

tff(tcon_Int_Oint___Groups_Ouminus_36,axiom,
    uminus(int) ).

tff(tcon_Int_Oint___Rings_Oring__1,axiom,
    ring_1(int) ).

tff(tcon_Int_Oint___Rings_Oabs__if,axiom,
    abs_if(int) ).

tff(tcon_Int_Oint___Power_Opower,axiom,
    power(int) ).

tff(tcon_Int_Oint___Num_Onumeral,axiom,
    numeral(int) ).

tff(tcon_Int_Oint___Groups_Ozero,axiom,
    zero(int) ).

tff(tcon_Int_Oint___Groups_Oplus,axiom,
    plus(int) ).

tff(tcon_Int_Oint___Rings_Oring,axiom,
    ring(int) ).

tff(tcon_Int_Oint___Rings_Oidom,axiom,
    idom(int) ).

tff(tcon_Int_Oint___Groups_Oone,axiom,
    one(int) ).

tff(tcon_Int_Oint___Rings_Odvd,axiom,
    dvd(int) ).

tff(tcon_Nat_Onat___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_37,axiom,
    condit6923001295902523014norder(nat) ).

tff(tcon_Nat_Onat___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_38,axiom,
    bit_un5681908812861735899ations(nat) ).

tff(tcon_Nat_Onat___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_39,axiom,
    semiri1453513574482234551roduct(nat) ).

tff(tcon_Nat_Onat___Euclidean__Division_Ounique__euclidean__semiring__with__nat_40,axiom,
    euclid5411537665997757685th_nat(nat) ).

tff(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__monoid__add__imp__le_41,axiom,
    ordere1937475149494474687imp_le(nat) ).

tff(tcon_Nat_Onat___Euclidean__Division_Ounique__euclidean__semiring_42,axiom,
    euclid3128863361964157862miring(nat) ).

tff(tcon_Nat_Onat___Euclidean__Division_Oeuclidean__semiring__cancel_43,axiom,
    euclid4440199948858584721cancel(nat) ).

tff(tcon_Nat_Onat___Divides_Ounique__euclidean__semiring__numeral_44,axiom,
    unique1627219031080169319umeral(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring__no__zero__divisors__cancel_45,axiom,
    semiri6575147826004484403cancel(nat) ).

tff(tcon_Nat_Onat___Groups_Ostrict__ordered__ab__semigroup__add_46,axiom,
    strict9044650504122735259up_add(nat) ).

tff(tcon_Nat_Onat___Groups_Oordered__cancel__comm__monoid__diff,axiom,
    ordere1170586879665033532d_diff(nat) ).

tff(tcon_Nat_Onat___Groups_Oordered__cancel__ab__semigroup__add_47,axiom,
    ordere580206878836729694up_add(nat) ).

tff(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add__imp__le_48,axiom,
    ordere2412721322843649153imp_le(nat) ).

tff(tcon_Nat_Onat___Bit__Operations_Osemiring__bit__operations_49,axiom,
    bit_se359711467146920520ations(nat) ).

tff(tcon_Nat_Onat___Rings_Olinordered__comm__semiring__strict_50,axiom,
    linord2810124833399127020strict(nat) ).

tff(tcon_Nat_Onat___Groups_Ostrict__ordered__comm__monoid__add_51,axiom,
    strict7427464778891057005id_add(nat) ).

tff(tcon_Nat_Onat___Groups_Oordered__cancel__comm__monoid__add_52,axiom,
    ordere8940638589300402666id_add(nat) ).

tff(tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add,axiom,
    canoni5634975068530333245id_add(nat) ).

tff(tcon_Nat_Onat___Euclidean__Division_Oeuclidean__semiring_53,axiom,
    euclid3725896446679973847miring(nat) ).

tff(tcon_Nat_Onat___Topological__Spaces_Otopological__space_54,axiom,
    topolo4958980785337419405_space(nat) ).

tff(tcon_Nat_Onat___Topological__Spaces_Olinorder__topology_55,axiom,
    topolo1944317154257567458pology(nat) ).

tff(tcon_Nat_Onat___Topological__Spaces_Odiscrete__topology_56,axiom,
    topolo8865339358273720382pology(nat) ).

tff(tcon_Nat_Onat___Quickcheck__Narrowing_Opartial__term__of_57,axiom,
    quickc6926020345158392990erm_of(nat) ).

tff(tcon_Nat_Onat___Limits_Otopological__comm__monoid__mult_58,axiom,
    topolo4987421752381908075d_mult(nat) ).

tff(tcon_Nat_Onat___Limits_Otopological__comm__monoid__add_59,axiom,
    topolo5987344860129210374id_add(nat) ).

tff(tcon_Nat_Onat___Groups_Olinordered__ab__semigroup__add_60,axiom,
    linord4140545234300271783up_add(nat) ).

tff(tcon_Nat_Onat___Topological__Spaces_Oorder__topology_61,axiom,
    topolo2564578578187576103pology(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring__1__no__zero__divisors_62,axiom,
    semiri2026040879449505780visors(nat) ).

tff(tcon_Nat_Onat___Rings_Olinordered__nonzero__semiring_63,axiom,
    linord181362715937106298miring(nat) ).

tff(tcon_Nat_Onat___Limits_Otopological__semigroup__mult_64,axiom,
    topolo4211221413907600880p_mult(nat) ).

tff(tcon_Nat_Onat___Rings_Olinordered__semiring__strict_65,axiom,
    linord8928482502909563296strict(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring__no__zero__divisors_66,axiom,
    semiri3467727345109120633visors(nat) ).

tff(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add_67,axiom,
    ordere6658533253407199908up_add(nat) ).

tff(tcon_Nat_Onat___Limits_Otopological__monoid__mult_68,axiom,
    topolo1898628316856586783d_mult(nat) ).

tff(tcon_Nat_Onat___Groups_Oordered__comm__monoid__add_69,axiom,
    ordere6911136660526730532id_add(nat) ).

tff(tcon_Nat_Onat___Groups_Ocancel__ab__semigroup__add_70,axiom,
    cancel2418104881723323429up_add(nat) ).

tff(tcon_Nat_Onat___Limits_Otopological__monoid__add_71,axiom,
    topolo6943815403480290642id_add(nat) ).

tff(tcon_Nat_Onat___Groups_Ocancel__comm__monoid__add_72,axiom,
    cancel1802427076303600483id_add(nat) ).

tff(tcon_Nat_Onat___Rings_Ocomm__semiring__1__cancel_73,axiom,
    comm_s4317794764714335236cancel(nat) ).

tff(tcon_Nat_Onat___Bit__Operations_Osemiring__bits_74,axiom,
    bit_semiring_bits(nat) ).

tff(tcon_Nat_Onat___Topological__Spaces_Ot2__space_75,axiom,
    topological_t2_space(nat) ).

tff(tcon_Nat_Onat___Topological__Spaces_Ot1__space_76,axiom,
    topological_t1_space(nat) ).

tff(tcon_Nat_Onat___Rings_Oordered__comm__semiring_77,axiom,
    ordere2520102378445227354miring(nat) ).

tff(tcon_Nat_Onat___Groups_Ocancel__semigroup__add_78,axiom,
    cancel_semigroup_add(nat) ).

tff(tcon_Nat_Onat___Rings_Olinordered__semiring_79,axiom,
    linordered_semiring(nat) ).

tff(tcon_Nat_Onat___Rings_Oordered__semiring__0_80,axiom,
    ordered_semiring_0(nat) ).

tff(tcon_Nat_Onat___Rings_Olinordered__semidom_81,axiom,
    linordered_semidom(nat) ).

tff(tcon_Nat_Onat___Groups_Oab__semigroup__mult_82,axiom,
    ab_semigroup_mult(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring__1__cancel_83,axiom,
    semiring_1_cancel(nat) ).

tff(tcon_Nat_Onat___Rings_Oalgebraic__semidom_84,axiom,
    algebraic_semidom(nat) ).

tff(tcon_Nat_Onat___Groups_Ocomm__monoid__mult_85,axiom,
    comm_monoid_mult(nat) ).

tff(tcon_Nat_Onat___Groups_Ocomm__monoid__diff,axiom,
    comm_monoid_diff(nat) ).

tff(tcon_Nat_Onat___Groups_Oab__semigroup__add_86,axiom,
    ab_semigroup_add(nat) ).

tff(tcon_Nat_Onat___Code__Evaluation_Oterm__of_87,axiom,
    code_term_of(nat) ).

tff(tcon_Nat_Onat___Rings_Oordered__semiring_88,axiom,
    ordered_semiring(nat) ).

tff(tcon_Nat_Onat___Parity_Osemiring__parity_89,axiom,
    semiring_parity(nat) ).

tff(tcon_Nat_Onat___Groups_Ocomm__monoid__add_90,axiom,
    comm_monoid_add(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring__modulo_91,axiom,
    semiring_modulo(nat) ).

tff(tcon_Nat_Onat___Rings_Ocomm__semiring__1_92,axiom,
    comm_semiring_1(nat) ).

tff(tcon_Nat_Onat___Rings_Ocomm__semiring__0_93,axiom,
    comm_semiring_0(nat) ).

tff(tcon_Nat_Onat___Groups_Osemigroup__mult_94,axiom,
    semigroup_mult(nat) ).

tff(tcon_Nat_Onat___Rings_Osemidom__modulo_95,axiom,
    semidom_modulo(nat) ).

tff(tcon_Nat_Onat___Rings_Osemidom__divide_96,axiom,
    semidom_divide(nat) ).

tff(tcon_Nat_Onat___Num_Osemiring__numeral_97,axiom,
    semiring_numeral(nat) ).

tff(tcon_Nat_Onat___Groups_Osemigroup__add_98,axiom,
    semigroup_add(nat) ).

tff(tcon_Nat_Onat___Rings_Ozero__less__one_99,axiom,
    zero_less_one(nat) ).

tff(tcon_Nat_Onat___Rings_Ocomm__semiring_100,axiom,
    comm_semiring(nat) ).

tff(tcon_Nat_Onat___Orderings_Owellorder,axiom,
    wellorder(nat) ).

tff(tcon_Nat_Onat___Orderings_Oorder__bot_101,axiom,
    order_bot(nat) ).

tff(tcon_Nat_Onat___Nat_Osemiring__char__0_102,axiom,
    semiring_char_0(nat) ).

tff(tcon_Nat_Onat___Rings_Ozero__neq__one_103,axiom,
    zero_neq_one(nat) ).

tff(tcon_Nat_Onat___Orderings_Opreorder_104,axiom,
    preorder(nat) ).

tff(tcon_Nat_Onat___Orderings_Olinorder_105,axiom,
    linorder(nat) ).

tff(tcon_Nat_Onat___Groups_Omonoid__mult_106,axiom,
    monoid_mult(nat) ).

tff(tcon_Nat_Onat___Groups_Omonoid__add_107,axiom,
    monoid_add(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring__1_108,axiom,
    semiring_1(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring__0_109,axiom,
    semiring_0(nat) ).

tff(tcon_Nat_Onat___Orderings_Ono__top_110,axiom,
    no_top(nat) ).

tff(tcon_Nat_Onat___GCD_Osemiring__gcd_111,axiom,
    semiring_gcd(nat) ).

tff(tcon_Nat_Onat___Rings_Omult__zero_112,axiom,
    mult_zero(nat) ).

tff(tcon_Nat_Onat___Orderings_Oorder_113,axiom,
    order(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring_114,axiom,
    semiring(nat) ).

tff(tcon_Nat_Onat___Rings_Osemidom_115,axiom,
    semidom(nat) ).

tff(tcon_Nat_Onat___Orderings_Oord_116,axiom,
    ord(nat) ).

tff(tcon_Nat_Onat___Power_Opower_117,axiom,
    power(nat) ).

tff(tcon_Nat_Onat___Num_Onumeral_118,axiom,
    numeral(nat) ).

tff(tcon_Nat_Onat___Groups_Ozero_119,axiom,
    zero(nat) ).

tff(tcon_Nat_Onat___Groups_Oplus_120,axiom,
    plus(nat) ).

tff(tcon_Nat_Onat___Groups_Oone_121,axiom,
    one(nat) ).

tff(tcon_Nat_Onat___Rings_Odvd_122,axiom,
    dvd(nat) ).

tff(tcon_Nat_Onat___Nat_Osize,axiom,
    size(nat) ).

tff(tcon_Num_Onum___Quickcheck__Narrowing_Opartial__term__of_123,axiom,
    quickc6926020345158392990erm_of(num) ).

tff(tcon_Num_Onum___Code__Evaluation_Oterm__of_124,axiom,
    code_term_of(num) ).

tff(tcon_Num_Onum___Orderings_Opreorder_125,axiom,
    preorder(num) ).

tff(tcon_Num_Onum___Orderings_Olinorder_126,axiom,
    linorder(num) ).

tff(tcon_Num_Onum___Orderings_Oorder_127,axiom,
    order(num) ).

tff(tcon_Num_Onum___Orderings_Oord_128,axiom,
    ord(num) ).

tff(tcon_Num_Onum___Groups_Oplus_129,axiom,
    plus(num) ).

tff(tcon_Num_Onum___Nat_Osize_130,axiom,
    size(num) ).

tff(tcon_Rat_Orat___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_131,axiom,
    semiri1453513574482234551roduct(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__ab__semigroup__monoid__add__imp__le_132,axiom,
    ordere1937475149494474687imp_le(rat) ).

tff(tcon_Rat_Orat___Rings_Osemiring__no__zero__divisors__cancel_133,axiom,
    semiri6575147826004484403cancel(rat) ).

tff(tcon_Rat_Orat___Groups_Ostrict__ordered__ab__semigroup__add_134,axiom,
    strict9044650504122735259up_add(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__cancel__ab__semigroup__add_135,axiom,
    ordere580206878836729694up_add(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__ab__semigroup__add__imp__le_136,axiom,
    ordere2412721322843649153imp_le(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__comm__semiring__strict_137,axiom,
    linord2810124833399127020strict(rat) ).

tff(tcon_Rat_Orat___Groups_Ostrict__ordered__comm__monoid__add_138,axiom,
    strict7427464778891057005id_add(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__cancel__comm__monoid__add_139,axiom,
    ordere8940638589300402666id_add(rat) ).

tff(tcon_Rat_Orat___Quickcheck__Narrowing_Opartial__term__of_140,axiom,
    quickc6926020345158392990erm_of(rat) ).

tff(tcon_Rat_Orat___Archimedean__Field_Oarchimedean__field,axiom,
    archim462609752435547400_field(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__semiring__1__strict_141,axiom,
    linord715952674999750819strict(rat) ).

tff(tcon_Rat_Orat___Orderings_Ounbounded__dense__linorder,axiom,
    unboun7993243217541854897norder(rat) ).

tff(tcon_Rat_Orat___Groups_Olinordered__ab__semigroup__add_142,axiom,
    linord4140545234300271783up_add(rat) ).

tff(tcon_Rat_Orat___Rings_Osemiring__1__no__zero__divisors_143,axiom,
    semiri2026040879449505780visors(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__nonzero__semiring_144,axiom,
    linord181362715937106298miring(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__semiring__strict_145,axiom,
    linord8928482502909563296strict(rat) ).

tff(tcon_Rat_Orat___Rings_Osemiring__no__zero__divisors_146,axiom,
    semiri3467727345109120633visors(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__ab__semigroup__add_147,axiom,
    ordere6658533253407199908up_add(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__ab__group__add__abs_148,axiom,
    ordere166539214618696060dd_abs(rat) ).

tff(tcon_Rat_Orat___Archimedean__Field_Ofloor__ceiling,axiom,
    archim2362893244070406136eiling(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__comm__monoid__add_149,axiom,
    ordere6911136660526730532id_add(rat) ).

tff(tcon_Rat_Orat___Groups_Olinordered__ab__group__add_150,axiom,
    linord5086331880401160121up_add(rat) ).

tff(tcon_Rat_Orat___Groups_Ocancel__ab__semigroup__add_151,axiom,
    cancel2418104881723323429up_add(rat) ).

tff(tcon_Rat_Orat___Rings_Oring__1__no__zero__divisors_152,axiom,
    ring_15535105094025558882visors(rat) ).

tff(tcon_Rat_Orat___Groups_Ocancel__comm__monoid__add_153,axiom,
    cancel1802427076303600483id_add(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__ring__strict_154,axiom,
    linord4710134922213307826strict(rat) ).

tff(tcon_Rat_Orat___Rings_Ocomm__semiring__1__cancel_155,axiom,
    comm_s4317794764714335236cancel(rat) ).

tff(tcon_Rat_Orat___Rings_Oordered__comm__semiring_156,axiom,
    ordere2520102378445227354miring(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__semiring__1_157,axiom,
    linord6961819062388156250ring_1(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__ab__group__add_158,axiom,
    ordered_ab_group_add(rat) ).

tff(tcon_Rat_Orat___Groups_Ocancel__semigroup__add_159,axiom,
    cancel_semigroup_add(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__semiring_160,axiom,
    linordered_semiring(rat) ).

tff(tcon_Rat_Orat___Rings_Oordered__semiring__0_161,axiom,
    ordered_semiring_0(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__semidom_162,axiom,
    linordered_semidom(rat) ).

tff(tcon_Rat_Orat___Orderings_Odense__linorder,axiom,
    dense_linorder(rat) ).

tff(tcon_Rat_Orat___Groups_Oab__semigroup__mult_163,axiom,
    ab_semigroup_mult(rat) ).

tff(tcon_Rat_Orat___Rings_Osemiring__1__cancel_164,axiom,
    semiring_1_cancel(rat) ).

tff(tcon_Rat_Orat___Groups_Ocomm__monoid__mult_165,axiom,
    comm_monoid_mult(rat) ).

tff(tcon_Rat_Orat___Groups_Oab__semigroup__add_166,axiom,
    ab_semigroup_add(rat) ).

tff(tcon_Rat_Orat___Fields_Olinordered__field,axiom,
    linordered_field(rat) ).

tff(tcon_Rat_Orat___Code__Evaluation_Oterm__of_167,axiom,
    code_term_of(rat) ).

tff(tcon_Rat_Orat___Rings_Oordered__semiring_168,axiom,
    ordered_semiring(rat) ).

tff(tcon_Rat_Orat___Rings_Oordered__ring__abs_169,axiom,
    ordered_ring_abs(rat) ).

tff(tcon_Rat_Orat___Groups_Ocomm__monoid__add_170,axiom,
    comm_monoid_add(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__ring_171,axiom,
    linordered_ring(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__idom_172,axiom,
    linordered_idom(rat) ).

tff(tcon_Rat_Orat___Rings_Ocomm__semiring__1_173,axiom,
    comm_semiring_1(rat) ).

tff(tcon_Rat_Orat___Rings_Ocomm__semiring__0_174,axiom,
    comm_semiring_0(rat) ).

tff(tcon_Rat_Orat___Orderings_Odense__order,axiom,
    dense_order(rat) ).

tff(tcon_Rat_Orat___Groups_Osemigroup__mult_175,axiom,
    semigroup_mult(rat) ).

tff(tcon_Rat_Orat___Rings_Osemidom__divide_176,axiom,
    semidom_divide(rat) ).

tff(tcon_Rat_Orat___Num_Osemiring__numeral_177,axiom,
    semiring_numeral(rat) ).

tff(tcon_Rat_Orat___Groups_Osemigroup__add_178,axiom,
    semigroup_add(rat) ).

tff(tcon_Rat_Orat___Fields_Ofield__abs__sgn,axiom,
    field_abs_sgn(rat) ).

tff(tcon_Rat_Orat___Fields_Odivision__ring,axiom,
    division_ring(rat) ).

tff(tcon_Rat_Orat___Rings_Ozero__less__one_179,axiom,
    zero_less_one(rat) ).

tff(tcon_Rat_Orat___Rings_Ocomm__semiring_180,axiom,
    comm_semiring(rat) ).

tff(tcon_Rat_Orat___Nat_Osemiring__char__0_181,axiom,
    semiring_char_0(rat) ).

tff(tcon_Rat_Orat___Groups_Oab__group__add_182,axiom,
    ab_group_add(rat) ).

tff(tcon_Rat_Orat___Fields_Ofield__char__0,axiom,
    field_char_0(rat) ).

tff(tcon_Rat_Orat___Rings_Ozero__neq__one_183,axiom,
    zero_neq_one(rat) ).

tff(tcon_Rat_Orat___Rings_Oordered__ring_184,axiom,
    ordered_ring(rat) ).

tff(tcon_Rat_Orat___Rings_Oidom__abs__sgn_185,axiom,
    idom_abs_sgn(rat) ).

tff(tcon_Rat_Orat___Orderings_Opreorder_186,axiom,
    preorder(rat) ).

tff(tcon_Rat_Orat___Orderings_Olinorder_187,axiom,
    linorder(rat) ).

tff(tcon_Rat_Orat___Groups_Omonoid__mult_188,axiom,
    monoid_mult(rat) ).

tff(tcon_Rat_Orat___Rings_Oidom__divide_189,axiom,
    idom_divide(rat) ).

tff(tcon_Rat_Orat___Rings_Ocomm__ring__1_190,axiom,
    comm_ring_1(rat) ).

tff(tcon_Rat_Orat___Groups_Omonoid__add_191,axiom,
    monoid_add(rat) ).

tff(tcon_Rat_Orat___Rings_Osemiring__1_192,axiom,
    semiring_1(rat) ).

tff(tcon_Rat_Orat___Rings_Osemiring__0_193,axiom,
    semiring_0(rat) ).

tff(tcon_Rat_Orat___Orderings_Ono__top_194,axiom,
    no_top(rat) ).

tff(tcon_Rat_Orat___Orderings_Ono__bot_195,axiom,
    no_bot(rat) ).

tff(tcon_Rat_Orat___Groups_Ogroup__add_196,axiom,
    group_add(rat) ).

tff(tcon_Rat_Orat___Rings_Omult__zero_197,axiom,
    mult_zero(rat) ).

tff(tcon_Rat_Orat___Rings_Ocomm__ring_198,axiom,
    comm_ring(rat) ).

tff(tcon_Rat_Orat___Orderings_Oorder_199,axiom,
    order(rat) ).

tff(tcon_Rat_Orat___Num_Oneg__numeral_200,axiom,
    neg_numeral(rat) ).

tff(tcon_Rat_Orat___Nat_Oring__char__0_201,axiom,
    ring_char_0(rat) ).

tff(tcon_Rat_Orat___Rings_Osemiring_202,axiom,
    semiring(rat) ).

tff(tcon_Rat_Orat___Fields_Oinverse,axiom,
    inverse(rat) ).

tff(tcon_Rat_Orat___Rings_Osemidom_203,axiom,
    semidom(rat) ).

tff(tcon_Rat_Orat___Orderings_Oord_204,axiom,
    ord(rat) ).

tff(tcon_Rat_Orat___Groups_Ouminus_205,axiom,
    uminus(rat) ).

tff(tcon_Rat_Orat___Rings_Oring__1_206,axiom,
    ring_1(rat) ).

tff(tcon_Rat_Orat___Rings_Oabs__if_207,axiom,
    abs_if(rat) ).

tff(tcon_Rat_Orat___Fields_Ofield,axiom,
    field(rat) ).

tff(tcon_Rat_Orat___Power_Opower_208,axiom,
    power(rat) ).

tff(tcon_Rat_Orat___Num_Onumeral_209,axiom,
    numeral(rat) ).

tff(tcon_Rat_Orat___Groups_Ozero_210,axiom,
    zero(rat) ).

tff(tcon_Rat_Orat___Groups_Oplus_211,axiom,
    plus(rat) ).

tff(tcon_Rat_Orat___Rings_Oring_212,axiom,
    ring(rat) ).

tff(tcon_Rat_Orat___Rings_Oidom_213,axiom,
    idom(rat) ).

tff(tcon_Rat_Orat___Groups_Oone_214,axiom,
    one(rat) ).

tff(tcon_Rat_Orat___Rings_Odvd_215,axiom,
    dvd(rat) ).

tff(tcon_Set_Oset___Quickcheck__Narrowing_Opartial__term__of_216,axiom,
    ! [A11: $tType] :
      ( typerep(A11)
     => quickc6926020345158392990erm_of(set(A11)) ) ).

tff(tcon_Set_Oset___Boolean__Algebras_Oboolean__algebra_217,axiom,
    ! [A11: $tType] : boolea8198339166811842893lgebra(set(A11)) ).

tff(tcon_Set_Oset___Code__Evaluation_Oterm__of_218,axiom,
    ! [A11: $tType] :
      ( typerep(A11)
     => code_term_of(set(A11)) ) ).

tff(tcon_Set_Oset___Orderings_Oorder__top_219,axiom,
    ! [A11: $tType] : order_top(set(A11)) ).

tff(tcon_Set_Oset___Orderings_Oorder__bot_220,axiom,
    ! [A11: $tType] : order_bot(set(A11)) ).

tff(tcon_Set_Oset___Orderings_Opreorder_221,axiom,
    ! [A11: $tType] : preorder(set(A11)) ).

tff(tcon_Set_Oset___Finite__Set_Ofinite_222,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => finite_finite(set(A11)) ) ).

tff(tcon_Set_Oset___Orderings_Oorder_223,axiom,
    ! [A11: $tType] : order(set(A11)) ).

tff(tcon_Set_Oset___Orderings_Oord_224,axiom,
    ! [A11: $tType] : ord(set(A11)) ).

tff(tcon_Set_Oset___Groups_Ouminus_225,axiom,
    ! [A11: $tType] : uminus(set(A11)) ).

tff(tcon_HOL_Obool___Topological__Spaces_Otopological__space_226,axiom,
    topolo4958980785337419405_space(bool) ).

tff(tcon_HOL_Obool___Topological__Spaces_Olinorder__topology_227,axiom,
    topolo1944317154257567458pology(bool) ).

tff(tcon_HOL_Obool___Topological__Spaces_Odiscrete__topology_228,axiom,
    topolo8865339358273720382pology(bool) ).

tff(tcon_HOL_Obool___Quickcheck__Narrowing_Opartial__term__of_229,axiom,
    quickc6926020345158392990erm_of(bool) ).

tff(tcon_HOL_Obool___Topological__Spaces_Oorder__topology_230,axiom,
    topolo2564578578187576103pology(bool) ).

tff(tcon_HOL_Obool___Boolean__Algebras_Oboolean__algebra_231,axiom,
    boolea8198339166811842893lgebra(bool) ).

tff(tcon_HOL_Obool___Topological__Spaces_Ot2__space_232,axiom,
    topological_t2_space(bool) ).

tff(tcon_HOL_Obool___Topological__Spaces_Ot1__space_233,axiom,
    topological_t1_space(bool) ).

tff(tcon_HOL_Obool___Code__Evaluation_Oterm__of_234,axiom,
    code_term_of(bool) ).

tff(tcon_HOL_Obool___Orderings_Oorder__top_235,axiom,
    order_top(bool) ).

tff(tcon_HOL_Obool___Orderings_Oorder__bot_236,axiom,
    order_bot(bool) ).

tff(tcon_HOL_Obool___Orderings_Opreorder_237,axiom,
    preorder(bool) ).

tff(tcon_HOL_Obool___Orderings_Olinorder_238,axiom,
    linorder(bool) ).

tff(tcon_HOL_Obool___Finite__Set_Ofinite_239,axiom,
    finite_finite(bool) ).

tff(tcon_HOL_Obool___Orderings_Oorder_240,axiom,
    order(bool) ).

tff(tcon_HOL_Obool___Orderings_Oord_241,axiom,
    ord(bool) ).

tff(tcon_HOL_Obool___Groups_Ouminus_242,axiom,
    uminus(bool) ).

tff(tcon_List_Olist___Quickcheck__Narrowing_Opartial__term__of_243,axiom,
    ! [A11: $tType] :
      ( typerep(A11)
     => quickc6926020345158392990erm_of(list(A11)) ) ).

tff(tcon_List_Olist___Code__Evaluation_Oterm__of_244,axiom,
    ! [A11: $tType] :
      ( typerep(A11)
     => code_term_of(list(A11)) ) ).

tff(tcon_List_Olist___Nat_Osize_245,axiom,
    ! [A11: $tType] : size(list(A11)) ).

tff(tcon_Real_Oreal___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_246,axiom,
    condit6923001295902523014norder(real) ).

tff(tcon_Real_Oreal___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_247,axiom,
    semiri1453513574482234551roduct(real) ).

tff(tcon_Real_Oreal___Groups_Oordered__ab__semigroup__monoid__add__imp__le_248,axiom,
    ordere1937475149494474687imp_le(real) ).

tff(tcon_Real_Oreal___Topological__Spaces_Olinear__continuum__topology,axiom,
    topolo8458572112393995274pology(real) ).

tff(tcon_Real_Oreal___Topological__Spaces_Ofirst__countable__topology,axiom,
    topolo3112930676232923870pology(real) ).

tff(tcon_Real_Oreal___Real__Vector__Spaces_Oreal__normed__div__algebra,axiom,
    real_V8999393235501362500lgebra(real) ).

tff(tcon_Real_Oreal___Real__Vector__Spaces_Oreal__normed__algebra__1,axiom,
    real_V2822296259951069270ebra_1(real) ).

tff(tcon_Real_Oreal___Rings_Osemiring__no__zero__divisors__cancel_249,axiom,
    semiri6575147826004484403cancel(real) ).

tff(tcon_Real_Oreal___Real__Vector__Spaces_Oreal__normed__algebra,axiom,
    real_V4412858255891104859lgebra(real) ).

tff(tcon_Real_Oreal___Real__Vector__Spaces_Oordered__real__vector,axiom,
    real_V5355595471888546746vector(real) ).

tff(tcon_Real_Oreal___Groups_Ostrict__ordered__ab__semigroup__add_250,axiom,
    strict9044650504122735259up_add(real) ).

tff(tcon_Real_Oreal___Groups_Oordered__cancel__ab__semigroup__add_251,axiom,
    ordere580206878836729694up_add(real) ).

tff(tcon_Real_Oreal___Groups_Oordered__ab__semigroup__add__imp__le_252,axiom,
    ordere2412721322843649153imp_le(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__comm__semiring__strict_253,axiom,
    linord2810124833399127020strict(real) ).

tff(tcon_Real_Oreal___Real__Vector__Spaces_Oreal__normed__vector,axiom,
    real_V822414075346904944vector(real) ).

tff(tcon_Real_Oreal___Groups_Ostrict__ordered__comm__monoid__add_254,axiom,
    strict7427464778891057005id_add(real) ).

tff(tcon_Real_Oreal___Groups_Oordered__cancel__comm__monoid__add_255,axiom,
    ordere8940638589300402666id_add(real) ).

tff(tcon_Real_Oreal___Topological__Spaces_Otopological__space_256,axiom,
    topolo4958980785337419405_space(real) ).

tff(tcon_Real_Oreal___Topological__Spaces_Olinorder__topology_257,axiom,
    topolo1944317154257567458pology(real) ).

tff(tcon_Real_Oreal___Real__Vector__Spaces_Oreal__normed__field,axiom,
    real_V3459762299906320749_field(real) ).

tff(tcon_Real_Oreal___Quickcheck__Narrowing_Opartial__term__of_258,axiom,
    quickc6926020345158392990erm_of(real) ).

tff(tcon_Real_Oreal___Real__Vector__Spaces_Oreal__div__algebra,axiom,
    real_V5047593784448816457lgebra(real) ).

tff(tcon_Real_Oreal___Archimedean__Field_Oarchimedean__field_259,axiom,
    archim462609752435547400_field(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__semiring__1__strict_260,axiom,
    linord715952674999750819strict(real) ).

tff(tcon_Real_Oreal___Real__Vector__Spaces_Ouniformity__dist,axiom,
    real_V768167426530841204y_dist(real) ).

tff(tcon_Real_Oreal___Orderings_Ounbounded__dense__linorder_261,axiom,
    unboun7993243217541854897norder(real) ).

tff(tcon_Real_Oreal___Limits_Otopological__comm__monoid__add_262,axiom,
    topolo5987344860129210374id_add(real) ).

tff(tcon_Real_Oreal___Groups_Olinordered__ab__semigroup__add_263,axiom,
    linord4140545234300271783up_add(real) ).

tff(tcon_Real_Oreal___Topological__Spaces_Oorder__topology_264,axiom,
    topolo2564578578187576103pology(real) ).

tff(tcon_Real_Oreal___Rings_Osemiring__1__no__zero__divisors_265,axiom,
    semiri2026040879449505780visors(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__nonzero__semiring_266,axiom,
    linord181362715937106298miring(real) ).

tff(tcon_Real_Oreal___Real__Vector__Spaces_Oreal__algebra__1,axiom,
    real_V2191834092415804123ebra_1(real) ).

tff(tcon_Real_Oreal___Real__Vector__Spaces_Ocomplete__space,axiom,
    real_V8037385150606011577_space(real) ).

tff(tcon_Real_Oreal___Limits_Otopological__semigroup__mult_267,axiom,
    topolo4211221413907600880p_mult(real) ).

tff(tcon_Real_Oreal___Topological__Spaces_Ouniform__space,axiom,
    topolo7287701948861334536_space(real) ).

tff(tcon_Real_Oreal___Topological__Spaces_Operfect__space,axiom,
    topolo8386298272705272623_space(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__semiring__strict_268,axiom,
    linord8928482502909563296strict(real) ).

tff(tcon_Real_Oreal___Rings_Osemiring__no__zero__divisors_269,axiom,
    semiri3467727345109120633visors(real) ).

tff(tcon_Real_Oreal___Real__Vector__Spaces_Oreal__algebra,axiom,
    real_V6157519004096292374lgebra(real) ).

tff(tcon_Real_Oreal___Real__Vector__Spaces_Ometric__space,axiom,
    real_V7819770556892013058_space(real) ).

tff(tcon_Real_Oreal___Limits_Otopological__ab__group__add,axiom,
    topolo1287966508704411220up_add(real) ).

tff(tcon_Real_Oreal___Groups_Oordered__ab__semigroup__add_270,axiom,
    ordere6658533253407199908up_add(real) ).

tff(tcon_Real_Oreal___Groups_Oordered__ab__group__add__abs_271,axiom,
    ordere166539214618696060dd_abs(real) ).

tff(tcon_Real_Oreal___Archimedean__Field_Ofloor__ceiling_272,axiom,
    archim2362893244070406136eiling(real) ).

tff(tcon_Real_Oreal___Real__Vector__Spaces_Oreal__vector,axiom,
    real_V4867850818363320053vector(real) ).

tff(tcon_Real_Oreal___Groups_Oordered__comm__monoid__add_273,axiom,
    ordere6911136660526730532id_add(real) ).

tff(tcon_Real_Oreal___Groups_Olinordered__ab__group__add_274,axiom,
    linord5086331880401160121up_add(real) ).

tff(tcon_Real_Oreal___Groups_Ocancel__ab__semigroup__add_275,axiom,
    cancel2418104881723323429up_add(real) ).

tff(tcon_Real_Oreal___Rings_Oring__1__no__zero__divisors_276,axiom,
    ring_15535105094025558882visors(real) ).

tff(tcon_Real_Oreal___Real__Vector__Spaces_Oreal__field,axiom,
    real_V7773925162809079976_field(real) ).

tff(tcon_Real_Oreal___Limits_Otopological__monoid__add_277,axiom,
    topolo6943815403480290642id_add(real) ).

tff(tcon_Real_Oreal___Groups_Ocancel__comm__monoid__add_278,axiom,
    cancel1802427076303600483id_add(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__ring__strict_279,axiom,
    linord4710134922213307826strict(real) ).

tff(tcon_Real_Oreal___Rings_Ocomm__semiring__1__cancel_280,axiom,
    comm_s4317794764714335236cancel(real) ).

tff(tcon_Real_Oreal___Limits_Otopological__group__add,axiom,
    topolo1633459387980952147up_add(real) ).

tff(tcon_Real_Oreal___Topological__Spaces_Ot2__space_281,axiom,
    topological_t2_space(real) ).

tff(tcon_Real_Oreal___Topological__Spaces_Ot1__space_282,axiom,
    topological_t1_space(real) ).

tff(tcon_Real_Oreal___Rings_Oordered__comm__semiring_283,axiom,
    ordere2520102378445227354miring(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__semiring__1_284,axiom,
    linord6961819062388156250ring_1(real) ).

tff(tcon_Real_Oreal___Groups_Oordered__ab__group__add_285,axiom,
    ordered_ab_group_add(real) ).

tff(tcon_Real_Oreal___Groups_Ocancel__semigroup__add_286,axiom,
    cancel_semigroup_add(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__semiring_287,axiom,
    linordered_semiring(real) ).

tff(tcon_Real_Oreal___Real__Vector__Spaces_Obanach,axiom,
    real_Vector_banach(real) ).

tff(tcon_Real_Oreal___Rings_Oordered__semiring__0_288,axiom,
    ordered_semiring_0(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__semidom_289,axiom,
    linordered_semidom(real) ).

tff(tcon_Real_Oreal___Orderings_Odense__linorder_290,axiom,
    dense_linorder(real) ).

tff(tcon_Real_Oreal___Groups_Oab__semigroup__mult_291,axiom,
    ab_semigroup_mult(real) ).

tff(tcon_Real_Oreal___Rings_Osemiring__1__cancel_292,axiom,
    semiring_1_cancel(real) ).

tff(tcon_Real_Oreal___Groups_Ocomm__monoid__mult_293,axiom,
    comm_monoid_mult(real) ).

tff(tcon_Real_Oreal___Groups_Oab__semigroup__add_294,axiom,
    ab_semigroup_add(real) ).

tff(tcon_Real_Oreal___Fields_Olinordered__field_295,axiom,
    linordered_field(real) ).

tff(tcon_Real_Oreal___Code__Evaluation_Oterm__of_296,axiom,
    code_term_of(real) ).

tff(tcon_Real_Oreal___Rings_Oordered__semiring_297,axiom,
    ordered_semiring(real) ).

tff(tcon_Real_Oreal___Rings_Oordered__ring__abs_298,axiom,
    ordered_ring_abs(real) ).

tff(tcon_Real_Oreal___Groups_Ocomm__monoid__add_299,axiom,
    comm_monoid_add(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__ring_300,axiom,
    linordered_ring(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__idom_301,axiom,
    linordered_idom(real) ).

tff(tcon_Real_Oreal___Rings_Ocomm__semiring__1_302,axiom,
    comm_semiring_1(real) ).

tff(tcon_Real_Oreal___Rings_Ocomm__semiring__0_303,axiom,
    comm_semiring_0(real) ).

tff(tcon_Real_Oreal___Orderings_Odense__order_304,axiom,
    dense_order(real) ).

tff(tcon_Real_Oreal___Groups_Osemigroup__mult_305,axiom,
    semigroup_mult(real) ).

tff(tcon_Real_Oreal___Rings_Osemidom__divide_306,axiom,
    semidom_divide(real) ).

tff(tcon_Real_Oreal___Num_Osemiring__numeral_307,axiom,
    semiring_numeral(real) ).

tff(tcon_Real_Oreal___Groups_Osemigroup__add_308,axiom,
    semigroup_add(real) ).

tff(tcon_Real_Oreal___Fields_Ofield__abs__sgn_309,axiom,
    field_abs_sgn(real) ).

tff(tcon_Real_Oreal___Fields_Odivision__ring_310,axiom,
    division_ring(real) ).

tff(tcon_Real_Oreal___Rings_Ozero__less__one_311,axiom,
    zero_less_one(real) ).

tff(tcon_Real_Oreal___Rings_Ocomm__semiring_312,axiom,
    comm_semiring(real) ).

tff(tcon_Real_Oreal___Nat_Osemiring__char__0_313,axiom,
    semiring_char_0(real) ).

tff(tcon_Real_Oreal___Groups_Oab__group__add_314,axiom,
    ab_group_add(real) ).

tff(tcon_Real_Oreal___Fields_Ofield__char__0_315,axiom,
    field_char_0(real) ).

tff(tcon_Real_Oreal___Rings_Ozero__neq__one_316,axiom,
    zero_neq_one(real) ).

tff(tcon_Real_Oreal___Rings_Oordered__ring_317,axiom,
    ordered_ring(real) ).

tff(tcon_Real_Oreal___Rings_Oidom__abs__sgn_318,axiom,
    idom_abs_sgn(real) ).

tff(tcon_Real_Oreal___Orderings_Opreorder_319,axiom,
    preorder(real) ).

tff(tcon_Real_Oreal___Orderings_Olinorder_320,axiom,
    linorder(real) ).

tff(tcon_Real_Oreal___Groups_Omonoid__mult_321,axiom,
    monoid_mult(real) ).

tff(tcon_Real_Oreal___Transcendental_Oln,axiom,
    ln(real) ).

tff(tcon_Real_Oreal___Rings_Oidom__divide_322,axiom,
    idom_divide(real) ).

tff(tcon_Real_Oreal___Rings_Ocomm__ring__1_323,axiom,
    comm_ring_1(real) ).

tff(tcon_Real_Oreal___Groups_Omonoid__add_324,axiom,
    monoid_add(real) ).

tff(tcon_Real_Oreal___Rings_Osemiring__1_325,axiom,
    semiring_1(real) ).

tff(tcon_Real_Oreal___Rings_Osemiring__0_326,axiom,
    semiring_0(real) ).

tff(tcon_Real_Oreal___Orderings_Ono__top_327,axiom,
    no_top(real) ).

tff(tcon_Real_Oreal___Orderings_Ono__bot_328,axiom,
    no_bot(real) ).

tff(tcon_Real_Oreal___Groups_Ogroup__add_329,axiom,
    group_add(real) ).

tff(tcon_Real_Oreal___Rings_Omult__zero_330,axiom,
    mult_zero(real) ).

tff(tcon_Real_Oreal___Rings_Ocomm__ring_331,axiom,
    comm_ring(real) ).

tff(tcon_Real_Oreal___Orderings_Oorder_332,axiom,
    order(real) ).

tff(tcon_Real_Oreal___Num_Oneg__numeral_333,axiom,
    neg_numeral(real) ).

tff(tcon_Real_Oreal___Nat_Oring__char__0_334,axiom,
    ring_char_0(real) ).

tff(tcon_Real_Oreal___Rings_Osemiring_335,axiom,
    semiring(real) ).

tff(tcon_Real_Oreal___Fields_Oinverse_336,axiom,
    inverse(real) ).

tff(tcon_Real_Oreal___Rings_Osemidom_337,axiom,
    semidom(real) ).

tff(tcon_Real_Oreal___Orderings_Oord_338,axiom,
    ord(real) ).

tff(tcon_Real_Oreal___Groups_Ouminus_339,axiom,
    uminus(real) ).

tff(tcon_Real_Oreal___Rings_Oring__1_340,axiom,
    ring_1(real) ).

tff(tcon_Real_Oreal___Rings_Oabs__if_341,axiom,
    abs_if(real) ).

tff(tcon_Real_Oreal___Fields_Ofield_342,axiom,
    field(real) ).

tff(tcon_Real_Oreal___Power_Opower_343,axiom,
    power(real) ).

tff(tcon_Real_Oreal___Num_Onumeral_344,axiom,
    numeral(real) ).

tff(tcon_Real_Oreal___Groups_Ozero_345,axiom,
    zero(real) ).

tff(tcon_Real_Oreal___Groups_Oplus_346,axiom,
    plus(real) ).

tff(tcon_Real_Oreal___Rings_Oring_347,axiom,
    ring(real) ).

tff(tcon_Real_Oreal___Rings_Oidom_348,axiom,
    idom(real) ).

tff(tcon_Real_Oreal___Groups_Oone_349,axiom,
    one(real) ).

tff(tcon_Real_Oreal___Rings_Odvd_350,axiom,
    dvd(real) ).

tff(tcon_Word_Oword___Bit__Operations_Osemiring__bit__operations_351,axiom,
    ! [A11: $tType] :
      ( type_len(A11)
     => bit_se359711467146920520ations(word(A11)) ) ).

tff(tcon_Word_Oword___Quickcheck__Narrowing_Opartial__term__of_352,axiom,
    ! [A11: $tType] :
      ( typerep(A11)
     => quickc6926020345158392990erm_of(word(A11)) ) ).

tff(tcon_Word_Oword___Bit__Comprehension_Obit__comprehension_353,axiom,
    ! [A11: $tType] :
      ( type_len(A11)
     => bit_bi6583157726757044596ension(word(A11)) ) ).

tff(tcon_Word_Oword___Bit__Operations_Oring__bit__operations_354,axiom,
    ! [A11: $tType] :
      ( type_len(A11)
     => bit_ri3973907225187159222ations(word(A11)) ) ).

tff(tcon_Word_Oword___Groups_Ocancel__ab__semigroup__add_355,axiom,
    ! [A11: $tType] :
      ( type_len(A11)
     => cancel2418104881723323429up_add(word(A11)) ) ).

tff(tcon_Word_Oword___Groups_Ocancel__comm__monoid__add_356,axiom,
    ! [A11: $tType] :
      ( type_len(A11)
     => cancel1802427076303600483id_add(word(A11)) ) ).

tff(tcon_Word_Oword___Rings_Ocomm__semiring__1__cancel_357,axiom,
    ! [A11: $tType] :
      ( type_len(A11)
     => comm_s4317794764714335236cancel(word(A11)) ) ).

tff(tcon_Word_Oword___Bit__Operations_Osemiring__bits_358,axiom,
    ! [A11: $tType] :
      ( type_len(A11)
     => bit_semiring_bits(word(A11)) ) ).

tff(tcon_Word_Oword___Groups_Ocancel__semigroup__add_359,axiom,
    ! [A11: $tType] :
      ( type_len(A11)
     => cancel_semigroup_add(word(A11)) ) ).

tff(tcon_Word_Oword___Least__significant__bit_Olsb_360,axiom,
    ! [A11: $tType] :
      ( type_len(A11)
     => least_6119777620449941438nt_lsb(word(A11)) ) ).

tff(tcon_Word_Oword___Groups_Oab__semigroup__mult_361,axiom,
    ! [A11: $tType] :
      ( type_len(A11)
     => ab_semigroup_mult(word(A11)) ) ).

tff(tcon_Word_Oword___Rings_Osemiring__1__cancel_362,axiom,
    ! [A11: $tType] :
      ( type_len(A11)
     => semiring_1_cancel(word(A11)) ) ).

tff(tcon_Word_Oword___Groups_Ocomm__monoid__mult_363,axiom,
    ! [A11: $tType] :
      ( type_len(A11)
     => comm_monoid_mult(word(A11)) ) ).

tff(tcon_Word_Oword___Groups_Oab__semigroup__add_364,axiom,
    ! [A11: $tType] :
      ( type_len(A11)
     => ab_semigroup_add(word(A11)) ) ).

tff(tcon_Word_Oword___Generic__set__bit_Oset__bit_365,axiom,
    ! [A11: $tType] :
      ( type_len(A11)
     => generic_set_set_bit(word(A11)) ) ).

tff(tcon_Word_Oword___Code__Evaluation_Oterm__of_366,axiom,
    ! [A11: $tType] :
      ( typerep(A11)
     => code_term_of(word(A11)) ) ).

tff(tcon_Word_Oword___Parity_Osemiring__parity_367,axiom,
    ! [A11: $tType] :
      ( type_len(A11)
     => semiring_parity(word(A11)) ) ).

tff(tcon_Word_Oword___Groups_Ocomm__monoid__add_368,axiom,
    ! [A11: $tType] :
      ( type_len(A11)
     => comm_monoid_add(word(A11)) ) ).

tff(tcon_Word_Oword___Rings_Osemiring__modulo_369,axiom,
    ! [A11: $tType] :
      ( type_len(A11)
     => semiring_modulo(word(A11)) ) ).

tff(tcon_Word_Oword___Rings_Ocomm__semiring__1_370,axiom,
    ! [A11: $tType] :
      ( type_len(A11)
     => comm_semiring_1(word(A11)) ) ).

tff(tcon_Word_Oword___Rings_Ocomm__semiring__0_371,axiom,
    ! [A11: $tType] :
      ( type_len(A11)
     => comm_semiring_0(word(A11)) ) ).

tff(tcon_Word_Oword___Groups_Osemigroup__mult_372,axiom,
    ! [A11: $tType] :
      ( type_len(A11)
     => semigroup_mult(word(A11)) ) ).

tff(tcon_Word_Oword___Num_Osemiring__numeral_373,axiom,
    ! [A11: $tType] :
      ( type_len(A11)
     => semiring_numeral(word(A11)) ) ).

tff(tcon_Word_Oword___Groups_Osemigroup__add_374,axiom,
    ! [A11: $tType] :
      ( type_len(A11)
     => semigroup_add(word(A11)) ) ).

tff(tcon_Word_Oword___Rings_Ocomm__semiring_375,axiom,
    ! [A11: $tType] :
      ( type_len(A11)
     => comm_semiring(word(A11)) ) ).

tff(tcon_Word_Oword___Orderings_Owellorder_376,axiom,
    ! [A11: $tType] :
      ( type_len(A11)
     => wellorder(word(A11)) ) ).

tff(tcon_Word_Oword___Groups_Oab__group__add_377,axiom,
    ! [A11: $tType] :
      ( type_len(A11)
     => ab_group_add(word(A11)) ) ).

tff(tcon_Word_Oword___Rings_Ozero__neq__one_378,axiom,
    ! [A11: $tType] :
      ( type_len(A11)
     => zero_neq_one(word(A11)) ) ).

tff(tcon_Word_Oword___Parity_Oring__parity_379,axiom,
    ! [A11: $tType] :
      ( type_len(A11)
     => ring_parity(word(A11)) ) ).

tff(tcon_Word_Oword___Orderings_Opreorder_380,axiom,
    ! [A11: $tType] :
      ( type_len(A11)
     => preorder(word(A11)) ) ).

tff(tcon_Word_Oword___Orderings_Olinorder_381,axiom,
    ! [A11: $tType] :
      ( type_len(A11)
     => linorder(word(A11)) ) ).

tff(tcon_Word_Oword___Groups_Omonoid__mult_382,axiom,
    ! [A11: $tType] :
      ( type_len(A11)
     => monoid_mult(word(A11)) ) ).

tff(tcon_Word_Oword___Rings_Ocomm__ring__1_383,axiom,
    ! [A11: $tType] :
      ( type_len(A11)
     => comm_ring_1(word(A11)) ) ).

tff(tcon_Word_Oword___Groups_Omonoid__add_384,axiom,
    ! [A11: $tType] :
      ( type_len(A11)
     => monoid_add(word(A11)) ) ).

tff(tcon_Word_Oword___Finite__Set_Ofinite_385,axiom,
    ! [A11: $tType] :
      ( type_len(A11)
     => finite_finite(word(A11)) ) ).

tff(tcon_Word_Oword___Rings_Osemiring__1_386,axiom,
    ! [A11: $tType] :
      ( type_len(A11)
     => semiring_1(word(A11)) ) ).

tff(tcon_Word_Oword___Rings_Osemiring__0_387,axiom,
    ! [A11: $tType] :
      ( type_len(A11)
     => semiring_0(word(A11)) ) ).

tff(tcon_Word_Oword___Groups_Ogroup__add_388,axiom,
    ! [A11: $tType] :
      ( type_len(A11)
     => group_add(word(A11)) ) ).

tff(tcon_Word_Oword___Rings_Omult__zero_389,axiom,
    ! [A11: $tType] :
      ( type_len(A11)
     => mult_zero(word(A11)) ) ).

tff(tcon_Word_Oword___Rings_Ocomm__ring_390,axiom,
    ! [A11: $tType] :
      ( type_len(A11)
     => comm_ring(word(A11)) ) ).

tff(tcon_Word_Oword___Orderings_Oorder_391,axiom,
    ! [A11: $tType] :
      ( type_len(A11)
     => order(word(A11)) ) ).

tff(tcon_Word_Oword___Num_Oneg__numeral_392,axiom,
    ! [A11: $tType] :
      ( type_len(A11)
     => neg_numeral(word(A11)) ) ).

tff(tcon_Word_Oword___Rings_Osemiring_393,axiom,
    ! [A11: $tType] :
      ( type_len(A11)
     => semiring(word(A11)) ) ).

tff(tcon_Word_Oword___Orderings_Oord_394,axiom,
    ! [A11: $tType] :
      ( type_len(A11)
     => ord(word(A11)) ) ).

tff(tcon_Word_Oword___Groups_Ouminus_395,axiom,
    ! [A11: $tType] :
      ( type_len(A11)
     => uminus(word(A11)) ) ).

tff(tcon_Word_Oword___Rings_Oring__1_396,axiom,
    ! [A11: $tType] :
      ( type_len(A11)
     => ring_1(word(A11)) ) ).

tff(tcon_Word_Oword___Power_Opower_397,axiom,
    ! [A11: $tType] :
      ( type_len(A11)
     => power(word(A11)) ) ).

tff(tcon_Word_Oword___Num_Onumeral_398,axiom,
    ! [A11: $tType] :
      ( type_len(A11)
     => numeral(word(A11)) ) ).

tff(tcon_Word_Oword___Groups_Ozero_399,axiom,
    ! [A11: $tType] :
      ( type_len(A11)
     => zero(word(A11)) ) ).

tff(tcon_Word_Oword___Groups_Oplus_400,axiom,
    ! [A11: $tType] :
      ( type_len(A11)
     => plus(word(A11)) ) ).

tff(tcon_Word_Oword___Rings_Oring_401,axiom,
    ! [A11: $tType] :
      ( type_len(A11)
     => ring(word(A11)) ) ).

tff(tcon_Word_Oword___Groups_Oone_402,axiom,
    ! [A11: $tType] :
      ( type_len(A11)
     => one(word(A11)) ) ).

tff(tcon_Word_Oword___Rings_Odvd_403,axiom,
    ! [A11: $tType] :
      ( type_len(A11)
     => dvd(word(A11)) ) ).

tff(tcon_Word_Oword___Nat_Osize_404,axiom,
    ! [A11: $tType] :
      ( type_len(A11)
     => size(word(A11)) ) ).

tff(tcon_String_Ochar___Quickcheck__Narrowing_Opartial__term__of_405,axiom,
    quickc6926020345158392990erm_of(char) ).

tff(tcon_String_Ochar___Code__Evaluation_Oterm__of_406,axiom,
    code_term_of(char) ).

tff(tcon_String_Ochar___Orderings_Opreorder_407,axiom,
    preorder(char) ).

tff(tcon_String_Ochar___Orderings_Olinorder_408,axiom,
    linorder(char) ).

tff(tcon_String_Ochar___Finite__Set_Ofinite_409,axiom,
    finite_finite(char) ).

tff(tcon_String_Ochar___Orderings_Oorder_410,axiom,
    order(char) ).

tff(tcon_String_Ochar___Orderings_Oord_411,axiom,
    ord(char) ).

tff(tcon_String_Ochar___Nat_Osize_412,axiom,
    size(char) ).

tff(tcon_Enum_Ofinite__1___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_413,axiom,
    condit6923001295902523014norder(finite_1) ).

tff(tcon_Enum_Ofinite__1___Groups_Oordered__ab__semigroup__monoid__add__imp__le_414,axiom,
    ordere1937475149494474687imp_le(finite_1) ).

tff(tcon_Enum_Ofinite__1___Rings_Osemiring__no__zero__divisors__cancel_415,axiom,
    semiri6575147826004484403cancel(finite_1) ).

tff(tcon_Enum_Ofinite__1___Groups_Ostrict__ordered__ab__semigroup__add_416,axiom,
    strict9044650504122735259up_add(finite_1) ).

tff(tcon_Enum_Ofinite__1___Groups_Oordered__cancel__comm__monoid__diff_417,axiom,
    ordere1170586879665033532d_diff(finite_1) ).

tff(tcon_Enum_Ofinite__1___Groups_Oordered__cancel__ab__semigroup__add_418,axiom,
    ordere580206878836729694up_add(finite_1) ).

tff(tcon_Enum_Ofinite__1___Groups_Oordered__ab__semigroup__add__imp__le_419,axiom,
    ordere2412721322843649153imp_le(finite_1) ).

tff(tcon_Enum_Ofinite__1___Rings_Olinordered__comm__semiring__strict_420,axiom,
    linord2810124833399127020strict(finite_1) ).

tff(tcon_Enum_Ofinite__1___Groups_Ostrict__ordered__comm__monoid__add_421,axiom,
    strict7427464778891057005id_add(finite_1) ).

tff(tcon_Enum_Ofinite__1___Groups_Oordered__cancel__comm__monoid__add_422,axiom,
    ordere8940638589300402666id_add(finite_1) ).

tff(tcon_Enum_Ofinite__1___Groups_Ocanonically__ordered__monoid__add_423,axiom,
    canoni5634975068530333245id_add(finite_1) ).

tff(tcon_Enum_Ofinite__1___Quickcheck__Narrowing_Opartial__term__of_424,axiom,
    quickc6926020345158392990erm_of(finite_1) ).

tff(tcon_Enum_Ofinite__1___Groups_Olinordered__ab__semigroup__add_425,axiom,
    linord4140545234300271783up_add(finite_1) ).

tff(tcon_Enum_Ofinite__1___Rings_Olinordered__semiring__strict_426,axiom,
    linord8928482502909563296strict(finite_1) ).

tff(tcon_Enum_Ofinite__1___Boolean__Algebras_Oboolean__algebra_427,axiom,
    boolea8198339166811842893lgebra(finite_1) ).

tff(tcon_Enum_Ofinite__1___Rings_Osemiring__no__zero__divisors_428,axiom,
    semiri3467727345109120633visors(finite_1) ).

tff(tcon_Enum_Ofinite__1___Groups_Oordered__ab__semigroup__add_429,axiom,
    ordere6658533253407199908up_add(finite_1) ).

tff(tcon_Enum_Ofinite__1___Groups_Oordered__ab__group__add__abs_430,axiom,
    ordere166539214618696060dd_abs(finite_1) ).

tff(tcon_Enum_Ofinite__1___Groups_Oordered__comm__monoid__add_431,axiom,
    ordere6911136660526730532id_add(finite_1) ).

tff(tcon_Enum_Ofinite__1___Groups_Olinordered__ab__group__add_432,axiom,
    linord5086331880401160121up_add(finite_1) ).

tff(tcon_Enum_Ofinite__1___Groups_Ocancel__ab__semigroup__add_433,axiom,
    cancel2418104881723323429up_add(finite_1) ).

tff(tcon_Enum_Ofinite__1___Groups_Ocancel__comm__monoid__add_434,axiom,
    cancel1802427076303600483id_add(finite_1) ).

tff(tcon_Enum_Ofinite__1___Rings_Olinordered__ring__strict_435,axiom,
    linord4710134922213307826strict(finite_1) ).

tff(tcon_Enum_Ofinite__1___Rings_Oordered__comm__semiring_436,axiom,
    ordere2520102378445227354miring(finite_1) ).

tff(tcon_Enum_Ofinite__1___Groups_Oordered__ab__group__add_437,axiom,
    ordered_ab_group_add(finite_1) ).

tff(tcon_Enum_Ofinite__1___Groups_Ocancel__semigroup__add_438,axiom,
    cancel_semigroup_add(finite_1) ).

tff(tcon_Enum_Ofinite__1___Rings_Olinordered__semiring_439,axiom,
    linordered_semiring(finite_1) ).

tff(tcon_Enum_Ofinite__1___Rings_Oordered__semiring__0_440,axiom,
    ordered_semiring_0(finite_1) ).

tff(tcon_Enum_Ofinite__1___Orderings_Odense__linorder_441,axiom,
    dense_linorder(finite_1) ).

tff(tcon_Enum_Ofinite__1___Groups_Oab__semigroup__mult_442,axiom,
    ab_semigroup_mult(finite_1) ).

tff(tcon_Enum_Ofinite__1___Groups_Ocomm__monoid__mult_443,axiom,
    comm_monoid_mult(finite_1) ).

tff(tcon_Enum_Ofinite__1___Groups_Ocomm__monoid__diff_444,axiom,
    comm_monoid_diff(finite_1) ).

tff(tcon_Enum_Ofinite__1___Groups_Oab__semigroup__add_445,axiom,
    ab_semigroup_add(finite_1) ).

tff(tcon_Enum_Ofinite__1___Code__Evaluation_Oterm__of_446,axiom,
    code_term_of(finite_1) ).

tff(tcon_Enum_Ofinite__1___Rings_Oordered__semiring_447,axiom,
    ordered_semiring(finite_1) ).

tff(tcon_Enum_Ofinite__1___Rings_Oordered__ring__abs_448,axiom,
    ordered_ring_abs(finite_1) ).

tff(tcon_Enum_Ofinite__1___Groups_Ocomm__monoid__add_449,axiom,
    comm_monoid_add(finite_1) ).

tff(tcon_Enum_Ofinite__1___Rings_Olinordered__ring_450,axiom,
    linordered_ring(finite_1) ).

tff(tcon_Enum_Ofinite__1___Rings_Ocomm__semiring__0_451,axiom,
    comm_semiring_0(finite_1) ).

tff(tcon_Enum_Ofinite__1___Orderings_Odense__order_452,axiom,
    dense_order(finite_1) ).

tff(tcon_Enum_Ofinite__1___Groups_Osemigroup__mult_453,axiom,
    semigroup_mult(finite_1) ).

tff(tcon_Enum_Ofinite__1___Groups_Osemigroup__add_454,axiom,
    semigroup_add(finite_1) ).

tff(tcon_Enum_Ofinite__1___Rings_Ocomm__semiring_455,axiom,
    comm_semiring(finite_1) ).

tff(tcon_Enum_Ofinite__1___Orderings_Owellorder_456,axiom,
    wellorder(finite_1) ).

tff(tcon_Enum_Ofinite__1___Orderings_Oorder__top_457,axiom,
    order_top(finite_1) ).

tff(tcon_Enum_Ofinite__1___Orderings_Oorder__bot_458,axiom,
    order_bot(finite_1) ).

tff(tcon_Enum_Ofinite__1___Groups_Oab__group__add_459,axiom,
    ab_group_add(finite_1) ).

tff(tcon_Enum_Ofinite__1___Rings_Oordered__ring_460,axiom,
    ordered_ring(finite_1) ).

tff(tcon_Enum_Ofinite__1___Orderings_Opreorder_461,axiom,
    preorder(finite_1) ).

tff(tcon_Enum_Ofinite__1___Orderings_Olinorder_462,axiom,
    linorder(finite_1) ).

tff(tcon_Enum_Ofinite__1___Groups_Omonoid__mult_463,axiom,
    monoid_mult(finite_1) ).

tff(tcon_Enum_Ofinite__1___Groups_Omonoid__add_464,axiom,
    monoid_add(finite_1) ).

tff(tcon_Enum_Ofinite__1___Finite__Set_Ofinite_465,axiom,
    finite_finite(finite_1) ).

tff(tcon_Enum_Ofinite__1___Type__Length_Olen0,axiom,
    type_len0(finite_1) ).

tff(tcon_Enum_Ofinite__1___Rings_Osemiring__0_466,axiom,
    semiring_0(finite_1) ).

tff(tcon_Enum_Ofinite__1___Groups_Ogroup__add_467,axiom,
    group_add(finite_1) ).

tff(tcon_Enum_Ofinite__1___Type__Length_Olen,axiom,
    type_len(finite_1) ).

tff(tcon_Enum_Ofinite__1___Rings_Omult__zero_468,axiom,
    mult_zero(finite_1) ).

tff(tcon_Enum_Ofinite__1___Rings_Ocomm__ring_469,axiom,
    comm_ring(finite_1) ).

tff(tcon_Enum_Ofinite__1___Orderings_Oorder_470,axiom,
    order(finite_1) ).

tff(tcon_Enum_Ofinite__1___Rings_Osemiring_471,axiom,
    semiring(finite_1) ).

tff(tcon_Enum_Ofinite__1___Fields_Oinverse_472,axiom,
    inverse(finite_1) ).

tff(tcon_Enum_Ofinite__1___Orderings_Oord_473,axiom,
    ord(finite_1) ).

tff(tcon_Enum_Ofinite__1___Groups_Ouminus_474,axiom,
    uminus(finite_1) ).

tff(tcon_Enum_Ofinite__1___Rings_Oabs__if_475,axiom,
    abs_if(finite_1) ).

tff(tcon_Enum_Ofinite__1___Power_Opower_476,axiom,
    power(finite_1) ).

tff(tcon_Enum_Ofinite__1___Groups_Ozero_477,axiom,
    zero(finite_1) ).

tff(tcon_Enum_Ofinite__1___Groups_Oplus_478,axiom,
    plus(finite_1) ).

tff(tcon_Enum_Ofinite__1___Rings_Oring_479,axiom,
    ring(finite_1) ).

tff(tcon_Enum_Ofinite__1___Groups_Oone_480,axiom,
    one(finite_1) ).

tff(tcon_Enum_Ofinite__1___Rings_Odvd_481,axiom,
    dvd(finite_1) ).

tff(tcon_Enum_Ofinite__2___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_482,axiom,
    condit6923001295902523014norder(finite_2) ).

tff(tcon_Enum_Ofinite__2___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_483,axiom,
    semiri1453513574482234551roduct(finite_2) ).

tff(tcon_Enum_Ofinite__2___Euclidean__Division_Ounique__euclidean__semiring_484,axiom,
    euclid3128863361964157862miring(finite_2) ).

tff(tcon_Enum_Ofinite__2___Euclidean__Division_Oeuclidean__semiring__cancel_485,axiom,
    euclid4440199948858584721cancel(finite_2) ).

tff(tcon_Enum_Ofinite__2___Rings_Osemiring__no__zero__divisors__cancel_486,axiom,
    semiri6575147826004484403cancel(finite_2) ).

tff(tcon_Enum_Ofinite__2___Euclidean__Division_Oeuclidean__semiring_487,axiom,
    euclid3725896446679973847miring(finite_2) ).

tff(tcon_Enum_Ofinite__2___Quickcheck__Narrowing_Opartial__term__of_488,axiom,
    quickc6926020345158392990erm_of(finite_2) ).

tff(tcon_Enum_Ofinite__2___Rings_Osemiring__1__no__zero__divisors_489,axiom,
    semiri2026040879449505780visors(finite_2) ).

tff(tcon_Enum_Ofinite__2___Rings_Osemiring__no__zero__divisors_490,axiom,
    semiri3467727345109120633visors(finite_2) ).

tff(tcon_Enum_Ofinite__2___Groups_Ocancel__ab__semigroup__add_491,axiom,
    cancel2418104881723323429up_add(finite_2) ).

tff(tcon_Enum_Ofinite__2___Rings_Oring__1__no__zero__divisors_492,axiom,
    ring_15535105094025558882visors(finite_2) ).

tff(tcon_Enum_Ofinite__2___Groups_Ocancel__comm__monoid__add_493,axiom,
    cancel1802427076303600483id_add(finite_2) ).

tff(tcon_Enum_Ofinite__2___Rings_Ocomm__semiring__1__cancel_494,axiom,
    comm_s4317794764714335236cancel(finite_2) ).

tff(tcon_Enum_Ofinite__2___Groups_Ocancel__semigroup__add_495,axiom,
    cancel_semigroup_add(finite_2) ).

tff(tcon_Enum_Ofinite__2___Groups_Oab__semigroup__mult_496,axiom,
    ab_semigroup_mult(finite_2) ).

tff(tcon_Enum_Ofinite__2___Rings_Osemiring__1__cancel_497,axiom,
    semiring_1_cancel(finite_2) ).

tff(tcon_Enum_Ofinite__2___Rings_Oalgebraic__semidom_498,axiom,
    algebraic_semidom(finite_2) ).

tff(tcon_Enum_Ofinite__2___Groups_Ocomm__monoid__mult_499,axiom,
    comm_monoid_mult(finite_2) ).

tff(tcon_Enum_Ofinite__2___Groups_Oab__semigroup__add_500,axiom,
    ab_semigroup_add(finite_2) ).

tff(tcon_Enum_Ofinite__2___Code__Evaluation_Oterm__of_501,axiom,
    code_term_of(finite_2) ).

tff(tcon_Enum_Ofinite__2___Groups_Ocomm__monoid__add_502,axiom,
    comm_monoid_add(finite_2) ).

tff(tcon_Enum_Ofinite__2___Rings_Osemiring__modulo_503,axiom,
    semiring_modulo(finite_2) ).

tff(tcon_Enum_Ofinite__2___Rings_Ocomm__semiring__1_504,axiom,
    comm_semiring_1(finite_2) ).

tff(tcon_Enum_Ofinite__2___Rings_Ocomm__semiring__0_505,axiom,
    comm_semiring_0(finite_2) ).

tff(tcon_Enum_Ofinite__2___Groups_Osemigroup__mult_506,axiom,
    semigroup_mult(finite_2) ).

tff(tcon_Enum_Ofinite__2___Rings_Osemidom__modulo_507,axiom,
    semidom_modulo(finite_2) ).

tff(tcon_Enum_Ofinite__2___Rings_Osemidom__divide_508,axiom,
    semidom_divide(finite_2) ).

tff(tcon_Enum_Ofinite__2___Num_Osemiring__numeral_509,axiom,
    semiring_numeral(finite_2) ).

tff(tcon_Enum_Ofinite__2___Groups_Osemigroup__add_510,axiom,
    semigroup_add(finite_2) ).

tff(tcon_Enum_Ofinite__2___Fields_Odivision__ring_511,axiom,
    division_ring(finite_2) ).

tff(tcon_Enum_Ofinite__2___Rings_Ocomm__semiring_512,axiom,
    comm_semiring(finite_2) ).

tff(tcon_Enum_Ofinite__2___Orderings_Owellorder_513,axiom,
    wellorder(finite_2) ).

tff(tcon_Enum_Ofinite__2___Orderings_Oorder__top_514,axiom,
    order_top(finite_2) ).

tff(tcon_Enum_Ofinite__2___Orderings_Oorder__bot_515,axiom,
    order_bot(finite_2) ).

tff(tcon_Enum_Ofinite__2___Groups_Oab__group__add_516,axiom,
    ab_group_add(finite_2) ).

tff(tcon_Enum_Ofinite__2___Rings_Ozero__neq__one_517,axiom,
    zero_neq_one(finite_2) ).

tff(tcon_Enum_Ofinite__2___Rings_Oidom__abs__sgn_518,axiom,
    idom_abs_sgn(finite_2) ).

tff(tcon_Enum_Ofinite__2___Orderings_Opreorder_519,axiom,
    preorder(finite_2) ).

tff(tcon_Enum_Ofinite__2___Orderings_Olinorder_520,axiom,
    linorder(finite_2) ).

tff(tcon_Enum_Ofinite__2___Groups_Omonoid__mult_521,axiom,
    monoid_mult(finite_2) ).

tff(tcon_Enum_Ofinite__2___Rings_Oidom__modulo_522,axiom,
    idom_modulo(finite_2) ).

tff(tcon_Enum_Ofinite__2___Rings_Oidom__divide_523,axiom,
    idom_divide(finite_2) ).

tff(tcon_Enum_Ofinite__2___Rings_Ocomm__ring__1_524,axiom,
    comm_ring_1(finite_2) ).

tff(tcon_Enum_Ofinite__2___Groups_Omonoid__add_525,axiom,
    monoid_add(finite_2) ).

tff(tcon_Enum_Ofinite__2___Finite__Set_Ofinite_526,axiom,
    finite_finite(finite_2) ).

tff(tcon_Enum_Ofinite__2___Type__Length_Olen0_527,axiom,
    type_len0(finite_2) ).

tff(tcon_Enum_Ofinite__2___Rings_Osemiring__1_528,axiom,
    semiring_1(finite_2) ).

tff(tcon_Enum_Ofinite__2___Rings_Osemiring__0_529,axiom,
    semiring_0(finite_2) ).

tff(tcon_Enum_Ofinite__2___Groups_Ogroup__add_530,axiom,
    group_add(finite_2) ).

tff(tcon_Enum_Ofinite__2___Type__Length_Olen_531,axiom,
    type_len(finite_2) ).

tff(tcon_Enum_Ofinite__2___Rings_Omult__zero_532,axiom,
    mult_zero(finite_2) ).

tff(tcon_Enum_Ofinite__2___Rings_Ocomm__ring_533,axiom,
    comm_ring(finite_2) ).

tff(tcon_Enum_Ofinite__2___Orderings_Oorder_534,axiom,
    order(finite_2) ).

tff(tcon_Enum_Ofinite__2___Num_Oneg__numeral_535,axiom,
    neg_numeral(finite_2) ).

tff(tcon_Enum_Ofinite__2___Rings_Osemiring_536,axiom,
    semiring(finite_2) ).

tff(tcon_Enum_Ofinite__2___Fields_Oinverse_537,axiom,
    inverse(finite_2) ).

tff(tcon_Enum_Ofinite__2___Rings_Osemidom_538,axiom,
    semidom(finite_2) ).

tff(tcon_Enum_Ofinite__2___Orderings_Oord_539,axiom,
    ord(finite_2) ).

tff(tcon_Enum_Ofinite__2___Groups_Ouminus_540,axiom,
    uminus(finite_2) ).

tff(tcon_Enum_Ofinite__2___Rings_Oring__1_541,axiom,
    ring_1(finite_2) ).

tff(tcon_Enum_Ofinite__2___Fields_Ofield_542,axiom,
    field(finite_2) ).

tff(tcon_Enum_Ofinite__2___Power_Opower_543,axiom,
    power(finite_2) ).

tff(tcon_Enum_Ofinite__2___Num_Onumeral_544,axiom,
    numeral(finite_2) ).

tff(tcon_Enum_Ofinite__2___Groups_Ozero_545,axiom,
    zero(finite_2) ).

tff(tcon_Enum_Ofinite__2___Groups_Oplus_546,axiom,
    plus(finite_2) ).

tff(tcon_Enum_Ofinite__2___Rings_Oring_547,axiom,
    ring(finite_2) ).

tff(tcon_Enum_Ofinite__2___Rings_Oidom_548,axiom,
    idom(finite_2) ).

tff(tcon_Enum_Ofinite__2___Groups_Oone_549,axiom,
    one(finite_2) ).

tff(tcon_Enum_Ofinite__2___Rings_Odvd_550,axiom,
    dvd(finite_2) ).

tff(tcon_Enum_Ofinite__3___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_551,axiom,
    condit6923001295902523014norder(finite_3) ).

tff(tcon_Enum_Ofinite__3___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_552,axiom,
    semiri1453513574482234551roduct(finite_3) ).

tff(tcon_Enum_Ofinite__3___Euclidean__Division_Ounique__euclidean__semiring_553,axiom,
    euclid3128863361964157862miring(finite_3) ).

tff(tcon_Enum_Ofinite__3___Euclidean__Division_Oeuclidean__semiring__cancel_554,axiom,
    euclid4440199948858584721cancel(finite_3) ).

tff(tcon_Enum_Ofinite__3___Rings_Osemiring__no__zero__divisors__cancel_555,axiom,
    semiri6575147826004484403cancel(finite_3) ).

tff(tcon_Enum_Ofinite__3___Euclidean__Division_Oeuclidean__semiring_556,axiom,
    euclid3725896446679973847miring(finite_3) ).

tff(tcon_Enum_Ofinite__3___Quickcheck__Narrowing_Opartial__term__of_557,axiom,
    quickc6926020345158392990erm_of(finite_3) ).

tff(tcon_Enum_Ofinite__3___Rings_Osemiring__1__no__zero__divisors_558,axiom,
    semiri2026040879449505780visors(finite_3) ).

tff(tcon_Enum_Ofinite__3___Rings_Osemiring__no__zero__divisors_559,axiom,
    semiri3467727345109120633visors(finite_3) ).

tff(tcon_Enum_Ofinite__3___Groups_Ocancel__ab__semigroup__add_560,axiom,
    cancel2418104881723323429up_add(finite_3) ).

tff(tcon_Enum_Ofinite__3___Rings_Oring__1__no__zero__divisors_561,axiom,
    ring_15535105094025558882visors(finite_3) ).

tff(tcon_Enum_Ofinite__3___Groups_Ocancel__comm__monoid__add_562,axiom,
    cancel1802427076303600483id_add(finite_3) ).

tff(tcon_Enum_Ofinite__3___Rings_Ocomm__semiring__1__cancel_563,axiom,
    comm_s4317794764714335236cancel(finite_3) ).

tff(tcon_Enum_Ofinite__3___Groups_Ocancel__semigroup__add_564,axiom,
    cancel_semigroup_add(finite_3) ).

tff(tcon_Enum_Ofinite__3___Groups_Oab__semigroup__mult_565,axiom,
    ab_semigroup_mult(finite_3) ).

tff(tcon_Enum_Ofinite__3___Rings_Osemiring__1__cancel_566,axiom,
    semiring_1_cancel(finite_3) ).

tff(tcon_Enum_Ofinite__3___Rings_Oalgebraic__semidom_567,axiom,
    algebraic_semidom(finite_3) ).

tff(tcon_Enum_Ofinite__3___Groups_Ocomm__monoid__mult_568,axiom,
    comm_monoid_mult(finite_3) ).

tff(tcon_Enum_Ofinite__3___Groups_Oab__semigroup__add_569,axiom,
    ab_semigroup_add(finite_3) ).

tff(tcon_Enum_Ofinite__3___Code__Evaluation_Oterm__of_570,axiom,
    code_term_of(finite_3) ).

tff(tcon_Enum_Ofinite__3___Groups_Ocomm__monoid__add_571,axiom,
    comm_monoid_add(finite_3) ).

tff(tcon_Enum_Ofinite__3___Rings_Osemiring__modulo_572,axiom,
    semiring_modulo(finite_3) ).

tff(tcon_Enum_Ofinite__3___Rings_Ocomm__semiring__1_573,axiom,
    comm_semiring_1(finite_3) ).

tff(tcon_Enum_Ofinite__3___Rings_Ocomm__semiring__0_574,axiom,
    comm_semiring_0(finite_3) ).

tff(tcon_Enum_Ofinite__3___Groups_Osemigroup__mult_575,axiom,
    semigroup_mult(finite_3) ).

tff(tcon_Enum_Ofinite__3___Rings_Osemidom__modulo_576,axiom,
    semidom_modulo(finite_3) ).

tff(tcon_Enum_Ofinite__3___Rings_Osemidom__divide_577,axiom,
    semidom_divide(finite_3) ).

tff(tcon_Enum_Ofinite__3___Num_Osemiring__numeral_578,axiom,
    semiring_numeral(finite_3) ).

tff(tcon_Enum_Ofinite__3___Groups_Osemigroup__add_579,axiom,
    semigroup_add(finite_3) ).

tff(tcon_Enum_Ofinite__3___Fields_Odivision__ring_580,axiom,
    division_ring(finite_3) ).

tff(tcon_Enum_Ofinite__3___Rings_Ocomm__semiring_581,axiom,
    comm_semiring(finite_3) ).

tff(tcon_Enum_Ofinite__3___Orderings_Owellorder_582,axiom,
    wellorder(finite_3) ).

tff(tcon_Enum_Ofinite__3___Orderings_Oorder__top_583,axiom,
    order_top(finite_3) ).

tff(tcon_Enum_Ofinite__3___Orderings_Oorder__bot_584,axiom,
    order_bot(finite_3) ).

tff(tcon_Enum_Ofinite__3___Groups_Oab__group__add_585,axiom,
    ab_group_add(finite_3) ).

tff(tcon_Enum_Ofinite__3___Rings_Ozero__neq__one_586,axiom,
    zero_neq_one(finite_3) ).

tff(tcon_Enum_Ofinite__3___Rings_Oidom__abs__sgn_587,axiom,
    idom_abs_sgn(finite_3) ).

tff(tcon_Enum_Ofinite__3___Orderings_Opreorder_588,axiom,
    preorder(finite_3) ).

tff(tcon_Enum_Ofinite__3___Orderings_Olinorder_589,axiom,
    linorder(finite_3) ).

tff(tcon_Enum_Ofinite__3___Groups_Omonoid__mult_590,axiom,
    monoid_mult(finite_3) ).

tff(tcon_Enum_Ofinite__3___Rings_Oidom__modulo_591,axiom,
    idom_modulo(finite_3) ).

tff(tcon_Enum_Ofinite__3___Rings_Oidom__divide_592,axiom,
    idom_divide(finite_3) ).

tff(tcon_Enum_Ofinite__3___Rings_Ocomm__ring__1_593,axiom,
    comm_ring_1(finite_3) ).

tff(tcon_Enum_Ofinite__3___Groups_Omonoid__add_594,axiom,
    monoid_add(finite_3) ).

tff(tcon_Enum_Ofinite__3___Finite__Set_Ofinite_595,axiom,
    finite_finite(finite_3) ).

tff(tcon_Enum_Ofinite__3___Type__Length_Olen0_596,axiom,
    type_len0(finite_3) ).

tff(tcon_Enum_Ofinite__3___Rings_Osemiring__1_597,axiom,
    semiring_1(finite_3) ).

tff(tcon_Enum_Ofinite__3___Rings_Osemiring__0_598,axiom,
    semiring_0(finite_3) ).

tff(tcon_Enum_Ofinite__3___Groups_Ogroup__add_599,axiom,
    group_add(finite_3) ).

tff(tcon_Enum_Ofinite__3___Type__Length_Olen_600,axiom,
    type_len(finite_3) ).

tff(tcon_Enum_Ofinite__3___Rings_Omult__zero_601,axiom,
    mult_zero(finite_3) ).

tff(tcon_Enum_Ofinite__3___Rings_Ocomm__ring_602,axiom,
    comm_ring(finite_3) ).

tff(tcon_Enum_Ofinite__3___Orderings_Oorder_603,axiom,
    order(finite_3) ).

tff(tcon_Enum_Ofinite__3___Num_Oneg__numeral_604,axiom,
    neg_numeral(finite_3) ).

tff(tcon_Enum_Ofinite__3___Rings_Osemiring_605,axiom,
    semiring(finite_3) ).

tff(tcon_Enum_Ofinite__3___Fields_Oinverse_606,axiom,
    inverse(finite_3) ).

tff(tcon_Enum_Ofinite__3___Rings_Osemidom_607,axiom,
    semidom(finite_3) ).

tff(tcon_Enum_Ofinite__3___Orderings_Oord_608,axiom,
    ord(finite_3) ).

tff(tcon_Enum_Ofinite__3___Groups_Ouminus_609,axiom,
    uminus(finite_3) ).

tff(tcon_Enum_Ofinite__3___Rings_Oring__1_610,axiom,
    ring_1(finite_3) ).

tff(tcon_Enum_Ofinite__3___Fields_Ofield_611,axiom,
    field(finite_3) ).

tff(tcon_Enum_Ofinite__3___Power_Opower_612,axiom,
    power(finite_3) ).

tff(tcon_Enum_Ofinite__3___Num_Onumeral_613,axiom,
    numeral(finite_3) ).

tff(tcon_Enum_Ofinite__3___Groups_Ozero_614,axiom,
    zero(finite_3) ).

tff(tcon_Enum_Ofinite__3___Groups_Oplus_615,axiom,
    plus(finite_3) ).

tff(tcon_Enum_Ofinite__3___Rings_Oring_616,axiom,
    ring(finite_3) ).

tff(tcon_Enum_Ofinite__3___Rings_Oidom_617,axiom,
    idom(finite_3) ).

tff(tcon_Enum_Ofinite__3___Groups_Oone_618,axiom,
    one(finite_3) ).

tff(tcon_Enum_Ofinite__3___Rings_Odvd_619,axiom,
    dvd(finite_3) ).

tff(tcon_Filter_Ofilter___Quickcheck__Narrowing_Opartial__term__of_620,axiom,
    ! [A11: $tType] :
      ( typerep(A11)
     => quickc6926020345158392990erm_of(filter(A11)) ) ).

tff(tcon_Filter_Ofilter___Code__Evaluation_Oterm__of_621,axiom,
    ! [A11: $tType] :
      ( typerep(A11)
     => code_term_of(filter(A11)) ) ).

tff(tcon_Filter_Ofilter___Orderings_Oorder__top_622,axiom,
    ! [A11: $tType] : order_top(filter(A11)) ).

tff(tcon_Filter_Ofilter___Orderings_Oorder__bot_623,axiom,
    ! [A11: $tType] : order_bot(filter(A11)) ).

tff(tcon_Filter_Ofilter___Orderings_Opreorder_624,axiom,
    ! [A11: $tType] : preorder(filter(A11)) ).

tff(tcon_Filter_Ofilter___Orderings_Oorder_625,axiom,
    ! [A11: $tType] : order(filter(A11)) ).

tff(tcon_Filter_Ofilter___Orderings_Oord_626,axiom,
    ! [A11: $tType] : ord(filter(A11)) ).

tff(tcon_Option_Ooption___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_627,axiom,
    ! [A11: $tType] :
      ( comple5582772986160207858norder(A11)
     => condit6923001295902523014norder(option(A11)) ) ).

tff(tcon_Option_Ooption___Quickcheck__Narrowing_Opartial__term__of_628,axiom,
    ! [A11: $tType] :
      ( typerep(A11)
     => quickc6926020345158392990erm_of(option(A11)) ) ).

tff(tcon_Option_Ooption___Code__Evaluation_Oterm__of_629,axiom,
    ! [A11: $tType] :
      ( typerep(A11)
     => code_term_of(option(A11)) ) ).

tff(tcon_Option_Ooption___Orderings_Owellorder_630,axiom,
    ! [A11: $tType] :
      ( wellorder(A11)
     => wellorder(option(A11)) ) ).

tff(tcon_Option_Ooption___Orderings_Oorder__top_631,axiom,
    ! [A11: $tType] :
      ( order_top(A11)
     => order_top(option(A11)) ) ).

tff(tcon_Option_Ooption___Orderings_Oorder__bot_632,axiom,
    ! [A11: $tType] :
      ( order(A11)
     => order_bot(option(A11)) ) ).

tff(tcon_Option_Ooption___Orderings_Opreorder_633,axiom,
    ! [A11: $tType] :
      ( preorder(A11)
     => preorder(option(A11)) ) ).

tff(tcon_Option_Ooption___Orderings_Olinorder_634,axiom,
    ! [A11: $tType] :
      ( linorder(A11)
     => linorder(option(A11)) ) ).

tff(tcon_Option_Ooption___Finite__Set_Ofinite_635,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => finite_finite(option(A11)) ) ).

tff(tcon_Option_Ooption___Orderings_Oorder_636,axiom,
    ! [A11: $tType] :
      ( order(A11)
     => order(option(A11)) ) ).

tff(tcon_Option_Ooption___Orderings_Oord_637,axiom,
    ! [A11: $tType] :
      ( preorder(A11)
     => ord(option(A11)) ) ).

tff(tcon_Option_Ooption___Nat_Osize_638,axiom,
    ! [A11: $tType] : size(option(A11)) ).

tff(tcon_Uint32_Ouint32___Bit__Operations_Osemiring__bit__operations_639,axiom,
    bit_se359711467146920520ations(uint32) ).

tff(tcon_Uint32_Ouint32___Quickcheck__Narrowing_Opartial__term__of_640,axiom,
    quickc6926020345158392990erm_of(uint32) ).

tff(tcon_Uint32_Ouint32___Bit__Comprehension_Obit__comprehension_641,axiom,
    bit_bi6583157726757044596ension(uint32) ).

tff(tcon_Uint32_Ouint32___Bit__Operations_Oring__bit__operations_642,axiom,
    bit_ri3973907225187159222ations(uint32) ).

tff(tcon_Uint32_Ouint32___Groups_Ocancel__ab__semigroup__add_643,axiom,
    cancel2418104881723323429up_add(uint32) ).

tff(tcon_Uint32_Ouint32___Groups_Ocancel__comm__monoid__add_644,axiom,
    cancel1802427076303600483id_add(uint32) ).

tff(tcon_Uint32_Ouint32___Rings_Ocomm__semiring__1__cancel_645,axiom,
    comm_s4317794764714335236cancel(uint32) ).

tff(tcon_Uint32_Ouint32___Bit__Operations_Osemiring__bits_646,axiom,
    bit_semiring_bits(uint32) ).

tff(tcon_Uint32_Ouint32___Groups_Ocancel__semigroup__add_647,axiom,
    cancel_semigroup_add(uint32) ).

tff(tcon_Uint32_Ouint32___Least__significant__bit_Olsb_648,axiom,
    least_6119777620449941438nt_lsb(uint32) ).

tff(tcon_Uint32_Ouint32___Groups_Oab__semigroup__mult_649,axiom,
    ab_semigroup_mult(uint32) ).

tff(tcon_Uint32_Ouint32___Rings_Osemiring__1__cancel_650,axiom,
    semiring_1_cancel(uint32) ).

tff(tcon_Uint32_Ouint32___Groups_Ocomm__monoid__mult_651,axiom,
    comm_monoid_mult(uint32) ).

tff(tcon_Uint32_Ouint32___Groups_Oab__semigroup__add_652,axiom,
    ab_semigroup_add(uint32) ).

tff(tcon_Uint32_Ouint32___Generic__set__bit_Oset__bit_653,axiom,
    generic_set_set_bit(uint32) ).

tff(tcon_Uint32_Ouint32___Code__Evaluation_Oterm__of_654,axiom,
    code_term_of(uint32) ).

tff(tcon_Uint32_Ouint32___Parity_Osemiring__parity_655,axiom,
    semiring_parity(uint32) ).

tff(tcon_Uint32_Ouint32___Groups_Ocomm__monoid__add_656,axiom,
    comm_monoid_add(uint32) ).

tff(tcon_Uint32_Ouint32___Rings_Osemiring__modulo_657,axiom,
    semiring_modulo(uint32) ).

tff(tcon_Uint32_Ouint32___Rings_Ocomm__semiring__1_658,axiom,
    comm_semiring_1(uint32) ).

tff(tcon_Uint32_Ouint32___Rings_Ocomm__semiring__0_659,axiom,
    comm_semiring_0(uint32) ).

tff(tcon_Uint32_Ouint32___Groups_Osemigroup__mult_660,axiom,
    semigroup_mult(uint32) ).

tff(tcon_Uint32_Ouint32___Num_Osemiring__numeral_661,axiom,
    semiring_numeral(uint32) ).

tff(tcon_Uint32_Ouint32___Groups_Osemigroup__add_662,axiom,
    semigroup_add(uint32) ).

tff(tcon_Uint32_Ouint32___Rings_Ocomm__semiring_663,axiom,
    comm_semiring(uint32) ).

tff(tcon_Uint32_Ouint32___Groups_Oab__group__add_664,axiom,
    ab_group_add(uint32) ).

tff(tcon_Uint32_Ouint32___Rings_Ozero__neq__one_665,axiom,
    zero_neq_one(uint32) ).

tff(tcon_Uint32_Ouint32___Parity_Oring__parity_666,axiom,
    ring_parity(uint32) ).

tff(tcon_Uint32_Ouint32___Orderings_Opreorder_667,axiom,
    preorder(uint32) ).

tff(tcon_Uint32_Ouint32___Orderings_Olinorder_668,axiom,
    linorder(uint32) ).

tff(tcon_Uint32_Ouint32___Groups_Omonoid__mult_669,axiom,
    monoid_mult(uint32) ).

tff(tcon_Uint32_Ouint32___Rings_Ocomm__ring__1_670,axiom,
    comm_ring_1(uint32) ).

tff(tcon_Uint32_Ouint32___Groups_Omonoid__add_671,axiom,
    monoid_add(uint32) ).

tff(tcon_Uint32_Ouint32___Rings_Osemiring__1_672,axiom,
    semiring_1(uint32) ).

tff(tcon_Uint32_Ouint32___Rings_Osemiring__0_673,axiom,
    semiring_0(uint32) ).

tff(tcon_Uint32_Ouint32___Groups_Ogroup__add_674,axiom,
    group_add(uint32) ).

tff(tcon_Uint32_Ouint32___Rings_Omult__zero_675,axiom,
    mult_zero(uint32) ).

tff(tcon_Uint32_Ouint32___Rings_Ocomm__ring_676,axiom,
    comm_ring(uint32) ).

tff(tcon_Uint32_Ouint32___Orderings_Oorder_677,axiom,
    order(uint32) ).

tff(tcon_Uint32_Ouint32___Num_Oneg__numeral_678,axiom,
    neg_numeral(uint32) ).

tff(tcon_Uint32_Ouint32___Rings_Osemiring_679,axiom,
    semiring(uint32) ).

tff(tcon_Uint32_Ouint32___Orderings_Oord_680,axiom,
    ord(uint32) ).

tff(tcon_Uint32_Ouint32___Groups_Ouminus_681,axiom,
    uminus(uint32) ).

tff(tcon_Uint32_Ouint32___Rings_Oring__1_682,axiom,
    ring_1(uint32) ).

tff(tcon_Uint32_Ouint32___Power_Opower_683,axiom,
    power(uint32) ).

tff(tcon_Uint32_Ouint32___Num_Onumeral_684,axiom,
    numeral(uint32) ).

tff(tcon_Uint32_Ouint32___Groups_Ozero_685,axiom,
    zero(uint32) ).

tff(tcon_Uint32_Ouint32___Groups_Oplus_686,axiom,
    plus(uint32) ).

tff(tcon_Uint32_Ouint32___Rings_Oring_687,axiom,
    ring(uint32) ).

tff(tcon_Uint32_Ouint32___Groups_Oone_688,axiom,
    one(uint32) ).

tff(tcon_Uint32_Ouint32___Rings_Odvd_689,axiom,
    dvd(uint32) ).

tff(tcon_Uint32_Ouint32___Nat_Osize_690,axiom,
    size(uint32) ).

tff(tcon_Complex_Ocomplex___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_691,axiom,
    semiri1453513574482234551roduct(complex) ).

tff(tcon_Complex_Ocomplex___Topological__Spaces_Ofirst__countable__topology_692,axiom,
    topolo3112930676232923870pology(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__div__algebra_693,axiom,
    real_V8999393235501362500lgebra(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__algebra__1_694,axiom,
    real_V2822296259951069270ebra_1(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Osemiring__no__zero__divisors__cancel_695,axiom,
    semiri6575147826004484403cancel(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__algebra_696,axiom,
    real_V4412858255891104859lgebra(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__vector_697,axiom,
    real_V822414075346904944vector(complex) ).

tff(tcon_Complex_Ocomplex___Topological__Spaces_Otopological__space_698,axiom,
    topolo4958980785337419405_space(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__field_699,axiom,
    real_V3459762299906320749_field(complex) ).

tff(tcon_Complex_Ocomplex___Quickcheck__Narrowing_Opartial__term__of_700,axiom,
    quickc6926020345158392990erm_of(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__div__algebra_701,axiom,
    real_V5047593784448816457lgebra(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Ouniformity__dist_702,axiom,
    real_V768167426530841204y_dist(complex) ).

tff(tcon_Complex_Ocomplex___Limits_Otopological__comm__monoid__add_703,axiom,
    topolo5987344860129210374id_add(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Osemiring__1__no__zero__divisors_704,axiom,
    semiri2026040879449505780visors(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__algebra__1_705,axiom,
    real_V2191834092415804123ebra_1(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Ocomplete__space_706,axiom,
    real_V8037385150606011577_space(complex) ).

tff(tcon_Complex_Ocomplex___Limits_Otopological__semigroup__mult_707,axiom,
    topolo4211221413907600880p_mult(complex) ).

tff(tcon_Complex_Ocomplex___Topological__Spaces_Ouniform__space_708,axiom,
    topolo7287701948861334536_space(complex) ).

tff(tcon_Complex_Ocomplex___Topological__Spaces_Operfect__space_709,axiom,
    topolo8386298272705272623_space(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Osemiring__no__zero__divisors_710,axiom,
    semiri3467727345109120633visors(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__algebra_711,axiom,
    real_V6157519004096292374lgebra(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Ometric__space_712,axiom,
    real_V7819770556892013058_space(complex) ).

tff(tcon_Complex_Ocomplex___Limits_Otopological__ab__group__add_713,axiom,
    topolo1287966508704411220up_add(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__vector_714,axiom,
    real_V4867850818363320053vector(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Ocancel__ab__semigroup__add_715,axiom,
    cancel2418104881723323429up_add(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Oring__1__no__zero__divisors_716,axiom,
    ring_15535105094025558882visors(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__field_717,axiom,
    real_V7773925162809079976_field(complex) ).

tff(tcon_Complex_Ocomplex___Limits_Otopological__monoid__add_718,axiom,
    topolo6943815403480290642id_add(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Ocancel__comm__monoid__add_719,axiom,
    cancel1802427076303600483id_add(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Ocomm__semiring__1__cancel_720,axiom,
    comm_s4317794764714335236cancel(complex) ).

tff(tcon_Complex_Ocomplex___Limits_Otopological__group__add_721,axiom,
    topolo1633459387980952147up_add(complex) ).

tff(tcon_Complex_Ocomplex___Topological__Spaces_Ot2__space_722,axiom,
    topological_t2_space(complex) ).

tff(tcon_Complex_Ocomplex___Topological__Spaces_Ot1__space_723,axiom,
    topological_t1_space(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Ocancel__semigroup__add_724,axiom,
    cancel_semigroup_add(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Obanach_725,axiom,
    real_Vector_banach(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Oab__semigroup__mult_726,axiom,
    ab_semigroup_mult(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Osemiring__1__cancel_727,axiom,
    semiring_1_cancel(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Ocomm__monoid__mult_728,axiom,
    comm_monoid_mult(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Oab__semigroup__add_729,axiom,
    ab_semigroup_add(complex) ).

tff(tcon_Complex_Ocomplex___Code__Evaluation_Oterm__of_730,axiom,
    code_term_of(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Ocomm__monoid__add_731,axiom,
    comm_monoid_add(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Ocomm__semiring__1_732,axiom,
    comm_semiring_1(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Ocomm__semiring__0_733,axiom,
    comm_semiring_0(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Osemigroup__mult_734,axiom,
    semigroup_mult(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Osemidom__divide_735,axiom,
    semidom_divide(complex) ).

tff(tcon_Complex_Ocomplex___Num_Osemiring__numeral_736,axiom,
    semiring_numeral(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Osemigroup__add_737,axiom,
    semigroup_add(complex) ).

tff(tcon_Complex_Ocomplex___Fields_Ofield__abs__sgn_738,axiom,
    field_abs_sgn(complex) ).

tff(tcon_Complex_Ocomplex___Fields_Odivision__ring_739,axiom,
    division_ring(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Ocomm__semiring_740,axiom,
    comm_semiring(complex) ).

tff(tcon_Complex_Ocomplex___Nat_Osemiring__char__0_741,axiom,
    semiring_char_0(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Oab__group__add_742,axiom,
    ab_group_add(complex) ).

tff(tcon_Complex_Ocomplex___Fields_Ofield__char__0_743,axiom,
    field_char_0(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Ozero__neq__one_744,axiom,
    zero_neq_one(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Oidom__abs__sgn_745,axiom,
    idom_abs_sgn(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Omonoid__mult_746,axiom,
    monoid_mult(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Oidom__divide_747,axiom,
    idom_divide(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Ocomm__ring__1_748,axiom,
    comm_ring_1(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Omonoid__add_749,axiom,
    monoid_add(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Osemiring__1_750,axiom,
    semiring_1(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Osemiring__0_751,axiom,
    semiring_0(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Ogroup__add_752,axiom,
    group_add(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Omult__zero_753,axiom,
    mult_zero(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Ocomm__ring_754,axiom,
    comm_ring(complex) ).

tff(tcon_Complex_Ocomplex___Num_Oneg__numeral_755,axiom,
    neg_numeral(complex) ).

tff(tcon_Complex_Ocomplex___Nat_Oring__char__0_756,axiom,
    ring_char_0(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Osemiring_757,axiom,
    semiring(complex) ).

tff(tcon_Complex_Ocomplex___Fields_Oinverse_758,axiom,
    inverse(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Osemidom_759,axiom,
    semidom(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Ouminus_760,axiom,
    uminus(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Oring__1_761,axiom,
    ring_1(complex) ).

tff(tcon_Complex_Ocomplex___Fields_Ofield_762,axiom,
    field(complex) ).

tff(tcon_Complex_Ocomplex___Power_Opower_763,axiom,
    power(complex) ).

tff(tcon_Complex_Ocomplex___Num_Onumeral_764,axiom,
    numeral(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Ozero_765,axiom,
    zero(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Oplus_766,axiom,
    plus(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Oring_767,axiom,
    ring(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Oidom_768,axiom,
    idom(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Oone_769,axiom,
    one(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Odvd_770,axiom,
    dvd(complex) ).

tff(tcon_Extended__Nat_Oenat___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_771,axiom,
    condit6923001295902523014norder(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Ostrict__ordered__ab__semigroup__add_772,axiom,
    strict9044650504122735259up_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Ostrict__ordered__comm__monoid__add_773,axiom,
    strict7427464778891057005id_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Ocanonically__ordered__monoid__add_774,axiom,
    canoni5634975068530333245id_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Quickcheck__Narrowing_Opartial__term__of_775,axiom,
    quickc6926020345158392990erm_of(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Olinordered__ab__semigroup__add_776,axiom,
    linord4140545234300271783up_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Olinordered__nonzero__semiring_777,axiom,
    linord181362715937106298miring(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Osemiring__no__zero__divisors_778,axiom,
    semiri3467727345109120633visors(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Oordered__ab__semigroup__add_779,axiom,
    ordere6658533253407199908up_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Oordered__comm__monoid__add_780,axiom,
    ordere6911136660526730532id_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Oordered__comm__semiring_781,axiom,
    ordere2520102378445227354miring(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Oab__semigroup__mult_782,axiom,
    ab_semigroup_mult(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Ocomm__monoid__mult_783,axiom,
    comm_monoid_mult(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Oab__semigroup__add_784,axiom,
    ab_semigroup_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Code__Evaluation_Oterm__of_785,axiom,
    code_term_of(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Oordered__semiring_786,axiom,
    ordered_semiring(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Ocomm__monoid__add_787,axiom,
    comm_monoid_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Ocomm__semiring__1_788,axiom,
    comm_semiring_1(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Ocomm__semiring__0_789,axiom,
    comm_semiring_0(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Osemigroup__mult_790,axiom,
    semigroup_mult(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Num_Osemiring__numeral_791,axiom,
    semiring_numeral(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Osemigroup__add_792,axiom,
    semigroup_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Ozero__less__one_793,axiom,
    zero_less_one(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Ocomm__semiring_794,axiom,
    comm_semiring(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Orderings_Owellorder_795,axiom,
    wellorder(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Orderings_Oorder__top_796,axiom,
    order_top(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Orderings_Oorder__bot_797,axiom,
    order_bot(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Nat_Osemiring__char__0_798,axiom,
    semiring_char_0(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Ozero__neq__one_799,axiom,
    zero_neq_one(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Orderings_Opreorder_800,axiom,
    preorder(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Orderings_Olinorder_801,axiom,
    linorder(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Omonoid__mult_802,axiom,
    monoid_mult(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Omonoid__add_803,axiom,
    monoid_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Osemiring__1_804,axiom,
    semiring_1(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Osemiring__0_805,axiom,
    semiring_0(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Omult__zero_806,axiom,
    mult_zero(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Orderings_Oorder_807,axiom,
    order(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Osemiring_808,axiom,
    semiring(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Orderings_Oord_809,axiom,
    ord(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Power_Opower_810,axiom,
    power(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Num_Onumeral_811,axiom,
    numeral(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Ozero_812,axiom,
    zero(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Oplus_813,axiom,
    plus(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Oone_814,axiom,
    one(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Odvd_815,axiom,
    dvd(extended_enat) ).

tff(tcon_Multiset_Omultiset___Quickcheck__Narrowing_Opartial__term__of_816,axiom,
    ! [A11: $tType] :
      ( typerep(A11)
     => quickc6926020345158392990erm_of(multiset(A11)) ) ).

tff(tcon_Multiset_Omultiset___Groups_Oordered__ab__semigroup__add_817,axiom,
    ! [A11: $tType] :
      ( preorder(A11)
     => ordere6658533253407199908up_add(multiset(A11)) ) ).

tff(tcon_Multiset_Omultiset___Groups_Ocancel__ab__semigroup__add_818,axiom,
    ! [A11: $tType] : cancel2418104881723323429up_add(multiset(A11)) ).

tff(tcon_Multiset_Omultiset___Groups_Ocancel__comm__monoid__add_819,axiom,
    ! [A11: $tType] : cancel1802427076303600483id_add(multiset(A11)) ).

tff(tcon_Multiset_Omultiset___Groups_Ocancel__semigroup__add_820,axiom,
    ! [A11: $tType] : cancel_semigroup_add(multiset(A11)) ).

tff(tcon_Multiset_Omultiset___Groups_Ocomm__monoid__diff_821,axiom,
    ! [A11: $tType] : comm_monoid_diff(multiset(A11)) ).

tff(tcon_Multiset_Omultiset___Groups_Oab__semigroup__add_822,axiom,
    ! [A11: $tType] : ab_semigroup_add(multiset(A11)) ).

tff(tcon_Multiset_Omultiset___Code__Evaluation_Oterm__of_823,axiom,
    ! [A11: $tType] :
      ( typerep(A11)
     => code_term_of(multiset(A11)) ) ).

tff(tcon_Multiset_Omultiset___Groups_Ocomm__monoid__add_824,axiom,
    ! [A11: $tType] : comm_monoid_add(multiset(A11)) ).

tff(tcon_Multiset_Omultiset___Groups_Osemigroup__add_825,axiom,
    ! [A11: $tType] : semigroup_add(multiset(A11)) ).

tff(tcon_Multiset_Omultiset___Orderings_Opreorder_826,axiom,
    ! [A11: $tType] :
      ( preorder(A11)
     => preorder(multiset(A11)) ) ).

tff(tcon_Multiset_Omultiset___Groups_Omonoid__add_827,axiom,
    ! [A11: $tType] : monoid_add(multiset(A11)) ).

tff(tcon_Multiset_Omultiset___Orderings_Oorder_828,axiom,
    ! [A11: $tType] :
      ( preorder(A11)
     => order(multiset(A11)) ) ).

tff(tcon_Multiset_Omultiset___Orderings_Oord_829,axiom,
    ! [A11: $tType] :
      ( preorder(A11)
     => ord(multiset(A11)) ) ).

tff(tcon_Multiset_Omultiset___Groups_Ozero_830,axiom,
    ! [A11: $tType] : zero(multiset(A11)) ).

tff(tcon_Multiset_Omultiset___Groups_Oplus_831,axiom,
    ! [A11: $tType] : plus(multiset(A11)) ).

tff(tcon_Multiset_Omultiset___Nat_Osize_832,axiom,
    ! [A11: $tType] : size(multiset(A11)) ).

tff(tcon_Numeral__Type_Obit0___Groups_Ocancel__ab__semigroup__add_833,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => cancel2418104881723323429up_add(numeral_bit0(A11)) ) ).

tff(tcon_Numeral__Type_Obit0___Groups_Ocancel__comm__monoid__add_834,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => cancel1802427076303600483id_add(numeral_bit0(A11)) ) ).

tff(tcon_Numeral__Type_Obit0___Rings_Ocomm__semiring__1__cancel_835,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => comm_s4317794764714335236cancel(numeral_bit0(A11)) ) ).

tff(tcon_Numeral__Type_Obit0___Groups_Ocancel__semigroup__add_836,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => cancel_semigroup_add(numeral_bit0(A11)) ) ).

tff(tcon_Numeral__Type_Obit0___Groups_Oab__semigroup__mult_837,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => ab_semigroup_mult(numeral_bit0(A11)) ) ).

tff(tcon_Numeral__Type_Obit0___Rings_Osemiring__1__cancel_838,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => semiring_1_cancel(numeral_bit0(A11)) ) ).

tff(tcon_Numeral__Type_Obit0___Groups_Ocomm__monoid__mult_839,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => comm_monoid_mult(numeral_bit0(A11)) ) ).

tff(tcon_Numeral__Type_Obit0___Groups_Oab__semigroup__add_840,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => ab_semigroup_add(numeral_bit0(A11)) ) ).

tff(tcon_Numeral__Type_Obit0___Code__Evaluation_Oterm__of_841,axiom,
    ! [A11: $tType] :
      ( typerep(A11)
     => code_term_of(numeral_bit0(A11)) ) ).

tff(tcon_Numeral__Type_Obit0___Groups_Ocomm__monoid__add_842,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => comm_monoid_add(numeral_bit0(A11)) ) ).

tff(tcon_Numeral__Type_Obit0___Rings_Ocomm__semiring__1_843,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => comm_semiring_1(numeral_bit0(A11)) ) ).

tff(tcon_Numeral__Type_Obit0___Rings_Ocomm__semiring__0_844,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => comm_semiring_0(numeral_bit0(A11)) ) ).

tff(tcon_Numeral__Type_Obit0___Groups_Osemigroup__mult_845,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => semigroup_mult(numeral_bit0(A11)) ) ).

tff(tcon_Numeral__Type_Obit0___Num_Osemiring__numeral_846,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => semiring_numeral(numeral_bit0(A11)) ) ).

tff(tcon_Numeral__Type_Obit0___Groups_Osemigroup__add_847,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => semigroup_add(numeral_bit0(A11)) ) ).

tff(tcon_Numeral__Type_Obit0___Rings_Ocomm__semiring_848,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => comm_semiring(numeral_bit0(A11)) ) ).

tff(tcon_Numeral__Type_Obit0___Orderings_Owellorder_849,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => wellorder(numeral_bit0(A11)) ) ).

tff(tcon_Numeral__Type_Obit0___Groups_Oab__group__add_850,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => ab_group_add(numeral_bit0(A11)) ) ).

tff(tcon_Numeral__Type_Obit0___Rings_Ozero__neq__one_851,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => zero_neq_one(numeral_bit0(A11)) ) ).

tff(tcon_Numeral__Type_Obit0___Orderings_Opreorder_852,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => preorder(numeral_bit0(A11)) ) ).

tff(tcon_Numeral__Type_Obit0___Orderings_Olinorder_853,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => linorder(numeral_bit0(A11)) ) ).

tff(tcon_Numeral__Type_Obit0___Groups_Omonoid__mult_854,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => monoid_mult(numeral_bit0(A11)) ) ).

tff(tcon_Numeral__Type_Obit0___Rings_Ocomm__ring__1_855,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => comm_ring_1(numeral_bit0(A11)) ) ).

tff(tcon_Numeral__Type_Obit0___Groups_Omonoid__add_856,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => monoid_add(numeral_bit0(A11)) ) ).

tff(tcon_Numeral__Type_Obit0___Finite__Set_Ofinite_857,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => finite_finite(numeral_bit0(A11)) ) ).

tff(tcon_Numeral__Type_Obit0___Type__Length_Olen0_858,axiom,
    ! [A11: $tType] :
      ( type_len0(A11)
     => type_len0(numeral_bit0(A11)) ) ).

tff(tcon_Numeral__Type_Obit0___Rings_Osemiring__1_859,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => semiring_1(numeral_bit0(A11)) ) ).

tff(tcon_Numeral__Type_Obit0___Rings_Osemiring__0_860,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => semiring_0(numeral_bit0(A11)) ) ).

tff(tcon_Numeral__Type_Obit0___Groups_Ogroup__add_861,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => group_add(numeral_bit0(A11)) ) ).

tff(tcon_Numeral__Type_Obit0___Type__Length_Olen_862,axiom,
    ! [A11: $tType] :
      ( type_len(A11)
     => type_len(numeral_bit0(A11)) ) ).

tff(tcon_Numeral__Type_Obit0___Rings_Omult__zero_863,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => mult_zero(numeral_bit0(A11)) ) ).

tff(tcon_Numeral__Type_Obit0___Rings_Ocomm__ring_864,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => comm_ring(numeral_bit0(A11)) ) ).

tff(tcon_Numeral__Type_Obit0___Orderings_Oorder_865,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => order(numeral_bit0(A11)) ) ).

tff(tcon_Numeral__Type_Obit0___Num_Oneg__numeral_866,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => neg_numeral(numeral_bit0(A11)) ) ).

tff(tcon_Numeral__Type_Obit0___Rings_Osemiring_867,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => semiring(numeral_bit0(A11)) ) ).

tff(tcon_Numeral__Type_Obit0___Orderings_Oord_868,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => ord(numeral_bit0(A11)) ) ).

tff(tcon_Numeral__Type_Obit0___Groups_Ouminus_869,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => uminus(numeral_bit0(A11)) ) ).

tff(tcon_Numeral__Type_Obit0___Rings_Oring__1_870,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => ring_1(numeral_bit0(A11)) ) ).

tff(tcon_Numeral__Type_Obit0___Power_Opower_871,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => power(numeral_bit0(A11)) ) ).

tff(tcon_Numeral__Type_Obit0___Num_Onumeral_872,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => numeral(numeral_bit0(A11)) ) ).

tff(tcon_Numeral__Type_Obit0___Groups_Ozero_873,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => zero(numeral_bit0(A11)) ) ).

tff(tcon_Numeral__Type_Obit0___Groups_Oplus_874,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => plus(numeral_bit0(A11)) ) ).

tff(tcon_Numeral__Type_Obit0___Rings_Oring_875,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => ring(numeral_bit0(A11)) ) ).

tff(tcon_Numeral__Type_Obit0___Groups_Oone_876,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => one(numeral_bit0(A11)) ) ).

tff(tcon_Numeral__Type_Obit0___Rings_Odvd_877,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => dvd(numeral_bit0(A11)) ) ).

tff(tcon_Numeral__Type_Obit1___Groups_Ocancel__ab__semigroup__add_878,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => cancel2418104881723323429up_add(numeral_bit1(A11)) ) ).

tff(tcon_Numeral__Type_Obit1___Groups_Ocancel__comm__monoid__add_879,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => cancel1802427076303600483id_add(numeral_bit1(A11)) ) ).

tff(tcon_Numeral__Type_Obit1___Rings_Ocomm__semiring__1__cancel_880,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => comm_s4317794764714335236cancel(numeral_bit1(A11)) ) ).

tff(tcon_Numeral__Type_Obit1___Groups_Ocancel__semigroup__add_881,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => cancel_semigroup_add(numeral_bit1(A11)) ) ).

tff(tcon_Numeral__Type_Obit1___Groups_Oab__semigroup__mult_882,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => ab_semigroup_mult(numeral_bit1(A11)) ) ).

tff(tcon_Numeral__Type_Obit1___Rings_Osemiring__1__cancel_883,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => semiring_1_cancel(numeral_bit1(A11)) ) ).

tff(tcon_Numeral__Type_Obit1___Groups_Ocomm__monoid__mult_884,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => comm_monoid_mult(numeral_bit1(A11)) ) ).

tff(tcon_Numeral__Type_Obit1___Groups_Oab__semigroup__add_885,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => ab_semigroup_add(numeral_bit1(A11)) ) ).

tff(tcon_Numeral__Type_Obit1___Code__Evaluation_Oterm__of_886,axiom,
    ! [A11: $tType] :
      ( typerep(A11)
     => code_term_of(numeral_bit1(A11)) ) ).

tff(tcon_Numeral__Type_Obit1___Groups_Ocomm__monoid__add_887,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => comm_monoid_add(numeral_bit1(A11)) ) ).

tff(tcon_Numeral__Type_Obit1___Rings_Ocomm__semiring__1_888,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => comm_semiring_1(numeral_bit1(A11)) ) ).

tff(tcon_Numeral__Type_Obit1___Rings_Ocomm__semiring__0_889,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => comm_semiring_0(numeral_bit1(A11)) ) ).

tff(tcon_Numeral__Type_Obit1___Groups_Osemigroup__mult_890,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => semigroup_mult(numeral_bit1(A11)) ) ).

tff(tcon_Numeral__Type_Obit1___Num_Osemiring__numeral_891,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => semiring_numeral(numeral_bit1(A11)) ) ).

tff(tcon_Numeral__Type_Obit1___Groups_Osemigroup__add_892,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => semigroup_add(numeral_bit1(A11)) ) ).

tff(tcon_Numeral__Type_Obit1___Rings_Ocomm__semiring_893,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => comm_semiring(numeral_bit1(A11)) ) ).

tff(tcon_Numeral__Type_Obit1___Orderings_Owellorder_894,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => wellorder(numeral_bit1(A11)) ) ).

tff(tcon_Numeral__Type_Obit1___Groups_Oab__group__add_895,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => ab_group_add(numeral_bit1(A11)) ) ).

tff(tcon_Numeral__Type_Obit1___Rings_Ozero__neq__one_896,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => zero_neq_one(numeral_bit1(A11)) ) ).

tff(tcon_Numeral__Type_Obit1___Orderings_Opreorder_897,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => preorder(numeral_bit1(A11)) ) ).

tff(tcon_Numeral__Type_Obit1___Orderings_Olinorder_898,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => linorder(numeral_bit1(A11)) ) ).

tff(tcon_Numeral__Type_Obit1___Groups_Omonoid__mult_899,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => monoid_mult(numeral_bit1(A11)) ) ).

tff(tcon_Numeral__Type_Obit1___Rings_Ocomm__ring__1_900,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => comm_ring_1(numeral_bit1(A11)) ) ).

tff(tcon_Numeral__Type_Obit1___Groups_Omonoid__add_901,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => monoid_add(numeral_bit1(A11)) ) ).

tff(tcon_Numeral__Type_Obit1___Finite__Set_Ofinite_902,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => finite_finite(numeral_bit1(A11)) ) ).

tff(tcon_Numeral__Type_Obit1___Type__Length_Olen0_903,axiom,
    ! [A11: $tType] :
      ( type_len0(A11)
     => type_len0(numeral_bit1(A11)) ) ).

tff(tcon_Numeral__Type_Obit1___Rings_Osemiring__1_904,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => semiring_1(numeral_bit1(A11)) ) ).

tff(tcon_Numeral__Type_Obit1___Rings_Osemiring__0_905,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => semiring_0(numeral_bit1(A11)) ) ).

tff(tcon_Numeral__Type_Obit1___Groups_Ogroup__add_906,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => group_add(numeral_bit1(A11)) ) ).

tff(tcon_Numeral__Type_Obit1___Type__Length_Olen_907,axiom,
    ! [A11: $tType] :
      ( type_len0(A11)
     => type_len(numeral_bit1(A11)) ) ).

tff(tcon_Numeral__Type_Obit1___Rings_Omult__zero_908,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => mult_zero(numeral_bit1(A11)) ) ).

tff(tcon_Numeral__Type_Obit1___Rings_Ocomm__ring_909,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => comm_ring(numeral_bit1(A11)) ) ).

tff(tcon_Numeral__Type_Obit1___Orderings_Oorder_910,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => order(numeral_bit1(A11)) ) ).

tff(tcon_Numeral__Type_Obit1___Num_Oneg__numeral_911,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => neg_numeral(numeral_bit1(A11)) ) ).

tff(tcon_Numeral__Type_Obit1___Rings_Osemiring_912,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => semiring(numeral_bit1(A11)) ) ).

tff(tcon_Numeral__Type_Obit1___Orderings_Oord_913,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => ord(numeral_bit1(A11)) ) ).

tff(tcon_Numeral__Type_Obit1___Groups_Ouminus_914,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => uminus(numeral_bit1(A11)) ) ).

tff(tcon_Numeral__Type_Obit1___Rings_Oring__1_915,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => ring_1(numeral_bit1(A11)) ) ).

tff(tcon_Numeral__Type_Obit1___Power_Opower_916,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => power(numeral_bit1(A11)) ) ).

tff(tcon_Numeral__Type_Obit1___Num_Onumeral_917,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => numeral(numeral_bit1(A11)) ) ).

tff(tcon_Numeral__Type_Obit1___Groups_Ozero_918,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => zero(numeral_bit1(A11)) ) ).

tff(tcon_Numeral__Type_Obit1___Groups_Oplus_919,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => plus(numeral_bit1(A11)) ) ).

tff(tcon_Numeral__Type_Obit1___Rings_Oring_920,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => ring(numeral_bit1(A11)) ) ).

tff(tcon_Numeral__Type_Obit1___Groups_Oone_921,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => one(numeral_bit1(A11)) ) ).

tff(tcon_Numeral__Type_Obit1___Rings_Odvd_922,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => dvd(numeral_bit1(A11)) ) ).

tff(tcon_Numeral__Type_Onum0___Quickcheck__Narrowing_Opartial__term__of_923,axiom,
    quickc6926020345158392990erm_of(numeral_num0) ).

tff(tcon_Numeral__Type_Onum0___Code__Evaluation_Oterm__of_924,axiom,
    code_term_of(numeral_num0) ).

tff(tcon_Numeral__Type_Onum0___Type__Length_Olen0_925,axiom,
    type_len0(numeral_num0) ).

tff(tcon_Numeral__Type_Onum1___Quickcheck__Narrowing_Opartial__term__of_926,axiom,
    quickc6926020345158392990erm_of(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Groups_Ocancel__ab__semigroup__add_927,axiom,
    cancel2418104881723323429up_add(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Groups_Ocancel__comm__monoid__add_928,axiom,
    cancel1802427076303600483id_add(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Groups_Ocancel__semigroup__add_929,axiom,
    cancel_semigroup_add(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Groups_Oab__semigroup__mult_930,axiom,
    ab_semigroup_mult(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Groups_Ocomm__monoid__mult_931,axiom,
    comm_monoid_mult(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Groups_Oab__semigroup__add_932,axiom,
    ab_semigroup_add(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Code__Evaluation_Oterm__of_933,axiom,
    code_term_of(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Groups_Ocomm__monoid__add_934,axiom,
    comm_monoid_add(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Rings_Ocomm__semiring__0_935,axiom,
    comm_semiring_0(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Groups_Osemigroup__mult_936,axiom,
    semigroup_mult(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Groups_Osemigroup__add_937,axiom,
    semigroup_add(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Rings_Ocomm__semiring_938,axiom,
    comm_semiring(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Orderings_Owellorder_939,axiom,
    wellorder(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Groups_Oab__group__add_940,axiom,
    ab_group_add(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Orderings_Opreorder_941,axiom,
    preorder(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Orderings_Olinorder_942,axiom,
    linorder(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Groups_Omonoid__mult_943,axiom,
    monoid_mult(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Groups_Omonoid__add_944,axiom,
    monoid_add(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Finite__Set_Ofinite_945,axiom,
    finite_finite(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Type__Length_Olen0_946,axiom,
    type_len0(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Rings_Osemiring__0_947,axiom,
    semiring_0(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Groups_Ogroup__add_948,axiom,
    group_add(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Type__Length_Olen_949,axiom,
    type_len(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Rings_Omult__zero_950,axiom,
    mult_zero(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Rings_Ocomm__ring_951,axiom,
    comm_ring(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Orderings_Oorder_952,axiom,
    order(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Rings_Osemiring_953,axiom,
    semiring(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Orderings_Oord_954,axiom,
    ord(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Groups_Ouminus_955,axiom,
    uminus(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Power_Opower_956,axiom,
    power(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Num_Onumeral_957,axiom,
    numeral(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Groups_Ozero_958,axiom,
    zero(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Groups_Oplus_959,axiom,
    plus(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Rings_Oring_960,axiom,
    ring(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Groups_Oone_961,axiom,
    one(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Rings_Odvd_962,axiom,
    dvd(numeral_num1) ).

tff(tcon_Product__Type_Oprod___Topological__Spaces_Otopological__space_963,axiom,
    ! [A11: $tType,A12: $tType] :
      ( ( topolo4958980785337419405_space(A11)
        & topolo4958980785337419405_space(A12) )
     => topolo4958980785337419405_space(product_prod(A11,A12)) ) ).

tff(tcon_Product__Type_Oprod___Quickcheck__Narrowing_Opartial__term__of_964,axiom,
    ! [A11: $tType,A12: $tType] :
      ( ( typerep(A11)
        & typerep(A12) )
     => quickc6926020345158392990erm_of(product_prod(A11,A12)) ) ).

tff(tcon_Product__Type_Oprod___Topological__Spaces_Ot2__space_965,axiom,
    ! [A11: $tType,A12: $tType] :
      ( ( topological_t2_space(A11)
        & topological_t2_space(A12) )
     => topological_t2_space(product_prod(A11,A12)) ) ).

tff(tcon_Product__Type_Oprod___Topological__Spaces_Ot1__space_966,axiom,
    ! [A11: $tType,A12: $tType] :
      ( ( topological_t1_space(A11)
        & topological_t1_space(A12) )
     => topological_t1_space(product_prod(A11,A12)) ) ).

tff(tcon_Product__Type_Oprod___Code__Evaluation_Oterm__of_967,axiom,
    ! [A11: $tType,A12: $tType] :
      ( ( typerep(A11)
        & typerep(A12) )
     => code_term_of(product_prod(A11,A12)) ) ).

tff(tcon_Product__Type_Oprod___Finite__Set_Ofinite_968,axiom,
    ! [A11: $tType,A12: $tType] :
      ( ( finite_finite(A11)
        & finite_finite(A12) )
     => finite_finite(product_prod(A11,A12)) ) ).

tff(tcon_Product__Type_Oprod___Nat_Osize_969,axiom,
    ! [A11: $tType,A12: $tType] : size(product_prod(A11,A12)) ).

tff(tcon_Code__Numeral_Ointeger___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_970,axiom,
    bit_un5681908812861735899ations(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_971,axiom,
    semiri1453513574482234551roduct(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Euclidean__Division_Ounique__euclidean__semiring__with__nat_972,axiom,
    euclid5411537665997757685th_nat(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Euclidean__Division_Ounique__euclidean__ring__with__nat_973,axiom,
    euclid8789492081693882211th_nat(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__semigroup__monoid__add__imp__le_974,axiom,
    ordere1937475149494474687imp_le(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Euclidean__Division_Ounique__euclidean__semiring_975,axiom,
    euclid3128863361964157862miring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__semiring__cancel_976,axiom,
    euclid4440199948858584721cancel(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Divides_Ounique__euclidean__semiring__numeral_977,axiom,
    unique1627219031080169319umeral(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__ring__cancel_978,axiom,
    euclid8851590272496341667cancel(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__no__zero__divisors__cancel_979,axiom,
    semiri6575147826004484403cancel(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ostrict__ordered__ab__semigroup__add_980,axiom,
    strict9044650504122735259up_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__cancel__ab__semigroup__add_981,axiom,
    ordere580206878836729694up_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__semigroup__add__imp__le_982,axiom,
    ordere2412721322843649153imp_le(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Bit__Operations_Osemiring__bit__operations_983,axiom,
    bit_se359711467146920520ations(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__comm__semiring__strict_984,axiom,
    linord2810124833399127020strict(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ostrict__ordered__comm__monoid__add_985,axiom,
    strict7427464778891057005id_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__cancel__comm__monoid__add_986,axiom,
    ordere8940638589300402666id_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__semiring_987,axiom,
    euclid3725896446679973847miring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Quickcheck__Narrowing_Opartial__term__of_988,axiom,
    quickc6926020345158392990erm_of(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring__1__strict_989,axiom,
    linord715952674999750819strict(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Olinordered__ab__semigroup__add_990,axiom,
    linord4140545234300271783up_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Bit__Operations_Oring__bit__operations_991,axiom,
    bit_ri3973907225187159222ations(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__1__no__zero__divisors_992,axiom,
    semiri2026040879449505780visors(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__nonzero__semiring_993,axiom,
    linord181362715937106298miring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__ring_994,axiom,
    euclid5891614535332579305n_ring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring__strict_995,axiom,
    linord8928482502909563296strict(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__no__zero__divisors_996,axiom,
    semiri3467727345109120633visors(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__semigroup__add_997,axiom,
    ordere6658533253407199908up_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__group__add__abs_998,axiom,
    ordere166539214618696060dd_abs(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__comm__monoid__add_999,axiom,
    ordere6911136660526730532id_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Olinordered__ab__group__add_1000,axiom,
    linord5086331880401160121up_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ocancel__ab__semigroup__add_1001,axiom,
    cancel2418104881723323429up_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oring__1__no__zero__divisors_1002,axiom,
    ring_15535105094025558882visors(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ocancel__comm__monoid__add_1003,axiom,
    cancel1802427076303600483id_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__ring__strict_1004,axiom,
    linord4710134922213307826strict(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring__1__cancel_1005,axiom,
    comm_s4317794764714335236cancel(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Bit__Operations_Osemiring__bits_1006,axiom,
    bit_semiring_bits(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oordered__comm__semiring_1007,axiom,
    ordere2520102378445227354miring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring__1_1008,axiom,
    linord6961819062388156250ring_1(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__group__add_1009,axiom,
    ordered_ab_group_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ocancel__semigroup__add_1010,axiom,
    cancel_semigroup_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring_1011,axiom,
    linordered_semiring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Least__significant__bit_Olsb_1012,axiom,
    least_6119777620449941438nt_lsb(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oordered__semiring__0_1013,axiom,
    ordered_semiring_0(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semidom_1014,axiom,
    linordered_semidom(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oab__semigroup__mult_1015,axiom,
    ab_semigroup_mult(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__1__cancel_1016,axiom,
    semiring_1_cancel(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oalgebraic__semidom_1017,axiom,
    algebraic_semidom(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ocomm__monoid__mult_1018,axiom,
    comm_monoid_mult(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oab__semigroup__add_1019,axiom,
    ab_semigroup_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Generic__set__bit_Oset__bit_1020,axiom,
    generic_set_set_bit(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Code__Evaluation_Oterm__of_1021,axiom,
    code_term_of(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oordered__semiring_1022,axiom,
    ordered_semiring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oordered__ring__abs_1023,axiom,
    ordered_ring_abs(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Parity_Osemiring__parity_1024,axiom,
    semiring_parity(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ocomm__monoid__add_1025,axiom,
    comm_monoid_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__modulo_1026,axiom,
    semiring_modulo(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__ring_1027,axiom,
    linordered_ring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__idom_1028,axiom,
    linordered_idom(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring__1_1029,axiom,
    comm_semiring_1(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring__0_1030,axiom,
    comm_semiring_0(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Osemigroup__mult_1031,axiom,
    semigroup_mult(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemidom__modulo_1032,axiom,
    semidom_modulo(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemidom__divide_1033,axiom,
    semidom_divide(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Num_Osemiring__numeral_1034,axiom,
    semiring_numeral(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Osemigroup__add_1035,axiom,
    semigroup_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ozero__less__one_1036,axiom,
    zero_less_one(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring_1037,axiom,
    comm_semiring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Nat_Osemiring__char__0_1038,axiom,
    semiring_char_0(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oab__group__add_1039,axiom,
    ab_group_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ozero__neq__one_1040,axiom,
    zero_neq_one(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oordered__ring_1041,axiom,
    ordered_ring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oidom__abs__sgn_1042,axiom,
    idom_abs_sgn(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Parity_Oring__parity_1043,axiom,
    ring_parity(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Orderings_Opreorder_1044,axiom,
    preorder(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Orderings_Olinorder_1045,axiom,
    linorder(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Omonoid__mult_1046,axiom,
    monoid_mult(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oidom__modulo_1047,axiom,
    idom_modulo(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oidom__divide_1048,axiom,
    idom_divide(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ocomm__ring__1_1049,axiom,
    comm_ring_1(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Omonoid__add_1050,axiom,
    monoid_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__1_1051,axiom,
    semiring_1(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__0_1052,axiom,
    semiring_0(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ogroup__add_1053,axiom,
    group_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Omult__zero_1054,axiom,
    mult_zero(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ocomm__ring_1055,axiom,
    comm_ring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Orderings_Oorder_1056,axiom,
    order(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Num_Oneg__numeral_1057,axiom,
    neg_numeral(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Nat_Oring__char__0_1058,axiom,
    ring_char_0(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring_1059,axiom,
    semiring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemidom_1060,axiom,
    semidom(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Orderings_Oord_1061,axiom,
    ord(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ouminus_1062,axiom,
    uminus(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oring__1_1063,axiom,
    ring_1(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oabs__if_1064,axiom,
    abs_if(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Power_Opower_1065,axiom,
    power(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Num_Onumeral_1066,axiom,
    numeral(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ozero_1067,axiom,
    zero(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oplus_1068,axiom,
    plus(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oring_1069,axiom,
    ring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oidom_1070,axiom,
    idom(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oone_1071,axiom,
    one(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Odvd_1072,axiom,
    dvd(code_integer) ).

tff(tcon_Heap__Time__Monad_OHeap___Quickcheck__Narrowing_Opartial__term__of_1073,axiom,
    ! [A11: $tType] :
      ( typerep(A11)
     => quickc6926020345158392990erm_of(heap_Time_Heap(A11)) ) ).

tff(tcon_Heap__Time__Monad_OHeap___Code__Evaluation_Oterm__of_1074,axiom,
    ! [A11: $tType] :
      ( typerep(A11)
     => code_term_of(heap_Time_Heap(A11)) ) ).

tff(tcon_Heap__Time__Monad_OHeap___Nat_Osize_1075,axiom,
    ! [A11: $tType] : size(heap_Time_Heap(A11)) ).

tff(tcon_VEBT__Definitions_OVEBT___Quickcheck__Narrowing_Opartial__term__of_1076,axiom,
    quickc6926020345158392990erm_of(vEBT_VEBT) ).

tff(tcon_VEBT__Definitions_OVEBT___Code__Evaluation_Oterm__of_1077,axiom,
    code_term_of(vEBT_VEBT) ).

tff(tcon_VEBT__Definitions_OVEBT___Nat_Osize_1078,axiom,
    size(vEBT_VEBT) ).

% Helper facts (24)
tff(help_If_2_1_T,axiom,
    ! [A: $tType,X: A,Y: A] : if(A,fFalse,X,Y) = Y ).

tff(help_If_1_1_T,axiom,
    ! [A: $tType,X: A,Y: A] : if(A,fTrue,X,Y) = X ).

tff(help_fEx_1_1_U,axiom,
    ! [A: $tType,P: fun(A,bool),X: A] :
      ( ~ pp(aa(A,bool,P,X))
      | pp(aa(fun(A,bool),bool,fEx(A),P)) ) ).

tff(help_fAll_1_1_U,axiom,
    ! [A: $tType,P: fun(A,bool),X: A] :
      ( ~ pp(fAll(A,P))
      | pp(aa(A,bool,P,X)) ) ).

tff(help_fNot_2_1_U,axiom,
    ! [P: bool] :
      ( pp(P)
      | pp(aa(bool,bool,fNot,P)) ) ).

tff(help_fNot_1_1_U,axiom,
    ! [P: bool] :
      ( ~ pp(aa(bool,bool,fNot,P))
      | ~ pp(P) ) ).

tff(help_COMBB_1_1_U,axiom,
    ! [C: $tType,B: $tType,A: $tType,P: fun(B,C),Q: fun(A,B),R: A] : aa(A,C,combb(B,C,A,P,Q),R) = aa(B,C,P,aa(A,B,Q,R)) ).

tff(help_COMBC_1_1_U,axiom,
    ! [A: $tType,C: $tType,B: $tType,P: fun(A,fun(B,C)),Q: B,R: A] : aa(A,C,combc(A,B,C,P,Q),R) = aa(B,C,aa(A,fun(B,C),P,R),Q) ).

tff(help_COMBS_1_1_U,axiom,
    ! [C: $tType,B: $tType,A: $tType,P: fun(A,fun(B,C)),Q: fun(A,B),R: A] : aa(A,C,combs(A,B,C,P,Q),R) = aa(B,C,aa(A,fun(B,C),P,R),aa(A,B,Q,R)) ).

tff(help_fTrue_1_1_U,axiom,
    pp(fTrue) ).

tff(help_fconj_3_1_U,axiom,
    ! [P: bool,Q: bool] :
      ( ~ pp(fconj(P,Q))
      | pp(Q) ) ).

tff(help_fconj_2_1_U,axiom,
    ! [P: bool,Q: bool] :
      ( ~ pp(fconj(P,Q))
      | pp(P) ) ).

tff(help_fconj_1_1_U,axiom,
    ! [P: bool,Q: bool] :
      ( ~ pp(P)
      | ~ pp(Q)
      | pp(fconj(P,Q)) ) ).

tff(help_fdisj_3_1_U,axiom,
    ! [P: bool,Q: bool] :
      ( ~ pp(fdisj(P,Q))
      | pp(P)
      | pp(Q) ) ).

tff(help_fdisj_2_1_U,axiom,
    ! [Q: bool,P: bool] :
      ( ~ pp(Q)
      | pp(fdisj(P,Q)) ) ).

tff(help_fdisj_1_1_U,axiom,
    ! [P: bool,Q: bool] :
      ( ~ pp(P)
      | pp(fdisj(P,Q)) ) ).

tff(help_fFalse_1_1_T,axiom,
    ! [P: bool] :
      ( ( P = fTrue )
      | ( P = fFalse ) ) ).

tff(help_fFalse_1_1_U,axiom,
    ~ pp(fFalse) ).

tff(help_fequal_2_1_T,axiom,
    ! [A: $tType,X: A,Y: A] :
      ( ( X != Y )
      | pp(aa(A,bool,fequal(A,X),Y)) ) ).

tff(help_fequal_1_1_T,axiom,
    ! [A: $tType,X: A,Y: A] :
      ( ~ pp(aa(A,bool,fequal(A,X),Y))
      | ( X = Y ) ) ).

tff(help_fChoice_1_1_T,axiom,
    ! [A: $tType,P: fun(A,bool)] : aa(A,bool,P,fChoice(A,P)) = aa(fun(A,bool),bool,fEx(A),P) ).

tff(help_fimplies_3_1_U,axiom,
    ! [P: bool,Q: bool] :
      ( ~ pp(aa(bool,bool,aa(bool,fun(bool,bool),fimplies,P),Q))
      | ~ pp(P)
      | pp(Q) ) ).

tff(help_fimplies_2_1_U,axiom,
    ! [Q: bool,P: bool] :
      ( ~ pp(Q)
      | pp(aa(bool,bool,aa(bool,fun(bool,bool),fimplies,P),Q)) ) ).

tff(help_fimplies_1_1_U,axiom,
    ! [P: bool,Q: bool] :
      ( pp(P)
      | pp(aa(bool,bool,aa(bool,fun(bool,bool),fimplies,P),Q)) ) ).

% Conjectures (1)
tff(conj_0,conjecture,
    pp(aa(nat,bool,ord_less_eq(nat,vEBT_VEBT_Tb2(aa(nat,nat,suc,aa(nat,nat,suc,na)))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,bit0(one2)))),aa(vEBT_VEBT,nat,vEBT_VEBT_cnt2,if(vEBT_VEBT,aa(nat,bool,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,aa(nat,nat,suc,na))),vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,na)),replicate(vEBT_VEBT,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,na))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),vEBT_vebt_buildup(aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,na))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_vebt_buildup(aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,na))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,na)),replicate(vEBT_VEBT,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,na))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_vebt_buildup(aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,na))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_vebt_buildup(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,na))),aa(num,nat,numeral_numeral(nat),bit0(one2))))))))))) ).

%------------------------------------------------------------------------------